LMFDB - Embedded newform 325.2.k.a.57.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [325,2,Mod(57,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("325.57"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.k (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.59513806569$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			57.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	325.57
Dual form			325.2.k.a.268.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +(-1.00000 + 1.00000i) q^{3} -1.00000 q^{4} +(1.00000 - 1.00000i) q^{6} +2.00000i q^{7} +3.00000 q^{8} +1.00000i q^{9} +(-1.00000 - 1.00000i) q^{11} +(1.00000 - 1.00000i) q^{12} +(-3.00000 - 2.00000i) q^{13} -2.00000i q^{14} -1.00000 q^{16} +(-1.00000 + 1.00000i) q^{17} -1.00000i q^{18} +(-5.00000 - 5.00000i) q^{19} +(-2.00000 - 2.00000i) q^{21} +(1.00000 + 1.00000i) q^{22} +(-3.00000 - 3.00000i) q^{23} +(-3.00000 + 3.00000i) q^{24} +(3.00000 + 2.00000i) q^{26} +(-4.00000 - 4.00000i) q^{27} -2.00000i q^{28} +(5.00000 - 5.00000i) q^{31} -5.00000 q^{32} +2.00000 q^{33} +(1.00000 - 1.00000i) q^{34} -1.00000i q^{36} +(5.00000 + 5.00000i) q^{38} +(5.00000 - 1.00000i) q^{39} +(-7.00000 + 7.00000i) q^{41} +(2.00000 + 2.00000i) q^{42} +(1.00000 + 1.00000i) q^{43} +(1.00000 + 1.00000i) q^{44} +(3.00000 + 3.00000i) q^{46} -6.00000i q^{47} +(1.00000 - 1.00000i) q^{48} +3.00000 q^{49} -2.00000i q^{51} +(3.00000 + 2.00000i) q^{52} +(-5.00000 + 5.00000i) q^{53} +(4.00000 + 4.00000i) q^{54} +6.00000i q^{56} +10.0000 q^{57} +(-7.00000 + 7.00000i) q^{59} -14.0000 q^{61} +(-5.00000 + 5.00000i) q^{62} -2.00000 q^{63} +7.00000 q^{64} -2.00000 q^{66} +4.00000 q^{67} +(1.00000 - 1.00000i) q^{68} +6.00000 q^{69} +(1.00000 - 1.00000i) q^{71} +3.00000i q^{72} +10.0000 q^{73} +(5.00000 + 5.00000i) q^{76} +(2.00000 - 2.00000i) q^{77} +(-5.00000 + 1.00000i) q^{78} -2.00000i q^{79} +5.00000 q^{81} +(7.00000 - 7.00000i) q^{82} +6.00000i q^{83} +(2.00000 + 2.00000i) q^{84} +(-1.00000 - 1.00000i) q^{86} +(-3.00000 - 3.00000i) q^{88} +(5.00000 - 5.00000i) q^{89} +(4.00000 - 6.00000i) q^{91} +(3.00000 + 3.00000i) q^{92} +10.0000i q^{93} +6.00000i q^{94} +(5.00000 - 5.00000i) q^{96} -2.00000 q^{97} -3.00000 q^{98} +(1.00000 - 1.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 6 q^{8} - 2 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{16} - 2 q^{17} - 10 q^{19} - 4 q^{21} + 2 q^{22} - 6 q^{23} - 6 q^{24} + 6 q^{26} - 8 q^{27} + 10 q^{31} - 10 q^{32}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 6 * q^8 - 2 * q^11 + 2 * q^12 - 6 * q^13 - 2 * q^16 - 2 * q^17 - 10 * q^19 - 4 * q^21 + 2 * q^22 - 6 * q^23 - 6 * q^24 + 6 * q^26 - 8 * q^27 + 10 * q^31 - 10 * q^32 + 4 * q^33 + 2 * q^34 + 10 * q^38 + 10 * q^39 - 14 * q^41 + 4 * q^42 + 2 * q^43 + 2 * q^44 + 6 * q^46 + 2 * q^48 + 6 * q^49 + 6 * q^52 - 10 * q^53 + 8 * q^54 + 20 * q^57 - 14 * q^59 - 28 * q^61 - 10 * q^62 - 4 * q^63 + 14 * q^64 - 4 * q^66 + 8 * q^67 + 2 * q^68 + 12 * q^69 + 2 * q^71 + 20 * q^73 + 10 * q^76 + 4 * q^77 - 10 * q^78 + 10 * q^81 + 14 * q^82 + 4 * q^84 - 2 * q^86 - 6 * q^88 + 10 * q^89 + 8 * q^91 + 6 * q^92 + 10 * q^96 - 4 * q^97 - 6 * q^98 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.00000			−0.707107			−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000			−0.707107			−0.353553	+	0.935414i	$0.615027\pi$
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	0.356822	+	0.934172i	$0.383860\pi$
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$	1.00000	−	1.00000i	0.408248	−	0.408248i
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	3.00000			1.06066
$9$			1.00000i			0.333333i
$10$		0			0
$11$	−1.00000	−	1.00000i	−0.301511	−	0.301511i	0.540094	−	0.841605i	$-0.318389\pi$
$11$	−1.00000	−	1.00000i	−0.301511	−	0.301511i	−0.841605	+	0.540094i	$0.818389\pi$
$12$	1.00000	−	1.00000i	0.288675	−	0.288675i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$14$		−	2.00000i		−	0.534522i
$15$		0			0
$16$	−1.00000			−0.250000
$17$	−1.00000	+	1.00000i	−0.242536	+	0.242536i	−0.817898	−	0.575363i	$-0.804861\pi$
$17$	−1.00000	+	1.00000i	−0.242536	+	0.242536i	0.575363	+	0.817898i	$0.304861\pi$
$18$		−	1.00000i		−	0.235702i
$19$	−5.00000	−	5.00000i	−1.14708	−	1.14708i	−0.987124	−	0.159954i	$-0.948865\pi$
$19$	−5.00000	−	5.00000i	−1.14708	−	1.14708i	−0.159954	−	0.987124i	$-0.551135\pi$
$20$		0			0
$21$	−2.00000	−	2.00000i	−0.436436	−	0.436436i
$22$	1.00000	+	1.00000i	0.213201	+	0.213201i
$23$	−3.00000	−	3.00000i	−0.625543	−	0.625543i	0.321400	−	0.946943i	$-0.395847\pi$
$23$	−3.00000	−	3.00000i	−0.625543	−	0.625543i	−0.946943	+	0.321400i	$0.895847\pi$
$24$	−3.00000	+	3.00000i	−0.612372	+	0.612372i
$25$		0			0
$26$	3.00000	+	2.00000i	0.588348	+	0.392232i
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$		−	2.00000i		−	0.377964i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	5.00000	−	5.00000i	0.898027	−	0.898027i	−0.0972349	−	0.995261i	$-0.531000\pi$
$31$	5.00000	−	5.00000i	0.898027	−	0.898027i	0.995261	+	0.0972349i	$0.0309998\pi$
$32$	−5.00000			−0.883883
$33$	2.00000			0.348155
$34$	1.00000	−	1.00000i	0.171499	−	0.171499i
$35$		0			0
$36$		−	1.00000i		−	0.166667i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	5.00000	+	5.00000i	0.811107	+	0.811107i
$39$	5.00000	−	1.00000i	0.800641	−	0.160128i
$40$		0			0
$41$	−7.00000	+	7.00000i	−1.09322	+	1.09322i	−0.0980332	+	0.995183i	$0.531255\pi$
$41$	−7.00000	+	7.00000i	−1.09322	+	1.09322i	−0.995183	+	0.0980332i	$0.968745\pi$
$42$	2.00000	+	2.00000i	0.308607	+	0.308607i
$43$	1.00000	+	1.00000i	0.152499	+	0.152499i	0.779233	−	0.626734i	$-0.215609\pi$
$43$	1.00000	+	1.00000i	0.152499	+	0.152499i	−0.626734	+	0.779233i	$0.715609\pi$
$44$	1.00000	+	1.00000i	0.150756	+	0.150756i
$45$		0			0
$46$	3.00000	+	3.00000i	0.442326	+	0.442326i
$47$		−	6.00000i		−	0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$		−	6.00000i		−	0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	1.00000	−	1.00000i	0.144338	−	0.144338i
$49$	3.00000			0.428571
$50$		0			0
$51$		−	2.00000i		−	0.280056i
$52$	3.00000	+	2.00000i	0.416025	+	0.277350i
$53$	−5.00000	+	5.00000i	−0.686803	+	0.686803i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−5.00000	+	5.00000i	−0.686803	+	0.686803i	0.274721	+	0.961524i	$0.411414\pi$
$54$	4.00000	+	4.00000i	0.544331	+	0.544331i
$55$		0			0
$56$			6.00000i			0.801784i
$57$	10.0000			1.32453
$58$		0			0
$59$	−7.00000	+	7.00000i	−0.911322	+	0.911322i	−0.996376	−	0.0850540i	$-0.972894\pi$
$59$	−7.00000	+	7.00000i	−0.911322	+	0.911322i	0.0850540	+	0.996376i	$0.472894\pi$
$60$		0			0
$61$	−14.0000			−1.79252			−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000			−1.79252			−0.896258	+	0.443533i	$0.853725\pi$
$62$	−5.00000	+	5.00000i	−0.635001	+	0.635001i
$63$	−2.00000			−0.251976
$64$	7.00000			0.875000
$65$		0			0
$66$	−2.00000			−0.246183
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$	1.00000	−	1.00000i	0.121268	−	0.121268i
$69$	6.00000			0.722315
$70$		0			0
$71$	1.00000	−	1.00000i	0.118678	−	0.118678i	−0.645273	−	0.763952i	$-0.723257\pi$
$71$	1.00000	−	1.00000i	0.118678	−	0.118678i	0.763952	+	0.645273i	$0.223257\pi$
$72$			3.00000i			0.353553i
$73$	10.0000			1.17041			0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000			1.17041			0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$		0			0
$76$	5.00000	+	5.00000i	0.573539	+	0.573539i
$77$	2.00000	−	2.00000i	0.227921	−	0.227921i
$78$	−5.00000	+	1.00000i	−0.566139	+	0.113228i
$79$		−	2.00000i		−	0.225018i	−0.993651	−	0.112509i	$-0.964111\pi$
$79$		−	2.00000i		−	0.225018i	0.993651	−	0.112509i	$-0.0358886\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$	7.00000	−	7.00000i	0.773021	−	0.773021i
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	2.00000	+	2.00000i	0.218218	+	0.218218i
$85$		0			0
$86$	−1.00000	−	1.00000i	−0.107833	−	0.107833i
$87$		0			0
$88$	−3.00000	−	3.00000i	−0.319801	−	0.319801i
$89$	5.00000	−	5.00000i	0.529999	−	0.529999i	−0.390573	−	0.920572i	$-0.627723\pi$
$89$	5.00000	−	5.00000i	0.529999	−	0.529999i	0.920572	+	0.390573i	$0.127723\pi$
$90$		0			0
$91$	4.00000	−	6.00000i	0.419314	−	0.628971i
$92$	3.00000	+	3.00000i	0.312772	+	0.312772i
$93$			10.0000i			1.03695i
$94$			6.00000i			0.618853i
$95$		0			0
$96$	5.00000	−	5.00000i	0.510310	−	0.510310i
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$	−3.00000			−0.303046
$99$	1.00000	−	1.00000i	0.100504	−	0.100504i
$100$		0			0
$101$			12.0000i			1.19404i	0.802225	+	0.597022i	$0.203650\pi$
$101$			12.0000i			1.19404i	−0.802225	+	0.597022i	$0.796350\pi$
$102$			2.00000i			0.198030i
$103$	−7.00000	−	7.00000i	−0.689730	−	0.689730i	0.272442	−	0.962172i	$-0.412169\pi$
$103$	−7.00000	−	7.00000i	−0.689730	−	0.689730i	−0.962172	+	0.272442i	$0.912169\pi$
$104$	−9.00000	−	6.00000i	−0.882523	−	0.588348i
$105$		0			0
$106$	5.00000	−	5.00000i	0.485643	−	0.485643i
$107$	−7.00000	−	7.00000i	−0.676716	−	0.676716i	0.282540	−	0.959256i	$-0.408823\pi$
$107$	−7.00000	−	7.00000i	−0.676716	−	0.676716i	−0.959256	+	0.282540i	$0.908823\pi$
$108$	4.00000	+	4.00000i	0.384900	+	0.384900i
$109$	9.00000	+	9.00000i	0.862044	+	0.862044i	0.991575	−	0.129532i	$-0.0413474\pi$
$109$	9.00000	+	9.00000i	0.862044	+	0.862044i	−0.129532	+	0.991575i	$0.541347\pi$
$110$		0			0
$111$		0			0
$112$		−	2.00000i		−	0.188982i
$113$	−5.00000	+	5.00000i	−0.470360	+	0.470360i	−0.902031	−	0.431671i	$-0.857924\pi$
$113$	−5.00000	+	5.00000i	−0.470360	+	0.470360i	0.431671	+	0.902031i	$0.357924\pi$
$114$	−10.0000			−0.936586
$115$		0			0
$116$		0			0
$117$	2.00000	−	3.00000i	0.184900	−	0.277350i
$118$	7.00000	−	7.00000i	0.644402	−	0.644402i
$119$	−2.00000	−	2.00000i	−0.183340	−	0.183340i
$120$		0			0
$121$		−	9.00000i		−	0.818182i
$122$	14.0000			1.26750
$123$		−	14.0000i		−	1.26234i
$124$	−5.00000	+	5.00000i	−0.449013	+	0.449013i
$125$		0			0
$126$	2.00000			0.178174
$127$	−9.00000	+	9.00000i	−0.798621	+	0.798621i	−0.982878	−	0.184257i	$-0.941012\pi$
$127$	−9.00000	+	9.00000i	−0.798621	+	0.798621i	0.184257	+	0.982878i	$0.441012\pi$
$128$	3.00000			0.265165
$129$	−2.00000			−0.176090
$130$		0			0
$131$	−20.0000			−1.74741			−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000			−1.74741			−0.873704	+	0.486458i	$0.838289\pi$
$132$	−2.00000			−0.174078
$133$	10.0000	−	10.0000i	0.867110	−	0.867110i
$134$	−4.00000			−0.345547
$135$		0			0
$136$	−3.00000	+	3.00000i	−0.257248	+	0.257248i
$137$			16.0000i			1.36697i	0.729964	+	0.683486i	$0.239537\pi$
$137$			16.0000i			1.36697i	−0.729964	+	0.683486i	$0.760463\pi$
$138$	−6.00000			−0.510754
$139$		−	14.0000i		−	1.18746i	−0.804663	−	0.593732i	$-0.797654\pi$
$139$		−	14.0000i		−	1.18746i	0.804663	−	0.593732i	$-0.202346\pi$
$140$		0			0
$141$	6.00000	+	6.00000i	0.505291	+	0.505291i
$142$	−1.00000	+	1.00000i	−0.0839181	+	0.0839181i
$143$	1.00000	+	5.00000i	0.0836242	+	0.418121i
$144$		−	1.00000i		−	0.0833333i
$145$		0			0
$146$	−10.0000			−0.827606
$147$	−3.00000	+	3.00000i	−0.247436	+	0.247436i
$148$		0			0
$149$	−3.00000	−	3.00000i	−0.245770	−	0.245770i	0.573462	−	0.819232i	$-0.305600\pi$
$149$	−3.00000	−	3.00000i	−0.245770	−	0.245770i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$	7.00000	+	7.00000i	0.569652	+	0.569652i	0.932031	−	0.362379i	$-0.118035\pi$
$151$	7.00000	+	7.00000i	0.569652	+	0.569652i	−0.362379	+	0.932031i	$0.618035\pi$
$152$	−15.0000	−	15.0000i	−1.21666	−	1.21666i
$153$	−1.00000	−	1.00000i	−0.0808452	−	0.0808452i
$154$	−2.00000	+	2.00000i	−0.161165	+	0.161165i
$155$		0			0
$156$	−5.00000	+	1.00000i	−0.400320	+	0.0800641i
$157$	−13.0000	−	13.0000i	−1.03751	−	1.03751i	−0.999268	−	0.0382445i	$-0.987823\pi$
$157$	−13.0000	−	13.0000i	−1.03751	−	1.03751i	−0.0382445	−	0.999268i	$-0.512177\pi$
$158$			2.00000i			0.159111i
$159$		−	10.0000i		−	0.793052i
$160$		0			0
$161$	6.00000	−	6.00000i	0.472866	−	0.472866i
$162$	−5.00000			−0.392837
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$	7.00000	−	7.00000i	0.546608	−	0.546608i
$165$		0			0
$166$		−	6.00000i		−	0.465690i
$167$			18.0000i			1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$			18.0000i			1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$	−6.00000	−	6.00000i	−0.462910	−	0.462910i
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$	5.00000	−	5.00000i	0.382360	−	0.382360i
$172$	−1.00000	−	1.00000i	−0.0762493	−	0.0762493i
$173$	11.0000	+	11.0000i	0.836315	+	0.836315i	0.988372	−	0.152057i	$-0.0485898\pi$
$173$	11.0000	+	11.0000i	0.836315	+	0.836315i	−0.152057	+	0.988372i	$0.548590\pi$
$174$		0			0
$175$		0			0
$176$	1.00000	+	1.00000i	0.0753778	+	0.0753778i
$177$		−	14.0000i		−	1.05230i
$178$	−5.00000	+	5.00000i	−0.374766	+	0.374766i
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$		−	8.00000i		−	0.594635i	−0.954779	−	0.297318i	$-0.903908\pi$
$181$		−	8.00000i		−	0.594635i	0.954779	−	0.297318i	$-0.0960920\pi$
$182$	−4.00000	+	6.00000i	−0.296500	+	0.444750i
$183$	14.0000	−	14.0000i	1.03491	−	1.03491i
$184$	−9.00000	−	9.00000i	−0.663489	−	0.663489i
$185$		0			0
$186$		−	10.0000i		−	0.733236i
$187$	2.00000			0.146254
$188$			6.00000i			0.437595i
$189$	8.00000	−	8.00000i	0.581914	−	0.581914i
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$	−7.00000	+	7.00000i	−0.505181	+	0.505181i
$193$	−18.0000			−1.29567			−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000			−1.29567			−0.647834	+	0.761781i	$0.724325\pi$
$194$	2.00000			0.143592
$195$		0			0
$196$	−3.00000			−0.214286
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$	−1.00000	+	1.00000i	−0.0710669	+	0.0710669i
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$	−4.00000	+	4.00000i	−0.282138	+	0.282138i
$202$		−	12.0000i		−	0.844317i
$203$		0			0
$204$			2.00000i			0.140028i
$205$		0			0
$206$	7.00000	+	7.00000i	0.487713	+	0.487713i
$207$	3.00000	−	3.00000i	0.208514	−	0.208514i
$208$	3.00000	+	2.00000i	0.208013	+	0.138675i
$209$			10.0000i			0.691714i
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$	5.00000	−	5.00000i	0.343401	−	0.343401i
$213$			2.00000i			0.137038i
$214$	7.00000	+	7.00000i	0.478510	+	0.478510i
$215$		0			0
$216$	−12.0000	−	12.0000i	−0.816497	−	0.816497i
$217$	10.0000	+	10.0000i	0.678844	+	0.678844i
$218$	−9.00000	−	9.00000i	−0.609557	−	0.609557i
$219$	−10.0000	+	10.0000i	−0.675737	+	0.675737i
$220$		0			0
$221$	5.00000	−	1.00000i	0.336336	−	0.0672673i
$222$		0			0
$223$			2.00000i			0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$			2.00000i			0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$		−	10.0000i		−	0.668153i
$225$		0			0
$226$	5.00000	−	5.00000i	0.332595	−	0.332595i
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$	−10.0000			−0.662266
$229$	−3.00000	+	3.00000i	−0.198246	+	0.198246i	−0.799248	−	0.601002i	$-0.794768\pi$
$229$	−3.00000	+	3.00000i	−0.198246	+	0.198246i	0.601002	+	0.799248i	$0.294768\pi$
$230$		0			0
$231$			4.00000i			0.263181i
$232$		0			0
$233$	−1.00000	−	1.00000i	−0.0655122	−	0.0655122i	0.673592	−	0.739104i	$-0.264751\pi$
$233$	−1.00000	−	1.00000i	−0.0655122	−	0.0655122i	−0.739104	+	0.673592i	$0.764751\pi$
$234$	−2.00000	+	3.00000i	−0.130744	+	0.196116i
$235$		0			0
$236$	7.00000	−	7.00000i	0.455661	−	0.455661i
$237$	2.00000	+	2.00000i	0.129914	+	0.129914i
$238$	2.00000	+	2.00000i	0.129641	+	0.129641i
$239$	3.00000	+	3.00000i	0.194054	+	0.194054i	0.797445	−	0.603391i	$-0.206184\pi$
$239$	3.00000	+	3.00000i	0.194054	+	0.194054i	−0.603391	+	0.797445i	$0.706184\pi$
$240$		0			0
$241$	17.0000	+	17.0000i	1.09507	+	1.09507i	0.994979	+	0.100088i	$0.0319123\pi$
$241$	17.0000	+	17.0000i	1.09507	+	1.09507i	0.100088	+	0.994979i	$0.468088\pi$
$242$			9.00000i			0.578542i
$243$	7.00000	−	7.00000i	0.449050	−	0.449050i
$244$	14.0000			0.896258
$245$		0			0
$246$			14.0000i			0.892607i
$247$	5.00000	+	25.0000i	0.318142	+	1.59071i
$248$	15.0000	−	15.0000i	0.952501	−	0.952501i
$249$	−6.00000	−	6.00000i	−0.380235	−	0.380235i
$250$		0			0
$251$			2.00000i			0.126239i	0.998006	+	0.0631194i	$0.0201049\pi$
$251$			2.00000i			0.126239i	−0.998006	+	0.0631194i	$0.979895\pi$
$252$	2.00000			0.125988
$253$			6.00000i			0.377217i
$254$	9.00000	−	9.00000i	0.564710	−	0.564710i
$255$		0			0
$256$	−17.0000			−1.06250
$257$	11.0000	−	11.0000i	0.686161	−	0.686161i	−0.275220	−	0.961381i	$-0.588751\pi$
$257$	11.0000	−	11.0000i	0.686161	−	0.686161i	0.961381	+	0.275220i	$0.0887507\pi$
$258$	2.00000			0.124515
$259$		0			0
$260$		0			0
$261$		0			0
$262$	20.0000			1.23560
$263$	−1.00000	+	1.00000i	−0.0616626	+	0.0616626i	−0.737266	−	0.675603i	$-0.763883\pi$
$263$	−1.00000	+	1.00000i	−0.0616626	+	0.0616626i	0.675603	+	0.737266i	$0.263883\pi$
$264$	6.00000			0.369274
$265$		0			0
$266$	−10.0000	+	10.0000i	−0.613139	+	0.613139i
$267$			10.0000i			0.611990i
$268$	−4.00000			−0.244339
$269$			12.0000i			0.731653i	0.930683	+	0.365826i	$0.119214\pi$
$269$			12.0000i			0.731653i	−0.930683	+	0.365826i	$0.880786\pi$
$270$		0			0
$271$	−9.00000	−	9.00000i	−0.546711	−	0.546711i	0.378777	−	0.925488i	$-0.376345\pi$
$271$	−9.00000	−	9.00000i	−0.546711	−	0.546711i	−0.925488	+	0.378777i	$0.876345\pi$
$272$	1.00000	−	1.00000i	0.0606339	−	0.0606339i
$273$	2.00000	+	10.0000i	0.121046	+	0.605228i
$274$		−	16.0000i		−	0.966595i
$275$		0			0
$276$	−6.00000			−0.361158
$277$	15.0000	−	15.0000i	0.901263	−	0.901263i	−0.0942828	−	0.995545i	$-0.530056\pi$
$277$	15.0000	−	15.0000i	0.901263	−	0.901263i	0.995545	+	0.0942828i	$0.0300558\pi$
$278$			14.0000i			0.839664i
$279$	5.00000	+	5.00000i	0.299342	+	0.299342i
$280$		0			0
$281$	1.00000	+	1.00000i	0.0596550	+	0.0596550i	0.736305	−	0.676650i	$-0.236569\pi$
$281$	1.00000	+	1.00000i	0.0596550	+	0.0596550i	−0.676650	+	0.736305i	$0.736569\pi$
$282$	−6.00000	−	6.00000i	−0.357295	−	0.357295i
$283$	9.00000	+	9.00000i	0.534994	+	0.534994i	0.922055	−	0.387060i	$-0.126509\pi$
$283$	9.00000	+	9.00000i	0.534994	+	0.534994i	−0.387060	+	0.922055i	$0.626509\pi$
$284$	−1.00000	+	1.00000i	−0.0593391	+	0.0593391i
$285$		0			0
$286$	−1.00000	−	5.00000i	−0.0591312	−	0.295656i
$287$	−14.0000	−	14.0000i	−0.826394	−	0.826394i
$288$		−	5.00000i		−	0.294628i
$289$			15.0000i			0.882353i
$290$		0			0
$291$	2.00000	−	2.00000i	0.117242	−	0.117242i
$292$	−10.0000			−0.585206
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$	3.00000	−	3.00000i	0.174964	−	0.174964i
$295$		0			0
$296$		0			0
$297$			8.00000i			0.464207i
$298$	3.00000	+	3.00000i	0.173785	+	0.173785i
$299$	3.00000	+	15.0000i	0.173494	+	0.867472i
$300$		0			0
$301$	−2.00000	+	2.00000i	−0.115278	+	0.115278i
$302$	−7.00000	−	7.00000i	−0.402805	−	0.402805i
$303$	−12.0000	−	12.0000i	−0.689382	−	0.689382i
$304$	5.00000	+	5.00000i	0.286770	+	0.286770i
$305$		0			0
$306$	1.00000	+	1.00000i	0.0571662	+	0.0571662i
$307$		−	18.0000i		−	1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$		−	18.0000i		−	1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$	−2.00000	+	2.00000i	−0.113961	+	0.113961i
$309$	14.0000			0.796432
$310$		0			0
$311$			6.00000i			0.340229i	0.985424	+	0.170114i	$0.0544137\pi$
$311$			6.00000i			0.340229i	−0.985424	+	0.170114i	$0.945586\pi$
$312$	15.0000	−	3.00000i	0.849208	−	0.169842i
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	−0.914130	−	0.405420i	$-0.867125\pi$
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	0.405420	+	0.914130i	$0.367125\pi$
$314$	13.0000	+	13.0000i	0.733632	+	0.733632i
$315$		0			0
$316$			2.00000i			0.112509i
$317$	14.0000			0.786318			0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000			0.786318			0.393159	+	0.919470i	$0.371382\pi$
$318$			10.0000i			0.560772i
$319$		0			0
$320$		0			0
$321$	14.0000			0.781404
$322$	−6.00000	+	6.00000i	−0.334367	+	0.334367i
$323$	10.0000			0.556415
$324$	−5.00000			−0.277778
$325$		0			0
$326$	4.00000			0.221540
$327$	−18.0000			−0.995402
$328$	−21.0000	+	21.0000i	−1.15953	+	1.15953i
$329$	12.0000			0.661581
$330$		0			0
$331$	−3.00000	+	3.00000i	−0.164895	+	0.164895i	−0.784731	−	0.619836i	$-0.787199\pi$
$331$	−3.00000	+	3.00000i	−0.164895	+	0.164895i	0.619836	+	0.784731i	$0.287199\pi$
$332$		−	6.00000i		−	0.329293i
$333$		0			0
$334$		−	18.0000i		−	0.984916i
$335$		0			0
$336$	2.00000	+	2.00000i	0.109109	+	0.109109i
$337$	−13.0000	+	13.0000i	−0.708155	+	0.708155i	−0.966147	−	0.257992i	$-0.916939\pi$
$337$	−13.0000	+	13.0000i	−0.708155	+	0.708155i	0.257992	+	0.966147i	$0.416939\pi$
$338$	−5.00000	−	12.0000i	−0.271964	−	0.652714i
$339$		−	10.0000i		−	0.543125i
$340$		0			0
$341$	−10.0000			−0.541530
$342$	−5.00000	+	5.00000i	−0.270369	+	0.270369i
$343$			20.0000i			1.07990i
$344$	3.00000	+	3.00000i	0.161749	+	0.161749i
$345$		0			0
$346$	−11.0000	−	11.0000i	−0.591364	−	0.591364i
$347$	−3.00000	−	3.00000i	−0.161048	−	0.161048i	0.621983	−	0.783031i	$-0.286327\pi$
$347$	−3.00000	−	3.00000i	−0.161048	−	0.161048i	−0.783031	+	0.621983i	$0.786327\pi$
$348$		0			0
$349$	9.00000	−	9.00000i	0.481759	−	0.481759i	−0.423934	−	0.905693i	$-0.639351\pi$
$349$	9.00000	−	9.00000i	0.481759	−	0.481759i	0.905693	+	0.423934i	$0.139351\pi$
$350$		0			0
$351$	4.00000	+	20.0000i	0.213504	+	1.06752i
$352$	5.00000	+	5.00000i	0.266501	+	0.266501i
$353$		−	12.0000i		−	0.638696i	−0.947638	−	0.319348i	$-0.896536\pi$
$353$		−	12.0000i		−	0.638696i	0.947638	−	0.319348i	$-0.103464\pi$
$354$			14.0000i			0.744092i
$355$		0			0
$356$	−5.00000	+	5.00000i	−0.264999	+	0.264999i
$357$	4.00000			0.211702
$358$	20.0000			1.05703
$359$	1.00000	−	1.00000i	0.0527780	−	0.0527780i	−0.680225	−	0.733003i	$-0.738118\pi$
$359$	1.00000	−	1.00000i	0.0527780	−	0.0527780i	0.733003	+	0.680225i	$0.238118\pi$
$360$		0			0
$361$			31.0000i			1.63158i
$362$			8.00000i			0.420471i
$363$	9.00000	+	9.00000i	0.472377	+	0.472377i
$364$	−4.00000	+	6.00000i	−0.209657	+	0.314485i
$365$		0			0
$366$	−14.0000	+	14.0000i	−0.731792	+	0.731792i
$367$	1.00000	+	1.00000i	0.0521996	+	0.0521996i	0.732725	−	0.680525i	$-0.238248\pi$
$367$	1.00000	+	1.00000i	0.0521996	+	0.0521996i	−0.680525	+	0.732725i	$0.738248\pi$
$368$	3.00000	+	3.00000i	0.156386	+	0.156386i
$369$	−7.00000	−	7.00000i	−0.364405	−	0.364405i
$370$		0			0
$371$	−10.0000	−	10.0000i	−0.519174	−	0.519174i
$372$		−	10.0000i		−	0.518476i
$373$	15.0000	−	15.0000i	0.776671	−	0.776671i	−0.202593	−	0.979263i	$-0.564937\pi$
$373$	15.0000	−	15.0000i	0.776671	−	0.776671i	0.979263	+	0.202593i	$0.0649367\pi$
$374$	−2.00000			−0.103418
$375$		0			0
$376$		−	18.0000i		−	0.928279i
$377$		0			0
$378$	−8.00000	+	8.00000i	−0.411476	+	0.411476i
$379$	−1.00000	−	1.00000i	−0.0513665	−	0.0513665i	0.680957	−	0.732323i	$-0.261564\pi$
$379$	−1.00000	−	1.00000i	−0.0513665	−	0.0513665i	−0.732323	+	0.680957i	$0.761564\pi$
$380$		0			0
$381$		−	18.0000i		−	0.922168i
$382$	−8.00000			−0.409316
$383$		−	30.0000i		−	1.53293i	−0.642287	−	0.766464i	$-0.722014\pi$
$383$		−	30.0000i		−	1.53293i	0.642287	−	0.766464i	$-0.277986\pi$
$384$	−3.00000	+	3.00000i	−0.153093	+	0.153093i
$385$		0			0
$386$	18.0000			0.916176
$387$	−1.00000	+	1.00000i	−0.0508329	+	0.0508329i
$388$	2.00000			0.101535
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$	6.00000			0.303433
$392$	9.00000			0.454569
$393$	20.0000	−	20.0000i	1.00887	−	1.00887i
$394$	6.00000			0.302276
$395$		0			0
$396$	−1.00000	+	1.00000i	−0.0502519	+	0.0502519i
$397$		−	16.0000i		−	0.803017i	−0.915855	−	0.401508i	$-0.868486\pi$
$397$		−	16.0000i		−	0.803017i	0.915855	−	0.401508i	$-0.131514\pi$
$398$	8.00000			0.401004
$399$			20.0000i			1.00125i
$400$		0			0
$401$	−11.0000	−	11.0000i	−0.549314	−	0.549314i	0.376929	−	0.926242i	$-0.376980\pi$
$401$	−11.0000	−	11.0000i	−0.549314	−	0.549314i	−0.926242	+	0.376929i	$0.876980\pi$
$402$	4.00000	−	4.00000i	0.199502	−	0.199502i
$403$	−25.0000	+	5.00000i	−1.24534	+	0.249068i
$404$		−	12.0000i		−	0.597022i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		−	6.00000i		−	0.297044i
$409$	−7.00000	−	7.00000i	−0.346128	−	0.346128i	0.512537	−	0.858665i	$-0.328706\pi$
$409$	−7.00000	−	7.00000i	−0.346128	−	0.346128i	−0.858665	+	0.512537i	$0.828706\pi$
$410$		0			0
$411$	−16.0000	−	16.0000i	−0.789222	−	0.789222i
$412$	7.00000	+	7.00000i	0.344865	+	0.344865i
$413$	−14.0000	−	14.0000i	−0.688895	−	0.688895i
$414$	−3.00000	+	3.00000i	−0.147442	+	0.147442i
$415$		0			0
$416$	15.0000	+	10.0000i	0.735436	+	0.490290i
$417$	14.0000	+	14.0000i	0.685583	+	0.685583i
$418$		−	10.0000i		−	0.489116i
$419$		−	38.0000i		−	1.85642i	−0.372055	−	0.928211i	$-0.621347\pi$
$419$		−	38.0000i		−	1.85642i	0.372055	−	0.928211i	$-0.378653\pi$
$420$		0			0
$421$	−11.0000	+	11.0000i	−0.536107	+	0.536107i	−0.922383	−	0.386276i	$-0.873761\pi$
$421$	−11.0000	+	11.0000i	−0.536107	+	0.536107i	0.386276	+	0.922383i	$0.373761\pi$
$422$	−4.00000			−0.194717
$423$	6.00000			0.291730
$424$	−15.0000	+	15.0000i	−0.728464	+	0.728464i
$425$		0			0
$426$		−	2.00000i		−	0.0969003i
$427$		−	28.0000i		−	1.35501i
$428$	7.00000	+	7.00000i	0.338358	+	0.338358i
$429$	−6.00000	−	4.00000i	−0.289683	−	0.193122i
$430$		0			0
$431$	13.0000	−	13.0000i	0.626188	−	0.626188i	−0.320919	−	0.947107i	$-0.603992\pi$
$431$	13.0000	−	13.0000i	0.626188	−	0.626188i	0.947107	+	0.320919i	$0.103992\pi$
$432$	4.00000	+	4.00000i	0.192450	+	0.192450i
$433$	−17.0000	−	17.0000i	−0.816968	−	0.816968i	0.168700	−	0.985668i	$-0.446043\pi$
$433$	−17.0000	−	17.0000i	−0.816968	−	0.816968i	−0.985668	+	0.168700i	$0.946043\pi$
$434$	−10.0000	−	10.0000i	−0.480015	−	0.480015i
$435$		0			0
$436$	−9.00000	−	9.00000i	−0.431022	−	0.431022i
$437$			30.0000i			1.43509i
$438$	10.0000	−	10.0000i	0.477818	−	0.477818i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$			3.00000i			0.142857i
$442$	−5.00000	+	1.00000i	−0.237826	+	0.0475651i
$443$	−25.0000	+	25.0000i	−1.18779	+	1.18779i	−0.210108	+	0.977678i	$0.567381\pi$
$443$	−25.0000	+	25.0000i	−1.18779	+	1.18779i	−0.977678	+	0.210108i	$0.932619\pi$
$444$		0			0
$445$		0			0
$446$		−	2.00000i		−	0.0947027i
$447$	6.00000			0.283790
$448$			14.0000i			0.661438i
$449$	−3.00000	+	3.00000i	−0.141579	+	0.141579i	−0.774344	−	0.632765i	$-0.781920\pi$
$449$	−3.00000	+	3.00000i	−0.141579	+	0.141579i	0.632765	+	0.774344i	$0.281920\pi$
$450$		0			0
$451$	14.0000			0.659234
$452$	5.00000	−	5.00000i	0.235180	−	0.235180i
$453$	−14.0000			−0.657777
$454$	−12.0000			−0.563188
$455$		0			0
$456$	30.0000			1.40488
$457$	−2.00000			−0.0935561			−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000			−0.0935561			−0.0467780	+	0.998905i	$0.514895\pi$
$458$	3.00000	−	3.00000i	0.140181	−	0.140181i
$459$	8.00000			0.373408
$460$		0			0
$461$	17.0000	−	17.0000i	0.791769	−	0.791769i	−0.190013	−	0.981782i	$-0.560853\pi$
$461$	17.0000	−	17.0000i	0.791769	−	0.791769i	0.981782	+	0.190013i	$0.0608529\pi$
$462$		−	4.00000i		−	0.186097i
$463$	24.0000			1.11537			0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000			1.11537			0.557687	+	0.830051i	$0.311689\pi$
$464$		0			0
$465$		0			0
$466$	1.00000	+	1.00000i	0.0463241	+	0.0463241i
$467$	−9.00000	+	9.00000i	−0.416470	+	0.416470i	−0.883985	−	0.467515i	$-0.845149\pi$
$467$	−9.00000	+	9.00000i	−0.416470	+	0.416470i	0.467515	+	0.883985i	$0.345149\pi$
$468$	−2.00000	+	3.00000i	−0.0924500	+	0.138675i
$469$			8.00000i			0.369406i
$470$		0			0
$471$	26.0000			1.19802
$472$	−21.0000	+	21.0000i	−0.966603	+	0.966603i
$473$		−	2.00000i		−	0.0919601i
$474$	−2.00000	−	2.00000i	−0.0918630	−	0.0918630i
$475$		0			0
$476$	2.00000	+	2.00000i	0.0916698	+	0.0916698i
$477$	−5.00000	−	5.00000i	−0.228934	−	0.228934i
$478$	−3.00000	−	3.00000i	−0.137217	−	0.137217i
$479$	−7.00000	+	7.00000i	−0.319838	+	0.319838i	−0.848705	−	0.528867i	$-0.822617\pi$
$479$	−7.00000	+	7.00000i	−0.319838	+	0.319838i	0.528867	+	0.848705i	$0.322617\pi$
$480$		0			0
$481$		0			0
$482$	−17.0000	−	17.0000i	−0.774329	−	0.774329i
$483$			12.0000i			0.546019i
$484$			9.00000i			0.409091i
$485$		0			0
$486$	−7.00000	+	7.00000i	−0.317526	+	0.317526i
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$	−42.0000			−1.90125
$489$	4.00000	−	4.00000i	0.180886	−	0.180886i
$490$		0			0
$491$		−	22.0000i		−	0.992846i	−0.868081	−	0.496423i	$-0.834646\pi$
$491$		−	22.0000i		−	0.992846i	0.868081	−	0.496423i	$-0.165354\pi$
$492$			14.0000i			0.631169i
$493$		0			0
$494$	−5.00000	−	25.0000i	−0.224961	−	1.12480i
$495$		0			0
$496$	−5.00000	+	5.00000i	−0.224507	+	0.224507i
$497$	2.00000	+	2.00000i	0.0897123	+	0.0897123i
$498$	6.00000	+	6.00000i	0.268866	+	0.268866i
$499$	3.00000	+	3.00000i	0.134298	+	0.134298i	0.771060	−	0.636762i	$-0.219727\pi$
$499$	3.00000	+	3.00000i	0.134298	+	0.134298i	−0.636762	+	0.771060i	$0.719727\pi$
$500$		0			0
$501$	−18.0000	−	18.0000i	−0.804181	−	0.804181i
$502$		−	2.00000i		−	0.0892644i
$503$	3.00000	−	3.00000i	0.133763	−	0.133763i	−0.637055	−	0.770818i	$-0.719848\pi$
$503$	3.00000	−	3.00000i	0.133763	−	0.133763i	0.770818	+	0.637055i	$0.219848\pi$
$504$	−6.00000			−0.267261
$505$		0			0
$506$		−	6.00000i		−	0.266733i
$507$	−17.0000	−	7.00000i	−0.754997	−	0.310881i
$508$	9.00000	−	9.00000i	0.399310	−	0.399310i
$509$	13.0000	+	13.0000i	0.576215	+	0.576215i	0.933858	−	0.357643i	$-0.116420\pi$
$509$	13.0000	+	13.0000i	0.576215	+	0.576215i	−0.357643	+	0.933858i	$0.616420\pi$
$510$		0			0
$511$			20.0000i			0.884748i
$512$	11.0000			0.486136
$513$			40.0000i			1.76604i
$514$	−11.0000	+	11.0000i	−0.485189	+	0.485189i
$515$		0			0
$516$	2.00000			0.0880451
$517$	−6.00000	+	6.00000i	−0.263880	+	0.263880i
$518$		0			0
$519$	−22.0000			−0.965693
$520$		0			0
$521$	−10.0000			−0.438108			−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000			−0.438108			−0.219054	+	0.975713i	$0.570297\pi$
$522$		0			0
$523$	−9.00000	+	9.00000i	−0.393543	+	0.393543i	−0.875948	−	0.482405i	$-0.839763\pi$
$523$	−9.00000	+	9.00000i	−0.393543	+	0.393543i	0.482405	+	0.875948i	$0.339763\pi$
$524$	20.0000			0.873704
$525$		0			0
$526$	1.00000	−	1.00000i	0.0436021	−	0.0436021i
$527$			10.0000i			0.435607i
$528$	−2.00000			−0.0870388
$529$		−	5.00000i		−	0.217391i
$530$		0			0
$531$	−7.00000	−	7.00000i	−0.303774	−	0.303774i
$532$	−10.0000	+	10.0000i	−0.433555	+	0.433555i
$533$	35.0000	−	7.00000i	1.51602	−	0.303204i
$534$		−	10.0000i		−	0.432742i
$535$		0			0
$536$	12.0000			0.518321
$537$	20.0000	−	20.0000i	0.863064	−	0.863064i
$538$		−	12.0000i		−	0.517357i
$539$	−3.00000	−	3.00000i	−0.129219	−	0.129219i
$540$		0			0
$541$	9.00000	+	9.00000i	0.386940	+	0.386940i	0.873595	−	0.486654i	$-0.161783\pi$
$541$	9.00000	+	9.00000i	0.386940	+	0.386940i	−0.486654	+	0.873595i	$0.661783\pi$
$542$	9.00000	+	9.00000i	0.386583	+	0.386583i
$543$	8.00000	+	8.00000i	0.343313	+	0.343313i
$544$	5.00000	−	5.00000i	0.214373	−	0.214373i
$545$		0			0
$546$	−2.00000	−	10.0000i	−0.0855921	−	0.427960i
$547$	9.00000	+	9.00000i	0.384812	+	0.384812i	0.872832	−	0.488020i	$-0.162281\pi$
$547$	9.00000	+	9.00000i	0.384812	+	0.384812i	−0.488020	+	0.872832i	$0.662281\pi$
$548$		−	16.0000i		−	0.683486i
$549$		−	14.0000i		−	0.597505i
$550$		0			0
$551$		0			0
$552$	18.0000			0.766131
$553$	4.00000			0.170097
$554$	−15.0000	+	15.0000i	−0.637289	+	0.637289i
$555$		0			0
$556$			14.0000i			0.593732i
$557$		−	24.0000i		−	1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$		−	24.0000i		−	1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$	−5.00000	−	5.00000i	−0.211667	−	0.211667i
$559$	−1.00000	−	5.00000i	−0.0422955	−	0.211477i
$560$		0			0
$561$	−2.00000	+	2.00000i	−0.0844401	+	0.0844401i
$562$	−1.00000	−	1.00000i	−0.0421825	−	0.0421825i
$563$	−15.0000	−	15.0000i	−0.632175	−	0.632175i	0.316438	−	0.948613i	$-0.397513\pi$
$563$	−15.0000	−	15.0000i	−0.632175	−	0.632175i	−0.948613	+	0.316438i	$0.897513\pi$
$564$	−6.00000	−	6.00000i	−0.252646	−	0.252646i
$565$		0			0
$566$	−9.00000	−	9.00000i	−0.378298	−	0.378298i
$567$			10.0000i			0.419961i
$568$	3.00000	−	3.00000i	0.125877	−	0.125877i
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$		−	6.00000i		−	0.251092i	−0.992088	−	0.125546i	$-0.959932\pi$
$571$		−	6.00000i		−	0.251092i	0.992088	−	0.125546i	$-0.0400683\pi$
$572$	−1.00000	−	5.00000i	−0.0418121	−	0.209061i
$573$	−8.00000	+	8.00000i	−0.334205	+	0.334205i
$574$	14.0000	+	14.0000i	0.584349	+	0.584349i
$575$		0			0
$576$			7.00000i			0.291667i
$577$	−46.0000			−1.91501			−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−46.0000			−1.91501			−0.957503	+	0.288425i	$0.906868\pi$
$578$		−	15.0000i		−	0.623918i
$579$	18.0000	−	18.0000i	0.748054	−	0.748054i
$580$		0			0
$581$	−12.0000			−0.497844
$582$	−2.00000	+	2.00000i	−0.0829027	+	0.0829027i
$583$	10.0000			0.414158
$584$	30.0000			1.24141
$585$		0			0
$586$	−6.00000			−0.247858
$587$	4.00000			0.165098			0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000			0.165098			0.0825488	+	0.996587i	$0.473694\pi$
$588$	3.00000	−	3.00000i	0.123718	−	0.123718i
$589$	−50.0000			−2.06021
$590$		0			0
$591$	6.00000	−	6.00000i	0.246807	−	0.246807i
$592$		0			0
$593$	−10.0000			−0.410651			−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000			−0.410651			−0.205325	+	0.978694i	$0.565825\pi$
$594$		−	8.00000i		−	0.328244i
$595$		0			0
$596$	3.00000	+	3.00000i	0.122885	+	0.122885i
$597$	8.00000	−	8.00000i	0.327418	−	0.327418i
$598$	−3.00000	−	15.0000i	−0.122679	−	0.613396i
$599$			30.0000i			1.22577i	0.790173	+	0.612883i	$0.209990\pi$
$599$			30.0000i			1.22577i	−0.790173	+	0.612883i	$0.790010\pi$
$600$		0			0
$601$	−38.0000			−1.55005			−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000			−1.55005			−0.775026	+	0.631929i	$0.782263\pi$
$602$	2.00000	−	2.00000i	0.0815139	−	0.0815139i
$603$			4.00000i			0.162893i
$604$	−7.00000	−	7.00000i	−0.284826	−	0.284826i
$605$		0			0
$606$	12.0000	+	12.0000i	0.487467	+	0.487467i
$607$	13.0000	+	13.0000i	0.527654	+	0.527654i	0.919872	−	0.392218i	$-0.128292\pi$
$607$	13.0000	+	13.0000i	0.527654	+	0.527654i	−0.392218	+	0.919872i	$0.628292\pi$
$608$	25.0000	+	25.0000i	1.01388	+	1.01388i
$609$		0			0
$610$		0			0
$611$	−12.0000	+	18.0000i	−0.485468	+	0.728202i
$612$	1.00000	+	1.00000i	0.0404226	+	0.0404226i
$613$		−	20.0000i		−	0.807792i	−0.914805	−	0.403896i	$-0.867656\pi$
$613$		−	20.0000i		−	0.807792i	0.914805	−	0.403896i	$-0.132344\pi$
$614$			18.0000i			0.726421i
$615$		0			0
$616$	6.00000	−	6.00000i	0.241747	−	0.241747i
$617$	22.0000			0.885687			0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000			0.885687			0.442843	+	0.896599i	$0.353970\pi$
$618$	−14.0000			−0.563163
$619$	25.0000	−	25.0000i	1.00483	−	1.00483i	0.00484658	−	0.999988i	$-0.498457\pi$
$619$	25.0000	−	25.0000i	1.00483	−	1.00483i	0.999988	−	0.00484658i	$-0.00154272\pi$
$620$		0			0
$621$			24.0000i			0.963087i
$622$		−	6.00000i		−	0.240578i
$623$	10.0000	+	10.0000i	0.400642	+	0.400642i
$624$	−5.00000	+	1.00000i	−0.200160	+	0.0400320i
$625$		0			0
$626$	9.00000	−	9.00000i	0.359712	−	0.359712i
$627$	−10.0000	−	10.0000i	−0.399362	−	0.399362i
$628$	13.0000	+	13.0000i	0.518756	+	0.518756i
$629$		0			0
$630$		0			0
$631$	11.0000	+	11.0000i	0.437903	+	0.437903i	0.891306	−	0.453403i	$-0.149790\pi$
$631$	11.0000	+	11.0000i	0.437903	+	0.437903i	−0.453403	+	0.891306i	$0.649790\pi$
$632$		−	6.00000i		−	0.238667i
$633$	−4.00000	+	4.00000i	−0.158986	+	0.158986i
$634$	−14.0000			−0.556011
$635$		0			0
$636$			10.0000i			0.396526i
$637$	−9.00000	−	6.00000i	−0.356593	−	0.237729i
$638$		0			0
$639$	1.00000	+	1.00000i	0.0395594	+	0.0395594i
$640$		0			0
$641$		−	24.0000i		−	0.947943i	−0.880540	−	0.473972i	$-0.842820\pi$
$641$		−	24.0000i		−	0.947943i	0.880540	−	0.473972i	$-0.157180\pi$
$642$	−14.0000			−0.552536
$643$		−	34.0000i		−	1.34083i	−0.741987	−	0.670415i	$-0.766116\pi$
$643$		−	34.0000i		−	1.34083i	0.741987	−	0.670415i	$-0.233884\pi$
$644$	−6.00000	+	6.00000i	−0.236433	+	0.236433i
$645$		0			0
$646$	−10.0000			−0.393445
$647$	−1.00000	+	1.00000i	−0.0393141	+	0.0393141i	−0.726491	−	0.687176i	$-0.758850\pi$
$647$	−1.00000	+	1.00000i	−0.0393141	+	0.0393141i	0.687176	+	0.726491i	$0.258850\pi$
$648$	15.0000			0.589256
$649$	14.0000			0.549548
$650$		0			0
$651$	−20.0000			−0.783862
$652$	4.00000			0.156652
$653$	−13.0000	+	13.0000i	−0.508729	+	0.508729i	−0.914136	−	0.405407i	$-0.867130\pi$
$653$	−13.0000	+	13.0000i	−0.508729	+	0.508729i	0.405407	+	0.914136i	$0.367130\pi$
$654$	18.0000			0.703856
$655$		0			0
$656$	7.00000	−	7.00000i	0.273304	−	0.273304i
$657$			10.0000i			0.390137i
$658$	−12.0000			−0.467809
$659$			26.0000i			1.01282i	0.862294	+	0.506408i	$0.169027\pi$
$659$			26.0000i			1.01282i	−0.862294	+	0.506408i	$0.830973\pi$
$660$		0			0
$661$	17.0000	+	17.0000i	0.661223	+	0.661223i	0.955668	−	0.294445i	$-0.0951348\pi$
$661$	17.0000	+	17.0000i	0.661223	+	0.661223i	−0.294445	+	0.955668i	$0.595135\pi$
$662$	3.00000	−	3.00000i	0.116598	−	0.116598i
$663$	−4.00000	+	6.00000i	−0.155347	+	0.233021i
$664$			18.0000i			0.698535i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		−	18.0000i		−	0.696441i
$669$	−2.00000	−	2.00000i	−0.0773245	−	0.0773245i
$670$		0			0
$671$	14.0000	+	14.0000i	0.540464	+	0.540464i
$672$	10.0000	+	10.0000i	0.385758	+	0.385758i
$673$	15.0000	+	15.0000i	0.578208	+	0.578208i	0.934409	−	0.356202i	$-0.115928\pi$
$673$	15.0000	+	15.0000i	0.578208	+	0.578208i	−0.356202	+	0.934409i	$0.615928\pi$
$674$	13.0000	−	13.0000i	0.500741	−	0.500741i
$675$		0			0
$676$	−5.00000	−	12.0000i	−0.192308	−	0.461538i
$677$	23.0000	+	23.0000i	0.883962	+	0.883962i	0.993935	−	0.109973i	$-0.0350764\pi$
$677$	23.0000	+	23.0000i	0.883962	+	0.883962i	−0.109973	+	0.993935i	$0.535076\pi$
$678$			10.0000i			0.384048i
$679$		−	4.00000i		−	0.153506i
$680$		0			0
$681$	−12.0000	+	12.0000i	−0.459841	+	0.459841i
$682$	10.0000			0.382920
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$	−5.00000	+	5.00000i	−0.191180	+	0.191180i
$685$		0			0
$686$		−	20.0000i		−	0.763604i
$687$		−	6.00000i		−	0.228914i
$688$	−1.00000	−	1.00000i	−0.0381246	−	0.0381246i
$689$	25.0000	−	5.00000i	0.952424	−	0.190485i
$690$		0			0
$691$	−3.00000	+	3.00000i	−0.114125	+	0.114125i	−0.761863	−	0.647738i	$-0.775715\pi$
$691$	−3.00000	+	3.00000i	−0.114125	+	0.114125i	0.647738	+	0.761863i	$0.275715\pi$
$692$	−11.0000	−	11.0000i	−0.418157	−	0.418157i
$693$	2.00000	+	2.00000i	0.0759737	+	0.0759737i
$694$	3.00000	+	3.00000i	0.113878	+	0.113878i
$695$		0			0
$696$		0			0
$697$		−	14.0000i		−	0.530288i
$698$	−9.00000	+	9.00000i	−0.340655	+	0.340655i
$699$	2.00000			0.0756469
$700$		0			0
$701$			12.0000i			0.453234i	0.973984	+	0.226617i	$0.0727665\pi$
$701$			12.0000i			0.453234i	−0.973984	+	0.226617i	$0.927233\pi$
$702$	−4.00000	−	20.0000i	−0.150970	−	0.754851i
$703$		0			0
$704$	−7.00000	−	7.00000i	−0.263822	−	0.263822i
$705$		0			0
$706$			12.0000i			0.451626i
$707$	−24.0000			−0.902613
$708$			14.0000i			0.526152i
$709$	29.0000	−	29.0000i	1.08912	−	1.08912i	0.0934984	−	0.995619i	$-0.470195\pi$
$709$	29.0000	−	29.0000i	1.08912	−	1.08912i	0.995619	−	0.0934984i	$-0.0298050\pi$
$710$		0			0
$711$	2.00000			0.0750059
$712$	15.0000	−	15.0000i	0.562149	−	0.562149i
$713$	−30.0000			−1.12351
$714$	−4.00000			−0.149696
$715$		0			0
$716$	20.0000			0.747435
$717$	−6.00000			−0.224074
$718$	−1.00000	+	1.00000i	−0.0373197	+	0.0373197i
$719$	−8.00000			−0.298350			−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000			−0.298350			−0.149175	+	0.988811i	$0.547662\pi$
$720$		0			0
$721$	14.0000	−	14.0000i	0.521387	−	0.521387i
$722$		−	31.0000i		−	1.15370i
$723$	−34.0000			−1.26447
$724$			8.00000i			0.297318i
$725$		0			0
$726$	−9.00000	−	9.00000i	−0.334021	−	0.334021i
$727$	35.0000	−	35.0000i	1.29808	−	1.29808i	0.368418	−	0.929660i	$-0.379900\pi$
$727$	35.0000	−	35.0000i	1.29808	−	1.29808i	0.929660	−	0.368418i	$-0.120100\pi$
$728$	12.0000	−	18.0000i	0.444750	−	0.667124i
$729$			29.0000i			1.07407i
$730$		0			0
$731$	−2.00000			−0.0739727
$732$	−14.0000	+	14.0000i	−0.517455	+	0.517455i
$733$		−	4.00000i		−	0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$		−	4.00000i		−	0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	−1.00000	−	1.00000i	−0.0369107	−	0.0369107i
$735$		0			0
$736$	15.0000	+	15.0000i	0.552907	+	0.552907i
$737$	−4.00000	−	4.00000i	−0.147342	−	0.147342i
$738$	7.00000	+	7.00000i	0.257674	+	0.257674i
$739$	−3.00000	+	3.00000i	−0.110357	+	0.110357i	−0.760129	−	0.649772i	$-0.774864\pi$
$739$	−3.00000	+	3.00000i	−0.110357	+	0.110357i	0.649772	+	0.760129i	$0.274864\pi$
$740$		0			0
$741$	−30.0000	−	20.0000i	−1.10208	−	0.734718i
$742$	10.0000	+	10.0000i	0.367112	+	0.367112i
$743$			34.0000i			1.24734i	0.781688	+	0.623670i	$0.214359\pi$
$743$			34.0000i			1.24734i	−0.781688	+	0.623670i	$0.785641\pi$
$744$			30.0000i			1.09985i
$745$		0			0
$746$	−15.0000	+	15.0000i	−0.549189	+	0.549189i
$747$	−6.00000			−0.219529
$748$	−2.00000			−0.0731272
$749$	14.0000	−	14.0000i	0.511549	−	0.511549i
$750$		0			0
$751$		−	50.0000i		−	1.82453i	−0.409605	−	0.912263i	$-0.634333\pi$
$751$		−	50.0000i		−	1.82453i	0.409605	−	0.912263i	$-0.365667\pi$
$752$			6.00000i			0.218797i
$753$	−2.00000	−	2.00000i	−0.0728841	−	0.0728841i
$754$		0			0
$755$		0			0
$756$	−8.00000	+	8.00000i	−0.290957	+	0.290957i
$757$	35.0000	+	35.0000i	1.27210	+	1.27210i	0.944986	+	0.327111i	$0.106075\pi$
$757$	35.0000	+	35.0000i	1.27210	+	1.27210i	0.327111	+	0.944986i	$0.393925\pi$
$758$	1.00000	+	1.00000i	0.0363216	+	0.0363216i
$759$	−6.00000	−	6.00000i	−0.217786	−	0.217786i
$760$		0			0
$761$	−7.00000	−	7.00000i	−0.253750	−	0.253750i	0.568756	−	0.822506i	$-0.307425\pi$
$761$	−7.00000	−	7.00000i	−0.253750	−	0.253750i	−0.822506	+	0.568756i	$0.807425\pi$
$762$			18.0000i			0.652071i
$763$	−18.0000	+	18.0000i	−0.651644	+	0.651644i
$764$	−8.00000			−0.289430
$765$		0			0
$766$			30.0000i			1.08394i
$767$	35.0000	−	7.00000i	1.26378	−	0.252755i
$768$	17.0000	−	17.0000i	0.613435	−	0.613435i
$769$	−15.0000	−	15.0000i	−0.540914	−	0.540914i	0.382883	−	0.923797i	$-0.374931\pi$
$769$	−15.0000	−	15.0000i	−0.540914	−	0.540914i	−0.923797	+	0.382883i	$0.874931\pi$
$770$		0			0
$771$			22.0000i			0.792311i
$772$	18.0000			0.647834
$773$		−	32.0000i		−	1.15096i	−0.817816	−	0.575480i	$-0.804815\pi$
$773$		−	32.0000i		−	1.15096i	0.817816	−	0.575480i	$-0.195185\pi$
$774$	1.00000	−	1.00000i	0.0359443	−	0.0359443i
$775$		0			0
$776$	−6.00000			−0.215387
$777$		0			0
$778$	18.0000			0.645331
$779$	70.0000			2.50801
$780$		0			0
$781$	−2.00000			−0.0715656
$782$	−6.00000			−0.214560
$783$		0			0
$784$	−3.00000			−0.107143
$785$		0			0
$786$	−20.0000	+	20.0000i	−0.713376	+	0.713376i
$787$			22.0000i			0.784215i	0.919919	+	0.392108i	$0.128254\pi$
$787$			22.0000i			0.784215i	−0.919919	+	0.392108i	$0.871746\pi$
$788$	6.00000			0.213741
$789$		−	2.00000i		−	0.0712019i
$790$		0			0
$791$	−10.0000	−	10.0000i	−0.355559	−	0.355559i
$792$	3.00000	−	3.00000i	0.106600	−	0.106600i
$793$	42.0000	+	28.0000i	1.49146	+	0.994309i
$794$			16.0000i			0.567819i
$795$		0			0
$796$	8.00000			0.283552
$797$	−17.0000	+	17.0000i	−0.602171	+	0.602171i	−0.940888	−	0.338717i	$-0.890007\pi$
$797$	−17.0000	+	17.0000i	−0.602171	+	0.602171i	0.338717	+	0.940888i	$0.390007\pi$
$798$		−	20.0000i		−	0.707992i
$799$	6.00000	+	6.00000i	0.212265	+	0.212265i
$800$		0			0
$801$	5.00000	+	5.00000i	0.176666	+	0.176666i
$802$	11.0000	+	11.0000i	0.388424	+	0.388424i
$803$	−10.0000	−	10.0000i	−0.352892	−	0.352892i
$804$	4.00000	−	4.00000i	0.141069	−	0.141069i
$805$		0			0
$806$	25.0000	−	5.00000i	0.880587	−	0.176117i
$807$	−12.0000	−	12.0000i	−0.422420	−	0.422420i
$808$			36.0000i			1.26648i
$809$			28.0000i			0.984428i	0.870474	+	0.492214i	$0.163812\pi$
$809$			28.0000i			0.984428i	−0.870474	+	0.492214i	$0.836188\pi$
$810$		0			0
$811$	−27.0000	+	27.0000i	−0.948098	+	0.948098i	−0.998718	−	0.0506198i	$-0.983880\pi$
$811$	−27.0000	+	27.0000i	−0.948098	+	0.948098i	0.0506198	+	0.998718i	$0.483880\pi$
$812$		0			0
$813$	18.0000			0.631288
$814$		0			0
$815$		0			0
$816$			2.00000i			0.0700140i
$817$		−	10.0000i		−	0.349856i
$818$	7.00000	+	7.00000i	0.244749	+	0.244749i
$819$	6.00000	+	4.00000i	0.209657	+	0.139771i
$820$		0			0
$821$	9.00000	−	9.00000i	0.314102	−	0.314102i	−0.532394	−	0.846496i	$-0.678708\pi$
$821$	9.00000	−	9.00000i	0.314102	−	0.314102i	0.846496	+	0.532394i	$0.178708\pi$
$822$	16.0000	+	16.0000i	0.558064	+	0.558064i
$823$	9.00000	+	9.00000i	0.313720	+	0.313720i	0.846349	−	0.532629i	$-0.178796\pi$
$823$	9.00000	+	9.00000i	0.313720	+	0.313720i	−0.532629	+	0.846349i	$0.678796\pi$
$824$	−21.0000	−	21.0000i	−0.731570	−	0.731570i
$825$		0			0
$826$	14.0000	+	14.0000i	0.487122	+	0.487122i
$827$			46.0000i			1.59958i	0.600282	+	0.799788i	$0.295055\pi$
$827$			46.0000i			1.59958i	−0.600282	+	0.799788i	$0.704945\pi$
$828$	−3.00000	+	3.00000i	−0.104257	+	0.104257i
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$			30.0000i			1.04069i
$832$	−21.0000	−	14.0000i	−0.728044	−	0.485363i
$833$	−3.00000	+	3.00000i	−0.103944	+	0.103944i
$834$	−14.0000	−	14.0000i	−0.484780	−	0.484780i
$835$		0			0
$836$		−	10.0000i		−	0.345857i
$837$	−40.0000			−1.38260
$838$			38.0000i			1.31269i
$839$	−35.0000	+	35.0000i	−1.20833	+	1.20833i	−0.236768	+	0.971566i	$0.576088\pi$
$839$	−35.0000	+	35.0000i	−1.20833	+	1.20833i	−0.971566	+	0.236768i	$0.923912\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	11.0000	−	11.0000i	0.379085	−	0.379085i
$843$	−2.00000			−0.0688837
$844$	−4.00000			−0.137686
$845$		0			0
$846$	−6.00000			−0.206284
$847$	18.0000			0.618487
$848$	5.00000	−	5.00000i	0.171701	−	0.171701i
$849$	−18.0000			−0.617758
$850$		0			0
$851$		0			0
$852$		−	2.00000i		−	0.0685189i
$853$	−26.0000			−0.890223			−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000			−0.890223			−0.445112	+	0.895475i	$0.646836\pi$
$854$			28.0000i			0.958140i
$855$		0			0
$856$	−21.0000	−	21.0000i	−0.717765	−	0.717765i
$857$	3.00000	−	3.00000i	0.102478	−	0.102478i	−0.654009	−	0.756487i	$-0.726914\pi$
$857$	3.00000	−	3.00000i	0.102478	−	0.102478i	0.756487	+	0.654009i	$0.226914\pi$
$858$	6.00000	+	4.00000i	0.204837	+	0.136558i
$859$		−	30.0000i		−	1.02359i	−0.859109	−	0.511793i	$-0.828981\pi$
$859$		−	30.0000i		−	1.02359i	0.859109	−	0.511793i	$-0.171019\pi$
$860$		0			0
$861$	28.0000			0.954237
$862$	−13.0000	+	13.0000i	−0.442782	+	0.442782i
$863$		−	30.0000i		−	1.02121i	−0.859815	−	0.510606i	$-0.829421\pi$
$863$		−	30.0000i		−	1.02121i	0.859815	−	0.510606i	$-0.170579\pi$
$864$	20.0000	+	20.0000i	0.680414	+	0.680414i
$865$		0			0
$866$	17.0000	+	17.0000i	0.577684	+	0.577684i
$867$	−15.0000	−	15.0000i	−0.509427	−	0.509427i
$868$	−10.0000	−	10.0000i	−0.339422	−	0.339422i
$869$	−2.00000	+	2.00000i	−0.0678454	+	0.0678454i
$870$		0			0
$871$	−12.0000	−	8.00000i	−0.406604	−	0.271070i
$872$	27.0000	+	27.0000i	0.914335	+	0.914335i
$873$		−	2.00000i		−	0.0676897i
$874$		−	30.0000i		−	1.01477i
$875$		0			0
$876$	10.0000	−	10.0000i	0.337869	−	0.337869i
$877$	−38.0000			−1.28317			−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000			−1.28317			−0.641584	+	0.767052i	$0.721723\pi$
$878$		0			0
$879$	−6.00000	+	6.00000i	−0.202375	+	0.202375i
$880$		0			0
$881$		−	52.0000i		−	1.75192i	−0.482380	−	0.875962i	$-0.660227\pi$
$881$		−	52.0000i		−	1.75192i	0.482380	−	0.875962i	$-0.339773\pi$
$882$		−	3.00000i		−	0.101015i
$883$	−39.0000	−	39.0000i	−1.31245	−	1.31245i	−0.919601	−	0.392853i	$-0.871488\pi$
$883$	−39.0000	−	39.0000i	−1.31245	−	1.31245i	−0.392853	−	0.919601i	$-0.628512\pi$
$884$	−5.00000	+	1.00000i	−0.168168	+	0.0336336i
$885$		0			0
$886$	25.0000	−	25.0000i	0.839891	−	0.839891i
$887$	1.00000	+	1.00000i	0.0335767	+	0.0335767i	0.723696	−	0.690119i	$-0.242442\pi$
$887$	1.00000	+	1.00000i	0.0335767	+	0.0335767i	−0.690119	+	0.723696i	$0.742442\pi$
$888$		0			0
$889$	−18.0000	−	18.0000i	−0.603701	−	0.603701i
$890$		0			0
$891$	−5.00000	−	5.00000i	−0.167506	−	0.167506i
$892$		−	2.00000i		−	0.0669650i
$893$	−30.0000	+	30.0000i	−1.00391	+	1.00391i
$894$	−6.00000			−0.200670
$895$		0			0
$896$			6.00000i			0.200446i
$897$	−18.0000	−	12.0000i	−0.601003	−	0.400668i
$898$	3.00000	−	3.00000i	0.100111	−	0.100111i
$899$		0			0
$900$		0			0
$901$		−	10.0000i		−	0.333148i
$902$	−14.0000			−0.466149
$903$		−	4.00000i		−	0.133112i
$904$	−15.0000	+	15.0000i	−0.498893	+	0.498893i
$905$		0			0
$906$	14.0000			0.465119
$907$	39.0000	−	39.0000i	1.29497	−	1.29497i	0.363303	−	0.931671i	$-0.381649\pi$
$907$	39.0000	−	39.0000i	1.29497	−	1.29497i	0.931671	−	0.363303i	$-0.118351\pi$
$908$	−12.0000			−0.398234
$909$	−12.0000			−0.398015
$910$		0			0
$911$	24.0000			0.795155			0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000			0.795155			0.397578	+	0.917568i	$0.369851\pi$
$912$	−10.0000			−0.331133
$913$	6.00000	−	6.00000i	0.198571	−	0.198571i
$914$	2.00000			0.0661541
$915$		0			0
$916$	3.00000	−	3.00000i	0.0991228	−	0.0991228i
$917$		−	40.0000i		−	1.32092i
$918$	−8.00000			−0.264039
$919$		−	10.0000i		−	0.329870i	−0.986304	−	0.164935i	$-0.947259\pi$
$919$		−	10.0000i		−	0.329870i	0.986304	−	0.164935i	$-0.0527414\pi$
$920$		0			0
$921$	18.0000	+	18.0000i	0.593120	+	0.593120i
$922$	−17.0000	+	17.0000i	−0.559865	+	0.559865i
$923$	−5.00000	+	1.00000i	−0.164577	+	0.0329154i
$924$		−	4.00000i		−	0.131590i
$925$		0			0
$926$	−24.0000			−0.788689
$927$	7.00000	−	7.00000i	0.229910	−	0.229910i
$928$		0			0
$929$	−19.0000	−	19.0000i	−0.623370	−	0.623370i	0.323022	−	0.946392i	$-0.395301\pi$
$929$	−19.0000	−	19.0000i	−0.623370	−	0.623370i	−0.946392	+	0.323022i	$0.895301\pi$
$930$		0			0
$931$	−15.0000	−	15.0000i	−0.491605	−	0.491605i
$932$	1.00000	+	1.00000i	0.0327561	+	0.0327561i
$933$	−6.00000	−	6.00000i	−0.196431	−	0.196431i
$934$	9.00000	−	9.00000i	0.294489	−	0.294489i
$935$		0			0
$936$	6.00000	−	9.00000i	0.196116	−	0.294174i
$937$	7.00000	+	7.00000i	0.228680	+	0.228680i	0.812141	−	0.583461i	$-0.198302\pi$
$937$	7.00000	+	7.00000i	0.228680	+	0.228680i	−0.583461	+	0.812141i	$0.698302\pi$
$938$		−	8.00000i		−	0.261209i
$939$		−	18.0000i		−	0.587408i
$940$		0			0
$941$	21.0000	−	21.0000i	0.684580	−	0.684580i	−0.276448	−	0.961029i	$-0.589157\pi$
$941$	21.0000	−	21.0000i	0.684580	−	0.684580i	0.961029	+	0.276448i	$0.0891575\pi$
$942$	−26.0000			−0.847126
$943$	42.0000			1.36771
$944$	7.00000	−	7.00000i	0.227831	−	0.227831i
$945$		0			0
$946$			2.00000i			0.0650256i
$947$		−	18.0000i		−	0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$		−	18.0000i		−	0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	−2.00000	−	2.00000i	−0.0649570	−	0.0649570i
$949$	−30.0000	−	20.0000i	−0.973841	−	0.649227i
$950$		0			0
$951$	−14.0000	+	14.0000i	−0.453981	+	0.453981i
$952$	−6.00000	−	6.00000i	−0.194461	−	0.194461i
$953$	−13.0000	−	13.0000i	−0.421111	−	0.421111i	0.464475	−	0.885586i	$-0.346243\pi$
$953$	−13.0000	−	13.0000i	−0.421111	−	0.421111i	−0.885586	+	0.464475i	$0.846243\pi$
$954$	5.00000	+	5.00000i	0.161881	+	0.161881i
$955$		0			0
$956$	−3.00000	−	3.00000i	−0.0970269	−	0.0970269i
$957$		0			0
$958$	7.00000	−	7.00000i	0.226160	−	0.226160i
$959$	−32.0000			−1.03333
$960$		0			0
$961$		−	19.0000i		−	0.612903i
$962$		0			0
$963$	7.00000	−	7.00000i	0.225572	−	0.225572i
$964$	−17.0000	−	17.0000i	−0.547533	−	0.547533i
$965$		0			0
$966$		−	12.0000i		−	0.386094i
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		−	27.0000i		−	0.867813i
$969$	−10.0000	+	10.0000i	−0.321246	+	0.321246i
$970$		0			0
$971$	−60.0000			−1.92549			−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000			−1.92549			−0.962746	+	0.270408i	$0.912841\pi$
$972$	−7.00000	+	7.00000i	−0.224525	+	0.224525i
$973$	28.0000			0.897639
$974$	16.0000			0.512673
$975$		0			0
$976$	14.0000			0.448129
$977$	−62.0000			−1.98356			−0.991778	−	0.127971i	$-0.959153\pi$
$977$	−62.0000			−1.98356			−0.991778	+	0.127971i	$0.959153\pi$
$978$	−4.00000	+	4.00000i	−0.127906	+	0.127906i
$979$	−10.0000			−0.319601
$980$		0			0
$981$	−9.00000	+	9.00000i	−0.287348	+	0.287348i
$982$			22.0000i			0.702048i
$983$	24.0000			0.765481			0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000			0.765481			0.382741	+	0.923856i	$0.374980\pi$
$984$		−	42.0000i		−	1.33891i
$985$		0			0
$986$		0			0
$987$	−12.0000	+	12.0000i	−0.381964	+	0.381964i
$988$	−5.00000	−	25.0000i	−0.159071	−	0.795356i
$989$		−	6.00000i		−	0.190789i
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	−25.0000	+	25.0000i	−0.793751	+	0.793751i
$993$		−	6.00000i		−	0.190404i
$994$	−2.00000	−	2.00000i	−0.0634361	−	0.0634361i
$995$		0			0
$996$	6.00000	+	6.00000i	0.190117	+	0.190117i
$997$	−9.00000	−	9.00000i	−0.285033	−	0.285033i	0.550079	−	0.835112i	$-0.314597\pi$
$997$	−9.00000	−	9.00000i	−0.285033	−	0.285033i	−0.835112	+	0.550079i	$0.814597\pi$
$998$	−3.00000	−	3.00000i	−0.0949633	−	0.0949633i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.2.k.a.57.1		2
5.2	odd	4		65.2.f.a.18.1	✓	2
5.3	odd	4		325.2.f.a.18.1		2
5.4	even	2		65.2.k.a.57.1	yes	2
13.8	odd	4		325.2.f.a.307.1		2
15.2	even	4		585.2.n.c.343.1		2
15.14	odd	2		585.2.w.b.577.1		2
20.7	even	4		1040.2.cd.b.993.1		2
20.19	odd	2		1040.2.bg.a.577.1		2
65.2	even	12		845.2.o.b.488.1		4
65.4	even	6		845.2.o.b.587.1		4
65.7	even	12		845.2.o.a.258.1		4
65.8	even	4	inner	325.2.k.a.268.1		2
65.9	even	6		845.2.o.a.587.1		4
65.12	odd	4		845.2.f.a.408.1		2
65.17	odd	12		845.2.t.b.418.1		4
65.19	odd	12		845.2.t.b.427.1		4
65.22	odd	12		845.2.t.a.418.1		4
65.24	odd	12		845.2.t.a.657.1		4
65.29	even	6		845.2.o.a.357.1		4
65.32	even	12		845.2.o.b.258.1		4
65.34	odd	4		65.2.f.a.47.1	yes	2
65.37	even	12		845.2.o.a.488.1		4
65.42	odd	12		845.2.t.a.188.1		4
65.44	odd	4		845.2.f.a.437.1		2
65.47	even	4		65.2.k.a.8.1	yes	2
65.49	even	6		845.2.o.b.357.1		4
65.54	odd	12		845.2.t.b.657.1		4
65.57	even	4		845.2.k.a.268.1		2
65.59	odd	12		845.2.t.a.427.1		4
65.62	odd	12		845.2.t.b.188.1		4
65.64	even	2		845.2.k.a.577.1		2
195.47	odd	4		585.2.w.b.73.1		2
195.164	even	4		585.2.n.c.307.1		2
260.47	odd	4		1040.2.bg.a.593.1		2
260.99	even	4		1040.2.cd.b.177.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.f.a.18.1	✓	2	5.2	odd	4
65.2.f.a.47.1	yes	2	65.34	odd	4
65.2.k.a.8.1	yes	2	65.47	even	4
65.2.k.a.57.1	yes	2	5.4	even	2
325.2.f.a.18.1		2	5.3	odd	4
325.2.f.a.307.1		2	13.8	odd	4
325.2.k.a.57.1		2	1.1	even	1	trivial
325.2.k.a.268.1		2	65.8	even	4	inner
585.2.n.c.307.1		2	195.164	even	4
585.2.n.c.343.1		2	15.2	even	4
585.2.w.b.73.1		2	195.47	odd	4
585.2.w.b.577.1		2	15.14	odd	2
845.2.f.a.408.1		2	65.12	odd	4
845.2.f.a.437.1		2	65.44	odd	4
845.2.k.a.268.1		2	65.57	even	4
845.2.k.a.577.1		2	65.64	even	2
845.2.o.a.258.1		4	65.7	even	12
845.2.o.a.357.1		4	65.29	even	6
845.2.o.a.488.1		4	65.37	even	12
845.2.o.a.587.1		4	65.9	even	6
845.2.o.b.258.1		4	65.32	even	12
845.2.o.b.357.1		4	65.49	even	6
845.2.o.b.488.1		4	65.2	even	12
845.2.o.b.587.1		4	65.4	even	6
845.2.t.a.188.1		4	65.42	odd	12
845.2.t.a.418.1		4	65.22	odd	12
845.2.t.a.427.1		4	65.59	odd	12
845.2.t.a.657.1		4	65.24	odd	12
845.2.t.b.188.1		4	65.62	odd	12
845.2.t.b.418.1		4	65.17	odd	12
845.2.t.b.427.1		4	65.19	odd	12
845.2.t.b.657.1		4	65.54	odd	12
1040.2.bg.a.577.1		2	20.19	odd	2
1040.2.bg.a.593.1		2	260.47	odd	4
1040.2.cd.b.177.1		2	260.99	even	4
1040.2.cd.b.993.1		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +(-1.00000 + 1.00000i) q^{3} -1.00000 q^{4} +(1.00000 - 1.00000i) q^{6} +2.00000i q^{7} +3.00000 q^{8} +1.00000i q^{9} +(-1.00000 - 1.00000i) q^{11} +(1.00000 - 1.00000i) q^{12} +(-3.00000 - 2.00000i) q^{13} -2.00000i q^{14} -1.00000 q^{16} +(-1.00000 + 1.00000i) q^{17} -1.00000i q^{18} +(-5.00000 - 5.00000i) q^{19} +(-2.00000 - 2.00000i) q^{21} +(1.00000 + 1.00000i) q^{22} +(-3.00000 - 3.00000i) q^{23} +(-3.00000 + 3.00000i) q^{24} +(3.00000 + 2.00000i) q^{26} +(-4.00000 - 4.00000i) q^{27} -2.00000i q^{28} +(5.00000 - 5.00000i) q^{31} -5.00000 q^{32} +2.00000 q^{33} +(1.00000 - 1.00000i) q^{34} -1.00000i q^{36} +(5.00000 + 5.00000i) q^{38} +(5.00000 - 1.00000i) q^{39} +(-7.00000 + 7.00000i) q^{41} +(2.00000 + 2.00000i) q^{42} +(1.00000 + 1.00000i) q^{43} +(1.00000 + 1.00000i) q^{44} +(3.00000 + 3.00000i) q^{46} -6.00000i q^{47} +(1.00000 - 1.00000i) q^{48} +3.00000 q^{49} -2.00000i q^{51} +(3.00000 + 2.00000i) q^{52} +(-5.00000 + 5.00000i) q^{53} +(4.00000 + 4.00000i) q^{54} +6.00000i q^{56} +10.0000 q^{57} +(-7.00000 + 7.00000i) q^{59} -14.0000 q^{61} +(-5.00000 + 5.00000i) q^{62} -2.00000 q^{63} +7.00000 q^{64} -2.00000 q^{66} +4.00000 q^{67} +(1.00000 - 1.00000i) q^{68} +6.00000 q^{69} +(1.00000 - 1.00000i) q^{71} +3.00000i q^{72} +10.0000 q^{73} +(5.00000 + 5.00000i) q^{76} +(2.00000 - 2.00000i) q^{77} +(-5.00000 + 1.00000i) q^{78} -2.00000i q^{79} +5.00000 q^{81} +(7.00000 - 7.00000i) q^{82} +6.00000i q^{83} +(2.00000 + 2.00000i) q^{84} +(-1.00000 - 1.00000i) q^{86} +(-3.00000 - 3.00000i) q^{88} +(5.00000 - 5.00000i) q^{89} +(4.00000 - 6.00000i) q^{91} +(3.00000 + 3.00000i) q^{92} +10.0000i q^{93} +6.00000i q^{94} +(5.00000 - 5.00000i) q^{96} -2.00000 q^{97} -3.00000 q^{98} +(1.00000 - 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 6 q^{8} - 2 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{16} - 2 q^{17} - 10 q^{19} - 4 q^{21} + 2 q^{22} - 6 q^{23} - 6 q^{24} + 6 q^{26} - 8 q^{27} + 10 q^{31} - 10 q^{32}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 6 * q^8 - 2 * q^11 + 2 * q^12 - 6 * q^13 - 2 * q^16 - 2 * q^17 - 10 * q^19 - 4 * q^21 + 2 * q^22 - 6 * q^23 - 6 * q^24 + 6 * q^26 - 8 * q^27 + 10 * q^31 - 10 * q^32 + 4 * q^33 + 2 * q^34 + 10 * q^38 + 10 * q^39 - 14 * q^41 + 4 * q^42 + 2 * q^43 + 2 * q^44 + 6 * q^46 + 2 * q^48 + 6 * q^49 + 6 * q^52 - 10 * q^53 + 8 * q^54 + 20 * q^57 - 14 * q^59 - 28 * q^61 - 10 * q^62 - 4 * q^63 + 14 * q^64 - 4 * q^66 + 8 * q^67 + 2 * q^68 + 12 * q^69 + 2 * q^71 + 20 * q^73 + 10 * q^76 + 4 * q^77 - 10 * q^78 + 10 * q^81 + 14 * q^82 + 4 * q^84 - 2 * q^86 - 6 * q^88 + 10 * q^89 + 8 * q^91 + 6 * q^92 + 10 * q^96 - 4 * q^97 - 6 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more