LMFDB - Embedded newform 1805.2.b.c.1084.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,2,Mod(1084,1805)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1805.1084");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1805 = 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1805.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$14.4129975648$
Analytic rank:	$0$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1084.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1805.1084
Dual form			1805.2.b.c.1084.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +2.00000i q^{7} +3.00000i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +2.00000i q^{7} +3.00000i q^{8} +3.00000 q^{9} +(2.00000 - 1.00000i) q^{10} -4.00000 q^{11} -2.00000i q^{13} -2.00000 q^{14} -1.00000 q^{16} +4.00000i q^{17} +3.00000i q^{18} +(-1.00000 - 2.00000i) q^{20} -4.00000i q^{22} +6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +2.00000 q^{26} +2.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} +5.00000i q^{32} -4.00000 q^{34} +(4.00000 - 2.00000i) q^{35} +3.00000 q^{36} +10.0000i q^{37} +(6.00000 - 3.00000i) q^{40} +10.0000 q^{41} -2.00000i q^{43} -4.00000 q^{44} +(-3.00000 - 6.00000i) q^{45} -6.00000 q^{46} -6.00000i q^{47} +3.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -2.00000i q^{52} +10.0000i q^{53} +(4.00000 + 8.00000i) q^{55} -6.00000 q^{56} -6.00000i q^{58} +2.00000 q^{61} +4.00000i q^{62} +6.00000i q^{63} -7.00000 q^{64} +(-4.00000 + 2.00000i) q^{65} -8.00000i q^{67} +4.00000i q^{68} +(2.00000 + 4.00000i) q^{70} -4.00000 q^{71} +9.00000i q^{72} -4.00000i q^{73} -10.0000 q^{74} -8.00000i q^{77} +4.00000 q^{79} +(1.00000 + 2.00000i) q^{80} +9.00000 q^{81} +10.0000i q^{82} +18.0000i q^{83} +(8.00000 - 4.00000i) q^{85} +2.00000 q^{86} -12.0000i q^{88} -2.00000 q^{89} +(6.00000 - 3.00000i) q^{90} +4.00000 q^{91} +6.00000i q^{92} +6.00000 q^{94} -6.00000i q^{97} +3.00000i q^{98} -12.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{5} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^4 - 2 * q^5 + 6 * q^9	$ 2 q + 2 q^{4} - 2 q^{5} + 6 q^{9} + 4 q^{10} - 8 q^{11} - 4 q^{14} - 2 q^{16} - 2 q^{20} - 6 q^{25} + 4 q^{26} - 12 q^{29} + 8 q^{31} - 8 q^{34} + 8 q^{35} + 6 q^{36} + 12 q^{40} + 20 q^{41} - 8 q^{44} - 6 q^{45} - 12 q^{46} + 6 q^{49} - 8 q^{50} + 8 q^{55} - 12 q^{56} + 4 q^{61} - 14 q^{64} - 8 q^{65} + 4 q^{70} - 8 q^{71} - 20 q^{74} + 8 q^{79} + 2 q^{80} + 18 q^{81} + 16 q^{85} + 4 q^{86} - 4 q^{89} + 12 q^{90} + 8 q^{91} + 12 q^{94} - 24 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^5 + 6 * q^9 + 4 * q^10 - 8 * q^11 - 4 * q^14 - 2 * q^16 - 2 * q^20 - 6 * q^25 + 4 * q^26 - 12 * q^29 + 8 * q^31 - 8 * q^34 + 8 * q^35 + 6 * q^36 + 12 * q^40 + 20 * q^41 - 8 * q^44 - 6 * q^45 - 12 * q^46 + 6 * q^49 - 8 * q^50 + 8 * q^55 - 12 * q^56 + 4 * q^61 - 14 * q^64 - 8 * q^65 + 4 * q^70 - 8 * q^71 - 20 * q^74 + 8 * q^79 + 2 * q^80 + 18 * q^81 + 16 * q^85 + 4 * q^86 - 4 * q^89 + 12 * q^90 + 8 * q^91 + 12 * q^94 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$.

$n$	$362$	$1446$
$\chi(n)$	$-1$	$1$

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 $

$2$			1.00000i			0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$			1.00000i			0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	1.00000			0.500000
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$			3.00000i			1.06066i
$9$	3.00000			1.00000
$10$	2.00000	−	1.00000i	0.632456	−	0.316228i
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−2.00000			−0.534522
$15$		0			0
$16$	−1.00000			−0.250000
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$			3.00000i			0.707107i
$19$		0			0
$19$		0			0
$20$	−1.00000	−	2.00000i	−0.223607	−	0.447214i
$21$		0			0
$22$		−	4.00000i		−	0.852803i
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$	2.00000			0.392232
$27$		0			0
$28$			2.00000i			0.377964i
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$			5.00000i			0.883883i
$33$		0			0
$34$	−4.00000			−0.685994
$35$	4.00000	−	2.00000i	0.676123	−	0.338062i
$36$	3.00000			0.500000
$37$			10.0000i			1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$			10.0000i			1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$		0			0
$39$		0			0
$40$	6.00000	−	3.00000i	0.948683	−	0.474342i
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$		0			0
$43$		−	2.00000i		−	0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$		−	2.00000i		−	0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	−4.00000			−0.603023
$45$	−3.00000	−	6.00000i	−0.447214	−	0.894427i
$46$	−6.00000			−0.884652
$47$		−	6.00000i		−	0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$		−	6.00000i		−	0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$	−4.00000	−	3.00000i	−0.565685	−	0.424264i
$51$		0			0
$52$		−	2.00000i		−	0.277350i
$53$			10.0000i			1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$			10.0000i			1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$		0			0
$55$	4.00000	+	8.00000i	0.539360	+	1.07872i
$56$	−6.00000			−0.801784
$57$		0			0
$58$		−	6.00000i		−	0.787839i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$			4.00000i			0.508001i
$63$			6.00000i			0.755929i
$64$	−7.00000			−0.875000
$65$	−4.00000	+	2.00000i	−0.496139	+	0.248069i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$			4.00000i			0.485071i
$69$		0			0
$70$	2.00000	+	4.00000i	0.239046	+	0.478091i
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$			9.00000i			1.06066i
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−10.0000			−1.16248
$75$		0			0
$76$		0			0
$77$		−	8.00000i		−	0.911685i
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$	1.00000	+	2.00000i	0.111803	+	0.223607i
$81$	9.00000			1.00000
$82$			10.0000i			1.10432i
$83$			18.0000i			1.97576i	0.155230	+	0.987878i	$0.450388\pi$
$83$			18.0000i			1.97576i	−0.155230	+	0.987878i	$0.549612\pi$
$84$		0			0
$85$	8.00000	−	4.00000i	0.867722	−	0.433861i
$86$	2.00000			0.215666
$87$		0			0
$88$		−	12.0000i		−	1.27920i
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$	6.00000	−	3.00000i	0.632456	−	0.316228i
$91$	4.00000			0.419314
$92$			6.00000i			0.625543i
$93$		0			0
$94$	6.00000			0.618853
$95$		0			0
$96$		0			0
$97$		−	6.00000i		−	0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$		−	6.00000i		−	0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$			3.00000i			0.303046i
$99$	−12.0000			−1.20605
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$			16.0000i			1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$			16.0000i			1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	6.00000			0.588348
$105$		0			0
$106$	−10.0000			−0.971286
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$	6.00000			0.574696			0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000			0.574696			0.287348	+	0.957826i	$0.407226\pi$
$110$	−8.00000	+	4.00000i	−0.762770	+	0.381385i
$111$		0			0
$112$		−	2.00000i		−	0.188982i
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$	12.0000	−	6.00000i	1.11901	−	0.559503i
$116$	−6.00000			−0.557086
$117$		−	6.00000i		−	0.554700i
$118$		0			0
$119$	−8.00000			−0.733359
$120$		0			0
$121$	5.00000			0.454545
$122$			2.00000i			0.181071i
$123$		0			0
$124$	4.00000			0.359211
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$	−6.00000			−0.534522
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$			3.00000i			0.265165i
$129$		0			0
$130$	−2.00000	−	4.00000i	−0.175412	−	0.350823i
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		0			0
$134$	8.00000			0.691095
$135$		0			0
$136$	−12.0000			−1.02899
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$	4.00000	−	2.00000i	0.338062	−	0.169031i
$141$		0			0
$142$		−	4.00000i		−	0.335673i
$143$			8.00000i			0.668994i
$144$	−3.00000			−0.250000
$145$	6.00000	+	12.0000i	0.498273	+	0.996546i
$146$	4.00000			0.331042
$147$		0			0
$148$			10.0000i			0.821995i
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$	−24.0000			−1.95309			−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000			−1.95309			−0.976546	+	0.215308i	$0.930924\pi$
$152$		0			0
$153$			12.0000i			0.970143i
$154$	8.00000			0.644658
$155$	−4.00000	−	8.00000i	−0.321288	−	0.642575i
$156$		0			0
$157$		−	8.00000i		−	0.638470i	−0.947676	−	0.319235i	$-0.896574\pi$
$157$		−	8.00000i		−	0.638470i	0.947676	−	0.319235i	$-0.103426\pi$
$158$			4.00000i			0.318223i
$159$		0			0
$160$	10.0000	−	5.00000i	0.790569	−	0.395285i
$161$	−12.0000			−0.945732
$162$			9.00000i			0.707107i
$163$		−	10.0000i		−	0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$		−	10.0000i		−	0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	10.0000			0.780869
$165$		0			0
$166$	−18.0000			−1.39707
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$	4.00000	+	8.00000i	0.306786	+	0.613572i
$171$		0			0
$172$		−	2.00000i		−	0.152499i
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$		0			0
$175$	−8.00000	−	6.00000i	−0.604743	−	0.453557i
$176$	4.00000			0.301511
$177$		0			0
$178$		−	2.00000i		−	0.149906i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−3.00000	−	6.00000i	−0.223607	−	0.447214i
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$			4.00000i			0.296500i
$183$		0			0
$184$	−18.0000			−1.32698
$185$	20.0000	−	10.0000i	1.47043	−	0.735215i
$186$		0			0
$187$		−	16.0000i		−	1.17004i
$188$		−	6.00000i		−	0.437595i
$189$		0			0
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$		−	26.0000i		−	1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$		−	26.0000i		−	1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$	6.00000			0.430775
$195$		0			0
$196$	3.00000			0.214286
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		−	12.0000i		−	0.852803i
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$	−12.0000	−	9.00000i	−0.848528	−	0.636396i
$201$		0			0
$202$		−	6.00000i		−	0.422159i
$203$		−	12.0000i		−	0.842235i
$204$		0			0
$205$	−10.0000	−	20.0000i	−0.698430	−	1.39686i
$206$	−16.0000			−1.11477
$207$			18.0000i			1.25109i
$208$			2.00000i			0.138675i
$209$		0			0
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$			10.0000i			0.686803i
$213$		0			0
$214$	−4.00000			−0.273434
$215$	−4.00000	+	2.00000i	−0.272798	+	0.136399i
$216$		0			0
$217$			8.00000i			0.543075i
$218$			6.00000i			0.406371i
$219$		0			0
$220$	4.00000	+	8.00000i	0.269680	+	0.539360i
$221$	8.00000			0.538138
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	−10.0000			−0.668153
$225$	−9.00000	+	12.0000i	−0.600000	+	0.800000i
$226$	−6.00000			−0.399114
$227$		−	20.0000i		−	1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$		−	20.0000i		−	1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$	6.00000	+	12.0000i	0.395628	+	0.791257i
$231$		0			0
$232$		−	18.0000i		−	1.18176i
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$	6.00000			0.392232
$235$	−12.0000	+	6.00000i	−0.782794	+	0.391397i
$236$		0			0
$237$		0			0
$238$		−	8.00000i		−	0.518563i
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$			5.00000i			0.321412i
$243$		0			0
$244$	2.00000			0.128037
$245$	−3.00000	−	6.00000i	−0.191663	−	0.383326i
$246$		0			0
$247$		0			0
$248$			12.0000i			0.762001i
$249$		0			0
$250$	−2.00000	+	11.0000i	−0.126491	+	0.695701i
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$			6.00000i			0.377964i
$253$		−	24.0000i		−	1.50887i
$254$		0			0
$255$		0			0
$256$	−17.0000			−1.06250
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$	−20.0000			−1.24274
$260$	−4.00000	+	2.00000i	−0.248069	+	0.124035i
$261$	−18.0000			−1.11417
$262$		−	12.0000i		−	0.741362i
$263$			6.00000i			0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$			6.00000i			0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$		0			0
$265$	20.0000	−	10.0000i	1.22859	−	0.614295i
$266$		0			0
$267$		0			0
$268$		−	8.00000i		−	0.488678i
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$		0			0
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$		−	4.00000i		−	0.242536i
$273$		0			0
$274$	−12.0000			−0.724947
$275$	12.0000	−	16.0000i	0.723627	−	0.964836i
$276$		0			0
$277$			8.00000i			0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$			8.00000i			0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$			4.00000i			0.239904i
$279$	12.0000			0.718421
$280$	6.00000	+	12.0000i	0.358569	+	0.717137i
$281$	26.0000			1.55103			0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000			1.55103			0.775515	+	0.631329i	$0.217490\pi$
$282$		0			0
$283$		−	14.0000i		−	0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$		−	14.0000i		−	0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	−4.00000			−0.237356
$285$		0			0
$286$	−8.00000			−0.473050
$287$			20.0000i			1.18056i
$288$			15.0000i			0.883883i
$289$	1.00000			0.0588235
$290$	−12.0000	+	6.00000i	−0.704664	+	0.352332i
$291$		0			0
$292$		−	4.00000i		−	0.234082i
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$		0			0
$295$		0			0
$296$	−30.0000			−1.74371
$297$		0			0
$298$			10.0000i			0.579284i
$299$	12.0000			0.693978
$300$		0			0
$301$	4.00000			0.230556
$302$		−	24.0000i		−	1.38104i
$303$		0			0
$304$		0			0
$305$	−2.00000	−	4.00000i	−0.114520	−	0.229039i
$306$	−12.0000			−0.685994
$307$			20.0000i			1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$			20.0000i			1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$		−	8.00000i		−	0.455842i
$309$		0			0
$310$	8.00000	−	4.00000i	0.454369	−	0.227185i
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$			24.0000i			1.35656i	0.734803	+	0.678280i	$0.237274\pi$
$313$			24.0000i			1.35656i	−0.734803	+	0.678280i	$0.762726\pi$
$314$	8.00000			0.451466
$315$	12.0000	−	6.00000i	0.676123	−	0.338062i
$316$	4.00000			0.225018
$317$		−	22.0000i		−	1.23564i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$		−	22.0000i		−	1.23564i	0.786318	−	0.617822i	$-0.211985\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$	7.00000	+	14.0000i	0.391312	+	0.782624i
$321$		0			0
$322$		−	12.0000i		−	0.668734i
$323$		0			0
$324$	9.00000			0.500000
$325$	8.00000	+	6.00000i	0.443760	+	0.332820i
$326$	10.0000			0.553849
$327$		0			0
$328$			30.0000i			1.65647i
$329$	12.0000			0.661581
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$			18.0000i			0.987878i
$333$			30.0000i			1.64399i
$334$	12.0000			0.656611
$335$	−16.0000	+	8.00000i	−0.874173	+	0.437087i
$336$		0			0
$337$		−	22.0000i		−	1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$		−	22.0000i		−	1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$			9.00000i			0.489535i
$339$		0			0
$340$	8.00000	−	4.00000i	0.433861	−	0.216930i
$341$	−16.0000			−0.866449
$342$		0			0
$343$			20.0000i			1.07990i
$344$	6.00000			0.323498
$345$		0			0
$346$	6.00000			0.322562
$347$		−	6.00000i		−	0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$		−	6.00000i		−	0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$	6.00000	−	8.00000i	0.320713	−	0.427618i
$351$		0			0
$352$		−	20.0000i		−	1.06600i
$353$		−	4.00000i		−	0.212899i	−0.994318	−	0.106449i	$-0.966052\pi$
$353$		−	4.00000i		−	0.212899i	0.994318	−	0.106449i	$-0.0339482\pi$
$354$		0			0
$355$	4.00000	+	8.00000i	0.212298	+	0.424596i
$356$	−2.00000			−0.106000
$357$		0			0
$358$		0			0
$359$	8.00000			0.422224			0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000			0.422224			0.211112	+	0.977462i	$0.432292\pi$
$360$	18.0000	−	9.00000i	0.948683	−	0.474342i
$361$		0			0
$362$			18.0000i			0.946059i
$363$		0			0
$364$	4.00000			0.209657
$365$	−8.00000	+	4.00000i	−0.418739	+	0.209370i
$366$		0			0
$367$		−	22.0000i		−	1.14839i	−0.818718	−	0.574195i	$-0.805315\pi$
$367$		−	22.0000i		−	1.14839i	0.818718	−	0.574195i	$-0.194685\pi$
$368$		−	6.00000i		−	0.312772i
$369$	30.0000			1.56174
$370$	10.0000	+	20.0000i	0.519875	+	1.03975i
$371$	−20.0000			−1.03835
$372$		0			0
$373$		−	6.00000i		−	0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$		−	6.00000i		−	0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	16.0000			0.827340
$375$		0			0
$376$	18.0000			0.928279
$377$			12.0000i			0.618031i
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$		0			0
$382$		−	24.0000i		−	1.22795i
$383$			12.0000i			0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$			12.0000i			0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$		0			0
$385$	−16.0000	+	8.00000i	−0.815436	+	0.407718i
$386$	26.0000			1.32337
$387$		−	6.00000i		−	0.304997i
$388$		−	6.00000i		−	0.304604i
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$			9.00000i			0.454569i
$393$		0			0
$394$	12.0000			0.604551
$395$	−4.00000	−	8.00000i	−0.201262	−	0.402524i
$396$	−12.0000			−0.603023
$397$		−	8.00000i		−	0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$		−	16.0000i		−	0.802008i
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		−	8.00000i		−	0.398508i
$404$	−6.00000			−0.298511
$405$	−9.00000	−	18.0000i	−0.447214	−	0.894427i
$406$	12.0000			0.595550
$407$		−	40.0000i		−	1.98273i
$408$		0			0
$409$	−14.0000			−0.692255			−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000			−0.692255			−0.346128	+	0.938187i	$0.612504\pi$
$410$	20.0000	−	10.0000i	0.987730	−	0.493865i
$411$		0			0
$412$			16.0000i			0.788263i
$413$		0			0
$414$	−18.0000			−0.884652
$415$	36.0000	−	18.0000i	1.76717	−	0.883585i
$416$	10.0000			0.490290
$417$		0			0
$418$		0			0
$419$	−36.0000			−1.75872			−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000			−1.75872			−0.879358	+	0.476162i	$0.842028\pi$
$420$		0			0
$421$	30.0000			1.46211			0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000			1.46211			0.731055	+	0.682318i	$0.239028\pi$
$422$		−	20.0000i		−	0.973585i
$423$		−	18.0000i		−	0.875190i
$424$	−30.0000			−1.45693
$425$	−16.0000	−	12.0000i	−0.776114	−	0.582086i
$426$		0			0
$427$			4.00000i			0.193574i
$428$			4.00000i			0.193347i
$429$		0			0
$430$	−2.00000	−	4.00000i	−0.0964486	−	0.192897i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		−	34.0000i		−	1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$		−	34.0000i		−	1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	−8.00000			−0.384012
$435$		0			0
$436$	6.00000			0.287348
$437$		0			0
$438$		0			0
$439$	20.0000			0.954548			0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000			0.954548			0.477274	+	0.878755i	$0.341625\pi$
$440$	−24.0000	+	12.0000i	−1.14416	+	0.572078i
$441$	9.00000			0.428571
$442$			8.00000i			0.380521i
$443$			2.00000i			0.0950229i	0.998871	+	0.0475114i	$0.0151291\pi$
$443$			2.00000i			0.0950229i	−0.998871	+	0.0475114i	$0.984871\pi$
$444$		0			0
$445$	2.00000	+	4.00000i	0.0948091	+	0.189618i
$446$	−16.0000			−0.757622
$447$		0			0
$448$		−	14.0000i		−	0.661438i
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$	−12.0000	−	9.00000i	−0.565685	−	0.424264i
$451$	−40.0000			−1.88353
$452$			6.00000i			0.282216i
$453$		0			0
$454$	20.0000			0.938647
$455$	−4.00000	−	8.00000i	−0.187523	−	0.375046i
$456$		0			0
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$			10.0000i			0.467269i
$459$		0			0
$460$	12.0000	−	6.00000i	0.559503	−	0.279751i
$461$	14.0000			0.652045			0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000			0.652045			0.326023	+	0.945362i	$0.394291\pi$
$462$		0			0
$463$		−	2.00000i		−	0.0929479i	−0.998920	−	0.0464739i	$-0.985202\pi$
$463$		−	2.00000i		−	0.0929479i	0.998920	−	0.0464739i	$-0.0147984\pi$
$464$	6.00000			0.278543
$465$		0			0
$466$	16.0000			0.741186
$467$		−	38.0000i		−	1.75843i	−0.476425	−	0.879215i	$-0.658068\pi$
$467$		−	38.0000i		−	1.75843i	0.476425	−	0.879215i	$-0.341932\pi$
$468$		−	6.00000i		−	0.277350i
$469$	16.0000			0.738811
$470$	−6.00000	−	12.0000i	−0.276759	−	0.553519i
$471$		0			0
$472$		0			0
$473$			8.00000i			0.367840i
$474$		0			0
$475$		0			0
$476$	−8.00000			−0.366679
$477$			30.0000i			1.37361i
$478$			16.0000i			0.731823i
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$	20.0000			0.911922
$482$			2.00000i			0.0910975i
$483$		0			0
$484$	5.00000			0.227273
$485$	−12.0000	+	6.00000i	−0.544892	+	0.272446i
$486$		0			0
$487$			20.0000i			0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$			20.0000i			0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$			6.00000i			0.271607i
$489$		0			0
$490$	6.00000	−	3.00000i	0.271052	−	0.135526i
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$		0			0
$493$		−	24.0000i		−	1.08091i
$494$		0			0
$495$	12.0000	+	24.0000i	0.539360	+	1.07872i
$496$	−4.00000			−0.179605
$497$		−	8.00000i		−	0.358849i
$498$		0			0
$499$	−4.00000			−0.179065			−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000			−0.179065			−0.0895323	+	0.995984i	$0.528537\pi$
$500$	11.0000	+	2.00000i	0.491935	+	0.0894427i
$501$		0			0
$502$			4.00000i			0.178529i
$503$		−	10.0000i		−	0.445878i	−0.974832	−	0.222939i	$-0.928435\pi$
$503$		−	10.0000i		−	0.445878i	0.974832	−	0.222939i	$-0.0715651\pi$
$504$	−18.0000			−0.801784
$505$	6.00000	+	12.0000i	0.266996	+	0.533993i
$506$	24.0000			1.06693
$507$		0			0
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$	8.00000			0.353899
$512$		−	11.0000i		−	0.486136i
$513$		0			0
$514$	−6.00000			−0.264649
$515$	32.0000	−	16.0000i	1.41009	−	0.705044i
$516$		0			0
$517$			24.0000i			1.05552i
$518$		−	20.0000i		−	0.878750i
$519$		0			0
$520$	−6.00000	−	12.0000i	−0.263117	−	0.526235i
$521$	−42.0000			−1.84005			−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000			−1.84005			−0.920027	+	0.391856i	$0.871833\pi$
$522$		−	18.0000i		−	0.787839i
$523$		−	44.0000i		−	1.92399i	−0.273075	−	0.961993i	$-0.588041\pi$
$523$		−	44.0000i		−	1.92399i	0.273075	−	0.961993i	$-0.411959\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$	−6.00000			−0.261612
$527$			16.0000i			0.696971i
$528$		0			0
$529$	−13.0000			−0.565217
$530$	10.0000	+	20.0000i	0.434372	+	0.868744i
$531$		0			0
$532$		0			0
$533$		−	20.0000i		−	0.866296i
$534$		0			0
$535$	8.00000	−	4.00000i	0.345870	−	0.172935i
$536$	24.0000			1.03664
$537$		0			0
$538$		−	18.0000i		−	0.776035i
$539$	−12.0000			−0.516877
$540$		0			0
$541$	−22.0000			−0.945854			−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000			−0.945854			−0.472927	+	0.881102i	$0.656803\pi$
$542$			24.0000i			1.03089i
$543$		0			0
$544$	−20.0000			−0.857493
$545$	−6.00000	−	12.0000i	−0.257012	−	0.514024i
$546$		0			0
$547$		−	28.0000i		−	1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$		−	28.0000i		−	1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$			12.0000i			0.512615i
$549$	6.00000			0.256074
$550$	16.0000	+	12.0000i	0.682242	+	0.511682i
$551$		0			0
$552$		0			0
$553$			8.00000i			0.340195i
$554$	−8.00000			−0.339887
$555$		0			0
$556$	4.00000			0.169638
$557$		−	16.0000i		−	0.677942i	−0.940797	−	0.338971i	$-0.889921\pi$
$557$		−	16.0000i		−	0.677942i	0.940797	−	0.338971i	$-0.110079\pi$
$558$			12.0000i			0.508001i
$559$	−4.00000			−0.169182
$560$	−4.00000	+	2.00000i	−0.169031	+	0.0845154i
$561$		0			0
$562$			26.0000i			1.09674i
$563$			12.0000i			0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$			12.0000i			0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$		0			0
$565$	12.0000	−	6.00000i	0.504844	−	0.252422i
$566$	14.0000			0.588464
$567$			18.0000i			0.755929i
$568$		−	12.0000i		−	0.503509i
$569$	42.0000			1.76073			0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000			1.76073			0.880366	+	0.474295i	$0.157297\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$			8.00000i			0.334497i
$573$		0			0
$574$	−20.0000			−0.834784
$575$	−24.0000	−	18.0000i	−1.00087	−	0.750652i
$576$	−21.0000			−0.875000
$577$			24.0000i			0.999133i	0.866276	+	0.499567i	$0.166507\pi$
$577$			24.0000i			0.999133i	−0.866276	+	0.499567i	$0.833493\pi$
$578$			1.00000i			0.0415945i
$579$		0			0
$580$	6.00000	+	12.0000i	0.249136	+	0.498273i
$581$	−36.0000			−1.49353
$582$		0			0
$583$		−	40.0000i		−	1.65663i
$584$	12.0000			0.496564
$585$	−12.0000	+	6.00000i	−0.496139	+	0.248069i
$586$	14.0000			0.578335
$587$			18.0000i			0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$			18.0000i			0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		−	10.0000i		−	0.410997i
$593$			8.00000i			0.328521i	0.986417	+	0.164260i	$0.0525237\pi$
$593$			8.00000i			0.328521i	−0.986417	+	0.164260i	$0.947476\pi$
$594$		0			0
$595$	8.00000	+	16.0000i	0.327968	+	0.655936i
$596$	10.0000			0.409616
$597$		0			0
$598$			12.0000i			0.490716i
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$	−14.0000			−0.571072			−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000			−0.571072			−0.285536	+	0.958368i	$0.592172\pi$
$602$			4.00000i			0.163028i
$603$		−	24.0000i		−	0.977356i
$604$	−24.0000			−0.976546
$605$	−5.00000	−	10.0000i	−0.203279	−	0.406558i
$606$		0			0
$607$			44.0000i			1.78590i	0.450151	+	0.892952i	$0.351370\pi$
$607$			44.0000i			1.78590i	−0.450151	+	0.892952i	$0.648630\pi$
$608$		0			0
$609$		0			0
$610$	4.00000	−	2.00000i	0.161955	−	0.0809776i
$611$	−12.0000			−0.485468
$612$			12.0000i			0.485071i
$613$			24.0000i			0.969351i	0.874694	+	0.484675i	$0.161062\pi$
$613$			24.0000i			0.969351i	−0.874694	+	0.484675i	$0.838938\pi$
$614$	−20.0000			−0.807134
$615$		0			0
$616$	24.0000			0.966988
$617$			12.0000i			0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$			12.0000i			0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$	−4.00000	−	8.00000i	−0.160644	−	0.321288i
$621$		0			0
$622$			24.0000i			0.962312i
$623$		−	4.00000i		−	0.160257i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−24.0000			−0.959233
$627$		0			0
$628$		−	8.00000i		−	0.319235i
$629$	−40.0000			−1.59490
$630$	6.00000	+	12.0000i	0.239046	+	0.478091i
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$			12.0000i			0.477334i
$633$		0			0
$634$	22.0000			0.873732
$635$		0			0
$636$		0			0
$637$		−	6.00000i		−	0.237729i
$638$			24.0000i			0.950169i
$639$	−12.0000			−0.474713
$640$	6.00000	−	3.00000i	0.237171	−	0.118585i
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$		0			0
$643$		−	38.0000i		−	1.49857i	−0.662246	−	0.749287i	$-0.730396\pi$
$643$		−	38.0000i		−	1.49857i	0.662246	−	0.749287i	$-0.269604\pi$
$644$	−12.0000			−0.472866
$645$		0			0
$646$		0			0
$647$			30.0000i			1.17942i	0.807614	+	0.589711i	$0.200758\pi$
$647$			30.0000i			1.17942i	−0.807614	+	0.589711i	$0.799242\pi$
$648$			27.0000i			1.06066i
$649$		0			0
$650$	−6.00000	+	8.00000i	−0.235339	+	0.313786i
$651$		0			0
$652$		−	10.0000i		−	0.391630i
$653$		−	16.0000i		−	0.626128i	−0.949732	−	0.313064i	$-0.898644\pi$
$653$		−	16.0000i		−	0.626128i	0.949732	−	0.313064i	$-0.101356\pi$
$654$		0			0
$655$	12.0000	+	24.0000i	0.468879	+	0.937758i
$656$	−10.0000			−0.390434
$657$		−	12.0000i		−	0.468165i
$658$			12.0000i			0.467809i
$659$	−16.0000			−0.623272			−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000			−0.623272			−0.311636	+	0.950202i	$0.600877\pi$
$660$		0			0
$661$	30.0000			1.16686			0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000			1.16686			0.583432	+	0.812162i	$0.301709\pi$
$662$			4.00000i			0.155464i
$663$		0			0
$664$	−54.0000			−2.09561
$665$		0			0
$666$	−30.0000			−1.16248
$667$		−	36.0000i		−	1.39393i
$668$		−	12.0000i		−	0.464294i
$669$		0			0
$670$	−8.00000	−	16.0000i	−0.309067	−	0.618134i
$671$	−8.00000			−0.308837
$672$		0			0
$673$			34.0000i			1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$			34.0000i			1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	22.0000			0.847408
$675$		0			0
$676$	9.00000			0.346154
$677$		−	18.0000i		−	0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$		−	18.0000i		−	0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$		0			0
$679$	12.0000			0.460518
$680$	12.0000	+	24.0000i	0.460179	+	0.920358i
$681$		0			0
$682$		−	16.0000i		−	0.612672i
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		0			0
$685$	24.0000	−	12.0000i	0.916993	−	0.458496i
$686$	−20.0000			−0.763604
$687$		0			0
$688$			2.00000i			0.0762493i
$689$	20.0000			0.761939
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		−	6.00000i		−	0.228086i
$693$		−	24.0000i		−	0.911685i
$694$	6.00000			0.227757
$695$	−4.00000	−	8.00000i	−0.151729	−	0.303457i
$696$		0			0
$697$			40.0000i			1.51511i
$698$		−	2.00000i		−	0.0757011i
$699$		0			0
$700$	−8.00000	−	6.00000i	−0.302372	−	0.226779i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$		0			0
$704$	28.0000			1.05529
$705$		0			0
$706$	4.00000			0.150542
$707$		−	12.0000i		−	0.451306i
$708$		0			0
$709$	38.0000			1.42712			0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000			1.42712			0.713560	+	0.700594i	$0.247082\pi$
$710$	−8.00000	+	4.00000i	−0.300235	+	0.150117i
$711$	12.0000			0.450035
$712$		−	6.00000i		−	0.224860i
$713$			24.0000i			0.898807i
$714$		0			0
$715$	16.0000	−	8.00000i	0.598366	−	0.299183i
$716$		0			0
$717$		0			0
$718$			8.00000i			0.298557i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	3.00000	+	6.00000i	0.111803	+	0.223607i
$721$	−32.0000			−1.19174
$722$		0			0
$723$		0			0
$724$	18.0000			0.668965
$725$	18.0000	−	24.0000i	0.668503	−	0.891338i
$726$		0			0
$727$			2.00000i			0.0741759i	0.999312	+	0.0370879i	$0.0118082\pi$
$727$			2.00000i			0.0741759i	−0.999312	+	0.0370879i	$0.988192\pi$
$728$			12.0000i			0.444750i
$729$	27.0000			1.00000
$730$	−4.00000	−	8.00000i	−0.148047	−	0.296093i
$731$	8.00000			0.295891
$732$		0			0
$733$			12.0000i			0.443230i	0.975134	+	0.221615i	$0.0711328\pi$
$733$			12.0000i			0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$734$	22.0000			0.812035
$735$		0			0
$736$	−30.0000			−1.10581
$737$			32.0000i			1.17874i
$738$			30.0000i			1.10432i
$739$	−4.00000			−0.147142			−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000			−0.147142			−0.0735712	+	0.997290i	$0.523440\pi$
$740$	20.0000	−	10.0000i	0.735215	−	0.367607i
$741$		0			0
$742$		−	20.0000i		−	0.734223i
$743$		−	4.00000i		−	0.146746i	−0.997305	−	0.0733729i	$-0.976624\pi$
$743$		−	4.00000i		−	0.146746i	0.997305	−	0.0733729i	$-0.0233763\pi$
$744$		0			0
$745$	−10.0000	−	20.0000i	−0.366372	−	0.732743i
$746$	6.00000			0.219676
$747$			54.0000i			1.97576i
$748$		−	16.0000i		−	0.585018i
$749$	−8.00000			−0.292314
$750$		0			0
$751$	40.0000			1.45962			0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000			1.45962			0.729810	+	0.683650i	$0.239608\pi$
$752$			6.00000i			0.218797i
$753$		0			0
$754$	−12.0000			−0.437014
$755$	24.0000	+	48.0000i	0.873449	+	1.74690i
$756$		0			0
$757$		−	4.00000i		−	0.145382i	−0.997354	−	0.0726912i	$-0.976841\pi$
$757$		−	4.00000i		−	0.145382i	0.997354	−	0.0726912i	$-0.0231588\pi$
$758$			28.0000i			1.01701i
$759$		0			0
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$			12.0000i			0.434429i
$764$	−24.0000			−0.868290
$765$	24.0000	−	12.0000i	0.867722	−	0.433861i
$766$	−12.0000			−0.433578
$767$		0			0
$768$		0			0
$769$	−26.0000			−0.937584			−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000			−0.937584			−0.468792	+	0.883309i	$0.655311\pi$
$770$	−8.00000	−	16.0000i	−0.288300	−	0.576600i
$771$		0			0
$772$		−	26.0000i		−	0.935760i
$773$			46.0000i			1.65451i	0.561830	+	0.827253i	$0.310097\pi$
$773$			46.0000i			1.65451i	−0.561830	+	0.827253i	$0.689903\pi$
$774$	6.00000			0.215666
$775$	−12.0000	+	16.0000i	−0.431053	+	0.574737i
$776$	18.0000			0.646162
$777$		0			0
$778$			30.0000i			1.07555i
$779$		0			0
$780$		0			0
$781$	16.0000			0.572525
$782$		−	24.0000i		−	0.858238i
$783$		0			0
$784$	−3.00000			−0.107143
$785$	−16.0000	+	8.00000i	−0.571064	+	0.285532i
$786$		0			0
$787$			44.0000i			1.56843i	0.620489	+	0.784215i	$0.286934\pi$
$787$			44.0000i			1.56843i	−0.620489	+	0.784215i	$0.713066\pi$
$788$		−	12.0000i		−	0.427482i
$789$		0

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1805 = 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1805.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +2.00000i q^{7} +3.00000i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +2.00000i q^{7} +3.00000i q^{8} +3.00000 q^{9} +(2.00000 - 1.00000i) q^{10} -4.00000 q^{11} -2.00000i q^{13} -2.00000 q^{14} -1.00000 q^{16} +4.00000i q^{17} +3.00000i q^{18} +(-1.00000 - 2.00000i) q^{20} -4.00000i q^{22} +6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +2.00000 q^{26} +2.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} +5.00000i q^{32} -4.00000 q^{34} +(4.00000 - 2.00000i) q^{35} +3.00000 q^{36} +10.0000i q^{37} +(6.00000 - 3.00000i) q^{40} +10.0000 q^{41} -2.00000i q^{43} -4.00000 q^{44} +(-3.00000 - 6.00000i) q^{45} -6.00000 q^{46} -6.00000i q^{47} +3.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -2.00000i q^{52} +10.0000i q^{53} +(4.00000 + 8.00000i) q^{55} -6.00000 q^{56} -6.00000i q^{58} +2.00000 q^{61} +4.00000i q^{62} +6.00000i q^{63} -7.00000 q^{64} +(-4.00000 + 2.00000i) q^{65} -8.00000i q^{67} +4.00000i q^{68} +(2.00000 + 4.00000i) q^{70} -4.00000 q^{71} +9.00000i q^{72} -4.00000i q^{73} -10.0000 q^{74} -8.00000i q^{77} +4.00000 q^{79} +(1.00000 + 2.00000i) q^{80} +9.00000 q^{81} +10.0000i q^{82} +18.0000i q^{83} +(8.00000 - 4.00000i) q^{85} +2.00000 q^{86} -12.0000i q^{88} -2.00000 q^{89} +(6.00000 - 3.00000i) q^{90} +4.00000 q^{91} +6.00000i q^{92} +6.00000 q^{94} -6.00000i q^{97} +3.00000i q^{98} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^5 + 6 * q^9	\( 2 q + 2 q^{4} - 2 q^{5} + 6 q^{9} + 4 q^{10} - 8 q^{11} - 4 q^{14} - 2 q^{16} - 2 q^{20} - 6 q^{25} + 4 q^{26} - 12 q^{29} + 8 q^{31} - 8 q^{34} + 8 q^{35} + 6 q^{36} + 12 q^{40} + 20 q^{41} - 8 q^{44} - 6 q^{45} - 12 q^{46} + 6 q^{49} - 8 q^{50} + 8 q^{55} - 12 q^{56} + 4 q^{61} - 14 q^{64} - 8 q^{65} + 4 q^{70} - 8 q^{71} - 20 q^{74} + 8 q^{79} + 2 q^{80} + 18 q^{81} + 16 q^{85} + 4 q^{86} - 4 q^{89} + 12 q^{90} + 8 q^{91} + 12 q^{94} - 24 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^5 + 6 * q^9 + 4 * q^10 - 8 * q^11 - 4 * q^14 - 2 * q^16 - 2 * q^20 - 6 * q^25 + 4 * q^26 - 12 * q^29 + 8 * q^31 - 8 * q^34 + 8 * q^35 + 6 * q^36 + 12 * q^40 + 20 * q^41 - 8 * q^44 - 6 * q^45 - 12 * q^46 + 6 * q^49 - 8 * q^50 + 8 * q^55 - 12 * q^56 + 4 * q^61 - 14 * q^64 - 8 * q^65 + 4 * q^70 - 8 * q^71 - 20 * q^74 + 8 * q^79 + 2 * q^80 + 18 * q^81 + 16 * q^85 + 4 * q^86 - 4 * q^89 + 12 * q^90 + 8 * q^91 + 12 * q^94 - 24 * q^99

Introduction

L-functions

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