LMFDB - Embedded newform 2900.2.j.a.2593.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2900,2,Mod(1757,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.1757"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.j (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$23.1566165862$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2593.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2900.2593
Dual form			2900.2.j.a.1757.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +(-3.00000 + 3.00000i) q^{7} -1.00000 q^{9} +(2.00000 + 2.00000i) q^{11} +(-2.00000 + 2.00000i) q^{13} +6.00000 q^{17} +(-4.00000 + 4.00000i) q^{19} +(-6.00000 - 6.00000i) q^{21} +(5.00000 + 5.00000i) q^{23} +4.00000i q^{27} +(-2.00000 + 5.00000i) q^{29} +(-6.00000 - 6.00000i) q^{31} +(-4.00000 + 4.00000i) q^{33} -6.00000i q^{37} +(-4.00000 - 4.00000i) q^{39} +(7.00000 - 7.00000i) q^{41} +4.00000i q^{43} +8.00000i q^{47} -11.0000i q^{49} +12.0000i q^{51} +(4.00000 + 4.00000i) q^{53} +(-8.00000 - 8.00000i) q^{57} -4.00000i q^{59} +(9.00000 + 9.00000i) q^{61} +(3.00000 - 3.00000i) q^{63} +(-3.00000 - 3.00000i) q^{67} +(-10.0000 + 10.0000i) q^{69} -12.0000i q^{71} -14.0000 q^{73} -12.0000 q^{77} +(-8.00000 + 8.00000i) q^{79} -11.0000 q^{81} +(3.00000 + 3.00000i) q^{83} +(-10.0000 - 4.00000i) q^{87} +(7.00000 - 7.00000i) q^{89} -12.0000i q^{91} +(12.0000 - 12.0000i) q^{93} -2.00000i q^{97} +(-2.00000 - 2.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{7} - 2 q^{9} + 4 q^{11} - 4 q^{13} + 12 q^{17} - 8 q^{19} - 12 q^{21} + 10 q^{23} - 4 q^{29} - 12 q^{31} - 8 q^{33} - 8 q^{39} + 14 q^{41} + 8 q^{53} - 16 q^{57} + 18 q^{61} + 6 q^{63} - 6 q^{67}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 6 * q^7 - 2 * q^9 + 4 * q^11 - 4 * q^13 + 12 * q^17 - 8 * q^19 - 12 * q^21 + 10 * q^23 - 4 * q^29 - 12 * q^31 - 8 * q^33 - 8 * q^39 + 14 * q^41 + 8 * q^53 - 16 * q^57 + 18 * q^61 + 6 * q^63 - 6 * q^67 - 20 * q^69 - 28 * q^73 - 24 * q^77 - 16 * q^79 - 22 * q^81 + 6 * q^83 - 20 * q^87 + 14 * q^89 + 24 * q^93 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{3}{4}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			2.00000i			1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$			2.00000i			1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.144370	+	0.989524i	$0.546115\pi$
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.989524	+	0.144370i	$0.953885\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	2.00000	+	2.00000i	0.603023	+	0.603023i	0.941113	−	0.338091i	$-0.109781\pi$
$11$	2.00000	+	2.00000i	0.603023	+	0.603023i	−0.338091	+	0.941113i	$0.609781\pi$
$12$		0			0
$13$	−2.00000	+	2.00000i	−0.554700	+	0.554700i	−0.927794	−	0.373094i	$-0.878297\pi$
$13$	−2.00000	+	2.00000i	−0.554700	+	0.554700i	0.373094	+	0.927794i	$0.378297\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$	−4.00000	+	4.00000i	−0.917663	+	0.917663i	−0.996859	−	0.0791961i	$-0.974765\pi$
$19$	−4.00000	+	4.00000i	−0.917663	+	0.917663i	0.0791961	+	0.996859i	$0.474765\pi$
$20$		0			0
$21$	−6.00000	−	6.00000i	−1.30931	−	1.30931i
$22$		0			0
$23$	5.00000	+	5.00000i	1.04257	+	1.04257i	0.999053	+	0.0435195i	$0.0138571\pi$
$23$	5.00000	+	5.00000i	1.04257	+	1.04257i	0.0435195	+	0.999053i	$0.486143\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	−2.00000	+	5.00000i	−0.371391	+	0.928477i
$29$	−2.00000	+	5.00000i	−0.371391	+	0.928477i
$30$		0			0
$31$	−6.00000	−	6.00000i	−1.07763	−	1.07763i	−0.996721	−	0.0809104i	$-0.974217\pi$
$31$	−6.00000	−	6.00000i	−1.07763	−	1.07763i	−0.0809104	−	0.996721i	$-0.525783\pi$
$32$		0			0
$33$	−4.00000	+	4.00000i	−0.696311	+	0.696311i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		0			0
$39$	−4.00000	−	4.00000i	−0.640513	−	0.640513i
$40$		0			0
$41$	7.00000	−	7.00000i	1.09322	−	1.09322i	0.0980332	−	0.995183i	$-0.468745\pi$
$41$	7.00000	−	7.00000i	1.09322	−	1.09322i	0.995183	−	0.0980332i	$-0.0312551\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$		−	11.0000i		−	1.57143i
$50$		0			0
$51$			12.0000i			1.68034i
$52$		0			0
$53$	4.00000	+	4.00000i	0.549442	+	0.549442i	0.926279	−	0.376837i	$-0.122988\pi$
$53$	4.00000	+	4.00000i	0.549442	+	0.549442i	−0.376837	+	0.926279i	$0.622988\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−8.00000	−	8.00000i	−1.05963	−	1.05963i
$58$		0			0
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		0			0
$61$	9.00000	+	9.00000i	1.15233	+	1.15233i	0.986084	+	0.166248i	$0.0531652\pi$
$61$	9.00000	+	9.00000i	1.15233	+	1.15233i	0.166248	+	0.986084i	$0.446835\pi$
$62$		0			0
$63$	3.00000	−	3.00000i	0.377964	−	0.377964i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.00000	−	3.00000i	−0.366508	−	0.366508i	0.499694	−	0.866202i	$-0.333446\pi$
$67$	−3.00000	−	3.00000i	−0.366508	−	0.366508i	−0.866202	+	0.499694i	$0.833446\pi$
$68$		0			0
$69$	−10.0000	+	10.0000i	−1.20386	+	1.20386i
$70$		0			0
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$		0			0
$73$	−14.0000			−1.63858			−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000			−1.63858			−0.819288	+	0.573382i	$0.805631\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.0000			−1.36753
$78$		0			0
$79$	−8.00000	+	8.00000i	−0.900070	+	0.900070i	−0.995442	−	0.0953714i	$-0.969596\pi$
$79$	−8.00000	+	8.00000i	−0.900070	+	0.900070i	0.0953714	+	0.995442i	$0.469596\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$	3.00000	+	3.00000i	0.329293	+	0.329293i	0.852318	−	0.523025i	$-0.175196\pi$
$83$	3.00000	+	3.00000i	0.329293	+	0.329293i	−0.523025	+	0.852318i	$0.675196\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−10.0000	−	4.00000i	−1.07211	−	0.428845i
$88$		0			0
$89$	7.00000	−	7.00000i	0.741999	−	0.741999i	−0.230964	−	0.972962i	$-0.574188\pi$
$89$	7.00000	−	7.00000i	0.741999	−	0.741999i	0.972962	+	0.230964i	$0.0741879\pi$
$90$		0			0
$91$		−	12.0000i		−	1.25794i
$92$		0			0
$93$	12.0000	−	12.0000i	1.24434	−	1.24434i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$		0			0
$99$	−2.00000	−	2.00000i	−0.201008	−	0.201008i
$100$		0			0
$101$	−9.00000	−	9.00000i	−0.895533	−	0.895533i	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	−9.00000	−	9.00000i	−0.895533	−	0.895533i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$		0			0
$103$	−9.00000	−	9.00000i	−0.886796	−	0.886796i	0.107418	−	0.994214i	$-0.465742\pi$
$103$	−9.00000	−	9.00000i	−0.886796	−	0.886796i	−0.994214	+	0.107418i	$0.965742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.00000	+	5.00000i	−0.483368	+	0.483368i	−0.906206	−	0.422837i	$-0.861034\pi$
$107$	−5.00000	+	5.00000i	−0.483368	+	0.483368i	0.422837	+	0.906206i	$0.361034\pi$
$108$		0			0
$109$	−18.0000			−1.72409			−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000			−1.72409			−0.862044	+	0.506834i	$0.830816\pi$
$110$		0			0
$111$	12.0000			1.13899
$112$		0			0
$113$	18.0000			1.69330			0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000			1.69330			0.846649	+	0.532152i	$0.178617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.00000	−	2.00000i	0.184900	−	0.184900i
$118$		0			0
$119$	−18.0000	+	18.0000i	−1.65006	+	1.65006i
$120$		0			0
$121$		−	3.00000i		−	0.272727i
$122$		0			0
$123$	14.0000	+	14.0000i	1.26234	+	1.26234i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$	−8.00000			−0.704361
$130$		0			0
$131$	4.00000	−	4.00000i	0.349482	−	0.349482i	−0.510435	−	0.859916i	$-0.670516\pi$
$131$	4.00000	−	4.00000i	0.349482	−	0.349482i	0.859916	+	0.510435i	$0.170516\pi$
$132$		0			0
$133$		−	24.0000i		−	2.08106i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000			0.512615			0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000			0.512615			0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−16.0000			−1.34744
$142$		0			0
$143$	−8.00000			−0.668994
$144$		0			0
$145$		0			0
$146$		0			0
$147$	22.0000			1.81453
$148$		0			0
$149$	4.00000			0.327693			0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000			0.327693			0.163846	+	0.986486i	$0.447610\pi$
$150$		0			0
$151$		−	12.0000i		−	0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$		−	12.0000i		−	0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	10.0000i		−	0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$		−	10.0000i		−	0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$		0			0
$159$	−8.00000	+	8.00000i	−0.634441	+	0.634441i
$160$		0			0
$161$	−30.0000			−2.36433
$162$		0			0
$163$	−6.00000			−0.469956			−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000			−0.469956			−0.234978	+	0.972001i	$0.575502\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13.0000	+	13.0000i	1.00597	+	1.00597i	0.999982	+	0.00598813i	$0.00190609\pi$
$167$	13.0000	+	13.0000i	1.00597	+	1.00597i	0.00598813	+	0.999982i	$0.498094\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$	4.00000	−	4.00000i	0.305888	−	0.305888i
$172$		0			0
$173$	−14.0000	+	14.0000i	−1.06440	+	1.06440i	−0.0666220	+	0.997778i	$0.521222\pi$
$173$	−14.0000	+	14.0000i	−1.06440	+	1.06440i	−0.997778	+	0.0666220i	$0.978778\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	8.00000			0.601317
$178$		0			0
$179$	8.00000			0.597948			0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000			0.597948			0.298974	+	0.954261i	$0.403356\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$	−18.0000	+	18.0000i	−1.33060	+	1.33060i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	12.0000	+	12.0000i	0.877527	+	0.877527i
$188$		0			0
$189$	−12.0000	−	12.0000i	−0.872872	−	0.872872i
$190$		0			0
$191$	12.0000	+	12.0000i	0.868290	+	0.868290i	0.992283	−	0.123994i	$-0.0395702\pi$
$191$	12.0000	+	12.0000i	0.868290	+	0.868290i	−0.123994	+	0.992283i	$0.539570\pi$
$192$		0			0
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.00000	+	2.00000i	−0.142494	+	0.142494i	−0.774755	−	0.632261i	$-0.782127\pi$
$197$	−2.00000	+	2.00000i	−0.142494	+	0.142494i	0.632261	+	0.774755i	$0.282127\pi$
$198$		0			0
$199$			8.00000i			0.567105i	0.958957	+	0.283552i	$0.0915130\pi$
$199$			8.00000i			0.567105i	−0.958957	+	0.283552i	$0.908487\pi$
$200$		0			0
$201$	6.00000	−	6.00000i	0.423207	−	0.423207i
$202$		0			0
$203$	−9.00000	−	21.0000i	−0.631676	−	1.47391i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−5.00000	−	5.00000i	−0.347524	−	0.347524i
$208$		0			0
$209$	−16.0000			−1.10674
$210$		0			0
$211$	18.0000	−	18.0000i	1.23917	−	1.23917i	0.278831	−	0.960340i	$-0.410053\pi$
$211$	18.0000	−	18.0000i	1.23917	−	1.23917i	0.960340	−	0.278831i	$-0.0899469\pi$
$212$		0			0
$213$	24.0000			1.64445
$214$		0			0
$215$		0			0
$216$		0			0
$217$	36.0000			2.44384
$218$		0			0
$219$		−	28.0000i		−	1.89206i
$220$		0			0
$221$	−12.0000	+	12.0000i	−0.807207	+	0.807207i
$222$		0			0
$223$	−17.0000	−	17.0000i	−1.13840	−	1.13840i	−0.988736	−	0.149668i	$-0.952180\pi$
$223$	−17.0000	−	17.0000i	−1.13840	−	1.13840i	−0.149668	−	0.988736i	$-0.547820\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.00000	+	5.00000i	−0.331862	+	0.331862i	−0.853293	−	0.521431i	$-0.825398\pi$
$227$	−5.00000	+	5.00000i	−0.331862	+	0.331862i	0.521431	+	0.853293i	$0.325398\pi$
$228$		0			0
$229$	3.00000	+	3.00000i	0.198246	+	0.198246i	0.799248	−	0.601002i	$-0.205232\pi$
$229$	3.00000	+	3.00000i	0.198246	+	0.198246i	−0.601002	+	0.799248i	$0.705232\pi$
$230$		0			0
$231$		−	24.0000i		−	1.57908i
$232$		0			0
$233$	−4.00000	−	4.00000i	−0.262049	−	0.262049i	0.563837	−	0.825886i	$-0.309325\pi$
$233$	−4.00000	−	4.00000i	−0.262049	−	0.262049i	−0.825886	+	0.563837i	$0.809325\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−16.0000	−	16.0000i	−1.03931	−	1.03931i
$238$		0			0
$239$		−	4.00000i		−	0.258738i	−0.991596	−	0.129369i	$-0.958705\pi$
$239$		−	4.00000i		−	0.258738i	0.991596	−	0.129369i	$-0.0412952\pi$
$240$		0			0
$241$			24.0000i			1.54598i	0.634421	+	0.772988i	$0.281239\pi$
$241$			24.0000i			1.54598i	−0.634421	+	0.772988i	$0.718761\pi$
$242$		0			0
$243$		−	10.0000i		−	0.641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	16.0000i		−	1.01806i
$248$		0			0
$249$	−6.00000	+	6.00000i	−0.380235	+	0.380235i
$250$		0			0
$251$	6.00000	+	6.00000i	0.378717	+	0.378717i	0.870639	−	0.491922i	$-0.163706\pi$
$251$	6.00000	+	6.00000i	0.378717	+	0.378717i	−0.491922	+	0.870639i	$0.663706\pi$
$252$		0			0
$253$			20.0000i			1.25739i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.0000	−	10.0000i	0.623783	−	0.623783i	−0.322714	−	0.946497i	$-0.604595\pi$
$257$	10.0000	−	10.0000i	0.623783	−	0.623783i	0.946497	+	0.322714i	$0.104595\pi$
$258$		0			0
$259$	18.0000	+	18.0000i	1.11847	+	1.11847i
$260$		0			0
$261$	2.00000	−	5.00000i	0.123797	−	0.309492i
$262$		0			0
$263$		−	18.0000i		−	1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$		−	18.0000i		−	1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	14.0000	+	14.0000i	0.856786	+	0.856786i
$268$		0			0
$269$	−7.00000	−	7.00000i	−0.426798	−	0.426798i	0.460738	−	0.887536i	$-0.347585\pi$
$269$	−7.00000	−	7.00000i	−0.426798	−	0.426798i	−0.887536	+	0.460738i	$0.847585\pi$
$270$		0			0
$271$	−16.0000	+	16.0000i	−0.971931	+	0.971931i	−0.999617	−	0.0276859i	$-0.991186\pi$
$271$	−16.0000	+	16.0000i	−0.971931	+	0.971931i	0.0276859	+	0.999617i	$0.491186\pi$
$272$		0			0
$273$	24.0000			1.45255
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−12.0000	+	12.0000i	−0.721010	+	0.721010i	−0.968811	−	0.247801i	$-0.920292\pi$
$277$	−12.0000	+	12.0000i	−0.721010	+	0.721010i	0.247801	+	0.968811i	$0.420292\pi$
$278$		0			0
$279$	6.00000	+	6.00000i	0.359211	+	0.359211i
$280$		0			0
$281$	−4.00000			−0.238620			−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000			−0.238620			−0.119310	+	0.992857i	$0.538068\pi$
$282$		0			0
$283$	−7.00000	+	7.00000i	−0.416107	+	0.416107i	−0.883859	−	0.467753i	$-0.845064\pi$
$283$	−7.00000	+	7.00000i	−0.416107	+	0.416107i	0.467753	+	0.883859i	$0.345064\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			42.0000i			2.47918i
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$	4.00000			0.234484
$292$		0			0
$293$		−	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
$296$		0			0
$297$	−8.00000	+	8.00000i	−0.464207	+	0.464207i
$298$		0			0
$299$	−20.0000			−1.15663
$300$		0			0
$301$	−12.0000	−	12.0000i	−0.691669	−	0.691669i
$302$		0			0
$303$	18.0000	−	18.0000i	1.03407	−	1.03407i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$	18.0000	−	18.0000i	1.02398	−	1.02398i
$310$		0			0
$311$	14.0000	+	14.0000i	0.793867	+	0.793867i	0.982121	−	0.188253i	$-0.0602826\pi$
$311$	14.0000	+	14.0000i	0.793867	+	0.793867i	−0.188253	+	0.982121i	$0.560283\pi$
$312$		0			0
$313$	−10.0000	−	10.0000i	−0.565233	−	0.565233i	0.365556	−	0.930789i	$-0.380879\pi$
$313$	−10.0000	−	10.0000i	−0.565233	−	0.565233i	−0.930789	+	0.365556i	$0.880879\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.00000i		−	0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$		−	2.00000i		−	0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$		0			0
$319$	−14.0000	+	6.00000i	−0.783850	+	0.335936i
$320$		0			0
$321$	−10.0000	−	10.0000i	−0.558146	−	0.558146i
$322$		0			0
$323$	−24.0000	+	24.0000i	−1.33540	+	1.33540i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	36.0000i		−	1.99080i
$328$		0			0
$329$	−24.0000	−	24.0000i	−1.32316	−	1.32316i
$330$		0			0
$331$	−12.0000	+	12.0000i	−0.659580	+	0.659580i	−0.955281	−	0.295701i	$-0.904447\pi$
$331$	−12.0000	+	12.0000i	−0.659580	+	0.659580i	0.295701	+	0.955281i	$0.404447\pi$
$332$		0			0
$333$			6.00000i			0.328798i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			26.0000i			1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$			26.0000i			1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$		0			0
$339$			36.0000i			1.95525i
$340$		0			0
$341$		−	24.0000i		−	1.29967i
$342$		0			0
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.0000	−	13.0000i	−0.697877	−	0.697877i	0.266076	−	0.963952i	$-0.414273\pi$
$347$	−13.0000	−	13.0000i	−0.697877	−	0.697877i	−0.963952	+	0.266076i	$0.914273\pi$
$348$		0			0
$349$			2.00000i			0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$			2.00000i			0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$		0			0
$351$	−8.00000	−	8.00000i	−0.427008	−	0.427008i
$352$		0			0
$353$	−8.00000	+	8.00000i	−0.425797	+	0.425797i	−0.887194	−	0.461397i	$-0.847348\pi$
$353$	−8.00000	+	8.00000i	−0.425797	+	0.425797i	0.461397	+	0.887194i	$0.347348\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−36.0000	−	36.0000i	−1.90532	−	1.90532i
$358$		0			0
$359$	18.0000	−	18.0000i	0.950004	−	0.950004i	−0.0488047	−	0.998808i	$-0.515541\pi$
$359$	18.0000	−	18.0000i	0.950004	−	0.950004i	0.998808	+	0.0488047i	$0.0155412\pi$
$360$		0			0
$361$		−	13.0000i		−	0.684211i
$362$		0			0
$363$	6.00000			0.314918
$364$		0			0
$365$		0			0
$366$		0			0
$367$	14.0000			0.730794			0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000			0.730794			0.365397	+	0.930852i	$0.380933\pi$
$368$		0			0
$369$	−7.00000	+	7.00000i	−0.364405	+	0.364405i
$370$		0			0
$371$	−24.0000			−1.24602
$372$		0			0
$373$	18.0000	+	18.0000i	0.932005	+	0.932005i	0.997831	−	0.0658264i	$-0.0209684\pi$
$373$	18.0000	+	18.0000i	0.932005	+	0.932005i	−0.0658264	+	0.997831i	$0.520968\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.00000	−	14.0000i	−0.309016	−	0.721037i
$378$		0			0
$379$	−6.00000	+	6.00000i	−0.308199	+	0.308199i	−0.844211	−	0.536011i	$-0.819930\pi$
$379$	−6.00000	+	6.00000i	−0.308199	+	0.308199i	0.536011	+	0.844211i	$0.319930\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.00000	−	3.00000i	0.153293	−	0.153293i	−0.626294	−	0.779587i	$-0.715429\pi$
$383$	3.00000	−	3.00000i	0.153293	−	0.153293i	0.779587	+	0.626294i	$0.215429\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	4.00000i		−	0.203331i
$388$		0			0
$389$	7.00000	+	7.00000i	0.354914	+	0.354914i	0.861934	−	0.507020i	$-0.169253\pi$
$389$	7.00000	+	7.00000i	0.354914	+	0.354914i	−0.507020	+	0.861934i	$0.669253\pi$
$390$		0			0
$391$	30.0000	+	30.0000i	1.51717	+	1.51717i
$392$		0			0
$393$	8.00000	+	8.00000i	0.403547	+	0.403547i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	2.00000	−	2.00000i	0.100377	−	0.100377i	−0.655135	−	0.755512i	$-0.727388\pi$
$397$	2.00000	−	2.00000i	0.100377	−	0.100377i	0.755512	+	0.655135i	$0.227388\pi$
$398$		0			0
$399$	48.0000			2.40301
$400$		0			0
$401$	12.0000			0.599251			0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000			0.599251			0.299626	+	0.954057i	$0.403138\pi$
$402$		0			0
$403$	24.0000			1.19553
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.0000	−	12.0000i	0.594818	−	0.594818i
$408$		0			0
$409$	11.0000	−	11.0000i	0.543915	−	0.543915i	−0.380759	−	0.924674i	$-0.624337\pi$
$409$	11.0000	−	11.0000i	0.543915	−	0.543915i	0.924674	+	0.380759i	$0.124337\pi$
$410$		0			0
$411$			12.0000i			0.591916i
$412$		0			0
$413$	12.0000	+	12.0000i	0.590481	+	0.590481i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.0000			0.781651			0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000			0.781651			0.390826	+	0.920465i	$0.372190\pi$
$420$		0			0
$421$	−9.00000	+	9.00000i	−0.438633	+	0.438633i	−0.891552	−	0.452919i	$-0.850383\pi$
$421$	−9.00000	+	9.00000i	−0.438633	+	0.438633i	0.452919	+	0.891552i	$0.350383\pi$
$422$		0			0
$423$		−	8.00000i		−	0.388973i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−54.0000			−2.61324
$428$		0			0
$429$		−	16.0000i		−	0.772487i
$430$		0			0
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$	−18.0000			−0.865025			−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000			−0.865025			−0.432512	+	0.901628i	$0.642373\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−40.0000			−1.91346
$438$		0			0
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$		0			0
$441$			11.0000i			0.523810i
$442$		0			0
$443$	−20.0000			−0.950229			−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000			−0.950229			−0.475114	+	0.879924i	$0.657593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			8.00000i			0.378387i
$448$		0			0
$449$	13.0000	−	13.0000i	0.613508	−	0.613508i	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	13.0000	−	13.0000i	0.613508	−	0.613508i	0.943858	+	0.330350i	$0.107167\pi$
$450$		0			0
$451$	28.0000			1.31847
$452$		0			0
$453$	24.0000			1.12762
$454$		0			0
$455$		0			0
$456$		0			0
$457$	20.0000	+	20.0000i	0.935561	+	0.935561i	0.998046	−	0.0624853i	$-0.0199027\pi$
$457$	20.0000	+	20.0000i	0.935561	+	0.935561i	−0.0624853	+	0.998046i	$0.519903\pi$
$458$		0			0
$459$			24.0000i			1.12022i
$460$		0			0
$461$	−25.0000	+	25.0000i	−1.16437	+	1.16437i	−0.180857	+	0.983509i	$0.557887\pi$
$461$	−25.0000	+	25.0000i	−1.16437	+	1.16437i	−0.983509	+	0.180857i	$0.942113\pi$
$462$		0			0
$463$	−15.0000	+	15.0000i	−0.697109	+	0.697109i	−0.963786	−	0.266677i	$-0.914074\pi$
$463$	−15.0000	+	15.0000i	−0.697109	+	0.697109i	0.266677	+	0.963786i	$0.414074\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.00000			−0.185098			−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000			−0.185098			−0.0925490	+	0.995708i	$0.529501\pi$
$468$		0			0
$469$	18.0000			0.831163
$470$		0			0
$471$	20.0000			0.921551
$472$		0			0
$473$	−8.00000	+	8.00000i	−0.367840	+	0.367840i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−4.00000	−	4.00000i	−0.183147	−	0.183147i
$478$		0			0
$479$	8.00000	+	8.00000i	0.365529	+	0.365529i	0.865844	−	0.500314i	$-0.166782\pi$
$479$	8.00000	+	8.00000i	0.365529	+	0.365529i	−0.500314	+	0.865844i	$0.666782\pi$
$480$		0			0
$481$	12.0000	+	12.0000i	0.547153	+	0.547153i
$482$		0			0
$483$		−	60.0000i		−	2.73009i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	3.00000	−	3.00000i	0.135943	−	0.135943i	−0.635861	−	0.771804i	$-0.719355\pi$
$487$	3.00000	−	3.00000i	0.135943	−	0.135943i	0.771804	+	0.635861i	$0.219355\pi$
$488$		0			0
$489$		−	12.0000i		−	0.542659i
$490$		0			0
$491$	−4.00000	+	4.00000i	−0.180517	+	0.180517i	−0.791581	−	0.611064i	$-0.790742\pi$
$491$	−4.00000	+	4.00000i	−0.180517	+	0.180517i	0.611064	+	0.791581i	$0.290742\pi$
$492$		0			0
$493$	−12.0000	+	30.0000i	−0.540453	+	1.35113i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	36.0000	+	36.0000i	1.61482	+	1.61482i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$	−26.0000	+	26.0000i	−1.16159	+	1.16159i
$502$		0			0
$503$	16.0000			0.713405			0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000			0.713405			0.356702	+	0.934218i	$0.383901\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−10.0000			−0.444116
$508$		0			0
$509$			24.0000i			1.06378i	0.846813	+	0.531891i	$0.178518\pi$
$509$			24.0000i			1.06378i	−0.846813	+	0.531891i	$0.821482\pi$
$510$		0			0
$511$	42.0000	−	42.0000i	1.85797	−	1.85797i
$512$		0			0
$513$	−16.0000	−	16.0000i	−0.706417	−	0.706417i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−16.0000	+	16.0000i	−0.703679	+	0.703679i
$518$		0			0
$519$	−28.0000	−	28.0000i	−1.22906	−	1.22906i
$520$		0			0
$521$			30.0000i			1.31432i	0.753749	+	0.657162i	$0.228243\pi$
$521$			30.0000i			1.31432i	−0.753749	+	0.657162i	$0.771757\pi$
$522$		0			0
$523$	23.0000	+	23.0000i	1.00572	+	1.00572i	0.999984	+	0.00573623i	$0.00182591\pi$
$523$	23.0000	+	23.0000i	1.00572	+	1.00572i	0.00573623	+	0.999984i	$0.498174\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−36.0000	−	36.0000i	−1.56818	−	1.56818i
$528$		0			0
$529$			27.0000i			1.17391i
$530$		0			0
$531$			4.00000i			0.173585i
$532$		0			0
$533$			28.0000i			1.21281i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			16.0000i			0.690451i
$538$		0			0
$539$	22.0000	−	22.0000i	0.947607	−	0.947607i
$540$		0			0
$541$	−1.00000	−	1.00000i	−0.0429934	−	0.0429934i	0.685283	−	0.728277i	$-0.259678\pi$
$541$	−1.00000	−	1.00000i	−0.0429934	−	0.0429934i	−0.728277	+	0.685283i	$0.759678\pi$
$542$		0			0
$543$			40.0000i			1.71656i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	7.00000	−	7.00000i	0.299298	−	0.299298i	−0.541441	−	0.840739i	$-0.682121\pi$
$547$	7.00000	−	7.00000i	0.299298	−	0.299298i	0.840739	+	0.541441i	$0.182121\pi$
$548$		0			0
$549$	−9.00000	−	9.00000i	−0.384111	−	0.384111i
$550$		0			0
$551$	−12.0000	−	28.0000i	−0.511217	−	1.19284i
$552$		0			0
$553$		−	48.0000i		−	2.04117i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.00000	+	8.00000i	0.338971	+	0.338971i	0.855980	−	0.517009i	$-0.172955\pi$
$557$	8.00000	+	8.00000i	0.338971	+	0.338971i	−0.517009	+	0.855980i	$0.672955\pi$
$558$		0			0
$559$	−8.00000	−	8.00000i	−0.338364	−	0.338364i
$560$		0			0
$561$	−24.0000	+	24.0000i	−1.01328	+	1.01328i
$562$		0			0
$563$	−10.0000			−0.421450			−0.210725	−	0.977545i	$-0.567582\pi$
$563$	−10.0000			−0.421450			−0.210725	+	0.977545i	$0.567582\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	33.0000	−	33.0000i	1.38587	−	1.38587i
$568$		0			0
$569$	9.00000	+	9.00000i	0.377300	+	0.377300i	0.870127	−	0.492827i	$-0.164037\pi$
$569$	9.00000	+	9.00000i	0.377300	+	0.377300i	−0.492827	+	0.870127i	$0.664037\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$		0			0
$573$	−24.0000	+	24.0000i	−1.00261	+	1.00261i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			30.0000i			1.24892i	0.781058	+	0.624458i	$0.214680\pi$
$577$			30.0000i			1.24892i	−0.781058	+	0.624458i	$0.785320\pi$
$578$		0			0
$579$	28.0000			1.16364
$580$		0			0
$581$	−18.0000			−0.746766
$582$		0			0
$583$			16.0000i			0.662652i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.00000	−	7.00000i	0.288921	−	0.288921i	−0.547733	−	0.836653i	$-0.684509\pi$
$587$	7.00000	−	7.00000i	0.288921	−	0.288921i	0.836653	+	0.547733i	$0.184509\pi$
$588$		0			0
$589$	48.0000			1.97781
$590$		0			0
$591$	−4.00000	−	4.00000i	−0.164538	−	0.164538i
$592$		0			0
$593$	4.00000	−	4.00000i	0.164260	−	0.164260i	−0.620191	−	0.784451i	$-0.712945\pi$
$593$	4.00000	−	4.00000i	0.164260	−	0.164260i	0.784451	+	0.620191i	$0.212945\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−16.0000			−0.654836
$598$		0			0
$599$	−12.0000	+	12.0000i	−0.490307	+	0.490307i	−0.908403	−	0.418096i	$-0.862698\pi$
$599$	−12.0000	+	12.0000i	−0.490307	+	0.490307i	0.418096	+	0.908403i	$0.362698\pi$
$600$		0			0
$601$	−11.0000	−	11.0000i	−0.448699	−	0.448699i	0.446223	−	0.894922i	$-0.352769\pi$
$601$	−11.0000	−	11.0000i	−0.448699	−	0.448699i	−0.894922	+	0.446223i	$0.852769\pi$
$602$		0			0
$603$	3.00000	+	3.00000i	0.122169	+	0.122169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			48.0000i			1.94826i	0.225989	+	0.974130i	$0.427439\pi$
$607$			48.0000i			1.94826i	−0.225989	+	0.974130i	$0.572561\pi$
$608$		0			0
$609$	42.0000	−	18.0000i	1.70193	−	0.729397i
$610$		0			0
$611$	−16.0000	−	16.0000i	−0.647291	−	0.647291i
$612$		0			0
$613$	−16.0000	+	16.0000i	−0.646234	+	0.646234i	−0.952081	−	0.305847i	$-0.901060\pi$
$613$	−16.0000	+	16.0000i	−0.646234	+	0.646234i	0.305847	+	0.952081i	$0.401060\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$		0			0
$619$	22.0000	+	22.0000i	0.884255	+	0.884255i	0.993964	−	0.109709i	$-0.0349919\pi$
$619$	22.0000	+	22.0000i	0.884255	+	0.884255i	−0.109709	+	0.993964i	$0.534992\pi$
$620$		0			0
$621$	−20.0000	+	20.0000i	−0.802572	+	0.802572i
$622$		0			0
$623$			42.0000i			1.68269i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	32.0000i		−	1.27796i
$628$		0			0
$629$		−	36.0000i		−	1.43541i
$630$		0			0
$631$		−	16.0000i		−	0.636950i	−0.947931	−	0.318475i	$-0.896829\pi$
$631$		−	16.0000i		−	0.636950i	0.947931	−	0.318475i	$-0.103171\pi$
$632$		0			0
$633$	36.0000	+	36.0000i	1.43087	+	1.43087i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	22.0000	+	22.0000i	0.871672	+	0.871672i
$638$		0			0
$639$			12.0000i			0.474713i
$640$		0			0
$641$	7.00000	+	7.00000i	0.276483	+	0.276483i	0.831703	−	0.555220i	$-0.187366\pi$
$641$	7.00000	+	7.00000i	0.276483	+	0.276483i	−0.555220	+	0.831703i	$0.687366\pi$
$642$		0			0
$643$	−9.00000	+	9.00000i	−0.354925	+	0.354925i	−0.861938	−	0.507013i	$-0.830750\pi$
$643$	−9.00000	+	9.00000i	−0.354925	+	0.354925i	0.507013	+	0.861938i	$0.330750\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	29.0000	+	29.0000i	1.14011	+	1.14011i	0.988430	+	0.151678i	$0.0484676\pi$
$647$	29.0000	+	29.0000i	1.14011	+	1.14011i	0.151678	+	0.988430i	$0.451532\pi$
$648$		0			0
$649$	8.00000	−	8.00000i	0.314027	−	0.314027i
$650$		0			0
$651$			72.0000i			2.82190i
$652$		0			0
$653$	−2.00000			−0.0782660			−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000			−0.0782660			−0.0391330	+	0.999234i	$0.512460\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	14.0000			0.546192
$658$		0			0
$659$	28.0000	−	28.0000i	1.09073	−	1.09073i	0.0952741	−	0.995451i	$-0.469627\pi$
$659$	28.0000	−	28.0000i	1.09073	−	1.09073i	0.995451	−	0.0952741i	$-0.0303727\pi$
$660$		0			0
$661$	24.0000			0.933492			0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000			0.933492			0.466746	+	0.884391i	$0.345426\pi$
$662$		0			0
$663$	−24.0000	−	24.0000i	−0.932083	−	0.932083i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−35.0000	+	15.0000i	−1.35521	+	0.580802i
$668$		0			0
$669$	34.0000	−	34.0000i	1.31452	−	1.31452i
$670$		0			0
$671$			36.0000i			1.38976i
$672$		0			0
$673$	−6.00000	+	6.00000i	−0.231283	+	0.231283i	−0.813228	−	0.581945i	$-0.802292\pi$
$673$	−6.00000	+	6.00000i	−0.231283	+	0.231283i	0.581945	+	0.813228i	$0.302292\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			18.0000i			0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$			18.0000i			0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$		0			0
$679$	6.00000	+	6.00000i	0.230259	+	0.230259i
$680$		0			0
$681$	−10.0000	−	10.0000i	−0.383201	−	0.383201i
$682$		0			0
$683$	23.0000	+	23.0000i	0.880071	+	0.880071i	0.993541	−	0.113471i	$-0.0361968\pi$
$683$	23.0000	+	23.0000i	0.880071	+	0.880071i	−0.113471	+	0.993541i	$0.536197\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−6.00000	+	6.00000i	−0.228914	+	0.228914i
$688$		0			0
$689$	−16.0000			−0.609551
$690$		0			0
$691$	16.0000			0.608669			0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000			0.608669			0.304334	+	0.952565i	$0.401566\pi$
$692$		0			0
$693$	12.0000			0.455842
$694$		0			0
$695$		0			0
$696$		0			0
$697$	42.0000	−	42.0000i	1.59086	−	1.59086i
$698$		0			0
$699$	8.00000	−	8.00000i	0.302588	−	0.302588i
$700$		0			0
$701$			2.00000i			0.0755390i	0.999286	+	0.0377695i	$0.0120253\pi$
$701$			2.00000i			0.0755390i	−0.999286	+	0.0377695i	$0.987975\pi$
$702$		0			0
$703$	24.0000	+	24.0000i	0.905177	+	0.905177i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	54.0000			2.03088
$708$		0			0
$709$	−26.0000			−0.976450			−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000			−0.976450			−0.488225	+	0.872718i	$0.662356\pi$
$710$		0			0
$711$	8.00000	−	8.00000i	0.300023	−	0.300023i
$712$		0			0
$713$		−	60.0000i		−	2.24702i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	8.00000			0.298765
$718$		0			0
$719$		−	24.0000i		−	0.895049i	−0.894272	−	0.447524i	$-0.852306\pi$
$719$		−	24.0000i		−	0.895049i	0.894272	−	0.447524i	$-0.147694\pi$
$720$		0			0
$721$	54.0000			2.01107
$722$		0			0
$723$	−48.0000			−1.78514
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−38.0000			−1.40934			−0.704671	−	0.709534i	$-0.748905\pi$
$727$	−38.0000			−1.40934			−0.704671	+	0.709534i	$0.748905\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$			24.0000i			0.887672i
$732$		0			0
$733$	2.00000			0.0738717			0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000			0.0738717			0.0369358	+	0.999318i	$0.488240\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	12.0000i		−	0.442026i
$738$		0			0
$739$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$739$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$740$		0			0
$741$	32.0000			1.17555
$742$		0			0
$743$	40.0000			1.46746			0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000			1.46746			0.733729	+	0.679442i	$0.237778\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−3.00000	−	3.00000i	−0.109764	−	0.109764i
$748$		0			0
$749$		−	30.0000i		−	1.09618i
$750$		0			0
$751$	−6.00000	+	6.00000i	−0.218943	+	0.218943i	−0.808053	−	0.589110i	$-0.799479\pi$
$751$	−6.00000	+	6.00000i	−0.218943	+	0.218943i	0.589110	+	0.808053i	$0.299479\pi$
$752$		0			0
$753$	−12.0000	+	12.0000i	−0.437304	+	0.437304i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	34.0000			1.23575			0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000			1.23575			0.617876	+	0.786276i	$0.287994\pi$
$758$		0			0
$759$	−40.0000			−1.45191
$760$		0			0
$761$	−38.0000			−1.37750			−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000			−1.37750			−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$	54.0000	−	54.0000i	1.95493	−	1.95493i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.00000	+	8.00000i	0.288863	+	0.288863i
$768$		0			0
$769$	29.0000	+	29.0000i	1.04577	+	1.04577i	0.998901	+	0.0468655i	$0.0149232\pi$
$769$	29.0000	+	29.0000i	1.04577	+	1.04577i	0.0468655	+	0.998901i	$0.485077\pi$
$770$		0			0
$771$	20.0000	+	20.0000i	0.720282	+	0.720282i
$772$		0			0
$773$		−	26.0000i		−	0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$		−	26.0000i		−	0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−36.0000	+	36.0000i	−1.29149	+	1.29149i
$778$		0			0
$779$			56.0000i			2.00641i
$780$		0			0
$781$	24.0000	−	24.0000i	0.858788	−	0.858788i
$782$		0			0
$783$	−20.0000	−	8.00000i	−0.714742	−	0.285897i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−27.0000	−	27.0000i	−0.962446	−	0.962446i	0.0368739	−	0.999320i	$-0.488260\pi$
$787$	−27.0000	−	27.0000i	−0.962446	−	0.962446i	−0.999320	+	0.0368739i	$0.988260\pi$
$788$		0			0
$789$	36.0000			1.28163
$790$		0			0
$791$	−54.0000	+	54.0000i	−1.92002	+	1.92002i
$792$		0			0
$793$	−36.0000			−1.27840
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−30.0000			−1.06265			−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000			−1.06265			−0.531327	+	0.847167i	$0.678307\pi$
$798$		0			0
$799$			48.0000i			1.69812i
$800$		0			0
$801$	−7.00000	+	7.00000i	−0.247333	+	0.247333i
$802$		0			0
$803$	−28.0000	−	28.0000i	−0.988099	−	0.988099i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	14.0000	−	14.0000i	0.492823	−	0.492823i
$808$		0			0
$809$	39.0000	+	39.0000i	1.37117	+	1.37117i	0.858717	+	0.512450i	$0.171262\pi$
$809$	39.0000	+	39.0000i	1.37117	+	1.37117i	0.512450	+	0.858717i	$0.328738\pi$
$810$		0			0
$811$		−	20.0000i		−	0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$		−	20.0000i		−	0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$		0			0
$813$	−32.0000	−	32.0000i	−1.12229	−	1.12229i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−16.0000	−	16.0000i	−0.559769	−	0.559769i
$818$		0			0
$819$			12.0000i			0.419314i
$820$		0			0
$821$		−	12.0000i		−	0.418803i	−0.977830	−	0.209401i	$-0.932848\pi$
$821$		−	12.0000i		−	0.418803i	0.977830	−	0.209401i	$-0.0671515\pi$
$822$		0			0
$823$		−	18.0000i		−	0.627441i	−0.949515	−	0.313720i	$-0.898425\pi$
$823$		−	18.0000i		−	0.627441i	0.949515	−	0.313720i	$-0.101575\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	12.0000i		−	0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$		−	12.0000i		−	0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$		0			0
$829$	−35.0000	+	35.0000i	−1.21560	+	1.21560i	−0.246443	+	0.969157i	$0.579262\pi$
$829$	−35.0000	+	35.0000i	−1.21560	+	1.21560i	−0.969157	+	0.246443i	$0.920738\pi$
$830$		0			0
$831$	−24.0000	−	24.0000i	−0.832551	−	0.832551i
$832$		0			0
$833$		−	66.0000i		−	2.28676i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	24.0000	−	24.0000i	0.829561	−	0.829561i
$838$		0			0
$839$	8.00000	+	8.00000i	0.276191	+	0.276191i	0.831586	−	0.555396i	$-0.187433\pi$
$839$	8.00000	+	8.00000i	0.276191	+	0.276191i	−0.555396	+	0.831586i	$0.687433\pi$
$840$		0			0
$841$	−21.0000	−	20.0000i	−0.724138	−	0.689655i
$842$		0			0
$843$		−	8.00000i		−	0.275535i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	9.00000	+	9.00000i	0.309244	+	0.309244i
$848$		0			0
$849$	−14.0000	−	14.0000i	−0.480479	−	0.480479i
$850$		0			0
$851$	30.0000	−	30.0000i	1.02839	−	1.02839i
$852$		0			0
$853$	26.0000			0.890223			0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000			0.890223			0.445112	+	0.895475i	$0.353164\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	34.0000	−	34.0000i	1.16142	−	1.16142i	0.177252	−	0.984165i	$-0.443279\pi$
$857$	34.0000	−	34.0000i	1.16142	−	1.16142i	0.984165	−	0.177252i	$-0.0567209\pi$
$858$		0			0
$859$	−38.0000	−	38.0000i	−1.29654	−	1.29654i	−0.930661	−	0.365882i	$-0.880767\pi$
$859$	−38.0000	−	38.0000i	−1.29654	−	1.29654i	−0.365882	−	0.930661i	$-0.619233\pi$
$860$		0			0
$861$	−84.0000			−2.86271
$862$		0			0
$863$	15.0000	−	15.0000i	0.510606	−	0.510606i	−0.404106	−	0.914712i	$-0.632417\pi$
$863$	15.0000	−	15.0000i	0.510606	−	0.510606i	0.914712	+	0.404106i	$0.132417\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			38.0000i			1.29055i
$868$		0			0
$869$	−32.0000			−1.08553
$870$		0			0
$871$	12.0000			0.406604
$872$		0			0
$873$			2.00000i			0.0676897i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−32.0000	+	32.0000i	−1.08056	+	1.08056i	−0.0841064	+	0.996457i	$0.526804\pi$
$877$	−32.0000	+	32.0000i	−1.08056	+	1.08056i	−0.996457	+	0.0841064i	$0.973196\pi$
$878$		0			0
$879$	28.0000			0.944417
$880$		0			0
$881$	−19.0000	−	19.0000i	−0.640126	−	0.640126i	0.310460	−	0.950586i	$-0.399517\pi$
$881$	−19.0000	−	19.0000i	−0.640126	−	0.640126i	−0.950586	+	0.310460i	$0.899517\pi$
$882$		0			0
$883$	−5.00000	+	5.00000i	−0.168263	+	0.168263i	−0.786216	−	0.617952i	$-0.787963\pi$
$883$	−5.00000	+	5.00000i	−0.168263	+	0.168263i	0.617952	+	0.786216i	$0.287963\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−56.0000			−1.88030			−0.940148	−	0.340766i	$-0.889313\pi$
$887$	−56.0000			−1.88030			−0.940148	+	0.340766i	$0.889313\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−22.0000	−	22.0000i	−0.737028	−	0.737028i
$892$		0			0
$893$	−32.0000	−	32.0000i	−1.07084	−	1.07084i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	40.0000i		−	1.33556i
$898$		0			0
$899$	42.0000	−	18.0000i	1.40078	−	0.600334i
$900$		0			0
$901$	24.0000	+	24.0000i	0.799556	+	0.799556i
$902$		0			0
$903$	24.0000	−	24.0000i	0.798670	−	0.798670i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	28.0000i		−	0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$		−	28.0000i		−	0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$		0			0
$909$	9.00000	+	9.00000i	0.298511	+	0.298511i
$910$		0			0
$911$	26.0000	−	26.0000i	0.861418	−	0.861418i	−0.130084	−	0.991503i	$-0.541525\pi$
$911$	26.0000	−	26.0000i	0.861418	−	0.861418i	0.991503	+	0.130084i	$0.0415249\pi$
$912$		0			0
$913$			12.0000i			0.397142i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			24.0000i			0.792550i
$918$		0			0
$919$		−	16.0000i		−	0.527791i	−0.964551	−	0.263896i	$-0.914993\pi$
$919$		−	16.0000i		−	0.527791i	0.964551	−	0.263896i	$-0.0850075\pi$
$920$		0			0
$921$			8.00000i			0.263609i
$922$		0			0
$923$	24.0000	+	24.0000i	0.789970	+	0.789970i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	9.00000	+	9.00000i	0.295599	+	0.295599i
$928$		0			0
$929$		−	46.0000i		−	1.50921i	−0.656179	−	0.754606i	$-0.727828\pi$
$929$		−	46.0000i		−	1.50921i	0.656179	−	0.754606i	$-0.272172\pi$
$930$		0			0
$931$	44.0000	+	44.0000i	1.44204	+	1.44204i
$932$		0			0
$933$	−28.0000	+	28.0000i	−0.916679	+	0.916679i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−40.0000	−	40.0000i	−1.30674	−	1.30674i	−0.923752	−	0.382991i	$-0.874894\pi$
$937$	−40.0000	−	40.0000i	−1.30674	−	1.30674i	−0.382991	−	0.923752i	$-0.625106\pi$
$938$		0			0
$939$	20.0000	−	20.0000i	0.652675	−	0.652675i
$940$		0			0
$941$			24.0000i			0.782378i	0.920310	+	0.391189i	$0.127936\pi$
$941$			24.0000i			0.782378i	−0.920310	+	0.391189i	$0.872064\pi$
$942$		0			0
$943$	70.0000			2.27951
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.0000			0.389948			0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000			0.389948			0.194974	+	0.980808i	$0.437538\pi$
$948$		0			0
$949$	28.0000	−	28.0000i	0.908918	−	0.908918i
$950$		0			0
$951$	4.00000			0.129709
$952$		0			0
$953$	−8.00000	−	8.00000i	−0.259145	−	0.259145i	0.565561	−	0.824706i	$-0.308660\pi$
$953$	−8.00000	−	8.00000i	−0.259145	−	0.259145i	−0.824706	+	0.565561i	$0.808660\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−12.0000	−	28.0000i	−0.387905	−	0.905111i
$958$		0			0
$959$	−18.0000	+	18.0000i	−0.581250	+	0.581250i
$960$		0			0
$961$			41.0000i			1.32258i
$962$		0			0
$963$	5.00000	−	5.00000i	0.161123	−	0.161123i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			2.00000i			0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$			2.00000i			0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$		0			0
$969$	−48.0000	−	48.0000i	−1.54198	−	1.54198i
$970$		0			0
$971$	36.0000	+	36.0000i	1.15529	+	1.15529i	0.985475	+	0.169820i	$0.0543186\pi$
$971$	36.0000	+	36.0000i	1.15529	+	1.15529i	0.169820	+	0.985475i	$0.445681\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.00000	+	2.00000i	−0.0639857	+	0.0639857i	−0.738375	−	0.674390i	$-0.764407\pi$
$977$	−2.00000	+	2.00000i	−0.0639857	+	0.0639857i	0.674390	+	0.738375i	$0.264407\pi$
$978$		0			0
$979$	28.0000			0.894884
$980$		0			0
$981$	18.0000			0.574696
$982$		0			0
$983$	−48.0000			−1.53096			−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000			−1.53096			−0.765481	+	0.643458i	$0.777499\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	48.0000	−	48.0000i	1.52786	−	1.52786i
$988$		0			0
$989$	−20.0000	+	20.0000i	−0.635963	+	0.635963i
$990$		0			0
$991$		−	32.0000i		−	1.01651i	−0.861206	−	0.508257i	$-0.830290\pi$
$991$		−	32.0000i		−	1.01651i	0.861206	−	0.508257i	$-0.169710\pi$
$992$		0			0
$993$	−24.0000	−	24.0000i	−0.761617	−	0.761617i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	26.0000			0.823428			0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000			0.823428			0.411714	+	0.911313i	$0.364930\pi$
$998$		0			0
$999$	24.0000			0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.j.a.2593.1	yes	2
5.2	odd	4		2900.2.s.b.157.1	yes	2
5.3	odd	4		2900.2.s.a.157.1	yes	2
5.4	even	2		2900.2.j.b.2593.1	yes	2
29.17	odd	4		2900.2.s.b.1293.1	yes	2
145.17	even	4	inner	2900.2.j.a.1757.1	✓	2
145.104	odd	4		2900.2.s.a.1293.1	yes	2
145.133	even	4		2900.2.j.b.1757.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2900.2.j.a.1757.1	✓	2	145.17	even	4	inner
2900.2.j.a.2593.1	yes	2	1.1	even	1	trivial
2900.2.j.b.1757.1	yes	2	145.133	even	4
2900.2.j.b.2593.1	yes	2	5.4	even	2
2900.2.s.a.157.1	yes	2	5.3	odd	4
2900.2.s.a.1293.1	yes	2	145.104	odd	4
2900.2.s.b.157.1	yes	2	5.2	odd	4
2900.2.s.b.1293.1	yes	2	29.17	odd	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:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +(-3.00000 + 3.00000i) q^{7} -1.00000 q^{9} +(2.00000 + 2.00000i) q^{11} +(-2.00000 + 2.00000i) q^{13} +6.00000 q^{17} +(-4.00000 + 4.00000i) q^{19} +(-6.00000 - 6.00000i) q^{21} +(5.00000 + 5.00000i) q^{23} +4.00000i q^{27} +(-2.00000 + 5.00000i) q^{29} +(-6.00000 - 6.00000i) q^{31} +(-4.00000 + 4.00000i) q^{33} -6.00000i q^{37} +(-4.00000 - 4.00000i) q^{39} +(7.00000 - 7.00000i) q^{41} +4.00000i q^{43} +8.00000i q^{47} -11.0000i q^{49} +12.0000i q^{51} +(4.00000 + 4.00000i) q^{53} +(-8.00000 - 8.00000i) q^{57} -4.00000i q^{59} +(9.00000 + 9.00000i) q^{61} +(3.00000 - 3.00000i) q^{63} +(-3.00000 - 3.00000i) q^{67} +(-10.0000 + 10.0000i) q^{69} -12.0000i q^{71} -14.0000 q^{73} -12.0000 q^{77} +(-8.00000 + 8.00000i) q^{79} -11.0000 q^{81} +(3.00000 + 3.00000i) q^{83} +(-10.0000 - 4.00000i) q^{87} +(7.00000 - 7.00000i) q^{89} -12.0000i q^{91} +(12.0000 - 12.0000i) q^{93} -2.00000i q^{97} +(-2.00000 - 2.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{7} - 2 q^{9} + 4 q^{11} - 4 q^{13} + 12 q^{17} - 8 q^{19} - 12 q^{21} + 10 q^{23} - 4 q^{29} - 12 q^{31} - 8 q^{33} - 8 q^{39} + 14 q^{41} + 8 q^{53} - 16 q^{57} + 18 q^{61} + 6 q^{63} - 6 q^{67}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 6 * q^7 - 2 * q^9 + 4 * q^11 - 4 * q^13 + 12 * q^17 - 8 * q^19 - 12 * q^21 + 10 * q^23 - 4 * q^29 - 12 * q^31 - 8 * q^33 - 8 * q^39 + 14 * q^41 + 8 * q^53 - 16 * q^57 + 18 * q^61 + 6 * q^63 - 6 * q^67 - 20 * q^69 - 28 * q^73 - 24 * q^77 - 16 * q^79 - 22 * q^81 + 6 * q^83 - 20 * q^87 + 14 * q^89 + 24 * q^93 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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