LMFDB - Embedded newform 3900.2.j.b.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3900,2,Mod(649,3900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3900.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3900.j (of order $2$, degree $1$, not 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:	$31.1416567883$
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 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3900.649
Dual form			3900.2.j.b.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -4.00000 q^{7} -1.00000 q^{9} +6.00000i q^{11} +(-2.00000 + 3.00000i) q^{13} +2.00000i q^{17} -4.00000i q^{21} -8.00000i q^{23} -1.00000i q^{27} -2.00000 q^{29} +8.00000i q^{31} -6.00000 q^{33} -8.00000 q^{37} +(-3.00000 - 2.00000i) q^{39} +2.00000i q^{41} -8.00000i q^{43} +6.00000 q^{47} +9.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -2.00000i q^{59} +2.00000 q^{61} +4.00000 q^{63} -4.00000 q^{67} +8.00000 q^{69} -6.00000i q^{71} -4.00000 q^{73} -24.0000i q^{77} +1.00000 q^{81} +14.0000 q^{83} -2.00000i q^{87} +6.00000i q^{89} +(8.00000 - 12.0000i) q^{91} -8.00000 q^{93} +12.0000 q^{97} -6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7} - 2 q^{9} - 4 q^{13} - 4 q^{29} - 12 q^{33} - 16 q^{37} - 6 q^{39} + 12 q^{47} + 18 q^{49} - 4 q^{51} + 4 q^{61} + 8 q^{63} - 8 q^{67} + 16 q^{69} - 8 q^{73} + 2 q^{81} + 28 q^{83} + 16 q^{91}+ \cdots + 24 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 - 2 * q^9 - 4 * q^13 - 4 * q^29 - 12 * q^33 - 16 * q^37 - 6 * q^39 + 12 * q^47 + 18 * q^49 - 4 * q^51 + 4 * q^61 + 8 * q^63 - 8 * q^67 + 16 * q^69 - 8 * q^73 + 2 * q^81 + 28 * q^83 + 16 * q^91 - 16 * q^93 + 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1951$	$3277$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−4.00000			−1.51186			−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000			−1.51186			−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			6.00000i			1.80907i	0.426401	+	0.904534i	$0.359781\pi$
$11$			6.00000i			1.80907i	−0.426401	+	0.904534i	$0.640219\pi$
$12$		0			0
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		−	4.00000i		−	0.872872i
$22$		0			0
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$			8.00000i			1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$			8.00000i			1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$		0			0
$33$	−6.00000			−1.04447
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−8.00000			−1.31519			−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000			−1.31519			−0.657596	+	0.753371i	$0.728427\pi$
$38$		0			0
$39$	−3.00000	−	2.00000i	−0.480384	−	0.320256i
$40$		0			0
$41$			2.00000i			0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$			2.00000i			0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$		0			0
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.00000			0.875190			0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000			0.875190			0.437595	+	0.899172i	$0.355830\pi$
$48$		0			0
$49$	9.00000			1.28571
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	2.00000i		−	0.260378i	−0.991489	−	0.130189i	$-0.958442\pi$
$59$		−	2.00000i		−	0.260378i	0.991489	−	0.130189i	$-0.0415584\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$	4.00000			0.503953
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$		0			0
$69$	8.00000			0.963087
$70$		0			0
$71$		−	6.00000i		−	0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$		−	6.00000i		−	0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	24.0000i		−	2.73505i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	14.0000			1.53670			0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000			1.53670			0.768350	+	0.640030i	$0.221078\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	2.00000i		−	0.214423i
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$	8.00000	−	12.0000i	0.838628	−	1.25794i
$92$		0			0
$93$	−8.00000			−0.829561
$94$		0			0
$95$		0			0
$96$		0			0
$97$	12.0000			1.21842			0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000			1.21842			0.609208	+	0.793011i	$0.291488\pi$
$98$		0			0
$99$		−	6.00000i		−	0.603023i
$100$		0			0
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	20.0000i		−	1.93347i	−0.255774	−	0.966736i	$-0.582330\pi$
$107$		−	20.0000i		−	1.93347i	0.255774	−	0.966736i	$-0.417670\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$		−	8.00000i		−	0.759326i
$112$		0			0
$113$		−	10.0000i		−	0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$		−	10.0000i		−	0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.00000	−	3.00000i	0.184900	−	0.277350i
$118$		0			0
$119$		−	8.00000i		−	0.733359i
$120$		0			0
$121$	−25.0000			−2.27273
$122$		0			0
$123$	−2.00000			−0.180334
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$		0			0
$129$	8.00000			0.704361
$130$		0			0
$131$	4.00000			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.0000			−0.854358			−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000			−0.854358			−0.427179	+	0.904167i	$0.640493\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$		0			0
$141$			6.00000i			0.505291i
$142$		0			0
$143$	−18.0000	−	12.0000i	−1.50524	−	1.00349i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			9.00000i			0.742307i
$148$		0			0
$149$		−	10.0000i		−	0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$		−	10.0000i		−	0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$		0			0
$151$		−	16.0000i		−	1.30206i	−0.759051	−	0.651031i	$-0.774337\pi$
$151$		−	16.0000i		−	1.30206i	0.759051	−	0.651031i	$-0.225663\pi$
$152$		0			0
$153$		−	2.00000i		−	0.161690i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			14.0000i			1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$			14.0000i			1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$			32.0000i			2.52195i
$162$		0			0
$163$	−8.00000			−0.626608			−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000			−0.626608			−0.313304	+	0.949653i	$0.601436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−18.0000			−1.39288			−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000			−1.39288			−0.696441	+	0.717614i	$0.745234\pi$
$168$		0			0
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	2.00000			0.150329
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$			2.00000i			0.147844i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−12.0000			−0.877527
$188$		0			0
$189$			4.00000i			0.290957i
$190$		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
$193$	−8.00000			−0.575853			−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000			−0.575853			−0.287926	+	0.957653i	$0.592966\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−26.0000			−1.85242			−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000			−1.85242			−0.926212	+	0.377004i	$0.876954\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		−	4.00000i		−	0.282138i
$202$		0			0
$203$	8.00000			0.561490
$204$		0			0
$205$		0			0
$206$		0			0
$207$			8.00000i			0.556038i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$	6.00000			0.411113
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	32.0000i		−	2.17230i
$218$		0			0
$219$		−	4.00000i		−	0.270295i
$220$		0			0
$221$	−6.00000	−	4.00000i	−0.403604	−	0.269069i
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.00000			−0.132745			−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000			−0.132745			−0.0663723	+	0.997795i	$0.521143\pi$
$228$		0			0
$229$		−	20.0000i		−	1.32164i	−0.750546	−	0.660819i	$-0.770209\pi$
$229$		−	20.0000i		−	1.32164i	0.750546	−	0.660819i	$-0.229791\pi$
$230$		0			0
$231$	24.0000			1.57908
$232$		0			0
$233$			18.0000i			1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$			18.0000i			1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	10.0000i		−	0.646846i	−0.946254	−	0.323423i	$-0.895166\pi$
$239$		−	10.0000i		−	0.646846i	0.946254	−	0.323423i	$-0.104834\pi$
$240$		0			0
$241$		−	20.0000i		−	1.28831i	−0.764894	−	0.644157i	$-0.777208\pi$
$241$		−	20.0000i		−	1.28831i	0.764894	−	0.644157i	$-0.222792\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$			14.0000i			0.887214i
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$	48.0000			3.01773
$254$		0			0
$255$		0			0
$256$		0			0
$257$			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
$259$	32.0000			1.98838
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−6.00000			−0.367194
$268$		0			0
$269$	22.0000			1.34136			0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000			1.34136			0.670682	+	0.741745i	$0.266002\pi$
$270$		0			0
$271$		−	20.0000i		−	1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$		−	20.0000i		−	1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$		0			0
$273$	12.0000	+	8.00000i	0.726273	+	0.484182i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	22.0000i		−	1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$		−	22.0000i		−	1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$		0			0
$279$		−	8.00000i		−	0.478947i
$280$		0			0
$281$			22.0000i			1.31241i	0.754583	+	0.656205i	$0.227839\pi$
$281$			22.0000i			1.31241i	−0.754583	+	0.656205i	$0.772161\pi$
$282$		0			0
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	8.00000i		−	0.472225i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$			12.0000i			0.703452i
$292$		0			0
$293$	−14.0000			−0.817889			−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000			−0.817889			−0.408944	+	0.912559i	$0.634103\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	6.00000			0.348155
$298$		0			0
$299$	24.0000	+	16.0000i	1.38796	+	0.925304i
$300$		0			0
$301$			32.0000i			1.84445i
$302$		0			0
$303$		−	10.0000i		−	0.574485i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$		0			0
$309$	4.00000			0.227552
$310$		0			0
$311$	−16.0000			−0.907277			−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000			−0.907277			−0.453638	+	0.891186i	$0.649874\pi$
$312$		0			0
$313$		−	14.0000i		−	0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$		−	14.0000i		−	0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$		−	12.0000i		−	0.671871i
$320$		0			0
$321$	20.0000			1.11629
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−4.00000			−0.221201
$328$		0			0
$329$	−24.0000			−1.32316
$330$		0			0
$331$			28.0000i			1.53902i	0.638635	+	0.769510i	$0.279499\pi$
$331$			28.0000i			1.53902i	−0.638635	+	0.769510i	$0.720501\pi$
$332$		0			0
$333$	8.00000			0.438397
$334$		0			0
$335$		0			0
$336$		0			0
$337$			14.0000i			0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$			14.0000i			0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$		0			0
$339$	10.0000			0.543125
$340$		0			0
$341$	−48.0000			−2.59935
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$			8.00000i			0.428230i	0.976808	+	0.214115i	$0.0686868\pi$
$349$			8.00000i			0.428230i	−0.976808	+	0.214115i	$0.931313\pi$
$350$		0			0
$351$	3.00000	+	2.00000i	0.160128	+	0.106752i
$352$		0			0
$353$	−10.0000			−0.532246			−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000			−0.532246			−0.266123	+	0.963939i	$0.585743\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	8.00000			0.423405
$358$		0			0
$359$			10.0000i			0.527780i	0.964553	+	0.263890i	$0.0850056\pi$
$359$			10.0000i			0.527780i	−0.964553	+	0.263890i	$0.914994\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$		−	25.0000i		−	1.31216i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			28.0000i			1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$			28.0000i			1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$		0			0
$369$		−	2.00000i		−	0.104116i
$370$		0			0
$371$		−	24.0000i		−	1.24602i
$372$		0			0
$373$			6.00000i			0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$			6.00000i			0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.00000	−	6.00000i	0.206010	−	0.309016i
$378$		0			0
$379$			16.0000i			0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$			16.0000i			0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$		0			0
$381$	8.00000			0.409852
$382$		0			0
$383$	−22.0000			−1.12415			−0.562074	−	0.827087i	$-0.689996\pi$
$383$	−22.0000			−1.12415			−0.562074	+	0.827087i	$0.689996\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			8.00000i			0.406663i
$388$		0			0
$389$	−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
$391$	16.0000			0.809155
$392$		0			0
$393$			4.00000i			0.201773i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	16.0000			0.803017			0.401508	−	0.915855i	$-0.368486\pi$
$397$	16.0000			0.803017			0.401508	+	0.915855i	$0.368486\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	6.00000i		−	0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$		−	6.00000i		−	0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$		0			0
$403$	−24.0000	−	16.0000i	−1.19553	−	0.797017i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	48.0000i		−	2.37927i
$408$		0			0
$409$		−	20.0000i		−	0.988936i	−0.869196	−	0.494468i	$-0.835363\pi$
$409$		−	20.0000i		−	0.988936i	0.869196	−	0.494468i	$-0.164637\pi$
$410$		0			0
$411$		−	10.0000i		−	0.493264i
$412$		0			0
$413$			8.00000i			0.393654i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			8.00000i			0.391762i
$418$		0			0
$419$	−28.0000			−1.36789			−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000			−1.36789			−0.683945	+	0.729534i	$0.739737\pi$
$420$		0			0
$421$			4.00000i			0.194948i	0.995238	+	0.0974740i	$0.0310763\pi$
$421$			4.00000i			0.194948i	−0.995238	+	0.0974740i	$0.968924\pi$
$422$		0			0
$423$	−6.00000			−0.291730
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−8.00000			−0.387147
$428$		0			0
$429$	12.0000	−	18.0000i	0.579365	−	0.869048i
$430$		0			0
$431$		−	10.0000i		−	0.481683i	−0.970564	−	0.240842i	$-0.922577\pi$
$431$		−	10.0000i		−	0.481683i	0.970564	−	0.240842i	$-0.0774234\pi$
$432$		0			0
$433$			22.0000i			1.05725i	0.848855	+	0.528626i	$0.177293\pi$
$433$			22.0000i			1.05725i	−0.848855	+	0.528626i	$0.822707\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−36.0000			−1.71819			−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000			−1.71819			−0.859093	+	0.511819i	$0.828972\pi$
$440$		0			0
$441$	−9.00000			−0.428571
$442$		0			0
$443$			20.0000i			0.950229i	0.879924	+	0.475114i	$0.157593\pi$
$443$			20.0000i			0.950229i	−0.879924	+	0.475114i	$0.842407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	10.0000			0.472984
$448$		0			0
$449$			30.0000i			1.41579i	0.706319	+	0.707894i	$0.250354\pi$
$449$			30.0000i			1.41579i	−0.706319	+	0.707894i	$0.749646\pi$
$450$		0			0
$451$	−12.0000			−0.565058
$452$		0			0
$453$	16.0000			0.751746
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.00000			−0.187112			−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000			−0.187112			−0.0935561	+	0.995614i	$0.529823\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$			6.00000i			0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$			6.00000i			0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$		0			0
$463$	12.0000			0.557687			0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000			0.557687			0.278844	+	0.960337i	$0.410049\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	12.0000i		−	0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$		−	12.0000i		−	0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$		0			0
$471$	−14.0000			−0.645086
$472$		0			0
$473$	48.0000			2.20704
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	6.00000i		−	0.274721i
$478$		0			0
$479$		−	18.0000i		−	0.822441i	−0.911536	−	0.411220i	$-0.865103\pi$
$479$		−	18.0000i		−	0.822441i	0.911536	−	0.411220i	$-0.134897\pi$
$480$		0			0
$481$	16.0000	−	24.0000i	0.729537	−	1.09431i
$482$		0			0
$483$	−32.0000			−1.45605
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		−	8.00000i		−	0.361773i
$490$		0			0
$491$	20.0000			0.902587			0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000			0.902587			0.451294	+	0.892375i	$0.350963\pi$
$492$		0			0
$493$		−	4.00000i		−	0.180151i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			24.0000i			1.07655i
$498$		0			0
$499$		−	4.00000i		−	0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$		−	4.00000i		−	0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$		0			0
$501$		−	18.0000i		−	0.804181i
$502$		0			0
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	12.0000	−	5.00000i	0.532939	−	0.222058i
$508$		0			0
$509$		−	30.0000i		−	1.32973i	−0.746965	−	0.664863i	$-0.768490\pi$
$509$		−	30.0000i		−	1.32973i	0.746965	−	0.664863i	$-0.231510\pi$
$510$		0			0
$511$	16.0000			0.707798
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			36.0000i			1.58328i
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$		−	8.00000i		−	0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$		−	8.00000i		−	0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−16.0000			−0.696971
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$			2.00000i			0.0867926i
$532$		0			0
$533$	−6.00000	−	4.00000i	−0.259889	−	0.173259i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			12.0000i			0.517838i
$538$		0			0
$539$			54.0000i			2.32594i
$540$		0			0
$541$			20.0000i			0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$			20.0000i			0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$		0			0
$543$			10.0000i			0.429141i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$		0			0
$549$	−2.00000			−0.0853579
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	30.0000			1.27114			0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000			1.27114			0.635570	+	0.772043i	$0.280765\pi$
$558$		0			0
$559$	24.0000	+	16.0000i	1.01509	+	0.676728i
$560$		0			0
$561$		−	12.0000i		−	0.506640i
$562$		0			0
$563$		−	4.00000i		−	0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$		−	4.00000i		−	0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−4.00000			−0.167984
$568$		0			0
$569$	−34.0000			−1.42535			−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000			−1.42535			−0.712677	+	0.701492i	$0.752517\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$		−	24.0000i		−	1.00261i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	40.0000			1.66522			0.832611	−	0.553858i	$-0.186845\pi$
$577$	40.0000			1.66522			0.832611	+	0.553858i	$0.186845\pi$
$578$		0			0
$579$		−	8.00000i		−	0.332469i
$580$		0			0
$581$	−56.0000			−2.32327
$582$		0			0
$583$	−36.0000			−1.49097
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.00000			0.0825488			0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000			0.0825488			0.0412744	+	0.999148i	$0.486858\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		−	26.0000i		−	1.06950i
$592$		0			0
$593$	−34.0000			−1.39621			−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000			−1.39621			−0.698106	+	0.715994i	$0.745974\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−14.0000			−0.571072			−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000			−0.571072			−0.285536	+	0.958368i	$0.592172\pi$
$602$		0			0
$603$	4.00000			0.162893
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	4.00000i		−	0.162355i	−0.996700	−	0.0811775i	$-0.974132\pi$
$607$		−	4.00000i		−	0.162355i	0.996700	−	0.0811775i	$-0.0258681\pi$
$608$		0			0
$609$			8.00000i			0.324176i
$610$		0			0
$611$	−12.0000	+	18.0000i	−0.485468	+	0.728202i
$612$		0			0
$613$	16.0000			0.646234			0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000			0.646234			0.323117	+	0.946359i	$0.395269\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.0000			−0.563619			−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000			−0.563619			−0.281809	+	0.959470i	$0.590935\pi$
$618$		0			0
$619$		−	20.0000i		−	0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$		−	20.0000i		−	0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$		0			0
$621$	−8.00000			−0.321029
$622$		0			0
$623$		−	24.0000i		−	0.961540i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	16.0000i		−	0.637962i
$630$		0			0
$631$			8.00000i			0.318475i	0.987240	+	0.159237i	$0.0509036\pi$
$631$			8.00000i			0.318475i	−0.987240	+	0.159237i	$0.949096\pi$
$632$		0			0
$633$			12.0000i			0.476957i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−18.0000	+	27.0000i	−0.713186	+	1.06978i
$638$		0			0
$639$			6.00000i			0.237356i
$640$		0			0
$641$	−42.0000			−1.65890			−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000			−1.65890			−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$	36.0000			1.41970			0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000			1.41970			0.709851	+	0.704352i	$0.248762\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			32.0000i			1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$			32.0000i			1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$	32.0000			1.25418
$652$		0			0
$653$		−	14.0000i		−	0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$		−	14.0000i		−	0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	4.00000			0.156055
$658$		0			0
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$		0			0
$661$		−	32.0000i		−	1.24466i	−0.782757	−	0.622328i	$-0.786187\pi$
$661$		−	32.0000i		−	1.24466i	0.782757	−	0.622328i	$-0.213813\pi$
$662$		0			0
$663$	4.00000	−	6.00000i	0.155347	−	0.233021i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			16.0000i			0.619522i
$668$		0			0
$669$		−	8.00000i		−	0.309298i
$670$		0			0
$671$			12.0000i			0.463255i
$672$		0			0
$673$			30.0000i			1.15642i	0.815890	+	0.578208i	$0.196248\pi$
$673$			30.0000i			1.15642i	−0.815890	+	0.578208i	$0.803752\pi$
$674$		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
$679$	−48.0000			−1.84207
$680$		0			0
$681$		−	2.00000i		−	0.0766402i
$682$		0			0
$683$	−6.00000			−0.229584			−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000			−0.229584			−0.114792	+	0.993390i	$0.536620\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	20.0000			0.763048
$688$		0			0
$689$	−18.0000	−	12.0000i	−0.685745	−	0.457164i
$690$		0			0
$691$			4.00000i			0.152167i	0.997101	+	0.0760836i	$0.0242416\pi$
$691$			4.00000i			0.152167i	−0.997101	+	0.0760836i	$0.975758\pi$
$692$		0			0
$693$			24.0000i			0.911685i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−4.00000			−0.151511
$698$		0			0
$699$	−18.0000			−0.680823
$700$		0			0
$701$	10.0000			0.377695			0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000			0.377695			0.188847	+	0.982006i	$0.439525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	40.0000			1.50435
$708$		0			0
$709$			36.0000i			1.35201i	0.736898	+	0.676004i	$0.236290\pi$
$709$			36.0000i			1.35201i	−0.736898	+	0.676004i	$0.763710\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	64.0000			2.39682
$714$		0			0
$715$		0			0
$716$		0			0
$717$	10.0000			0.373457
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$			16.0000i			0.595871i
$722$		0			0
$723$	20.0000			0.743808
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	8.00000i		−	0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$		−	8.00000i		−	0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	16.0000			0.591781
$732$		0			0
$733$	−20.0000			−0.738717			−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000			−0.738717			−0.369358	+	0.929287i	$0.620423\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	24.0000i		−	0.884051i
$738$		0			0
$739$			28.0000i			1.03000i	0.857191	+	0.514998i	$0.172207\pi$
$739$			28.0000i			1.03000i	−0.857191	+	0.514998i	$0.827793\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	22.0000			0.807102			0.403551	−	0.914957i	$-0.367776\pi$
$743$	22.0000			0.807102			0.403551	+	0.914957i	$0.367776\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−14.0000			−0.512233
$748$		0			0
$749$			80.0000i			2.92314i
$750$		0			0
$751$	20.0000			0.729810			0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000			0.729810			0.364905	+	0.931045i	$0.381101\pi$
$752$		0			0
$753$			20.0000i			0.728841i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	2.00000i		−	0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$		−	2.00000i		−	0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$		0			0
$759$			48.0000i			1.74229i
$760$		0			0
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		0			0
$763$		−	16.0000i		−	0.579239i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.00000	+	4.00000i	0.216647	+	0.144432i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−6.00000			−0.216085
$772$		0			0
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			32.0000i			1.14799i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$			2.00000i			0.0714742i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−44.0000			−1.56843			−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000			−1.56843			−0.784215	+	0.620489i	$0.786934\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			40.0000i			1.42224i
$792$		0			0
$793$	−4.00000	+	6.00000i	−0.142044	+	0.213066i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	10.0000i		−	0.354218i	−0.984191	−	0.177109i	$-0.943325\pi$
$797$		−	10.0000i		−	0.354218i	0.984191	−	0.177109i	$-0.0566745\pi$
$798$		0			0
$799$			12.0000i			0.424529i
$800$		0			0
$801$		−	6.00000i		−	0.212000i
$802$		0			0
$803$		−	24.0000i		−	0.846942i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			22.0000i			0.774437i
$808$		0			0
$809$	−22.0000			−0.773479			−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000			−0.773479			−0.386739	+	0.922189i	$0.626399\pi$
$810$		0			0
$811$		−	40.0000i		−	1.40459i	−0.711886	−	0.702295i	$-0.752159\pi$
$811$		−	40.0000i		−	1.40459i	0.711886	−	0.702295i	$-0.247841\pi$
$812$		0			0
$813$	20.0000			0.701431
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−8.00000	+	12.0000i	−0.279543	+	0.419314i
$820$		0			0
$821$		−	30.0000i		−	1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$		−	30.0000i		−	1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$		0			0
$823$		−	36.0000i		−	1.25488i	−0.778664	−	0.627441i	$-0.784103\pi$
$823$		−	36.0000i		−	1.25488i	0.778664	−	0.627441i	$-0.215897\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.0000			−0.625921			−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000			−0.625921			−0.312961	+	0.949766i	$0.601321\pi$
$828$		0			0
$829$	54.0000			1.87550			0.937749	−	0.347314i	$-0.112906\pi$
$829$	54.0000			1.87550			0.937749	+	0.347314i	$0.112906\pi$
$830$		0			0
$831$	22.0000			0.763172
$832$		0			0
$833$			18.0000i			0.623663i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	8.00000			0.276520
$838$		0			0
$839$		−	10.0000i		−	0.345238i	−0.984989	−	0.172619i	$-0.944777\pi$
$839$		−	10.0000i		−	0.345238i	0.984989	−	0.172619i	$-0.0552230\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	−22.0000			−0.757720
$844$		0			0
$845$		0			0
$846$		0			0
$847$	100.000			3.43604
$848$		0			0
$849$	−4.00000			−0.137280
$850$		0			0
$851$			64.0000i			2.19389i
$852$		0			0
$853$	−16.0000			−0.547830			−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000			−0.547830			−0.273915	+	0.961754i	$0.588319\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			22.0000i			0.751506i	0.926720	+	0.375753i	$0.122616\pi$
$857$			22.0000i			0.751506i	−0.926720	+	0.375753i	$0.877384\pi$
$858$		0			0
$859$	−52.0000			−1.77422			−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000			−1.77422			−0.887109	+	0.461561i	$0.847290\pi$
$860$		0			0
$861$	8.00000			0.272639
$862$		0			0
$863$	34.0000			1.15737			0.578687	−	0.815550i	$-0.303565\pi$
$863$	34.0000			1.15737			0.578687	+	0.815550i	$0.303565\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			13.0000i			0.441503i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	8.00000	−	12.0000i	0.271070	−	0.406604i
$872$		0			0
$873$	−12.0000			−0.406138
$874$		0			0
$875$		0			0
$876$		0			0
$877$	16.0000			0.540282			0.270141	−	0.962821i	$-0.412930\pi$
$877$	16.0000			0.540282			0.270141	+	0.962821i	$0.412930\pi$
$878$		0			0
$879$		−	14.0000i		−	0.472208i
$880$		0			0
$881$	42.0000			1.41502			0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000			1.41502			0.707508	+	0.706705i	$0.249819\pi$
$882$		0			0
$883$		−	12.0000i		−	0.403832i	−0.979403	−	0.201916i	$-0.935283\pi$
$883$		−	12.0000i		−	0.403832i	0.979403	−	0.201916i	$-0.0647168\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	24.0000i		−	0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$		−	24.0000i		−	0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$		0			0
$889$			32.0000i			1.07325i
$890$		0			0
$891$			6.00000i			0.201008i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−16.0000	+	24.0000i	−0.534224	+	0.801337i
$898$		0			0
$899$		−	16.0000i		−	0.533630i
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$	−32.0000			−1.06489
$904$		0			0
$905$		0			0
$906$		0			0
$907$			44.0000i			1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$			44.0000i			1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$		0			0
$909$	10.0000			0.331679
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$			84.0000i			2.77999i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−16.0000			−0.528367
$918$		0			0
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$		0			0
$921$			20.0000i			0.659022i
$922$		0			0
$923$	18.0000	+	12.0000i	0.592477	+	0.394985i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			4.00000i			0.131377i
$928$		0			0
$929$			30.0000i			0.984268i	0.870519	+	0.492134i	$0.163783\pi$
$929$			30.0000i			0.984268i	−0.870519	+	0.492134i	$0.836217\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	16.0000i		−	0.523816i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			50.0000i			1.63343i	0.577042	+	0.816714i	$0.304207\pi$
$937$			50.0000i			1.63343i	−0.577042	+	0.816714i	$0.695793\pi$
$938$		0			0
$939$	14.0000			0.456873
$940$		0			0
$941$		−	38.0000i		−	1.23876i	−0.785090	−	0.619382i	$-0.787383\pi$
$941$		−	38.0000i		−	1.23876i	0.785090	−	0.619382i	$-0.212617\pi$
$942$		0			0
$943$	16.0000			0.521032
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−38.0000			−1.23483			−0.617417	−	0.786636i	$-0.711821\pi$
$947$	−38.0000			−1.23483			−0.617417	+	0.786636i	$0.711821\pi$
$948$		0			0
$949$	8.00000	−	12.0000i	0.259691	−	0.389536i
$950$		0			0
$951$		−	18.0000i		−	0.583690i
$952$		0			0
$953$		−	54.0000i		−	1.74923i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$		−	54.0000i		−	1.74923i	0.484817	−	0.874616i	$-0.338886\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	12.0000			0.387905
$958$		0			0
$959$	40.0000			1.29167
$960$		0			0
$961$	−33.0000			−1.06452
$962$		0			0
$963$			20.0000i			0.644491i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−8.00000			−0.257263			−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000			−0.257263			−0.128631	+	0.991692i	$0.541058\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$	−32.0000			−1.02587
$974$		0			0
$975$		0			0
$976$		0			0
$977$	30.0000			0.959785			0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000			0.959785			0.479893	+	0.877327i	$0.340676\pi$
$978$		0			0
$979$	−36.0000			−1.15056
$980$		0			0
$981$		−	4.00000i		−	0.127710i
$982$		0			0
$983$	−18.0000			−0.574111			−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000			−0.574111			−0.287055	+	0.957914i	$0.592676\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	24.0000i		−	0.763928i
$988$		0			0
$989$	−64.0000			−2.03508
$990$		0			0
$991$	−44.0000			−1.39771			−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000			−1.39771			−0.698853	+	0.715265i	$0.746306\pi$
$992$		0			0
$993$	−28.0000			−0.888553
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	6.00000i		−	0.190022i	−0.995476	−	0.0950110i	$-0.969711\pi$
$997$		−	6.00000i		−	0.190022i	0.995476	−	0.0950110i	$-0.0302886\pi$
$998$		0			0
$999$			8.00000i			0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3900.2.j.b.649.2		2
5.2	odd	4		156.2.b.b.25.1	✓	2
5.3	odd	4		3900.2.c.a.3301.2		2
5.4	even	2		3900.2.j.e.649.1		2
13.12	even	2		3900.2.j.e.649.2		2
15.2	even	4		468.2.b.c.181.2		2
20.7	even	4		624.2.c.d.337.1		2
35.27	even	4		7644.2.e.b.4705.2		2
40.27	even	4		2496.2.c.i.961.2		2
40.37	odd	4		2496.2.c.b.961.2		2
60.47	odd	4		1872.2.c.h.1585.2		2
65.2	even	12		2028.2.i.a.529.1		2
65.7	even	12		2028.2.i.d.2005.1		2
65.12	odd	4		156.2.b.b.25.2	yes	2
65.17	odd	12		2028.2.q.e.361.2		4
65.22	odd	12		2028.2.q.e.361.1		4
65.32	even	12		2028.2.i.a.2005.1		2
65.37	even	12		2028.2.i.d.529.1		2
65.38	odd	4		3900.2.c.a.3301.1		2
65.42	odd	12		2028.2.q.e.1837.1		4
65.47	even	4		2028.2.a.f.1.1		1
65.57	even	4		2028.2.a.d.1.1		1
65.62	odd	12		2028.2.q.e.1837.2		4
65.64	even	2	inner	3900.2.j.b.649.1		2
195.47	odd	4		6084.2.a.d.1.1		1
195.77	even	4		468.2.b.c.181.1		2
195.122	odd	4		6084.2.a.n.1.1		1
260.47	odd	4		8112.2.a.l.1.1		1
260.187	odd	4		8112.2.a.d.1.1		1
260.207	even	4		624.2.c.d.337.2		2
455.272	even	4		7644.2.e.b.4705.1		2
520.77	odd	4		2496.2.c.b.961.1		2
520.467	even	4		2496.2.c.i.961.1		2
780.467	odd	4		1872.2.c.h.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.b.b.25.1	✓	2	5.2	odd	4
156.2.b.b.25.2	yes	2	65.12	odd	4
468.2.b.c.181.1		2	195.77	even	4
468.2.b.c.181.2		2	15.2	even	4
624.2.c.d.337.1		2	20.7	even	4
624.2.c.d.337.2		2	260.207	even	4
1872.2.c.h.1585.1		2	780.467	odd	4
1872.2.c.h.1585.2		2	60.47	odd	4
2028.2.a.d.1.1		1	65.57	even	4
2028.2.a.f.1.1		1	65.47	even	4
2028.2.i.a.529.1		2	65.2	even	12
2028.2.i.a.2005.1		2	65.32	even	12
2028.2.i.d.529.1		2	65.37	even	12
2028.2.i.d.2005.1		2	65.7	even	12
2028.2.q.e.361.1		4	65.22	odd	12
2028.2.q.e.361.2		4	65.17	odd	12
2028.2.q.e.1837.1		4	65.42	odd	12
2028.2.q.e.1837.2		4	65.62	odd	12
2496.2.c.b.961.1		2	520.77	odd	4
2496.2.c.b.961.2		2	40.37	odd	4
2496.2.c.i.961.1		2	520.467	even	4
2496.2.c.i.961.2		2	40.27	even	4
3900.2.c.a.3301.1		2	65.38	odd	4
3900.2.c.a.3301.2		2	5.3	odd	4
3900.2.j.b.649.1		2	65.64	even	2	inner
3900.2.j.b.649.2		2	1.1	even	1	trivial
3900.2.j.e.649.1		2	5.4	even	2
3900.2.j.e.649.2		2	13.12	even	2
6084.2.a.d.1.1		1	195.47	odd	4
6084.2.a.n.1.1		1	195.122	odd	4
7644.2.e.b.4705.1		2	455.272	even	4
7644.2.e.b.4705.2		2	35.27	even	4
8112.2.a.d.1.1		1	260.187	odd	4
8112.2.a.l.1.1		1	260.47	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3900.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -4.00000 q^{7} -1.00000 q^{9} +6.00000i q^{11} +(-2.00000 + 3.00000i) q^{13} +2.00000i q^{17} -4.00000i q^{21} -8.00000i q^{23} -1.00000i q^{27} -2.00000 q^{29} +8.00000i q^{31} -6.00000 q^{33} -8.00000 q^{37} +(-3.00000 - 2.00000i) q^{39} +2.00000i q^{41} -8.00000i q^{43} +6.00000 q^{47} +9.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -2.00000i q^{59} +2.00000 q^{61} +4.00000 q^{63} -4.00000 q^{67} +8.00000 q^{69} -6.00000i q^{71} -4.00000 q^{73} -24.0000i q^{77} +1.00000 q^{81} +14.0000 q^{83} -2.00000i q^{87} +6.00000i q^{89} +(8.00000 - 12.0000i) q^{91} -8.00000 q^{93} +12.0000 q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7} - 2 q^{9} - 4 q^{13} - 4 q^{29} - 12 q^{33} - 16 q^{37} - 6 q^{39} + 12 q^{47} + 18 q^{49} - 4 q^{51} + 4 q^{61} + 8 q^{63} - 8 q^{67} + 16 q^{69} - 8 q^{73} + 2 q^{81} + 28 q^{83} + 16 q^{91}+ \cdots + 24 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 - 2 * q^9 - 4 * q^13 - 4 * q^29 - 12 * q^33 - 16 * q^37 - 6 * q^39 + 12 * q^47 + 18 * q^49 - 4 * q^51 + 4 * q^61 + 8 * q^63 - 8 * q^67 + 16 * q^69 - 8 * q^73 + 2 * q^81 + 28 * q^83 + 16 * q^91 - 16 * q^93 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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