LMFDB - Embedded newform 3900.2.h.b.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3900,2,Mod(1249,3900)]

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, 0]))

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

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

chi := DirichletCharacter("3900.1249");

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.h (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:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3900.1249
Dual form			3900.2.h.b.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{13} +2.00000i q^{17} +2.00000 q^{19} +2.00000 q^{21} -1.00000i q^{27} +6.00000 q^{29} -10.0000 q^{31} -4.00000i q^{33} +10.0000i q^{37} +1.00000 q^{39} +8.00000 q^{41} -4.00000i q^{43} -4.00000i q^{47} +3.00000 q^{49} -2.00000 q^{51} +10.0000i q^{53} +2.00000i q^{57} +8.00000 q^{59} -14.0000 q^{61} +2.00000i q^{63} +2.00000i q^{67} +16.0000 q^{71} +10.0000i q^{73} +8.00000i q^{77} +16.0000 q^{79} +1.00000 q^{81} +6.00000i q^{87} +4.00000 q^{89} -2.00000 q^{91} -10.0000i q^{93} -2.00000i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 8 q^{11} + 4 q^{19} + 4 q^{21} + 12 q^{29} - 20 q^{31} + 2 q^{39} + 16 q^{41} + 6 q^{49} - 4 q^{51} + 16 q^{59} - 28 q^{61} + 32 q^{71} + 32 q^{79} + 2 q^{81} + 8 q^{89} - 4 q^{91} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 8 * q^11 + 4 * q^19 + 4 * q^21 + 12 * q^29 - 20 * q^31 + 2 * q^39 + 16 * q^41 + 6 * q^49 - 4 * q^51 + 16 * q^59 - 28 * q^61 + 32 * q^71 + 32 * q^79 + 2 * q^81 + 8 * q^89 - 4 * q^91 + 8 * q^99

Display 100 coefficients 10 coefficients

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
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−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
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	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
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	− 4.00000i	− 0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	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
$39$	1.00000	0.160128
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	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
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.00000i	0.911685i
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000i	0.643268i
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	− 10.0000i	− 1.03695i
$94$	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
$99$	4.00000	0.402015
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	8.00000i	0.721336i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	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
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.00000i	0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$	8.00000i	0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	4.00000i	0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.00000i	0.247436i
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 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
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.00000i	0.601317i
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	− 14.0000i	− 1.03491i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	16.0000i	1.09630i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	20.0000i	1.35769i
$218$	0	0
$219$	−10.0000	−0.675737
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	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
$237$	16.0000i	1.03931i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.00000i	− 0.127257i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	0	0
$259$	20.0000	1.24274
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.00000i	0.244796i
$268$	0	0
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	− 2.00000i	− 0.121046i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	0	0
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 16.0000i	− 0.944450i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	− 8.00000i	− 0.467365i	−0.972313	−	0.233682i	$-0.924922\pi$
$293$	− 8.00000i	− 0.467365i	0.972313	−	0.233682i	$-0.0750776\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.00000i	0.232104i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	10.0000i	0.574485i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.0000i	1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$	22.0000i	1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	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
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	4.00000i	0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.00000i	0.110600i
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	0	0
$333$	− 10.0000i	− 0.547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 30.0000i	− 1.63420i	−0.576493	−	0.817102i	$-0.695579\pi$
$337$	− 30.0000i	− 1.63420i	0.576493	−	0.817102i	$-0.304421\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	16.0000i	0.851594i	0.904819	+	0.425797i	$0.140006\pi$
$353$	16.0000i	0.851594i	−0.904819	+	0.425797i	$0.859994\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.00000i	0.211702i
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.00000i	0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	4.00000i	0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$	0	0
$369$	−8.00000	−0.416463
$370$	0	0
$371$	20.0000	1.03835
$372$	0	0
$373$	18.0000i	0.932005i	0.884783	+	0.466002i	$0.154306\pi$
$373$	18.0000i	0.932005i	−0.884783	+	0.466002i	$0.845694\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.00000i	− 0.309016i
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	4.00000i	0.201773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	10.0000i	0.498135i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 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	$0.612504\pi$
$410$	0	0
$411$	−8.00000	−0.394611
$412$	0	0
$413$	− 16.0000i	− 0.787309i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	4.00000i	0.194487i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	28.0000i	1.35501i
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	0	0
$433$	18.0000i	0.865025i	0.901628	+	0.432512i	$0.142373\pi$
$433$	18.0000i	0.865025i	−0.901628	+	0.432512i	$0.857627\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	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
$441$	−3.00000	−0.142857
$442$	0	0
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	−32.0000	−1.50682
$452$	0	0
$453$	18.0000i	0.845714i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.0000i	− 0.457869i
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 26.0000i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$	− 26.0000i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 32.0000i	− 1.43540i
$498$	0	0
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.00000i	− 0.0444116i
$508$	0	0
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	0	0
$513$	− 2.00000i	− 0.0883022i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 20.0000i	− 0.871214i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	− 8.00000i	− 0.346518i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 12.0000i	− 0.517838i
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	0	0
$543$	2.00000i	0.0858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	32.0000i	1.36822i	0.729378	+	0.684111i	$0.239809\pi$
$547$	32.0000i	1.36822i	−0.729378	+	0.684111i	$0.760191\pi$
$548$	0	0
$549$	14.0000	0.597505
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.0000i	1.01691i	0.861088	+	0.508456i	$0.169784\pi$
$557$	24.0000i	1.01691i	−0.861088	+	0.508456i	$0.830216\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	− 28.0000i	− 1.18006i	−0.807382	−	0.590030i	$-0.799116\pi$
$563$	− 28.0000i	− 1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	0	0
$573$	− 8.00000i	− 0.334205i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 30.0000i	− 1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$	− 30.0000i	− 1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 40.0000i	− 1.65663i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.00000i	− 0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$	− 8.00000i	− 0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	36.0000i	1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$	36.0000i	1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4.00000i	− 0.163709i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	− 2.00000i	− 0.0814463i
$604$	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
$609$	12.0000	0.486265
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 48.0000i	− 1.93241i	−0.257780	−	0.966204i	$-0.582991\pi$
$617$	− 48.0000i	− 1.93241i	0.257780	−	0.966204i	$-0.417009\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 8.00000i	− 0.320513i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 8.00000i	− 0.319489i
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	0	0
$633$	− 12.0000i	− 0.476957i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.00000i	− 0.118864i
$638$	0	0
$639$	−16.0000	−0.632950
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	− 26.0000i	− 1.02534i	−0.858586	−	0.512670i	$-0.828656\pi$
$643$	− 26.0000i	− 1.02534i	0.858586	−	0.512670i	$-0.171344\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	0	0
$649$	−32.0000	−1.25611
$650$	0	0
$651$	−20.0000	−0.783862
$652$	0	0
$653$	46.0000i	1.80012i	0.435767	+	0.900060i	$0.356477\pi$
$653$	46.0000i	1.80012i	−0.435767	+	0.900060i	$0.643523\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	2.00000i	0.0776736i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	56.0000	2.16186
$672$	0	0
$673$	− 22.0000i	− 0.848038i	−0.905653	−	0.424019i	$-0.860619\pi$
$673$	− 22.0000i	− 0.848038i	0.905653	−	0.424019i	$-0.139381\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.0000i	1.46046i	0.683202	+	0.730229i	$0.260587\pi$
$677$	38.0000i	1.46046i	−0.683202	+	0.730229i	$0.739413\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	− 20.0000i	− 0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$	− 20.0000i	− 0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.00000i	0.0763048i
$688$	0	0
$689$	10.0000	0.380970
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	0	0
$693$	− 8.00000i	− 0.303895i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.0000i	0.606043i
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	20.0000i	0.754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 20.0000i	− 0.752177i
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.0000i	0.896296i
$718$	0	0
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	− 2.00000i	− 0.0743808i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	44.0000i	1.63187i	0.578144	+	0.815935i	$0.303777\pi$
$727$	44.0000i	1.63187i	−0.578144	+	0.815935i	$0.696223\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	−22.0000	−0.809283	−0.404642	−	0.914475i	$-0.632604\pi$
$739$	−22.0000	−0.809283	−0.404642	+	0.914475i	$0.632604\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	− 20.0000i	− 0.728841i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.0000i	0.654221i	0.944986	+	0.327111i	$0.106075\pi$
$757$	18.0000i	0.654221i	−0.944986	+	0.327111i	$0.893925\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.0000	1.45000	0.724999	−	0.688749i	$-0.241840\pi$
$761$	40.0000	1.45000	0.724999	+	0.688749i	$0.241840\pi$
$762$	0	0
$763$	− 4.00000i	− 0.144810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 8.00000i	− 0.288863i
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	0	0
$773$	48.0000i	1.72644i	0.504828	+	0.863220i	$0.331556\pi$
$773$	48.0000i	1.72644i	−0.504828	+	0.863220i	$0.668444\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.0000i	0.717496i
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	−64.0000	−2.29010
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 50.0000i	− 1.78231i	−0.453701	−	0.891154i	$-0.649897\pi$
$787$	− 50.0000i	− 1.78231i	0.453701	−	0.891154i	$-0.350103\pi$
$788$	0	0
$789$	−16.0000	−0.569615
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	14.0000i	0.497155i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 26.0000i	− 0.920967i	−0.887668	−	0.460484i	$-0.847676\pi$
$797$	− 26.0000i	− 0.920967i	0.887668	−	0.460484i	$-0.152324\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	− 40.0000i	− 1.41157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.0000i	0.492823i
$808$	0	0
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	6.00000	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	0	0
$813$	− 10.0000i	− 0.350715i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.00000i	− 0.279885i
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	0	0
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	0	0
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.0000i	0.345651i
$838$	0	0
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	− 8.00000i	− 0.275535i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	0	0
$852$	0	0
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	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
$861$	16.0000	0.545279
$862$	0	0
$863$	− 4.00000i	− 0.136162i	−0.997680	−	0.0680808i	$-0.978312\pi$
$863$	− 4.00000i	− 0.136162i	0.997680	−	0.0680808i	$-0.0216876\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	0	0
$879$	8.00000	0.269833
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	32.0000i	1.07689i	0.842662	+	0.538443i	$0.180987\pi$
$883$	32.0000i	1.07689i	−0.842662	+	0.538443i	$0.819013\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	− 8.00000i	− 0.267710i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−60.0000	−2.00111
$900$	0	0
$901$	−20.0000	−0.666297
$902$	0	0
$903$	− 8.00000i	− 0.266223i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 56.0000i	− 1.85945i	−0.368255	−	0.929725i	$-0.620045\pi$
$907$	− 56.0000i	− 1.85945i	0.368255	−	0.929725i	$-0.379955\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−56.0000	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.00000i	− 0.264183i
$918$	0	0
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	0	0
$921$	−22.0000	−0.724925
$922$	0	0
$923$	− 16.0000i	− 0.526646i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	−16.0000	−0.524943	−0.262471	−	0.964940i	$-0.584538\pi$
$929$	−16.0000	−0.524943	−0.262471	+	0.964940i	$0.584538\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	24.0000i	0.785725i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 34.0000i	− 1.11073i	−0.831606	−	0.555366i	$-0.812578\pi$
$937$	− 34.0000i	− 1.11073i	0.831606	−	0.555366i	$-0.187422\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.00000i	0.259965i	0.991516	+	0.129983i	$0.0414921\pi$
$947$	8.00000i	0.259965i	−0.991516	+	0.129983i	$0.958508\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	42.0000i	1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$	42.0000i	1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 24.0000i	− 0.775810i
$958$	0	0
$959$	16.0000	0.516667
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	22.0000i	0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$967$	22.0000i	0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 12.0000i	− 0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$	− 12.0000i	− 0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	0	0
$993$	2.00000i	0.0634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	22.0000i	0.696747i	0.937356	+	0.348373i	$0.113266\pi$
$997$	22.0000i	0.696747i	−0.937356	+	0.348373i	$0.886734\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3900.2.h.b.1249.2		2
5.2	odd	4		3900.2.a.m.1.1		1
5.3	odd	4		156.2.a.a.1.1	✓	1
5.4	even	2	inner	3900.2.h.b.1249.1		2
15.8	even	4		468.2.a.d.1.1		1
20.3	even	4		624.2.a.e.1.1		1
35.13	even	4		7644.2.a.k.1.1		1
40.3	even	4		2496.2.a.o.1.1		1
40.13	odd	4		2496.2.a.bc.1.1		1
45.13	odd	12		4212.2.i.l.2809.1		2
45.23	even	12		4212.2.i.b.2809.1		2
45.38	even	12		4212.2.i.b.1405.1		2
45.43	odd	12		4212.2.i.l.1405.1		2
60.23	odd	4		1872.2.a.s.1.1		1
65.3	odd	12		2028.2.i.e.529.1		2
65.8	even	4		2028.2.b.a.337.1		2
65.18	even	4		2028.2.b.a.337.2		2
65.23	odd	12		2028.2.i.g.529.1		2
65.28	even	12		2028.2.q.h.1837.2		4
65.33	even	12		2028.2.q.h.361.1		4
65.38	odd	4		2028.2.a.c.1.1		1
65.43	odd	12		2028.2.i.g.2005.1		2
65.48	odd	12		2028.2.i.e.2005.1		2
65.58	even	12		2028.2.q.h.361.2		4
65.63	even	12		2028.2.q.h.1837.1		4
120.53	even	4		7488.2.a.c.1.1		1
120.83	odd	4		7488.2.a.d.1.1		1
195.8	odd	4		6084.2.b.j.4393.2		2
195.38	even	4		6084.2.a.b.1.1		1
195.83	odd	4		6084.2.b.j.4393.1		2
260.103	even	4		8112.2.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.a.a.1.1	✓	1	5.3	odd	4
468.2.a.d.1.1		1	15.8	even	4
624.2.a.e.1.1		1	20.3	even	4
1872.2.a.s.1.1		1	60.23	odd	4
2028.2.a.c.1.1		1	65.38	odd	4
2028.2.b.a.337.1		2	65.8	even	4
2028.2.b.a.337.2		2	65.18	even	4
2028.2.i.e.529.1		2	65.3	odd	12
2028.2.i.e.2005.1		2	65.48	odd	12
2028.2.i.g.529.1		2	65.23	odd	12
2028.2.i.g.2005.1		2	65.43	odd	12
2028.2.q.h.361.1		4	65.33	even	12
2028.2.q.h.361.2		4	65.58	even	12
2028.2.q.h.1837.1		4	65.63	even	12
2028.2.q.h.1837.2		4	65.28	even	12
2496.2.a.o.1.1		1	40.3	even	4
2496.2.a.bc.1.1		1	40.13	odd	4
3900.2.a.m.1.1		1	5.2	odd	4
3900.2.h.b.1249.1		2	5.4	even	2	inner
3900.2.h.b.1249.2		2	1.1	even	1	trivial
4212.2.i.b.1405.1		2	45.38	even	12
4212.2.i.b.2809.1		2	45.23	even	12
4212.2.i.l.1405.1		2	45.43	odd	12
4212.2.i.l.2809.1		2	45.13	odd	12
6084.2.a.b.1.1		1	195.38	even	4
6084.2.b.j.4393.1		2	195.83	odd	4
6084.2.b.j.4393.2		2	195.8	odd	4
7488.2.a.c.1.1		1	120.53	even	4
7488.2.a.d.1.1		1	120.83	odd	4
7644.2.a.k.1.1		1	35.13	even	4
8112.2.a.bi.1.1		1	260.103	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{13} +2.00000i q^{17} +2.00000 q^{19} +2.00000 q^{21} -1.00000i q^{27} +6.00000 q^{29} -10.0000 q^{31} -4.00000i q^{33} +10.0000i q^{37} +1.00000 q^{39} +8.00000 q^{41} -4.00000i q^{43} -4.00000i q^{47} +3.00000 q^{49} -2.00000 q^{51} +10.0000i q^{53} +2.00000i q^{57} +8.00000 q^{59} -14.0000 q^{61} +2.00000i q^{63} +2.00000i q^{67} +16.0000 q^{71} +10.0000i q^{73} +8.00000i q^{77} +16.0000 q^{79} +1.00000 q^{81} +6.00000i q^{87} +4.00000 q^{89} -2.00000 q^{91} -10.0000i q^{93} -2.00000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 8 q^{11} + 4 q^{19} + 4 q^{21} + 12 q^{29} - 20 q^{31} + 2 q^{39} + 16 q^{41} + 6 q^{49} - 4 q^{51} + 16 q^{59} - 28 q^{61} + 32 q^{71} + 32 q^{79} + 2 q^{81} + 8 q^{89} - 4 q^{91} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 8 * q^11 + 4 * q^19 + 4 * q^21 + 12 * q^29 - 20 * q^31 + 2 * q^39 + 16 * q^41 + 6 * q^49 - 4 * q^51 + 16 * q^59 - 28 * q^61 + 32 * q^71 + 32 * q^79 + 2 * q^81 + 8 * q^89 - 4 * q^91 + 8 * q^99

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