LMFDB - Embedded newform 5202.2.a.bm.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5202,2,Mod(1,5202)] 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("5202.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5202, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5202 = 2 \cdot 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5202.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,-6,0,3,3,0,-6,-6,0,6,3,0,3,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.5381791315$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1734)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	5202.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.120615 q^{5} +1.69459 q^{7} +1.00000 q^{8} -0.120615 q^{10} -6.10607 q^{11} -1.75877 q^{13} +1.69459 q^{14} +1.00000 q^{16} +4.82295 q^{19} -0.120615 q^{20} -6.10607 q^{22} -6.00000 q^{23} -4.98545 q^{25} -1.75877 q^{26} +1.69459 q^{28} +3.90167 q^{29} -7.29086 q^{31} +1.00000 q^{32} -0.204393 q^{35} +3.67499 q^{37} +4.82295 q^{38} -0.120615 q^{40} +3.14796 q^{41} +6.73917 q^{43} -6.10607 q^{44} -6.00000 q^{46} -5.43376 q^{47} -4.12836 q^{49} -4.98545 q^{50} -1.75877 q^{52} -4.71688 q^{53} +0.736482 q^{55} +1.69459 q^{56} +3.90167 q^{58} -2.12061 q^{59} -10.6946 q^{61} -7.29086 q^{62} +1.00000 q^{64} +0.212134 q^{65} -5.88713 q^{67} -0.204393 q^{70} -15.4047 q^{71} -12.5963 q^{73} +3.67499 q^{74} +4.82295 q^{76} -10.3473 q^{77} +13.9932 q^{79} -0.120615 q^{80} +3.14796 q^{82} +14.8871 q^{83} +6.73917 q^{86} -6.10607 q^{88} -16.4979 q^{89} -2.98040 q^{91} -6.00000 q^{92} -5.43376 q^{94} -0.581719 q^{95} -9.08647 q^{97} -4.12836 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} + 3 q^{8} - 6 q^{10} - 6 q^{11} + 6 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{19} - 6 q^{20} - 6 q^{22} - 18 q^{23} + 3 q^{25} + 6 q^{26} + 3 q^{28} - 6 q^{31} + 3 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 + 3 * q^8 - 6 * q^10 - 6 * q^11 + 6 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^19 - 6 * q^20 - 6 * q^22 - 18 * q^23 + 3 * q^25 + 6 * q^26 + 3 * q^28 - 6 * q^31 + 3 * q^32 + 6 * q^37 - 6 * q^38 - 6 * q^40 - 6 * q^41 + 6 * q^43 - 6 * q^44 - 18 * q^46 + 6 * q^49 + 3 * q^50 + 6 * q^52 - 6 * q^53 - 3 * q^55 + 3 * q^56 - 12 * q^59 - 30 * q^61 - 6 * q^62 + 3 * q^64 - 24 * q^65 + 12 * q^67 + 6 * q^71 - 24 * q^73 + 6 * q^74 - 6 * q^76 - 30 * q^77 - 6 * q^80 - 6 * q^82 + 15 * q^83 + 6 * q^86 - 6 * q^88 - 24 * q^89 - 6 * q^91 - 18 * q^92 + 30 * q^95 - 12 * q^97 + 6 * q^98

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.120615	−0.0539406	−0.0269703	−	0.999636i	$-0.508586\pi$
$5$	−0.120615	−0.0539406	−0.0269703	+	0.999636i	$0.508586\pi$
$6$	0	0
$7$	1.69459	0.640496	0.320248	−	0.947334i	$-0.396234\pi$
$7$	1.69459	0.640496	0.320248	+	0.947334i	$0.396234\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.120615	−0.0381417
$11$	−6.10607	−1.84105	−0.920524	−	0.390686i	$-0.872238\pi$
$11$	−6.10607	−1.84105	−0.920524	+	0.390686i	$0.872238\pi$
$12$	0	0
$13$	−1.75877	−0.487795	−0.243898	−	0.969801i	$-0.578426\pi$
$13$	−1.75877	−0.487795	−0.243898	+	0.969801i	$0.578426\pi$
$14$	1.69459	0.452899
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	4.82295	1.10646	0.553230	−	0.833028i	$-0.313395\pi$
$19$	4.82295	1.10646	0.553230	+	0.833028i	$0.313395\pi$
$20$	−0.120615	−0.0269703
$21$	0	0
$22$	−6.10607	−1.30182
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−4.98545	−0.997090
$26$	−1.75877	−0.344923
$27$	0	0
$28$	1.69459	0.320248
$29$	3.90167	0.724523	0.362261	−	0.932077i	$-0.382005\pi$
$29$	3.90167	0.724523	0.362261	+	0.932077i	$0.382005\pi$
$30$	0	0
$31$	−7.29086	−1.30948	−0.654738	−	0.755855i	$-0.727221\pi$
$31$	−7.29086	−1.30948	−0.654738	+	0.755855i	$0.727221\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	−0.204393	−0.0345487
$36$	0	0
$37$	3.67499	0.604165	0.302083	−	0.953282i	$-0.402318\pi$
$37$	3.67499	0.604165	0.302083	+	0.953282i	$0.402318\pi$
$38$	4.82295	0.782386
$39$	0	0
$40$	−0.120615	−0.0190709
$41$	3.14796	0.491628	0.245814	−	0.969317i	$-0.420945\pi$
$41$	3.14796	0.491628	0.245814	+	0.969317i	$0.420945\pi$
$42$	0	0
$43$	6.73917	1.02771	0.513857	−	0.857876i	$-0.328216\pi$
$43$	6.73917	1.02771	0.513857	+	0.857876i	$0.328216\pi$
$44$	−6.10607	−0.920524
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−5.43376	−0.792596	−0.396298	−	0.918122i	$-0.629705\pi$
$47$	−5.43376	−0.792596	−0.396298	+	0.918122i	$0.629705\pi$
$48$	0	0
$49$	−4.12836	−0.589765
$50$	−4.98545	−0.705049
$51$	0	0
$52$	−1.75877	−0.243898
$53$	−4.71688	−0.647913	−0.323957	−	0.946072i	$-0.605013\pi$
$53$	−4.71688	−0.647913	−0.323957	+	0.946072i	$0.605013\pi$
$54$	0	0
$55$	0.736482	0.0993072
$56$	1.69459	0.226449
$57$	0	0
$58$	3.90167	0.512315
$59$	−2.12061	−0.276081	−0.138040	−	0.990427i	$-0.544080\pi$
$59$	−2.12061	−0.276081	−0.138040	+	0.990427i	$0.544080\pi$
$60$	0	0
$61$	−10.6946	−1.36930	−0.684651	−	0.728871i	$-0.740045\pi$
$61$	−10.6946	−1.36930	−0.684651	+	0.728871i	$0.740045\pi$
$62$	−7.29086	−0.925940
$63$	0	0
$64$	1.00000	0.125000
$65$	0.212134	0.0263119
$66$	0	0
$67$	−5.88713	−0.719227	−0.359613	−	0.933101i	$-0.617091\pi$
$67$	−5.88713	−0.719227	−0.359613	+	0.933101i	$0.617091\pi$
$68$	0	0
$69$	0	0
$70$	−0.204393	−0.0244296
$71$	−15.4047	−1.82820	−0.914099	−	0.405492i	$-0.867100\pi$
$71$	−15.4047	−1.82820	−0.914099	+	0.405492i	$0.867100\pi$
$72$	0	0
$73$	−12.5963	−1.47428	−0.737141	−	0.675739i	$-0.763825\pi$
$73$	−12.5963	−1.47428	−0.737141	+	0.675739i	$0.763825\pi$
$74$	3.67499	0.427209
$75$	0	0
$76$	4.82295	0.553230
$77$	−10.3473	−1.17918
$78$	0	0
$79$	13.9932	1.57436	0.787179	−	0.616725i	$-0.211541\pi$
$79$	13.9932	1.57436	0.787179	+	0.616725i	$0.211541\pi$
$80$	−0.120615	−0.0134851
$81$	0	0
$82$	3.14796	0.347634
$83$	14.8871	1.63407	0.817037	−	0.576585i	$-0.195615\pi$
$83$	14.8871	1.63407	0.817037	+	0.576585i	$0.195615\pi$
$84$	0	0
$85$	0	0
$86$	6.73917	0.726703
$87$	0	0
$88$	−6.10607	−0.650909
$89$	−16.4979	−1.74878	−0.874389	−	0.485225i	$-0.838738\pi$
$89$	−16.4979	−1.74878	−0.874389	+	0.485225i	$0.838738\pi$
$90$	0	0
$91$	−2.98040	−0.312431
$92$	−6.00000	−0.625543
$93$	0	0
$94$	−5.43376	−0.560450
$95$	−0.581719	−0.0596831
$96$	0	0
$97$	−9.08647	−0.922591	−0.461295	−	0.887247i	$-0.652615\pi$
$97$	−9.08647	−0.922591	−0.461295	+	0.887247i	$0.652615\pi$
$98$	−4.12836	−0.417027
$99$	0	0
$100$	−4.98545	−0.498545
$101$	−2.24897	−0.223781	−0.111890	−	0.993721i	$-0.535691\pi$
$101$	−2.24897	−0.223781	−0.111890	+	0.993721i	$0.535691\pi$
$102$	0	0
$103$	18.3259	1.80571	0.902854	−	0.429947i	$-0.141468\pi$
$103$	18.3259	1.80571	0.902854	+	0.429947i	$0.141468\pi$
$104$	−1.75877	−0.172462
$105$	0	0
$106$	−4.71688	−0.458144
$107$	−14.1138	−1.36443	−0.682217	−	0.731150i	$-0.738984\pi$
$107$	−14.1138	−1.36443	−0.682217	+	0.731150i	$0.738984\pi$
$108$	0	0
$109$	−6.69459	−0.641226	−0.320613	−	0.947210i	$-0.603889\pi$
$109$	−6.69459	−0.641226	−0.320613	+	0.947210i	$0.603889\pi$
$110$	0.736482	0.0702208
$111$	0	0
$112$	1.69459	0.160124
$113$	4.58172	0.431012	0.215506	−	0.976503i	$-0.430860\pi$
$113$	4.58172	0.431012	0.215506	+	0.976503i	$0.430860\pi$
$114$	0	0
$115$	0.723689	0.0674843
$116$	3.90167	0.362261
$117$	0	0
$118$	−2.12061	−0.195218
$119$	0	0
$120$	0	0
$121$	26.2841	2.38946
$122$	−10.6946	−0.968243
$123$	0	0
$124$	−7.29086	−0.654738
$125$	1.20439	0.107724
$126$	0	0
$127$	−4.30541	−0.382043	−0.191022	−	0.981586i	$-0.561180\pi$
$127$	−4.30541	−0.382043	−0.191022	+	0.981586i	$0.561180\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.212134	0.0186054
$131$	15.0155	1.31191	0.655954	−	0.754801i	$-0.272266\pi$
$131$	15.0155	1.31191	0.655954	+	0.754801i	$0.272266\pi$
$132$	0	0
$133$	8.17293	0.708683
$134$	−5.88713	−0.508570
$135$	0	0
$136$	0	0
$137$	6.82295	0.582924	0.291462	−	0.956582i	$-0.405858\pi$
$137$	6.82295	0.582924	0.291462	+	0.956582i	$0.405858\pi$
$138$	0	0
$139$	−16.9067	−1.43401	−0.717005	−	0.697068i	$-0.754488\pi$
$139$	−16.9067	−1.43401	−0.717005	+	0.697068i	$0.754488\pi$
$140$	−0.204393	−0.0172744
$141$	0	0
$142$	−15.4047	−1.29273
$143$	10.7392	0.898055
$144$	0	0
$145$	−0.470599	−0.0390812
$146$	−12.5963	−1.04247
$147$	0	0
$148$	3.67499	0.302083
$149$	−11.2831	−0.924349	−0.462175	−	0.886789i	$-0.652931\pi$
$149$	−11.2831	−0.924349	−0.462175	+	0.886789i	$0.652931\pi$
$150$	0	0
$151$	23.1702	1.88557	0.942784	−	0.333404i	$-0.108197\pi$
$151$	23.1702	1.88557	0.942784	+	0.333404i	$0.108197\pi$
$152$	4.82295	0.391193
$153$	0	0
$154$	−10.3473	−0.833809
$155$	0.879385	0.0706339
$156$	0	0
$157$	5.38919	0.430104	0.215052	−	0.976603i	$-0.431008\pi$
$157$	5.38919	0.430104	0.215052	+	0.976603i	$0.431008\pi$
$158$	13.9932	1.11324
$159$	0	0
$160$	−0.120615	−0.00953543
$161$	−10.1676	−0.801316
$162$	0	0
$163$	−17.8871	−1.40103	−0.700514	−	0.713639i	$-0.747046\pi$
$163$	−17.8871	−1.40103	−0.700514	+	0.713639i	$0.747046\pi$
$164$	3.14796	0.245814
$165$	0	0
$166$	14.8871	1.15547
$167$	−5.84255	−0.452110	−0.226055	−	0.974115i	$-0.572583\pi$
$167$	−5.84255	−0.452110	−0.226055	+	0.974115i	$0.572583\pi$
$168$	0	0
$169$	−9.90673	−0.762056
$170$	0	0
$171$	0	0
$172$	6.73917	0.513857
$173$	5.41147	0.411427	0.205713	−	0.978612i	$-0.434049\pi$
$173$	5.41147	0.411427	0.205713	+	0.978612i	$0.434049\pi$
$174$	0	0
$175$	−8.44831	−0.638632
$176$	−6.10607	−0.460262
$177$	0	0
$178$	−16.4979	−1.23657
$179$	−7.24123	−0.541235	−0.270617	−	0.962687i	$-0.587228\pi$
$179$	−7.24123	−0.541235	−0.270617	+	0.962687i	$0.587228\pi$
$180$	0	0
$181$	−0.157451	−0.0117033	−0.00585163	−	0.999983i	$-0.501863\pi$
$181$	−0.157451	−0.0117033	−0.00585163	+	0.999983i	$0.501863\pi$
$182$	−2.98040	−0.220922
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−0.443258	−0.0325890
$186$	0	0
$187$	0	0
$188$	−5.43376	−0.396298
$189$	0	0
$190$	−0.581719	−0.0422023
$191$	6.90673	0.499753	0.249877	−	0.968278i	$-0.419610\pi$
$191$	6.90673	0.499753	0.249877	+	0.968278i	$0.419610\pi$
$192$	0	0
$193$	−11.5253	−0.829608	−0.414804	−	0.909911i	$-0.636150\pi$
$193$	−11.5253	−0.829608	−0.414804	+	0.909911i	$0.636150\pi$
$194$	−9.08647	−0.652370
$195$	0	0
$196$	−4.12836	−0.294883
$197$	−6.71419	−0.478366	−0.239183	−	0.970974i	$-0.576880\pi$
$197$	−6.71419	−0.478366	−0.239183	+	0.970974i	$0.576880\pi$
$198$	0	0
$199$	7.22937	0.512476	0.256238	−	0.966614i	$-0.417517\pi$
$199$	7.22937	0.512476	0.256238	+	0.966614i	$0.417517\pi$
$200$	−4.98545	−0.352525
$201$	0	0
$202$	−2.24897	−0.158237
$203$	6.61175	0.464054
$204$	0	0
$205$	−0.379690	−0.0265187
$206$	18.3259	1.27683
$207$	0	0
$208$	−1.75877	−0.121949
$209$	−29.4492	−2.03705
$210$	0	0
$211$	10.5371	0.725407	0.362703	−	0.931905i	$-0.381854\pi$
$211$	10.5371	0.725407	0.362703	+	0.931905i	$0.381854\pi$
$212$	−4.71688	−0.323957
$213$	0	0
$214$	−14.1138	−0.964800
$215$	−0.812843	−0.0554355
$216$	0	0
$217$	−12.3550	−0.838715
$218$	−6.69459	−0.453415
$219$	0	0
$220$	0.736482	0.0496536
$221$	0	0
$222$	0	0
$223$	−11.0027	−0.736795	−0.368397	−	0.929668i	$-0.620093\pi$
$223$	−11.0027	−0.736795	−0.368397	+	0.929668i	$0.620093\pi$
$224$	1.69459	0.113225
$225$	0	0
$226$	4.58172	0.304771
$227$	−0.758770	−0.0503614	−0.0251807	−	0.999683i	$-0.508016\pi$
$227$	−0.758770	−0.0503614	−0.0251807	+	0.999683i	$0.508016\pi$
$228$	0	0
$229$	24.5817	1.62441	0.812203	−	0.583375i	$-0.198268\pi$
$229$	24.5817	1.62441	0.812203	+	0.583375i	$0.198268\pi$
$230$	0.723689	0.0477186
$231$	0	0
$232$	3.90167	0.256157
$233$	−5.67499	−0.371781	−0.185891	−	0.982570i	$-0.559517\pi$
$233$	−5.67499	−0.371781	−0.185891	+	0.982570i	$0.559517\pi$
$234$	0	0
$235$	0.655392	0.0427531
$236$	−2.12061	−0.138040
$237$	0	0
$238$	0	0
$239$	−25.5175	−1.65059	−0.825296	−	0.564700i	$-0.808992\pi$
$239$	−25.5175	−1.65059	−0.825296	+	0.564700i	$0.808992\pi$
$240$	0	0
$241$	20.4320	1.31614	0.658071	−	0.752956i	$-0.271373\pi$
$241$	20.4320	1.31614	0.658071	+	0.752956i	$0.271373\pi$
$242$	26.2841	1.68960
$243$	0	0
$244$	−10.6946	−0.684651
$245$	0.497941	0.0318123
$246$	0	0
$247$	−8.48246	−0.539726
$248$	−7.29086	−0.462970
$249$	0	0
$250$	1.20439	0.0761725
$251$	−7.50299	−0.473585	−0.236792	−	0.971560i	$-0.576096\pi$
$251$	−7.50299	−0.473585	−0.236792	+	0.971560i	$0.576096\pi$
$252$	0	0
$253$	36.6364	2.30331
$254$	−4.30541	−0.270145
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.3851	1.14683	0.573414	−	0.819265i	$-0.305618\pi$
$257$	18.3851	1.14683	0.573414	+	0.819265i	$0.305618\pi$
$258$	0	0
$259$	6.22762	0.386965
$260$	0.212134	0.0131560
$261$	0	0
$262$	15.0155	0.927660
$263$	3.01960	0.186197	0.0930983	−	0.995657i	$-0.470323\pi$
$263$	3.01960	0.186197	0.0930983	+	0.995657i	$0.470323\pi$
$264$	0	0
$265$	0.568926	0.0349488
$266$	8.17293	0.501115
$267$	0	0
$268$	−5.88713	−0.359613
$269$	10.8452	0.661246	0.330623	−	0.943763i	$-0.392741\pi$
$269$	10.8452	0.661246	0.330623	+	0.943763i	$0.392741\pi$
$270$	0	0
$271$	−17.6159	−1.07009	−0.535044	−	0.844824i	$-0.679705\pi$
$271$	−17.6159	−1.07009	−0.535044	+	0.844824i	$0.679705\pi$
$272$	0	0
$273$	0	0
$274$	6.82295	0.412189
$275$	30.4415	1.83569
$276$	0	0
$277$	1.10338	0.0662956	0.0331478	−	0.999450i	$-0.489447\pi$
$277$	1.10338	0.0662956	0.0331478	+	0.999450i	$0.489447\pi$
$278$	−16.9067	−1.01400
$279$	0	0
$280$	−0.204393	−0.0122148
$281$	−6.28581	−0.374980	−0.187490	−	0.982267i	$-0.560035\pi$
$281$	−6.28581	−0.374980	−0.187490	+	0.982267i	$0.560035\pi$
$282$	0	0
$283$	3.27631	0.194757	0.0973783	−	0.995247i	$-0.468954\pi$
$283$	3.27631	0.194757	0.0973783	+	0.995247i	$0.468954\pi$
$284$	−15.4047	−0.914099
$285$	0	0
$286$	10.7392	0.635020
$287$	5.33450	0.314886
$288$	0	0
$289$	0	0
$290$	−0.470599	−0.0276346
$291$	0	0
$292$	−12.5963	−0.737141
$293$	−5.76651	−0.336883	−0.168442	−	0.985712i	$-0.553873\pi$
$293$	−5.76651	−0.336883	−0.168442	+	0.985712i	$0.553873\pi$
$294$	0	0
$295$	0.255777	0.0148919
$296$	3.67499	0.213605
$297$	0	0
$298$	−11.2831	−0.653614
$299$	10.5526	0.610274
$300$	0	0
$301$	11.4201	0.658246
$302$	23.1702	1.33330
$303$	0	0
$304$	4.82295	0.276615
$305$	1.28993	0.0738609
$306$	0	0
$307$	−1.26083	−0.0719594	−0.0359797	−	0.999353i	$-0.511455\pi$
$307$	−1.26083	−0.0719594	−0.0359797	+	0.999353i	$0.511455\pi$
$308$	−10.3473	−0.589592
$309$	0	0
$310$	0.879385	0.0499457
$311$	−4.77837	−0.270957	−0.135478	−	0.990780i	$-0.543257\pi$
$311$	−4.77837	−0.270957	−0.135478	+	0.990780i	$0.543257\pi$
$312$	0	0
$313$	−6.25671	−0.353650	−0.176825	−	0.984242i	$-0.556583\pi$
$313$	−6.25671	−0.353650	−0.176825	+	0.984242i	$0.556583\pi$
$314$	5.38919	0.304129
$315$	0	0
$316$	13.9932	0.787179
$317$	26.1925	1.47112	0.735560	−	0.677460i	$-0.236919\pi$
$317$	26.1925	1.47112	0.735560	+	0.677460i	$0.236919\pi$
$318$	0	0
$319$	−23.8239	−1.33388
$320$	−0.120615	−0.00674257
$321$	0	0
$322$	−10.1676	−0.566616
$323$	0	0
$324$	0	0
$325$	8.76827	0.486376
$326$	−17.8871	−0.990676
$327$	0	0
$328$	3.14796	0.173817
$329$	−9.20801	−0.507654
$330$	0	0
$331$	−12.5817	−0.691554	−0.345777	−	0.938317i	$-0.612385\pi$
$331$	−12.5817	−0.691554	−0.345777	+	0.938317i	$0.612385\pi$
$332$	14.8871	0.817037
$333$	0	0
$334$	−5.84255	−0.319690
$335$	0.710074	0.0387955
$336$	0	0
$337$	−23.2175	−1.26474	−0.632369	−	0.774667i	$-0.717917\pi$
$337$	−23.2175	−1.26474	−0.632369	+	0.774667i	$0.717917\pi$
$338$	−9.90673	−0.538855
$339$	0	0
$340$	0	0
$341$	44.5185	2.41081
$342$	0	0
$343$	−18.8580	−1.01824
$344$	6.73917	0.363352
$345$	0	0
$346$	5.41147	0.290923
$347$	10.7706	0.578198	0.289099	−	0.957299i	$-0.406644\pi$
$347$	10.7706	0.578198	0.289099	+	0.957299i	$0.406644\pi$
$348$	0	0
$349$	−29.9864	−1.60513	−0.802567	−	0.596562i	$-0.796533\pi$
$349$	−29.9864	−1.60513	−0.802567	+	0.596562i	$0.796533\pi$
$350$	−8.44831	−0.451581
$351$	0	0
$352$	−6.10607	−0.325454
$353$	29.6459	1.57789	0.788946	−	0.614463i	$-0.210627\pi$
$353$	29.6459	1.57789	0.788946	+	0.614463i	$0.210627\pi$
$354$	0	0
$355$	1.85803	0.0986140
$356$	−16.4979	−0.874389
$357$	0	0
$358$	−7.24123	−0.382711
$359$	18.7101	0.987480	0.493740	−	0.869610i	$-0.335629\pi$
$359$	18.7101	0.987480	0.493740	+	0.869610i	$0.335629\pi$
$360$	0	0
$361$	4.26083	0.224254
$362$	−0.157451	−0.00827546
$363$	0	0
$364$	−2.98040	−0.156215
$365$	1.51930	0.0795236
$366$	0	0
$367$	5.84018	0.304855	0.152428	−	0.988315i	$-0.451291\pi$
$367$	5.84018	0.304855	0.152428	+	0.988315i	$0.451291\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	−0.443258	−0.0230439
$371$	−7.99319	−0.414986
$372$	0	0
$373$	−10.6209	−0.549930	−0.274965	−	0.961454i	$-0.588666\pi$
$373$	−10.6209	−0.549930	−0.274965	+	0.961454i	$0.588666\pi$
$374$	0	0
$375$	0	0
$376$	−5.43376	−0.280225
$377$	−6.86215	−0.353419
$378$	0	0
$379$	−0.340489	−0.0174898	−0.00874488	−	0.999962i	$-0.502784\pi$
$379$	−0.340489	−0.0174898	−0.00874488	+	0.999962i	$0.502784\pi$
$380$	−0.581719	−0.0298415
$381$	0	0
$382$	6.90673	0.353379
$383$	30.8776	1.57777	0.788887	−	0.614539i	$-0.210658\pi$
$383$	30.8776	1.57777	0.788887	+	0.614539i	$0.210658\pi$
$384$	0	0
$385$	1.24804	0.0636058
$386$	−11.5253	−0.586621
$387$	0	0
$388$	−9.08647	−0.461295
$389$	3.25166	0.164866	0.0824328	−	0.996597i	$-0.473731\pi$
$389$	3.25166	0.164866	0.0824328	+	0.996597i	$0.473731\pi$
$390$	0	0
$391$	0	0
$392$	−4.12836	−0.208513
$393$	0	0
$394$	−6.71419	−0.338256
$395$	−1.68779	−0.0849217
$396$	0	0
$397$	−18.3405	−0.920483	−0.460241	−	0.887794i	$-0.652237\pi$
$397$	−18.3405	−0.920483	−0.460241	+	0.887794i	$0.652237\pi$
$398$	7.22937	0.362376
$399$	0	0
$400$	−4.98545	−0.249273
$401$	18.7101	0.934337	0.467168	−	0.884168i	$-0.345274\pi$
$401$	18.7101	0.934337	0.467168	+	0.884168i	$0.345274\pi$
$402$	0	0
$403$	12.8229	0.638757
$404$	−2.24897	−0.111890
$405$	0	0
$406$	6.61175	0.328136
$407$	−22.4397	−1.11230
$408$	0	0
$409$	−5.94862	−0.294140	−0.147070	−	0.989126i	$-0.546984\pi$
$409$	−5.94862	−0.294140	−0.147070	+	0.989126i	$0.546984\pi$
$410$	−0.379690	−0.0187515
$411$	0	0
$412$	18.3259	0.902854
$413$	−3.59358	−0.176828
$414$	0	0
$415$	−1.79561	−0.0881429
$416$	−1.75877	−0.0862308
$417$	0	0
$418$	−29.4492	−1.44041
$419$	12.8990	0.630157	0.315078	−	0.949066i	$-0.397969\pi$
$419$	12.8990	0.630157	0.315078	+	0.949066i	$0.397969\pi$
$420$	0	0
$421$	−27.2472	−1.32795	−0.663974	−	0.747756i	$-0.731131\pi$
$421$	−27.2472	−1.32795	−0.663974	+	0.747756i	$0.731131\pi$
$422$	10.5371	0.512940
$423$	0	0
$424$	−4.71688	−0.229072
$425$	0	0
$426$	0	0
$427$	−18.1230	−0.877032
$428$	−14.1138	−0.682217
$429$	0	0
$430$	−0.812843	−0.0391988
$431$	12.0547	0.580654	0.290327	−	0.956928i	$-0.406236\pi$
$431$	12.0547	0.580654	0.290327	+	0.956928i	$0.406236\pi$
$432$	0	0
$433$	−10.6031	−0.509551	−0.254776	−	0.967000i	$-0.582002\pi$
$433$	−10.6031	−0.509551	−0.254776	+	0.967000i	$0.582002\pi$
$434$	−12.3550	−0.593061
$435$	0	0
$436$	−6.69459	−0.320613
$437$	−28.9377	−1.38428
$438$	0	0
$439$	7.09327	0.338543	0.169272	−	0.985569i	$-0.445858\pi$
$439$	7.09327	0.338543	0.169272	+	0.985569i	$0.445858\pi$
$440$	0.736482	0.0351104
$441$	0	0
$442$	0	0
$443$	0.206148	0.00979437	0.00489718	−	0.999988i	$-0.498441\pi$
$443$	0.206148	0.00979437	0.00489718	+	0.999988i	$0.498441\pi$
$444$	0	0
$445$	1.98990	0.0943301
$446$	−11.0027	−0.520992
$447$	0	0
$448$	1.69459	0.0800620
$449$	−33.1343	−1.56371	−0.781853	−	0.623463i	$-0.785725\pi$
$449$	−33.1343	−1.56371	−0.781853	+	0.623463i	$0.785725\pi$
$450$	0	0
$451$	−19.2216	−0.905111
$452$	4.58172	0.215506
$453$	0	0
$454$	−0.758770	−0.0356109
$455$	0.359480	0.0168527
$456$	0	0
$457$	−11.9463	−0.558822	−0.279411	−	0.960172i	$-0.590139\pi$
$457$	−11.9463	−0.558822	−0.279411	+	0.960172i	$0.590139\pi$
$458$	24.5817	1.14863
$459$	0	0
$460$	0.723689	0.0337422
$461$	−25.5185	−1.18851	−0.594257	−	0.804275i	$-0.702554\pi$
$461$	−25.5185	−1.18851	−0.594257	+	0.804275i	$0.702554\pi$
$462$	0	0
$463$	−36.5449	−1.69838	−0.849192	−	0.528084i	$-0.822911\pi$
$463$	−36.5449	−1.69838	−0.849192	+	0.528084i	$0.822911\pi$
$464$	3.90167	0.181131
$465$	0	0
$466$	−5.67499	−0.262889
$467$	−20.7246	−0.959021	−0.479511	−	0.877536i	$-0.659186\pi$
$467$	−20.7246	−0.959021	−0.479511	+	0.877536i	$0.659186\pi$
$468$	0	0
$469$	−9.97628	−0.460662
$470$	0.655392	0.0302310
$471$	0	0
$472$	−2.12061	−0.0976092
$473$	−41.1498	−1.89207
$474$	0	0
$475$	−24.0446	−1.10324
$476$	0	0
$477$	0	0
$478$	−25.5175	−1.16715
$479$	31.6168	1.44461	0.722304	−	0.691575i	$-0.243083\pi$
$479$	31.6168	1.44461	0.722304	+	0.691575i	$0.243083\pi$
$480$	0	0
$481$	−6.46347	−0.294709
$482$	20.4320	0.930652
$483$	0	0
$484$	26.2841	1.19473
$485$	1.09596	0.0497651
$486$	0	0
$487$	21.2841	0.964472	0.482236	−	0.876041i	$-0.339825\pi$
$487$	21.2841	0.964472	0.482236	+	0.876041i	$0.339825\pi$
$488$	−10.6946	−0.484121
$489$	0	0
$490$	0.497941	0.0224947
$491$	19.5253	0.881164	0.440582	−	0.897712i	$-0.354772\pi$
$491$	19.5253	0.881164	0.440582	+	0.897712i	$0.354772\pi$
$492$	0	0
$493$	0	0
$494$	−8.48246	−0.381644
$495$	0	0
$496$	−7.29086	−0.327369
$497$	−26.1046	−1.17095
$498$	0	0
$499$	22.5972	1.01159	0.505795	−	0.862654i	$-0.331199\pi$
$499$	22.5972	1.01159	0.505795	+	0.862654i	$0.331199\pi$
$500$	1.20439	0.0538621
$501$	0	0
$502$	−7.50299	−0.334875
$503$	13.3054	0.593259	0.296629	−	0.954993i	$-0.404137\pi$
$503$	13.3054	0.593259	0.296629	+	0.954993i	$0.404137\pi$
$504$	0	0
$505$	0.271259	0.0120709
$506$	36.6364	1.62869
$507$	0	0
$508$	−4.30541	−0.191022
$509$	−40.9760	−1.81623	−0.908114	−	0.418724i	$-0.862478\pi$
$509$	−40.9760	−1.81623	−0.908114	+	0.418724i	$0.862478\pi$
$510$	0	0
$511$	−21.3455	−0.944271
$512$	1.00000	0.0441942
$513$	0	0
$514$	18.3851	0.810931
$515$	−2.21038	−0.0974009
$516$	0	0
$517$	33.1789	1.45921
$518$	6.22762	0.273626
$519$	0	0
$520$	0.212134	0.00930268
$521$	39.0951	1.71279	0.856395	−	0.516322i	$-0.172699\pi$
$521$	39.0951	1.71279	0.856395	+	0.516322i	$0.172699\pi$
$522$	0	0
$523$	18.0702	0.790153	0.395077	−	0.918648i	$-0.370718\pi$
$523$	18.0702	0.790153	0.395077	+	0.918648i	$0.370718\pi$
$524$	15.0155	0.655954
$525$	0	0
$526$	3.01960	0.131661
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0.568926	0.0247125
$531$	0	0
$532$	8.17293	0.354342
$533$	−5.53653	−0.239814
$534$	0	0
$535$	1.70233	0.0735983
$536$	−5.88713	−0.254285
$537$	0	0
$538$	10.8452	0.467571
$539$	25.2080	1.08579
$540$	0	0
$541$	−22.5270	−0.968513	−0.484256	−	0.874926i	$-0.660910\pi$
$541$	−22.5270	−0.968513	−0.484256	+	0.874926i	$0.660910\pi$
$542$	−17.6159	−0.756666
$543$	0	0
$544$	0	0
$545$	0.807467	0.0345881
$546$	0	0
$547$	2.53714	0.108480	0.0542402	−	0.998528i	$-0.482726\pi$
$547$	2.53714	0.108480	0.0542402	+	0.998528i	$0.482726\pi$
$548$	6.82295	0.291462
$549$	0	0
$550$	30.4415	1.29803
$551$	18.8176	0.801656
$552$	0	0
$553$	23.7128	1.00837
$554$	1.10338	0.0468781
$555$	0	0
$556$	−16.9067	−0.717005
$557$	−24.9317	−1.05639	−0.528195	−	0.849123i	$-0.677131\pi$
$557$	−24.9317	−1.05639	−0.528195	+	0.849123i	$0.677131\pi$
$558$	0	0
$559$	−11.8527	−0.501314
$560$	−0.204393	−0.00863718
$561$	0	0
$562$	−6.28581	−0.265151
$563$	31.3354	1.32063	0.660316	−	0.750988i	$-0.270423\pi$
$563$	31.3354	1.32063	0.660316	+	0.750988i	$0.270423\pi$
$564$	0	0
$565$	−0.552623	−0.0232490
$566$	3.27631	0.137714
$567$	0	0
$568$	−15.4047	−0.646365
$569$	22.9121	0.960525	0.480263	−	0.877125i	$-0.340541\pi$
$569$	22.9121	0.960525	0.480263	+	0.877125i	$0.340541\pi$
$570$	0	0
$571$	−10.5425	−0.441191	−0.220595	−	0.975365i	$-0.570800\pi$
$571$	−10.5425	−0.441191	−0.220595	+	0.975365i	$0.570800\pi$
$572$	10.7392	0.449027
$573$	0	0
$574$	5.33450	0.222658
$575$	29.9127	1.24745
$576$	0	0
$577$	30.9418	1.28812	0.644062	−	0.764973i	$-0.277248\pi$
$577$	30.9418	1.28812	0.644062	+	0.764973i	$0.277248\pi$
$578$	0	0
$579$	0	0
$580$	−0.470599	−0.0195406
$581$	25.2276	1.04662
$582$	0	0
$583$	28.8016	1.19284
$584$	−12.5963	−0.521237
$585$	0	0
$586$	−5.76651	−0.238212
$587$	18.7861	0.775386	0.387693	−	0.921789i	$-0.373272\pi$
$587$	18.7861	0.775386	0.387693	+	0.921789i	$0.373272\pi$
$588$	0	0
$589$	−35.1634	−1.44888
$590$	0.255777	0.0105302
$591$	0	0
$592$	3.67499	0.151041
$593$	15.8817	0.652185	0.326093	−	0.945338i	$-0.394268\pi$
$593$	15.8817	0.652185	0.326093	+	0.945338i	$0.394268\pi$
$594$	0	0
$595$	0	0
$596$	−11.2831	−0.462175
$597$	0	0
$598$	10.5526	0.431529
$599$	−2.86215	−0.116944	−0.0584721	−	0.998289i	$-0.518623\pi$
$599$	−2.86215	−0.116944	−0.0584721	+	0.998289i	$0.518623\pi$
$600$	0	0
$601$	−16.7929	−0.684997	−0.342499	−	0.939518i	$-0.611273\pi$
$601$	−16.7929	−0.684997	−0.342499	+	0.939518i	$0.611273\pi$
$602$	11.4201	0.465451
$603$	0	0
$604$	23.1702	0.942784
$605$	−3.17024	−0.128889
$606$	0	0
$607$	−11.7151	−0.475502	−0.237751	−	0.971326i	$-0.576410\pi$
$607$	−11.7151	−0.475502	−0.237751	+	0.971326i	$0.576410\pi$
$608$	4.82295	0.195596
$609$	0	0
$610$	1.28993	0.0522276
$611$	9.55674	0.386624
$612$	0	0
$613$	−14.8675	−0.600494	−0.300247	−	0.953862i	$-0.597069\pi$
$613$	−14.8675	−0.600494	−0.300247	+	0.953862i	$0.597069\pi$
$614$	−1.26083	−0.0508830
$615$	0	0
$616$	−10.3473	−0.416904
$617$	49.0607	1.97511	0.987554	−	0.157280i	$-0.0502725\pi$
$617$	49.0607	1.97511	0.987554	+	0.157280i	$0.0502725\pi$
$618$	0	0
$619$	−4.17293	−0.167724	−0.0838622	−	0.996477i	$-0.526726\pi$
$619$	−4.17293	−0.167724	−0.0838622	+	0.996477i	$0.526726\pi$
$620$	0.879385	0.0353170
$621$	0	0
$622$	−4.77837	−0.191595
$623$	−27.9573	−1.12009
$624$	0	0
$625$	24.7820	0.991280
$626$	−6.25671	−0.250068
$627$	0	0
$628$	5.38919	0.215052
$629$	0	0
$630$	0	0
$631$	16.2267	0.645974	0.322987	−	0.946403i	$-0.395313\pi$
$631$	16.2267	0.645974	0.322987	+	0.946403i	$0.395313\pi$
$632$	13.9932	0.556619
$633$	0	0
$634$	26.1925	1.04024
$635$	0.519296	0.0206076
$636$	0	0
$637$	7.26083	0.287685
$638$	−23.8239	−0.943197
$639$	0	0
$640$	−0.120615	−0.00476772
$641$	5.83244	0.230368	0.115184	−	0.993344i	$-0.463254\pi$
$641$	5.83244	0.230368	0.115184	+	0.993344i	$0.463254\pi$
$642$	0	0
$643$	−5.14796	−0.203016	−0.101508	−	0.994835i	$-0.532367\pi$
$643$	−5.14796	−0.203016	−0.101508	+	0.994835i	$0.532367\pi$
$644$	−10.1676	−0.400658
$645$	0	0
$646$	0	0
$647$	5.59121	0.219813	0.109907	−	0.993942i	$-0.464945\pi$
$647$	5.59121	0.219813	0.109907	+	0.993942i	$0.464945\pi$
$648$	0	0
$649$	12.9486	0.508278
$650$	8.76827	0.343920
$651$	0	0
$652$	−17.8871	−0.700514
$653$	15.8716	0.621105	0.310553	−	0.950556i	$-0.399486\pi$
$653$	15.8716	0.621105	0.310553	+	0.950556i	$0.399486\pi$
$654$	0	0
$655$	−1.81109	−0.0707651
$656$	3.14796	0.122907
$657$	0	0
$658$	−9.20801	−0.358966
$659$	−26.0033	−1.01294	−0.506472	−	0.862256i	$-0.669051\pi$
$659$	−26.0033	−1.01294	−0.506472	+	0.862256i	$0.669051\pi$
$660$	0	0
$661$	11.7980	0.458888	0.229444	−	0.973322i	$-0.426309\pi$
$661$	11.7980	0.458888	0.229444	+	0.973322i	$0.426309\pi$
$662$	−12.5817	−0.489002
$663$	0	0
$664$	14.8871	0.577733
$665$	−0.985776	−0.0382268
$666$	0	0
$667$	−23.4100	−0.906441
$668$	−5.84255	−0.226055
$669$	0	0
$670$	0.710074	0.0274326
$671$	65.3019	2.52095
$672$	0	0
$673$	47.9026	1.84651	0.923255	−	0.384188i	$-0.125519\pi$
$673$	47.9026	1.84651	0.923255	+	0.384188i	$0.125519\pi$
$674$	−23.2175	−0.894305
$675$	0	0
$676$	−9.90673	−0.381028
$677$	−40.4303	−1.55386	−0.776930	−	0.629586i	$-0.783224\pi$
$677$	−40.4303	−1.55386	−0.776930	+	0.629586i	$0.783224\pi$
$678$	0	0
$679$	−15.3979	−0.590916
$680$	0	0
$681$	0	0
$682$	44.5185	1.70470
$683$	−8.08141	−0.309227	−0.154613	−	0.987975i	$-0.549413\pi$
$683$	−8.08141	−0.309227	−0.154613	+	0.987975i	$0.549413\pi$
$684$	0	0
$685$	−0.822948	−0.0314432
$686$	−18.8580	−0.720003
$687$	0	0
$688$	6.73917	0.256928
$689$	8.29591	0.316049
$690$	0	0
$691$	−2.34049	−0.0890364	−0.0445182	−	0.999009i	$-0.514175\pi$
$691$	−2.34049	−0.0890364	−0.0445182	+	0.999009i	$0.514175\pi$
$692$	5.41147	0.205713
$693$	0	0
$694$	10.7706	0.408848
$695$	2.03920	0.0773513
$696$	0	0
$697$	0	0
$698$	−29.9864	−1.13500
$699$	0	0
$700$	−8.44831	−0.319316
$701$	−19.5534	−0.738523	−0.369262	−	0.929325i	$-0.620389\pi$
$701$	−19.5534	−0.738523	−0.369262	+	0.929325i	$0.620389\pi$
$702$	0	0
$703$	17.7243	0.668485
$704$	−6.10607	−0.230131
$705$	0	0
$706$	29.6459	1.11574
$707$	−3.81109	−0.143331
$708$	0	0
$709$	35.2472	1.32374	0.661868	−	0.749620i	$-0.269764\pi$
$709$	35.2472	1.32374	0.661868	+	0.749620i	$0.269764\pi$
$710$	1.85803	0.0697306
$711$	0	0
$712$	−16.4979	−0.618286
$713$	43.7452	1.63827
$714$	0	0
$715$	−1.29530	−0.0484416
$716$	−7.24123	−0.270617
$717$	0	0
$718$	18.7101	0.698254
$719$	−11.6013	−0.432656	−0.216328	−	0.976321i	$-0.569408\pi$
$719$	−11.6013	−0.432656	−0.216328	+	0.976321i	$0.569408\pi$
$720$	0	0
$721$	31.0550	1.15655
$722$	4.26083	0.158572
$723$	0	0
$724$	−0.157451	−0.00585163
$725$	−19.4516	−0.722415
$726$	0	0
$727$	41.7083	1.54688	0.773438	−	0.633872i	$-0.218535\pi$
$727$	41.7083	1.54688	0.773438	+	0.633872i	$0.218535\pi$
$728$	−2.98040	−0.110461
$729$	0	0
$730$	1.51930	0.0562317
$731$	0	0
$732$	0	0
$733$	−46.2877	−1.70967	−0.854837	−	0.518896i	$-0.826343\pi$
$733$	−46.2877	−1.70967	−0.854837	+	0.518896i	$0.826343\pi$
$734$	5.84018	0.215565
$735$	0	0
$736$	−6.00000	−0.221163
$737$	35.9472	1.32413
$738$	0	0
$739$	31.2080	1.14801	0.574003	−	0.818853i	$-0.305390\pi$
$739$	31.2080	1.14801	0.574003	+	0.818853i	$0.305390\pi$
$740$	−0.443258	−0.0162945
$741$	0	0
$742$	−7.99319	−0.293439
$743$	30.5134	1.11943	0.559714	−	0.828686i	$-0.310911\pi$
$743$	30.5134	1.11943	0.559714	+	0.828686i	$0.310911\pi$
$744$	0	0
$745$	1.36091	0.0498599
$746$	−10.6209	−0.388859
$747$	0	0
$748$	0	0
$749$	−23.9172	−0.873914
$750$	0	0
$751$	30.4320	1.11048	0.555240	−	0.831690i	$-0.312626\pi$
$751$	30.4320	1.11048	0.555240	+	0.831690i	$0.312626\pi$
$752$	−5.43376	−0.198149
$753$	0	0
$754$	−6.86215	−0.249905
$755$	−2.79467	−0.101709
$756$	0	0
$757$	17.5567	0.638111	0.319055	−	0.947736i	$-0.396634\pi$
$757$	17.5567	0.638111	0.319055	+	0.947736i	$0.396634\pi$
$758$	−0.340489	−0.0123671
$759$	0	0
$760$	−0.581719	−0.0211012
$761$	4.69459	0.170179	0.0850894	−	0.996373i	$-0.472882\pi$
$761$	4.69459	0.170179	0.0850894	+	0.996373i	$0.472882\pi$
$762$	0	0
$763$	−11.3446	−0.410702
$764$	6.90673	0.249877
$765$	0	0
$766$	30.8776	1.11565
$767$	3.72967	0.134671
$768$	0	0
$769$	2.36184	0.0851703	0.0425851	−	0.999093i	$-0.486441\pi$
$769$	2.36184	0.0851703	0.0425851	+	0.999093i	$0.486441\pi$
$770$	1.24804	0.0449761
$771$	0	0
$772$	−11.5253	−0.414804
$773$	−29.4561	−1.05946	−0.529730	−	0.848166i	$-0.677707\pi$
$773$	−29.4561	−1.05946	−0.529730	+	0.848166i	$0.677707\pi$
$774$	0	0
$775$	36.3482	1.30567
$776$	−9.08647	−0.326185
$777$	0	0
$778$	3.25166	0.116578
$779$	15.1824	0.543967
$780$	0	0
$781$	94.0619	3.36580
$782$	0	0
$783$	0	0
$784$	−4.12836	−0.147441
$785$	−0.650015	−0.0232000
$786$	0	0
$787$	−55.6715	−1.98447	−0.992237	−	0.124361i	$-0.960312\pi$
$787$	−55.6715	−1.98447	−0.992237	+	0.124361i	$0.960312\pi$
$788$	−6.71419	−0.239183
$789$	0	0
$790$	−1.68779	−0.0600487
$791$	7.76415	0.276061
$792$	0	0
$793$	18.8093	0.667939
$794$	−18.3405	−0.650880
$795$	0	0
$796$	7.22937	0.256238
$797$	19.7802	0.700652	0.350326	−	0.936628i	$-0.386071\pi$
$797$	19.7802	0.700652	0.350326	+	0.936628i	$0.386071\pi$
$798$	0	0
$799$	0	0
$800$	−4.98545	−0.176262
$801$	0	0
$802$	18.7101	0.660676
$803$	76.9136	2.71422
$804$	0	0
$805$	1.22636	0.0432234
$806$	12.8229	0.451669
$807$	0	0
$808$	−2.24897	−0.0791185
$809$	−20.9222	−0.735586	−0.367793	−	0.929908i	$-0.619886\pi$
$809$	−20.9222	−0.735586	−0.367793	+	0.929908i	$0.619886\pi$
$810$	0	0
$811$	−0.896622	−0.0314846	−0.0157423	−	0.999876i	$-0.505011\pi$
$811$	−0.896622	−0.0314846	−0.0157423	+	0.999876i	$0.505011\pi$
$812$	6.61175	0.232027
$813$	0	0
$814$	−22.4397	−0.786513
$815$	2.15745	0.0755722
$816$	0	0
$817$	32.5027	1.13712
$818$	−5.94862	−0.207988
$819$	0	0
$820$	−0.379690	−0.0132593
$821$	37.6851	1.31522	0.657609	−	0.753359i	$-0.271568\pi$
$821$	37.6851	1.31522	0.657609	+	0.753359i	$0.271568\pi$
$822$	0	0
$823$	−22.1607	−0.772475	−0.386238	−	0.922399i	$-0.626226\pi$
$823$	−22.1607	−0.772475	−0.386238	+	0.922399i	$0.626226\pi$
$824$	18.3259	0.638414
$825$	0	0
$826$	−3.59358	−0.125037
$827$	−30.8016	−1.07108	−0.535538	−	0.844511i	$-0.679891\pi$
$827$	−30.8016	−1.07108	−0.535538	+	0.844511i	$0.679891\pi$
$828$	0	0
$829$	18.5716	0.645019	0.322509	−	0.946566i	$-0.395474\pi$
$829$	18.5716	0.645019	0.322509	+	0.946566i	$0.395474\pi$
$830$	−1.79561	−0.0623264
$831$	0	0
$832$	−1.75877	−0.0609744
$833$	0	0
$834$	0	0
$835$	0.704698	0.0243871
$836$	−29.4492	−1.01852
$837$	0	0
$838$	12.8990	0.445588
$839$	27.9763	0.965848	0.482924	−	0.875662i	$-0.339575\pi$
$839$	27.9763	0.965848	0.482924	+	0.875662i	$0.339575\pi$
$840$	0	0
$841$	−13.7769	−0.475067
$842$	−27.2472	−0.939001
$843$	0	0
$844$	10.5371	0.362703
$845$	1.19490	0.0411057
$846$	0	0
$847$	44.5408	1.53044
$848$	−4.71688	−0.161978
$849$	0	0
$850$	0	0
$851$	−22.0500	−0.755863
$852$	0	0
$853$	10.6500	0.364650	0.182325	−	0.983238i	$-0.441638\pi$
$853$	10.6500	0.364650	0.182325	+	0.983238i	$0.441638\pi$
$854$	−18.1230	−0.620156
$855$	0	0
$856$	−14.1138	−0.482400
$857$	−42.3560	−1.44685	−0.723426	−	0.690402i	$-0.757434\pi$
$857$	−42.3560	−1.44685	−0.723426	+	0.690402i	$0.757434\pi$
$858$	0	0
$859$	52.2039	1.78117	0.890587	−	0.454813i	$-0.150294\pi$
$859$	52.2039	1.78117	0.890587	+	0.454813i	$0.150294\pi$
$860$	−0.812843	−0.0277177
$861$	0	0
$862$	12.0547	0.410584
$863$	−48.2330	−1.64187	−0.820935	−	0.571022i	$-0.806547\pi$
$863$	−48.2330	−1.64187	−0.820935	+	0.571022i	$0.806547\pi$
$864$	0	0
$865$	−0.652704	−0.0221926
$866$	−10.6031	−0.360307
$867$	0	0
$868$	−12.3550	−0.419357
$869$	−85.4434	−2.89847
$870$	0	0
$871$	10.3541	0.350835
$872$	−6.69459	−0.226708
$873$	0	0
$874$	−28.9377	−0.978832
$875$	2.04096	0.0689969
$876$	0	0
$877$	−29.4884	−0.995754	−0.497877	−	0.867248i	$-0.665887\pi$
$877$	−29.4884	−0.995754	−0.497877	+	0.867248i	$0.665887\pi$
$878$	7.09327	0.239386
$879$	0	0
$880$	0.736482	0.0248268
$881$	−6.92034	−0.233152	−0.116576	−	0.993182i	$-0.537192\pi$
$881$	−6.92034	−0.233152	−0.116576	+	0.993182i	$0.537192\pi$
$882$	0	0
$883$	−19.0405	−0.640762	−0.320381	−	0.947289i	$-0.603811\pi$
$883$	−19.0405	−0.640762	−0.320381	+	0.947289i	$0.603811\pi$
$884$	0	0
$885$	0	0
$886$	0.206148	0.00692566
$887$	7.36009	0.247128	0.123564	−	0.992337i	$-0.460568\pi$
$887$	7.36009	0.247128	0.123564	+	0.992337i	$0.460568\pi$
$888$	0	0
$889$	−7.29591	−0.244697
$890$	1.98990	0.0667014
$891$	0	0
$892$	−11.0027	−0.368397
$893$	−26.2068	−0.876976
$894$	0	0
$895$	0.873399	0.0291945
$896$	1.69459	0.0566124
$897$	0	0
$898$	−33.1343	−1.10571
$899$	−28.4466	−0.948746
$900$	0	0
$901$	0	0
$902$	−19.2216	−0.640010
$903$	0	0
$904$	4.58172	0.152386
$905$	0.0189910	0.000631281 0
$906$	0	0
$907$	42.7837	1.42061	0.710306	−	0.703894i	$-0.248557\pi$
$907$	42.7837	1.42061	0.710306	+	0.703894i	$0.248557\pi$
$908$	−0.758770	−0.0251807
$909$	0	0
$910$	0.359480	0.0119167
$911$	−0.508045	−0.0168323	−0.00841615	−	0.999965i	$-0.502679\pi$
$911$	−0.508045	−0.0168323	−0.00841615	+	0.999965i	$0.502679\pi$
$912$	0	0
$913$	−90.9018	−3.00841
$914$	−11.9463	−0.395147
$915$	0	0
$916$	24.5817	0.812203
$917$	25.4451	0.840272
$918$	0	0
$919$	8.20801	0.270757	0.135379	−	0.990794i	$-0.456775\pi$
$919$	8.20801	0.270757	0.135379	+	0.990794i	$0.456775\pi$
$920$	0.723689	0.0238593
$921$	0	0
$922$	−25.5185	−0.840406
$923$	27.0933	0.891786
$924$	0	0
$925$	−18.3215	−0.602407
$926$	−36.5449	−1.20094
$927$	0	0
$928$	3.90167	0.128079
$929$	−1.85616	−0.0608987	−0.0304494	−	0.999536i	$-0.509694\pi$
$929$	−1.85616	−0.0608987	−0.0304494	+	0.999536i	$0.509694\pi$
$930$	0	0
$931$	−19.9108	−0.652552
$932$	−5.67499	−0.185891
$933$	0	0
$934$	−20.7246	−0.678130
$935$	0	0
$936$	0	0
$937$	−8.40104	−0.274450	−0.137225	−	0.990540i	$-0.543818\pi$
$937$	−8.40104	−0.274450	−0.137225	+	0.990540i	$0.543818\pi$
$938$	−9.97628	−0.325737
$939$	0	0
$940$	0.655392	0.0213765
$941$	32.9459	1.07401	0.537003	−	0.843580i	$-0.319556\pi$
$941$	32.9459	1.07401	0.537003	+	0.843580i	$0.319556\pi$
$942$	0	0
$943$	−18.8877	−0.615069
$944$	−2.12061	−0.0690201
$945$	0	0
$946$	−41.1498	−1.33790
$947$	19.6182	0.637507	0.318753	−	0.947838i	$-0.396736\pi$
$947$	19.6182	0.637507	0.318753	+	0.947838i	$0.396736\pi$
$948$	0	0
$949$	22.1539	0.719147
$950$	−24.0446	−0.780109
$951$	0	0
$952$	0	0
$953$	51.5931	1.67126	0.835632	−	0.549290i	$-0.185102\pi$
$953$	51.5931	1.67126	0.835632	+	0.549290i	$0.185102\pi$
$954$	0	0
$955$	−0.833053	−0.0269570
$956$	−25.5175	−0.825296
$957$	0	0
$958$	31.6168	1.02149
$959$	11.5621	0.373360
$960$	0	0
$961$	22.1566	0.714730
$962$	−6.46347	−0.208391
$963$	0	0
$964$	20.4320	0.658071
$965$	1.39012	0.0447495
$966$	0	0
$967$	−39.7475	−1.27819	−0.639097	−	0.769126i	$-0.720692\pi$
$967$	−39.7475	−1.27819	−0.639097	+	0.769126i	$0.720692\pi$
$968$	26.2841	0.844801
$969$	0	0
$970$	1.09596	0.0351892
$971$	−39.5398	−1.26889	−0.634447	−	0.772967i	$-0.718772\pi$
$971$	−39.5398	−1.26889	−0.634447	+	0.772967i	$0.718772\pi$
$972$	0	0
$973$	−28.6500	−0.918477
$974$	21.2841	0.681985
$975$	0	0
$976$	−10.6946	−0.342326
$977$	−10.3696	−0.331752	−0.165876	−	0.986147i	$-0.553045\pi$
$977$	−10.3696	−0.331752	−0.165876	+	0.986147i	$0.553045\pi$
$978$	0	0
$979$	100.738	3.21959
$980$	0.497941	0.0159061
$981$	0	0
$982$	19.5253	0.623077
$983$	−3.18243	−0.101504	−0.0507519	−	0.998711i	$-0.516162\pi$
$983$	−3.18243	−0.101504	−0.0507519	+	0.998711i	$0.516162\pi$
$984$	0	0
$985$	0.809831	0.0258034
$986$	0	0
$987$	0	0
$988$	−8.48246	−0.269863
$989$	−40.4350	−1.28576
$990$	0	0
$991$	−13.6946	−0.435023	−0.217512	−	0.976058i	$-0.569794\pi$
$991$	−13.6946	−0.435023	−0.217512	+	0.976058i	$0.569794\pi$
$992$	−7.29086	−0.231485
$993$	0	0
$994$	−26.1046	−0.827989
$995$	−0.871969	−0.0276433
$996$	0	0
$997$	0.964918	0.0305593	0.0152796	−	0.999883i	$-0.495136\pi$
$997$	0.964918	0.0305593	0.0152796	+	0.999883i	$0.495136\pi$
$998$	22.5972	0.715302
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5202.2.a.bm.1.3		3
3.2	odd	2		1734.2.a.q.1.1	yes	3
17.16	even	2		5202.2.a.bp.1.1		3
51.2	odd	8		1734.2.f.n.1483.6		12
51.8	odd	8		1734.2.f.n.829.1		12
51.26	odd	8		1734.2.f.n.829.6		12
51.32	odd	8		1734.2.f.n.1483.1		12
51.38	odd	4		1734.2.b.j.577.3		6
51.47	odd	4		1734.2.b.j.577.4		6
51.50	odd	2		1734.2.a.p.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1734.2.a.p.1.3	✓	3	51.50	odd	2
1734.2.a.q.1.1	yes	3	3.2	odd	2
1734.2.b.j.577.3		6	51.38	odd	4
1734.2.b.j.577.4		6	51.47	odd	4
1734.2.f.n.829.1		12	51.8	odd	8
1734.2.f.n.829.6		12	51.26	odd	8
1734.2.f.n.1483.1		12	51.32	odd	8
1734.2.f.n.1483.6		12	51.2	odd	8
5202.2.a.bm.1.3		3	1.1	even	1	trivial
5202.2.a.bp.1.1		3	17.16	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5202.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.120615 q^{5} +1.69459 q^{7} +1.00000 q^{8} -0.120615 q^{10} -6.10607 q^{11} -1.75877 q^{13} +1.69459 q^{14} +1.00000 q^{16} +4.82295 q^{19} -0.120615 q^{20} -6.10607 q^{22} -6.00000 q^{23} -4.98545 q^{25} -1.75877 q^{26} +1.69459 q^{28} +3.90167 q^{29} -7.29086 q^{31} +1.00000 q^{32} -0.204393 q^{35} +3.67499 q^{37} +4.82295 q^{38} -0.120615 q^{40} +3.14796 q^{41} +6.73917 q^{43} -6.10607 q^{44} -6.00000 q^{46} -5.43376 q^{47} -4.12836 q^{49} -4.98545 q^{50} -1.75877 q^{52} -4.71688 q^{53} +0.736482 q^{55} +1.69459 q^{56} +3.90167 q^{58} -2.12061 q^{59} -10.6946 q^{61} -7.29086 q^{62} +1.00000 q^{64} +0.212134 q^{65} -5.88713 q^{67} -0.204393 q^{70} -15.4047 q^{71} -12.5963 q^{73} +3.67499 q^{74} +4.82295 q^{76} -10.3473 q^{77} +13.9932 q^{79} -0.120615 q^{80} +3.14796 q^{82} +14.8871 q^{83} +6.73917 q^{86} -6.10607 q^{88} -16.4979 q^{89} -2.98040 q^{91} -6.00000 q^{92} -5.43376 q^{94} -0.581719 q^{95} -9.08647 q^{97} -4.12836 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} + 3 q^{8} - 6 q^{10} - 6 q^{11} + 6 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{19} - 6 q^{20} - 6 q^{22} - 18 q^{23} + 3 q^{25} + 6 q^{26} + 3 q^{28} - 6 q^{31} + 3 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 + 3 * q^8 - 6 * q^10 - 6 * q^11 + 6 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^19 - 6 * q^20 - 6 * q^22 - 18 * q^23 + 3 * q^25 + 6 * q^26 + 3 * q^28 - 6 * q^31 + 3 * q^32 + 6 * q^37 - 6 * q^38 - 6 * q^40 - 6 * q^41 + 6 * q^43 - 6 * q^44 - 18 * q^46 + 6 * q^49 + 3 * q^50 + 6 * q^52 - 6 * q^53 - 3 * q^55 + 3 * q^56 - 12 * q^59 - 30 * q^61 - 6 * q^62 + 3 * q^64 - 24 * q^65 + 12 * q^67 + 6 * q^71 - 24 * q^73 + 6 * q^74 - 6 * q^76 - 30 * q^77 - 6 * q^80 - 6 * q^82 + 15 * q^83 + 6 * q^86 - 6 * q^88 - 24 * q^89 - 6 * q^91 - 18 * q^92 + 30 * q^95 - 12 * q^97 + 6 * q^98

Introduction

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