LMFDB - Embedded newform 7350.2.a.ch.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7350.a (trivial)

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:	yes
Analytic conductor:	$58.6900454856$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -7.00000 q^{19} -5.00000 q^{22} -3.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} +6.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +5.00000 q^{37} -7.00000 q^{38} +1.00000 q^{39} +9.00000 q^{41} -10.0000 q^{43} -5.00000 q^{44} -3.00000 q^{46} -13.0000 q^{47} +1.00000 q^{48} -2.00000 q^{51} +1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} -7.00000 q^{57} -4.00000 q^{59} +2.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -5.00000 q^{66} -6.00000 q^{67} -2.00000 q^{68} -3.00000 q^{69} -2.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +5.00000 q^{74} -7.00000 q^{76} +1.00000 q^{78} -14.0000 q^{79} +1.00000 q^{81} +9.00000 q^{82} -10.0000 q^{83} -10.0000 q^{86} -5.00000 q^{88} -10.0000 q^{89} -3.00000 q^{92} +6.00000 q^{93} -13.0000 q^{94} +1.00000 q^{96} +8.00000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	0	0
$22$	−5.00000	−1.06600
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	−5.00000	−0.870388
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	−7.00000	−1.13555
$39$	1.00000	0.160128
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	−3.00000	−0.442326
$47$	−13.0000	−1.89624	−0.948122	−	0.317905i	$-0.897021\pi$
$47$	−13.0000	−1.89624	−0.948122	+	0.317905i	$0.897021\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	−2.00000	−0.280056
$52$	1.00000	0.138675
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.00000	−0.615457
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	−2.00000	−0.242536
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	1.00000	0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	5.00000	0.581238
$75$	0	0
$76$	−7.00000	−0.802955
$77$	0	0
$78$	1.00000	0.113228
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	0	0
$86$	−10.0000	−1.07833
$87$	0	0
$88$	−5.00000	−0.533002
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	−3.00000	−0.312772
$93$	6.00000	0.622171
$94$	−13.0000	−1.34085
$95$	0	0
$96$	1.00000	0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	−2.00000	−0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	1.00000	0.0971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	5.00000	0.474579
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−7.00000	−0.655610
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	−4.00000	−0.368230
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	2.00000	0.181071
$123$	9.00000	0.811503
$124$	6.00000	0.538816
$125$	0	0
$126$	0	0
$127$	−9.00000	−0.798621	−0.399310	−	0.916816i	$-0.630750\pi$
$127$	−9.00000	−0.798621	−0.399310	+	0.916816i	$0.630750\pi$
$128$	1.00000	0.0883883
$129$	−10.0000	−0.880451
$130$	0	0
$131$	17.0000	1.48530	0.742648	−	0.669681i	$-0.233569\pi$
$131$	17.0000	1.48530	0.742648	+	0.669681i	$0.233569\pi$
$132$	−5.00000	−0.435194
$133$	0	0
$134$	−6.00000	−0.518321
$135$	0	0
$136$	−2.00000	−0.171499
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	−3.00000	−0.255377
$139$	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
$141$	−13.0000	−1.09480
$142$	−2.00000	−0.167836
$143$	−5.00000	−0.418121
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	5.00000	0.410997
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	−7.00000	−0.567775
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	1.00000	0.0800641
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	−14.0000	−1.11378
$159$	1.00000	0.0793052
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−10.0000	−0.776151
$167$	−19.0000	−1.47026	−0.735132	−	0.677924i	$-0.762880\pi$
$167$	−19.0000	−1.47026	−0.735132	+	0.677924i	$0.762880\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−7.00000	−0.535303
$172$	−10.0000	−0.762493
$173$	7.00000	0.532200	0.266100	−	0.963945i	$-0.414265\pi$
$173$	7.00000	0.532200	0.266100	+	0.963945i	$0.414265\pi$
$174$	0	0
$175$	0	0
$176$	−5.00000	−0.376889
$177$	−4.00000	−0.300658
$178$	−10.0000	−0.749532
$179$	−11.0000	−0.822179	−0.411089	−	0.911595i	$-0.634852\pi$
$179$	−11.0000	−0.822179	−0.411089	+	0.911595i	$0.634852\pi$
$180$	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
$183$	2.00000	0.147844
$184$	−3.00000	−0.221163
$185$	0	0
$186$	6.00000	0.439941
$187$	10.0000	0.731272
$188$	−13.0000	−0.948122
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	1.00000	0.0721688
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	0	0
$197$	27.0000	1.92367	0.961835	−	0.273629i	$-0.0882242\pi$
$197$	27.0000	1.92367	0.961835	+	0.273629i	$0.0882242\pi$
$198$	−5.00000	−0.355335
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	−8.00000	−0.562878
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	0	0
$207$	−3.00000	−0.208514
$208$	1.00000	0.0693375
$209$	35.0000	2.42100
$210$	0	0
$211$	19.0000	1.30801	0.654007	−	0.756489i	$-0.273087\pi$
$211$	19.0000	1.30801	0.654007	+	0.756489i	$0.273087\pi$
$212$	1.00000	0.0686803
$213$	−2.00000	−0.137038
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	−18.0000	−1.21911
$219$	4.00000	0.270295
$220$	0	0
$221$	−2.00000	−0.134535
$222$	5.00000	0.335578
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	−7.00000	−0.463586
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−4.00000	−0.260378
$237$	−14.0000	−0.909398
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	14.0000	0.899954
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	9.00000	0.573819
$247$	−7.00000	−0.445399
$248$	6.00000	0.381000
$249$	−10.0000	−0.633724
$250$	0	0
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	0	0
$253$	15.0000	0.943042
$254$	−9.00000	−0.564710
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	−10.0000	−0.622573
$259$	0	0
$260$	0	0
$261$	0	0
$262$	17.0000	1.05026
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−5.00000	−0.307729
$265$	0	0
$266$	0	0
$267$	−10.0000	−0.611990
$268$	−6.00000	−0.366508
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	4.00000	0.241649
$275$	0	0
$276$	−3.00000	−0.180579
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	8.00000	0.479808
$279$	6.00000	0.359211
$280$	0	0
$281$	−11.0000	−0.656205	−0.328102	−	0.944642i	$-0.606409\pi$
$281$	−11.0000	−0.656205	−0.328102	+	0.944642i	$0.606409\pi$
$282$	−13.0000	−0.774139
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	−5.00000	−0.295656
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	8.00000	0.468968
$292$	4.00000	0.234082
$293$	1.00000	0.0584206	0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000	0.0584206	0.0292103	+	0.999573i	$0.490701\pi$
$294$	0	0
$295$	0	0
$296$	5.00000	0.290619
$297$	−5.00000	−0.290129
$298$	−6.00000	−0.347571
$299$	−3.00000	−0.173494
$300$	0	0
$301$	0	0
$302$	−22.0000	−1.26596
$303$	−8.00000	−0.459588
$304$	−7.00000	−0.401478
$305$	0	0
$306$	−2.00000	−0.114332
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	26.0000	1.47432	0.737162	−	0.675716i	$-0.236165\pi$
$311$	26.0000	1.47432	0.737162	+	0.675716i	$0.236165\pi$
$312$	1.00000	0.0566139
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	13.0000	0.733632
$315$	0	0
$316$	−14.0000	−0.787562
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	1.00000	0.0560772
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	14.0000	0.778981
$324$	1.00000	0.0555556
$325$	0	0
$326$	12.0000	0.664619
$327$	−18.0000	−0.995402
$328$	9.00000	0.496942
$329$	0	0
$330$	0	0
$331$	−15.0000	−0.824475	−0.412237	−	0.911077i	$-0.635253\pi$
$331$	−15.0000	−0.824475	−0.412237	+	0.911077i	$0.635253\pi$
$332$	−10.0000	−0.548821
$333$	5.00000	0.273998
$334$	−19.0000	−1.03963
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−12.0000	−0.652714
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−30.0000	−1.62459
$342$	−7.00000	−0.378517
$343$	0	0
$344$	−10.0000	−0.539164
$345$	0	0
$346$	7.00000	0.376322
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−5.00000	−0.266501
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−11.0000	−0.581368
$359$	−28.0000	−1.47778	−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000	−1.47778	−0.738892	+	0.673824i	$0.764651\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	2.00000	0.105118
$363$	14.0000	0.734809
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	−37.0000	−1.93138	−0.965692	−	0.259690i	$-0.916380\pi$
$367$	−37.0000	−1.93138	−0.965692	+	0.259690i	$0.916380\pi$
$368$	−3.00000	−0.156386
$369$	9.00000	0.468521
$370$	0	0
$371$	0	0
$372$	6.00000	0.311086
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	10.0000	0.517088
$375$	0	0
$376$	−13.0000	−0.670424
$377$	0	0
$378$	0	0
$379$	−1.00000	−0.0513665	−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000	−0.0513665	−0.0256833	+	0.999670i	$0.508176\pi$
$380$	0	0
$381$	−9.00000	−0.461084
$382$	−16.0000	−0.818631
$383$	−9.00000	−0.459879	−0.229939	−	0.973205i	$-0.573853\pi$
$383$	−9.00000	−0.459879	−0.229939	+	0.973205i	$0.573853\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	18.0000	0.916176
$387$	−10.0000	−0.508329
$388$	8.00000	0.406138
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	17.0000	0.857537
$394$	27.0000	1.36024
$395$	0	0
$396$	−5.00000	−0.251259
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	14.0000	0.701757
$399$	0	0
$400$	0	0
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	−6.00000	−0.299253
$403$	6.00000	0.298881
$404$	−8.00000	−0.398015
$405$	0	0
$406$	0	0
$407$	−25.0000	−1.23920
$408$	−2.00000	−0.0990148
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	0	0
$413$	0	0
$414$	−3.00000	−0.147442
$415$	0	0
$416$	1.00000	0.0490290
$417$	8.00000	0.391762
$418$	35.0000	1.71191
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	19.0000	0.924906
$423$	−13.0000	−0.632082
$424$	1.00000	0.0485643
$425$	0	0
$426$	−2.00000	−0.0969003
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−5.00000	−0.241402
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	1.00000	0.0481125
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	0	0
$435$	0	0
$436$	−18.0000	−0.862044
$437$	21.0000	1.00457
$438$	4.00000	0.191127
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	−2.00000	−0.0951303
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	5.00000	0.237289
$445$	0	0
$446$	16.0000	0.757622
$447$	−6.00000	−0.283790
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	−45.0000	−2.11897
$452$	−6.00000	−0.282216
$453$	−22.0000	−1.03365
$454$	14.0000	0.657053
$455$	0	0
$456$	−7.00000	−0.327805
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	4.00000	0.186908
$459$	−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
$463$	15.0000	0.697109	0.348555	−	0.937288i	$-0.386673\pi$
$463$	15.0000	0.697109	0.348555	+	0.937288i	$0.386673\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.00000	0.0925490	0.0462745	−	0.998929i	$-0.485265\pi$
$467$	2.00000	0.0925490	0.0462745	+	0.998929i	$0.485265\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	0	0
$471$	13.0000	0.599008
$472$	−4.00000	−0.184115
$473$	50.0000	2.29900
$474$	−14.0000	−0.643041
$475$	0	0
$476$	0	0
$477$	1.00000	0.0457869
$478$	20.0000	0.914779
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	5.00000	0.227980
$482$	1.00000	0.0455488
$483$	0	0
$484$	14.0000	0.636364
$485$	0	0
$486$	1.00000	0.0453609
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	2.00000	0.0905357
$489$	12.0000	0.542659
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	9.00000	0.405751
$493$	0	0
$494$	−7.00000	−0.314945
$495$	0	0
$496$	6.00000	0.269408
$497$	0	0
$498$	−10.0000	−0.448111
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	−19.0000	−0.848857
$502$	3.00000	0.133897
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	0	0
$506$	15.0000	0.666831
$507$	−12.0000	−0.532939
$508$	−9.00000	−0.399310
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−7.00000	−0.309058
$514$	−10.0000	−0.441081
$515$	0	0
$516$	−10.0000	−0.440225
$517$	65.0000	2.85870
$518$	0	0
$519$	7.00000	0.307266
$520$	0	0
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	0	0
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	17.0000	0.742648
$525$	0	0
$526$	−24.0000	−1.04645
$527$	−12.0000	−0.522728
$528$	−5.00000	−0.217597
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	9.00000	0.389833
$534$	−10.0000	−0.432742
$535$	0	0
$536$	−6.00000	−0.259161
$537$	−11.0000	−0.474685
$538$	−14.0000	−0.603583
$539$	0	0
$540$	0	0
$541$	4.00000	0.171973	0.0859867	−	0.996296i	$-0.472596\pi$
$541$	4.00000	0.171973	0.0859867	+	0.996296i	$0.472596\pi$
$542$	−8.00000	−0.343629
$543$	2.00000	0.0858282
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	14.0000	0.598597	0.299298	−	0.954160i	$-0.403247\pi$
$547$	14.0000	0.598597	0.299298	+	0.954160i	$0.403247\pi$
$548$	4.00000	0.170872
$549$	2.00000	0.0853579
$550$	0	0
$551$	0	0
$552$	−3.00000	−0.127688
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	8.00000	0.339276
$557$	39.0000	1.65248	0.826242	−	0.563316i	$-0.190475\pi$
$557$	39.0000	1.65248	0.826242	+	0.563316i	$0.190475\pi$
$558$	6.00000	0.254000
$559$	−10.0000	−0.422955
$560$	0	0
$561$	10.0000	0.422200
$562$	−11.0000	−0.464007
$563$	30.0000	1.26435	0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000	1.26435	0.632175	+	0.774826i	$0.282163\pi$
$564$	−13.0000	−0.547399
$565$	0	0
$566$	26.0000	1.09286
$567$	0	0
$568$	−2.00000	−0.0839181
$569$	−3.00000	−0.125767	−0.0628833	−	0.998021i	$-0.520030\pi$
$569$	−3.00000	−0.125767	−0.0628833	+	0.998021i	$0.520030\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	−5.00000	−0.209061
$573$	−16.0000	−0.668410
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	24.0000	0.999133	0.499567	−	0.866276i	$-0.333493\pi$
$577$	24.0000	0.999133	0.499567	+	0.866276i	$0.333493\pi$
$578$	−13.0000	−0.540729
$579$	18.0000	0.748054
$580$	0	0
$581$	0	0
$582$	8.00000	0.331611
$583$	−5.00000	−0.207079
$584$	4.00000	0.165521
$585$	0	0
$586$	1.00000	0.0413096
$587$	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
$589$	−42.0000	−1.73058
$590$	0	0
$591$	27.0000	1.11063
$592$	5.00000	0.205499
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	−5.00000	−0.205152
$595$	0	0
$596$	−6.00000	−0.245770
$597$	14.0000	0.572982
$598$	−3.00000	−0.122679
$599$	−28.0000	−1.14405	−0.572024	−	0.820237i	$-0.693842\pi$
$599$	−28.0000	−1.14405	−0.572024	+	0.820237i	$0.693842\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	−6.00000	−0.244339
$604$	−22.0000	−0.895167
$605$	0	0
$606$	−8.00000	−0.324978
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	−7.00000	−0.283887
$609$	0	0
$610$	0	0
$611$	−13.0000	−0.525924
$612$	−2.00000	−0.0808452
$613$	−19.0000	−0.767403	−0.383701	−	0.923457i	$-0.625351\pi$
$613$	−19.0000	−0.767403	−0.383701	+	0.923457i	$0.625351\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	−15.0000	−0.602901	−0.301450	−	0.953482i	$-0.597471\pi$
$619$	−15.0000	−0.602901	−0.301450	+	0.953482i	$0.597471\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	26.0000	1.04251
$623$	0	0
$624$	1.00000	0.0400320
$625$	0	0
$626$	−10.0000	−0.399680
$627$	35.0000	1.39777
$628$	13.0000	0.518756
$629$	−10.0000	−0.398726
$630$	0	0
$631$	18.0000	0.716569	0.358284	−	0.933613i	$-0.383362\pi$
$631$	18.0000	0.716569	0.358284	+	0.933613i	$0.383362\pi$
$632$	−14.0000	−0.556890
$633$	19.0000	0.755182
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	1.00000	0.0396526
$637$	0	0
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	−12.0000	−0.473602
$643$	38.0000	1.49857	0.749287	−	0.662246i	$-0.230396\pi$
$643$	38.0000	1.49857	0.749287	+	0.662246i	$0.230396\pi$
$644$	0	0
$645$	0	0
$646$	14.0000	0.550823
$647$	−1.00000	−0.0393141	−0.0196570	−	0.999807i	$-0.506257\pi$
$647$	−1.00000	−0.0393141	−0.0196570	+	0.999807i	$0.506257\pi$
$648$	1.00000	0.0392837
$649$	20.0000	0.785069
$650$	0	0
$651$	0	0
$652$	12.0000	0.469956
$653$	−5.00000	−0.195665	−0.0978326	−	0.995203i	$-0.531191\pi$
$653$	−5.00000	−0.195665	−0.0978326	+	0.995203i	$0.531191\pi$
$654$	−18.0000	−0.703856
$655$	0	0
$656$	9.00000	0.351391
$657$	4.00000	0.156055
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−40.0000	−1.55582	−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000	−1.55582	−0.777910	+	0.628376i	$0.783720\pi$
$662$	−15.0000	−0.582992
$663$	−2.00000	−0.0776736
$664$	−10.0000	−0.388075
$665$	0	0
$666$	5.00000	0.193746
$667$	0	0
$668$	−19.0000	−0.735132
$669$	16.0000	0.618596
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	−36.0000	−1.38770	−0.693849	−	0.720121i	$-0.744086\pi$
$673$	−36.0000	−1.38770	−0.693849	+	0.720121i	$0.744086\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−12.0000	−0.461538
$677$	33.0000	1.26829	0.634147	−	0.773213i	$-0.281352\pi$
$677$	33.0000	1.26829	0.634147	+	0.773213i	$0.281352\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	0	0
$681$	14.0000	0.536481
$682$	−30.0000	−1.14876
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	−7.00000	−0.267652
$685$	0	0
$686$	0	0
$687$	4.00000	0.152610
$688$	−10.0000	−0.381246
$689$	1.00000	0.0380970
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	7.00000	0.266100
$693$	0	0
$694$	16.0000	0.607352
$695$	0	0
$696$	0	0
$697$	−18.0000	−0.681799
$698$	24.0000	0.908413
$699$	0	0
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	1.00000	0.0377426
$703$	−35.0000	−1.32005
$704$	−5.00000	−0.188445
$705$	0	0
$706$	0	0
$707$	0	0
$708$	−4.00000	−0.150329
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	−14.0000	−0.525041
$712$	−10.0000	−0.374766
$713$	−18.0000	−0.674105
$714$	0	0
$715$	0	0
$716$	−11.0000	−0.411089
$717$	20.0000	0.746914
$718$	−28.0000	−1.04495
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	0	0
$721$	0	0
$722$	30.0000	1.11648
$723$	1.00000	0.0371904
$724$	2.00000	0.0743294
$725$	0	0
$726$	14.0000	0.519589
$727$	53.0000	1.96566	0.982831	−	0.184510i	$-0.0590699\pi$
$727$	53.0000	1.96566	0.982831	+	0.184510i	$0.0590699\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	20.0000	0.739727
$732$	2.00000	0.0739221
$733$	−21.0000	−0.775653	−0.387826	−	0.921732i	$-0.626774\pi$
$733$	−21.0000	−0.775653	−0.387826	+	0.921732i	$0.626774\pi$
$734$	−37.0000	−1.36569
$735$	0	0
$736$	−3.00000	−0.110581
$737$	30.0000	1.10506
$738$	9.00000	0.331295
$739$	47.0000	1.72892	0.864461	−	0.502699i	$-0.167660\pi$
$739$	47.0000	1.72892	0.864461	+	0.502699i	$0.167660\pi$
$740$	0	0
$741$	−7.00000	−0.257151
$742$	0	0
$743$	31.0000	1.13728	0.568640	−	0.822587i	$-0.307470\pi$
$743$	31.0000	1.13728	0.568640	+	0.822587i	$0.307470\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	−6.00000	−0.219676
$747$	−10.0000	−0.365881
$748$	10.0000	0.365636
$749$	0	0
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	−13.0000	−0.474061
$753$	3.00000	0.109326
$754$	0	0
$755$	0	0
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	−1.00000	−0.0363216
$759$	15.0000	0.544466
$760$	0	0
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	−9.00000	−0.326036
$763$	0	0
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−9.00000	−0.325183
$767$	−4.00000	−0.144432
$768$	1.00000	0.0360844
$769$	−51.0000	−1.83911	−0.919554	−	0.392965i	$-0.871449\pi$
$769$	−51.0000	−1.83911	−0.919554	+	0.392965i	$0.871449\pi$
$770$	0	0
$771$	−10.0000	−0.360141
$772$	18.0000	0.647834
$773$	−37.0000	−1.33080	−0.665399	−	0.746488i	$-0.731738\pi$
$773$	−37.0000	−1.33080	−0.665399	+	0.746488i	$0.731738\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	8.00000	0.287183
$777$	0	0
$778$	6.00000	0.215110
$779$	−63.0000	−2.25721
$780$	0	0
$781$	10.0000	0.357828
$782$	6.00000	0.214560
$783$	0	0
$784$	0	0
$785$	0	0
$786$	17.0000	0.606370
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	27.0000	0.961835
$789$	−24.0000	−0.854423
$790$	0	0
$791$	0	0
$792$	−5.00000	−0.177667
$793$	2.00000	0.0710221
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	14.0000	0.496217
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	26.0000	0.919814
$800$	0	0
$801$	−10.0000	−0.353333
$802$	−27.0000	−0.953403
$803$	−20.0000	−0.705785
$804$	−6.00000	−0.211604
$805$	0	0
$806$	6.00000	0.211341
$807$	−14.0000	−0.492823
$808$	−8.00000	−0.281439
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	−11.0000	−0.386262	−0.193131	−	0.981173i	$-0.561864\pi$
$811$	−11.0000	−0.386262	−0.193131	+	0.981173i	$0.561864\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	−25.0000	−0.876250
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	70.0000	2.44899
$818$	−10.0000	−0.349642
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	4.00000	0.139516
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	−3.00000	−0.104257
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	1.00000	0.0346688
$833$	0	0
$834$	8.00000	0.277017
$835$	0	0
$836$	35.0000	1.21050
$837$	6.00000	0.207390
$838$	−3.00000	−0.103633
$839$	−2.00000	−0.0690477	−0.0345238	−	0.999404i	$-0.510991\pi$
$839$	−2.00000	−0.0690477	−0.0345238	+	0.999404i	$0.510991\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−20.0000	−0.689246
$843$	−11.0000	−0.378860
$844$	19.0000	0.654007
$845$	0	0
$846$	−13.0000	−0.446949
$847$	0	0
$848$	1.00000	0.0343401
$849$	26.0000	0.892318
$850$	0	0
$851$	−15.0000	−0.514193
$852$	−2.00000	−0.0685189
$853$	−49.0000	−1.67773	−0.838864	−	0.544341i	$-0.816780\pi$
$853$	−49.0000	−1.67773	−0.838864	+	0.544341i	$0.816780\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−56.0000	−1.91292	−0.956462	−	0.291858i	$-0.905727\pi$
$857$	−56.0000	−1.91292	−0.956462	+	0.291858i	$0.905727\pi$
$858$	−5.00000	−0.170697
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	0	0
$862$	18.0000	0.613082
$863$	−15.0000	−0.510606	−0.255303	−	0.966861i	$-0.582175\pi$
$863$	−15.0000	−0.510606	−0.255303	+	0.966861i	$0.582175\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	4.00000	0.135926
$867$	−13.0000	−0.441503
$868$	0	0
$869$	70.0000	2.37459
$870$	0	0
$871$	−6.00000	−0.203302
$872$	−18.0000	−0.609557
$873$	8.00000	0.270759
$874$	21.0000	0.710336
$875$	0	0
$876$	4.00000	0.135147
$877$	27.0000	0.911725	0.455863	−	0.890050i	$-0.349331\pi$
$877$	27.0000	0.911725	0.455863	+	0.890050i	$0.349331\pi$
$878$	0	0
$879$	1.00000	0.0337292
$880$	0	0
$881$	3.00000	0.101073	0.0505363	−	0.998722i	$-0.483907\pi$
$881$	3.00000	0.101073	0.0505363	+	0.998722i	$0.483907\pi$
$882$	0	0
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	6.00000	0.201574
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	5.00000	0.167789
$889$	0	0
$890$	0	0
$891$	−5.00000	−0.167506
$892$	16.0000	0.535720
$893$	91.0000	3.04520
$894$	−6.00000	−0.200670
$895$	0	0
$896$	0	0
$897$	−3.00000	−0.100167
$898$	9.00000	0.300334
$899$	0	0
$900$	0	0
$901$	−2.00000	−0.0666297
$902$	−45.0000	−1.49834
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−22.0000	−0.730901
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	14.0000	0.464606
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	−7.00000	−0.231793
$913$	50.0000	1.65476
$914$	38.0000	1.25693
$915$	0	0
$916$	4.00000	0.132164
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	12.0000	0.395199
$923$	−2.00000	−0.0658308
$924$	0	0
$925$	0	0
$926$	15.0000	0.492931
$927$	0	0
$928$	0	0
$929$	−33.0000	−1.08269	−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	26.0000	0.851202
$934$	2.00000	0.0654420
$935$	0	0
$936$	1.00000	0.0326860
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	13.0000	0.423563
$943$	−27.0000	−0.879241
$944$	−4.00000	−0.130189
$945$	0	0
$946$	50.0000	1.62564
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−14.0000	−0.454699
$949$	4.00000	0.129845
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	1.00000	0.0323762
$955$	0	0
$956$	20.0000	0.646846
$957$	0	0
$958$	−8.00000	−0.258468
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	5.00000	0.161206
$963$	−12.0000	−0.386695
$964$	1.00000	0.0322078
$965$	0	0
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	14.0000	0.449977
$969$	14.0000	0.449745
$970$	0	0
$971$	−39.0000	−1.25157	−0.625785	−	0.779996i	$-0.715221\pi$
$971$	−39.0000	−1.25157	−0.625785	+	0.779996i	$0.715221\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−24.0000	−0.769010
$975$	0	0
$976$	2.00000	0.0640184
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	12.0000	0.383718
$979$	50.0000	1.59801
$980$	0	0
$981$	−18.0000	−0.574696
$982$	24.0000	0.765871
$983$	33.0000	1.05254	0.526268	−	0.850319i	$-0.323591\pi$
$983$	33.0000	1.05254	0.526268	+	0.850319i	$0.323591\pi$
$984$	9.00000	0.286910
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−7.00000	−0.222700
$989$	30.0000	0.953945
$990$	0	0
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	6.00000	0.190500
$993$	−15.0000	−0.476011
$994$	0	0
$995$	0	0
$996$	−10.0000	−0.316862
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−28.0000	−0.886325
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.ch.1.1		1
5.2	odd	4		1470.2.g.a.589.2		2
5.3	odd	4		1470.2.g.a.589.1		2
5.4	even	2		7350.2.a.b.1.1		1
7.3	odd	6		1050.2.i.f.751.1		2
7.5	odd	6		1050.2.i.f.151.1		2
7.6	odd	2		7350.2.a.bn.1.1		1
35.2	odd	12		1470.2.n.i.949.2		4
35.3	even	12		210.2.n.a.79.2	yes	4
35.12	even	12		210.2.n.a.109.2	yes	4
35.13	even	4		1470.2.g.f.589.1		2
35.17	even	12		210.2.n.a.79.1	✓	4
35.18	odd	12		1470.2.n.i.79.2		4
35.19	odd	6		1050.2.i.o.151.1		2
35.23	odd	12		1470.2.n.i.949.1		4
35.24	odd	6		1050.2.i.o.751.1		2
35.27	even	4		1470.2.g.f.589.2		2
35.32	odd	12		1470.2.n.i.79.1		4
35.33	even	12		210.2.n.a.109.1	yes	4
35.34	odd	2		7350.2.a.t.1.1		1
105.17	odd	12		630.2.u.c.289.2		4
105.38	odd	12		630.2.u.c.289.1		4
105.47	odd	12		630.2.u.c.109.1		4
105.68	odd	12		630.2.u.c.109.2		4
140.3	odd	12		1680.2.di.a.289.2		4
140.47	odd	12		1680.2.di.a.529.2		4
140.87	odd	12		1680.2.di.a.289.1		4
140.103	odd	12		1680.2.di.a.529.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.n.a.79.1	✓	4	35.17	even	12
210.2.n.a.79.2	yes	4	35.3	even	12
210.2.n.a.109.1	yes	4	35.33	even	12
210.2.n.a.109.2	yes	4	35.12	even	12
630.2.u.c.109.1		4	105.47	odd	12
630.2.u.c.109.2		4	105.68	odd	12
630.2.u.c.289.1		4	105.38	odd	12
630.2.u.c.289.2		4	105.17	odd	12
1050.2.i.f.151.1		2	7.5	odd	6
1050.2.i.f.751.1		2	7.3	odd	6
1050.2.i.o.151.1		2	35.19	odd	6
1050.2.i.o.751.1		2	35.24	odd	6
1470.2.g.a.589.1		2	5.3	odd	4
1470.2.g.a.589.2		2	5.2	odd	4
1470.2.g.f.589.1		2	35.13	even	4
1470.2.g.f.589.2		2	35.27	even	4
1470.2.n.i.79.1		4	35.32	odd	12
1470.2.n.i.79.2		4	35.18	odd	12
1470.2.n.i.949.1		4	35.23	odd	12
1470.2.n.i.949.2		4	35.2	odd	12
1680.2.di.a.289.1		4	140.87	odd	12
1680.2.di.a.289.2		4	140.3	odd	12
1680.2.di.a.529.1		4	140.103	odd	12
1680.2.di.a.529.2		4	140.47	odd	12
7350.2.a.b.1.1		1	5.4	even	2
7350.2.a.t.1.1		1	35.34	odd	2
7350.2.a.bn.1.1		1	7.6	odd	2
7350.2.a.ch.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

Introduction

L-functions

Modular forms

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Fields

Representations

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