LMFDB - Embedded newform 350.6.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,6,Mod(1,350)] 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("350.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,4,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.1343369345$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	350.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} -162.000 q^{9} -187.000 q^{11} +144.000 q^{12} -627.000 q^{13} +196.000 q^{14} +256.000 q^{16} -1813.00 q^{17} -648.000 q^{18} +258.000 q^{19} +441.000 q^{21} -748.000 q^{22} -2970.00 q^{23} +576.000 q^{24} -2508.00 q^{26} -3645.00 q^{27} +784.000 q^{28} +1299.00 q^{29} +1916.00 q^{31} +1024.00 q^{32} -1683.00 q^{33} -7252.00 q^{34} -2592.00 q^{36} -6578.00 q^{37} +1032.00 q^{38} -5643.00 q^{39} +6676.00 q^{41} +1764.00 q^{42} -3178.00 q^{43} -2992.00 q^{44} -11880.0 q^{46} +22001.0 q^{47} +2304.00 q^{48} +2401.00 q^{49} -16317.0 q^{51} -10032.0 q^{52} -26168.0 q^{53} -14580.0 q^{54} +3136.00 q^{56} +2322.00 q^{57} +5196.00 q^{58} +3932.00 q^{59} -48740.0 q^{61} +7664.00 q^{62} -7938.00 q^{63} +4096.00 q^{64} -6732.00 q^{66} +44832.0 q^{67} -29008.0 q^{68} -26730.0 q^{69} +63736.0 q^{71} -10368.0 q^{72} -60470.0 q^{73} -26312.0 q^{74} +4128.00 q^{76} -9163.00 q^{77} -22572.0 q^{78} -43721.0 q^{79} +6561.00 q^{81} +26704.0 q^{82} -97276.0 q^{83} +7056.00 q^{84} -12712.0 q^{86} +11691.0 q^{87} -11968.0 q^{88} +45560.0 q^{89} -30723.0 q^{91} -47520.0 q^{92} +17244.0 q^{93} +88004.0 q^{94} +9216.00 q^{96} +57295.0 q^{97} +9604.00 q^{98} +30294.0 q^{99} +O(q^{100})\)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	9.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	9.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	64.0000	0.353553
$9$	−162.000	−0.666667
$10$	0	0
$11$	−187.000	−0.465972	−0.232986	−	0.972480i	$-0.574850\pi$
$11$	−187.000	−0.465972	−0.232986	+	0.972480i	$0.574850\pi$
$12$	144.000	0.288675
$13$	−627.000	−1.02899	−0.514493	−	0.857495i	$-0.672020\pi$
$13$	−627.000	−1.02899	−0.514493	+	0.857495i	$0.672020\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	−1813.00	−1.52151	−0.760756	−	0.649038i	$-0.775172\pi$
$17$	−1813.00	−1.52151	−0.760756	+	0.649038i	$0.775172\pi$
$18$	−648.000	−0.471405
$19$	258.000	0.163959	0.0819796	−	0.996634i	$-0.473876\pi$
$19$	258.000	0.163959	0.0819796	+	0.996634i	$0.473876\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	−748.000	−0.329492
$23$	−2970.00	−1.17068	−0.585338	−	0.810789i	$-0.699038\pi$
$23$	−2970.00	−1.17068	−0.585338	+	0.810789i	$0.699038\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	−2508.00	−0.727602
$27$	−3645.00	−0.962250
$28$	784.000	0.188982
$29$	1299.00	0.286823	0.143412	−	0.989663i	$-0.454193\pi$
$29$	1299.00	0.286823	0.143412	+	0.989663i	$0.454193\pi$
$30$	0	0
$31$	1916.00	0.358089	0.179045	−	0.983841i	$-0.442699\pi$
$31$	1916.00	0.358089	0.179045	+	0.983841i	$0.442699\pi$
$32$	1024.00	0.176777
$33$	−1683.00	−0.269029
$34$	−7252.00	−1.07587
$35$	0	0
$36$	−2592.00	−0.333333
$37$	−6578.00	−0.789932	−0.394966	−	0.918696i	$-0.629244\pi$
$37$	−6578.00	−0.789932	−0.394966	+	0.918696i	$0.629244\pi$
$38$	1032.00	0.115937
$39$	−5643.00	−0.594085
$40$	0	0
$41$	6676.00	0.620236	0.310118	−	0.950698i	$-0.399632\pi$
$41$	6676.00	0.620236	0.310118	+	0.950698i	$0.399632\pi$
$42$	1764.00	0.154303
$43$	−3178.00	−0.262109	−0.131055	−	0.991375i	$-0.541836\pi$
$43$	−3178.00	−0.262109	−0.131055	+	0.991375i	$0.541836\pi$
$44$	−2992.00	−0.232986
$45$	0	0
$46$	−11880.0	−0.827793
$47$	22001.0	1.45277	0.726387	−	0.687286i	$-0.241198\pi$
$47$	22001.0	1.45277	0.726387	+	0.687286i	$0.241198\pi$
$48$	2304.00	0.144338
$49$	2401.00	0.142857
$50$	0	0
$51$	−16317.0	−0.878446
$52$	−10032.0	−0.514493
$53$	−26168.0	−1.27962	−0.639810	−	0.768533i	$-0.720987\pi$
$53$	−26168.0	−1.27962	−0.639810	+	0.768533i	$0.720987\pi$
$54$	−14580.0	−0.680414
$55$	0	0
$56$	3136.00	0.133631
$57$	2322.00	0.0946619
$58$	5196.00	0.202815
$59$	3932.00	0.147056	0.0735281	−	0.997293i	$-0.476574\pi$
$59$	3932.00	0.147056	0.0735281	+	0.997293i	$0.476574\pi$
$60$	0	0
$61$	−48740.0	−1.67711	−0.838554	−	0.544819i	$-0.816598\pi$
$61$	−48740.0	−1.67711	−0.838554	+	0.544819i	$0.816598\pi$
$62$	7664.00	0.253207
$63$	−7938.00	−0.251976
$64$	4096.00	0.125000
$65$	0	0
$66$	−6732.00	−0.190232
$67$	44832.0	1.22012	0.610058	−	0.792357i	$-0.291146\pi$
$67$	44832.0	1.22012	0.610058	+	0.792357i	$0.291146\pi$
$68$	−29008.0	−0.760756
$69$	−26730.0	−0.675890
$70$	0	0
$71$	63736.0	1.50051	0.750255	−	0.661148i	$-0.229931\pi$
$71$	63736.0	1.50051	0.750255	+	0.661148i	$0.229931\pi$
$72$	−10368.0	−0.235702
$73$	−60470.0	−1.32811	−0.664053	−	0.747685i	$-0.731165\pi$
$73$	−60470.0	−1.32811	−0.664053	+	0.747685i	$0.731165\pi$
$74$	−26312.0	−0.558566
$75$	0	0
$76$	4128.00	0.0819796
$77$	−9163.00	−0.176121
$78$	−22572.0	−0.420081
$79$	−43721.0	−0.788174	−0.394087	−	0.919073i	$-0.628939\pi$
$79$	−43721.0	−0.788174	−0.394087	+	0.919073i	$0.628939\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	26704.0	0.438573
$83$	−97276.0	−1.54992	−0.774962	−	0.632008i	$-0.782231\pi$
$83$	−97276.0	−1.54992	−0.774962	+	0.632008i	$0.782231\pi$
$84$	7056.00	0.109109
$85$	0	0
$86$	−12712.0	−0.185339
$87$	11691.0	0.165597
$88$	−11968.0	−0.164746
$89$	45560.0	0.609689	0.304845	−	0.952402i	$-0.401395\pi$
$89$	45560.0	0.609689	0.304845	+	0.952402i	$0.401395\pi$
$90$	0	0
$91$	−30723.0	−0.388920
$92$	−47520.0	−0.585338
$93$	17244.0	0.206743
$94$	88004.0	1.02727
$95$	0	0
$96$	9216.00	0.102062
$97$	57295.0	0.618283	0.309142	−	0.951016i	$-0.399958\pi$
$97$	57295.0	0.618283	0.309142	+	0.951016i	$0.399958\pi$
$98$	9604.00	0.101015
$99$	30294.0	0.310648
$100$	0	0
$101$	−44970.0	−0.438651	−0.219326	−	0.975652i	$-0.570386\pi$
$101$	−44970.0	−0.438651	−0.219326	+	0.975652i	$0.570386\pi$
$102$	−65268.0	−0.621155
$103$	101405.	0.941817	0.470908	−	0.882182i	$-0.343926\pi$
$103$	101405.	0.941817	0.470908	+	0.882182i	$0.343926\pi$
$104$	−40128.0	−0.363801
$105$	0	0
$106$	−104672.	−0.904828
$107$	166002.	1.40170	0.700848	−	0.713311i	$-0.252805\pi$
$107$	166002.	1.40170	0.700848	+	0.713311i	$0.252805\pi$
$108$	−58320.0	−0.481125
$109$	8289.00	0.0668245	0.0334123	−	0.999442i	$-0.489363\pi$
$109$	8289.00	0.0668245	0.0334123	+	0.999442i	$0.489363\pi$
$110$	0	0
$111$	−59202.0	−0.456067
$112$	12544.0	0.0944911
$113$	−263206.	−1.93910	−0.969549	−	0.244898i	$-0.921245\pi$
$113$	−263206.	−1.93910	−0.969549	+	0.244898i	$0.921245\pi$
$114$	9288.00	0.0669360
$115$	0	0
$116$	20784.0	0.143412
$117$	101574.	0.685990
$118$	15728.0	0.103984
$119$	−88837.0	−0.575078
$120$	0	0
$121$	−126082.	−0.782870
$122$	−194960.	−1.18589
$123$	60084.0	0.358093
$124$	30656.0	0.179045
$125$	0	0
$126$	−31752.0	−0.178174
$127$	−30052.0	−0.165335	−0.0826674	−	0.996577i	$-0.526344\pi$
$127$	−30052.0	−0.165335	−0.0826674	+	0.996577i	$0.526344\pi$
$128$	16384.0	0.0883883
$129$	−28602.0	−0.151329
$130$	0	0
$131$	−120050.	−0.611201	−0.305600	−	0.952160i	$-0.598857\pi$
$131$	−120050.	−0.611201	−0.305600	+	0.952160i	$0.598857\pi$
$132$	−26928.0	−0.134515
$133$	12642.0	0.0619707
$134$	179328.	0.862752
$135$	0	0
$136$	−116032.	−0.537936
$137$	−31776.0	−0.144643	−0.0723216	−	0.997381i	$-0.523041\pi$
$137$	−31776.0	−0.144643	−0.0723216	+	0.997381i	$0.523041\pi$
$138$	−106920.	−0.477927
$139$	−200162.	−0.878708	−0.439354	−	0.898314i	$-0.644793\pi$
$139$	−200162.	−0.878708	−0.439354	+	0.898314i	$0.644793\pi$
$140$	0	0
$141$	198009.	0.838759
$142$	254944.	1.06102
$143$	117249.	0.479478
$144$	−41472.0	−0.166667
$145$	0	0
$146$	−241880.	−0.939113
$147$	21609.0	0.0824786
$148$	−105248.	−0.394966
$149$	309642.	1.14260	0.571300	−	0.820741i	$-0.306439\pi$
$149$	309642.	1.14260	0.571300	+	0.820741i	$0.306439\pi$
$150$	0	0
$151$	−208657.	−0.744716	−0.372358	−	0.928089i	$-0.621451\pi$
$151$	−208657.	−0.744716	−0.372358	+	0.928089i	$0.621451\pi$
$152$	16512.0	0.0579683
$153$	293706.	1.01434
$154$	−36652.0	−0.124536
$155$	0	0
$156$	−90288.0	−0.297042
$157$	−36010.0	−0.116593	−0.0582967	−	0.998299i	$-0.518567\pi$
$157$	−36010.0	−0.116593	−0.0582967	+	0.998299i	$0.518567\pi$
$158$	−174884.	−0.557324
$159$	−235512.	−0.738789
$160$	0	0
$161$	−145530.	−0.442474
$162$	26244.0	0.0785674
$163$	−175670.	−0.517879	−0.258940	−	0.965893i	$-0.583373\pi$
$163$	−175670.	−0.517879	−0.258940	+	0.965893i	$0.583373\pi$
$164$	106816.	0.310118
$165$	0	0
$166$	−389104.	−1.09596
$167$	157413.	0.436767	0.218383	−	0.975863i	$-0.429922\pi$
$167$	157413.	0.436767	0.218383	+	0.975863i	$0.429922\pi$
$168$	28224.0	0.0771517
$169$	21836.0	0.0588107
$170$	0	0
$171$	−41796.0	−0.109306
$172$	−50848.0	−0.131055
$173$	23471.0	0.0596233	0.0298117	−	0.999556i	$-0.490509\pi$
$173$	23471.0	0.0596233	0.0298117	+	0.999556i	$0.490509\pi$
$174$	46764.0	0.117095
$175$	0	0
$176$	−47872.0	−0.116493
$177$	35388.0	0.0849030
$178$	182240.	0.431116
$179$	612228.	1.42817	0.714086	−	0.700058i	$-0.246842\pi$
$179$	612228.	1.42817	0.714086	+	0.700058i	$0.246842\pi$
$180$	0	0
$181$	528832.	1.19983	0.599917	−	0.800062i	$-0.295200\pi$
$181$	528832.	1.19983	0.599917	+	0.800062i	$0.295200\pi$
$182$	−122892.	−0.275008
$183$	−438660.	−0.968279
$184$	−190080.	−0.413897
$185$	0	0
$186$	68976.0	0.146189
$187$	339031.	0.708982
$188$	352016.	0.726387
$189$	−178605.	−0.363696
$190$	0	0
$191$	540369.	1.07178	0.535892	−	0.844287i	$-0.319976\pi$
$191$	540369.	1.07178	0.535892	+	0.844287i	$0.319976\pi$
$192$	36864.0	0.0721688
$193$	−960320.	−1.85576	−0.927882	−	0.372874i	$-0.878372\pi$
$193$	−960320.	−1.85576	−0.927882	+	0.372874i	$0.878372\pi$
$194$	229180.	0.437192
$195$	0	0
$196$	38416.0	0.0714286
$197$	−761944.	−1.39881	−0.699403	−	0.714728i	$-0.746551\pi$
$197$	−761944.	−1.39881	−0.699403	+	0.714728i	$0.746551\pi$
$198$	121176.	0.219661
$199$	125084.	0.223908	0.111954	−	0.993713i	$-0.464289\pi$
$199$	125084.	0.223908	0.111954	+	0.993713i	$0.464289\pi$
$200$	0	0
$201$	403488.	0.704434
$202$	−179880.	−0.310173
$203$	63651.0	0.108409
$204$	−261072.	−0.439223
$205$	0	0
$206$	405620.	0.665965
$207$	481140.	0.780451
$208$	−160512.	−0.257246
$209$	−48246.0	−0.0764004
$210$	0	0
$211$	627547.	0.970376	0.485188	−	0.874410i	$-0.338751\pi$
$211$	627547.	0.970376	0.485188	+	0.874410i	$0.338751\pi$
$212$	−418688.	−0.639810
$213$	573624.	0.866320
$214$	664008.	0.991149
$215$	0	0
$216$	−233280.	−0.340207
$217$	93884.0	0.135345
$218$	33156.0	0.0472521
$219$	−544230.	−0.766782
$220$	0	0
$221$	1.13675e6	1.56561
$222$	−236808.	−0.322488
$223$	1.22110e6	1.64433	0.822166	−	0.569248i	$-0.192766\pi$
$223$	1.22110e6	1.64433	0.822166	+	0.569248i	$0.192766\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	−1.05282e6	−1.37115
$227$	−390547.	−0.503047	−0.251524	−	0.967851i	$-0.580932\pi$
$227$	−390547.	−0.503047	−0.251524	+	0.967851i	$0.580932\pi$
$228$	37152.0	0.0473309
$229$	−712124.	−0.897360	−0.448680	−	0.893692i	$-0.648106\pi$
$229$	−712124.	−0.897360	−0.448680	+	0.893692i	$0.648106\pi$
$230$	0	0
$231$	−82467.0	−0.101683
$232$	83136.0	0.101407
$233$	−561576.	−0.677671	−0.338835	−	0.940846i	$-0.610033\pi$
$233$	−561576.	−0.677671	−0.338835	+	0.940846i	$0.610033\pi$
$234$	406296.	0.485068
$235$	0	0
$236$	62912.0	0.0735281
$237$	−393489.	−0.455053
$238$	−355348.	−0.406641
$239$	−1.36084e6	−1.54103	−0.770515	−	0.637421i	$-0.780001\pi$
$239$	−1.36084e6	−1.54103	−0.770515	+	0.637421i	$0.780001\pi$
$240$	0	0
$241$	530050.	0.587860	0.293930	−	0.955827i	$-0.405037\pi$
$241$	530050.	0.587860	0.293930	+	0.955827i	$0.405037\pi$
$242$	−504328.	−0.553573
$243$	944784.	1.02640
$244$	−779840.	−0.838554
$245$	0	0
$246$	240336.	0.253210
$247$	−161766.	−0.168712
$248$	122624.	0.126604
$249$	−875484.	−0.894849
$250$	0	0
$251$	990330.	0.992192	0.496096	−	0.868268i	$-0.334766\pi$
$251$	990330.	0.992192	0.496096	+	0.868268i	$0.334766\pi$
$252$	−127008.	−0.125988
$253$	555390.	0.545503
$254$	−120208.	−0.116909
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.81643e6	1.71548	0.857740	−	0.514083i	$-0.171868\pi$
$257$	1.81643e6	1.71548	0.857740	+	0.514083i	$0.171868\pi$
$258$	−114408.	−0.107006
$259$	−322322.	−0.298566
$260$	0	0
$261$	−210438.	−0.191215
$262$	−480200.	−0.432184
$263$	1.95847e6	1.74594	0.872968	−	0.487777i	$-0.162192\pi$
$263$	1.95847e6	1.74594	0.872968	+	0.487777i	$0.162192\pi$
$264$	−107712.	−0.0951162
$265$	0	0
$266$	50568.0	0.0438199
$267$	410040.	0.352004
$268$	717312.	0.610058
$269$	218034.	0.183715	0.0918573	−	0.995772i	$-0.470720\pi$
$269$	218034.	0.183715	0.0918573	+	0.995772i	$0.470720\pi$
$270$	0	0
$271$	1.26265e6	1.04438	0.522192	−	0.852828i	$-0.325114\pi$
$271$	1.26265e6	1.04438	0.522192	+	0.852828i	$0.325114\pi$
$272$	−464128.	−0.380378
$273$	−276507.	−0.224543
$274$	−127104.	−0.102278
$275$	0	0
$276$	−427680.	−0.337945
$277$	1.10264e6	0.863443	0.431721	−	0.902007i	$-0.357906\pi$
$277$	1.10264e6	0.863443	0.431721	+	0.902007i	$0.357906\pi$
$278$	−800648.	−0.621340
$279$	−310392.	−0.238726
$280$	0	0
$281$	−998213.	−0.754149	−0.377075	−	0.926183i	$-0.623070\pi$
$281$	−998213.	−0.754149	−0.377075	+	0.926183i	$0.623070\pi$
$282$	792036.	0.593092
$283$	386371.	0.286773	0.143387	−	0.989667i	$-0.454201\pi$
$283$	386371.	0.286773	0.143387	+	0.989667i	$0.454201\pi$
$284$	1.01978e6	0.750255
$285$	0	0
$286$	468996.	0.339042
$287$	327124.	0.234427
$288$	−165888.	−0.117851
$289$	1.86711e6	1.31500
$290$	0	0
$291$	515655.	0.356966
$292$	−967520.	−0.664053
$293$	783571.	0.533224	0.266612	−	0.963804i	$-0.414096\pi$
$293$	783571.	0.533224	0.266612	+	0.963804i	$0.414096\pi$
$294$	86436.0	0.0583212
$295$	0	0
$296$	−420992.	−0.279283
$297$	681615.	0.448382
$298$	1.23857e6	0.807940
$299$	1.86219e6	1.20461
$300$	0	0
$301$	−155722.	−0.0990681
$302$	−834628.	−0.526594
$303$	−404730.	−0.253255
$304$	66048.0	0.0409898
$305$	0	0
$306$	1.17482e6	0.717248
$307$	−2.81773e6	−1.70629	−0.853147	−	0.521670i	$-0.825309\pi$
$307$	−2.81773e6	−1.70629	−0.853147	+	0.521670i	$0.825309\pi$
$308$	−146608.	−0.0880604
$309$	912645.	0.543758
$310$	0	0
$311$	−847398.	−0.496806	−0.248403	−	0.968657i	$-0.579906\pi$
$311$	−847398.	−0.496806	−0.248403	+	0.968657i	$0.579906\pi$
$312$	−361152.	−0.210041
$313$	−364955.	−0.210561	−0.105281	−	0.994443i	$-0.533574\pi$
$313$	−364955.	−0.210561	−0.105281	+	0.994443i	$0.533574\pi$
$314$	−144040.	−0.0824440
$315$	0	0
$316$	−699536.	−0.394087
$317$	−1.93744e6	−1.08288	−0.541439	−	0.840740i	$-0.682120\pi$
$317$	−1.93744e6	−1.08288	−0.541439	+	0.840740i	$0.682120\pi$
$318$	−942048.	−0.522402
$319$	−242913.	−0.133652
$320$	0	0
$321$	1.49402e6	0.809270
$322$	−582120.	−0.312876
$323$	−467754.	−0.249466
$324$	104976.	0.0555556
$325$	0	0
$326$	−702680.	−0.366196
$327$	74601.0	0.0385812
$328$	427264.	0.219286
$329$	1.07805e6	0.549097
$330$	0	0
$331$	−61460.0	−0.0308335	−0.0154167	−	0.999881i	$-0.504907\pi$
$331$	−61460.0	−0.0308335	−0.0154167	+	0.999881i	$0.504907\pi$
$332$	−1.55642e6	−0.774962
$333$	1.06564e6	0.526621
$334$	629652.	0.308841
$335$	0	0
$336$	112896.	0.0545545
$337$	3.74116e6	1.79445	0.897225	−	0.441574i	$-0.145580\pi$
$337$	3.74116e6	1.79445	0.897225	+	0.441574i	$0.145580\pi$
$338$	87344.0	0.0415854
$339$	−2.36885e6	−1.11954
$340$	0	0
$341$	−358292.	−0.166860
$342$	−167184.	−0.0772911
$343$	117649.	0.0539949
$344$	−203392.	−0.0926697
$345$	0	0
$346$	93884.0	0.0421601
$347$	−211334.	−0.0942206	−0.0471103	−	0.998890i	$-0.515001\pi$
$347$	−211334.	−0.0942206	−0.0471103	+	0.998890i	$0.515001\pi$
$348$	187056.	0.0827987
$349$	3.39558e6	1.49228	0.746140	−	0.665789i	$-0.231905\pi$
$349$	3.39558e6	1.49228	0.746140	+	0.665789i	$0.231905\pi$
$350$	0	0
$351$	2.28542e6	0.990142
$352$	−191488.	−0.0823730
$353$	3.88094e6	1.65768	0.828838	−	0.559489i	$-0.189003\pi$
$353$	3.88094e6	1.65768	0.828838	+	0.559489i	$0.189003\pi$
$354$	141552.	0.0600355
$355$	0	0
$356$	728960.	0.304845
$357$	−799533.	−0.332021
$358$	2.44891e6	1.00987
$359$	3.24210e6	1.32767	0.663836	−	0.747878i	$-0.268927\pi$
$359$	3.24210e6	1.32767	0.663836	+	0.747878i	$0.268927\pi$
$360$	0	0
$361$	−2.40954e6	−0.973117
$362$	2.11533e6	0.848411
$363$	−1.13474e6	−0.451990
$364$	−491568.	−0.194460
$365$	0	0
$366$	−1.75464e6	−0.684676
$367$	−1.44430e6	−0.559749	−0.279874	−	0.960037i	$-0.590293\pi$
$367$	−1.44430e6	−0.559749	−0.279874	+	0.960037i	$0.590293\pi$
$368$	−760320.	−0.292669
$369$	−1.08151e6	−0.413490
$370$	0	0
$371$	−1.28223e6	−0.483651
$372$	275904.	0.103371
$373$	−3.43542e6	−1.27852	−0.639260	−	0.768991i	$-0.720759\pi$
$373$	−3.43542e6	−1.27852	−0.639260	+	0.768991i	$0.720759\pi$
$374$	1.35612e6	0.501326
$375$	0	0
$376$	1.40806e6	0.513633
$377$	−814473.	−0.295137
$378$	−714420.	−0.257172
$379$	−1.68635e6	−0.603044	−0.301522	−	0.953459i	$-0.597495\pi$
$379$	−1.68635e6	−0.603044	−0.301522	+	0.953459i	$0.597495\pi$
$380$	0	0
$381$	−270468.	−0.0954560
$382$	2.16148e6	0.757865
$383$	−2.64354e6	−0.920850	−0.460425	−	0.887699i	$-0.652303\pi$
$383$	−2.64354e6	−0.920850	−0.460425	+	0.887699i	$0.652303\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	−3.84128e6	−1.31222
$387$	514836.	0.174740
$388$	916720.	0.309142
$389$	−452099.	−0.151481	−0.0757407	−	0.997128i	$-0.524132\pi$
$389$	−452099.	−0.151481	−0.0757407	+	0.997128i	$0.524132\pi$
$390$	0	0
$391$	5.38461e6	1.78120
$392$	153664.	0.0505076
$393$	−1.08045e6	−0.352877
$394$	−3.04778e6	−0.989105
$395$	0	0
$396$	484704.	0.155324
$397$	1.95530e6	0.622641	0.311321	−	0.950305i	$-0.399229\pi$
$397$	1.95530e6	0.622641	0.311321	+	0.950305i	$0.399229\pi$
$398$	500336.	0.158327
$399$	113778.	0.0357788
$400$	0	0
$401$	−4.76737e6	−1.48053	−0.740266	−	0.672314i	$-0.765300\pi$
$401$	−4.76737e6	−1.48053	−0.740266	+	0.672314i	$0.765300\pi$
$402$	1.61395e6	0.498110
$403$	−1.20133e6	−0.368469
$404$	−719520.	−0.219326
$405$	0	0
$406$	254604.	0.0766567
$407$	1.23009e6	0.368086
$408$	−1.04429e6	−0.310577
$409$	−4.13199e6	−1.22138	−0.610690	−	0.791870i	$-0.709108\pi$
$409$	−4.13199e6	−1.22138	−0.610690	+	0.791870i	$0.709108\pi$
$410$	0	0
$411$	−285984.	−0.0835097
$412$	1.62248e6	0.470908
$413$	192668.	0.0555820
$414$	1.92456e6	0.551862
$415$	0	0
$416$	−642048.	−0.181901
$417$	−1.80146e6	−0.507322
$418$	−192984.	−0.0540232
$419$	190512.	0.0530136	0.0265068	−	0.999649i	$-0.491562\pi$
$419$	190512.	0.0530136	0.0265068	+	0.999649i	$0.491562\pi$
$420$	0	0
$421$	−5.19186e6	−1.42764	−0.713818	−	0.700332i	$-0.753035\pi$
$421$	−5.19186e6	−1.42764	−0.713818	+	0.700332i	$0.753035\pi$
$422$	2.51019e6	0.686160
$423$	−3.56416e6	−0.968515
$424$	−1.67475e6	−0.452414
$425$	0	0
$426$	2.29450e6	0.612581
$427$	−2.38826e6	−0.633887
$428$	2.65603e6	0.700848
$429$	1.05524e6	0.276827
$430$	0	0
$431$	−4.21781e6	−1.09369	−0.546845	−	0.837234i	$-0.684171\pi$
$431$	−4.21781e6	−1.09369	−0.546845	+	0.837234i	$0.684171\pi$
$432$	−933120.	−0.240563
$433$	4.86027e6	1.24578	0.622890	−	0.782310i	$-0.285959\pi$
$433$	4.86027e6	1.24578	0.622890	+	0.782310i	$0.285959\pi$
$434$	375536.	0.0957034
$435$	0	0
$436$	132624.	0.0334123
$437$	−766260.	−0.191943
$438$	−2.17692e6	−0.542197
$439$	−2.03113e6	−0.503011	−0.251505	−	0.967856i	$-0.580926\pi$
$439$	−2.03113e6	−0.503011	−0.251505	+	0.967856i	$0.580926\pi$
$440$	0	0
$441$	−388962.	−0.0952381
$442$	4.54700e6	1.10706
$443$	−2.84199e6	−0.688038	−0.344019	−	0.938963i	$-0.611789\pi$
$443$	−2.84199e6	−0.688038	−0.344019	+	0.938963i	$0.611789\pi$
$444$	−947232.	−0.228034
$445$	0	0
$446$	4.88440e6	1.16272
$447$	2.78678e6	0.659680
$448$	200704.	0.0472456
$449$	−4.59682e6	−1.07607	−0.538037	−	0.842921i	$-0.680834\pi$
$449$	−4.59682e6	−1.07607	−0.538037	+	0.842921i	$0.680834\pi$
$450$	0	0
$451$	−1.24841e6	−0.289012
$452$	−4.21130e6	−0.969549
$453$	−1.87791e6	−0.429962
$454$	−1.56219e6	−0.355708
$455$	0	0
$456$	148608.	0.0334680
$457$	−4.93367e6	−1.10504	−0.552522	−	0.833498i	$-0.686335\pi$
$457$	−4.93367e6	−1.10504	−0.552522	+	0.833498i	$0.686335\pi$
$458$	−2.84850e6	−0.634530
$459$	6.60838e6	1.46408
$460$	0	0
$461$	−4.75667e6	−1.04244	−0.521220	−	0.853422i	$-0.674523\pi$
$461$	−4.75667e6	−1.04244	−0.521220	+	0.853422i	$0.674523\pi$
$462$	−329868.	−0.0719011
$463$	−4.08619e6	−0.885862	−0.442931	−	0.896556i	$-0.646061\pi$
$463$	−4.08619e6	−0.885862	−0.442931	+	0.896556i	$0.646061\pi$
$464$	332544.	0.0717058
$465$	0	0
$466$	−2.24630e6	−0.479186
$467$	4.15932e6	0.882531	0.441266	−	0.897377i	$-0.354530\pi$
$467$	4.15932e6	0.882531	0.441266	+	0.897377i	$0.354530\pi$
$468$	1.62518e6	0.342995
$469$	2.19677e6	0.461160
$470$	0	0
$471$	−324090.	−0.0673152
$472$	251648.	0.0519922
$473$	594286.	0.122136
$474$	−1.57396e6	−0.321771
$475$	0	0
$476$	−1.42139e6	−0.287539
$477$	4.23922e6	0.853080
$478$	−5.44335e6	−1.08967
$479$	−3.36040e6	−0.669195	−0.334597	−	0.942361i	$-0.608600\pi$
$479$	−3.36040e6	−0.669195	−0.334597	+	0.942361i	$0.608600\pi$
$480$	0	0
$481$	4.12441e6	0.812828
$482$	2.12020e6	0.415680
$483$	−1.30977e6	−0.255463
$484$	−2.01731e6	−0.391435
$485$	0	0
$486$	3.77914e6	0.725775
$487$	−7.05243e6	−1.34746	−0.673730	−	0.738977i	$-0.735309\pi$
$487$	−7.05243e6	−1.34746	−0.673730	+	0.738977i	$0.735309\pi$
$488$	−3.11936e6	−0.592947
$489$	−1.58103e6	−0.298998
$490$	0	0
$491$	−83937.0	−0.0157127	−0.00785633	−	0.999969i	$-0.502501\pi$
$491$	−83937.0	−0.0157127	−0.00785633	+	0.999969i	$0.502501\pi$
$492$	961344.	0.179047
$493$	−2.35509e6	−0.436405
$494$	−647064.	−0.119297
$495$	0	0
$496$	490496.	0.0895223
$497$	3.12306e6	0.567140
$498$	−3.50194e6	−0.632754
$499$	−7.10526e6	−1.27741	−0.638703	−	0.769454i	$-0.720529\pi$
$499$	−7.10526e6	−1.27741	−0.638703	+	0.769454i	$0.720529\pi$
$500$	0	0
$501$	1.41672e6	0.252167
$502$	3.96132e6	0.701586
$503$	−2.89147e6	−0.509564	−0.254782	−	0.966999i	$-0.582004\pi$
$503$	−2.89147e6	−0.509564	−0.254782	+	0.966999i	$0.582004\pi$
$504$	−508032.	−0.0890871
$505$	0	0
$506$	2.22156e6	0.385729
$507$	196524.	0.0339544
$508$	−480832.	−0.0826674
$509$	1.03548e6	0.177153	0.0885764	−	0.996069i	$-0.471768\pi$
$509$	1.03548e6	0.177153	0.0885764	+	0.996069i	$0.471768\pi$
$510$	0	0
$511$	−2.96303e6	−0.501977
$512$	262144.	0.0441942
$513$	−940410.	−0.157770
$514$	7.26572e6	1.21303
$515$	0	0
$516$	−457632.	−0.0756645
$517$	−4.11419e6	−0.676952
$518$	−1.28929e6	−0.211118
$519$	211239.	0.0344236
$520$	0	0
$521$	−7.49715e6	−1.21005	−0.605023	−	0.796208i	$-0.706836\pi$
$521$	−7.49715e6	−1.21005	−0.605023	+	0.796208i	$0.706836\pi$
$522$	−841752.	−0.135210
$523$	−3.53223e6	−0.564670	−0.282335	−	0.959316i	$-0.591109\pi$
$523$	−3.53223e6	−0.564670	−0.282335	+	0.959316i	$0.591109\pi$
$524$	−1.92080e6	−0.305600
$525$	0	0
$526$	7.83390e6	1.23456
$527$	−3.47371e6	−0.544837
$528$	−430848.	−0.0672573
$529$	2.38456e6	0.370483
$530$	0	0
$531$	−636984.	−0.0980375
$532$	202272.	0.0309854
$533$	−4.18585e6	−0.638213
$534$	1.64016e6	0.248905
$535$	0	0
$536$	2.86925e6	0.431376
$537$	5.51005e6	0.824556
$538$	872136.	0.129906
$539$	−448987.	−0.0665674
$540$	0	0
$541$	−4.99188e6	−0.733281	−0.366641	−	0.930363i	$-0.619492\pi$
$541$	−4.99188e6	−0.733281	−0.366641	+	0.930363i	$0.619492\pi$
$542$	5.05061e6	0.738491
$543$	4.75949e6	0.692725
$544$	−1.85651e6	−0.268968
$545$	0	0
$546$	−1.10603e6	−0.158776
$547$	−5.12634e6	−0.732553	−0.366277	−	0.930506i	$-0.619368\pi$
$547$	−5.12634e6	−0.732553	−0.366277	+	0.930506i	$0.619368\pi$
$548$	−508416.	−0.0723216
$549$	7.89588e6	1.11807
$550$	0	0
$551$	335142.	0.0470273
$552$	−1.71072e6	−0.238963
$553$	−2.14233e6	−0.297902
$554$	4.41055e6	0.610546
$555$	0	0
$556$	−3.20259e6	−0.439354
$557$	−8.86866e6	−1.21121	−0.605606	−	0.795765i	$-0.707069\pi$
$557$	−8.86866e6	−1.21121	−0.605606	+	0.795765i	$0.707069\pi$
$558$	−1.24157e6	−0.168805
$559$	1.99261e6	0.269707
$560$	0	0
$561$	3.05128e6	0.409331
$562$	−3.99285e6	−0.533264
$563$	−9.07277e6	−1.20634	−0.603169	−	0.797613i	$-0.706096\pi$
$563$	−9.07277e6	−1.20634	−0.603169	+	0.797613i	$0.706096\pi$
$564$	3.16814e6	0.419379
$565$	0	0
$566$	1.54548e6	0.202779
$567$	321489.	0.0419961
$568$	4.07910e6	0.530510
$569$	2.08310e6	0.269730	0.134865	−	0.990864i	$-0.456940\pi$
$569$	2.08310e6	0.269730	0.134865	+	0.990864i	$0.456940\pi$
$570$	0	0
$571$	−5.46368e6	−0.701286	−0.350643	−	0.936509i	$-0.614037\pi$
$571$	−5.46368e6	−0.701286	−0.350643	+	0.936509i	$0.614037\pi$
$572$	1.87598e6	0.239739
$573$	4.86332e6	0.618794
$574$	1.30850e6	0.165765
$575$	0	0
$576$	−663552.	−0.0833333
$577$	−7.66246e6	−0.958140	−0.479070	−	0.877777i	$-0.659026\pi$
$577$	−7.66246e6	−0.958140	−0.479070	+	0.877777i	$0.659026\pi$
$578$	7.46845e6	0.929845
$579$	−8.64288e6	−1.07143
$580$	0	0
$581$	−4.76652e6	−0.585816
$582$	2.06262e6	0.252413
$583$	4.89342e6	0.596267
$584$	−3.87008e6	−0.469556
$585$	0	0
$586$	3.13428e6	0.377046
$587$	1.57465e7	1.88620	0.943100	−	0.332510i	$-0.107896\pi$
$587$	1.57465e7	1.88620	0.943100	+	0.332510i	$0.107896\pi$
$588$	345744.	0.0412393
$589$	494328.	0.0587120
$590$	0	0
$591$	−6.85750e6	−0.807601
$592$	−1.68397e6	−0.197483
$593$	1.62409e7	1.89658	0.948292	−	0.317398i	$-0.102809\pi$
$593$	1.62409e7	1.89658	0.948292	+	0.317398i	$0.102809\pi$
$594$	2.72646e6	0.317054
$595$	0	0
$596$	4.95427e6	0.571300
$597$	1.12576e6	0.129273
$598$	7.44876e6	0.851787
$599$	−1.90793e6	−0.217268	−0.108634	−	0.994082i	$-0.534648\pi$
$599$	−1.90793e6	−0.217268	−0.108634	+	0.994082i	$0.534648\pi$
$600$	0	0
$601$	3.52970e6	0.398613	0.199306	−	0.979937i	$-0.436131\pi$
$601$	3.52970e6	0.398613	0.199306	+	0.979937i	$0.436131\pi$
$602$	−622888.	−0.0700517
$603$	−7.26278e6	−0.813411
$604$	−3.33851e6	−0.372358
$605$	0	0
$606$	−1.61892e6	−0.179079
$607$	−3.37799e6	−0.372123	−0.186061	−	0.982538i	$-0.559572\pi$
$607$	−3.37799e6	−0.372123	−0.186061	+	0.982538i	$0.559572\pi$
$608$	264192.	0.0289842
$609$	572859.	0.0625899
$610$	0	0
$611$	−1.37946e7	−1.49488
$612$	4.69930e6	0.507171
$613$	−1.20412e6	−0.129425	−0.0647127	−	0.997904i	$-0.520613\pi$
$613$	−1.20412e6	−0.129425	−0.0647127	+	0.997904i	$0.520613\pi$
$614$	−1.12709e7	−1.20653
$615$	0	0
$616$	−586432.	−0.0622681
$617$	5.47330e6	0.578810	0.289405	−	0.957207i	$-0.406543\pi$
$617$	5.47330e6	0.578810	0.289405	+	0.957207i	$0.406543\pi$
$618$	3.65058e6	0.384495
$619$	3.22662e6	0.338471	0.169236	−	0.985576i	$-0.445870\pi$
$619$	3.22662e6	0.338471	0.169236	+	0.985576i	$0.445870\pi$
$620$	0	0
$621$	1.08256e7	1.12648
$622$	−3.38959e6	−0.351295
$623$	2.23244e6	0.230441
$624$	−1.44461e6	−0.148521
$625$	0	0
$626$	−1.45982e6	−0.148889
$627$	−434214.	−0.0441098
$628$	−576160.	−0.0582967
$629$	1.19259e7	1.20189
$630$	0	0
$631$	1.36282e7	1.36259	0.681297	−	0.732007i	$-0.261416\pi$
$631$	1.36282e7	1.36259	0.681297	+	0.732007i	$0.261416\pi$
$632$	−2.79814e6	−0.278662
$633$	5.64792e6	0.560247
$634$	−7.74975e6	−0.765711
$635$	0	0
$636$	−3.76819e6	−0.369394
$637$	−1.50543e6	−0.146998
$638$	−971652.	−0.0945059
$639$	−1.03252e7	−1.00034
$640$	0	0
$641$	−1.92472e7	−1.85021	−0.925106	−	0.379710i	$-0.876024\pi$
$641$	−1.92472e7	−1.85021	−0.925106	+	0.379710i	$0.876024\pi$
$642$	5.97607e6	0.572240
$643$	1.28399e7	1.22472	0.612358	−	0.790580i	$-0.290221\pi$
$643$	1.28399e7	1.22472	0.612358	+	0.790580i	$0.290221\pi$
$644$	−2.32848e6	−0.221237
$645$	0	0
$646$	−1.87102e6	−0.176399
$647$	−2.00233e7	−1.88050	−0.940251	−	0.340481i	$-0.889410\pi$
$647$	−2.00233e7	−1.88050	−0.940251	+	0.340481i	$0.889410\pi$
$648$	419904.	0.0392837
$649$	−735284.	−0.0685241
$650$	0	0
$651$	844956.	0.0781415
$652$	−2.81072e6	−0.258940
$653$	7.23655e6	0.664124	0.332062	−	0.943258i	$-0.392256\pi$
$653$	7.23655e6	0.664124	0.332062	+	0.943258i	$0.392256\pi$
$654$	298404.	0.0272810
$655$	0	0
$656$	1.70906e6	0.155059
$657$	9.79614e6	0.885404
$658$	4.31220e6	0.388270
$659$	1.42474e7	1.27798	0.638989	−	0.769216i	$-0.279353\pi$
$659$	1.42474e7	1.27798	0.638989	+	0.769216i	$0.279353\pi$
$660$	0	0
$661$	1.49265e7	1.32878	0.664391	−	0.747385i	$-0.268691\pi$
$661$	1.49265e7	1.32878	0.664391	+	0.747385i	$0.268691\pi$
$662$	−245840.	−0.0218026
$663$	1.02308e7	0.903908
$664$	−6.22566e6	−0.547981
$665$	0	0
$666$	4.26254e6	0.372377
$667$	−3.85803e6	−0.335777
$668$	2.51861e6	0.218383
$669$	1.09899e7	0.949355
$670$	0	0
$671$	9.11438e6	0.781485
$672$	451584.	0.0385758
$673$	1.55062e7	1.31967	0.659837	−	0.751409i	$-0.270625\pi$
$673$	1.55062e7	1.31967	0.659837	+	0.751409i	$0.270625\pi$
$674$	1.49646e7	1.26887
$675$	0	0
$676$	349376.	0.0294053
$677$	7.80065e6	0.654122	0.327061	−	0.945003i	$-0.393942\pi$
$677$	7.80065e6	0.654122	0.327061	+	0.945003i	$0.393942\pi$
$678$	−9.47542e6	−0.791633
$679$	2.80745e6	0.233689
$680$	0	0
$681$	−3.51492e6	−0.290434
$682$	−1.43317e6	−0.117988
$683$	−1.58547e7	−1.30049	−0.650243	−	0.759727i	$-0.725333\pi$
$683$	−1.58547e7	−1.30049	−0.650243	+	0.759727i	$0.725333\pi$
$684$	−668736.	−0.0546531
$685$	0	0
$686$	470596.	0.0381802
$687$	−6.40912e6	−0.518091
$688$	−813568.	−0.0655274
$689$	1.64073e7	1.31671
$690$	0	0
$691$	2.03656e7	1.62257	0.811284	−	0.584652i	$-0.198769\pi$
$691$	2.03656e7	1.62257	0.811284	+	0.584652i	$0.198769\pi$
$692$	375536.	0.0298117
$693$	1.48441e6	0.117414
$694$	−845336.	−0.0666240
$695$	0	0
$696$	748224.	0.0585475
$697$	−1.21036e7	−0.943696
$698$	1.35823e7	1.05520
$699$	−5.05418e6	−0.391253
$700$	0	0
$701$	2.48036e7	1.90643	0.953213	−	0.302300i	$-0.0977543\pi$
$701$	2.48036e7	1.90643	0.953213	+	0.302300i	$0.0977543\pi$
$702$	9.14166e6	0.700136
$703$	−1.69712e6	−0.129517
$704$	−765952.	−0.0582465
$705$	0	0
$706$	1.55237e7	1.17215
$707$	−2.20353e6	−0.165795
$708$	566208.	0.0424515
$709$	1.81917e7	1.35912	0.679560	−	0.733620i	$-0.262171\pi$
$709$	1.81917e7	1.35912	0.679560	+	0.733620i	$0.262171\pi$
$710$	0	0
$711$	7.08280e6	0.525450
$712$	2.91584e6	0.215558
$713$	−5.69052e6	−0.419207
$714$	−3.19813e6	−0.234774
$715$	0	0
$716$	9.79565e6	0.714086
$717$	−1.22475e7	−0.889715
$718$	1.29684e7	0.938806
$719$	−1.66202e7	−1.19899	−0.599493	−	0.800380i	$-0.704631\pi$
$719$	−1.66202e7	−1.19899	−0.599493	+	0.800380i	$0.704631\pi$
$720$	0	0
$721$	4.96884e6	0.355973
$722$	−9.63814e6	−0.688098
$723$	4.77045e6	0.339401
$724$	8.46131e6	0.599917
$725$	0	0
$726$	−4.53895e6	−0.319605
$727$	1.57591e7	1.10585	0.552925	−	0.833231i	$-0.313511\pi$
$727$	1.57591e7	1.10585	0.552925	+	0.833231i	$0.313511\pi$
$728$	−1.96627e6	−0.137504
$729$	6.90873e6	0.481481
$730$	0	0
$731$	5.76171e6	0.398803
$732$	−7.01856e6	−0.484139
$733$	2.15238e6	0.147965	0.0739827	−	0.997260i	$-0.476429\pi$
$733$	2.15238e6	0.147965	0.0739827	+	0.997260i	$0.476429\pi$
$734$	−5.77721e6	−0.395802
$735$	0	0
$736$	−3.04128e6	−0.206948
$737$	−8.38358e6	−0.568540
$738$	−4.32605e6	−0.292382
$739$	−2.27267e7	−1.53083	−0.765413	−	0.643540i	$-0.777465\pi$
$739$	−2.27267e7	−1.53083	−0.765413	+	0.643540i	$0.777465\pi$
$740$	0	0
$741$	−1.45589e6	−0.0974057
$742$	−5.12893e6	−0.341993
$743$	−1.21153e7	−0.805123	−0.402561	−	0.915393i	$-0.631880\pi$
$743$	−1.21153e7	−0.805123	−0.402561	+	0.915393i	$0.631880\pi$
$744$	1.10362e6	0.0730947
$745$	0	0
$746$	−1.37417e7	−0.904050
$747$	1.57587e7	1.03328
$748$	5.42450e6	0.354491
$749$	8.13410e6	0.529791
$750$	0	0
$751$	−2.07590e7	−1.34310	−0.671549	−	0.740961i	$-0.734371\pi$
$751$	−2.07590e7	−1.34310	−0.671549	+	0.740961i	$0.734371\pi$
$752$	5.63226e6	0.363193
$753$	8.91297e6	0.572842
$754$	−3.25789e6	−0.208693
$755$	0	0
$756$	−2.85768e6	−0.181848
$757$	1.86222e7	1.18111	0.590556	−	0.806997i	$-0.298909\pi$
$757$	1.86222e7	1.18111	0.590556	+	0.806997i	$0.298909\pi$
$758$	−6.74539e6	−0.426417
$759$	4.99851e6	0.314946
$760$	0	0
$761$	2.65336e7	1.66087	0.830434	−	0.557117i	$-0.188093\pi$
$761$	2.65336e7	1.66087	0.830434	+	0.557117i	$0.188093\pi$
$762$	−1.08187e6	−0.0674976
$763$	406161.	0.0252573
$764$	8.64590e6	0.535892
$765$	0	0
$766$	−1.05742e7	−0.651139
$767$	−2.46536e6	−0.151319
$768$	589824.	0.0360844
$769$	−2.01595e7	−1.22931	−0.614657	−	0.788794i	$-0.710706\pi$
$769$	−2.01595e7	−1.22931	−0.614657	+	0.788794i	$0.710706\pi$
$770$	0	0
$771$	1.63479e7	0.990433
$772$	−1.53651e7	−0.927882
$773$	5.86488e6	0.353029	0.176514	−	0.984298i	$-0.443518\pi$
$773$	5.86488e6	0.353029	0.176514	+	0.984298i	$0.443518\pi$
$774$	2.05934e6	0.123560
$775$	0	0
$776$	3.66688e6	0.218596
$777$	−2.90090e6	−0.172377
$778$	−1.80840e6	−0.107114
$779$	1.72241e6	0.101693
$780$	0	0
$781$	−1.19186e7	−0.699196
$782$	2.15384e7	1.25950
$783$	−4.73486e6	−0.275996
$784$	614656.	0.0357143
$785$	0	0
$786$	−4.32180e6	−0.249522
$787$	1.63347e6	0.0940100	0.0470050	−	0.998895i	$-0.485032\pi$
$787$	1.63347e6	0.0940100	0.0470050	+	0.998895i	$0.485032\pi$
$788$	−1.21911e7	−0.699403
$789$	1.76263e7	1.00802
$790$	0	0
$791$	−1.28971e7	−0.732910
$792$	1.93882e6	0.109831
$793$	3.05600e7	1.72572
$794$	7.82121e6	0.440274
$795$	0	0
$796$	2.00134e6	0.111954
$797$	−2.07673e7	−1.15807	−0.579034	−	0.815303i	$-0.696570\pi$
$797$	−2.07673e7	−1.15807	−0.579034	+	0.815303i	$0.696570\pi$
$798$	455112.	0.0252994
$799$	−3.98878e7	−2.21041
$800$	0	0
$801$	−7.38072e6	−0.406460
$802$	−1.90695e7	−1.04689
$803$	1.13079e7	0.618860
$804$	6.45581e6	0.352217
$805$	0	0
$806$	−4.80533e6	−0.260547
$807$	1.96231e6	0.106068
$808$	−2.87808e6	−0.155087
$809$	3.53936e6	0.190131	0.0950656	−	0.995471i	$-0.469694\pi$
$809$	3.53936e6	0.190131	0.0950656	+	0.995471i	$0.469694\pi$
$810$	0	0
$811$	−2.11480e7	−1.12906	−0.564530	−	0.825412i	$-0.690943\pi$
$811$	−2.11480e7	−1.12906	−0.564530	+	0.825412i	$0.690943\pi$
$812$	1.01842e6	0.0542045
$813$	1.13639e7	0.602976
$814$	4.92034e6	0.260276
$815$	0	0
$816$	−4.17715e6	−0.219611
$817$	−819924.	−0.0429753
$818$	−1.65280e7	−0.863646
$819$	4.97713e6	0.259280
$820$	0	0
$821$	265389.	0.0137412	0.00687061	−	0.999976i	$-0.497813\pi$
$821$	265389.	0.0137412	0.00687061	+	0.999976i	$0.497813\pi$
$822$	−1.14394e6	−0.0590503
$823$	−3.09261e7	−1.59157	−0.795785	−	0.605579i	$-0.792942\pi$
$823$	−3.09261e7	−1.59157	−0.795785	+	0.605579i	$0.792942\pi$
$824$	6.48992e6	0.332982
$825$	0	0
$826$	770672.	0.0393024
$827$	2.84152e7	1.44473	0.722367	−	0.691510i	$-0.243054\pi$
$827$	2.84152e7	1.44473	0.722367	+	0.691510i	$0.243054\pi$
$828$	7.69824e6	0.390225
$829$	−3.33547e7	−1.68566	−0.842832	−	0.538177i	$-0.819113\pi$
$829$	−3.33547e7	−1.68566	−0.842832	+	0.538177i	$0.819113\pi$
$830$	0	0
$831$	9.92374e6	0.498509
$832$	−2.56819e6	−0.128623
$833$	−4.35301e6	−0.217359
$834$	−7.20583e6	−0.358731
$835$	0	0
$836$	−771936.	−0.0382002
$837$	−6.98382e6	−0.344572
$838$	762048.	0.0374863
$839$	−5.66205e6	−0.277695	−0.138848	−	0.990314i	$-0.544340\pi$
$839$	−5.66205e6	−0.277695	−0.138848	+	0.990314i	$0.544340\pi$
$840$	0	0
$841$	−1.88237e7	−0.917732
$842$	−2.07674e7	−1.00949
$843$	−8.98392e6	−0.435408
$844$	1.00408e7	0.485188
$845$	0	0
$846$	−1.42566e7	−0.684844
$847$	−6.17802e6	−0.295897
$848$	−6.69901e6	−0.319905
$849$	3.47734e6	0.165569
$850$	0	0
$851$	1.95367e7	0.924754
$852$	9.17798e6	0.433160
$853$	−2.19983e7	−1.03518	−0.517592	−	0.855628i	$-0.673171\pi$
$853$	−2.19983e7	−1.03518	−0.517592	+	0.855628i	$0.673171\pi$
$854$	−9.55304e6	−0.448226
$855$	0	0
$856$	1.06241e7	0.495574
$857$	2.17568e7	1.01191	0.505956	−	0.862559i	$-0.331140\pi$
$857$	2.17568e7	1.01191	0.505956	+	0.862559i	$0.331140\pi$
$858$	4.22096e6	0.195746
$859$	−4.09384e7	−1.89299	−0.946494	−	0.322721i	$-0.895402\pi$
$859$	−4.09384e7	−1.89299	−0.946494	+	0.322721i	$0.895402\pi$
$860$	0	0
$861$	2.94412e6	0.135347
$862$	−1.68712e7	−0.773355
$863$	−5.65597e6	−0.258512	−0.129256	−	0.991611i	$-0.541259\pi$
$863$	−5.65597e6	−0.258512	−0.129256	+	0.991611i	$0.541259\pi$
$864$	−3.73248e6	−0.170103
$865$	0	0
$866$	1.94411e7	0.880899
$867$	1.68040e7	0.759216
$868$	1.50214e6	0.0676725
$869$	8.17583e6	0.367267
$870$	0	0
$871$	−2.81097e7	−1.25548
$872$	530496.	0.0236260
$873$	−9.28179e6	−0.412189
$874$	−3.06504e6	−0.135724
$875$	0	0
$876$	−8.70768e6	−0.383391
$877$	−2.61067e7	−1.14618	−0.573089	−	0.819493i	$-0.694255\pi$
$877$	−2.61067e7	−1.14618	−0.573089	+	0.819493i	$0.694255\pi$
$878$	−8.12454e6	−0.355682
$879$	7.05214e6	0.307857
$880$	0	0
$881$	1.44294e6	0.0626339	0.0313170	−	0.999510i	$-0.490030\pi$
$881$	1.44294e6	0.0626339	0.0313170	+	0.999510i	$0.490030\pi$
$882$	−1.55585e6	−0.0673435
$883$	1.52432e7	0.657921	0.328960	−	0.944344i	$-0.393302\pi$
$883$	1.52432e7	0.657921	0.328960	+	0.944344i	$0.393302\pi$
$884$	1.81880e7	0.782807
$885$	0	0
$886$	−1.13679e7	−0.486517
$887$	3.31500e7	1.41473	0.707366	−	0.706847i	$-0.249883\pi$
$887$	3.31500e7	1.41473	0.707366	+	0.706847i	$0.249883\pi$
$888$	−3.78893e6	−0.161244
$889$	−1.47255e6	−0.0624907
$890$	0	0
$891$	−1.22691e6	−0.0517747
$892$	1.95376e7	0.822166
$893$	5.67626e6	0.238195
$894$	1.11471e7	0.466464
$895$	0	0
$896$	802816.	0.0334077
$897$	1.67597e7	0.695481
$898$	−1.83873e7	−0.760899
$899$	2.48888e6	0.102708
$900$	0	0
$901$	4.74426e7	1.94696
$902$	−4.99365e6	−0.204363
$903$	−1.40150e6	−0.0571970
$904$	−1.68452e7	−0.685575
$905$	0	0
$906$	−7.51165e6	−0.304029
$907$	−1.16963e7	−0.472096	−0.236048	−	0.971741i	$-0.575852\pi$
$907$	−1.16963e7	−0.472096	−0.236048	+	0.971741i	$0.575852\pi$
$908$	−6.24875e6	−0.251524
$909$	7.28514e6	0.292434
$910$	0	0
$911$	2.89321e7	1.15501	0.577503	−	0.816389i	$-0.304027\pi$
$911$	2.89321e7	1.15501	0.577503	+	0.816389i	$0.304027\pi$
$912$	594432.	0.0236655
$913$	1.81906e7	0.722221
$914$	−1.97347e7	−0.781384
$915$	0	0
$916$	−1.13940e7	−0.448680
$917$	−5.88245e6	−0.231012
$918$	2.64335e7	1.03526
$919$	−4.57838e7	−1.78823	−0.894115	−	0.447838i	$-0.852194\pi$
$919$	−4.57838e7	−1.78823	−0.894115	+	0.447838i	$0.852194\pi$
$920$	0	0
$921$	−2.53596e7	−0.985129
$922$	−1.90267e7	−0.737116
$923$	−3.99625e7	−1.54400
$924$	−1.31947e6	−0.0508417
$925$	0	0
$926$	−1.63448e7	−0.626399
$927$	−1.64276e7	−0.627878
$928$	1.33018e6	0.0507036
$929$	2.46947e7	0.938782	0.469391	−	0.882990i	$-0.344474\pi$
$929$	2.46947e7	0.938782	0.469391	+	0.882990i	$0.344474\pi$
$930$	0	0
$931$	619458.	0.0234227
$932$	−8.98522e6	−0.338835
$933$	−7.62658e6	−0.286831
$934$	1.66373e7	0.624044
$935$	0	0
$936$	6.50074e6	0.242534
$937$	1.98926e7	0.740187	0.370094	−	0.928994i	$-0.379326\pi$
$937$	1.98926e7	0.740187	0.370094	+	0.928994i	$0.379326\pi$
$938$	8.78707e6	0.326090
$939$	−3.28460e6	−0.121568
$940$	0	0
$941$	−3.73454e7	−1.37488	−0.687438	−	0.726243i	$-0.741265\pi$
$941$	−3.73454e7	−1.37488	−0.687438	+	0.726243i	$0.741265\pi$
$942$	−1.29636e6	−0.0475991
$943$	−1.98277e7	−0.726095
$944$	1.00659e6	0.0367641
$945$	0	0
$946$	2.37714e6	0.0863630
$947$	−5.10396e7	−1.84941	−0.924703	−	0.380689i	$-0.875687\pi$
$947$	−5.10396e7	−1.84941	−0.924703	+	0.380689i	$0.875687\pi$
$948$	−6.29582e6	−0.227526
$949$	3.79147e7	1.36660
$950$	0	0
$951$	−1.74369e7	−0.625200
$952$	−5.68557e6	−0.203321
$953$	−254832.	−0.00908912	−0.00454456	−	0.999990i	$-0.501447\pi$
$953$	−254832.	−0.00908912	−0.00454456	+	0.999990i	$0.501447\pi$
$954$	1.69569e7	0.603218
$955$	0	0
$956$	−2.17734e7	−0.770515
$957$	−2.18622e6	−0.0771638
$958$	−1.34416e7	−0.473192
$959$	−1.55702e6	−0.0546700
$960$	0	0
$961$	−2.49581e7	−0.871772
$962$	1.64976e7	0.574756
$963$	−2.68923e7	−0.934464
$964$	8.48080e6	0.293930
$965$	0	0
$966$	−5.23908e6	−0.180639
$967$	−1.17012e7	−0.402405	−0.201203	−	0.979550i	$-0.564485\pi$
$967$	−1.17012e7	−0.402405	−0.201203	+	0.979550i	$0.564485\pi$
$968$	−8.06925e6	−0.276786
$969$	−4.20979e6	−0.144029
$970$	0	0
$971$	3.59080e7	1.22220	0.611101	−	0.791553i	$-0.290727\pi$
$971$	3.59080e7	1.22220	0.611101	+	0.791553i	$0.290727\pi$
$972$	1.51165e7	0.513200
$973$	−9.80794e6	−0.332120
$974$	−2.82097e7	−0.952799
$975$	0	0
$976$	−1.24774e7	−0.419277
$977$	5.50592e7	1.84541	0.922706	−	0.385504i	$-0.125972\pi$
$977$	5.50592e7	1.84541	0.922706	+	0.385504i	$0.125972\pi$
$978$	−6.32412e6	−0.211423
$979$	−8.51972e6	−0.284098
$980$	0	0
$981$	−1.34282e6	−0.0445497
$982$	−335748.	−0.0111105
$983$	1.81317e7	0.598488	0.299244	−	0.954177i	$-0.403266\pi$
$983$	1.81317e7	0.598488	0.299244	+	0.954177i	$0.403266\pi$
$984$	3.84538e6	0.126605
$985$	0	0
$986$	−9.42035e6	−0.308585
$987$	9.70244e6	0.317021
$988$	−2.58826e6	−0.0843558
$989$	9.43866e6	0.306845
$990$	0	0
$991$	−2.02908e7	−0.656318	−0.328159	−	0.944623i	$-0.606428\pi$
$991$	−2.02908e7	−0.656318	−0.328159	+	0.944623i	$0.606428\pi$
$992$	1.96198e6	0.0633018
$993$	−553140.	−0.0178017
$994$	1.24923e7	0.401028
$995$	0	0
$996$	−1.40077e7	−0.447425
$997$	−4.75390e7	−1.51465	−0.757325	−	0.653038i	$-0.773494\pi$
$997$	−4.75390e7	−1.51465	−0.757325	+	0.653038i	$0.773494\pi$
$998$	−2.84211e7	−0.903262
$999$	2.39768e7	0.760112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.l.1.1		1
5.2	odd	4		350.6.c.g.99.2		2
5.3	odd	4		350.6.c.g.99.1		2
5.4	even	2		70.6.a.b.1.1	✓	1
15.14	odd	2		630.6.a.i.1.1		1
20.19	odd	2		560.6.a.e.1.1		1
35.34	odd	2		490.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.b.1.1	✓	1	5.4	even	2
350.6.a.l.1.1		1	1.1	even	1	trivial
350.6.c.g.99.1		2	5.3	odd	4
350.6.c.g.99.2		2	5.2	odd	4
490.6.a.g.1.1		1	35.34	odd	2
560.6.a.e.1.1		1	20.19	odd	2
630.6.a.i.1.1		1	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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