LMFDB - Embedded newform 240.8.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,8,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("240.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	240.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:	$74.9724061162$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+27.0000 q^{3} -125.000 q^{5} +988.000 q^{7} +729.000 q^{9} +O(q^{10})$

$q+27.0000 q^{3} -125.000 q^{5} +988.000 q^{7} +729.000 q^{9} +8040.00 q^{11} -3334.00 q^{13} -3375.00 q^{15} +6582.00 q^{17} +27436.0 q^{19} +26676.0 q^{21} -48600.0 q^{23} +15625.0 q^{25} +19683.0 q^{27} -132414. q^{29} -254408. q^{31} +217080. q^{33} -123500. q^{35} +519434. q^{37} -90018.0 q^{39} +92394.0 q^{41} +234532. q^{43} -91125.0 q^{45} +1.27764e6 q^{47} +152601. q^{49} +177714. q^{51} -835278. q^{53} -1.00500e6 q^{55} +740772. q^{57} +3.06876e6 q^{59} -1.00933e6 q^{61} +720252. q^{63} +416750. q^{65} -3.08217e6 q^{67} -1.31220e6 q^{69} +3.66672e6 q^{71} +1.12287e6 q^{73} +421875. q^{75} +7.94352e6 q^{77} +4.12881e6 q^{79} +531441. q^{81} -4.58656e6 q^{83} -822750. q^{85} -3.57518e6 q^{87} -5.76368e6 q^{89} -3.29399e6 q^{91} -6.86902e6 q^{93} -3.42950e6 q^{95} +6.74755e6 q^{97} +5.86116e6 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	988.000	1.08871	0.544357	−	0.838854i	$-0.316774\pi$
$7$	988.000	1.08871	0.544357	+	0.838854i	$0.316774\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	8040.00	1.82130	0.910650	−	0.413178i	$-0.135581\pi$
$11$	8040.00	1.82130	0.910650	+	0.413178i	$0.135581\pi$
$12$	0	0
$13$	−3334.00	−0.420885	−0.210443	−	0.977606i	$-0.567491\pi$
$13$	−3334.00	−0.420885	−0.210443	+	0.977606i	$0.567491\pi$
$14$	0	0
$15$	−3375.00	−0.258199
$16$	0	0
$17$	6582.00	0.324928	0.162464	−	0.986715i	$-0.448056\pi$
$17$	6582.00	0.324928	0.162464	+	0.986715i	$0.448056\pi$
$18$	0	0
$19$	27436.0	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	27436.0	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	26676.0	0.628569
$22$	0	0
$23$	−48600.0	−0.832892	−0.416446	−	0.909160i	$-0.636725\pi$
$23$	−48600.0	−0.832892	−0.416446	+	0.909160i	$0.636725\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−132414.	−1.00819	−0.504093	−	0.863649i	$-0.668173\pi$
$29$	−132414.	−1.00819	−0.504093	+	0.863649i	$0.668173\pi$
$30$	0	0
$31$	−254408.	−1.53379	−0.766893	−	0.641775i	$-0.778198\pi$
$31$	−254408.	−1.53379	−0.766893	+	0.641775i	$0.778198\pi$
$32$	0	0
$33$	217080.	1.05153
$34$	0	0
$35$	−123500.	−0.486888
$36$	0	0
$37$	519434.	1.68587	0.842935	−	0.538015i	$-0.180825\pi$
$37$	519434.	1.68587	0.842935	+	0.538015i	$0.180825\pi$
$38$	0	0
$39$	−90018.0	−0.242998
$40$	0	0
$41$	92394.0	0.209363	0.104682	−	0.994506i	$-0.466618\pi$
$41$	92394.0	0.209363	0.104682	+	0.994506i	$0.466618\pi$
$42$	0	0
$43$	234532.	0.449845	0.224922	−	0.974377i	$-0.427787\pi$
$43$	234532.	0.449845	0.224922	+	0.974377i	$0.427787\pi$
$44$	0	0
$45$	−91125.0	−0.149071
$46$	0	0
$47$	1.27764e6	1.79501	0.897503	−	0.441008i	$-0.145379\pi$
$47$	1.27764e6	1.79501	0.897503	+	0.441008i	$0.145379\pi$
$48$	0	0
$49$	152601.	0.185298
$50$	0	0
$51$	177714.	0.187597
$52$	0	0
$53$	−835278.	−0.770665	−0.385332	−	0.922778i	$-0.625913\pi$
$53$	−835278.	−0.770665	−0.385332	+	0.922778i	$0.625913\pi$
$54$	0	0
$55$	−1.00500e6	−0.814510
$56$	0	0
$57$	740772.	0.529813
$58$	0	0
$59$	3.06876e6	1.94527	0.972637	−	0.232329i	$-0.0746346\pi$
$59$	3.06876e6	1.94527	0.972637	+	0.232329i	$0.0746346\pi$
$60$	0	0
$61$	−1.00933e6	−0.569349	−0.284675	−	0.958624i	$-0.591886\pi$
$61$	−1.00933e6	−0.569349	−0.284675	+	0.958624i	$0.591886\pi$
$62$	0	0
$63$	720252.	0.362905
$64$	0	0
$65$	416750.	0.188226
$66$	0	0
$67$	−3.08217e6	−1.25197	−0.625987	−	0.779834i	$-0.715304\pi$
$67$	−3.08217e6	−1.25197	−0.625987	+	0.779834i	$0.715304\pi$
$68$	0	0
$69$	−1.31220e6	−0.480871
$70$	0	0
$71$	3.66672e6	1.21583	0.607916	−	0.794001i	$-0.292006\pi$
$71$	3.66672e6	1.21583	0.607916	+	0.794001i	$0.292006\pi$
$72$	0	0
$73$	1.12287e6	0.337830	0.168915	−	0.985631i	$-0.445974\pi$
$73$	1.12287e6	0.337830	0.168915	+	0.985631i	$0.445974\pi$
$74$	0	0
$75$	421875.	0.115470
$76$	0	0
$77$	7.94352e6	1.98288
$78$	0	0
$79$	4.12881e6	0.942171	0.471086	−	0.882087i	$-0.343862\pi$
$79$	4.12881e6	0.942171	0.471086	+	0.882087i	$0.343862\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−4.58656e6	−0.880468	−0.440234	−	0.897883i	$-0.645104\pi$
$83$	−4.58656e6	−0.880468	−0.440234	+	0.897883i	$0.645104\pi$
$84$	0	0
$85$	−822750.	−0.145312
$86$	0	0
$87$	−3.57518e6	−0.582077
$88$	0	0
$89$	−5.76368e6	−0.866632	−0.433316	−	0.901242i	$-0.642657\pi$
$89$	−5.76368e6	−0.866632	−0.433316	+	0.901242i	$0.642657\pi$
$90$	0	0
$91$	−3.29399e6	−0.458224
$92$	0	0
$93$	−6.86902e6	−0.885532
$94$	0	0
$95$	−3.42950e6	−0.410391
$96$	0	0
$97$	6.74755e6	0.750663	0.375332	−	0.926891i	$-0.377529\pi$
$97$	6.74755e6	0.750663	0.375332	+	0.926891i	$0.377529\pi$
$98$	0	0
$99$	5.86116e6	0.607100
$100$	0	0
$101$	5.70974e6	0.551431	0.275716	−	0.961239i	$-0.411085\pi$
$101$	5.70974e6	0.551431	0.275716	+	0.961239i	$0.411085\pi$
$102$	0	0
$103$	6.76769e6	0.610254	0.305127	−	0.952312i	$-0.401301\pi$
$103$	6.76769e6	0.610254	0.305127	+	0.952312i	$0.401301\pi$
$104$	0	0
$105$	−3.33450e6	−0.281105
$106$	0	0
$107$	−1.76452e6	−0.139246	−0.0696229	−	0.997573i	$-0.522180\pi$
$107$	−1.76452e6	−0.139246	−0.0696229	+	0.997573i	$0.522180\pi$
$108$	0	0
$109$	643790.	0.0476158	0.0238079	−	0.999717i	$-0.492421\pi$
$109$	643790.	0.0476158	0.0238079	+	0.999717i	$0.492421\pi$
$110$	0	0
$111$	1.40247e7	0.973338
$112$	0	0
$113$	1.42571e7	0.929515	0.464757	−	0.885438i	$-0.346142\pi$
$113$	1.42571e7	0.929515	0.464757	+	0.885438i	$0.346142\pi$
$114$	0	0
$115$	6.07500e6	0.372481
$116$	0	0
$117$	−2.43049e6	−0.140295
$118$	0	0
$119$	6.50302e6	0.353753
$120$	0	0
$121$	4.51544e7	2.31714
$122$	0	0
$123$	2.49464e6	0.120876
$124$	0	0
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	4.41682e6	0.191336	0.0956680	−	0.995413i	$-0.469501\pi$
$127$	4.41682e6	0.191336	0.0956680	+	0.995413i	$0.469501\pi$
$128$	0	0
$129$	6.33236e6	0.259718
$130$	0	0
$131$	3.58480e7	1.39320	0.696602	−	0.717457i	$-0.254694\pi$
$131$	3.58480e7	1.39320	0.696602	+	0.717457i	$0.254694\pi$
$132$	0	0
$133$	2.71068e7	0.999072
$134$	0	0
$135$	−2.46038e6	−0.0860663
$136$	0	0
$137$	−1.54837e7	−0.514463	−0.257231	−	0.966350i	$-0.582810\pi$
$137$	−1.54837e7	−0.514463	−0.257231	+	0.966350i	$0.582810\pi$
$138$	0	0
$139$	4.61360e7	1.45710	0.728548	−	0.684995i	$-0.240196\pi$
$139$	4.61360e7	1.45710	0.728548	+	0.684995i	$0.240196\pi$
$140$	0	0
$141$	3.44963e7	1.03635
$142$	0	0
$143$	−2.68054e7	−0.766559
$144$	0	0
$145$	1.65517e7	0.450875
$146$	0	0
$147$	4.12023e6	0.106982
$148$	0	0
$149$	4.89613e7	1.21255	0.606276	−	0.795254i	$-0.292662\pi$
$149$	4.89613e7	1.21255	0.606276	+	0.795254i	$0.292662\pi$
$150$	0	0
$151$	−6.36953e7	−1.50553	−0.752763	−	0.658292i	$-0.771279\pi$
$151$	−6.36953e7	−1.50553	−0.752763	+	0.658292i	$0.771279\pi$
$152$	0	0
$153$	4.79828e6	0.108309
$154$	0	0
$155$	3.18010e7	0.685930
$156$	0	0
$157$	−9.10586e7	−1.87790	−0.938949	−	0.344056i	$-0.888199\pi$
$157$	−9.10586e7	−1.87790	−0.938949	+	0.344056i	$0.888199\pi$
$158$	0	0
$159$	−2.25525e7	−0.444944
$160$	0	0
$161$	−4.80168e7	−0.906781
$162$	0	0
$163$	3.38098e7	0.611485	0.305743	−	0.952114i	$-0.401095\pi$
$163$	3.38098e7	0.611485	0.305743	+	0.952114i	$0.401095\pi$
$164$	0	0
$165$	−2.71350e7	−0.470258
$166$	0	0
$167$	6.76806e7	1.12449	0.562246	−	0.826970i	$-0.309937\pi$
$167$	6.76806e7	1.12449	0.562246	+	0.826970i	$0.309937\pi$
$168$	0	0
$169$	−5.16330e7	−0.822855
$170$	0	0
$171$	2.00008e7	0.305888
$172$	0	0
$173$	4.02346e7	0.590796	0.295398	−	0.955374i	$-0.404548\pi$
$173$	4.02346e7	0.590796	0.295398	+	0.955374i	$0.404548\pi$
$174$	0	0
$175$	1.54375e7	0.217743
$176$	0	0
$177$	8.28565e7	1.12310
$178$	0	0
$179$	1.27582e8	1.66266	0.831329	−	0.555781i	$-0.187581\pi$
$179$	1.27582e8	1.66266	0.831329	+	0.555781i	$0.187581\pi$
$180$	0	0
$181$	−7.47231e7	−0.936656	−0.468328	−	0.883555i	$-0.655143\pi$
$181$	−7.47231e7	−0.936656	−0.468328	+	0.883555i	$0.655143\pi$
$182$	0	0
$183$	−2.72519e7	−0.328714
$184$	0	0
$185$	−6.49292e7	−0.753944
$186$	0	0
$187$	5.29193e7	0.591791
$188$	0	0
$189$	1.94468e7	0.209523
$190$	0	0
$191$	−5.82415e7	−0.604806	−0.302403	−	0.953180i	$-0.597789\pi$
$191$	−5.82415e7	−0.604806	−0.302403	+	0.953180i	$0.597789\pi$
$192$	0	0
$193$	1.89230e8	1.89470	0.947348	−	0.320207i	$-0.103752\pi$
$193$	1.89230e8	1.89470	0.947348	+	0.320207i	$0.103752\pi$
$194$	0	0
$195$	1.12522e7	0.108672
$196$	0	0
$197$	6.15975e7	0.574025	0.287013	−	0.957927i	$-0.407338\pi$
$197$	6.15975e7	0.574025	0.287013	+	0.957927i	$0.407338\pi$
$198$	0	0
$199$	−1.35983e8	−1.22321	−0.611604	−	0.791164i	$-0.709475\pi$
$199$	−1.35983e8	−1.22321	−0.611604	+	0.791164i	$0.709475\pi$
$200$	0	0
$201$	−8.32186e7	−0.722827
$202$	0	0
$203$	−1.30825e8	−1.09763
$204$	0	0
$205$	−1.15492e7	−0.0936301
$206$	0	0
$207$	−3.54294e7	−0.277631
$208$	0	0
$209$	2.20585e8	1.67134
$210$	0	0
$211$	−2.50058e8	−1.83254	−0.916268	−	0.400565i	$-0.868814\pi$
$211$	−2.50058e8	−1.83254	−0.916268	+	0.400565i	$0.868814\pi$
$212$	0	0
$213$	9.90014e7	0.701961
$214$	0	0
$215$	−2.93165e7	−0.201177
$216$	0	0
$217$	−2.51355e8	−1.66985
$218$	0	0
$219$	3.03174e7	0.195046
$220$	0	0
$221$	−2.19444e7	−0.136757
$222$	0	0
$223$	2.34027e8	1.41319	0.706594	−	0.707619i	$-0.250231\pi$
$223$	2.34027e8	1.41319	0.706594	+	0.707619i	$0.250231\pi$
$224$	0	0
$225$	1.13906e7	0.0666667
$226$	0	0
$227$	−3.27245e8	−1.85687	−0.928437	−	0.371489i	$-0.878847\pi$
$227$	−3.27245e8	−1.85687	−0.928437	+	0.371489i	$0.878847\pi$
$228$	0	0
$229$	−2.42853e8	−1.33635	−0.668174	−	0.744005i	$-0.732924\pi$
$229$	−2.42853e8	−1.33635	−0.668174	+	0.744005i	$0.732924\pi$
$230$	0	0
$231$	2.14475e8	1.14481
$232$	0	0
$233$	−1.35169e8	−0.700052	−0.350026	−	0.936740i	$-0.613827\pi$
$233$	−1.35169e8	−0.700052	−0.350026	+	0.936740i	$0.613827\pi$
$234$	0	0
$235$	−1.59705e8	−0.802751
$236$	0	0
$237$	1.11478e8	0.543963
$238$	0	0
$239$	−6.79202e7	−0.321815	−0.160908	−	0.986969i	$-0.551442\pi$
$239$	−6.79202e7	−0.321815	−0.160908	+	0.986969i	$0.551442\pi$
$240$	0	0
$241$	2.14382e7	0.0986574	0.0493287	−	0.998783i	$-0.484292\pi$
$241$	2.14382e7	0.0986574	0.0493287	+	0.998783i	$0.484292\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−1.90751e7	−0.0828679
$246$	0	0
$247$	−9.14716e7	−0.386231
$248$	0	0
$249$	−1.23837e8	−0.508338
$250$	0	0
$251$	−8.01294e7	−0.319841	−0.159920	−	0.987130i	$-0.551124\pi$
$251$	−8.01294e7	−0.319841	−0.159920	+	0.987130i	$0.551124\pi$
$252$	0	0
$253$	−3.90744e8	−1.51695
$254$	0	0
$255$	−2.22142e7	−0.0838960
$256$	0	0
$257$	−1.83529e7	−0.0674432	−0.0337216	−	0.999431i	$-0.510736\pi$
$257$	−1.83529e7	−0.0674432	−0.0337216	+	0.999431i	$0.510736\pi$
$258$	0	0
$259$	5.13201e8	1.83543
$260$	0	0
$261$	−9.65298e7	−0.336062
$262$	0	0
$263$	2.18329e8	0.740060	0.370030	−	0.929020i	$-0.379347\pi$
$263$	2.18329e8	0.740060	0.370030	+	0.929020i	$0.379347\pi$
$264$	0	0
$265$	1.04410e8	0.344652
$266$	0	0
$267$	−1.55619e8	−0.500350
$268$	0	0
$269$	−9.06729e7	−0.284017	−0.142009	−	0.989865i	$-0.545356\pi$
$269$	−9.06729e7	−0.284017	−0.142009	+	0.989865i	$0.545356\pi$
$270$	0	0
$271$	−2.14335e8	−0.654185	−0.327092	−	0.944992i	$-0.606069\pi$
$271$	−2.14335e8	−0.654185	−0.327092	+	0.944992i	$0.606069\pi$
$272$	0	0
$273$	−8.89378e7	−0.264556
$274$	0	0
$275$	1.25625e8	0.364260
$276$	0	0
$277$	−5.21702e7	−0.147483	−0.0737417	−	0.997277i	$-0.523494\pi$
$277$	−5.21702e7	−0.147483	−0.0737417	+	0.997277i	$0.523494\pi$
$278$	0	0
$279$	−1.85463e8	−0.511262
$280$	0	0
$281$	2.69514e8	0.724619	0.362310	−	0.932058i	$-0.381988\pi$
$281$	2.69514e8	0.724619	0.362310	+	0.932058i	$0.381988\pi$
$282$	0	0
$283$	−5.34114e8	−1.40082	−0.700408	−	0.713743i	$-0.746999\pi$
$283$	−5.34114e8	−1.40082	−0.700408	+	0.713743i	$0.746999\pi$
$284$	0	0
$285$	−9.25965e7	−0.236940
$286$	0	0
$287$	9.12853e7	0.227937
$288$	0	0
$289$	−3.67016e8	−0.894422
$290$	0	0
$291$	1.82184e8	0.433396
$292$	0	0
$293$	−4.12548e8	−0.958159	−0.479079	−	0.877772i	$-0.659029\pi$
$293$	−4.12548e8	−0.958159	−0.479079	+	0.877772i	$0.659029\pi$
$294$	0	0
$295$	−3.83595e8	−0.869953
$296$	0	0
$297$	1.58251e8	0.350509
$298$	0	0
$299$	1.62032e8	0.350552
$300$	0	0
$301$	2.31718e8	0.489752
$302$	0	0
$303$	1.54163e8	0.318369
$304$	0	0
$305$	1.26166e8	0.254621
$306$	0	0
$307$	−3.01332e8	−0.594375	−0.297187	−	0.954819i	$-0.596049\pi$
$307$	−3.01332e8	−0.594375	−0.297187	+	0.954819i	$0.596049\pi$
$308$	0	0
$309$	1.82728e8	0.352330
$310$	0	0
$311$	5.89748e8	1.11175	0.555873	−	0.831268i	$-0.312384\pi$
$311$	5.89748e8	1.11175	0.555873	+	0.831268i	$0.312384\pi$
$312$	0	0
$313$	−2.16634e8	−0.399320	−0.199660	−	0.979865i	$-0.563984\pi$
$313$	−2.16634e8	−0.399320	−0.199660	+	0.979865i	$0.563984\pi$
$314$	0	0
$315$	−9.00315e7	−0.162296
$316$	0	0
$317$	−1.47956e8	−0.260871	−0.130436	−	0.991457i	$-0.541638\pi$
$317$	−1.47956e8	−0.260871	−0.130436	+	0.991457i	$0.541638\pi$
$318$	0	0
$319$	−1.06461e9	−1.83621
$320$	0	0
$321$	−4.76419e7	−0.0803937
$322$	0	0
$323$	1.80584e8	0.298174
$324$	0	0
$325$	−5.20938e7	−0.0841771
$326$	0	0
$327$	1.73823e7	0.0274910
$328$	0	0
$329$	1.26231e9	1.95425
$330$	0	0
$331$	−3.67114e8	−0.556421	−0.278210	−	0.960520i	$-0.589741\pi$
$331$	−3.67114e8	−0.556421	−0.278210	+	0.960520i	$0.589741\pi$
$332$	0	0
$333$	3.78667e8	0.561957
$334$	0	0
$335$	3.85271e8	0.559899
$336$	0	0
$337$	6.12445e8	0.871691	0.435845	−	0.900022i	$-0.356449\pi$
$337$	6.12445e8	0.871691	0.435845	+	0.900022i	$0.356449\pi$
$338$	0	0
$339$	3.84941e8	0.536656
$340$	0	0
$341$	−2.04544e9	−2.79349
$342$	0	0
$343$	−6.62891e8	−0.886977
$344$	0	0
$345$	1.64025e8	0.215052
$346$	0	0
$347$	5.51097e8	0.708067	0.354034	−	0.935233i	$-0.384810\pi$
$347$	5.51097e8	0.708067	0.354034	+	0.935233i	$0.384810\pi$
$348$	0	0
$349$	−7.09429e6	−0.00893347	−0.00446673	−	0.999990i	$-0.501422\pi$
$349$	−7.09429e6	−0.00893347	−0.00446673	+	0.999990i	$0.501422\pi$
$350$	0	0
$351$	−6.56231e7	−0.0809994
$352$	0	0
$353$	1.31396e8	0.158990	0.0794952	−	0.996835i	$-0.474669\pi$
$353$	1.31396e8	0.158990	0.0794952	+	0.996835i	$0.474669\pi$
$354$	0	0
$355$	−4.58340e8	−0.543737
$356$	0	0
$357$	1.75581e8	0.204240
$358$	0	0
$359$	−3.06664e8	−0.349810	−0.174905	−	0.984585i	$-0.555962\pi$
$359$	−3.06664e8	−0.349810	−0.174905	+	0.984585i	$0.555962\pi$
$360$	0	0
$361$	−1.41138e8	−0.157895
$362$	0	0
$363$	1.21917e9	1.33780
$364$	0	0
$365$	−1.40358e8	−0.151082
$366$	0	0
$367$	1.32444e9	1.39863	0.699313	−	0.714815i	$-0.253489\pi$
$367$	1.32444e9	1.39863	0.699313	+	0.714815i	$0.253489\pi$
$368$	0	0
$369$	6.73552e7	0.0697877
$370$	0	0
$371$	−8.25255e8	−0.839034
$372$	0	0
$373$	−1.76524e9	−1.76126	−0.880630	−	0.473805i	$-0.842880\pi$
$373$	−1.76524e9	−1.76126	−0.880630	+	0.473805i	$0.842880\pi$
$374$	0	0
$375$	−5.27344e7	−0.0516398
$376$	0	0
$377$	4.41468e8	0.424331
$378$	0	0
$379$	−9.79148e8	−0.923871	−0.461935	−	0.886914i	$-0.652845\pi$
$379$	−9.79148e8	−0.923871	−0.461935	+	0.886914i	$0.652845\pi$
$380$	0	0
$381$	1.19254e8	0.110468
$382$	0	0
$383$	−1.53116e9	−1.39259	−0.696297	−	0.717754i	$-0.745170\pi$
$383$	−1.53116e9	−1.39259	−0.696297	+	0.717754i	$0.745170\pi$
$384$	0	0
$385$	−9.92940e8	−0.886769
$386$	0	0
$387$	1.70974e8	0.149948
$388$	0	0
$389$	1.13175e9	0.974825	0.487413	−	0.873172i	$-0.337941\pi$
$389$	1.13175e9	0.974825	0.487413	+	0.873172i	$0.337941\pi$
$390$	0	0
$391$	−3.19885e8	−0.270630
$392$	0	0
$393$	9.67895e8	0.804367
$394$	0	0
$395$	−5.16101e8	−0.421352
$396$	0	0
$397$	−2.19817e8	−0.176317	−0.0881584	−	0.996106i	$-0.528098\pi$
$397$	−2.19817e8	−0.176317	−0.0881584	+	0.996106i	$0.528098\pi$
$398$	0	0
$399$	7.31883e8	0.576815
$400$	0	0
$401$	9.88276e8	0.765373	0.382686	−	0.923878i	$-0.374999\pi$
$401$	9.88276e8	0.765373	0.382686	+	0.923878i	$0.374999\pi$
$402$	0	0
$403$	8.48196e8	0.645548
$404$	0	0
$405$	−6.64301e7	−0.0496904
$406$	0	0
$407$	4.17625e9	3.07048
$408$	0	0
$409$	1.94597e9	1.40638	0.703192	−	0.711000i	$-0.251757\pi$
$409$	1.94597e9	1.40638	0.703192	+	0.711000i	$0.251757\pi$
$410$	0	0
$411$	−4.18061e8	−0.297025
$412$	0	0
$413$	3.03193e9	2.11785
$414$	0	0
$415$	5.73320e8	0.393757
$416$	0	0
$417$	1.24567e9	0.841255
$418$	0	0
$419$	−1.07893e9	−0.716549	−0.358275	−	0.933616i	$-0.616635\pi$
$419$	−1.07893e9	−0.716549	−0.358275	+	0.933616i	$0.616635\pi$
$420$	0	0
$421$	−2.53361e9	−1.65483	−0.827414	−	0.561592i	$-0.810189\pi$
$421$	−2.53361e9	−1.65483	−0.827414	+	0.561592i	$0.810189\pi$
$422$	0	0
$423$	9.31400e8	0.598335
$424$	0	0
$425$	1.02844e8	0.0649855
$426$	0	0
$427$	−9.97218e8	−0.619859
$428$	0	0
$429$	−7.23745e8	−0.442573
$430$	0	0
$431$	−1.84499e9	−1.11000	−0.555000	−	0.831850i	$-0.687282\pi$
$431$	−1.84499e9	−1.11000	−0.555000	+	0.831850i	$0.687282\pi$
$432$	0	0
$433$	6.06893e8	0.359256	0.179628	−	0.983735i	$-0.442511\pi$
$433$	6.06893e8	0.359256	0.179628	+	0.983735i	$0.442511\pi$
$434$	0	0
$435$	4.46897e8	0.260313
$436$	0	0
$437$	−1.33339e9	−0.764314
$438$	0	0
$439$	2.71714e9	1.53280	0.766400	−	0.642364i	$-0.222046\pi$
$439$	2.71714e9	1.53280	0.766400	+	0.642364i	$0.222046\pi$
$440$	0	0
$441$	1.11246e8	0.0617661
$442$	0	0
$443$	2.85842e9	1.56212	0.781058	−	0.624458i	$-0.214680\pi$
$443$	2.85842e9	1.56212	0.781058	+	0.624458i	$0.214680\pi$
$444$	0	0
$445$	7.20460e8	0.387570
$446$	0	0
$447$	1.32195e9	0.700068
$448$	0	0
$449$	−7.25038e7	−0.0378006	−0.0189003	−	0.999821i	$-0.506017\pi$
$449$	−7.25038e7	−0.0378006	−0.0189003	+	0.999821i	$0.506017\pi$
$450$	0	0
$451$	7.42848e8	0.381313
$452$	0	0
$453$	−1.71977e9	−0.869215
$454$	0	0
$455$	4.11749e8	0.204924
$456$	0	0
$457$	1.29110e9	0.632778	0.316389	−	0.948629i	$-0.397529\pi$
$457$	1.29110e9	0.632778	0.316389	+	0.948629i	$0.397529\pi$
$458$	0	0
$459$	1.29554e8	0.0625324
$460$	0	0
$461$	1.25755e9	0.597822	0.298911	−	0.954281i	$-0.403377\pi$
$461$	1.25755e9	0.597822	0.298911	+	0.954281i	$0.403377\pi$
$462$	0	0
$463$	−1.06437e9	−0.498380	−0.249190	−	0.968455i	$-0.580164\pi$
$463$	−1.06437e9	−0.498380	−0.249190	+	0.968455i	$0.580164\pi$
$464$	0	0
$465$	8.58627e8	0.396022
$466$	0	0
$467$	2.43398e9	1.10588	0.552940	−	0.833221i	$-0.313506\pi$
$467$	2.43398e9	1.10588	0.552940	+	0.833221i	$0.313506\pi$
$468$	0	0
$469$	−3.04519e9	−1.36304
$470$	0	0
$471$	−2.45858e9	−1.08420
$472$	0	0
$473$	1.88564e9	0.819302
$474$	0	0
$475$	4.28687e8	0.183533
$476$	0	0
$477$	−6.08918e8	−0.256888
$478$	0	0
$479$	1.38642e9	0.576396	0.288198	−	0.957571i	$-0.406944\pi$
$479$	1.38642e9	0.576396	0.288198	+	0.957571i	$0.406944\pi$
$480$	0	0
$481$	−1.73179e9	−0.709558
$482$	0	0
$483$	−1.29645e9	−0.523531
$484$	0	0
$485$	−8.43444e8	−0.335707
$486$	0	0
$487$	−2.08323e9	−0.817310	−0.408655	−	0.912689i	$-0.634002\pi$
$487$	−2.08323e9	−0.817310	−0.408655	+	0.912689i	$0.634002\pi$
$488$	0	0
$489$	9.12865e8	0.353041
$490$	0	0
$491$	−3.32501e8	−0.126767	−0.0633837	−	0.997989i	$-0.520189\pi$
$491$	−3.32501e8	−0.126767	−0.0633837	+	0.997989i	$0.520189\pi$
$492$	0	0
$493$	−8.71549e8	−0.327588
$494$	0	0
$495$	−7.32645e8	−0.271503
$496$	0	0
$497$	3.62272e9	1.32369
$498$	0	0
$499$	−4.21560e8	−0.151882	−0.0759412	−	0.997112i	$-0.524196\pi$
$499$	−4.21560e8	−0.151882	−0.0759412	+	0.997112i	$0.524196\pi$
$500$	0	0
$501$	1.82738e9	0.649226
$502$	0	0
$503$	8.08436e8	0.283242	0.141621	−	0.989921i	$-0.454769\pi$
$503$	8.08436e8	0.283242	0.141621	+	0.989921i	$0.454769\pi$
$504$	0	0
$505$	−7.13717e8	−0.246608
$506$	0	0
$507$	−1.39409e9	−0.475076
$508$	0	0
$509$	−1.45321e8	−0.0488445	−0.0244223	−	0.999702i	$-0.507775\pi$
$509$	−1.45321e8	−0.0488445	−0.0244223	+	0.999702i	$0.507775\pi$
$510$	0	0
$511$	1.10939e9	0.367800
$512$	0	0
$513$	5.40023e8	0.176604
$514$	0	0
$515$	−8.45962e8	−0.272914
$516$	0	0
$517$	1.02722e10	3.26925
$518$	0	0
$519$	1.08633e9	0.341096
$520$	0	0
$521$	−1.93158e8	−0.0598383	−0.0299192	−	0.999552i	$-0.509525\pi$
$521$	−1.93158e8	−0.0598383	−0.0299192	+	0.999552i	$0.509525\pi$
$522$	0	0
$523$	2.19458e9	0.670802	0.335401	−	0.942075i	$-0.391128\pi$
$523$	2.19458e9	0.670802	0.335401	+	0.942075i	$0.391128\pi$
$524$	0	0
$525$	4.16813e8	0.125714
$526$	0	0
$527$	−1.67451e9	−0.498370
$528$	0	0
$529$	−1.04287e9	−0.306290
$530$	0	0
$531$	2.23713e9	0.648425
$532$	0	0
$533$	−3.08042e8	−0.0881179
$534$	0	0
$535$	2.20564e8	0.0622727
$536$	0	0
$537$	3.44471e9	0.959936
$538$	0	0
$539$	1.22691e9	0.337484
$540$	0	0
$541$	−6.29137e9	−1.70826	−0.854131	−	0.520058i	$-0.825910\pi$
$541$	−6.29137e9	−1.70826	−0.854131	+	0.520058i	$0.825910\pi$
$542$	0	0
$543$	−2.01752e9	−0.540779
$544$	0	0
$545$	−8.04738e7	−0.0212945
$546$	0	0
$547$	−2.61205e9	−0.682379	−0.341190	−	0.939994i	$-0.610830\pi$
$547$	−2.61205e9	−0.682379	−0.341190	+	0.939994i	$0.610830\pi$
$548$	0	0
$549$	−7.35802e8	−0.189783
$550$	0	0
$551$	−3.63291e9	−0.925175
$552$	0	0
$553$	4.07926e9	1.02576
$554$	0	0
$555$	−1.75309e9	−0.435290
$556$	0	0
$557$	−6.03816e9	−1.48051	−0.740255	−	0.672327i	$-0.765295\pi$
$557$	−6.03816e9	−1.48051	−0.740255	+	0.672327i	$0.765295\pi$
$558$	0	0
$559$	−7.81930e8	−0.189333
$560$	0	0
$561$	1.42882e9	0.341671
$562$	0	0
$563$	−8.16188e8	−0.192757	−0.0963787	−	0.995345i	$-0.530726\pi$
$563$	−8.16188e8	−0.192757	−0.0963787	+	0.995345i	$0.530726\pi$
$564$	0	0
$565$	−1.78214e9	−0.415692
$566$	0	0
$567$	5.25064e8	0.120968
$568$	0	0
$569$	−3.35516e9	−0.763520	−0.381760	−	0.924261i	$-0.624682\pi$
$569$	−3.35516e9	−0.763520	−0.381760	+	0.924261i	$0.624682\pi$
$570$	0	0
$571$	−6.86780e9	−1.54380	−0.771900	−	0.635744i	$-0.780694\pi$
$571$	−6.86780e9	−1.54380	−0.771900	+	0.635744i	$0.780694\pi$
$572$	0	0
$573$	−1.57252e9	−0.349185
$574$	0	0
$575$	−7.59375e8	−0.166578
$576$	0	0
$577$	−2.92115e9	−0.633051	−0.316525	−	0.948584i	$-0.602516\pi$
$577$	−2.92115e9	−0.633051	−0.316525	+	0.948584i	$0.602516\pi$
$578$	0	0
$579$	5.10921e9	1.09390
$580$	0	0
$581$	−4.53152e9	−0.958577
$582$	0	0
$583$	−6.71564e9	−1.40361
$584$	0	0
$585$	3.03811e8	0.0627419
$586$	0	0
$587$	5.24861e9	1.07105	0.535526	−	0.844519i	$-0.320113\pi$
$587$	5.24861e9	1.07105	0.535526	+	0.844519i	$0.320113\pi$
$588$	0	0
$589$	−6.97994e9	−1.40750
$590$	0	0
$591$	1.66313e9	0.331414
$592$	0	0
$593$	−3.02365e9	−0.595443	−0.297722	−	0.954653i	$-0.596227\pi$
$593$	−3.02365e9	−0.595443	−0.297722	+	0.954653i	$0.596227\pi$
$594$	0	0
$595$	−8.12877e8	−0.158203
$596$	0	0
$597$	−3.67155e9	−0.706219
$598$	0	0
$599$	8.11227e8	0.154223	0.0771114	−	0.997022i	$-0.475430\pi$
$599$	8.11227e8	0.154223	0.0771114	+	0.997022i	$0.475430\pi$
$600$	0	0
$601$	−7.55714e9	−1.42003	−0.710014	−	0.704188i	$-0.751311\pi$
$601$	−7.55714e9	−1.42003	−0.710014	+	0.704188i	$0.751311\pi$
$602$	0	0
$603$	−2.24690e9	−0.417324
$604$	0	0
$605$	−5.64430e9	−1.03625
$606$	0	0
$607$	3.10246e9	0.563048	0.281524	−	0.959554i	$-0.409160\pi$
$607$	3.10246e9	0.563048	0.281524	+	0.959554i	$0.409160\pi$
$608$	0	0
$609$	−3.53228e9	−0.633715
$610$	0	0
$611$	−4.25965e9	−0.755492
$612$	0	0
$613$	−1.82934e9	−0.320762	−0.160381	−	0.987055i	$-0.551272\pi$
$613$	−1.82934e9	−0.320762	−0.160381	+	0.987055i	$0.551272\pi$
$614$	0	0
$615$	−3.11830e8	−0.0540573
$616$	0	0
$617$	1.14156e10	1.95659	0.978294	−	0.207220i	$-0.0664416\pi$
$617$	1.14156e10	1.95659	0.978294	+	0.207220i	$0.0664416\pi$
$618$	0	0
$619$	5.65578e9	0.958462	0.479231	−	0.877689i	$-0.340916\pi$
$619$	5.65578e9	0.958462	0.479231	+	0.877689i	$0.340916\pi$
$620$	0	0
$621$	−9.56594e8	−0.160290
$622$	0	0
$623$	−5.69451e9	−0.943514
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	5.95581e9	0.964949
$628$	0	0
$629$	3.41891e9	0.547786
$630$	0	0
$631$	−3.93985e9	−0.624277	−0.312138	−	0.950037i	$-0.601045\pi$
$631$	−3.93985e9	−0.624277	−0.312138	+	0.950037i	$0.601045\pi$
$632$	0	0
$633$	−6.75157e9	−1.05802
$634$	0	0
$635$	−5.52102e8	−0.0855681
$636$	0	0
$637$	−5.08772e8	−0.0779893
$638$	0	0
$639$	2.67304e9	0.405277
$640$	0	0
$641$	5.45282e9	0.817745	0.408873	−	0.912591i	$-0.365922\pi$
$641$	5.45282e9	0.817745	0.408873	+	0.912591i	$0.365922\pi$
$642$	0	0
$643$	−6.55463e7	−0.00972322	−0.00486161	−	0.999988i	$-0.501548\pi$
$643$	−6.55463e7	−0.00972322	−0.00486161	+	0.999988i	$0.501548\pi$
$644$	0	0
$645$	−7.91546e8	−0.116149
$646$	0	0
$647$	5.34631e9	0.776048	0.388024	−	0.921649i	$-0.373158\pi$
$647$	5.34631e9	0.776048	0.388024	+	0.921649i	$0.373158\pi$
$648$	0	0
$649$	2.46728e10	3.54293
$650$	0	0
$651$	−6.78659e9	−0.964091
$652$	0	0
$653$	−3.07490e9	−0.432151	−0.216075	−	0.976377i	$-0.569326\pi$
$653$	−3.07490e9	−0.432151	−0.216075	+	0.976377i	$0.569326\pi$
$654$	0	0
$655$	−4.48100e9	−0.623060
$656$	0	0
$657$	8.18569e8	0.112610
$658$	0	0
$659$	−9.34559e9	−1.27206	−0.636031	−	0.771664i	$-0.719425\pi$
$659$	−9.34559e9	−1.27206	−0.636031	+	0.771664i	$0.719425\pi$
$660$	0	0
$661$	−3.45515e9	−0.465331	−0.232666	−	0.972557i	$-0.574745\pi$
$661$	−3.45515e9	−0.465331	−0.232666	+	0.972557i	$0.574745\pi$
$662$	0	0
$663$	−5.92498e8	−0.0789569
$664$	0	0
$665$	−3.38835e9	−0.446799
$666$	0	0
$667$	6.43532e9	0.839711
$668$	0	0
$669$	6.31874e9	0.815904
$670$	0	0
$671$	−8.11501e9	−1.03696
$672$	0	0
$673$	4.36251e9	0.551676	0.275838	−	0.961204i	$-0.411045\pi$
$673$	4.36251e9	0.551676	0.275838	+	0.961204i	$0.411045\pi$
$674$	0	0
$675$	3.07547e8	0.0384900
$676$	0	0
$677$	−1.58108e9	−0.195836	−0.0979182	−	0.995194i	$-0.531218\pi$
$677$	−1.58108e9	−0.195836	−0.0979182	+	0.995194i	$0.531218\pi$
$678$	0	0
$679$	6.66658e9	0.817258
$680$	0	0
$681$	−8.83562e9	−1.07207
$682$	0	0
$683$	6.54644e9	0.786200	0.393100	−	0.919496i	$-0.371403\pi$
$683$	6.54644e9	0.786200	0.393100	+	0.919496i	$0.371403\pi$
$684$	0	0
$685$	1.93547e9	0.230075
$686$	0	0
$687$	−6.55704e9	−0.771541
$688$	0	0
$689$	2.78482e9	0.324362
$690$	0	0
$691$	−1.13916e10	−1.31345	−0.656724	−	0.754131i	$-0.728058\pi$
$691$	−1.13916e10	−1.31345	−0.656724	+	0.754131i	$0.728058\pi$
$692$	0	0
$693$	5.79083e9	0.660959
$694$	0	0
$695$	−5.76700e9	−0.651633
$696$	0	0
$697$	6.08137e8	0.0680279
$698$	0	0
$699$	−3.64955e9	−0.404175
$700$	0	0
$701$	−9.60390e9	−1.05301	−0.526507	−	0.850171i	$-0.676499\pi$
$701$	−9.60390e9	−1.05301	−0.526507	+	0.850171i	$0.676499\pi$
$702$	0	0
$703$	1.42512e10	1.54706
$704$	0	0
$705$	−4.31204e9	−0.463469
$706$	0	0
$707$	5.64122e9	0.600351
$708$	0	0
$709$	−1.93713e9	−0.204126	−0.102063	−	0.994778i	$-0.532544\pi$
$709$	−1.93713e9	−0.204126	−0.102063	+	0.994778i	$0.532544\pi$
$710$	0	0
$711$	3.00990e9	0.314057
$712$	0	0
$713$	1.23642e10	1.27748
$714$	0	0
$715$	3.35067e9	0.342816
$716$	0	0
$717$	−1.83385e9	−0.185800
$718$	0	0
$719$	−7.40065e9	−0.742538	−0.371269	−	0.928525i	$-0.621077\pi$
$719$	−7.40065e9	−0.742538	−0.371269	+	0.928525i	$0.621077\pi$
$720$	0	0
$721$	6.68648e9	0.664392
$722$	0	0
$723$	5.78833e8	0.0569599
$724$	0	0
$725$	−2.06897e9	−0.201637
$726$	0	0
$727$	−4.30049e8	−0.0415095	−0.0207547	−	0.999785i	$-0.506607\pi$
$727$	−4.30049e8	−0.0415095	−0.0207547	+	0.999785i	$0.506607\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.54369e9	0.146167
$732$	0	0
$733$	−1.13354e8	−0.0106309	−0.00531547	−	0.999986i	$-0.501692\pi$
$733$	−1.13354e8	−0.0106309	−0.00531547	+	0.999986i	$0.501692\pi$
$734$	0	0
$735$	−5.15028e8	−0.0478438
$736$	0	0
$737$	−2.47807e10	−2.28022
$738$	0	0
$739$	−5.90138e9	−0.537896	−0.268948	−	0.963155i	$-0.586676\pi$
$739$	−5.90138e9	−0.537896	−0.268948	+	0.963155i	$0.586676\pi$
$740$	0	0
$741$	−2.46973e9	−0.222991
$742$	0	0
$743$	−2.64871e9	−0.236905	−0.118453	−	0.992960i	$-0.537793\pi$
$743$	−2.64871e9	−0.236905	−0.118453	+	0.992960i	$0.537793\pi$
$744$	0	0
$745$	−6.12016e9	−0.542270
$746$	0	0
$747$	−3.34360e9	−0.293489
$748$	0	0
$749$	−1.74334e9	−0.151599
$750$	0	0
$751$	1.44933e10	1.24861	0.624305	−	0.781181i	$-0.285382\pi$
$751$	1.44933e10	1.24861	0.624305	+	0.781181i	$0.285382\pi$
$752$	0	0
$753$	−2.16349e9	−0.184660
$754$	0	0
$755$	7.96191e9	0.673291
$756$	0	0
$757$	−2.02665e9	−0.169802	−0.0849012	−	0.996389i	$-0.527057\pi$
$757$	−2.02665e9	−0.169802	−0.0849012	+	0.996389i	$0.527057\pi$
$758$	0	0
$759$	−1.05501e10	−0.875810
$760$	0	0
$761$	−1.92349e10	−1.58214	−0.791069	−	0.611727i	$-0.790475\pi$
$761$	−1.92349e10	−1.58214	−0.791069	+	0.611727i	$0.790475\pi$
$762$	0	0
$763$	6.36065e8	0.0518400
$764$	0	0
$765$	−5.99785e8	−0.0484374
$766$	0	0
$767$	−1.02312e10	−0.818738
$768$	0	0
$769$	6.35399e9	0.503854	0.251927	−	0.967746i	$-0.418936\pi$
$769$	6.35399e9	0.503854	0.251927	+	0.967746i	$0.418936\pi$
$770$	0	0
$771$	−4.95527e8	−0.0389383
$772$	0	0
$773$	−3.27598e9	−0.255102	−0.127551	−	0.991832i	$-0.540712\pi$
$773$	−3.27598e9	−0.255102	−0.127551	+	0.991832i	$0.540712\pi$
$774$	0	0
$775$	−3.97513e9	−0.306757
$776$	0	0
$777$	1.38564e10	1.05969
$778$	0	0
$779$	2.53492e9	0.192125
$780$	0	0
$781$	2.94804e10	2.21440
$782$	0	0
$783$	−2.60630e9	−0.194026
$784$	0	0
$785$	1.13823e10	0.839822
$786$	0	0
$787$	1.21823e10	0.890875	0.445437	−	0.895313i	$-0.353048\pi$
$787$	1.21823e10	0.890875	0.445437	+	0.895313i	$0.353048\pi$
$788$	0	0
$789$	5.89489e9	0.427274
$790$	0	0
$791$	1.40860e10	1.01198
$792$	0	0
$793$	3.36511e9	0.239631
$794$	0	0
$795$	2.81906e9	0.198985
$796$	0	0
$797$	2.40548e10	1.68306	0.841528	−	0.540214i	$-0.181657\pi$
$797$	2.40548e10	1.68306	0.841528	+	0.540214i	$0.181657\pi$
$798$	0	0
$799$	8.40943e9	0.583247
$800$	0	0
$801$	−4.20172e9	−0.288877
$802$	0	0
$803$	9.02784e9	0.615290
$804$	0	0
$805$	6.00210e9	0.405525
$806$	0	0
$807$	−2.44817e9	−0.163977
$808$	0	0
$809$	7.44389e8	0.0494288	0.0247144	−	0.999695i	$-0.492132\pi$
$809$	7.44389e8	0.0494288	0.0247144	+	0.999695i	$0.492132\pi$
$810$	0	0
$811$	−1.18071e10	−0.777264	−0.388632	−	0.921393i	$-0.627052\pi$
$811$	−1.18071e10	−0.777264	−0.388632	+	0.921393i	$0.627052\pi$
$812$	0	0
$813$	−5.78704e9	−0.377694
$814$	0	0
$815$	−4.22623e9	−0.273465
$816$	0	0
$817$	6.43462e9	0.412806
$818$	0	0
$819$	−2.40132e9	−0.152741
$820$	0	0
$821$	−2.93131e10	−1.84867	−0.924336	−	0.381579i	$-0.875381\pi$
$821$	−2.93131e10	−1.84867	−0.924336	+	0.381579i	$0.875381\pi$
$822$	0	0
$823$	2.03889e10	1.27496	0.637478	−	0.770469i	$-0.279978\pi$
$823$	2.03889e10	1.27496	0.637478	+	0.770469i	$0.279978\pi$
$824$	0	0
$825$	3.39188e9	0.210306
$826$	0	0
$827$	−2.41235e10	−1.48310	−0.741550	−	0.670898i	$-0.765909\pi$
$827$	−2.41235e10	−1.48310	−0.741550	+	0.670898i	$0.765909\pi$
$828$	0	0
$829$	−6.32929e9	−0.385846	−0.192923	−	0.981214i	$-0.561797\pi$
$829$	−6.32929e9	−0.385846	−0.192923	+	0.981214i	$0.561797\pi$
$830$	0	0
$831$	−1.40859e9	−0.0851496
$832$	0	0
$833$	1.00442e9	0.0602085
$834$	0	0
$835$	−8.46008e9	−0.502888
$836$	0	0
$837$	−5.00751e9	−0.295177
$838$	0	0
$839$	2.01943e10	1.18049	0.590243	−	0.807225i	$-0.299032\pi$
$839$	2.01943e10	1.18049	0.590243	+	0.807225i	$0.299032\pi$
$840$	0	0
$841$	2.83591e8	0.0164402
$842$	0	0
$843$	7.27689e9	0.418359
$844$	0	0
$845$	6.45412e9	0.367992
$846$	0	0
$847$	4.46126e10	2.52270
$848$	0	0
$849$	−1.44211e10	−0.808762
$850$	0	0
$851$	−2.52445e10	−1.40415
$852$	0	0
$853$	−1.07171e10	−0.591229	−0.295614	−	0.955307i	$-0.595524\pi$
$853$	−1.07171e10	−0.591229	−0.295614	+	0.955307i	$0.595524\pi$
$854$	0	0
$855$	−2.50011e9	−0.136797
$856$	0	0
$857$	8.85297e9	0.480458	0.240229	−	0.970716i	$-0.422777\pi$
$857$	8.85297e9	0.480458	0.240229	+	0.970716i	$0.422777\pi$
$858$	0	0
$859$	2.37823e10	1.28020	0.640100	−	0.768292i	$-0.278893\pi$
$859$	2.37823e10	1.28020	0.640100	+	0.768292i	$0.278893\pi$
$860$	0	0
$861$	2.46470e9	0.131599
$862$	0	0
$863$	2.99137e10	1.58428	0.792140	−	0.610340i	$-0.208967\pi$
$863$	2.99137e10	1.58428	0.792140	+	0.610340i	$0.208967\pi$
$864$	0	0
$865$	−5.02932e9	−0.264212
$866$	0	0
$867$	−9.90943e9	−0.516395
$868$	0	0
$869$	3.31956e10	1.71598
$870$	0	0
$871$	1.02760e10	0.526937
$872$	0	0
$873$	4.91897e9	0.250221
$874$	0	0
$875$	−1.92969e9	−0.0973775
$876$	0	0
$877$	9.04648e9	0.452878	0.226439	−	0.974025i	$-0.427292\pi$
$877$	9.04648e9	0.452878	0.226439	+	0.974025i	$0.427292\pi$
$878$	0	0
$879$	−1.11388e10	−0.553193
$880$	0	0
$881$	−4.04413e9	−0.199255	−0.0996275	−	0.995025i	$-0.531765\pi$
$881$	−4.04413e9	−0.199255	−0.0996275	+	0.995025i	$0.531765\pi$
$882$	0	0
$883$	−2.37982e10	−1.16328	−0.581638	−	0.813448i	$-0.697588\pi$
$883$	−2.37982e10	−1.16328	−0.581638	+	0.813448i	$0.697588\pi$
$884$	0	0
$885$	−1.03571e10	−0.502268
$886$	0	0
$887$	−7.17840e9	−0.345378	−0.172689	−	0.984976i	$-0.555246\pi$
$887$	−7.17840e9	−0.345378	−0.172689	+	0.984976i	$0.555246\pi$
$888$	0	0
$889$	4.36382e9	0.208310
$890$	0	0
$891$	4.27279e9	0.202367
$892$	0	0
$893$	3.50533e10	1.64721
$894$	0	0
$895$	−1.59477e10	−0.743563
$896$	0	0
$897$	4.37487e9	0.202391
$898$	0	0
$899$	3.36872e10	1.54634
$900$	0	0
$901$	−5.49780e9	−0.250410
$902$	0	0
$903$	6.25638e9	0.282759
$904$	0	0
$905$	9.34039e9	0.418885
$906$	0	0
$907$	1.61336e10	0.717969	0.358985	−	0.933343i	$-0.383123\pi$
$907$	1.61336e10	0.717969	0.358985	+	0.933343i	$0.383123\pi$
$908$	0	0
$909$	4.16240e9	0.183810
$910$	0	0
$911$	−9.90491e9	−0.434047	−0.217023	−	0.976166i	$-0.569635\pi$
$911$	−9.90491e9	−0.434047	−0.217023	+	0.976166i	$0.569635\pi$
$912$	0	0
$913$	−3.68759e10	−1.60360
$914$	0	0
$915$	3.40649e9	0.147005
$916$	0	0
$917$	3.54178e10	1.51680
$918$	0	0
$919$	1.07988e10	0.458958	0.229479	−	0.973314i	$-0.426298\pi$
$919$	1.07988e10	0.458958	0.229479	+	0.973314i	$0.426298\pi$
$920$	0	0
$921$	−8.13595e9	−0.343162
$922$	0	0
$923$	−1.22248e10	−0.511726
$924$	0	0
$925$	8.11616e9	0.337174
$926$	0	0
$927$	4.93365e9	0.203418
$928$	0	0
$929$	−1.24984e10	−0.511444	−0.255722	−	0.966750i	$-0.582313\pi$
$929$	−1.24984e10	−0.511444	−0.255722	+	0.966750i	$0.582313\pi$
$930$	0	0
$931$	4.18676e9	0.170041
$932$	0	0
$933$	1.59232e10	0.641866
$934$	0	0
$935$	−6.61491e9	−0.264657
$936$	0	0
$937$	4.89405e9	0.194348	0.0971739	−	0.995267i	$-0.469020\pi$
$937$	4.89405e9	0.194348	0.0971739	+	0.995267i	$0.469020\pi$
$938$	0	0
$939$	−5.84912e9	−0.230548
$940$	0	0
$941$	1.32199e10	0.517209	0.258604	−	0.965983i	$-0.416737\pi$
$941$	1.32199e10	0.517209	0.258604	+	0.965983i	$0.416737\pi$
$942$	0	0
$943$	−4.49035e9	−0.174377
$944$	0	0
$945$	−2.43085e9	−0.0937016
$946$	0	0
$947$	−6.88319e9	−0.263369	−0.131684	−	0.991292i	$-0.542039\pi$
$947$	−6.88319e9	−0.263369	−0.131684	+	0.991292i	$0.542039\pi$
$948$	0	0
$949$	−3.74364e9	−0.142188
$950$	0	0
$951$	−3.99482e9	−0.150614
$952$	0	0
$953$	−2.56888e10	−0.961433	−0.480717	−	0.876876i	$-0.659623\pi$
$953$	−2.56888e10	−0.961433	−0.480717	+	0.876876i	$0.659623\pi$
$954$	0	0
$955$	7.28019e9	0.270477
$956$	0	0
$957$	−2.87444e10	−1.06014
$958$	0	0
$959$	−1.52979e10	−0.560103
$960$	0	0
$961$	3.72108e10	1.35250
$962$	0	0
$963$	−1.28633e9	−0.0464153
$964$	0	0
$965$	−2.36537e10	−0.847334
$966$	0	0
$967$	1.17717e9	0.0418647	0.0209323	−	0.999781i	$-0.493337\pi$
$967$	1.17717e9	0.0418647	0.0209323	+	0.999781i	$0.493337\pi$
$968$	0	0
$969$	4.87576e9	0.172151
$970$	0	0
$971$	3.53867e9	0.124043	0.0620215	−	0.998075i	$-0.480245\pi$
$971$	3.53867e9	0.124043	0.0620215	+	0.998075i	$0.480245\pi$
$972$	0	0
$973$	4.55824e10	1.58636
$974$	0	0
$975$	−1.40653e9	−0.0485997
$976$	0	0
$977$	−9.98892e9	−0.342679	−0.171340	−	0.985212i	$-0.554810\pi$
$977$	−9.98892e9	−0.342679	−0.171340	+	0.985212i	$0.554810\pi$
$978$	0	0
$979$	−4.63400e10	−1.57840
$980$	0	0
$981$	4.69323e8	0.0158719
$982$	0	0
$983$	−4.58129e10	−1.53833	−0.769167	−	0.639048i	$-0.779329\pi$
$983$	−4.58129e10	−1.53833	−0.769167	+	0.639048i	$0.779329\pi$
$984$	0	0
$985$	−7.69968e9	−0.256712
$986$	0	0
$987$	3.40823e10	1.12829
$988$	0	0
$989$	−1.13983e10	−0.374672
$990$	0	0
$991$	−2.31266e10	−0.754840	−0.377420	−	0.926042i	$-0.623189\pi$
$991$	−2.31266e10	−0.754840	−0.377420	+	0.926042i	$0.623189\pi$
$992$	0	0
$993$	−9.91208e9	−0.321250
$994$	0	0
$995$	1.69979e10	0.547035
$996$	0	0
$997$	4.79092e10	1.53104	0.765519	−	0.643413i	$-0.222482\pi$
$997$	4.79092e10	1.53104	0.765519	+	0.643413i	$0.222482\pi$
$998$	0	0
$999$	1.02240e10	0.324446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.a.j.1.1		1
4.3	odd	2		30.8.a.d.1.1	✓	1
12.11	even	2		90.8.a.c.1.1		1
20.3	even	4		150.8.c.a.49.1		2
20.7	even	4		150.8.c.a.49.2		2
20.19	odd	2		150.8.a.i.1.1		1
60.23	odd	4		450.8.c.r.199.2		2
60.47	odd	4		450.8.c.r.199.1		2
60.59	even	2		450.8.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.8.a.d.1.1	✓	1	4.3	odd	2
90.8.a.c.1.1		1	12.11	even	2
150.8.a.i.1.1		1	20.19	odd	2
150.8.c.a.49.1		2	20.3	even	4
150.8.c.a.49.2		2	20.7	even	4
240.8.a.j.1.1		1	1.1	even	1	trivial
450.8.a.x.1.1		1	60.59	even	2
450.8.c.r.199.1		2	60.47	odd	4
450.8.c.r.199.2		2	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more