LMFDB - Embedded newform 800.5.e.e.399.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,5,Mod(399,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("800.399");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	800.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$82.6959704671$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			399.20
Character	$\chi$	$=$	800.399
Dual form			800.5.e.e.399.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.715641i q^{3} -25.0983 q^{7} +80.4879 q^{9} +156.704 q^{11} -301.788 q^{13} +123.427i q^{17} +322.113 q^{19} -17.9613i q^{21} -45.9062 q^{23} +115.567i q^{27} -1084.15i q^{29} +1624.75i q^{31} +112.144i q^{33} +895.053 q^{37} -215.972i q^{39} -1097.66 q^{41} -950.337i q^{43} +1343.83 q^{47} -1771.08 q^{49} -88.3294 q^{51} -709.918 q^{53} +230.517i q^{57} +4466.60 q^{59} -933.943i q^{61} -2020.10 q^{63} -4019.28i q^{67} -32.8523i q^{69} -3551.94i q^{71} +5298.05i q^{73} -3933.01 q^{77} +11461.9i q^{79} +6436.81 q^{81} +6588.95i q^{83} +775.861 q^{87} +6280.53 q^{89} +7574.35 q^{91} -1162.73 q^{93} +14960.7i q^{97} +12612.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 864 q^{9} - 384 q^{11} - 1408 q^{19} - 4416 q^{41} + 4960 q^{49} + 35584 q^{51} + 28032 q^{59} + 20768 q^{81} - 13632 q^{89} - 49152 q^{91} + 5248 q^{99}+O(q^{100}) $ 32 * q - 864 * q^9 - 384 * q^11 - 1408 * q^19 - 4416 * q^41 + 4960 * q^49 + 35584 * q^51 + 28032 * q^59 + 20768 * q^81 - 13632 * q^89 - 49152 * q^91 + 5248 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.715641i	0.0795156i	0.999209	+	0.0397578i	$0.0126586\pi$
$3$	0.715641i	0.0795156i	−0.999209	+	0.0397578i	$0.987341\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−25.0983	−0.512209	−0.256105	−	0.966649i	$-0.582439\pi$
$7$	−25.0983	−0.512209	−0.256105	+	0.966649i	$0.582439\pi$
$8$	0	0
$9$	80.4879	0.993677
$10$	0	0
$11$	156.704	1.29508	0.647539	−	0.762033i	$-0.275798\pi$
$11$	156.704	1.29508	0.647539	+	0.762033i	$0.275798\pi$
$12$	0	0
$13$	−301.788	−1.78573	−0.892863	−	0.450328i	$-0.851307\pi$
$13$	−301.788	−1.78573	−0.892863	+	0.450328i	$0.851307\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	123.427i	0.427083i	0.976934	+	0.213542i	$0.0684999\pi$
$17$	123.427i	0.427083i	−0.976934	+	0.213542i	$0.931500\pi$
$18$	0	0
$19$	322.113	0.892280	0.446140	−	0.894963i	$-0.352798\pi$
$19$	322.113	0.892280	0.446140	+	0.894963i	$0.352798\pi$
$20$	0	0
$21$	− 17.9613i	− 0.0407286i
$22$	0	0
$23$	−45.9062	−0.0867792	−0.0433896	−	0.999058i	$-0.513816\pi$
$23$	−45.9062	−0.0867792	−0.0433896	+	0.999058i	$0.513816\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	115.567i	0.158528i
$28$	0	0
$29$	− 1084.15i	− 1.28912i	−0.764554	−	0.644560i	$-0.777041\pi$
$29$	− 1084.15i	− 1.28912i	0.764554	−	0.644560i	$-0.222959\pi$
$30$	0	0
$31$	1624.75i	1.69068i	0.534227	+	0.845341i	$0.320603\pi$
$31$	1624.75i	1.69068i	−0.534227	+	0.845341i	$0.679397\pi$
$32$	0	0
$33$	112.144i	0.102979i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	895.053	0.653801	0.326900	−	0.945059i	$-0.393996\pi$
$37$	895.053	0.653801	0.326900	+	0.945059i	$0.393996\pi$
$38$	0	0
$39$	− 215.972i	− 0.141993i
$40$	0	0
$41$	−1097.66	−0.652978	−0.326489	−	0.945201i	$-0.605866\pi$
$41$	−1097.66	−0.652978	−0.326489	+	0.945201i	$0.605866\pi$
$42$	0	0
$43$	− 950.337i	− 0.513974i	−0.966415	−	0.256987i	$-0.917270\pi$
$43$	− 950.337i	− 0.513974i	0.966415	−	0.256987i	$-0.0827297\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1343.83	0.608341	0.304171	−	0.952618i	$-0.401621\pi$
$47$	1343.83	0.608341	0.304171	+	0.952618i	$0.401621\pi$
$48$	0	0
$49$	−1771.08	−0.737642
$50$	0	0
$51$	−88.3294	−0.0339598
$52$	0	0
$53$	−709.918	−0.252730	−0.126365	−	0.991984i	$-0.540331\pi$
$53$	−709.918	−0.252730	−0.126365	+	0.991984i	$0.540331\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	230.517i	0.0709502i
$58$	0	0
$59$	4466.60	1.28314	0.641569	−	0.767065i	$-0.278284\pi$
$59$	4466.60	1.28314	0.641569	+	0.767065i	$0.278284\pi$
$60$	0	0
$61$	− 933.943i	− 0.250993i	−0.992094	−	0.125496i	$-0.959948\pi$
$61$	− 933.943i	− 0.250993i	0.992094	−	0.125496i	$-0.0400523\pi$
$62$	0	0
$63$	−2020.10	−0.508971
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4019.28i	− 0.895362i	−0.894193	−	0.447681i	$-0.852250\pi$
$67$	− 4019.28i	− 0.895362i	0.894193	−	0.447681i	$-0.147750\pi$
$68$	0	0
$69$	− 32.8523i	− 0.00690030i
$70$	0	0
$71$	− 3551.94i	− 0.704610i	−0.935885	−	0.352305i	$-0.885398\pi$
$71$	− 3551.94i	− 0.704610i	0.935885	−	0.352305i	$-0.114602\pi$
$72$	0	0
$73$	5298.05i	0.994192i	0.867695	+	0.497096i	$0.165600\pi$
$73$	5298.05i	0.994192i	−0.867695	+	0.497096i	$0.834400\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3933.01	−0.663351
$78$	0	0
$79$	11461.9i	1.83654i	0.395952	+	0.918271i	$0.370414\pi$
$79$	11461.9i	1.83654i	−0.395952	+	0.918271i	$0.629586\pi$
$80$	0	0
$81$	6436.81	0.981072
$82$	0	0
$83$	6588.95i	0.956445i	0.878239	+	0.478223i	$0.158719\pi$
$83$	6588.95i	0.956445i	−0.878239	+	0.478223i	$0.841281\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	775.861	0.102505
$88$	0	0
$89$	6280.53	0.792896	0.396448	−	0.918057i	$-0.370243\pi$
$89$	6280.53	0.792896	0.396448	+	0.918057i	$0.370243\pi$
$90$	0	0
$91$	7574.35	0.914666
$92$	0	0
$93$	−1162.73	−0.134436
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14960.7i	1.59004i	0.606580	+	0.795022i	$0.292541\pi$
$97$	14960.7i	1.59004i	−0.606580	+	0.795022i	$0.707459\pi$
$98$	0	0
$99$	12612.8	1.28689
$100$	0	0
$101$	4782.58i	0.468835i	0.972136	+	0.234417i	$0.0753182\pi$
$101$	4782.58i	0.468835i	−0.972136	+	0.234417i	$0.924682\pi$
$102$	0	0
$103$	18366.9	1.73126	0.865629	−	0.500686i	$-0.166919\pi$
$103$	18366.9	1.73126	0.865629	+	0.500686i	$0.166919\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14286.0i	− 1.24779i	−0.781507	−	0.623897i	$-0.785549\pi$
$107$	− 14286.0i	− 1.24779i	0.781507	−	0.623897i	$-0.214451\pi$
$108$	0	0
$109$	5247.94i	0.441709i	0.975307	+	0.220854i	$0.0708845\pi$
$109$	5247.94i	0.441709i	−0.975307	+	0.220854i	$0.929115\pi$
$110$	0	0
$111$	640.536i	0.0519874i
$112$	0	0
$113$	15287.0i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$113$	15287.0i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−24290.3	−1.77444
$118$	0	0
$119$	− 3097.80i	− 0.218756i
$120$	0	0
$121$	9915.25	0.677225
$122$	0	0
$123$	− 785.527i	− 0.0519219i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18194.5	1.12806	0.564031	−	0.825754i	$-0.309250\pi$
$127$	18194.5	1.12806	0.564031	+	0.825754i	$0.309250\pi$
$128$	0	0
$129$	680.100	0.0408689
$130$	0	0
$131$	−15447.1	−0.900127	−0.450063	−	0.892997i	$-0.648599\pi$
$131$	−15447.1	−0.900127	−0.450063	+	0.892997i	$0.648599\pi$
$132$	0	0
$133$	−8084.48	−0.457034
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13714.2i	− 0.730683i	−0.930874	−	0.365342i	$-0.880952\pi$
$137$	− 13714.2i	− 0.730683i	0.930874	−	0.365342i	$-0.119048\pi$
$138$	0	0
$139$	19270.4	0.997382	0.498691	−	0.866780i	$-0.333814\pi$
$139$	19270.4	0.997382	0.498691	+	0.866780i	$0.333814\pi$
$140$	0	0
$141$	961.696i	0.0483726i
$142$	0	0
$143$	−47291.5	−2.31265
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1267.46i	− 0.0586540i
$148$	0	0
$149$	2282.56i	0.102813i	0.998678	+	0.0514067i	$0.0163705\pi$
$149$	2282.56i	0.102813i	−0.998678	+	0.0514067i	$0.983630\pi$
$150$	0	0
$151$	− 14813.2i	− 0.649671i	−0.945770	−	0.324836i	$-0.894691\pi$
$151$	− 14813.2i	− 0.649671i	0.945770	−	0.324836i	$-0.105309\pi$
$152$	0	0
$153$	9934.37i	0.424383i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	25139.3	1.01989	0.509945	−	0.860207i	$-0.329666\pi$
$157$	25139.3	1.01989	0.509945	+	0.860207i	$0.329666\pi$
$158$	0	0
$159$	− 508.046i	− 0.0200960i
$160$	0	0
$161$	1152.17	0.0444491
$162$	0	0
$163$	45411.2i	1.70918i	0.519304	+	0.854589i	$0.326191\pi$
$163$	45411.2i	1.70918i	−0.519304	+	0.854589i	$0.673809\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12439.3	−0.446030	−0.223015	−	0.974815i	$-0.571590\pi$
$167$	−12439.3	−0.446030	−0.223015	+	0.974815i	$0.571590\pi$
$168$	0	0
$169$	62514.9	2.18882
$170$	0	0
$171$	25926.2	0.886639
$172$	0	0
$173$	−14118.1	−0.471718	−0.235859	−	0.971787i	$-0.575790\pi$
$173$	−14118.1	−0.471718	−0.235859	+	0.971787i	$0.575790\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3196.48i	0.102029i
$178$	0	0
$179$	40847.7	1.27486	0.637429	−	0.770509i	$-0.279998\pi$
$179$	40847.7	1.27486	0.637429	+	0.770509i	$0.279998\pi$
$180$	0	0
$181$	33131.0i	1.01129i	0.862741	+	0.505647i	$0.168746\pi$
$181$	33131.0i	1.01129i	−0.862741	+	0.505647i	$0.831254\pi$
$182$	0	0
$183$	668.368	0.0199578
$184$	0	0
$185$	0	0
$186$	0	0
$187$	19341.5i	0.553105i
$188$	0	0
$189$	− 2900.54i	− 0.0811998i
$190$	0	0
$191$	62084.2i	1.70182i	0.525310	+	0.850911i	$0.323949\pi$
$191$	62084.2i	1.70182i	−0.525310	+	0.850911i	$0.676051\pi$
$192$	0	0
$193$	37684.7i	1.01170i	0.862623	+	0.505848i	$0.168820\pi$
$193$	37684.7i	1.01170i	−0.862623	+	0.505848i	$0.831180\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	65101.2	1.67748	0.838738	−	0.544535i	$-0.183294\pi$
$197$	65101.2	1.67748	0.838738	+	0.544535i	$0.183294\pi$
$198$	0	0
$199$	35595.6i	0.898857i	0.893316	+	0.449428i	$0.148372\pi$
$199$	35595.6i	0.898857i	−0.893316	+	0.449428i	$0.851628\pi$
$200$	0	0
$201$	2876.36	0.0711952
$202$	0	0
$203$	27210.3i	0.660299i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3694.89	−0.0862305
$208$	0	0
$209$	50476.5	1.15557
$210$	0	0
$211$	38176.8	0.857502	0.428751	−	0.903423i	$-0.358954\pi$
$211$	38176.8	0.857502	0.428751	+	0.903423i	$0.358954\pi$
$212$	0	0
$213$	2541.91	0.0560275
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 40778.3i	− 0.865983i
$218$	0	0
$219$	−3791.50	−0.0790538
$220$	0	0
$221$	− 37248.8i	− 0.762653i
$222$	0	0
$223$	26454.9	0.531981	0.265991	−	0.963976i	$-0.414301\pi$
$223$	26454.9	0.531981	0.265991	+	0.963976i	$0.414301\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	64747.5i	1.25653i	0.778001	+	0.628263i	$0.216234\pi$
$227$	64747.5i	1.25653i	−0.778001	+	0.628263i	$0.783766\pi$
$228$	0	0
$229$	61366.2i	1.17020i	0.810963	+	0.585098i	$0.198944\pi$
$229$	61366.2i	1.17020i	−0.810963	+	0.585098i	$0.801056\pi$
$230$	0	0
$231$	− 2814.62i	− 0.0527467i
$232$	0	0
$233$	− 9362.16i	− 0.172450i	−0.996276	−	0.0862252i	$-0.972520\pi$
$233$	− 9362.16i	− 0.172450i	0.996276	−	0.0862252i	$-0.0274805\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8202.57	−0.146034
$238$	0	0
$239$	− 78207.5i	− 1.36915i	−0.728940	−	0.684577i	$-0.759987\pi$
$239$	− 78207.5i	− 1.36915i	0.728940	−	0.684577i	$-0.240013\pi$
$240$	0	0
$241$	−3956.80	−0.0681256	−0.0340628	−	0.999420i	$-0.510845\pi$
$241$	−3956.80	−0.0681256	−0.0340628	+	0.999420i	$0.510845\pi$
$242$	0	0
$243$	13967.4i	0.236539i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−97209.8	−1.59337
$248$	0	0
$249$	−4715.32	−0.0760524
$250$	0	0
$251$	23807.3	0.377888	0.188944	−	0.981988i	$-0.439494\pi$
$251$	23807.3	0.377888	0.188944	+	0.981988i	$0.439494\pi$
$252$	0	0
$253$	−7193.70	−0.112386
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 80394.1i	− 1.21719i	−0.793482	−	0.608594i	$-0.791734\pi$
$257$	− 80394.1i	− 1.21719i	0.793482	−	0.608594i	$-0.208266\pi$
$258$	0	0
$259$	−22464.3	−0.334883
$260$	0	0
$261$	− 87260.9i	− 1.28097i
$262$	0	0
$263$	−45150.6	−0.652757	−0.326379	−	0.945239i	$-0.605828\pi$
$263$	−45150.6	−0.652757	−0.326379	+	0.945239i	$0.605828\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4494.60i	0.0630476i
$268$	0	0
$269$	24807.6i	0.342831i	0.985199	+	0.171415i	$0.0548340\pi$
$269$	24807.6i	0.342831i	−0.985199	+	0.171415i	$0.945166\pi$
$270$	0	0
$271$	− 10825.0i	− 0.147398i	−0.997281	−	0.0736989i	$-0.976520\pi$
$271$	− 10825.0i	− 0.147398i	0.997281	−	0.0736989i	$-0.0234804\pi$
$272$	0	0
$273$	5420.51i	0.0727302i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−89707.4	−1.16915	−0.584573	−	0.811341i	$-0.698738\pi$
$277$	−89707.4	−1.16915	−0.584573	+	0.811341i	$0.698738\pi$
$278$	0	0
$279$	130772.i	1.67999i
$280$	0	0
$281$	29909.4	0.378787	0.189393	−	0.981901i	$-0.439348\pi$
$281$	29909.4	0.378787	0.189393	+	0.981901i	$0.439348\pi$
$282$	0	0
$283$	10330.5i	0.128988i	0.997918	+	0.0644941i	$0.0205434\pi$
$283$	10330.5i	0.128988i	−0.997918	+	0.0644941i	$0.979457\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	27549.2	0.334461
$288$	0	0
$289$	68286.8	0.817600
$290$	0	0
$291$	−10706.5	−0.126433
$292$	0	0
$293$	40558.7	0.472443	0.236221	−	0.971699i	$-0.424091\pi$
$293$	40558.7	0.472443	0.236221	+	0.971699i	$0.424091\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	18109.9i	0.205307i
$298$	0	0
$299$	13853.9	0.154964
$300$	0	0
$301$	23851.8i	0.263262i
$302$	0	0
$303$	−3422.61	−0.0372797
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 111914.i	− 1.18743i	−0.804675	−	0.593716i	$-0.797660\pi$
$307$	− 111914.i	− 1.18743i	0.804675	−	0.593716i	$-0.202340\pi$
$308$	0	0
$309$	13144.1i	0.137662i
$310$	0	0
$311$	− 75585.9i	− 0.781484i	−0.920500	−	0.390742i	$-0.872218\pi$
$311$	− 75585.9i	− 0.781484i	0.920500	−	0.390742i	$-0.127782\pi$
$312$	0	0
$313$	84909.1i	0.866694i	0.901227	+	0.433347i	$0.142667\pi$
$313$	84909.1i	0.866694i	−0.901227	+	0.433347i	$0.857333\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	84609.1	0.841974	0.420987	−	0.907067i	$-0.361684\pi$
$317$	84609.1	0.841974	0.420987	+	0.907067i	$0.361684\pi$
$318$	0	0
$319$	− 169891.i	− 1.66951i
$320$	0	0
$321$	10223.6	0.0992191
$322$	0	0
$323$	39757.5i	0.381078i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3755.64	−0.0351227
$328$	0	0
$329$	−33727.7	−0.311598
$330$	0	0
$331$	−109784.	−1.00203	−0.501017	−	0.865437i	$-0.667041\pi$
$331$	−109784.	−1.00203	−0.501017	+	0.865437i	$0.667041\pi$
$332$	0	0
$333$	72040.9	0.649667
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 9734.87i	− 0.0857177i	−0.999081	−	0.0428588i	$-0.986353\pi$
$337$	− 9734.87i	− 0.0857177i	0.999081	−	0.0428588i	$-0.0136466\pi$
$338$	0	0
$339$	−10940.0	−0.0951956
$340$	0	0
$341$	254605.i	2.18956i
$342$	0	0
$343$	104712.	0.890036
$344$	0	0
$345$	0	0
$346$	0	0
$347$	74541.3i	0.619068i	0.950888	+	0.309534i	$0.100173\pi$
$347$	74541.3i	0.619068i	−0.950888	+	0.309534i	$0.899827\pi$
$348$	0	0
$349$	− 10246.4i	− 0.0841241i	−0.999115	−	0.0420621i	$-0.986607\pi$
$349$	− 10246.4i	− 0.0841241i	0.999115	−	0.0420621i	$-0.0133927\pi$
$350$	0	0
$351$	− 34876.8i	− 0.283089i
$352$	0	0
$353$	− 238080.i	− 1.91062i	−0.295611	−	0.955308i	$-0.595523\pi$
$353$	− 238080.i	− 1.91062i	0.295611	−	0.955308i	$-0.404477\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2216.91	0.0173945
$358$	0	0
$359$	98659.7i	0.765510i	0.923850	+	0.382755i	$0.125025\pi$
$359$	98659.7i	0.765510i	−0.923850	+	0.382755i	$0.874975\pi$
$360$	0	0
$361$	−26564.1	−0.203836
$362$	0	0
$363$	7095.75i	0.0538500i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	155343.	1.15335	0.576674	−	0.816974i	$-0.304350\pi$
$367$	155343.	1.15335	0.576674	+	0.816974i	$0.304350\pi$
$368$	0	0
$369$	−88348.0	−0.648849
$370$	0	0
$371$	17817.7	0.129451
$372$	0	0
$373$	−91535.5	−0.657918	−0.328959	−	0.944344i	$-0.606698\pi$
$373$	−91535.5	−0.657918	−0.328959	+	0.944344i	$0.606698\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	327183.i	2.30201i
$378$	0	0
$379$	−155561.	−1.08299	−0.541494	−	0.840705i	$-0.682141\pi$
$379$	−155561.	−1.08299	−0.541494	+	0.840705i	$0.682141\pi$
$380$	0	0
$381$	13020.7i	0.0896985i
$382$	0	0
$383$	−165482.	−1.12811	−0.564057	−	0.825736i	$-0.690760\pi$
$383$	−165482.	−1.12811	−0.564057	+	0.825736i	$0.690760\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 76490.6i	− 0.510724i
$388$	0	0
$389$	− 246155.i	− 1.62671i	−0.581771	−	0.813353i	$-0.697640\pi$
$389$	− 246155.i	− 1.62671i	0.581771	−	0.813353i	$-0.302360\pi$
$390$	0	0
$391$	− 5666.06i	− 0.0370619i
$392$	0	0
$393$	− 11054.6i	− 0.0715742i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−143019.	−0.907429	−0.453715	−	0.891147i	$-0.649901\pi$
$397$	−143019.	−0.907429	−0.453715	+	0.891147i	$0.649901\pi$
$398$	0	0
$399$	− 5785.58i	− 0.0363414i
$400$	0	0
$401$	43425.1	0.270055	0.135027	−	0.990842i	$-0.456888\pi$
$401$	43425.1	0.270055	0.135027	+	0.990842i	$0.456888\pi$
$402$	0	0
$403$	− 490328.i	− 3.01910i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	140259.	0.846722
$408$	0	0
$409$	−289995.	−1.73358	−0.866791	−	0.498671i	$-0.833822\pi$
$409$	−289995.	−1.73358	−0.866791	+	0.498671i	$0.833822\pi$
$410$	0	0
$411$	9814.43	0.0581007
$412$	0	0
$413$	−112104.	−0.657235
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13790.7i	0.0793075i
$418$	0	0
$419$	67134.3	0.382399	0.191199	−	0.981551i	$-0.438762\pi$
$419$	67134.3	0.382399	0.191199	+	0.981551i	$0.438762\pi$
$420$	0	0
$421$	− 336530.i	− 1.89872i	−0.314197	−	0.949358i	$-0.601735\pi$
$421$	− 336530.i	− 1.89872i	0.314197	−	0.949358i	$-0.398265\pi$
$422$	0	0
$423$	108162.	0.604495
$424$	0	0
$425$	0	0
$426$	0	0
$427$	23440.4i	0.128561i
$428$	0	0
$429$	− 33843.7i	− 0.183892i
$430$	0	0
$431$	204940.i	1.10325i	0.834094	+	0.551623i	$0.185991\pi$
$431$	204940.i	1.10325i	−0.834094	+	0.551623i	$0.814009\pi$
$432$	0	0
$433$	59164.1i	0.315560i	0.987474	+	0.157780i	$0.0504337\pi$
$433$	59164.1i	0.315560i	−0.987474	+	0.157780i	$0.949566\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−14787.0	−0.0774314
$438$	0	0
$439$	− 234276.i	− 1.21562i	−0.794082	−	0.607811i	$-0.792048\pi$
$439$	− 234276.i	− 1.21562i	0.794082	−	0.607811i	$-0.207952\pi$
$440$	0	0
$441$	−142550.	−0.732978
$442$	0	0
$443$	23778.7i	0.121166i	0.998163	+	0.0605830i	$0.0192960\pi$
$443$	23778.7i	0.121166i	−0.998163	+	0.0605830i	$0.980704\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1633.49	−0.00817528
$448$	0	0
$449$	−232928.	−1.15539	−0.577695	−	0.816253i	$-0.696048\pi$
$449$	−232928.	−1.15539	−0.577695	+	0.816253i	$0.696048\pi$
$450$	0	0
$451$	−172007.	−0.845657
$452$	0	0
$453$	10600.9	0.0516590
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 167826.i	− 0.803576i	−0.915733	−	0.401788i	$-0.868389\pi$
$457$	− 167826.i	− 0.803576i	0.915733	−	0.401788i	$-0.131611\pi$
$458$	0	0
$459$	−14264.1	−0.0677048
$460$	0	0
$461$	22126.3i	0.104113i	0.998644	+	0.0520567i	$0.0165776\pi$
$461$	22126.3i	0.104113i	−0.998644	+	0.0520567i	$0.983422\pi$
$462$	0	0
$463$	174200.	0.812618	0.406309	−	0.913736i	$-0.366816\pi$
$463$	174200.	0.812618	0.406309	+	0.913736i	$0.366816\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 174200.i	− 0.798757i	−0.916786	−	0.399379i	$-0.869226\pi$
$467$	− 174200.i	− 0.798757i	0.916786	−	0.399379i	$-0.130774\pi$
$468$	0	0
$469$	100877.i	0.458613i
$470$	0	0
$471$	17990.7i	0.0810973i
$472$	0	0
$473$	− 148922.i	− 0.665635i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−57139.8	−0.251132
$478$	0	0
$479$	266904.i	1.16328i	0.813447	+	0.581640i	$0.197589\pi$
$479$	266904.i	1.16328i	−0.813447	+	0.581640i	$0.802411\pi$
$480$	0	0
$481$	−270116.	−1.16751
$482$	0	0
$483$	824.536i	0.00353440i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−183710.	−0.774596	−0.387298	−	0.921955i	$-0.626592\pi$
$487$	−183710.	−0.774596	−0.387298	+	0.921955i	$0.626592\pi$
$488$	0	0
$489$	−32498.1	−0.135906
$490$	0	0
$491$	−154481.	−0.640783	−0.320392	−	0.947285i	$-0.603814\pi$
$491$	−154481.	−0.640783	−0.320392	+	0.947285i	$0.603814\pi$
$492$	0	0
$493$	133813.	0.550561
$494$	0	0
$495$	0	0
$496$	0	0
$497$	89147.5i	0.360908i
$498$	0	0
$499$	−171039.	−0.686899	−0.343449	−	0.939171i	$-0.611595\pi$
$499$	−171039.	−0.686899	−0.343449	+	0.939171i	$0.611595\pi$
$500$	0	0
$501$	− 8902.09i	− 0.0354663i
$502$	0	0
$503$	165859.	0.655545	0.327772	−	0.944757i	$-0.393702\pi$
$503$	165859.	0.655545	0.327772	+	0.944757i	$0.393702\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	44738.2i	0.174045i
$508$	0	0
$509$	− 369386.i	− 1.42575i	−0.701289	−	0.712877i	$-0.747392\pi$
$509$	− 369386.i	− 1.42575i	0.701289	−	0.712877i	$-0.252608\pi$
$510$	0	0
$511$	− 132972.i	− 0.509235i
$512$	0	0
$513$	37225.7i	0.141452i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	210583.	0.787849
$518$	0	0
$519$	− 10103.5i	− 0.0375090i
$520$	0	0
$521$	61240.8	0.225613	0.112807	−	0.993617i	$-0.464016\pi$
$521$	61240.8	0.225613	0.112807	+	0.993617i	$0.464016\pi$
$522$	0	0
$523$	− 12048.3i	− 0.0440475i	−0.999757	−	0.0220238i	$-0.992989\pi$
$523$	− 12048.3i	− 0.0440475i	0.999757	−	0.0220238i	$-0.00701095\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−200537.	−0.722061
$528$	0	0
$529$	−277734.	−0.992469
$530$	0	0
$531$	359507.	1.27502
$532$	0	0
$533$	331259.	1.16604
$534$	0	0
$535$	0	0
$536$	0	0
$537$	29232.3i	0.101371i
$538$	0	0
$539$	−277536.	−0.955303
$540$	0	0
$541$	− 15058.6i	− 0.0514505i	−0.999669	−	0.0257252i	$-0.991810\pi$
$541$	− 15058.6i	− 0.0514505i	0.999669	−	0.0257252i	$-0.00818950\pi$
$542$	0	0
$543$	−23709.9	−0.0804137
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 89318.5i	− 0.298515i	−0.988798	−	0.149258i	$-0.952312\pi$
$547$	− 89318.5i	− 0.298515i	0.988798	−	0.149258i	$-0.0476884\pi$
$548$	0	0
$549$	− 75171.1i	− 0.249406i
$550$	0	0
$551$	− 349219.i	− 1.15026i
$552$	0	0
$553$	− 287673.i	− 0.940694i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−323886.	−1.04396	−0.521978	−	0.852959i	$-0.674806\pi$
$557$	−323886.	−1.04396	−0.521978	+	0.852959i	$0.674806\pi$
$558$	0	0
$559$	286800.i	0.917816i
$560$	0	0
$561$	−13841.6	−0.0439805
$562$	0	0
$563$	− 174858.i	− 0.551655i	−0.961207	−	0.275827i	$-0.911048\pi$
$563$	− 174858.i	− 0.551655i	0.961207	−	0.275827i	$-0.0889518\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−161553.	−0.502514
$568$	0	0
$569$	−137244.	−0.423906	−0.211953	−	0.977280i	$-0.567982\pi$
$569$	−137244.	−0.423906	−0.211953	+	0.977280i	$0.567982\pi$
$570$	0	0
$571$	30212.9	0.0926661	0.0463331	−	0.998926i	$-0.485246\pi$
$571$	30212.9	0.0926661	0.0463331	+	0.998926i	$0.485246\pi$
$572$	0	0
$573$	−44429.9	−0.135321
$574$	0	0
$575$	0	0
$576$	0	0
$577$	157140.i	0.471991i	0.971754	+	0.235996i	$0.0758351\pi$
$577$	157140.i	0.471991i	−0.971754	+	0.235996i	$0.924165\pi$
$578$	0	0
$579$	−26968.7	−0.0804456
$580$	0	0
$581$	− 165371.i	− 0.489900i
$582$	0	0
$583$	−111247.	−0.327305
$584$	0	0
$585$	0	0
$586$	0	0
$587$	454782.i	1.31986i	0.751328	+	0.659929i	$0.229413\pi$
$587$	454782.i	1.31986i	−0.751328	+	0.659929i	$0.770587\pi$
$588$	0	0
$589$	523352.i	1.50856i
$590$	0	0
$591$	46589.1i	0.133386i
$592$	0	0
$593$	− 311501.i	− 0.885830i	−0.896564	−	0.442915i	$-0.853944\pi$
$593$	− 311501.i	− 0.885830i	0.896564	−	0.442915i	$-0.146056\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−25473.7	−0.0714732
$598$	0	0
$599$	− 456248.i	− 1.27159i	−0.771858	−	0.635795i	$-0.780672\pi$
$599$	− 456248.i	− 1.27159i	0.771858	−	0.635795i	$-0.219328\pi$
$600$	0	0
$601$	76983.7	0.213133	0.106566	−	0.994306i	$-0.466014\pi$
$601$	76983.7	0.213133	0.106566	+	0.994306i	$0.466014\pi$
$602$	0	0
$603$	− 323503.i	− 0.889700i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	114126.	0.309747	0.154873	−	0.987934i	$-0.450503\pi$
$607$	114126.	0.309747	0.154873	+	0.987934i	$0.450503\pi$
$608$	0	0
$609$	−19472.8	−0.0525041
$610$	0	0
$611$	−405550.	−1.08633
$612$	0	0
$613$	−258067.	−0.686770	−0.343385	−	0.939195i	$-0.611574\pi$
$613$	−258067.	−0.686770	−0.343385	+	0.939195i	$0.611574\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 622513.i	− 1.63523i	−0.575767	−	0.817614i	$-0.695296\pi$
$617$	− 622513.i	− 1.63523i	0.575767	−	0.817614i	$-0.304704\pi$
$618$	0	0
$619$	−490064.	−1.27900	−0.639502	−	0.768790i	$-0.720859\pi$
$619$	−490064.	−1.27900	−0.639502	+	0.768790i	$0.720859\pi$
$620$	0	0
$621$	− 5305.25i	− 0.0137570i
$622$	0	0
$623$	−157630.	−0.406129
$624$	0	0
$625$	0	0
$626$	0	0
$627$	36123.1i	0.0918860i
$628$	0	0
$629$	110474.i	0.279227i
$630$	0	0
$631$	260622.i	0.654565i	0.944927	+	0.327283i	$0.106133\pi$
$631$	260622.i	0.654565i	−0.944927	+	0.327283i	$0.893867\pi$
$632$	0	0
$633$	27320.9i	0.0681848i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	534490.	1.31723
$638$	0	0
$639$	− 285888.i	− 0.700155i
$640$	0	0
$641$	309582.	0.753459	0.376730	−	0.926323i	$-0.377049\pi$
$641$	309582.	0.753459	0.376730	+	0.926323i	$0.377049\pi$
$642$	0	0
$643$	227221.i	0.549575i	0.961505	+	0.274788i	$0.0886076\pi$
$643$	227221.i	0.549575i	−0.961505	+	0.274788i	$0.911392\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	464773.	1.11028	0.555140	−	0.831757i	$-0.312665\pi$
$647$	464773.	1.11028	0.555140	+	0.831757i	$0.312665\pi$
$648$	0	0
$649$	699936.	1.66176
$650$	0	0
$651$	29182.6	0.0688592
$652$	0	0
$653$	413098.	0.968784	0.484392	−	0.874851i	$-0.339041\pi$
$653$	413098.	0.968784	0.484392	+	0.874851i	$0.339041\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	426429.i	0.987906i
$658$	0	0
$659$	336357.	0.774514	0.387257	−	0.921972i	$-0.373423\pi$
$659$	336357.	0.774514	0.387257	+	0.921972i	$0.373423\pi$
$660$	0	0
$661$	345461.i	0.790673i	0.918536	+	0.395336i	$0.129372\pi$
$661$	345461.i	0.790673i	−0.918536	+	0.395336i	$0.870628\pi$
$662$	0	0
$663$	26656.7	0.0606429
$664$	0	0
$665$	0	0
$666$	0	0
$667$	49769.2i	0.111869i
$668$	0	0
$669$	18932.2i	0.0423008i
$670$	0	0
$671$	− 146353.i	− 0.325055i
$672$	0	0
$673$	462062.i	1.02016i	0.860126	+	0.510082i	$0.170385\pi$
$673$	462062.i	1.02016i	−0.860126	+	0.510082i	$0.829615\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38158.5	−0.0832556	−0.0416278	−	0.999133i	$-0.513254\pi$
$677$	−38158.5	−0.0832556	−0.0416278	+	0.999133i	$0.513254\pi$
$678$	0	0
$679$	− 375488.i	− 0.814436i
$680$	0	0
$681$	−46336.0	−0.0999134
$682$	0	0
$683$	653710.i	1.40134i	0.713486	+	0.700670i	$0.247115\pi$
$683$	653710.i	1.40134i	−0.713486	+	0.700670i	$0.752885\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−43916.2	−0.0930488
$688$	0	0
$689$	214245.	0.451306
$690$	0	0
$691$	−55946.9	−0.117171	−0.0585854	−	0.998282i	$-0.518659\pi$
$691$	−55946.9	−0.117171	−0.0585854	+	0.998282i	$0.518659\pi$
$692$	0	0
$693$	−316559.	−0.659156
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 135480.i	− 0.278876i
$698$	0	0
$699$	6699.94	0.0137125
$700$	0	0
$701$	604760.i	1.23069i	0.788260	+	0.615343i	$0.210982\pi$
$701$	604760.i	1.23069i	−0.788260	+	0.615343i	$0.789018\pi$
$702$	0	0
$703$	288308.	0.583373
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 120035.i	− 0.240142i
$708$	0	0
$709$	− 188395.i	− 0.374781i	−0.982285	−	0.187391i	$-0.939997\pi$
$709$	− 188395.i	− 0.374781i	0.982285	−	0.187391i	$-0.0600030\pi$
$710$	0	0
$711$	922540.i	1.82493i
$712$	0	0
$713$	− 74585.9i	− 0.146716i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	55968.4	0.108869
$718$	0	0
$719$	161045.i	0.311522i	0.987795	+	0.155761i	$0.0497830\pi$
$719$	161045.i	0.311522i	−0.987795	+	0.155761i	$0.950217\pi$
$720$	0	0
$721$	−460978.	−0.886766
$722$	0	0
$723$	− 2831.65i	− 0.00541705i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	433372.	0.819960	0.409980	−	0.912095i	$-0.365536\pi$
$727$	433372.	0.819960	0.409980	+	0.912095i	$0.365536\pi$
$728$	0	0
$729$	511386.	0.962263
$730$	0	0
$731$	117297.	0.219509
$732$	0	0
$733$	−643138.	−1.19701	−0.598503	−	0.801121i	$-0.704237\pi$
$733$	−643138.	−1.19701	−0.598503	+	0.801121i	$0.704237\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 629838.i	− 1.15956i
$738$	0	0
$739$	412921.	0.756098	0.378049	−	0.925786i	$-0.376595\pi$
$739$	412921.	0.756098	0.378049	+	0.925786i	$0.376595\pi$
$740$	0	0
$741$	− 69567.3i	− 0.126698i
$742$	0	0
$743$	−969654.	−1.75646	−0.878231	−	0.478236i	$-0.841276\pi$
$743$	−969654.	−1.75646	−0.878231	+	0.478236i	$0.841276\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	530331.i	0.950398i
$748$	0	0
$749$	358553.i	0.639131i
$750$	0	0
$751$	− 858378.i	− 1.52194i	−0.648784	−	0.760972i	$-0.724722\pi$
$751$	− 858378.i	− 1.52194i	0.648784	−	0.760972i	$-0.275278\pi$
$752$	0	0
$753$	17037.5i	0.0300480i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	155580.	0.271495	0.135747	−	0.990743i	$-0.456656\pi$
$757$	155580.	0.271495	0.135747	+	0.990743i	$0.456656\pi$
$758$	0	0
$759$	− 5148.10i	− 0.00893642i
$760$	0	0
$761$	216729.	0.374237	0.187119	−	0.982337i	$-0.440085\pi$
$761$	216729.	0.374237	0.187119	+	0.982337i	$0.440085\pi$
$762$	0	0
$763$	− 131714.i	− 0.226247i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.34797e6	−2.29133
$768$	0	0
$769$	413439.	0.699132	0.349566	−	0.936912i	$-0.386329\pi$
$769$	413439.	0.699132	0.349566	+	0.936912i	$0.386329\pi$
$770$	0	0
$771$	57533.3	0.0967855
$772$	0	0
$773$	515813.	0.863244	0.431622	−	0.902055i	$-0.357942\pi$
$773$	515813.	0.863244	0.431622	+	0.902055i	$0.357942\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 16076.3i	− 0.0266284i
$778$	0	0
$779$	−353569.	−0.582639
$780$	0	0
$781$	− 556604.i	− 0.912524i
$782$	0	0
$783$	125292.	0.204362
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 694926.i	− 1.12199i	−0.827819	−	0.560995i	$-0.810419\pi$
$787$	− 694926.i	− 1.12199i	0.827819	−	0.560995i	$-0.189581\pi$
$788$	0	0
$789$	− 32311.6i	− 0.0519044i
$790$	0	0
$791$	− 383676.i	− 0.613214i
$792$	0	0
$793$	281853.i	0.448204i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	287395.	0.452442	0.226221	−	0.974076i	$-0.427363\pi$
$797$	287395.	0.452442	0.226221	+	0.974076i	$0.427363\pi$
$798$	0	0
$799$	165864.i	0.259812i
$800$	0	0
$801$	505506.	0.787883
$802$	0	0
$803$	830228.i	1.28756i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−17753.3	−0.0272604
$808$	0	0
$809$	104931.	0.160328	0.0801638	−	0.996782i	$-0.474456\pi$
$809$	104931.	0.160328	0.0801638	+	0.996782i	$0.474456\pi$
$810$	0	0
$811$	731396.	1.11202	0.556008	−	0.831177i	$-0.312332\pi$
$811$	731396.	1.11202	0.556008	+	0.831177i	$0.312332\pi$
$812$	0	0
$813$	7746.84	0.0117204
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 306116.i	− 0.458608i
$818$	0	0
$819$	609643.	0.908883
$820$	0	0
$821$	− 160376.i	− 0.237932i	−0.992898	−	0.118966i	$-0.962042\pi$
$821$	− 160376.i	− 0.237932i	0.992898	−	0.118966i	$-0.0379579\pi$
$822$	0	0
$823$	−639468.	−0.944103	−0.472052	−	0.881571i	$-0.656486\pi$
$823$	−639468.	−0.944103	−0.472052	+	0.881571i	$0.656486\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	392782.i	0.574302i	0.957885	+	0.287151i	$0.0927082\pi$
$827$	392782.i	0.574302i	−0.957885	+	0.287151i	$0.907292\pi$
$828$	0	0
$829$	− 749281.i	− 1.09027i	−0.838347	−	0.545137i	$-0.816478\pi$
$829$	− 749281.i	− 1.09027i	0.838347	−	0.545137i	$-0.183522\pi$
$830$	0	0
$831$	− 64198.3i	− 0.0929654i
$832$	0	0
$833$	− 218599.i	− 0.315034i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−187767.	−0.268021
$838$	0	0
$839$	1.05604e6i	1.50023i	0.661308	+	0.750115i	$0.270002\pi$
$839$	1.05604e6i	1.50023i	−0.661308	+	0.750115i	$0.729998\pi$
$840$	0	0
$841$	−468099.	−0.661829
$842$	0	0
$843$	21404.4i	0.0301195i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−248855.	−0.346881
$848$	0	0
$849$	−7392.95	−0.0102566
$850$	0	0
$851$	−41088.5	−0.0567363
$852$	0	0
$853$	−166621.	−0.228999	−0.114499	−	0.993423i	$-0.536526\pi$
$853$	−166621.	−0.228999	−0.114499	+	0.993423i	$0.536526\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	834744.i	1.13656i	0.822836	+	0.568279i	$0.192391\pi$
$857$	834744.i	1.13656i	−0.822836	+	0.568279i	$0.807609\pi$
$858$	0	0
$859$	639041.	0.866048	0.433024	−	0.901382i	$-0.357446\pi$
$859$	639041.	0.866048	0.433024	+	0.901382i	$0.357446\pi$
$860$	0	0
$861$	19715.4i	0.0265949i
$862$	0	0
$863$	1.38716e6	1.86254	0.931268	−	0.364335i	$-0.118704\pi$
$863$	1.38716e6	1.86254	0.931268	+	0.364335i	$0.118704\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	48868.8i	0.0650120i
$868$	0	0
$869$	1.79612e6i	2.37846i
$870$	0	0
$871$	1.21297e6i	1.59887i
$872$	0	0
$873$	1.20416e6i	1.57999i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−853159.	−1.10925	−0.554627	−	0.832099i	$-0.687139\pi$
$877$	−853159.	−1.10925	−0.554627	+	0.832099i	$0.687139\pi$
$878$	0	0
$879$	29025.5i	0.0375666i
$880$	0	0
$881$	1.26731e6	1.63279	0.816396	−	0.577493i	$-0.195969\pi$
$881$	1.26731e6	1.63279	0.816396	+	0.577493i	$0.195969\pi$
$882$	0	0
$883$	324445.i	0.416121i	0.978116	+	0.208061i	$0.0667152\pi$
$883$	324445.i	0.416121i	−0.978116	+	0.208061i	$0.933285\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−186805.	−0.237433	−0.118716	−	0.992928i	$-0.537878\pi$
$887$	−186805.	−0.237433	−0.118716	+	0.992928i	$0.537878\pi$
$888$	0	0
$889$	−456650.	−0.577803
$890$	0	0
$891$	1.00868e6	1.27056
$892$	0	0
$893$	432864.	0.542811
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9914.43i	0.0123221i
$898$	0	0
$899$	1.76147e6	2.17949
$900$	0	0
$901$	− 87623.1i	− 0.107937i
$902$	0	0
$903$	−17069.3	−0.0209334
$904$	0	0
$905$	0	0
$906$	0	0
$907$	49214.0i	0.0598238i	0.999553	+	0.0299119i	$0.00952268\pi$
$907$	49214.0i	0.0598238i	−0.999553	+	0.0299119i	$0.990477\pi$
$908$	0	0
$909$	384940.i	0.465870i
$910$	0	0
$911$	285157.i	0.343595i	0.985132	+	0.171797i	$0.0549574\pi$
$911$	285157.i	0.343595i	−0.985132	+	0.171797i	$0.945043\pi$
$912$	0	0
$913$	1.03252e6i	1.23867i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	387695.	0.461053
$918$	0	0
$919$	698642.i	0.827225i	0.910453	+	0.413613i	$0.135733\pi$
$919$	698642.i	0.827225i	−0.910453	+	0.413613i	$0.864267\pi$
$920$	0	0
$921$	80090.4	0.0944194
$922$	0	0
$923$	1.07193e6i	1.25824i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.47831e6	1.72031
$928$	0	0
$929$	1.24579e6	1.44349	0.721746	−	0.692158i	$-0.243340\pi$
$929$	1.24579e6	1.44349	0.721746	+	0.692158i	$0.243340\pi$
$930$	0	0
$931$	−570487.	−0.658183
$932$	0	0
$933$	54092.3	0.0621402
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 253678.i	− 0.288937i	−0.989509	−	0.144469i	$-0.953853\pi$
$937$	− 253678.i	− 0.288937i	0.989509	−	0.144469i	$-0.0461473\pi$
$938$	0	0
$939$	−60764.4	−0.0689157
$940$	0	0
$941$	− 1.11582e6i	− 1.26013i	−0.776543	−	0.630064i	$-0.783029\pi$
$941$	− 1.11582e6i	− 1.26013i	0.776543	−	0.630064i	$-0.216971\pi$
$942$	0	0
$943$	50389.2	0.0566649
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 875095.i	− 0.975787i	−0.872903	−	0.487893i	$-0.837765\pi$
$947$	− 875095.i	− 0.975787i	0.872903	−	0.487893i	$-0.162235\pi$
$948$	0	0
$949$	− 1.59889e6i	− 1.77536i
$950$	0	0
$951$	60549.7i	0.0669501i
$952$	0	0
$953$	1.16693e6i	1.28487i	0.766342	+	0.642433i	$0.222075\pi$
$953$	1.16693e6i	1.28487i	−0.766342	+	0.642433i	$0.777925\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	121581.	0.132752
$958$	0	0
$959$	344202.i	0.374263i
$960$	0	0
$961$	−1.71628e6	−1.85840
$962$	0	0
$963$	− 1.14985e6i	− 1.23990i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	764848.	0.817942	0.408971	−	0.912547i	$-0.365888\pi$
$967$	764848.	0.817942	0.408971	+	0.912547i	$0.365888\pi$
$968$	0	0
$969$	−28452.1	−0.0303016
$970$	0	0
$971$	148874.	0.157899	0.0789497	−	0.996879i	$-0.474843\pi$
$971$	148874.	0.157899	0.0789497	+	0.996879i	$0.474843\pi$
$972$	0	0
$973$	−483654.	−0.510868
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 99829.5i	− 0.104585i	−0.998632	−	0.0522925i	$-0.983347\pi$
$977$	− 99829.5i	− 0.104585i	0.998632	−	0.0522925i	$-0.0166528\pi$
$978$	0	0
$979$	984186.	1.02686
$980$	0	0
$981$	422395.i	0.438916i
$982$	0	0
$983$	−329283.	−0.340771	−0.170385	−	0.985378i	$-0.554501\pi$
$983$	−329283.	−0.340771	−0.170385	+	0.985378i	$0.554501\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24136.9i	− 0.0247769i
$988$	0	0
$989$	43626.4i	0.0446022i
$990$	0	0
$991$	− 40937.9i	− 0.0416848i	−0.999783	−	0.0208424i	$-0.993365\pi$
$991$	− 40937.9i	− 0.0416848i	0.999783	−	0.0208424i	$-0.00663483\pi$
$992$	0	0
$993$	− 78565.8i	− 0.0796774i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.03701e6	−1.04326	−0.521630	−	0.853172i	$-0.674676\pi$
$997$	−1.03701e6	−1.04326	−0.521630	+	0.853172i	$0.674676\pi$
$998$	0	0
$999$	103439.i	0.103646i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.5.e.e.399.20		32
4.3	odd	2		200.5.e.e.99.18		32
5.2	odd	4		160.5.g.a.111.8		16
5.3	odd	4		800.5.g.h.751.10		16
5.4	even	2	inner	800.5.e.e.399.13		32
8.3	odd	2	inner	800.5.e.e.399.14		32
8.5	even	2		200.5.e.e.99.16		32
15.2	even	4		1440.5.g.a.271.4		16
20.3	even	4		200.5.g.h.51.15		16
20.7	even	4		40.5.g.a.11.2	yes	16
20.19	odd	2		200.5.e.e.99.15		32
40.3	even	4		800.5.g.h.751.9		16
40.13	odd	4		200.5.g.h.51.16		16
40.19	odd	2	inner	800.5.e.e.399.19		32
40.27	even	4		160.5.g.a.111.7		16
40.29	even	2		200.5.e.e.99.17		32
40.37	odd	4		40.5.g.a.11.1	✓	16
60.47	odd	4		360.5.g.a.91.15		16
120.77	even	4		360.5.g.a.91.16		16
120.107	odd	4		1440.5.g.a.271.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.g.a.11.1	✓	16	40.37	odd	4
40.5.g.a.11.2	yes	16	20.7	even	4
160.5.g.a.111.7		16	40.27	even	4
160.5.g.a.111.8		16	5.2	odd	4
200.5.e.e.99.15		32	20.19	odd	2
200.5.e.e.99.16		32	8.5	even	2
200.5.e.e.99.17		32	40.29	even	2
200.5.e.e.99.18		32	4.3	odd	2
200.5.g.h.51.15		16	20.3	even	4
200.5.g.h.51.16		16	40.13	odd	4
360.5.g.a.91.15		16	60.47	odd	4
360.5.g.a.91.16		16	120.77	even	4
800.5.e.e.399.13		32	5.4	even	2	inner
800.5.e.e.399.14		32	8.3	odd	2	inner
800.5.e.e.399.19		32	40.19	odd	2	inner
800.5.e.e.399.20		32	1.1	even	1	trivial
800.5.g.h.751.9		16	40.3	even	4
800.5.g.h.751.10		16	5.3	odd	4
1440.5.g.a.271.4		16	15.2	even	4
1440.5.g.a.271.13		16	120.107	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	800.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.715641i q^{3} -25.0983 q^{7} +80.4879 q^{9} +156.704 q^{11} -301.788 q^{13} +123.427i q^{17} +322.113 q^{19} -17.9613i q^{21} -45.9062 q^{23} +115.567i q^{27} -1084.15i q^{29} +1624.75i q^{31} +112.144i q^{33} +895.053 q^{37} -215.972i q^{39} -1097.66 q^{41} -950.337i q^{43} +1343.83 q^{47} -1771.08 q^{49} -88.3294 q^{51} -709.918 q^{53} +230.517i q^{57} +4466.60 q^{59} -933.943i q^{61} -2020.10 q^{63} -4019.28i q^{67} -32.8523i q^{69} -3551.94i q^{71} +5298.05i q^{73} -3933.01 q^{77} +11461.9i q^{79} +6436.81 q^{81} +6588.95i q^{83} +775.861 q^{87} +6280.53 q^{89} +7574.35 q^{91} -1162.73 q^{93} +14960.7i q^{97} +12612.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 864 q^{9} - 384 q^{11} - 1408 q^{19} - 4416 q^{41} + 4960 q^{49} + 35584 q^{51} + 28032 q^{59} + 20768 q^{81} - 13632 q^{89} - 49152 q^{91} + 5248 q^{99}+O(q^{100}) \) 32 * q - 864 * q^9 - 384 * q^11 - 1408 * q^19 - 4416 * q^41 + 4960 * q^49 + 35584 * q^51 + 28032 * q^59 + 20768 * q^81 - 13632 * q^89 - 49152 * q^91 + 5248 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more