LMFDB - Embedded newform 800.5.e.e.399.22

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			399.22
Character	$\chi$	$=$	800.399
Dual form			800.5.e.e.399.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.51805i q^{3} +50.0881 q^{7} +60.5872 q^{9} -158.856 q^{11} -97.3070 q^{13} -285.416i q^{17} +414.217 q^{19} +226.301i q^{21} -546.228 q^{23} +639.698i q^{27} -1162.39i q^{29} -369.759i q^{31} -717.721i q^{33} -1423.26 q^{37} -439.638i q^{39} -2901.21 q^{41} -905.493i q^{43} +3025.75 q^{47} +107.819 q^{49} +1289.52 q^{51} -1605.79 q^{53} +1871.45i q^{57} -1809.93 q^{59} +1119.29i q^{61} +3034.70 q^{63} +3916.26i q^{67} -2467.89i q^{69} -4224.11i q^{71} +4667.66i q^{73} -7956.81 q^{77} +6812.31i q^{79} +2017.38 q^{81} +3166.94i q^{83} +5251.74 q^{87} -10721.2 q^{89} -4873.93 q^{91} +1670.59 q^{93} -10051.4i q^{97} -9624.66 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	4.51805i	0.502006i	0.967986	+	0.251003i	$0.0807603\pi$
$3$	4.51805i	0.502006i	−0.967986	+	0.251003i	$0.919240\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	50.0881	1.02221	0.511103	−	0.859519i	$-0.329237\pi$
$7$	50.0881	1.02221	0.511103	+	0.859519i	$0.329237\pi$
$8$	0	0
$9$	60.5872	0.747990
$10$	0	0
$11$	−158.856	−1.31286	−0.656431	−	0.754386i	$-0.727935\pi$
$11$	−158.856	−1.31286	−0.656431	+	0.754386i	$0.727935\pi$
$12$	0	0
$13$	−97.3070	−0.575781	−0.287891	−	0.957663i	$-0.592954\pi$
$13$	−97.3070	−0.575781	−0.287891	+	0.957663i	$0.592954\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 285.416i	− 0.987598i	−0.869576	−	0.493799i	$-0.835608\pi$
$17$	− 285.416i	− 0.987598i	0.869576	−	0.493799i	$-0.164392\pi$
$18$	0	0
$19$	414.217	1.14741	0.573707	−	0.819061i	$-0.305505\pi$
$19$	414.217	1.14741	0.573707	+	0.819061i	$0.305505\pi$
$20$	0	0
$21$	226.301i	0.513153i
$22$	0	0
$23$	−546.228	−1.03257	−0.516284	−	0.856418i	$-0.672685\pi$
$23$	−546.228	−1.03257	−0.516284	+	0.856418i	$0.672685\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	639.698i	0.877501i
$28$	0	0
$29$	− 1162.39i	− 1.38215i	−0.722781	−	0.691077i	$-0.757137\pi$
$29$	− 1162.39i	− 1.38215i	0.722781	−	0.691077i	$-0.242863\pi$
$30$	0	0
$31$	− 369.759i	− 0.384765i	−0.981320	−	0.192382i	$-0.938379\pi$
$31$	− 369.759i	− 0.384765i	0.981320	−	0.192382i	$-0.0616214\pi$
$32$	0	0
$33$	− 717.721i	− 0.659064i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1423.26	−1.03964	−0.519818	−	0.854277i	$-0.674000\pi$
$37$	−1423.26	−1.03964	−0.519818	+	0.854277i	$0.674000\pi$
$38$	0	0
$39$	− 439.638i	− 0.289045i
$40$	0	0
$41$	−2901.21	−1.72588	−0.862942	−	0.505303i	$-0.831381\pi$
$41$	−2901.21	−1.72588	−0.862942	+	0.505303i	$0.831381\pi$
$42$	0	0
$43$	− 905.493i	− 0.489720i	−0.969558	−	0.244860i	$-0.921258\pi$
$43$	− 905.493i	− 0.489720i	0.969558	−	0.244860i	$-0.0787421\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3025.75	1.36974	0.684870	−	0.728665i	$-0.259859\pi$
$47$	3025.75	1.36974	0.684870	+	0.728665i	$0.259859\pi$
$48$	0	0
$49$	107.819	0.0449058
$50$	0	0
$51$	1289.52	0.495780
$52$	0	0
$53$	−1605.79	−0.571661	−0.285830	−	0.958280i	$-0.592269\pi$
$53$	−1605.79	−0.571661	−0.285830	+	0.958280i	$0.592269\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1871.45i	0.576008i
$58$	0	0
$59$	−1809.93	−0.519946	−0.259973	−	0.965616i	$-0.583714\pi$
$59$	−1809.93	−0.519946	−0.259973	+	0.965616i	$0.583714\pi$
$60$	0	0
$61$	1119.29i	0.300804i	0.988625	+	0.150402i	$0.0480567\pi$
$61$	1119.29i	0.300804i	−0.988625	+	0.150402i	$0.951943\pi$
$62$	0	0
$63$	3034.70	0.764600
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3916.26i	0.872412i	0.899847	+	0.436206i	$0.143678\pi$
$67$	3916.26i	0.872412i	−0.899847	+	0.436206i	$0.856322\pi$
$68$	0	0
$69$	− 2467.89i	− 0.518355i
$70$	0	0
$71$	− 4224.11i	− 0.837951i	−0.907998	−	0.418975i	$-0.862389\pi$
$71$	− 4224.11i	− 0.837951i	0.907998	−	0.418975i	$-0.137611\pi$
$72$	0	0
$73$	4667.66i	0.875897i	0.899000	+	0.437949i	$0.144295\pi$
$73$	4667.66i	0.875897i	−0.899000	+	0.437949i	$0.855705\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7956.81	−1.34202
$78$	0	0
$79$	6812.31i	1.09154i	0.837935	+	0.545771i	$0.183763\pi$
$79$	6812.31i	1.09154i	−0.837935	+	0.545771i	$0.816237\pi$
$80$	0	0
$81$	2017.38	0.307480
$82$	0	0
$83$	3166.94i	0.459709i	0.973225	+	0.229855i	$0.0738251\pi$
$83$	3166.94i	0.459709i	−0.973225	+	0.229855i	$0.926175\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5251.74	0.693849
$88$	0	0
$89$	−10721.2	−1.35351	−0.676756	−	0.736207i	$-0.736615\pi$
$89$	−10721.2	−1.35351	−0.676756	+	0.736207i	$0.736615\pi$
$90$	0	0
$91$	−4873.93	−0.588567
$92$	0	0
$93$	1670.59	0.193154
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10051.4i	− 1.06828i	−0.845396	−	0.534140i	$-0.820636\pi$
$97$	− 10051.4i	− 1.06828i	0.845396	−	0.534140i	$-0.179364\pi$
$98$	0	0
$99$	−9624.66	−0.982008
$100$	0	0
$101$	− 15016.7i	− 1.47208i	−0.676935	−	0.736042i	$-0.736692\pi$
$101$	− 15016.7i	− 1.47208i	0.676935	−	0.736042i	$-0.263308\pi$
$102$	0	0
$103$	13177.3	1.24209	0.621045	−	0.783775i	$-0.286708\pi$
$103$	13177.3	1.24209	0.621045	+	0.783775i	$0.286708\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13284.8i	− 1.16035i	−0.814493	−	0.580173i	$-0.802985\pi$
$107$	− 13284.8i	− 1.16035i	0.814493	−	0.580173i	$-0.197015\pi$
$108$	0	0
$109$	− 19211.6i	− 1.61700i	−0.588493	−	0.808502i	$-0.700279\pi$
$109$	− 19211.6i	− 1.61700i	0.588493	−	0.808502i	$-0.299721\pi$
$110$	0	0
$111$	− 6430.36i	− 0.521903i
$112$	0	0
$113$	− 20975.3i	− 1.64268i	−0.570442	−	0.821338i	$-0.693228\pi$
$113$	− 20975.3i	− 1.64268i	0.570442	−	0.821338i	$-0.306772\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5895.56	−0.430679
$118$	0	0
$119$	− 14295.9i	− 1.00953i
$120$	0	0
$121$	10594.3	0.723607
$122$	0	0
$123$	− 13107.8i	− 0.866403i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10769.3	−0.667696	−0.333848	−	0.942627i	$-0.608347\pi$
$127$	−10769.3	−0.667696	−0.333848	+	0.942627i	$0.608347\pi$
$128$	0	0
$129$	4091.06	0.245842
$130$	0	0
$131$	−15667.6	−0.912975	−0.456487	−	0.889730i	$-0.650893\pi$
$131$	−15667.6	−0.912975	−0.456487	+	0.889730i	$0.650893\pi$
$132$	0	0
$133$	20747.3	1.17289
$134$	0	0
$135$	0	0
$136$	0	0
$137$	25231.8i	1.34433i	0.740399	+	0.672167i	$0.234636\pi$
$137$	25231.8i	1.34433i	−0.740399	+	0.672167i	$0.765364\pi$
$138$	0	0
$139$	−21388.1	−1.10699	−0.553494	−	0.832853i	$-0.686706\pi$
$139$	−21388.1	−1.10699	−0.553494	+	0.832853i	$0.686706\pi$
$140$	0	0
$141$	13670.5i	0.687617i
$142$	0	0
$143$	15457.8	0.755922
$144$	0	0
$145$	0	0
$146$	0	0
$147$	487.131i	0.0225429i
$148$	0	0
$149$	− 40319.2i	− 1.81610i	−0.418863	−	0.908049i	$-0.637571\pi$
$149$	− 40319.2i	− 1.81610i	0.418863	−	0.908049i	$-0.362429\pi$
$150$	0	0
$151$	28388.2i	1.24504i	0.782603	+	0.622521i	$0.213891\pi$
$151$	28388.2i	1.24504i	−0.782603	+	0.622521i	$0.786109\pi$
$152$	0	0
$153$	− 17292.5i	− 0.738714i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4835.50	−0.196174	−0.0980872	−	0.995178i	$-0.531272\pi$
$157$	−4835.50	−0.196174	−0.0980872	+	0.995178i	$0.531272\pi$
$158$	0	0
$159$	− 7255.06i	− 0.286977i
$160$	0	0
$161$	−27359.5	−1.05550
$162$	0	0
$163$	− 26938.1i	− 1.01389i	−0.861977	−	0.506947i	$-0.830774\pi$
$163$	− 26938.1i	− 1.01389i	0.861977	−	0.506947i	$-0.169226\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24228.8	0.868760	0.434380	−	0.900730i	$-0.356968\pi$
$167$	24228.8	0.868760	0.434380	+	0.900730i	$0.356968\pi$
$168$	0	0
$169$	−19092.3	−0.668476
$170$	0	0
$171$	25096.2	0.858255
$172$	0	0
$173$	15853.9	0.529716	0.264858	−	0.964287i	$-0.414675\pi$
$173$	15853.9	0.529716	0.264858	+	0.964287i	$0.414675\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 8177.36i	− 0.261016i
$178$	0	0
$179$	5340.06	0.166663	0.0833316	−	0.996522i	$-0.473444\pi$
$179$	5340.06	0.166663	0.0833316	+	0.996522i	$0.473444\pi$
$180$	0	0
$181$	− 16455.4i	− 0.502286i	−0.967950	−	0.251143i	$-0.919193\pi$
$181$	− 16455.4i	− 0.502286i	0.967950	−	0.251143i	$-0.0808065\pi$
$182$	0	0
$183$	−5057.01	−0.151005
$184$	0	0
$185$	0	0
$186$	0	0
$187$	45340.1i	1.29658i
$188$	0	0
$189$	32041.3i	0.896987i
$190$	0	0
$191$	− 36286.3i	− 0.994663i	−0.867561	−	0.497331i	$-0.834313\pi$
$191$	− 36286.3i	− 0.994663i	0.867561	−	0.497331i	$-0.165687\pi$
$192$	0	0
$193$	1934.56i	0.0519358i	0.999663	+	0.0259679i	$0.00826676\pi$
$193$	1934.56i	0.0519358i	−0.999663	+	0.0259679i	$0.991733\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	72587.7	1.87038	0.935192	−	0.354142i	$-0.115227\pi$
$197$	72587.7	1.87038	0.935192	+	0.354142i	$0.115227\pi$
$198$	0	0
$199$	− 77717.2i	− 1.96251i	−0.192723	−	0.981253i	$-0.561732\pi$
$199$	− 77717.2i	− 1.96251i	0.192723	−	0.981253i	$-0.438268\pi$
$200$	0	0
$201$	−17693.9	−0.437956
$202$	0	0
$203$	− 58222.0i	− 1.41285i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−33094.5	−0.772351
$208$	0	0
$209$	−65800.9	−1.50640
$210$	0	0
$211$	−35806.9	−0.804271	−0.402135	−	0.915580i	$-0.631732\pi$
$211$	−35806.9	−0.804271	−0.402135	+	0.915580i	$0.631732\pi$
$212$	0	0
$213$	19084.7	0.420656
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 18520.5i	− 0.393309i
$218$	0	0
$219$	−21088.7	−0.439705
$220$	0	0
$221$	27773.0i	0.568640i
$222$	0	0
$223$	66087.5	1.32895	0.664477	−	0.747309i	$-0.268655\pi$
$223$	66087.5	1.32895	0.664477	+	0.747309i	$0.268655\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6504.85i	− 0.126237i	−0.998006	−	0.0631184i	$-0.979895\pi$
$227$	− 6504.85i	− 0.126237i	0.998006	−	0.0631184i	$-0.0201046\pi$
$228$	0	0
$229$	− 5180.89i	− 0.0987946i	−0.998779	−	0.0493973i	$-0.984270\pi$
$229$	− 5180.89i	− 0.0987946i	0.998779	−	0.0493973i	$-0.0157301\pi$
$230$	0	0
$231$	− 35949.3i	− 0.673700i
$232$	0	0
$233$	− 45055.0i	− 0.829910i	−0.909842	−	0.414955i	$-0.863797\pi$
$233$	− 45055.0i	− 0.829910i	0.909842	−	0.414955i	$-0.136203\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−30778.4	−0.547960
$238$	0	0
$239$	− 71745.9i	− 1.25603i	−0.778200	−	0.628017i	$-0.783867\pi$
$239$	− 71745.9i	− 1.25603i	0.778200	−	0.628017i	$-0.216133\pi$
$240$	0	0
$241$	−13213.7	−0.227504	−0.113752	−	0.993509i	$-0.536287\pi$
$241$	−13213.7	−0.227504	−0.113752	+	0.993509i	$0.536287\pi$
$242$	0	0
$243$	60930.2i	1.03186i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−40306.2	−0.660660
$248$	0	0
$249$	−14308.4	−0.230777
$250$	0	0
$251$	9203.91	0.146091	0.0730457	−	0.997329i	$-0.476728\pi$
$251$	9203.91	0.146091	0.0730457	+	0.997329i	$0.476728\pi$
$252$	0	0
$253$	86771.8	1.35562
$254$	0	0
$255$	0	0
$256$	0	0
$257$	32215.5i	0.487751i	0.969807	+	0.243876i	$0.0784189\pi$
$257$	32215.5i	0.487751i	−0.969807	+	0.243876i	$0.921581\pi$
$258$	0	0
$259$	−71288.4	−1.06272
$260$	0	0
$261$	− 70426.0i	− 1.03384i
$262$	0	0
$263$	−90235.1	−1.30456	−0.652280	−	0.757978i	$-0.726187\pi$
$263$	−90235.1	−1.30456	−0.652280	+	0.757978i	$0.726187\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 48438.8i	− 0.679471i
$268$	0	0
$269$	10715.9i	0.148090i	0.997255	+	0.0740448i	$0.0235908\pi$
$269$	10715.9i	0.148090i	−0.997255	+	0.0740448i	$0.976409\pi$
$270$	0	0
$271$	− 40139.7i	− 0.546557i	−0.961935	−	0.273278i	$-0.911892\pi$
$271$	− 40139.7i	− 0.546557i	0.961935	−	0.273278i	$-0.0881080\pi$
$272$	0	0
$273$	− 22020.6i	− 0.295464i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−9941.96	−0.129572	−0.0647862	−	0.997899i	$-0.520637\pi$
$277$	−9941.96	−0.129572	−0.0647862	+	0.997899i	$0.520637\pi$
$278$	0	0
$279$	− 22402.7i	− 0.287800i
$280$	0	0
$281$	3041.90	0.0385241	0.0192621	−	0.999814i	$-0.493868\pi$
$281$	3041.90	0.0385241	0.0192621	+	0.999814i	$0.493868\pi$
$282$	0	0
$283$	63175.8i	0.788821i	0.918934	+	0.394410i	$0.129051\pi$
$283$	63175.8i	0.788821i	−0.918934	+	0.394410i	$0.870949\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−145316.	−1.76421
$288$	0	0
$289$	2058.84	0.0246506
$290$	0	0
$291$	45412.9	0.536282
$292$	0	0
$293$	−50125.3	−0.583877	−0.291939	−	0.956437i	$-0.594300\pi$
$293$	−50125.3	−0.583877	−0.291939	+	0.956437i	$0.594300\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 101620.i	− 1.15204i
$298$	0	0
$299$	53151.9	0.594533
$300$	0	0
$301$	− 45354.4i	− 0.500595i
$302$	0	0
$303$	67846.4	0.738995
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 49587.2i	− 0.526129i	−0.964778	−	0.263065i	$-0.915267\pi$
$307$	− 49587.2i	− 0.526129i	0.964778	−	0.263065i	$-0.0847332\pi$
$308$	0	0
$309$	59535.9i	0.623537i
$310$	0	0
$311$	122692.i	1.26852i	0.773120	+	0.634260i	$0.218695\pi$
$311$	122692.i	1.26852i	−0.773120	+	0.634260i	$0.781305\pi$
$312$	0	0
$313$	178246.i	1.81942i	0.415248	+	0.909708i	$0.363695\pi$
$313$	178246.i	1.81942i	−0.415248	+	0.909708i	$0.636305\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9020.39	0.0897650	0.0448825	−	0.998992i	$-0.485709\pi$
$317$	9020.39	0.0897650	0.0448825	+	0.998992i	$0.485709\pi$
$318$	0	0
$319$	184653.i	1.81458i
$320$	0	0
$321$	60021.4	0.582500
$322$	0	0
$323$	− 118224.i	− 1.13318i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	86799.1	0.811745
$328$	0	0
$329$	151554.	1.40016
$330$	0	0
$331$	−70357.4	−0.642175	−0.321088	−	0.947049i	$-0.604048\pi$
$331$	−70357.4	−0.642175	−0.321088	+	0.947049i	$0.604048\pi$
$332$	0	0
$333$	−86231.4	−0.777637
$334$	0	0
$335$	0	0
$336$	0	0
$337$	148629.i	1.30871i	0.756186	+	0.654356i	$0.227060\pi$
$337$	148629.i	1.30871i	−0.756186	+	0.654356i	$0.772940\pi$
$338$	0	0
$339$	94767.6	0.824633
$340$	0	0
$341$	58738.5i	0.505143i
$342$	0	0
$343$	−114861.	−0.976303
$344$	0	0
$345$	0	0
$346$	0	0
$347$	94414.8i	0.784117i	0.919940	+	0.392059i	$0.128237\pi$
$347$	94414.8i	0.784117i	−0.919940	+	0.392059i	$0.871763\pi$
$348$	0	0
$349$	− 72082.1i	− 0.591802i	−0.955219	−	0.295901i	$-0.904380\pi$
$349$	− 72082.1i	− 0.591802i	0.955219	−	0.295901i	$-0.0956198\pi$
$350$	0	0
$351$	− 62247.1i	− 0.505249i
$352$	0	0
$353$	− 31879.4i	− 0.255836i	−0.991785	−	0.127918i	$-0.959171\pi$
$353$	− 31879.4i	− 0.255836i	0.991785	−	0.127918i	$-0.0408294\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	64589.8	0.506789
$358$	0	0
$359$	− 34649.8i	− 0.268851i	−0.990924	−	0.134426i	$-0.957081\pi$
$359$	− 34649.8i	− 0.268851i	0.990924	−	0.134426i	$-0.0429189\pi$
$360$	0	0
$361$	41254.3	0.316559
$362$	0	0
$363$	47865.7i	0.363255i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	90234.4	0.669946	0.334973	−	0.942228i	$-0.391273\pi$
$367$	90234.4	0.669946	0.334973	+	0.942228i	$0.391273\pi$
$368$	0	0
$369$	−175776.	−1.29094
$370$	0	0
$371$	−80431.2	−0.584355
$372$	0	0
$373$	−18934.5	−0.136093	−0.0680466	−	0.997682i	$-0.521677\pi$
$373$	−18934.5	−0.136093	−0.0680466	+	0.997682i	$0.521677\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	113109.i	0.795818i
$378$	0	0
$379$	−146129.	−1.01732	−0.508661	−	0.860967i	$-0.669859\pi$
$379$	−146129.	−1.01732	−0.508661	+	0.860967i	$0.669859\pi$
$380$	0	0
$381$	− 48656.1i	− 0.335187i
$382$	0	0
$383$	−63738.0	−0.434511	−0.217255	−	0.976115i	$-0.569710\pi$
$383$	−63738.0	−0.434511	−0.217255	+	0.976115i	$0.569710\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 54861.3i	− 0.366306i
$388$	0	0
$389$	− 73071.2i	− 0.482889i	−0.970415	−	0.241444i	$-0.922379\pi$
$389$	− 73071.2i	− 0.482889i	0.970415	−	0.241444i	$-0.0776212\pi$
$390$	0	0
$391$	155902.i	1.01976i
$392$	0	0
$393$	− 70786.8i	− 0.458318i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17918.3	0.113689	0.0568443	−	0.998383i	$-0.481896\pi$
$397$	17918.3	0.113689	0.0568443	+	0.998383i	$0.481896\pi$
$398$	0	0
$399$	93737.5i	0.588799i
$400$	0	0
$401$	−66769.8	−0.415232	−0.207616	−	0.978210i	$-0.566570\pi$
$401$	−66769.8	−0.415232	−0.207616	+	0.978210i	$0.566570\pi$
$402$	0	0
$403$	35980.1i	0.221540i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	226094.	1.36490
$408$	0	0
$409$	291009.	1.73964	0.869822	−	0.493366i	$-0.164234\pi$
$409$	291009.	1.73964	0.869822	+	0.493366i	$0.164234\pi$
$410$	0	0
$411$	−113999.	−0.674863
$412$	0	0
$413$	−90656.0	−0.531492
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 96632.6i	− 0.555714i
$418$	0	0
$419$	199703.	1.13751	0.568757	−	0.822506i	$-0.307424\pi$
$419$	199703.	1.13751	0.568757	+	0.822506i	$0.307424\pi$
$420$	0	0
$421$	116540.i	0.657525i	0.944413	+	0.328763i	$0.106632\pi$
$421$	116540.i	0.657525i	−0.944413	+	0.328763i	$0.893368\pi$
$422$	0	0
$423$	183322.	1.02455
$424$	0	0
$425$	0	0
$426$	0	0
$427$	56063.1i	0.307483i
$428$	0	0
$429$	69839.3i	0.379477i
$430$	0	0
$431$	− 57475.2i	− 0.309404i	−0.987961	−	0.154702i	$-0.950558\pi$
$431$	− 57475.2i	− 0.309404i	0.987961	−	0.154702i	$-0.0494417\pi$
$432$	0	0
$433$	− 78282.3i	− 0.417530i	−0.977966	−	0.208765i	$-0.933056\pi$
$433$	− 78282.3i	− 0.417530i	0.977966	−	0.208765i	$-0.0669444\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−226257.	−1.18478
$438$	0	0
$439$	248919.i	1.29160i	0.763506	+	0.645801i	$0.223476\pi$
$439$	248919.i	1.29160i	−0.763506	+	0.645801i	$0.776524\pi$
$440$	0	0
$441$	6532.44	0.0335891
$442$	0	0
$443$	305178.i	1.55505i	0.628849	+	0.777527i	$0.283526\pi$
$443$	305178.i	1.55505i	−0.628849	+	0.777527i	$0.716474\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	182164.	0.911692
$448$	0	0
$449$	106754.	0.529529	0.264765	−	0.964313i	$-0.414706\pi$
$449$	106754.	0.529529	0.264765	+	0.964313i	$0.414706\pi$
$450$	0	0
$451$	460876.	2.26585
$452$	0	0
$453$	−128259.	−0.625018
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 212418.i	− 1.01709i	−0.861036	−	0.508543i	$-0.830184\pi$
$457$	− 212418.i	− 1.01709i	0.861036	−	0.508543i	$-0.169816\pi$
$458$	0	0
$459$	182580.	0.866618
$460$	0	0
$461$	377548.i	1.77652i	0.459340	+	0.888261i	$0.348086\pi$
$461$	377548.i	1.77652i	−0.459340	+	0.888261i	$0.651914\pi$
$462$	0	0
$463$	−64579.0	−0.301252	−0.150626	−	0.988591i	$-0.548129\pi$
$463$	−64579.0	−0.301252	−0.150626	+	0.988591i	$0.548129\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	100635.i	0.461439i	0.973020	+	0.230720i	$0.0741080\pi$
$467$	100635.i	0.461439i	−0.973020	+	0.230720i	$0.925892\pi$
$468$	0	0
$469$	196158.i	0.891785i
$470$	0	0
$471$	− 21847.0i	− 0.0984806i
$472$	0	0
$473$	143843.i	0.642936i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−97290.6	−0.427597
$478$	0	0
$479$	188868.i	0.823168i	0.911372	+	0.411584i	$0.135024\pi$
$479$	188868.i	0.823168i	−0.911372	+	0.411584i	$0.864976\pi$
$480$	0	0
$481$	138493.	0.598603
$482$	0	0
$483$	− 123612.i	− 0.529866i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−105446.	−0.444604	−0.222302	−	0.974978i	$-0.571357\pi$
$487$	−105446.	−0.444604	−0.222302	+	0.974978i	$0.571357\pi$
$488$	0	0
$489$	121708.	0.508980
$490$	0	0
$491$	248385.	1.03030	0.515149	−	0.857101i	$-0.327737\pi$
$491$	248385.	1.03030	0.515149	+	0.857101i	$0.327737\pi$
$492$	0	0
$493$	−331765.	−1.36501
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 211578.i	− 0.856558i
$498$	0	0
$499$	194138.	0.779668	0.389834	−	0.920885i	$-0.372532\pi$
$499$	194138.	0.779668	0.389834	+	0.920885i	$0.372532\pi$
$500$	0	0
$501$	109467.i	0.436122i
$502$	0	0
$503$	−323180.	−1.27735	−0.638673	−	0.769478i	$-0.720516\pi$
$503$	−323180.	−1.27735	−0.638673	+	0.769478i	$0.720516\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 86260.2i	− 0.335579i
$508$	0	0
$509$	148319.i	0.572481i	0.958158	+	0.286240i	$0.0924056\pi$
$509$	148319.i	0.572481i	−0.958158	+	0.286240i	$0.907594\pi$
$510$	0	0
$511$	233794.i	0.895348i
$512$	0	0
$513$	264974.i	1.00686i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−480660.	−1.79828
$518$	0	0
$519$	71628.6i	0.265920i
$520$	0	0
$521$	88345.8	0.325470	0.162735	−	0.986670i	$-0.447969\pi$
$521$	88345.8	0.325470	0.162735	+	0.986670i	$0.447969\pi$
$522$	0	0
$523$	− 116531.i	− 0.426028i	−0.977049	−	0.213014i	$-0.931672\pi$
$523$	− 116531.i	− 0.426028i	0.977049	−	0.213014i	$-0.0683279\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−105535.	−0.379993
$528$	0	0
$529$	18524.3	0.0661959
$530$	0	0
$531$	−109659.	−0.388914
$532$	0	0
$533$	282308.	0.993732
$534$	0	0
$535$	0	0
$536$	0	0
$537$	24126.7i	0.0836659i
$538$	0	0
$539$	−17127.7	−0.0589551
$540$	0	0
$541$	− 468624.i	− 1.60114i	−0.599237	−	0.800572i	$-0.704529\pi$
$541$	− 468624.i	− 1.60114i	0.599237	−	0.800572i	$-0.295471\pi$
$542$	0	0
$543$	74346.4	0.252151
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 362111.i	− 1.21023i	−0.796139	−	0.605114i	$-0.793128\pi$
$547$	− 362111.i	− 1.21023i	0.796139	−	0.605114i	$-0.206872\pi$
$548$	0	0
$549$	67814.7i	0.224998i
$550$	0	0
$551$	− 481482.i	− 1.58590i
$552$	0	0
$553$	341216.i	1.11578i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	498265.	1.60602	0.803008	−	0.595968i	$-0.203231\pi$
$557$	498265.	1.60602	0.803008	+	0.595968i	$0.203231\pi$
$558$	0	0
$559$	88110.9i	0.281972i
$560$	0	0
$561$	−204849.	−0.650890
$562$	0	0
$563$	245981.i	0.776041i	0.921651	+	0.388021i	$0.126841\pi$
$563$	245981.i	0.776041i	−0.921651	+	0.388021i	$0.873159\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	101047.	0.314308
$568$	0	0
$569$	−385802.	−1.19163	−0.595813	−	0.803123i	$-0.703170\pi$
$569$	−385802.	−1.19163	−0.595813	+	0.803123i	$0.703170\pi$
$570$	0	0
$571$	14171.0	0.0434640	0.0217320	−	0.999764i	$-0.493082\pi$
$571$	14171.0	0.0434640	0.0217320	+	0.999764i	$0.493082\pi$
$572$	0	0
$573$	163943.	0.499326
$574$	0	0
$575$	0	0
$576$	0	0
$577$	109691.i	0.329474i	0.986338	+	0.164737i	$0.0526775\pi$
$577$	109691.i	0.329474i	−0.986338	+	0.164737i	$0.947323\pi$
$578$	0	0
$579$	−8740.42	−0.0260720
$580$	0	0
$581$	158626.i	0.469918i
$582$	0	0
$583$	255091.	0.750512
$584$	0	0
$585$	0	0
$586$	0	0
$587$	377043.i	1.09425i	0.837052	+	0.547123i	$0.184277\pi$
$587$	377043.i	1.09425i	−0.837052	+	0.547123i	$0.815723\pi$
$588$	0	0
$589$	− 153160.i	− 0.441485i
$590$	0	0
$591$	327955.i	0.938943i
$592$	0	0
$593$	− 27615.5i	− 0.0785314i	−0.999229	−	0.0392657i	$-0.987498\pi$
$593$	− 27615.5i	− 0.0785314i	0.999229	−	0.0392657i	$-0.0125019\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	351130.	0.985189
$598$	0	0
$599$	170000.i	0.473802i	0.971534	+	0.236901i	$0.0761316\pi$
$599$	170000.i	0.473802i	−0.971534	+	0.236901i	$0.923868\pi$
$600$	0	0
$601$	297213.	0.822846	0.411423	−	0.911445i	$-0.365032\pi$
$601$	297213.	0.822846	0.411423	+	0.911445i	$0.365032\pi$
$602$	0	0
$603$	237275.i	0.652556i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23984.9	−0.0650969	−0.0325485	−	0.999470i	$-0.510362\pi$
$607$	−23984.9	−0.0650969	−0.0325485	+	0.999470i	$0.510362\pi$
$608$	0	0
$609$	263050.	0.709257
$610$	0	0
$611$	−294427.	−0.788671
$612$	0	0
$613$	−291516.	−0.775785	−0.387893	−	0.921705i	$-0.626797\pi$
$613$	−291516.	−0.775785	−0.387893	+	0.921705i	$0.626797\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	111058.i	0.291729i	0.989305	+	0.145864i	$0.0465963\pi$
$617$	111058.i	0.291729i	−0.989305	+	0.145864i	$0.953404\pi$
$618$	0	0
$619$	−300205.	−0.783496	−0.391748	−	0.920073i	$-0.628129\pi$
$619$	−300205.	−0.783496	−0.391748	+	0.920073i	$0.628129\pi$
$620$	0	0
$621$	− 349421.i	− 0.906079i
$622$	0	0
$623$	−537003.	−1.38357
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 297292.i	− 0.756220i
$628$	0	0
$629$	406221.i	1.02674i
$630$	0	0
$631$	150496.i	0.377979i	0.981979	+	0.188989i	$0.0605212\pi$
$631$	150496.i	0.377979i	−0.981979	+	0.188989i	$0.939479\pi$
$632$	0	0
$633$	− 161778.i	− 0.403749i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−10491.5	−0.0258559
$638$	0	0
$639$	− 255927.i	− 0.626779i
$640$	0	0
$641$	−341612.	−0.831414	−0.415707	−	0.909499i	$-0.636466\pi$
$641$	−341612.	−0.831414	−0.415707	+	0.909499i	$0.636466\pi$
$642$	0	0
$643$	− 89782.7i	− 0.217156i	−0.994088	−	0.108578i	$-0.965370\pi$
$643$	− 89782.7i	− 0.217156i	0.994088	−	0.108578i	$-0.0346297\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−569707.	−1.36095	−0.680477	−	0.732770i	$-0.738227\pi$
$647$	−569707.	−1.36095	−0.680477	+	0.732770i	$0.738227\pi$
$648$	0	0
$649$	287519.	0.682617
$650$	0	0
$651$	83676.7	0.197443
$652$	0	0
$653$	258929.	0.607233	0.303616	−	0.952794i	$-0.401806\pi$
$653$	258929.	0.607233	0.303616	+	0.952794i	$0.401806\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	282800.i	0.655163i
$658$	0	0
$659$	−43971.3	−0.101251	−0.0506255	−	0.998718i	$-0.516121\pi$
$659$	−43971.3	−0.101251	−0.0506255	+	0.998718i	$0.516121\pi$
$660$	0	0
$661$	216610.i	0.495765i	0.968790	+	0.247882i	$0.0797347\pi$
$661$	216610.i	0.495765i	−0.968790	+	0.247882i	$0.920265\pi$
$662$	0	0
$663$	−125480.	−0.285461
$664$	0	0
$665$	0	0
$666$	0	0
$667$	634931.i	1.42717i
$668$	0	0
$669$	298587.i	0.667142i
$670$	0	0
$671$	− 177806.i	− 0.394914i
$672$	0	0
$673$	− 614134.i	− 1.35592i	−0.735101	−	0.677958i	$-0.762865\pi$
$673$	− 614134.i	− 1.35592i	0.735101	−	0.677958i	$-0.237135\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	710586.	1.55038	0.775192	−	0.631725i	$-0.217653\pi$
$677$	710586.	1.55038	0.775192	+	0.631725i	$0.217653\pi$
$678$	0	0
$679$	− 503458.i	− 1.09200i
$680$	0	0
$681$	29389.3	0.0633716
$682$	0	0
$683$	187636.i	0.402231i	0.979568	+	0.201116i	$0.0644567\pi$
$683$	187636.i	0.402231i	−0.979568	+	0.201116i	$0.935543\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	23407.5	0.0495955
$688$	0	0
$689$	156255.	0.329151
$690$	0	0
$691$	−628570.	−1.31643	−0.658215	−	0.752830i	$-0.728688\pi$
$691$	−628570.	−1.31643	−0.658215	+	0.752830i	$0.728688\pi$
$692$	0	0
$693$	−482081.	−1.00382
$694$	0	0
$695$	0	0
$696$	0	0
$697$	828051.i	1.70448i
$698$	0	0
$699$	203561.	0.416619
$700$	0	0
$701$	− 149078.i	− 0.303374i	−0.988429	−	0.151687i	$-0.951529\pi$
$701$	− 149078.i	− 0.303374i	0.988429	−	0.151687i	$-0.0484705\pi$
$702$	0	0
$703$	−589538.	−1.19289
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 752160.i	− 1.50477i
$708$	0	0
$709$	− 47510.9i	− 0.0945151i	−0.998883	−	0.0472575i	$-0.984952\pi$
$709$	− 47510.9i	− 0.0945151i	0.998883	−	0.0472575i	$-0.0150481\pi$
$710$	0	0
$711$	412739.i	0.816462i
$712$	0	0
$713$	201973.i	0.397296i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	324152.	0.630536
$718$	0	0
$719$	− 843806.i	− 1.63224i	−0.577880	−	0.816122i	$-0.696120\pi$
$719$	− 843806.i	− 1.63224i	0.577880	−	0.816122i	$-0.303880\pi$
$720$	0	0
$721$	660028.	1.26967
$722$	0	0
$723$	− 59700.0i	− 0.114208i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−792065.	−1.49862	−0.749311	−	0.662218i	$-0.769615\pi$
$727$	−792065.	−1.49862	−0.749311	+	0.662218i	$0.769615\pi$
$728$	0	0
$729$	−111878.	−0.210519
$730$	0	0
$731$	−258442.	−0.483647
$732$	0	0
$733$	−4047.83	−0.00753380	−0.00376690	−	0.999993i	$-0.501199\pi$
$733$	−4047.83	−0.00753380	−0.00376690	+	0.999993i	$0.501199\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 622123.i	− 1.14536i
$738$	0	0
$739$	895023.	1.63887	0.819437	−	0.573170i	$-0.194286\pi$
$739$	895023.	1.63887	0.819437	+	0.573170i	$0.194286\pi$
$740$	0	0
$741$	− 182105.i	− 0.331655i
$742$	0	0
$743$	616018.	1.11588	0.557938	−	0.829883i	$-0.311593\pi$
$743$	616018.	1.11588	0.557938	+	0.829883i	$0.311593\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	191876.i	0.343858i
$748$	0	0
$749$	− 665410.i	− 1.18611i
$750$	0	0
$751$	232448.i	0.412141i	0.978537	+	0.206071i	$0.0660677\pi$
$751$	232448.i	0.412141i	−0.978537	+	0.206071i	$0.933932\pi$
$752$	0	0
$753$	41583.7i	0.0733387i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−431771.	−0.753462	−0.376731	−	0.926323i	$-0.622952\pi$
$757$	−431771.	−0.753462	−0.376731	+	0.926323i	$0.622952\pi$
$758$	0	0
$759$	392040.i	0.680528i
$760$	0	0
$761$	−368807.	−0.636840	−0.318420	−	0.947950i	$-0.603152\pi$
$761$	−368807.	−0.636840	−0.318420	+	0.947950i	$0.603152\pi$
$762$	0	0
$763$	− 962274.i	− 1.65291i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	176119.	0.299375
$768$	0	0
$769$	−850748.	−1.43863	−0.719313	−	0.694686i	$-0.755544\pi$
$769$	−850748.	−1.43863	−0.719313	+	0.694686i	$0.755544\pi$
$770$	0	0
$771$	−145551.	−0.244854
$772$	0	0
$773$	967445.	1.61908	0.809538	−	0.587067i	$-0.199718\pi$
$773$	967445.	1.61908	0.809538	+	0.587067i	$0.199718\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 322085.i	− 0.533492i
$778$	0	0
$779$	−1.20173e6	−1.98030
$780$	0	0
$781$	671026.i	1.10011i
$782$	0	0
$783$	743580.	1.21284
$784$	0	0
$785$	0	0
$786$	0	0
$787$	434908.i	0.702178i	0.936342	+	0.351089i	$0.114189\pi$
$787$	434908.i	0.702178i	−0.936342	+	0.351089i	$0.885811\pi$
$788$	0	0
$789$	− 407687.i	− 0.654896i
$790$	0	0
$791$	− 1.05061e6i	− 1.67915i
$792$	0	0
$793$	− 108915.i	− 0.173197i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−648541.	−1.02099	−0.510494	−	0.859881i	$-0.670537\pi$
$797$	−648541.	−1.02099	−0.510494	+	0.859881i	$0.670537\pi$
$798$	0	0
$799$	− 863598.i	− 1.35275i
$800$	0	0
$801$	−649566.	−1.01241
$802$	0	0
$803$	− 741487.i	− 1.14993i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−48415.0	−0.0743418
$808$	0	0
$809$	−338083.	−0.516567	−0.258284	−	0.966069i	$-0.583157\pi$
$809$	−338083.	−0.516567	−0.258284	+	0.966069i	$0.583157\pi$
$810$	0	0
$811$	941405.	1.43131	0.715657	−	0.698452i	$-0.246128\pi$
$811$	941405.	1.43131	0.715657	+	0.698452i	$0.246128\pi$
$812$	0	0
$813$	181353.	0.274375
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 375070.i	− 0.561912i
$818$	0	0
$819$	−295298.	−0.440243
$820$	0	0
$821$	− 702066.i	− 1.04158i	−0.853686	−	0.520789i	$-0.825638\pi$
$821$	− 702066.i	− 1.04158i	0.853686	−	0.520789i	$-0.174362\pi$
$822$	0	0
$823$	736652.	1.08758	0.543792	−	0.839220i	$-0.316988\pi$
$823$	736652.	1.08758	0.543792	+	0.839220i	$0.316988\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 698186.i	− 1.02085i	−0.859923	−	0.510423i	$-0.829489\pi$
$827$	− 698186.i	− 1.02085i	0.859923	−	0.510423i	$-0.170511\pi$
$828$	0	0
$829$	609784.i	0.887293i	0.896202	+	0.443647i	$0.146315\pi$
$829$	609784.i	0.887293i	−0.896202	+	0.443647i	$0.853685\pi$
$830$	0	0
$831$	− 44918.3i	− 0.0650461i
$832$	0	0
$833$	− 30773.2i	− 0.0443488i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	236534.	0.337631
$838$	0	0
$839$	− 245795.i	− 0.349179i	−0.984641	−	0.174590i	$-0.944140\pi$
$839$	− 245795.i	− 0.349179i	0.984641	−	0.174590i	$-0.0558599\pi$
$840$	0	0
$841$	−643872.	−0.910348
$842$	0	0
$843$	13743.5i	0.0193393i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	530650.	0.739676
$848$	0	0
$849$	−285432.	−0.395992
$850$	0	0
$851$	777425.	1.07349
$852$	0	0
$853$	482861.	0.663627	0.331813	−	0.943345i	$-0.392340\pi$
$853$	482861.	0.663627	0.331813	+	0.943345i	$0.392340\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 275704.i	− 0.375389i	−0.982227	−	0.187695i	$-0.939898\pi$
$857$	− 275704.i	− 0.375389i	0.982227	−	0.187695i	$-0.0601015\pi$
$858$	0	0
$859$	−874456.	−1.18509	−0.592546	−	0.805537i	$-0.701877\pi$
$859$	−874456.	−1.18509	−0.592546	+	0.805537i	$0.701877\pi$
$860$	0	0
$861$	− 656546.i	− 0.885643i
$862$	0	0
$863$	−207383.	−0.278453	−0.139227	−	0.990261i	$-0.544462\pi$
$863$	−207383.	−0.278453	−0.139227	+	0.990261i	$0.544462\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9301.96i	0.0123747i
$868$	0	0
$869$	− 1.08218e6i	− 1.43304i
$870$	0	0
$871$	− 381080.i	− 0.502319i
$872$	0	0
$873$	− 608989.i	− 0.799063i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−287580.	−0.373903	−0.186952	−	0.982369i	$-0.559861\pi$
$877$	−287580.	−0.373903	−0.186952	+	0.982369i	$0.559861\pi$
$878$	0	0
$879$	− 226468.i	− 0.293110i
$880$	0	0
$881$	72878.9	0.0938966	0.0469483	−	0.998897i	$-0.485050\pi$
$881$	72878.9	0.0938966	0.0469483	+	0.998897i	$0.485050\pi$
$882$	0	0
$883$	587119.i	0.753017i	0.926413	+	0.376509i	$0.122875\pi$
$883$	587119.i	0.753017i	−0.926413	+	0.376509i	$0.877125\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	525528.	0.667958	0.333979	−	0.942581i	$-0.391609\pi$
$887$	525528.	0.667958	0.333979	+	0.942581i	$0.391609\pi$
$888$	0	0
$889$	−539413.	−0.682523
$890$	0	0
$891$	−320473.	−0.403679
$892$	0	0
$893$	1.25332e6	1.57166
$894$	0	0
$895$	0	0
$896$	0	0
$897$	240143.i	0.298459i
$898$	0	0
$899$	−429804.	−0.531804
$900$	0	0
$901$	458319.i	0.564571i
$902$	0	0
$903$	204914.	0.251302
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.45695e6i	1.77105i	0.464596	+	0.885523i	$0.346199\pi$
$907$	1.45695e6i	1.77105i	−0.464596	+	0.885523i	$0.653801\pi$
$908$	0	0
$909$	− 909822.i	− 1.10111i
$910$	0	0
$911$	− 546104.i	− 0.658020i	−0.944326	−	0.329010i	$-0.893285\pi$
$911$	− 546104.i	− 0.658020i	0.944326	−	0.329010i	$-0.106715\pi$
$912$	0	0
$913$	− 503088.i	− 0.603535i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−784758.	−0.933248
$918$	0	0
$919$	1.30937e6i	1.55036i	0.631740	+	0.775180i	$0.282341\pi$
$919$	1.30937e6i	1.55036i	−0.631740	+	0.775180i	$0.717659\pi$
$920$	0	0
$921$	224037.	0.264120
$922$	0	0
$923$	411036.i	0.482476i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	798378.	0.929072
$928$	0	0
$929$	−731556.	−0.847649	−0.423824	−	0.905744i	$-0.639313\pi$
$929$	−731556.	−0.847649	−0.423824	+	0.905744i	$0.639313\pi$
$930$	0	0
$931$	44660.3	0.0515255
$932$	0	0
$933$	−554331.	−0.636804
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 368844.i	− 0.420111i	−0.977690	−	0.210055i	$-0.932636\pi$
$937$	− 368844.i	− 0.420111i	0.977690	−	0.210055i	$-0.0673644\pi$
$938$	0	0
$939$	−805327.	−0.913358
$940$	0	0
$941$	1.05710e6i	1.19381i	0.802310	+	0.596907i	$0.203604\pi$
$941$	1.05710e6i	1.19381i	−0.802310	+	0.596907i	$0.796396\pi$
$942$	0	0
$943$	1.58472e6	1.78209
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.05246e6i	− 1.17356i	−0.809748	−	0.586778i	$-0.800396\pi$
$947$	− 1.05246e6i	− 1.17356i	0.809748	−	0.586778i	$-0.199604\pi$
$948$	0	0
$949$	− 454196.i	− 0.504325i
$950$	0	0
$951$	40754.6i	0.0450625i
$952$	0	0
$953$	65088.8i	0.0716672i	0.999358	+	0.0358336i	$0.0114086\pi$
$953$	65088.8i	0.0716672i	−0.999358	+	0.0358336i	$0.988591\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−834273.	−0.910928
$958$	0	0
$959$	1.26381e6i	1.37419i
$960$	0	0
$961$	786799.	0.851956
$962$	0	0
$963$	− 804889.i	− 0.867927i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	481804.	0.515250	0.257625	−	0.966245i	$-0.417060\pi$
$967$	481804.	0.515250	0.257625	+	0.966245i	$0.417060\pi$
$968$	0	0
$969$	534142.	0.568865
$970$	0	0
$971$	−439500.	−0.466144	−0.233072	−	0.972459i	$-0.574878\pi$
$971$	−439500.	−0.466144	−0.233072	+	0.972459i	$0.574878\pi$
$972$	0	0
$973$	−1.07129e6	−1.13157
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.37613e6i	1.44169i	0.693098	+	0.720844i	$0.256245\pi$
$977$	1.37613e6i	1.44169i	−0.693098	+	0.720844i	$0.743755\pi$
$978$	0	0
$979$	1.70313e6	1.77697
$980$	0	0
$981$	− 1.16398e6i	− 1.20950i
$982$	0	0
$983$	−767551.	−0.794328	−0.397164	−	0.917748i	$-0.630006\pi$
$983$	−767551.	−0.794328	−0.397164	+	0.917748i	$0.630006\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	684730.i	0.702887i
$988$	0	0
$989$	494606.i	0.505670i
$990$	0	0
$991$	1.85016e6i	1.88392i	0.335722	+	0.941961i	$0.391020\pi$
$991$	1.85016e6i	1.88392i	−0.335722	+	0.941961i	$0.608980\pi$
$992$	0	0
$993$	− 317878.i	− 0.322376i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	337598.	0.339633	0.169816	−	0.985476i	$-0.445683\pi$
$997$	337598.	0.339633	0.169816	+	0.985476i	$0.445683\pi$
$998$	0	0
$999$	− 910457.i	− 0.912281i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.5.e.e.399.22		32
4.3	odd	2		200.5.e.e.99.7		32
5.2	odd	4		160.5.g.a.111.9		16
5.3	odd	4		800.5.g.h.751.7		16
5.4	even	2	inner	800.5.e.e.399.11		32
8.3	odd	2	inner	800.5.e.e.399.12		32
8.5	even	2		200.5.e.e.99.25		32
15.2	even	4		1440.5.g.a.271.15		16
20.3	even	4		200.5.g.h.51.4		16
20.7	even	4		40.5.g.a.11.13	✓	16
20.19	odd	2		200.5.e.e.99.26		32
40.3	even	4		800.5.g.h.751.8		16
40.13	odd	4		200.5.g.h.51.3		16
40.19	odd	2	inner	800.5.e.e.399.21		32
40.27	even	4		160.5.g.a.111.10		16
40.29	even	2		200.5.e.e.99.8		32
40.37	odd	4		40.5.g.a.11.14	yes	16
60.47	odd	4		360.5.g.a.91.4		16
120.77	even	4		360.5.g.a.91.3		16
120.107	odd	4		1440.5.g.a.271.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.g.a.11.13	✓	16	20.7	even	4
40.5.g.a.11.14	yes	16	40.37	odd	4
160.5.g.a.111.9		16	5.2	odd	4
160.5.g.a.111.10		16	40.27	even	4
200.5.e.e.99.7		32	4.3	odd	2
200.5.e.e.99.8		32	40.29	even	2
200.5.e.e.99.25		32	8.5	even	2
200.5.e.e.99.26		32	20.19	odd	2
200.5.g.h.51.3		16	40.13	odd	4
200.5.g.h.51.4		16	20.3	even	4
360.5.g.a.91.3		16	120.77	even	4
360.5.g.a.91.4		16	60.47	odd	4
800.5.e.e.399.11		32	5.4	even	2	inner
800.5.e.e.399.12		32	8.3	odd	2	inner
800.5.e.e.399.21		32	40.19	odd	2	inner
800.5.e.e.399.22		32	1.1	even	1	trivial
800.5.g.h.751.7		16	5.3	odd	4
800.5.g.h.751.8		16	40.3	even	4
1440.5.g.a.271.2		16	120.107	odd	4
1440.5.g.a.271.15		16	15.2	even	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)

\(f(q)\)	\(=\)	\(q+4.51805i q^{3} +50.0881 q^{7} +60.5872 q^{9} -158.856 q^{11} -97.3070 q^{13} -285.416i q^{17} +414.217 q^{19} +226.301i q^{21} -546.228 q^{23} +639.698i q^{27} -1162.39i q^{29} -369.759i q^{31} -717.721i q^{33} -1423.26 q^{37} -439.638i q^{39} -2901.21 q^{41} -905.493i q^{43} +3025.75 q^{47} +107.819 q^{49} +1289.52 q^{51} -1605.79 q^{53} +1871.45i q^{57} -1809.93 q^{59} +1119.29i q^{61} +3034.70 q^{63} +3916.26i q^{67} -2467.89i q^{69} -4224.11i q^{71} +4667.66i q^{73} -7956.81 q^{77} +6812.31i q^{79} +2017.38 q^{81} +3166.94i q^{83} +5251.74 q^{87} -10721.2 q^{89} -4873.93 q^{91} +1670.59 q^{93} -10051.4i q^{97} -9624.66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 864 q^{9} - 384 q^{11} - 1408 q^{19} - 4416 q^{41} + 4960 q^{49} + 35584 q^{51} + 28032 q^{59} + 20768 q^{81} - 13632 q^{89} - 49152 q^{91} + 5248 q^{99}+O(q^{100}) \) 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

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