LMFDB - Embedded newform 800.4.c.o.449.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(449,800)]

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([0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.c (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:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1135425807366400.12
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 52x^{6} + 664x^{4} + 337x^{2} + 36 $ x^8 + 52x^6 + 664x^4 + 337*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.5
Root			$-0.610728i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.4.c.o.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.221457i q^{3} -22.7992i q^{7} +26.9510 q^{9} +O(q^{10})$	$q+0.221457i q^{3} -22.7992i q^{7} +26.9510 q^{9} -66.8474 q^{11} +32.9510i q^{13} +35.9510i q^{17} -44.0482 q^{19} +5.04904 q^{21} +139.010i q^{23} +11.9478i q^{27} -134.853 q^{29} +229.321 q^{31} -14.8038i q^{33} +79.1471i q^{37} -7.29722 q^{39} +384.804 q^{41} -251.234i q^{43} +241.490i q^{47} -176.804 q^{49} -7.96159 q^{51} +222.000i q^{53} -9.75478i q^{57} +552.707 q^{59} +494.951 q^{61} -614.460i q^{63} -574.631i q^{67} -30.7847 q^{69} -654.533 q^{71} +1131.17i q^{73} +1524.07i q^{77} -179.504 q^{79} +725.030 q^{81} +810.131i q^{83} -29.8641i q^{87} +29.7548 q^{89} +751.256 q^{91} +50.7847i q^{93} +383.510i q^{97} -1801.60 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 192 q^{9}+O(q^{10}) $ 8 * q - 192 * q^9	$ 8 q - 192 q^{9} + 448 q^{21} + 144 q^{29} + 1448 q^{41} + 216 q^{49} + 3552 q^{61} - 6768 q^{69} + 14360 q^{81} - 1800 q^{89}+O(q^{100}) $ 8 * q - 192 * q^9 + 448 * q^21 + 144 * q^29 + 1448 * q^41 + 216 * q^49 + 3552 * q^61 - 6768 * q^69 + 14360 * q^81 - 1800 * q^89

Display 100 coefficients 10 coefficients

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
$3$	0.221457i	0.0426194i	0.999773	+	0.0213097i	$0.00678360\pi$
$3$	0.221457i	0.0426194i	−0.999773	+	0.0213097i	$0.993216\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 22.7992i	− 1.23104i	−0.788121	−	0.615521i	$-0.788946\pi$
$7$	− 22.7992i	− 1.23104i	0.788121	−	0.615521i	$-0.211054\pi$
$8$	0	0
$9$	26.9510	0.998184
$10$	0	0
$11$	−66.8474	−1.83230	−0.916148	−	0.400840i	$-0.868718\pi$
$11$	−66.8474	−1.83230	−0.916148	+	0.400840i	$0.868718\pi$
$12$	0	0
$13$	32.9510i	0.702996i	0.936189	+	0.351498i	$0.114328\pi$
$13$	32.9510i	0.702996i	−0.936189	+	0.351498i	$0.885672\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	35.9510i	0.512905i	0.966557	+	0.256453i	$0.0825537\pi$
$17$	35.9510i	0.512905i	−0.966557	+	0.256453i	$0.917446\pi$
$18$	0	0
$19$	−44.0482	−0.531861	−0.265930	−	0.963992i	$-0.585679\pi$
$19$	−44.0482	−0.531861	−0.265930	+	0.963992i	$0.585679\pi$
$20$	0	0
$21$	5.04904	0.0524663
$22$	0	0
$23$	139.010i	1.26024i	0.776497	+	0.630121i	$0.216995\pi$
$23$	139.010i	1.26024i	−0.776497	+	0.630121i	$0.783005\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	11.9478i	0.0851614i
$28$	0	0
$29$	−134.853	−0.863502	−0.431751	−	0.901993i	$-0.642104\pi$
$29$	−134.853	−0.863502	−0.431751	+	0.901993i	$0.642104\pi$
$30$	0	0
$31$	229.321	1.32862	0.664310	−	0.747457i	$-0.268725\pi$
$31$	229.321	1.32862	0.664310	+	0.747457i	$0.268725\pi$
$32$	0	0
$33$	− 14.8038i	− 0.0780914i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	79.1471i	0.351668i	0.984420	+	0.175834i	$0.0562622\pi$
$37$	79.1471i	0.351668i	−0.984420	+	0.175834i	$0.943738\pi$
$38$	0	0
$39$	−7.29722	−0.0299613
$40$	0	0
$41$	384.804	1.46576	0.732881	−	0.680357i	$-0.238175\pi$
$41$	384.804	1.46576	0.732881	+	0.680357i	$0.238175\pi$
$42$	0	0
$43$	− 251.234i	− 0.890997i	−0.895283	−	0.445498i	$-0.853027\pi$
$43$	− 251.234i	− 0.890997i	0.895283	−	0.445498i	$-0.146973\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	241.490i	0.749467i	0.927133	+	0.374733i	$0.122266\pi$
$47$	241.490i	0.749467i	−0.927133	+	0.374733i	$0.877734\pi$
$48$	0	0
$49$	−176.804	−0.515463
$50$	0	0
$51$	−7.96159	−0.0218597
$52$	0	0
$53$	222.000i	0.575359i	0.957727	+	0.287680i	$0.0928838\pi$
$53$	222.000i	0.575359i	−0.957727	+	0.287680i	$0.907116\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 9.75478i	− 0.0226676i
$58$	0	0
$59$	552.707	1.21960	0.609799	−	0.792556i	$-0.291250\pi$
$59$	552.707	1.21960	0.609799	+	0.792556i	$0.291250\pi$
$60$	0	0
$61$	494.951	1.03888	0.519442	−	0.854505i	$-0.326140\pi$
$61$	494.951	1.03888	0.519442	+	0.854505i	$0.326140\pi$
$62$	0	0
$63$	− 614.460i	− 1.22881i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 574.631i	− 1.04780i	−0.851781	−	0.523898i	$-0.824477\pi$
$67$	− 574.631i	− 1.04780i	0.851781	−	0.523898i	$-0.175523\pi$
$68$	0	0
$69$	−30.7847	−0.0537107
$70$	0	0
$71$	−654.533	−1.09407	−0.547034	−	0.837110i	$-0.684243\pi$
$71$	−654.533	−1.09407	−0.547034	+	0.837110i	$0.684243\pi$
$72$	0	0
$73$	1131.17i	1.81360i	0.421558	+	0.906801i	$0.361483\pi$
$73$	1131.17i	1.81360i	−0.421558	+	0.906801i	$0.638517\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1524.07i	2.25563i
$78$	0	0
$79$	−179.504	−0.255643	−0.127821	−	0.991797i	$-0.540798\pi$
$79$	−179.504	−0.255643	−0.127821	+	0.991797i	$0.540798\pi$
$80$	0	0
$81$	725.030	0.994554
$82$	0	0
$83$	810.131i	1.07137i	0.844419	+	0.535683i	$0.179946\pi$
$83$	810.131i	1.07137i	−0.844419	+	0.535683i	$0.820054\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 29.8641i	− 0.0368019i
$88$	0	0
$89$	29.7548	0.0354382	0.0177191	−	0.999843i	$-0.494360\pi$
$89$	29.7548	0.0354382	0.0177191	+	0.999843i	$0.494360\pi$
$90$	0	0
$91$	751.256	0.865418
$92$	0	0
$93$	50.7847i	0.0566250i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	383.510i	0.401438i	0.979649	+	0.200719i	$0.0643278\pi$
$97$	383.510i	0.401438i	−0.979649	+	0.200719i	$0.935672\pi$
$98$	0	0
$99$	−1801.60	−1.82897
$100$	0	0
$101$	1272.26	1.25342	0.626708	−	0.779254i	$-0.284402\pi$
$101$	1272.26	1.25342	0.626708	+	0.779254i	$0.284402\pi$
$102$	0	0
$103$	1423.51i	1.36177i	0.732391	+	0.680884i	$0.238404\pi$
$103$	1423.51i	1.36177i	−0.732391	+	0.680884i	$0.761596\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1663.68i	1.50312i	0.659665	+	0.751560i	$0.270698\pi$
$107$	1663.68i	1.50312i	−0.659665	+	0.751560i	$0.729302\pi$
$108$	0	0
$109$	−1847.87	−1.62380	−0.811899	−	0.583798i	$-0.801566\pi$
$109$	−1847.87	−1.62380	−0.811899	+	0.583798i	$0.801566\pi$
$110$	0	0
$111$	−17.5277	−0.0149879
$112$	0	0
$113$	1063.39i	0.885270i	0.896702	+	0.442635i	$0.145956\pi$
$113$	1063.39i	0.885270i	−0.896702	+	0.442635i	$0.854044\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	888.060i	0.701719i
$118$	0	0
$119$	819.653	0.631408
$120$	0	0
$121$	3137.58	2.35731
$122$	0	0
$123$	85.2175i	0.0624699i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2120.07i	− 1.48131i	−0.671887	−	0.740653i	$-0.734516\pi$
$127$	− 2120.07i	− 1.48131i	0.671887	−	0.740653i	$-0.265484\pi$
$128$	0	0
$129$	55.6376	0.0379738
$130$	0	0
$131$	−675.329	−0.450410	−0.225205	−	0.974311i	$-0.572305\pi$
$131$	−675.329	−0.450410	−0.225205	+	0.974311i	$0.572305\pi$
$132$	0	0
$133$	1004.26i	0.654743i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1089.92i	− 0.679695i	−0.940481	−	0.339848i	$-0.889625\pi$
$137$	− 1089.92i	− 0.679695i	0.940481	−	0.339848i	$-0.110375\pi$
$138$	0	0
$139$	1429.48	0.872283	0.436141	−	0.899878i	$-0.356345\pi$
$139$	1429.48	0.872283	0.436141	+	0.899878i	$0.356345\pi$
$140$	0	0
$141$	−53.4797	−0.0319418
$142$	0	0
$143$	− 2202.69i	− 1.28810i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 39.1544i	− 0.0219687i
$148$	0	0
$149$	−2040.36	−1.12183	−0.560916	−	0.827873i	$-0.689551\pi$
$149$	−2040.36	−1.12183	−0.560916	+	0.827873i	$0.689551\pi$
$150$	0	0
$151$	174.232	0.0938995	0.0469498	−	0.998897i	$-0.485050\pi$
$151$	174.232	0.0938995	0.0469498	+	0.998897i	$0.485050\pi$
$152$	0	0
$153$	968.913i	0.511974i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3494.72i	1.77649i	0.459367	+	0.888247i	$0.348076\pi$
$157$	3494.72i	1.77649i	−0.459367	+	0.888247i	$0.651924\pi$
$158$	0	0
$159$	−49.1634	−0.0245215
$160$	0	0
$161$	3169.31	1.55141
$162$	0	0
$163$	− 2989.55i	− 1.43656i	−0.695752	−	0.718282i	$-0.744929\pi$
$163$	− 2989.55i	− 1.43656i	0.695752	−	0.718282i	$-0.255071\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2215.51i	1.02659i	0.858211	+	0.513297i	$0.171576\pi$
$167$	2215.51i	1.02659i	−0.858211	+	0.513297i	$0.828424\pi$
$168$	0	0
$169$	1111.23	0.505796
$170$	0	0
$171$	−1187.14	−0.530895
$172$	0	0
$173$	829.646i	0.364606i	0.983242	+	0.182303i	$0.0583552\pi$
$173$	829.646i	0.364606i	−0.983242	+	0.182303i	$0.941645\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	122.401i	0.0519785i
$178$	0	0
$179$	700.164	0.292362	0.146181	−	0.989258i	$-0.453302\pi$
$179$	700.164	0.292362	0.146181	+	0.989258i	$0.453302\pi$
$180$	0	0
$181$	−1157.22	−0.475222	−0.237611	−	0.971360i	$-0.576364\pi$
$181$	−1157.22	−0.475222	−0.237611	+	0.971360i	$0.576364\pi$
$182$	0	0
$183$	109.610i	0.0442767i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2403.23i	− 0.939794i
$188$	0	0
$189$	272.401	0.104837
$190$	0	0
$191$	2306.05	0.873613	0.436807	−	0.899555i	$-0.356109\pi$
$191$	2306.05	0.873613	0.436807	+	0.899555i	$0.356109\pi$
$192$	0	0
$193$	567.461i	0.211641i	0.994385	+	0.105820i	$0.0337469\pi$
$193$	567.461i	0.211641i	−0.994385	+	0.105820i	$0.966253\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4301.74i	1.55577i	0.628407	+	0.777885i	$0.283707\pi$
$197$	4301.74i	1.55577i	−0.628407	+	0.777885i	$0.716293\pi$
$198$	0	0
$199$	−258.531	−0.0920945	−0.0460473	−	0.998939i	$-0.514662\pi$
$199$	−258.531	−0.0920945	−0.0460473	+	0.998939i	$0.514662\pi$
$200$	0	0
$201$	127.256	0.0446564
$202$	0	0
$203$	3074.54i	1.06301i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3746.45i	1.25795i
$208$	0	0
$209$	2944.51	0.974526
$210$	0	0
$211$	3083.01	1.00589	0.502946	−	0.864318i	$-0.332250\pi$
$211$	3083.01	1.00589	0.502946	+	0.864318i	$0.332250\pi$
$212$	0	0
$213$	− 144.951i	− 0.0466285i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5228.33i	− 1.63559i
$218$	0	0
$219$	−250.505	−0.0772947
$220$	0	0
$221$	−1184.62	−0.360570
$222$	0	0
$223$	5701.49i	1.71211i	0.516888	+	0.856053i	$0.327091\pi$
$223$	5701.49i	1.71211i	−0.516888	+	0.856053i	$0.672909\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1893.20i	− 0.553551i	−0.960935	−	0.276775i	$-0.910734\pi$
$227$	− 1893.20i	− 0.553551i	0.960935	−	0.276775i	$-0.0892658\pi$
$228$	0	0
$229$	−3968.92	−1.14530	−0.572650	−	0.819800i	$-0.694085\pi$
$229$	−3968.92	−1.14530	−0.572650	+	0.819800i	$0.694085\pi$
$230$	0	0
$231$	−337.516	−0.0961337
$232$	0	0
$233$	3215.84i	0.904192i	0.891969	+	0.452096i	$0.149324\pi$
$233$	3215.84i	0.904192i	−0.891969	+	0.452096i	$0.850676\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 39.7524i	− 0.0108953i
$238$	0	0
$239$	−6465.86	−1.74997	−0.874983	−	0.484154i	$-0.839128\pi$
$239$	−6465.86	−1.74997	−0.874983	+	0.484154i	$0.839128\pi$
$240$	0	0
$241$	2418.68	0.646476	0.323238	−	0.946318i	$-0.395229\pi$
$241$	2418.68	0.646476	0.323238	+	0.946318i	$0.395229\pi$
$242$	0	0
$243$	483.154i	0.127549i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1451.43i	− 0.373896i
$248$	0	0
$249$	−179.409	−0.0456610
$250$	0	0
$251$	−5331.27	−1.34066	−0.670332	−	0.742061i	$-0.733848\pi$
$251$	−5331.27	−1.34066	−0.670332	+	0.742061i	$0.733848\pi$
$252$	0	0
$253$	− 9292.45i	− 2.30914i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6579.19i	− 1.59688i	−0.602073	−	0.798441i	$-0.705658\pi$
$257$	− 6579.19i	− 1.59688i	0.602073	−	0.798441i	$-0.294342\pi$
$258$	0	0
$259$	1804.49	0.432918
$260$	0	0
$261$	−3634.41	−0.861933
$262$	0	0
$263$	− 5228.27i	− 1.22581i	−0.790155	−	0.612907i	$-0.790000\pi$
$263$	− 5228.27i	− 1.22581i	0.790155	−	0.612907i	$-0.210000\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.58940i	0.00151036i
$268$	0	0
$269$	335.667	0.0760818	0.0380409	−	0.999276i	$-0.487888\pi$
$269$	335.667	0.0760818	0.0380409	+	0.999276i	$0.487888\pi$
$270$	0	0
$271$	−5998.13	−1.34450	−0.672252	−	0.740323i	$-0.734673\pi$
$271$	−5998.13	−1.34450	−0.672252	+	0.740323i	$0.734673\pi$
$272$	0	0
$273$	166.371i	0.0368836i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1386.93i	0.300839i	0.988622	+	0.150420i	$0.0480625\pi$
$277$	1386.93i	0.300839i	−0.988622	+	0.150420i	$0.951938\pi$
$278$	0	0
$279$	6180.42	1.32621
$280$	0	0
$281$	−5596.57	−1.18813	−0.594063	−	0.804419i	$-0.702477\pi$
$281$	−5596.57	−1.18813	−0.594063	+	0.804419i	$0.702477\pi$
$282$	0	0
$283$	− 1305.37i	− 0.274191i	−0.990558	−	0.137095i	$-0.956223\pi$
$283$	− 1305.37i	− 0.274191i	0.990558	−	0.137095i	$-0.0437767\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 8773.22i	− 1.80441i
$288$	0	0
$289$	3620.53	0.736928
$290$	0	0
$291$	−84.9309	−0.0171091
$292$	0	0
$293$	− 3603.03i	− 0.718400i	−0.933261	−	0.359200i	$-0.883050\pi$
$293$	− 3603.03i	− 0.718400i	0.933261	−	0.359200i	$-0.116950\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 798.681i	− 0.156041i
$298$	0	0
$299$	−4580.51	−0.885945
$300$	0	0
$301$	−5727.94	−1.09685
$302$	0	0
$303$	281.752i	0.0534199i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9456.83i	1.75808i	0.476750	+	0.879039i	$0.341815\pi$
$307$	9456.83i	1.75808i	−0.476750	+	0.879039i	$0.658185\pi$
$308$	0	0
$309$	−315.245	−0.0580378
$310$	0	0
$311$	6805.49	1.24085	0.620424	−	0.784266i	$-0.286960\pi$
$311$	6805.49	1.24085	0.620424	+	0.784266i	$0.286960\pi$
$312$	0	0
$313$	− 3229.55i	− 0.583210i	−0.956539	−	0.291605i	$-0.905811\pi$
$313$	− 3229.55i	− 0.583210i	0.956539	−	0.291605i	$-0.0941893\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1536.59i	0.272251i	0.990692	+	0.136125i	$0.0434650\pi$
$317$	1536.59i	0.272251i	−0.990692	+	0.136125i	$0.956535\pi$
$318$	0	0
$319$	9014.57	1.58219
$320$	0	0
$321$	−368.433	−0.0640621
$322$	0	0
$323$	− 1583.58i	− 0.272794i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 409.224i	− 0.0692053i
$328$	0	0
$329$	5505.78	0.922625
$330$	0	0
$331$	7492.66	1.24421	0.622105	−	0.782934i	$-0.286278\pi$
$331$	7492.66	1.24421	0.622105	+	0.782934i	$0.286278\pi$
$332$	0	0
$333$	2133.09i	0.351029i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 10345.3i	− 1.67224i	−0.548546	−	0.836120i	$-0.684819\pi$
$337$	− 10345.3i	− 1.67224i	0.548546	−	0.836120i	$-0.315181\pi$
$338$	0	0
$339$	−235.496	−0.0377297
$340$	0	0
$341$	−15329.5	−2.43443
$342$	0	0
$343$	− 3789.14i	− 0.596485i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3077.69i	0.476136i	0.971248	+	0.238068i	$0.0765141\pi$
$347$	3077.69i	0.476136i	−0.971248	+	0.238068i	$0.923486\pi$
$348$	0	0
$349$	2568.28	0.393917	0.196958	−	0.980412i	$-0.436894\pi$
$349$	2568.28	0.393917	0.196958	+	0.980412i	$0.436894\pi$
$350$	0	0
$351$	−393.692	−0.0598682
$352$	0	0
$353$	− 6843.39i	− 1.03183i	−0.856639	−	0.515916i	$-0.827452\pi$
$353$	− 6843.39i	− 1.03183i	0.856639	−	0.515916i	$-0.172548\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	181.518i	0.0269102i
$358$	0	0
$359$	−7296.40	−1.07267	−0.536336	−	0.844005i	$-0.680192\pi$
$359$	−7296.40	−1.07267	−0.536336	+	0.844005i	$0.680192\pi$
$360$	0	0
$361$	−4918.75	−0.717124
$362$	0	0
$363$	694.838i	0.100467i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11304.9i	1.60793i	0.594675	+	0.803966i	$0.297281\pi$
$367$	11304.9i	1.60793i	−0.594675	+	0.803966i	$0.702719\pi$
$368$	0	0
$369$	10370.8	1.46310
$370$	0	0
$371$	5061.42	0.708291
$372$	0	0
$373$	1924.45i	0.267142i	0.991039	+	0.133571i	$0.0426445\pi$
$373$	1924.45i	0.267142i	−0.991039	+	0.133571i	$0.957356\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4443.53i	− 0.607038i
$378$	0	0
$379$	−4477.79	−0.606883	−0.303441	−	0.952850i	$-0.598136\pi$
$379$	−4477.79	−0.606883	−0.303441	+	0.952850i	$0.598136\pi$
$380$	0	0
$381$	469.505	0.0631324
$382$	0	0
$383$	175.452i	0.0234078i	0.999932	+	0.0117039i	$0.00372556\pi$
$383$	175.452i	0.0234078i	−0.999932	+	0.0117039i	$0.996274\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 6771.00i	− 0.889378i
$388$	0	0
$389$	1208.94	0.157573	0.0787864	−	0.996892i	$-0.474895\pi$
$389$	1208.94	0.157573	0.0787864	+	0.996892i	$0.474895\pi$
$390$	0	0
$391$	−4997.54	−0.646384
$392$	0	0
$393$	− 149.556i	− 0.0191962i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7449.92i	− 0.941815i	−0.882182	−	0.470908i	$-0.843926\pi$
$397$	− 7449.92i	− 0.941815i	0.882182	−	0.470908i	$-0.156074\pi$
$398$	0	0
$399$	−222.401	−0.0279047
$400$	0	0
$401$	−2421.19	−0.301517	−0.150759	−	0.988571i	$-0.548172\pi$
$401$	−2421.19	−0.301517	−0.150759	+	0.988571i	$0.548172\pi$
$402$	0	0
$403$	7556.34i	0.934015i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5290.78i	− 0.644359i
$408$	0	0
$409$	−9305.41	−1.12499	−0.562497	−	0.826799i	$-0.690159\pi$
$409$	−9305.41	−1.12499	−0.562497	+	0.826799i	$0.690159\pi$
$410$	0	0
$411$	241.371	0.0289682
$412$	0	0
$413$	− 12601.3i	− 1.50138i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	316.569i	0.0371762i
$418$	0	0
$419$	15744.4	1.83572	0.917859	−	0.396907i	$-0.129916\pi$
$419$	15744.4	1.83572	0.917859	+	0.396907i	$0.129916\pi$
$420$	0	0
$421$	−7645.39	−0.885068	−0.442534	−	0.896752i	$-0.645920\pi$
$421$	−7645.39	−0.885068	−0.442534	+	0.896752i	$0.645920\pi$
$422$	0	0
$423$	6508.39i	0.748106i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 11284.5i	− 1.27891i
$428$	0	0
$429$	487.800	0.0548979
$430$	0	0
$431$	−6915.12	−0.772829	−0.386415	−	0.922325i	$-0.626287\pi$
$431$	−6915.12	−0.772829	−0.386415	+	0.922325i	$0.626287\pi$
$432$	0	0
$433$	− 14836.3i	− 1.64662i	−0.567592	−	0.823310i	$-0.692125\pi$
$433$	− 14836.3i	− 1.64662i	0.567592	−	0.823310i	$-0.307875\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6123.13i	− 0.670273i
$438$	0	0
$439$	−15194.2	−1.65189	−0.825944	−	0.563752i	$-0.809357\pi$
$439$	−15194.2	−1.65189	−0.825944	+	0.563752i	$0.809357\pi$
$440$	0	0
$441$	−4765.03	−0.514527
$442$	0	0
$443$	− 4707.62i	− 0.504889i	−0.967611	−	0.252444i	$-0.918766\pi$
$443$	− 4707.62i	− 0.504889i	0.967611	−	0.252444i	$-0.0812345\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 451.852i	− 0.0478118i
$448$	0	0
$449$	2980.66	0.313287	0.156644	−	0.987655i	$-0.449933\pi$
$449$	2980.66	0.313287	0.156644	+	0.987655i	$0.449933\pi$
$450$	0	0
$451$	−25723.1	−2.68571
$452$	0	0
$453$	38.5850i	0.00400194i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 4677.53i	− 0.478787i	−0.970923	−	0.239393i	$-0.923051\pi$
$457$	− 4677.53i	− 0.478787i	0.970923	−	0.239393i	$-0.0769486\pi$
$458$	0	0
$459$	−429.535	−0.0436797
$460$	0	0
$461$	2410.16	0.243497	0.121749	−	0.992561i	$-0.461150\pi$
$461$	2410.16	0.243497	0.121749	+	0.992561i	$0.461150\pi$
$462$	0	0
$463$	− 4926.69i	− 0.494520i	−0.968949	−	0.247260i	$-0.920470\pi$
$463$	− 4926.69i	− 0.494520i	0.968949	−	0.247260i	$-0.0795302\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1492.64i	− 0.147904i	−0.997262	−	0.0739521i	$-0.976439\pi$
$467$	− 1492.64i	− 0.147904i	0.997262	−	0.0739521i	$-0.0235612\pi$
$468$	0	0
$469$	−13101.1	−1.28988
$470$	0	0
$471$	−773.931	−0.0757131
$472$	0	0
$473$	16794.4i	1.63257i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5983.11i	0.574314i
$478$	0	0
$479$	11609.5	1.10742	0.553708	−	0.832711i	$-0.313212\pi$
$479$	11609.5	1.10742	0.553708	+	0.832711i	$0.313212\pi$
$480$	0	0
$481$	−2607.97	−0.247221
$482$	0	0
$483$	701.867i	0.0661202i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8199.43i	0.762940i	0.924381	+	0.381470i	$0.124582\pi$
$487$	8199.43i	0.762940i	−0.924381	+	0.381470i	$0.875418\pi$
$488$	0	0
$489$	662.057	0.0612255
$490$	0	0
$491$	6788.22	0.623927	0.311963	−	0.950094i	$-0.399013\pi$
$491$	6788.22	0.623927	0.311963	+	0.950094i	$0.399013\pi$
$492$	0	0
$493$	− 4848.09i	− 0.442894i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14922.8i	1.34684i
$498$	0	0
$499$	−18758.2	−1.68283	−0.841415	−	0.540390i	$-0.818277\pi$
$499$	−18758.2	−1.68283	−0.841415	+	0.540390i	$0.818277\pi$
$500$	0	0
$501$	−490.640	−0.0437528
$502$	0	0
$503$	13192.9i	1.16947i	0.811226	+	0.584733i	$0.198801\pi$
$503$	13192.9i	1.16947i	−0.811226	+	0.584733i	$0.801199\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	246.091i	0.0215567i
$508$	0	0
$509$	22232.9	1.93607	0.968033	−	0.250823i	$-0.0807014\pi$
$509$	22232.9	1.93607	0.968033	+	0.250823i	$0.0807014\pi$
$510$	0	0
$511$	25789.7	2.23262
$512$	0	0
$513$	− 526.280i	− 0.0452940i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16143.0i	− 1.37325i
$518$	0	0
$519$	−183.731	−0.0155393
$520$	0	0
$521$	−6609.16	−0.555763	−0.277881	−	0.960615i	$-0.589632\pi$
$521$	−6609.16	−0.555763	−0.277881	+	0.960615i	$0.589632\pi$
$522$	0	0
$523$	− 13834.2i	− 1.15665i	−0.815808	−	0.578323i	$-0.803708\pi$
$523$	− 13834.2i	− 1.15665i	0.815808	−	0.578323i	$-0.196292\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8244.30i	0.681456i
$528$	0	0
$529$	−7156.73	−0.588208
$530$	0	0
$531$	14896.0	1.21738
$532$	0	0
$533$	12679.7i	1.03043i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	155.056i	0.0124603i
$538$	0	0
$539$	11818.9	0.944481
$540$	0	0
$541$	20781.9	1.65154	0.825771	−	0.564005i	$-0.190740\pi$
$541$	20781.9	1.65154	0.825771	+	0.564005i	$0.190740\pi$
$542$	0	0
$543$	− 256.273i	− 0.0202537i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9805.41i	0.766452i	0.923655	+	0.383226i	$0.125187\pi$
$547$	9805.41i	0.766452i	−0.923655	+	0.383226i	$0.874813\pi$
$548$	0	0
$549$	13339.4	1.03700
$550$	0	0
$551$	5940.03	0.459263
$552$	0	0
$553$	4092.55i	0.314707i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11660.5i	0.887023i	0.896269	+	0.443512i	$0.146267\pi$
$557$	11660.5i	0.887023i	−0.896269	+	0.443512i	$0.853733\pi$
$558$	0	0
$559$	8278.41	0.626367
$560$	0	0
$561$	532.212	0.0400535
$562$	0	0
$563$	9368.59i	0.701313i	0.936504	+	0.350656i	$0.114042\pi$
$563$	9368.59i	0.701313i	−0.936504	+	0.350656i	$0.885958\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 16530.1i	− 1.22434i
$568$	0	0
$569$	−9830.77	−0.724301	−0.362150	−	0.932120i	$-0.617957\pi$
$569$	−9830.77	−0.724301	−0.362150	+	0.932120i	$0.617957\pi$
$570$	0	0
$571$	−6529.64	−0.478559	−0.239279	−	0.970951i	$-0.576911\pi$
$571$	−6529.64	−0.478559	−0.239279	+	0.970951i	$0.576911\pi$
$572$	0	0
$573$	510.691i	0.0372329i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14167.7i	1.02220i	0.859522	+	0.511098i	$0.170761\pi$
$577$	14167.7i	1.02220i	−0.859522	+	0.511098i	$0.829239\pi$
$578$	0	0
$579$	−125.668	−0.00902001
$580$	0	0
$581$	18470.3	1.31890
$582$	0	0
$583$	− 14840.1i	− 1.05423i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18531.5i	1.30303i	0.758638	+	0.651513i	$0.225865\pi$
$587$	18531.5i	1.30303i	−0.758638	+	0.651513i	$0.774135\pi$
$588$	0	0
$589$	−10101.2	−0.706641
$590$	0	0
$591$	−952.651	−0.0663060
$592$	0	0
$593$	− 1492.11i	− 0.103328i	−0.998665	−	0.0516642i	$-0.983547\pi$
$593$	− 1492.11i	− 0.103328i	0.998665	−	0.0516642i	$-0.0164525\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 57.2536i	− 0.00392501i
$598$	0	0
$599$	9668.23	0.659488	0.329744	−	0.944070i	$-0.393038\pi$
$599$	9668.23	0.659488	0.329744	+	0.944070i	$0.393038\pi$
$600$	0	0
$601$	6895.64	0.468018	0.234009	−	0.972234i	$-0.424815\pi$
$601$	6895.64	0.468018	0.234009	+	0.972234i	$0.424815\pi$
$602$	0	0
$603$	− 15486.8i	− 1.04589i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 11614.6i	− 0.776640i	−0.921525	−	0.388320i	$-0.873056\pi$
$607$	− 11614.6i	− 0.776640i	0.921525	−	0.388320i	$-0.126944\pi$
$608$	0	0
$609$	−680.878	−0.0453047
$610$	0	0
$611$	−7957.33	−0.526872
$612$	0	0
$613$	22009.2i	1.45015i	0.688670	+	0.725075i	$0.258195\pi$
$613$	22009.2i	1.45015i	−0.688670	+	0.725075i	$0.741805\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10913.5i	0.712089i	0.934469	+	0.356045i	$0.115875\pi$
$617$	10913.5i	0.712089i	−0.934469	+	0.356045i	$0.884125\pi$
$618$	0	0
$619$	1100.19	0.0714385	0.0357193	−	0.999362i	$-0.488628\pi$
$619$	1100.19	0.0714385	0.0357193	+	0.999362i	$0.488628\pi$
$620$	0	0
$621$	−1660.86	−0.107324
$622$	0	0
$623$	− 678.385i	− 0.0436259i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	652.082i	0.0415337i
$628$	0	0
$629$	−2845.42	−0.180372
$630$	0	0
$631$	−19436.0	−1.22621	−0.613103	−	0.790003i	$-0.710079\pi$
$631$	−19436.0	−1.22621	−0.613103	+	0.790003i	$0.710079\pi$
$632$	0	0
$633$	682.754i	0.0428705i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5825.86i	− 0.362369i
$638$	0	0
$639$	−17640.3	−1.09208
$640$	0	0
$641$	16001.9	0.986014	0.493007	−	0.870025i	$-0.335898\pi$
$641$	16001.9	0.986014	0.493007	+	0.870025i	$0.335898\pi$
$642$	0	0
$643$	7965.67i	0.488547i	0.969706	+	0.244273i	$0.0785494\pi$
$643$	7965.67i	0.488547i	−0.969706	+	0.244273i	$0.921451\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 4065.35i	− 0.247025i	−0.992343	−	0.123513i	$-0.960584\pi$
$647$	− 4065.35i	− 0.247025i	0.992343	−	0.123513i	$-0.0394159\pi$
$648$	0	0
$649$	−36947.0	−2.23466
$650$	0	0
$651$	1157.85	0.0697078
$652$	0	0
$653$	2625.87i	0.157363i	0.996900	+	0.0786816i	$0.0250710\pi$
$653$	2625.87i	0.157363i	−0.996900	+	0.0786816i	$0.974929\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30486.0i	1.81031i
$658$	0	0
$659$	−6099.81	−0.360569	−0.180284	−	0.983615i	$-0.557702\pi$
$659$	−6099.81	−0.360569	−0.180284	+	0.983615i	$0.557702\pi$
$660$	0	0
$661$	21701.3	1.27698	0.638490	−	0.769630i	$-0.279559\pi$
$661$	21701.3	1.27698	0.638490	+	0.769630i	$0.279559\pi$
$662$	0	0
$663$	− 262.342i	− 0.0153673i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 18745.9i	− 1.08822i
$668$	0	0
$669$	−1262.63	−0.0729690
$670$	0	0
$671$	−33086.2	−1.90354
$672$	0	0
$673$	− 27560.5i	− 1.57857i	−0.614024	−	0.789287i	$-0.710450\pi$
$673$	− 27560.5i	− 1.57857i	0.614024	−	0.789287i	$-0.289550\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 22643.1i	− 1.28545i	−0.766099	−	0.642723i	$-0.777805\pi$
$677$	− 22643.1i	− 1.28545i	0.766099	−	0.642723i	$-0.222195\pi$
$678$	0	0
$679$	8743.71	0.494187
$680$	0	0
$681$	419.262	0.0235920
$682$	0	0
$683$	27407.8i	1.53548i	0.640763	+	0.767739i	$0.278618\pi$
$683$	27407.8i	1.53548i	−0.640763	+	0.767739i	$0.721382\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 878.945i	− 0.0488120i
$688$	0	0
$689$	−7315.11	−0.404475
$690$	0	0
$691$	24189.2	1.33170	0.665848	−	0.746088i	$-0.268070\pi$
$691$	24189.2	1.33170	0.665848	+	0.746088i	$0.268070\pi$
$692$	0	0
$693$	41075.1i	2.25154i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13834.1i	0.751797i
$698$	0	0
$699$	−712.171	−0.0385361
$700$	0	0
$701$	28135.4	1.51592	0.757961	−	0.652300i	$-0.226196\pi$
$701$	28135.4	1.51592	0.757961	+	0.652300i	$0.226196\pi$
$702$	0	0
$703$	− 3486.29i	− 0.187038i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 29006.6i	− 1.54301i
$708$	0	0
$709$	−20075.8	−1.06342	−0.531708	−	0.846928i	$-0.678450\pi$
$709$	−20075.8	−1.06342	−0.531708	+	0.846928i	$0.678450\pi$
$710$	0	0
$711$	−4837.80	−0.255178
$712$	0	0
$713$	31877.8i	1.67438i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1431.91i	− 0.0745825i
$718$	0	0
$719$	−17789.9	−0.922742	−0.461371	−	0.887207i	$-0.652642\pi$
$719$	−17789.9	−0.922742	−0.461371	+	0.887207i	$0.652642\pi$
$720$	0	0
$721$	32454.8	1.67639
$722$	0	0
$723$	535.633i	0.0275524i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28884.8i	1.47356i	0.676132	+	0.736781i	$0.263655\pi$
$727$	28884.8i	1.47356i	−0.676132	+	0.736781i	$0.736345\pi$
$728$	0	0
$729$	19468.8	0.989118
$730$	0	0
$731$	9032.11	0.456997
$732$	0	0
$733$	− 22393.4i	− 1.12840i	−0.825637	−	0.564202i	$-0.809184\pi$
$733$	− 22393.4i	− 1.12840i	0.825637	−	0.564202i	$-0.190816\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38412.6i	1.91987i
$738$	0	0
$739$	3465.65	0.172511	0.0862557	−	0.996273i	$-0.472510\pi$
$739$	3465.65	0.172511	0.0862557	+	0.996273i	$0.472510\pi$
$740$	0	0
$741$	321.429	0.0159352
$742$	0	0
$743$	12870.3i	0.635485i	0.948177	+	0.317742i	$0.102925\pi$
$743$	12870.3i	0.635485i	−0.948177	+	0.317742i	$0.897075\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	21833.8i	1.06942i
$748$	0	0
$749$	37930.5	1.85040
$750$	0	0
$751$	19763.7	0.960304	0.480152	−	0.877185i	$-0.340581\pi$
$751$	19763.7	0.960304	0.480152	+	0.877185i	$0.340581\pi$
$752$	0	0
$753$	− 1180.65i	− 0.0571383i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10045.4i	− 0.482306i	−0.970487	−	0.241153i	$-0.922475\pi$
$757$	− 10045.4i	− 0.482306i	0.970487	−	0.241153i	$-0.0775255\pi$
$758$	0	0
$759$	2057.88	0.0984140
$760$	0	0
$761$	−28857.9	−1.37464	−0.687318	−	0.726357i	$-0.741212\pi$
$761$	−28857.9	−1.37464	−0.687318	+	0.726357i	$0.741212\pi$
$762$	0	0
$763$	42130.0i	1.99896i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18212.2i	0.857373i
$768$	0	0
$769$	−7056.72	−0.330913	−0.165456	−	0.986217i	$-0.552910\pi$
$769$	−7056.72	−0.330913	−0.165456	+	0.986217i	$0.552910\pi$
$770$	0	0
$771$	1457.01	0.0680582
$772$	0	0
$773$	33312.1i	1.55000i	0.631960	+	0.775001i	$0.282251\pi$
$773$	33312.1i	1.55000i	−0.631960	+	0.775001i	$0.717749\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	399.617i	0.0184507i
$778$	0	0
$779$	−16949.9	−0.779581
$780$	0	0
$781$	43753.9	2.00466
$782$	0	0
$783$	− 1611.20i	− 0.0735370i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	15240.9i	0.690318i	0.938544	+	0.345159i	$0.112175\pi$
$787$	15240.9i	0.690318i	−0.938544	+	0.345159i	$0.887825\pi$
$788$	0	0
$789$	1157.84	0.0522435
$790$	0	0
$791$	24244.5	1.08980
$792$	0	0
$793$	16309.1i	0.730332i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20742.7i	0.921886i	0.887430	+	0.460943i	$0.152489\pi$
$797$	20742.7i	0.921886i	−0.887430	+	0.460943i	$0.847511\pi$
$798$	0	0
$799$	−8681.80	−0.384405
$800$	0	0
$801$	801.920	0.0353738
$802$	0	0
$803$	− 75615.5i	− 3.32306i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	74.3359i	0.00324256i
$808$	0	0
$809$	−24441.1	−1.06218	−0.531089	−	0.847316i	$-0.678217\pi$
$809$	−24441.1	−1.06218	−0.531089	+	0.847316i	$0.678217\pi$
$810$	0	0
$811$	−18571.3	−0.804104	−0.402052	−	0.915617i	$-0.631703\pi$
$811$	−18571.3	−0.804104	−0.402052	+	0.915617i	$0.631703\pi$
$812$	0	0
$813$	− 1328.33i	− 0.0573020i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11066.4i	0.473886i
$818$	0	0
$819$	20247.1	0.863846
$820$	0	0
$821$	1293.62	0.0549912	0.0274956	−	0.999622i	$-0.491247\pi$
$821$	1293.62	0.0549912	0.0274956	+	0.999622i	$0.491247\pi$
$822$	0	0
$823$	− 10159.6i	− 0.430305i	−0.976580	−	0.215152i	$-0.930975\pi$
$823$	− 10159.6i	− 0.430305i	0.976580	−	0.215152i	$-0.0690248\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	304.104i	0.0127869i	0.999980	+	0.00639343i	$0.00203510\pi$
$827$	304.104i	0.0127869i	−0.999980	+	0.00639343i	$0.997965\pi$
$828$	0	0
$829$	−9680.47	−0.405569	−0.202784	−	0.979223i	$-0.564999\pi$
$829$	−9680.47	−0.405569	−0.202784	+	0.979223i	$0.564999\pi$
$830$	0	0
$831$	−307.145	−0.0128216
$832$	0	0
$833$	− 6356.27i	− 0.264384i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2739.88i	0.113147i
$838$	0	0
$839$	28033.4	1.15354	0.576770	−	0.816907i	$-0.304313\pi$
$839$	28033.4	1.15354	0.576770	+	0.816907i	$0.304313\pi$
$840$	0	0
$841$	−6203.70	−0.254365
$842$	0	0
$843$	− 1239.40i	− 0.0506372i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 71534.3i	− 2.90194i
$848$	0	0
$849$	289.083	0.0116859
$850$	0	0
$851$	−11002.2	−0.443186
$852$	0	0
$853$	− 1303.27i	− 0.0523132i	−0.999658	−	0.0261566i	$-0.991673\pi$
$853$	− 1303.27i	− 0.0523132i	0.999658	−	0.0261566i	$-0.00832685\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17269.4i	0.688344i	0.938907	+	0.344172i	$0.111840\pi$
$857$	17269.4i	0.688344i	−0.938907	+	0.344172i	$0.888160\pi$
$858$	0	0
$859$	34492.8	1.37006	0.685029	−	0.728515i	$-0.259789\pi$
$859$	34492.8	1.37006	0.685029	+	0.728515i	$0.259789\pi$
$860$	0	0
$861$	1942.89	0.0769031
$862$	0	0
$863$	− 9348.29i	− 0.368737i	−0.982857	−	0.184368i	$-0.940976\pi$
$863$	− 9348.29i	− 0.368737i	0.982857	−	0.184368i	$-0.0590239\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	801.791i	0.0314075i
$868$	0	0
$869$	11999.4	0.468413
$870$	0	0
$871$	18934.6	0.736597
$872$	0	0
$873$	10335.9i	0.400709i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 44386.3i	− 1.70903i	−0.519426	−	0.854516i	$-0.673854\pi$
$877$	− 44386.3i	− 1.70903i	0.519426	−	0.854516i	$-0.326146\pi$
$878$	0	0
$879$	797.916	0.0306178
$880$	0	0
$881$	−17267.9	−0.660354	−0.330177	−	0.943919i	$-0.607108\pi$
$881$	−17267.9	−0.660354	−0.330177	+	0.943919i	$0.607108\pi$
$882$	0	0
$883$	− 21218.0i	− 0.808655i	−0.914614	−	0.404328i	$-0.867506\pi$
$883$	− 21218.0i	− 0.808655i	0.914614	−	0.404328i	$-0.132494\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19106.0i	0.723243i	0.932325	+	0.361621i	$0.117777\pi$
$887$	19106.0i	0.723243i	−0.932325	+	0.361621i	$0.882223\pi$
$888$	0	0
$889$	−48336.0	−1.82355
$890$	0	0
$891$	−48466.4	−1.82232
$892$	0	0
$893$	− 10637.2i	− 0.398612i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1014.39i	− 0.0377585i
$898$	0	0
$899$	−30924.6	−1.14727
$900$	0	0
$901$	−7981.11	−0.295105
$902$	0	0
$903$	− 1268.49i	− 0.0467473i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 7041.23i	− 0.257773i	−0.991659	−	0.128887i	$-0.958860\pi$
$907$	− 7041.23i	− 0.257773i	0.991659	−	0.128887i	$-0.0411403\pi$
$908$	0	0
$909$	34288.7	1.25114
$910$	0	0
$911$	4329.40	0.157453	0.0787263	−	0.996896i	$-0.474915\pi$
$911$	4329.40	0.157453	0.0787263	+	0.996896i	$0.474915\pi$
$912$	0	0
$913$	− 54155.1i	− 1.96306i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15397.0i	0.554474i
$918$	0	0
$919$	6812.79	0.244541	0.122270	−	0.992497i	$-0.460982\pi$
$919$	6812.79	0.244541	0.122270	+	0.992497i	$0.460982\pi$
$920$	0	0
$921$	−2094.28	−0.0749282
$922$	0	0
$923$	− 21567.5i	− 0.769126i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	38364.8i	1.35930i
$928$	0	0
$929$	17406.0	0.614716	0.307358	−	0.951594i	$-0.400555\pi$
$929$	17406.0	0.614716	0.307358	+	0.951594i	$0.400555\pi$
$930$	0	0
$931$	7787.89	0.274155
$932$	0	0
$933$	1507.12i	0.0528842i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 24803.5i	− 0.864775i	−0.901688	−	0.432387i	$-0.857671\pi$
$937$	− 24803.5i	− 0.864775i	0.901688	−	0.432387i	$-0.142329\pi$
$938$	0	0
$939$	715.206	0.0248561
$940$	0	0
$941$	31855.3	1.10356	0.551781	−	0.833989i	$-0.313948\pi$
$941$	31855.3	1.10356	0.551781	+	0.833989i	$0.313948\pi$
$942$	0	0
$943$	53491.5i	1.84721i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1411.03i	− 0.0484185i	−0.999707	−	0.0242093i	$-0.992293\pi$
$947$	− 1411.03i	− 0.0484185i	0.999707	−	0.0242093i	$-0.00770680\pi$
$948$	0	0
$949$	−37273.0	−1.27496
$950$	0	0
$951$	−340.288	−0.0116032
$952$	0	0
$953$	30090.1i	1.02278i	0.859348	+	0.511392i	$0.170870\pi$
$953$	30090.1i	1.02278i	−0.859348	+	0.511392i	$0.829130\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1996.34i	0.0674320i
$958$	0	0
$959$	−24849.3	−0.836733
$960$	0	0
$961$	22797.0	0.765232
$962$	0	0
$963$	44837.7i	1.50039i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 5840.40i	− 0.194224i	−0.995273	−	0.0971120i	$-0.969039\pi$
$967$	− 5840.40i	− 0.194224i	0.995273	−	0.0971120i	$-0.0309605\pi$
$968$	0	0
$969$	350.694	0.0116263
$970$	0	0
$971$	20428.7	0.675166	0.337583	−	0.941296i	$-0.390391\pi$
$971$	20428.7	0.675166	0.337583	+	0.941296i	$0.390391\pi$
$972$	0	0
$973$	− 32591.1i	− 1.07382i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 39986.7i	− 1.30940i	−0.755887	−	0.654702i	$-0.772794\pi$
$977$	− 39986.7i	− 1.30940i	0.755887	−	0.654702i	$-0.227206\pi$
$978$	0	0
$979$	−1989.03	−0.0649333
$980$	0	0
$981$	−49801.9	−1.62085
$982$	0	0
$983$	1096.54i	0.0355790i	0.999842	+	0.0177895i	$0.00566288\pi$
$983$	1096.54i	0.0355790i	−0.999842	+	0.0177895i	$0.994337\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1219.29i	0.0393217i
$988$	0	0
$989$	34924.0	1.12287
$990$	0	0
$991$	−12697.6	−0.407016	−0.203508	−	0.979073i	$-0.565234\pi$
$991$	−12697.6	−0.407016	−0.203508	+	0.979073i	$0.565234\pi$
$992$	0	0
$993$	1659.30i	0.0530275i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 54735.3i	− 1.73870i	−0.494197	−	0.869350i	$-0.664538\pi$
$997$	− 54735.3i	− 1.73870i	0.494197	−	0.869350i	$-0.335462\pi$
$998$	0	0
$999$	−945.635	−0.0299485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.c.o.449.5		8
4.3	odd	2	inner	800.4.c.o.449.4		8
5.2	odd	4		800.4.a.ba.1.3	yes	4
5.3	odd	4		800.4.a.bb.1.2	yes	4
5.4	even	2	inner	800.4.c.o.449.3		8
20.3	even	4		800.4.a.bb.1.3	yes	4
20.7	even	4		800.4.a.ba.1.2	✓	4
20.19	odd	2	inner	800.4.c.o.449.6		8
40.3	even	4		1600.4.a.cw.1.2		4
40.13	odd	4		1600.4.a.cw.1.3		4
40.27	even	4		1600.4.a.cx.1.3		4
40.37	odd	4		1600.4.a.cx.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.4.a.ba.1.2	✓	4	20.7	even	4
800.4.a.ba.1.3	yes	4	5.2	odd	4
800.4.a.bb.1.2	yes	4	5.3	odd	4
800.4.a.bb.1.3	yes	4	20.3	even	4
800.4.c.o.449.3		8	5.4	even	2	inner
800.4.c.o.449.4		8	4.3	odd	2	inner
800.4.c.o.449.5		8	1.1	even	1	trivial
800.4.c.o.449.6		8	20.19	odd	2	inner
1600.4.a.cw.1.2		4	40.3	even	4
1600.4.a.cw.1.3		4	40.13	odd	4
1600.4.a.cx.1.2		4	40.37	odd	4
1600.4.a.cx.1.3		4	40.27	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.221457i q^{3} -22.7992i q^{7} +26.9510 q^{9} +O(q^{10})\)	\(q+0.221457i q^{3} -22.7992i q^{7} +26.9510 q^{9} -66.8474 q^{11} +32.9510i q^{13} +35.9510i q^{17} -44.0482 q^{19} +5.04904 q^{21} +139.010i q^{23} +11.9478i q^{27} -134.853 q^{29} +229.321 q^{31} -14.8038i q^{33} +79.1471i q^{37} -7.29722 q^{39} +384.804 q^{41} -251.234i q^{43} +241.490i q^{47} -176.804 q^{49} -7.96159 q^{51} +222.000i q^{53} -9.75478i q^{57} +552.707 q^{59} +494.951 q^{61} -614.460i q^{63} -574.631i q^{67} -30.7847 q^{69} -654.533 q^{71} +1131.17i q^{73} +1524.07i q^{77} -179.504 q^{79} +725.030 q^{81} +810.131i q^{83} -29.8641i q^{87} +29.7548 q^{89} +751.256 q^{91} +50.7847i q^{93} +383.510i q^{97} -1801.60 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 192 q^{9}+O(q^{10}) \) 8 * q - 192 * q^9	\( 8 q - 192 q^{9} + 448 q^{21} + 144 q^{29} + 1448 q^{41} + 216 q^{49} + 3552 q^{61} - 6768 q^{69} + 14360 q^{81} - 1800 q^{89}+O(q^{100}) \) 8 * q - 192 * q^9 + 448 * q^21 + 144 * q^29 + 1448 * q^41 + 216 * q^49 + 3552 * q^61 - 6768 * q^69 + 14360 * q^81 - 1800 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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