LMFDB - Embedded newform 800.3.h.j.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,3,Mod(799,800)]

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, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("800.799");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	800.h (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:	$21.7984211488$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	800.799
Dual form			800.3.h.j.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.763932 q^{3} +12.1803 q^{7} -8.41641 q^{9} +O(q^{10})$	$q+0.763932 q^{3} +12.1803 q^{7} -8.41641 q^{9} +5.52786i q^{11} +20.4721i q^{13} +1.05573i q^{17} +12.0000i q^{19} +9.30495 q^{21} -31.5967 q^{23} -13.3050 q^{27} +44.8328 q^{29} +27.4164i q^{31} +4.22291i q^{33} -23.5279i q^{37} +15.6393i q^{39} +8.69505 q^{41} +26.6525 q^{43} -44.5410 q^{47} +99.3607 q^{49} +0.806504i q^{51} +0.695048i q^{53} +9.16718i q^{57} +94.6099i q^{59} -33.1935 q^{61} -102.515 q^{63} +82.2067 q^{67} -24.1378 q^{69} -122.249i q^{71} +132.164i q^{73} +67.3313i q^{77} +134.833i q^{79} +65.5836 q^{81} -19.4590 q^{83} +34.2492 q^{87} +30.0000 q^{89} +249.358i q^{91} +20.9443i q^{93} -12.3344i q^{97} -46.5248i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 4 q^{7} + 20 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 4 * q^7 + 20 * q^9	$ 4 q + 12 q^{3} + 4 q^{7} + 20 q^{9} - 88 q^{21} - 28 q^{23} + 72 q^{27} + 72 q^{29} + 160 q^{41} + 44 q^{43} - 44 q^{47} + 308 q^{49} + 64 q^{61} - 580 q^{63} - 20 q^{67} + 136 q^{69} + 316 q^{81} - 212 q^{83} - 24 q^{87} + 120 q^{89}+O(q^{100}) $ 4 * q + 12 * q^3 + 4 * q^7 + 20 * q^9 - 88 * q^21 - 28 * q^23 + 72 * q^27 + 72 * q^29 + 160 * q^41 + 44 * q^43 - 44 * q^47 + 308 * q^49 + 64 * q^61 - 580 * q^63 - 20 * q^67 + 136 * q^69 + 316 * q^81 - 212 * q^83 - 24 * q^87 + 120 * 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
$3$	0.763932	0.254644	0.127322	−	0.991861i	$-0.459362\pi$
$3$	0.763932	0.254644	0.127322	+	0.991861i	$0.459362\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	12.1803	1.74005	0.870024	−	0.493009i	$-0.164103\pi$
$7$	12.1803	1.74005	0.870024	+	0.493009i	$0.164103\pi$
$8$	0	0
$9$	−8.41641	−0.935156
$10$	0	0
$11$	5.52786i	0.502533i	0.967918	+	0.251267i	$0.0808471\pi$
$11$	5.52786i	0.502533i	−0.967918	+	0.251267i	$0.919153\pi$
$12$	0	0
$13$	20.4721i	1.57478i	0.616455	+	0.787390i	$0.288568\pi$
$13$	20.4721i	1.57478i	−0.616455	+	0.787390i	$0.711432\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.05573i	0.0621017i	0.999518	+	0.0310508i	$0.00988537\pi$
$17$	1.05573i	0.0621017i	−0.999518	+	0.0310508i	$0.990115\pi$
$18$	0	0
$19$	12.0000i	0.631579i	0.948829	+	0.315789i	$0.102269\pi$
$19$	12.0000i	0.631579i	−0.948829	+	0.315789i	$0.897731\pi$
$20$	0	0
$21$	9.30495	0.443093
$22$	0	0
$23$	−31.5967	−1.37377	−0.686886	−	0.726765i	$-0.741023\pi$
$23$	−31.5967	−1.37377	−0.686886	+	0.726765i	$0.741023\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−13.3050	−0.492776
$28$	0	0
$29$	44.8328	1.54596	0.772980	−	0.634431i	$-0.218765\pi$
$29$	44.8328	1.54596	0.772980	+	0.634431i	$0.218765\pi$
$30$	0	0
$31$	27.4164i	0.884400i	0.896916	+	0.442200i	$0.145802\pi$
$31$	27.4164i	0.884400i	−0.896916	+	0.442200i	$0.854198\pi$
$32$	0	0
$33$	4.22291i	0.127967i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 23.5279i	− 0.635888i	−0.948109	−	0.317944i	$-0.897008\pi$
$37$	− 23.5279i	− 0.635888i	0.948109	−	0.317944i	$-0.102992\pi$
$38$	0	0
$39$	15.6393i	0.401008i
$40$	0	0
$41$	8.69505	0.212074	0.106037	−	0.994362i	$-0.466184\pi$
$41$	8.69505	0.212074	0.106037	+	0.994362i	$0.466184\pi$
$42$	0	0
$43$	26.6525	0.619825	0.309913	−	0.950765i	$-0.399700\pi$
$43$	26.6525	0.619825	0.309913	+	0.950765i	$0.399700\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−44.5410	−0.947681	−0.473841	−	0.880611i	$-0.657133\pi$
$47$	−44.5410	−0.947681	−0.473841	+	0.880611i	$0.657133\pi$
$48$	0	0
$49$	99.3607	2.02777
$50$	0	0
$51$	0.806504i	0.0158138i
$52$	0	0
$53$	0.695048i	0.0131141i	0.999979	+	0.00655706i	$0.00208719\pi$
$53$	0.695048i	0.0131141i	−0.999979	+	0.00655706i	$0.997913\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9.16718i	0.160828i
$58$	0	0
$59$	94.6099i	1.60356i	0.597621	+	0.801779i	$0.296113\pi$
$59$	94.6099i	1.60356i	−0.597621	+	0.801779i	$0.703887\pi$
$60$	0	0
$61$	−33.1935	−0.544156	−0.272078	−	0.962275i	$-0.587711\pi$
$61$	−33.1935	−0.544156	−0.272078	+	0.962275i	$0.587711\pi$
$62$	0	0
$63$	−102.515	−1.62722
$64$	0	0
$65$	0	0
$66$	0	0
$67$	82.2067	1.22696	0.613482	−	0.789708i	$-0.289768\pi$
$67$	82.2067	1.22696	0.613482	+	0.789708i	$0.289768\pi$
$68$	0	0
$69$	−24.1378	−0.349823
$70$	0	0
$71$	− 122.249i	− 1.72182i	−0.508757	−	0.860910i	$-0.669895\pi$
$71$	− 122.249i	− 1.72182i	0.508757	−	0.860910i	$-0.330105\pi$
$72$	0	0
$73$	132.164i	1.81047i	0.424915	+	0.905233i	$0.360304\pi$
$73$	132.164i	1.81047i	−0.424915	+	0.905233i	$0.639696\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	67.3313i	0.874432i
$78$	0	0
$79$	134.833i	1.70674i	0.521302	+	0.853372i	$0.325447\pi$
$79$	134.833i	1.70674i	−0.521302	+	0.853372i	$0.674553\pi$
$80$	0	0
$81$	65.5836	0.809674
$82$	0	0
$83$	−19.4590	−0.234446	−0.117223	−	0.993106i	$-0.537399\pi$
$83$	−19.4590	−0.234446	−0.117223	+	0.993106i	$0.537399\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	34.2492	0.393669
$88$	0	0
$89$	30.0000	0.337079	0.168539	−	0.985695i	$-0.446095\pi$
$89$	30.0000	0.337079	0.168539	+	0.985695i	$0.446095\pi$
$90$	0	0
$91$	249.358i	2.74019i
$92$	0	0
$93$	20.9443i	0.225207i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 12.3344i	− 0.127158i	−0.997977	−	0.0635792i	$-0.979748\pi$
$97$	− 12.3344i	− 0.127158i	0.997977	−	0.0635792i	$-0.0202516\pi$
$98$	0	0
$99$	− 46.5248i	− 0.469947i
$100$	0	0
$101$	105.554	1.04509	0.522545	−	0.852611i	$-0.324983\pi$
$101$	105.554	1.04509	0.522545	+	0.852611i	$0.324983\pi$
$102$	0	0
$103$	−100.987	−0.980455	−0.490227	−	0.871595i	$-0.663086\pi$
$103$	−100.987	−0.980455	−0.490227	+	0.871595i	$0.663086\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	22.8754	0.213789	0.106894	−	0.994270i	$-0.465909\pi$
$107$	22.8754	0.213789	0.106894	+	0.994270i	$0.465909\pi$
$108$	0	0
$109$	35.5279	0.325944	0.162972	−	0.986631i	$-0.447892\pi$
$109$	35.5279	0.325944	0.162972	+	0.986631i	$0.447892\pi$
$110$	0	0
$111$	− 17.9737i	− 0.161925i
$112$	0	0
$113$	− 70.7214i	− 0.625853i	−0.949777	−	0.312926i	$-0.898691\pi$
$113$	− 70.7214i	− 0.625853i	0.949777	−	0.312926i	$-0.101309\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 172.302i	− 1.47267i
$118$	0	0
$119$	12.8591i	0.108060i
$120$	0	0
$121$	90.4427	0.747460
$122$	0	0
$123$	6.64243	0.0540035
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−148.541	−1.16961	−0.584807	−	0.811172i	$-0.698830\pi$
$127$	−148.541	−1.16961	−0.584807	+	0.811172i	$0.698830\pi$
$128$	0	0
$129$	20.3607	0.157835
$130$	0	0
$131$	34.9180i	0.266549i	0.991079	+	0.133275i	$0.0425492\pi$
$131$	34.9180i	0.266549i	−0.991079	+	0.133275i	$0.957451\pi$
$132$	0	0
$133$	146.164i	1.09898i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 100.610i	− 0.734379i	−0.930146	−	0.367189i	$-0.880320\pi$
$137$	− 100.610i	− 0.734379i	0.930146	−	0.367189i	$-0.119680\pi$
$138$	0	0
$139$	− 198.610i	− 1.42885i	−0.699713	−	0.714424i	$-0.746689\pi$
$139$	− 198.610i	− 1.42885i	0.699713	−	0.714424i	$-0.253311\pi$
$140$	0	0
$141$	−34.0263	−0.241321
$142$	0	0
$143$	−113.167	−0.791379
$144$	0	0
$145$	0	0
$146$	0	0
$147$	75.9048	0.516359
$148$	0	0
$149$	203.082	1.36297	0.681483	−	0.731834i	$-0.261335\pi$
$149$	203.082	1.36297	0.681483	+	0.731834i	$0.261335\pi$
$150$	0	0
$151$	− 113.803i	− 0.753665i	−0.926281	−	0.376832i	$-0.877013\pi$
$151$	− 113.803i	− 0.753665i	0.926281	−	0.376832i	$-0.122987\pi$
$152$	0	0
$153$	− 8.88544i	− 0.0580748i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	144.918i	0.923044i	0.887129	+	0.461522i	$0.152697\pi$
$157$	144.918i	0.923044i	−0.887129	+	0.461522i	$0.847303\pi$
$158$	0	0
$159$	0.530970i	0.00333943i
$160$	0	0
$161$	−384.859	−2.39043
$162$	0	0
$163$	203.374	1.24769	0.623846	−	0.781547i	$-0.285569\pi$
$163$	203.374	1.24769	0.623846	+	0.781547i	$0.285569\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−88.7639	−0.531521	−0.265760	−	0.964039i	$-0.585623\pi$
$167$	−88.7639	−0.531521	−0.265760	+	0.964039i	$0.585623\pi$
$168$	0	0
$169$	−250.108	−1.47993
$170$	0	0
$171$	− 100.997i	− 0.590625i
$172$	0	0
$173$	− 135.580i	− 0.783702i	−0.920029	−	0.391851i	$-0.871835\pi$
$173$	− 135.580i	− 0.783702i	0.920029	−	0.391851i	$-0.128165\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	72.2755i	0.408336i
$178$	0	0
$179$	203.830i	1.13871i	0.822091	+	0.569357i	$0.192808\pi$
$179$	203.830i	1.13871i	−0.822091	+	0.569357i	$0.807192\pi$
$180$	0	0
$181$	22.6687	0.125242	0.0626208	−	0.998037i	$-0.480054\pi$
$181$	22.6687	0.125242	0.0626208	+	0.998037i	$0.480054\pi$
$182$	0	0
$183$	−25.3576	−0.138566
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.83592	−0.0312081
$188$	0	0
$189$	−162.059	−0.857454
$190$	0	0
$191$	− 150.748i	− 0.789255i	−0.918841	−	0.394627i	$-0.870874\pi$
$191$	− 150.748i	− 0.789255i	0.918841	−	0.394627i	$-0.129126\pi$
$192$	0	0
$193$	− 210.498i	− 1.09067i	−0.838220	−	0.545333i	$-0.816403\pi$
$193$	− 210.498i	− 1.09067i	0.838220	−	0.545333i	$-0.183597\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	97.0232i	0.492504i	0.969206	+	0.246252i	$0.0791990\pi$
$197$	97.0232i	0.492504i	−0.969206	+	0.246252i	$0.920801\pi$
$198$	0	0
$199$	− 104.892i	− 0.527094i	−0.964647	−	0.263547i	$-0.915108\pi$
$199$	− 104.892i	− 0.527094i	0.964647	−	0.263547i	$-0.0848924\pi$
$200$	0	0
$201$	62.8003	0.312439
$202$	0	0
$203$	546.079	2.69004
$204$	0	0
$205$	0	0
$206$	0	0
$207$	265.931	1.28469
$208$	0	0
$209$	−66.3344	−0.317389
$210$	0	0
$211$	240.584i	1.14021i	0.821573	+	0.570103i	$0.193097\pi$
$211$	240.584i	1.14021i	−0.821573	+	0.570103i	$0.806903\pi$
$212$	0	0
$213$	− 93.3901i	− 0.438451i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	333.941i	1.53890i
$218$	0	0
$219$	100.964i	0.461025i
$220$	0	0
$221$	−21.6130	−0.0977964
$222$	0	0
$223$	18.2918	0.0820260	0.0410130	−	0.999159i	$-0.486941\pi$
$223$	18.2918	0.0820260	0.0410130	+	0.999159i	$0.486941\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−92.4559	−0.407295	−0.203647	−	0.979044i	$-0.565280\pi$
$227$	−92.4559	−0.407295	−0.203647	+	0.979044i	$0.565280\pi$
$228$	0	0
$229$	−165.161	−0.721227	−0.360613	−	0.932715i	$-0.617433\pi$
$229$	−165.161	−0.721227	−0.360613	+	0.932715i	$0.617433\pi$
$230$	0	0
$231$	51.4365i	0.222669i
$232$	0	0
$233$	37.3313i	0.160220i	0.996786	+	0.0801100i	$0.0255272\pi$
$233$	37.3313i	0.160220i	−0.996786	+	0.0801100i	$0.974473\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	103.003i	0.434612i
$238$	0	0
$239$	− 397.495i	− 1.66316i	−0.555405	−	0.831580i	$-0.687437\pi$
$239$	− 397.495i	− 1.66316i	0.555405	−	0.831580i	$-0.312563\pi$
$240$	0	0
$241$	237.915	0.987199	0.493599	−	0.869689i	$-0.335681\pi$
$241$	237.915	0.987199	0.493599	+	0.869689i	$0.335681\pi$
$242$	0	0
$243$	169.846	0.698955
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−245.666	−0.994598
$248$	0	0
$249$	−14.8653	−0.0597002
$250$	0	0
$251$	− 277.803i	− 1.10679i	−0.832920	−	0.553393i	$-0.813333\pi$
$251$	− 277.803i	− 1.10679i	0.832920	−	0.553393i	$-0.186667\pi$
$252$	0	0
$253$	− 174.663i	− 0.690366i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 446.498i	− 1.73735i	−0.495384	−	0.868674i	$-0.664973\pi$
$257$	− 446.498i	− 1.73735i	0.495384	−	0.868674i	$-0.335027\pi$
$258$	0	0
$259$	− 286.577i	− 1.10648i
$260$	0	0
$261$	−377.331	−1.44571
$262$	0	0
$263$	−105.039	−0.399390	−0.199695	−	0.979858i	$-0.563995\pi$
$263$	−105.039	−0.399390	−0.199695	+	0.979858i	$0.563995\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	22.9180	0.0858351
$268$	0	0
$269$	−504.354	−1.87492	−0.937462	−	0.348088i	$-0.886831\pi$
$269$	−504.354	−1.87492	−0.937462	+	0.348088i	$0.886831\pi$
$270$	0	0
$271$	− 164.413i	− 0.606691i	−0.952881	−	0.303346i	$-0.901896\pi$
$271$	− 164.413i	− 0.606691i	0.952881	−	0.303346i	$-0.0981037\pi$
$272$	0	0
$273$	190.492i	0.697774i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 315.580i	− 1.13928i	−0.821894	−	0.569640i	$-0.807083\pi$
$277$	− 315.580i	− 1.13928i	0.821894	−	0.569640i	$-0.192917\pi$
$278$	0	0
$279$	− 230.748i	− 0.827053i
$280$	0	0
$281$	−207.410	−0.738115	−0.369057	−	0.929407i	$-0.620319\pi$
$281$	−207.410	−0.738115	−0.369057	+	0.929407i	$0.620319\pi$
$282$	0	0
$283$	−438.895	−1.55087	−0.775434	−	0.631429i	$-0.782469\pi$
$283$	−438.895	−1.55087	−0.775434	+	0.631429i	$0.782469\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	105.909	0.369020
$288$	0	0
$289$	287.885	0.996143
$290$	0	0
$291$	− 9.42262i	− 0.0323801i
$292$	0	0
$293$	− 272.918i	− 0.931461i	−0.884927	−	0.465730i	$-0.845792\pi$
$293$	− 272.918i	− 0.931461i	0.884927	−	0.465730i	$-0.154208\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 73.5480i	− 0.247636i
$298$	0	0
$299$	− 646.853i	− 2.16339i
$300$	0	0
$301$	324.636	1.07853
$302$	0	0
$303$	80.6362	0.266126
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−251.459	−0.819085	−0.409542	−	0.912291i	$-0.634312\pi$
$307$	−251.459	−0.819085	−0.409542	+	0.912291i	$0.634312\pi$
$308$	0	0
$309$	−77.1471	−0.249667
$310$	0	0
$311$	352.912i	1.13476i	0.823455	+	0.567382i	$0.192044\pi$
$311$	352.912i	1.13476i	−0.823455	+	0.567382i	$0.807956\pi$
$312$	0	0
$313$	− 435.220i	− 1.39048i	−0.718778	−	0.695239i	$-0.755298\pi$
$313$	− 435.220i	− 1.39048i	0.718778	−	0.695239i	$-0.244702\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 41.4164i	− 0.130651i	−0.997864	−	0.0653256i	$-0.979191\pi$
$317$	− 41.4164i	− 0.130651i	0.997864	−	0.0653256i	$-0.0208086\pi$
$318$	0	0
$319$	247.830i	0.776896i
$320$	0	0
$321$	17.4752	0.0544400
$322$	0	0
$323$	−12.6687	−0.0392221
$324$	0	0
$325$	0	0
$326$	0	0
$327$	27.1409	0.0829996
$328$	0	0
$329$	−542.525	−1.64901
$330$	0	0
$331$	− 103.416i	− 0.312436i	−0.987723	−	0.156218i	$-0.950070\pi$
$331$	− 103.416i	− 0.312436i	0.987723	−	0.156218i	$-0.0499303\pi$
$332$	0	0
$333$	198.020i	0.594655i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 106.997i	− 0.317498i	−0.987319	−	0.158749i	$-0.949254\pi$
$337$	− 106.997i	− 0.317498i	0.987319	−	0.158749i	$-0.0507461\pi$
$338$	0	0
$339$	− 54.0263i	− 0.159370i
$340$	0	0
$341$	−151.554	−0.444440
$342$	0	0
$343$	613.410	1.78837
$344$	0	0
$345$	0	0
$346$	0	0
$347$	511.426	1.47385	0.736926	−	0.675974i	$-0.236277\pi$
$347$	511.426	1.47385	0.736926	+	0.675974i	$0.236277\pi$
$348$	0	0
$349$	63.1672	0.180995	0.0904974	−	0.995897i	$-0.471154\pi$
$349$	63.1672	0.180995	0.0904974	+	0.995897i	$0.471154\pi$
$350$	0	0
$351$	− 272.381i	− 0.776014i
$352$	0	0
$353$	62.7740i	0.177830i	0.996039	+	0.0889150i	$0.0283399\pi$
$353$	62.7740i	0.177830i	−0.996039	+	0.0889150i	$0.971660\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	9.82350i	0.0275168i
$358$	0	0
$359$	176.105i	0.490544i	0.969454	+	0.245272i	$0.0788772\pi$
$359$	176.105i	0.490544i	−0.969454	+	0.245272i	$0.921123\pi$
$360$	0	0
$361$	217.000	0.601108
$362$	0	0
$363$	69.0921	0.190336
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−190.259	−0.518418	−0.259209	−	0.965821i	$-0.583462\pi$
$367$	−190.259	−0.518418	−0.259209	+	0.965821i	$0.583462\pi$
$368$	0	0
$369$	−73.1811	−0.198323
$370$	0	0
$371$	8.46592i	0.0228192i
$372$	0	0
$373$	− 525.076i	− 1.40771i	−0.710344	−	0.703855i	$-0.751460\pi$
$373$	− 525.076i	− 1.40771i	0.710344	−	0.703855i	$-0.248540\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	917.823i	2.43455i
$378$	0	0
$379$	530.217i	1.39899i	0.714638	+	0.699494i	$0.246591\pi$
$379$	530.217i	1.39899i	−0.714638	+	0.699494i	$0.753409\pi$
$380$	0	0
$381$	−113.475	−0.297835
$382$	0	0
$383$	165.177	0.431272	0.215636	−	0.976474i	$-0.430818\pi$
$383$	165.177	0.431272	0.215636	+	0.976474i	$0.430818\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−224.318	−0.579633
$388$	0	0
$389$	532.079	1.36781	0.683906	−	0.729570i	$-0.260280\pi$
$389$	532.079	1.36781	0.683906	+	0.729570i	$0.260280\pi$
$390$	0	0
$391$	− 33.3576i	− 0.0853135i
$392$	0	0
$393$	26.6749i	0.0678752i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	512.407i	1.29070i	0.763888	+	0.645349i	$0.223288\pi$
$397$	512.407i	1.29070i	−0.763888	+	0.645349i	$0.776712\pi$
$398$	0	0
$399$	111.659i	0.279848i
$400$	0	0
$401$	22.3282	0.0556812	0.0278406	−	0.999612i	$-0.491137\pi$
$401$	22.3282	0.0556812	0.0278406	+	0.999612i	$0.491137\pi$
$402$	0	0
$403$	−561.272	−1.39274
$404$	0	0
$405$	0	0
$406$	0	0
$407$	130.059	0.319555
$408$	0	0
$409$	42.4659	0.103829	0.0519143	−	0.998652i	$-0.483468\pi$
$409$	42.4659	0.103829	0.0519143	+	0.998652i	$0.483468\pi$
$410$	0	0
$411$	− 76.8591i	− 0.187005i
$412$	0	0
$413$	1152.38i	2.79027i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 151.724i	− 0.363848i
$418$	0	0
$419$	284.381i	0.678713i	0.940658	+	0.339357i	$0.110209\pi$
$419$	284.381i	0.678713i	−0.940658	+	0.339357i	$0.889791\pi$
$420$	0	0
$421$	201.246	0.478019	0.239010	−	0.971017i	$-0.423177\pi$
$421$	201.246	0.478019	0.239010	+	0.971017i	$0.423177\pi$
$422$	0	0
$423$	374.875	0.886230
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−404.308	−0.946857
$428$	0	0
$429$	−86.4520	−0.201520
$430$	0	0
$431$	81.9086i	0.190043i	0.995475	+	0.0950216i	$0.0302920\pi$
$431$	81.9086i	0.190043i	−0.995475	+	0.0950216i	$0.969708\pi$
$432$	0	0
$433$	− 195.325i	− 0.451097i	−0.974232	−	0.225549i	$-0.927583\pi$
$433$	− 195.325i	− 0.451097i	0.974232	−	0.225549i	$-0.0724174\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 379.161i	− 0.867645i
$438$	0	0
$439$	466.780i	1.06328i	0.846970	+	0.531640i	$0.178424\pi$
$439$	466.780i	1.06328i	−0.846970	+	0.531640i	$0.821576\pi$
$440$	0	0
$441$	−836.260	−1.89628
$442$	0	0
$443$	−376.508	−0.849906	−0.424953	−	0.905215i	$-0.639709\pi$
$443$	−376.508	−0.849906	−0.424953	+	0.905215i	$0.639709\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	155.141	0.347071
$448$	0	0
$449$	247.029	0.550177	0.275088	−	0.961419i	$-0.411293\pi$
$449$	247.029	0.550177	0.275088	+	0.961419i	$0.411293\pi$
$450$	0	0
$451$	48.0650i	0.106574i
$452$	0	0
$453$	− 86.9381i	− 0.191916i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	229.003i	0.501101i	0.968104	+	0.250550i	$0.0806116\pi$
$457$	229.003i	0.501101i	−0.968104	+	0.250550i	$0.919388\pi$
$458$	0	0
$459$	− 14.0464i	− 0.0306022i
$460$	0	0
$461$	62.7740	0.136169	0.0680846	−	0.997680i	$-0.478311\pi$
$461$	62.7740	0.136169	0.0680846	+	0.997680i	$0.478311\pi$
$462$	0	0
$463$	204.075	0.440767	0.220383	−	0.975413i	$-0.429269\pi$
$463$	204.075	0.440767	0.220383	+	0.975413i	$0.429269\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	412.371	0.883021	0.441510	−	0.897256i	$-0.354443\pi$
$467$	412.371	0.883021	0.441510	+	0.897256i	$0.354443\pi$
$468$	0	0
$469$	1001.30	2.13498
$470$	0	0
$471$	110.707i	0.235048i
$472$	0	0
$473$	147.331i	0.311483i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5.84981i	− 0.0122638i
$478$	0	0
$479$	− 638.597i	− 1.33319i	−0.745421	−	0.666594i	$-0.767751\pi$
$479$	− 638.597i	− 1.33319i	0.745421	−	0.666594i	$-0.232249\pi$
$480$	0	0
$481$	481.666	1.00138
$482$	0	0
$483$	−294.006	−0.608709
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−24.8692	−0.0510661	−0.0255330	−	0.999674i	$-0.508128\pi$
$487$	−24.8692	−0.0510661	−0.0255330	+	0.999674i	$0.508128\pi$
$488$	0	0
$489$	155.364	0.317717
$490$	0	0
$491$	− 320.584i	− 0.652920i	−0.945211	−	0.326460i	$-0.894144\pi$
$491$	− 320.584i	− 0.652920i	0.945211	−	0.326460i	$-0.105856\pi$
$492$	0	0
$493$	47.3313i	0.0960066i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1489.04i	− 2.99605i
$498$	0	0
$499$	168.774i	0.338224i	0.985597	+	0.169112i	$0.0540900\pi$
$499$	168.774i	0.338224i	−0.985597	+	0.169112i	$0.945910\pi$
$500$	0	0
$501$	−67.8096	−0.135349
$502$	0	0
$503$	−662.089	−1.31628	−0.658140	−	0.752895i	$-0.728657\pi$
$503$	−662.089	−1.31628	−0.658140	+	0.752895i	$0.728657\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−191.066	−0.376856
$508$	0	0
$509$	−651.823	−1.28060	−0.640298	−	0.768126i	$-0.721189\pi$
$509$	−651.823	−1.28060	−0.640298	+	0.768126i	$0.721189\pi$
$510$	0	0
$511$	1609.80i	3.15030i
$512$	0	0
$513$	− 159.659i	− 0.311227i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 246.217i	− 0.476241i
$518$	0	0
$519$	− 103.574i	− 0.199565i
$520$	0	0
$521$	750.984	1.44143	0.720714	−	0.693232i	$-0.243814\pi$
$521$	750.984	1.44143	0.720714	+	0.693232i	$0.243814\pi$
$522$	0	0
$523$	−328.128	−0.627395	−0.313698	−	0.949523i	$-0.601568\pi$
$523$	−328.128	−0.627395	−0.313698	+	0.949523i	$0.601568\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28.9443	−0.0549227
$528$	0	0
$529$	469.354	0.887249
$530$	0	0
$531$	− 796.276i	− 1.49958i
$532$	0	0
$533$	178.006i	0.333970i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	155.712i	0.289967i
$538$	0	0
$539$	549.252i	1.01902i
$540$	0	0
$541$	238.563	0.440968	0.220484	−	0.975391i	$-0.429236\pi$
$541$	238.563	0.440968	0.220484	+	0.975391i	$0.429236\pi$
$542$	0	0
$543$	17.3174	0.0318920
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−341.453	−0.624228	−0.312114	−	0.950045i	$-0.601037\pi$
$547$	−341.453	−0.624228	−0.312114	+	0.950045i	$0.601037\pi$
$548$	0	0
$549$	279.370	0.508871
$550$	0	0
$551$	537.994i	0.976395i
$552$	0	0
$553$	1642.31i	2.96982i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 349.574i	− 0.627602i	−0.949489	−	0.313801i	$-0.898398\pi$
$557$	− 349.574i	− 0.627602i	0.949489	−	0.313801i	$-0.101602\pi$
$558$	0	0
$559$	545.633i	0.976088i
$560$	0	0
$561$	−4.45825	−0.00794696
$562$	0	0
$563$	−746.778	−1.32643	−0.663213	−	0.748431i	$-0.730808\pi$
$563$	−746.778	−1.32643	−0.663213	+	0.748431i	$0.730808\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	798.830	1.40887
$568$	0	0
$569$	660.906	1.16152	0.580761	−	0.814074i	$-0.302755\pi$
$569$	660.906	1.16152	0.580761	+	0.814074i	$0.302755\pi$
$570$	0	0
$571$	529.240i	0.926865i	0.886132	+	0.463432i	$0.153382\pi$
$571$	529.240i	0.926865i	−0.886132	+	0.463432i	$0.846618\pi$
$572$	0	0
$573$	− 115.161i	− 0.200979i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	677.830i	1.17475i	0.809316	+	0.587374i	$0.199838\pi$
$577$	677.830i	1.17475i	−0.809316	+	0.587374i	$0.800162\pi$
$578$	0	0
$579$	− 160.807i	− 0.277731i
$580$	0	0
$581$	−237.017	−0.407947
$582$	0	0
$583$	−3.84213	−0.00659028
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−80.6137	−0.137332	−0.0686659	−	0.997640i	$-0.521874\pi$
$587$	−80.6137	−0.137332	−0.0686659	+	0.997640i	$0.521874\pi$
$588$	0	0
$589$	−328.997	−0.558569
$590$	0	0
$591$	74.1191i	0.125413i
$592$	0	0
$593$	347.548i	0.586084i	0.956100	+	0.293042i	$0.0946676\pi$
$593$	347.548i	0.586084i	−0.956100	+	0.293042i	$0.905332\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 80.1301i	− 0.134221i
$598$	0	0
$599$	− 78.2817i	− 0.130687i	−0.997863	−	0.0653437i	$-0.979186\pi$
$599$	− 78.2817i	− 0.130687i	0.997863	−	0.0653437i	$-0.0208144\pi$
$600$	0	0
$601$	−39.3050	−0.0653993	−0.0326996	−	0.999465i	$-0.510410\pi$
$601$	−39.3050	−0.0653993	−0.0326996	+	0.999465i	$0.510410\pi$
$602$	0	0
$603$	−691.885	−1.14740
$604$	0	0
$605$	0	0
$606$	0	0
$607$	432.443	0.712427	0.356214	−	0.934405i	$-0.384068\pi$
$607$	432.443	0.712427	0.356214	+	0.934405i	$0.384068\pi$
$608$	0	0
$609$	417.167	0.685004
$610$	0	0
$611$	− 911.850i	− 1.49239i
$612$	0	0
$613$	498.413i	0.813072i	0.913635	+	0.406536i	$0.133263\pi$
$613$	498.413i	0.813072i	−0.913635	+	0.406536i	$0.866737\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	611.325i	0.990802i	0.868664	+	0.495401i	$0.164979\pi$
$617$	611.325i	0.990802i	−0.868664	+	0.495401i	$0.835021\pi$
$618$	0	0
$619$	490.663i	0.792670i	0.918106	+	0.396335i	$0.129718\pi$
$619$	490.663i	0.792670i	−0.918106	+	0.396335i	$0.870282\pi$
$620$	0	0
$621$	420.393	0.676962
$622$	0	0
$623$	365.410	0.586533
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−50.6749	−0.0808213
$628$	0	0
$629$	24.8390	0.0394897
$630$	0	0
$631$	104.636i	0.165826i	0.996557	+	0.0829130i	$0.0264224\pi$
$631$	104.636i	0.165826i	−0.996557	+	0.0829130i	$0.973578\pi$
$632$	0	0
$633$	183.790i	0.290347i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2034.13i	3.19329i
$638$	0	0
$639$	1028.90i	1.61017i
$640$	0	0
$641$	150.020	0.234041	0.117020	−	0.993130i	$-0.462666\pi$
$641$	150.020	0.234041	0.117020	+	0.993130i	$0.462666\pi$
$642$	0	0
$643$	646.810	1.00593	0.502963	−	0.864308i	$-0.332243\pi$
$643$	646.810	1.00593	0.502963	+	0.864308i	$0.332243\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	694.450	1.07334	0.536669	−	0.843793i	$-0.319682\pi$
$647$	694.450	1.07334	0.536669	+	0.843793i	$0.319682\pi$
$648$	0	0
$649$	−522.991	−0.805841
$650$	0	0
$651$	255.108i	0.391872i
$652$	0	0
$653$	− 698.361i	− 1.06947i	−0.845021	−	0.534733i	$-0.820412\pi$
$653$	− 698.361i	− 1.06947i	0.845021	−	0.534733i	$-0.179588\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1112.35i	− 1.69307i
$658$	0	0
$659$	834.282i	1.26598i	0.774159	+	0.632991i	$0.218173\pi$
$659$	834.282i	1.26598i	−0.774159	+	0.632991i	$0.781827\pi$
$660$	0	0
$661$	978.584	1.48046	0.740230	−	0.672354i	$-0.234717\pi$
$661$	978.584	1.48046	0.740230	+	0.672354i	$0.234717\pi$
$662$	0	0
$663$	−16.5109	−0.0249033
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1416.57	−2.12379
$668$	0	0
$669$	13.9737	0.0208874
$670$	0	0
$671$	− 183.489i	− 0.273456i
$672$	0	0
$673$	− 682.669i	− 1.01437i	−0.861838	−	0.507183i	$-0.830687\pi$
$673$	− 682.669i	− 1.01437i	0.861838	−	0.507183i	$-0.169313\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 948.067i	− 1.40039i	−0.713950	−	0.700197i	$-0.753096\pi$
$677$	− 948.067i	− 1.40039i	0.713950	−	0.700197i	$-0.246904\pi$
$678$	0	0
$679$	− 150.237i	− 0.221262i
$680$	0	0
$681$	−70.6300	−0.103715
$682$	0	0
$683$	−341.688	−0.500275	−0.250138	−	0.968210i	$-0.580476\pi$
$683$	−341.688	−0.500275	−0.250138	+	0.968210i	$0.580476\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−126.172	−0.183656
$688$	0	0
$689$	−14.2291	−0.0206518
$690$	0	0
$691$	215.416i	0.311746i	0.987777	+	0.155873i	$0.0498190\pi$
$691$	215.416i	0.311746i	−0.987777	+	0.155873i	$0.950181\pi$
$692$	0	0
$693$	− 566.687i	− 0.817731i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.17961i	0.0131702i
$698$	0	0
$699$	28.5185i	0.0407991i
$700$	0	0
$701$	−1137.40	−1.62254	−0.811272	−	0.584668i	$-0.801225\pi$
$701$	−1137.40	−1.62254	−0.811272	+	0.584668i	$0.801225\pi$
$702$	0	0
$703$	282.334	0.401614
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1285.69	1.81851
$708$	0	0
$709$	853.149	1.20331	0.601656	−	0.798755i	$-0.294508\pi$
$709$	853.149	1.20331	0.601656	+	0.798755i	$0.294508\pi$
$710$	0	0
$711$	− 1134.81i	− 1.59607i
$712$	0	0
$713$	− 866.269i	− 1.21496i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 303.659i	− 0.423514i
$718$	0	0
$719$	− 696.616i	− 0.968868i	−0.874828	−	0.484434i	$-0.839026\pi$
$719$	− 696.616i	− 0.968868i	0.874828	−	0.484434i	$-0.160974\pi$
$720$	0	0
$721$	−1230.05	−1.70604
$722$	0	0
$723$	181.751	0.251384
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−588.265	−0.809168	−0.404584	−	0.914501i	$-0.632584\pi$
$727$	−588.265	−0.809168	−0.404584	+	0.914501i	$0.632584\pi$
$728$	0	0
$729$	−460.502	−0.631689
$730$	0	0
$731$	28.1378i	0.0384922i
$732$	0	0
$733$	1106.86i	1.51004i	0.655702	+	0.755020i	$0.272373\pi$
$733$	1106.86i	1.51004i	−0.655702	+	0.755020i	$0.727627\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	454.427i	0.616590i
$738$	0	0
$739$	− 720.053i	− 0.974361i	−0.873301	−	0.487180i	$-0.838025\pi$
$739$	− 720.053i	− 0.974361i	0.873301	−	0.487180i	$-0.161975\pi$
$740$	0	0
$741$	−187.672	−0.253268
$742$	0	0
$743$	317.793	0.427716	0.213858	−	0.976865i	$-0.431397\pi$
$743$	317.793	0.427716	0.213858	+	0.976865i	$0.431397\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	163.775	0.219243
$748$	0	0
$749$	278.630	0.372003
$750$	0	0
$751$	606.512i	0.807606i	0.914846	+	0.403803i	$0.132312\pi$
$751$	606.512i	0.807606i	−0.914846	+	0.403803i	$0.867688\pi$
$752$	0	0
$753$	− 212.223i	− 0.281837i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 359.463i	− 0.474852i	−0.971406	−	0.237426i	$-0.923696\pi$
$757$	− 359.463i	− 0.474852i	0.971406	−	0.237426i	$-0.0763037\pi$
$758$	0	0
$759$	− 133.430i	− 0.175797i
$760$	0	0
$761$	239.876	0.315212	0.157606	−	0.987502i	$-0.449622\pi$
$761$	239.876	0.315212	0.157606	+	0.987502i	$0.449622\pi$
$762$	0	0
$763$	432.741	0.567158
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1936.87	−2.52525
$768$	0	0
$769$	768.885	0.999851	0.499926	−	0.866068i	$-0.333361\pi$
$769$	768.885	0.999851	0.499926	+	0.866068i	$0.333361\pi$
$770$	0	0
$771$	− 341.094i	− 0.442405i
$772$	0	0
$773$	− 33.5743i	− 0.0434337i	−0.999764	−	0.0217169i	$-0.993087\pi$
$773$	− 33.5743i	− 0.0434337i	0.999764	−	0.0217169i	$-0.00691324\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 218.926i	− 0.281758i
$778$	0	0
$779$	104.341i	0.133942i
$780$	0	0
$781$	675.777	0.865272
$782$	0	0
$783$	−596.498	−0.761812
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−426.738	−0.542233	−0.271117	−	0.962546i	$-0.587393\pi$
$787$	−426.738	−0.542233	−0.271117	+	0.962546i	$0.587393\pi$
$788$	0	0
$789$	−80.2430	−0.101702
$790$	0	0
$791$	− 861.410i	− 1.08901i
$792$	0	0
$793$	− 679.542i	− 0.856925i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 194.203i	− 0.243667i	−0.992551	−	0.121834i	$-0.961123\pi$
$797$	− 194.203i	− 0.243667i	0.992551	−	0.121834i	$-0.0388774\pi$
$798$	0	0
$799$	− 47.0232i	− 0.0588526i
$800$	0	0
$801$	−252.492	−0.315221
$802$	0	0
$803$	−730.585	−0.909819
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−385.293	−0.477438
$808$	0	0
$809$	−1446.33	−1.78780	−0.893899	−	0.448269i	$-0.852041\pi$
$809$	−1446.33	−1.78780	−0.893899	+	0.448269i	$0.852041\pi$
$810$	0	0
$811$	23.3112i	0.0287437i	0.999897	+	0.0143719i	$0.00457486\pi$
$811$	23.3112i	0.0287437i	−0.999897	+	0.0143719i	$0.995425\pi$
$812$	0	0
$813$	− 125.601i	− 0.154490i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	319.830i	0.391468i
$818$	0	0
$819$	− 2098.70i	− 2.56251i
$820$	0	0
$821$	−701.416	−0.854344	−0.427172	−	0.904170i	$-0.640490\pi$
$821$	−701.416	−0.854344	−0.427172	+	0.904170i	$0.640490\pi$
$822$	0	0
$823$	270.961	0.329235	0.164618	−	0.986357i	$-0.447361\pi$
$823$	270.961	0.329235	0.164618	+	0.986357i	$0.447361\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1035.10	1.25163	0.625815	−	0.779971i	$-0.284766\pi$
$827$	1035.10	1.25163	0.625815	+	0.779971i	$0.284766\pi$
$828$	0	0
$829$	332.735	0.401369	0.200685	−	0.979656i	$-0.435683\pi$
$829$	332.735	0.401369	0.200685	+	0.979656i	$0.435683\pi$
$830$	0	0
$831$	− 241.082i	− 0.290111i
$832$	0	0
$833$	104.898i	0.125928i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 364.774i	− 0.435811i
$838$	0	0
$839$	− 102.177i	− 0.121784i	−0.998144	−	0.0608918i	$-0.980606\pi$
$839$	− 102.177i	− 0.121784i	0.998144	−	0.0608918i	$-0.0193945\pi$
$840$	0	0
$841$	1168.98	1.38999
$842$	0	0
$843$	−158.447	−0.187956
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1101.62	1.30062
$848$	0	0
$849$	−335.286	−0.394919
$850$	0	0
$851$	743.404i	0.873565i
$852$	0	0
$853$	358.401i	0.420165i	0.977684	+	0.210083i	$0.0673733\pi$
$853$	358.401i	0.420165i	−0.977684	+	0.210083i	$0.932627\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 33.3839i	− 0.0389544i	−0.999810	−	0.0194772i	$-0.993800\pi$
$857$	− 33.3839i	− 0.0389544i	0.999810	−	0.0194772i	$-0.00620017\pi$
$858$	0	0
$859$	− 989.483i	− 1.15190i	−0.817485	−	0.575950i	$-0.804632\pi$
$859$	− 989.483i	− 1.15190i	0.817485	−	0.575950i	$-0.195368\pi$
$860$	0	0
$861$	80.9070	0.0939686
$862$	0	0
$863$	767.341	0.889156	0.444578	−	0.895740i	$-0.353354\pi$
$863$	767.341	0.889156	0.444578	+	0.895740i	$0.353354\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	219.925	0.253662
$868$	0	0
$869$	−745.337	−0.857696
$870$	0	0
$871$	1682.95i	1.93220i
$872$	0	0
$873$	103.811i	0.118913i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1013.80i	1.15598i	0.816043	+	0.577992i	$0.196163\pi$
$877$	1013.80i	1.15598i	−0.816043	+	0.577992i	$0.803837\pi$
$878$	0	0
$879$	− 208.491i	− 0.237191i
$880$	0	0
$881$	−280.748	−0.318669	−0.159335	−	0.987225i	$-0.550935\pi$
$881$	−280.748	−0.318669	−0.159335	+	0.987225i	$0.550935\pi$
$882$	0	0
$883$	1096.87	1.24221	0.621104	−	0.783728i	$-0.286685\pi$
$883$	1096.87	1.24221	0.621104	+	0.783728i	$0.286685\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1164.82	−1.31321	−0.656605	−	0.754235i	$-0.728008\pi$
$887$	−1164.82	−1.31321	−0.656605	+	0.754235i	$0.728008\pi$
$888$	0	0
$889$	−1809.28	−2.03519
$890$	0	0
$891$	362.537i	0.406888i
$892$	0	0
$893$	− 534.492i	− 0.598536i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 494.152i	− 0.550894i
$898$	0	0
$899$	1229.15i	1.36725i
$900$	0	0
$901$	−0.733782	−0.000814408 0
$902$	0	0
$903$	248.000	0.274640
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1132.82	1.24897	0.624485	−	0.781036i	$-0.285309\pi$
$907$	1132.82	1.24897	0.624485	+	0.781036i	$0.285309\pi$
$908$	0	0
$909$	−888.387	−0.977323
$910$	0	0
$911$	− 1793.23i	− 1.96842i	−0.177014	−	0.984208i	$-0.556644\pi$
$911$	− 1793.23i	− 1.96842i	0.177014	−	0.984208i	$-0.443356\pi$
$912$	0	0
$913$	− 107.567i	− 0.117817i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	425.313i	0.463809i
$918$	0	0
$919$	205.390i	0.223493i	0.993737	+	0.111747i	$0.0356445\pi$
$919$	205.390i	0.223493i	−0.993737	+	0.111747i	$0.964356\pi$
$920$	0	0
$921$	−192.098	−0.208575
$922$	0	0
$923$	2502.70	2.71149
$924$	0	0
$925$	0	0
$926$	0	0
$927$	849.946	0.916879
$928$	0	0
$929$	736.237	0.792505	0.396252	−	0.918142i	$-0.370311\pi$
$929$	736.237	0.792505	0.396252	+	0.918142i	$0.370311\pi$
$930$	0	0
$931$	1192.33i	1.28070i
$932$	0	0
$933$	269.601i	0.288961i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	40.3344i	0.0430463i	0.999768	+	0.0215231i	$0.00685156\pi$
$937$	40.3344i	0.0430463i	−0.999768	+	0.0215231i	$0.993148\pi$
$938$	0	0
$939$	− 332.478i	− 0.354077i
$940$	0	0
$941$	447.115	0.475148	0.237574	−	0.971369i	$-0.423648\pi$
$941$	447.115	0.475148	0.237574	+	0.971369i	$0.423648\pi$
$942$	0	0
$943$	−274.735	−0.291342
$944$	0	0
$945$	0	0
$946$	0	0
$947$	144.553	0.152644	0.0763218	−	0.997083i	$-0.475682\pi$
$947$	144.553	0.152644	0.0763218	+	0.997083i	$0.475682\pi$
$948$	0	0
$949$	−2705.68	−2.85109
$950$	0	0
$951$	− 31.6393i	− 0.0332695i
$952$	0	0
$953$	432.006i	0.453312i	0.973975	+	0.226656i	$0.0727793\pi$
$953$	432.006i	0.453312i	−0.973975	+	0.226656i	$0.927221\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	189.325i	0.197832i
$958$	0	0
$959$	− 1225.46i	− 1.27785i
$960$	0	0
$961$	209.341	0.217836
$962$	0	0
$963$	−192.529	−0.199926
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1472.95	1.52322	0.761610	−	0.648035i	$-0.224409\pi$
$967$	1472.95	1.52322	0.761610	+	0.648035i	$0.224409\pi$
$968$	0	0
$969$	−9.67805	−0.00998767
$970$	0	0
$971$	− 308.190i	− 0.317395i	−0.987327	−	0.158697i	$-0.949271\pi$
$971$	− 308.190i	− 0.317395i	0.987327	−	0.158697i	$-0.0507294\pi$
$972$	0	0
$973$	− 2419.14i	− 2.48627i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1376.78i	1.40919i	0.709609	+	0.704596i	$0.248872\pi$
$977$	1376.78i	1.40919i	−0.709609	+	0.704596i	$0.751128\pi$
$978$	0	0
$979$	165.836i	0.169393i
$980$	0	0
$981$	−299.017	−0.304808
$982$	0	0
$983$	−1250.38	−1.27200	−0.636000	−	0.771689i	$-0.719412\pi$
$983$	−1250.38	−1.27200	−0.636000	+	0.771689i	$0.719412\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−414.452	−0.419911
$988$	0	0
$989$	−842.132	−0.851498
$990$	0	0
$991$	− 1727.44i	− 1.74313i	−0.490278	−	0.871566i	$-0.663105\pi$
$991$	− 1727.44i	− 1.74313i	0.490278	−	0.871566i	$-0.336895\pi$
$992$	0	0
$993$	− 79.0031i	− 0.0795600i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1131.90i	− 1.13530i	−0.823269	−	0.567651i	$-0.807852\pi$
$997$	− 1131.90i	− 1.13530i	0.823269	−	0.567651i	$-0.192148\pi$
$998$	0	0
$999$	313.037i	0.313350i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.3.h.j.799.2		4
4.3	odd	2		800.3.h.c.799.3		4
5.2	odd	4		160.3.b.a.31.2	✓	4
5.3	odd	4		800.3.b.d.351.3		4
5.4	even	2		800.3.h.c.799.4		4
8.3	odd	2		1600.3.h.m.1599.2		4
8.5	even	2		1600.3.h.d.1599.3		4
15.2	even	4		1440.3.e.b.991.4		4
20.3	even	4		800.3.b.d.351.2		4
20.7	even	4		160.3.b.a.31.3	yes	4
20.19	odd	2	inner	800.3.h.j.799.1		4
40.3	even	4		1600.3.b.n.1151.3		4
40.13	odd	4		1600.3.b.n.1151.2		4
40.19	odd	2		1600.3.h.d.1599.4		4
40.27	even	4		320.3.b.b.191.2		4
40.29	even	2		1600.3.h.m.1599.1		4
40.37	odd	4		320.3.b.b.191.3		4
60.47	odd	4		1440.3.e.b.991.3		4
80.27	even	4		1280.3.g.d.1151.1		4
80.37	odd	4		1280.3.g.a.1151.3		4
80.67	even	4		1280.3.g.a.1151.4		4
80.77	odd	4		1280.3.g.d.1151.2		4
120.77	even	4		2880.3.e.a.2431.2		4
120.107	odd	4		2880.3.e.a.2431.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.b.a.31.2	✓	4	5.2	odd	4
160.3.b.a.31.3	yes	4	20.7	even	4
320.3.b.b.191.2		4	40.27	even	4
320.3.b.b.191.3		4	40.37	odd	4
800.3.b.d.351.2		4	20.3	even	4
800.3.b.d.351.3		4	5.3	odd	4
800.3.h.c.799.3		4	4.3	odd	2
800.3.h.c.799.4		4	5.4	even	2
800.3.h.j.799.1		4	20.19	odd	2	inner
800.3.h.j.799.2		4	1.1	even	1	trivial
1280.3.g.a.1151.3		4	80.37	odd	4
1280.3.g.a.1151.4		4	80.67	even	4
1280.3.g.d.1151.1		4	80.27	even	4
1280.3.g.d.1151.2		4	80.77	odd	4
1440.3.e.b.991.3		4	60.47	odd	4
1440.3.e.b.991.4		4	15.2	even	4
1600.3.b.n.1151.2		4	40.13	odd	4
1600.3.b.n.1151.3		4	40.3	even	4
1600.3.h.d.1599.3		4	8.5	even	2
1600.3.h.d.1599.4		4	40.19	odd	2
1600.3.h.m.1599.1		4	40.29	even	2
1600.3.h.m.1599.2		4	8.3	odd	2
2880.3.e.a.2431.1		4	120.107	odd	4
2880.3.e.a.2431.2		4	120.77	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	800.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.763932 q^{3} +12.1803 q^{7} -8.41641 q^{9} +O(q^{10})\)	\(q+0.763932 q^{3} +12.1803 q^{7} -8.41641 q^{9} +5.52786i q^{11} +20.4721i q^{13} +1.05573i q^{17} +12.0000i q^{19} +9.30495 q^{21} -31.5967 q^{23} -13.3050 q^{27} +44.8328 q^{29} +27.4164i q^{31} +4.22291i q^{33} -23.5279i q^{37} +15.6393i q^{39} +8.69505 q^{41} +26.6525 q^{43} -44.5410 q^{47} +99.3607 q^{49} +0.806504i q^{51} +0.695048i q^{53} +9.16718i q^{57} +94.6099i q^{59} -33.1935 q^{61} -102.515 q^{63} +82.2067 q^{67} -24.1378 q^{69} -122.249i q^{71} +132.164i q^{73} +67.3313i q^{77} +134.833i q^{79} +65.5836 q^{81} -19.4590 q^{83} +34.2492 q^{87} +30.0000 q^{89} +249.358i q^{91} +20.9443i q^{93} -12.3344i q^{97} -46.5248i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 4 q^{7} + 20 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 4 * q^7 + 20 * q^9	\( 4 q + 12 q^{3} + 4 q^{7} + 20 q^{9} - 88 q^{21} - 28 q^{23} + 72 q^{27} + 72 q^{29} + 160 q^{41} + 44 q^{43} - 44 q^{47} + 308 q^{49} + 64 q^{61} - 580 q^{63} - 20 q^{67} + 136 q^{69} + 316 q^{81} - 212 q^{83} - 24 q^{87} + 120 q^{89}+O(q^{100}) \) 4 * q + 12 * q^3 + 4 * q^7 + 20 * q^9 - 88 * q^21 - 28 * q^23 + 72 * q^27 + 72 * q^29 + 160 * q^41 + 44 * q^43 - 44 * q^47 + 308 * q^49 + 64 * q^61 - 580 * q^63 - 20 * q^67 + 136 * q^69 + 316 * q^81 - 212 * q^83 - 24 * q^87 + 120 * q^89

Introduction

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