LMFDB - Embedded newform 800.3.g.h.751.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,3,Mod(751,800)]

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, 0]))

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

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

chi := DirichletCharacter("800.751");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	800.g (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:	$8$
Coefficient field:	8.0.53824000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 6x^{6} + 36x^{4} - 96x^{2} + 256 $ x^8 - 6x^6 + 36x^4 - 96*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			751.2
Root			$1.48020 + 1.34500i$ of defining polynomial
Character	$\chi$	$=$	800.751
Dual form			800.3.g.h.751.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.79002 q^{3} +7.67752i q^{7} +13.9443 q^{9} +0.472136 q^{11} +4.10995i q^{13} -2.26154 q^{17} +26.3607 q^{19} -36.7754i q^{21} -9.73249i q^{23} -23.6832 q^{27} -41.6971i q^{29} +22.0104i q^{31} -2.26154 q^{33} +51.7449i q^{37} -19.6867i q^{39} +15.0557 q^{41} -9.31310 q^{43} +8.76226i q^{47} -9.94427 q^{49} +10.8328 q^{51} +39.9002i q^{53} -126.268 q^{57} -77.1935 q^{59} +14.7650i q^{61} +107.057i q^{63} -75.8395 q^{67} +46.6188i q^{69} +81.0705i q^{71} -83.4249 q^{73} +3.62483i q^{77} +100.757i q^{79} -12.0557 q^{81} +0.266939 q^{83} +199.730i q^{87} -85.5542 q^{89} -31.5542 q^{91} -105.430i q^{93} +99.1297 q^{97} +6.58359 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 40 q^{9} - 32 q^{11} + 32 q^{19} + 192 q^{41} - 8 q^{49} - 128 q^{51} - 224 q^{59} - 168 q^{81} - 112 q^{89} + 320 q^{91} + 160 q^{99}+O(q^{100}) $ 8 * q + 40 * q^9 - 32 * q^11 + 32 * q^19 + 192 * q^41 - 8 * q^49 - 128 * q^51 - 224 * q^59 - 168 * q^81 - 112 * q^89 + 320 * q^91 + 160 * 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
$3$	−4.79002	−1.59667	−0.798336	−	0.602212i	$-0.794286\pi$
$3$	−4.79002	−1.59667	−0.798336	+	0.602212i	$0.794286\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.67752i	1.09679i	0.836220	+	0.548394i	$0.184761\pi$
$7$	7.67752i	1.09679i	−0.836220	+	0.548394i	$0.815239\pi$
$8$	0	0
$9$	13.9443	1.54936
$10$	0	0
$11$	0.472136	0.0429215	0.0214607	−	0.999770i	$-0.493168\pi$
$11$	0.472136	0.0429215	0.0214607	+	0.999770i	$0.493168\pi$
$12$	0	0
$13$	4.10995i	0.316150i	0.987427	+	0.158075i	$0.0505287\pi$
$13$	4.10995i	0.316150i	−0.987427	+	0.158075i	$0.949471\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.26154	−0.133032	−0.0665159	−	0.997785i	$-0.521188\pi$
$17$	−2.26154	−0.133032	−0.0665159	+	0.997785i	$0.521188\pi$
$18$	0	0
$19$	26.3607	1.38740	0.693702	−	0.720262i	$-0.255978\pi$
$19$	26.3607	1.38740	0.693702	+	0.720262i	$0.255978\pi$
$20$	0	0
$21$	− 36.7754i	− 1.75121i
$22$	0	0
$23$	− 9.73249i	− 0.423152i	−0.977362	−	0.211576i	$-0.932140\pi$
$23$	− 9.73249i	− 0.423152i	0.977362	−	0.211576i	$-0.0678595\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−23.6832	−0.877154
$28$	0	0
$29$	− 41.6971i	− 1.43783i	−0.695097	−	0.718916i	$-0.744639\pi$
$29$	− 41.6971i	− 1.43783i	0.695097	−	0.718916i	$-0.255361\pi$
$30$	0	0
$31$	22.0104i	0.710013i	0.934864	+	0.355007i	$0.115521\pi$
$31$	22.0104i	0.710013i	−0.934864	+	0.355007i	$0.884479\pi$
$32$	0	0
$33$	−2.26154	−0.0685315
$34$	0	0
$35$	0	0
$36$	0	0
$37$	51.7449i	1.39851i	0.714872	+	0.699256i	$0.246485\pi$
$37$	51.7449i	1.39851i	−0.714872	+	0.699256i	$0.753515\pi$
$38$	0	0
$39$	− 19.6867i	− 0.504787i
$40$	0	0
$41$	15.0557	0.367213	0.183606	−	0.983000i	$-0.441223\pi$
$41$	15.0557	0.367213	0.183606	+	0.983000i	$0.441223\pi$
$42$	0	0
$43$	−9.31310	−0.216584	−0.108292	−	0.994119i	$-0.534538\pi$
$43$	−9.31310	−0.216584	−0.108292	+	0.994119i	$0.534538\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.76226i	0.186431i	0.995646	+	0.0932156i	$0.0297146\pi$
$47$	8.76226i	0.186431i	−0.995646	+	0.0932156i	$0.970285\pi$
$48$	0	0
$49$	−9.94427	−0.202944
$50$	0	0
$51$	10.8328	0.212408
$52$	0	0
$53$	39.9002i	0.752834i	0.926450	+	0.376417i	$0.122844\pi$
$53$	39.9002i	0.752834i	−0.926450	+	0.376417i	$0.877156\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−126.268	−2.21523
$58$	0	0
$59$	−77.1935	−1.30836	−0.654182	−	0.756337i	$-0.726987\pi$
$59$	−77.1935	−1.30836	−0.654182	+	0.756337i	$0.726987\pi$
$60$	0	0
$61$	14.7650i	0.242050i	0.992649	+	0.121025i	$0.0386181\pi$
$61$	14.7650i	0.242050i	−0.992649	+	0.121025i	$0.961382\pi$
$62$	0	0
$63$	107.057i	1.69932i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−75.8395	−1.13193	−0.565966	−	0.824428i	$-0.691497\pi$
$67$	−75.8395	−1.13193	−0.565966	+	0.824428i	$0.691497\pi$
$68$	0	0
$69$	46.6188i	0.675635i
$70$	0	0
$71$	81.0705i	1.14184i	0.821006	+	0.570919i	$0.193413\pi$
$71$	81.0705i	1.14184i	−0.821006	+	0.570919i	$0.806587\pi$
$72$	0	0
$73$	−83.4249	−1.14281	−0.571403	−	0.820669i	$-0.693601\pi$
$73$	−83.4249	−1.14281	−0.571403	+	0.820669i	$0.693601\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.62483i	0.0470757i
$78$	0	0
$79$	100.757i	1.27541i	0.770281	+	0.637704i	$0.220116\pi$
$79$	100.757i	1.27541i	−0.770281	+	0.637704i	$0.779884\pi$
$80$	0	0
$81$	−12.0557	−0.148836
$82$	0	0
$83$	0.266939	0.00321613	0.00160806	−	0.999999i	$-0.499488\pi$
$83$	0.266939	0.00321613	0.00160806	+	0.999999i	$0.499488\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	199.730i	2.29575i
$88$	0	0
$89$	−85.5542	−0.961283	−0.480641	−	0.876917i	$-0.659596\pi$
$89$	−85.5542	−0.961283	−0.480641	+	0.876917i	$0.659596\pi$
$90$	0	0
$91$	−31.5542	−0.346749
$92$	0	0
$93$	− 105.430i	− 1.13366i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	99.1297	1.02196	0.510978	−	0.859594i	$-0.329284\pi$
$97$	99.1297	1.02196	0.510978	+	0.859594i	$0.329284\pi$
$98$	0	0
$99$	6.58359	0.0665009
$100$	0	0
$101$	105.405i	1.04361i	0.853065	+	0.521805i	$0.174741\pi$
$101$	105.405i	1.04361i	−0.853065	+	0.521805i	$0.825259\pi$
$102$	0	0
$103$	− 101.863i	− 0.988958i	−0.869190	−	0.494479i	$-0.835359\pi$
$103$	− 101.863i	− 0.988958i	0.869190	−	0.494479i	$-0.164641\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−32.9962	−0.308376	−0.154188	−	0.988042i	$-0.549276\pi$
$107$	−32.9962	−0.308376	−0.154188	+	0.988042i	$0.549276\pi$
$108$	0	0
$109$	181.554i	1.66563i	0.553552	+	0.832814i	$0.313272\pi$
$109$	181.554i	1.66563i	−0.553552	+	0.832814i	$0.686728\pi$
$110$	0	0
$111$	− 247.859i	− 2.23296i
$112$	0	0
$113$	76.6403	0.678233	0.339116	−	0.940744i	$-0.389872\pi$
$113$	76.6403	0.678233	0.339116	+	0.940744i	$0.389872\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	57.3102i	0.489831i
$118$	0	0
$119$	− 17.3630i	− 0.145908i
$120$	0	0
$121$	−120.777	−0.998158
$122$	0	0
$123$	−72.1172	−0.586319
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 142.962i	− 1.12569i	−0.826564	−	0.562843i	$-0.809708\pi$
$127$	− 142.962i	− 1.12569i	0.826564	−	0.562843i	$-0.190292\pi$
$128$	0	0
$129$	44.6099	0.345813
$130$	0	0
$131$	−110.138	−0.840746	−0.420373	−	0.907351i	$-0.638101\pi$
$131$	−110.138	−0.840746	−0.420373	+	0.907351i	$0.638101\pi$
$132$	0	0
$133$	202.385i	1.52169i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−225.398	−1.64524	−0.822620	−	0.568592i	$-0.807488\pi$
$137$	−225.398	−1.64524	−0.822620	+	0.568592i	$0.807488\pi$
$138$	0	0
$139$	97.4164	0.700837	0.350419	−	0.936593i	$-0.386039\pi$
$139$	97.4164	0.700837	0.350419	+	0.936593i	$0.386039\pi$
$140$	0	0
$141$	− 41.9714i	− 0.297669i
$142$	0	0
$143$	1.94045i	0.0135696i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	47.6332	0.324036
$148$	0	0
$149$	− 191.397i	− 1.28454i	−0.766477	−	0.642271i	$-0.777992\pi$
$149$	− 191.397i	− 1.28454i	0.766477	−	0.642271i	$-0.222008\pi$
$150$	0	0
$151$	− 68.3549i	− 0.452682i	−0.974048	−	0.226341i	$-0.927324\pi$
$151$	− 68.3549i	− 0.452682i	0.974048	−	0.226341i	$-0.0726763\pi$
$152$	0	0
$153$	−31.5355	−0.206115
$154$	0	0
$155$	0	0
$156$	0	0
$157$	155.235i	0.988756i	0.869247	+	0.494378i	$0.164604\pi$
$157$	155.235i	0.988756i	−0.869247	+	0.494378i	$0.835396\pi$
$158$	0	0
$159$	− 191.123i	− 1.20203i
$160$	0	0
$161$	74.7214	0.464108
$162$	0	0
$163$	−90.4765	−0.555070	−0.277535	−	0.960715i	$-0.589518\pi$
$163$	−90.4765	−0.555070	−0.277535	+	0.960715i	$0.589518\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 100.778i	− 0.603461i	−0.953393	−	0.301730i	$-0.902436\pi$
$167$	− 100.778i	− 0.603461i	0.953393	−	0.301730i	$-0.0975642\pi$
$168$	0	0
$169$	152.108	0.900049
$170$	0	0
$171$	367.580	2.14959
$172$	0	0
$173$	54.6556i	0.315928i	0.987445	+	0.157964i	$0.0504930\pi$
$173$	54.6556i	0.315928i	−0.987445	+	0.157964i	$0.949507\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	369.758	2.08903
$178$	0	0
$179$	−106.807	−0.596684	−0.298342	−	0.954459i	$-0.596434\pi$
$179$	−106.807	−0.596684	−0.298342	+	0.954459i	$0.596434\pi$
$180$	0	0
$181$	144.778i	0.799879i	0.916542	+	0.399939i	$0.130969\pi$
$181$	144.778i	0.799879i	−0.916542	+	0.399939i	$0.869031\pi$
$182$	0	0
$183$	− 70.7248i	− 0.386474i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.06775	−0.00570992
$188$	0	0
$189$	− 181.828i	− 0.962052i
$190$	0	0
$191$	210.809i	1.10371i	0.833939	+	0.551857i	$0.186081\pi$
$191$	210.809i	1.10371i	−0.833939	+	0.551857i	$0.813919\pi$
$192$	0	0
$193$	108.176	0.560496	0.280248	−	0.959928i	$-0.409583\pi$
$193$	108.176	0.560496	0.280248	+	0.959928i	$0.409583\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 308.300i	− 1.56497i	−0.622667	−	0.782487i	$-0.713951\pi$
$197$	− 308.300i	− 1.56497i	0.622667	−	0.782487i	$-0.286049\pi$
$198$	0	0
$199$	− 247.859i	− 1.24552i	−0.782412	−	0.622761i	$-0.786011\pi$
$199$	− 247.859i	− 1.24552i	0.782412	−	0.622761i	$-0.213989\pi$
$200$	0	0
$201$	363.272	1.80733
$202$	0	0
$203$	320.130	1.57700
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 135.712i	− 0.655616i
$208$	0	0
$209$	12.4458	0.0595494
$210$	0	0
$211$	−230.584	−1.09281	−0.546407	−	0.837520i	$-0.684005\pi$
$211$	−230.584	−1.09281	−0.546407	+	0.837520i	$0.684005\pi$
$212$	0	0
$213$	− 388.329i	− 1.82314i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−168.985	−0.778734
$218$	0	0
$219$	399.607	1.82469
$220$	0	0
$221$	− 9.29480i	− 0.0420579i
$222$	0	0
$223$	177.782i	0.797229i	0.917118	+	0.398615i	$0.130509\pi$
$223$	177.782i	0.797229i	−0.917118	+	0.398615i	$0.869491\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	203.301	0.895601	0.447800	−	0.894134i	$-0.352208\pi$
$227$	203.301	0.895601	0.447800	+	0.894134i	$0.352208\pi$
$228$	0	0
$229$	81.0705i	0.354020i	0.984209	+	0.177010i	$0.0566425\pi$
$229$	81.0705i	0.354020i	−0.984209	+	0.177010i	$0.943358\pi$
$230$	0	0
$231$	− 17.3630i	− 0.0751645i
$232$	0	0
$233$	−56.2864	−0.241573	−0.120786	−	0.992679i	$-0.538542\pi$
$233$	−56.2864	−0.241573	−0.120786	+	0.992679i	$0.538542\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 482.629i	− 2.03641i
$238$	0	0
$239$	360.235i	1.50726i	0.657300	+	0.753629i	$0.271699\pi$
$239$	360.235i	1.50726i	−0.657300	+	0.753629i	$0.728301\pi$
$240$	0	0
$241$	−197.495	−0.819483	−0.409741	−	0.912202i	$-0.634381\pi$
$241$	−197.495	−0.819483	−0.409741	+	0.912202i	$0.634381\pi$
$242$	0	0
$243$	270.896	1.11480
$244$	0	0
$245$	0	0
$246$	0	0
$247$	108.341i	0.438627i
$248$	0	0
$249$	−1.27864	−0.00513510
$250$	0	0
$251$	178.361	0.710600	0.355300	−	0.934752i	$-0.384379\pi$
$251$	178.361	0.710600	0.355300	+	0.934752i	$0.384379\pi$
$252$	0	0
$253$	− 4.59506i	− 0.0181623i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	453.183	1.76336	0.881679	−	0.471849i	$-0.156413\pi$
$257$	453.183	1.76336	0.881679	+	0.471849i	$0.156413\pi$
$258$	0	0
$259$	−397.272	−1.53387
$260$	0	0
$261$	− 581.436i	− 2.22772i
$262$	0	0
$263$	21.0921i	0.0801981i	0.999196	+	0.0400990i	$0.0127673\pi$
$263$	21.0921i	0.0801981i	−0.999196	+	0.0400990i	$0.987233\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	409.806	1.53485
$268$	0	0
$269$	− 92.4148i	− 0.343549i	−0.985136	−	0.171775i	$-0.945050\pi$
$269$	− 92.4148i	− 0.343549i	0.985136	−	0.171775i	$-0.0549501\pi$
$270$	0	0
$271$	188.799i	0.696675i	0.937369	+	0.348337i	$0.113254\pi$
$271$	188.799i	0.696675i	−0.937369	+	0.348337i	$0.886746\pi$
$272$	0	0
$273$	151.145	0.553645
$274$	0	0
$275$	0	0
$276$	0	0
$277$	70.1251i	0.253159i	0.991956	+	0.126580i	$0.0403999\pi$
$277$	70.1251i	0.253159i	−0.991956	+	0.126580i	$0.959600\pi$
$278$	0	0
$279$	306.919i	1.10007i
$280$	0	0
$281$	−506.269	−1.80167	−0.900835	−	0.434161i	$-0.857045\pi$
$281$	−506.269	−1.80167	−0.900835	+	0.434161i	$0.857045\pi$
$282$	0	0
$283$	−501.350	−1.77156	−0.885778	−	0.464110i	$-0.846374\pi$
$283$	−501.350	−1.77156	−0.885778	+	0.464110i	$0.846374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	115.591i	0.402755i
$288$	0	0
$289$	−283.885	−0.982303
$290$	0	0
$291$	−474.833	−1.63173
$292$	0	0
$293$	− 324.026i	− 1.10589i	−0.833218	−	0.552945i	$-0.813504\pi$
$293$	− 324.026i	− 1.10589i	0.833218	−	0.552945i	$-0.186496\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−11.1817	−0.0376487
$298$	0	0
$299$	40.0000	0.133779
$300$	0	0
$301$	− 71.5015i	− 0.237546i
$302$	0	0
$303$	− 504.890i	− 1.66630i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.84697	0.0320748	0.0160374	−	0.999871i	$-0.494895\pi$
$307$	9.84697	0.0320748	0.0160374	+	0.999871i	$0.494895\pi$
$308$	0	0
$309$	487.924i	1.57904i
$310$	0	0
$311$	200.417i	0.644429i	0.946667	+	0.322214i	$0.104427\pi$
$311$	200.417i	0.644429i	−0.946667	+	0.322214i	$0.895573\pi$
$312$	0	0
$313$	−261.582	−0.835727	−0.417863	−	0.908510i	$-0.637221\pi$
$313$	−261.582	−0.835727	−0.417863	+	0.908510i	$0.637221\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	322.341i	1.01685i	0.861107	+	0.508425i	$0.169772\pi$
$317$	322.341i	1.01685i	−0.861107	+	0.508425i	$0.830228\pi$
$318$	0	0
$319$	− 19.6867i	− 0.0617138i
$320$	0	0
$321$	158.053	0.492376
$322$	0	0
$323$	−59.6157	−0.184569
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 869.645i	− 2.65946i
$328$	0	0
$329$	−67.2724	−0.204475
$330$	0	0
$331$	61.0883	0.184557	0.0922783	−	0.995733i	$-0.470585\pi$
$331$	61.0883	0.184557	0.0922783	+	0.995733i	$0.470585\pi$
$332$	0	0
$333$	721.545i	2.16680i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	114.960	0.341129	0.170564	−	0.985347i	$-0.445441\pi$
$337$	114.960	0.341129	0.170564	+	0.985347i	$0.445441\pi$
$338$	0	0
$339$	−367.108	−1.08292
$340$	0	0
$341$	10.3919i	0.0304748i
$342$	0	0
$343$	299.851i	0.874201i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−143.967	−0.414892	−0.207446	−	0.978246i	$-0.566515\pi$
$347$	−143.967	−0.414892	−0.207446	+	0.978246i	$0.566515\pi$
$348$	0	0
$349$	35.9526i	0.103016i	0.998673	+	0.0515080i	$0.0164028\pi$
$349$	35.9526i	0.103016i	−0.998673	+	0.0515080i	$0.983597\pi$
$350$	0	0
$351$	− 97.3365i	− 0.277312i
$352$	0	0
$353$	252.536	0.715400	0.357700	−	0.933837i	$-0.383561\pi$
$353$	252.536	0.715400	0.357700	+	0.933837i	$0.383561\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	83.1691i	0.232967i
$358$	0	0
$359$	594.152i	1.65502i	0.561452	+	0.827509i	$0.310243\pi$
$359$	594.152i	1.65502i	−0.561452	+	0.827509i	$0.689757\pi$
$360$	0	0
$361$	333.885	0.924890
$362$	0	0
$363$	578.524	1.59373
$364$	0	0
$365$	0	0
$366$	0	0
$367$	88.9062i	0.242251i	0.992637	+	0.121126i	$0.0386504\pi$
$367$	88.9062i	0.242251i	−0.992637	+	0.121126i	$0.961350\pi$
$368$	0	0
$369$	209.941	0.568946
$370$	0	0
$371$	−306.334	−0.825699
$372$	0	0
$373$	83.6542i	0.224274i	0.993693	+	0.112137i	$0.0357696\pi$
$373$	83.6542i	0.224274i	−0.993693	+	0.112137i	$0.964230\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	171.373	0.454570
$378$	0	0
$379$	−135.135	−0.356556	−0.178278	−	0.983980i	$-0.557053\pi$
$379$	−135.135	−0.356556	−0.178278	+	0.983980i	$0.557053\pi$
$380$	0	0
$381$	684.791i	1.79735i
$382$	0	0
$383$	− 498.526i	− 1.30164i	−0.759234	−	0.650818i	$-0.774426\pi$
$383$	− 498.526i	− 1.30164i	0.759234	−	0.650818i	$-0.225574\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−129.864	−0.335567
$388$	0	0
$389$	− 308.420i	− 0.792854i	−0.918066	−	0.396427i	$-0.870250\pi$
$389$	− 308.420i	− 0.792854i	0.918066	−	0.396427i	$-0.129750\pi$
$390$	0	0
$391$	22.0104i	0.0562926i
$392$	0	0
$393$	527.562	1.34240
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 297.682i	− 0.749828i	−0.927060	−	0.374914i	$-0.877672\pi$
$397$	− 297.682i	− 0.749828i	0.927060	−	0.374914i	$-0.122328\pi$
$398$	0	0
$399$	− 969.426i	− 2.42964i
$400$	0	0
$401$	148.663	0.370729	0.185365	−	0.982670i	$-0.440653\pi$
$401$	148.663	0.370729	0.185365	+	0.982670i	$0.440653\pi$
$402$	0	0
$403$	−90.4616	−0.224470
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.4306i	0.0600261i
$408$	0	0
$409$	−442.387	−1.08163	−0.540815	−	0.841141i	$-0.681884\pi$
$409$	−442.387	−1.08163	−0.540815	+	0.841141i	$0.681884\pi$
$410$	0	0
$411$	1079.66	2.62691
$412$	0	0
$413$	− 592.654i	− 1.43500i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−466.626	−1.11901
$418$	0	0
$419$	536.184	1.27968	0.639838	−	0.768510i	$-0.279001\pi$
$419$	536.184	1.27968	0.639838	+	0.768510i	$0.279001\pi$
$420$	0	0
$421$	514.582i	1.22228i	0.791521	+	0.611142i	$0.209290\pi$
$421$	514.582i	1.22228i	−0.791521	+	0.611142i	$0.790710\pi$
$422$	0	0
$423$	122.183i	0.288850i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−113.359	−0.265477
$428$	0	0
$429$	− 9.29480i	− 0.0216662i
$430$	0	0
$431$	86.8151i	0.201427i	0.994915	+	0.100714i	$0.0321126\pi$
$431$	86.8151i	0.201427i	−0.994915	+	0.100714i	$0.967887\pi$
$432$	0	0
$433$	−494.017	−1.14092	−0.570458	−	0.821327i	$-0.693234\pi$
$433$	−494.017	−1.14092	−0.570458	+	0.821327i	$0.693234\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 256.555i	− 0.587082i
$438$	0	0
$439$	− 524.700i	− 1.19522i	−0.801789	−	0.597608i	$-0.796118\pi$
$439$	− 524.700i	− 1.19522i	0.801789	−	0.597608i	$-0.203882\pi$
$440$	0	0
$441$	−138.666	−0.314435
$442$	0	0
$443$	−496.575	−1.12094	−0.560469	−	0.828176i	$-0.689379\pi$
$443$	−496.575	−1.12094	−0.560469	+	0.828176i	$0.689379\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	916.795i	2.05099i
$448$	0	0
$449$	332.158	0.739773	0.369886	−	0.929077i	$-0.379397\pi$
$449$	332.158	0.739773	0.369886	+	0.929077i	$0.379397\pi$
$450$	0	0
$451$	7.10835	0.0157613
$452$	0	0
$453$	327.421i	0.722785i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−252.788	−0.553147	−0.276574	−	0.960993i	$-0.589199\pi$
$457$	−252.788	−0.553147	−0.276574	+	0.960993i	$0.589199\pi$
$458$	0	0
$459$	53.5604	0.116689
$460$	0	0
$461$	− 832.716i	− 1.80633i	−0.429299	−	0.903163i	$-0.641239\pi$
$461$	− 832.716i	− 1.80633i	0.429299	−	0.903163i	$-0.358761\pi$
$462$	0	0
$463$	854.205i	1.84494i	0.386074	+	0.922468i	$0.373831\pi$
$463$	854.205i	1.84494i	−0.386074	+	0.922468i	$0.626169\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	102.726	0.219970	0.109985	−	0.993933i	$-0.464920\pi$
$467$	102.726	0.219970	0.109985	+	0.993933i	$0.464920\pi$
$468$	0	0
$469$	− 582.259i	− 1.24149i
$470$	0	0
$471$	− 743.577i	− 1.57872i
$472$	0	0
$473$	−4.39705	−0.00929608
$474$	0	0
$475$	0	0
$476$	0	0
$477$	556.379i	1.16641i
$478$	0	0
$479$	268.772i	0.561111i	0.959838	+	0.280556i	$0.0905187\pi$
$479$	268.772i	0.561111i	−0.959838	+	0.280556i	$0.909481\pi$
$480$	0	0
$481$	−212.669	−0.442139
$482$	0	0
$483$	−357.917	−0.741028
$484$	0	0
$485$	0	0
$486$	0	0
$487$	504.065i	1.03504i	0.855671	+	0.517520i	$0.173145\pi$
$487$	504.065i	1.03504i	−0.855671	+	0.517520i	$0.826855\pi$
$488$	0	0
$489$	433.384	0.886266
$490$	0	0
$491$	−130.020	−0.264807	−0.132403	−	0.991196i	$-0.542269\pi$
$491$	−130.020	−0.264807	−0.132403	+	0.991196i	$0.542269\pi$
$492$	0	0
$493$	94.2997i	0.191277i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−622.421	−1.25236
$498$	0	0
$499$	664.184	1.33103	0.665515	−	0.746384i	$-0.268212\pi$
$499$	664.184	1.33103	0.665515	+	0.746384i	$0.268212\pi$
$500$	0	0
$501$	482.728i	0.963529i
$502$	0	0
$503$	550.015i	1.09347i	0.837306	+	0.546735i	$0.184129\pi$
$503$	550.015i	1.09347i	−0.837306	+	0.546735i	$0.815871\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−728.602	−1.43708
$508$	0	0
$509$	349.843i	0.687314i	0.939095	+	0.343657i	$0.111666\pi$
$509$	349.843i	0.687314i	−0.939095	+	0.343657i	$0.888334\pi$
$510$	0	0
$511$	− 640.496i	− 1.25342i
$512$	0	0
$513$	−624.304	−1.21697
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.13698i	0.00800189i
$518$	0	0
$519$	− 261.801i	− 0.504434i
$520$	0	0
$521$	29.7771	0.0571537	0.0285769	−	0.999592i	$-0.490902\pi$
$521$	29.7771	0.0571537	0.0285769	+	0.999592i	$0.490902\pi$
$522$	0	0
$523$	603.023	1.15301	0.576504	−	0.817094i	$-0.304416\pi$
$523$	603.023	1.15301	0.576504	+	0.817094i	$0.304416\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 49.7774i	− 0.0944543i
$528$	0	0
$529$	434.279	0.820943
$530$	0	0
$531$	−1076.41	−2.02713
$532$	0	0
$533$	61.8782i	0.116094i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	511.605	0.952710
$538$	0	0
$539$	−4.69505	−0.00871066
$540$	0	0
$541$	− 163.368i	− 0.301974i	−0.988536	−	0.150987i	$-0.951755\pi$
$541$	− 163.368i	− 0.301974i	0.988536	−	0.150987i	$-0.0482451\pi$
$542$	0	0
$543$	− 693.490i	− 1.27714i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−524.218	−0.958350	−0.479175	−	0.877719i	$-0.659064\pi$
$547$	−524.218	−0.958350	−0.479175	+	0.877719i	$0.659064\pi$
$548$	0	0
$549$	205.888i	0.375023i
$550$	0	0
$551$	− 1099.16i	− 1.99485i
$552$	0	0
$553$	−773.566	−1.39885
$554$	0	0
$555$	0	0
$556$	0	0
$557$	589.515i	1.05837i	0.848505	+	0.529187i	$0.177503\pi$
$557$	589.515i	1.05837i	−0.848505	+	0.529187i	$0.822497\pi$
$558$	0	0
$559$	− 38.2763i	− 0.0684728i
$560$	0	0
$561$	5.11456	0.00911687
$562$	0	0
$563$	557.763	0.990698	0.495349	−	0.868694i	$-0.335040\pi$
$563$	557.763	0.990698	0.495349	+	0.868694i	$0.335040\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 92.5581i	− 0.163242i
$568$	0	0
$569$	542.715	0.953805	0.476903	−	0.878956i	$-0.341759\pi$
$569$	542.715	0.953805	0.476903	+	0.878956i	$0.341759\pi$
$570$	0	0
$571$	476.695	0.834842	0.417421	−	0.908713i	$-0.362934\pi$
$571$	476.695	0.834842	0.417421	+	0.908713i	$0.362934\pi$
$572$	0	0
$573$	− 1009.78i	− 1.76227i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−865.659	−1.50027	−0.750137	−	0.661282i	$-0.770013\pi$
$577$	−865.659	−1.50027	−0.750137	+	0.661282i	$0.770013\pi$
$578$	0	0
$579$	−518.164	−0.894929
$580$	0	0
$581$	2.04943i	0.00352741i
$582$	0	0
$583$	18.8383i	0.0323127i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	567.091	0.966083	0.483041	−	0.875597i	$-0.339532\pi$
$587$	567.091	0.966083	0.483041	+	0.875597i	$0.339532\pi$
$588$	0	0
$589$	580.209i	0.985075i
$590$	0	0
$591$	1476.76i	2.49875i
$592$	0	0
$593$	421.774	0.711254	0.355627	−	0.934628i	$-0.384267\pi$
$593$	421.774	0.711254	0.355627	+	0.934628i	$0.384267\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1187.25i	1.98869i
$598$	0	0
$599$	120.444i	0.201075i	0.994933	+	0.100538i	$0.0320563\pi$
$599$	120.444i	0.201075i	−0.994933	+	0.100538i	$0.967944\pi$
$600$	0	0
$601$	340.158	0.565986	0.282993	−	0.959122i	$-0.408673\pi$
$601$	340.158	0.565986	0.282993	+	0.959122i	$0.408673\pi$
$602$	0	0
$603$	−1057.53	−1.75377
$604$	0	0
$605$	0	0
$606$	0	0
$607$	115.621i	0.190479i	0.995454	+	0.0952396i	$0.0303617\pi$
$607$	115.621i	0.190479i	−0.995454	+	0.0952396i	$0.969638\pi$
$608$	0	0
$609$	−1533.43	−2.51795
$610$	0	0
$611$	−36.0124	−0.0589401
$612$	0	0
$613$	1095.63i	1.78733i	0.448738	+	0.893663i	$0.351874\pi$
$613$	1095.63i	1.78733i	−0.448738	+	0.893663i	$0.648126\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	716.775	1.16171	0.580855	−	0.814007i	$-0.302718\pi$
$617$	716.775	1.16171	0.580855	+	0.814007i	$0.302718\pi$
$618$	0	0
$619$	118.584	0.191573	0.0957864	−	0.995402i	$-0.469463\pi$
$619$	118.584	0.191573	0.0957864	+	0.995402i	$0.469463\pi$
$620$	0	0
$621$	230.496i	0.371169i
$622$	0	0
$623$	− 656.844i	− 1.05432i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−59.6157	−0.0950809
$628$	0	0
$629$	− 117.023i	− 0.186046i
$630$	0	0
$631$	1168.62i	1.85201i	0.377515	+	0.926004i	$0.376779\pi$
$631$	1168.62i	1.85201i	−0.377515	+	0.926004i	$0.623221\pi$
$632$	0	0
$633$	1104.50	1.74486
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 40.8704i	− 0.0641608i
$638$	0	0
$639$	1130.47i	1.76912i
$640$	0	0
$641$	−244.158	−0.380902	−0.190451	−	0.981697i	$-0.560995\pi$
$641$	−244.158	−0.380902	−0.190451	+	0.981697i	$0.560995\pi$
$642$	0	0
$643$	−868.847	−1.35124	−0.675620	−	0.737250i	$-0.736124\pi$
$643$	−868.847	−1.35124	−0.675620	+	0.737250i	$0.736124\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 35.1333i	− 0.0543019i	−0.999631	−	0.0271510i	$-0.991357\pi$
$647$	− 35.1333i	− 0.0543019i	0.999631	−	0.0271510i	$-0.00864348\pi$
$648$	0	0
$649$	−36.4458	−0.0561569
$650$	0	0
$651$	809.443	1.24338
$652$	0	0
$653$	580.068i	0.888313i	0.895949	+	0.444157i	$0.146497\pi$
$653$	580.068i	0.888313i	−0.895949	+	0.444157i	$0.853503\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1163.30	−1.77062
$658$	0	0
$659$	−992.788	−1.50651	−0.753253	−	0.657731i	$-0.771517\pi$
$659$	−992.788	−1.50651	−0.753253	+	0.657731i	$0.771517\pi$
$660$	0	0
$661$	− 719.647i	− 1.08872i	−0.838850	−	0.544362i	$-0.816772\pi$
$661$	− 719.647i	− 1.08872i	0.838850	−	0.544362i	$-0.183228\pi$
$662$	0	0
$663$	44.5223i	0.0671528i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−405.817	−0.608421
$668$	0	0
$669$	− 851.580i	− 1.27291i
$670$	0	0
$671$	6.97110i	0.0103891i
$672$	0	0
$673$	−189.591	−0.281711	−0.140855	−	0.990030i	$-0.544985\pi$
$673$	−189.591	−0.281711	−0.140855	+	0.990030i	$0.544985\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 327.192i	− 0.483297i	−0.970364	−	0.241649i	$-0.922312\pi$
$677$	− 327.192i	− 0.483297i	0.970364	−	0.241649i	$-0.0776882\pi$
$678$	0	0
$679$	761.070i	1.12087i
$680$	0	0
$681$	−973.817	−1.42998
$682$	0	0
$683$	673.539	0.986148	0.493074	−	0.869987i	$-0.335873\pi$
$683$	673.539	0.986148	0.493074	+	0.869987i	$0.335873\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 388.329i	− 0.565254i
$688$	0	0
$689$	−163.988	−0.238008
$690$	0	0
$691$	−237.036	−0.343033	−0.171516	−	0.985181i	$-0.554867\pi$
$691$	−237.036	−0.343033	−0.171516	+	0.985181i	$0.554867\pi$
$692$	0	0
$693$	50.5456i	0.0729374i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−34.0491	−0.0488510
$698$	0	0
$699$	269.613	0.385712
$700$	0	0
$701$	476.306i	0.679466i	0.940522	+	0.339733i	$0.110337\pi$
$701$	476.306i	0.679466i	−0.940522	+	0.339733i	$0.889663\pi$
$702$	0	0
$703$	1364.03i	1.94030i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−809.246	−1.14462
$708$	0	0
$709$	964.778i	1.36076i	0.732860	+	0.680380i	$0.238185\pi$
$709$	964.778i	1.36076i	−0.732860	+	0.680380i	$0.761815\pi$
$710$	0	0
$711$	1404.99i	1.97607i
$712$	0	0
$713$	214.216	0.300443
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1725.53i	− 2.40660i
$718$	0	0
$719$	− 899.974i	− 1.25170i	−0.779943	−	0.625851i	$-0.784752\pi$
$719$	− 899.974i	− 1.25170i	0.779943	−	0.625851i	$-0.215248\pi$
$720$	0	0
$721$	782.053	1.08468
$722$	0	0
$723$	946.006	1.30845
$724$	0	0
$725$	0	0
$726$	0	0
$727$	908.888i	1.25019i	0.780549	+	0.625095i	$0.214940\pi$
$727$	908.888i	1.25019i	−0.780549	+	0.625095i	$0.785060\pi$
$728$	0	0
$729$	−1189.09	−1.63113
$730$	0	0
$731$	21.0619	0.0288125
$732$	0	0
$733$	433.283i	0.591109i	0.955326	+	0.295554i	$0.0955043\pi$
$733$	433.283i	0.591109i	−0.955326	+	0.295554i	$0.904496\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−35.8065	−0.0485842
$738$	0	0
$739$	−886.138	−1.19910	−0.599552	−	0.800336i	$-0.704655\pi$
$739$	−886.138	−1.19910	−0.599552	+	0.800336i	$0.704655\pi$
$740$	0	0
$741$	− 518.955i	− 0.700344i
$742$	0	0
$743$	− 895.305i	− 1.20499i	−0.798124	−	0.602493i	$-0.794174\pi$
$743$	− 895.305i	− 1.20499i	0.798124	−	0.602493i	$-0.205826\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.72226	0.00498295
$748$	0	0
$749$	− 253.329i	− 0.338223i
$750$	0	0
$751$	− 1297.13i	− 1.72720i	−0.504176	−	0.863601i	$-0.668204\pi$
$751$	− 1297.13i	− 1.72720i	0.504176	−	0.863601i	$-0.331796\pi$
$752$	0	0
$753$	−854.351	−1.13460
$754$	0	0
$755$	0	0
$756$	0	0
$757$	421.694i	0.557060i	0.960428	+	0.278530i	$0.0898471\pi$
$757$	421.694i	0.557060i	−0.960428	+	0.278530i	$0.910153\pi$
$758$	0	0
$759$	22.0104i	0.0289992i
$760$	0	0
$761$	1415.76	1.86039	0.930196	−	0.367063i	$-0.119637\pi$
$761$	1415.76	1.86039	0.930196	+	0.367063i	$0.119637\pi$
$762$	0	0
$763$	−1393.88	−1.82684
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 317.261i	− 0.413639i
$768$	0	0
$769$	414.210	0.538635	0.269318	−	0.963051i	$-0.413202\pi$
$769$	414.210	0.538635	0.269318	+	0.963051i	$0.413202\pi$
$770$	0	0
$771$	−2170.76	−2.81551
$772$	0	0
$773$	− 727.056i	− 0.940564i	−0.882516	−	0.470282i	$-0.844152\pi$
$773$	− 727.056i	− 0.940564i	0.882516	−	0.470282i	$-0.155848\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1902.94	2.44909
$778$	0	0
$779$	396.879	0.509473
$780$	0	0
$781$	38.2763i	0.0490094i
$782$	0	0
$783$	987.519i	1.26120i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1030.11	1.30890	0.654451	−	0.756104i	$-0.272900\pi$
$787$	1030.11	1.30890	0.654451	+	0.756104i	$0.272900\pi$
$788$	0	0
$789$	− 101.032i	− 0.128050i
$790$	0	0
$791$	588.407i	0.743878i
$792$	0	0
$793$	−60.6835	−0.0765239
$794$	0	0
$795$	0	0
$796$	0	0
$797$	371.202i	0.465750i	0.972507	+	0.232875i	$0.0748132\pi$
$797$	371.202i	0.465750i	−0.972507	+	0.232875i	$0.925187\pi$
$798$	0	0
$799$	− 19.8162i	− 0.0248013i
$800$	0	0
$801$	−1192.99	−1.48938
$802$	0	0
$803$	−39.3879	−0.0490509
$804$	0	0
$805$	0	0
$806$	0	0
$807$	442.668i	0.548536i
$808$	0	0
$809$	−77.7771	−0.0961398	−0.0480699	−	0.998844i	$-0.515307\pi$
$809$	−77.7771	−0.0961398	−0.0480699	+	0.998844i	$0.515307\pi$
$810$	0	0
$811$	1407.90	1.73600	0.868000	−	0.496564i	$-0.165405\pi$
$811$	1407.90	1.73600	0.868000	+	0.496564i	$0.165405\pi$
$812$	0	0
$813$	− 904.350i	− 1.11236i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−245.500	−0.300489
$818$	0	0
$819$	−440.000	−0.537241
$820$	0	0
$821$	− 820.275i	− 0.999116i	−0.866280	−	0.499558i	$-0.833496\pi$
$821$	− 820.275i	− 0.999116i	0.866280	−	0.499558i	$-0.166504\pi$
$822$	0	0
$823$	− 1385.41i	− 1.68336i	−0.539976	−	0.841681i	$-0.681567\pi$
$823$	− 1385.41i	− 1.68336i	0.539976	−	0.841681i	$-0.318433\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	258.898	0.313057	0.156529	−	0.987673i	$-0.449970\pi$
$827$	258.898	0.313057	0.156529	+	0.987673i	$0.449970\pi$
$828$	0	0
$829$	− 1299.31i	− 1.56732i	−0.621190	−	0.783660i	$-0.713351\pi$
$829$	− 1299.31i	− 1.56732i	0.621190	−	0.783660i	$-0.286649\pi$
$830$	0	0
$831$	− 335.901i	− 0.404213i
$832$	0	0
$833$	22.4894	0.0269980
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 521.276i	− 0.622791i
$838$	0	0
$839$	− 804.961i	− 0.959429i	−0.877425	−	0.479715i	$-0.840740\pi$
$839$	− 804.961i	− 0.959429i	0.877425	−	0.479715i	$-0.159260\pi$
$840$	0	0
$841$	−897.650	−1.06736
$842$	0	0
$843$	2425.04	2.87668
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 927.268i	− 1.09477i
$848$	0	0
$849$	2401.48	2.82859
$850$	0	0
$851$	503.607	0.591782
$852$	0	0
$853$	− 470.811i	− 0.551948i	−0.961165	−	0.275974i	$-0.911000\pi$
$853$	− 470.811i	− 0.551948i	0.961165	−	0.275974i	$-0.0890003\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	266.105	0.310508	0.155254	−	0.987875i	$-0.450380\pi$
$857$	266.105	0.310508	0.155254	+	0.987875i	$0.450380\pi$
$858$	0	0
$859$	1442.02	1.67872	0.839360	−	0.543576i	$-0.182930\pi$
$859$	1442.02	1.67872	0.839360	+	0.543576i	$0.182930\pi$
$860$	0	0
$861$	− 553.681i	− 0.643067i
$862$	0	0
$863$	− 364.495i	− 0.422358i	−0.977447	−	0.211179i	$-0.932270\pi$
$863$	− 364.495i	− 0.422358i	0.977447	−	0.211179i	$-0.0677303\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1359.82	1.56842
$868$	0	0
$869$	47.5711i	0.0547424i
$870$	0	0
$871$	− 311.696i	− 0.357860i
$872$	0	0
$873$	1382.29	1.58338
$874$	0	0
$875$	0	0
$876$	0	0
$877$	77.4289i	0.0882883i	0.999025	+	0.0441442i	$0.0140561\pi$
$877$	77.4289i	0.0882883i	−0.999025	+	0.0441442i	$0.985944\pi$
$878$	0	0
$879$	1552.09i	1.76574i
$880$	0	0
$881$	724.932	0.822851	0.411426	−	0.911443i	$-0.365031\pi$
$881$	724.932	0.822851	0.411426	+	0.911443i	$0.365031\pi$
$882$	0	0
$883$	−493.342	−0.558711	−0.279356	−	0.960188i	$-0.590121\pi$
$883$	−493.342	−0.558711	−0.279356	+	0.960188i	$0.590121\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1514.47i	− 1.70741i	−0.520758	−	0.853705i	$-0.674351\pi$
$887$	− 1514.47i	− 1.70741i	0.520758	−	0.853705i	$-0.325649\pi$
$888$	0	0
$889$	1097.59	1.23464
$890$	0	0
$891$	−5.69194	−0.00638826
$892$	0	0
$893$	230.979i	0.258655i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−191.601	−0.213602
$898$	0	0
$899$	917.771	1.02088
$900$	0	0
$901$	− 90.2359i	− 0.100151i
$902$	0	0
$903$	342.493i	0.379284i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	851.570	0.938887	0.469443	−	0.882963i	$-0.344455\pi$
$907$	851.570	0.938887	0.469443	+	0.882963i	$0.344455\pi$
$908$	0	0
$909$	1469.79i	1.61693i
$910$	0	0
$911$	− 517.728i	− 0.568308i	−0.958779	−	0.284154i	$-0.908287\pi$
$911$	− 517.728i	− 0.568308i	0.958779	−	0.284154i	$-0.0917127\pi$
$912$	0	0
$913$	0.126031	0.000138041 0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 845.585i	− 0.922121i
$918$	0	0
$919$	− 657.730i	− 0.715701i	−0.933779	−	0.357851i	$-0.883510\pi$
$919$	− 657.730i	− 0.715701i	0.933779	−	0.357851i	$-0.116490\pi$
$920$	0	0
$921$	−47.1672	−0.0512130
$922$	0	0
$923$	−333.195	−0.360992
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1420.40i	− 1.53226i
$928$	0	0
$929$	−1206.28	−1.29847	−0.649237	−	0.760586i	$-0.724912\pi$
$929$	−1206.28	−1.29847	−0.649237	+	0.760586i	$0.724912\pi$
$930$	0	0
$931$	−262.138	−0.281566
$932$	0	0
$933$	− 960.003i	− 1.02894i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	216.730	0.231302	0.115651	−	0.993290i	$-0.463105\pi$
$937$	216.730	0.231302	0.115651	+	0.993290i	$0.463105\pi$
$938$	0	0
$939$	1252.98	1.33438
$940$	0	0
$941$	749.322i	0.796304i	0.917320	+	0.398152i	$0.130348\pi$
$941$	749.322i	0.796304i	−0.917320	+	0.398152i	$0.869652\pi$
$942$	0	0
$943$	− 146.530i	− 0.155387i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−114.160	−0.120549	−0.0602743	−	0.998182i	$-0.519198\pi$
$947$	−114.160	−0.120549	−0.0602743	+	0.998182i	$0.519198\pi$
$948$	0	0
$949$	− 342.872i	− 0.361298i
$950$	0	0
$951$	− 1544.02i	− 1.62358i
$952$	0	0
$953$	−820.680	−0.861154	−0.430577	−	0.902554i	$-0.641690\pi$
$953$	−820.680	−0.861154	−0.430577	+	0.902554i	$0.641690\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	94.2997i	0.0985368i
$958$	0	0
$959$	− 1730.50i	− 1.80448i
$960$	0	0
$961$	476.542	0.495881
$962$	0	0
$963$	−460.109	−0.477787
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1108.56i	1.14639i	0.819420	+	0.573194i	$0.194296\pi$
$967$	1108.56i	1.14639i	−0.819420	+	0.573194i	$0.805704\pi$
$968$	0	0
$969$	285.560	0.294696
$970$	0	0
$971$	1372.41	1.41340	0.706698	−	0.707516i	$-0.250184\pi$
$971$	1372.41	1.41340	0.706698	+	0.707516i	$0.250184\pi$
$972$	0	0
$973$	747.916i	0.768670i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	927.224	0.949052	0.474526	−	0.880241i	$-0.342619\pi$
$977$	927.224	0.949052	0.474526	+	0.880241i	$0.342619\pi$
$978$	0	0
$979$	−40.3932	−0.0412597
$980$	0	0
$981$	2531.63i	2.58066i
$982$	0	0
$983$	354.106i	0.360230i	0.983646	+	0.180115i	$0.0576469\pi$
$983$	354.106i	0.360230i	−0.983646	+	0.180115i	$0.942353\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	322.236	0.326480
$988$	0	0
$989$	90.6396i	0.0916478i
$990$	0	0
$991$	537.545i	0.542427i	0.962519	+	0.271213i	$0.0874249\pi$
$991$	537.545i	0.542427i	−0.962519	+	0.271213i	$0.912575\pi$
$992$	0	0
$993$	−292.614	−0.294677
$994$	0	0
$995$	0	0
$996$	0	0
$997$	441.759i	0.443088i	0.975150	+	0.221544i	$0.0711096\pi$
$997$	441.759i	0.443088i	−0.975150	+	0.221544i	$0.928890\pi$
$998$	0	0
$999$	− 1225.48i	− 1.22671i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.3.g.h.751.2		8
4.3	odd	2		200.3.g.h.51.6		8
5.2	odd	4		160.3.e.c.79.7		8
5.3	odd	4		160.3.e.c.79.2		8
5.4	even	2	inner	800.3.g.h.751.7		8
8.3	odd	2	inner	800.3.g.h.751.1		8
8.5	even	2		200.3.g.h.51.5		8
15.2	even	4		1440.3.p.g.559.8		8
15.8	even	4		1440.3.p.g.559.2		8
20.3	even	4		40.3.e.c.19.7	yes	8
20.7	even	4		40.3.e.c.19.2	yes	8
20.19	odd	2		200.3.g.h.51.3		8
40.3	even	4		160.3.e.c.79.1		8
40.13	odd	4		40.3.e.c.19.1	✓	8
40.19	odd	2	inner	800.3.g.h.751.8		8
40.27	even	4		160.3.e.c.79.8		8
40.29	even	2		200.3.g.h.51.4		8
40.37	odd	4		40.3.e.c.19.8	yes	8
60.23	odd	4		360.3.p.g.19.2		8
60.47	odd	4		360.3.p.g.19.7		8
80.3	even	4		1280.3.h.m.1279.1		16
80.13	odd	4		1280.3.h.m.1279.13		16
80.27	even	4		1280.3.h.m.1279.3		16
80.37	odd	4		1280.3.h.m.1279.15		16
80.43	even	4		1280.3.h.m.1279.16		16
80.53	odd	4		1280.3.h.m.1279.4		16
80.67	even	4		1280.3.h.m.1279.14		16
80.77	odd	4		1280.3.h.m.1279.2		16
120.53	even	4		360.3.p.g.19.8		8
120.77	even	4		360.3.p.g.19.1		8
120.83	odd	4		1440.3.p.g.559.7		8
120.107	odd	4		1440.3.p.g.559.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.3.e.c.19.1	✓	8	40.13	odd	4
40.3.e.c.19.2	yes	8	20.7	even	4
40.3.e.c.19.7	yes	8	20.3	even	4
40.3.e.c.19.8	yes	8	40.37	odd	4
160.3.e.c.79.1		8	40.3	even	4
160.3.e.c.79.2		8	5.3	odd	4
160.3.e.c.79.7		8	5.2	odd	4
160.3.e.c.79.8		8	40.27	even	4
200.3.g.h.51.3		8	20.19	odd	2
200.3.g.h.51.4		8	40.29	even	2
200.3.g.h.51.5		8	8.5	even	2
200.3.g.h.51.6		8	4.3	odd	2
360.3.p.g.19.1		8	120.77	even	4
360.3.p.g.19.2		8	60.23	odd	4
360.3.p.g.19.7		8	60.47	odd	4
360.3.p.g.19.8		8	120.53	even	4
800.3.g.h.751.1		8	8.3	odd	2	inner
800.3.g.h.751.2		8	1.1	even	1	trivial
800.3.g.h.751.7		8	5.4	even	2	inner
800.3.g.h.751.8		8	40.19	odd	2	inner
1280.3.h.m.1279.1		16	80.3	even	4
1280.3.h.m.1279.2		16	80.77	odd	4
1280.3.h.m.1279.3		16	80.27	even	4
1280.3.h.m.1279.4		16	80.53	odd	4
1280.3.h.m.1279.13		16	80.13	odd	4
1280.3.h.m.1279.14		16	80.67	even	4
1280.3.h.m.1279.15		16	80.37	odd	4
1280.3.h.m.1279.16		16	80.43	even	4
1440.3.p.g.559.1		8	120.107	odd	4
1440.3.p.g.559.2		8	15.8	even	4
1440.3.p.g.559.7		8	120.83	odd	4
1440.3.p.g.559.8		8	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	800.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.79002 q^{3} +7.67752i q^{7} +13.9443 q^{9} +0.472136 q^{11} +4.10995i q^{13} -2.26154 q^{17} +26.3607 q^{19} -36.7754i q^{21} -9.73249i q^{23} -23.6832 q^{27} -41.6971i q^{29} +22.0104i q^{31} -2.26154 q^{33} +51.7449i q^{37} -19.6867i q^{39} +15.0557 q^{41} -9.31310 q^{43} +8.76226i q^{47} -9.94427 q^{49} +10.8328 q^{51} +39.9002i q^{53} -126.268 q^{57} -77.1935 q^{59} +14.7650i q^{61} +107.057i q^{63} -75.8395 q^{67} +46.6188i q^{69} +81.0705i q^{71} -83.4249 q^{73} +3.62483i q^{77} +100.757i q^{79} -12.0557 q^{81} +0.266939 q^{83} +199.730i q^{87} -85.5542 q^{89} -31.5542 q^{91} -105.430i q^{93} +99.1297 q^{97} +6.58359 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 40 q^{9} - 32 q^{11} + 32 q^{19} + 192 q^{41} - 8 q^{49} - 128 q^{51} - 224 q^{59} - 168 q^{81} - 112 q^{89} + 320 q^{91} + 160 q^{99}+O(q^{100}) \) 8 * q + 40 * q^9 - 32 * q^11 + 32 * q^19 + 192 * q^41 - 8 * q^49 - 128 * q^51 - 224 * q^59 - 168 * q^81 - 112 * q^89 + 320 * q^91 + 160 * q^99

Introduction

L-functions

Modular forms

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