LMFDB - Embedded newform 800.6.d.d.401.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(401,800)]

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, 1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("800.401");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.d (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:	$128.307055850$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} - 130 x^{17} + 144 x^{16} + 1560 x^{15} - 12320 x^{14} - 56128 x^{13} + \cdots + 11\!\cdots\!24 $ x^20 - x^19 - 130x^17 + 144x^16 + 1560x^15 - 12320x^14 - 56128x^13 + 672256x^12 - 4183040x^11 + 25763840x^10 - 133857280x^9 + 688390144x^8 - 1839202304x^7 - 12918456320x^6 + 52344913920x^5 + 154618822656x^4 - 4466765987840x^3 - 35184372088832x + 1125899906842624
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{99}\cdot 5^{4}\cdot 31 $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.7
Root			$-5.25771 + 2.08722i$ of defining polynomial
Character	$\chi$	$=$	800.401
Dual form			800.6.d.d.401.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.4354i q^{3} -66.0164 q^{7} +134.101 q^{9} -141.942i q^{11} +246.793i q^{13} +297.407 q^{17} +174.593i q^{19} +688.910i q^{21} -1576.89 q^{23} -3935.22i q^{27} -922.162i q^{29} +6194.91 q^{31} -1481.22 q^{33} +13956.6i q^{37} +2575.40 q^{39} -3008.44 q^{41} +16270.0i q^{43} +1004.93 q^{47} -12448.8 q^{49} -3103.57i q^{51} -22990.4i q^{53} +1821.96 q^{57} -31893.8i q^{59} +21370.1i q^{61} -8852.89 q^{63} +38648.6i q^{67} +16455.5i q^{69} +30998.5 q^{71} +79573.5 q^{73} +9370.46i q^{77} -23301.9 q^{79} -8479.17 q^{81} +66705.7i q^{83} -9623.18 q^{87} +66188.9 q^{89} -16292.4i q^{91} -64646.7i q^{93} +12550.8 q^{97} -19034.6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 196 q^{7} - 1620 q^{9} + 7184 q^{23} - 7160 q^{31} - 2836 q^{33} - 22452 q^{39} - 5804 q^{41} - 44180 q^{47} + 62652 q^{49} + 43696 q^{57} + 1240 q^{63} + 7724 q^{71} - 105136 q^{73} + 7780 q^{79}+ \cdots - 73688 q^{97}+O(q^{100}) $ 20 * q + 196 * q^7 - 1620 * q^9 + 7184 * q^23 - 7160 * q^31 - 2836 * q^33 - 22452 * q^39 - 5804 * q^41 - 44180 * q^47 + 62652 * q^49 + 43696 * q^57 + 1240 * q^63 + 7724 * q^71 - 105136 * q^73 + 7780 * q^79 + 96984 * q^81 + 106188 * q^87 - 3160 * q^89 - 73688 * q^97

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
$3$	− 10.4354i	− 0.669434i	−0.942319	−	0.334717i	$-0.891359\pi$
$3$	− 10.4354i	− 0.669434i	0.942319	−	0.334717i	$-0.108641\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−66.0164	−0.509221	−0.254611	−	0.967044i	$-0.581947\pi$
$7$	−66.0164	−0.509221	−0.254611	+	0.967044i	$0.581947\pi$
$8$	0	0
$9$	134.101	0.551858
$10$	0	0
$11$	− 141.942i	− 0.353694i	−0.984238	−	0.176847i	$-0.943410\pi$
$11$	− 141.942i	− 0.353694i	0.984238	−	0.176847i	$-0.0565898\pi$
$12$	0	0
$13$	246.793i	0.405018i	0.979280	+	0.202509i	$0.0649096\pi$
$13$	246.793i	0.405018i	−0.979280	+	0.202509i	$0.935090\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	297.407	0.249591	0.124795	−	0.992182i	$-0.460173\pi$
$17$	297.407	0.249591	0.124795	+	0.992182i	$0.460173\pi$
$18$	0	0
$19$	174.593i	0.110954i	0.998460	+	0.0554770i	$0.0176680\pi$
$19$	174.593i	0.110954i	−0.998460	+	0.0554770i	$0.982332\pi$
$20$	0	0
$21$	688.910i	0.340890i
$22$	0	0
$23$	−1576.89	−0.621558	−0.310779	−	0.950482i	$-0.600590\pi$
$23$	−1576.89	−0.621558	−0.310779	+	0.950482i	$0.600590\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 3935.22i	− 1.03887i
$28$	0	0
$29$	− 922.162i	− 0.203616i	−0.994804	−	0.101808i	$-0.967537\pi$
$29$	− 922.162i	− 0.203616i	0.994804	−	0.101808i	$-0.0324628\pi$
$30$	0	0
$31$	6194.91	1.15779	0.578897	−	0.815401i	$-0.303484\pi$
$31$	6194.91	1.15779	0.578897	+	0.815401i	$0.303484\pi$
$32$	0	0
$33$	−1481.22	−0.236775
$34$	0	0
$35$	0	0
$36$	0	0
$37$	13956.6i	1.67601i	0.545666	+	0.838003i	$0.316277\pi$
$37$	13956.6i	1.67601i	−0.545666	+	0.838003i	$0.683723\pi$
$38$	0	0
$39$	2575.40	0.271133
$40$	0	0
$41$	−3008.44	−0.279500	−0.139750	−	0.990187i	$-0.544630\pi$
$41$	−3008.44	−0.279500	−0.139750	+	0.990187i	$0.544630\pi$
$42$	0	0
$43$	16270.0i	1.34189i	0.741508	+	0.670944i	$0.234111\pi$
$43$	16270.0i	1.34189i	−0.741508	+	0.670944i	$0.765889\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1004.93	0.0663578	0.0331789	−	0.999449i	$-0.489437\pi$
$47$	1004.93	0.0663578	0.0331789	+	0.999449i	$0.489437\pi$
$48$	0	0
$49$	−12448.8	−0.740694
$50$	0	0
$51$	− 3103.57i	− 0.167085i
$52$	0	0
$53$	− 22990.4i	− 1.12424i	−0.827057	−	0.562118i	$-0.809987\pi$
$53$	− 22990.4i	− 1.12424i	0.827057	−	0.562118i	$-0.190013\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1821.96	0.0742764
$58$	0	0
$59$	− 31893.8i	− 1.19282i	−0.802678	−	0.596412i	$-0.796592\pi$
$59$	− 31893.8i	− 1.19282i	0.802678	−	0.596412i	$-0.203408\pi$
$60$	0	0
$61$	21370.1i	0.735328i	0.929959	+	0.367664i	$0.119842\pi$
$61$	21370.1i	0.735328i	−0.929959	+	0.367664i	$0.880158\pi$
$62$	0	0
$63$	−8852.89	−0.281018
$64$	0	0
$65$	0	0
$66$	0	0
$67$	38648.6i	1.05183i	0.850536	+	0.525916i	$0.176277\pi$
$67$	38648.6i	1.05183i	−0.850536	+	0.525916i	$0.823723\pi$
$68$	0	0
$69$	16455.5i	0.416092i
$70$	0	0
$71$	30998.5	0.729784	0.364892	−	0.931050i	$-0.381106\pi$
$71$	30998.5	0.729784	0.364892	+	0.931050i	$0.381106\pi$
$72$	0	0
$73$	79573.5	1.74768	0.873838	−	0.486217i	$-0.161623\pi$
$73$	79573.5	1.74768	0.873838	+	0.486217i	$0.161623\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9370.46i	0.180109i
$78$	0	0
$79$	−23301.9	−0.420072	−0.210036	−	0.977694i	$-0.567358\pi$
$79$	−23301.9	−0.420072	−0.210036	+	0.977694i	$0.567358\pi$
$80$	0	0
$81$	−8479.17	−0.143596
$82$	0	0
$83$	66705.7i	1.06284i	0.847109	+	0.531420i	$0.178341\pi$
$83$	66705.7i	1.06284i	−0.847109	+	0.531420i	$0.821659\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9623.18	−0.136308
$88$	0	0
$89$	66188.9	0.885748	0.442874	−	0.896584i	$-0.353959\pi$
$89$	66188.9	0.885748	0.442874	+	0.896584i	$0.353959\pi$
$90$	0	0
$91$	− 16292.4i	− 0.206244i
$92$	0	0
$93$	− 64646.7i	− 0.775066i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12550.8	0.135439	0.0677194	−	0.997704i	$-0.478428\pi$
$97$	12550.8	0.135439	0.0677194	+	0.997704i	$0.478428\pi$
$98$	0	0
$99$	− 19034.6i	− 0.195189i
$100$	0	0
$101$	94627.3i	0.923024i	0.887134	+	0.461512i	$0.152693\pi$
$101$	94627.3i	0.923024i	−0.887134	+	0.461512i	$0.847307\pi$
$102$	0	0
$103$	−74704.4	−0.693830	−0.346915	−	0.937897i	$-0.612771\pi$
$103$	−74704.4	−0.693830	−0.346915	+	0.937897i	$0.612771\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	95390.0i	0.805459i	0.915319	+	0.402729i	$0.131938\pi$
$107$	95390.0i	0.805459i	−0.915319	+	0.402729i	$0.868062\pi$
$108$	0	0
$109$	− 166091.i	− 1.33900i	−0.742814	−	0.669498i	$-0.766509\pi$
$109$	− 166091.i	− 1.33900i	0.742814	−	0.669498i	$-0.233491\pi$
$110$	0	0
$111$	145643.	1.12198
$112$	0	0
$113$	174815.	1.28790	0.643951	−	0.765067i	$-0.277294\pi$
$113$	174815.	1.28790	0.643951	+	0.765067i	$0.277294\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	33095.3i	0.223512i
$118$	0	0
$119$	−19633.7	−0.127097
$120$	0	0
$121$	140904.	0.874901
$122$	0	0
$123$	31394.4i	0.187107i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−94893.2	−0.522067	−0.261033	−	0.965330i	$-0.584063\pi$
$127$	−94893.2	−0.522067	−0.261033	+	0.965330i	$0.584063\pi$
$128$	0	0
$129$	169785.	0.898307
$130$	0	0
$131$	− 263177.i	− 1.33989i	−0.742409	−	0.669947i	$-0.766317\pi$
$131$	− 263177.i	− 1.33989i	0.742409	−	0.669947i	$-0.233683\pi$
$132$	0	0
$133$	− 11526.0i	− 0.0565001i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12040.5	−0.0548080	−0.0274040	−	0.999624i	$-0.508724\pi$
$137$	−12040.5	−0.0548080	−0.0274040	+	0.999624i	$0.508724\pi$
$138$	0	0
$139$	357698.i	1.57029i	0.619312	+	0.785145i	$0.287412\pi$
$139$	357698.i	1.57029i	−0.619312	+	0.785145i	$0.712588\pi$
$140$	0	0
$141$	− 10486.9i	− 0.0444222i
$142$	0	0
$143$	35030.2	0.143253
$144$	0	0
$145$	0	0
$146$	0	0
$147$	129909.i	0.495846i
$148$	0	0
$149$	− 361479.i	− 1.33388i	−0.745111	−	0.666940i	$-0.767604\pi$
$149$	− 361479.i	− 1.33388i	0.745111	−	0.666940i	$-0.232396\pi$
$150$	0	0
$151$	337938.	1.20613	0.603067	−	0.797691i	$-0.293945\pi$
$151$	337938.	1.20613	0.603067	+	0.797691i	$0.293945\pi$
$152$	0	0
$153$	39882.7	0.137739
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 155005.i	− 0.501876i	−0.968003	−	0.250938i	$-0.919261\pi$
$157$	− 155005.i	− 0.501876i	0.968003	−	0.250938i	$-0.0807391\pi$
$158$	0	0
$159$	−239915.	−0.752602
$160$	0	0
$161$	104100.	0.316510
$162$	0	0
$163$	− 126397.i	− 0.372621i	−0.982491	−	0.186311i	$-0.940347\pi$
$163$	− 126397.i	− 0.372621i	0.982491	−	0.186311i	$-0.0596531\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−709796.	−1.96944	−0.984719	−	0.174152i	$-0.944282\pi$
$167$	−709796.	−1.96944	−0.984719	+	0.174152i	$0.944282\pi$
$168$	0	0
$169$	310386.	0.835960
$170$	0	0
$171$	23413.2i	0.0612308i
$172$	0	0
$173$	− 395365.i	− 1.00435i	−0.864767	−	0.502173i	$-0.832534\pi$
$173$	− 395365.i	− 1.00435i	0.864767	−	0.502173i	$-0.167466\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−332826.	−0.798517
$178$	0	0
$179$	265918.i	0.620318i	0.950685	+	0.310159i	$0.100382\pi$
$179$	265918.i	0.620318i	−0.950685	+	0.310159i	$0.899618\pi$
$180$	0	0
$181$	− 836091.i	− 1.89696i	−0.316843	−	0.948478i	$-0.602623\pi$
$181$	− 836091.i	− 1.89696i	0.316843	−	0.948478i	$-0.397377\pi$
$182$	0	0
$183$	223006.	0.492254
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 42214.4i	− 0.0882788i
$188$	0	0
$189$	259789.i	0.529013i
$190$	0	0
$191$	770184.	1.52761	0.763803	−	0.645450i	$-0.223330\pi$
$191$	770184.	1.52761	0.763803	+	0.645450i	$0.223330\pi$
$192$	0	0
$193$	694883.	1.34282	0.671411	−	0.741085i	$-0.265688\pi$
$193$	694883.	1.34282	0.671411	+	0.741085i	$0.265688\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 304179.i	− 0.558424i	−0.960229	−	0.279212i	$-0.909927\pi$
$197$	− 304179.i	− 0.558424i	0.960229	−	0.279212i	$-0.0900733\pi$
$198$	0	0
$199$	575412.	1.03002	0.515011	−	0.857184i	$-0.327788\pi$
$199$	575412.	1.03002	0.515011	+	0.857184i	$0.327788\pi$
$200$	0	0
$201$	403315.	0.704133
$202$	0	0
$203$	60877.8i	0.103686i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−211463.	−0.343011
$208$	0	0
$209$	24782.0	0.0392438
$210$	0	0
$211$	− 154406.i	− 0.238758i	−0.992849	−	0.119379i	$-0.961910\pi$
$211$	− 154406.i	− 0.238758i	0.992849	−	0.119379i	$-0.0380903\pi$
$212$	0	0
$213$	− 323483.i	− 0.488543i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−408966.	−0.589573
$218$	0	0
$219$	− 830385.i	− 1.16995i
$220$	0	0
$221$	73398.0i	0.101089i
$222$	0	0
$223$	817613.	1.10100	0.550498	−	0.834836i	$-0.314438\pi$
$223$	817613.	1.10100	0.550498	+	0.834836i	$0.314438\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 45598.7i	− 0.0587338i	−0.999569	−	0.0293669i	$-0.990651\pi$
$227$	− 45598.7i	− 0.0587338i	0.999569	−	0.0293669i	$-0.00934912\pi$
$228$	0	0
$229$	633888.i	0.798774i	0.916783	+	0.399387i	$0.130777\pi$
$229$	633888.i	0.798774i	−0.916783	+	0.399387i	$0.869223\pi$
$230$	0	0
$231$	97785.0	0.120571
$232$	0	0
$233$	245572.	0.296339	0.148169	−	0.988962i	$-0.452662\pi$
$233$	245572.	0.296339	0.148169	+	0.988962i	$0.452662\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	243166.i	0.281211i
$238$	0	0
$239$	1.13159e6	1.28142	0.640711	−	0.767782i	$-0.278639\pi$
$239$	1.13159e6	1.28142	0.640711	+	0.767782i	$0.278639\pi$
$240$	0	0
$241$	1.56424e6	1.73484	0.867421	−	0.497575i	$-0.165776\pi$
$241$	1.56424e6	1.73484	0.867421	+	0.497575i	$0.165776\pi$
$242$	0	0
$243$	− 867775.i	− 0.942739i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−43088.4	−0.0449384
$248$	0	0
$249$	696104.	0.711501
$250$	0	0
$251$	− 1.45223e6i	− 1.45496i	−0.686130	−	0.727479i	$-0.740692\pi$
$251$	− 1.45223e6i	− 1.45496i	0.686130	−	0.727479i	$-0.259308\pi$
$252$	0	0
$253$	223826.i	0.219841i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.67551e6	−1.58240	−0.791198	−	0.611560i	$-0.790542\pi$
$257$	−1.67551e6	−1.58240	−0.791198	+	0.611560i	$0.790542\pi$
$258$	0	0
$259$	− 921364.i	− 0.853458i
$260$	0	0
$261$	− 123663.i	− 0.112367i
$262$	0	0
$263$	977406.	0.871336	0.435668	−	0.900107i	$-0.356512\pi$
$263$	977406.	0.871336	0.435668	+	0.900107i	$0.356512\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 690711.i	− 0.592950i
$268$	0	0
$269$	− 15731.4i	− 0.0132552i	−0.999978	−	0.00662760i	$-0.997890\pi$
$269$	− 15731.4i	− 0.0132552i	0.999978	−	0.00662760i	$-0.00210964\pi$
$270$	0	0
$271$	861364.	0.712465	0.356233	−	0.934397i	$-0.384061\pi$
$271$	861364.	0.712465	0.356233	+	0.934397i	$0.384061\pi$
$272$	0	0
$273$	−170018.	−0.138067
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.16950e6i	1.69887i	0.527693	+	0.849435i	$0.323057\pi$
$277$	2.16950e6i	1.69887i	−0.527693	+	0.849435i	$0.676943\pi$
$278$	0	0
$279$	830746.	0.638937
$280$	0	0
$281$	1.81796e6	1.37347	0.686736	−	0.726907i	$-0.259043\pi$
$281$	1.81796e6	1.37347	0.686736	+	0.726907i	$0.259043\pi$
$282$	0	0
$283$	− 998020.i	− 0.740753i	−0.928882	−	0.370376i	$-0.879229\pi$
$283$	− 998020.i	− 0.740753i	0.928882	−	0.370376i	$-0.120771\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	198606.	0.142327
$288$	0	0
$289$	−1.33141e6	−0.937704
$290$	0	0
$291$	− 130974.i	− 0.0906674i
$292$	0	0
$293$	399603.i	0.271932i	0.990714	+	0.135966i	$0.0434138\pi$
$293$	399603.i	0.271932i	−0.990714	+	0.135966i	$0.956586\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−558571.	−0.367441
$298$	0	0
$299$	− 389165.i	− 0.251742i
$300$	0	0
$301$	− 1.07409e6i	− 0.683318i
$302$	0	0
$303$	987479.	0.617904
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 368112.i	− 0.222912i	−0.993769	−	0.111456i	$-0.964449\pi$
$307$	− 368112.i	− 0.222912i	0.993769	−	0.111456i	$-0.0355515\pi$
$308$	0	0
$309$	779573.i	0.464473i
$310$	0	0
$311$	1.13519e6	0.665530	0.332765	−	0.943010i	$-0.392018\pi$
$311$	1.13519e6	0.665530	0.332765	+	0.943010i	$0.392018\pi$
$312$	0	0
$313$	334474.	0.192975	0.0964877	−	0.995334i	$-0.469239\pi$
$313$	334474.	0.192975	0.0964877	+	0.995334i	$0.469239\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	837898.i	0.468321i	0.972198	+	0.234160i	$0.0752341\pi$
$317$	837898.i	0.468321i	−0.972198	+	0.234160i	$0.924766\pi$
$318$	0	0
$319$	−130893.	−0.0720179
$320$	0	0
$321$	995437.	0.539202
$322$	0	0
$323$	51925.2i	0.0276931i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.73323e6	−0.896370
$328$	0	0
$329$	−66341.9	−0.0337908
$330$	0	0
$331$	2.30318e6i	1.15547i	0.816225	+	0.577734i	$0.196063\pi$
$331$	2.30318e6i	1.15547i	−0.816225	+	0.577734i	$0.803937\pi$
$332$	0	0
$333$	1.87160e6i	0.924917i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−698751.	−0.335156	−0.167578	−	0.985859i	$-0.553595\pi$
$337$	−698751.	−0.335156	−0.167578	+	0.985859i	$0.553595\pi$
$338$	0	0
$339$	− 1.82427e6i	− 0.862166i
$340$	0	0
$341$	− 879315.i	− 0.409505i
$342$	0	0
$343$	1.93136e6	0.886398
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.30033e6i	1.47141i	0.677301	+	0.735706i	$0.263149\pi$
$347$	3.30033e6i	1.47141i	−0.677301	+	0.735706i	$0.736851\pi$
$348$	0	0
$349$	− 121442.i	− 0.0533711i	−0.999644	−	0.0266855i	$-0.991505\pi$
$349$	− 121442.i	− 0.0533711i	0.999644	−	0.0266855i	$-0.00849528\pi$
$350$	0	0
$351$	971186.	0.420760
$352$	0	0
$353$	−56609.0	−0.0241796	−0.0120898	−	0.999927i	$-0.503848\pi$
$353$	−56609.0	−0.0241796	−0.0120898	+	0.999927i	$0.503848\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	204887.i	0.0850831i
$358$	0	0
$359$	446755.	0.182950	0.0914751	−	0.995807i	$-0.470842\pi$
$359$	446755.	0.182950	0.0914751	+	0.995807i	$0.470842\pi$
$360$	0	0
$361$	2.44562e6	0.987689
$362$	0	0
$363$	− 1.47039e6i	− 0.585688i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	304081.	0.117849	0.0589243	−	0.998262i	$-0.481233\pi$
$367$	304081.	0.117849	0.0589243	+	0.998262i	$0.481233\pi$
$368$	0	0
$369$	−403436.	−0.154244
$370$	0	0
$371$	1.51774e6i	0.572484i
$372$	0	0
$373$	− 932701.i	− 0.347112i	−0.984824	−	0.173556i	$-0.944474\pi$
$373$	− 932701.i	− 0.347112i	0.984824	−	0.173556i	$-0.0555258\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	227583.	0.0824683
$378$	0	0
$379$	760659.i	0.272014i	0.990708	+	0.136007i	$0.0434270\pi$
$379$	760659.i	0.272014i	−0.990708	+	0.136007i	$0.956573\pi$
$380$	0	0
$381$	990254.i	0.349489i
$382$	0	0
$383$	1.80993e6	0.630470	0.315235	−	0.949014i	$-0.397917\pi$
$383$	1.80993e6	0.630470	0.315235	+	0.949014i	$0.397917\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.18183e6i	0.740532i
$388$	0	0
$389$	− 4.14597e6i	− 1.38916i	−0.719416	−	0.694579i	$-0.755591\pi$
$389$	− 4.14597e6i	− 1.38916i	0.719416	−	0.694579i	$-0.244409\pi$
$390$	0	0
$391$	−468977.	−0.155135
$392$	0	0
$393$	−2.74637e6	−0.896971
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.67426e6i	− 0.533145i	−0.963815	−	0.266573i	$-0.914109\pi$
$397$	− 1.67426e6i	− 0.533145i	0.963815	−	0.266573i	$-0.0858912\pi$
$398$	0	0
$399$	−120279.	−0.0378231
$400$	0	0
$401$	−2.20886e6	−0.685975	−0.342987	−	0.939340i	$-0.611439\pi$
$401$	−2.20886e6	−0.685975	−0.342987	+	0.939340i	$0.611439\pi$
$402$	0	0
$403$	1.52886e6i	0.468927i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.98102e6	0.592793
$408$	0	0
$409$	1.59010e6	0.470019	0.235010	−	0.971993i	$-0.424488\pi$
$409$	1.59010e6	0.470019	0.235010	+	0.971993i	$0.424488\pi$
$410$	0	0
$411$	125648.i	0.0366904i
$412$	0	0
$413$	2.10551e6i	0.607411i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.73274e6	1.05121
$418$	0	0
$419$	4.24535e6i	1.18135i	0.806909	+	0.590675i	$0.201139\pi$
$419$	4.24535e6i	1.18135i	−0.806909	+	0.590675i	$0.798861\pi$
$420$	0	0
$421$	− 5.17330e6i	− 1.42253i	−0.702922	−	0.711267i	$-0.748122\pi$
$421$	− 5.17330e6i	− 1.42253i	0.702922	−	0.711267i	$-0.251878\pi$
$422$	0	0
$423$	134763.	0.0366200
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.41077e6i	− 0.374445i
$428$	0	0
$429$	− 365556.i	− 0.0958982i
$430$	0	0
$431$	−1.02248e6	−0.265132	−0.132566	−	0.991174i	$-0.542322\pi$
$431$	−1.02248e6	−0.265132	−0.132566	+	0.991174i	$0.542322\pi$
$432$	0	0
$433$	2.18503e6	0.560063	0.280031	−	0.959991i	$-0.409655\pi$
$433$	2.18503e6	0.560063	0.280031	+	0.959991i	$0.409655\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 275314.i	− 0.0689643i
$438$	0	0
$439$	−1.22033e6	−0.302216	−0.151108	−	0.988517i	$-0.548284\pi$
$439$	−1.22033e6	−0.302216	−0.151108	+	0.988517i	$0.548284\pi$
$440$	0	0
$441$	−1.66941e6	−0.408758
$442$	0	0
$443$	− 509711.i	− 0.123400i	−0.998095	−	0.0617000i	$-0.980348\pi$
$443$	− 509711.i	− 0.123400i	0.998095	−	0.0617000i	$-0.0196522\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.77219e6	−0.892945
$448$	0	0
$449$	3.54089e6	0.828890	0.414445	−	0.910074i	$-0.363976\pi$
$449$	3.54089e6	0.828890	0.414445	+	0.910074i	$0.363976\pi$
$450$	0	0
$451$	427022.i	0.0988573i
$452$	0	0
$453$	− 3.52654e6i	− 0.807427i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.06559e6	0.238671	0.119335	−	0.992854i	$-0.461924\pi$
$457$	1.06559e6	0.238671	0.119335	+	0.992854i	$0.461924\pi$
$458$	0	0
$459$	− 1.17036e6i	− 0.259292i
$460$	0	0
$461$	6.92975e6i	1.51868i	0.650696	+	0.759339i	$0.274477\pi$
$461$	6.92975e6i	1.51868i	−0.650696	+	0.759339i	$0.725523\pi$
$462$	0	0
$463$	−7.21004e6	−1.56310	−0.781548	−	0.623845i	$-0.785569\pi$
$463$	−7.21004e6	−1.56310	−0.781548	+	0.623845i	$0.785569\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.00541e6i	− 0.637693i	−0.947806	−	0.318846i	$-0.896705\pi$
$467$	− 3.00541e6i	− 0.637693i	0.947806	−	0.318846i	$-0.103295\pi$
$468$	0	0
$469$	− 2.55144e6i	− 0.535615i
$470$	0	0
$471$	−1.61755e6	−0.335973
$472$	0	0
$473$	2.30939e6	0.474618
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3.08305e6i	− 0.620418i
$478$	0	0
$479$	−2.47740e6	−0.493353	−0.246676	−	0.969098i	$-0.579338\pi$
$479$	−2.47740e6	−0.493353	−0.246676	+	0.969098i	$0.579338\pi$
$480$	0	0
$481$	−3.44439e6	−0.678813
$482$	0	0
$483$	− 1.08633e6i	− 0.211883i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.58130e6	0.875320	0.437660	−	0.899141i	$-0.355807\pi$
$487$	4.58130e6	0.875320	0.437660	+	0.899141i	$0.355807\pi$
$488$	0	0
$489$	−1.31901e6	−0.249446
$490$	0	0
$491$	− 8.23983e6i	− 1.54246i	−0.636555	−	0.771231i	$-0.719641\pi$
$491$	− 8.23983e6i	− 1.54246i	0.636555	−	0.771231i	$-0.280359\pi$
$492$	0	0
$493$	− 274257.i	− 0.0508207i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.04641e6	−0.371622
$498$	0	0
$499$	4.49557e6i	0.808228i	0.914709	+	0.404114i	$0.132420\pi$
$499$	4.49557e6i	0.808228i	−0.914709	+	0.404114i	$0.867580\pi$
$500$	0	0
$501$	7.40704e6i	1.31841i
$502$	0	0
$503$	−6.28876e6	−1.10827	−0.554134	−	0.832427i	$-0.686951\pi$
$503$	−6.28876e6	−1.10827	−0.554134	+	0.832427i	$0.686951\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 3.23902e6i	− 0.559620i
$508$	0	0
$509$	8.26032e6i	1.41320i	0.707615	+	0.706598i	$0.249771\pi$
$509$	8.26032e6i	1.41320i	−0.707615	+	0.706598i	$0.750229\pi$
$510$	0	0
$511$	−5.25315e6	−0.889954
$512$	0	0
$513$	687063.	0.115266
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 142642.i	− 0.0234704i
$518$	0	0
$519$	−4.12581e6	−0.672344
$520$	0	0
$521$	−1.09563e7	−1.76835	−0.884176	−	0.467154i	$-0.845279\pi$
$521$	−1.09563e7	−1.76835	−0.884176	+	0.467154i	$0.845279\pi$
$522$	0	0
$523$	9.74999e6i	1.55865i	0.626617	+	0.779327i	$0.284439\pi$
$523$	9.74999e6i	1.55865i	−0.626617	+	0.779327i	$0.715561\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.84241e6	0.288975
$528$	0	0
$529$	−3.94977e6	−0.613666
$530$	0	0
$531$	− 4.27701e6i	− 0.658269i
$532$	0	0
$533$	− 742461.i	− 0.113202i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.77497e6	0.415262
$538$	0	0
$539$	1.76701e6i	0.261979i
$540$	0	0
$541$	1.86030e6i	0.273268i	0.990622	+	0.136634i	$0.0436285\pi$
$541$	1.86030e6i	0.273268i	−0.990622	+	0.136634i	$0.956372\pi$
$542$	0	0
$543$	−8.72499e6	−1.26989
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 725841.i	− 0.103723i	−0.998654	−	0.0518613i	$-0.983485\pi$
$547$	− 725841.i	− 0.103723i	0.998654	−	0.0518613i	$-0.0165154\pi$
$548$	0	0
$549$	2.86576e6i	0.405796i
$550$	0	0
$551$	161003.	0.0225920
$552$	0	0
$553$	1.53831e6	0.213910
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.72234e6i	− 0.644939i	−0.946580	−	0.322470i	$-0.895487\pi$
$557$	− 4.72234e6i	− 0.644939i	0.946580	−	0.322470i	$-0.104513\pi$
$558$	0	0
$559$	−4.01533e6	−0.543490
$560$	0	0
$561$	−440526.	−0.0590969
$562$	0	0
$563$	4.74239e6i	0.630560i	0.948999	+	0.315280i	$0.102098\pi$
$563$	4.74239e6i	0.630560i	−0.948999	+	0.315280i	$0.897902\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	559764.	0.0731219
$568$	0	0
$569$	−2.99470e6	−0.387768	−0.193884	−	0.981024i	$-0.562109\pi$
$569$	−2.99470e6	−0.387768	−0.193884	+	0.981024i	$0.562109\pi$
$570$	0	0
$571$	1.01416e7i	1.30172i	0.759198	+	0.650860i	$0.225591\pi$
$571$	1.01416e7i	1.30172i	−0.759198	+	0.650860i	$0.774409\pi$
$572$	0	0
$573$	− 8.03722e6i	− 1.02263i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.80870e6	0.476253	0.238126	−	0.971234i	$-0.423467\pi$
$577$	3.80870e6	0.476253	0.238126	+	0.971234i	$0.423467\pi$
$578$	0	0
$579$	− 7.25142e6i	− 0.898931i
$580$	0	0
$581$	− 4.40367e6i	− 0.541220i
$582$	0	0
$583$	−3.26330e6	−0.397635
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.40054e6i	− 0.407335i	−0.979040	−	0.203668i	$-0.934714\pi$
$587$	− 3.40054e6i	− 0.407335i	0.979040	−	0.203668i	$-0.0652862\pi$
$588$	0	0
$589$	1.08159e6i	0.128462i
$590$	0	0
$591$	−3.17425e6	−0.373828
$592$	0	0
$593$	−1.27800e7	−1.49243	−0.746216	−	0.665704i	$-0.768132\pi$
$593$	−1.27800e7	−1.49243	−0.746216	+	0.665704i	$0.768132\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.00468e6i	− 0.689532i
$598$	0	0
$599$	614365.	0.0699616	0.0349808	−	0.999388i	$-0.488863\pi$
$599$	614365.	0.0699616	0.0349808	+	0.999388i	$0.488863\pi$
$600$	0	0
$601$	−1.12864e7	−1.27459	−0.637294	−	0.770621i	$-0.719946\pi$
$601$	−1.12864e7	−1.27459	−0.637294	+	0.770621i	$0.719946\pi$
$602$	0	0
$603$	5.18283e6i	0.580462i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.81019e6	−0.199413	−0.0997065	−	0.995017i	$-0.531790\pi$
$607$	−1.81019e6	−0.199413	−0.0997065	+	0.995017i	$0.531790\pi$
$608$	0	0
$609$	635287.	0.0694108
$610$	0	0
$611$	248010.i	0.0268761i
$612$	0	0
$613$	− 1.08728e7i	− 1.16867i	−0.811513	−	0.584335i	$-0.801355\pi$
$613$	− 1.08728e7i	− 1.16867i	0.811513	−	0.584335i	$-0.198645\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.60152e6	0.698122	0.349061	−	0.937100i	$-0.386501\pi$
$617$	6.60152e6	0.698122	0.349061	+	0.937100i	$0.386501\pi$
$618$	0	0
$619$	8.83255e6i	0.926530i	0.886220	+	0.463265i	$0.153322\pi$
$619$	8.83255e6i	0.926530i	−0.886220	+	0.463265i	$0.846678\pi$
$620$	0	0
$621$	6.20541e6i	0.645716i
$622$	0	0
$623$	−4.36955e6	−0.451042
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 258611.i	− 0.0262711i
$628$	0	0
$629$	4.15079e6i	0.418316i
$630$	0	0
$631$	−1.48464e7	−1.48439	−0.742194	−	0.670185i	$-0.766215\pi$
$631$	−1.48464e7	−1.48439	−0.742194	+	0.670185i	$0.766215\pi$
$632$	0	0
$633$	−1.61129e6	−0.159833
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.07229e6i	− 0.299995i
$638$	0	0
$639$	4.15694e6	0.402737
$640$	0	0
$641$	−1.42105e7	−1.36604	−0.683022	−	0.730398i	$-0.739335\pi$
$641$	−1.42105e7	−1.36604	−0.683022	+	0.730398i	$0.739335\pi$
$642$	0	0
$643$	7.89022e6i	0.752595i	0.926499	+	0.376298i	$0.122803\pi$
$643$	7.89022e6i	0.752595i	−0.926499	+	0.376298i	$0.877197\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.41387e7	1.32785	0.663924	−	0.747800i	$-0.268890\pi$
$647$	1.41387e7	1.32785	0.663924	+	0.747800i	$0.268890\pi$
$648$	0	0
$649$	−4.52706e6	−0.421895
$650$	0	0
$651$	4.26774e6i	0.394680i
$652$	0	0
$653$	2.09667e7i	1.92419i	0.272720	+	0.962094i	$0.412077\pi$
$653$	2.09667e7i	1.92419i	−0.272720	+	0.962094i	$0.587923\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.06709e7	0.964469
$658$	0	0
$659$	− 5.17814e6i	− 0.464473i	−0.972659	−	0.232236i	$-0.925396\pi$
$659$	− 5.17814e6i	− 0.464473i	0.972659	−	0.232236i	$-0.0746043\pi$
$660$	0	0
$661$	− 1.15802e7i	− 1.03089i	−0.856923	−	0.515444i	$-0.827627\pi$
$661$	− 1.15802e7i	− 1.03089i	0.856923	−	0.515444i	$-0.172373\pi$
$662$	0	0
$663$	765941.	0.0676724
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.45415e6i	0.126559i
$668$	0	0
$669$	− 8.53216e6i	− 0.737045i
$670$	0	0
$671$	3.03330e6	0.260081
$672$	0	0
$673$	3.44202e6	0.292938	0.146469	−	0.989215i	$-0.453209\pi$
$673$	3.44202e6	0.292938	0.146469	+	0.989215i	$0.453209\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.81661e7i	1.52332i	0.647980	+	0.761658i	$0.275614\pi$
$677$	1.81661e7i	1.52332i	−0.647980	+	0.761658i	$0.724386\pi$
$678$	0	0
$679$	−828560.	−0.0689683
$680$	0	0
$681$	−475843.	−0.0393184
$682$	0	0
$683$	− 1.58711e7i	− 1.30184i	−0.759148	−	0.650918i	$-0.774384\pi$
$683$	− 1.58711e7i	− 1.30184i	0.759148	−	0.650918i	$-0.225616\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.61490e6	0.534726
$688$	0	0
$689$	5.67388e6	0.455336
$690$	0	0
$691$	7.37470e6i	0.587556i	0.955874	+	0.293778i	$0.0949127\pi$
$691$	7.37470e6i	0.587556i	−0.955874	+	0.293778i	$0.905087\pi$
$692$	0	0
$693$	1.25659e6i	0.0993943i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−894730.	−0.0697605
$698$	0	0
$699$	− 2.56265e6i	− 0.198379i
$700$	0	0
$701$	2.16654e7i	1.66522i	0.553857	+	0.832612i	$0.313155\pi$
$701$	2.16654e7i	1.66522i	−0.553857	+	0.832612i	$0.686845\pi$
$702$	0	0
$703$	−2.43673e6	−0.185960
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.24695e6i	− 0.470023i
$708$	0	0
$709$	− 5.46373e6i	− 0.408201i	−0.978950	−	0.204100i	$-0.934573\pi$
$709$	− 5.46373e6i	− 0.408201i	0.978950	−	0.204100i	$-0.0654269\pi$
$710$	0	0
$711$	−3.12482e6	−0.231820
$712$	0	0
$713$	−9.76869e6	−0.719635
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.18086e7i	− 0.857829i
$718$	0	0
$719$	−1.42857e7	−1.03058	−0.515288	−	0.857017i	$-0.672315\pi$
$719$	−1.42857e7	−1.03058	−0.515288	+	0.857017i	$0.672315\pi$
$720$	0	0
$721$	4.93171e6	0.353313
$722$	0	0
$723$	− 1.63235e7i	− 1.16136i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.55756e6	−0.249641	−0.124821	−	0.992179i	$-0.539836\pi$
$727$	−3.55756e6	−0.249641	−0.124821	+	0.992179i	$0.539836\pi$
$728$	0	0
$729$	−1.11161e7	−0.774697
$730$	0	0
$731$	4.83881e6i	0.334923i
$732$	0	0
$733$	1.61816e7i	1.11240i	0.831048	+	0.556201i	$0.187742\pi$
$733$	1.61816e7i	1.11240i	−0.831048	+	0.556201i	$0.812258\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.48584e6	0.372027
$738$	0	0
$739$	1.16551e7i	0.785063i	0.919739	+	0.392531i	$0.128401\pi$
$739$	1.16551e7i	0.785063i	−0.919739	+	0.392531i	$0.871599\pi$
$740$	0	0
$741$	449647.i	0.0300833i
$742$	0	0
$743$	2.41917e7	1.60766	0.803829	−	0.594860i	$-0.202792\pi$
$743$	2.41917e7	1.60766	0.803829	+	0.594860i	$0.202792\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.94532e6i	0.586536i
$748$	0	0
$749$	− 6.29730e6i	− 0.410157i
$750$	0	0
$751$	−2.13480e7	−1.38120	−0.690601	−	0.723236i	$-0.742654\pi$
$751$	−2.13480e7	−1.38120	−0.690601	+	0.723236i	$0.742654\pi$
$752$	0	0
$753$	−1.51546e7	−0.973998
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.68129e7i	1.06636i	0.846002	+	0.533179i	$0.179003\pi$
$757$	1.68129e7i	1.06636i	−0.846002	+	0.533179i	$0.820997\pi$
$758$	0	0
$759$	2.33572e6	0.147169
$760$	0	0
$761$	1.07818e7	0.674888	0.337444	−	0.941346i	$-0.390438\pi$
$761$	1.07818e7	0.674888	0.337444	+	0.941346i	$0.390438\pi$
$762$	0	0
$763$	1.09647e7i	0.681845i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.87117e6	0.483116
$768$	0	0
$769$	−3.03763e6	−0.185233	−0.0926166	−	0.995702i	$-0.529523\pi$
$769$	−3.03763e6	−0.185233	−0.0926166	+	0.995702i	$0.529523\pi$
$770$	0	0
$771$	1.74847e7i	1.05931i
$772$	0	0
$773$	2.73040e7i	1.64353i	0.569828	+	0.821764i	$0.307010\pi$
$773$	2.73040e7i	1.64353i	−0.569828	+	0.821764i	$0.692990\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−9.61485e6	−0.571334
$778$	0	0
$779$	− 525252.i	− 0.0310116i
$780$	0	0
$781$	− 4.39997e6i	− 0.258120i
$782$	0	0
$783$	−3.62891e6	−0.211530
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.58229e7i	0.910644i	0.890327	+	0.455322i	$0.150476\pi$
$787$	1.58229e7i	0.910644i	−0.890327	+	0.455322i	$0.849524\pi$
$788$	0	0
$789$	− 1.01997e7i	− 0.583302i
$790$	0	0
$791$	−1.15407e7	−0.655827
$792$	0	0
$793$	−5.27398e6	−0.297821
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.53728e6i	0.308781i	0.988010	+	0.154390i	$0.0493414\pi$
$797$	5.53728e6i	0.308781i	−0.988010	+	0.154390i	$0.950659\pi$
$798$	0	0
$799$	298874.	0.0165623
$800$	0	0
$801$	8.87602e6	0.488807
$802$	0	0
$803$	− 1.12948e7i	− 0.618143i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−164164.	−0.00887348
$808$	0	0
$809$	−4.59562e6	−0.246873	−0.123436	−	0.992352i	$-0.539391\pi$
$809$	−4.59562e6	−0.246873	−0.123436	+	0.992352i	$0.539391\pi$
$810$	0	0
$811$	3.14901e6i	0.168121i	0.996461	+	0.0840604i	$0.0267889\pi$
$811$	3.14901e6i	0.168121i	−0.996461	+	0.0840604i	$0.973211\pi$
$812$	0	0
$813$	− 8.98872e6i	− 0.476949i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.84063e6	−0.148888
$818$	0	0
$819$	− 2.18483e6i	− 0.113817i
$820$	0	0
$821$	− 1.98853e7i	− 1.02961i	−0.857306	−	0.514807i	$-0.827864\pi$
$821$	− 1.98853e7i	− 1.02961i	0.857306	−	0.514807i	$-0.172136\pi$
$822$	0	0
$823$	1.53764e7	0.791328	0.395664	−	0.918395i	$-0.370515\pi$
$823$	1.53764e7	0.791328	0.395664	+	0.918395i	$0.370515\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.95740e6i	− 0.455426i	−0.973728	−	0.227713i	$-0.926875\pi$
$827$	− 8.95740e6i	− 0.455426i	0.973728	−	0.227713i	$-0.0731248\pi$
$828$	0	0
$829$	2.95693e7i	1.49436i	0.664623	+	0.747179i	$0.268592\pi$
$829$	2.95693e7i	1.49436i	−0.664623	+	0.747179i	$0.731408\pi$
$830$	0	0
$831$	2.26397e7	1.13728
$832$	0	0
$833$	−3.70237e6	−0.184870
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.43784e7i	− 1.20279i
$838$	0	0
$839$	1.26360e7	0.619732	0.309866	−	0.950780i	$-0.399716\pi$
$839$	1.26360e7	0.619732	0.309866	+	0.950780i	$0.399716\pi$
$840$	0	0
$841$	1.96608e7	0.958540
$842$	0	0
$843$	− 1.89713e7i	− 0.919449i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9.30194e6	−0.445518
$848$	0	0
$849$	−1.04148e7	−0.495885
$850$	0	0
$851$	− 2.20080e7i	− 1.04173i
$852$	0	0
$853$	− 6.41059e6i	− 0.301665i	−0.988559	−	0.150833i	$-0.951805\pi$
$853$	− 6.41059e6i	− 0.301665i	0.988559	−	0.150833i	$-0.0481954\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.57309e7	−0.731646	−0.365823	−	0.930684i	$-0.619212\pi$
$857$	−1.57309e7	−0.731646	−0.365823	+	0.930684i	$0.619212\pi$
$858$	0	0
$859$	− 3.33475e7i	− 1.54199i	−0.636844	−	0.770993i	$-0.719760\pi$
$859$	− 3.33475e7i	− 1.54199i	0.636844	−	0.770993i	$-0.280240\pi$
$860$	0	0
$861$	− 2.07254e6i	− 0.0952787i
$862$	0	0
$863$	1.62060e6	0.0740709	0.0370355	−	0.999314i	$-0.488209\pi$
$863$	1.62060e6	0.0740709	0.0370355	+	0.999314i	$0.488209\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.38938e7i	0.627732i
$868$	0	0
$869$	3.30751e6i	0.148577i
$870$	0	0
$871$	−9.53820e6	−0.426011
$872$	0	0
$873$	1.68308e6	0.0747430
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.93461e6i	0.260551i	0.991478	+	0.130276i	$0.0415863\pi$
$877$	5.93461e6i	0.260551i	−0.991478	+	0.130276i	$0.958414\pi$
$878$	0	0
$879$	4.17004e6	0.182041
$880$	0	0
$881$	7.17940e6	0.311637	0.155818	−	0.987786i	$-0.450199\pi$
$881$	7.17940e6	0.311637	0.155818	+	0.987786i	$0.450199\pi$
$882$	0	0
$883$	5.72329e6i	0.247027i	0.992343	+	0.123513i	$0.0394162\pi$
$883$	5.72329e6i	0.247027i	−0.992343	+	0.123513i	$0.960584\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.46048e7	0.623285	0.311643	−	0.950199i	$-0.399121\pi$
$887$	1.46048e7	0.623285	0.311643	+	0.950199i	$0.399121\pi$
$888$	0	0
$889$	6.26451e6	0.265847
$890$	0	0
$891$	1.20355e6i	0.0507889i
$892$	0	0
$893$	175454.i	0.00736266i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.06111e6	−0.168525
$898$	0	0
$899$	− 5.71272e6i	− 0.235745i
$900$	0	0
$901$	− 6.83751e6i	− 0.280599i
$902$	0	0
$903$	−1.12086e7	−0.457437
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 5.43818e6i	− 0.219501i	−0.993959	−	0.109750i	$-0.964995\pi$
$907$	− 5.43818e6i	− 0.219501i	0.993959	−	0.109750i	$-0.0350051\pi$
$908$	0	0
$909$	1.26897e7i	0.509378i
$910$	0	0
$911$	3.71033e7	1.48121	0.740605	−	0.671941i	$-0.234539\pi$
$911$	3.71033e7	1.48121	0.740605	+	0.671941i	$0.234539\pi$
$912$	0	0
$913$	9.46831e6	0.375920
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.73740e7i	0.682302i
$918$	0	0
$919$	2.24895e7	0.878398	0.439199	−	0.898390i	$-0.355262\pi$
$919$	2.24895e7	0.878398	0.439199	+	0.898390i	$0.355262\pi$
$920$	0	0
$921$	−3.84142e6	−0.149225
$922$	0	0
$923$	7.65021e6i	0.295576i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.00180e7	−0.382895
$928$	0	0
$929$	2.80271e7	1.06547	0.532733	−	0.846283i	$-0.321165\pi$
$929$	2.80271e7	1.06547	0.532733	+	0.846283i	$0.321165\pi$
$930$	0	0
$931$	− 2.17348e6i	− 0.0821830i
$932$	0	0
$933$	− 1.18462e7i	− 0.445528i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−62078.6	−0.00230990	−0.00115495	−	0.999999i	$-0.500368\pi$
$937$	−62078.6	−0.00230990	−0.00115495	+	0.999999i	$0.500368\pi$
$938$	0	0
$939$	− 3.49039e6i	− 0.129184i
$940$	0	0
$941$	− 4.45412e6i	− 0.163979i	−0.996633	−	0.0819894i	$-0.973873\pi$
$941$	− 4.45412e6i	− 0.163979i	0.996633	−	0.0819894i	$-0.0261274\pi$
$942$	0	0
$943$	4.74397e6	0.173725
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.59793e7i	− 1.66605i	−0.553237	−	0.833024i	$-0.686608\pi$
$947$	− 4.59793e7i	− 1.66605i	0.553237	−	0.833024i	$-0.313392\pi$
$948$	0	0
$949$	1.96382e7i	0.707841i
$950$	0	0
$951$	8.74385e6	0.313510
$952$	0	0
$953$	2.32064e7	0.827706	0.413853	−	0.910344i	$-0.364183\pi$
$953$	2.32064e7	0.827706	0.413853	+	0.910344i	$0.364183\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.36593e6i	0.0482112i
$958$	0	0
$959$	794872.	0.0279094
$960$	0	0
$961$	9.74779e6	0.340485
$962$	0	0
$963$	1.27919e7i	0.444499i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.33110e7	−1.83337	−0.916686	−	0.399607i	$-0.869147\pi$
$967$	−5.33110e7	−1.83337	−0.916686	+	0.399607i	$0.869147\pi$
$968$	0	0
$969$	541863.	0.0185387
$970$	0	0
$971$	− 2.49744e7i	− 0.850056i	−0.905180	−	0.425028i	$-0.860264\pi$
$971$	− 2.49744e7i	− 0.850056i	0.905180	−	0.425028i	$-0.139736\pi$
$972$	0	0
$973$	− 2.36139e7i	− 0.799625i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.97806e7	−1.33332	−0.666661	−	0.745361i	$-0.732277\pi$
$977$	−3.97806e7	−1.33332	−0.666661	+	0.745361i	$0.732277\pi$
$978$	0	0
$979$	− 9.39495e6i	− 0.313284i
$980$	0	0
$981$	− 2.22730e7i	− 0.738936i
$982$	0	0
$983$	3.80176e7	1.25488	0.627439	−	0.778666i	$-0.284103\pi$
$983$	3.80176e7	1.25488	0.627439	+	0.778666i	$0.284103\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	692308.i	0.0226207i
$988$	0	0
$989$	− 2.56560e7i	− 0.834061i
$990$	0	0
$991$	4.34863e7	1.40659	0.703296	−	0.710897i	$-0.251711\pi$
$991$	4.34863e7	1.40659	0.703296	+	0.710897i	$0.251711\pi$
$992$	0	0
$993$	2.40347e7	0.773510
$994$	0	0
$995$	0	0
$996$	0	0
$997$	9.27058e6i	0.295372i	0.989034	+	0.147686i	$0.0471825\pi$
$997$	9.27058e6i	0.295372i	−0.989034	+	0.147686i	$0.952818\pi$
$998$	0	0
$999$	5.49223e7	1.74115

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.d.d.401.7		20
4.3	odd	2		200.6.d.c.101.18	yes	20
5.2	odd	4		800.6.f.d.49.14		40
5.3	odd	4		800.6.f.d.49.27		40
5.4	even	2		800.6.d.b.401.14		20
8.3	odd	2		200.6.d.c.101.17	✓	20
8.5	even	2	inner	800.6.d.d.401.14		20
20.3	even	4		200.6.f.d.149.25		40
20.7	even	4		200.6.f.d.149.16		40
20.19	odd	2		200.6.d.d.101.3	yes	20
40.3	even	4		200.6.f.d.149.15		40
40.13	odd	4		800.6.f.d.49.13		40
40.19	odd	2		200.6.d.d.101.4	yes	20
40.27	even	4		200.6.f.d.149.26		40
40.29	even	2		800.6.d.b.401.7		20
40.37	odd	4		800.6.f.d.49.28		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.6.d.c.101.17	✓	20	8.3	odd	2
200.6.d.c.101.18	yes	20	4.3	odd	2
200.6.d.d.101.3	yes	20	20.19	odd	2
200.6.d.d.101.4	yes	20	40.19	odd	2
200.6.f.d.149.15		40	40.3	even	4
200.6.f.d.149.16		40	20.7	even	4
200.6.f.d.149.25		40	20.3	even	4
200.6.f.d.149.26		40	40.27	even	4
800.6.d.b.401.7		20	40.29	even	2
800.6.d.b.401.14		20	5.4	even	2
800.6.d.d.401.7		20	1.1	even	1	trivial
800.6.d.d.401.14		20	8.5	even	2	inner
800.6.f.d.49.13		40	40.13	odd	4
800.6.f.d.49.14		40	5.2	odd	4
800.6.f.d.49.27		40	5.3	odd	4
800.6.f.d.49.28		40	40.37	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-10.4354i q^{3} -66.0164 q^{7} +134.101 q^{9} -141.942i q^{11} +246.793i q^{13} +297.407 q^{17} +174.593i q^{19} +688.910i q^{21} -1576.89 q^{23} -3935.22i q^{27} -922.162i q^{29} +6194.91 q^{31} -1481.22 q^{33} +13956.6i q^{37} +2575.40 q^{39} -3008.44 q^{41} +16270.0i q^{43} +1004.93 q^{47} -12448.8 q^{49} -3103.57i q^{51} -22990.4i q^{53} +1821.96 q^{57} -31893.8i q^{59} +21370.1i q^{61} -8852.89 q^{63} +38648.6i q^{67} +16455.5i q^{69} +30998.5 q^{71} +79573.5 q^{73} +9370.46i q^{77} -23301.9 q^{79} -8479.17 q^{81} +66705.7i q^{83} -9623.18 q^{87} +66188.9 q^{89} -16292.4i q^{91} -64646.7i q^{93} +12550.8 q^{97} -19034.6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 196 q^{7} - 1620 q^{9} + 7184 q^{23} - 7160 q^{31} - 2836 q^{33} - 22452 q^{39} - 5804 q^{41} - 44180 q^{47} + 62652 q^{49} + 43696 q^{57} + 1240 q^{63} + 7724 q^{71} - 105136 q^{73} + 7780 q^{79}+ \cdots - 73688 q^{97}+O(q^{100}) \) 20 * q + 196 * q^7 - 1620 * q^9 + 7184 * q^23 - 7160 * q^31 - 2836 * q^33 - 22452 * q^39 - 5804 * q^41 - 44180 * q^47 + 62652 * q^49 + 43696 * q^57 + 1240 * q^63 + 7724 * q^71 - 105136 * q^73 + 7780 * q^79 + 96984 * q^81 + 106188 * q^87 - 3160 * q^89 - 73688 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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