LMFDB - Embedded newform 800.6.f.d.49.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(49,800)]

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, 1]))

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

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.f (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:	$40$
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.13
Character	$\chi$	$=$	800.49
Dual form			800.6.f.d.49.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.4354 q^{3} +66.0164i q^{7} -134.101 q^{9} +141.942i q^{11} +246.793 q^{13} -297.407i q^{17} +174.593i q^{19} -688.910i q^{21} -1576.89i q^{23} +3935.22 q^{27} -922.162i q^{29} +6194.91 q^{31} -1481.22i q^{33} -13956.6 q^{37} -2575.40 q^{39} -3008.44 q^{41} +16270.0 q^{43} -1004.93i q^{47} +12448.8 q^{49} +3103.57i q^{51} -22990.4 q^{53} -1821.96i q^{57} -31893.8i q^{59} -21370.1i q^{61} -8852.89i q^{63} -38648.6 q^{67} +16455.5i q^{69} +30998.5 q^{71} +79573.5i q^{73} -9370.46 q^{77} +23301.9 q^{79} -8479.17 q^{81} +66705.7 q^{83} +9623.18i q^{87} -66188.9 q^{89} +16292.4i q^{91} -64646.7 q^{93} -12550.8i q^{97} -19034.6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 3240 q^{9} - 14320 q^{31} + 44904 q^{39} - 11608 q^{41} - 125304 q^{49} + 15448 q^{71} - 15560 q^{79} + 193968 q^{81} + 6320 q^{89}+O(q^{100}) $ 40 * q + 3240 * q^9 - 14320 * q^31 + 44904 * q^39 - 11608 * q^41 - 125304 * q^49 + 15448 * q^71 - 15560 * q^79 + 193968 * q^81 + 6320 * q^89

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
$3$	−10.4354	−0.669434	−0.334717	−	0.942319i	$-0.608641\pi$
$3$	−10.4354	−0.669434	−0.334717	+	0.942319i	$0.608641\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	66.0164i	0.509221i	0.967044	+	0.254611i	$0.0819473\pi$
$7$	66.0164i	0.509221i	−0.967044	+	0.254611i	$0.918053\pi$
$8$	0	0
$9$	−134.101	−0.551858
$10$	0	0
$11$	141.942i	0.353694i	0.984238	+	0.176847i	$0.0565898\pi$
$11$	141.942i	0.353694i	−0.984238	+	0.176847i	$0.943410\pi$
$12$	0	0
$13$	246.793	0.405018	0.202509	−	0.979280i	$-0.435090\pi$
$13$	246.793	0.405018	0.202509	+	0.979280i	$0.435090\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 297.407i	− 0.249591i	−0.992182	−	0.124795i	$-0.960173\pi$
$17$	− 297.407i	− 0.249591i	0.992182	−	0.124795i	$-0.0398275\pi$
$18$	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
$23$	− 1576.89i	− 0.621558i	−0.950482	−	0.310779i	$-0.899410\pi$
$23$	− 1576.89i	− 0.621558i	0.950482	−	0.310779i	$-0.100590\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3935.22	1.03887
$28$	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
$33$	− 1481.22i	− 0.236775i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−13956.6	−1.67601	−0.838003	−	0.545666i	$-0.816277\pi$
$37$	−13956.6	−1.67601	−0.838003	+	0.545666i	$0.816277\pi$
$38$	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
$43$	16270.0	1.34189	0.670944	−	0.741508i	$-0.265889\pi$
$43$	16270.0	1.34189	0.670944	+	0.741508i	$0.265889\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1004.93i	− 0.0663578i	−0.999449	−	0.0331789i	$-0.989437\pi$
$47$	− 1004.93i	− 0.0663578i	0.999449	−	0.0331789i	$-0.0105631\pi$
$48$	0	0
$49$	12448.8	0.740694
$50$	0	0
$51$	3103.57i	0.167085i
$52$	0	0
$53$	−22990.4	−1.12424	−0.562118	−	0.827057i	$-0.690013\pi$
$53$	−22990.4	−1.12424	−0.562118	+	0.827057i	$0.690013\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1821.96i	− 0.0742764i
$58$	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	$-0.880158\pi$
$61$	− 21370.1i	− 0.735328i	0.929959	−	0.367664i	$-0.119842\pi$
$62$	0	0
$63$	− 8852.89i	− 0.281018i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−38648.6	−1.05183	−0.525916	−	0.850536i	$-0.676277\pi$
$67$	−38648.6	−1.05183	−0.525916	+	0.850536i	$0.676277\pi$
$68$	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
$73$	79573.5i	1.74768i	0.486217	+	0.873838i	$0.338377\pi$
$73$	79573.5i	1.74768i	−0.486217	+	0.873838i	$0.661623\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9370.46	−0.180109
$78$	0	0
$79$	23301.9	0.420072	0.210036	−	0.977694i	$-0.432642\pi$
$79$	23301.9	0.420072	0.210036	+	0.977694i	$0.432642\pi$
$80$	0	0
$81$	−8479.17	−0.143596
$82$	0	0
$83$	66705.7	1.06284	0.531420	−	0.847109i	$-0.321659\pi$
$83$	66705.7	1.06284	0.531420	+	0.847109i	$0.321659\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9623.18i	0.136308i
$88$	0	0
$89$	−66188.9	−0.885748	−0.442874	−	0.896584i	$-0.646041\pi$
$89$	−66188.9	−0.885748	−0.442874	+	0.896584i	$0.646041\pi$
$90$	0	0
$91$	16292.4i	0.206244i
$92$	0	0
$93$	−64646.7	−0.775066
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 12550.8i	− 0.135439i	−0.997704	−	0.0677194i	$-0.978428\pi$
$97$	− 12550.8i	− 0.135439i	0.997704	−	0.0677194i	$-0.0215723\pi$
$98$	0	0
$99$	− 19034.6i	− 0.195189i
$100$	0	0
$101$	− 94627.3i	− 0.923024i	−0.887134	−	0.461512i	$-0.847307\pi$
$101$	− 94627.3i	− 0.923024i	0.887134	−	0.461512i	$-0.152693\pi$
$102$	0	0
$103$	− 74704.4i	− 0.693830i	−0.937897	−	0.346915i	$-0.887229\pi$
$103$	− 74704.4i	− 0.693830i	0.937897	−	0.346915i	$-0.112771\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−95390.0	−0.805459	−0.402729	−	0.915319i	$-0.631938\pi$
$107$	−95390.0	−0.805459	−0.402729	+	0.915319i	$0.631938\pi$
$108$	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.i	1.28790i	0.765067	+	0.643951i	$0.222706\pi$
$113$	174815.i	1.28790i	−0.765067	+	0.643951i	$0.777294\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−33095.3	−0.223512
$118$	0	0
$119$	19633.7	0.127097
$120$	0	0
$121$	140904.	0.874901
$122$	0	0
$123$	31394.4	0.187107
$124$	0	0
$125$	0	0
$126$	0	0
$127$	94893.2i	0.522067i	0.965330	+	0.261033i	$0.0840633\pi$
$127$	94893.2i	0.522067i	−0.965330	+	0.261033i	$0.915937\pi$
$128$	0	0
$129$	−169785.	−0.898307
$130$	0	0
$131$	263177.i	1.33989i	0.742409	+	0.669947i	$0.233683\pi$
$131$	263177.i	1.33989i	−0.742409	+	0.669947i	$0.766317\pi$
$132$	0	0
$133$	−11526.0	−0.0565001
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12040.5i	0.0548080i	0.999624	+	0.0274040i	$0.00872406\pi$
$137$	12040.5i	0.0548080i	−0.999624	+	0.0274040i	$0.991276\pi$
$138$	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
$143$	35030.2i	0.143253i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−129909.	−0.495846
$148$	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
$153$	39882.7i	0.137739i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	155005.	0.501876	0.250938	−	0.968003i	$-0.419261\pi$
$157$	155005.	0.501876	0.250938	+	0.968003i	$0.419261\pi$
$158$	0	0
$159$	239915.	0.752602
$160$	0	0
$161$	104100.	0.316510
$162$	0	0
$163$	−126397.	−0.372621	−0.186311	−	0.982491i	$-0.559653\pi$
$163$	−126397.	−0.372621	−0.186311	+	0.982491i	$0.559653\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	709796.i	1.96944i	0.174152	+	0.984719i	$0.444282\pi$
$167$	709796.i	1.96944i	−0.174152	+	0.984719i	$0.555718\pi$
$168$	0	0
$169$	−310386.	−0.835960
$170$	0	0
$171$	− 23413.2i	− 0.0612308i
$172$	0	0
$173$	−395365.	−1.00435	−0.502173	−	0.864767i	$-0.667466\pi$
$173$	−395365.	−1.00435	−0.502173	+	0.864767i	$0.667466\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	332826.i	0.798517i
$178$	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	$0.397377\pi$
$181$	836091.i	1.89696i	−0.316843	+	0.948478i	$0.602623\pi$
$182$	0	0
$183$	223006.i	0.492254i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	42214.4	0.0882788
$188$	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.i	1.34282i	0.741085	+	0.671411i	$0.234312\pi$
$193$	694883.i	1.34282i	−0.741085	+	0.671411i	$0.765688\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	304179.	0.558424	0.279212	−	0.960229i	$-0.409927\pi$
$197$	304179.	0.558424	0.279212	+	0.960229i	$0.409927\pi$
$198$	0	0
$199$	−575412.	−1.03002	−0.515011	−	0.857184i	$-0.672212\pi$
$199$	−575412.	−1.03002	−0.515011	+	0.857184i	$0.672212\pi$
$200$	0	0
$201$	403315.	0.704133
$202$	0	0
$203$	60877.8	0.103686
$204$	0	0
$205$	0	0
$206$	0	0
$207$	211463.i	0.343011i
$208$	0	0
$209$	−24782.0	−0.0392438
$210$	0	0
$211$	154406.i	0.238758i	0.992849	+	0.119379i	$0.0380903\pi$
$211$	154406.i	0.238758i	−0.992849	+	0.119379i	$0.961910\pi$
$212$	0	0
$213$	−323483.	−0.488543
$214$	0	0
$215$	0	0
$216$	0	0
$217$	408966.i	0.589573i
$218$	0	0
$219$	− 830385.i	− 1.16995i
$220$	0	0
$221$	− 73398.0i	− 0.101089i
$222$	0	0
$223$	817613.i	1.10100i	0.834836	+	0.550498i	$0.185562\pi$
$223$	817613.i	1.10100i	−0.834836	+	0.550498i	$0.814438\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	45598.7	0.0587338	0.0293669	−	0.999569i	$-0.490651\pi$
$227$	45598.7	0.0587338	0.0293669	+	0.999569i	$0.490651\pi$
$228$	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.i	0.296339i	0.988962	+	0.148169i	$0.0473381\pi$
$233$	245572.i	0.296339i	−0.988962	+	0.148169i	$0.952662\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−243166.	−0.281211
$238$	0	0
$239$	−1.13159e6	−1.28142	−0.640711	−	0.767782i	$-0.721361\pi$
$239$	−1.13159e6	−1.28142	−0.640711	+	0.767782i	$0.721361\pi$
$240$	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.	−0.942739
$244$	0	0
$245$	0	0
$246$	0	0
$247$	43088.4i	0.0449384i
$248$	0	0
$249$	−696104.	−0.711501
$250$	0	0
$251$	1.45223e6i	1.45496i	0.686130	+	0.727479i	$0.259308\pi$
$251$	1.45223e6i	1.45496i	−0.686130	+	0.727479i	$0.740692\pi$
$252$	0	0
$253$	223826.	0.219841
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.67551e6i	1.58240i	0.611560	+	0.791198i	$0.290542\pi$
$257$	1.67551e6i	1.58240i	−0.611560	+	0.791198i	$0.709458\pi$
$258$	0	0
$259$	− 921364.i	− 0.853458i
$260$	0	0
$261$	123663.i	0.112367i
$262$	0	0
$263$	977406.i	0.871336i	0.900107	+	0.435668i	$0.143488\pi$
$263$	977406.i	0.871336i	−0.900107	+	0.435668i	$0.856512\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	690711.	0.592950
$268$	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.i	− 0.138067i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.16950e6	−1.69887	−0.849435	−	0.527693i	$-0.823057\pi$
$277$	−2.16950e6	−1.69887	−0.849435	+	0.527693i	$0.823057\pi$
$278$	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.	−0.740753	−0.370376	−	0.928882i	$-0.620771\pi$
$283$	−998020.	−0.740753	−0.370376	+	0.928882i	$0.620771\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 198606.i	− 0.142327i
$288$	0	0
$289$	1.33141e6	0.937704
$290$	0	0
$291$	130974.i	0.0906674i
$292$	0	0
$293$	399603.	0.271932	0.135966	−	0.990714i	$-0.456586\pi$
$293$	399603.	0.271932	0.135966	+	0.990714i	$0.456586\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	558571.i	0.367441i
$298$	0	0
$299$	− 389165.i	− 0.251742i
$300$	0	0
$301$	1.07409e6i	0.683318i
$302$	0	0
$303$	987479.i	0.617904i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	368112.	0.222912	0.111456	−	0.993769i	$-0.464449\pi$
$307$	368112.	0.222912	0.111456	+	0.993769i	$0.464449\pi$
$308$	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.i	0.192975i	0.995334	+	0.0964877i	$0.0307608\pi$
$313$	334474.i	0.192975i	−0.995334	+	0.0964877i	$0.969239\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−837898.	−0.468321	−0.234160	−	0.972198i	$-0.575234\pi$
$317$	−837898.	−0.468321	−0.234160	+	0.972198i	$0.575234\pi$
$318$	0	0
$319$	130893.	0.0720179
$320$	0	0
$321$	995437.	0.539202
$322$	0	0
$323$	51925.2	0.0276931
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.73323e6i	0.896370i
$328$	0	0
$329$	66341.9	0.0337908
$330$	0	0
$331$	− 2.30318e6i	− 1.15547i	−0.816225	−	0.577734i	$-0.803937\pi$
$331$	− 2.30318e6i	− 1.15547i	0.816225	−	0.577734i	$-0.196063\pi$
$332$	0	0
$333$	1.87160e6	0.924917
$334$	0	0
$335$	0	0
$336$	0	0
$337$	698751.i	0.335156i	0.985859	+	0.167578i	$0.0535947\pi$
$337$	698751.i	0.335156i	−0.985859	+	0.167578i	$0.946405\pi$
$338$	0	0
$339$	− 1.82427e6i	− 0.862166i
$340$	0	0
$341$	879315.i	0.409505i
$342$	0	0
$343$	1.93136e6i	0.886398i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.30033e6	−1.47141	−0.735706	−	0.677301i	$-0.763149\pi$
$347$	−3.30033e6	−1.47141	−0.735706	+	0.677301i	$0.763149\pi$
$348$	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
$353$	− 56609.0i	− 0.0241796i	−0.999927	−	0.0120898i	$-0.996152\pi$
$353$	− 56609.0i	− 0.0241796i	0.999927	−	0.0120898i	$-0.00384839\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−204887.	−0.0850831
$358$	0	0
$359$	−446755.	−0.182950	−0.0914751	−	0.995807i	$-0.529158\pi$
$359$	−446755.	−0.182950	−0.0914751	+	0.995807i	$0.529158\pi$
$360$	0	0
$361$	2.44562e6	0.987689
$362$	0	0
$363$	−1.47039e6	−0.585688
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 304081.i	− 0.117849i	−0.998262	−	0.0589243i	$-0.981233\pi$
$367$	− 304081.i	− 0.117849i	0.998262	−	0.0589243i	$-0.0187671\pi$
$368$	0	0
$369$	403436.	0.154244
$370$	0	0
$371$	− 1.51774e6i	− 0.572484i
$372$	0	0
$373$	−932701.	−0.347112	−0.173556	−	0.984824i	$-0.555526\pi$
$373$	−932701.	−0.347112	−0.173556	+	0.984824i	$0.555526\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 227583.i	− 0.0824683i
$378$	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
$383$	1.80993e6i	0.630470i	0.949014	+	0.315235i	$0.102083\pi$
$383$	1.80993e6i	0.630470i	−0.949014	+	0.315235i	$0.897917\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.18183e6	−0.740532
$388$	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
$393$	− 2.74637e6i	− 0.896971i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.67426e6	0.533145	0.266573	−	0.963815i	$-0.414109\pi$
$397$	1.67426e6	0.533145	0.266573	+	0.963815i	$0.414109\pi$
$398$	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
$403$	1.52886e6	0.468927
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.98102e6i	− 0.592793i
$408$	0	0
$409$	−1.59010e6	−0.470019	−0.235010	−	0.971993i	$-0.575512\pi$
$409$	−1.59010e6	−0.470019	−0.235010	+	0.971993i	$0.575512\pi$
$410$	0	0
$411$	− 125648.i	− 0.0366904i
$412$	0	0
$413$	2.10551e6	0.607411
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 3.73274e6i	− 1.05121i
$418$	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	$0.251878\pi$
$421$	5.17330e6i	1.42253i	−0.702922	+	0.711267i	$0.748122\pi$
$422$	0	0
$423$	134763.i	0.0366200i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.41077e6	0.374445
$428$	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
$433$	2.18503e6i	0.560063i	0.959991	+	0.280031i	$0.0903449\pi$
$433$	2.18503e6i	0.560063i	−0.959991	+	0.280031i	$0.909655\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	275314.	0.0689643
$438$	0	0
$439$	1.22033e6	0.302216	0.151108	−	0.988517i	$-0.451716\pi$
$439$	1.22033e6	0.302216	0.151108	+	0.988517i	$0.451716\pi$
$440$	0	0
$441$	−1.66941e6	−0.408758
$442$	0	0
$443$	−509711.	−0.123400	−0.0617000	−	0.998095i	$-0.519652\pi$
$443$	−509711.	−0.123400	−0.0617000	+	0.998095i	$0.519652\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.77219e6i	0.892945i
$448$	0	0
$449$	−3.54089e6	−0.828890	−0.414445	−	0.910074i	$-0.636024\pi$
$449$	−3.54089e6	−0.828890	−0.414445	+	0.910074i	$0.636024\pi$
$450$	0	0
$451$	− 427022.i	− 0.0988573i
$452$	0	0
$453$	−3.52654e6	−0.807427
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1.06559e6i	− 0.238671i	−0.992854	−	0.119335i	$-0.961924\pi$
$457$	− 1.06559e6i	− 0.238671i	0.992854	−	0.119335i	$-0.0380764\pi$
$458$	0	0
$459$	− 1.17036e6i	− 0.259292i
$460$	0	0
$461$	− 6.92975e6i	− 1.51868i	−0.650696	−	0.759339i	$-0.725523\pi$
$461$	− 6.92975e6i	− 1.51868i	0.650696	−	0.759339i	$-0.274477\pi$
$462$	0	0
$463$	− 7.21004e6i	− 1.56310i	−0.623845	−	0.781548i	$-0.714431\pi$
$463$	− 7.21004e6i	− 1.56310i	0.623845	−	0.781548i	$-0.285569\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.00541e6	0.637693	0.318846	−	0.947806i	$-0.396705\pi$
$467$	3.00541e6	0.637693	0.318846	+	0.947806i	$0.396705\pi$
$468$	0	0
$469$	− 2.55144e6i	− 0.535615i
$470$	0	0
$471$	−1.61755e6	−0.335973
$472$	0	0
$473$	2.30939e6i	0.474618i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.08305e6	0.620418
$478$	0	0
$479$	2.47740e6	0.493353	0.246676	−	0.969098i	$-0.420662\pi$
$479$	2.47740e6	0.493353	0.246676	+	0.969098i	$0.420662\pi$
$480$	0	0
$481$	−3.44439e6	−0.678813
$482$	0	0
$483$	−1.08633e6	−0.211883
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.58130e6i	− 0.875320i	−0.899141	−	0.437660i	$-0.855807\pi$
$487$	− 4.58130e6i	− 0.875320i	0.899141	−	0.437660i	$-0.144193\pi$
$488$	0	0
$489$	1.31901e6	0.249446
$490$	0	0
$491$	8.23983e6i	1.54246i	0.636555	+	0.771231i	$0.280359\pi$
$491$	8.23983e6i	1.54246i	−0.636555	+	0.771231i	$0.719641\pi$
$492$	0	0
$493$	−274257.	−0.0508207
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.04641e6i	0.371622i
$498$	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
$503$	− 6.28876e6i	− 1.10827i	−0.832427	−	0.554134i	$-0.813049\pi$
$503$	− 6.28876e6i	− 1.10827i	0.832427	−	0.554134i	$-0.186951\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.23902e6	0.559620
$508$	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.i	0.115266i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	142642.	0.0234704
$518$	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
$523$	9.74999e6	1.55865	0.779327	−	0.626617i	$-0.215561\pi$
$523$	9.74999e6	1.55865	0.779327	+	0.626617i	$0.215561\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.84241e6i	− 0.288975i
$528$	0	0
$529$	3.94977e6	0.613666
$530$	0	0
$531$	4.27701e6i	0.658269i
$532$	0	0
$533$	−742461.	−0.113202
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.77497e6i	− 0.415262i
$538$	0	0
$539$	1.76701e6i	0.261979i
$540$	0	0
$541$	− 1.86030e6i	− 0.273268i	−0.990622	−	0.136634i	$-0.956372\pi$
$541$	− 1.86030e6i	− 0.273268i	0.990622	−	0.136634i	$-0.0436285\pi$
$542$	0	0
$543$	− 8.72499e6i	− 1.26989i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	725841.	0.103723	0.0518613	−	0.998654i	$-0.483485\pi$
$547$	725841.	0.103723	0.0518613	+	0.998654i	$0.483485\pi$
$548$	0	0
$549$	2.86576e6i	0.405796i
$550$	0	0
$551$	161003.	0.0225920
$552$	0	0
$553$	1.53831e6i	0.213910i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.72234e6	0.644939	0.322470	−	0.946580i	$-0.395487\pi$
$557$	4.72234e6	0.644939	0.322470	+	0.946580i	$0.395487\pi$
$558$	0	0
$559$	4.01533e6	0.543490
$560$	0	0
$561$	−440526.	−0.0590969
$562$	0	0
$563$	4.74239e6	0.630560	0.315280	−	0.948999i	$-0.397902\pi$
$563$	4.74239e6	0.630560	0.315280	+	0.948999i	$0.397902\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 559764.i	− 0.0731219i
$568$	0	0
$569$	2.99470e6	0.387768	0.193884	−	0.981024i	$-0.437891\pi$
$569$	2.99470e6	0.387768	0.193884	+	0.981024i	$0.437891\pi$
$570$	0	0
$571$	− 1.01416e7i	− 1.30172i	−0.759198	−	0.650860i	$-0.774409\pi$
$571$	− 1.01416e7i	− 1.30172i	0.759198	−	0.650860i	$-0.225591\pi$
$572$	0	0
$573$	−8.03722e6	−1.02263
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 3.80870e6i	− 0.476253i	−0.971234	−	0.238126i	$-0.923467\pi$
$577$	− 3.80870e6i	− 0.476253i	0.971234	−	0.238126i	$-0.0765333\pi$
$578$	0	0
$579$	− 7.25142e6i	− 0.898931i
$580$	0	0
$581$	4.40367e6i	0.541220i
$582$	0	0
$583$	− 3.26330e6i	− 0.397635i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.40054e6	0.407335	0.203668	−	0.979040i	$-0.434714\pi$
$587$	3.40054e6	0.407335	0.203668	+	0.979040i	$0.434714\pi$
$588$	0	0
$589$	1.08159e6i	0.128462i
$590$	0	0
$591$	−3.17425e6	−0.373828
$592$	0	0
$593$	− 1.27800e7i	− 1.49243i	−0.665704	−	0.746216i	$-0.731868\pi$
$593$	− 1.27800e7i	− 1.49243i	0.665704	−	0.746216i	$-0.268132\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.00468e6	0.689532
$598$	0	0
$599$	−614365.	−0.0699616	−0.0349808	−	0.999388i	$-0.511137\pi$
$599$	−614365.	−0.0699616	−0.0349808	+	0.999388i	$0.511137\pi$
$600$	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
$603$	5.18283e6	0.580462
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.81019e6i	0.199413i	0.995017	+	0.0997065i	$0.0317904\pi$
$607$	1.81019e6i	0.199413i	−0.995017	+	0.0997065i	$0.968210\pi$
$608$	0	0
$609$	−635287.	−0.0694108
$610$	0	0
$611$	− 248010.i	− 0.0268761i
$612$	0	0
$613$	−1.08728e7	−1.16867	−0.584335	−	0.811513i	$-0.698645\pi$
$613$	−1.08728e7	−1.16867	−0.584335	+	0.811513i	$0.698645\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.60152e6i	− 0.698122i	−0.937100	−	0.349061i	$-0.886501\pi$
$617$	− 6.60152e6i	− 0.698122i	0.937100	−	0.349061i	$-0.113499\pi$
$618$	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
$623$	− 4.36955e6i	− 0.451042i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	258611.	0.0262711
$628$	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
$633$	− 1.61129e6i	− 0.159833i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.07229e6	0.299995
$638$	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
$643$	7.89022e6	0.752595	0.376298	−	0.926499i	$-0.377197\pi$
$643$	7.89022e6	0.752595	0.376298	+	0.926499i	$0.377197\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.41387e7i	− 1.32785i	−0.747800	−	0.663924i	$-0.768890\pi$
$647$	− 1.41387e7i	− 1.32785i	0.747800	−	0.663924i	$-0.231110\pi$
$648$	0	0
$649$	4.52706e6	0.421895
$650$	0	0
$651$	− 4.26774e6i	− 0.394680i
$652$	0	0
$653$	2.09667e7	1.92419	0.962094	−	0.272720i	$-0.0879232\pi$
$653$	2.09667e7	1.92419	0.962094	+	0.272720i	$0.0879232\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.06709e7i	− 0.964469i
$658$	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	$0.172373\pi$
$661$	1.15802e7i	1.03089i	−0.856923	+	0.515444i	$0.827627\pi$
$662$	0	0
$663$	765941.i	0.0676724i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.45415e6	−0.126559
$668$	0	0
$669$	− 8.53216e6i	− 0.737045i
$670$	0	0
$671$	3.03330e6	0.260081
$672$	0	0
$673$	3.44202e6i	0.292938i	0.989215	+	0.146469i	$0.0467909\pi$
$673$	3.44202e6i	0.292938i	−0.989215	+	0.146469i	$0.953209\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.81661e7	−1.52332	−0.761658	−	0.647980i	$-0.775614\pi$
$677$	−1.81661e7	−1.52332	−0.761658	+	0.647980i	$0.775614\pi$
$678$	0	0
$679$	828560.	0.0689683
$680$	0	0
$681$	−475843.	−0.0393184
$682$	0	0
$683$	−1.58711e7	−1.30184	−0.650918	−	0.759148i	$-0.725616\pi$
$683$	−1.58711e7	−1.30184	−0.650918	+	0.759148i	$0.725616\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 6.61490e6i	− 0.534726i
$688$	0	0
$689$	−5.67388e6	−0.455336
$690$	0	0
$691$	− 7.37470e6i	− 0.587556i	−0.955874	−	0.293778i	$-0.905087\pi$
$691$	− 7.37470e6i	− 0.587556i	0.955874	−	0.293778i	$-0.0949127\pi$
$692$	0	0
$693$	1.25659e6	0.0993943
$694$	0	0
$695$	0	0
$696$	0	0
$697$	894730.i	0.0697605i
$698$	0	0
$699$	− 2.56265e6i	− 0.198379i
$700$	0	0
$701$	− 2.16654e7i	− 1.66522i	−0.553857	−	0.832612i	$-0.686845\pi$
$701$	− 2.16654e7i	− 1.66522i	0.553857	−	0.832612i	$-0.313155\pi$
$702$	0	0
$703$	− 2.43673e6i	− 0.185960i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.24695e6	0.470023
$708$	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
$713$	− 9.76869e6i	− 0.719635i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.18086e7	0.857829
$718$	0	0
$719$	1.42857e7	1.03058	0.515288	−	0.857017i	$-0.327685\pi$
$719$	1.42857e7	1.03058	0.515288	+	0.857017i	$0.327685\pi$
$720$	0	0
$721$	4.93171e6	0.353313
$722$	0	0
$723$	−1.63235e7	−1.16136
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.55756e6i	0.249641i	0.992179	+	0.124821i	$0.0398355\pi$
$727$	3.55756e6i	0.249641i	−0.992179	+	0.124821i	$0.960164\pi$
$728$	0	0
$729$	1.11161e7	0.774697
$730$	0	0
$731$	− 4.83881e6i	− 0.334923i
$732$	0	0
$733$	1.61816e7	1.11240	0.556201	−	0.831048i	$-0.312258\pi$
$733$	1.61816e7	1.11240	0.556201	+	0.831048i	$0.312258\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.48584e6i	− 0.372027i
$738$	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
$743$	2.41917e7i	1.60766i	0.594860	+	0.803829i	$0.297208\pi$
$743$	2.41917e7i	1.60766i	−0.594860	+	0.803829i	$0.702792\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.94532e6	−0.586536
$748$	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
$753$	− 1.51546e7i	− 0.973998i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.68129e7	−1.06636	−0.533179	−	0.846002i	$-0.679003\pi$
$757$	−1.68129e7	−1.06636	−0.533179	+	0.846002i	$0.679003\pi$
$758$	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
$763$	1.09647e7	0.681845
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 7.87117e6i	− 0.483116i
$768$	0	0
$769$	3.03763e6	0.185233	0.0926166	−	0.995702i	$-0.470477\pi$
$769$	3.03763e6	0.185233	0.0926166	+	0.995702i	$0.470477\pi$
$770$	0	0
$771$	− 1.74847e7i	− 1.05931i
$772$	0	0
$773$	2.73040e7	1.64353	0.821764	−	0.569828i	$-0.192990\pi$
$773$	2.73040e7	1.64353	0.821764	+	0.569828i	$0.192990\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.61485e6i	0.571334i
$778$	0	0
$779$	− 525252.i	− 0.0310116i
$780$	0	0
$781$	4.39997e6i	0.258120i
$782$	0	0
$783$	− 3.62891e6i	− 0.211530i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.58229e7	−0.910644	−0.455322	−	0.890327i	$-0.650476\pi$
$787$	−1.58229e7	−0.910644	−0.455322	+	0.890327i	$0.650476\pi$
$788$	0	0
$789$	− 1.01997e7i	− 0.583302i
$790$	0	0
$791$	−1.15407e7	−0.655827
$792$	0	0
$793$	− 5.27398e6i	− 0.297821i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.53728e6	−0.308781	−0.154390	−	0.988010i	$-0.549341\pi$
$797$	−5.53728e6	−0.308781	−0.154390	+	0.988010i	$0.549341\pi$
$798$	0	0
$799$	−298874.	−0.0165623
$800$	0	0
$801$	8.87602e6	0.488807
$802$	0	0
$803$	−1.12948e7	−0.618143
$804$	0	0
$805$	0	0
$806$	0	0
$807$	164164.i	0.00887348i
$808$	0	0
$809$	4.59562e6	0.246873	0.123436	−	0.992352i	$-0.460609\pi$
$809$	4.59562e6	0.246873	0.123436	+	0.992352i	$0.460609\pi$
$810$	0	0
$811$	− 3.14901e6i	− 0.168121i	−0.996461	−	0.0840604i	$-0.973211\pi$
$811$	− 3.14901e6i	− 0.168121i	0.996461	−	0.0840604i	$-0.0267889\pi$
$812$	0	0
$813$	−8.98872e6	−0.476949
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.84063e6i	0.148888i
$818$	0	0
$819$	− 2.18483e6i	− 0.113817i
$820$	0	0
$821$	1.98853e7i	1.02961i	0.857306	+	0.514807i	$0.172136\pi$
$821$	1.98853e7i	1.02961i	−0.857306	+	0.514807i	$0.827864\pi$
$822$	0	0
$823$	1.53764e7i	0.791328i	0.918395	+	0.395664i	$0.129485\pi$
$823$	1.53764e7i	0.791328i	−0.918395	+	0.395664i	$0.870515\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.95740e6	0.455426	0.227713	−	0.973728i	$-0.426875\pi$
$827$	8.95740e6	0.455426	0.227713	+	0.973728i	$0.426875\pi$
$828$	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
$833$	− 3.70237e6i	− 0.184870i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.43784e7	1.20279
$838$	0	0
$839$	−1.26360e7	−0.619732	−0.309866	−	0.950780i	$-0.600284\pi$
$839$	−1.26360e7	−0.619732	−0.309866	+	0.950780i	$0.600284\pi$
$840$	0	0
$841$	1.96608e7	0.958540
$842$	0	0
$843$	−1.89713e7	−0.919449
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.30194e6i	0.445518i
$848$	0	0
$849$	1.04148e7	0.495885
$850$	0	0
$851$	2.20080e7i	1.04173i
$852$	0	0
$853$	−6.41059e6	−0.301665	−0.150833	−	0.988559i	$-0.548195\pi$
$853$	−6.41059e6	−0.301665	−0.150833	+	0.988559i	$0.548195\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.57309e7i	0.731646i	0.930684	+	0.365823i	$0.119212\pi$
$857$	1.57309e7i	0.731646i	−0.930684	+	0.365823i	$0.880788\pi$
$858$	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
$863$	1.62060e6i	0.0740709i	0.999314	+	0.0370355i	$0.0117915\pi$
$863$	1.62060e6i	0.0740709i	−0.999314	+	0.0370355i	$0.988209\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.38938e7	−0.627732
$868$	0	0
$869$	3.30751e6i	0.148577i
$870$	0	0
$871$	−9.53820e6	−0.426011
$872$	0	0
$873$	1.68308e6i	0.0747430i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5.93461e6	−0.260551	−0.130276	−	0.991478i	$-0.541586\pi$
$877$	−5.93461e6	−0.260551	−0.130276	+	0.991478i	$0.541586\pi$
$878$	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
$883$	5.72329e6	0.247027	0.123513	−	0.992343i	$-0.460584\pi$
$883$	5.72329e6	0.247027	0.123513	+	0.992343i	$0.460584\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.46048e7i	− 0.623285i	−0.950199	−	0.311643i	$-0.899121\pi$
$887$	− 1.46048e7i	− 0.623285i	0.950199	−	0.311643i	$-0.100879\pi$
$888$	0	0
$889$	−6.26451e6	−0.265847
$890$	0	0
$891$	− 1.20355e6i	− 0.0507889i
$892$	0	0
$893$	175454.	0.00736266
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.06111e6i	0.168525i
$898$	0	0
$899$	− 5.71272e6i	− 0.235745i
$900$	0	0
$901$	6.83751e6i	0.280599i
$902$	0	0
$903$	− 1.12086e7i	− 0.457437i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.43818e6	0.219501	0.109750	−	0.993959i	$-0.464995\pi$
$907$	5.43818e6	0.219501	0.109750	+	0.993959i	$0.464995\pi$
$908$	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
$913$	9.46831e6i	0.375920i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.73740e7	−0.682302
$918$	0	0
$919$	−2.24895e7	−0.878398	−0.439199	−	0.898390i	$-0.644738\pi$
$919$	−2.24895e7	−0.878398	−0.439199	+	0.898390i	$0.644738\pi$
$920$	0	0
$921$	−3.84142e6	−0.149225
$922$	0	0
$923$	7.65021e6	0.295576
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.00180e7i	0.382895i
$928$	0	0
$929$	−2.80271e7	−1.06547	−0.532733	−	0.846283i	$-0.678835\pi$
$929$	−2.80271e7	−1.06547	−0.532733	+	0.846283i	$0.678835\pi$
$930$	0	0
$931$	2.17348e6i	0.0821830i
$932$	0	0
$933$	−1.18462e7	−0.445528
$934$	0	0
$935$	0	0
$936$	0	0
$937$	62078.6i	0.00230990i	0.999999	+	0.00115495i	$0.000367632\pi$
$937$	62078.6i	0.00230990i	−0.999999	+	0.00115495i	$0.999632\pi$
$938$	0	0
$939$	− 3.49039e6i	− 0.129184i
$940$	0	0
$941$	4.45412e6i	0.163979i	0.996633	+	0.0819894i	$0.0261274\pi$
$941$	4.45412e6i	0.163979i	−0.996633	+	0.0819894i	$0.973873\pi$
$942$	0	0
$943$	4.74397e6i	0.173725i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.59793e7	1.66605	0.833024	−	0.553237i	$-0.186608\pi$
$947$	4.59793e7	1.66605	0.833024	+	0.553237i	$0.186608\pi$
$948$	0	0
$949$	1.96382e7i	0.707841i
$950$	0	0
$951$	8.74385e6	0.313510
$952$	0	0
$953$	2.32064e7i	0.827706i	0.910344	+	0.413853i	$0.135817\pi$
$953$	2.32064e7i	0.827706i	−0.910344	+	0.413853i	$0.864183\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.36593e6	−0.0482112
$958$	0	0
$959$	−794872.	−0.0279094
$960$	0	0
$961$	9.74779e6	0.340485
$962$	0	0
$963$	1.27919e7	0.444499
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5.33110e7i	1.83337i	0.399607	+	0.916686i	$0.369147\pi$
$967$	5.33110e7i	1.83337i	−0.399607	+	0.916686i	$0.630853\pi$
$968$	0	0
$969$	−541863.	−0.0185387
$970$	0	0
$971$	2.49744e7i	0.850056i	0.905180	+	0.425028i	$0.139736\pi$
$971$	2.49744e7i	0.850056i	−0.905180	+	0.425028i	$0.860264\pi$
$972$	0	0
$973$	−2.36139e7	−0.799625
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.97806e7i	1.33332i	0.745361	+	0.666661i	$0.232277\pi$
$977$	3.97806e7i	1.33332i	−0.745361	+	0.666661i	$0.767723\pi$
$978$	0	0
$979$	− 9.39495e6i	− 0.313284i
$980$	0	0
$981$	2.22730e7i	0.738936i
$982$	0	0
$983$	3.80176e7i	1.25488i	0.778666	+	0.627439i	$0.215897\pi$
$983$	3.80176e7i	1.25488i	−0.778666	+	0.627439i	$0.784103\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−692308.	−0.0226207
$988$	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
$993$	2.40347e7i	0.773510i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−9.27058e6	−0.295372	−0.147686	−	0.989034i	$-0.547182\pi$
$997$	−9.27058e6	−0.295372	−0.147686	+	0.989034i	$0.547182\pi$
$998$	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.f.d.49.13		40
4.3	odd	2		200.6.f.d.149.15		40
5.2	odd	4		800.6.d.d.401.14		20
5.3	odd	4		800.6.d.b.401.7		20
5.4	even	2	inner	800.6.f.d.49.28		40
8.3	odd	2		200.6.f.d.149.25		40
8.5	even	2	inner	800.6.f.d.49.27		40
20.3	even	4		200.6.d.d.101.4	yes	20
20.7	even	4		200.6.d.c.101.17	✓	20
20.19	odd	2		200.6.f.d.149.26		40
40.3	even	4		200.6.d.d.101.3	yes	20
40.13	odd	4		800.6.d.b.401.14		20
40.19	odd	2		200.6.f.d.149.16		40
40.27	even	4		200.6.d.c.101.18	yes	20
40.29	even	2	inner	800.6.f.d.49.14		40
40.37	odd	4		800.6.d.d.401.7		20

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

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-10.4354 q^{3} +66.0164i q^{7} -134.101 q^{9} +141.942i q^{11} +246.793 q^{13} -297.407i q^{17} +174.593i q^{19} -688.910i q^{21} -1576.89i q^{23} +3935.22 q^{27} -922.162i q^{29} +6194.91 q^{31} -1481.22i q^{33} -13956.6 q^{37} -2575.40 q^{39} -3008.44 q^{41} +16270.0 q^{43} -1004.93i q^{47} +12448.8 q^{49} +3103.57i q^{51} -22990.4 q^{53} -1821.96i q^{57} -31893.8i q^{59} -21370.1i q^{61} -8852.89i q^{63} -38648.6 q^{67} +16455.5i q^{69} +30998.5 q^{71} +79573.5i q^{73} -9370.46 q^{77} +23301.9 q^{79} -8479.17 q^{81} +66705.7 q^{83} +9623.18i q^{87} -66188.9 q^{89} +16292.4i q^{91} -64646.7 q^{93} -12550.8i q^{97} -19034.6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 3240 q^{9} - 14320 q^{31} + 44904 q^{39} - 11608 q^{41} - 125304 q^{49} + 15448 q^{71} - 15560 q^{79} + 193968 q^{81} + 6320 q^{89}+O(q^{100}) \) 40 * q + 3240 * q^9 - 14320 * q^31 + 44904 * q^39 - 11608 * q^41 - 125304 * q^49 + 15448 * q^71 - 15560 * q^79 + 193968 * q^81 + 6320 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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