LMFDB - Embedded newform 208.10.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,10,Mod(1,208)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.a (trivial)

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:	yes
Analytic conductor:	$107.127453922$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2119705.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 376x + 1820 $ x^3 - x^2 - 376*x + 1820
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-21.0141$ of defining polynomial
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-85.7459 q^{3} +1787.19 q^{5} +751.989 q^{7} -12330.6 q^{9} -58551.0 q^{11} -28561.0 q^{13} -153244. q^{15} +234846. q^{17} +103190. q^{19} -64480.0 q^{21} -1.08806e6 q^{23} +1.24092e6 q^{25} +2.74504e6 q^{27} +3.17511e6 q^{29} +6.17201e6 q^{31} +5.02051e6 q^{33} +1.34395e6 q^{35} -1.56181e7 q^{37} +2.44899e6 q^{39} -2.01455e7 q^{41} +2.69024e7 q^{43} -2.20372e7 q^{45} +3.03118e7 q^{47} -3.97881e7 q^{49} -2.01371e7 q^{51} +6.56149e7 q^{53} -1.04642e8 q^{55} -8.84808e6 q^{57} -1.12025e8 q^{59} +2.44277e7 q^{61} -9.27250e6 q^{63} -5.10439e7 q^{65} -1.29055e8 q^{67} +9.32967e7 q^{69} +1.12546e8 q^{71} -1.77758e8 q^{73} -1.06404e8 q^{75} -4.40297e7 q^{77} +6.05015e8 q^{79} +7.32791e6 q^{81} -2.01755e8 q^{83} +4.19714e8 q^{85} -2.72253e8 q^{87} +8.62460e8 q^{89} -2.14776e7 q^{91} -5.29225e8 q^{93} +1.84419e8 q^{95} -2.15081e8 q^{97} +7.21971e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 156 q^{3} - 1272 q^{5} - 17058 q^{7} + 42273 q^{9} - 73974 q^{11} - 85683 q^{13} - 393756 q^{15} + 374976 q^{17} - 418338 q^{19} - 284694 q^{21} - 1026168 q^{23} + 3337287 q^{25} - 4218588 q^{27}+ \cdots + 2480087466 q^{99}+O(q^{100}) $ 3 * q - 156 * q^3 - 1272 * q^5 - 17058 * q^7 + 42273 * q^9 - 73974 * q^11 - 85683 * q^13 - 393756 * q^15 + 374976 * q^17 - 418338 * q^19 - 284694 * q^21 - 1026168 * q^23 + 3337287 * q^25 - 4218588 * q^27 - 3075834 * q^29 - 9286482 * q^31 - 25456452 * q^33 + 31802592 * q^35 - 17647776 * q^37 + 4455516 * q^39 - 47257110 * q^41 + 60023760 * q^43 - 81112122 * q^45 + 40824726 * q^47 + 45921177 * q^49 + 27559188 * q^51 + 8072046 * q^53 + 36910236 * q^55 + 245224476 * q^57 - 138035310 * q^59 + 264203886 * q^61 - 436094838 * q^63 + 36329592 * q^65 - 203167074 * q^67 - 397637544 * q^69 - 123067110 * q^71 - 433013250 * q^73 + 884469216 * q^75 + 380043264 * q^77 - 406418748 * q^79 + 100121427 * q^81 - 105365610 * q^83 + 5268186 * q^85 + 12753768 * q^87 + 1365375798 * q^89 + 487193538 * q^91 - 95909856 * q^93 + 85600692 * q^95 + 669691662 * q^97 + 2480087466 * q^99

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
$3$	−85.7459	−0.611178	−0.305589	−	0.952163i	$-0.598853\pi$
$3$	−85.7459	−0.611178	−0.305589	+	0.952163i	$0.598853\pi$
$4$	0	0
$5$	1787.19	1.27881	0.639404	−	0.768871i	$-0.279181\pi$
$5$	1787.19	1.27881	0.639404	+	0.768871i	$0.279181\pi$
$6$	0	0
$7$	751.989	0.118378	0.0591889	−	0.998247i	$-0.481149\pi$
$7$	751.989	0.118378	0.0591889	+	0.998247i	$0.481149\pi$
$8$	0	0
$9$	−12330.6	−0.626461
$10$	0	0
$11$	−58551.0	−1.20578	−0.602889	−	0.797825i	$-0.705984\pi$
$11$	−58551.0	−1.20578	−0.602889	+	0.797825i	$0.705984\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	−153244.	−0.781580
$16$	0	0
$17$	234846.	0.681966	0.340983	−	0.940069i	$-0.389240\pi$
$17$	234846.	0.681966	0.340983	+	0.940069i	$0.389240\pi$
$18$	0	0
$19$	103190.	0.181654	0.0908269	−	0.995867i	$-0.471049\pi$
$19$	103190.	0.181654	0.0908269	+	0.995867i	$0.471049\pi$
$20$	0	0
$21$	−64480.0	−0.0723499
$22$	0	0
$23$	−1.08806e6	−0.810732	−0.405366	−	0.914154i	$-0.632856\pi$
$23$	−1.08806e6	−0.810732	−0.405366	+	0.914154i	$0.632856\pi$
$24$	0	0
$25$	1.24092e6	0.635350
$26$	0	0
$27$	2.74504e6	0.994058
$28$	0	0
$29$	3.17511e6	0.833620	0.416810	−	0.908994i	$-0.363148\pi$
$29$	3.17511e6	0.833620	0.416810	+	0.908994i	$0.363148\pi$
$30$	0	0
$31$	6.17201e6	1.20033	0.600163	−	0.799878i	$-0.295102\pi$
$31$	6.17201e6	1.20033	0.600163	+	0.799878i	$0.295102\pi$
$32$	0	0
$33$	5.02051e6	0.736945
$34$	0	0
$35$	1.34395e6	0.151382
$36$	0	0
$37$	−1.56181e7	−1.37000	−0.684998	−	0.728545i	$-0.740197\pi$
$37$	−1.56181e7	−1.37000	−0.684998	+	0.728545i	$0.740197\pi$
$38$	0	0
$39$	2.44899e6	0.169510
$40$	0	0
$41$	−2.01455e7	−1.11340	−0.556700	−	0.830714i	$-0.687933\pi$
$41$	−2.01455e7	−1.11340	−0.556700	+	0.830714i	$0.687933\pi$
$42$	0	0
$43$	2.69024e7	1.20001	0.600003	−	0.799998i	$-0.295166\pi$
$43$	2.69024e7	1.20001	0.600003	+	0.799998i	$0.295166\pi$
$44$	0	0
$45$	−2.20372e7	−0.801123
$46$	0	0
$47$	3.03118e7	0.906090	0.453045	−	0.891488i	$-0.350338\pi$
$47$	3.03118e7	0.906090	0.453045	+	0.891488i	$0.350338\pi$
$48$	0	0
$49$	−3.97881e7	−0.985987
$50$	0	0
$51$	−2.01371e7	−0.416803
$52$	0	0
$53$	6.56149e7	1.14225	0.571125	−	0.820863i	$-0.306507\pi$
$53$	6.56149e7	1.14225	0.571125	+	0.820863i	$0.306507\pi$
$54$	0	0
$55$	−1.04642e8	−1.54196
$56$	0	0
$57$	−8.84808e6	−0.111023
$58$	0	0
$59$	−1.12025e8	−1.20359	−0.601796	−	0.798650i	$-0.705548\pi$
$59$	−1.12025e8	−1.20359	−0.601796	+	0.798650i	$0.705548\pi$
$60$	0	0
$61$	2.44277e7	0.225891	0.112945	−	0.993601i	$-0.463971\pi$
$61$	2.44277e7	0.225891	0.112945	+	0.993601i	$0.463971\pi$
$62$	0	0
$63$	−9.27250e6	−0.0741591
$64$	0	0
$65$	−5.10439e7	−0.354678
$66$	0	0
$67$	−1.29055e8	−0.782419	−0.391209	−	0.920302i	$-0.627943\pi$
$67$	−1.29055e8	−0.782419	−0.391209	+	0.920302i	$0.627943\pi$
$68$	0	0
$69$	9.32967e7	0.495502
$70$	0	0
$71$	1.12546e8	0.525616	0.262808	−	0.964848i	$-0.415351\pi$
$71$	1.12546e8	0.525616	0.262808	+	0.964848i	$0.415351\pi$
$72$	0	0
$73$	−1.77758e8	−0.732617	−0.366308	−	0.930493i	$-0.619378\pi$
$73$	−1.77758e8	−0.732617	−0.366308	+	0.930493i	$0.619378\pi$
$74$	0	0
$75$	−1.06404e8	−0.388312
$76$	0	0
$77$	−4.40297e7	−0.142737
$78$	0	0
$79$	6.05015e8	1.74761	0.873805	−	0.486277i	$-0.161645\pi$
$79$	6.05015e8	1.74761	0.873805	+	0.486277i	$0.161645\pi$
$80$	0	0
$81$	7.32791e6	0.0189146
$82$	0	0
$83$	−2.01755e8	−0.466630	−0.233315	−	0.972401i	$-0.574957\pi$
$83$	−2.01755e8	−0.466630	−0.233315	+	0.972401i	$0.574957\pi$
$84$	0	0
$85$	4.19714e8	0.872103
$86$	0	0
$87$	−2.72253e8	−0.509491
$88$	0	0
$89$	8.62460e8	1.45708	0.728541	−	0.685002i	$-0.240199\pi$
$89$	8.62460e8	1.45708	0.728541	+	0.685002i	$0.240199\pi$
$90$	0	0
$91$	−2.14776e7	−0.0328321
$92$	0	0
$93$	−5.29225e8	−0.733613
$94$	0	0
$95$	1.84419e8	0.232300
$96$	0	0
$97$	−2.15081e8	−0.246677	−0.123339	−	0.992365i	$-0.539360\pi$
$97$	−2.15081e8	−0.246677	−0.123339	+	0.992365i	$0.539360\pi$
$98$	0	0
$99$	7.21971e8	0.755373
$100$	0	0
$101$	8.80961e8	0.842385	0.421192	−	0.906971i	$-0.361612\pi$
$101$	8.80961e8	0.842385	0.421192	+	0.906971i	$0.361612\pi$
$102$	0	0
$103$	1.67438e9	1.46584	0.732918	−	0.680317i	$-0.238158\pi$
$103$	1.67438e9	1.46584	0.732918	+	0.680317i	$0.238158\pi$
$104$	0	0
$105$	−1.15238e8	−0.0925217
$106$	0	0
$107$	3.53576e8	0.260769	0.130385	−	0.991463i	$-0.458379\pi$
$107$	3.53576e8	0.260769	0.130385	+	0.991463i	$0.458379\pi$
$108$	0	0
$109$	1.69143e9	1.14772	0.573858	−	0.818955i	$-0.305446\pi$
$109$	1.69143e9	1.14772	0.573858	+	0.818955i	$0.305446\pi$
$110$	0	0
$111$	1.33918e9	0.837312
$112$	0	0
$113$	2.01468e9	1.16239	0.581196	−	0.813763i	$-0.302585\pi$
$113$	2.01468e9	1.16239	0.581196	+	0.813763i	$0.302585\pi$
$114$	0	0
$115$	−1.94457e9	−1.03677
$116$	0	0
$117$	3.52175e8	0.173749
$118$	0	0
$119$	1.76601e8	0.0807296
$120$	0	0
$121$	1.07027e9	0.453901
$122$	0	0
$123$	1.72740e9	0.680486
$124$	0	0
$125$	−1.27285e9	−0.466317
$126$	0	0
$127$	3.44437e8	0.117488	0.0587439	−	0.998273i	$-0.481290\pi$
$127$	3.44437e8	0.117488	0.0587439	+	0.998273i	$0.481290\pi$
$128$	0	0
$129$	−2.30677e9	−0.733418
$130$	0	0
$131$	1.13152e9	0.335692	0.167846	−	0.985813i	$-0.446319\pi$
$131$	1.13152e9	0.335692	0.167846	+	0.985813i	$0.446319\pi$
$132$	0	0
$133$	7.75974e7	0.0215038
$134$	0	0
$135$	4.90590e9	1.27121
$136$	0	0
$137$	−1.20622e8	−0.0292540	−0.0146270	−	0.999893i	$-0.504656\pi$
$137$	−1.20622e8	−0.0292540	−0.0146270	+	0.999893i	$0.504656\pi$
$138$	0	0
$139$	−5.00754e9	−1.13778	−0.568889	−	0.822414i	$-0.692627\pi$
$139$	−5.00754e9	−1.13778	−0.568889	+	0.822414i	$0.692627\pi$
$140$	0	0
$141$	−2.59911e9	−0.553782
$142$	0	0
$143$	1.67228e9	0.334423
$144$	0	0
$145$	5.67453e9	1.06604
$146$	0	0
$147$	3.41167e9	0.602614
$148$	0	0
$149$	−8.79381e9	−1.46164	−0.730818	−	0.682573i	$-0.760861\pi$
$149$	−8.79381e9	−1.46164	−0.730818	+	0.682573i	$0.760861\pi$
$150$	0	0
$151$	1.89322e9	0.296350	0.148175	−	0.988961i	$-0.452660\pi$
$151$	1.89322e9	0.296350	0.148175	+	0.988961i	$0.452660\pi$
$152$	0	0
$153$	−2.89580e9	−0.427225
$154$	0	0
$155$	1.10306e10	1.53499
$156$	0	0
$157$	1.42967e10	1.87797	0.938984	−	0.343961i	$-0.111769\pi$
$157$	1.42967e10	1.87797	0.938984	+	0.343961i	$0.111769\pi$
$158$	0	0
$159$	−5.62621e9	−0.698118
$160$	0	0
$161$	−8.18208e8	−0.0959726
$162$	0	0
$163$	−1.35312e10	−1.50139	−0.750694	−	0.660650i	$-0.770281\pi$
$163$	−1.35312e10	−1.50139	−0.750694	+	0.660650i	$0.770281\pi$
$164$	0	0
$165$	8.97260e9	0.942412
$166$	0	0
$167$	8.67633e9	0.863201	0.431600	−	0.902065i	$-0.357949\pi$
$167$	8.67633e9	0.863201	0.431600	+	0.902065i	$0.357949\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−1.27239e9	−0.113799
$172$	0	0
$173$	2.02999e10	1.72301	0.861504	−	0.507750i	$-0.169523\pi$
$173$	2.02999e10	1.72301	0.861504	+	0.507750i	$0.169523\pi$
$174$	0	0
$175$	9.33156e8	0.0752113
$176$	0	0
$177$	9.60565e9	0.735609
$178$	0	0
$179$	1.08759e10	0.791817	0.395908	−	0.918290i	$-0.370430\pi$
$179$	1.08759e10	0.791817	0.395908	+	0.918290i	$0.370430\pi$
$180$	0	0
$181$	1.52561e10	1.05655	0.528275	−	0.849073i	$-0.322839\pi$
$181$	1.52561e10	1.05655	0.528275	+	0.849073i	$0.322839\pi$
$182$	0	0
$183$	−2.09458e9	−0.138060
$184$	0	0
$185$	−2.79124e10	−1.75196
$186$	0	0
$187$	−1.37505e10	−0.822300
$188$	0	0
$189$	2.06424e9	0.117674
$190$	0	0
$191$	−1.20771e9	−0.0656616	−0.0328308	−	0.999461i	$-0.510452\pi$
$191$	−1.20771e9	−0.0656616	−0.0328308	+	0.999461i	$0.510452\pi$
$192$	0	0
$193$	1.46443e10	0.759730	0.379865	−	0.925042i	$-0.375970\pi$
$193$	1.46443e10	0.759730	0.379865	+	0.925042i	$0.375970\pi$
$194$	0	0
$195$	4.37681e9	0.216771
$196$	0	0
$197$	3.80476e10	1.79982	0.899911	−	0.436075i	$-0.143632\pi$
$197$	3.80476e10	1.79982	0.899911	+	0.436075i	$0.143632\pi$
$198$	0	0
$199$	2.69086e10	1.21633	0.608166	−	0.793810i	$-0.291905\pi$
$199$	2.69086e10	1.21633	0.608166	+	0.793810i	$0.291905\pi$
$200$	0	0
$201$	1.10660e10	0.478197
$202$	0	0
$203$	2.38765e9	0.0986821
$204$	0	0
$205$	−3.60038e10	−1.42382
$206$	0	0
$207$	1.34165e10	0.507892
$208$	0	0
$209$	−6.04185e9	−0.219034
$210$	0	0
$211$	2.87876e10	0.999849	0.499925	−	0.866069i	$-0.333361\pi$
$211$	2.87876e10	0.999849	0.499925	+	0.866069i	$0.333361\pi$
$212$	0	0
$213$	−9.65038e9	−0.321245
$214$	0	0
$215$	4.80797e10	1.53458
$216$	0	0
$217$	4.64129e9	0.142092
$218$	0	0
$219$	1.52420e10	0.447760
$220$	0	0
$221$	−6.70743e9	−0.189143
$222$	0	0
$223$	−5.58460e10	−1.51224	−0.756120	−	0.654433i	$-0.772907\pi$
$223$	−5.58460e10	−1.51224	−0.756120	+	0.654433i	$0.772907\pi$
$224$	0	0
$225$	−1.53013e10	−0.398022
$226$	0	0
$227$	2.12255e10	0.530567	0.265284	−	0.964170i	$-0.414534\pi$
$227$	2.12255e10	0.530567	0.265284	+	0.964170i	$0.414534\pi$
$228$	0	0
$229$	5.16671e10	1.24152	0.620761	−	0.784000i	$-0.286824\pi$
$229$	5.16671e10	1.24152	0.620761	+	0.784000i	$0.286824\pi$
$230$	0	0
$231$	3.77537e9	0.0872380
$232$	0	0
$233$	−6.99428e10	−1.55468	−0.777341	−	0.629080i	$-0.783432\pi$
$233$	−6.99428e10	−1.55468	−0.777341	+	0.629080i	$0.783432\pi$
$234$	0	0
$235$	5.41729e10	1.15871
$236$	0	0
$237$	−5.18776e10	−1.06810
$238$	0	0
$239$	−9.50597e10	−1.88454	−0.942271	−	0.334850i	$-0.891314\pi$
$239$	−9.50597e10	−1.88454	−0.942271	+	0.334850i	$0.891314\pi$
$240$	0	0
$241$	5.96117e10	1.13829	0.569147	−	0.822235i	$-0.307273\pi$
$241$	5.96117e10	1.13829	0.569147	+	0.822235i	$0.307273\pi$
$242$	0	0
$243$	−5.46589e10	−1.00562
$244$	0	0
$245$	−7.11089e10	−1.26089
$246$	0	0
$247$	−2.94720e9	−0.0503817
$248$	0	0
$249$	1.72997e10	0.285194
$250$	0	0
$251$	6.63592e10	1.05528	0.527642	−	0.849467i	$-0.323076\pi$
$251$	6.63592e10	1.05528	0.527642	+	0.849467i	$0.323076\pi$
$252$	0	0
$253$	6.37070e10	0.977563
$254$	0	0
$255$	−3.59887e10	−0.533011
$256$	0	0
$257$	−3.60015e10	−0.514779	−0.257390	−	0.966308i	$-0.582862\pi$
$257$	−3.60015e10	−0.514779	−0.257390	+	0.966308i	$0.582862\pi$
$258$	0	0
$259$	−1.17446e10	−0.162177
$260$	0	0
$261$	−3.91512e10	−0.522231
$262$	0	0
$263$	−5.53571e10	−0.713465	−0.356732	−	0.934207i	$-0.616109\pi$
$263$	−5.53571e10	−0.713465	−0.356732	+	0.934207i	$0.616109\pi$
$264$	0	0
$265$	1.17266e11	1.46072
$266$	0	0
$267$	−7.39525e10	−0.890537
$268$	0	0
$269$	9.87231e10	1.14956	0.574782	−	0.818306i	$-0.305087\pi$
$269$	9.87231e10	1.14956	0.574782	+	0.818306i	$0.305087\pi$
$270$	0	0
$271$	1.57825e11	1.77752	0.888760	−	0.458372i	$-0.151567\pi$
$271$	1.57825e11	1.77752	0.888760	+	0.458372i	$0.151567\pi$
$272$	0	0
$273$	1.84161e9	0.0200663
$274$	0	0
$275$	−7.26570e10	−0.766091
$276$	0	0
$277$	2.52506e10	0.257700	0.128850	−	0.991664i	$-0.458872\pi$
$277$	2.52506e10	0.257700	0.128850	+	0.991664i	$0.458872\pi$
$278$	0	0
$279$	−7.61049e10	−0.751958
$280$	0	0
$281$	−7.44249e10	−0.712098	−0.356049	−	0.934467i	$-0.615876\pi$
$281$	−7.44249e10	−0.712098	−0.356049	+	0.934467i	$0.615876\pi$
$282$	0	0
$283$	1.08399e11	1.00458	0.502291	−	0.864699i	$-0.332491\pi$
$283$	1.08399e11	1.00458	0.502291	+	0.864699i	$0.332491\pi$
$284$	0	0
$285$	−1.58132e10	−0.141977
$286$	0	0
$287$	−1.51492e10	−0.131802
$288$	0	0
$289$	−6.34353e10	−0.534922
$290$	0	0
$291$	1.84423e10	0.150764
$292$	0	0
$293$	4.84627e10	0.384152	0.192076	−	0.981380i	$-0.438478\pi$
$293$	4.84627e10	0.384152	0.192076	+	0.981380i	$0.438478\pi$
$294$	0	0
$295$	−2.00209e11	−1.53916
$296$	0	0
$297$	−1.60725e11	−1.19861
$298$	0	0
$299$	3.10761e10	0.224857
$300$	0	0
$301$	2.02303e10	0.142054
$302$	0	0
$303$	−7.55388e10	−0.514847
$304$	0	0
$305$	4.36569e10	0.288871
$306$	0	0
$307$	8.71117e10	0.559698	0.279849	−	0.960044i	$-0.409716\pi$
$307$	8.71117e10	0.559698	0.279849	+	0.960044i	$0.409716\pi$
$308$	0	0
$309$	−1.43571e11	−0.895887
$310$	0	0
$311$	1.81111e10	0.109780	0.0548900	−	0.998492i	$-0.482519\pi$
$311$	1.81111e10	0.109780	0.0548900	+	0.998492i	$0.482519\pi$
$312$	0	0
$313$	−2.50444e11	−1.47490	−0.737449	−	0.675403i	$-0.763970\pi$
$313$	−2.50444e11	−1.47490	−0.737449	+	0.675403i	$0.763970\pi$
$314$	0	0
$315$	−1.65717e10	−0.0948352
$316$	0	0
$317$	−1.62467e11	−0.903644	−0.451822	−	0.892108i	$-0.649226\pi$
$317$	−1.62467e11	−0.903644	−0.451822	+	0.892108i	$0.649226\pi$
$318$	0	0
$319$	−1.85906e11	−1.00516
$320$	0	0
$321$	−3.03177e10	−0.159376
$322$	0	0
$323$	2.42336e10	0.123882
$324$	0	0
$325$	−3.54419e10	−0.176214
$326$	0	0
$327$	−1.45033e11	−0.701459
$328$	0	0
$329$	2.27941e10	0.107261
$330$	0	0
$331$	6.54231e10	0.299575	0.149787	−	0.988718i	$-0.452141\pi$
$331$	6.54231e10	0.299575	0.149787	+	0.988718i	$0.452141\pi$
$332$	0	0
$333$	1.92581e11	0.858249
$334$	0	0
$335$	−2.30646e11	−1.00056
$336$	0	0
$337$	−8.37766e10	−0.353825	−0.176912	−	0.984227i	$-0.556611\pi$
$337$	−8.37766e10	−0.353825	−0.176912	+	0.984227i	$0.556611\pi$
$338$	0	0
$339$	−1.72751e11	−0.710429
$340$	0	0
$341$	−3.61378e11	−1.44733
$342$	0	0
$343$	−6.02657e10	−0.235097
$344$	0	0
$345$	1.66739e11	0.633652
$346$	0	0
$347$	−5.18275e11	−1.91901	−0.959506	−	0.281689i	$-0.909105\pi$
$347$	−5.18275e11	−1.91901	−0.959506	+	0.281689i	$0.909105\pi$
$348$	0	0
$349$	2.77982e11	1.00300	0.501502	−	0.865157i	$-0.332781\pi$
$349$	2.77982e11	1.00300	0.501502	+	0.865157i	$0.332781\pi$
$350$	0	0
$351$	−7.84011e10	−0.275702
$352$	0	0
$353$	−1.60940e11	−0.551669	−0.275835	−	0.961205i	$-0.588954\pi$
$353$	−1.60940e11	−0.551669	−0.275835	+	0.961205i	$0.588954\pi$
$354$	0	0
$355$	2.01141e11	0.672162
$356$	0	0
$357$	−1.51429e10	−0.0493402
$358$	0	0
$359$	1.45229e11	0.461455	0.230728	−	0.973018i	$-0.425889\pi$
$359$	1.45229e11	0.461455	0.230728	+	0.973018i	$0.425889\pi$
$360$	0	0
$361$	−3.12040e11	−0.967002
$362$	0	0
$363$	−9.17717e10	−0.277414
$364$	0	0
$365$	−3.17688e11	−0.936876
$366$	0	0
$367$	5.88415e11	1.69311	0.846557	−	0.532298i	$-0.178671\pi$
$367$	5.88415e11	1.69311	0.846557	+	0.532298i	$0.178671\pi$
$368$	0	0
$369$	2.48407e11	0.697502
$370$	0	0
$371$	4.93416e10	0.135217
$372$	0	0
$373$	2.92744e11	0.783065	0.391533	−	0.920164i	$-0.371945\pi$
$373$	2.92744e11	0.783065	0.391533	+	0.920164i	$0.371945\pi$
$374$	0	0
$375$	1.09142e11	0.285003
$376$	0	0
$377$	−9.06844e10	−0.231205
$378$	0	0
$379$	−1.16451e11	−0.289912	−0.144956	−	0.989438i	$-0.546304\pi$
$379$	−1.16451e11	−0.289912	−0.144956	+	0.989438i	$0.546304\pi$
$380$	0	0
$381$	−2.95340e10	−0.0718059
$382$	0	0
$383$	3.18071e11	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$383$	3.18071e11	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$384$	0	0
$385$	−7.86894e10	−0.182534
$386$	0	0
$387$	−3.31724e11	−0.751757
$388$	0	0
$389$	−1.88045e11	−0.416379	−0.208190	−	0.978088i	$-0.566757\pi$
$389$	−1.88045e11	−0.416379	−0.208190	+	0.978088i	$0.566757\pi$
$390$	0	0
$391$	−2.55526e11	−0.552892
$392$	0	0
$393$	−9.70231e10	−0.205168
$394$	0	0
$395$	1.08128e12	2.23486
$396$	0	0
$397$	−8.16657e11	−1.64999	−0.824997	−	0.565137i	$-0.808823\pi$
$397$	−8.16657e11	−1.64999	−0.824997	+	0.565137i	$0.808823\pi$
$398$	0	0
$399$	−6.65366e9	−0.0131426
$400$	0	0
$401$	9.64580e11	1.86290	0.931448	−	0.363875i	$-0.118546\pi$
$401$	9.64580e11	1.86290	0.931448	+	0.363875i	$0.118546\pi$
$402$	0	0
$403$	−1.76279e11	−0.332911
$404$	0	0
$405$	1.30964e10	0.0241882
$406$	0	0
$407$	9.14453e11	1.65191
$408$	0	0
$409$	4.92782e11	0.870763	0.435381	−	0.900246i	$-0.356614\pi$
$409$	4.92782e11	0.870763	0.435381	+	0.900246i	$0.356614\pi$
$410$	0	0
$411$	1.03429e10	0.0178794
$412$	0	0
$413$	−8.42412e10	−0.142478
$414$	0	0
$415$	−3.60574e11	−0.596730
$416$	0	0
$417$	4.29376e11	0.695385
$418$	0	0
$419$	−1.06792e11	−0.169268	−0.0846338	−	0.996412i	$-0.526972\pi$
$419$	−1.06792e11	−0.169268	−0.0846338	+	0.996412i	$0.526972\pi$
$420$	0	0
$421$	−9.06902e10	−0.140699	−0.0703495	−	0.997522i	$-0.522411\pi$
$421$	−9.06902e10	−0.140699	−0.0703495	+	0.997522i	$0.522411\pi$
$422$	0	0
$423$	−3.73764e11	−0.567630
$424$	0	0
$425$	2.91424e11	0.433287
$426$	0	0
$427$	1.83694e10	0.0267404
$428$	0	0
$429$	−1.43391e11	−0.204392
$430$	0	0
$431$	−5.39153e11	−0.752600	−0.376300	−	0.926498i	$-0.622804\pi$
$431$	−5.39153e11	−0.752600	−0.376300	+	0.926498i	$0.622804\pi$
$432$	0	0
$433$	4.68512e11	0.640509	0.320254	−	0.947332i	$-0.396232\pi$
$433$	4.68512e11	0.640509	0.320254	+	0.947332i	$0.396232\pi$
$434$	0	0
$435$	−4.86568e11	−0.651541
$436$	0	0
$437$	−1.12276e11	−0.147273
$438$	0	0
$439$	−3.60014e11	−0.462624	−0.231312	−	0.972880i	$-0.574302\pi$
$439$	−3.60014e11	−0.462624	−0.231312	+	0.972880i	$0.574302\pi$
$440$	0	0
$441$	4.90613e11	0.617682
$442$	0	0
$443$	−1.35127e12	−1.66697	−0.833483	−	0.552546i	$-0.813656\pi$
$443$	−1.35127e12	−1.66697	−0.833483	+	0.552546i	$0.813656\pi$
$444$	0	0
$445$	1.54138e12	1.86333
$446$	0	0
$447$	7.54034e11	0.893320
$448$	0	0
$449$	1.10673e12	1.28509	0.642547	−	0.766246i	$-0.277878\pi$
$449$	1.10673e12	1.28509	0.642547	+	0.766246i	$0.277878\pi$
$450$	0	0
$451$	1.17954e12	1.34251
$452$	0	0
$453$	−1.62336e11	−0.181123
$454$	0	0
$455$	−3.83844e10	−0.0419859
$456$	0	0
$457$	−1.50037e12	−1.60908	−0.804538	−	0.593901i	$-0.797587\pi$
$457$	−1.50037e12	−1.60908	−0.804538	+	0.593901i	$0.797587\pi$
$458$	0	0
$459$	6.44661e11	0.677913
$460$	0	0
$461$	1.25558e12	1.29476	0.647382	−	0.762166i	$-0.275864\pi$
$461$	1.25558e12	1.29476	0.647382	+	0.762166i	$0.275864\pi$
$462$	0	0
$463$	4.59412e11	0.464610	0.232305	−	0.972643i	$-0.425373\pi$
$463$	4.59412e11	0.464610	0.232305	+	0.972643i	$0.425373\pi$
$464$	0	0
$465$	−9.45825e11	−0.938151
$466$	0	0
$467$	1.97470e12	1.92121	0.960605	−	0.277917i	$-0.0896441\pi$
$467$	1.97470e12	1.92121	0.960605	+	0.277917i	$0.0896441\pi$
$468$	0	0
$469$	−9.70481e10	−0.0926210
$470$	0	0
$471$	−1.22589e12	−1.14777
$472$	0	0
$473$	−1.57516e12	−1.44694
$474$	0	0
$475$	1.28050e11	0.115414
$476$	0	0
$477$	−8.09073e11	−0.715575
$478$	0	0
$479$	−1.97850e12	−1.71723	−0.858613	−	0.512625i	$-0.828673\pi$
$479$	−1.97850e12	−1.71723	−0.858613	+	0.512625i	$0.828673\pi$
$480$	0	0
$481$	4.46067e11	0.379968
$482$	0	0
$483$	7.01580e10	0.0586564
$484$	0	0
$485$	−3.84390e11	−0.315453
$486$	0	0
$487$	−8.70430e11	−0.701219	−0.350609	−	0.936522i	$-0.614026\pi$
$487$	−8.70430e11	−0.701219	−0.350609	+	0.936522i	$0.614026\pi$
$488$	0	0
$489$	1.16025e12	0.917616
$490$	0	0
$491$	9.15629e11	0.710973	0.355486	−	0.934681i	$-0.384315\pi$
$491$	9.15629e11	0.710973	0.355486	+	0.934681i	$0.384315\pi$
$492$	0	0
$493$	7.45662e11	0.568501
$494$	0	0
$495$	1.29030e12	0.965977
$496$	0	0
$497$	8.46335e10	0.0622212
$498$	0	0
$499$	1.04001e12	0.750906	0.375453	−	0.926841i	$-0.377487\pi$
$499$	1.04001e12	0.750906	0.375453	+	0.926841i	$0.377487\pi$
$500$	0	0
$501$	−7.43960e11	−0.527570
$502$	0	0
$503$	−1.96238e11	−0.136687	−0.0683435	−	0.997662i	$-0.521771\pi$
$503$	−1.96238e11	−0.136687	−0.0683435	+	0.997662i	$0.521771\pi$
$504$	0	0
$505$	1.57444e12	1.07725
$506$	0	0
$507$	−6.99456e10	−0.0470137
$508$	0	0
$509$	−9.33446e11	−0.616395	−0.308197	−	0.951322i	$-0.599726\pi$
$509$	−9.33446e11	−0.616395	−0.308197	+	0.951322i	$0.599726\pi$
$510$	0	0
$511$	−1.33672e11	−0.0867256
$512$	0	0
$513$	2.83259e11	0.180574
$514$	0	0
$515$	2.99243e12	1.87452
$516$	0	0
$517$	−1.77479e12	−1.09254
$518$	0	0
$519$	−1.74064e12	−1.05307
$520$	0	0
$521$	−2.60930e12	−1.55151	−0.775754	−	0.631036i	$-0.782630\pi$
$521$	−2.60930e12	−1.55151	−0.775754	+	0.631036i	$0.782630\pi$
$522$	0	0
$523$	−1.76867e12	−1.03368	−0.516842	−	0.856081i	$-0.672893\pi$
$523$	−1.76867e12	−1.03368	−0.516842	+	0.856081i	$0.672893\pi$
$524$	0	0
$525$	−8.00144e10	−0.0459675
$526$	0	0
$527$	1.44947e12	0.818582
$528$	0	0
$529$	−6.17280e11	−0.342714
$530$	0	0
$531$	1.38133e12	0.754003
$532$	0	0
$533$	5.75376e11	0.308802
$534$	0	0
$535$	6.31908e11	0.333474
$536$	0	0
$537$	−9.32560e11	−0.483941
$538$	0	0
$539$	2.32964e12	1.18888
$540$	0	0
$541$	3.95324e12	1.98411	0.992056	−	0.125800i	$-0.0401499\pi$
$541$	3.95324e12	1.98411	0.992056	+	0.125800i	$0.0401499\pi$
$542$	0	0
$543$	−1.30815e12	−0.645740
$544$	0	0
$545$	3.02290e12	1.46771
$546$	0	0
$547$	2.49418e12	1.19120	0.595601	−	0.803281i	$-0.296914\pi$
$547$	2.49418e12	1.19120	0.595601	+	0.803281i	$0.296914\pi$
$548$	0	0
$549$	−3.01209e11	−0.141512
$550$	0	0
$551$	3.27639e11	0.151430
$552$	0	0
$553$	4.54965e11	0.206878
$554$	0	0
$555$	2.39338e12	1.07076
$556$	0	0
$557$	2.22532e12	0.979588	0.489794	−	0.871838i	$-0.337072\pi$
$557$	2.22532e12	0.979588	0.489794	+	0.871838i	$0.337072\pi$
$558$	0	0
$559$	−7.68360e11	−0.332822
$560$	0	0
$561$	1.17905e12	0.502572
$562$	0	0
$563$	3.00484e12	1.26047	0.630237	−	0.776403i	$-0.282958\pi$
$563$	3.00484e12	1.26047	0.630237	+	0.776403i	$0.282958\pi$
$564$	0	0
$565$	3.60061e12	1.48648
$566$	0	0
$567$	5.51051e9	0.00223907
$568$	0	0
$569$	3.35545e12	1.34198	0.670989	−	0.741467i	$-0.265870\pi$
$569$	3.35545e12	1.34198	0.670989	+	0.741467i	$0.265870\pi$
$570$	0	0
$571$	−2.15636e12	−0.848903	−0.424451	−	0.905451i	$-0.639533\pi$
$571$	−2.15636e12	−0.848903	−0.424451	+	0.905451i	$0.639533\pi$
$572$	0	0
$573$	1.03556e11	0.0401310
$574$	0	0
$575$	−1.35019e12	−0.515098
$576$	0	0
$577$	−2.02642e12	−0.761095	−0.380547	−	0.924761i	$-0.624264\pi$
$577$	−2.02642e12	−0.761095	−0.380547	+	0.924761i	$0.624264\pi$
$578$	0	0
$579$	−1.25569e12	−0.464331
$580$	0	0
$581$	−1.51717e11	−0.0552386
$582$	0	0
$583$	−3.84182e12	−1.37730
$584$	0	0
$585$	6.29404e11	0.222192
$586$	0	0
$587$	1.74926e12	0.608112	0.304056	−	0.952654i	$-0.401659\pi$
$587$	1.74926e12	0.608112	0.304056	+	0.952654i	$0.401659\pi$
$588$	0	0
$589$	6.36887e11	0.218044
$590$	0	0
$591$	−3.26243e12	−1.10001
$592$	0	0
$593$	2.36850e12	0.786551	0.393275	−	0.919421i	$-0.371342\pi$
$593$	2.36850e12	0.786551	0.393275	+	0.919421i	$0.371342\pi$
$594$	0	0
$595$	3.15620e11	0.103238
$596$	0	0
$597$	−2.30730e12	−0.743396
$598$	0	0
$599$	−3.95956e12	−1.25668	−0.628342	−	0.777937i	$-0.716266\pi$
$599$	−3.95956e12	−1.25668	−0.628342	+	0.777937i	$0.716266\pi$
$600$	0	0
$601$	−1.46479e12	−0.457974	−0.228987	−	0.973429i	$-0.573541\pi$
$601$	−1.46479e12	−0.457974	−0.228987	+	0.973429i	$0.573541\pi$
$602$	0	0
$603$	1.59133e12	0.490155
$604$	0	0
$605$	1.91278e12	0.580452
$606$	0	0
$607$	−2.37878e12	−0.711221	−0.355610	−	0.934634i	$-0.615727\pi$
$607$	−2.37878e12	−0.711221	−0.355610	+	0.934634i	$0.615727\pi$
$608$	0	0
$609$	−2.04731e11	−0.0603124
$610$	0	0
$611$	−8.65735e11	−0.251304
$612$	0	0
$613$	−3.04291e12	−0.870397	−0.435198	−	0.900335i	$-0.643322\pi$
$613$	−3.04291e12	−0.870397	−0.435198	+	0.900335i	$0.643322\pi$
$614$	0	0
$615$	3.08718e12	0.870211
$616$	0	0
$617$	−6.91186e12	−1.92005	−0.960024	−	0.279918i	$-0.909693\pi$
$617$	−6.91186e12	−1.92005	−0.960024	+	0.279918i	$0.909693\pi$
$618$	0	0
$619$	9.28522e10	0.0254205	0.0127103	−	0.999919i	$-0.495954\pi$
$619$	9.28522e10	0.0254205	0.0127103	+	0.999919i	$0.495954\pi$
$620$	0	0
$621$	−2.98677e12	−0.805914
$622$	0	0
$623$	6.48561e11	0.172486
$624$	0	0
$625$	−4.69849e12	−1.23168
$626$	0	0
$627$	5.18064e11	0.133869
$628$	0	0
$629$	−3.66784e12	−0.934290
$630$	0	0
$631$	−3.44921e12	−0.866139	−0.433070	−	0.901360i	$-0.642570\pi$
$631$	−3.44921e12	−0.866139	−0.433070	+	0.901360i	$0.642570\pi$
$632$	0	0
$633$	−2.46842e12	−0.611086
$634$	0	0
$635$	6.15573e11	0.150244
$636$	0	0
$637$	1.13639e12	0.273464
$638$	0	0
$639$	−1.38777e12	−0.329278
$640$	0	0
$641$	1.38534e12	0.324111	0.162056	−	0.986782i	$-0.448188\pi$
$641$	1.38534e12	0.324111	0.162056	+	0.986782i	$0.448188\pi$
$642$	0	0
$643$	1.95260e12	0.450468	0.225234	−	0.974305i	$-0.427685\pi$
$643$	1.95260e12	0.450468	0.225234	+	0.974305i	$0.427685\pi$
$644$	0	0
$645$	−4.12264e12	−0.937900
$646$	0	0
$647$	−7.01375e11	−0.157355	−0.0786775	−	0.996900i	$-0.525070\pi$
$647$	−7.01375e11	−0.157355	−0.0786775	+	0.996900i	$0.525070\pi$
$648$	0	0
$649$	6.55915e12	1.45126
$650$	0	0
$651$	−3.97971e11	−0.0868435
$652$	0	0
$653$	2.16551e12	0.466070	0.233035	−	0.972468i	$-0.425134\pi$
$653$	2.16551e12	0.466070	0.233035	+	0.972468i	$0.425134\pi$
$654$	0	0
$655$	2.02224e12	0.429285
$656$	0	0
$657$	2.19187e12	0.458956
$658$	0	0
$659$	−6.99434e12	−1.44465	−0.722325	−	0.691554i	$-0.756926\pi$
$659$	−6.99434e12	−1.44465	−0.722325	+	0.691554i	$0.756926\pi$
$660$	0	0
$661$	−7.13364e12	−1.45347	−0.726733	−	0.686920i	$-0.758962\pi$
$661$	−7.13364e12	−1.45347	−0.726733	+	0.686920i	$0.758962\pi$
$662$	0	0
$663$	5.75135e11	0.115600
$664$	0	0
$665$	1.38681e11	0.0274992
$666$	0	0
$667$	−3.45471e12	−0.675843
$668$	0	0
$669$	4.78857e12	0.924248
$670$	0	0
$671$	−1.43027e12	−0.272374
$672$	0	0
$673$	−5.90336e12	−1.10925	−0.554627	−	0.832099i	$-0.687139\pi$
$673$	−5.90336e12	−1.10925	−0.554627	+	0.832099i	$0.687139\pi$
$674$	0	0
$675$	3.40637e12	0.631574
$676$	0	0
$677$	4.09197e12	0.748657	0.374329	−	0.927296i	$-0.377873\pi$
$677$	4.09197e12	0.748657	0.374329	+	0.927296i	$0.377873\pi$
$678$	0	0
$679$	−1.61738e11	−0.0292011
$680$	0	0
$681$	−1.82000e12	−0.324271
$682$	0	0
$683$	1.90878e12	0.335631	0.167816	−	0.985818i	$-0.446329\pi$
$683$	1.90878e12	0.335631	0.167816	+	0.985818i	$0.446329\pi$
$684$	0	0
$685$	−2.15575e11	−0.0374102
$686$	0	0
$687$	−4.43024e12	−0.758791
$688$	0	0
$689$	−1.87403e12	−0.316803
$690$	0	0
$691$	1.26665e12	0.211352	0.105676	−	0.994401i	$-0.466299\pi$
$691$	1.26665e12	0.211352	0.105676	+	0.994401i	$0.466299\pi$
$692$	0	0
$693$	5.42914e11	0.0894194
$694$	0	0
$695$	−8.94941e12	−1.45500
$696$	0	0
$697$	−4.73109e12	−0.759301
$698$	0	0
$699$	5.99731e12	0.950188
$700$	0	0
$701$	−5.95358e12	−0.931208	−0.465604	−	0.884993i	$-0.654163\pi$
$701$	−5.95358e12	−0.931208	−0.465604	+	0.884993i	$0.654163\pi$
$702$	0	0
$703$	−1.61162e12	−0.248865
$704$	0	0
$705$	−4.64510e12	−0.708181
$706$	0	0
$707$	6.62473e11	0.0997196
$708$	0	0
$709$	8.83324e11	0.131284	0.0656420	−	0.997843i	$-0.479090\pi$
$709$	8.83324e11	0.131284	0.0656420	+	0.997843i	$0.479090\pi$
$710$	0	0
$711$	−7.46022e12	−1.09481
$712$	0	0
$713$	−6.71552e12	−0.973143
$714$	0	0
$715$	2.98867e12	0.427662
$716$	0	0
$717$	8.15099e12	1.15179
$718$	0	0
$719$	−2.53369e12	−0.353568	−0.176784	−	0.984250i	$-0.556569\pi$
$719$	−2.53369e12	−0.353568	−0.176784	+	0.984250i	$0.556569\pi$
$720$	0	0
$721$	1.25911e12	0.173522
$722$	0	0
$723$	−5.11146e12	−0.695701
$724$	0	0
$725$	3.94006e12	0.529641
$726$	0	0
$727$	8.73905e12	1.16027	0.580135	−	0.814520i	$-0.303000\pi$
$727$	8.73905e12	1.16027	0.580135	+	0.814520i	$0.303000\pi$
$728$	0	0
$729$	4.54255e12	0.595697
$730$	0	0
$731$	6.31792e12	0.818363
$732$	0	0
$733$	−2.45358e12	−0.313930	−0.156965	−	0.987604i	$-0.550171\pi$
$733$	−2.45358e12	−0.313930	−0.156965	+	0.987604i	$0.550171\pi$
$734$	0	0
$735$	6.09730e12	0.770627
$736$	0	0
$737$	7.55632e12	0.943423
$738$	0	0
$739$	−1.44149e13	−1.77792	−0.888960	−	0.457985i	$-0.848571\pi$
$739$	−1.44149e13	−1.77792	−0.888960	+	0.457985i	$0.848571\pi$
$740$	0	0
$741$	2.52710e11	0.0307922
$742$	0	0
$743$	−2.25852e12	−0.271878	−0.135939	−	0.990717i	$-0.543405\pi$
$743$	−2.25852e12	−0.271878	−0.135939	+	0.990717i	$0.543405\pi$
$744$	0	0
$745$	−1.57162e13	−1.86915
$746$	0	0
$747$	2.48777e12	0.292325
$748$	0	0
$749$	2.65885e11	0.0308693
$750$	0	0
$751$	−5.94360e12	−0.681821	−0.340910	−	0.940096i	$-0.610735\pi$
$751$	−5.94360e12	−0.681821	−0.340910	+	0.940096i	$0.610735\pi$
$752$	0	0
$753$	−5.69003e12	−0.644967
$754$	0	0
$755$	3.38354e12	0.378975
$756$	0	0
$757$	−5.15480e12	−0.570533	−0.285266	−	0.958448i	$-0.592082\pi$
$757$	−5.15480e12	−0.570533	−0.285266	+	0.958448i	$0.592082\pi$
$758$	0	0
$759$	−5.46261e12	−0.597465
$760$	0	0
$761$	6.04018e11	0.0652858	0.0326429	−	0.999467i	$-0.489608\pi$
$761$	6.04018e11	0.0652858	0.0326429	+	0.999467i	$0.489608\pi$
$762$	0	0
$763$	1.27193e12	0.135864
$764$	0	0
$765$	−5.17534e12	−0.546339
$766$	0	0
$767$	3.19953e12	0.333816
$768$	0	0
$769$	1.50501e12	0.155193	0.0775963	−	0.996985i	$-0.475275\pi$
$769$	1.50501e12	0.155193	0.0775963	+	0.996985i	$0.475275\pi$
$770$	0	0
$771$	3.08698e12	0.314622
$772$	0	0
$773$	7.53437e12	0.758996	0.379498	−	0.925193i	$-0.376097\pi$
$773$	7.53437e12	0.758996	0.379498	+	0.925193i	$0.376097\pi$
$774$	0	0
$775$	7.65896e12	0.762627
$776$	0	0
$777$	1.00705e12	0.0991191
$778$	0	0
$779$	−2.07881e12	−0.202253
$780$	0	0
$781$	−6.58970e12	−0.633776
$782$	0	0
$783$	8.71581e12	0.828667
$784$	0	0
$785$	2.55509e13	2.40156
$786$	0	0
$787$	4.10896e12	0.381809	0.190904	−	0.981609i	$-0.438858\pi$
$787$	4.10896e12	0.381809	0.190904	+	0.981609i	$0.438858\pi$
$788$	0	0
$789$	4.74665e12	0.436054
$790$	0	0
$791$	1.51502e12	0.137601
$792$	0	0
$793$	−6.97680e11	−0.0626508
$794$	0	0
$795$	−1.00551e13	−0.892759
$796$	0	0
$797$	−3.53460e12	−0.310297	−0.155149	−	0.987891i	$-0.549586\pi$
$797$	−3.53460e12	−0.310297	−0.155149	+	0.987891i	$0.549586\pi$
$798$	0	0
$799$	7.11860e12	0.617922
$800$	0	0
$801$	−1.06347e13	−0.912806
$802$	0	0
$803$	1.04079e13	0.883373
$804$	0	0
$805$	−1.46229e12	−0.122731
$806$	0	0
$807$	−8.46510e12	−0.702589
$808$	0	0
$809$	8.96434e12	0.735783	0.367892	−	0.929869i	$-0.380080\pi$
$809$	8.96434e12	0.735783	0.367892	+	0.929869i	$0.380080\pi$
$810$	0	0
$811$	8.61534e12	0.699324	0.349662	−	0.936876i	$-0.386296\pi$
$811$	8.61534e12	0.699324	0.349662	+	0.936876i	$0.386296\pi$
$812$	0	0
$813$	−1.35329e13	−1.08638
$814$	0	0
$815$	−2.41829e13	−1.91999
$816$	0	0
$817$	2.77605e12	0.217986
$818$	0	0
$819$	2.64832e11	0.0205680
$820$	0	0
$821$	2.09873e13	1.61218	0.806088	−	0.591796i	$-0.201581\pi$
$821$	2.09873e13	1.61218	0.806088	+	0.591796i	$0.201581\pi$
$822$	0	0
$823$	−1.47480e13	−1.12056	−0.560280	−	0.828304i	$-0.689306\pi$
$823$	−1.47480e13	−1.12056	−0.560280	+	0.828304i	$0.689306\pi$
$824$	0	0
$825$	6.23004e12	0.468218
$826$	0	0
$827$	−1.00085e13	−0.744037	−0.372018	−	0.928225i	$-0.621334\pi$
$827$	−1.00085e13	−0.744037	−0.372018	+	0.928225i	$0.621334\pi$
$828$	0	0
$829$	8.15300e12	0.599546	0.299773	−	0.954011i	$-0.403089\pi$
$829$	8.15300e12	0.599546	0.299773	+	0.954011i	$0.403089\pi$
$830$	0	0
$831$	−2.16514e12	−0.157500
$832$	0	0
$833$	−9.34407e12	−0.672409
$834$	0	0
$835$	1.55062e13	1.10387
$836$	0	0
$837$	1.69424e13	1.19319
$838$	0	0
$839$	−1.94338e13	−1.35403	−0.677017	−	0.735968i	$-0.736728\pi$
$839$	−1.94338e13	−1.35403	−0.677017	+	0.735968i	$0.736728\pi$
$840$	0	0
$841$	−4.42580e12	−0.305077
$842$	0	0
$843$	6.38163e12	0.435219
$844$	0	0
$845$	1.45786e12	0.0983698
$846$	0	0
$847$	8.04834e11	0.0537318
$848$	0	0
$849$	−9.29475e12	−0.613979
$850$	0	0
$851$	1.69934e13	1.11070
$852$	0	0
$853$	−8.40777e12	−0.543764	−0.271882	−	0.962331i	$-0.587646\pi$
$853$	−8.40777e12	−0.543764	−0.271882	+	0.962331i	$0.587646\pi$
$854$	0	0
$855$	−2.27400e12	−0.145527
$856$	0	0
$857$	1.57510e13	0.997458	0.498729	−	0.866758i	$-0.333800\pi$
$857$	1.57510e13	0.997458	0.498729	+	0.866758i	$0.333800\pi$
$858$	0	0
$859$	−3.45790e12	−0.216692	−0.108346	−	0.994113i	$-0.534555\pi$
$859$	−3.45790e12	−0.216692	−0.108346	+	0.994113i	$0.534555\pi$
$860$	0	0
$861$	1.29898e12	0.0805544
$862$	0	0
$863$	−5.82263e12	−0.357331	−0.178665	−	0.983910i	$-0.557178\pi$
$863$	−5.82263e12	−0.357331	−0.178665	+	0.983910i	$0.557178\pi$
$864$	0	0
$865$	3.62798e13	2.20340
$866$	0	0
$867$	5.43932e12	0.326933
$868$	0	0
$869$	−3.54243e13	−2.10723
$870$	0	0
$871$	3.68595e12	0.217004
$872$	0	0
$873$	2.65208e12	0.154534
$874$	0	0
$875$	−9.57168e11	−0.0552016
$876$	0	0
$877$	−5.98999e12	−0.341923	−0.170961	−	0.985278i	$-0.554687\pi$
$877$	−5.98999e12	−0.341923	−0.170961	+	0.985278i	$0.554687\pi$
$878$	0	0
$879$	−4.15548e12	−0.234785
$880$	0	0
$881$	−2.57327e13	−1.43911	−0.719554	−	0.694437i	$-0.755654\pi$
$881$	−2.57327e13	−1.43911	−0.719554	+	0.694437i	$0.755654\pi$
$882$	0	0
$883$	3.22167e13	1.78344	0.891719	−	0.452590i	$-0.149500\pi$
$883$	3.22167e13	1.78344	0.891719	+	0.452590i	$0.149500\pi$
$884$	0	0
$885$	1.71671e13	0.940702
$886$	0	0
$887$	1.54777e13	0.839557	0.419779	−	0.907627i	$-0.362108\pi$
$887$	1.54777e13	0.839557	0.419779	+	0.907627i	$0.362108\pi$
$888$	0	0
$889$	2.59012e11	0.0139079
$890$	0	0
$891$	−4.29057e11	−0.0228068
$892$	0	0
$893$	3.12786e12	0.164595
$894$	0	0
$895$	1.94372e13	1.01258
$896$	0	0
$897$	−2.66465e12	−0.137427
$898$	0	0
$899$	1.95968e13	1.00062
$900$	0	0
$901$	1.54094e13	0.778975
$902$	0	0
$903$	−1.73467e12	−0.0868203
$904$	0	0
$905$	2.72655e13	1.35112
$906$	0	0
$907$	4.09617e12	0.200977	0.100488	−	0.994938i	$-0.467960\pi$
$907$	4.09617e12	0.200977	0.100488	+	0.994938i	$0.467960\pi$
$908$	0	0
$909$	−1.08628e13	−0.527721
$910$	0	0
$911$	3.19819e13	1.53841	0.769204	−	0.639003i	$-0.220653\pi$
$911$	3.19819e13	1.53841	0.769204	+	0.639003i	$0.220653\pi$
$912$	0	0
$913$	1.18130e13	0.562652
$914$	0	0
$915$	−3.74340e12	−0.176552
$916$	0	0
$917$	8.50889e11	0.0397384
$918$	0	0
$919$	3.05636e13	1.41347	0.706733	−	0.707481i	$-0.250168\pi$
$919$	3.05636e13	1.41347	0.706733	+	0.707481i	$0.250168\pi$
$920$	0	0
$921$	−7.46947e12	−0.342075
$922$	0	0
$923$	−3.21443e12	−0.145780
$924$	0	0
$925$	−1.93807e13	−0.870427
$926$	0	0
$927$	−2.06461e13	−0.918289
$928$	0	0
$929$	−1.96670e13	−0.866300	−0.433150	−	0.901322i	$-0.642598\pi$
$929$	−1.96670e13	−0.866300	−0.433150	+	0.901322i	$0.642598\pi$
$930$	0	0
$931$	−4.10572e12	−0.179108
$932$	0	0
$933$	−1.55295e12	−0.0670952
$934$	0	0
$935$	−2.45747e13	−1.05156
$936$	0	0
$937$	5.88474e12	0.249401	0.124701	−	0.992194i	$-0.460203\pi$
$937$	5.88474e12	0.249401	0.124701	+	0.992194i	$0.460203\pi$
$938$	0	0
$939$	2.14746e13	0.901425
$940$	0	0
$941$	−1.79868e13	−0.747827	−0.373914	−	0.927464i	$-0.621984\pi$
$941$	−1.79868e13	−0.747827	−0.373914	+	0.927464i	$0.621984\pi$
$942$	0	0
$943$	2.19195e13	0.902669
$944$	0	0
$945$	3.68918e12	0.150483
$946$	0	0
$947$	1.42787e13	0.576917	0.288459	−	0.957492i	$-0.406857\pi$
$947$	1.42787e13	0.576917	0.288459	+	0.957492i	$0.406857\pi$
$948$	0	0
$949$	5.07695e12	0.203191
$950$	0	0
$951$	1.39308e13	0.552287
$952$	0	0
$953$	−4.85651e11	−0.0190724	−0.00953622	−	0.999955i	$-0.503036\pi$
$953$	−4.85651e11	−0.0190724	−0.00953622	+	0.999955i	$0.503036\pi$
$954$	0	0
$955$	−2.15840e12	−0.0839686
$956$	0	0
$957$	1.59407e13	0.614333
$958$	0	0
$959$	−9.07066e10	−0.00346302
$960$	0	0
$961$	1.16541e13	0.440783
$962$	0	0
$963$	−4.35982e12	−0.163362
$964$	0	0
$965$	2.61720e13	0.971549
$966$	0	0
$967$	−3.93014e13	−1.44540	−0.722702	−	0.691160i	$-0.757100\pi$
$967$	−3.93014e13	−1.44540	−0.722702	+	0.691160i	$0.757100\pi$
$968$	0	0
$969$	−2.07794e12	−0.0757138
$970$	0	0
$971$	−7.17694e12	−0.259091	−0.129546	−	0.991573i	$-0.541352\pi$
$971$	−7.17694e12	−0.259091	−0.129546	+	0.991573i	$0.541352\pi$
$972$	0	0
$973$	−3.76561e12	−0.134688
$974$	0	0
$975$	3.03899e12	0.107698
$976$	0	0
$977$	2.36882e13	0.831775	0.415888	−	0.909416i	$-0.363471\pi$
$977$	2.36882e13	0.831775	0.415888	+	0.909416i	$0.363471\pi$
$978$	0	0
$979$	−5.04979e13	−1.75692
$980$	0	0
$981$	−2.08564e13	−0.718999
$982$	0	0
$983$	−2.27685e12	−0.0777755	−0.0388877	−	0.999244i	$-0.512381\pi$
$983$	−2.27685e12	−0.0777755	−0.0388877	+	0.999244i	$0.512381\pi$
$984$	0	0
$985$	6.79982e13	2.30163
$986$	0	0
$987$	−1.95450e12	−0.0655555
$988$	0	0
$989$	−2.92714e13	−0.972883
$990$	0	0
$991$	−5.41684e13	−1.78408	−0.892041	−	0.451955i	$-0.850727\pi$
$991$	−5.41684e13	−1.78408	−0.892041	+	0.451955i	$0.850727\pi$
$992$	0	0
$993$	−5.60976e12	−0.183094
$994$	0	0
$995$	4.80908e13	1.55546
$996$	0	0
$997$	2.39920e12	0.0769020	0.0384510	−	0.999260i	$-0.487758\pi$
$997$	2.39920e12	0.0769020	0.0384510	+	0.999260i	$0.487758\pi$
$998$	0	0
$999$	−4.28722e13	−1.36185

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.d.1.2		3
4.3	odd	2		26.10.a.e.1.2	✓	3
12.11	even	2		234.10.a.k.1.1		3
52.51	odd	2		338.10.a.e.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.10.a.e.1.2	✓	3	4.3	odd	2
208.10.a.d.1.2		3	1.1	even	1	trivial
234.10.a.k.1.1		3	12.11	even	2
338.10.a.e.1.2		3	52.51	odd	2

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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q-85.7459 q^{3} +1787.19 q^{5} +751.989 q^{7} -12330.6 q^{9} -58551.0 q^{11} -28561.0 q^{13} -153244. q^{15} +234846. q^{17} +103190. q^{19} -64480.0 q^{21} -1.08806e6 q^{23} +1.24092e6 q^{25} +2.74504e6 q^{27} +3.17511e6 q^{29} +6.17201e6 q^{31} +5.02051e6 q^{33} +1.34395e6 q^{35} -1.56181e7 q^{37} +2.44899e6 q^{39} -2.01455e7 q^{41} +2.69024e7 q^{43} -2.20372e7 q^{45} +3.03118e7 q^{47} -3.97881e7 q^{49} -2.01371e7 q^{51} +6.56149e7 q^{53} -1.04642e8 q^{55} -8.84808e6 q^{57} -1.12025e8 q^{59} +2.44277e7 q^{61} -9.27250e6 q^{63} -5.10439e7 q^{65} -1.29055e8 q^{67} +9.32967e7 q^{69} +1.12546e8 q^{71} -1.77758e8 q^{73} -1.06404e8 q^{75} -4.40297e7 q^{77} +6.05015e8 q^{79} +7.32791e6 q^{81} -2.01755e8 q^{83} +4.19714e8 q^{85} -2.72253e8 q^{87} +8.62460e8 q^{89} -2.14776e7 q^{91} -5.29225e8 q^{93} +1.84419e8 q^{95} -2.15081e8 q^{97} +7.21971e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 156 q^{3} - 1272 q^{5} - 17058 q^{7} + 42273 q^{9} - 73974 q^{11} - 85683 q^{13} - 393756 q^{15} + 374976 q^{17} - 418338 q^{19} - 284694 q^{21} - 1026168 q^{23} + 3337287 q^{25} - 4218588 q^{27}+ \cdots + 2480087466 q^{99}+O(q^{100}) \) 3 * q - 156 * q^3 - 1272 * q^5 - 17058 * q^7 + 42273 * q^9 - 73974 * q^11 - 85683 * q^13 - 393756 * q^15 + 374976 * q^17 - 418338 * q^19 - 284694 * q^21 - 1026168 * q^23 + 3337287 * q^25 - 4218588 * q^27 - 3075834 * q^29 - 9286482 * q^31 - 25456452 * q^33 + 31802592 * q^35 - 17647776 * q^37 + 4455516 * q^39 - 47257110 * q^41 + 60023760 * q^43 - 81112122 * q^45 + 40824726 * q^47 + 45921177 * q^49 + 27559188 * q^51 + 8072046 * q^53 + 36910236 * q^55 + 245224476 * q^57 - 138035310 * q^59 + 264203886 * q^61 - 436094838 * q^63 + 36329592 * q^65 - 203167074 * q^67 - 397637544 * q^69 - 123067110 * q^71 - 433013250 * q^73 + 884469216 * q^75 + 380043264 * q^77 - 406418748 * q^79 + 100121427 * q^81 - 105365610 * q^83 + 5268186 * q^85 + 12753768 * q^87 + 1365375798 * q^89 + 487193538 * q^91 - 95909856 * q^93 + 85600692 * q^95 + 669691662 * q^97 + 2480087466 * q^99

Introduction

L-functions

Modular forms

Varieties

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