LMFDB - Embedded newform 208.10.a.h.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:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 $ x^5 - 1438x^3 - 4164x^2 + 396957*x - 59580
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-27.7188$ 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)

$f(q)$	$=$	$q-194.269 q^{3} -920.299 q^{5} -5359.02 q^{7} +18057.4 q^{9} +O(q^{10})$	$q-194.269 q^{3} -920.299 q^{5} -5359.02 q^{7} +18057.4 q^{9} -79284.0 q^{11} +28561.0 q^{13} +178785. q^{15} +452068. q^{17} -212533. q^{19} +1.04109e6 q^{21} +759566. q^{23} -1.10618e6 q^{25} +315803. q^{27} -900101. q^{29} -2.27141e6 q^{31} +1.54024e7 q^{33} +4.93189e6 q^{35} -4.70433e6 q^{37} -5.54851e6 q^{39} +3.39775e7 q^{41} +2.33244e7 q^{43} -1.66182e7 q^{45} +5.14121e7 q^{47} -1.16346e7 q^{49} -8.78228e7 q^{51} +1.01005e8 q^{53} +7.29650e7 q^{55} +4.12885e7 q^{57} -1.32234e8 q^{59} -1.23648e8 q^{61} -9.67699e7 q^{63} -2.62846e7 q^{65} +2.15282e8 q^{67} -1.47560e8 q^{69} +2.06198e8 q^{71} +3.44444e8 q^{73} +2.14896e8 q^{75} +4.24884e8 q^{77} -5.03324e7 q^{79} -4.16775e8 q^{81} -8.20266e7 q^{83} -4.16038e8 q^{85} +1.74862e8 q^{87} -6.17891e8 q^{89} -1.53059e8 q^{91} +4.41265e8 q^{93} +1.95594e8 q^{95} -9.91253e8 q^{97} -1.43166e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9}+O(q^{10}) $ 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9	$ 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9} - 121746 q^{11} + 142805 q^{13} - 105973 q^{15} - 495669 q^{17} + 840738 q^{19} - 1599467 q^{21} + 592152 q^{23} + 1670362 q^{25} - 6847883 q^{27} + 10678182 q^{29} - 12885296 q^{31} + 17278298 q^{33} - 8380731 q^{35} + 7171823 q^{37} - 4598321 q^{39} + 9294012 q^{41} - 12831975 q^{43} + 26135198 q^{45} - 43354215 q^{47} + 25249488 q^{49} - 16905901 q^{51} + 93231780 q^{53} - 99448846 q^{55} + 90173382 q^{57} - 246496182 q^{59} - 132232612 q^{61} + 416955202 q^{63} + 51495483 q^{65} + 369388534 q^{67} - 579986760 q^{69} - 212150457 q^{71} - 252729806 q^{73} + 752457788 q^{75} + 449666118 q^{77} + 1247271728 q^{79} - 317713115 q^{81} - 1696894296 q^{83} - 775363765 q^{85} + 614530466 q^{87} - 753854382 q^{89} - 288437539 q^{91} - 892784668 q^{93} - 1442632962 q^{95} + 3824606 q^{97} - 3016199848 q^{99}+O(q^{100}) $ 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9 - 121746 * q^11 + 142805 * q^13 - 105973 * q^15 - 495669 * q^17 + 840738 * q^19 - 1599467 * q^21 + 592152 * q^23 + 1670362 * q^25 - 6847883 * q^27 + 10678182 * q^29 - 12885296 * q^31 + 17278298 * q^33 - 8380731 * q^35 + 7171823 * q^37 - 4598321 * q^39 + 9294012 * q^41 - 12831975 * q^43 + 26135198 * q^45 - 43354215 * q^47 + 25249488 * q^49 - 16905901 * q^51 + 93231780 * q^53 - 99448846 * q^55 + 90173382 * q^57 - 246496182 * q^59 - 132232612 * q^61 + 416955202 * q^63 + 51495483 * q^65 + 369388534 * q^67 - 579986760 * q^69 - 212150457 * q^71 - 252729806 * q^73 + 752457788 * q^75 + 449666118 * q^77 + 1247271728 * q^79 - 317713115 * q^81 - 1696894296 * q^83 - 775363765 * q^85 + 614530466 * q^87 - 753854382 * q^89 - 288437539 * q^91 - 892784668 * q^93 - 1442632962 * q^95 + 3824606 * q^97 - 3016199848 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−194.269	−1.38471	−0.692353	−	0.721559i	$-0.743426\pi$
$3$	−194.269	−1.38471	−0.692353	+	0.721559i	$0.743426\pi$
$4$	0	0
$5$	−920.299	−0.658512	−0.329256	−	0.944241i	$-0.606798\pi$
$5$	−920.299	−0.658512	−0.329256	+	0.944241i	$0.606798\pi$
$6$	0	0
$7$	−5359.02	−0.843614	−0.421807	−	0.906686i	$-0.638604\pi$
$7$	−5359.02	−0.843614	−0.421807	+	0.906686i	$0.638604\pi$
$8$	0	0
$9$	18057.4	0.917411
$10$	0	0
$11$	−79284.0	−1.63275	−0.816373	−	0.577525i	$-0.804019\pi$
$11$	−79284.0	−1.63275	−0.816373	+	0.577525i	$0.804019\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	178785.	0.911846
$16$	0	0
$17$	452068.	1.31276	0.656378	−	0.754432i	$-0.272088\pi$
$17$	452068.	1.31276	0.656378	+	0.754432i	$0.272088\pi$
$18$	0	0
$19$	−212533.	−0.374141	−0.187070	−	0.982347i	$-0.559899\pi$
$19$	−212533.	−0.374141	−0.187070	+	0.982347i	$0.559899\pi$
$20$	0	0
$21$	1.04109e6	1.16816
$22$	0	0
$23$	759566.	0.565966	0.282983	−	0.959125i	$-0.408676\pi$
$23$	759566.	0.565966	0.282983	+	0.959125i	$0.408676\pi$
$24$	0	0
$25$	−1.10618e6	−0.566362
$26$	0	0
$27$	315803.	0.114361
$28$	0	0
$29$	−900101.	−0.236320	−0.118160	−	0.992995i	$-0.537700\pi$
$29$	−900101.	−0.236320	−0.118160	+	0.992995i	$0.537700\pi$
$30$	0	0
$31$	−2.27141e6	−0.441742	−0.220871	−	0.975303i	$-0.570890\pi$
$31$	−2.27141e6	−0.441742	−0.220871	+	0.975303i	$0.570890\pi$
$32$	0	0
$33$	1.54024e7	2.26087
$34$	0	0
$35$	4.93189e6	0.555530
$36$	0	0
$37$	−4.70433e6	−0.412658	−0.206329	−	0.978483i	$-0.566152\pi$
$37$	−4.70433e6	−0.412658	−0.206329	+	0.978483i	$0.566152\pi$
$38$	0	0
$39$	−5.54851e6	−0.384048
$40$	0	0
$41$	3.39775e7	1.87786	0.938932	−	0.344102i	$-0.111817\pi$
$41$	3.39775e7	1.87786	0.938932	+	0.344102i	$0.111817\pi$
$42$	0	0
$43$	2.33244e7	1.04041	0.520203	−	0.854042i	$-0.325856\pi$
$43$	2.33244e7	1.04041	0.520203	+	0.854042i	$0.325856\pi$
$44$	0	0
$45$	−1.66182e7	−0.604126
$46$	0	0
$47$	5.14121e7	1.53683	0.768413	−	0.639954i	$-0.221046\pi$
$47$	5.14121e7	1.53683	0.768413	+	0.639954i	$0.221046\pi$
$48$	0	0
$49$	−1.16346e7	−0.288315
$50$	0	0
$51$	−8.78228e7	−1.81778
$52$	0	0
$53$	1.01005e8	1.75834	0.879170	−	0.476509i	$-0.158098\pi$
$53$	1.01005e8	1.75834	0.879170	+	0.476509i	$0.158098\pi$
$54$	0	0
$55$	7.29650e7	1.07518
$56$	0	0
$57$	4.12885e7	0.518075
$58$	0	0
$59$	−1.32234e8	−1.42073	−0.710363	−	0.703836i	$-0.751469\pi$
$59$	−1.32234e8	−1.42073	−0.710363	+	0.703836i	$0.751469\pi$
$60$	0	0
$61$	−1.23648e8	−1.14341	−0.571706	−	0.820459i	$-0.693718\pi$
$61$	−1.23648e8	−1.14341	−0.571706	+	0.820459i	$0.693718\pi$
$62$	0	0
$63$	−9.67699e7	−0.773941
$64$	0	0
$65$	−2.62846e7	−0.182638
$66$	0	0
$67$	2.15282e8	1.30518	0.652592	−	0.757710i	$-0.273682\pi$
$67$	2.15282e8	1.30518	0.652592	+	0.757710i	$0.273682\pi$
$68$	0	0
$69$	−1.47560e8	−0.783696
$70$	0	0
$71$	2.06198e8	0.962992	0.481496	−	0.876448i	$-0.340094\pi$
$71$	2.06198e8	0.962992	0.481496	+	0.876448i	$0.340094\pi$
$72$	0	0
$73$	3.44444e8	1.41960	0.709800	−	0.704403i	$-0.248785\pi$
$73$	3.44444e8	1.41960	0.709800	+	0.704403i	$0.248785\pi$
$74$	0	0
$75$	2.14896e8	0.784245
$76$	0	0
$77$	4.24884e8	1.37741
$78$	0	0
$79$	−5.03324e7	−0.145387	−0.0726935	−	0.997354i	$-0.523159\pi$
$79$	−5.03324e7	−0.145387	−0.0726935	+	0.997354i	$0.523159\pi$
$80$	0	0
$81$	−4.16775e8	−1.07577
$82$	0	0
$83$	−8.20266e7	−0.189716	−0.0948579	−	0.995491i	$-0.530240\pi$
$83$	−8.20266e7	−0.189716	−0.0948579	+	0.995491i	$0.530240\pi$
$84$	0	0
$85$	−4.16038e8	−0.864465
$86$	0	0
$87$	1.74862e8	0.327233
$88$	0	0
$89$	−6.17891e8	−1.04390	−0.521948	−	0.852977i	$-0.674794\pi$
$89$	−6.17891e8	−1.04390	−0.521948	+	0.852977i	$0.674794\pi$
$90$	0	0
$91$	−1.53059e8	−0.233976
$92$	0	0
$93$	4.41265e8	0.611682
$94$	0	0
$95$	1.95594e8	0.246376
$96$	0	0
$97$	−9.91253e8	−1.13687	−0.568436	−	0.822727i	$-0.692451\pi$
$97$	−9.91253e8	−1.13687	−0.568436	+	0.822727i	$0.692451\pi$
$98$	0	0
$99$	−1.43166e9	−1.49790
$100$	0	0
$101$	−1.15157e9	−1.10115	−0.550573	−	0.834787i	$-0.685591\pi$
$101$	−1.15157e9	−1.10115	−0.550573	+	0.834787i	$0.685591\pi$
$102$	0	0
$103$	−1.28814e9	−1.12770	−0.563852	−	0.825876i	$-0.690681\pi$
$103$	−1.28814e9	−1.12770	−0.563852	+	0.825876i	$0.690681\pi$
$104$	0	0
$105$	−9.58114e8	−0.769246
$106$	0	0
$107$	−7.90577e8	−0.583065	−0.291533	−	0.956561i	$-0.594165\pi$
$107$	−7.90577e8	−0.583065	−0.291533	+	0.956561i	$0.594165\pi$
$108$	0	0
$109$	−4.40070e8	−0.298609	−0.149304	−	0.988791i	$-0.547703\pi$
$109$	−4.40070e8	−0.298609	−0.149304	+	0.988791i	$0.547703\pi$
$110$	0	0
$111$	9.13906e8	0.571410
$112$	0	0
$113$	−8.28687e8	−0.478121	−0.239060	−	0.971005i	$-0.576839\pi$
$113$	−8.28687e8	−0.478121	−0.239060	+	0.971005i	$0.576839\pi$
$114$	0	0
$115$	−6.99027e8	−0.372695
$116$	0	0
$117$	5.15737e8	0.254444
$118$	0	0
$119$	−2.42264e9	−1.10746
$120$	0	0
$121$	3.92801e9	1.66586
$122$	0	0
$123$	−6.60077e9	−2.60029
$124$	0	0
$125$	2.81547e9	1.03147
$126$	0	0
$127$	2.09036e9	0.713025	0.356513	−	0.934291i	$-0.383966\pi$
$127$	2.09036e9	0.713025	0.356513	+	0.934291i	$0.383966\pi$
$128$	0	0
$129$	−4.53121e9	−1.44066
$130$	0	0
$131$	−1.23787e8	−0.0367243	−0.0183621	−	0.999831i	$-0.505845\pi$
$131$	−1.23787e8	−0.0367243	−0.0183621	+	0.999831i	$0.505845\pi$
$132$	0	0
$133$	1.13897e9	0.315630
$134$	0	0
$135$	−2.90633e8	−0.0753084
$136$	0	0
$137$	−1.31278e9	−0.318383	−0.159192	−	0.987248i	$-0.550889\pi$
$137$	−1.31278e9	−0.318383	−0.159192	+	0.987248i	$0.550889\pi$
$138$	0	0
$139$	4.82994e9	1.09743	0.548713	−	0.836011i	$-0.315118\pi$
$139$	4.82994e9	1.09743	0.548713	+	0.836011i	$0.315118\pi$
$140$	0	0
$141$	−9.98777e9	−2.12805
$142$	0	0
$143$	−2.26443e9	−0.452842
$144$	0	0
$145$	8.28361e8	0.155619
$146$	0	0
$147$	2.26023e9	0.399232
$148$	0	0
$149$	−4.39244e9	−0.730075	−0.365037	−	0.930993i	$-0.618944\pi$
$149$	−4.39244e9	−0.730075	−0.365037	+	0.930993i	$0.618944\pi$
$150$	0	0
$151$	4.70301e9	0.736172	0.368086	−	0.929792i	$-0.380013\pi$
$151$	4.70301e9	0.736172	0.368086	+	0.929792i	$0.380013\pi$
$152$	0	0
$153$	8.16318e9	1.20434
$154$	0	0
$155$	2.09038e9	0.290892
$156$	0	0
$157$	−5.28793e9	−0.694605	−0.347302	−	0.937753i	$-0.612902\pi$
$157$	−5.28793e9	−0.694605	−0.347302	+	0.937753i	$0.612902\pi$
$158$	0	0
$159$	−1.96222e10	−2.43478
$160$	0	0
$161$	−4.07052e9	−0.477457
$162$	0	0
$163$	−6.72980e9	−0.746721	−0.373360	−	0.927686i	$-0.621794\pi$
$163$	−6.72980e9	−0.746721	−0.373360	+	0.927686i	$0.621794\pi$
$164$	0	0
$165$	−1.41748e10	−1.48881
$166$	0	0
$167$	−5.87782e9	−0.584780	−0.292390	−	0.956299i	$-0.594450\pi$
$167$	−5.87782e9	−0.584780	−0.292390	+	0.956299i	$0.594450\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−3.83779e9	−0.343241
$172$	0	0
$173$	−1.40328e10	−1.19107	−0.595534	−	0.803330i	$-0.703059\pi$
$173$	−1.40328e10	−1.19107	−0.595534	+	0.803330i	$0.703059\pi$
$174$	0	0
$175$	5.92801e9	0.477791
$176$	0	0
$177$	2.56890e10	1.96729
$178$	0	0
$179$	2.48592e10	1.80987	0.904936	−	0.425548i	$-0.139919\pi$
$179$	2.48592e10	1.80987	0.904936	+	0.425548i	$0.139919\pi$
$180$	0	0
$181$	1.35989e10	0.941784	0.470892	−	0.882191i	$-0.343932\pi$
$181$	1.35989e10	0.941784	0.470892	+	0.882191i	$0.343932\pi$
$182$	0	0
$183$	2.40209e10	1.58329
$184$	0	0
$185$	4.32939e9	0.271740
$186$	0	0
$187$	−3.58418e10	−2.14340
$188$	0	0
$189$	−1.69240e9	−0.0964770
$190$	0	0
$191$	1.65985e9	0.0902440	0.0451220	−	0.998981i	$-0.485632\pi$
$191$	1.65985e9	0.0902440	0.0451220	+	0.998981i	$0.485632\pi$
$192$	0	0
$193$	3.09499e10	1.60565	0.802827	−	0.596212i	$-0.203328\pi$
$193$	3.09499e10	1.60565	0.802827	+	0.596212i	$0.203328\pi$
$194$	0	0
$195$	5.10629e9	0.252900
$196$	0	0
$197$	−3.04341e10	−1.43967	−0.719835	−	0.694145i	$-0.755783\pi$
$197$	−3.04341e10	−1.43967	−0.719835	+	0.694145i	$0.755783\pi$
$198$	0	0
$199$	−4.44812e8	−0.0201066	−0.0100533	−	0.999949i	$-0.503200\pi$
$199$	−4.44812e8	−0.0201066	−0.0100533	+	0.999949i	$0.503200\pi$
$200$	0	0
$201$	−4.18227e10	−1.80730
$202$	0	0
$203$	4.82365e9	0.199363
$204$	0	0
$205$	−3.12695e10	−1.23660
$206$	0	0
$207$	1.37158e10	0.519223
$208$	0	0
$209$	1.68505e10	0.610877
$210$	0	0
$211$	−4.88078e9	−0.169519	−0.0847595	−	0.996401i	$-0.527012\pi$
$211$	−4.88078e9	−0.169519	−0.0847595	+	0.996401i	$0.527012\pi$
$212$	0	0
$213$	−4.00579e10	−1.33346
$214$	0	0
$215$	−2.14654e10	−0.685120
$216$	0	0
$217$	1.21725e10	0.372660
$218$	0	0
$219$	−6.69148e10	−1.96573
$220$	0	0
$221$	1.29115e10	0.364093
$222$	0	0
$223$	−2.47486e10	−0.670162	−0.335081	−	0.942189i	$-0.608764\pi$
$223$	−2.47486e10	−0.670162	−0.335081	+	0.942189i	$0.608764\pi$
$224$	0	0
$225$	−1.99747e10	−0.519587
$226$	0	0
$227$	−2.62443e10	−0.656022	−0.328011	−	0.944674i	$-0.606378\pi$
$227$	−2.62443e10	−0.656022	−0.328011	+	0.944674i	$0.606378\pi$
$228$	0	0
$229$	−4.15262e10	−0.997844	−0.498922	−	0.866647i	$-0.666271\pi$
$229$	−4.15262e10	−0.997844	−0.498922	+	0.866647i	$0.666271\pi$
$230$	0	0
$231$	−8.25418e10	−1.90731
$232$	0	0
$233$	3.67402e10	0.816657	0.408329	−	0.912835i	$-0.366112\pi$
$233$	3.67402e10	0.816657	0.408329	+	0.912835i	$0.366112\pi$
$234$	0	0
$235$	−4.73145e10	−1.01202
$236$	0	0
$237$	9.77802e9	0.201318
$238$	0	0
$239$	4.35921e10	0.864206	0.432103	−	0.901824i	$-0.357772\pi$
$239$	4.35921e10	0.864206	0.432103	+	0.901824i	$0.357772\pi$
$240$	0	0
$241$	−1.04473e10	−0.199493	−0.0997464	−	0.995013i	$-0.531803\pi$
$241$	−1.04473e10	−0.199493	−0.0997464	+	0.995013i	$0.531803\pi$
$242$	0	0
$243$	7.47504e10	1.37526
$244$	0	0
$245$	1.07073e10	0.189859
$246$	0	0
$247$	−6.07015e9	−0.103768
$248$	0	0
$249$	1.59352e10	0.262701
$250$	0	0
$251$	−8.69402e10	−1.38258	−0.691288	−	0.722580i	$-0.742956\pi$
$251$	−8.69402e10	−1.38258	−0.691288	+	0.722580i	$0.742956\pi$
$252$	0	0
$253$	−6.02214e10	−0.924078
$254$	0	0
$255$	8.08232e10	1.19703
$256$	0	0
$257$	4.05492e10	0.579807	0.289903	−	0.957056i	$-0.406377\pi$
$257$	4.05492e10	0.579807	0.289903	+	0.957056i	$0.406377\pi$
$258$	0	0
$259$	2.52106e10	0.348124
$260$	0	0
$261$	−1.62535e10	−0.216802
$262$	0	0
$263$	−1.55920e10	−0.200956	−0.100478	−	0.994939i	$-0.532037\pi$
$263$	−1.55920e10	−0.200956	−0.100478	+	0.994939i	$0.532037\pi$
$264$	0	0
$265$	−9.29550e10	−1.15789
$266$	0	0
$267$	1.20037e11	1.44549
$268$	0	0
$269$	1.34814e11	1.56982	0.784911	−	0.619608i	$-0.212708\pi$
$269$	1.34814e11	1.56982	0.784911	+	0.619608i	$0.212708\pi$
$270$	0	0
$271$	−1.08155e10	−0.121810	−0.0609052	−	0.998144i	$-0.519399\pi$
$271$	−1.08155e10	−0.121810	−0.0609052	+	0.998144i	$0.519399\pi$
$272$	0	0
$273$	2.97346e10	0.323989
$274$	0	0
$275$	8.77021e10	0.924725
$276$	0	0
$277$	1.05265e11	1.07429	0.537147	−	0.843488i	$-0.319502\pi$
$277$	1.05265e11	1.07429	0.537147	+	0.843488i	$0.319502\pi$
$278$	0	0
$279$	−4.10158e10	−0.405259
$280$	0	0
$281$	5.90656e10	0.565140	0.282570	−	0.959247i	$-0.408813\pi$
$281$	5.90656e10	0.565140	0.282570	+	0.959247i	$0.408813\pi$
$282$	0	0
$283$	1.45893e11	1.35206	0.676030	−	0.736875i	$-0.263699\pi$
$283$	1.45893e11	1.35206	0.676030	+	0.736875i	$0.263699\pi$
$284$	0	0
$285$	−3.79978e10	−0.341158
$286$	0	0
$287$	−1.82086e11	−1.58419
$288$	0	0
$289$	8.57779e10	0.723328
$290$	0	0
$291$	1.92570e11	1.57423
$292$	0	0
$293$	−7.69633e9	−0.0610070	−0.0305035	−	0.999535i	$-0.509711\pi$
$293$	−7.69633e9	−0.0610070	−0.0305035	+	0.999535i	$0.509711\pi$
$294$	0	0
$295$	1.21695e11	0.935565
$296$	0	0
$297$	−2.50382e10	−0.186723
$298$	0	0
$299$	2.16940e10	0.156971
$300$	0	0
$301$	−1.24996e11	−0.877702
$302$	0	0
$303$	2.23715e11	1.52476
$304$	0	0
$305$	1.13793e11	0.752950
$306$	0	0
$307$	−7.83929e10	−0.503679	−0.251839	−	0.967769i	$-0.581035\pi$
$307$	−7.83929e10	−0.503679	−0.251839	+	0.967769i	$0.581035\pi$
$308$	0	0
$309$	2.50245e11	1.56154
$310$	0	0
$311$	2.02455e11	1.22717	0.613587	−	0.789627i	$-0.289726\pi$
$311$	2.02455e11	1.22717	0.613587	+	0.789627i	$0.289726\pi$
$312$	0	0
$313$	4.65893e10	0.274370	0.137185	−	0.990545i	$-0.456195\pi$
$313$	4.65893e10	0.274370	0.137185	+	0.990545i	$0.456195\pi$
$314$	0	0
$315$	8.90572e10	0.509649
$316$	0	0
$317$	−1.19232e10	−0.0663170	−0.0331585	−	0.999450i	$-0.510557\pi$
$317$	−1.19232e10	−0.0663170	−0.0331585	+	0.999450i	$0.510557\pi$
$318$	0	0
$319$	7.13636e10	0.385850
$320$	0	0
$321$	1.53585e11	0.807374
$322$	0	0
$323$	−9.60794e10	−0.491155
$324$	0	0
$325$	−3.15935e10	−0.157081
$326$	0	0
$327$	8.54918e10	0.413485
$328$	0	0
$329$	−2.75518e11	−1.29649
$330$	0	0
$331$	−1.99404e11	−0.913076	−0.456538	−	0.889704i	$-0.650911\pi$
$331$	−1.99404e11	−0.913076	−0.456538	+	0.889704i	$0.650911\pi$
$332$	0	0
$333$	−8.49480e10	−0.378577
$334$	0	0
$335$	−1.98124e11	−0.859479
$336$	0	0
$337$	3.29370e11	1.39107	0.695536	−	0.718491i	$-0.255167\pi$
$337$	3.29370e11	1.39107	0.695536	+	0.718491i	$0.255167\pi$
$338$	0	0
$339$	1.60988e11	0.662057
$340$	0	0
$341$	1.80087e11	0.721252
$342$	0	0
$343$	2.78605e11	1.08684
$344$	0	0
$345$	1.35799e11	0.516073
$346$	0	0
$347$	2.16834e10	0.0802868	0.0401434	−	0.999194i	$-0.487219\pi$
$347$	2.16834e10	0.0802868	0.0401434	+	0.999194i	$0.487219\pi$
$348$	0	0
$349$	1.43336e11	0.517178	0.258589	−	0.965987i	$-0.416742\pi$
$349$	1.43336e11	0.517178	0.258589	+	0.965987i	$0.416742\pi$
$350$	0	0
$351$	9.01966e9	0.0317182
$352$	0	0
$353$	−8.38423e10	−0.287393	−0.143697	−	0.989622i	$-0.545899\pi$
$353$	−8.38423e10	−0.287393	−0.143697	+	0.989622i	$0.545899\pi$
$354$	0	0
$355$	−1.89764e11	−0.634142
$356$	0	0
$357$	4.70644e11	1.53351
$358$	0	0
$359$	−1.26213e11	−0.401033	−0.200516	−	0.979690i	$-0.564262\pi$
$359$	−1.26213e11	−0.401033	−0.200516	+	0.979690i	$0.564262\pi$
$360$	0	0
$361$	−2.77517e11	−0.860019
$362$	0	0
$363$	−7.63090e11	−2.30673
$364$	0	0
$365$	−3.16992e11	−0.934824
$366$	0	0
$367$	6.56218e11	1.88821	0.944106	−	0.329642i	$-0.106928\pi$
$367$	6.56218e11	1.88821	0.944106	+	0.329642i	$0.106928\pi$
$368$	0	0
$369$	6.13546e11	1.72277
$370$	0	0
$371$	−5.41289e11	−1.48336
$372$	0	0
$373$	−5.52730e11	−1.47851	−0.739253	−	0.673428i	$-0.764821\pi$
$373$	−5.52730e11	−1.47851	−0.739253	+	0.673428i	$0.764821\pi$
$374$	0	0
$375$	−5.46958e11	−1.42828
$376$	0	0
$377$	−2.57078e10	−0.0655433
$378$	0	0
$379$	3.07664e11	0.765950	0.382975	−	0.923759i	$-0.374900\pi$
$379$	3.07664e11	0.765950	0.382975	+	0.923759i	$0.374900\pi$
$380$	0	0
$381$	−4.06092e11	−0.987330
$382$	0	0
$383$	−1.31812e11	−0.313012	−0.156506	−	0.987677i	$-0.550023\pi$
$383$	−1.31812e11	−0.313012	−0.156506	+	0.987677i	$0.550023\pi$
$384$	0	0
$385$	−3.91021e11	−0.907040
$386$	0	0
$387$	4.21179e11	0.954480
$388$	0	0
$389$	1.31742e11	0.291709	0.145855	−	0.989306i	$-0.453407\pi$
$389$	1.31742e11	0.291709	0.145855	+	0.989306i	$0.453407\pi$
$390$	0	0
$391$	3.43376e11	0.742975
$392$	0	0
$393$	2.40479e10	0.0508523
$394$	0	0
$395$	4.63208e10	0.0957391
$396$	0	0
$397$	4.44767e11	0.898619	0.449309	−	0.893376i	$-0.351670\pi$
$397$	4.44767e11	0.898619	0.449309	+	0.893376i	$0.351670\pi$
$398$	0	0
$399$	−2.21266e11	−0.437055
$400$	0	0
$401$	−8.36539e11	−1.61561	−0.807805	−	0.589450i	$-0.799345\pi$
$401$	−8.36539e11	−1.61561	−0.807805	+	0.589450i	$0.799345\pi$
$402$	0	0
$403$	−6.48738e10	−0.122517
$404$	0	0
$405$	3.83557e11	0.708406
$406$	0	0
$407$	3.72979e11	0.673766
$408$	0	0
$409$	2.11826e11	0.374304	0.187152	−	0.982331i	$-0.440074\pi$
$409$	2.11826e11	0.374304	0.187152	+	0.982331i	$0.440074\pi$
$410$	0	0
$411$	2.55033e11	0.440867
$412$	0	0
$413$	7.08646e11	1.19854
$414$	0	0
$415$	7.54890e10	0.124930
$416$	0	0
$417$	−9.38307e11	−1.51961
$418$	0	0
$419$	−5.30153e11	−0.840308	−0.420154	−	0.907453i	$-0.638024\pi$
$419$	−5.30153e11	−0.840308	−0.420154	+	0.907453i	$0.638024\pi$
$420$	0	0
$421$	3.41563e10	0.0529909	0.0264955	−	0.999649i	$-0.491565\pi$
$421$	3.41563e10	0.0529909	0.0264955	+	0.999649i	$0.491565\pi$
$422$	0	0
$423$	9.28369e11	1.40990
$424$	0	0
$425$	−5.00067e11	−0.743495
$426$	0	0
$427$	6.62631e11	0.964598
$428$	0	0
$429$	4.39909e11	0.627054
$430$	0	0
$431$	−8.63019e11	−1.20468	−0.602342	−	0.798238i	$-0.705765\pi$
$431$	−8.63019e11	−1.20468	−0.602342	+	0.798238i	$0.705765\pi$
$432$	0	0
$433$	−3.23681e11	−0.442509	−0.221254	−	0.975216i	$-0.571015\pi$
$433$	−3.23681e11	−0.442509	−0.221254	+	0.975216i	$0.571015\pi$
$434$	0	0
$435$	−1.60925e11	−0.215487
$436$	0	0
$437$	−1.61433e11	−0.211751
$438$	0	0
$439$	1.45809e12	1.87368	0.936838	−	0.349763i	$-0.113738\pi$
$439$	1.45809e12	1.87368	0.936838	+	0.349763i	$0.113738\pi$
$440$	0	0
$441$	−2.10090e11	−0.264503
$442$	0	0
$443$	−1.49027e12	−1.83844	−0.919219	−	0.393748i	$-0.871178\pi$
$443$	−1.49027e12	−1.83844	−0.919219	+	0.393748i	$0.871178\pi$
$444$	0	0
$445$	5.68645e11	0.687418
$446$	0	0
$447$	8.53314e11	1.01094
$448$	0	0
$449$	3.04750e11	0.353862	0.176931	−	0.984223i	$-0.443383\pi$
$449$	3.04750e11	0.353862	0.176931	+	0.984223i	$0.443383\pi$
$450$	0	0
$451$	−2.69387e12	−3.06608
$452$	0	0
$453$	−9.13648e11	−1.01938
$454$	0	0
$455$	1.40860e11	0.154076
$456$	0	0
$457$	−1.23809e11	−0.132779	−0.0663896	−	0.997794i	$-0.521148\pi$
$457$	−1.23809e11	−0.132779	−0.0663896	+	0.997794i	$0.521148\pi$
$458$	0	0
$459$	1.42765e11	0.150129
$460$	0	0
$461$	3.16152e11	0.326018	0.163009	−	0.986625i	$-0.447880\pi$
$461$	3.16152e11	0.326018	0.163009	+	0.986625i	$0.447880\pi$
$462$	0	0
$463$	−1.73371e12	−1.75332	−0.876659	−	0.481112i	$-0.840233\pi$
$463$	−1.73371e12	−1.75332	−0.876659	+	0.481112i	$0.840233\pi$
$464$	0	0
$465$	−4.06095e11	−0.402800
$466$	0	0
$467$	5.67889e11	0.552506	0.276253	−	0.961085i	$-0.410907\pi$
$467$	5.67889e11	0.552506	0.276253	+	0.961085i	$0.410907\pi$
$468$	0	0
$469$	−1.15370e12	−1.10107
$470$	0	0
$471$	1.02728e12	0.961823
$472$	0	0
$473$	−1.84926e12	−1.69872
$474$	0	0
$475$	2.35099e11	0.211899
$476$	0	0
$477$	1.82389e12	1.61312
$478$	0	0
$479$	−9.31010e11	−0.808062	−0.404031	−	0.914745i	$-0.632391\pi$
$479$	−9.31010e11	−0.808062	−0.404031	+	0.914745i	$0.632391\pi$
$480$	0	0
$481$	−1.34360e11	−0.114451
$482$	0	0
$483$	7.90776e11	0.661137
$484$	0	0
$485$	9.12249e11	0.748644
$486$	0	0
$487$	4.49592e10	0.0362192	0.0181096	−	0.999836i	$-0.494235\pi$
$487$	4.49592e10	0.0362192	0.0181096	+	0.999836i	$0.494235\pi$
$488$	0	0
$489$	1.30739e12	1.03399
$490$	0	0
$491$	−4.47956e11	−0.347831	−0.173916	−	0.984761i	$-0.555642\pi$
$491$	−4.47956e11	−0.347831	−0.173916	+	0.984761i	$0.555642\pi$
$492$	0	0
$493$	−4.06907e11	−0.310230
$494$	0	0
$495$	1.31756e12	0.986385
$496$	0	0
$497$	−1.10502e12	−0.812394
$498$	0	0
$499$	−7.27774e11	−0.525465	−0.262733	−	0.964869i	$-0.584624\pi$
$499$	−7.27774e11	−0.525465	−0.262733	+	0.964869i	$0.584624\pi$
$500$	0	0
$501$	1.14188e12	0.809748
$502$	0	0
$503$	−6.23613e11	−0.434369	−0.217185	−	0.976131i	$-0.569687\pi$
$503$	−6.23613e11	−0.434369	−0.217185	+	0.976131i	$0.569687\pi$
$504$	0	0
$505$	1.05979e12	0.725118
$506$	0	0
$507$	−1.58471e11	−0.106516
$508$	0	0
$509$	1.18958e12	0.785534	0.392767	−	0.919638i	$-0.371518\pi$
$509$	1.18958e12	0.785534	0.392767	+	0.919638i	$0.371518\pi$
$510$	0	0
$511$	−1.84588e12	−1.19760
$512$	0	0
$513$	−6.71186e10	−0.0427873
$514$	0	0
$515$	1.18547e12	0.742607
$516$	0	0
$517$	−4.07616e12	−2.50925
$518$	0	0
$519$	2.72613e12	1.64928
$520$	0	0
$521$	2.26245e12	1.34527	0.672635	−	0.739974i	$-0.265162\pi$
$521$	2.26245e12	1.34527	0.672635	+	0.739974i	$0.265162\pi$
$522$	0	0
$523$	−1.15246e12	−0.673549	−0.336775	−	0.941585i	$-0.609336\pi$
$523$	−1.15246e12	−0.673549	−0.336775	+	0.941585i	$0.609336\pi$
$524$	0	0
$525$	−1.15163e12	−0.661600
$526$	0	0
$527$	−1.02683e12	−0.579899
$528$	0	0
$529$	−1.22421e12	−0.679683
$530$	0	0
$531$	−2.38781e12	−1.30339
$532$	0	0
$533$	9.70432e11	0.520826
$534$	0	0
$535$	7.27567e11	0.383955
$536$	0	0
$537$	−4.82936e12	−2.50614
$538$	0	0
$539$	9.22434e11	0.470745
$540$	0	0
$541$	1.31455e12	0.659766	0.329883	−	0.944022i	$-0.392991\pi$
$541$	1.31455e12	0.659766	0.329883	+	0.944022i	$0.392991\pi$
$542$	0	0
$543$	−2.64185e12	−1.30409
$544$	0	0
$545$	4.04995e11	0.196637
$546$	0	0
$547$	−1.63148e12	−0.779182	−0.389591	−	0.920988i	$-0.627384\pi$
$547$	−1.63148e12	−0.779182	−0.389591	+	0.920988i	$0.627384\pi$
$548$	0	0
$549$	−2.23276e12	−1.04898
$550$	0	0
$551$	1.91301e11	0.0884168
$552$	0	0
$553$	2.69732e11	0.122651
$554$	0	0
$555$	−8.41066e11	−0.376280
$556$	0	0
$557$	−2.20811e12	−0.972013	−0.486007	−	0.873955i	$-0.661547\pi$
$557$	−2.20811e12	−0.972013	−0.486007	+	0.873955i	$0.661547\pi$
$558$	0	0
$559$	6.66169e11	0.288557
$560$	0	0
$561$	6.96295e12	2.96797
$562$	0	0
$563$	1.16602e12	0.489125	0.244563	−	0.969634i	$-0.421356\pi$
$563$	1.16602e12	0.489125	0.244563	+	0.969634i	$0.421356\pi$
$564$	0	0
$565$	7.62639e11	0.314848
$566$	0	0
$567$	2.23350e12	0.907533
$568$	0	0
$569$	−2.63455e12	−1.05366	−0.526831	−	0.849970i	$-0.676620\pi$
$569$	−2.63455e12	−1.05366	−0.526831	+	0.849970i	$0.676620\pi$
$570$	0	0
$571$	1.65379e12	0.651055	0.325527	−	0.945533i	$-0.394458\pi$
$571$	1.65379e12	0.651055	0.325527	+	0.945533i	$0.394458\pi$
$572$	0	0
$573$	−3.22457e11	−0.124961
$574$	0	0
$575$	−8.40213e11	−0.320541
$576$	0	0
$577$	1.77685e12	0.667358	0.333679	−	0.942687i	$-0.391710\pi$
$577$	1.77685e12	0.667358	0.333679	+	0.942687i	$0.391710\pi$
$578$	0	0
$579$	−6.01261e12	−2.22336
$580$	0	0
$581$	4.39582e11	0.160047
$582$	0	0
$583$	−8.00811e12	−2.87092
$584$	0	0
$585$	−4.74632e11	−0.167554
$586$	0	0
$587$	−4.61417e12	−1.60406	−0.802032	−	0.597281i	$-0.796248\pi$
$587$	−4.61417e12	−1.60406	−0.802032	+	0.597281i	$0.796248\pi$
$588$	0	0
$589$	4.82750e11	0.165273
$590$	0	0
$591$	5.91241e12	1.99352
$592$	0	0
$593$	−1.60396e12	−0.532655	−0.266328	−	0.963883i	$-0.585810\pi$
$593$	−1.60396e12	−0.532655	−0.266328	+	0.963883i	$0.585810\pi$
$594$	0	0
$595$	2.22955e12	0.729275
$596$	0	0
$597$	8.64132e10	0.0278417
$598$	0	0
$599$	3.56445e12	1.13128	0.565642	−	0.824651i	$-0.308628\pi$
$599$	3.56445e12	1.13128	0.565642	+	0.824651i	$0.308628\pi$
$600$	0	0
$601$	−4.23106e12	−1.32286	−0.661431	−	0.750006i	$-0.730051\pi$
$601$	−4.23106e12	−1.32286	−0.661431	+	0.750006i	$0.730051\pi$
$602$	0	0
$603$	3.88744e12	1.19739
$604$	0	0
$605$	−3.61494e12	−1.09699
$606$	0	0
$607$	−2.69700e12	−0.806366	−0.403183	−	0.915119i	$-0.632096\pi$
$607$	−2.69700e12	−0.806366	−0.403183	+	0.915119i	$0.632096\pi$
$608$	0	0
$609$	−9.37086e11	−0.276059
$610$	0	0
$611$	1.46838e12	0.426239
$612$	0	0
$613$	−4.29368e12	−1.22817	−0.614084	−	0.789240i	$-0.710475\pi$
$613$	−4.29368e12	−1.22817	−0.614084	+	0.789240i	$0.710475\pi$
$614$	0	0
$615$	6.07468e12	1.71232
$616$	0	0
$617$	2.63888e12	0.733054	0.366527	−	0.930407i	$-0.380547\pi$
$617$	2.63888e12	0.733054	0.366527	+	0.930407i	$0.380547\pi$
$618$	0	0
$619$	2.36449e12	0.647335	0.323667	−	0.946171i	$-0.395084\pi$
$619$	2.36449e12	0.647335	0.323667	+	0.946171i	$0.395084\pi$
$620$	0	0
$621$	2.39873e11	0.0647247
$622$	0	0
$623$	3.31129e12	0.880645
$624$	0	0
$625$	−4.30574e11	−0.112872
$626$	0	0
$627$	−3.27352e12	−0.845885
$628$	0	0
$629$	−2.12668e12	−0.541719
$630$	0	0
$631$	4.08896e12	1.02679	0.513393	−	0.858153i	$-0.328388\pi$
$631$	4.08896e12	1.02679	0.513393	+	0.858153i	$0.328388\pi$
$632$	0	0
$633$	9.48184e11	0.234734
$634$	0	0
$635$	−1.92376e12	−0.469536
$636$	0	0
$637$	−3.32295e11	−0.0799642
$638$	0	0
$639$	3.72341e12	0.883459
$640$	0	0
$641$	−3.55153e12	−0.830911	−0.415456	−	0.909613i	$-0.636378\pi$
$641$	−3.55153e12	−0.830911	−0.415456	+	0.909613i	$0.636378\pi$
$642$	0	0
$643$	−6.85000e12	−1.58031	−0.790153	−	0.612910i	$-0.789999\pi$
$643$	−6.85000e12	−1.58031	−0.790153	+	0.612910i	$0.789999\pi$
$644$	0	0
$645$	4.17007e12	0.948690
$646$	0	0
$647$	4.28931e12	0.962317	0.481159	−	0.876634i	$-0.340216\pi$
$647$	4.28931e12	0.962317	0.481159	+	0.876634i	$0.340216\pi$
$648$	0	0
$649$	1.04841e13	2.31968
$650$	0	0
$651$	−2.36474e12	−0.516024
$652$	0	0
$653$	8.81913e12	1.89809	0.949045	−	0.315142i	$-0.102052\pi$
$653$	8.81913e12	1.89809	0.949045	+	0.315142i	$0.102052\pi$
$654$	0	0
$655$	1.13921e11	0.0241834
$656$	0	0
$657$	6.21977e12	1.30236
$658$	0	0
$659$	−5.76538e12	−1.19081	−0.595406	−	0.803425i	$-0.703009\pi$
$659$	−5.76538e12	−1.19081	−0.595406	+	0.803425i	$0.703009\pi$
$660$	0	0
$661$	−4.77184e12	−0.972254	−0.486127	−	0.873888i	$-0.661591\pi$
$661$	−4.77184e12	−0.972254	−0.486127	+	0.873888i	$0.661591\pi$
$662$	0	0
$663$	−2.50831e12	−0.504162
$664$	0	0
$665$	−1.04819e12	−0.207846
$666$	0	0
$667$	−6.83686e11	−0.133749
$668$	0	0
$669$	4.80789e12	0.927977
$670$	0	0
$671$	9.80330e12	1.86690
$672$	0	0
$673$	1.38206e10	0.00259692	0.00129846	−	0.999999i	$-0.499587\pi$
$673$	1.38206e10	0.00259692	0.00129846	+	0.999999i	$0.499587\pi$
$674$	0	0
$675$	−3.49334e11	−0.0647700
$676$	0	0
$677$	−2.70699e12	−0.495265	−0.247633	−	0.968854i	$-0.579653\pi$
$677$	−2.70699e12	−0.495265	−0.247633	+	0.968854i	$0.579653\pi$
$678$	0	0
$679$	5.31214e12	0.959082
$680$	0	0
$681$	5.09845e12	0.908398
$682$	0	0
$683$	3.77305e12	0.663436	0.331718	−	0.943379i	$-0.392372\pi$
$683$	3.77305e12	0.663436	0.331718	+	0.943379i	$0.392372\pi$
$684$	0	0
$685$	1.20815e12	0.209659
$686$	0	0
$687$	8.06725e12	1.38172
$688$	0	0
$689$	2.88481e12	0.487676
$690$	0	0
$691$	1.42326e12	0.237484	0.118742	−	0.992925i	$-0.462114\pi$
$691$	1.42326e12	0.237484	0.118742	+	0.992925i	$0.462114\pi$
$692$	0	0
$693$	7.67231e12	1.26365
$694$	0	0
$695$	−4.44499e12	−0.722668
$696$	0	0
$697$	1.53602e13	2.46518
$698$	0	0
$699$	−7.13748e12	−1.13083
$700$	0	0
$701$	−4.18953e12	−0.655291	−0.327645	−	0.944801i	$-0.606255\pi$
$701$	−4.18953e12	−0.655291	−0.327645	+	0.944801i	$0.606255\pi$
$702$	0	0
$703$	9.99825e11	0.154392
$704$	0	0
$705$	9.19173e12	1.40135
$706$	0	0
$707$	6.17129e12	0.928942
$708$	0	0
$709$	−5.30584e12	−0.788580	−0.394290	−	0.918986i	$-0.629009\pi$
$709$	−5.30584e12	−0.788580	−0.394290	+	0.918986i	$0.629009\pi$
$710$	0	0
$711$	−9.08872e11	−0.133380
$712$	0	0
$713$	−1.72529e12	−0.250011
$714$	0	0
$715$	2.08395e12	0.298202
$716$	0	0
$717$	−8.46859e12	−1.19667
$718$	0	0
$719$	−2.67496e12	−0.373282	−0.186641	−	0.982428i	$-0.559760\pi$
$719$	−2.67496e12	−0.373282	−0.186641	+	0.982428i	$0.559760\pi$
$720$	0	0
$721$	6.90316e12	0.951347
$722$	0	0
$723$	2.02958e12	0.276239
$724$	0	0
$725$	9.95669e11	0.133843
$726$	0	0
$727$	8.33355e12	1.10643	0.553217	−	0.833037i	$-0.313400\pi$
$727$	8.33355e12	1.10643	0.553217	+	0.833037i	$0.313400\pi$
$728$	0	0
$729$	−6.31830e12	−0.828564
$730$	0	0
$731$	1.05442e13	1.36580
$732$	0	0
$733$	1.11038e11	0.0142071	0.00710355	−	0.999975i	$-0.497739\pi$
$733$	1.11038e11	0.0142071	0.00710355	+	0.999975i	$0.497739\pi$
$734$	0	0
$735$	−2.08009e12	−0.262899
$736$	0	0
$737$	−1.70684e13	−2.13103
$738$	0	0
$739$	−1.02565e13	−1.26502	−0.632511	−	0.774551i	$-0.717976\pi$
$739$	−1.02565e13	−1.26502	−0.632511	+	0.774551i	$0.717976\pi$
$740$	0	0
$741$	1.17924e12	0.143688
$742$	0	0
$743$	−3.90792e12	−0.470431	−0.235215	−	0.971943i	$-0.575580\pi$
$743$	−3.90792e12	−0.470431	−0.235215	+	0.971943i	$0.575580\pi$
$744$	0	0
$745$	4.04235e12	0.480763
$746$	0	0
$747$	−1.48119e12	−0.174047
$748$	0	0
$749$	4.23671e12	0.491882
$750$	0	0
$751$	2.63552e12	0.302333	0.151167	−	0.988508i	$-0.451697\pi$
$751$	2.63552e12	0.302333	0.151167	+	0.988508i	$0.451697\pi$
$752$	0	0
$753$	1.68898e13	1.91446
$754$	0	0
$755$	−4.32817e12	−0.484778
$756$	0	0
$757$	−1.02796e13	−1.13774	−0.568870	−	0.822428i	$-0.692619\pi$
$757$	−1.02796e13	−1.13774	−0.568870	+	0.822428i	$0.692619\pi$
$758$	0	0
$759$	1.16992e13	1.27958
$760$	0	0
$761$	1.47695e12	0.159638	0.0798190	−	0.996809i	$-0.474566\pi$
$761$	1.47695e12	0.159638	0.0798190	+	0.996809i	$0.474566\pi$
$762$	0	0
$763$	2.35834e12	0.251910
$764$	0	0
$765$	−7.51256e12	−0.793070
$766$	0	0
$767$	−3.77674e12	−0.394038
$768$	0	0
$769$	−3.06990e12	−0.316560	−0.158280	−	0.987394i	$-0.550595\pi$
$769$	−3.06990e12	−0.316560	−0.158280	+	0.987394i	$0.550595\pi$
$770$	0	0
$771$	−7.87745e12	−0.802862
$772$	0	0
$773$	1.22536e13	1.23440	0.617200	−	0.786807i	$-0.288267\pi$
$773$	1.22536e13	1.23440	0.617200	+	0.786807i	$0.288267\pi$
$774$	0	0
$775$	2.51258e12	0.250186
$776$	0	0
$777$	−4.89764e12	−0.482050
$778$	0	0
$779$	−7.22134e12	−0.702585
$780$	0	0
$781$	−1.63482e13	−1.57232
$782$	0	0
$783$	−2.84255e11	−0.0270259
$784$	0	0
$785$	4.86648e12	0.457405
$786$	0	0
$787$	−1.16840e13	−1.08569	−0.542847	−	0.839832i	$-0.682653\pi$
$787$	−1.16840e13	−1.08569	−0.542847	+	0.839832i	$0.682653\pi$
$788$	0	0
$789$	3.02904e12	0.278265
$790$	0	0
$791$	4.44095e12	0.403349
$792$	0	0
$793$	−3.53151e12	−0.317125
$794$	0	0
$795$	1.80583e13	1.60333
$796$	0	0
$797$	−1.05711e13	−0.928024	−0.464012	−	0.885829i	$-0.653590\pi$
$797$	−1.05711e13	−0.928024	−0.464012	+	0.885829i	$0.653590\pi$
$798$	0	0
$799$	2.32418e13	2.01748
$800$	0	0
$801$	−1.11575e13	−0.957682
$802$	0	0
$803$	−2.73089e13	−2.31785
$804$	0	0
$805$	3.74610e12	0.314411
$806$	0	0
$807$	−2.61902e13	−2.17374
$808$	0	0
$809$	−9.18043e12	−0.753520	−0.376760	−	0.926311i	$-0.622962\pi$
$809$	−9.18043e12	−0.753520	−0.376760	+	0.926311i	$0.622962\pi$
$810$	0	0
$811$	1.38738e12	0.112616	0.0563081	−	0.998413i	$-0.482067\pi$
$811$	1.38738e12	0.112616	0.0563081	+	0.998413i	$0.482067\pi$
$812$	0	0
$813$	2.10111e12	0.168672
$814$	0	0
$815$	6.19343e12	0.491725
$816$	0	0
$817$	−4.95721e12	−0.389258
$818$	0	0
$819$	−2.76385e12	−0.214653
$820$	0	0
$821$	1.09862e12	0.0843924	0.0421962	−	0.999109i	$-0.486565\pi$
$821$	1.09862e12	0.0843924	0.0421962	+	0.999109i	$0.486565\pi$
$822$	0	0
$823$	1.94330e13	1.47653	0.738263	−	0.674513i	$-0.235646\pi$
$823$	1.94330e13	1.47653	0.738263	+	0.674513i	$0.235646\pi$
$824$	0	0
$825$	−1.70378e13	−1.28047
$826$	0	0
$827$	2.56778e12	0.190890	0.0954448	−	0.995435i	$-0.469573\pi$
$827$	2.56778e12	0.190890	0.0954448	+	0.995435i	$0.469573\pi$
$828$	0	0
$829$	1.15684e13	0.850703	0.425352	−	0.905028i	$-0.360150\pi$
$829$	1.15684e13	0.850703	0.425352	+	0.905028i	$0.360150\pi$
$830$	0	0
$831$	−2.04496e13	−1.48758
$832$	0	0
$833$	−5.25961e12	−0.378487
$834$	0	0
$835$	5.40935e12	0.385085
$836$	0	0
$837$	−7.17320e11	−0.0505182
$838$	0	0
$839$	1.07035e13	0.745754	0.372877	−	0.927881i	$-0.378371\pi$
$839$	1.07035e13	0.745754	0.372877	+	0.927881i	$0.378371\pi$
$840$	0	0
$841$	−1.36970e13	−0.944153
$842$	0	0
$843$	−1.14746e13	−0.782553
$844$	0	0
$845$	−7.50716e11	−0.0506548
$846$	0	0
$847$	−2.10503e13	−1.40534
$848$	0	0
$849$	−2.83425e13	−1.87220
$850$	0	0
$851$	−3.57325e12	−0.233550
$852$	0	0
$853$	1.85417e13	1.19917	0.599583	−	0.800313i	$-0.295333\pi$
$853$	1.85417e13	1.19917	0.599583	+	0.800313i	$0.295333\pi$
$854$	0	0
$855$	3.53191e12	0.226028
$856$	0	0
$857$	8.38633e12	0.531078	0.265539	−	0.964100i	$-0.414450\pi$
$857$	8.38633e12	0.531078	0.265539	+	0.964100i	$0.414450\pi$
$858$	0	0
$859$	−2.49383e12	−0.156278	−0.0781390	−	0.996942i	$-0.524898\pi$
$859$	−2.49383e12	−0.156278	−0.0781390	+	0.996942i	$0.524898\pi$
$860$	0	0
$861$	3.53736e13	2.19364
$862$	0	0
$863$	1.05758e12	0.0649029	0.0324515	−	0.999473i	$-0.489669\pi$
$863$	1.05758e12	0.0649029	0.0324515	+	0.999473i	$0.489669\pi$
$864$	0	0
$865$	1.29143e13	0.784332
$866$	0	0
$867$	−1.66640e13	−1.00160
$868$	0	0
$869$	3.99056e12	0.237380
$870$	0	0
$871$	6.14868e12	0.361993
$872$	0	0
$873$	−1.78995e13	−1.04298
$874$	0	0
$875$	−1.50881e13	−0.870161
$876$	0	0
$877$	2.46623e13	1.40778	0.703891	−	0.710308i	$-0.251444\pi$
$877$	2.46623e13	1.40778	0.703891	+	0.710308i	$0.251444\pi$
$878$	0	0
$879$	1.49516e12	0.0844767
$880$	0	0
$881$	−7.21428e12	−0.403461	−0.201731	−	0.979441i	$-0.564657\pi$
$881$	−7.21428e12	−0.403461	−0.201731	+	0.979441i	$0.564657\pi$
$882$	0	0
$883$	5.65890e12	0.313263	0.156631	−	0.987657i	$-0.449937\pi$
$883$	5.65890e12	0.313263	0.156631	+	0.987657i	$0.449937\pi$
$884$	0	0
$885$	−2.36416e13	−1.29548
$886$	0	0
$887$	−1.70857e13	−0.926778	−0.463389	−	0.886155i	$-0.653367\pi$
$887$	−1.70857e13	−0.926778	−0.463389	+	0.886155i	$0.653367\pi$
$888$	0	0
$889$	−1.12023e13	−0.601518
$890$	0	0
$891$	3.30436e13	1.75646
$892$	0	0
$893$	−1.09268e13	−0.574989
$894$	0	0
$895$	−2.28779e13	−1.19182
$896$	0	0
$897$	−4.21446e12	−0.217358
$898$	0	0
$899$	2.04450e12	0.104392
$900$	0	0
$901$	4.56613e13	2.30827
$902$	0	0
$903$	2.42828e13	1.21536
$904$	0	0
$905$	−1.25151e13	−0.620176
$906$	0	0
$907$	−4.31310e12	−0.211620	−0.105810	−	0.994386i	$-0.533744\pi$
$907$	−4.31310e12	−0.211620	−0.105810	+	0.994386i	$0.533744\pi$
$908$	0	0
$909$	−2.07944e13	−1.01020
$910$	0	0
$911$	−3.59972e13	−1.73155	−0.865777	−	0.500430i	$-0.833175\pi$
$911$	−3.59972e13	−1.73155	−0.865777	+	0.500430i	$0.833175\pi$
$912$	0	0
$913$	6.50340e12	0.309758
$914$	0	0
$915$	−2.21064e13	−1.04261
$916$	0	0
$917$	6.63375e11	0.0309811
$918$	0	0
$919$	1.78486e13	0.825437	0.412719	−	0.910859i	$-0.364579\pi$
$919$	1.78486e13	0.825437	0.412719	+	0.910859i	$0.364579\pi$
$920$	0	0
$921$	1.52293e13	0.697447
$922$	0	0
$923$	5.88923e12	0.267086
$924$	0	0
$925$	5.20382e12	0.233714
$926$	0	0
$927$	−2.32604e13	−1.03457
$928$	0	0
$929$	−2.28123e13	−1.00484	−0.502422	−	0.864622i	$-0.667558\pi$
$929$	−2.28123e13	−1.00484	−0.502422	+	0.864622i	$0.667558\pi$
$930$	0	0
$931$	2.47272e12	0.107870
$932$	0	0
$933$	−3.93307e13	−1.69928
$934$	0	0
$935$	3.29852e13	1.41145
$936$	0	0
$937$	9.14211e12	0.387452	0.193726	−	0.981056i	$-0.437943\pi$
$937$	9.14211e12	0.387452	0.193726	+	0.981056i	$0.437943\pi$
$938$	0	0
$939$	−9.05084e12	−0.379922
$940$	0	0
$941$	2.42073e13	1.00645	0.503225	−	0.864155i	$-0.332147\pi$
$941$	2.42073e13	1.00645	0.503225	+	0.864155i	$0.332147\pi$
$942$	0	0
$943$	2.58082e13	1.06281
$944$	0	0
$945$	1.55751e12	0.0635312
$946$	0	0
$947$	−1.63239e13	−0.659550	−0.329775	−	0.944060i	$-0.606973\pi$
$947$	−1.63239e13	−0.659550	−0.329775	+	0.944060i	$0.606973\pi$
$948$	0	0
$949$	9.83767e12	0.393726
$950$	0	0
$951$	2.31630e12	0.0918295
$952$	0	0
$953$	2.42382e13	0.951880	0.475940	−	0.879478i	$-0.342108\pi$
$953$	2.42382e13	0.951880	0.475940	+	0.879478i	$0.342108\pi$
$954$	0	0
$955$	−1.52756e12	−0.0594268
$956$	0	0
$957$	−1.38637e13	−0.534289
$958$	0	0
$959$	7.03523e12	0.268593
$960$	0	0
$961$	−2.12803e13	−0.804864
$962$	0	0
$963$	−1.42758e13	−0.534910
$964$	0	0
$965$	−2.84832e13	−1.05734
$966$	0	0
$967$	1.11387e13	0.409654	0.204827	−	0.978798i	$-0.434337\pi$
$967$	1.11387e13	0.409654	0.204827	+	0.978798i	$0.434337\pi$
$968$	0	0
$969$	1.86652e13	0.680106
$970$	0	0
$971$	3.10112e13	1.11952	0.559760	−	0.828655i	$-0.310893\pi$
$971$	3.10112e13	1.11952	0.559760	+	0.828655i	$0.310893\pi$
$972$	0	0
$973$	−2.58837e13	−0.925804
$974$	0	0
$975$	6.13763e12	0.217510
$976$	0	0
$977$	1.16626e13	0.409516	0.204758	−	0.978813i	$-0.434359\pi$
$977$	1.16626e13	0.409516	0.204758	+	0.978813i	$0.434359\pi$
$978$	0	0
$979$	4.89889e13	1.70442
$980$	0	0
$981$	−7.94651e12	−0.273947
$982$	0	0
$983$	1.20376e13	0.411198	0.205599	−	0.978636i	$-0.434086\pi$
$983$	1.20376e13	0.411198	0.205599	+	0.978636i	$0.434086\pi$
$984$	0	0
$985$	2.80085e13	0.948040
$986$	0	0
$987$	5.35246e13	1.79526
$988$	0	0
$989$	1.77164e13	0.588834
$990$	0	0
$991$	−4.45908e13	−1.46863	−0.734317	−	0.678806i	$-0.762498\pi$
$991$	−4.45908e13	−1.46863	−0.734317	+	0.678806i	$0.762498\pi$
$992$	0	0
$993$	3.87379e13	1.26434
$994$	0	0
$995$	4.09360e11	0.0132404
$996$	0	0
$997$	5.03829e13	1.61493	0.807467	−	0.589912i	$-0.200838\pi$
$997$	5.03829e13	1.61493	0.807467	+	0.589912i	$0.200838\pi$
$998$	0	0
$999$	−1.48564e12	−0.0471922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.h.1.2		5
4.3	odd	2		13.10.a.b.1.1	✓	5
12.11	even	2		117.10.a.e.1.5		5
20.19	odd	2		325.10.a.b.1.5		5
52.51	odd	2		169.10.a.b.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.a.b.1.1	✓	5	4.3	odd	2
117.10.a.e.1.5		5	12.11	even	2
169.10.a.b.1.5		5	52.51	odd	2
208.10.a.h.1.2		5	1.1	even	1	trivial
325.10.a.b.1.5		5	20.19	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)

\(f(q)\)	\(=\)	\(q-194.269 q^{3} -920.299 q^{5} -5359.02 q^{7} +18057.4 q^{9} +O(q^{10})\)	\(q-194.269 q^{3} -920.299 q^{5} -5359.02 q^{7} +18057.4 q^{9} -79284.0 q^{11} +28561.0 q^{13} +178785. q^{15} +452068. q^{17} -212533. q^{19} +1.04109e6 q^{21} +759566. q^{23} -1.10618e6 q^{25} +315803. q^{27} -900101. q^{29} -2.27141e6 q^{31} +1.54024e7 q^{33} +4.93189e6 q^{35} -4.70433e6 q^{37} -5.54851e6 q^{39} +3.39775e7 q^{41} +2.33244e7 q^{43} -1.66182e7 q^{45} +5.14121e7 q^{47} -1.16346e7 q^{49} -8.78228e7 q^{51} +1.01005e8 q^{53} +7.29650e7 q^{55} +4.12885e7 q^{57} -1.32234e8 q^{59} -1.23648e8 q^{61} -9.67699e7 q^{63} -2.62846e7 q^{65} +2.15282e8 q^{67} -1.47560e8 q^{69} +2.06198e8 q^{71} +3.44444e8 q^{73} +2.14896e8 q^{75} +4.24884e8 q^{77} -5.03324e7 q^{79} -4.16775e8 q^{81} -8.20266e7 q^{83} -4.16038e8 q^{85} +1.74862e8 q^{87} -6.17891e8 q^{89} -1.53059e8 q^{91} +4.41265e8 q^{93} +1.95594e8 q^{95} -9.91253e8 q^{97} -1.43166e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9}+O(q^{10}) \) 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9	\( 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9} - 121746 q^{11} + 142805 q^{13} - 105973 q^{15} - 495669 q^{17} + 840738 q^{19} - 1599467 q^{21} + 592152 q^{23} + 1670362 q^{25} - 6847883 q^{27} + 10678182 q^{29} - 12885296 q^{31} + 17278298 q^{33} - 8380731 q^{35} + 7171823 q^{37} - 4598321 q^{39} + 9294012 q^{41} - 12831975 q^{43} + 26135198 q^{45} - 43354215 q^{47} + 25249488 q^{49} - 16905901 q^{51} + 93231780 q^{53} - 99448846 q^{55} + 90173382 q^{57} - 246496182 q^{59} - 132232612 q^{61} + 416955202 q^{63} + 51495483 q^{65} + 369388534 q^{67} - 579986760 q^{69} - 212150457 q^{71} - 252729806 q^{73} + 752457788 q^{75} + 449666118 q^{77} + 1247271728 q^{79} - 317713115 q^{81} - 1696894296 q^{83} - 775363765 q^{85} + 614530466 q^{87} - 753854382 q^{89} - 288437539 q^{91} - 892784668 q^{93} - 1442632962 q^{95} + 3824606 q^{97} - 3016199848 q^{99}+O(q^{100}) \) 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9 - 121746 * q^11 + 142805 * q^13 - 105973 * q^15 - 495669 * q^17 + 840738 * q^19 - 1599467 * q^21 + 592152 * q^23 + 1670362 * q^25 - 6847883 * q^27 + 10678182 * q^29 - 12885296 * q^31 + 17278298 * q^33 - 8380731 * q^35 + 7171823 * q^37 - 4598321 * q^39 + 9294012 * q^41 - 12831975 * q^43 + 26135198 * q^45 - 43354215 * q^47 + 25249488 * q^49 - 16905901 * q^51 + 93231780 * q^53 - 99448846 * q^55 + 90173382 * q^57 - 246496182 * q^59 - 132232612 * q^61 + 416955202 * q^63 + 51495483 * q^65 + 369388534 * q^67 - 579986760 * q^69 - 212150457 * q^71 - 252729806 * q^73 + 752457788 * q^75 + 449666118 * q^77 + 1247271728 * q^79 - 317713115 * q^81 - 1696894296 * q^83 - 775363765 * q^85 + 614530466 * q^87 - 753854382 * q^89 - 288437539 * q^91 - 892784668 * q^93 - 1442632962 * q^95 + 3824606 * q^97 - 3016199848 * q^99

Introduction

L-functions

Modular forms

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