LMFDB - Embedded newform 208.10.a.h.1.1

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			1.1
Root			$16.7176$ 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)

$f(q)$	$=$	$q-250.479 q^{3} +1555.58 q^{5} +8329.39 q^{7} +43056.6 q^{9} +O(q^{10})$	$q-250.479 q^{3} +1555.58 q^{5} +8329.39 q^{7} +43056.6 q^{9} -30751.5 q^{11} +28561.0 q^{13} -389640. q^{15} -637455. q^{17} -105834. q^{19} -2.08634e6 q^{21} +511169. q^{23} +466710. q^{25} -5.85460e6 q^{27} +781868. q^{29} +2.83285e6 q^{31} +7.70259e6 q^{33} +1.29571e7 q^{35} +1.22183e7 q^{37} -7.15392e6 q^{39} -6.83367e6 q^{41} -3.84656e7 q^{43} +6.69781e7 q^{45} +1.30402e7 q^{47} +2.90252e7 q^{49} +1.59669e8 q^{51} -2.42871e7 q^{53} -4.78364e7 q^{55} +2.65092e7 q^{57} -1.63738e8 q^{59} +1.90751e7 q^{61} +3.58636e8 q^{63} +4.44290e7 q^{65} +7.22869e7 q^{67} -1.28037e8 q^{69} -2.65461e7 q^{71} +2.42850e8 q^{73} -1.16901e8 q^{75} -2.56141e8 q^{77} +4.64290e8 q^{79} +6.18969e8 q^{81} -5.46643e8 q^{83} -9.91613e8 q^{85} -1.95841e8 q^{87} +3.65672e8 q^{89} +2.37896e8 q^{91} -7.09568e8 q^{93} -1.64634e8 q^{95} +9.98914e7 q^{97} -1.32405e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−250.479	−1.78536	−0.892679	−	0.450693i	$-0.851177\pi$
$3$	−250.479	−1.78536	−0.892679	+	0.450693i	$0.851177\pi$
$4$	0	0
$5$	1555.58	1.11308	0.556542	−	0.830820i	$-0.312128\pi$
$5$	1555.58	1.11308	0.556542	+	0.830820i	$0.312128\pi$
$6$	0	0
$7$	8329.39	1.31121	0.655605	−	0.755104i	$-0.272414\pi$
$7$	8329.39	1.31121	0.655605	+	0.755104i	$0.272414\pi$
$8$	0	0
$9$	43056.6	2.18750
$10$	0	0
$11$	−30751.5	−0.633284	−0.316642	−	0.948545i	$-0.602555\pi$
$11$	−30751.5	−0.633284	−0.316642	+	0.948545i	$0.602555\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	−389640.	−1.98725
$16$	0	0
$17$	−637455.	−1.85110	−0.925549	−	0.378629i	$-0.876396\pi$
$17$	−637455.	−1.85110	−0.925549	+	0.378629i	$0.876396\pi$
$18$	0	0
$19$	−105834.	−0.186310	−0.0931548	−	0.995652i	$-0.529695\pi$
$19$	−105834.	−0.186310	−0.0931548	+	0.995652i	$0.529695\pi$
$20$	0	0
$21$	−2.08634e6	−2.34098
$22$	0	0
$23$	511169.	0.380881	0.190441	−	0.981699i	$-0.439008\pi$
$23$	511169.	0.380881	0.190441	+	0.981699i	$0.439008\pi$
$24$	0	0
$25$	466710.	0.238955
$26$	0	0
$27$	−5.85460e6	−2.12012
$28$	0	0
$29$	781868.	0.205278	0.102639	−	0.994719i	$-0.467271\pi$
$29$	781868.	0.205278	0.102639	+	0.994719i	$0.467271\pi$
$30$	0	0
$31$	2.83285e6	0.550929	0.275464	−	0.961311i	$-0.411168\pi$
$31$	2.83285e6	0.550929	0.275464	+	0.961311i	$0.411168\pi$
$32$	0	0
$33$	7.70259e6	1.13064
$34$	0	0
$35$	1.29571e7	1.45949
$36$	0	0
$37$	1.22183e7	1.07178	0.535889	−	0.844289i	$-0.319977\pi$
$37$	1.22183e7	1.07178	0.535889	+	0.844289i	$0.319977\pi$
$38$	0	0
$39$	−7.15392e6	−0.495169
$40$	0	0
$41$	−6.83367e6	−0.377682	−0.188841	−	0.982008i	$-0.560473\pi$
$41$	−6.83367e6	−0.377682	−0.188841	+	0.982008i	$0.560473\pi$
$42$	0	0
$43$	−3.84656e7	−1.71579	−0.857896	−	0.513823i	$-0.828229\pi$
$43$	−3.84656e7	−1.71579	−0.857896	+	0.513823i	$0.828229\pi$
$44$	0	0
$45$	6.69781e7	2.43487
$46$	0	0
$47$	1.30402e7	0.389802	0.194901	−	0.980823i	$-0.437561\pi$
$47$	1.30402e7	0.389802	0.194901	+	0.980823i	$0.437561\pi$
$48$	0	0
$49$	2.90252e7	0.719272
$50$	0	0
$51$	1.59669e8	3.30487
$52$	0	0
$53$	−2.42871e7	−0.422800	−0.211400	−	0.977400i	$-0.567802\pi$
$53$	−2.42871e7	−0.422800	−0.211400	+	0.977400i	$0.567802\pi$
$54$	0	0
$55$	−4.78364e7	−0.704898
$56$	0	0
$57$	2.65092e7	0.332629
$58$	0	0
$59$	−1.63738e8	−1.75920	−0.879599	−	0.475716i	$-0.842189\pi$
$59$	−1.63738e8	−1.75920	−0.879599	+	0.475716i	$0.842189\pi$
$60$	0	0
$61$	1.90751e7	0.176394	0.0881968	−	0.996103i	$-0.471890\pi$
$61$	1.90751e7	0.176394	0.0881968	+	0.996103i	$0.471890\pi$
$62$	0	0
$63$	3.58636e8	2.86828
$64$	0	0
$65$	4.44290e7	0.308714
$66$	0	0
$67$	7.22869e7	0.438251	0.219125	−	0.975697i	$-0.429680\pi$
$67$	7.22869e7	0.438251	0.219125	+	0.975697i	$0.429680\pi$
$68$	0	0
$69$	−1.28037e8	−0.680009
$70$	0	0
$71$	−2.65461e7	−0.123976	−0.0619880	−	0.998077i	$-0.519744\pi$
$71$	−2.65461e7	−0.123976	−0.0619880	+	0.998077i	$0.519744\pi$
$72$	0	0
$73$	2.42850e8	1.00089	0.500445	−	0.865769i	$-0.333170\pi$
$73$	2.42850e8	1.00089	0.500445	+	0.865769i	$0.333170\pi$
$74$	0	0
$75$	−1.16901e8	−0.426621
$76$	0	0
$77$	−2.56141e8	−0.830369
$78$	0	0
$79$	4.64290e8	1.34112	0.670560	−	0.741855i	$-0.266054\pi$
$79$	4.64290e8	1.34112	0.670560	+	0.741855i	$0.266054\pi$
$80$	0	0
$81$	6.18969e8	1.59767
$82$	0	0
$83$	−5.46643e8	−1.26431	−0.632153	−	0.774843i	$-0.717829\pi$
$83$	−5.46643e8	−1.26431	−0.632153	+	0.774843i	$0.717829\pi$
$84$	0	0
$85$	−9.91613e8	−2.06043
$86$	0	0
$87$	−1.95841e8	−0.366495
$88$	0	0
$89$	3.65672e8	0.617783	0.308892	−	0.951097i	$-0.400042\pi$
$89$	3.65672e8	0.617783	0.308892	+	0.951097i	$0.400042\pi$
$90$	0	0
$91$	2.37896e8	0.363664
$92$	0	0
$93$	−7.09568e8	−0.983605
$94$	0	0
$95$	−1.64634e8	−0.207378
$96$	0	0
$97$	9.98914e7	0.114566	0.0572830	−	0.998358i	$-0.481756\pi$
$97$	9.98914e7	0.114566	0.0572830	+	0.998358i	$0.481756\pi$
$98$	0	0
$99$	−1.32405e9	−1.38531
$100$	0	0
$101$	−6.76155e8	−0.646547	−0.323273	−	0.946306i	$-0.604783\pi$
$101$	−6.76155e8	−0.646547	−0.323273	+	0.946306i	$0.604783\pi$
$102$	0	0
$103$	1.73267e8	0.151687	0.0758434	−	0.997120i	$-0.475835\pi$
$103$	1.73267e8	0.151687	0.0758434	+	0.997120i	$0.475835\pi$
$104$	0	0
$105$	−3.24547e9	−2.60571
$106$	0	0
$107$	−1.23684e9	−0.912191	−0.456095	−	0.889931i	$-0.650752\pi$
$107$	−1.23684e9	−0.912191	−0.456095	+	0.889931i	$0.650752\pi$
$108$	0	0
$109$	−1.04481e9	−0.708956	−0.354478	−	0.935064i	$-0.615341\pi$
$109$	−1.04481e9	−0.708956	−0.354478	+	0.935064i	$0.615341\pi$
$110$	0	0
$111$	−3.06044e9	−1.91351
$112$	0	0
$113$	−1.17492e9	−0.677884	−0.338942	−	0.940807i	$-0.610069\pi$
$113$	−1.17492e9	−0.677884	−0.338942	+	0.940807i	$0.610069\pi$
$114$	0	0
$115$	7.95166e8	0.423953
$116$	0	0
$117$	1.22974e9	0.606704
$118$	0	0
$119$	−5.30961e9	−2.42718
$120$	0	0
$121$	−1.41230e9	−0.598951
$122$	0	0
$123$	1.71169e9	0.674298
$124$	0	0
$125$	−2.31224e9	−0.847106
$126$	0	0
$127$	−6.34961e8	−0.216586	−0.108293	−	0.994119i	$-0.534538\pi$
$127$	−6.34961e8	−0.216586	−0.108293	+	0.994119i	$0.534538\pi$
$128$	0	0
$129$	9.63482e9	3.06330
$130$	0	0
$131$	5.68611e9	1.68692	0.843459	−	0.537193i	$-0.180515\pi$
$131$	5.68611e9	1.68692	0.843459	+	0.537193i	$0.180515\pi$
$132$	0	0
$133$	−8.81536e8	−0.244291
$134$	0	0
$135$	−9.10731e9	−2.35987
$136$	0	0
$137$	4.47162e9	1.08448	0.542241	−	0.840223i	$-0.317576\pi$
$137$	4.47162e9	1.08448	0.542241	+	0.840223i	$0.317576\pi$
$138$	0	0
$139$	8.63285e9	1.96150	0.980748	−	0.195275i	$-0.0625601\pi$
$139$	8.63285e9	1.96150	0.980748	+	0.195275i	$0.0625601\pi$
$140$	0	0
$141$	−3.26629e9	−0.695936
$142$	0	0
$143$	−8.78292e8	−0.175641
$144$	0	0
$145$	1.21626e9	0.228492
$146$	0	0
$147$	−7.27020e9	−1.28416
$148$	0	0
$149$	−6.88630e9	−1.14458	−0.572292	−	0.820050i	$-0.693946\pi$
$149$	−6.88630e9	−1.14458	−0.572292	+	0.820050i	$0.693946\pi$
$150$	0	0
$151$	−4.01063e9	−0.627792	−0.313896	−	0.949457i	$-0.601634\pi$
$151$	−4.01063e9	−0.627792	−0.313896	+	0.949457i	$0.601634\pi$
$152$	0	0
$153$	−2.74467e10	−4.04928
$154$	0	0
$155$	4.40672e9	0.613230
$156$	0	0
$157$	9.39913e9	1.23464	0.617318	−	0.786714i	$-0.288219\pi$
$157$	9.39913e9	1.23464	0.617318	+	0.786714i	$0.288219\pi$
$158$	0	0
$159$	6.08341e9	0.754849
$160$	0	0
$161$	4.25773e9	0.499415
$162$	0	0
$163$	−3.72144e9	−0.412920	−0.206460	−	0.978455i	$-0.566194\pi$
$163$	−3.72144e9	−0.412920	−0.206460	+	0.978455i	$0.566194\pi$
$164$	0	0
$165$	1.19820e10	1.25850
$166$	0	0
$167$	4.48607e9	0.446315	0.223158	−	0.974782i	$-0.428364\pi$
$167$	4.48607e9	0.446315	0.223158	+	0.974782i	$0.428364\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−4.55687e9	−0.407553
$172$	0	0
$173$	−1.69286e10	−1.43686	−0.718428	−	0.695602i	$-0.755138\pi$
$173$	−1.69286e10	−1.43686	−0.718428	+	0.695602i	$0.755138\pi$
$174$	0	0
$175$	3.88741e9	0.313321
$176$	0	0
$177$	4.10128e10	3.14080
$178$	0	0
$179$	−1.23127e10	−0.896428	−0.448214	−	0.893926i	$-0.647940\pi$
$179$	−1.23127e10	−0.896428	−0.448214	+	0.893926i	$0.647940\pi$
$180$	0	0
$181$	−2.43748e10	−1.68806	−0.844030	−	0.536295i	$-0.819823\pi$
$181$	−2.43748e10	−1.68806	−0.844030	+	0.536295i	$0.819823\pi$
$182$	0	0
$183$	−4.77791e9	−0.314926
$184$	0	0
$185$	1.90066e10	1.19298
$186$	0	0
$187$	1.96027e10	1.17227
$188$	0	0
$189$	−4.87653e10	−2.77992
$190$	0	0
$191$	8.67702e9	0.471759	0.235880	−	0.971782i	$-0.424203\pi$
$191$	8.67702e9	0.471759	0.235880	+	0.971782i	$0.424203\pi$
$192$	0	0
$193$	−2.33509e10	−1.21142	−0.605712	−	0.795684i	$-0.707112\pi$
$193$	−2.33509e10	−1.21142	−0.605712	+	0.795684i	$0.707112\pi$
$194$	0	0
$195$	−1.11285e10	−0.551165
$196$	0	0
$197$	−2.62332e9	−0.124095	−0.0620473	−	0.998073i	$-0.519763\pi$
$197$	−2.62332e9	−0.124095	−0.0620473	+	0.998073i	$0.519763\pi$
$198$	0	0
$199$	−8.36775e9	−0.378242	−0.189121	−	0.981954i	$-0.560564\pi$
$199$	−8.36775e9	−0.378242	−0.189121	+	0.981954i	$0.560564\pi$
$200$	0	0
$201$	−1.81063e10	−0.782435
$202$	0	0
$203$	6.51249e9	0.269163
$204$	0	0
$205$	−1.06303e10	−0.420392
$206$	0	0
$207$	2.20092e10	0.833179
$208$	0	0
$209$	3.25456e9	0.117987
$210$	0	0
$211$	−3.66318e10	−1.27229	−0.636147	−	0.771568i	$-0.719473\pi$
$211$	−3.66318e10	−1.27229	−0.636147	+	0.771568i	$0.719473\pi$
$212$	0	0
$213$	6.64923e9	0.221342
$214$	0	0
$215$	−5.98364e10	−1.90982
$216$	0	0
$217$	2.35959e10	0.722383
$218$	0	0
$219$	−6.08289e10	−1.78695
$220$	0	0
$221$	−1.82063e10	−0.513402
$222$	0	0
$223$	−6.22383e10	−1.68533	−0.842667	−	0.538435i	$-0.819016\pi$
$223$	−6.22383e10	−1.68533	−0.842667	+	0.538435i	$0.819016\pi$
$224$	0	0
$225$	2.00949e10	0.522715
$226$	0	0
$227$	−4.00531e10	−1.00120	−0.500599	−	0.865679i	$-0.666887\pi$
$227$	−4.00531e10	−1.00120	−0.500599	+	0.865679i	$0.666887\pi$
$228$	0	0
$229$	4.61622e10	1.10924	0.554621	−	0.832103i	$-0.312863\pi$
$229$	4.61622e10	1.10924	0.554621	+	0.832103i	$0.312863\pi$
$230$	0	0
$231$	6.41579e10	1.48251
$232$	0	0
$233$	−2.82042e10	−0.626920	−0.313460	−	0.949601i	$-0.601488\pi$
$233$	−2.82042e10	−0.626920	−0.313460	+	0.949601i	$0.601488\pi$
$234$	0	0
$235$	2.02851e10	0.433882
$236$	0	0
$237$	−1.16295e11	−2.39438
$238$	0	0
$239$	−7.93923e9	−0.157394	−0.0786969	−	0.996899i	$-0.525076\pi$
$239$	−7.93923e9	−0.157394	−0.0786969	+	0.996899i	$0.525076\pi$
$240$	0	0
$241$	−4.33069e10	−0.826953	−0.413476	−	0.910515i	$-0.635686\pi$
$241$	−4.33069e10	−0.826953	−0.413476	+	0.910515i	$0.635686\pi$
$242$	0	0
$243$	−3.98026e10	−0.732290
$244$	0	0
$245$	4.51511e10	0.800610
$246$	0	0
$247$	−3.02273e9	−0.0516730
$248$	0	0
$249$	1.36922e11	2.25724
$250$	0	0
$251$	−3.48447e10	−0.554121	−0.277061	−	0.960852i	$-0.589360\pi$
$251$	−3.48447e10	−0.554121	−0.277061	+	0.960852i	$0.589360\pi$
$252$	0	0
$253$	−1.57192e10	−0.241206
$254$	0	0
$255$	2.48378e11	3.67860
$256$	0	0
$257$	6.17149e10	0.882452	0.441226	−	0.897396i	$-0.354544\pi$
$257$	6.17149e10	0.882452	0.441226	+	0.897396i	$0.354544\pi$
$258$	0	0
$259$	1.01771e11	1.40533
$260$	0	0
$261$	3.36646e10	0.449046
$262$	0	0
$263$	1.78558e10	0.230132	0.115066	−	0.993358i	$-0.463292\pi$
$263$	1.78558e10	0.230132	0.115066	+	0.993358i	$0.463292\pi$
$264$	0	0
$265$	−3.77806e10	−0.470611
$266$	0	0
$267$	−9.15930e10	−1.10296
$268$	0	0
$269$	−5.24399e10	−0.610628	−0.305314	−	0.952252i	$-0.598761\pi$
$269$	−5.24399e10	−0.610628	−0.305314	+	0.952252i	$0.598761\pi$
$270$	0	0
$271$	5.03988e10	0.567621	0.283811	−	0.958880i	$-0.408401\pi$
$271$	5.03988e10	0.567621	0.283811	+	0.958880i	$0.408401\pi$
$272$	0	0
$273$	−5.95879e10	−0.649271
$274$	0	0
$275$	−1.43520e10	−0.151327
$276$	0	0
$277$	−1.58677e11	−1.61941	−0.809703	−	0.586840i	$-0.800372\pi$
$277$	−1.58677e11	−1.61941	−0.809703	+	0.586840i	$0.800372\pi$
$278$	0	0
$279$	1.21973e11	1.20516
$280$	0	0
$281$	3.04536e10	0.291381	0.145690	−	0.989330i	$-0.453460\pi$
$281$	3.04536e10	0.291381	0.145690	+	0.989330i	$0.453460\pi$
$282$	0	0
$283$	−4.21589e10	−0.390706	−0.195353	−	0.980733i	$-0.562585\pi$
$283$	−4.21589e10	−0.390706	−0.195353	+	0.980733i	$0.562585\pi$
$284$	0	0
$285$	4.12373e10	0.370244
$286$	0	0
$287$	−5.69204e10	−0.495221
$288$	0	0
$289$	2.87761e11	2.42656
$290$	0	0
$291$	−2.50207e10	−0.204541
$292$	0	0
$293$	9.73493e10	0.771665	0.385832	−	0.922569i	$-0.373914\pi$
$293$	9.73493e10	0.771665	0.385832	+	0.922569i	$0.373914\pi$
$294$	0	0
$295$	−2.54707e11	−1.95813
$296$	0	0
$297$	1.80037e11	1.34264
$298$	0	0
$299$	1.45995e10	0.105637
$300$	0	0
$301$	−3.20395e11	−2.24976
$302$	0	0
$303$	1.69362e11	1.15432
$304$	0	0
$305$	2.96729e10	0.196341
$306$	0	0
$307$	1.62669e11	1.04516	0.522578	−	0.852591i	$-0.324970\pi$
$307$	1.62669e11	1.04516	0.522578	+	0.852591i	$0.324970\pi$
$308$	0	0
$309$	−4.33996e10	−0.270815
$310$	0	0
$311$	−1.80301e11	−1.09289	−0.546445	−	0.837495i	$-0.684019\pi$
$311$	−1.80301e11	−1.09289	−0.546445	+	0.837495i	$0.684019\pi$
$312$	0	0
$313$	−6.08927e10	−0.358605	−0.179302	−	0.983794i	$-0.557384\pi$
$313$	−6.08927e10	−0.358605	−0.179302	+	0.983794i	$0.557384\pi$
$314$	0	0
$315$	5.57887e11	3.19263
$316$	0	0
$317$	4.75152e10	0.264281	0.132140	−	0.991231i	$-0.457815\pi$
$317$	4.75152e10	0.264281	0.132140	+	0.991231i	$0.457815\pi$
$318$	0	0
$319$	−2.40436e10	−0.129999
$320$	0	0
$321$	3.09802e11	1.62859
$322$	0	0
$323$	6.74646e10	0.344877
$324$	0	0
$325$	1.33297e10	0.0662743
$326$	0	0
$327$	2.61704e11	1.26574
$328$	0	0
$329$	1.08617e11	0.511112
$330$	0	0
$331$	8.96236e10	0.410390	0.205195	−	0.978721i	$-0.434217\pi$
$331$	8.96236e10	0.410390	0.205195	+	0.978721i	$0.434217\pi$
$332$	0	0
$333$	5.26081e11	2.34452
$334$	0	0
$335$	1.12448e11	0.487810
$336$	0	0
$337$	2.05747e11	0.868957	0.434478	−	0.900682i	$-0.356933\pi$
$337$	2.05747e11	0.868957	0.434478	+	0.900682i	$0.356933\pi$
$338$	0	0
$339$	2.94293e11	1.21027
$340$	0	0
$341$	−8.71142e10	−0.348894
$342$	0	0
$343$	−9.43587e10	−0.368094
$344$	0	0
$345$	−1.99172e11	−0.756907
$346$	0	0
$347$	−3.41311e11	−1.26377	−0.631884	−	0.775063i	$-0.717718\pi$
$347$	−3.41311e11	−1.26377	−0.631884	+	0.775063i	$0.717718\pi$
$348$	0	0
$349$	−4.15352e11	−1.49866	−0.749328	−	0.662198i	$-0.769624\pi$
$349$	−4.15352e11	−1.49866	−0.749328	+	0.662198i	$0.769624\pi$
$350$	0	0
$351$	−1.67213e11	−0.588015
$352$	0	0
$353$	−1.00300e11	−0.343806	−0.171903	−	0.985114i	$-0.554992\pi$
$353$	−1.00300e11	−0.343806	−0.171903	+	0.985114i	$0.554992\pi$
$354$	0	0
$355$	−4.12946e10	−0.137996
$356$	0	0
$357$	1.32995e12	4.33338
$358$	0	0
$359$	3.35856e11	1.06716	0.533578	−	0.845751i	$-0.320847\pi$
$359$	3.35856e11	1.06716	0.533578	+	0.845751i	$0.320847\pi$
$360$	0	0
$361$	−3.11487e11	−0.965289
$362$	0	0
$363$	3.53750e11	1.06934
$364$	0	0
$365$	3.77774e11	1.11407
$366$	0	0
$367$	−3.43252e11	−0.987679	−0.493839	−	0.869553i	$-0.664407\pi$
$367$	−3.43252e11	−0.987679	−0.493839	+	0.869553i	$0.664407\pi$
$368$	0	0
$369$	−2.94235e11	−0.826182
$370$	0	0
$371$	−2.02297e11	−0.554379
$372$	0	0
$373$	−3.20544e11	−0.857428	−0.428714	−	0.903440i	$-0.641033\pi$
$373$	−3.20544e11	−0.857428	−0.428714	+	0.903440i	$0.641033\pi$
$374$	0	0
$375$	5.79167e11	1.51239
$376$	0	0
$377$	2.23309e10	0.0569339
$378$	0	0
$379$	−5.59129e11	−1.39199	−0.695994	−	0.718047i	$-0.745036\pi$
$379$	−5.59129e11	−1.39199	−0.695994	+	0.718047i	$0.745036\pi$
$380$	0	0
$381$	1.59044e11	0.386683
$382$	0	0
$383$	1.14158e11	0.271089	0.135544	−	0.990771i	$-0.456722\pi$
$383$	1.14158e11	0.271089	0.135544	+	0.990771i	$0.456722\pi$
$384$	0	0
$385$	−3.98448e11	−0.924270
$386$	0	0
$387$	−1.65620e12	−3.75330
$388$	0	0
$389$	−1.10883e11	−0.245522	−0.122761	−	0.992436i	$-0.539175\pi$
$389$	−1.10883e11	−0.245522	−0.122761	+	0.992436i	$0.539175\pi$
$390$	0	0
$391$	−3.25847e11	−0.705048
$392$	0	0
$393$	−1.42425e12	−3.01175
$394$	0	0
$395$	7.22242e11	1.49278
$396$	0	0
$397$	2.31687e11	0.468107	0.234053	−	0.972224i	$-0.424801\pi$
$397$	2.31687e11	0.468107	0.234053	+	0.972224i	$0.424801\pi$
$398$	0	0
$399$	2.20806e11	0.436147
$400$	0	0
$401$	5.44480e10	0.105156	0.0525778	−	0.998617i	$-0.483256\pi$
$401$	5.44480e10	0.105156	0.0525778	+	0.998617i	$0.483256\pi$
$402$	0	0
$403$	8.09089e10	0.152800
$404$	0	0
$405$	9.62857e11	1.77834
$406$	0	0
$407$	−3.75732e11	−0.678740
$408$	0	0
$409$	2.35793e11	0.416654	0.208327	−	0.978059i	$-0.433198\pi$
$409$	2.35793e11	0.416654	0.208327	+	0.978059i	$0.433198\pi$
$410$	0	0
$411$	−1.12005e12	−1.93619
$412$	0	0
$413$	−1.36384e12	−2.30668
$414$	0	0
$415$	−8.50348e11	−1.40728
$416$	0	0
$417$	−2.16235e12	−3.50197
$418$	0	0
$419$	−5.43639e11	−0.861682	−0.430841	−	0.902428i	$-0.641783\pi$
$419$	−5.43639e11	−0.861682	−0.430841	+	0.902428i	$0.641783\pi$
$420$	0	0
$421$	2.70622e11	0.419849	0.209925	−	0.977718i	$-0.432678\pi$
$421$	2.70622e11	0.419849	0.209925	+	0.977718i	$0.432678\pi$
$422$	0	0
$423$	5.61467e11	0.852693
$424$	0	0
$425$	−2.97506e11	−0.442329
$426$	0	0
$427$	1.58884e11	0.231289
$428$	0	0
$429$	2.19994e11	0.313583
$430$	0	0
$431$	1.07553e12	1.50132	0.750662	−	0.660687i	$-0.229735\pi$
$431$	1.07553e12	1.50132	0.750662	+	0.660687i	$0.229735\pi$
$432$	0	0
$433$	−2.63085e11	−0.359667	−0.179833	−	0.983697i	$-0.557556\pi$
$433$	−2.63085e11	−0.359667	−0.179833	+	0.983697i	$0.557556\pi$
$434$	0	0
$435$	−3.04647e11	−0.407939
$436$	0	0
$437$	−5.40992e10	−0.0709618
$438$	0	0
$439$	8.20045e11	1.05377	0.526887	−	0.849935i	$-0.323359\pi$
$439$	8.20045e11	1.05377	0.526887	+	0.849935i	$0.323359\pi$
$440$	0	0
$441$	1.24973e12	1.57341
$442$	0	0
$443$	1.77977e10	0.0219557	0.0109779	−	0.999940i	$-0.496506\pi$
$443$	1.77977e10	0.0219557	0.0109779	+	0.999940i	$0.496506\pi$
$444$	0	0
$445$	5.68832e11	0.687645
$446$	0	0
$447$	1.72487e12	2.04349
$448$	0	0
$449$	1.15873e12	1.34547	0.672734	−	0.739885i	$-0.265120\pi$
$449$	1.15873e12	1.34547	0.672734	+	0.739885i	$0.265120\pi$
$450$	0	0
$451$	2.10145e11	0.239180
$452$	0	0
$453$	1.00458e12	1.12083
$454$	0	0
$455$	3.70066e11	0.404789
$456$	0	0
$457$	−9.23583e11	−0.990497	−0.495249	−	0.868751i	$-0.664923\pi$
$457$	−9.23583e11	−0.990497	−0.495249	+	0.868751i	$0.664923\pi$
$458$	0	0
$459$	3.73204e12	3.92455
$460$	0	0
$461$	6.32718e11	0.652464	0.326232	−	0.945290i	$-0.394221\pi$
$461$	6.32718e11	0.652464	0.326232	+	0.945290i	$0.394221\pi$
$462$	0	0
$463$	−2.14607e11	−0.217035	−0.108517	−	0.994095i	$-0.534610\pi$
$463$	−2.14607e11	−0.217035	−0.108517	+	0.994095i	$0.534610\pi$
$464$	0	0
$465$	−1.10379e12	−1.09483
$466$	0	0
$467$	−5.49922e11	−0.535026	−0.267513	−	0.963554i	$-0.586202\pi$
$467$	−5.49922e11	−0.535026	−0.267513	+	0.963554i	$0.586202\pi$
$468$	0	0
$469$	6.02106e11	0.574639
$470$	0	0
$471$	−2.35428e12	−2.20427
$472$	0	0
$473$	1.18287e12	1.08658
$474$	0	0
$475$	−4.93939e10	−0.0445197
$476$	0	0
$477$	−1.04572e12	−0.924875
$478$	0	0
$479$	−1.60843e12	−1.39602	−0.698010	−	0.716088i	$-0.745931\pi$
$479$	−1.60843e12	−1.39602	−0.698010	+	0.716088i	$0.745931\pi$
$480$	0	0
$481$	3.48968e11	0.297258
$482$	0	0
$483$	−1.06647e12	−0.891635
$484$	0	0
$485$	1.55389e11	0.127521
$486$	0	0
$487$	1.40030e12	1.12808	0.564040	−	0.825747i	$-0.309246\pi$
$487$	1.40030e12	1.12808	0.564040	+	0.825747i	$0.309246\pi$
$488$	0	0
$489$	9.32141e11	0.737211
$490$	0	0
$491$	−1.31274e12	−1.01932	−0.509661	−	0.860376i	$-0.670229\pi$
$491$	−1.31274e12	−1.01932	−0.509661	+	0.860376i	$0.670229\pi$
$492$	0	0
$493$	−4.98406e11	−0.379990
$494$	0	0
$495$	−2.05967e12	−1.54197
$496$	0	0
$497$	−2.21113e11	−0.162559
$498$	0	0
$499$	3.50247e11	0.252885	0.126442	−	0.991974i	$-0.459644\pi$
$499$	3.50247e11	0.252885	0.126442	+	0.991974i	$0.459644\pi$
$500$	0	0
$501$	−1.12366e12	−0.796833
$502$	0	0
$503$	−2.62962e12	−1.83163	−0.915815	−	0.401601i	$-0.868454\pi$
$503$	−2.62962e12	−1.83163	−0.915815	+	0.401601i	$0.868454\pi$
$504$	0	0
$505$	−1.05181e12	−0.719661
$506$	0	0
$507$	−2.04323e11	−0.137335
$508$	0	0
$509$	4.07673e11	0.269204	0.134602	−	0.990900i	$-0.457024\pi$
$509$	4.07673e11	0.269204	0.134602	+	0.990900i	$0.457024\pi$
$510$	0	0
$511$	2.02280e12	1.31238
$512$	0	0
$513$	6.19617e11	0.394998
$514$	0	0
$515$	2.69531e11	0.168840
$516$	0	0
$517$	−4.01005e11	−0.246855
$518$	0	0
$519$	4.24025e12	2.56530
$520$	0	0
$521$	−1.90638e12	−1.13355	−0.566773	−	0.823874i	$-0.691808\pi$
$521$	−1.90638e12	−1.13355	−0.566773	+	0.823874i	$0.691808\pi$
$522$	0	0
$523$	6.84980e11	0.400332	0.200166	−	0.979762i	$-0.435852\pi$
$523$	6.84980e11	0.400332	0.200166	+	0.979762i	$0.435852\pi$
$524$	0	0
$525$	−9.73713e11	−0.559389
$526$	0	0
$527$	−1.80581e12	−1.01982
$528$	0	0
$529$	−1.53986e12	−0.854930
$530$	0	0
$531$	−7.04999e12	−3.84825
$532$	0	0
$533$	−1.95177e11	−0.104750
$534$	0	0
$535$	−1.92400e12	−1.01534
$536$	0	0
$537$	3.08407e12	1.60044
$538$	0	0
$539$	−8.92568e11	−0.455503
$540$	0	0
$541$	−1.99962e12	−1.00360	−0.501798	−	0.864985i	$-0.667328\pi$
$541$	−1.99962e12	−1.00360	−0.501798	+	0.864985i	$0.667328\pi$
$542$	0	0
$543$	6.10538e12	3.01379
$544$	0	0
$545$	−1.62529e12	−0.789128
$546$	0	0
$547$	−2.19967e12	−1.05055	−0.525273	−	0.850933i	$-0.676037\pi$
$547$	−2.19967e12	−1.05055	−0.525273	+	0.850933i	$0.676037\pi$
$548$	0	0
$549$	8.21310e11	0.385861
$550$	0	0
$551$	−8.27485e10	−0.0382453
$552$	0	0
$553$	3.86726e12	1.75849
$554$	0	0
$555$	−4.76076e12	−2.12989
$556$	0	0
$557$	2.10846e12	0.928148	0.464074	−	0.885797i	$-0.346387\pi$
$557$	2.10846e12	0.928148	0.464074	+	0.885797i	$0.346387\pi$
$558$	0	0
$559$	−1.09862e12	−0.475875
$560$	0	0
$561$	−4.91005e12	−2.09292
$562$	0	0
$563$	−2.49136e12	−1.04508	−0.522540	−	0.852615i	$-0.675015\pi$
$563$	−2.49136e12	−1.04508	−0.522540	+	0.852615i	$0.675015\pi$
$564$	0	0
$565$	−1.82769e12	−0.754542
$566$	0	0
$567$	5.15564e12	2.09488
$568$	0	0
$569$	6.91496e11	0.276557	0.138278	−	0.990393i	$-0.455843\pi$
$569$	6.91496e11	0.276557	0.138278	+	0.990393i	$0.455843\pi$
$570$	0	0
$571$	2.70238e11	0.106386	0.0531930	−	0.998584i	$-0.483060\pi$
$571$	2.70238e11	0.106386	0.0531930	+	0.998584i	$0.483060\pi$
$572$	0	0
$573$	−2.17341e12	−0.842259
$574$	0	0
$575$	2.38568e11	0.0910136
$576$	0	0
$577$	−3.66686e12	−1.37722	−0.688610	−	0.725132i	$-0.741779\pi$
$577$	−3.66686e12	−1.37722	−0.688610	+	0.725132i	$0.741779\pi$
$578$	0	0
$579$	5.84891e12	2.16283
$580$	0	0
$581$	−4.55321e12	−1.65777
$582$	0	0
$583$	7.46864e11	0.267752
$584$	0	0
$585$	1.91296e12	0.675313
$586$	0	0
$587$	−1.16732e12	−0.405806	−0.202903	−	0.979199i	$-0.565038\pi$
$587$	−1.16732e12	−0.405806	−0.202903	+	0.979199i	$0.565038\pi$
$588$	0	0
$589$	−2.99812e11	−0.102643
$590$	0	0
$591$	6.57085e11	0.221553
$592$	0	0
$593$	3.14463e12	1.04430	0.522148	−	0.852855i	$-0.325131\pi$
$593$	3.14463e12	1.04430	0.522148	+	0.852855i	$0.325131\pi$
$594$	0	0
$595$	−8.25954e12	−2.70165
$596$	0	0
$597$	2.09594e12	0.675297
$598$	0	0
$599$	−2.09251e12	−0.664120	−0.332060	−	0.943258i	$-0.607744\pi$
$599$	−2.09251e12	−0.664120	−0.332060	+	0.943258i	$0.607744\pi$
$600$	0	0
$601$	4.34608e12	1.35882	0.679410	−	0.733758i	$-0.262236\pi$
$601$	4.34608e12	1.35882	0.679410	+	0.733758i	$0.262236\pi$
$602$	0	0
$603$	3.11243e12	0.958675
$604$	0	0
$605$	−2.19694e12	−0.666683
$606$	0	0
$607$	−1.73690e12	−0.519310	−0.259655	−	0.965701i	$-0.583609\pi$
$607$	−1.73690e12	−0.519310	−0.259655	+	0.965701i	$0.583609\pi$
$608$	0	0
$609$	−1.63124e12	−0.480552
$610$	0	0
$611$	3.72441e11	0.108112
$612$	0	0
$613$	4.02259e12	1.15062	0.575312	−	0.817934i	$-0.304881\pi$
$613$	4.02259e12	1.15062	0.575312	+	0.817934i	$0.304881\pi$
$614$	0	0
$615$	2.66267e12	0.750551
$616$	0	0
$617$	−5.03083e12	−1.39752	−0.698758	−	0.715358i	$-0.746263\pi$
$617$	−5.03083e12	−1.39752	−0.698758	+	0.715358i	$0.746263\pi$
$618$	0	0
$619$	4.77626e12	1.30761	0.653807	−	0.756661i	$-0.273171\pi$
$619$	4.77626e12	1.30761	0.653807	+	0.756661i	$0.273171\pi$
$620$	0	0
$621$	−2.99269e12	−0.807513
$622$	0	0
$623$	3.04582e12	0.810044
$624$	0	0
$625$	−4.50842e12	−1.18186
$626$	0	0
$627$	−8.15198e11	−0.210649
$628$	0	0
$629$	−7.78864e12	−1.98396
$630$	0	0
$631$	3.04417e12	0.764428	0.382214	−	0.924074i	$-0.375162\pi$
$631$	3.04417e12	0.764428	0.382214	+	0.924074i	$0.375162\pi$
$632$	0	0
$633$	9.17549e12	2.27150
$634$	0	0
$635$	−9.87734e11	−0.241078
$636$	0	0
$637$	8.28989e11	0.199490
$638$	0	0
$639$	−1.14298e12	−0.271198
$640$	0	0
$641$	−1.53417e12	−0.358932	−0.179466	−	0.983764i	$-0.557437\pi$
$641$	−1.53417e12	−0.358932	−0.179466	+	0.983764i	$0.557437\pi$
$642$	0	0
$643$	−3.17074e12	−0.731494	−0.365747	−	0.930714i	$-0.619186\pi$
$643$	−3.17074e12	−0.731494	−0.365747	+	0.930714i	$0.619186\pi$
$644$	0	0
$645$	1.49878e13	3.40971
$646$	0	0
$647$	6.75140e12	1.51469	0.757347	−	0.653013i	$-0.226495\pi$
$647$	6.75140e12	1.51469	0.757347	+	0.653013i	$0.226495\pi$
$648$	0	0
$649$	5.03517e12	1.11407
$650$	0	0
$651$	−5.91027e12	−1.28971
$652$	0	0
$653$	−3.96591e12	−0.853559	−0.426780	−	0.904356i	$-0.640352\pi$
$653$	−3.96591e12	−0.853559	−0.426780	+	0.904356i	$0.640352\pi$
$654$	0	0
$655$	8.84520e12	1.87768
$656$	0	0
$657$	1.04563e13	2.18945
$658$	0	0
$659$	1.08932e12	0.224993	0.112497	−	0.993652i	$-0.464115\pi$
$659$	1.08932e12	0.224993	0.112497	+	0.993652i	$0.464115\pi$
$660$	0	0
$661$	−9.01775e12	−1.83735	−0.918674	−	0.395017i	$-0.870739\pi$
$661$	−9.01775e12	−1.83735	−0.918674	+	0.395017i	$0.870739\pi$
$662$	0	0
$663$	4.56030e12	0.916606
$664$	0	0
$665$	−1.37130e12	−0.271916
$666$	0	0
$667$	3.99667e11	0.0781865
$668$	0	0
$669$	1.55894e13	3.00893
$670$	0	0
$671$	−5.86587e11	−0.111707
$672$	0	0
$673$	6.77812e12	1.27362	0.636812	−	0.771019i	$-0.280253\pi$
$673$	6.77812e12	1.27362	0.636812	+	0.771019i	$0.280253\pi$
$674$	0	0
$675$	−2.73240e12	−0.506613
$676$	0	0
$677$	−9.77011e12	−1.78752	−0.893759	−	0.448548i	$-0.851941\pi$
$677$	−9.77011e12	−1.78752	−0.893759	+	0.448548i	$0.851941\pi$
$678$	0	0
$679$	8.32035e11	0.150220
$680$	0	0
$681$	1.00325e13	1.78750
$682$	0	0
$683$	−7.99180e12	−1.40524	−0.702621	−	0.711564i	$-0.747987\pi$
$683$	−7.99180e12	−1.40524	−0.702621	+	0.711564i	$0.747987\pi$
$684$	0	0
$685$	6.95597e12	1.20712
$686$	0	0
$687$	−1.15626e13	−1.98040
$688$	0	0
$689$	−6.93664e11	−0.117264
$690$	0	0
$691$	7.13265e12	1.19014	0.595072	−	0.803672i	$-0.297123\pi$
$691$	7.13265e12	1.19014	0.595072	+	0.803672i	$0.297123\pi$
$692$	0	0
$693$	−1.10286e13	−1.81643
$694$	0	0
$695$	1.34291e13	2.18331
$696$	0	0
$697$	4.35616e12	0.699127
$698$	0	0
$699$	7.06456e12	1.11928
$700$	0	0
$701$	1.19764e13	1.87325	0.936626	−	0.350330i	$-0.113931\pi$
$701$	1.19764e13	1.87325	0.936626	+	0.350330i	$0.113931\pi$
$702$	0	0
$703$	−1.29312e12	−0.199683
$704$	0	0
$705$	−5.08099e12	−0.774635
$706$	0	0
$707$	−5.63196e12	−0.847759
$708$	0	0
$709$	−6.13692e12	−0.912099	−0.456050	−	0.889954i	$-0.650736\pi$
$709$	−6.13692e12	−0.912099	−0.456050	+	0.889954i	$0.650736\pi$
$710$	0	0
$711$	1.99908e13	2.93371
$712$	0	0
$713$	1.44806e12	0.209838
$714$	0	0
$715$	−1.36626e12	−0.195504
$716$	0	0
$717$	1.98861e12	0.281004
$718$	0	0
$719$	−4.74258e12	−0.661812	−0.330906	−	0.943664i	$-0.607354\pi$
$719$	−4.74258e12	−0.661812	−0.330906	+	0.943664i	$0.607354\pi$
$720$	0	0
$721$	1.44321e12	0.198893
$722$	0	0
$723$	1.08475e13	1.47641
$724$	0	0
$725$	3.64905e11	0.0490523
$726$	0	0
$727$	9.67795e12	1.28493	0.642464	−	0.766316i	$-0.277912\pi$
$727$	9.67795e12	1.28493	0.642464	+	0.766316i	$0.277912\pi$
$728$	0	0
$729$	−2.21347e12	−0.290268
$730$	0	0
$731$	2.45201e13	3.17610
$732$	0	0
$733$	1.50341e13	1.92357	0.961787	−	0.273798i	$-0.0882801\pi$
$733$	1.50341e13	1.92357	0.961787	+	0.273798i	$0.0882801\pi$
$734$	0	0
$735$	−1.13094e13	−1.42937
$736$	0	0
$737$	−2.22293e12	−0.277537
$738$	0	0
$739$	2.14842e12	0.264983	0.132492	−	0.991184i	$-0.457702\pi$
$739$	2.14842e12	0.264983	0.132492	+	0.991184i	$0.457702\pi$
$740$	0	0
$741$	7.57131e11	0.0922548
$742$	0	0
$743$	−5.22310e12	−0.628751	−0.314376	−	0.949299i	$-0.601795\pi$
$743$	−5.22310e12	−0.628751	−0.314376	+	0.949299i	$0.601795\pi$
$744$	0	0
$745$	−1.07122e13	−1.27402
$746$	0	0
$747$	−2.35366e13	−2.76567
$748$	0	0
$749$	−1.03021e13	−1.19607
$750$	0	0
$751$	−1.65523e13	−1.89880	−0.949399	−	0.314071i	$-0.898307\pi$
$751$	−1.65523e13	−1.89880	−0.949399	+	0.314071i	$0.898307\pi$
$752$	0	0
$753$	8.72786e12	0.989305
$754$	0	0
$755$	−6.23886e12	−0.698786
$756$	0	0
$757$	1.59809e13	1.76877	0.884384	−	0.466761i	$-0.154579\pi$
$757$	1.59809e13	1.76877	0.884384	+	0.466761i	$0.154579\pi$
$758$	0	0
$759$	3.93733e12	0.430639
$760$	0	0
$761$	3.31655e12	0.358473	0.179236	−	0.983806i	$-0.442637\pi$
$761$	3.31655e12	0.358473	0.179236	+	0.983806i	$0.442637\pi$
$762$	0	0
$763$	−8.70266e12	−0.929591
$764$	0	0
$765$	−4.26955e13	−4.50719
$766$	0	0
$767$	−4.67651e12	−0.487914
$768$	0	0
$769$	−2.26842e12	−0.233913	−0.116956	−	0.993137i	$-0.537314\pi$
$769$	−2.26842e12	−0.233913	−0.116956	+	0.993137i	$0.537314\pi$
$770$	0	0
$771$	−1.54583e13	−1.57549
$772$	0	0
$773$	1.45998e13	1.47075	0.735375	−	0.677661i	$-0.237006\pi$
$773$	1.45998e13	1.47075	0.735375	+	0.677661i	$0.237006\pi$
$774$	0	0
$775$	1.32212e12	0.131647
$776$	0	0
$777$	−2.54916e13	−2.50901
$778$	0	0
$779$	7.23237e11	0.0703659
$780$	0	0
$781$	8.16330e11	0.0785120
$782$	0	0
$783$	−4.57752e12	−0.435214
$784$	0	0
$785$	1.46211e13	1.37425
$786$	0	0
$787$	1.26923e13	1.17938	0.589692	−	0.807628i	$-0.299249\pi$
$787$	1.26923e13	1.17938	0.589692	+	0.807628i	$0.299249\pi$
$788$	0	0
$789$	−4.47249e12	−0.410869
$790$	0	0
$791$	−9.78638e12	−0.888849
$792$	0	0
$793$	5.44804e11	0.0489228
$794$	0	0
$795$	9.46324e12	0.840210
$796$	0	0
$797$	−1.22392e13	−1.07446	−0.537230	−	0.843436i	$-0.680529\pi$
$797$	−1.22392e13	−1.07446	−0.537230	+	0.843436i	$0.680529\pi$
$798$	0	0
$799$	−8.31254e12	−0.721561
$800$	0	0
$801$	1.57446e13	1.35140
$802$	0	0
$803$	−7.46801e12	−0.633847
$804$	0	0
$805$	6.62325e12	0.555891
$806$	0	0
$807$	1.31351e13	1.09019
$808$	0	0
$809$	−4.89425e12	−0.401714	−0.200857	−	0.979621i	$-0.564373\pi$
$809$	−4.89425e12	−0.401714	−0.200857	+	0.979621i	$0.564373\pi$
$810$	0	0
$811$	3.11178e12	0.252590	0.126295	−	0.991993i	$-0.459691\pi$
$811$	3.11178e12	0.252590	0.126295	+	0.991993i	$0.459691\pi$
$812$	0	0
$813$	−1.26238e13	−1.01341
$814$	0	0
$815$	−5.78900e12	−0.459615
$816$	0	0
$817$	4.07098e12	0.319669
$818$	0	0
$819$	1.02430e13	0.795517
$820$	0	0
$821$	3.48502e12	0.267708	0.133854	−	0.991001i	$-0.457265\pi$
$821$	3.48502e12	0.267708	0.133854	+	0.991001i	$0.457265\pi$
$822$	0	0
$823$	4.38954e12	0.333518	0.166759	−	0.985998i	$-0.446670\pi$
$823$	4.38954e12	0.333518	0.166759	+	0.985998i	$0.446670\pi$
$824$	0	0
$825$	3.59487e12	0.270172
$826$	0	0
$827$	1.21660e13	0.904424	0.452212	−	0.891910i	$-0.350635\pi$
$827$	1.21660e13	0.904424	0.452212	+	0.891910i	$0.350635\pi$
$828$	0	0
$829$	−1.35524e12	−0.0996600	−0.0498300	−	0.998758i	$-0.515868\pi$
$829$	−1.35524e12	−0.0996600	−0.0498300	+	0.998758i	$0.515868\pi$
$830$	0	0
$831$	3.97453e13	2.89122
$832$	0	0
$833$	−1.85023e13	−1.33144
$834$	0	0
$835$	6.97845e12	0.496786
$836$	0	0
$837$	−1.65852e13	−1.16803
$838$	0	0
$839$	1.12215e13	0.781847	0.390924	−	0.920423i	$-0.372156\pi$
$839$	1.12215e13	0.781847	0.390924	+	0.920423i	$0.372156\pi$
$840$	0	0
$841$	−1.38958e13	−0.957861
$842$	0	0
$843$	−7.62799e12	−0.520219
$844$	0	0
$845$	1.26894e12	0.0856218
$846$	0	0
$847$	−1.17636e13	−0.785351
$848$	0	0
$849$	1.05599e13	0.697551
$850$	0	0
$851$	6.24564e12	0.408220
$852$	0	0
$853$	7.91778e11	0.0512074	0.0256037	−	0.999672i	$-0.491849\pi$
$853$	7.91778e11	0.0512074	0.0256037	+	0.999672i	$0.491849\pi$
$854$	0	0
$855$	−7.08858e12	−0.453640
$856$	0	0
$857$	1.76782e13	1.11950	0.559751	−	0.828661i	$-0.310897\pi$
$857$	1.76782e13	1.11950	0.559751	+	0.828661i	$0.310897\pi$
$858$	0	0
$859$	−4.03561e12	−0.252895	−0.126448	−	0.991973i	$-0.540358\pi$
$859$	−4.03561e12	−0.252895	−0.126448	+	0.991973i	$0.540358\pi$
$860$	0	0
$861$	1.42573e13	0.884147
$862$	0	0
$863$	7.74540e11	0.0475330	0.0237665	−	0.999718i	$-0.492434\pi$
$863$	7.74540e11	0.0475330	0.0237665	+	0.999718i	$0.492434\pi$
$864$	0	0
$865$	−2.63338e13	−1.59934
$866$	0	0
$867$	−7.20780e13	−4.33228
$868$	0	0
$869$	−1.42776e13	−0.849311
$870$	0	0
$871$	2.06459e12	0.121549
$872$	0	0
$873$	4.30099e12	0.250613
$874$	0	0
$875$	−1.92596e13	−1.11073
$876$	0	0
$877$	−2.57530e13	−1.47004	−0.735020	−	0.678046i	$-0.762827\pi$
$877$	−2.57530e13	−1.47004	−0.735020	+	0.678046i	$0.762827\pi$
$878$	0	0
$879$	−2.43839e13	−1.37770
$880$	0	0
$881$	1.93336e12	0.108124	0.0540620	−	0.998538i	$-0.482783\pi$
$881$	1.93336e12	0.108124	0.0540620	+	0.998538i	$0.482783\pi$
$882$	0	0
$883$	−2.94334e13	−1.62936	−0.814681	−	0.579910i	$-0.803088\pi$
$883$	−2.94334e13	−1.62936	−0.814681	+	0.579910i	$0.803088\pi$
$884$	0	0
$885$	6.37988e13	3.49597
$886$	0	0
$887$	−4.07138e12	−0.220844	−0.110422	−	0.993885i	$-0.535220\pi$
$887$	−4.07138e12	−0.220844	−0.110422	+	0.993885i	$0.535220\pi$
$888$	0	0
$889$	−5.28884e12	−0.283990
$890$	0	0
$891$	−1.90342e13	−1.01178
$892$	0	0
$893$	−1.38010e12	−0.0726238
$894$	0	0
$895$	−1.91534e13	−0.997799
$896$	0	0
$897$	−3.65687e12	−0.188601
$898$	0	0
$899$	2.21491e12	0.113094
$900$	0	0
$901$	1.54819e13	0.782643
$902$	0	0
$903$	8.02523e13	4.01664
$904$	0	0
$905$	−3.79170e13	−1.87895
$906$	0	0
$907$	5.11710e12	0.251068	0.125534	−	0.992089i	$-0.459936\pi$
$907$	5.11710e12	0.251068	0.125534	+	0.992089i	$0.459936\pi$
$908$	0	0
$909$	−2.91130e13	−1.41432
$910$	0	0
$911$	1.65675e13	0.796937	0.398468	−	0.917182i	$-0.369542\pi$
$911$	1.65675e13	0.796937	0.398468	+	0.917182i	$0.369542\pi$
$912$	0	0
$913$	1.68101e13	0.800665
$914$	0	0
$915$	−7.43243e12	−0.350539
$916$	0	0
$917$	4.73618e13	2.21191
$918$	0	0
$919$	−2.46210e13	−1.13864	−0.569319	−	0.822117i	$-0.692793\pi$
$919$	−2.46210e13	−1.13864	−0.569319	+	0.822117i	$0.692793\pi$
$920$	0	0
$921$	−4.07451e13	−1.86598
$922$	0	0
$923$	−7.58182e11	−0.0343848
$924$	0	0
$925$	5.70242e12	0.256107
$926$	0	0
$927$	7.46028e12	0.331815
$928$	0	0
$929$	2.10967e13	0.929272	0.464636	−	0.885502i	$-0.346185\pi$
$929$	2.10967e13	0.929272	0.464636	+	0.885502i	$0.346185\pi$
$930$	0	0
$931$	−3.07186e12	−0.134007
$932$	0	0
$933$	4.51616e13	1.95120
$934$	0	0
$935$	3.04935e13	1.30484
$936$	0	0
$937$	−3.29230e13	−1.39531	−0.697656	−	0.716433i	$-0.745774\pi$
$937$	−3.29230e13	−1.39531	−0.697656	+	0.716433i	$0.745774\pi$
$938$	0	0
$939$	1.52523e13	0.640238
$940$	0	0
$941$	3.85496e13	1.60275	0.801376	−	0.598161i	$-0.204102\pi$
$941$	3.85496e13	1.60275	0.801376	+	0.598161i	$0.204102\pi$
$942$	0	0
$943$	−3.49316e12	−0.143852
$944$	0	0
$945$	−7.58583e13	−3.09428
$946$	0	0
$947$	−1.22394e13	−0.494521	−0.247261	−	0.968949i	$-0.579530\pi$
$947$	−1.22394e13	−0.494521	−0.247261	+	0.968949i	$0.579530\pi$
$948$	0	0
$949$	6.93605e12	0.277597
$950$	0	0
$951$	−1.19015e13	−0.471836
$952$	0	0
$953$	3.26425e13	1.28194	0.640968	−	0.767568i	$-0.278533\pi$
$953$	3.26425e13	1.28194	0.640968	+	0.767568i	$0.278533\pi$
$954$	0	0
$955$	1.34978e13	0.525108
$956$	0	0
$957$	6.02241e12	0.232095
$958$	0	0
$959$	3.72459e13	1.42198
$960$	0	0
$961$	−1.84146e13	−0.696478
$962$	0	0
$963$	−5.32540e13	−1.99542
$964$	0	0
$965$	−3.63243e13	−1.34842
$966$	0	0
$967$	1.28552e13	0.472780	0.236390	−	0.971658i	$-0.424036\pi$
$967$	1.28552e13	0.472780	0.236390	+	0.971658i	$0.424036\pi$
$968$	0	0
$969$	−1.68984e13	−0.615729
$970$	0	0
$971$	4.30846e13	1.55538	0.777689	−	0.628650i	$-0.216392\pi$
$971$	4.30846e13	1.55538	0.777689	+	0.628650i	$0.216392\pi$
$972$	0	0
$973$	7.19064e13	2.57193
$974$	0	0
$975$	−3.33880e12	−0.118323
$976$	0	0
$977$	3.58870e13	1.26012	0.630059	−	0.776547i	$-0.283031\pi$
$977$	3.58870e13	1.26012	0.630059	+	0.776547i	$0.283031\pi$
$978$	0	0
$979$	−1.12449e13	−0.391233
$980$	0	0
$981$	−4.49861e13	−1.55084
$982$	0	0
$983$	−4.49637e12	−0.153593	−0.0767964	−	0.997047i	$-0.524469\pi$
$983$	−4.49637e12	−0.153593	−0.0767964	+	0.997047i	$0.524469\pi$
$984$	0	0
$985$	−4.08078e12	−0.138128
$986$	0	0
$987$	−2.72062e13	−0.912518
$988$	0	0
$989$	−1.96624e13	−0.653513
$990$	0	0
$991$	−1.56221e13	−0.514526	−0.257263	−	0.966341i	$-0.582821\pi$
$991$	−1.56221e13	−0.514526	−0.257263	+	0.966341i	$0.582821\pi$
$992$	0	0
$993$	−2.24488e13	−0.732693
$994$	0	0
$995$	−1.30167e13	−0.421015
$996$	0	0
$997$	−3.76783e13	−1.20771	−0.603856	−	0.797094i	$-0.706370\pi$
$997$	−3.76783e13	−1.20771	−0.603856	+	0.797094i	$0.706370\pi$
$998$	0	0
$999$	−7.15335e13	−2.27230

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.h.1.1		5
4.3	odd	2		13.10.a.b.1.4	✓	5
12.11	even	2		117.10.a.e.1.2		5
20.19	odd	2		325.10.a.b.1.2		5
52.51	odd	2		169.10.a.b.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.a.b.1.4	✓	5	4.3	odd	2
117.10.a.e.1.2		5	12.11	even	2
169.10.a.b.1.2		5	52.51	odd	2
208.10.a.h.1.1		5	1.1	even	1	trivial
325.10.a.b.1.2		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)

\(f(q)\)	\(=\)	\(q-250.479 q^{3} +1555.58 q^{5} +8329.39 q^{7} +43056.6 q^{9} +O(q^{10})\)	\(q-250.479 q^{3} +1555.58 q^{5} +8329.39 q^{7} +43056.6 q^{9} -30751.5 q^{11} +28561.0 q^{13} -389640. q^{15} -637455. q^{17} -105834. q^{19} -2.08634e6 q^{21} +511169. q^{23} +466710. q^{25} -5.85460e6 q^{27} +781868. q^{29} +2.83285e6 q^{31} +7.70259e6 q^{33} +1.29571e7 q^{35} +1.22183e7 q^{37} -7.15392e6 q^{39} -6.83367e6 q^{41} -3.84656e7 q^{43} +6.69781e7 q^{45} +1.30402e7 q^{47} +2.90252e7 q^{49} +1.59669e8 q^{51} -2.42871e7 q^{53} -4.78364e7 q^{55} +2.65092e7 q^{57} -1.63738e8 q^{59} +1.90751e7 q^{61} +3.58636e8 q^{63} +4.44290e7 q^{65} +7.22869e7 q^{67} -1.28037e8 q^{69} -2.65461e7 q^{71} +2.42850e8 q^{73} -1.16901e8 q^{75} -2.56141e8 q^{77} +4.64290e8 q^{79} +6.18969e8 q^{81} -5.46643e8 q^{83} -9.91613e8 q^{85} -1.95841e8 q^{87} +3.65672e8 q^{89} +2.37896e8 q^{91} -7.09568e8 q^{93} -1.64634e8 q^{95} +9.98914e7 q^{97} -1.32405e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9}+O(q^{10}) \) 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

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