LMFDB - Embedded newform 208.10.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,10,Mod(1,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [8,0,-141] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$107.127453922$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3 x^{7} - 135356 x^{6} - 24398 x^{5} + 5213582205 x^{4} + 598076469 x^{3} + \cdots + 15\!\cdots\!64 $ x^8 - 3x^7 - 135356x^6 - 24398x^5 + 5213582205x^4 + 598076469x^3 - 44131402568826x^2 + 60107523849756*x + 1558753465556664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-109.049$ 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)

$f(q)$	$=$	$q-127.049 q^{3} +393.414 q^{5} -6601.13 q^{7} -3541.59 q^{9} -55090.9 q^{11} +28561.0 q^{13} -49982.8 q^{15} -229326. q^{17} -370189. q^{19} +838666. q^{21} -1.83233e6 q^{23} -1.79835e6 q^{25} +2.95066e6 q^{27} +1.20415e6 q^{29} -5.78451e6 q^{31} +6.99924e6 q^{33} -2.59698e6 q^{35} -839222. q^{37} -3.62864e6 q^{39} +2.52785e6 q^{41} -1.28588e6 q^{43} -1.39331e6 q^{45} -1.25359e7 q^{47} +3.22131e6 q^{49} +2.91356e7 q^{51} -9.21329e7 q^{53} -2.16735e7 q^{55} +4.70321e7 q^{57} -1.25593e8 q^{59} +1.80842e7 q^{61} +2.33785e7 q^{63} +1.12363e7 q^{65} -2.32656e8 q^{67} +2.32796e8 q^{69} -1.13475e8 q^{71} -5.38049e7 q^{73} +2.28478e8 q^{75} +3.63662e8 q^{77} -5.37073e8 q^{79} -3.05169e8 q^{81} -1.78667e8 q^{83} -9.02201e7 q^{85} -1.52986e8 q^{87} +9.93346e8 q^{89} -1.88535e8 q^{91} +7.34915e8 q^{93} -1.45638e8 q^{95} +1.12190e9 q^{97} +1.95109e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 141 q^{3} + 2051 q^{5} + 2417 q^{7} + 115741 q^{9} + 53118 q^{11} + 228488 q^{13} + 464555 q^{15} + 433095 q^{17} + 434954 q^{19} + 906875 q^{21} + 1124296 q^{23} + 5966065 q^{25} - 7820643 q^{27}+ \cdots + 641626736 q^{99}+O(q^{100}) $ 8 * q - 141 * q^3 + 2051 * q^5 + 2417 * q^7 + 115741 * q^9 + 53118 * q^11 + 228488 * q^13 + 464555 * q^15 + 433095 * q^17 + 434954 * q^19 + 906875 * q^21 + 1124296 * q^23 + 5966065 * q^25 - 7820643 * q^27 - 1913432 * q^29 - 5220148 * q^31 - 1626882 * q^33 - 950707 * q^35 - 419409 * q^37 - 4027101 * q^39 + 12148046 * q^41 - 32143187 * q^43 + 18893544 * q^45 - 57898411 * q^47 + 104302637 * q^49 - 198521477 * q^51 + 273106866 * q^53 - 141214662 * q^55 + 253032058 * q^57 - 50734502 * q^59 + 139520018 * q^61 + 20344158 * q^63 + 58578611 * q^65 - 528144786 * q^67 + 483506576 * q^69 - 334814973 * q^71 + 630864344 * q^73 - 2265796264 * q^75 + 939664146 * q^77 - 865037608 * q^79 + 3194281040 * q^81 - 1520229952 * q^83 + 2130158385 * q^85 - 1167867006 * q^87 + 894700240 * q^89 + 69031937 * q^91 + 3803209548 * q^93 + 2695926542 * q^95 + 980232164 * q^97 + 641626736 * 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
$3$	−127.049	−0.905576	−0.452788	−	0.891618i	$-0.649571\pi$
$3$	−127.049	−0.905576	−0.452788	+	0.891618i	$0.649571\pi$
$4$	0	0
$5$	393.414	0.281504	0.140752	−	0.990045i	$-0.455048\pi$
$5$	393.414	0.281504	0.140752	+	0.990045i	$0.455048\pi$
$6$	0	0
$7$	−6601.13	−1.03915	−0.519574	−	0.854426i	$-0.673909\pi$
$7$	−6601.13	−1.03915	−0.519574	+	0.854426i	$0.673909\pi$
$8$	0	0
$9$	−3541.59	−0.179931
$10$	0	0
$11$	−55090.9	−1.13452	−0.567261	−	0.823538i	$-0.691997\pi$
$11$	−55090.9	−1.13452	−0.567261	+	0.823538i	$0.691997\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	−49982.8	−0.254924
$16$	0	0
$17$	−229326.	−0.665937	−0.332969	−	0.942938i	$-0.608050\pi$
$17$	−229326.	−0.665937	−0.332969	+	0.942938i	$0.608050\pi$
$18$	0	0
$19$	−370189.	−0.651677	−0.325839	−	0.945425i	$-0.605647\pi$
$19$	−370189.	−0.651677	−0.325839	+	0.945425i	$0.605647\pi$
$20$	0	0
$21$	838666.	0.941027
$22$	0	0
$23$	−1.83233e6	−1.36530	−0.682651	−	0.730744i	$-0.739173\pi$
$23$	−1.83233e6	−1.36530	−0.682651	+	0.730744i	$0.739173\pi$
$24$	0	0
$25$	−1.79835e6	−0.920755
$26$	0	0
$27$	2.95066e6	1.06852
$28$	0	0
$29$	1.20415e6	0.316148	0.158074	−	0.987427i	$-0.449472\pi$
$29$	1.20415e6	0.316148	0.158074	+	0.987427i	$0.449472\pi$
$30$	0	0
$31$	−5.78451e6	−1.12496	−0.562482	−	0.826809i	$-0.690154\pi$
$31$	−5.78451e6	−1.12496	−0.562482	+	0.826809i	$0.690154\pi$
$32$	0	0
$33$	6.99924e6	1.02740
$34$	0	0
$35$	−2.59698e6	−0.292524
$36$	0	0
$37$	−839222.	−0.0736154	−0.0368077	−	0.999322i	$-0.511719\pi$
$37$	−839222.	−0.0736154	−0.0368077	+	0.999322i	$0.511719\pi$
$38$	0	0
$39$	−3.62864e6	−0.251162
$40$	0	0
$41$	2.52785e6	0.139709	0.0698545	−	0.997557i	$-0.477746\pi$
$41$	2.52785e6	0.139709	0.0698545	+	0.997557i	$0.477746\pi$
$42$	0	0
$43$	−1.28588e6	−0.0573579	−0.0286790	−	0.999589i	$-0.509130\pi$
$43$	−1.28588e6	−0.0573579	−0.0286790	+	0.999589i	$0.509130\pi$
$44$	0	0
$45$	−1.39331e6	−0.0506515
$46$	0	0
$47$	−1.25359e7	−0.374728	−0.187364	−	0.982291i	$-0.559994\pi$
$47$	−1.25359e7	−0.374728	−0.187364	+	0.982291i	$0.559994\pi$
$48$	0	0
$49$	3.22131e6	0.0798271
$50$	0	0
$51$	2.91356e7	0.603057
$52$	0	0
$53$	−9.21329e7	−1.60389	−0.801943	−	0.597401i	$-0.796200\pi$
$53$	−9.21329e7	−1.60389	−0.801943	+	0.597401i	$0.796200\pi$
$54$	0	0
$55$	−2.16735e7	−0.319373
$56$	0	0
$57$	4.70321e7	0.590144
$58$	0	0
$59$	−1.25593e8	−1.34937	−0.674686	−	0.738105i	$-0.735721\pi$
$59$	−1.25593e8	−1.34937	−0.674686	+	0.738105i	$0.735721\pi$
$60$	0	0
$61$	1.80842e7	0.167231	0.0836153	−	0.996498i	$-0.473353\pi$
$61$	1.80842e7	0.167231	0.0836153	+	0.996498i	$0.473353\pi$
$62$	0	0
$63$	2.33785e7	0.186975
$64$	0	0
$65$	1.12363e7	0.0780752
$66$	0	0
$67$	−2.32656e8	−1.41052	−0.705258	−	0.708951i	$-0.749169\pi$
$67$	−2.32656e8	−1.41052	−0.705258	+	0.708951i	$0.749169\pi$
$68$	0	0
$69$	2.32796e8	1.23639
$70$	0	0
$71$	−1.13475e8	−0.529955	−0.264978	−	0.964255i	$-0.585365\pi$
$71$	−1.13475e8	−0.529955	−0.264978	+	0.964255i	$0.585365\pi$
$72$	0	0
$73$	−5.38049e7	−0.221753	−0.110876	−	0.993834i	$-0.535366\pi$
$73$	−5.38049e7	−0.221753	−0.110876	+	0.993834i	$0.535366\pi$
$74$	0	0
$75$	2.28478e8	0.833814
$76$	0	0
$77$	3.63662e8	1.17894
$78$	0	0
$79$	−5.37073e8	−1.55136	−0.775678	−	0.631129i	$-0.782592\pi$
$79$	−5.37073e8	−1.55136	−0.775678	+	0.631129i	$0.782592\pi$
$80$	0	0
$81$	−3.05169e8	−0.787693
$82$	0	0
$83$	−1.78667e8	−0.413231	−0.206615	−	0.978422i	$-0.566245\pi$
$83$	−1.78667e8	−0.413231	−0.206615	+	0.978422i	$0.566245\pi$
$84$	0	0
$85$	−9.02201e7	−0.187464
$86$	0	0
$87$	−1.52986e8	−0.286296
$88$	0	0
$89$	9.93346e8	1.67821	0.839104	−	0.543972i	$-0.183080\pi$
$89$	9.93346e8	1.67821	0.839104	+	0.543972i	$0.183080\pi$
$90$	0	0
$91$	−1.88535e8	−0.288208
$92$	0	0
$93$	7.34915e8	1.01874
$94$	0	0
$95$	−1.45638e8	−0.183450
$96$	0	0
$97$	1.12190e9	1.28672	0.643358	−	0.765566i	$-0.277541\pi$
$97$	1.12190e9	1.28672	0.643358	+	0.765566i	$0.277541\pi$
$98$	0	0
$99$	1.95109e8	0.204136
$100$	0	0
$101$	1.71691e9	1.64172	0.820862	−	0.571126i	$-0.193493\pi$
$101$	1.71691e9	1.64172	0.820862	+	0.571126i	$0.193493\pi$
$102$	0	0
$103$	1.60945e9	1.40899	0.704497	−	0.709707i	$-0.251173\pi$
$103$	1.60945e9	1.40899	0.704497	+	0.709707i	$0.251173\pi$
$104$	0	0
$105$	3.29943e8	0.264903
$106$	0	0
$107$	1.93658e9	1.42826	0.714131	−	0.700012i	$-0.246822\pi$
$107$	1.93658e9	1.42826	0.714131	+	0.700012i	$0.246822\pi$
$108$	0	0
$109$	6.89772e8	0.468044	0.234022	−	0.972231i	$-0.424811\pi$
$109$	6.89772e8	0.468044	0.234022	+	0.972231i	$0.424811\pi$
$110$	0	0
$111$	1.06622e8	0.0666644
$112$	0	0
$113$	−1.96773e9	−1.13531	−0.567653	−	0.823268i	$-0.692149\pi$
$113$	−1.96773e9	−1.13531	−0.567653	+	0.823268i	$0.692149\pi$
$114$	0	0
$115$	−7.20866e8	−0.384339
$116$	0	0
$117$	−1.01151e8	−0.0499040
$118$	0	0
$119$	1.51381e9	0.692007
$120$	0	0
$121$	6.77062e8	0.287140
$122$	0	0
$123$	−3.21161e8	−0.126517
$124$	0	0
$125$	−1.47588e9	−0.540701
$126$	0	0
$127$	−8.12096e8	−0.277007	−0.138503	−	0.990362i	$-0.544229\pi$
$127$	−8.12096e8	−0.277007	−0.138503	+	0.990362i	$0.544229\pi$
$128$	0	0
$129$	1.63370e8	0.0519420
$130$	0	0
$131$	5.70328e9	1.69201	0.846007	−	0.533173i	$-0.179000\pi$
$131$	5.70328e9	1.69201	0.846007	+	0.533173i	$0.179000\pi$
$132$	0	0
$133$	2.44367e9	0.677189
$134$	0	0
$135$	1.16083e9	0.300792
$136$	0	0
$137$	−7.92802e8	−0.192275	−0.0961374	−	0.995368i	$-0.530649\pi$
$137$	−7.92802e8	−0.192275	−0.0961374	+	0.995368i	$0.530649\pi$
$138$	0	0
$139$	−6.17846e8	−0.140383	−0.0701913	−	0.997534i	$-0.522361\pi$
$139$	−6.17846e8	−0.140383	−0.0701913	+	0.997534i	$0.522361\pi$
$140$	0	0
$141$	1.59267e9	0.339345
$142$	0	0
$143$	−1.57345e9	−0.314660
$144$	0	0
$145$	4.73730e8	0.0889970
$146$	0	0
$147$	−4.09264e8	−0.0722896
$148$	0	0
$149$	−2.22284e9	−0.369462	−0.184731	−	0.982789i	$-0.559141\pi$
$149$	−2.22284e9	−0.369462	−0.184731	+	0.982789i	$0.559141\pi$
$150$	0	0
$151$	−6.02833e8	−0.0943627	−0.0471814	−	0.998886i	$-0.515024\pi$
$151$	−6.02833e8	−0.0943627	−0.0471814	+	0.998886i	$0.515024\pi$
$152$	0	0
$153$	8.12179e8	0.119823
$154$	0	0
$155$	−2.27571e9	−0.316682
$156$	0	0
$157$	−4.04266e9	−0.531030	−0.265515	−	0.964107i	$-0.585542\pi$
$157$	−4.04266e9	−0.531030	−0.265515	+	0.964107i	$0.585542\pi$
$158$	0	0
$159$	1.17054e10	1.45244
$160$	0	0
$161$	1.20955e10	1.41875
$162$	0	0
$163$	1.70965e10	1.89698	0.948490	−	0.316806i	$-0.102611\pi$
$163$	1.70965e10	1.89698	0.948490	+	0.316806i	$0.102611\pi$
$164$	0	0
$165$	2.75360e9	0.289216
$166$	0	0
$167$	4.00765e9	0.398718	0.199359	−	0.979926i	$-0.436114\pi$
$167$	4.00765e9	0.398718	0.199359	+	0.979926i	$0.436114\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	1.31106e9	0.117257
$172$	0	0
$173$	−1.48580e10	−1.26111	−0.630556	−	0.776144i	$-0.717173\pi$
$173$	−1.48580e10	−1.26111	−0.630556	+	0.776144i	$0.717173\pi$
$174$	0	0
$175$	1.18711e10	0.956800
$176$	0	0
$177$	1.59565e10	1.22196
$178$	0	0
$179$	1.80486e9	0.131403	0.0657015	−	0.997839i	$-0.479071\pi$
$179$	1.80486e9	0.131403	0.0657015	+	0.997839i	$0.479071\pi$
$180$	0	0
$181$	−2.12646e10	−1.47266	−0.736331	−	0.676621i	$-0.763444\pi$
$181$	−2.12646e10	−1.47266	−0.736331	+	0.676621i	$0.763444\pi$
$182$	0	0
$183$	−2.29758e9	−0.151440
$184$	0	0
$185$	−3.30162e8	−0.0207231
$186$	0	0
$187$	1.26338e10	0.755520
$188$	0	0
$189$	−1.94777e10	−1.11035
$190$	0	0
$191$	−3.38980e9	−0.184299	−0.0921497	−	0.995745i	$-0.529374\pi$
$191$	−3.38980e9	−0.184299	−0.0921497	+	0.995745i	$0.529374\pi$
$192$	0	0
$193$	4.51338e9	0.234150	0.117075	−	0.993123i	$-0.462648\pi$
$193$	4.51338e9	0.234150	0.117075	+	0.993123i	$0.462648\pi$
$194$	0	0
$195$	−1.42756e9	−0.0707031
$196$	0	0
$197$	5.43748e8	0.0257217	0.0128608	−	0.999917i	$-0.495906\pi$
$197$	5.43748e8	0.0257217	0.0128608	+	0.999917i	$0.495906\pi$
$198$	0	0
$199$	1.11070e10	0.502064	0.251032	−	0.967979i	$-0.419230\pi$
$199$	1.11070e10	0.502064	0.251032	+	0.967979i	$0.419230\pi$
$200$	0	0
$201$	2.95587e10	1.27733
$202$	0	0
$203$	−7.94877e9	−0.328524
$204$	0	0
$205$	9.94493e8	0.0393287
$206$	0	0
$207$	6.48937e9	0.245661
$208$	0	0
$209$	2.03941e10	0.739342
$210$	0	0
$211$	3.97386e9	0.138020	0.0690099	−	0.997616i	$-0.478016\pi$
$211$	3.97386e9	0.138020	0.0690099	+	0.997616i	$0.478016\pi$
$212$	0	0
$213$	1.44169e10	0.479915
$214$	0	0
$215$	−5.05884e8	−0.0161465
$216$	0	0
$217$	3.81843e10	1.16900
$218$	0	0
$219$	6.83585e9	0.200814
$220$	0	0
$221$	−6.54978e9	−0.184698
$222$	0	0
$223$	1.65882e10	0.449187	0.224593	−	0.974453i	$-0.427895\pi$
$223$	1.65882e10	0.449187	0.224593	+	0.974453i	$0.427895\pi$
$224$	0	0
$225$	6.36902e9	0.165673
$226$	0	0
$227$	1.03298e10	0.258212	0.129106	−	0.991631i	$-0.458789\pi$
$227$	1.03298e10	0.258212	0.129106	+	0.991631i	$0.458789\pi$
$228$	0	0
$229$	−5.51827e8	−0.0132600	−0.00663000	−	0.999978i	$-0.502110\pi$
$229$	−5.51827e8	−0.0132600	−0.00663000	+	0.999978i	$0.502110\pi$
$230$	0	0
$231$	−4.62029e10	−1.06762
$232$	0	0
$233$	−1.38078e10	−0.306919	−0.153460	−	0.988155i	$-0.549041\pi$
$233$	−1.38078e10	−0.306919	−0.153460	+	0.988155i	$0.549041\pi$
$234$	0	0
$235$	−4.93181e9	−0.105487
$236$	0	0
$237$	6.82345e10	1.40487
$238$	0	0
$239$	1.60685e10	0.318556	0.159278	−	0.987234i	$-0.449083\pi$
$239$	1.60685e10	0.318556	0.159278	+	0.987234i	$0.449083\pi$
$240$	0	0
$241$	1.30321e10	0.248850	0.124425	−	0.992229i	$-0.460291\pi$
$241$	1.30321e10	0.248850	0.124425	+	0.992229i	$0.460291\pi$
$242$	0	0
$243$	−1.93065e10	−0.355202
$244$	0	0
$245$	1.26731e9	0.0224717
$246$	0	0
$247$	−1.05730e10	−0.180743
$248$	0	0
$249$	2.26994e10	0.374212
$250$	0	0
$251$	−1.00894e11	−1.60448	−0.802240	−	0.597001i	$-0.796359\pi$
$251$	−1.00894e11	−1.60448	−0.802240	+	0.597001i	$0.796359\pi$
$252$	0	0
$253$	1.00945e11	1.54897
$254$	0	0
$255$	1.14624e10	0.169763
$256$	0	0
$257$	1.72654e10	0.246876	0.123438	−	0.992352i	$-0.460608\pi$
$257$	1.72654e10	0.246876	0.123438	+	0.992352i	$0.460608\pi$
$258$	0	0
$259$	5.53981e9	0.0764973
$260$	0	0
$261$	−4.26461e9	−0.0568849
$262$	0	0
$263$	−7.95217e10	−1.02491	−0.512454	−	0.858715i	$-0.671264\pi$
$263$	−7.95217e10	−1.02491	−0.512454	+	0.858715i	$0.671264\pi$
$264$	0	0
$265$	−3.62464e10	−0.451501
$266$	0	0
$267$	−1.26203e11	−1.51974
$268$	0	0
$269$	−5.26188e10	−0.612711	−0.306356	−	0.951917i	$-0.599110\pi$
$269$	−5.26188e10	−0.612711	−0.306356	+	0.951917i	$0.599110\pi$
$270$	0	0
$271$	6.71009e10	0.755730	0.377865	−	0.925861i	$-0.376658\pi$
$271$	6.71009e10	0.755730	0.377865	+	0.925861i	$0.376658\pi$
$272$	0	0
$273$	2.39531e10	0.260994
$274$	0	0
$275$	9.90728e10	1.04462
$276$	0	0
$277$	1.61100e8	0.00164413	0.000822064	−	1.00000i	$-0.499738\pi$
$277$	1.61100e8	0.00164413	0.000822064		1.00000i	$0.499738\pi$
$278$	0	0
$279$	2.04864e10	0.202416
$280$	0	0
$281$	−1.01446e11	−0.970635	−0.485317	−	0.874338i	$-0.661296\pi$
$281$	−1.01446e11	−0.970635	−0.485317	+	0.874338i	$0.661296\pi$
$282$	0	0
$283$	8.27056e10	0.766471	0.383235	−	0.923651i	$-0.374810\pi$
$283$	8.27056e10	0.766471	0.383235	+	0.923651i	$0.374810\pi$
$284$	0	0
$285$	1.85031e10	0.166128
$286$	0	0
$287$	−1.66867e10	−0.145178
$288$	0	0
$289$	−6.59974e10	−0.556528
$290$	0	0
$291$	−1.42536e11	−1.16522
$292$	0	0
$293$	−2.01297e11	−1.59563	−0.797817	−	0.602899i	$-0.794012\pi$
$293$	−2.01297e11	−1.59563	−0.797817	+	0.602899i	$0.794012\pi$
$294$	0	0
$295$	−4.94101e10	−0.379854
$296$	0	0
$297$	−1.62554e11	−1.21226
$298$	0	0
$299$	−5.23333e10	−0.378667
$300$	0	0
$301$	8.48828e9	0.0596033
$302$	0	0
$303$	−2.18131e11	−1.48671
$304$	0	0
$305$	7.11459e9	0.0470761
$306$	0	0
$307$	−1.04206e11	−0.669527	−0.334764	−	0.942302i	$-0.608656\pi$
$307$	−1.04206e11	−0.669527	−0.334764	+	0.942302i	$0.608656\pi$
$308$	0	0
$309$	−2.04478e11	−1.27595
$310$	0	0
$311$	2.71489e11	1.64562	0.822811	−	0.568315i	$-0.192404\pi$
$311$	2.71489e11	1.64562	0.822811	+	0.568315i	$0.192404\pi$
$312$	0	0
$313$	3.31671e11	1.95325	0.976625	−	0.214951i	$-0.0689590\pi$
$313$	3.31671e11	1.95325	0.976625	+	0.214951i	$0.0689590\pi$
$314$	0	0
$315$	9.19743e9	0.0526343
$316$	0	0
$317$	−2.92495e11	−1.62686	−0.813432	−	0.581660i	$-0.802404\pi$
$317$	−2.92495e11	−1.62686	−0.813432	+	0.581660i	$0.802404\pi$
$318$	0	0
$319$	−6.63378e10	−0.358677
$320$	0	0
$321$	−2.46040e11	−1.29340
$322$	0	0
$323$	8.48941e10	0.433976
$324$	0	0
$325$	−5.13627e10	−0.255372
$326$	0	0
$327$	−8.76348e10	−0.423849
$328$	0	0
$329$	8.27512e10	0.389397
$330$	0	0
$331$	−3.44902e11	−1.57932	−0.789660	−	0.613545i	$-0.789743\pi$
$331$	−3.44902e11	−1.57932	−0.789660	+	0.613545i	$0.789743\pi$
$332$	0	0
$333$	2.97218e9	0.0132457
$334$	0	0
$335$	−9.15302e10	−0.397066
$336$	0	0
$337$	−3.97412e11	−1.67844	−0.839221	−	0.543791i	$-0.816989\pi$
$337$	−3.97412e11	−1.67844	−0.839221	+	0.543791i	$0.816989\pi$
$338$	0	0
$339$	2.49998e11	1.02811
$340$	0	0
$341$	3.18674e11	1.27630
$342$	0	0
$343$	2.45115e11	0.956195
$344$	0	0
$345$	9.15851e10	0.348048
$346$	0	0
$347$	−2.53853e11	−0.939940	−0.469970	−	0.882682i	$-0.655735\pi$
$347$	−2.53853e11	−0.939940	−0.469970	+	0.882682i	$0.655735\pi$
$348$	0	0
$349$	−1.65473e11	−0.597053	−0.298526	−	0.954401i	$-0.596495\pi$
$349$	−1.65473e11	−0.597053	−0.298526	+	0.954401i	$0.596495\pi$
$350$	0	0
$351$	8.42737e10	0.296354
$352$	0	0
$353$	−6.87614e10	−0.235699	−0.117850	−	0.993031i	$-0.537600\pi$
$353$	−6.87614e10	−0.235699	−0.117850	+	0.993031i	$0.537600\pi$
$354$	0	0
$355$	−4.46428e10	−0.149185
$356$	0	0
$357$	−1.92328e11	−0.626665
$358$	0	0
$359$	3.28858e11	1.04492	0.522461	−	0.852663i	$-0.325014\pi$
$359$	3.28858e11	1.04492	0.522461	+	0.852663i	$0.325014\pi$
$360$	0	0
$361$	−1.85648e11	−0.575316
$362$	0	0
$363$	−8.60199e10	−0.260027
$364$	0	0
$365$	−2.11676e10	−0.0624243
$366$	0	0
$367$	−2.96263e11	−0.852472	−0.426236	−	0.904612i	$-0.640161\pi$
$367$	−2.96263e11	−0.852472	−0.426236	+	0.904612i	$0.640161\pi$
$368$	0	0
$369$	−8.95262e9	−0.0251380
$370$	0	0
$371$	6.08181e11	1.66667
$372$	0	0
$373$	4.74369e11	1.26890	0.634448	−	0.772965i	$-0.281227\pi$
$373$	4.74369e11	1.26890	0.634448	+	0.772965i	$0.281227\pi$
$374$	0	0
$375$	1.87509e11	0.489646
$376$	0	0
$377$	3.43918e10	0.0876837
$378$	0	0
$379$	9.65146e10	0.240279	0.120140	−	0.992757i	$-0.461666\pi$
$379$	9.65146e10	0.240279	0.120140	+	0.992757i	$0.461666\pi$
$380$	0	0
$381$	1.03176e11	0.250851
$382$	0	0
$383$	−8.69705e10	−0.206527	−0.103264	−	0.994654i	$-0.532929\pi$
$383$	−8.69705e10	−0.206527	−0.103264	+	0.994654i	$0.532929\pi$
$384$	0	0
$385$	1.43070e11	0.331875
$386$	0	0
$387$	4.55407e9	0.0103205
$388$	0	0
$389$	−2.50943e10	−0.0555651	−0.0277826	−	0.999614i	$-0.508845\pi$
$389$	−2.50943e10	−0.0555651	−0.0277826	+	0.999614i	$0.508845\pi$
$390$	0	0
$391$	4.20202e11	0.909206
$392$	0	0
$393$	−7.24595e11	−1.53225
$394$	0	0
$395$	−2.11292e11	−0.436713
$396$	0	0
$397$	−8.58270e11	−1.73407	−0.867035	−	0.498247i	$-0.833977\pi$
$397$	−8.58270e11	−1.73407	−0.867035	+	0.498247i	$0.833977\pi$
$398$	0	0
$399$	−3.10465e11	−0.613246
$400$	0	0
$401$	8.65573e10	0.167168	0.0835842	−	0.996501i	$-0.473363\pi$
$401$	8.65573e10	0.167168	0.0835842	+	0.996501i	$0.473363\pi$
$402$	0	0
$403$	−1.65211e11	−0.312009
$404$	0	0
$405$	−1.20058e11	−0.221739
$406$	0	0
$407$	4.62335e10	0.0835183
$408$	0	0
$409$	2.18678e11	0.386412	0.193206	−	0.981158i	$-0.438111\pi$
$409$	2.18678e11	0.386412	0.193206	+	0.981158i	$0.438111\pi$
$410$	0	0
$411$	1.00725e11	0.174119
$412$	0	0
$413$	8.29056e11	1.40220
$414$	0	0
$415$	−7.02900e10	−0.116326
$416$	0	0
$417$	7.84966e10	0.127127
$418$	0	0
$419$	−8.33942e11	−1.32182	−0.660910	−	0.750465i	$-0.729830\pi$
$419$	−8.33942e11	−1.32182	−0.660910	+	0.750465i	$0.729830\pi$
$420$	0	0
$421$	2.17094e11	0.336805	0.168403	−	0.985718i	$-0.446139\pi$
$421$	2.17094e11	0.336805	0.168403	+	0.985718i	$0.446139\pi$
$422$	0	0
$423$	4.43971e10	0.0674253
$424$	0	0
$425$	4.12409e11	0.613165
$426$	0	0
$427$	−1.19376e11	−0.173777
$428$	0	0
$429$	1.99905e11	0.284948
$430$	0	0
$431$	8.80280e11	1.22878	0.614388	−	0.789004i	$-0.289403\pi$
$431$	8.80280e11	1.22878	0.614388	+	0.789004i	$0.289403\pi$
$432$	0	0
$433$	1.70204e11	0.232689	0.116344	−	0.993209i	$-0.462882\pi$
$433$	1.70204e11	0.232689	0.116344	+	0.993209i	$0.462882\pi$
$434$	0	0
$435$	−6.01869e10	−0.0805936
$436$	0	0
$437$	6.78310e11	0.889737
$438$	0	0
$439$	2.76051e11	0.354731	0.177366	−	0.984145i	$-0.443242\pi$
$439$	2.76051e11	0.354731	0.177366	+	0.984145i	$0.443242\pi$
$440$	0	0
$441$	−1.14086e10	−0.0143634
$442$	0	0
$443$	5.01079e11	0.618144	0.309072	−	0.951039i	$-0.399982\pi$
$443$	5.01079e11	0.618144	0.309072	+	0.951039i	$0.399982\pi$
$444$	0	0
$445$	3.90796e11	0.472422
$446$	0	0
$447$	2.82409e11	0.334576
$448$	0	0
$449$	−5.79974e11	−0.673441	−0.336721	−	0.941605i	$-0.609318\pi$
$449$	−5.79974e11	−0.673441	−0.336721	+	0.941605i	$0.609318\pi$
$450$	0	0
$451$	−1.39262e11	−0.158503
$452$	0	0
$453$	7.65892e10	0.0854527
$454$	0	0
$455$	−7.41723e10	−0.0811317
$456$	0	0
$457$	1.35952e12	1.45802	0.729011	−	0.684502i	$-0.239980\pi$
$457$	1.35952e12	1.45802	0.729011	+	0.684502i	$0.239980\pi$
$458$	0	0
$459$	−6.76663e11	−0.711566
$460$	0	0
$461$	9.28421e11	0.957394	0.478697	−	0.877980i	$-0.341109\pi$
$461$	9.28421e11	0.957394	0.478697	+	0.877980i	$0.341109\pi$
$462$	0	0
$463$	−1.52150e12	−1.53871	−0.769357	−	0.638819i	$-0.779423\pi$
$463$	−1.52150e12	−1.53871	−0.769357	+	0.638819i	$0.779423\pi$
$464$	0	0
$465$	2.89126e11	0.286780
$466$	0	0
$467$	−8.06341e11	−0.784500	−0.392250	−	0.919859i	$-0.628303\pi$
$467$	−8.06341e11	−0.784500	−0.392250	+	0.919859i	$0.628303\pi$
$468$	0	0
$469$	1.53579e12	1.46573
$470$	0	0
$471$	5.13616e11	0.480888
$472$	0	0
$473$	7.08405e10	0.0650738
$474$	0	0
$475$	6.65730e11	0.600036
$476$	0	0
$477$	3.26297e11	0.288589
$478$	0	0
$479$	−8.16554e10	−0.0708721	−0.0354360	−	0.999372i	$-0.511282\pi$
$479$	−8.16554e10	−0.0708721	−0.0354360	+	0.999372i	$0.511282\pi$
$480$	0	0
$481$	−2.39690e10	−0.0204173
$482$	0	0
$483$	−1.53672e12	−1.28479
$484$	0	0
$485$	4.41372e11	0.362216
$486$	0	0
$487$	−2.93731e11	−0.236630	−0.118315	−	0.992976i	$-0.537749\pi$
$487$	−2.93731e11	−0.236630	−0.118315	+	0.992976i	$0.537749\pi$
$488$	0	0
$489$	−2.17209e12	−1.71786
$490$	0	0
$491$	8.52285e10	0.0661787	0.0330894	−	0.999452i	$-0.489465\pi$
$491$	8.52285e10	0.0661787	0.0330894	+	0.999452i	$0.489465\pi$
$492$	0	0
$493$	−2.76143e11	−0.210535
$494$	0	0
$495$	7.67588e10	0.0574652
$496$	0	0
$497$	7.49066e11	0.550702
$498$	0	0
$499$	−1.87390e12	−1.35299	−0.676495	−	0.736448i	$-0.736502\pi$
$499$	−1.87390e12	−1.35299	−0.676495	+	0.736448i	$0.736502\pi$
$500$	0	0
$501$	−5.09168e11	−0.361070
$502$	0	0
$503$	2.29233e12	1.59669	0.798346	−	0.602199i	$-0.205709\pi$
$503$	2.29233e12	1.59669	0.798346	+	0.602199i	$0.205709\pi$
$504$	0	0
$505$	6.75455e11	0.462152
$506$	0	0
$507$	−1.03638e11	−0.0696597
$508$	0	0
$509$	5.51785e11	0.364368	0.182184	−	0.983264i	$-0.441683\pi$
$509$	5.51785e11	0.364368	0.182184	+	0.983264i	$0.441683\pi$
$510$	0	0
$511$	3.55173e11	0.230434
$512$	0	0
$513$	−1.09230e12	−0.696329
$514$	0	0
$515$	6.33179e11	0.396638
$516$	0	0
$517$	6.90615e11	0.425137
$518$	0	0
$519$	1.88769e12	1.14203
$520$	0	0
$521$	−2.10966e12	−1.25442	−0.627210	−	0.778850i	$-0.715803\pi$
$521$	−2.10966e12	−1.25442	−0.627210	+	0.778850i	$0.715803\pi$
$522$	0	0
$523$	1.93741e12	1.13231	0.566153	−	0.824300i	$-0.308431\pi$
$523$	1.93741e12	1.13231	0.566153	+	0.824300i	$0.308431\pi$
$524$	0	0
$525$	−1.50822e12	−0.866456
$526$	0	0
$527$	1.32654e12	0.749156
$528$	0	0
$529$	1.55629e12	0.864052
$530$	0	0
$531$	4.44799e11	0.242794
$532$	0	0
$533$	7.21980e10	0.0387483
$534$	0	0
$535$	7.61877e11	0.402062
$536$	0	0
$537$	−2.29306e11	−0.118995
$538$	0	0
$539$	−1.77465e11	−0.0905657
$540$	0	0
$541$	−2.25790e12	−1.13323	−0.566614	−	0.823983i	$-0.691747\pi$
$541$	−2.25790e12	−1.13323	−0.566614	+	0.823983i	$0.691747\pi$
$542$	0	0
$543$	2.70164e12	1.33361
$544$	0	0
$545$	2.71366e11	0.131756
$546$	0	0
$547$	−1.90541e12	−0.910007	−0.455003	−	0.890490i	$-0.650362\pi$
$547$	−1.90541e12	−0.910007	−0.455003	+	0.890490i	$0.650362\pi$
$548$	0	0
$549$	−6.40469e10	−0.0300900
$550$	0	0
$551$	−4.45764e11	−0.206026
$552$	0	0
$553$	3.54529e12	1.61209
$554$	0	0
$555$	4.19467e10	0.0187663
$556$	0	0
$557$	2.62651e12	1.15620	0.578098	−	0.815968i	$-0.303795\pi$
$557$	2.62651e12	1.15620	0.578098	+	0.815968i	$0.303795\pi$
$558$	0	0
$559$	−3.67261e10	−0.0159082
$560$	0	0
$561$	−1.60511e12	−0.684181
$562$	0	0
$563$	1.43306e12	0.601141	0.300570	−	0.953760i	$-0.402823\pi$
$563$	1.43306e12	0.601141	0.300570	+	0.953760i	$0.402823\pi$
$564$	0	0
$565$	−7.74134e11	−0.319594
$566$	0	0
$567$	2.01446e12	0.818529
$568$	0	0
$569$	−7.00796e11	−0.280276	−0.140138	−	0.990132i	$-0.544755\pi$
$569$	−7.00796e11	−0.280276	−0.140138	+	0.990132i	$0.544755\pi$
$570$	0	0
$571$	−2.44640e12	−0.963084	−0.481542	−	0.876423i	$-0.659923\pi$
$571$	−2.44640e12	−0.963084	−0.481542	+	0.876423i	$0.659923\pi$
$572$	0	0
$573$	4.30670e11	0.166897
$574$	0	0
$575$	3.29518e12	1.25711
$576$	0	0
$577$	3.67506e12	1.38030	0.690150	−	0.723666i	$-0.257545\pi$
$577$	3.67506e12	1.38030	0.690150	+	0.723666i	$0.257545\pi$
$578$	0	0
$579$	−5.73420e11	−0.212041
$580$	0	0
$581$	1.17940e12	0.429407
$582$	0	0
$583$	5.07569e12	1.81964
$584$	0	0
$585$	−3.97944e10	−0.0140482
$586$	0	0
$587$	5.17892e11	0.180039	0.0900197	−	0.995940i	$-0.471307\pi$
$587$	5.17892e11	0.180039	0.0900197	+	0.995940i	$0.471307\pi$
$588$	0	0
$589$	2.14136e12	0.733114
$590$	0	0
$591$	−6.90825e10	−0.0232929
$592$	0	0
$593$	−3.70659e12	−1.23092	−0.615459	−	0.788169i	$-0.711029\pi$
$593$	−3.70659e12	−1.23092	−0.615459	+	0.788169i	$0.711029\pi$
$594$	0	0
$595$	5.95555e11	0.194803
$596$	0	0
$597$	−1.41114e12	−0.454657
$598$	0	0
$599$	8.24633e11	0.261722	0.130861	−	0.991401i	$-0.458226\pi$
$599$	8.24633e11	0.261722	0.130861	+	0.991401i	$0.458226\pi$
$600$	0	0
$601$	1.63144e12	0.510077	0.255038	−	0.966931i	$-0.417912\pi$
$601$	1.63144e12	0.510077	0.255038	+	0.966931i	$0.417912\pi$
$602$	0	0
$603$	8.23972e11	0.253796
$604$	0	0
$605$	2.66366e11	0.0808312
$606$	0	0
$607$	8.25132e11	0.246703	0.123351	−	0.992363i	$-0.460636\pi$
$607$	8.25132e11	0.246703	0.123351	+	0.992363i	$0.460636\pi$
$608$	0	0
$609$	1.00988e12	0.297504
$610$	0	0
$611$	−3.58038e11	−0.103931
$612$	0	0
$613$	−6.84543e12	−1.95807	−0.979036	−	0.203685i	$-0.934708\pi$
$613$	−6.84543e12	−1.95807	−0.979036	+	0.203685i	$0.934708\pi$
$614$	0	0
$615$	−1.26349e11	−0.0356151
$616$	0	0
$617$	−4.51738e12	−1.25488	−0.627441	−	0.778664i	$-0.715898\pi$
$617$	−4.51738e12	−1.25488	−0.627441	+	0.778664i	$0.715898\pi$
$618$	0	0
$619$	−5.18691e12	−1.42004	−0.710020	−	0.704181i	$-0.751314\pi$
$619$	−5.18691e12	−1.42004	−0.710020	+	0.704181i	$0.751314\pi$
$620$	0	0
$621$	−5.40659e12	−1.45885
$622$	0	0
$623$	−6.55721e12	−1.74390
$624$	0	0
$625$	2.93177e12	0.768546
$626$	0	0
$627$	−2.59104e12	−0.669531
$628$	0	0
$629$	1.92455e11	0.0490233
$630$	0	0
$631$	−2.56557e12	−0.644245	−0.322123	−	0.946698i	$-0.604396\pi$
$631$	−2.56557e12	−0.644245	−0.322123	+	0.946698i	$0.604396\pi$
$632$	0	0
$633$	−5.04874e11	−0.124987
$634$	0	0
$635$	−3.19490e11	−0.0779786
$636$	0	0
$637$	9.20039e10	0.0221401
$638$	0	0
$639$	4.01883e11	0.0953556
$640$	0	0
$641$	−6.90157e12	−1.61468	−0.807341	−	0.590085i	$-0.799094\pi$
$641$	−6.90157e12	−1.61468	−0.807341	+	0.590085i	$0.799094\pi$
$642$	0	0
$643$	−2.57207e12	−0.593381	−0.296690	−	0.954974i	$-0.595883\pi$
$643$	−2.57207e12	−0.593381	−0.296690	+	0.954974i	$0.595883\pi$
$644$	0	0
$645$	6.42720e10	0.0146219
$646$	0	0
$647$	5.68315e12	1.27503	0.637514	−	0.770439i	$-0.279963\pi$
$647$	5.68315e12	1.27503	0.637514	+	0.770439i	$0.279963\pi$
$648$	0	0
$649$	6.91904e12	1.53089
$650$	0	0
$651$	−4.85127e12	−1.05862
$652$	0	0
$653$	−2.44912e12	−0.527109	−0.263555	−	0.964644i	$-0.584895\pi$
$653$	−2.44912e12	−0.527109	−0.263555	+	0.964644i	$0.584895\pi$
$654$	0	0
$655$	2.24375e12	0.476309
$656$	0	0
$657$	1.90555e11	0.0399003
$658$	0	0
$659$	9.46263e11	0.195446	0.0977231	−	0.995214i	$-0.468844\pi$
$659$	9.46263e11	0.195446	0.0977231	+	0.995214i	$0.468844\pi$
$660$	0	0
$661$	−3.98418e11	−0.0811768	−0.0405884	−	0.999176i	$-0.512923\pi$
$661$	−3.98418e11	−0.0811768	−0.0405884	+	0.999176i	$0.512923\pi$
$662$	0	0
$663$	8.32142e11	0.167258
$664$	0	0
$665$	9.61374e11	0.190632
$666$	0	0
$667$	−2.20641e12	−0.431638
$668$	0	0
$669$	−2.10751e12	−0.406773
$670$	0	0
$671$	−9.96277e11	−0.189727
$672$	0	0
$673$	1.52054e12	0.285712	0.142856	−	0.989743i	$-0.454371\pi$
$673$	1.52054e12	0.285712	0.142856	+	0.989743i	$0.454371\pi$
$674$	0	0
$675$	−5.30632e12	−0.983844
$676$	0	0
$677$	1.72963e11	0.0316450	0.0158225	−	0.999875i	$-0.494963\pi$
$677$	1.72963e11	0.0316450	0.0158225	+	0.999875i	$0.494963\pi$
$678$	0	0
$679$	−7.40583e12	−1.33709
$680$	0	0
$681$	−1.31239e12	−0.233831
$682$	0	0
$683$	−1.41533e11	−0.0248865	−0.0124432	−	0.999923i	$-0.503961\pi$
$683$	−1.41533e11	−0.0248865	−0.0124432	+	0.999923i	$0.503961\pi$
$684$	0	0
$685$	−3.11900e11	−0.0541262
$686$	0	0
$687$	7.01090e10	0.0120079
$688$	0	0
$689$	−2.63141e12	−0.444838
$690$	0	0
$691$	−3.45405e12	−0.576339	−0.288169	−	0.957579i	$-0.593047\pi$
$691$	−3.45405e12	−0.576339	−0.288169	+	0.957579i	$0.593047\pi$
$692$	0	0
$693$	−1.28794e12	−0.212128
$694$	0	0
$695$	−2.43069e11	−0.0395183
$696$	0	0
$697$	−5.79702e11	−0.0930374
$698$	0	0
$699$	1.75427e12	0.277939
$700$	0	0
$701$	−8.53804e12	−1.33545	−0.667724	−	0.744409i	$-0.732732\pi$
$701$	−8.53804e12	−1.33545	−0.667724	+	0.744409i	$0.732732\pi$
$702$	0	0
$703$	3.10671e11	0.0479735
$704$	0	0
$705$	6.26580e11	0.0955269
$706$	0	0
$707$	−1.13335e13	−1.70599
$708$	0	0
$709$	−1.16700e12	−0.173445	−0.0867227	−	0.996232i	$-0.527639\pi$
$709$	−1.16700e12	−0.173445	−0.0867227	+	0.996232i	$0.527639\pi$
$710$	0	0
$711$	1.90209e12	0.279138
$712$	0	0
$713$	1.05991e13	1.53592
$714$	0	0
$715$	−6.19018e11	−0.0885781
$716$	0	0
$717$	−2.04149e12	−0.288477
$718$	0	0
$719$	7.92647e12	1.10611	0.553057	−	0.833144i	$-0.313461\pi$
$719$	7.92647e12	1.10611	0.553057	+	0.833144i	$0.313461\pi$
$720$	0	0
$721$	−1.06242e13	−1.46415
$722$	0	0
$723$	−1.65571e12	−0.225353
$724$	0	0
$725$	−2.16549e12	−0.291095
$726$	0	0
$727$	−1.43334e13	−1.90302	−0.951511	−	0.307615i	$-0.900469\pi$
$727$	−1.43334e13	−1.90302	−0.951511	+	0.307615i	$0.900469\pi$
$728$	0	0
$729$	8.45950e12	1.10936
$730$	0	0
$731$	2.94886e11	0.0381968
$732$	0	0
$733$	1.23061e13	1.57453	0.787266	−	0.616613i	$-0.211496\pi$
$733$	1.23061e13	1.57453	0.787266	+	0.616613i	$0.211496\pi$
$734$	0	0
$735$	−1.61010e11	−0.0203498
$736$	0	0
$737$	1.28172e13	1.60026
$738$	0	0
$739$	−1.24191e13	−1.53175	−0.765876	−	0.642988i	$-0.777695\pi$
$739$	−1.24191e13	−1.53175	−0.765876	+	0.642988i	$0.777695\pi$
$740$	0	0
$741$	1.34328e12	0.163676
$742$	0	0
$743$	5.23605e12	0.630310	0.315155	−	0.949040i	$-0.397943\pi$
$743$	5.23605e12	0.630310	0.315155	+	0.949040i	$0.397943\pi$
$744$	0	0
$745$	−8.74497e11	−0.104005
$746$	0	0
$747$	6.32764e11	0.0743531
$748$	0	0
$749$	−1.27836e13	−1.48417
$750$	0	0
$751$	1.07482e13	1.23298	0.616491	−	0.787362i	$-0.288554\pi$
$751$	1.07482e13	1.23298	0.616491	+	0.787362i	$0.288554\pi$
$752$	0	0
$753$	1.28185e13	1.45298
$754$	0	0
$755$	−2.37163e11	−0.0265635
$756$	0	0
$757$	−4.80201e12	−0.531486	−0.265743	−	0.964044i	$-0.585617\pi$
$757$	−4.80201e12	−0.531486	−0.265743	+	0.964044i	$0.585617\pi$
$758$	0	0
$759$	−1.28249e13	−1.40271
$760$	0	0
$761$	−6.81874e12	−0.737010	−0.368505	−	0.929626i	$-0.620130\pi$
$761$	−6.81874e12	−0.737010	−0.368505	+	0.929626i	$0.620130\pi$
$762$	0	0
$763$	−4.55328e12	−0.486366
$764$	0	0
$765$	3.19523e11	0.0337307
$766$	0	0
$767$	−3.58706e12	−0.374248
$768$	0	0
$769$	−2.29037e12	−0.236177	−0.118088	−	0.993003i	$-0.537677\pi$
$769$	−2.29037e12	−0.236177	−0.118088	+	0.993003i	$0.537677\pi$
$770$	0	0
$771$	−2.19356e12	−0.223565
$772$	0	0
$773$	3.12369e12	0.314674	0.157337	−	0.987545i	$-0.449709\pi$
$773$	3.12369e12	0.314674	0.157337	+	0.987545i	$0.449709\pi$
$774$	0	0
$775$	1.04026e13	1.03582
$776$	0	0
$777$	−7.03827e11	−0.0692741
$778$	0	0
$779$	−9.35784e11	−0.0910452
$780$	0	0
$781$	6.25146e12	0.601246
$782$	0	0
$783$	3.55304e12	0.337810
$784$	0	0
$785$	−1.59044e12	−0.149487
$786$	0	0
$787$	−1.60343e13	−1.48993	−0.744963	−	0.667106i	$-0.767533\pi$
$787$	−1.60343e13	−1.48993	−0.744963	+	0.667106i	$0.767533\pi$
$788$	0	0
$789$	1.01031e13	0.928133
$790$	0	0
$791$	1.29893e13	1.17975
$792$	0	0
$793$	5.16504e11	0.0463814
$794$	0	0
$795$	4.60506e12	0.408868
$796$	0	0
$797$	1.66365e13	1.46050	0.730248	−	0.683182i	$-0.239404\pi$
$797$	1.66365e13	1.46050	0.730248	+	0.683182i	$0.239404\pi$
$798$	0	0
$799$	2.87481e12	0.249545
$800$	0	0
$801$	−3.51802e12	−0.301962
$802$	0	0
$803$	2.96416e12	0.251583
$804$	0	0
$805$	4.75853e12	0.399384
$806$	0	0
$807$	6.68516e12	0.554857
$808$	0	0
$809$	−2.02416e13	−1.66141	−0.830705	−	0.556713i	$-0.812062\pi$
$809$	−2.02416e13	−1.66141	−0.830705	+	0.556713i	$0.812062\pi$
$810$	0	0
$811$	−4.06322e12	−0.329820	−0.164910	−	0.986309i	$-0.552733\pi$
$811$	−4.06322e12	−0.329820	−0.164910	+	0.986309i	$0.552733\pi$
$812$	0	0
$813$	−8.52510e12	−0.684372
$814$	0	0
$815$	6.72600e12	0.534008
$816$	0	0
$817$	4.76020e11	0.0373789
$818$	0	0
$819$	6.67713e11	0.0518576
$820$	0	0
$821$	1.32353e13	1.01669	0.508345	−	0.861154i	$-0.330258\pi$
$821$	1.32353e13	1.01669	0.508345	+	0.861154i	$0.330258\pi$
$822$	0	0
$823$	7.48381e12	0.568622	0.284311	−	0.958732i	$-0.408235\pi$
$823$	7.48381e12	0.568622	0.284311	+	0.958732i	$0.408235\pi$
$824$	0	0
$825$	−1.25871e13	−0.945981
$826$	0	0
$827$	1.17986e13	0.877112	0.438556	−	0.898704i	$-0.355490\pi$
$827$	1.17986e13	0.877112	0.438556	+	0.898704i	$0.355490\pi$
$828$	0	0
$829$	−7.42428e12	−0.545958	−0.272979	−	0.962020i	$-0.588009\pi$
$829$	−7.42428e12	−0.545958	−0.272979	+	0.962020i	$0.588009\pi$
$830$	0	0
$831$	−2.04675e10	−0.00148888
$832$	0	0
$833$	−7.38731e11	−0.0531599
$834$	0	0
$835$	1.57667e12	0.112241
$836$	0	0
$837$	−1.70681e13	−1.20204
$838$	0	0
$839$	2.76973e13	1.92978	0.964892	−	0.262645i	$-0.0845949\pi$
$839$	2.76973e13	1.92978	0.964892	+	0.262645i	$0.0845949\pi$
$840$	0	0
$841$	−1.30572e13	−0.900050
$842$	0	0
$843$	1.28886e13	0.878984
$844$	0	0
$845$	3.20920e11	0.0216542
$846$	0	0
$847$	−4.46937e12	−0.298381
$848$	0	0
$849$	−1.05076e13	−0.694098
$850$	0	0
$851$	1.53773e12	0.100507
$852$	0	0
$853$	−1.03384e13	−0.668626	−0.334313	−	0.942462i	$-0.608504\pi$
$853$	−1.03384e13	−0.668626	−0.334313	+	0.942462i	$0.608504\pi$
$854$	0	0
$855$	5.15789e11	0.0330084
$856$	0	0
$857$	1.12259e13	0.710895	0.355448	−	0.934696i	$-0.384328\pi$
$857$	1.12259e13	0.710895	0.355448	+	0.934696i	$0.384328\pi$
$858$	0	0
$859$	2.01153e13	1.26054	0.630271	−	0.776375i	$-0.282944\pi$
$859$	2.01153e13	1.26054	0.630271	+	0.776375i	$0.282944\pi$
$860$	0	0
$861$	2.12002e12	0.131470
$862$	0	0
$863$	−2.51003e13	−1.54039	−0.770193	−	0.637811i	$-0.779840\pi$
$863$	−2.51003e13	−1.54039	−0.770193	+	0.637811i	$0.779840\pi$
$864$	0	0
$865$	−5.84535e12	−0.355008
$866$	0	0
$867$	8.38490e12	0.503978
$868$	0	0
$869$	2.95878e13	1.76005
$870$	0	0
$871$	−6.64489e12	−0.391207
$872$	0	0
$873$	−3.97332e12	−0.231520
$874$	0	0
$875$	9.74250e12	0.561868
$876$	0	0
$877$	2.73527e13	1.56135	0.780677	−	0.624934i	$-0.214874\pi$
$877$	2.73527e13	1.56135	0.780677	+	0.624934i	$0.214874\pi$
$878$	0	0
$879$	2.55746e13	1.44497
$880$	0	0
$881$	−1.27163e13	−0.711164	−0.355582	−	0.934645i	$-0.615717\pi$
$881$	−1.27163e13	−0.711164	−0.355582	+	0.934645i	$0.615717\pi$
$882$	0	0
$883$	−2.49170e13	−1.37935	−0.689673	−	0.724121i	$-0.742246\pi$
$883$	−2.49170e13	−1.37935	−0.689673	+	0.724121i	$0.742246\pi$
$884$	0	0
$885$	6.27749e12	0.343987
$886$	0	0
$887$	3.04068e13	1.64936	0.824679	−	0.565602i	$-0.191356\pi$
$887$	3.04068e13	1.64936	0.824679	+	0.565602i	$0.191356\pi$
$888$	0	0
$889$	5.36075e12	0.287851
$890$	0	0
$891$	1.68120e13	0.893655
$892$	0	0
$893$	4.64066e12	0.244202
$894$	0	0
$895$	7.10058e11	0.0369905
$896$	0	0
$897$	6.64888e12	0.342912
$898$	0	0
$899$	−6.96543e12	−0.355655
$900$	0	0
$901$	2.11285e13	1.06809
$902$	0	0
$903$	−1.07843e12	−0.0539753
$904$	0	0
$905$	−8.36579e12	−0.414561
$906$	0	0
$907$	1.73147e13	0.849538	0.424769	−	0.905302i	$-0.360355\pi$
$907$	1.73147e13	0.849538	0.424769	+	0.905302i	$0.360355\pi$
$908$	0	0
$909$	−6.08057e12	−0.295398
$910$	0	0
$911$	−3.93024e12	−0.189054	−0.0945271	−	0.995522i	$-0.530134\pi$
$911$	−3.93024e12	−0.189054	−0.0945271	+	0.995522i	$0.530134\pi$
$912$	0	0
$913$	9.84292e12	0.468819
$914$	0	0
$915$	−9.03901e11	−0.0426310
$916$	0	0
$917$	−3.76481e13	−1.75825
$918$	0	0
$919$	−2.25315e13	−1.04200	−0.521002	−	0.853555i	$-0.674442\pi$
$919$	−2.25315e13	−1.04200	−0.521002	+	0.853555i	$0.674442\pi$
$920$	0	0
$921$	1.32392e13	0.606308
$922$	0	0
$923$	−3.24097e12	−0.146983
$924$	0	0
$925$	1.50921e12	0.0677818
$926$	0	0
$927$	−5.70000e12	−0.253522
$928$	0	0
$929$	−4.28132e12	−0.188585	−0.0942925	−	0.995545i	$-0.530059\pi$
$929$	−4.28132e12	−0.188585	−0.0942925	+	0.995545i	$0.530059\pi$
$930$	0	0
$931$	−1.19250e12	−0.0520216
$932$	0	0
$933$	−3.44923e13	−1.49024
$934$	0	0
$935$	4.97031e12	0.212682
$936$	0	0
$937$	4.29988e13	1.82233	0.911167	−	0.412038i	$-0.135183\pi$
$937$	4.29988e13	1.82233	0.911167	+	0.412038i	$0.135183\pi$
$938$	0	0
$939$	−4.21384e13	−1.76882
$940$	0	0
$941$	3.11345e13	1.29446	0.647231	−	0.762294i	$-0.275927\pi$
$941$	3.11345e13	1.29446	0.647231	+	0.762294i	$0.275927\pi$
$942$	0	0
$943$	−4.63187e12	−0.190745
$944$	0	0
$945$	−7.66279e12	−0.312568
$946$	0	0
$947$	1.90723e13	0.770599	0.385299	−	0.922792i	$-0.374098\pi$
$947$	1.90723e13	0.770599	0.385299	+	0.922792i	$0.374098\pi$
$948$	0	0
$949$	−1.53672e12	−0.0615031
$950$	0	0
$951$	3.71611e13	1.47325
$952$	0	0
$953$	−2.95033e13	−1.15865	−0.579325	−	0.815097i	$-0.696684\pi$
$953$	−2.95033e13	−1.15865	−0.579325	+	0.815097i	$0.696684\pi$
$954$	0	0
$955$	−1.33360e12	−0.0518811
$956$	0	0
$957$	8.42815e12	0.324809
$958$	0	0
$959$	5.23339e12	0.199802
$960$	0	0
$961$	7.02091e12	0.265545
$962$	0	0
$963$	−6.85856e12	−0.256989
$964$	0	0
$965$	1.77563e12	0.0659142
$966$	0	0
$967$	8.45368e12	0.310904	0.155452	−	0.987843i	$-0.450317\pi$
$967$	8.45368e12	0.310904	0.155452	+	0.987843i	$0.450317\pi$
$968$	0	0
$969$	−1.07857e13	−0.392999
$970$	0	0
$971$	−3.62956e13	−1.31029	−0.655145	−	0.755503i	$-0.727393\pi$
$971$	−3.62956e13	−1.31029	−0.655145	+	0.755503i	$0.727393\pi$
$972$	0	0
$973$	4.07848e12	0.145878
$974$	0	0
$975$	6.52557e12	0.231258
$976$	0	0
$977$	2.90066e12	0.101852	0.0509261	−	0.998702i	$-0.483783\pi$
$977$	2.90066e12	0.101852	0.0509261	+	0.998702i	$0.483783\pi$
$978$	0	0
$979$	−5.47244e13	−1.90396
$980$	0	0
$981$	−2.44289e12	−0.0842158
$982$	0	0
$983$	2.46610e13	0.842403	0.421202	−	0.906967i	$-0.361608\pi$
$983$	2.46610e13	0.842403	0.421202	+	0.906967i	$0.361608\pi$
$984$	0	0
$985$	2.13918e11	0.00724076
$986$	0	0
$987$	−1.05134e13	−0.352629
$988$	0	0
$989$	2.35616e12	0.0783109
$990$	0	0
$991$	−5.76916e13	−1.90012	−0.950060	−	0.312067i	$-0.898979\pi$
$991$	−5.76916e13	−1.90012	−0.950060	+	0.312067i	$0.898979\pi$
$992$	0	0
$993$	4.38194e13	1.43019
$994$	0	0
$995$	4.36966e12	0.141333
$996$	0	0
$997$	5.52453e13	1.77079	0.885395	−	0.464839i	$-0.153888\pi$
$997$	5.52453e13	1.77079	0.885395	+	0.464839i	$0.153888\pi$
$998$	0	0
$999$	−2.47626e12	−0.0786594

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.m.1.3		8
4.3	odd	2		104.10.a.d.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.d.1.6	✓	8	4.3	odd	2
208.10.a.m.1.3		8	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-127.049 q^{3} +393.414 q^{5} -6601.13 q^{7} -3541.59 q^{9} -55090.9 q^{11} +28561.0 q^{13} -49982.8 q^{15} -229326. q^{17} -370189. q^{19} +838666. q^{21} -1.83233e6 q^{23} -1.79835e6 q^{25} +2.95066e6 q^{27} +1.20415e6 q^{29} -5.78451e6 q^{31} +6.99924e6 q^{33} -2.59698e6 q^{35} -839222. q^{37} -3.62864e6 q^{39} +2.52785e6 q^{41} -1.28588e6 q^{43} -1.39331e6 q^{45} -1.25359e7 q^{47} +3.22131e6 q^{49} +2.91356e7 q^{51} -9.21329e7 q^{53} -2.16735e7 q^{55} +4.70321e7 q^{57} -1.25593e8 q^{59} +1.80842e7 q^{61} +2.33785e7 q^{63} +1.12363e7 q^{65} -2.32656e8 q^{67} +2.32796e8 q^{69} -1.13475e8 q^{71} -5.38049e7 q^{73} +2.28478e8 q^{75} +3.63662e8 q^{77} -5.37073e8 q^{79} -3.05169e8 q^{81} -1.78667e8 q^{83} -9.02201e7 q^{85} -1.52986e8 q^{87} +9.93346e8 q^{89} -1.88535e8 q^{91} +7.34915e8 q^{93} -1.45638e8 q^{95} +1.12190e9 q^{97} +1.95109e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 141 q^{3} + 2051 q^{5} + 2417 q^{7} + 115741 q^{9} + 53118 q^{11} + 228488 q^{13} + 464555 q^{15} + 433095 q^{17} + 434954 q^{19} + 906875 q^{21} + 1124296 q^{23} + 5966065 q^{25} - 7820643 q^{27}+ \cdots + 641626736 q^{99}+O(q^{100}) \) 8 * q - 141 * q^3 + 2051 * q^5 + 2417 * q^7 + 115741 * q^9 + 53118 * q^11 + 228488 * q^13 + 464555 * q^15 + 433095 * q^17 + 434954 * q^19 + 906875 * q^21 + 1124296 * q^23 + 5966065 * q^25 - 7820643 * q^27 - 1913432 * q^29 - 5220148 * q^31 - 1626882 * q^33 - 950707 * q^35 - 419409 * q^37 - 4027101 * q^39 + 12148046 * q^41 - 32143187 * q^43 + 18893544 * q^45 - 57898411 * q^47 + 104302637 * q^49 - 198521477 * q^51 + 273106866 * q^53 - 141214662 * q^55 + 253032058 * q^57 - 50734502 * q^59 + 139520018 * q^61 + 20344158 * q^63 + 58578611 * q^65 - 528144786 * q^67 + 483506576 * q^69 - 334814973 * q^71 + 630864344 * q^73 - 2265796264 * q^75 + 939664146 * q^77 - 865037608 * q^79 + 3194281040 * q^81 - 1520229952 * q^83 + 2130158385 * q^85 - 1167867006 * q^87 + 894700240 * q^89 + 69031937 * q^91 + 3803209548 * q^93 + 2695926542 * q^95 + 980232164 * q^97 + 641626736 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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