LMFDB - Embedded newform 384.10.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, 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("384.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	384.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:	$197.773761087$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2070x^{2} - 13768x + 561570 $ x^4 - 2070x^2 - 13768x + 561570
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-34.7879$ of defining polynomial
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-81.0000 q^{3} -2550.24 q^{5} -11484.8 q^{7} +6561.00 q^{9} +O(q^{10})$	$q-81.0000 q^{3} -2550.24 q^{5} -11484.8 q^{7} +6561.00 q^{9} +17694.9 q^{11} -2878.66 q^{13} +206569. q^{15} -465200. q^{17} -409191. q^{19} +930265. q^{21} -374507. q^{23} +4.55058e6 q^{25} -531441. q^{27} -3.81849e6 q^{29} -2.57794e6 q^{31} -1.43328e6 q^{33} +2.92889e7 q^{35} -1.07373e7 q^{37} +233171. q^{39} +1.50461e7 q^{41} +1.21149e7 q^{43} -1.67321e7 q^{45} +4.65533e7 q^{47} +9.15461e7 q^{49} +3.76812e7 q^{51} +1.10710e8 q^{53} -4.51261e7 q^{55} +3.31445e7 q^{57} +9.19329e7 q^{59} +1.64770e8 q^{61} -7.53515e7 q^{63} +7.34126e6 q^{65} +2.68898e8 q^{67} +3.03350e7 q^{69} -3.57888e8 q^{71} -7.40257e7 q^{73} -3.68597e8 q^{75} -2.03221e8 q^{77} -9.17558e7 q^{79} +4.30467e7 q^{81} +1.86443e8 q^{83} +1.18637e9 q^{85} +3.09298e8 q^{87} -6.37166e8 q^{89} +3.30607e7 q^{91} +2.08813e8 q^{93} +1.04353e9 q^{95} -5.12367e7 q^{97} +1.16096e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 324 q^{3} - 240 q^{5} - 4840 q^{7} + 26244 q^{9}+O(q^{10}) $ 4 * q - 324 * q^3 - 240 * q^5 - 4840 * q^7 + 26244 * q^9	$ 4 q - 324 q^{3} - 240 q^{5} - 4840 q^{7} + 26244 q^{9} + 99664 q^{11} - 60840 q^{13} + 19440 q^{15} - 434952 q^{17} - 631776 q^{19} + 392040 q^{21} - 749392 q^{23} + 5991532 q^{25} - 2125764 q^{27} - 7908544 q^{29} - 11351240 q^{31} - 8072784 q^{33} + 25567008 q^{35} - 13592920 q^{37} + 4928040 q^{39} - 18838888 q^{41} - 14177920 q^{43} - 1574640 q^{45} + 37779120 q^{47} + 9409332 q^{49} + 35231112 q^{51} + 115336512 q^{53} + 184580544 q^{55} + 51173856 q^{57} + 115028080 q^{59} + 173228648 q^{61} - 31755240 q^{63} - 328077984 q^{65} + 231785104 q^{67} + 60700752 q^{69} - 197476208 q^{71} + 44629400 q^{73} - 485314092 q^{75} - 308117920 q^{77} + 355774584 q^{79} + 172186884 q^{81} + 607613328 q^{83} + 1087351392 q^{85} + 640592064 q^{87} - 1157146424 q^{89} + 847629840 q^{91} + 919450440 q^{93} + 329699328 q^{95} - 1599536472 q^{97} + 653895504 q^{99}+O(q^{100}) $ 4 * q - 324 * q^3 - 240 * q^5 - 4840 * q^7 + 26244 * q^9 + 99664 * q^11 - 60840 * q^13 + 19440 * q^15 - 434952 * q^17 - 631776 * q^19 + 392040 * q^21 - 749392 * q^23 + 5991532 * q^25 - 2125764 * q^27 - 7908544 * q^29 - 11351240 * q^31 - 8072784 * q^33 + 25567008 * q^35 - 13592920 * q^37 + 4928040 * q^39 - 18838888 * q^41 - 14177920 * q^43 - 1574640 * q^45 + 37779120 * q^47 + 9409332 * q^49 + 35231112 * q^51 + 115336512 * q^53 + 184580544 * q^55 + 51173856 * q^57 + 115028080 * q^59 + 173228648 * q^61 - 31755240 * q^63 - 328077984 * q^65 + 231785104 * q^67 + 60700752 * q^69 - 197476208 * q^71 + 44629400 * q^73 - 485314092 * q^75 - 308117920 * q^77 + 355774584 * q^79 + 172186884 * q^81 + 607613328 * q^83 + 1087351392 * q^85 + 640592064 * q^87 - 1157146424 * q^89 + 847629840 * q^91 + 919450440 * q^93 + 329699328 * q^95 - 1599536472 * q^97 + 653895504 * 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
$3$	−81.0000	−0.577350
$3$	−81.0000	−0.577350
$4$	0	0
$5$	−2550.24	−1.82480	−0.912401	−	0.409298i	$-0.865774\pi$
$5$	−2550.24	−1.82480	−0.912401	+	0.409298i	$0.865774\pi$
$6$	0	0
$7$	−11484.8	−1.80793	−0.903963	−	0.427610i	$-0.859356\pi$
$7$	−11484.8	−1.80793	−0.903963	+	0.427610i	$0.859356\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	17694.9	0.364401	0.182201	−	0.983261i	$-0.441678\pi$
$11$	17694.9	0.364401	0.182201	+	0.983261i	$0.441678\pi$
$12$	0	0
$13$	−2878.66	−0.0279541	−0.0139770	−	0.999902i	$-0.504449\pi$
$13$	−2878.66	−0.0279541	−0.0139770	+	0.999902i	$0.504449\pi$
$14$	0	0
$15$	206569.	1.05355
$16$	0	0
$17$	−465200.	−1.35089	−0.675444	−	0.737411i	$-0.736048\pi$
$17$	−465200.	−1.35089	−0.675444	+	0.737411i	$0.736048\pi$
$18$	0	0
$19$	−409191.	−0.720335	−0.360168	−	0.932888i	$-0.617281\pi$
$19$	−409191.	−0.720335	−0.360168	+	0.932888i	$0.617281\pi$
$20$	0	0
$21$	930265.	1.04381
$22$	0	0
$23$	−374507.	−0.279051	−0.139526	−	0.990218i	$-0.544558\pi$
$23$	−374507.	−0.279051	−0.139526	+	0.990218i	$0.544558\pi$
$24$	0	0
$25$	4.55058e6	2.32990
$26$	0	0
$27$	−531441.	−0.192450
$28$	0	0
$29$	−3.81849e6	−1.00254	−0.501269	−	0.865291i	$-0.667133\pi$
$29$	−3.81849e6	−1.00254	−0.501269	+	0.865291i	$0.667133\pi$
$30$	0	0
$31$	−2.57794e6	−0.501355	−0.250678	−	0.968071i	$-0.580653\pi$
$31$	−2.57794e6	−0.501355	−0.250678	+	0.968071i	$0.580653\pi$
$32$	0	0
$33$	−1.43328e6	−0.210387
$34$	0	0
$35$	2.92889e7	3.29911
$36$	0	0
$37$	−1.07373e7	−0.941858	−0.470929	−	0.882171i	$-0.656081\pi$
$37$	−1.07373e7	−0.941858	−0.470929	+	0.882171i	$0.656081\pi$
$38$	0	0
$39$	233171.	0.0161393
$40$	0	0
$41$	1.50461e7	0.831564	0.415782	−	0.909464i	$-0.363508\pi$
$41$	1.50461e7	0.831564	0.415782	+	0.909464i	$0.363508\pi$
$42$	0	0
$43$	1.21149e7	0.540394	0.270197	−	0.962805i	$-0.412911\pi$
$43$	1.21149e7	0.540394	0.270197	+	0.962805i	$0.412911\pi$
$44$	0	0
$45$	−1.67321e7	−0.608267
$46$	0	0
$47$	4.65533e7	1.39159	0.695793	−	0.718242i	$-0.255053\pi$
$47$	4.65533e7	1.39159	0.695793	+	0.718242i	$0.255053\pi$
$48$	0	0
$49$	9.15461e7	2.26860
$50$	0	0
$51$	3.76812e7	0.779936
$52$	0	0
$53$	1.10710e8	1.92728	0.963639	−	0.267209i	$-0.0861014\pi$
$53$	1.10710e8	1.92728	0.963639	+	0.267209i	$0.0861014\pi$
$54$	0	0
$55$	−4.51261e7	−0.664960
$56$	0	0
$57$	3.31445e7	0.415886
$58$	0	0
$59$	9.19329e7	0.987726	0.493863	−	0.869540i	$-0.335584\pi$
$59$	9.19329e7	0.987726	0.493863	+	0.869540i	$0.335584\pi$
$60$	0	0
$61$	1.64770e8	1.52368	0.761838	−	0.647767i	$-0.224297\pi$
$61$	1.64770e8	1.52368	0.761838	+	0.647767i	$0.224297\pi$
$62$	0	0
$63$	−7.53515e7	−0.602642
$64$	0	0
$65$	7.34126e6	0.0510106
$66$	0	0
$67$	2.68898e8	1.63024	0.815119	−	0.579294i	$-0.196672\pi$
$67$	2.68898e8	1.63024	0.815119	+	0.579294i	$0.196672\pi$
$68$	0	0
$69$	3.03350e7	0.161110
$70$	0	0
$71$	−3.57888e8	−1.67142	−0.835708	−	0.549174i	$-0.814942\pi$
$71$	−3.57888e8	−1.67142	−0.835708	+	0.549174i	$0.814942\pi$
$72$	0	0
$73$	−7.40257e7	−0.305091	−0.152546	−	0.988296i	$-0.548747\pi$
$73$	−7.40257e7	−0.305091	−0.152546	+	0.988296i	$0.548747\pi$
$74$	0	0
$75$	−3.68597e8	−1.34517
$76$	0	0
$77$	−2.03221e8	−0.658811
$78$	0	0
$79$	−9.17558e7	−0.265040	−0.132520	−	0.991180i	$-0.542307\pi$
$79$	−9.17558e7	−0.265040	−0.132520	+	0.991180i	$0.542307\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	1.86443e8	0.431215	0.215608	−	0.976480i	$-0.430827\pi$
$83$	1.86443e8	0.431215	0.215608	+	0.976480i	$0.430827\pi$
$84$	0	0
$85$	1.18637e9	2.46510
$86$	0	0
$87$	3.09298e8	0.578816
$88$	0	0
$89$	−6.37166e8	−1.07646	−0.538230	−	0.842798i	$-0.680907\pi$
$89$	−6.37166e8	−1.07646	−0.538230	+	0.842798i	$0.680907\pi$
$90$	0	0
$91$	3.30607e7	0.0505389
$92$	0	0
$93$	2.08813e8	0.289458
$94$	0	0
$95$	1.04353e9	1.31447
$96$	0	0
$97$	−5.12367e7	−0.0587636	−0.0293818	−	0.999568i	$-0.509354\pi$
$97$	−5.12367e7	−0.0587636	−0.0293818	+	0.999568i	$0.509354\pi$
$98$	0	0
$99$	1.16096e8	0.121467
$100$	0	0
$101$	−3.28576e8	−0.314188	−0.157094	−	0.987584i	$-0.550213\pi$
$101$	−3.28576e8	−0.314188	−0.157094	+	0.987584i	$0.550213\pi$
$102$	0	0
$103$	−2.13814e9	−1.87184	−0.935918	−	0.352218i	$-0.885428\pi$
$103$	−2.13814e9	−1.87184	−0.935918	+	0.352218i	$0.885428\pi$
$104$	0	0
$105$	−2.37240e9	−1.90474
$106$	0	0
$107$	−1.38054e9	−1.01817	−0.509087	−	0.860715i	$-0.670017\pi$
$107$	−1.38054e9	−1.01817	−0.509087	+	0.860715i	$0.670017\pi$
$108$	0	0
$109$	−1.42057e9	−0.963926	−0.481963	−	0.876192i	$-0.660076\pi$
$109$	−1.42057e9	−0.963926	−0.481963	+	0.876192i	$0.660076\pi$
$110$	0	0
$111$	8.69718e8	0.543782
$112$	0	0
$113$	4.63697e8	0.267535	0.133768	−	0.991013i	$-0.457292\pi$
$113$	4.63697e8	0.267535	0.133768	+	0.991013i	$0.457292\pi$
$114$	0	0
$115$	9.55081e8	0.509213
$116$	0	0
$117$	−1.88869e7	−0.00931802
$118$	0	0
$119$	5.34271e9	2.44231
$120$	0	0
$121$	−2.04484e9	−0.867212
$122$	0	0
$123$	−1.21873e9	−0.480104
$124$	0	0
$125$	−6.62414e9	−2.42680
$126$	0	0
$127$	3.20157e9	1.09206	0.546029	−	0.837766i	$-0.316139\pi$
$127$	3.20157e9	1.09206	0.546029	+	0.837766i	$0.316139\pi$
$128$	0	0
$129$	−9.81304e8	−0.311997
$130$	0	0
$131$	4.24459e9	1.25926	0.629630	−	0.776895i	$-0.283207\pi$
$131$	4.24459e9	1.25926	0.629630	+	0.776895i	$0.283207\pi$
$132$	0	0
$133$	4.69946e9	1.30231
$134$	0	0
$135$	1.35530e9	0.351183
$136$	0	0
$137$	−1.87982e9	−0.455905	−0.227952	−	0.973672i	$-0.573203\pi$
$137$	−1.87982e9	−0.455905	−0.227952	+	0.973672i	$0.573203\pi$
$138$	0	0
$139$	3.99062e9	0.906720	0.453360	−	0.891327i	$-0.350225\pi$
$139$	3.99062e9	0.906720	0.453360	+	0.891327i	$0.350225\pi$
$140$	0	0
$141$	−3.77082e9	−0.803433
$142$	0	0
$143$	−5.09374e7	−0.0101865
$144$	0	0
$145$	9.73807e9	1.82943
$146$	0	0
$147$	−7.41523e9	−1.30978
$148$	0	0
$149$	−8.67985e9	−1.44269	−0.721346	−	0.692574i	$-0.756476\pi$
$149$	−8.67985e9	−1.44269	−0.721346	+	0.692574i	$0.756476\pi$
$150$	0	0
$151$	−4.40680e9	−0.689806	−0.344903	−	0.938638i	$-0.612088\pi$
$151$	−4.40680e9	−0.689806	−0.344903	+	0.938638i	$0.612088\pi$
$152$	0	0
$153$	−3.05218e9	−0.450296
$154$	0	0
$155$	6.57436e9	0.914874
$156$	0	0
$157$	7.13694e9	0.937484	0.468742	−	0.883335i	$-0.344707\pi$
$157$	7.13694e9	0.937484	0.468742	+	0.883335i	$0.344707\pi$
$158$	0	0
$159$	−8.96748e9	−1.11271
$160$	0	0
$161$	4.30112e9	0.504504
$162$	0	0
$163$	9.33307e8	0.103557	0.0517786	−	0.998659i	$-0.483511\pi$
$163$	9.33307e8	0.103557	0.0517786	+	0.998659i	$0.483511\pi$
$164$	0	0
$165$	3.65521e9	0.383915
$166$	0	0
$167$	1.62183e10	1.61355	0.806773	−	0.590862i	$-0.201212\pi$
$167$	1.62183e10	1.61355	0.806773	+	0.590862i	$0.201212\pi$
$168$	0	0
$169$	−1.05962e10	−0.999219
$170$	0	0
$171$	−2.68470e9	−0.240112
$172$	0	0
$173$	6.78681e9	0.576047	0.288024	−	0.957623i	$-0.407002\pi$
$173$	6.78681e9	0.576047	0.288024	+	0.957623i	$0.407002\pi$
$174$	0	0
$175$	−5.22624e10	−4.21229
$176$	0	0
$177$	−7.44656e9	−0.570264
$178$	0	0
$179$	6.04635e9	0.440205	0.220102	−	0.975477i	$-0.429361\pi$
$179$	6.04635e9	0.440205	0.220102	+	0.975477i	$0.429361\pi$
$180$	0	0
$181$	1.73661e10	1.20268	0.601339	−	0.798994i	$-0.294634\pi$
$181$	1.73661e10	1.20268	0.601339	+	0.798994i	$0.294634\pi$
$182$	0	0
$183$	−1.33463e10	−0.879695
$184$	0	0
$185$	2.73825e10	1.71870
$186$	0	0
$187$	−8.23165e9	−0.492266
$188$	0	0
$189$	6.10347e9	0.347936
$190$	0	0
$191$	−1.55901e10	−0.847616	−0.423808	−	0.905752i	$-0.639307\pi$
$191$	−1.55901e10	−0.847616	−0.423808	+	0.905752i	$0.639307\pi$
$192$	0	0
$193$	−2.58307e10	−1.34007	−0.670036	−	0.742329i	$-0.733721\pi$
$193$	−2.58307e10	−1.34007	−0.670036	+	0.742329i	$0.733721\pi$
$194$	0	0
$195$	−5.94642e8	−0.0294510
$196$	0	0
$197$	−3.09030e10	−1.46185	−0.730924	−	0.682459i	$-0.760911\pi$
$197$	−3.09030e10	−1.46185	−0.730924	+	0.682459i	$0.760911\pi$
$198$	0	0
$199$	−1.28574e10	−0.581183	−0.290592	−	0.956847i	$-0.593852\pi$
$199$	−1.28574e10	−0.581183	−0.290592	+	0.956847i	$0.593852\pi$
$200$	0	0
$201$	−2.17807e10	−0.941218
$202$	0	0
$203$	4.38545e10	1.81252
$204$	0	0
$205$	−3.83711e10	−1.51744
$206$	0	0
$207$	−2.45714e9	−0.0930171
$208$	0	0
$209$	−7.24058e9	−0.262491
$210$	0	0
$211$	2.77349e10	0.963288	0.481644	−	0.876367i	$-0.340040\pi$
$211$	2.77349e10	0.963288	0.481644	+	0.876367i	$0.340040\pi$
$212$	0	0
$213$	2.89889e10	0.964993
$214$	0	0
$215$	−3.08958e10	−0.986112
$216$	0	0
$217$	2.96070e10	0.906413
$218$	0	0
$219$	5.99608e9	0.176145
$220$	0	0
$221$	1.33915e9	0.0377628
$222$	0	0
$223$	4.93029e10	1.33506	0.667530	−	0.744583i	$-0.267351\pi$
$223$	4.93029e10	1.33506	0.667530	+	0.744583i	$0.267351\pi$
$224$	0	0
$225$	2.98564e10	0.776633
$226$	0	0
$227$	2.66250e10	0.665538	0.332769	−	0.943008i	$-0.392017\pi$
$227$	2.66250e10	0.665538	0.332769	+	0.943008i	$0.392017\pi$
$228$	0	0
$229$	−6.63083e10	−1.59334	−0.796670	−	0.604415i	$-0.793407\pi$
$229$	−6.63083e10	−1.59334	−0.796670	+	0.604415i	$0.793407\pi$
$230$	0	0
$231$	1.64609e10	0.380365
$232$	0	0
$233$	5.24513e10	1.16588	0.582941	−	0.812514i	$-0.301902\pi$
$233$	5.24513e10	1.16588	0.582941	+	0.812514i	$0.301902\pi$
$234$	0	0
$235$	−1.18722e11	−2.53937
$236$	0	0
$237$	7.43222e9	0.153021
$238$	0	0
$239$	2.83125e10	0.561291	0.280646	−	0.959811i	$-0.409451\pi$
$239$	2.83125e10	0.561291	0.280646	+	0.959811i	$0.409451\pi$
$240$	0	0
$241$	−8.73233e10	−1.66745	−0.833726	−	0.552178i	$-0.813797\pi$
$241$	−8.73233e10	−1.66745	−0.833726	+	0.552178i	$0.813797\pi$
$242$	0	0
$243$	−3.48678e9	−0.0641500
$244$	0	0
$245$	−2.33464e11	−4.13974
$246$	0	0
$247$	1.17792e9	0.0201363
$248$	0	0
$249$	−1.51019e10	−0.248962
$250$	0	0
$251$	3.44391e10	0.547671	0.273836	−	0.961776i	$-0.411708\pi$
$251$	3.44391e10	0.547671	0.273836	+	0.961776i	$0.411708\pi$
$252$	0	0
$253$	−6.62684e9	−0.101687
$254$	0	0
$255$	−9.60960e10	−1.42323
$256$	0	0
$257$	−1.18147e11	−1.68937	−0.844685	−	0.535264i	$-0.820212\pi$
$257$	−1.18147e11	−1.68937	−0.844685	+	0.535264i	$0.820212\pi$
$258$	0	0
$259$	1.23315e11	1.70281
$260$	0	0
$261$	−2.50531e10	−0.334180
$262$	0	0
$263$	−3.09809e9	−0.0399294	−0.0199647	−	0.999801i	$-0.506355\pi$
$263$	−3.09809e9	−0.0399294	−0.0199647	+	0.999801i	$0.506355\pi$
$264$	0	0
$265$	−2.82336e11	−3.51690
$266$	0	0
$267$	5.16105e10	0.621494
$268$	0	0
$269$	8.92323e10	1.03905	0.519526	−	0.854455i	$-0.326109\pi$
$269$	8.92323e10	1.03905	0.519526	+	0.854455i	$0.326109\pi$
$270$	0	0
$271$	1.33152e11	1.49964	0.749821	−	0.661641i	$-0.230140\pi$
$271$	1.33152e11	1.49964	0.749821	+	0.661641i	$0.230140\pi$
$272$	0	0
$273$	−2.67792e9	−0.0291786
$274$	0	0
$275$	8.05220e10	0.849019
$276$	0	0
$277$	−1.36458e11	−1.39264	−0.696322	−	0.717730i	$-0.745181\pi$
$277$	−1.36458e11	−1.39264	−0.696322	+	0.717730i	$0.745181\pi$
$278$	0	0
$279$	−1.69139e10	−0.167118
$280$	0	0
$281$	5.62851e10	0.538537	0.269268	−	0.963065i	$-0.413218\pi$
$281$	5.62851e10	0.538537	0.269268	+	0.963065i	$0.413218\pi$
$282$	0	0
$283$	−1.30168e10	−0.120633	−0.0603163	−	0.998179i	$-0.519211\pi$
$283$	−1.30168e10	−0.120633	−0.0603163	+	0.998179i	$0.519211\pi$
$284$	0	0
$285$	−8.45262e10	−0.758909
$286$	0	0
$287$	−1.72801e11	−1.50341
$288$	0	0
$289$	9.78231e10	0.824900
$290$	0	0
$291$	4.15017e9	0.0339272
$292$	0	0
$293$	2.06089e11	1.63362	0.816810	−	0.576907i	$-0.195741\pi$
$293$	2.06089e11	1.63362	0.816810	+	0.576907i	$0.195741\pi$
$294$	0	0
$295$	−2.34451e11	−1.80240
$296$	0	0
$297$	−9.40377e9	−0.0701291
$298$	0	0
$299$	1.07808e9	0.00780062
$300$	0	0
$301$	−1.39136e11	−0.976993
$302$	0	0
$303$	2.66147e10	0.181397
$304$	0	0
$305$	−4.20202e11	−2.78041
$306$	0	0
$307$	−6.28066e10	−0.403536	−0.201768	−	0.979433i	$-0.564669\pi$
$307$	−6.28066e10	−0.403536	−0.201768	+	0.979433i	$0.564669\pi$
$308$	0	0
$309$	1.73189e11	1.08071
$310$	0	0
$311$	2.85384e10	0.172985	0.0864925	−	0.996253i	$-0.472434\pi$
$311$	2.85384e10	0.172985	0.0864925	+	0.996253i	$0.472434\pi$
$312$	0	0
$313$	1.03679e10	0.0610578	0.0305289	−	0.999534i	$-0.490281\pi$
$313$	1.03679e10	0.0610578	0.0305289	+	0.999534i	$0.490281\pi$
$314$	0	0
$315$	1.92164e11	1.09970
$316$	0	0
$317$	1.10284e11	0.613400	0.306700	−	0.951806i	$-0.400775\pi$
$317$	1.10284e11	0.613400	0.306700	+	0.951806i	$0.400775\pi$
$318$	0	0
$319$	−6.75677e10	−0.365327
$320$	0	0
$321$	1.11824e11	0.587844
$322$	0	0
$323$	1.90356e11	0.973093
$324$	0	0
$325$	−1.30996e10	−0.0651301
$326$	0	0
$327$	1.15066e11	0.556523
$328$	0	0
$329$	−5.34654e11	−2.51589
$330$	0	0
$331$	2.32144e11	1.06299	0.531497	−	0.847060i	$-0.321630\pi$
$331$	2.32144e11	1.06299	0.531497	+	0.847060i	$0.321630\pi$
$332$	0	0
$333$	−7.04471e10	−0.313953
$334$	0	0
$335$	−6.85753e11	−2.97486
$336$	0	0
$337$	1.90284e11	0.803652	0.401826	−	0.915716i	$-0.368376\pi$
$337$	1.90284e11	0.803652	0.401826	+	0.915716i	$0.368376\pi$
$338$	0	0
$339$	−3.75595e10	−0.154462
$340$	0	0
$341$	−4.56163e10	−0.182695
$342$	0	0
$343$	−5.87933e11	−2.29353
$344$	0	0
$345$	−7.73615e10	−0.293994
$346$	0	0
$347$	2.80544e11	1.03877	0.519383	−	0.854541i	$-0.326162\pi$
$347$	2.80544e11	1.03877	0.519383	+	0.854541i	$0.326162\pi$
$348$	0	0
$349$	−5.24413e10	−0.189217	−0.0946083	−	0.995515i	$-0.530160\pi$
$349$	−5.24413e10	−0.189217	−0.0946083	+	0.995515i	$0.530160\pi$
$350$	0	0
$351$	1.52984e9	0.00537976
$352$	0	0
$353$	5.36154e11	1.83782	0.918911	−	0.394466i	$-0.129070\pi$
$353$	5.36154e11	1.83782	0.918911	+	0.394466i	$0.129070\pi$
$354$	0	0
$355$	9.12699e11	3.05000
$356$	0	0
$357$	−4.32759e11	−1.41007
$358$	0	0
$359$	−1.53244e11	−0.486920	−0.243460	−	0.969911i	$-0.578282\pi$
$359$	−1.53244e11	−0.486920	−0.243460	+	0.969911i	$0.578282\pi$
$360$	0	0
$361$	−1.55250e11	−0.481117
$362$	0	0
$363$	1.65632e11	0.500685
$364$	0	0
$365$	1.88783e11	0.556731
$366$	0	0
$367$	3.74040e11	1.07627	0.538134	−	0.842859i	$-0.319129\pi$
$367$	3.74040e11	1.07627	0.538134	+	0.842859i	$0.319129\pi$
$368$	0	0
$369$	9.87173e10	0.277188
$370$	0	0
$371$	−1.27147e12	−3.48437
$372$	0	0
$373$	2.01779e11	0.539742	0.269871	−	0.962896i	$-0.413019\pi$
$373$	2.01779e11	0.539742	0.269871	+	0.962896i	$0.413019\pi$
$374$	0	0
$375$	5.36555e11	1.40111
$376$	0	0
$377$	1.09921e10	0.0280250
$378$	0	0
$379$	6.79938e11	1.69275	0.846375	−	0.532588i	$-0.178780\pi$
$379$	6.79938e11	1.69275	0.846375	+	0.532588i	$0.178780\pi$
$380$	0	0
$381$	−2.59327e11	−0.630500
$382$	0	0
$383$	4.62580e11	1.09848	0.549240	−	0.835664i	$-0.314917\pi$
$383$	4.62580e11	1.09848	0.549240	+	0.835664i	$0.314917\pi$
$384$	0	0
$385$	5.18262e11	1.20220
$386$	0	0
$387$	7.94857e10	0.180131
$388$	0	0
$389$	−5.13044e11	−1.13601	−0.568004	−	0.823026i	$-0.692284\pi$
$389$	−5.13044e11	−1.13601	−0.568004	+	0.823026i	$0.692284\pi$
$390$	0	0
$391$	1.74220e11	0.376967
$392$	0	0
$393$	−3.43812e11	−0.727034
$394$	0	0
$395$	2.33999e11	0.483646
$396$	0	0
$397$	−7.14443e11	−1.44348	−0.721739	−	0.692165i	$-0.756657\pi$
$397$	−7.14443e11	−1.44348	−0.721739	+	0.692165i	$0.756657\pi$
$398$	0	0
$399$	−3.80656e11	−0.751891
$400$	0	0
$401$	−2.80010e11	−0.540785	−0.270392	−	0.962750i	$-0.587153\pi$
$401$	−2.80010e11	−0.540785	−0.270392	+	0.962750i	$0.587153\pi$
$402$	0	0
$403$	7.42101e9	0.0140149
$404$	0	0
$405$	−1.09779e11	−0.202756
$406$	0	0
$407$	−1.89994e11	−0.343214
$408$	0	0
$409$	1.09398e11	0.193309	0.0966546	−	0.995318i	$-0.469186\pi$
$409$	1.09398e11	0.193309	0.0966546	+	0.995318i	$0.469186\pi$
$410$	0	0
$411$	1.52266e11	0.263217
$412$	0	0
$413$	−1.05583e12	−1.78574
$414$	0	0
$415$	−4.75473e11	−0.786882
$416$	0	0
$417$	−3.23240e11	−0.523495
$418$	0	0
$419$	−6.69404e10	−0.106102	−0.0530512	−	0.998592i	$-0.516895\pi$
$419$	−6.69404e10	−0.106102	−0.0530512	+	0.998592i	$0.516895\pi$
$420$	0	0
$421$	−4.27317e11	−0.662949	−0.331475	−	0.943464i	$-0.607546\pi$
$421$	−4.27317e11	−0.662949	−0.331475	+	0.943464i	$0.607546\pi$
$422$	0	0
$423$	3.05436e11	0.463862
$424$	0	0
$425$	−2.11693e12	−3.14743
$426$	0	0
$427$	−1.89234e12	−2.75469
$428$	0	0
$429$	4.12593e9	0.00588118
$430$	0	0
$431$	−4.00618e11	−0.559220	−0.279610	−	0.960114i	$-0.590205\pi$
$431$	−4.00618e11	−0.559220	−0.279610	+	0.960114i	$0.590205\pi$
$432$	0	0
$433$	3.69071e11	0.504562	0.252281	−	0.967654i	$-0.418819\pi$
$433$	3.69071e11	0.504562	0.252281	+	0.967654i	$0.418819\pi$
$434$	0	0
$435$	−7.88783e11	−1.05622
$436$	0	0
$437$	1.53245e11	0.201011
$438$	0	0
$439$	−1.02355e12	−1.31528	−0.657640	−	0.753332i	$-0.728445\pi$
$439$	−1.02355e12	−1.31528	−0.657640	+	0.753332i	$0.728445\pi$
$440$	0	0
$441$	6.00634e11	0.756199
$442$	0	0
$443$	2.64264e11	0.326002	0.163001	−	0.986626i	$-0.447883\pi$
$443$	2.64264e11	0.326002	0.163001	+	0.986626i	$0.447883\pi$
$444$	0	0
$445$	1.62493e12	1.96432
$446$	0	0
$447$	7.03068e11	0.832939
$448$	0	0
$449$	−5.41761e11	−0.629070	−0.314535	−	0.949246i	$-0.601849\pi$
$449$	−5.41761e11	−0.629070	−0.314535	+	0.949246i	$0.601849\pi$
$450$	0	0
$451$	2.66238e11	0.303023
$452$	0	0
$453$	3.56951e11	0.398260
$454$	0	0
$455$	−8.43126e10	−0.0922234
$456$	0	0
$457$	−5.87461e10	−0.0630022	−0.0315011	−	0.999504i	$-0.510029\pi$
$457$	−5.87461e10	−0.0630022	−0.0315011	+	0.999504i	$0.510029\pi$
$458$	0	0
$459$	2.47226e11	0.259979
$460$	0	0
$461$	1.57233e12	1.62139	0.810697	−	0.585466i	$-0.199089\pi$
$461$	1.57233e12	1.62139	0.810697	+	0.585466i	$0.199089\pi$
$462$	0	0
$463$	9.14079e11	0.924420	0.462210	−	0.886771i	$-0.347057\pi$
$463$	9.14079e11	0.924420	0.462210	+	0.886771i	$0.347057\pi$
$464$	0	0
$465$	−5.32523e11	−0.528202
$466$	0	0
$467$	−6.14555e11	−0.597909	−0.298954	−	0.954267i	$-0.596638\pi$
$467$	−6.14555e11	−0.597909	−0.298954	+	0.954267i	$0.596638\pi$
$468$	0	0
$469$	−3.08823e12	−2.94735
$470$	0	0
$471$	−5.78092e11	−0.541257
$472$	0	0
$473$	2.14371e11	0.196920
$474$	0	0
$475$	−1.86206e12	−1.67831
$476$	0	0
$477$	7.26366e11	0.642426
$478$	0	0
$479$	9.71639e11	0.843325	0.421663	−	0.906753i	$-0.361447\pi$
$479$	9.71639e11	0.843325	0.421663	+	0.906753i	$0.361447\pi$
$480$	0	0
$481$	3.09089e10	0.0263288
$482$	0	0
$483$	−3.48391e11	−0.291276
$484$	0	0
$485$	1.30666e11	0.107232
$486$	0	0
$487$	−2.35678e12	−1.89862	−0.949312	−	0.314335i	$-0.898218\pi$
$487$	−2.35678e12	−1.89862	−0.949312	+	0.314335i	$0.898218\pi$
$488$	0	0
$489$	−7.55979e10	−0.0597888
$490$	0	0
$491$	−7.35095e11	−0.570791	−0.285395	−	0.958410i	$-0.592125\pi$
$491$	−7.35095e11	−0.570791	−0.285395	+	0.958410i	$0.592125\pi$
$492$	0	0
$493$	1.77636e12	1.35432
$494$	0	0
$495$	−2.96072e11	−0.221653
$496$	0	0
$497$	4.11026e12	3.02180
$498$	0	0
$499$	−2.92617e11	−0.211275	−0.105637	−	0.994405i	$-0.533688\pi$
$499$	−2.92617e11	−0.211275	−0.105637	+	0.994405i	$0.533688\pi$
$500$	0	0
$501$	−1.31368e12	−0.931581
$502$	0	0
$503$	2.02398e12	1.40978	0.704890	−	0.709317i	$-0.250996\pi$
$503$	2.02398e12	1.40978	0.704890	+	0.709317i	$0.250996\pi$
$504$	0	0
$505$	8.37948e11	0.573332
$506$	0	0
$507$	8.58293e11	0.576899
$508$	0	0
$509$	7.65483e11	0.505482	0.252741	−	0.967534i	$-0.418668\pi$
$509$	7.65483e11	0.505482	0.252741	+	0.967534i	$0.418668\pi$
$510$	0	0
$511$	8.50168e11	0.551583
$512$	0	0
$513$	2.17461e11	0.138629
$514$	0	0
$515$	5.45275e12	3.41573
$516$	0	0
$517$	8.23755e11	0.507096
$518$	0	0
$519$	−5.49731e11	−0.332581
$520$	0	0
$521$	2.50936e12	1.49208	0.746041	−	0.665900i	$-0.231952\pi$
$521$	2.50936e12	1.49208	0.746041	+	0.665900i	$0.231952\pi$
$522$	0	0
$523$	−9.38072e10	−0.0548250	−0.0274125	−	0.999624i	$-0.508727\pi$
$523$	−9.38072e10	−0.0548250	−0.0274125	+	0.999624i	$0.508727\pi$
$524$	0	0
$525$	4.23325e12	2.43196
$526$	0	0
$527$	1.19926e12	0.677275
$528$	0	0
$529$	−1.66090e12	−0.922130
$530$	0	0
$531$	6.03171e11	0.329242
$532$	0	0
$533$	−4.33125e10	−0.0232456
$534$	0	0
$535$	3.52071e12	1.85797
$536$	0	0
$537$	−4.89755e11	−0.254152
$538$	0	0
$539$	1.61990e12	0.826680
$540$	0	0
$541$	1.38883e12	0.697044	0.348522	−	0.937301i	$-0.386684\pi$
$541$	1.38883e12	0.697044	0.348522	+	0.937301i	$0.386684\pi$
$542$	0	0
$543$	−1.40666e12	−0.694366
$544$	0	0
$545$	3.62279e12	1.75897
$546$	0	0
$547$	−2.50687e11	−0.119726	−0.0598631	−	0.998207i	$-0.519066\pi$
$547$	−2.50687e11	−0.119726	−0.0598631	+	0.998207i	$0.519066\pi$
$548$	0	0
$549$	1.08105e12	0.507892
$550$	0	0
$551$	1.56249e12	0.722164
$552$	0	0
$553$	1.05379e12	0.479173
$554$	0	0
$555$	−2.21799e12	−0.992294
$556$	0	0
$557$	−2.07853e11	−0.0914973	−0.0457487	−	0.998953i	$-0.514567\pi$
$557$	−2.07853e11	−0.0914973	−0.0457487	+	0.998953i	$0.514567\pi$
$558$	0	0
$559$	−3.48746e10	−0.0151062
$560$	0	0
$561$	6.66764e11	0.284210
$562$	0	0
$563$	−1.85431e12	−0.777850	−0.388925	−	0.921269i	$-0.627153\pi$
$563$	−1.85431e12	−0.777850	−0.388925	+	0.921269i	$0.627153\pi$
$564$	0	0
$565$	−1.18254e12	−0.488199
$566$	0	0
$567$	−4.94381e11	−0.200881
$568$	0	0
$569$	−1.99208e12	−0.796712	−0.398356	−	0.917231i	$-0.630419\pi$
$569$	−1.99208e12	−0.796712	−0.398356	+	0.917231i	$0.630419\pi$
$570$	0	0
$571$	1.49143e12	0.587139	0.293570	−	0.955938i	$-0.405157\pi$
$571$	1.49143e12	0.587139	0.293570	+	0.955938i	$0.405157\pi$
$572$	0	0
$573$	1.26280e12	0.489372
$574$	0	0
$575$	−1.70422e12	−0.650162
$576$	0	0
$577$	1.82633e12	0.685942	0.342971	−	0.939346i	$-0.388567\pi$
$577$	1.82633e12	0.685942	0.342971	+	0.939346i	$0.388567\pi$
$578$	0	0
$579$	2.09229e12	0.773691
$580$	0	0
$581$	−2.14125e12	−0.779605
$582$	0	0
$583$	1.95899e12	0.702303
$584$	0	0
$585$	4.81660e10	0.0170035
$586$	0	0
$587$	2.12345e12	0.738193	0.369096	−	0.929391i	$-0.379667\pi$
$587$	2.12345e12	0.738193	0.369096	+	0.929391i	$0.379667\pi$
$588$	0	0
$589$	1.05487e12	0.361144
$590$	0	0
$591$	2.50314e12	0.843998
$592$	0	0
$593$	2.09754e12	0.696569	0.348284	−	0.937389i	$-0.386764\pi$
$593$	2.09754e12	0.696569	0.348284	+	0.937389i	$0.386764\pi$
$594$	0	0
$595$	−1.36252e13	−4.45672
$596$	0	0
$597$	1.04145e12	0.335546
$598$	0	0
$599$	2.89930e12	0.920178	0.460089	−	0.887873i	$-0.347818\pi$
$599$	2.89930e12	0.920178	0.460089	+	0.887873i	$0.347818\pi$
$600$	0	0
$601$	−4.36479e12	−1.36467	−0.682337	−	0.731038i	$-0.739036\pi$
$601$	−4.36479e12	−1.36467	−0.682337	+	0.731038i	$0.739036\pi$
$602$	0	0
$603$	1.76424e12	0.543412
$604$	0	0
$605$	5.21483e12	1.58249
$606$	0	0
$607$	−2.16703e12	−0.647913	−0.323956	−	0.946072i	$-0.605013\pi$
$607$	−2.16703e12	−0.647913	−0.323956	+	0.946072i	$0.605013\pi$
$608$	0	0
$609$	−3.55221e12	−1.04646
$610$	0	0
$611$	−1.34011e11	−0.0389005
$612$	0	0
$613$	−1.83561e12	−0.525058	−0.262529	−	0.964924i	$-0.584557\pi$
$613$	−1.83561e12	−0.525058	−0.262529	+	0.964924i	$0.584557\pi$
$614$	0	0
$615$	3.10806e12	0.876094
$616$	0	0
$617$	−7.04193e11	−0.195618	−0.0978089	−	0.995205i	$-0.531183\pi$
$617$	−7.04193e11	−0.195618	−0.0978089	+	0.995205i	$0.531183\pi$
$618$	0	0
$619$	−6.20869e12	−1.69978	−0.849889	−	0.526962i	$-0.823331\pi$
$619$	−6.20869e12	−1.69978	−0.849889	+	0.526962i	$0.823331\pi$
$620$	0	0
$621$	1.99028e11	0.0537035
$622$	0	0
$623$	7.31770e12	1.94616
$624$	0	0
$625$	8.00526e12	2.09853
$626$	0	0
$627$	5.86487e11	0.151549
$628$	0	0
$629$	4.99497e12	1.27235
$630$	0	0
$631$	1.83096e12	0.459776	0.229888	−	0.973217i	$-0.426164\pi$
$631$	1.83096e12	0.459776	0.229888	+	0.973217i	$0.426164\pi$
$632$	0	0
$633$	−2.24653e12	−0.556155
$634$	0	0
$635$	−8.16475e12	−1.99279
$636$	0	0
$637$	−2.63530e11	−0.0634165
$638$	0	0
$639$	−2.34810e12	−0.557139
$640$	0	0
$641$	9.57030e11	0.223905	0.111953	−	0.993714i	$-0.464290\pi$
$641$	9.57030e11	0.223905	0.111953	+	0.993714i	$0.464290\pi$
$642$	0	0
$643$	−1.17043e12	−0.270019	−0.135010	−	0.990844i	$-0.543107\pi$
$643$	−1.17043e12	−0.270019	−0.135010	+	0.990844i	$0.543107\pi$
$644$	0	0
$645$	2.50256e12	0.569332
$646$	0	0
$647$	7.13180e12	1.60004	0.800018	−	0.599976i	$-0.204823\pi$
$647$	7.13180e12	1.60004	0.800018	+	0.599976i	$0.204823\pi$
$648$	0	0
$649$	1.62674e12	0.359929
$650$	0	0
$651$	−2.39817e12	−0.523318
$652$	0	0
$653$	−1.45995e12	−0.314217	−0.157108	−	0.987581i	$-0.550217\pi$
$653$	−1.45995e12	−0.314217	−0.157108	+	0.987581i	$0.550217\pi$
$654$	0	0
$655$	−1.08247e13	−2.29790
$656$	0	0
$657$	−4.85683e11	−0.101697
$658$	0	0
$659$	−3.16319e12	−0.653343	−0.326672	−	0.945138i	$-0.605927\pi$
$659$	−3.16319e12	−0.653343	−0.326672	+	0.945138i	$0.605927\pi$
$660$	0	0
$661$	−5.94931e12	−1.21216	−0.606080	−	0.795404i	$-0.707259\pi$
$661$	−5.94931e12	−1.21216	−0.606080	+	0.795404i	$0.707259\pi$
$662$	0	0
$663$	−1.08471e11	−0.0218024
$664$	0	0
$665$	−1.19847e13	−2.37646
$666$	0	0
$667$	1.43005e12	0.279760
$668$	0	0
$669$	−3.99354e12	−0.770798
$670$	0	0
$671$	2.91558e12	0.555230
$672$	0	0
$673$	6.63233e12	1.24623	0.623115	−	0.782130i	$-0.285867\pi$
$673$	6.63233e12	1.24623	0.623115	+	0.782130i	$0.285867\pi$
$674$	0	0
$675$	−2.41837e12	−0.448389
$676$	0	0
$677$	2.68780e12	0.491754	0.245877	−	0.969301i	$-0.420924\pi$
$677$	2.68780e12	0.491754	0.245877	+	0.969301i	$0.420924\pi$
$678$	0	0
$679$	5.88441e11	0.106240
$680$	0	0
$681$	−2.15662e12	−0.384249
$682$	0	0
$683$	2.14691e11	0.0377503	0.0188751	−	0.999822i	$-0.493992\pi$
$683$	2.14691e11	0.0377503	0.0188751	+	0.999822i	$0.493992\pi$
$684$	0	0
$685$	4.79399e12	0.831935
$686$	0	0
$687$	5.37097e12	0.919915
$688$	0	0
$689$	−3.18695e11	−0.0538752
$690$	0	0
$691$	5.18271e12	0.864780	0.432390	−	0.901687i	$-0.357670\pi$
$691$	5.18271e12	0.864780	0.432390	+	0.901687i	$0.357670\pi$
$692$	0	0
$693$	−1.33333e12	−0.219604
$694$	0	0
$695$	−1.01770e13	−1.65458
$696$	0	0
$697$	−6.99943e12	−1.12335
$698$	0	0
$699$	−4.24856e12	−0.673122
$700$	0	0
$701$	5.92911e12	0.927381	0.463691	−	0.885997i	$-0.346525\pi$
$701$	5.92911e12	0.927381	0.463691	+	0.885997i	$0.346525\pi$
$702$	0	0
$703$	4.39359e12	0.678454
$704$	0	0
$705$	9.61648e12	1.46611
$706$	0	0
$707$	3.77362e12	0.568030
$708$	0	0
$709$	4.63915e12	0.689493	0.344747	−	0.938696i	$-0.387965\pi$
$709$	4.63915e12	0.689493	0.344747	+	0.938696i	$0.387965\pi$
$710$	0	0
$711$	−6.02010e11	−0.0883467
$712$	0	0
$713$	9.65456e11	0.139904
$714$	0	0
$715$	1.29903e11	0.0185883
$716$	0	0
$717$	−2.29332e12	−0.324062
$718$	0	0
$719$	−3.00203e12	−0.418924	−0.209462	−	0.977817i	$-0.567171\pi$
$719$	−3.00203e12	−0.418924	−0.209462	+	0.977817i	$0.567171\pi$
$720$	0	0
$721$	2.45560e13	3.38414
$722$	0	0
$723$	7.07319e12	0.962704
$724$	0	0
$725$	−1.73764e13	−2.33581
$726$	0	0
$727$	−9.19643e11	−0.122100	−0.0610498	−	0.998135i	$-0.519445\pi$
$727$	−9.19643e11	−0.122100	−0.0610498	+	0.998135i	$0.519445\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−5.63584e12	−0.730012
$732$	0	0
$733$	−1.03255e13	−1.32113	−0.660564	−	0.750769i	$-0.729683\pi$
$733$	−1.03255e13	−1.32113	−0.660564	+	0.750769i	$0.729683\pi$
$734$	0	0
$735$	1.89106e13	2.39008
$736$	0	0
$737$	4.75811e12	0.594061
$738$	0	0
$739$	−3.25490e12	−0.401456	−0.200728	−	0.979647i	$-0.564331\pi$
$739$	−3.25490e12	−0.401456	−0.200728	+	0.979647i	$0.564331\pi$
$740$	0	0
$741$	−9.54116e10	−0.0116257
$742$	0	0
$743$	−3.84135e12	−0.462418	−0.231209	−	0.972904i	$-0.574268\pi$
$743$	−3.84135e12	−0.462418	−0.231209	+	0.972904i	$0.574268\pi$
$744$	0	0
$745$	2.21357e13	2.63263
$746$	0	0
$747$	1.22325e12	0.143738
$748$	0	0
$749$	1.58552e13	1.84078
$750$	0	0
$751$	−1.42668e13	−1.63662	−0.818311	−	0.574776i	$-0.805089\pi$
$751$	−1.42668e13	−1.63662	−0.818311	+	0.574776i	$0.805089\pi$
$752$	0	0
$753$	−2.78957e12	−0.316198
$754$	0	0
$755$	1.12384e13	1.25876
$756$	0	0
$757$	−2.85396e12	−0.315876	−0.157938	−	0.987449i	$-0.550485\pi$
$757$	−2.85396e12	−0.315876	−0.157938	+	0.987449i	$0.550485\pi$
$758$	0	0
$759$	5.36774e11	0.0587089
$760$	0	0
$761$	−2.19569e12	−0.237323	−0.118662	−	0.992935i	$-0.537860\pi$
$761$	−2.19569e12	−0.237323	−0.118662	+	0.992935i	$0.537860\pi$
$762$	0	0
$763$	1.63149e13	1.74271
$764$	0	0
$765$	7.78378e12	0.821701
$766$	0	0
$767$	−2.64643e11	−0.0276110
$768$	0	0
$769$	5.45095e12	0.562087	0.281044	−	0.959695i	$-0.409319\pi$
$769$	5.45095e12	0.562087	0.281044	+	0.959695i	$0.409319\pi$
$770$	0	0
$771$	9.56993e12	0.975358
$772$	0	0
$773$	6.19087e12	0.623654	0.311827	−	0.950139i	$-0.399059\pi$
$773$	6.19087e12	0.623654	0.311827	+	0.950139i	$0.399059\pi$
$774$	0	0
$775$	−1.17311e13	−1.16811
$776$	0	0
$777$	−9.98850e12	−0.983118
$778$	0	0
$779$	−6.15672e12	−0.599005
$780$	0	0
$781$	−6.33278e12	−0.609066
$782$	0	0
$783$	2.02930e12	0.192939
$784$	0	0
$785$	−1.82009e13	−1.71072
$786$	0	0
$787$	−1.77687e12	−0.165109	−0.0825543	−	0.996587i	$-0.526308\pi$
$787$	−1.77687e12	−0.165109	−0.0825543	+	0.996587i	$0.526308\pi$
$788$	0	0
$789$	2.50945e11	0.0230533
$790$	0	0
$791$	−5.32545e12	−0.483684
$792$	0	0
$793$	−4.74315e11	−0.0425929
$794$	0	0
$795$	2.28692e13	2.03048
$796$	0	0
$797$	9.53311e12	0.836898	0.418449	−	0.908240i	$-0.362574\pi$
$797$	9.53311e12	0.836898	0.418449	+	0.908240i	$0.362574\pi$
$798$	0	0
$799$	−2.16566e13	−1.87988
$800$	0	0
$801$	−4.18045e12	−0.358820
$802$	0	0
$803$	−1.30988e12	−0.111176
$804$	0	0
$805$	−1.09689e13	−0.920620
$806$	0	0
$807$	−7.22782e12	−0.599897
$808$	0	0
$809$	−4.71178e12	−0.386738	−0.193369	−	0.981126i	$-0.561941\pi$
$809$	−4.71178e12	−0.386738	−0.193369	+	0.981126i	$0.561941\pi$
$810$	0	0
$811$	9.33679e12	0.757885	0.378943	−	0.925420i	$-0.376288\pi$
$811$	9.33679e12	0.757885	0.378943	+	0.925420i	$0.376288\pi$
$812$	0	0
$813$	−1.07854e13	−0.865818
$814$	0	0
$815$	−2.38015e12	−0.188971
$816$	0	0
$817$	−4.95729e12	−0.389265
$818$	0	0
$819$	2.16911e11	0.0168463
$820$	0	0
$821$	−9.62688e11	−0.0739505	−0.0369753	−	0.999316i	$-0.511772\pi$
$821$	−9.62688e11	−0.0739505	−0.0369753	+	0.999316i	$0.511772\pi$
$822$	0	0
$823$	−2.35265e12	−0.178755	−0.0893774	−	0.995998i	$-0.528488\pi$
$823$	−2.35265e12	−0.178755	−0.0893774	+	0.995998i	$0.528488\pi$
$824$	0	0
$825$	−6.52228e12	−0.490181
$826$	0	0
$827$	−2.49308e13	−1.85337	−0.926685	−	0.375838i	$-0.877355\pi$
$827$	−2.49308e13	−1.85337	−0.926685	+	0.375838i	$0.877355\pi$
$828$	0	0
$829$	−4.92371e12	−0.362073	−0.181037	−	0.983476i	$-0.557945\pi$
$829$	−4.92371e12	−0.362073	−0.181037	+	0.983476i	$0.557945\pi$
$830$	0	0
$831$	1.10531e13	0.804043
$832$	0	0
$833$	−4.25872e13	−3.06462
$834$	0	0
$835$	−4.13605e13	−2.94440
$836$	0	0
$837$	1.37002e12	0.0964858
$838$	0	0
$839$	−1.58161e13	−1.10197	−0.550987	−	0.834514i	$-0.685748\pi$
$839$	−1.58161e13	−1.10197	−0.550987	+	0.834514i	$0.685748\pi$
$840$	0	0
$841$	7.37525e10	0.00508388
$842$	0	0
$843$	−4.55910e12	−0.310924
$844$	0	0
$845$	2.70229e13	1.82338
$846$	0	0
$847$	2.34845e13	1.56785
$848$	0	0
$849$	1.05436e12	0.0696473
$850$	0	0
$851$	4.02117e12	0.262827
$852$	0	0
$853$	−2.37094e13	−1.53338	−0.766690	−	0.642017i	$-0.778098\pi$
$853$	−2.37094e13	−1.53338	−0.766690	+	0.642017i	$0.778098\pi$
$854$	0	0
$855$	6.84663e12	0.438156
$856$	0	0
$857$	1.88590e13	1.19427	0.597137	−	0.802139i	$-0.296305\pi$
$857$	1.88590e13	1.19427	0.597137	+	0.802139i	$0.296305\pi$
$858$	0	0
$859$	−2.73506e13	−1.71395	−0.856974	−	0.515360i	$-0.827658\pi$
$859$	−2.73506e13	−1.71395	−0.856974	+	0.515360i	$0.827658\pi$
$860$	0	0
$861$	1.39968e13	0.867992
$862$	0	0
$863$	5.40502e12	0.331703	0.165851	−	0.986151i	$-0.446963\pi$
$863$	5.40502e12	0.331703	0.165851	+	0.986151i	$0.446963\pi$
$864$	0	0
$865$	−1.73080e13	−1.05117
$866$	0	0
$867$	−7.92367e12	−0.476256
$868$	0	0
$869$	−1.62361e12	−0.0965810
$870$	0	0
$871$	−7.74065e11	−0.0455718
$872$	0	0
$873$	−3.36164e11	−0.0195879
$874$	0	0
$875$	7.60766e13	4.38748
$876$	0	0
$877$	5.41805e12	0.309275	0.154638	−	0.987971i	$-0.450579\pi$
$877$	5.41805e12	0.309275	0.154638	+	0.987971i	$0.450579\pi$
$878$	0	0
$879$	−1.66932e13	−0.943171
$880$	0	0
$881$	−7.92850e12	−0.443404	−0.221702	−	0.975114i	$-0.571161\pi$
$881$	−7.92850e12	−0.443404	−0.221702	+	0.975114i	$0.571161\pi$
$882$	0	0
$883$	2.68918e13	1.48866	0.744331	−	0.667811i	$-0.232769\pi$
$883$	2.68918e13	1.48866	0.744331	+	0.667811i	$0.232769\pi$
$884$	0	0
$885$	1.89905e13	1.04062
$886$	0	0
$887$	2.82440e13	1.53204	0.766020	−	0.642817i	$-0.222234\pi$
$887$	2.82440e13	1.53204	0.766020	+	0.642817i	$0.222234\pi$
$888$	0	0
$889$	−3.67692e13	−1.97436
$890$	0	0
$891$	7.61706e11	0.0404890
$892$	0	0
$893$	−1.90492e13	−1.00241
$894$	0	0
$895$	−1.54196e13	−0.803286
$896$	0	0
$897$	−8.73242e10	−0.00450369
$898$	0	0
$899$	9.84386e12	0.502628
$900$	0	0
$901$	−5.15021e13	−2.60354
$902$	0	0
$903$	1.12700e13	0.564067
$904$	0	0
$905$	−4.42877e13	−2.19465
$906$	0	0
$907$	−1.28245e13	−0.629226	−0.314613	−	0.949220i	$-0.601875\pi$
$907$	−1.28245e13	−0.629226	−0.314613	+	0.949220i	$0.601875\pi$
$908$	0	0
$909$	−2.15579e12	−0.104729
$910$	0	0
$911$	−9.36953e12	−0.450697	−0.225349	−	0.974278i	$-0.572352\pi$
$911$	−9.36953e12	−0.450697	−0.225349	+	0.974278i	$0.572352\pi$
$912$	0	0
$913$	3.29908e12	0.157135
$914$	0	0
$915$	3.40363e13	1.60527
$916$	0	0
$917$	−4.87481e13	−2.27665
$918$	0	0
$919$	−9.04580e12	−0.418338	−0.209169	−	0.977880i	$-0.567076\pi$
$919$	−9.04580e12	−0.418338	−0.209169	+	0.977880i	$0.567076\pi$
$920$	0	0
$921$	5.08734e12	0.232982
$922$	0	0
$923$	1.03024e12	0.0467229
$924$	0	0
$925$	−4.88608e13	−2.19443
$926$	0	0
$927$	−1.40283e13	−0.623945
$928$	0	0
$929$	1.59777e13	0.703789	0.351895	−	0.936040i	$-0.385538\pi$
$929$	1.59777e13	0.703789	0.351895	+	0.936040i	$0.385538\pi$
$930$	0	0
$931$	−3.74598e13	−1.63415
$932$	0	0
$933$	−2.31161e12	−0.0998729
$934$	0	0
$935$	2.09927e13	0.898287
$936$	0	0
$937$	−5.20573e12	−0.220624	−0.110312	−	0.993897i	$-0.535185\pi$
$937$	−5.20573e12	−0.220624	−0.110312	+	0.993897i	$0.535185\pi$
$938$	0	0
$939$	−8.39800e11	−0.0352517
$940$	0	0
$941$	−1.35801e13	−0.564612	−0.282306	−	0.959324i	$-0.591099\pi$
$941$	−1.35801e13	−0.564612	−0.282306	+	0.959324i	$0.591099\pi$
$942$	0	0
$943$	−5.63485e12	−0.232049
$944$	0	0
$945$	−1.55653e13	−0.634913
$946$	0	0
$947$	1.11344e13	0.449874	0.224937	−	0.974373i	$-0.427782\pi$
$947$	1.11344e13	0.449874	0.224937	+	0.974373i	$0.427782\pi$
$948$	0	0
$949$	2.13095e11	0.00852854
$950$	0	0
$951$	−8.93297e12	−0.354147
$952$	0	0
$953$	3.70642e13	1.45558	0.727791	−	0.685799i	$-0.240547\pi$
$953$	3.70642e13	1.45558	0.727791	+	0.685799i	$0.240547\pi$
$954$	0	0
$955$	3.97585e13	1.54673
$956$	0	0
$957$	5.47299e12	0.210921
$958$	0	0
$959$	2.15893e13	0.824242
$960$	0	0
$961$	−1.97938e13	−0.748643
$962$	0	0
$963$	−9.05773e12	−0.339392
$964$	0	0
$965$	6.58744e13	2.44537
$966$	0	0
$967$	−2.56420e13	−0.943047	−0.471524	−	0.881853i	$-0.656296\pi$
$967$	−2.56420e13	−0.943047	−0.471524	+	0.881853i	$0.656296\pi$
$968$	0	0
$969$	−1.54188e13	−0.561815
$970$	0	0
$971$	4.47900e13	1.61694	0.808471	−	0.588536i	$-0.200296\pi$
$971$	4.47900e13	1.61694	0.808471	+	0.588536i	$0.200296\pi$
$972$	0	0
$973$	−4.58313e13	−1.63928
$974$	0	0
$975$	1.06107e12	0.0376029
$976$	0	0
$977$	−4.70660e12	−0.165265	−0.0826326	−	0.996580i	$-0.526333\pi$
$977$	−4.70660e12	−0.165265	−0.0826326	+	0.996580i	$0.526333\pi$
$978$	0	0
$979$	−1.12746e13	−0.392263
$980$	0	0
$981$	−9.32036e12	−0.321309
$982$	0	0
$983$	4.45816e12	0.152288	0.0761438	−	0.997097i	$-0.475739\pi$
$983$	4.45816e12	0.152288	0.0761438	+	0.997097i	$0.475739\pi$
$984$	0	0
$985$	7.88099e13	2.66758
$986$	0	0
$987$	4.33069e13	1.45255
$988$	0	0
$989$	−4.53710e12	−0.150798
$990$	0	0
$991$	−1.02988e13	−0.339201	−0.169600	−	0.985513i	$-0.554248\pi$
$991$	−1.02988e13	−0.339201	−0.169600	+	0.985513i	$0.554248\pi$
$992$	0	0
$993$	−1.88036e13	−0.613720
$994$	0	0
$995$	3.27893e13	1.06054
$996$	0	0
$997$	4.40551e13	1.41211	0.706054	−	0.708158i	$-0.250474\pi$
$997$	4.40551e13	1.41211	0.706054	+	0.708158i	$0.250474\pi$
$998$	0	0
$999$	5.70622e12	0.181261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.a.b.1.1	✓	4
4.3	odd	2		384.10.a.f.1.1	yes	4
8.3	odd	2		384.10.a.c.1.4	yes	4
8.5	even	2		384.10.a.g.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.a.b.1.1	✓	4	1.1	even	1	trivial
384.10.a.c.1.4	yes	4	8.3	odd	2
384.10.a.f.1.1	yes	4	4.3	odd	2
384.10.a.g.1.4	yes	4	8.5	even	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q-81.0000 q^{3} -2550.24 q^{5} -11484.8 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q-81.0000 q^{3} -2550.24 q^{5} -11484.8 q^{7} +6561.00 q^{9} +17694.9 q^{11} -2878.66 q^{13} +206569. q^{15} -465200. q^{17} -409191. q^{19} +930265. q^{21} -374507. q^{23} +4.55058e6 q^{25} -531441. q^{27} -3.81849e6 q^{29} -2.57794e6 q^{31} -1.43328e6 q^{33} +2.92889e7 q^{35} -1.07373e7 q^{37} +233171. q^{39} +1.50461e7 q^{41} +1.21149e7 q^{43} -1.67321e7 q^{45} +4.65533e7 q^{47} +9.15461e7 q^{49} +3.76812e7 q^{51} +1.10710e8 q^{53} -4.51261e7 q^{55} +3.31445e7 q^{57} +9.19329e7 q^{59} +1.64770e8 q^{61} -7.53515e7 q^{63} +7.34126e6 q^{65} +2.68898e8 q^{67} +3.03350e7 q^{69} -3.57888e8 q^{71} -7.40257e7 q^{73} -3.68597e8 q^{75} -2.03221e8 q^{77} -9.17558e7 q^{79} +4.30467e7 q^{81} +1.86443e8 q^{83} +1.18637e9 q^{85} +3.09298e8 q^{87} -6.37166e8 q^{89} +3.30607e7 q^{91} +2.08813e8 q^{93} +1.04353e9 q^{95} -5.12367e7 q^{97} +1.16096e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 324 q^{3} - 240 q^{5} - 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) 4 * q - 324 * q^3 - 240 * q^5 - 4840 * q^7 + 26244 * q^9	\( 4 q - 324 q^{3} - 240 q^{5} - 4840 q^{7} + 26244 q^{9} + 99664 q^{11} - 60840 q^{13} + 19440 q^{15} - 434952 q^{17} - 631776 q^{19} + 392040 q^{21} - 749392 q^{23} + 5991532 q^{25} - 2125764 q^{27} - 7908544 q^{29} - 11351240 q^{31} - 8072784 q^{33} + 25567008 q^{35} - 13592920 q^{37} + 4928040 q^{39} - 18838888 q^{41} - 14177920 q^{43} - 1574640 q^{45} + 37779120 q^{47} + 9409332 q^{49} + 35231112 q^{51} + 115336512 q^{53} + 184580544 q^{55} + 51173856 q^{57} + 115028080 q^{59} + 173228648 q^{61} - 31755240 q^{63} - 328077984 q^{65} + 231785104 q^{67} + 60700752 q^{69} - 197476208 q^{71} + 44629400 q^{73} - 485314092 q^{75} - 308117920 q^{77} + 355774584 q^{79} + 172186884 q^{81} + 607613328 q^{83} + 1087351392 q^{85} + 640592064 q^{87} - 1157146424 q^{89} + 847629840 q^{91} + 919450440 q^{93} + 329699328 q^{95} - 1599536472 q^{97} + 653895504 q^{99}+O(q^{100}) \) 4 * q - 324 * q^3 - 240 * q^5 - 4840 * q^7 + 26244 * q^9 + 99664 * q^11 - 60840 * q^13 + 19440 * q^15 - 434952 * q^17 - 631776 * q^19 + 392040 * q^21 - 749392 * q^23 + 5991532 * q^25 - 2125764 * q^27 - 7908544 * q^29 - 11351240 * q^31 - 8072784 * q^33 + 25567008 * q^35 - 13592920 * q^37 + 4928040 * q^39 - 18838888 * q^41 - 14177920 * q^43 - 1574640 * q^45 + 37779120 * q^47 + 9409332 * q^49 + 35231112 * q^51 + 115336512 * q^53 + 184580544 * q^55 + 51173856 * q^57 + 115028080 * q^59 + 173228648 * q^61 - 31755240 * q^63 - 328077984 * q^65 + 231785104 * q^67 + 60700752 * q^69 - 197476208 * q^71 + 44629400 * q^73 - 485314092 * q^75 - 308117920 * q^77 + 355774584 * q^79 + 172186884 * q^81 + 607613328 * q^83 + 1087351392 * q^85 + 640592064 * q^87 - 1157146424 * q^89 + 847629840 * q^91 + 919450440 * q^93 + 329699328 * q^95 - 1599536472 * q^97 + 653895504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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