LMFDB - Embedded newform 234.10.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [234,10,Mod(1,234)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("234.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 234 = 2 \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	234.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-48,0,768,1272] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$120.518385662$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2119705.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 376x + 1820 $ x^3 - x^2 - 376*x + 1820
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.12938$ of defining polynomial
Character	$\chi$	$=$	234.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-16.0000 q^{2} +256.000 q^{4} +716.175 q^{5} +6927.79 q^{7} -4096.00 q^{8} -11458.8 q^{10} +64797.9 q^{11} -28561.0 q^{13} -110845. q^{14} +65536.0 q^{16} +52936.2 q^{17} +811860. q^{19} +183341. q^{20} -1.03677e6 q^{22} +1.17347e6 q^{23} -1.44022e6 q^{25} +456976. q^{26} +1.77351e6 q^{28} +3.27842e6 q^{29} +7.47282e6 q^{31} -1.04858e6 q^{32} -846979. q^{34} +4.96151e6 q^{35} -1.26693e6 q^{37} -1.29898e7 q^{38} -2.93345e6 q^{40} +2.30350e7 q^{41} -2.89799e7 q^{43} +1.65883e7 q^{44} -1.87755e7 q^{46} -9.91630e6 q^{47} +7.64069e6 q^{49} +2.30435e7 q^{50} -7.31162e6 q^{52} +1.08085e8 q^{53} +4.64066e7 q^{55} -2.83762e7 q^{56} -5.24548e7 q^{58} -8.73536e7 q^{59} +1.08586e8 q^{61} -1.19565e8 q^{62} +1.67772e7 q^{64} -2.04547e7 q^{65} -2.48447e8 q^{67} +1.35517e7 q^{68} -7.93842e7 q^{70} -2.90630e8 q^{71} -3.58455e8 q^{73} +2.02709e7 q^{74} +2.07836e8 q^{76} +4.48906e8 q^{77} +4.06821e8 q^{79} +4.69352e7 q^{80} -3.68561e8 q^{82} -4.70848e8 q^{83} +3.79116e7 q^{85} +4.63678e8 q^{86} -2.65412e8 q^{88} -3.65116e8 q^{89} -1.97865e8 q^{91} +3.00409e8 q^{92} +1.58661e8 q^{94} +5.81433e8 q^{95} +8.96766e8 q^{97} -1.22251e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 48 q^{2} + 768 q^{4} + 1272 q^{5} + 17058 q^{7} - 12288 q^{8} - 20352 q^{10} - 73974 q^{11} - 85683 q^{13} - 272928 q^{14} + 196608 q^{16} - 374976 q^{17} + 418338 q^{19} + 325632 q^{20} + 1183584 q^{22}+ \cdots - 734738832 q^{98}+O(q^{100}) $ 3 * q - 48 * q^2 + 768 * q^4 + 1272 * q^5 + 17058 * q^7 - 12288 * q^8 - 20352 * q^10 - 73974 * q^11 - 85683 * q^13 - 272928 * q^14 + 196608 * q^16 - 374976 * q^17 + 418338 * q^19 + 325632 * q^20 + 1183584 * q^22 - 1026168 * q^23 + 3337287 * q^25 + 1370928 * q^26 + 4366848 * q^28 + 3075834 * q^29 + 9286482 * q^31 - 3145728 * q^32 + 5999616 * q^34 + 31802592 * q^35 - 17647776 * q^37 - 6693408 * q^38 - 5210112 * q^40 + 47257110 * q^41 - 60023760 * q^43 - 18937344 * q^44 + 16418688 * q^46 + 40824726 * q^47 + 45921177 * q^49 - 53396592 * q^50 - 21934848 * q^52 - 8072046 * q^53 - 36910236 * q^55 - 69869568 * q^56 - 49213344 * q^58 - 138035310 * q^59 + 264203886 * q^61 - 148583712 * q^62 + 50331648 * q^64 - 36329592 * q^65 + 203167074 * q^67 - 95993856 * q^68 - 508841472 * q^70 - 123067110 * q^71 - 433013250 * q^73 + 282364416 * q^74 + 107094528 * q^76 - 380043264 * q^77 + 406418748 * q^79 + 83361792 * q^80 - 756113760 * q^82 - 105365610 * q^83 + 5268186 * q^85 + 960380160 * q^86 + 302997504 * q^88 - 1365375798 * q^89 - 487193538 * q^91 - 262699008 * q^92 - 653195616 * q^94 + 85600692 * q^95 + 669691662 * q^97 - 734738832 * q^98

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 $

$2$	−16.0000	−0.707107
$2$	−16.0000	−0.707107
$3$	0	0
$3$	0	0
$4$	256.000	0.500000
$5$	716.175	0.512453	0.256227	−	0.966617i	$-0.417521\pi$
$5$	716.175	0.512453	0.256227	+	0.966617i	$0.417521\pi$
$6$	0	0
$7$	6927.79	1.09057	0.545285	−	0.838251i	$-0.316421\pi$
$7$	6927.79	1.09057	0.545285	+	0.838251i	$0.316421\pi$
$8$	−4096.00	−0.353553
$9$	0	0
$10$	−11458.8	−0.362359
$11$	64797.9	1.33442	0.667212	−	0.744868i	$-0.267488\pi$
$11$	64797.9	1.33442	0.667212	+	0.744868i	$0.267488\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	−110845.	−0.771150
$15$	0	0
$16$	65536.0	0.250000
$17$	52936.2	0.153721	0.0768604	−	0.997042i	$-0.475510\pi$
$17$	52936.2	0.153721	0.0768604	+	0.997042i	$0.475510\pi$
$18$	0	0
$19$	811860.	1.42919	0.714595	−	0.699539i	$-0.246611\pi$
$19$	811860.	1.42919	0.714595	+	0.699539i	$0.246611\pi$
$20$	183341.	0.256227
$21$	0	0
$22$	−1.03677e6	−0.943580
$23$	1.17347e6	0.874374	0.437187	−	0.899371i	$-0.355975\pi$
$23$	1.17347e6	0.874374	0.437187	+	0.899371i	$0.355975\pi$
$24$	0	0
$25$	−1.44022e6	−0.737392
$26$	456976.	0.196116
$27$	0	0
$28$	1.77351e6	0.545285
$29$	3.27842e6	0.860744	0.430372	−	0.902652i	$-0.358382\pi$
$29$	3.27842e6	0.860744	0.430372	+	0.902652i	$0.358382\pi$
$30$	0	0
$31$	7.47282e6	1.45331	0.726653	−	0.687005i	$-0.241075\pi$
$31$	7.47282e6	1.45331	0.726653	+	0.687005i	$0.241075\pi$
$32$	−1.04858e6	−0.176777
$33$	0	0
$34$	−846979.	−0.108697
$35$	4.96151e6	0.558866
$36$	0	0
$37$	−1.26693e6	−0.111134	−0.0555668	−	0.998455i	$-0.517697\pi$
$37$	−1.26693e6	−0.111134	−0.0555668	+	0.998455i	$0.517697\pi$
$38$	−1.29898e7	−1.01059
$39$	0	0
$40$	−2.93345e6	−0.181180
$41$	2.30350e7	1.27310	0.636548	−	0.771237i	$-0.280362\pi$
$41$	2.30350e7	1.27310	0.636548	+	0.771237i	$0.280362\pi$
$42$	0	0
$43$	−2.89799e7	−1.29267	−0.646336	−	0.763053i	$-0.723699\pi$
$43$	−2.89799e7	−1.29267	−0.646336	+	0.763053i	$0.723699\pi$
$44$	1.65883e7	0.667212
$45$	0	0
$46$	−1.87755e7	−0.618276
$47$	−9.91630e6	−0.296421	−0.148211	−	0.988956i	$-0.547351\pi$
$47$	−9.91630e6	−0.296421	−0.148211	+	0.988956i	$0.547351\pi$
$48$	0	0
$49$	7.64069e6	0.189343
$50$	2.30435e7	0.521415
$51$	0	0
$52$	−7.31162e6	−0.138675
$53$	1.08085e8	1.88158	0.940791	−	0.338986i	$-0.110084\pi$
$53$	1.08085e8	1.88158	0.940791	+	0.338986i	$0.110084\pi$
$54$	0	0
$55$	4.64066e7	0.683829
$56$	−2.83762e7	−0.385575
$57$	0	0
$58$	−5.24548e7	−0.608638
$59$	−8.73536e7	−0.938527	−0.469263	−	0.883058i	$-0.655480\pi$
$59$	−8.73536e7	−0.938527	−0.469263	+	0.883058i	$0.655480\pi$
$60$	0	0
$61$	1.08586e8	1.00413	0.502065	−	0.864830i	$-0.332574\pi$
$61$	1.08586e8	1.00413	0.502065	+	0.864830i	$0.332574\pi$
$62$	−1.19565e8	−1.02764
$63$	0	0
$64$	1.67772e7	0.125000
$65$	−2.04547e7	−0.142129
$66$	0	0
$67$	−2.48447e8	−1.50625	−0.753124	−	0.657878i	$-0.771454\pi$
$67$	−2.48447e8	−1.50625	−0.753124	+	0.657878i	$0.771454\pi$
$68$	1.35517e7	0.0768604
$69$	0	0
$70$	−7.93842e7	−0.395178
$71$	−2.90630e8	−1.35731	−0.678653	−	0.734459i	$-0.737436\pi$
$71$	−2.90630e8	−1.35731	−0.678653	+	0.734459i	$0.737436\pi$
$72$	0	0
$73$	−3.58455e8	−1.47734	−0.738672	−	0.674065i	$-0.764547\pi$
$73$	−3.58455e8	−1.47734	−0.738672	+	0.674065i	$0.764547\pi$
$74$	2.02709e7	0.0785834
$75$	0	0
$76$	2.07836e8	0.714595
$77$	4.48906e8	1.45528
$78$	0	0
$79$	4.06821e8	1.17512	0.587559	−	0.809181i	$-0.300089\pi$
$79$	4.06821e8	1.17512	0.587559	+	0.809181i	$0.300089\pi$
$80$	4.69352e7	0.128113
$81$	0	0
$82$	−3.68561e8	−0.900215
$83$	−4.70848e8	−1.08900	−0.544502	−	0.838759i	$-0.683281\pi$
$83$	−4.70848e8	−1.08900	−0.544502	+	0.838759i	$0.683281\pi$
$84$	0	0
$85$	3.79116e7	0.0787747
$86$	4.63678e8	0.914057
$87$	0	0
$88$	−2.65412e8	−0.471790
$89$	−3.65116e8	−0.616845	−0.308422	−	0.951249i	$-0.599801\pi$
$89$	−3.65116e8	−0.616845	−0.308422	+	0.951249i	$0.599801\pi$
$90$	0	0
$91$	−1.97865e8	−0.302470
$92$	3.00409e8	0.437187
$93$	0	0
$94$	1.58661e8	0.209601
$95$	5.81433e8	0.732392
$96$	0	0
$97$	8.96766e8	1.02850	0.514252	−	0.857639i	$-0.328069\pi$
$97$	8.96766e8	1.02850	0.514252	+	0.857639i	$0.328069\pi$
$98$	−1.22251e8	−0.133886
$99$	0	0
$100$	−3.68696e8	−0.368696
$101$	1.62603e8	0.155482	0.0777412	−	0.996974i	$-0.475229\pi$
$101$	1.62603e8	0.155482	0.0777412	+	0.996974i	$0.475229\pi$
$102$	0	0
$103$	−2.32078e8	−0.203173	−0.101586	−	0.994827i	$-0.532392\pi$
$103$	−2.32078e8	−0.203173	−0.101586	+	0.994827i	$0.532392\pi$
$104$	1.16986e8	0.0980581
$105$	0	0
$106$	−1.72936e9	−1.33048
$107$	−1.84672e9	−1.36199	−0.680994	−	0.732289i	$-0.738452\pi$
$107$	−1.84672e9	−1.36199	−0.680994	+	0.732289i	$0.738452\pi$
$108$	0	0
$109$	1.10794e9	0.751791	0.375896	−	0.926662i	$-0.377335\pi$
$109$	1.10794e9	0.751791	0.375896	+	0.926662i	$0.377335\pi$
$110$	−7.42506e8	−0.483540
$111$	0	0
$112$	4.54020e8	0.272643
$113$	−2.55084e9	−1.47174	−0.735868	−	0.677125i	$-0.763226\pi$
$113$	−2.55084e9	−1.47174	−0.735868	+	0.677125i	$0.763226\pi$
$114$	0	0
$115$	8.40411e8	0.448076
$116$	8.39276e8	0.430372
$117$	0	0
$118$	1.39766e9	0.663638
$119$	3.66731e8	0.167643
$120$	0	0
$121$	1.84082e9	0.780686
$122$	−1.73738e9	−0.710027
$123$	0	0
$124$	1.91304e9	0.726653
$125$	−2.43023e9	−0.890332
$126$	0	0
$127$	−1.39169e9	−0.474706	−0.237353	−	0.971423i	$-0.576280\pi$
$127$	−1.39169e9	−0.474706	−0.237353	+	0.971423i	$0.576280\pi$
$128$	−2.68435e8	−0.0883883
$129$	0	0
$130$	3.27275e8	0.100500
$131$	4.17723e8	0.123927	0.0619637	−	0.998078i	$-0.480264\pi$
$131$	4.17723e8	0.123927	0.0619637	+	0.998078i	$0.480264\pi$
$132$	0	0
$133$	5.62439e9	1.55863
$134$	3.97515e9	1.06508
$135$	0	0
$136$	−2.16827e8	−0.0543485
$137$	3.17262e9	0.769441	0.384721	−	0.923033i	$-0.374298\pi$
$137$	3.17262e9	0.769441	0.384721	+	0.923033i	$0.374298\pi$
$138$	0	0
$139$	7.49108e8	0.170207	0.0851036	−	0.996372i	$-0.472878\pi$
$139$	7.49108e8	0.170207	0.0851036	+	0.996372i	$0.472878\pi$
$140$	1.27015e9	0.279433
$141$	0	0
$142$	4.65008e9	0.959761
$143$	−1.85069e9	−0.370103
$144$	0	0
$145$	2.34792e9	0.441091
$146$	5.73527e9	1.04464
$147$	0	0
$148$	−3.24335e8	−0.0555668
$149$	3.94098e9	0.655038	0.327519	−	0.944845i	$-0.393787\pi$
$149$	3.94098e9	0.655038	0.327519	+	0.944845i	$0.393787\pi$
$150$	0	0
$151$	8.63414e9	1.35152	0.675760	−	0.737121i	$-0.263815\pi$
$151$	8.63414e9	1.35152	0.675760	+	0.737121i	$0.263815\pi$
$152$	−3.32538e9	−0.505295
$153$	0	0
$154$	−7.18250e9	−1.02904
$155$	5.35185e9	0.744751
$156$	0	0
$157$	1.08229e10	1.42166	0.710829	−	0.703365i	$-0.248320\pi$
$157$	1.08229e10	1.42166	0.710829	+	0.703365i	$0.248320\pi$
$158$	−6.50914e9	−0.830934
$159$	0	0
$160$	−7.50964e8	−0.0905898
$161$	8.12956e9	0.953566
$162$	0	0
$163$	−5.62787e8	−0.0624453	−0.0312226	−	0.999512i	$-0.509940\pi$
$163$	−5.62787e8	−0.0624453	−0.0312226	+	0.999512i	$0.509940\pi$
$164$	5.89697e9	0.636548
$165$	0	0
$166$	7.53358e9	0.770043
$167$	−7.18451e9	−0.714781	−0.357390	−	0.933955i	$-0.616333\pi$
$167$	−7.18451e9	−0.714781	−0.357390	+	0.933955i	$0.616333\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	−6.06585e8	−0.0557021
$171$	0	0
$172$	−7.41884e9	−0.646336
$173$	7.72366e9	0.655565	0.327783	−	0.944753i	$-0.393699\pi$
$173$	7.72366e9	0.655565	0.327783	+	0.944753i	$0.393699\pi$
$174$	0	0
$175$	−9.97753e9	−0.804178
$176$	4.24659e9	0.333606
$177$	0	0
$178$	5.84186e9	0.436175
$179$	2.20856e10	1.60794	0.803972	−	0.594667i	$-0.202716\pi$
$179$	2.20856e10	1.60794	0.803972	+	0.594667i	$0.202716\pi$
$180$	0	0
$181$	−8.63359e8	−0.0597913	−0.0298956	−	0.999553i	$-0.509517\pi$
$181$	−8.63359e8	−0.0597913	−0.0298956	+	0.999553i	$0.509517\pi$
$182$	3.16583e9	0.213878
$183$	0	0
$184$	−4.80654e9	−0.309138
$185$	−9.07345e8	−0.0569508
$186$	0	0
$187$	3.43015e9	0.205129
$188$	−2.53857e9	−0.148211
$189$	0	0
$190$	−9.30293e9	−0.517880
$191$	6.46347e9	0.351411	0.175706	−	0.984443i	$-0.443779\pi$
$191$	6.46347e9	0.351411	0.175706	+	0.984443i	$0.443779\pi$
$192$	0	0
$193$	−3.91408e9	−0.203059	−0.101529	−	0.994833i	$-0.532374\pi$
$193$	−3.91408e9	−0.203059	−0.101529	+	0.994833i	$0.532374\pi$
$194$	−1.43483e10	−0.727263
$195$	0	0
$196$	1.95602e9	0.0946717
$197$	1.33095e10	0.629600	0.314800	−	0.949158i	$-0.398062\pi$
$197$	1.33095e10	0.629600	0.314800	+	0.949158i	$0.398062\pi$
$198$	0	0
$199$	4.22698e10	1.91069	0.955347	−	0.295488i	$-0.0954821\pi$
$199$	4.22698e10	1.91069	0.955347	+	0.295488i	$0.0954821\pi$
$200$	5.89914e9	0.260707
$201$	0	0
$202$	−2.60164e9	−0.109943
$203$	2.27122e10	0.938702
$204$	0	0
$205$	1.64971e10	0.652402
$206$	3.71324e9	0.143665
$207$	0	0
$208$	−1.87177e9	−0.0693375
$209$	5.26068e10	1.90714
$210$	0	0
$211$	−4.36607e10	−1.51642	−0.758211	−	0.652010i	$-0.773926\pi$
$211$	−4.36607e10	−1.51642	−0.758211	+	0.652010i	$0.773926\pi$
$212$	2.76697e10	0.940791
$213$	0	0
$214$	2.95475e10	0.963071
$215$	−2.07546e10	−0.662434
$216$	0	0
$217$	5.17702e10	1.58493
$218$	−1.77271e10	−0.531597
$219$	0	0
$220$	1.18801e10	0.341915
$221$	−1.51191e9	−0.0426345
$222$	0	0
$223$	3.96069e10	1.07250	0.536252	−	0.844058i	$-0.319840\pi$
$223$	3.96069e10	1.07250	0.536252	+	0.844058i	$0.319840\pi$
$224$	−7.26432e9	−0.192787
$225$	0	0
$226$	4.08134e10	1.04067
$227$	5.88274e9	0.147049	0.0735247	−	0.997293i	$-0.476575\pi$
$227$	5.88274e9	0.147049	0.0735247	+	0.997293i	$0.476575\pi$
$228$	0	0
$229$	2.02060e10	0.485536	0.242768	−	0.970084i	$-0.421945\pi$
$229$	2.02060e10	0.485536	0.242768	+	0.970084i	$0.421945\pi$
$230$	−1.34466e10	−0.316837
$231$	0	0
$232$	−1.34284e10	−0.304319
$233$	−5.28051e10	−1.17375	−0.586873	−	0.809679i	$-0.699641\pi$
$233$	−5.28051e10	−1.17375	−0.586873	+	0.809679i	$0.699641\pi$
$234$	0	0
$235$	−7.10181e9	−0.151902
$236$	−2.23625e10	−0.469263
$237$	0	0
$238$	−5.86769e9	−0.118542
$239$	1.98291e10	0.393109	0.196555	−	0.980493i	$-0.437025\pi$
$239$	1.98291e10	0.393109	0.196555	+	0.980493i	$0.437025\pi$
$240$	0	0
$241$	−9.13883e10	−1.74507	−0.872537	−	0.488548i	$-0.837527\pi$
$241$	−9.13883e10	−1.74507	−0.872537	+	0.488548i	$0.837527\pi$
$242$	−2.94531e10	−0.552029
$243$	0	0
$244$	2.77980e10	0.502065
$245$	5.47207e9	0.0970296
$246$	0	0
$247$	−2.31875e10	−0.396386
$248$	−3.06087e10	−0.513821
$249$	0	0
$250$	3.88836e10	0.629560
$251$	4.66320e9	0.0741570	0.0370785	−	0.999312i	$-0.488195\pi$
$251$	4.66320e9	0.0741570	0.0370785	+	0.999312i	$0.488195\pi$
$252$	0	0
$253$	7.60384e10	1.16679
$254$	2.22670e10	0.335668
$255$	0	0
$256$	4.29497e9	0.0625000
$257$	1.20163e11	1.71819	0.859094	−	0.511817i	$-0.171028\pi$
$257$	1.20163e11	1.71819	0.859094	+	0.511817i	$0.171028\pi$
$258$	0	0
$259$	−8.77704e9	−0.121199
$260$	−5.23640e9	−0.0710644
$261$	0	0
$262$	−6.68356e9	−0.0876299
$263$	4.25793e9	0.0548779	0.0274389	−	0.999623i	$-0.491265\pi$
$263$	4.25793e9	0.0548779	0.0274389	+	0.999623i	$0.491265\pi$
$264$	0	0
$265$	7.74076e10	0.964223
$266$	−8.99903e10	−1.10212
$267$	0	0
$268$	−6.36023e10	−0.753124
$269$	−1.29999e11	−1.51375	−0.756874	−	0.653561i	$-0.773274\pi$
$269$	−1.29999e11	−1.51375	−0.756874	+	0.653561i	$0.773274\pi$
$270$	0	0
$271$	9.56827e10	1.07763	0.538817	−	0.842423i	$-0.318871\pi$
$271$	9.56827e10	1.07763	0.538817	+	0.842423i	$0.318871\pi$
$272$	3.46923e9	0.0384302
$273$	0	0
$274$	−5.07619e10	−0.544077
$275$	−9.33231e10	−0.983993
$276$	0	0
$277$	−7.37939e10	−0.753115	−0.376558	−	0.926393i	$-0.622892\pi$
$277$	−7.37939e10	−0.753115	−0.376558	+	0.926393i	$0.622892\pi$
$278$	−1.19857e10	−0.120355
$279$	0	0
$280$	−2.03223e10	−0.197589
$281$	−4.75391e9	−0.0454854	−0.0227427	−	0.999741i	$-0.507240\pi$
$281$	−4.75391e9	−0.0454854	−0.0227427	+	0.999741i	$0.507240\pi$
$282$	0	0
$283$	1.13785e11	1.05450	0.527251	−	0.849710i	$-0.323223\pi$
$283$	1.13785e11	1.05450	0.527251	+	0.849710i	$0.323223\pi$
$284$	−7.44013e10	−0.678653
$285$	0	0
$286$	2.96111e10	0.261702
$287$	1.59582e11	1.38840
$288$	0	0
$289$	−1.15786e11	−0.976370
$290$	−3.75668e10	−0.311898
$291$	0	0
$292$	−9.17644e10	−0.738672
$293$	−6.88493e10	−0.545752	−0.272876	−	0.962049i	$-0.587975\pi$
$293$	−6.88493e10	−0.545752	−0.272876	+	0.962049i	$0.587975\pi$
$294$	0	0
$295$	−6.25604e10	−0.480951
$296$	5.18935e9	0.0392917
$297$	0	0
$298$	−6.30558e10	−0.463182
$299$	−3.35155e10	−0.242508
$300$	0	0
$301$	−2.00766e11	−1.40975
$302$	−1.38146e11	−0.955669
$303$	0	0
$304$	5.32060e10	0.357297
$305$	7.77666e10	0.514569
$306$	0	0
$307$	−1.19876e11	−0.770209	−0.385105	−	0.922873i	$-0.625835\pi$
$307$	−1.19876e11	−0.770209	−0.385105	+	0.922873i	$0.625835\pi$
$308$	1.14920e11	0.727641
$309$	0	0
$310$	−8.56296e10	−0.526619
$311$	1.92486e11	1.16675	0.583375	−	0.812203i	$-0.301732\pi$
$311$	1.92486e11	1.16675	0.583375	+	0.812203i	$0.301732\pi$
$312$	0	0
$313$	−8.85346e10	−0.521391	−0.260695	−	0.965421i	$-0.583952\pi$
$313$	−8.85346e10	−0.521391	−0.260695	+	0.965421i	$0.583952\pi$
$314$	−1.73166e11	−1.00526
$315$	0	0
$316$	1.04146e11	0.587559
$317$	−9.22613e10	−0.513160	−0.256580	−	0.966523i	$-0.582596\pi$
$317$	−9.22613e10	−0.513160	−0.256580	+	0.966523i	$0.582596\pi$
$318$	0	0
$319$	2.12435e11	1.14860
$320$	1.20154e10	0.0640566
$321$	0	0
$322$	−1.30073e11	−0.674273
$323$	4.29767e10	0.219696
$324$	0	0
$325$	4.11341e10	0.204516
$326$	9.00458e9	0.0441555
$327$	0	0
$328$	−9.43515e10	−0.450108
$329$	−6.86981e10	−0.323268
$330$	0	0
$331$	1.73234e11	0.793246	0.396623	−	0.917981i	$-0.370182\pi$
$331$	1.73234e11	0.793246	0.396623	+	0.917981i	$0.370182\pi$
$332$	−1.20537e11	−0.544502
$333$	0	0
$334$	1.14952e11	0.505426
$335$	−1.77931e11	−0.771881
$336$	0	0
$337$	5.97509e10	0.252354	0.126177	−	0.992008i	$-0.459729\pi$
$337$	5.97509e10	0.252354	0.126177	+	0.992008i	$0.459729\pi$
$338$	−1.30517e10	−0.0543928
$339$	0	0
$340$	9.70536e9	0.0393873
$341$	4.84223e11	1.93933
$342$	0	0
$343$	−2.26628e11	−0.884078
$344$	1.18702e11	0.457028
$345$	0	0
$346$	−1.23579e11	−0.463555
$347$	1.05729e11	0.391481	0.195740	−	0.980656i	$-0.437289\pi$
$347$	1.05729e11	0.391481	0.195740	+	0.980656i	$0.437289\pi$
$348$	0	0
$349$	−3.04810e11	−1.09980	−0.549901	−	0.835230i	$-0.685335\pi$
$349$	−3.04810e11	−1.09980	−0.549901	+	0.835230i	$0.685335\pi$
$350$	1.59641e11	0.568639
$351$	0	0
$352$	−6.79455e10	−0.235895
$353$	−1.32310e11	−0.453530	−0.226765	−	0.973950i	$-0.572815\pi$
$353$	−1.32310e11	−0.453530	−0.226765	+	0.973950i	$0.572815\pi$
$354$	0	0
$355$	−2.08142e11	−0.695556
$356$	−9.34697e10	−0.308422
$357$	0	0
$358$	−3.53370e11	−1.13699
$359$	−3.72672e11	−1.18414	−0.592069	−	0.805887i	$-0.701689\pi$
$359$	−3.72672e11	−1.18414	−0.592069	+	0.805887i	$0.701689\pi$
$360$	0	0
$361$	3.36428e11	1.04258
$362$	1.38137e10	0.0422788
$363$	0	0
$364$	−5.06534e10	−0.151235
$365$	−2.56716e11	−0.757069
$366$	0	0
$367$	−4.80734e11	−1.38327	−0.691635	−	0.722247i	$-0.743109\pi$
$367$	−4.80734e11	−1.38327	−0.691635	+	0.722247i	$0.743109\pi$
$368$	7.69046e10	0.218593
$369$	0	0
$370$	1.45175e10	0.0402703
$371$	7.48789e11	2.05200
$372$	0	0
$373$	−6.45327e10	−0.172620	−0.0863098	−	0.996268i	$-0.527507\pi$
$373$	−6.45327e10	−0.172620	−0.0863098	+	0.996268i	$0.527507\pi$
$374$	−5.48824e10	−0.145048
$375$	0	0
$376$	4.06172e10	0.104801
$377$	−9.36350e10	−0.238727
$378$	0	0
$379$	−4.64545e11	−1.15652	−0.578258	−	0.815854i	$-0.696267\pi$
$379$	−4.64545e11	−1.15652	−0.578258	+	0.815854i	$0.696267\pi$
$380$	1.48847e11	0.366196
$381$	0	0
$382$	−1.03416e11	−0.248485
$383$	3.32737e11	0.790145	0.395073	−	0.918650i	$-0.370719\pi$
$383$	3.32737e11	0.790145	0.395073	+	0.918650i	$0.370719\pi$
$384$	0	0
$385$	3.21495e11	0.745764
$386$	6.26253e10	0.143584
$387$	0	0
$388$	2.29572e11	0.514252
$389$	−5.73386e11	−1.26962	−0.634810	−	0.772668i	$-0.718922\pi$
$389$	−5.73386e11	−1.26962	−0.634810	+	0.772668i	$0.718922\pi$
$390$	0	0
$391$	6.21191e10	0.134409
$392$	−3.12963e10	−0.0669430
$393$	0	0
$394$	−2.12953e11	−0.445195
$395$	2.91355e11	0.602193
$396$	0	0
$397$	5.39945e11	1.09092	0.545459	−	0.838137i	$-0.316355\pi$
$397$	5.39945e11	1.09092	0.545459	+	0.838137i	$0.316355\pi$
$398$	−6.76316e11	−1.35106
$399$	0	0
$400$	−9.43862e10	−0.184348
$401$	3.41860e11	0.660236	0.330118	−	0.943940i	$-0.392912\pi$
$401$	3.41860e11	0.660236	0.330118	+	0.943940i	$0.392912\pi$
$402$	0	0
$403$	−2.13431e11	−0.403075
$404$	4.16263e10	0.0777412
$405$	0	0
$406$	−3.63396e11	−0.663762
$407$	−8.20945e10	−0.148299
$408$	0	0
$409$	4.81409e11	0.850666	0.425333	−	0.905037i	$-0.360157\pi$
$409$	4.81409e11	0.850666	0.425333	+	0.905037i	$0.360157\pi$
$410$	−2.63954e11	−0.461318
$411$	0	0
$412$	−5.94119e10	−0.101586
$413$	−6.05167e11	−1.02353
$414$	0	0
$415$	−3.37210e11	−0.558064
$416$	2.99484e10	0.0490290
$417$	0	0
$418$	−8.41708e11	−1.34855
$419$	−4.42064e11	−0.700683	−0.350342	−	0.936622i	$-0.613935\pi$
$419$	−4.42064e11	−0.700683	−0.350342	+	0.936622i	$0.613935\pi$
$420$	0	0
$421$	−4.81550e11	−0.747088	−0.373544	−	0.927613i	$-0.621857\pi$
$421$	−4.81550e11	−0.747088	−0.373544	+	0.927613i	$0.621857\pi$
$422$	6.98572e11	1.07227
$423$	0	0
$424$	−4.42715e11	−0.665240
$425$	−7.62397e10	−0.113352
$426$	0	0
$427$	7.52262e11	1.09507
$428$	−4.72760e11	−0.680994
$429$	0	0
$430$	3.32074e11	0.468411
$431$	1.16295e11	0.162335	0.0811676	−	0.996700i	$-0.474135\pi$
$431$	1.16295e11	0.162335	0.0811676	+	0.996700i	$0.474135\pi$
$432$	0	0
$433$	8.57306e11	1.17203	0.586017	−	0.810298i	$-0.300695\pi$
$433$	8.57306e11	1.17203	0.586017	+	0.810298i	$0.300695\pi$
$434$	−8.28323e11	−1.12072
$435$	0	0
$436$	2.83633e11	0.375896
$437$	9.52694e11	1.24965
$438$	0	0
$439$	1.03627e12	1.33163	0.665814	−	0.746118i	$-0.268084\pi$
$439$	1.03627e12	1.33163	0.665814	+	0.746118i	$0.268084\pi$
$440$	−1.90081e11	−0.241770
$441$	0	0
$442$	2.41906e10	0.0301471
$443$	1.13736e12	1.40307	0.701535	−	0.712635i	$-0.252498\pi$
$443$	1.13736e12	1.40307	0.701535	+	0.712635i	$0.252498\pi$
$444$	0	0
$445$	−2.61487e11	−0.316104
$446$	−6.33710e11	−0.758375
$447$	0	0
$448$	1.16229e11	0.136321
$449$	4.47904e11	0.520088	0.260044	−	0.965597i	$-0.416263\pi$
$449$	4.47904e11	0.520088	0.260044	+	0.965597i	$0.416263\pi$
$450$	0	0
$451$	1.49262e12	1.69885
$452$	−6.53015e11	−0.735868
$453$	0	0
$454$	−9.41238e10	−0.103980
$455$	−1.41706e11	−0.155002
$456$	0	0
$457$	1.48228e12	1.58968	0.794838	−	0.606822i	$-0.207556\pi$
$457$	1.48228e12	1.58968	0.794838	+	0.606822i	$0.207556\pi$
$458$	−3.23297e11	−0.343326
$459$	0	0
$460$	2.15145e11	0.224038
$461$	−1.10053e11	−0.113487	−0.0567435	−	0.998389i	$-0.518072\pi$
$461$	−1.10053e11	−0.113487	−0.0567435	+	0.998389i	$0.518072\pi$
$462$	0	0
$463$	−1.90086e11	−0.192236	−0.0961181	−	0.995370i	$-0.530643\pi$
$463$	−1.90086e11	−0.192236	−0.0961181	+	0.995370i	$0.530643\pi$
$464$	2.14855e11	0.215186
$465$	0	0
$466$	8.44882e11	0.829964
$467$	8.85267e11	0.861288	0.430644	−	0.902522i	$-0.358286\pi$
$467$	8.85267e11	0.861288	0.430644	+	0.902522i	$0.358286\pi$
$468$	0	0
$469$	−1.72119e12	−1.64267
$470$	1.13629e11	0.107411
$471$	0	0
$472$	3.57800e11	0.331819
$473$	−1.87783e12	−1.72497
$474$	0	0
$475$	−1.16926e12	−1.05387
$476$	9.38831e10	0.0838216
$477$	0	0
$478$	−3.17266e11	−0.277970
$479$	−9.62538e10	−0.0835426	−0.0417713	−	0.999127i	$-0.513300\pi$
$479$	−9.62538e10	−0.0835426	−0.0417713	+	0.999127i	$0.513300\pi$
$480$	0	0
$481$	3.61848e10	0.0308229
$482$	1.46221e12	1.23395
$483$	0	0
$484$	4.71249e11	0.390343
$485$	6.42241e11	0.527060
$486$	0	0
$487$	−1.05656e12	−0.851163	−0.425581	−	0.904920i	$-0.639930\pi$
$487$	−1.05656e12	−0.851163	−0.425581	+	0.904920i	$0.639930\pi$
$488$	−4.44769e11	−0.355014
$489$	0	0
$490$	−8.75531e10	−0.0686103
$491$	3.08706e11	0.239706	0.119853	−	0.992792i	$-0.461758\pi$
$491$	3.08706e11	0.239706	0.119853	+	0.992792i	$0.461758\pi$
$492$	0	0
$493$	1.73547e11	0.132314
$494$	3.71000e11	0.280287
$495$	0	0
$496$	4.89739e11	0.363327
$497$	−2.01342e12	−1.48024
$498$	0	0
$499$	−2.72504e11	−0.196752	−0.0983762	−	0.995149i	$-0.531365\pi$
$499$	−2.72504e11	−0.196752	−0.0983762	+	0.995149i	$0.531365\pi$
$500$	−6.22138e11	−0.445166
$501$	0	0
$502$	−7.46112e10	−0.0524369
$503$	1.67621e12	1.16754	0.583769	−	0.811920i	$-0.301577\pi$
$503$	1.67621e12	1.16754	0.583769	+	0.811920i	$0.301577\pi$
$504$	0	0
$505$	1.16452e11	0.0796774
$506$	−1.21662e12	−0.825042
$507$	0	0
$508$	−3.56272e11	−0.237353
$509$	1.08327e11	0.0715331	0.0357666	−	0.999360i	$-0.488613\pi$
$509$	1.08327e11	0.0715331	0.0357666	+	0.999360i	$0.488613\pi$
$510$	0	0
$511$	−2.48330e12	−1.61115
$512$	−6.87195e10	−0.0441942
$513$	0	0
$514$	−1.92260e12	−1.21494
$515$	−1.66208e11	−0.104117
$516$	0	0
$517$	−6.42555e11	−0.395552
$518$	1.40433e11	0.0857007
$519$	0	0
$520$	8.37823e10	0.0502502
$521$	6.57658e11	0.391048	0.195524	−	0.980699i	$-0.437359\pi$
$521$	6.57658e11	0.391048	0.195524	+	0.980699i	$0.437359\pi$
$522$	0	0
$523$	2.28923e11	0.133793	0.0668964	−	0.997760i	$-0.478690\pi$
$523$	2.28923e11	0.133793	0.0668964	+	0.997760i	$0.478690\pi$
$524$	1.06937e11	0.0619637
$525$	0	0
$526$	−6.81268e10	−0.0388045
$527$	3.95583e11	0.223403
$528$	0	0
$529$	−4.24118e11	−0.235470
$530$	−1.23852e12	−0.681808
$531$	0	0
$532$	1.43984e12	0.779316
$533$	−6.57904e11	−0.353094
$534$	0	0
$535$	−1.32257e12	−0.697955
$536$	1.01764e12	0.532539
$537$	0	0
$538$	2.07998e12	1.07038
$539$	4.95101e11	0.252664
$540$	0	0
$541$	1.87557e12	0.941336	0.470668	−	0.882310i	$-0.344013\pi$
$541$	1.87557e12	0.941336	0.470668	+	0.882310i	$0.344013\pi$
$542$	−1.53092e12	−0.762003
$543$	0	0
$544$	−5.55076e10	−0.0271742
$545$	7.93479e11	0.385258
$546$	0	0
$547$	−2.73493e12	−1.30618	−0.653089	−	0.757281i	$-0.726527\pi$
$547$	−2.73493e12	−1.30618	−0.653089	+	0.757281i	$0.726527\pi$
$548$	8.12191e11	0.384721
$549$	0	0
$550$	1.49317e12	0.695788
$551$	2.66162e12	1.23017
$552$	0	0
$553$	2.81837e12	1.28155
$554$	1.18070e12	0.532533
$555$	0	0
$556$	1.91772e11	0.0851036
$557$	2.61842e12	1.15263	0.576316	−	0.817227i	$-0.304490\pi$
$557$	2.61842e12	1.15263	0.576316	+	0.817227i	$0.304490\pi$
$558$	0	0
$559$	8.27694e11	0.358523
$560$	3.25158e11	0.139717
$561$	0	0
$562$	7.60625e10	0.0321630
$563$	3.12388e11	0.131041	0.0655203	−	0.997851i	$-0.479129\pi$
$563$	3.12388e11	0.131041	0.0655203	+	0.997851i	$0.479129\pi$
$564$	0	0
$565$	−1.82685e12	−0.754196
$566$	−1.82056e12	−0.745645
$567$	0	0
$568$	1.19042e12	0.479880
$569$	1.39773e12	0.559006	0.279503	−	0.960145i	$-0.409830\pi$
$569$	1.39773e12	0.559006	0.279503	+	0.960145i	$0.409830\pi$
$570$	0	0
$571$	2.54454e12	1.00172	0.500861	−	0.865527i	$-0.333017\pi$
$571$	2.54454e12	1.00172	0.500861	+	0.865527i	$0.333017\pi$
$572$	−4.73777e11	−0.185051
$573$	0	0
$574$	−2.55331e12	−0.981748
$575$	−1.69005e12	−0.644756
$576$	0	0
$577$	1.47191e12	0.552828	0.276414	−	0.961039i	$-0.410854\pi$
$577$	1.47191e12	0.552828	0.276414	+	0.961039i	$0.410854\pi$
$578$	1.85257e12	0.690398
$579$	0	0
$580$	6.01068e11	0.220545
$581$	−3.26194e12	−1.18764
$582$	0	0
$583$	7.00367e12	2.51083
$584$	1.46823e12	0.522320
$585$	0	0
$586$	1.10159e12	0.385905
$587$	2.93835e12	1.02149	0.510743	−	0.859733i	$-0.329370\pi$
$587$	2.93835e12	1.02149	0.510743	+	0.859733i	$0.329370\pi$
$588$	0	0
$589$	6.06688e12	2.07705
$590$	1.00097e12	0.340084
$591$	0	0
$592$	−8.30297e10	−0.0277834
$593$	−2.96605e12	−0.984991	−0.492495	−	0.870315i	$-0.663915\pi$
$593$	−2.96605e12	−0.984991	−0.492495	+	0.870315i	$0.663915\pi$
$594$	0	0
$595$	2.62643e11	0.0859093
$596$	1.00889e12	0.327519
$597$	0	0
$598$	5.36248e11	0.171479
$599$	−5.13235e11	−0.162890	−0.0814452	−	0.996678i	$-0.525954\pi$
$599$	−5.13235e11	−0.162890	−0.0814452	+	0.996678i	$0.525954\pi$
$600$	0	0
$601$	−1.16242e12	−0.363436	−0.181718	−	0.983351i	$-0.558166\pi$
$601$	−1.16242e12	−0.363436	−0.181718	+	0.983351i	$0.558166\pi$
$602$	3.21226e12	0.996843
$603$	0	0
$604$	2.21034e12	0.675760
$605$	1.31835e12	0.400065
$606$	0	0
$607$	−1.30452e12	−0.390034	−0.195017	−	0.980800i	$-0.562476\pi$
$607$	−1.30452e12	−0.390034	−0.195017	+	0.980800i	$0.562476\pi$
$608$	−8.51296e11	−0.252647
$609$	0	0
$610$	−1.24427e12	−0.363856
$611$	2.83220e11	0.0822125
$612$	0	0
$613$	1.51535e12	0.433453	0.216727	−	0.976232i	$-0.430462\pi$
$613$	1.51535e12	0.433453	0.216727	+	0.976232i	$0.430462\pi$
$614$	1.91801e12	0.544620
$615$	0	0
$616$	−1.83872e12	−0.514520
$617$	−4.34398e12	−1.20672	−0.603358	−	0.797471i	$-0.706171\pi$
$617$	−4.34398e12	−1.20672	−0.603358	+	0.797471i	$0.706171\pi$
$618$	0	0
$619$	−3.76078e12	−1.02960	−0.514802	−	0.857309i	$-0.672135\pi$
$619$	−3.76078e12	−1.02960	−0.514802	+	0.857309i	$0.672135\pi$
$620$	1.37007e12	0.372376
$621$	0	0
$622$	−3.07978e12	−0.825016
$623$	−2.52945e12	−0.672713
$624$	0	0
$625$	1.07246e12	0.281139
$626$	1.41655e12	0.368679
$627$	0	0
$628$	2.77066e12	0.710829
$629$	−6.70665e10	−0.0170835
$630$	0	0
$631$	4.80012e12	1.20537	0.602684	−	0.797980i	$-0.294098\pi$
$631$	4.80012e12	1.20537	0.602684	+	0.797980i	$0.294098\pi$
$632$	−1.66634e12	−0.415467
$633$	0	0
$634$	1.47618e12	0.362859
$635$	−9.96692e11	−0.243265
$636$	0	0
$637$	−2.18226e11	−0.0525144
$638$	−3.39896e12	−0.812181
$639$	0	0
$640$	−1.92247e11	−0.0452949
$641$	2.46922e12	0.577695	0.288848	−	0.957375i	$-0.406728\pi$
$641$	2.46922e12	0.577695	0.288848	+	0.957375i	$0.406728\pi$
$642$	0	0
$643$	−5.12941e12	−1.18336	−0.591682	−	0.806171i	$-0.701536\pi$
$643$	−5.12941e12	−1.18336	−0.591682	+	0.806171i	$0.701536\pi$
$644$	2.08117e12	0.476783
$645$	0	0
$646$	−6.87628e11	−0.155349
$647$	−1.17603e12	−0.263846	−0.131923	−	0.991260i	$-0.542115\pi$
$647$	−1.17603e12	−0.263846	−0.131923	+	0.991260i	$0.542115\pi$
$648$	0	0
$649$	−5.66033e12	−1.25239
$650$	−6.58145e11	−0.144614
$651$	0	0
$652$	−1.44073e11	−0.0312226
$653$	3.83054e12	0.824424	0.412212	−	0.911088i	$-0.364756\pi$
$653$	3.83054e12	0.824424	0.412212	+	0.911088i	$0.364756\pi$
$654$	0	0
$655$	2.99162e11	0.0635070
$656$	1.50962e12	0.318274
$657$	0	0
$658$	1.09917e12	0.228585
$659$	−7.92312e12	−1.63649	−0.818243	−	0.574873i	$-0.805051\pi$
$659$	−7.92312e12	−1.63649	−0.818243	+	0.574873i	$0.805051\pi$
$660$	0	0
$661$	8.08129e11	0.164655	0.0823273	−	0.996605i	$-0.473765\pi$
$661$	8.08129e11	0.164655	0.0823273	+	0.996605i	$0.473765\pi$
$662$	−2.77175e12	−0.560910
$663$	0	0
$664$	1.92860e12	0.385021
$665$	4.02805e12	0.798725
$666$	0	0
$667$	3.84713e12	0.752612
$668$	−1.83923e12	−0.357390
$669$	0	0
$670$	2.84690e12	0.545803
$671$	7.03615e12	1.33993
$672$	0	0
$673$	−2.18500e12	−0.410568	−0.205284	−	0.978702i	$-0.565812\pi$
$673$	−2.18500e12	−0.410568	−0.205284	+	0.978702i	$0.565812\pi$
$674$	−9.56014e11	−0.178441
$675$	0	0
$676$	2.08827e11	0.0384615
$677$	5.92825e12	1.08462	0.542310	−	0.840179i	$-0.317550\pi$
$677$	5.92825e12	1.08462	0.542310	+	0.840179i	$0.317550\pi$
$678$	0	0
$679$	6.21261e12	1.12166
$680$	−1.55286e11	−0.0278510
$681$	0	0
$682$	−7.74757e12	−1.37131
$683$	−9.91825e12	−1.74398	−0.871990	−	0.489523i	$-0.837171\pi$
$683$	−9.91825e12	−1.74398	−0.871990	+	0.489523i	$0.837171\pi$
$684$	0	0
$685$	2.27215e12	0.394303
$686$	3.62605e12	0.625137
$687$	0	0
$688$	−1.89922e12	−0.323168
$689$	−3.08701e12	−0.521857
$690$	0	0
$691$	−7.28392e12	−1.21539	−0.607693	−	0.794172i	$-0.707905\pi$
$691$	−7.28392e12	−1.21539	−0.607693	+	0.794172i	$0.707905\pi$
$692$	1.97726e12	0.327783
$693$	0	0
$694$	−1.69166e12	−0.276819
$695$	5.36493e11	0.0872232
$696$	0	0
$697$	1.21939e12	0.195701
$698$	4.87696e12	0.777678
$699$	0	0
$700$	−2.55425e12	−0.402089
$701$	−7.97899e12	−1.24801	−0.624003	−	0.781422i	$-0.714495\pi$
$701$	−7.97899e12	−1.24801	−0.624003	+	0.781422i	$0.714495\pi$
$702$	0	0
$703$	−1.02857e12	−0.158831
$704$	1.08713e12	0.166803
$705$	0	0
$706$	2.11696e12	0.320694
$707$	1.12648e12	0.169564
$708$	0	0
$709$	3.06000e12	0.454793	0.227396	−	0.973802i	$-0.426979\pi$
$709$	3.06000e12	0.454793	0.227396	+	0.973802i	$0.426979\pi$
$710$	3.33027e12	0.491832
$711$	0	0
$712$	1.49552e12	0.218088
$713$	8.76914e12	1.27073
$714$	0	0
$715$	−1.32542e12	−0.189660
$716$	5.65392e12	0.803972
$717$	0	0
$718$	5.96276e12	0.837312
$719$	−1.05167e13	−1.46758	−0.733789	−	0.679377i	$-0.762250\pi$
$719$	−1.05167e13	−1.46758	−0.733789	+	0.679377i	$0.762250\pi$
$720$	0	0
$721$	−1.60779e12	−0.221574
$722$	−5.38285e12	−0.737216
$723$	0	0
$724$	−2.21020e11	−0.0298956
$725$	−4.72164e12	−0.634705
$726$	0	0
$727$	−8.09614e12	−1.07491	−0.537456	−	0.843292i	$-0.680615\pi$
$727$	−8.09614e12	−1.07491	−0.537456	+	0.843292i	$0.680615\pi$
$728$	8.10454e11	0.106939
$729$	0	0
$730$	4.10746e12	0.535329
$731$	−1.53408e12	−0.198710
$732$	0	0
$733$	3.20844e12	0.410512	0.205256	−	0.978708i	$-0.434197\pi$
$733$	3.20844e12	0.410512	0.205256	+	0.978708i	$0.434197\pi$
$734$	7.69174e12	0.978120
$735$	0	0
$736$	−1.23047e12	−0.154569
$737$	−1.60988e13	−2.00997
$738$	0	0
$739$	−7.62960e12	−0.941026	−0.470513	−	0.882393i	$-0.655931\pi$
$739$	−7.62960e12	−0.941026	−0.470513	+	0.882393i	$0.655931\pi$
$740$	−2.32280e11	−0.0284754
$741$	0	0
$742$	−1.19806e13	−1.45098
$743$	1.41259e12	0.170046	0.0850229	−	0.996379i	$-0.472904\pi$
$743$	1.41259e12	0.170046	0.0850229	+	0.996379i	$0.472904\pi$
$744$	0	0
$745$	2.82243e12	0.335676
$746$	1.03252e12	0.122060
$747$	0	0
$748$	8.78119e11	0.102564
$749$	−1.27937e13	−1.48534
$750$	0	0
$751$	−1.49642e13	−1.71662	−0.858311	−	0.513130i	$-0.828486\pi$
$751$	−1.49642e13	−1.71662	−0.858311	+	0.513130i	$0.828486\pi$
$752$	−6.49875e11	−0.0741053
$753$	0	0
$754$	1.49816e12	0.168806
$755$	6.18355e12	0.692591
$756$	0	0
$757$	−1.41035e12	−0.156097	−0.0780487	−	0.996950i	$-0.524869\pi$
$757$	−1.41035e12	−0.156097	−0.0780487	+	0.996950i	$0.524869\pi$
$758$	7.43272e12	0.817780
$759$	0	0
$760$	−2.38155e12	−0.258940
$761$	−9.06064e12	−0.979327	−0.489664	−	0.871911i	$-0.662880\pi$
$761$	−9.06064e12	−0.979327	−0.489664	+	0.871911i	$0.662880\pi$
$762$	0	0
$763$	7.67558e12	0.819881
$764$	1.65465e12	0.175706
$765$	0	0
$766$	−5.32380e12	−0.558717
$767$	2.49490e12	0.260300
$768$	0	0
$769$	−6.70677e12	−0.691584	−0.345792	−	0.938311i	$-0.612390\pi$
$769$	−6.70677e12	−0.691584	−0.345792	+	0.938311i	$0.612390\pi$
$770$	−5.14393e12	−0.527335
$771$	0	0
$772$	−1.00201e12	−0.101529
$773$	1.17340e13	1.18206	0.591029	−	0.806650i	$-0.298722\pi$
$773$	1.17340e13	1.18206	0.591029	+	0.806650i	$0.298722\pi$
$774$	0	0
$775$	−1.07625e13	−1.07166
$776$	−3.67315e12	−0.363631
$777$	0	0
$778$	9.17417e12	0.897757
$779$	1.87012e13	1.81950
$780$	0	0
$781$	−1.88322e13	−1.81122
$782$	−9.93905e11	−0.0950418
$783$	0	0
$784$	5.00740e11	0.0473359
$785$	7.75109e12	0.728533
$786$	0	0
$787$	1.08434e12	0.100758	0.0503790	−	0.998730i	$-0.483957\pi$
$787$	1.08434e12	0.100758	0.0503790	+	0.998730i	$0.483957\pi$
$788$	3.40724e12	0.314800
$789$	0	0
$790$	−4.66168e12	−0.425815
$791$	−1.76717e13	−1.60503
$792$	0	0
$793$	−3.10133e12	−0.278496
$794$	−8.63912e12	−0.771396
$795$	0	0
$796$	1.08211e13	0.955347
$797$	−4.17303e12	−0.366344	−0.183172	−	0.983081i	$-0.558637\pi$
$797$	−4.17303e12	−0.366344	−0.183172	+	0.983081i	$0.558637\pi$
$798$	0	0
$799$	−5.24931e11	−0.0455661
$800$	1.51018e12	0.130354
$801$	0	0
$802$	−5.46976e12	−0.466857
$803$	−2.32271e13	−1.97140
$804$	0	0
$805$	5.82219e12	0.488658
$806$	3.41490e12	0.285017
$807$	0	0
$808$	−6.66020e11	−0.0549713
$809$	−2.03832e13	−1.67303	−0.836515	−	0.547944i	$-0.815411\pi$
$809$	−2.03832e13	−1.67303	−0.836515	+	0.547944i	$0.815411\pi$
$810$	0	0
$811$	3.12648e12	0.253782	0.126891	−	0.991917i	$-0.459500\pi$
$811$	3.12648e12	0.253782	0.126891	+	0.991917i	$0.459500\pi$
$812$	5.81433e12	0.469351
$813$	0	0
$814$	1.31351e12	0.104863
$815$	−4.03054e11	−0.0320003
$816$	0	0
$817$	−2.35276e13	−1.84747
$818$	−7.70254e12	−0.601512
$819$	0	0
$820$	4.22326e12	0.326201
$821$	−1.57719e12	−0.121155	−0.0605773	−	0.998164i	$-0.519294\pi$
$821$	−1.57719e12	−0.121155	−0.0605773	+	0.998164i	$0.519294\pi$
$822$	0	0
$823$	2.74085e12	0.208250	0.104125	−	0.994564i	$-0.466796\pi$
$823$	2.74085e12	0.208250	0.104125	+	0.994564i	$0.466796\pi$
$824$	9.50590e11	0.0718325
$825$	0	0
$826$	9.68268e12	0.723744
$827$	1.51185e13	1.12391	0.561957	−	0.827166i	$-0.310048\pi$
$827$	1.51185e13	1.12391	0.561957	+	0.827166i	$0.310048\pi$
$828$	0	0
$829$	−1.10159e13	−0.810070	−0.405035	−	0.914301i	$-0.632741\pi$
$829$	−1.10159e13	−0.810070	−0.405035	+	0.914301i	$0.632741\pi$
$830$	5.39536e12	0.394611
$831$	0	0
$832$	−4.79174e11	−0.0346688
$833$	4.04469e11	0.0291060
$834$	0	0
$835$	−5.14536e12	−0.366292
$836$	1.34673e13	0.953572
$837$	0	0
$838$	7.07302e12	0.495458
$839$	1.56657e13	1.09149	0.545747	−	0.837950i	$-0.316246\pi$
$839$	1.56657e13	1.09149	0.545747	+	0.837950i	$0.316246\pi$
$840$	0	0
$841$	−3.75909e12	−0.259120
$842$	7.70479e12	0.528271
$843$	0	0
$844$	−1.11771e13	−0.758211
$845$	5.84206e11	0.0394195
$846$	0	0
$847$	1.27528e13	0.851393
$848$	7.08345e12	0.470396
$849$	0	0
$850$	1.21983e12	0.0801523
$851$	−1.48671e12	−0.0971724
$852$	0	0
$853$	−9.67583e12	−0.625774	−0.312887	−	0.949790i	$-0.601296\pi$
$853$	−9.67583e12	−0.625774	−0.312887	+	0.949790i	$0.601296\pi$
$854$	−1.20362e13	−0.774334
$855$	0	0
$856$	7.56416e12	0.481536
$857$	−1.90306e13	−1.20514	−0.602571	−	0.798065i	$-0.705857\pi$
$857$	−1.90306e13	−1.20514	−0.602571	+	0.798065i	$0.705857\pi$
$858$	0	0
$859$	−1.52060e13	−0.952895	−0.476447	−	0.879203i	$-0.658076\pi$
$859$	−1.52060e13	−0.952895	−0.476447	+	0.879203i	$0.658076\pi$
$860$	−5.31319e12	−0.331217
$861$	0	0
$862$	−1.86072e12	−0.114788
$863$	9.61352e12	0.589975	0.294988	−	0.955501i	$-0.404685\pi$
$863$	9.61352e12	0.589975	0.294988	+	0.955501i	$0.404685\pi$
$864$	0	0
$865$	5.53149e12	0.335946
$866$	−1.37169e13	−0.828754
$867$	0	0
$868$	1.32532e13	0.792466
$869$	2.63611e13	1.56811
$870$	0	0
$871$	7.09589e12	0.417758
$872$	−4.53813e12	−0.265798
$873$	0	0
$874$	−1.52431e13	−0.883633
$875$	−1.68361e13	−0.970969
$876$	0	0
$877$	−1.61211e13	−0.920227	−0.460114	−	0.887860i	$-0.652191\pi$
$877$	−1.61211e13	−0.920227	−0.460114	+	0.887860i	$0.652191\pi$
$878$	−1.65803e13	−0.941603
$879$	0	0
$880$	3.04130e12	0.170957
$881$	−1.55403e13	−0.869099	−0.434549	−	0.900648i	$-0.643092\pi$
$881$	−1.55403e13	−0.869099	−0.434549	+	0.900648i	$0.643092\pi$
$882$	0	0
$883$	1.81773e13	1.00625	0.503124	−	0.864214i	$-0.332184\pi$
$883$	1.81773e13	1.00625	0.503124	+	0.864214i	$0.332184\pi$
$884$	−3.87049e11	−0.0213172
$885$	0	0
$886$	−1.81977e13	−0.992121
$887$	−2.45772e13	−1.33314	−0.666572	−	0.745441i	$-0.732239\pi$
$887$	−2.45772e13	−1.33314	−0.666572	+	0.745441i	$0.732239\pi$
$888$	0	0
$889$	−9.64132e12	−0.517701
$890$	4.18379e12	0.223519
$891$	0	0
$892$	1.01394e13	0.536252
$893$	−8.05064e12	−0.423642
$894$	0	0
$895$	1.58172e13	0.823996
$896$	−1.85966e12	−0.0963937
$897$	0	0
$898$	−7.16647e12	−0.367758
$899$	2.44991e13	1.25092
$900$	0	0
$901$	5.72160e12	0.289238
$902$	−2.38819e13	−1.20127
$903$	0	0
$904$	1.04482e13	0.520337
$905$	−6.18316e11	−0.0306402
$906$	0	0
$907$	2.50906e13	1.23106	0.615529	−	0.788115i	$-0.288943\pi$
$907$	2.50906e13	1.23106	0.615529	+	0.788115i	$0.288943\pi$
$908$	1.50598e12	0.0735247
$909$	0	0
$910$	2.26729e12	0.109603
$911$	1.70571e13	0.820490	0.410245	−	0.911975i	$-0.365443\pi$
$911$	1.70571e13	0.820490	0.410245	+	0.911975i	$0.365443\pi$
$912$	0	0
$913$	−3.05100e13	−1.45319
$914$	−2.37165e13	−1.12407
$915$	0	0
$916$	5.17275e12	0.242768
$917$	2.89390e12	0.135151
$918$	0	0
$919$	3.78241e13	1.74924	0.874620	−	0.484809i	$-0.161111\pi$
$919$	3.78241e13	1.74924	0.874620	+	0.484809i	$0.161111\pi$
$920$	−3.44232e12	−0.158419
$921$	0	0
$922$	1.76084e12	0.0802474
$923$	8.30069e12	0.376449
$924$	0	0
$925$	1.82466e12	0.0819491
$926$	3.04137e12	0.135932
$927$	0	0
$928$	−3.43767e12	−0.152159
$929$	2.88455e13	1.27059	0.635297	−	0.772268i	$-0.280878\pi$
$929$	2.88455e13	1.27059	0.635297	+	0.772268i	$0.280878\pi$
$930$	0	0
$931$	6.20317e12	0.270608
$932$	−1.35181e13	−0.586873
$933$	0	0
$934$	−1.41643e13	−0.609022
$935$	2.45659e12	0.105119
$936$	0	0
$937$	6.13062e12	0.259822	0.129911	−	0.991526i	$-0.458531\pi$
$937$	6.13062e12	0.259822	0.129911	+	0.991526i	$0.458531\pi$
$938$	2.75390e13	1.16154
$939$	0	0
$940$	−1.81806e12	−0.0759510
$941$	6.77593e12	0.281719	0.140859	−	0.990030i	$-0.455013\pi$
$941$	6.77593e12	0.281719	0.140859	+	0.990030i	$0.455013\pi$
$942$	0	0
$943$	2.70309e13	1.11316
$944$	−5.72480e12	−0.234632
$945$	0	0
$946$	3.00453e13	1.21974
$947$	2.51244e13	1.01513	0.507565	−	0.861614i	$-0.330546\pi$
$947$	2.51244e13	1.01513	0.507565	+	0.861614i	$0.330546\pi$
$948$	0	0
$949$	1.02378e13	0.409741
$950$	1.87081e13	0.745200
$951$	0	0
$952$	−1.50213e12	−0.0592708
$953$	−6.21828e12	−0.244204	−0.122102	−	0.992518i	$-0.538963\pi$
$953$	−6.21828e12	−0.244204	−0.122102	+	0.992518i	$0.538963\pi$
$954$	0	0
$955$	4.62898e12	0.180082
$956$	5.07626e12	0.196555
$957$	0	0
$958$	1.54006e12	0.0590735
$959$	2.19793e13	0.839130
$960$	0	0
$961$	2.94035e13	1.11210
$962$	−5.78958e11	−0.0217951
$963$	0	0
$964$	−2.33954e13	−0.872537
$965$	−2.80317e12	−0.104058
$966$	0	0
$967$	9.31427e12	0.342554	0.171277	−	0.985223i	$-0.445211\pi$
$967$	9.31427e12	0.342554	0.171277	+	0.985223i	$0.445211\pi$
$968$	−7.53999e12	−0.276014
$969$	0	0
$970$	−1.02759e13	−0.372688
$971$	−2.14377e13	−0.773912	−0.386956	−	0.922098i	$-0.626473\pi$
$971$	−2.14377e13	−0.773912	−0.386956	+	0.922098i	$0.626473\pi$
$972$	0	0
$973$	5.18967e12	0.185623
$974$	1.69049e13	0.601863
$975$	0	0
$976$	7.11630e12	0.251032
$977$	1.22818e13	0.431258	0.215629	−	0.976475i	$-0.430820\pi$
$977$	1.22818e13	0.431258	0.215629	+	0.976475i	$0.430820\pi$
$978$	0	0
$979$	−2.36587e13	−0.823132
$980$	1.40085e12	0.0485148
$981$	0	0
$982$	−4.93930e12	−0.169498
$983$	−5.75537e13	−1.96599	−0.982997	−	0.183622i	$-0.941218\pi$
$983$	−5.75537e13	−1.96599	−0.982997	+	0.183622i	$0.941218\pi$
$984$	0	0
$985$	9.53196e12	0.322641
$986$	−2.77675e12	−0.0935602
$987$	0	0
$988$	−5.93601e12	−0.198193
$989$	−3.40070e13	−1.13028
$990$	0	0
$991$	3.89816e13	1.28389	0.641945	−	0.766751i	$-0.278128\pi$
$991$	3.89816e13	1.28389	0.641945	+	0.766751i	$0.278128\pi$
$992$	−7.83582e12	−0.256911
$993$	0	0
$994$	3.22148e13	1.04669
$995$	3.02725e13	0.979141
$996$	0	0
$997$	−3.28468e13	−1.05285	−0.526423	−	0.850223i	$-0.676467\pi$
$997$	−3.28468e13	−1.05285	−0.526423	+	0.850223i	$0.676467\pi$
$998$	4.36006e12	0.139125
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	234.10.a.k.1.2		3
3.2	odd	2		26.10.a.e.1.3	✓	3
12.11	even	2		208.10.a.d.1.1		3
39.38	odd	2		338.10.a.e.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.10.a.e.1.3	✓	3	3.2	odd	2
208.10.a.d.1.1		3	12.11	even	2
234.10.a.k.1.2		3	1.1	even	1	trivial
338.10.a.e.1.3		3	39.38	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	234.a (trivial)

\(f(q)\)	\(=\)	\(q-16.0000 q^{2} +256.000 q^{4} +716.175 q^{5} +6927.79 q^{7} -4096.00 q^{8} -11458.8 q^{10} +64797.9 q^{11} -28561.0 q^{13} -110845. q^{14} +65536.0 q^{16} +52936.2 q^{17} +811860. q^{19} +183341. q^{20} -1.03677e6 q^{22} +1.17347e6 q^{23} -1.44022e6 q^{25} +456976. q^{26} +1.77351e6 q^{28} +3.27842e6 q^{29} +7.47282e6 q^{31} -1.04858e6 q^{32} -846979. q^{34} +4.96151e6 q^{35} -1.26693e6 q^{37} -1.29898e7 q^{38} -2.93345e6 q^{40} +2.30350e7 q^{41} -2.89799e7 q^{43} +1.65883e7 q^{44} -1.87755e7 q^{46} -9.91630e6 q^{47} +7.64069e6 q^{49} +2.30435e7 q^{50} -7.31162e6 q^{52} +1.08085e8 q^{53} +4.64066e7 q^{55} -2.83762e7 q^{56} -5.24548e7 q^{58} -8.73536e7 q^{59} +1.08586e8 q^{61} -1.19565e8 q^{62} +1.67772e7 q^{64} -2.04547e7 q^{65} -2.48447e8 q^{67} +1.35517e7 q^{68} -7.93842e7 q^{70} -2.90630e8 q^{71} -3.58455e8 q^{73} +2.02709e7 q^{74} +2.07836e8 q^{76} +4.48906e8 q^{77} +4.06821e8 q^{79} +4.69352e7 q^{80} -3.68561e8 q^{82} -4.70848e8 q^{83} +3.79116e7 q^{85} +4.63678e8 q^{86} -2.65412e8 q^{88} -3.65116e8 q^{89} -1.97865e8 q^{91} +3.00409e8 q^{92} +1.58661e8 q^{94} +5.81433e8 q^{95} +8.96766e8 q^{97} -1.22251e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 48 q^{2} + 768 q^{4} + 1272 q^{5} + 17058 q^{7} - 12288 q^{8} - 20352 q^{10} - 73974 q^{11} - 85683 q^{13} - 272928 q^{14} + 196608 q^{16} - 374976 q^{17} + 418338 q^{19} + 325632 q^{20} + 1183584 q^{22}+ \cdots - 734738832 q^{98}+O(q^{100}) \) 3 * q - 48 * q^2 + 768 * q^4 + 1272 * q^5 + 17058 * q^7 - 12288 * q^8 - 20352 * q^10 - 73974 * q^11 - 85683 * q^13 - 272928 * q^14 + 196608 * q^16 - 374976 * q^17 + 418338 * q^19 + 325632 * q^20 + 1183584 * q^22 - 1026168 * q^23 + 3337287 * q^25 + 1370928 * q^26 + 4366848 * q^28 + 3075834 * q^29 + 9286482 * q^31 - 3145728 * q^32 + 5999616 * q^34 + 31802592 * q^35 - 17647776 * q^37 - 6693408 * q^38 - 5210112 * q^40 + 47257110 * q^41 - 60023760 * q^43 - 18937344 * q^44 + 16418688 * q^46 + 40824726 * q^47 + 45921177 * q^49 - 53396592 * q^50 - 21934848 * q^52 - 8072046 * q^53 - 36910236 * q^55 - 69869568 * q^56 - 49213344 * q^58 - 138035310 * q^59 + 264203886 * q^61 - 148583712 * q^62 + 50331648 * q^64 - 36329592 * q^65 + 203167074 * q^67 - 95993856 * q^68 - 508841472 * q^70 - 123067110 * q^71 - 433013250 * q^73 + 282364416 * q^74 + 107094528 * q^76 - 380043264 * q^77 + 406418748 * q^79 + 83361792 * q^80 - 756113760 * q^82 - 105365610 * q^83 + 5268186 * q^85 + 960380160 * q^86 + 302997504 * q^88 - 1365375798 * q^89 - 487193538 * q^91 - 262699008 * q^92 - 653195616 * q^94 + 85600692 * q^95 + 669691662 * q^97 - 734738832 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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