LMFDB - Embedded newform 104.10.f.a.25.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [104,10,Mod(25,104)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("104.25"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 104 = 2^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	104.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$53.5637269610$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			25.16
Character	$\chi$	$=$	104.25
Dual form			104.10.f.a.25.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-33.7913 q^{3} -2480.68i q^{5} +3601.42i q^{7} -18541.1 q^{9} +12056.4i q^{11} +(8433.48 + 102632. i) q^{13} +83825.4i q^{15} -8068.71 q^{17} +851829. i q^{19} -121697. i q^{21} +1.55232e6 q^{23} -4.20066e6 q^{25} +1.29164e6 q^{27} -729125. q^{29} -7.56456e6i q^{31} -407403. i q^{33} +8.93399e6 q^{35} -1.66123e7i q^{37} +(-284978. - 3.46807e6i) q^{39} +1.25789e7i q^{41} +1.11091e7 q^{43} +4.59947e7i q^{45} +3.07308e7i q^{47} +2.73834e7 q^{49} +272652. q^{51} -1.42693e7 q^{53} +2.99082e7 q^{55} -2.87844e7i q^{57} +6.69658e7i q^{59} +1.93936e8 q^{61} -6.67745e7i q^{63} +(2.54598e8 - 2.09208e7i) q^{65} -1.24743e8i q^{67} -5.24549e7 q^{69} -3.04874e8i q^{71} +1.02678e8i q^{73} +1.41946e8 q^{75} -4.34203e7 q^{77} +3.87253e8 q^{79} +3.21299e8 q^{81} +5.61698e8i q^{83} +2.00159e7i q^{85} +2.46381e7 q^{87} +9.09198e8i q^{89} +(-3.69622e8 + 3.03725e7i) q^{91} +2.55616e8i q^{93} +2.11312e9 q^{95} -6.23976e8i q^{97} -2.23540e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} + 3171556 q^{23} - 13526722 q^{25} - 3694974 q^{27} + 8833508 q^{29} - 8281126 q^{35} - 12056860 q^{39} + 89959038 q^{43} - 172344874 q^{49}+ \cdots + 1741143356 q^{95}+O(q^{100}) $ 32 * q - 162 * q^3 + 223074 * q^9 + 66270 * q^13 - 487902 * q^17 + 3171556 * q^23 - 13526722 * q^25 - 3694974 * q^27 + 8833508 * q^29 - 8281126 * q^35 - 12056860 * q^39 + 89959038 * q^43 - 172344874 * q^49 + 51206562 * q^51 + 139521112 * q^53 - 84550176 * q^55 + 372036836 * q^61 + 241471810 * q^65 - 486052516 * q^69 - 604473728 * q^75 - 1666197144 * q^77 + 1165982916 * q^79 + 859636544 * q^81 - 1738691296 * q^87 - 291718146 * q^91 + 1741143356 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/104\mathbb{Z}\right)^\times$.

$n$	$41$	$53$	$79$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	−33.7913			−0.240857			−0.120428	−	0.992722i	$-0.538427\pi$
$3$	−33.7913			−0.240857			−0.120428	+	0.992722i	$0.538427\pi$
$4$		0			0
$5$		−	2480.68i		−	1.77503i	−0.460777	−	0.887516i	$-0.652429\pi$
$5$		−	2480.68i		−	1.77503i	0.460777	−	0.887516i	$-0.347571\pi$
$6$		0			0
$7$			3601.42i			0.566935i	0.958982	+	0.283467i	$0.0914847\pi$
$7$			3601.42i			0.566935i	−0.958982	+	0.283467i	$0.908515\pi$
$8$		0			0
$9$	−18541.1			−0.941988
$10$		0			0
$11$			12056.4i			0.248286i	0.992264	+	0.124143i	$0.0396181\pi$
$11$			12056.4i			0.248286i	−0.992264	+	0.124143i	$0.960382\pi$
$12$		0			0
$13$	8433.48	+	102632.i	0.0818958	+	0.996641i
$13$	8433.48	+	102632.i	0.0818958	+	0.996641i
$14$		0			0
$15$			83825.4i			0.427529i
$16$		0			0
$17$	−8068.71			−0.0234306			−0.0117153	−	0.999931i	$-0.503729\pi$
$17$	−8068.71			−0.0234306			−0.0117153	+	0.999931i	$0.503729\pi$
$18$		0			0
$19$			851829.i			1.49955i	0.661692	+	0.749776i	$0.269839\pi$
$19$			851829.i			1.49955i	−0.661692	+	0.749776i	$0.730161\pi$
$20$		0			0
$21$		−	121697.i		−	0.136550i
$22$		0			0
$23$	1.55232e6			1.15666			0.578331	−	0.815802i	$-0.303704\pi$
$23$	1.55232e6			1.15666			0.578331	+	0.815802i	$0.303704\pi$
$24$		0			0
$25$	−4.20066e6			−2.15074
$26$		0			0
$27$	1.29164e6			0.467741
$28$		0			0
$29$	−729125.			−0.191430			−0.0957152	−	0.995409i	$-0.530514\pi$
$29$	−729125.			−0.191430			−0.0957152	+	0.995409i	$0.530514\pi$
$30$		0			0
$31$		−	7.56456e6i		−	1.47115i	−0.677444	−	0.735574i	$-0.736913\pi$
$31$		−	7.56456e6i		−	1.47115i	0.677444	−	0.735574i	$-0.263087\pi$
$32$		0			0
$33$		−	407403.i		−	0.0598013i
$34$		0			0
$35$	8.93399e6			1.00633
$36$		0			0
$37$		−	1.66123e7i		−	1.45721i	−0.684935	−	0.728604i	$-0.740169\pi$
$37$		−	1.66123e7i		−	1.45721i	0.684935	−	0.728604i	$-0.259831\pi$
$38$		0			0
$39$	−284978.	−	3.46807e6i	−0.0197252	−	0.240048i
$40$		0			0
$41$			1.25789e7i			0.695208i	0.937641	+	0.347604i	$0.113005\pi$
$41$			1.25789e7i			0.695208i	−0.937641	+	0.347604i	$0.886995\pi$
$42$		0			0
$43$	1.11091e7			0.495529			0.247765	−	0.968820i	$-0.420304\pi$
$43$	1.11091e7			0.495529			0.247765	+	0.968820i	$0.420304\pi$
$44$		0			0
$45$			4.59947e7i			1.67206i
$46$		0			0
$47$			3.07308e7i			0.918615i	0.888277	+	0.459307i	$0.151902\pi$
$47$			3.07308e7i			0.918615i	−0.888277	+	0.459307i	$0.848098\pi$
$48$		0			0
$49$	2.73834e7			0.678585
$50$		0			0
$51$	272652.			0.00564343
$52$		0			0
$53$	−1.42693e7			−0.248405			−0.124203	−	0.992257i	$-0.539637\pi$
$53$	−1.42693e7			−0.248405			−0.124203	+	0.992257i	$0.539637\pi$
$54$		0			0
$55$	2.99082e7			0.440715
$56$		0			0
$57$		−	2.87844e7i		−	0.361177i
$58$		0			0
$59$			6.69658e7i			0.719481i	0.933052	+	0.359740i	$0.117135\pi$
$59$			6.69658e7i			0.719481i	−0.933052	+	0.359740i	$0.882865\pi$
$60$		0			0
$61$	1.93936e8			1.79339			0.896696	−	0.442648i	$-0.145961\pi$
$61$	1.93936e8			1.79339			0.896696	+	0.442648i	$0.145961\pi$
$62$		0			0
$63$		−	6.67745e7i		−	0.534046i
$64$		0			0
$65$	2.54598e8	−	2.09208e7i	1.76907	−	0.145368i
$66$		0			0
$67$		−	1.24743e8i		−	0.756273i	−0.925750	−	0.378136i	$-0.876565\pi$
$67$		−	1.24743e8i		−	0.756273i	0.925750	−	0.378136i	$-0.123435\pi$
$68$		0			0
$69$	−5.24549e7			−0.278590
$70$		0			0
$71$		−	3.04874e8i		−	1.42383i	−0.702265	−	0.711915i	$-0.747828\pi$
$71$		−	3.04874e8i		−	1.42383i	0.702265	−	0.711915i	$-0.252172\pi$
$72$		0			0
$73$			1.02678e8i			0.423180i	0.977358	+	0.211590i	$0.0678642\pi$
$73$			1.02678e8i			0.423180i	−0.977358	+	0.211590i	$0.932136\pi$
$74$		0			0
$75$	1.41946e8			0.518020
$76$		0			0
$77$	−4.34203e7			−0.140762
$78$		0			0
$79$	3.87253e8			1.11859			0.559297	−	0.828967i	$-0.311071\pi$
$79$	3.87253e8			1.11859			0.559297	+	0.828967i	$0.311071\pi$
$80$		0			0
$81$	3.21299e8			0.829329
$82$		0			0
$83$			5.61698e8i			1.29913i	0.760307	+	0.649563i	$0.225048\pi$
$83$			5.61698e8i			1.29913i	−0.760307	+	0.649563i	$0.774952\pi$
$84$		0			0
$85$			2.00159e7i			0.0415901i
$86$		0			0
$87$	2.46381e7			0.0461073
$88$		0			0
$89$			9.09198e8i			1.53604i	0.640424	+	0.768021i	$0.278759\pi$
$89$			9.09198e8i			1.53604i	−0.640424	+	0.768021i	$0.721241\pi$
$90$		0			0
$91$	−3.69622e8	+	3.03725e7i	−0.565030	+	0.0464296i
$92$		0			0
$93$			2.55616e8i			0.354336i
$94$		0			0
$95$	2.11312e9			2.66175
$96$		0			0
$97$		−	6.23976e8i		−	0.715640i	−0.933791	−	0.357820i	$-0.883520\pi$
$97$		−	6.23976e8i		−	0.715640i	0.933791	−	0.357820i	$-0.116480\pi$
$98$		0			0
$99$		−	2.23540e8i		−	0.233882i
$100$		0			0
$101$	6.69015e8			0.639719			0.319860	−	0.947465i	$-0.396364\pi$
$101$	6.69015e8			0.639719			0.319860	+	0.947465i	$0.396364\pi$
$102$		0			0
$103$	1.21287e9			1.06181			0.530905	−	0.847432i	$-0.321852\pi$
$103$	1.21287e9			1.06181			0.530905	+	0.847432i	$0.321852\pi$
$104$		0			0
$105$	−3.01891e8			−0.242381
$106$		0			0
$107$	−1.14055e9			−0.841176			−0.420588	−	0.907252i	$-0.638176\pi$
$107$	−1.14055e9			−0.841176			−0.420588	+	0.907252i	$0.638176\pi$
$108$		0			0
$109$			9.82861e7i			0.0666919i	0.999444	+	0.0333459i	$0.0106163\pi$
$109$			9.82861e7i			0.0666919i	−0.999444	+	0.0333459i	$0.989384\pi$
$110$		0			0
$111$			5.61351e8i			0.350979i
$112$		0			0
$113$	−2.48449e8			−0.143346			−0.0716728	−	0.997428i	$-0.522834\pi$
$113$	−2.48449e8			−0.143346			−0.0716728	+	0.997428i	$0.522834\pi$
$114$		0			0
$115$		−	3.85082e9i		−	2.05311i
$116$		0			0
$117$	−1.56366e8	−	1.90292e9i	−0.0771449	−	0.938824i
$118$		0			0
$119$		−	2.90588e7i		−	0.0132836i
$120$		0			0
$121$	2.21259e9			0.938354
$122$		0			0
$123$		−	4.25057e8i		−	0.167446i
$124$		0			0
$125$			5.57542e9i			2.04260i
$126$		0			0
$127$	1.57466e9			0.537118			0.268559	−	0.963263i	$-0.413453\pi$
$127$	1.57466e9			0.537118			0.268559	+	0.963263i	$0.413453\pi$
$128$		0			0
$129$	−3.75389e8			−0.119352
$130$		0			0
$131$	−3.33624e9			−0.989774			−0.494887	−	0.868957i	$-0.664790\pi$
$131$	−3.33624e9			−0.989774			−0.494887	+	0.868957i	$0.664790\pi$
$132$		0			0
$133$	−3.06780e9			−0.850147
$134$		0			0
$135$		−	3.20416e9i		−	0.830255i
$136$		0			0
$137$			5.69379e9i			1.38089i	0.723386	+	0.690444i	$0.242585\pi$
$137$			5.69379e9i			1.38089i	−0.723386	+	0.690444i	$0.757415\pi$
$138$		0			0
$139$	−1.81683e9			−0.412809			−0.206404	−	0.978467i	$-0.566176\pi$
$139$	−1.81683e9			−0.412809			−0.206404	+	0.978467i	$0.566176\pi$
$140$		0			0
$141$		−	1.03843e9i		−	0.221255i
$142$		0			0
$143$	−1.23738e9	+	1.01678e8i	−0.247452	+	0.0203336i
$144$		0			0
$145$			1.80873e9i			0.339795i
$146$		0			0
$147$	−9.25319e8			−0.163442
$148$		0			0
$149$			2.39994e9i			0.398898i	0.979908	+	0.199449i	$0.0639153\pi$
$149$			2.39994e9i			0.398898i	−0.979908	+	0.199449i	$0.936085\pi$
$150$		0			0
$151$		−	2.74361e9i		−	0.429463i	−0.976673	−	0.214731i	$-0.931112\pi$
$151$		−	2.74361e9i		−	0.429463i	0.976673	−	0.214731i	$-0.0688876\pi$
$152$		0			0
$153$	1.49603e8			0.0220714
$154$		0			0
$155$	−1.87653e10			−2.61133
$156$		0			0
$157$	−9.62221e9			−1.26394			−0.631970	−	0.774993i	$-0.717753\pi$
$157$	−9.62221e9			−1.26394			−0.631970	+	0.774993i	$0.717753\pi$
$158$		0			0
$159$	4.82177e8			0.0598301
$160$		0			0
$161$			5.59057e9i			0.655752i
$162$		0			0
$163$		−	9.96850e9i		−	1.10608i	−0.833156	−	0.553039i	$-0.813468\pi$
$163$		−	9.96850e9i		−	1.10608i	0.833156	−	0.553039i	$-0.186532\pi$
$164$		0			0
$165$	−1.01064e9			−0.106149
$166$		0			0
$167$			4.84870e9i			0.482393i	0.970476	+	0.241197i	$0.0775399\pi$
$167$			4.84870e9i			0.482393i	−0.970476	+	0.241197i	$0.922460\pi$
$168$		0			0
$169$	−1.04623e10	+	1.73109e9i	−0.986586	+	0.163241i
$170$		0			0
$171$		−	1.57939e10i		−	1.41256i
$172$		0			0
$173$	1.79702e10			1.52527			0.762635	−	0.646830i	$-0.223905\pi$
$173$	1.79702e10			1.52527			0.762635	+	0.646830i	$0.223905\pi$
$174$		0			0
$175$		−	1.51284e10i		−	1.21933i
$176$		0			0
$177$		−	2.26286e9i		−	0.173292i
$178$		0			0
$179$	2.31558e10			1.68586			0.842930	−	0.538023i	$-0.180829\pi$
$179$	2.31558e10			1.68586			0.842930	+	0.538023i	$0.180829\pi$
$180$		0			0
$181$	−1.46060e10			−1.01153			−0.505765	−	0.862671i	$-0.668790\pi$
$181$	−1.46060e10			−1.01153			−0.505765	+	0.862671i	$0.668790\pi$
$182$		0			0
$183$	−6.55336e9			−0.431951
$184$		0			0
$185$	−4.12098e10			−2.58659
$186$		0			0
$187$		−	9.72799e7i		−	0.00581749i
$188$		0			0
$189$			4.65175e9i			0.265179i
$190$		0			0
$191$	1.09778e10			0.596851			0.298425	−	0.954433i	$-0.403539\pi$
$191$	1.09778e10			0.596851			0.298425	+	0.954433i	$0.403539\pi$
$192$		0			0
$193$			1.91123e10i			0.991527i	0.868458	+	0.495763i	$0.165112\pi$
$193$			1.91123e10i			0.991527i	−0.868458	+	0.495763i	$0.834888\pi$
$194$		0			0
$195$	−8.60319e9	+	7.06940e8i	−0.426092	+	0.0350128i
$196$		0			0
$197$			2.16512e10i			1.02420i	0.858927	+	0.512098i	$0.171132\pi$
$197$			2.16512e10i			1.02420i	−0.858927	+	0.512098i	$0.828868\pi$
$198$		0			0
$199$	3.01847e10			1.36442			0.682210	−	0.731156i	$-0.261019\pi$
$199$	3.01847e10			1.36442			0.682210	+	0.731156i	$0.261019\pi$
$200$		0			0
$201$			4.21521e9i			0.182153i
$202$		0			0
$203$		−	2.62589e9i		−	0.108529i
$204$		0			0
$205$	3.12042e10			1.23402
$206$		0			0
$207$	−2.87818e10			−1.08956
$208$		0			0
$209$	−1.02700e10			−0.372317
$210$		0			0
$211$	−1.82448e10			−0.633676			−0.316838	−	0.948480i	$-0.602621\pi$
$211$	−1.82448e10			−0.633676			−0.316838	+	0.948480i	$0.602621\pi$
$212$		0			0
$213$			1.03021e10i			0.342939i
$214$		0			0
$215$		−	2.75580e10i		−	0.879580i
$216$		0			0
$217$	2.72432e10			0.834045
$218$		0			0
$219$		−	3.46963e9i		−	0.101926i
$220$		0			0
$221$	−6.80473e7	−	8.28110e8i	−0.00191887	−	0.0233519i
$222$		0			0
$223$			2.84507e10i			0.770409i	0.922831	+	0.385205i	$0.125869\pi$
$223$			2.84507e10i			0.770409i	−0.922831	+	0.385205i	$0.874131\pi$
$224$		0			0
$225$	7.78851e10			2.02597
$226$		0			0
$227$			5.14986e10i			1.28730i	0.765321	+	0.643649i	$0.222580\pi$
$227$			5.14986e10i			1.28730i	−0.765321	+	0.643649i	$0.777420\pi$
$228$		0			0
$229$		−	2.56379e10i		−	0.616061i	−0.951377	−	0.308030i	$-0.900330\pi$
$229$		−	2.56379e10i		−	0.616061i	0.951377	−	0.308030i	$-0.0996698\pi$
$230$		0			0
$231$	1.46723e9			0.0339035
$232$		0			0
$233$	2.07583e10			0.461414			0.230707	−	0.973023i	$-0.425896\pi$
$233$	2.07583e10			0.461414			0.230707	+	0.973023i	$0.425896\pi$
$234$		0			0
$235$	7.62334e10			1.63057
$236$		0			0
$237$	−1.30858e10			−0.269421
$238$		0			0
$239$		−	9.37885e10i		−	1.85934i	−0.368392	−	0.929671i	$-0.620091\pi$
$239$		−	9.37885e10i		−	1.85934i	0.368392	−	0.929671i	$-0.379909\pi$
$240$		0			0
$241$			1.05800e10i			0.202026i	0.994885	+	0.101013i	$0.0322084\pi$
$241$			1.05800e10i			0.202026i	−0.994885	+	0.101013i	$0.967792\pi$
$242$		0			0
$243$	−3.62805e10			−0.667491
$244$		0			0
$245$		−	6.79294e10i		−	1.20451i
$246$		0			0
$247$	−8.74251e10	+	7.18389e9i	−1.49451	+	0.122807i
$248$		0			0
$249$		−	1.89805e10i		−	0.312904i
$250$		0			0
$251$	1.07571e11			1.71066			0.855328	−	0.518088i	$-0.173356\pi$
$251$	1.07571e11			1.71066			0.855328	+	0.518088i	$0.173356\pi$
$252$		0			0
$253$			1.87155e10i			0.287183i
$254$		0			0
$255$		−	6.76363e8i		−	0.0100173i
$256$		0			0
$257$	4.66181e10			0.666585			0.333292	−	0.942824i	$-0.391840\pi$
$257$	4.66181e10			0.666585			0.333292	+	0.942824i	$0.391840\pi$
$258$		0			0
$259$	5.98279e10			0.826142
$260$		0			0
$261$	1.35188e10			0.180325
$262$		0			0
$263$	−2.21429e10			−0.285387			−0.142693	−	0.989767i	$-0.545576\pi$
$263$	−2.21429e10			−0.285387			−0.142693	+	0.989767i	$0.545576\pi$
$264$		0			0
$265$			3.53976e10i			0.440927i
$266$		0			0
$267$		−	3.07230e10i		−	0.369966i
$268$		0			0
$269$	1.09267e11			1.27234			0.636172	−	0.771548i	$-0.280517\pi$
$269$	1.09267e11			1.27234			0.636172	+	0.771548i	$0.280517\pi$
$270$		0			0
$271$			1.68151e11i			1.89381i	0.321508	+	0.946907i	$0.395810\pi$
$271$			1.68151e11i			1.89381i	−0.321508	+	0.946907i	$0.604190\pi$
$272$		0			0
$273$	1.24900e10	−	1.02633e9i	0.136091	−	0.0111829i
$274$		0			0
$275$		−	5.06450e10i		−	0.533998i
$276$		0			0
$277$	−7.45812e10			−0.761150			−0.380575	−	0.924750i	$-0.624274\pi$
$277$	−7.45812e10			−0.761150			−0.380575	+	0.924750i	$0.624274\pi$
$278$		0			0
$279$			1.40256e11i			1.38580i
$280$		0			0
$281$			8.30956e10i			0.795059i	0.917589	+	0.397530i	$0.130132\pi$
$281$			8.30956e10i			0.795059i	−0.917589	+	0.397530i	$0.869868\pi$
$282$		0			0
$283$	1.94259e11			1.80029			0.900146	−	0.435588i	$-0.143459\pi$
$283$	1.94259e11			1.80029			0.900146	+	0.435588i	$0.143459\pi$
$284$		0			0
$285$	−7.14050e10			−0.641101
$286$		0			0
$287$	−4.53019e10			−0.394137
$288$		0			0
$289$	−1.18523e11			−0.999451
$290$		0			0
$291$			2.10849e10i			0.172367i
$292$		0			0
$293$		−	4.16649e10i		−	0.330268i	−0.986271	−	0.165134i	$-0.947194\pi$
$293$		−	4.16649e10i		−	0.330268i	0.986271	−	0.165134i	$-0.0528056\pi$
$294$		0			0
$295$	1.66121e11			1.27710
$296$		0			0
$297$			1.55726e10i			0.116134i
$298$		0			0
$299$	1.30915e10	+	1.59318e11i	0.0947258	+	1.15278i
$300$		0			0
$301$			4.00084e10i			0.280933i
$302$		0			0
$303$	−2.26069e10			−0.154081
$304$		0			0
$305$		−	4.81095e11i		−	3.18333i
$306$		0			0
$307$		−	1.32639e11i		−	0.852215i	−0.904672	−	0.426108i	$-0.859884\pi$
$307$		−	1.32639e11i		−	0.852215i	0.904672	−	0.426108i	$-0.140116\pi$
$308$		0			0
$309$	−4.09844e10			−0.255744
$310$		0			0
$311$	−2.26899e11			−1.37534			−0.687672	−	0.726021i	$-0.741367\pi$
$311$	−2.26899e11			−1.37534			−0.687672	+	0.726021i	$0.741367\pi$
$312$		0			0
$313$	−1.17724e11			−0.693294			−0.346647	−	0.937996i	$-0.612680\pi$
$313$	−1.17724e11			−0.693294			−0.346647	+	0.937996i	$0.612680\pi$
$314$		0			0
$315$	−1.65646e11			−0.947948
$316$		0			0
$317$		−	4.09676e10i		−	0.227863i	−0.993489	−	0.113931i	$-0.963656\pi$
$317$		−	4.09676e10i		−	0.227863i	0.993489	−	0.113931i	$-0.0363444\pi$
$318$		0			0
$319$		−	8.79065e9i		−	0.0475295i
$320$		0			0
$321$	3.85406e10			0.202603
$322$		0			0
$323$		−	6.87316e9i		−	0.0351354i
$324$		0			0
$325$	−3.54262e10	−	4.31123e11i	−0.176136	−	2.14351i
$326$		0			0
$327$		−	3.32121e9i		−	0.0160632i
$328$		0			0
$329$	−1.10675e11			−0.520795
$330$		0			0
$331$			3.46906e10i			0.158850i	0.996841	+	0.0794248i	$0.0253084\pi$
$331$			3.46906e10i			0.158850i	−0.996841	+	0.0794248i	$0.974692\pi$
$332$		0			0
$333$			3.08011e11i			1.37267i
$334$		0			0
$335$	−3.09447e11			−1.34241
$336$		0			0
$337$	2.95493e11			1.24800			0.623998	−	0.781426i	$-0.285507\pi$
$337$	2.95493e11			1.24800			0.623998	+	0.781426i	$0.285507\pi$
$338$		0			0
$339$	8.39542e9			0.0345258
$340$		0			0
$341$	9.12017e10			0.365265
$342$		0			0
$343$			2.43949e11i			0.951648i
$344$		0			0
$345$			1.30124e11i			0.494506i
$346$		0			0
$347$	−3.63709e11			−1.34670			−0.673351	−	0.739323i	$-0.735146\pi$
$347$	−3.63709e11			−1.34670			−0.673351	+	0.739323i	$0.735146\pi$
$348$		0			0
$349$		−	1.68090e11i		−	0.606494i	−0.952912	−	0.303247i	$-0.901929\pi$
$349$		−	1.68090e11i		−	0.606494i	0.952912	−	0.303247i	$-0.0980708\pi$
$350$		0			0
$351$	1.08930e10	+	1.32564e11i	0.0383060	+	0.466170i
$352$		0			0
$353$			2.07923e11i			0.712715i	0.934350	+	0.356357i	$0.115981\pi$
$353$			2.07923e11i			0.712715i	−0.934350	+	0.356357i	$0.884019\pi$
$354$		0			0
$355$	−7.56296e11			−2.52734
$356$		0			0
$357$			9.81935e8i			0.00319945i
$358$		0			0
$359$			1.38162e11i			0.438998i	0.975613	+	0.219499i	$0.0704422\pi$
$359$			1.38162e11i			0.438998i	−0.975613	+	0.219499i	$0.929558\pi$
$360$		0			0
$361$	−4.02925e11			−1.24865
$362$		0			0
$363$	−7.47662e10			−0.226009
$364$		0			0
$365$	2.54712e11			0.751159
$366$		0			0
$367$	6.66426e11			1.91758			0.958792	−	0.284108i	$-0.0916973\pi$
$367$	6.66426e11			1.91758			0.958792	+	0.284108i	$0.0916973\pi$
$368$		0			0
$369$		−	2.33227e11i		−	0.654878i
$370$		0			0
$371$		−	5.13897e10i		−	0.140830i
$372$		0			0
$373$	−5.96945e11			−1.59678			−0.798390	−	0.602141i	$-0.794314\pi$
$373$	−5.96945e11			−1.59678			−0.798390	+	0.602141i	$0.794314\pi$
$374$		0			0
$375$		−	1.88401e11i		−	0.491973i
$376$		0			0
$377$	−6.14906e9	−	7.48317e10i	−0.0156774	−	0.190787i
$378$		0			0
$379$			3.71080e11i			0.923828i	0.886925	+	0.461914i	$0.152837\pi$
$379$			3.71080e11i			0.923828i	−0.886925	+	0.461914i	$0.847163\pi$
$380$		0			0
$381$	−5.32097e10			−0.129369
$382$		0			0
$383$			7.23332e11i			1.71768i	0.512242	+	0.858841i	$0.328815\pi$
$383$			7.23332e11i			1.71768i	−0.512242	+	0.858841i	$0.671185\pi$
$384$		0			0
$385$			1.07712e11i			0.249857i
$386$		0			0
$387$	−2.05975e11			−0.466782
$388$		0			0
$389$	6.25929e10			0.138596			0.0692982	−	0.997596i	$-0.477924\pi$
$389$	6.25929e10			0.138596			0.0692982	+	0.997596i	$0.477924\pi$
$390$		0			0
$391$	−1.25252e10			−0.0271013
$392$		0			0
$393$	1.12736e11			0.238394
$394$		0			0
$395$		−	9.60651e11i		−	1.98554i
$396$		0			0
$397$			3.39383e10i			0.0685699i	0.999412	+	0.0342849i	$0.0109154\pi$
$397$			3.39383e10i			0.0685699i	−0.999412	+	0.0342849i	$0.989085\pi$
$398$		0			0
$399$	1.03665e11			0.204764
$400$		0			0
$401$		−	2.01721e11i		−	0.389584i	−0.980845	−	0.194792i	$-0.937597\pi$
$401$		−	2.01721e11i		−	0.389584i	0.980845	−	0.194792i	$-0.0624031\pi$
$402$		0			0
$403$	7.76368e11	−	6.37956e10i	1.46621	−	0.120481i
$404$		0			0
$405$		−	7.97041e11i		−	1.47209i
$406$		0			0
$407$	2.00285e11			0.361804
$408$		0			0
$409$		−	9.33986e11i		−	1.65039i	−0.564851	−	0.825193i	$-0.691066\pi$
$409$		−	9.33986e11i		−	1.65039i	0.564851	−	0.825193i	$-0.308934\pi$
$410$		0			0
$411$		−	1.92400e11i		−	0.332596i
$412$		0			0
$413$	−2.41172e11			−0.407899
$414$		0			0
$415$	1.39339e12			2.30599
$416$		0			0
$417$	6.13932e10			0.0994278
$418$		0			0
$419$	−6.88583e10			−0.109142			−0.0545711	−	0.998510i	$-0.517379\pi$
$419$	−6.88583e10			−0.109142			−0.0545711	+	0.998510i	$0.517379\pi$
$420$		0			0
$421$			1.15439e12i			1.79095i	0.445111	+	0.895475i	$0.353164\pi$
$421$			1.15439e12i			1.79095i	−0.445111	+	0.895475i	$0.646836\pi$
$422$		0			0
$423$		−	5.69784e11i		−	0.865324i
$424$		0			0
$425$	3.38939e10			0.0503931
$426$		0			0
$427$			6.98447e11i			1.01674i
$428$		0			0
$429$	4.18126e10	−	3.43582e9i	0.0596005	−	0.00489748i
$430$		0			0
$431$		−	8.01718e10i		−	0.111911i	−0.998433	−	0.0559557i	$-0.982179\pi$
$431$		−	8.01718e10i		−	0.111911i	0.998433	−	0.0559557i	$-0.0178206\pi$
$432$		0			0
$433$	−1.16534e11			−0.159315			−0.0796573	−	0.996822i	$-0.525383\pi$
$433$	−1.16534e11			−0.159315			−0.0796573	+	0.996822i	$0.525383\pi$
$434$		0			0
$435$		−	6.11192e10i		−	0.0818420i
$436$		0			0
$437$			1.32231e12i			1.73447i
$438$		0			0
$439$	−8.15734e11			−1.04823			−0.524117	−	0.851646i	$-0.675604\pi$
$439$	−8.15734e11			−1.04823			−0.524117	+	0.851646i	$0.675604\pi$
$440$		0			0
$441$	−5.07719e11			−0.639219
$442$		0			0
$443$	5.07254e10			0.0625761			0.0312880	−	0.999510i	$-0.490039\pi$
$443$	5.07254e10			0.0625761			0.0312880	+	0.999510i	$0.490039\pi$
$444$		0			0
$445$	2.25543e12			2.72652
$446$		0			0
$447$		−	8.10970e10i		−	0.0960774i
$448$		0			0
$449$			1.49521e12i			1.73618i	0.496408	+	0.868089i	$0.334652\pi$
$449$			1.49521e12i			1.73618i	−0.496408	+	0.868089i	$0.665348\pi$
$450$		0			0
$451$	−1.51657e11			−0.172610
$452$		0			0
$453$			9.27100e10i			0.103439i
$454$		0			0
$455$	7.53446e10	+	9.16915e11i	0.0824140	+	1.00295i
$456$		0			0
$457$			1.07544e12i			1.15335i	0.816973	+	0.576676i	$0.195650\pi$
$457$			1.07544e12i			1.15335i	−0.816973	+	0.576676i	$0.804350\pi$
$458$		0			0
$459$	−1.04219e10			−0.0109595
$460$		0			0
$461$		−	5.71601e11i		−	0.589439i	−0.955584	−	0.294719i	$-0.904774\pi$
$461$		−	5.71601e11i		−	0.589439i	0.955584	−	0.294719i	$-0.0952261\pi$
$462$		0			0
$463$		−	1.28589e12i		−	1.30044i	−0.759745	−	0.650221i	$-0.774676\pi$
$463$		−	1.28589e12i		−	1.30044i	0.759745	−	0.650221i	$-0.225324\pi$
$464$		0			0
$465$	6.34103e11			0.628958
$466$		0			0
$467$	5.26066e11			0.511816			0.255908	−	0.966701i	$-0.417626\pi$
$467$	5.26066e11			0.511816			0.255908	+	0.966701i	$0.417626\pi$
$468$		0			0
$469$	4.49251e11			0.428757
$470$		0			0
$471$	3.25147e11			0.304429
$472$		0			0
$473$			1.33936e11i			0.123033i
$474$		0			0
$475$		−	3.57825e12i		−	3.22514i
$476$		0			0
$477$	2.64569e11			0.233995
$478$		0			0
$479$			1.34343e12i			1.16602i	0.812467	+	0.583008i	$0.198124\pi$
$479$			1.34343e12i			1.16602i	−0.812467	+	0.583008i	$0.801876\pi$
$480$		0			0
$481$	1.70496e12	−	1.40099e11i	1.45231	−	0.119339i
$482$		0			0
$483$		−	1.88912e11i		−	0.157942i
$484$		0			0
$485$	−1.54789e12			−1.27028
$486$		0			0
$487$			1.01560e12i			0.818167i	0.912497	+	0.409083i	$0.134151\pi$
$487$			1.01560e12i			0.818167i	−0.912497	+	0.409083i	$0.865849\pi$
$488$		0			0
$489$			3.36848e11i			0.266406i
$490$		0			0
$491$	8.74388e11			0.678949			0.339475	−	0.940615i	$-0.389751\pi$
$491$	8.74388e11			0.678949			0.339475	+	0.940615i	$0.389751\pi$
$492$		0			0
$493$	5.88310e9			0.00448534
$494$		0			0
$495$	−5.54532e11			−0.415149
$496$		0			0
$497$	1.09798e12			0.807219
$498$		0			0
$499$		−	1.06049e12i		−	0.765689i	−0.923813	−	0.382844i	$-0.874945\pi$
$499$		−	1.06049e12i		−	0.765689i	0.923813	−	0.382844i	$-0.125055\pi$
$500$		0			0
$501$		−	1.63844e11i		−	0.116188i
$502$		0			0
$503$	−7.38730e11			−0.514553			−0.257276	−	0.966338i	$-0.582825\pi$
$503$	−7.38730e11			−0.514553			−0.257276	+	0.966338i	$0.582825\pi$
$504$		0			0
$505$		−	1.65961e12i		−	1.13552i
$506$		0			0
$507$	3.53533e11	−	5.84959e10i	0.237626	−	0.0393178i
$508$		0			0
$509$			8.32382e11i			0.549658i	0.961493	+	0.274829i	$0.0886213\pi$
$509$			8.32382e11i			0.549658i	−0.961493	+	0.274829i	$0.911379\pi$
$510$		0			0
$511$	−3.69788e11			−0.239916
$512$		0			0
$513$			1.10026e12i			0.701402i
$514$		0			0
$515$		−	3.00874e12i		−	1.88474i
$516$		0			0
$517$	−3.70504e11			−0.228079
$518$		0			0
$519$	−6.07238e11			−0.367372
$520$		0			0
$521$	−1.50790e12			−0.896611			−0.448305	−	0.893880i	$-0.647972\pi$
$521$	−1.50790e12			−0.896611			−0.448305	+	0.893880i	$0.647972\pi$
$522$		0			0
$523$	5.89491e11			0.344524			0.172262	−	0.985051i	$-0.444892\pi$
$523$	5.89491e11			0.344524			0.172262	+	0.985051i	$0.444892\pi$
$524$		0			0
$525$			5.11206e11i			0.293683i
$526$		0			0
$527$			6.10363e10i			0.0344699i
$528$		0			0
$529$	6.08550e11			0.337867
$530$		0			0
$531$		−	1.24162e12i		−	0.677742i
$532$		0			0
$533$	−1.29100e12	+	1.06084e11i	−0.692873	+	0.0569346i
$534$		0			0
$535$			2.82934e12i			1.49311i
$536$		0			0
$537$	−7.82465e11			−0.406051
$538$		0			0
$539$			3.30146e11i			0.168483i
$540$		0			0
$541$			2.19081e12i			1.09956i	0.835311	+	0.549778i	$0.185288\pi$
$541$			2.19081e12i			1.09956i	−0.835311	+	0.549778i	$0.814712\pi$
$542$		0			0
$543$	4.93557e11			0.243634
$544$		0			0
$545$	2.43817e11			0.118380
$546$		0			0
$547$	−7.33131e10			−0.0350138			−0.0175069	−	0.999847i	$-0.505573\pi$
$547$	−7.33131e10			−0.0350138			−0.0175069	+	0.999847i	$0.505573\pi$
$548$		0			0
$549$	−3.59580e12			−1.68935
$550$		0			0
$551$		−	6.21090e11i		−	0.287060i
$552$		0			0
$553$			1.39466e12i			0.634170i
$554$		0			0
$555$	1.39253e12			0.622998
$556$		0			0
$557$		−	1.03789e12i		−	0.456880i	−0.973558	−	0.228440i	$-0.926637\pi$
$557$		−	1.03789e12i		−	0.456880i	0.973558	−	0.228440i	$-0.0733626\pi$
$558$		0			0
$559$	9.36880e10	+	1.14015e12i	0.0405818	+	0.493865i
$560$		0			0
$561$			3.28721e9i			0.00140118i
$562$		0			0
$563$	2.43180e12			1.02009			0.510046	−	0.860147i	$-0.329628\pi$
$563$	2.43180e12			1.02009			0.510046	+	0.860147i	$0.329628\pi$
$564$		0			0
$565$			6.16324e11i			0.254443i
$566$		0			0
$567$			1.15713e12i			0.470175i
$568$		0			0
$569$	−2.12643e12			−0.850446			−0.425223	−	0.905089i	$-0.639804\pi$
$569$	−2.12643e12			−0.850446			−0.425223	+	0.905089i	$0.639804\pi$
$570$		0			0
$571$	−1.01100e12			−0.398004			−0.199002	−	0.979999i	$-0.563770\pi$
$571$	−1.01100e12			−0.398004			−0.199002	+	0.979999i	$0.563770\pi$
$572$		0			0
$573$	−3.70954e11			−0.143756
$574$		0			0
$575$	−6.52078e12			−2.48768
$576$		0			0
$577$		−	1.09472e12i		−	0.411161i	−0.978640	−	0.205581i	$-0.934092\pi$
$577$		−	1.09472e12i		−	0.411161i	0.978640	−	0.205581i	$-0.0659083\pi$
$578$		0			0
$579$		−	6.45828e11i		−	0.238816i
$580$		0			0
$581$	−2.02291e12			−0.736520
$582$		0			0
$583$		−	1.72037e11i		−	0.0616755i
$584$		0			0
$585$	−4.72054e12	+	3.87895e11i	−1.66644	+	0.136935i
$586$		0			0
$587$		−	3.19306e12i		−	1.11003i	−0.831839	−	0.555017i	$-0.812712\pi$
$587$		−	3.19306e12i		−	1.11003i	0.831839	−	0.555017i	$-0.187288\pi$
$588$		0			0
$589$	6.44372e12			2.20606
$590$		0			0
$591$		−	7.31621e11i		−	0.246685i
$592$		0			0
$593$			1.90468e12i			0.632521i	0.948672	+	0.316261i	$0.102427\pi$
$593$			1.90468e12i			0.632521i	−0.948672	+	0.316261i	$0.897573\pi$
$594$		0			0
$595$	−7.20857e10			−0.0235789
$596$		0			0
$597$	−1.01998e12			−0.328630
$598$		0			0
$599$	−7.04878e11			−0.223714			−0.111857	−	0.993724i	$-0.535680\pi$
$599$	−7.04878e11			−0.223714			−0.111857	+	0.993724i	$0.535680\pi$
$600$		0			0
$601$	−2.01041e12			−0.628565			−0.314282	−	0.949330i	$-0.601764\pi$
$601$	−2.01041e12			−0.628565			−0.314282	+	0.949330i	$0.601764\pi$
$602$		0			0
$603$			2.31287e12i			0.712400i
$604$		0			0
$605$		−	5.48873e12i		−	1.66561i
$606$		0			0
$607$	−2.44941e12			−0.732338			−0.366169	−	0.930548i	$-0.619331\pi$
$607$	−2.44941e12			−0.732338			−0.366169	+	0.930548i	$0.619331\pi$
$608$		0			0
$609$			8.87321e10i			0.0261398i
$610$		0			0
$611$	−3.15397e12	+	2.59168e11i	−0.915529	+	0.0752307i
$612$		0			0
$613$		−	3.35213e12i		−	0.958846i	−0.877584	−	0.479423i	$-0.840846\pi$
$613$		−	3.35213e12i		−	0.958846i	0.877584	−	0.479423i	$-0.159154\pi$
$614$		0			0
$615$	−1.05443e12			−0.297221
$616$		0			0
$617$		−	5.56801e12i		−	1.54674i	−0.633956	−	0.773369i	$-0.718570\pi$
$617$		−	5.56801e12i		−	1.54674i	0.633956	−	0.773369i	$-0.281430\pi$
$618$		0			0
$619$			1.68860e12i			0.462295i	0.972919	+	0.231147i	$0.0742480\pi$
$619$			1.68860e12i			0.462295i	−0.972919	+	0.231147i	$0.925752\pi$
$620$		0			0
$621$	2.00505e12			0.541018
$622$		0			0
$623$	−3.27441e12			−0.870836
$624$		0			0
$625$	5.62644e12			1.47494
$626$		0			0
$627$	3.47037e11			0.0896752
$628$		0			0
$629$			1.34040e11i			0.0341433i
$630$		0			0
$631$			3.30581e12i			0.830130i	0.909792	+	0.415065i	$0.136241\pi$
$631$			3.30581e12i			0.830130i	−0.909792	+	0.415065i	$0.863759\pi$
$632$		0			0
$633$	6.16514e11			0.152625
$634$		0			0
$635$		−	3.90623e12i		−	0.953401i
$636$		0			0
$637$	2.30937e11	+	2.81042e12i	0.0555733	+	0.676306i
$638$		0			0
$639$			5.65272e12i			1.34123i
$640$		0			0
$641$	1.40932e12			0.329722			0.164861	−	0.986317i	$-0.447282\pi$
$641$	1.40932e12			0.329722			0.164861	+	0.986317i	$0.447282\pi$
$642$		0			0
$643$		−	5.28783e12i		−	1.21991i	−0.792436	−	0.609955i	$-0.791187\pi$
$643$		−	5.28783e12i		−	1.21991i	0.792436	−	0.609955i	$-0.208813\pi$
$644$		0			0
$645$			9.31222e11i			0.211853i
$646$		0			0
$647$	6.17036e12			1.38434			0.692168	−	0.721737i	$-0.256656\pi$
$647$	6.17036e12			1.38434			0.692168	+	0.721737i	$0.256656\pi$
$648$		0			0
$649$	−8.07369e11			−0.178637
$650$		0			0
$651$	−9.20582e11			−0.200885
$652$		0			0
$653$	−3.45993e12			−0.744661			−0.372330	−	0.928100i	$-0.621441\pi$
$653$	−3.45993e12			−0.744661			−0.372330	+	0.928100i	$0.621441\pi$
$654$		0			0
$655$			8.27614e12i			1.75688i
$656$		0			0
$657$		−	1.90377e12i		−	0.398631i
$658$		0			0
$659$	7.43406e12			1.53547			0.767736	−	0.640766i	$-0.221383\pi$
$659$	7.43406e12			1.53547			0.767736	+	0.640766i	$0.221383\pi$
$660$		0			0
$661$		−	2.27635e12i		−	0.463802i	−0.972739	−	0.231901i	$-0.925506\pi$
$661$		−	2.27635e12i		−	0.463802i	0.972739	−	0.231901i	$-0.0744945\pi$
$662$		0			0
$663$	2.29941e9	+	2.79829e10i	0.000462173	+	0.00562447i
$664$		0			0
$665$			7.61023e12i			1.50904i
$666$		0			0
$667$	−1.13184e12			−0.221420
$668$		0			0
$669$		−	9.61387e11i		−	0.185558i
$670$		0			0
$671$			2.33818e12i			0.445274i
$672$		0			0
$673$	5.42300e12			1.01899			0.509497	−	0.860472i	$-0.329831\pi$
$673$	5.42300e12			1.01899			0.509497	+	0.860472i	$0.329831\pi$
$674$		0			0
$675$	−5.42575e12			−1.00599
$676$		0			0
$677$	−6.13007e12			−1.12154			−0.560772	−	0.827970i	$-0.689496\pi$
$677$	−6.13007e12			−1.12154			−0.560772	+	0.827970i	$0.689496\pi$
$678$		0			0
$679$	2.24720e12			0.405721
$680$		0			0
$681$		−	1.74020e12i		−	0.310054i
$682$		0			0
$683$		−	2.12778e12i		−	0.374140i	−0.982347	−	0.187070i	$-0.940101\pi$
$683$		−	2.12778e12i		−	0.374140i	0.982347	−	0.187070i	$-0.0598991\pi$
$684$		0			0
$685$	1.41245e13			2.45112
$686$		0			0
$687$			8.66339e11i			0.148382i
$688$		0			0
$689$	−1.20340e11	−	1.46449e12i	−0.0203434	−	0.247571i
$690$		0			0
$691$			1.99702e12i			0.333220i	0.986023	+	0.166610i	$0.0532821\pi$
$691$			1.99702e12i			0.333220i	−0.986023	+	0.166610i	$0.946718\pi$
$692$		0			0
$693$	8.05063e11			0.132596
$694$		0			0
$695$			4.50699e12i			0.732749i
$696$		0			0
$697$		−	1.01495e11i		−	0.0162892i
$698$		0			0
$699$	−7.01450e11			−0.111135
$700$		0			0
$701$	−5.33488e11			−0.0834437			−0.0417218	−	0.999129i	$-0.513284\pi$
$701$	−5.33488e11			−0.0834437			−0.0417218	+	0.999129i	$0.513284\pi$
$702$		0			0
$703$	1.41508e13			2.18516
$704$		0			0
$705$	−2.57602e12			−0.392734
$706$		0			0
$707$			2.40941e12i			0.362679i
$708$		0			0
$709$		−	2.15946e12i		−	0.320950i	−0.987040	−	0.160475i	$-0.948697\pi$
$709$		−	2.15946e12i		−	0.320950i	0.987040	−	0.160475i	$-0.0513027\pi$
$710$		0			0
$711$	−7.18011e12			−1.05370
$712$		0			0
$713$		−	1.17426e13i		−	1.70162i
$714$		0			0
$715$	2.52230e11	+	3.06955e12i	0.0360927	+	0.439235i
$716$		0			0
$717$			3.16923e12i			0.447835i
$718$		0			0
$719$	1.06545e13			1.48680			0.743400	−	0.668847i	$-0.233212\pi$
$719$	1.06545e13			1.48680			0.743400	+	0.668847i	$0.233212\pi$
$720$		0			0
$721$			4.36805e12i			0.601976i
$722$		0			0
$723$		−	3.57511e11i		−	0.0486594i
$724$		0			0
$725$	3.06281e12			0.411717
$726$		0			0
$727$	1.18120e13			1.56827			0.784134	−	0.620592i	$-0.213108\pi$
$727$	1.18120e13			1.56827			0.784134	+	0.620592i	$0.213108\pi$
$728$		0			0
$729$	−5.09817e12			−0.668560
$730$		0			0
$731$	−8.96357e10			−0.0116106
$732$		0			0
$733$		−	1.04467e13i		−	1.33663i	−0.743878	−	0.668315i	$-0.767016\pi$
$733$		−	1.04467e13i		−	1.33663i	0.743878	−	0.668315i	$-0.232984\pi$
$734$		0			0
$735$			2.29542e12i			0.290115i
$736$		0			0
$737$	1.50395e12			0.187772
$738$		0			0
$739$		−	1.25115e13i		−	1.54315i	−0.636137	−	0.771576i	$-0.719469\pi$
$739$		−	1.25115e13i		−	1.54315i	0.636137	−	0.771576i	$-0.280531\pi$
$740$		0			0
$741$	2.95421e12	−	2.42753e11i	0.359964	−	0.0295789i
$742$		0			0
$743$			1.13486e13i			1.36614i	0.730354	+	0.683069i	$0.239355\pi$
$743$			1.13486e13i			1.36614i	−0.730354	+	0.683069i	$0.760645\pi$
$744$		0			0
$745$	5.95349e12			0.708057
$746$		0			0
$747$		−	1.04145e13i		−	1.22376i
$748$		0			0
$749$		−	4.10760e12i		−	0.476892i
$750$		0			0
$751$	9.95796e12			1.14233			0.571164	−	0.820836i	$-0.306492\pi$
$751$	9.95796e12			1.14233			0.571164	+	0.820836i	$0.306492\pi$
$752$		0			0
$753$	−3.63495e12			−0.412023
$754$		0			0
$755$	−6.80602e12			−0.762310
$756$		0			0
$757$	−2.18826e12			−0.242197			−0.121098	−	0.992641i	$-0.538642\pi$
$757$	−2.18826e12			−0.242197			−0.121098	+	0.992641i	$0.538642\pi$
$758$		0			0
$759$		−	6.32420e11i		−	0.0691700i
$760$		0			0
$761$		−	1.14282e13i		−	1.23523i	−0.786481	−	0.617615i	$-0.788099\pi$
$761$		−	1.14282e13i		−	1.23523i	0.786481	−	0.617615i	$-0.211901\pi$
$762$		0			0
$763$	−3.53970e11			−0.0378099
$764$		0			0
$765$		−	3.71118e11i		−	0.0391774i
$766$		0			0
$767$	−6.87285e12	+	5.64755e11i	−0.717064	+	0.0589225i
$768$		0			0
$769$		−	9.31548e12i		−	0.960587i	−0.877108	−	0.480293i	$-0.840530\pi$
$769$		−	9.31548e12i		−	0.960587i	0.877108	−	0.480293i	$-0.159470\pi$
$770$		0			0
$771$	−1.57528e12			−0.160552
$772$		0			0
$773$			6.03026e12i			0.607475i	0.952756	+	0.303737i	$0.0982345\pi$
$773$			6.03026e12i			0.607475i	−0.952756	+	0.303737i	$0.901765\pi$
$774$		0			0
$775$			3.17762e13i			3.16405i
$776$		0			0
$777$	−2.02166e12			−0.198982
$778$		0			0
$779$	−1.07151e13			−1.04250
$780$		0			0
$781$	3.67570e12			0.353517
$782$		0			0
$783$	−9.41769e11			−0.0895399
$784$		0			0
$785$			2.38697e13i			2.24353i
$786$		0			0
$787$		−	6.85370e12i		−	0.636852i	−0.947948	−	0.318426i	$-0.896846\pi$
$787$		−	6.85370e12i		−	0.636852i	0.947948	−	0.318426i	$-0.103154\pi$
$788$		0			0
$789$	7.48237e11			0.0687373
$790$		0			0
$791$		−	8.94771e11i		−	0.0812676i
$792$		0			0
$793$	1.63556e12	+	1.99041e13i	0.146871	+	1.78737i
$794$		0			0
$795$		−	1.19613e12i		−	0.106200i
$796$		0			0
$797$	−1.81697e13			−1.59509			−0.797545	−	0.603260i	$-0.793868\pi$
$797$	−1.81697e13			−1.59509			−0.797545	+	0.603260i	$0.793868\pi$
$798$		0			0
$799$		−	2.47958e11i		−	0.0215237i
$800$		0			0
$801$		−	1.68576e13i		−	1.44693i
$802$		0			0
$803$	−1.23793e12			−0.105070
$804$		0			0
$805$	1.38684e13			1.16398
$806$		0			0
$807$	−3.69228e12			−0.306453
$808$		0			0
$809$	1.88038e13			1.54339			0.771697	−	0.635990i	$-0.219408\pi$
$809$	1.88038e13			1.54339			0.771697	+	0.635990i	$0.219408\pi$
$810$		0			0
$811$		−	1.23796e13i		−	1.00487i	−0.864614	−	0.502436i	$-0.832437\pi$
$811$		−	1.23796e13i		−	1.00487i	0.864614	−	0.502436i	$-0.167563\pi$
$812$		0			0
$813$		−	5.68203e12i		−	0.456138i
$814$		0			0
$815$	−2.47287e13			−1.96332
$816$		0			0
$817$			9.46302e12i			0.743071i
$818$		0			0
$819$	6.85322e12	−	5.63142e11i	0.532252	−	0.0437361i
$820$		0			0
$821$			1.66782e13i			1.28117i	0.767889	+	0.640584i	$0.221307\pi$
$821$			1.66782e13i			1.28117i	−0.767889	+	0.640584i	$0.778693\pi$
$822$		0			0
$823$	2.24352e11			0.0170463			0.00852315	−	0.999964i	$-0.497287\pi$
$823$	2.24352e11			0.0170463			0.00852315	+	0.999964i	$0.497287\pi$
$824$		0			0
$825$			1.71136e12i			0.128617i
$826$		0			0
$827$			4.52668e12i			0.336516i	0.985743	+	0.168258i	$0.0538141\pi$
$827$			4.52668e12i			0.336516i	−0.985743	+	0.168258i	$0.946186\pi$
$828$		0			0
$829$	−2.17452e13			−1.59907			−0.799537	−	0.600616i	$-0.794922\pi$
$829$	−2.17452e13			−1.59907			−0.799537	+	0.600616i	$0.794922\pi$
$830$		0			0
$831$	2.52019e12			0.183328
$832$		0			0
$833$	−2.20948e11			−0.0158997
$834$		0			0
$835$	1.20281e13			0.856263
$836$		0			0
$837$		−	9.77072e12i		−	0.688116i
$838$		0			0
$839$		−	2.03862e13i		−	1.42039i	−0.704006	−	0.710194i	$-0.748607\pi$
$839$		−	2.03862e13i		−	1.42039i	0.704006	−	0.710194i	$-0.251393\pi$
$840$		0			0
$841$	−1.39755e13			−0.963354
$842$		0			0
$843$		−	2.80791e12i		−	0.191495i
$844$		0			0
$845$	4.29429e12	+	2.59535e13i	0.289759	+	1.75122i
$846$		0			0
$847$			7.96847e12i			0.531985i
$848$		0			0
$849$	−6.56427e12			−0.433613
$850$		0			0
$851$		−	2.57876e13i		−	1.68550i
$852$		0			0
$853$		−	2.18038e13i		−	1.41014i	−0.709140	−	0.705068i	$-0.750916\pi$
$853$		−	2.18038e13i		−	1.41014i	0.709140	−	0.705068i	$-0.249084\pi$
$854$		0			0
$855$	−3.91796e13			−2.50734
$856$		0			0
$857$	1.63788e13			1.03722			0.518608	−	0.855012i	$-0.326451\pi$
$857$	1.63788e13			1.03722			0.518608	+	0.855012i	$0.326451\pi$
$858$		0			0
$859$	−6.46193e12			−0.404942			−0.202471	−	0.979288i	$-0.564897\pi$
$859$	−6.46193e12			−0.404942			−0.202471	+	0.979288i	$0.564897\pi$
$860$		0			0
$861$	1.53081e12			0.0949307
$862$		0			0
$863$			5.52060e12i			0.338796i	0.985548	+	0.169398i	$0.0541823\pi$
$863$			5.52060e12i			0.338796i	−0.985548	+	0.169398i	$0.945818\pi$
$864$		0			0
$865$		−	4.45785e13i		−	2.70740i
$866$		0			0
$867$	4.00504e12			0.240725
$868$		0			0
$869$			4.66889e12i			0.277731i
$870$		0			0
$871$	1.28026e13	−	1.05201e12i	0.753732	−	0.0619356i
$872$		0			0
$873$			1.15692e13i			0.674125i
$874$		0			0
$875$	−2.00795e13			−1.15802
$876$		0			0
$877$			9.43230e12i			0.538418i	0.963082	+	0.269209i	$0.0867622\pi$
$877$			9.43230e12i			0.538418i	−0.963082	+	0.269209i	$0.913238\pi$
$878$		0			0
$879$			1.40791e12i			0.0795472i
$880$		0			0
$881$	−3.28469e13			−1.83697			−0.918485	−	0.395455i	$-0.870587\pi$
$881$	−3.28469e13			−1.83697			−0.918485	+	0.395455i	$0.870587\pi$
$882$		0			0
$883$	7.28752e12			0.403419			0.201710	−	0.979445i	$-0.435350\pi$
$883$	7.28752e12			0.403419			0.201710	+	0.979445i	$0.435350\pi$
$884$		0			0
$885$	−5.61344e12			−0.307599
$886$		0			0
$887$	2.78898e12			0.151283			0.0756413	−	0.997135i	$-0.475900\pi$
$887$	2.78898e12			0.151283			0.0756413	+	0.997135i	$0.475900\pi$
$888$		0			0
$889$			5.67101e12i			0.304511i
$890$		0			0
$891$			3.87372e12i			0.205911i
$892$		0			0
$893$	−2.61774e13			−1.37751
$894$		0			0
$895$		−	5.74423e13i		−	2.99246i
$896$		0			0
$897$	−4.42378e11	−	5.38357e12i	−0.0228154	−	0.277654i
$898$		0			0
$899$			5.51551e12i			0.281623i
$900$		0			0
$901$	1.15135e11			0.00582029
$902$		0			0
$903$		−	1.35194e12i		−	0.0676645i
$904$		0			0
$905$			3.62330e13i			1.79550i
$906$		0			0
$907$	2.96724e13			1.45586			0.727930	−	0.685651i	$-0.240482\pi$
$907$	2.96724e13			1.45586			0.727930	+	0.685651i	$0.240482\pi$
$908$		0			0
$909$	−1.24043e13			−0.602608
$910$		0			0
$911$	−2.08157e13			−1.00128			−0.500642	−	0.865654i	$-0.666903\pi$
$911$	−2.08157e13			−1.00128			−0.500642	+	0.865654i	$0.666903\pi$
$912$		0			0
$913$	−6.77208e12			−0.322555
$914$		0			0
$915$			1.62568e13i			0.766726i
$916$		0			0
$917$		−	1.20152e13i		−	0.561137i
$918$		0			0
$919$	1.89789e13			0.877712			0.438856	−	0.898557i	$-0.355384\pi$
$919$	1.89789e13			0.877712			0.438856	+	0.898557i	$0.355384\pi$
$920$		0			0
$921$			4.48205e12i			0.205262i
$922$		0			0
$923$	3.12899e13	−	2.57115e12i	1.41905	−	0.116606i
$924$		0			0
$925$			6.97826e13i			3.13407i
$926$		0			0
$927$	−2.24880e13			−1.00021
$928$		0			0
$929$			1.34519e13i			0.592535i	0.955105	+	0.296267i	$0.0957420\pi$
$929$			1.34519e13i			0.592535i	−0.955105	+	0.296267i	$0.904258\pi$
$930$		0			0
$931$			2.33259e13i			1.01757i
$932$		0			0
$933$	7.66722e12			0.331261
$934$		0			0
$935$	−2.41321e11			−0.0103262
$936$		0			0
$937$	1.61070e13			0.682631			0.341316	−	0.939949i	$-0.389128\pi$
$937$	1.61070e13			0.682631			0.341316	+	0.939949i	$0.389128\pi$
$938$		0			0
$939$	3.97806e12			0.166985
$940$		0			0
$941$		−	1.21020e13i		−	0.503157i	−0.967837	−	0.251578i	$-0.919050\pi$
$941$		−	1.21020e13i		−	0.503157i	0.967837	−	0.251578i	$-0.0809496\pi$
$942$		0			0
$943$			1.95265e13i			0.804121i
$944$		0			0
$945$	1.15395e13			0.470700
$946$		0			0
$947$		−	2.33927e13i		−	0.945162i	−0.881287	−	0.472581i	$-0.843323\pi$
$947$		−	2.33927e13i		−	0.945162i	0.881287	−	0.472581i	$-0.156677\pi$
$948$		0			0
$949$	−1.05381e13	+	8.65935e11i	−0.421759	+	0.0346567i
$950$		0			0
$951$			1.38435e12i			0.0548823i
$952$		0			0
$953$	−2.85315e13			−1.12049			−0.560243	−	0.828328i	$-0.689292\pi$
$953$	−2.85315e13			−1.12049			−0.560243	+	0.828328i	$0.689292\pi$
$954$		0			0
$955$		−	2.72325e13i		−	1.05943i
$956$		0			0
$957$			2.97047e11i			0.0114478i
$958$		0			0
$959$	−2.05057e13			−0.782873
$960$		0			0
$961$	−3.07830e13			−1.16428
$962$		0			0
$963$	2.11471e13			0.792378
$964$		0			0
$965$	4.74115e13			1.75999
$966$		0			0
$967$		−	1.02571e13i		−	0.377230i	−0.982051	−	0.188615i	$-0.939600\pi$
$967$		−	1.02571e13i		−	0.377230i	0.982051	−	0.188615i	$-0.0603998\pi$
$968$		0			0
$969$			2.32253e11i			0.00846261i
$970$		0			0
$971$	−6.57160e12			−0.237238			−0.118619	−	0.992940i	$-0.537847\pi$
$971$	−6.57160e12			−0.237238			−0.118619	+	0.992940i	$0.537847\pi$
$972$		0			0
$973$		−	6.54319e12i		−	0.234036i
$974$		0			0
$975$	1.19710e12	+	1.45682e13i	0.0424237	+	0.516280i
$976$		0			0
$977$			8.02624e12i			0.281829i	0.990022	+	0.140915i	$0.0450043\pi$
$977$			8.02624e12i			0.281829i	−0.990022	+	0.140915i	$0.954996\pi$
$978$		0			0
$979$	−1.09617e13			−0.381378
$980$		0			0
$981$		−	1.82234e12i		−	0.0628229i
$982$		0			0
$983$		−	1.37031e12i		−	0.0468089i	−0.999726	−	0.0234045i	$-0.992549\pi$
$983$		−	1.37031e12i		−	0.0468089i	0.999726	−	0.0234045i	$-0.00745055\pi$
$984$		0			0
$985$	5.37097e13			1.81798
$986$		0			0
$987$	3.73984e12			0.125437
$988$		0			0
$989$	1.72448e13			0.573160
$990$		0			0
$991$	2.44907e13			0.806621			0.403310	−	0.915063i	$-0.367859\pi$
$991$	2.44907e13			0.806621			0.403310	+	0.915063i	$0.367859\pi$
$992$		0			0
$993$		−	1.17224e12i		−	0.0382600i
$994$		0			0
$995$		−	7.48787e13i		−	2.42189i
$996$		0			0
$997$	2.41809e13			0.775075			0.387538	−	0.921854i	$-0.373326\pi$
$997$	2.41809e13			0.775075			0.387538	+	0.921854i	$0.373326\pi$
$998$		0			0
$999$		−	2.14571e13i		−	0.681596i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	104.10.f.a.25.16	yes	32
4.3	odd	2		208.10.f.d.129.17		32
13.12	even	2	inner	104.10.f.a.25.15	✓	32
52.51	odd	2		208.10.f.d.129.18		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.15	✓	32	13.12	even	2	inner
104.10.f.a.25.16	yes	32	1.1	even	1	trivial
208.10.f.d.129.17		32	4.3	odd	2
208.10.f.d.129.18		32	52.51	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:	\( N \)	\(=\)	\( 104 = 2^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	104.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-33.7913 q^{3} -2480.68i q^{5} +3601.42i q^{7} -18541.1 q^{9} +12056.4i q^{11} +(8433.48 + 102632. i) q^{13} +83825.4i q^{15} -8068.71 q^{17} +851829. i q^{19} -121697. i q^{21} +1.55232e6 q^{23} -4.20066e6 q^{25} +1.29164e6 q^{27} -729125. q^{29} -7.56456e6i q^{31} -407403. i q^{33} +8.93399e6 q^{35} -1.66123e7i q^{37} +(-284978. - 3.46807e6i) q^{39} +1.25789e7i q^{41} +1.11091e7 q^{43} +4.59947e7i q^{45} +3.07308e7i q^{47} +2.73834e7 q^{49} +272652. q^{51} -1.42693e7 q^{53} +2.99082e7 q^{55} -2.87844e7i q^{57} +6.69658e7i q^{59} +1.93936e8 q^{61} -6.67745e7i q^{63} +(2.54598e8 - 2.09208e7i) q^{65} -1.24743e8i q^{67} -5.24549e7 q^{69} -3.04874e8i q^{71} +1.02678e8i q^{73} +1.41946e8 q^{75} -4.34203e7 q^{77} +3.87253e8 q^{79} +3.21299e8 q^{81} +5.61698e8i q^{83} +2.00159e7i q^{85} +2.46381e7 q^{87} +9.09198e8i q^{89} +(-3.69622e8 + 3.03725e7i) q^{91} +2.55616e8i q^{93} +2.11312e9 q^{95} -6.23976e8i q^{97} -2.23540e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} + 3171556 q^{23} - 13526722 q^{25} - 3694974 q^{27} + 8833508 q^{29} - 8281126 q^{35} - 12056860 q^{39} + 89959038 q^{43} - 172344874 q^{49}+ \cdots + 1741143356 q^{95}+O(q^{100}) \) 32 * q - 162 * q^3 + 223074 * q^9 + 66270 * q^13 - 487902 * q^17 + 3171556 * q^23 - 13526722 * q^25 - 3694974 * q^27 + 8833508 * q^29 - 8281126 * q^35 - 12056860 * q^39 + 89959038 * q^43 - 172344874 * q^49 + 51206562 * q^51 + 139521112 * q^53 - 84550176 * q^55 + 372036836 * q^61 + 241471810 * q^65 - 486052516 * q^69 - 604473728 * q^75 - 1666197144 * q^77 + 1165982916 * q^79 + 859636544 * q^81 - 1738691296 * q^87 - 291718146 * q^91 + 1741143356 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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