LMFDB - Embedded newform 104.10.f.a.25.7

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			25.7
Character	$\chi$	$=$	104.25
Dual form			104.10.f.a.25.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-178.824 q^{3} -872.510i q^{5} +12429.2i q^{7} +12295.0 q^{9} +20342.9i q^{11} +(65751.3 - 79254.4i) q^{13} +156026. i q^{15} +182499. q^{17} -354678. i q^{19} -2.22264e6i q^{21} -1.23045e6 q^{23} +1.19185e6 q^{25} +1.32115e6 q^{27} +4.17940e6 q^{29} +3.16248e6i q^{31} -3.63780e6i q^{33} +1.08446e7 q^{35} -1.27577e6i q^{37} +(-1.17579e7 + 1.41726e7i) q^{39} +7.40821e6i q^{41} +1.11930e7 q^{43} -1.07275e7i q^{45} +944086. i q^{47} -1.14131e8 q^{49} -3.26353e7 q^{51} -9.68577e7 q^{53} +1.77494e7 q^{55} +6.34249e7i q^{57} +1.05521e8i q^{59} -1.66607e6 q^{61} +1.52817e8i q^{63} +(-6.91503e7 - 5.73686e7i) q^{65} +2.19175e8i q^{67} +2.20034e8 q^{69} +5.98629e7i q^{71} +3.96875e7i q^{73} -2.13132e8 q^{75} -2.52846e8 q^{77} -2.20092e7 q^{79} -4.78256e8 q^{81} -6.19116e8i q^{83} -1.59232e8i q^{85} -7.47377e8 q^{87} +6.79975e7i q^{89} +(9.85068e8 + 8.17235e8i) q^{91} -5.65527e8i q^{93} -3.09460e8 q^{95} +1.66575e9i q^{97} +2.50117e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−178.824			−1.27462			−0.637309	−	0.770608i	$-0.719953\pi$
$3$	−178.824			−1.27462			−0.637309	+	0.770608i	$0.719953\pi$
$4$		0			0
$5$		−	872.510i		−	0.624317i	−0.950030	−	0.312159i	$-0.898948\pi$
$5$		−	872.510i		−	0.624317i	0.950030	−	0.312159i	$-0.101052\pi$
$6$		0			0
$7$			12429.2i			1.95660i	0.207198	+	0.978299i	$0.433565\pi$
$7$			12429.2i			1.95660i	−0.207198	+	0.978299i	$0.566435\pi$
$8$		0			0
$9$	12295.0			0.624652
$10$		0			0
$11$			20342.9i			0.418934i	0.977816	+	0.209467i	$0.0671729\pi$
$11$			20342.9i			0.418934i	−0.977816	+	0.209467i	$0.932827\pi$
$12$		0			0
$13$	65751.3	−	79254.4i	0.638497	−	0.769624i
$13$	65751.3	−	79254.4i	0.638497	−	0.769624i
$14$		0			0
$15$			156026.i			0.795766i
$16$		0			0
$17$	182499.			0.529958			0.264979	−	0.964254i	$-0.414635\pi$
$17$	182499.			0.529958			0.264979	+	0.964254i	$0.414635\pi$
$18$		0			0
$19$		−	354678.i		−	0.624371i	−0.950021	−	0.312186i	$-0.898939\pi$
$19$		−	354678.i		−	0.624371i	0.950021	−	0.312186i	$-0.101061\pi$
$20$		0			0
$21$		−	2.22264e6i		−	2.49392i
$22$		0			0
$23$	−1.23045e6			−0.916830			−0.458415	−	0.888738i	$-0.651583\pi$
$23$	−1.23045e6			−0.916830			−0.458415	+	0.888738i	$0.651583\pi$
$24$		0			0
$25$	1.19185e6			0.610228
$26$		0			0
$27$	1.32115e6			0.478425
$28$		0			0
$29$	4.17940e6			1.09729			0.548647	−	0.836054i	$-0.315143\pi$
$29$	4.17940e6			1.09729			0.548647	+	0.836054i	$0.315143\pi$
$30$		0			0
$31$			3.16248e6i			0.615035i	0.951542	+	0.307517i	$0.0994983\pi$
$31$			3.16248e6i			0.615035i	−0.951542	+	0.307517i	$0.900502\pi$
$32$		0			0
$33$		−	3.63780e6i		−	0.533981i
$34$		0			0
$35$	1.08446e7			1.22154
$36$		0			0
$37$		−	1.27577e6i		−	0.111909i	−0.998433	−	0.0559545i	$-0.982180\pi$
$37$		−	1.27577e6i		−	0.111909i	0.998433	−	0.0559545i	$-0.0178202\pi$
$38$		0			0
$39$	−1.17579e7	+	1.41726e7i	−0.813841	+	0.980977i
$40$		0			0
$41$			7.40821e6i			0.409436i	0.978821	+	0.204718i	$0.0656277\pi$
$41$			7.40821e6i			0.409436i	−0.978821	+	0.204718i	$0.934372\pi$
$42$		0			0
$43$	1.11930e7			0.499274			0.249637	−	0.968339i	$-0.419689\pi$
$43$	1.11930e7			0.499274			0.249637	+	0.968339i	$0.419689\pi$
$44$		0			0
$45$		−	1.07275e7i		−	0.389981i
$46$		0			0
$47$			944086.i			0.0282209i	0.999900	+	0.0141105i	$0.00449165\pi$
$47$			944086.i			0.0282209i	−0.999900	+	0.0141105i	$0.995508\pi$
$48$		0			0
$49$	−1.14131e8			−2.82828
$50$		0			0
$51$	−3.26353e7			−0.675494
$52$		0			0
$53$	−9.68577e7			−1.68614			−0.843068	−	0.537806i	$-0.819253\pi$
$53$	−9.68577e7			−1.68614			−0.843068	+	0.537806i	$0.819253\pi$
$54$		0			0
$55$	1.77494e7			0.261548
$56$		0			0
$57$			6.34249e7i			0.795835i
$58$		0			0
$59$			1.05521e8i			1.13372i	0.823813	+	0.566861i	$0.191842\pi$
$59$			1.05521e8i			1.13372i	−0.823813	+	0.566861i	$0.808158\pi$
$60$		0			0
$61$	−1.66607e6			−0.0154067			−0.00770336	−	0.999970i	$-0.502452\pi$
$61$	−1.66607e6			−0.0154067			−0.00770336	+	0.999970i	$0.502452\pi$
$62$		0			0
$63$			1.52817e8i			1.22219i
$64$		0			0
$65$	−6.91503e7	−	5.73686e7i	−0.480489	−	0.398625i
$66$		0			0
$67$			2.19175e8i			1.32878i	0.747385	+	0.664391i	$0.231309\pi$
$67$			2.19175e8i			1.32878i	−0.747385	+	0.664391i	$0.768691\pi$
$68$		0			0
$69$	2.20034e8			1.16861
$70$		0			0
$71$			5.98629e7i			0.279573i	0.990182	+	0.139786i	$0.0446416\pi$
$71$			5.98629e7i			0.279573i	−0.990182	+	0.139786i	$0.955358\pi$
$72$		0			0
$73$			3.96875e7i			0.163569i	0.996650	+	0.0817845i	$0.0260619\pi$
$73$			3.96875e7i			0.163569i	−0.996650	+	0.0817845i	$0.973938\pi$
$74$		0			0
$75$	−2.13132e8			−0.777808
$76$		0			0
$77$	−2.52846e8			−0.819686
$78$		0			0
$79$	−2.20092e7			−0.0635744			−0.0317872	−	0.999495i	$-0.510120\pi$
$79$	−2.20092e7			−0.0635744			−0.0317872	+	0.999495i	$0.510120\pi$
$80$		0			0
$81$	−4.78256e8			−1.23446
$82$		0			0
$83$		−	6.19116e8i		−	1.43193i	−0.698138	−	0.715963i	$-0.745988\pi$
$83$		−	6.19116e8i		−	1.43193i	0.698138	−	0.715963i	$-0.254012\pi$
$84$		0			0
$85$		−	1.59232e8i		−	0.330862i
$86$		0			0
$87$	−7.47377e8			−1.39863
$88$		0			0
$89$			6.79975e7i			0.114878i	0.998349	+	0.0574392i	$0.0182935\pi$
$89$			6.79975e7i			0.114878i	−0.998349	+	0.0574392i	$0.981706\pi$
$90$		0			0
$91$	9.85068e8	+	8.17235e8i	1.50584	+	1.24928i
$92$		0			0
$93$		−	5.65527e8i		−	0.783935i
$94$		0			0
$95$	−3.09460e8			−0.389806
$96$		0			0
$97$			1.66575e9i			1.91045i	0.295879	+	0.955225i	$0.404387\pi$
$97$			1.66575e9i			1.91045i	−0.295879	+	0.955225i	$0.595613\pi$
$98$		0			0
$99$			2.50117e8i			0.261688i
$100$		0			0
$101$	9.66630e8			0.924302			0.462151	−	0.886801i	$-0.347078\pi$
$101$	9.66630e8			0.924302			0.462151	+	0.886801i	$0.347078\pi$
$102$		0			0
$103$	−5.37076e8			−0.470185			−0.235092	−	0.971973i	$-0.575539\pi$
$103$	−5.37076e8			−0.470185			−0.235092	+	0.971973i	$0.575539\pi$
$104$		0			0
$105$	−1.93927e9			−1.55699
$106$		0			0
$107$	−7.30059e8			−0.538432			−0.269216	−	0.963080i	$-0.586765\pi$
$107$	−7.30059e8			−0.538432			−0.269216	+	0.963080i	$0.586765\pi$
$108$		0			0
$109$			2.08808e9i			1.41686i	0.705780	+	0.708431i	$0.250597\pi$
$109$			2.08808e9i			1.41686i	−0.705780	+	0.708431i	$0.749403\pi$
$110$		0			0
$111$			2.28138e8i			0.142641i
$112$		0			0
$113$	−3.12325e9			−1.80200			−0.900998	−	0.433823i	$-0.857164\pi$
$113$	−3.12325e9			−1.80200			−0.900998	+	0.433823i	$0.857164\pi$
$114$		0			0
$115$			1.07358e9i			0.572393i
$116$		0			0
$117$	8.08414e8	−	9.74436e8i	0.398839	−	0.480747i
$118$		0			0
$119$			2.26832e9i			1.03691i
$120$		0			0
$121$	1.94411e9			0.824494
$122$		0			0
$123$		−	1.32477e9i		−	0.521874i
$124$		0			0
$125$		−	2.74402e9i		−	1.00529i
$126$		0			0
$127$	1.71367e9			0.584534			0.292267	−	0.956337i	$-0.405590\pi$
$127$	1.71367e9			0.584534			0.292267	+	0.956337i	$0.405590\pi$
$128$		0			0
$129$	−2.00158e9			−0.636384
$130$		0			0
$131$	−6.41271e9			−1.90248			−0.951241	−	0.308449i	$-0.900190\pi$
$131$	−6.41271e9			−1.90248			−0.951241	+	0.308449i	$0.900190\pi$
$132$		0			0
$133$	4.40836e9			1.22164
$134$		0			0
$135$		−	1.15271e9i		−	0.298689i
$136$		0			0
$137$		−	5.19325e9i		−	1.25950i	−0.776800	−	0.629748i	$-0.783158\pi$
$137$		−	5.19325e9i		−	1.25950i	0.776800	−	0.629748i	$-0.216842\pi$
$138$		0			0
$139$	3.07823e9			0.699414			0.349707	−	0.936859i	$-0.386281\pi$
$139$	3.07823e9			0.699414			0.349707	+	0.936859i	$0.386281\pi$
$140$		0			0
$141$		−	1.68825e8i		−	0.0359709i
$142$		0			0
$143$	1.61227e9	+	1.33757e9i	0.322422	+	0.267488i
$144$		0			0
$145$		−	3.64657e9i		−	0.685059i
$146$		0			0
$147$	2.04094e10			3.60497
$148$		0			0
$149$		−	3.58609e9i		−	0.596050i	−0.954558	−	0.298025i	$-0.903672\pi$
$149$		−	3.58609e9i		−	0.596050i	0.954558	−	0.298025i	$-0.0963278\pi$
$150$		0			0
$151$			3.75641e9i			0.588000i	0.955805	+	0.294000i	$0.0949865\pi$
$151$			3.75641e9i			0.588000i	−0.955805	+	0.294000i	$0.905013\pi$
$152$		0			0
$153$	2.24384e9			0.331039
$154$		0			0
$155$	2.75929e9			0.383977
$156$		0			0
$157$	−1.24409e10			−1.63419			−0.817096	−	0.576502i	$-0.804417\pi$
$157$	−1.24409e10			−1.63419			−0.817096	+	0.576502i	$0.804417\pi$
$158$		0			0
$159$	1.73205e10			2.14918
$160$		0			0
$161$		−	1.52935e10i		−	1.79387i
$162$		0			0
$163$		−	6.63286e9i		−	0.735964i	−0.929833	−	0.367982i	$-0.880049\pi$
$163$		−	6.63286e9i		−	0.735964i	0.929833	−	0.367982i	$-0.119951\pi$
$164$		0			0
$165$	−3.17401e9			−0.333374
$166$		0			0
$167$			5.92980e9i			0.589951i	0.955505	+	0.294976i	$0.0953115\pi$
$167$			5.92980e9i			0.589951i	−0.955505	+	0.294976i	$0.904689\pi$
$168$		0			0
$169$	−1.95804e9	−	1.04222e10i	−0.184642	−	0.982806i
$170$		0			0
$171$		−	4.36078e9i		−	0.390015i
$172$		0			0
$173$	1.14700e10			0.973548			0.486774	−	0.873528i	$-0.338173\pi$
$173$	1.14700e10			0.973548			0.486774	+	0.873528i	$0.338173\pi$
$174$		0			0
$175$			1.48138e10i			1.19397i
$176$		0			0
$177$		−	1.88698e10i		−	1.44506i
$178$		0			0
$179$	9.33032e9			0.679294			0.339647	−	0.940553i	$-0.389692\pi$
$179$	9.33032e9			0.679294			0.339647	+	0.940553i	$0.389692\pi$
$180$		0			0
$181$	−1.42971e10			−0.990133			−0.495066	−	0.868855i	$-0.664856\pi$
$181$	−1.42971e10			−0.990133			−0.495066	+	0.868855i	$0.664856\pi$
$182$		0			0
$183$	2.97934e8			0.0196377
$184$		0			0
$185$	−1.11312e9			−0.0698667
$186$		0			0
$187$			3.71257e9i			0.222017i
$188$		0			0
$189$			1.64208e10i			0.936085i
$190$		0			0
$191$	−2.06764e10			−1.12415			−0.562075	−	0.827087i	$-0.689997\pi$
$191$	−2.06764e10			−1.12415			−0.562075	+	0.827087i	$0.689997\pi$
$192$		0			0
$193$		−	2.82166e9i		−	0.146385i	−0.997318	−	0.0731927i	$-0.976681\pi$
$193$		−	2.82166e9i		−	0.146385i	0.997318	−	0.0731927i	$-0.0233188\pi$
$194$		0			0
$195$	1.23657e10	+	1.02589e10i	0.612441	+	0.508095i
$196$		0			0
$197$			3.11937e10i			1.47560i	0.675018	+	0.737801i	$0.264136\pi$
$197$			3.11937e10i			1.47560i	−0.675018	+	0.737801i	$0.735864\pi$
$198$		0			0
$199$	1.59157e10			0.719426			0.359713	−	0.933063i	$-0.382875\pi$
$199$	1.59157e10			0.719426			0.359713	+	0.933063i	$0.382875\pi$
$200$		0			0
$201$		−	3.91937e10i		−	1.69369i
$202$		0			0
$203$			5.19466e10i			2.14696i
$204$		0			0
$205$	6.46373e9			0.255618
$206$		0			0
$207$	−1.51284e10			−0.572700
$208$		0			0
$209$	7.21518e9			0.261570
$210$		0			0
$211$	−5.45855e10			−1.89586			−0.947931	−	0.318476i	$-0.896829\pi$
$211$	−5.45855e10			−1.89586			−0.947931	+	0.318476i	$0.896829\pi$
$212$		0			0
$213$		−	1.07049e10i		−	0.356349i
$214$		0			0
$215$		−	9.76602e9i		−	0.311705i
$216$		0			0
$217$	−3.93070e10			−1.20338
$218$		0			0
$219$		−	7.09708e9i		−	0.208488i
$220$		0			0
$221$	1.19996e10	−	1.44639e10i	0.338377	−	0.407868i
$222$		0			0
$223$		−	5.64556e10i		−	1.52875i	−0.644775	−	0.764373i	$-0.723049\pi$
$223$		−	5.64556e10i		−	1.52875i	0.644775	−	0.764373i	$-0.276951\pi$
$224$		0			0
$225$	1.46539e10			0.381181
$226$		0			0
$227$			1.65635e10i			0.414034i	0.978337	+	0.207017i	$0.0663756\pi$
$227$			1.65635e10i			0.414034i	−0.978337	+	0.207017i	$0.933624\pi$
$228$		0			0
$229$			4.42815e10i			1.06405i	0.846728	+	0.532026i	$0.178569\pi$
$229$			4.42815e10i			1.06405i	−0.846728	+	0.532026i	$0.821431\pi$
$230$		0			0
$231$	4.52149e10			1.04479
$232$		0			0
$233$	−5.13087e10			−1.14049			−0.570243	−	0.821476i	$-0.693151\pi$
$233$	−5.13087e10			−1.14049			−0.570243	+	0.821476i	$0.693151\pi$
$234$		0			0
$235$	8.23724e8			0.0176188
$236$		0			0
$237$	3.93577e9			0.0810331
$238$		0			0
$239$		−	3.70720e10i		−	0.734946i	−0.930034	−	0.367473i	$-0.880223\pi$
$239$		−	3.70720e10i		−	0.734946i	0.930034	−	0.367473i	$-0.119777\pi$
$240$		0			0
$241$		−	1.68658e10i		−	0.322055i	−0.986950	−	0.161028i	$-0.948519\pi$
$241$		−	1.68658e10i		−	0.322055i	0.986950	−	0.161028i	$-0.0514809\pi$
$242$		0			0
$243$	5.95195e10			1.09504
$244$		0			0
$245$			9.95805e10i			1.76574i
$246$		0			0
$247$	−2.81098e10	−	2.33205e10i	−0.480531	−	0.398659i
$248$		0			0
$249$			1.10713e11i			1.82516i
$250$		0			0
$251$	−6.18959e10			−0.984306			−0.492153	−	0.870509i	$-0.663790\pi$
$251$	−6.18959e10			−0.984306			−0.492153	+	0.870509i	$0.663790\pi$
$252$		0			0
$253$		−	2.50309e10i		−	0.384092i
$254$		0			0
$255$			2.84746e10i			0.421722i
$256$		0			0
$257$	−3.86249e10			−0.552291			−0.276145	−	0.961116i	$-0.589057\pi$
$257$	−3.86249e10			−0.552291			−0.276145	+	0.961116i	$0.589057\pi$
$258$		0			0
$259$	1.58568e10			0.218961
$260$		0			0
$261$	5.13859e10			0.685427
$262$		0			0
$263$	−9.89872e10			−1.27579			−0.637894	−	0.770124i	$-0.720194\pi$
$263$	−9.89872e10			−1.27579			−0.637894	+	0.770124i	$0.720194\pi$
$264$		0			0
$265$			8.45093e10i			1.05268i
$266$		0			0
$267$		−	1.21596e10i		−	0.146426i
$268$		0			0
$269$	1.49995e11			1.74659			0.873293	−	0.487195i	$-0.161980\pi$
$269$	1.49995e11			1.74659			0.873293	+	0.487195i	$0.161980\pi$
$270$		0			0
$271$			1.41916e11i			1.59834i	0.601103	+	0.799172i	$0.294728\pi$
$271$			1.41916e11i			1.59834i	−0.601103	+	0.799172i	$0.705272\pi$
$272$		0			0
$273$	−1.76154e11	−	1.46141e11i	−1.91938	−	1.59236i
$274$		0			0
$275$			2.42457e10i			0.255645i
$276$		0			0
$277$	−5.28360e10			−0.539226			−0.269613	−	0.962969i	$-0.586896\pi$
$277$	−5.28360e10			−0.539226			−0.269613	+	0.962969i	$0.586896\pi$
$278$		0			0
$279$			3.88828e10i			0.384183i
$280$		0			0
$281$			3.28951e10i			0.314740i	0.987540	+	0.157370i	$0.0503016\pi$
$281$			3.28951e10i			0.314740i	−0.987540	+	0.157370i	$0.949698\pi$
$282$		0			0
$283$	−1.71935e11			−1.59340			−0.796702	−	0.604372i	$-0.793424\pi$
$283$	−1.71935e11			−1.59340			−0.796702	+	0.604372i	$0.793424\pi$
$284$		0			0
$285$	5.53389e10			0.496853
$286$		0			0
$287$	−9.20780e10			−0.801101
$288$		0			0
$289$	−8.52819e10			−0.719145
$290$		0			0
$291$		−	2.97875e11i		−	2.43510i
$292$		0			0
$293$		−	1.80544e11i		−	1.43113i	−0.698548	−	0.715563i	$-0.746170\pi$
$293$		−	1.80544e11i		−	1.43113i	0.698548	−	0.715563i	$-0.253830\pi$
$294$		0			0
$295$	9.20684e10			0.707802
$296$		0			0
$297$			2.68759e10i			0.200429i
$298$		0			0
$299$	−8.09037e10	+	9.75187e10i	−0.585394	+	0.705615i
$300$		0			0
$301$			1.39120e11i			0.976879i
$302$		0			0
$303$	−1.72857e11			−1.17813
$304$		0			0
$305$			1.45367e9i			0.00961868i
$306$		0			0
$307$			2.74566e11i			1.76410i	0.471155	+	0.882050i	$0.343837\pi$
$307$			2.74566e11i			1.76410i	−0.471155	+	0.882050i	$0.656163\pi$
$308$		0			0
$309$	9.60422e10			0.599306
$310$		0			0
$311$	2.51067e11			1.52183			0.760917	−	0.648849i	$-0.224749\pi$
$311$	2.51067e11			1.52183			0.760917	+	0.648849i	$0.224749\pi$
$312$		0			0
$313$	1.97868e11			1.16527			0.582633	−	0.812735i	$-0.302022\pi$
$313$	1.97868e11			1.16527			0.582633	+	0.812735i	$0.302022\pi$
$314$		0			0
$315$	1.33335e11			0.763036
$316$		0			0
$317$			7.71521e10i			0.429122i	0.976711	+	0.214561i	$0.0688321\pi$
$317$			7.71521e10i			0.429122i	−0.976711	+	0.214561i	$0.931168\pi$
$318$		0			0
$319$			8.50211e10i			0.459694i
$320$		0			0
$321$	1.30552e11			0.686296
$322$		0			0
$323$		−	6.47285e10i		−	0.330890i
$324$		0			0
$325$	7.83658e10	−	9.44596e10i	0.389629	−	0.469646i
$326$		0			0
$327$		−	3.73399e11i		−	1.80596i
$328$		0			0
$329$	−1.17342e10			−0.0552170
$330$		0			0
$331$		−	2.43934e10i		−	0.111698i	−0.998439	−	0.0558491i	$-0.982213\pi$
$331$		−	2.43934e10i		−	0.111698i	0.998439	−	0.0558491i	$-0.0177866\pi$
$332$		0			0
$333$		−	1.56856e10i		−	0.0699042i
$334$		0			0
$335$	1.91232e11			0.829581
$336$		0			0
$337$	2.20289e11			0.930373			0.465187	−	0.885213i	$-0.345987\pi$
$337$	2.20289e11			0.930373			0.465187	+	0.885213i	$0.345987\pi$
$338$		0			0
$339$	5.58512e11			2.29686
$340$		0			0
$341$	−6.43340e10			−0.257659
$342$		0			0
$343$		−	9.16995e11i		−	3.57720i
$344$		0			0
$345$		−	1.91982e11i		−	0.729583i
$346$		0			0
$347$	−9.62069e10			−0.356224			−0.178112	−	0.984010i	$-0.556999\pi$
$347$	−9.62069e10			−0.356224			−0.178112	+	0.984010i	$0.556999\pi$
$348$		0			0
$349$			3.83556e11i			1.38393i	0.721931	+	0.691965i	$0.243255\pi$
$349$			3.83556e11i			1.38393i	−0.721931	+	0.691965i	$0.756745\pi$
$350$		0			0
$351$	8.68670e10	−	1.04707e11i	0.305473	−	0.368207i
$352$		0			0
$353$			3.25469e11i			1.11564i	0.829963	+	0.557818i	$0.188361\pi$
$353$			3.25469e11i			1.11564i	−0.829963	+	0.557818i	$0.811639\pi$
$354$		0			0
$355$	5.22309e10			0.174542
$356$		0			0
$357$		−	4.05630e11i		−	1.32167i
$358$		0			0
$359$		−	1.48039e11i		−	0.470382i	−0.971949	−	0.235191i	$-0.924428\pi$
$359$		−	1.48039e11i		−	0.470382i	0.971949	−	0.235191i	$-0.0755716\pi$
$360$		0			0
$361$	1.96891e11			0.610160
$362$		0			0
$363$	−3.47654e11			−1.05092
$364$		0			0
$365$	3.46277e10			0.102119
$366$		0			0
$367$	−6.02521e11			−1.73370			−0.866851	−	0.498567i	$-0.833860\pi$
$367$	−6.02521e11			−1.73370			−0.866851	+	0.498567i	$0.833860\pi$
$368$		0			0
$369$			9.10841e10i			0.255755i
$370$		0			0
$371$		−	1.20386e12i		−	3.29909i
$372$		0			0
$373$	3.39160e10			0.0907224			0.0453612	−	0.998971i	$-0.485556\pi$
$373$	3.39160e10			0.0907224			0.0453612	+	0.998971i	$0.485556\pi$
$374$		0			0
$375$			4.90697e11i			1.28137i
$376$		0			0
$377$	2.74801e11	−	3.31236e11i	0.700619	−	0.844504i
$378$		0			0
$379$			5.05542e11i			1.25858i	0.777170	+	0.629290i	$0.216654\pi$
$379$			5.05542e11i			1.25858i	−0.777170	+	0.629290i	$0.783346\pi$
$380$		0			0
$381$	−3.06445e11			−0.745058
$382$		0			0
$383$			4.79295e11i			1.13817i	0.822278	+	0.569087i	$0.192703\pi$
$383$			4.79295e11i			1.13817i	−0.822278	+	0.569087i	$0.807297\pi$
$384$		0			0
$385$			2.20610e11i			0.511744i
$386$		0			0
$387$	1.37619e11			0.311873
$388$		0			0
$389$	2.51652e11			0.557220			0.278610	−	0.960404i	$-0.410126\pi$
$389$	2.51652e11			0.557220			0.278610	+	0.960404i	$0.410126\pi$
$390$		0			0
$391$	−2.24556e11			−0.485881
$392$		0			0
$393$	1.14675e12			2.42494
$394$		0			0
$395$			1.92032e10i			0.0396906i
$396$		0			0
$397$			4.05274e11i			0.818826i	0.912349	+	0.409413i	$0.134267\pi$
$397$			4.05274e11i			0.818826i	−0.912349	+	0.409413i	$0.865733\pi$
$398$		0			0
$399$	−7.88320e11			−1.55713
$400$		0			0
$401$			9.16320e11i			1.76969i	0.465885	+	0.884845i	$0.345736\pi$
$401$			9.16320e11i			1.76969i	−0.465885	+	0.884845i	$0.654264\pi$
$402$		0			0
$403$	2.50640e11	+	2.07937e11i	0.473346	+	0.392698i
$404$		0			0
$405$			4.17283e11i			0.770696i
$406$		0			0
$407$	2.59529e10			0.0468825
$408$		0			0
$409$		−	4.36484e11i		−	0.771283i	−0.922649	−	0.385641i	$-0.873980\pi$
$409$		−	4.36484e11i		−	0.771283i	0.922649	−	0.385641i	$-0.126020\pi$
$410$		0			0
$411$			9.28678e11i			1.60538i
$412$		0			0
$413$	−1.31155e12			−2.21824
$414$		0			0
$415$	−5.40185e11			−0.893976
$416$		0			0
$417$	−5.50461e11			−0.891486
$418$		0			0
$419$	5.55256e11			0.880096			0.440048	−	0.897974i	$-0.354961\pi$
$419$	5.55256e11			0.880096			0.440048	+	0.897974i	$0.354961\pi$
$420$		0			0
$421$		−	1.10385e12i		−	1.71254i	−0.516532	−	0.856268i	$-0.672777\pi$
$421$		−	1.10385e12i		−	1.71254i	0.516532	−	0.856268i	$-0.327223\pi$
$422$		0			0
$423$			1.16076e10i			0.0176283i
$424$		0			0
$425$	2.17512e11			0.323395
$426$		0			0
$427$		−	2.07080e10i		−	0.0301448i
$428$		0			0
$429$	−2.88312e11	−	2.39190e11i	−0.410965	−	0.340946i
$430$		0			0
$431$		−	6.45409e11i		−	0.900922i	−0.892796	−	0.450461i	$-0.851260\pi$
$431$		−	6.45409e11i		−	0.900922i	0.892796	−	0.450461i	$-0.148740\pi$
$432$		0			0
$433$	5.94512e11			0.812765			0.406383	−	0.913703i	$-0.366790\pi$
$433$	5.94512e11			0.812765			0.406383	+	0.913703i	$0.366790\pi$
$434$		0			0
$435$			6.52094e11i			0.873189i
$436$		0			0
$437$			4.36414e11i			0.572443i
$438$		0			0
$439$	−5.14469e11			−0.661102			−0.330551	−	0.943788i	$-0.607235\pi$
$439$	−5.14469e11			−0.661102			−0.330551	+	0.943788i	$0.607235\pi$
$440$		0			0
$441$	−1.40325e12			−1.76669
$442$		0			0
$443$	−6.20213e11			−0.765110			−0.382555	−	0.923933i	$-0.624956\pi$
$443$	−6.20213e11			−0.765110			−0.382555	+	0.923933i	$0.624956\pi$
$444$		0			0
$445$	5.93285e10			0.0717205
$446$		0			0
$447$			6.41278e11i			0.759736i
$448$		0			0
$449$		−	1.46150e9i		−	0.00169703i	−1.00000		0.000848514i	$-0.999730\pi$
$449$		−	1.46150e9i		−	0.00169703i	1.00000		0.000848514i	$-0.000270090\pi$
$450$		0			0
$451$	−1.50704e11			−0.171527
$452$		0			0
$453$		−	6.71737e11i		−	0.749475i
$454$		0			0
$455$	7.13045e11	−	8.59482e11i	0.779949	−	0.940124i
$456$		0			0
$457$			2.62753e11i			0.281790i	0.990025	+	0.140895i	$0.0449979\pi$
$457$			2.62753e11i			0.281790i	−0.990025	+	0.140895i	$0.955002\pi$
$458$		0			0
$459$	2.41108e11			0.253545
$460$		0			0
$461$		−	5.56713e11i		−	0.574087i	−0.957918	−	0.287043i	$-0.907328\pi$
$461$		−	5.56713e11i		−	0.574087i	0.957918	−	0.287043i	$-0.0926724\pi$
$462$		0			0
$463$			2.92683e11i			0.295994i	0.988988	+	0.147997i	$0.0472825\pi$
$463$			2.92683e11i			0.295994i	−0.988988	+	0.147997i	$0.952717\pi$
$464$		0			0
$465$	−4.93428e11			−0.489424
$466$		0			0
$467$	1.47515e12			1.43519			0.717595	−	0.696461i	$-0.245243\pi$
$467$	1.47515e12			1.43519			0.717595	+	0.696461i	$0.245243\pi$
$468$		0			0
$469$	−2.72416e12			−2.59989
$470$		0			0
$471$	2.22473e12			2.08297
$472$		0			0
$473$			2.27699e11i			0.209163i
$474$		0			0
$475$		−	4.22724e11i		−	0.381009i
$476$		0			0
$477$	−1.19087e12			−1.05325
$478$		0			0
$479$			4.83412e11i			0.419573i	0.977747	+	0.209787i	$0.0672769\pi$
$479$			4.83412e11i			0.419573i	−0.977747	+	0.209787i	$0.932723\pi$
$480$		0			0
$481$	−1.01111e11	−	8.38836e10i	−0.0861278	−	0.0714536i
$482$		0			0
$483$			2.73485e12i			2.28650i
$484$		0			0
$485$	1.45338e12			1.19273
$486$		0			0
$487$			1.27915e11i			0.103049i	0.998672	+	0.0515243i	$0.0164080\pi$
$487$			1.27915e11i			0.103049i	−0.998672	+	0.0515243i	$0.983592\pi$
$488$		0			0
$489$			1.18612e12i			0.938074i
$490$		0			0
$491$	1.76741e12			1.37237			0.686186	−	0.727426i	$-0.259284\pi$
$491$	1.76741e12			1.37237			0.686186	+	0.727426i	$0.259284\pi$
$492$		0			0
$493$	7.62738e11			0.581519
$494$		0			0
$495$	2.18229e11			0.163376
$496$		0			0
$497$	−7.44047e11			−0.547011
$498$		0			0
$499$			9.65216e11i			0.696903i	0.937327	+	0.348451i	$0.113292\pi$
$499$			9.65216e11i			0.696903i	−0.937327	+	0.348451i	$0.886708\pi$
$500$		0			0
$501$		−	1.06039e12i		−	0.751963i
$502$		0			0
$503$	−2.40140e12			−1.67266			−0.836331	−	0.548224i	$-0.815304\pi$
$503$	−2.40140e12			−1.67266			−0.836331	+	0.548224i	$0.815304\pi$
$504$		0			0
$505$		−	8.43394e11i		−	0.577058i
$506$		0			0
$507$	3.50144e11	+	1.86373e12i	0.235348	+	1.25270i
$508$		0			0
$509$		−	2.19424e12i		−	1.44895i	−0.689300	−	0.724476i	$-0.742082\pi$
$509$		−	2.19424e12i		−	1.44895i	0.689300	−	0.724476i	$-0.257918\pi$
$510$		0			0
$511$	−4.93284e11			−0.320039
$512$		0			0
$513$		−	4.68581e11i		−	0.298715i
$514$		0			0
$515$			4.68604e11i			0.293544i
$516$		0			0
$517$	−1.92054e10			−0.0118227
$518$		0			0
$519$	−2.05112e12			−1.24090
$520$		0			0
$521$	−4.95469e11			−0.294609			−0.147305	−	0.989091i	$-0.547060\pi$
$521$	−4.95469e11			−0.294609			−0.147305	+	0.989091i	$0.547060\pi$
$522$		0			0
$523$	−2.61822e12			−1.53020			−0.765100	−	0.643912i	$-0.777310\pi$
$523$	−2.61822e12			−1.53020			−0.765100	+	0.643912i	$0.777310\pi$
$524$		0			0
$525$		−	2.64906e12i		−	1.52186i
$526$		0			0
$527$			5.77150e11i			0.325942i
$528$		0			0
$529$	−2.87143e11			−0.159422
$530$		0			0
$531$			1.29739e12i			0.708182i
$532$		0			0
$533$	5.87133e11	+	4.87099e11i	0.315111	+	0.261424i
$534$		0			0
$535$			6.36983e11i			0.336152i
$536$		0			0
$537$	−1.66849e12			−0.865841
$538$		0			0
$539$		−	2.32176e12i		−	1.18486i
$540$		0			0
$541$			2.57741e12i			1.29359i	0.762665	+	0.646793i	$0.223890\pi$
$541$			2.57741e12i			1.29359i	−0.762665	+	0.646793i	$0.776110\pi$
$542$		0			0
$543$	2.55666e12			1.26204
$544$		0			0
$545$	1.82187e12			0.884571
$546$		0			0
$547$	1.86894e12			0.892589			0.446295	−	0.894886i	$-0.352743\pi$
$547$	1.86894e12			0.892589			0.446295	+	0.894886i	$0.352743\pi$
$548$		0			0
$549$	−2.04844e10			−0.00962385
$550$		0			0
$551$		−	1.48234e12i		−	0.685119i
$552$		0			0
$553$		−	2.73556e11i		−	0.124390i
$554$		0			0
$555$	1.99053e11			0.0890534
$556$		0			0
$557$		−	1.81436e12i		−	0.798684i	−0.916802	−	0.399342i	$-0.869239\pi$
$557$		−	1.81436e12i		−	0.798684i	0.916802	−	0.399342i	$-0.130761\pi$
$558$		0			0
$559$	7.35956e11	−	8.87097e11i	0.318785	−	0.384254i
$560$		0			0
$561$		−	6.63896e11i		−	0.282987i
$562$		0			0
$563$	5.15732e11			0.216340			0.108170	−	0.994132i	$-0.465501\pi$
$563$	5.15732e11			0.216340			0.108170	+	0.994132i	$0.465501\pi$
$564$		0			0
$565$			2.72507e12i			1.12502i
$566$		0			0
$567$		−	5.94433e12i		−	2.41535i
$568$		0			0
$569$	−1.32815e12			−0.531180			−0.265590	−	0.964086i	$-0.585567\pi$
$569$	−1.32815e12			−0.531180			−0.265590	+	0.964086i	$0.585567\pi$
$570$		0			0
$571$	3.17589e12			1.25027			0.625133	−	0.780518i	$-0.285045\pi$
$571$	3.17589e12			1.25027			0.625133	+	0.780518i	$0.285045\pi$
$572$		0			0
$573$	3.69743e12			1.43286
$574$		0			0
$575$	−1.46652e12			−0.559476
$576$		0			0
$577$			4.86230e12i			1.82621i	0.407725	+	0.913105i	$0.366322\pi$
$577$			4.86230e12i			1.82621i	−0.407725	+	0.913105i	$0.633678\pi$
$578$		0			0
$579$			5.04581e11i			0.186585i
$580$		0			0
$581$	7.69511e12			2.80170
$582$		0			0
$583$		−	1.97037e12i		−	0.706380i
$584$		0			0
$585$	−8.50205e11	−	7.05349e11i	−0.300139	−	0.249002i
$586$		0			0
$587$			2.28065e12i			0.792843i	0.918069	+	0.396421i	$0.129748\pi$
$587$			2.28065e12i			0.792843i	−0.918069	+	0.396421i	$0.870252\pi$
$588$		0			0
$589$	1.12166e12			0.384010
$590$		0			0
$591$		−	5.57819e12i		−	1.88083i
$592$		0			0
$593$		−	2.89812e11i		−	0.0962433i	−0.998841	−	0.0481216i	$-0.984676\pi$
$593$		−	2.89812e11i		−	0.0962433i	0.998841	−	0.0481216i	$-0.0153235\pi$
$594$		0			0
$595$	1.97913e12			0.647363
$596$		0			0
$597$	−2.84610e12			−0.916993
$598$		0			0
$599$	1.19267e12			0.378528			0.189264	−	0.981926i	$-0.439390\pi$
$599$	1.19267e12			0.378528			0.189264	+	0.981926i	$0.439390\pi$
$600$		0			0
$601$	1.34698e12			0.421140			0.210570	−	0.977579i	$-0.432468\pi$
$601$	1.34698e12			0.421140			0.210570	+	0.977579i	$0.432468\pi$
$602$		0			0
$603$			2.69476e12i			0.830027i
$604$		0			0
$605$		−	1.69626e12i		−	0.514746i
$606$		0			0
$607$	1.07605e11			0.0321724			0.0160862	−	0.999871i	$-0.494879\pi$
$607$	1.07605e11			0.0321724			0.0160862	+	0.999871i	$0.494879\pi$
$608$		0			0
$609$		−	9.28929e12i		−	2.73656i
$610$		0			0
$611$	7.48230e10	+	6.20749e10i	0.0217195	+	0.0180190i
$612$		0			0
$613$			7.68586e11i			0.219847i	0.993940	+	0.109923i	$0.0350606\pi$
$613$			7.68586e11i			0.219847i	−0.993940	+	0.109923i	$0.964939\pi$
$614$		0			0
$615$	−1.15587e12			−0.325815
$616$		0			0
$617$		−	1.45026e12i		−	0.402868i	−0.979502	−	0.201434i	$-0.935440\pi$
$617$		−	1.45026e12i		−	0.402868i	0.979502	−	0.201434i	$-0.0645602\pi$
$618$		0			0
$619$		−	4.23653e12i		−	1.15985i	−0.814669	−	0.579926i	$-0.803081\pi$
$619$		−	4.23653e12i		−	1.15985i	0.814669	−	0.579926i	$-0.196919\pi$
$620$		0			0
$621$	−1.62560e12			−0.438635
$622$		0			0
$623$	−8.45154e11			−0.224771
$624$		0			0
$625$	−6.63500e10			−0.0173933
$626$		0			0
$627$	−1.29025e12			−0.333403
$628$		0			0
$629$		−	2.32827e11i		−	0.0593070i
$630$		0			0
$631$		−	4.40245e12i		−	1.10551i	−0.833344	−	0.552754i	$-0.813577\pi$
$631$		−	4.40245e12i		−	1.10551i	0.833344	−	0.552754i	$-0.186423\pi$
$632$		0			0
$633$	9.76121e12			2.41650
$634$		0			0
$635$		−	1.49519e12i		−	0.364935i
$636$		0			0
$637$	−7.50427e12	+	9.04540e12i	−1.80585	+	2.17671i
$638$		0			0
$639$			7.36016e11i			0.174636i
$640$		0			0
$641$	2.29586e12			0.537136			0.268568	−	0.963261i	$-0.413450\pi$
$641$	2.29586e12			0.537136			0.268568	+	0.963261i	$0.413450\pi$
$642$		0			0
$643$			6.50802e12i			1.50141i	0.660637	+	0.750705i	$0.270286\pi$
$643$			6.50802e12i			1.50141i	−0.660637	+	0.750705i	$0.729714\pi$
$644$		0			0
$645$			1.74640e12i			0.397306i
$646$		0			0
$647$	−3.71987e12			−0.834561			−0.417280	−	0.908778i	$-0.637017\pi$
$647$	−3.71987e12			−0.834561			−0.417280	+	0.908778i	$0.637017\pi$
$648$		0			0
$649$	−2.14661e12			−0.474955
$650$		0			0
$651$	7.02904e12			1.53385
$652$		0			0
$653$	5.52818e12			1.18980			0.594898	−	0.803801i	$-0.297192\pi$
$653$	5.52818e12			1.18980			0.594898	+	0.803801i	$0.297192\pi$
$654$		0			0
$655$			5.59515e12i			1.18775i
$656$		0			0
$657$			4.87960e11i			0.102174i
$658$		0			0
$659$	−8.04400e11			−0.166145			−0.0830726	−	0.996543i	$-0.526473\pi$
$659$	−8.04400e11			−0.166145			−0.0830726	+	0.996543i	$0.526473\pi$
$660$		0			0
$661$			3.35982e11i			0.0684557i	0.999414	+	0.0342278i	$0.0108972\pi$
$661$			3.35982e11i			0.0684557i	−0.999414	+	0.0342278i	$0.989103\pi$
$662$		0			0
$663$	−2.14581e12	+	2.58649e12i	−0.431301	+	0.519876i
$664$		0			0
$665$		−	3.84633e12i		−	0.762693i
$666$		0			0
$667$	−5.14255e12			−1.00603
$668$		0			0
$669$			1.00956e13i			1.94857i
$670$		0			0
$671$		−	3.38928e10i		−	0.00645440i
$672$		0			0
$673$	−6.72025e12			−1.26275			−0.631376	−	0.775477i	$-0.717509\pi$
$673$	−6.72025e12			−1.26275			−0.631376	+	0.775477i	$0.717509\pi$
$674$		0			0
$675$	1.57461e12			0.291948
$676$		0			0
$677$	−7.28330e11			−0.133254			−0.0666268	−	0.997778i	$-0.521224\pi$
$677$	−7.28330e11			−0.133254			−0.0666268	+	0.997778i	$0.521224\pi$
$678$		0			0
$679$	−2.07039e13			−3.73798
$680$		0			0
$681$		−	2.96196e12i		−	0.527736i
$682$		0			0
$683$			2.33811e12i			0.411123i	0.978644	+	0.205561i	$0.0659020\pi$
$683$			2.33811e12i			0.411123i	−0.978644	+	0.205561i	$0.934098\pi$
$684$		0			0
$685$	−4.53116e12			−0.786324
$686$		0			0
$687$		−	7.91860e12i		−	1.35626i
$688$		0			0
$689$	−6.36852e12	+	7.67640e12i	−1.07659	+	1.29769i
$690$		0			0
$691$			1.87535e12i			0.312918i	0.987684	+	0.156459i	$0.0500080\pi$
$691$			1.87535e12i			0.312918i	−0.987684	+	0.156459i	$0.949992\pi$
$692$		0			0
$693$	−3.10875e12			−0.512019
$694$		0			0
$695$		−	2.68578e12i		−	0.436656i
$696$		0			0
$697$			1.35199e12i			0.216984i
$698$		0			0
$699$	9.17523e12			1.45368
$700$		0			0
$701$	7.63818e11			0.119470			0.0597350	−	0.998214i	$-0.480974\pi$
$701$	7.63818e11			0.119470			0.0597350	+	0.998214i	$0.480974\pi$
$702$		0			0
$703$	−4.52488e11			−0.0698727
$704$		0			0
$705$	−1.47302e11			−0.0224573
$706$		0			0
$707$			1.20144e13i			1.80849i
$708$		0			0
$709$			3.19158e12i			0.474348i	0.971467	+	0.237174i	$0.0762212\pi$
$709$			3.19158e12i			0.474348i	−0.971467	+	0.237174i	$0.923779\pi$
$710$		0			0
$711$	−2.70604e11			−0.0397119
$712$		0			0
$713$		−	3.89127e12i		−	0.563883i
$714$		0			0
$715$	1.16704e12	−	1.40672e12i	0.166998	−	0.201293i
$716$		0			0
$717$			6.62937e12i			0.936776i
$718$		0			0
$719$	−3.34548e12			−0.466852			−0.233426	−	0.972375i	$-0.574994\pi$
$719$	−3.34548e12			−0.466852			−0.233426	+	0.972375i	$0.574994\pi$
$720$		0			0
$721$		−	6.67542e12i		−	0.919962i
$722$		0			0
$723$			3.01601e12i			0.410498i
$724$		0			0
$725$	4.98123e12			0.669600
$726$		0			0
$727$	−7.76320e12			−1.03071			−0.515354	−	0.856977i	$-0.672340\pi$
$727$	−7.76320e12			−1.03071			−0.515354	+	0.856977i	$0.672340\pi$
$728$		0			0
$729$	−1.23001e12			−0.161300
$730$		0			0
$731$	2.04272e12			0.264594
$732$		0			0
$733$			4.28627e12i			0.548417i	0.961670	+	0.274209i	$0.0884159\pi$
$733$			4.28627e12i			0.548417i	−0.961670	+	0.274209i	$0.911584\pi$
$734$		0			0
$735$		−	1.78074e13i		−	2.25065i
$736$		0			0
$737$	−4.45865e12			−0.556672
$738$		0			0
$739$		−	2.44262e12i		−	0.301270i	−0.988589	−	0.150635i	$-0.951868\pi$
$739$		−	2.44262e12i		−	0.301270i	0.988589	−	0.150635i	$-0.0481319\pi$
$740$		0			0
$741$	5.02671e12	+	4.17027e12i	0.612494	+	0.508139i
$742$		0			0
$743$			7.65598e12i			0.921618i	0.887499	+	0.460809i	$0.152441\pi$
$743$			7.65598e12i			0.921618i	−0.887499	+	0.460809i	$0.847559\pi$
$744$		0			0
$745$	−3.12889e12			−0.372124
$746$		0			0
$747$		−	7.61206e12i		−	0.894457i
$748$		0			0
$749$		−	9.07404e12i		−	1.05350i
$750$		0			0
$751$	7.61452e12			0.873500			0.436750	−	0.899583i	$-0.356129\pi$
$751$	7.61452e12			0.873500			0.436750	+	0.899583i	$0.356129\pi$
$752$		0			0
$753$	1.10685e13			1.25461
$754$		0			0
$755$	3.27751e12			0.367098
$756$		0			0
$757$	3.87823e12			0.429242			0.214621	−	0.976697i	$-0.431148\pi$
$757$	3.87823e12			0.429242			0.214621	+	0.976697i	$0.431148\pi$
$758$		0			0
$759$			4.47613e12i			0.489570i
$760$		0			0
$761$			7.72485e12i			0.834947i	0.908689	+	0.417473i	$0.137084\pi$
$761$			7.72485e12i			0.834947i	−0.908689	+	0.417473i	$0.862916\pi$
$762$		0			0
$763$	−2.59531e13			−2.77223
$764$		0			0
$765$		−	1.95777e12i		−	0.206673i
$766$		0			0
$767$	8.36304e12	+	6.93817e12i	0.872540	+	0.723879i
$768$		0			0
$769$			1.32730e13i			1.36868i	0.729164	+	0.684339i	$0.239909\pi$
$769$			1.32730e13i			1.36868i	−0.729164	+	0.684339i	$0.760091\pi$
$770$		0			0
$771$	6.90705e12			0.703960
$772$		0			0
$773$		−	4.57301e12i		−	0.460675i	−0.973111	−	0.230337i	$-0.926017\pi$
$773$		−	4.57301e12i		−	0.460675i	0.973111	−	0.230337i	$-0.0739829\pi$
$774$		0			0
$775$			3.76920e12i			0.375312i
$776$		0			0
$777$	−2.83558e12			−0.279092
$778$		0			0
$779$	2.62753e12			0.255640
$780$		0			0
$781$	−1.21778e12			−0.117123
$782$		0			0
$783$	5.52160e12			0.524973
$784$		0			0
$785$			1.08548e13i			1.02025i
$786$		0			0
$787$			1.76744e13i			1.64232i	0.570700	+	0.821159i	$0.306672\pi$
$787$			1.76744e13i			1.64232i	−0.570700	+	0.821159i	$0.693328\pi$
$788$		0			0
$789$	1.77013e13			1.62614
$790$		0			0
$791$		−	3.88195e13i		−	3.52578i
$792$		0			0
$793$	−1.09547e11	+	1.32044e11i	−0.00983715	+	0.0118574i
$794$		0			0
$795$		−	1.51123e13i		−	1.34177i
$796$		0			0
$797$	1.08571e13			0.953125			0.476563	−	0.879140i	$-0.341883\pi$
$797$	1.08571e13			0.953125			0.476563	+	0.879140i	$0.341883\pi$
$798$		0			0
$799$			1.72295e11i			0.0149559i
$800$		0			0
$801$			8.36032e11i			0.0717590i
$802$		0			0
$803$	−8.07359e11			−0.0685247
$804$		0			0
$805$	−1.33437e13			−1.11994
$806$		0			0
$807$	−2.68226e13			−2.22623
$808$		0			0
$809$	7.43378e12			0.610157			0.305078	−	0.952327i	$-0.401317\pi$
$809$	7.43378e12			0.610157			0.305078	+	0.952327i	$0.401317\pi$
$810$		0			0
$811$		−	2.89877e12i		−	0.235299i	−0.993055	−	0.117649i	$-0.962464\pi$
$811$		−	2.89877e12i		−	0.235299i	0.993055	−	0.117649i	$-0.0375359\pi$
$812$		0			0
$813$		−	2.53780e13i		−	2.03728i
$814$		0			0
$815$	−5.78724e12			−0.459475
$816$		0			0
$817$		−	3.96992e12i		−	0.311733i
$818$		0			0
$819$	1.21115e13	+	1.00479e13i	0.940630	+	0.780368i
$820$		0			0
$821$		−	9.41158e11i		−	0.0722967i	−0.999346	−	0.0361484i	$-0.988491\pi$
$821$		−	9.41158e11i		−	0.0722967i	0.999346	−	0.0361484i	$-0.0115089\pi$
$822$		0			0
$823$	−1.18579e13			−0.900963			−0.450482	−	0.892786i	$-0.648748\pi$
$823$	−1.18579e13			−0.900963			−0.450482	+	0.892786i	$0.648748\pi$
$824$		0			0
$825$		−	4.33572e12i		−	0.325850i
$826$		0			0
$827$			7.69904e12i			0.572350i	0.958177	+	0.286175i	$0.0923839\pi$
$827$			7.69904e12i			0.572350i	−0.958177	+	0.286175i	$0.907616\pi$
$828$		0			0
$829$	−6.20673e12			−0.456423			−0.228211	−	0.973612i	$-0.573288\pi$
$829$	−6.20673e12			−0.456423			−0.228211	+	0.973612i	$0.573288\pi$
$830$		0			0
$831$	9.44835e12			0.687308
$832$		0			0
$833$	−2.08288e13			−1.49887
$834$		0			0
$835$	5.17381e12			0.368316
$836$		0			0
$837$			4.17809e12i			0.294248i
$838$		0			0
$839$		−	6.25209e12i		−	0.435609i	−0.975992	−	0.217804i	$-0.930111\pi$
$839$		−	6.25209e12i		−	0.435609i	0.975992	−	0.217804i	$-0.0698895\pi$
$840$		0			0
$841$	2.96024e12			0.204054
$842$		0			0
$843$		−	5.88243e12i		−	0.401174i
$844$		0			0
$845$	−9.09344e12	+	1.70841e12i	−0.613582	+	0.115275i
$846$		0			0
$847$			2.41638e13i			1.61320i
$848$		0			0
$849$	3.07462e13			2.03098
$850$		0			0
$851$			1.56977e12i			0.102602i
$852$		0			0
$853$		−	1.05576e13i		−	0.682804i	−0.939917	−	0.341402i	$-0.889098\pi$
$853$		−	1.05576e13i		−	0.682804i	0.939917	−	0.341402i	$-0.110902\pi$
$854$		0			0
$855$	−3.80482e12			−0.243493
$856$		0			0
$857$	−1.84183e13			−1.16637			−0.583184	−	0.812340i	$-0.698194\pi$
$857$	−1.84183e13			−1.16637			−0.583184	+	0.812340i	$0.698194\pi$
$858$		0			0
$859$	7.81498e12			0.489732			0.244866	−	0.969557i	$-0.421256\pi$
$859$	7.81498e12			0.489732			0.244866	+	0.969557i	$0.421256\pi$
$860$		0			0
$861$	1.64658e13			1.02110
$862$		0			0
$863$		−	2.64440e13i		−	1.62285i	−0.584456	−	0.811425i	$-0.698692\pi$
$863$		−	2.64440e13i		−	1.62285i	0.584456	−	0.811425i	$-0.301308\pi$
$864$		0			0
$865$		−	1.00077e13i		−	0.607803i
$866$		0			0
$867$	1.52505e13			0.916636
$868$		0			0
$869$		−	4.47731e11i		−	0.0266335i
$870$		0			0
$871$	1.73706e13	+	1.44110e13i	1.02266	+	0.848424i
$872$		0			0
$873$			2.04804e13i			1.19337i
$874$		0			0
$875$	3.41060e13			1.96695
$876$		0			0
$877$		−	4.20549e12i		−	0.240059i	−0.992770	−	0.120030i	$-0.961701\pi$
$877$		−	4.20549e12i		−	0.240059i	0.992770	−	0.120030i	$-0.0382990\pi$
$878$		0			0
$879$			3.22855e13i			1.82414i
$880$		0			0
$881$	−2.01404e13			−1.12636			−0.563180	−	0.826334i	$-0.690422\pi$
$881$	−2.01404e13			−1.12636			−0.563180	+	0.826334i	$0.690422\pi$
$882$		0			0
$883$	6.51095e12			0.360430			0.180215	−	0.983627i	$-0.442321\pi$
$883$	6.51095e12			0.360430			0.180215	+	0.983627i	$0.442321\pi$
$884$		0			0
$885$	−1.64641e13			−0.902177
$886$		0			0
$887$	3.19150e13			1.73116			0.865582	−	0.500767i	$-0.166949\pi$
$887$	3.19150e13			1.73116			0.865582	+	0.500767i	$0.166949\pi$
$888$		0			0
$889$			2.12995e13i			1.14370i
$890$		0			0
$891$		−	9.72911e12i		−	0.517158i
$892$		0			0
$893$	3.34846e11			0.0176203
$894$		0			0
$895$		−	8.14080e12i		−	0.424095i
$896$		0			0
$897$	1.44675e13	−	1.74387e13i	0.746154	−	0.899389i
$898$		0			0
$899$			1.32173e13i			0.674874i
$900$		0			0
$901$	−1.76765e13			−0.893581
$902$		0			0
$903$		−	2.48780e13i		−	1.24515i
$904$		0			0
$905$			1.24743e13i			0.618157i
$906$		0			0
$907$	7.63795e12			0.374752			0.187376	−	0.982288i	$-0.440002\pi$
$907$	7.63795e12			0.374752			0.187376	+	0.982288i	$0.440002\pi$
$908$		0			0
$909$	1.18847e13			0.577368
$910$		0			0
$911$	2.29839e13			1.10558			0.552791	−	0.833320i	$-0.313563\pi$
$911$	2.29839e13			1.10558			0.552791	+	0.833320i	$0.313563\pi$
$912$		0			0
$913$	1.25946e13			0.599883
$914$		0			0
$915$		−	2.59950e11i		−	0.0122601i
$916$		0			0
$917$		−	7.97047e13i		−	3.72239i
$918$		0			0
$919$	−3.29468e13			−1.52368			−0.761840	−	0.647765i	$-0.775704\pi$
$919$	−3.29468e13			−1.52368			−0.761840	+	0.647765i	$0.775704\pi$
$920$		0			0
$921$		−	4.90989e13i		−	2.24856i
$922$		0			0
$923$	4.74440e12	+	3.93606e12i	0.215166	+	0.178506i
$924$		0			0
$925$		−	1.52053e12i		−	0.0682900i
$926$		0			0
$927$	−6.60337e12			−0.293702
$928$		0			0
$929$		−	3.70200e13i		−	1.63067i	−0.578991	−	0.815334i	$-0.696553\pi$
$929$		−	3.70200e13i		−	1.63067i	0.578991	−	0.815334i	$-0.303447\pi$
$930$		0			0
$931$			4.04798e13i			1.76589i
$932$		0			0
$933$	−4.48968e13			−1.93976
$934$		0			0
$935$	3.23925e12			0.138609
$936$		0			0
$937$	2.43607e13			1.03243			0.516216	−	0.856458i	$-0.327340\pi$
$937$	2.43607e13			1.03243			0.516216	+	0.856458i	$0.327340\pi$
$938$		0			0
$939$	−3.53835e13			−1.48527
$940$		0			0
$941$		−	1.87203e13i		−	0.778323i	−0.921169	−	0.389162i	$-0.872765\pi$
$941$		−	1.87203e13i		−	0.778323i	0.921169	−	0.389162i	$-0.127235\pi$
$942$		0			0
$943$		−	9.11543e12i		−	0.375383i
$944$		0			0
$945$	1.43273e13			0.584414
$946$		0			0
$947$			2.86292e13i			1.15674i	0.815776	+	0.578369i	$0.196310\pi$
$947$			2.86292e13i			1.15674i	−0.815776	+	0.578369i	$0.803690\pi$
$948$		0			0
$949$	3.14541e12	+	2.60951e12i	0.125887	+	0.104438i
$950$		0			0
$951$		−	1.37966e13i		−	0.546967i
$952$		0			0
$953$	−2.64752e13			−1.03973			−0.519865	−	0.854248i	$-0.674018\pi$
$953$	−2.64752e13			−1.03973			−0.519865	+	0.854248i	$0.674018\pi$
$954$		0			0
$955$			1.80403e13i			0.701825i
$956$		0			0
$957$		−	1.52038e13i		−	0.585934i
$958$		0			0
$959$	6.45479e13			2.46433
$960$		0			0
$961$	1.64384e13			0.621732
$962$		0			0
$963$	−8.97610e12			−0.336333
$964$		0			0
$965$	−2.46193e12			−0.0913908
$966$		0			0
$967$		−	1.18419e13i		−	0.435513i	−0.976003	−	0.217756i	$-0.930126\pi$
$967$		−	1.18419e13i		−	0.435513i	0.976003	−	0.217756i	$-0.0698739\pi$
$968$		0			0
$969$			1.15750e13i			0.421759i
$970$		0			0
$971$	−2.88614e13			−1.04191			−0.520956	−	0.853584i	$-0.674424\pi$
$971$	−2.88614e13			−1.04191			−0.520956	+	0.853584i	$0.674424\pi$
$972$		0			0
$973$			3.82599e13i			1.36847i
$974$		0			0
$975$	−1.40137e13	+	1.68916e13i	−0.496629	+	0.598620i
$976$		0			0
$977$			3.45215e13i			1.21217i	0.795400	+	0.606085i	$0.207261\pi$
$977$			3.45215e13i			1.21217i	−0.795400	+	0.606085i	$0.792739\pi$
$978$		0			0
$979$	−1.38327e12			−0.0481265
$980$		0			0
$981$			2.56730e13i			0.885046i
$982$		0			0
$983$			3.79621e13i			1.29676i	0.761317	+	0.648380i	$0.224553\pi$
$983$			3.79621e13i			1.29676i	−0.761317	+	0.648380i	$0.775447\pi$
$984$		0			0
$985$	2.72168e13			0.921243
$986$		0			0
$987$	2.09836e12			0.0703806
$988$		0			0
$989$	−1.37725e13			−0.457750
$990$		0			0
$991$	−2.05074e13			−0.675428			−0.337714	−	0.941249i	$-0.609654\pi$
$991$	−2.05074e13			−0.675428			−0.337714	+	0.941249i	$0.609654\pi$
$992$		0			0
$993$			4.36213e12i			0.142373i
$994$		0			0
$995$		−	1.38866e13i		−	0.449150i
$996$		0			0
$997$	−5.02628e13			−1.61108			−0.805542	−	0.592538i	$-0.798126\pi$
$997$	−5.02628e13			−1.61108			−0.805542	+	0.592538i	$0.798126\pi$
$998$		0			0
$999$		−	1.68548e12i		−	0.0535400i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	104.10.f.a.25.7	✓	32
4.3	odd	2		208.10.f.d.129.26		32
13.12	even	2	inner	104.10.f.a.25.8	yes	32
52.51	odd	2		208.10.f.d.129.25		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.7	✓	32	1.1	even	1	trivial
104.10.f.a.25.8	yes	32	13.12	even	2	inner
208.10.f.d.129.25		32	52.51	odd	2
208.10.f.d.129.26		32	4.3	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)

\(f(q)\)	\(=\)	\(q-178.824 q^{3} -872.510i q^{5} +12429.2i q^{7} +12295.0 q^{9} +20342.9i q^{11} +(65751.3 - 79254.4i) q^{13} +156026. i q^{15} +182499. q^{17} -354678. i q^{19} -2.22264e6i q^{21} -1.23045e6 q^{23} +1.19185e6 q^{25} +1.32115e6 q^{27} +4.17940e6 q^{29} +3.16248e6i q^{31} -3.63780e6i q^{33} +1.08446e7 q^{35} -1.27577e6i q^{37} +(-1.17579e7 + 1.41726e7i) q^{39} +7.40821e6i q^{41} +1.11930e7 q^{43} -1.07275e7i q^{45} +944086. i q^{47} -1.14131e8 q^{49} -3.26353e7 q^{51} -9.68577e7 q^{53} +1.77494e7 q^{55} +6.34249e7i q^{57} +1.05521e8i q^{59} -1.66607e6 q^{61} +1.52817e8i q^{63} +(-6.91503e7 - 5.73686e7i) q^{65} +2.19175e8i q^{67} +2.20034e8 q^{69} +5.98629e7i q^{71} +3.96875e7i q^{73} -2.13132e8 q^{75} -2.52846e8 q^{77} -2.20092e7 q^{79} -4.78256e8 q^{81} -6.19116e8i q^{83} -1.59232e8i q^{85} -7.47377e8 q^{87} +6.79975e7i q^{89} +(9.85068e8 + 8.17235e8i) q^{91} -5.65527e8i q^{93} -3.09460e8 q^{95} +1.66575e9i q^{97} +2.50117e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} + 3171556 q^{23} - 13526722 q^{25} - 3694974 q^{27} + 8833508 q^{29} - 8281126 q^{35} - 12056860 q^{39} + 89959038 q^{43} - 172344874 q^{49}+ \cdots + 1741143356 q^{95}+O(q^{100}) \) 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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