LMFDB - Embedded newform 208.10.f.d.129.23

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,10,Mod(129,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("208.129"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,162] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$107.127453922$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.23
Character	$\chi$	$=$	208.129
Dual form			208.10.f.d.129.24

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+144.011 q^{3} +1019.43i q^{5} +5657.65i q^{7} +1056.24 q^{9} -66157.5i q^{11} +(43262.5 - 93449.7i) q^{13} +146809. i q^{15} +100620. q^{17} -188252. i q^{19} +814765. i q^{21} -2.22375e6 q^{23} +913897. q^{25} -2.68246e6 q^{27} -5.75947e6 q^{29} +7.19142e6i q^{31} -9.52743e6i q^{33} -5.76755e6 q^{35} +3.71695e6i q^{37} +(6.23029e6 - 1.34578e7i) q^{39} -2.77654e6i q^{41} +1.44942e7 q^{43} +1.07676e6i q^{45} -6.34145e7i q^{47} +8.34464e6 q^{49} +1.44903e7 q^{51} -8.43154e7 q^{53} +6.74427e7 q^{55} -2.71104e7i q^{57} -1.12066e8i q^{59} -1.70009e8 q^{61} +5.97583e6i q^{63} +(9.52651e7 + 4.41029e7i) q^{65} +4.94626e7i q^{67} -3.20245e8 q^{69} -1.17412e8i q^{71} -2.58227e8i q^{73} +1.31611e8 q^{75} +3.74296e8 q^{77} +4.68955e8 q^{79} -4.07095e8 q^{81} -5.71870e8i q^{83} +1.02574e8i q^{85} -8.29428e8 q^{87} +1.75654e8i q^{89} +(5.28706e8 + 2.44764e8i) q^{91} +1.03565e9i q^{93} +1.91909e8 q^{95} -1.58656e8i q^{97} -6.98782e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	144.011			1.02648			0.513240	−	0.858245i	$-0.328445\pi$
$3$	144.011			1.02648			0.513240	+	0.858245i	$0.328445\pi$
$4$		0			0
$5$			1019.43i			0.729442i	0.931117	+	0.364721i	$0.118836\pi$
$5$			1019.43i			0.729442i	−0.931117	+	0.364721i	$0.881164\pi$
$6$		0			0
$7$			5657.65i			0.890625i	0.895375	+	0.445312i	$0.146907\pi$
$7$			5657.65i			0.890625i	−0.895375	+	0.445312i	$0.853093\pi$
$8$		0			0
$9$	1056.24			0.0536625
$10$		0			0
$11$		−	66157.5i		−	1.36242i	−0.732086	−	0.681212i	$-0.761453\pi$
$11$		−	66157.5i		−	1.36242i	0.732086	−	0.681212i	$-0.238547\pi$
$12$		0			0
$13$	43262.5	−	93449.7i	0.420113	−	0.907472i
$13$	43262.5	−	93449.7i	0.420113	−	0.907472i
$14$		0			0
$15$			146809.i			0.748758i
$16$		0			0
$17$	100620.			0.292188			0.146094	−	0.989271i	$-0.453330\pi$
$17$	100620.			0.292188			0.146094	+	0.989271i	$0.453330\pi$
$18$		0			0
$19$		−	188252.i		−	0.331396i	−0.986177	−	0.165698i	$-0.947012\pi$
$19$		−	188252.i		−	0.331396i	0.986177	−	0.165698i	$-0.0529878\pi$
$20$		0			0
$21$			814765.i			0.914209i
$22$		0			0
$23$	−2.22375e6			−1.65696			−0.828478	−	0.560022i	$-0.810793\pi$
$23$	−2.22375e6			−1.65696			−0.828478	+	0.560022i	$0.810793\pi$
$24$		0			0
$25$	913897.			0.467915
$26$		0			0
$27$	−2.68246e6			−0.971397
$28$		0			0
$29$	−5.75947e6			−1.51214			−0.756069	−	0.654492i	$-0.772882\pi$
$29$	−5.75947e6			−1.51214			−0.756069	+	0.654492i	$0.772882\pi$
$30$		0			0
$31$			7.19142e6i			1.39858i	0.714838	+	0.699290i	$0.246500\pi$
$31$			7.19142e6i			1.39858i	−0.714838	+	0.699290i	$0.753500\pi$
$32$		0			0
$33$		−	9.52743e6i		−	1.39850i
$34$		0			0
$35$	−5.76755e6			−0.649658
$36$		0			0
$37$			3.71695e6i			0.326046i	0.986622	+	0.163023i	$0.0521244\pi$
$37$			3.71695e6i			0.326046i	−0.986622	+	0.163023i	$0.947876\pi$
$38$		0			0
$39$	6.23029e6	−	1.34578e7i	0.431238	−	0.931502i
$40$		0			0
$41$		−	2.77654e6i		−	0.153454i	−0.997052	−	0.0767268i	$-0.975553\pi$
$41$		−	2.77654e6i		−	0.153454i	0.997052	−	0.0767268i	$-0.0244469\pi$
$42$		0			0
$43$	1.44942e7			0.646526			0.323263	−	0.946309i	$-0.395220\pi$
$43$	1.44942e7			0.646526			0.323263	+	0.946309i	$0.395220\pi$
$44$		0			0
$45$			1.07676e6i			0.0391437i
$46$		0			0
$47$		−	6.34145e7i		−	1.89561i	−0.318854	−	0.947804i	$-0.603298\pi$
$47$		−	6.34145e7i		−	1.89561i	0.318854	−	0.947804i	$-0.396702\pi$
$48$		0			0
$49$	8.34464e6			0.206788
$50$		0			0
$51$	1.44903e7			0.299925
$52$		0			0
$53$	−8.43154e7			−1.46779			−0.733897	−	0.679260i	$-0.762301\pi$
$53$	−8.43154e7			−1.46779			−0.733897	+	0.679260i	$0.762301\pi$
$54$		0			0
$55$	6.74427e7			0.993808
$56$		0			0
$57$		−	2.71104e7i		−	0.340172i
$58$		0			0
$59$		−	1.12066e8i		−	1.20404i	−0.798483	−	0.602018i	$-0.794364\pi$
$59$		−	1.12066e8i		−	1.20404i	0.798483	−	0.602018i	$-0.205636\pi$
$60$		0			0
$61$	−1.70009e8			−1.57213			−0.786063	−	0.618146i	$-0.787884\pi$
$61$	−1.70009e8			−1.57213			−0.786063	+	0.618146i	$0.787884\pi$
$62$		0			0
$63$			5.97583e6i			0.0477932i
$64$		0			0
$65$	9.52651e7	+	4.41029e7i	0.661947	+	0.306448i
$66$		0			0
$67$			4.94626e7i			0.299875i	0.988695	+	0.149937i	$0.0479072\pi$
$67$			4.94626e7i			0.299875i	−0.988695	+	0.149937i	$0.952093\pi$
$68$		0			0
$69$	−3.20245e8			−1.70083
$70$		0			0
$71$		−	1.17412e8i		−	0.548341i	−0.961681	−	0.274171i	$-0.911597\pi$
$71$		−	1.17412e8i		−	0.548341i	0.961681	−	0.274171i	$-0.0884033\pi$
$72$		0			0
$73$		−	2.58227e8i		−	1.06426i	−0.846662	−	0.532131i	$-0.821391\pi$
$73$		−	2.58227e8i		−	1.06426i	0.846662	−	0.532131i	$-0.178609\pi$
$74$		0			0
$75$	1.31611e8			0.480306
$76$		0			0
$77$	3.74296e8			1.21341
$78$		0			0
$79$	4.68955e8			1.35459			0.677297	−	0.735710i	$-0.263151\pi$
$79$	4.68955e8			1.35459			0.677297	+	0.735710i	$0.263151\pi$
$80$		0			0
$81$	−4.07095e8			−1.05078
$82$		0			0
$83$		−	5.71870e8i		−	1.32265i	−0.750098	−	0.661327i	$-0.769994\pi$
$83$		−	5.71870e8i		−	1.32265i	0.750098	−	0.661327i	$-0.230006\pi$
$84$		0			0
$85$			1.02574e8i			0.213134i
$86$		0			0
$87$	−8.29428e8			−1.55218
$88$		0			0
$89$			1.75654e8i			0.296758i	0.988931	+	0.148379i	$0.0474056\pi$
$89$			1.75654e8i			0.296758i	−0.988931	+	0.148379i	$0.952594\pi$
$90$		0			0
$91$	5.28706e8	+	2.44764e8i	0.808216	+	0.374163i
$92$		0			0
$93$			1.03565e9i			1.43561i
$94$		0			0
$95$	1.91909e8			0.241734
$96$		0			0
$97$		−	1.58656e8i		−	0.181963i	−0.995853	−	0.0909817i	$-0.971000\pi$
$97$		−	1.58656e8i		−	0.181963i	0.995853	−	0.0909817i	$-0.0290005\pi$
$98$		0			0
$99$		−	6.98782e7i		−	0.0731111i
$100$		0			0
$101$	−1.60681e9			−1.53644			−0.768222	−	0.640183i	$-0.778859\pi$
$101$	−1.60681e9			−1.53644			−0.768222	+	0.640183i	$0.778859\pi$
$102$		0			0
$103$	−4.56431e8			−0.399583			−0.199792	−	0.979838i	$-0.564027\pi$
$103$	−4.56431e8			−0.399583			−0.199792	+	0.979838i	$0.564027\pi$
$104$		0			0
$105$	−8.30592e8			−0.666862
$106$		0			0
$107$	2.54620e9			1.87787			0.938936	−	0.344091i	$-0.111813\pi$
$107$	2.54620e9			1.87787			0.938936	+	0.344091i	$0.111813\pi$
$108$		0			0
$109$		−	1.44182e9i		−	0.978345i	−0.872187	−	0.489173i	$-0.837299\pi$
$109$		−	1.44182e9i		−	0.978345i	0.872187	−	0.489173i	$-0.162701\pi$
$110$		0			0
$111$			5.35282e8i			0.334680i
$112$		0			0
$113$	4.84320e8			0.279434			0.139717	−	0.990191i	$-0.455381\pi$
$113$	4.84320e8			0.279434			0.139717	+	0.990191i	$0.455381\pi$
$114$		0			0
$115$		−	2.26695e9i		−	1.20865i
$116$		0			0
$117$	4.56956e7	−	9.87053e7i	0.0225443	−	0.0486972i
$118$		0			0
$119$			5.69270e8i			0.260230i
$120$		0			0
$121$	−2.01887e9			−0.856198
$122$		0			0
$123$		−	3.99854e8i		−	0.157517i
$124$		0			0
$125$			2.92272e9i			1.07076i
$126$		0			0
$127$	7.84228e8			0.267501			0.133751	−	0.991015i	$-0.457298\pi$
$127$	7.84228e8			0.267501			0.133751	+	0.991015i	$0.457298\pi$
$128$		0			0
$129$	2.08733e9			0.663646
$130$		0			0
$131$	−4.48111e9			−1.32943			−0.664713	−	0.747099i	$-0.731446\pi$
$131$	−4.48111e9			−1.32943			−0.664713	+	0.747099i	$0.731446\pi$
$132$		0			0
$133$	1.06506e9			0.295150
$134$		0			0
$135$		−	2.73457e9i		−	0.708577i
$136$		0			0
$137$		−	3.14080e9i		−	0.761723i	−0.924632	−	0.380862i	$-0.875627\pi$
$137$		−	3.14080e9i		−	0.761723i	0.924632	−	0.380862i	$-0.124373\pi$
$138$		0			0
$139$	−6.22133e8			−0.141357			−0.0706784	−	0.997499i	$-0.522516\pi$
$139$	−6.22133e8			−0.141357			−0.0706784	+	0.997499i	$0.522516\pi$
$140$		0			0
$141$		−	9.13241e9i		−	1.94580i
$142$		0			0
$143$	−6.18240e9	−	2.86214e9i	−1.23636	−	0.572372i
$144$		0			0
$145$		−	5.87135e9i		−	1.10302i
$146$		0			0
$147$	1.20172e9			0.212264
$148$		0			0
$149$			4.92841e8i			0.0819159i	0.999161	+	0.0409580i	$0.0130410\pi$
$149$			4.92841e8i			0.0819159i	−0.999161	+	0.0409580i	$0.986959\pi$
$150$		0			0
$151$			1.47293e9i			0.230561i	0.993333	+	0.115281i	$0.0367767\pi$
$151$			1.47293e9i			0.230561i	−0.993333	+	0.115281i	$0.963223\pi$
$152$		0			0
$153$	1.06278e8			0.0156795
$154$		0			0
$155$	−7.33112e9			−1.02018
$156$		0			0
$157$	1.22713e10			1.61192			0.805960	−	0.591970i	$-0.201650\pi$
$157$	1.22713e10			1.61192			0.805960	+	0.591970i	$0.201650\pi$
$158$		0			0
$159$	−1.21424e10			−1.50666
$160$		0			0
$161$		−	1.25812e10i		−	1.47573i
$162$		0			0
$163$		−	8.06073e9i		−	0.894397i	−0.894435	−	0.447198i	$-0.852422\pi$
$163$		−	8.06073e9i		−	0.894397i	0.894435	−	0.447198i	$-0.147578\pi$
$164$		0			0
$165$	9.71250e9			1.02012
$166$		0			0
$167$		−	1.49991e10i		−	1.49224i	−0.665809	−	0.746122i	$-0.731914\pi$
$167$		−	1.49991e10i		−	1.49224i	0.665809	−	0.746122i	$-0.268086\pi$
$168$		0			0
$169$	−6.86121e9	−	8.08574e9i	−0.647010	−	0.762482i
$170$		0			0
$171$		−	1.98839e8i		−	0.0177836i
$172$		0			0
$173$	−1.28624e10			−1.09173			−0.545863	−	0.837874i	$-0.683798\pi$
$173$	−1.28624e10			−1.09173			−0.545863	+	0.837874i	$0.683798\pi$
$174$		0			0
$175$			5.17050e9i			0.416737i
$176$		0			0
$177$		−	1.61387e10i		−	1.23592i
$178$		0			0
$179$	−4.35641e9			−0.317169			−0.158584	−	0.987345i	$-0.550693\pi$
$179$	−4.35641e9			−0.317169			−0.158584	+	0.987345i	$0.550693\pi$
$180$		0			0
$181$	−8.91267e9			−0.617240			−0.308620	−	0.951185i	$-0.599867\pi$
$181$	−8.91267e9			−0.617240			−0.308620	+	0.951185i	$0.599867\pi$
$182$		0			0
$183$	−2.44832e10			−1.61376
$184$		0			0
$185$	−3.78915e9			−0.237831
$186$		0			0
$187$		−	6.65674e9i		−	0.398084i
$188$		0			0
$189$		−	1.51764e10i		−	0.865150i
$190$		0			0
$191$	3.13880e9			0.170653			0.0853264	−	0.996353i	$-0.472807\pi$
$191$	3.13880e9			0.170653			0.0853264	+	0.996353i	$0.472807\pi$
$192$		0			0
$193$		−	3.53426e9i		−	0.183354i	−0.995789	−	0.0916769i	$-0.970777\pi$
$193$		−	3.53426e9i		−	0.183354i	0.995789	−	0.0916769i	$-0.0292227\pi$
$194$		0			0
$195$	1.37192e10	+	6.35131e9i	0.679476	+	0.314563i
$196$		0			0
$197$			3.09388e10i			1.46355i	0.681549	+	0.731773i	$0.261307\pi$
$197$			3.09388e10i			1.46355i	−0.681549	+	0.731773i	$0.738693\pi$
$198$		0			0
$199$	2.71870e10			1.22892			0.614459	−	0.788949i	$-0.289375\pi$
$199$	2.71870e10			1.22892			0.614459	+	0.788949i	$0.289375\pi$
$200$		0			0
$201$			7.12317e9i			0.307816i
$202$		0			0
$203$		−	3.25850e10i		−	1.34675i
$204$		0			0
$205$	2.83048e9			0.111935
$206$		0			0
$207$	−2.34881e9			−0.0889164
$208$		0			0
$209$	−1.24543e10			−0.451502
$210$		0			0
$211$	−4.36809e10			−1.51712			−0.758561	−	0.651602i	$-0.774097\pi$
$211$	−4.36809e10			−1.51712			−0.758561	+	0.651602i	$0.774097\pi$
$212$		0			0
$213$		−	1.69087e10i		−	0.562862i
$214$		0			0
$215$			1.47758e10i			0.471603i
$216$		0			0
$217$	−4.06865e10			−1.24561
$218$		0			0
$219$		−	3.71876e10i		−	1.09244i
$220$		0			0
$221$	4.35305e9	−	9.40287e9i	0.122752	−	0.265152i
$222$		0			0
$223$		−	1.84977e10i		−	0.500895i	−0.968130	−	0.250448i	$-0.919422\pi$
$223$		−	1.84977e10i		−	0.500895i	0.968130	−	0.250448i	$-0.0805778\pi$
$224$		0			0
$225$	9.65294e8			0.0251095
$226$		0			0
$227$			6.94482e10i			1.73598i	0.496582	+	0.867990i	$0.334588\pi$
$227$			6.94482e10i			1.73598i	−0.496582	+	0.867990i	$0.665412\pi$
$228$		0			0
$229$			4.16396e10i			1.00057i	0.865861	+	0.500285i	$0.166771\pi$
$229$			4.16396e10i			1.00057i	−0.865861	+	0.500285i	$0.833229\pi$
$230$		0			0
$231$	5.39028e10			1.24554
$232$		0			0
$233$	1.25595e10			0.279171			0.139585	−	0.990210i	$-0.455423\pi$
$233$	1.25595e10			0.279171			0.139585	+	0.990210i	$0.455423\pi$
$234$		0			0
$235$	6.46464e10			1.38273
$236$		0			0
$237$	6.75348e10			1.39046
$238$		0			0
$239$			5.12144e9i			0.101532i	0.998711	+	0.0507658i	$0.0161662\pi$
$239$			5.12144e9i			0.101532i	−0.998711	+	0.0507658i	$0.983834\pi$
$240$		0			0
$241$			1.76949e9i			0.0337887i	0.999857	+	0.0168944i	$0.00537790\pi$
$241$			1.76949e9i			0.0337887i	−0.999857	+	0.0168944i	$0.994622\pi$
$242$		0			0
$243$	−5.82731e9			−0.107211
$244$		0			0
$245$			8.50674e9i			0.150840i
$246$		0			0
$247$	−1.75921e10	−	8.14424e9i	−0.300733	−	0.139224i
$248$		0			0
$249$		−	8.23557e10i		−	1.35768i
$250$		0			0
$251$	4.22358e10			0.671660			0.335830	−	0.941923i	$-0.390983\pi$
$251$	4.22358e10			0.671660			0.335830	+	0.941923i	$0.390983\pi$
$252$		0			0
$253$			1.47118e11i			2.25748i
$254$		0			0
$255$			1.47718e10i			0.218778i
$256$		0			0
$257$	−8.08463e10			−1.15601			−0.578004	−	0.816034i	$-0.696168\pi$
$257$	−8.08463e10			−1.15601			−0.578004	+	0.816034i	$0.696168\pi$
$258$		0			0
$259$	−2.10292e10			−0.290384
$260$		0			0
$261$	−6.08337e9			−0.0811451
$262$		0			0
$263$	1.29543e11			1.66960			0.834802	−	0.550550i	$-0.185582\pi$
$263$	1.29543e11			1.66960			0.834802	+	0.550550i	$0.185582\pi$
$264$		0			0
$265$		−	8.59533e10i		−	1.07067i
$266$		0			0
$267$			2.52961e10i			0.304617i
$268$		0			0
$269$	−6.65588e10			−0.775033			−0.387516	−	0.921863i	$-0.626667\pi$
$269$	−6.65588e10			−0.775033			−0.387516	+	0.921863i	$0.626667\pi$
$270$		0			0
$271$		−	4.22523e10i		−	0.475870i	−0.971281	−	0.237935i	$-0.923529\pi$
$271$		−	4.22523e10i		−	0.475870i	0.971281	−	0.237935i	$-0.0764706\pi$
$272$		0			0
$273$	7.61396e10	+	3.52488e10i	0.829619	+	0.384071i
$274$		0			0
$275$		−	6.04611e10i		−	0.637498i
$276$		0			0
$277$	−8.10641e10			−0.827313			−0.413656	−	0.910433i	$-0.635748\pi$
$277$	−8.10641e10			−0.827313			−0.413656	+	0.910433i	$0.635748\pi$
$278$		0			0
$279$			7.59586e9i			0.0750513i
$280$		0			0
$281$			1.39877e11i			1.33834i	0.743108	+	0.669172i	$0.233351\pi$
$281$			1.39877e11i			1.33834i	−0.743108	+	0.669172i	$0.766649\pi$
$282$		0			0
$283$	−7.96402e10			−0.738063			−0.369031	−	0.929417i	$-0.620310\pi$
$283$	−7.96402e10			−0.738063			−0.369031	+	0.929417i	$0.620310\pi$
$284$		0			0
$285$	2.76370e10			0.248136
$286$		0			0
$287$	1.57087e10			0.136670
$288$		0			0
$289$	−1.08464e11			−0.914626
$290$		0			0
$291$		−	2.28483e10i		−	0.186782i
$292$		0			0
$293$			6.84760e10i			0.542792i	0.962468	+	0.271396i	$0.0874854\pi$
$293$			6.84760e10i			0.542792i	−0.962468	+	0.271396i	$0.912515\pi$
$294$		0			0
$295$	1.14243e11			0.878273
$296$		0			0
$297$			1.77465e11i			1.32345i
$298$		0			0
$299$	−9.62050e10	+	2.07809e11i	−0.696109	+	1.50364i
$300$		0			0
$301$			8.20030e10i			0.575812i
$302$		0			0
$303$	−2.31398e11			−1.57713
$304$		0			0
$305$		−	1.73311e11i		−	1.14677i
$306$		0			0
$307$		−	6.22329e10i		−	0.399851i	−0.979811	−	0.199925i	$-0.935930\pi$
$307$		−	6.22329e10i		−	0.399851i	0.979811	−	0.199925i	$-0.0640699\pi$
$308$		0			0
$309$	−6.57312e10			−0.410165
$310$		0			0
$311$	1.23651e11			0.749504			0.374752	−	0.927125i	$-0.377728\pi$
$311$	1.23651e11			0.749504			0.374752	+	0.927125i	$0.377728\pi$
$312$		0			0
$313$	−1.63911e11			−0.965289			−0.482644	−	0.875816i	$-0.660324\pi$
$313$	−1.63911e11			−0.965289			−0.482644	+	0.875816i	$0.660324\pi$
$314$		0			0
$315$	−6.09191e9			−0.0348623
$316$		0			0
$317$		−	2.49803e11i		−	1.38941i	−0.719294	−	0.694706i	$-0.755534\pi$
$317$		−	2.49803e11i		−	1.38941i	0.719294	−	0.694706i	$-0.244466\pi$
$318$		0			0
$319$			3.81032e11i			2.06017i
$320$		0			0
$321$	3.66682e11			1.92760
$322$		0			0
$323$		−	1.89418e10i		−	0.0968300i
$324$		0			0
$325$	3.95374e10	−	8.54034e10i	0.196577	−	0.424620i
$326$		0			0
$327$		−	2.07638e11i		−	1.00425i
$328$		0			0
$329$	3.58777e11			1.68827
$330$		0			0
$331$			9.19306e10i			0.420953i	0.977599	+	0.210477i	$0.0675016\pi$
$331$			9.19306e10i			0.420953i	−0.977599	+	0.210477i	$0.932498\pi$
$332$		0			0
$333$			3.92599e9i			0.0174964i
$334$		0			0
$335$	−5.04234e10			−0.218741
$336$		0			0
$337$	1.66525e11			0.703306			0.351653	−	0.936130i	$-0.385620\pi$
$337$	1.66525e11			0.703306			0.351653	+	0.936130i	$0.385620\pi$
$338$		0			0
$339$	6.97475e10			0.286834
$340$		0			0
$341$	4.75767e11			1.90546
$342$		0			0
$343$			2.75517e11i			1.07479i
$344$		0			0
$345$		−	3.26466e11i		−	1.24066i
$346$		0			0
$347$	2.63149e10			0.0974361			0.0487180	−	0.998813i	$-0.484486\pi$
$347$	2.63149e10			0.0974361			0.0487180	+	0.998813i	$0.484486\pi$
$348$		0			0
$349$			5.07968e11i			1.83283i	0.400231	+	0.916414i	$0.368930\pi$
$349$			5.07968e11i			1.83283i	−0.400231	+	0.916414i	$0.631070\pi$
$350$		0			0
$351$	−1.16050e11	+	2.50675e11i	−0.408097	+	0.881515i
$352$		0			0
$353$			1.01111e11i			0.346588i	0.984870	+	0.173294i	$0.0554411\pi$
$353$			1.01111e11i			0.346588i	−0.984870	+	0.173294i	$0.944559\pi$
$354$		0			0
$355$	1.19693e11			0.399983
$356$		0			0
$357$			8.19812e10i			0.267121i
$358$		0			0
$359$		−	8.98227e10i		−	0.285405i	−0.989766	−	0.142702i	$-0.954421\pi$
$359$		−	8.98227e10i		−	0.285405i	0.989766	−	0.142702i	$-0.0455792\pi$
$360$		0			0
$361$	2.87249e11			0.890176
$362$		0			0
$363$	−2.90740e11			−0.878870
$364$		0			0
$365$	2.63243e11			0.776317
$366$		0			0
$367$	−2.00813e11			−0.577823			−0.288911	−	0.957356i	$-0.593293\pi$
$367$	−2.00813e11			−0.577823			−0.288911	+	0.957356i	$0.593293\pi$
$368$		0			0
$369$		−	2.93270e9i		−	0.00823471i
$370$		0			0
$371$		−	4.77027e11i		−	1.30725i
$372$		0			0
$373$	−1.14957e11			−0.307500			−0.153750	−	0.988110i	$-0.549135\pi$
$373$	−1.14957e11			−0.307500			−0.153750	+	0.988110i	$0.549135\pi$
$374$		0			0
$375$			4.20904e11i			1.09911i
$376$		0			0
$377$	−2.49169e11	+	5.38221e11i	−0.635269	+	1.37222i
$378$		0			0
$379$			7.25204e10i			0.180544i	0.995917	+	0.0902721i	$0.0287737\pi$
$379$			7.25204e10i			0.180544i	−0.995917	+	0.0902721i	$0.971226\pi$
$380$		0			0
$381$	1.12938e11			0.274585
$382$		0			0
$383$			5.58942e11i			1.32731i	0.748039	+	0.663655i	$0.230996\pi$
$383$			5.58942e11i			1.32731i	−0.748039	+	0.663655i	$0.769004\pi$
$384$		0			0
$385$			3.81567e11i			0.885110i
$386$		0			0
$387$	1.53093e10			0.0346942
$388$		0			0
$389$	−6.00696e11			−1.33009			−0.665045	−	0.746803i	$-0.731588\pi$
$389$	−6.00696e11			−1.33009			−0.665045	+	0.746803i	$0.731588\pi$
$390$		0			0
$391$	−2.23753e11			−0.484142
$392$		0			0
$393$	−6.45330e11			−1.36463
$394$		0			0
$395$			4.78064e11i			0.988097i
$396$		0			0
$397$		−	7.85491e11i		−	1.58703i	−0.608554	−	0.793513i	$-0.708250\pi$
$397$		−	7.85491e11i		−	1.58703i	0.608554	−	0.793513i	$-0.291750\pi$
$398$		0			0
$399$	1.53381e11			0.302966
$400$		0			0
$401$			3.29034e11i			0.635465i	0.948180	+	0.317732i	$0.102921\pi$
$401$			3.29034e11i			0.635465i	−0.948180	+	0.317732i	$0.897079\pi$
$402$		0			0
$403$	6.72037e11	+	3.11119e11i	1.26917	+	0.587562i
$404$		0			0
$405$		−	4.15003e11i		−	0.766485i
$406$		0			0
$407$	2.45904e11			0.444212
$408$		0			0
$409$			7.87247e11i			1.39109i	0.718481	+	0.695547i	$0.244838\pi$
$409$			7.87247e11i			1.39109i	−0.718481	+	0.695547i	$0.755162\pi$
$410$		0			0
$411$		−	4.52310e11i		−	0.781894i
$412$		0			0
$413$	6.34029e11			1.07234
$414$		0			0
$415$	5.82979e11			0.964798
$416$		0			0
$417$	−8.95942e10			−0.145100
$418$		0			0
$419$	1.50859e11			0.239116			0.119558	−	0.992827i	$-0.461852\pi$
$419$	1.50859e11			0.239116			0.119558	+	0.992827i	$0.461852\pi$
$420$		0			0
$421$		−	2.40864e11i		−	0.373682i	−0.982390	−	0.186841i	$-0.940175\pi$
$421$		−	2.40864e11i		−	0.373682i	0.982390	−	0.186841i	$-0.0598250\pi$
$422$		0			0
$423$		−	6.69809e10i		−	0.101723i
$424$		0			0
$425$	9.19558e10			0.136719
$426$		0			0
$427$		−	9.61851e11i		−	1.40017i
$428$		0			0
$429$	−8.90336e11	−	4.12180e11i	−1.26910	−	0.587529i
$430$		0			0
$431$		−	6.22528e11i		−	0.868983i	−0.900676	−	0.434492i	$-0.856928\pi$
$431$		−	6.22528e11i		−	0.868983i	0.900676	−	0.434492i	$-0.143072\pi$
$432$		0			0
$433$	−3.18425e11			−0.435324			−0.217662	−	0.976024i	$-0.569843\pi$
$433$	−3.18425e11			−0.435324			−0.217662	+	0.976024i	$0.569843\pi$
$434$		0			0
$435$		−	8.45540e11i		−	1.13222i
$436$		0			0
$437$			4.18625e11i			0.549109i
$438$		0			0
$439$	−3.56306e11			−0.457860			−0.228930	−	0.973443i	$-0.573523\pi$
$439$	−3.56306e11			−0.457860			−0.228930	+	0.973443i	$0.573523\pi$
$440$		0			0
$441$	8.81394e9			0.0110968
$442$		0			0
$443$	−8.26027e11			−1.01901			−0.509504	−	0.860468i	$-0.670171\pi$
$443$	−8.26027e11			−1.01901			−0.509504	+	0.860468i	$0.670171\pi$
$444$		0			0
$445$	−1.79066e11			−0.216468
$446$		0			0
$447$			7.09746e10i			0.0840851i
$448$		0			0
$449$		−	2.78166e11i		−	0.322994i	−0.986873	−	0.161497i	$-0.948368\pi$
$449$		−	2.78166e11i		−	0.322994i	0.986873	−	0.161497i	$-0.0516323\pi$
$450$		0			0
$451$	−1.83689e11			−0.209069
$452$		0			0
$453$			2.12119e11i			0.236667i
$454$		0			0
$455$	−2.49519e11	+	5.38976e11i	−0.272930	+	0.589547i
$456$		0			0
$457$			3.13216e11i			0.335909i	0.985795	+	0.167954i	$0.0537161\pi$
$457$			3.13216e11i			0.335909i	−0.985795	+	0.167954i	$0.946284\pi$
$458$		0			0
$459$	−2.69908e11			−0.283830
$460$		0			0
$461$		−	1.62785e12i		−	1.67865i	−0.543629	−	0.839326i	$-0.682950\pi$
$461$		−	1.62785e12i		−	1.67865i	0.543629	−	0.839326i	$-0.317050\pi$
$462$		0			0
$463$		−	1.20183e12i		−	1.21542i	−0.794158	−	0.607711i	$-0.792088\pi$
$463$		−	1.20183e12i		−	1.21542i	0.794158	−	0.607711i	$-0.207912\pi$
$464$		0			0
$465$	−1.05576e12			−1.04720
$466$		0			0
$467$	4.12465e11			0.401293			0.200646	−	0.979664i	$-0.435696\pi$
$467$	4.12465e11			0.401293			0.200646	+	0.979664i	$0.435696\pi$
$468$		0			0
$469$	−2.79842e11			−0.267076
$470$		0			0
$471$	1.76721e12			1.65460
$472$		0			0
$473$		−	9.58900e11i		−	0.880842i
$474$		0			0
$475$		−	1.72043e11i		−	0.155065i
$476$		0			0
$477$	−8.90572e10			−0.0787656
$478$		0			0
$479$		−	2.80958e11i		−	0.243855i	−0.992539	−	0.121927i	$-0.961092\pi$
$479$		−	2.80958e11i		−	0.243855i	0.992539	−	0.121927i	$-0.0389075\pi$
$480$		0			0
$481$	3.47348e11	+	1.60804e11i	0.295877	+	0.136976i
$482$		0			0
$483$		−	1.81183e12i		−	1.51480i
$484$		0			0
$485$	1.61738e11			0.132732
$486$		0			0
$487$			7.68961e11i			0.619475i	0.950822	+	0.309738i	$0.100241\pi$
$487$			7.68961e11i			0.619475i	−0.950822	+	0.309738i	$0.899759\pi$
$488$		0			0
$489$		−	1.16084e12i		−	0.918081i
$490$		0			0
$491$	−1.31018e12			−1.01734			−0.508670	−	0.860962i	$-0.669863\pi$
$491$	−1.31018e12			−1.01734			−0.508670	+	0.860962i	$0.669863\pi$
$492$		0			0
$493$	−5.79515e11			−0.441828
$494$		0			0
$495$	7.12356e10			0.0533303
$496$		0			0
$497$	6.64277e11			0.488366
$498$		0			0
$499$		−	1.80186e12i		−	1.30097i	−0.759519	−	0.650486i	$-0.774565\pi$
$499$		−	1.80186e12i		−	1.30097i	0.759519	−	0.650486i	$-0.225435\pi$
$500$		0			0
$501$		−	2.16003e12i		−	1.53176i
$502$		0			0
$503$	4.33301e11			0.301810			0.150905	−	0.988548i	$-0.451781\pi$
$503$	4.33301e11			0.301810			0.150905	+	0.988548i	$0.451781\pi$
$504$		0			0
$505$		−	1.63802e12i		−	1.12075i
$506$		0			0
$507$	−9.88092e11	−	1.16444e12i	−0.664143	−	0.782673i
$508$		0			0
$509$			1.74228e12i			1.15050i	0.817977	+	0.575251i	$0.195096\pi$
$509$			1.74228e12i			1.15050i	−0.817977	+	0.575251i	$0.804904\pi$
$510$		0			0
$511$	1.46096e12			0.947858
$512$		0			0
$513$			5.04978e11i			0.321918i
$514$		0			0
$515$		−	4.65297e11i		−	0.291473i
$516$		0			0
$517$	−4.19535e12			−2.58262
$518$		0			0
$519$	−1.85233e12			−1.12064
$520$		0			0
$521$	−1.05717e12			−0.628601			−0.314300	−	0.949324i	$-0.601770\pi$
$521$	−1.05717e12			−0.628601			−0.314300	+	0.949324i	$0.601770\pi$
$522$		0			0
$523$	1.45993e12			0.853246			0.426623	−	0.904430i	$-0.359703\pi$
$523$	1.45993e12			0.853246			0.426623	+	0.904430i	$0.359703\pi$
$524$		0			0
$525$			7.44611e11i			0.427772i
$526$		0			0
$527$			7.23597e11i			0.408648i
$528$		0			0
$529$	3.14392e12			1.74550
$530$		0			0
$531$		−	1.18368e11i		−	0.0646116i
$532$		0			0
$533$	−2.59467e11	−	1.20120e11i	−0.139255	−	0.0644679i
$534$		0			0
$535$			2.59567e12i			1.36980i
$536$		0			0
$537$	−6.27373e11			−0.325568
$538$		0			0
$539$		−	5.52061e11i		−	0.281733i
$540$		0			0
$541$		−	1.20235e12i		−	0.603455i	−0.953394	−	0.301727i	$-0.902437\pi$
$541$		−	1.20235e12i		−	0.603455i	0.953394	−	0.301727i	$-0.0975633\pi$
$542$		0			0
$543$	−1.28352e12			−0.633585
$544$		0			0
$545$	1.46983e12			0.713646
$546$		0			0
$547$	−1.37979e12			−0.658978			−0.329489	−	0.944159i	$-0.606876\pi$
$547$	−1.37979e12			−0.658978			−0.329489	+	0.944159i	$0.606876\pi$
$548$		0			0
$549$	−1.79570e11			−0.0843643
$550$		0			0
$551$			1.08423e12i			0.501117i
$552$		0			0
$553$			2.65318e12i			1.20643i
$554$		0			0
$555$	−5.45680e11			−0.244129
$556$		0			0
$557$			7.03748e11i			0.309791i	0.987931	+	0.154895i	$0.0495041\pi$
$557$			7.03748e11i			0.309791i	−0.987931	+	0.154895i	$0.950496\pi$
$558$		0			0
$559$	6.27055e11	−	1.35448e12i	0.271614	−	0.586704i
$560$		0			0
$561$		−	9.58645e11i		−	0.408625i
$562$		0			0
$563$	−3.48720e12			−1.46282			−0.731408	−	0.681940i	$-0.761136\pi$
$563$	−3.48720e12			−1.46282			−0.731408	+	0.681940i	$0.761136\pi$
$564$		0			0
$565$			4.93728e11i			0.203831i
$566$		0			0
$567$		−	2.30320e12i		−	0.935853i
$568$		0			0
$569$	5.14453e11			0.205750			0.102875	−	0.994694i	$-0.467196\pi$
$569$	5.14453e11			0.205750			0.102875	+	0.994694i	$0.467196\pi$
$570$		0			0
$571$	−1.66430e11			−0.0655191			−0.0327596	−	0.999463i	$-0.510430\pi$
$571$	−1.66430e11			−0.0655191			−0.0327596	+	0.999463i	$0.510430\pi$
$572$		0			0
$573$	4.52022e11			0.175172
$574$		0			0
$575$	−2.03228e12			−0.775315
$576$		0			0
$577$		−	3.87826e11i		−	0.145662i	−0.997344	−	0.0728310i	$-0.976797\pi$
$577$		−	3.87826e11i		−	0.145662i	0.997344	−	0.0728310i	$-0.0232034\pi$
$578$		0			0
$579$		−	5.08973e11i		−	0.188209i
$580$		0			0
$581$	3.23544e12			1.17799
$582$		0			0
$583$			5.57810e12i			1.99976i
$584$		0			0
$585$	1.00623e11	+	4.65832e10i	0.0355218	+	0.0164448i
$586$		0			0
$587$			3.36701e12i			1.17050i	0.810852	+	0.585252i	$0.199004\pi$
$587$			3.36701e12i			1.17050i	−0.810852	+	0.585252i	$0.800996\pi$
$588$		0			0
$589$	1.35380e12			0.463484
$590$		0			0
$591$			4.45554e12i			1.50230i
$592$		0			0
$593$		−	3.59554e12i		−	1.19404i	−0.802227	−	0.597019i	$-0.796352\pi$
$593$		−	3.59554e12i		−	1.19404i	0.802227	−	0.597019i	$-0.203648\pi$
$594$		0			0
$595$	−5.80328e11			−0.189822
$596$		0			0
$597$	3.91523e12			1.26146
$598$		0			0
$599$	3.88170e12			1.23197			0.615987	−	0.787756i	$-0.288757\pi$
$599$	3.88170e12			1.23197			0.615987	+	0.787756i	$0.288757\pi$
$600$		0			0
$601$	3.39593e12			1.06175			0.530877	−	0.847449i	$-0.321863\pi$
$601$	3.39593e12			1.06175			0.530877	+	0.847449i	$0.321863\pi$
$602$		0			0
$603$			5.22443e10i			0.0160920i
$604$		0			0
$605$		−	2.05809e12i		−	0.624546i
$606$		0			0
$607$	5.99535e12			1.79252			0.896262	−	0.443524i	$-0.146272\pi$
$607$	5.99535e12			1.79252			0.896262	+	0.443524i	$0.146272\pi$
$608$		0			0
$609$		−	4.69261e12i		−	1.38241i
$610$		0			0
$611$	−5.92607e12	−	2.74347e12i	−1.72021	−	0.796370i
$612$		0			0
$613$			6.77602e12i			1.93822i	0.246632	+	0.969109i	$0.420676\pi$
$613$			6.77602e12i			1.93822i	−0.246632	+	0.969109i	$0.579324\pi$
$614$		0			0
$615$	4.07621e11			0.114900
$616$		0			0
$617$			2.93800e12i			0.816148i	0.912949	+	0.408074i	$0.133799\pi$
$617$			2.93800e12i			0.816148i	−0.912949	+	0.408074i	$0.866201\pi$
$618$		0			0
$619$		−	4.03266e12i		−	1.10404i	−0.833832	−	0.552019i	$-0.813858\pi$
$619$		−	4.03266e12i		−	1.10404i	0.833832	−	0.552019i	$-0.186142\pi$
$620$		0			0
$621$	5.96513e12			1.60956
$622$		0			0
$623$	−9.93788e11			−0.264300
$624$		0			0
$625$	−1.19454e12			−0.313140
$626$		0			0
$627$	−1.79355e12			−0.463458
$628$		0			0
$629$			3.73997e11i			0.0952666i
$630$		0			0
$631$			5.08917e12i			1.27795i	0.769226	+	0.638977i	$0.220642\pi$
$631$			5.08917e12i			1.27795i	−0.769226	+	0.638977i	$0.779358\pi$
$632$		0			0
$633$	−6.29054e12			−1.55730
$634$		0			0
$635$			7.99462e11i			0.195126i
$636$		0			0
$637$	3.61010e11	−	7.79805e11i	0.0868744	−	0.187654i
$638$		0			0
$639$		−	1.24016e11i		−	0.0294254i
$640$		0			0
$641$	1.72593e12			0.403797			0.201898	−	0.979406i	$-0.435289\pi$
$641$	1.72593e12			0.403797			0.201898	+	0.979406i	$0.435289\pi$
$642$		0			0
$643$		−	1.01796e12i		−	0.234844i	−0.993082	−	0.117422i	$-0.962537\pi$
$643$		−	1.01796e12i		−	0.234844i	0.993082	−	0.117422i	$-0.0374630\pi$
$644$		0			0
$645$			2.12787e12i			0.484091i
$646$		0			0
$647$	5.28525e11			0.118576			0.0592879	−	0.998241i	$-0.481117\pi$
$647$	5.28525e11			0.118576			0.0592879	+	0.998241i	$0.481117\pi$
$648$		0			0
$649$	−7.41400e12			−1.64041
$650$		0			0
$651$	−5.85932e12			−1.27859
$652$		0			0
$653$	4.61467e12			0.993187			0.496594	−	0.867983i	$-0.334584\pi$
$653$	4.61467e12			0.993187			0.496594	+	0.867983i	$0.334584\pi$
$654$		0			0
$655$		−	4.56815e12i		−	0.969739i
$656$		0			0
$657$		−	2.72749e11i		−	0.0571110i
$658$		0			0
$659$	6.06886e12			1.25350			0.626748	−	0.779222i	$-0.284386\pi$
$659$	6.06886e12			1.25350			0.626748	+	0.779222i	$0.284386\pi$
$660$		0			0
$661$			2.26704e12i			0.461904i	0.972965	+	0.230952i	$0.0741841\pi$
$661$			2.26704e12i			0.461904i	−0.972965	+	0.230952i	$0.925816\pi$
$662$		0			0
$663$	6.26888e11	−	1.35412e12i	0.126003	−	0.272174i
$664$		0			0
$665$			1.08575e12i			0.215294i
$666$		0			0
$667$	1.28076e13			2.50554
$668$		0			0
$669$		−	2.66388e12i		−	0.514159i
$670$		0			0
$671$			1.12474e13i			2.14190i
$672$		0			0
$673$	−9.67448e12			−1.81786			−0.908929	−	0.416951i	$-0.863099\pi$
$673$	−9.67448e12			−1.81786			−0.908929	+	0.416951i	$0.863099\pi$
$674$		0			0
$675$	−2.45149e12			−0.454531
$676$		0			0
$677$	−7.02561e12			−1.28539			−0.642695	−	0.766122i	$-0.722184\pi$
$677$	−7.02561e12			−1.28539			−0.642695	+	0.766122i	$0.722184\pi$
$678$		0			0
$679$	8.97620e11			0.162061
$680$		0			0
$681$			1.00013e13i			1.78195i
$682$		0			0
$683$			1.24988e12i			0.219774i	0.993944	+	0.109887i	$0.0350489\pi$
$683$			1.24988e12i			0.219774i	−0.993944	+	0.109887i	$0.964951\pi$
$684$		0			0
$685$	3.20181e12			0.555633
$686$		0			0
$687$			5.99658e12i			1.02707i
$688$		0			0
$689$	−3.64769e12	+	7.87925e12i	−0.616640	+	1.33198i
$690$		0			0
$691$		−	1.07396e13i		−	1.79200i	−0.444055	−	0.896000i	$-0.646460\pi$
$691$		−	1.07396e13i		−	1.79200i	0.444055	−	0.896000i	$-0.353540\pi$
$692$		0			0
$693$	3.95346e11			0.0651145
$694$		0			0
$695$		−	6.34218e11i		−	0.103112i
$696$		0			0
$697$		−	2.79375e11i		−	0.0448373i
$698$		0			0
$699$	1.80871e12			0.286563
$700$		0			0
$701$	−4.89863e12			−0.766203			−0.383101	−	0.923706i	$-0.625144\pi$
$701$	−4.89863e12			−0.766203			−0.383101	+	0.923706i	$0.625144\pi$
$702$		0			0
$703$	6.99722e11			0.108050
$704$		0			0
$705$	9.30981e12			1.41935
$706$		0			0
$707$		−	9.09074e12i		−	1.36840i
$708$		0			0
$709$			6.17750e12i			0.918131i	0.888402	+	0.459065i	$0.151816\pi$
$709$			6.17750e12i			0.918131i	−0.888402	+	0.459065i	$0.848184\pi$
$710$		0			0
$711$	4.95328e11			0.0726909
$712$		0			0
$713$		−	1.59919e13i		−	2.31738i
$714$		0			0
$715$	2.91774e12	−	6.30250e12i	0.417512	−	0.901853i
$716$		0			0
$717$			7.37545e11i			0.104220i
$718$		0			0
$719$	−1.50484e11			−0.0209996			−0.0104998	−	0.999945i	$-0.503342\pi$
$719$	−1.50484e11			−0.0209996			−0.0104998	+	0.999945i	$0.503342\pi$
$720$		0			0
$721$		−	2.58232e12i		−	0.355879i
$722$		0			0
$723$			2.54827e11i			0.0346835i
$724$		0			0
$725$	−5.26356e12			−0.707552
$726$		0			0
$727$	1.33111e13			1.76729			0.883644	−	0.468159i	$-0.155082\pi$
$727$	1.33111e13			1.76729			0.883644	+	0.468159i	$0.155082\pi$
$728$		0			0
$729$	7.17365e12			0.940733
$730$		0			0
$731$	1.45840e12			0.188907
$732$		0			0
$733$			9.04695e12i			1.15754i	0.815492	+	0.578768i	$0.196466\pi$
$733$			9.04695e12i			1.15754i	−0.815492	+	0.578768i	$0.803534\pi$
$734$		0			0
$735$			1.22507e12i			0.154834i
$736$		0			0
$737$	3.27232e12			0.408557
$738$		0			0
$739$			8.38312e12i			1.03397i	0.855996	+	0.516983i	$0.172945\pi$
$739$			8.38312e12i			1.03397i	−0.855996	+	0.516983i	$0.827055\pi$
$740$		0			0
$741$	−2.53346e12	−	1.17286e12i	−0.308696	−	0.142911i
$742$		0			0
$743$			9.05951e12i			1.09057i	0.838249	+	0.545287i	$0.183579\pi$
$743$			9.05951e12i			1.09057i	−0.838249	+	0.545287i	$0.816421\pi$
$744$		0			0
$745$	−5.02414e11			−0.0597529
$746$		0			0
$747$		−	6.04032e11i		−	0.0709769i
$748$		0			0
$749$			1.44055e13i			1.67248i
$750$		0			0
$751$	−5.44464e12			−0.624583			−0.312291	−	0.949986i	$-0.601096\pi$
$751$	−5.44464e12			−0.624583			−0.312291	+	0.949986i	$0.601096\pi$
$752$		0			0
$753$	6.08243e12			0.689446
$754$		0			0
$755$	−1.50154e12			−0.168181
$756$		0			0
$757$	−9.76848e12			−1.08117			−0.540587	−	0.841288i	$-0.681798\pi$
$757$	−9.76848e12			−1.08117			−0.540587	+	0.841288i	$0.681798\pi$
$758$		0			0
$759$			2.11866e13i			2.31725i
$760$		0			0
$761$		−	1.04489e13i		−	1.12938i	−0.825305	−	0.564688i	$-0.808997\pi$
$761$		−	1.04489e13i		−	1.12938i	0.825305	−	0.564688i	$-0.191003\pi$
$762$		0			0
$763$	8.15731e12			0.871338
$764$		0			0
$765$			1.08343e11i			0.0114373i
$766$		0			0
$767$	−1.04725e13	−	4.84825e12i	−1.09263	−	0.505831i
$768$		0			0
$769$		−	1.33503e11i		−	0.0137664i	−0.999976	−	0.00688322i	$-0.997809\pi$
$769$		−	1.33503e11i		−	0.0137664i	0.999976	−	0.00688322i	$-0.00219101\pi$
$770$		0			0
$771$	−1.16428e13			−1.18662
$772$		0			0
$773$		−	5.29031e12i		−	0.532934i	−0.963844	−	0.266467i	$-0.914144\pi$
$773$		−	5.29031e12i		−	0.532934i	0.963844	−	0.266467i	$-0.0858563\pi$
$774$		0			0
$775$			6.57222e12i			0.654416i
$776$		0			0
$777$	−3.02844e12			−0.298074
$778$		0			0
$779$	−5.22689e11			−0.0508540
$780$		0			0
$781$	−7.76771e12			−0.747073
$782$		0			0
$783$	1.54496e13			1.46889
$784$		0			0
$785$			1.25097e13i			1.17580i
$786$		0			0
$787$		−	3.47869e12i		−	0.323243i	−0.986853	−	0.161621i	$-0.948328\pi$
$787$		−	3.47869e12i		−	0.323243i	0.986853	−	0.161621i	$-0.0516723\pi$
$788$		0			0
$789$	1.86557e13			1.71382
$790$		0			0
$791$			2.74011e12i			0.248871i
$792$		0			0
$793$	−7.35501e12	+	1.58873e13i	−0.660471	+	1.42666i
$794$		0			0
$795$		−	1.23782e13i		−	1.09902i
$796$		0			0
$797$	3.96304e12			0.347909			0.173955	−	0.984754i	$-0.444345\pi$
$797$	3.96304e12			0.347909			0.173955	+	0.984754i	$0.444345\pi$
$798$		0			0
$799$		−	6.38074e12i		−	0.553873i
$800$		0			0
$801$			1.85533e11i			0.0159248i
$802$		0			0
$803$	−1.70837e13			−1.44998
$804$		0			0
$805$	1.28256e13			1.07646
$806$		0			0
$807$	−9.58521e12			−0.795556
$808$		0			0
$809$	9.83579e12			0.807311			0.403656	−	0.914911i	$-0.367739\pi$
$809$	9.83579e12			0.807311			0.403656	+	0.914911i	$0.367739\pi$
$810$		0			0
$811$			4.13429e12i			0.335588i	0.985822	+	0.167794i	$0.0536644\pi$
$811$			4.13429e12i			0.335588i	−0.985822	+	0.167794i	$0.946336\pi$
$812$		0			0
$813$		−	6.08481e12i		−	0.488472i
$814$		0			0
$815$	8.21732e12			0.652410
$816$		0			0
$817$		−	2.72856e12i		−	0.214256i
$818$		0			0
$819$	5.58440e11	+	2.58529e11i	0.0433709	+	0.0200785i
$820$		0			0
$821$			9.84105e12i			0.755957i	0.925814	+	0.377979i	$0.123381\pi$
$821$			9.84105e12i			0.755957i	−0.925814	+	0.377979i	$0.876619\pi$
$822$		0			0
$823$	8.39381e12			0.637764			0.318882	−	0.947794i	$-0.396693\pi$
$823$	8.39381e12			0.637764			0.318882	+	0.947794i	$0.396693\pi$
$824$		0			0
$825$		−	8.70708e12i		−	0.654380i
$826$		0			0
$827$		−	1.61555e13i		−	1.20101i	−0.799621	−	0.600505i	$-0.794966\pi$
$827$		−	1.61555e13i		−	1.20101i	0.799621	−	0.600505i	$-0.205034\pi$
$828$		0			0
$829$	−2.38584e13			−1.75447			−0.877234	−	0.480063i	$-0.840614\pi$
$829$	−2.38584e13			−1.75447			−0.877234	+	0.480063i	$0.840614\pi$
$830$		0			0
$831$	−1.16741e13			−0.849220
$832$		0			0
$833$	8.39634e11			0.0604209
$834$		0			0
$835$	1.52904e13			1.08850
$836$		0			0
$837$		−	1.92907e13i		−	1.35858i
$838$		0			0
$839$			2.05540e13i			1.43208i	0.698057	+	0.716042i	$0.254048\pi$
$839$			2.05540e13i			1.43208i	−0.698057	+	0.716042i	$0.745952\pi$
$840$		0			0
$841$	1.86643e13			1.28656
$842$		0			0
$843$			2.01438e13i			1.37378i
$844$		0			0
$845$	8.24281e12	−	6.99449e12i	0.556186	−	0.471956i
$846$		0			0
$847$		−	1.14221e13i		−	0.762551i
$848$		0			0
$849$	−1.14691e13			−0.757607
$850$		0			0
$851$		−	8.26556e12i		−	0.540243i
$852$		0			0
$853$			9.96261e12i			0.644321i	0.946685	+	0.322161i	$0.104409\pi$
$853$			9.96261e12i			0.644321i	−0.946685	+	0.322161i	$0.895591\pi$
$854$		0			0
$855$	2.02701e11			0.0129721
$856$		0			0
$857$	−2.03354e12			−0.128777			−0.0643884	−	0.997925i	$-0.520510\pi$
$857$	−2.03354e12			−0.128777			−0.0643884	+	0.997925i	$0.520510\pi$
$858$		0			0
$859$	1.31123e13			0.821690			0.410845	−	0.911705i	$-0.365234\pi$
$859$	1.31123e13			0.821690			0.410845	+	0.911705i	$0.365234\pi$
$860$		0			0
$861$	2.26223e12			0.140289
$862$		0			0
$863$		−	3.19813e13i		−	1.96267i	−0.192307	−	0.981335i	$-0.561597\pi$
$863$		−	3.19813e13i		−	1.96267i	0.192307	−	0.981335i	$-0.438403\pi$
$864$		0			0
$865$		−	1.31122e13i		−	0.796351i
$866$		0			0
$867$	−1.56200e13			−0.938846
$868$		0			0
$869$		−	3.10249e13i		−	1.84553i
$870$		0			0
$871$	4.62227e12	+	2.13987e12i	0.272128	+	0.125981i
$872$		0			0
$873$		−	1.67579e11i		−	0.00976462i
$874$		0			0
$875$	−1.65357e13			−0.953643
$876$		0			0
$877$		−	2.83695e13i		−	1.61939i	−0.586848	−	0.809697i	$-0.699631\pi$
$877$		−	2.83695e13i		−	1.61939i	0.586848	−	0.809697i	$-0.300369\pi$
$878$		0			0
$879$			9.86131e12i			0.557166i
$880$		0			0
$881$	2.70171e13			1.51094			0.755469	−	0.655184i	$-0.227409\pi$
$881$	2.70171e13			1.51094			0.755469	+	0.655184i	$0.227409\pi$
$882$		0			0
$883$	−2.67416e13			−1.48035			−0.740174	−	0.672416i	$-0.765257\pi$
$883$	−2.67416e13			−1.48035			−0.740174	+	0.672416i	$0.765257\pi$
$884$		0			0
$885$	1.64522e13			0.901531
$886$		0			0
$887$	7.44336e12			0.403750			0.201875	−	0.979411i	$-0.435297\pi$
$887$	7.44336e12			0.403750			0.201875	+	0.979411i	$0.435297\pi$
$888$		0			0
$889$			4.43689e12i			0.238243i
$890$		0			0
$891$			2.69324e13i			1.43161i
$892$		0			0
$893$	−1.19379e13			−0.628198
$894$		0			0
$895$		−	4.44104e12i		−	0.231356i
$896$		0			0
$897$	−1.38546e13	+	2.99268e13i	−0.714543	+	1.54346i
$898$		0			0
$899$		−	4.14187e13i		−	2.11484i
$900$		0			0
$901$	−8.48377e12			−0.428872
$902$		0			0
$903$			1.18094e13i			0.591060i
$904$		0			0
$905$		−	9.08580e12i		−	0.450241i
$906$		0			0
$907$	−3.19922e13			−1.56968			−0.784841	−	0.619697i	$-0.787256\pi$
$907$	−3.19922e13			−1.56968			−0.784841	+	0.619697i	$0.787256\pi$
$908$		0			0
$909$	−1.69717e12			−0.0824495
$910$		0			0
$911$	−2.16462e13			−1.04123			−0.520617	−	0.853790i	$-0.674298\pi$
$911$	−2.16462e13			−1.04123			−0.520617	+	0.853790i	$0.674298\pi$
$912$		0			0
$913$	−3.78335e13			−1.80201
$914$		0			0
$915$		−	2.49588e13i		−	1.17714i
$916$		0			0
$917$		−	2.53525e13i		−	1.18402i
$918$		0			0
$919$	−2.07241e13			−0.958422			−0.479211	−	0.877700i	$-0.659077\pi$
$919$	−2.07241e13			−0.958422			−0.479211	+	0.877700i	$0.659077\pi$
$920$		0			0
$921$		−	8.96224e12i		−	0.410439i
$922$		0			0
$923$	−1.09722e13	−	5.07955e12i	−0.497604	−	0.230366i
$924$		0			0
$925$			3.39691e12i			0.152562i
$926$		0			0
$927$	−4.82100e11			−0.0214427
$928$		0			0
$929$		−	8.15082e12i		−	0.359030i	−0.983755	−	0.179515i	$-0.942547\pi$
$929$		−	8.15082e12i		−	0.359030i	0.983755	−	0.179515i	$-0.0574528\pi$
$930$		0			0
$931$		−	1.57089e12i		−	0.0685288i
$932$		0			0
$933$	1.78071e13			0.769352
$934$		0			0
$935$	6.78605e12			0.290379
$936$		0			0
$937$	−8.09462e12			−0.343058			−0.171529	−	0.985179i	$-0.554871\pi$
$937$	−8.09462e12			−0.343058			−0.171529	+	0.985179i	$0.554871\pi$
$938$		0			0
$939$	−2.36050e13			−0.990850
$940$		0			0
$941$		−	4.10705e13i		−	1.70756i	−0.520633	−	0.853780i	$-0.674304\pi$
$941$		−	4.10705e13i		−	1.70756i	0.520633	−	0.853780i	$-0.325696\pi$
$942$		0			0
$943$			6.17434e12i			0.254266i
$944$		0			0
$945$	1.54712e13			0.631076
$946$		0			0
$947$		−	1.12577e13i		−	0.454855i	−0.973795	−	0.227428i	$-0.926969\pi$
$947$		−	1.12577e13i		−	0.454855i	0.973795	−	0.227428i	$-0.0730315\pi$
$948$		0			0
$949$	−2.41312e13	−	1.11715e13i	−0.965788	−	0.447111i
$950$		0			0
$951$		−	3.59745e13i		−	1.42620i
$952$		0			0
$953$	4.29747e13			1.68770			0.843849	−	0.536580i	$-0.180284\pi$
$953$	4.29747e13			1.68770			0.843849	+	0.536580i	$0.180284\pi$
$954$		0			0
$955$			3.19977e12i			0.124481i
$956$		0			0
$957$			5.48729e13i			2.11473i
$958$		0			0
$959$	1.77695e13			0.678409
$960$		0			0
$961$	−2.52769e13			−0.956025
$962$		0			0
$963$	2.68940e12			0.100771
$964$		0			0
$965$	3.60291e12			0.133746
$966$		0			0
$967$			1.48377e13i			0.545692i	0.962058	+	0.272846i	$0.0879649\pi$
$967$			1.48377e13i			0.545692i	−0.962058	+	0.272846i	$0.912035\pi$
$968$		0			0
$969$		−	2.72783e12i		−	0.0993941i
$970$		0			0
$971$	2.51875e13			0.909282			0.454641	−	0.890675i	$-0.349768\pi$
$971$	2.51875e13			0.909282			0.454641	+	0.890675i	$0.349768\pi$
$972$		0			0
$973$		−	3.51981e12i		−	0.125896i
$974$		0			0
$975$	5.69384e12	−	1.22991e13i	0.201783	−	0.435864i
$976$		0			0
$977$		−	1.45041e13i		−	0.509289i	−0.967035	−	0.254644i	$-0.918042\pi$
$977$		−	1.45041e13i		−	0.509289i	0.967035	−	0.254644i	$-0.0819584\pi$
$978$		0			0
$979$	1.16208e13			0.404311
$980$		0			0
$981$		−	1.52291e12i		−	0.0525005i
$982$		0			0
$983$		−	4.23394e13i		−	1.44628i	−0.690699	−	0.723142i	$-0.742697\pi$
$983$		−	4.23394e13i		−	1.44628i	0.690699	−	0.723142i	$-0.257303\pi$
$984$		0			0
$985$	−3.15399e13			−1.06757
$986$		0			0
$987$	5.16679e13			1.73298
$988$		0			0
$989$	−3.22315e13			−1.07126
$990$		0			0
$991$	−2.45013e13			−0.806972			−0.403486	−	0.914986i	$-0.632202\pi$
$991$	−2.45013e13			−0.806972			−0.403486	+	0.914986i	$0.632202\pi$
$992$		0			0
$993$			1.32390e13i			0.432100i
$994$		0			0
$995$			2.77151e13i			0.896423i
$996$		0			0
$997$	6.54322e11			0.0209731			0.0104866	−	0.999945i	$-0.496662\pi$
$997$	6.54322e11			0.0209731			0.0104866	+	0.999945i	$0.496662\pi$
$998$		0			0
$999$		−	9.97057e12i		−	0.316720i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.f.d.129.23		32
4.3	odd	2		104.10.f.a.25.10	yes	32
13.12	even	2	inner	208.10.f.d.129.24		32
52.51	odd	2		104.10.f.a.25.9	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.9	✓	32	52.51	odd	2
104.10.f.a.25.10	yes	32	4.3	odd	2
208.10.f.d.129.23		32	1.1	even	1	trivial
208.10.f.d.129.24		32	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+144.011 q^{3} +1019.43i q^{5} +5657.65i q^{7} +1056.24 q^{9} -66157.5i q^{11} +(43262.5 - 93449.7i) q^{13} +146809. i q^{15} +100620. q^{17} -188252. i q^{19} +814765. i q^{21} -2.22375e6 q^{23} +913897. q^{25} -2.68246e6 q^{27} -5.75947e6 q^{29} +7.19142e6i q^{31} -9.52743e6i q^{33} -5.76755e6 q^{35} +3.71695e6i q^{37} +(6.23029e6 - 1.34578e7i) q^{39} -2.77654e6i q^{41} +1.44942e7 q^{43} +1.07676e6i q^{45} -6.34145e7i q^{47} +8.34464e6 q^{49} +1.44903e7 q^{51} -8.43154e7 q^{53} +6.74427e7 q^{55} -2.71104e7i q^{57} -1.12066e8i q^{59} -1.70009e8 q^{61} +5.97583e6i q^{63} +(9.52651e7 + 4.41029e7i) q^{65} +4.94626e7i q^{67} -3.20245e8 q^{69} -1.17412e8i q^{71} -2.58227e8i q^{73} +1.31611e8 q^{75} +3.74296e8 q^{77} +4.68955e8 q^{79} -4.07095e8 q^{81} -5.71870e8i q^{83} +1.02574e8i q^{85} -8.29428e8 q^{87} +1.75654e8i q^{89} +(5.28706e8 + 2.44764e8i) q^{91} +1.03565e9i q^{93} +1.91909e8 q^{95} -1.58656e8i q^{97} -6.98782e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} - 3171556 q^{23} - 13526722 q^{25} + 3694974 q^{27} + 8833508 q^{29} + 8281126 q^{35} + 12056860 q^{39} - 89959038 q^{43} - 172344874 q^{49}+ \cdots - 1741143356 q^{95}+O(q^{100}) \) 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

Related objects

Downloads

Learn more