LMFDB - Embedded newform 208.10.f.d.129.5

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			129.5
Character	$\chi$	$=$	208.129
Dual form			208.10.f.d.129.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-195.153 q^{3} +2044.04i q^{5} +11607.1i q^{7} +18401.8 q^{9} +67982.7i q^{11} +(-80500.9 - 64219.2i) q^{13} -398902. i q^{15} -601985. q^{17} -695208. i q^{19} -2.26516e6i q^{21} +514539. q^{23} -2.22499e6 q^{25} +250023. q^{27} -4.65059e6 q^{29} +7.34256e6i q^{31} -1.32671e7i q^{33} -2.37254e7 q^{35} +6.51698e6i q^{37} +(1.57100e7 + 1.25326e7i) q^{39} -3.94333e6i q^{41} -1.49061e7 q^{43} +3.76142e7i q^{45} -2.16467e7i q^{47} -9.43711e7 q^{49} +1.17479e8 q^{51} +1.51204e7 q^{53} -1.38960e8 q^{55} +1.35672e8i q^{57} +4.81722e7i q^{59} +9.62097e7 q^{61} +2.13592e8i q^{63} +(1.31267e8 - 1.64547e8i) q^{65} +1.42051e8i q^{67} -1.00414e8 q^{69} +9.40346e7i q^{71} -2.75593e8i q^{73} +4.34214e8 q^{75} -7.89082e8 q^{77} -5.49152e8 q^{79} -4.10996e8 q^{81} +3.10281e8i q^{83} -1.23048e9i q^{85} +9.07579e8 q^{87} +6.74619e8i q^{89} +(7.45399e8 - 9.34381e8i) q^{91} -1.43293e9i q^{93} +1.42104e9 q^{95} +6.74712e8i q^{97} +1.25101e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−195.153			−1.39101			−0.695505	−	0.718521i	$-0.744819\pi$
$3$	−195.153			−1.39101			−0.695505	+	0.718521i	$0.744819\pi$
$4$		0			0
$5$			2044.04i			1.46260i	0.682057	+	0.731299i	$0.261086\pi$
$5$			2044.04i			1.46260i	−0.682057	+	0.731299i	$0.738914\pi$
$6$		0			0
$7$			11607.1i			1.82718i	0.406632	+	0.913592i	$0.366703\pi$
$7$			11607.1i			1.82718i	−0.406632	+	0.913592i	$0.633297\pi$
$8$		0			0
$9$	18401.8			0.934910
$10$		0			0
$11$			67982.7i			1.40001i	0.714137	+	0.700006i	$0.246819\pi$
$11$			67982.7i			1.40001i	−0.714137	+	0.700006i	$0.753181\pi$
$12$		0			0
$13$	−80500.9	−	64219.2i	−0.781728	−	0.623620i
$13$	−80500.9	−	64219.2i	−0.781728	−	0.623620i
$14$		0			0
$15$		−	398902.i		−	2.03449i
$16$		0			0
$17$	−601985.			−1.74810			−0.874049	−	0.485838i	$-0.838514\pi$
$17$	−601985.			−1.74810			−0.874049	+	0.485838i	$0.838514\pi$
$18$		0			0
$19$		−	695208.i		−	1.22384i	−0.790921	−	0.611919i	$-0.790398\pi$
$19$		−	695208.i		−	1.22384i	0.790921	−	0.611919i	$-0.209602\pi$
$20$		0			0
$21$		−	2.26516e6i		−	2.54163i
$22$		0			0
$23$	514539.			0.383392			0.191696	−	0.981454i	$-0.438601\pi$
$23$	514539.			0.383392			0.191696	+	0.981454i	$0.438601\pi$
$24$		0			0
$25$	−2.22499e6			−1.13920
$26$		0			0
$27$	250023.			0.0905405
$28$		0			0
$29$	−4.65059e6			−1.22100			−0.610502	−	0.792015i	$-0.709032\pi$
$29$	−4.65059e6			−1.22100			−0.610502	+	0.792015i	$0.709032\pi$
$30$		0			0
$31$			7.34256e6i			1.42797i	0.700160	+	0.713986i	$0.253112\pi$
$31$			7.34256e6i			1.42797i	−0.700160	+	0.713986i	$0.746888\pi$
$32$		0			0
$33$		−	1.32671e7i		−	1.94743i
$34$		0			0
$35$	−2.37254e7			−2.67244
$36$		0			0
$37$			6.51698e6i			0.571661i	0.958280	+	0.285830i	$0.0922694\pi$
$37$			6.51698e6i			0.571661i	−0.958280	+	0.285830i	$0.907731\pi$
$38$		0			0
$39$	1.57100e7	+	1.25326e7i	1.08739	+	0.867462i
$40$		0			0
$41$		−	3.94333e6i		−	0.217939i	−0.994045	−	0.108970i	$-0.965245\pi$
$41$		−	3.94333e6i		−	0.217939i	0.994045	−	0.108970i	$-0.0347551\pi$
$42$		0			0
$43$	−1.49061e7			−0.664898			−0.332449	−	0.943121i	$-0.607875\pi$
$43$	−1.49061e7			−0.664898			−0.332449	+	0.943121i	$0.607875\pi$
$44$		0			0
$45$			3.76142e7i			1.36740i
$46$		0			0
$47$		−	2.16467e7i		−	0.647069i	−0.946216	−	0.323535i	$-0.895129\pi$
$47$		−	2.16467e7i		−	0.647069i	0.946216	−	0.323535i	$-0.104871\pi$
$48$		0			0
$49$	−9.43711e7			−2.33860
$50$		0			0
$51$	1.17479e8			2.43162
$52$		0			0
$53$	1.51204e7			0.263221			0.131611	−	0.991301i	$-0.457985\pi$
$53$	1.51204e7			0.263221			0.131611	+	0.991301i	$0.457985\pi$
$54$		0			0
$55$	−1.38960e8			−2.04765
$56$		0			0
$57$			1.35672e8i			1.70237i
$58$		0			0
$59$			4.81722e7i			0.517562i	0.965936	+	0.258781i	$0.0833208\pi$
$59$			4.81722e7i			0.517562i	−0.965936	+	0.258781i	$0.916679\pi$
$60$		0			0
$61$	9.62097e7			0.889682			0.444841	−	0.895610i	$-0.353260\pi$
$61$	9.62097e7			0.889682			0.444841	+	0.895610i	$0.353260\pi$
$62$		0			0
$63$			2.13592e8i			1.70825i
$64$		0			0
$65$	1.31267e8	−	1.64547e8i	0.912106	−	1.14335i
$66$		0			0
$67$			1.42051e8i			0.861209i	0.902541	+	0.430605i	$0.141700\pi$
$67$			1.42051e8i			0.861209i	−0.902541	+	0.430605i	$0.858300\pi$
$68$		0			0
$69$	−1.00414e8			−0.533302
$70$		0			0
$71$			9.40346e7i			0.439162i	0.975594	+	0.219581i	$0.0704691\pi$
$71$			9.40346e7i			0.439162i	−0.975594	+	0.219581i	$0.929531\pi$
$72$		0			0
$73$		−	2.75593e8i		−	1.13583i	−0.823086	−	0.567917i	$-0.807749\pi$
$73$		−	2.75593e8i		−	1.13583i	0.823086	−	0.567917i	$-0.192251\pi$
$74$		0			0
$75$	4.34214e8			1.58463
$76$		0			0
$77$	−7.89082e8			−2.55808
$78$		0			0
$79$	−5.49152e8			−1.58625			−0.793124	−	0.609060i	$-0.791547\pi$
$79$	−5.49152e8			−1.58625			−0.793124	+	0.609060i	$0.791547\pi$
$80$		0			0
$81$	−4.10996e8			−1.06085
$82$		0			0
$83$			3.10281e8i			0.717636i	0.933408	+	0.358818i	$0.116820\pi$
$83$			3.10281e8i			0.717636i	−0.933408	+	0.358818i	$0.883180\pi$
$84$		0			0
$85$		−	1.23048e9i		−	2.55677i
$86$		0			0
$87$	9.07579e8			1.69843
$88$		0			0
$89$			6.74619e8i			1.13973i	0.821737	+	0.569867i	$0.193005\pi$
$89$			6.74619e8i			1.13973i	−0.821737	+	0.569867i	$0.806995\pi$
$90$		0			0
$91$	7.45399e8	−	9.34381e8i	1.13947	−	1.42836i
$92$		0			0
$93$		−	1.43293e9i		−	1.98632i
$94$		0			0
$95$	1.42104e9			1.78998
$96$		0			0
$97$			6.74712e8i			0.773830i	0.922115	+	0.386915i	$0.126459\pi$
$97$			6.74712e8i			0.773830i	−0.922115	+	0.386915i	$0.873541\pi$
$98$		0			0
$99$			1.25101e9i			1.30888i
$100$		0			0
$101$	4.74573e8			0.453792			0.226896	−	0.973919i	$-0.427142\pi$
$101$	4.74573e8			0.453792			0.226896	+	0.973919i	$0.427142\pi$
$102$		0			0
$103$	−1.05069e9			−0.919830			−0.459915	−	0.887963i	$-0.652120\pi$
$103$	−1.05069e9			−0.919830			−0.459915	+	0.887963i	$0.652120\pi$
$104$		0			0
$105$	4.63009e9			3.71739
$106$		0			0
$107$	1.07039e9			0.789430			0.394715	−	0.918804i	$-0.370843\pi$
$107$	1.07039e9			0.789430			0.394715	+	0.918804i	$0.370843\pi$
$108$		0			0
$109$		−	1.87142e9i		−	1.26985i	−0.772574	−	0.634925i	$-0.781031\pi$
$109$		−	1.87142e9i		−	1.26985i	0.772574	−	0.634925i	$-0.218969\pi$
$110$		0			0
$111$		−	1.27181e9i		−	0.795186i
$112$		0			0
$113$	−8.69209e8			−0.501500			−0.250750	−	0.968052i	$-0.580677\pi$
$113$	−8.69209e8			−0.501500			−0.250750	+	0.968052i	$0.580677\pi$
$114$		0			0
$115$			1.05174e9i			0.560748i
$116$		0			0
$117$	−1.48136e9	−	1.18175e9i	−0.730845	−	0.583029i
$118$		0			0
$119$		−	6.98730e9i		−	3.19410i
$120$		0			0
$121$	−2.26370e9			−0.960031
$122$		0			0
$123$			7.69554e8i			0.303156i
$124$		0			0
$125$		−	5.55706e8i		−	0.203587i
$126$		0			0
$127$	−2.19867e9			−0.749968			−0.374984	−	0.927031i	$-0.622352\pi$
$127$	−2.19867e9			−0.749968			−0.374984	+	0.927031i	$0.622352\pi$
$128$		0			0
$129$	2.90897e9			0.924880
$130$		0			0
$131$	3.63577e9			1.07864			0.539319	−	0.842101i	$-0.318682\pi$
$131$	3.63577e9			1.07864			0.539319	+	0.842101i	$0.318682\pi$
$132$		0			0
$133$	8.06935e9			2.23618
$134$		0			0
$135$			5.11058e8i			0.132424i
$136$		0			0
$137$			4.33206e9i			1.05064i	0.850906	+	0.525318i	$0.176054\pi$
$137$			4.33206e9i			1.05064i	−0.850906	+	0.525318i	$0.823946\pi$
$138$		0			0
$139$	−4.88274e9			−1.10942			−0.554712	−	0.832043i	$-0.687171\pi$
$139$	−4.88274e9			−1.10942			−0.554712	+	0.832043i	$0.687171\pi$
$140$		0			0
$141$			4.22442e9i			0.900080i
$142$		0			0
$143$	4.36580e9	−	5.47267e9i	0.873075	−	1.09443i
$144$		0			0
$145$		−	9.50601e9i		−	1.78584i
$146$		0			0
$147$	1.84168e10			3.25302
$148$		0			0
$149$			1.69092e9i			0.281052i	0.990077	+	0.140526i	$0.0448793\pi$
$149$			1.69092e9i			0.281052i	−0.990077	+	0.140526i	$0.955121\pi$
$150$		0			0
$151$			3.83276e9i			0.599951i	0.953947	+	0.299975i	$0.0969785\pi$
$151$			3.83276e9i			0.599951i	−0.953947	+	0.299975i	$0.903021\pi$
$152$		0			0
$153$	−1.10776e10			−1.63431
$154$		0			0
$155$	−1.50085e10			−2.08855
$156$		0			0
$157$	6.45848e9			0.848363			0.424182	−	0.905577i	$-0.360562\pi$
$157$	6.45848e9			0.848363			0.424182	+	0.905577i	$0.360562\pi$
$158$		0			0
$159$	−2.95079e9			−0.366144
$160$		0			0
$161$			5.97230e9i			0.700527i
$162$		0			0
$163$			8.73717e9i			0.969452i	0.874666	+	0.484726i	$0.161081\pi$
$163$			8.73717e9i			0.969452i	−0.874666	+	0.484726i	$0.838919\pi$
$164$		0			0
$165$	2.71185e10			2.84831
$166$		0			0
$167$			8.80744e9i			0.876245i	0.898915	+	0.438123i	$0.144356\pi$
$167$			8.80744e9i			0.876245i	−0.898915	+	0.438123i	$0.855644\pi$
$168$		0			0
$169$	2.35628e9	+	1.03394e10i	0.222197	+	0.975002i
$170$		0			0
$171$		−	1.27931e10i		−	1.14418i
$172$		0			0
$173$	−1.55139e10			−1.31678			−0.658390	−	0.752677i	$-0.728762\pi$
$173$	−1.55139e10			−1.31678			−0.658390	+	0.752677i	$0.728762\pi$
$174$		0			0
$175$		−	2.58257e10i		−	2.08152i
$176$		0			0
$177$		−	9.40097e9i		−	0.719935i
$178$		0			0
$179$	1.27802e10			0.930463			0.465231	−	0.885189i	$-0.345971\pi$
$179$	1.27802e10			0.930463			0.465231	+	0.885189i	$0.345971\pi$
$180$		0			0
$181$	2.24749e10			1.55648			0.778242	−	0.627965i	$-0.216112\pi$
$181$	2.24749e10			1.55648			0.778242	+	0.627965i	$0.216112\pi$
$182$		0			0
$183$	−1.87756e10			−1.23756
$184$		0			0
$185$	−1.33210e10			−0.836110
$186$		0			0
$187$		−	4.09246e10i		−	2.44736i
$188$		0			0
$189$			2.90204e9i			0.165434i
$190$		0			0
$191$	−1.53268e10			−0.833298			−0.416649	−	0.909068i	$-0.636796\pi$
$191$	−1.53268e10			−0.833298			−0.416649	+	0.909068i	$0.636796\pi$
$192$		0			0
$193$			1.67399e10i			0.868449i	0.900805	+	0.434224i	$0.142977\pi$
$193$			1.67399e10i			0.868449i	−0.900805	+	0.434224i	$0.857023\pi$
$194$		0			0
$195$	−2.56172e10	+	3.21120e10i	−1.26875	+	1.59042i
$196$		0			0
$197$		−	2.63345e10i		−	1.24574i	−0.782326	−	0.622870i	$-0.785967\pi$
$197$		−	2.63345e10i		−	1.24574i	0.782326	−	0.622870i	$-0.214033\pi$
$198$		0			0
$199$	2.22579e10			1.00611			0.503056	−	0.864254i	$-0.332209\pi$
$199$	2.22579e10			1.00611			0.503056	+	0.864254i	$0.332209\pi$
$200$		0			0
$201$		−	2.77218e10i		−	1.19795i
$202$		0			0
$203$		−	5.39799e10i		−	2.23100i
$204$		0			0
$205$	8.06034e9			0.318758
$206$		0			0
$207$	9.46846e9			0.358437
$208$		0			0
$209$	4.72622e10			1.71339
$210$		0			0
$211$	3.62511e10			1.25907			0.629536	−	0.776971i	$-0.283245\pi$
$211$	3.62511e10			1.25907			0.629536	+	0.776971i	$0.283245\pi$
$212$		0			0
$213$		−	1.83512e10i		−	0.610879i
$214$		0			0
$215$		−	3.04686e10i		−	0.972479i
$216$		0			0
$217$	−8.52258e10			−2.60917
$218$		0			0
$219$			5.37828e10i			1.57996i
$220$		0			0
$221$	4.84603e10	+	3.86590e10i	1.36654	+	1.09015i
$222$		0			0
$223$			7.54110e9i			0.204203i	0.994774	+	0.102102i	$0.0325567\pi$
$223$			7.54110e9i			0.204203i	−0.994774	+	0.102102i	$0.967443\pi$
$224$		0			0
$225$	−4.09439e10			−1.06505
$226$		0			0
$227$			4.71271e10i			1.17803i	0.808124	+	0.589013i	$0.200483\pi$
$227$			4.71271e10i			1.17803i	−0.808124	+	0.589013i	$0.799517\pi$
$228$		0			0
$229$		−	2.35079e10i		−	0.564877i	−0.959285	−	0.282438i	$-0.908857\pi$
$229$		−	2.35079e10i		−	0.564877i	0.959285	−	0.282438i	$-0.0911432\pi$
$230$		0			0
$231$	1.53992e11			3.55831
$232$		0			0
$233$	2.92724e10			0.650664			0.325332	−	0.945600i	$-0.394524\pi$
$233$	2.92724e10			0.650664			0.325332	+	0.945600i	$0.394524\pi$
$234$		0			0
$235$	4.42468e10			0.946403
$236$		0			0
$237$	1.07169e11			2.20649
$238$		0			0
$239$		−	1.26138e10i		−	0.250067i	−0.992153	−	0.125033i	$-0.960096\pi$
$239$		−	1.26138e10i		−	0.250067i	0.992153	−	0.125033i	$-0.0399038\pi$
$240$		0			0
$241$			1.81575e10i			0.346720i	0.984858	+	0.173360i	$0.0554625\pi$
$241$			1.81575e10i			0.346720i	−0.984858	+	0.173360i	$0.944538\pi$
$242$		0			0
$243$	7.52861e10			1.38512
$244$		0			0
$245$		−	1.92899e11i		−	3.42044i
$246$		0			0
$247$	−4.46457e10	+	5.59649e10i	−0.763210	+	0.956708i
$248$		0			0
$249$		−	6.05524e10i		−	0.998239i
$250$		0			0
$251$	−3.15257e10			−0.501341			−0.250671	−	0.968072i	$-0.580651\pi$
$251$	−3.15257e10			−0.501341			−0.250671	+	0.968072i	$0.580651\pi$
$252$		0			0
$253$			3.49797e10i			0.536753i
$254$		0			0
$255$			2.40133e11i			3.55649i
$256$		0			0
$257$	−2.90207e10			−0.414962			−0.207481	−	0.978239i	$-0.566527\pi$
$257$	−2.90207e10			−0.414962			−0.207481	+	0.978239i	$0.566527\pi$
$258$		0			0
$259$	−7.56432e10			−1.04453
$260$		0			0
$261$	−8.55794e10			−1.14153
$262$		0			0
$263$	−6.88736e10			−0.887671			−0.443835	−	0.896108i	$-0.646382\pi$
$263$	−6.88736e10			−0.887671			−0.443835	+	0.896108i	$0.646382\pi$
$264$		0			0
$265$			3.09067e10i			0.384987i
$266$		0			0
$267$		−	1.31654e11i		−	1.58538i
$268$		0			0
$269$	−2.75452e10			−0.320746			−0.160373	−	0.987056i	$-0.551270\pi$
$269$	−2.75452e10			−0.320746			−0.160373	+	0.987056i	$0.551270\pi$
$270$		0			0
$271$		−	5.81652e10i		−	0.655091i	−0.944835	−	0.327545i	$-0.893779\pi$
$271$		−	5.81652e10i		−	0.655091i	0.944835	−	0.327545i	$-0.106221\pi$
$272$		0			0
$273$	−1.45467e11	+	1.82348e11i	−1.58501	+	1.98686i
$274$		0			0
$275$		−	1.51261e11i		−	1.59489i
$276$		0			0
$277$	−8.12480e10			−0.829190			−0.414595	−	0.910006i	$-0.636077\pi$
$277$	−8.12480e10			−0.829190			−0.414595	+	0.910006i	$0.636077\pi$
$278$		0			0
$279$			1.35117e11i			1.33503i
$280$		0			0
$281$			1.13980e11i			1.09056i	0.838254	+	0.545280i	$0.183577\pi$
$281$			1.13980e11i			1.09056i	−0.838254	+	0.545280i	$0.816423\pi$
$282$		0			0
$283$	−1.13569e11			−1.05250			−0.526248	−	0.850331i	$-0.676402\pi$
$283$	−1.13569e11			−1.05250			−0.526248	+	0.850331i	$0.676402\pi$
$284$		0			0
$285$	−2.77320e11			−2.48989
$286$		0			0
$287$	4.57706e10			0.398215
$288$		0			0
$289$	2.43798e11			2.05585
$290$		0			0
$291$		−	1.31672e11i		−	1.07641i
$292$		0			0
$293$		−	7.70666e10i		−	0.610888i	−0.952210	−	0.305444i	$-0.901195\pi$
$293$		−	7.70666e10i		−	0.610888i	0.952210	−	0.305444i	$-0.0988049\pi$
$294$		0			0
$295$	−9.84661e10			−0.756986
$296$		0			0
$297$			1.69972e10i			0.126758i
$298$		0			0
$299$	−4.14208e10	−	3.30433e10i	−0.299708	−	0.239091i
$300$		0			0
$301$		−	1.73016e11i		−	1.21489i
$302$		0			0
$303$	−9.26146e10			−0.631230
$304$		0			0
$305$			1.96657e11i			1.30125i
$306$		0			0
$307$			1.69749e11i			1.09065i	0.838225	+	0.545325i	$0.183594\pi$
$307$			1.69749e11i			1.09065i	−0.838225	+	0.545325i	$0.816406\pi$
$308$		0			0
$309$	2.05046e11			1.27949
$310$		0			0
$311$	−2.68747e10			−0.162900			−0.0814502	−	0.996677i	$-0.525955\pi$
$311$	−2.68747e10			−0.162900			−0.0814502	+	0.996677i	$0.525955\pi$
$312$		0			0
$313$	−3.11862e11			−1.83659			−0.918295	−	0.395896i	$-0.870434\pi$
$313$	−3.11862e11			−1.83659			−0.918295	+	0.395896i	$0.870434\pi$
$314$		0			0
$315$	−4.36591e11			−2.49849
$316$		0			0
$317$			1.50743e11i			0.838437i	0.907885	+	0.419218i	$0.137696\pi$
$317$			1.50743e11i			0.838437i	−0.907885	+	0.419218i	$0.862304\pi$
$318$		0			0
$319$		−	3.16160e11i		−	1.70942i
$320$		0			0
$321$	−2.08890e11			−1.09811
$322$		0			0
$323$			4.18505e11i			2.13939i
$324$		0			0
$325$	1.79114e11	+	1.42887e11i	0.890541	+	0.710425i
$326$		0			0
$327$			3.65214e11i			1.76637i
$328$		0			0
$329$	2.51255e11			1.18232
$330$		0			0
$331$			3.22419e11i			1.47637i	0.674600	+	0.738183i	$0.264316\pi$
$331$			3.22419e11i			1.47637i	−0.674600	+	0.738183i	$0.735684\pi$
$332$		0			0
$333$			1.19924e11i			0.534452i
$334$		0			0
$335$	−2.90359e11			−1.25960
$336$		0			0
$337$	1.42221e11			0.600662			0.300331	−	0.953835i	$-0.402903\pi$
$337$	1.42221e11			0.600662			0.300331	+	0.953835i	$0.402903\pi$
$338$		0			0
$339$	1.69629e11			0.697592
$340$		0			0
$341$	−4.99167e11			−1.99918
$342$		0			0
$343$		−	6.26986e11i		−	2.44587i
$344$		0			0
$345$		−	2.05251e11i		−	0.780007i
$346$		0			0
$347$	3.18456e11			1.17914			0.589571	−	0.807716i	$-0.299297\pi$
$347$	3.18456e11			1.17914			0.589571	+	0.807716i	$0.299297\pi$
$348$		0			0
$349$		−	1.82982e11i		−	0.660229i	−0.943941	−	0.330114i	$-0.892913\pi$
$349$		−	1.82982e11i		−	0.660229i	0.943941	−	0.330114i	$-0.107087\pi$
$350$		0			0
$351$	−2.01271e10	−	1.60563e10i	−0.0707781	−	0.0564629i
$352$		0			0
$353$			3.06629e11i			1.05106i	0.850775	+	0.525529i	$0.176133\pi$
$353$			3.06629e11i			1.05106i	−0.850775	+	0.525529i	$0.823867\pi$
$354$		0			0
$355$	−1.92211e11			−0.642318
$356$		0			0
$357$			1.36360e12i			4.44302i
$358$		0			0
$359$			3.13878e11i			0.997324i	0.866797	+	0.498662i	$0.166175\pi$
$359$			3.13878e11i			0.997324i	−0.866797	+	0.498662i	$0.833825\pi$
$360$		0			0
$361$	−1.60627e11			−0.497779
$362$		0			0
$363$	4.41769e11			1.33541
$364$		0			0
$365$	5.63324e11			1.66127
$366$		0			0
$367$	3.91992e11			1.12792			0.563962	−	0.825801i	$-0.309277\pi$
$367$	3.91992e11			1.12792			0.563962	+	0.825801i	$0.309277\pi$
$368$		0			0
$369$		−	7.25645e10i		−	0.203754i
$370$		0			0
$371$			1.75504e11i			0.480954i
$372$		0			0
$373$	−2.97656e11			−0.796205			−0.398102	−	0.917341i	$-0.630331\pi$
$373$	−2.97656e11			−0.796205			−0.398102	+	0.917341i	$0.630331\pi$
$374$		0			0
$375$			1.08448e11i			0.283192i
$376$		0			0
$377$	3.74377e11	+	2.98657e11i	0.954493	+	0.761443i
$378$		0			0
$379$		−	4.59920e11i		−	1.14500i	−0.819904	−	0.572501i	$-0.805973\pi$
$379$		−	4.59920e11i		−	1.14500i	0.819904	−	0.572501i	$-0.194027\pi$
$380$		0			0
$381$	4.29077e11			1.04321
$382$		0			0
$383$		−	7.05491e10i		−	0.167532i	−0.996485	−	0.0837659i	$-0.973305\pi$
$383$		−	7.05491e10i		−	0.167532i	0.996485	−	0.0837659i	$-0.0266948\pi$
$384$		0			0
$385$		−	1.61292e12i		−	3.74144i
$386$		0			0
$387$	−2.74299e11			−0.621620
$388$		0			0
$389$	3.67385e11			0.813483			0.406742	−	0.913543i	$-0.366665\pi$
$389$	3.67385e11			0.813483			0.406742	+	0.913543i	$0.366665\pi$
$390$		0			0
$391$	−3.09745e11			−0.670206
$392$		0			0
$393$	−7.09533e11			−1.50040
$394$		0			0
$395$		−	1.12249e12i		−	2.32004i
$396$		0			0
$397$			5.05825e11i			1.02198i	0.859586	+	0.510991i	$0.170722\pi$
$397$			5.05825e11i			1.02198i	−0.859586	+	0.510991i	$0.829278\pi$
$398$		0			0
$399$	−1.57476e12			−3.11055
$400$		0			0
$401$		−	3.63956e11i		−	0.702909i	−0.936205	−	0.351454i	$-0.885687\pi$
$401$		−	3.63956e11i		−	0.702909i	0.936205	−	0.351454i	$-0.114313\pi$
$402$		0			0
$403$	4.71533e11	−	5.91082e11i	0.890512	−	1.11629i
$404$		0			0
$405$		−	8.40094e11i		−	1.55160i
$406$		0			0
$407$	−4.43042e11			−0.800332
$408$		0			0
$409$		−	5.23141e11i		−	0.924409i	−0.886773	−	0.462204i	$-0.847059\pi$
$409$		−	5.23141e11i		−	0.924409i	0.886773	−	0.462204i	$-0.152941\pi$
$410$		0			0
$411$		−	8.45416e11i		−	1.46144i
$412$		0			0
$413$	−5.59140e11			−0.945682
$414$		0			0
$415$	−6.34228e11			−1.04961
$416$		0			0
$417$	9.52884e11			1.54322
$418$		0			0
$419$	1.21423e12			1.92458			0.962292	−	0.272018i	$-0.0876911\pi$
$419$	1.21423e12			1.92458			0.962292	+	0.272018i	$0.0876911\pi$
$420$		0			0
$421$		−	6.82454e11i		−	1.05878i	−0.848380	−	0.529388i	$-0.822422\pi$
$421$		−	6.82454e11i		−	1.05878i	0.848380	−	0.529388i	$-0.177578\pi$
$422$		0			0
$423$		−	3.98339e11i		−	0.604952i
$424$		0			0
$425$	1.33941e12			1.99142
$426$		0			0
$427$			1.11672e12i			1.62561i
$428$		0			0
$429$	−8.52000e11	+	1.06801e12i	−1.21446	+	1.52236i
$430$		0			0
$431$			4.54494e11i			0.634425i	0.948355	+	0.317212i	$0.102747\pi$
$431$			4.54494e11i			0.634425i	−0.948355	+	0.317212i	$0.897253\pi$
$432$		0			0
$433$	−4.74468e10			−0.0648651			−0.0324326	−	0.999474i	$-0.510325\pi$
$433$	−4.74468e10			−0.0648651			−0.0324326	+	0.999474i	$0.510325\pi$
$434$		0			0
$435$			1.85513e12i			2.48412i
$436$		0			0
$437$		−	3.57712e11i		−	0.469209i
$438$		0			0
$439$	1.52136e12			1.95498			0.977491	−	0.210978i	$-0.0676649\pi$
$439$	1.52136e12			1.95498			0.977491	+	0.210978i	$0.0676649\pi$
$440$		0			0
$441$	−1.73660e12			−2.18638
$442$		0			0
$443$	1.37527e12			1.69656			0.848281	−	0.529546i	$-0.177638\pi$
$443$	1.37527e12			1.69656			0.848281	+	0.529546i	$0.177638\pi$
$444$		0			0
$445$	−1.37895e12			−1.66697
$446$		0			0
$447$		−	3.29990e11i		−	0.390946i
$448$		0			0
$449$			1.28293e12i			1.48969i	0.667240	+	0.744843i	$0.267476\pi$
$449$			1.28293e12i			1.48969i	−0.667240	+	0.744843i	$0.732524\pi$
$450$		0			0
$451$	2.68078e11			0.305117
$452$		0			0
$453$		−	7.47977e11i		−	0.834538i
$454$		0			0
$455$	1.90992e12	+	1.52363e12i	2.08912	+	1.66659i
$456$		0			0
$457$			4.86475e11i			0.521720i	0.965377	+	0.260860i	$0.0840061\pi$
$457$			4.86475e11i			0.521720i	−0.965377	+	0.260860i	$0.915994\pi$
$458$		0			0
$459$	−1.50510e11			−0.158274
$460$		0			0
$461$			2.39829e11i			0.247314i	0.992325	+	0.123657i	$0.0394622\pi$
$461$			2.39829e11i			0.247314i	−0.992325	+	0.123657i	$0.960538\pi$
$462$		0			0
$463$		−	4.21468e11i		−	0.426236i	−0.977026	−	0.213118i	$-0.931638\pi$
$463$		−	4.21468e11i		−	0.426236i	0.977026	−	0.213118i	$-0.0683619\pi$
$464$		0			0
$465$	2.92896e12			2.90520
$466$		0			0
$467$	5.16039e11			0.502061			0.251031	−	0.967979i	$-0.419231\pi$
$467$	5.16039e11			0.502061			0.251031	+	0.967979i	$0.419231\pi$
$468$		0			0
$469$	−1.64880e12			−1.57359
$470$		0			0
$471$	−1.26039e12			−1.18008
$472$		0			0
$473$		−	1.01335e12i		−	0.930864i
$474$		0			0
$475$			1.54683e12i			1.39419i
$476$		0			0
$477$	2.78243e11			0.246088
$478$		0			0
$479$			1.02884e12i			0.892975i	0.894790	+	0.446488i	$0.147325\pi$
$479$			1.02884e12i			0.892975i	−0.894790	+	0.446488i	$0.852675\pi$
$480$		0			0
$481$	4.18515e11	−	5.24622e11i	0.356499	−	0.446883i
$482$		0			0
$483$		−	1.16551e12i		−	0.974441i
$484$		0			0
$485$	−1.37914e12			−1.13180
$486$		0			0
$487$			1.26918e12i			1.02245i	0.859447	+	0.511224i	$0.170808\pi$
$487$			1.26918e12i			1.02245i	−0.859447	+	0.511224i	$0.829192\pi$
$488$		0			0
$489$		−	1.70509e12i		−	1.34852i
$490$		0			0
$491$	4.17282e11			0.324013			0.162006	−	0.986790i	$-0.448203\pi$
$491$	4.17282e11			0.324013			0.162006	+	0.986790i	$0.448203\pi$
$492$		0			0
$493$	2.79959e12			2.13444
$494$		0			0
$495$	−2.55711e12			−1.91437
$496$		0			0
$497$	−1.09147e12			−0.802431
$498$		0			0
$499$		−	1.90982e11i		−	0.137892i	−0.997620	−	0.0689460i	$-0.978036\pi$
$499$		−	1.90982e11i		−	0.137892i	0.997620	−	0.0689460i	$-0.0219636\pi$
$500$		0			0
$501$		−	1.71880e12i		−	1.21887i
$502$		0			0
$503$	−1.05092e12			−0.732007			−0.366004	−	0.930613i	$-0.619274\pi$
$503$	−1.05092e12			−0.732007			−0.366004	+	0.930613i	$0.619274\pi$
$504$		0			0
$505$			9.70049e11i			0.663716i
$506$		0			0
$507$	−4.59837e11	−	2.01777e12i	−0.309078	−	1.35624i
$508$		0			0
$509$		−	1.89691e12i		−	1.25261i	−0.779577	−	0.626306i	$-0.784566\pi$
$509$		−	1.89691e12i		−	1.25261i	0.779577	−	0.626306i	$-0.215434\pi$
$510$		0			0
$511$	3.19883e12			2.07538
$512$		0			0
$513$		−	1.73818e11i		−	0.110807i
$514$		0			0
$515$		−	2.14766e12i		−	1.34534i
$516$		0			0
$517$	1.47160e12			0.905904
$518$		0			0
$519$	3.02759e12			1.83166
$520$		0			0
$521$	1.54410e12			0.918131			0.459065	−	0.888403i	$-0.348184\pi$
$521$	1.54410e12			0.918131			0.459065	+	0.888403i	$0.348184\pi$
$522$		0			0
$523$	−1.17561e12			−0.687078			−0.343539	−	0.939138i	$-0.611626\pi$
$523$	−1.17561e12			−0.687078			−0.343539	+	0.939138i	$0.611626\pi$
$524$		0			0
$525$			5.03997e12i			2.89542i
$526$		0			0
$527$		−	4.42011e12i		−	2.49624i
$528$		0			0
$529$	−1.53640e12			−0.853011
$530$		0			0
$531$			8.86457e11i			0.483874i
$532$		0			0
$533$	−2.53237e11	+	3.17441e11i	−0.135911	+	0.170369i
$534$		0			0
$535$			2.18792e12i			1.15462i
$536$		0			0
$537$	−2.49410e12			−1.29428
$538$		0			0
$539$		−	6.41560e12i		−	3.27407i
$540$		0			0
$541$			1.04469e12i			0.524324i	0.965024	+	0.262162i	$0.0844355\pi$
$541$			1.04469e12i			0.524324i	−0.965024	+	0.262162i	$0.915564\pi$
$542$		0			0
$543$	−4.38606e12			−2.16509
$544$		0			0
$545$	3.82527e12			1.85728
$546$		0			0
$547$	−4.68950e11			−0.223967			−0.111983	−	0.993710i	$-0.535720\pi$
$547$	−4.68950e11			−0.223967			−0.111983	+	0.993710i	$0.535720\pi$
$548$		0			0
$549$	1.77044e12			0.831772
$550$		0			0
$551$			3.23313e12i			1.49431i
$552$		0			0
$553$		−	6.37407e12i		−	2.89837i
$554$		0			0
$555$	2.59964e12			1.16304
$556$		0			0
$557$		−	2.07740e12i		−	0.914475i	−0.889345	−	0.457238i	$-0.848839\pi$
$557$		−	2.07740e12i		−	0.914475i	0.889345	−	0.457238i	$-0.151161\pi$
$558$		0			0
$559$	1.19995e12	+	9.57255e11i	0.519769	+	0.414643i
$560$		0			0
$561$			7.98657e12i			3.40430i
$562$		0			0
$563$	−4.37887e12			−1.83685			−0.918427	−	0.395590i	$-0.870540\pi$
$563$	−4.37887e12			−1.83685			−0.918427	+	0.395590i	$0.870540\pi$
$564$		0			0
$565$		−	1.77670e12i		−	0.733494i
$566$		0			0
$567$		−	4.77047e12i		−	1.93837i
$568$		0			0
$569$	8.35971e11			0.334338			0.167169	−	0.985928i	$-0.446537\pi$
$569$	8.35971e11			0.334338			0.167169	+	0.985928i	$0.446537\pi$
$570$		0			0
$571$	1.78534e12			0.702845			0.351422	−	0.936217i	$-0.385698\pi$
$571$	1.78534e12			0.702845			0.351422	+	0.936217i	$0.385698\pi$
$572$		0			0
$573$	2.99107e12			1.15913
$574$		0			0
$575$	−1.14484e12			−0.436758
$576$		0			0
$577$			2.36766e12i			0.889258i	0.895715	+	0.444629i	$0.146664\pi$
$577$			2.36766e12i			0.889258i	−0.895715	+	0.444629i	$0.853336\pi$
$578$		0			0
$579$		−	3.26684e12i		−	1.20802i
$580$		0			0
$581$	−3.60146e12			−1.31125
$582$		0			0
$583$			1.02792e12i			0.368513i
$584$		0			0
$585$	2.41555e12	−	3.02797e12i	0.852737	−	1.06893i
$586$		0			0
$587$		−	2.04346e12i		−	0.710387i	−0.934793	−	0.355193i	$-0.884415\pi$
$587$		−	2.04346e12i		−	0.710387i	0.934793	−	0.355193i	$-0.115585\pi$
$588$		0			0
$589$	5.10461e12			1.74761
$590$		0			0
$591$			5.13927e12i			1.73284i
$592$		0			0
$593$		−	1.34412e12i		−	0.446368i	−0.974776	−	0.223184i	$-0.928355\pi$
$593$		−	1.34412e12i		−	0.446368i	0.974776	−	0.223184i	$-0.0716450\pi$
$594$		0			0
$595$	1.42823e13			4.67168
$596$		0			0
$597$	−4.34371e12			−1.39951
$598$		0			0
$599$	1.32024e12			0.419017			0.209509	−	0.977807i	$-0.432814\pi$
$599$	1.32024e12			0.419017			0.209509	+	0.977807i	$0.432814\pi$
$600$		0			0
$601$	1.75457e12			0.548573			0.274286	−	0.961648i	$-0.411558\pi$
$601$	1.75457e12			0.548573			0.274286	+	0.961648i	$0.411558\pi$
$602$		0			0
$603$			2.61401e12i			0.805153i
$604$		0			0
$605$		−	4.62711e12i		−	1.40414i
$606$		0			0
$607$	−3.85812e12			−1.15352			−0.576762	−	0.816912i	$-0.695684\pi$
$607$	−3.85812e12			−1.15352			−0.576762	+	0.816912i	$0.695684\pi$
$608$		0			0
$609$			1.05344e13i			3.10334i
$610$		0			0
$611$	−1.39013e12	+	1.74258e12i	−0.403525	+	0.505832i
$612$		0			0
$613$		−	1.91721e12i		−	0.548399i	−0.961673	−	0.274200i	$-0.911587\pi$
$613$		−	1.91721e12i		−	0.548399i	0.961673	−	0.274200i	$-0.0884129\pi$
$614$		0			0
$615$	−1.57300e12			−0.443395
$616$		0			0
$617$		−	1.66156e12i		−	0.461566i	−0.973005	−	0.230783i	$-0.925871\pi$
$617$		−	1.66156e12i		−	0.461566i	0.973005	−	0.230783i	$-0.0741288\pi$
$618$		0			0
$619$		−	2.72781e12i		−	0.746804i	−0.927670	−	0.373402i	$-0.878191\pi$
$619$		−	2.72781e12i		−	0.746804i	0.927670	−	0.373402i	$-0.121809\pi$
$620$		0			0
$621$	1.28647e11			0.0347125
$622$		0			0
$623$	−7.83037e12			−2.08251
$624$		0			0
$625$	−3.20980e12			−0.841429
$626$		0			0
$627$	−9.22337e12			−2.38334
$628$		0			0
$629$		−	3.92312e12i		−	0.999319i
$630$		0			0
$631$			6.38718e12i			1.60390i	0.597391	+	0.801950i	$0.296204\pi$
$631$			6.38718e12i			1.60390i	−0.597391	+	0.801950i	$0.703796\pi$
$632$		0			0
$633$	−7.07453e12			−1.75138
$634$		0			0
$635$		−	4.49417e12i		−	1.09690i
$636$		0			0
$637$	7.59695e12	+	6.06044e12i	1.82815	+	1.45840i
$638$		0			0
$639$			1.73041e12i			0.410577i
$640$		0			0
$641$	−1.20312e12			−0.281479			−0.140740	−	0.990047i	$-0.544948\pi$
$641$	−1.20312e12			−0.281479			−0.140740	+	0.990047i	$0.544948\pi$
$642$		0			0
$643$		−	1.34420e12i		−	0.310110i	−0.987906	−	0.155055i	$-0.950445\pi$
$643$		−	1.34420e12i		−	0.310110i	0.987906	−	0.155055i	$-0.0495554\pi$
$644$		0			0
$645$			5.94606e12i			1.35273i
$646$		0			0
$647$	4.23542e12			0.950226			0.475113	−	0.879925i	$-0.342407\pi$
$647$	4.23542e12			0.950226			0.475113	+	0.879925i	$0.342407\pi$
$648$		0			0
$649$	−3.27488e12			−0.724593
$650$		0			0
$651$	1.66321e13			3.62938
$652$		0			0
$653$	1.97296e11			0.0424628			0.0212314	−	0.999775i	$-0.493241\pi$
$653$	1.97296e11			0.0424628			0.0212314	+	0.999775i	$0.493241\pi$
$654$		0			0
$655$			7.43168e12i			1.57762i
$656$		0			0
$657$		−	5.07141e12i		−	1.06190i
$658$		0			0
$659$	−7.35592e12			−1.51933			−0.759666	−	0.650313i	$-0.774638\pi$
$659$	−7.35592e12			−1.51933			−0.759666	+	0.650313i	$0.774638\pi$
$660$		0			0
$661$		−	3.96002e11i		−	0.0806847i	−0.999186	−	0.0403424i	$-0.987155\pi$
$661$		−	3.96002e11i		−	0.0806847i	0.999186	−	0.0403424i	$-0.0128449\pi$
$662$		0			0
$663$	−9.45720e12	−	7.54444e12i	−1.90087	−	1.51641i
$664$		0			0
$665$			1.64941e13i			3.27063i
$666$		0			0
$667$	−2.39291e12			−0.468123
$668$		0			0
$669$		−	1.47167e12i		−	0.284049i
$670$		0			0
$671$			6.54060e12i			1.24556i
$672$		0			0
$673$	−9.06698e12			−1.70371			−0.851854	−	0.523780i	$-0.824521\pi$
$673$	−9.06698e12			−1.70371			−0.851854	+	0.523780i	$0.824521\pi$
$674$		0			0
$675$	−5.56299e11			−0.103143
$676$		0			0
$677$	8.65021e12			1.58262			0.791312	−	0.611413i	$-0.209398\pi$
$677$	8.65021e12			1.58262			0.791312	+	0.611413i	$0.209398\pi$
$678$		0			0
$679$	−7.83144e12			−1.41393
$680$		0			0
$681$		−	9.19702e12i		−	1.63865i
$682$		0			0
$683$			4.27465e12i			0.751635i	0.926694	+	0.375818i	$0.122638\pi$
$683$			4.27465e12i			0.751635i	−0.926694	+	0.375818i	$0.877362\pi$
$684$		0			0
$685$	−8.85492e12			−1.53666
$686$		0			0
$687$			4.58764e12i			0.785749i
$688$		0			0
$689$	−1.21720e12	−	9.71018e11i	−0.205767	−	0.164150i
$690$		0			0
$691$			1.85871e12i			0.310142i	0.987903	+	0.155071i	$0.0495607\pi$
$691$			1.85871e12i			0.310142i	−0.987903	+	0.155071i	$0.950439\pi$
$692$		0			0
$693$	−1.45206e13			−2.39157
$694$		0			0
$695$		−	9.98054e12i		−	1.62264i
$696$		0			0
$697$			2.37383e12i			0.380979i
$698$		0			0
$699$	−5.71261e12			−0.905080
$700$		0			0
$701$	7.75993e11			0.121374			0.0606871	−	0.998157i	$-0.480671\pi$
$701$	7.75993e11			0.121374			0.0606871	+	0.998157i	$0.480671\pi$
$702$		0			0
$703$	4.53066e12			0.699620
$704$		0			0
$705$	−8.63490e12			−1.31646
$706$		0			0
$707$			5.50842e12i			0.829162i
$708$		0			0
$709$			1.12555e13i			1.67285i	0.548080	+	0.836426i	$0.315359\pi$
$709$			1.12555e13i			1.67285i	−0.548080	+	0.836426i	$0.684641\pi$
$710$		0			0
$711$	−1.01054e13			−1.48300
$712$		0			0
$713$			3.77803e12i			0.547473i
$714$		0			0
$715$	1.11864e13	+	8.92388e12i	1.60071	+	1.27696i
$716$		0			0
$717$			2.46163e12i			0.347845i
$718$		0			0
$719$	3.31877e12			0.463124			0.231562	−	0.972820i	$-0.425616\pi$
$719$	3.31877e12			0.463124			0.231562	+	0.972820i	$0.425616\pi$
$720$		0			0
$721$		−	1.21955e13i		−	1.68070i
$722$		0			0
$723$		−	3.54350e12i		−	0.482292i
$724$		0			0
$725$	1.03475e13			1.39096
$726$		0			0
$727$	−7.03586e12			−0.934141			−0.467070	−	0.884220i	$-0.654691\pi$
$727$	−7.03586e12			−0.934141			−0.467070	+	0.884220i	$0.654691\pi$
$728$		0			0
$729$	−6.60270e12			−0.865860
$730$		0			0
$731$	8.97323e12			1.16231
$732$		0			0
$733$		−	4.64783e12i		−	0.594679i	−0.954772	−	0.297340i	$-0.903901\pi$
$733$		−	4.64783e12i		−	0.594679i	0.954772	−	0.297340i	$-0.0960993\pi$
$734$		0			0
$735$			3.76448e13i			4.75786i
$736$		0			0
$737$	−9.65704e12			−1.20570
$738$		0			0
$739$		−	7.23266e11i		−	0.0892068i	−0.999005	−	0.0446034i	$-0.985798\pi$
$739$		−	7.23266e11i		−	0.0892068i	0.999005	−	0.0446034i	$-0.0142024\pi$
$740$		0			0
$741$	8.71277e12	−	1.09217e13i	1.06163	−	1.33079i
$742$		0			0
$743$		−	1.36661e13i		−	1.64511i	−0.568689	−	0.822553i	$-0.692549\pi$
$743$		−	1.36661e13i		−	1.64511i	0.568689	−	0.822553i	$-0.307451\pi$
$744$		0			0
$745$	−3.45632e12			−0.411066
$746$		0			0
$747$			5.70974e12i			0.670925i
$748$		0			0
$749$			1.24241e13i			1.44243i
$750$		0			0
$751$	−7.46716e12			−0.856596			−0.428298	−	0.903638i	$-0.640887\pi$
$751$	−7.46716e12			−0.856596			−0.428298	+	0.903638i	$0.640887\pi$
$752$		0			0
$753$	6.15235e12			0.697371
$754$		0			0
$755$	−7.83434e12			−0.877488
$756$		0			0
$757$	−7.27732e12			−0.805453			−0.402726	−	0.915320i	$-0.631937\pi$
$757$	−7.27732e12			−0.805453			−0.402726	+	0.915320i	$0.631937\pi$
$758$		0			0
$759$		−	6.82641e12i		−	0.746629i
$760$		0			0
$761$		−	8.06232e12i		−	0.871424i	−0.900086	−	0.435712i	$-0.856497\pi$
$761$		−	8.06232e12i		−	0.871424i	0.900086	−	0.435712i	$-0.143503\pi$
$762$		0			0
$763$	2.17218e13			2.32025
$764$		0			0
$765$		−	2.26432e13i		−	2.39035i
$766$		0			0
$767$	3.09358e12	−	3.87791e12i	0.322762	−	0.404593i
$768$		0			0
$769$		−	1.05335e13i		−	1.08618i	−0.839673	−	0.543092i	$-0.817254\pi$
$769$		−	1.05335e13i		−	1.08618i	0.839673	−	0.543092i	$-0.182746\pi$
$770$		0			0
$771$	5.66348e12			0.577217
$772$		0			0
$773$		−	8.85431e12i		−	0.891963i	−0.895042	−	0.445982i	$-0.852855\pi$
$773$		−	8.85431e12i		−	0.891963i	0.895042	−	0.445982i	$-0.147145\pi$
$774$		0			0
$775$		−	1.63371e13i		−	1.62674i
$776$		0			0
$777$	1.47620e13			1.45295
$778$		0			0
$779$	−2.74144e12			−0.266722
$780$		0			0
$781$	−6.39273e12			−0.614832
$782$		0			0
$783$	−1.16276e12			−0.110550
$784$		0			0
$785$			1.32014e13i			1.24082i
$786$		0			0
$787$			4.28652e12i			0.398308i	0.979968	+	0.199154i	$0.0638194\pi$
$787$			4.28652e12i			0.398308i	−0.979968	+	0.199154i	$0.936181\pi$
$788$		0			0
$789$	1.34409e13			1.23476
$790$		0			0
$791$		−	1.00890e13i		−	0.916334i
$792$		0			0
$793$	−7.74497e12	−	6.17851e12i	−0.695489	−	0.554823i
$794$		0			0
$795$		−	6.03155e12i		−	0.535521i
$796$		0			0
$797$	−4.26252e12			−0.374200			−0.187100	−	0.982341i	$-0.559909\pi$
$797$	−4.26252e12			−0.374200			−0.187100	+	0.982341i	$0.559909\pi$
$798$		0			0
$799$			1.30310e13i			1.13114i
$800$		0			0
$801$			1.24142e13i			1.06555i
$802$		0			0
$803$	1.87355e13			1.59018
$804$		0			0
$805$	−1.22076e13			−1.02459
$806$		0			0
$807$	5.37555e12			0.446161
$808$		0			0
$809$	2.81433e11			0.0230997			0.0115498	−	0.999933i	$-0.496323\pi$
$809$	2.81433e11			0.0230997			0.0115498	+	0.999933i	$0.496323\pi$
$810$		0			0
$811$			5.82312e12i			0.472674i	0.971671	+	0.236337i	$0.0759469\pi$
$811$			5.82312e12i			0.472674i	−0.971671	+	0.236337i	$0.924053\pi$
$812$		0			0
$813$			1.13511e13i			0.911238i
$814$		0			0
$815$	−1.78592e13			−1.41792
$816$		0			0
$817$			1.03628e13i			0.813727i
$818$		0			0
$819$	1.37167e13	−	1.71943e13i	1.06530	−	1.33539i
$820$		0			0
$821$			9.70362e12i			0.745400i	0.927952	+	0.372700i	$0.121568\pi$
$821$			9.70362e12i			0.745400i	−0.927952	+	0.372700i	$0.878432\pi$
$822$		0			0
$823$	−2.07448e13			−1.57620			−0.788098	−	0.615550i	$-0.788934\pi$
$823$	−2.07448e13			−1.57620			−0.788098	+	0.615550i	$0.788934\pi$
$824$		0			0
$825$			2.95191e13i			2.21850i
$826$		0			0
$827$		−	1.79156e13i		−	1.33185i	−0.746017	−	0.665927i	$-0.768036\pi$
$827$		−	1.79156e13i		−	1.33185i	0.746017	−	0.665927i	$-0.231964\pi$
$828$		0			0
$829$	1.33128e13			0.978983			0.489491	−	0.872008i	$-0.337182\pi$
$829$	1.33128e13			0.978983			0.489491	+	0.872008i	$0.337182\pi$
$830$		0			0
$831$	1.58558e13			1.15341
$832$		0			0
$833$	5.68100e13			4.08811
$834$		0			0
$835$	−1.80028e13			−1.28160
$836$		0			0
$837$			1.83581e12i			0.129289i
$838$		0			0
$839$			2.48694e13i			1.73275i	0.499394	+	0.866375i	$0.333556\pi$
$839$			2.48694e13i			1.73275i	−0.499394	+	0.866375i	$0.666444\pi$
$840$		0			0
$841$	7.12086e12			0.490852
$842$		0			0
$843$		−	2.22436e13i		−	1.51698i
$844$		0			0
$845$	−2.11342e13	+	4.81635e12i	−1.42604	+	0.324985i
$846$		0			0
$847$		−	2.62750e13i		−	1.75415i
$848$		0			0
$849$	2.21634e13			1.46403
$850$		0			0
$851$			3.35324e12i			0.219170i
$852$		0			0
$853$			6.73331e12i			0.435470i	0.976008	+	0.217735i	$0.0698668\pi$
$853$			6.73331e12i			0.435470i	−0.976008	+	0.217735i	$0.930133\pi$
$854$		0			0
$855$	2.61497e13			1.67347
$856$		0			0
$857$	1.35518e13			0.858191			0.429096	−	0.903259i	$-0.358832\pi$
$857$	1.35518e13			0.858191			0.429096	+	0.903259i	$0.358832\pi$
$858$		0			0
$859$	−9.79357e12			−0.613722			−0.306861	−	0.951754i	$-0.599279\pi$
$859$	−9.79357e12			−0.613722			−0.306861	+	0.951754i	$0.599279\pi$
$860$		0			0
$861$	−8.93228e12			−0.553922
$862$		0			0
$863$		−	3.81725e12i		−	0.234262i	−0.993116	−	0.117131i	$-0.962630\pi$
$863$		−	3.81725e12i		−	0.234262i	0.993116	−	0.117131i	$-0.0373697\pi$
$864$		0			0
$865$		−	3.17111e13i		−	1.92592i
$866$		0			0
$867$	−4.75781e13			−2.85970
$868$		0			0
$869$		−	3.73329e13i		−	2.22076i
$870$		0			0
$871$	9.12242e12	−	1.14353e13i	0.537067	−	0.673231i
$872$		0			0
$873$			1.24159e13i			0.723461i
$874$		0			0
$875$	6.45013e12			0.371991
$876$		0			0
$877$		−	1.20369e13i		−	0.687094i	−0.939135	−	0.343547i	$-0.888371\pi$
$877$		−	1.20369e13i		−	0.687094i	0.939135	−	0.343547i	$-0.111629\pi$
$878$		0			0
$879$			1.50398e13i			0.849752i
$880$		0			0
$881$	1.22937e13			0.687530			0.343765	−	0.939056i	$-0.388298\pi$
$881$	1.22937e13			0.687530			0.343765	+	0.939056i	$0.388298\pi$
$882$		0			0
$883$	2.75030e12			0.152250			0.0761249	−	0.997098i	$-0.475745\pi$
$883$	2.75030e12			0.152250			0.0761249	+	0.997098i	$0.475745\pi$
$884$		0			0
$885$	1.92160e13			1.05298
$886$		0			0
$887$	4.87054e12			0.264193			0.132096	−	0.991237i	$-0.457829\pi$
$887$	4.87054e12			0.264193			0.132096	+	0.991237i	$0.457829\pi$
$888$		0			0
$889$		−	2.55201e13i		−	1.37033i
$890$		0			0
$891$		−	2.79406e13i		−	1.48521i
$892$		0			0
$893$	−1.50490e13			−0.791908
$894$		0			0
$895$			2.61233e13i			1.36089i
$896$		0			0
$897$	8.08341e12	+	6.44851e12i	0.416897	+	0.332578i
$898$		0			0
$899$		−	3.41472e13i		−	1.74356i
$900$		0			0
$901$	−9.10224e12			−0.460137
$902$		0			0
$903$			3.37647e13i			1.68993i
$904$		0			0
$905$			4.59397e13i			2.27651i
$906$		0			0
$907$	1.27220e13			0.624198			0.312099	−	0.950050i	$-0.398968\pi$
$907$	1.27220e13			0.624198			0.312099	+	0.950050i	$0.398968\pi$
$908$		0			0
$909$	8.73302e12			0.424255
$910$		0			0
$911$	−2.41400e13			−1.16120			−0.580598	−	0.814190i	$-0.697181\pi$
$911$	−2.41400e13			−1.16120			−0.580598	+	0.814190i	$0.697181\pi$
$912$		0			0
$913$	−2.10938e13			−1.00470
$914$		0			0
$915$		−	3.83782e13i		−	1.81005i
$916$		0			0
$917$			4.22008e13i			1.97087i
$918$		0			0
$919$	−2.45604e13			−1.13584			−0.567918	−	0.823085i	$-0.692251\pi$
$919$	−2.45604e13			−1.13584			−0.567918	+	0.823085i	$0.692251\pi$
$920$		0			0
$921$		−	3.31271e13i		−	1.51710i
$922$		0			0
$923$	6.03883e12	−	7.56987e12i	0.273870	−	0.343305i
$924$		0			0
$925$		−	1.45002e13i		−	0.651233i
$926$		0			0
$927$	−1.93347e13			−0.859959
$928$		0			0
$929$		−	2.29424e13i		−	1.01057i	−0.862952	−	0.505287i	$-0.831387\pi$
$929$		−	2.29424e13i		−	1.01057i	0.862952	−	0.505287i	$-0.168613\pi$
$930$		0			0
$931$			6.56076e13i			2.86207i
$932$		0			0
$933$	5.24469e12			0.226596
$934$		0			0
$935$	8.36517e13			3.57950
$936$		0			0
$937$	−3.18986e13			−1.35190			−0.675949	−	0.736949i	$-0.736266\pi$
$937$	−3.18986e13			−1.35190			−0.675949	+	0.736949i	$0.736266\pi$
$938$		0			0
$939$	6.08609e13			2.55472
$940$		0			0
$941$			3.41587e13i			1.42020i	0.704102	+	0.710098i	$0.251350\pi$
$941$			3.41587e13i			1.42020i	−0.704102	+	0.710098i	$0.748650\pi$
$942$		0			0
$943$		−	2.02899e12i		−	0.0835561i
$944$		0			0
$945$	−5.93190e12			−0.241964
$946$		0			0
$947$			2.79320e13i			1.12857i	0.825581	+	0.564284i	$0.190848\pi$
$947$			2.79320e13i			1.12857i	−0.825581	+	0.564284i	$0.809152\pi$
$948$		0			0
$949$	−1.76983e13	+	2.21855e13i	−0.708329	+	0.887913i
$950$		0			0
$951$		−	2.94180e13i		−	1.16627i
$952$		0			0
$953$	−2.75447e13			−1.08173			−0.540866	−	0.841109i	$-0.681903\pi$
$953$	−2.75447e13			−1.08173			−0.540866	+	0.841109i	$0.681903\pi$
$954$		0			0
$955$		−	3.13286e13i		−	1.21878i
$956$		0			0
$957$			6.16997e13i			2.37782i
$958$		0			0
$959$	−5.02826e13			−1.91970
$960$		0			0
$961$	−2.74735e13			−1.03910
$962$		0			0
$963$	1.96971e13			0.738046
$964$		0			0
$965$	−3.42170e13			−1.27019
$966$		0			0
$967$			6.06675e12i			0.223119i	0.993758	+	0.111560i	$0.0355846\pi$
$967$			6.06675e12i			0.223119i	−0.993758	+	0.111560i	$0.964415\pi$
$968$		0			0
$969$		−	8.16727e13i		−	2.97591i
$970$		0			0
$971$	3.40239e13			1.22828			0.614139	−	0.789198i	$-0.289503\pi$
$971$	3.40239e13			1.22828			0.614139	+	0.789198i	$0.289503\pi$
$972$		0			0
$973$		−	5.66745e13i		−	2.02712i
$974$		0			0
$975$	−3.49546e13	−	2.78849e13i	−1.23875	−	0.988208i
$976$		0			0
$977$		−	4.23784e13i		−	1.48806i	−0.668148	−	0.744028i	$-0.732913\pi$
$977$		−	4.23784e13i		−	1.48806i	0.668148	−	0.744028i	$-0.267087\pi$
$978$		0			0
$979$	−4.58625e13			−1.59564
$980$		0			0
$981$		−	3.44376e13i		−	1.18720i
$982$		0			0
$983$			2.66982e13i			0.911992i	0.889982	+	0.455996i	$0.150717\pi$
$983$			2.66982e13i			0.911992i	−0.889982	+	0.455996i	$0.849283\pi$
$984$		0			0
$985$	5.38289e13			1.82202
$986$		0			0
$987$	−4.90333e13			−1.64461
$988$		0			0
$989$	−7.66974e12			−0.254916
$990$		0			0
$991$	−4.43518e13			−1.46076			−0.730382	−	0.683039i	$-0.760658\pi$
$991$	−4.43518e13			−1.46076			−0.730382	+	0.683039i	$0.760658\pi$
$992$		0			0
$993$		−	6.29211e13i		−	2.05364i
$994$		0			0
$995$			4.54962e13i			1.47154i
$996$		0			0
$997$	2.32994e13			0.746821			0.373410	−	0.927666i	$-0.378188\pi$
$997$	2.32994e13			0.746821			0.373410	+	0.927666i	$0.378188\pi$
$998$		0			0
$999$			1.62939e12i			0.0517585i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.f.d.129.5		32
4.3	odd	2		104.10.f.a.25.28	yes	32
13.12	even	2	inner	208.10.f.d.129.6		32
52.51	odd	2		104.10.f.a.25.27	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.27	✓	32	52.51	odd	2
104.10.f.a.25.28	yes	32	4.3	odd	2
208.10.f.d.129.5		32	1.1	even	1	trivial
208.10.f.d.129.6		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)

\(f(q)\)	\(=\)	\(q-195.153 q^{3} +2044.04i q^{5} +11607.1i q^{7} +18401.8 q^{9} +67982.7i q^{11} +(-80500.9 - 64219.2i) q^{13} -398902. i q^{15} -601985. q^{17} -695208. i q^{19} -2.26516e6i q^{21} +514539. q^{23} -2.22499e6 q^{25} +250023. q^{27} -4.65059e6 q^{29} +7.34256e6i q^{31} -1.32671e7i q^{33} -2.37254e7 q^{35} +6.51698e6i q^{37} +(1.57100e7 + 1.25326e7i) q^{39} -3.94333e6i q^{41} -1.49061e7 q^{43} +3.76142e7i q^{45} -2.16467e7i q^{47} -9.43711e7 q^{49} +1.17479e8 q^{51} +1.51204e7 q^{53} -1.38960e8 q^{55} +1.35672e8i q^{57} +4.81722e7i q^{59} +9.62097e7 q^{61} +2.13592e8i q^{63} +(1.31267e8 - 1.64547e8i) q^{65} +1.42051e8i q^{67} -1.00414e8 q^{69} +9.40346e7i q^{71} -2.75593e8i q^{73} +4.34214e8 q^{75} -7.89082e8 q^{77} -5.49152e8 q^{79} -4.10996e8 q^{81} +3.10281e8i q^{83} -1.23048e9i q^{85} +9.07579e8 q^{87} +6.74619e8i q^{89} +(7.45399e8 - 9.34381e8i) q^{91} -1.43293e9i q^{93} +1.42104e9 q^{95} +6.74712e8i q^{97} +1.25101e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} - 3171556 q^{23} - 13526722 q^{25} + 3694974 q^{27} + 8833508 q^{29} + 8281126 q^{35} + 12056860 q^{39} - 89959038 q^{43} - 172344874 q^{49}+ \cdots - 1741143356 q^{95}+O(q^{100}) \) 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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