LMFDB - Embedded newform 208.10.f.d.129.12

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			129.12
Character	$\chi$	$=$	208.129
Dual form			208.10.f.d.129.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-117.861 q^{3} -287.112i q^{5} -10358.2i q^{7} -5791.89 q^{9} +82910.8i q^{11} +(6945.91 - 102744. i) q^{13} +33839.1i q^{15} +305577. q^{17} -379376. i q^{19} +1.22082e6i q^{21} -815623. q^{23} +1.87069e6 q^{25} +3.00248e6 q^{27} +5.94311e6 q^{29} +2.09996e6i q^{31} -9.77191e6i q^{33} -2.97396e6 q^{35} +219223. i q^{37} +(-818648. + 1.21094e7i) q^{39} +3.13033e7i q^{41} -2.58799e7 q^{43} +1.66292e6i q^{45} -5.79575e6i q^{47} -6.69388e7 q^{49} -3.60155e7 q^{51} +1.97834e7 q^{53} +2.38046e7 q^{55} +4.47134e7i q^{57} +2.88970e7i q^{59} +2.14079e8 q^{61} +5.99936e7i q^{63} +(-2.94989e7 - 1.99425e6i) q^{65} +2.28111e8i q^{67} +9.61298e7 q^{69} -2.43008e8i q^{71} -1.63522e8i q^{73} -2.20481e8 q^{75} +8.58807e8 q^{77} -1.95194e8 q^{79} -2.39873e8 q^{81} -3.87329e8i q^{83} -8.77347e7i q^{85} -7.00458e8 q^{87} +4.30960e8i q^{89} +(-1.06424e9 - 7.19471e7i) q^{91} -2.47503e8i q^{93} -1.08923e8 q^{95} -1.04747e9i q^{97} -4.80210e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−117.861			−0.840084			−0.420042	−	0.907505i	$-0.637985\pi$
$3$	−117.861			−0.840084			−0.420042	+	0.907505i	$0.637985\pi$
$4$		0			0
$5$		−	287.112i		−	0.205440i	−0.994710	−	0.102720i	$-0.967245\pi$
$5$		−	287.112i		−	0.205440i	0.994710	−	0.102720i	$-0.0327546\pi$
$6$		0			0
$7$		−	10358.2i		−	1.63058i	−0.579050	−	0.815292i	$-0.696576\pi$
$7$		−	10358.2i		−	1.63058i	0.579050	−	0.815292i	$-0.303424\pi$
$8$		0			0
$9$	−5791.89			−0.294258
$10$		0			0
$11$			82910.8i			1.70743i	0.520737	+	0.853717i	$0.325657\pi$
$11$			82910.8i			1.70743i	−0.520737	+	0.853717i	$0.674343\pi$
$12$		0			0
$13$	6945.91	−	102744.i	0.0674503	−	0.997723i
$13$	6945.91	−	102744.i	0.0674503	−	0.997723i
$14$		0			0
$15$			33839.1i			0.172587i
$16$		0			0
$17$	305577.			0.887361			0.443681	−	0.896185i	$-0.353672\pi$
$17$	305577.			0.887361			0.443681	+	0.896185i	$0.353672\pi$
$18$		0			0
$19$		−	379376.i		−	0.667849i	−0.942600	−	0.333925i	$-0.891627\pi$
$19$		−	379376.i		−	0.667849i	0.942600	−	0.333925i	$-0.108373\pi$
$20$		0			0
$21$			1.22082e6i			1.36983i
$22$		0			0
$23$	−815623.			−0.607735			−0.303868	−	0.952714i	$-0.598278\pi$
$23$	−815623.			−0.607735			−0.303868	+	0.952714i	$0.598278\pi$
$24$		0			0
$25$	1.87069e6			0.957794
$26$		0			0
$27$	3.00248e6			1.08729
$28$		0			0
$29$	5.94311e6			1.56035			0.780176	−	0.625560i	$-0.215129\pi$
$29$	5.94311e6			1.56035			0.780176	+	0.625560i	$0.215129\pi$
$30$		0			0
$31$			2.09996e6i			0.408398i	0.978929	+	0.204199i	$0.0654590\pi$
$31$			2.09996e6i			0.408398i	−0.978929	+	0.204199i	$0.934541\pi$
$32$		0			0
$33$		−	9.77191e6i		−	1.43439i
$34$		0			0
$35$	−2.97396e6			−0.334988
$36$		0			0
$37$			219223.i			0.0192300i	0.999954	+	0.00961498i	$0.00306059\pi$
$37$			219223.i			0.0192300i	−0.999954	+	0.00961498i	$0.996939\pi$
$38$		0			0
$39$	−818648.	+	1.21094e7i	−0.0566639	+	0.838171i
$40$		0			0
$41$			3.13033e7i			1.73006i	0.501716	+	0.865032i	$0.332702\pi$
$41$			3.13033e7i			1.73006i	−0.501716	+	0.865032i	$0.667298\pi$
$42$		0			0
$43$	−2.58799e7			−1.15440			−0.577198	−	0.816604i	$-0.695854\pi$
$43$	−2.58799e7			−1.15440			−0.577198	+	0.816604i	$0.695854\pi$
$44$		0			0
$45$			1.66292e6i			0.0604525i
$46$		0			0
$47$		−	5.79575e6i		−	0.173248i	−0.996241	−	0.0866242i	$-0.972392\pi$
$47$		−	5.79575e6i		−	0.173248i	0.996241	−	0.0866242i	$-0.0276079\pi$
$48$		0			0
$49$	−6.69388e7			−1.65881
$50$		0			0
$51$	−3.60155e7			−0.745458
$52$		0			0
$53$	1.97834e7			0.344397			0.172199	−	0.985062i	$-0.444913\pi$
$53$	1.97834e7			0.344397			0.172199	+	0.985062i	$0.444913\pi$
$54$		0			0
$55$	2.38046e7			0.350776
$56$		0			0
$57$			4.47134e7i			0.561050i
$58$		0			0
$59$			2.88970e7i			0.310469i	0.987878	+	0.155235i	$0.0496133\pi$
$59$			2.88970e7i			0.310469i	−0.987878	+	0.155235i	$0.950387\pi$
$60$		0			0
$61$	2.14079e8			1.97965			0.989826	−	0.142283i	$-0.0454444\pi$
$61$	2.14079e8			1.97965			0.989826	+	0.142283i	$0.0454444\pi$
$62$		0			0
$63$			5.99936e7i			0.479813i
$64$		0			0
$65$	−2.94989e7	−	1.99425e6i	−0.204972	−	0.0138570i
$66$		0			0
$67$			2.28111e8i			1.38296i	0.722396	+	0.691480i	$0.243041\pi$
$67$			2.28111e8i			1.38296i	−0.722396	+	0.691480i	$0.756959\pi$
$68$		0			0
$69$	9.61298e7			0.510549
$70$		0			0
$71$		−	2.43008e8i		−	1.13490i	−0.823408	−	0.567450i	$-0.807930\pi$
$71$		−	2.43008e8i		−	1.13490i	0.823408	−	0.567450i	$-0.192070\pi$
$72$		0			0
$73$		−	1.63522e8i		−	0.673944i	−0.941515	−	0.336972i	$-0.890597\pi$
$73$		−	1.63522e8i		−	0.673944i	0.941515	−	0.336972i	$-0.109403\pi$
$74$		0			0
$75$	−2.20481e8			−0.804628
$76$		0			0
$77$	8.58807e8			2.78412
$78$		0			0
$79$	−1.95194e8			−0.563826			−0.281913	−	0.959440i	$-0.590969\pi$
$79$	−1.95194e8			−0.563826			−0.281913	+	0.959440i	$0.590969\pi$
$80$		0			0
$81$	−2.39873e8			−0.619154
$82$		0			0
$83$		−	3.87329e8i		−	0.895837i	−0.894074	−	0.447919i	$-0.852165\pi$
$83$		−	3.87329e8i		−	0.895837i	0.894074	−	0.447919i	$-0.147835\pi$
$84$		0			0
$85$		−	8.77347e7i		−	0.182300i
$86$		0			0
$87$	−7.00458e8			−1.31083
$88$		0			0
$89$			4.30960e8i			0.728085i	0.931382	+	0.364042i	$0.118604\pi$
$89$			4.30960e8i			0.728085i	−0.931382	+	0.364042i	$0.881396\pi$
$90$		0			0
$91$	−1.06424e9	−	7.19471e7i	−1.62687	−	0.109983i
$92$		0			0
$93$		−	2.47503e8i		−	0.343089i
$94$		0			0
$95$	−1.08923e8			−0.137203
$96$		0			0
$97$		−	1.04747e9i		−	1.20135i	−0.799494	−	0.600674i	$-0.794899\pi$
$97$		−	1.04747e9i		−	1.20135i	0.799494	−	0.600674i	$-0.205101\pi$
$98$		0			0
$99$		−	4.80210e8i		−	0.502427i
$100$		0			0
$101$	3.14680e8			0.300900			0.150450	−	0.988618i	$-0.451928\pi$
$101$	3.14680e8			0.300900			0.150450	+	0.988618i	$0.451928\pi$
$102$		0			0
$103$	−8.34709e8			−0.730748			−0.365374	−	0.930861i	$-0.619059\pi$
$103$	−8.34709e8			−0.730748			−0.365374	+	0.930861i	$0.619059\pi$
$104$		0			0
$105$	3.50513e8			0.281418
$106$		0			0
$107$	2.44048e9			1.79990			0.899951	−	0.435991i	$-0.143602\pi$
$107$	2.44048e9			1.79990			0.899951	+	0.435991i	$0.143602\pi$
$108$		0			0
$109$		−	2.79659e9i		−	1.89762i	−0.315850	−	0.948809i	$-0.602290\pi$
$109$		−	2.79659e9i		−	1.89762i	0.315850	−	0.948809i	$-0.397710\pi$
$110$		0			0
$111$		−	2.58378e7i		−	0.0161548i
$112$		0			0
$113$	1.81100e9			1.04488			0.522438	−	0.852678i	$-0.325023\pi$
$113$	1.81100e9			1.04488			0.522438	+	0.852678i	$0.325023\pi$
$114$		0			0
$115$			2.34175e8i			0.124853i
$116$		0			0
$117$	−4.02299e7	+	5.95080e8i	−0.0198478	+	0.293588i
$118$		0			0
$119$		−	3.16523e9i		−	1.44692i
$120$		0			0
$121$	−4.51625e9			−1.91533
$122$		0			0
$123$		−	3.68942e9i		−	1.45340i
$124$		0			0
$125$		−	1.09786e9i		−	0.402210i
$126$		0			0
$127$	2.74230e9			0.935402			0.467701	−	0.883887i	$-0.345082\pi$
$127$	2.74230e9			0.935402			0.467701	+	0.883887i	$0.345082\pi$
$128$		0			0
$129$	3.05022e9			0.969791
$130$		0			0
$131$	−3.22571e9			−0.956983			−0.478492	−	0.878092i	$-0.658816\pi$
$131$	−3.22571e9			−0.956983			−0.478492	+	0.878092i	$0.658816\pi$
$132$		0			0
$133$	−3.92965e9			−1.08898
$134$		0			0
$135$		−	8.62048e8i		−	0.223372i
$136$		0			0
$137$		−	2.75061e9i		−	0.667093i	−0.942734	−	0.333547i	$-0.891755\pi$
$137$		−	2.75061e9i		−	0.667093i	0.942734	−	0.333547i	$-0.108245\pi$
$138$		0			0
$139$	−6.79518e9			−1.54395			−0.771977	−	0.635650i	$-0.780732\pi$
$139$	−6.79518e9			−1.54395			−0.771977	+	0.635650i	$0.780732\pi$
$140$		0			0
$141$			6.83090e8i			0.145543i
$142$		0			0
$143$	8.51855e9	+	5.75890e8i	1.70355	+	0.115167i
$144$		0			0
$145$		−	1.70633e9i		−	0.320559i
$146$		0			0
$147$	7.88945e9			1.39354
$148$		0			0
$149$		−	8.86856e9i		−	1.47406i	−0.675861	−	0.737029i	$-0.736228\pi$
$149$		−	8.86856e9i		−	1.47406i	0.675861	−	0.737029i	$-0.263772\pi$
$150$		0			0
$151$			2.24777e9i			0.351848i	0.984404	+	0.175924i	$0.0562913\pi$
$151$			2.24777e9i			0.351848i	−0.984404	+	0.175924i	$0.943709\pi$
$152$		0			0
$153$	−1.76987e9			−0.261113
$154$		0			0
$155$	6.02923e8			0.0839014
$156$		0			0
$157$	2.44684e9			0.321408			0.160704	−	0.987003i	$-0.448624\pi$
$157$	2.44684e9			0.321408			0.160704	+	0.987003i	$0.448624\pi$
$158$		0			0
$159$	−2.33168e9			−0.289323
$160$		0			0
$161$			8.44840e9i			0.990964i
$162$		0			0
$163$		−	1.31982e10i		−	1.46443i	−0.681072	−	0.732217i	$-0.738486\pi$
$163$		−	1.31982e10i		−	1.46443i	0.681072	−	0.732217i	$-0.261514\pi$
$164$		0			0
$165$	−2.80563e9			−0.294681
$166$		0			0
$167$		−	6.48176e8i		−	0.0644865i	−0.999480	−	0.0322432i	$-0.989735\pi$
$167$		−	6.48176e8i		−	0.0644865i	0.999480	−	0.0322432i	$-0.0102651\pi$
$168$		0			0
$169$	−1.05080e10	−	1.42730e9i	−0.990901	−	0.134593i
$170$		0			0
$171$			2.19730e9i			0.196520i
$172$		0			0
$173$	−9.23110e9			−0.783513			−0.391756	−	0.920069i	$-0.628132\pi$
$173$	−9.23110e9			−0.783513			−0.391756	+	0.920069i	$0.628132\pi$
$174$		0			0
$175$		−	1.93770e10i		−	1.56176i
$176$		0			0
$177$		−	3.40582e9i		−	0.260820i
$178$		0			0
$179$	1.62540e10			1.18338			0.591688	−	0.806167i	$-0.298462\pi$
$179$	1.62540e10			1.18338			0.591688	+	0.806167i	$0.298462\pi$
$180$		0			0
$181$	−1.76021e10			−1.21902			−0.609509	−	0.792779i	$-0.708633\pi$
$181$	−1.76021e10			−1.21902			−0.609509	+	0.792779i	$0.708633\pi$
$182$		0			0
$183$	−2.52314e10			−1.66307
$184$		0			0
$185$	6.29415e7			0.00395061
$186$		0			0
$187$			2.53356e10i			1.51511i
$188$		0			0
$189$		−	3.11004e10i		−	1.77291i
$190$		0			0
$191$	2.47938e10			1.34801			0.674004	−	0.738727i	$-0.264573\pi$
$191$	2.47938e10			1.34801			0.674004	+	0.738727i	$0.264573\pi$
$192$		0			0
$193$			2.32103e10i			1.20413i	0.798448	+	0.602064i	$0.205655\pi$
$193$			2.32103e10i			1.20413i	−0.798448	+	0.602064i	$0.794345\pi$
$194$		0			0
$195$	3.47675e9	+	2.35043e8i	0.172194	+	0.0116411i
$196$		0			0
$197$		−	1.83364e10i		−	0.867393i	−0.901059	−	0.433697i	$-0.857209\pi$
$197$		−	1.83364e10i		−	0.867393i	0.901059	−	0.433697i	$-0.142791\pi$
$198$		0			0
$199$	2.38275e10			1.07706			0.538529	−	0.842607i	$-0.318980\pi$
$199$	2.38275e10			1.07706			0.538529	+	0.842607i	$0.318980\pi$
$200$		0			0
$201$		−	2.68853e10i		−	1.16180i
$202$		0			0
$203$		−	6.15599e10i		−	2.54429i
$204$		0			0
$205$	8.98753e9			0.355425
$206$		0			0
$207$	4.72400e9			0.178831
$208$		0			0
$209$	3.14543e10			1.14031
$210$		0			0
$211$	−1.40683e10			−0.488618			−0.244309	−	0.969697i	$-0.578561\pi$
$211$	−1.40683e10			−0.488618			−0.244309	+	0.969697i	$0.578561\pi$
$212$		0			0
$213$			2.86410e10i			0.953412i
$214$		0			0
$215$			7.43043e9i			0.237160i
$216$		0			0
$217$	2.17518e10			0.665928
$218$		0			0
$219$			1.92728e10i			0.566170i
$220$		0			0
$221$	2.12251e9	−	3.13961e10i	0.0598528	−	0.885340i
$222$		0			0
$223$		−	5.01267e10i		−	1.35737i	−0.734430	−	0.678684i	$-0.762551\pi$
$223$		−	5.01267e10i		−	1.35737i	0.734430	−	0.678684i	$-0.237449\pi$
$224$		0			0
$225$	−1.08348e10			−0.281839
$226$		0			0
$227$		−	3.02939e10i		−	0.757250i	−0.925550	−	0.378625i	$-0.876397\pi$
$227$		−	3.02939e10i		−	0.757250i	0.925550	−	0.378625i	$-0.123603\pi$
$228$		0			0
$229$		−	3.87640e10i		−	0.931470i	−0.884924	−	0.465735i	$-0.845790\pi$
$229$		−	3.87640e10i		−	0.931470i	0.884924	−	0.465735i	$-0.154210\pi$
$230$		0			0
$231$	−1.01219e11			−2.33889
$232$		0			0
$233$	4.42353e10			0.983258			0.491629	−	0.870805i	$-0.336402\pi$
$233$	4.42353e10			0.983258			0.491629	+	0.870805i	$0.336402\pi$
$234$		0			0
$235$	−1.66403e9			−0.0355922
$236$		0			0
$237$	2.30057e10			0.473661
$238$		0			0
$239$			7.17928e10i			1.42328i	0.702544	+	0.711640i	$0.252047\pi$
$239$			7.17928e10i			1.42328i	−0.702544	+	0.711640i	$0.747953\pi$
$240$		0			0
$241$			1.31018e10i			0.250181i	0.992145	+	0.125091i	$0.0399222\pi$
$241$			1.31018e10i			0.250181i	−0.992145	+	0.125091i	$0.960078\pi$
$242$		0			0
$243$	−3.08264e10			−0.567145
$244$		0			0
$245$			1.92189e10i			0.340786i
$246$		0			0
$247$	−3.89784e10	−	2.63511e9i	−0.666328	−	0.0450466i
$248$		0			0
$249$			4.56509e10i			0.752579i
$250$		0			0
$251$	−7.85551e10			−1.24923			−0.624615	−	0.780933i	$-0.714744\pi$
$251$	−7.85551e10			−1.24923			−0.624615	+	0.780933i	$0.714744\pi$
$252$		0			0
$253$		−	6.76240e10i		−	1.03767i
$254$		0			0
$255$			1.03405e10i			0.153147i
$256$		0			0
$257$	−7.88264e10			−1.12713			−0.563564	−	0.826073i	$-0.690570\pi$
$257$	−7.88264e10			−1.12713			−0.563564	+	0.826073i	$0.690570\pi$
$258$		0			0
$259$	2.27076e9			0.0313561
$260$		0			0
$261$	−3.44218e10			−0.459147
$262$		0			0
$263$	−5.85081e10			−0.754076			−0.377038	−	0.926198i	$-0.623057\pi$
$263$	−5.85081e10			−0.754076			−0.377038	+	0.926198i	$0.623057\pi$
$264$		0			0
$265$		−	5.68004e9i		−	0.0707530i
$266$		0			0
$267$		−	5.07932e10i		−	0.611653i
$268$		0			0
$269$	−9.27738e9			−0.108029			−0.0540145	−	0.998540i	$-0.517202\pi$
$269$	−9.27738e9			−0.108029			−0.0540145	+	0.998540i	$0.517202\pi$
$270$		0			0
$271$		−	1.74708e10i		−	0.196766i	−0.995149	−	0.0983830i	$-0.968633\pi$
$271$		−	1.74708e10i		−	0.196766i	0.995149	−	0.0983830i	$-0.0313670\pi$
$272$		0			0
$273$	1.25432e11	+	8.47973e9i	1.36671	+	0.0923953i
$274$		0			0
$275$			1.55101e11i			1.63537i
$276$		0			0
$277$	1.69004e11			1.72480			0.862401	−	0.506226i	$-0.168960\pi$
$277$	1.69004e11			1.72480			0.862401	+	0.506226i	$0.168960\pi$
$278$		0			0
$279$		−	1.21627e10i		−	0.120175i
$280$		0			0
$281$			7.28605e10i			0.697130i	0.937285	+	0.348565i	$0.113331\pi$
$281$			7.28605e10i			0.697130i	−0.937285	+	0.348565i	$0.886669\pi$
$282$		0			0
$283$	1.43889e11			1.33349			0.666744	−	0.745286i	$-0.267687\pi$
$283$	1.43889e11			1.33349			0.666744	+	0.745286i	$0.267687\pi$
$284$		0			0
$285$	1.28377e10			0.115262
$286$		0			0
$287$	3.24246e11			2.82102
$288$		0			0
$289$	−2.52106e10			−0.212590
$290$		0			0
$291$			1.23455e11i			1.00923i
$292$		0			0
$293$			4.33136e10i			0.343336i	0.985155	+	0.171668i	$0.0549157\pi$
$293$			4.33136e10i			0.343336i	−0.985155	+	0.171668i	$0.945084\pi$
$294$		0			0
$295$	8.29666e9			0.0637829
$296$		0			0
$297$			2.48938e11i			1.85647i
$298$		0			0
$299$	−5.66524e9	+	8.38001e10i	−0.0409919	+	0.606351i
$300$		0			0
$301$			2.68070e11i			1.88234i
$302$		0			0
$303$	−3.70883e10			−0.252782
$304$		0			0
$305$		−	6.14644e10i		−	0.406700i
$306$		0			0
$307$		−	1.41537e11i		−	0.909382i	−0.890649	−	0.454691i	$-0.849750\pi$
$307$		−	1.41537e11i		−	0.909382i	0.890649	−	0.454691i	$-0.150250\pi$
$308$		0			0
$309$	9.83793e10			0.613890
$310$		0			0
$311$	−1.18766e11			−0.719895			−0.359947	−	0.932973i	$-0.617205\pi$
$311$	−1.18766e11			−0.719895			−0.359947	+	0.932973i	$0.617205\pi$
$312$		0			0
$313$	−2.74722e11			−1.61787			−0.808936	−	0.587897i	$-0.799956\pi$
$313$	−2.74722e11			−1.61787			−0.808936	+	0.587897i	$0.799956\pi$
$314$		0			0
$315$	1.72248e10			0.0985730
$316$		0			0
$317$			8.89417e9i			0.0494696i	0.999694	+	0.0247348i	$0.00787414\pi$
$317$			8.89417e9i			0.0494696i	−0.999694	+	0.0247348i	$0.992126\pi$
$318$		0			0
$319$			4.92748e11i			2.66420i
$320$		0			0
$321$	−2.87637e11			−1.51207
$322$		0			0
$323$		−	1.15929e11i		−	0.592624i
$324$		0			0
$325$	1.29937e10	−	1.92202e11i	0.0646035	−	0.955613i
$326$		0			0
$327$			3.29607e11i			1.59416i
$328$		0			0
$329$	−6.00335e10			−0.282496
$330$		0			0
$331$		−	5.22620e10i		−	0.239310i	−0.992816	−	0.119655i	$-0.961821\pi$
$331$		−	5.22620e10i		−	0.239310i	0.992816	−	0.119655i	$-0.0381788\pi$
$332$		0			0
$333$		−	1.26972e9i		−	0.00565858i
$334$		0			0
$335$	6.54933e10			0.284115
$336$		0			0
$337$	3.04679e10			0.128679			0.0643396	−	0.997928i	$-0.479506\pi$
$337$	3.04679e10			0.128679			0.0643396	+	0.997928i	$0.479506\pi$
$338$		0			0
$339$	−2.13445e11			−0.877783
$340$		0			0
$341$	−1.74109e11			−0.697313
$342$		0			0
$343$			2.75375e11i			1.07424i
$344$		0			0
$345$		−	2.76000e10i		−	0.104887i
$346$		0			0
$347$	1.07323e11			0.397384			0.198692	−	0.980062i	$-0.436331\pi$
$347$	1.07323e11			0.397384			0.198692	+	0.980062i	$0.436331\pi$
$348$		0			0
$349$			1.84031e11i			0.664015i	0.943277	+	0.332007i	$0.107726\pi$
$349$			1.84031e11i			0.664015i	−0.943277	+	0.332007i	$0.892274\pi$
$350$		0			0
$351$	2.08550e10	−	3.08486e11i	0.0733378	−	1.08481i
$352$		0			0
$353$			5.22950e11i			1.79256i	0.443489	+	0.896280i	$0.353741\pi$
$353$			5.22950e11i			1.79256i	−0.443489	+	0.896280i	$0.646259\pi$
$354$		0			0
$355$	−6.97704e10			−0.233154
$356$		0			0
$357$			3.73056e11i			1.21553i
$358$		0			0
$359$		−	4.49510e11i		−	1.42828i	−0.700001	−	0.714142i	$-0.746817\pi$
$359$		−	4.49510e11i		−	1.42828i	0.700001	−	0.714142i	$-0.253183\pi$
$360$		0			0
$361$	1.78762e11			0.553977
$362$		0			0
$363$	5.32288e11			1.60904
$364$		0			0
$365$	−4.69491e10			−0.138455
$366$		0			0
$367$	1.79671e11			0.516988			0.258494	−	0.966013i	$-0.416774\pi$
$367$	1.79671e11			0.516988			0.258494	+	0.966013i	$0.416774\pi$
$368$		0			0
$369$		−	1.81305e11i		−	0.509086i
$370$		0			0
$371$		−	2.04920e11i		−	0.561569i
$372$		0			0
$373$	1.70024e10			0.0454801			0.0227400	−	0.999741i	$-0.492761\pi$
$373$	1.70024e10			0.0454801			0.0227400	+	0.999741i	$0.492761\pi$
$374$		0			0
$375$			1.29395e11i			0.337890i
$376$		0			0
$377$	4.12803e10	−	6.10617e11i	0.105246	−	1.55680i
$378$		0			0
$379$			2.57542e11i			0.641168i	0.947220	+	0.320584i	$0.103879\pi$
$379$			2.57542e11i			0.641168i	−0.947220	+	0.320584i	$0.896121\pi$
$380$		0			0
$381$	−3.23209e11			−0.785817
$382$		0			0
$383$			1.66718e11i			0.395901i	0.980212	+	0.197951i	$0.0634286\pi$
$383$			1.66718e11i			0.395901i	−0.980212	+	0.197951i	$0.936571\pi$
$384$		0			0
$385$		−	2.46573e11i		−	0.571969i
$386$		0			0
$387$	1.49894e11			0.339691
$388$		0			0
$389$	−5.97102e11			−1.32213			−0.661066	−	0.750327i	$-0.729896\pi$
$389$	−5.97102e11			−1.32213			−0.661066	+	0.750327i	$0.729896\pi$
$390$		0			0
$391$	−2.49236e11			−0.539281
$392$		0			0
$393$	3.80184e11			0.803947
$394$		0			0
$395$			5.60425e10i			0.115832i
$396$		0			0
$397$		−	1.02153e11i		−	0.206392i	−0.994661	−	0.103196i	$-0.967093\pi$
$397$		−	1.02153e11i		−	0.206392i	0.994661	−	0.103196i	$-0.0329068\pi$
$398$		0			0
$399$	4.63151e11			0.914839
$400$		0			0
$401$		−	5.24543e10i		−	0.101305i	−0.998716	−	0.0506526i	$-0.983870\pi$
$401$		−	5.24543e10i		−	0.101305i	0.998716	−	0.0506526i	$-0.0161301\pi$
$402$		0			0
$403$	2.15758e11	+	1.45861e10i	0.407468	+	0.0275466i
$404$		0			0
$405$			6.88702e10i			0.127199i
$406$		0			0
$407$	−1.81760e10			−0.0328339
$408$		0			0
$409$		−	2.40354e11i		−	0.424714i	−0.977192	−	0.212357i	$-0.931886\pi$
$409$		−	2.40354e11i		−	0.424714i	0.977192	−	0.212357i	$-0.0681140\pi$
$410$		0			0
$411$			3.24188e11i			0.560414i
$412$		0			0
$413$	2.99321e11			0.506246
$414$		0			0
$415$	−1.11207e11			−0.184041
$416$		0			0
$417$	8.00884e11			1.29705
$418$		0			0
$419$	−5.72554e11			−0.907514			−0.453757	−	0.891125i	$-0.649917\pi$
$419$	−5.72554e11			−0.907514			−0.453757	+	0.891125i	$0.649917\pi$
$420$		0			0
$421$			2.53296e11i			0.392970i	0.980507	+	0.196485i	$0.0629526\pi$
$421$			2.53296e11i			0.392970i	−0.980507	+	0.196485i	$0.937047\pi$
$422$		0			0
$423$			3.35683e10i			0.0509798i
$424$		0			0
$425$	5.71640e11			0.849910
$426$		0			0
$427$		−	2.21747e12i		−	3.22799i
$428$		0			0
$429$	−1.00400e12	−	6.78748e10i	−1.43112	−	0.0967499i
$430$		0			0
$431$		−	9.69705e10i		−	0.135360i	−0.997707	−	0.0676802i	$-0.978440\pi$
$431$		−	9.69705e10i		−	0.135360i	0.997707	−	0.0676802i	$-0.0215598\pi$
$432$		0			0
$433$	9.47354e10			0.129514			0.0647570	−	0.997901i	$-0.479373\pi$
$433$	9.47354e10			0.129514			0.0647570	+	0.997901i	$0.479373\pi$
$434$		0			0
$435$			2.01110e11i			0.269297i
$436$		0			0
$437$			3.09428e11i			0.405876i
$438$		0			0
$439$	−4.03403e11			−0.518380			−0.259190	−	0.965826i	$-0.583456\pi$
$439$	−4.03403e11			−0.518380			−0.259190	+	0.965826i	$0.583456\pi$
$440$		0			0
$441$	3.87702e11			0.488118
$442$		0			0
$443$	−2.78717e11			−0.343833			−0.171916	−	0.985112i	$-0.554996\pi$
$443$	−2.78717e11			−0.343833			−0.171916	+	0.985112i	$0.554996\pi$
$444$		0			0
$445$	1.23734e11			0.149578
$446$		0			0
$447$			1.04525e12i			1.23833i
$448$		0			0
$449$		−	2.01740e11i		−	0.234252i	−0.993117	−	0.117126i	$-0.962632\pi$
$449$		−	2.01740e11i		−	0.234252i	0.993117	−	0.117126i	$-0.0373681\pi$
$450$		0			0
$451$	−2.59538e12			−2.95397
$452$		0			0
$453$		−	2.64923e11i		−	0.295582i
$454$		0			0
$455$	−2.06568e10	+	3.05555e11i	−0.0225950	+	0.334225i
$456$		0			0
$457$		−	4.52715e11i		−	0.485515i	−0.970087	−	0.242757i	$-0.921948\pi$
$457$		−	4.52715e11i		−	0.485515i	0.970087	−	0.242757i	$-0.0780519\pi$
$458$		0			0
$459$	9.17490e11			0.964816
$460$		0			0
$461$			2.70174e11i			0.278605i	0.990250	+	0.139303i	$0.0444861\pi$
$461$			2.70174e11i			0.278605i	−0.990250	+	0.139303i	$0.955514\pi$
$462$		0			0
$463$		−	4.37353e11i		−	0.442301i	−0.975240	−	0.221151i	$-0.929019\pi$
$463$		−	4.37353e11i		−	0.442301i	0.975240	−	0.221151i	$-0.0709812\pi$
$464$		0			0
$465$	−7.10608e10			−0.0704842
$466$		0			0
$467$	−1.18691e12			−1.15476			−0.577382	−	0.816474i	$-0.695926\pi$
$467$	−1.18691e12			−1.15476			−0.577382	+	0.816474i	$0.695926\pi$
$468$		0			0
$469$	2.36282e12			2.25503
$470$		0			0
$471$	−2.88386e11			−0.270010
$472$		0			0
$473$		−	2.14573e12i		−	1.97106i
$474$		0			0
$475$		−	7.09695e11i		−	0.639662i
$476$		0			0
$477$	−1.14583e11			−0.101342
$478$		0			0
$479$			5.14115e11i			0.446222i	0.974793	+	0.223111i	$0.0716212\pi$
$479$			5.14115e11i			0.446222i	−0.974793	+	0.223111i	$0.928379\pi$
$480$		0			0
$481$	2.25238e10	+	1.52270e9i	0.0191862	+	0.00129707i
$482$		0			0
$483$		−	9.95733e11i		−	0.832493i
$484$		0			0
$485$	−3.00741e11			−0.246805
$486$		0			0
$487$			9.09851e11i			0.732976i	0.930423	+	0.366488i	$0.119440\pi$
$487$			9.09851e11i			0.732976i	−0.930423	+	0.366488i	$0.880560\pi$
$488$		0			0
$489$			1.55554e12i			1.23025i
$490$		0			0
$491$	4.39330e11			0.341134			0.170567	−	0.985346i	$-0.445440\pi$
$491$	4.39330e11			0.341134			0.170567	+	0.985346i	$0.445440\pi$
$492$		0			0
$493$	1.81608e12			1.38460
$494$		0			0
$495$	−1.37874e11			−0.103219
$496$		0			0
$497$	−2.51713e12			−1.85055
$498$		0			0
$499$		−	1.24276e12i		−	0.897291i	−0.893710	−	0.448646i	$-0.851907\pi$
$499$		−	1.24276e12i		−	0.897291i	0.893710	−	0.448646i	$-0.148093\pi$
$500$		0			0
$501$			7.63943e10i			0.0541741i
$502$		0			0
$503$	−1.47132e11			−0.102483			−0.0512415	−	0.998686i	$-0.516318\pi$
$503$	−1.47132e11			−0.102483			−0.0512415	+	0.998686i	$0.516318\pi$
$504$		0			0
$505$		−	9.03482e10i		−	0.0618170i
$506$		0			0
$507$	1.23848e12	+	1.68222e11i	0.832440	+	0.113070i
$508$		0			0
$509$		−	1.91001e12i		−	1.26126i	−0.776082	−	0.630632i	$-0.782796\pi$
$509$		−	1.91001e12i		−	1.26126i	0.776082	−	0.630632i	$-0.217204\pi$
$510$		0			0
$511$	−1.69380e12			−1.09892
$512$		0			0
$513$		−	1.13907e12i		−	0.726143i
$514$		0			0
$515$			2.39655e11i			0.150125i
$516$		0			0
$517$	4.80530e11			0.295810
$518$		0			0
$519$	1.08798e12			0.658217
$520$		0			0
$521$	−1.36628e12			−0.812401			−0.406201	−	0.913784i	$-0.633147\pi$
$521$	−1.36628e12			−0.812401			−0.406201	+	0.913784i	$0.633147\pi$
$522$		0			0
$523$	2.32317e12			1.35776			0.678881	−	0.734249i	$-0.262465\pi$
$523$	2.32317e12			1.35776			0.678881	+	0.734249i	$0.262465\pi$
$524$		0			0
$525$			2.28379e12i			1.31201i
$526$		0			0
$527$			6.41700e11i			0.362397i
$528$		0			0
$529$	−1.13591e12			−0.630658
$530$		0			0
$531$		−	1.67368e11i		−	0.0913582i
$532$		0			0
$533$	3.21621e12	+	2.17430e11i	1.72612	+	0.116693i
$534$		0			0
$535$		−	7.00691e11i		−	0.369772i
$536$		0			0
$537$	−1.91571e12			−0.994135
$538$		0			0
$539$		−	5.54995e12i		−	2.83230i
$540$		0			0
$541$		−	3.14940e12i		−	1.58067i	−0.612677	−	0.790333i	$-0.709907\pi$
$541$		−	3.14940e12i		−	1.58067i	0.612677	−	0.790333i	$-0.290093\pi$
$542$		0			0
$543$	2.07459e12			1.02408
$544$		0			0
$545$	−8.02932e11			−0.389847
$546$		0			0
$547$	2.53983e12			1.21300			0.606501	−	0.795083i	$-0.292573\pi$
$547$	2.53983e12			1.21300			0.606501	+	0.795083i	$0.292573\pi$
$548$		0			0
$549$	−1.23992e12			−0.582529
$550$		0			0
$551$		−	2.25467e12i		−	1.04208i
$552$		0			0
$553$			2.02186e12i			0.919365i
$554$		0			0
$555$	−7.41832e9			−0.00331885
$556$		0			0
$557$		−	6.71608e11i		−	0.295643i	−0.989014	−	0.147821i	$-0.952774\pi$
$557$		−	6.71608e11i		−	0.295643i	0.989014	−	0.147821i	$-0.0472261\pi$
$558$		0			0
$559$	−1.79760e11	+	2.65900e12i	−0.0778644	+	1.15177i
$560$		0			0
$561$		−	2.98607e12i		−	1.27282i
$562$		0			0
$563$	1.47508e12			0.618768			0.309384	−	0.950937i	$-0.399877\pi$
$563$	1.47508e12			0.618768			0.309384	+	0.950937i	$0.399877\pi$
$564$		0			0
$565$		−	5.19958e11i		−	0.214659i
$566$		0			0
$567$			2.48465e12i			1.00958i
$568$		0			0
$569$	8.74932e11			0.349920			0.174960	−	0.984576i	$-0.444020\pi$
$569$	8.74932e11			0.349920			0.174960	+	0.984576i	$0.444020\pi$
$570$		0			0
$571$	−3.22112e12			−1.26807			−0.634037	−	0.773302i	$-0.718603\pi$
$571$	−3.22112e12			−1.26807			−0.634037	+	0.773302i	$0.718603\pi$
$572$		0			0
$573$	−2.92221e12			−1.13244
$574$		0			0
$575$	−1.52578e12			−0.582085
$576$		0			0
$577$		−	2.87776e12i		−	1.08085i	−0.841394	−	0.540423i	$-0.818264\pi$
$577$		−	2.87776e12i		−	1.08085i	0.841394	−	0.540423i	$-0.181736\pi$
$578$		0			0
$579$		−	2.73558e12i		−	1.01157i
$580$		0			0
$581$	−4.01204e12			−1.46074
$582$		0			0
$583$			1.64026e12i			0.588035i
$584$		0			0
$585$	1.70854e11	+	1.15505e10i	0.0603149	+	0.00407754i
$586$		0			0
$587$		−	1.67814e12i		−	0.583387i	−0.956512	−	0.291693i	$-0.905781\pi$
$587$		−	1.67814e12i		−	0.583387i	0.956512	−	0.291693i	$-0.0942187\pi$
$588$		0			0
$589$	7.96674e11			0.272748
$590$		0			0
$591$			2.16114e12i			0.728683i
$592$		0			0
$593$			2.55499e12i			0.848481i	0.905549	+	0.424241i	$0.139459\pi$
$593$			2.55499e12i			0.848481i	−0.905549	+	0.424241i	$0.860541\pi$
$594$		0			0
$595$	−9.08774e11			−0.297255
$596$		0			0
$597$	−2.80832e12			−0.904820
$598$		0			0
$599$	4.54852e12			1.44361			0.721804	−	0.692098i	$-0.243313\pi$
$599$	4.54852e12			1.44361			0.721804	+	0.692098i	$0.243313\pi$
$600$		0			0
$601$	1.13936e12			0.356227			0.178113	−	0.984010i	$-0.443001\pi$
$601$	1.13936e12			0.356227			0.178113	+	0.984010i	$0.443001\pi$
$602$		0			0
$603$		−	1.32119e12i		−	0.406947i
$604$		0			0
$605$			1.29667e12i			0.393486i
$606$		0			0
$607$	−2.74311e12			−0.820151			−0.410075	−	0.912052i	$-0.634498\pi$
$607$	−2.74311e12			−0.820151			−0.410075	+	0.912052i	$0.634498\pi$
$608$		0			0
$609$			7.25549e12i			2.13742i
$610$		0			0
$611$	−5.95476e11	−	4.02567e10i	−0.172854	−	0.0116856i
$612$		0			0
$613$		−	5.31737e12i		−	1.52098i	−0.649348	−	0.760492i	$-0.724958\pi$
$613$		−	5.31737e12i		−	1.52098i	0.649348	−	0.760492i	$-0.275042\pi$
$614$		0			0
$615$	−1.05928e12			−0.298587
$616$		0			0
$617$			3.03575e11i			0.0843302i	0.999111	+	0.0421651i	$0.0134256\pi$
$617$			3.03575e11i			0.0843302i	−0.999111	+	0.0421651i	$0.986574\pi$
$618$		0			0
$619$			1.76719e12i			0.483810i	0.970300	+	0.241905i	$0.0777723\pi$
$619$			1.76719e12i			0.483810i	−0.970300	+	0.241905i	$0.922228\pi$
$620$		0			0
$621$	−2.44890e12			−0.660782
$622$		0			0
$623$	4.46397e12			1.18720
$624$		0			0
$625$	3.33849e12			0.875164
$626$		0			0
$627$	−3.70723e12			−0.957955
$628$		0			0
$629$			6.69895e10i			0.0170639i
$630$		0			0
$631$		−	8.00234e11i		−	0.200949i	−0.994940	−	0.100474i	$-0.967964\pi$
$631$		−	8.00234e11i		−	0.200949i	0.994940	−	0.100474i	$-0.0320360\pi$
$632$		0			0
$633$	1.65809e12			0.410480
$634$		0			0
$635$		−	7.87346e11i		−	0.192169i
$636$		0			0
$637$	−4.64951e11	+	6.87754e12i	−0.111887	+	1.65503i
$638$		0			0
$639$			1.40747e12i			0.333954i
$640$		0			0
$641$	5.13894e12			1.20230			0.601150	−	0.799137i	$-0.294710\pi$
$641$	5.13894e12			1.20230			0.601150	+	0.799137i	$0.294710\pi$
$642$		0			0
$643$			1.58114e12i			0.364772i	0.983227	+	0.182386i	$0.0583820\pi$
$643$			1.58114e12i			0.364772i	−0.983227	+	0.182386i	$0.941618\pi$
$644$		0			0
$645$		−	8.75754e11i		−	0.199234i
$646$		0			0
$647$	7.12841e12			1.59928			0.799638	−	0.600482i	$-0.205025\pi$
$647$	7.12841e12			1.59928			0.799638	+	0.600482i	$0.205025\pi$
$648$		0			0
$649$	−2.39587e12			−0.530106
$650$		0			0
$651$	−2.56368e12			−0.559435
$652$		0			0
$653$	2.58426e12			0.556194			0.278097	−	0.960553i	$-0.410296\pi$
$653$	2.58426e12			0.556194			0.278097	+	0.960553i	$0.410296\pi$
$654$		0			0
$655$			9.26138e11i			0.196603i
$656$		0			0
$657$			9.47102e11i			0.198314i
$658$		0			0
$659$	9.50790e12			1.96381			0.981907	−	0.189365i	$-0.0606430\pi$
$659$	9.50790e12			1.96381			0.981907	+	0.189365i	$0.0606430\pi$
$660$		0			0
$661$		−	4.92881e12i		−	1.00423i	−0.864799	−	0.502117i	$-0.832554\pi$
$661$		−	4.92881e12i		−	1.00423i	0.864799	−	0.502117i	$-0.167446\pi$
$662$		0			0
$663$	−2.50160e11	+	3.70036e12i	−0.0502814	+	0.743761i
$664$		0			0
$665$			1.12825e12i			0.223721i
$666$		0			0
$667$	−4.84734e12			−0.948281
$668$		0			0
$669$			5.90797e12i			1.14030i
$670$		0			0
$671$			1.77494e13i			3.38012i
$672$		0			0
$673$	7.54694e12			1.41809			0.709044	−	0.705164i	$-0.249127\pi$
$673$	7.54694e12			1.41809			0.709044	+	0.705164i	$0.249127\pi$
$674$		0			0
$675$	5.61672e12			1.04140
$676$		0			0
$677$	−4.71143e12			−0.861994			−0.430997	−	0.902353i	$-0.641838\pi$
$677$	−4.71143e12			−0.861994			−0.430997	+	0.902353i	$0.641838\pi$
$678$		0			0
$679$	−1.08499e13			−1.95890
$680$		0			0
$681$			3.57046e12i			0.636154i
$682$		0			0
$683$		−	6.72669e12i		−	1.18279i	−0.806381	−	0.591396i	$-0.798577\pi$
$683$		−	6.72669e12i		−	1.18279i	0.806381	−	0.591396i	$-0.201423\pi$
$684$		0			0
$685$	−7.89732e11			−0.137048
$686$		0			0
$687$			4.56875e12i			0.782513i
$688$		0			0
$689$	1.37414e11	−	2.03262e12i	0.0232297	−	0.343613i
$690$		0			0
$691$			1.34071e12i			0.223709i	0.993725	+	0.111854i	$0.0356790\pi$
$691$			1.34071e12i			0.223709i	−0.993725	+	0.111854i	$0.964321\pi$
$692$		0			0
$693$	−4.97411e12			−0.819249
$694$		0			0
$695$			1.95098e12i			0.317190i
$696$		0			0
$697$			9.56556e12i			1.53519i
$698$		0			0
$699$	−5.21360e12			−0.826019
$700$		0			0
$701$	6.31380e12			0.987551			0.493776	−	0.869589i	$-0.335616\pi$
$701$	6.31380e12			0.987551			0.493776	+	0.869589i	$0.335616\pi$
$702$		0			0
$703$	8.31680e10			0.0128427
$704$		0			0
$705$	1.96123e11			0.0299004
$706$		0			0
$707$		−	3.25952e12i		−	0.490643i
$708$		0			0
$709$			1.13652e13i			1.68915i	0.535439	+	0.844574i	$0.320146\pi$
$709$			1.13652e13i			1.68915i	−0.535439	+	0.844574i	$0.679854\pi$
$710$		0			0
$711$	1.13054e12			0.165910
$712$		0			0
$713$		−	1.71278e12i		−	0.248198i
$714$		0			0
$715$	1.65345e11	−	2.44577e12i	0.0236599	−	0.349977i
$716$		0			0
$717$		−	8.46154e12i		−	1.19568i
$718$		0			0
$719$	−1.06987e13			−1.49297			−0.746484	−	0.665403i	$-0.768260\pi$
$719$	−1.06987e13			−1.49297			−0.746484	+	0.665403i	$0.768260\pi$
$720$		0			0
$721$			8.64609e12i			1.19155i
$722$		0			0
$723$		−	1.54419e12i		−	0.210174i
$724$		0			0
$725$	1.11177e13			1.49450
$726$		0			0
$727$	−4.51319e12			−0.599210			−0.299605	−	0.954063i	$-0.596855\pi$
$727$	−4.51319e12			−0.599210			−0.299605	+	0.954063i	$0.596855\pi$
$728$		0			0
$729$	8.35463e12			1.09560
$730$		0			0
$731$	−7.90831e12			−1.02437
$732$		0			0
$733$			6.87256e12i			0.879328i	0.898162	+	0.439664i	$0.144902\pi$
$733$			6.87256e12i			0.879328i	−0.898162	+	0.439664i	$0.855098\pi$
$734$		0			0
$735$		−	2.26515e12i		−	0.286289i
$736$		0			0
$737$	−1.89128e13			−2.36131
$738$		0			0
$739$		−	2.20989e11i		−	0.0272565i	−0.999907	−	0.0136283i	$-0.995662\pi$
$739$		−	2.20989e11i		−	0.0272565i	0.999907	−	0.0136283i	$-0.00433815\pi$
$740$		0			0
$741$	4.59402e12	+	3.10575e11i	0.559772	+	0.0378430i
$742$		0			0
$743$			1.40022e13i			1.68557i	0.538247	+	0.842787i	$0.319087\pi$
$743$			1.40022e13i			1.68557i	−0.538247	+	0.842787i	$0.680913\pi$
$744$		0			0
$745$	−2.54626e12			−0.302831
$746$		0			0
$747$			2.24337e12i			0.263608i
$748$		0			0
$749$		−	2.52790e13i		−	2.93489i
$750$		0			0
$751$	1.92803e12			0.221174			0.110587	−	0.993866i	$-0.464727\pi$
$751$	1.92803e12			0.221174			0.110587	+	0.993866i	$0.464727\pi$
$752$		0			0
$753$	9.25854e12			1.04946
$754$		0			0
$755$	6.45359e11			0.0722837
$756$		0			0
$757$	−1.75178e12			−0.193887			−0.0969433	−	0.995290i	$-0.530907\pi$
$757$	−1.75178e12			−0.193887			−0.0969433	+	0.995290i	$0.530907\pi$
$758$		0			0
$759$			7.97020e12i			0.871728i
$760$		0			0
$761$			9.26118e12i			1.00100i	0.865736	+	0.500502i	$0.166851\pi$
$761$			9.26118e12i			1.00100i	−0.865736	+	0.500502i	$0.833149\pi$
$762$		0			0
$763$	−2.89676e13			−3.09423
$764$		0			0
$765$			5.08149e11i			0.0536432i
$766$		0			0
$767$	2.96898e12	+	2.00716e11i	0.309762	+	0.0209412i
$768$		0			0
$769$		−	5.38976e12i		−	0.555778i	−0.960613	−	0.277889i	$-0.910365\pi$
$769$		−	5.38976e12i		−	0.555778i	0.960613	−	0.277889i	$-0.0896347\pi$
$770$		0			0
$771$	9.29053e12			0.946882
$772$		0			0
$773$			1.17249e13i			1.18114i	0.806988	+	0.590568i	$0.201096\pi$
$773$			1.17249e13i			1.18114i	−0.806988	+	0.590568i	$0.798904\pi$
$774$		0			0
$775$			3.92838e12i			0.391161i
$776$		0			0
$777$	−2.67633e11			−0.0263418
$778$		0			0
$779$	1.18757e13			1.15542
$780$		0			0
$781$	2.01480e13			1.93777
$782$		0			0
$783$	1.78441e13			1.69655
$784$		0			0
$785$		−	7.02515e11i		−	0.0660301i
$786$		0			0
$787$		−	4.02466e12i		−	0.373976i	−0.982362	−	0.186988i	$-0.940127\pi$
$787$		−	4.02466e12i		−	0.373976i	0.982362	−	0.186988i	$-0.0598725\pi$
$788$		0			0
$789$	6.89580e12			0.633488
$790$		0			0
$791$		−	1.87587e13i		−	1.70376i
$792$		0			0
$793$	1.48697e12	−	2.19952e13i	0.133528	−	1.97514i
$794$		0			0
$795$			6.69453e11i			0.0594385i
$796$		0			0
$797$	−4.71670e12			−0.414072			−0.207036	−	0.978333i	$-0.566382\pi$
$797$	−4.71670e12			−0.414072			−0.207036	+	0.978333i	$0.566382\pi$
$798$		0			0
$799$		−	1.77105e12i		−	0.153734i
$800$		0			0
$801$		−	2.49607e12i		−	0.214245i
$802$		0			0
$803$	1.35578e13			1.15071
$804$		0			0
$805$	2.42563e12			0.203584
$806$		0			0
$807$	1.09344e12			0.0907534
$808$		0			0
$809$	−2.01556e13			−1.65435			−0.827175	−	0.561944i	$-0.810053\pi$
$809$	−2.01556e13			−1.65435			−0.827175	+	0.561944i	$0.810053\pi$
$810$		0			0
$811$		−	1.98588e13i		−	1.61198i	−0.591930	−	0.805989i	$-0.701634\pi$
$811$		−	1.98588e13i		−	1.61198i	0.591930	−	0.805989i	$-0.298366\pi$
$812$		0			0
$813$			2.05911e12i			0.165300i
$814$		0			0
$815$	−3.78935e12			−0.300854
$816$		0			0
$817$			9.81822e12i			0.770963i
$818$		0			0
$819$	6.16396e12	+	4.16710e11i	0.478721	+	0.0323635i
$820$		0			0
$821$			8.60654e12i			0.661127i	0.943784	+	0.330563i	$0.107239\pi$
$821$			8.60654e12i			0.661127i	−0.943784	+	0.330563i	$0.892761\pi$
$822$		0			0
$823$	1.69297e13			1.28633			0.643163	−	0.765729i	$-0.277622\pi$
$823$	1.69297e13			1.28633			0.643163	+	0.765729i	$0.277622\pi$
$824$		0			0
$825$		−	1.82802e13i		−	1.37385i
$826$		0			0
$827$		−	2.31778e13i		−	1.72305i	−0.507716	−	0.861524i	$-0.669510\pi$
$827$		−	2.31778e13i		−	1.72305i	0.507716	−	0.861524i	$-0.330490\pi$
$828$		0			0
$829$	−1.39793e13			−1.02799			−0.513996	−	0.857793i	$-0.671835\pi$
$829$	−1.39793e13			−1.02799			−0.513996	+	0.857793i	$0.671835\pi$
$830$		0			0
$831$	−1.99190e13			−1.44898
$832$		0			0
$833$	−2.04550e13			−1.47196
$834$		0			0
$835$	−1.86099e11			−0.0132481
$836$		0			0
$837$			6.30510e12i			0.444045i
$838$		0			0
$839$		−	1.58497e13i		−	1.10431i	−0.833740	−	0.552157i	$-0.813805\pi$
$839$		−	1.58497e13i		−	1.10431i	0.833740	−	0.552157i	$-0.186195\pi$
$840$		0			0
$841$	2.08134e13			1.43470
$842$		0			0
$843$		−	8.58738e12i		−	0.585648i
$844$		0			0
$845$	−4.09793e11	+	3.01697e12i	−0.0276509	+	0.203571i
$846$		0			0
$847$			4.67802e13i			3.12311i
$848$		0			0
$849$	−1.69589e13			−1.12024
$850$		0			0
$851$		−	1.78804e11i		−	0.0116867i
$852$		0			0
$853$		−	2.79835e13i		−	1.80980i	−0.425619	−	0.904902i	$-0.639944\pi$
$853$		−	2.79835e13i		−	1.80980i	0.425619	−	0.904902i	$-0.360056\pi$
$854$		0			0
$855$	6.30871e11			0.0403732
$856$		0			0
$857$	−1.50277e13			−0.951652			−0.475826	−	0.879539i	$-0.657851\pi$
$857$	−1.50277e13			−0.951652			−0.475826	+	0.879539i	$0.657851\pi$
$858$		0			0
$859$	1.05497e13			0.661108			0.330554	−	0.943787i	$-0.392764\pi$
$859$	1.05497e13			0.661108			0.330554	+	0.943787i	$0.392764\pi$
$860$		0			0
$861$	−3.82158e13			−2.36989
$862$		0			0
$863$			8.04536e12i			0.493739i	0.969049	+	0.246869i	$0.0794018\pi$
$863$			8.04536e12i			0.493739i	−0.969049	+	0.246869i	$0.920598\pi$
$864$		0			0
$865$			2.65035e12i			0.160965i
$866$		0			0
$867$	2.97134e12			0.178594
$868$		0			0
$869$		−	1.61837e13i		−	0.962695i
$870$		0			0
$871$	2.34369e13	+	1.58444e12i	1.37981	+	0.0932810i
$872$		0			0
$873$			6.06683e12i			0.353507i
$874$		0			0
$875$	−1.13719e13			−0.655837
$876$		0			0
$877$			1.57718e13i			0.900294i	0.892955	+	0.450147i	$0.148628\pi$
$877$			1.57718e13i			0.900294i	−0.892955	+	0.450147i	$0.851372\pi$
$878$		0			0
$879$		−	5.10496e12i		−	0.288431i
$880$		0			0
$881$	−3.27966e12			−0.183416			−0.0917080	−	0.995786i	$-0.529233\pi$
$881$	−3.27966e12			−0.183416			−0.0917080	+	0.995786i	$0.529233\pi$
$882$		0			0
$883$	−9.55819e12			−0.529118			−0.264559	−	0.964370i	$-0.585226\pi$
$883$	−9.55819e12			−0.529118			−0.264559	+	0.964370i	$0.585226\pi$
$884$		0			0
$885$	−9.77849e11			−0.0535830
$886$		0			0
$887$	−4.23026e12			−0.229462			−0.114731	−	0.993397i	$-0.536601\pi$
$887$	−4.23026e12			−0.229462			−0.114731	+	0.993397i	$0.536601\pi$
$888$		0			0
$889$		−	2.84053e13i		−	1.52525i
$890$		0			0
$891$		−	1.98880e13i		−	1.05716i
$892$		0			0
$893$	−2.19877e12			−0.115704
$894$		0			0
$895$		−	4.66672e12i		−	0.243113i
$896$		0			0
$897$	6.67709e11	−	9.87673e12i	0.0344367	−	0.509386i
$898$		0			0
$899$			1.24803e13i			0.637245i
$900$		0			0
$901$	6.04535e12			0.305605
$902$		0			0
$903$		−	3.15948e13i		−	1.58133i
$904$		0			0
$905$			5.05376e12i			0.250435i
$906$		0			0
$907$	1.50413e13			0.737991			0.368996	−	0.929431i	$-0.379702\pi$
$907$	1.50413e13			0.737991			0.368996	+	0.929431i	$0.379702\pi$
$908$		0			0
$909$	−1.82259e12			−0.0885424
$910$		0			0
$911$	1.03485e13			0.497787			0.248894	−	0.968531i	$-0.419933\pi$
$911$	1.03485e13			0.497787			0.248894	+	0.968531i	$0.419933\pi$
$912$		0			0
$913$	3.21138e13			1.52958
$914$		0			0
$915$			7.24423e12i			0.341662i
$916$		0			0
$917$			3.34126e13i			1.56044i
$918$		0			0
$919$	−1.34997e13			−0.624314			−0.312157	−	0.950031i	$-0.601051\pi$
$919$	−1.34997e13			−0.624314			−0.312157	+	0.950031i	$0.601051\pi$
$920$		0			0
$921$			1.66816e13i			0.763958i
$922$		0			0
$923$	−2.49675e13	−	1.68791e12i	−1.13232	−	0.0765494i
$924$		0			0
$925$			4.10099e11i			0.0184184i
$926$		0			0
$927$	4.83454e12			0.215029
$928$		0			0
$929$		−	2.57332e13i		−	1.13351i	−0.823888	−	0.566753i	$-0.808199\pi$
$929$		−	2.57332e13i		−	1.13351i	0.823888	−	0.566753i	$-0.191801\pi$
$930$		0			0
$931$			2.53950e13i			1.10783i
$932$		0			0
$933$	1.39978e13			0.604772
$934$		0			0
$935$	7.27415e12			0.311265
$936$		0			0
$937$	4.16059e13			1.76330			0.881652	−	0.471900i	$-0.156432\pi$
$937$	4.16059e13			1.76330			0.881652	+	0.471900i	$0.156432\pi$
$938$		0			0
$939$	3.23789e13			1.35915
$940$		0			0
$941$		−	4.56087e12i		−	0.189625i	−0.995495	−	0.0948123i	$-0.969775\pi$
$941$		−	4.56087e12i		−	0.189625i	0.995495	−	0.0948123i	$-0.0302251\pi$
$942$		0			0
$943$		−	2.55317e13i		−	1.05142i
$944$		0			0
$945$	−8.92927e12			−0.364228
$946$		0			0
$947$		−	2.64086e13i		−	1.06702i	−0.845795	−	0.533508i	$-0.820874\pi$
$947$		−	2.64086e13i		−	1.06702i	0.845795	−	0.533508i	$-0.179126\pi$
$948$		0			0
$949$	−1.68009e13	−	1.13581e12i	−0.672409	−	0.0454577i
$950$		0			0
$951$		−	1.04827e12i		−	0.0415587i
$952$		0			0
$953$	−5.77362e12			−0.226741			−0.113370	−	0.993553i	$-0.536165\pi$
$953$	−5.77362e12			−0.226741			−0.113370	+	0.993553i	$0.536165\pi$
$954$		0			0
$955$		−	7.11858e12i		−	0.276935i
$956$		0			0
$957$		−	5.80755e13i		−	2.23815i
$958$		0			0
$959$	−2.84914e13			−1.08775
$960$		0			0
$961$	2.20298e13			0.833211
$962$		0			0
$963$	−1.41350e13			−0.529636
$964$		0			0
$965$	6.66394e12			0.247376
$966$		0			0
$967$			1.21724e13i			0.447669i	0.974627	+	0.223834i	$0.0718574\pi$
$967$			1.21724e13i			0.447669i	−0.974627	+	0.223834i	$0.928143\pi$
$968$		0			0
$969$			1.36634e13i			0.497854i
$970$		0			0
$971$	3.82669e13			1.38146			0.690728	−	0.723115i	$-0.257290\pi$
$971$	3.82669e13			1.38146			0.690728	+	0.723115i	$0.257290\pi$
$972$		0			0
$973$			7.03859e13i			2.51755i
$974$		0			0
$975$	−1.53144e12	+	2.26530e13i	−0.0542724	+	0.802796i
$976$		0			0
$977$		−	4.96320e13i		−	1.74275i	−0.490614	−	0.871377i	$-0.663227\pi$
$977$		−	4.96320e13i		−	1.74275i	0.490614	−	0.871377i	$-0.336773\pi$
$978$		0			0
$979$	−3.57312e13			−1.24316
$980$		0			0
$981$			1.61975e13i			0.558390i
$982$		0			0
$983$		−	2.22486e13i		−	0.759997i	−0.924987	−	0.379998i	$-0.875925\pi$
$983$		−	2.22486e13i		−	0.759997i	0.924987	−	0.379998i	$-0.124075\pi$
$984$		0			0
$985$	−5.26459e12			−0.178198
$986$		0			0
$987$	7.07559e12			0.237320
$988$		0			0
$989$	2.11083e13			0.701568
$990$		0			0
$991$	−4.38186e13			−1.44320			−0.721601	−	0.692310i	$-0.756593\pi$
$991$	−4.38186e13			−1.44320			−0.721601	+	0.692310i	$0.756593\pi$
$992$		0			0
$993$			6.15963e12i			0.201040i
$994$		0			0
$995$		−	6.84114e12i		−	0.221271i
$996$		0			0
$997$	−6.60845e12			−0.211822			−0.105911	−	0.994376i	$-0.533776\pi$
$997$	−6.60845e12			−0.211822			−0.105911	+	0.994376i	$0.533776\pi$
$998$		0			0
$999$			6.58214e11i			0.0209085i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.f.d.129.12		32
4.3	odd	2		104.10.f.a.25.21	✓	32
13.12	even	2	inner	208.10.f.d.129.11		32
52.51	odd	2		104.10.f.a.25.22	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.21	✓	32	4.3	odd	2
104.10.f.a.25.22	yes	32	52.51	odd	2
208.10.f.d.129.11		32	13.12	even	2	inner
208.10.f.d.129.12		32	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-117.861 q^{3} -287.112i q^{5} -10358.2i q^{7} -5791.89 q^{9} +82910.8i q^{11} +(6945.91 - 102744. i) q^{13} +33839.1i q^{15} +305577. q^{17} -379376. i q^{19} +1.22082e6i q^{21} -815623. q^{23} +1.87069e6 q^{25} +3.00248e6 q^{27} +5.94311e6 q^{29} +2.09996e6i q^{31} -9.77191e6i q^{33} -2.97396e6 q^{35} +219223. i q^{37} +(-818648. + 1.21094e7i) q^{39} +3.13033e7i q^{41} -2.58799e7 q^{43} +1.66292e6i q^{45} -5.79575e6i q^{47} -6.69388e7 q^{49} -3.60155e7 q^{51} +1.97834e7 q^{53} +2.38046e7 q^{55} +4.47134e7i q^{57} +2.88970e7i q^{59} +2.14079e8 q^{61} +5.99936e7i q^{63} +(-2.94989e7 - 1.99425e6i) q^{65} +2.28111e8i q^{67} +9.61298e7 q^{69} -2.43008e8i q^{71} -1.63522e8i q^{73} -2.20481e8 q^{75} +8.58807e8 q^{77} -1.95194e8 q^{79} -2.39873e8 q^{81} -3.87329e8i q^{83} -8.77347e7i q^{85} -7.00458e8 q^{87} +4.30960e8i q^{89} +(-1.06424e9 - 7.19471e7i) q^{91} -2.47503e8i q^{93} -1.08923e8 q^{95} -1.04747e9i q^{97} -4.80210e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} - 3171556 q^{23} - 13526722 q^{25} + 3694974 q^{27} + 8833508 q^{29} + 8281126 q^{35} + 12056860 q^{39} - 89959038 q^{43} - 172344874 q^{49}+ \cdots - 1741143356 q^{95}+O(q^{100}) \) 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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