LMFDB - Embedded newform 208.12.f.b.129.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,12,Mod(129,208)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("208.129");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	208.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$159.815381556$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18433 x^{10} + 121088056 x^{8} + 340607607312 x^{6} + 380893885719552 x^{4} + \cdots + 14\!\cdots\!56 $ x^12 + 18433x^10 + 121088056x^8 + 340607607312x^6 + 380893885719552x^4 + 134825856231997440*x^2 + 1497425476589715456
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{31}\cdot 3^{4}\cdot 13^{4} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.4
Root			$-3.38741i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.12.f.b.129.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-494.534 q^{3} +9091.88i q^{5} +45027.4i q^{7} +67417.3 q^{9} +O(q^{10})$	$q-494.534 q^{3} +9091.88i q^{5} +45027.4i q^{7} +67417.3 q^{9} -810639. i q^{11} +(1.11466e6 - 741417. i) q^{13} -4.49625e6i q^{15} -851428. q^{17} +7.56605e6i q^{19} -2.22676e7i q^{21} -3.69275e6 q^{23} -3.38341e7 q^{25} +5.42651e7 q^{27} -9.33625e7 q^{29} +2.61508e8i q^{31} +4.00889e8i q^{33} -4.09384e8 q^{35} +5.38221e8i q^{37} +(-5.51236e8 + 3.66656e8i) q^{39} -6.71292e8i q^{41} +1.09294e9 q^{43} +6.12950e8i q^{45} +1.79698e9i q^{47} -5.01432e7 q^{49} +4.21060e8 q^{51} -5.82497e9 q^{53} +7.37023e9 q^{55} -3.74167e9i q^{57} +4.74847e9i q^{59} +6.50495e9 q^{61} +3.03563e9i q^{63} +(6.74087e9 + 1.01343e10i) q^{65} -8.11940e9i q^{67} +1.82619e9 q^{69} +3.16640e9i q^{71} -8.22937e9i q^{73} +1.67321e10 q^{75} +3.65010e10 q^{77} +3.75500e9 q^{79} -3.87787e10 q^{81} +6.60119e9i q^{83} -7.74108e9i q^{85} +4.61710e10 q^{87} +5.68420e10i q^{89} +(3.33841e10 + 5.01902e10i) q^{91} -1.29325e11i q^{93} -6.87896e10 q^{95} +1.62006e11i q^{97} -5.46511e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 488 q^{3} + 654644 q^{9}+O(q^{10}) $ 12 * q + 488 * q^3 + 654644 * q^9	$ 12 q + 488 q^{3} + 654644 q^{9} + 3208868 q^{13} + 12198768 q^{17} - 5810592 q^{23} + 6102388 q^{25} + 52613336 q^{27} - 244463112 q^{29} + 562027560 q^{35} + 2199109744 q^{39} - 2294519976 q^{43} - 3573617796 q^{49} - 7713246552 q^{51} - 4602062760 q^{53} + 6178744976 q^{55} - 13775649944 q^{61} - 7598401512 q^{65} - 25419983328 q^{69} - 68016370832 q^{75} - 80478036048 q^{77} - 18046097296 q^{79} - 132677486692 q^{81} - 94507900752 q^{87} - 104793638664 q^{91} - 145093149648 q^{95}+O(q^{100}) $ 12 * q + 488 * q^3 + 654644 * q^9 + 3208868 * q^13 + 12198768 * q^17 - 5810592 * q^23 + 6102388 * q^25 + 52613336 * q^27 - 244463112 * q^29 + 562027560 * q^35 + 2199109744 * q^39 - 2294519976 * q^43 - 3573617796 * q^49 - 7713246552 * q^51 - 4602062760 * q^53 + 6178744976 * q^55 - 13775649944 * q^61 - 7598401512 * q^65 - 25419983328 * q^69 - 68016370832 * q^75 - 80478036048 * q^77 - 18046097296 * q^79 - 132677486692 * q^81 - 94507900752 * q^87 - 104793638664 * q^91 - 145093149648 * q^95

Display 100 coefficients 10 coefficients

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
$3$	−494.534			−1.17498			−0.587489	−	0.809232i	$-0.699883\pi$
$3$	−494.534			−1.17498			−0.587489	+	0.809232i	$0.699883\pi$
$4$		0			0
$5$			9091.88i			1.30112i	0.759453	+	0.650562i	$0.225467\pi$
$5$			9091.88i			1.30112i	−0.759453	+	0.650562i	$0.774533\pi$
$6$		0			0
$7$			45027.4i			1.01260i	0.862357	+	0.506300i	$0.168987\pi$
$7$			45027.4i			1.01260i	−0.862357	+	0.506300i	$0.831013\pi$
$8$		0			0
$9$	67417.3			0.380573
$10$		0			0
$11$		−	810639.i		−	1.51764i	−0.651303	−	0.758818i	$-0.725777\pi$
$11$		−	810639.i		−	1.51764i	0.651303	−	0.758818i	$-0.274223\pi$
$12$		0			0
$13$	1.11466e6	−	741417.i	0.832632	−	0.553827i
$13$	1.11466e6	−	741417.i	0.832632	−	0.553827i
$14$		0			0
$15$		−	4.49625e6i		−	1.52879i
$16$		0			0
$17$	−851428.			−0.145438			−0.0727192	−	0.997352i	$-0.523168\pi$
$17$	−851428.			−0.145438			−0.0727192	+	0.997352i	$0.523168\pi$
$18$		0			0
$19$			7.56605e6i			0.701011i	0.936561	+	0.350505i	$0.113990\pi$
$19$			7.56605e6i			0.701011i	−0.936561	+	0.350505i	$0.886010\pi$
$20$		0			0
$21$		−	2.22676e7i		−	1.18978i
$22$		0			0
$23$	−3.69275e6			−0.119632			−0.0598159	−	0.998209i	$-0.519051\pi$
$23$	−3.69275e6			−0.119632			−0.0598159	+	0.998209i	$0.519051\pi$
$24$		0			0
$25$	−3.38341e7			−0.692923
$26$		0			0
$27$	5.42651e7			0.727813
$28$		0			0
$29$	−9.33625e7			−0.845247			−0.422623	−	0.906305i	$-0.638891\pi$
$29$	−9.33625e7			−0.845247			−0.422623	+	0.906305i	$0.638891\pi$
$30$		0			0
$31$			2.61508e8i			1.64057i	0.571953	+	0.820286i	$0.306186\pi$
$31$			2.61508e8i			1.64057i	−0.571953	+	0.820286i	$0.693814\pi$
$32$		0			0
$33$			4.00889e8i			1.78319i
$34$		0			0
$35$	−4.09384e8			−1.31752
$36$		0			0
$37$			5.38221e8i			1.27600i	0.770036	+	0.638000i	$0.220238\pi$
$37$			5.38221e8i			1.27600i	−0.770036	+	0.638000i	$0.779762\pi$
$38$		0			0
$39$	−5.51236e8	+	3.66656e8i	−0.978324	+	0.650734i
$40$		0			0
$41$		−	6.71292e8i		−	0.904899i	−0.891790	−	0.452449i	$-0.850550\pi$
$41$		−	6.71292e8i		−	0.904899i	0.891790	−	0.452449i	$-0.149450\pi$
$42$		0			0
$43$	1.09294e9			1.13375			0.566876	−	0.823803i	$-0.308152\pi$
$43$	1.09294e9			1.13375			0.566876	+	0.823803i	$0.308152\pi$
$44$		0			0
$45$			6.12950e8i			0.495172i
$46$		0			0
$47$			1.79698e9i			1.14289i	0.820641	+	0.571445i	$0.193617\pi$
$47$			1.79698e9i			1.14289i	−0.820641	+	0.571445i	$0.806383\pi$
$48$		0			0
$49$	−5.01432e7			−0.0253591
$50$		0			0
$51$	4.21060e8			0.170887
$52$		0			0
$53$	−5.82497e9			−1.91327			−0.956636	−	0.291287i	$-0.905917\pi$
$53$	−5.82497e9			−1.91327			−0.956636	+	0.291287i	$0.905917\pi$
$54$		0			0
$55$	7.37023e9			1.97463
$56$		0			0
$57$		−	3.74167e9i		−	0.823672i
$58$		0			0
$59$			4.74847e9i			0.864704i	0.901705	+	0.432352i	$0.142316\pi$
$59$			4.74847e9i			0.864704i	−0.901705	+	0.432352i	$0.857684\pi$
$60$		0			0
$61$	6.50495e9			0.986120			0.493060	−	0.869995i	$-0.335878\pi$
$61$	6.50495e9			0.986120			0.493060	+	0.869995i	$0.335878\pi$
$62$		0			0
$63$			3.03563e9i			0.385368i
$64$		0			0
$65$	6.74087e9	+	1.01343e10i	0.720597	+	1.08336i
$66$		0			0
$67$		−	8.11940e9i		−	0.734704i	−0.930082	−	0.367352i	$-0.880264\pi$
$67$		−	8.11940e9i		−	0.734704i	0.930082	−	0.367352i	$-0.119736\pi$
$68$		0			0
$69$	1.82619e9			0.140565
$70$		0			0
$71$			3.16640e9i			0.208279i	0.994563	+	0.104139i	$0.0332088\pi$
$71$			3.16640e9i			0.208279i	−0.994563	+	0.104139i	$0.966791\pi$
$72$		0			0
$73$		−	8.22937e9i		−	0.464613i	−0.972643	−	0.232306i	$-0.925373\pi$
$73$		−	8.22937e9i		−	0.464613i	0.972643	−	0.232306i	$-0.0746272\pi$
$74$		0			0
$75$	1.67321e10			0.814169
$76$		0			0
$77$	3.65010e10			1.53676
$78$		0			0
$79$	3.75500e9			0.137297			0.0686485	−	0.997641i	$-0.478131\pi$
$79$	3.75500e9			0.137297			0.0686485	+	0.997641i	$0.478131\pi$
$80$		0			0
$81$	−3.87787e10			−1.23574
$82$		0			0
$83$			6.60119e9i			0.183947i	0.995761	+	0.0919735i	$0.0293175\pi$
$83$			6.60119e9i			0.183947i	−0.995761	+	0.0919735i	$0.970682\pi$
$84$		0			0
$85$		−	7.74108e9i		−	0.189233i
$86$		0			0
$87$	4.61710e10			0.993146
$88$		0			0
$89$			5.68420e10i			1.07901i	0.841983	+	0.539503i	$0.181388\pi$
$89$			5.68420e10i			1.07901i	−0.841983	+	0.539503i	$0.818612\pi$
$90$		0			0
$91$	3.33841e10	+	5.01902e10i	0.560805	+	0.843123i
$92$		0			0
$93$		−	1.29325e11i		−	1.92764i
$94$		0			0
$95$	−6.87896e10			−0.912101
$96$		0			0
$97$			1.62006e11i			1.91551i	0.287578	+	0.957757i	$0.407150\pi$
$97$			1.62006e11i			1.91551i	−0.287578	+	0.957757i	$0.592850\pi$
$98$		0			0
$99$		−	5.46511e10i		−	0.577571i
$100$		0			0
$101$	1.31282e11			1.24290			0.621450	−	0.783454i	$-0.286544\pi$
$101$	1.31282e11			1.24290			0.621450	+	0.783454i	$0.286544\pi$
$102$		0			0
$103$	5.50452e10			0.467859			0.233929	−	0.972254i	$-0.424842\pi$
$103$	5.50452e10			0.467859			0.233929	+	0.972254i	$0.424842\pi$
$104$		0			0
$105$	2.02454e11			1.54805
$106$		0			0
$107$	−1.85072e11			−1.27564			−0.637822	−	0.770184i	$-0.720164\pi$
$107$	−1.85072e11			−1.27564			−0.637822	+	0.770184i	$0.720164\pi$
$108$		0			0
$109$		−	2.74414e10i		−	0.170829i	−0.996346	−	0.0854143i	$-0.972779\pi$
$109$		−	2.74414e10i		−	0.170829i	0.996346	−	0.0854143i	$-0.0272214\pi$
$110$		0			0
$111$		−	2.66169e11i		−	1.49927i
$112$		0			0
$113$	−1.64428e11			−0.839547			−0.419773	−	0.907629i	$-0.637890\pi$
$113$	−1.64428e11			−0.839547			−0.419773	+	0.907629i	$0.637890\pi$
$114$		0			0
$115$		−	3.35740e10i		−	0.155656i
$116$		0			0
$117$	7.51472e10	−	4.99844e10i	0.316877	−	0.210771i
$118$		0			0
$119$		−	3.83376e10i		−	0.147271i
$120$		0			0
$121$	−3.71823e11			−1.30322
$122$		0			0
$123$			3.31977e11i			1.06324i
$124$		0			0
$125$			1.36324e11i			0.399546i
$126$		0			0
$127$	−5.48728e11			−1.47379			−0.736897	−	0.676005i	$-0.763710\pi$
$127$	−5.48728e11			−1.47379			−0.736897	+	0.676005i	$0.763710\pi$
$128$		0			0
$129$	−5.40494e11			−1.33213
$130$		0			0
$131$	−3.22742e11			−0.730910			−0.365455	−	0.930829i	$-0.619087\pi$
$131$	−3.22742e11			−0.730910			−0.365455	+	0.930829i	$0.619087\pi$
$132$		0			0
$133$	−3.40680e11			−0.709843
$134$		0			0
$135$			4.93372e11i			0.946975i
$136$		0			0
$137$			1.94892e11i			0.345009i	0.985009	+	0.172505i	$0.0551859\pi$
$137$			1.94892e11i			0.345009i	−0.985009	+	0.172505i	$0.944814\pi$
$138$		0			0
$139$	6.64756e11			1.08663			0.543314	−	0.839530i	$-0.317169\pi$
$139$	6.64756e11			1.08663			0.543314	+	0.839530i	$0.317169\pi$
$140$		0			0
$141$		−	8.88667e11i		−	1.34287i
$142$		0			0
$143$	−6.01021e11	−	9.03584e11i	−0.840508	−	1.26363i
$144$		0			0
$145$		−	8.48840e11i		−	1.09977i
$146$		0			0
$147$	2.47976e10			0.0297964
$148$		0			0
$149$			4.29539e11i			0.479157i	0.970877	+	0.239578i	$0.0770092\pi$
$149$			4.29539e11i			0.479157i	−0.970877	+	0.239578i	$0.922991\pi$
$150$		0			0
$151$		−	1.20700e12i		−	1.25122i	−0.780136	−	0.625610i	$-0.784850\pi$
$151$		−	1.20700e12i		−	1.25122i	0.780136	−	0.625610i	$-0.215150\pi$
$152$		0			0
$153$	−5.74010e10			−0.0553499
$154$		0			0
$155$	−2.37760e12			−2.13459
$156$		0			0
$157$	−9.61496e10			−0.0804450			−0.0402225	−	0.999191i	$-0.512807\pi$
$157$	−9.61496e10			−0.0804450			−0.0402225	+	0.999191i	$0.512807\pi$
$158$		0			0
$159$	2.88065e12			2.24805
$160$		0			0
$161$		−	1.66275e11i		−	0.121139i
$162$		0			0
$163$			1.50626e12i			1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$163$			1.50626e12i			1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$164$		0			0
$165$	−3.64483e12			−2.32015
$166$		0			0
$167$		−	1.03474e12i		−	0.616441i	−0.951315	−	0.308221i	$-0.900266\pi$
$167$		−	1.03474e12i		−	0.616441i	0.951315	−	0.308221i	$-0.0997335\pi$
$168$		0			0
$169$	6.92762e11	−	1.65285e12i	0.386551	−	0.922268i
$170$		0			0
$171$			5.10083e11i			0.266786i
$172$		0			0
$173$	−1.11263e11			−0.0545881			−0.0272941	−	0.999627i	$-0.508689\pi$
$173$	−1.11263e11			−0.0545881			−0.0272941	+	0.999627i	$0.508689\pi$
$174$		0			0
$175$		−	1.52346e12i		−	0.701653i
$176$		0			0
$177$		−	2.34828e12i		−	1.01601i
$178$		0			0
$179$	−2.06269e12			−0.838964			−0.419482	−	0.907764i	$-0.637788\pi$
$179$	−2.06269e12			−0.838964			−0.419482	+	0.907764i	$0.637788\pi$
$180$		0			0
$181$	3.46724e12			1.32663			0.663317	−	0.748338i	$-0.269148\pi$
$181$	3.46724e12			1.32663			0.663317	+	0.748338i	$0.269148\pi$
$182$		0			0
$183$	−3.21692e12			−1.15867
$184$		0			0
$185$	−4.89344e12			−1.66023
$186$		0			0
$187$			6.90200e11i			0.220722i
$188$		0			0
$189$			2.44342e12i			0.736984i
$190$		0			0
$191$	4.63569e12			1.31957			0.659783	−	0.751456i	$-0.270648\pi$
$191$	4.63569e12			1.31957			0.659783	+	0.751456i	$0.270648\pi$
$192$		0			0
$193$		−	1.13783e12i		−	0.305852i	−0.988238	−	0.152926i	$-0.951130\pi$
$193$		−	1.13783e12i		−	0.305852i	0.988238	−	0.152926i	$-0.0488696\pi$
$194$		0			0
$195$	−3.33359e12	−	5.01177e12i	−0.846686	−	1.27292i
$196$		0			0
$197$		−	2.32298e12i		−	0.557803i	−0.960320	−	0.278901i	$-0.910030\pi$
$197$		−	2.32298e12i		−	0.557803i	0.960320	−	0.278901i	$-0.0899702\pi$
$198$		0			0
$199$	−2.94354e11			−0.0668618			−0.0334309	−	0.999441i	$-0.510643\pi$
$199$	−2.94354e11			−0.0668618			−0.0334309	+	0.999441i	$0.510643\pi$
$200$		0			0
$201$			4.01532e12i			0.863261i
$202$		0			0
$203$		−	4.20387e12i		−	0.855897i
$204$		0			0
$205$	6.10330e12			1.17739
$206$		0			0
$207$	−2.48955e11			−0.0455286
$208$		0			0
$209$	6.13333e12			1.06388
$210$		0			0
$211$	−8.00209e11			−0.131719			−0.0658597	−	0.997829i	$-0.520979\pi$
$211$	−8.00209e11			−0.131719			−0.0658597	+	0.997829i	$0.520979\pi$
$212$		0			0
$213$		−	1.56589e12i		−	0.244723i
$214$		0			0
$215$			9.93684e12i			1.47515i
$216$		0			0
$217$	−1.17750e13			−1.66124
$218$		0			0
$219$			4.06971e12i			0.545910i
$220$		0			0
$221$	−9.49050e11	+	6.31263e11i	−0.121097	+	0.0805477i
$222$		0			0
$223$		−	1.48493e13i		−	1.80314i	−0.432631	−	0.901571i	$-0.642415\pi$
$223$		−	1.48493e13i		−	1.80314i	0.432631	−	0.901571i	$-0.357585\pi$
$224$		0			0
$225$	−2.28100e12			−0.263707
$226$		0			0
$227$			1.57088e13i			1.72982i	0.501928	+	0.864910i	$0.332624\pi$
$227$			1.57088e13i			1.72982i	−0.501928	+	0.864910i	$0.667376\pi$
$228$		0			0
$229$			2.72068e12i			0.285485i	0.989760	+	0.142742i	$0.0455920\pi$
$229$			2.72068e12i			0.285485i	−0.989760	+	0.142742i	$0.954408\pi$
$230$		0			0
$231$	−1.80510e13			−1.80566
$232$		0			0
$233$	−8.42380e12			−0.803620			−0.401810	−	0.915723i	$-0.631619\pi$
$233$	−8.42380e12			−0.803620			−0.401810	+	0.915723i	$0.631619\pi$
$234$		0			0
$235$	−1.63379e13			−1.48704
$236$		0			0
$237$	−1.85698e12			−0.161321
$238$		0			0
$239$		−	2.27825e13i		−	1.88979i	−0.327374	−	0.944895i	$-0.606164\pi$
$239$		−	2.27825e13i		−	1.88979i	0.327374	−	0.944895i	$-0.393836\pi$
$240$		0			0
$241$			6.32011e12i			0.500761i	0.968147	+	0.250381i	$0.0805558\pi$
$241$			6.32011e12i			0.500761i	−0.968147	+	0.250381i	$0.919444\pi$
$242$		0			0
$243$	9.56452e12			0.724150
$244$		0			0
$245$		−	4.55896e11i		−	0.0329953i
$246$		0			0
$247$	5.60960e12	+	8.43356e12i	0.388239	+	0.583684i
$248$		0			0
$249$		−	3.26452e12i		−	0.216134i
$250$		0			0
$251$	−5.09290e12			−0.322671			−0.161335	−	0.986900i	$-0.551580\pi$
$251$	−5.09290e12			−0.322671			−0.161335	+	0.986900i	$0.551580\pi$
$252$		0			0
$253$			2.99349e12i			0.181558i
$254$		0			0
$255$			3.82823e12i			0.222345i
$256$		0			0
$257$	−2.23129e13			−1.24144			−0.620718	−	0.784034i	$-0.713159\pi$
$257$	−2.23129e13			−1.24144			−0.620718	+	0.784034i	$0.713159\pi$
$258$		0			0
$259$	−2.42347e13			−1.29208
$260$		0			0
$261$	−6.29425e12			−0.321678
$262$		0			0
$263$	−1.02420e12			−0.0501911			−0.0250956	−	0.999685i	$-0.507989\pi$
$263$	−1.02420e12			−0.0501911			−0.0250956	+	0.999685i	$0.507989\pi$
$264$		0			0
$265$		−	5.29599e13i		−	2.48940i
$266$		0			0
$267$		−	2.81103e13i		−	1.26781i
$268$		0			0
$269$	−3.39292e13			−1.46871			−0.734355	−	0.678766i	$-0.762515\pi$
$269$	−3.39292e13			−1.46871			−0.734355	+	0.678766i	$0.762515\pi$
$270$		0			0
$271$		−	4.37089e13i		−	1.81651i	−0.418414	−	0.908257i	$-0.637414\pi$
$271$		−	4.37089e13i		−	1.81651i	0.418414	−	0.908257i	$-0.362586\pi$
$272$		0			0
$273$	−1.65096e13	−	2.48208e13i	−0.658934	−	0.990651i
$274$		0			0
$275$			2.74272e13i			1.05160i
$276$		0			0
$277$	1.69270e13			0.623651			0.311826	−	0.950139i	$-0.399060\pi$
$277$	1.69270e13			0.623651			0.311826	+	0.950139i	$0.399060\pi$
$278$		0			0
$279$			1.76302e13i			0.624357i
$280$		0			0
$281$		−	4.31608e13i		−	1.46962i	−0.678274	−	0.734809i	$-0.737272\pi$
$281$		−	4.31608e13i		−	1.46962i	0.678274	−	0.734809i	$-0.262728\pi$
$282$		0			0
$283$	4.77458e13			1.56354			0.781772	−	0.623565i	$-0.214316\pi$
$283$	4.77458e13			1.56354			0.781772	+	0.623565i	$0.214316\pi$
$284$		0			0
$285$	3.40188e13			1.07170
$286$		0			0
$287$	3.02265e13			0.916301
$288$		0			0
$289$	−3.35470e13			−0.978848
$290$		0			0
$291$		−	8.01173e13i		−	2.25069i
$292$		0			0
$293$		−	1.64453e13i		−	0.444908i	−0.974943	−	0.222454i	$-0.928593\pi$
$293$		−	1.64453e13i		−	0.444908i	0.974943	−	0.222454i	$-0.0714067\pi$
$294$		0			0
$295$	−4.31725e13			−1.12509
$296$		0			0
$297$		−	4.39894e13i		−	1.10456i
$298$		0			0
$299$	−4.11615e12	+	2.73787e12i	−0.0996093	+	0.0662553i
$300$		0			0
$301$			4.92121e13i			1.14804i
$302$		0			0
$303$	−6.49232e13			−1.46038
$304$		0			0
$305$			5.91422e13i			1.28306i
$306$		0			0
$307$		−	5.79282e12i		−	0.121235i	−0.998161	−	0.0606176i	$-0.980693\pi$
$307$		−	5.79282e12i		−	0.121235i	0.998161	−	0.0606176i	$-0.0193070\pi$
$308$		0			0
$309$	−2.72217e13			−0.549724
$310$		0			0
$311$	−6.88588e13			−1.34208			−0.671039	−	0.741422i	$-0.734152\pi$
$311$	−6.88588e13			−1.34208			−0.671039	+	0.741422i	$0.734152\pi$
$312$		0			0
$313$	−2.94167e13			−0.553477			−0.276738	−	0.960945i	$-0.589254\pi$
$313$	−2.94167e13			−0.553477			−0.276738	+	0.960945i	$0.589254\pi$
$314$		0			0
$315$	−2.75996e13			−0.501411
$316$		0			0
$317$			2.48320e13i			0.435698i	0.975983	+	0.217849i	$0.0699040\pi$
$317$			2.48320e13i			0.435698i	−0.975983	+	0.217849i	$0.930096\pi$
$318$		0			0
$319$			7.56832e13i			1.28278i
$320$		0			0
$321$	9.15244e13			1.49885
$322$		0			0
$323$		−	6.44195e12i		−	0.101954i
$324$		0			0
$325$	−3.77134e13	+	2.50852e13i	−0.576949	+	0.383759i
$326$		0			0
$327$			1.35707e13i			0.200720i
$328$		0			0
$329$	−8.09132e13			−1.15729
$330$		0			0
$331$			3.37169e13i			0.466438i	0.972424	+	0.233219i	$0.0749259\pi$
$331$			3.37169e13i			0.466438i	−0.972424	+	0.233219i	$0.925074\pi$
$332$		0			0
$333$			3.62854e13i			0.485611i
$334$		0			0
$335$	7.38206e13			0.955941
$336$		0			0
$337$	−8.98229e13			−1.12570			−0.562850	−	0.826559i	$-0.690295\pi$
$337$	−8.98229e13			−1.12570			−0.562850	+	0.826559i	$0.690295\pi$
$338$		0			0
$339$	8.13154e13			0.986449
$340$		0			0
$341$	2.11988e14			2.48979
$342$		0			0
$343$			8.67761e13i			0.986922i
$344$		0			0
$345$			1.66035e13i			0.182892i
$346$		0			0
$347$	−8.75394e13			−0.934096			−0.467048	−	0.884232i	$-0.654682\pi$
$347$	−8.75394e13			−0.934096			−0.467048	+	0.884232i	$0.654682\pi$
$348$		0			0
$349$			3.87065e13i			0.400169i	0.979779	+	0.200085i	$0.0641217\pi$
$349$			3.87065e13i			0.400169i	−0.979779	+	0.200085i	$0.935878\pi$
$350$		0			0
$351$	6.04870e13	−	4.02331e13i	0.606000	−	0.403083i
$352$		0			0
$353$			3.24210e13i			0.314822i	0.987533	+	0.157411i	$0.0503148\pi$
$353$			3.24210e13i			0.314822i	−0.987533	+	0.157411i	$0.949685\pi$
$354$		0			0
$355$	−2.87885e13			−0.270996
$356$		0			0
$357$			1.89593e13i			0.173040i
$358$		0			0
$359$			1.45534e14i			1.28809i	0.764990	+	0.644043i	$0.222744\pi$
$359$			1.45534e14i			1.28809i	−0.764990	+	0.644043i	$0.777256\pi$
$360$		0			0
$361$	5.92451e13			0.508584
$362$		0			0
$363$	1.83879e14			1.53125
$364$		0			0
$365$	7.48205e13			0.604518
$366$		0			0
$367$	−6.95668e13			−0.545430			−0.272715	−	0.962095i	$-0.587922\pi$
$367$	−6.95668e13			−0.545430			−0.272715	+	0.962095i	$0.587922\pi$
$368$		0			0
$369$		−	4.52567e13i		−	0.344380i
$370$		0			0
$371$		−	2.62284e14i		−	1.93738i
$372$		0			0
$373$	1.20982e14			0.867604			0.433802	−	0.901008i	$-0.357172\pi$
$373$	1.20982e14			0.867604			0.433802	+	0.901008i	$0.357172\pi$
$374$		0			0
$375$		−	6.74168e13i		−	0.469457i
$376$		0			0
$377$	−1.04067e14	+	6.92206e13i	−0.703779	+	0.468120i
$378$		0			0
$379$			3.22918e13i			0.212117i	0.994360	+	0.106059i	$0.0338231\pi$
$379$			3.22918e13i			0.212117i	−0.994360	+	0.106059i	$0.966177\pi$
$380$		0			0
$381$	2.71365e14			1.73168
$382$		0			0
$383$			6.60313e13i			0.409408i	0.978824	+	0.204704i	$0.0656232\pi$
$383$			6.60313e13i			0.409408i	−0.978824	+	0.204704i	$0.934377\pi$
$384$		0			0
$385$			3.31862e14i			1.99951i
$386$		0			0
$387$	7.36828e13			0.431475
$388$		0			0
$389$	1.13990e14			0.648851			0.324426	−	0.945911i	$-0.394829\pi$
$389$	1.13990e14			0.648851			0.324426	+	0.945911i	$0.394829\pi$
$390$		0			0
$391$	3.14411e12			0.0173991
$392$		0			0
$393$	1.59607e14			0.858803
$394$		0			0
$395$			3.41400e13i			0.178640i
$396$		0			0
$397$		−	2.68777e14i		−	1.36787i	−0.729543	−	0.683935i	$-0.760267\pi$
$397$		−	2.68777e14i		−	1.36787i	0.729543	−	0.683935i	$-0.239733\pi$
$398$		0			0
$399$	1.68478e14			0.834050
$400$		0			0
$401$		−	3.43265e14i		−	1.65324i	−0.562762	−	0.826619i	$-0.690261\pi$
$401$		−	3.43265e14i		−	1.65324i	0.562762	−	0.826619i	$-0.309739\pi$
$402$		0			0
$403$	1.93886e14	+	2.91492e14i	0.908593	+	1.36599i
$404$		0			0
$405$		−	3.52572e14i		−	1.60785i
$406$		0			0
$407$	4.36303e14			1.93650
$408$		0			0
$409$			2.66850e14i			1.15289i	0.817135	+	0.576446i	$0.195561\pi$
$409$			2.66850e14i			1.15289i	−0.817135	+	0.576446i	$0.804439\pi$
$410$		0			0
$411$		−	9.63807e13i		−	0.405378i
$412$		0			0
$413$	−2.13811e14			−0.875599
$414$		0			0
$415$	−6.00172e13			−0.239338
$416$		0			0
$417$	−3.28745e14			−1.27676
$418$		0			0
$419$	2.13925e14			0.809253			0.404626	−	0.914482i	$-0.367402\pi$
$419$	2.13925e14			0.809253			0.404626	+	0.914482i	$0.367402\pi$
$420$		0			0
$421$			1.24393e14i			0.458398i	0.973380	+	0.229199i	$0.0736107\pi$
$421$			1.24393e14i			0.458398i	−0.973380	+	0.229199i	$0.926389\pi$
$422$		0			0
$423$			1.21147e14i			0.434953i
$424$		0			0
$425$	2.88073e13			0.100777
$426$		0			0
$427$			2.92901e14i			0.998545i
$428$		0			0
$429$	2.97226e14	+	4.46854e14i	0.987578	+	1.48474i
$430$		0			0
$431$		−	3.23913e14i		−	1.04907i	−0.851389	−	0.524535i	$-0.824239\pi$
$431$		−	3.23913e14i		−	1.04907i	0.851389	−	0.524535i	$-0.175761\pi$
$432$		0			0
$433$	−2.35022e13			−0.0742036			−0.0371018	−	0.999311i	$-0.511813\pi$
$433$	−2.35022e13			−0.0742036			−0.0371018	+	0.999311i	$0.511813\pi$
$434$		0			0
$435$			4.19781e14i			1.29221i
$436$		0			0
$437$		−	2.79395e13i		−	0.0838632i
$438$		0			0
$439$	−4.02014e14			−1.17676			−0.588379	−	0.808586i	$-0.700233\pi$
$439$	−4.02014e14			−1.17676			−0.588379	+	0.808586i	$0.700233\pi$
$440$		0			0
$441$	−3.38052e12			−0.00965098
$442$		0			0
$443$	3.08170e14			0.858164			0.429082	−	0.903265i	$-0.358837\pi$
$443$	3.08170e14			0.858164			0.429082	+	0.903265i	$0.358837\pi$
$444$		0			0
$445$	−5.16800e14			−1.40392
$446$		0			0
$447$		−	2.12422e14i		−	0.562999i
$448$		0			0
$449$		−	1.99777e14i		−	0.516643i	−0.966059	−	0.258322i	$-0.916831\pi$
$449$		−	1.99777e14i		−	0.516643i	0.966059	−	0.258322i	$-0.0831694\pi$
$450$		0			0
$451$	−5.44175e14			−1.37331
$452$		0			0
$453$			5.96902e14i			1.47016i
$454$		0			0
$455$	−4.56323e14	+	3.03524e14i	−1.09701	+	0.729677i
$456$		0			0
$457$		−	4.57271e13i		−	0.107309i	−0.998560	−	0.0536543i	$-0.982913\pi$
$457$		−	4.57271e13i		−	0.107309i	0.998560	−	0.0536543i	$-0.0170869\pi$
$458$		0			0
$459$	−4.62028e13			−0.105852
$460$		0			0
$461$			3.40488e14i			0.761635i	0.924650	+	0.380817i	$0.124357\pi$
$461$			3.40488e14i			0.761635i	−0.924650	+	0.380817i	$0.875643\pi$
$462$		0			0
$463$		−	3.63954e14i		−	0.794971i	−0.917608	−	0.397486i	$-0.869883\pi$
$463$		−	3.63954e14i		−	0.794971i	0.917608	−	0.397486i	$-0.130117\pi$
$464$		0			0
$465$	1.17580e15			2.50809
$466$		0			0
$467$	1.10274e14			0.229738			0.114869	−	0.993381i	$-0.463355\pi$
$467$	1.10274e14			0.229738			0.114869	+	0.993381i	$0.463355\pi$
$468$		0			0
$469$	3.65596e14			0.743962
$470$		0			0
$471$	4.75493e13			0.0945211
$472$		0			0
$473$		−	8.85976e14i		−	1.72062i
$474$		0			0
$475$		−	2.55991e14i		−	0.485746i
$476$		0			0
$477$	−3.92704e14			−0.728139
$478$		0			0
$479$			1.66407e14i			0.301526i	0.988570	+	0.150763i	$0.0481731\pi$
$479$			1.66407e14i			0.301526i	−0.988570	+	0.150763i	$0.951827\pi$
$480$		0			0
$481$	3.99046e14	+	5.99932e14i	0.706684	+	1.06244i
$482$		0			0
$483$			8.22287e13i			0.142336i
$484$		0			0
$485$	−1.47293e15			−2.49232
$486$		0			0
$487$		−	4.21283e13i		−	0.0696891i	−0.999393	−	0.0348446i	$-0.988906\pi$
$487$		−	4.21283e13i		−	0.0696891i	0.999393	−	0.0348446i	$-0.0110936\pi$
$488$		0			0
$489$		−	7.44897e14i		−	1.20475i
$490$		0			0
$491$	−3.56448e13			−0.0563700			−0.0281850	−	0.999603i	$-0.508973\pi$
$491$	−3.56448e13			−0.0563700			−0.0281850	+	0.999603i	$0.508973\pi$
$492$		0			0
$493$	7.94914e13			0.122931
$494$		0			0
$495$	4.96881e14			0.751491
$496$		0			0
$497$	−1.42575e14			−0.210903
$498$		0			0
$499$		−	2.70054e14i		−	0.390748i	−0.980729	−	0.195374i	$-0.937408\pi$
$499$		−	2.70054e14i		−	0.390748i	0.980729	−	0.195374i	$-0.0625921\pi$
$500$		0			0
$501$			5.11716e14i			0.724305i
$502$		0			0
$503$	4.93264e14			0.683054			0.341527	−	0.939872i	$-0.389056\pi$
$503$	4.93264e14			0.683054			0.341527	+	0.939872i	$0.389056\pi$
$504$		0			0
$505$			1.19360e15i			1.61717i
$506$		0			0
$507$	−3.42595e14	+	8.17392e14i	−0.454189	+	1.08364i
$508$		0			0
$509$		−	7.49682e14i		−	0.972589i	−0.873795	−	0.486294i	$-0.838348\pi$
$509$		−	7.49682e14i		−	0.972589i	0.873795	−	0.486294i	$-0.161652\pi$
$510$		0			0
$511$	3.70548e14			0.470467
$512$		0			0
$513$			4.10573e14i			0.510205i
$514$		0			0
$515$			5.00464e14i			0.608742i
$516$		0			0
$517$	1.45670e15			1.73449
$518$		0			0
$519$	5.50235e13			0.0641398
$520$		0			0
$521$	1.28499e15			1.46654			0.733270	−	0.679938i	$-0.237993\pi$
$521$	1.28499e15			1.46654			0.733270	+	0.679938i	$0.237993\pi$
$522$		0			0
$523$	−1.71007e14			−0.191097			−0.0955486	−	0.995425i	$-0.530461\pi$
$523$	−1.71007e14			−0.191097			−0.0955486	+	0.995425i	$0.530461\pi$
$524$		0			0
$525$			7.53405e14i			0.824427i
$526$		0			0
$527$		−	2.22655e14i		−	0.238602i
$528$		0			0
$529$	−9.39173e14			−0.985688
$530$		0			0
$531$			3.20129e14i			0.329083i
$532$		0			0
$533$	−4.97707e14	−	7.48260e14i	−0.501157	−	0.753447i
$534$		0			0
$535$		−	1.68265e15i		−	1.65977i
$536$		0			0
$537$	1.02007e15			0.985764
$538$		0			0
$539$			4.06480e13i			0.0384859i
$540$		0			0
$541$		−	1.59371e15i		−	1.47851i	−0.673424	−	0.739256i	$-0.735177\pi$
$541$		−	1.59371e15i		−	1.47851i	0.673424	−	0.739256i	$-0.264823\pi$
$542$		0			0
$543$	−1.71467e15			−1.55877
$544$		0			0
$545$	2.49494e14			0.222269
$546$		0			0
$547$	−1.12365e15			−0.981073			−0.490537	−	0.871421i	$-0.663199\pi$
$547$	−1.12365e15			−0.981073			−0.490537	+	0.871421i	$0.663199\pi$
$548$		0			0
$549$	4.38546e14			0.375290
$550$		0			0
$551$		−	7.06386e14i		−	0.592527i
$552$		0			0
$553$			1.69078e14i			0.139027i
$554$		0			0
$555$	2.41997e15			1.95074
$556$		0			0
$557$			5.45822e14i			0.431368i	0.976463	+	0.215684i	$0.0691981\pi$
$557$			5.45822e14i			0.431368i	−0.976463	+	0.215684i	$0.930802\pi$
$558$		0			0
$559$	1.21825e15	−	8.10321e14i	0.943998	−	0.627903i
$560$		0			0
$561$		−	3.41328e14i		−	0.259344i
$562$		0			0
$563$	1.07491e15			0.800897			0.400448	−	0.916319i	$-0.368854\pi$
$563$	1.07491e15			0.800897			0.400448	+	0.916319i	$0.368854\pi$
$564$		0			0
$565$		−	1.49496e15i		−	1.09235i
$566$		0			0
$567$		−	1.74611e15i		−	1.25131i
$568$		0			0
$569$	−3.71485e14			−0.261110			−0.130555	−	0.991441i	$-0.541676\pi$
$569$	−3.71485e14			−0.261110			−0.130555	+	0.991441i	$0.541676\pi$
$570$		0			0
$571$	−5.89713e14			−0.406576			−0.203288	−	0.979119i	$-0.565163\pi$
$571$	−5.89713e14			−0.406576			−0.203288	+	0.979119i	$0.565163\pi$
$572$		0			0
$573$	−2.29251e15			−1.55046
$574$		0			0
$575$	1.24941e14			0.0828956
$576$		0			0
$577$		−	4.47961e14i		−	0.291590i	−0.989315	−	0.145795i	$-0.953426\pi$
$577$		−	4.47961e14i		−	0.291590i	0.989315	−	0.145795i	$-0.0465740\pi$
$578$		0			0
$579$			5.62695e14i			0.359369i
$580$		0			0
$581$	−2.97235e14			−0.186265
$582$		0			0
$583$			4.72195e15i			2.90365i
$584$		0			0
$585$	4.54452e14	+	6.83229e14i	0.274240	+	0.412296i
$586$		0			0
$587$		−	2.90281e14i		−	0.171913i	−0.996299	−	0.0859567i	$-0.972605\pi$
$587$		−	2.90281e14i		−	0.171913i	0.996299	−	0.0859567i	$-0.0273947\pi$
$588$		0			0
$589$	−1.97858e15			−1.15006
$590$		0			0
$591$			1.14879e15i			0.655406i
$592$		0			0
$593$			1.53135e15i			0.857581i	0.903404	+	0.428790i	$0.141060\pi$
$593$			1.53135e15i			0.857581i	−0.903404	+	0.428790i	$0.858940\pi$
$594$		0			0
$595$	3.48561e14			0.191618
$596$		0			0
$597$	1.45568e14			0.0785611
$598$		0			0
$599$	2.34047e14			0.124009			0.0620047	−	0.998076i	$-0.480251\pi$
$599$	2.34047e14			0.124009			0.0620047	+	0.998076i	$0.480251\pi$
$600$		0			0
$601$	−2.19750e15			−1.14319			−0.571597	−	0.820535i	$-0.693676\pi$
$601$	−2.19750e15			−1.14319			−0.571597	+	0.820535i	$0.693676\pi$
$602$		0			0
$603$		−	5.47388e14i		−	0.279608i
$604$		0			0
$605$		−	3.38057e15i		−	1.69565i
$606$		0			0
$607$	−3.02645e15			−1.49072			−0.745360	−	0.666662i	$-0.767723\pi$
$607$	−3.02645e15			−1.49072			−0.745360	+	0.666662i	$0.767723\pi$
$608$		0			0
$609$			2.07896e15i			1.00566i
$610$		0			0
$611$	1.33231e15	+	2.00301e15i	0.632963	+	0.951606i
$612$		0			0
$613$			1.31524e15i			0.613724i	0.951754	+	0.306862i	$0.0992791\pi$
$613$			1.31524e15i			0.613724i	−0.951754	+	0.306862i	$0.900721\pi$
$614$		0			0
$615$	−3.01829e15			−1.38340
$616$		0			0
$617$			7.90565e14i			0.355934i	0.984036	+	0.177967i	$0.0569520\pi$
$617$			7.90565e14i			0.355934i	−0.984036	+	0.177967i	$0.943048\pi$
$618$		0			0
$619$		−	5.75601e13i		−	0.0254579i	−0.999919	−	0.0127290i	$-0.995948\pi$
$619$		−	5.75601e13i		−	0.0254579i	0.999919	−	0.0127290i	$-0.00405186\pi$
$620$		0			0
$621$	−2.00387e14			−0.0870696
$622$		0			0
$623$	−2.55945e15			−1.09260
$624$		0			0
$625$	−2.89150e15			−1.21278
$626$		0			0
$627$	−3.03315e15			−1.25003
$628$		0			0
$629$		−	4.58256e14i		−	0.185579i
$630$		0			0
$631$		−	3.34887e15i		−	1.33271i	−0.745633	−	0.666356i	$-0.767853\pi$
$631$		−	3.34887e15i		−	1.33271i	0.745633	−	0.666356i	$-0.232147\pi$
$632$		0			0
$633$	3.95731e14			0.154767
$634$		0			0
$635$		−	4.98897e15i		−	1.91759i
$636$		0			0
$637$	−5.58925e13	+	3.71771e13i	−0.0211148	+	0.0140446i
$638$		0			0
$639$			2.13470e14i			0.0792652i
$640$		0			0
$641$	−5.29783e14			−0.193365			−0.0966827	−	0.995315i	$-0.530823\pi$
$641$	−5.29783e14			−0.193365			−0.0966827	+	0.995315i	$0.530823\pi$
$642$		0			0
$643$			1.01506e15i			0.364194i	0.983281	+	0.182097i	$0.0582885\pi$
$643$			1.01506e15i			0.364194i	−0.983281	+	0.182097i	$0.941712\pi$
$644$		0			0
$645$		−	4.91411e15i		−	1.73327i
$646$		0			0
$647$	−1.92293e14			−0.0666793			−0.0333396	−	0.999444i	$-0.510614\pi$
$647$	−1.92293e14			−0.0666793			−0.0333396	+	0.999444i	$0.510614\pi$
$648$		0			0
$649$	3.84929e15			1.31231
$650$		0			0
$651$	5.82316e15			1.95192
$652$		0			0
$653$	−2.34263e15			−0.772114			−0.386057	−	0.922475i	$-0.626163\pi$
$653$	−2.34263e15			−0.772114			−0.386057	+	0.922475i	$0.626163\pi$
$654$		0			0
$655$		−	2.93433e15i		−	0.951004i
$656$		0			0
$657$		−	5.54802e14i		−	0.176819i
$658$		0			0
$659$	−5.91488e15			−1.85386			−0.926928	−	0.375239i	$-0.877560\pi$
$659$	−5.91488e15			−1.85386			−0.926928	+	0.375239i	$0.877560\pi$
$660$		0			0
$661$		−	4.81720e15i		−	1.48486i	−0.669922	−	0.742432i	$-0.733672\pi$
$661$		−	4.81720e15i		−	1.48486i	0.669922	−	0.742432i	$-0.266328\pi$
$662$		0			0
$663$	4.69338e14	−	3.12181e14i	0.142286	−	0.0946417i
$664$		0			0
$665$		−	3.09742e15i		−	0.923594i
$666$		0			0
$667$	3.44764e14			0.101118
$668$		0			0
$669$			7.34350e15i			2.11865i
$670$		0			0
$671$		−	5.27316e15i		−	1.49657i
$672$		0			0
$673$	−1.84391e15			−0.514821			−0.257410	−	0.966302i	$-0.582869\pi$
$673$	−1.84391e15			−0.514821			−0.257410	+	0.966302i	$0.582869\pi$
$674$		0			0
$675$	−1.83601e15			−0.504318
$676$		0			0
$677$	−2.85026e15			−0.770276			−0.385138	−	0.922859i	$-0.625846\pi$
$677$	−2.85026e15			−0.770276			−0.385138	+	0.922859i	$0.625846\pi$
$678$		0			0
$679$	−7.29470e15			−1.93965
$680$		0			0
$681$		−	7.76854e15i		−	2.03250i
$682$		0			0
$683$		−	4.69918e15i		−	1.20978i	−0.796308	−	0.604892i	$-0.793216\pi$
$683$		−	4.69918e15i		−	1.20978i	0.796308	−	0.604892i	$-0.206784\pi$
$684$		0			0
$685$	−1.77193e15			−0.448899
$686$		0			0
$687$		−	1.34547e15i		−	0.335438i
$688$		0			0
$689$	−6.49285e15	+	4.31873e15i	−1.59305	+	1.05962i
$690$		0			0
$691$			2.34559e15i			0.566399i	0.959061	+	0.283199i	$0.0913958\pi$
$691$			2.34559e15i			0.566399i	−0.959061	+	0.283199i	$0.908604\pi$
$692$		0			0
$693$	2.46080e15			0.584848
$694$		0			0
$695$			6.04388e15i			1.41384i
$696$		0			0
$697$			5.71556e14i			0.131607i
$698$		0			0
$699$	4.16586e15			0.944236
$700$		0			0
$701$	2.09935e15			0.468421			0.234210	−	0.972186i	$-0.424750\pi$
$701$	2.09935e15			0.468421			0.234210	+	0.972186i	$0.424750\pi$
$702$		0			0
$703$	−4.07221e15			−0.894490
$704$		0			0
$705$	8.07965e15			1.74724
$706$		0			0
$707$			5.91127e15i			1.25856i
$708$		0			0
$709$		−	6.10587e15i		−	1.27995i	−0.768396	−	0.639975i	$-0.778945\pi$
$709$		−	6.10587e15i		−	1.27995i	0.768396	−	0.639975i	$-0.221055\pi$
$710$		0			0
$711$	2.53152e14			0.0522515
$712$		0			0
$713$		−	9.65683e14i		−	0.196265i
$714$		0			0
$715$	8.21528e15	−	5.46441e15i	1.64414	−	1.09360i
$716$		0			0
$717$			1.12667e16i			2.22046i
$718$		0			0
$719$	−5.91981e15			−1.14894			−0.574472	−	0.818524i	$-0.694793\pi$
$719$	−5.91981e15			−1.14894			−0.574472	+	0.818524i	$0.694793\pi$
$720$		0			0
$721$			2.47854e15i			0.473754i
$722$		0			0
$723$		−	3.12551e15i		−	0.588384i
$724$		0			0
$725$	3.15884e15			0.585690
$726$		0			0
$727$	−2.61053e15			−0.476748			−0.238374	−	0.971173i	$-0.576614\pi$
$727$	−2.61053e15			−0.476748			−0.238374	+	0.971173i	$0.576614\pi$
$728$		0			0
$729$	2.13955e15			0.384876
$730$		0			0
$731$	−9.30556e14			−0.164891
$732$		0			0
$733$			9.10402e15i			1.58914i	0.607175	+	0.794568i	$0.292303\pi$
$733$			9.10402e15i			1.58914i	−0.607175	+	0.794568i	$0.707697\pi$
$734$		0			0
$735$			2.25456e14i			0.0387688i
$736$		0			0
$737$	−6.58190e15			−1.11501
$738$		0			0
$739$			2.00804e15i			0.335141i	0.985860	+	0.167570i	$0.0535921\pi$
$739$			2.00804e15i			0.335141i	−0.985860	+	0.167570i	$0.946408\pi$
$740$		0			0
$741$	−2.77414e15	−	4.17068e15i	−0.456172	−	0.685815i
$742$		0			0
$743$			8.48953e15i			1.37545i	0.725971	+	0.687725i	$0.241391\pi$
$743$			8.48953e15i			1.37545i	−0.725971	+	0.687725i	$0.758609\pi$
$744$		0			0
$745$	−3.90531e15			−0.623442
$746$		0			0
$747$			4.45035e14i			0.0700052i
$748$		0			0
$749$		−	8.33331e15i		−	1.29172i
$750$		0			0
$751$	5.75001e15			0.878312			0.439156	−	0.898411i	$-0.355278\pi$
$751$	5.75001e15			0.878312			0.439156	+	0.898411i	$0.355278\pi$
$752$		0			0
$753$	2.51861e15			0.379131
$754$		0			0
$755$	1.09739e16			1.62799
$756$		0			0
$757$	4.57782e15			0.669317			0.334658	−	0.942339i	$-0.391379\pi$
$757$	4.57782e15			0.669317			0.334658	+	0.942339i	$0.391379\pi$
$758$		0			0
$759$		−	1.48038e15i		−	0.213326i
$760$		0			0
$761$		−	3.71156e15i		−	0.527158i	−0.964638	−	0.263579i	$-0.915097\pi$
$761$		−	3.71156e15i		−	0.527158i	0.964638	−	0.263579i	$-0.0849029\pi$
$762$		0			0
$763$	1.23562e15			0.172981
$764$		0			0
$765$		−	5.21883e14i		−	0.0720170i
$766$		0			0
$767$	3.52059e15	+	5.29291e15i	0.478896	+	0.719980i
$768$		0			0
$769$		−	2.37640e15i		−	0.318658i	−0.987226	−	0.159329i	$-0.949067\pi$
$769$		−	2.37640e15i		−	0.318658i	0.987226	−	0.159329i	$-0.0509331\pi$
$770$		0			0
$771$	1.10345e16			1.45866
$772$		0			0
$773$			4.48843e15i			0.584934i	0.956276	+	0.292467i	$0.0944762\pi$
$773$			4.48843e15i			0.584934i	−0.956276	+	0.292467i	$0.905524\pi$
$774$		0			0
$775$		−	8.84788e15i		−	1.13679i
$776$		0			0
$777$	1.19849e16			1.51816
$778$		0			0
$779$	5.07903e15			0.634344
$780$		0			0
$781$	2.56681e15			0.316091
$782$		0			0
$783$	−5.06633e15			−0.615182
$784$		0			0
$785$		−	8.74180e14i		−	0.104669i
$786$		0			0
$787$		−	1.79432e15i		−	0.211856i	−0.994374	−	0.105928i	$-0.966219\pi$
$787$		−	1.79432e15i		−	0.211856i	0.994374	−	0.105928i	$-0.0337812\pi$
$788$		0			0
$789$	5.06501e14			0.0589735
$790$		0			0
$791$		−	7.40378e15i		−	0.850125i
$792$		0			0
$793$	7.25079e15	−	4.82288e15i	0.821074	−	0.546140i
$794$		0			0
$795$			2.61905e16i			2.92499i
$796$		0			0
$797$	1.29926e15			0.143111			0.0715556	−	0.997437i	$-0.477204\pi$
$797$	1.29926e15			0.143111			0.0715556	+	0.997437i	$0.477204\pi$
$798$		0			0
$799$		−	1.53000e15i		−	0.166220i
$800$		0			0
$801$			3.83213e15i			0.410641i
$802$		0			0
$803$	−6.67105e15			−0.705113
$804$		0			0
$805$	1.51175e15			0.157617
$806$		0			0
$807$	1.67791e16			1.72570
$808$		0			0
$809$	−1.71565e16			−1.74065			−0.870326	−	0.492476i	$-0.836092\pi$
$809$	−1.71565e16			−1.74065			−0.870326	+	0.492476i	$0.836092\pi$
$810$		0			0
$811$			8.99602e15i			0.900401i	0.892928	+	0.450200i	$0.148647\pi$
$811$			8.99602e15i			0.900401i	−0.892928	+	0.450200i	$0.851353\pi$
$812$		0			0
$813$			2.16155e16i			2.13436i
$814$		0			0
$815$	−1.36947e16			−1.33409
$816$		0			0
$817$			8.26921e15i			0.794772i
$818$		0			0
$819$	2.25067e15	+	3.38369e15i	0.213427	+	0.320870i
$820$		0			0
$821$			1.60428e16i			1.50104i	0.660848	+	0.750520i	$0.270197\pi$
$821$			1.60428e16i			1.50104i	−0.660848	+	0.750520i	$0.729803\pi$
$822$		0			0
$823$	5.49308e15			0.507128			0.253564	−	0.967319i	$-0.418397\pi$
$823$	5.49308e15			0.507128			0.253564	+	0.967319i	$0.418397\pi$
$824$		0			0
$825$		−	1.35637e16i		−	1.23561i
$826$		0			0
$827$			8.14813e15i			0.732450i	0.930526	+	0.366225i	$0.119350\pi$
$827$			8.14813e15i			0.732450i	−0.930526	+	0.366225i	$0.880650\pi$
$828$		0			0
$829$	1.98376e16			1.75971			0.879853	−	0.475246i	$-0.157641\pi$
$829$	1.98376e16			1.75971			0.879853	+	0.475246i	$0.157641\pi$
$830$		0			0
$831$	−8.37100e15			−0.732776
$832$		0			0
$833$	4.26934e13			0.00368819
$834$		0			0
$835$	9.40775e15			0.802066
$836$		0			0
$837$			1.41907e16i			1.19403i
$838$		0			0
$839$		−	2.34908e15i		−	0.195077i	−0.995232	−	0.0975387i	$-0.968903\pi$
$839$		−	2.34908e15i		−	0.195077i	0.995232	−	0.0975387i	$-0.0310970\pi$
$840$		0			0
$841$	−3.48395e15			−0.285558
$842$		0			0
$843$			2.13445e16i			1.72677i
$844$		0			0
$845$	1.50275e16	+	6.29850e15i	1.19998	+	0.502951i
$846$		0			0
$847$		−	1.67423e16i		−	1.31964i
$848$		0			0
$849$	−2.36120e16			−1.83713
$850$		0			0
$851$		−	1.98751e15i		−	0.152650i
$852$		0			0
$853$		−	1.56183e16i		−	1.18417i	−0.805875	−	0.592086i	$-0.798305\pi$
$853$		−	1.56183e16i		−	1.18417i	0.805875	−	0.592086i	$-0.201695\pi$
$854$		0			0
$855$	−4.63761e15			−0.347121
$856$		0			0
$857$	1.24481e16			0.919834			0.459917	−	0.887962i	$-0.347879\pi$
$857$	1.24481e16			0.919834			0.459917	+	0.887962i	$0.347879\pi$
$858$		0			0
$859$	1.17701e16			0.858651			0.429326	−	0.903150i	$-0.358751\pi$
$859$	1.17701e16			0.858651			0.429326	+	0.903150i	$0.358751\pi$
$860$		0			0
$861$	−1.49481e16			−1.07663
$862$		0			0
$863$		−	4.09282e15i		−	0.291047i	−0.989355	−	0.145524i	$-0.953513\pi$
$863$		−	4.09282e15i		−	0.291047i	0.989355	−	0.145524i	$-0.0464867\pi$
$864$		0			0
$865$		−	1.01159e15i		−	0.0710259i
$866$		0			0
$867$	1.65901e16			1.15012
$868$		0			0
$869$		−	3.04395e15i		−	0.208367i
$870$		0			0
$871$	−6.01986e15	−	9.05035e15i	−0.406899	−	0.611738i
$872$		0			0
$873$			1.09220e16i			0.728993i
$874$		0			0
$875$	−6.13831e15			−0.404580
$876$		0			0
$877$		−	1.66555e16i		−	1.08408i	−0.840354	−	0.542038i	$-0.817653\pi$
$877$		−	1.66555e16i		−	1.08408i	0.840354	−	0.542038i	$-0.182347\pi$
$878$		0			0
$879$			8.13278e15i			0.522757i
$880$		0			0
$881$	2.33295e16			1.48094			0.740472	−	0.672087i	$-0.234602\pi$
$881$	2.33295e16			1.48094			0.740472	+	0.672087i	$0.234602\pi$
$882$		0			0
$883$	−5.66211e14			−0.0354972			−0.0177486	−	0.999842i	$-0.505650\pi$
$883$	−5.66211e14			−0.0354972			−0.0177486	+	0.999842i	$0.505650\pi$
$884$		0			0
$885$	2.13503e16			1.32195
$886$		0			0
$887$	−8.53270e15			−0.521803			−0.260901	−	0.965365i	$-0.584020\pi$
$887$	−8.53270e15			−0.521803			−0.260901	+	0.965365i	$0.584020\pi$
$888$		0			0
$889$		−	2.47078e16i		−	1.49236i
$890$		0			0
$891$			3.14355e16i			1.87540i
$892$		0			0
$893$	−1.35960e16			−0.801177
$894$		0			0
$895$		−	1.87538e16i		−	1.09160i
$896$		0			0
$897$	2.03558e15	−	1.35397e15i	0.117039	−	0.0778486i
$898$		0			0
$899$		−	2.44150e16i		−	1.38669i
$900$		0			0
$901$	4.95954e15			0.278263
$902$		0			0
$903$		−	2.43371e16i		−	1.34892i
$904$		0			0
$905$			3.15237e16i			1.72612i
$906$		0			0
$907$	−9.92538e15			−0.536917			−0.268458	−	0.963291i	$-0.586514\pi$
$907$	−9.92538e15			−0.536917			−0.268458	+	0.963291i	$0.586514\pi$
$908$		0			0
$909$	8.85065e15			0.473014
$910$		0			0
$911$	−1.06854e16			−0.564208			−0.282104	−	0.959384i	$-0.591032\pi$
$911$	−1.06854e16			−0.564208			−0.282104	+	0.959384i	$0.591032\pi$
$912$		0			0
$913$	5.35118e15			0.279165
$914$		0			0
$915$		−	2.92478e16i		−	1.50757i
$916$		0			0
$917$		−	1.45323e16i		−	0.740119i
$918$		0			0
$919$	6.16827e15			0.310405			0.155202	−	0.987883i	$-0.450397\pi$
$919$	6.16827e15			0.310405			0.155202	+	0.987883i	$0.450397\pi$
$920$		0			0
$921$			2.86475e15i			0.142449i
$922$		0			0
$923$	2.34762e15	+	3.52945e15i	0.115350	+	0.173420i
$924$		0			0
$925$		−	1.82102e16i		−	0.884170i
$926$		0			0
$927$	3.71100e15			0.178054
$928$		0			0
$929$			1.71984e16i			0.815460i	0.913103	+	0.407730i	$0.133680\pi$
$929$			1.71984e16i			0.815460i	−0.913103	+	0.407730i	$0.866320\pi$
$930$		0			0
$931$		−	3.79386e14i		−	0.0177770i
$932$		0			0
$933$	3.40531e16			1.57691
$934$		0			0
$935$	−6.27522e15			−0.287187
$936$		0			0
$937$	−1.38540e16			−0.626626			−0.313313	−	0.949650i	$-0.601439\pi$
$937$	−1.38540e16			−0.626626			−0.313313	+	0.949650i	$0.601439\pi$
$938$		0			0
$939$	1.45476e16			0.650323
$940$		0			0
$941$			3.01742e16i			1.33319i	0.745419	+	0.666597i	$0.232250\pi$
$941$			3.01742e16i			1.33319i	−0.745419	+	0.666597i	$0.767750\pi$
$942$		0			0
$943$			2.47891e15i			0.108255i
$944$		0			0
$945$	−2.22153e16			−0.958907
$946$		0			0
$947$			1.47817e16i			0.630665i	0.948981	+	0.315333i	$0.102116\pi$
$947$			1.47817e16i			0.630665i	−0.948981	+	0.315333i	$0.897884\pi$
$948$		0			0
$949$	−6.10140e15	−	9.17293e15i	−0.257315	−	0.386851i
$950$		0			0
$951$		−	1.22803e16i		−	0.511935i
$952$		0			0
$953$	−2.28957e16			−0.943501			−0.471750	−	0.881732i	$-0.656378\pi$
$953$	−2.28957e16			−0.943501			−0.471750	+	0.881732i	$0.656378\pi$
$954$		0			0
$955$			4.21471e16i			1.71692i
$956$		0			0
$957$		−	3.74280e16i		−	1.50723i
$958$		0			0
$959$	−8.77548e15			−0.349356
$960$		0			0
$961$	−4.29779e16			−1.69148
$962$		0			0
$963$	−1.24770e16			−0.485475
$964$		0			0
$965$	1.03450e16			0.397951
$966$		0			0
$967$			3.65782e16i			1.39116i	0.718448	+	0.695580i	$0.244853\pi$
$967$			3.65782e16i			1.39116i	−0.718448	+	0.695580i	$0.755147\pi$
$968$		0			0
$969$			3.18577e15i			0.119793i
$970$		0			0
$971$	−2.24568e16			−0.834915			−0.417458	−	0.908696i	$-0.637079\pi$
$971$	−2.24568e16			−0.834915			−0.417458	+	0.908696i	$0.637079\pi$
$972$		0			0
$973$			2.99323e16i			1.10032i
$974$		0			0
$975$	1.86506e16	−	1.24055e16i	0.677903	−	0.450909i
$976$		0			0
$977$		−	2.77848e16i		−	0.998590i	−0.866432	−	0.499295i	$-0.833592\pi$
$977$		−	2.77848e16i		−	0.998590i	0.866432	−	0.499295i	$-0.166408\pi$
$978$		0			0
$979$	4.60783e16			1.63754
$980$		0			0
$981$		−	1.85003e15i		−	0.0650127i
$982$		0			0
$983$			6.91991e15i			0.240467i	0.992746	+	0.120234i	$0.0383644\pi$
$983$			6.91991e15i			0.240467i	−0.992746	+	0.120234i	$0.961636\pi$
$984$		0			0
$985$	2.11202e16			0.725770
$986$		0			0
$987$	4.00144e16			1.35979
$988$		0			0
$989$	−4.03594e15			−0.135633
$990$		0			0
$991$	−4.49174e16			−1.49283			−0.746414	−	0.665482i	$-0.768226\pi$
$991$	−4.49174e16			−1.49283			−0.746414	+	0.665482i	$0.768226\pi$
$992$		0			0
$993$		−	1.66742e16i		−	0.548054i
$994$		0			0
$995$		−	2.67623e15i		−	0.0869954i
$996$		0			0
$997$	−4.98182e16			−1.60164			−0.800819	−	0.598907i	$-0.795602\pi$
$997$	−4.98182e16			−1.60164			−0.800819	+	0.598907i	$0.795602\pi$
$998$		0			0
$999$			2.92066e16i			0.928690i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.12.f.b.129.4		12
4.3	odd	2		13.12.b.a.12.7	yes	12
12.11	even	2		117.12.b.b.64.6		12
13.12	even	2	inner	208.12.f.b.129.3		12
52.31	even	4		169.12.a.e.1.7		12
52.47	even	4		169.12.a.e.1.6		12
52.51	odd	2		13.12.b.a.12.6	✓	12
156.155	even	2		117.12.b.b.64.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.b.a.12.6	✓	12	52.51	odd	2
13.12.b.a.12.7	yes	12	4.3	odd	2
117.12.b.b.64.6		12	12.11	even	2
117.12.b.b.64.7		12	156.155	even	2
169.12.a.e.1.6		12	52.47	even	4
169.12.a.e.1.7		12	52.31	even	4
208.12.f.b.129.3		12	13.12	even	2	inner
208.12.f.b.129.4		12	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-494.534 q^{3} +9091.88i q^{5} +45027.4i q^{7} +67417.3 q^{9} +O(q^{10})\)	\(q-494.534 q^{3} +9091.88i q^{5} +45027.4i q^{7} +67417.3 q^{9} -810639. i q^{11} +(1.11466e6 - 741417. i) q^{13} -4.49625e6i q^{15} -851428. q^{17} +7.56605e6i q^{19} -2.22676e7i q^{21} -3.69275e6 q^{23} -3.38341e7 q^{25} +5.42651e7 q^{27} -9.33625e7 q^{29} +2.61508e8i q^{31} +4.00889e8i q^{33} -4.09384e8 q^{35} +5.38221e8i q^{37} +(-5.51236e8 + 3.66656e8i) q^{39} -6.71292e8i q^{41} +1.09294e9 q^{43} +6.12950e8i q^{45} +1.79698e9i q^{47} -5.01432e7 q^{49} +4.21060e8 q^{51} -5.82497e9 q^{53} +7.37023e9 q^{55} -3.74167e9i q^{57} +4.74847e9i q^{59} +6.50495e9 q^{61} +3.03563e9i q^{63} +(6.74087e9 + 1.01343e10i) q^{65} -8.11940e9i q^{67} +1.82619e9 q^{69} +3.16640e9i q^{71} -8.22937e9i q^{73} +1.67321e10 q^{75} +3.65010e10 q^{77} +3.75500e9 q^{79} -3.87787e10 q^{81} +6.60119e9i q^{83} -7.74108e9i q^{85} +4.61710e10 q^{87} +5.68420e10i q^{89} +(3.33841e10 + 5.01902e10i) q^{91} -1.29325e11i q^{93} -6.87896e10 q^{95} +1.62006e11i q^{97} -5.46511e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 488 q^{3} + 654644 q^{9}+O(q^{10}) \) 12 * q + 488 * q^3 + 654644 * q^9	\( 12 q + 488 q^{3} + 654644 q^{9} + 3208868 q^{13} + 12198768 q^{17} - 5810592 q^{23} + 6102388 q^{25} + 52613336 q^{27} - 244463112 q^{29} + 562027560 q^{35} + 2199109744 q^{39} - 2294519976 q^{43} - 3573617796 q^{49} - 7713246552 q^{51} - 4602062760 q^{53} + 6178744976 q^{55} - 13775649944 q^{61} - 7598401512 q^{65} - 25419983328 q^{69} - 68016370832 q^{75} - 80478036048 q^{77} - 18046097296 q^{79} - 132677486692 q^{81} - 94507900752 q^{87} - 104793638664 q^{91} - 145093149648 q^{95}+O(q^{100}) \) 12 * q + 488 * q^3 + 654644 * q^9 + 3208868 * q^13 + 12198768 * q^17 - 5810592 * q^23 + 6102388 * q^25 + 52613336 * q^27 - 244463112 * q^29 + 562027560 * q^35 + 2199109744 * q^39 - 2294519976 * q^43 - 3573617796 * q^49 - 7713246552 * q^51 - 4602062760 * q^53 + 6178744976 * q^55 - 13775649944 * q^61 - 7598401512 * q^65 - 25419983328 * q^69 - 68016370832 * q^75 - 80478036048 * q^77 - 18046097296 * q^79 - 132677486692 * q^81 - 94507900752 * q^87 - 104793638664 * q^91 - 145093149648 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more