LMFDB - Embedded newform 208.12.f.b.129.9

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			129.9
Root			$85.7436i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.12.f.b.129.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+584.699 q^{3} -6975.15i q^{5} +41147.2i q^{7} +164726. q^{9} +O(q^{10})$	$q+584.699 q^{3} -6975.15i q^{5} +41147.2i q^{7} +164726. q^{9} +502259. i q^{11} +(1.13182e6 - 714936. i) q^{13} -4.07837e6i q^{15} -7.46014e6 q^{17} -9.86577e6i q^{19} +2.40587e7i q^{21} -3.09915e7 q^{23} +175380. q^{25} -7.26244e6 q^{27} -1.95223e8 q^{29} -1.15138e7i q^{31} +2.93670e8i q^{33} +2.87008e8 q^{35} -1.48877e8i q^{37} +(6.61777e8 - 4.18023e8i) q^{39} -9.79851e8i q^{41} +6.27084e6 q^{43} -1.14899e9i q^{45} +3.75374e8i q^{47} +2.84233e8 q^{49} -4.36194e9 q^{51} -9.10670e8 q^{53} +3.50333e9 q^{55} -5.76851e9i q^{57} -1.75161e9i q^{59} -8.61653e8 q^{61} +6.77803e9i q^{63} +(-4.98679e9 - 7.89465e9i) q^{65} -1.06499e10i q^{67} -1.81207e10 q^{69} -3.89999e9i q^{71} -6.08778e9i q^{73} +1.02544e8 q^{75} -2.06665e10 q^{77} -3.82480e9 q^{79} -3.34271e10 q^{81} -5.34883e10i q^{83} +5.20356e10i q^{85} -1.14147e11 q^{87} -3.76634e10i q^{89} +(2.94176e10 + 4.65714e10i) q^{91} -6.73209e9i q^{93} -6.88153e10 q^{95} +1.11844e11i q^{97} +8.27351e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	584.699			1.38920			0.694601	−	0.719395i	$-0.255581\pi$
$3$	584.699			1.38920			0.694601	+	0.719395i	$0.255581\pi$
$4$		0			0
$5$		−	6975.15i		−	0.998202i	−0.866544	−	0.499101i	$-0.833664\pi$
$5$		−	6975.15i		−	0.998202i	0.866544	−	0.499101i	$-0.166336\pi$
$6$		0			0
$7$			41147.2i			0.925340i	0.886531	+	0.462670i	$0.153108\pi$
$7$			41147.2i			0.925340i	−0.886531	+	0.462670i	$0.846892\pi$
$8$		0			0
$9$	164726.			0.929884
$10$		0			0
$11$			502259.i			0.940302i	0.882586	+	0.470151i	$0.155801\pi$
$11$			502259.i			0.940302i	−0.882586	+	0.470151i	$0.844199\pi$
$12$		0			0
$13$	1.13182e6	−	714936.i	0.845455	−	0.534046i
$13$	1.13182e6	−	714936.i	0.845455	−	0.534046i
$14$		0			0
$15$		−	4.07837e6i		−	1.38671i
$16$		0			0
$17$	−7.46014e6			−1.27432			−0.637159	−	0.770732i	$-0.719890\pi$
$17$	−7.46014e6			−1.27432			−0.637159	+	0.770732i	$0.719890\pi$
$18$		0			0
$19$		−	9.86577e6i		−	0.914084i	−0.889445	−	0.457042i	$-0.848909\pi$
$19$		−	9.86577e6i		−	0.914084i	0.889445	−	0.457042i	$-0.151091\pi$
$20$		0			0
$21$			2.40587e7i			1.28548i
$22$		0			0
$23$	−3.09915e7			−1.00401			−0.502007	−	0.864864i	$-0.667405\pi$
$23$	−3.09915e7			−1.00401			−0.502007	+	0.864864i	$0.667405\pi$
$24$		0			0
$25$	175380.			0.00359177
$26$		0			0
$27$	−7.26244e6			−0.0974051
$28$		0			0
$29$	−1.95223e8			−1.76743			−0.883715	−	0.468025i	$-0.844966\pi$
$29$	−1.95223e8			−1.76743			−0.883715	+	0.468025i	$0.844966\pi$
$30$		0			0
$31$		−	1.15138e7i		−	0.0722317i	−0.999348	−	0.0361159i	$-0.988501\pi$
$31$		−	1.15138e7i		−	0.0722317i	0.999348	−	0.0361159i	$-0.0114985\pi$
$32$		0			0
$33$			2.93670e8i			1.30627i
$34$		0			0
$35$	2.87008e8			0.923677
$36$		0			0
$37$		−	1.48877e8i		−	0.352954i	−0.984305	−	0.176477i	$-0.943530\pi$
$37$		−	1.48877e8i		−	0.352954i	0.984305	−	0.176477i	$-0.0564701\pi$
$38$		0			0
$39$	6.61777e8	−	4.18023e8i	1.17451	−	0.741898i
$40$		0			0
$41$		−	9.79851e8i		−	1.32084i	−0.750898	−	0.660418i	$-0.770379\pi$
$41$		−	9.79851e8i		−	1.32084i	0.750898	−	0.660418i	$-0.229621\pi$
$42$		0			0
$43$	6.27084e6			0.00650503			0.00325251	−	0.999995i	$-0.498965\pi$
$43$	6.27084e6			0.00650503			0.00325251	+	0.999995i	$0.498965\pi$
$44$		0			0
$45$		−	1.14899e9i		−	0.928213i
$46$		0			0
$47$			3.75374e8i			0.238741i	0.992850	+	0.119370i	$0.0380876\pi$
$47$			3.75374e8i			0.238741i	−0.992850	+	0.119370i	$0.961912\pi$
$48$		0			0
$49$	2.84233e8			0.143746
$50$		0			0
$51$	−4.36194e9			−1.77029
$52$		0			0
$53$	−9.10670e8			−0.299119			−0.149559	−	0.988753i	$-0.547786\pi$
$53$	−9.10670e8			−0.299119			−0.149559	+	0.988753i	$0.547786\pi$
$54$		0			0
$55$	3.50333e9			0.938612
$56$		0			0
$57$		−	5.76851e9i		−	1.26985i
$58$		0			0
$59$		−	1.75161e9i		−	0.318972i	−0.987200	−	0.159486i	$-0.949016\pi$
$59$		−	1.75161e9i		−	0.318972i	0.987200	−	0.159486i	$-0.0509836\pi$
$60$		0			0
$61$	−8.61653e8			−0.130623			−0.0653113	−	0.997865i	$-0.520804\pi$
$61$	−8.61653e8			−0.130623			−0.0653113	+	0.997865i	$0.520804\pi$
$62$		0			0
$63$			6.77803e9i			0.860459i
$64$		0			0
$65$	−4.98679e9	−	7.89465e9i	−0.533086	−	0.843936i
$66$		0			0
$67$		−	1.06499e10i		−	0.963685i	−0.876258	−	0.481842i	$-0.839968\pi$
$67$		−	1.06499e10i		−	0.963685i	0.876258	−	0.481842i	$-0.160032\pi$
$68$		0			0
$69$	−1.81207e10			−1.39478
$70$		0			0
$71$		−	3.89999e9i		−	0.256533i	−0.991740	−	0.128266i	$-0.959059\pi$
$71$		−	3.89999e9i		−	0.256533i	0.991740	−	0.128266i	$-0.0409412\pi$
$72$		0			0
$73$		−	6.08778e9i		−	0.343703i	−0.985123	−	0.171851i	$-0.945025\pi$
$73$		−	6.08778e9i		−	0.343703i	0.985123	−	0.171851i	$-0.0549749\pi$
$74$		0			0
$75$	1.02544e8			0.00498970
$76$		0			0
$77$	−2.06665e10			−0.870099
$78$		0			0
$79$	−3.82480e9			−0.139849			−0.0699245	−	0.997552i	$-0.522276\pi$
$79$	−3.82480e9			−0.139849			−0.0699245	+	0.997552i	$0.522276\pi$
$80$		0			0
$81$	−3.34271e10			−1.06520
$82$		0			0
$83$		−	5.34883e10i		−	1.49049i	−0.666791	−	0.745245i	$-0.732333\pi$
$83$		−	5.34883e10i		−	1.49049i	0.666791	−	0.745245i	$-0.267667\pi$
$84$		0			0
$85$			5.20356e10i			1.27203i
$86$		0			0
$87$	−1.14147e11			−2.45532
$88$		0			0
$89$		−	3.76634e10i		−	0.714949i	−0.933923	−	0.357474i	$-0.883638\pi$
$89$		−	3.76634e10i		−	0.714949i	0.933923	−	0.357474i	$-0.116362\pi$
$90$		0			0
$91$	2.94176e10	+	4.65714e10i	0.494174	+	0.782334i
$92$		0			0
$93$		−	6.73209e9i		−	0.100344i
$94$		0			0
$95$	−6.88153e10			−0.912441
$96$		0			0
$97$			1.11844e11i			1.32242i	0.750200	+	0.661211i	$0.229957\pi$
$97$			1.11844e11i			1.32242i	−0.750200	+	0.661211i	$0.770043\pi$
$98$		0			0
$99$			8.27351e10i			0.874372i
$100$		0			0
$101$	7.42332e9			0.0702798			0.0351399	−	0.999382i	$-0.488812\pi$
$101$	7.42332e9			0.0702798			0.0351399	+	0.999382i	$0.488812\pi$
$102$		0			0
$103$	3.31079e10			0.281402			0.140701	−	0.990052i	$-0.455064\pi$
$103$	3.31079e10			0.281402			0.140701	+	0.990052i	$0.455064\pi$
$104$		0			0
$105$	1.67813e11			1.28317
$106$		0			0
$107$	−6.53250e10			−0.450265			−0.225133	−	0.974328i	$-0.572282\pi$
$107$	−6.53250e10			−0.450265			−0.225133	+	0.974328i	$0.572282\pi$
$108$		0			0
$109$		−	2.02213e11i		−	1.25882i	−0.777074	−	0.629409i	$-0.783297\pi$
$109$		−	2.02213e11i		−	1.25882i	0.777074	−	0.629409i	$-0.216703\pi$
$110$		0			0
$111$		−	8.70483e10i		−	0.490325i
$112$		0			0
$113$	−2.27423e11			−1.16119			−0.580593	−	0.814194i	$-0.697179\pi$
$113$	−2.27423e11			−1.16119			−0.580593	+	0.814194i	$0.697179\pi$
$114$		0			0
$115$			2.16170e11i			1.00221i
$116$		0			0
$117$	1.86441e11	−	1.17769e11i	0.786176	−	0.496601i
$118$		0			0
$119$		−	3.06964e11i		−	1.17918i
$120$		0			0
$121$	3.30481e10			0.115831
$122$		0			0
$123$		−	5.72918e11i		−	1.83491i
$124$		0			0
$125$		−	3.41807e11i		−	1.00179i
$126$		0			0
$127$	−1.72065e11			−0.462139			−0.231070	−	0.972937i	$-0.574223\pi$
$127$	−1.72065e11			−0.462139			−0.231070	+	0.972937i	$0.574223\pi$
$128$		0			0
$129$	3.66655e9			0.00903680
$130$		0			0
$131$	6.28677e11			1.42376			0.711878	−	0.702303i	$-0.247845\pi$
$131$	6.28677e11			1.42376			0.711878	+	0.702303i	$0.247845\pi$
$132$		0			0
$133$	4.05949e11			0.845839
$134$		0			0
$135$			5.06566e10i			0.0972300i
$136$		0			0
$137$			6.09740e11i			1.07940i	0.841858	+	0.539699i	$0.181462\pi$
$137$			6.09740e11i			1.07940i	−0.841858	+	0.539699i	$0.818538\pi$
$138$		0			0
$139$	1.62064e11			0.264915			0.132457	−	0.991189i	$-0.457713\pi$
$139$	1.62064e11			0.264915			0.132457	+	0.991189i	$0.457713\pi$
$140$		0			0
$141$			2.19481e11i			0.331659i
$142$		0			0
$143$	3.59083e11	+	5.68469e11i	0.502165	+	0.794984i
$144$		0			0
$145$			1.36171e12i			1.76425i
$146$		0			0
$147$	1.66191e11			0.199692
$148$		0			0
$149$		−	5.51858e11i		−	0.615606i	−0.951450	−	0.307803i	$-0.900406\pi$
$149$		−	5.51858e11i		−	0.615606i	0.951450	−	0.307803i	$-0.0995938\pi$
$150$		0			0
$151$			5.14744e11i			0.533603i	0.963751	+	0.266802i	$0.0859668\pi$
$151$			5.14744e11i			0.533603i	−0.963751	+	0.266802i	$0.914033\pi$
$152$		0			0
$153$	−1.22888e12			−1.18497
$154$		0			0
$155$	−8.03102e10			−0.0721019
$156$		0			0
$157$	−1.32671e12			−1.11001			−0.555007	−	0.831846i	$-0.687284\pi$
$157$	−1.32671e12			−1.11001			−0.555007	+	0.831846i	$0.687284\pi$
$158$		0			0
$159$	−5.32468e11			−0.415536
$160$		0			0
$161$		−	1.27521e12i		−	0.929054i
$162$		0			0
$163$			9.87427e11i			0.672161i	0.941833	+	0.336081i	$0.109101\pi$
$163$			9.87427e11i			0.672161i	−0.941833	+	0.336081i	$0.890899\pi$
$164$		0			0
$165$	2.04839e12			1.30392
$166$		0			0
$167$			2.47608e12i			1.47511i	0.675289	+	0.737554i	$0.264019\pi$
$167$			2.47608e12i			1.47511i	−0.675289	+	0.737554i	$0.735981\pi$
$168$		0			0
$169$	7.69894e11	−	1.61836e12i	0.429590	−	0.903024i
$170$		0			0
$171$		−	1.62515e12i		−	0.849992i
$172$		0			0
$173$	−1.64885e12			−0.808961			−0.404480	−	0.914547i	$-0.632548\pi$
$173$	−1.64885e12			−0.808961			−0.404480	+	0.914547i	$0.632548\pi$
$174$		0			0
$175$			7.21638e9i			0.00332361i
$176$		0			0
$177$		−	1.02417e12i		−	0.443116i
$178$		0			0
$179$	−3.06710e12			−1.24749			−0.623745	−	0.781628i	$-0.714390\pi$
$179$	−3.06710e12			−1.24749			−0.623745	+	0.781628i	$0.714390\pi$
$180$		0			0
$181$	−4.80558e12			−1.83871			−0.919356	−	0.393426i	$-0.871290\pi$
$181$	−4.80558e12			−1.83871			−0.919356	+	0.393426i	$0.871290\pi$
$182$		0			0
$183$	−5.03808e11			−0.181461
$184$		0			0
$185$	−1.03844e12			−0.352320
$186$		0			0
$187$		−	3.74692e12i		−	1.19824i
$188$		0			0
$189$		−	2.98829e11i		−	0.0901328i
$190$		0			0
$191$	−5.15032e12			−1.46606			−0.733028	−	0.680198i	$-0.761894\pi$
$191$	−5.15032e12			−1.46606			−0.733028	+	0.680198i	$0.761894\pi$
$192$		0			0
$193$			6.22855e12i			1.67426i	0.547007	+	0.837128i	$0.315767\pi$
$193$			6.22855e12i			1.67426i	−0.547007	+	0.837128i	$0.684233\pi$
$194$		0			0
$195$	−2.91577e12	−	4.61600e12i	−0.740565	−	1.17240i
$196$		0			0
$197$		−	6.41855e12i		−	1.54125i	−0.637290	−	0.770624i	$-0.719944\pi$
$197$		−	6.41855e12i		−	1.54125i	0.637290	−	0.770624i	$-0.280056\pi$
$198$		0			0
$199$	2.10530e12			0.478213			0.239107	−	0.970993i	$-0.423145\pi$
$199$	2.10530e12			0.478213			0.239107	+	0.970993i	$0.423145\pi$
$200$		0			0
$201$		−	6.22700e12i		−	1.33875i
$202$		0			0
$203$		−	8.03289e12i		−	1.63547i
$204$		0			0
$205$	−6.83461e12			−1.31846
$206$		0			0
$207$	−5.10511e12			−0.933616
$208$		0			0
$209$	4.95517e12			0.859516
$210$		0			0
$211$	−2.19678e12			−0.361604			−0.180802	−	0.983520i	$-0.557869\pi$
$211$	−2.19678e12			−0.361604			−0.180802	+	0.983520i	$0.557869\pi$
$212$		0			0
$213$		−	2.28032e12i		−	0.356376i
$214$		0			0
$215$		−	4.37401e10i		−	0.00649333i
$216$		0			0
$217$	4.73759e11			0.0668389
$218$		0			0
$219$		−	3.55952e12i		−	0.477473i
$220$		0			0
$221$	−8.44357e12	+	5.33352e12i	−1.07738	+	0.680544i
$222$		0			0
$223$		−	1.63090e13i		−	1.98038i	−0.139714	−	0.990192i	$-0.544618\pi$
$223$		−	1.63090e13i		−	1.98038i	0.139714	−	0.990192i	$-0.455382\pi$
$224$		0			0
$225$	2.88896e10			0.00333993
$226$		0			0
$227$			9.59550e12i			1.05664i	0.849046	+	0.528318i	$0.177177\pi$
$227$			9.59550e12i			1.05664i	−0.849046	+	0.528318i	$0.822823\pi$
$228$		0			0
$229$		−	4.08570e12i		−	0.428717i	−0.976755	−	0.214359i	$-0.931234\pi$
$229$		−	4.08570e12i		−	0.428717i	0.976755	−	0.214359i	$-0.0687661\pi$
$230$		0			0
$231$	−1.20837e13			−1.20874
$232$		0			0
$233$	−1.65283e13			−1.57678			−0.788391	−	0.615174i	$-0.789086\pi$
$233$	−1.65283e13			−1.57678			−0.788391	+	0.615174i	$0.789086\pi$
$234$		0			0
$235$	2.61829e12			0.238312
$236$		0			0
$237$	−2.23636e12			−0.194279
$238$		0			0
$239$			8.34003e12i			0.691798i	0.938272	+	0.345899i	$0.112426\pi$
$239$			8.34003e12i			0.691798i	−0.938272	+	0.345899i	$0.887574\pi$
$240$		0			0
$241$			4.72462e12i			0.374346i	0.982327	+	0.187173i	$0.0599325\pi$
$241$			4.72462e12i			0.374346i	−0.982327	+	0.187173i	$0.940068\pi$
$242$		0			0
$243$	−1.82583e13			−1.38237
$244$		0			0
$245$		−	1.98257e12i		−	0.143488i
$246$		0			0
$247$	−7.05339e12	−	1.11663e13i	−0.488163	−	0.772817i
$248$		0			0
$249$		−	3.12745e13i		−	2.07059i
$250$		0			0
$251$	5.12472e11			0.0324687			0.0162343	−	0.999868i	$-0.494832\pi$
$251$	5.12472e11			0.0324687			0.0162343	+	0.999868i	$0.494832\pi$
$252$		0			0
$253$		−	1.55657e13i		−	0.944076i
$254$		0			0
$255$			3.04252e13i			1.76710i
$256$		0			0
$257$	3.04623e13			1.69484			0.847422	−	0.530919i	$-0.178153\pi$
$257$	3.04623e13			1.69484			0.847422	+	0.530919i	$0.178153\pi$
$258$		0			0
$259$	6.12588e12			0.326602
$260$		0			0
$261$	−3.21584e13			−1.64351
$262$		0			0
$263$	−1.09841e13			−0.538280			−0.269140	−	0.963101i	$-0.586739\pi$
$263$	−1.09841e13			−0.538280			−0.269140	+	0.963101i	$0.586739\pi$
$264$		0			0
$265$			6.35206e12i			0.298581i
$266$		0			0
$267$		−	2.20218e13i		−	0.993209i
$268$		0			0
$269$	−2.33921e12			−0.101258			−0.0506292	−	0.998718i	$-0.516123\pi$
$269$	−2.33921e12			−0.101258			−0.0506292	+	0.998718i	$0.516123\pi$
$270$		0			0
$271$		−	1.85439e13i		−	0.770673i	−0.922776	−	0.385336i	$-0.874085\pi$
$271$		−	1.85439e13i		−	0.770673i	0.922776	−	0.385336i	$-0.125915\pi$
$272$		0			0
$273$	1.72005e13	+	2.72303e13i	0.686508	+	1.08682i
$274$		0			0
$275$			8.80858e10i			0.00337735i
$276$		0			0
$277$	−3.72493e13			−1.37240			−0.686199	−	0.727414i	$-0.740722\pi$
$277$	−3.72493e13			−1.37240			−0.686199	+	0.727414i	$0.740722\pi$
$278$		0			0
$279$		−	1.89662e12i		−	0.0671671i
$280$		0			0
$281$			1.88208e13i			0.640844i	0.947275	+	0.320422i	$0.103825\pi$
$281$			1.88208e13i			0.640844i	−0.947275	+	0.320422i	$0.896175\pi$
$282$		0			0
$283$	4.20998e13			1.37865			0.689326	−	0.724452i	$-0.257907\pi$
$283$	4.20998e13			1.37865			0.689326	+	0.724452i	$0.257907\pi$
$284$		0			0
$285$	−4.02362e13			−1.26757
$286$		0			0
$287$	4.03182e13			1.22222
$288$		0			0
$289$	2.13817e13			0.623886
$290$		0			0
$291$			6.53954e13i			1.83711i
$292$		0			0
$293$			1.79710e13i			0.486183i	0.970003	+	0.243092i	$0.0781616\pi$
$293$			1.79710e13i			0.486183i	−0.970003	+	0.243092i	$0.921838\pi$
$294$		0			0
$295$	−1.22178e13			−0.318398
$296$		0			0
$297$		−	3.64762e12i		−	0.0915902i
$298$		0			0
$299$	−3.50770e13	+	2.21569e13i	−0.848849	+	0.536190i
$300$		0			0
$301$			2.58028e11i			0.00601936i
$302$		0			0
$303$	4.34041e12			0.0976329
$304$		0			0
$305$			6.01016e12i			0.130388i
$306$		0			0
$307$		−	3.65709e13i		−	0.765375i	−0.923878	−	0.382687i	$-0.874999\pi$
$307$		−	3.65709e13i		−	0.765375i	0.923878	−	0.382687i	$-0.125001\pi$
$308$		0			0
$309$	1.93582e13			0.390924
$310$		0			0
$311$	1.93821e12			0.0377763			0.0188881	−	0.999822i	$-0.493987\pi$
$311$	1.93821e12			0.0377763			0.0188881	+	0.999822i	$0.493987\pi$
$312$		0			0
$313$	5.34605e13			1.00586			0.502932	−	0.864326i	$-0.332255\pi$
$313$	5.34605e13			1.00586			0.502932	+	0.864326i	$0.332255\pi$
$314$		0			0
$315$	4.72778e13			0.858912
$316$		0			0
$317$		−	9.39479e13i		−	1.64839i	−0.566302	−	0.824197i	$-0.691627\pi$
$317$		−	9.39479e13i		−	1.64839i	0.566302	−	0.824197i	$-0.308373\pi$
$318$		0			0
$319$		−	9.80525e13i		−	1.66192i
$320$		0			0
$321$	−3.81955e13			−0.625510
$322$		0			0
$323$			7.36000e13i			1.16483i
$324$		0			0
$325$	1.98499e11	−	1.25385e11i	0.00303668	−	0.00191817i
$326$		0			0
$327$		−	1.18234e14i		−	1.74875i
$328$		0			0
$329$	−1.54456e13			−0.220916
$330$		0			0
$331$		−	3.47342e13i		−	0.480511i	−0.970710	−	0.240256i	$-0.922769\pi$
$331$		−	3.47342e13i		−	0.480511i	0.970710	−	0.240256i	$-0.0772312\pi$
$332$		0			0
$333$		−	2.45239e13i		−	0.328206i
$334$		0			0
$335$	−7.42848e13			−0.961953
$336$		0			0
$337$	1.05870e14			1.32681			0.663404	−	0.748261i	$-0.269111\pi$
$337$	1.05870e14			1.32681			0.663404	+	0.748261i	$0.269111\pi$
$338$		0			0
$339$	−1.32974e14			−1.61312
$340$		0			0
$341$	5.78288e12			0.0679197
$342$		0			0
$343$			9.30569e13i			1.05835i
$344$		0			0
$345$			1.26395e14i			1.39227i
$346$		0			0
$347$	−5.93660e13			−0.633470			−0.316735	−	0.948514i	$-0.602587\pi$
$347$	−5.93660e13			−0.633470			−0.316735	+	0.948514i	$0.602587\pi$
$348$		0			0
$349$		−	6.40637e13i		−	0.662327i	−0.943573	−	0.331163i	$-0.892559\pi$
$349$		−	6.40637e13i		−	0.662327i	0.943573	−	0.331163i	$-0.107441\pi$
$350$		0			0
$351$	−8.21981e12	+	5.19218e12i	−0.0823517	+	0.0520188i
$352$		0			0
$353$			1.64500e14i			1.59736i	0.601753	+	0.798682i	$0.294469\pi$
$353$			1.64500e14i			1.59736i	−0.601753	+	0.798682i	$0.705531\pi$
$354$		0			0
$355$	−2.72030e13			−0.256072
$356$		0			0
$357$		−	1.79482e14i		−	1.63812i
$358$		0			0
$359$			2.22544e14i			1.96968i	0.173456	+	0.984842i	$0.444506\pi$
$359$			2.22544e14i			1.96968i	−0.173456	+	0.984842i	$0.555494\pi$
$360$		0			0
$361$	1.91568e13			0.164450
$362$		0			0
$363$	1.93232e13			0.160913
$364$		0			0
$365$	−4.24632e13			−0.343085
$366$		0			0
$367$	1.20237e14			0.942703			0.471352	−	0.881945i	$-0.343766\pi$
$367$	1.20237e14			0.942703			0.471352	+	0.881945i	$0.343766\pi$
$368$		0			0
$369$		−	1.61407e14i		−	1.22822i
$370$		0			0
$371$		−	3.74715e13i		−	0.276786i
$372$		0			0
$373$	3.19771e13			0.229319			0.114660	−	0.993405i	$-0.463422\pi$
$373$	3.19771e13			0.229319			0.114660	+	0.993405i	$0.463422\pi$
$374$		0			0
$375$		−	1.99854e14i		−	1.39169i
$376$		0			0
$377$	−2.20958e14	+	1.39572e14i	−1.49428	+	0.943889i
$378$		0			0
$379$		−	1.65062e14i		−	1.08425i	−0.840297	−	0.542126i	$-0.817619\pi$
$379$		−	1.65062e14i		−	1.08425i	0.840297	−	0.542126i	$-0.182381\pi$
$380$		0			0
$381$	−1.00606e14			−0.642005
$382$		0			0
$383$			2.85775e13i			0.177187i	0.996068	+	0.0885933i	$0.0282372\pi$
$383$			2.85775e13i			0.177187i	−0.996068	+	0.0885933i	$0.971763\pi$
$384$		0			0
$385$			1.44152e14i			0.868535i
$386$		0			0
$387$	1.03297e12			0.00604892
$388$		0			0
$389$	−1.12606e14			−0.640970			−0.320485	−	0.947254i	$-0.603846\pi$
$389$	−1.12606e14			−0.640970			−0.320485	+	0.947254i	$0.603846\pi$
$390$		0			0
$391$	2.31201e14			1.27943
$392$		0			0
$393$	3.67587e14			1.97789
$394$		0			0
$395$			2.66786e13i			0.139598i
$396$		0			0
$397$			1.21589e14i			0.618792i	0.950933	+	0.309396i	$0.100127\pi$
$397$			1.21589e14i			0.618792i	−0.950933	+	0.309396i	$0.899873\pi$
$398$		0			0
$399$	2.37358e14			1.17504
$400$		0			0
$401$		−	1.10050e14i		−	0.530022i	−0.964245	−	0.265011i	$-0.914624\pi$
$401$		−	1.10050e14i		−	0.530022i	0.964245	−	0.265011i	$-0.0853756\pi$
$402$		0			0
$403$	−8.23160e12	−	1.30316e13i	−0.0385751	−	0.0610687i
$404$		0			0
$405$			2.33159e14i			1.06328i
$406$		0			0
$407$	7.47747e13			0.331884
$408$		0			0
$409$		−	7.76040e13i		−	0.335279i	−0.985848	−	0.167639i	$-0.946386\pi$
$409$		−	7.76040e13i		−	0.335279i	0.985848	−	0.167639i	$-0.0536144\pi$
$410$		0			0
$411$			3.56515e14i			1.49950i
$412$		0			0
$413$	7.20740e13			0.295157
$414$		0			0
$415$	−3.73089e14			−1.48781
$416$		0			0
$417$	9.47589e13			0.368020
$418$		0			0
$419$	3.99718e14			1.51208			0.756042	−	0.654523i	$-0.227130\pi$
$419$	3.99718e14			1.51208			0.756042	+	0.654523i	$0.227130\pi$
$420$		0			0
$421$		−	2.77154e14i		−	1.02134i	−0.859777	−	0.510669i	$-0.829398\pi$
$421$		−	2.77154e14i		−	1.02134i	0.859777	−	0.510669i	$-0.170602\pi$
$422$		0			0
$423$			6.18340e13i			0.222001i
$424$		0			0
$425$	−1.30836e12			−0.00457706
$426$		0			0
$427$		−	3.54546e13i		−	0.120870i
$428$		0			0
$429$	2.09955e14	+	3.32383e14i	0.697609	+	1.10439i
$430$		0			0
$431$			2.85786e14i			0.925584i	0.886467	+	0.462792i	$0.153152\pi$
$431$			2.85786e14i			0.925584i	−0.886467	+	0.462792i	$0.846848\pi$
$432$		0			0
$433$	5.24142e14			1.65488			0.827438	−	0.561557i	$-0.189798\pi$
$433$	5.24142e14			1.65488			0.827438	+	0.561557i	$0.189798\pi$
$434$		0			0
$435$			7.96191e14i			2.45091i
$436$		0			0
$437$			3.05755e14i			0.917753i
$438$		0			0
$439$	4.13032e13			0.120901			0.0604503	−	0.998171i	$-0.480746\pi$
$439$	4.13032e13			0.120901			0.0604503	+	0.998171i	$0.480746\pi$
$440$		0			0
$441$	4.68206e13			0.133667
$442$		0			0
$443$	2.63437e14			0.733596			0.366798	−	0.930301i	$-0.380454\pi$
$443$	2.63437e14			0.733596			0.366798	+	0.930301i	$0.380454\pi$
$444$		0			0
$445$	−2.62708e14			−0.713664
$446$		0			0
$447$		−	3.22671e14i		−	0.855202i
$448$		0			0
$449$			1.58354e14i			0.409520i	0.978812	+	0.204760i	$0.0656413\pi$
$449$			1.58354e14i			0.409520i	−0.978812	+	0.204760i	$0.934359\pi$
$450$		0			0
$451$	4.92139e14			1.24199
$452$		0			0
$453$			3.00971e14i			0.741283i
$454$		0			0
$455$	3.24843e14	−	2.05192e14i	0.780927	−	0.493286i
$456$		0			0
$457$		−	4.54600e14i		−	1.06682i	−0.845857	−	0.533409i	$-0.820911\pi$
$457$		−	4.54600e14i		−	1.06682i	0.845857	−	0.533409i	$-0.179089\pi$
$458$		0			0
$459$	5.41788e13			0.124125
$460$		0			0
$461$		−	3.25732e14i		−	0.728627i	−0.931276	−	0.364314i	$-0.881304\pi$
$461$		−	3.25732e14i		−	0.728627i	0.931276	−	0.364314i	$-0.118696\pi$
$462$		0			0
$463$			8.44698e14i			1.84504i	0.385948	+	0.922521i	$0.373874\pi$
$463$			8.44698e14i			1.84504i	−0.385948	+	0.922521i	$0.626126\pi$
$464$		0			0
$465$	−4.69573e13			−0.100164
$466$		0			0
$467$	8.12404e13			0.169250			0.0846252	−	0.996413i	$-0.473031\pi$
$467$	8.12404e13			0.169250			0.0846252	+	0.996413i	$0.473031\pi$
$468$		0			0
$469$	4.38215e14			0.891736
$470$		0			0
$471$	−7.75727e14			−1.54203
$472$		0			0
$473$			3.14958e12i			0.00611669i
$474$		0			0
$475$		−	1.73025e12i		−	0.00328318i
$476$		0			0
$477$	−1.50011e14			−0.278146
$478$		0			0
$479$		−	5.45937e14i		−	0.989229i	−0.869112	−	0.494615i	$-0.835309\pi$
$479$		−	5.45937e14i		−	0.989229i	0.869112	−	0.494615i	$-0.164691\pi$
$480$		0			0
$481$	−1.06438e14	−	1.68503e14i	−0.188494	−	0.298407i
$482$		0			0
$483$		−	7.45617e14i		−	1.29064i
$484$		0			0
$485$	7.80132e14			1.32004
$486$		0			0
$487$			1.00389e14i			0.166065i	0.996547	+	0.0830323i	$0.0264605\pi$
$487$			1.00389e14i			0.166065i	−0.996547	+	0.0830323i	$0.973540\pi$
$488$		0			0
$489$			5.77348e14i			0.933768i
$490$		0			0
$491$	8.79448e14			1.39079			0.695396	−	0.718627i	$-0.255229\pi$
$491$	8.79448e14			1.39079			0.695396	+	0.718627i	$0.255229\pi$
$492$		0			0
$493$	1.45639e15			2.25227
$494$		0			0
$495$	5.77090e14			0.872801
$496$		0			0
$497$	1.60474e14			0.237380
$498$		0			0
$499$			1.77108e14i			0.256263i	0.991757	+	0.128132i	$0.0408980\pi$
$499$			1.77108e14i			0.256263i	−0.991757	+	0.128132i	$0.959102\pi$
$500$		0			0
$501$			1.44776e15i			2.04922i
$502$		0			0
$503$	−1.60263e14			−0.221927			−0.110963	−	0.993824i	$-0.535394\pi$
$503$	−1.60263e14			−0.221927			−0.110963	+	0.993824i	$0.535394\pi$
$504$		0			0
$505$		−	5.17788e13i		−	0.0701535i
$506$		0			0
$507$	4.50156e14	−	9.46256e14i	0.596787	−	1.25448i
$508$		0			0
$509$		−	1.43136e14i		−	0.185695i	−0.995680	−	0.0928475i	$-0.970403\pi$
$509$		−	1.43136e14i		−	0.185695i	0.995680	−	0.0928475i	$-0.0295969\pi$
$510$		0			0
$511$	2.50495e14			0.318042
$512$		0			0
$513$			7.16495e13i			0.0890365i
$514$		0			0
$515$		−	2.30933e14i		−	0.280896i
$516$		0			0
$517$	−1.88535e14			−0.224488
$518$		0			0
$519$	−9.64081e14			−1.12381
$520$		0			0
$521$	−4.38313e14			−0.500238			−0.250119	−	0.968215i	$-0.580470\pi$
$521$	−4.38313e14			−0.500238			−0.250119	+	0.968215i	$0.580470\pi$
$522$		0			0
$523$	−5.54155e14			−0.619258			−0.309629	−	0.950857i	$-0.600205\pi$
$523$	−5.54155e14			−0.619258			−0.309629	+	0.950857i	$0.600205\pi$
$524$		0			0
$525$			4.21941e12i			0.00461717i
$526$		0			0
$527$			8.58942e13i			0.0920462i
$528$		0			0
$529$	7.66378e12			0.00804334
$530$		0			0
$531$		−	2.88537e14i		−	0.296607i
$532$		0			0
$533$	−7.00531e14	−	1.10902e15i	−0.705387	−	1.11671i
$534$		0			0
$535$			4.55652e14i			0.449456i
$536$		0			0
$537$	−1.79333e15			−1.73302
$538$		0			0
$539$			1.42758e14i			0.135165i
$540$		0			0
$541$		−	2.25293e14i		−	0.209008i	−0.994524	−	0.104504i	$-0.966675\pi$
$541$		−	2.25293e14i		−	0.209008i	0.994524	−	0.104504i	$-0.0333254\pi$
$542$		0			0
$543$	−2.80982e15			−2.55435
$544$		0			0
$545$	−1.41047e15			−1.25656
$546$		0			0
$547$	−1.58459e15			−1.38352			−0.691761	−	0.722127i	$-0.743165\pi$
$547$	−1.58459e15			−1.38352			−0.691761	+	0.722127i	$0.743165\pi$
$548$		0			0
$549$	−1.41937e14			−0.121464
$550$		0			0
$551$			1.92603e15i			1.61558i
$552$		0			0
$553$		−	1.57380e14i		−	0.129408i
$554$		0			0
$555$	−6.07175e14			−0.489443
$556$		0			0
$557$		−	1.23568e15i		−	0.976569i	−0.872684	−	0.488285i	$-0.837623\pi$
$557$		−	1.23568e15i		−	0.976569i	0.872684	−	0.488285i	$-0.162377\pi$
$558$		0			0
$559$	7.09749e12	−	4.48325e12i	0.00549971	−	0.00347398i
$560$		0			0
$561$		−	2.19082e15i		−	1.66460i
$562$		0			0
$563$	1.43646e15			1.07028			0.535141	−	0.844763i	$-0.320258\pi$
$563$	1.43646e15			1.07028			0.535141	+	0.844763i	$0.320258\pi$
$564$		0			0
$565$			1.58631e15i			1.15910i
$566$		0			0
$567$		−	1.37543e15i		−	0.985672i
$568$		0			0
$569$	−8.37393e14			−0.588589			−0.294294	−	0.955715i	$-0.595085\pi$
$569$	−8.37393e14			−0.588589			−0.294294	+	0.955715i	$0.595085\pi$
$570$		0			0
$571$	−7.46251e13			−0.0514501			−0.0257251	−	0.999669i	$-0.508189\pi$
$571$	−7.46251e13			−0.0514501			−0.0257251	+	0.999669i	$0.508189\pi$
$572$		0			0
$573$	−3.01139e15			−2.03665
$574$		0			0
$575$	−5.43528e12			−0.00360619
$576$		0			0
$577$		−	3.14336e14i		−	0.204610i	−0.994753	−	0.102305i	$-0.967378\pi$
$577$		−	3.14336e14i		−	0.204610i	0.994753	−	0.102305i	$-0.0326218\pi$
$578$		0			0
$579$			3.64183e15i			2.32588i
$580$		0			0
$581$	2.20089e15			1.37921
$582$		0			0
$583$		−	4.57392e14i		−	0.281262i
$584$		0			0
$585$	−8.21454e14	−	1.30046e15i	−0.495708	−	0.784762i
$586$		0			0
$587$		−	1.92615e15i		−	1.14073i	−0.821393	−	0.570363i	$-0.806802\pi$
$587$		−	1.92615e15i		−	1.14073i	0.821393	−	0.570363i	$-0.193198\pi$
$588$		0			0
$589$	−1.13592e14			−0.0660259
$590$		0			0
$591$		−	3.75292e15i		−	2.14111i
$592$		0			0
$593$			1.12115e15i			0.627858i	0.949446	+	0.313929i	$0.101645\pi$
$593$			1.12115e15i			0.627858i	−0.949446	+	0.313929i	$0.898355\pi$
$594$		0			0
$595$	−2.14112e15			−1.17706
$596$		0			0
$597$	1.23097e15			0.664335
$598$		0			0
$599$	2.62899e13			0.0139297			0.00696485	−	0.999976i	$-0.497783\pi$
$599$	2.62899e13			0.0139297			0.00696485	+	0.999976i	$0.497783\pi$
$600$		0			0
$601$	−5.52388e14			−0.287366			−0.143683	−	0.989624i	$-0.545895\pi$
$601$	−5.52388e14			−0.287366			−0.143683	+	0.989624i	$0.545895\pi$
$602$		0			0
$603$		−	1.75432e15i		−	0.896115i
$604$		0			0
$605$		−	2.30515e14i		−	0.115623i
$606$		0			0
$607$	3.37460e15			1.66220			0.831102	−	0.556120i	$-0.187711\pi$
$607$	3.37460e15			1.66220			0.831102	+	0.556120i	$0.187711\pi$
$608$		0			0
$609$		−	4.69682e15i		−	2.27200i
$610$		0			0
$611$	2.68369e14	+	4.24858e14i	0.127499	+	0.201845i
$612$		0			0
$613$			2.18972e15i			1.02178i	0.859647	+	0.510889i	$0.170684\pi$
$613$			2.18972e15i			1.02178i	−0.859647	+	0.510889i	$0.829316\pi$
$614$		0			0
$615$	−3.99619e15			−1.83161
$616$		0			0
$617$			1.86091e15i			0.837832i	0.908025	+	0.418916i	$0.137590\pi$
$617$			1.86091e15i			0.837832i	−0.908025	+	0.418916i	$0.862410\pi$
$618$		0			0
$619$			2.34987e15i			1.03931i	0.854376	+	0.519656i	$0.173940\pi$
$619$			2.34987e15i			1.03931i	−0.854376	+	0.519656i	$0.826060\pi$
$620$		0			0
$621$	2.25074e14			0.0977960
$622$		0			0
$623$	1.54975e15			0.661571
$624$		0			0
$625$	−2.37559e15			−0.996395
$626$		0			0
$627$	2.89728e15			1.19404
$628$		0			0
$629$			1.11064e15i			0.449776i
$630$		0			0
$631$		−	1.80725e15i		−	0.719211i	−0.933104	−	0.359606i	$-0.882911\pi$
$631$		−	1.80725e15i		−	0.719211i	0.933104	−	0.359606i	$-0.117089\pi$
$632$		0			0
$633$	−1.28446e15			−0.502341
$634$		0			0
$635$			1.20018e15i			0.461309i
$636$		0			0
$637$	3.21702e14	−	2.03208e14i	0.121531	−	0.0767670i
$638$		0			0
$639$		−	6.42431e14i		−	0.238546i
$640$		0			0
$641$	−1.18548e15			−0.432686			−0.216343	−	0.976317i	$-0.569413\pi$
$641$	−1.18548e15			−0.432686			−0.216343	+	0.976317i	$0.569413\pi$
$642$		0			0
$643$		−	3.97110e15i		−	1.42479i	−0.701778	−	0.712395i	$-0.747610\pi$
$643$		−	3.97110e15i		−	1.42479i	0.701778	−	0.712395i	$-0.252390\pi$
$644$		0			0
$645$		−	2.55748e13i		−	0.00902056i
$646$		0			0
$647$	−4.87109e14			−0.168909			−0.0844544	−	0.996427i	$-0.526915\pi$
$647$	−4.87109e14			−0.168909			−0.0844544	+	0.996427i	$0.526915\pi$
$648$		0			0
$649$	8.79763e14			0.299930
$650$		0			0
$651$	2.77007e14			0.0928528
$652$		0			0
$653$	4.18439e15			1.37914			0.689572	−	0.724217i	$-0.257799\pi$
$653$	4.18439e15			1.37914			0.689572	+	0.724217i	$0.257799\pi$
$654$		0			0
$655$		−	4.38512e15i		−	1.42120i
$656$		0			0
$657$		−	1.00282e15i		−	0.319604i
$658$		0			0
$659$	−1.28338e15			−0.402239			−0.201119	−	0.979567i	$-0.564458\pi$
$659$	−1.28338e15			−0.402239			−0.201119	+	0.979567i	$0.564458\pi$
$660$		0			0
$661$			3.13791e15i			0.967237i	0.875279	+	0.483618i	$0.160678\pi$
$661$			3.13791e15i			0.967237i	−0.875279	+	0.483618i	$0.839322\pi$
$662$		0			0
$663$	−4.93695e15	+	3.11851e15i	−1.49670	+	0.945414i
$664$		0			0
$665$		−	2.83156e15i		−	0.844318i
$666$		0			0
$667$	6.05026e15			1.77452
$668$		0			0
$669$		−	9.53583e15i		−	2.75115i
$670$		0			0
$671$		−	4.32772e14i		−	0.122825i
$672$		0			0
$673$	3.90884e15			1.09135			0.545677	−	0.837996i	$-0.316273\pi$
$673$	3.90884e15			1.09135			0.545677	+	0.837996i	$0.316273\pi$
$674$		0			0
$675$	−1.27368e12			−0.000349857
$676$		0			0
$677$	−4.93850e15			−1.33462			−0.667310	−	0.744780i	$-0.732554\pi$
$677$	−4.93850e15			−1.33462			−0.667310	+	0.744780i	$0.732554\pi$
$678$		0			0
$679$	−4.60209e15			−1.22369
$680$		0			0
$681$			5.61048e15i			1.46788i
$682$		0			0
$683$			6.64858e15i			1.71165i	0.517265	+	0.855825i	$0.326950\pi$
$683$			6.64858e15i			1.71165i	−0.517265	+	0.855825i	$0.673050\pi$
$684$		0			0
$685$	4.25303e15			1.07746
$686$		0			0
$687$		−	2.38890e15i		−	0.595575i
$688$		0			0
$689$	−1.03072e15	+	6.51070e14i	−0.252892	+	0.159743i
$690$		0			0
$691$			6.86206e15i			1.65701i	0.559981	+	0.828505i	$0.310808\pi$
$691$			6.86206e15i			1.65701i	−0.559981	+	0.828505i	$0.689192\pi$
$692$		0			0
$693$	−3.40432e15			−0.809092
$694$		0			0
$695$		−	1.13042e15i		−	0.264439i
$696$		0			0
$697$			7.30982e15i			1.68316i
$698$		0			0
$699$	−9.66411e15			−2.19047
$700$		0			0
$701$	2.83494e15			0.632550			0.316275	−	0.948668i	$-0.397568\pi$
$701$	2.83494e15			0.632550			0.316275	+	0.948668i	$0.397568\pi$
$702$		0			0
$703$	−1.46879e15			−0.322630
$704$		0			0
$705$	1.53091e15			0.331063
$706$		0			0
$707$			3.05449e14i			0.0650327i
$708$		0			0
$709$		−	3.69606e15i		−	0.774792i	−0.921914	−	0.387396i	$-0.873375\pi$
$709$		−	3.69606e15i		−	0.774792i	0.921914	−	0.387396i	$-0.126625\pi$
$710$		0			0
$711$	−6.30045e14			−0.130043
$712$		0			0
$713$			3.56829e14i			0.0725216i
$714$		0			0
$715$	3.96515e15	−	2.50466e15i	0.793555	−	0.501262i
$716$		0			0
$717$			4.87641e15i			0.961048i
$718$		0			0
$719$	1.68960e15			0.327925			0.163963	−	0.986467i	$-0.447572\pi$
$719$	1.68960e15			0.327925			0.163963	+	0.986467i	$0.447572\pi$
$720$		0			0
$721$			1.36230e15i			0.260392i
$722$		0			0
$723$			2.76248e15i			0.520042i
$724$		0			0
$725$	−3.42381e13			−0.00634821
$726$		0			0
$727$	2.09058e15			0.381792			0.190896	−	0.981610i	$-0.438861\pi$
$727$	2.09058e15			0.381792			0.190896	+	0.981610i	$0.438861\pi$
$728$		0			0
$729$	−4.75409e15			−0.855197
$730$		0			0
$731$	−4.67813e13			−0.00828947
$732$		0			0
$733$			6.81895e15i			1.19027i	0.803626	+	0.595135i	$0.202902\pi$
$733$			6.81895e15i			1.19027i	−0.803626	+	0.595135i	$0.797098\pi$
$734$		0			0
$735$		−	1.15921e15i		−	0.199333i
$736$		0			0
$737$	5.34901e15			0.906155
$738$		0			0
$739$			2.39909e15i			0.400407i	0.979754	+	0.200203i	$0.0641603\pi$
$739$			2.39909e15i			0.400407i	−0.979754	+	0.200203i	$0.935840\pi$
$740$		0			0
$741$	−4.12411e15	−	6.52894e15i	−0.678157	−	1.07360i
$742$		0			0
$743$			6.02647e15i			0.976393i	0.872734	+	0.488197i	$0.162345\pi$
$743$			6.02647e15i			0.976393i	−0.872734	+	0.488197i	$0.837655\pi$
$744$		0			0
$745$	−3.84929e15			−0.614500
$746$		0			0
$747$		−	8.81092e15i		−	1.38598i
$748$		0			0
$749$		−	2.68794e15i		−	0.416648i
$750$		0			0
$751$	−6.17649e15			−0.943457			−0.471729	−	0.881744i	$-0.656370\pi$
$751$	−6.17649e15			−0.943457			−0.471729	+	0.881744i	$0.656370\pi$
$752$		0			0
$753$	2.99642e14			0.0451056
$754$		0			0
$755$	3.59042e15			0.532644
$756$		0			0
$757$	−7.54770e15			−1.10354			−0.551769	−	0.833997i	$-0.686047\pi$
$757$	−7.54770e15			−1.10354			−0.551769	+	0.833997i	$0.686047\pi$
$758$		0			0
$759$		−	9.10128e15i		−	1.31151i
$760$		0			0
$761$		−	4.10819e15i		−	0.583492i	−0.956496	−	0.291746i	$-0.905764\pi$
$761$		−	4.10819e15i		−	0.583492i	0.956496	−	0.291746i	$-0.0942362\pi$
$762$		0			0
$763$	8.32050e15			1.16484
$764$		0			0
$765$			8.57162e15i			1.18284i
$766$		0			0
$767$	−1.25229e15	−	1.98252e15i	−0.170346	−	0.269676i
$768$		0			0
$769$		−	1.20391e16i		−	1.61435i	−0.590313	−	0.807175i	$-0.700996\pi$
$769$		−	1.20391e16i		−	1.61435i	0.590313	−	0.807175i	$-0.299004\pi$
$770$		0			0
$771$	1.78113e16			2.35448
$772$		0			0
$773$		−	8.71824e15i		−	1.13617i	−0.822971	−	0.568083i	$-0.807685\pi$
$773$		−	8.71824e15i		−	1.13617i	0.822971	−	0.568083i	$-0.192315\pi$
$774$		0			0
$775$		−	2.01928e12i		−	0.000259440i
$776$		0			0
$777$	3.58179e15			0.453717
$778$		0			0
$779$	−9.66699e15			−1.20736
$780$		0			0
$781$	1.95880e15			0.241218
$782$		0			0
$783$	1.41780e15			0.172157
$784$		0			0
$785$			9.25401e15i			1.10802i
$786$		0			0
$787$			3.02774e15i			0.357484i	0.983896	+	0.178742i	$0.0572028\pi$
$787$			3.02774e15i			0.357484i	−0.983896	+	0.178742i	$0.942797\pi$
$788$		0			0
$789$	−6.42240e15			−0.747780
$790$		0			0
$791$		−	9.35781e15i		−	1.07449i
$792$		0			0
$793$	−9.75240e14	+	6.16027e14i	−0.110436	+	0.0697585i
$794$		0			0
$795$			3.71404e15i			0.414790i
$796$		0			0
$797$	2.22908e15			0.245531			0.122765	−	0.992436i	$-0.460824\pi$
$797$	2.22908e15			0.245531			0.122765	+	0.992436i	$0.460824\pi$
$798$		0			0
$799$		−	2.80035e15i		−	0.304232i
$800$		0			0
$801$		−	6.20415e15i		−	0.664819i
$802$		0			0
$803$	3.05764e15			0.323184
$804$		0			0
$805$	−8.89481e15			−0.927384
$806$		0			0
$807$	−1.36773e15			−0.140668
$808$		0			0
$809$	1.47557e16			1.49707			0.748536	−	0.663095i	$-0.230757\pi$
$809$	1.47557e16			1.49707			0.748536	+	0.663095i	$0.230757\pi$
$810$		0			0
$811$			1.63665e16i			1.63811i	0.573717	+	0.819054i	$0.305501\pi$
$811$			1.63665e16i			1.63811i	−0.573717	+	0.819054i	$0.694499\pi$
$812$		0			0
$813$		−	1.08426e16i		−	1.07062i
$814$		0			0
$815$	6.88746e15			0.670953
$816$		0			0
$817$		−	6.18667e13i		−	0.00594614i
$818$		0			0
$819$	4.84585e15	+	7.67154e15i	0.459525	+	0.727480i
$820$		0			0
$821$			1.10022e16i			1.02942i	0.857366	+	0.514708i	$0.172100\pi$
$821$			1.10022e16i			1.02942i	−0.857366	+	0.514708i	$0.827900\pi$
$822$		0			0
$823$	−1.46654e16			−1.35393			−0.676964	−	0.736016i	$-0.736705\pi$
$823$	−1.46654e16			−1.35393			−0.676964	+	0.736016i	$0.736705\pi$
$824$		0			0
$825$			5.15037e13i			0.00469183i
$826$		0			0
$827$		−	1.06804e16i		−	0.960083i	−0.877246	−	0.480041i	$-0.840622\pi$
$827$		−	1.06804e16i		−	0.960083i	0.877246	−	0.480041i	$-0.159378\pi$
$828$		0			0
$829$	−7.59988e15			−0.674150			−0.337075	−	0.941478i	$-0.609438\pi$
$829$	−7.59988e15			−0.674150			−0.337075	+	0.941478i	$0.609438\pi$
$830$		0			0
$831$	−2.17797e16			−1.90654
$832$		0			0
$833$	−2.12042e15			−0.183178
$834$		0			0
$835$	1.72710e16			1.47246
$836$		0			0
$837$			8.36180e13i			0.00703574i
$838$		0			0
$839$		−	6.28372e15i		−	0.521827i	−0.965362	−	0.260913i	$-0.915976\pi$
$839$		−	6.28372e15i		−	0.521827i	0.965362	−	0.260913i	$-0.0840237\pi$
$840$		0			0
$841$	2.59116e16			2.12381
$842$		0			0
$843$			1.10045e16i			0.890263i
$844$		0			0
$845$	−1.12883e16	−	5.37012e15i	−0.901401	−	0.428817i
$846$		0			0
$847$			1.35984e15i			0.107183i
$848$		0			0
$849$	2.46157e16			1.91523
$850$		0			0
$851$			4.61392e15i			0.354371i
$852$		0			0
$853$			9.24439e14i			0.0700904i	0.999386	+	0.0350452i	$0.0111575\pi$
$853$			9.24439e14i			0.0700904i	−0.999386	+	0.0350452i	$0.988842\pi$
$854$		0			0
$855$	−1.13357e16			−0.848465
$856$		0			0
$857$	1.59547e16			1.17895			0.589473	−	0.807788i	$-0.299336\pi$
$857$	1.59547e16			1.17895			0.589473	+	0.807788i	$0.299336\pi$
$858$		0			0
$859$	2.27490e16			1.65958			0.829792	−	0.558072i	$-0.188459\pi$
$859$	2.27490e16			1.65958			0.829792	+	0.558072i	$0.188459\pi$
$860$		0			0
$861$	2.35740e16			1.69791
$862$		0			0
$863$			2.36296e16i			1.68034i	0.542325	+	0.840168i	$0.317544\pi$
$863$			2.36296e16i			1.68034i	−0.542325	+	0.840168i	$0.682456\pi$
$864$		0			0
$865$			1.15010e16i			0.807507i
$866$		0			0
$867$	1.25019e16			0.866704
$868$		0			0
$869$		−	1.92104e15i		−	0.131500i
$870$		0			0
$871$	−7.61401e15	−	1.20538e16i	−0.514652	−	0.814752i
$872$		0			0
$873$			1.84237e16i			1.22970i
$874$		0			0
$875$	1.40644e16			0.926994
$876$		0			0
$877$			2.52370e16i			1.64263i	0.570476	+	0.821314i	$0.306759\pi$
$877$			2.52370e16i			1.64263i	−0.570476	+	0.821314i	$0.693241\pi$
$878$		0			0
$879$			1.05076e16i			0.675407i
$880$		0			0
$881$	−1.37790e16			−0.874683			−0.437341	−	0.899296i	$-0.644080\pi$
$881$	−1.37790e16			−0.874683			−0.437341	+	0.899296i	$0.644080\pi$
$882$		0			0
$883$	1.42704e16			0.894647			0.447324	−	0.894372i	$-0.352377\pi$
$883$	1.42704e16			0.894647			0.447324	+	0.894372i	$0.352377\pi$
$884$		0			0
$885$	−7.14372e15			−0.442320
$886$		0			0
$887$	1.01571e16			0.621142			0.310571	−	0.950550i	$-0.399480\pi$
$887$	1.01571e16			0.621142			0.310571	+	0.950550i	$0.399480\pi$
$888$		0			0
$889$		−	7.08001e15i		−	0.427636i
$890$		0			0
$891$		−	1.67890e16i		−	1.00161i
$892$		0			0
$893$	3.70336e15			0.218229
$894$		0			0
$895$			2.13935e16i			1.24525i
$896$		0			0
$897$	−2.05095e16	+	1.29551e16i	−1.17922	+	0.744876i
$898$		0			0
$899$			2.24775e15i			0.127665i
$900$		0			0
$901$	6.79372e15			0.381172
$902$		0			0
$903$			1.50869e14i			0.00836211i
$904$		0			0
$905$			3.35197e16i			1.83541i
$906$		0			0
$907$	−3.34242e16			−1.80809			−0.904046	−	0.427436i	$-0.859417\pi$
$907$	−3.34242e16			−1.80809			−0.904046	+	0.427436i	$0.859417\pi$
$908$		0			0
$909$	1.22282e15			0.0653521
$910$		0			0
$911$	4.10924e15			0.216975			0.108488	−	0.994098i	$-0.465399\pi$
$911$	4.10924e15			0.216975			0.108488	+	0.994098i	$0.465399\pi$
$912$		0			0
$913$	2.68649e16			1.40151
$914$		0			0
$915$			3.51414e15i			0.181135i
$916$		0			0
$917$			2.58683e16i			1.31746i
$918$		0			0
$919$	3.44855e16			1.73541			0.867704	−	0.497082i	$-0.165595\pi$
$919$	3.44855e16			1.73541			0.867704	+	0.497082i	$0.165595\pi$
$920$		0			0
$921$		−	2.13830e16i		−	1.06326i
$922$		0			0
$923$	−2.78824e15	−	4.41411e15i	−0.137000	−	0.216887i
$924$		0			0
$925$		−	2.61100e13i		−	0.00126773i
$926$		0			0
$927$	5.45374e15			0.261671
$928$		0			0
$929$			2.06856e16i			0.980804i	0.871496	+	0.490402i	$0.163150\pi$
$929$			2.06856e16i			0.980804i	−0.871496	+	0.490402i	$0.836850\pi$
$930$		0			0
$931$		−	2.80418e15i		−	0.131396i
$932$		0			0
$933$	1.13327e15			0.0524789
$934$		0			0
$935$	−2.61353e16			−1.19609
$936$		0			0
$937$	−9.81694e15			−0.444026			−0.222013	−	0.975044i	$-0.571263\pi$
$937$	−9.81694e15			−0.444026			−0.222013	+	0.975044i	$0.571263\pi$
$938$		0			0
$939$	3.12583e16			1.39735
$940$		0			0
$941$		−	2.46580e16i		−	1.08947i	−0.838609	−	0.544734i	$-0.816631\pi$
$941$		−	2.46580e16i		−	1.08947i	0.838609	−	0.544734i	$-0.183369\pi$
$942$		0			0
$943$			3.03671e16i			1.32614i
$944$		0			0
$945$	−2.08438e15			−0.0899708
$946$		0			0
$947$			4.65104e15i			0.198438i	0.995066	+	0.0992190i	$0.0316344\pi$
$947$			4.65104e15i			0.198438i	−0.995066	+	0.0992190i	$0.968366\pi$
$948$		0			0
$949$	−4.35237e15	−	6.89029e15i	−0.183553	−	0.290585i
$950$		0			0
$951$		−	5.49313e16i		−	2.28995i
$952$		0			0
$953$	2.07030e15			0.0853142			0.0426571	−	0.999090i	$-0.486418\pi$
$953$	2.07030e15			0.0853142			0.0426571	+	0.999090i	$0.486418\pi$
$954$		0			0
$955$			3.59243e16i			1.46342i
$956$		0			0
$957$		−	5.73312e16i		−	2.30874i
$958$		0			0
$959$	−2.50891e16			−0.998811
$960$		0			0
$961$	2.52759e16			0.994783
$962$		0			0
$963$	−1.07607e16			−0.418694
$964$		0			0
$965$	4.34451e16			1.67125
$966$		0			0
$967$		−	1.22729e16i		−	0.466770i	−0.972384	−	0.233385i	$-0.925020\pi$
$967$		−	1.22729e16i		−	0.466770i	0.972384	−	0.233385i	$-0.0749802\pi$
$968$		0			0
$969$			4.30339e16i			1.61819i
$970$		0			0
$971$	1.50255e16			0.558629			0.279314	−	0.960200i	$-0.409893\pi$
$971$	1.50255e16			0.558629			0.279314	+	0.960200i	$0.409893\pi$
$972$		0			0
$973$			6.66850e15i			0.245136i
$974$		0			0
$975$	1.16062e14	−	7.33126e13i	0.00421857	−	0.00266473i
$976$		0			0
$977$		−	3.35526e16i		−	1.20589i	−0.797784	−	0.602943i	$-0.793994\pi$
$977$		−	3.35526e16i		−	1.20589i	0.797784	−	0.602943i	$-0.206006\pi$
$978$		0			0
$979$	1.89168e16			0.672268
$980$		0			0
$981$		−	3.33098e16i		−	1.17056i
$982$		0			0
$983$		−	4.83809e16i		−	1.68124i	−0.541626	−	0.840619i	$-0.682191\pi$
$983$		−	4.83809e16i		−	1.68124i	0.541626	−	0.840619i	$-0.317809\pi$
$984$		0			0
$985$	−4.47704e16			−1.53848
$986$		0			0
$987$	−9.03104e15			−0.306898
$988$		0			0
$989$	−1.94343e14			−0.00653114
$990$		0			0
$991$	−4.75906e15			−0.158167			−0.0790835	−	0.996868i	$-0.525199\pi$
$991$	−4.75906e15			−0.158167			−0.0790835	+	0.996868i	$0.525199\pi$
$992$		0			0
$993$		−	2.03091e16i		−	0.667527i
$994$		0			0
$995$		−	1.46848e16i		−	0.477354i
$996$		0			0
$997$	2.40308e16			0.772582			0.386291	−	0.922377i	$-0.373756\pi$
$997$	2.40308e16			0.772582			0.386291	+	0.922377i	$0.373756\pi$
$998$		0			0
$999$			1.08121e15i			0.0343795i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.12.f.b.129.9		12
4.3	odd	2		13.12.b.a.12.1	✓	12
12.11	even	2		117.12.b.b.64.12		12
13.12	even	2	inner	208.12.f.b.129.10		12
52.31	even	4		169.12.a.e.1.1		12
52.47	even	4		169.12.a.e.1.12		12
52.51	odd	2		13.12.b.a.12.12	yes	12
156.155	even	2		117.12.b.b.64.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.b.a.12.1	✓	12	4.3	odd	2
13.12.b.a.12.12	yes	12	52.51	odd	2
117.12.b.b.64.1		12	156.155	even	2
117.12.b.b.64.12		12	12.11	even	2
169.12.a.e.1.1		12	52.31	even	4
169.12.a.e.1.12		12	52.47	even	4
208.12.f.b.129.9		12	1.1	even	1	trivial
208.12.f.b.129.10		12	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+584.699 q^{3} -6975.15i q^{5} +41147.2i q^{7} +164726. q^{9} +O(q^{10})\)	\(q+584.699 q^{3} -6975.15i q^{5} +41147.2i q^{7} +164726. q^{9} +502259. i q^{11} +(1.13182e6 - 714936. i) q^{13} -4.07837e6i q^{15} -7.46014e6 q^{17} -9.86577e6i q^{19} +2.40587e7i q^{21} -3.09915e7 q^{23} +175380. q^{25} -7.26244e6 q^{27} -1.95223e8 q^{29} -1.15138e7i q^{31} +2.93670e8i q^{33} +2.87008e8 q^{35} -1.48877e8i q^{37} +(6.61777e8 - 4.18023e8i) q^{39} -9.79851e8i q^{41} +6.27084e6 q^{43} -1.14899e9i q^{45} +3.75374e8i q^{47} +2.84233e8 q^{49} -4.36194e9 q^{51} -9.10670e8 q^{53} +3.50333e9 q^{55} -5.76851e9i q^{57} -1.75161e9i q^{59} -8.61653e8 q^{61} +6.77803e9i q^{63} +(-4.98679e9 - 7.89465e9i) q^{65} -1.06499e10i q^{67} -1.81207e10 q^{69} -3.89999e9i q^{71} -6.08778e9i q^{73} +1.02544e8 q^{75} -2.06665e10 q^{77} -3.82480e9 q^{79} -3.34271e10 q^{81} -5.34883e10i q^{83} +5.20356e10i q^{85} -1.14147e11 q^{87} -3.76634e10i q^{89} +(2.94176e10 + 4.65714e10i) q^{91} -6.73209e9i q^{93} -6.88153e10 q^{95} +1.11844e11i q^{97} +8.27351e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 488 q^{3} + 654644 q^{9}+O(q^{10}) \) 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

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