LMFDB - Embedded newform 272.10.b.c.33.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [272,10,Mod(33,272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(272, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("272.33");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	272.b (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:	$140.089747437$
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} + 122690 x^{10} + 5157152560 x^{8} + 87983684680032 x^{6} + \cdots + 20\!\cdots\!28 $ x^12 + 122690x^10 + 5157152560x^8 + 87983684680032x^6 + 612743619071665152x^4 + 1335826553351738886144*x^2 + 203949399568932198678528
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			33.8
Root			$59.5904i$ of defining polynomial
Character	$\chi$	$=$	272.33
Dual form			272.10.b.c.33.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+59.5904i q^{3} +633.821i q^{5} -10932.1i q^{7} +16132.0 q^{9} +O(q^{10})$	$q+59.5904i q^{3} +633.821i q^{5} -10932.1i q^{7} +16132.0 q^{9} -26182.2i q^{11} -132465. q^{13} -37769.6 q^{15} +(97124.1 + 330386. i) q^{17} +834785. q^{19} +651448. q^{21} +1.27202e6i q^{23} +1.55140e6 q^{25} +2.13423e6i q^{27} -6.34953e6i q^{29} -8.52001e6i q^{31} +1.56021e6 q^{33} +6.92899e6 q^{35} +8.05093e6i q^{37} -7.89366e6i q^{39} +2.36819e7i q^{41} -2.89782e6 q^{43} +1.02248e7i q^{45} -2.63358e7 q^{47} -7.91572e7 q^{49} +(-1.96878e7 + 5.78766e6i) q^{51} +2.45299e6 q^{53} +1.65948e7 q^{55} +4.97452e7i q^{57} +8.84531e6 q^{59} -6.18808e7i q^{61} -1.76356e8i q^{63} -8.39593e7i q^{65} -1.46096e8 q^{67} -7.58002e7 q^{69} -2.08285e8i q^{71} +7.44027e7i q^{73} +9.24483e7i q^{75} -2.86226e8 q^{77} -1.47713e8i q^{79} +1.90346e8 q^{81} +5.69010e7 q^{83} +(-2.09405e8 + 6.15593e7i) q^{85} +3.78371e8 q^{87} +1.03356e9 q^{89} +1.44812e9i q^{91} +5.07710e8 q^{93} +5.29104e8i q^{95} -1.03075e9i q^{97} -4.22370e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 9184 q^{9}+O(q^{10}) $ 12 * q - 9184 * q^9	$ 12 q - 9184 q^{9} - 63204 q^{13} + 243480 q^{15} - 105960 q^{17} - 1110672 q^{19} - 172580 q^{21} - 4441796 q^{25} - 6557404 q^{33} - 3519864 q^{35} - 10004616 q^{43} + 112552440 q^{47} + 121354720 q^{49} + 52506472 q^{51} + 76804272 q^{53} - 300732568 q^{55} - 11618904 q^{59} + 304208752 q^{67} - 211308236 q^{69} + 138357828 q^{77} - 363335792 q^{81} + 845042136 q^{83} - 388949632 q^{85} - 610860648 q^{87} - 938223804 q^{89} + 1635779524 q^{93}+O(q^{100}) $ 12 * q - 9184 * q^9 - 63204 * q^13 + 243480 * q^15 - 105960 * q^17 - 1110672 * q^19 - 172580 * q^21 - 4441796 * q^25 - 6557404 * q^33 - 3519864 * q^35 - 10004616 * q^43 + 112552440 * q^47 + 121354720 * q^49 + 52506472 * q^51 + 76804272 * q^53 - 300732568 * q^55 - 11618904 * q^59 + 304208752 * q^67 - 211308236 * q^69 + 138357828 * q^77 - 363335792 * q^81 + 845042136 * q^83 - 388949632 * q^85 - 610860648 * q^87 - 938223804 * q^89 + 1635779524 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/272\mathbb{Z}\right)^\times$.

$n$	$69$	$239$	$241$
$\chi(n)$	$1$	$1$	$-1$

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
$3$			59.5904i			0.424747i	0.977189	+	0.212374i	$0.0681194\pi$
$3$			59.5904i			0.424747i	−0.977189	+	0.212374i	$0.931881\pi$
$4$		0			0
$5$			633.821i			0.453525i	0.973950	+	0.226763i	$0.0728142\pi$
$5$			633.821i			0.453525i	−0.973950	+	0.226763i	$0.927186\pi$
$6$		0			0
$7$		−	10932.1i		−	1.72093i	−0.509512	−	0.860463i	$-0.670174\pi$
$7$		−	10932.1i		−	1.72093i	0.509512	−	0.860463i	$-0.329826\pi$
$8$		0			0
$9$	16132.0			0.819590
$10$		0			0
$11$		−	26182.2i		−	0.539186i	−0.962974	−	0.269593i	$-0.913111\pi$
$11$		−	26182.2i		−	0.539186i	0.962974	−	0.269593i	$-0.0868891\pi$
$12$		0			0
$13$	−132465.			−1.28634			−0.643172	−	0.765722i	$-0.722382\pi$
$13$	−132465.			−1.28634			−0.643172	+	0.765722i	$0.722382\pi$
$14$		0			0
$15$	−37769.6			−0.192634
$16$		0			0
$17$	97124.1	+	330386.i	0.282038	+	0.959403i
$17$	97124.1	+	330386.i	0.282038	+	0.959403i
$18$		0			0
$19$	834785.			1.46955			0.734774	−	0.678312i	$-0.237288\pi$
$19$	834785.			1.46955			0.734774	+	0.678312i	$0.237288\pi$
$20$		0			0
$21$	651448.			0.730959
$22$		0			0
$23$			1.27202e6i			0.947805i	0.880577	+	0.473902i	$0.157155\pi$
$23$			1.27202e6i			0.947805i	−0.880577	+	0.473902i	$0.842845\pi$
$24$		0			0
$25$	1.55140e6			0.794315
$26$		0			0
$27$			2.13423e6i			0.772866i
$28$		0			0
$29$		−	6.34953e6i		−	1.66706i	−0.552476	−	0.833529i	$-0.686317\pi$
$29$		−	6.34953e6i		−	1.66706i	0.552476	−	0.833529i	$-0.313683\pi$
$30$		0			0
$31$		−	8.52001e6i		−	1.65696i	−0.560018	−	0.828480i	$-0.689206\pi$
$31$		−	8.52001e6i		−	1.65696i	0.560018	−	0.828480i	$-0.310794\pi$
$32$		0			0
$33$	1.56021e6			0.229018
$34$		0			0
$35$	6.92899e6			0.780484
$36$		0			0
$37$			8.05093e6i			0.706217i	0.935582	+	0.353109i	$0.114875\pi$
$37$			8.05093e6i			0.706217i	−0.935582	+	0.353109i	$0.885125\pi$
$38$		0			0
$39$		−	7.89366e6i		−	0.546371i
$40$		0			0
$41$			2.36819e7i			1.30885i	0.756127	+	0.654425i	$0.227089\pi$
$41$			2.36819e7i			1.30885i	−0.756127	+	0.654425i	$0.772911\pi$
$42$		0			0
$43$	−2.89782e6			−0.129260			−0.0646298	−	0.997909i	$-0.520587\pi$
$43$	−2.89782e6			−0.129260			−0.0646298	+	0.997909i	$0.520587\pi$
$44$		0			0
$45$			1.02248e7i			0.371705i
$46$		0			0
$47$	−2.63358e7			−0.787238			−0.393619	−	0.919274i	$-0.628777\pi$
$47$	−2.63358e7			−0.787238			−0.393619	+	0.919274i	$0.628777\pi$
$48$		0			0
$49$	−7.91572e7			−1.96159
$50$		0			0
$51$	−1.96878e7	+	5.78766e6i	−0.407504	+	0.119795i
$52$		0			0
$53$	2.45299e6			0.0427027			0.0213513	−	0.999772i	$-0.493203\pi$
$53$	2.45299e6			0.0427027			0.0213513	+	0.999772i	$0.493203\pi$
$54$		0			0
$55$	1.65948e7			0.244534
$56$		0			0
$57$			4.97452e7i			0.624186i
$58$		0			0
$59$	8.84531e6			0.0950340			0.0475170	−	0.998870i	$-0.484869\pi$
$59$	8.84531e6			0.0950340			0.0475170	+	0.998870i	$0.484869\pi$
$60$		0			0
$61$		−	6.18808e7i		−	0.572231i	−0.958195	−	0.286116i	$-0.907636\pi$
$61$		−	6.18808e7i		−	0.572231i	0.958195	−	0.286116i	$-0.0923642\pi$
$62$		0			0
$63$		−	1.76356e8i		−	1.41045i
$64$		0			0
$65$		−	8.39593e7i		−	0.583389i
$66$		0			0
$67$	−1.46096e8			−0.885730			−0.442865	−	0.896588i	$-0.646038\pi$
$67$	−1.46096e8			−0.885730			−0.442865	+	0.896588i	$0.646038\pi$
$68$		0			0
$69$	−7.58002e7			−0.402577
$70$		0			0
$71$		−	2.08285e8i		−	0.972737i	−0.873754	−	0.486369i	$-0.838321\pi$
$71$		−	2.08285e8i		−	0.972737i	0.873754	−	0.486369i	$-0.161679\pi$
$72$		0			0
$73$			7.44027e7i			0.306645i	0.988176	+	0.153323i	$0.0489974\pi$
$73$			7.44027e7i			0.306645i	−0.988176	+	0.153323i	$0.951003\pi$
$74$		0			0
$75$			9.24483e7i			0.337383i
$76$		0			0
$77$	−2.86226e8			−0.927899
$78$		0			0
$79$		−	1.47713e8i		−	0.426676i	−0.976978	−	0.213338i	$-0.931566\pi$
$79$		−	1.47713e8i		−	0.426676i	0.976978	−	0.213338i	$-0.0684335\pi$
$80$		0			0
$81$	1.90346e8			0.491317
$82$		0			0
$83$	5.69010e7			0.131604			0.0658019	−	0.997833i	$-0.479039\pi$
$83$	5.69010e7			0.131604			0.0658019	+	0.997833i	$0.479039\pi$
$84$		0			0
$85$	−2.09405e8	+	6.15593e7i	−0.435114	+	0.127911i
$86$		0			0
$87$	3.78371e8			0.708078
$88$		0			0
$89$	1.03356e9			1.74614			0.873071	−	0.487592i	$-0.162125\pi$
$89$	1.03356e9			1.74614			0.873071	+	0.487592i	$0.162125\pi$
$90$		0			0
$91$			1.44812e9i			2.21370i
$92$		0			0
$93$	5.07710e8			0.703789
$94$		0			0
$95$			5.29104e8i			0.666477i
$96$		0			0
$97$		−	1.03075e9i		−	1.18217i	−0.806609	−	0.591086i	$-0.798699\pi$
$97$		−	1.03075e9i		−	1.18217i	0.806609	−	0.591086i	$-0.201301\pi$
$98$		0			0
$99$		−	4.22370e8i		−	0.441911i
$100$		0			0
$101$	−4.52386e8			−0.432576			−0.216288	−	0.976330i	$-0.569395\pi$
$101$	−4.52386e8			−0.432576			−0.216288	+	0.976330i	$0.569395\pi$
$102$		0			0
$103$	2.13433e8			0.186850			0.0934250	−	0.995626i	$-0.470218\pi$
$103$	2.13433e8			0.186850			0.0934250	+	0.995626i	$0.470218\pi$
$104$		0			0
$105$			4.12901e8i			0.331508i
$106$		0			0
$107$		−	1.25133e9i		−	0.922881i	−0.887171	−	0.461440i	$-0.847333\pi$
$107$		−	1.25133e9i		−	0.922881i	0.887171	−	0.461440i	$-0.152667\pi$
$108$		0			0
$109$		−	9.41035e8i		−	0.638538i	−0.947664	−	0.319269i	$-0.896563\pi$
$109$		−	9.41035e8i		−	0.638538i	0.947664	−	0.319269i	$-0.103437\pi$
$110$		0			0
$111$	−4.79758e8			−0.299964
$112$		0			0
$113$			7.72792e7i			0.0445871i	0.999751	+	0.0222936i	$0.00709685\pi$
$113$			7.72792e7i			0.0445871i	−0.999751	+	0.0222936i	$0.992903\pi$
$114$		0			0
$115$	−8.06233e8			−0.429854
$116$		0			0
$117$	−2.13693e9			−1.05427
$118$		0			0
$119$	3.61181e9	−	1.06177e9i	1.65106	−	0.485366i
$120$		0			0
$121$	1.67244e9			0.709279
$122$		0			0
$123$	−1.41122e9			−0.555930
$124$		0			0
$125$			2.22124e9i			0.813767i
$126$		0			0
$127$	−3.54892e9			−1.21054			−0.605271	−	0.796020i	$-0.706935\pi$
$127$	−3.54892e9			−1.21054			−0.605271	+	0.796020i	$0.706935\pi$
$128$		0			0
$129$		−	1.72682e8i		−	0.0549026i
$130$		0			0
$131$		−	2.85977e9i		−	0.848419i	−0.905564	−	0.424209i	$-0.860552\pi$
$131$		−	2.85977e9i		−	0.848419i	0.905564	−	0.424209i	$-0.139448\pi$
$132$		0			0
$133$		−	9.12596e9i		−	2.52898i
$134$		0			0
$135$	−1.35272e9			−0.350514
$136$		0			0
$137$	−1.35821e8			−0.0329400			−0.0164700	−	0.999864i	$-0.505243\pi$
$137$	−1.35821e8			−0.0329400			−0.0164700	+	0.999864i	$0.505243\pi$
$138$		0			0
$139$		−	4.19737e8i		−	0.0953698i	−0.998862	−	0.0476849i	$-0.984816\pi$
$139$		−	4.19737e8i		−	0.0953698i	0.998862	−	0.0476849i	$-0.0151843\pi$
$140$		0			0
$141$		−	1.56936e9i		−	0.334377i
$142$		0			0
$143$			3.46823e9i			0.693578i
$144$		0			0
$145$	4.02447e9			0.756053
$146$		0			0
$147$		−	4.71701e9i		−	0.833179i
$148$		0			0
$149$	−8.05680e9			−1.33913			−0.669567	−	0.742751i	$-0.733520\pi$
$149$	−8.05680e9			−1.33913			−0.669567	+	0.742751i	$0.733520\pi$
$150$		0			0
$151$	5.71660e9			0.894832			0.447416	−	0.894326i	$-0.352344\pi$
$151$	5.71660e9			0.894832			0.447416	+	0.894326i	$0.352344\pi$
$152$		0			0
$153$	1.56681e9	+	5.32978e9i	0.231155	+	0.786317i
$154$		0			0
$155$	5.40016e9			0.751474
$156$		0			0
$157$	−3.73112e9			−0.490107			−0.245053	−	0.969510i	$-0.578806\pi$
$157$	−3.73112e9			−0.490107			−0.245053	+	0.969510i	$0.578806\pi$
$158$		0			0
$159$			1.46175e8i			0.0181378i
$160$		0			0
$161$	1.39059e10			1.63110
$162$		0			0
$163$		−	3.76537e9i		−	0.417795i	−0.977938	−	0.208898i	$-0.933012\pi$
$163$		−	3.76537e9i		−	0.417795i	0.977938	−	0.208898i	$-0.0669876\pi$
$164$		0			0
$165$			9.88891e8i			0.103865i
$166$		0			0
$167$		−	7.70007e9i		−	0.766074i	−0.923733	−	0.383037i	$-0.874878\pi$
$167$		−	7.70007e9i		−	0.766074i	0.923733	−	0.383037i	$-0.125122\pi$
$168$		0			0
$169$	6.94255e9			0.654680
$170$		0			0
$171$	1.34667e10			1.20443
$172$		0			0
$173$			5.96147e9i			0.505995i	0.967467	+	0.252997i	$0.0814164\pi$
$173$			5.96147e9i			0.505995i	−0.967467	+	0.252997i	$0.918584\pi$
$174$		0			0
$175$		−	1.69600e10i		−	1.36696i
$176$		0			0
$177$			5.27095e8i			0.0403654i
$178$		0			0
$179$	−4.90200e9			−0.356890			−0.178445	−	0.983950i	$-0.557107\pi$
$179$	−4.90200e9			−0.356890			−0.178445	+	0.983950i	$0.557107\pi$
$180$		0			0
$181$		−	6.71648e9i		−	0.465144i	−0.972579	−	0.232572i	$-0.925286\pi$
$181$		−	6.71648e9i		−	0.465144i	0.972579	−	0.232572i	$-0.0747142\pi$
$182$		0			0
$183$	3.68750e9			0.243054
$184$		0			0
$185$	−5.10285e9			−0.320287
$186$		0			0
$187$	8.65022e9	−	2.54292e9i	0.517297	−	0.152071i
$188$		0			0
$189$	2.33316e10			1.33005
$190$		0			0
$191$	1.84682e10			1.00409			0.502046	−	0.864841i	$-0.332581\pi$
$191$	1.84682e10			1.00409			0.502046	+	0.864841i	$0.332581\pi$
$192$		0			0
$193$		−	3.37714e10i		−	1.75203i	−0.482287	−	0.876013i	$-0.660194\pi$
$193$		−	3.37714e10i		−	1.75203i	0.482287	−	0.876013i	$-0.339806\pi$
$194$		0			0
$195$	5.00316e9			0.247793
$196$		0			0
$197$		−	6.25236e9i		−	0.295765i	−0.989005	−	0.147882i	$-0.952754\pi$
$197$		−	6.25236e9i		−	0.295765i	0.989005	−	0.147882i	$-0.0472457\pi$
$198$		0			0
$199$		−	1.46173e10i		−	0.660735i	−0.943852	−	0.330368i	$-0.892827\pi$
$199$		−	1.46173e10i		−	0.660735i	0.943852	−	0.330368i	$-0.107173\pi$
$200$		0			0
$201$		−	8.70591e9i		−	0.376211i
$202$		0			0
$203$	−6.94137e10			−2.86889
$204$		0			0
$205$	−1.50101e10			−0.593597
$206$		0			0
$207$			2.05202e10i			0.776811i
$208$		0			0
$209$		−	2.18565e10i		−	0.792359i
$210$		0			0
$211$		−	2.69240e10i		−	0.935122i	−0.883961	−	0.467561i	$-0.845133\pi$
$211$		−	2.69240e10i		−	0.935122i	0.883961	−	0.467561i	$-0.154867\pi$
$212$		0			0
$213$	1.24118e10			0.413167
$214$		0			0
$215$		−	1.83670e9i		−	0.0586225i
$216$		0			0
$217$	−9.31416e10			−2.85151
$218$		0			0
$219$	−4.43369e9			−0.130247
$220$		0			0
$221$	−1.28656e10	−	4.37646e10i	−0.362797	−	1.23412i
$222$		0			0
$223$	−1.09042e10			−0.295273			−0.147636	−	0.989042i	$-0.547166\pi$
$223$	−1.09042e10			−0.295273			−0.147636	+	0.989042i	$0.547166\pi$
$224$		0			0
$225$	2.50271e10			0.651012
$226$		0			0
$227$		−	5.73714e10i		−	1.43410i	−0.697022	−	0.717050i	$-0.745492\pi$
$227$		−	5.73714e10i		−	1.43410i	0.697022	−	0.717050i	$-0.254508\pi$
$228$		0			0
$229$	4.85190e10			1.16587			0.582937	−	0.812517i	$-0.301903\pi$
$229$	4.85190e10			1.16587			0.582937	+	0.812517i	$0.301903\pi$
$230$		0			0
$231$		−	1.70563e10i		−	0.394123i
$232$		0			0
$233$		−	1.92873e9i		−	0.0428717i	−0.999770	−	0.0214358i	$-0.993176\pi$
$233$		−	1.92873e9i		−	0.0428717i	0.999770	−	0.0214358i	$-0.00682376\pi$
$234$		0			0
$235$		−	1.66922e10i		−	0.357032i
$236$		0			0
$237$	8.80230e9			0.181229
$238$		0			0
$239$	9.39248e10			1.86204			0.931021	−	0.364965i	$-0.118919\pi$
$239$	9.39248e10			1.86204			0.931021	+	0.364965i	$0.118919\pi$
$240$		0			0
$241$		−	4.52039e10i		−	0.863175i	−0.902071	−	0.431588i	$-0.857954\pi$
$241$		−	4.52039e10i		−	0.863175i	0.902071	−	0.431588i	$-0.142046\pi$
$242$		0			0
$243$			5.33508e10i			0.981551i
$244$		0			0
$245$		−	5.01715e10i		−	0.889630i
$246$		0			0
$247$	−1.10580e11			−1.89034
$248$		0			0
$249$			3.39075e9i			0.0558983i
$250$		0			0
$251$	4.87193e10			0.774763			0.387382	−	0.921919i	$-0.373380\pi$
$251$	4.87193e10			0.774763			0.387382	+	0.921919i	$0.373380\pi$
$252$		0			0
$253$	3.33043e10			0.511043
$254$		0			0
$255$	−3.66834e9	−	1.24785e10i	−0.0543299	−	0.184813i
$256$		0			0
$257$	8.25999e10			1.18108			0.590542	−	0.807007i	$-0.298914\pi$
$257$	8.25999e10			1.18108			0.590542	+	0.807007i	$0.298914\pi$
$258$		0			0
$259$	8.80135e10			1.21535
$260$		0			0
$261$		−	1.02431e11i		−	1.36630i
$262$		0			0
$263$	−3.46665e10			−0.446795			−0.223398	−	0.974727i	$-0.571715\pi$
$263$	−3.46665e10			−0.446795			−0.223398	+	0.974727i	$0.571715\pi$
$264$		0			0
$265$			1.55476e9i			0.0193667i
$266$		0			0
$267$			6.15901e10i			0.741669i
$268$		0			0
$269$			1.26130e11i			1.46871i	0.678768	+	0.734353i	$0.262514\pi$
$269$			1.26130e11i			1.46871i	−0.678768	+	0.734353i	$0.737486\pi$
$270$		0			0
$271$	−5.63928e10			−0.635129			−0.317564	−	0.948237i	$-0.602865\pi$
$271$	−5.63928e10			−0.635129			−0.317564	+	0.948237i	$0.602865\pi$
$272$		0			0
$273$	−8.62942e10			−0.940264
$274$		0			0
$275$		−	4.06189e10i		−	0.428283i
$276$		0			0
$277$			4.76405e10i			0.486203i	0.970001	+	0.243102i	$0.0781648\pi$
$277$			4.76405e10i			0.486203i	−0.970001	+	0.243102i	$0.921835\pi$
$278$		0			0
$279$		−	1.37445e11i		−	1.35803i
$280$		0			0
$281$	1.14333e11			1.09394			0.546969	−	0.837153i	$-0.315782\pi$
$281$	1.14333e11			1.09394			0.546969	+	0.837153i	$0.315782\pi$
$282$		0			0
$283$		−	2.54590e10i		−	0.235941i	−0.993017	−	0.117970i	$-0.962361\pi$
$283$		−	2.54590e10i		−	0.235941i	0.993017	−	0.117970i	$-0.0376388\pi$
$284$		0			0
$285$	−3.15295e10			−0.283084
$286$		0			0
$287$	2.58893e11			2.25243
$288$		0			0
$289$	−9.97217e10	+	6.41769e10i	−0.840910	+	0.541176i
$290$		0			0
$291$	6.14228e10			0.502124
$292$		0			0
$293$	4.43928e10			0.351891			0.175946	−	0.984400i	$-0.443702\pi$
$293$	4.43928e10			0.351891			0.175946	+	0.984400i	$0.443702\pi$
$294$		0			0
$295$			5.60634e9i			0.0431003i
$296$		0			0
$297$	5.58787e10			0.416718
$298$		0			0
$299$		−	1.68499e11i		−	1.21920i
$300$		0			0
$301$			3.16792e10i			0.222446i
$302$		0			0
$303$		−	2.69578e10i		−	0.183735i
$304$		0			0
$305$	3.92213e10			0.259521
$306$		0			0
$307$	−1.77005e11			−1.13727			−0.568635	−	0.822590i	$-0.692528\pi$
$307$	−1.77005e11			−1.13727			−0.568635	+	0.822590i	$0.692528\pi$
$308$		0			0
$309$			1.27185e10i			0.0793640i
$310$		0			0
$311$		−	1.16929e11i		−	0.708764i	−0.935101	−	0.354382i	$-0.884691\pi$
$311$		−	1.16929e11i		−	0.708764i	0.935101	−	0.354382i	$-0.115309\pi$
$312$		0			0
$313$		−	6.60894e10i		−	0.389208i	−0.980882	−	0.194604i	$-0.937658\pi$
$313$		−	6.60894e10i		−	0.389208i	0.980882	−	0.194604i	$-0.0623422\pi$
$314$		0			0
$315$	1.11778e11			0.639677
$316$		0			0
$317$		−	2.15865e11i		−	1.20065i	−0.799757	−	0.600324i	$-0.795038\pi$
$317$		−	2.15865e11i		−	1.20065i	0.799757	−	0.600324i	$-0.204962\pi$
$318$		0			0
$319$	−1.66245e11			−0.898854
$320$		0			0
$321$	7.45674e10			0.391991
$322$		0			0
$323$	8.10778e10	+	2.75801e11i	0.414468	+	1.40989i
$324$		0			0
$325$	−2.05506e11			−1.02176
$326$		0			0
$327$	5.60767e10			0.271217
$328$		0			0
$329$			2.87905e11i			1.35478i
$330$		0			0
$331$	−2.19673e11			−1.00589			−0.502946	−	0.864318i	$-0.667751\pi$
$331$	−2.19673e11			−1.00589			−0.502946	+	0.864318i	$0.667751\pi$
$332$		0			0
$333$			1.29877e11i			0.578808i
$334$		0			0
$335$		−	9.25987e10i		−	0.401701i
$336$		0			0
$337$		−	3.65398e11i		−	1.54323i	−0.636088	−	0.771617i	$-0.719448\pi$
$337$		−	3.65398e11i		−	1.54323i	0.636088	−	0.771617i	$-0.280552\pi$
$338$		0			0
$339$	−4.60510e9			−0.0189383
$340$		0			0
$341$	−2.23072e11			−0.893410
$342$		0			0
$343$			4.24205e11i			1.65482i
$344$		0			0
$345$		−	4.80438e10i		−	0.182579i
$346$		0			0
$347$			3.18226e11i			1.17829i	0.808026	+	0.589147i	$0.200536\pi$
$347$			3.18226e11i			1.17829i	−0.808026	+	0.589147i	$0.799464\pi$
$348$		0			0
$349$	−1.05189e11			−0.379537			−0.189769	−	0.981829i	$-0.560774\pi$
$349$	−1.05189e11			−0.379537			−0.189769	+	0.981829i	$0.560774\pi$
$350$		0			0
$351$		−	2.82711e11i		−	0.994171i
$352$		0			0
$353$	2.74596e11			0.941258			0.470629	−	0.882331i	$-0.344027\pi$
$353$	2.74596e11			0.941258			0.470629	+	0.882331i	$0.344027\pi$
$354$		0			0
$355$	1.32015e11			0.441161
$356$		0			0
$357$	6.32713e10	+	2.15229e11i	0.206158	+	0.701284i
$358$		0			0
$359$	−3.36158e11			−1.06812			−0.534058	−	0.845448i	$-0.679334\pi$
$359$	−3.36158e11			−1.06812			−0.534058	+	0.845448i	$0.679334\pi$
$360$		0			0
$361$	3.74179e11			1.15957
$362$		0			0
$363$			9.96614e10i			0.301264i
$364$		0			0
$365$	−4.71580e10			−0.139071
$366$		0			0
$367$			4.27664e11i			1.23057i	0.788306	+	0.615283i	$0.210958\pi$
$367$			4.27664e11i			1.23057i	−0.788306	+	0.615283i	$0.789042\pi$
$368$		0			0
$369$			3.82037e11i			1.07272i
$370$		0			0
$371$		−	2.68164e10i		−	0.0734882i
$372$		0			0
$373$	−6.59774e11			−1.76484			−0.882420	−	0.470463i	$-0.844087\pi$
$373$	−6.59774e11			−1.76484			−0.882420	+	0.470463i	$0.844087\pi$
$374$		0			0
$375$	−1.32364e11			−0.345645
$376$		0			0
$377$			8.41093e11i			2.14441i
$378$		0			0
$379$		−	3.76630e11i		−	0.937645i	−0.883292	−	0.468823i	$-0.844678\pi$
$379$		−	3.76630e11i		−	0.937645i	0.883292	−	0.468823i	$-0.155322\pi$
$380$		0			0
$381$		−	2.11482e11i		−	0.514174i
$382$		0			0
$383$	2.37194e11			0.563261			0.281630	−	0.959523i	$-0.409125\pi$
$383$	2.37194e11			0.563261			0.281630	+	0.959523i	$0.409125\pi$
$384$		0			0
$385$		−	1.81416e11i		−	0.420826i
$386$		0			0
$387$	−4.67475e10			−0.105940
$388$		0			0
$389$	−3.64753e10			−0.0807656			−0.0403828	−	0.999184i	$-0.512858\pi$
$389$	−3.64753e10			−0.0807656			−0.0403828	+	0.999184i	$0.512858\pi$
$390$		0			0
$391$	−4.20258e11	+	1.23544e11i	−0.909327	+	0.267317i
$392$		0			0
$393$	1.70415e11			0.360363
$394$		0			0
$395$	9.36238e10			0.193508
$396$		0			0
$397$		−	5.75210e11i		−	1.16217i	−0.813844	−	0.581084i	$-0.802629\pi$
$397$		−	5.75210e11i		−	1.16217i	0.813844	−	0.581084i	$-0.197371\pi$
$398$		0			0
$399$	5.43819e11			1.07418
$400$		0			0
$401$		−	3.39949e11i		−	0.656545i	−0.944583	−	0.328273i	$-0.893534\pi$
$401$		−	3.39949e11i		−	0.656545i	0.944583	−	0.328273i	$-0.106466\pi$
$402$		0			0
$403$			1.12860e12i			2.13142i
$404$		0			0
$405$			1.20646e11i			0.222825i
$406$		0			0
$407$	2.10791e11			0.380782
$408$		0			0
$409$	−1.12478e11			−0.198752			−0.0993760	−	0.995050i	$-0.531685\pi$
$409$	−1.12478e11			−0.198752			−0.0993760	+	0.995050i	$0.531685\pi$
$410$		0			0
$411$		−	8.09362e9i		−	0.0139912i
$412$		0			0
$413$		−	9.66978e10i		−	0.163547i
$414$		0			0
$415$			3.60650e10i			0.0596857i
$416$		0			0
$417$	2.50123e10			0.0405081
$418$		0			0
$419$		−	2.86731e10i		−	0.0454477i	−0.999742	−	0.0227239i	$-0.992766\pi$
$419$		−	2.86731e10i		−	0.0454477i	0.999742	−	0.0227239i	$-0.00723385\pi$
$420$		0			0
$421$	7.41818e11			1.15087			0.575437	−	0.817846i	$-0.304832\pi$
$421$	7.41818e11			1.15087			0.575437	+	0.817846i	$0.304832\pi$
$422$		0			0
$423$	−4.24849e11			−0.645212
$424$		0			0
$425$	1.50678e11	+	5.12559e11i	0.224027	+	0.762068i
$426$		0			0
$427$	−6.76487e11			−0.984768
$428$		0			0
$429$	−2.06673e11			−0.294595
$430$		0			0
$431$			9.85196e10i			0.137523i	0.997633	+	0.0687614i	$0.0219047\pi$
$431$			9.85196e10i			0.137523i	−0.997633	+	0.0687614i	$0.978095\pi$
$432$		0			0
$433$	−6.29756e11			−0.860948			−0.430474	−	0.902603i	$-0.641654\pi$
$433$	−6.29756e11			−0.860948			−0.430474	+	0.902603i	$0.641654\pi$
$434$		0			0
$435$			2.39819e11i			0.321131i
$436$		0			0
$437$			1.06186e12i			1.39284i
$438$		0			0
$439$		−	3.19248e10i		−	0.0410240i	−0.999790	−	0.0205120i	$-0.993470\pi$
$439$		−	3.19248e10i		−	0.0410240i	0.999790	−	0.0205120i	$-0.00652962\pi$
$440$		0			0
$441$	−1.27696e12			−1.60770
$442$		0			0
$443$	−9.76504e11			−1.20464			−0.602320	−	0.798255i	$-0.705757\pi$
$443$	−9.76504e11			−1.20464			−0.602320	+	0.798255i	$0.705757\pi$
$444$		0			0
$445$			6.55091e11i			0.791920i
$446$		0			0
$447$		−	4.80108e11i		−	0.568794i
$448$		0			0
$449$			3.18417e11i			0.369733i	0.982764	+	0.184867i	$0.0591853\pi$
$449$			3.18417e11i			0.369733i	−0.982764	+	0.184867i	$0.940815\pi$
$450$		0			0
$451$	6.20044e11			0.705713
$452$		0			0
$453$			3.40654e11i			0.380077i
$454$		0			0
$455$	−9.17851e11			−1.00397
$456$		0			0
$457$	5.17641e11			0.555144			0.277572	−	0.960705i	$-0.410470\pi$
$457$	5.17641e11			0.555144			0.277572	+	0.960705i	$0.410470\pi$
$458$		0			0
$459$	−7.05119e11	+	2.07285e11i	−0.741490	+	0.217977i
$460$		0			0
$461$	−1.03511e12			−1.06741			−0.533704	−	0.845671i	$-0.679200\pi$
$461$	−1.03511e12			−1.06741			−0.533704	+	0.845671i	$0.679200\pi$
$462$		0			0
$463$	4.59441e11			0.464638			0.232319	−	0.972640i	$-0.425369\pi$
$463$	4.59441e11			0.464638			0.232319	+	0.972640i	$0.425369\pi$
$464$		0			0
$465$			3.21797e11i			0.319186i
$466$		0			0
$467$	1.31601e12			1.28036			0.640182	−	0.768223i	$-0.278859\pi$
$467$	1.31601e12			1.28036			0.640182	+	0.768223i	$0.278859\pi$
$468$		0			0
$469$			1.59714e12i			1.52428i
$470$		0			0
$471$		−	2.22339e11i		−	0.208171i
$472$		0			0
$473$			7.58711e10i			0.0696949i
$474$		0			0
$475$	1.29508e12			1.16728
$476$		0			0
$477$	3.95717e10			0.0349987
$478$		0			0
$479$			2.08121e12i			1.80637i	0.429251	+	0.903185i	$0.358777\pi$
$479$			2.08121e12i			1.80637i	−0.429251	+	0.903185i	$0.641223\pi$
$480$		0			0
$481$		−	1.06647e12i		−	0.908438i
$482$		0			0
$483$			8.28655e11i			0.692806i
$484$		0			0
$485$	6.53311e11			0.536145
$486$		0			0
$487$		−	7.18370e11i		−	0.578719i	−0.957220	−	0.289360i	$-0.906558\pi$
$487$		−	7.18370e11i		−	0.578719i	0.957220	−	0.289360i	$-0.0934424\pi$
$488$		0			0
$489$	2.24380e11			0.177457
$490$		0			0
$491$	−9.78925e11			−0.760121			−0.380060	−	0.924962i	$-0.624097\pi$
$491$	−9.78925e11			−0.760121			−0.380060	+	0.924962i	$0.624097\pi$
$492$		0			0
$493$	2.09780e12	−	6.16693e11i	1.59938	−	0.470173i
$494$		0			0
$495$	2.67707e11			0.200418
$496$		0			0
$497$	−2.27699e12			−1.67401
$498$		0			0
$499$		−	1.70045e12i		−	1.22775i	−0.789402	−	0.613876i	$-0.789609\pi$
$499$		−	1.70045e12i		−	1.22775i	0.789402	−	0.613876i	$-0.210391\pi$
$500$		0			0
$501$	4.58850e11			0.325388
$502$		0			0
$503$		−	1.03194e11i		−	0.0718787i	−0.999354	−	0.0359393i	$-0.988558\pi$
$503$		−	1.03194e11i		−	0.0718787i	0.999354	−	0.0359393i	$-0.0114423\pi$
$504$		0			0
$505$		−	2.86731e11i		−	0.196184i
$506$		0			0
$507$			4.13709e11i			0.278073i
$508$		0			0
$509$	7.49118e11			0.494675			0.247338	−	0.968929i	$-0.420444\pi$
$509$	7.49118e11			0.494675			0.247338	+	0.968929i	$0.420444\pi$
$510$		0			0
$511$	8.13378e11			0.527714
$512$		0			0
$513$			1.78162e12i			1.13576i
$514$		0			0
$515$			1.35278e11i			0.0847412i
$516$		0			0
$517$			6.89528e11i			0.424467i
$518$		0			0
$519$	−3.55246e11			−0.214920
$520$		0			0
$521$		−	1.51334e12i		−	0.899843i	−0.893068	−	0.449921i	$-0.851452\pi$
$521$		−	1.51334e12i		−	0.899843i	0.893068	−	0.449921i	$-0.148548\pi$
$522$		0			0
$523$	−1.87630e12			−1.09659			−0.548295	−	0.836285i	$-0.684723\pi$
$523$	−1.87630e12			−1.09659			−0.548295	+	0.836285i	$0.684723\pi$
$524$		0			0
$525$	1.01065e12			0.580611
$526$		0			0
$527$	2.81489e12	−	8.27498e11i	1.58969	−	0.467325i
$528$		0			0
$529$	1.83116e11			0.101666
$530$		0			0
$531$	1.42692e11			0.0778889
$532$		0			0
$533$		−	3.13703e12i		−	1.68363i
$534$		0			0
$535$	7.93121e11			0.418550
$536$		0			0
$537$		−	2.92112e11i		−	0.151588i
$538$		0			0
$539$			2.07251e12i			1.05766i
$540$		0			0
$541$			2.51866e12i			1.26410i	0.774927	+	0.632050i	$0.217786\pi$
$541$			2.51866e12i			1.26410i	−0.774927	+	0.632050i	$0.782214\pi$
$542$		0			0
$543$	4.00237e11			0.197569
$544$		0			0
$545$	5.96448e11			0.289593
$546$		0			0
$547$			2.33711e12i			1.11618i	0.829779	+	0.558091i	$0.188466\pi$
$547$			2.33711e12i			1.11618i	−0.829779	+	0.558091i	$0.811534\pi$
$548$		0			0
$549$		−	9.98260e11i		−	0.468995i
$550$		0			0
$551$		−	5.30050e12i		−	2.44982i
$552$		0			0
$553$	−1.61482e12			−0.734278
$554$		0			0
$555$		−	3.04081e11i		−	0.136041i
$556$		0			0
$557$	7.35788e11			0.323895			0.161948	−	0.986799i	$-0.448222\pi$
$557$	7.35788e11			0.323895			0.161948	+	0.986799i	$0.448222\pi$
$558$		0			0
$559$	3.83860e11			0.166272
$560$		0			0
$561$	1.51534e11	+	5.15470e11i	0.0645916	+	0.219720i
$562$		0			0
$563$	−2.80471e12			−1.17652			−0.588262	−	0.808670i	$-0.700188\pi$
$563$	−2.80471e12			−1.17652			−0.588262	+	0.808670i	$0.700188\pi$
$564$		0			0
$565$	−4.89812e10			−0.0202214
$566$		0			0
$567$		−	2.08089e12i		−	0.845521i
$568$		0			0
$569$	−7.41576e11			−0.296586			−0.148293	−	0.988943i	$-0.547378\pi$
$569$	−7.41576e11			−0.296586			−0.148293	+	0.988943i	$0.547378\pi$
$570$		0			0
$571$			2.37109e12i			0.933440i	0.884405	+	0.466720i	$0.154564\pi$
$571$			2.37109e12i			0.933440i	−0.884405	+	0.466720i	$0.845436\pi$
$572$		0			0
$573$			1.10052e12i			0.426485i
$574$		0			0
$575$			1.97341e12i			0.752855i
$576$		0			0
$577$	1.64213e12			0.616759			0.308380	−	0.951263i	$-0.400213\pi$
$577$	1.64213e12			0.616759			0.308380	+	0.951263i	$0.400213\pi$
$578$		0			0
$579$	2.01245e12			0.744168
$580$		0			0
$581$		−	6.22047e11i		−	0.226481i
$582$		0			0
$583$		−	6.42247e10i		−	0.0230247i
$584$		0			0
$585$		−	1.35443e12i		−	0.478140i
$586$		0			0
$587$	−4.40209e12			−1.53034			−0.765168	−	0.643830i	$-0.777344\pi$
$587$	−4.40209e12			−1.53034			−0.765168	+	0.643830i	$0.777344\pi$
$588$		0			0
$589$		−	7.11238e12i		−	2.43498i
$590$		0			0
$591$	3.72581e11			0.125625
$592$		0			0
$593$	−3.23487e12			−1.07426			−0.537132	−	0.843498i	$-0.680492\pi$
$593$	−3.23487e12			−1.07426			−0.537132	+	0.843498i	$0.680492\pi$
$594$		0			0
$595$	6.72972e11	+	2.28924e12i	0.220126	+	0.748799i
$596$		0			0
$597$	8.71049e11			0.280645
$598$		0			0
$599$	−4.90816e12			−1.55775			−0.778875	−	0.627179i	$-0.784209\pi$
$599$	−4.90816e12			−1.55775			−0.778875	+	0.627179i	$0.784209\pi$
$600$		0			0
$601$			1.05961e12i			0.331292i	0.986185	+	0.165646i	$0.0529710\pi$
$601$			1.05961e12i			0.331292i	−0.986185	+	0.165646i	$0.947029\pi$
$602$		0			0
$603$	−2.35682e12			−0.725936
$604$		0			0
$605$			1.06003e12i			0.321676i
$606$		0			0
$607$			5.76988e11i			0.172511i	0.996273	+	0.0862556i	$0.0274902\pi$
$607$			5.76988e11i			0.172511i	−0.996273	+	0.0862556i	$0.972510\pi$
$608$		0			0
$609$		−	4.13639e12i		−	1.21855i
$610$		0			0
$611$	3.48858e12			1.01266
$612$		0			0
$613$	−5.09644e12			−1.45779			−0.728895	−	0.684625i	$-0.759966\pi$
$613$	−5.09644e12			−1.45779			−0.728895	+	0.684625i	$0.759966\pi$
$614$		0			0
$615$		−	8.94458e11i		−	0.252128i
$616$		0			0
$617$		−	5.92498e12i		−	1.64590i	−0.568113	−	0.822951i	$-0.692326\pi$
$617$		−	5.92498e12i		−	1.64590i	0.568113	−	0.822951i	$-0.307674\pi$
$618$		0			0
$619$			3.09980e12i			0.848644i	0.905511	+	0.424322i	$0.139487\pi$
$619$			3.09980e12i			0.848644i	−0.905511	+	0.424322i	$0.860513\pi$
$620$		0			0
$621$	−2.71478e12			−0.732526
$622$		0			0
$623$		−	1.12990e13i		−	3.00498i
$624$		0			0
$625$	1.62220e12			0.425251
$626$		0			0
$627$	1.30244e12			0.336552
$628$		0			0
$629$	−2.65991e12	+	7.81939e11i	−0.677547	+	0.199180i
$630$		0			0
$631$	3.86695e11			0.0971040			0.0485520	−	0.998821i	$-0.484539\pi$
$631$	3.86695e11			0.0971040			0.0485520	+	0.998821i	$0.484539\pi$
$632$		0			0
$633$	1.60441e12			0.397191
$634$		0			0
$635$		−	2.24938e12i		−	0.549011i
$636$		0			0
$637$	1.04856e13			2.52328
$638$		0			0
$639$		−	3.36005e12i		−	0.797245i
$640$		0			0
$641$			5.83791e12i			1.36583i	0.730499	+	0.682914i	$0.239288\pi$
$641$			5.83791e12i			1.36583i	−0.730499	+	0.682914i	$0.760712\pi$
$642$		0			0
$643$		−	4.53130e12i		−	1.04538i	−0.852523	−	0.522689i	$-0.824929\pi$
$643$		−	4.53130e12i		−	1.04538i	0.852523	−	0.522689i	$-0.175071\pi$
$644$		0			0
$645$	1.09449e11			0.0248997
$646$		0			0
$647$	8.18289e12			1.83585			0.917925	−	0.396754i	$-0.129863\pi$
$647$	8.18289e12			1.83585			0.917925	+	0.396754i	$0.129863\pi$
$648$		0			0
$649$		−	2.31589e11i		−	0.0512410i
$650$		0			0
$651$		−	5.55034e12i		−	1.21117i
$652$		0			0
$653$			2.43949e12i			0.525036i	0.964927	+	0.262518i	$0.0845529\pi$
$653$			2.43949e12i			0.525036i	−0.964927	+	0.262518i	$0.915447\pi$
$654$		0			0
$655$	1.81258e12			0.384779
$656$		0			0
$657$			1.20026e12i			0.251323i
$658$		0			0
$659$	1.61269e12			0.333094			0.166547	−	0.986034i	$-0.446738\pi$
$659$	1.61269e12			0.333094			0.166547	+	0.986034i	$0.446738\pi$
$660$		0			0
$661$	7.30795e11			0.148898			0.0744491	−	0.997225i	$-0.476280\pi$
$661$	7.30795e11			0.148898			0.0744491	+	0.997225i	$0.476280\pi$
$662$		0			0
$663$	2.60795e12	−	7.66664e11i	0.524190	−	0.154097i
$664$		0			0
$665$	5.78422e12			1.14696
$666$		0			0
$667$	8.07674e12			1.58005
$668$		0			0
$669$		−	6.49787e11i		−	0.125416i
$670$		0			0
$671$	−1.62017e12			−0.308539
$672$		0			0
$673$		−	4.80522e12i		−	0.902911i	−0.892294	−	0.451456i	$-0.850905\pi$
$673$		−	4.80522e12i		−	0.902911i	0.892294	−	0.451456i	$-0.149095\pi$
$674$		0			0
$675$			3.31103e12i			0.613899i
$676$		0			0
$677$			2.48514e12i			0.454675i	0.973816	+	0.227337i	$0.0730020\pi$
$677$			2.48514e12i			0.454675i	−0.973816	+	0.227337i	$0.926998\pi$
$678$		0			0
$679$	−1.12683e13			−2.03443
$680$		0			0
$681$	3.41878e12			0.609130
$682$		0			0
$683$		−	8.09671e11i		−	0.142369i	−0.997463	−	0.0711845i	$-0.977322\pi$
$683$		−	8.09671e11i		−	0.142369i	0.997463	−	0.0711845i	$-0.0226779\pi$
$684$		0			0
$685$		−	8.60861e10i		−	0.0149391i
$686$		0			0
$687$			2.89126e12i			0.495202i
$688$		0			0
$689$	−3.24937e11			−0.0549303
$690$		0			0
$691$		−	3.01668e12i		−	0.503359i	−0.967811	−	0.251679i	$-0.919017\pi$
$691$		−	3.01668e12i		−	0.503359i	0.967811	−	0.251679i	$-0.0809829\pi$
$692$		0			0
$693$	−4.61740e12			−0.760497
$694$		0			0
$695$	2.66038e11			0.0432526
$696$		0			0
$697$	−7.82417e12	+	2.30009e12i	−1.25571	+	0.369145i
$698$		0			0
$699$	1.14934e11			0.0182096
$700$		0			0
$701$	−4.71932e12			−0.738156			−0.369078	−	0.929398i	$-0.620326\pi$
$701$	−4.71932e12			−0.738156			−0.369078	+	0.929398i	$0.620326\pi$
$702$		0			0
$703$			6.72080e12i			1.03782i
$704$		0			0
$705$	9.94693e11			0.151648
$706$		0			0
$707$			4.94552e12i			0.744432i
$708$		0			0
$709$		−	8.19568e12i		−	1.21808i	−0.793138	−	0.609041i	$-0.791554\pi$
$709$		−	8.19568e12i		−	1.21808i	0.793138	−	0.609041i	$-0.208446\pi$
$710$		0			0
$711$		−	2.38291e12i		−	0.349699i
$712$		0			0
$713$	1.08376e13			1.57048
$714$		0			0
$715$	−2.19824e12			−0.314555
$716$		0			0
$717$			5.59701e12i			0.790897i
$718$		0			0
$719$		−	1.16808e13i		−	1.63001i	−0.579452	−	0.815006i	$-0.696733\pi$
$719$		−	1.16808e13i		−	1.63001i	0.579452	−	0.815006i	$-0.303267\pi$
$720$		0			0
$721$		−	2.33327e12i		−	0.321555i
$722$		0			0
$723$	2.69372e12			0.366631
$724$		0			0
$725$		−	9.85064e12i		−	1.32417i
$726$		0			0
$727$	5.18587e12			0.688520			0.344260	−	0.938874i	$-0.388130\pi$
$727$	5.18587e12			0.688520			0.344260	+	0.938874i	$0.388130\pi$
$728$		0			0
$729$	5.67392e11			0.0744063
$730$		0			0
$731$	−2.81448e11	−	9.57397e11i	−0.0364561	−	0.124012i
$732$		0			0
$733$	9.00169e10			0.0115175			0.00575873	−	0.999983i	$-0.498167\pi$
$733$	9.00169e10			0.0115175			0.00575873	+	0.999983i	$0.498167\pi$
$734$		0			0
$735$	2.98974e12			0.377868
$736$		0			0
$737$			3.82511e12i			0.477573i
$738$		0			0
$739$	−1.27401e13			−1.57135			−0.785677	−	0.618638i	$-0.787685\pi$
$739$	−1.27401e13			−1.57135			−0.785677	+	0.618638i	$0.787685\pi$
$740$		0			0
$741$		−	6.58951e12i		−	0.802918i
$742$		0			0
$743$			7.98450e11i			0.0961165i	0.998845	+	0.0480583i	$0.0153033\pi$
$743$			7.98450e11i			0.0961165i	−0.998845	+	0.0480583i	$0.984697\pi$
$744$		0			0
$745$		−	5.10657e12i		−	0.607332i
$746$		0			0
$747$	9.17926e11			0.107861
$748$		0			0
$749$	−1.36797e13			−1.58821
$750$		0			0
$751$		−	9.56992e12i		−	1.09781i	−0.835883	−	0.548907i	$-0.815044\pi$
$751$		−	9.56992e12i		−	1.09781i	0.835883	−	0.548907i	$-0.184956\pi$
$752$		0			0
$753$			2.90320e12i			0.329078i
$754$		0			0
$755$			3.62330e12i			0.405829i
$756$		0			0
$757$	1.18435e13			1.31084			0.655418	−	0.755267i	$-0.272493\pi$
$757$	1.18435e13			1.31084			0.655418	+	0.755267i	$0.272493\pi$
$758$		0			0
$759$			1.98461e12i			0.217064i
$760$		0			0
$761$	7.12456e12			0.770064			0.385032	−	0.922903i	$-0.374190\pi$
$761$	7.12456e12			0.770064			0.385032	+	0.922903i	$0.374190\pi$
$762$		0			0
$763$	−1.02875e13			−1.09888
$764$		0			0
$765$	−3.37813e12	+	9.93074e11i	−0.356615	+	0.104835i
$766$		0			0
$767$	−1.17170e12			−0.122246
$768$		0			0
$769$	−1.51364e13			−1.56082			−0.780411	−	0.625267i	$-0.784990\pi$
$769$	−1.51364e13			−1.56082			−0.780411	+	0.625267i	$0.784990\pi$
$770$		0			0
$771$			4.92216e12i			0.501662i
$772$		0			0
$773$	−6.55361e12			−0.660196			−0.330098	−	0.943947i	$-0.607082\pi$
$773$	−6.55361e12			−0.660196			−0.330098	+	0.943947i	$0.607082\pi$
$774$		0			0
$775$		−	1.32179e13i		−	1.31615i
$776$		0			0
$777$			5.24476e12i			0.516216i
$778$		0			0
$779$			1.97693e13i			1.92342i
$780$		0			0
$781$	−5.45335e12			−0.524486
$782$		0			0
$783$	1.35514e13			1.28841
$784$		0			0
$785$		−	2.36486e12i		−	0.222276i
$786$		0			0
$787$			1.72441e13i			1.60234i	0.598439	+	0.801168i	$0.295788\pi$
$787$			1.72441e13i			1.60234i	−0.598439	+	0.801168i	$0.704212\pi$
$788$		0			0
$789$		−	2.06579e12i		−	0.189775i
$790$		0			0
$791$	8.44824e11			0.0767312
$792$		0			0
$793$			8.19706e12i			0.736086i
$794$		0			0
$795$	−9.26487e10			−0.00822597
$796$		0			0
$797$	1.92951e13			1.69389			0.846944	−	0.531682i	$-0.178440\pi$
$797$	1.92951e13			1.69389			0.846944	+	0.531682i	$0.178440\pi$
$798$		0			0
$799$	−2.55784e12	−	8.70097e12i	−0.222031	−	0.755278i
$800$		0			0
$801$	1.66733e13			1.43112
$802$		0			0
$803$	1.94803e12			0.165339
$804$		0			0
$805$			8.81382e12i			0.739746i
$806$		0			0
$807$	−7.51616e12			−0.623829
$808$		0			0
$809$			3.75180e12i			0.307944i	0.988075	+	0.153972i	$0.0492065\pi$
$809$			3.75180e12i			0.307944i	−0.988075	+	0.153972i	$0.950794\pi$
$810$		0			0
$811$		−	4.21763e12i		−	0.342353i	−0.985240	−	0.171177i	$-0.945243\pi$
$811$		−	4.21763e12i		−	0.342353i	0.985240	−	0.171177i	$-0.0547569\pi$
$812$		0			0
$813$		−	3.36047e12i		−	0.269769i
$814$		0			0
$815$	2.38657e12			0.189481
$816$		0			0
$817$	−2.41905e12			−0.189953
$818$		0			0
$819$			2.33611e13i			1.81433i
$820$		0			0
$821$			2.89064e12i			0.222050i	0.993818	+	0.111025i	$0.0354133\pi$
$821$			2.89064e12i			0.222050i	−0.993818	+	0.111025i	$0.964587\pi$
$822$		0			0
$823$		−	1.85820e11i		−	0.0141186i	−0.999975	−	0.00705931i	$-0.997753\pi$
$823$		−	1.85820e11i		−	0.0141186i	0.999975	−	0.00705931i	$-0.00224707\pi$
$824$		0			0
$825$	2.42050e12			0.181912
$826$		0			0
$827$			2.19451e13i			1.63141i	0.578470	+	0.815704i	$0.303650\pi$
$827$			2.19451e13i			1.63141i	−0.578470	+	0.815704i	$0.696350\pi$
$828$		0			0
$829$	7.80531e12			0.573977			0.286989	−	0.957934i	$-0.407346\pi$
$829$	7.80531e12			0.573977			0.286989	+	0.957934i	$0.407346\pi$
$830$		0			0
$831$	−2.83892e12			−0.206513
$832$		0			0
$833$	−7.68807e12	−	2.61524e13i	−0.553242	−	1.88196i
$834$		0			0
$835$	4.88047e12			0.347434
$836$		0			0
$837$	1.81836e13			1.28061
$838$		0			0
$839$		−	1.83657e13i		−	1.27962i	−0.768535	−	0.639808i	$-0.779014\pi$
$839$		−	1.83657e13i		−	1.27962i	0.768535	−	0.639808i	$-0.220986\pi$
$840$		0			0
$841$	−2.58094e13			−1.77908
$842$		0			0
$843$			6.81314e12i			0.464647i
$844$		0			0
$845$			4.40033e12i			0.296914i
$846$		0			0
$847$		−	1.82833e13i		−	1.22062i
$848$		0			0
$849$	1.51711e12			0.100215
$850$		0			0
$851$	−1.02409e13			−0.669356
$852$		0			0
$853$			1.15587e13i			0.747544i	0.927521	+	0.373772i	$0.121936\pi$
$853$			1.15587e13i			0.747544i	−0.927521	+	0.373772i	$0.878064\pi$
$854$		0			0
$855$			8.53551e12i			0.546238i
$856$		0			0
$857$			3.67551e12i			0.232757i	0.993205	+	0.116379i	$0.0371286\pi$
$857$			3.67551e12i			0.232757i	−0.993205	+	0.116379i	$0.962871\pi$
$858$		0			0
$859$	2.36948e13			1.48485			0.742426	−	0.669928i	$-0.233675\pi$
$859$	2.36948e13			1.48485			0.742426	+	0.669928i	$0.233675\pi$
$860$		0			0
$861$			1.54275e13i			0.956715i
$862$		0			0
$863$	−8.18769e12			−0.502473			−0.251237	−	0.967926i	$-0.580837\pi$
$863$	−8.18769e12			−0.502473			−0.251237	+	0.967926i	$0.580837\pi$
$864$		0			0
$865$	−3.77850e12			−0.229481
$866$		0			0
$867$	−3.82432e12	−	5.94245e12i	−0.229863	−	0.357174i
$868$		0			0
$869$	−3.86746e12			−0.230058
$870$		0			0
$871$	1.93526e13			1.13935
$872$		0			0
$873$		−	1.66281e13i		−	0.968896i
$874$		0			0
$875$	2.42828e13			1.40043
$876$		0			0
$877$			1.34379e13i			0.767064i	0.923528	+	0.383532i	$0.125292\pi$
$877$			1.34379e13i			0.767064i	−0.923528	+	0.383532i	$0.874708\pi$
$878$		0			0
$879$			2.64538e12i			0.149465i
$880$		0			0
$881$			3.19613e13i			1.78745i	0.448618	+	0.893724i	$0.351916\pi$
$881$			3.19613e13i			1.78745i	−0.448618	+	0.893724i	$0.648084\pi$
$882$		0			0
$883$	−9.21484e12			−0.510111			−0.255056	−	0.966926i	$-0.582094\pi$
$883$	−9.21484e12			−0.510111			−0.255056	+	0.966926i	$0.582094\pi$
$884$		0			0
$885$	−3.34084e11			−0.0183067
$886$		0			0
$887$		−	5.95887e12i		−	0.323227i	−0.986854	−	0.161614i	$-0.948330\pi$
$887$		−	5.95887e12i		−	0.323227i	0.986854	−	0.161614i	$-0.0516698\pi$
$888$		0			0
$889$			3.87972e13i			2.08325i
$890$		0			0
$891$		−	4.98368e12i		−	0.264911i
$892$		0			0
$893$	−2.19847e13			−1.15688
$894$		0			0
$895$		−	3.10699e12i		−	0.161859i
$896$		0			0
$897$	1.00409e13			0.517853
$898$		0			0
$899$	−5.40981e13			−2.76225
$900$		0			0
$901$	2.38245e11	+	8.10434e11i	0.0120438	+	0.0409691i
$902$		0			0
$903$	−1.88778e12			−0.0944834
$904$		0			0
$905$	4.25704e12			0.210955
$906$		0			0
$907$			3.49523e13i			1.71491i	0.514555	+	0.857457i	$0.327957\pi$
$907$			3.49523e13i			1.71491i	−0.514555	+	0.857457i	$0.672043\pi$
$908$		0			0
$909$	−7.29788e12			−0.354535
$910$		0			0
$911$			1.82282e12i			0.0876819i	0.999039	+	0.0438410i	$0.0139595\pi$
$911$			1.82282e12i			0.0876819i	−0.999039	+	0.0438410i	$0.986041\pi$
$912$		0			0
$913$		−	1.48979e12i		−	0.0709589i
$914$		0			0
$915$			2.33721e12i			0.110231i
$916$		0			0
$917$	−3.12633e13			−1.46007
$918$		0			0
$919$	1.16854e13			0.540411			0.270206	−	0.962803i	$-0.412908\pi$
$919$	1.16854e13			0.540411			0.270206	+	0.962803i	$0.412908\pi$
$920$		0			0
$921$		−	1.05478e13i		−	0.483052i
$922$		0			0
$923$			2.75905e13i			1.25127i
$924$		0			0
$925$			1.24902e13i			0.560959i
$926$		0			0
$927$	3.44309e12			0.153140
$928$		0			0
$929$		−	2.03224e13i		−	0.895166i	−0.894242	−	0.447583i	$-0.852285\pi$
$929$		−	2.03224e13i		−	0.895166i	0.894242	−	0.447583i	$-0.147715\pi$
$930$		0			0
$931$	−6.60793e13			−2.88265
$932$		0			0
$933$	6.96787e12			0.301046
$934$		0			0
$935$	1.61176e12	+	5.48269e12i	0.0689679	+	0.234607i
$936$		0			0
$937$	−1.03126e13			−0.437061			−0.218530	−	0.975830i	$-0.570126\pi$
$937$	−1.03126e13			−0.437061			−0.218530	+	0.975830i	$0.570126\pi$
$938$		0			0
$939$	3.93829e12			0.165315
$940$		0			0
$941$			9.97481e12i			0.414717i	0.978265	+	0.207358i	$0.0664866\pi$
$941$			9.97481e12i			0.414717i	−0.978265	+	0.207358i	$0.933513\pi$
$942$		0			0
$943$	−3.01239e13			−1.24053
$944$		0			0
$945$			1.47881e13i			0.603209i
$946$		0			0
$947$		−	7.55330e12i		−	0.305184i	−0.988289	−	0.152592i	$-0.951238\pi$
$947$		−	7.55330e12i		−	0.305184i	0.988289	−	0.152592i	$-0.0487620\pi$
$948$		0			0
$949$		−	9.85578e12i		−	0.394451i
$950$		0			0
$951$	1.28635e13			0.509972
$952$		0			0
$953$	5.11924e12			0.201042			0.100521	−	0.994935i	$-0.467949\pi$
$953$	5.11924e12			0.201042			0.100521	+	0.994935i	$0.467949\pi$
$954$		0			0
$955$			1.17055e13i			0.455381i
$956$		0			0
$957$		−	9.90657e12i		−	0.381786i
$958$		0			0
$959$			1.48481e12i			0.0566874i
$960$		0			0
$961$	−4.61509e13			−1.74552
$962$		0			0
$963$		−	2.01865e13i		−	0.756384i
$964$		0			0
$965$	2.14050e13			0.794588
$966$		0			0
$967$	2.96090e13			1.08894			0.544471	−	0.838780i	$-0.316731\pi$
$967$	2.96090e13			1.08894			0.544471	+	0.838780i	$0.316731\pi$
$968$		0			0
$969$	−1.64351e13	+	4.83146e12i	−0.598846	+	0.176044i
$970$		0			0
$971$	2.36780e13			0.854790			0.427395	−	0.904065i	$-0.359431\pi$
$971$	2.36780e13			0.854790			0.427395	+	0.904065i	$0.359431\pi$
$972$		0			0
$973$	−4.58861e12			−0.164124
$974$		0			0
$975$		−	1.22462e13i		−	0.433990i
$976$		0			0
$977$	1.30708e13			0.458961			0.229480	−	0.973313i	$-0.426297\pi$
$977$	1.30708e13			0.458961			0.229480	+	0.973313i	$0.426297\pi$
$978$		0			0
$979$		−	2.70608e13i		−	0.941496i
$980$		0			0
$981$		−	1.51808e13i		−	0.523339i
$982$		0			0
$983$		−	1.05637e13i		−	0.360850i	−0.983589	−	0.180425i	$-0.942253\pi$
$983$		−	1.05637e13i		−	0.360850i	0.983589	−	0.180425i	$-0.0577473\pi$
$984$		0			0
$985$	3.96288e12			0.134137
$986$		0			0
$987$	−1.71564e13			−0.575438
$988$		0			0
$989$		−	3.68608e12i		−	0.122513i
$990$		0			0
$991$			4.60965e13i			1.51823i	0.650958	+	0.759113i	$0.274367\pi$
$991$			4.60965e13i			1.51823i	−0.650958	+	0.759113i	$0.725633\pi$
$992$		0			0
$993$		−	1.30904e13i		−	0.427250i
$994$		0			0
$995$	9.26473e12			0.299660
$996$		0			0
$997$			2.60737e13i			0.835748i	0.908505	+	0.417874i	$0.137225\pi$
$997$			2.60737e13i			0.835748i	−0.908505	+	0.417874i	$0.862775\pi$
$998$		0			0
$999$	−1.71825e13			−0.545811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.b.c.33.8		12
4.3	odd	2		17.10.b.a.16.7	✓	12
12.11	even	2		153.10.d.b.118.5		12
17.16	even	2	inner	272.10.b.c.33.5		12
68.47	odd	4		289.10.a.c.1.6		12
68.55	odd	4		289.10.a.c.1.5		12
68.67	odd	2		17.10.b.a.16.8	yes	12
204.203	even	2		153.10.d.b.118.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.b.a.16.7	✓	12	4.3	odd	2
17.10.b.a.16.8	yes	12	68.67	odd	2
153.10.d.b.118.5		12	12.11	even	2
153.10.d.b.118.6		12	204.203	even	2
272.10.b.c.33.5		12	17.16	even	2	inner
272.10.b.c.33.8		12	1.1	even	1	trivial
289.10.a.c.1.5		12	68.55	odd	4
289.10.a.c.1.6		12	68.47	odd	4

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:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	272.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+59.5904i q^{3} +633.821i q^{5} -10932.1i q^{7} +16132.0 q^{9} +O(q^{10})\)	\(q+59.5904i q^{3} +633.821i q^{5} -10932.1i q^{7} +16132.0 q^{9} -26182.2i q^{11} -132465. q^{13} -37769.6 q^{15} +(97124.1 + 330386. i) q^{17} +834785. q^{19} +651448. q^{21} +1.27202e6i q^{23} +1.55140e6 q^{25} +2.13423e6i q^{27} -6.34953e6i q^{29} -8.52001e6i q^{31} +1.56021e6 q^{33} +6.92899e6 q^{35} +8.05093e6i q^{37} -7.89366e6i q^{39} +2.36819e7i q^{41} -2.89782e6 q^{43} +1.02248e7i q^{45} -2.63358e7 q^{47} -7.91572e7 q^{49} +(-1.96878e7 + 5.78766e6i) q^{51} +2.45299e6 q^{53} +1.65948e7 q^{55} +4.97452e7i q^{57} +8.84531e6 q^{59} -6.18808e7i q^{61} -1.76356e8i q^{63} -8.39593e7i q^{65} -1.46096e8 q^{67} -7.58002e7 q^{69} -2.08285e8i q^{71} +7.44027e7i q^{73} +9.24483e7i q^{75} -2.86226e8 q^{77} -1.47713e8i q^{79} +1.90346e8 q^{81} +5.69010e7 q^{83} +(-2.09405e8 + 6.15593e7i) q^{85} +3.78371e8 q^{87} +1.03356e9 q^{89} +1.44812e9i q^{91} +5.07710e8 q^{93} +5.29104e8i q^{95} -1.03075e9i q^{97} -4.22370e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 9184 q^{9}+O(q^{10}) \) 12 * q - 9184 * q^9	\( 12 q - 9184 q^{9} - 63204 q^{13} + 243480 q^{15} - 105960 q^{17} - 1110672 q^{19} - 172580 q^{21} - 4441796 q^{25} - 6557404 q^{33} - 3519864 q^{35} - 10004616 q^{43} + 112552440 q^{47} + 121354720 q^{49} + 52506472 q^{51} + 76804272 q^{53} - 300732568 q^{55} - 11618904 q^{59} + 304208752 q^{67} - 211308236 q^{69} + 138357828 q^{77} - 363335792 q^{81} + 845042136 q^{83} - 388949632 q^{85} - 610860648 q^{87} - 938223804 q^{89} + 1635779524 q^{93}+O(q^{100}) \) 12 * q - 9184 * q^9 - 63204 * q^13 + 243480 * q^15 - 105960 * q^17 - 1110672 * q^19 - 172580 * q^21 - 4441796 * q^25 - 6557404 * q^33 - 3519864 * q^35 - 10004616 * q^43 + 112552440 * q^47 + 121354720 * q^49 + 52506472 * q^51 + 76804272 * q^53 - 300732568 * q^55 - 11618904 * q^59 + 304208752 * q^67 - 211308236 * q^69 + 138357828 * q^77 - 363335792 * q^81 + 845042136 * q^83 - 388949632 * q^85 - 610860648 * q^87 - 938223804 * q^89 + 1635779524 * q^93

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