LMFDB - Embedded newform 272.12.b.c.33.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [272,12,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, 12, names="a")

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

chi := DirichletCharacter("272.33");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 12 $
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:	$208.989345112$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2012924 x^{14} + 1580196076372 x^{12} + \cdots + 11\!\cdots\!16 $ x^16 + 2012924x^14 + 1580196076372x^12 + 612185482864471968x^10 + 122089588908147616630464x^8 + 12002063596127081323758696960x^6 + 538160293280924897115168650612736x^4 + 8663850655413946592880080714919542784*x^2 + 11159369979872736421907749802317490159616
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{59}\cdot 3^{4}\cdot 17^{2} $
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			33.9
Root			$37.5132i$ of defining polynomial
Character	$\chi$	$=$	272.33
Dual form			272.12.b.c.33.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+37.5132i q^{3} -7015.33i q^{5} +40939.8i q^{7} +175740. q^{9} +O(q^{10})$	$q+37.5132i q^{3} -7015.33i q^{5} +40939.8i q^{7} +175740. q^{9} +436092. i q^{11} +954432. q^{13} +263168. q^{15} +(1.84007e6 + 5.55752e6i) q^{17} +2.78988e6 q^{19} -1.53578e6 q^{21} +4.14357e7i q^{23} -386743. q^{25} +1.32379e7i q^{27} +1.18734e8i q^{29} -1.77240e8i q^{31} -1.63592e7 q^{33} +2.87206e8 q^{35} +5.80565e8i q^{37} +3.58038e7i q^{39} +1.07835e8i q^{41} +1.42735e7 q^{43} -1.23287e9i q^{45} -1.65181e9 q^{47} +3.01260e8 q^{49} +(-2.08480e8 + 6.90271e7i) q^{51} -1.69980e9 q^{53} +3.05933e9 q^{55} +1.04658e8i q^{57} -1.97623e9 q^{59} -7.84931e9i q^{61} +7.19475e9i q^{63} -6.69565e9i q^{65} -1.92771e10 q^{67} -1.55439e9 q^{69} -4.81174e9i q^{71} -1.24976e10i q^{73} -1.45080e7i q^{75} -1.78535e10 q^{77} +2.94120e10i q^{79} +3.06352e10 q^{81} -2.55654e9 q^{83} +(3.89878e10 - 1.29087e10i) q^{85} -4.45410e9 q^{87} -9.01332e10 q^{89} +3.90742e10i q^{91} +6.64885e9 q^{93} -1.95720e10i q^{95} -1.01766e10i q^{97} +7.66387e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 1191496 q^{9}+O(q^{10}) $ 16 * q - 1191496 * q^9	$ 16 q - 1191496 q^{9} - 1045192 q^{13} + 928176 q^{15} + 3554632 q^{17} - 4588736 q^{19} + 66662344 q^{21} - 58748400 q^{25} + 724766552 q^{33} - 1999765296 q^{35} + 2666979472 q^{43} - 1869667792 q^{47} - 5944064168 q^{49} - 8437689968 q^{51} - 5942183760 q^{53} + 11128140752 q^{55} - 7494118800 q^{59} - 17007290816 q^{67} + 13676754040 q^{69} + 32130668824 q^{77} + 145538020840 q^{81} + 112706231184 q^{83} + 77452876928 q^{85} + 368269123632 q^{87} - 89466414808 q^{89} - 57312497768 q^{93}+O(q^{100}) $ 16 * q - 1191496 * q^9 - 1045192 * q^13 + 928176 * q^15 + 3554632 * q^17 - 4588736 * q^19 + 66662344 * q^21 - 58748400 * q^25 + 724766552 * q^33 - 1999765296 * q^35 + 2666979472 * q^43 - 1869667792 * q^47 - 5944064168 * q^49 - 8437689968 * q^51 - 5942183760 * q^53 + 11128140752 * q^55 - 7494118800 * q^59 - 17007290816 * q^67 + 13676754040 * q^69 + 32130668824 * q^77 + 145538020840 * q^81 + 112706231184 * q^83 + 77452876928 * q^85 + 368269123632 * q^87 - 89466414808 * q^89 - 57312497768 * 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
$3$			37.5132i			0.0891287i	0.999007	+	0.0445643i	$0.0141900\pi$
$3$			37.5132i			0.0891287i	−0.999007	+	0.0445643i	$0.985810\pi$
$4$		0			0
$5$		−	7015.33i		−	1.00395i	−0.864881	−	0.501976i	$-0.832606\pi$
$5$		−	7015.33i		−	1.00395i	0.864881	−	0.501976i	$-0.167394\pi$
$6$		0			0
$7$			40939.8i			0.920675i	0.887744	+	0.460338i	$0.152272\pi$
$7$			40939.8i			0.920675i	−0.887744	+	0.460338i	$0.847728\pi$
$8$		0			0
$9$	175740.			0.992056
$10$		0			0
$11$			436092.i			0.816429i	0.912886	+	0.408214i	$0.133848\pi$
$11$			436092.i			0.816429i	−0.912886	+	0.408214i	$0.866152\pi$
$12$		0			0
$13$	954432.			0.712946			0.356473	−	0.934306i	$-0.383979\pi$
$13$	954432.			0.712946			0.356473	+	0.934306i	$0.383979\pi$
$14$		0			0
$15$	263168.			0.0894810
$16$		0			0
$17$	1.84007e6	+	5.55752e6i	0.314316	+	0.949319i
$17$	1.84007e6	+	5.55752e6i	0.314316	+	0.949319i
$18$		0			0
$19$	2.78988e6			0.258489			0.129244	−	0.991613i	$-0.458745\pi$
$19$	2.78988e6			0.258489			0.129244	+	0.991613i	$0.458745\pi$
$20$		0			0
$21$	−1.53578e6			−0.0820586
$22$		0			0
$23$			4.14357e7i			1.34237i	0.741291	+	0.671184i	$0.234214\pi$
$23$			4.14357e7i			1.34237i	−0.741291	+	0.671184i	$0.765786\pi$
$24$		0			0
$25$	−386743.			−0.00792050
$26$		0			0
$27$			1.32379e7i			0.177549i
$28$		0			0
$29$			1.18734e8i			1.07495i	0.843281	+	0.537473i	$0.180621\pi$
$29$			1.18734e8i			1.07495i	−0.843281	+	0.537473i	$0.819379\pi$
$30$		0			0
$31$		−	1.77240e8i		−	1.11192i	−0.831210	−	0.555959i	$-0.812351\pi$
$31$		−	1.77240e8i		−	1.11192i	0.831210	−	0.555959i	$-0.187649\pi$
$32$		0			0
$33$	−1.63592e7			−0.0727672
$34$		0			0
$35$	2.87206e8			0.924314
$36$		0			0
$37$			5.80565e8i			1.37639i	0.725526	+	0.688195i	$0.241596\pi$
$37$			5.80565e8i			1.37639i	−0.725526	+	0.688195i	$0.758404\pi$
$38$		0			0
$39$			3.58038e7i			0.0635439i
$40$		0			0
$41$			1.07835e8i			0.145361i	0.997355	+	0.0726804i	$0.0231553\pi$
$41$			1.07835e8i			0.145361i	−0.997355	+	0.0726804i	$0.976845\pi$
$42$		0			0
$43$	1.42735e7			0.0148065			0.00740326	−	0.999973i	$-0.497643\pi$
$43$	1.42735e7			0.0148065			0.00740326	+	0.999973i	$0.497643\pi$
$44$		0			0
$45$		−	1.23287e9i		−	0.995977i
$46$		0			0
$47$	−1.65181e9			−1.05056			−0.525281	−	0.850929i	$-0.676040\pi$
$47$	−1.65181e9			−1.05056			−0.525281	+	0.850929i	$0.676040\pi$
$48$		0			0
$49$	3.01260e8			0.152357
$50$		0			0
$51$	−2.08480e8	+	6.90271e7i	−0.0846115	+	0.0280145i
$52$		0			0
$53$	−1.69980e9			−0.558318			−0.279159	−	0.960245i	$-0.590056\pi$
$53$	−1.69980e9			−0.558318			−0.279159	+	0.960245i	$0.590056\pi$
$54$		0			0
$55$	3.05933e9			0.819656
$56$		0			0
$57$			1.04658e8i			0.0230387i
$58$		0			0
$59$	−1.97623e9			−0.359876			−0.179938	−	0.983678i	$-0.557590\pi$
$59$	−1.97623e9			−0.359876			−0.179938	+	0.983678i	$0.557590\pi$
$60$		0			0
$61$		−	7.84931e9i		−	1.18992i	−0.803756	−	0.594960i	$-0.797168\pi$
$61$		−	7.84931e9i		−	1.18992i	0.803756	−	0.594960i	$-0.202832\pi$
$62$		0			0
$63$			7.19475e9i			0.913361i
$64$		0			0
$65$		−	6.69565e9i		−	0.715763i
$66$		0			0
$67$	−1.92771e10			−1.74434			−0.872169	−	0.489205i	$-0.837287\pi$
$67$	−1.92771e10			−1.74434			−0.872169	+	0.489205i	$0.837287\pi$
$68$		0			0
$69$	−1.55439e9			−0.119643
$70$		0			0
$71$		−	4.81174e9i		−	0.316505i	−0.987399	−	0.158253i	$-0.949414\pi$
$71$		−	4.81174e9i		−	0.316505i	0.987399	−	0.158253i	$-0.0505860\pi$
$72$		0			0
$73$		−	1.24976e10i		−	0.705586i	−0.935701	−	0.352793i	$-0.885232\pi$
$73$		−	1.24976e10i		−	0.705586i	0.935701	−	0.352793i	$-0.114768\pi$
$74$		0			0
$75$		−	1.45080e7i		−	0.000705944i
$76$		0			0
$77$	−1.78535e10			−0.751666
$78$		0			0
$79$			2.94120e10i			1.07541i	0.843132	+	0.537706i	$0.180709\pi$
$79$			2.94120e10i			1.07541i	−0.843132	+	0.537706i	$0.819291\pi$
$80$		0			0
$81$	3.06352e10			0.976231
$82$		0			0
$83$	−2.55654e9			−0.0712399			−0.0356199	−	0.999365i	$-0.511341\pi$
$83$	−2.55654e9			−0.0712399			−0.0356199	+	0.999365i	$0.511341\pi$
$84$		0			0
$85$	3.89878e10	−	1.29087e10i	0.953071	−	0.315558i
$86$		0			0
$87$	−4.45410e9			−0.0958085
$88$		0			0
$89$	−9.01332e10			−1.71096			−0.855480	−	0.517837i	$-0.826738\pi$
$89$	−9.01332e10			−1.71096			−0.855480	+	0.517837i	$0.826738\pi$
$90$		0			0
$91$			3.90742e10i			0.656391i
$92$		0			0
$93$	6.64885e9			0.0991038
$94$		0			0
$95$		−	1.95720e10i		−	0.259510i
$96$		0			0
$97$		−	1.01766e10i		−	0.120325i	−0.998189	−	0.0601626i	$-0.980838\pi$
$97$		−	1.01766e10i		−	0.120325i	0.998189	−	0.0601626i	$-0.0191619\pi$
$98$		0			0
$99$			7.66387e10i			0.809943i
$100$		0			0
$101$	6.71725e10			0.635951			0.317976	−	0.948099i	$-0.396997\pi$
$101$	6.71725e10			0.635951			0.317976	+	0.948099i	$0.396997\pi$
$102$		0			0
$103$	5.26906e10			0.447846			0.223923	−	0.974607i	$-0.428114\pi$
$103$	5.26906e10			0.447846			0.223923	+	0.974607i	$0.428114\pi$
$104$		0			0
$105$			1.07740e10i			0.0823829i
$106$		0			0
$107$		−	2.30745e11i		−	1.59046i	−0.606309	−	0.795229i	$-0.707350\pi$
$107$		−	2.30745e11i		−	1.59046i	0.606309	−	0.795229i	$-0.292650\pi$
$108$		0			0
$109$			2.37657e11i			1.47946i	0.672901	+	0.739732i	$0.265048\pi$
$109$			2.37657e11i			1.47946i	−0.672901	+	0.739732i	$0.734952\pi$
$110$		0			0
$111$	−2.17789e10			−0.122676
$112$		0			0
$113$			1.18308e11i			0.604062i	0.953298	+	0.302031i	$0.0976646\pi$
$113$			1.18308e11i			0.604062i	−0.953298	+	0.302031i	$0.902335\pi$
$114$		0			0
$115$	2.90685e11			1.34767
$116$		0			0
$117$	1.67732e11			0.707282
$118$		0			0
$119$	−2.27524e11	+	7.53322e10i	−0.874014	+	0.289383i
$120$		0			0
$121$	9.51354e10			0.333444
$122$		0			0
$123$	−4.04523e9			−0.0129558
$124$		0			0
$125$		−	3.39832e11i		−	0.996001i
$126$		0			0
$127$	5.94266e11			1.59610			0.798050	−	0.602591i	$-0.205865\pi$
$127$	5.94266e11			1.59610			0.798050	+	0.602591i	$0.205865\pi$
$128$		0			0
$129$			5.35444e8i			0.00131969i
$130$		0			0
$131$			3.66561e11i			0.830145i	0.909788	+	0.415073i	$0.136244\pi$
$131$			3.66561e11i			0.830145i	−0.909788	+	0.415073i	$0.863756\pi$
$132$		0			0
$133$			1.14217e11i			0.237984i
$134$		0			0
$135$	9.28684e10			0.178251
$136$		0			0
$137$	1.59466e11			0.282297			0.141148	−	0.989988i	$-0.454921\pi$
$137$	1.59466e11			0.282297			0.141148	+	0.989988i	$0.454921\pi$
$138$		0			0
$139$		−	7.52929e11i		−	1.23076i	−0.788231	−	0.615379i	$-0.789003\pi$
$139$		−	7.52929e11i		−	1.23076i	0.788231	−	0.615379i	$-0.210997\pi$
$140$		0			0
$141$		−	6.19647e10i		−	0.0936352i
$142$		0			0
$143$			4.16220e11i			0.582069i
$144$		0			0
$145$	8.32959e11			1.07919
$146$		0			0
$147$			1.13012e10i			0.0135794i
$148$		0			0
$149$	−1.63907e12			−1.82840			−0.914202	−	0.405259i	$-0.867181\pi$
$149$	−1.63907e12			−1.82840			−0.914202	+	0.405259i	$0.867181\pi$
$150$		0			0
$151$	−1.57160e12			−1.62917			−0.814587	−	0.580041i	$-0.803037\pi$
$151$	−1.57160e12			−1.62917			−0.814587	+	0.580041i	$0.803037\pi$
$152$		0			0
$153$	3.23374e11	+	9.76677e11i	0.311819	+	0.941777i
$154$		0			0
$155$	−1.24340e12			−1.11631
$156$		0			0
$157$	−1.24309e12			−1.04005			−0.520027	−	0.854150i	$-0.674078\pi$
$157$	−1.24309e12			−1.04005			−0.520027	+	0.854150i	$0.674078\pi$
$158$		0			0
$159$		−	6.37651e10i		−	0.0497621i
$160$		0			0
$161$	−1.69637e12			−1.23588
$162$		0			0
$163$		−	1.76482e12i		−	1.20135i	−0.799494	−	0.600675i	$-0.794899\pi$
$163$		−	1.76482e12i		−	1.20135i	0.799494	−	0.600675i	$-0.205101\pi$
$164$		0			0
$165$			1.14765e11i			0.0730548i
$166$		0			0
$167$			2.00184e12i			1.19258i	0.802768	+	0.596292i	$0.203360\pi$
$167$			2.00184e12i			1.19258i	−0.802768	+	0.596292i	$0.796640\pi$
$168$		0			0
$169$	−8.81221e11			−0.491709
$170$		0			0
$171$	4.90293e11			0.256435
$172$		0			0
$173$			3.54453e11i			0.173902i	0.996213	+	0.0869511i	$0.0277124\pi$
$173$			3.54453e11i			0.173902i	−0.996213	+	0.0869511i	$0.972288\pi$
$174$		0			0
$175$		−	1.58332e10i		−	0.00729221i
$176$		0			0
$177$		−	7.41349e10i		−	0.0320752i
$178$		0			0
$179$	−3.84094e11			−0.156223			−0.0781116	−	0.996945i	$-0.524889\pi$
$179$	−3.84094e11			−0.156223			−0.0781116	+	0.996945i	$0.524889\pi$
$180$		0			0
$181$			3.59801e12i			1.37667i	0.725392	+	0.688336i	$0.241659\pi$
$181$			3.59801e12i			1.37667i	−0.725392	+	0.688336i	$0.758341\pi$
$182$		0			0
$183$	2.94453e11			0.106056
$184$		0			0
$185$	4.07285e12			1.38183
$186$		0			0
$187$	−2.42359e12	+	8.02441e11i	−0.775051	+	0.256616i
$188$		0			0
$189$	−5.41958e11			−0.163465
$190$		0			0
$191$	5.54417e12			1.57817			0.789084	−	0.614285i	$-0.210556\pi$
$191$	5.54417e12			1.57817			0.789084	+	0.614285i	$0.210556\pi$
$192$		0			0
$193$			6.31838e11i			0.169840i	0.996388	+	0.0849202i	$0.0270635\pi$
$193$			6.31838e11i			0.169840i	−0.996388	+	0.0849202i	$0.972936\pi$
$194$		0			0
$195$	2.51176e11			0.0637951
$196$		0			0
$197$		−	7.74159e12i		−	1.85894i	−0.368895	−	0.929471i	$-0.620264\pi$
$197$		−	7.74159e12i		−	1.85894i	0.368895	−	0.929471i	$-0.379736\pi$
$198$		0			0
$199$			1.47357e12i			0.334718i	0.985896	+	0.167359i	$0.0535239\pi$
$199$			1.47357e12i			0.334718i	−0.985896	+	0.167359i	$0.946476\pi$
$200$		0			0
$201$		−	7.23147e11i		−	0.155471i
$202$		0			0
$203$	−4.86095e12			−0.989676
$204$		0			0
$205$	7.56496e11			0.145935
$206$		0			0
$207$			7.28190e12i			1.33170i
$208$		0			0
$209$			1.21665e12i			0.211038i
$210$		0			0
$211$			1.93592e11i			0.0318664i	0.999873	+	0.0159332i	$0.00507191\pi$
$211$			1.93592e11i			0.0318664i	−0.999873	+	0.0159332i	$0.994928\pi$
$212$		0			0
$213$	1.80504e11			0.0282097
$214$		0			0
$215$		−	1.00133e11i		−	0.0148650i
$216$		0			0
$217$	7.25617e12			1.02372
$218$		0			0
$219$	4.68825e11			0.0628880
$220$		0			0
$221$	1.75622e12	+	5.30427e12i	0.224090	+	0.676812i
$222$		0			0
$223$	8.10132e12			0.983737			0.491869	−	0.870669i	$-0.336314\pi$
$223$	8.10132e12			0.983737			0.491869	+	0.870669i	$0.336314\pi$
$224$		0			0
$225$	−6.79661e10			−0.00785758
$226$		0			0
$227$			1.47294e13i			1.62197i	0.585065	+	0.810987i	$0.301069\pi$
$227$			1.47294e13i			1.62197i	−0.585065	+	0.810987i	$0.698931\pi$
$228$		0			0
$229$	−6.60411e12			−0.692977			−0.346489	−	0.938054i	$-0.612626\pi$
$229$	−6.60411e12			−0.692977			−0.346489	+	0.938054i	$0.612626\pi$
$230$		0			0
$231$		−	6.69743e11i		−	0.0669950i
$232$		0			0
$233$			1.89377e13i			1.80663i	0.428979	+	0.903314i	$0.358873\pi$
$233$			1.89377e13i			1.80663i	−0.428979	+	0.903314i	$0.641127\pi$
$234$		0			0
$235$			1.15880e13i			1.05471i
$236$		0			0
$237$	−1.10334e12			−0.0958501
$238$		0			0
$239$	−8.13918e12			−0.675138			−0.337569	−	0.941301i	$-0.609605\pi$
$239$	−8.13918e12			−0.675138			−0.337569	+	0.941301i	$0.609605\pi$
$240$		0			0
$241$		−	5.41957e12i		−	0.429409i	−0.976679	−	0.214704i	$-0.931121\pi$
$241$		−	5.41957e12i		−	0.429409i	0.976679	−	0.214704i	$-0.0688788\pi$
$242$		0			0
$243$			3.49428e12i			0.264560i
$244$		0			0
$245$		−	2.11344e12i		−	0.152960i
$246$		0			0
$247$	2.66275e12			0.184288
$248$		0			0
$249$		−	9.59041e10i		−	0.00634952i
$250$		0			0
$251$	−1.61292e13			−1.02190			−0.510950	−	0.859610i	$-0.670706\pi$
$251$	−1.61292e13			−1.02190			−0.510950	+	0.859610i	$0.670706\pi$
$252$		0			0
$253$	−1.80698e13			−1.09595
$254$		0			0
$255$	4.84248e11	+	1.46256e12i	0.0281253	+	0.0849459i
$256$		0			0
$257$	−5.15196e12			−0.286642			−0.143321	−	0.989676i	$-0.545778\pi$
$257$	−5.15196e12			−0.286642			−0.143321	+	0.989676i	$0.545778\pi$
$258$		0			0
$259$	−2.37682e13			−1.26721
$260$		0			0
$261$			2.08663e13i			1.06641i
$262$		0			0
$263$	−1.77857e12			−0.0871592			−0.0435796	−	0.999050i	$-0.513876\pi$
$263$	−1.77857e12			−0.0871592			−0.0435796	+	0.999050i	$0.513876\pi$
$264$		0			0
$265$			1.19247e13i			0.560525i
$266$		0			0
$267$		−	3.38119e12i		−	0.152496i
$268$		0			0
$269$			1.78034e13i			0.770663i	0.922778	+	0.385331i	$0.125913\pi$
$269$			1.78034e13i			0.770663i	−0.922778	+	0.385331i	$0.874087\pi$
$270$		0			0
$271$	3.26277e13			1.35599			0.677993	−	0.735069i	$-0.262850\pi$
$271$	3.26277e13			1.35599			0.677993	+	0.735069i	$0.262850\pi$
$272$		0			0
$273$	−1.46580e12			−0.0585033
$274$		0			0
$275$		−	1.68656e11i		−	0.00646652i
$276$		0			0
$277$			2.71764e13i			1.00128i	0.865657	+	0.500638i	$0.166901\pi$
$277$			2.71764e13i			1.00128i	−0.865657	+	0.500638i	$0.833099\pi$
$278$		0			0
$279$		−	3.11481e13i		−	1.10309i
$280$		0			0
$281$	−3.57016e13			−1.21563			−0.607817	−	0.794077i	$-0.707954\pi$
$281$	−3.57016e13			−1.21563			−0.607817	+	0.794077i	$0.707954\pi$
$282$		0			0
$283$		−	1.04843e13i		−	0.343333i	−0.985155	−	0.171666i	$-0.945085\pi$
$283$		−	1.04843e13i		−	0.343333i	0.985155	−	0.171666i	$-0.0549151\pi$
$284$		0			0
$285$	7.34207e11			0.0231298
$286$		0			0
$287$	−4.41473e12			−0.133830
$288$		0			0
$289$	−2.75002e13	+	2.04525e13i	−0.802411	+	0.596771i
$290$		0			0
$291$	3.81756e11			0.0107244
$292$		0			0
$293$	7.03512e13			1.90327			0.951633	−	0.307238i	$-0.0994047\pi$
$293$	7.03512e13			1.90327			0.951633	+	0.307238i	$0.0994047\pi$
$294$		0			0
$295$			1.38639e13i			0.361298i
$296$		0			0
$297$	−5.77295e12			−0.144956
$298$		0			0
$299$			3.95475e13i			0.957035i
$300$		0			0
$301$			5.84353e11i			0.0136320i
$302$		0			0
$303$			2.51986e12i			0.0566815i
$304$		0			0
$305$	−5.50655e13			−1.19462
$306$		0			0
$307$	−3.93815e13			−0.824197			−0.412099	−	0.911139i	$-0.635204\pi$
$307$	−3.93815e13			−0.824197			−0.412099	+	0.911139i	$0.635204\pi$
$308$		0			0
$309$			1.97659e12i			0.0399159i
$310$		0			0
$311$			8.38957e13i			1.63515i	0.575823	+	0.817575i	$0.304682\pi$
$311$			8.38957e13i			1.63515i	−0.575823	+	0.817575i	$0.695318\pi$
$312$		0			0
$313$			5.07613e13i			0.955078i	0.878610	+	0.477539i	$0.158471\pi$
$313$			5.07613e13i			0.955078i	−0.878610	+	0.477539i	$0.841529\pi$
$314$		0			0
$315$	5.04735e13			0.916971
$316$		0			0
$317$			7.52575e12i			0.132046i	0.997818	+	0.0660228i	$0.0210310\pi$
$317$			7.52575e12i			0.132046i	−0.997818	+	0.0660228i	$0.978969\pi$
$318$		0			0
$319$	−5.17790e13			−0.877617
$320$		0			0
$321$	8.65600e12			0.141755
$322$		0			0
$323$	5.13359e12	+	1.55048e13i	0.0812470	+	0.245388i
$324$		0			0
$325$	−3.69120e11			−0.00564689
$326$		0			0
$327$	−8.91527e12			−0.131863
$328$		0			0
$329$		−	6.76247e13i		−	0.967226i
$330$		0			0
$331$	−4.92457e13			−0.681262			−0.340631	−	0.940197i	$-0.610641\pi$
$331$	−4.92457e13			−0.681262			−0.340631	+	0.940197i	$0.610641\pi$
$332$		0			0
$333$			1.02028e14i			1.36546i
$334$		0			0
$335$			1.35235e14i			1.75123i
$336$		0			0
$337$		−	1.20199e13i		−	0.150638i	−0.997159	−	0.0753190i	$-0.976002\pi$
$337$		−	1.20199e13i		−	0.150638i	0.997159	−	0.0753190i	$-0.0239975\pi$
$338$		0			0
$339$	−4.43810e12			−0.0538393
$340$		0			0
$341$	7.72930e13			0.907802
$342$		0			0
$343$			9.32849e13i			1.06095i
$344$		0			0
$345$			1.09045e13i			0.120116i
$346$		0			0
$347$			3.76384e13i			0.401624i	0.979630	+	0.200812i	$0.0643580\pi$
$347$			3.76384e13i			0.401624i	−0.979630	+	0.200812i	$0.935642\pi$
$348$		0			0
$349$	−4.41594e13			−0.456545			−0.228273	−	0.973597i	$-0.573308\pi$
$349$	−4.41594e13			−0.456545			−0.228273	+	0.973597i	$0.573308\pi$
$350$		0			0
$351$			1.26347e13i			0.126583i
$352$		0			0
$353$	1.03528e14			1.00530			0.502652	−	0.864489i	$-0.332358\pi$
$353$	1.03528e14			1.00530			0.502652	+	0.864489i	$0.332358\pi$
$354$		0			0
$355$	−3.37559e13			−0.317756
$356$		0			0
$357$	−2.82595e12	−	8.53515e12i	−0.0257923	−	0.0778997i
$358$		0			0
$359$	7.07192e13			0.625918			0.312959	−	0.949767i	$-0.398680\pi$
$359$	7.07192e13			0.625918			0.312959	+	0.949767i	$0.398680\pi$
$360$		0			0
$361$	−1.08707e14			−0.933184
$362$		0			0
$363$			3.56884e12i			0.0297194i
$364$		0			0
$365$	−8.76747e13			−0.708375
$366$		0			0
$367$		−	1.59083e13i		−	0.124727i	−0.998053	−	0.0623636i	$-0.980136\pi$
$367$		−	1.59083e13i		−	0.124727i	0.998053	−	0.0623636i	$-0.0198639\pi$
$368$		0			0
$369$			1.89508e13i			0.144206i
$370$		0			0
$371$		−	6.95896e13i		−	0.514029i
$372$		0			0
$373$	−1.58474e13			−0.113647			−0.0568237	−	0.998384i	$-0.518097\pi$
$373$	−1.58474e13			−0.113647			−0.0568237	+	0.998384i	$0.518097\pi$
$374$		0			0
$375$	1.27482e13			0.0887722
$376$		0			0
$377$			1.13324e14i			0.766378i
$378$		0			0
$379$			6.78416e12i			0.0445636i	0.999752	+	0.0222818i	$0.00709311\pi$
$379$			6.78416e12i			0.0445636i	−0.999752	+	0.0222818i	$0.992907\pi$
$380$		0			0
$381$			2.22928e13i			0.142258i
$382$		0			0
$383$	8.16812e13			0.506441			0.253221	−	0.967409i	$-0.418510\pi$
$383$	8.16812e13			0.506441			0.253221	+	0.967409i	$0.418510\pi$
$384$		0			0
$385$			1.25248e14i			0.754637i
$386$		0			0
$387$	2.50842e12			0.0146889
$388$		0			0
$389$	1.02293e14			0.582270			0.291135	−	0.956682i	$-0.405967\pi$
$389$	1.02293e14			0.582270			0.291135	+	0.956682i	$0.405967\pi$
$390$		0			0
$391$	−2.30280e14	+	7.62447e13i	−1.27433	+	0.421927i
$392$		0			0
$393$	−1.37509e13			−0.0739897
$394$		0			0
$395$	2.06335e14			1.07966
$396$		0			0
$397$			4.33650e13i			0.220694i	0.993893	+	0.110347i	$0.0351963\pi$
$397$			4.33650e13i			0.220694i	−0.993893	+	0.110347i	$0.964804\pi$
$398$		0			0
$399$	−4.28466e12			−0.0212112
$400$		0			0
$401$		−	1.20106e13i		−	0.0578459i	−0.999582	−	0.0289229i	$-0.990792\pi$
$401$		−	1.20106e13i		−	0.0578459i	0.999582	−	0.0289229i	$-0.00920774\pi$
$402$		0			0
$403$		−	1.69164e14i		−	0.792737i
$404$		0			0
$405$		−	2.14916e14i		−	0.980090i
$406$		0			0
$407$	−2.53180e14			−1.12372
$408$		0			0
$409$	2.37591e14			1.02648			0.513242	−	0.858244i	$-0.328444\pi$
$409$	2.37591e14			1.02648			0.513242	+	0.858244i	$0.328444\pi$
$410$		0			0
$411$			5.98209e12i			0.0251607i
$412$		0			0
$413$		−	8.09066e13i		−	0.331328i
$414$		0			0
$415$			1.79350e13i			0.0715215i
$416$		0			0
$417$	2.82448e13			0.109696
$418$		0			0
$419$			4.63527e14i			1.75347i	0.480974	+	0.876735i	$0.340283\pi$
$419$			4.63527e14i			1.75347i	−0.480974	+	0.876735i	$0.659717\pi$
$420$		0			0
$421$	−5.04223e14			−1.85811			−0.929055	−	0.369942i	$-0.879378\pi$
$421$	−5.04223e14			−1.85811			−0.929055	+	0.369942i	$0.879378\pi$
$422$		0			0
$423$	−2.90288e14			−1.04222
$424$		0			0
$425$	−7.11636e11	−	2.14933e12i	−0.00248954	−	0.00751908i
$426$		0			0
$427$	3.21349e14			1.09553
$428$		0			0
$429$	−1.56138e13			−0.0518791
$430$		0			0
$431$		−	3.94622e14i		−	1.27807i	−0.769176	−	0.639037i	$-0.779333\pi$
$431$		−	3.94622e14i		−	1.27807i	0.769176	−	0.639037i	$-0.220667\pi$
$432$		0			0
$433$	−4.14797e14			−1.30964			−0.654820	−	0.755785i	$-0.727256\pi$
$433$	−4.14797e14			−1.30964			−0.654820	+	0.755785i	$0.727256\pi$
$434$		0			0
$435$			3.12470e13i			0.0961872i
$436$		0			0
$437$			1.15601e14i			0.346987i
$438$		0			0
$439$		−	3.31859e14i		−	0.971402i	−0.874125	−	0.485701i	$-0.838564\pi$
$439$		−	3.31859e14i		−	0.971402i	0.874125	−	0.485701i	$-0.161436\pi$
$440$		0			0
$441$	5.29434e13			0.151147
$442$		0			0
$443$	−3.44788e13			−0.0960135			−0.0480067	−	0.998847i	$-0.515287\pi$
$443$	−3.44788e13			−0.0960135			−0.0480067	+	0.998847i	$0.515287\pi$
$444$		0			0
$445$			6.32314e14i			1.71772i
$446$		0			0
$447$		−	6.14867e13i		−	0.162963i
$448$		0			0
$449$		−	2.93233e14i		−	0.758329i	−0.925329	−	0.379164i	$-0.876211\pi$
$449$		−	2.93233e14i		−	0.758329i	0.925329	−	0.379164i	$-0.123789\pi$
$450$		0			0
$451$	−4.70258e13			−0.118677
$452$		0			0
$453$		−	5.89556e13i		−	0.145206i
$454$		0			0
$455$	2.74119e14			0.658986
$456$		0			0
$457$	4.64192e14			1.08933			0.544664	−	0.838654i	$-0.316657\pi$
$457$	4.64192e14			1.08933			0.544664	+	0.838654i	$0.316657\pi$
$458$		0			0
$459$	−7.35700e13	+	2.43587e13i	−0.168551	+	0.0558065i
$460$		0			0
$461$	6.23883e14			1.39556			0.697779	−	0.716313i	$-0.254172\pi$
$461$	6.23883e14			1.39556			0.697779	+	0.716313i	$0.254172\pi$
$462$		0			0
$463$	−2.75054e14			−0.600790			−0.300395	−	0.953815i	$-0.597119\pi$
$463$	−2.75054e14			−0.600790			−0.300395	+	0.953815i	$0.597119\pi$
$464$		0			0
$465$		−	4.66439e13i		−	0.0994955i
$466$		0			0
$467$	5.49799e14			1.14541			0.572705	−	0.819761i	$-0.305894\pi$
$467$	5.49799e14			1.14541			0.572705	+	0.819761i	$0.305894\pi$
$468$		0			0
$469$		−	7.89201e14i		−	1.60597i
$470$		0			0
$471$		−	4.66325e13i		−	0.0926987i
$472$		0			0
$473$			6.22454e12i			0.0120885i
$474$		0			0
$475$	−1.07897e12			−0.00204736
$476$		0			0
$477$	−2.98723e14			−0.553883
$478$		0			0
$479$			8.02043e14i			1.45329i	0.687013	+	0.726645i	$0.258922\pi$
$479$			8.02043e14i			1.45329i	−0.687013	+	0.726645i	$0.741078\pi$
$480$		0			0
$481$			5.54109e14i			0.981291i
$482$		0			0
$483$		−	6.36362e13i		−	0.110153i
$484$		0			0
$485$	−7.13920e13			−0.120801
$486$		0			0
$487$		−	1.07735e15i		−	1.78216i	−0.453843	−	0.891082i	$-0.649947\pi$
$487$		−	1.07735e15i		−	1.78216i	0.453843	−	0.891082i	$-0.350053\pi$
$488$		0			0
$489$	6.62042e13			0.107075
$490$		0			0
$491$	−5.30247e14			−0.838552			−0.419276	−	0.907859i	$-0.637716\pi$
$491$	−5.30247e14			−0.838552			−0.419276	+	0.907859i	$0.637716\pi$
$492$		0			0
$493$	−6.59867e14	+	2.18480e14i	−1.02047	+	0.337872i
$494$		0			0
$495$	5.37646e14			0.813144
$496$		0			0
$497$	1.96991e14			0.291398
$498$		0			0
$499$		−	4.36944e14i		−	0.632227i	−0.948721	−	0.316113i	$-0.897622\pi$
$499$		−	4.36944e14i		−	0.632227i	0.948721	−	0.316113i	$-0.102378\pi$
$500$		0			0
$501$	−7.50955e13			−0.106293
$502$		0			0
$503$		−	4.09919e14i		−	0.567642i	−0.958877	−	0.283821i	$-0.908398\pi$
$503$		−	4.09919e14i		−	0.567642i	0.958877	−	0.283821i	$-0.0916022\pi$
$504$		0			0
$505$		−	4.71237e14i		−	0.638465i
$506$		0			0
$507$		−	3.30574e13i		−	0.0438253i
$508$		0			0
$509$	1.07378e15			1.39305			0.696524	−	0.717533i	$-0.254729\pi$
$509$	1.07378e15			1.39305			0.696524	+	0.717533i	$0.254729\pi$
$510$		0			0
$511$	5.11648e14			0.649616
$512$		0			0
$513$			3.69323e13i			0.0458945i
$514$		0			0
$515$		−	3.69642e14i		−	0.449616i
$516$		0			0
$517$		−	7.20341e14i		−	0.857709i
$518$		0			0
$519$	−1.32967e13			−0.0154997
$520$		0			0
$521$		−	1.25271e15i		−	1.42970i	−0.699278	−	0.714849i	$-0.746495\pi$
$521$		−	1.25271e15i		−	1.42970i	0.699278	−	0.714849i	$-0.253505\pi$
$522$		0			0
$523$	4.63073e14			0.517476			0.258738	−	0.965948i	$-0.416693\pi$
$523$	4.63073e14			0.517476			0.258738	+	0.965948i	$0.416693\pi$
$524$		0			0
$525$	5.93954e11			0.000649945
$526$		0			0
$527$	9.85016e14	−	3.26135e14i	1.05556	−	0.349493i
$528$		0			0
$529$	−7.64106e14			−0.801951
$530$		0			0
$531$	−3.47303e14			−0.357017
$532$		0			0
$533$			1.02921e14i			0.103634i
$534$		0			0
$535$	−1.61876e15			−1.59674
$536$		0			0
$537$		−	1.44086e13i		−	0.0139240i
$538$		0			0
$539$			1.31377e14i			0.124389i
$540$		0			0
$541$		−	3.93255e14i		−	0.364829i	−0.983222	−	0.182415i	$-0.941609\pi$
$541$		−	3.93255e14i		−	0.364829i	0.983222	−	0.182415i	$-0.0583913\pi$
$542$		0			0
$543$	−1.34973e14			−0.122701
$544$		0			0
$545$	1.66724e15			1.48531
$546$		0			0
$547$		−	1.15470e15i		−	1.00818i	−0.863652	−	0.504089i	$-0.831828\pi$
$547$		−	1.15470e15i		−	1.00818i	0.863652	−	0.504089i	$-0.168172\pi$
$548$		0			0
$549$		−	1.37944e15i		−	1.18047i
$550$		0			0
$551$			3.31254e14i			0.277861i
$552$		0			0
$553$	−1.20412e15			−0.990105
$554$		0			0
$555$			1.52786e14i			0.123161i
$556$		0			0
$557$	6.77959e14			0.535796			0.267898	−	0.963447i	$-0.413671\pi$
$557$	6.77959e14			0.535796			0.267898	+	0.963447i	$0.413671\pi$
$558$		0			0
$559$	1.36230e13			0.0105562
$560$		0			0
$561$	−3.01022e13	−	9.09167e13i	−0.0228719	−	0.0690793i
$562$		0			0
$563$	1.64475e15			1.22547			0.612735	−	0.790289i	$-0.290069\pi$
$563$	1.64475e15			1.22547			0.612735	+	0.790289i	$0.290069\pi$
$564$		0			0
$565$	8.29968e14			0.606450
$566$		0			0
$567$			1.25420e15i			0.898792i
$568$		0			0
$569$	6.00657e14			0.422191			0.211096	−	0.977465i	$-0.432297\pi$
$569$	6.00657e14			0.422191			0.211096	+	0.977465i	$0.432297\pi$
$570$		0			0
$571$		−	6.83687e14i		−	0.471367i	−0.971830	−	0.235683i	$-0.924267\pi$
$571$		−	6.83687e14i		−	0.471367i	0.971830	−	0.235683i	$-0.0757328\pi$
$572$		0			0
$573$			2.07980e14i			0.140660i
$574$		0			0
$575$		−	1.60250e13i		−	0.0106322i
$576$		0			0
$577$	2.27361e15			1.47995			0.739977	−	0.672632i	$-0.234836\pi$
$577$	2.27361e15			1.47995			0.739977	+	0.672632i	$0.234836\pi$
$578$		0			0
$579$	−2.37023e13			−0.0151376
$580$		0			0
$581$		−	1.04664e14i		−	0.0655888i
$582$		0			0
$583$		−	7.41271e14i		−	0.455827i
$584$		0			0
$585$		−	1.17669e15i		−	0.710078i
$586$		0			0
$587$	1.59694e15			0.945755			0.472877	−	0.881128i	$-0.343215\pi$
$587$	1.59694e15			0.945755			0.472877	+	0.881128i	$0.343215\pi$
$588$		0			0
$589$		−	4.94479e14i		−	0.287418i
$590$		0			0
$591$	2.90412e14			0.165685
$592$		0			0
$593$	−2.87894e15			−1.61225			−0.806123	−	0.591748i	$-0.798438\pi$
$593$	−2.87894e15			−1.61225			−0.806123	+	0.591748i	$0.798438\pi$
$594$		0			0
$595$	5.28480e14	+	1.59615e15i	0.290526	+	0.877468i
$596$		0			0
$597$	−5.52783e13			−0.0298330
$598$		0			0
$599$	−4.13434e14			−0.219058			−0.109529	−	0.993984i	$-0.534934\pi$
$599$	−4.13434e14			−0.219058			−0.109529	+	0.993984i	$0.534934\pi$
$600$		0			0
$601$		−	2.17849e15i		−	1.13330i	−0.823958	−	0.566652i	$-0.808239\pi$
$601$		−	2.17849e15i		−	1.13330i	0.823958	−	0.566652i	$-0.191761\pi$
$602$		0			0
$603$	−3.38776e15			−1.73048
$604$		0			0
$605$		−	6.67407e14i		−	0.334762i
$606$		0			0
$607$			3.25356e15i			1.60258i	0.598273	+	0.801292i	$0.295854\pi$
$607$			3.25356e15i			1.60258i	−0.598273	+	0.801292i	$0.704146\pi$
$608$		0			0
$609$		−	1.82350e14i		−	0.0882085i
$610$		0			0
$611$	−1.57654e15			−0.748993
$612$		0			0
$613$	1.72107e15			0.803095			0.401547	−	0.915838i	$-0.368472\pi$
$613$	1.72107e15			0.803095			0.401547	+	0.915838i	$0.368472\pi$
$614$		0			0
$615$			2.83786e13i			0.0130070i
$616$		0			0
$617$			4.72457e14i			0.212713i	0.994328	+	0.106357i	$0.0339185\pi$
$617$			4.72457e14i			0.212713i	−0.994328	+	0.106357i	$0.966082\pi$
$618$		0			0
$619$		−	3.90001e15i		−	1.72491i	−0.506131	−	0.862456i	$-0.668925\pi$
$619$		−	3.90001e15i		−	1.72491i	0.506131	−	0.862456i	$-0.331075\pi$
$620$		0			0
$621$	−5.48522e14			−0.238336
$622$		0			0
$623$		−	3.69003e15i		−	1.57524i
$624$		0			0
$625$	−2.40292e15			−1.00786
$626$		0			0
$627$	−4.56403e13			−0.0188095
$628$		0			0
$629$	−3.22650e15	+	1.06828e15i	−1.30663	+	0.432621i
$630$		0			0
$631$	2.31182e15			0.920011			0.460006	−	0.887916i	$-0.347847\pi$
$631$	2.31182e15			0.920011			0.460006	+	0.887916i	$0.347847\pi$
$632$		0			0
$633$	−7.26225e12			−0.00284021
$634$		0			0
$635$		−	4.16897e15i		−	1.60241i
$636$		0			0
$637$	2.87532e14			0.108622
$638$		0			0
$639$		−	8.45613e14i		−	0.313991i
$640$		0			0
$641$			3.34975e15i			1.22263i	0.791389	+	0.611313i	$0.209358\pi$
$641$			3.34975e15i			1.22263i	−0.791389	+	0.611313i	$0.790642\pi$
$642$		0			0
$643$		−	3.23818e15i		−	1.16182i	−0.813966	−	0.580912i	$-0.802696\pi$
$643$		−	3.23818e15i		−	1.16182i	0.813966	−	0.580912i	$-0.197304\pi$
$644$		0			0
$645$	3.75631e12			0.00132490
$646$		0			0
$647$	−1.93334e15			−0.670401			−0.335201	−	0.942147i	$-0.608804\pi$
$647$	−1.93334e15			−0.670401			−0.335201	+	0.942147i	$0.608804\pi$
$648$		0			0
$649$		−	8.61820e14i		−	0.293813i
$650$		0			0
$651$			2.72202e14i			0.0912424i
$652$		0			0
$653$			3.67243e15i			1.21041i	0.796071	+	0.605203i	$0.206908\pi$
$653$			3.67243e15i			1.21041i	−0.796071	+	0.605203i	$0.793092\pi$
$654$		0			0
$655$	2.57155e15			0.833426
$656$		0			0
$657$		−	2.19632e15i		−	0.699981i
$658$		0			0
$659$	3.41018e15			1.06883			0.534414	−	0.845223i	$-0.320532\pi$
$659$	3.41018e15			1.06883			0.534414	+	0.845223i	$0.320532\pi$
$660$		0			0
$661$	3.62601e15			1.11769			0.558845	−	0.829272i	$-0.311245\pi$
$661$	3.62601e15			1.11769			0.558845	+	0.829272i	$0.311245\pi$
$662$		0			0
$663$	−1.98980e14	+	6.58816e13i	−0.0603234	+	0.0199728i
$664$		0			0
$665$	8.01272e14			0.238925
$666$		0			0
$667$	−4.91983e15			−1.44297
$668$		0			0
$669$			3.03907e14i			0.0876792i
$670$		0			0
$671$	3.42302e15			0.971484
$672$		0			0
$673$			2.54005e15i			0.709184i	0.935021	+	0.354592i	$0.115380\pi$
$673$			2.54005e15i			0.709184i	−0.935021	+	0.354592i	$0.884620\pi$
$674$		0			0
$675$		−	5.11967e12i		−	0.00140628i
$676$		0			0
$677$		−	5.36364e15i		−	1.44951i	−0.689005	−	0.724757i	$-0.741952\pi$
$677$		−	5.36364e15i		−	1.44951i	0.689005	−	0.724757i	$-0.258048\pi$
$678$		0			0
$679$	4.16627e14			0.110780
$680$		0			0
$681$	−5.52548e14			−0.144564
$682$		0			0
$683$			1.42743e15i			0.367486i	0.982974	+	0.183743i	$0.0588214\pi$
$683$			1.42743e15i			0.367486i	−0.982974	+	0.183743i	$0.941179\pi$
$684$		0			0
$685$		−	1.11871e15i		−	0.283412i
$686$		0			0
$687$		−	2.47741e14i		−	0.0617641i
$688$		0			0
$689$	−1.62235e15			−0.398050
$690$		0			0
$691$		−	1.20192e15i		−	0.290233i	−0.989415	−	0.145117i	$-0.953644\pi$
$691$		−	1.20192e15i		−	0.290233i	0.989415	−	0.145117i	$-0.0463558\pi$
$692$		0			0
$693$	−3.13757e15			−0.745695
$694$		0			0
$695$	−5.28205e15			−1.23562
$696$		0			0
$697$	−5.99293e14	+	1.98424e14i	−0.137994	+	0.0456892i
$698$		0			0
$699$	−7.10413e14			−0.161022
$700$		0			0
$701$	−1.33740e15			−0.298409			−0.149205	−	0.988806i	$-0.547671\pi$
$701$	−1.33740e15			−0.298409			−0.149205	+	0.988806i	$0.547671\pi$
$702$		0			0
$703$			1.61971e15i			0.355781i
$704$		0			0
$705$	−4.34703e14			−0.0940053
$706$		0			0
$707$			2.75003e15i			0.585504i
$708$		0			0
$709$			6.70214e15i			1.40494i	0.711711	+	0.702472i	$0.247921\pi$
$709$			6.70214e15i			1.40494i	−0.711711	+	0.702472i	$0.752079\pi$
$710$		0			0
$711$			5.16885e15i			1.06687i
$712$		0			0
$713$	7.34407e15			1.49260
$714$		0			0
$715$	2.91992e15			0.584370
$716$		0			0
$717$		−	3.05327e14i		−	0.0601741i
$718$		0			0
$719$			5.55050e15i			1.07727i	0.842540	+	0.538633i	$0.181059\pi$
$719$			5.55050e15i			1.07727i	−0.842540	+	0.538633i	$0.818941\pi$
$720$		0			0
$721$			2.15714e15i			0.412320i
$722$		0			0
$723$	2.03306e14			0.0382726
$724$		0			0
$725$		−	4.59196e13i		−	0.00851411i
$726$		0			0
$727$	7.21071e15			1.31686			0.658428	−	0.752643i	$-0.271222\pi$
$727$	7.21071e15			1.31686			0.658428	+	0.752643i	$0.271222\pi$
$728$		0			0
$729$	5.29585e15			0.952652
$730$		0			0
$731$	2.62642e13	+	7.93251e13i	0.00465392	+	0.0140561i
$732$		0			0
$733$	−1.98314e15			−0.346163			−0.173082	−	0.984907i	$-0.555372\pi$
$733$	−1.98314e15			−0.346163			−0.173082	+	0.984907i	$0.555372\pi$
$734$		0			0
$735$	7.92820e13			0.0136331
$736$		0			0
$737$		−	8.40660e15i		−	1.42413i
$738$		0			0
$739$	2.59631e15			0.433323			0.216662	−	0.976247i	$-0.430483\pi$
$739$	2.59631e15			0.433323			0.216662	+	0.976247i	$0.430483\pi$
$740$		0			0
$741$			9.98885e13i			0.0164254i
$742$		0			0
$743$			1.13755e15i			0.184304i	0.995745	+	0.0921518i	$0.0293745\pi$
$743$			1.13755e15i			0.184304i	−0.995745	+	0.0921518i	$0.970625\pi$
$744$		0			0
$745$			1.14986e16i			1.83563i
$746$		0			0
$747$	−4.49286e14			−0.0706740
$748$		0			0
$749$	9.44667e15			1.46430
$750$		0			0
$751$			1.21818e16i			1.86077i	0.366586	+	0.930384i	$0.380527\pi$
$751$			1.21818e16i			1.86077i	−0.366586	+	0.930384i	$0.619473\pi$
$752$		0			0
$753$		−	6.05060e14i		−	0.0910806i
$754$		0			0
$755$			1.10253e16i			1.63561i
$756$		0			0
$757$	−1.32645e15			−0.193938			−0.0969689	−	0.995287i	$-0.530915\pi$
$757$	−1.32645e15			−0.193938			−0.0969689	+	0.995287i	$0.530915\pi$
$758$		0			0
$759$		−	6.77855e14i		−	0.0976804i
$760$		0			0
$761$	3.04705e15			0.432777			0.216389	−	0.976307i	$-0.430572\pi$
$761$	3.04705e15			0.432777			0.216389	+	0.976307i	$0.430572\pi$
$762$		0			0
$763$	−9.72962e15			−1.36211
$764$		0			0
$765$	6.85171e15	−	2.26858e15i	0.945500	−	0.313051i
$766$		0			0
$767$	−1.88618e15			−0.256572
$768$		0			0
$769$	−5.94860e15			−0.797664			−0.398832	−	0.917024i	$-0.630584\pi$
$769$	−5.94860e15			−0.797664			−0.398832	+	0.917024i	$0.630584\pi$
$770$		0			0
$771$		−	1.93267e14i		−	0.0255481i
$772$		0			0
$773$	−1.28250e16			−1.67136			−0.835679	−	0.549218i	$-0.814926\pi$
$773$	−1.28250e16			−1.67136			−0.835679	+	0.549218i	$0.814926\pi$
$774$		0			0
$775$			6.85464e13i			0.00880695i
$776$		0			0
$777$		−	8.91622e14i		−	0.112945i
$778$		0			0
$779$			3.00846e14i			0.0375741i
$780$		0			0
$781$	2.09836e15			0.258404
$782$		0			0
$783$	−1.57179e15			−0.190856
$784$		0			0
$785$			8.72072e15i			1.04417i
$786$		0			0
$787$		−	1.44859e16i		−	1.71035i	−0.518337	−	0.855177i	$-0.673449\pi$
$787$		−	1.44859e16i		−	1.71035i	0.518337	−	0.855177i	$-0.326551\pi$
$788$		0			0
$789$		−	6.67197e13i		−	0.00776839i
$790$		0			0
$791$	−4.84349e15			−0.556145
$792$		0			0
$793$		−	7.49163e15i		−	0.848348i
$794$		0			0
$795$	−4.47333e14			−0.0499588
$796$		0			0
$797$	−7.71817e15			−0.850146			−0.425073	−	0.905159i	$-0.639752\pi$
$797$	−7.71817e15			−0.850146			−0.425073	+	0.905159i	$0.639752\pi$
$798$		0			0
$799$	−3.03945e15	−	9.17996e15i	−0.330208	−	0.997318i
$800$		0			0
$801$	−1.58400e16			−1.69737
$802$		0			0
$803$	5.45010e15			0.576061
$804$		0			0
$805$			1.19006e16i			1.24077i
$806$		0			0
$807$	−6.67861e14			−0.0686882
$808$		0			0
$809$		−	1.64116e15i		−	0.166508i	−0.996528	−	0.0832540i	$-0.973469\pi$
$809$		−	1.64116e15i		−	0.166508i	0.996528	−	0.0832540i	$-0.0265313\pi$
$810$		0			0
$811$			9.52939e15i			0.953784i	0.878962	+	0.476892i	$0.158237\pi$
$811$			9.52939e15i			0.953784i	−0.878962	+	0.476892i	$0.841763\pi$
$812$		0			0
$813$			1.22397e15i			0.120857i
$814$		0			0
$815$	−1.23808e16			−1.20610
$816$		0			0
$817$	3.98213e13			0.00382731
$818$		0			0
$819$			6.86690e15i			0.651177i
$820$		0			0
$821$		−	1.50797e16i		−	1.41093i	−0.708746	−	0.705464i	$-0.750739\pi$
$821$		−	1.50797e16i		−	1.41093i	0.708746	−	0.705464i	$-0.249261\pi$
$822$		0			0
$823$			1.05599e16i			0.974902i	0.873150	+	0.487451i	$0.162073\pi$
$823$			1.05599e16i			0.974902i	−0.873150	+	0.487451i	$0.837927\pi$
$824$		0			0
$825$	6.32681e12			0.000576353
$826$		0			0
$827$		−	1.10007e16i		−	0.988871i	−0.869214	−	0.494435i	$-0.835375\pi$
$827$		−	1.10007e16i		−	0.988871i	0.869214	−	0.494435i	$-0.164625\pi$
$828$		0			0
$829$	−3.29931e15			−0.292667			−0.146334	−	0.989235i	$-0.546747\pi$
$829$	−3.29931e15			−0.292667			−0.146334	+	0.989235i	$0.546747\pi$
$830$		0			0
$831$	−1.01948e15			−0.0892424
$832$		0			0
$833$	5.54341e14	+	1.67426e15i	0.0478883	+	0.144636i
$834$		0			0
$835$	1.40436e16			1.19730
$836$		0			0
$837$	2.34629e15			0.197420
$838$		0			0
$839$			7.12405e15i			0.591611i	0.955248	+	0.295806i	$0.0955881\pi$
$839$			7.12405e15i			0.591611i	−0.955248	+	0.295806i	$0.904412\pi$
$840$		0			0
$841$	−1.89729e15			−0.155509
$842$		0			0
$843$		−	1.33928e15i		−	0.108348i
$844$		0			0
$845$			6.18205e15i			0.493652i
$846$		0			0
$847$			3.89482e15i			0.306993i
$848$		0			0
$849$	3.93301e14			0.0306008
$850$		0			0
$851$	−2.40561e16			−1.84762
$852$		0			0
$853$			2.31162e16i			1.75266i	0.481713	+	0.876329i	$0.340015\pi$
$853$			2.31162e16i			1.75266i	−0.481713	+	0.876329i	$0.659985\pi$
$854$		0			0
$855$		−	3.43957e15i		−	0.257449i
$856$		0			0
$857$			2.55454e16i			1.88764i	0.330466	+	0.943818i	$0.392794\pi$
$857$			2.55454e16i			1.88764i	−0.330466	+	0.943818i	$0.607206\pi$
$858$		0			0
$859$	8.70194e14			0.0634825			0.0317412	−	0.999496i	$-0.489895\pi$
$859$	8.70194e14			0.0634825			0.0317412	+	0.999496i	$0.489895\pi$
$860$		0			0
$861$		−	1.65611e14i		−	0.0119281i
$862$		0			0
$863$	−7.24118e15			−0.514932			−0.257466	−	0.966287i	$-0.582888\pi$
$863$	−7.24118e15			−0.514932			−0.257466	+	0.966287i	$0.582888\pi$
$864$		0			0
$865$	2.48661e15			0.174590
$866$		0			0
$867$	−7.67239e14	−	1.03162e15i	−0.0531894	−	0.0715179i
$868$		0			0
$869$	−1.28263e16			−0.877998
$870$		0			0
$871$	−1.83987e16			−1.24362
$872$		0			0
$873$		−	1.78843e15i		−	0.119369i
$874$		0			0
$875$	1.39127e16			0.916993
$876$		0			0
$877$			4.70621e15i			0.306319i	0.988202	+	0.153159i	$0.0489448\pi$
$877$			4.70621e15i			0.306319i	−0.988202	+	0.153159i	$0.951055\pi$
$878$		0			0
$879$			2.63910e15i			0.169636i
$880$		0			0
$881$		−	6.34030e15i		−	0.402478i	−0.979542	−	0.201239i	$-0.935503\pi$
$881$		−	6.34030e15i		−	0.402478i	0.979542	−	0.201239i	$-0.0644968\pi$
$882$		0			0
$883$	2.54085e16			1.59292			0.796461	−	0.604690i	$-0.206703\pi$
$883$	2.54085e16			1.59292			0.796461	+	0.604690i	$0.206703\pi$
$884$		0			0
$885$	−5.20081e14			−0.0322020
$886$		0			0
$887$			4.30148e15i			0.263050i	0.991313	+	0.131525i	$0.0419874\pi$
$887$			4.30148e15i			0.263050i	−0.991313	+	0.131525i	$0.958013\pi$
$888$		0			0
$889$			2.43291e16i			1.46949i
$890$		0			0
$891$			1.33598e16i			0.797023i
$892$		0			0
$893$	−4.60835e15			−0.271558
$894$		0			0
$895$			2.69455e15i			0.156841i
$896$		0			0
$897$	−1.48356e15			−0.0852993
$898$		0			0
$899$	2.10445e16			1.19525
$900$		0			0
$901$	−3.12776e15	−	9.44669e15i	−0.175488	−	0.530022i
$902$		0			0
$903$	−2.19209e13			−0.00121500
$904$		0			0
$905$	2.52413e16			1.38211
$906$		0			0
$907$		−	8.06988e15i		−	0.436543i	−0.975888	−	0.218272i	$-0.929958\pi$
$907$		−	8.06988e15i		−	0.436543i	0.975888	−	0.218272i	$-0.0700419\pi$
$908$		0			0
$909$	1.18049e16			0.630899
$910$		0			0
$911$			2.04923e16i			1.08203i	0.841013	+	0.541016i	$0.181960\pi$
$911$			2.04923e16i			1.08203i	−0.841013	+	0.541016i	$0.818040\pi$
$912$		0			0
$913$		−	1.11489e15i		−	0.0581623i
$914$		0			0
$915$		−	2.06568e15i		−	0.106475i
$916$		0			0
$917$	−1.50069e16			−0.764294
$918$		0			0
$919$	−1.55836e16			−0.784210			−0.392105	−	0.919921i	$-0.628253\pi$
$919$	−1.55836e16			−0.784210			−0.392105	+	0.919921i	$0.628253\pi$
$920$		0			0
$921$		−	1.47733e15i		−	0.0734596i
$922$		0			0
$923$		−	4.59247e15i		−	0.225651i
$924$		0			0
$925$		−	2.24529e14i		−	0.0109017i
$926$		0			0
$927$	9.25984e15			0.444288
$928$		0			0
$929$		−	2.26357e16i		−	1.07327i	−0.843816	−	0.536633i	$-0.819696\pi$
$929$		−	2.26357e16i		−	1.07327i	0.843816	−	0.536633i	$-0.180304\pi$
$930$		0			0
$931$	8.40481e14			0.0393826
$932$		0			0
$933$	−3.14720e15			−0.145739
$934$		0			0
$935$	5.62939e15	+	1.70023e16i	0.257631	+	0.778114i
$936$		0			0
$937$	2.85278e16			1.29033			0.645164	−	0.764044i	$-0.276789\pi$
$937$	2.85278e16			1.29033			0.645164	+	0.764044i	$0.276789\pi$
$938$		0			0
$939$	−1.90422e15			−0.0851249
$940$		0			0
$941$			1.88306e15i			0.0831995i	0.999134	+	0.0415997i	$0.0132454\pi$
$941$			1.88306e15i			0.0831995i	−0.999134	+	0.0415997i	$0.986755\pi$
$942$		0			0
$943$	−4.46820e15			−0.195128
$944$		0			0
$945$			3.80201e15i			0.164111i
$946$		0			0
$947$			1.78705e16i			0.762451i	0.924482	+	0.381226i	$0.124498\pi$
$947$			1.78705e16i			0.762451i	−0.924482	+	0.381226i	$0.875502\pi$
$948$		0			0
$949$		−	1.19281e16i		−	0.503045i
$950$		0			0
$951$	−2.82315e14			−0.0117691
$952$		0			0
$953$	1.70042e15			0.0700721			0.0350361	−	0.999386i	$-0.488845\pi$
$953$	1.70042e15			0.0700721			0.0350361	+	0.999386i	$0.488845\pi$
$954$		0			0
$955$		−	3.88942e16i		−	1.58441i
$956$		0			0
$957$		−	1.94240e15i		−	0.0782208i
$958$		0			0
$959$			6.52852e15i			0.259904i
$960$		0			0
$961$	−6.00559e15			−0.236362
$962$		0			0
$963$		−	4.05511e16i		−	1.57782i
$964$		0			0
$965$	4.43256e15			0.170512
$966$		0			0
$967$	2.07223e16			0.788121			0.394060	−	0.919085i	$-0.371070\pi$
$967$	2.07223e16			0.788121			0.394060	+	0.919085i	$0.371070\pi$
$968$		0			0
$969$	−5.81636e14	+	1.92578e14i	−0.0218711	+	0.00724144i
$970$		0			0
$971$	−7.81060e15			−0.290388			−0.145194	−	0.989403i	$-0.546381\pi$
$971$	−7.81060e15			−0.290388			−0.145194	+	0.989403i	$0.546381\pi$
$972$		0			0
$973$	3.08248e16			1.13313
$974$		0			0
$975$		−	1.38469e13i		−	0.000503299i
$976$		0			0
$977$	1.83369e16			0.659031			0.329516	−	0.944150i	$-0.393115\pi$
$977$	1.83369e16			0.659031			0.329516	+	0.944150i	$0.393115\pi$
$978$		0			0
$979$		−	3.93064e16i		−	1.39688i
$980$		0			0
$981$			4.17657e16i			1.46771i
$982$		0			0
$983$		−	2.01380e15i		−	0.0699797i	−0.999388	−	0.0349898i	$-0.988860\pi$
$983$		−	2.01380e15i		−	0.0699797i	0.999388	−	0.0349898i	$-0.0111399\pi$
$984$		0			0
$985$	−5.43098e16			−1.86629
$986$		0			0
$987$	2.53682e15			0.0862076
$988$		0			0
$989$			5.91431e14i			0.0198758i
$990$		0			0
$991$			3.19132e16i			1.06063i	0.847800	+	0.530317i	$0.177927\pi$
$991$			3.19132e16i			1.06063i	−0.847800	+	0.530317i	$0.822073\pi$
$992$		0			0
$993$		−	1.84736e15i		−	0.0607200i
$994$		0			0
$995$	1.03376e16			0.336041
$996$		0			0
$997$		−	4.30322e16i		−	1.38347i	−0.722150	−	0.691736i	$-0.756846\pi$
$997$		−	4.30322e16i		−	1.38347i	0.722150	−	0.691736i	$-0.243154\pi$
$998$		0			0
$999$	−7.68547e15			−0.244377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.12.b.c.33.9		16
4.3	odd	2		17.12.b.a.16.13	✓	16
12.11	even	2		153.12.d.b.118.4		16
17.16	even	2	inner	272.12.b.c.33.8		16
68.67	odd	2		17.12.b.a.16.14	yes	16
204.203	even	2		153.12.d.b.118.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.12.b.a.16.13	✓	16	4.3	odd	2
17.12.b.a.16.14	yes	16	68.67	odd	2
153.12.d.b.118.3		16	204.203	even	2
153.12.d.b.118.4		16	12.11	even	2
272.12.b.c.33.8		16	17.16	even	2	inner
272.12.b.c.33.9		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	272.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+37.5132i q^{3} -7015.33i q^{5} +40939.8i q^{7} +175740. q^{9} +O(q^{10})\)	\(q+37.5132i q^{3} -7015.33i q^{5} +40939.8i q^{7} +175740. q^{9} +436092. i q^{11} +954432. q^{13} +263168. q^{15} +(1.84007e6 + 5.55752e6i) q^{17} +2.78988e6 q^{19} -1.53578e6 q^{21} +4.14357e7i q^{23} -386743. q^{25} +1.32379e7i q^{27} +1.18734e8i q^{29} -1.77240e8i q^{31} -1.63592e7 q^{33} +2.87206e8 q^{35} +5.80565e8i q^{37} +3.58038e7i q^{39} +1.07835e8i q^{41} +1.42735e7 q^{43} -1.23287e9i q^{45} -1.65181e9 q^{47} +3.01260e8 q^{49} +(-2.08480e8 + 6.90271e7i) q^{51} -1.69980e9 q^{53} +3.05933e9 q^{55} +1.04658e8i q^{57} -1.97623e9 q^{59} -7.84931e9i q^{61} +7.19475e9i q^{63} -6.69565e9i q^{65} -1.92771e10 q^{67} -1.55439e9 q^{69} -4.81174e9i q^{71} -1.24976e10i q^{73} -1.45080e7i q^{75} -1.78535e10 q^{77} +2.94120e10i q^{79} +3.06352e10 q^{81} -2.55654e9 q^{83} +(3.89878e10 - 1.29087e10i) q^{85} -4.45410e9 q^{87} -9.01332e10 q^{89} +3.90742e10i q^{91} +6.64885e9 q^{93} -1.95720e10i q^{95} -1.01766e10i q^{97} +7.66387e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1191496 q^{9}+O(q^{10}) \) 16 * q - 1191496 * q^9	\( 16 q - 1191496 q^{9} - 1045192 q^{13} + 928176 q^{15} + 3554632 q^{17} - 4588736 q^{19} + 66662344 q^{21} - 58748400 q^{25} + 724766552 q^{33} - 1999765296 q^{35} + 2666979472 q^{43} - 1869667792 q^{47} - 5944064168 q^{49} - 8437689968 q^{51} - 5942183760 q^{53} + 11128140752 q^{55} - 7494118800 q^{59} - 17007290816 q^{67} + 13676754040 q^{69} + 32130668824 q^{77} + 145538020840 q^{81} + 112706231184 q^{83} + 77452876928 q^{85} + 368269123632 q^{87} - 89466414808 q^{89} - 57312497768 q^{93}+O(q^{100}) \) 16 * q - 1191496 * q^9 - 1045192 * q^13 + 928176 * q^15 + 3554632 * q^17 - 4588736 * q^19 + 66662344 * q^21 - 58748400 * q^25 + 724766552 * q^33 - 1999765296 * q^35 + 2666979472 * q^43 - 1869667792 * q^47 - 5944064168 * q^49 - 8437689968 * q^51 - 5942183760 * q^53 + 11128140752 * q^55 - 7494118800 * q^59 - 17007290816 * q^67 + 13676754040 * q^69 + 32130668824 * q^77 + 145538020840 * q^81 + 112706231184 * q^83 + 77452876928 * q^85 + 368269123632 * q^87 - 89466414808 * q^89 - 57312497768 * q^93

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