LMFDB - Embedded newform 12.21.c.b.5.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [12,21,Mod(5,12)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(12, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 21, names="a")

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

chi := DirichletCharacter("12.5");

S:= CuspForms(chi, 21);

N := Newforms(S);

Level:	$ N $	$=$	$ 12 = 2^{2} \cdot 3 $
Weight:	$ k $	$=$	$ 21 $
Character orbit:	$[\chi]$	$=$	12.c (of order $2$, degree $1$, 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:	$30.4216518123$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 24769850x^{4} + 131733035896000x^{2} + 250851218720256000 $ x^6 + 24769850x^4 + 131733035896000x^2 + 250851218720256000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{22}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5.5
Root			$-43.6454i$ of defining polynomial
Character	$\chi$	$=$	12.5
Dual form			12.21.c.b.5.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(19623.0 - 55693.1i) q^{3} +7.88530e6i q^{5} +3.14494e8 q^{7} +(-2.71666e9 - 2.18573e9i) q^{9} +O(q^{10})$	\(q+(19623.0 - 55693.1i) q^{3} +7.88530e6i q^{5} +3.14494e8 q^{7} +(-2.71666e9 - 2.18573e9i) q^{9} +3.97631e10i q^{11} -2.60857e11 q^{13} +(4.39157e11 + 1.54733e11i) q^{15} -4.22183e10i q^{17} +1.21520e12 q^{19} +(6.17132e12 - 1.75151e13i) q^{21} +4.94987e13i q^{23} +3.31895e13 q^{25} +(-1.75039e14 + 1.08408e14i) q^{27} +4.36286e14i q^{29} +3.04488e14 q^{31} +(2.21453e15 + 7.80272e14i) q^{33} +2.47988e15i q^{35} +1.82925e15 q^{37} +(-5.11881e15 + 1.45280e16i) q^{39} +2.29227e16i q^{41} +1.37879e16 q^{43} +(1.72352e16 - 2.14217e16i) q^{45} +4.50977e16i q^{47} +1.91142e16 q^{49} +(-2.35127e15 - 8.28452e14i) q^{51} -1.96578e17i q^{53} -3.13544e17 q^{55} +(2.38459e16 - 6.76783e16i) q^{57} -4.53809e17i q^{59} +1.15524e17 q^{61} +(-8.54372e17 - 6.87400e17i) q^{63} -2.05694e18i q^{65} +7.02341e17 q^{67} +(2.75674e18 + 9.71314e17i) q^{69} +4.29885e18i q^{71} -7.58366e18 q^{73} +(6.51278e17 - 1.84843e18i) q^{75} +1.25052e19i q^{77} +1.37396e19 q^{79} +(2.60280e18 + 1.18758e19i) q^{81} -9.70385e18i q^{83} +3.32904e17 q^{85} +(2.42981e19 + 8.56125e18i) q^{87} -1.74459e18i q^{89} -8.20381e19 q^{91} +(5.97497e18 - 1.69579e19i) q^{93} +9.58222e18i q^{95} -1.87844e19 q^{97} +(8.69115e19 - 1.08023e20i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 84378 q^{3} - 145040532 q^{7} - 7747974234 q^{9}+O(q^{10}) $ 6 * q - 84378 * q^3 - 145040532 * q^7 - 7747974234 * q^9	$ 6 q - 84378 q^{3} - 145040532 q^{7} - 7747974234 q^{9} - 366963002772 q^{13} - 534244714560 q^{15} + 12201993657804 q^{19} + 10561619781804 q^{21} - 236482695022170 q^{25} - 269341388965818 q^{27} - 647531494989396 q^{31} - 233666770697280 q^{33} - 11\!\cdots\!16 q^{37}+ \cdots + 47\!\cdots\!80 q^{99}+O(q^{100}) $ 6 * q - 84378 * q^3 - 145040532 * q^7 - 7747974234 * q^9 - 366963002772 * q^13 - 534244714560 * q^15 + 12201993657804 * q^19 + 10561619781804 * q^21 - 236482695022170 * q^25 - 269341388965818 * q^27 - 647531494989396 * q^31 - 233666770697280 * q^33 - 11617802966744916 * q^37 - 18953557962425556 * q^39 - 74673968433381492 * q^43 + 16986459376321920 * q^45 + 199908182522509650 * q^49 - 21650764841061120 * q^51 - 342407748800488320 * q^55 + 395526902614004748 * q^57 + 1829222818097529324 * q^61 + 1916686598498095788 * q^63 + 354895661781418188 * q^67 + 1403390676183557760 * q^69 - 20036153441939082036 * q^73 + 2838112082798091270 * q^75 - 18182911195481706708 * q^79 + 20248226439018709446 * q^81 - 70523384934991772160 * q^85 + 101352340532658519360 * q^87 - 143360708053521287784 * q^91 + 132455306774032049388 * q^93 - 192284635486467464052 * q^97 + 475048517452037815680 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/12\mathbb{Z}\right)^\times$.

$n$	$5$	$7$
$\chi(n)$	$-1$	$1$

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
$3$	19623.0	−	55693.1i	0.332318	−	0.943168i
$3$	19623.0	−	55693.1i	0.332318	−	0.943168i
$4$		0			0
$5$			7.88530e6i			0.807455i	0.914879	+	0.403727i	$0.132286\pi$
$5$			7.88530e6i			0.807455i	−0.914879	+	0.403727i	$0.867714\pi$
$6$		0			0
$7$	3.14494e8			1.11335			0.556675	−	0.830730i	$-0.312077\pi$
$7$	3.14494e8			1.11335			0.556675	+	0.830730i	$0.312077\pi$
$8$		0			0
$9$	−2.71666e9	−	2.18573e9i	−0.779130	−	0.626863i
$10$		0			0
$11$			3.97631e10i			1.53304i	0.642221	+	0.766520i	$0.278013\pi$
$11$			3.97631e10i			1.53304i	−0.642221	+	0.766520i	$0.721987\pi$
$12$		0			0
$13$	−2.60857e11			−1.89221			−0.946106	−	0.323857i	$-0.895020\pi$
$13$	−2.60857e11			−1.89221			−0.946106	+	0.323857i	$0.895020\pi$
$14$		0			0
$15$	4.39157e11	+	1.54733e11i	0.761565	+	0.268331i
$16$		0			0
$17$		−	4.22183e10i		−	0.0209417i	−0.999945	−	0.0104709i	$-0.996667\pi$
$17$		−	4.22183e10i		−	0.0209417i	0.999945	−	0.0104709i	$-0.00333304\pi$
$18$		0			0
$19$	1.21520e12			0.198204			0.0991019	−	0.995077i	$-0.468403\pi$
$19$	1.21520e12			0.198204			0.0991019	+	0.995077i	$0.468403\pi$
$20$		0			0
$21$	6.17132e12	−	1.75151e13i	0.369986	−	1.05008i
$22$		0			0
$23$			4.94987e13i			1.19486i	0.801923	+	0.597428i	$0.203811\pi$
$23$			4.94987e13i			1.19486i	−0.801923	+	0.597428i	$0.796189\pi$
$24$		0			0
$25$	3.31895e13			0.348017
$26$		0			0
$27$	−1.75039e14	+	1.08408e14i	−0.850155	+	0.526532i
$28$		0			0
$29$			4.36286e14i			1.03703i	0.855069	+	0.518515i	$0.173515\pi$
$29$			4.36286e14i			1.03703i	−0.855069	+	0.518515i	$0.826485\pi$
$30$		0			0
$31$	3.04488e14			0.371495			0.185747	−	0.982598i	$-0.440529\pi$
$31$	3.04488e14			0.371495			0.185747	+	0.982598i	$0.440529\pi$
$32$		0			0
$33$	2.21453e15	+	7.80272e14i	1.44591	+	0.509456i
$34$		0			0
$35$			2.47988e15i			0.898980i
$36$		0			0
$37$	1.82925e15			0.380414			0.190207	−	0.981744i	$-0.439084\pi$
$37$	1.82925e15			0.380414			0.190207	+	0.981744i	$0.439084\pi$
$38$		0			0
$39$	−5.11881e15	+	1.45280e16i	−0.628815	+	1.78467i
$40$		0			0
$41$			2.29227e16i			1.70776i	0.520467	+	0.853882i	$0.325758\pi$
$41$			2.29227e16i			1.70776i	−0.520467	+	0.853882i	$0.674242\pi$
$42$		0			0
$43$	1.37879e16			0.637991			0.318996	−	0.947756i	$-0.396654\pi$
$43$	1.37879e16			0.637991			0.318996	+	0.947756i	$0.396654\pi$
$44$		0			0
$45$	1.72352e16	−	2.14217e16i	0.506163	−	0.629112i
$46$		0			0
$47$			4.50977e16i			0.857386i	0.903450	+	0.428693i	$0.141026\pi$
$47$			4.50977e16i			0.857386i	−0.903450	+	0.428693i	$0.858974\pi$
$48$		0			0
$49$	1.91142e16			0.239549
$50$		0			0
$51$	−2.35127e15	−	8.28452e14i	−0.0197515	−	0.00695930i
$52$		0			0
$53$		−	1.96578e17i		−	1.12403i	−0.827128	−	0.562013i	$-0.810027\pi$
$53$		−	1.96578e17i		−	1.12403i	0.827128	−	0.562013i	$-0.189973\pi$
$54$		0			0
$55$	−3.13544e17			−1.23786
$56$		0			0
$57$	2.38459e16	−	6.76783e16i	0.0658666	−	0.186939i
$58$		0			0
$59$		−	4.53809e17i		−	0.887877i	−0.896057	−	0.443939i	$-0.853581\pi$
$59$		−	4.53809e17i		−	0.887877i	0.896057	−	0.443939i	$-0.146419\pi$
$60$		0			0
$61$	1.15524e17			0.161948			0.0809739	−	0.996716i	$-0.474197\pi$
$61$	1.15524e17			0.161948			0.0809739	+	0.996716i	$0.474197\pi$
$62$		0			0
$63$	−8.54372e17	−	6.87400e17i	−0.867445	−	0.697918i
$64$		0			0
$65$		−	2.05694e18i		−	1.52788i
$66$		0			0
$67$	7.02341e17			0.385301			0.192650	−	0.981267i	$-0.438292\pi$
$67$	7.02341e17			0.385301			0.192650	+	0.981267i	$0.438292\pi$
$68$		0			0
$69$	2.75674e18	+	9.71314e17i	1.12695	+	0.397072i
$70$		0			0
$71$			4.29885e18i			1.32059i	0.751005	+	0.660297i	$0.229569\pi$
$71$			4.29885e18i			1.32059i	−0.751005	+	0.660297i	$0.770431\pi$
$72$		0			0
$73$	−7.58366e18			−1.76462			−0.882309	−	0.470671i	$-0.844012\pi$
$73$	−7.58366e18			−1.76462			−0.882309	+	0.470671i	$0.844012\pi$
$74$		0			0
$75$	6.51278e17	−	1.84843e18i	0.115652	−	0.328238i
$76$		0			0
$77$			1.25052e19i			1.70681i
$78$		0			0
$79$	1.37396e19			1.45112			0.725561	−	0.688158i	$-0.241580\pi$
$79$	1.37396e19			1.45112			0.725561	+	0.688158i	$0.241580\pi$
$80$		0			0
$81$	2.60280e18	+	1.18758e19i	0.214087	+	0.976815i
$82$		0			0
$83$		−	9.70385e18i		−	0.625408i	−0.949851	−	0.312704i	$-0.898765\pi$
$83$		−	9.70385e18i		−	0.625408i	0.949851	−	0.312704i	$-0.101235\pi$
$84$		0			0
$85$	3.32904e17			0.0169095
$86$		0			0
$87$	2.42981e19	+	8.56125e18i	0.978092	+	0.344623i
$88$		0			0
$89$		−	1.74459e18i		−	0.0559491i	−0.999609	−	0.0279746i	$-0.991094\pi$
$89$		−	1.74459e18i		−	0.0559491i	0.999609	−	0.0279746i	$-0.00890575\pi$
$90$		0			0
$91$	−8.20381e19			−2.10669
$92$		0			0
$93$	5.97497e18	−	1.69579e19i	0.123454	−	0.350382i
$94$		0			0
$95$			9.58222e18i			0.160041i
$96$		0			0
$97$	−1.87844e19			−0.254730			−0.127365	−	0.991856i	$-0.540652\pi$
$97$	−1.87844e19			−0.254730			−0.127365	+	0.991856i	$0.540652\pi$
$98$		0			0
$99$	8.69115e19	−	1.08023e20i	0.961005	−	1.19444i
$100$		0			0
$101$		−	1.96513e20i		−	1.77901i	−0.456929	−	0.889503i	$-0.651051\pi$
$101$		−	1.96513e20i		−	1.77901i	0.456929	−	0.889503i	$-0.348949\pi$
$102$		0			0
$103$	−1.83050e20			−1.36207			−0.681033	−	0.732253i	$-0.738469\pi$
$103$	−1.83050e20			−1.36207			−0.681033	+	0.732253i	$0.738469\pi$
$104$		0			0
$105$	1.38112e20	+	4.86627e19i	0.847889	+	0.298747i
$106$		0			0
$107$			4.04717e18i			0.0205738i	0.999947	+	0.0102869i	$0.00327447\pi$
$107$			4.04717e18i			0.0205738i	−0.999947	+	0.0102869i	$0.996726\pi$
$108$		0			0
$109$	−4.91912e19			−0.207789			−0.103894	−	0.994588i	$-0.533130\pi$
$109$	−4.91912e19			−0.207789			−0.103894	+	0.994588i	$0.533130\pi$
$110$		0			0
$111$	3.58954e19	−	1.01877e20i	0.126418	−	0.358794i
$112$		0			0
$113$			3.77461e20i			1.11196i	0.831197	+	0.555978i	$0.187656\pi$
$113$			3.77461e20i			1.11196i	−0.831197	+	0.555978i	$0.812344\pi$
$114$		0			0
$115$	−3.90312e20			−0.964792
$116$		0			0
$117$	7.08661e20	+	5.70165e20i	1.47428	+	1.18616i
$118$		0			0
$119$		−	1.32774e19i		−	0.0233154i
$120$		0			0
$121$	−9.08353e20			−1.35021
$122$		0			0
$123$	1.27664e21	+	4.49813e20i	1.61071	+	0.567520i
$124$		0			0
$125$			1.01371e21i			1.08846i
$126$		0			0
$127$	2.28501e20			0.209340			0.104670	−	0.994507i	$-0.466621\pi$
$127$	2.28501e20			0.209340			0.104670	+	0.994507i	$0.466621\pi$
$128$		0			0
$129$	2.70561e20	−	7.67893e20i	0.212016	−	0.601733i
$130$		0			0
$131$		−	1.24134e21i		−	0.834021i	−0.908902	−	0.417011i	$-0.863078\pi$
$131$		−	1.24134e21i		−	0.834021i	0.908902	−	0.417011i	$-0.136922\pi$
$132$		0			0
$133$	3.82173e20			0.220670
$134$		0			0
$135$	−8.54832e20	−	1.38024e21i	−0.425151	−	0.686462i
$136$		0			0
$137$			3.86182e21i			1.65801i	0.559245	+	0.829003i	$0.311091\pi$
$137$			3.86182e21i			1.65801i	−0.559245	+	0.829003i	$0.688909\pi$
$138$		0			0
$139$	−4.88069e21			−1.81273			−0.906365	−	0.422495i	$-0.861154\pi$
$139$	−4.88069e21			−1.81273			−0.906365	+	0.422495i	$0.861154\pi$
$140$		0			0
$141$	2.51163e21	+	8.84954e20i	0.808658	+	0.284924i
$142$		0			0
$143$		−	1.03725e22i		−	2.90083i
$144$		0			0
$145$	−3.44024e21			−0.837354
$146$		0			0
$147$	3.75078e20	−	1.06453e21i	0.0796064	−	0.225935i
$148$		0			0
$149$		−	2.88939e21i		−	0.535726i	−0.963457	−	0.267863i	$-0.913683\pi$
$149$		−	2.88939e21i		−	0.535726i	0.963457	−	0.267863i	$-0.0863175\pi$
$150$		0			0
$151$	7.94630e21			1.28942			0.644712	−	0.764426i	$-0.276977\pi$
$151$	7.94630e21			1.28942			0.644712	+	0.764426i	$0.276977\pi$
$152$		0			0
$153$	−9.22781e19	+	1.14693e20i	−0.0131276	+	0.0163163i
$154$		0			0
$155$			2.40098e21i			0.299965i
$156$		0			0
$157$	5.61168e21			0.616732			0.308366	−	0.951268i	$-0.400218\pi$
$157$	5.61168e21			0.616732			0.308366	+	0.951268i	$0.400218\pi$
$158$		0			0
$159$	−1.09480e22	−	3.85746e21i	−1.06015	−	0.373534i
$160$		0			0
$161$			1.55670e22i			1.33029i
$162$		0			0
$163$	−1.22271e22			−0.923522			−0.461761	−	0.887004i	$-0.652782\pi$
$163$	−1.22271e22			−0.923522			−0.461761	+	0.887004i	$0.652782\pi$
$164$		0			0
$165$	−6.15268e21	+	1.74622e22i	−0.411363	+	1.16751i
$166$		0			0
$167$			9.09020e21i			0.538777i	0.963032	+	0.269388i	$0.0868215\pi$
$167$			9.09020e21i			0.538777i	−0.963032	+	0.269388i	$0.913178\pi$
$168$		0			0
$169$	4.90417e22			2.58047
$170$		0			0
$171$	−3.30128e21	−	2.65611e21i	−0.154426	−	0.124247i
$172$		0			0
$173$			2.38674e22i			0.993905i	0.867778	+	0.496952i	$0.165548\pi$
$173$			2.38674e22i			0.993905i	−0.867778	+	0.496952i	$0.834452\pi$
$174$		0			0
$175$	1.04379e22			0.387465
$176$		0			0
$177$	−2.52740e22	−	8.90510e21i	−0.837417	−	0.295057i
$178$		0			0
$179$		−	3.83110e22i		−	1.13447i	−0.823556	−	0.567235i	$-0.808013\pi$
$179$		−	3.83110e22i		−	1.13447i	0.823556	−	0.567235i	$-0.191987\pi$
$180$		0			0
$181$	8.47085e21			0.224461			0.112231	−	0.993682i	$-0.464200\pi$
$181$	8.47085e21			0.224461			0.112231	+	0.993682i	$0.464200\pi$
$182$		0			0
$183$	2.26694e21	−	6.43391e21i	0.0538182	−	0.152744i
$184$		0			0
$185$			1.44242e22i			0.307167i
$186$		0			0
$187$	1.67873e21			0.0321044
$188$		0			0
$189$	−5.50488e22	+	3.40938e22i	−0.946520	+	0.586215i
$190$		0			0
$191$		−	8.13879e22i		−	1.25958i	−0.776765	−	0.629790i	$-0.783141\pi$
$191$		−	8.13879e22i		−	1.25958i	0.776765	−	0.629790i	$-0.216859\pi$
$192$		0			0
$193$	3.66575e21			0.0511199			0.0255600	−	0.999673i	$-0.491863\pi$
$193$	3.66575e21			0.0511199			0.0255600	+	0.999673i	$0.491863\pi$
$194$		0			0
$195$	−1.14557e23	−	4.03634e22i	−1.44104	−	0.507740i
$196$		0			0
$197$		−	1.16574e23i		−	1.32416i	−0.749434	−	0.662079i	$-0.769674\pi$
$197$		−	1.16574e23i		−	1.32416i	0.749434	−	0.662079i	$-0.230326\pi$
$198$		0			0
$199$	1.49109e23			1.53099			0.765495	−	0.643441i	$-0.222494\pi$
$199$	1.49109e23			1.53099			0.765495	+	0.643441i	$0.222494\pi$
$200$		0			0
$201$	1.37821e22	−	3.91155e22i	0.128042	−	0.363403i
$202$		0			0
$203$			1.37209e23i			1.15458i
$204$		0			0
$205$	−1.80753e23			−1.37894
$206$		0			0
$207$	1.08191e23	−	1.34471e23i	0.749010	−	0.930948i
$208$		0			0
$209$			4.83201e22i			0.303854i
$210$		0			0
$211$	−1.34740e22			−0.0770319			−0.0385160	−	0.999258i	$-0.512263\pi$
$211$	−1.34740e22			−0.0770319			−0.0385160	+	0.999258i	$0.512263\pi$
$212$		0			0
$213$	2.39416e23	+	8.43565e22i	1.24554	+	0.438856i
$214$		0			0
$215$			1.08722e23i			0.515149i
$216$		0			0
$217$	9.57596e22			0.413604
$218$		0			0
$219$	−1.48814e23	+	4.22358e23i	−0.586414	+	1.66433i
$220$		0			0
$221$			1.10130e22i			0.0396261i
$222$		0			0
$223$	−3.02159e23			−0.993542			−0.496771	−	0.867882i	$-0.665481\pi$
$223$	−3.02159e23			−0.993542			−0.496771	+	0.867882i	$0.665481\pi$
$224$		0			0
$225$	−9.01645e22	−	7.25434e22i	−0.271151	−	0.218159i
$226$		0			0
$227$			6.03994e23i			1.66255i	0.555864	+	0.831273i	$0.312387\pi$
$227$			6.03994e23i			1.66255i	−0.555864	+	0.831273i	$0.687613\pi$
$228$		0			0
$229$	1.65876e23			0.418243			0.209122	−	0.977890i	$-0.432940\pi$
$229$	1.65876e23			0.418243			0.209122	+	0.977890i	$0.432940\pi$
$230$		0			0
$231$	6.96456e23	+	2.45391e23i	1.60981	+	0.567203i
$232$		0			0
$233$		−	6.42971e23i		−	1.36343i	−0.731619	−	0.681714i	$-0.761235\pi$
$233$		−	6.42971e23i		−	1.36343i	0.731619	−	0.681714i	$-0.238765\pi$
$234$		0			0
$235$	−3.55609e23			−0.692300
$236$		0			0
$237$	2.69613e23	−	7.65202e23i	0.482234	−	1.36865i
$238$		0			0
$239$		−	3.82997e23i		−	0.629819i	−0.949122	−	0.314910i	$-0.898026\pi$
$239$		−	3.82997e23i		−	0.629819i	0.949122	−	0.314910i	$-0.101974\pi$
$240$		0			0
$241$	−6.74584e23			−1.02062			−0.510312	−	0.859990i	$-0.670470\pi$
$241$	−6.74584e23			−1.02062			−0.510312	+	0.859990i	$0.670470\pi$
$242$		0			0
$243$	7.12474e23	+	8.80811e22i	0.992445	+	0.122693i
$244$		0			0
$245$			1.50721e23i			0.193425i
$246$		0			0
$247$	−3.16994e23			−0.375044
$248$		0			0
$249$	−5.40438e23	−	1.90419e23i	−0.589864	−	0.207834i
$250$		0			0
$251$		−	6.65686e23i		−	0.670706i	−0.942093	−	0.335353i	$-0.891144\pi$
$251$		−	6.65686e23i		−	0.670706i	0.942093	−	0.335353i	$-0.108856\pi$
$252$		0			0
$253$	−1.96822e24			−1.83176
$254$		0			0
$255$	6.53259e21	−	1.85405e22i	0.00561932	−	0.0159485i
$256$		0			0
$257$		−	4.34825e23i		−	0.345926i	−0.984928	−	0.172963i	$-0.944666\pi$
$257$		−	4.34825e23i		−	0.345926i	0.984928	−	0.172963i	$-0.0553341\pi$
$258$		0			0
$259$	5.75288e23			0.423534
$260$		0			0
$261$	9.53605e23	−	1.18524e24i	0.650075	−	0.807980i
$262$		0			0
$263$			8.26368e23i			0.521934i	0.965348	+	0.260967i	$0.0840414\pi$
$263$			8.26368e23i			0.521934i	−0.965348	+	0.260967i	$0.915959\pi$
$264$		0			0
$265$	1.55008e24			0.907601
$266$		0			0
$267$	−9.71617e22	−	3.42342e22i	−0.0527694	−	0.0185929i
$268$		0			0
$269$		−	6.31337e23i		−	0.318228i	−0.987260	−	0.159114i	$-0.949136\pi$
$269$		−	6.31337e23i		−	0.318228i	0.987260	−	0.159114i	$-0.0508637\pi$
$270$		0			0
$271$	1.79695e24			0.841092			0.420546	−	0.907271i	$-0.361838\pi$
$271$	1.79695e24			0.841092			0.420546	+	0.907271i	$0.361838\pi$
$272$		0			0
$273$	−1.60984e24	+	4.56896e24i	−0.700092	+	1.98697i
$274$		0			0
$275$			1.31972e24i			0.533524i
$276$		0			0
$277$	1.29147e24			0.485609			0.242805	−	0.970075i	$-0.421933\pi$
$277$	1.29147e24			0.485609			0.242805	+	0.970075i	$0.421933\pi$
$278$		0			0
$279$	−8.27189e23	−	6.65529e23i	−0.289443	−	0.232876i
$280$		0			0
$281$		−	2.22419e23i		−	0.0724618i	−0.999343	−	0.0362309i	$-0.988465\pi$
$281$		−	2.22419e23i		−	0.0724618i	0.999343	−	0.0362309i	$-0.0115352\pi$
$282$		0			0
$283$	−3.24331e24			−0.984291			−0.492145	−	0.870513i	$-0.663787\pi$
$283$	−3.24331e24			−0.984291			−0.492145	+	0.870513i	$0.663787\pi$
$284$		0			0
$285$	5.33663e23	+	1.88032e23i	0.150945	+	0.0531843i
$286$		0			0
$287$			7.20906e24i			1.90134i
$288$		0			0
$289$	4.06245e24			0.999561
$290$		0			0
$291$	−3.68607e23	+	1.04616e24i	−0.0846514	+	0.240253i
$292$		0			0
$293$			3.65876e24i			0.784618i	0.919834	+	0.392309i	$0.128324\pi$
$293$			3.65876e24i			0.784618i	−0.919834	+	0.392309i	$0.871676\pi$
$294$		0			0
$295$	3.57842e24			0.716920
$296$		0			0
$297$	−4.31065e24	−	6.96011e24i	−0.807195	−	1.30332i
$298$		0			0
$299$		−	1.29121e25i		−	2.26092i
$300$		0			0
$301$	4.33622e24			0.710308
$302$		0			0
$303$	−1.09444e25	−	3.85618e24i	−1.67790	−	0.591195i
$304$		0			0
$305$			9.10944e23i			0.130766i
$306$		0			0
$307$	5.70264e24			0.766818			0.383409	−	0.923579i	$-0.374750\pi$
$307$	5.70264e24			0.766818			0.383409	+	0.923579i	$0.374750\pi$
$308$		0			0
$309$	−3.59200e24	+	1.01946e25i	−0.452639	+	1.28466i
$310$		0			0
$311$		−	1.22152e25i		−	1.44310i	−0.692362	−	0.721550i	$-0.743430\pi$
$311$		−	1.22152e25i		−	1.44310i	0.692362	−	0.721550i	$-0.256570\pi$
$312$		0			0
$313$	3.82642e24			0.423984			0.211992	−	0.977271i	$-0.432005\pi$
$313$	3.82642e24			0.423984			0.211992	+	0.977271i	$0.432005\pi$
$314$		0			0
$315$	5.42036e24	−	6.73698e24i	0.563537	−	0.700422i
$316$		0			0
$317$		−	2.21600e24i		−	0.216260i	−0.994137	−	0.108130i	$-0.965514\pi$
$317$		−	2.21600e24i		−	0.216260i	0.994137	−	0.108130i	$-0.0344863\pi$
$318$		0			0
$319$	−1.73481e25			−1.58981
$320$		0			0
$321$	2.25399e23	+	7.94177e22i	0.0194045	+	0.00683703i
$322$		0			0
$323$		−	5.13038e22i		−	0.00415072i
$324$		0			0
$325$	−8.65773e24			−0.658522
$326$		0			0
$327$	−9.65279e23	+	2.73961e24i	−0.0690519	+	0.195980i
$328$		0			0
$329$			1.41830e25i			0.954571i
$330$		0			0
$331$	9.62467e24			0.609686			0.304843	−	0.952403i	$-0.401396\pi$
$331$	9.62467e24			0.609686			0.304843	+	0.952403i	$0.401396\pi$
$332$		0			0
$333$	−4.96945e24	−	3.99826e24i	−0.296392	−	0.238467i
$334$		0			0
$335$			5.53817e24i			0.311113i
$336$		0			0
$337$	3.07755e25			1.62894			0.814472	−	0.580203i	$-0.197027\pi$
$337$	3.07755e25			1.62894			0.814472	+	0.580203i	$0.197027\pi$
$338$		0			0
$339$	2.10220e25	+	7.40693e24i	1.04876	+	0.369523i
$340$		0			0
$341$			1.21074e25i			0.569516i
$342$		0			0
$343$	−1.90829e25			−0.846648
$344$		0			0
$345$	−7.65910e24	+	2.17377e25i	−0.320617	+	0.909960i
$346$		0			0
$347$		−	1.37027e25i		−	0.541390i	−0.962665	−	0.270695i	$-0.912747\pi$
$347$		−	1.37027e25i		−	0.541390i	0.962665	−	0.270695i	$-0.0872534\pi$
$348$		0			0
$349$	−1.38931e24			−0.0518258			−0.0259129	−	0.999664i	$-0.508249\pi$
$349$	−1.38931e24			−0.0518258			−0.0259129	+	0.999664i	$0.508249\pi$
$350$		0			0
$351$	4.56603e25	−	2.82791e25i	1.60867	−	0.996311i
$352$		0			0
$353$			4.90483e25i			1.63259i	0.577637	+	0.816294i	$0.303975\pi$
$353$			4.90483e25i			1.63259i	−0.577637	+	0.816294i	$0.696025\pi$
$354$		0			0
$355$	−3.38977e25			−1.06632
$356$		0			0
$357$	−7.39460e23	−	2.60543e23i	−0.0219904	−	0.00774814i
$358$		0			0
$359$			3.44222e25i			0.968042i	0.875056	+	0.484021i	$0.160824\pi$
$359$			3.44222e25i			0.968042i	−0.875056	+	0.484021i	$0.839176\pi$
$360$		0			0
$361$	−3.61133e25			−0.960715
$362$		0			0
$363$	−1.78246e25	+	5.05890e25i	−0.448698	+	1.27347i
$364$		0			0
$365$		−	5.97995e25i		−	1.42485i
$366$		0			0
$367$	−4.18693e25			−0.944573			−0.472286	−	0.881445i	$-0.656571\pi$
$367$	−4.18693e25			−0.944573			−0.472286	+	0.881445i	$0.656571\pi$
$368$		0			0
$369$	5.01030e25	−	6.22732e25i	1.07053	−	1.33057i
$370$		0			0
$371$		−	6.18227e25i		−	1.25144i
$372$		0			0
$373$	1.40813e25			0.270119			0.135060	−	0.990837i	$-0.456877\pi$
$373$	1.40813e25			0.270119			0.135060	+	0.990837i	$0.456877\pi$
$374$		0			0
$375$	5.64566e25	+	1.98921e25i	1.02660	+	0.361715i
$376$		0			0
$377$		−	1.13808e26i		−	1.96228i
$378$		0			0
$379$	1.25926e25			0.205931			0.102966	−	0.994685i	$-0.467167\pi$
$379$	1.25926e25			0.205931			0.102966	+	0.994685i	$0.467167\pi$
$380$		0			0
$381$	4.48389e24	−	1.27260e25i	0.0695673	−	0.197442i
$382$		0			0
$383$			4.83103e25i			0.711299i	0.934619	+	0.355650i	$0.115740\pi$
$383$			4.83103e25i			0.711299i	−0.934619	+	0.355650i	$0.884260\pi$
$384$		0			0
$385$	−9.86076e25			−1.37817
$386$		0			0
$387$	−3.74571e25	−	3.01368e25i	−0.497078	−	0.399933i
$388$		0			0
$389$		−	1.44091e25i		−	0.181611i	−0.995869	−	0.0908053i	$-0.971056\pi$
$389$		−	1.44091e25i		−	0.181611i	0.995869	−	0.0908053i	$-0.0289441\pi$
$390$		0			0
$391$	2.08975e24			0.0250223
$392$		0			0
$393$	−6.91340e25	−	2.43588e25i	−0.786622	−	0.277160i
$394$		0			0
$395$			1.08341e26i			1.17172i
$396$		0			0
$397$	1.38683e26			1.42600			0.712998	−	0.701166i	$-0.247337\pi$
$397$	1.38683e26			1.42600			0.712998	+	0.701166i	$0.247337\pi$
$398$		0			0
$399$	7.49940e24	−	2.12844e25i	0.0733326	−	0.208129i
$400$		0			0
$401$			8.37420e25i			0.778933i	0.921041	+	0.389466i	$0.127341\pi$
$401$			8.37420e25i			0.778933i	−0.921041	+	0.389466i	$0.872659\pi$
$402$		0			0
$403$	−7.94279e25			−0.702947
$404$		0			0
$405$	−9.36441e25	+	2.05238e25i	−0.788733	+	0.172865i
$406$		0			0
$407$			7.27367e25i			0.583189i
$408$		0			0
$409$	8.72264e25			0.665909			0.332955	−	0.942943i	$-0.391954\pi$
$409$	8.72264e25			0.665909			0.332955	+	0.942943i	$0.391954\pi$
$410$		0			0
$411$	2.15077e26	+	7.57805e25i	1.56378	+	0.550985i
$412$		0			0
$413$		−	1.42720e26i		−	0.988518i
$414$		0			0
$415$	7.65178e25			0.504988
$416$		0			0
$417$	−9.57739e25	+	2.71821e26i	−0.602402	+	1.70971i
$418$		0			0
$419$			1.38280e26i			0.829123i	0.910021	+	0.414562i	$0.136065\pi$
$419$			1.38280e26i			0.829123i	−0.910021	+	0.414562i	$0.863935\pi$
$420$		0			0
$421$	1.29653e26			0.741243			0.370621	−	0.928784i	$-0.379145\pi$
$421$	1.29653e26			0.741243			0.370621	+	0.928784i	$0.379145\pi$
$422$		0			0
$423$	9.85717e25	−	1.22515e26i	0.537463	−	0.668015i
$424$		0			0
$425$		−	1.40121e24i		−	0.00728807i
$426$		0			0
$427$	3.63317e25			0.180305
$428$		0			0
$429$	−5.77677e26	−	2.03540e26i	−2.73597	−	0.963999i
$430$		0			0
$431$		−	1.78679e25i		−	0.0807796i	−0.999184	−	0.0403898i	$-0.987140\pi$
$431$		−	1.78679e25i		−	0.0807796i	0.999184	−	0.0403898i	$-0.0128600\pi$
$432$		0			0
$433$	2.48047e26			1.07067			0.535333	−	0.844641i	$-0.320186\pi$
$433$	2.48047e26			1.07067			0.535333	+	0.844641i	$0.320186\pi$
$434$		0			0
$435$	−6.75080e25	+	1.91598e26i	−0.278268	+	0.789765i
$436$		0			0
$437$			6.01509e25i			0.236825i
$438$		0			0
$439$	2.11522e26			0.795629			0.397814	−	0.917466i	$-0.369769\pi$
$439$	2.11522e26			0.795629			0.397814	+	0.917466i	$0.369769\pi$
$440$		0			0
$441$	−5.19266e25	−	4.17785e25i	−0.186640	−	0.150164i
$442$		0			0
$443$		−	1.31175e26i		−	0.450624i	−0.974287	−	0.225312i	$-0.927660\pi$
$443$		−	1.31175e26i		−	0.450624i	0.974287	−	0.225312i	$-0.0723402\pi$
$444$		0			0
$445$	1.37566e25			0.0451764
$446$		0			0
$447$	−1.60919e26	−	5.66985e25i	−0.505279	−	0.178031i
$448$		0			0
$449$		−	1.44183e26i		−	0.432962i	−0.976287	−	0.216481i	$-0.930542\pi$
$449$		−	1.44183e26i		−	0.432962i	0.976287	−	0.216481i	$-0.0694580\pi$
$450$		0			0
$451$	−9.11479e26			−2.61807
$452$		0			0
$453$	1.55931e26	−	4.42554e26i	0.428498	−	1.21614i
$454$		0			0
$455$		−	6.46895e26i		−	1.70106i
$456$		0			0
$457$	6.53992e26			1.64593			0.822963	−	0.568095i	$-0.192319\pi$
$457$	6.53992e26			1.64593			0.822963	+	0.568095i	$0.192319\pi$
$458$		0			0
$459$	4.57682e24	+	7.38987e24i	0.0110265	+	0.0178037i
$460$		0			0
$461$			2.66400e26i			0.614505i	0.951628	+	0.307252i	$0.0994096\pi$
$461$			2.66400e26i			0.614505i	−0.951628	+	0.307252i	$0.900590\pi$
$462$		0			0
$463$	−6.82915e25			−0.150854			−0.0754270	−	0.997151i	$-0.524032\pi$
$463$	−6.82915e25			−0.150854			−0.0754270	+	0.997151i	$0.524032\pi$
$464$		0			0
$465$	1.33718e26	+	4.71144e25i	0.282918	+	0.0996838i
$466$		0			0
$467$		−	1.97599e26i		−	0.400512i	−0.979744	−	0.200256i	$-0.935823\pi$
$467$		−	1.97599e26i		−	0.400512i	0.979744	−	0.200256i	$-0.0641774\pi$
$468$		0			0
$469$	2.20882e26			0.428975
$470$		0			0
$471$	1.10118e26	−	3.12532e26i	0.204951	−	0.581682i
$472$		0			0
$473$			5.48251e26i			0.978066i
$474$		0			0
$475$	4.03319e25			0.0689783
$476$		0			0
$477$	−4.29668e26	+	5.34036e26i	−0.704610	+	0.875763i
$478$		0			0
$479$			4.59905e26i			0.723291i	0.932316	+	0.361645i	$0.117785\pi$
$479$			4.59905e26i			0.723291i	−0.932316	+	0.361645i	$0.882215\pi$
$480$		0			0
$481$	−4.77174e26			−0.719823
$482$		0			0
$483$	8.66977e26	+	3.05472e26i	1.25469	+	0.442080i
$484$		0			0
$485$		−	1.48121e26i		−	0.205683i
$486$		0			0
$487$	−9.34278e26			−1.24505			−0.622525	−	0.782600i	$-0.713893\pi$
$487$	−9.34278e26			−1.24505			−0.622525	+	0.782600i	$0.713893\pi$
$488$		0			0
$489$	−2.39933e26	+	6.80965e26i	−0.306903	+	0.871036i
$490$		0			0
$491$		−	2.02005e26i		−	0.248054i	−0.992279	−	0.124027i	$-0.960419\pi$
$491$		−	2.02005e26i		−	0.248054i	0.992279	−	0.124027i	$-0.0395809\pi$
$492$		0			0
$493$	1.84193e25			0.0217172
$494$		0			0
$495$	8.51791e26	+	6.85323e26i	0.964453	+	0.775968i
$496$		0			0
$497$			1.35196e27i			1.47028i
$498$		0			0
$499$	3.69198e26			0.385703			0.192852	−	0.981228i	$-0.438226\pi$
$499$	3.69198e26			0.385703			0.192852	+	0.981228i	$0.438226\pi$
$500$		0			0
$501$	5.06262e26	+	1.78377e26i	0.508157	+	0.179045i
$502$		0			0
$503$			5.90526e25i			0.0569586i	0.999594	+	0.0284793i	$0.00906647\pi$
$503$			5.90526e25i			0.0569586i	−0.999594	+	0.0284793i	$0.990934\pi$
$504$		0			0
$505$	1.54956e27			1.43647
$506$		0			0
$507$	9.62346e26	−	2.73128e27i	0.857534	−	2.43381i
$508$		0			0
$509$		−	4.58339e26i		−	0.392653i	−0.980539	−	0.196327i	$-0.937099\pi$
$509$		−	4.58339e26i		−	0.392653i	0.980539	−	0.196327i	$-0.0629013\pi$
$510$		0			0
$511$	−2.38502e27			−1.96464
$512$		0			0
$513$	−2.12708e26	+	1.31738e26i	−0.168504	+	0.104361i
$514$		0			0
$515$		−	1.44341e27i		−	1.09981i
$516$		0			0
$517$	−1.79323e27			−1.31441
$518$		0			0
$519$	1.32925e27	+	4.68351e26i	0.937419	+	0.330292i
$520$		0			0
$521$			2.35400e27i			1.59746i	0.601689	+	0.798730i	$0.294495\pi$
$521$			2.35400e27i			1.59746i	−0.601689	+	0.798730i	$0.705505\pi$
$522$		0			0
$523$	−9.14081e26			−0.596991			−0.298496	−	0.954411i	$-0.596485\pi$
$523$	−9.14081e26			−0.596991			−0.298496	+	0.954411i	$0.596485\pi$
$524$		0			0
$525$	2.04823e26	−	5.81319e26i	0.128761	−	0.365444i
$526$		0			0
$527$		−	1.28550e25i		−	0.00777974i
$528$		0			0
$529$	−7.33966e26			−0.427680
$530$		0			0
$531$	−9.91906e26	+	1.23284e27i	−0.556577	+	0.691772i
$532$		0			0
$533$		−	5.97957e27i		−	3.23145i
$534$		0			0
$535$	−3.19131e25			−0.0166124
$536$		0			0
$537$	−2.13366e27	−	7.51777e26i	−1.07000	−	0.377004i
$538$		0			0
$539$			7.60038e26i			0.367238i
$540$		0			0
$541$	2.73537e27			1.27363			0.636815	−	0.771017i	$-0.280251\pi$
$541$	2.73537e27			1.27363			0.636815	+	0.771017i	$0.280251\pi$
$542$		0			0
$543$	1.66224e26	−	4.71768e26i	0.0745925	−	0.211705i
$544$		0			0
$545$		−	3.87887e26i		−	0.167780i
$546$		0			0
$547$	5.08524e26			0.212050			0.106025	−	0.994363i	$-0.466188\pi$
$547$	5.08524e26			0.212050			0.106025	+	0.994363i	$0.466188\pi$
$548$		0			0
$549$	−3.13840e26	−	2.52506e26i	−0.126178	−	0.101519i
$550$		0			0
$551$			5.30175e26i			0.205543i
$552$		0			0
$553$	4.32103e27			1.61561
$554$		0			0
$555$	8.03328e26	+	2.83046e26i	0.289710	+	0.102077i
$556$		0			0
$557$		−	2.93532e26i		−	0.102118i	−0.998696	−	0.0510592i	$-0.983740\pi$
$557$		−	2.93532e26i		−	0.102118i	0.998696	−	0.0510592i	$-0.0162597\pi$
$558$		0			0
$559$	−3.59669e27			−1.20721
$560$		0			0
$561$	3.29418e25	−	9.34938e25i	0.0106689	−	0.0302799i
$562$		0			0
$563$			4.55421e27i			1.42341i	0.702480	+	0.711704i	$0.252076\pi$
$563$			4.55421e27i			1.42341i	−0.702480	+	0.711704i	$0.747924\pi$
$564$		0			0
$565$	−2.97639e27			−0.897854
$566$		0			0
$567$	8.18563e26	+	3.73486e27i	0.238354	+	1.08754i
$568$		0			0
$569$			3.19119e27i			0.897077i	0.893763	+	0.448539i	$0.148055\pi$
$569$			3.19119e27i			0.897077i	−0.893763	+	0.448539i	$0.851945\pi$
$570$		0			0
$571$	4.81689e27			1.30739			0.653695	−	0.756758i	$-0.273218\pi$
$571$	4.81689e27			1.30739			0.653695	+	0.756758i	$0.273218\pi$
$572$		0			0
$573$	−4.53274e27	−	1.59708e27i	−1.18800	−	0.418581i
$574$		0			0
$575$			1.64284e27i			0.415830i
$576$		0			0
$577$	2.62360e27			0.641416			0.320708	−	0.947178i	$-0.396079\pi$
$577$	2.62360e27			0.641416			0.320708	+	0.947178i	$0.396079\pi$
$578$		0			0
$579$	7.19332e25	−	2.04157e26i	0.0169881	−	0.0482147i
$580$		0			0
$581$		−	3.05180e27i		−	0.696298i
$582$		0			0
$583$	7.81656e27			1.72318
$584$		0			0
$585$	−4.49592e27	+	5.58800e27i	−0.957768	+	1.19041i
$586$		0			0
$587$			6.79955e27i			1.39991i	0.714188	+	0.699954i	$0.246796\pi$
$587$			6.79955e27i			1.39991i	−0.714188	+	0.699954i	$0.753204\pi$
$588$		0			0
$589$	3.70014e26			0.0736317
$590$		0			0
$591$	−6.49237e27	−	2.28754e27i	−1.24890	−	0.440041i
$592$		0			0
$593$			1.81972e27i			0.338422i	0.985580	+	0.169211i	$0.0541219\pi$
$593$			1.81972e27i			0.338422i	−0.985580	+	0.169211i	$0.945878\pi$
$594$		0			0
$595$	1.04696e26			0.0188262
$596$		0			0
$597$	2.92597e27	−	8.30433e27i	0.508775	−	1.44398i
$598$		0			0
$599$		−	6.03711e27i		−	1.01522i	−0.861587	−	0.507611i	$-0.830529\pi$
$599$		−	6.03711e27i		−	1.01522i	0.861587	−	0.507611i	$-0.169471\pi$
$600$		0			0
$601$	9.95061e26			0.161847			0.0809236	−	0.996720i	$-0.474213\pi$
$601$	9.95061e26			0.161847			0.0809236	+	0.996720i	$0.474213\pi$
$602$		0			0
$603$	−1.90802e27	−	1.53513e27i	−0.300199	−	0.241531i
$604$		0			0
$605$		−	7.16263e27i		−	1.09023i
$606$		0			0
$607$	3.64181e27			0.536328			0.268164	−	0.963373i	$-0.413583\pi$
$607$	3.64181e27			0.536328			0.268164	+	0.963373i	$0.413583\pi$
$608$		0			0
$609$	7.64161e27	+	2.69246e27i	1.08896	+	0.383686i
$610$		0			0
$611$		−	1.17641e28i		−	1.62236i
$612$		0			0
$613$	−9.51510e27			−1.27001			−0.635007	−	0.772506i	$-0.719003\pi$
$613$	−9.51510e27			−1.27001			−0.635007	+	0.772506i	$0.719003\pi$
$614$		0			0
$615$	−3.54691e27	+	1.00667e28i	−0.458247	+	1.30057i
$616$		0			0
$617$			3.01402e26i			0.0376959i	0.999822	+	0.0188479i	$0.00599984\pi$
$617$			3.01402e26i			0.0376959i	−0.999822	+	0.0188479i	$0.994000\pi$
$618$		0			0
$619$	−2.69164e27			−0.325919			−0.162960	−	0.986633i	$-0.552104\pi$
$619$	−2.69164e27			−0.325919			−0.162960	+	0.986633i	$0.552104\pi$
$620$		0			0
$621$	−5.36607e27	−	8.66422e27i	−0.629130	−	1.01581i
$622$		0			0
$623$		−	5.48663e26i		−	0.0622910i
$624$		0			0
$625$	−4.82821e27			−0.530867
$626$		0			0
$627$	2.69110e27	+	9.48187e26i	0.286585	+	0.100976i
$628$		0			0
$629$		−	7.72279e25i		−	0.00796651i
$630$		0			0
$631$	2.80463e27			0.280273			0.140137	−	0.990132i	$-0.455246\pi$
$631$	2.80463e27			0.280273			0.140137	+	0.990132i	$0.455246\pi$
$632$		0			0
$633$	−2.64400e26	+	7.50406e26i	−0.0255991	+	0.0726540i
$634$		0			0
$635$			1.80180e27i			0.169032i
$636$		0			0
$637$	−4.98607e27			−0.453278
$638$		0			0
$639$	9.39615e27	−	1.16785e28i	0.827830	−	1.02891i
$640$		0			0
$641$			1.61326e28i			1.37760i	0.724949	+	0.688802i	$0.241863\pi$
$641$			1.61326e28i			1.37760i	−0.724949	+	0.688802i	$0.758137\pi$
$642$		0			0
$643$	−2.04162e28			−1.68992			−0.844960	−	0.534830i	$-0.820376\pi$
$643$	−2.04162e28			−1.68992			−0.844960	+	0.534830i	$0.820376\pi$
$644$		0			0
$645$	6.05507e27	+	2.13345e27i	0.485872	+	0.171193i
$646$		0			0
$647$		−	5.06577e27i		−	0.394096i	−0.980394	−	0.197048i	$-0.936864\pi$
$647$		−	5.06577e27i		−	0.394096i	0.980394	−	0.197048i	$-0.0631356\pi$
$648$		0			0
$649$	1.80448e28			1.36115
$650$		0			0
$651$	1.87909e27	−	5.33315e27i	0.137448	−	0.390098i
$652$		0			0
$653$			3.27157e27i			0.232073i	0.993245	+	0.116036i	$0.0370189\pi$
$653$			3.27157e27i			0.232073i	−0.993245	+	0.116036i	$0.962981\pi$
$654$		0			0
$655$	9.78832e27			0.673434
$656$		0			0
$657$	2.06022e28	+	1.65759e28i	1.37487	+	1.10617i
$658$		0			0
$659$			1.74575e28i			1.13013i	0.825046	+	0.565065i	$0.191149\pi$
$659$			1.74575e28i			1.13013i	−0.825046	+	0.565065i	$0.808851\pi$
$660$		0			0
$661$	1.12548e28			0.706841			0.353421	−	0.935465i	$-0.385018\pi$
$661$	1.12548e28			0.706841			0.353421	+	0.935465i	$0.385018\pi$
$662$		0			0
$663$	6.13346e26	+	2.16108e26i	0.0373741	+	0.0131685i
$664$		0			0
$665$			3.01355e27i			0.178181i
$666$		0			0
$667$	−2.15956e28			−1.23910
$668$		0			0
$669$	−5.92926e27	+	1.68281e28i	−0.330172	+	0.937076i
$670$		0			0
$671$			4.59361e27i			0.248272i
$672$		0			0
$673$	−3.27012e28			−1.71559			−0.857795	−	0.513992i	$-0.828166\pi$
$673$	−3.27012e28			−1.71559			−0.857795	+	0.513992i	$0.828166\pi$
$674$		0			0
$675$	−5.80947e27	+	3.59802e27i	−0.295868	+	0.183242i
$676$		0			0
$677$		−	2.43840e28i		−	1.20564i	−0.797877	−	0.602820i	$-0.794044\pi$
$677$		−	2.43840e28i		−	1.20564i	0.797877	−	0.602820i	$-0.205956\pi$
$678$		0			0
$679$	−5.90759e27			−0.283604
$680$		0			0
$681$	3.36383e28	+	1.18522e28i	1.56806	+	0.552494i
$682$		0			0
$683$			2.34113e27i			0.105979i	0.998595	+	0.0529893i	$0.0168749\pi$
$683$			2.34113e27i			0.105979i	−0.998595	+	0.0529893i	$0.983125\pi$
$684$		0			0
$685$	−3.04516e28			−1.33876
$686$		0			0
$687$	3.25499e27	−	9.23815e27i	0.138990	−	0.394473i
$688$		0			0
$689$			5.12789e28i			2.12690i
$690$		0			0
$691$	3.45709e28			1.39293			0.696467	−	0.717589i	$-0.254754\pi$
$691$	3.45709e28			1.39293			0.696467	+	0.717589i	$0.254754\pi$
$692$		0			0
$693$	2.73332e28	−	3.39725e28i	1.06993	−	1.32983i
$694$		0			0
$695$		−	3.84857e28i		−	1.46370i
$696$		0			0
$697$	9.67760e26			0.0357635
$698$		0			0
$699$	−3.58091e28	−	1.26170e28i	−1.28594	−	0.453091i
$700$		0			0
$701$		−	4.29230e28i		−	1.49799i	−0.662575	−	0.748996i	$-0.730536\pi$
$701$		−	4.29230e28i		−	1.49799i	0.662575	−	0.748996i	$-0.269464\pi$
$702$		0			0
$703$	2.22291e27			0.0753994
$704$		0			0
$705$	−6.97813e27	+	1.98050e28i	−0.230063	+	0.652955i
$706$		0			0
$707$		−	6.18021e28i		−	1.98066i
$708$		0			0
$709$	−2.16726e27			−0.0675225			−0.0337612	−	0.999430i	$-0.510749\pi$
$709$	−2.16726e27			−0.0675225			−0.0337612	+	0.999430i	$0.510749\pi$
$710$		0			0
$711$	−3.73259e28	−	3.00312e28i	−1.13061	−	0.909654i
$712$		0			0
$713$			1.50718e28i			0.443883i
$714$		0			0
$715$	8.17902e28			2.34229
$716$		0			0
$717$	−2.13303e28	−	7.51555e27i	−0.594025	−	0.209300i
$718$		0			0
$719$			4.22557e28i			1.14445i	0.820097	+	0.572225i	$0.193919\pi$
$719$			4.22557e28i			1.14445i	−0.820097	+	0.572225i	$0.806081\pi$
$720$		0			0
$721$	−5.75682e28			−1.51646
$722$		0			0
$723$	−1.32374e28	+	3.75697e28i	−0.339171	+	0.962619i
$724$		0			0
$725$			1.44801e28i			0.360904i
$726$		0			0
$727$	−3.23940e28			−0.785454			−0.392727	−	0.919655i	$-0.628468\pi$
$727$	−3.23940e28			−0.785454			−0.392727	+	0.919655i	$0.628468\pi$
$728$		0			0
$729$	1.88864e28	−	3.79515e28i	0.445527	−	0.895268i
$730$		0			0
$731$		−	5.82104e26i		−	0.0133606i
$732$		0			0
$733$	1.03772e28			0.231761			0.115881	−	0.993263i	$-0.463031\pi$
$733$	1.03772e28			0.231761			0.115881	+	0.993263i	$0.463031\pi$
$734$		0			0
$735$	8.39411e27	+	2.95760e27i	0.182432	+	0.0642785i
$736$		0			0
$737$			2.79272e28i			0.590681i
$738$		0			0
$739$	3.58121e28			0.737201			0.368600	−	0.929588i	$-0.379837\pi$
$739$	3.58121e28			0.737201			0.368600	+	0.929588i	$0.379837\pi$
$740$		0			0
$741$	−6.22038e27	+	1.76544e28i	−0.124634	+	0.353729i
$742$		0			0
$743$		−	9.76572e27i		−	0.190465i	−0.995455	−	0.0952327i	$-0.969640\pi$
$743$		−	9.76572e27i		−	0.190465i	0.995455	−	0.0952327i	$-0.0303595\pi$
$744$		0			0
$745$	2.27837e28			0.432574
$746$		0			0
$747$	−2.12100e28	+	2.63621e28i	−0.392045	+	0.487274i
$748$		0			0
$749$			1.27281e27i			0.0229058i
$750$		0			0
$751$	6.68019e28			1.17055			0.585274	−	0.810835i	$-0.300987\pi$
$751$	6.68019e28			1.17055			0.585274	+	0.810835i	$0.300987\pi$
$752$		0			0
$753$	−3.70741e28	−	1.30628e28i	−0.632588	−	0.222887i
$754$		0			0
$755$			6.26590e28i			1.04115i
$756$		0			0
$757$	8.19780e28			1.32660			0.663298	−	0.748356i	$-0.269156\pi$
$757$	8.19780e28			1.32660			0.663298	+	0.748356i	$0.269156\pi$
$758$		0			0
$759$	−3.86225e28	+	1.09616e29i	−0.608726	+	1.72766i
$760$		0			0
$761$		−	6.37328e28i		−	0.978399i	−0.872172	−	0.489200i	$-0.837289\pi$
$761$		−	6.37328e28i		−	0.978399i	0.872172	−	0.489200i	$-0.162711\pi$
$762$		0			0
$763$	−1.54703e28			−0.231342
$764$		0			0
$765$	−9.04387e26	−	7.27640e26i	−0.0131747	−	0.0105999i
$766$		0			0
$767$			1.18379e29i			1.68005i
$768$		0			0
$769$	9.10526e28			1.25901			0.629505	−	0.776997i	$-0.283258\pi$
$769$	9.10526e28			1.25901			0.629505	+	0.776997i	$0.283258\pi$
$770$		0			0
$771$	−2.42168e28	−	8.53258e27i	−0.326266	−	0.114957i
$772$		0			0
$773$		−	1.28958e29i		−	1.69299i	−0.532397	−	0.846495i	$-0.678709\pi$
$773$		−	1.28958e29i		−	1.69299i	0.532397	−	0.846495i	$-0.321291\pi$
$774$		0			0
$775$	1.01058e28			0.129287
$776$		0			0
$777$	1.12889e28	−	3.20396e28i	0.140748	−	0.399463i
$778$		0			0
$779$			2.78557e28i			0.338485i
$780$		0			0
$781$	−1.70936e29			−2.02452
$782$		0			0
$783$	−4.72970e28	−	7.63672e28i	−0.546030	−	0.881636i
$784$		0			0
$785$			4.42498e28i			0.497983i
$786$		0			0
$787$	4.05364e28			0.444731			0.222365	−	0.974963i	$-0.428622\pi$
$787$	4.05364e28			0.444731			0.222365	+	0.974963i	$0.428622\pi$
$788$		0			0
$789$	4.60230e28	+	1.62158e28i	0.492271	+	0.173448i
$790$		0			0
$791$			1.18709e29i			1.23800i
$792$		0			0
$793$	−3.01354e28			−0.306440
$794$		0			0
$795$	3.04172e28	−	8.63286e28i	0.301612	−	0.856019i
$796$		0			0
$797$			2.40130e28i			0.232200i	0.993238	+	0.116100i	$0.0370394\pi$
$797$			2.40130e28i			0.232200i	−0.993238	+	0.116100i	$0.962961\pi$
$798$		0			0
$799$	1.90395e27			0.0179551
$800$		0			0
$801$	−3.81321e27	+	4.73946e27i	−0.0350724	+	0.0435917i
$802$		0			0
$803$		−	3.01550e29i		−	2.70523i
$804$		0			0
$805$	−1.22751e29			−1.07415
$806$		0			0
$807$	−3.51611e28	−	1.23887e28i	−0.300142	−	0.105753i
$808$		0			0
$809$		−	8.90967e28i		−	0.741953i	−0.928642	−	0.370976i	$-0.879023\pi$
$809$		−	8.90967e28i		−	0.741953i	0.928642	−	0.370976i	$-0.120977\pi$
$810$		0			0
$811$	−9.30289e28			−0.755804			−0.377902	−	0.925846i	$-0.623354\pi$
$811$	−9.30289e28			−0.755804			−0.377902	+	0.925846i	$0.623354\pi$
$812$		0			0
$813$	3.52616e28	−	1.00078e29i	0.279510	−	0.793291i
$814$		0			0
$815$		−	9.64143e28i		−	0.745702i
$816$		0			0
$817$	1.67551e28			0.126452
$818$		0			0
$819$	2.22869e29	+	1.79313e29i	1.64139	+	1.32061i
$820$		0			0
$821$		−	8.90419e28i		−	0.639975i	−0.947422	−	0.319987i	$-0.896321\pi$
$821$		−	8.90419e28i		−	0.639975i	0.947422	−	0.319987i	$-0.103679\pi$
$822$		0			0
$823$	1.62755e29			1.14166			0.570828	−	0.821069i	$-0.306622\pi$
$823$	1.62755e29			1.14166			0.570828	+	0.821069i	$0.306622\pi$
$824$		0			0
$825$	7.34991e28	+	2.58968e28i	0.503202	+	0.177299i
$826$		0			0
$827$			1.17100e29i			0.782532i	0.920278	+	0.391266i	$0.127963\pi$
$827$			1.17100e29i			0.782532i	−0.920278	+	0.391266i	$0.872037\pi$
$828$		0			0
$829$	−2.25814e29			−1.47301			−0.736505	−	0.676432i	$-0.763525\pi$
$829$	−2.25814e29			−1.47301			−0.736505	+	0.676432i	$0.763525\pi$
$830$		0			0
$831$	2.53426e28	−	7.19260e28i	0.161377	−	0.458011i
$832$		0			0
$833$		−	8.06968e26i		−	0.00501656i
$834$		0			0
$835$	−7.16790e28			−0.435038
$836$		0			0
$837$	−5.32974e28	+	3.30090e28i	−0.315828	+	0.195604i
$838$		0			0
$839$			1.46775e29i			0.849244i	0.905371	+	0.424622i	$0.139593\pi$
$839$			1.46775e29i			0.849244i	−0.905371	+	0.424622i	$0.860407\pi$
$840$		0			0
$841$	−1.33506e28			−0.0754296
$842$		0			0
$843$	−1.23872e28	−	4.36454e27i	−0.0683436	−	0.0240803i
$844$		0			0
$845$			3.86708e29i			2.08361i
$846$		0			0
$847$	−2.85671e29			−1.50326
$848$		0			0
$849$	−6.36435e28	+	1.80630e29i	−0.327097	+	0.928351i
$850$		0			0
$851$			9.05456e28i			0.454539i
$852$		0			0
$853$	3.33935e29			1.63747			0.818733	−	0.574175i	$-0.194677\pi$
$853$	3.33935e29			1.63747			0.818733	+	0.574175i	$0.194677\pi$
$854$		0			0
$855$	2.09442e28	−	2.60316e28i	0.100323	−	0.124692i
$856$		0			0
$857$		−	1.34597e29i		−	0.629835i	−0.949119	−	0.314917i	$-0.898023\pi$
$857$		−	1.34597e29i		−	0.629835i	0.949119	−	0.314917i	$-0.101977\pi$
$858$		0			0
$859$	8.10994e28			0.370754			0.185377	−	0.982667i	$-0.440649\pi$
$859$	8.10994e28			0.370754			0.185377	+	0.982667i	$0.440649\pi$
$860$		0			0
$861$	4.01495e29	+	1.41464e29i	1.79328	+	0.631849i
$862$		0			0
$863$		−	3.61943e29i		−	1.57955i	−0.613399	−	0.789773i	$-0.710198\pi$
$863$		−	3.61943e29i		−	1.57955i	0.613399	−	0.789773i	$-0.289802\pi$
$864$		0			0
$865$	−1.88202e29			−0.802533
$866$		0			0
$867$	7.97175e28	−	2.26250e29i	0.332172	−	0.942754i
$868$		0			0
$869$			5.46330e29i			2.22463i
$870$		0			0
$871$	−1.83211e29			−0.729071
$872$		0			0
$873$	5.10308e28	+	4.10578e28i	0.198468	+	0.159681i
$874$		0			0
$875$			3.18806e29i			1.21184i
$876$		0			0
$877$	3.84403e29			1.42821			0.714103	−	0.700041i	$-0.246835\pi$
$877$	3.84403e29			1.42821			0.714103	+	0.700041i	$0.246835\pi$
$878$		0			0
$879$	2.03768e29	+	7.17960e28i	0.740026	+	0.260742i
$880$		0			0
$881$		−	4.68099e29i		−	1.66180i	−0.556423	−	0.830899i	$-0.687826\pi$
$881$		−	4.68099e29i		−	1.66180i	0.556423	−	0.830899i	$-0.312174\pi$
$882$		0			0
$883$	−4.07985e29			−1.41592			−0.707958	−	0.706255i	$-0.750383\pi$
$883$	−4.07985e29			−1.41592			−0.707958	+	0.706255i	$0.750383\pi$
$884$		0			0
$885$	7.02194e28	−	1.99293e29i	0.238245	−	0.676176i
$886$		0			0
$887$		−	3.42604e29i		−	1.13647i	−0.822868	−	0.568233i	$-0.807627\pi$
$887$		−	3.42604e29i		−	1.13647i	0.822868	−	0.568233i	$-0.192373\pi$
$888$		0			0
$889$	7.18623e28			0.233069
$890$		0			0
$891$	−4.72218e29	+	1.03495e29i	−1.49749	+	0.328203i
$892$		0			0
$893$			5.48028e28i			0.169937i
$894$		0			0
$895$	3.02093e29			0.916033
$896$		0			0
$897$	−7.19115e29	−	2.53375e29i	−2.13243	−	0.751344i
$898$		0			0
$899$			1.32844e29i			0.385251i
$900$		0			0
$901$	−8.29921e27			−0.0235390
$902$		0			0
$903$	8.50898e28	−	2.41498e29i	0.236048	−	0.669939i
$904$		0			0
$905$			6.67952e28i			0.181242i
$906$		0			0
$907$	−2.26455e29			−0.601048			−0.300524	−	0.953774i	$-0.597162\pi$
$907$	−2.26455e29			−0.601048			−0.300524	+	0.953774i	$0.597162\pi$
$908$		0			0
$909$	−4.29525e29	+	5.33859e29i	−1.11519	+	1.38608i
$910$		0			0
$911$		−	3.84836e29i		−	0.977444i	−0.872440	−	0.488722i	$-0.837463\pi$
$911$		−	3.84836e29i		−	0.977444i	0.872440	−	0.488722i	$-0.162537\pi$
$912$		0			0
$913$	3.85855e29			0.958775
$914$		0			0
$915$	5.07333e28	+	1.78755e28i	0.123334	+	0.0434557i
$916$		0			0
$917$		−	3.90393e29i		−	0.928558i
$918$		0			0
$919$	3.54701e29			0.825481			0.412740	−	0.910849i	$-0.364572\pi$
$919$	3.54701e29			0.825481			0.412740	+	0.910849i	$0.364572\pi$
$920$		0			0
$921$	1.11903e29	−	3.17598e29i	0.254827	−	0.723238i
$922$		0			0
$923$		−	1.12139e30i		−	2.49884i
$924$		0			0
$925$	6.07119e28			0.132390
$926$		0			0
$927$	4.97285e29	+	4.00099e29i	1.06123	+	0.853828i
$928$		0			0
$929$			6.71970e29i			1.40344i	0.712454	+	0.701719i	$0.247584\pi$
$929$			6.71970e29i			1.40344i	−0.712454	+	0.701719i	$0.752416\pi$
$930$		0			0
$931$	2.32275e28			0.0474795
$932$		0			0
$933$	−6.80302e29	−	2.39699e29i	−1.36108	−	0.479568i
$934$		0			0
$935$			1.32373e28i			0.0259229i
$936$		0			0
$937$	1.21549e29			0.232999			0.116500	−	0.993191i	$-0.462833\pi$
$937$	1.21549e29			0.232999			0.116500	+	0.993191i	$0.462833\pi$
$938$		0			0
$939$	7.50859e28	−	2.13105e29i	0.140897	−	0.399888i
$940$		0			0
$941$			5.28961e29i			0.971690i	0.874045	+	0.485845i	$0.161488\pi$
$941$			5.28961e29i			0.971690i	−0.874045	+	0.485845i	$0.838512\pi$
$942$		0			0
$943$	−1.13465e30			−2.04053
$944$		0			0
$945$	−2.68840e29	−	4.34076e29i	−0.473342	−	0.764272i
$946$		0			0
$947$			8.81348e29i			1.51931i	0.650324	+	0.759657i	$0.274633\pi$
$947$			8.81348e29i			1.51931i	−0.650324	+	0.759657i	$0.725367\pi$
$948$		0			0
$949$	1.97826e30			3.33903
$950$		0			0
$951$	−1.23416e29	−	4.34845e28i	−0.203969	−	0.0718670i
$952$		0			0
$953$			1.77584e28i			0.0287392i	0.999897	+	0.0143696i	$0.00457414\pi$
$953$			1.77584e28i			0.0287392i	−0.999897	+	0.0143696i	$0.995426\pi$
$954$		0			0
$955$	6.41768e29			1.01705
$956$		0			0
$957$	−3.40422e29	+	9.66167e29i	−0.528321	+	1.49945i
$958$		0			0
$959$			1.21452e30i			1.84594i
$960$		0			0
$961$	−5.79078e29			−0.861991
$962$		0			0
$963$	8.84604e27	−	1.09948e28i	0.0128969	−	0.0160296i
$964$		0			0
$965$			2.89056e28i			0.0412770i
$966$		0			0
$967$	7.57488e29			1.05952			0.529762	−	0.848146i	$-0.322281\pi$
$967$	7.57488e29			1.05952			0.529762	+	0.848146i	$0.322281\pi$
$968$		0			0
$969$	−2.85726e27	−	1.00673e27i	−0.00391483	−	0.00137936i
$970$		0			0
$971$			5.31893e28i			0.0713892i	0.999363	+	0.0356946i	$0.0113644\pi$
$971$			5.31893e28i			0.0713892i	−0.999363	+	0.0356946i	$0.988636\pi$
$972$		0			0
$973$	−1.53495e30			−2.01820
$974$		0			0
$975$	−1.69891e29	+	4.82176e29i	−0.218839	+	0.621097i
$976$		0			0
$977$		−	1.36450e30i		−	1.72197i	−0.508626	−	0.860987i	$-0.669847\pi$
$977$		−	1.36450e30i		−	1.72197i	0.508626	−	0.860987i	$-0.330153\pi$
$978$		0			0
$979$	6.93703e28			0.0857722
$980$		0			0
$981$	1.33636e29	+	1.07519e29i	0.161894	+	0.130255i
$982$		0			0
$983$		−	9.44107e28i		−	0.112069i	−0.998429	−	0.0560346i	$-0.982154\pi$
$983$		−	9.44107e28i		−	0.112069i	0.998429	−	0.0560346i	$-0.0178457\pi$
$984$		0			0
$985$	9.19221e29			1.06920
$986$		0			0
$987$	7.89893e29	+	2.78313e29i	0.900320	+	0.317221i
$988$		0			0
$989$			6.82485e29i			0.762308i
$990$		0			0
$991$	−1.59171e29			−0.174232			−0.0871159	−	0.996198i	$-0.527765\pi$
$991$	−1.59171e29			−0.174232			−0.0871159	+	0.996198i	$0.527765\pi$
$992$		0			0
$993$	1.88865e29	−	5.36028e29i	0.202609	−	0.575036i
$994$		0			0
$995$			1.17577e30i			1.23621i
$996$		0			0
$997$	−1.93257e28			−0.0199151			−0.00995756	−	0.999950i	$-0.503170\pi$
$997$	−1.93257e28			−0.0199151			−0.00995756	+	0.999950i	$0.503170\pi$
$998$		0			0
$999$	−3.20191e29	+	1.98306e29i	−0.323411	+	0.200300i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.21.c.b.5.5	✓	6
3.2	odd	2	inner	12.21.c.b.5.6	yes	6
4.3	odd	2		48.21.e.d.17.2		6
12.11	even	2		48.21.e.d.17.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.21.c.b.5.5	✓	6	1.1	even	1	trivial
12.21.c.b.5.6	yes	6	3.2	odd	2	inner
48.21.e.d.17.1		6	12.11	even	2
48.21.e.d.17.2		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 21 \)
Character orbit:	\([\chi]\)	\(=\)	12.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(19623.0 - 55693.1i) q^{3} +7.88530e6i q^{5} +3.14494e8 q^{7} +(-2.71666e9 - 2.18573e9i) q^{9} +O(q^{10})\)	\(q+(19623.0 - 55693.1i) q^{3} +7.88530e6i q^{5} +3.14494e8 q^{7} +(-2.71666e9 - 2.18573e9i) q^{9} +3.97631e10i q^{11} -2.60857e11 q^{13} +(4.39157e11 + 1.54733e11i) q^{15} -4.22183e10i q^{17} +1.21520e12 q^{19} +(6.17132e12 - 1.75151e13i) q^{21} +4.94987e13i q^{23} +3.31895e13 q^{25} +(-1.75039e14 + 1.08408e14i) q^{27} +4.36286e14i q^{29} +3.04488e14 q^{31} +(2.21453e15 + 7.80272e14i) q^{33} +2.47988e15i q^{35} +1.82925e15 q^{37} +(-5.11881e15 + 1.45280e16i) q^{39} +2.29227e16i q^{41} +1.37879e16 q^{43} +(1.72352e16 - 2.14217e16i) q^{45} +4.50977e16i q^{47} +1.91142e16 q^{49} +(-2.35127e15 - 8.28452e14i) q^{51} -1.96578e17i q^{53} -3.13544e17 q^{55} +(2.38459e16 - 6.76783e16i) q^{57} -4.53809e17i q^{59} +1.15524e17 q^{61} +(-8.54372e17 - 6.87400e17i) q^{63} -2.05694e18i q^{65} +7.02341e17 q^{67} +(2.75674e18 + 9.71314e17i) q^{69} +4.29885e18i q^{71} -7.58366e18 q^{73} +(6.51278e17 - 1.84843e18i) q^{75} +1.25052e19i q^{77} +1.37396e19 q^{79} +(2.60280e18 + 1.18758e19i) q^{81} -9.70385e18i q^{83} +3.32904e17 q^{85} +(2.42981e19 + 8.56125e18i) q^{87} -1.74459e18i q^{89} -8.20381e19 q^{91} +(5.97497e18 - 1.69579e19i) q^{93} +9.58222e18i q^{95} -1.87844e19 q^{97} +(8.69115e19 - 1.08023e20i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 84378 q^{3} - 145040532 q^{7} - 7747974234 q^{9}+O(q^{10}) \) 6 * q - 84378 * q^3 - 145040532 * q^7 - 7747974234 * q^9	\( 6 q - 84378 q^{3} - 145040532 q^{7} - 7747974234 q^{9} - 366963002772 q^{13} - 534244714560 q^{15} + 12201993657804 q^{19} + 10561619781804 q^{21} - 236482695022170 q^{25} - 269341388965818 q^{27} - 647531494989396 q^{31} - 233666770697280 q^{33} - 11\!\cdots\!16 q^{37}+ \cdots + 47\!\cdots\!80 q^{99}+O(q^{100}) \) 6 * q - 84378 * q^3 - 145040532 * q^7 - 7747974234 * q^9 - 366963002772 * q^13 - 534244714560 * q^15 + 12201993657804 * q^19 + 10561619781804 * q^21 - 236482695022170 * q^25 - 269341388965818 * q^27 - 647531494989396 * q^31 - 233666770697280 * q^33 - 11617802966744916 * q^37 - 18953557962425556 * q^39 - 74673968433381492 * q^43 + 16986459376321920 * q^45 + 199908182522509650 * q^49 - 21650764841061120 * q^51 - 342407748800488320 * q^55 + 395526902614004748 * q^57 + 1829222818097529324 * q^61 + 1916686598498095788 * q^63 + 354895661781418188 * q^67 + 1403390676183557760 * q^69 - 20036153441939082036 * q^73 + 2838112082798091270 * q^75 - 18182911195481706708 * q^79 + 20248226439018709446 * q^81 - 70523384934991772160 * q^85 + 101352340532658519360 * q^87 - 143360708053521287784 * q^91 + 132455306774032049388 * q^93 - 192284635486467464052 * q^97 + 475048517452037815680 * q^99

Introduction

L-functions

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