LMFDB - Embedded newform 12.21.c.b.5.2

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			5.2
Root			$-4127.93i$ of defining polynomial
Character	$\chi$	$=$	12.5
Dual form			12.21.c.b.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-53241.6 + 25536.6i) q^{3} -9.13793e6i q^{5} +9.41859e7 q^{7} +(2.18255e9 - 2.71922e9i) q^{9} +O(q^{10})$	\(q+(-53241.6 + 25536.6i) q^{3} -9.13793e6i q^{5} +9.41859e7 q^{7} +(2.18255e9 - 2.71922e9i) q^{9} +4.84994e10i q^{11} +8.27115e10 q^{13} +(2.33352e11 + 4.86518e11i) q^{15} -2.05730e12i q^{17} -4.83065e12 q^{19} +(-5.01461e12 + 2.40518e12i) q^{21} -3.14021e13i q^{23} +1.18656e13 q^{25} +(-4.67630e13 + 2.00510e14i) q^{27} -2.13432e14i q^{29} -1.22827e15 q^{31} +(-1.23851e15 - 2.58218e15i) q^{33} -8.60664e14i q^{35} -7.97589e15 q^{37} +(-4.40369e15 + 2.11217e15i) q^{39} -5.40680e15i q^{41} -1.38665e16 q^{43} +(-2.48480e16 - 1.99440e16i) q^{45} -4.76439e16i q^{47} -7.09213e16 q^{49} +(5.25364e16 + 1.09534e17i) q^{51} +2.36037e16i q^{53} +4.43184e17 q^{55} +(2.57192e17 - 1.23358e17i) q^{57} +9.32852e17i q^{59} +4.44133e17 q^{61} +(2.05565e17 - 2.56112e17i) q^{63} -7.55812e17i q^{65} -7.05775e16 q^{67} +(8.01903e17 + 1.67190e18i) q^{69} -2.01022e18i q^{71} -2.63350e18 q^{73} +(-6.31744e17 + 3.03007e17i) q^{75} +4.56795e18i q^{77} -1.54177e19 q^{79} +(-2.63061e18 - 1.18697e19i) q^{81} -2.96520e19i q^{83} -1.87995e19 q^{85} +(5.45033e18 + 1.13635e19i) q^{87} +4.76598e19i q^{89} +7.79026e18 q^{91} +(6.53951e19 - 3.13658e19i) q^{93} +4.41422e19i q^{95} -2.44765e18 q^{97} +(1.31880e20 + 1.05852e20i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−53241.6	+	25536.6i	−0.901651	+	0.432464i
$3$	−53241.6	+	25536.6i	−0.901651	+	0.432464i
$4$		0			0
$5$		−	9.13793e6i		−	0.935724i	−0.883801	−	0.467862i	$-0.845024\pi$
$5$		−	9.13793e6i		−	0.935724i	0.883801	−	0.467862i	$-0.154976\pi$
$6$		0			0
$7$	9.41859e7			0.333430			0.166715	−	0.986005i	$-0.446684\pi$
$7$	9.41859e7			0.333430			0.166715	+	0.986005i	$0.446684\pi$
$8$		0			0
$9$	2.18255e9	−	2.71922e9i	0.625949	−	0.779864i
$10$		0			0
$11$			4.84994e10i			1.86986i	0.354832	+	0.934930i	$0.384538\pi$
$11$			4.84994e10i			1.86986i	−0.354832	+	0.934930i	$0.615462\pi$
$12$		0			0
$13$	8.27115e10			0.599974			0.299987	−	0.953943i	$-0.403018\pi$
$13$	8.27115e10			0.599974			0.299987	+	0.953943i	$0.403018\pi$
$14$		0			0
$15$	2.33352e11	+	4.86518e11i	0.404667	+	0.843697i
$16$		0			0
$17$		−	2.05730e12i		−	1.02049i	−0.860029	−	0.510245i	$-0.829555\pi$
$17$		−	2.05730e12i		−	1.02049i	0.860029	−	0.510245i	$-0.170445\pi$
$18$		0			0
$19$	−4.83065e12			−0.787898			−0.393949	−	0.919132i	$-0.628891\pi$
$19$	−4.83065e12			−0.787898			−0.393949	+	0.919132i	$0.628891\pi$
$20$		0			0
$21$	−5.01461e12	+	2.40518e12i	−0.300638	+	0.144197i
$22$		0			0
$23$		−	3.14021e13i		−	0.758020i	−0.925393	−	0.379010i	$-0.876265\pi$
$23$		−	3.14021e13i		−	0.758020i	0.925393	−	0.379010i	$-0.123735\pi$
$24$		0			0
$25$	1.18656e13			0.124420
$26$		0			0
$27$	−4.67630e13	+	2.00510e14i	−0.227125	+	0.973866i
$28$		0			0
$29$		−	2.13432e14i		−	0.507318i	−0.967294	−	0.253659i	$-0.918366\pi$
$29$		−	2.13432e14i		−	0.507318i	0.967294	−	0.253659i	$-0.0816341\pi$
$30$		0			0
$31$	−1.22827e15			−1.49857			−0.749285	−	0.662248i	$-0.769603\pi$
$31$	−1.22827e15			−1.49857			−0.749285	+	0.662248i	$0.769603\pi$
$32$		0			0
$33$	−1.23851e15	−	2.58218e15i	−0.808648	−	1.68596i
$34$		0			0
$35$		−	8.60664e14i		−	0.311999i
$36$		0			0
$37$	−7.97589e15			−1.65868			−0.829339	−	0.558746i	$-0.811283\pi$
$37$	−7.97589e15			−1.65868			−0.829339	+	0.558746i	$0.811283\pi$
$38$		0			0
$39$	−4.40369e15	+	2.11217e15i	−0.540967	+	0.259467i
$40$		0			0
$41$		−	5.40680e15i		−	0.402811i	−0.979508	−	0.201406i	$-0.935449\pi$
$41$		−	5.40680e15i		−	0.402811i	0.979508	−	0.201406i	$-0.0645509\pi$
$42$		0			0
$43$	−1.38665e16			−0.641629			−0.320814	−	0.947142i	$-0.603957\pi$
$43$	−1.38665e16			−0.641629			−0.320814	+	0.947142i	$0.603957\pi$
$44$		0			0
$45$	−2.48480e16	−	1.99440e16i	−0.729737	−	0.585716i
$46$		0			0
$47$		−	4.76439e16i		−	0.905793i	−0.891563	−	0.452896i	$-0.850391\pi$
$47$		−	4.76439e16i		−	0.905793i	0.891563	−	0.452896i	$-0.149609\pi$
$48$		0			0
$49$	−7.09213e16			−0.888824
$50$		0			0
$51$	5.25364e16	+	1.09534e17i	0.441325	+	0.920125i
$52$		0			0
$53$			2.36037e16i			0.134965i	0.997720	+	0.0674824i	$0.0214967\pi$
$53$			2.36037e16i			0.134965i	−0.997720	+	0.0674824i	$0.978503\pi$
$54$		0			0
$55$	4.43184e17			1.74967
$56$		0			0
$57$	2.57192e17	−	1.23358e17i	0.710409	−	0.340738i
$58$		0			0
$59$			9.32852e17i			1.82512i	0.408938	+	0.912562i	$0.365899\pi$
$59$			9.32852e17i			1.82512i	−0.408938	+	0.912562i	$0.634101\pi$
$60$		0			0
$61$	4.44133e17			0.622607			0.311304	−	0.950310i	$-0.399234\pi$
$61$	4.44133e17			0.622607			0.311304	+	0.950310i	$0.399234\pi$
$62$		0			0
$63$	2.05565e17	−	2.56112e17i	0.208711	−	0.260030i
$64$		0			0
$65$		−	7.55812e17i		−	0.561410i
$66$		0			0
$67$	−7.05775e16			−0.0387184			−0.0193592	−	0.999813i	$-0.506163\pi$
$67$	−7.05775e16			−0.0387184			−0.0193592	+	0.999813i	$0.506163\pi$
$68$		0			0
$69$	8.01903e17	+	1.67190e18i	0.327817	+	0.683470i
$70$		0			0
$71$		−	2.01022e18i		−	0.617533i	−0.951138	−	0.308767i	$-0.900084\pi$
$71$		−	2.01022e18i		−	0.617533i	0.951138	−	0.308767i	$-0.0999162\pi$
$72$		0			0
$73$	−2.63350e18			−0.612781			−0.306390	−	0.951906i	$-0.599121\pi$
$73$	−2.63350e18			−0.612781			−0.306390	+	0.951906i	$0.599121\pi$
$74$		0			0
$75$	−6.31744e17	+	3.03007e17i	−0.112183	+	0.0538071i
$76$		0			0
$77$			4.56795e18i			0.623468i
$78$		0			0
$79$	−1.54177e19			−1.62835			−0.814177	−	0.580617i	$-0.802811\pi$
$79$	−1.54177e19			−1.62835			−0.814177	+	0.580617i	$0.802811\pi$
$80$		0			0
$81$	−2.63061e18	−	1.18697e19i	−0.216375	−	0.976310i
$82$		0			0
$83$		−	2.96520e19i		−	1.91105i	−0.294907	−	0.955526i	$-0.595289\pi$
$83$		−	2.96520e19i		−	1.91105i	0.294907	−	0.955526i	$-0.404711\pi$
$84$		0			0
$85$	−1.87995e19			−0.954897
$86$		0			0
$87$	5.45033e18	+	1.13635e19i	0.219397	+	0.457424i
$88$		0			0
$89$			4.76598e19i			1.52845i	0.644947	+	0.764227i	$0.276879\pi$
$89$			4.76598e19i			1.52845i	−0.644947	+	0.764227i	$0.723121\pi$
$90$		0			0
$91$	7.79026e18			0.200050
$92$		0			0
$93$	6.53951e19	−	3.13658e19i	1.35119	−	0.648078i
$94$		0			0
$95$			4.41422e19i			0.737255i
$96$		0			0
$97$	−2.44765e18			−0.0331919			−0.0165959	−	0.999862i	$-0.505283\pi$
$97$	−2.44765e18			−0.0331919			−0.0165959	+	0.999862i	$0.505283\pi$
$98$		0			0
$99$	1.31880e20	+	1.05852e20i	1.45824	+	1.17044i
$100$		0			0
$101$		−	1.51774e20i		−	1.37399i	−0.726661	−	0.686996i	$-0.758929\pi$
$101$		−	1.51774e20i		−	1.37399i	0.726661	−	0.686996i	$-0.241071\pi$
$102$		0			0
$103$	−2.78320e18			−0.0207097			−0.0103548	−	0.999946i	$-0.503296\pi$
$103$	−2.78320e18			−0.0207097			−0.0103548	+	0.999946i	$0.503296\pi$
$104$		0			0
$105$	2.19784e19	+	4.58231e19i	0.134928	+	0.281314i
$106$		0			0
$107$			1.25237e20i			0.636641i	0.947983	+	0.318321i	$0.103119\pi$
$107$			1.25237e20i			0.636641i	−0.947983	+	0.318321i	$0.896881\pi$
$108$		0			0
$109$	−3.14931e20			−1.33030			−0.665152	−	0.746708i	$-0.731633\pi$
$109$	−3.14931e20			−1.33030			−0.665152	+	0.746708i	$0.731633\pi$
$110$		0			0
$111$	4.24649e20	−	2.03677e20i	1.49555	−	0.717319i
$112$		0			0
$113$			2.01905e20i			0.594789i	0.954755	+	0.297394i	$0.0961176\pi$
$113$			2.01905e20i			0.594789i	−0.954755	+	0.297394i	$0.903882\pi$
$114$		0			0
$115$	−2.86951e20			−0.709298
$116$		0			0
$117$	1.80522e20	−	2.24911e20i	0.375553	−	0.467898i
$118$		0			0
$119$		−	1.93769e20i		−	0.340262i
$120$		0			0
$121$	−1.67944e21			−2.49638
$122$		0			0
$123$	1.38071e20	+	2.87867e20i	0.174201	+	0.363195i
$124$		0			0
$125$		−	9.79888e20i		−	1.05215i
$126$		0			0
$127$	−1.10159e21			−1.00921			−0.504607	−	0.863349i	$-0.668363\pi$
$127$	−1.10159e21			−1.00921			−0.504607	+	0.863349i	$0.668363\pi$
$128$		0			0
$129$	7.38277e20	−	3.54104e20i	0.578525	−	0.277481i
$130$		0			0
$131$		−	2.39385e20i		−	0.160837i	−0.996761	−	0.0804183i	$-0.974374\pi$
$131$		−	2.39385e20i		−	0.160837i	0.996761	−	0.0804183i	$-0.0256256\pi$
$132$		0			0
$133$	−4.54979e20			−0.262709
$134$		0			0
$135$	1.83225e21	+	4.27317e20i	0.911270	+	0.212526i
$136$		0			0
$137$			1.54298e21i			0.662454i	0.943551	+	0.331227i	$0.107463\pi$
$137$			1.54298e21i			0.662454i	−0.943551	+	0.331227i	$0.892537\pi$
$138$		0			0
$139$	3.66375e21			1.36075			0.680375	−	0.732864i	$-0.261817\pi$
$139$	3.66375e21			1.36075			0.680375	+	0.732864i	$0.261817\pi$
$140$		0			0
$141$	1.21666e21	+	2.53664e21i	0.391723	+	0.816709i
$142$		0			0
$143$			4.01146e21i			1.12187i
$144$		0			0
$145$	−1.95033e21			−0.474710
$146$		0			0
$147$	3.77596e21	−	1.81109e21i	0.801409	−	0.384385i
$148$		0			0
$149$		−	7.36490e21i		−	1.36554i	−0.730634	−	0.682769i	$-0.760775\pi$
$149$		−	7.36490e21i		−	1.36554i	0.730634	−	0.682769i	$-0.239225\pi$
$150$		0			0
$151$	6.67699e21			1.08346			0.541728	−	0.840554i	$-0.317770\pi$
$151$	6.67699e21			1.08346			0.541728	+	0.840554i	$0.317770\pi$
$152$		0			0
$153$	−5.59424e21	−	4.49016e21i	−0.795842	−	0.638775i
$154$		0			0
$155$			1.12239e22i			1.40225i
$156$		0			0
$157$	−8.65746e21			−0.951468			−0.475734	−	0.879589i	$-0.657817\pi$
$157$	−8.65746e21			−0.951468			−0.475734	+	0.879589i	$0.657817\pi$
$158$		0			0
$159$	−6.02757e20	−	1.25670e21i	−0.0583674	−	0.121691i
$160$		0			0
$161$		−	2.95764e21i		−	0.252747i
$162$		0			0
$163$	1.58484e22			1.19704			0.598520	−	0.801108i	$-0.295756\pi$
$163$	1.58484e22			1.19704			0.598520	+	0.801108i	$0.295756\pi$
$164$		0			0
$165$	−2.35958e22	+	1.13174e22i	−1.57760	+	0.756671i
$166$		0			0
$167$		−	1.54088e22i		−	0.913282i	−0.889651	−	0.456641i	$-0.849052\pi$
$167$		−	1.54088e22i		−	0.913282i	0.889651	−	0.456641i	$-0.150948\pi$
$168$		0			0
$169$	−1.21638e22			−0.640031
$170$		0			0
$171$	−1.05431e22	+	1.31356e22i	−0.493184	+	0.614453i
$172$		0			0
$173$		−	3.72030e22i		−	1.54923i	−0.632431	−	0.774616i	$-0.717943\pi$
$173$		−	3.72030e22i		−	1.54923i	0.632431	−	0.774616i	$-0.282057\pi$
$174$		0			0
$175$	1.11757e21			0.0414854
$176$		0			0
$177$	−2.38218e22	−	4.96665e22i	−0.789301	−	1.64563i
$178$		0			0
$179$			4.06417e22i			1.20349i	0.798689	+	0.601743i	$0.205527\pi$
$179$			4.06417e22i			1.20349i	−0.798689	+	0.601743i	$0.794473\pi$
$180$		0			0
$181$	1.56792e22			0.415468			0.207734	−	0.978185i	$-0.433391\pi$
$181$	1.56792e22			0.415468			0.207734	+	0.978185i	$0.433391\pi$
$182$		0			0
$183$	−2.36463e22	+	1.13416e22i	−0.561375	+	0.269255i
$184$		0			0
$185$			7.28832e22i			1.55207i
$186$		0			0
$187$	9.97777e22			1.90817
$188$		0			0
$189$	−4.40441e21	+	1.88852e22i	−0.0757303	+	0.324716i
$190$		0			0
$191$		−	9.46697e22i		−	1.46513i	−0.680695	−	0.732567i	$-0.738322\pi$
$191$		−	9.46697e22i		−	1.46513i	0.680695	−	0.732567i	$-0.261678\pi$
$192$		0			0
$193$	1.17058e23			1.63241			0.816205	−	0.577763i	$-0.196074\pi$
$193$	1.17058e23			1.63241			0.816205	+	0.577763i	$0.196074\pi$
$194$		0			0
$195$	1.93009e22	+	4.02407e22i	0.242790	+	0.506196i
$196$		0			0
$197$			1.29353e23i			1.46931i	0.678440	+	0.734656i	$0.262656\pi$
$197$			1.29353e23i			1.46931i	−0.678440	+	0.734656i	$0.737344\pi$
$198$		0			0
$199$	1.95774e20			0.00201013			0.00100507	−	0.999999i	$-0.499680\pi$
$199$	1.95774e20			0.00201013			0.00100507	+	0.999999i	$0.499680\pi$
$200$		0			0
$201$	3.75766e21	−	1.80231e21i	0.0349105	−	0.0167443i
$202$		0			0
$203$		−	2.01023e22i		−	0.169155i
$204$		0			0
$205$	−4.94070e22			−0.376920
$206$		0			0
$207$	−8.53892e22	−	6.85367e22i	−0.591152	−	0.474482i
$208$		0			0
$209$		−	2.34284e23i		−	1.47326i
$210$		0			0
$211$	−1.16727e23			−0.667342			−0.333671	−	0.942690i	$-0.608287\pi$
$211$	−1.16727e23			−0.667342			−0.333671	+	0.942690i	$0.608287\pi$
$212$		0			0
$213$	5.13342e22	+	1.07027e23i	0.267061	+	0.556799i
$214$		0			0
$215$			1.26712e23i			0.600388i
$216$		0			0
$217$	−1.15686e23			−0.499669
$218$		0			0
$219$	1.40212e23	−	6.72506e22i	0.552514	−	0.265006i
$220$		0			0
$221$		−	1.70162e23i		−	0.612267i
$222$		0			0
$223$	−4.72610e23			−1.55401			−0.777006	−	0.629494i	$-0.783262\pi$
$223$	−4.72610e23			−1.55401			−0.777006	+	0.629494i	$0.783262\pi$
$224$		0			0
$225$	2.58973e22	−	3.22651e22i	0.0778805	−	0.0970305i
$226$		0			0
$227$			2.25654e23i			0.621134i	0.950552	+	0.310567i	$0.100519\pi$
$227$			2.25654e23i			0.621134i	−0.950552	+	0.310567i	$0.899481\pi$
$228$		0			0
$229$	−1.48234e23			−0.373760			−0.186880	−	0.982383i	$-0.559837\pi$
$229$	−1.48234e23			−0.373760			−0.186880	+	0.982383i	$0.559837\pi$
$230$		0			0
$231$	−1.16650e23	−	2.43205e23i	−0.269628	−	0.562151i
$232$		0			0
$233$			2.83760e23i			0.601717i	0.953669	+	0.300859i	$0.0972732\pi$
$233$			2.83760e23i			0.601717i	−0.953669	+	0.300859i	$0.902727\pi$
$234$		0			0
$235$	−4.35367e23			−0.847572
$236$		0			0
$237$	8.20863e23	−	3.93715e23i	1.46821	−	0.704204i
$238$		0			0
$239$		−	8.99167e23i		−	1.47864i	−0.673356	−	0.739318i	$-0.735148\pi$
$239$		−	8.99167e23i		−	1.47864i	0.673356	−	0.739318i	$-0.264852\pi$
$240$		0			0
$241$	2.51482e23			0.380484			0.190242	−	0.981737i	$-0.439073\pi$
$241$	2.51482e23			0.380484			0.190242	+	0.981737i	$0.439073\pi$
$242$		0			0
$243$	4.43168e23	+	5.64783e23i	0.617314	+	0.786717i
$244$		0			0
$245$			6.48074e23i			0.831694i
$246$		0			0
$247$	−3.99551e23			−0.472718
$248$		0			0
$249$	7.57210e23	+	1.57872e24i	0.826462	+	1.72310i
$250$		0			0
$251$			2.19407e23i			0.221061i	0.993873	+	0.110531i	$0.0352550\pi$
$251$			2.19407e23i			0.221061i	−0.993873	+	0.110531i	$0.964745\pi$
$252$		0			0
$253$	1.52298e24			1.41739
$254$		0			0
$255$	1.00091e24	−	4.80074e23i	0.860984	−	0.412959i
$256$		0			0
$257$			5.39353e23i			0.429083i	0.976715	+	0.214542i	$0.0688258\pi$
$257$			5.39353e23i			0.429083i	−0.976715	+	0.214542i	$0.931174\pi$
$258$		0			0
$259$	−7.51216e23			−0.553054
$260$		0			0
$261$	−5.80369e23	−	4.65827e23i	−0.395639	−	0.317555i
$262$		0			0
$263$		−	1.11179e23i		−	0.0702204i	−0.999383	−	0.0351102i	$-0.988822\pi$
$263$		−	1.11179e23i		−	0.0702204i	0.999383	−	0.0351102i	$-0.0111782\pi$
$264$		0			0
$265$	2.15689e23			0.126290
$266$		0			0
$267$	−1.21707e24	−	2.53749e24i	−0.661002	−	1.37813i
$268$		0			0
$269$		−	8.77533e23i		−	0.442324i	−0.975237	−	0.221162i	$-0.929015\pi$
$269$		−	8.77533e23i		−	0.442324i	0.975237	−	0.221162i	$-0.0709850\pi$
$270$		0			0
$271$	−2.08081e24			−0.973955			−0.486977	−	0.873415i	$-0.661901\pi$
$271$	−2.08081e24			−0.973955			−0.486977	+	0.873415i	$0.661901\pi$
$272$		0			0
$273$	−4.14766e23	+	1.98937e23i	−0.180375	+	0.0865143i
$274$		0			0
$275$			5.75474e23i			0.232648i
$276$		0			0
$277$	−1.36813e24			−0.514434			−0.257217	−	0.966354i	$-0.582805\pi$
$277$	−1.36813e24			−0.514434			−0.257217	+	0.966354i	$0.582805\pi$
$278$		0			0
$279$	−2.68076e24	+	3.33993e24i	−0.938029	+	1.16868i
$280$		0			0
$281$		−	2.29577e24i		−	0.747939i	−0.927441	−	0.373969i	$-0.877996\pi$
$281$		−	2.29577e24i		−	0.747939i	0.927441	−	0.373969i	$-0.122004\pi$
$282$		0			0
$283$	2.08917e24			0.634031			0.317015	−	0.948420i	$-0.397319\pi$
$283$	2.08917e24			0.634031			0.317015	+	0.948420i	$0.397319\pi$
$284$		0			0
$285$	−1.12724e24	−	2.35020e24i	−0.318836	−	0.664747i
$286$		0			0
$287$		−	5.09244e23i		−	0.134310i
$288$		0			0
$289$	−1.68251e23			−0.0413981
$290$		0			0
$291$	1.30317e23	−	6.25046e22i	0.0299275	−	0.0143543i
$292$		0			0
$293$			3.39712e24i			0.728509i	0.931299	+	0.364255i	$0.118676\pi$
$293$			3.39712e24i			0.728509i	−0.931299	+	0.364255i	$0.881324\pi$
$294$		0			0
$295$	8.52434e24			1.70781
$296$		0			0
$297$	−9.72462e24	−	2.26797e24i	−1.82099	−	0.424692i
$298$		0			0
$299$		−	2.59732e24i		−	0.454793i
$300$		0			0
$301$	−1.30603e24			−0.213939
$302$		0			0
$303$	3.87579e24	+	8.08070e24i	0.594202	+	1.23886i
$304$		0			0
$305$		−	4.05845e24i		−	0.582589i
$306$		0			0
$307$	7.27021e24			0.977604			0.488802	−	0.872395i	$-0.337434\pi$
$307$	7.27021e24			0.977604			0.488802	+	0.872395i	$0.337434\pi$
$308$		0			0
$309$	1.48182e23	−	7.10735e22i	0.0186729	−	0.00895618i
$310$		0			0
$311$			4.18008e24i			0.493834i	0.969037	+	0.246917i	$0.0794175\pi$
$311$			4.18008e24i			0.493834i	−0.969037	+	0.246917i	$0.920582\pi$
$312$		0			0
$313$	1.19844e25			1.32792			0.663960	−	0.747768i	$-0.268874\pi$
$313$	1.19844e25			1.32792			0.663960	+	0.747768i	$0.268874\pi$
$314$		0			0
$315$	−2.34033e24	−	1.87844e24i	−0.243317	−	0.195296i
$316$		0			0
$317$			1.08398e25i			1.05786i	0.848665	+	0.528931i	$0.177407\pi$
$317$			1.08398e25i			1.05786i	−0.848665	+	0.528931i	$0.822593\pi$
$318$		0			0
$319$	1.03513e25			0.948614
$320$		0			0
$321$	−3.19812e24	−	6.66782e24i	−0.275325	−	0.574028i
$322$		0			0
$323$			9.93810e24i			0.804041i
$324$		0			0
$325$	9.81422e23			0.0746487
$326$		0			0
$327$	1.67675e25	−	8.04227e24i	1.19947	−	0.575309i
$328$		0			0
$329$		−	4.48738e24i		−	0.302019i
$330$		0			0
$331$	−2.32686e25			−1.47398			−0.736989	−	0.675905i	$-0.763753\pi$
$331$	−2.32686e25			−1.47398			−0.736989	+	0.675905i	$0.763753\pi$
$332$		0			0
$333$	−1.74078e25	+	2.16882e25i	−1.03825	+	1.29354i
$334$		0			0
$335$			6.44932e23i			0.0362298i
$336$		0			0
$337$	−2.21281e25			−1.17124			−0.585618	−	0.810587i	$-0.699148\pi$
$337$	−2.21281e25			−1.17124			−0.585618	+	0.810587i	$0.699148\pi$
$338$		0			0
$339$	−5.15596e24	−	1.07497e25i	−0.257225	−	0.536292i
$340$		0			0
$341$		−	5.95703e25i		−	2.80212i
$342$		0			0
$343$	−1.41951e25			−0.629792
$344$		0			0
$345$	1.52777e25	−	7.32774e24i	0.639539	−	0.306746i
$346$		0			0
$347$		−	9.83190e24i		−	0.388457i	−0.980956	−	0.194228i	$-0.937780\pi$
$347$		−	9.83190e24i		−	0.388457i	0.980956	−	0.194228i	$-0.0622203\pi$
$348$		0			0
$349$	−1.11748e25			−0.416855			−0.208428	−	0.978038i	$-0.566835\pi$
$349$	−1.11748e25			−0.416855			−0.208428	+	0.978038i	$0.566835\pi$
$350$		0			0
$351$	−3.86784e24	+	1.65845e25i	−0.136269	+	0.584294i
$352$		0			0
$353$			4.48096e25i			1.49150i	0.666224	+	0.745752i	$0.267910\pi$
$353$			4.48096e25i			1.49150i	−0.666224	+	0.745752i	$0.732090\pi$
$354$		0			0
$355$	−1.83693e25			−0.577841
$356$		0			0
$357$	4.94819e24	+	1.03165e25i	0.147151	+	0.306798i
$358$		0			0
$359$			1.51664e25i			0.426518i	0.976996	+	0.213259i	$0.0684078\pi$
$359$			1.51664e25i			0.426518i	−0.976996	+	0.213259i	$0.931592\pi$
$360$		0			0
$361$	−1.42548e25			−0.379217
$362$		0			0
$363$	8.94159e25	−	4.28871e25i	2.25086	−	1.07959i
$364$		0			0
$365$			2.40648e25i			0.573394i
$366$		0			0
$367$	−2.56826e25			−0.579400			−0.289700	−	0.957118i	$-0.593555\pi$
$367$	−2.56826e25			−0.579400			−0.289700	+	0.957118i	$0.593555\pi$
$368$		0			0
$369$	−1.47023e25	−	1.18006e25i	−0.314138	−	0.252139i
$370$		0			0
$371$			2.22313e24i			0.0450014i
$372$		0			0
$373$	5.59496e25			1.07327			0.536635	−	0.843814i	$-0.319695\pi$
$373$	5.59496e25			1.07327			0.536635	+	0.843814i	$0.319695\pi$
$374$		0			0
$375$	2.50230e25	+	5.21708e25i	0.455016	+	0.948670i
$376$		0			0
$377$		−	1.76533e25i		−	0.304378i
$378$		0			0
$379$	−2.06930e25			−0.338401			−0.169200	−	0.985582i	$-0.554119\pi$
$379$	−2.06930e25			−0.338401			−0.169200	+	0.985582i	$0.554119\pi$
$380$		0			0
$381$	5.86505e25	−	2.81309e25i	0.909959	−	0.436449i
$382$		0			0
$383$		−	4.88188e25i		−	0.718785i	−0.933186	−	0.359393i	$-0.882984\pi$
$383$		−	4.88188e25i		−	0.718785i	0.933186	−	0.359393i	$-0.117016\pi$
$384$		0			0
$385$	4.17417e25			0.583394
$386$		0			0
$387$	−3.02644e25	+	3.77061e25i	−0.401627	+	0.500383i
$388$		0			0
$389$			4.04595e25i			0.509947i	0.966948	+	0.254973i	$0.0820667\pi$
$389$			4.04595e25i			0.509947i	−0.966948	+	0.254973i	$0.917933\pi$
$390$		0			0
$391$	−6.46036e25			−0.773551
$392$		0			0
$393$	6.11309e24	+	1.27453e25i	0.0695561	+	0.145018i
$394$		0			0
$395$			1.40886e26i			1.52369i
$396$		0			0
$397$	1.41452e26			1.45447			0.727234	−	0.686389i	$-0.240805\pi$
$397$	1.41452e26			1.45447			0.727234	+	0.686389i	$0.240805\pi$
$398$		0			0
$399$	2.42238e25	−	1.16186e25i	0.236872	−	0.113612i
$400$		0			0
$401$			5.21137e24i			0.0484739i	0.999706	+	0.0242369i	$0.00771562\pi$
$401$			5.21137e24i			0.0484739i	−0.999706	+	0.0242369i	$0.992284\pi$
$402$		0			0
$403$	−1.01592e26			−0.899103
$404$		0			0
$405$	−1.08464e26	+	2.40383e25i	−0.913557	+	0.202467i
$406$		0			0
$407$		−	3.86826e26i		−	3.10150i
$408$		0			0
$409$	3.12081e25			0.238251			0.119125	−	0.992879i	$-0.461991\pi$
$409$	3.12081e25			0.238251			0.119125	+	0.992879i	$0.461991\pi$
$410$		0			0
$411$	−3.94025e25	−	8.21509e25i	−0.286487	−	0.597302i
$412$		0			0
$413$			8.78614e25i			0.608552i
$414$		0			0
$415$	−2.70958e26			−1.78822
$416$		0			0
$417$	−1.95064e26	+	9.35597e25i	−1.22692	+	0.588475i
$418$		0			0
$419$			3.29411e25i			0.197514i	0.995112	+	0.0987570i	$0.0314866\pi$
$419$			3.29411e25i			0.197514i	−0.995112	+	0.0987570i	$0.968513\pi$
$420$		0			0
$421$	1.64212e26			0.938822			0.469411	−	0.882980i	$-0.344466\pi$
$421$	1.64212e26			0.938822			0.469411	+	0.882980i	$0.344466\pi$
$422$		0			0
$423$	−1.29554e26	−	1.03985e26i	−0.706395	−	0.566980i
$424$		0			0
$425$		−	2.44111e25i		−	0.126969i
$426$		0			0
$427$	4.18310e25			0.207596
$428$		0			0
$429$	−1.02439e26	−	2.13576e26i	−0.485168	−	1.01153i
$430$		0			0
$431$			2.02005e26i			0.913248i	0.889660	+	0.456624i	$0.150942\pi$
$431$			2.02005e26i			0.913248i	−0.889660	+	0.456624i	$0.849058\pi$
$432$		0			0
$433$	2.58219e26			1.11457			0.557287	−	0.830320i	$-0.311842\pi$
$433$	2.58219e26			1.11457			0.557287	+	0.830320i	$0.311842\pi$
$434$		0			0
$435$	1.03839e26	−	4.98048e25i	0.428023	−	0.205295i
$436$		0			0
$437$			1.51693e26i			0.597242i
$438$		0			0
$439$	1.94566e26			0.731851			0.365925	−	0.930644i	$-0.380753\pi$
$439$	1.94566e26			0.731851			0.365925	+	0.930644i	$0.380753\pi$
$440$		0			0
$441$	−1.54789e26	+	1.92850e26i	−0.556359	+	0.693162i
$442$		0			0
$443$			3.07352e25i			0.105584i	0.998606	+	0.0527920i	$0.0168120\pi$
$443$			3.07352e25i			0.105584i	−0.998606	+	0.0527920i	$0.983188\pi$
$444$		0			0
$445$	4.35513e26			1.43021
$446$		0			0
$447$	1.88074e26	+	3.92119e26i	0.590547	+	1.23124i
$448$		0			0
$449$		−	3.30402e26i		−	0.992155i	−0.868278	−	0.496077i	$-0.834773\pi$
$449$		−	3.30402e26i		−	0.992155i	0.868278	−	0.496077i	$-0.165227\pi$
$450$		0			0
$451$	2.62226e26			0.753201
$452$		0			0
$453$	−3.55494e26	+	1.70508e26i	−0.976900	+	0.468556i
$454$		0			0
$455$		−	7.11868e25i		−	0.187191i
$456$		0			0
$457$	−8.55834e25			−0.215391			−0.107696	−	0.994184i	$-0.534347\pi$
$457$	−8.55834e25			−0.215391			−0.107696	+	0.994184i	$0.534347\pi$
$458$		0			0
$459$	4.12510e26	+	9.62055e25i	0.993819	+	0.231778i
$460$		0			0
$461$			1.27454e26i			0.293998i	0.989137	+	0.146999i	$0.0469614\pi$
$461$			1.27454e26i			0.293998i	−0.989137	+	0.146999i	$0.953039\pi$
$462$		0			0
$463$	−2.01692e26			−0.445531			−0.222765	−	0.974872i	$-0.571508\pi$
$463$	−2.01692e26			−0.445531			−0.222765	+	0.974872i	$0.571508\pi$
$464$		0			0
$465$	−2.86619e26	−	5.97576e26i	−0.606422	−	1.26434i
$466$		0			0
$467$			3.57889e26i			0.725404i	0.931905	+	0.362702i	$0.118146\pi$
$467$			3.57889e26i			0.725404i	−0.931905	+	0.362702i	$0.881854\pi$
$468$		0			0
$469$	−6.64740e24			−0.0129099
$470$		0			0
$471$	4.60937e26	−	2.21082e26i	0.857892	−	0.411476i
$472$		0			0
$473$		−	6.72518e26i		−	1.19976i
$474$		0			0
$475$	−5.73186e25			−0.0980301
$476$		0			0
$477$	6.41834e25	+	5.15162e25i	0.105254	+	0.0844811i
$478$		0			0
$479$			8.52653e26i			1.34096i	0.741925	+	0.670482i	$0.233913\pi$
$479$			8.52653e26i			1.34096i	−0.741925	+	0.670482i	$0.766087\pi$
$480$		0			0
$481$	−6.59698e26			−0.995164
$482$		0			0
$483$	7.55279e25	+	1.57469e26i	0.109304	+	0.227890i
$484$		0			0
$485$			2.23665e25i			0.0310585i
$486$		0			0
$487$	−4.31528e26			−0.575068			−0.287534	−	0.957770i	$-0.592835\pi$
$487$	−4.31528e26			−0.575068			−0.287534	+	0.957770i	$0.592835\pi$
$488$		0			0
$489$	−8.43792e26	+	4.04713e26i	−1.07931	+	0.517677i
$490$		0			0
$491$		−	4.42203e26i		−	0.543009i	−0.962437	−	0.271504i	$-0.912479\pi$
$491$		−	4.42203e26i		−	0.543009i	0.962437	−	0.271504i	$-0.0875211\pi$
$492$		0			0
$493$	−4.39094e26			−0.517713
$494$		0			0
$495$	9.67271e26	−	1.20511e27i	1.09521	−	1.36451i
$496$		0			0
$497$		−	1.89334e26i		−	0.205904i
$498$		0			0
$499$	−1.46237e27			−1.52775			−0.763874	−	0.645366i	$-0.776705\pi$
$499$	−1.46237e27			−1.52775			−0.763874	+	0.645366i	$0.776705\pi$
$500$		0			0
$501$	3.93489e26	+	8.20390e26i	0.394962	+	0.823462i
$502$		0			0
$503$		−	1.22286e27i		−	1.17950i	−0.807586	−	0.589750i	$-0.799226\pi$
$503$		−	1.22286e27i		−	1.17950i	0.807586	−	0.589750i	$-0.200774\pi$
$504$		0			0
$505$	−1.38690e27			−1.28568
$506$		0			0
$507$	6.47618e26	−	3.10621e26i	0.577085	−	0.276791i
$508$		0			0
$509$		−	1.09032e25i		−	0.00934060i	−0.999989	−	0.00467030i	$-0.998513\pi$
$509$		−	1.09032e25i		−	0.00934060i	0.999989	−	0.00467030i	$-0.00148661\pi$
$510$		0			0
$511$	−2.48039e26			−0.204320
$512$		0			0
$513$	2.25896e26	−	9.68596e26i	0.178951	−	0.767307i
$514$		0			0
$515$			2.54327e25i			0.0193785i
$516$		0			0
$517$	2.31070e27			1.69371
$518$		0			0
$519$	9.50037e26	+	1.98075e27i	0.669988	+	1.39687i
$520$		0			0
$521$			1.08949e27i			0.739345i	0.929162	+	0.369672i	$0.120530\pi$
$521$			1.08949e27i			0.739345i	−0.929162	+	0.369672i	$0.879470\pi$
$522$		0			0
$523$	−6.97251e26			−0.455378			−0.227689	−	0.973734i	$-0.573117\pi$
$523$	−6.97251e26			−0.455378			−0.227689	+	0.973734i	$0.573117\pi$
$524$		0			0
$525$	−5.95013e25	+	2.85390e25i	−0.0374053	+	0.0179409i
$526$		0			0
$527$			2.52692e27i			1.52927i
$528$		0			0
$529$	7.30062e26			0.425405
$530$		0			0
$531$	2.53663e27	+	2.03600e27i	1.42335	+	1.14244i
$532$		0			0
$533$		−	4.47205e26i		−	0.241676i
$534$		0			0
$535$	1.14441e27			0.595721
$536$		0			0
$537$	−1.03785e27	−	2.16383e27i	−0.520465	−	1.08513i
$538$		0			0
$539$		−	3.43964e27i		−	1.66198i
$540$		0			0
$541$	3.16917e27			1.47562			0.737808	−	0.675011i	$-0.235861\pi$
$541$	3.16917e27			1.47562			0.737808	+	0.675011i	$0.235861\pi$
$542$		0			0
$543$	−8.34785e26	+	4.00393e26i	−0.374607	+	0.179675i
$544$		0			0
$545$			2.87782e27i			1.24480i
$546$		0			0
$547$	−2.04862e27			−0.854257			−0.427129	−	0.904191i	$-0.640475\pi$
$547$	−2.04862e27			−0.854257			−0.427129	+	0.904191i	$0.640475\pi$
$548$		0			0
$549$	9.69342e26	−	1.20769e27i	0.389721	−	0.485549i
$550$		0			0
$551$			1.03102e27i			0.399715i
$552$		0			0
$553$	−1.45213e27			−0.542943
$554$		0			0
$555$	−1.86119e27	−	3.88042e27i	−0.671213	−	1.39942i
$556$		0			0
$557$			8.11783e26i			0.282415i	0.989980	+	0.141208i	$0.0450985\pi$
$557$			8.11783e26i			0.282415i	−0.989980	+	0.141208i	$0.954902\pi$
$558$		0			0
$559$	−1.14692e27			−0.384961
$560$		0			0
$561$	−5.31232e27	+	2.54798e27i	−1.72051	+	0.825216i
$562$		0			0
$563$		−	9.28796e26i		−	0.290293i	−0.989410	−	0.145146i	$-0.953635\pi$
$563$		−	9.28796e26i		−	0.290293i	0.989410	−	0.145146i	$-0.0463653\pi$
$564$		0			0
$565$	1.84499e27			0.556558
$566$		0			0
$567$	−2.47766e26	−	1.11795e27i	−0.0721459	−	0.325532i
$568$		0			0
$569$		−	5.08444e27i		−	1.42929i	−0.699487	−	0.714646i	$-0.746588\pi$
$569$		−	5.08444e27i		−	1.42929i	0.699487	−	0.714646i	$-0.253412\pi$
$570$		0			0
$571$	4.76771e26			0.129404			0.0647021	−	0.997905i	$-0.479390\pi$
$571$	4.76771e26			0.129404			0.0647021	+	0.997905i	$0.479390\pi$
$572$		0			0
$573$	2.41754e27	+	5.04037e27i	0.633618	+	1.32104i
$574$		0			0
$575$		−	3.72605e26i		−	0.0943128i
$576$		0			0
$577$	5.75491e26			0.140696			0.0703478	−	0.997523i	$-0.477589\pi$
$577$	5.75491e26			0.140696			0.0703478	+	0.997523i	$0.477589\pi$
$578$		0			0
$579$	−6.23237e27	+	2.98927e27i	−1.47186	+	0.705959i
$580$		0			0
$581$		−	2.79280e27i		−	0.637203i
$582$		0			0
$583$	−1.14476e27			−0.252365
$584$		0			0
$585$	−2.05522e27	−	1.64960e27i	−0.437824	−	0.351415i
$586$		0			0
$587$			1.46477e27i			0.301570i	0.988567	+	0.150785i	$0.0481802\pi$
$587$			1.46477e27i			0.301570i	−0.988567	+	0.150785i	$0.951820\pi$
$588$		0			0
$589$	5.93335e27			1.18072
$590$		0			0
$591$	−3.30323e27	−	6.88695e27i	−0.635424	−	1.32481i
$592$		0			0
$593$			8.19503e26i			0.152406i	0.997092	+	0.0762032i	$0.0242798\pi$
$593$			8.19503e26i			0.152406i	−0.997092	+	0.0762032i	$0.975720\pi$
$594$		0			0
$595$	−1.77064e27			−0.318392
$596$		0			0
$597$	−1.04233e25	+	4.99940e24i	−0.00181244	+	0.000869311i
$598$		0			0
$599$		−	9.67401e26i		−	0.162682i	−0.996686	−	0.0813408i	$-0.974080\pi$
$599$		−	9.67401e26i		−	0.162682i	0.996686	−	0.0813408i	$-0.0259202\pi$
$600$		0			0
$601$	1.25843e27			0.204683			0.102342	−	0.994749i	$-0.467366\pi$
$601$	1.25843e27			0.204683			0.102342	+	0.994749i	$0.467366\pi$
$602$		0			0
$603$	−1.54039e26	+	1.91915e26i	−0.0242358	+	0.0301951i
$604$		0			0
$605$			1.53466e28i			2.33592i
$606$		0			0
$607$	1.15815e27			0.170561			0.0852804	−	0.996357i	$-0.472821\pi$
$607$	1.15815e27			0.170561			0.0852804	+	0.996357i	$0.472821\pi$
$608$		0			0
$609$	5.13344e26	+	1.07028e27i	0.0731536	+	0.152519i
$610$		0			0
$611$		−	3.94070e27i		−	0.543452i
$612$		0			0
$613$	1.44535e28			1.92916			0.964582	−	0.263782i	$-0.0849698\pi$
$613$	1.44535e28			1.92916			0.964582	+	0.263782i	$0.0849698\pi$
$614$		0			0
$615$	2.63051e27	−	1.26168e27i	0.339851	−	0.163005i
$616$		0			0
$617$		−	1.28226e28i		−	1.60371i	−0.597519	−	0.801854i	$-0.703847\pi$
$617$		−	1.28226e28i		−	1.60371i	0.597519	−	0.801854i	$-0.296153\pi$
$618$		0			0
$619$	6.14168e27			0.743671			0.371835	−	0.928299i	$-0.378729\pi$
$619$	6.14168e27			0.743671			0.371835	+	0.928299i	$0.378729\pi$
$620$		0			0
$621$	6.29645e27	+	1.46846e27i	0.738210	+	0.172165i
$622$		0			0
$623$			4.48888e27i			0.509633i
$624$		0			0
$625$	−7.82256e27			−0.860100
$626$		0			0
$627$	5.98280e27	+	1.24736e28i	0.637132	+	1.32837i
$628$		0			0
$629$			1.64088e28i			1.69266i
$630$		0			0
$631$	−4.26100e27			−0.425813			−0.212906	−	0.977073i	$-0.568293\pi$
$631$	−4.26100e27			−0.425813			−0.212906	+	0.977073i	$0.568293\pi$
$632$		0			0
$633$	6.21475e27	−	2.98082e27i	0.601709	−	0.288601i
$634$		0			0
$635$			1.00663e28i			0.944346i
$636$		0			0
$637$	−5.86601e27			−0.533271
$638$		0			0
$639$	−5.46623e27	−	4.38741e27i	−0.481592	−	0.386544i
$640$		0			0
$641$		−	1.86999e28i		−	1.59683i	−0.602105	−	0.798417i	$-0.705671\pi$
$641$		−	1.86999e28i		−	1.59683i	0.602105	−	0.798417i	$-0.294329\pi$
$642$		0			0
$643$	1.53495e28			1.27053			0.635264	−	0.772295i	$-0.280891\pi$
$643$	1.53495e28			1.27053			0.635264	+	0.772295i	$0.280891\pi$
$644$		0			0
$645$	−3.23578e27	−	6.74633e27i	−0.259646	−	0.541340i
$646$		0			0
$647$		−	1.15099e28i		−	0.895422i	−0.894178	−	0.447711i	$-0.852239\pi$
$647$		−	1.15099e28i		−	0.895422i	0.894178	−	0.447711i	$-0.147761\pi$
$648$		0			0
$649$	−4.52427e28			−3.41273
$650$		0			0
$651$	6.15929e27	−	2.95422e27i	0.450527	−	0.216089i
$652$		0			0
$653$		−	7.75748e27i		−	0.550286i	−0.961403	−	0.275143i	$-0.911275\pi$
$653$		−	7.75748e27i		−	0.550286i	0.961403	−	0.275143i	$-0.0887252\pi$
$654$		0			0
$655$	−2.18749e27			−0.150499
$656$		0			0
$657$	−5.74775e27	+	7.16106e27i	−0.383570	+	0.477885i
$658$		0			0
$659$		−	1.64362e28i		−	1.06401i	−0.846740	−	0.532007i	$-0.821438\pi$
$659$		−	1.64362e28i		−	1.06401i	0.846740	−	0.532007i	$-0.178562\pi$
$660$		0			0
$661$	−1.21574e28			−0.763532			−0.381766	−	0.924259i	$-0.624684\pi$
$661$	−1.21574e28			−0.763532			−0.381766	+	0.924259i	$0.624684\pi$
$662$		0			0
$663$	4.34537e27	+	9.05972e27i	0.264784	+	0.552051i
$664$		0			0
$665$			4.15757e27i			0.245823i
$666$		0			0
$667$	−6.70223e27			−0.384557
$668$		0			0
$669$	2.51625e28	−	1.20688e28i	1.40118	−	0.672054i
$670$		0			0
$671$			2.15401e28i			1.16419i
$672$		0			0
$673$	−2.35753e28			−1.23682			−0.618411	−	0.785855i	$-0.712223\pi$
$673$	−2.35753e28			−1.23682			−0.618411	+	0.785855i	$0.712223\pi$
$674$		0			0
$675$	−5.54871e26	+	2.37918e27i	−0.0282588	+	0.121168i
$676$		0			0
$677$		−	9.98592e27i		−	0.493743i	−0.969048	−	0.246872i	$-0.920597\pi$
$677$		−	9.98592e27i		−	0.493743i	0.969048	−	0.246872i	$-0.0794026\pi$
$678$		0			0
$679$	−2.30534e26			−0.0110672
$680$		0			0
$681$	−5.76244e27	−	1.20142e28i	−0.268618	−	0.560046i
$682$		0			0
$683$			4.31745e28i			1.95443i	0.212253	+	0.977215i	$0.431920\pi$
$683$			4.31745e28i			1.95443i	−0.212253	+	0.977215i	$0.568080\pi$
$684$		0			0
$685$	1.40997e28			0.619874
$686$		0			0
$687$	7.89220e27	−	3.78538e27i	0.337001	−	0.161638i
$688$		0			0
$689$			1.95229e27i			0.0809754i
$690$		0			0
$691$	1.46372e28			0.589762			0.294881	−	0.955534i	$-0.404720\pi$
$691$	1.46372e28			0.589762			0.294881	+	0.955534i	$0.404720\pi$
$692$		0			0
$693$	1.24213e28	+	9.96979e27i	0.486220	+	0.390260i
$694$		0			0
$695$		−	3.34791e28i		−	1.27329i
$696$		0			0
$697$	−1.11234e28			−0.411065
$698$		0			0
$699$	−7.24627e27	−	1.51078e28i	−0.260221	−	0.542539i
$700$		0			0
$701$			5.10282e26i			0.0178086i	0.999960	+	0.00890430i	$0.00283437\pi$
$701$			5.10282e26i			0.0178086i	−0.999960	+	0.00890430i	$0.997166\pi$
$702$		0			0
$703$	3.85288e28			1.30687
$704$		0			0
$705$	2.31796e28	−	1.11178e28i	0.764214	−	0.366545i
$706$		0			0
$707$		−	1.42950e28i		−	0.458131i
$708$		0			0
$709$	−1.00425e28			−0.312880			−0.156440	−	0.987687i	$-0.550002\pi$
$709$	−1.00425e28			−0.312880			−0.156440	+	0.987687i	$0.550002\pi$
$710$		0			0
$711$	−3.36499e28	+	4.19241e28i	−1.01927	+	1.26989i
$712$		0			0
$713$			3.85703e28i			1.13595i
$714$		0			0
$715$	3.66564e28			1.04976
$716$		0			0
$717$	2.29617e28	+	4.78731e28i	0.639457	+	1.33321i
$718$		0			0
$719$		−	2.89944e28i		−	0.785282i	−0.919692	−	0.392641i	$-0.871562\pi$
$719$		−	2.89944e28i		−	0.785282i	0.919692	−	0.392641i	$-0.128438\pi$
$720$		0			0
$721$	−2.62138e26			−0.00690523
$722$		0			0
$723$	−1.33893e28	+	6.42199e27i	−0.343064	+	0.164546i
$724$		0			0
$725$		−	2.53250e27i		−	0.0631204i
$726$		0			0
$727$	3.89297e28			0.943925			0.471962	−	0.881619i	$-0.343546\pi$
$727$	3.89297e28			0.943925			0.471962	+	0.881619i	$0.343546\pi$
$728$		0			0
$729$	−3.80176e28	−	1.87529e28i	−0.896829	−	0.442378i
$730$		0			0
$731$			2.85276e28i			0.654775i
$732$		0			0
$733$	7.40517e28			1.65385			0.826924	−	0.562314i	$-0.190089\pi$
$733$	7.40517e28			1.65385			0.826924	+	0.562314i	$0.190089\pi$
$734$		0			0
$735$	−1.65496e28	−	3.45045e28i	−0.359678	−	0.749898i
$736$		0			0
$737$		−	3.42296e27i		−	0.0723981i
$738$		0			0
$739$	−4.85352e28			−0.999109			−0.499554	−	0.866283i	$-0.666503\pi$
$739$	−4.85352e28			−0.999109			−0.499554	+	0.866283i	$0.666503\pi$
$740$		0			0
$741$	2.12727e28	−	1.02032e28i	0.426227	−	0.204434i
$742$		0			0
$743$			2.20589e28i			0.430225i	0.976589	+	0.215113i	$0.0690119\pi$
$743$			2.20589e28i			0.430225i	−0.976589	+	0.215113i	$0.930988\pi$
$744$		0			0
$745$	−6.72999e28			−1.27777
$746$		0			0
$747$	−8.06301e28	−	6.47169e28i	−1.49036	−	1.19622i
$748$		0			0
$749$			1.17956e28i			0.212276i
$750$		0			0
$751$	3.27229e28			0.573393			0.286697	−	0.958021i	$-0.407443\pi$
$751$	3.27229e28			0.573393			0.286697	+	0.958021i	$0.407443\pi$
$752$		0			0
$753$	−5.60290e27	−	1.16816e28i	−0.0956011	−	0.199320i
$754$		0			0
$755$		−	6.10139e28i		−	1.01382i
$756$		0			0
$757$	−3.64685e28			−0.590146			−0.295073	−	0.955475i	$-0.595344\pi$
$757$	−3.64685e28			−0.590146			−0.295073	+	0.955475i	$0.595344\pi$
$758$		0			0
$759$	−8.10861e28	+	3.88918e28i	−1.27799	+	0.612971i
$760$		0			0
$761$		−	8.41834e28i		−	1.29235i	−0.763189	−	0.646175i	$-0.776368\pi$
$761$		−	8.41834e28i		−	1.29235i	0.763189	−	0.646175i	$-0.223632\pi$
$762$		0			0
$763$	−2.96621e28			−0.443564
$764$		0			0
$765$	−4.10308e28	+	5.11198e28i	−0.597717	+	0.744689i
$766$		0			0
$767$			7.71576e28i			1.09503i
$768$		0			0
$769$	3.97870e28			0.550146			0.275073	−	0.961423i	$-0.411298\pi$
$769$	3.97870e28			0.550146			0.275073	+	0.961423i	$0.411298\pi$
$770$		0			0
$771$	−1.37732e28	−	2.87160e28i	−0.185563	−	0.386884i
$772$		0			0
$773$		−	8.52228e28i		−	1.11882i	−0.828890	−	0.559411i	$-0.811027\pi$
$773$		−	8.52228e28i		−	1.11882i	0.828890	−	0.559411i	$-0.188973\pi$
$774$		0			0
$775$	−1.45742e28			−0.186452
$776$		0			0
$777$	3.99960e28	−	1.91835e28i	0.498661	−	0.239176i
$778$		0			0
$779$			2.61184e28i			0.317374i
$780$		0			0
$781$	9.74944e28			1.15470
$782$		0			0
$783$	4.27954e28	+	9.98074e27i	0.494060	+	0.115225i
$784$		0			0
$785$			7.91113e28i			0.890312i
$786$		0			0
$787$	9.17633e28			1.00675			0.503375	−	0.864068i	$-0.332091\pi$
$787$	9.17633e28			1.00675			0.503375	+	0.864068i	$0.332091\pi$
$788$		0			0
$789$	2.83912e27	+	5.91932e27i	0.0303678	+	0.0633143i
$790$		0			0
$791$			1.90166e28i			0.198321i
$792$		0			0
$793$	3.67349e28			0.373548
$794$		0			0
$795$	−1.14836e28	+	5.50795e27i	−0.113869	+	0.0546158i
$796$		0			0
$797$			1.50476e29i			1.45508i	0.686068	+	0.727538i	$0.259335\pi$
$797$			1.50476e29i			1.45508i	−0.686068	+	0.727538i	$0.740665\pi$
$798$		0			0
$799$	−9.80178e28			−0.924351
$800$		0			0
$801$	1.29597e29	+	1.04020e29i	1.19199	+	0.956735i
$802$		0			0
$803$		−	1.27723e29i		−	1.14581i
$804$		0			0
$805$	−2.70267e28			−0.236502
$806$		0			0
$807$	2.24092e28	+	4.67213e28i	0.191289	+	0.398822i
$808$		0			0
$809$		−	5.97949e28i		−	0.497942i	−0.968511	−	0.248971i	$-0.919908\pi$
$809$		−	5.97949e28i		−	0.497942i	0.968511	−	0.248971i	$-0.0800924\pi$
$810$		0			0
$811$	−3.67445e28			−0.298527			−0.149264	−	0.988797i	$-0.547690\pi$
$811$	−3.67445e28			−0.298527			−0.149264	+	0.988797i	$0.547690\pi$
$812$		0			0
$813$	1.10785e29	−	5.31367e28i	0.878167	−	0.421201i
$814$		0			0
$815$		−	1.44821e29i		−	1.12010i
$816$		0			0
$817$	6.69845e28			0.505538
$818$		0			0
$819$	1.70026e28	−	2.11834e28i	0.125221	−	0.156011i
$820$		0			0
$821$			1.55554e29i			1.11802i	0.829161	+	0.559010i	$0.188819\pi$
$821$			1.55554e29i			1.11802i	−0.829161	+	0.559010i	$0.811181\pi$
$822$		0			0
$823$	8.88643e28			0.623346			0.311673	−	0.950189i	$-0.399111\pi$
$823$	8.88643e28			0.623346			0.311673	+	0.950189i	$0.399111\pi$
$824$		0			0
$825$	−1.46956e28	−	3.06392e28i	−0.100612	−	0.209767i
$826$		0			0
$827$			7.07122e28i			0.472541i	0.971687	+	0.236271i	$0.0759252\pi$
$827$			7.07122e28i			0.472541i	−0.971687	+	0.236271i	$0.924075\pi$
$828$		0			0
$829$	−2.69500e29			−1.75798			−0.878988	−	0.476844i	$-0.841781\pi$
$829$	−2.69500e29			−1.75798			−0.878988	+	0.476844i	$0.841781\pi$
$830$		0			0
$831$	7.28414e28	−	3.49374e28i	0.463840	−	0.222474i
$832$		0			0
$833$			1.45906e29i			0.907035i
$834$		0			0
$835$	−1.40805e29			−0.854580
$836$		0			0
$837$	5.74376e28	−	2.46281e29i	0.340363	−	1.45941i
$838$		0			0
$839$		−	1.64698e29i		−	0.952948i	−0.879189	−	0.476474i	$-0.841915\pi$
$839$		−	1.64698e29i		−	0.952948i	0.879189	−	0.476474i	$-0.158085\pi$
$840$		0			0
$841$	1.31441e29			0.742628
$842$		0			0
$843$	5.86262e28	+	1.22231e29i	0.323457	+	0.674380i
$844$		0			0
$845$			1.11152e29i			0.598893i
$846$		0			0
$847$	−1.58179e29			−0.832368
$848$		0			0
$849$	−1.11231e29	+	5.33503e28i	−0.571674	+	0.274196i
$850$		0			0
$851$			2.50460e29i			1.25731i
$852$		0			0
$853$	−2.86869e29			−1.40667			−0.703336	−	0.710857i	$-0.748307\pi$
$853$	−2.86869e29			−1.40667			−0.703336	+	0.710857i	$0.748307\pi$
$854$		0			0
$855$	1.20032e29	+	9.63426e28i	0.574959	+	0.461484i
$856$		0			0
$857$		−	3.72372e29i		−	1.74248i	−0.490854	−	0.871242i	$-0.663315\pi$
$857$		−	3.72372e29i		−	1.74248i	0.490854	−	0.871242i	$-0.336685\pi$
$858$		0			0
$859$	−1.07125e29			−0.489733			−0.244866	−	0.969557i	$-0.578744\pi$
$859$	−1.07125e29			−0.489733			−0.244866	+	0.969557i	$0.578744\pi$
$860$		0			0
$861$	1.30043e28	+	2.71130e28i	0.0580841	+	0.121100i
$862$		0			0
$863$		−	4.57374e28i		−	0.199601i	−0.995007	−	0.0998007i	$-0.968179\pi$
$863$		−	4.57374e28i		−	0.199601i	0.995007	−	0.0998007i	$-0.0318205\pi$
$864$		0			0
$865$	−3.39958e29			−1.44965
$866$		0			0
$867$	8.95797e27	−	4.29656e27i	0.0373266	−	0.0179032i
$868$		0			0
$869$		−	7.47748e29i		−	3.04479i
$870$		0			0
$871$	−5.83757e27			−0.0232301
$872$		0			0
$873$	−5.34212e27	+	6.65569e27i	−0.0207764	+	0.0258851i
$874$		0			0
$875$		−	9.22916e28i		−	0.350818i
$876$		0			0
$877$	1.32420e29			0.491992			0.245996	−	0.969271i	$-0.420885\pi$
$877$	1.32420e29			0.491992			0.245996	+	0.969271i	$0.420885\pi$
$878$		0			0
$879$	−8.67509e28	−	1.80868e29i	−0.315054	−	0.656861i
$880$		0			0
$881$			2.71575e29i			0.964120i	0.876138	+	0.482060i	$0.160111\pi$
$881$			2.71575e29i			0.964120i	−0.876138	+	0.482060i	$0.839889\pi$
$882$		0			0
$883$	3.04016e29			1.05509			0.527545	−	0.849527i	$-0.323113\pi$
$883$	3.04016e29			1.05509			0.527545	+	0.849527i	$0.323113\pi$
$884$		0			0
$885$	−4.53849e29	+	2.17682e29i	−1.53985	+	0.738568i
$886$		0			0
$887$			5.84550e29i			1.93903i	0.245024	+	0.969517i	$0.421204\pi$
$887$			5.84550e29i			1.93903i	−0.245024	+	0.969517i	$0.578796\pi$
$888$		0			0
$889$	−1.03754e29			−0.336503
$890$		0			0
$891$	5.75671e29	−	1.27583e29i	1.82556	−	0.404590i
$892$		0			0
$893$			2.30151e29i			0.713672i
$894$		0			0
$895$	3.71381e29			1.12613
$896$		0			0
$897$	6.63266e28	+	1.38285e29i	0.196682	+	0.410064i
$898$		0			0
$899$			2.62153e29i			0.760252i
$900$		0			0
$901$	4.85598e28			0.137730
$902$		0			0
$903$	6.95352e28	−	3.33516e28i	0.192898	−	0.0925208i
$904$		0			0
$905$		−	1.43275e29i		−	0.388764i
$906$		0			0
$907$	3.05742e29			0.811489			0.405744	−	0.913987i	$-0.367012\pi$
$907$	3.05742e29			0.811489			0.405744	+	0.913987i	$0.367012\pi$
$908$		0			0
$909$	−4.12707e29	−	3.31255e29i	−1.07153	−	0.860049i
$910$		0			0
$911$			9.64612e28i			0.245002i	0.992468	+	0.122501i	$0.0390914\pi$
$911$			9.64612e28i			0.245002i	−0.992468	+	0.122501i	$0.960909\pi$
$912$		0			0
$913$	1.43810e30			3.57340
$914$		0			0
$915$	1.03639e29	+	2.16079e29i	0.251949	+	0.525292i
$916$		0			0
$917$		−	2.25467e28i		−	0.0536278i
$918$		0			0
$919$	−5.73885e29			−1.33558			−0.667790	−	0.744350i	$-0.732759\pi$
$919$	−5.73885e29			−1.33558			−0.667790	+	0.744350i	$0.732759\pi$
$920$		0			0
$921$	−3.87077e29	+	1.85656e29i	−0.881458	+	0.422779i
$922$		0			0
$923$		−	1.66268e29i		−	0.370504i
$924$		0			0
$925$	−9.46387e28			−0.206372
$926$		0			0
$927$	−6.07448e27	+	7.56813e27i	−0.0129632	+	0.0161507i
$928$		0			0
$929$			4.89420e29i			1.02217i	0.859529	+	0.511087i	$0.170757\pi$
$929$			4.89420e29i			1.02217i	−0.859529	+	0.511087i	$0.829243\pi$
$930$		0			0
$931$	3.42596e29			0.700303
$932$		0			0
$933$	−1.06745e29	−	2.22554e29i	−0.213566	−	0.445266i
$934$		0			0
$935$		−	9.11762e29i		−	1.78552i
$936$		0			0
$937$	−7.68642e28			−0.147343			−0.0736713	−	0.997283i	$-0.523472\pi$
$937$	−7.68642e28			−0.147343			−0.0736713	+	0.997283i	$0.523472\pi$
$938$		0			0
$939$	−6.38067e29	+	3.06040e29i	−1.19732	+	0.574278i
$940$		0			0
$941$		−	4.69011e29i		−	0.861563i	−0.902456	−	0.430782i	$-0.858238\pi$
$941$		−	4.69011e29i		−	0.861563i	0.902456	−	0.430782i	$-0.141762\pi$
$942$		0			0
$943$	−1.69785e29			−0.305339
$944$		0			0
$945$	1.72572e29	+	4.02472e28i	0.303845	+	0.0708627i
$946$		0			0
$947$		−	3.62658e28i		−	0.0625170i	−0.999511	−	0.0312585i	$-0.990048\pi$
$947$		−	3.62658e28i		−	0.0625170i	0.999511	−	0.0312585i	$-0.00995150\pi$
$948$		0			0
$949$	−2.17821e29			−0.367652
$950$		0			0
$951$	−2.76812e29	−	5.77129e29i	−0.457488	−	0.953823i
$952$		0			0
$953$			1.19636e30i			1.93612i	0.250723	+	0.968059i	$0.419332\pi$
$953$			1.19636e30i			1.93612i	−0.250723	+	0.968059i	$0.580668\pi$
$954$		0			0
$955$	−8.65086e29			−1.37096
$956$		0			0
$957$	−5.51121e29	+	2.64338e29i	−0.855319	+	0.410241i
$958$		0			0
$959$			1.45327e29i			0.220882i
$960$		0			0
$961$	8.36858e29			1.24571
$962$		0			0
$963$	3.40546e29	+	2.73336e29i	0.496493	+	0.398505i
$964$		0			0
$965$		−	1.06967e30i		−	1.52749i
$966$		0			0
$967$	5.90646e29			0.826158			0.413079	−	0.910695i	$-0.364453\pi$
$967$	5.90646e29			0.826158			0.413079	+	0.910695i	$0.364453\pi$
$968$		0			0
$969$	−2.53785e29	−	5.29120e29i	−0.347719	−	0.724965i
$970$		0			0
$971$		−	5.94347e29i		−	0.797715i	−0.917013	−	0.398858i	$-0.869407\pi$
$971$		−	5.94347e29i		−	0.797715i	0.917013	−	0.398858i	$-0.130593\pi$
$972$		0			0
$973$	3.45074e29			0.453715
$974$		0			0
$975$	−5.22525e28	+	2.50622e28i	−0.0673071	+	0.0322829i
$976$		0			0
$977$		−	6.05223e29i		−	0.763783i	−0.924207	−	0.381892i	$-0.875273\pi$
$977$		−	6.05223e29i		−	0.763783i	0.924207	−	0.381892i	$-0.124727\pi$
$978$		0			0
$979$	−2.31147e30			−2.85800
$980$		0			0
$981$	−6.87354e29	+	8.56367e29i	−0.832703	+	1.03746i
$982$		0			0
$983$			1.00285e30i			1.19042i	0.803569	+	0.595212i	$0.202932\pi$
$983$			1.00285e30i			1.19042i	−0.803569	+	0.595212i	$0.797068\pi$
$984$		0			0
$985$	1.18202e30			1.37487
$986$		0			0
$987$	1.14592e29	+	2.38915e29i	0.130612	+	0.272316i
$988$		0			0
$989$			4.35439e29i			0.486367i
$990$		0			0
$991$	−1.00336e30			−1.09829			−0.549147	−	0.835726i	$-0.685047\pi$
$991$	−1.00336e30			−1.09829			−0.549147	+	0.835726i	$0.685047\pi$
$992$		0			0
$993$	1.23886e30	−	5.94201e29i	1.32901	−	0.637442i
$994$		0			0
$995$		−	1.78897e27i		−	0.00188093i
$996$		0			0
$997$	−8.96618e29			−0.923966			−0.461983	−	0.886889i	$-0.652862\pi$
$997$	−8.96618e29			−0.923966			−0.461983	+	0.886889i	$0.652862\pi$
$998$		0			0
$999$	3.72977e29	−	1.59925e30i	0.376727	−	1.61533i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.21.c.b.5.2	yes	6
3.2	odd	2	inner	12.21.c.b.5.1	✓	6
4.3	odd	2		48.21.e.d.17.5		6
12.11	even	2		48.21.e.d.17.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.21.c.b.5.1	✓	6	3.2	odd	2	inner
12.21.c.b.5.2	yes	6	1.1	even	1	trivial
48.21.e.d.17.5		6	4.3	odd	2
48.21.e.d.17.6		6	12.11	even	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)

\(f(q)\)	\(=\)	\(q+(-53241.6 + 25536.6i) q^{3} -9.13793e6i q^{5} +9.41859e7 q^{7} +(2.18255e9 - 2.71922e9i) q^{9} +O(q^{10})\)	\(q+(-53241.6 + 25536.6i) q^{3} -9.13793e6i q^{5} +9.41859e7 q^{7} +(2.18255e9 - 2.71922e9i) q^{9} +4.84994e10i q^{11} +8.27115e10 q^{13} +(2.33352e11 + 4.86518e11i) q^{15} -2.05730e12i q^{17} -4.83065e12 q^{19} +(-5.01461e12 + 2.40518e12i) q^{21} -3.14021e13i q^{23} +1.18656e13 q^{25} +(-4.67630e13 + 2.00510e14i) q^{27} -2.13432e14i q^{29} -1.22827e15 q^{31} +(-1.23851e15 - 2.58218e15i) q^{33} -8.60664e14i q^{35} -7.97589e15 q^{37} +(-4.40369e15 + 2.11217e15i) q^{39} -5.40680e15i q^{41} -1.38665e16 q^{43} +(-2.48480e16 - 1.99440e16i) q^{45} -4.76439e16i q^{47} -7.09213e16 q^{49} +(5.25364e16 + 1.09534e17i) q^{51} +2.36037e16i q^{53} +4.43184e17 q^{55} +(2.57192e17 - 1.23358e17i) q^{57} +9.32852e17i q^{59} +4.44133e17 q^{61} +(2.05565e17 - 2.56112e17i) q^{63} -7.55812e17i q^{65} -7.05775e16 q^{67} +(8.01903e17 + 1.67190e18i) q^{69} -2.01022e18i q^{71} -2.63350e18 q^{73} +(-6.31744e17 + 3.03007e17i) q^{75} +4.56795e18i q^{77} -1.54177e19 q^{79} +(-2.63061e18 - 1.18697e19i) q^{81} -2.96520e19i q^{83} -1.87995e19 q^{85} +(5.45033e18 + 1.13635e19i) q^{87} +4.76598e19i q^{89} +7.79026e18 q^{91} +(6.53951e19 - 3.13658e19i) q^{93} +4.41422e19i q^{95} -2.44765e18 q^{97} +(1.31880e20 + 1.05852e20i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 84378 q^{3} - 145040532 q^{7} - 7747974234 q^{9}+O(q^{10}) \) 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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