LMFDB - Embedded newform 12.21.c.b.5.4

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			5.4
Root			$-2779.95i$ of defining polynomial
Character	$\chi$	$=$	12.5
Dual form			12.21.c.b.5.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8570.43 + 58423.7i) q^{3} +1.60830e7i q^{5} -4.81200e8 q^{7} +(-3.33988e9 - 1.00143e9i) q^{9} +O(q^{10})$	\(q+(-8570.43 + 58423.7i) q^{3} +1.60830e7i q^{5} -4.81200e8 q^{7} +(-3.33988e9 - 1.00143e9i) q^{9} +1.87057e10i q^{11} -5.33557e9 q^{13} +(-9.39631e11 - 1.37839e11i) q^{15} +1.04428e12i q^{17} +9.71645e12 q^{19} +(4.12409e12 - 2.81135e13i) q^{21} +4.89004e13i q^{23} -1.63296e14 q^{25} +(8.71317e13 - 1.86546e14i) q^{27} -3.58206e14i q^{29} +6.00017e14 q^{31} +(-1.09286e15 - 1.60316e14i) q^{33} -7.73916e15i q^{35} +3.37740e14 q^{37} +(4.57281e13 - 3.11724e14i) q^{39} -1.97416e16i q^{41} -3.72584e16 q^{43} +(1.61061e16 - 5.37154e16i) q^{45} +2.49297e16i q^{47} +1.51761e17 q^{49} +(-6.10105e16 - 8.94990e15i) q^{51} +8.92803e16i q^{53} -3.00844e17 q^{55} +(-8.32742e16 + 5.67671e17i) q^{57} +4.77272e17i q^{59} +3.54954e17 q^{61} +(1.60715e18 + 4.81890e17i) q^{63} -8.58121e16i q^{65} -4.54315e17 q^{67} +(-2.85694e18 - 4.19098e17i) q^{69} +2.82474e18i q^{71} +1.99089e17 q^{73} +(1.39952e18 - 9.54039e18i) q^{75} -9.00117e18i q^{77} -7.41338e18 q^{79} +(1.01519e19 + 6.68933e18i) q^{81} -1.89710e19i q^{83} -1.67951e19 q^{85} +(2.09277e19 + 3.06998e18i) q^{87} -2.85392e19i q^{89} +2.56748e18 q^{91} +(-5.14241e18 + 3.50552e19i) q^{93} +1.56270e20i q^{95} -7.49102e19 q^{97} +(1.87325e19 - 6.24747e19i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−8570.43	+	58423.7i	−0.145141	+	0.989411i
$3$	−8570.43	+	58423.7i	−0.145141	+	0.989411i
$4$		0			0
$5$			1.60830e7i			1.64690i	0.567387	+	0.823451i	$0.307954\pi$
$5$			1.60830e7i			1.64690i	−0.567387	+	0.823451i	$0.692046\pi$
$6$		0			0
$7$	−4.81200e8			−1.70351			−0.851756	−	0.523938i	$-0.824462\pi$
$7$	−4.81200e8			−1.70351			−0.851756	+	0.523938i	$0.824462\pi$
$8$		0			0
$9$	−3.33988e9	−	1.00143e9i	−0.957868	−	0.287208i
$10$		0			0
$11$			1.87057e10i			0.721185i	0.932724	+	0.360592i	$0.117425\pi$
$11$			1.87057e10i			0.721185i	−0.932724	+	0.360592i	$0.882575\pi$
$12$		0			0
$13$	−5.33557e9			−0.0387032			−0.0193516	−	0.999813i	$-0.506160\pi$
$13$	−5.33557e9			−0.0387032			−0.0193516	+	0.999813i	$0.506160\pi$
$14$		0			0
$15$	−9.39631e11	−	1.37839e11i	−1.62946	−	0.239033i
$16$		0			0
$17$			1.04428e12i			0.517996i	0.965878	+	0.258998i	$0.0833923\pi$
$17$			1.04428e12i			0.517996i	−0.965878	+	0.258998i	$0.916608\pi$
$18$		0			0
$19$	9.71645e12			1.58479			0.792395	−	0.610009i	$-0.208834\pi$
$19$	9.71645e12			1.58479			0.792395	+	0.610009i	$0.208834\pi$
$20$		0			0
$21$	4.12409e12	−	2.81135e13i	0.247250	−	1.68547i
$22$		0			0
$23$			4.89004e13i			1.18041i	0.807252	+	0.590207i	$0.200954\pi$
$23$			4.89004e13i			1.18041i	−0.807252	+	0.590207i	$0.799046\pi$
$24$		0			0
$25$	−1.63296e14			−1.71229
$26$		0			0
$27$	8.71317e13	−	1.86546e14i	0.423193	−	0.906040i
$28$		0			0
$29$		−	3.58206e14i		−	0.851438i	−0.904855	−	0.425719i	$-0.860021\pi$
$29$		−	3.58206e14i		−	0.851438i	0.904855	−	0.425719i	$-0.139979\pi$
$30$		0			0
$31$	6.00017e14			0.732060			0.366030	−	0.930603i	$-0.380717\pi$
$31$	6.00017e14			0.732060			0.366030	+	0.930603i	$0.380717\pi$
$32$		0			0
$33$	−1.09286e15	−	1.60316e14i	−0.713548	−	0.104673i
$34$		0			0
$35$		−	7.73916e15i		−	2.80552i
$36$		0			0
$37$	3.37740e14			0.0702368			0.0351184	−	0.999383i	$-0.488819\pi$
$37$	3.37740e14			0.0702368			0.0351184	+	0.999383i	$0.488819\pi$
$38$		0			0
$39$	4.57281e13	−	3.11724e14i	0.00561743	−	0.0382934i
$40$		0			0
$41$		−	1.97416e16i		−	1.47077i	−0.677652	−	0.735383i	$-0.737002\pi$
$41$		−	1.97416e16i		−	1.47077i	0.677652	−	0.735383i	$-0.262998\pi$
$42$		0			0
$43$	−3.72584e16			−1.72401			−0.862004	−	0.506901i	$-0.830791\pi$
$43$	−3.72584e16			−1.72401			−0.862004	+	0.506901i	$0.830791\pi$
$44$		0			0
$45$	1.61061e16	−	5.37154e16i	0.473004	−	1.57752i
$46$		0			0
$47$			2.49297e16i			0.473957i	0.971515	+	0.236978i	$0.0761571\pi$
$47$			2.49297e16i			0.473957i	−0.971515	+	0.236978i	$0.923843\pi$
$48$		0			0
$49$	1.51761e17			1.90195
$50$		0			0
$51$	−6.10105e16	−	8.94990e15i	−0.512511	−	0.0751824i
$52$		0			0
$53$			8.92803e16i			0.510501i	0.966875	+	0.255251i	$0.0821580\pi$
$53$			8.92803e16i			0.510501i	−0.966875	+	0.255251i	$0.917842\pi$
$54$		0			0
$55$	−3.00844e17			−1.18772
$56$		0			0
$57$	−8.32742e16	+	5.67671e17i	−0.230018	+	1.56801i
$58$		0			0
$59$			4.77272e17i			0.933783i	0.884315	+	0.466891i	$0.154626\pi$
$59$			4.77272e17i			0.933783i	−0.884315	+	0.466891i	$0.845374\pi$
$60$		0			0
$61$	3.54954e17			0.497593			0.248796	−	0.968556i	$-0.419965\pi$
$61$	3.54954e17			0.497593			0.248796	+	0.968556i	$0.419965\pi$
$62$		0			0
$63$	1.60715e18	+	4.81890e17i	1.63174	+	0.489263i
$64$		0			0
$65$		−	8.58121e16i		−	0.0637404i
$66$		0			0
$67$	−4.54315e17			−0.249235			−0.124618	−	0.992205i	$-0.539770\pi$
$67$	−4.54315e17			−0.249235			−0.124618	+	0.992205i	$0.539770\pi$
$68$		0			0
$69$	−2.85694e18	−	4.19098e17i	−1.16791	−	0.171326i
$70$		0			0
$71$			2.82474e18i			0.867751i	0.900973	+	0.433875i	$0.142854\pi$
$71$			2.82474e18i			0.867751i	−0.900973	+	0.433875i	$0.857146\pi$
$72$		0			0
$73$	1.99089e17			0.0463254			0.0231627	−	0.999732i	$-0.492626\pi$
$73$	1.99089e17			0.0463254			0.0231627	+	0.999732i	$0.492626\pi$
$74$		0			0
$75$	1.39952e18	−	9.54039e18i	0.248523	−	1.69416i
$76$		0			0
$77$		−	9.00117e18i		−	1.22855i
$78$		0			0
$79$	−7.41338e18			−0.782971			−0.391485	−	0.920184i	$-0.628039\pi$
$79$	−7.41338e18			−0.782971			−0.391485	+	0.920184i	$0.628039\pi$
$80$		0			0
$81$	1.01519e19	+	6.68933e18i	0.835023	+	0.550215i
$82$		0			0
$83$		−	1.89710e19i		−	1.22267i	−0.791372	−	0.611335i	$-0.790633\pi$
$83$		−	1.89710e19i		−	1.22267i	0.791372	−	0.611335i	$-0.209367\pi$
$84$		0			0
$85$	−1.67951e19			−0.853088
$86$		0			0
$87$	2.09277e19	+	3.06998e18i	0.842422	+	0.123579i
$88$		0			0
$89$		−	2.85392e19i		−	0.915255i	−0.889144	−	0.457628i	$-0.848699\pi$
$89$		−	2.85392e19i		−	0.915255i	0.889144	−	0.457628i	$-0.151301\pi$
$90$		0			0
$91$	2.56748e18			0.0659314
$92$		0			0
$93$	−5.14241e18	+	3.50552e19i	−0.106252	+	0.724308i
$94$		0			0
$95$			1.56270e20i			2.60999i
$96$		0			0
$97$	−7.49102e19			−1.01584			−0.507918	−	0.861405i	$-0.669585\pi$
$97$	−7.49102e19			−1.01584			−0.507918	+	0.861405i	$0.669585\pi$
$98$		0			0
$99$	1.87325e19	−	6.24747e19i	0.207130	−	0.690800i
$100$		0			0
$101$			3.01671e19i			0.273099i	0.990633	+	0.136549i	$0.0436013\pi$
$101$			3.01671e19i			0.273099i	−0.990633	+	0.136549i	$0.956399\pi$
$102$		0			0
$103$	−1.48341e19			−0.110380			−0.0551898	−	0.998476i	$-0.517576\pi$
$103$	−1.48341e19			−0.110380			−0.0551898	+	0.998476i	$0.517576\pi$
$104$		0			0
$105$	4.52150e20	+	6.63279e19i	2.77581	+	0.407196i
$106$		0			0
$107$			2.04358e20i			1.03885i	0.854515	+	0.519427i	$0.173854\pi$
$107$			2.04358e20i			1.03885i	−0.854515	+	0.519427i	$0.826146\pi$
$108$		0			0
$109$	−3.01856e20			−1.27507			−0.637536	−	0.770421i	$-0.720046\pi$
$109$	−3.01856e20			−1.27507			−0.637536	+	0.770421i	$0.720046\pi$
$110$		0			0
$111$	−2.89457e18	+	1.97320e19i	−0.0101942	+	0.0694931i
$112$		0			0
$113$		−	5.63871e20i		−	1.66110i	−0.556946	−	0.830549i	$-0.688027\pi$
$113$		−	5.63871e20i		−	1.66110i	0.556946	−	0.830549i	$-0.311973\pi$
$114$		0			0
$115$	−7.86467e20			−1.94403
$116$		0			0
$117$	1.78202e19	+	5.34322e18i	0.0370726	+	0.0111159i
$118$		0			0
$119$		−	5.02506e20i		−	0.882412i
$120$		0			0
$121$	3.22848e20			0.479893
$122$		0			0
$123$	1.15338e21	+	1.69194e20i	1.45519	+	0.213468i
$124$		0			0
$125$		−	1.09250e21i		−	1.17307i
$126$		0			0
$127$	−1.43517e20			−0.131482			−0.0657411	−	0.997837i	$-0.520941\pi$
$127$	−1.43517e20			−0.131482			−0.0657411	+	0.997837i	$0.520941\pi$
$128$		0			0
$129$	3.19320e20	−	2.17677e21i	0.250224	−	1.70575i
$130$		0			0
$131$			1.63035e21i			1.09539i	0.836679	+	0.547694i	$0.184494\pi$
$131$			1.63035e21i			1.09539i	−0.836679	+	0.547694i	$0.815506\pi$
$132$		0			0
$133$	−4.67556e21			−2.69971
$134$		0			0
$135$	3.00022e21	+	1.40134e21i	1.49216	+	0.696957i
$136$		0			0
$137$			3.57571e21i			1.53517i	0.640946	+	0.767586i	$0.278542\pi$
$137$			3.57571e21i			1.53517i	−0.640946	+	0.767586i	$0.721458\pi$
$138$		0			0
$139$	7.48588e20			0.278032			0.139016	−	0.990290i	$-0.455606\pi$
$139$	7.48588e20			0.278032			0.139016	+	0.990290i	$0.455606\pi$
$140$		0			0
$141$	−1.45649e21	−	2.13658e20i	−0.468938	−	0.0687906i
$142$		0			0
$143$		−	9.98054e19i		−	0.0279122i
$144$		0			0
$145$	5.76104e21			1.40224
$146$		0			0
$147$	−1.30066e21	+	8.86646e21i	−0.276052	+	1.88181i
$148$		0			0
$149$		−	3.62062e21i		−	0.671306i	−0.941986	−	0.335653i	$-0.891043\pi$
$149$		−	3.62062e21i		−	0.671306i	0.941986	−	0.335653i	$-0.108957\pi$
$150$		0			0
$151$	−3.48705e21			−0.565833			−0.282917	−	0.959145i	$-0.591302\pi$
$151$	−3.48705e21			−0.565833			−0.282917	+	0.959145i	$0.591302\pi$
$152$		0			0
$153$	1.04577e21	−	3.48776e21i	0.148773	−	0.496172i
$154$		0			0
$155$			9.65009e21i			1.20563i
$156$		0			0
$157$	6.71199e21			0.737657			0.368828	−	0.929497i	$-0.379759\pi$
$157$	6.71199e21			0.737657			0.368828	+	0.929497i	$0.379759\pi$
$158$		0			0
$159$	−5.21609e21	−	7.65170e20i	−0.505095	−	0.0740947i
$160$		0			0
$161$		−	2.35309e22i		−	2.01085i
$162$		0			0
$163$	−2.02470e21			−0.152927			−0.0764635	−	0.997072i	$-0.524363\pi$
$163$	−2.02470e21			−0.152927			−0.0764635	+	0.997072i	$0.524363\pi$
$164$		0			0
$165$	2.57836e21	−	1.75764e22i	0.172387	−	1.17514i
$166$		0			0
$167$			1.62223e22i			0.961499i	0.876858	+	0.480750i	$0.159635\pi$
$167$			1.62223e22i			0.961499i	−0.876858	+	0.480750i	$0.840365\pi$
$168$		0			0
$169$	−1.89765e22			−0.998502
$170$		0			0
$171$	−3.24518e22	−	9.73038e21i	−1.51802	−	0.455165i
$172$		0			0
$173$		−	1.45621e22i		−	0.606404i	−0.952926	−	0.303202i	$-0.901944\pi$
$173$		−	1.45621e22i		−	0.606404i	0.952926	−	0.303202i	$-0.0980557\pi$
$174$		0			0
$175$	7.85783e22			2.91690
$176$		0			0
$177$	−2.78840e22	−	4.09043e21i	−0.923895	−	0.135530i
$178$		0			0
$179$		−	2.77928e22i		−	0.823005i	−0.911409	−	0.411502i	$-0.865004\pi$
$179$		−	2.77928e22i		−	0.823005i	0.911409	−	0.411502i	$-0.134996\pi$
$180$		0			0
$181$	4.36088e22			1.15555			0.577774	−	0.816197i	$-0.303921\pi$
$181$	4.36088e22			1.15555			0.577774	+	0.816197i	$0.303921\pi$
$182$		0			0
$183$	−3.04211e21	+	2.07378e22i	−0.0722211	+	0.492324i
$184$		0			0
$185$			5.43188e21i			0.115673i
$186$		0			0
$187$	−1.95339e22			−0.373571
$188$		0			0
$189$	−4.19278e22	+	8.97657e22i	−0.720914	+	1.54345i
$190$		0			0
$191$			1.20506e23i			1.86498i	0.361201	+	0.932488i	$0.382367\pi$
$191$			1.20506e23i			1.86498i	−0.361201	+	0.932488i	$0.617633\pi$
$192$		0			0
$193$	−6.61303e22			−0.922205			−0.461102	−	0.887347i	$-0.652546\pi$
$193$	−6.61303e22			−0.922205			−0.461102	+	0.887347i	$0.652546\pi$
$194$		0			0
$195$	5.01346e21	+	7.35447e20i	0.0630655	+	0.00925135i
$196$		0			0
$197$		−	8.83998e22i		−	1.00413i	−0.864830	−	0.502064i	$-0.832574\pi$
$197$		−	8.83998e22i		−	1.00413i	0.864830	−	0.502064i	$-0.167426\pi$
$198$		0			0
$199$	4.52480e22			0.464588			0.232294	−	0.972646i	$-0.425377\pi$
$199$	4.52480e22			0.464588			0.232294	+	0.972646i	$0.425377\pi$
$200$		0			0
$201$	3.89368e21	−	2.65428e22i	0.0361743	−	0.246596i
$202$		0			0
$203$			1.72369e23i			1.45043i
$204$		0			0
$205$	3.17504e23			2.42221
$206$		0			0
$207$	4.89705e22	−	1.63321e23i	0.339024	−	1.13068i
$208$		0			0
$209$			1.81753e23i			1.14293i
$210$		0			0
$211$	−2.25496e23			−1.28918			−0.644592	−	0.764527i	$-0.722973\pi$
$211$	−2.25496e23			−1.28918			−0.644592	+	0.764527i	$0.722973\pi$
$212$		0			0
$213$	−1.65032e23	−	2.42092e22i	−0.858562	−	0.125946i
$214$		0			0
$215$		−	5.99228e23i		−	2.83927i
$216$		0			0
$217$	−2.88728e23			−1.24707
$218$		0			0
$219$	−1.70628e21	+	1.16315e22i	−0.00672372	+	0.0458349i
$220$		0			0
$221$		−	5.57181e21i		−	0.0200481i
$222$		0			0
$223$	−9.39688e22			−0.308983			−0.154492	−	0.987994i	$-0.549374\pi$
$223$	−9.39688e22			−0.308983			−0.154492	+	0.987994i	$0.549374\pi$
$224$		0			0
$225$	5.45391e23	+	1.63530e23i	1.64015	+	0.491783i
$226$		0			0
$227$		−	3.32067e23i		−	0.914043i	−0.889456	−	0.457022i	$-0.848916\pi$
$227$		−	3.32067e23i		−	0.914043i	0.889456	−	0.457022i	$-0.151084\pi$
$228$		0			0
$229$	−3.35912e23			−0.846974			−0.423487	−	0.905902i	$-0.639194\pi$
$229$	−3.35912e23			−0.846974			−0.423487	+	0.905902i	$0.639194\pi$
$230$		0			0
$231$	5.25882e23	+	7.71439e22i	1.21554	+	0.178313i
$232$		0			0
$233$		−	1.37105e23i		−	0.290734i	−0.989378	−	0.145367i	$-0.953564\pi$
$233$		−	1.37105e23i		−	0.290734i	0.989378	−	0.145367i	$-0.0464362\pi$
$234$		0			0
$235$	−4.00945e23			−0.780561
$236$		0			0
$237$	6.35359e22	−	4.33118e23i	0.113641	−	0.774680i
$238$		0			0
$239$		−	1.74733e22i		−	0.0287340i	−0.999897	−	0.0143670i	$-0.995427\pi$
$239$		−	1.74733e22i		−	0.0287340i	0.999897	−	0.0143670i	$-0.00457332\pi$
$240$		0			0
$241$	3.32783e23			0.503490			0.251745	−	0.967794i	$-0.418996\pi$
$241$	3.32783e23			0.503490			0.251745	+	0.967794i	$0.418996\pi$
$242$		0			0
$243$	−4.77822e23	+	5.35783e23i	−0.665585	+	0.746322i
$244$		0			0
$245$			2.44078e24i			3.13233i
$246$		0			0
$247$	−5.18428e22			−0.0613365
$248$		0			0
$249$	1.10836e24	+	1.62590e23i	1.20972	+	0.177459i
$250$		0			0
$251$		−	5.25673e23i		−	0.529638i	−0.964298	−	0.264819i	$-0.914688\pi$
$251$		−	5.25673e23i		−	0.529638i	0.964298	−	0.264819i	$-0.0853121\pi$
$252$		0			0
$253$	−9.14715e23			−0.851296
$254$		0			0
$255$	1.43941e23	−	9.81234e23i	0.123818	−	0.844055i
$256$		0			0
$257$			1.22060e24i			0.971053i	0.874222	+	0.485526i	$0.161372\pi$
$257$			1.22060e24i			0.971053i	−0.874222	+	0.485526i	$0.838628\pi$
$258$		0			0
$259$	−1.62520e23			−0.119649
$260$		0			0
$261$	−3.58719e23	+	1.19637e24i	−0.244540	+	0.815565i
$262$		0			0
$263$		−	4.80284e23i		−	0.303348i	−0.988431	−	0.151674i	$-0.951534\pi$
$263$		−	4.80284e23i		−	0.303348i	0.988431	−	0.151674i	$-0.0484664\pi$
$264$		0			0
$265$	−1.43590e24			−0.840746
$266$		0			0
$267$	1.66737e24	+	2.44594e23i	0.905563	+	0.132841i
$268$		0			0
$269$		−	2.47474e24i		−	1.24741i	−0.781662	−	0.623703i	$-0.785628\pi$
$269$		−	2.47474e24i		−	1.24741i	0.781662	−	0.623703i	$-0.214372\pi$
$270$		0			0
$271$	1.75378e24			0.820884			0.410442	−	0.911887i	$-0.365374\pi$
$271$	1.75378e24			0.820884			0.410442	+	0.911887i	$0.365374\pi$
$272$		0			0
$273$	−2.20044e22	+	1.50002e23i	−0.00956936	+	0.0652333i
$274$		0			0
$275$		−	3.05457e24i		−	1.23488i
$276$		0			0
$277$	2.64821e24			0.995760			0.497880	−	0.867246i	$-0.334112\pi$
$277$	2.64821e24			0.995760			0.497880	+	0.867246i	$0.334112\pi$
$278$		0			0
$279$	−2.00398e24	−	6.00877e23i	−0.701217	−	0.210254i
$280$		0			0
$281$		−	5.58572e24i		−	1.81977i	−0.414864	−	0.909883i	$-0.636171\pi$
$281$		−	5.58572e24i		−	1.81977i	0.414864	−	0.909883i	$-0.363829\pi$
$282$		0			0
$283$	3.79049e22			0.0115035			0.00575176	−	0.999983i	$-0.498169\pi$
$283$	3.79049e22			0.0115035			0.00575176	+	0.999983i	$0.498169\pi$
$284$		0			0
$285$	−9.12987e24	−	1.33930e24i	−2.58236	−	0.378817i
$286$		0			0
$287$			9.49965e24i			2.50547i
$288$		0			0
$289$	2.97372e24			0.731680
$290$		0			0
$291$	6.42013e23	−	4.37654e24i	0.147440	−	1.00508i
$292$		0			0
$293$		−	7.41495e23i		−	0.159013i	−0.996834	−	0.0795064i	$-0.974666\pi$
$293$		−	7.41495e23i		−	0.159013i	0.996834	−	0.0795064i	$-0.0253344\pi$
$294$		0			0
$295$	−7.67598e24			−1.53785
$296$		0			0
$297$	3.48946e24	+	1.62986e24i	0.653422	+	0.305200i
$298$		0			0
$299$		−	2.60912e23i		−	0.0456858i
$300$		0			0
$301$	1.79287e25			2.93687
$302$		0			0
$303$	−1.76247e24	−	2.58545e23i	−0.270207	−	0.0396378i
$304$		0			0
$305$			5.70874e24i			0.819487i
$306$		0			0
$307$	6.70979e24			0.902246			0.451123	−	0.892462i	$-0.351024\pi$
$307$	6.70979e24			0.902246			0.451123	+	0.892462i	$0.351024\pi$
$308$		0			0
$309$	1.27135e23	−	8.66663e23i	0.0160206	−	0.109211i
$310$		0			0
$311$			8.37382e24i			0.989281i	0.869098	+	0.494641i	$0.164700\pi$
$311$			8.37382e24i			0.989281i	−0.869098	+	0.494641i	$0.835300\pi$
$312$		0			0
$313$	−1.55066e25			−1.71819			−0.859097	−	0.511813i	$-0.828974\pi$
$313$	−1.55066e25			−1.71819			−0.859097	+	0.511813i	$0.828974\pi$
$314$		0			0
$315$	−7.75025e24	+	2.58478e25i	−0.805768	+	2.68732i
$316$		0			0
$317$		−	1.14692e25i		−	1.11928i	−0.828736	−	0.559640i	$-0.810939\pi$
$317$		−	1.14692e25i		−	1.11928i	0.828736	−	0.559640i	$-0.189061\pi$
$318$		0			0
$319$	6.70049e24			0.614044
$320$		0			0
$321$	−1.19394e25	−	1.75144e24i	−1.02785	−	0.150780i
$322$		0			0
$323$			1.01467e25i			0.820914i
$324$		0			0
$325$	8.71280e23			0.0662711
$326$		0			0
$327$	2.58703e24	−	1.76355e25i	0.185065	−	1.26157i
$328$		0			0
$329$		−	1.19962e25i		−	0.807391i
$330$		0			0
$331$	2.44780e25			1.55058			0.775292	−	0.631603i	$-0.217603\pi$
$331$	2.44780e25			1.55058			0.775292	+	0.631603i	$0.217603\pi$
$332$		0			0
$333$	−1.12801e24	−	3.38224e23i	−0.0672776	−	0.0201726i
$334$		0			0
$335$		−	7.30677e24i		−	0.410466i
$336$		0			0
$337$	−1.69206e25			−0.895608			−0.447804	−	0.894132i	$-0.647794\pi$
$337$	−1.69206e25			−0.895608			−0.447804	+	0.894132i	$0.647794\pi$
$338$		0			0
$339$	3.29434e25	+	4.83262e24i	1.64351	+	0.241093i
$340$		0			0
$341$			1.12237e25i			0.527950i
$342$		0			0
$343$	−3.46315e25			−1.53649
$344$		0			0
$345$	6.74036e24	−	4.59483e25i	0.282158	−	1.92344i
$346$		0			0
$347$		−	4.12025e25i		−	1.62791i	−0.580931	−	0.813953i	$-0.697311\pi$
$347$		−	4.12025e25i		−	1.62791i	0.580931	−	0.813953i	$-0.302689\pi$
$348$		0			0
$349$	−1.76388e25			−0.657982			−0.328991	−	0.944333i	$-0.606709\pi$
$349$	−1.76388e25			−0.657982			−0.328991	+	0.944333i	$0.606709\pi$
$350$		0			0
$351$	−4.64897e23	+	9.95327e23i	−0.0163789	+	0.0350667i
$352$		0			0
$353$			7.71074e24i			0.256654i	0.991732	+	0.128327i	$0.0409608\pi$
$353$			7.71074e24i			0.256654i	−0.991732	+	0.128327i	$0.959039\pi$
$354$		0			0
$355$	−4.54304e25			−1.42910
$356$		0			0
$357$	2.93583e25	+	4.30669e24i	0.873068	+	0.128074i
$358$		0			0
$359$			3.04463e25i			0.856230i	0.903724	+	0.428115i	$0.140822\pi$
$359$			3.04463e25i			0.856230i	−0.903724	+	0.428115i	$0.859178\pi$
$360$		0			0
$361$	5.68194e25			1.51156
$362$		0			0
$363$	−2.76694e24	+	1.88620e25i	−0.0696521	+	0.474811i
$364$		0			0
$365$			3.20196e24i			0.0762934i
$366$		0			0
$367$	−2.14499e25			−0.483910			−0.241955	−	0.970287i	$-0.577789\pi$
$367$	−2.14499e25			−0.483910			−0.241955	+	0.970287i	$0.577789\pi$
$368$		0			0
$369$	−1.97699e25	+	6.59345e25i	−0.422416	+	1.40880i
$370$		0			0
$371$		−	4.29617e25i		−	0.869645i
$372$		0			0
$373$	−1.38708e25			−0.266081			−0.133041	−	0.991111i	$-0.542474\pi$
$373$	−1.38708e25			−0.266081			−0.133041	+	0.991111i	$0.542474\pi$
$374$		0			0
$375$	6.38282e25	+	9.36324e24i	1.16065	+	0.170260i
$376$		0			0
$377$			1.91123e24i			0.0329534i
$378$		0			0
$379$	−9.48779e25			−1.55158			−0.775788	−	0.630994i	$-0.782647\pi$
$379$	−9.48779e25			−1.55158			−0.775788	+	0.630994i	$0.782647\pi$
$380$		0			0
$381$	1.23001e24	−	8.38482e24i	0.0190835	−	0.130090i
$382$		0			0
$383$			9.45553e25i			1.39219i	0.717950	+	0.696094i	$0.245081\pi$
$383$			9.45553e25i			1.39219i	−0.717950	+	0.696094i	$0.754919\pi$
$384$		0			0
$385$	1.44766e26			2.02330
$386$		0			0
$387$	1.24439e26	+	3.73118e25i	1.65137	+	0.495149i
$388$		0			0
$389$			1.34761e26i			1.69851i	0.527984	+	0.849254i	$0.322948\pi$
$389$			1.34761e26i			1.69851i	−0.527984	+	0.849254i	$0.677052\pi$
$390$		0			0
$391$	−5.10655e25			−0.611449
$392$		0			0
$393$	−9.52512e25	−	1.39728e25i	−1.08379	−	0.158986i
$394$		0			0
$395$		−	1.19230e26i		−	1.28948i
$396$		0			0
$397$	−1.54400e26			−1.58760			−0.793800	−	0.608179i	$-0.791900\pi$
$397$	−1.54400e26			−1.58760			−0.793800	+	0.608179i	$0.791900\pi$
$398$		0			0
$399$	4.00715e25	−	2.73163e26i	0.391838	−	2.67112i
$400$		0			0
$401$			3.39724e25i			0.315997i	0.987439	+	0.157998i	$0.0505041\pi$
$401$			3.39724e25i			0.315997i	−0.987439	+	0.157998i	$0.949496\pi$
$402$		0			0
$403$	−3.20143e24			−0.0283331
$404$		0			0
$405$	−1.07585e26	+	1.63274e26i	−0.906151	+	1.37520i
$406$		0			0
$407$			6.31765e24i			0.0506537i
$408$		0			0
$409$	1.16784e26			0.891559			0.445780	−	0.895143i	$-0.352926\pi$
$409$	1.16784e26			0.891559			0.445780	+	0.895143i	$0.352926\pi$
$410$		0			0
$411$	−2.08907e26	−	3.06454e25i	−1.51892	−	0.222816i
$412$		0			0
$413$		−	2.29663e26i		−	1.59071i
$414$		0			0
$415$	3.05111e26			2.01362
$416$		0			0
$417$	−6.41572e24	+	4.37353e25i	−0.0403538	+	0.275088i
$418$		0			0
$419$		−	1.69805e26i		−	1.01814i	−0.860724	−	0.509072i	$-0.829989\pi$
$419$		−	1.69805e26i		−	1.01814i	0.860724	−	0.509072i	$-0.170011\pi$
$420$		0			0
$421$	2.13330e26			1.21964			0.609820	−	0.792540i	$-0.291242\pi$
$421$	2.13330e26			1.21964			0.609820	+	0.792540i	$0.291242\pi$
$422$		0			0
$423$	2.49654e25	−	8.32623e25i	0.136124	−	0.453988i
$424$		0			0
$425$		−	1.70527e26i		−	0.886958i
$426$		0			0
$427$	−1.70804e26			−0.847656
$428$		0			0
$429$	5.83101e24	+	8.55376e23i	0.0276166	+	0.00405120i
$430$		0			0
$431$		−	9.26717e25i		−	0.418962i	−0.977813	−	0.209481i	$-0.932823\pi$
$431$		−	9.26717e25i		−	0.418962i	0.977813	−	0.209481i	$-0.0671774\pi$
$432$		0			0
$433$	−2.54463e25			−0.109836			−0.0549181	−	0.998491i	$-0.517490\pi$
$433$	−2.54463e25			−0.109836			−0.0549181	+	0.998491i	$0.517490\pi$
$434$		0			0
$435$	−4.93746e25	+	3.36581e26i	−0.203522	+	1.38739i
$436$		0			0
$437$			4.75138e26i			1.87071i
$438$		0			0
$439$	−3.93650e26			−1.48069			−0.740347	−	0.672225i	$-0.765339\pi$
$439$	−3.93650e26			−1.48069			−0.740347	+	0.672225i	$0.765339\pi$
$440$		0			0
$441$	−5.06864e26	−	1.51979e26i	−1.82182	−	0.546257i
$442$		0			0
$443$			4.19684e26i			1.44173i	0.693073	+	0.720867i	$0.256256\pi$
$443$			4.19684e26i			1.44173i	−0.693073	+	0.720867i	$0.743744\pi$
$444$		0			0
$445$	4.58997e26			1.50734
$446$		0			0
$447$	2.11530e26	+	3.10303e25i	0.664197	+	0.0974340i
$448$		0			0
$449$		−	4.83697e25i		−	0.145248i	−0.997359	−	0.0726240i	$-0.976863\pi$
$449$		−	4.83697e25i		−	0.145248i	0.997359	−	0.0726240i	$-0.0231373\pi$
$450$		0			0
$451$	3.69280e26			1.06069
$452$		0			0
$453$	2.98855e25	−	2.03726e26i	0.0821256	−	0.559841i
$454$		0			0
$455$			4.12928e25i			0.108583i
$456$		0			0
$457$	2.30411e26			0.579884			0.289942	−	0.957044i	$-0.406364\pi$
$457$	2.30411e26			0.579884			0.289942	+	0.957044i	$0.406364\pi$
$458$		0			0
$459$	1.94805e26	+	9.09895e25i	0.469325	+	0.219212i
$460$		0			0
$461$			6.55608e26i			1.51229i	0.654404	+	0.756145i	$0.272919\pi$
$461$			6.55608e26i			1.51229i	−0.654404	+	0.756145i	$0.727081\pi$
$462$		0			0
$463$	3.26500e26			0.721230			0.360615	−	0.932715i	$-0.382567\pi$
$463$	3.26500e26			0.721230			0.360615	+	0.932715i	$0.382567\pi$
$464$		0			0
$465$	−5.63794e26	−	8.27055e25i	−1.19286	−	0.174987i
$466$		0			0
$467$			4.32880e25i			0.0877401i	0.999037	+	0.0438701i	$0.0139688\pi$
$467$			4.32880e25i			0.0877401i	−0.999037	+	0.0438701i	$0.986031\pi$
$468$		0			0
$469$	2.18617e26			0.424575
$470$		0			0
$471$	−5.75246e25	+	3.92139e26i	−0.107064	+	0.729846i
$472$		0			0
$473$		−	6.96943e26i		−	1.24333i
$474$		0			0
$475$	−1.58666e27			−2.71362
$476$		0			0
$477$	8.94082e25	−	2.98185e26i	0.146620	−	0.488993i
$478$		0			0
$479$		−	2.05281e26i		−	0.322845i	−0.986885	−	0.161422i	$-0.948392\pi$
$479$		−	2.05281e26i		−	0.322845i	0.986885	−	0.161422i	$-0.0516081\pi$
$480$		0			0
$481$	−1.80203e24			−0.00271839
$482$		0			0
$483$	1.37476e27	+	2.01670e26i	1.98956	+	0.291857i
$484$		0			0
$485$		−	1.20478e27i		−	1.67298i
$486$		0			0
$487$	−1.18860e27			−1.58396			−0.791982	−	0.610544i	$-0.790951\pi$
$487$	−1.18860e27			−1.58396			−0.791982	+	0.610544i	$0.790951\pi$
$488$		0			0
$489$	1.73525e25	−	1.18290e26i	0.0221960	−	0.151308i
$490$		0			0
$491$		−	2.45846e26i		−	0.301890i	−0.988542	−	0.150945i	$-0.951768\pi$
$491$		−	2.45846e26i		−	0.301890i	0.988542	−	0.150945i	$-0.0482317\pi$
$492$		0			0
$493$	3.74066e26			0.441041
$494$		0			0
$495$	1.00478e27	+	3.01275e26i	1.13768	+	0.341123i
$496$		0			0
$497$		−	1.35926e27i		−	1.47822i
$498$		0			0
$499$	4.77987e25			0.0499356			0.0249678	−	0.999688i	$-0.492052\pi$
$499$	4.77987e25			0.0499356			0.0249678	+	0.999688i	$0.492052\pi$
$500$		0			0
$501$	−9.47770e26	−	1.39032e26i	−0.951318	−	0.139553i
$502$		0			0
$503$			1.48856e27i			1.43578i	0.696159	+	0.717888i	$0.254891\pi$
$503$			1.48856e27i			1.43578i	−0.696159	+	0.717888i	$0.745109\pi$
$504$		0			0
$505$	−4.85178e26			−0.449767
$506$		0			0
$507$	1.62637e26	−	1.10868e27i	0.144924	−	0.987929i
$508$		0			0
$509$			9.90006e26i			0.848125i	0.905633	+	0.424062i	$0.139396\pi$
$509$			9.90006e26i			0.848125i	−0.905633	+	0.424062i	$0.860604\pi$
$510$		0			0
$511$	−9.58018e25			−0.0789159
$512$		0			0
$513$	8.46611e26	−	1.81256e27i	0.670672	−	1.43588i
$514$		0			0
$515$		−	2.38577e26i		−	0.181784i
$516$		0			0
$517$	−4.66327e26			−0.341810
$518$		0			0
$519$	8.50770e26	+	1.24803e26i	0.599983	+	0.0880141i
$520$		0			0
$521$			1.97097e27i			1.33753i	0.743474	+	0.668765i	$0.233177\pi$
$521$			1.97097e27i			1.33753i	−0.743474	+	0.668765i	$0.766823\pi$
$522$		0			0
$523$	1.13162e27			0.739066			0.369533	−	0.929218i	$-0.379518\pi$
$523$	1.13162e27			0.739066			0.369533	+	0.929218i	$0.379518\pi$
$524$		0			0
$525$	−6.73450e26	+	4.59083e27i	−0.423362	+	2.88602i
$526$		0			0
$527$			6.26584e26i			0.379204i
$528$		0			0
$529$	−6.75094e26			−0.393376
$530$		0			0
$531$	4.77956e26	−	1.59403e27i	0.268190	−	0.894441i
$532$		0			0
$533$			1.05333e26i			0.0569234i
$534$		0			0
$535$	−3.28670e27			−1.71089
$536$		0			0
$537$	1.62376e27	+	2.38196e26i	0.814290	+	0.119452i
$538$		0			0
$539$			2.83880e27i			1.37166i
$540$		0			0
$541$	−9.63653e26			−0.448692			−0.224346	−	0.974510i	$-0.572025\pi$
$541$	−9.63653e26			−0.448692			−0.224346	+	0.974510i	$0.572025\pi$
$542$		0			0
$543$	−3.73746e26	+	2.54779e27i	−0.167717	+	1.14331i
$544$		0			0
$545$		−	4.85476e27i		−	2.09992i
$546$		0			0
$547$	2.56681e27			1.07034			0.535169	−	0.844745i	$-0.320248\pi$
$547$	2.56681e27			1.07034			0.535169	+	0.844745i	$0.320248\pi$
$548$		0			0
$549$	−1.18550e27	−	3.55463e26i	−0.476628	−	0.142913i
$550$		0			0
$551$		−	3.48049e27i		−	1.34935i
$552$		0			0
$553$	3.56732e27			1.33380
$554$		0			0
$555$	−3.17350e26	−	4.65535e25i	−0.114448	−	0.0167889i
$556$		0			0
$557$			5.14140e27i			1.78867i	0.447401	+	0.894333i	$0.352350\pi$
$557$			5.14140e27i			1.78867i	−0.447401	+	0.894333i	$0.647650\pi$
$558$		0			0
$559$	1.98795e26			0.0667247
$560$		0			0
$561$	1.67414e26	−	1.14124e27i	0.0542204	−	0.369615i
$562$		0			0
$563$		−	7.15409e26i		−	0.223599i	−0.993731	−	0.111800i	$-0.964339\pi$
$563$		−	7.15409e26i		−	0.223599i	0.993731	−	0.111800i	$-0.0356615\pi$
$564$		0			0
$565$	9.06875e27			2.73567
$566$		0			0
$567$	−4.88511e27	−	3.21891e27i	−1.42247	−	0.937298i
$568$		0			0
$569$		−	3.38383e27i		−	0.951230i	−0.879654	−	0.475615i	$-0.842226\pi$
$569$		−	3.38383e27i		−	0.951230i	0.879654	−	0.475615i	$-0.157774\pi$
$570$		0			0
$571$	−5.61966e27			−1.52528			−0.762639	−	0.646824i	$-0.776097\pi$
$571$	−5.61966e27			−1.52528			−0.762639	+	0.646824i	$0.776097\pi$
$572$		0			0
$573$	−7.04038e27	−	1.03278e27i	−1.84523	−	0.270685i
$574$		0			0
$575$		−	7.98526e27i		−	2.02121i
$576$		0			0
$577$	−4.26035e27			−1.04157			−0.520783	−	0.853689i	$-0.674360\pi$
$577$	−4.26035e27			−1.04157			−0.520783	+	0.853689i	$0.674360\pi$
$578$		0			0
$579$	5.66765e26	−	3.86358e27i	0.133850	−	0.912440i
$580$		0			0
$581$			9.12884e27i			2.08283i
$582$		0			0
$583$	−1.67005e27			−0.368166
$584$		0			0
$585$	−8.59351e25	+	2.86602e26i	−0.0183068	+	0.0610549i
$586$		0			0
$587$			2.60334e27i			0.535981i	0.963422	+	0.267991i	$0.0863596\pi$
$587$			2.60334e27i			0.535981i	−0.963422	+	0.267991i	$0.913640\pi$
$588$		0			0
$589$	5.83004e27			1.16016
$590$		0			0
$591$	5.16465e27	+	7.57625e26i	0.993496	+	0.145740i
$592$		0			0
$593$		−	8.11452e26i		−	0.150909i	−0.997149	−	0.0754546i	$-0.975959\pi$
$593$		−	8.11452e26i		−	0.150909i	0.997149	−	0.0754546i	$-0.0240408\pi$
$594$		0			0
$595$	8.08182e27			1.45325
$596$		0			0
$597$	−3.87795e26	+	2.64356e27i	−0.0674308	+	0.459669i
$598$		0			0
$599$		−	8.37348e27i		−	1.40811i	−0.710143	−	0.704057i	$-0.751370\pi$
$599$		−	8.37348e27i		−	1.40811i	0.710143	−	0.704057i	$-0.248630\pi$
$600$		0			0
$601$	−3.20952e27			−0.522030			−0.261015	−	0.965335i	$-0.584057\pi$
$601$	−3.20952e27			−0.522030			−0.261015	+	0.965335i	$0.584057\pi$
$602$		0			0
$603$	1.51736e27	+	4.54967e26i	0.238735	+	0.0715824i
$604$		0			0
$605$			5.19237e27i			0.790336i
$606$		0			0
$607$	1.22148e28			1.79887			0.899437	−	0.437050i	$-0.143977\pi$
$607$	1.22148e28			1.79887			0.899437	+	0.437050i	$0.143977\pi$
$608$		0			0
$609$	−1.00704e28	−	1.47728e27i	−1.43508	−	0.210518i
$610$		0			0
$611$		−	1.33014e26i		−	0.0183437i
$612$		0			0
$613$	−7.87295e27			−1.05083			−0.525416	−	0.850846i	$-0.676090\pi$
$613$	−7.87295e27			−1.05083			−0.525416	+	0.850846i	$0.676090\pi$
$614$		0			0
$615$	−2.72115e27	+	1.85498e28i	−0.351562	+	2.39656i
$616$		0			0
$617$			2.55214e27i			0.319193i	0.987182	+	0.159597i	$0.0510193\pi$
$617$			2.55214e27i			0.319193i	−0.987182	+	0.159597i	$0.948981\pi$
$618$		0			0
$619$	−9.50025e27			−1.15035			−0.575173	−	0.818032i	$-0.695065\pi$
$619$	−9.50025e27			−1.15035			−0.575173	+	0.818032i	$0.695065\pi$
$620$		0			0
$621$	9.12215e27	+	4.26077e27i	1.06950	+	0.499543i
$622$		0			0
$623$			1.37331e28i			1.55915i
$624$		0			0
$625$	1.99762e27			0.219641
$626$		0			0
$627$	−1.06187e28	−	1.55770e27i	−1.13082	−	0.165885i
$628$		0			0
$629$			3.52693e26i			0.0363824i
$630$		0			0
$631$	9.05582e27			0.904971			0.452485	−	0.891772i	$-0.350538\pi$
$631$	9.05582e27			0.904971			0.452485	+	0.891772i	$0.350538\pi$
$632$		0			0
$633$	1.93260e27	−	1.31743e28i	0.187113	−	1.27553i
$634$		0			0
$635$		−	2.30819e27i		−	0.216538i
$636$		0			0
$637$	−8.09733e26			−0.0736118
$638$		0			0
$639$	2.82879e27	−	9.43429e27i	0.249225	−	0.831191i
$640$		0			0
$641$		−	6.60021e27i		−	0.563609i	−0.959472	−	0.281804i	$-0.909067\pi$
$641$		−	6.60021e27i		−	0.563609i	0.959472	−	0.281804i	$-0.0909329\pi$
$642$		0			0
$643$	−1.07321e28			−0.888332			−0.444166	−	0.895945i	$-0.646500\pi$
$643$	−1.07321e28			−0.888332			−0.444166	+	0.895945i	$0.646500\pi$
$644$		0			0
$645$	3.50091e28	+	5.13564e27i	2.80921	+	0.412095i
$646$		0			0
$647$		−	1.01457e27i		−	0.0789298i	−0.999221	−	0.0394649i	$-0.987435\pi$
$647$		−	1.01457e27i		−	0.0789298i	0.999221	−	0.0394649i	$-0.0125653\pi$
$648$		0			0
$649$	−8.92770e27			−0.673430
$650$		0			0
$651$	2.47453e27	−	1.68686e28i	0.181001	−	1.23387i
$652$		0			0
$653$			1.32813e28i			0.942129i	0.882099	+	0.471064i	$0.156130\pi$
$653$			1.32813e28i			0.942129i	−0.882099	+	0.471064i	$0.843870\pi$
$654$		0			0
$655$	−2.62210e28			−1.80400
$656$		0			0
$657$	−6.64934e26	−	1.99375e26i	−0.0443736	−	0.0133050i
$658$		0			0
$659$			1.07577e28i			0.696408i	0.937419	+	0.348204i	$0.113208\pi$
$659$			1.07577e28i			0.696408i	−0.937419	+	0.348204i	$0.886792\pi$
$660$		0			0
$661$	−8.22112e27			−0.516317			−0.258159	−	0.966103i	$-0.583116\pi$
$661$	−8.22112e27			−0.516317			−0.258159	+	0.966103i	$0.583116\pi$
$662$		0			0
$663$	3.25526e26	+	4.77528e25i	0.0198358	+	0.00290980i
$664$		0			0
$665$		−	7.51971e28i		−	4.44616i
$666$		0			0
$667$	1.75164e28			1.00505
$668$		0			0
$669$	8.05353e26	−	5.49001e27i	0.0448461	−	0.305711i
$670$		0			0
$671$			6.63966e27i			0.358856i
$672$		0			0
$673$	−1.33316e28			−0.699408			−0.349704	−	0.936860i	$-0.613718\pi$
$673$	−1.33316e28			−0.699408			−0.349704	+	0.936860i	$0.613718\pi$
$674$		0			0
$675$	−1.42283e28	+	3.04622e28i	−0.724628	+	1.55140i
$676$		0			0
$677$		−	9.14077e27i		−	0.451956i	−0.974132	−	0.225978i	$-0.927442\pi$
$677$		−	9.14077e27i		−	0.451956i	0.974132	−	0.225978i	$-0.0725577\pi$
$678$		0			0
$679$	3.60468e28			1.73049
$680$		0			0
$681$	1.94006e28	+	2.84596e27i	0.904364	+	0.132665i
$682$		0			0
$683$			2.67166e28i			1.20941i	0.796449	+	0.604705i	$0.206709\pi$
$683$			2.67166e28i			1.20941i	−0.796449	+	0.604705i	$0.793291\pi$
$684$		0			0
$685$	−5.75083e28			−2.52828
$686$		0			0
$687$	2.87891e27	−	1.96252e28i	0.122931	−	0.838006i
$688$		0			0
$689$		−	4.76361e26i		−	0.0197580i
$690$		0			0
$691$	3.33913e26			0.0134540			0.00672702	−	0.999977i	$-0.497859\pi$
$691$	3.33913e26			0.0134540			0.00672702	+	0.999977i	$0.497859\pi$
$692$		0			0
$693$	−9.01407e27	+	3.00628e28i	−0.352849	+	1.17679i
$694$		0			0
$695$			1.20396e28i			0.457892i
$696$		0			0
$697$	2.06157e28			0.761850
$698$		0			0
$699$	8.01021e27	+	1.17505e27i	0.287655	+	0.0421974i
$700$		0			0
$701$			3.31262e28i			1.15609i	0.816005	+	0.578044i	$0.196184\pi$
$701$			3.31262e28i			1.15609i	−0.816005	+	0.578044i	$0.803816\pi$
$702$		0			0
$703$	3.28163e27			0.111311
$704$		0			0
$705$	3.43628e27	−	2.34247e28i	0.113291	−	0.772295i
$706$		0			0
$707$		−	1.45164e28i		−	0.465227i
$708$		0			0
$709$	1.71861e28			0.535447			0.267723	−	0.963496i	$-0.413729\pi$
$709$	1.71861e28			0.535447			0.267723	+	0.963496i	$0.413729\pi$
$710$		0			0
$711$	2.47598e28	+	7.42401e27i	0.749983	+	0.224876i
$712$		0			0
$713$			2.93411e28i			0.864133i
$714$		0			0
$715$	1.60517e27			0.0459686
$716$		0			0
$717$	1.02086e27	+	1.49754e26i	0.0284297	+	0.00417048i
$718$		0			0
$719$			2.90362e28i			0.786414i	0.919450	+	0.393207i	$0.128634\pi$
$719$			2.90362e28i			0.786414i	−0.919450	+	0.393207i	$0.871366\pi$
$720$		0			0
$721$	7.13816e27			0.188033
$722$		0			0
$723$	−2.85210e27	+	1.94424e28i	−0.0730771	+	0.498159i
$724$		0			0
$725$			5.84938e28i			1.45791i
$726$		0			0
$727$	6.69591e27			0.162355			0.0811774	−	0.996700i	$-0.474132\pi$
$727$	6.69591e27			0.162355			0.0811774	+	0.996700i	$0.474132\pi$
$728$		0			0
$729$	−2.72073e28	−	3.25080e28i	−0.641815	−	0.766859i
$730$		0			0
$731$		−	3.89080e28i		−	0.893029i
$732$		0			0
$733$	−6.28219e27			−0.140304			−0.0701522	−	0.997536i	$-0.522348\pi$
$733$	−6.28219e27			−0.140304			−0.0701522	+	0.997536i	$0.522348\pi$
$734$		0			0
$735$	−1.42599e29	−	2.09185e28i	−3.09916	−	0.454630i
$736$		0			0
$737$		−	8.49828e27i		−	0.179745i
$738$		0			0
$739$	1.70998e28			0.352004			0.176002	−	0.984390i	$-0.443683\pi$
$739$	1.70998e28			0.352004			0.176002	+	0.984390i	$0.443683\pi$
$740$		0			0
$741$	4.44315e26	−	3.02885e27i	0.00890244	−	0.0606870i
$742$		0			0
$743$		−	6.49079e28i		−	1.26593i	−0.774181	−	0.632964i	$-0.781838\pi$
$743$		−	6.49079e28i		−	1.26593i	0.774181	−	0.632964i	$-0.218162\pi$
$744$		0			0
$745$	5.82306e28			1.10558
$746$		0			0
$747$	−1.89982e28	+	6.33608e28i	−0.351161	+	1.17116i
$748$		0			0
$749$		−	9.83371e28i		−	1.76970i
$750$		0			0
$751$	5.16400e28			0.904871			0.452436	−	0.891797i	$-0.350555\pi$
$751$	5.16400e28			0.904871			0.452436	+	0.891797i	$0.350555\pi$
$752$		0			0
$753$	3.07118e28	+	4.50525e27i	0.524029	+	0.0768721i
$754$		0			0
$755$		−	5.60823e28i		−	0.931872i
$756$		0			0
$757$	−1.22224e28			−0.197788			−0.0988938	−	0.995098i	$-0.531530\pi$
$757$	−1.22224e28			−0.197788			−0.0988938	+	0.995098i	$0.531530\pi$
$758$		0			0
$759$	7.83950e27	−	5.34411e28i	0.123558	−	0.842282i
$760$		0			0
$761$		−	8.29744e27i		−	0.127379i	−0.997970	−	0.0636894i	$-0.979713\pi$
$761$		−	8.29744e27i		−	0.127379i	0.997970	−	0.0636894i	$-0.0202867\pi$
$762$		0			0
$763$	1.45253e29			2.17210
$764$		0			0
$765$	5.60937e28	+	1.68192e28i	0.817146	+	0.245014i
$766$		0			0
$767$		−	2.54652e27i		−	0.0361404i
$768$		0			0
$769$	−1.10858e29			−1.53286			−0.766429	−	0.642328i	$-0.777969\pi$
$769$	−1.10858e29			−1.53286			−0.766429	+	0.642328i	$0.777969\pi$
$770$		0			0
$771$	−7.13121e28	−	1.04611e28i	−0.960770	−	0.140940i
$772$		0			0
$773$			1.08745e29i			1.42762i	0.700338	+	0.713812i	$0.253033\pi$
$773$			1.08745e29i			1.42762i	−0.700338	+	0.713812i	$0.746967\pi$
$774$		0			0
$775$	−9.79806e28			−1.25350
$776$		0			0
$777$	1.39287e27	−	9.49504e27i	0.0173660	−	0.118382i
$778$		0			0
$779$		−	1.91818e29i		−	2.33085i
$780$		0			0
$781$	−5.28387e28			−0.625808
$782$		0			0
$783$	−6.68217e28	−	3.12111e28i	−0.771436	−	0.360323i
$784$		0			0
$785$			1.07949e29i			1.21485i
$786$		0			0
$787$	3.50361e28			0.384387			0.192194	−	0.981357i	$-0.438440\pi$
$787$	3.50361e28			0.384387			0.192194	+	0.981357i	$0.438440\pi$
$788$		0			0
$789$	2.80600e28	+	4.11625e27i	0.300136	+	0.0440282i
$790$		0			0
$791$			2.71335e29i			2.82970i
$792$		0			0
$793$	−1.89388e27			−0.0192585
$794$		0			0
$795$	1.23063e28	−	8.38905e28i	0.122027	−	0.831843i
$796$		0			0
$797$			1.49596e29i			1.44656i	0.690556	+	0.723279i	$0.257366\pi$
$797$			1.49596e29i			1.44656i	−0.690556	+	0.723279i	$0.742634\pi$
$798$		0			0
$799$	−2.60335e28			−0.245508
$800$		0			0
$801$	−2.85801e28	+	9.53176e28i	−0.262869	+	0.876694i
$802$		0			0
$803$			3.72410e27i			0.0334092i
$804$		0			0
$805$	3.78448e29			3.31167
$806$		0			0
$807$	1.44584e29	+	2.12096e28i	1.23420	+	0.181050i
$808$		0			0
$809$		−	1.43178e29i		−	1.19231i	−0.802868	−	0.596157i	$-0.796694\pi$
$809$		−	1.43178e29i		−	1.19231i	0.802868	−	0.596157i	$-0.203306\pi$
$810$		0			0
$811$	−3.41614e28			−0.277541			−0.138770	−	0.990325i	$-0.544315\pi$
$811$	−3.41614e28			−0.277541			−0.138770	+	0.990325i	$0.544315\pi$
$812$		0			0
$813$	−1.50306e28	+	1.02462e29i	−0.119144	+	0.812192i
$814$		0			0
$815$		−	3.25633e28i		−	0.251856i
$816$		0			0
$817$	−3.62019e29			−2.73219
$818$		0			0
$819$	−8.57506e27	−	2.57116e27i	−0.0631536	−	0.0189361i
$820$		0			0
$821$		−	1.61422e29i		−	1.16020i	−0.814546	−	0.580099i	$-0.803014\pi$
$821$		−	1.61422e29i		−	1.16020i	0.814546	−	0.580099i	$-0.196986\pi$
$822$		0			0
$823$	9.10280e28			0.638523			0.319262	−	0.947667i	$-0.396565\pi$
$823$	9.10280e28			0.638523			0.319262	+	0.947667i	$0.396565\pi$
$824$		0			0
$825$	1.78459e29	+	2.61790e28i	1.22180	+	0.179231i
$826$		0			0
$827$			1.48114e29i			0.989787i	0.868954	+	0.494894i	$0.164793\pi$
$827$			1.48114e29i			0.989787i	−0.868954	+	0.494894i	$0.835207\pi$
$828$		0			0
$829$	−1.97309e29			−1.28707			−0.643534	−	0.765418i	$-0.722532\pi$
$829$	−1.97309e29			−1.28707			−0.643534	+	0.765418i	$0.722532\pi$
$830$		0			0
$831$	−2.26963e28	+	1.54718e29i	−0.144526	+	0.985216i
$832$		0			0
$833$			1.58481e29i			0.985204i
$834$		0			0
$835$	−2.60904e29			−1.58350
$836$		0			0
$837$	5.22805e28	−	1.11930e29i	0.309803	−	0.663275i
$838$		0			0
$839$		−	2.24184e29i		−	1.29713i	−0.761158	−	0.648567i	$-0.775369\pi$
$839$		−	2.24184e29i		−	1.29713i	0.761158	−	0.648567i	$-0.224631\pi$
$840$		0			0
$841$	4.86830e28			0.275053
$842$		0			0
$843$	3.26338e29	+	4.78720e28i	1.80050	+	0.264123i
$844$		0			0
$845$		−	3.05200e29i		−	1.64444i
$846$		0			0
$847$	−1.55354e29			−0.817503
$848$		0			0
$849$	−3.24861e26	+	2.21454e27i	−0.00166963	+	0.0113817i
$850$		0			0
$851$			1.65156e28i			0.0829084i
$852$		0			0
$853$	−8.45167e28			−0.414431			−0.207215	−	0.978295i	$-0.566440\pi$
$853$	−8.45167e28			−0.414431			−0.207215	+	0.978295i	$0.566440\pi$
$854$		0			0
$855$	1.56494e29	−	5.21923e29i	0.749612	−	2.50003i
$856$		0			0
$857$		−	7.10879e28i		−	0.332650i	−0.986071	−	0.166325i	$-0.946810\pi$
$857$		−	7.10879e28i		−	0.332650i	0.986071	−	0.166325i	$-0.0531901\pi$
$858$		0			0
$859$	5.63791e28			0.257743			0.128871	−	0.991661i	$-0.458865\pi$
$859$	5.63791e28			0.257743			0.128871	+	0.991661i	$0.458865\pi$
$860$		0			0
$861$	−5.55005e29	−	8.14161e28i	−2.47894	−	0.363646i
$862$		0			0
$863$			6.69006e28i			0.291959i	0.989288	+	0.145980i	$0.0466334\pi$
$863$			6.69006e28i			0.291959i	−0.989288	+	0.145980i	$0.953367\pi$
$864$		0			0
$865$	2.34202e29			0.998688
$866$		0			0
$867$	−2.54861e28	+	1.73736e29i	−0.106197	+	0.723933i
$868$		0			0
$869$		−	1.38672e29i		−	0.564667i
$870$		0			0
$871$	2.42403e27			0.00964621
$872$		0			0
$873$	2.50191e29	+	7.50176e28i	0.973038	+	0.291757i
$874$		0			0
$875$			5.25713e29i			1.99834i
$876$		0			0
$877$	1.57214e29			0.584112			0.292056	−	0.956401i	$-0.405661\pi$
$877$	1.57214e29			0.584112			0.292056	+	0.956401i	$0.405661\pi$
$878$		0			0
$879$	4.33209e28	+	6.35493e27i	0.157329	+	0.0230793i
$880$		0			0
$881$		−	2.07529e29i		−	0.736749i	−0.929678	−	0.368374i	$-0.879914\pi$
$881$		−	2.07529e29i		−	0.736749i	0.929678	−	0.368374i	$-0.120086\pi$
$882$		0			0
$883$	−3.33591e29			−1.15773			−0.578866	−	0.815423i	$-0.696504\pi$
$883$	−3.33591e29			−1.15773			−0.578866	+	0.815423i	$0.696504\pi$
$884$		0			0
$885$	6.57865e28	−	4.48459e29i	0.223205	−	1.52156i
$886$		0			0
$887$		−	1.89508e29i		−	0.628623i	−0.949320	−	0.314312i	$-0.898226\pi$
$887$		−	1.89508e29i		−	0.628623i	0.949320	−	0.314312i	$-0.101774\pi$
$888$		0			0
$889$	6.90605e28			0.223982
$890$		0			0
$891$	−1.25128e29	+	1.89899e29i	−0.396807	+	0.602206i
$892$		0			0
$893$			2.42228e29i			0.751122i
$894$		0			0
$895$	4.46993e29			1.35541
$896$		0			0
$897$	1.52434e28	+	2.23612e27i	0.0452020	+	0.00663089i
$898$		0			0
$899$		−	2.14930e29i		−	0.623304i
$900$		0			0
$901$	−9.32333e28			−0.264437
$902$		0			0
$903$	−1.53657e29	+	1.04746e30i	−0.426260	+	2.90577i
$904$		0			0
$905$			7.01361e29i			1.90308i
$906$		0			0
$907$	−3.69710e29			−0.981268			−0.490634	−	0.871366i	$-0.663235\pi$
$907$	−3.69710e29			−0.981268			−0.490634	+	0.871366i	$0.663235\pi$
$908$		0			0
$909$	3.02103e28	−	1.00755e29i	0.0784362	−	0.261593i
$910$		0			0
$911$			5.26294e29i			1.33673i	0.743832	+	0.668367i	$0.233006\pi$
$911$			5.26294e29i			1.33673i	−0.743832	+	0.668367i	$0.766994\pi$
$912$		0			0
$913$	3.54865e29			0.881770
$914$		0			0
$915$	−3.33526e29	−	4.89264e28i	−0.810809	−	0.118941i
$916$		0			0
$917$		−	7.84525e29i		−	1.86601i
$918$		0			0
$919$	−4.45724e29			−1.03732			−0.518658	−	0.854982i	$-0.673568\pi$
$919$	−4.45724e29			−1.03732			−0.518658	+	0.854982i	$0.673568\pi$
$920$		0			0
$921$	−5.75058e28	+	3.92011e29i	−0.130953	+	0.892692i
$922$		0			0
$923$		−	1.50716e28i		−	0.0335848i
$924$		0			0
$925$	−5.51517e28			−0.120266
$926$		0			0
$927$	4.95441e28	+	1.48553e28i	0.105729	+	0.0317019i
$928$		0			0
$929$		−	1.03332e29i		−	0.215814i	−0.994161	−	0.107907i	$-0.965585\pi$
$929$		−	1.03332e29i		−	0.215814i	0.994161	−	0.107907i	$-0.0344148\pi$
$930$		0			0
$931$	1.47458e30			3.01420
$932$		0			0
$933$	−4.89230e29	−	7.17673e28i	−0.978806	−	0.143585i
$934$		0			0
$935$		−	3.14164e29i		−	0.615234i
$936$		0			0
$937$	8.26293e29			1.58394			0.791969	−	0.610561i	$-0.209056\pi$
$937$	8.26293e29			1.58394			0.791969	+	0.610561i	$0.209056\pi$
$938$		0			0
$939$	1.32898e29	−	9.05951e29i	0.249380	−	1.70000i
$940$		0			0
$941$			4.14645e29i			0.761694i	0.924638	+	0.380847i	$0.124368\pi$
$941$			4.14645e29i			0.761694i	−0.924638	+	0.380847i	$0.875632\pi$
$942$		0			0
$943$	9.65371e29			1.73611
$944$		0			0
$945$	−1.44370e30	−	6.74326e29i	−2.54191	−	1.18728i
$946$		0			0
$947$			2.99301e29i			0.515952i	0.966151	+	0.257976i	$0.0830555\pi$
$947$			2.99301e29i			0.515952i	−0.966151	+	0.257976i	$0.916945\pi$
$948$		0			0
$949$	−1.06225e27			−0.00179294
$950$		0			0
$951$	6.70071e29	+	9.82956e28i	1.10743	+	0.162453i
$952$		0			0
$953$			1.23259e30i			1.99476i	0.0723423	+	0.997380i	$0.476953\pi$
$953$			1.23259e30i			1.99476i	−0.0723423	+	0.997380i	$0.523047\pi$
$954$		0			0
$955$	−1.93809e30			−3.07143
$956$		0			0
$957$	−5.74261e28	+	3.91467e29i	−0.0891230	+	0.607542i
$958$		0			0
$959$		−	1.72063e30i		−	2.61518i
$960$		0			0
$961$	−3.11770e29			−0.464088
$962$		0			0
$963$	2.04651e29	−	6.82532e29i	0.298367	−	0.995084i
$964$		0			0
$965$		−	1.06358e30i		−	1.51878i
$966$		0			0
$967$	−5.67913e29			−0.794360			−0.397180	−	0.917741i	$-0.630011\pi$
$967$	−5.67913e29			−0.794360			−0.397180	+	0.917741i	$0.630011\pi$
$968$		0			0
$969$	−5.92806e29	−	8.69612e28i	−0.812222	−	0.119148i
$970$		0			0
$971$		−	9.65495e28i		−	0.129586i	−0.997899	−	0.0647930i	$-0.979361\pi$
$971$		−	9.65495e28i		−	0.129586i	0.997899	−	0.0647930i	$-0.0206387\pi$
$972$		0			0
$973$	−3.60220e29			−0.473631
$974$		0			0
$975$	−7.46724e27	+	5.09034e28i	−0.00961865	+	0.0655693i
$976$		0			0
$977$			3.45880e29i			0.436495i	0.975893	+	0.218248i	$0.0700340\pi$
$977$			3.45880e29i			0.436495i	−0.975893	+	0.218248i	$0.929966\pi$
$978$		0			0
$979$	5.33846e29			0.660068
$980$		0			0
$981$	1.00816e30	+	3.02288e29i	1.22135	+	0.366211i
$982$		0			0
$983$			4.06792e29i			0.482877i	0.970416	+	0.241439i	$0.0776193\pi$
$983$			4.06792e29i			0.482877i	−0.970416	+	0.241439i	$0.922381\pi$
$984$		0			0
$985$	1.42174e30			1.65370
$986$		0			0
$987$	7.00862e29	+	1.02812e29i	0.798842	+	0.117186i
$988$		0			0
$989$		−	1.82195e30i		−	2.03504i
$990$		0			0
$991$	−1.03319e30			−1.13095			−0.565474	−	0.824766i	$-0.691307\pi$
$991$	−1.03319e30			−1.13095			−0.565474	+	0.824766i	$0.691307\pi$
$992$		0			0
$993$	−2.09787e29	+	1.43009e30i	−0.225053	+	1.53417i
$994$		0			0
$995$			7.27725e29i			0.765132i
$996$		0			0
$997$	1.36725e30			1.40895			0.704475	−	0.709729i	$-0.251183\pi$
$997$	1.36725e30			1.40895			0.704475	+	0.709729i	$0.251183\pi$
$998$		0			0
$999$	2.94278e28	−	6.30038e28i	0.0297237	−	0.0636373i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.21.c.b.5.4	yes	6
3.2	odd	2	inner	12.21.c.b.5.3	✓	6
4.3	odd	2		48.21.e.d.17.3		6
12.11	even	2		48.21.e.d.17.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.21.c.b.5.3	✓	6	3.2	odd	2	inner
12.21.c.b.5.4	yes	6	1.1	even	1	trivial
48.21.e.d.17.3		6	4.3	odd	2
48.21.e.d.17.4		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)

\(f(q)\)	\(=\)	\(q+(-8570.43 + 58423.7i) q^{3} +1.60830e7i q^{5} -4.81200e8 q^{7} +(-3.33988e9 - 1.00143e9i) q^{9} +O(q^{10})\)	\(q+(-8570.43 + 58423.7i) q^{3} +1.60830e7i q^{5} -4.81200e8 q^{7} +(-3.33988e9 - 1.00143e9i) q^{9} +1.87057e10i q^{11} -5.33557e9 q^{13} +(-9.39631e11 - 1.37839e11i) q^{15} +1.04428e12i q^{17} +9.71645e12 q^{19} +(4.12409e12 - 2.81135e13i) q^{21} +4.89004e13i q^{23} -1.63296e14 q^{25} +(8.71317e13 - 1.86546e14i) q^{27} -3.58206e14i q^{29} +6.00017e14 q^{31} +(-1.09286e15 - 1.60316e14i) q^{33} -7.73916e15i q^{35} +3.37740e14 q^{37} +(4.57281e13 - 3.11724e14i) q^{39} -1.97416e16i q^{41} -3.72584e16 q^{43} +(1.61061e16 - 5.37154e16i) q^{45} +2.49297e16i q^{47} +1.51761e17 q^{49} +(-6.10105e16 - 8.94990e15i) q^{51} +8.92803e16i q^{53} -3.00844e17 q^{55} +(-8.32742e16 + 5.67671e17i) q^{57} +4.77272e17i q^{59} +3.54954e17 q^{61} +(1.60715e18 + 4.81890e17i) q^{63} -8.58121e16i q^{65} -4.54315e17 q^{67} +(-2.85694e18 - 4.19098e17i) q^{69} +2.82474e18i q^{71} +1.99089e17 q^{73} +(1.39952e18 - 9.54039e18i) q^{75} -9.00117e18i q^{77} -7.41338e18 q^{79} +(1.01519e19 + 6.68933e18i) q^{81} -1.89710e19i q^{83} -1.67951e19 q^{85} +(2.09277e19 + 3.06998e18i) q^{87} -2.85392e19i q^{89} +2.56748e18 q^{91} +(-5.14241e18 + 3.50552e19i) q^{93} +1.56270e20i q^{95} -7.49102e19 q^{97} +(1.87325e19 - 6.24747e19i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 84378 q^{3} - 145040532 q^{7} - 7747974234 q^{9}+O(q^{10}) \) 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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