LMFDB - Embedded newform 12.19.c.a.5.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [12,19,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, 19, names="a")

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

chi := DirichletCharacter("12.5");

S:= CuspForms(chi, 19);

N := Newforms(S);

Level:	$ N $	$=$	$ 12 = 2^{2} \cdot 3 $
Weight:	$ k $	$=$	$ 19 $
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:	$24.6463365252$
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} + 132762x^{4} + 1042140330x^{2} + 1430023595000 $ x^6 + 132762x^4 + 1042140330x^2 + 1430023595000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{19}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5.2
Root			$-352.821i$ of defining polynomial
Character	$\chi$	$=$	12.5
Dual form			12.19.c.a.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-14327.9 + 13495.6i) q^{3} -480487. i q^{5} -7.51877e6 q^{7} +(2.31580e7 - 3.86728e8i) q^{9} +O(q^{10})$	\(q+(-14327.9 + 13495.6i) q^{3} -480487. i q^{5} -7.51877e6 q^{7} +(2.31580e7 - 3.86728e8i) q^{9} -2.15795e9i q^{11} -1.26300e9 q^{13} +(6.48446e9 + 6.88437e9i) q^{15} +1.24913e11i q^{17} +1.23848e11 q^{19} +(1.07728e11 - 1.01470e11i) q^{21} +1.00868e11i q^{23} +3.58383e12 q^{25} +(4.88732e12 + 5.85353e12i) q^{27} +2.73950e13i q^{29} +3.40498e13 q^{31} +(2.91228e13 + 3.09189e13i) q^{33} +3.61267e12i q^{35} +1.11682e14 q^{37} +(1.80962e13 - 1.70450e13i) q^{39} +3.61209e14i q^{41} -1.64305e14 q^{43} +(-1.85818e14 - 1.11271e13i) q^{45} -1.71188e15i q^{47} -1.57188e15 q^{49} +(-1.68578e15 - 1.78975e15i) q^{51} -2.70671e15i q^{53} -1.03687e15 q^{55} +(-1.77448e15 + 1.67140e15i) q^{57} +6.75230e15i q^{59} +7.85189e15 q^{61} +(-1.74120e14 + 2.90772e15i) q^{63} +6.06856e14i q^{65} +3.25108e16 q^{67} +(-1.36127e15 - 1.44522e15i) q^{69} +4.47595e16i q^{71} +7.24004e16 q^{73} +(-5.13488e16 + 4.83659e16i) q^{75} +1.62251e16i q^{77} +2.22980e16 q^{79} +(-1.49022e17 - 1.79117e16i) q^{81} -2.78963e17i q^{83} +6.00192e16 q^{85} +(-3.69712e17 - 3.92513e17i) q^{87} +6.88956e15i q^{89} +9.49624e15 q^{91} +(-4.87863e17 + 4.59522e17i) q^{93} -5.95072e16i q^{95} +1.33916e18 q^{97} +(-8.34538e17 - 4.99737e16i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 23934 q^{3} + 11024364 q^{7} - 34078986 q^{9}+O(q^{10}) $ 6 * q + 23934 * q^3 + 11024364 * q^7 - 34078986 * q^9	$ 6 q + 23934 q^{3} + 11024364 q^{7} - 34078986 q^{9} + 1775593596 q^{13} - 9856918080 q^{15} - 202034532804 q^{19} - 200158320036 q^{21} - 5958025101930 q^{25} - 8167776022674 q^{27} - 56059471942836 q^{31} - 51478992417600 q^{33} - 167596105515108 q^{37} - 73688099161044 q^{39} + 448591926775836 q^{43} - 12918361749120 q^{45} + 38\!\cdots\!10 q^{49}+ \cdots - 46\!\cdots\!60 q^{99}+O(q^{100}) $ 6 * q + 23934 * q^3 + 11024364 * q^7 - 34078986 * q^9 + 1775593596 * q^13 - 9856918080 * q^15 - 202034532804 * q^19 - 200158320036 * q^21 - 5958025101930 * q^25 - 8167776022674 * q^27 - 56059471942836 * q^31 - 51478992417600 * q^33 - 167596105515108 * q^37 - 73688099161044 * q^39 + 448591926775836 * q^43 - 12918361749120 * q^45 + 3815313399549810 * q^49 + 579278365205760 * q^51 + 8271035021335680 * q^55 - 7540820577807636 * q^57 + 13350472196682684 * q^61 - 40748979535300404 * q^63 + 37560450307637244 * q^67 - 131632980577038720 * q^69 + 149418174175674348 * q^73 - 287990904570521010 * q^75 + 646522133618739468 * q^79 - 656120326335021594 * q^81 + 1416345514989166080 * q^85 - 1306513723216363200 * q^87 + 2966957008358949816 * q^91 - 2834708598841722564 * q^93 + 2984598304382259084 * q^97 - 4681229544239276160 * 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
$3$	−14327.9	+	13495.6i	−0.727934	+	0.685648i
$3$	−14327.9	+	13495.6i	−0.727934	+	0.685648i
$4$		0			0
$5$		−	480487.i		−	0.246009i	−0.992406	−	0.123005i	$-0.960747\pi$
$5$		−	480487.i		−	0.246009i	0.992406	−	0.123005i	$-0.0392530\pi$
$6$		0			0
$7$	−7.51877e6			−0.186322			−0.0931611	−	0.995651i	$-0.529697\pi$
$7$	−7.51877e6			−0.186322			−0.0931611	+	0.995651i	$0.529697\pi$
$8$		0			0
$9$	2.31580e7	−	3.86728e8i	0.0597748	−	0.998212i
$10$		0			0
$11$		−	2.15795e9i		−	0.915181i	−0.889163	−	0.457590i	$-0.848713\pi$
$11$		−	2.15795e9i		−	0.915181i	0.889163	−	0.457590i	$-0.151287\pi$
$12$		0			0
$13$	−1.26300e9			−0.119101			−0.0595503	−	0.998225i	$-0.518967\pi$
$13$	−1.26300e9			−0.119101			−0.0595503	+	0.998225i	$0.518967\pi$
$14$		0			0
$15$	6.48446e9	+	6.88437e9i	0.168676	+	0.179078i
$16$		0			0
$17$			1.24913e11i			1.05334i	0.850070	+	0.526670i	$0.176560\pi$
$17$			1.24913e11i			1.05334i	−0.850070	+	0.526670i	$0.823440\pi$
$18$		0			0
$19$	1.23848e11			0.383801			0.191900	−	0.981414i	$-0.438535\pi$
$19$	1.23848e11			0.383801			0.191900	+	0.981414i	$0.438535\pi$
$20$		0			0
$21$	1.07728e11	−	1.01470e11i	0.135630	−	0.127751i
$22$		0			0
$23$			1.00868e11i			0.0560017i	0.999608	+	0.0280009i	$0.00891412\pi$
$23$			1.00868e11i			0.0560017i	−0.999608	+	0.0280009i	$0.991086\pi$
$24$		0			0
$25$	3.58383e12			0.939479
$26$		0			0
$27$	4.88732e12	+	5.85353e12i	0.640909	+	0.767616i
$28$		0			0
$29$			2.73950e13i			1.88838i	0.329404	+	0.944189i	$0.393152\pi$
$29$			2.73950e13i			1.88838i	−0.329404	+	0.944189i	$0.606848\pi$
$30$		0			0
$31$	3.40498e13			1.28783			0.643916	−	0.765096i	$-0.277309\pi$
$31$	3.40498e13			1.28783			0.643916	+	0.765096i	$0.277309\pi$
$32$		0			0
$33$	2.91228e13	+	3.09189e13i	0.627491	+	0.666191i
$34$		0			0
$35$			3.61267e12i			0.0458370i
$36$		0			0
$37$	1.11682e14			0.859344			0.429672	−	0.902985i	$-0.358629\pi$
$37$	1.11682e14			0.859344			0.429672	+	0.902985i	$0.358629\pi$
$38$		0			0
$39$	1.80962e13	−	1.70450e13i	0.0866974	−	0.0816611i
$40$		0			0
$41$			3.61209e14i			1.10333i	0.834067	+	0.551663i	$0.186007\pi$
$41$			3.61209e14i			1.10333i	−0.834067	+	0.551663i	$0.813993\pi$
$42$		0			0
$43$	−1.64305e14			−0.326915			−0.163457	−	0.986550i	$-0.552265\pi$
$43$	−1.64305e14			−0.326915			−0.163457	+	0.986550i	$0.552265\pi$
$44$		0			0
$45$	−1.85818e14	−	1.11271e13i	−0.245569	−	0.0147052i
$46$		0			0
$47$		−	1.71188e15i		−	1.52965i	−0.644239	−	0.764824i	$-0.722826\pi$
$47$		−	1.71188e15i		−	1.52965i	0.644239	−	0.764824i	$-0.277174\pi$
$48$		0			0
$49$	−1.57188e15			−0.965284
$50$		0			0
$51$	−1.68578e15	−	1.78975e15i	−0.722221	−	0.766762i
$52$		0			0
$53$		−	2.70671e15i		−	0.820273i	−0.912024	−	0.410137i	$-0.865481\pi$
$53$		−	2.70671e15i		−	0.820273i	0.912024	−	0.410137i	$-0.134519\pi$
$54$		0			0
$55$	−1.03687e15			−0.225143
$56$		0			0
$57$	−1.77448e15	+	1.67140e15i	−0.279382	+	0.263152i
$58$		0			0
$59$			6.75230e15i			0.779441i	0.920933	+	0.389721i	$0.127428\pi$
$59$			6.75230e15i			0.779441i	−0.920933	+	0.389721i	$0.872572\pi$
$60$		0			0
$61$	7.85189e15			0.671438			0.335719	−	0.941962i	$-0.391021\pi$
$61$	7.85189e15			0.671438			0.335719	+	0.941962i	$0.391021\pi$
$62$		0			0
$63$	−1.74120e14	+	2.90772e15i	−0.0111374	+	0.185989i
$64$		0			0
$65$			6.06856e14i			0.0292999i
$66$		0			0
$67$	3.25108e16			1.19496			0.597481	−	0.801883i	$-0.296168\pi$
$67$	3.25108e16			1.19496			0.597481	+	0.801883i	$0.296168\pi$
$68$		0			0
$69$	−1.36127e15	−	1.44522e15i	−0.0383974	−	0.0407655i
$70$		0			0
$71$			4.47595e16i			0.976247i	0.872774	+	0.488124i	$0.162319\pi$
$71$			4.47595e16i			0.976247i	−0.872774	+	0.488124i	$0.837681\pi$
$72$		0			0
$73$	7.24004e16			1.22980			0.614901	−	0.788605i	$-0.289196\pi$
$73$	7.24004e16			1.22980			0.614901	+	0.788605i	$0.289196\pi$
$74$		0			0
$75$	−5.13488e16	+	4.83659e16i	−0.683879	+	0.644152i
$76$		0			0
$77$			1.62251e16i			0.170518i
$78$		0			0
$79$	2.22980e16			0.186047			0.0930233	−	0.995664i	$-0.470347\pi$
$79$	2.22980e16			0.186047			0.0930233	+	0.995664i	$0.470347\pi$
$80$		0			0
$81$	−1.49022e17	−	1.79117e16i	−0.992854	−	0.119336i
$82$		0			0
$83$		−	2.78963e17i		−	1.49226i	−0.665803	−	0.746128i	$-0.731911\pi$
$83$		−	2.78963e17i		−	1.49226i	0.665803	−	0.746128i	$-0.268089\pi$
$84$		0			0
$85$	6.00192e16			0.259132
$86$		0			0
$87$	−3.69712e17	−	3.92513e17i	−1.29476	−	1.37461i
$88$		0			0
$89$			6.88956e15i			0.0196644i	0.999952	+	0.00983222i	$0.00312974\pi$
$89$			6.88956e15i			0.0196644i	−0.999952	+	0.00983222i	$0.996870\pi$
$90$		0			0
$91$	9.49624e15			0.0221911
$92$		0			0
$93$	−4.87863e17	+	4.59522e17i	−0.937456	+	0.882999i
$94$		0			0
$95$		−	5.95072e16i		−	0.0944186i
$96$		0			0
$97$	1.33916e18			1.76152			0.880759	−	0.473566i	$-0.157033\pi$
$97$	1.33916e18			1.76152			0.880759	+	0.473566i	$0.157033\pi$
$98$		0			0
$99$	−8.34538e17	−	4.99737e16i	−0.913544	−	0.0547047i
$100$		0			0
$101$			8.89791e17i			0.813571i	0.913524	+	0.406786i	$0.133350\pi$
$101$			8.89791e17i			0.813571i	−0.913524	+	0.406786i	$0.866650\pi$
$102$		0			0
$103$	1.15740e18			0.887053			0.443526	−	0.896261i	$-0.353727\pi$
$103$	1.15740e18			0.887053			0.443526	+	0.896261i	$0.353727\pi$
$104$		0			0
$105$	−4.87552e16	−	5.17621e16i	−0.0314280	−	0.0333663i
$106$		0			0
$107$		−	4.87181e17i		−	0.264994i	−0.991183	−	0.132497i	$-0.957700\pi$
$107$		−	4.87181e17i		−	0.264994i	0.991183	−	0.132497i	$-0.0422995\pi$
$108$		0			0
$109$	3.52034e18			1.62086			0.810432	−	0.585832i	$-0.199232\pi$
$109$	3.52034e18			1.62086			0.810432	+	0.585832i	$0.199232\pi$
$110$		0			0
$111$	−1.60017e18	+	1.50721e18i	−0.625545	+	0.589207i
$112$		0			0
$113$			2.78297e18i			0.926409i	0.886251	+	0.463205i	$0.153300\pi$
$113$			2.78297e18i			0.926409i	−0.886251	+	0.463205i	$0.846700\pi$
$114$		0			0
$115$	4.84656e16			0.0137769
$116$		0			0
$117$	−2.92486e16	+	4.88438e17i	−0.00711922	+	0.118888i
$118$		0			0
$119$		−	9.39196e17i		−	0.196261i
$120$		0			0
$121$	9.03179e17			0.162445
$122$		0			0
$123$	−4.87474e18	−	5.17538e18i	−0.756493	−	0.803149i
$124$		0			0
$125$		−	3.55489e18i		−	0.477130i
$126$		0			0
$127$	−1.32334e19			−1.53971			−0.769853	−	0.638221i	$-0.779671\pi$
$127$	−1.32334e19			−1.53971			−0.769853	+	0.638221i	$0.779671\pi$
$128$		0			0
$129$	2.35415e18	−	2.21739e18i	0.237972	−	0.224148i
$130$		0			0
$131$		−	1.97916e19i		−	1.74196i	−0.491317	−	0.870981i	$-0.663484\pi$
$131$		−	1.97916e19i		−	1.74196i	0.491317	−	0.870981i	$-0.336516\pi$
$132$		0			0
$133$	−9.31184e17			−0.0715107
$134$		0			0
$135$	2.81255e18	−	2.34829e18i	0.188841	−	0.157670i
$136$		0			0
$137$			2.06967e19i			1.21735i	0.793419	+	0.608676i	$0.208299\pi$
$137$			2.06967e19i			1.21735i	−0.793419	+	0.608676i	$0.791701\pi$
$138$		0			0
$139$	−2.09458e19			−1.08134			−0.540672	−	0.841233i	$-0.681830\pi$
$139$	−2.09458e19			−1.08134			−0.540672	+	0.841233i	$0.681830\pi$
$140$		0			0
$141$	2.31028e19	+	2.45276e19i	1.04880	+	1.11348i
$142$		0			0
$143$			2.72550e18i			0.108999i
$144$		0			0
$145$	1.31629e19			0.464558
$146$		0			0
$147$	2.25218e19	−	2.12135e19i	0.702663	−	0.661845i
$148$		0			0
$149$			3.22462e19i			0.890844i	0.895321	+	0.445422i	$0.146946\pi$
$149$			3.22462e19i			0.890844i	−0.895321	+	0.445422i	$0.853054\pi$
$150$		0			0
$151$	−1.25802e19			−0.308245			−0.154122	−	0.988052i	$-0.549255\pi$
$151$	−1.25802e19			−0.308245			−0.154122	+	0.988052i	$0.549255\pi$
$152$		0			0
$153$	4.83075e19	+	2.89274e18i	1.05146	+	0.0629632i
$154$		0			0
$155$		−	1.63605e19i		−	0.316818i
$156$		0			0
$157$	−1.09989e19			−0.189780			−0.0948902	−	0.995488i	$-0.530250\pi$
$157$	−1.09989e19			−0.189780			−0.0948902	+	0.995488i	$0.530250\pi$
$158$		0			0
$159$	3.65286e19	+	3.87815e19i	0.562418	+	0.597104i
$160$		0			0
$161$		−	7.58401e17i		−	0.0104344i
$162$		0			0
$163$	−5.94894e19			−0.732405			−0.366203	−	0.930535i	$-0.619342\pi$
$163$	−5.94894e19			−0.732405			−0.366203	+	0.930535i	$0.619342\pi$
$164$		0			0
$165$	1.48561e19	−	1.39931e19i	0.163889	−	0.154369i
$166$		0			0
$167$			7.67423e19i			0.759603i	0.925068	+	0.379801i	$0.124008\pi$
$167$			7.67423e19i			0.759603i	−0.925068	+	0.379801i	$0.875992\pi$
$168$		0			0
$169$	−1.10860e20			−0.985815
$170$		0			0
$171$	2.86807e18	−	4.78954e19i	0.0229416	−	0.383115i
$172$		0			0
$173$		−	1.95775e20i		−	1.41040i	−0.709010	−	0.705199i	$-0.750858\pi$
$173$		−	1.95775e20i		−	1.41040i	0.709010	−	0.705199i	$-0.249142\pi$
$174$		0			0
$175$	−2.69460e19			−0.175046
$176$		0			0
$177$	−9.11263e19	−	9.67464e19i	−0.534422	−	0.567382i
$178$		0			0
$179$		−	1.73746e20i		−	0.920952i	−0.887672	−	0.460476i	$-0.847679\pi$
$179$		−	1.73746e20i		−	0.920952i	0.887672	−	0.460476i	$-0.152321\pi$
$180$		0			0
$181$	−9.94182e19			−0.476825			−0.238412	−	0.971164i	$-0.576627\pi$
$181$	−9.94182e19			−0.476825			−0.238412	+	0.971164i	$0.576627\pi$
$182$		0			0
$183$	−1.12501e20	+	1.05966e20i	−0.488762	+	0.460370i
$184$		0			0
$185$		−	5.36616e19i		−	0.211407i
$186$		0			0
$187$	2.69557e20			0.963997
$188$		0			0
$189$	−3.67466e19	−	4.40114e19i	−0.119416	−	0.143024i
$190$		0			0
$191$			6.12526e20i			1.81061i	0.424765	+	0.905303i	$0.360357\pi$
$191$			6.12526e20i			1.81061i	−0.424765	+	0.905303i	$0.639643\pi$
$192$		0			0
$193$	2.32773e20			0.626493			0.313246	−	0.949672i	$-0.398583\pi$
$193$	2.32773e20			0.626493			0.313246	+	0.949672i	$0.398583\pi$
$194$		0			0
$195$	−8.18989e18	−	8.69499e18i	−0.0200894	−	0.0213284i
$196$		0			0
$197$			2.37396e20i			0.531223i	0.964080	+	0.265612i	$0.0855739\pi$
$197$			2.37396e20i			0.531223i	−0.964080	+	0.265612i	$0.914426\pi$
$198$		0			0
$199$	8.55694e20			1.74840			0.874200	−	0.485566i	$-0.161386\pi$
$199$	8.55694e20			1.74840			0.874200	+	0.485566i	$0.161386\pi$
$200$		0			0
$201$	−4.65812e20	+	4.38752e20i	−0.869853	+	0.819323i
$202$		0			0
$203$		−	2.05977e20i		−	0.351847i
$204$		0			0
$205$	1.73556e20			0.271429
$206$		0			0
$207$	3.90083e19	+	2.33589e18i	0.0559016	+	0.00334749i
$208$		0			0
$209$		−	2.67257e20i		−	0.351247i
$210$		0			0
$211$	8.21314e20			0.990757			0.495378	−	0.868677i	$-0.335029\pi$
$211$	8.21314e20			0.990757			0.495378	+	0.868677i	$0.335029\pi$
$212$		0			0
$213$	−6.04056e20	−	6.41310e20i	−0.669362	−	0.710643i
$214$		0			0
$215$			7.89463e19i			0.0804240i
$216$		0			0
$217$	−2.56013e20			−0.239952
$218$		0			0
$219$	−1.03735e21	+	9.77086e20i	−0.895214	+	0.843210i
$220$		0			0
$221$		−	1.57766e20i		−	0.125454i
$222$		0			0
$223$	−2.41155e20			−0.176829			−0.0884143	−	0.996084i	$-0.528180\pi$
$223$	−2.41155e20			−0.176829			−0.0884143	+	0.996084i	$0.528180\pi$
$224$		0			0
$225$	8.29943e19	−	1.38597e21i	0.0561572	−	0.937800i
$226$		0			0
$227$			3.46060e20i			0.216232i	0.994138	+	0.108116i	$0.0344817\pi$
$227$			3.46060e20i			0.216232i	−0.994138	+	0.108116i	$0.965518\pi$
$228$		0			0
$229$	−1.49149e21			−0.861194			−0.430597	−	0.902544i	$-0.641697\pi$
$229$	−1.49149e21			−0.861194			−0.430597	+	0.902544i	$0.641697\pi$
$230$		0			0
$231$	−2.18968e20	−	2.32472e20i	−0.116916	−	0.124126i
$232$		0			0
$233$		−	2.56942e21i		−	1.26950i	−0.772719	−	0.634748i	$-0.781104\pi$
$233$		−	2.56942e21i		−	1.26950i	0.772719	−	0.634748i	$-0.218896\pi$
$234$		0			0
$235$	−8.22534e20			−0.376308
$236$		0			0
$237$	−3.19484e20	+	3.00925e20i	−0.135430	+	0.127562i
$238$		0			0
$239$		−	6.84505e20i		−	0.269027i	−0.990912	−	0.134513i	$-0.957053\pi$
$239$		−	6.84505e20i		−	0.269027i	0.990912	−	0.134513i	$-0.0429471\pi$
$240$		0			0
$241$	−6.57244e19			−0.0239648			−0.0119824	−	0.999928i	$-0.503814\pi$
$241$	−6.57244e19			−0.0239648			−0.0119824	+	0.999928i	$0.503814\pi$
$242$		0			0
$243$	2.37690e21	−	1.75451e21i	0.804554	−	0.593879i
$244$		0			0
$245$			7.55268e20i			0.237469i
$246$		0			0
$247$	−1.56420e20			−0.0457110
$248$		0			0
$249$	3.76477e21	+	3.99695e21i	1.02316	+	1.08626i
$250$		0			0
$251$			5.18722e21i			1.31181i	0.754843	+	0.655906i	$0.227713\pi$
$251$			5.18722e21i			1.31181i	−0.754843	+	0.655906i	$0.772287\pi$
$252$		0			0
$253$	2.17667e20			0.0512517
$254$		0			0
$255$	−8.59951e20	+	8.09996e20i	−0.188631	+	0.177673i
$256$		0			0
$257$			6.90847e21i			1.41248i	0.707970	+	0.706242i	$0.249611\pi$
$257$			6.90847e21i			1.41248i	−0.707970	+	0.706242i	$0.750389\pi$
$258$		0			0
$259$	−8.39711e20			−0.160115
$260$		0			0
$261$	1.05944e22	+	6.34413e20i	1.88500	+	0.112877i
$262$		0			0
$263$		−	9.64443e21i		−	1.60204i	−0.598634	−	0.801022i	$-0.704290\pi$
$263$		−	9.64443e21i		−	1.60204i	0.598634	−	0.801022i	$-0.295710\pi$
$264$		0			0
$265$	−1.30054e21			−0.201795
$266$		0			0
$267$	−9.29788e19	−	9.87131e19i	−0.0134829	−	0.0143144i
$268$		0			0
$269$			2.57652e21i			0.349352i	0.984626	+	0.174676i	$0.0558878\pi$
$269$			2.57652e21i			0.349352i	−0.984626	+	0.174676i	$0.944112\pi$
$270$		0			0
$271$	−7.26423e21			−0.921438			−0.460719	−	0.887546i	$-0.652408\pi$
$271$	−7.26423e21			−0.921438			−0.460719	+	0.887546i	$0.652408\pi$
$272$		0			0
$273$	−1.36061e20	+	1.28157e20i	−0.0161537	+	0.0152153i
$274$		0			0
$275$		−	7.73372e21i		−	0.859793i
$276$		0			0
$277$	−4.78426e21			−0.498307			−0.249153	−	0.968464i	$-0.580152\pi$
$277$	−4.78426e21			−0.498307			−0.249153	+	0.968464i	$0.580152\pi$
$278$		0			0
$279$	7.88524e20	−	1.31680e22i	0.0769799	−	1.28553i
$280$		0			0
$281$			7.07892e21i			0.648053i	0.946048	+	0.324026i	$0.105037\pi$
$281$			7.07892e21i			0.648053i	−0.946048	+	0.324026i	$0.894963\pi$
$282$		0			0
$283$	−7.98806e21			−0.686062			−0.343031	−	0.939324i	$-0.611454\pi$
$283$	−7.98806e21			−0.686062			−0.343031	+	0.939324i	$0.611454\pi$
$284$		0			0
$285$	8.03086e20	+	8.52615e20i	0.0647379	+	0.0687304i
$286$		0			0
$287$		−	2.71585e21i		−	0.205574i
$288$		0			0
$289$	−1.54029e21			−0.109527
$290$		0			0
$291$	−1.91874e22	+	1.80728e22i	−1.28227	+	1.20778i
$292$		0			0
$293$		−	1.58907e22i		−	0.998469i	−0.866467	−	0.499234i	$-0.833615\pi$
$293$		−	1.58907e22i		−	0.998469i	0.866467	−	0.499234i	$-0.166385\pi$
$294$		0			0
$295$	3.24439e21			0.191750
$296$		0			0
$297$	1.26316e22	−	1.05466e22i	0.702508	−	0.586548i
$298$		0			0
$299$		−	1.27396e20i		−	0.00666984i
$300$		0			0
$301$	1.23537e21			0.0609115
$302$		0			0
$303$	−1.20083e22	−	1.27489e22i	−0.557823	−	0.592226i
$304$		0			0
$305$		−	3.77273e21i		−	0.165180i
$306$		0			0
$307$	3.15756e22			1.30349			0.651743	−	0.758440i	$-0.274038\pi$
$307$	3.15756e22			1.30349			0.651743	+	0.758440i	$0.274038\pi$
$308$		0			0
$309$	−1.65832e22	+	1.56198e22i	−0.645716	+	0.608206i
$310$		0			0
$311$			2.78075e22i			1.02169i	0.859673	+	0.510844i	$0.170667\pi$
$311$			2.78075e22i			1.02169i	−0.859673	+	0.510844i	$0.829333\pi$
$312$		0			0
$313$	1.67499e22			0.580917			0.290458	−	0.956888i	$-0.406192\pi$
$313$	1.67499e22			0.580917			0.290458	+	0.956888i	$0.406192\pi$
$314$		0			0
$315$	1.39712e21	+	8.36622e19i	0.0457550	+	0.00273990i
$316$		0			0
$317$			2.05387e22i			0.635388i	0.948193	+	0.317694i	$0.102908\pi$
$317$			2.05387e22i			0.635388i	−0.948193	+	0.317694i	$0.897092\pi$
$318$		0			0
$319$	5.91169e22			1.72821
$320$		0			0
$321$	6.57480e21	+	6.98029e21i	0.181693	+	0.192898i
$322$		0			0
$323$			1.54703e22i			0.404273i
$324$		0			0
$325$	−4.52639e21			−0.111893
$326$		0			0
$327$	−5.04392e22	+	4.75092e22i	−1.17988	+	1.11134i
$328$		0			0
$329$			1.28712e22i			0.285008i
$330$		0			0
$331$	4.35767e22			0.913697			0.456849	−	0.889544i	$-0.348978\pi$
$331$	4.35767e22			0.913697			0.456849	+	0.889544i	$0.348978\pi$
$332$		0			0
$333$	2.58633e21	−	4.31905e22i	0.0513671	−	0.857807i
$334$		0			0
$335$		−	1.56210e22i		−	0.293972i
$336$		0			0
$337$	−3.85298e22			−0.687271			−0.343636	−	0.939103i	$-0.611658\pi$
$337$	−3.85298e22			−0.687271			−0.343636	+	0.939103i	$0.611658\pi$
$338$		0			0
$339$	−3.75579e22	−	3.98742e22i	−0.635190	−	0.674364i
$340$		0			0
$341$		−	7.34777e22i		−	1.17860i
$342$		0			0
$343$	2.40623e22			0.366176
$344$		0			0
$345$	−6.94411e20	+	6.54072e20i	−0.0100287	+	0.00944612i
$346$		0			0
$347$		−	5.85290e22i		−	0.802428i	−0.915984	−	0.401214i	$-0.868588\pi$
$347$		−	5.85290e22i		−	0.802428i	0.915984	−	0.401214i	$-0.131412\pi$
$348$		0			0
$349$	−1.31139e23			−1.70727			−0.853635	−	0.520871i	$-0.825607\pi$
$349$	−1.31139e23			−1.70727			−0.853635	+	0.520871i	$0.825607\pi$
$350$		0			0
$351$	−6.17270e21	−	7.39303e21i	−0.0763328	−	0.0914237i
$352$		0			0
$353$		−	1.31042e22i		−	0.153971i	−0.997032	−	0.0769855i	$-0.975470\pi$
$353$		−	1.31042e22i		−	0.153971i	0.997032	−	0.0769855i	$-0.0245295\pi$
$354$		0			0
$355$	2.15063e22			0.240166
$356$		0			0
$357$	1.26750e22	+	1.34567e22i	0.134566	+	0.142865i
$358$		0			0
$359$			9.56845e22i			0.966032i	0.875612	+	0.483016i	$0.160459\pi$
$359$			9.56845e22i			0.966032i	−0.875612	+	0.483016i	$0.839541\pi$
$360$		0			0
$361$	−8.87891e22			−0.852697
$362$		0			0
$363$	−1.29407e22	+	1.21889e22i	−0.118249	+	0.111380i
$364$		0			0
$365$		−	3.47874e22i		−	0.302542i
$366$		0			0
$367$	6.93037e22			0.573801			0.286901	−	0.957960i	$-0.407375\pi$
$367$	6.93037e22			0.573801			0.286901	+	0.957960i	$0.407375\pi$
$368$		0			0
$369$	1.39690e23	+	8.36488e21i	1.10135	+	0.0659511i
$370$		0			0
$371$			2.03511e22i			0.152835i
$372$		0			0
$373$	−1.43201e23			−1.02463			−0.512315	−	0.858798i	$-0.671212\pi$
$373$	−1.43201e23			−1.02463			−0.512315	+	0.858798i	$0.671212\pi$
$374$		0			0
$375$	4.79754e22	+	5.09342e22i	0.327143	+	0.347319i
$376$		0			0
$377$		−	3.45999e22i		−	0.224907i
$378$		0			0
$379$	8.21686e22			0.509276			0.254638	−	0.967036i	$-0.418044\pi$
$379$	8.21686e22			0.509276			0.254638	+	0.967036i	$0.418044\pi$
$380$		0			0
$381$	1.89607e23	−	1.78593e23i	1.12080	−	1.05570i
$382$		0			0
$383$			2.13126e23i			1.20184i	0.799309	+	0.600920i	$0.205199\pi$
$383$			2.13126e23i			1.20184i	−0.799309	+	0.600920i	$0.794801\pi$
$384$		0			0
$385$	7.79596e21			0.0419491
$386$		0			0
$387$	−3.80497e21	+	6.35413e22i	−0.0195413	+	0.326330i
$388$		0			0
$389$		−	3.46020e21i		−	0.0169650i	−0.999964	−	0.00848251i	$-0.997300\pi$
$389$		−	3.46020e21i		−	0.0169650i	0.999964	−	0.00848251i	$-0.00270010\pi$
$390$		0			0
$391$	−1.25997e22			−0.0589889
$392$		0			0
$393$	2.67099e23	+	2.83572e23i	1.19437	+	1.26803i
$394$		0			0
$395$		−	1.07139e22i		−	0.0457692i
$396$		0			0
$397$	−1.92387e23			−0.785346			−0.392673	−	0.919678i	$-0.628450\pi$
$397$	−1.92387e23			−0.785346			−0.392673	+	0.919678i	$0.628450\pi$
$398$		0			0
$399$	1.33419e22	−	1.25669e22i	0.0520550	−	0.0490311i
$400$		0			0
$401$			1.81975e23i			0.678753i	0.940651	+	0.339377i	$0.110216\pi$
$401$			1.81975e23i			0.678753i	−0.940651	+	0.339377i	$0.889784\pi$
$402$		0			0
$403$	−4.30050e22			−0.153382
$404$		0			0
$405$	−8.60632e21	+	7.16031e22i	−0.0293577	+	0.244251i
$406$		0			0
$407$		−	2.41004e23i		−	0.786455i
$408$		0			0
$409$	−3.70958e21			−0.0115828			−0.00579142	−	0.999983i	$-0.501843\pi$
$409$	−3.70958e21			−0.0115828			−0.00579142	+	0.999983i	$0.501843\pi$
$410$		0			0
$411$	−2.79314e23	−	2.96541e23i	−0.834674	−	0.886151i
$412$		0			0
$413$		−	5.07690e22i		−	0.145227i
$414$		0			0
$415$	−1.34038e23			−0.367109
$416$		0			0
$417$	3.00110e23	−	2.82676e23i	0.787147	−	0.741421i
$418$		0			0
$419$		−	6.24414e23i		−	1.56872i	−0.620304	−	0.784362i	$-0.712991\pi$
$419$		−	6.24414e23i		−	1.56872i	0.620304	−	0.784362i	$-0.287009\pi$
$420$		0			0
$421$	−3.41310e23			−0.821504			−0.410752	−	0.911747i	$-0.634734\pi$
$421$	−3.41310e23			−0.821504			−0.410752	+	0.911747i	$0.634734\pi$
$422$		0			0
$423$	−6.62030e23	−	3.96436e22i	−1.52691	−	0.0914345i
$424$		0			0
$425$			4.47669e23i			0.989592i
$426$		0			0
$427$	−5.90366e22			−0.125104
$428$		0			0
$429$	−3.67822e22	−	3.90507e22i	−0.0747347	−	0.0793438i
$430$		0			0
$431$		−	6.12494e23i		−	1.19346i	−0.802444	−	0.596728i	$-0.796467\pi$
$431$		−	6.12494e23i		−	1.19346i	0.802444	−	0.596728i	$-0.203533\pi$
$432$		0			0
$433$	4.86998e23			0.910199			0.455099	−	0.890441i	$-0.349604\pi$
$433$	4.86998e23			0.910199			0.455099	+	0.890441i	$0.349604\pi$
$434$		0			0
$435$	−1.88597e23	+	1.77642e23i	−0.338168	+	0.318523i
$436$		0			0
$437$			1.24922e22i			0.0214935i
$438$		0			0
$439$	5.44441e23			0.899022			0.449511	−	0.893275i	$-0.351598\pi$
$439$	5.44441e23			0.899022			0.449511	+	0.893275i	$0.351598\pi$
$440$		0			0
$441$	−3.64016e22	+	6.07890e23i	−0.0576997	+	0.963558i
$442$		0			0
$443$			8.16756e23i			1.24296i	0.783428	+	0.621482i	$0.213469\pi$
$443$			8.16756e23i			1.24296i	−0.783428	+	0.621482i	$0.786531\pi$
$444$		0			0
$445$	3.31034e21			0.00483763
$446$		0			0
$447$	−4.35182e23	−	4.62021e23i	−0.610805	−	0.648476i
$448$		0			0
$449$			1.00337e24i			1.35283i	0.736521	+	0.676415i	$0.236467\pi$
$449$			1.00337e24i			1.35283i	−0.736521	+	0.676415i	$0.763533\pi$
$450$		0			0
$451$	7.79471e23			1.00974
$452$		0			0
$453$	1.80248e23	−	1.69778e23i	0.224382	−	0.211347i
$454$		0			0
$455$		−	4.56282e21i		−	0.00545922i
$456$		0			0
$457$	1.17157e24			1.34748			0.673742	−	0.738967i	$-0.264686\pi$
$457$	1.17157e24			1.34748			0.673742	+	0.738967i	$0.264686\pi$
$458$		0			0
$459$	−7.31185e23	+	6.10492e23i	−0.808562	+	0.675096i
$460$		0			0
$461$		−	1.72767e24i		−	1.83719i	−0.395205	−	0.918593i	$-0.629326\pi$
$461$		−	1.72767e24i		−	1.83719i	0.395205	−	0.918593i	$-0.370674\pi$
$462$		0			0
$463$	6.32762e23			0.647160			0.323580	−	0.946201i	$-0.395113\pi$
$463$	6.32762e23			0.647160			0.323580	+	0.946201i	$0.395113\pi$
$464$		0			0
$465$	2.20794e23	+	2.34411e23i	0.217226	+	0.230623i
$466$		0			0
$467$			1.01552e24i			0.961248i	0.876927	+	0.480624i	$0.159590\pi$
$467$			1.01552e24i			0.961248i	−0.876927	+	0.480624i	$0.840410\pi$
$468$		0			0
$469$	−2.44441e23			−0.222648
$470$		0			0
$471$	1.57591e23	−	1.48436e23i	0.138148	−	0.130123i
$472$		0			0
$473$			3.54561e23i			0.299186i
$474$		0			0
$475$	4.43850e23			0.360573
$476$		0			0
$477$	−1.04676e24	−	6.26819e22i	−0.818806	−	0.0490317i
$478$		0			0
$479$			1.84506e23i			0.138992i	0.997582	+	0.0694961i	$0.0221392\pi$
$479$			1.84506e23i			0.138992i	−0.997582	+	0.0694961i	$0.977861\pi$
$480$		0			0
$481$	−1.41055e23			−0.102348
$482$		0			0
$483$	1.02351e22	+	1.08663e22i	0.00715430	+	0.00759553i
$484$		0			0
$485$		−	6.43449e23i		−	0.433349i
$486$		0			0
$487$	−4.72656e23			−0.306750			−0.153375	−	0.988168i	$-0.549014\pi$
$487$	−4.72656e23			−0.306750			−0.153375	+	0.988168i	$0.549014\pi$
$488$		0			0
$489$	8.52360e23	−	8.02846e23i	0.533142	−	0.502172i
$490$		0			0
$491$			5.50609e23i			0.331979i	0.986128	+	0.165989i	$0.0530817\pi$
$491$			5.50609e23i			0.331979i	−0.986128	+	0.165989i	$0.946918\pi$
$492$		0			0
$493$	−3.42200e24			−1.98911
$494$		0			0
$495$	−2.40117e22	+	4.00985e23i	−0.0134579	+	0.224740i
$496$		0			0
$497$		−	3.36536e23i		−	0.181897i
$498$		0			0
$499$	−5.90972e23			−0.308079			−0.154040	−	0.988065i	$-0.549228\pi$
$499$	−5.90972e23			−0.308079			−0.154040	+	0.988065i	$0.549228\pi$
$500$		0			0
$501$	−1.03568e24	−	1.09956e24i	−0.520820	−	0.552940i
$502$		0			0
$503$		−	2.94518e24i		−	1.42889i	−0.699690	−	0.714446i	$-0.746679\pi$
$503$		−	2.94518e24i		−	1.42889i	0.699690	−	0.714446i	$-0.253321\pi$
$504$		0			0
$505$	4.27533e23			0.200146
$506$		0			0
$507$	1.58840e24	−	1.49613e24i	0.717608	−	0.675922i
$508$		0			0
$509$		−	1.65616e24i		−	0.722175i	−0.932532	−	0.361088i	$-0.882406\pi$
$509$		−	1.65616e24i		−	0.722175i	0.932532	−	0.361088i	$-0.117594\pi$
$510$		0			0
$511$	−5.44362e23			−0.229139
$512$		0			0
$513$	6.05284e23	+	7.24948e23i	0.245982	+	0.294612i
$514$		0			0
$515$		−	5.56117e23i		−	0.218223i
$516$		0			0
$517$	−3.69414e24			−1.39990
$518$		0			0
$519$	2.64210e24	+	2.80504e24i	0.967036	+	1.02668i
$520$		0			0
$521$			3.11801e24i			1.10240i	0.834374	+	0.551198i	$0.185829\pi$
$521$			3.11801e24i			1.10240i	−0.834374	+	0.551198i	$0.814171\pi$
$522$		0			0
$523$	−1.44569e23			−0.0493811			−0.0246906	−	0.999695i	$-0.507860\pi$
$523$	−1.44569e23			−0.0493811			−0.0246906	+	0.999695i	$0.507860\pi$
$524$		0			0
$525$	3.86080e23	−	3.63653e23i	0.127422	−	0.120020i
$526$		0			0
$527$			4.25328e24i			1.35653i
$528$		0			0
$529$	3.23398e24			0.996864
$530$		0			0
$531$	2.61130e24	+	1.56370e23i	0.778048	+	0.0465910i
$532$		0			0
$533$		−	4.56208e23i		−	0.131407i
$534$		0			0
$535$	−2.34084e23			−0.0651910
$536$		0			0
$537$	2.34480e24	+	2.48941e24i	0.631448	+	0.670392i
$538$		0			0
$539$			3.39204e24i			0.883409i
$540$		0			0
$541$	−6.05433e23			−0.152507			−0.0762536	−	0.997088i	$-0.524296\pi$
$541$	−6.05433e23			−0.152507			−0.0762536	+	0.997088i	$0.524296\pi$
$542$		0			0
$543$	1.42446e24	−	1.34171e24i	0.347097	−	0.326934i
$544$		0			0
$545$		−	1.69148e24i		−	0.398747i
$546$		0			0
$547$	−2.24143e23			−0.0511257			−0.0255629	−	0.999673i	$-0.508138\pi$
$547$	−2.24143e23			−0.0511257			−0.0255629	+	0.999673i	$0.508138\pi$
$548$		0			0
$549$	1.81834e23	−	3.03654e24i	0.0401351	−	0.670237i
$550$		0			0
$551$			3.39281e24i			0.724761i
$552$		0			0
$553$	−1.67653e23			−0.0346646
$554$		0			0
$555$	7.24196e23	+	7.68860e23i	0.144950	+	0.153890i
$556$		0			0
$557$		−	7.02796e24i		−	1.36186i	−0.732348	−	0.680930i	$-0.761576\pi$
$557$		−	7.02796e24i		−	1.36186i	0.732348	−	0.680930i	$-0.238424\pi$
$558$		0			0
$559$	2.07518e23			0.0389358
$560$		0			0
$561$	−3.86219e24	+	3.63783e24i	−0.701726	+	0.660962i
$562$		0			0
$563$		−	8.37178e24i		−	1.47313i	−0.676365	−	0.736567i	$-0.736446\pi$
$563$		−	8.37178e24i		−	1.47313i	0.676365	−	0.736567i	$-0.263554\pi$
$564$		0			0
$565$	1.33718e24			0.227905
$566$		0			0
$567$	1.12046e24	+	1.34674e23i	0.184991	+	0.0222349i
$568$		0			0
$569$			9.12956e24i			1.46029i	0.683291	+	0.730146i	$0.260548\pi$
$569$			9.12956e24i			1.46029i	−0.683291	+	0.730146i	$0.739452\pi$
$570$		0			0
$571$	−7.81895e24			−1.21178			−0.605890	−	0.795549i	$-0.707183\pi$
$571$	−7.81895e24			−1.21178			−0.605890	+	0.795549i	$0.707183\pi$
$572$		0			0
$573$	−8.26641e24	−	8.77622e24i	−1.24144	−	1.31800i
$574$		0			0
$575$			3.61492e23i			0.0526125i
$576$		0			0
$577$	8.24528e24			1.16312			0.581558	−	0.813505i	$-0.302443\pi$
$577$	8.24528e24			1.16312			0.581558	+	0.813505i	$0.302443\pi$
$578$		0			0
$579$	−3.33515e24	+	3.14141e24i	−0.456045	+	0.429553i
$580$		0			0
$581$			2.09746e24i			0.278040i
$582$		0			0
$583$	−5.84093e24			−0.750698
$584$		0			0
$585$	2.34688e23	+	1.40536e22i	0.0292475	+	0.00175139i
$586$		0			0
$587$		−	1.05528e25i		−	1.27533i	−0.770313	−	0.637666i	$-0.779900\pi$
$587$		−	1.05528e25i		−	1.27533i	0.770313	−	0.637666i	$-0.220100\pi$
$588$		0			0
$589$	4.21699e24			0.494271
$590$		0			0
$591$	−3.20380e24	−	3.40139e24i	−0.364232	−	0.386695i
$592$		0			0
$593$		−	4.62289e24i		−	0.509826i	−0.966964	−	0.254913i	$-0.917953\pi$
$593$		−	4.62289e24i		−	0.509826i	0.966964	−	0.254913i	$-0.0820468\pi$
$594$		0			0
$595$	−4.51271e23			−0.0482820
$596$		0			0
$597$	−1.22603e25	+	1.15481e25i	−1.27272	+	1.19879i
$598$		0			0
$599$		−	6.50172e23i		−	0.0654918i	−0.999464	−	0.0327459i	$-0.989575\pi$
$599$		−	6.50172e23i		−	0.0654918i	0.999464	−	0.0327459i	$-0.0104252\pi$
$600$		0			0
$601$	1.62307e25			1.58660			0.793299	−	0.608833i	$-0.208362\pi$
$601$	1.62307e25			1.58660			0.793299	+	0.608833i	$0.208362\pi$
$602$		0			0
$603$	7.52884e23	−	1.25728e25i	0.0714286	−	1.19283i
$604$		0			0
$605$		−	4.33965e23i		−	0.0399629i
$606$		0			0
$607$	−6.35704e24			−0.568273			−0.284136	−	0.958784i	$-0.591707\pi$
$607$	−6.35704e24			−0.568273			−0.284136	+	0.958784i	$0.591707\pi$
$608$		0			0
$609$	2.77978e24	+	2.95122e24i	0.241243	+	0.256121i
$610$		0			0
$611$			2.16211e24i			0.182182i
$612$		0			0
$613$	1.28760e25			1.05351			0.526754	−	0.850018i	$-0.323409\pi$
$613$	1.28760e25			1.05351			0.526754	+	0.850018i	$0.323409\pi$
$614$		0			0
$615$	−2.48670e24	+	2.34225e24i	−0.197582	+	0.186104i
$616$		0			0
$617$			2.46163e25i			1.89958i	0.312895	+	0.949788i	$0.398701\pi$
$617$			2.46163e25i			1.89958i	−0.312895	+	0.949788i	$0.601299\pi$
$618$		0			0
$619$	1.30807e25			0.980426			0.490213	−	0.871603i	$-0.336919\pi$
$619$	1.30807e25			0.980426			0.490213	+	0.871603i	$0.336919\pi$
$620$		0			0
$621$	−5.90432e23	+	4.92972e23i	−0.0429878	+	0.0358920i
$622$		0			0
$623$		−	5.18011e22i		−	0.00366392i
$624$		0			0
$625$	1.19631e25			0.822101
$626$		0			0
$627$	3.60680e24	+	3.82924e24i	0.240832	+	0.255685i
$628$		0			0
$629$			1.39506e25i			0.905182i
$630$		0			0
$631$	1.35039e24			0.0851522			0.0425761	−	0.999093i	$-0.486444\pi$
$631$	1.35039e24			0.0851522			0.0425761	+	0.999093i	$0.486444\pi$
$632$		0			0
$633$	−1.17677e25	+	1.10841e25i	−0.721205	+	0.679310i
$634$		0			0
$635$			6.35847e24i			0.378782i
$636$		0			0
$637$	1.98529e24			0.114966
$638$		0			0
$639$	1.73097e25	+	1.03654e24i	0.974502	+	0.0583550i
$640$		0			0
$641$		−	1.86667e25i		−	1.02175i	−0.859654	−	0.510877i	$-0.829321\pi$
$641$		−	1.86667e25i		−	1.02175i	0.859654	−	0.510877i	$-0.170679\pi$
$642$		0			0
$643$	−3.14994e25			−1.67650			−0.838251	−	0.545284i	$-0.816422\pi$
$643$	−3.14994e25			−1.67650			−0.838251	+	0.545284i	$0.816422\pi$
$644$		0			0
$645$	−1.06543e24	−	1.13114e24i	−0.0551425	−	0.0585434i
$646$		0			0
$647$		−	1.16484e25i		−	0.586309i	−0.956065	−	0.293154i	$-0.905295\pi$
$647$		−	1.16484e25i		−	0.586309i	0.956065	−	0.293154i	$-0.0947050\pi$
$648$		0			0
$649$	1.45711e25			0.713330
$650$		0			0
$651$	3.66813e24	−	3.45504e24i	0.174669	−	0.164522i
$652$		0			0
$653$		−	3.43332e25i		−	1.59036i	−0.606373	−	0.795180i	$-0.707376\pi$
$653$		−	3.43332e25i		−	1.59036i	0.606373	−	0.795180i	$-0.292624\pi$
$654$		0			0
$655$	−9.50959e24			−0.428539
$656$		0			0
$657$	1.67665e24	−	2.79992e25i	0.0735111	−	1.22760i
$658$		0			0
$659$			1.87695e25i			0.800726i	0.916357	+	0.400363i	$0.131116\pi$
$659$			1.87695e25i			0.800726i	−0.916357	+	0.400363i	$0.868884\pi$
$660$		0			0
$661$	−4.29889e25			−1.78461			−0.892306	−	0.451432i	$-0.850913\pi$
$661$	−4.29889e25			−1.78461			−0.892306	+	0.451432i	$0.850913\pi$
$662$		0			0
$663$	2.12915e24	+	2.26046e24i	0.0860170	+	0.0913219i
$664$		0			0
$665$			4.47421e23i			0.0175923i
$666$		0			0
$667$	−2.76327e24			−0.105752
$668$		0			0
$669$	3.45525e24	−	3.25453e24i	0.128720	−	0.121242i
$670$		0			0
$671$		−	1.69440e25i		−	0.614487i
$672$		0			0
$673$	−3.73138e25			−1.31745			−0.658725	−	0.752384i	$-0.728904\pi$
$673$	−3.73138e25			−1.31745			−0.658725	+	0.752384i	$0.728904\pi$
$674$		0			0
$675$	1.75153e25	+	2.09781e25i	0.602121	+	0.721160i
$676$		0			0
$677$			1.79692e24i			0.0601492i	0.999548	+	0.0300746i	$0.00957449\pi$
$677$			1.79692e24i			0.0601492i	−0.999548	+	0.0300746i	$0.990426\pi$
$678$		0			0
$679$	−1.00688e25			−0.328210
$680$		0			0
$681$	−4.67029e24	−	4.95832e24i	−0.148259	−	0.157402i
$682$		0			0
$683$			2.03248e25i			0.628407i	0.949356	+	0.314203i	$0.101737\pi$
$683$			2.03248e25i			0.628407i	−0.949356	+	0.314203i	$0.898263\pi$
$684$		0			0
$685$	9.94449e24			0.299480
$686$		0			0
$687$	2.13699e25	−	2.01285e25i	0.626892	−	0.590475i
$688$		0			0
$689$			3.41858e24i			0.0976951i
$690$		0			0
$691$	1.32210e24			0.0368096			0.0184048	−	0.999831i	$-0.494141\pi$
$691$	1.32210e24			0.0368096			0.0184048	+	0.999831i	$0.494141\pi$
$692$		0			0
$693$	6.27471e24	+	3.75741e23i	0.170214	+	0.0101927i
$694$		0			0
$695$			1.00642e25i			0.266021i
$696$		0			0
$697$	−4.51199e25			−1.16218
$698$		0			0
$699$	3.46758e25	+	3.68144e25i	0.870427	+	0.924109i
$700$		0			0
$701$		−	2.83982e25i		−	0.694751i	−0.937726	−	0.347376i	$-0.887073\pi$
$701$		−	2.83982e25i		−	0.694751i	0.937726	−	0.347376i	$-0.112927\pi$
$702$		0			0
$703$	1.38316e25			0.329817
$704$		0			0
$705$	1.17852e25	−	1.11006e25i	0.273927	−	0.258014i
$706$		0			0
$707$		−	6.69014e24i		−	0.151586i
$708$		0			0
$709$	6.59263e25			1.45627			0.728136	−	0.685432i	$-0.240387\pi$
$709$	6.59263e25			1.45627			0.728136	+	0.685432i	$0.240387\pi$
$710$		0			0
$711$	5.16376e23	−	8.62325e24i	0.0111209	−	0.185714i
$712$		0			0
$713$			3.43452e24i			0.0721208i
$714$		0			0
$715$	1.30956e24			0.0268147
$716$		0			0
$717$	9.23780e24	+	9.80753e24i	0.184458	+	0.195834i
$718$		0			0
$719$			5.64358e25i			1.09899i	0.835496	+	0.549496i	$0.185180\pi$
$719$			5.64358e25i			1.09899i	−0.835496	+	0.549496i	$0.814820\pi$
$720$		0			0
$721$	−8.70225e24			−0.165278
$722$		0			0
$723$	9.41694e23	−	8.86991e23i	0.0174448	−	0.0164314i
$724$		0			0
$725$			9.81789e25i			1.77409i
$726$		0			0
$727$	4.29830e25			0.757682			0.378841	−	0.925462i	$-0.376323\pi$
$727$	4.29830e25			0.757682			0.378841	+	0.925462i	$0.376323\pi$
$728$		0			0
$729$	−1.03780e25	+	5.72162e25i	−0.178470	+	0.983945i
$730$		0			0
$731$		−	2.05239e25i		−	0.344353i
$732$		0			0
$733$	−6.75848e25			−1.10640			−0.553201	−	0.833048i	$-0.686594\pi$
$733$	−6.75848e25			−1.10640			−0.553201	+	0.833048i	$0.686594\pi$
$734$		0			0
$735$	−1.01928e25	−	1.08214e25i	−0.162820	−	0.172861i
$736$		0			0
$737$		−	7.01565e25i		−	1.09361i
$738$		0			0
$739$	1.25275e26			1.90574			0.952868	−	0.303384i	$-0.0981165\pi$
$739$	1.25275e26			1.90574			0.952868	+	0.303384i	$0.0981165\pi$
$740$		0			0
$741$	2.24118e24	−	2.11099e24i	0.0332745	−	0.0313416i
$742$		0			0
$743$		−	4.27774e25i		−	0.619891i	−0.950754	−	0.309945i	$-0.899689\pi$
$743$		−	4.27774e25i		−	0.619891i	0.950754	−	0.309945i	$-0.100311\pi$
$744$		0			0
$745$	1.54939e25			0.219156
$746$		0			0
$747$	−1.07883e26	−	6.46021e24i	−1.48959	−	0.0891993i
$748$		0			0
$749$			3.66300e24i			0.0493743i
$750$		0			0
$751$	−1.06753e26			−1.40482			−0.702408	−	0.711775i	$-0.747892\pi$
$751$	−1.06753e26			−1.40482			−0.702408	+	0.711775i	$0.747892\pi$
$752$		0			0
$753$	−7.00047e25	−	7.43221e25i	−0.899441	−	0.954912i
$754$		0			0
$755$			6.04463e24i			0.0758311i
$756$		0			0
$757$	5.91571e25			0.724676			0.362338	−	0.932047i	$-0.381979\pi$
$757$	5.91571e25			0.724676			0.362338	+	0.932047i	$0.381979\pi$
$758$		0			0
$759$	−3.11872e24	+	2.93755e24i	−0.0373078	+	0.0351406i
$760$		0			0
$761$		−	1.22980e26i		−	1.43673i	−0.695669	−	0.718363i	$-0.744892\pi$
$761$		−	1.22980e26i		−	1.43673i	0.695669	−	0.718363i	$-0.255108\pi$
$762$		0			0
$763$	−2.64687e25			−0.302003
$764$		0			0
$765$	1.38992e24	−	2.32111e25i	0.0154895	−	0.258668i
$766$		0			0
$767$		−	8.52817e24i		−	0.0928320i
$768$		0			0
$769$	−6.01098e25			−0.639158			−0.319579	−	0.947560i	$-0.603541\pi$
$769$	−6.01098e25			−0.639158			−0.319579	+	0.947560i	$0.603541\pi$
$770$		0			0
$771$	−9.32339e25	−	9.89840e25i	−0.968466	−	1.02819i
$772$		0			0
$773$			1.93106e25i			0.195966i	0.995188	+	0.0979831i	$0.0312391\pi$
$773$			1.93106e25i			0.195966i	−0.995188	+	0.0979831i	$0.968761\pi$
$774$		0			0
$775$	1.22029e26			1.20989
$776$		0			0
$777$	1.20313e25	−	1.13324e25i	0.116553	−	0.109782i
$778$		0			0
$779$			4.47350e25i			0.423458i
$780$		0			0
$781$	9.65886e25			0.893443
$782$		0			0
$783$	−1.60357e26	+	1.33888e26i	−1.44955	+	1.21028i
$784$		0			0
$785$			5.28481e24i			0.0466877i
$786$		0			0
$787$	−8.27472e25			−0.714465			−0.357232	−	0.934016i	$-0.616280\pi$
$787$	−8.27472e25			−0.714465			−0.357232	+	0.934016i	$0.616280\pi$
$788$		0			0
$789$	1.30157e26	+	1.38185e26i	1.09844	+	1.16618i
$790$		0			0
$791$		−	2.09245e25i		−	0.172611i
$792$		0			0
$793$	−9.91696e24			−0.0799687
$794$		0			0
$795$	1.86340e25	−	1.75515e25i	0.146893	−	0.138360i
$796$		0			0
$797$			1.66543e26i			1.28351i	0.766909	+	0.641756i	$0.221794\pi$
$797$			1.66543e26i			1.28351i	−0.766909	+	0.641756i	$0.778206\pi$
$798$		0			0
$799$	2.13836e26			1.61124
$800$		0			0
$801$	2.66439e24	+	1.59548e23i	0.0196293	+	0.00117544i
$802$		0			0
$803$		−	1.56236e26i		−	1.12549i
$804$		0			0
$805$	−3.64402e23			−0.00256695
$806$		0			0
$807$	−3.47717e25	−	3.69162e25i	−0.239532	−	0.254305i
$808$		0			0
$809$			1.68906e26i			1.13791i	0.822368	+	0.568955i	$0.192652\pi$
$809$			1.68906e26i			1.13791i	−0.822368	+	0.568955i	$0.807348\pi$
$810$		0			0
$811$	1.07788e26			0.710201			0.355101	−	0.934828i	$-0.384447\pi$
$811$	1.07788e26			0.710201			0.355101	+	0.934828i	$0.384447\pi$
$812$		0			0
$813$	1.04081e26	−	9.80352e25i	0.670746	−	0.631782i
$814$		0			0
$815$			2.85839e25i			0.180178i
$816$		0			0
$817$	−2.03488e25			−0.125470
$818$		0			0
$819$	2.19914e23	−	3.67246e24i	0.00132647	−	0.0221514i
$820$		0			0
$821$			1.24368e26i			0.733871i	0.930246	+	0.366935i	$0.119593\pi$
$821$			1.24368e26i			0.733871i	−0.930246	+	0.366935i	$0.880407\pi$
$822$		0			0
$823$	−2.82257e26			−1.62947			−0.814736	−	0.579832i	$-0.803118\pi$
$823$	−2.82257e26			−1.62947			−0.814736	+	0.579832i	$0.803118\pi$
$824$		0			0
$825$	1.04371e26	+	1.10808e26i	0.589515	+	0.625873i
$826$		0			0
$827$		−	1.67546e26i		−	0.925942i	−0.886374	−	0.462971i	$-0.846783\pi$
$827$		−	1.67546e26i		−	0.925942i	0.886374	−	0.462971i	$-0.153217\pi$
$828$		0			0
$829$	2.61415e26			1.41364			0.706821	−	0.707392i	$-0.250129\pi$
$829$	2.61415e26			1.41364			0.706821	+	0.707392i	$0.250129\pi$
$830$		0			0
$831$	6.85485e25	−	6.45665e25i	0.362734	−	0.341663i
$832$		0			0
$833$		−	1.96349e26i		−	1.01677i
$834$		0			0
$835$	3.68737e25			0.186869
$836$		0			0
$837$	1.66412e26	+	1.99312e26i	0.825384	+	0.988561i
$838$		0			0
$839$			2.87799e26i			1.39711i	0.715556	+	0.698556i	$0.246174\pi$
$839$			2.87799e26i			1.39711i	−0.715556	+	0.698556i	$0.753826\pi$
$840$		0			0
$841$	−5.40028e26			−2.56597
$842$		0			0
$843$	−9.55343e25	−	1.01426e26i	−0.444336	−	0.471739i
$844$		0			0
$845$			5.32669e25i			0.242520i
$846$		0			0
$847$	−6.79080e24			−0.0302670
$848$		0			0
$849$	1.14452e26	−	1.07804e26i	0.499407	−	0.470397i
$850$		0			0
$851$			1.12651e25i			0.0481247i
$852$		0			0
$853$	−2.27550e26			−0.951779			−0.475889	−	0.879505i	$-0.657874\pi$
$853$	−2.27550e26			−0.951779			−0.475889	+	0.879505i	$0.657874\pi$
$854$		0			0
$855$	−2.30131e25	−	1.37807e24i	−0.0942497	−	0.00564385i
$856$		0			0
$857$		−	1.35115e26i		−	0.541846i	−0.962601	−	0.270923i	$-0.912671\pi$
$857$		−	1.35115e26i		−	0.541846i	0.962601	−	0.270923i	$-0.0873289\pi$
$858$		0			0
$859$	−1.32554e26			−0.520540			−0.260270	−	0.965536i	$-0.583811\pi$
$859$	−1.32554e26			−0.520540			−0.260270	+	0.965536i	$0.583811\pi$
$860$		0			0
$861$	3.66520e25	+	3.89125e25i	0.140952	+	0.149644i
$862$		0			0
$863$		−	4.61257e26i		−	1.73718i	−0.495527	−	0.868592i	$-0.665025\pi$
$863$		−	4.61257e26i		−	1.73718i	0.495527	−	0.868592i	$-0.334975\pi$
$864$		0			0
$865$	−9.40671e25			−0.346971
$866$		0			0
$867$	2.20691e25	−	2.07871e25i	0.0797283	−	0.0750969i
$868$		0			0
$869$		−	4.81179e25i		−	0.170266i
$870$		0			0
$871$	−4.10612e25			−0.142321
$872$		0			0
$873$	3.10122e25	−	5.17890e26i	0.105294	−	1.75837i
$874$		0			0
$875$			2.67284e25i			0.0888999i
$876$		0			0
$877$	−4.28270e26			−1.39547			−0.697737	−	0.716354i	$-0.745810\pi$
$877$	−4.28270e26			−1.39547			−0.697737	+	0.716354i	$0.745810\pi$
$878$		0			0
$879$	2.14455e26	+	2.27681e26i	0.684598	+	0.726819i
$880$		0			0
$881$		−	1.62543e26i		−	0.508377i	−0.967155	−	0.254188i	$-0.918192\pi$
$881$		−	1.62543e26i		−	0.508377i	0.967155	−	0.254188i	$-0.0818083\pi$
$882$		0			0
$883$	1.59780e26			0.489638			0.244819	−	0.969569i	$-0.421271\pi$
$883$	1.59780e26			0.489638			0.244819	+	0.969569i	$0.421271\pi$
$884$		0			0
$885$	−4.64853e25	+	4.37850e25i	−0.139581	+	0.131473i
$886$		0			0
$887$			9.50081e25i			0.279543i	0.990184	+	0.139771i	$0.0446367\pi$
$887$			9.50081e25i			0.279543i	−0.990184	+	0.139771i	$0.955363\pi$
$888$		0			0
$889$	9.94989e25			0.286882
$890$		0			0
$891$	−3.86525e25	+	3.21582e26i	−0.109214	+	0.908641i
$892$		0			0
$893$		−	2.12012e26i		−	0.587081i
$894$		0			0
$895$	−8.34825e25			−0.226563
$896$		0			0
$897$	1.71929e24	+	1.82532e24i	0.00457316	+	0.00485520i
$898$		0			0
$899$			9.32793e26i			2.43191i
$900$		0			0
$901$	3.38104e26			0.864027
$902$		0			0
$903$	−1.77003e25	+	1.66721e25i	−0.0443395	+	0.0417638i
$904$		0			0
$905$			4.77691e25i			0.117303i
$906$		0			0
$907$	6.04962e26			1.45634			0.728169	−	0.685397i	$-0.240371\pi$
$907$	6.04962e26			1.45634			0.728169	+	0.685397i	$0.240371\pi$
$908$		0			0
$909$	3.44107e26	+	2.06058e25i	0.812117	+	0.0486311i
$910$		0			0
$911$			1.97760e26i			0.457586i	0.973475	+	0.228793i	$0.0734780\pi$
$911$			1.97760e26i			0.457586i	−0.973475	+	0.228793i	$0.926522\pi$
$912$		0			0
$913$	−6.01987e26			−1.36568
$914$		0			0
$915$	5.09152e25	+	5.40554e25i	0.113255	+	0.120240i
$916$		0			0
$917$			1.48808e26i			0.324566i
$918$		0			0
$919$	8.70654e26			1.86211			0.931057	−	0.364874i	$-0.118888\pi$
$919$	8.70654e26			1.86211			0.931057	+	0.364874i	$0.118888\pi$
$920$		0			0
$921$	−4.52413e26	+	4.26132e26i	−0.948852	+	0.893732i
$922$		0			0
$923$		−	5.65314e25i		−	0.116272i
$924$		0			0
$925$	4.00249e26			0.807336
$926$		0			0
$927$	2.68031e25	−	4.47600e26i	0.0530234	−	0.885467i
$928$		0			0
$929$		−	7.41037e25i		−	0.143780i	−0.997413	−	0.0718900i	$-0.977097\pi$
$929$		−	7.41037e25i		−	0.143780i	0.997413	−	0.0718900i	$-0.0229031\pi$
$930$		0			0
$931$	−1.94674e26			−0.370477
$932$		0			0
$933$	−3.75279e26	−	3.98424e26i	−0.700518	−	0.743722i
$934$		0			0
$935$		−	1.29518e26i		−	0.237152i
$936$		0			0
$937$	3.85595e26			0.692588			0.346294	−	0.938126i	$-0.387440\pi$
$937$	3.85595e26			0.692588			0.346294	+	0.938126i	$0.387440\pi$
$938$		0			0
$939$	−2.39992e26	+	2.26050e26i	−0.422869	+	0.398304i
$940$		0			0
$941$		−	1.14490e26i		−	0.197906i	−0.995092	−	0.0989531i	$-0.968451\pi$
$941$		−	1.14490e26i		−	0.197906i	0.995092	−	0.0989531i	$-0.0315494\pi$
$942$		0			0
$943$	−3.64343e25			−0.0617882
$944$		0			0
$945$	−2.11469e25	+	1.76563e25i	−0.0351852	+	0.0293774i
$946$		0			0
$947$		−	2.92739e26i		−	0.477893i	−0.971033	−	0.238947i	$-0.923198\pi$
$947$		−	2.92739e26i		−	0.477893i	0.971033	−	0.238947i	$-0.0768021\pi$
$948$		0			0
$949$	−9.14419e25			−0.146470
$950$		0			0
$951$	−2.77182e26	−	2.94276e26i	−0.435652	−	0.462520i
$952$		0			0
$953$		−	2.50518e26i		−	0.386369i	−0.981162	−	0.193185i	$-0.938118\pi$
$953$		−	2.50518e26i		−	0.386369i	0.981162	−	0.193185i	$-0.0618817\pi$
$954$		0			0
$955$	2.94311e26			0.445426
$956$		0			0
$957$	−8.47023e26	+	7.97819e26i	−1.25802	+	1.18494i
$958$		0			0
$959$		−	1.55614e26i		−	0.226820i
$960$		0			0
$961$	4.60334e26			0.658511
$962$		0			0
$963$	−1.88406e26	−	1.12821e25i	−0.264520	−	0.0158400i
$964$		0			0
$965$		−	1.11844e26i		−	0.154123i
$966$		0			0
$967$	−1.16478e27			−1.57546			−0.787730	−	0.616020i	$-0.788744\pi$
$967$	−1.16478e27			−1.57546			−0.787730	+	0.616020i	$0.788744\pi$
$968$		0			0
$969$	−2.08780e26	−	2.21657e26i	−0.277189	−	0.294284i
$970$		0			0
$971$		−	9.82477e25i		−	0.128041i	−0.997949	−	0.0640205i	$-0.979608\pi$
$971$		−	9.82477e25i		−	0.128041i	0.997949	−	0.0640205i	$-0.0203923\pi$
$972$		0			0
$973$	1.57487e26			0.201478
$974$		0			0
$975$	6.48537e25	−	6.10863e25i	0.0814504	−	0.0767189i
$976$		0			0
$977$			6.27958e26i			0.774247i	0.922028	+	0.387124i	$0.126531\pi$
$977$			6.27958e26i			0.774247i	−0.922028	+	0.387124i	$0.873469\pi$
$978$		0			0
$979$	1.48673e25			0.0179965
$980$		0			0
$981$	8.15241e25	−	1.36141e27i	0.0968869	−	1.61797i
$982$		0			0
$983$		−	4.85842e26i		−	0.566910i	−0.958986	−	0.283455i	$-0.908519\pi$
$983$		−	4.85842e26i		−	0.566910i	0.958986	−	0.283455i	$-0.0914806\pi$
$984$		0			0
$985$	1.14066e26			0.130686
$986$		0			0
$987$	−1.73705e26	−	1.84418e26i	−0.195415	−	0.207467i
$988$		0			0
$989$		−	1.65731e25i		−	0.0183078i
$990$		0			0
$991$	−1.26805e26			−0.137554			−0.0687771	−	0.997632i	$-0.521910\pi$
$991$	−1.26805e26			−0.137554			−0.0687771	+	0.997632i	$0.521910\pi$
$992$		0			0
$993$	−6.24363e26	+	5.88094e26i	−0.665111	+	0.626474i
$994$		0			0
$995$		−	4.11150e26i		−	0.430123i
$996$		0			0
$997$	3.20009e26			0.328780			0.164390	−	0.986395i	$-0.447434\pi$
$997$	3.20009e26			0.328780			0.164390	+	0.986395i	$0.447434\pi$
$998$		0			0
$999$	5.45825e26	+	6.53734e26i	0.550762	+	0.659647i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.19.c.a.5.2	yes	6
3.2	odd	2	inner	12.19.c.a.5.1	✓	6
4.3	odd	2		48.19.e.c.17.5		6
12.11	even	2		48.19.e.c.17.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.19.c.a.5.1	✓	6	3.2	odd	2	inner
12.19.c.a.5.2	yes	6	1.1	even	1	trivial
48.19.e.c.17.5		6	4.3	odd	2
48.19.e.c.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:	\( k \)	\(=\)	\( 19 \)
Character orbit:	\([\chi]\)	\(=\)	12.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-14327.9 + 13495.6i) q^{3} -480487. i q^{5} -7.51877e6 q^{7} +(2.31580e7 - 3.86728e8i) q^{9} +O(q^{10})\)	\(q+(-14327.9 + 13495.6i) q^{3} -480487. i q^{5} -7.51877e6 q^{7} +(2.31580e7 - 3.86728e8i) q^{9} -2.15795e9i q^{11} -1.26300e9 q^{13} +(6.48446e9 + 6.88437e9i) q^{15} +1.24913e11i q^{17} +1.23848e11 q^{19} +(1.07728e11 - 1.01470e11i) q^{21} +1.00868e11i q^{23} +3.58383e12 q^{25} +(4.88732e12 + 5.85353e12i) q^{27} +2.73950e13i q^{29} +3.40498e13 q^{31} +(2.91228e13 + 3.09189e13i) q^{33} +3.61267e12i q^{35} +1.11682e14 q^{37} +(1.80962e13 - 1.70450e13i) q^{39} +3.61209e14i q^{41} -1.64305e14 q^{43} +(-1.85818e14 - 1.11271e13i) q^{45} -1.71188e15i q^{47} -1.57188e15 q^{49} +(-1.68578e15 - 1.78975e15i) q^{51} -2.70671e15i q^{53} -1.03687e15 q^{55} +(-1.77448e15 + 1.67140e15i) q^{57} +6.75230e15i q^{59} +7.85189e15 q^{61} +(-1.74120e14 + 2.90772e15i) q^{63} +6.06856e14i q^{65} +3.25108e16 q^{67} +(-1.36127e15 - 1.44522e15i) q^{69} +4.47595e16i q^{71} +7.24004e16 q^{73} +(-5.13488e16 + 4.83659e16i) q^{75} +1.62251e16i q^{77} +2.22980e16 q^{79} +(-1.49022e17 - 1.79117e16i) q^{81} -2.78963e17i q^{83} +6.00192e16 q^{85} +(-3.69712e17 - 3.92513e17i) q^{87} +6.88956e15i q^{89} +9.49624e15 q^{91} +(-4.87863e17 + 4.59522e17i) q^{93} -5.95072e16i q^{95} +1.33916e18 q^{97} +(-8.34538e17 - 4.99737e16i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 23934 q^{3} + 11024364 q^{7} - 34078986 q^{9}+O(q^{10}) \) 6 * q + 23934 * q^3 + 11024364 * q^7 - 34078986 * q^9	\( 6 q + 23934 q^{3} + 11024364 q^{7} - 34078986 q^{9} + 1775593596 q^{13} - 9856918080 q^{15} - 202034532804 q^{19} - 200158320036 q^{21} - 5958025101930 q^{25} - 8167776022674 q^{27} - 56059471942836 q^{31} - 51478992417600 q^{33} - 167596105515108 q^{37} - 73688099161044 q^{39} + 448591926775836 q^{43} - 12918361749120 q^{45} + 38\!\cdots\!10 q^{49}+ \cdots - 46\!\cdots\!60 q^{99}+O(q^{100}) \) 6 * q + 23934 * q^3 + 11024364 * q^7 - 34078986 * q^9 + 1775593596 * q^13 - 9856918080 * q^15 - 202034532804 * q^19 - 200158320036 * q^21 - 5958025101930 * q^25 - 8167776022674 * q^27 - 56059471942836 * q^31 - 51478992417600 * q^33 - 167596105515108 * q^37 - 73688099161044 * q^39 + 448591926775836 * q^43 - 12918361749120 * q^45 + 3815313399549810 * q^49 + 579278365205760 * q^51 + 8271035021335680 * q^55 - 7540820577807636 * q^57 + 13350472196682684 * q^61 - 40748979535300404 * q^63 + 37560450307637244 * q^67 - 131632980577038720 * q^69 + 149418174175674348 * q^73 - 287990904570521010 * q^75 + 646522133618739468 * q^79 - 656120326335021594 * q^81 + 1416345514989166080 * q^85 - 1306513723216363200 * q^87 + 2966957008358949816 * q^91 - 2834708598841722564 * q^93 + 2984598304382259084 * q^97 - 4681229544239276160 * q^99

Introduction

L-functions

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