LMFDB - Embedded newform 48.23.e.b.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,23,Mod(17,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 23, names="a")

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

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 23);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 23 $
Character orbit:	$[\chi]$	$=$	48.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$147.219568724$
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} + 126474x^{4} + 3861674040x^{2} + 9831214131200 $ x^6 + 126474x^4 + 3861674040x^2 + 9831214131200
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{33}\cdot 3^{22} $
Twist minimal:	no (minimal twist has level 3)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$281.771i$ of defining polynomial
Character	$\chi$	$=$	48.17
Dual form			48.23.e.b.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-169408. + 51788.6i) q^{3} +6.36810e7i q^{5} +2.60308e9 q^{7} +(2.60170e10 - 1.75468e10i) q^{9} +O(q^{10})$	\(q+(-169408. + 51788.6i) q^{3} +6.36810e7i q^{5} +2.60308e9 q^{7} +(2.60170e10 - 1.75468e10i) q^{9} -2.74891e10i q^{11} -6.09922e9 q^{13} +(-3.29794e12 - 1.07881e13i) q^{15} +8.12449e12i q^{17} -7.54682e13 q^{19} +(-4.40982e14 + 1.34810e14i) q^{21} -1.66353e15i q^{23} -1.67108e15 q^{25} +(-3.49875e15 + 4.31994e15i) q^{27} +1.26684e16i q^{29} -1.69379e16 q^{31} +(1.42362e15 + 4.65686e15i) q^{33} +1.65766e17i q^{35} +1.59704e17 q^{37} +(1.03325e15 - 3.15870e14i) q^{39} +4.83588e17i q^{41} -4.91968e17 q^{43} +(1.11740e18 + 1.65678e18i) q^{45} -7.73166e17i q^{47} +2.86619e18 q^{49} +(-4.20756e17 - 1.37635e18i) q^{51} -5.90402e18i q^{53} +1.75053e18 q^{55} +(1.27849e19 - 3.90839e18i) q^{57} +2.68658e19i q^{59} +2.43398e19 q^{61} +(6.77241e19 - 4.56756e19i) q^{63} -3.88404e17i q^{65} +1.30406e20 q^{67} +(8.61517e19 + 2.81815e20i) q^{69} +4.07363e20i q^{71} -4.04873e20 q^{73} +(2.83094e20 - 8.65427e19i) q^{75} -7.15562e19i q^{77} -1.11027e21 q^{79} +(3.68993e20 - 9.13027e20i) q^{81} -1.31903e19i q^{83} -5.17375e20 q^{85} +(-6.56080e20 - 2.14613e21i) q^{87} -2.62916e20i q^{89} -1.58767e19 q^{91} +(2.86941e21 - 8.77187e20i) q^{93} -4.80589e21i q^{95} -1.92665e21 q^{97} +(-4.82344e20 - 7.15181e20i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 86670 q^{3} + 3447063060 q^{7} + 57339715158 q^{9}+O(q^{10}) $ 6 * q - 86670 * q^3 + 3447063060 * q^7 + 57339715158 * q^9	$ 6 q - 86670 q^{3} + 3447063060 q^{7} + 57339715158 q^{9} + 2025132496860 q^{13} - 2628031314240 q^{15} - 100485688668636 q^{19} - 789079193287812 q^{21} + 13\!\cdots\!90 q^{25}+ \cdots + 60\!\cdots\!60 q^{99}+O(q^{100}) $ 6 * q - 86670 * q^3 + 3447063060 * q^7 + 57339715158 * q^9 + 2025132496860 * q^13 - 2628031314240 * q^15 - 100485688668636 * q^19 - 789079193287812 * q^21 + 1396569263732790 * q^25 + 10731657558901410 * q^27 - 114109662401153292 * q^31 - 174850022773609920 * q^33 + 683782874080661820 * q^37 + 1038045209126993748 * q^39 - 273273330169769340 * q^43 + 3228622736179443840 * q^45 - 8294807529216677070 * q^49 - 14237699192275302144 * q^51 + 40874587160137441920 * q^55 + 43721391850797114540 * q^57 + 121649250388145492892 * q^61 + 184494903535705683060 * q^63 - 75701571086175260700 * q^67 - 233446021721582921856 * q^69 - 57178481309533986900 * q^73 + 333973626119358339330 * q^75 - 384537395891327736396 * q^79 + 3109769787236091157926 * q^81 - 6772579853411844011520 * q^85 - 4995406256507248236480 * q^87 + 2806742501496183911304 * q^91 + 2087220280517252053020 * q^93 + 1565231688506989945740 * q^97 + 6095367699696122981760 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−169408.	+	51788.6i	−0.956312	+	0.292348i
$3$	−169408.	+	51788.6i	−0.956312	+	0.292348i
$4$		0			0
$5$			6.36810e7i			1.30419i	0.758139	+	0.652093i	$0.226109\pi$
$5$			6.36810e7i			1.30419i	−0.758139	+	0.652093i	$0.773891\pi$
$6$		0			0
$7$	2.60308e9			1.31646			0.658232	−	0.752816i	$-0.271305\pi$
$7$	2.60308e9			1.31646			0.658232	+	0.752816i	$0.271305\pi$
$8$		0			0
$9$	2.60170e10	−	1.75468e10i	0.829065	−	0.559152i
$10$		0			0
$11$		−	2.74891e10i		−	0.0963475i	−0.998839	−	0.0481737i	$-0.984660\pi$
$11$		−	2.74891e10i		−	0.0963475i	0.998839	−	0.0481737i	$-0.0153401\pi$
$12$		0			0
$13$	−6.09922e9			−0.00340328			−0.00170164	−	0.999999i	$-0.500542\pi$
$13$	−6.09922e9			−0.00340328			−0.00170164	+	0.999999i	$0.500542\pi$
$14$		0			0
$15$	−3.29794e12	−	1.07881e13i	−0.381276	−	1.24721i
$16$		0			0
$17$			8.12449e12i			0.237060i	0.992950	+	0.118530i	$0.0378182\pi$
$17$			8.12449e12i			0.237060i	−0.992950	+	0.118530i	$0.962182\pi$
$18$		0			0
$19$	−7.54682e13			−0.647850			−0.323925	−	0.946083i	$-0.605003\pi$
$19$	−7.54682e13			−0.647850			−0.323925	+	0.946083i	$0.605003\pi$
$20$		0			0
$21$	−4.40982e14	+	1.34810e14i	−1.25895	+	0.384865i
$22$		0			0
$23$		−	1.66353e15i		−	1.74592i	−0.487794	−	0.872959i	$-0.662198\pi$
$23$		−	1.66353e15i		−	1.74592i	0.487794	−	0.872959i	$-0.337802\pi$
$24$		0			0
$25$	−1.67108e15			−0.700901
$26$		0			0
$27$	−3.49875e15	+	4.31994e15i	−0.629378	+	0.777099i
$28$		0			0
$29$			1.26684e16i			1.03835i	0.854667	+	0.519176i	$0.173761\pi$
$29$			1.26684e16i			1.03835i	−0.854667	+	0.519176i	$0.826239\pi$
$30$		0			0
$31$	−1.69379e16			−0.666622			−0.333311	−	0.942817i	$-0.608166\pi$
$31$	−1.69379e16			−0.666622			−0.333311	+	0.942817i	$0.608166\pi$
$32$		0			0
$33$	1.42362e15	+	4.65686e15i	0.0281670	+	0.0921382i
$34$		0			0
$35$			1.65766e17i			1.71691i
$36$		0			0
$37$	1.59704e17			0.897628			0.448814	−	0.893625i	$-0.351847\pi$
$37$	1.59704e17			0.897628			0.448814	+	0.893625i	$0.351847\pi$
$38$		0			0
$39$	1.03325e15	−	3.15870e14i	0.00325459	−	0.000994940i
$40$		0			0
$41$			4.83588e17i			0.878726i	0.898310	+	0.439363i	$0.144796\pi$
$41$			4.83588e17i			0.878726i	−0.898310	+	0.439363i	$0.855204\pi$
$42$		0			0
$43$	−4.91968e17			−0.529400			−0.264700	−	0.964331i	$-0.585273\pi$
$43$	−4.91968e17			−0.529400			−0.264700	+	0.964331i	$0.585273\pi$
$44$		0			0
$45$	1.11740e18	+	1.65678e18i	0.729238	+	1.08126i
$46$		0			0
$47$		−	7.73166e17i		−	0.312749i	−0.987698	−	0.156375i	$-0.950019\pi$
$47$		−	7.73166e17i		−	0.312749i	0.987698	−	0.156375i	$-0.0499807\pi$
$48$		0			0
$49$	2.86619e18			0.733075
$50$		0			0
$51$	−4.20756e17	−	1.37635e18i	−0.0693039	−	0.226703i
$52$		0			0
$53$		−	5.90402e18i		−	0.636962i	−0.947929	−	0.318481i	$-0.896827\pi$
$53$		−	5.90402e18i		−	0.636962i	0.947929	−	0.318481i	$-0.103173\pi$
$54$		0			0
$55$	1.75053e18			0.125655
$56$		0			0
$57$	1.27849e19	−	3.90839e18i	0.619547	−	0.189398i
$58$		0			0
$59$			2.68658e19i			0.890897i	0.895308	+	0.445449i	$0.146956\pi$
$59$			2.68658e19i			0.890897i	−0.895308	+	0.445449i	$0.853044\pi$
$60$		0			0
$61$	2.43398e19			0.559357			0.279678	−	0.960094i	$-0.409772\pi$
$61$	2.43398e19			0.559357			0.279678	+	0.960094i	$0.409772\pi$
$62$		0			0
$63$	6.77241e19	−	4.56756e19i	1.09143	−	0.736102i
$64$		0			0
$65$		−	3.88404e17i		−	0.00443850i
$66$		0			0
$67$	1.30406e20			1.06776			0.533882	−	0.845559i	$-0.320733\pi$
$67$	1.30406e20			1.06776			0.533882	+	0.845559i	$0.320733\pi$
$68$		0			0
$69$	8.61517e19	+	2.81815e20i	0.510415	+	1.66964i
$70$		0			0
$71$			4.07363e20i			1.76254i	0.472609	+	0.881272i	$0.343312\pi$
$71$			4.07363e20i			1.76254i	−0.472609	+	0.881272i	$0.656688\pi$
$72$		0			0
$73$	−4.04873e20			−1.29053			−0.645263	−	0.763960i	$-0.723252\pi$
$73$	−4.04873e20			−1.29053			−0.645263	+	0.763960i	$0.723252\pi$
$74$		0			0
$75$	2.83094e20	−	8.65427e19i	0.670280	−	0.204907i
$76$		0			0
$77$		−	7.15562e19i		−	0.126838i
$78$		0			0
$79$	−1.11027e21			−1.48433			−0.742167	−	0.670215i	$-0.766202\pi$
$79$	−1.11027e21			−1.48433			−0.742167	+	0.670215i	$0.766202\pi$
$80$		0			0
$81$	3.68993e20	−	9.13027e20i	0.374699	−	0.927147i
$82$		0			0
$83$		−	1.31903e19i		−	0.0102422i	−0.999987	−	0.00512112i	$-0.998370\pi$
$83$		−	1.31903e19i		−	0.0102422i	0.999987	−	0.00512112i	$-0.00163011\pi$
$84$		0			0
$85$	−5.17375e20			−0.309170
$86$		0			0
$87$	−6.56080e20	−	2.14613e21i	−0.303560	−	0.992989i
$88$		0			0
$89$		−	2.62916e20i		−	0.0947384i	−0.998877	−	0.0473692i	$-0.984916\pi$
$89$		−	2.62916e20i		−	0.0947384i	0.998877	−	0.0473692i	$-0.0150837\pi$
$90$		0			0
$91$	−1.58767e19			−0.00448029
$92$		0			0
$93$	2.86941e21	−	8.77187e20i	0.637499	−	0.194886i
$94$		0			0
$95$		−	4.80589e21i		−	0.844917i
$96$		0			0
$97$	−1.92665e21			−0.269348			−0.134674	−	0.990890i	$-0.542999\pi$
$97$	−1.92665e21			−0.269348			−0.134674	+	0.990890i	$0.542999\pi$
$98$		0			0
$99$	−4.82344e20	−	7.15181e20i	−0.0538728	−	0.0798784i
$100$		0			0
$101$			1.07496e22i			0.963512i	0.876305	+	0.481756i	$0.160001\pi$
$101$			1.07496e22i			0.963512i	−0.876305	+	0.481756i	$0.839999\pi$
$102$		0			0
$103$	−1.33938e22			−0.967595			−0.483797	−	0.875180i	$-0.660743\pi$
$103$	−1.33938e22			−0.967595			−0.483797	+	0.875180i	$0.660743\pi$
$104$		0			0
$105$	−8.58481e21	−	2.80821e22i	−0.501936	−	1.64190i
$106$		0			0
$107$			2.24306e22i			1.06566i	0.846221	+	0.532832i	$0.178872\pi$
$107$			2.24306e22i			1.06566i	−0.846221	+	0.532832i	$0.821128\pi$
$108$		0			0
$109$	−2.63116e22			−1.01966			−0.509831	−	0.860275i	$-0.670292\pi$
$109$	−2.63116e22			−1.01966			−0.509831	+	0.860275i	$0.670292\pi$
$110$		0			0
$111$	−2.70551e22	+	8.27083e21i	−0.858412	+	0.262420i
$112$		0			0
$113$			4.24878e22i			1.10765i	0.832634	+	0.553823i	$0.186832\pi$
$113$			4.24878e22i			1.10765i	−0.832634	+	0.553823i	$0.813168\pi$
$114$		0			0
$115$	1.05935e23			2.27700
$116$		0			0
$117$	−1.58683e20	+	1.07022e20i	−0.00282154	+	0.00190295i
$118$		0			0
$119$			2.11487e22i			0.312081i
$120$		0			0
$121$	8.06471e22			0.990717
$122$		0			0
$123$	−2.50443e22	−	8.19237e22i	−0.256894	−	0.840336i
$124$		0			0
$125$			4.54114e22i			0.390081i
$126$		0			0
$127$	5.07164e22			0.365853			0.182927	−	0.983127i	$-0.441443\pi$
$127$	5.07164e22			0.365853			0.182927	+	0.983127i	$0.441443\pi$
$128$		0			0
$129$	8.33433e22	−	2.54783e22i	0.506272	−	0.154769i
$130$		0			0
$131$			8.53623e22i			0.437806i	0.975747	+	0.218903i	$0.0702478\pi$
$131$			8.53623e22i			0.437806i	−0.975747	+	0.218903i	$0.929752\pi$
$132$		0			0
$133$	−1.96450e23			−0.852870
$134$		0			0
$135$	−2.75098e23	−	2.22804e23i	−1.01348	−	0.820827i
$136$		0			0
$137$		−	5.86882e23i		−	1.83918i	−0.392878	−	0.919590i	$-0.628521\pi$
$137$		−	5.86882e23i		−	1.83918i	0.392878	−	0.919590i	$-0.371479\pi$
$138$		0			0
$139$	−5.05196e23			−1.34989			−0.674943	−	0.737870i	$-0.735832\pi$
$139$	−5.05196e23			−1.34989			−0.674943	+	0.737870i	$0.735832\pi$
$140$		0			0
$141$	4.00412e22	+	1.30980e23i	0.0914316	+	0.299086i
$142$		0			0
$143$			1.67662e20i			0.000327897i
$144$		0			0
$145$	−8.06738e23			−1.35420
$146$		0			0
$147$	−4.85555e23	+	1.48436e23i	−0.701048	+	0.214313i
$148$		0			0
$149$			1.20360e24i			1.49773i	0.662723	+	0.748865i	$0.269401\pi$
$149$			1.20360e24i			1.49773i	−0.662723	+	0.748865i	$0.730599\pi$
$150$		0			0
$151$	−9.34709e22			−0.100445			−0.0502227	−	0.998738i	$-0.515993\pi$
$151$	−9.34709e22			−0.100445			−0.0502227	+	0.998738i	$0.515993\pi$
$152$		0			0
$153$	1.42559e23	+	2.11374e23i	0.132552	+	0.196538i
$154$		0			0
$155$		−	1.07862e24i		−	0.869400i
$156$		0			0
$157$	−2.65459e24			−1.85824			−0.929118	−	0.369783i	$-0.879432\pi$
$157$	−2.65459e24			−1.85824			−0.929118	+	0.369783i	$0.879432\pi$
$158$		0			0
$159$	3.05761e23	+	1.00019e24i	0.186214	+	0.609134i
$160$		0			0
$161$		−	4.33029e24i		−	2.29844i
$162$		0			0
$163$	−1.54342e24			−0.715187			−0.357593	−	0.933877i	$-0.616403\pi$
$163$	−1.54342e24			−0.715187			−0.357593	+	0.933877i	$0.616403\pi$
$164$		0			0
$165$	−2.96553e23	+	9.06574e22i	−0.120165	+	0.0367350i
$166$		0			0
$167$		−	2.90075e24i		−	1.02951i	−0.857338	−	0.514754i	$-0.827883\pi$
$167$		−	2.90075e24i		−	1.02951i	0.857338	−	0.514754i	$-0.172117\pi$
$168$		0			0
$169$	−3.21180e24			−0.999988
$170$		0			0
$171$	−1.96345e24	+	1.32422e24i	−0.537110	+	0.362246i
$172$		0			0
$173$			5.31439e23i			0.127922i	0.997952	+	0.0639611i	$0.0203734\pi$
$173$			5.31439e23i			0.127922i	−0.997952	+	0.0639611i	$0.979627\pi$
$174$		0			0
$175$	−4.34995e24			−0.922710
$176$		0			0
$177$	−1.39134e24	−	4.55128e24i	−0.260452	−	0.851976i
$178$		0			0
$179$		−	3.21962e24i		−	0.532624i	−0.963887	−	0.266312i	$-0.914195\pi$
$179$		−	3.21962e24i		−	0.532624i	0.963887	−	0.266312i	$-0.0858052\pi$
$180$		0			0
$181$	2.29263e24			0.335637			0.167818	−	0.985818i	$-0.446328\pi$
$181$	2.29263e24			0.335637			0.167818	+	0.985818i	$0.446328\pi$
$182$		0			0
$183$	−4.12335e24	+	1.26052e24i	−0.534919	+	0.163527i
$184$		0			0
$185$			1.01701e25i			1.17067i
$186$		0			0
$187$	2.23335e23			0.0228401
$188$		0			0
$189$	−9.10753e24	+	1.12451e25i	−0.828553	+	1.02302i
$190$		0			0
$191$		−	1.89558e24i		−	0.153595i	−0.997047	−	0.0767973i	$-0.975531\pi$
$191$		−	1.89558e24i		−	0.153595i	0.997047	−	0.0767973i	$-0.0244694\pi$
$192$		0			0
$193$	5.59039e24			0.403935			0.201967	−	0.979392i	$-0.435266\pi$
$193$	5.59039e24			0.403935			0.201967	+	0.979392i	$0.435266\pi$
$194$		0			0
$195$	2.01149e22	+	6.57986e22i	0.00129759	+	0.00424459i
$196$		0			0
$197$		−	2.86286e25i		−	1.65072i	−0.564610	−	0.825358i	$-0.690973\pi$
$197$		−	2.86286e25i		−	1.65072i	0.564610	−	0.825358i	$-0.309027\pi$
$198$		0			0
$199$	−3.46992e25			−1.79034			−0.895171	−	0.445722i	$-0.852947\pi$
$199$	−3.46992e25			−1.79034			−0.895171	+	0.445722i	$0.852947\pi$
$200$		0			0
$201$	−2.20918e25	+	6.75355e24i	−1.02112	+	0.312159i
$202$		0			0
$203$			3.29769e25i			1.36695i
$204$		0			0
$205$	−3.07954e25			−1.14602
$206$		0			0
$207$	−2.91895e25	−	4.32799e25i	−0.976233	−	1.44748i
$208$		0			0
$209$			2.07455e24i			0.0624187i
$210$		0			0
$211$	3.16564e24			0.0857738			0.0428869	−	0.999080i	$-0.486344\pi$
$211$	3.16564e24			0.0857738			0.0428869	+	0.999080i	$0.486344\pi$
$212$		0			0
$213$	−2.10968e25	−	6.90105e25i	−0.515276	−	1.68554i
$214$		0			0
$215$		−	3.13290e25i		−	0.690436i
$216$		0			0
$217$	−4.40906e25			−0.877584
$218$		0			0
$219$	6.85886e25	−	2.09678e25i	1.23415	−	0.377283i
$220$		0			0
$221$		−	4.95530e22i		−	0.000806780i
$222$		0			0
$223$	−1.01616e26			−1.49833			−0.749165	−	0.662383i	$-0.769545\pi$
$223$	−1.01616e26			−1.49833			−0.749165	+	0.662383i	$0.769545\pi$
$224$		0			0
$225$	−4.34764e25	+	2.93220e25i	−0.581093	+	0.391910i
$226$		0			0
$227$		−	2.81242e25i		−	0.341032i	−0.985355	−	0.170516i	$-0.945457\pi$
$227$		−	2.81242e25i		−	0.341032i	0.985355	−	0.170516i	$-0.0545435\pi$
$228$		0			0
$229$	2.37376e25			0.261364			0.130682	−	0.991424i	$-0.458283\pi$
$229$	2.37376e25			0.261364			0.130682	+	0.991424i	$0.458283\pi$
$230$		0			0
$231$	3.70579e24	+	1.21222e25i	0.0370808	+	0.121297i
$232$		0			0
$233$			5.50447e25i			0.500957i	0.968122	+	0.250479i	$0.0805880\pi$
$233$			5.50447e25i			0.500957i	−0.968122	+	0.250479i	$0.919412\pi$
$234$		0			0
$235$	4.92360e25			0.407883
$236$		0			0
$237$	1.88089e26	−	5.74994e25i	1.41949	−	0.433942i
$238$		0			0
$239$			5.89602e25i			0.405679i	0.979212	+	0.202839i	$0.0650169\pi$
$239$			5.89602e25i			0.405679i	−0.979212	+	0.202839i	$0.934983\pi$
$240$		0			0
$241$	1.23102e26			0.772818			0.386409	−	0.922328i	$-0.373715\pi$
$241$	1.23102e26			0.772818			0.386409	+	0.922328i	$0.373715\pi$
$242$		0			0
$243$	−1.52259e25	+	1.73783e26i	−0.0872797	+	0.996184i
$244$		0			0
$245$			1.82522e26i			0.956066i
$246$		0			0
$247$	4.60297e23			0.00220481
$248$		0			0
$249$	6.83106e23	+	2.23454e24i	0.00299430	+	0.00979478i
$250$		0			0
$251$			2.15323e26i			0.864332i	0.901794	+	0.432166i	$0.142250\pi$
$251$			2.15323e26i			0.864332i	−0.901794	+	0.432166i	$0.857750\pi$
$252$		0			0
$253$	−4.57288e25			−0.168215
$254$		0			0
$255$	8.76474e25	−	2.67941e25i	0.295663	−	0.0903852i
$256$		0			0
$257$			6.83596e25i			0.211609i	0.994387	+	0.105805i	$0.0337418\pi$
$257$			6.83596e25i			0.211609i	−0.994387	+	0.105805i	$0.966258\pi$
$258$		0			0
$259$	4.15721e26			1.18169
$260$		0			0
$261$	2.22290e26	+	3.29594e26i	0.580596	+	0.860862i
$262$		0			0
$263$		−	6.08879e26i		−	1.46224i	−0.682250	−	0.731119i	$-0.738999\pi$
$263$		−	6.08879e26i		−	1.46224i	0.682250	−	0.731119i	$-0.261001\pi$
$264$		0			0
$265$	3.75974e26			0.830717
$266$		0			0
$267$	1.36160e25	+	4.45399e25i	0.0276966	+	0.0905995i
$268$		0			0
$269$		−	6.14042e26i		−	1.15060i	−0.817944	−	0.575298i	$-0.804886\pi$
$269$		−	6.14042e26i		−	1.15060i	0.817944	−	0.575298i	$-0.195114\pi$
$270$		0			0
$271$	8.66689e26			1.49693			0.748464	−	0.663175i	$-0.230792\pi$
$271$	8.66689e26			1.49693			0.748464	+	0.663175i	$0.230792\pi$
$272$		0			0
$273$	2.68964e24	−	8.22233e23i	0.00428455	−	0.00130980i
$274$		0			0
$275$			4.59364e25i			0.0675300i
$276$		0			0
$277$	−1.44952e26			−0.196764			−0.0983821	−	0.995149i	$-0.531367\pi$
$277$	−1.44952e26			−0.196764			−0.0983821	+	0.995149i	$0.531367\pi$
$278$		0			0
$279$	−4.40671e26	+	2.97205e26i	−0.552674	+	0.372743i
$280$		0			0
$281$		−	1.01210e27i		−	1.17343i	−0.809795	−	0.586713i	$-0.800422\pi$
$281$		−	1.01210e27i		−	1.17343i	0.809795	−	0.586713i	$-0.199578\pi$
$282$		0			0
$283$	1.22710e27			1.31592			0.657960	−	0.753053i	$-0.271419\pi$
$283$	1.22710e27			1.31592			0.657960	+	0.753053i	$0.271419\pi$
$284$		0			0
$285$	2.48890e26	+	8.14155e26i	0.247010	+	0.808004i
$286$		0			0
$287$			1.25882e27i			1.15681i
$288$		0			0
$289$	1.10856e27			0.943803
$290$		0			0
$291$	3.26390e26	−	9.97784e25i	0.257581	−	0.0787433i
$292$		0			0
$293$			1.23894e27i			0.906790i	0.891310	+	0.453395i	$0.149787\pi$
$293$			1.23894e27i			0.906790i	−0.891310	+	0.453395i	$0.850213\pi$
$294$		0			0
$295$	−1.71084e27			−1.16190
$296$		0			0
$297$	1.18751e26	+	9.61774e25i	0.0748715	+	0.0606390i
$298$		0			0
$299$			1.01462e25i			0.00594184i
$300$		0			0
$301$	−1.28063e27			−0.696936
$302$		0			0
$303$	−5.56706e26	−	1.82107e27i	−0.281681	−	0.921418i
$304$		0			0
$305$			1.54998e27i			0.729505i
$306$		0			0
$307$	1.07440e26			0.0470590			0.0235295	−	0.999723i	$-0.492510\pi$
$307$	1.07440e26			0.0470590			0.0235295	+	0.999723i	$0.492510\pi$
$308$		0			0
$309$	2.26901e27	−	6.93644e26i	0.925322	−	0.282874i
$310$		0			0
$311$		−	1.57800e27i		−	0.599436i	−0.954028	−	0.299718i	$-0.903107\pi$
$311$		−	1.57800e27i		−	0.599436i	0.954028	−	0.299718i	$-0.0968927\pi$
$312$		0			0
$313$	2.12037e27			0.750625			0.375313	−	0.926898i	$-0.377535\pi$
$313$	2.12037e27			0.750625			0.375313	+	0.926898i	$0.377535\pi$
$314$		0			0
$315$	2.90867e27	+	4.31274e27i	0.960014	+	1.42343i
$316$		0			0
$317$			2.48006e27i			0.763501i	0.924265	+	0.381750i	$0.124679\pi$
$317$			2.48006e27i			0.763501i	−0.924265	+	0.381750i	$0.875321\pi$
$318$		0			0
$319$	3.48243e26			0.100043
$320$		0			0
$321$	−1.16165e27	−	3.79992e27i	−0.311544	−	1.01911i
$322$		0			0
$323$		−	6.13141e26i		−	0.153579i
$324$		0			0
$325$	1.01923e25			0.00238536
$326$		0			0
$327$	4.45739e27	−	1.36264e27i	0.975114	−	0.298096i
$328$		0			0
$329$		−	2.01261e27i		−	0.411723i
$330$		0			0
$331$	−1.14762e27			−0.219630			−0.109815	−	0.993952i	$-0.535026\pi$
$331$	−1.14762e27			−0.219630			−0.109815	+	0.993952i	$0.535026\pi$
$332$		0			0
$333$	4.15500e27	−	2.80229e27i	0.744192	−	0.501910i
$334$		0			0
$335$			8.30439e27i			1.39256i
$336$		0			0
$337$	9.17701e26			0.144136			0.0720679	−	0.997400i	$-0.477040\pi$
$337$	9.17701e26			0.144136			0.0720679	+	0.997400i	$0.477040\pi$
$338$		0			0
$339$	−2.20038e27	−	7.19776e27i	−0.323818	−	1.05926i
$340$		0			0
$341$			4.65606e26i			0.0642274i
$342$		0			0
$343$	−2.71665e27			−0.351397
$344$		0			0
$345$	−1.79462e28	+	5.48622e27i	−2.17752	+	0.665677i
$346$		0			0
$347$		−	6.91877e27i		−	0.787779i	−0.919158	−	0.393890i	$-0.871129\pi$
$347$		−	6.91877e27i		−	0.787779i	0.919158	−	0.393890i	$-0.128871\pi$
$348$		0			0
$349$	−1.49774e28			−1.60087			−0.800437	−	0.599417i	$-0.795399\pi$
$349$	−1.49774e28			−1.60087			−0.800437	+	0.599417i	$0.795399\pi$
$350$		0			0
$351$	2.13396e25	−	2.63482e25i	0.00214195	−	0.00264468i
$352$		0			0
$353$			4.56597e27i			0.430538i	0.976555	+	0.215269i	$0.0690628\pi$
$353$			4.56597e27i			0.430538i	−0.976555	+	0.215269i	$0.930937\pi$
$354$		0			0
$355$	−2.59413e28			−2.29869
$356$		0			0
$357$	−1.09526e27	−	3.58275e27i	−0.0912361	−	0.298446i
$358$		0			0
$359$			8.52502e26i			0.0667816i	0.999442	+	0.0333908i	$0.0106306\pi$
$359$			8.52502e26i			0.0667816i	−0.999442	+	0.0333908i	$0.989369\pi$
$360$		0			0
$361$	−7.87453e27			−0.580291
$362$		0			0
$363$	−1.36622e28	+	4.17660e27i	−0.947435	+	0.289634i
$364$		0			0
$365$		−	2.57827e28i		−	1.68309i
$366$		0			0
$367$	4.97496e27			0.305818			0.152909	−	0.988240i	$-0.451136\pi$
$367$	4.97496e27			0.305818			0.152909	+	0.988240i	$0.451136\pi$
$368$		0			0
$369$	8.48542e27	+	1.25815e28i	0.491341	+	0.728521i
$370$		0			0
$371$		−	1.53686e28i		−	0.838537i
$372$		0			0
$373$	7.77252e27			0.399728			0.199864	−	0.979824i	$-0.435950\pi$
$373$	7.77252e27			0.399728			0.199864	+	0.979824i	$0.435950\pi$
$374$		0			0
$375$	−2.35179e27	−	7.69304e27i	−0.114039	−	0.373039i
$376$		0			0
$377$		−	7.72675e25i		−	0.00353380i
$378$		0			0
$379$	−1.81644e28			−0.783771			−0.391885	−	0.920014i	$-0.628177\pi$
$379$	−1.81644e28			−0.783771			−0.391885	+	0.920014i	$0.628177\pi$
$380$		0			0
$381$	−8.59175e27	+	2.62653e27i	−0.349870	+	0.106956i
$382$		0			0
$383$			2.52550e28i			0.970871i	0.874273	+	0.485435i	$0.161339\pi$
$383$			2.52550e28i			0.970871i	−0.874273	+	0.485435i	$0.838661\pi$
$384$		0			0
$385$	4.55676e27			0.165420
$386$		0			0
$387$	−1.27995e28	+	8.63245e27i	−0.438907	+	0.296015i
$388$		0			0
$389$		−	2.07224e28i		−	0.671419i	−0.941966	−	0.335709i	$-0.891024\pi$
$389$		−	2.07224e28i		−	0.671419i	0.941966	−	0.335709i	$-0.108976\pi$
$390$		0			0
$391$	1.35153e28			0.413887
$392$		0			0
$393$	−4.42079e27	−	1.44610e28i	−0.127992	−	0.418679i
$394$		0			0
$395$		−	7.07033e28i		−	1.93585i
$396$		0			0
$397$	−5.09414e28			−1.31940			−0.659698	−	0.751531i	$-0.729316\pi$
$397$	−5.09414e28			−1.31940			−0.659698	+	0.751531i	$0.729316\pi$
$398$		0			0
$399$	3.32801e28	−	1.01738e28i	0.815610	−	0.249335i
$400$		0			0
$401$			9.15869e27i			0.212444i	0.994342	+	0.106222i	$0.0338755\pi$
$401$			9.15869e27i			0.212444i	−0.994342	+	0.106222i	$0.966124\pi$
$402$		0			0
$403$	1.03308e26			0.00226870
$404$		0			0
$405$	5.81424e28	+	2.34978e28i	1.20917	+	0.488677i
$406$		0			0
$407$		−	4.39011e27i		−	0.0864841i
$408$		0			0
$409$	−8.43773e28			−1.57496			−0.787479	−	0.616341i	$-0.788614\pi$
$409$	−8.43773e28			−1.57496			−0.787479	+	0.616341i	$0.788614\pi$
$410$		0			0
$411$	3.03938e28	+	9.94224e28i	0.537681	+	1.75883i
$412$		0			0
$413$			6.99338e28i			1.17283i
$414$		0			0
$415$	8.39970e26			0.0133578
$416$		0			0
$417$	8.55842e28	−	2.61634e28i	1.29091	−	0.394637i
$418$		0			0
$419$			6.84490e28i			0.979519i	0.871857	+	0.489760i	$0.162915\pi$
$419$			6.84490e28i			0.979519i	−0.871857	+	0.489760i	$0.837085\pi$
$420$		0			0
$421$	5.64315e28			0.766334			0.383167	−	0.923679i	$-0.374833\pi$
$421$	5.64315e28			0.766334			0.383167	+	0.923679i	$0.374833\pi$
$422$		0			0
$423$	−1.35666e28	−	2.01154e28i	−0.174874	−	0.259290i
$424$		0			0
$425$		−	1.35767e28i		−	0.166155i
$426$		0			0
$427$	6.33584e28			0.736372
$428$		0			0
$429$	−8.68296e24	−	2.84032e25i	−9.58600e−5	−	0.000313572i
$430$		0			0
$431$			1.34048e29i			1.40608i	0.711151	+	0.703040i	$0.248174\pi$
$431$			1.34048e29i			1.40608i	−0.711151	+	0.703040i	$0.751826\pi$
$432$		0			0
$433$	5.41506e28			0.539804			0.269902	−	0.962888i	$-0.413009\pi$
$433$	5.41506e28			0.539804			0.269902	+	0.962888i	$0.413009\pi$
$434$		0			0
$435$	1.36668e29	−	4.17798e28i	1.29504	−	0.395899i
$436$		0			0
$437$			1.25543e29i			1.13109i
$438$		0			0
$439$	−9.15985e28			−0.784836			−0.392418	−	0.919787i	$-0.628361\pi$
$439$	−9.15985e28			−0.784836			−0.392418	+	0.919787i	$0.628361\pi$
$440$		0			0
$441$	7.45696e28	−	5.02924e28i	0.607767	−	0.409900i
$442$		0			0
$443$		−	1.52046e29i		−	1.17905i	−0.807749	−	0.589526i	$-0.799315\pi$
$443$		−	1.52046e29i		−	1.17905i	0.807749	−	0.589526i	$-0.200685\pi$
$444$		0			0
$445$	1.67427e28			0.123557
$446$		0			0
$447$	−6.23327e28	−	2.03899e29i	−0.437858	−	1.43230i
$448$		0			0
$449$		−	2.11279e29i		−	1.41301i	−0.707707	−	0.706506i	$-0.750270\pi$
$449$		−	2.11279e29i		−	1.41301i	0.707707	−	0.706506i	$-0.249730\pi$
$450$		0			0
$451$	1.32934e28			0.0846630
$452$		0			0
$453$	1.58347e28	−	4.84072e27i	0.0960571	−	0.0293650i
$454$		0			0
$455$		−	1.01105e27i		−	0.00584313i
$456$		0			0
$457$	−1.25653e29			−0.691982			−0.345991	−	0.938238i	$-0.612457\pi$
$457$	−1.25653e29			−0.691982			−0.345991	+	0.938238i	$0.612457\pi$
$458$		0			0
$459$	−3.50973e28	−	2.84256e28i	−0.184219	−	0.149200i
$460$		0			0
$461$			8.00496e28i			0.400543i	0.979740	+	0.200271i	$0.0641824\pi$
$461$			8.00496e28i			0.400543i	−0.979740	+	0.200271i	$0.935818\pi$
$462$		0			0
$463$	−1.02477e29			−0.488918			−0.244459	−	0.969660i	$-0.578610\pi$
$463$	−1.02477e29			−0.488918			−0.244459	+	0.969660i	$0.578610\pi$
$464$		0			0
$465$	5.58601e28	+	1.82726e29i	0.254167	+	0.831417i
$466$		0			0
$467$			2.34893e29i			1.01949i	0.860325	+	0.509746i	$0.170261\pi$
$467$			2.34893e29i			1.01949i	−0.860325	+	0.509746i	$0.829739\pi$
$468$		0			0
$469$	3.39457e29			1.40567
$470$		0			0
$471$	4.49708e29	−	1.37477e29i	1.77705	−	0.543251i
$472$		0			0
$473$			1.35237e28i			0.0510064i
$474$		0			0
$475$	1.26113e29			0.454079
$476$		0			0
$477$	−1.03597e29	−	1.53605e29i	−0.356158	−	0.528083i
$478$		0			0
$479$			2.40723e29i			0.790365i	0.918603	+	0.395182i	$0.129319\pi$
$479$			2.40723e29i			0.790365i	−0.918603	+	0.395182i	$0.870681\pi$
$480$		0			0
$481$	−9.74068e26			−0.00305487
$482$		0			0
$483$	2.24259e29	+	7.33585e29i	0.671943	+	2.19802i
$484$		0			0
$485$		−	1.22691e29i		−	0.351280i
$486$		0			0
$487$	1.92933e29			0.527943			0.263971	−	0.964531i	$-0.414968\pi$
$487$	1.92933e29			0.527943			0.263971	+	0.964531i	$0.414968\pi$
$488$		0			0
$489$	2.61467e29	−	7.99313e28i	0.683942	−	0.209083i
$490$		0			0
$491$		−	2.56167e29i		−	0.640658i	−0.947306	−	0.320329i	$-0.896206\pi$
$491$		−	2.56167e29i		−	0.640658i	0.947306	−	0.320329i	$-0.103794\pi$
$492$		0			0
$493$	−1.02925e29			−0.246152
$494$		0			0
$495$	4.55434e28	−	3.07161e28i	0.104176	−	0.0702602i
$496$		0			0
$497$			1.06040e30i			2.32032i
$498$		0			0
$499$	−3.69485e29			−0.773554			−0.386777	−	0.922173i	$-0.626412\pi$
$499$	−3.69485e29			−0.773554			−0.386777	+	0.922173i	$0.626412\pi$
$500$		0			0
$501$	1.50226e29	+	4.91410e29i	0.300975	+	0.984531i
$502$		0			0
$503$		−	1.47551e29i		−	0.282940i	−0.989943	−	0.141470i	$-0.954817\pi$
$503$		−	1.47551e29i		−	0.282940i	0.989943	−	0.141470i	$-0.0451829\pi$
$504$		0			0
$505$	−6.84545e29			−1.25660
$506$		0			0
$507$	5.44104e29	−	1.66335e29i	0.956301	−	0.292345i
$508$		0			0
$509$		−	9.75884e29i		−	1.64249i	−0.570576	−	0.821245i	$-0.693280\pi$
$509$		−	9.75884e29i		−	1.64249i	0.570576	−	0.821245i	$-0.306720\pi$
$510$		0			0
$511$	−1.05392e30			−1.69893
$512$		0			0
$513$	2.64045e29	−	3.26018e29i	0.407743	−	0.503443i
$514$		0			0
$515$		−	8.52928e29i		−	1.26192i
$516$		0			0
$517$	−2.12536e28			−0.0301326
$518$		0			0
$519$	−2.75225e28	−	9.00299e28i	−0.0373978	−	0.122334i
$520$		0			0
$521$		−	5.11220e29i		−	0.665876i	−0.942949	−	0.332938i	$-0.891960\pi$
$521$		−	5.11220e29i		−	0.665876i	0.942949	−	0.332938i	$-0.108040\pi$
$522$		0			0
$523$	−7.39983e29			−0.924066			−0.462033	−	0.886863i	$-0.652880\pi$
$523$	−7.39983e29			−0.924066			−0.462033	+	0.886863i	$0.652880\pi$
$524$		0			0
$525$	7.36915e29	−	2.25277e29i	0.882399	−	0.269752i
$526$		0			0
$527$		−	1.37611e29i		−	0.158029i
$528$		0			0
$529$	−1.85948e30			−2.04823
$530$		0			0
$531$	4.71408e29	+	6.98966e29i	0.498147	+	0.738612i
$532$		0			0
$533$		−	2.94951e27i		−	0.00299055i
$534$		0			0
$535$	−1.42840e30			−1.38982
$536$		0			0
$537$	1.66739e29	+	5.45428e29i	0.155712	+	0.509355i
$538$		0			0
$539$		−	7.87889e28i		−	0.0706299i
$540$		0			0
$541$	1.14061e28			0.00981676			0.00490838	−	0.999988i	$-0.498438\pi$
$541$	1.14061e28			0.00981676			0.00490838	+	0.999988i	$0.498438\pi$
$542$		0			0
$543$	−3.88389e29	+	1.18732e29i	−0.320974	+	0.0981227i
$544$		0			0
$545$		−	1.67555e30i		−	1.32983i
$546$		0			0
$547$	−8.07862e29			−0.615852			−0.307926	−	0.951410i	$-0.599635\pi$
$547$	−8.07862e29			−0.615852			−0.307926	+	0.951410i	$0.599635\pi$
$548$		0			0
$549$	6.33247e29	−	4.27085e29i	0.463743	−	0.312765i
$550$		0			0
$551$		−	9.56063e29i		−	0.672696i
$552$		0			0
$553$	−2.89013e30			−1.95407
$554$		0			0
$555$	−5.26694e29	−	1.72289e30i	−0.342244	−	1.11953i
$556$		0			0
$557$			2.05598e30i			1.28414i	0.766647	+	0.642069i	$0.221924\pi$
$557$			2.05598e30i			1.28414i	−0.766647	+	0.642069i	$0.778076\pi$
$558$		0			0
$559$	3.00062e27			0.00180169
$560$		0			0
$561$	−3.78346e28	+	1.15662e28i	−0.0218423	+	0.00667726i
$562$		0			0
$563$			3.18768e30i			1.76963i	0.465944	+	0.884814i	$0.345715\pi$
$563$			3.18768e30i			1.76963i	−0.465944	+	0.884814i	$0.654285\pi$
$564$		0			0
$565$	−2.70566e30			−1.44458
$566$		0			0
$567$	9.60516e29	−	2.37668e30i	0.493277	−	1.22055i
$568$		0			0
$569$		−	2.11635e28i		−	0.0104557i	−0.999986	−	0.00522784i	$-0.998336\pi$
$569$		−	2.11635e28i		−	0.0104557i	0.999986	−	0.00522784i	$-0.00166408\pi$
$570$		0			0
$571$	2.03671e29			0.0968124			0.0484062	−	0.998828i	$-0.484586\pi$
$571$	2.03671e29			0.0968124			0.0484062	+	0.998828i	$0.484586\pi$
$572$		0			0
$573$	9.81696e28	+	3.21127e29i	0.0449031	+	0.146884i
$574$		0			0
$575$			2.77988e30i			1.22372i
$576$		0			0
$577$	−7.10098e29			−0.300874			−0.150437	−	0.988620i	$-0.548068\pi$
$577$	−7.10098e29			−0.300874			−0.150437	+	0.988620i	$0.548068\pi$
$578$		0			0
$579$	−9.47055e29	+	2.89518e29i	−0.386288	+	0.118090i
$580$		0			0
$581$		−	3.43353e28i		−	0.0134835i
$582$		0			0
$583$	−1.62296e29			−0.0613697
$584$		0			0
$585$	−6.81523e27	−	1.01051e28i	−0.00248180	−	0.00367981i
$586$		0			0
$587$		−	4.23524e30i		−	1.48545i	−0.669594	−	0.742727i	$-0.733532\pi$
$587$		−	4.23524e30i		−	1.48545i	0.669594	−	0.742727i	$-0.266468\pi$
$588$		0			0
$589$	1.27827e30			0.431871
$590$		0			0
$591$	1.48264e30	+	4.84992e30i	0.482583	+	1.57860i
$592$		0			0
$593$		−	1.89823e30i		−	0.595316i	−0.954673	−	0.297658i	$-0.903795\pi$
$593$		−	1.89823e30i		−	0.595316i	0.954673	−	0.297658i	$-0.0962055\pi$
$594$		0			0
$595$	−1.34677e30			−0.407011
$596$		0			0
$597$	5.87832e30	−	1.79702e30i	1.71213	−	0.523403i
$598$		0			0
$599$		−	1.20877e29i		−	0.0339352i	−0.999856	−	0.0169676i	$-0.994599\pi$
$599$		−	1.20877e29i		−	0.0339352i	0.999856	−	0.0169676i	$-0.00540121\pi$
$600$		0			0
$601$	4.57436e30			1.23797			0.618987	−	0.785401i	$-0.287543\pi$
$601$	4.57436e30			1.23797			0.618987	+	0.785401i	$0.287543\pi$
$602$		0			0
$603$	3.39277e30	−	2.28821e30i	0.885246	−	0.597042i
$604$		0			0
$605$			5.13568e30i			1.29208i
$606$		0			0
$607$	1.54217e30			0.374159			0.187080	−	0.982345i	$-0.440098\pi$
$607$	1.54217e30			0.374159			0.187080	+	0.982345i	$0.440098\pi$
$608$		0			0
$609$	−1.70783e30	−	5.58655e30i	−0.399626	−	1.30723i
$610$		0			0
$611$			4.71571e27i			0.00106437i
$612$		0			0
$613$	1.07192e30			0.233398			0.116699	−	0.993167i	$-0.462769\pi$
$613$	1.07192e30			0.233398			0.116699	+	0.993167i	$0.462769\pi$
$614$		0			0
$615$	5.21698e30	−	1.59485e30i	1.09595	−	0.335037i
$616$		0			0
$617$			2.87077e30i			0.581918i	0.956736	+	0.290959i	$0.0939743\pi$
$617$			2.87077e30i			0.581918i	−0.956736	+	0.290959i	$0.906026\pi$
$618$		0			0
$619$	−7.00164e30			−1.36963			−0.684814	−	0.728718i	$-0.740116\pi$
$619$	−7.00164e30			−1.36963			−0.684814	+	0.728718i	$0.740116\pi$
$620$		0			0
$621$	7.18634e30	+	5.82027e30i	1.35675	+	1.09884i
$622$		0			0
$623$		−	6.84390e29i		−	0.124720i
$624$		0			0
$625$	−6.87600e30			−1.20964
$626$		0			0
$627$	−1.07438e29	−	3.51445e29i	−0.0182480	−	0.0596917i
$628$		0			0
$629$			1.29751e30i			0.212791i
$630$		0			0
$631$	6.42277e30			1.01718			0.508592	−	0.861008i	$-0.330166\pi$
$631$	6.42277e30			1.01718			0.508592	+	0.861008i	$0.330166\pi$
$632$		0			0
$633$	−5.36284e29	+	1.63944e29i	−0.0820265	+	0.0250758i
$634$		0			0
$635$			3.22967e30i			0.477141i
$636$		0			0
$637$	−1.74815e28			−0.00249486
$638$		0			0
$639$	7.14791e30	+	1.05984e31i	0.985530	+	1.46126i
$640$		0			0
$641$		−	5.36151e29i		−	0.0714248i	−0.999362	−	0.0357124i	$-0.988630\pi$
$641$		−	5.36151e29i		−	0.0714248i	0.999362	−	0.0357124i	$-0.0113700\pi$
$642$		0			0
$643$	−1.20075e31			−1.54573			−0.772864	−	0.634572i	$-0.781177\pi$
$643$	−1.20075e31			−1.54573			−0.772864	+	0.634572i	$0.781177\pi$
$644$		0			0
$645$	1.62248e30	+	5.30738e30i	0.201848	+	0.660273i
$646$		0			0
$647$		−	5.19437e30i		−	0.624577i	−0.949987	−	0.312288i	$-0.898904\pi$
$647$		−	5.19437e30i		−	0.624577i	0.949987	−	0.312288i	$-0.101096\pi$
$648$		0			0
$649$	7.38515e29			0.0858357
$650$		0			0
$651$	7.46929e30	−	2.28339e30i	0.839244	−	0.256560i
$652$		0			0
$653$		−	1.52058e31i		−	1.65183i	−0.563794	−	0.825916i	$-0.690659\pi$
$653$		−	1.52058e31i		−	1.65183i	0.563794	−	0.825916i	$-0.309341\pi$
$654$		0			0
$655$	−5.43595e30			−0.570980
$656$		0			0
$657$	−1.05336e31	+	7.10421e30i	−1.06993	+	0.721600i
$658$		0			0
$659$			5.32045e30i			0.522648i	0.965251	+	0.261324i	$0.0841591\pi$
$659$			5.32045e30i			0.522648i	−0.965251	+	0.261324i	$0.915841\pi$
$660$		0			0
$661$	8.18313e30			0.777505			0.388753	−	0.921342i	$-0.372906\pi$
$661$	8.18313e30			0.777505			0.388753	+	0.921342i	$0.372906\pi$
$662$		0			0
$663$	2.56628e27	+	8.39467e27i	0.000235860	+	0.000771533i
$664$		0			0
$665$		−	1.25101e31i		−	1.11230i
$666$		0			0
$667$	2.10743e31			1.81288
$668$		0			0
$669$	1.72145e31	−	5.26253e30i	1.43287	−	0.438034i
$670$		0			0
$671$		−	6.69078e29i		−	0.0538926i
$672$		0			0
$673$	−2.13511e31			−1.66439			−0.832194	−	0.554484i	$-0.812916\pi$
$673$	−2.13511e31			−1.66439			−0.832194	+	0.554484i	$0.812916\pi$
$674$		0			0
$675$	5.84669e30	−	7.21896e30i	0.441132	−	0.544669i
$676$		0			0
$677$			1.73132e31i			1.26445i	0.774783	+	0.632227i	$0.217859\pi$
$677$			1.73132e31i			1.26445i	−0.774783	+	0.632227i	$0.782141\pi$
$678$		0			0
$679$	−5.01522e30			−0.354587
$680$		0			0
$681$	1.45651e30	+	4.76445e30i	0.0997000	+	0.326133i
$682$		0			0
$683$			7.74068e30i			0.513040i	0.966539	+	0.256520i	$0.0825760\pi$
$683$			7.74068e30i			0.513040i	−0.966539	+	0.256520i	$0.917424\pi$
$684$		0			0
$685$	3.73732e31			2.39863
$686$		0			0
$687$	−4.02133e30	+	1.22934e30i	−0.249946	+	0.0764093i
$688$		0			0
$689$			3.60099e28i			0.00216776i
$690$		0			0
$691$	1.61533e31			0.941897			0.470948	−	0.882161i	$-0.343912\pi$
$691$	1.61533e31			0.941897			0.470948	+	0.882161i	$0.343912\pi$
$692$		0			0
$693$	−1.25558e30	−	1.86167e30i	−0.0709216	−	0.105157i
$694$		0			0
$695$		−	3.21714e31i		−	1.76050i
$696$		0			0
$697$	−3.92891e30			−0.208311
$698$		0			0
$699$	−2.85069e30	−	9.32501e30i	−0.146454	−	0.479071i
$700$		0			0
$701$			4.58185e30i			0.228109i	0.993475	+	0.114054i	$0.0363838\pi$
$701$			4.58185e30i			0.228109i	−0.993475	+	0.114054i	$0.963616\pi$
$702$		0			0
$703$	−1.20526e31			−0.581528
$704$		0			0
$705$	−8.34095e30	+	2.54986e30i	−0.390064	+	0.119244i
$706$		0			0
$707$			2.79820e31i			1.26843i
$708$		0			0
$709$	1.19148e31			0.523573			0.261787	−	0.965126i	$-0.415688\pi$
$709$	1.19148e31			0.523573			0.261787	+	0.965126i	$0.415688\pi$
$710$		0			0
$711$	−2.88859e31	+	1.94817e31i	−1.23061	+	0.829968i
$712$		0			0
$713$			2.81766e31i			1.16387i
$714$		0			0
$715$	−1.06769e28			−0.000427639
$716$		0			0
$717$	−3.05347e30	−	9.98832e30i	−0.118599	−	0.387955i
$718$		0			0
$719$			2.53712e31i			0.955702i	0.878441	+	0.477851i	$0.158584\pi$
$719$			2.53712e31i			0.955702i	−0.878441	+	0.477851i	$0.841416\pi$
$720$		0			0
$721$	−3.48650e31			−1.27380
$722$		0			0
$723$	−2.08544e31	+	6.37527e30i	−0.739055	+	0.225932i
$724$		0			0
$725$		−	2.11699e31i		−	0.727782i
$726$		0			0
$727$	−1.92745e31			−0.642844			−0.321422	−	0.946936i	$-0.604161\pi$
$727$	−1.92745e31			−0.642844			−0.321422	+	0.946936i	$0.604161\pi$
$728$		0			0
$729$	−6.42061e30	−	3.02288e31i	−0.207766	−	0.978179i
$730$		0			0
$731$		−	3.99699e30i		−	0.125500i
$732$		0			0
$733$	−2.69272e31			−0.820441			−0.410221	−	0.911986i	$-0.634548\pi$
$733$	−2.69272e31			−0.820441			−0.410221	+	0.911986i	$0.634548\pi$
$734$		0			0
$735$	−9.45254e30	−	3.09206e31i	−0.279504	−	0.914298i
$736$		0			0
$737$		−	3.58474e30i		−	0.102876i
$738$		0			0
$739$	2.50535e31			0.697879			0.348939	−	0.937145i	$-0.386542\pi$
$739$	2.50535e31			0.697879			0.348939	+	0.937145i	$0.386542\pi$
$740$		0			0
$741$	−7.79779e28	+	2.38381e28i	−0.00210849	+	0.000644572i
$742$		0			0
$743$		−	4.91616e31i		−	1.29047i	−0.763982	−	0.645237i	$-0.776759\pi$
$743$		−	4.91616e31i		−	1.29047i	0.763982	−	0.645237i	$-0.223241\pi$
$744$		0			0
$745$	−7.66464e31			−1.95332
$746$		0			0
$747$	−2.31447e29	−	3.43171e29i	−0.00572697	−	0.00849149i
$748$		0			0
$749$			5.83887e31i			1.40291i
$750$		0			0
$751$	3.14313e30			0.0733370			0.0366685	−	0.999327i	$-0.488325\pi$
$751$	3.14313e30			0.0733370			0.0366685	+	0.999327i	$0.488325\pi$
$752$		0			0
$753$	−1.11513e31	−	3.64775e31i	−0.252686	−	0.826571i
$754$		0			0
$755$		−	5.95231e30i		−	0.130999i
$756$		0			0
$757$	7.54271e31			1.61240			0.806200	−	0.591643i	$-0.201520\pi$
$757$	7.54271e31			1.61240			0.806200	+	0.591643i	$0.201520\pi$
$758$		0			0
$759$	7.74682e30	−	2.36823e30i	0.160866	−	0.0491772i
$760$		0			0
$761$		−	6.42228e31i		−	1.29556i	−0.761827	−	0.647781i	$-0.775697\pi$
$761$		−	6.42228e31i		−	1.29556i	0.761827	−	0.647781i	$-0.224303\pi$
$762$		0			0
$763$	−6.84912e31			−1.34235
$764$		0			0
$765$	−1.34605e31	+	9.07826e30i	−0.256322	+	0.172873i
$766$		0			0
$767$		−	1.63860e29i		−	0.00303197i
$768$		0			0
$769$	−5.92659e31			−1.06565			−0.532825	−	0.846225i	$-0.678870\pi$
$769$	−5.92659e31			−1.06565			−0.532825	+	0.846225i	$0.678870\pi$
$770$		0			0
$771$	−3.54024e30	−	1.15806e31i	−0.0618636	−	0.202365i
$772$		0			0
$773$			1.03663e31i			0.176055i	0.996118	+	0.0880273i	$0.0280563\pi$
$773$			1.03663e31i			0.176055i	−0.996118	+	0.0880273i	$0.971944\pi$
$774$		0			0
$775$	2.83045e31			0.467236
$776$		0			0
$777$	−7.04264e31	+	2.15296e31i	−1.13007	+	0.345466i
$778$		0			0
$779$		−	3.64955e31i		−	0.569283i
$780$		0			0
$781$	1.11980e31			0.169817
$782$		0			0
$783$	−5.47269e31	−	4.43237e31i	−0.806903	−	0.653517i
$784$		0			0
$785$		−	1.69047e32i		−	2.42349i
$786$		0			0
$787$	−2.60427e31			−0.363048			−0.181524	−	0.983387i	$-0.558103\pi$
$787$	−2.60427e31			−0.363048			−0.181524	+	0.983387i	$0.558103\pi$
$788$		0			0
$789$	3.15330e31	+	1.03149e32i	0.427482	+	1.39836i
$790$		0			0
$791$			1.10599e32i			1.45818i
$792$		0			0
$793$	−1.48454e29			−0.00190364
$794$		0			0
$795$	−6.36929e31	+	1.94711e31i	−0.794424	+	0.242858i
$796$		0			0
$797$		−	2.42229e31i		−	0.293889i	−0.989145	−	0.146945i	$-0.953056\pi$
$797$		−	2.42229e31i		−	0.293889i	0.989145	−	0.146945i	$-0.0469439\pi$
$798$		0			0
$799$	6.28158e30			0.0741403
$800$		0			0
$801$	−4.61332e30	−	6.84026e30i	−0.0529731	−	0.0785444i
$802$		0			0
$803$			1.11296e31i			0.124339i
$804$		0			0
$805$	2.75757e32			2.99759
$806$		0			0
$807$	3.18003e31	+	1.04023e32i	0.336374	+	1.10033i
$808$		0			0
$809$			1.57502e32i			1.62126i	0.585561	+	0.810628i	$0.300874\pi$
$809$			1.57502e32i			1.62126i	−0.585561	+	0.810628i	$0.699126\pi$
$810$		0			0
$811$	1.02490e32			1.02672			0.513361	−	0.858173i	$-0.328400\pi$
$811$	1.02490e32			1.02672			0.513361	+	0.858173i	$0.328400\pi$
$812$		0			0
$813$	−1.46824e32	+	4.48846e31i	−1.43153	+	0.437624i
$814$		0			0
$815$		−	9.82862e31i		−	0.932736i
$816$		0			0
$817$	3.71280e31			0.342972
$818$		0			0
$819$	−4.13064e29	+	2.78585e29i	−0.00371445	+	0.00250516i
$820$		0			0
$821$			1.81814e32i			1.59167i	0.605512	+	0.795836i	$0.292968\pi$
$821$			1.81814e32i			1.59167i	−0.605512	+	0.795836i	$0.707032\pi$
$822$		0			0
$823$	9.73130e31			0.829417			0.414708	−	0.909954i	$-0.363884\pi$
$823$	9.73130e31			0.829417			0.414708	+	0.909954i	$0.363884\pi$
$824$		0			0
$825$	−2.37898e30	−	7.78198e30i	−0.0197423	−	0.0645798i
$826$		0			0
$827$		−	1.29618e31i		−	0.104738i	−0.998628	−	0.0523689i	$-0.983323\pi$
$827$		−	1.29618e31i		−	0.104738i	0.998628	−	0.0523689i	$-0.0166772\pi$
$828$		0			0
$829$	1.84340e32			1.45050			0.725252	−	0.688484i	$-0.241723\pi$
$829$	1.84340e32			1.45050			0.725252	+	0.688484i	$0.241723\pi$
$830$		0			0
$831$	2.45560e31	−	7.50684e30i	0.188168	−	0.0575236i
$832$		0			0
$833$			2.32863e31i			0.173783i
$834$		0			0
$835$	1.84723e32			1.34267
$836$		0			0
$837$	5.92614e31	−	7.31705e31i	0.419558	−	0.518032i
$838$		0			0
$839$			1.79761e32i			1.23969i	0.784724	+	0.619845i	$0.212805\pi$
$839$			1.79761e32i			1.23969i	−0.784724	+	0.619845i	$0.787195\pi$
$840$		0			0
$841$	−1.16367e31			−0.0781758
$842$		0			0
$843$	5.24154e31	+	1.71458e32i	0.343049	+	1.12216i
$844$		0			0
$845$		−	2.04531e32i		−	1.30417i
$846$		0			0
$847$	2.09931e32			1.30424
$848$		0			0
$849$	−2.07881e32	+	6.35498e31i	−1.25843	+	0.384707i
$850$		0			0
$851$		−	2.65672e32i		−	1.56718i
$852$		0			0
$853$	1.78802e32			1.02786			0.513930	−	0.857832i	$-0.328189\pi$
$853$	1.78802e32			1.02786			0.513930	+	0.857832i	$0.328189\pi$
$854$		0			0
$855$	−8.43278e31	−	1.25035e32i	−0.472436	−	0.700491i
$856$		0			0
$857$		−	1.90019e32i		−	1.03755i	−0.854912	−	0.518773i	$-0.826389\pi$
$857$		−	1.90019e32i		−	1.03755i	0.854912	−	0.518773i	$-0.173611\pi$
$858$		0			0
$859$	−1.06484e32			−0.566711			−0.283355	−	0.959015i	$-0.591448\pi$
$859$	−1.06484e32			−0.566711			−0.283355	+	0.959015i	$0.591448\pi$
$860$		0			0
$861$	−6.51924e31	−	2.13254e32i	−0.338191	−	1.10627i
$862$		0			0
$863$		−	6.46783e31i		−	0.327070i	−0.986538	−	0.163535i	$-0.947710\pi$
$863$		−	6.46783e31i		−	0.327070i	0.986538	−	0.163535i	$-0.0522896\pi$
$864$		0			0
$865$	−3.38425e31			−0.166834
$866$		0			0
$867$	−1.87798e32	+	5.74105e31i	−0.902570	+	0.275919i
$868$		0			0
$869$			3.05204e31i			0.143012i
$870$		0			0
$871$	−7.95375e29			−0.00363389
$872$		0			0
$873$	−5.01256e31	+	3.38065e31i	−0.223307	+	0.150606i
$874$		0			0
$875$			1.18209e32i			0.513527i
$876$		0			0
$877$	−8.00076e31			−0.338950			−0.169475	−	0.985534i	$-0.554207\pi$
$877$	−8.00076e31			−0.338950			−0.169475	+	0.985534i	$0.554207\pi$
$878$		0			0
$879$	−6.41629e31	−	2.09886e32i	−0.265098	−	0.867174i
$880$		0			0
$881$			4.43743e32i			1.78812i	0.447948	+	0.894060i	$0.352155\pi$
$881$			4.43743e32i			1.78812i	−0.447948	+	0.894060i	$0.647845\pi$
$882$		0			0
$883$	1.57479e32			0.618949			0.309474	−	0.950908i	$-0.399847\pi$
$883$	1.57479e32			0.618949			0.309474	+	0.950908i	$0.399847\pi$
$884$		0			0
$885$	2.89830e32	−	8.86019e31i	1.11113	−	0.339678i
$886$		0			0
$887$			3.61043e32i			1.35020i	0.737725	+	0.675102i	$0.235900\pi$
$887$			3.61043e32i			1.35020i	−0.737725	+	0.675102i	$0.764100\pi$
$888$		0			0
$889$	1.32019e32			0.481633
$890$		0			0
$891$	−2.50983e31	−	1.01433e31i	−0.0893282	−	0.0361013i
$892$		0			0
$893$			5.83494e31i			0.202615i
$894$		0			0
$895$	2.05028e32			0.694641
$896$		0			0
$897$	−5.25458e29	−	1.71885e30i	−0.00173708	−	0.00568225i
$898$		0			0
$899$		−	2.14576e32i		−	0.692189i
$900$		0			0
$901$	4.79672e31			0.150998
$902$		0			0
$903$	2.16949e32	−	6.63221e31i	0.666488	−	0.203748i
$904$		0			0
$905$			1.45997e32i			0.437733i
$906$		0			0
$907$	−9.61787e31			−0.281448			−0.140724	−	0.990049i	$-0.544943\pi$
$907$	−9.61787e31			−0.281448			−0.140724	+	0.990049i	$0.544943\pi$
$908$		0			0
$909$	1.88621e32	+	2.79672e32i	0.538749	+	0.798814i
$910$		0			0
$911$			4.61504e32i			1.28669i	0.765577	+	0.643344i	$0.222454\pi$
$911$			4.61504e32i			1.28669i	−0.765577	+	0.643344i	$0.777546\pi$
$912$		0			0
$913$	−3.62589e29			−0.000986815
$914$		0			0
$915$	−8.02713e31	−	2.62579e32i	−0.213269	−	0.697634i
$916$		0			0
$917$			2.22205e32i			0.576355i
$918$		0			0
$919$	−1.82792e32			−0.462899			−0.231450	−	0.972847i	$-0.574347\pi$
$919$	−1.82792e32			−0.462899			−0.231450	+	0.972847i	$0.574347\pi$
$920$		0			0
$921$	−1.82011e31	+	5.56415e30i	−0.0450031	+	0.0137576i
$922$		0			0
$923$		−	2.48460e30i		−	0.00599842i
$924$		0			0
$925$	−2.66877e32			−0.629148
$926$		0			0
$927$	−3.48465e32	+	2.35017e32i	−0.802199	+	0.541032i
$928$		0			0
$929$		−	7.69269e32i		−	1.72944i	−0.502255	−	0.864719i	$-0.667496\pi$
$929$		−	7.69269e32i		−	1.72944i	0.502255	−	0.864719i	$-0.332504\pi$
$930$		0			0
$931$	−2.16306e32			−0.474923
$932$		0			0
$933$	8.17224e31	+	2.67326e32i	0.175244	+	0.573248i
$934$		0			0
$935$			1.42222e31i			0.0297878i
$936$		0			0
$937$	5.05414e31			0.103398			0.0516989	−	0.998663i	$-0.483536\pi$
$937$	5.05414e31			0.103398			0.0516989	+	0.998663i	$0.483536\pi$
$938$		0			0
$939$	−3.59207e32	+	1.09811e32i	−0.717832	+	0.219444i
$940$		0			0
$941$		−	6.49725e32i		−	1.26836i	−0.773184	−	0.634182i	$-0.781337\pi$
$941$		−	6.49725e32i		−	1.26836i	0.773184	−	0.634182i	$-0.218663\pi$
$942$		0			0
$943$	8.04463e32			1.53418
$944$		0			0
$945$	−7.16101e32	−	5.79976e32i	−1.33421	−	1.08059i
$946$		0			0
$947$		−	4.73368e31i		−	0.0861685i	−0.999071	−	0.0430843i	$-0.986282\pi$
$947$		−	4.73368e31i		−	0.0861685i	0.999071	−	0.0430843i	$-0.0137184\pi$
$948$		0			0
$949$	2.46941e30			0.00439202
$950$		0			0
$951$	−1.28438e32	−	4.20141e32i	−0.223208	−	0.730145i
$952$		0			0
$953$			2.95640e32i			0.502044i	0.967981	+	0.251022i	$0.0807666\pi$
$953$			2.95640e32i			0.502044i	−0.967981	+	0.251022i	$0.919233\pi$
$954$		0			0
$955$	1.20713e32			0.200316
$956$		0			0
$957$	−5.89951e31	+	1.80350e31i	−0.0956720	+	0.0292473i
$958$		0			0
$959$		−	1.52770e33i		−	2.42121i
$960$		0			0
$961$	−3.58700e32			−0.555615
$962$		0			0
$963$	3.93585e32	+	5.83577e32i	0.595867	+	0.883505i
$964$		0			0
$965$			3.56001e32i			0.526806i
$966$		0			0
$967$	−7.29083e32			−1.05459			−0.527297	−	0.849681i	$-0.676795\pi$
$967$	−7.29083e32			−1.05459			−0.527297	+	0.849681i	$0.676795\pi$
$968$		0			0
$969$	3.17537e31	+	1.03871e32i	0.0448985	+	0.146870i
$970$		0			0
$971$			8.36746e32i			1.15660i	0.815825	+	0.578299i	$0.196283\pi$
$971$			8.36746e32i			1.15660i	−0.815825	+	0.578299i	$0.803717\pi$
$972$		0			0
$973$	−1.31507e33			−1.77708
$974$		0			0
$975$	−1.72665e30	+	5.27843e29i	−0.00228115	+	0.000697355i
$976$		0			0
$977$		−	4.51728e32i		−	0.583494i	−0.956495	−	0.291747i	$-0.905763\pi$
$977$		−	4.51728e32i		−	0.583494i	0.956495	−	0.291747i	$-0.0942366\pi$
$978$		0			0
$979$	−7.22730e30			−0.00912781
$980$		0			0
$981$	−6.84548e32	+	4.61684e32i	−0.845366	+	0.570145i
$982$		0			0
$983$			1.89765e32i			0.229154i	0.993414	+	0.114577i	$0.0365513\pi$
$983$			1.89765e32i			0.229154i	−0.993414	+	0.114577i	$0.963449\pi$
$984$		0			0
$985$	1.82310e33			2.15284
$986$		0			0
$987$	1.04230e32	+	3.40952e32i	0.120366	+	0.393736i
$988$		0			0
$989$			8.18403e32i			0.924289i
$990$		0			0
$991$	−1.41092e33			−1.55844			−0.779222	−	0.626748i	$-0.784385\pi$
$991$	−1.41092e33			−1.55844			−0.779222	+	0.626748i	$0.784385\pi$
$992$		0			0
$993$	1.94416e32	−	5.94337e31i	0.210035	−	0.0642083i
$994$		0			0
$995$		−	2.20968e33i		−	2.33494i
$996$		0			0
$997$	−5.89392e32			−0.609197			−0.304598	−	0.952481i	$-0.598522\pi$
$997$	−5.89392e32			−0.609197			−0.304598	+	0.952481i	$0.598522\pi$
$998$		0			0
$999$	−5.58764e32	+	6.89911e32i	−0.564947	+	0.697545i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.23.e.b.17.2		6
3.2	odd	2	inner	48.23.e.b.17.1		6
4.3	odd	2		3.23.b.a.2.6	yes	6
12.11	even	2		3.23.b.a.2.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.23.b.a.2.1	✓	6	12.11	even	2
3.23.b.a.2.6	yes	6	4.3	odd	2
48.23.e.b.17.1		6	3.2	odd	2	inner
48.23.e.b.17.2		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 23 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-169408. + 51788.6i) q^{3} +6.36810e7i q^{5} +2.60308e9 q^{7} +(2.60170e10 - 1.75468e10i) q^{9} +O(q^{10})\)	\(q+(-169408. + 51788.6i) q^{3} +6.36810e7i q^{5} +2.60308e9 q^{7} +(2.60170e10 - 1.75468e10i) q^{9} -2.74891e10i q^{11} -6.09922e9 q^{13} +(-3.29794e12 - 1.07881e13i) q^{15} +8.12449e12i q^{17} -7.54682e13 q^{19} +(-4.40982e14 + 1.34810e14i) q^{21} -1.66353e15i q^{23} -1.67108e15 q^{25} +(-3.49875e15 + 4.31994e15i) q^{27} +1.26684e16i q^{29} -1.69379e16 q^{31} +(1.42362e15 + 4.65686e15i) q^{33} +1.65766e17i q^{35} +1.59704e17 q^{37} +(1.03325e15 - 3.15870e14i) q^{39} +4.83588e17i q^{41} -4.91968e17 q^{43} +(1.11740e18 + 1.65678e18i) q^{45} -7.73166e17i q^{47} +2.86619e18 q^{49} +(-4.20756e17 - 1.37635e18i) q^{51} -5.90402e18i q^{53} +1.75053e18 q^{55} +(1.27849e19 - 3.90839e18i) q^{57} +2.68658e19i q^{59} +2.43398e19 q^{61} +(6.77241e19 - 4.56756e19i) q^{63} -3.88404e17i q^{65} +1.30406e20 q^{67} +(8.61517e19 + 2.81815e20i) q^{69} +4.07363e20i q^{71} -4.04873e20 q^{73} +(2.83094e20 - 8.65427e19i) q^{75} -7.15562e19i q^{77} -1.11027e21 q^{79} +(3.68993e20 - 9.13027e20i) q^{81} -1.31903e19i q^{83} -5.17375e20 q^{85} +(-6.56080e20 - 2.14613e21i) q^{87} -2.62916e20i q^{89} -1.58767e19 q^{91} +(2.86941e21 - 8.77187e20i) q^{93} -4.80589e21i q^{95} -1.92665e21 q^{97} +(-4.82344e20 - 7.15181e20i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 86670 q^{3} + 3447063060 q^{7} + 57339715158 q^{9}+O(q^{10}) \) 6 * q - 86670 * q^3 + 3447063060 * q^7 + 57339715158 * q^9	\( 6 q - 86670 q^{3} + 3447063060 q^{7} + 57339715158 q^{9} + 2025132496860 q^{13} - 2628031314240 q^{15} - 100485688668636 q^{19} - 789079193287812 q^{21} + 13\!\cdots\!90 q^{25}+ \cdots + 60\!\cdots\!60 q^{99}+O(q^{100}) \) 6 * q - 86670 * q^3 + 3447063060 * q^7 + 57339715158 * q^9 + 2025132496860 * q^13 - 2628031314240 * q^15 - 100485688668636 * q^19 - 789079193287812 * q^21 + 1396569263732790 * q^25 + 10731657558901410 * q^27 - 114109662401153292 * q^31 - 174850022773609920 * q^33 + 683782874080661820 * q^37 + 1038045209126993748 * q^39 - 273273330169769340 * q^43 + 3228622736179443840 * q^45 - 8294807529216677070 * q^49 - 14237699192275302144 * q^51 + 40874587160137441920 * q^55 + 43721391850797114540 * q^57 + 121649250388145492892 * q^61 + 184494903535705683060 * q^63 - 75701571086175260700 * q^67 - 233446021721582921856 * q^69 - 57178481309533986900 * q^73 + 333973626119358339330 * q^75 - 384537395891327736396 * q^79 + 3109769787236091157926 * q^81 - 6772579853411844011520 * q^85 - 4995406256507248236480 * q^87 + 2806742501496183911304 * q^91 + 2087220280517252053020 * q^93 + 1565231688506989945740 * q^97 + 6095367699696122981760 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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