LMFDB - Embedded newform 1386.2.e.e.307.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(307,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1386.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.e (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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.6679465984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 61x^{4} + 88x^{2} + 36 $ x^8 + 14x^6 + 61x^4 + 88*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 462)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			307.5
Root			$-0.829319i$ of defining polynomial
Character	$\chi$	$=$	1386.307
Dual form			1386.2.e.e.307.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -4.14155i q^{5} +(0.717333 - 2.54665i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} -4.14155i q^{5} +(0.717333 - 2.54665i) q^{7} -1.00000i q^{8} +4.14155 q^{10} +(0.170681 - 3.31223i) q^{11} +3.09330 q^{13} +(2.54665 + 0.717333i) q^{14} +1.00000 q^{16} -3.57621 q^{17} +3.91758 q^{19} +4.14155i q^{20} +(3.31223 + 0.170681i) q^{22} -7.71776 q^{23} -12.1524 q^{25} +3.09330i q^{26} +(-0.717333 + 2.54665i) q^{28} +1.65864i q^{29} +9.23485i q^{31} +1.00000i q^{32} -3.57621i q^{34} +(-10.5471 - 2.97087i) q^{35} -0.869330 q^{37} +3.91758i q^{38} -4.14155 q^{40} +5.91758 q^{41} -10.2831i q^{43} +(-0.170681 + 3.31223i) q^{44} -7.71776i q^{46} -6.76601i q^{47} +(-5.97087 - 3.65359i) q^{49} -12.1524i q^{50} -3.09330 q^{52} +1.88261 q^{53} +(-13.7178 - 0.706884i) q^{55} +(-2.54665 - 0.717333i) q^{56} -1.65864 q^{58} +10.1866i q^{59} -3.37640 q^{61} -9.23485 q^{62} -1.00000 q^{64} -12.8111i q^{65} +9.84524 q^{67} +3.57621 q^{68} +(2.97087 - 10.5471i) q^{70} -1.33817 q^{71} -5.57621 q^{73} -0.869330i q^{74} -3.91758 q^{76} +(-8.31266 - 2.81064i) q^{77} +10.8484i q^{79} -4.14155i q^{80} +5.91758i q^{82} +5.67271 q^{83} +14.8111i q^{85} +10.2831 q^{86} +(-3.31223 - 0.170681i) q^{88} -10.6245i q^{89} +(2.21893 - 7.87756i) q^{91} +7.71776 q^{92} +6.76601 q^{94} -16.2248i q^{95} +4.52797i q^{97} +(3.65359 - 5.97087i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 4 q^{10} + 8 q^{11} + 8 q^{14} + 8 q^{16} + 12 q^{17} + 4 q^{19} + 4 q^{22} + 8 q^{23} - 16 q^{25} - 8 q^{35} + 16 q^{37} - 4 q^{40} + 20 q^{41} - 8 q^{44} - 12 q^{49} - 40 q^{55} - 8 q^{56} + 56 q^{61} - 20 q^{62} - 8 q^{64} + 16 q^{67} - 12 q^{68} - 12 q^{70} - 8 q^{71} - 4 q^{73} - 4 q^{76} + 20 q^{77} - 4 q^{83} + 24 q^{86} - 4 q^{88} + 20 q^{91} - 8 q^{92} - 20 q^{94} + 20 q^{98}+O(q^{100}) $ 8 * q - 8 * q^4 + 4 * q^10 + 8 * q^11 + 8 * q^14 + 8 * q^16 + 12 * q^17 + 4 * q^19 + 4 * q^22 + 8 * q^23 - 16 * q^25 - 8 * q^35 + 16 * q^37 - 4 * q^40 + 20 * q^41 - 8 * q^44 - 12 * q^49 - 40 * q^55 - 8 * q^56 + 56 * q^61 - 20 * q^62 - 8 * q^64 + 16 * q^67 - 12 * q^68 - 12 * q^70 - 8 * q^71 - 4 * q^73 - 4 * q^76 + 20 * q^77 - 4 * q^83 + 24 * q^86 - 4 * q^88 + 20 * q^91 - 8 * q^92 - 20 * q^94 + 20 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	4.14155i		−	1.85216i	−0.377331	−	0.926079i	$-0.623158\pi$
$5$		−	4.14155i		−	1.85216i	0.377331	−	0.926079i	$-0.376842\pi$
$6$		0			0
$7$	0.717333	−	2.54665i	0.271126	−	0.962544i
$7$	0.717333	−	2.54665i	0.271126	−	0.962544i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	4.14155			1.30967
$11$	0.170681	−	3.31223i	0.0514623	−	0.998675i
$11$	0.170681	−	3.31223i	0.0514623	−	0.998675i
$12$		0			0
$13$	3.09330			0.857928			0.428964	−	0.903322i	$-0.358879\pi$
$13$	3.09330			0.857928			0.428964	+	0.903322i	$0.358879\pi$
$14$	2.54665	+	0.717333i	0.680621	+	0.191715i
$15$		0			0
$16$	1.00000			0.250000
$17$	−3.57621			−0.867359			−0.433680	−	0.901067i	$-0.642785\pi$
$17$	−3.57621			−0.867359			−0.433680	+	0.901067i	$0.642785\pi$
$18$		0			0
$19$	3.91758			0.898754			0.449377	−	0.893342i	$-0.351646\pi$
$19$	3.91758			0.898754			0.449377	+	0.893342i	$0.351646\pi$
$20$			4.14155i			0.926079i
$21$		0			0
$22$	3.31223	+	0.170681i	0.706170	+	0.0363893i
$23$	−7.71776			−1.60926			−0.804632	−	0.593773i	$-0.797638\pi$
$23$	−7.71776			−1.60926			−0.804632	+	0.593773i	$0.797638\pi$
$24$		0			0
$25$	−12.1524			−2.43049
$26$			3.09330i			0.606647i
$27$		0			0
$28$	−0.717333	+	2.54665i	−0.135563	+	0.481272i
$29$			1.65864i			0.308001i	0.988071	+	0.154001i	$0.0492158\pi$
$29$			1.65864i			0.308001i	−0.988071	+	0.154001i	$0.950784\pi$
$30$		0			0
$31$			9.23485i			1.65863i	0.558783	+	0.829314i	$0.311269\pi$
$31$			9.23485i			1.65863i	−0.558783	+	0.829314i	$0.688731\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	3.57621i		−	0.613316i
$35$	−10.5471	−	2.97087i	−1.78278	−	0.502168i
$36$		0			0
$37$	−0.869330			−0.142917			−0.0714585	−	0.997444i	$-0.522765\pi$
$37$	−0.869330			−0.142917			−0.0714585	+	0.997444i	$0.522765\pi$
$38$			3.91758i			0.635515i
$39$		0			0
$40$	−4.14155			−0.654836
$41$	5.91758			0.924170			0.462085	−	0.886836i	$-0.347101\pi$
$41$	5.91758			0.924170			0.462085	+	0.886836i	$0.347101\pi$
$42$		0			0
$43$		−	10.2831i		−	1.56816i	−0.620661	−	0.784079i	$-0.713136\pi$
$43$		−	10.2831i		−	1.56816i	0.620661	−	0.784079i	$-0.286864\pi$
$44$	−0.170681	+	3.31223i	−0.0257311	+	0.499337i
$45$		0			0
$46$		−	7.71776i		−	1.13792i
$47$		−	6.76601i		−	0.986924i	−0.869767	−	0.493462i	$-0.835731\pi$
$47$		−	6.76601i		−	0.986924i	0.869767	−	0.493462i	$-0.164269\pi$
$48$		0			0
$49$	−5.97087	−	3.65359i	−0.852981	−	0.521942i
$50$		−	12.1524i		−	1.71861i
$51$		0			0
$52$	−3.09330			−0.428964
$53$	1.88261			0.258596			0.129298	−	0.991606i	$-0.458728\pi$
$53$	1.88261			0.258596			0.129298	+	0.991606i	$0.458728\pi$
$54$		0			0
$55$	−13.7178	−	0.706884i	−1.84970	−	0.0953162i
$56$	−2.54665	−	0.717333i	−0.340311	−	0.0958576i
$57$		0			0
$58$	−1.65864			−0.217790
$59$			10.1866i			1.32618i	0.748538	+	0.663092i	$0.230756\pi$
$59$			10.1866i			1.32618i	−0.748538	+	0.663092i	$0.769244\pi$
$60$		0			0
$61$	−3.37640			−0.432304			−0.216152	−	0.976360i	$-0.569351\pi$
$61$	−3.37640			−0.432304			−0.216152	+	0.976360i	$0.569351\pi$
$62$	−9.23485			−1.17283
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	12.8111i		−	1.58902i
$66$		0			0
$67$	9.84524			1.20279			0.601394	−	0.798953i	$-0.294612\pi$
$67$	9.84524			1.20279			0.601394	+	0.798953i	$0.294612\pi$
$68$	3.57621			0.433680
$69$		0			0
$70$	2.97087	−	10.5471i	0.355087	−	1.26062i
$71$	−1.33817			−0.158812			−0.0794060	−	0.996842i	$-0.525302\pi$
$71$	−1.33817			−0.158812			−0.0794060	+	0.996842i	$0.525302\pi$
$72$		0			0
$73$	−5.57621			−0.652647			−0.326323	−	0.945258i	$-0.605810\pi$
$73$	−5.57621			−0.652647			−0.326323	+	0.945258i	$0.605810\pi$
$74$		−	0.869330i		−	0.101058i
$75$		0			0
$76$	−3.91758			−0.449377
$77$	−8.31266	−	2.81064i	−0.947316	−	0.320302i
$78$		0			0
$79$			10.8484i			1.22054i	0.792192	+	0.610272i	$0.208940\pi$
$79$			10.8484i			1.22054i	−0.792192	+	0.610272i	$0.791060\pi$
$80$		−	4.14155i		−	0.463039i
$81$		0			0
$82$			5.91758i			0.653487i
$83$	5.67271			0.622660			0.311330	−	0.950302i	$-0.399225\pi$
$83$	5.67271			0.622660			0.311330	+	0.950302i	$0.399225\pi$
$84$		0			0
$85$			14.8111i			1.60649i
$86$	10.2831			1.10885
$87$		0			0
$88$	−3.31223	−	0.170681i	−0.353085	−	0.0181947i
$89$		−	10.6245i		−	1.12619i	−0.826392	−	0.563095i	$-0.809610\pi$
$89$		−	10.6245i		−	1.12619i	0.826392	−	0.563095i	$-0.190390\pi$
$90$		0			0
$91$	2.21893	−	7.87756i	0.232607	−	0.825793i
$92$	7.71776			0.804632
$93$		0			0
$94$	6.76601			0.697861
$95$		−	16.2248i		−	1.66463i
$96$		0			0
$97$			4.52797i			0.459746i	0.973221	+	0.229873i	$0.0738310\pi$
$97$			4.52797i			0.459746i	−0.973221	+	0.229873i	$0.926169\pi$
$98$	3.65359	−	5.97087i	0.369069	−	0.603149i
$99$		0			0
$100$	12.1524			1.21524
$101$	−1.37554			−0.136871			−0.0684357	−	0.997656i	$-0.521801\pi$
$101$	−1.37554			−0.136871			−0.0684357	+	0.997656i	$0.521801\pi$
$102$		0			0
$103$			0.0140682i			0.00138618i	1.00000		0.000693089i	$0.000220617\pi$
$103$			0.0140682i			0.00138618i	−1.00000		0.000693089i	$0.999779\pi$
$104$		−	3.09330i		−	0.303323i
$105$		0			0
$106$			1.88261i			0.182855i
$107$			1.37554i			0.132978i	0.997787	+	0.0664892i	$0.0211798\pi$
$107$			1.37554i			0.132978i	−0.997787	+	0.0664892i	$0.978820\pi$
$108$		0			0
$109$		−	13.1315i		−	1.25777i	−0.777497	−	0.628886i	$-0.783511\pi$
$109$		−	13.1315i		−	1.25777i	0.777497	−	0.628886i	$-0.216489\pi$
$110$	0.706884	−	13.7178i	0.0673987	−	1.30794i
$111$		0			0
$112$	0.717333	−	2.54665i	0.0677816	−	0.240636i
$113$	7.30718			0.687402			0.343701	−	0.939079i	$-0.388319\pi$
$113$	7.30718			0.687402			0.343701	+	0.939079i	$0.388319\pi$
$114$		0			0
$115$			31.9635i			2.98061i
$116$		−	1.65864i		−	0.154001i
$117$		0			0
$118$	−10.1866			−0.937753
$119$	−2.56533	+	9.10737i	−0.235164	+	0.834871i
$120$		0			0
$121$	−10.9417	−	1.13067i	−0.994703	−	0.102788i
$122$		−	3.37640i		−	0.305685i
$123$		0			0
$124$		−	9.23485i		−	0.829314i
$125$			29.6221i			2.64948i
$126$		0			0
$127$		−	19.4564i		−	1.72648i	−0.504795	−	0.863239i	$-0.668432\pi$
$127$		−	19.4564i		−	1.72648i	0.504795	−	0.863239i	$-0.331568\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	12.8111			1.12360
$131$	14.2108			1.24160			0.620800	−	0.783969i	$-0.286808\pi$
$131$	14.2108			1.24160			0.620800	+	0.783969i	$0.286808\pi$
$132$		0			0
$133$	2.81021	−	9.97670i	0.243676	−	0.865090i
$134$			9.84524i			0.850500i
$135$		0			0
$136$			3.57621i			0.306658i
$137$	−13.4938			−1.15285			−0.576426	−	0.817149i	$-0.695553\pi$
$137$	−13.4938			−1.15285			−0.576426	+	0.817149i	$0.695553\pi$
$138$		0			0
$139$	14.5521			1.23430			0.617148	−	0.786847i	$-0.288288\pi$
$139$	14.5521			1.23430			0.617148	+	0.786847i	$0.288288\pi$
$140$	10.5471	+	2.97087i	0.891391	+	0.251084i
$141$		0			0
$142$		−	1.33817i		−	0.112297i
$143$	0.527968	−	10.2457i	0.0441509	−	0.856791i
$144$		0			0
$145$	6.86933			0.570467
$146$		−	5.57621i		−	0.461491i
$147$		0			0
$148$	0.869330			0.0714585
$149$			7.24892i			0.593855i	0.954900	+	0.296927i	$0.0959619\pi$
$149$			7.24892i			0.593855i	−0.954900	+	0.296927i	$0.904038\pi$
$150$		0			0
$151$		−	10.3422i		−	0.841638i	−0.907145	−	0.420819i	$-0.861743\pi$
$151$		−	10.3422i		−	0.841638i	0.907145	−	0.420819i	$-0.138257\pi$
$152$		−	3.91758i		−	0.317757i
$153$		0			0
$154$	2.81064	−	8.31266i	0.226487	−	0.669853i
$155$	38.2466			3.07204
$156$		0			0
$157$		−	13.0801i		−	1.04391i	−0.852974	−	0.521953i	$-0.825204\pi$
$157$		−	13.0801i		−	1.04391i	0.852974	−	0.521953i	$-0.174796\pi$
$158$	−10.8484			−0.863055
$159$		0			0
$160$	4.14155			0.327418
$161$	−5.53620	+	19.6545i	−0.436314	+	1.54899i
$162$		0			0
$163$	−3.75513			−0.294124			−0.147062	−	0.989127i	$-0.546982\pi$
$163$	−3.75513			−0.294124			−0.147062	+	0.989127i	$0.546982\pi$
$164$	−5.91758			−0.462085
$165$		0			0
$166$			5.67271i			0.440287i
$167$	16.9658			1.31285			0.656427	−	0.754389i	$-0.272067\pi$
$167$	16.9658			1.31285			0.656427	+	0.754389i	$0.272067\pi$
$168$		0			0
$169$	−3.43148			−0.263960
$170$	−14.8111			−1.13596
$171$		0			0
$172$			10.2831i			0.784079i
$173$	−3.75513			−0.285497			−0.142749	−	0.989759i	$-0.545594\pi$
$173$	−3.75513			−0.285497			−0.142749	+	0.989759i	$0.545594\pi$
$174$		0			0
$175$	−8.71733	+	30.9480i	−0.658968	+	2.33945i
$176$	0.170681	−	3.31223i	0.0128656	−	0.249669i
$177$		0			0
$178$	10.6245			0.796337
$179$	−18.6627			−1.39491			−0.697457	−	0.716626i	$-0.745685\pi$
$179$	−18.6627			−1.39491			−0.697457	+	0.716626i	$0.745685\pi$
$180$		0			0
$181$		−	14.9216i		−	1.10912i	−0.832145	−	0.554558i	$-0.812887\pi$
$181$		−	14.9216i		−	1.10912i	0.832145	−	0.554558i	$-0.187113\pi$
$182$	7.87756	+	2.21893i	0.583924	+	0.164478i
$183$		0			0
$184$			7.71776i			0.568961i
$185$			3.60037i			0.264705i
$186$		0			0
$187$	−0.610392	+	11.8452i	−0.0446363	+	0.866210i
$188$			6.76601i			0.493462i
$189$		0			0
$190$	16.2248			1.17707
$191$	10.5871			0.766055			0.383028	−	0.923737i	$-0.374881\pi$
$191$	10.5871			0.766055			0.383028	+	0.923737i	$0.374881\pi$
$192$		0			0
$193$			11.4355i			0.823147i	0.911377	+	0.411574i	$0.135021\pi$
$193$			11.4355i			0.823147i	−0.911377	+	0.411574i	$0.864979\pi$
$194$	−4.52797			−0.325089
$195$		0			0
$196$	5.97087	+	3.65359i	0.426491	+	0.260971i
$197$		−	13.2107i		−	0.941223i	−0.882341	−	0.470611i	$-0.844033\pi$
$197$		−	13.2107i		−	0.941223i	0.882341	−	0.470611i	$-0.155967\pi$
$198$		0			0
$199$		−	14.0077i		−	0.992979i	−0.868043	−	0.496489i	$-0.834622\pi$
$199$		−	14.0077i		−	0.992979i	0.868043	−	0.496489i	$-0.165378\pi$
$200$			12.1524i			0.859306i
$201$		0			0
$202$		−	1.37554i		−	0.0967826i
$203$	4.22397	+	1.18979i	0.296465	+	0.0835072i
$204$		0			0
$205$		−	24.5079i		−	1.71171i
$206$	−0.0140682			−0.000980176
$207$		0			0
$208$	3.09330			0.214482
$209$	0.668656	−	12.9759i	0.0462519	−	0.897563i
$210$		0			0
$211$			7.33903i			0.505240i	0.967566	+	0.252620i	$0.0812922\pi$
$211$			7.33903i			0.505240i	−0.967566	+	0.252620i	$0.918708\pi$
$212$	−1.88261			−0.129298
$213$		0			0
$214$	−1.37554			−0.0940300
$215$	−42.5880			−2.90447
$216$		0			0
$217$	23.5180	+	6.62446i	1.59650	+	0.449698i
$218$	13.1315			0.889380
$219$		0			0
$220$	13.7178	+	0.706884i	0.924851	+	0.0476581i
$221$	−11.0623			−0.744132
$222$		0			0
$223$		−	7.42146i		−	0.496978i	−0.968635	−	0.248489i	$-0.920066\pi$
$223$		−	7.42146i		−	0.496978i	0.968635	−	0.248489i	$-0.0799339\pi$
$224$	2.54665	+	0.717333i	0.170155	+	0.0479288i
$225$		0			0
$226$			7.30718i			0.486067i
$227$	8.46201			0.561644			0.280822	−	0.959760i	$-0.409393\pi$
$227$	8.46201			0.561644			0.280822	+	0.959760i	$0.409393\pi$
$228$		0			0
$229$		−	1.03178i		−	0.0681817i	−0.999419	−	0.0340909i	$-0.989146\pi$
$229$		−	1.03178i		−	0.0681817i	0.999419	−	0.0340909i	$-0.0108536\pi$
$230$	−31.9635			−2.10761
$231$		0			0
$232$	1.65864			0.108895
$233$		−	0.944064i		−	0.0618477i	−0.999522	−	0.0309238i	$-0.990155\pi$
$233$		−	0.944064i		−	0.0618477i	0.999522	−	0.0309238i	$-0.00984493\pi$
$234$		0			0
$235$	−28.0218			−1.82794
$236$		−	10.1866i		−	0.663092i
$237$		0			0
$238$	−9.10737	−	2.56533i	−0.590343	−	0.166286i
$239$			20.9394i			1.35446i	0.735772	+	0.677229i	$0.236819\pi$
$239$			20.9394i			1.35446i	−0.735772	+	0.677229i	$0.763181\pi$
$240$		0			0
$241$	−17.8593			−1.15042			−0.575210	−	0.818006i	$-0.695080\pi$
$241$	−17.8593			−1.15042			−0.575210	+	0.818006i	$0.695080\pi$
$242$	1.13067	−	10.9417i	0.0726822	−	0.703361i
$243$		0			0
$244$	3.37640			0.216152
$245$	−15.1315	+	24.7286i	−0.966718	+	1.57986i
$246$		0			0
$247$	12.1183			0.771066
$248$	9.23485			0.586414
$249$		0			0
$250$	−29.6221			−1.87347
$251$			8.96582i			0.565918i	0.959132	+	0.282959i	$0.0913160\pi$
$251$			8.96582i			0.565918i	−0.959132	+	0.282959i	$0.908684\pi$
$252$		0			0
$253$	−1.31728	+	25.5630i	−0.0828164	+	1.60713i
$254$	19.4564			1.22080
$255$		0			0
$256$	1.00000			0.0625000
$257$			8.52797i			0.531960i	0.963979	+	0.265980i	$0.0856955\pi$
$257$			8.52797i			0.531960i	−0.963979	+	0.265980i	$0.914304\pi$
$258$		0			0
$259$	−0.623599	+	2.21388i	−0.0387486	+	0.137564i
$260$			12.8111i			0.794509i
$261$		0			0
$262$			14.2108i			0.877944i
$263$			31.2871i			1.92925i	0.263629	+	0.964624i	$0.415080\pi$
$263$			31.2871i			1.92925i	−0.263629	+	0.964624i	$0.584920\pi$
$264$		0			0
$265$		−	7.79692i		−	0.478961i
$266$	9.97670	+	2.81021i	0.611711	+	0.172305i
$267$		0			0
$268$	−9.84524			−0.601394
$269$		−	7.57535i		−	0.461877i	−0.972968	−	0.230939i	$-0.925820\pi$
$269$		−	7.57535i		−	0.461877i	0.972968	−	0.230939i	$-0.0741797\pi$
$270$		0			0
$271$	14.9667			0.909161			0.454581	−	0.890706i	$-0.349789\pi$
$271$	14.9667			0.909161			0.454581	+	0.890706i	$0.349789\pi$
$272$	−3.57621			−0.216840
$273$		0			0
$274$		−	13.4938i		−	0.815190i
$275$	−2.07419	+	40.2516i	−0.125078	+	2.42727i
$276$		0			0
$277$		−	29.2498i		−	1.75745i	−0.477329	−	0.878725i	$-0.658395\pi$
$277$		−	29.2498i		−	1.75745i	0.477329	−	0.878725i	$-0.341605\pi$
$278$			14.5521i			0.872779i
$279$		0			0
$280$	−2.97087	+	10.5471i	−0.177543	+	0.630309i
$281$		−	15.4938i		−	0.924282i	−0.886806	−	0.462141i	$-0.847081\pi$
$281$		−	15.4938i		−	0.924282i	0.886806	−	0.462141i	$-0.152919\pi$
$282$		0			0
$283$	0.855262			0.0508401			0.0254200	−	0.999677i	$-0.491908\pi$
$283$	0.855262			0.0508401			0.0254200	+	0.999677i	$0.491908\pi$
$284$	1.33817			0.0794060
$285$		0			0
$286$	10.2457	+	0.527968i	0.605843	+	0.0312194i
$287$	4.24487	−	15.0700i	0.250567	−	0.889554i
$288$		0			0
$289$	−4.21069			−0.247688
$290$			6.86933i			0.403381i
$291$		0			0
$292$	5.57621			0.326323
$293$	10.0319			0.586067			0.293033	−	0.956102i	$-0.405335\pi$
$293$	10.0319			0.586067			0.293033	+	0.956102i	$0.405335\pi$
$294$		0			0
$295$	42.1883			2.45630
$296$			0.869330i			0.0505288i
$297$		0			0
$298$	−7.24892			−0.419919
$299$	−23.8734			−1.38063
$300$		0			0
$301$	−26.1875	−	7.37640i	−1.50942	−	0.425169i
$302$	10.3422			0.595128
$303$		0			0
$304$	3.91758			0.224688
$305$			13.9835i			0.800695i
$306$		0			0
$307$	−2.55213			−0.145658			−0.0728288	−	0.997344i	$-0.523203\pi$
$307$	−2.55213			−0.145658			−0.0728288	+	0.997344i	$0.523203\pi$
$308$	8.31266	+	2.81064i	0.473658	+	0.160151i
$309$		0			0
$310$			38.2466i			2.17226i
$311$		−	15.6136i		−	0.885365i	−0.896678	−	0.442682i	$-0.854027\pi$
$311$		−	15.6136i		−	0.885365i	0.896678	−	0.442682i	$-0.145973\pi$
$312$		0			0
$313$			20.0435i			1.13293i	0.824087	+	0.566463i	$0.191689\pi$
$313$			20.0435i			1.13293i	−0.824087	+	0.566463i	$0.808311\pi$
$314$	13.0801			0.738153
$315$		0			0
$316$		−	10.8484i		−	0.610272i
$317$	2.56533			0.144084			0.0720418	−	0.997402i	$-0.477049\pi$
$317$	2.56533			0.144084			0.0720418	+	0.997402i	$0.477049\pi$
$318$		0			0
$319$	5.49379	+	0.283098i	0.307593	+	0.0158504i
$320$			4.14155i			0.231520i
$321$		0			0
$322$	−19.6545	−	5.53620i	−1.09530	−	0.308521i
$323$	−14.0101			−0.779542
$324$		0			0
$325$	−37.5911			−2.08518
$326$		−	3.75513i		−	0.207977i
$327$		0			0
$328$		−	5.91758i		−	0.326743i
$329$	−17.2307	−	4.85348i	−0.949957	−	0.267581i
$330$		0			0
$331$	8.35145			0.459037			0.229519	−	0.973304i	$-0.426285\pi$
$331$	8.35145			0.459037			0.229519	+	0.973304i	$0.426285\pi$
$332$	−5.67271			−0.311330
$333$		0			0
$334$			16.9658i			0.928328i
$335$		−	40.7746i		−	2.22775i
$336$		0			0
$337$			9.93164i			0.541011i	0.962718	+	0.270506i	$0.0871909\pi$
$337$			9.93164i			0.541011i	−0.962718	+	0.270506i	$0.912809\pi$
$338$		−	3.43148i		−	0.186648i
$339$		0			0
$340$		−	14.8111i		−	0.803243i
$341$	30.5880	+	1.57621i	1.65643	+	0.0853568i
$342$		0			0
$343$	−13.5875	+	12.5849i	−0.733657	+	0.679520i
$344$	−10.2831			−0.554427
$345$		0			0
$346$		−	3.75513i		−	0.201877i
$347$		−	31.2007i		−	1.67494i	−0.546480	−	0.837472i	$-0.684033\pi$
$347$		−	31.2007i		−	1.67494i	0.546480	−	0.837472i	$-0.315967\pi$
$348$		0			0
$349$	28.7154			1.53710			0.768551	−	0.639789i	$-0.220978\pi$
$349$	28.7154			1.53710			0.768551	+	0.639789i	$0.220978\pi$
$350$	−30.9480	−	8.71733i	−1.65424	−	0.465961i
$351$		0			0
$352$	3.31223	+	0.170681i	0.176542	+	0.00909733i
$353$			5.05065i			0.268819i	0.990926	+	0.134409i	$0.0429137\pi$
$353$			5.05065i			0.268819i	−0.990926	+	0.134409i	$0.957086\pi$
$354$		0			0
$355$			5.54211i			0.294145i
$356$			10.6245i			0.563095i
$357$		0			0
$358$		−	18.6627i		−	0.986354i
$359$		−	21.3973i		−	1.12931i	−0.825328	−	0.564653i	$-0.809010\pi$
$359$		−	21.3973i		−	1.12931i	0.825328	−	0.564653i	$-0.190990\pi$
$360$		0			0
$361$	−3.65260			−0.192242
$362$	14.9216			0.784263
$363$		0			0
$364$	−2.21893	+	7.87756i	−0.116303	+	0.412897i
$365$			23.0942i			1.20880i
$366$		0			0
$367$		−	36.3455i		−	1.89722i	−0.316449	−	0.948609i	$-0.602491\pi$
$367$		−	36.3455i		−	1.89722i	0.316449	−	0.948609i	$-0.397509\pi$
$368$	−7.71776			−0.402316
$369$		0			0
$370$	−3.60037			−0.187175
$371$	1.35046	−	4.79435i	0.0701123	−	0.248910i
$372$		0			0
$373$			18.1292i			0.938695i	0.883014	+	0.469347i	$0.155511\pi$
$373$			18.1292i			0.938695i	−0.883014	+	0.469347i	$0.844489\pi$
$374$	−11.8452	−	0.610392i	−0.612503	−	0.0315626i
$375$		0			0
$376$	−6.76601			−0.348930
$377$			5.13067i			0.264243i
$378$		0			0
$379$	−2.87942			−0.147906			−0.0739530	−	0.997262i	$-0.523561\pi$
$379$	−2.87942			−0.147906			−0.0739530	+	0.997262i	$0.523561\pi$
$380$			16.2248i			0.832316i
$381$		0			0
$382$			10.5871i			0.541683i
$383$		−	17.7255i		−	0.905728i	−0.891580	−	0.452864i	$-0.850402\pi$
$383$		−	17.7255i		−	0.905728i	0.891580	−	0.452864i	$-0.149598\pi$
$384$		0			0
$385$	−11.6404	+	34.4273i	−0.593249	+	1.75458i
$386$	−11.4355			−0.582053
$387$		0			0
$388$		−	4.52797i		−	0.229873i
$389$	9.43467			0.478357			0.239178	−	0.970976i	$-0.423122\pi$
$389$	9.43467			0.478357			0.239178	+	0.970976i	$0.423122\pi$
$390$		0			0
$391$	27.6004			1.39581
$392$	−3.65359	+	5.97087i	−0.184534	+	0.301574i
$393$		0			0
$394$	13.2107			0.665545
$395$	44.9293			2.26064
$396$		0			0
$397$			16.4238i			0.824286i	0.911119	+	0.412143i	$0.135220\pi$
$397$			16.4238i			0.824286i	−0.911119	+	0.412143i	$0.864780\pi$
$398$	14.0077			0.702142
$399$		0			0
$400$	−12.1524			−0.607621
$401$	15.7551			0.786774			0.393387	−	0.919373i	$-0.371303\pi$
$401$	15.7551			0.786774			0.393387	+	0.919373i	$0.371303\pi$
$402$		0			0
$403$			28.5662i			1.42298i
$404$	1.37554			0.0684357
$405$		0			0
$406$	−1.18979	+	4.22397i	−0.0590485	+	0.209632i
$407$	−0.148378	+	2.87942i	−0.00735483	+	0.142728i
$408$		0			0
$409$	7.19662			0.355850			0.177925	−	0.984044i	$-0.443062\pi$
$409$	7.19662			0.355850			0.177925	+	0.984044i	$0.443062\pi$
$410$	24.5079			1.21036
$411$		0			0
$412$		−	0.0140682i		−	0.000693089i
$413$	25.9417	+	7.30718i	1.27651	+	0.359563i
$414$		0			0
$415$		−	23.4938i		−	1.15326i
$416$			3.09330i			0.151662i
$417$		0			0
$418$	12.9759	+	0.668656i	0.634673	+	0.0327050i
$419$			15.6004i			0.762128i	0.924549	+	0.381064i	$0.124442\pi$
$419$			15.6004i			0.762128i	−0.924549	+	0.381064i	$0.875558\pi$
$420$		0			0
$421$	−16.0701			−0.783208			−0.391604	−	0.920134i	$-0.628080\pi$
$421$	−16.0701			−0.783208			−0.391604	+	0.920134i	$0.628080\pi$
$422$	−7.33903			−0.357259
$423$		0			0
$424$		−	1.88261i		−	0.0914277i
$425$	43.4597			2.10810
$426$		0			0
$427$	−2.42200	+	8.59852i	−0.117209	+	0.416111i
$428$		−	1.37554i		−	0.0664892i
$429$		0			0
$430$		−	42.5880i		−	2.05377i
$431$			17.3555i			0.835985i	0.908450	+	0.417993i	$0.137266\pi$
$431$			17.3555i			0.835985i	−0.908450	+	0.417993i	$0.862734\pi$
$432$		0			0
$433$			30.8711i			1.48357i	0.670639	+	0.741784i	$0.266020\pi$
$433$			30.8711i			1.48357i	−0.670639	+	0.741784i	$0.733980\pi$
$434$	−6.62446	+	23.5180i	−0.317984	+	1.12890i
$435$		0			0
$436$			13.1315i			0.628886i
$437$	−30.2349			−1.44633
$438$		0			0
$439$	19.3245			0.922309			0.461154	−	0.887320i	$-0.347435\pi$
$439$	19.3245			0.922309			0.461154	+	0.887320i	$0.347435\pi$
$440$	−0.706884	+	13.7178i	−0.0336994	+	0.653969i
$441$		0			0
$442$		−	11.0623i		−	0.526181i
$443$	7.71862			0.366723			0.183361	−	0.983046i	$-0.441302\pi$
$443$	7.71862			0.366723			0.183361	+	0.983046i	$0.441302\pi$
$444$		0			0
$445$	−44.0017			−2.08588
$446$	7.42146			0.351416
$447$		0			0
$448$	−0.717333	+	2.54665i	−0.0338908	+	0.120318i
$449$	10.1765			0.480259			0.240130	−	0.970741i	$-0.422810\pi$
$449$	10.1765			0.480259			0.240130	+	0.970741i	$0.422810\pi$
$450$		0			0
$451$	1.01002	−	19.6004i	0.0475599	−	0.922946i
$452$	−7.30718			−0.343701
$453$		0			0
$454$			8.46201i			0.397142i
$455$	−32.6253	−	9.18979i	−1.52950	−	0.430824i
$456$		0			0
$457$		−	4.80097i		−	0.224580i	−0.993675	−	0.112290i	$-0.964181\pi$
$457$		−	4.80097i		−	0.224580i	0.993675	−	0.112290i	$-0.0358186\pi$
$458$	1.03178			0.0482118
$459$		0			0
$460$		−	31.9635i		−	1.49031i
$461$	−39.6386			−1.84615			−0.923077	−	0.384616i	$-0.874334\pi$
$461$	−39.6386			−1.84615			−0.923077	+	0.384616i	$0.874334\pi$
$462$		0			0
$463$	37.5474			1.74498			0.872488	−	0.488636i	$-0.162505\pi$
$463$	37.5474			1.74498			0.872488	+	0.488636i	$0.162505\pi$
$464$			1.65864i			0.0770003i
$465$		0			0
$466$	0.944064			0.0437329
$467$			24.5179i			1.13455i	0.823528	+	0.567276i	$0.192003\pi$
$467$			24.5179i			1.13455i	−0.823528	+	0.567276i	$0.807997\pi$
$468$		0			0
$469$	7.06231	−	25.0724i	0.326107	−	1.15774i
$470$		−	28.0218i		−	1.29255i
$471$		0			0
$472$	10.1866			0.468877
$473$	−34.0600	−	1.75513i	−1.56608	−	0.0807010i
$474$		0			0
$475$	−47.6081			−2.18441
$476$	2.56533	−	9.10737i	0.117582	−	0.417436i
$477$		0			0
$478$	−20.9394			−0.957746
$479$	11.2272			0.512982			0.256491	−	0.966547i	$-0.417434\pi$
$479$	11.2272			0.512982			0.256491	+	0.966547i	$0.417434\pi$
$480$		0			0
$481$	−2.68910			−0.122612
$482$		−	17.8593i		−	0.813469i
$483$		0			0
$484$	10.9417	+	1.13067i	0.497352	+	0.0513941i
$485$	18.7528			0.851521
$486$		0			0
$487$	24.9394			1.13011			0.565056	−	0.825052i	$-0.308854\pi$
$487$	24.9394			1.13011			0.565056	+	0.825052i	$0.308854\pi$
$488$			3.37640i			0.152842i
$489$		0			0
$490$	−24.7286	−	15.1315i	−1.11713	−	0.683573i
$491$		−	14.9977i		−	0.676835i	−0.940996	−	0.338418i	$-0.890108\pi$
$491$		−	14.9977i		−	0.676835i	0.940996	−	0.338418i	$-0.109892\pi$
$492$		0			0
$493$		−	5.93164i		−	0.267148i
$494$			12.1183i			0.545226i
$495$		0			0
$496$			9.23485i			0.414657i
$497$	−0.959915	+	3.40786i	−0.0430581	+	0.152863i
$498$		0			0
$499$	12.2147			0.546807			0.273403	−	0.961899i	$-0.411851\pi$
$499$	12.2147			0.546807			0.273403	+	0.961899i	$0.411851\pi$
$500$		−	29.6221i		−	1.32474i
$501$		0			0
$502$	−8.96582			−0.400164
$503$	40.7328			1.81618			0.908092	−	0.418770i	$-0.137539\pi$
$503$	40.7328			1.81618			0.908092	+	0.418770i	$0.137539\pi$
$504$		0			0
$505$			5.69687i			0.253507i
$506$	−25.5630	−	1.31728i	−1.13641	−	0.0585601i
$507$		0			0
$508$			19.4564i			0.863239i
$509$			35.5771i			1.57693i	0.615082	+	0.788463i	$0.289123\pi$
$509$			35.5771i			1.57693i	−0.615082	+	0.788463i	$0.710877\pi$
$510$		0			0
$511$	−4.00000	+	14.2007i	−0.176950	+	0.628201i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−8.52797			−0.376153
$515$	0.0582640			0.00256742
$516$		0			0
$517$	−22.4106	−	1.15483i	−0.985616	−	0.0507893i
$518$	−2.21388	−	0.623599i	−0.0972724	−	0.0273994i
$519$		0			0
$520$	−12.8111			−0.561802
$521$		−	24.1565i		−	1.05831i	−0.848524	−	0.529157i	$-0.822508\pi$
$521$		−	24.1565i		−	1.05831i	0.848524	−	0.529157i	$-0.177492\pi$
$522$		0			0
$523$	8.53037			0.373007			0.186504	−	0.982454i	$-0.440284\pi$
$523$	8.53037			0.373007			0.186504	+	0.982454i	$0.440284\pi$
$524$	−14.2108			−0.620800
$525$		0			0
$526$	−31.2871			−1.36418
$527$		−	33.0258i		−	1.43863i
$528$		0			0
$529$	36.5639			1.58973
$530$	7.79692			0.338677
$531$		0			0
$532$	−2.81021	+	9.97670i	−0.121838	+	0.432545i
$533$	18.3049			0.792871
$534$		0			0
$535$	5.69687			0.246297
$536$		−	9.84524i		−	0.425250i
$537$		0			0
$538$	7.57535			0.326597
$539$	−13.1207	+	19.1533i	−0.565146	+	0.824991i
$540$		0			0
$541$			7.49293i			0.322146i	0.986942	+	0.161073i	$0.0514955\pi$
$541$			7.49293i			0.322146i	−0.986942	+	0.161073i	$0.948505\pi$
$542$			14.9667i			0.642874i
$543$		0			0
$544$		−	3.57621i		−	0.153329i
$545$	−54.3849			−2.32959
$546$		0			0
$547$		−	3.57381i		−	0.152805i	−0.997077	−	0.0764026i	$-0.975657\pi$
$547$		−	3.57381i		−	0.152805i	0.997077	−	0.0764026i	$-0.0243434\pi$
$548$	13.4938			0.576426
$549$		0			0
$550$	−40.2516	−	2.07419i	−1.71634	−	0.0884437i
$551$			6.49784i			0.276817i
$552$		0			0
$553$	27.6272	+	7.78193i	1.17483	+	0.330922i
$554$	29.2498			1.24270
$555$		0			0
$556$	−14.5521			−0.617148
$557$		−	29.0194i		−	1.22959i	−0.788686	−	0.614796i	$-0.789238\pi$
$557$		−	29.0194i		−	1.22959i	0.788686	−	0.614796i	$-0.210762\pi$
$558$		0			0
$559$		−	31.8087i		−	1.34537i
$560$	−10.5471	−	2.97087i	−0.445696	−	0.125542i
$561$		0			0
$562$	15.4938			0.653566
$563$	15.2566			0.642989			0.321495	−	0.946911i	$-0.395815\pi$
$563$	15.2566			0.642989			0.321495	+	0.946911i	$0.395815\pi$
$564$		0			0
$565$		−	30.2631i		−	1.27318i
$566$			0.855262i			0.0359493i
$567$		0			0
$568$			1.33817i			0.0561485i
$569$			27.6740i			1.16016i	0.814561	+	0.580078i	$0.196978\pi$
$569$			27.6740i			1.16016i	−0.814561	+	0.580078i	$0.803022\pi$
$570$		0			0
$571$		−	3.97996i		−	0.166556i	−0.996526	−	0.0832782i	$-0.973461\pi$
$571$		−	3.97996i		−	0.166556i	0.996526	−	0.0832782i	$-0.0265390\pi$
$572$	−0.527968	+	10.2457i	−0.0220755	+	0.428396i
$573$		0			0
$574$	15.0700	+	4.24487i	0.629010	+	0.177177i
$575$	93.7896			3.91130
$576$		0			0
$577$			20.0382i			0.834202i	0.908860	+	0.417101i	$0.136954\pi$
$577$			20.0382i			0.834202i	−0.908860	+	0.417101i	$0.863046\pi$
$578$		−	4.21069i		−	0.175142i
$579$		0			0
$580$	−6.86933			−0.285233
$581$	4.06922	−	14.4464i	0.168820	−	0.599338i
$582$		0			0
$583$	0.321326	−	6.23564i	0.0133080	−	0.258254i
$584$			5.57621i			0.230745i
$585$		0			0
$586$			10.0319i			0.414412i
$587$		−	22.3796i		−	0.923705i	−0.886957	−	0.461852i	$-0.847185\pi$
$587$		−	22.3796i		−	0.923705i	0.886957	−	0.461852i	$-0.152815\pi$
$588$		0			0
$589$			36.1782i			1.49070i
$590$			42.1883i			1.73687i
$591$		0			0
$592$	−0.869330			−0.0357293
$593$	29.7492			1.22165			0.610826	−	0.791765i	$-0.290838\pi$
$593$	29.7492			1.22165			0.610826	+	0.791765i	$0.290838\pi$
$594$		0			0
$595$	37.7186	+	10.6245i	1.54631	+	0.435560i
$596$		−	7.24892i		−	0.296927i
$597$		0			0
$598$		−	23.8734i		−	0.976255i
$599$	−24.8484			−1.01528			−0.507640	−	0.861569i	$-0.669482\pi$
$599$	−24.8484			−1.01528			−0.507640	+	0.861569i	$0.669482\pi$
$600$		0			0
$601$	−22.6805			−0.925156			−0.462578	−	0.886579i	$-0.653075\pi$
$601$	−22.6805			−0.925156			−0.462578	+	0.886579i	$0.653075\pi$
$602$	7.37640	−	26.1875i	0.300640	−	1.06732i
$603$		0			0
$604$			10.3422i			0.420819i
$605$	−4.68272	+	45.3157i	−0.190380	+	1.84235i
$606$		0			0
$607$	−26.2075			−1.06373			−0.531865	−	0.846829i	$-0.678509\pi$
$607$	−26.2075			−1.06373			−0.531865	+	0.846829i	$0.678509\pi$
$608$			3.91758i			0.158879i
$609$		0			0
$610$	−13.9835			−0.566177
$611$		−	20.9293i		−	0.846710i
$612$		0			0
$613$		−	1.56301i		−	0.0631293i	−0.999502	−	0.0315646i	$-0.989951\pi$
$613$		−	1.56301i		−	0.0631293i	0.999502	−	0.0315646i	$-0.0100490\pi$
$614$		−	2.55213i		−	0.102996i
$615$		0			0
$616$	−2.81064	+	8.31266i	−0.113244	+	0.334927i
$617$	6.87571			0.276806			0.138403	−	0.990376i	$-0.455803\pi$
$617$	6.87571			0.276806			0.138403	+	0.990376i	$0.455803\pi$
$618$		0			0
$619$		−	5.37726i		−	0.216130i	−0.994144	−	0.108065i	$-0.965534\pi$
$619$		−	5.37726i		−	0.216130i	0.994144	−	0.108065i	$-0.0344655\pi$
$620$	−38.2466			−1.53602
$621$		0			0
$622$	15.6136			0.626048
$623$	−27.0568	−	7.62127i	−1.08401	−	0.305340i
$624$		0			0
$625$	61.9194			2.47677
$626$	−20.0435			−0.801100
$627$		0			0
$628$			13.0801i			0.521953i
$629$	3.10891			0.123960
$630$		0			0
$631$	4.31123			0.171628			0.0858138	−	0.996311i	$-0.472651\pi$
$631$	4.31123			0.171628			0.0858138	+	0.996311i	$0.472651\pi$
$632$	10.8484			0.431527
$633$		0			0
$634$			2.56533i			0.101882i
$635$	−80.5797			−3.19771
$636$		0			0
$637$	−18.4697	−	11.3017i	−0.731796	−	0.447788i
$638$	−0.283098	+	5.49379i	−0.0112080	+	0.217501i
$639$		0			0
$640$	−4.14155			−0.163709
$641$	−0.869330			−0.0343365			−0.0171682	−	0.999853i	$-0.505465\pi$
$641$	−0.869330			−0.0343365			−0.0171682	+	0.999853i	$0.505465\pi$
$642$		0			0
$643$			0.955731i			0.0376903i	0.999822	+	0.0188452i	$0.00599896\pi$
$643$			0.955731i			0.0376903i	−0.999822	+	0.0188452i	$0.994001\pi$
$644$	5.53620	−	19.6545i	0.218157	−	0.774494i
$645$		0			0
$646$		−	14.0101i		−	0.551220i
$647$		−	25.9668i		−	1.02086i	−0.859920	−	0.510429i	$-0.829487\pi$
$647$		−	25.9668i		−	1.02086i	0.859920	−	0.510429i	$-0.170513\pi$
$648$		0			0
$649$	33.7404	+	1.73866i	1.32443	+	0.0682484i
$650$		−	37.5911i		−	1.47445i
$651$		0			0
$652$	3.75513			0.147062
$653$	−2.93855			−0.114994			−0.0574971	−	0.998346i	$-0.518312\pi$
$653$	−2.93855			−0.114994			−0.0574971	+	0.998346i	$0.518312\pi$
$654$		0			0
$655$		−	58.8546i		−	2.29964i
$656$	5.91758			0.231043
$657$		0			0
$658$	4.85348	−	17.2307i	0.189208	−	0.671721i
$659$			16.5763i			0.645720i	0.946447	+	0.322860i	$0.104644\pi$
$659$			16.5763i			0.645720i	−0.946447	+	0.322860i	$0.895356\pi$
$660$		0			0
$661$		−	1.87935i		−	0.0730982i	−0.999332	−	0.0365491i	$-0.988363\pi$
$661$		−	1.87935i		−	0.0730982i	0.999332	−	0.0365491i	$-0.0116365\pi$
$662$			8.35145i			0.324588i
$663$		0			0
$664$		−	5.67271i		−	0.220144i
$665$	−41.3190	−	11.6386i	−1.60228	−	0.451326i
$666$		0			0
$667$		−	12.8010i		−	0.495656i
$668$	−16.9658			−0.656427
$669$		0			0
$670$	40.7746			1.57526
$671$	−0.576288	+	11.1834i	−0.0222473	+	0.431731i
$672$		0			0
$673$		−	51.6158i		−	1.98964i	−0.101647	−	0.994821i	$-0.532411\pi$
$673$		−	51.6158i		−	1.98964i	0.101647	−	0.994821i	$-0.467589\pi$
$674$	−9.93164			−0.382553
$675$		0			0
$676$	3.43148			0.131980
$677$	−38.3633			−1.47442			−0.737210	−	0.675664i	$-0.763857\pi$
$677$	−38.3633			−1.47442			−0.737210	+	0.675664i	$0.763857\pi$
$678$		0			0
$679$	11.5312	+	3.24806i	0.442525	+	0.124649i
$680$	14.8111			0.567978
$681$		0			0
$682$	−1.57621	+	30.5880i	−0.0603564	+	1.17127i
$683$	14.4697			0.553668			0.276834	−	0.960918i	$-0.410715\pi$
$683$	14.4697			0.553668			0.276834	+	0.960918i	$0.410715\pi$
$684$		0			0
$685$			55.8852i			2.13526i
$686$	−12.5849	−	13.5875i	−0.480493	−	0.518774i
$687$		0			0
$688$		−	10.2831i		−	0.392039i
$689$	5.82349			0.221857
$690$		0			0
$691$		−	8.64622i		−	0.328918i	−0.986384	−	0.164459i	$-0.947412\pi$
$691$		−	8.64622i		−	0.328918i	0.986384	−	0.164459i	$-0.0525878\pi$
$692$	3.75513			0.142749
$693$		0			0
$694$	31.2007			1.18436
$695$		−	60.2684i		−	2.28611i
$696$		0			0
$697$	−21.1625			−0.801588
$698$			28.7154i			1.08690i
$699$		0			0
$700$	8.71733	−	30.9480i	0.329484	−	1.16972i
$701$		−	4.37959i		−	0.165415i	−0.996574	−	0.0827074i	$-0.973643\pi$
$701$		−	4.37959i		−	0.165415i	0.996574	−	0.0827074i	$-0.0263567\pi$
$702$		0			0
$703$	−3.40567			−0.128447
$704$	−0.170681	+	3.31223i	−0.00643278	+	0.124834i
$705$		0			0
$706$	−5.05065			−0.190084
$707$	−0.986720	+	3.50302i	−0.0371094	+	0.131745i
$708$		0			0
$709$	−12.4214			−0.466495			−0.233247	−	0.972417i	$-0.574935\pi$
$709$	−12.4214			−0.466495			−0.233247	+	0.972417i	$0.574935\pi$
$710$	−5.54211			−0.207992
$711$		0			0
$712$	−10.6245			−0.398168
$713$		−	71.2724i		−	2.66917i
$714$		0			0
$715$	−42.4332	−	2.18661i	−1.58691	−	0.0817744i
$716$	18.6627			0.697457
$717$		0			0
$718$	21.3973			0.798540
$719$		−	10.7862i		−	0.402257i	−0.979565	−	0.201129i	$-0.935539\pi$
$719$		−	10.7862i		−	0.402257i	0.979565	−	0.201129i	$-0.0644609\pi$
$720$		0			0
$721$	0.0358267	+	0.0100916i	0.00133426	+	0.000375829i
$722$		−	3.65260i		−	0.135936i
$723$		0			0
$724$			14.9216i			0.554558i
$725$		−	20.1565i		−	0.748593i
$726$		0			0
$727$		−	15.4215i		−	0.571950i	−0.958237	−	0.285975i	$-0.907683\pi$
$727$		−	15.4215i		−	0.571950i	0.958237	−	0.285975i	$-0.0923175\pi$
$728$	−7.87756	−	2.21893i	−0.291962	−	0.0822389i
$729$		0			0
$730$	−23.0942			−0.854753
$731$			36.7746i			1.36016i
$732$		0			0
$733$	−44.1026			−1.62897			−0.814484	−	0.580186i	$-0.802980\pi$
$733$	−44.1026			−1.62897			−0.814484	+	0.580186i	$0.802980\pi$
$734$	36.3455			1.34154
$735$		0			0
$736$		−	7.71776i		−	0.284481i
$737$	1.68040	−	32.6097i	0.0618982	−	1.20119i
$738$		0			0
$739$			12.8975i			0.474441i	0.971456	+	0.237220i	$0.0762364\pi$
$739$			12.8975i			0.474441i	−0.971456	+	0.237220i	$0.923764\pi$
$740$		−	3.60037i		−	0.132352i
$741$		0			0
$742$	4.79435	+	1.35046i	0.176006	+	0.0495769i
$743$			8.11825i			0.297830i	0.988850	+	0.148915i	$0.0475780\pi$
$743$			8.11825i			0.297830i	−0.988850	+	0.148915i	$0.952422\pi$
$744$		0			0
$745$	30.0218			1.09991
$746$	−18.1292			−0.663757
$747$		0			0
$748$	0.610392	−	11.8452i	0.0223181	−	0.433105i
$749$	3.50302	+	0.986720i	0.127998	+	0.0360539i
$750$		0			0
$751$	12.4915			0.455820			0.227910	−	0.973682i	$-0.426811\pi$
$751$	12.4915			0.455820			0.227910	+	0.973682i	$0.426811\pi$
$752$		−	6.76601i		−	0.246731i
$753$		0			0
$754$	−5.13067			−0.186848
$755$	−42.8328			−1.55885
$756$		0			0
$757$	19.4837			0.708147			0.354074	−	0.935218i	$-0.384796\pi$
$757$	19.4837			0.708147			0.354074	+	0.935218i	$0.384796\pi$
$758$		−	2.87942i		−	0.104585i
$759$		0			0
$760$	−16.2248			−0.588537
$761$	31.6044			1.14566			0.572828	−	0.819675i	$-0.305846\pi$
$761$	31.6044			1.14566			0.572828	+	0.819675i	$0.305846\pi$
$762$		0			0
$763$	−33.4414	−	9.41967i	−1.21066	−	0.341015i
$764$	−10.5871			−0.383028
$765$		0			0
$766$	17.7255			0.640447
$767$			31.5103i			1.13777i
$768$		0			0
$769$	30.3491			1.09441			0.547207	−	0.836997i	$-0.315691\pi$
$769$	30.3491			1.09441			0.547207	+	0.836997i	$0.315691\pi$
$770$	−34.4273	−	11.6404i	−1.24067	−	0.419490i
$771$		0			0
$772$		−	11.4355i		−	0.411574i
$773$		−	31.1010i		−	1.11862i	−0.828957	−	0.559312i	$-0.811065\pi$
$773$		−	31.1010i		−	1.11862i	0.828957	−	0.559312i	$-0.188935\pi$
$774$		0			0
$775$		−	112.226i		−	4.03127i
$776$	4.52797			0.162545
$777$		0			0
$778$			9.43467i			0.338249i
$779$	23.1826			0.830601
$780$		0			0
$781$	−0.228401	+	4.43234i	−0.00817282	+	0.158602i
$782$			27.6004i			0.986987i
$783$		0			0
$784$	−5.97087	−	3.65359i	−0.213245	−	0.130485i
$785$	−54.1719			−1.93348
$786$		0			0
$787$	32.6503			1.16386			0.581930	−	0.813239i	$-0.302298\pi$
$787$	32.6503			1.16386			0.581930	+	0.813239i	$0.302298\pi$
$788$			13.2107i			0.470611i
$789$		0			0
$790$			44.9293i			1.59851i
$791$	5.24168	−	18.6089i	0.186373	−	0.661655i
$792$		0			0
$793$	−10.4442			−0.370886
$794$	−16.4238			−0.582858
$795$		0			0
$796$			14.0077i			0.496489i
$797$			24.4400i			0.865710i	0.901464	+	0.432855i	$0.142494\pi$
$797$			24.4400i			0.865710i	−0.901464	+	0.432855i	$0.857506\pi$
$798$		0			0
$799$			24.1967i			0.856018i
$800$		−	12.1524i		−	0.429653i
$801$		0			0
$802$			15.7551i			0.556333i
$803$	−0.951754	+	18.4697i	−0.0335867	+	0.651782i
$804$		0			0
$805$	81.3999	+	22.9285i	2.86897	+	0.808122i
$806$	−28.5662			−1.00620
$807$		0			0
$808$			1.37554i			0.0483913i
$809$		−	23.7387i		−	0.834607i	−0.908767	−	0.417303i	$-0.862975\pi$
$809$		−	23.7387i		−	0.834607i	0.908767	−	0.417303i	$-0.137025\pi$
$810$		0			0
$811$	−9.56612			−0.335912			−0.167956	−	0.985794i	$-0.553717\pi$
$811$	−9.56612			−0.335912			−0.167956	+	0.985794i	$0.553717\pi$
$812$	−4.22397	−	1.18979i	−0.148232	−	0.0417536i
$813$		0			0
$814$	−2.87942	−	0.148378i	−0.100924	−	0.00520065i
$815$			15.5521i			0.544765i
$816$		0			0
$817$		−	40.2848i		−	1.40939i
$818$			7.19662i			0.251624i
$819$		0			0
$820$			24.5079i			0.855854i
$821$		−	27.7387i		−	0.968086i	−0.875044	−	0.484043i	$-0.839168\pi$
$821$		−	27.7387i		−	0.968086i	0.875044	−	0.484043i	$-0.160832\pi$
$822$		0			0
$823$	11.1041			0.387065			0.193532	−	0.981094i	$-0.438006\pi$
$823$	11.1041			0.387065			0.193532	+	0.981094i	$0.438006\pi$
$824$	0.0140682			0.000490088
$825$		0			0
$826$	−7.30718	+	25.9417i	−0.254249	+	0.902628i
$827$		−	49.0995i		−	1.70736i	−0.520802	−	0.853678i	$-0.674367\pi$
$827$		−	49.0995i		−	1.70736i	0.520802	−	0.853678i	$-0.325633\pi$
$828$		0			0
$829$			44.4409i			1.54350i	0.635929	+	0.771748i	$0.280617\pi$
$829$			44.4409i			1.54350i	−0.635929	+	0.771748i	$0.719383\pi$
$830$	23.4938			0.815481
$831$		0			0
$832$	−3.09330			−0.107241
$833$	21.3531	+	13.0660i	0.739841	+	0.452711i
$834$		0			0
$835$		−	70.2648i		−	2.43161i
$836$	−0.668656	+	12.9759i	−0.0231260	+	0.448781i
$837$		0			0
$838$	−15.6004			−0.538906
$839$		−	0.412833i		−	0.0142526i	−0.999975	−	0.00712629i	$-0.997732\pi$
$839$		−	0.412833i		−	0.0142526i	0.999975	−	0.00712629i	$-0.00226839\pi$
$840$		0			0
$841$	26.2489			0.905135
$842$		−	16.0701i		−	0.553811i
$843$		0			0
$844$		−	7.33903i		−	0.252620i
$845$			14.2116i			0.488895i
$846$		0			0
$847$	−10.7283	+	27.0537i	−0.368628	+	0.929577i
$848$	1.88261			0.0646491
$849$		0			0
$850$			43.4597i			1.49065i
$851$	6.70929			0.229991
$852$		0			0
$853$	19.4028			0.664340			0.332170	−	0.943220i	$-0.392219\pi$
$853$	19.4028			0.664340			0.332170	+	0.943220i	$0.392219\pi$
$854$	−8.59852	−	2.42200i	−0.294235	−	0.0828792i
$855$		0			0
$856$	1.37554			0.0470150
$857$	8.86164			0.302708			0.151354	−	0.988480i	$-0.451637\pi$
$857$	8.86164			0.302708			0.151354	+	0.988480i	$0.451637\pi$
$858$		0			0
$859$		−	9.40897i		−	0.321030i	−0.987033	−	0.160515i	$-0.948685\pi$
$859$		−	9.40897i		−	0.321030i	0.987033	−	0.160515i	$-0.0513155\pi$
$860$	42.5880			1.45224
$861$		0			0
$862$	−17.3555			−0.591131
$863$	39.2715			1.33682			0.668409	−	0.743794i	$-0.266975\pi$
$863$	39.2715			1.33682			0.668409	+	0.743794i	$0.266975\pi$
$864$		0			0
$865$			15.5521i			0.528786i
$866$	−30.8711			−1.04904
$867$		0			0
$868$	−23.5180	−	6.62446i	−0.798251	−	0.224849i
$869$	35.9325	+	1.85162i	1.21893	+	0.0628120i
$870$		0			0
$871$	30.4543			1.03191
$872$	−13.1315			−0.444690
$873$		0			0
$874$		−	30.2349i		−	1.02271i
$875$	75.4372	+	21.2489i	2.55024	+	0.718345i
$876$		0			0
$877$		−	29.1113i		−	0.983020i	−0.870872	−	0.491510i	$-0.836445\pi$
$877$		−	29.1113i		−	0.983020i	0.870872	−	0.491510i	$-0.163555\pi$
$878$			19.3245i			0.652171i
$879$		0			0
$880$	−13.7178	−	0.706884i	−0.462426	−	0.0238291i
$881$			36.7210i			1.23716i	0.785722	+	0.618580i	$0.212292\pi$
$881$			36.7210i			1.23716i	−0.785722	+	0.618580i	$0.787708\pi$
$882$		0			0
$883$	40.9394			1.37772			0.688860	−	0.724894i	$-0.258111\pi$
$883$	40.9394			1.37772			0.688860	+	0.724894i	$0.258111\pi$
$884$	11.0623			0.372066
$885$		0			0
$886$			7.71862i			0.259312i
$887$	9.08407			0.305013			0.152507	−	0.988302i	$-0.451265\pi$
$887$	9.08407			0.305013			0.152507	+	0.988302i	$0.451265\pi$
$888$		0			0
$889$	−49.5487	−	13.9567i	−1.66181	−	0.468094i
$890$		−	44.0017i		−	1.47494i
$891$		0			0
$892$			7.42146i			0.248489i
$893$		−	26.5064i		−	0.887001i
$894$		0			0
$895$			77.2924i			2.58360i
$896$	−2.54665	−	0.717333i	−0.0850777	−	0.0239644i
$897$		0			0
$898$			10.1765i			0.339595i
$899$	−15.3173			−0.510860
$900$		0			0
$901$	−6.73262			−0.224296
$902$	19.6004	+	1.01002i	0.652621	+	0.0336299i
$903$		0			0
$904$		−	7.30718i		−	0.243033i
$905$	−61.7986			−2.05426
$906$		0			0
$907$	38.6044			1.28184			0.640919	−	0.767608i	$-0.278553\pi$
$907$	38.6044			1.28184			0.640919	+	0.767608i	$0.278553\pi$
$908$	−8.46201			−0.280822
$909$		0			0
$910$	9.18979	−	32.6253i	0.304639	−	1.08152i
$911$	−22.6618			−0.750820			−0.375410	−	0.926859i	$-0.622498\pi$
$911$	−22.6618			−0.750820			−0.375410	+	0.926859i	$0.622498\pi$
$912$		0			0
$913$	0.968223	−	18.7893i	0.0320435	−	0.621835i
$914$	4.80097			0.158802
$915$		0			0
$916$			1.03178i			0.0340909i
$917$	10.1938	−	36.1899i	0.336630	−	1.19509i
$918$		0			0
$919$		−	0.294051i		−	0.00969984i	−0.999988	−	0.00484992i	$-0.998456\pi$
$919$		−	0.294051i		−	0.00969984i	0.999988	−	0.00484992i	$-0.00154378\pi$
$920$	31.9635			1.05381
$921$		0			0
$922$		−	39.6386i		−	1.30543i
$923$	−4.13938			−0.136249
$924$		0			0
$925$	10.5645			0.347358
$926$			37.5474i			1.23388i
$927$		0			0
$928$	−1.65864			−0.0544475
$929$		−	7.42543i		−	0.243621i	−0.992553	−	0.121810i	$-0.961130\pi$
$929$		−	7.42543i		−	0.243621i	0.992553	−	0.121810i	$-0.0388700\pi$
$930$		0			0
$931$	−23.3913	−	14.3132i	−0.766620	−	0.469097i
$932$			0.944064i			0.0309238i
$933$		0			0
$934$	−24.5179			−0.802249
$935$	49.0577	+	2.52797i	1.60436	+	0.0826734i
$936$		0			0
$937$	−31.9976			−1.04532			−0.522658	−	0.852542i	$-0.675060\pi$
$937$	−31.9976			−1.04532			−0.522658	+	0.852542i	$0.675060\pi$
$938$	25.0724	+	7.06231i	0.818643	+	0.230593i
$939$		0			0
$940$	28.0218			0.913969
$941$	10.4033			0.339139			0.169570	−	0.985518i	$-0.445762\pi$
$941$	10.4033			0.339139			0.169570	+	0.985518i	$0.445762\pi$
$942$		0			0
$943$	−45.6705			−1.48723
$944$			10.1866i			0.331546i
$945$		0			0
$946$	1.75513	−	34.0600i	0.0570642	−	1.10739i
$947$	−20.6944			−0.672477			−0.336239	−	0.941777i	$-0.609155\pi$
$947$	−20.6944			−0.672477			−0.336239	+	0.941777i	$0.609155\pi$
$948$		0			0
$949$	−17.2489			−0.559924
$950$		−	47.6081i		−	1.54461i
$951$		0			0
$952$	9.10737	+	2.56533i	0.295172	+	0.0831430i
$953$			43.1324i			1.39720i	0.715515	+	0.698598i	$0.246192\pi$
$953$			43.1324i			1.39720i	−0.715515	+	0.698598i	$0.753808\pi$
$954$		0			0
$955$		−	43.8470i		−	1.41885i
$956$		−	20.9394i		−	0.677229i
$957$		0			0
$958$			11.2272i			0.362733i
$959$	−9.67954	+	34.3640i	−0.312568	+	1.10967i
$960$		0			0
$961$	−54.2825			−1.75105
$962$		−	2.68910i		−	0.0867001i
$963$		0			0
$964$	17.8593			0.575210
$965$	47.3608			1.52460
$966$		0			0
$967$			32.0062i			1.02925i	0.857416	+	0.514624i	$0.172069\pi$
$967$			32.0062i			1.02925i	−0.857416	+	0.514624i	$0.827931\pi$
$968$	−1.13067	+	10.9417i	−0.0363411	+	0.351681i
$969$		0			0
$970$			18.7528i			0.602116i
$971$			38.4560i			1.23411i	0.786919	+	0.617057i	$0.211675\pi$
$971$			38.4560i			1.23411i	−0.786919	+	0.617057i	$0.788325\pi$
$972$		0			0
$973$	10.4387	−	37.0592i	0.334650	−	1.18806i
$974$			24.9394i			0.799110i
$975$		0			0
$976$	−3.37640			−0.108076
$977$	39.8771			1.27578			0.637891	−	0.770127i	$-0.279807\pi$
$977$	39.8771			1.27578			0.637891	+	0.770127i	$0.279807\pi$
$978$		0			0
$979$	−35.1907	−	1.81339i	−1.12470	−	0.0579563i
$980$	15.1315	−	24.7286i	0.483359	−	0.789928i
$981$		0			0
$982$	14.9977			0.478595
$983$			46.8641i			1.49473i	0.664413	+	0.747366i	$0.268682\pi$
$983$			46.8641i			1.49473i	−0.664413	+	0.747366i	$0.731318\pi$
$984$		0			0
$985$	−54.7127			−1.74329
$986$	5.93164			0.188902
$987$		0			0
$988$	−12.1183			−0.385533
$989$			79.3625i			2.52358i
$990$		0			0
$991$	−26.2631			−0.834274			−0.417137	−	0.908844i	$-0.636966\pi$
$991$	−26.2631			−0.834274			−0.417137	+	0.908844i	$0.636966\pi$
$992$	−9.23485			−0.293207
$993$		0			0
$994$	−3.40786	−	0.959915i	−0.108091	−	0.0304467i
$995$	−58.0135			−1.83915
$996$		0			0
$997$	−33.5630			−1.06295			−0.531476	−	0.847074i	$-0.678362\pi$
$997$	−33.5630			−1.06295			−0.531476	+	0.847074i	$0.678362\pi$
$998$			12.2147i			0.386651i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.e.e.307.5		8
3.2	odd	2		462.2.e.a.307.4	✓	8
7.6	odd	2		1386.2.e.a.307.8		8
11.10	odd	2		1386.2.e.a.307.1		8
12.11	even	2		3696.2.q.b.769.8		8
21.20	even	2		462.2.e.b.307.1	yes	8
33.32	even	2		462.2.e.b.307.8	yes	8
77.76	even	2	inner	1386.2.e.e.307.4		8
84.83	odd	2		3696.2.q.c.769.1		8
132.131	odd	2		3696.2.q.c.769.8		8
231.230	odd	2		462.2.e.a.307.5	yes	8
924.923	even	2		3696.2.q.b.769.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.e.a.307.4	✓	8	3.2	odd	2
462.2.e.a.307.5	yes	8	231.230	odd	2
462.2.e.b.307.1	yes	8	21.20	even	2
462.2.e.b.307.8	yes	8	33.32	even	2
1386.2.e.a.307.1		8	11.10	odd	2
1386.2.e.a.307.8		8	7.6	odd	2
1386.2.e.e.307.4		8	77.76	even	2	inner
1386.2.e.e.307.5		8	1.1	even	1	trivial
3696.2.q.b.769.1		8	924.923	even	2
3696.2.q.b.769.8		8	12.11	even	2
3696.2.q.c.769.1		8	84.83	odd	2
3696.2.q.c.769.8		8	132.131	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -4.14155i q^{5} +(0.717333 - 2.54665i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -4.14155i q^{5} +(0.717333 - 2.54665i) q^{7} -1.00000i q^{8} +4.14155 q^{10} +(0.170681 - 3.31223i) q^{11} +3.09330 q^{13} +(2.54665 + 0.717333i) q^{14} +1.00000 q^{16} -3.57621 q^{17} +3.91758 q^{19} +4.14155i q^{20} +(3.31223 + 0.170681i) q^{22} -7.71776 q^{23} -12.1524 q^{25} +3.09330i q^{26} +(-0.717333 + 2.54665i) q^{28} +1.65864i q^{29} +9.23485i q^{31} +1.00000i q^{32} -3.57621i q^{34} +(-10.5471 - 2.97087i) q^{35} -0.869330 q^{37} +3.91758i q^{38} -4.14155 q^{40} +5.91758 q^{41} -10.2831i q^{43} +(-0.170681 + 3.31223i) q^{44} -7.71776i q^{46} -6.76601i q^{47} +(-5.97087 - 3.65359i) q^{49} -12.1524i q^{50} -3.09330 q^{52} +1.88261 q^{53} +(-13.7178 - 0.706884i) q^{55} +(-2.54665 - 0.717333i) q^{56} -1.65864 q^{58} +10.1866i q^{59} -3.37640 q^{61} -9.23485 q^{62} -1.00000 q^{64} -12.8111i q^{65} +9.84524 q^{67} +3.57621 q^{68} +(2.97087 - 10.5471i) q^{70} -1.33817 q^{71} -5.57621 q^{73} -0.869330i q^{74} -3.91758 q^{76} +(-8.31266 - 2.81064i) q^{77} +10.8484i q^{79} -4.14155i q^{80} +5.91758i q^{82} +5.67271 q^{83} +14.8111i q^{85} +10.2831 q^{86} +(-3.31223 - 0.170681i) q^{88} -10.6245i q^{89} +(2.21893 - 7.87756i) q^{91} +7.71776 q^{92} +6.76601 q^{94} -16.2248i q^{95} +4.52797i q^{97} +(3.65359 - 5.97087i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 4 q^{10} + 8 q^{11} + 8 q^{14} + 8 q^{16} + 12 q^{17} + 4 q^{19} + 4 q^{22} + 8 q^{23} - 16 q^{25} - 8 q^{35} + 16 q^{37} - 4 q^{40} + 20 q^{41} - 8 q^{44} - 12 q^{49} - 40 q^{55} - 8 q^{56} + 56 q^{61} - 20 q^{62} - 8 q^{64} + 16 q^{67} - 12 q^{68} - 12 q^{70} - 8 q^{71} - 4 q^{73} - 4 q^{76} + 20 q^{77} - 4 q^{83} + 24 q^{86} - 4 q^{88} + 20 q^{91} - 8 q^{92} - 20 q^{94} + 20 q^{98}+O(q^{100}) \) 8 * q - 8 * q^4 + 4 * q^10 + 8 * q^11 + 8 * q^14 + 8 * q^16 + 12 * q^17 + 4 * q^19 + 4 * q^22 + 8 * q^23 - 16 * q^25 - 8 * q^35 + 16 * q^37 - 4 * q^40 + 20 * q^41 - 8 * q^44 - 12 * q^49 - 40 * q^55 - 8 * q^56 + 56 * q^61 - 20 * q^62 - 8 * q^64 + 16 * q^67 - 12 * q^68 - 12 * q^70 - 8 * q^71 - 4 * q^73 - 4 * q^76 + 20 * q^77 - 4 * q^83 + 24 * q^86 - 4 * q^88 + 20 * q^91 - 8 * q^92 - 20 * q^94 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more