LMFDB - Embedded newform 24.9.e.a.17.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [24,9,Mod(17,24)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("24.17"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(24, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	24.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$9.77708664147$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 78x^{6} + 144x^{5} + 2079x^{4} + 936x^{3} - 658x^{2} + 2884x + 30633 $ x^8 - 4x^7 - 78x^6 + 144x^5 + 2079x^4 + 936x^3 - 658x^2 + 2884*x + 30633
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{42}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.4
Root			$7.73966 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	24.17
Dual form			24.9.e.a.17.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.95410 + 80.8484i) q^{3} +404.296i q^{5} -262.245 q^{7} +(-6511.91 - 801.062i) q^{9} -6268.43i q^{11} -43933.9 q^{13} +(-32686.7 - 2002.92i) q^{15} +91180.4i q^{17} -170581. q^{19} +(1299.19 - 21202.1i) q^{21} -234950. i q^{23} +227170. q^{25} +(97025.2 - 522509. i) q^{27} +1.23399e6i q^{29} +1.50803e6 q^{31} +(506792. + 31054.4i) q^{33} -106025. i q^{35} +295815. q^{37} +(217653. - 3.55198e6i) q^{39} +2.45609e6i q^{41} -3.32545e6 q^{43} +(323866. - 2.63274e6i) q^{45} +5.63483e6i q^{47} -5.69603e6 q^{49} +(-7.37179e6 - 451717. i) q^{51} +4.50039e6i q^{53} +2.53430e6 q^{55} +(845078. - 1.37912e7i) q^{57} +9.47475e6i q^{59} -5.61098e6 q^{61} +(1.70772e6 + 210075. i) q^{63} -1.77623e7i q^{65} +3.14398e7 q^{67} +(1.89953e7 + 1.16397e6i) q^{69} +1.69223e7i q^{71} +5.16633e6 q^{73} +(-1.12542e6 + 1.83663e7i) q^{75} +1.64387e6i q^{77} +5.76774e6 q^{79} +(4.17633e7 + 1.04329e7i) q^{81} -4.72276e7i q^{83} -3.68639e7 q^{85} +(-9.97664e7 - 6.11333e6i) q^{87} -6.72340e7i q^{89} +1.15215e7 q^{91} +(-7.47092e6 + 1.21922e8i) q^{93} -6.89654e7i q^{95} -1.36244e8 q^{97} +(-5.02140e6 + 4.08195e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 56 q^{3} + 1584 q^{7} + 328 q^{9} + 25232 q^{13} + 38336 q^{15} + 157936 q^{19} + 30480 q^{21} - 579704 q^{25} - 276040 q^{27} + 805552 q^{31} + 102848 q^{33} - 3985008 q^{37} - 2297104 q^{39} + 6962672 q^{43}+ \cdots - 369701504 q^{99}+O(q^{100}) $ 8 * q + 56 * q^3 + 1584 * q^7 + 328 * q^9 + 25232 * q^13 + 38336 * q^15 + 157936 * q^19 + 30480 * q^21 - 579704 * q^25 - 276040 * q^27 + 805552 * q^31 + 102848 * q^33 - 3985008 * q^37 - 2297104 * q^39 + 6962672 * q^43 + 8670592 * q^45 - 5884520 * q^49 - 15590144 * q^51 + 27101312 * q^55 + 36756688 * q^57 - 51583600 * q^61 - 69759312 * q^63 + 58200688 * q^67 + 94226048 * q^69 - 116854768 * q^73 - 143181896 * q^75 + 172454576 * q^79 + 194700040 * q^81 - 264333824 * q^85 - 242851008 * q^87 + 382128480 * q^91 + 313470352 * q^93 - 337326704 * q^97 - 369701504 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$.

$n$	$7$	$13$	$17$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−4.95410	+	80.8484i	−0.0611617	+	0.998128i
$3$	−4.95410	+	80.8484i	−0.0611617	+	0.998128i
$4$		0			0
$5$			404.296i			0.646873i	0.946250	+	0.323437i	$0.104838\pi$
$5$			404.296i			0.646873i	−0.946250	+	0.323437i	$0.895162\pi$
$6$		0			0
$7$	−262.245			−0.109223			−0.0546117	−	0.998508i	$-0.517392\pi$
$7$	−262.245			−0.109223			−0.0546117	+	0.998508i	$0.517392\pi$
$8$		0			0
$9$	−6511.91	−	801.062i	−0.992518	−	0.122094i
$10$		0			0
$11$		−	6268.43i		−	0.428142i	−0.976818	−	0.214071i	$-0.931328\pi$
$11$		−	6268.43i		−	0.428142i	0.976818	−	0.214071i	$-0.0686724\pi$
$12$		0			0
$13$	−43933.9			−1.53825			−0.769124	−	0.639100i	$-0.779307\pi$
$13$	−43933.9			−1.53825			−0.769124	+	0.639100i	$0.779307\pi$
$14$		0			0
$15$	−32686.7	−	2002.92i	−0.645662	−	0.0395639i
$16$		0			0
$17$			91180.4i			1.09171i	0.837881	+	0.545853i	$0.183794\pi$
$17$			91180.4i			1.09171i	−0.837881	+	0.545853i	$0.816206\pi$
$18$		0			0
$19$	−170581.			−1.30893			−0.654467	−	0.756091i	$-0.727107\pi$
$19$	−170581.			−1.30893			−0.654467	+	0.756091i	$0.727107\pi$
$20$		0			0
$21$	1299.19	−	21202.1i	0.00668029	−	0.109019i
$22$		0			0
$23$		−	234950.i		−	0.839584i	−0.907620	−	0.419792i	$-0.862103\pi$
$23$		−	234950.i		−	0.839584i	0.907620	−	0.419792i	$-0.137897\pi$
$24$		0			0
$25$	227170.			0.581555
$26$		0			0
$27$	97025.2	−	522509.i	0.182570	−	0.983193i
$28$		0			0
$29$			1.23399e6i			1.74470i	0.488880	+	0.872351i	$0.337405\pi$
$29$			1.23399e6i			1.74470i	−0.488880	+	0.872351i	$0.662595\pi$
$30$		0			0
$31$	1.50803e6			1.63291			0.816456	−	0.577408i	$-0.195936\pi$
$31$	1.50803e6			1.63291			0.816456	+	0.577408i	$0.195936\pi$
$32$		0			0
$33$	506792.	+	31054.4i	0.427341	+	0.0261859i
$34$		0			0
$35$		−	106025.i		−	0.0706537i
$36$		0			0
$37$	295815.			0.157838			0.0789192	−	0.996881i	$-0.474853\pi$
$37$	295815.			0.157838			0.0789192	+	0.996881i	$0.474853\pi$
$38$		0			0
$39$	217653.	−	3.55198e6i	0.0940819	−	1.53537i
$40$		0			0
$41$			2.45609e6i			0.869177i	0.900629	+	0.434589i	$0.143106\pi$
$41$			2.45609e6i			0.869177i	−0.900629	+	0.434589i	$0.856894\pi$
$42$		0			0
$43$	−3.32545e6			−0.972694			−0.486347	−	0.873766i	$-0.661671\pi$
$43$	−3.32545e6			−0.972694			−0.486347	+	0.873766i	$0.661671\pi$
$44$		0			0
$45$	323866.	−	2.63274e6i	0.0789797	−	0.642034i
$46$		0			0
$47$			5.63483e6i			1.15475i	0.816478	+	0.577377i	$0.195924\pi$
$47$			5.63483e6i			1.15475i	−0.816478	+	0.577377i	$0.804076\pi$
$48$		0			0
$49$	−5.69603e6			−0.988070
$50$		0			0
$51$	−7.37179e6	−	451717.i	−1.08966	−	0.0667707i
$52$		0			0
$53$			4.50039e6i			0.570357i	0.958474	+	0.285178i	$0.0920528\pi$
$53$			4.50039e6i			0.570357i	−0.958474	+	0.285178i	$0.907947\pi$
$54$		0			0
$55$	2.53430e6			0.276954
$56$		0			0
$57$	845078.	−	1.37912e7i	0.0800566	−	1.30648i
$58$		0			0
$59$			9.47475e6i			0.781916i	0.920409	+	0.390958i	$0.127856\pi$
$59$			9.47475e6i			0.781916i	−0.920409	+	0.390958i	$0.872144\pi$
$60$		0			0
$61$	−5.61098e6			−0.405247			−0.202623	−	0.979257i	$-0.564947\pi$
$61$	−5.61098e6			−0.405247			−0.202623	+	0.979257i	$0.564947\pi$
$62$		0			0
$63$	1.70772e6	+	210075.i	0.108406	+	0.0133356i
$64$		0			0
$65$		−	1.77623e7i		−	0.995052i
$66$		0			0
$67$	3.14398e7			1.56020			0.780101	−	0.625654i	$-0.215168\pi$
$67$	3.14398e7			1.56020			0.780101	+	0.625654i	$0.215168\pi$
$68$		0			0
$69$	1.89953e7	+	1.16397e6i	0.838013	+	0.0513504i
$70$		0			0
$71$			1.69223e7i			0.665925i	0.942940	+	0.332962i	$0.108048\pi$
$71$			1.69223e7i			0.665925i	−0.942940	+	0.332962i	$0.891952\pi$
$72$		0			0
$73$	5.16633e6			0.181924			0.0909622	−	0.995854i	$-0.471006\pi$
$73$	5.16633e6			0.181924			0.0909622	+	0.995854i	$0.471006\pi$
$74$		0			0
$75$	−1.12542e6	+	1.83663e7i	−0.0355689	+	0.580466i
$76$		0			0
$77$			1.64387e6i			0.0467632i
$78$		0			0
$79$	5.76774e6			0.148080			0.0740401	−	0.997255i	$-0.476411\pi$
$79$	5.76774e6			0.148080			0.0740401	+	0.997255i	$0.476411\pi$
$80$		0			0
$81$	4.17633e7	+	1.04329e7i	0.970186	+	0.242362i
$82$		0			0
$83$		−	4.72276e7i		−	0.995139i	−0.867424	−	0.497570i	$-0.834226\pi$
$83$		−	4.72276e7i		−	0.995139i	0.867424	−	0.497570i	$-0.165774\pi$
$84$		0			0
$85$	−3.68639e7			−0.706196
$86$		0			0
$87$	−9.97664e7	−	6.11333e6i	−1.74144	−	0.106709i
$88$		0			0
$89$		−	6.72340e7i		−	1.07159i	−0.844348	−	0.535795i	$-0.820012\pi$
$89$		−	6.72340e7i		−	1.07159i	0.844348	−	0.535795i	$-0.179988\pi$
$90$		0			0
$91$	1.15215e7			0.168013
$92$		0			0
$93$	−7.47092e6	+	1.21922e8i	−0.0998717	+	1.62985i
$94$		0			0
$95$		−	6.89654e7i		−	0.846714i
$96$		0			0
$97$	−1.36244e8			−1.53897			−0.769484	−	0.638666i	$-0.779486\pi$
$97$	−1.36244e8			−1.53897			−0.769484	+	0.638666i	$0.779486\pi$
$98$		0			0
$99$	−5.02140e6	+	4.08195e7i	−0.0522738	+	0.424939i
$100$		0			0
$101$			1.62745e8i			1.56395i	0.623310	+	0.781975i	$0.285788\pi$
$101$			1.62745e8i			1.56395i	−0.623310	+	0.781975i	$0.714212\pi$
$102$		0			0
$103$	4.49972e7			0.399794			0.199897	−	0.979817i	$-0.435939\pi$
$103$	4.49972e7			0.399794			0.199897	+	0.979817i	$0.435939\pi$
$104$		0			0
$105$	8.57192e6	+	525257.i	0.0705214	+	0.00432130i
$106$		0			0
$107$		−	1.57223e8i		−	1.19944i	−0.800208	−	0.599722i	$-0.795278\pi$
$107$		−	1.57223e8i		−	1.19944i	0.800208	−	0.599722i	$-0.204722\pi$
$108$		0			0
$109$	−2.51315e7			−0.178038			−0.0890189	−	0.996030i	$-0.528373\pi$
$109$	−2.51315e7			−0.178038			−0.0890189	+	0.996030i	$0.528373\pi$
$110$		0			0
$111$	−1.46550e6	+	2.39161e7i	−0.00965367	+	0.157543i
$112$		0			0
$113$		−	1.93202e8i		−	1.18495i	−0.805591	−	0.592473i	$-0.798152\pi$
$113$		−	1.93202e8i		−	1.18495i	0.805591	−	0.592473i	$-0.201848\pi$
$114$		0			0
$115$	9.49894e7			0.543105
$116$		0			0
$117$	2.86094e8	+	3.51938e7i	1.52674	+	0.187812i
$118$		0			0
$119$		−	2.39116e7i		−	0.119240i
$120$		0			0
$121$	1.75066e8			0.816694
$122$		0			0
$123$	−1.98571e8	−	1.21677e7i	−0.867550	−	0.0531604i
$124$		0			0
$125$			2.49772e8i			1.02307i
$126$		0			0
$127$	2.48708e8			0.956037			0.478019	−	0.878350i	$-0.341355\pi$
$127$	2.48708e8			0.956037			0.478019	+	0.878350i	$0.341355\pi$
$128$		0			0
$129$	1.64746e7	−	2.68857e8i	0.0594917	−	0.970873i
$130$		0			0
$131$			4.72372e8i			1.60398i	0.597337	+	0.801990i	$0.296225\pi$
$131$			4.72372e8i			1.60398i	−0.597337	+	0.801990i	$0.703775\pi$
$132$		0			0
$133$	4.47342e7			0.142966
$134$		0			0
$135$	2.11248e8	+	3.92269e7i	0.636001	+	0.118100i
$136$		0			0
$137$		−	4.20841e8i		−	1.19464i	−0.802005	−	0.597318i	$-0.796233\pi$
$137$		−	4.20841e8i		−	1.19464i	0.802005	−	0.597318i	$-0.203767\pi$
$138$		0			0
$139$	−2.44286e8			−0.654394			−0.327197	−	0.944956i	$-0.606104\pi$
$139$	−2.44286e8			−0.654394			−0.327197	+	0.944956i	$0.606104\pi$
$140$		0			0
$141$	−4.55567e8	−	2.79155e7i	−1.15259	−	0.0706267i
$142$		0			0
$143$			2.75397e8i			0.658589i
$144$		0			0
$145$	−4.98899e8			−1.12860
$146$		0			0
$147$	2.82187e7	−	4.60515e8i	0.0604321	−	0.986220i
$148$		0			0
$149$			2.23041e8i			0.452522i	0.974067	+	0.226261i	$0.0726502\pi$
$149$			2.23041e8i			0.452522i	−0.974067	+	0.226261i	$0.927350\pi$
$150$		0			0
$151$	−1.82742e8			−0.351505			−0.175752	−	0.984434i	$-0.556236\pi$
$151$	−1.82742e8			−0.351505			−0.175752	+	0.984434i	$0.556236\pi$
$152$		0			0
$153$	7.30411e7	−	5.93759e8i	0.133291	−	1.08354i
$154$		0			0
$155$			6.09690e8i			1.05629i
$156$		0			0
$157$	−2.78972e8			−0.459158			−0.229579	−	0.973290i	$-0.573735\pi$
$157$	−2.78972e8			−0.459158			−0.229579	+	0.973290i	$0.573735\pi$
$158$		0			0
$159$	−3.63849e8	−	2.22954e7i	−0.569289	−	0.0348840i
$160$		0			0
$161$			6.16146e7i			0.0917023i
$162$		0			0
$163$	−6.21653e8			−0.880638			−0.440319	−	0.897841i	$-0.645135\pi$
$163$	−6.21653e8			−0.880638			−0.440319	+	0.897841i	$0.645135\pi$
$164$		0			0
$165$	−1.25552e7	+	2.04894e8i	−0.0169390	+	0.276435i
$166$		0			0
$167$			1.14731e7i			0.0147508i	0.999973	+	0.00737539i	$0.00234768\pi$
$167$			1.14731e7i			0.0147508i	−0.999973	+	0.00737539i	$0.997652\pi$
$168$		0			0
$169$	1.11446e9			1.36621
$170$		0			0
$171$	1.11081e9	+	1.36646e8i	1.29914	+	0.159813i
$172$		0			0
$173$			9.13885e8i			1.02025i	0.860100	+	0.510125i	$0.170401\pi$
$173$			9.13885e8i			1.02025i	−0.860100	+	0.510125i	$0.829599\pi$
$174$		0			0
$175$	−5.95742e7			−0.0635194
$176$		0			0
$177$	−7.66018e8	−	4.69389e7i	−0.780452	−	0.0478233i
$178$		0			0
$179$		−	1.46335e9i		−	1.42540i	−0.701468	−	0.712701i	$-0.747472\pi$
$179$		−	1.46335e9i		−	1.42540i	0.701468	−	0.712701i	$-0.252528\pi$
$180$		0			0
$181$	1.75899e9			1.63889			0.819445	−	0.573158i	$-0.194282\pi$
$181$	1.75899e9			1.63889			0.819445	+	0.573158i	$0.194282\pi$
$182$		0			0
$183$	2.77974e7	−	4.53639e8i	0.0247856	−	0.404488i
$184$		0			0
$185$			1.19597e8i			0.102102i
$186$		0			0
$187$	5.71558e8			0.467406
$188$		0			0
$189$	−2.54444e7	+	1.37026e8i	−0.0199409	+	0.107388i
$190$		0			0
$191$		−	9.59184e8i		−	0.720723i	−0.932813	−	0.360361i	$-0.882653\pi$
$191$		−	9.59184e8i		−	0.720723i	0.932813	−	0.360361i	$-0.117347\pi$
$192$		0			0
$193$	−4.76277e6			−0.00343266			−0.00171633	−	0.999999i	$-0.500546\pi$
$193$	−4.76277e6			−0.00343266			−0.00171633	+	0.999999i	$0.500546\pi$
$194$		0			0
$195$	1.43605e9	+	8.79962e7i	0.993189	+	0.0608591i
$196$		0			0
$197$		−	5.50726e8i		−	0.365654i	−0.983145	−	0.182827i	$-0.941475\pi$
$197$		−	5.50726e8i		−	0.365654i	0.983145	−	0.182827i	$-0.0585248\pi$
$198$		0			0
$199$	−4.82710e8			−0.307804			−0.153902	−	0.988086i	$-0.549184\pi$
$199$	−4.82710e8			−0.307804			−0.153902	+	0.988086i	$0.549184\pi$
$200$		0			0
$201$	−1.55756e8	+	2.54186e9i	−0.0954246	+	1.55728i
$202$		0			0
$203$		−	3.23609e8i		−	0.190562i
$204$		0			0
$205$	−9.92986e8			−0.562248
$206$		0			0
$207$	−1.88210e8	+	1.52998e9i	−0.102509	+	0.833303i
$208$		0			0
$209$			1.06928e9i			0.560410i
$210$		0			0
$211$	−1.68845e9			−0.851843			−0.425921	−	0.904760i	$-0.640050\pi$
$211$	−1.68845e9			−0.851843			−0.425921	+	0.904760i	$0.640050\pi$
$212$		0			0
$213$	−1.36814e9	−	8.38346e7i	−0.664678	−	0.0407291i
$214$		0			0
$215$		−	1.34446e9i		−	0.629210i
$216$		0			0
$217$	−3.95473e8			−0.178352
$218$		0			0
$219$	−2.55945e7	+	4.17690e8i	−0.0111268	+	0.181584i
$220$		0			0
$221$		−	4.00591e9i		−	1.67932i
$222$		0			0
$223$	−2.06196e9			−0.833797			−0.416899	−	0.908953i	$-0.636883\pi$
$223$	−2.06196e9			−0.833797			−0.416899	+	0.908953i	$0.636883\pi$
$224$		0			0
$225$	−1.47931e9	−	1.81977e8i	−0.577204	−	0.0710046i
$226$		0			0
$227$			5.26909e8i			0.198441i	0.995065	+	0.0992206i	$0.0316350\pi$
$227$			5.26909e8i			0.198441i	−0.995065	+	0.0992206i	$0.968365\pi$
$228$		0			0
$229$	7.93492e8			0.288536			0.144268	−	0.989539i	$-0.453917\pi$
$229$	7.93492e8			0.288536			0.144268	+	0.989539i	$0.453917\pi$
$230$		0			0
$231$	−1.32904e8	−	8.14388e6i	−0.0466756	−	0.00286012i
$232$		0			0
$233$		−	1.80359e9i		−	0.611947i	−0.952040	−	0.305974i	$-0.901018\pi$
$233$		−	1.80359e9i		−	0.611947i	0.952040	−	0.305974i	$-0.0989819\pi$
$234$		0			0
$235$	−2.27814e9			−0.746979
$236$		0			0
$237$	−2.85739e7	+	4.66312e8i	−0.00905684	+	0.147803i
$238$		0			0
$239$			4.41034e9i			1.35170i	0.737039	+	0.675851i	$0.236224\pi$
$239$			4.41034e9i			1.35170i	−0.737039	+	0.675851i	$0.763776\pi$
$240$		0			0
$241$	2.25090e9			0.667248			0.333624	−	0.942706i	$-0.391728\pi$
$241$	2.25090e9			0.667248			0.333624	+	0.942706i	$0.391728\pi$
$242$		0			0
$243$	−1.05038e9	+	3.32481e9i	−0.301247	+	0.953546i
$244$		0			0
$245$		−	2.30288e9i		−	0.639156i
$246$		0			0
$247$	7.49431e9			2.01346
$248$		0			0
$249$	3.81828e9	+	2.33970e8i	0.993276	+	0.0608644i
$250$		0			0
$251$		−	2.04067e8i		−	0.0514135i	−0.999670	−	0.0257068i	$-0.991816\pi$
$251$		−	2.04067e8i		−	0.0514135i	0.999670	−	0.0257068i	$-0.00818362\pi$
$252$		0			0
$253$	−1.47277e9			−0.359462
$254$		0			0
$255$	1.82627e8	−	2.98038e9i	0.0431922	−	0.704874i
$256$		0			0
$257$			6.51675e9i			1.49382i	0.664925	+	0.746910i	$0.268464\pi$
$257$			6.51675e9i			1.49382i	−0.664925	+	0.746910i	$0.731536\pi$
$258$		0			0
$259$	−7.75760e7			−0.0172397
$260$		0			0
$261$	9.88506e8	−	8.03566e9i	0.213018	−	1.73165i
$262$		0			0
$263$			1.32086e9i			0.276079i	0.990427	+	0.138039i	$0.0440800\pi$
$263$			1.32086e9i			0.276079i	−0.990427	+	0.138039i	$0.955920\pi$
$264$		0			0
$265$	−1.81949e9			−0.368948
$266$		0			0
$267$	5.43576e9	+	3.33084e8i	1.06958	+	0.0655404i
$268$		0			0
$269$		−	5.62436e9i		−	1.07415i	−0.843535	−	0.537073i	$-0.819530\pi$
$269$		−	5.62436e9i		−	1.07415i	0.843535	−	0.537073i	$-0.180470\pi$
$270$		0			0
$271$	9.52211e9			1.76545			0.882726	−	0.469888i	$-0.155706\pi$
$271$	9.52211e9			1.76545			0.882726	+	0.469888i	$0.155706\pi$
$272$		0			0
$273$	−5.70785e7	+	9.31491e8i	−0.0102759	+	0.167698i
$274$		0			0
$275$		−	1.42400e9i		−	0.248988i
$276$		0			0
$277$	−2.92789e9			−0.497320			−0.248660	−	0.968591i	$-0.579990\pi$
$277$	−2.92789e9			−0.497320			−0.248660	+	0.968591i	$0.579990\pi$
$278$		0			0
$279$	−9.82015e9	−	1.20802e9i	−1.62070	−	0.199369i
$280$		0			0
$281$			1.04540e10i			1.67671i	0.545125	+	0.838355i	$0.316482\pi$
$281$			1.04540e10i			1.67671i	−0.545125	+	0.838355i	$0.683518\pi$
$282$		0			0
$283$	−8.32937e9			−1.29857			−0.649287	−	0.760544i	$-0.724933\pi$
$283$	−8.32937e9			−1.29857			−0.649287	+	0.760544i	$0.724933\pi$
$284$		0			0
$285$	5.57574e9	+	3.41661e8i	0.845129	+	0.0517865i
$286$		0			0
$287$		−	6.44097e8i		−	0.0949345i
$288$		0			0
$289$	−1.33811e9			−0.191823
$290$		0			0
$291$	6.74965e8	−	1.10151e10i	0.0941259	−	1.53609i
$292$		0			0
$293$			8.11907e9i			1.10163i	0.834627	+	0.550815i	$0.185683\pi$
$293$			8.11907e9i			1.10163i	−0.834627	+	0.550815i	$0.814317\pi$
$294$		0			0
$295$	−3.83060e9			−0.505800
$296$		0			0
$297$	−3.27531e9	−	6.08196e8i	−0.420946	−	0.0781660i
$298$		0			0
$299$			1.03223e10i			1.29149i
$300$		0			0
$301$	8.72083e8			0.106241
$302$		0			0
$303$	−1.31577e10	−	8.06256e8i	−1.56102	−	0.0956539i
$304$		0			0
$305$		−	2.26850e9i		−	0.262143i
$306$		0			0
$307$	1.37771e9			0.155097			0.0775485	−	0.996989i	$-0.475291\pi$
$307$	1.37771e9			0.155097			0.0775485	+	0.996989i	$0.475291\pi$
$308$		0			0
$309$	−2.22921e8	+	3.63795e9i	−0.0244521	+	0.399046i
$310$		0			0
$311$		−	9.39351e9i		−	1.00412i	−0.864832	−	0.502061i	$-0.832575\pi$
$311$		−	9.39351e9i		−	1.00412i	0.864832	−	0.502061i	$-0.167425\pi$
$312$		0			0
$313$	−1.17651e10			−1.22580			−0.612901	−	0.790160i	$-0.709997\pi$
$313$	−1.17651e10			−1.22580			−0.612901	+	0.790160i	$0.709997\pi$
$314$		0			0
$315$	−8.49323e7	+	6.90424e8i	−0.00862642	+	0.0701251i
$316$		0			0
$317$			3.52509e9i			0.349086i	0.984650	+	0.174543i	$0.0558449\pi$
$317$			3.52509e9i			0.349086i	−0.984650	+	0.174543i	$0.944155\pi$
$318$		0			0
$319$	7.73521e9			0.746981
$320$		0			0
$321$	1.27112e10	+	7.78897e8i	1.19720	+	0.0733601i
$322$		0			0
$323$		−	1.55537e10i		−	1.42897i
$324$		0			0
$325$	−9.98046e9			−0.894575
$326$		0			0
$327$	1.24504e8	−	2.03184e9i	0.0108891	−	0.177704i
$328$		0			0
$329$		−	1.47771e9i		−	0.126126i
$330$		0			0
$331$	1.23925e10			1.03240			0.516198	−	0.856469i	$-0.327347\pi$
$331$	1.23925e10			1.03240			0.516198	+	0.856469i	$0.327347\pi$
$332$		0			0
$333$	−1.92632e9	−	2.36966e8i	−0.156658	−	0.0192712i
$334$		0			0
$335$			1.27110e10i			1.00925i
$336$		0			0
$337$	5.59022e9			0.433420			0.216710	−	0.976236i	$-0.430467\pi$
$337$	5.59022e9			0.433420			0.216710	+	0.976236i	$0.430467\pi$
$338$		0			0
$339$	1.56201e10	+	9.57143e8i	1.18273	+	0.0724733i
$340$		0			0
$341$		−	9.45297e9i		−	0.699119i
$342$		0			0
$343$	3.00555e9			0.217144
$344$		0			0
$345$	−4.70587e8	+	7.67974e9i	−0.0332172	+	0.542088i
$346$		0			0
$347$			1.14727e10i			0.791314i	0.918398	+	0.395657i	$0.129483\pi$
$347$			1.14727e10i			0.791314i	−0.918398	+	0.395657i	$0.870517\pi$
$348$		0			0
$349$	2.10078e10			1.41605			0.708027	−	0.706185i	$-0.249585\pi$
$349$	2.10078e10			1.41605			0.708027	+	0.706185i	$0.249585\pi$
$350$		0			0
$351$	−4.26269e9	+	2.29559e10i	−0.280838	+	1.51239i
$352$		0			0
$353$		−	1.15235e10i		−	0.742141i	−0.928605	−	0.371071i	$-0.878991\pi$
$353$		−	1.15235e10i		−	0.742141i	0.928605	−	0.371071i	$-0.121009\pi$
$354$		0			0
$355$	−6.84160e9			−0.430769
$356$		0			0
$357$	1.93322e9	+	1.18461e8i	0.119017	+	0.00729292i
$358$		0			0
$359$		−	2.50612e10i		−	1.50878i	−0.656429	−	0.754388i	$-0.727934\pi$
$359$		−	2.50612e10i		−	1.50878i	0.656429	−	0.754388i	$-0.272066\pi$
$360$		0			0
$361$	1.21145e10			0.713306
$362$		0			0
$363$	−8.67293e8	+	1.41538e10i	−0.0499504	+	0.815165i
$364$		0			0
$365$			2.08873e9i			0.117682i
$366$		0			0
$367$	1.05056e10			0.579103			0.289552	−	0.957162i	$-0.406494\pi$
$367$	1.05056e10			0.579103			0.289552	+	0.957162i	$0.406494\pi$
$368$		0			0
$369$	1.96748e9	−	1.59938e10i	0.106122	−	0.862674i
$370$		0			0
$371$		−	1.18021e9i		−	0.0622963i
$372$		0			0
$373$	−2.38053e10			−1.22981			−0.614905	−	0.788601i	$-0.710806\pi$
$373$	−2.38053e10			−1.22981			−0.614905	+	0.788601i	$0.710806\pi$
$374$		0			0
$375$	−2.01936e10	−	1.23739e9i	−1.02115	−	0.0625725i
$376$		0			0
$377$		−	5.42142e10i		−	2.68378i
$378$		0			0
$379$	−3.46936e10			−1.68148			−0.840740	−	0.541438i	$-0.817880\pi$
$379$	−3.46936e10			−1.68148			−0.840740	+	0.541438i	$0.817880\pi$
$380$		0			0
$381$	−1.23212e9	+	2.01076e10i	−0.0584729	+	0.954247i
$382$		0			0
$383$			2.09822e10i			0.975116i	0.873091	+	0.487558i	$0.162112\pi$
$383$			2.09822e10i			0.975116i	−0.873091	+	0.487558i	$0.837888\pi$
$384$		0			0
$385$	−6.64609e8			−0.0302498
$386$		0			0
$387$	2.16550e10	+	2.66389e9i	0.965417	+	0.118761i
$388$		0			0
$389$			2.63670e10i			1.15149i	0.817628	+	0.575747i	$0.195289\pi$
$389$			2.63670e10i			1.15149i	−0.817628	+	0.575747i	$0.804711\pi$
$390$		0			0
$391$	2.14229e10			0.916580
$392$		0			0
$393$	−3.81905e10	−	2.34018e9i	−1.60098	−	0.0981022i
$394$		0			0
$395$			2.33187e9i			0.0957892i
$396$		0			0
$397$	3.66748e9			0.147641			0.0738204	−	0.997272i	$-0.476481\pi$
$397$	3.66748e9			0.147641			0.0738204	+	0.997272i	$0.476481\pi$
$398$		0			0
$399$	−2.21618e8	+	3.61669e9i	−0.00874405	+	0.142698i
$400$		0			0
$401$			1.06928e10i			0.413536i	0.978390	+	0.206768i	$0.0662945\pi$
$401$			1.06928e10i			0.413536i	−0.978390	+	0.206768i	$0.933706\pi$
$402$		0			0
$403$	−6.62536e10			−2.51182
$404$		0			0
$405$	−4.21797e9	+	1.68847e10i	−0.156778	+	0.627587i
$406$		0			0
$407$		−	1.85429e9i		−	0.0675773i
$408$		0			0
$409$	−1.60158e10			−0.572343			−0.286172	−	0.958178i	$-0.592383\pi$
$409$	−1.60158e10			−0.572343			−0.286172	+	0.958178i	$0.592383\pi$
$410$		0			0
$411$	3.40243e10	+	2.08489e9i	1.19240	+	0.0730660i
$412$		0			0
$413$		−	2.48471e9i		−	0.0854035i
$414$		0			0
$415$	1.90939e10			0.643729
$416$		0			0
$417$	1.21022e9	−	1.97501e10i	0.0400239	−	0.653169i
$418$		0			0
$419$		−	2.09916e10i		−	0.681067i	−0.940232	−	0.340534i	$-0.889392\pi$
$419$		−	2.09916e10i		−	0.681067i	0.940232	−	0.340534i	$-0.110608\pi$
$420$		0			0
$421$	4.25670e10			1.35502			0.677508	−	0.735516i	$-0.263060\pi$
$421$	4.25670e10			1.35502			0.677508	+	0.735516i	$0.263060\pi$
$422$		0			0
$423$	4.51384e9	−	3.66935e10i	0.140989	−	1.14611i
$424$		0			0
$425$			2.07134e10i			0.634887i
$426$		0			0
$427$	1.47145e9			0.0442624
$428$		0			0
$429$	−2.22654e10	−	1.36434e9i	−0.657356	−	0.0402804i
$430$		0			0
$431$		−	1.65345e10i		−	0.479160i	−0.970877	−	0.239580i	$-0.922990\pi$
$431$		−	1.65345e10i		−	0.479160i	0.970877	−	0.239580i	$-0.0770098\pi$
$432$		0			0
$433$	1.02436e10			0.291408			0.145704	−	0.989328i	$-0.453455\pi$
$433$	1.02436e10			0.291408			0.145704	+	0.989328i	$0.453455\pi$
$434$		0			0
$435$	2.47159e9	−	4.03351e10i	0.0690272	−	1.12649i
$436$		0			0
$437$			4.00781e10i			1.09896i
$438$		0			0
$439$	2.29220e10			0.617156			0.308578	−	0.951199i	$-0.400147\pi$
$439$	2.29220e10			0.617156			0.308578	+	0.951199i	$0.400147\pi$
$440$		0			0
$441$	3.70920e10	+	4.56287e9i	0.980678	+	0.120638i
$442$		0			0
$443$			4.45008e10i			1.15546i	0.816230	+	0.577728i	$0.196060\pi$
$443$			4.45008e10i			1.15546i	−0.816230	+	0.577728i	$0.803940\pi$
$444$		0			0
$445$	2.71824e10			0.693184
$446$		0			0
$447$	−1.80325e10	−	1.10497e9i	−0.451674	−	0.0276770i
$448$		0			0
$449$		−	1.30866e10i		−	0.321989i	−0.986955	−	0.160994i	$-0.948530\pi$
$449$		−	1.30866e10i		−	0.321989i	0.986955	−	0.160994i	$-0.0514701\pi$
$450$		0			0
$451$	1.53958e10			0.372132
$452$		0			0
$453$	9.05323e8	−	1.47744e10i	0.0214986	−	0.350846i
$454$		0			0
$455$			4.65808e9i			0.108683i
$456$		0			0
$457$	5.79860e10			1.32941			0.664705	−	0.747106i	$-0.268557\pi$
$457$	5.79860e10			1.32941			0.664705	+	0.747106i	$0.268557\pi$
$458$		0			0
$459$	4.76426e10	+	8.84680e9i	1.07336	+	0.199313i
$460$		0			0
$461$			7.76440e10i			1.71911i	0.511042	+	0.859556i	$0.329260\pi$
$461$			7.76440e10i			1.71911i	−0.511042	+	0.859556i	$0.670740\pi$
$462$		0			0
$463$	−4.16807e10			−0.907008			−0.453504	−	0.891254i	$-0.649826\pi$
$463$	−4.16807e10			−0.907008			−0.453504	+	0.891254i	$0.649826\pi$
$464$		0			0
$465$	−4.92924e10	−	3.02046e9i	−1.05431	−	0.0646043i
$466$		0			0
$467$		−	7.38367e10i		−	1.55240i	−0.630484	−	0.776202i	$-0.717144\pi$
$467$		−	7.38367e10i		−	1.55240i	0.630484	−	0.776202i	$-0.282856\pi$
$468$		0			0
$469$	−8.24494e9			−0.170410
$470$		0			0
$471$	1.38205e9	−	2.25544e10i	0.0280829	−	0.458298i
$472$		0			0
$473$			2.08453e10i			0.416452i
$474$		0			0
$475$	−3.87510e10			−0.761216
$476$		0			0
$477$	3.60509e9	−	2.93061e10i	0.0696374	−	0.566089i
$478$		0			0
$479$			8.49464e10i			1.61363i	0.590807	+	0.806813i	$0.298810\pi$
$479$			8.49464e10i			1.61363i	−0.590807	+	0.806813i	$0.701190\pi$
$480$		0			0
$481$	−1.29963e10			−0.242795
$482$		0			0
$483$	−4.98144e9	−	3.05245e8i	−0.0915306	−	0.00560867i
$484$		0			0
$485$		−	5.50828e10i		−	0.995517i
$486$		0			0
$487$	−2.37054e10			−0.421435			−0.210718	−	0.977547i	$-0.567580\pi$
$487$	−2.37054e10			−0.421435			−0.210718	+	0.977547i	$0.567580\pi$
$488$		0			0
$489$	3.07973e9	−	5.02596e10i	0.0538613	−	0.878989i
$490$		0			0
$491$			3.67026e10i			0.631497i	0.948843	+	0.315748i	$0.102256\pi$
$491$			3.67026e10i			0.631497i	−0.948843	+	0.315748i	$0.897744\pi$
$492$		0			0
$493$	−1.12516e11			−1.90470
$494$		0			0
$495$	−1.65032e10	−	2.03013e9i	−0.274882	−	0.0338145i
$496$		0			0
$497$		−	4.43779e9i		−	0.0727346i
$498$		0			0
$499$	4.12874e10			0.665910			0.332955	−	0.942943i	$-0.391954\pi$
$499$	4.12874e10			0.665910			0.332955	+	0.942943i	$0.391954\pi$
$500$		0			0
$501$	−9.27581e8	−	5.68389e7i	−0.0147232	−	0.000902183i
$502$		0			0
$503$		−	6.97550e10i		−	1.08969i	−0.838537	−	0.544845i	$-0.816588\pi$
$503$		−	6.97550e10i		−	1.08969i	0.838537	−	0.544845i	$-0.183412\pi$
$504$		0			0
$505$	−6.57972e10			−1.01168
$506$		0			0
$507$	−5.52113e9	+	9.01020e10i	−0.0835596	+	1.36365i
$508$		0			0
$509$		−	7.34702e10i		−	1.09456i	−0.836949	−	0.547281i	$-0.815663\pi$
$509$		−	7.34702e10i		−	1.09456i	0.836949	−	0.547281i	$-0.184337\pi$
$510$		0			0
$511$	−1.35485e9			−0.0198704
$512$		0			0
$513$	−1.65507e10	+	8.91304e10i	−0.238972	+	1.28693i
$514$		0			0
$515$			1.81922e10i			0.258616i
$516$		0			0
$517$	3.53215e10			0.494399
$518$		0			0
$519$	−7.38861e10	−	4.52748e9i	−1.01834	−	0.0624003i
$520$		0			0
$521$		−	2.70948e9i		−	0.0367736i	−0.999831	−	0.0183868i	$-0.994147\pi$
$521$		−	2.70948e9i		−	0.0367736i	0.999831	−	0.0183868i	$-0.00585302\pi$
$522$		0			0
$523$	4.70244e10			0.628516			0.314258	−	0.949338i	$-0.398244\pi$
$523$	4.70244e10			0.628516			0.314258	+	0.949338i	$0.398244\pi$
$524$		0			0
$525$	2.95137e8	−	4.81648e9i	0.00388496	−	0.0634005i
$526$		0			0
$527$			1.37503e11i			1.78266i
$528$		0			0
$529$	2.31094e10			0.295098
$530$		0			0
$531$	7.58986e9	−	6.16988e10i	0.0954676	−	0.776066i
$532$		0			0
$533$		−	1.07905e11i		−	1.33701i
$534$		0			0
$535$	6.35645e10			0.775889
$536$		0			0
$537$	1.18310e11	+	7.24960e9i	1.42273	+	0.0871800i
$538$		0			0
$539$			3.57052e10i			0.423035i
$540$		0			0
$541$	5.39641e10			0.629965			0.314982	−	0.949098i	$-0.398001\pi$
$541$	5.39641e10			0.629965			0.314982	+	0.949098i	$0.398001\pi$
$542$		0			0
$543$	−8.71423e9	+	1.42212e11i	−0.100237	+	1.63582i
$544$		0			0
$545$		−	1.01606e10i		−	0.115168i
$546$		0			0
$547$	−8.48341e10			−0.947592			−0.473796	−	0.880635i	$-0.657117\pi$
$547$	−8.48341e10			−0.947592			−0.473796	+	0.880635i	$0.657117\pi$
$548$		0			0
$549$	3.65382e10	+	4.49474e9i	0.402215	+	0.0494784i
$550$		0			0
$551$		−	2.10497e11i		−	2.28370i
$552$		0			0
$553$	−1.51256e9			−0.0161738
$554$		0			0
$555$	−9.66919e9	−	5.92494e8i	−0.101910	−	0.00624470i
$556$		0			0
$557$		−	1.22606e11i		−	1.27377i	−0.770958	−	0.636886i	$-0.780222\pi$
$557$		−	1.22606e11i		−	1.27377i	0.770958	−	0.636886i	$-0.219778\pi$
$558$		0			0
$559$	1.46100e11			1.49624
$560$		0			0
$561$	−2.83156e9	+	4.62095e10i	−0.0285873	+	0.466531i
$562$		0			0
$563$		−	9.59755e10i		−	0.955271i	−0.878558	−	0.477636i	$-0.841494\pi$
$563$		−	9.59755e10i		−	0.955271i	0.878558	−	0.477636i	$-0.158506\pi$
$564$		0			0
$565$	7.81109e10			0.766510
$566$		0			0
$567$	−1.09522e10	−	2.73598e9i	−0.105967	−	0.0264716i
$568$		0			0
$569$			1.49524e11i			1.42647i	0.700927	+	0.713233i	$0.252770\pi$
$569$			1.49524e11i			1.42647i	−0.700927	+	0.713233i	$0.747230\pi$
$570$		0			0
$571$	4.89683e10			0.460650			0.230325	−	0.973114i	$-0.426021\pi$
$571$	4.89683e10			0.460650			0.230325	+	0.973114i	$0.426021\pi$
$572$		0			0
$573$	7.75484e10	+	4.75189e9i	0.719374	+	0.0440807i
$574$		0			0
$575$		−	5.33736e10i		−	0.488264i
$576$		0			0
$577$	−1.19225e11			−1.07563			−0.537815	−	0.843063i	$-0.680750\pi$
$577$	−1.19225e11			−1.07563			−0.537815	+	0.843063i	$0.680750\pi$
$578$		0			0
$579$	2.35953e7	−	3.85062e8i	0.000209947	−	0.00342623i
$580$		0			0
$581$			1.23852e10i			0.108692i
$582$		0			0
$583$	2.82104e10			0.244194
$584$		0			0
$585$	−1.42287e10	+	1.15667e11i	−0.121490	+	0.987607i
$586$		0			0
$587$			1.31496e11i			1.10754i	0.832668	+	0.553772i	$0.186812\pi$
$587$			1.31496e11i			1.10754i	−0.832668	+	0.553772i	$0.813188\pi$
$588$		0			0
$589$	−2.57242e11			−2.13737
$590$		0			0
$591$	4.45253e10	+	2.72835e9i	0.364969	+	0.0223640i
$592$		0			0
$593$			9.66587e10i			0.781668i	0.920461	+	0.390834i	$0.127813\pi$
$593$			9.66587e10i			0.781668i	−0.920461	+	0.390834i	$0.872187\pi$
$594$		0			0
$595$	9.66738e9			0.0771331
$596$		0			0
$597$	2.39140e9	−	3.90263e10i	0.0188258	−	0.307228i
$598$		0			0
$599$		−	3.95546e10i		−	0.307248i	−0.988129	−	0.153624i	$-0.950905\pi$
$599$		−	3.95546e10i		−	0.307248i	0.988129	−	0.153624i	$-0.0490945\pi$
$600$		0			0
$601$	5.19533e10			0.398213			0.199106	−	0.979978i	$-0.436196\pi$
$601$	5.19533e10			0.398213			0.199106	+	0.979978i	$0.436196\pi$
$602$		0			0
$603$	−2.04733e11	−	2.51852e10i	−1.54853	−	0.190492i
$604$		0			0
$605$			7.07783e10i			0.528298i
$606$		0			0
$607$	2.94897e10			0.217228			0.108614	−	0.994084i	$-0.465359\pi$
$607$	2.94897e10			0.217228			0.108614	+	0.994084i	$0.465359\pi$
$608$		0			0
$609$	2.61633e10	+	1.60319e9i	0.190205	+	0.0116551i
$610$		0			0
$611$		−	2.47560e11i		−	1.77630i
$612$		0			0
$613$	−8.56628e10			−0.606667			−0.303333	−	0.952884i	$-0.598100\pi$
$613$	−8.56628e10			−0.606667			−0.303333	+	0.952884i	$0.598100\pi$
$614$		0			0
$615$	4.91935e9	−	8.02813e10i	0.0343880	−	0.561195i
$616$		0			0
$617$			1.79832e10i			0.124087i	0.998073	+	0.0620435i	$0.0197617\pi$
$617$			1.79832e10i			0.124087i	−0.998073	+	0.0620435i	$0.980238\pi$
$618$		0			0
$619$	−1.14708e11			−0.781325			−0.390663	−	0.920534i	$-0.627754\pi$
$619$	−1.14708e11			−0.781325			−0.390663	+	0.920534i	$0.627754\pi$
$620$		0			0
$621$	−1.22764e11	−	2.27961e10i	−0.825473	−	0.153283i
$622$		0			0
$623$			1.76318e10i			0.117043i
$624$		0			0
$625$	−1.22435e10			−0.0802392
$626$		0			0
$627$	−8.64494e10	−	5.29731e9i	−0.559361	−	0.0342756i
$628$		0			0
$629$			2.69725e10i			0.172313i
$630$		0			0
$631$	−2.70187e11			−1.70431			−0.852153	−	0.523293i	$-0.824703\pi$
$631$	−2.70187e11			−1.70431			−0.852153	+	0.523293i	$0.824703\pi$
$632$		0			0
$633$	8.36477e9	−	1.36509e11i	0.0521002	−	0.850248i
$634$		0			0
$635$			1.00552e11i			0.618435i
$636$		0			0
$637$	2.50249e11			1.51990
$638$		0			0
$639$	1.35558e10	−	1.10196e11i	0.0813057	−	0.660943i
$640$		0			0
$641$			6.37085e10i			0.377368i	0.982038	+	0.188684i	$0.0604222\pi$
$641$			6.37085e10i			0.377368i	−0.982038	+	0.188684i	$0.939578\pi$
$642$		0			0
$643$	2.43207e11			1.42276			0.711380	−	0.702808i	$-0.248071\pi$
$643$	2.43207e11			1.42276			0.711380	+	0.702808i	$0.248071\pi$
$644$		0			0
$645$	1.08698e11	+	6.66061e9i	0.628032	+	0.0384836i
$646$		0			0
$647$			2.32556e11i			1.32712i	0.748122	+	0.663562i	$0.230956\pi$
$647$			2.32556e11i			1.32712i	−0.748122	+	0.663562i	$0.769044\pi$
$648$		0			0
$649$	5.93919e10			0.334771
$650$		0			0
$651$	1.95921e9	−	3.19734e10i	0.0109083	−	0.178018i
$652$		0			0
$653$			4.88765e10i			0.268811i	0.990926	+	0.134406i	$0.0429125\pi$
$653$			4.88765e10i			0.268811i	−0.990926	+	0.134406i	$0.957087\pi$
$654$		0			0
$655$	−1.90978e11			−1.03757
$656$		0			0
$657$	−3.36427e10	−	4.13855e9i	−0.180563	−	0.0222120i
$658$		0			0
$659$			2.07990e11i			1.10281i	0.834238	+	0.551404i	$0.185908\pi$
$659$			2.07990e11i			1.10281i	−0.834238	+	0.551404i	$0.814092\pi$
$660$		0			0
$661$	−2.17392e11			−1.13877			−0.569387	−	0.822069i	$-0.692820\pi$
$661$	−2.17392e11			−1.13877			−0.569387	+	0.822069i	$0.692820\pi$
$662$		0			0
$663$	3.23871e11	+	1.98457e10i	1.67617	+	0.102710i
$664$		0			0
$665$			1.80859e10i			0.0924810i
$666$		0			0
$667$	2.89927e11			1.46482
$668$		0			0
$669$	1.02151e10	−	1.66706e11i	0.0509965	−	0.832236i
$670$		0			0
$671$			3.51720e10i			0.173503i
$672$		0			0
$673$	2.28012e11			1.11147			0.555735	−	0.831359i	$-0.312437\pi$
$673$	2.28012e11			1.11147			0.555735	+	0.831359i	$0.312437\pi$
$674$		0			0
$675$	2.20412e10	−	1.18698e11i	0.106174	−	0.571781i
$676$		0			0
$677$		−	2.16200e11i		−	1.02920i	−0.857429	−	0.514602i	$-0.827940\pi$
$677$		−	2.16200e11i		−	1.02920i	0.857429	−	0.514602i	$-0.172060\pi$
$678$		0			0
$679$	3.57293e10			0.168091
$680$		0			0
$681$	−4.25997e10	−	2.61036e9i	−0.198070	−	0.0121370i
$682$		0			0
$683$			1.43240e11i			0.658236i	0.944289	+	0.329118i	$0.106751\pi$
$683$			1.43240e11i			0.658236i	−0.944289	+	0.329118i	$0.893249\pi$
$684$		0			0
$685$	1.70144e11			0.772778
$686$		0			0
$687$	−3.93104e9	+	6.41525e10i	−0.0176474	+	0.287996i
$688$		0			0
$689$		−	1.97720e11i		−	0.877350i
$690$		0			0
$691$	6.27484e10			0.275226			0.137613	−	0.990486i	$-0.456057\pi$
$691$	6.27484e10			0.275226			0.137613	+	0.990486i	$0.456057\pi$
$692$		0			0
$693$	1.31684e9	−	1.07047e10i	0.00570952	−	0.0464133i
$694$		0			0
$695$		−	9.87638e10i		−	0.423310i
$696$		0			0
$697$	−2.23947e11			−0.948886
$698$		0			0
$699$	1.45817e11	+	8.93516e9i	0.610802	+	0.0374277i
$700$		0			0
$701$			2.88764e11i			1.19583i	0.801558	+	0.597917i	$0.204005\pi$
$701$			2.88764e11i			1.19583i	−0.801558	+	0.597917i	$0.795995\pi$
$702$		0			0
$703$	−5.04605e10			−0.206600
$704$		0			0
$705$	1.12861e10	−	1.84184e11i	0.0456865	−	0.745581i
$706$		0			0
$707$		−	4.26792e10i		−	0.170820i
$708$		0			0
$709$	2.10091e11			0.831425			0.415712	−	0.909496i	$-0.363532\pi$
$709$	2.10091e11			0.831425			0.415712	+	0.909496i	$0.363532\pi$
$710$		0			0
$711$	−3.75590e10	−	4.62031e9i	−0.146972	−	0.0180798i
$712$		0			0
$713$		−	3.54311e11i		−	1.37097i
$714$		0			0
$715$	−1.11342e11			−0.426024
$716$		0			0
$717$	−3.56569e11	−	2.18493e10i	−1.34917	−	0.0826724i
$718$		0			0
$719$			4.62917e11i			1.73216i	0.499905	+	0.866080i	$0.333368\pi$
$719$			4.62917e11i			1.73216i	−0.499905	+	0.866080i	$0.666632\pi$
$720$		0			0
$721$	−1.18003e10			−0.0436669
$722$		0			0
$723$	−1.11512e10	+	1.81981e11i	−0.0408101	+	0.665999i
$724$		0			0
$725$			2.80326e11i			1.01464i
$726$		0			0
$727$	−4.83930e11			−1.73239			−0.866193	−	0.499710i	$-0.833440\pi$
$727$	−4.83930e11			−1.73239			−0.866193	+	0.499710i	$0.833440\pi$
$728$		0			0
$729$	−2.63602e11	−	1.01393e11i	−0.933336	−	0.359003i
$730$		0			0
$731$		−	3.03216e11i		−	1.06190i
$732$		0			0
$733$	4.31584e11			1.49503			0.747515	−	0.664245i	$-0.231247\pi$
$733$	4.31584e11			1.49503			0.747515	+	0.664245i	$0.231247\pi$
$734$		0			0
$735$	1.86184e11	+	1.14087e10i	0.637960	+	0.0390919i
$736$		0			0
$737$		−	1.97078e11i		−	0.667988i
$738$		0			0
$739$	−2.00585e11			−0.672545			−0.336273	−	0.941765i	$-0.609166\pi$
$739$	−2.00585e11			−0.672545			−0.336273	+	0.941765i	$0.609166\pi$
$740$		0			0
$741$	−3.71276e10	+	6.05903e11i	−0.123147	+	2.00969i
$742$		0			0
$743$		−	2.97739e11i		−	0.976967i	−0.872573	−	0.488483i	$-0.837550\pi$
$743$		−	2.97739e11i		−	0.976967i	0.872573	−	0.488483i	$-0.162450\pi$
$744$		0			0
$745$	−9.01745e10			−0.292724
$746$		0			0
$747$	−3.78322e10	+	3.07542e11i	−0.121501	+	0.987694i
$748$		0			0
$749$			4.12309e10i			0.131007i
$750$		0			0
$751$	−2.84301e11			−0.893756			−0.446878	−	0.894595i	$-0.647464\pi$
$751$	−2.84301e11			−0.893756			−0.446878	+	0.894595i	$0.647464\pi$
$752$		0			0
$753$	1.64985e10	+	1.01097e9i	0.0513173	+	0.00314454i
$754$		0			0
$755$		−	7.38819e10i		−	0.227379i
$756$		0			0
$757$	−6.47181e11			−1.97080			−0.985399	−	0.170259i	$-0.945540\pi$
$757$	−6.47181e11			−1.97080			−0.985399	+	0.170259i	$0.945540\pi$
$758$		0			0
$759$	7.29625e9	−	1.19071e11i	0.0219853	−	0.358789i
$760$		0			0
$761$			7.43573e10i			0.221710i	0.993837	+	0.110855i	$0.0353589\pi$
$761$			7.43573e10i			0.221710i	−0.993837	+	0.110855i	$0.964641\pi$
$762$		0			0
$763$	6.59061e9			0.0194459
$764$		0			0
$765$	2.40054e11	+	2.95302e10i	0.700912	+	0.0862226i
$766$		0			0
$767$		−	4.16263e11i		−	1.20278i
$768$		0			0
$769$	2.37455e11			0.679011			0.339505	−	0.940604i	$-0.389740\pi$
$769$	2.37455e11			0.679011			0.339505	+	0.940604i	$0.389740\pi$
$770$		0			0
$771$	−5.26868e11	−	3.22846e10i	−1.49102	−	0.0913646i
$772$		0			0
$773$			1.17994e11i			0.330478i	0.986254	+	0.165239i	$0.0528396\pi$
$773$			1.17994e11i			0.330478i	−0.986254	+	0.165239i	$0.947160\pi$
$774$		0			0
$775$	3.42579e11			0.949628
$776$		0			0
$777$	3.84319e8	−	6.27189e9i	0.00105441	−	0.0172074i
$778$		0			0
$779$		−	4.18963e11i		−	1.13769i
$780$		0			0
$781$	1.06076e11			0.285111
$782$		0			0
$783$	6.44773e11	+	1.19729e11i	1.71538	+	0.318530i
$784$		0			0
$785$		−	1.12787e11i		−	0.297017i
$786$		0			0
$787$	2.65668e11			0.692533			0.346267	−	0.938136i	$-0.387449\pi$
$787$	2.65668e11			0.692533			0.346267	+	0.938136i	$0.387449\pi$
$788$		0			0
$789$	−1.06789e11	−	6.54366e9i	−0.275562	−	0.0168854i
$790$		0			0
$791$			5.06664e10i			0.129424i
$792$		0			0
$793$	2.46512e11			0.623370
$794$		0			0
$795$	9.01393e9	−	1.47103e11i	0.0225655	−	0.368258i
$796$		0			0
$797$		−	2.08533e11i		−	0.516822i	−0.966035	−	0.258411i	$-0.916801\pi$
$797$		−	2.08533e11i		−	0.516822i	0.966035	−	0.258411i	$-0.0831988\pi$
$798$		0			0
$799$	−5.13786e11			−1.26065
$800$		0			0
$801$	−5.38586e10	+	4.37822e11i	−0.130835	+	1.06357i
$802$		0			0
$803$		−	3.23848e10i		−	0.0778896i
$804$		0			0
$805$	−2.49105e10			−0.0593197
$806$		0			0
$807$	4.54720e11	+	2.78636e10i	1.07214	+	0.0656967i
$808$		0			0
$809$			1.23180e11i			0.287571i	0.989609	+	0.143785i	$0.0459275\pi$
$809$			1.23180e11i			0.287571i	−0.989609	+	0.143785i	$0.954073\pi$
$810$		0			0
$811$	3.65240e11			0.844296			0.422148	−	0.906527i	$-0.361276\pi$
$811$	3.65240e11			0.844296			0.422148	+	0.906527i	$0.361276\pi$
$812$		0			0
$813$	−4.71735e10	+	7.69847e11i	−0.107978	+	1.76215i
$814$		0			0
$815$		−	2.51332e11i		−	0.569661i
$816$		0			0
$817$	5.67260e11			1.27319
$818$		0			0
$819$	−7.50268e10	−	9.22940e9i	−0.166756	−	0.0205134i
$820$		0			0
$821$			3.33541e11i			0.734137i	0.930194	+	0.367068i	$0.119639\pi$
$821$			3.33541e11i			0.734137i	−0.930194	+	0.367068i	$0.880361\pi$
$822$		0			0
$823$	6.81722e11			1.48596			0.742982	−	0.669312i	$-0.233411\pi$
$823$	6.81722e11			1.48596			0.742982	+	0.669312i	$0.233411\pi$
$824$		0			0
$825$	1.15128e11	+	7.05463e9i	0.248522	+	0.0152286i
$826$		0			0
$827$			5.04791e11i			1.07917i	0.841931	+	0.539585i	$0.181419\pi$
$827$			5.04791e11i			1.07917i	−0.841931	+	0.539585i	$0.818581\pi$
$828$		0			0
$829$	2.16100e11			0.457548			0.228774	−	0.973480i	$-0.426528\pi$
$829$	2.16100e11			0.457548			0.228774	+	0.973480i	$0.426528\pi$
$830$		0			0
$831$	1.45051e10	−	2.36715e11i	0.0304169	−	0.496388i
$832$		0			0
$833$		−	5.19366e11i		−	1.07868i
$834$		0			0
$835$	−4.63852e9			−0.00954188
$836$		0			0
$837$	1.46317e11	−	7.87958e11i	0.298121	−	1.60547i
$838$		0			0
$839$			2.04905e11i			0.413529i	0.978391	+	0.206764i	$0.0662933\pi$
$839$			2.04905e11i			0.413529i	−0.978391	+	0.206764i	$0.933707\pi$
$840$		0			0
$841$	−1.02250e12			−2.04398
$842$		0			0
$843$	−8.45190e11	−	5.17902e10i	−1.67357	−	0.102550i
$844$		0			0
$845$			4.50570e11i			0.883763i
$846$		0			0
$847$	−4.59101e10			−0.0892021
$848$		0			0
$849$	4.12646e10	−	6.73416e11i	0.0794230	−	1.29614i
$850$		0			0
$851$		−	6.95017e10i		−	0.132519i
$852$		0			0
$853$	4.13139e11			0.780368			0.390184	−	0.920737i	$-0.372411\pi$
$853$	4.13139e11			0.780368			0.390184	+	0.920737i	$0.372411\pi$
$854$		0			0
$855$	−5.52455e10	+	4.49097e11i	−0.103379	+	0.840379i
$856$		0			0
$857$		−	6.67723e11i		−	1.23787i	−0.785444	−	0.618933i	$-0.787565\pi$
$857$		−	6.67723e11i		−	1.23787i	0.785444	−	0.618933i	$-0.212435\pi$
$858$		0			0
$859$	2.74330e11			0.503850			0.251925	−	0.967747i	$-0.418936\pi$
$859$	2.74330e11			0.503850			0.251925	+	0.967747i	$0.418936\pi$
$860$		0			0
$861$	5.20742e10	+	3.19092e9i	0.0947567	+	0.00580636i
$862$		0			0
$863$		−	1.22070e11i		−	0.220072i	−0.993928	−	0.110036i	$-0.964903\pi$
$863$		−	1.22070e11i		−	0.220072i	0.993928	−	0.110036i	$-0.0350967\pi$
$864$		0			0
$865$	−3.69480e11			−0.659973
$866$		0			0
$867$	6.62913e9	−	1.08184e11i	0.0117322	−	0.191464i
$868$		0			0
$869$		−	3.61547e10i		−	0.0633994i
$870$		0			0
$871$	−1.38127e12			−2.39998
$872$		0			0
$873$	8.87207e11	+	1.09140e11i	1.52745	+	0.187899i
$874$		0			0
$875$		−	6.55015e10i		−	0.111743i
$876$		0			0
$877$	−8.27364e11			−1.39862			−0.699308	−	0.714820i	$-0.746508\pi$
$877$	−8.27364e11			−1.39862			−0.699308	+	0.714820i	$0.746508\pi$
$878$		0			0
$879$	−6.56414e11	−	4.02227e10i	−1.09957	−	0.0673776i
$880$		0			0
$881$			1.05827e12i			1.75669i	0.478031	+	0.878343i	$0.341351\pi$
$881$			1.05827e12i			1.75669i	−0.478031	+	0.878343i	$0.658649\pi$
$882$		0			0
$883$	1.68354e11			0.276937			0.138469	−	0.990367i	$-0.455782\pi$
$883$	1.68354e11			0.276937			0.138469	+	0.990367i	$0.455782\pi$
$884$		0			0
$885$	1.89772e10	−	3.09698e11i	0.0309356	−	0.504854i
$886$		0			0
$887$			1.20459e12i			1.94600i	0.230796	+	0.973002i	$0.425867\pi$
$887$			1.20459e12i			1.94600i	−0.230796	+	0.973002i	$0.574133\pi$
$888$		0			0
$889$	−6.52225e10			−0.104422
$890$		0			0
$891$	6.53979e10	−	2.61791e11i	0.103765	−	0.415378i
$892$		0			0
$893$		−	9.61197e11i		−	1.51149i
$894$		0			0
$895$	5.91628e11			0.922054
$896$		0			0
$897$	−8.34539e11	−	5.11376e10i	−1.28907	−	0.0789897i
$898$		0			0
$899$			1.86090e12i			2.84894i
$900$		0			0
$901$	−4.10347e11			−0.622662
$902$		0			0
$903$	−4.32039e9	+	7.05065e10i	−0.00649788	+	0.106042i
$904$		0			0
$905$			7.11154e11i			1.06015i
$906$		0			0
$907$	7.52950e10			0.111259			0.0556297	−	0.998451i	$-0.482283\pi$
$907$	7.52950e10			0.111259			0.0556297	+	0.998451i	$0.482283\pi$
$908$		0			0
$909$	1.30369e11	−	1.05978e12i	0.190950	−	1.55225i
$910$		0			0
$911$		−	9.36161e11i		−	1.35918i	−0.733592	−	0.679590i	$-0.762158\pi$
$911$		−	9.36161e11i		−	1.35918i	0.733592	−	0.679590i	$-0.237842\pi$
$912$		0			0
$913$	−2.96043e11			−0.426061
$914$		0			0
$915$	1.83404e11	+	1.12384e10i	0.261652	+	0.0160331i
$916$		0			0
$917$		−	1.23877e11i		−	0.175192i
$918$		0			0
$919$	2.46228e11			0.345204			0.172602	−	0.984992i	$-0.444783\pi$
$919$	2.46228e11			0.345204			0.172602	+	0.984992i	$0.444783\pi$
$920$		0			0
$921$	−6.82530e9	+	1.11385e11i	−0.00948601	+	0.154807i
$922$		0			0
$923$		−	7.43461e11i		−	1.02436i
$924$		0			0
$925$	6.72002e10			0.0917917
$926$		0			0
$927$	−2.93018e11	−	3.60455e10i	−0.396803	−	0.0488127i
$928$		0			0
$929$			3.89557e11i			0.523007i	0.965202	+	0.261504i	$0.0842184\pi$
$929$			3.89557e11i			0.523007i	−0.965202	+	0.261504i	$0.915782\pi$
$930$		0			0
$931$	9.71637e11			1.29332
$932$		0			0
$933$	7.59450e11	+	4.65364e10i	1.00224	+	0.0614138i
$934$		0			0
$935$			2.31079e11i			0.302352i
$936$		0			0
$937$	−6.84516e9			−0.00888026			−0.00444013	−	0.999990i	$-0.501413\pi$
$937$	−6.84516e9			−0.00888026			−0.00444013	+	0.999990i	$0.501413\pi$
$938$		0			0
$939$	5.82857e10	−	9.51193e11i	0.0749721	−	1.22351i
$940$		0			0
$941$		−	7.50450e11i		−	0.957113i	−0.878057	−	0.478557i	$-0.841160\pi$
$941$		−	7.50450e11i		−	0.957113i	0.878057	−	0.478557i	$-0.158840\pi$
$942$		0			0
$943$	5.77058e11			0.729748
$944$		0			0
$945$	−5.53989e10	−	1.02871e10i	−0.0694662	−	0.0128992i
$946$		0			0
$947$			6.00249e10i			0.0746331i	0.999303	+	0.0373165i	$0.0118810\pi$
$947$			6.00249e10i			0.0746331i	−0.999303	+	0.0373165i	$0.988119\pi$
$948$		0			0
$949$	−2.26977e11			−0.279845
$950$		0			0
$951$	−2.84998e11	−	1.74636e10i	−0.348433	−	0.0213507i
$952$		0			0
$953$			5.03889e11i			0.610890i	0.952210	+	0.305445i	$0.0988052\pi$
$953$			5.03889e11i			0.610890i	−0.952210	+	0.305445i	$0.901195\pi$
$954$		0			0
$955$	3.87794e11			0.466216
$956$		0			0
$957$	−3.83210e10	+	6.25379e11i	−0.0456866	+	0.745582i
$958$		0			0
$959$			1.10363e11i			0.130482i
$960$		0			0
$961$	1.42126e12			1.66640
$962$		0			0
$963$	−1.25945e11	+	1.02382e12i	−0.146446	+	1.19047i
$964$		0			0
$965$		−	1.92557e9i		−	0.00222050i
$966$		0			0
$967$	−1.04596e11			−0.119622			−0.0598110	−	0.998210i	$-0.519050\pi$
$967$	−1.04596e11			−0.119622			−0.0598110	+	0.998210i	$0.519050\pi$
$968$		0			0
$969$	1.25749e12	+	7.70545e10i	1.42630	+	0.0873983i
$970$		0			0
$971$		−	1.18109e12i		−	1.32864i	−0.747449	−	0.664319i	$-0.768722\pi$
$971$		−	1.18109e12i		−	1.32864i	0.747449	−	0.664319i	$-0.231278\pi$
$972$		0			0
$973$	6.40629e10			0.0714751
$974$		0			0
$975$	4.94442e10	−	8.06904e11i	0.0547138	−	0.892901i
$976$		0			0
$977$			3.68979e11i			0.404971i	0.979285	+	0.202486i	$0.0649019\pi$
$977$			3.68979e11i			0.404971i	−0.979285	+	0.202486i	$0.935098\pi$
$978$		0			0
$979$	−4.21452e11			−0.458793
$980$		0			0
$981$	1.63654e11	+	2.01319e10i	0.176706	+	0.0217374i
$982$		0			0
$983$			4.42608e11i			0.474030i	0.971506	+	0.237015i	$0.0761690\pi$
$983$			4.42608e11i			0.474030i	−0.971506	+	0.237015i	$0.923831\pi$
$984$		0			0
$985$	2.22656e11			0.236532
$986$		0			0
$987$	1.19470e11	+	7.32071e9i	0.125890	+	0.00771409i
$988$		0			0
$989$			7.81314e11i			0.816659i
$990$		0			0
$991$	−4.61329e11			−0.478317			−0.239159	−	0.970981i	$-0.576872\pi$
$991$	−4.61329e11			−0.478317			−0.239159	+	0.970981i	$0.576872\pi$
$992$		0			0
$993$	−6.13936e10	+	1.00191e12i	−0.0631431	+	1.03046i
$994$		0			0
$995$		−	1.95158e11i		−	0.199110i
$996$		0			0
$997$	4.87409e11			0.493302			0.246651	−	0.969104i	$-0.420670\pi$
$997$	4.87409e11			0.493302			0.246651	+	0.969104i	$0.420670\pi$
$998$		0			0
$999$	2.87015e10	−	1.54566e11i	0.0288166	−	0.155186i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.9.e.a.17.4	yes	8
3.2	odd	2	inner	24.9.e.a.17.3	✓	8
4.3	odd	2		48.9.e.e.17.5		8
8.3	odd	2		192.9.e.j.65.4		8
8.5	even	2		192.9.e.i.65.5		8
12.11	even	2		48.9.e.e.17.6		8
24.5	odd	2		192.9.e.i.65.6		8
24.11	even	2		192.9.e.j.65.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.9.e.a.17.3	✓	8	3.2	odd	2	inner
24.9.e.a.17.4	yes	8	1.1	even	1	trivial
48.9.e.e.17.5		8	4.3	odd	2
48.9.e.e.17.6		8	12.11	even	2
192.9.e.i.65.5		8	8.5	even	2
192.9.e.i.65.6		8	24.5	odd	2
192.9.e.j.65.3		8	24.11	even	2
192.9.e.j.65.4		8	8.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	24.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.95410 + 80.8484i) q^{3} +404.296i q^{5} -262.245 q^{7} +(-6511.91 - 801.062i) q^{9} -6268.43i q^{11} -43933.9 q^{13} +(-32686.7 - 2002.92i) q^{15} +91180.4i q^{17} -170581. q^{19} +(1299.19 - 21202.1i) q^{21} -234950. i q^{23} +227170. q^{25} +(97025.2 - 522509. i) q^{27} +1.23399e6i q^{29} +1.50803e6 q^{31} +(506792. + 31054.4i) q^{33} -106025. i q^{35} +295815. q^{37} +(217653. - 3.55198e6i) q^{39} +2.45609e6i q^{41} -3.32545e6 q^{43} +(323866. - 2.63274e6i) q^{45} +5.63483e6i q^{47} -5.69603e6 q^{49} +(-7.37179e6 - 451717. i) q^{51} +4.50039e6i q^{53} +2.53430e6 q^{55} +(845078. - 1.37912e7i) q^{57} +9.47475e6i q^{59} -5.61098e6 q^{61} +(1.70772e6 + 210075. i) q^{63} -1.77623e7i q^{65} +3.14398e7 q^{67} +(1.89953e7 + 1.16397e6i) q^{69} +1.69223e7i q^{71} +5.16633e6 q^{73} +(-1.12542e6 + 1.83663e7i) q^{75} +1.64387e6i q^{77} +5.76774e6 q^{79} +(4.17633e7 + 1.04329e7i) q^{81} -4.72276e7i q^{83} -3.68639e7 q^{85} +(-9.97664e7 - 6.11333e6i) q^{87} -6.72340e7i q^{89} +1.15215e7 q^{91} +(-7.47092e6 + 1.21922e8i) q^{93} -6.89654e7i q^{95} -1.36244e8 q^{97} +(-5.02140e6 + 4.08195e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 56 q^{3} + 1584 q^{7} + 328 q^{9} + 25232 q^{13} + 38336 q^{15} + 157936 q^{19} + 30480 q^{21} - 579704 q^{25} - 276040 q^{27} + 805552 q^{31} + 102848 q^{33} - 3985008 q^{37} - 2297104 q^{39} + 6962672 q^{43}+ \cdots - 369701504 q^{99}+O(q^{100}) \) 8 * q + 56 * q^3 + 1584 * q^7 + 328 * q^9 + 25232 * q^13 + 38336 * q^15 + 157936 * q^19 + 30480 * q^21 - 579704 * q^25 - 276040 * q^27 + 805552 * q^31 + 102848 * q^33 - 3985008 * q^37 - 2297104 * q^39 + 6962672 * q^43 + 8670592 * q^45 - 5884520 * q^49 - 15590144 * q^51 + 27101312 * q^55 + 36756688 * q^57 - 51583600 * q^61 - 69759312 * q^63 + 58200688 * q^67 + 94226048 * q^69 - 116854768 * q^73 - 143181896 * q^75 + 172454576 * q^79 + 194700040 * q^81 - 264333824 * q^85 - 242851008 * q^87 + 382128480 * q^91 + 313470352 * q^93 - 337326704 * q^97 - 369701504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more