LMFDB - Embedded newform 24.9.e.a.17.2

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			17.2
Root			$1.26427 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	24.17
Dual form			24.9.e.a.17.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-76.4245 + 26.8384i) q^{3} +295.589i q^{5} -843.136 q^{7} +(5120.40 - 4102.23i) q^{9} -26059.3i q^{11} +22818.6 q^{13} +(-7933.16 - 22590.3i) q^{15} -84406.1i q^{17} -6534.32 q^{19} +(64436.3 - 22628.5i) q^{21} -227233. i q^{23} +303252. q^{25} +(-281226. + 450934. i) q^{27} +678039. i q^{29} -1.71504e6 q^{31} +(699391. + 1.99157e6i) q^{33} -249222. i q^{35} +1.40444e6 q^{37} +(-1.74390e6 + 612415. i) q^{39} -3.78445e6i q^{41} +2.61805e6 q^{43} +(1.21257e6 + 1.51354e6i) q^{45} -7.44156e6i q^{47} -5.05392e6 q^{49} +(2.26533e6 + 6.45069e6i) q^{51} -9.84213e6i q^{53} +7.70286e6 q^{55} +(499382. - 175371. i) q^{57} -6.41059e6i q^{59} -7.03941e6 q^{61} +(-4.31719e6 + 3.45874e6i) q^{63} +6.74493e6i q^{65} +1.35782e7 q^{67} +(6.09857e6 + 1.73661e7i) q^{69} +3.94753e7i q^{71} -5.36079e7 q^{73} +(-2.31759e7 + 8.13881e6i) q^{75} +2.19716e7i q^{77} +4.45020e7 q^{79} +(9.39020e6 - 4.20101e7i) q^{81} -7.91003e6i q^{83} +2.49496e7 q^{85} +(-1.81975e7 - 5.18188e7i) q^{87} -2.17407e7i q^{89} -1.92392e7 q^{91} +(1.31071e8 - 4.60290e7i) q^{93} -1.93148e6i q^{95} -2.69154e7 q^{97} +(-1.06901e8 - 1.33434e8i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−76.4245	+	26.8384i	−0.943512	+	0.331339i
$3$	−76.4245	+	26.8384i	−0.943512	+	0.331339i
$4$		0			0
$5$			295.589i			0.472943i	0.971638	+	0.236472i	$0.0759910\pi$
$5$			295.589i			0.472943i	−0.971638	+	0.236472i	$0.924009\pi$
$6$		0			0
$7$	−843.136			−0.351161			−0.175580	−	0.984465i	$-0.556180\pi$
$7$	−843.136			−0.351161			−0.175580	+	0.984465i	$0.556180\pi$
$8$		0			0
$9$	5120.40	−	4102.23i	0.780429	−	0.625244i
$10$		0			0
$11$		−	26059.3i		−	1.77989i	−0.456072	−	0.889943i	$-0.650744\pi$
$11$		−	26059.3i		−	1.77989i	0.456072	−	0.889943i	$-0.349256\pi$
$12$		0			0
$13$	22818.6			0.798942			0.399471	−	0.916746i	$-0.369194\pi$
$13$	22818.6			0.798942			0.399471	+	0.916746i	$0.369194\pi$
$14$		0			0
$15$	−7933.16	−	22590.3i	−0.156704	−	0.446227i
$16$		0			0
$17$		−	84406.1i		−	1.01060i	−0.862944	−	0.505299i	$-0.831382\pi$
$17$		−	84406.1i		−	1.01060i	0.862944	−	0.505299i	$-0.168618\pi$
$18$		0			0
$19$	−6534.32			−0.0501402			−0.0250701	−	0.999686i	$-0.507981\pi$
$19$	−6534.32			−0.0501402			−0.0250701	+	0.999686i	$0.507981\pi$
$20$		0			0
$21$	64436.3	−	22628.5i	0.331324	−	0.116353i
$22$		0			0
$23$		−	227233.i		−	0.812007i	−0.913872	−	0.406003i	$-0.866922\pi$
$23$		−	227233.i		−	0.812007i	0.913872	−	0.406003i	$-0.133078\pi$
$24$		0			0
$25$	303252.			0.776325
$26$		0			0
$27$	−281226.	+	450934.i	−0.529177	+	0.848512i
$28$		0			0
$29$			678039.i			0.958656i	0.877636	+	0.479328i	$0.159120\pi$
$29$			678039.i			0.958656i	−0.877636	+	0.479328i	$0.840880\pi$
$30$		0			0
$31$	−1.71504e6			−1.85707			−0.928534	−	0.371248i	$-0.878930\pi$
$31$	−1.71504e6			−1.85707			−0.928534	+	0.371248i	$0.878930\pi$
$32$		0			0
$33$	699391.	+	1.99157e6i	0.589745	+	1.67934i
$34$		0			0
$35$		−	249222.i		−	0.166079i
$36$		0			0
$37$	1.40444e6			0.749372			0.374686	−	0.927152i	$-0.377750\pi$
$37$	1.40444e6			0.749372			0.374686	+	0.927152i	$0.377750\pi$
$38$		0			0
$39$	−1.74390e6	+	612415.i	−0.753811	+	0.264720i
$40$		0			0
$41$		−	3.78445e6i		−	1.33927i	−0.742691	−	0.669635i	$-0.766451\pi$
$41$		−	3.78445e6i		−	1.33927i	0.742691	−	0.669635i	$-0.233549\pi$
$42$		0			0
$43$	2.61805e6			0.765781			0.382890	−	0.923794i	$-0.374929\pi$
$43$	2.61805e6			0.765781			0.382890	+	0.923794i	$0.374929\pi$
$44$		0			0
$45$	1.21257e6	+	1.51354e6i	0.295705	+	0.369099i
$46$		0			0
$47$		−	7.44156e6i		−	1.52501i	−0.646982	−	0.762505i	$-0.723969\pi$
$47$		−	7.44156e6i		−	1.52501i	0.646982	−	0.762505i	$-0.276031\pi$
$48$		0			0
$49$	−5.05392e6			−0.876686
$50$		0			0
$51$	2.26533e6	+	6.45069e6i	0.334850	+	0.953511i
$52$		0			0
$53$		−	9.84213e6i		−	1.24734i	−0.781687	−	0.623671i	$-0.785640\pi$
$53$		−	9.84213e6i		−	1.24734i	0.781687	−	0.623671i	$-0.214360\pi$
$54$		0			0
$55$	7.70286e6			0.841785
$56$		0			0
$57$	499382.	−	175371.i	0.0473079	−	0.0166134i
$58$		0			0
$59$		−	6.41059e6i		−	0.529042i	−0.964380	−	0.264521i	$-0.914786\pi$
$59$		−	6.41059e6i		−	0.529042i	0.964380	−	0.264521i	$-0.0852138\pi$
$60$		0			0
$61$	−7.03941e6			−0.508413			−0.254207	−	0.967150i	$-0.581814\pi$
$61$	−7.03941e6			−0.508413			−0.254207	+	0.967150i	$0.581814\pi$
$62$		0			0
$63$	−4.31719e6	+	3.45874e6i	−0.274056	+	0.219561i
$64$		0			0
$65$			6.74493e6i			0.377854i
$66$		0			0
$67$	1.35782e7			0.673817			0.336908	−	0.941537i	$-0.390619\pi$
$67$	1.35782e7			0.673817			0.336908	+	0.941537i	$0.390619\pi$
$68$		0			0
$69$	6.09857e6	+	1.73661e7i	0.269049	+	0.766138i
$70$		0			0
$71$			3.94753e7i			1.55343i	0.629850	+	0.776716i	$0.283116\pi$
$71$			3.94753e7i			1.55343i	−0.629850	+	0.776716i	$0.716884\pi$
$72$		0			0
$73$	−5.36079e7			−1.88772			−0.943859	−	0.330348i	$-0.892834\pi$
$73$	−5.36079e7			−1.88772			−0.943859	+	0.330348i	$0.892834\pi$
$74$		0			0
$75$	−2.31759e7	+	8.13881e6i	−0.732472	+	0.257226i
$76$		0			0
$77$			2.19716e7i			0.625026i
$78$		0			0
$79$	4.45020e7			1.14254			0.571270	−	0.820763i	$-0.306451\pi$
$79$	4.45020e7			1.14254			0.571270	+	0.820763i	$0.306451\pi$
$80$		0			0
$81$	9.39020e6	−	4.20101e7i	0.218140	−	0.975918i
$82$		0			0
$83$		−	7.91003e6i		−	0.166673i	−0.996521	−	0.0833366i	$-0.973442\pi$
$83$		−	7.91003e6i		−	0.166673i	0.996521	−	0.0833366i	$-0.0265577\pi$
$84$		0			0
$85$	2.49496e7			0.477955
$86$		0			0
$87$	−1.81975e7	−	5.18188e7i	−0.317640	−	0.904503i
$88$		0			0
$89$		−	2.17407e7i		−	0.346508i	−0.984877	−	0.173254i	$-0.944572\pi$
$89$		−	2.17407e7i		−	0.346508i	0.984877	−	0.173254i	$-0.0554281\pi$
$90$		0			0
$91$	−1.92392e7			−0.280557
$92$		0			0
$93$	1.31071e8	−	4.60290e7i	1.75217	−	0.615318i
$94$		0			0
$95$		−	1.93148e6i		−	0.0237135i
$96$		0			0
$97$	−2.69154e7			−0.304028			−0.152014	−	0.988378i	$-0.548576\pi$
$97$	−2.69154e7			−0.304028			−0.152014	+	0.988378i	$0.548576\pi$
$98$		0			0
$99$	−1.06901e8	−	1.33434e8i	−1.11286	−	1.38908i
$100$		0			0
$101$			1.04904e8i			1.00810i	0.863674	+	0.504051i	$0.168158\pi$
$101$			1.04904e8i			1.00810i	−0.863674	+	0.504051i	$0.831842\pi$
$102$		0			0
$103$	−1.36257e7			−0.121063			−0.0605313	−	0.998166i	$-0.519279\pi$
$103$	−1.36257e7			−0.121063			−0.0605313	+	0.998166i	$0.519279\pi$
$104$		0			0
$105$	6.68874e6	+	1.90467e7i	0.0550284	+	0.156697i
$106$		0			0
$107$			7.80372e6i			0.0595342i	0.999557	+	0.0297671i	$0.00947656\pi$
$107$			7.80372e6i			0.0595342i	−0.999557	+	0.0297671i	$0.990523\pi$
$108$		0			0
$109$	1.58365e8			1.12190			0.560950	−	0.827850i	$-0.310436\pi$
$109$	1.58365e8			1.12190			0.560950	+	0.827850i	$0.310436\pi$
$110$		0			0
$111$	−1.07334e8	+	3.76931e7i	−0.707042	+	0.248296i
$112$		0			0
$113$			3.68590e7i			0.226063i	0.993591	+	0.113032i	$0.0360561\pi$
$113$			3.68590e7i			0.226063i	−0.993591	+	0.113032i	$0.963944\pi$
$114$		0			0
$115$	6.71676e7			0.384033
$116$		0			0
$117$	1.16840e8	−	9.36070e7i	0.623518	−	0.499534i
$118$		0			0
$119$			7.11659e7i			0.354882i
$120$		0			0
$121$	−4.64729e8			−2.16799
$122$		0			0
$123$	1.01569e8	+	2.89225e8i	0.443752	+	1.26362i
$124$		0			0
$125$			2.05103e8i			0.840101i
$126$		0			0
$127$	9.88841e7			0.380112			0.190056	−	0.981773i	$-0.439133\pi$
$127$	9.88841e7			0.380112			0.190056	+	0.981773i	$0.439133\pi$
$128$		0			0
$129$	−2.00083e8	+	7.02644e7i	−0.722523	+	0.253733i
$130$		0			0
$131$			2.29100e8i			0.777930i	0.921253	+	0.388965i	$0.127167\pi$
$131$			2.29100e8i			0.777930i	−0.921253	+	0.388965i	$0.872833\pi$
$132$		0			0
$133$	5.50932e6			0.0176073
$134$		0			0
$135$	−1.33291e8	−	8.31275e7i	−0.401298	−	0.250270i
$136$		0			0
$137$		−	5.36141e8i		−	1.52194i	−0.648788	−	0.760969i	$-0.724724\pi$
$137$		−	5.36141e8i		−	1.52194i	0.648788	−	0.760969i	$-0.275276\pi$
$138$		0			0
$139$	2.91861e8			0.781838			0.390919	−	0.920425i	$-0.372157\pi$
$139$	2.91861e8			0.781838			0.390919	+	0.920425i	$0.372157\pi$
$140$		0			0
$141$	1.99720e8	+	5.68717e8i	0.505295	+	1.43887i
$142$		0			0
$143$		−	5.94637e8i		−	1.42203i
$144$		0			0
$145$	−2.00421e8			−0.453390
$146$		0			0
$147$	3.86243e8	−	1.35639e8i	0.827164	−	0.290480i
$148$		0			0
$149$		−	7.97219e8i		−	1.61746i	−0.588182	−	0.808728i	$-0.700156\pi$
$149$		−	7.97219e8i		−	1.61746i	0.588182	−	0.808728i	$-0.299844\pi$
$150$		0			0
$151$	4.74739e7			0.0913160			0.0456580	−	0.998957i	$-0.485462\pi$
$151$	4.74739e7			0.0913160			0.0456580	+	0.998957i	$0.485462\pi$
$152$		0			0
$153$	−3.46253e8	−	4.32193e8i	−0.631870	−	0.788700i
$154$		0			0
$155$		−	5.06948e8i		−	0.878287i
$156$		0			0
$157$	7.82849e8			1.28848			0.644242	−	0.764822i	$-0.277173\pi$
$157$	7.82849e8			1.28848			0.644242	+	0.764822i	$0.277173\pi$
$158$		0			0
$159$	2.64147e8	+	7.52180e8i	0.413293	+	1.17688i
$160$		0			0
$161$			1.91588e8i			0.285145i
$162$		0			0
$163$	−4.30721e8			−0.610162			−0.305081	−	0.952326i	$-0.598684\pi$
$163$	−4.30721e8			−0.610162			−0.305081	+	0.952326i	$0.598684\pi$
$164$		0			0
$165$	−5.88687e8	+	2.06733e8i	−0.794234	+	0.278916i
$166$		0			0
$167$			1.06559e8i			0.137001i	0.997651	+	0.0685004i	$0.0218214\pi$
$167$			1.06559e8i			0.137001i	−0.997651	+	0.0685004i	$0.978179\pi$
$168$		0			0
$169$	−2.95043e8			−0.361692
$170$		0			0
$171$	−3.34583e7	+	2.68053e7i	−0.0391309	+	0.0313499i
$172$		0			0
$173$			4.00012e8i			0.446569i	0.974753	+	0.223285i	$0.0716779\pi$
$173$			4.00012e8i			0.446569i	−0.974753	+	0.223285i	$0.928322\pi$
$174$		0			0
$175$	−2.55683e8			−0.272615
$176$		0			0
$177$	1.72050e8	+	4.89926e8i	0.175292	+	0.499157i
$178$		0			0
$179$			5.23159e8i			0.509591i	0.966995	+	0.254796i	$0.0820082\pi$
$179$			5.23159e8i			0.509591i	−0.966995	+	0.254796i	$0.917992\pi$
$180$		0			0
$181$	5.28083e8			0.492026			0.246013	−	0.969267i	$-0.420879\pi$
$181$	5.28083e8			0.492026			0.246013	+	0.969267i	$0.420879\pi$
$182$		0			0
$183$	5.37983e8	−	1.88927e8i	0.479694	−	0.168457i
$184$		0			0
$185$			4.15139e8i			0.354410i
$186$		0			0
$187$	−2.19957e9			−1.79875
$188$		0			0
$189$	2.37112e8	−	3.80199e8i	0.185826	−	0.297964i
$190$		0			0
$191$			7.37738e8i			0.554330i	0.960822	+	0.277165i	$0.0893949\pi$
$191$			7.37738e8i			0.554330i	−0.960822	+	0.277165i	$0.910605\pi$
$192$		0			0
$193$	1.92399e9			1.38667			0.693337	−	0.720613i	$-0.256140\pi$
$193$	1.92399e9			1.38667			0.693337	+	0.720613i	$0.256140\pi$
$194$		0			0
$195$	−1.81023e8	−	5.15478e8i	−0.125198	−	0.356510i
$196$		0			0
$197$			2.02664e9i			1.34559i	0.739831	+	0.672793i	$0.234906\pi$
$197$			2.02664e9i			1.34559i	−0.739831	+	0.672793i	$0.765094\pi$
$198$		0			0
$199$	−5.91241e8			−0.377010			−0.188505	−	0.982072i	$-0.560364\pi$
$199$	−5.91241e8			−0.377010			−0.188505	+	0.982072i	$0.560364\pi$
$200$		0			0
$201$	−1.03770e9	+	3.64417e8i	−0.635754	+	0.223262i
$202$		0			0
$203$		−	5.71679e8i		−	0.336642i
$204$		0			0
$205$	1.11864e9			0.633398
$206$		0			0
$207$	−9.32160e8	−	1.16352e9i	−0.507702	−	0.633714i
$208$		0			0
$209$			1.70280e8i			0.0892438i
$210$		0			0
$211$	−1.43041e9			−0.721656			−0.360828	−	0.932632i	$-0.617506\pi$
$211$	−1.43041e9			−0.721656			−0.360828	+	0.932632i	$0.617506\pi$
$212$		0			0
$213$	−1.05946e9	−	3.01688e9i	−0.514713	−	1.46568i
$214$		0			0
$215$			7.73869e8i			0.362171i
$216$		0			0
$217$	1.44601e9			0.652129
$218$		0			0
$219$	4.09695e9	−	1.43875e9i	1.78108	−	0.625474i
$220$		0			0
$221$		−	1.92603e9i		−	0.807409i
$222$		0			0
$223$	2.90210e9			1.17353			0.586764	−	0.809758i	$-0.300402\pi$
$223$	2.90210e9			1.17353			0.586764	+	0.809758i	$0.300402\pi$
$224$		0			0
$225$	1.55277e9	−	1.24401e9i	0.605867	−	0.485393i
$226$		0			0
$227$		−	7.54053e8i		−	0.283987i	−0.989868	−	0.141993i	$-0.954649\pi$
$227$		−	7.54053e8i		−	0.283987i	0.989868	−	0.141993i	$-0.0453512\pi$
$228$		0			0
$229$	−4.65000e9			−1.69087			−0.845436	−	0.534076i	$-0.820659\pi$
$229$	−4.65000e9			−1.69087			−0.845436	+	0.534076i	$0.820659\pi$
$230$		0			0
$231$	−5.89682e8	−	1.67916e9i	−0.207095	−	0.589719i
$232$		0			0
$233$		−	4.92943e8i		−	0.167253i	−0.996497	−	0.0836264i	$-0.973350\pi$
$233$		−	4.92943e8i		−	0.167253i	0.996497	−	0.0836264i	$-0.0266502\pi$
$234$		0			0
$235$	2.19965e9			0.721243
$236$		0			0
$237$	−3.40104e9	+	1.19436e9i	−1.07800	+	0.378567i
$238$		0			0
$239$			2.01507e9i			0.617587i	0.951129	+	0.308794i	$0.0999253\pi$
$239$			2.01507e9i			0.617587i	−0.951129	+	0.308794i	$0.900075\pi$
$240$		0			0
$241$	1.02858e9			0.304908			0.152454	−	0.988311i	$-0.451282\pi$
$241$	1.02858e9			0.304908			0.152454	+	0.988311i	$0.451282\pi$
$242$		0			0
$243$	4.09843e8	+	3.46261e9i	0.117542	+	0.993068i
$244$		0			0
$245$		−	1.49389e9i		−	0.414623i
$246$		0			0
$247$	−1.49104e8			−0.0400591
$248$		0			0
$249$	2.12293e8	+	6.04520e8i	0.0552253	+	0.157258i
$250$		0			0
$251$			6.55871e9i			1.65243i	0.563353	+	0.826216i	$0.309511\pi$
$251$			6.55871e9i			1.65243i	−0.563353	+	0.826216i	$0.690489\pi$
$252$		0			0
$253$	−5.92153e9			−1.44528
$254$		0			0
$255$	−1.90676e9	+	6.69607e8i	−0.450956	+	0.158365i
$256$		0			0
$257$			5.48219e9i			1.25667i	0.777943	+	0.628335i	$0.216263\pi$
$257$			5.48219e9i			1.25667i	−0.777943	+	0.628335i	$0.783737\pi$
$258$		0			0
$259$	−1.18414e9			−0.263150
$260$		0			0
$261$	2.78147e9	+	3.47183e9i	0.599394	+	0.748163i
$262$		0			0
$263$		−	7.06484e9i		−	1.47666i	−0.674442	−	0.738328i	$-0.735616\pi$
$263$		−	7.06484e9i		−	1.47666i	0.674442	−	0.738328i	$-0.264384\pi$
$264$		0			0
$265$	2.90923e9			0.589922
$266$		0			0
$267$	5.83485e8	+	1.66152e9i	0.114811	+	0.326934i
$268$		0			0
$269$			2.54565e8i			0.0486172i	0.999705	+	0.0243086i	$0.00773843\pi$
$269$			2.54565e8i			0.0486172i	−0.999705	+	0.0243086i	$0.992262\pi$
$270$		0			0
$271$	3.98030e9			0.737969			0.368985	−	0.929435i	$-0.379705\pi$
$271$	3.98030e9			0.737969			0.368985	+	0.929435i	$0.379705\pi$
$272$		0			0
$273$	1.47034e9	−	5.16350e8i	0.264709	−	0.0929594i
$274$		0			0
$275$		−	7.90254e9i		−	1.38177i
$276$		0			0
$277$	−1.65715e9			−0.281478			−0.140739	−	0.990047i	$-0.544948\pi$
$277$	−1.65715e9			−0.281478			−0.140739	+	0.990047i	$0.544948\pi$
$278$		0			0
$279$	−8.78169e9	+	7.03549e9i	−1.44931	+	1.16112i
$280$		0			0
$281$		−	3.69038e9i		−	0.591896i	−0.955204	−	0.295948i	$-0.904364\pi$
$281$		−	3.69038e9i		−	0.591896i	0.955204	−	0.295948i	$-0.0956355\pi$
$282$		0			0
$283$	1.37735e9			0.214733			0.107367	−	0.994220i	$-0.465758\pi$
$283$	1.37735e9			0.214733			0.107367	+	0.994220i	$0.465758\pi$
$284$		0			0
$285$	5.18378e7	+	1.47612e8i	0.00785719	+	0.0223739i
$286$		0			0
$287$			3.19081e9i			0.470299i
$288$		0			0
$289$	−1.48639e8			−0.0213079
$290$		0			0
$291$	2.05699e9	−	7.22367e8i	0.286854	−	0.100736i
$292$		0			0
$293$		−	1.13174e10i		−	1.53559i	−0.640698	−	0.767793i	$-0.721355\pi$
$293$		−	1.13174e10i		−	1.53559i	0.640698	−	0.767793i	$-0.278645\pi$
$294$		0			0
$295$	1.89490e9			0.250207
$296$		0			0
$297$	1.17510e10	+	7.32856e9i	1.51025	+	0.941874i
$298$		0			0
$299$		−	5.18513e9i		−	0.648746i
$300$		0			0
$301$	−2.20738e9			−0.268912
$302$		0			0
$303$	−2.81545e9	−	8.01720e9i	−0.334024	−	0.951157i
$304$		0			0
$305$		−	2.08077e9i		−	0.240450i
$306$		0			0
$307$	−1.20343e10			−1.35478			−0.677390	−	0.735624i	$-0.736889\pi$
$307$	−1.20343e10			−1.35478			−0.677390	+	0.735624i	$0.736889\pi$
$308$		0			0
$309$	1.04134e9	−	3.65692e8i	0.114224	−	0.0401127i
$310$		0			0
$311$		−	4.90375e9i		−	0.524188i	−0.965042	−	0.262094i	$-0.915587\pi$
$311$		−	4.90375e9i		−	0.524188i	0.965042	−	0.262094i	$-0.0844130\pi$
$312$		0			0
$313$	5.18630e9			0.540357			0.270178	−	0.962810i	$-0.412917\pi$
$313$	5.18630e9			0.540357			0.270178	+	0.962810i	$0.412917\pi$
$314$		0			0
$315$	−1.02237e9	−	1.27612e9i	−0.103840	−	0.129613i
$316$		0			0
$317$		−	3.07294e9i		−	0.304311i	−0.988357	−	0.152155i	$-0.951379\pi$
$317$		−	3.07294e9i		−	0.304311i	0.988357	−	0.152155i	$-0.0486214\pi$
$318$		0			0
$319$	1.76692e10			1.70630
$320$		0			0
$321$	−2.09440e8	−	5.96395e8i	−0.0197260	−	0.0561712i
$322$		0			0
$323$			5.51537e8i			0.0506716i
$324$		0			0
$325$	6.91978e9			0.620239
$326$		0			0
$327$	−1.21030e10	+	4.25028e9i	−1.05853	+	0.371729i
$328$		0			0
$329$			6.27425e9i			0.535523i
$330$		0			0
$331$	−9.73266e9			−0.810811			−0.405405	−	0.914137i	$-0.632870\pi$
$331$	−9.73266e9			−0.810811			−0.405405	+	0.914137i	$0.632870\pi$
$332$		0			0
$333$	7.19131e9	−	5.76135e9i	0.584832	−	0.468541i
$334$		0			0
$335$			4.01356e9i			0.318677i
$336$		0			0
$337$	−1.52628e10			−1.18336			−0.591678	−	0.806174i	$-0.701534\pi$
$337$	−1.52628e10			−1.18336			−0.591678	+	0.806174i	$0.701534\pi$
$338$		0			0
$339$	−9.89238e8	−	2.81693e9i	−0.0749035	−	0.213293i
$340$		0			0
$341$			4.46928e10i			3.30537i
$342$		0			0
$343$	9.12166e9			0.659018
$344$		0			0
$345$	−5.13325e9	+	1.80267e9i	−0.362340	+	0.127245i
$346$		0			0
$347$			1.25598e10i			0.866296i	0.901323	+	0.433148i	$0.142597\pi$
$347$			1.25598e10i			0.866296i	−0.901323	+	0.433148i	$0.857403\pi$
$348$		0			0
$349$	−7.95406e9			−0.536151			−0.268075	−	0.963398i	$-0.586388\pi$
$349$	−7.95406e9			−0.536151			−0.268075	+	0.963398i	$0.586388\pi$
$350$		0			0
$351$	−6.41718e9	+	1.02897e10i	−0.422782	+	0.677912i
$352$		0			0
$353$		−	1.95049e10i		−	1.25616i	−0.778148	−	0.628082i	$-0.783840\pi$
$353$		−	1.95049e10i		−	1.25616i	0.778148	−	0.628082i	$-0.216160\pi$
$354$		0			0
$355$	−1.16685e10			−0.734685
$356$		0			0
$357$	−1.90998e9	−	5.43882e9i	−0.117586	−	0.334835i
$358$		0			0
$359$			2.51611e10i			1.51479i	0.652957	+	0.757395i	$0.273528\pi$
$359$			2.51611e10i			1.51479i	−0.652957	+	0.757395i	$0.726472\pi$
$360$		0			0
$361$	−1.69409e10			−0.997486
$362$		0			0
$363$	3.55167e10	−	1.24726e10i	2.04553	−	0.718341i
$364$		0			0
$365$		−	1.58459e10i		−	0.892783i
$366$		0			0
$367$	2.38106e10			1.31252			0.656259	−	0.754536i	$-0.272138\pi$
$367$	2.38106e10			1.31252			0.656259	+	0.754536i	$0.272138\pi$
$368$		0			0
$369$	−1.55247e10	−	1.93779e10i	−0.837370	−	1.04520i
$370$		0			0
$371$			8.29826e9i			0.438017i
$372$		0			0
$373$	1.41525e10			0.731134			0.365567	−	0.930785i	$-0.380875\pi$
$373$	1.41525e10			0.731134			0.365567	+	0.930785i	$0.380875\pi$
$374$		0			0
$375$	−5.50464e9	−	1.56749e10i	−0.278358	−	0.792645i
$376$		0			0
$377$			1.54719e10i			0.765910i
$378$		0			0
$379$	1.84700e10			0.895180			0.447590	−	0.894239i	$-0.352282\pi$
$379$	1.84700e10			0.895180			0.447590	+	0.894239i	$0.352282\pi$
$380$		0			0
$381$	−7.55716e9	+	2.65389e9i	−0.358640	+	0.125946i
$382$		0			0
$383$		−	2.52881e10i		−	1.17523i	−0.809142	−	0.587613i	$-0.800068\pi$
$383$		−	2.52881e10i		−	1.17523i	0.809142	−	0.587613i	$-0.199932\pi$
$384$		0			0
$385$	−6.49456e9			−0.295602
$386$		0			0
$387$	1.34055e10	−	1.07398e10i	0.597638	−	0.478800i
$388$		0			0
$389$		−	1.87703e10i		−	0.819733i	−0.912146	−	0.409866i	$-0.865575\pi$
$389$		−	1.87703e10i		−	0.819733i	0.912146	−	0.409866i	$-0.134425\pi$
$390$		0			0
$391$	−1.91798e10			−0.820612
$392$		0			0
$393$	−6.14869e9	−	1.75089e10i	−0.257758	−	0.733986i
$394$		0			0
$395$			1.31543e10i			0.540356i
$396$		0			0
$397$	2.42988e10			0.978187			0.489094	−	0.872231i	$-0.337328\pi$
$397$	2.42988e10			0.978187			0.489094	+	0.872231i	$0.337328\pi$
$398$		0			0
$399$	−4.21047e8	+	1.47862e8i	−0.0166127	+	0.00583397i
$400$		0			0
$401$			1.27648e10i			0.493669i	0.969058	+	0.246835i	$0.0793904\pi$
$401$			1.27648e10i			0.493669i	−0.969058	+	0.246835i	$0.920610\pi$
$402$		0			0
$403$	−3.91348e10			−1.48369
$404$		0			0
$405$	1.24177e10	+	2.77564e9i	0.461553	+	0.103168i
$406$		0			0
$407$		−	3.65988e10i		−	1.33380i
$408$		0			0
$409$	4.95148e9			0.176947			0.0884733	−	0.996079i	$-0.471801\pi$
$409$	4.95148e9			0.176947			0.0884733	+	0.996079i	$0.471801\pi$
$410$		0			0
$411$	1.43892e10	+	4.09743e10i	0.504277	+	1.43597i
$412$		0			0
$413$			5.40500e9i			0.185779i
$414$		0			0
$415$	2.33812e9			0.0788269
$416$		0			0
$417$	−2.23053e10	+	7.83309e9i	−0.737673	+	0.259053i
$418$		0			0
$419$		−	2.05827e10i		−	0.667799i	−0.942609	−	0.333900i	$-0.891635\pi$
$419$		−	2.05827e10i		−	0.667799i	0.942609	−	0.333900i	$-0.108365\pi$
$420$		0			0
$421$	4.70625e10			1.49812			0.749060	−	0.662502i	$-0.230505\pi$
$421$	4.70625e10			1.49812			0.749060	+	0.662502i	$0.230505\pi$
$422$		0			0
$423$	−3.05270e10	−	3.81038e10i	−0.953503	−	1.19016i
$424$		0			0
$425$		−	2.55963e10i		−	0.784552i
$426$		0			0
$427$	5.93518e9			0.178535
$428$		0			0
$429$	1.59591e10	+	4.54448e10i	0.471172	+	1.34170i
$430$		0			0
$431$			6.30963e10i			1.82850i	0.405150	+	0.914250i	$0.367219\pi$
$431$			6.30963e10i			1.82850i	−0.405150	+	0.914250i	$0.632781\pi$
$432$		0			0
$433$	5.98569e9			0.170280			0.0851398	−	0.996369i	$-0.472866\pi$
$433$	5.98569e9			0.170280			0.0851398	+	0.996369i	$0.472866\pi$
$434$		0			0
$435$	1.53171e10	−	5.37899e9i	0.427778	−	0.150226i
$436$		0			0
$437$			1.48481e9i			0.0407142i
$438$		0			0
$439$	−4.69975e10			−1.26537			−0.632683	−	0.774411i	$-0.718046\pi$
$439$	−4.69975e10			−1.26537			−0.632683	+	0.774411i	$0.718046\pi$
$440$		0			0
$441$	−2.58781e10	+	2.07323e10i	−0.684192	+	0.548143i
$442$		0			0
$443$		−	3.33997e10i		−	0.867218i	−0.901101	−	0.433609i	$-0.857240\pi$
$443$		−	3.33997e10i		−	0.867218i	0.901101	−	0.433609i	$-0.142760\pi$
$444$		0			0
$445$	6.42631e9			0.163878
$446$		0			0
$447$	2.13961e10	+	6.09270e10i	0.535926	+	1.52609i
$448$		0			0
$449$			1.25339e10i			0.308391i	0.988040	+	0.154195i	$0.0492786\pi$
$449$			1.25339e10i			0.308391i	−0.988040	+	0.154195i	$0.950721\pi$
$450$		0			0
$451$	−9.86203e10			−2.38375
$452$		0			0
$453$	−3.62816e9	+	1.27412e9i	−0.0861577	+	0.0302565i
$454$		0			0
$455$		−	5.68690e9i		−	0.132687i
$456$		0			0
$457$	−5.49070e9			−0.125882			−0.0629409	−	0.998017i	$-0.520048\pi$
$457$	−5.49070e9			−0.125882			−0.0629409	+	0.998017i	$0.520048\pi$
$458$		0			0
$459$	3.80616e10	+	2.37372e10i	0.857504	+	0.534785i
$460$		0			0
$461$			3.42257e10i			0.757790i	0.925440	+	0.378895i	$0.123696\pi$
$461$			3.42257e10i			0.757790i	−0.925440	+	0.378895i	$0.876304\pi$
$462$		0			0
$463$	7.56677e10			1.64659			0.823297	−	0.567611i	$-0.192132\pi$
$463$	7.56677e10			1.64659			0.823297	+	0.567611i	$0.192132\pi$
$464$		0			0
$465$	1.36057e10	+	3.87432e10i	0.291011	+	0.828674i
$466$		0			0
$467$			1.38885e10i			0.292003i	0.989284	+	0.146002i	$0.0466405\pi$
$467$			1.38885e10i			0.292003i	−0.989284	+	0.146002i	$0.953359\pi$
$468$		0			0
$469$	−1.14482e10			−0.236618
$470$		0			0
$471$	−5.98288e10	+	2.10104e10i	−1.21570	+	0.426925i
$472$		0			0
$473$		−	6.82246e10i		−	1.36300i
$474$		0			0
$475$	−1.98154e9			−0.0389251
$476$		0			0
$477$	−4.03747e10	−	5.03956e10i	−0.779893	−	0.973463i
$478$		0			0
$479$		−	7.89474e10i		−	1.49967i	−0.661625	−	0.749835i	$-0.730133\pi$
$479$		−	7.89474e10i		−	1.49967i	0.661625	−	0.749835i	$-0.269867\pi$
$480$		0			0
$481$	3.20474e10			0.598705
$482$		0			0
$483$	−5.14193e9	−	1.46420e10i	−0.0944795	−	0.269037i
$484$		0			0
$485$		−	7.95590e9i		−	0.143788i
$486$		0			0
$487$	4.98983e10			0.887095			0.443548	−	0.896251i	$-0.353720\pi$
$487$	4.98983e10			0.887095			0.443548	+	0.896251i	$0.353720\pi$
$488$		0			0
$489$	3.29176e10	−	1.15599e10i	0.575695	−	0.202170i
$490$		0			0
$491$		−	5.37198e10i		−	0.924290i	−0.886804	−	0.462145i	$-0.847080\pi$
$491$		−	5.37198e10i		−	0.924290i	0.886804	−	0.462145i	$-0.152920\pi$
$492$		0			0
$493$	5.72306e10			0.968815
$494$		0			0
$495$	3.94417e10	−	3.15989e10i	0.656954	−	0.526321i
$496$		0			0
$497$		−	3.32831e10i		−	0.545504i
$498$		0			0
$499$	−3.84919e10			−0.620822			−0.310411	−	0.950602i	$-0.600467\pi$
$499$	−3.84919e10			−0.620822			−0.310411	+	0.950602i	$0.600467\pi$
$500$		0			0
$501$	−2.85987e9	−	8.14369e9i	−0.0453937	−	0.129262i
$502$		0			0
$503$			2.52386e10i			0.394270i	0.980376	+	0.197135i	$0.0631637\pi$
$503$			2.52386e10i			0.394270i	−0.980376	+	0.197135i	$0.936836\pi$
$504$		0			0
$505$	−3.10084e10			−0.476775
$506$		0			0
$507$	2.25485e10	−	7.91849e9i	0.341260	−	0.119842i
$508$		0			0
$509$		−	2.21872e10i		−	0.330546i	−0.986248	−	0.165273i	$-0.947149\pi$
$509$		−	2.21872e10i		−	0.330546i	0.986248	−	0.165273i	$-0.0528505\pi$
$510$		0			0
$511$	4.51988e10			0.662892
$512$		0			0
$513$	1.83762e9	−	2.94655e9i	0.0265330	−	0.0425445i
$514$		0			0
$515$		−	4.02761e9i		−	0.0572557i
$516$		0			0
$517$	−1.93922e11			−2.71434
$518$		0			0
$519$	−1.07357e10	−	3.05707e10i	−0.147966	−	0.421343i
$520$		0			0
$521$			9.85762e10i			1.33789i	0.743311	+	0.668946i	$0.233254\pi$
$521$			9.85762e10i			1.33789i	−0.743311	+	0.668946i	$0.766746\pi$
$522$		0			0
$523$	1.15879e11			1.54881			0.774406	−	0.632689i	$-0.218049\pi$
$523$	1.15879e11			1.54881			0.774406	+	0.632689i	$0.218049\pi$
$524$		0			0
$525$	1.95404e10	−	6.86213e9i	0.257215	−	0.0903278i
$526$		0			0
$527$			1.44760e11i			1.87675i
$528$		0			0
$529$	2.66763e10			0.340645
$530$		0			0
$531$	−2.62977e10	−	3.28248e10i	−0.330780	−	0.412880i
$532$		0			0
$533$		−	8.63559e10i		−	1.07000i
$534$		0			0
$535$	−2.30670e9			−0.0281563
$536$		0			0
$537$	−1.40408e10	−	3.99822e10i	−0.168847	−	0.480805i
$538$		0			0
$539$			1.31702e11i			1.56040i
$540$		0			0
$541$	−6.56153e10			−0.765978			−0.382989	−	0.923753i	$-0.625105\pi$
$541$	−6.56153e10			−0.765978			−0.382989	+	0.923753i	$0.625105\pi$
$542$		0			0
$543$	−4.03585e10	+	1.41729e10i	−0.464232	+	0.163027i
$544$		0			0
$545$			4.68111e10i			0.530595i
$546$		0			0
$547$	1.02875e10			0.114911			0.0574556	−	0.998348i	$-0.481701\pi$
$547$	1.02875e10			0.114911			0.0574556	+	0.998348i	$0.481701\pi$
$548$		0			0
$549$	−3.60445e10	+	2.88772e10i	−0.396780	+	0.317882i
$550$		0			0
$551$		−	4.43052e9i		−	0.0480672i
$552$		0			0
$553$	−3.75213e10			−0.401215
$554$		0			0
$555$	−1.11417e10	−	3.17268e10i	−0.117430	−	0.334390i
$556$		0			0
$557$			1.04225e11i			1.08280i	0.840764	+	0.541402i	$0.182106\pi$
$557$			1.04225e11i			1.08280i	−0.840764	+	0.541402i	$0.817894\pi$
$558$		0			0
$559$	5.97402e10			0.611814
$560$		0			0
$561$	1.68101e11	−	5.90329e10i	1.69714	−	0.595995i
$562$		0			0
$563$		−	5.86461e9i		−	0.0583721i	−0.999574	−	0.0291860i	$-0.990708\pi$
$563$		−	5.86461e9i		−	0.0583721i	0.999574	−	0.0291860i	$-0.00929153\pi$
$564$		0			0
$565$	−1.08951e10			−0.106915
$566$		0			0
$567$	−7.91722e9	+	3.54202e10i	−0.0766021	+	0.342704i
$568$		0			0
$569$			1.06465e11i			1.01568i	0.861450	+	0.507842i	$0.169557\pi$
$569$			1.06465e11i			1.01568i	−0.861450	+	0.507842i	$0.830443\pi$
$570$		0			0
$571$	5.77376e10			0.543143			0.271572	−	0.962418i	$-0.412457\pi$
$571$	5.77376e10			0.543143			0.271572	+	0.962418i	$0.412457\pi$
$572$		0			0
$573$	−1.97997e10	−	5.63812e10i	−0.183671	−	0.523017i
$574$		0			0
$575$		−	6.89088e10i		−	0.630381i
$576$		0			0
$577$	4.53645e10			0.409273			0.204636	−	0.978838i	$-0.434399\pi$
$577$	4.53645e10			0.409273			0.204636	+	0.978838i	$0.434399\pi$
$578$		0			0
$579$	−1.47040e11	+	5.16370e10i	−1.30834	+	0.459459i
$580$		0			0
$581$			6.66924e9i			0.0585291i
$582$		0			0
$583$	−2.56479e11			−2.22013
$584$		0			0
$585$	2.76692e10	+	3.45367e10i	0.236251	+	0.294888i
$586$		0			0
$587$			1.19249e11i			1.00439i	0.864753	+	0.502197i	$0.167475\pi$
$587$			1.19249e11i			1.00439i	−0.864753	+	0.502197i	$0.832525\pi$
$588$		0			0
$589$	1.12066e10			0.0931137
$590$		0			0
$591$	−5.43918e10	−	1.54885e11i	−0.445845	−	1.26958i
$592$		0			0
$593$			1.51133e11i			1.22219i	0.791556	+	0.611097i	$0.209271\pi$
$593$			1.51133e11i			1.22219i	−0.791556	+	0.611097i	$0.790729\pi$
$594$		0			0
$595$	−2.10359e10			−0.167839
$596$		0			0
$597$	4.51853e10	−	1.58680e10i	0.355713	−	0.124918i
$598$		0			0
$599$		−	1.78703e11i		−	1.38811i	−0.719922	−	0.694055i	$-0.755823\pi$
$599$		−	1.78703e11i		−	1.38811i	0.719922	−	0.694055i	$-0.244177\pi$
$600$		0			0
$601$	2.28123e11			1.74852			0.874262	−	0.485455i	$-0.161346\pi$
$601$	2.28123e11			1.74852			0.874262	+	0.485455i	$0.161346\pi$
$602$		0			0
$603$	6.95256e10	−	5.57007e10i	0.525866	−	0.421300i
$604$		0			0
$605$		−	1.37369e11i		−	1.02534i
$606$		0			0
$607$	1.41216e11			1.04023			0.520113	−	0.854097i	$-0.325890\pi$
$607$	1.41216e11			1.04023			0.520113	+	0.854097i	$0.325890\pi$
$608$		0			0
$609$	1.53430e10	+	4.36903e10i	0.111543	+	0.317626i
$610$		0			0
$611$		−	1.69806e11i		−	1.21839i
$612$		0			0
$613$	−1.11995e11			−0.793150			−0.396575	−	0.918002i	$-0.629801\pi$
$613$	−1.11995e11			−0.793150			−0.396575	+	0.918002i	$0.629801\pi$
$614$		0			0
$615$	−8.54918e10	+	3.00227e10i	−0.597619	+	0.209869i
$616$		0			0
$617$		−	2.66321e11i		−	1.83766i	−0.394658	−	0.918828i	$-0.629137\pi$
$617$		−	2.66321e11i		−	1.83766i	0.394658	−	0.918828i	$-0.370863\pi$
$618$		0			0
$619$	4.11190e10			0.280078			0.140039	−	0.990146i	$-0.455277\pi$
$619$	4.11190e10			0.280078			0.140039	+	0.990146i	$0.455277\pi$
$620$		0			0
$621$	1.02467e11	+	6.39038e10i	0.688997	+	0.429695i
$622$		0			0
$623$			1.83303e10i			0.121680i
$624$		0			0
$625$	5.78316e10			0.379005
$626$		0			0
$627$	−4.57005e9	−	1.30135e10i	−0.0295699	−	0.0842026i
$628$		0			0
$629$		−	1.18544e11i		−	0.757314i
$630$		0			0
$631$	−8.58535e10			−0.541553			−0.270776	−	0.962642i	$-0.587280\pi$
$631$	−8.58535e10			−0.541553			−0.270776	+	0.962642i	$0.587280\pi$
$632$		0			0
$633$	1.09318e11	−	3.83900e10i	0.680891	−	0.239113i
$634$		0			0
$635$			2.92291e10i			0.179771i
$636$		0			0
$637$	−1.15323e11			−0.700422
$638$		0			0
$639$	1.61937e11	+	2.02129e11i	0.971275	+	1.21234i
$640$		0			0
$641$		−	1.12986e11i		−	0.669256i	−0.942350	−	0.334628i	$-0.891389\pi$
$641$		−	1.12986e11i		−	0.669256i	0.942350	−	0.334628i	$-0.108611\pi$
$642$		0			0
$643$	8.53586e10			0.499348			0.249674	−	0.968330i	$-0.419677\pi$
$643$	8.53586e10			0.499348			0.249674	+	0.968330i	$0.419677\pi$
$644$		0			0
$645$	−2.07694e10	−	5.91425e10i	−0.120001	−	0.341712i
$646$		0			0
$647$		−	5.35767e10i		−	0.305745i	−0.988246	−	0.152872i	$-0.951148\pi$
$647$		−	5.35767e10i		−	0.305745i	0.988246	−	0.152872i	$-0.0488523\pi$
$648$		0			0
$649$	−1.67056e11			−0.941634
$650$		0			0
$651$	−1.10511e11	+	3.88087e10i	−0.615291	+	0.216076i
$652$		0			0
$653$			3.03189e11i			1.66748i	0.552156	+	0.833741i	$0.313805\pi$
$653$			3.03189e11i			1.66748i	−0.552156	+	0.833741i	$0.686195\pi$
$654$		0			0
$655$	−6.77196e10			−0.367916
$656$		0			0
$657$	−2.74494e11	+	2.19912e11i	−1.47323	+	1.18028i
$658$		0			0
$659$		−	1.05655e11i		−	0.560206i	−0.959970	−	0.280103i	$-0.909631\pi$
$659$		−	1.05655e11i		−	0.560206i	0.959970	−	0.280103i	$-0.0903687\pi$
$660$		0			0
$661$	9.52659e10			0.499036			0.249518	−	0.968370i	$-0.419728\pi$
$661$	9.52659e10			0.499036			0.249518	+	0.968370i	$0.419728\pi$
$662$		0			0
$663$	5.16916e10	+	1.47196e11i	0.267526	+	0.761800i
$664$		0			0
$665$			1.62850e9i			0.00832723i
$666$		0			0
$667$	1.54073e11			0.778435
$668$		0			0
$669$	−2.21792e11	+	7.78879e10i	−1.10724	+	0.388835i
$670$		0			0
$671$			1.83442e11i			0.904917i
$672$		0			0
$673$	1.22063e10			0.0595008			0.0297504	−	0.999557i	$-0.490529\pi$
$673$	1.22063e10			0.0595008			0.0297504	+	0.999557i	$0.490529\pi$
$674$		0			0
$675$	−8.52824e10	+	1.36747e11i	−0.410813	+	0.658721i
$676$		0			0
$677$			9.96131e10i			0.474200i	0.971485	+	0.237100i	$0.0761969\pi$
$677$			9.96131e10i			0.474200i	−0.971485	+	0.237100i	$0.923803\pi$
$678$		0			0
$679$	2.26933e10			0.106763
$680$		0			0
$681$	2.02376e10	+	5.76281e10i	0.0940959	+	0.267945i
$682$		0			0
$683$			6.00211e10i			0.275817i	0.990445	+	0.137909i	$0.0440380\pi$
$683$			6.00211e10i			0.275817i	−0.990445	+	0.137909i	$0.955962\pi$
$684$		0			0
$685$	1.58478e11			0.719790
$686$		0			0
$687$	3.55374e11	−	1.24799e11i	1.59536	−	0.560252i
$688$		0			0
$689$		−	2.24584e11i		−	0.996554i
$690$		0			0
$691$	−3.50694e11			−1.53821			−0.769107	−	0.639120i	$-0.779299\pi$
$691$	−3.50694e11			−1.53821			−0.769107	+	0.639120i	$0.779299\pi$
$692$		0			0
$693$	9.01323e10	+	1.12503e11i	0.390794	+	0.487788i
$694$		0			0
$695$			8.62710e10i			0.369765i
$696$		0			0
$697$	−3.19431e11			−1.35346
$698$		0			0
$699$	1.32298e10	+	3.76729e10i	0.0554173	+	0.157805i
$700$		0			0
$701$		−	4.54851e11i		−	1.88364i	−0.336123	−	0.941818i	$-0.609116\pi$
$701$		−	4.54851e11i		−	1.88364i	0.336123	−	0.941818i	$-0.390884\pi$
$702$		0			0
$703$	−9.17709e9			−0.0375737
$704$		0			0
$705$	−1.68107e11	+	5.90351e10i	−0.680501	+	0.238976i
$706$		0			0
$707$		−	8.84480e10i		−	0.354006i
$708$		0			0
$709$	2.96653e11			1.17399			0.586995	−	0.809591i	$-0.300311\pi$
$709$	2.96653e11			1.17399			0.586995	+	0.809591i	$0.300311\pi$
$710$		0			0
$711$	2.27868e11	−	1.82557e11i	0.891671	−	0.714366i
$712$		0			0
$713$			3.89713e11i			1.50795i
$714$		0			0
$715$	1.75768e11			0.672537
$716$		0			0
$717$	−5.40813e10	−	1.54001e11i	−0.204631	−	0.582701i
$718$		0			0
$719$			3.07407e10i			0.115027i	0.998345	+	0.0575134i	$0.0183172\pi$
$719$			3.07407e10i			0.115027i	−0.998345	+	0.0575134i	$0.981683\pi$
$720$		0			0
$721$	1.14883e10			0.0425124
$722$		0			0
$723$	−7.86085e10	+	2.76054e10i	−0.287685	+	0.101028i
$724$		0			0
$725$			2.05617e11i			0.744228i
$726$		0			0
$727$	−1.46690e11			−0.525124			−0.262562	−	0.964915i	$-0.584567\pi$
$727$	−1.46690e11			−0.525124			−0.262562	+	0.964915i	$0.584567\pi$
$728$		0			0
$729$	−1.24253e11	−	2.53629e11i	−0.439944	−	0.898025i
$730$		0			0
$731$		−	2.20980e11i		−	0.773896i
$732$		0			0
$733$	−1.69832e11			−0.588308			−0.294154	−	0.955758i	$-0.595038\pi$
$733$	−1.69832e11			−0.588308			−0.294154	+	0.955758i	$0.595038\pi$
$734$		0			0
$735$	4.00936e10	+	1.14169e11i	0.137381	+	0.391201i
$736$		0			0
$737$		−	3.53837e11i		−	1.19932i
$738$		0			0
$739$	3.52927e11			1.18333			0.591667	−	0.806182i	$-0.298470\pi$
$739$	3.52927e11			1.18333			0.591667	+	0.806182i	$0.298470\pi$
$740$		0			0
$741$	1.13952e10	−	4.00172e9i	0.0377962	−	0.0132731i
$742$		0			0
$743$			9.72332e10i			0.319050i	0.987194	+	0.159525i	$0.0509963\pi$
$743$			9.72332e10i			0.319050i	−0.987194	+	0.159525i	$0.949004\pi$
$744$		0			0
$745$	2.35650e11			0.764965
$746$		0			0
$747$	−3.24487e10	−	4.05025e10i	−0.104211	−	0.130077i
$748$		0			0
$749$		−	6.57960e9i		−	0.0209061i
$750$		0			0
$751$	−3.61629e11			−1.13685			−0.568425	−	0.822735i	$-0.692447\pi$
$751$	−3.61629e11			−1.13685			−0.568425	+	0.822735i	$0.692447\pi$
$752$		0			0
$753$	−1.76026e11	−	5.01246e11i	−0.547515	−	1.55909i
$754$		0			0
$755$			1.40328e10i			0.0431873i
$756$		0			0
$757$	3.48181e11			1.06028			0.530141	−	0.847909i	$-0.322139\pi$
$757$	3.48181e11			1.06028			0.530141	+	0.847909i	$0.322139\pi$
$758$		0			0
$759$	4.52550e11	−	1.58925e11i	1.36364	−	0.478877i
$760$		0			0
$761$			3.78907e11i			1.12978i	0.825166	+	0.564890i	$0.191081\pi$
$761$			3.78907e11i			1.12978i	−0.825166	+	0.564890i	$0.808919\pi$
$762$		0			0
$763$	−1.33524e11			−0.393967
$764$		0			0
$765$	1.27752e11	−	1.02349e11i	0.373010	−	0.298839i
$766$		0			0
$767$		−	1.46281e11i		−	0.422674i
$768$		0			0
$769$	−3.07319e11			−0.878788			−0.439394	−	0.898294i	$-0.644807\pi$
$769$	−3.07319e11			−0.878788			−0.439394	+	0.898294i	$0.644807\pi$
$770$		0			0
$771$	−1.47133e11	−	4.18973e11i	−0.416384	−	1.18568i
$772$		0			0
$773$			9.66999e10i			0.270837i	0.990788	+	0.135418i	$0.0432379\pi$
$773$			9.66999e10i			0.270837i	−0.990788	+	0.135418i	$0.956762\pi$
$774$		0			0
$775$	−5.20089e11			−1.44169
$776$		0			0
$777$	9.04971e10	−	3.17804e10i	0.248285	−	0.0871918i
$778$		0			0
$779$			2.47288e10i			0.0671512i
$780$		0			0
$781$	1.02870e12			2.76493
$782$		0			0
$783$	−3.05751e11	−	1.90682e11i	−0.813430	−	0.507298i
$784$		0			0
$785$			2.31402e11i			0.609380i
$786$		0			0
$787$	4.67767e11			1.21936			0.609679	−	0.792649i	$-0.291299\pi$
$787$	4.67767e11			1.21936			0.609679	+	0.792649i	$0.291299\pi$
$788$		0			0
$789$	1.89609e11	+	5.39926e11i	0.489273	+	1.39324i
$790$		0			0
$791$		−	3.10772e10i		−	0.0793845i
$792$		0			0
$793$	−1.60629e11			−0.406193
$794$		0			0
$795$	−2.22336e11	+	7.80792e10i	−0.556598	+	0.195464i
$796$		0			0
$797$			3.49248e11i			0.865566i	0.901498	+	0.432783i	$0.142468\pi$
$797$			3.49248e11i			0.865566i	−0.901498	+	0.432783i	$0.857532\pi$
$798$		0			0
$799$	−6.28114e11			−1.54117
$800$		0			0
$801$	−8.91851e10	−	1.11321e11i	−0.216652	−	0.270425i
$802$		0			0
$803$			1.39698e12i			3.35992i
$804$		0			0
$805$	−5.66314e10			−0.134857
$806$		0			0
$807$	−6.83214e9	−	1.94550e10i	−0.0161088	−	0.0458709i
$808$		0			0
$809$		−	4.82005e11i		−	1.12527i	−0.826705	−	0.562636i	$-0.809787\pi$
$809$		−	4.82005e11i		−	1.12527i	0.826705	−	0.562636i	$-0.190213\pi$
$810$		0			0
$811$	−3.66723e11			−0.847726			−0.423863	−	0.905726i	$-0.639326\pi$
$811$	−3.66723e11			−0.847726			−0.423863	+	0.905726i	$0.639326\pi$
$812$		0			0
$813$	−3.04192e11	+	1.06825e11i	−0.696283	+	0.244518i
$814$		0			0
$815$		−	1.27317e11i		−	0.288572i
$816$		0			0
$817$	−1.71072e10			−0.0383964
$818$		0			0
$819$	−9.85122e10	+	7.89235e10i	−0.218955	+	0.175417i
$820$		0			0
$821$			6.21129e11i			1.36713i	0.729891	+	0.683563i	$0.239571\pi$
$821$			6.21129e11i			1.36713i	−0.729891	+	0.683563i	$0.760429\pi$
$822$		0			0
$823$	2.26082e11			0.492795			0.246397	−	0.969169i	$-0.420753\pi$
$823$	2.26082e11			0.492795			0.246397	+	0.969169i	$0.420753\pi$
$824$		0			0
$825$	2.12092e11	+	6.03947e11i	0.457834	+	1.30372i
$826$		0			0
$827$		−	6.30674e11i		−	1.34829i	−0.738600	−	0.674144i	$-0.764513\pi$
$827$		−	6.30674e11i		−	1.34829i	0.738600	−	0.674144i	$-0.235487\pi$
$828$		0			0
$829$	−5.87179e11			−1.24323			−0.621617	−	0.783322i	$-0.713524\pi$
$829$	−5.87179e11			−1.24323			−0.621617	+	0.783322i	$0.713524\pi$
$830$		0			0
$831$	1.26647e11	−	4.44754e10i	0.265577	−	0.0932644i
$832$		0			0
$833$			4.26582e11i			0.885977i
$834$		0			0
$835$	−3.14976e10			−0.0647936
$836$		0			0
$837$	4.82314e11	−	7.73370e11i	0.982717	−	1.57574i
$838$		0			0
$839$			2.32236e11i			0.468685i	0.972154	+	0.234343i	$0.0752938\pi$
$839$			2.32236e11i			0.468685i	−0.972154	+	0.234343i	$0.924706\pi$
$840$		0			0
$841$	4.05096e10			0.0809793
$842$		0			0
$843$	9.90439e10	+	2.82035e11i	0.196118	+	0.558461i
$844$		0			0
$845$		−	8.72116e10i		−	0.171060i
$846$		0			0
$847$	3.91830e11			0.761314
$848$		0			0
$849$	−1.05263e11	+	3.69660e10i	−0.202603	+	0.0711494i
$850$		0			0
$851$		−	3.19136e11i		−	0.608495i
$852$		0			0
$853$	−2.90901e11			−0.549476			−0.274738	−	0.961519i	$-0.588591\pi$
$853$	−2.90901e11			−0.549476			−0.274738	+	0.961519i	$0.588591\pi$
$854$		0			0
$855$	−7.92335e9	−	9.88992e9i	−0.0148267	−	0.0185067i
$856$		0			0
$857$		−	7.52371e11i		−	1.39479i	−0.716687	−	0.697395i	$-0.754343\pi$
$857$		−	7.52371e11i		−	1.39479i	0.716687	−	0.697395i	$-0.245657\pi$
$858$		0			0
$859$	8.20314e11			1.50663			0.753316	−	0.657658i	$-0.228453\pi$
$859$	8.20314e11			1.50663			0.753316	+	0.657658i	$0.228453\pi$
$860$		0			0
$861$	−8.56364e10	−	2.43856e11i	−0.155828	−	0.443732i
$862$		0			0
$863$		−	3.62283e11i		−	0.653138i	−0.945173	−	0.326569i	$-0.894108\pi$
$863$		−	3.62283e11i		−	0.653138i	0.945173	−	0.326569i	$-0.105892\pi$
$864$		0			0
$865$	−1.18239e11			−0.211202
$866$		0			0
$867$	1.13597e10	−	3.98924e9i	0.0201043	−	0.00706015i
$868$		0			0
$869$		−	1.15969e12i		−	2.03359i
$870$		0			0
$871$	3.09834e11			0.538340
$872$		0			0
$873$	−1.37817e11	+	1.10413e11i	−0.237272	+	0.190092i
$874$		0			0
$875$		−	1.72930e11i		−	0.295010i
$876$		0			0
$877$	−3.24793e11			−0.549046			−0.274523	−	0.961581i	$-0.588520\pi$
$877$	−3.24793e11			−0.549046			−0.274523	+	0.961581i	$0.588520\pi$
$878$		0			0
$879$	3.03740e11	+	8.64923e11i	0.508799	+	1.44884i
$880$		0			0
$881$		−	3.90012e11i		−	0.647402i	−0.946159	−	0.323701i	$-0.895073\pi$
$881$		−	3.90012e11i		−	0.647402i	0.946159	−	0.323701i	$-0.104927\pi$
$882$		0			0
$883$	1.04920e12			1.72591			0.862953	−	0.505283i	$-0.168612\pi$
$883$	1.04920e12			1.72591			0.862953	+	0.505283i	$0.168612\pi$
$884$		0			0
$885$	−1.44817e11	+	5.08562e10i	−0.236073	+	0.0829032i
$886$		0			0
$887$			4.31684e11i			0.697383i	0.937238	+	0.348691i	$0.113374\pi$
$887$			4.31684e11i			0.697383i	−0.937238	+	0.348691i	$0.886626\pi$
$888$		0			0
$889$	−8.33728e10			−0.133480
$890$		0			0
$891$	−1.09475e12	−	2.44702e11i	−1.73702	−	0.388264i
$892$		0			0
$893$			4.86255e10i			0.0764643i
$894$		0			0
$895$	−1.54640e11			−0.241008
$896$		0			0
$897$	1.39161e11	+	3.96271e11i	0.214955	+	0.612100i
$898$		0			0
$899$		−	1.16286e12i		−	1.78029i
$900$		0			0
$901$	−8.30736e11			−1.26056
$902$		0			0
$903$	1.68697e11	−	5.92425e10i	0.253722	−	0.0891010i
$904$		0			0
$905$			1.56096e11i			0.232700i
$906$		0			0
$907$	−5.00137e11			−0.739027			−0.369513	−	0.929225i	$-0.620476\pi$
$907$	−5.00137e11			−0.739027			−0.369513	+	0.929225i	$0.620476\pi$
$908$		0			0
$909$	4.30338e11	+	5.37148e11i	0.630310	+	0.786753i
$910$		0			0
$911$		−	5.09319e11i		−	0.739463i	−0.929139	−	0.369732i	$-0.879450\pi$
$911$		−	5.09319e11i		−	0.739463i	0.929139	−	0.369732i	$-0.120550\pi$
$912$		0			0
$913$	−2.06130e11			−0.296659
$914$		0			0
$915$	5.58447e10	+	1.59022e11i	0.0796705	+	0.226868i
$916$		0			0
$917$		−	1.93163e11i		−	0.273178i
$918$		0			0
$919$	9.61924e11			1.34859			0.674293	−	0.738464i	$-0.264448\pi$
$919$	9.61924e11			1.34859			0.674293	+	0.738464i	$0.264448\pi$
$920$		0			0
$921$	9.19718e11	−	3.22983e11i	1.27825	−	0.448891i
$922$		0			0
$923$			9.00771e11i			1.24110i
$924$		0			0
$925$	4.25900e11			0.581756
$926$		0			0
$927$	−6.97690e10	+	5.58957e10i	−0.0944808	+	0.0756936i
$928$		0			0
$929$			2.64711e11i			0.355393i	0.984085	+	0.177697i	$0.0568646\pi$
$929$			2.64711e11i			0.355393i	−0.984085	+	0.177697i	$0.943135\pi$
$930$		0			0
$931$	3.30239e10			0.0439572
$932$		0			0
$933$	1.31609e11	+	3.74767e11i	0.173684	+	0.494577i
$934$		0			0
$935$		−	6.50168e11i		−	0.850706i
$936$		0			0
$937$	−2.30761e11			−0.299367			−0.149683	−	0.988734i	$-0.547825\pi$
$937$	−2.30761e11			−0.299367			−0.149683	+	0.988734i	$0.547825\pi$
$938$		0			0
$939$	−3.96360e11	+	1.39192e11i	−0.509833	+	0.179041i
$940$		0			0
$941$		−	3.62828e11i		−	0.462746i	−0.972865	−	0.231373i	$-0.925678\pi$
$941$		−	3.62828e11i		−	0.462746i	0.972865	−	0.231373i	$-0.0743217\pi$
$942$		0			0
$943$	−8.59952e11			−1.08750
$944$		0			0
$945$	1.12383e11	+	7.00878e10i	0.140920	+	0.0878851i
$946$		0			0
$947$			5.79208e11i			0.720169i	0.932920	+	0.360085i	$0.117252\pi$
$947$			5.79208e11i			0.720169i	−0.932920	+	0.360085i	$0.882748\pi$
$948$		0			0
$949$	−1.22326e12			−1.50818
$950$		0			0
$951$	8.24729e10	+	2.34848e11i	0.100830	+	0.287121i
$952$		0			0
$953$			3.01850e11i			0.365949i	0.983118	+	0.182974i	$0.0585725\pi$
$953$			3.01850e11i			0.365949i	−0.983118	+	0.182974i	$0.941427\pi$
$954$		0			0
$955$	−2.18067e11			−0.262167
$956$		0			0
$957$	−1.35036e12	+	4.74214e11i	−1.60991	+	0.565363i
$958$		0			0
$959$			4.52040e11i			0.534445i
$960$		0			0
$961$	2.08847e12			2.44870
$962$		0			0
$963$	3.20126e10	+	3.99581e10i	0.0372234	+	0.0464622i
$964$		0			0
$965$			5.68713e11i			0.655818i
$966$		0			0
$967$	7.16225e11			0.819112			0.409556	−	0.912285i	$-0.365684\pi$
$967$	7.16225e11			0.819112			0.409556	+	0.912285i	$0.365684\pi$
$968$		0			0
$969$	−1.48024e10	−	4.21509e10i	−0.0167895	−	0.0478092i
$970$		0			0
$971$		−	4.93024e10i		−	0.0554615i	−0.999615	−	0.0277307i	$-0.991172\pi$
$971$		−	4.93024e10i		−	0.0554615i	0.999615	−	0.0277307i	$-0.00882810\pi$
$972$		0			0
$973$	−2.46078e11			−0.274551
$974$		0			0
$975$	−5.28840e11	+	1.85716e11i	−0.585202	+	0.205509i
$976$		0			0
$977$			1.33140e12i			1.46127i	0.682769	+	0.730634i	$0.260775\pi$
$977$			1.33140e12i			1.46127i	−0.682769	+	0.730634i	$0.739225\pi$
$978$		0			0
$979$	−5.66547e11			−0.616744
$980$		0			0
$981$	8.10893e11	−	6.49651e11i	0.875564	−	0.701461i
$982$		0			0
$983$			1.09952e12i			1.17758i	0.808287	+	0.588789i	$0.200395\pi$
$983$			1.09952e12i			1.17758i	−0.808287	+	0.588789i	$0.799605\pi$
$984$		0			0
$985$	−5.99053e11			−0.636386
$986$		0			0
$987$	−1.68391e11	−	4.79506e11i	−0.177440	−	0.505273i
$988$		0			0
$989$		−	5.94907e11i		−	0.621819i
$990$		0			0
$991$	−8.46177e11			−0.877337			−0.438669	−	0.898649i	$-0.644550\pi$
$991$	−8.46177e11			−0.877337			−0.438669	+	0.898649i	$0.644550\pi$
$992$		0			0
$993$	7.43813e11	−	2.61209e11i	0.765009	−	0.268653i
$994$		0			0
$995$		−	1.74765e11i		−	0.178304i
$996$		0			0
$997$	−8.25610e11			−0.835592			−0.417796	−	0.908541i	$-0.637197\pi$
$997$	−8.25610e11			−0.835592			−0.417796	+	0.908541i	$0.637197\pi$
$998$		0			0
$999$	−3.94966e11	+	6.33311e11i	−0.396550	+	0.635851i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.9.e.a.17.2	yes	8
3.2	odd	2	inner	24.9.e.a.17.1	✓	8
4.3	odd	2		48.9.e.e.17.7		8
8.3	odd	2		192.9.e.j.65.2		8
8.5	even	2		192.9.e.i.65.7		8
12.11	even	2		48.9.e.e.17.8		8
24.5	odd	2		192.9.e.i.65.8		8
24.11	even	2		192.9.e.j.65.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.9.e.a.17.1	✓	8	3.2	odd	2	inner
24.9.e.a.17.2	yes	8	1.1	even	1	trivial
48.9.e.e.17.7		8	4.3	odd	2
48.9.e.e.17.8		8	12.11	even	2
192.9.e.i.65.7		8	8.5	even	2
192.9.e.i.65.8		8	24.5	odd	2
192.9.e.j.65.1		8	24.11	even	2
192.9.e.j.65.2		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)

\(f(q)\)	\(=\)	\(q+(-76.4245 + 26.8384i) q^{3} +295.589i q^{5} -843.136 q^{7} +(5120.40 - 4102.23i) q^{9} -26059.3i q^{11} +22818.6 q^{13} +(-7933.16 - 22590.3i) q^{15} -84406.1i q^{17} -6534.32 q^{19} +(64436.3 - 22628.5i) q^{21} -227233. i q^{23} +303252. q^{25} +(-281226. + 450934. i) q^{27} +678039. i q^{29} -1.71504e6 q^{31} +(699391. + 1.99157e6i) q^{33} -249222. i q^{35} +1.40444e6 q^{37} +(-1.74390e6 + 612415. i) q^{39} -3.78445e6i q^{41} +2.61805e6 q^{43} +(1.21257e6 + 1.51354e6i) q^{45} -7.44156e6i q^{47} -5.05392e6 q^{49} +(2.26533e6 + 6.45069e6i) q^{51} -9.84213e6i q^{53} +7.70286e6 q^{55} +(499382. - 175371. i) q^{57} -6.41059e6i q^{59} -7.03941e6 q^{61} +(-4.31719e6 + 3.45874e6i) q^{63} +6.74493e6i q^{65} +1.35782e7 q^{67} +(6.09857e6 + 1.73661e7i) q^{69} +3.94753e7i q^{71} -5.36079e7 q^{73} +(-2.31759e7 + 8.13881e6i) q^{75} +2.19716e7i q^{77} +4.45020e7 q^{79} +(9.39020e6 - 4.20101e7i) q^{81} -7.91003e6i q^{83} +2.49496e7 q^{85} +(-1.81975e7 - 5.18188e7i) q^{87} -2.17407e7i q^{89} -1.92392e7 q^{91} +(1.31071e8 - 4.60290e7i) q^{93} -1.93148e6i q^{95} -2.69154e7 q^{97} +(-1.06901e8 - 1.33434e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 56 q^{3} + 1584 q^{7} + 328 q^{9} + 25232 q^{13} + 38336 q^{15} + 157936 q^{19} + 30480 q^{21} - 579704 q^{25} - 276040 q^{27} + 805552 q^{31} + 102848 q^{33} - 3985008 q^{37} - 2297104 q^{39} + 6962672 q^{43}+ \cdots - 369701504 q^{99}+O(q^{100}) \) 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

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Database

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