LMFDB - Embedded newform 234.8.b.c.181.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [234,8,Mod(181,234)] 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("234.181"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 234 = 2 \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	234.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$73.0980959633$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 17770x^{8} + 98320641x^{6} + 176057788072x^{4} + 109845194658832x^{2} + 14762086704451584 $ x^10 + 17770x^8 + 98320641x^6 + 176057788072x^4 + 109845194658832x^2 + 14762086704451584
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.9
Root			$36.7123i$ of defining polynomial
Character	$\chi$	$=$	234.181
Dual form			234.8.b.c.181.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.00000i q^{2} -64.0000 q^{4} +319.050i q^{5} +1008.64i q^{7} -512.000i q^{8} -2552.40 q^{10} -7261.68i q^{11} +(7921.28 + 43.0248i) q^{13} -8069.13 q^{14} +4096.00 q^{16} +25483.8 q^{17} +8741.60i q^{19} -20419.2i q^{20} +58093.4 q^{22} +83505.4 q^{23} -23668.0 q^{25} +(-344.198 + 63370.2i) q^{26} -64553.0i q^{28} +197882. q^{29} -138208. i q^{31} +32768.0i q^{32} +203870. i q^{34} -321807. q^{35} -61869.9i q^{37} -69932.8 q^{38} +163354. q^{40} -784542. i q^{41} -440828. q^{43} +464747. i q^{44} +668043. i q^{46} -180772. i q^{47} -193813. q^{49} -189344. i q^{50} +(-506962. - 2753.59i) q^{52} +594626. q^{53} +2.31684e6 q^{55} +516424. q^{56} +1.58306e6i q^{58} -43491.6i q^{59} +607137. q^{61} +1.10566e6 q^{62} -262144. q^{64} +(-13727.1 + 2.52729e6i) q^{65} +3.33027e6i q^{67} -1.63096e6 q^{68} -2.57446e6i q^{70} +3.42469e6i q^{71} -2.18369e6i q^{73} +494959. q^{74} -559463. i q^{76} +7.32442e6 q^{77} -4.16881e6 q^{79} +1.30683e6i q^{80} +6.27634e6 q^{82} -415486. i q^{83} +8.13060e6i q^{85} -3.52662e6i q^{86} -3.71798e6 q^{88} +2.04571e6i q^{89} +(-43396.5 + 7.98972e6i) q^{91} -5.34435e6 q^{92} +1.44618e6 q^{94} -2.78901e6 q^{95} +1.09309e6i q^{97} -1.55050e6i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 640 q^{4} - 1136 q^{10} + 3432 q^{13} - 3792 q^{14} + 40960 q^{16} + 6918 q^{17} + 17280 q^{22} - 94164 q^{23} - 330788 q^{25} - 118896 q^{26} + 131304 q^{29} - 873450 q^{35} - 602976 q^{38} + 72704 q^{40}+ \cdots - 16547484 q^{95}+O(q^{100}) $ 10 * q - 640 * q^4 - 1136 * q^10 + 3432 * q^13 - 3792 * q^14 + 40960 * q^16 + 6918 * q^17 + 17280 * q^22 - 94164 * q^23 - 330788 * q^25 - 118896 * q^26 + 131304 * q^29 - 873450 * q^35 - 602976 * q^38 + 72704 * q^40 + 2740774 * q^43 - 3083904 * q^49 - 219648 * q^52 + 3605916 * q^53 + 3999840 * q^55 + 242688 * q^56 + 11727656 * q^61 + 4210560 * q^62 - 2621440 * q^64 + 7481034 * q^65 - 442752 * q^68 - 18145392 * q^74 + 7147920 * q^77 - 28449228 * q^79 + 17868800 * q^82 - 1105920 * q^88 - 20225194 * q^91 + 6026496 * q^92 + 8240976 * q^94 - 16547484 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/234\mathbb{Z}\right)^\times$.

$n$	$145$	$209$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$			8.00000i			0.707107i
$2$			8.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−64.0000			−0.500000
$5$			319.050i			1.14147i	0.821135	+	0.570734i	$0.193341\pi$
$5$			319.050i			1.14147i	−0.821135	+	0.570734i	$0.806659\pi$
$6$		0			0
$7$			1008.64i			1.11146i	0.831363	+	0.555729i	$0.187561\pi$
$7$			1008.64i			1.11146i	−0.831363	+	0.555729i	$0.812439\pi$
$8$		−	512.000i		−	0.353553i
$9$		0			0
$10$	−2552.40			−0.807140
$11$		−	7261.68i		−	1.64499i	−0.568775	−	0.822493i	$-0.692582\pi$
$11$		−	7261.68i		−	1.64499i	0.568775	−	0.822493i	$-0.307418\pi$
$12$		0			0
$13$	7921.28	+	43.0248i	0.999985	+	0.00543146i
$13$	7921.28	+	43.0248i	0.999985	+	0.00543146i
$14$	−8069.13			−0.785920
$15$		0			0
$16$	4096.00			0.250000
$17$	25483.8			1.25803			0.629017	−	0.777392i	$-0.283458\pi$
$17$	25483.8			1.25803			0.629017	+	0.777392i	$0.283458\pi$
$18$		0			0
$19$			8741.60i			0.292384i	0.989256	+	0.146192i	$0.0467017\pi$
$19$			8741.60i			0.292384i	−0.989256	+	0.146192i	$0.953298\pi$
$20$		−	20419.2i		−	0.570734i
$21$		0			0
$22$	58093.4			1.16318
$23$	83505.4			1.43109			0.715546	−	0.698566i	$-0.246178\pi$
$23$	83505.4			1.43109			0.715546	+	0.698566i	$0.246178\pi$
$24$		0			0
$25$	−23668.0			−0.302950
$26$	−344.198	+	63370.2i	−0.00384062	+	0.707096i
$27$		0			0
$28$		−	64553.0i		−	0.555729i
$29$	197882.			1.50666			0.753328	−	0.657644i	$-0.228447\pi$
$29$	197882.			1.50666			0.753328	+	0.657644i	$0.228447\pi$
$30$		0			0
$31$		−	138208.i		−	0.833234i	−0.909082	−	0.416617i	$-0.863216\pi$
$31$		−	138208.i		−	0.833234i	0.909082	−	0.416617i	$-0.136784\pi$
$32$			32768.0i			0.176777i
$33$		0			0
$34$			203870.i			0.889564i
$35$	−321807.			−1.26870
$36$		0			0
$37$		−	61869.9i		−	0.200805i	−0.994947	−	0.100402i	$-0.967987\pi$
$37$		−	61869.9i		−	0.200805i	0.994947	−	0.100402i	$-0.0320130\pi$
$38$	−69932.8			−0.206747
$39$		0			0
$40$	163354.			0.403570
$41$		−	784542.i		−	1.77776i	−0.458141	−	0.888880i	$-0.651485\pi$
$41$		−	784542.i		−	1.77776i	0.458141	−	0.888880i	$-0.348515\pi$
$42$		0			0
$43$	−440828.			−0.845531			−0.422765	−	0.906239i	$-0.638941\pi$
$43$	−440828.			−0.845531			−0.422765	+	0.906239i	$0.638941\pi$
$44$			464747.i			0.822493i
$45$		0			0
$46$			668043.i			1.01193i
$47$		−	180772.i		−	0.253974i	−0.991904	−	0.126987i	$-0.959469\pi$
$47$		−	180772.i		−	0.253974i	0.991904	−	0.126987i	$-0.0405306\pi$
$48$		0			0
$49$	−193813.			−0.235340
$50$		−	189344.i		−	0.214218i
$51$		0			0
$52$	−506962.	−	2753.59i	−0.499993	−	0.00271573i
$53$	594626.			0.548628			0.274314	−	0.961640i	$-0.411549\pi$
$53$	594626.			0.548628			0.274314	+	0.961640i	$0.411549\pi$
$54$		0			0
$55$	2.31684e6			1.87770
$56$	516424.			0.392960
$57$		0			0
$58$			1.58306e6i			1.06537i
$59$		−	43491.6i		−	0.0275691i	−0.999905	−	0.0137846i	$-0.995612\pi$
$59$		−	43491.6i		−	0.0275691i	0.999905	−	0.0137846i	$-0.00438790\pi$
$60$		0			0
$61$	607137.			0.342478			0.171239	−	0.985230i	$-0.445223\pi$
$61$	607137.			0.342478			0.171239	+	0.985230i	$0.445223\pi$
$62$	1.10566e6			0.589185
$63$		0			0
$64$	−262144.			−0.125000
$65$	−13727.1	+	2.52729e6i	−0.00619984	+	1.14145i
$66$		0			0
$67$			3.33027e6i			1.35275i	0.736557	+	0.676375i	$0.236450\pi$
$67$			3.33027e6i			1.35275i	−0.736557	+	0.676375i	$0.763550\pi$
$68$	−1.63096e6			−0.629017
$69$		0			0
$70$		−	2.57446e6i		−	0.897103i
$71$			3.42469e6i			1.13558i	0.823174	+	0.567789i	$0.192201\pi$
$71$			3.42469e6i			1.13558i	−0.823174	+	0.567789i	$0.807799\pi$
$72$		0			0
$73$		−	2.18369e6i		−	0.656993i	−0.944505	−	0.328497i	$-0.893458\pi$
$73$		−	2.18369e6i		−	0.656993i	0.944505	−	0.328497i	$-0.106542\pi$
$74$	494959.			0.141990
$75$		0			0
$76$		−	559463.i		−	0.146192i
$77$	7.32442e6			1.82834
$78$		0			0
$79$	−4.16881e6			−0.951300			−0.475650	−	0.879635i	$-0.657787\pi$
$79$	−4.16881e6			−0.951300			−0.475650	+	0.879635i	$0.657787\pi$
$80$			1.30683e6i			0.285367i
$81$		0			0
$82$	6.27634e6			1.25707
$83$		−	415486.i		−	0.0797596i	−0.999204	−	0.0398798i	$-0.987302\pi$
$83$		−	415486.i		−	0.0797596i	0.999204	−	0.0398798i	$-0.0126975\pi$
$84$		0			0
$85$			8.13060e6i			1.43601i
$86$		−	3.52662e6i		−	0.597881i
$87$		0			0
$88$	−3.71798e6			−0.581591
$89$			2.04571e6i			0.307595i	0.988102	+	0.153798i	$0.0491504\pi$
$89$			2.04571e6i			0.307595i	−0.988102	+	0.153798i	$0.950850\pi$
$90$		0			0
$91$	−43396.5	+	7.98972e6i	−0.00603685	+	1.11144i
$92$	−5.34435e6			−0.715546
$93$		0			0
$94$	1.44618e6			0.179587
$95$	−2.78901e6			−0.333747
$96$		0			0
$97$			1.09309e6i			0.121606i	0.998150	+	0.0608030i	$0.0193662\pi$
$97$			1.09309e6i			0.121606i	−0.998150	+	0.0608030i	$0.980634\pi$
$98$		−	1.55050e6i		−	0.166411i
$99$		0			0
$100$	1.51475e6			0.151475
$101$	1.95752e7			1.89052			0.945261	−	0.326316i	$-0.105807\pi$
$101$	1.95752e7			1.89052			0.945261	+	0.326316i	$0.105807\pi$
$102$		0			0
$103$	2.91690e6			0.263022			0.131511	−	0.991315i	$-0.458017\pi$
$103$	2.91690e6			0.263022			0.131511	+	0.991315i	$0.458017\pi$
$104$	22028.7	−	4.05570e6i	0.00192031	−	0.353548i
$105$		0			0
$106$			4.75701e6i			0.387939i
$107$	−1.05834e7			−0.835187			−0.417593	−	0.908634i	$-0.637126\pi$
$107$	−1.05834e7			−0.835187			−0.417593	+	0.908634i	$0.637126\pi$
$108$		0			0
$109$		−	1.12416e7i		−	0.831448i	−0.909491	−	0.415724i	$-0.863528\pi$
$109$		−	1.12416e7i		−	0.831448i	0.909491	−	0.415724i	$-0.136472\pi$
$110$			1.85347e7i			1.32774i
$111$		0			0
$112$			4.13139e6i			0.277865i
$113$	−1.68098e7			−1.09594			−0.547971	−	0.836497i	$-0.684600\pi$
$113$	−1.68098e7			−1.09594			−0.547971	+	0.836497i	$0.684600\pi$
$114$		0			0
$115$			2.66424e7i			1.63355i
$116$	−1.26645e7			−0.753328
$117$		0			0
$118$	347933.			0.0194943
$119$			2.57040e7i			1.39825i
$120$		0			0
$121$	−3.32448e7			−1.70598
$122$			4.85709e6i			0.242168i
$123$		0			0
$124$			8.84531e6i			0.416617i
$125$			1.73745e7i			0.795660i
$126$		0			0
$127$	4.03893e7			1.74966			0.874829	−	0.484432i	$-0.160974\pi$
$127$	4.03893e7			1.74966			0.874829	+	0.484432i	$0.160974\pi$
$128$		−	2.09715e6i		−	0.0883883i
$129$		0			0
$130$	−2.02183e7	−	109816.i	−0.807128	−	0.00438395i
$131$	−2.38235e7			−0.925882			−0.462941	−	0.886389i	$-0.653206\pi$
$131$	−2.38235e7			−0.925882			−0.462941	+	0.886389i	$0.653206\pi$
$132$		0			0
$133$	−8.81714e6			−0.324973
$134$	−2.66422e7			−0.956539
$135$		0			0
$136$		−	1.30477e7i		−	0.444782i
$137$		−	4.63404e7i		−	1.53971i	−0.638221	−	0.769853i	$-0.720330\pi$
$137$		−	4.63404e7i		−	1.53971i	0.638221	−	0.769853i	$-0.279670\pi$
$138$		0			0
$139$	1.67521e7			0.529074			0.264537	−	0.964376i	$-0.414781\pi$
$139$	1.67521e7			0.529074			0.264537	+	0.964376i	$0.414781\pi$
$140$	2.05956e7			0.634348
$141$		0			0
$142$	−2.73975e7			−0.802975
$143$	312432.	−	5.75218e7i	0.00893469	−	1.64496i
$144$		0			0
$145$			6.31344e7i			1.71980i
$146$	1.74695e7			0.464565
$147$		0			0
$148$			3.95968e6i			0.100402i
$149$			5.71941e7i			1.41644i	0.705991	+	0.708221i	$0.250502\pi$
$149$			5.71941e7i			1.41644i	−0.705991	+	0.708221i	$0.749498\pi$
$150$		0			0
$151$			5.99491e7i			1.41698i	0.705721	+	0.708490i	$0.250623\pi$
$151$			5.99491e7i			1.41698i	−0.705721	+	0.708490i	$0.749377\pi$
$152$	4.47570e6			0.103373
$153$		0			0
$154$			5.85954e7i			1.29283i
$155$	4.40953e7			0.951110
$156$		0			0
$157$	−3.11279e7			−0.641950			−0.320975	−	0.947088i	$-0.604011\pi$
$157$	−3.11279e7			−0.641950			−0.320975	+	0.947088i	$0.604011\pi$
$158$		−	3.33505e7i		−	0.672671i
$159$		0			0
$160$	−1.04546e7			−0.201785
$161$			8.42270e7i			1.59060i
$162$		0			0
$163$			2.52525e7i			0.456717i	0.973577	+	0.228359i	$0.0733359\pi$
$163$			2.52525e7i			0.456717i	−0.973577	+	0.228359i	$0.926664\pi$
$164$			5.02107e7i			0.888880i
$165$		0			0
$166$	3.32389e6			0.0563986
$167$			6.63359e6i			0.110215i	0.998480	+	0.0551075i	$0.0175502\pi$
$167$			6.63359e6i			0.110215i	−0.998480	+	0.0551075i	$0.982450\pi$
$168$		0			0
$169$	6.27448e7	+	681623.i	0.999941	+	0.0108628i
$170$	−6.50448e7			−1.01541
$171$		0			0
$172$	2.82130e7			0.422765
$173$	−2.40285e7			−0.352830			−0.176415	−	0.984316i	$-0.556450\pi$
$173$	−2.40285e7			−0.352830			−0.176415	+	0.984316i	$0.556450\pi$
$174$		0			0
$175$		−	2.38725e7i		−	0.336717i
$176$		−	2.97438e7i		−	0.411247i
$177$		0			0
$178$	−1.63657e7			−0.217503
$179$	6.80314e7			0.886592			0.443296	−	0.896375i	$-0.353809\pi$
$179$	6.80314e7			0.886592			0.443296	+	0.896375i	$0.353809\pi$
$180$		0			0
$181$	1.78681e7			0.223977			0.111988	−	0.993710i	$-0.464278\pi$
$181$	1.78681e7			0.223977			0.111988	+	0.993710i	$0.464278\pi$
$182$	−6.39178e7	−	347172.i	−0.785908	−	0.00426870i
$183$		0			0
$184$		−	4.27548e7i		−	0.505967i
$185$	1.97396e7			0.229212
$186$		0			0
$187$		−	1.85055e8i		−	2.06945i
$188$			1.15694e7i			0.126987i
$189$		0			0
$190$		−	2.23121e7i		−	0.235995i
$191$	1.26330e8			1.31187			0.655934	−	0.754819i	$-0.272275\pi$
$191$	1.26330e8			1.31187			0.655934	+	0.754819i	$0.272275\pi$
$192$		0			0
$193$		−	8.24125e7i		−	0.825168i	−0.910920	−	0.412584i	$-0.864626\pi$
$193$		−	8.24125e7i		−	0.825168i	0.910920	−	0.412584i	$-0.135374\pi$
$194$	−8.74473e6			−0.0859885
$195$		0			0
$196$	1.24040e7			0.117670
$197$		−	1.65240e8i		−	1.53987i	−0.638124	−	0.769933i	$-0.720289\pi$
$197$		−	1.65240e8i		−	1.53987i	0.638124	−	0.769933i	$-0.279711\pi$
$198$		0			0
$199$	−7.27913e7			−0.654778			−0.327389	−	0.944890i	$-0.606169\pi$
$199$	−7.27913e7			−0.654778			−0.327389	+	0.944890i	$0.606169\pi$
$200$			1.21180e7i			0.107109i
$201$		0			0
$202$			1.56602e8i			1.33680i
$203$			1.99592e8i			1.67459i
$204$		0			0
$205$	2.50308e8			2.02926
$206$			2.33352e7i			0.185985i
$207$		0			0
$208$	3.24456e7	+	176229.i	0.249996	+	0.00135787i
$209$	6.34787e7			0.480968
$210$		0			0
$211$	−1.76488e8			−1.29338			−0.646692	−	0.762751i	$-0.723848\pi$
$211$	−1.76488e8			−1.29338			−0.646692	+	0.762751i	$0.723848\pi$
$212$	−3.80561e7			−0.274314
$213$		0			0
$214$		−	8.46675e7i		−	0.590566i
$215$		−	1.40646e8i		−	0.965147i
$216$		0			0
$217$	1.39402e8			0.926105
$218$	8.99328e7			0.587923
$219$		0			0
$220$	−1.48278e8			−0.938850
$221$	2.01864e8	+	1.09643e6i	1.25802	+	0.00683296i
$222$		0			0
$223$			1.41494e8i			0.854421i	0.904152	+	0.427211i	$0.140504\pi$
$223$			1.41494e8i			0.854421i	−0.904152	+	0.427211i	$0.859496\pi$
$224$	−3.30511e7			−0.196480
$225$		0			0
$226$		−	1.34478e8i		−	0.774948i
$227$		−	2.61188e8i		−	1.48205i	−0.671477	−	0.741025i	$-0.734340\pi$
$227$		−	2.61188e8i		−	1.48205i	0.671477	−	0.741025i	$-0.265660\pi$
$228$		0			0
$229$		−	7.84396e7i		−	0.431630i	−0.976434	−	0.215815i	$-0.930759\pi$
$229$		−	7.84396e7i		−	0.431630i	0.976434	−	0.215815i	$-0.0692408\pi$
$230$	−2.13139e8			−1.15509
$231$		0			0
$232$		−	1.01316e8i		−	0.532684i
$233$	−2.79209e8			−1.44605			−0.723027	−	0.690820i	$-0.757250\pi$
$233$	−2.79209e8			−1.44605			−0.723027	+	0.690820i	$0.757250\pi$
$234$		0			0
$235$	5.76754e7			0.289903
$236$			2.78346e6i			0.0137846i
$237$		0			0
$238$	−2.05632e8			−0.988714
$239$			2.18569e8i			1.03561i	0.855499	+	0.517804i	$0.173250\pi$
$239$			2.18569e8i			1.03561i	−0.855499	+	0.517804i	$0.826750\pi$
$240$		0			0
$241$			3.20066e7i			0.147292i	0.997284	+	0.0736461i	$0.0234635\pi$
$241$			3.20066e7i			0.147292i	−0.997284	+	0.0736461i	$0.976536\pi$
$242$		−	2.65958e8i		−	1.20631i
$243$		0			0
$244$	−3.88568e7			−0.171239
$245$		−	6.18361e7i		−	0.268634i
$246$		0			0
$247$	−376105.	+	6.92447e7i	−0.00158807	+	0.292380i
$248$	−7.07624e7			−0.294593
$249$		0			0
$250$	−1.38996e8			−0.562617
$251$	3.12385e8			1.24690			0.623452	−	0.781862i	$-0.285730\pi$
$251$	3.12385e8			1.24690			0.623452	+	0.781862i	$0.285730\pi$
$252$		0			0
$253$		−	6.06389e8i		−	2.35413i
$254$			3.23114e8i			1.23719i
$255$		0			0
$256$	1.67772e7			0.0625000
$257$	3.60093e8			1.32327			0.661636	−	0.749826i	$-0.269863\pi$
$257$	3.60093e8			1.32327			0.661636	+	0.749826i	$0.269863\pi$
$258$		0			0
$259$	6.24045e7			0.223186
$260$	878532.	−	1.61746e8i	0.00309992	−	0.570726i
$261$		0			0
$262$		−	1.90588e8i		−	0.654698i
$263$	−3.17357e8			−1.07573			−0.537865	−	0.843031i	$-0.680769\pi$
$263$	−3.17357e8			−1.07573			−0.537865	+	0.843031i	$0.680769\pi$
$264$		0			0
$265$			1.89715e8i			0.626242i
$266$		−	7.05371e7i		−	0.229790i
$267$		0			0
$268$		−	2.13137e8i		−	0.676375i
$269$	−2.13959e8			−0.670189			−0.335095	−	0.942184i	$-0.608768\pi$
$269$	−2.13959e8			−0.670189			−0.335095	+	0.942184i	$0.608768\pi$
$270$		0			0
$271$		−	8.60078e7i		−	0.262510i	−0.991349	−	0.131255i	$-0.958099\pi$
$271$		−	8.60078e7i		−	0.262510i	0.991349	−	0.131255i	$-0.0419006\pi$
$272$	1.04381e8			0.314508
$273$		0			0
$274$	3.70723e8			1.08874
$275$			1.71869e8i			0.498349i
$276$		0			0
$277$	6.28154e7			0.177577			0.0887886	−	0.996050i	$-0.471700\pi$
$277$	6.28154e7			0.177577			0.0887886	+	0.996050i	$0.471700\pi$
$278$			1.34016e8i			0.374112i
$279$		0			0
$280$			1.64765e8i			0.448551i
$281$		−	3.98236e8i		−	1.07070i	−0.844630	−	0.535351i	$-0.820179\pi$
$281$		−	3.98236e8i		−	1.07070i	0.844630	−	0.535351i	$-0.179821\pi$
$282$		0			0
$283$	−7.71807e6			−0.0202421			−0.0101211	−	0.999949i	$-0.503222\pi$
$283$	−7.71807e6			−0.0202421			−0.0101211	+	0.999949i	$0.503222\pi$
$284$		−	2.19180e8i		−	0.567789i
$285$		0			0
$286$	4.60174e8	+	2.49946e6i	1.16316	+	0.00631778i
$287$	7.91321e8			1.97591
$288$		0			0
$289$	2.39083e8			0.582649
$290$	−5.05075e8			−1.21608
$291$		0			0
$292$			1.39756e8i			0.328497i
$293$		−	2.50651e8i		−	0.582147i	−0.956701	−	0.291073i	$-0.905988\pi$
$293$		−	2.50651e8i		−	0.582147i	0.956701	−	0.291073i	$-0.0940123\pi$
$294$		0			0
$295$	1.38760e7			0.0314693
$296$	−3.16774e7			−0.0709951
$297$		0			0
$298$	−4.57552e8			−1.00158
$299$	6.61470e8	+	3.59280e6i	1.43107	+	0.00777292i
$300$		0			0
$301$		−	4.44637e8i		−	0.939773i
$302$	−4.79593e8			−1.00196
$303$		0			0
$304$			3.58056e7i			0.0730960i
$305$			1.93707e8i			0.390927i
$306$		0			0
$307$			5.43078e8i			1.07122i	0.844466	+	0.535609i	$0.179918\pi$
$307$			5.43078e8i			1.07122i	−0.844466	+	0.535609i	$0.820082\pi$
$308$	−4.68763e8			−0.914168
$309$		0			0
$310$			3.52762e8i			0.672537i
$311$	−3.55055e8			−0.669320			−0.334660	−	0.942339i	$-0.608621\pi$
$311$	−3.55055e8			−0.669320			−0.334660	+	0.942339i	$0.608621\pi$
$312$		0			0
$313$	−8.33333e8			−1.53608			−0.768039	−	0.640403i	$-0.778767\pi$
$313$	−8.33333e8			−1.53608			−0.768039	+	0.640403i	$0.778767\pi$
$314$		−	2.49023e8i		−	0.453927i
$315$		0			0
$316$	2.66804e8			0.475650
$317$		−	7.90347e8i		−	1.39351i	−0.717308	−	0.696756i	$-0.754626\pi$
$317$		−	7.90347e8i		−	1.39351i	0.717308	−	0.696756i	$-0.245374\pi$
$318$		0			0
$319$		−	1.43696e9i		−	2.47843i
$320$		−	8.36371e7i		−	0.142684i
$321$		0			0
$322$	−6.73816e8			−1.12472
$323$			2.22769e8i			0.367829i
$324$		0			0
$325$	−1.87481e8	−	1.01831e6i	−0.302946	−	0.00164546i
$326$	−2.02020e8			−0.322948
$327$		0			0
$328$	−4.01686e8			−0.628533
$329$	1.82334e8			0.282281
$330$		0			0
$331$			8.53316e8i			1.29334i	0.762771	+	0.646669i	$0.223838\pi$
$331$			8.53316e8i			1.29334i	−0.762771	+	0.646669i	$0.776162\pi$
$332$			2.65911e7i			0.0398798i
$333$		0			0
$334$	−5.30687e7			−0.0779338
$335$	−1.06252e9			−1.54412
$336$		0			0
$337$	2.21108e8			0.314702			0.157351	−	0.987543i	$-0.449705\pi$
$337$	2.21108e8			0.314702			0.157351	+	0.987543i	$0.449705\pi$
$338$	−5.45298e6	+	5.01959e8i	−0.00768114	+	0.707065i
$339$		0			0
$340$		−	5.20358e8i		−	0.718003i
$341$	−1.00362e9			−1.37066
$342$		0			0
$343$			6.35171e8i			0.849887i
$344$			2.25704e8i			0.298940i
$345$		0			0
$346$		−	1.92228e8i		−	0.249488i
$347$	−5.58923e8			−0.718123			−0.359061	−	0.933314i	$-0.616903\pi$
$347$	−5.58923e8			−0.718123			−0.359061	+	0.933314i	$0.616903\pi$
$348$		0			0
$349$		−	1.52400e9i		−	1.91909i	−0.281558	−	0.959544i	$-0.590851\pi$
$349$		−	1.52400e9i		−	1.91909i	0.281558	−	0.959544i	$-0.409149\pi$
$350$	1.90980e8			0.238095
$351$		0			0
$352$	2.37951e8			0.290795
$353$			1.09584e9i			1.32597i	0.748632	+	0.662986i	$0.230711\pi$
$353$			1.09584e9i			1.32597i	−0.748632	+	0.662986i	$0.769289\pi$
$354$		0			0
$355$	−1.09265e9			−1.29623
$356$		−	1.30926e8i		−	0.153798i
$357$		0			0
$358$			5.44251e8i			0.626915i
$359$			5.47766e8i			0.624834i	0.949945	+	0.312417i	$0.101139\pi$
$359$			5.47766e8i			0.624834i	−0.949945	+	0.312417i	$0.898861\pi$
$360$		0			0
$361$	8.17456e8			0.914512
$362$			1.42945e8i			0.158375i
$363$		0			0
$364$	2.77738e6	−	5.11342e8i	0.00301842	−	0.555721i
$365$	6.96707e8			0.749937
$366$		0			0
$367$	4.33710e8			0.458003			0.229001	−	0.973426i	$-0.426454\pi$
$367$	4.33710e8			0.458003			0.229001	+	0.973426i	$0.426454\pi$
$368$	3.42038e8			0.357773
$369$		0			0
$370$			1.57917e8i			0.162077i
$371$			5.99764e8i			0.609778i
$372$		0			0
$373$	4.38624e8			0.437635			0.218817	−	0.975766i	$-0.429780\pi$
$373$	4.38624e8			0.437635			0.218817	+	0.975766i	$0.429780\pi$
$374$	1.48044e9			1.46332
$375$		0			0
$376$	−9.25553e7			−0.0897933
$377$	1.56748e9	+	8.51385e6i	1.50663	+	0.00818335i
$378$		0			0
$379$		−	4.68589e8i		−	0.442135i	−0.975259	−	0.221067i	$-0.929046\pi$
$379$		−	4.68589e8i		−	0.442135i	0.975259	−	0.221067i	$-0.0709540\pi$
$380$	1.78497e8			0.166874
$381$		0			0
$382$			1.01064e9i			0.927630i
$383$		−	1.15286e9i		−	1.04853i	−0.851555	−	0.524265i	$-0.824340\pi$
$383$		−	1.15286e9i		−	1.04853i	0.851555	−	0.524265i	$-0.175660\pi$
$384$		0			0
$385$			2.33686e9i			2.08699i
$386$	6.59300e8			0.583482
$387$		0			0
$388$		−	6.99578e7i		−	0.0608030i
$389$	7.44354e8			0.641145			0.320572	−	0.947224i	$-0.396125\pi$
$389$	7.44354e8			0.641145			0.320572	+	0.947224i	$0.396125\pi$
$390$		0			0
$391$	2.12803e9			1.80036
$392$			9.92323e7i			0.0832054i
$393$		0			0
$394$	1.32192e9			1.08885
$395$		−	1.33006e9i		−	1.08588i
$396$		0			0
$397$			1.02930e9i			0.825610i	0.910819	+	0.412805i	$0.135451\pi$
$397$			1.02930e9i			0.825610i	−0.910819	+	0.412805i	$0.864549\pi$
$398$		−	5.82331e8i		−	0.462998i
$399$		0			0
$400$	−9.69441e7			−0.0757376
$401$			9.81488e8i			0.760116i	0.924963	+	0.380058i	$0.124096\pi$
$401$			9.81488e8i			0.760116i	−0.924963	+	0.380058i	$0.875904\pi$
$402$		0			0
$403$	5.94636e6	−	1.09478e9i	0.00452568	−	0.833222i
$404$	−1.25281e9			−0.945261
$405$		0			0
$406$	−1.59674e9			−1.18411
$407$	−4.49279e8			−0.330321
$408$		0			0
$409$			2.27543e9i			1.64449i	0.569132	+	0.822246i	$0.307279\pi$
$409$			2.27543e9i			1.64449i	−0.569132	+	0.822246i	$0.692721\pi$
$410$			2.00247e9i			1.43490i
$411$		0			0
$412$	−1.86682e8			−0.131511
$413$	4.38674e7			0.0306420
$414$		0			0
$415$	1.32561e8			0.0910431
$416$	−1.40984e6	+	2.59564e8i	−0.000960156	+	0.176774i
$417$		0			0
$418$			5.07829e8i			0.340096i
$419$	7.89660e8			0.524435			0.262217	−	0.965009i	$-0.415546\pi$
$419$	7.89660e8			0.524435			0.262217	+	0.965009i	$0.415546\pi$
$420$		0			0
$421$			2.16620e9i			1.41485i	0.706787	+	0.707426i	$0.250144\pi$
$421$			2.16620e9i			1.41485i	−0.706787	+	0.707426i	$0.749856\pi$
$422$		−	1.41191e9i		−	0.914561i
$423$		0			0
$424$		−	3.04448e8i		−	0.193969i
$425$	−6.03150e8			−0.381122
$426$		0			0
$427$			6.12383e8i			0.380650i
$428$	6.77340e8			0.417593
$429$		0			0
$430$	1.12517e9			0.682462
$431$			1.22325e9i			0.735947i	0.929836	+	0.367973i	$0.119948\pi$
$431$			1.22325e9i			0.735947i	−0.929836	+	0.367973i	$0.880052\pi$
$432$		0			0
$433$	−4.78952e8			−0.283520			−0.141760	−	0.989901i	$-0.545276\pi$
$433$	−4.78952e8			−0.283520			−0.141760	+	0.989901i	$0.545276\pi$
$434$			1.11522e9i			0.654855i
$435$		0			0
$436$			7.19462e8i			0.415724i
$437$			7.29971e8i			0.418428i
$438$		0			0
$439$	2.02013e9			1.13960			0.569802	−	0.821782i	$-0.307020\pi$
$439$	2.02013e9			1.13960			0.569802	+	0.821782i	$0.307020\pi$
$440$		−	1.18622e9i		−	0.663868i
$441$		0			0
$442$	−8.77146e6	+	1.61491e9i	−0.00483164	+	0.889551i
$443$	7.36847e8			0.402684			0.201342	−	0.979521i	$-0.435470\pi$
$443$	7.36847e8			0.402684			0.201342	+	0.979521i	$0.435470\pi$
$444$		0			0
$445$	−6.52685e8			−0.351110
$446$	−1.13195e9			−0.604167
$447$		0			0
$448$		−	2.64409e8i		−	0.138932i
$449$			1.76768e9i			0.921596i	0.887505	+	0.460798i	$0.152437\pi$
$449$			1.76768e9i			0.921596i	−0.887505	+	0.460798i	$0.847563\pi$
$450$		0			0
$451$	−5.69709e9			−2.92439
$452$	1.07583e9			0.547971
$453$		0			0
$454$	2.08951e9			1.04797
$455$	−2.54912e9	−	1.38457e7i	−1.26868	−	0.00689087i
$456$		0			0
$457$		−	2.18099e8i		−	0.106892i	−0.998571	−	0.0534462i	$-0.982979\pi$
$457$		−	2.18099e8i		−	0.106892i	0.998571	−	0.0534462i	$-0.0170206\pi$
$458$	6.27517e8			0.305208
$459$		0			0
$460$		−	1.70511e9i		−	0.816773i
$461$		−	7.87784e8i		−	0.374502i	−0.982312	−	0.187251i	$-0.940042\pi$
$461$		−	7.87784e8i		−	0.374502i	0.982312	−	0.187251i	$-0.0599577\pi$
$462$		0			0
$463$			1.57223e9i			0.736178i	0.929791	+	0.368089i	$0.119988\pi$
$463$			1.57223e9i			0.736178i	−0.929791	+	0.368089i	$0.880012\pi$
$464$	8.10527e8			0.376664
$465$		0			0
$466$		−	2.23368e9i		−	1.02251i
$467$	9.07954e8			0.412529			0.206265	−	0.978496i	$-0.433869\pi$
$467$	9.07954e8			0.412529			0.206265	+	0.978496i	$0.433869\pi$
$468$		0			0
$469$	−3.35905e9			−1.50353
$470$			4.61403e8i			0.204992i
$471$		0			0
$472$	−2.22677e7			−0.00974716
$473$			3.20115e9i			1.39089i
$474$		0			0
$475$		−	2.06896e8i		−	0.0885778i
$476$		−	1.64505e9i		−	0.699126i
$477$		0			0
$478$	−1.74855e9			−0.732285
$479$		−	2.23883e9i		−	0.930779i	−0.885106	−	0.465390i	$-0.845914\pi$
$479$		−	2.23883e9i		−	0.930779i	0.885106	−	0.465390i	$-0.154086\pi$
$480$		0			0
$481$	2.66194e6	−	4.90089e8i	0.00109066	−	0.200802i
$482$	−2.56053e8			−0.104151
$483$		0			0
$484$	2.12766e9			0.852991
$485$	−3.48751e8			−0.138810
$486$		0			0
$487$		−	3.07668e9i		−	1.20706i	−0.797338	−	0.603532i	$-0.793759\pi$
$487$		−	3.07668e9i		−	1.20706i	0.797338	−	0.603532i	$-0.206241\pi$
$488$		−	3.10854e8i		−	0.121084i
$489$		0			0
$490$	4.94689e8			0.189953
$491$	−2.48221e9			−0.946354			−0.473177	−	0.880967i	$-0.656893\pi$
$491$	−2.48221e9			−0.946354			−0.473177	+	0.880967i	$0.656893\pi$
$492$		0			0
$493$	5.04279e9			1.89543
$494$	−5.53957e8	−	3.00884e6i	−0.206744	−	0.00112294i
$495$		0			0
$496$		−	5.66100e8i		−	0.208308i
$497$	−3.45428e9			−1.26215
$498$		0			0
$499$			2.24551e8i			0.0809026i	0.999182	+	0.0404513i	$0.0128796\pi$
$499$			2.24551e8i			0.0809026i	−0.999182	+	0.0404513i	$0.987120\pi$
$500$		−	1.11197e9i		−	0.397830i
$501$		0			0
$502$			2.49908e9i			0.881694i
$503$	−6.52953e8			−0.228767			−0.114384	−	0.993437i	$-0.536489\pi$
$503$	−6.52953e8			−0.228767			−0.114384	+	0.993437i	$0.536489\pi$
$504$		0			0
$505$			6.24547e9i			2.15797i
$506$	4.85111e9			1.66462
$507$		0			0
$508$	−2.58491e9			−0.874829
$509$		−	1.82791e8i		−	0.0614389i	−0.999528	−	0.0307195i	$-0.990220\pi$
$509$		−	1.82791e8i		−	0.0614389i	0.999528	−	0.0307195i	$-0.00977985\pi$
$510$		0			0
$511$	2.20256e9			0.730221
$512$			1.34218e8i			0.0441942i
$513$		0			0
$514$			2.88074e9i			0.935694i
$515$			9.30639e8i			0.300231i
$516$		0			0
$517$	−1.31271e9			−0.417784
$518$			4.99236e8i			0.157816i
$519$		0			0
$520$	1.29397e9	+	7.02826e6i	0.403564	+	0.00219198i
$521$	−5.23398e9			−1.62144			−0.810718	−	0.585437i	$-0.800923\pi$
$521$	−5.23398e9			−1.62144			−0.810718	+	0.585437i	$0.800923\pi$
$522$		0			0
$523$	−3.93597e9			−1.20308			−0.601542	−	0.798841i	$-0.705447\pi$
$523$	−3.93597e9			−1.20308			−0.601542	+	0.798841i	$0.705447\pi$
$524$	1.52470e9			0.462941
$525$		0			0
$526$		−	2.53886e9i		−	0.760655i
$527$		−	3.52206e9i		−	1.04824i
$528$		0			0
$529$	3.56833e9			1.04802
$530$	−1.51772e9			−0.442820
$531$		0			0
$532$	5.64297e8			0.162486
$533$	3.37548e7	−	6.21458e9i	0.00965583	−	1.77773i
$534$		0			0
$535$		−	3.37665e9i		−	0.953340i
$536$	1.70510e9			0.478270
$537$		0			0
$538$		−	1.71167e9i		−	0.473895i
$539$			1.40741e9i			0.387132i
$540$		0			0
$541$		−	1.96466e9i		−	0.533455i	−0.963772	−	0.266727i	$-0.914058\pi$
$541$		−	1.96466e9i		−	0.533455i	0.963772	−	0.266727i	$-0.0859424\pi$
$542$	6.88063e8			0.185622
$543$		0			0
$544$			8.35052e8i			0.222391i
$545$	3.58663e9			0.949072
$546$		0			0
$547$	−6.37181e8			−0.166459			−0.0832295	−	0.996530i	$-0.526523\pi$
$547$	−6.37181e8			−0.166459			−0.0832295	+	0.996530i	$0.526523\pi$
$548$			2.96579e9i			0.769853i
$549$		0			0
$550$	−1.37495e9			−0.352386
$551$			1.72981e9i			0.440522i
$552$		0			0
$553$		−	4.20483e9i		−	1.05733i
$554$			5.02524e8i			0.125566i
$555$		0			0
$556$	−1.07213e9			−0.264537
$557$			5.17753e9i			1.26949i	0.772721	+	0.634745i	$0.218895\pi$
$557$			5.17753e9i			1.26949i	−0.772721	+	0.634745i	$0.781105\pi$
$558$		0			0
$559$	−3.49192e9	−	1.89665e7i	−0.845518	−	0.00459247i
$560$	−1.31812e9			−0.317174
$561$		0			0
$562$	3.18589e9			0.757101
$563$	−1.67722e9			−0.396105			−0.198052	−	0.980191i	$-0.563462\pi$
$563$	−1.67722e9			−0.396105			−0.198052	+	0.980191i	$0.563462\pi$
$564$		0			0
$565$		−	5.36317e9i		−	1.25098i
$566$		−	6.17446e7i		−	0.0143133i
$567$		0			0
$568$	1.75344e9			0.401487
$569$	4.19299e9			0.954181			0.477090	−	0.878854i	$-0.341691\pi$
$569$	4.19299e9			0.954181			0.477090	+	0.878854i	$0.341691\pi$
$570$		0			0
$571$	−8.30446e8			−0.186674			−0.0933372	−	0.995635i	$-0.529753\pi$
$571$	−8.30446e8			−0.186674			−0.0933372	+	0.995635i	$0.529753\pi$
$572$	−1.99956e7	+	3.68139e9i	−0.00446734	+	0.822481i
$573$		0			0
$574$			6.33057e9i			1.39718i
$575$	−1.97641e9			−0.433550
$576$		0			0
$577$		−	7.17557e9i		−	1.55504i	−0.628859	−	0.777520i	$-0.716478\pi$
$577$		−	7.17557e9i		−	1.55504i	0.628859	−	0.777520i	$-0.283522\pi$
$578$			1.91267e9i			0.411995i
$579$		0			0
$580$		−	4.04060e9i		−	0.859901i
$581$	4.19076e8			0.0886495
$582$		0			0
$583$		−	4.31798e9i		−	0.902486i
$584$	−1.11805e9			−0.232282
$585$		0			0
$586$	2.00521e9			0.411640
$587$		−	5.83672e9i		−	1.19106i	−0.803331	−	0.595532i	$-0.796941\pi$
$587$		−	5.83672e9i		−	1.19106i	0.803331	−	0.595532i	$-0.203059\pi$
$588$		0			0
$589$	1.20816e9			0.243624
$590$			1.11008e8i			0.0222522i
$591$		0			0
$592$		−	2.53419e8i		−	0.0502011i
$593$			1.90799e9i			0.375737i	0.982194	+	0.187869i	$0.0601579\pi$
$593$			1.90799e9i			0.375737i	−0.982194	+	0.187869i	$0.939842\pi$
$594$		0			0
$595$	−8.20085e9			−1.59606
$596$		−	3.66042e9i		−	0.708221i
$597$		0			0
$598$	−2.87424e7	+	5.29176e9i	−0.00549628	+	1.01192i
$599$	−4.94369e9			−0.939847			−0.469924	−	0.882707i	$-0.655719\pi$
$599$	−4.94369e9			−0.939847			−0.469924	+	0.882707i	$0.655719\pi$
$600$		0			0
$601$	1.86163e9			0.349810			0.174905	−	0.984585i	$-0.444038\pi$
$601$	1.86163e9			0.349810			0.174905	+	0.984585i	$0.444038\pi$
$602$	3.55710e9			0.664520
$603$		0			0
$604$		−	3.83675e9i		−	0.708490i
$605$		−	1.06067e10i		−	1.94732i
$606$		0			0
$607$	−2.86458e8			−0.0519878			−0.0259939	−	0.999662i	$-0.508275\pi$
$607$	−2.86458e8			−0.0519878			−0.0259939	+	0.999662i	$0.508275\pi$
$608$	−2.86445e8			−0.0516867
$609$		0			0
$610$	−1.54966e9			−0.276427
$611$	7.77768e6	−	1.43195e9i	0.00137945	−	0.253970i
$612$		0			0
$613$		−	6.75234e9i		−	1.18398i	−0.805947	−	0.591988i	$-0.798343\pi$
$613$		−	6.75234e9i		−	1.18398i	0.805947	−	0.591988i	$-0.201657\pi$
$614$	−4.34462e9			−0.757466
$615$		0			0
$616$		−	3.75010e9i		−	0.646414i
$617$		−	2.16934e8i		−	0.0371817i	−0.999827	−	0.0185909i	$-0.994082\pi$
$617$		−	2.16934e8i		−	0.0371817i	0.999827	−	0.0185909i	$-0.00591799\pi$
$618$		0			0
$619$			8.84627e9i			1.49914i	0.661924	+	0.749571i	$0.269740\pi$
$619$			8.84627e9i			1.49914i	−0.661924	+	0.749571i	$0.730260\pi$
$620$	−2.82210e9			−0.475555
$621$		0			0
$622$		−	2.84044e9i		−	0.473281i
$623$	−2.06339e9			−0.341879
$624$		0			0
$625$	−7.39240e9			−1.21117
$626$		−	6.66666e9i		−	1.08617i
$627$		0			0
$628$	1.99219e9			0.320975
$629$		−	1.57668e9i		−	0.252619i
$630$		0			0
$631$			7.03063e9i			1.11402i	0.830507	+	0.557008i	$0.188051\pi$
$631$			7.03063e9i			1.11402i	−0.830507	+	0.557008i	$0.811949\pi$
$632$			2.13443e9i			0.336335i
$633$		0			0
$634$	6.32278e9			0.985362
$635$			1.28862e10i			1.99718i
$636$		0			0
$637$	−1.53525e9	−	8.33876e6i	−0.235337	−	0.00127824i
$638$	1.14957e10			1.75252
$639$		0			0
$640$	6.69097e8			0.100893
$641$	6.42851e8			0.0964068			0.0482034	−	0.998838i	$-0.484650\pi$
$641$	6.42851e8			0.0964068			0.0482034	+	0.998838i	$0.484650\pi$
$642$		0			0
$643$			1.02399e10i			1.51900i	0.650507	+	0.759501i	$0.274557\pi$
$643$			1.02399e10i			1.51900i	−0.650507	+	0.759501i	$0.725443\pi$
$644$		−	5.39053e9i		−	0.795299i
$645$		0			0
$646$	−1.78215e9			−0.260094
$647$	−1.08730e10			−1.57829			−0.789144	−	0.614208i	$-0.789476\pi$
$647$	−1.08730e10			−1.57829			−0.789144	+	0.614208i	$0.789476\pi$
$648$		0			0
$649$	−3.15822e8			−0.0453509
$650$	8.14648e6	−	1.49985e9i	0.00116352	−	0.214215i
$651$		0			0
$652$		−	1.61616e9i		−	0.228359i
$653$	3.68093e9			0.517322			0.258661	−	0.965968i	$-0.416719\pi$
$653$	3.68093e9			0.517322			0.258661	+	0.965968i	$0.416719\pi$
$654$		0			0
$655$		−	7.60089e9i		−	1.05687i
$656$		−	3.21349e9i		−	0.444440i
$657$		0			0
$658$			1.45867e9i			0.199603i
$659$	1.41639e10			1.92790			0.963951	−	0.266078i	$-0.0857280\pi$
$659$	1.41639e10			1.92790			0.963951	+	0.266078i	$0.0857280\pi$
$660$		0			0
$661$			5.78170e9i			0.778665i	0.921097	+	0.389332i	$0.127294\pi$
$661$			5.78170e9i			0.778665i	−0.921097	+	0.389332i	$0.872706\pi$
$662$	−6.82653e9			−0.914528
$663$		0			0
$664$	−2.12729e8			−0.0281993
$665$		−	2.81311e9i		−	0.370946i
$666$		0			0
$667$	1.65243e10			2.15616
$668$		−	4.24549e8i		−	0.0551075i
$669$		0			0
$670$		−	8.50019e9i		−	1.09186i
$671$		−	4.40883e9i		−	0.563371i
$672$		0			0
$673$	−1.96225e8			−0.0248143			−0.0124071	−	0.999923i	$-0.503949\pi$
$673$	−1.96225e8			−0.0248143			−0.0124071	+	0.999923i	$0.503949\pi$
$674$			1.76886e9i			0.222528i
$675$		0			0
$676$	−4.01567e9	−	4.36238e7i	−0.499970	−	0.00543138i
$677$	−2.79910e9			−0.346703			−0.173351	−	0.984860i	$-0.555460\pi$
$677$	−2.79910e9			−0.346703			−0.173351	+	0.984860i	$0.555460\pi$
$678$		0			0
$679$	−1.10254e9			−0.135160
$680$	4.16287e9			0.507705
$681$		0			0
$682$		−	8.02897e9i		−	0.969202i
$683$		−	6.64284e9i		−	0.797776i	−0.917000	−	0.398888i	$-0.869396\pi$
$683$		−	6.64284e9i		−	0.797776i	0.917000	−	0.398888i	$-0.130604\pi$
$684$		0			0
$685$	1.47849e10			1.75753
$686$	−5.08137e9			−0.600961
$687$		0			0
$688$	−1.80563e9			−0.211383
$689$	4.71020e9	+	2.55836e7i	0.548620	+	0.00297985i
$690$		0			0
$691$		−	5.54998e9i		−	0.639909i	−0.947433	−	0.319955i	$-0.896332\pi$
$691$		−	5.54998e9i		−	0.639909i	0.947433	−	0.319955i	$-0.103668\pi$
$692$	1.53782e9			0.176415
$693$		0			0
$694$		−	4.47138e9i		−	0.507789i
$695$			5.34475e9i			0.603921i
$696$		0			0
$697$		−	1.99931e10i		−	2.23648i
$698$	1.21920e10			1.35700
$699$		0			0
$700$			1.52784e9i			0.168358i
$701$	1.09055e10			1.19572			0.597862	−	0.801599i	$-0.296017\pi$
$701$	1.09055e10			1.19572			0.597862	+	0.801599i	$0.296017\pi$
$702$		0			0
$703$	5.40842e8			0.0587120
$704$			1.90360e9i			0.205623i
$705$		0			0
$706$	−8.76669e9			−0.937604
$707$			1.97444e10i			2.10124i
$708$		0			0
$709$		−	4.89719e9i		−	0.516042i	−0.966139	−	0.258021i	$-0.916930\pi$
$709$		−	4.89719e9i		−	0.516042i	0.966139	−	0.258021i	$-0.0830704\pi$
$710$		−	8.74118e9i		−	0.916571i
$711$		0			0
$712$	1.04741e9			0.108751
$713$		−	1.15411e10i		−	1.19243i
$714$		0			0
$715$	1.83523e10	+	9.96815e7i	1.87767	+	0.0101987i
$716$	−4.35401e9			−0.443296
$717$		0			0
$718$	−4.38213e9			−0.441824
$719$	−3.27973e9			−0.329069			−0.164535	−	0.986371i	$-0.552612\pi$
$719$	−3.27973e9			−0.329069			−0.164535	+	0.986371i	$0.552612\pi$
$720$		0			0
$721$			2.94211e9i			0.292338i
$722$			6.53965e9i			0.646657i
$723$		0			0
$724$	−1.14356e9			−0.111988
$725$	−4.68348e9			−0.456442
$726$		0			0
$727$	−7.76073e9			−0.749087			−0.374543	−	0.927209i	$-0.622201\pi$
$727$	−7.76073e9			−0.749087			−0.374543	+	0.927209i	$0.622201\pi$
$728$	4.09074e9	+	2.22190e7i	0.392954	+	0.00213435i
$729$		0			0
$730$			5.57365e9i			0.530286i
$731$	−1.12340e10			−1.06371
$732$		0			0
$733$			1.82989e10i			1.71618i	0.513502	+	0.858088i	$0.328348\pi$
$733$			1.82989e10i			1.71618i	−0.513502	+	0.858088i	$0.671652\pi$
$734$			3.46968e9i			0.323857i
$735$		0			0
$736$			2.73631e9i			0.252984i
$737$	2.41833e10			2.22526
$738$		0			0
$739$			7.96616e8i			0.0726095i	0.999341	+	0.0363048i	$0.0115587\pi$
$739$			7.96616e8i			0.0726095i	−0.999341	+	0.0363048i	$0.988441\pi$
$740$	−1.26334e9			−0.114606
$741$		0			0
$742$	−4.79811e9			−0.431178
$743$			4.75982e7i			0.00425726i	0.999998	+	0.00212863i	$0.000677564\pi$
$743$			4.75982e7i			0.00425726i	−0.999998	+	0.00212863i	$0.999322\pi$
$744$		0			0
$745$	−1.82478e10			−1.61682
$746$			3.50899e9i			0.309454i
$747$		0			0
$748$			1.18435e10i			1.03472i
$749$		−	1.06749e10i		−	0.928276i
$750$		0			0
$751$	−1.41304e10			−1.21735			−0.608673	−	0.793421i	$-0.708298\pi$
$751$	−1.41304e10			−1.21735			−0.608673	+	0.793421i	$0.708298\pi$
$752$		−	7.40443e8i		−	0.0634935i
$753$		0			0
$754$	−6.81108e7	+	1.25399e10i	−0.00578650	+	1.06535i
$755$	−1.91268e10			−1.61744
$756$		0			0
$757$	−1.05665e10			−0.885314			−0.442657	−	0.896691i	$-0.645964\pi$
$757$	−1.05665e10			−0.885314			−0.442657	+	0.896691i	$0.645964\pi$
$758$	3.74871e9			0.312636
$759$		0			0
$760$			1.42797e9i			0.117997i
$761$			1.95804e10i			1.61055i	0.592901	+	0.805275i	$0.297982\pi$
$761$			1.95804e10i			1.61055i	−0.592901	+	0.805275i	$0.702018\pi$
$762$		0			0
$763$	1.13387e10			0.924120
$764$	−8.08512e9			−0.655934
$765$		0			0
$766$	9.22288e9			0.741423
$767$	1.87122e6	−	3.44509e8i	0.000149741	−	0.0275687i
$768$		0			0
$769$			1.01771e10i			0.807012i	0.914977	+	0.403506i	$0.132209\pi$
$769$			1.01771e10i			0.807012i	−0.914977	+	0.403506i	$0.867791\pi$
$770$	−1.86949e10			−1.47572
$771$		0			0
$772$			5.27440e9i			0.412584i
$773$			2.31166e9i			0.180009i	0.995941	+	0.0900047i	$0.0286882\pi$
$773$			2.31166e9i			0.180009i	−0.995941	+	0.0900047i	$0.971312\pi$
$774$		0			0
$775$			3.27110e9i			0.252429i
$776$	5.59663e8			0.0429942
$777$		0			0
$778$			5.95484e9i			0.453358i
$779$	6.85816e9			0.519788
$780$		0			0
$781$	2.48690e10			1.86801
$782$			1.70243e10i			1.27305i
$783$		0			0
$784$	−7.93858e8			−0.0588351
$785$		−	9.93137e9i		−	0.732766i
$786$		0			0
$787$		−	2.61033e9i		−	0.190890i	−0.995435	−	0.0954450i	$-0.969573\pi$
$787$		−	2.61033e9i		−	0.190890i	0.995435	−	0.0954450i	$-0.0304274\pi$
$788$			1.05754e10i			0.769933i
$789$		0			0
$790$	1.06405e10			0.767833
$791$		−	1.69550e10i		−	1.21810i
$792$		0			0
$793$	4.80930e9	+	2.61219e7i	0.342473	+	0.00186015i
$794$	−8.23440e9			−0.583795
$795$		0			0
$796$	4.65865e9			0.327389
$797$	2.30473e10			1.61256			0.806279	−	0.591536i	$-0.201478\pi$
$797$	2.30473e10			1.61256			0.806279	+	0.591536i	$0.201478\pi$
$798$		0			0
$799$		−	4.60675e9i		−	0.319508i
$800$		−	7.75553e8i		−	0.0535546i
$801$		0			0
$802$	−7.85191e9			−0.537483
$803$	−1.58573e10			−1.08075
$804$		0			0
$805$	−2.68726e10			−1.81562
$806$	8.75827e9	+	4.75709e7i	0.589177	+	0.00320014i
$807$		0			0
$808$		−	1.00225e10i		−	0.668400i
$809$	2.25879e10			1.49988			0.749938	−	0.661508i	$-0.230083\pi$
$809$	2.25879e10			1.49988			0.749938	+	0.661508i	$0.230083\pi$
$810$		0			0
$811$			2.33114e10i			1.53460i	0.641290	+	0.767299i	$0.278400\pi$
$811$			2.33114e10i			1.53460i	−0.641290	+	0.767299i	$0.721600\pi$
$812$		−	1.27739e10i		−	0.837293i
$813$		0			0
$814$		−	3.59423e9i		−	0.233572i
$815$	−8.05681e9			−0.521328
$816$		0			0
$817$		−	3.85354e9i		−	0.247220i
$818$	−1.82034e10			−1.16283
$819$		0			0
$820$	−1.60197e10			−1.01463
$821$			1.14906e10i			0.724673i	0.932047	+	0.362337i	$0.118021\pi$
$821$			1.14906e10i			0.724673i	−0.932047	+	0.362337i	$0.881979\pi$
$822$		0			0
$823$	−1.06159e9			−0.0663833			−0.0331916	−	0.999449i	$-0.510567\pi$
$823$	−1.06159e9			−0.0663833			−0.0331916	+	0.999449i	$0.510567\pi$
$824$		−	1.49346e9i		−	0.0929923i
$825$		0			0
$826$			3.50939e8i			0.0216671i
$827$			6.50961e9i			0.400208i	0.979775	+	0.200104i	$0.0641280\pi$
$827$			6.50961e9i			0.400208i	−0.979775	+	0.200104i	$0.935872\pi$
$828$		0			0
$829$	9.62128e9			0.586532			0.293266	−	0.956031i	$-0.405258\pi$
$829$	9.62128e9			0.586532			0.293266	+	0.956031i	$0.405258\pi$
$830$			1.06049e9i			0.0643772i
$831$		0			0
$832$	−2.07652e9	−	1.12787e7i	−0.124998	−	0.000678933i
$833$	−4.93908e9			−0.296066
$834$		0			0
$835$	−2.11645e9			−0.125807
$836$	−4.06264e9			−0.240484
$837$		0			0
$838$			6.31728e9i			0.370831i
$839$			2.21131e10i			1.29265i	0.763061	+	0.646327i	$0.223696\pi$
$839$			2.21131e10i			1.29265i	−0.763061	+	0.646327i	$0.776304\pi$
$840$		0			0
$841$	2.19076e10			1.27001
$842$	−1.73296e10			−1.00045
$843$		0			0
$844$	1.12953e10			0.646692
$845$	−2.17472e8	+	2.00187e10i	−0.0123995	+	1.14140i
$846$		0			0
$847$		−	3.35320e10i		−	1.89613i
$848$	2.43559e9			0.137157
$849$		0			0
$850$		−	4.82520e9i		−	0.269494i
$851$		−	5.16647e9i		−	0.287370i
$852$		0			0
$853$			2.61062e10i			1.44019i	0.693873	+	0.720097i	$0.255903\pi$
$853$			2.61062e10i			1.44019i	−0.693873	+	0.720097i	$0.744097\pi$
$854$	−4.89906e9			−0.269160
$855$		0			0
$856$			5.41872e9i			0.295283i
$857$	−4.81819e9			−0.261487			−0.130744	−	0.991416i	$-0.541736\pi$
$857$	−4.81819e9			−0.261487			−0.130744	+	0.991416i	$0.541736\pi$
$858$		0			0
$859$	−2.95711e10			−1.59181			−0.795905	−	0.605421i	$-0.793005\pi$
$859$	−2.95711e10			−1.59181			−0.795905	+	0.605421i	$0.793005\pi$
$860$			9.00136e9i			0.482573i
$861$		0			0
$862$	−9.78604e9			−0.520393
$863$		−	1.65072e10i		−	0.874248i	−0.899401	−	0.437124i	$-0.855997\pi$
$863$		−	1.65072e10i		−	0.874248i	0.899401	−	0.437124i	$-0.144003\pi$
$864$		0			0
$865$		−	7.66630e9i		−	0.402744i
$866$		−	3.83162e9i		−	0.200479i
$867$		0			0
$868$	−8.92174e9			−0.463053
$869$			3.02726e10i			1.56488i
$870$		0			0
$871$	−1.43284e8	+	2.63800e10i	−0.00734741	+	1.35273i
$872$	−5.75570e9			−0.293961
$873$		0			0
$874$	−5.83977e9			−0.295873
$875$	−1.75246e10			−0.884343
$876$		0			0
$877$		−	1.45908e9i		−	0.0730434i	−0.999333	−	0.0365217i	$-0.988372\pi$
$877$		−	1.45908e9i		−	0.0730434i	0.999333	−	0.0365217i	$-0.0116278\pi$
$878$			1.61611e10i			0.805822i
$879$		0			0
$880$	9.48977e9			0.469425
$881$	2.84404e10			1.40127			0.700633	−	0.713522i	$-0.252901\pi$
$881$	2.84404e10			1.40127			0.700633	+	0.713522i	$0.252901\pi$
$882$		0			0
$883$	1.81412e10			0.886754			0.443377	−	0.896335i	$-0.353780\pi$
$883$	1.81412e10			0.886754			0.443377	+	0.896335i	$0.353780\pi$
$884$	−1.29193e10	−	7.01717e7i	−0.629008	−	0.00341648i
$885$		0			0
$886$			5.89478e9i			0.284741i
$887$	−2.33252e10			−1.12226			−0.561130	−	0.827728i	$-0.689633\pi$
$887$	−2.33252e10			−1.12226			−0.561130	+	0.827728i	$0.689633\pi$
$888$		0			0
$889$			4.07383e10i			1.94467i
$890$		−	5.22148e9i		−	0.248273i
$891$		0			0
$892$		−	9.05564e9i		−	0.427211i
$893$	1.58024e9			0.0742579
$894$		0			0
$895$			2.17054e10i			1.01202i
$896$	2.11527e9			0.0982400
$897$		0			0
$898$	−1.41414e10			−0.651667
$899$		−	2.73489e10i		−	1.25540i
$900$		0			0
$901$	1.51533e10			0.690193
$902$		−	4.55767e10i		−	2.06786i
$903$		0			0
$904$			8.60661e9i			0.387474i
$905$			5.70081e9i			0.255662i
$906$		0			0
$907$	−1.35577e9			−0.0603340			−0.0301670	−	0.999545i	$-0.509604\pi$
$907$	−1.35577e9			−0.0603340			−0.0301670	+	0.999545i	$0.509604\pi$
$908$			1.67160e10i			0.741025i
$909$		0			0
$910$	1.10765e8	−	2.03930e10i	0.00487258	−	0.897090i
$911$	1.32864e10			0.582227			0.291114	−	0.956689i	$-0.405974\pi$
$911$	1.32864e10			0.582227			0.291114	+	0.956689i	$0.405974\pi$
$912$		0			0
$913$	−3.01712e9			−0.131203
$914$	1.74479e9			0.0755843
$915$		0			0
$916$			5.02014e9i			0.215815i
$917$		−	2.40293e10i		−	1.02908i
$918$		0			0
$919$	−3.60465e10			−1.53200			−0.766000	−	0.642840i	$-0.777756\pi$
$919$	−3.60465e10			−1.53200			−0.766000	+	0.642840i	$0.777756\pi$
$920$	1.36409e10			0.577546
$921$		0			0
$922$	6.30227e9			0.264813
$923$	−1.47346e8	+	2.71279e10i	−0.00616785	+	1.13556i
$924$		0			0
$925$			1.46434e9i			0.0608338i
$926$	−1.25779e10			−0.520557
$927$		0			0
$928$			6.48421e9i			0.266342i
$929$		−	4.64342e10i		−	1.90013i	−0.312054	−	0.950064i	$-0.601017\pi$
$929$		−	4.64342e10i		−	1.90013i	0.312054	−	0.950064i	$-0.398983\pi$
$930$		0			0
$931$		−	1.69424e9i		−	0.0688098i
$932$	1.78694e10			0.723027
$933$		0			0
$934$			7.26363e9i			0.291702i
$935$	5.90418e10			2.36221
$936$		0			0
$937$	3.86501e10			1.53484			0.767418	−	0.641148i	$-0.221541\pi$
$937$	3.86501e10			1.53484			0.767418	+	0.641148i	$0.221541\pi$
$938$		−	2.68724e10i		−	1.06315i
$939$		0			0
$940$	−3.69122e9			−0.144952
$941$			2.74177e10i			1.07267i	0.844005	+	0.536336i	$0.180192\pi$
$941$			2.74177e10i			1.07267i	−0.844005	+	0.536336i	$0.819808\pi$
$942$		0			0
$943$		−	6.55135e10i		−	2.54414i
$944$		−	1.78142e8i		−	0.00689229i
$945$		0			0
$946$	−2.56092e10			−0.983506
$947$		−	1.39667e10i		−	0.534403i	−0.963641	−	0.267202i	$-0.913901\pi$
$947$		−	1.39667e10i		−	0.534403i	0.963641	−	0.267202i	$-0.0860990\pi$
$948$		0			0
$949$	9.39528e7	−	1.72976e10i	0.00356844	−	0.656984i
$950$	1.65517e9			0.0626340
$951$		0			0
$952$	1.31604e10			0.494357
$953$	−2.75080e10			−1.02952			−0.514759	−	0.857335i	$-0.672119\pi$
$953$	−2.75080e10			−1.02952			−0.514759	+	0.857335i	$0.672119\pi$
$954$		0			0
$955$			4.03056e10i			1.49746i
$956$		−	1.39884e10i		−	0.517804i
$957$		0			0
$958$	1.79106e10			0.658160
$959$	4.67408e10			1.71132
$960$		0			0
$961$	8.41119e9			0.305721
$962$	3.92071e9	+	2.12955e7i	0.141988	+	0.000771215i
$963$		0			0
$964$		−	2.04842e9i		−	0.0736461i
$965$	2.62937e10			0.941903
$966$		0			0
$967$		−	1.15208e10i		−	0.409722i	−0.978791	−	0.204861i	$-0.934326\pi$
$967$		−	1.15208e10i		−	0.409722i	0.978791	−	0.204861i	$-0.0656742\pi$
$968$			1.70213e10i			0.603156i
$969$		0			0
$970$		−	2.79001e9i		−	0.0981531i
$971$	−1.91388e10			−0.670885			−0.335443	−	0.942061i	$-0.608886\pi$
$971$	−1.91388e10			−0.670885			−0.335443	+	0.942061i	$0.608886\pi$
$972$		0			0
$973$			1.68968e10i			0.588044i
$974$	2.46134e10			0.853524
$975$		0			0
$976$	2.48683e9			0.0856194
$977$			3.56631e10i			1.22345i	0.791069	+	0.611727i	$0.209525\pi$
$977$			3.56631e10i			1.22345i	−0.791069	+	0.611727i	$0.790475\pi$
$978$		0			0
$979$	1.48553e10			0.505990
$980$			3.95751e9i			0.134317i
$981$		0			0
$982$		−	1.98577e10i		−	0.669173i
$983$			2.95255e9i			0.0991424i	0.998771	+	0.0495712i	$0.0157855\pi$
$983$			2.95255e9i			0.0991424i	−0.998771	+	0.0495712i	$0.984215\pi$
$984$		0			0
$985$	5.27198e10			1.75771
$986$			4.03423e10i			1.34027i
$987$		0			0
$988$	2.40708e7	−	4.43166e9i	0.000794036	−	0.146190i
$989$	−3.68115e10			−1.21003
$990$		0			0
$991$	−6.91025e9			−0.225547			−0.112773	−	0.993621i	$-0.535973\pi$
$991$	−6.91025e9			−0.225547			−0.112773	+	0.993621i	$0.535973\pi$
$992$	4.52880e9			0.147296
$993$		0			0
$994$		−	2.76342e10i		−	0.892473i
$995$		−	2.32241e10i		−	0.747408i
$996$		0			0
$997$	9.16218e9			0.292797			0.146398	−	0.989226i	$-0.453232\pi$
$997$	9.16218e9			0.292797			0.146398	+	0.989226i	$0.453232\pi$
$998$	−1.79640e9			−0.0572068
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	234.8.b.c.181.9		10
3.2	odd	2		26.8.b.a.25.2	✓	10
12.11	even	2		208.8.f.c.129.7		10
13.12	even	2	inner	234.8.b.c.181.2		10
39.5	even	4		338.8.a.k.1.2		5
39.8	even	4		338.8.a.l.1.2		5
39.38	odd	2		26.8.b.a.25.7	yes	10
156.155	even	2		208.8.f.c.129.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.b.a.25.2	✓	10	3.2	odd	2
26.8.b.a.25.7	yes	10	39.38	odd	2
208.8.f.c.129.7		10	12.11	even	2
208.8.f.c.129.8		10	156.155	even	2
234.8.b.c.181.2		10	13.12	even	2	inner
234.8.b.c.181.9		10	1.1	even	1	trivial
338.8.a.k.1.2		5	39.5	even	4
338.8.a.l.1.2		5	39.8	even	4

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:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	234.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{2} -64.0000 q^{4} +319.050i q^{5} +1008.64i q^{7} -512.000i q^{8} -2552.40 q^{10} -7261.68i q^{11} +(7921.28 + 43.0248i) q^{13} -8069.13 q^{14} +4096.00 q^{16} +25483.8 q^{17} +8741.60i q^{19} -20419.2i q^{20} +58093.4 q^{22} +83505.4 q^{23} -23668.0 q^{25} +(-344.198 + 63370.2i) q^{26} -64553.0i q^{28} +197882. q^{29} -138208. i q^{31} +32768.0i q^{32} +203870. i q^{34} -321807. q^{35} -61869.9i q^{37} -69932.8 q^{38} +163354. q^{40} -784542. i q^{41} -440828. q^{43} +464747. i q^{44} +668043. i q^{46} -180772. i q^{47} -193813. q^{49} -189344. i q^{50} +(-506962. - 2753.59i) q^{52} +594626. q^{53} +2.31684e6 q^{55} +516424. q^{56} +1.58306e6i q^{58} -43491.6i q^{59} +607137. q^{61} +1.10566e6 q^{62} -262144. q^{64} +(-13727.1 + 2.52729e6i) q^{65} +3.33027e6i q^{67} -1.63096e6 q^{68} -2.57446e6i q^{70} +3.42469e6i q^{71} -2.18369e6i q^{73} +494959. q^{74} -559463. i q^{76} +7.32442e6 q^{77} -4.16881e6 q^{79} +1.30683e6i q^{80} +6.27634e6 q^{82} -415486. i q^{83} +8.13060e6i q^{85} -3.52662e6i q^{86} -3.71798e6 q^{88} +2.04571e6i q^{89} +(-43396.5 + 7.98972e6i) q^{91} -5.34435e6 q^{92} +1.44618e6 q^{94} -2.78901e6 q^{95} +1.09309e6i q^{97} -1.55050e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 640 q^{4} - 1136 q^{10} + 3432 q^{13} - 3792 q^{14} + 40960 q^{16} + 6918 q^{17} + 17280 q^{22} - 94164 q^{23} - 330788 q^{25} - 118896 q^{26} + 131304 q^{29} - 873450 q^{35} - 602976 q^{38} + 72704 q^{40}+ \cdots - 16547484 q^{95}+O(q^{100}) \) 10 * q - 640 * q^4 - 1136 * q^10 + 3432 * q^13 - 3792 * q^14 + 40960 * q^16 + 6918 * q^17 + 17280 * q^22 - 94164 * q^23 - 330788 * q^25 - 118896 * q^26 + 131304 * q^29 - 873450 * q^35 - 602976 * q^38 + 72704 * q^40 + 2740774 * q^43 - 3083904 * q^49 - 219648 * q^52 + 3605916 * q^53 + 3999840 * q^55 + 242688 * q^56 + 11727656 * q^61 + 4210560 * q^62 - 2621440 * q^64 + 7481034 * q^65 - 442752 * q^68 - 18145392 * q^74 + 7147920 * q^77 - 28449228 * q^79 + 17868800 * q^82 - 1105920 * q^88 - 20225194 * q^91 + 6026496 * q^92 + 8240976 * q^94 - 16547484 * q^95

Introduction

L-functions

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