LMFDB - Embedded newform 117.12.b.e.64.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [117,12,Mod(64,117)] 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("117.64"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	117.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,-12288] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$89.8961521255$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 6409296870 x^{18} + \cdots + 25\!\cdots\!25 $ x^20 + 6409296870x^18 + 17460287982676083483x^16 + 26392149053720370272125789560x^14 + 24215610104843504114416671743892774381x^12 + 13839630572253649178735195612887353780030462966x^10 + 4849802296649618808485542975805992917218278575137043861x^8 + 981057650674091966578301529937541537485364653820592498442841208x^6 + 100416219118870395304462250703510810777937137112006877452917623636189331x^4 + 3875204156204591214055683105335029276015591666716438132245219226371146915476294*x^2 + 2550263913310843773340520395852312847495333848669179683402503783746251301536536208025
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{56}\cdot 3^{31}\cdot 5^{4}\cdot 11^{4}\cdot 13^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.6
Root			$-37188.3i$ of defining polynomial
Character	$\chi$	$=$	117.64
Dual form			117.12.b.e.64.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-38.6066i q^{2} +557.529 q^{4} +1195.10i q^{5} +73143.0i q^{7} -100591. i q^{8} +46138.6 q^{10} -954939. i q^{11} +(-1.02805e6 + 857481. i) q^{13} +2.82380e6 q^{14} -2.74164e6 q^{16} +2.94883e6 q^{17} +1.05100e7i q^{19} +666301. i q^{20} -3.68670e7 q^{22} -5.56429e7 q^{23} +4.73999e7 q^{25} +(3.31044e7 + 3.96895e7i) q^{26} +4.07793e7i q^{28} +4.50346e7 q^{29} -1.42399e8i q^{31} -1.00164e8i q^{32} -1.13844e8i q^{34} -8.74129e7 q^{35} -4.72518e8i q^{37} +4.05756e8 q^{38} +1.20216e8 q^{40} -7.78714e7i q^{41} +4.31061e8 q^{43} -5.32407e8i q^{44} +2.14819e9i q^{46} -5.63078e8i q^{47} -3.37257e9 q^{49} -1.82995e9i q^{50} +(-5.73168e8 + 4.78071e8i) q^{52} +1.65808e9 q^{53} +1.14124e9 q^{55} +7.35750e9 q^{56} -1.73864e9i q^{58} -3.87972e9i q^{59} +1.04030e10 q^{61} -5.49754e9 q^{62} -9.48188e9 q^{64} +(-1.02477e9 - 1.22862e9i) q^{65} -1.38881e10i q^{67} +1.64406e9 q^{68} +3.37472e9i q^{70} -1.13127e10i q^{71} -1.01627e10i q^{73} -1.82423e10 q^{74} +5.85964e9i q^{76} +6.98471e10 q^{77} -2.77723e9 q^{79} -3.27653e9i q^{80} -3.00635e9 q^{82} -3.88719e10i q^{83} +3.52414e9i q^{85} -1.66418e10i q^{86} -9.60580e10 q^{88} +2.56648e9i q^{89} +(-6.27187e10 - 7.51946e10i) q^{91} -3.10226e10 q^{92} -2.17385e10 q^{94} -1.25605e10 q^{95} +3.20279e10i q^{97} +1.30203e11i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12288 q^{4} - 2212240 q^{10} - 1708564 q^{13} + 32533888 q^{16} - 89284144 q^{22} + 209701380 q^{25} + 309455680 q^{40} + 2936996032 q^{43} - 10965350212 q^{49} + 4282707936 q^{52} + 6622126880 q^{55}+ \cdots + 330546354704 q^{94}+O(q^{100}) $ 20 * q - 12288 * q^4 - 2212240 * q^10 - 1708564 * q^13 + 32533888 * q^16 - 89284144 * q^22 + 209701380 * q^25 + 309455680 * q^40 + 2936996032 * q^43 - 10965350212 * q^49 + 4282707936 * q^52 + 6622126880 * q^55 + 3358771336 * q^61 - 43877642752 * q^64 - 148998409616 * q^79 + 319857511952 * q^82 - 250888459328 * q^88 + 62924651088 * q^91 + 330546354704 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	38.6066i		−	0.853094i	−0.904465	−	0.426547i	$-0.859730\pi$
$2$		−	38.6066i		−	0.853094i	0.904465	−	0.426547i	$-0.140270\pi$
$3$		0			0
$3$		0			0
$4$	557.529			0.272231
$5$			1195.10i			0.171028i	0.996337	+	0.0855141i	$0.0272533\pi$
$5$			1195.10i			0.171028i	−0.996337	+	0.0855141i	$0.972747\pi$
$6$		0			0
$7$			73143.0i			1.64488i	0.568854	+	0.822438i	$0.307387\pi$
$7$			73143.0i			1.64488i	−0.568854	+	0.822438i	$0.692613\pi$
$8$		−	100591.i		−	1.08533i
$9$		0			0
$10$	46138.6			0.145903
$11$		−	954939.i		−	1.78779i	−0.448279	−	0.893894i	$-0.647963\pi$
$11$		−	954939.i		−	1.78779i	0.448279	−	0.893894i	$-0.352037\pi$
$12$		0			0
$13$	−1.02805e6	+	857481.i	−0.767937	+	0.640525i
$13$	−1.02805e6	+	857481.i	−0.767937	+	0.640525i
$14$	2.82380e6			1.40323
$15$		0			0
$16$	−2.74164e6			−0.653659
$17$	2.94883e6			0.503711			0.251855	−	0.967765i	$-0.418959\pi$
$17$	2.94883e6			0.503711			0.251855	+	0.967765i	$0.418959\pi$
$18$		0			0
$19$			1.05100e7i			0.973775i	0.873465	+	0.486888i	$0.161868\pi$
$19$			1.05100e7i			0.973775i	−0.873465	+	0.486888i	$0.838132\pi$
$20$			666301.i			0.0465592i
$21$		0			0
$22$	−3.68670e7			−1.52515
$23$	−5.56429e7			−1.80263			−0.901316	−	0.433162i	$-0.857398\pi$
$23$	−5.56429e7			−1.80263			−0.901316	+	0.433162i	$0.857398\pi$
$24$		0			0
$25$	4.73999e7			0.970749
$26$	3.31044e7	+	3.96895e7i	0.546428	+	0.655122i
$27$		0			0
$28$			4.07793e7i			0.447787i
$29$	4.50346e7			0.407716			0.203858	−	0.979000i	$-0.434652\pi$
$29$	4.50346e7			0.407716			0.203858	+	0.979000i	$0.434652\pi$
$30$		0			0
$31$		−	1.42399e8i		−	0.893340i	−0.894699	−	0.446670i	$-0.852610\pi$
$31$		−	1.42399e8i		−	0.893340i	0.894699	−	0.446670i	$-0.147390\pi$
$32$		−	1.00164e8i		−	0.527700i
$33$		0			0
$34$		−	1.13844e8i		−	0.429712i
$35$	−8.74129e7			−0.281320
$36$		0			0
$37$		−	4.72518e8i		−	1.12023i	−0.828413	−	0.560117i	$-0.810756\pi$
$37$		−	4.72518e8i		−	1.12023i	0.828413	−	0.560117i	$-0.189244\pi$
$38$	4.05756e8			0.830721
$39$		0			0
$40$	1.20216e8			0.185623
$41$		−	7.78714e7i		−	0.104970i	−0.998622	−	0.0524852i	$-0.983286\pi$
$41$		−	7.78714e7i		−	0.104970i	0.998622	−	0.0524852i	$-0.0167142\pi$
$42$		0			0
$43$	4.31061e8			0.447159			0.223580	−	0.974686i	$-0.428226\pi$
$43$	4.31061e8			0.447159			0.223580	+	0.974686i	$0.428226\pi$
$44$		−	5.32407e8i		−	0.486691i
$45$		0			0
$46$			2.14819e9i			1.53781i
$47$		−	5.63078e8i		−	0.358121i	−0.983838	−	0.179061i	$-0.942694\pi$
$47$		−	5.63078e8i		−	0.358121i	0.983838	−	0.179061i	$-0.0573059\pi$
$48$		0			0
$49$	−3.37257e9			−1.70562
$50$		−	1.82995e9i		−	0.828140i
$51$		0			0
$52$	−5.73168e8	+	4.78071e8i	−0.209056	+	0.174371i
$53$	1.65808e9			0.544613			0.272307	−	0.962211i	$-0.412214\pi$
$53$	1.65808e9			0.544613			0.272307	+	0.962211i	$0.412214\pi$
$54$		0			0
$55$	1.14124e9			0.305762
$56$	7.35750e9			1.78524
$57$		0			0
$58$		−	1.73864e9i		−	0.347820i
$59$		−	3.87972e9i		−	0.706503i	−0.935528	−	0.353252i	$-0.885076\pi$
$59$		−	3.87972e9i		−	0.706503i	0.935528	−	0.353252i	$-0.114924\pi$
$60$		0			0
$61$	1.04030e10			1.57704			0.788522	−	0.615006i	$-0.210846\pi$
$61$	1.04030e10			1.57704			0.788522	+	0.615006i	$0.210846\pi$
$62$	−5.49754e9			−0.762103
$63$		0			0
$64$	−9.48188e9			−1.10384
$65$	−1.02477e9	−	1.22862e9i	−0.109548	−	0.131339i
$66$		0			0
$67$		−	1.38881e10i		−	1.25670i	−0.777930	−	0.628351i	$-0.783730\pi$
$67$		−	1.38881e10i		−	1.25670i	0.777930	−	0.628351i	$-0.216270\pi$
$68$	1.64406e9			0.137126
$69$		0			0
$70$			3.37472e9i			0.239993i
$71$		−	1.13127e10i		−	0.744127i	−0.928207	−	0.372063i	$-0.878650\pi$
$71$		−	1.13127e10i		−	0.744127i	0.928207	−	0.372063i	$-0.121350\pi$
$72$		0			0
$73$		−	1.01627e10i		−	0.573766i	−0.957966	−	0.286883i	$-0.907381\pi$
$73$		−	1.01627e10i		−	0.573766i	0.957966	−	0.286883i	$-0.0926191\pi$
$74$	−1.82423e10			−0.955665
$75$		0			0
$76$			5.85964e9i			0.265092i
$77$	6.98471e10			2.94069
$78$		0			0
$79$	−2.77723e9			−0.101546			−0.0507730	−	0.998710i	$-0.516168\pi$
$79$	−2.77723e9			−0.101546			−0.0507730	+	0.998710i	$0.516168\pi$
$80$		−	3.27653e9i		−	0.111794i
$81$		0			0
$82$	−3.00635e9			−0.0895495
$83$		−	3.88719e10i		−	1.08319i	−0.840639	−	0.541597i	$-0.817820\pi$
$83$		−	3.88719e10i		−	1.08319i	0.840639	−	0.541597i	$-0.182180\pi$
$84$		0			0
$85$			3.52414e9i			0.0861487i
$86$		−	1.66418e10i		−	0.381469i
$87$		0			0
$88$	−9.60580e10			−1.94034
$89$			2.56648e9i			0.0487184i	0.999703	+	0.0243592i	$0.00775453\pi$
$89$			2.56648e9i			0.0487184i	−0.999703	+	0.0243592i	$0.992245\pi$
$90$		0			0
$91$	−6.27187e10	−	7.51946e10i	−1.05358	−	1.26316i
$92$	−3.10226e10			−0.490732
$93$		0			0
$94$	−2.17385e10			−0.305511
$95$	−1.25605e10			−0.166543
$96$		0			0
$97$			3.20279e10i			0.378691i	0.981911	+	0.189345i	$0.0606366\pi$
$97$			3.20279e10i			0.378691i	−0.981911	+	0.189345i	$0.939363\pi$
$98$			1.30203e11i			1.45505i
$99$		0			0
$100$	2.64268e10			0.264268
$101$	−1.10243e11			−1.04372			−0.521861	−	0.853030i	$-0.674762\pi$
$101$	−1.10243e11			−1.04372			−0.521861	+	0.853030i	$0.674762\pi$
$102$		0			0
$103$	−1.94652e10			−0.165445			−0.0827227	−	0.996573i	$-0.526362\pi$
$103$	−1.94652e10			−0.165445			−0.0827227	+	0.996573i	$0.526362\pi$
$104$	8.62546e10	+	1.03412e11i	0.695183	+	0.833467i
$105$		0			0
$106$		−	6.40128e10i		−	0.464606i
$107$	−2.13566e11			−1.47205			−0.736023	−	0.676957i	$-0.763298\pi$
$107$	−2.13566e11			−1.47205			−0.736023	+	0.676957i	$0.763298\pi$
$108$		0			0
$109$		−	2.44818e11i		−	1.52404i	−0.647552	−	0.762021i	$-0.724207\pi$
$109$		−	2.44818e11i		−	1.52404i	0.647552	−	0.762021i	$-0.275793\pi$
$110$		−	4.40596e10i		−	0.260844i
$111$		0			0
$112$		−	2.00532e11i		−	1.07519i
$113$	2.16778e11			1.10684			0.553419	−	0.832903i	$-0.313323\pi$
$113$	2.16778e11			1.10684			0.553419	+	0.832903i	$0.313323\pi$
$114$		0			0
$115$		−	6.64987e10i		−	0.308301i
$116$	2.51081e10			0.110993
$117$		0			0
$118$	−1.49783e11			−0.602713
$119$			2.15686e11i			0.828542i
$120$		0			0
$121$	−6.26597e11			−2.19618
$122$		−	4.01624e11i		−	1.34537i
$123$		0			0
$124$		−	7.93915e10i		−	0.243195i
$125$			1.15002e11i			0.337054i
$126$		0			0
$127$	3.69355e11			0.992026			0.496013	−	0.868315i	$-0.334797\pi$
$127$	3.69355e11			0.992026			0.496013	+	0.868315i	$0.334797\pi$
$128$			1.60927e11i			0.413976i
$129$		0			0
$130$	−4.74328e10	+	3.95630e10i	−0.112044	+	0.0934546i
$131$	−6.58983e11			−1.49239			−0.746195	−	0.665727i	$-0.768121\pi$
$131$	−6.58983e11			−1.49239			−0.746195	+	0.665727i	$0.768121\pi$
$132$		0			0
$133$	−7.68734e11			−1.60174
$134$	−5.36174e11			−1.07208
$135$		0			0
$136$		−	2.96625e11i		−	0.546693i
$137$			8.45092e11i			1.49603i	0.663681	+	0.748016i	$0.268993\pi$
$137$			8.45092e11i			1.49603i	−0.663681	+	0.748016i	$0.731007\pi$
$138$		0			0
$139$	−9.38004e11			−1.53329			−0.766643	−	0.642074i	$-0.778074\pi$
$139$	−9.38004e11			−1.53329			−0.766643	+	0.642074i	$0.778074\pi$
$140$	−4.87353e10			−0.0765842
$141$		0			0
$142$	−4.36747e11			−0.634810
$143$	8.18842e11	+	9.81725e11i	1.14512	+	1.37291i
$144$		0			0
$145$			5.38207e10i			0.0697310i
$146$	−3.92349e11			−0.489476
$147$		0			0
$148$		−	2.63443e11i		−	0.304963i
$149$		−	1.14328e12i		−	1.27535i	−0.770305	−	0.637675i	$-0.779896\pi$
$149$		−	1.14328e12i		−	1.27535i	0.770305	−	0.637675i	$-0.220104\pi$
$150$		0			0
$151$		−	1.13990e11i		−	0.118166i	−0.998253	−	0.0590832i	$-0.981182\pi$
$151$		−	1.13990e11i		−	0.118166i	0.998253	−	0.0590832i	$-0.0188177\pi$
$152$	1.05721e12			1.05687
$153$		0			0
$154$		−	2.69656e12i		−	2.50868i
$155$	1.70180e11			0.152786
$156$		0			0
$157$	−4.68773e10			−0.0392207			−0.0196103	−	0.999808i	$-0.506243\pi$
$157$	−4.68773e10			−0.0392207			−0.0196103	+	0.999808i	$0.506243\pi$
$158$			1.07219e11i			0.0866282i
$159$		0			0
$160$	1.19706e11			0.0902516
$161$		−	4.06989e12i		−	2.96511i
$162$		0			0
$163$		−	8.59928e11i		−	0.585370i	−0.956209	−	0.292685i	$-0.905451\pi$
$163$		−	8.59928e11i		−	0.585370i	0.956209	−	0.292685i	$-0.0945487\pi$
$164$		−	4.34156e10i		−	0.0285762i
$165$		0			0
$166$	−1.50071e12			−0.924065
$167$		−	1.12243e12i		−	0.668682i	−0.942452	−	0.334341i	$-0.891486\pi$
$167$		−	1.12243e12i		−	0.668682i	0.942452	−	0.334341i	$-0.108514\pi$
$168$		0			0
$169$	3.21613e11	−	1.76307e12i	0.179455	−	0.983766i
$170$	1.36055e11			0.0734929
$171$		0			0
$172$	2.40329e11			0.121731
$173$	2.62834e12			1.28952			0.644760	−	0.764385i	$-0.276957\pi$
$173$	2.62834e12			1.28952			0.644760	+	0.764385i	$0.276957\pi$
$174$		0			0
$175$			3.46697e12i			1.59676i
$176$			2.61810e12i			1.16860i
$177$		0			0
$178$	9.90830e10			0.0415613
$179$	2.34555e12			0.954012			0.477006	−	0.878900i	$-0.341722\pi$
$179$	2.34555e12			0.954012			0.477006	+	0.878900i	$0.341722\pi$
$180$		0			0
$181$	−1.18326e12			−0.452739			−0.226369	−	0.974042i	$-0.572686\pi$
$181$	−1.18326e12			−0.452739			−0.226369	+	0.974042i	$0.572686\pi$
$182$	−2.90301e12	+	2.42136e12i	−1.07760	+	0.898806i
$183$		0			0
$184$			5.59716e12i			1.95645i
$185$	5.64705e11			0.191592
$186$		0			0
$187$		−	2.81596e12i		−	0.900527i
$188$		−	3.13933e11i		−	0.0974918i
$189$		0			0
$190$			4.84918e11i			0.142077i
$191$	−4.82060e12			−1.37220			−0.686100	−	0.727507i	$-0.740679\pi$
$191$	−4.82060e12			−1.37220			−0.686100	+	0.727507i	$0.740679\pi$
$192$		0			0
$193$		−	2.21550e12i		−	0.595535i	−0.954638	−	0.297768i	$-0.903758\pi$
$193$		−	2.21550e12i		−	0.595535i	0.954638	−	0.297768i	$-0.0962420\pi$
$194$	1.23649e12			0.323059
$195$		0			0
$196$	−1.88030e12			−0.464323
$197$		−	5.49523e12i		−	1.31954i	−0.751469	−	0.659769i	$-0.770654\pi$
$197$		−	5.49523e12i		−	1.31954i	0.751469	−	0.659769i	$-0.229346\pi$
$198$		0			0
$199$	2.56932e12			0.583614			0.291807	−	0.956477i	$-0.405743\pi$
$199$	2.56932e12			0.583614			0.291807	+	0.956477i	$0.405743\pi$
$200$		−	4.76798e12i		−	1.05359i
$201$		0			0
$202$			4.25613e12i			0.890393i
$203$			3.29397e12i			0.670643i
$204$		0			0
$205$	9.30638e10			0.0179529
$206$			7.51487e11i			0.141141i
$207$		0			0
$208$	2.81855e12	−	2.35091e12i	0.501969	−	0.418685i
$209$	1.00364e13			1.74090
$210$		0			0
$211$	8.22813e12			1.35440			0.677201	−	0.735798i	$-0.263193\pi$
$211$	8.22813e12			1.35440			0.677201	+	0.735798i	$0.263193\pi$
$212$	9.24428e11			0.148261
$213$		0			0
$214$			8.24506e12i			1.25579i
$215$			5.15159e11i			0.0764769i
$216$		0			0
$217$	1.04155e13			1.46943
$218$	−9.45158e12			−1.30015
$219$		0			0
$220$	6.36277e11			0.0832380
$221$	−3.03155e12	+	2.52857e12i	−0.386818	+	0.322639i
$222$		0			0
$223$			6.72248e12i			0.816305i	0.912914	+	0.408153i	$0.133827\pi$
$223$			6.72248e12i			0.816305i	−0.912914	+	0.408153i	$0.866173\pi$
$224$	7.32630e12			0.868001
$225$		0			0
$226$		−	8.36907e12i		−	0.944237i
$227$			1.02259e13i			1.12605i	0.826440	+	0.563025i	$0.190362\pi$
$227$			1.02259e13i			1.12605i	−0.826440	+	0.563025i	$0.809638\pi$
$228$		0			0
$229$		−	1.16258e12i		−	0.121990i	−0.998138	−	0.0609952i	$-0.980573\pi$
$229$		−	1.16258e12i		−	0.121990i	0.998138	−	0.0609952i	$-0.0194274\pi$
$230$	−2.56729e12			−0.263010
$231$		0			0
$232$		−	4.53007e12i		−	0.442507i
$233$	−1.36620e13			−1.30334			−0.651668	−	0.758504i	$-0.725931\pi$
$233$	−1.36620e13			−1.30334			−0.651668	+	0.758504i	$0.725931\pi$
$234$		0			0
$235$	6.72932e11			0.0612489
$236$		−	2.16306e12i		−	0.192332i
$237$		0			0
$238$	8.32692e12			0.706824
$239$		−	7.78795e11i		−	0.0646003i	−0.999478	−	0.0323002i	$-0.989717\pi$
$239$		−	7.78795e11i		−	0.0646003i	0.999478	−	0.0323002i	$-0.0102832\pi$
$240$		0			0
$241$			2.26102e13i			1.79147i	0.444585	+	0.895736i	$0.353351\pi$
$241$			2.26102e13i			1.79147i	−0.444585	+	0.895736i	$0.646649\pi$
$242$			2.41908e13i			1.87355i
$243$		0			0
$244$	5.79997e12			0.429321
$245$		−	4.03054e12i		−	0.291709i
$246$		0			0
$247$	−9.01214e12	−	1.08048e13i	−0.623727	−	0.747798i
$248$	−1.43240e13			−0.969571
$249$		0			0
$250$	4.43983e12			0.287539
$251$	−1.87662e13			−1.18897			−0.594485	−	0.804107i	$-0.702644\pi$
$251$	−1.87662e13			−1.18897			−0.594485	+	0.804107i	$0.702644\pi$
$252$		0			0
$253$			5.31356e13i			3.22272i
$254$		−	1.42595e13i		−	0.846291i
$255$		0			0
$256$	−1.32060e13			−0.750676
$257$	3.08184e13			1.71466			0.857330	−	0.514768i	$-0.172122\pi$
$257$	3.08184e13			1.71466			0.857330	+	0.514768i	$0.172122\pi$
$258$		0			0
$259$	3.45614e13			1.84265
$260$	−5.71341e11	−	6.84991e11i	−0.0298223	−	0.0357546i
$261$		0			0
$262$			2.54411e13i			1.27315i
$263$	5.00200e12			0.245125			0.122562	−	0.992461i	$-0.460889\pi$
$263$	5.00200e12			0.245125			0.122562	+	0.992461i	$0.460889\pi$
$264$		0			0
$265$			1.98156e12i			0.0931442i
$266$			2.96782e13i			1.36643i
$267$		0			0
$268$		−	7.74304e12i		−	0.342113i
$269$	7.05544e12			0.305412			0.152706	−	0.988272i	$-0.451201\pi$
$269$	7.05544e12			0.305412			0.152706	+	0.988272i	$0.451201\pi$
$270$		0			0
$271$		−	4.50472e13i		−	1.87213i	−0.351825	−	0.936066i	$-0.614439\pi$
$271$		−	4.50472e13i		−	1.87213i	0.351825	−	0.936066i	$-0.385561\pi$
$272$	−8.08465e12			−0.329255
$273$		0			0
$274$	3.26261e13			1.27625
$275$		−	4.52640e13i		−	1.73549i
$276$		0			0
$277$	−3.50464e13			−1.29123			−0.645616	−	0.763662i	$-0.723399\pi$
$277$	−3.50464e13			−1.29123			−0.645616	+	0.763662i	$0.723399\pi$
$278$			3.62131e13i			1.30804i
$279$		0			0
$280$			8.79292e12i			0.305326i
$281$			3.65086e12i			0.124311i	0.998066	+	0.0621556i	$0.0197975\pi$
$281$			3.65086e12i			0.124311i	−0.998066	+	0.0621556i	$0.980202\pi$
$282$		0			0
$283$	4.13087e13			1.35275			0.676373	−	0.736559i	$-0.263551\pi$
$283$	4.13087e13			1.35275			0.676373	+	0.736559i	$0.263551\pi$
$284$		−	6.30718e12i		−	0.202574i
$285$		0			0
$286$	3.79011e13	−	3.16127e13i	1.17122	−	0.976897i
$287$	5.69574e12			0.172663
$288$		0			0
$289$	−2.55763e13			−0.746276
$290$	2.07784e12			0.0594870
$291$		0			0
$292$		−	5.66602e12i		−	0.156197i
$293$		−	5.85460e13i		−	1.58389i	−0.610592	−	0.791946i	$-0.709068\pi$
$293$		−	5.85460e13i		−	1.58389i	0.610592	−	0.791946i	$-0.290932\pi$
$294$		0			0
$295$	4.63664e12			0.120832
$296$	−4.75309e13			−1.21583
$297$		0			0
$298$	−4.41383e13			−1.08799
$299$	5.72037e13	−	4.77128e13i	1.38431	−	1.15463i
$300$		0			0
$301$			3.15291e13i			0.735522i
$302$	−4.40077e12			−0.100807
$303$		0			0
$304$		−	2.88147e13i		−	0.636517i
$305$			1.24326e13i			0.269719i
$306$		0			0
$307$			3.50235e13i			0.732992i	0.930420	+	0.366496i	$0.119443\pi$
$307$			3.50235e13i			0.732992i	−0.930420	+	0.366496i	$0.880557\pi$
$308$	3.89418e13			0.800547
$309$		0			0
$310$		−	6.57008e12i		−	0.130341i
$311$	5.98876e13			1.16723			0.583613	−	0.812032i	$-0.301639\pi$
$311$	5.98876e13			1.16723			0.583613	+	0.812032i	$0.301639\pi$
$312$		0			0
$313$	1.21122e12			0.0227892			0.0113946	−	0.999935i	$-0.496373\pi$
$313$	1.21122e12			0.0227892			0.0113946	+	0.999935i	$0.496373\pi$
$314$			1.80978e12i			0.0334589i
$315$		0			0
$316$	−1.54839e12			−0.0276440
$317$			3.05253e12i			0.0535593i	0.999641	+	0.0267796i	$0.00852524\pi$
$317$			3.05253e12i			0.0535593i	−0.999641	+	0.0267796i	$0.991475\pi$
$318$		0			0
$319$		−	4.30053e13i		−	0.728910i
$320$		−	1.13318e13i		−	0.188787i
$321$		0			0
$322$	−1.57125e14			−2.52951
$323$			3.09923e13i			0.490501i
$324$		0			0
$325$	−4.87294e13	+	4.06445e13i	−0.745475	+	0.621789i
$326$	−3.31989e13			−0.499375
$327$		0			0
$328$	−7.83313e12			−0.113928
$329$	4.11852e13			0.589066
$330$		0			0
$331$		−	4.33747e13i		−	0.600043i	−0.953932	−	0.300021i	$-0.903006\pi$
$331$		−	4.33747e13i		−	0.600043i	0.953932	−	0.300021i	$-0.0969938\pi$
$332$		−	2.16722e13i		−	0.294879i
$333$		0			0
$334$	−4.33333e13			−0.570449
$335$	1.65976e13			0.214932
$336$		0			0
$337$	1.89342e13			0.237292			0.118646	−	0.992937i	$-0.462145\pi$
$337$	1.89342e13			0.237292			0.118646	+	0.992937i	$0.462145\pi$
$338$	−6.80660e13	−	1.24164e13i	−0.839245	−	0.153092i
$339$		0			0
$340$			1.96481e12i			0.0234524i
$341$	−1.35982e14			−1.59710
$342$		0			0
$343$		−	1.02052e14i		−	1.16066i
$344$		−	4.33607e13i		−	0.485316i
$345$		0			0
$346$		−	1.01471e14i		−	1.10008i
$347$	−3.14756e13			−0.335862			−0.167931	−	0.985799i	$-0.553709\pi$
$347$	−3.14756e13			−0.335862			−0.167931	+	0.985799i	$0.553709\pi$
$348$		0			0
$349$		−	4.21755e13i		−	0.436034i	−0.975945	−	0.218017i	$-0.930041\pi$
$349$		−	4.21755e13i		−	0.436034i	0.975945	−	0.218017i	$-0.0699589\pi$
$350$	1.33848e14			1.36219
$351$		0			0
$352$	−9.56506e13			−0.943415
$353$			1.16131e14i			1.12768i	0.825883	+	0.563842i	$0.190677\pi$
$353$			1.16131e14i			1.12768i	−0.825883	+	0.563842i	$0.809323\pi$
$354$		0			0
$355$	1.35198e13			0.127267
$356$			1.43089e12i			0.0132627i
$357$		0			0
$358$		−	9.05539e13i		−	0.813861i
$359$		−	9.68116e13i		−	0.856856i	−0.903576	−	0.428428i	$-0.859068\pi$
$359$		−	9.68116e13i		−	0.856856i	0.903576	−	0.428428i	$-0.140932\pi$
$360$		0			0
$361$	6.02977e12			0.0517620
$362$			4.56816e13i			0.386229i
$363$		0			0
$364$	−3.49675e13	−	4.19232e13i	−0.286819	−	0.343872i
$365$	1.21454e13			0.0981302
$366$		0			0
$367$	8.37029e13			0.656262			0.328131	−	0.944632i	$-0.393581\pi$
$367$	8.37029e13			0.656262			0.328131	+	0.944632i	$0.393581\pi$
$368$	1.52553e14			1.17831
$369$		0			0
$370$		−	2.18013e13i		−	0.163446i
$371$			1.21277e14i			0.895821i
$372$		0			0
$373$	−1.96708e13			−0.141066			−0.0705331	−	0.997509i	$-0.522470\pi$
$373$	−1.96708e13			−0.141066			−0.0705331	+	0.997509i	$0.522470\pi$
$374$	−1.08715e14			−0.768234
$375$		0			0
$376$	−5.66404e13			−0.388681
$377$	−4.62979e13	+	3.86164e13i	−0.313100	+	0.261152i
$378$		0			0
$379$		−	1.84123e14i		−	1.20946i	−0.796430	−	0.604731i	$-0.793280\pi$
$379$		−	1.84123e14i		−	1.20946i	0.796430	−	0.604731i	$-0.206720\pi$
$380$	−7.00284e12			−0.0453382
$381$		0			0
$382$			1.86107e14i			1.17062i
$383$		−	1.03120e14i		−	0.639364i	−0.947525	−	0.319682i	$-0.896424\pi$
$383$		−	1.03120e14i		−	0.639364i	0.947525	−	0.319682i	$-0.103576\pi$
$384$		0			0
$385$			8.34740e13i			0.502941i
$386$	−8.55331e13			−0.508047
$387$		0			0
$388$			1.78565e13i			0.103091i
$389$	−2.40454e14			−1.36870			−0.684352	−	0.729151i	$-0.739915\pi$
$389$	−2.40454e14			−1.36870			−0.684352	+	0.729151i	$0.739915\pi$
$390$		0			0
$391$	−1.64082e14			−0.908005
$392$			3.39249e14i			1.85116i
$393$		0			0
$394$	−2.12152e14			−1.12569
$395$		−	3.31905e12i		−	0.0173672i
$396$		0			0
$397$			7.22425e13i			0.367659i	0.982958	+	0.183829i	$0.0588494\pi$
$397$			7.22425e13i			0.367659i	−0.982958	+	0.183829i	$0.941151\pi$
$398$		−	9.91927e13i		−	0.497878i
$399$		0			0
$400$	−1.29954e14			−0.634539
$401$			2.33817e14i			1.12611i	0.826418	+	0.563057i	$0.190375\pi$
$401$			2.33817e14i			1.12611i	−0.826418	+	0.563057i	$0.809625\pi$
$402$		0			0
$403$	1.22104e14	+	1.46393e14i	0.572207	+	0.686029i
$404$	−6.14640e13			−0.284134
$405$		0			0
$406$	1.27169e14			0.572121
$407$	−4.51226e14			−2.00274
$408$		0			0
$409$		−	2.40388e14i		−	1.03857i	−0.854602	−	0.519283i	$-0.826199\pi$
$409$		−	2.40388e14i		−	1.03857i	0.854602	−	0.519283i	$-0.173801\pi$
$410$		−	3.59288e12i		−	0.0153155i
$411$		0			0
$412$	−1.08524e13			−0.0450394
$413$	2.83774e14			1.16211
$414$		0			0
$415$	4.64556e13			0.185257
$416$	8.58888e13	+	1.02974e14i	0.338005	+	0.405240i
$417$		0			0
$418$		−	3.87473e14i		−	1.48515i
$419$	4.02827e14			1.52385			0.761923	−	0.647668i	$-0.224256\pi$
$419$	4.02827e14			1.52385			0.761923	+	0.647668i	$0.224256\pi$
$420$		0			0
$421$			3.20869e14i			1.18243i	0.806513	+	0.591216i	$0.201352\pi$
$421$			3.20869e14i			1.18243i	−0.806513	+	0.591216i	$0.798648\pi$
$422$		−	3.17660e14i		−	1.15543i
$423$		0			0
$424$		−	1.66787e14i		−	0.591086i
$425$	1.39774e14			0.488977
$426$		0			0
$427$			7.60905e14i			2.59404i
$428$	−1.19069e14			−0.400736
$429$		0			0
$430$	1.98886e13			0.0652419
$431$			2.22214e14i			0.719690i	0.933012	+	0.359845i	$0.117170\pi$
$431$			2.22214e14i			0.719690i	−0.933012	+	0.359845i	$0.882830\pi$
$432$		0			0
$433$	1.95046e14			0.615818			0.307909	−	0.951416i	$-0.400371\pi$
$433$	1.95046e14			0.615818			0.307909	+	0.951416i	$0.400371\pi$
$434$		−	4.02106e14i		−	1.25357i
$435$		0			0
$436$		−	1.36493e14i		−	0.414892i
$437$		−	5.84808e14i		−	1.75536i
$438$		0			0
$439$	4.58811e14			1.34301			0.671505	−	0.741000i	$-0.265648\pi$
$439$	4.58811e14			1.34301			0.671505	+	0.741000i	$0.265648\pi$
$440$		−	1.14799e14i		−	0.331854i
$441$		0			0
$442$	9.76194e13	+	1.17038e14i	0.275241	+	0.329992i
$443$	1.55808e14			0.433879			0.216939	−	0.976185i	$-0.430393\pi$
$443$	1.55808e14			0.433879			0.216939	+	0.976185i	$0.430393\pi$
$444$		0			0
$445$	−3.06719e12			−0.00833222
$446$	2.59532e14			0.696385
$447$		0			0
$448$		−	6.93533e14i		−	1.81567i
$449$			5.03741e14i			1.30273i	0.758767	+	0.651363i	$0.225802\pi$
$449$			5.03741e14i			1.30273i	−0.758767	+	0.651363i	$0.774198\pi$
$450$		0			0
$451$	−7.43624e13			−0.187665
$452$	1.20860e14			0.301316
$453$		0			0
$454$	3.94786e14			0.960626
$455$	8.98648e13	−	7.49549e13i	0.216036	−	0.180193i
$456$		0			0
$457$		−	8.67205e13i		−	0.203508i	−0.994810	−	0.101754i	$-0.967554\pi$
$457$		−	8.67205e13i		−	0.203508i	0.994810	−	0.101754i	$-0.0324455\pi$
$458$	−4.48831e13			−0.104069
$459$		0			0
$460$		−	3.70750e13i		−	0.0839291i
$461$			1.01192e14i			0.226356i	0.993575	+	0.113178i	$0.0361031\pi$
$461$			1.01192e14i			0.226356i	−0.993575	+	0.113178i	$0.963897\pi$
$462$		0			0
$463$			4.25093e14i			0.928514i	0.885700	+	0.464257i	$0.153679\pi$
$463$			4.25093e14i			0.928514i	−0.885700	+	0.464257i	$0.846321\pi$
$464$	−1.23469e14			−0.266507
$465$		0			0
$466$			5.27444e14i			1.11187i
$467$	−9.02962e14			−1.88116			−0.940582	−	0.339565i	$-0.889720\pi$
$467$	−9.02962e14			−1.88116			−0.940582	+	0.339565i	$0.889720\pi$
$468$		0			0
$469$	1.01582e15			2.06712
$470$		−	2.59796e13i		−	0.0522510i
$471$		0			0
$472$	−3.90263e14			−0.766791
$473$		−	4.11637e14i		−	0.799426i
$474$		0			0
$475$			4.98174e14i			0.945292i
$476$			1.20251e14i			0.225555i
$477$		0			0
$478$	−3.00666e13			−0.0551101
$479$			5.85187e14i			1.06035i	0.847888	+	0.530175i	$0.177874\pi$
$479$			5.85187e14i			1.06035i	−0.847888	+	0.530175i	$0.822126\pi$
$480$		0			0
$481$	4.05175e14	+	4.85772e14i	0.717538	+	0.860270i
$482$	8.72902e14			1.52829
$483$		0			0
$484$	−3.49346e14			−0.597870
$485$	−3.82765e13			−0.0647668
$486$		0			0
$487$			3.79384e14i			0.627582i	0.949492	+	0.313791i	$0.101599\pi$
$487$			3.79384e14i			0.627582i	−0.949492	+	0.313791i	$0.898401\pi$
$488$		−	1.04644e15i		−	1.71162i
$489$		0			0
$490$	−1.55606e14			−0.248855
$491$	−4.15557e14			−0.657177			−0.328589	−	0.944473i	$-0.606573\pi$
$491$	−4.15557e14			−0.657177			−0.328589	+	0.944473i	$0.606573\pi$
$492$		0			0
$493$	1.32800e14			0.205371
$494$	−4.17138e14	+	3.47928e14i	−0.637942	+	0.532098i
$495$		0			0
$496$			3.90407e14i			0.583940i
$497$	8.27447e14			1.22400
$498$		0			0
$499$			6.43898e14i			0.931674i	0.884871	+	0.465837i	$0.154247\pi$
$499$			6.43898e14i			0.931674i	−0.884871	+	0.465837i	$0.845753\pi$
$500$			6.41168e13i			0.0917566i
$501$		0			0
$502$			7.24500e14i			1.01430i
$503$	2.25226e14			0.311885			0.155943	−	0.987766i	$-0.450159\pi$
$503$	2.25226e14			0.311885			0.155943	+	0.987766i	$0.450159\pi$
$504$		0			0
$505$		−	1.31752e14i		−	0.178506i
$506$	2.05139e15			2.74928
$507$		0			0
$508$	2.05926e14			0.270060
$509$		−	1.26379e15i		−	1.63955i	−0.572684	−	0.819777i	$-0.694098\pi$
$509$		−	1.26379e15i		−	1.63955i	0.572684	−	0.819777i	$-0.305902\pi$
$510$		0			0
$511$	7.43332e14			0.943774
$512$			8.39420e14i			1.05437i
$513$		0			0
$514$		−	1.18979e15i		−	1.46276i
$515$		−	2.32628e13i		−	0.0282959i
$516$		0			0
$517$	−5.37705e14			−0.640245
$518$		−	1.33430e15i		−	1.57195i
$519$		0			0
$520$	−1.23588e14	+	1.03083e14i	−0.142546	+	0.118896i
$521$	−1.46094e15			−1.66734			−0.833671	−	0.552262i	$-0.813765\pi$
$521$	−1.46094e15			−1.66734			−0.833671	+	0.552262i	$0.813765\pi$
$522$		0			0
$523$	1.24984e15			1.39668			0.698338	−	0.715768i	$-0.253923\pi$
$523$	1.24984e15			1.39668			0.698338	+	0.715768i	$0.253923\pi$
$524$	−3.67403e14			−0.406275
$525$		0			0
$526$		−	1.93110e14i		−	0.209114i
$527$		−	4.19910e14i		−	0.449985i
$528$		0			0
$529$	2.14333e15			2.24948
$530$	7.65015e13			0.0794608
$531$		0			0
$532$	−4.28592e14			−0.436043
$533$	6.67732e13	+	8.00557e13i	0.0672361	+	0.0806106i
$534$		0			0
$535$		−	2.55232e14i		−	0.251761i
$536$	−1.39702e15			−1.36394
$537$		0			0
$538$		−	2.72387e14i		−	0.260545i
$539$			3.22060e15i			3.04928i
$540$		0			0
$541$		−	1.09182e15i		−	1.01290i	−0.862270	−	0.506448i	$-0.830958\pi$
$541$		−	1.09182e15i		−	1.01290i	0.862270	−	0.506448i	$-0.169042\pi$
$542$	−1.73912e15			−1.59710
$543$		0			0
$544$		−	2.95367e14i		−	0.265808i
$545$	2.92581e14			0.260654
$546$		0			0
$547$	−1.18659e15			−1.03603			−0.518015	−	0.855372i	$-0.673329\pi$
$547$	−1.18659e15			−1.03603			−0.518015	+	0.855372i	$0.673329\pi$
$548$			4.71163e14i			0.407266i
$549$		0			0
$550$	−1.74749e15			−1.48054
$551$			4.73315e14i			0.397024i
$552$		0			0
$553$		−	2.03135e14i		−	0.167030i
$554$			1.35302e15i			1.10154i
$555$		0			0
$556$	−5.22965e14			−0.417408
$557$			1.51714e15i			1.19901i	0.800373	+	0.599503i	$0.204635\pi$
$557$			1.51714e15i			1.19901i	−0.800373	+	0.599503i	$0.795365\pi$
$558$		0			0
$559$	−4.43152e14	+	3.69627e14i	−0.343390	+	0.286417i
$560$	2.39655e14			0.183888
$561$		0			0
$562$	1.40947e14			0.106049
$563$	1.29636e15			0.965896			0.482948	−	0.875649i	$-0.339566\pi$
$563$	1.29636e15			0.965896			0.482948	+	0.875649i	$0.339566\pi$
$564$		0			0
$565$			2.59071e14i			0.189301i
$566$		−	1.59479e15i		−	1.15402i
$567$		0			0
$568$	−1.13796e15			−0.807625
$569$	−1.31170e15			−0.921968			−0.460984	−	0.887408i	$-0.652503\pi$
$569$	−1.31170e15			−0.921968			−0.460984	+	0.887408i	$0.652503\pi$
$570$		0			0
$571$	7.54286e14			0.520041			0.260020	−	0.965603i	$-0.416271\pi$
$571$	7.54286e14			0.520041			0.260020	+	0.965603i	$0.416271\pi$
$572$	4.56529e14	+	5.47341e14i	0.311738	+	0.373749i
$573$		0			0
$574$		−	2.19893e14i		−	0.147298i
$575$	−2.63747e15			−1.74990
$576$		0			0
$577$			1.43380e15i			0.933298i	0.884443	+	0.466649i	$0.154539\pi$
$577$			1.43380e15i			0.933298i	−0.884443	+	0.466649i	$0.845461\pi$
$578$			9.87414e14i			0.636643i
$579$		0			0
$580$			3.00066e13i			0.0189829i
$581$	2.84320e15			1.78172
$582$		0			0
$583$		−	1.58337e15i		−	0.973653i
$584$	−1.02228e15			−0.622727
$585$		0			0
$586$	−2.26026e15			−1.35121
$587$		−	3.26552e15i		−	1.93394i	−0.254891	−	0.966970i	$-0.582040\pi$
$587$		−	3.26552e15i		−	1.93394i	0.254891	−	0.966970i	$-0.417960\pi$
$588$		0			0
$589$	1.49661e15			0.869913
$590$		−	1.79005e14i		−	0.103081i
$591$		0			0
$592$			1.29548e15i			0.732252i
$593$		−	1.00603e15i		−	0.563392i	−0.959504	−	0.281696i	$-0.909103\pi$
$593$		−	1.00603e15i		−	0.563392i	0.959504	−	0.281696i	$-0.0908970\pi$
$594$		0			0
$595$	−2.57766e14			−0.141704
$596$		−	6.37414e14i		−	0.347190i
$597$		0			0
$598$	−1.84203e15	−	2.20844e15i	−0.985008	−	1.18094i
$599$	−1.00989e15			−0.535090			−0.267545	−	0.963545i	$-0.586212\pi$
$599$	−1.00989e15			−0.535090			−0.267545	+	0.963545i	$0.586212\pi$
$600$		0			0
$601$	−2.17878e15			−1.13345			−0.566726	−	0.823907i	$-0.691790\pi$
$601$	−2.17878e15			−1.13345			−0.566726	+	0.823907i	$0.691790\pi$
$602$	1.21723e15			0.627469
$603$		0			0
$604$		−	6.35529e13i		−	0.0321686i
$605$		−	7.48844e14i		−	0.375610i
$606$		0			0
$607$	−3.53335e15			−1.74040			−0.870201	−	0.492698i	$-0.836011\pi$
$607$	−3.53335e15			−1.74040			−0.870201	+	0.492698i	$0.836011\pi$
$608$	1.05273e15			0.513861
$609$		0			0
$610$	4.79980e14			0.230096
$611$	4.82829e14	+	5.78872e14i	0.229386	+	0.275015i
$612$		0			0
$613$			2.55011e15i			1.18994i	0.803746	+	0.594972i	$0.202837\pi$
$613$			2.55011e15i			1.18994i	−0.803746	+	0.594972i	$0.797163\pi$
$614$	1.35214e15			0.625311
$615$		0			0
$616$		−	7.02596e15i		−	3.19163i
$617$			3.01365e15i			1.35683i	0.734681	+	0.678413i	$0.237332\pi$
$617$			3.01365e15i			1.35683i	−0.734681	+	0.678413i	$0.762668\pi$
$618$		0			0
$619$		−	9.49187e14i		−	0.419811i	−0.977722	−	0.209905i	$-0.932684\pi$
$619$		−	9.49187e14i		−	0.419811i	0.977722	−	0.209905i	$-0.0673156\pi$
$620$	9.48805e13			0.0415932
$621$		0			0
$622$		−	2.31206e15i		−	0.995753i
$623$	−1.87720e14			−0.0801357
$624$		0			0
$625$	2.17701e15			0.913104
$626$		−	4.67612e13i		−	0.0194414i
$627$		0			0
$628$	−2.61355e13			−0.0106771
$629$		−	1.39338e15i		−	0.564274i
$630$		0			0
$631$			1.91627e15i			0.762599i	0.924452	+	0.381299i	$0.124523\pi$
$631$			1.91627e15i			0.762599i	−0.924452	+	0.381299i	$0.875477\pi$
$632$			2.79363e14i			0.110211i
$633$		0			0
$634$	1.17848e14			0.0456911
$635$			4.41414e14i			0.169665i
$636$		0			0
$637$	3.46717e15	−	2.89191e15i	1.30981	−	1.09249i
$638$	−1.66029e15			−0.621828
$639$		0			0
$640$	−1.92324e14			−0.0708016
$641$	−4.86008e15			−1.77388			−0.886940	−	0.461886i	$-0.847173\pi$
$641$	−4.86008e15			−1.77388			−0.886940	+	0.461886i	$0.847173\pi$
$642$		0			0
$643$			4.30752e15i			1.54549i	0.634716	+	0.772746i	$0.281117\pi$
$643$			4.30752e15i			1.54549i	−0.634716	+	0.772746i	$0.718883\pi$
$644$		−	2.26908e15i		−	0.807194i
$645$		0			0
$646$	1.19651e15			0.418443
$647$	3.57668e15			1.24024			0.620121	−	0.784506i	$-0.287084\pi$
$647$	3.57668e15			1.24024			0.620121	+	0.784506i	$0.287084\pi$
$648$		0			0
$649$	−3.70490e15			−1.26308
$650$	1.56915e15	+	1.88128e15i	0.530444	+	0.635960i
$651$		0			0
$652$		−	4.79435e14i		−	0.159356i
$653$	2.82974e15			0.932662			0.466331	−	0.884610i	$-0.345576\pi$
$653$	2.82974e15			0.932662			0.466331	+	0.884610i	$0.345576\pi$
$654$		0			0
$655$		−	7.87549e14i		−	0.255241i
$656$			2.13496e14i			0.0686148i
$657$		0			0
$658$		−	1.59002e15i		−	0.502528i
$659$	1.92910e15			0.604622			0.302311	−	0.953209i	$-0.402242\pi$
$659$	1.92910e15			0.604622			0.302311	+	0.953209i	$0.402242\pi$
$660$		0			0
$661$		−	2.68418e15i		−	0.827379i	−0.910418	−	0.413689i	$-0.864240\pi$
$661$		−	2.68418e15i		−	0.827379i	0.910418	−	0.413689i	$-0.135760\pi$
$662$	−1.67455e15			−0.511893
$663$		0			0
$664$	−3.91015e15			−1.17562
$665$		−	9.18711e14i		−	0.273943i
$666$		0			0
$667$	−2.50586e15			−0.734962
$668$		−	6.25789e14i		−	0.182036i
$669$		0			0
$670$		−	6.40779e14i		−	0.183357i
$671$		−	9.93422e15i		−	2.81942i
$672$		0			0
$673$	7.27068e14			0.202998			0.101499	−	0.994836i	$-0.467636\pi$
$673$	7.27068e14			0.202998			0.101499	+	0.994836i	$0.467636\pi$
$674$		−	7.30987e14i		−	0.202433i
$675$		0			0
$676$	1.79309e14	−	9.82961e14i	0.0488534	−	0.267812i
$677$	5.46656e15			1.47733			0.738664	−	0.674074i	$-0.235457\pi$
$677$	5.46656e15			1.47733			0.738664	+	0.674074i	$0.235457\pi$
$678$		0			0
$679$	−2.34262e15			−0.622899
$680$	3.54495e14			0.0935000
$681$		0			0
$682$			5.24981e15i			1.36248i
$683$			1.25630e15i			0.323431i	0.986837	+	0.161715i	$0.0517026\pi$
$683$			1.25630e15i			0.323431i	−0.986837	+	0.161715i	$0.948297\pi$
$684$		0			0
$685$	−1.00997e15			−0.255864
$686$	−3.93988e15			−0.990149
$687$		0			0
$688$	−1.18182e15			−0.292290
$689$	−1.70459e15	+	1.42177e15i	−0.418229	+	0.348838i
$690$		0			0
$691$			2.40753e15i			0.581357i	0.956821	+	0.290678i	$0.0938810\pi$
$691$			2.40753e15i			0.581357i	−0.956821	+	0.290678i	$0.906119\pi$
$692$	1.46538e15			0.351048
$693$		0			0
$694$			1.21516e15i			0.286522i
$695$		−	1.12100e15i		−	0.262235i
$696$		0			0
$697$		−	2.29630e14i		−	0.0528747i
$698$	−1.62825e15			−0.371978
$699$		0			0
$700$			1.93294e15i			0.434689i
$701$	7.44821e14			0.166189			0.0830946	−	0.996542i	$-0.473520\pi$
$701$	7.44821e14			0.166189			0.0830946	+	0.996542i	$0.473520\pi$
$702$		0			0
$703$	4.96617e15			1.09086
$704$			9.05462e15i			1.97343i
$705$		0			0
$706$	4.48343e15			0.962020
$707$		−	8.06353e15i		−	1.71680i
$708$		0			0
$709$			1.01890e15i			0.213588i	0.994281	+	0.106794i	$0.0340585\pi$
$709$			1.01890e15i			0.213588i	−0.994281	+	0.106794i	$0.965941\pi$
$710$		−	5.21954e14i		−	0.108570i
$711$		0			0
$712$	2.58164e14			0.0528756
$713$			7.92349e15i			1.61036i
$714$		0			0
$715$	−1.17326e15	+	9.78595e14i	−0.234806	+	0.195848i
$716$	1.30772e15			0.259712
$717$		0			0
$718$	−3.73757e15			−0.730978
$719$	6.45032e15			1.25191			0.625954	−	0.779860i	$-0.284710\pi$
$719$	6.45032e15			1.25191			0.625954	+	0.779860i	$0.284710\pi$
$720$		0			0
$721$		−	1.42375e15i		−	0.272137i
$722$		−	2.32789e14i		−	0.0441579i
$723$		0			0
$724$	−6.59702e14			−0.123250
$725$	2.13464e15			0.395790
$726$		0			0
$727$	1.04558e16			1.90948			0.954742	−	0.297434i	$-0.0961309\pi$
$727$	1.04558e16			1.90948			0.954742	+	0.297434i	$0.0961309\pi$
$728$	−7.56388e15	+	6.30892e15i	−1.37095	+	1.14349i
$729$		0			0
$730$		−	4.68894e14i		−	0.0837142i
$731$	1.27113e15			0.225239
$732$		0			0
$733$			3.93592e15i			0.687028i	0.939148	+	0.343514i	$0.111617\pi$
$733$			3.93592e15i			0.687028i	−0.939148	+	0.343514i	$0.888383\pi$
$734$		−	3.23149e15i		−	0.559853i
$735$		0			0
$736$			5.57342e15i			0.951249i
$737$	−1.32623e16			−2.24672
$738$		0			0
$739$		−	8.65402e15i		−	1.44435i	−0.691708	−	0.722177i	$-0.743142\pi$
$739$		−	8.65402e15i		−	1.44435i	0.691708	−	0.722177i	$-0.256858\pi$
$740$	3.14839e14			0.0521573
$741$		0			0
$742$	4.68209e15			0.764220
$743$			7.58900e15i			1.22955i	0.788702	+	0.614775i	$0.210753\pi$
$743$			7.58900e15i			1.22955i	−0.788702	+	0.614775i	$0.789247\pi$
$744$		0			0
$745$	1.36633e15			0.218121
$746$			7.59422e14i			0.120343i
$747$		0			0
$748$		−	1.56998e15i		−	0.245152i
$749$		−	1.56208e16i		−	2.42133i
$750$		0			0
$751$	1.56895e15			0.239656			0.119828	−	0.992795i	$-0.461766\pi$
$751$	1.56895e15			0.239656			0.119828	+	0.992795i	$0.461766\pi$
$752$			1.54376e15i			0.234089i
$753$		0			0
$754$	1.49085e15	+	1.78740e15i	0.222787	+	0.267104i
$755$	1.36229e14			0.0202098
$756$		0			0
$757$	5.35536e15			0.783000			0.391500	−	0.920178i	$-0.371956\pi$
$757$	5.35536e15			0.783000			0.391500	+	0.920178i	$0.371956\pi$
$758$	−7.10837e15			−1.03179
$759$		0			0
$760$			1.26347e15i			0.180755i
$761$			8.40352e14i			0.119356i	0.998218	+	0.0596782i	$0.0190074\pi$
$761$			8.40352e14i			0.119356i	−0.998218	+	0.0596782i	$0.980993\pi$
$762$		0			0
$763$	1.79067e16			2.50686
$764$	−2.68763e15			−0.373556
$765$		0			0
$766$	−3.98110e15			−0.545437
$767$	3.32679e15	+	3.98854e15i	0.452533	+	0.542550i
$768$		0			0
$769$			9.57018e15i			1.28329i	0.767001	+	0.641646i	$0.221748\pi$
$769$			9.57018e15i			1.28329i	−0.767001	+	0.641646i	$0.778252\pi$
$770$	3.22265e15			0.429056
$771$		0			0
$772$		−	1.23521e15i		−	0.162123i
$773$			7.42721e15i			0.967918i	0.875091	+	0.483959i	$0.160802\pi$
$773$			7.42721e15i			0.967918i	−0.875091	+	0.483959i	$0.839198\pi$
$774$		0			0
$775$		−	6.74968e15i		−	0.867210i
$776$	3.22171e15			0.411005
$777$		0			0
$778$			9.28313e15i			1.16763i
$779$	8.18430e14			0.102218
$780$		0			0
$781$	−1.08030e16			−1.33034
$782$			6.33464e15i			0.774613i
$783$		0			0
$784$	9.24638e15			1.11489
$785$		−	5.60229e13i		−	0.00670784i
$786$		0			0
$787$			7.03450e15i			0.830563i	0.909693	+	0.415281i	$0.136317\pi$
$787$			7.03450e15i			0.830563i	−0.909693	+	0.415281i	$0.863683\pi$
$788$		−	3.06375e15i		−	0.359219i
$789$		0			0
$790$	−1.28137e14			−0.0148159
$791$			1.58558e16i			1.82061i
$792$		0			0
$793$	−1.06948e16	+	8.92037e15i	−1.21107	+	1.01014i
$794$	2.78904e15			0.313647
$795$		0			0
$796$	1.43247e15			0.158878
$797$	3.04148e15			0.335015			0.167508	−	0.985871i	$-0.446428\pi$
$797$	3.04148e15			0.335015			0.167508	+	0.985871i	$0.446428\pi$
$798$		0			0
$799$		−	1.66042e15i		−	0.180390i
$800$		−	4.74776e15i		−	0.512264i
$801$		0			0
$802$	9.02689e15			0.960681
$803$	−9.70479e15			−1.02577
$804$		0			0
$805$	4.86391e15			0.507117
$806$	5.65174e15	−	4.71403e15i	0.585247	−	0.488146i
$807$		0			0
$808$			1.10895e16i			1.13279i
$809$	−7.49530e15			−0.760452			−0.380226	−	0.924894i	$-0.624154\pi$
$809$	−7.49530e15			−0.760452			−0.380226	+	0.924894i	$0.624154\pi$
$810$		0			0
$811$			6.73584e15i			0.674181i	0.941472	+	0.337091i	$0.109443\pi$
$811$			6.73584e15i			0.674181i	−0.941472	+	0.337091i	$0.890557\pi$
$812$			1.83648e15i			0.182570i
$813$		0			0
$814$			1.74203e16i			1.70853i
$815$	1.02770e15			0.100115
$816$		0			0
$817$			4.53046e15i			0.435433i
$818$	−9.28055e15			−0.885994
$819$		0			0
$820$	5.18858e13			0.00488734
$821$		−	2.93109e15i		−	0.274246i	−0.990554	−	0.137123i	$-0.956214\pi$
$821$		−	2.93109e15i		−	0.274246i	0.990554	−	0.137123i	$-0.0437856\pi$
$822$		0			0
$823$	−1.35274e16			−1.24887			−0.624433	−	0.781078i	$-0.714670\pi$
$823$	−1.35274e16			−1.24887			−0.624433	+	0.781078i	$0.714670\pi$
$824$			1.95802e15i			0.179563i
$825$		0			0
$826$		−	1.09556e16i		−	0.991389i
$827$		−	1.95134e16i		−	1.75409i	−0.480404	−	0.877047i	$-0.659510\pi$
$827$		−	1.95134e16i		−	1.75409i	0.480404	−	0.877047i	$-0.340490\pi$
$828$		0			0
$829$	6.36321e15			0.564451			0.282226	−	0.959348i	$-0.408927\pi$
$829$	6.36321e15			0.564451			0.282226	+	0.959348i	$0.408927\pi$
$830$		−	1.79349e15i		−	0.158041i
$831$		0			0
$832$	9.74785e15	−	8.13054e15i	0.847677	−	0.707035i
$833$	−9.94513e15			−0.859138
$834$		0			0
$835$	1.34142e15			0.114364
$836$	5.59560e15			0.473928
$837$		0			0
$838$		−	1.55518e16i		−	1.29998i
$839$		−	1.96609e16i		−	1.63272i	−0.577543	−	0.816360i	$-0.695989\pi$
$839$		−	1.96609e16i		−	1.63272i	0.577543	−	0.816360i	$-0.304011\pi$
$840$		0			0
$841$	−1.01724e16			−0.833768
$842$	1.23877e16			1.00873
$843$		0			0
$844$	4.58743e15			0.368711
$845$	2.10703e15	+	3.84358e14i	0.168252	+	0.0306920i
$846$		0			0
$847$		−	4.58312e16i		−	3.61245i
$848$	−4.54587e15			−0.355991
$849$		0			0
$850$		−	5.39621e15i		−	0.417143i
$851$			2.62923e16i			2.01937i
$852$		0			0
$853$		−	6.46765e15i		−	0.490373i	−0.969476	−	0.245187i	$-0.921151\pi$
$853$		−	6.46765e15i		−	0.490373i	0.969476	−	0.245187i	$-0.0788493\pi$
$854$	2.93760e16			2.21296
$855$		0			0
$856$			2.14827e16i			1.59766i
$857$	2.12502e16			1.57025			0.785125	−	0.619337i	$-0.212599\pi$
$857$	2.12502e16			1.57025			0.785125	+	0.619337i	$0.212599\pi$
$858$		0			0
$859$	−9.52147e15			−0.694611			−0.347305	−	0.937752i	$-0.612903\pi$
$859$	−9.52147e15			−0.694611			−0.347305	+	0.937752i	$0.612903\pi$
$860$			2.87216e14i			0.0208194i
$861$		0			0
$862$	8.57891e15			0.613963
$863$			1.71361e15i			0.121858i	0.998142	+	0.0609288i	$0.0194063\pi$
$863$			1.71361e15i			0.121858i	−0.998142	+	0.0609288i	$0.980594\pi$
$864$		0			0
$865$			3.14112e15i			0.220544i
$866$		−	7.53005e15i		−	0.525351i
$867$		0			0
$868$	5.80693e15			0.400026
$869$			2.65208e15i			0.181543i
$870$		0			0
$871$	1.19088e16	+	1.42777e16i	0.804949	+	0.965068i
$872$	−2.46264e16			−1.65409
$873$		0			0
$874$	−2.25775e16			−1.49748
$875$	−8.41157e15			−0.554412
$876$		0			0
$877$		−	2.10433e16i		−	1.36967i	−0.728698	−	0.684835i	$-0.759874\pi$
$877$		−	2.10433e16i		−	1.36967i	0.728698	−	0.684835i	$-0.240126\pi$
$878$		−	1.77132e16i		−	1.14571i
$879$		0			0
$880$	−3.12889e15			−0.199864
$881$	2.15127e16			1.36561			0.682805	−	0.730600i	$-0.260760\pi$
$881$	2.15127e16			1.36561			0.682805	+	0.730600i	$0.260760\pi$
$882$		0			0
$883$	−1.10643e16			−0.693646			−0.346823	−	0.937931i	$-0.612740\pi$
$883$	−1.10643e16			−0.693646			−0.346823	+	0.937931i	$0.612740\pi$
$884$	−1.69018e15	+	1.40975e15i	−0.105304	+	0.0878324i
$885$		0			0
$886$		−	6.01521e15i		−	0.370139i
$887$	−1.92655e16			−1.17815			−0.589074	−	0.808079i	$-0.700507\pi$
$887$	−1.92655e16			−1.17815			−0.589074	+	0.808079i	$0.700507\pi$
$888$		0			0
$889$			2.70157e16i			1.63176i
$890$			1.18414e14i			0.00710816i
$891$		0			0
$892$			3.74798e15i			0.222224i
$893$	5.91796e15			0.348730
$894$		0			0
$895$			2.80316e15i			0.163163i
$896$	−1.17707e16			−0.680940
$897$		0			0
$898$	1.94478e16			1.11135
$899$		−	6.41288e15i		−	0.364229i
$900$		0			0
$901$	4.88940e15			0.274327
$902$			2.87088e15i			0.160096i
$903$		0			0
$904$		−	2.18059e16i		−	1.20129i
$905$		−	1.41411e15i		−	0.0774311i
$906$		0			0
$907$	−2.07996e16			−1.12516			−0.562579	−	0.826743i	$-0.690191\pi$
$907$	−2.07996e16			−1.12516			−0.562579	+	0.826743i	$0.690191\pi$
$908$			5.70122e15i			0.306546i
$909$		0			0
$910$	−2.89375e15	−	3.46938e15i	−0.153721	−	0.184299i
$911$	2.26315e16			1.19499			0.597493	−	0.801874i	$-0.296164\pi$
$911$	2.26315e16			1.19499			0.597493	+	0.801874i	$0.296164\pi$
$912$		0			0
$913$	−3.71203e16			−1.93652
$914$	−3.34798e15			−0.173612
$915$		0			0
$916$		−	6.48170e14i		−	0.0332096i
$917$		−	4.82000e16i		−	2.45480i
$918$		0			0
$919$	−3.68428e16			−1.85403			−0.927016	−	0.375021i	$-0.877636\pi$
$919$	−3.68428e16			−1.85403			−0.927016	+	0.375021i	$0.877636\pi$
$920$	−6.68915e15			−0.334609
$921$		0			0
$922$	3.90670e15			0.193103
$923$	9.70046e15	+	1.16301e16i	0.476632	+	0.571443i
$924$		0			0
$925$		−	2.23973e16i		−	1.08747i
$926$	1.64114e16			0.792110
$927$		0			0
$928$		−	4.51085e15i		−	0.215152i
$929$			3.26468e16i			1.54794i	0.633222	+	0.773971i	$0.281732\pi$
$929$			3.26468e16i			1.54794i	−0.633222	+	0.773971i	$0.718268\pi$
$930$		0			0
$931$		−	3.54457e16i		−	1.66089i
$932$	−7.61697e15			−0.354809
$933$		0			0
$934$			3.48603e16i			1.60481i
$935$	3.36534e15			0.154016
$936$		0			0
$937$	−3.37155e16			−1.52497			−0.762485	−	0.647005i	$-0.776021\pi$
$937$	−3.37155e16			−1.52497			−0.762485	+	0.647005i	$0.776021\pi$
$938$		−	3.92173e16i		−	1.76345i
$939$		0			0
$940$	3.75180e14			0.0166739
$941$		−	1.00494e16i		−	0.444017i	−0.975045	−	0.222008i	$-0.928739\pi$
$941$		−	1.00494e16i		−	0.444017i	0.975045	−	0.222008i	$-0.0712612\pi$
$942$		0			0
$943$			4.33299e15i			0.189223i
$944$			1.06368e16i			0.461812i
$945$		0			0
$946$	−1.58919e16			−0.681985
$947$			1.95408e16i			0.833714i	0.908972	+	0.416857i	$0.136868\pi$
$947$			1.95408e16i			0.833714i	−0.908972	+	0.416857i	$0.863132\pi$
$948$		0			0
$949$	8.71435e15	+	1.04478e16i	0.367511	+	0.440616i
$950$	1.92328e16			0.806422
$951$		0			0
$952$	2.16960e16			0.899243
$953$	1.63671e16			0.674468			0.337234	−	0.941421i	$-0.390509\pi$
$953$	1.63671e16			0.674468			0.337234	+	0.941421i	$0.390509\pi$
$954$		0			0
$955$		−	5.76108e15i		−	0.234685i
$956$		−	4.34201e14i		−	0.0175862i
$957$		0			0
$958$	2.25921e16			0.904577
$959$	−6.18125e16			−2.46079
$960$		0			0
$961$	5.13106e15			0.201943
$962$	1.87540e16	−	1.56425e16i	0.733891	−	0.612127i
$963$		0			0
$964$			1.26058e16i			0.487695i
$965$	2.64774e15			0.101853
$966$		0			0
$967$		−	1.41778e16i		−	0.539218i	−0.962970	−	0.269609i	$-0.913106\pi$
$967$		−	1.41778e16i		−	0.539218i	0.962970	−	0.269609i	$-0.0868945\pi$
$968$			6.30298e16i			2.38359i
$969$		0			0
$970$			1.47773e15i			0.0552522i
$971$	−4.69091e16			−1.74402			−0.872010	−	0.489488i	$-0.837184\pi$
$971$	−4.69091e16			−1.74402			−0.872010	+	0.489488i	$0.837184\pi$
$972$		0			0
$973$		−	6.86084e16i		−	2.52207i
$974$	1.46467e16			0.535386
$975$		0			0
$976$	−2.85213e16			−1.03085
$977$			2.72451e16i			0.979194i	0.871949	+	0.489597i	$0.162856\pi$
$977$			2.72451e16i			0.979194i	−0.871949	+	0.489597i	$0.837144\pi$
$978$		0			0
$979$	2.45083e15			0.0870981
$980$		−	2.24715e15i		−	0.0794123i
$981$		0			0
$982$			1.60433e16i			0.560634i
$983$			9.74058e15i			0.338486i	0.985574	+	0.169243i	$0.0541322\pi$
$983$			9.74058e15i			0.338486i	−0.985574	+	0.169243i	$0.945868\pi$
$984$		0			0
$985$	6.56733e15			0.225678
$986$		−	5.12694e15i		−	0.175201i
$987$		0			0
$988$	−5.02453e15	−	6.02401e15i	−0.169798	−	0.203574i
$989$	−2.39855e16			−0.806063
$990$		0			0
$991$	−2.38400e16			−0.792320			−0.396160	−	0.918181i	$-0.629657\pi$
$991$	−2.38400e16			−0.792320			−0.396160	+	0.918181i	$0.629657\pi$
$992$	−1.42632e16			−0.471416
$993$		0			0
$994$		−	3.19449e16i		−	1.04418i
$995$			3.07058e15i			0.0998146i
$996$		0			0
$997$	2.99777e16			0.963774			0.481887	−	0.876233i	$-0.339952\pi$
$997$	2.99777e16			0.963774			0.481887	+	0.876233i	$0.339952\pi$
$998$	2.48587e16			0.794805
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.12.b.e.64.6	yes	20
3.2	odd	2	inner	117.12.b.e.64.16	yes	20
13.12	even	2	inner	117.12.b.e.64.15	yes	20
39.38	odd	2	inner	117.12.b.e.64.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.12.b.e.64.5	✓	20	39.38	odd	2	inner
117.12.b.e.64.6	yes	20	1.1	even	1	trivial
117.12.b.e.64.15	yes	20	13.12	even	2	inner
117.12.b.e.64.16	yes	20	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-38.6066i q^{2} +557.529 q^{4} +1195.10i q^{5} +73143.0i q^{7} -100591. i q^{8} +46138.6 q^{10} -954939. i q^{11} +(-1.02805e6 + 857481. i) q^{13} +2.82380e6 q^{14} -2.74164e6 q^{16} +2.94883e6 q^{17} +1.05100e7i q^{19} +666301. i q^{20} -3.68670e7 q^{22} -5.56429e7 q^{23} +4.73999e7 q^{25} +(3.31044e7 + 3.96895e7i) q^{26} +4.07793e7i q^{28} +4.50346e7 q^{29} -1.42399e8i q^{31} -1.00164e8i q^{32} -1.13844e8i q^{34} -8.74129e7 q^{35} -4.72518e8i q^{37} +4.05756e8 q^{38} +1.20216e8 q^{40} -7.78714e7i q^{41} +4.31061e8 q^{43} -5.32407e8i q^{44} +2.14819e9i q^{46} -5.63078e8i q^{47} -3.37257e9 q^{49} -1.82995e9i q^{50} +(-5.73168e8 + 4.78071e8i) q^{52} +1.65808e9 q^{53} +1.14124e9 q^{55} +7.35750e9 q^{56} -1.73864e9i q^{58} -3.87972e9i q^{59} +1.04030e10 q^{61} -5.49754e9 q^{62} -9.48188e9 q^{64} +(-1.02477e9 - 1.22862e9i) q^{65} -1.38881e10i q^{67} +1.64406e9 q^{68} +3.37472e9i q^{70} -1.13127e10i q^{71} -1.01627e10i q^{73} -1.82423e10 q^{74} +5.85964e9i q^{76} +6.98471e10 q^{77} -2.77723e9 q^{79} -3.27653e9i q^{80} -3.00635e9 q^{82} -3.88719e10i q^{83} +3.52414e9i q^{85} -1.66418e10i q^{86} -9.60580e10 q^{88} +2.56648e9i q^{89} +(-6.27187e10 - 7.51946e10i) q^{91} -3.10226e10 q^{92} -2.17385e10 q^{94} -1.25605e10 q^{95} +3.20279e10i q^{97} +1.30203e11i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12288 q^{4} - 2212240 q^{10} - 1708564 q^{13} + 32533888 q^{16} - 89284144 q^{22} + 209701380 q^{25} + 309455680 q^{40} + 2936996032 q^{43} - 10965350212 q^{49} + 4282707936 q^{52} + 6622126880 q^{55}+ \cdots + 330546354704 q^{94}+O(q^{100}) \) 20 * q - 12288 * q^4 - 2212240 * q^10 - 1708564 * q^13 + 32533888 * q^16 - 89284144 * q^22 + 209701380 * q^25 + 309455680 * q^40 + 2936996032 * q^43 - 10965350212 * q^49 + 4282707936 * q^52 + 6622126880 * q^55 + 3358771336 * q^61 - 43877642752 * q^64 - 148998409616 * q^79 + 319857511952 * q^82 - 250888459328 * q^88 + 62924651088 * q^91 + 330546354704 * q^94

Introduction

L-functions

Modular forms

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Database

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