LMFDB - Embedded newform 234.8.b.c.181.3

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			181.3
Root			$13.5100i$ of defining polynomial
Character	$\chi$	$=$	234.181
Dual form			234.8.b.c.181.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000i q^{2} -64.0000 q^{4} +6.61448i q^{5} +1604.67i q^{7} +512.000i q^{8} +52.9158 q^{10} +1173.55i q^{11} +(-3780.46 + 6961.08i) q^{13} +12837.3 q^{14} +4096.00 q^{16} +28053.1 q^{17} +13825.5i q^{19} -423.327i q^{20} +9388.37 q^{22} -94424.9 q^{23} +78081.2 q^{25} +(55688.6 + 30243.7i) q^{26} -102699. i q^{28} -159421. q^{29} +113805. i q^{31} -32768.0i q^{32} -224425. i q^{34} -10614.0 q^{35} -402800. i q^{37} +110604. q^{38} -3386.61 q^{40} -583062. i q^{41} +88507.4 q^{43} -75107.0i q^{44} +755399. i q^{46} +112810. i q^{47} -1.75141e6 q^{49} -624650. i q^{50} +(241949. - 445509. i) q^{52} -548744. q^{53} -7762.40 q^{55} -821589. q^{56} +1.27536e6i q^{58} +1.69495e6i q^{59} +1.33588e6 q^{61} +910442. q^{62} -262144. q^{64} +(-46043.9 - 25005.8i) q^{65} -719239. i q^{67} -1.79540e6 q^{68} +84912.2i q^{70} -1.85217e6i q^{71} +1.50153e6i q^{73} -3.22240e6 q^{74} -884833. i q^{76} -1.88315e6 q^{77} -6.07672e6 q^{79} +27092.9i q^{80} -4.66450e6 q^{82} +661334. i q^{83} +185557. i q^{85} -708059. i q^{86} -600856. q^{88} -1.03091e7i q^{89} +(-1.11702e7 - 6.06637e6i) q^{91} +6.04319e6 q^{92} +902482. q^{94} -91448.6 q^{95} +6.14220e6i q^{97} +1.40113e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			6.61448i			0.0236647i	0.999930	+	0.0118323i	$0.00376644\pi$
$5$			6.61448i			0.0236647i	−0.999930	+	0.0118323i	$0.996234\pi$
$6$		0			0
$7$			1604.67i			1.76824i	0.467260	+	0.884120i	$0.345241\pi$
$7$			1604.67i			1.76824i	−0.467260	+	0.884120i	$0.654759\pi$
$8$			512.000i			0.353553i
$9$		0			0
$10$	52.9158			0.0167335
$11$			1173.55i			0.265843i	0.991127	+	0.132922i	$0.0424359\pi$
$11$			1173.55i			0.265843i	−0.991127	+	0.132922i	$0.957564\pi$
$12$		0			0
$13$	−3780.46	+	6961.08i	−0.477247	+	0.878769i
$13$	−3780.46	+	6961.08i	−0.477247	+	0.878769i
$14$	12837.3			1.25033
$15$		0			0
$16$	4096.00			0.250000
$17$	28053.1			1.38487			0.692437	−	0.721478i	$-0.256537\pi$
$17$	28053.1			1.38487			0.692437	+	0.721478i	$0.256537\pi$
$18$		0			0
$19$			13825.5i			0.462428i	0.972903	+	0.231214i	$0.0742697\pi$
$19$			13825.5i			0.462428i	−0.972903	+	0.231214i	$0.925730\pi$
$20$		−	423.327i		−	0.0118323i
$21$		0			0
$22$	9388.37			0.187980
$23$	−94424.9			−1.61823			−0.809113	−	0.587653i	$-0.800052\pi$
$23$	−94424.9			−1.61823			−0.809113	+	0.587653i	$0.800052\pi$
$24$		0			0
$25$	78081.2			0.999440
$26$	55688.6	+	30243.7i	0.621384	+	0.337464i
$27$		0			0
$28$		−	102699.i		−	0.884120i
$29$	−159421.			−1.21381			−0.606906	−	0.794774i	$-0.707590\pi$
$29$	−159421.			−1.21381			−0.606906	+	0.794774i	$0.707590\pi$
$30$		0			0
$31$			113805.i			0.686114i	0.939315	+	0.343057i	$0.111462\pi$
$31$			113805.i			0.686114i	−0.939315	+	0.343057i	$0.888538\pi$
$32$		−	32768.0i		−	0.176777i
$33$		0			0
$34$		−	224425.i		−	0.979254i
$35$	−10614.0			−0.0418448
$36$		0			0
$37$		−	402800.i		−	1.30732i	−0.756787	−	0.653662i	$-0.773232\pi$
$37$		−	402800.i		−	1.30732i	0.756787	−	0.653662i	$-0.226768\pi$
$38$	110604.			0.326986
$39$		0			0
$40$	−3386.61			−0.00836673
$41$		−	583062.i		−	1.32121i	−0.750734	−	0.660604i	$-0.770300\pi$
$41$		−	583062.i		−	1.32121i	0.750734	−	0.660604i	$-0.229700\pi$
$42$		0			0
$43$	88507.4			0.169762			0.0848809	−	0.996391i	$-0.472949\pi$
$43$	88507.4			0.169762			0.0848809	+	0.996391i	$0.472949\pi$
$44$		−	75107.0i		−	0.132922i
$45$		0			0
$46$			755399.i			1.14426i
$47$			112810.i			0.158491i	0.996855	+	0.0792457i	$0.0252512\pi$
$47$			112810.i			0.158491i	−0.996855	+	0.0792457i	$0.974749\pi$
$48$		0			0
$49$	−1.75141e6			−2.12667
$50$		−	624650.i		−	0.706711i
$51$		0			0
$52$	241949.	−	445509.i	0.238623	−	0.439385i
$53$	−548744.			−0.506296			−0.253148	−	0.967428i	$-0.581466\pi$
$53$	−548744.			−0.506296			−0.253148	+	0.967428i	$0.581466\pi$
$54$		0			0
$55$	−7762.40			−0.00629110
$56$	−821589.			−0.625167
$57$		0			0
$58$			1.27536e6i			0.858294i
$59$			1.69495e6i			1.07442i	0.843448	+	0.537211i	$0.180522\pi$
$59$			1.69495e6i			1.07442i	−0.843448	+	0.537211i	$0.819478\pi$
$60$		0			0
$61$	1.33588e6			0.753554			0.376777	−	0.926304i	$-0.377032\pi$
$61$	1.33588e6			0.753554			0.376777	+	0.926304i	$0.377032\pi$
$62$	910442.			0.485156
$63$		0			0
$64$	−262144.			−0.125000
$65$	−46043.9	−	25005.8i	−0.0207958	−	0.0112939i
$66$		0			0
$67$		−	719239.i		−	0.292154i	−0.989273	−	0.146077i	$-0.953335\pi$
$67$		−	719239.i		−	0.292154i	0.989273	−	0.146077i	$-0.0466647\pi$
$68$	−1.79540e6			−0.692437
$69$		0			0
$70$			84912.2i			0.0295888i
$71$		−	1.85217e6i		−	0.614153i	−0.951685	−	0.307077i	$-0.900649\pi$
$71$		−	1.85217e6i		−	0.614153i	0.951685	−	0.307077i	$-0.0993508\pi$
$72$		0			0
$73$			1.50153e6i			0.451756i	0.974156	+	0.225878i	$0.0725250\pi$
$73$			1.50153e6i			0.451756i	−0.974156	+	0.225878i	$0.927475\pi$
$74$	−3.22240e6			−0.924417
$75$		0			0
$76$		−	884833.i		−	0.231214i
$77$	−1.88315e6			−0.470075
$78$		0			0
$79$	−6.07672e6			−1.38667			−0.693337	−	0.720614i	$-0.743860\pi$
$79$	−6.07672e6			−1.38667			−0.693337	+	0.720614i	$0.743860\pi$
$80$			27092.9i			0.00591617i
$81$		0			0
$82$	−4.66450e6			−0.934235
$83$			661334.i			0.126954i	0.997983	+	0.0634771i	$0.0202190\pi$
$83$			661334.i			0.126954i	−0.997983	+	0.0634771i	$0.979781\pi$
$84$		0			0
$85$			185557.i			0.0327726i
$86$		−	708059.i		−	0.120040i
$87$		0			0
$88$	−600856.			−0.0939898
$89$		−	1.03091e7i		−	1.55008i	−0.631910	−	0.775041i	$-0.717729\pi$
$89$		−	1.03091e7i		−	1.55008i	0.631910	−	0.775041i	$-0.282271\pi$
$90$		0			0
$91$	−1.11702e7	−	6.06637e6i	−1.55388	−	0.843887i
$92$	6.04319e6			0.809113
$93$		0			0
$94$	902482.			0.112070
$95$	−91448.6			−0.0109432
$96$		0			0
$97$			6.14220e6i			0.683318i	0.939824	+	0.341659i	$0.110989\pi$
$97$			6.14220e6i			0.683318i	−0.939824	+	0.341659i	$0.889011\pi$
$98$			1.40113e7i			1.50379i
$99$		0			0
$100$	−4.99720e6			−0.499720
$101$	5.85424e6			0.565387			0.282693	−	0.959210i	$-0.408772\pi$
$101$	5.85424e6			0.565387			0.282693	+	0.959210i	$0.408772\pi$
$102$		0			0
$103$	1.80551e6			0.162806			0.0814031	−	0.996681i	$-0.474060\pi$
$103$	1.80551e6			0.162806			0.0814031	+	0.996681i	$0.474060\pi$
$104$	−3.56407e6	−	1.93559e6i	−0.310692	−	0.168732i
$105$		0			0
$106$			4.38996e6i			0.358005i
$107$	−1.32024e7			−1.04186			−0.520931	−	0.853599i	$-0.674415\pi$
$107$	−1.32024e7			−1.04186			−0.520931	+	0.853599i	$0.674415\pi$
$108$		0			0
$109$			2.36217e7i			1.74710i	0.486734	+	0.873550i	$0.338188\pi$
$109$			2.36217e7i			1.74710i	−0.486734	+	0.873550i	$0.661812\pi$
$110$			62099.2i			0.00444848i
$111$		0			0
$112$			6.57271e6i			0.442060i
$113$	−1.03738e7			−0.676340			−0.338170	−	0.941085i	$-0.609808\pi$
$113$	−1.03738e7			−0.676340			−0.338170	+	0.941085i	$0.609808\pi$
$114$		0			0
$115$		−	624571.i		−	0.0382948i
$116$	1.02029e7			0.606906
$117$		0			0
$118$	1.35596e7			0.759732
$119$			4.50159e7i			2.44879i
$120$		0			0
$121$	1.81100e7			0.929327
$122$		−	1.06871e7i		−	0.532843i
$123$		0			0
$124$		−	7.28354e6i		−	0.343057i
$125$			1.03322e6i			0.0473161i
$126$		0			0
$127$	−3.07208e7			−1.33082			−0.665409	−	0.746479i	$-0.731743\pi$
$127$	−3.07208e7			−1.33082			−0.665409	+	0.746479i	$0.731743\pi$
$128$			2.09715e6i			0.0883883i
$129$		0			0
$130$	−200046.	+	368351.i	−0.00798598	+	0.0147048i
$131$	−1.62980e7			−0.633411			−0.316706	−	0.948524i	$-0.602577\pi$
$131$	−1.62980e7			−0.633411			−0.316706	+	0.948524i	$0.602577\pi$
$132$		0			0
$133$	−2.21853e7			−0.817683
$134$	−5.75391e6			−0.206584
$135$		0			0
$136$			1.43632e7i			0.489627i
$137$			3.22644e7i			1.07202i	0.844213	+	0.536008i	$0.180068\pi$
$137$			3.22644e7i			1.07202i	−0.844213	+	0.536008i	$0.819932\pi$
$138$		0			0
$139$	−3.77383e7			−1.19188			−0.595938	−	0.803030i	$-0.703220\pi$
$139$	−3.77383e7			−1.19188			−0.595938	+	0.803030i	$0.703220\pi$
$140$	679298.			0.0209224
$141$		0			0
$142$	−1.48174e7			−0.434272
$143$	−8.16915e6	−	4.43654e6i	−0.233615	−	0.126873i
$144$		0			0
$145$		−	1.05448e6i		−	0.0287245i
$146$	1.20122e7			0.319439
$147$		0			0
$148$			2.57792e7i			0.653662i
$149$		−	1.69205e7i		−	0.419045i	−0.977804	−	0.209523i	$-0.932809\pi$
$149$		−	1.69205e7i		−	0.419045i	0.977804	−	0.209523i	$-0.0671910\pi$
$150$		0			0
$151$			3.87829e7i			0.916687i	0.888775	+	0.458343i	$0.151557\pi$
$151$			3.87829e7i			0.916687i	−0.888775	+	0.458343i	$0.848443\pi$
$152$	−7.07867e6			−0.163493
$153$		0			0
$154$			1.50652e7i			0.332393i
$155$	−752763.			−0.0162367
$156$		0			0
$157$	−4.86546e7			−1.00340			−0.501701	−	0.865041i	$-0.667292\pi$
$157$	−4.86546e7			−1.00340			−0.501701	+	0.865041i	$0.667292\pi$
$158$			4.86137e7i			0.980526i
$159$		0			0
$160$	216743.			0.00418336
$161$		−	1.51520e8i		−	2.86141i
$162$		0			0
$163$		−	8.82607e7i		−	1.59629i	−0.602468	−	0.798143i	$-0.705816\pi$
$163$		−	8.82607e7i		−	1.59629i	0.602468	−	0.798143i	$-0.294184\pi$
$164$			3.73160e7i			0.660604i
$165$		0			0
$166$	5.29067e6			0.0897702
$167$			7.18201e7i			1.19327i	0.802513	+	0.596634i	$0.203496\pi$
$167$			7.18201e7i			1.19327i	−0.802513	+	0.596634i	$0.796504\pi$
$168$		0			0
$169$	−3.41648e7	−	5.26322e7i	−0.544471	−	0.838779i
$170$	1.48446e6			0.0231737
$171$		0			0
$172$	−5.66447e6			−0.0848809
$173$	6.00357e7			0.881553			0.440776	−	0.897617i	$-0.354703\pi$
$173$	6.00357e7			0.881553			0.440776	+	0.897617i	$0.354703\pi$
$174$		0			0
$175$			1.25294e8i			1.76725i
$176$			4.80685e6i			0.0664609i
$177$		0			0
$178$	−8.24726e7			−1.09607
$179$	−1.34440e8			−1.75203			−0.876015	−	0.482284i	$-0.839807\pi$
$179$	−1.34440e8			−1.75203			−0.876015	+	0.482284i	$0.839807\pi$
$180$		0			0
$181$	1.55862e8			1.95374			0.976868	−	0.213843i	$-0.0685980\pi$
$181$	1.55862e8			1.95374			0.976868	+	0.213843i	$0.0685980\pi$
$182$	−4.85310e7	+	8.93616e7i	−0.596718	+	1.09876i
$183$		0			0
$184$		−	4.83455e7i		−	0.572129i
$185$	2.66431e6			0.0309374
$186$		0			0
$187$			3.29217e7i			0.368160i
$188$		−	7.21985e6i		−	0.0792457i
$189$		0			0
$190$			731589.i			0.00773801i
$191$	2.23323e7			0.231909			0.115954	−	0.993255i	$-0.463007\pi$
$191$	2.23323e7			0.231909			0.115954	+	0.993255i	$0.463007\pi$
$192$		0			0
$193$		−	2.69730e6i		−	0.0270072i	−0.999909	−	0.0135036i	$-0.995702\pi$
$193$		−	2.69730e6i		−	0.0270072i	0.999909	−	0.0135036i	$-0.00429845\pi$
$194$	4.91376e7			0.483179
$195$		0			0
$196$	1.12090e8			1.06334
$197$		−	7.30124e7i		−	0.680401i	−0.940353	−	0.340200i	$-0.889505\pi$
$197$		−	7.30124e7i		−	0.680401i	0.940353	−	0.340200i	$-0.110495\pi$
$198$		0			0
$199$	1.10593e8			0.994817			0.497409	−	0.867516i	$-0.334285\pi$
$199$	1.10593e8			0.994817			0.497409	+	0.867516i	$0.334285\pi$
$200$			3.99776e7i			0.353355i
$201$		0			0
$202$		−	4.68339e7i		−	0.399789i
$203$		−	2.55817e8i		−	2.14631i
$204$		0			0
$205$	3.85665e6			0.0312660
$206$		−	1.44441e7i		−	0.115121i
$207$		0			0
$208$	−1.54848e7	+	2.85126e7i	−0.119312	+	0.219692i
$209$	−1.62249e7			−0.122933
$210$		0			0
$211$	−7.79932e7			−0.571568			−0.285784	−	0.958294i	$-0.592254\pi$
$211$	−7.79932e7			−0.571568			−0.285784	+	0.958294i	$0.592254\pi$
$212$	3.51196e7			0.253148
$213$		0			0
$214$			1.05619e8i			0.736708i
$215$			585430.i			0.00401736i
$216$		0			0
$217$	−1.82619e8			−1.21322
$218$	1.88973e8			1.23539
$219$		0			0
$220$	496793.			0.00314555
$221$	−1.06054e8	+	1.95280e8i	−0.660926	+	1.21698i
$222$		0			0
$223$		−	2.81334e8i		−	1.69885i	−0.527709	−	0.849425i	$-0.676949\pi$
$223$		−	2.81334e8i		−	1.69885i	0.527709	−	0.849425i	$-0.323051\pi$
$224$	5.25817e7			0.312584
$225$		0			0
$226$			8.29908e7i			0.478245i
$227$		−	2.87156e8i		−	1.62940i	−0.579882	−	0.814700i	$-0.696901\pi$
$227$		−	2.87156e8i		−	1.62940i	0.579882	−	0.814700i	$-0.303099\pi$
$228$		0			0
$229$		−	1.77698e8i		−	0.977819i	−0.872335	−	0.488909i	$-0.837395\pi$
$229$		−	1.77698e8i		−	0.977819i	0.872335	−	0.488909i	$-0.162605\pi$
$230$	−4.99657e6			−0.0270785
$231$		0			0
$232$		−	8.16233e7i		−	0.429147i
$233$	1.38051e8			0.714979			0.357490	−	0.933917i	$-0.383633\pi$
$233$	1.38051e8			0.714979			0.357490	+	0.933917i	$0.383633\pi$
$234$		0			0
$235$	−746181.			−0.00375065
$236$		−	1.08477e8i		−	0.537211i
$237$		0			0
$238$	3.60127e8			1.73156
$239$		−	6.38380e7i		−	0.302473i	−0.988498	−	0.151237i	$-0.951674\pi$
$239$		−	6.38380e7i		−	0.302473i	0.988498	−	0.151237i	$-0.0483255\pi$
$240$		0			0
$241$		−	1.77813e8i		−	0.818285i	−0.912470	−	0.409143i	$-0.865828\pi$
$241$		−	1.77813e8i		−	0.818285i	0.912470	−	0.409143i	$-0.134172\pi$
$242$		−	1.44880e8i		−	0.657134i
$243$		0			0
$244$	−8.54966e7			−0.376777
$245$		−	1.15846e7i		−	0.0503271i
$246$		0			0
$247$	−9.62406e7	−	5.22668e7i	−0.406367	−	0.220692i
$248$	−5.82683e7			−0.242578
$249$		0			0
$250$	8.26578e6			0.0334575
$251$	2.57165e8			1.02649			0.513244	−	0.858243i	$-0.328444\pi$
$251$	2.57165e8			1.02649			0.513244	+	0.858243i	$0.328444\pi$
$252$		0			0
$253$		−	1.10812e8i		−	0.430195i
$254$			2.45766e8i			0.941031i
$255$		0			0
$256$	1.67772e7			0.0625000
$257$	−2.24906e8			−0.826486			−0.413243	−	0.910621i	$-0.635604\pi$
$257$	−2.24906e8			−0.826486			−0.413243	+	0.910621i	$0.635604\pi$
$258$		0			0
$259$	6.46359e8			2.31166
$260$	2.94681e6	+	1.60037e6i	0.0103979	+	0.00564694i
$261$		0			0
$262$			1.30384e8i			0.447890i
$263$	−4.78020e8			−1.62032			−0.810161	−	0.586208i	$-0.800620\pi$
$263$	−4.78020e8			−1.62032			−0.810161	+	0.586208i	$0.800620\pi$
$264$		0			0
$265$		−	3.62966e6i		−	0.0119813i
$266$			1.77483e8i			0.578189i
$267$		0			0
$268$			4.60313e7i			0.146077i
$269$	−1.76243e8			−0.552049			−0.276025	−	0.961151i	$-0.589017\pi$
$269$	−1.76243e8			−0.552049			−0.276025	+	0.961151i	$0.589017\pi$
$270$		0			0
$271$			1.88337e8i			0.574834i	0.957806	+	0.287417i	$0.0927966\pi$
$271$			1.88337e8i			0.574834i	−0.957806	+	0.287417i	$0.907203\pi$
$272$	1.14906e8			0.346219
$273$		0			0
$274$	2.58115e8			0.758030
$275$			9.16320e7i			0.265695i
$276$		0			0
$277$	−3.15057e7			−0.0890656			−0.0445328	−	0.999008i	$-0.514180\pi$
$277$	−3.15057e7			−0.0890656			−0.0445328	+	0.999008i	$0.514180\pi$
$278$			3.01907e8i			0.842784i
$279$		0			0
$280$		−	5.43438e6i		−	0.0147944i
$281$			2.67569e8i			0.719388i	0.933070	+	0.359694i	$0.117119\pi$
$281$			2.67569e8i			0.719388i	−0.933070	+	0.359694i	$0.882881\pi$
$282$		0			0
$283$	1.31279e8			0.344303			0.172152	−	0.985070i	$-0.444928\pi$
$283$	1.31279e8			0.344303			0.172152	+	0.985070i	$0.444928\pi$
$284$			1.18539e8i			0.307077i
$285$		0			0
$286$	−3.54924e7	+	6.53532e7i	−0.0897126	+	0.165191i
$287$	9.35620e8			2.33621
$288$		0			0
$289$	3.76640e8			0.917876
$290$	−8.43587e6			−0.0203113
$291$		0			0
$292$		−	9.60978e7i		−	0.225878i
$293$		−	5.02530e8i		−	1.16715i	−0.812060	−	0.583574i	$-0.801654\pi$
$293$		−	5.02530e8i		−	1.16715i	0.812060	−	0.583574i	$-0.198346\pi$
$294$		0			0
$295$	−1.12112e7			−0.0254259
$296$	2.06233e8			0.462209
$297$		0			0
$298$	−1.35364e8			−0.296310
$299$	3.56969e8	−	6.57299e8i	0.772293	−	1.42205i
$300$		0			0
$301$			1.42025e8i			0.300180i
$302$	3.10263e8			0.648196
$303$		0			0
$304$			5.66293e7i			0.115607i
$305$			8.83618e6i			0.0178326i
$306$		0			0
$307$			4.04464e7i			0.0797802i	0.999204	+	0.0398901i	$0.0127008\pi$
$307$			4.04464e7i			0.0797802i	−0.999204	+	0.0398901i	$0.987299\pi$
$308$	1.20522e8			0.235038
$309$		0			0
$310$			6.02210e6i			0.0114811i
$311$	−5.80479e7			−0.109427			−0.0547136	−	0.998502i	$-0.517425\pi$
$311$	−5.80479e7			−0.109427			−0.0547136	+	0.998502i	$0.517425\pi$
$312$		0			0
$313$	1.87305e8			0.345259			0.172629	−	0.984987i	$-0.444774\pi$
$313$	1.87305e8			0.345259			0.172629	+	0.984987i	$0.444774\pi$
$314$			3.89237e8i			0.709513i
$315$		0			0
$316$	3.88910e8			0.693337
$317$		−	6.09607e7i		−	0.107484i	−0.998555	−	0.0537419i	$-0.982885\pi$
$317$		−	6.09607e7i		−	0.107484i	0.998555	−	0.0537419i	$-0.0171148\pi$
$318$		0			0
$319$		−	1.87087e8i		−	0.322684i
$320$		−	1.73395e6i		−	0.00295808i
$321$		0			0
$322$	−1.21216e9			−2.02332
$323$			3.87849e8i			0.640404i
$324$		0			0
$325$	−2.95183e8	+	5.43530e8i	−0.476979	+	0.878277i
$326$	−7.06085e8			−1.12874
$327$		0			0
$328$	2.98528e8			0.467118
$329$	−1.81023e8			−0.280251
$330$		0			0
$331$		−	4.54468e8i		−	0.688820i	−0.938819	−	0.344410i	$-0.888079\pi$
$331$		−	4.54468e8i		−	0.688820i	0.938819	−	0.344410i	$-0.111921\pi$
$332$		−	4.23254e7i		−	0.0634771i
$333$		0			0
$334$	5.74561e8			0.843769
$335$	4.75739e6			0.00691372
$336$		0			0
$337$	−2.05246e8			−0.292125			−0.146063	−	0.989275i	$-0.546660\pi$
$337$	−2.05246e8			−0.292125			−0.146063	+	0.989275i	$0.546660\pi$
$338$	−4.21057e8	+	2.73318e8i	−0.593107	+	0.384999i
$339$		0			0
$340$		−	1.18756e7i		−	0.0163863i
$341$	−1.33556e8			−0.182399
$342$		0			0
$343$		−	1.48891e9i		−	1.99223i
$344$			4.53158e7i			0.0600198i
$345$		0			0
$346$		−	4.80286e8i		−	0.623352i
$347$	−2.99289e8			−0.384537			−0.192268	−	0.981342i	$-0.561584\pi$
$347$	−2.99289e8			−0.384537			−0.192268	+	0.981342i	$0.561584\pi$
$348$		0			0
$349$		−	1.03098e8i		−	0.129826i	−0.997891	−	0.0649128i	$-0.979323\pi$
$349$		−	1.03098e8i		−	0.129826i	0.997891	−	0.0649128i	$-0.0206769\pi$
$350$	1.00235e9			1.24963
$351$		0			0
$352$	3.84548e7			0.0469949
$353$		−	3.79231e8i		−	0.458873i	−0.973324	−	0.229436i	$-0.926312\pi$
$353$		−	3.79231e8i		−	0.458873i	0.973324	−	0.229436i	$-0.0736883\pi$
$354$		0			0
$355$	1.22511e7			0.0145337
$356$			6.59781e8i			0.775041i
$357$		0			0
$358$			1.07552e9i			1.23887i
$359$			1.29204e9i			1.47382i	0.675988	+	0.736912i	$0.263717\pi$
$359$			1.29204e9i			1.47382i	−0.675988	+	0.736912i	$0.736283\pi$
$360$		0			0
$361$	7.02727e8			0.786161
$362$		−	1.24690e9i		−	1.38150i
$363$		0			0
$364$	7.14893e8	+	3.88248e8i	0.776938	+	0.421943i
$365$	−9.93183e6			−0.0106907
$366$		0			0
$367$	−3.21664e8			−0.339681			−0.169841	−	0.985472i	$-0.554325\pi$
$367$	−3.21664e8			−0.339681			−0.169841	+	0.985472i	$0.554325\pi$
$368$	−3.86764e8			−0.404556
$369$		0			0
$370$		−	2.13145e7i		−	0.0218760i
$371$		−	8.80551e8i		−	0.895253i
$372$		0			0
$373$	−5.95386e8			−0.594043			−0.297021	−	0.954871i	$-0.595993\pi$
$373$	−5.95386e8			−0.594043			−0.297021	+	0.954871i	$0.595993\pi$
$374$	2.63373e8			0.260328
$375$		0			0
$376$	−5.77588e7			−0.0560352
$377$	6.02683e8	−	1.10974e9i	0.579287	−	1.06666i
$378$		0			0
$379$			1.84954e9i			1.74512i	0.488503	+	0.872562i	$0.337543\pi$
$379$			1.84954e9i			1.74512i	−0.488503	+	0.872562i	$0.662457\pi$
$380$	5.85271e6			0.00547160
$381$		0			0
$382$		−	1.78659e8i		−	0.163984i
$383$			1.94010e9i			1.76453i	0.470757	+	0.882263i	$0.343981\pi$
$383$			1.94010e9i			1.76453i	−0.470757	+	0.882263i	$0.656019\pi$
$384$		0			0
$385$		−	1.24561e7i		−	0.0111242i
$386$	−2.15784e7			−0.0190969
$387$		0			0
$388$		−	3.93101e8i		−	0.341659i
$389$	−1.63159e9			−1.40536			−0.702680	−	0.711506i	$-0.748014\pi$
$389$	−1.63159e9			−1.40536			−0.702680	+	0.711506i	$0.748014\pi$
$390$		0			0
$391$	−2.64891e9			−2.24104
$392$		−	8.96721e8i		−	0.751893i
$393$		0			0
$394$	−5.84099e8			−0.481116
$395$		−	4.01943e7i		−	0.0328152i
$396$		0			0
$397$			1.46137e9i			1.17218i	0.810246	+	0.586090i	$0.199334\pi$
$397$			1.46137e9i			1.17218i	−0.810246	+	0.586090i	$0.800666\pi$
$398$		−	8.84747e8i		−	0.703442i
$399$		0			0
$400$	3.19821e8			0.249860
$401$			2.15957e8i			0.167248i	0.996497	+	0.0836240i	$0.0266495\pi$
$401$			2.15957e8i			0.167248i	−0.996497	+	0.0836240i	$0.973351\pi$
$402$		0			0
$403$	−7.92208e8	−	4.30236e8i	−0.602936	−	0.327446i
$404$	−3.74671e8			−0.282693
$405$		0			0
$406$	−2.04653e9			−1.51767
$407$	4.72704e8			0.347543
$408$		0			0
$409$			2.10034e9i			1.51795i	0.651120	+	0.758975i	$0.274300\pi$
$409$			2.10034e9i			1.51795i	−0.651120	+	0.758975i	$0.725700\pi$
$410$		−	3.08532e7i		−	0.0221084i
$411$		0			0
$412$	−1.15553e8			−0.0814031
$413$	−2.71983e9			−1.89984
$414$		0			0
$415$	−4.37438e6			−0.00300433
$416$	2.28101e8	+	1.23878e8i	0.155346	+	0.0843661i
$417$		0			0
$418$			1.29799e8i			0.0869270i
$419$	−4.33365e8			−0.287809			−0.143905	−	0.989592i	$-0.545966\pi$
$419$	−4.33365e8			−0.287809			−0.143905	+	0.989592i	$0.545966\pi$
$420$		0			0
$421$		−	2.60776e9i		−	1.70326i	−0.524145	−	0.851629i	$-0.675615\pi$
$421$		−	2.60776e9i		−	1.70326i	0.524145	−	0.851629i	$-0.324385\pi$
$422$			6.23946e8i			0.404160i
$423$		0			0
$424$		−	2.80957e8i		−	0.179003i
$425$	2.19042e9			1.38410
$426$		0			0
$427$			2.14365e9i			1.33246i
$428$	8.44954e8			0.520931
$429$		0			0
$430$	4.68344e6			0.00284070
$431$			6.82317e8i			0.410503i	0.978709	+	0.205251i	$0.0658012\pi$
$431$			6.82317e8i			0.410503i	−0.978709	+	0.205251i	$0.934199\pi$
$432$		0			0
$433$	−2.49592e9			−1.47749			−0.738743	−	0.673987i	$-0.764580\pi$
$433$	−2.49592e9			−1.47749			−0.738743	+	0.673987i	$0.764580\pi$
$434$			1.46096e9i			0.857873i
$435$		0			0
$436$		−	1.51179e9i		−	0.873550i
$437$		−	1.30547e9i		−	0.748312i
$438$		0			0
$439$	1.51778e9			0.856216			0.428108	−	0.903727i	$-0.359180\pi$
$439$	1.51778e9			0.856216			0.428108	+	0.903727i	$0.359180\pi$
$440$		−	3.97435e6i		−	0.00222424i
$441$		0			0
$442$	1.56224e9	+	8.48430e8i	0.860538	+	0.467345i
$443$	−2.56533e9			−1.40194			−0.700971	−	0.713190i	$-0.747250\pi$
$443$	−2.56533e9			−1.40194			−0.700971	+	0.713190i	$0.747250\pi$
$444$		0			0
$445$	6.81892e7			0.0366822
$446$	−2.25067e9			−1.20127
$447$		0			0
$448$		−	4.20653e8i		−	0.221030i
$449$		−	2.22454e9i		−	1.15978i	−0.814693	−	0.579892i	$-0.803095\pi$
$449$		−	2.22454e9i		−	1.15978i	0.814693	−	0.579892i	$-0.196905\pi$
$450$		0			0
$451$	6.84251e8			0.351235
$452$	6.63926e8			0.338170
$453$		0			0
$454$	−2.29725e9			−1.15216
$455$	4.01259e7	−	7.38851e7i	0.0199703	−	0.0367720i
$456$		0			0
$457$		−	1.58220e9i		−	0.775453i	−0.921775	−	0.387726i	$-0.873261\pi$
$457$		−	1.58220e9i		−	0.775453i	0.921775	−	0.387726i	$-0.126739\pi$
$458$	−1.42158e9			−0.691422
$459$		0			0
$460$			3.99726e7i			0.0191474i
$461$			1.33292e9i			0.633652i	0.948484	+	0.316826i	$0.102617\pi$
$461$			1.33292e9i			0.633652i	−0.948484	+	0.316826i	$0.897383\pi$
$462$		0			0
$463$		−	2.03271e9i		−	0.951790i	−0.879502	−	0.475895i	$-0.842124\pi$
$463$		−	2.03271e9i		−	0.951790i	0.879502	−	0.475895i	$-0.157876\pi$
$464$	−6.52987e8			−0.303453
$465$		0			0
$466$		−	1.10441e9i		−	0.505567i
$467$	−3.10675e7			−0.0141155			−0.00705777	−	0.999975i	$-0.502247\pi$
$467$	−3.10675e7			−0.0141155			−0.00705777	+	0.999975i	$0.502247\pi$
$468$		0			0
$469$	1.15414e9			0.516598
$470$			5.96945e6i			0.00265211i
$471$		0			0
$472$	−8.67815e8			−0.379866
$473$			1.03867e8i			0.0451300i
$474$		0			0
$475$			1.07951e9i			0.462169i
$476$		−	2.88102e9i		−	1.22440i
$477$		0			0
$478$	−5.10704e8			−0.213881
$479$			2.95620e9i			1.22902i	0.788908	+	0.614512i	$0.210647\pi$
$479$			2.95620e9i			1.22902i	−0.788908	+	0.614512i	$0.789353\pi$
$480$		0			0
$481$	2.80392e9	+	1.52277e9i	1.14884	+	0.623916i
$482$	−1.42251e9			−0.578615
$483$		0			0
$484$	−1.15904e9			−0.464664
$485$	−4.06274e7			−0.0161705
$486$		0			0
$487$			3.14356e8i			0.123330i	0.998097	+	0.0616652i	$0.0196411\pi$
$487$			3.14356e8i			0.123330i	−0.998097	+	0.0616652i	$0.980359\pi$
$488$			6.83973e8i			0.266422i
$489$		0			0
$490$	−9.26772e7			−0.0355866
$491$	1.74300e9			0.664525			0.332262	−	0.943187i	$-0.392188\pi$
$491$	1.74300e9			0.664525			0.332262	+	0.943187i	$0.392188\pi$
$492$		0			0
$493$	−4.47225e9			−1.68098
$494$	−4.18134e8	+	7.69925e8i	−0.156053	+	0.287345i
$495$		0			0
$496$			4.66146e8i			0.171529i
$497$	2.97211e9			1.08597
$498$		0			0
$499$			3.47846e9i			1.25324i	0.779324	+	0.626621i	$0.215563\pi$
$499$			3.47846e9i			1.25324i	−0.779324	+	0.626621i	$0.784437\pi$
$500$		−	6.61263e7i		−	0.0236581i
$501$		0			0
$502$		−	2.05732e9i		−	0.725837i
$503$	2.26923e9			0.795043			0.397521	−	0.917593i	$-0.369870\pi$
$503$	2.26923e9			0.795043			0.397521	+	0.917593i	$0.369870\pi$
$504$		0			0
$505$			3.87227e7i			0.0133797i
$506$	−8.86496e8			−0.304194
$507$		0			0
$508$	1.96613e9			0.665409
$509$			3.74316e9i			1.25813i	0.777352	+	0.629065i	$0.216562\pi$
$509$			3.74316e9i			1.25813i	−0.777352	+	0.629065i	$0.783438\pi$
$510$		0			0
$511$	−2.40945e9			−0.798812
$512$		−	1.34218e8i		−	0.0441942i
$513$		0			0
$514$			1.79925e9i			0.584414i
$515$			1.19425e7i			0.00385275i
$516$		0			0
$517$	−1.32388e8			−0.0421339
$518$		−	5.17087e9i		−	1.63459i
$519$		0			0
$520$	1.28030e7	−	2.35745e7i	0.00399299	−	0.00735242i
$521$	1.68654e9			0.522475			0.261238	−	0.965275i	$-0.415869\pi$
$521$	1.68654e9			0.522475			0.261238	+	0.965275i	$0.415869\pi$
$522$		0			0
$523$	4.11403e9			1.25751			0.628755	−	0.777604i	$-0.283565\pi$
$523$	4.11403e9			1.25751			0.628755	+	0.777604i	$0.283565\pi$
$524$	1.04307e9			0.316706
$525$		0			0
$526$			3.82416e9i			1.14574i
$527$			3.19260e9i			0.950182i
$528$		0			0
$529$	5.51124e9			1.61865
$530$	−2.90373e7			−0.00847208
$531$		0			0
$532$	1.41986e9			0.408842
$533$	4.05874e9	+	2.20424e9i	1.16104	+	0.630542i
$534$		0			0
$535$		−	8.73271e7i		−	0.0246553i
$536$	3.68250e8			0.103292
$537$		0			0
$538$			1.40994e9i			0.390358i
$539$		−	2.05536e9i		−	0.565362i
$540$		0			0
$541$			5.21199e9i			1.41519i	0.706620	+	0.707593i	$0.250219\pi$
$541$			5.21199e9i			1.41519i	−0.706620	+	0.707593i	$0.749781\pi$
$542$	1.50669e9			0.406469
$543$		0			0
$544$		−	9.19245e8i		−	0.244813i
$545$	−1.56245e8			−0.0413446
$546$		0			0
$547$	−5.06174e9			−1.32234			−0.661172	−	0.750235i	$-0.729941\pi$
$547$	−5.06174e9			−1.32234			−0.661172	+	0.750235i	$0.729941\pi$
$548$		−	2.06492e9i		−	0.536008i
$549$		0			0
$550$	7.33056e8			0.187874
$551$		−	2.20407e9i		−	0.561300i
$552$		0			0
$553$		−	9.75110e9i		−	2.45197i
$554$			2.52046e8i			0.0629789i
$555$		0			0
$556$	2.41525e9			0.595938
$557$			5.64001e9i			1.38289i	0.722430	+	0.691444i	$0.243025\pi$
$557$			5.64001e9i			1.38289i	−0.722430	+	0.691444i	$0.756975\pi$
$558$		0			0
$559$	−3.34598e8	+	6.16107e8i	−0.0810182	+	0.149181i
$560$	−4.34750e7			−0.0104612
$561$		0			0
$562$	2.14055e9			0.508684
$563$	−1.03881e9			−0.245334			−0.122667	−	0.992448i	$-0.539145\pi$
$563$	−1.03881e9			−0.245334			−0.122667	+	0.992448i	$0.539145\pi$
$564$		0			0
$565$		−	6.86176e7i		−	0.0160054i
$566$		−	1.05023e9i		−	0.243459i
$567$		0			0
$568$	9.48311e8			0.217136
$569$	1.36295e9			0.310160			0.155080	−	0.987902i	$-0.450436\pi$
$569$	1.36295e9			0.310160			0.155080	+	0.987902i	$0.450436\pi$
$570$		0			0
$571$	7.59504e9			1.70727			0.853637	−	0.520868i	$-0.174392\pi$
$571$	7.59504e9			1.70727			0.853637	+	0.520868i	$0.174392\pi$
$572$	5.22826e8	+	2.83939e8i	0.116808	+	0.0634364i
$573$		0			0
$574$		−	7.48496e9i		−	1.65195i
$575$	−7.37281e9			−1.61732
$576$		0			0
$577$		−	9.14980e8i		−	0.198288i	−0.995073	−	0.0991439i	$-0.968390\pi$
$577$		−	9.14980e8i		−	0.198288i	0.995073	−	0.0991439i	$-0.0316104\pi$
$578$		−	3.01312e9i		−	0.649036i
$579$		0			0
$580$			6.74870e7i			0.0143622i
$581$	−1.06122e9			−0.224486
$582$		0			0
$583$		−	6.43977e8i		−	0.134595i
$584$	−7.68782e8			−0.159720
$585$		0			0
$586$	−4.02024e9			−0.825298
$587$			3.69625e9i			0.754272i	0.926158	+	0.377136i	$0.123091\pi$
$587$			3.69625e9i			0.754272i	−0.926158	+	0.377136i	$0.876909\pi$
$588$		0			0
$589$	−1.57342e9			−0.317278
$590$			8.96898e7i			0.0179788i
$591$		0			0
$592$		−	1.64987e9i		−	0.326831i
$593$			4.56755e9i			0.899480i	0.893159	+	0.449740i	$0.148483\pi$
$593$			4.56755e9i			0.899480i	−0.893159	+	0.449740i	$0.851517\pi$
$594$		0			0
$595$	−2.97757e8			−0.0579498
$596$			1.08291e9i			0.209523i
$597$		0			0
$598$	−5.25839e9	−	2.85576e9i	−1.00554	−	0.546093i
$599$	−1.63940e9			−0.311668			−0.155834	−	0.987783i	$-0.549806\pi$
$599$	−1.63940e9			−0.311668			−0.155834	+	0.987783i	$0.549806\pi$
$600$		0			0
$601$	−4.92573e9			−0.925572			−0.462786	−	0.886470i	$-0.653150\pi$
$601$	−4.92573e9			−0.925572			−0.462786	+	0.886470i	$0.653150\pi$
$602$	1.13620e9			0.212259
$603$		0			0
$604$		−	2.48211e9i		−	0.458343i
$605$			1.19788e8i			0.0219922i
$606$		0			0
$607$	7.26751e9			1.31894			0.659470	−	0.751731i	$-0.270781\pi$
$607$	7.26751e9			1.31894			0.659470	+	0.751731i	$0.270781\pi$
$608$	4.53035e8			0.0817464
$609$		0			0
$610$	7.06894e7			0.0126096
$611$	−7.85281e8	−	4.26474e8i	−0.139277	−	0.0756395i
$612$		0			0
$613$			3.16466e9i			0.554900i	0.960740	+	0.277450i	$0.0894893\pi$
$613$			3.16466e9i			0.554900i	−0.960740	+	0.277450i	$0.910511\pi$
$614$	3.23571e8			0.0564131
$615$		0			0
$616$		−	9.64172e8i		−	0.166197i
$617$		−	1.11473e10i		−	1.91060i	−0.295634	−	0.955301i	$-0.595531\pi$
$617$		−	1.11473e10i		−	1.91060i	0.295634	−	0.955301i	$-0.404469\pi$
$618$		0			0
$619$			8.69140e9i			1.47290i	0.676493	+	0.736449i	$0.263499\pi$
$619$			8.69140e9i			1.47290i	−0.676493	+	0.736449i	$0.736501\pi$
$620$	4.81768e7			0.00811834
$621$		0			0
$622$			4.64384e8i			0.0773767i
$623$	1.65426e10			2.74092
$624$		0			0
$625$	6.09326e9			0.998320
$626$		−	1.49844e9i		−	0.244135i
$627$		0			0
$628$	3.11390e9			0.501701
$629$		−	1.12998e10i		−	1.81048i
$630$		0			0
$631$		−	3.41348e9i		−	0.540872i	−0.962738	−	0.270436i	$-0.912832\pi$
$631$		−	3.41348e9i		−	0.540872i	0.962738	−	0.270436i	$-0.0871679\pi$
$632$		−	3.11128e9i		−	0.490263i
$633$		0			0
$634$	−4.87686e8			−0.0760025
$635$		−	2.03202e8i		−	0.0314934i
$636$		0			0
$637$	6.62112e9	−	1.21917e10i	1.01495	−	1.86886i
$638$	−1.49670e9			−0.228172
$639$		0			0
$640$	−1.38716e7			−0.00209168
$641$	1.93523e9			0.290221			0.145110	−	0.989415i	$-0.453646\pi$
$641$	1.93523e9			0.290221			0.145110	+	0.989415i	$0.453646\pi$
$642$		0			0
$643$			7.98936e9i			1.18515i	0.805515	+	0.592575i	$0.201889\pi$
$643$			7.98936e9i			1.18515i	−0.805515	+	0.592575i	$0.798111\pi$
$644$			9.69730e9i			1.43071i
$645$		0			0
$646$	3.10279e9			0.452834
$647$	8.47178e9			1.22973			0.614865	−	0.788632i	$-0.289211\pi$
$647$	8.47178e9			1.22973			0.614865	+	0.788632i	$0.289211\pi$
$648$		0			0
$649$	−1.98910e9			−0.285628
$650$	4.34824e9	+	2.36146e9i	0.621036	+	0.337275i
$651$		0			0
$652$			5.64868e9i			0.798143i
$653$	−8.72038e9			−1.22557			−0.612787	−	0.790248i	$-0.709952\pi$
$653$	−8.72038e9			−1.22557			−0.612787	+	0.790248i	$0.709952\pi$
$654$		0			0
$655$		−	1.07803e8i		−	0.0149895i
$656$		−	2.38822e9i		−	0.330302i
$657$		0			0
$658$			1.44818e9i			0.198167i
$659$	4.03937e9			0.549813			0.274907	−	0.961471i	$-0.411353\pi$
$659$	4.03937e9			0.549813			0.274907	+	0.961471i	$0.411353\pi$
$660$		0			0
$661$		−	3.29803e9i		−	0.444170i	−0.975027	−	0.222085i	$-0.928714\pi$
$661$		−	3.29803e9i		−	0.444170i	0.975027	−	0.222085i	$-0.0712862\pi$
$662$	−3.63575e9			−0.487069
$663$		0			0
$664$	−3.38603e8			−0.0448851
$665$		−	1.46744e8i		−	0.0193502i
$666$		0			0
$667$	1.50533e10			1.96422
$668$		−	4.59649e9i		−	0.596634i
$669$		0			0
$670$		−	3.80591e7i		−	0.00488874i
$671$			1.56772e9i			0.200327i
$672$		0			0
$673$	−5.27449e9			−0.667003			−0.333502	−	0.942750i	$-0.608230\pi$
$673$	−5.27449e9			−0.667003			−0.333502	+	0.942750i	$0.608230\pi$
$674$			1.64196e9i			0.206564i
$675$		0			0
$676$	2.18655e9	+	3.36846e9i	0.272236	+	0.419390i
$677$	−1.02661e10			−1.27158			−0.635791	−	0.771861i	$-0.719326\pi$
$677$	−1.02661e10			−1.27158			−0.635791	+	0.771861i	$0.719326\pi$
$678$		0			0
$679$	−9.85617e9			−1.20827
$680$	−9.50051e7			−0.0115869
$681$		0			0
$682$			1.06845e9i			0.128976i
$683$			3.81750e9i			0.458465i	0.973372	+	0.229233i	$0.0736216\pi$
$683$			3.81750e9i			0.458465i	−0.973372	+	0.229233i	$0.926378\pi$
$684$		0			0
$685$	−2.13412e8			−0.0253689
$686$	−1.19113e10			−1.40872
$687$		0			0
$688$	3.62526e8			0.0424404
$689$	2.07451e9	−	3.81985e9i	0.241628	−	0.444918i
$690$		0			0
$691$		−	1.41493e10i		−	1.63140i	−0.578475	−	0.815700i	$-0.696352\pi$
$691$		−	1.41493e10i		−	1.63140i	0.578475	−	0.815700i	$-0.303648\pi$
$692$	−3.84229e9			−0.440776
$693$		0			0
$694$			2.39431e9i			0.271909i
$695$		−	2.49619e8i		−	0.0282054i
$696$		0			0
$697$		−	1.63567e10i		−	1.82971i
$698$	−8.24782e8			−0.0918006
$699$		0			0
$700$		−	8.01883e9i		−	0.883625i
$701$	−2.63084e9			−0.288457			−0.144228	−	0.989544i	$-0.546070\pi$
$701$	−2.63084e9			−0.288457			−0.144228	+	0.989544i	$0.546070\pi$
$702$		0			0
$703$	5.56892e9			0.604543
$704$		−	3.07638e8i		−	0.0332304i
$705$		0			0
$706$	−3.03385e9			−0.324472
$707$			9.39409e9i			0.999740i
$708$		0			0
$709$			1.48321e10i			1.56293i	0.623947	+	0.781466i	$0.285528\pi$
$709$			1.48321e10i			1.56293i	−0.623947	+	0.781466i	$0.714472\pi$
$710$		−	9.80092e7i		−	0.0102769i
$711$		0			0
$712$	5.27825e9			0.548037
$713$		−	1.07461e10i		−	1.11029i
$714$		0			0
$715$	2.93454e7	−	5.40347e7i	0.00300241	−	0.00552843i
$716$	8.60413e9			0.876015
$717$		0			0
$718$	1.03363e10			1.04215
$719$	4.09811e9			0.411180			0.205590	−	0.978638i	$-0.434089\pi$
$719$	4.09811e9			0.411180			0.205590	+	0.978638i	$0.434089\pi$
$720$		0			0
$721$			2.89725e9i			0.287880i
$722$		−	5.62181e9i		−	0.555899i
$723$		0			0
$724$	−9.97518e9			−0.976868
$725$	−1.24478e10			−1.21313
$726$		0			0
$727$	1.07923e10			1.04171			0.520853	−	0.853646i	$-0.325614\pi$
$727$	1.07923e10			1.04171			0.520853	+	0.853646i	$0.325614\pi$
$728$	3.10598e9	−	5.71914e9i	0.298359	−	0.549378i
$729$		0			0
$730$			7.94546e7i			0.00755943i
$731$	2.48291e9			0.235099
$732$		0			0
$733$			4.31755e9i			0.404924i	0.979290	+	0.202462i	$0.0648942\pi$
$733$			4.31755e9i			0.404924i	−0.979290	+	0.202462i	$0.935106\pi$
$734$			2.57331e9i			0.240191i
$735$		0			0
$736$			3.09411e9i			0.286065i
$737$	8.44060e8			0.0776671
$738$		0			0
$739$		−	1.59219e10i		−	1.45124i	−0.688097	−	0.725619i	$-0.741554\pi$
$739$		−	1.59219e10i		−	1.45124i	0.688097	−	0.725619i	$-0.258446\pi$
$740$	−1.70516e8			−0.0154687
$741$		0			0
$742$	−7.04441e9			−0.633040
$743$			7.46206e9i			0.667418i	0.942676	+	0.333709i	$0.108300\pi$
$743$			7.46206e9i			0.667418i	−0.942676	+	0.333709i	$0.891700\pi$
$744$		0			0
$745$	1.11920e8			0.00991657
$746$			4.76309e9i			0.420052i
$747$		0			0
$748$		−	2.10699e9i		−	0.184080i
$749$		−	2.11855e10i		−	1.84226i
$750$		0			0
$751$	−7.78292e9			−0.670507			−0.335253	−	0.942128i	$-0.608822\pi$
$751$	−7.78292e9			−0.670507			−0.335253	+	0.942128i	$0.608822\pi$
$752$			4.62071e8i			0.0396229i
$753$		0			0
$754$	−8.87792e9	−	4.82146e9i	−0.754243	−	0.409618i
$755$	−2.56529e8			−0.0216931
$756$		0			0
$757$	−1.36187e10			−1.14104			−0.570518	−	0.821285i	$-0.693257\pi$
$757$	−1.36187e10			−1.14104			−0.570518	+	0.821285i	$0.693257\pi$
$758$	1.47963e10			1.23399
$759$		0			0
$760$		−	4.68217e7i		−	0.00386901i
$761$			7.93048e9i			0.652309i	0.945317	+	0.326154i	$0.105753\pi$
$761$			7.93048e9i			0.652309i	−0.945317	+	0.326154i	$0.894247\pi$
$762$		0			0
$763$	−3.79049e10			−3.08929
$764$	−1.42927e9			−0.115954
$765$		0			0
$766$	1.55208e10			1.24771
$767$	−1.17987e10	−	6.40769e9i	−0.944170	−	0.512765i
$768$		0			0
$769$			1.80037e10i			1.42764i	0.700329	+	0.713820i	$0.253036\pi$
$769$			1.80037e10i			1.42764i	−0.700329	+	0.713820i	$0.746964\pi$
$770$	−9.96484e7			−0.00786598
$771$		0			0
$772$			1.72627e8i			0.0135036i
$773$			1.79933e10i			1.40114i	0.713583	+	0.700571i	$0.247071\pi$
$773$			1.79933e10i			1.40114i	−0.713583	+	0.700571i	$0.752929\pi$
$774$		0			0
$775$			8.88606e9i			0.685730i
$776$	−3.14480e9			−0.241589
$777$		0			0
$778$			1.30527e10i			0.993740i
$779$	8.06114e9			0.610963
$780$		0			0
$781$	2.17361e9			0.163269
$782$			2.11913e10i			1.58465i
$783$		0			0
$784$	−7.17377e9			−0.531669
$785$		−	3.21825e8i		−	0.0237452i
$786$		0			0
$787$			4.61318e9i			0.337356i	0.985671	+	0.168678i	$0.0539498\pi$
$787$			4.61318e9i			0.337356i	−0.985671	+	0.168678i	$0.946050\pi$
$788$			4.67279e9i			0.340200i
$789$		0			0
$790$	−3.21554e8			−0.0232038
$791$		−	1.66465e10i		−	1.19593i
$792$		0			0
$793$	−5.05026e9	+	9.29920e9i	−0.359631	+	0.662200i
$794$	1.16910e10			0.828857
$795$		0			0
$796$	−7.07798e9			−0.497409
$797$	−2.80281e9			−0.196106			−0.0980528	−	0.995181i	$-0.531261\pi$
$797$	−2.80281e9			−0.196106			−0.0980528	+	0.995181i	$0.531261\pi$
$798$		0			0
$799$			3.16468e9i			0.219491i
$800$		−	2.55857e9i		−	0.176678i
$801$		0			0
$802$	1.72765e9			0.118262
$803$	−1.76211e9			−0.120096
$804$		0			0
$805$	1.00223e9			0.0677144
$806$	−3.44189e9	+	6.33766e9i	−0.231539	+	0.426340i
$807$		0			0
$808$			2.99737e9i			0.199894i
$809$	−8.36628e9			−0.555537			−0.277768	−	0.960648i	$-0.589595\pi$
$809$	−8.36628e9			−0.555537			−0.277768	+	0.960648i	$0.589595\pi$
$810$		0			0
$811$		−	4.54358e9i		−	0.299106i	−0.988754	−	0.149553i	$-0.952217\pi$
$811$		−	4.54358e9i		−	0.299106i	0.988754	−	0.149553i	$-0.0477835\pi$
$812$			1.63723e10i			1.07316i
$813$		0			0
$814$		−	3.78163e9i		−	0.245750i
$815$	5.83798e8			0.0377756
$816$		0			0
$817$			1.22366e9i			0.0785025i
$818$	1.68027e10			1.07335
$819$		0			0
$820$	−2.46826e8			−0.0156330
$821$		−	1.50748e10i		−	0.950717i	−0.879792	−	0.475358i	$-0.842318\pi$
$821$		−	1.50748e10i		−	0.950717i	0.879792	−	0.475358i	$-0.157682\pi$
$822$		0			0
$823$	1.67438e10			1.04702			0.523508	−	0.852021i	$-0.324623\pi$
$823$	1.67438e10			1.04702			0.523508	+	0.852021i	$0.324623\pi$
$824$			9.24424e8i			0.0575607i
$825$		0			0
$826$			2.17586e10i			1.34339i
$827$			1.05833e10i			0.650656i	0.945601	+	0.325328i	$0.105475\pi$
$827$			1.05833e10i			0.650656i	−0.945601	+	0.325328i	$0.894525\pi$
$828$		0			0
$829$	−2.53716e10			−1.54670			−0.773352	−	0.633976i	$-0.781422\pi$
$829$	−2.53716e10			−1.54670			−0.773352	+	0.633976i	$0.781422\pi$
$830$			3.49950e7i			0.00212438i
$831$		0			0
$832$	9.91025e8	−	1.82481e9i	0.0596558	−	0.109846i
$833$	−4.91325e10			−2.94518
$834$		0			0
$835$	−4.75053e8			−0.0282383
$836$	1.03839e9			0.0614667
$837$		0			0
$838$			3.46692e9i			0.203512i
$839$		−	8.37014e9i		−	0.489290i	−0.969613	−	0.244645i	$-0.921329\pi$
$839$		−	8.37014e9i		−	0.489290i	0.969613	−	0.244645i	$-0.0786714\pi$
$840$		0			0
$841$	8.16504e9			0.473339
$842$	−2.08621e10			−1.20439
$843$		0			0
$844$	4.99156e9			0.285784
$845$	3.48134e8	−	2.25982e8i	0.0198494	−	0.0128847i
$846$		0			0
$847$			2.90604e10i			1.64327i
$848$	−2.24766e9			−0.126574
$849$		0			0
$850$		−	1.75234e10i		−	0.978705i
$851$			3.80343e10i			2.11554i
$852$		0			0
$853$			8.05268e9i			0.444241i	0.975019	+	0.222121i	$0.0712979\pi$
$853$			8.05268e9i			0.444241i	−0.975019	+	0.222121i	$0.928702\pi$
$854$	1.71492e10			0.942195
$855$		0			0
$856$		−	6.75964e9i		−	0.368354i
$857$	2.21962e10			1.20461			0.602304	−	0.798267i	$-0.294249\pi$
$857$	2.21962e10			1.20461			0.602304	+	0.798267i	$0.294249\pi$
$858$		0			0
$859$	−4.25915e8			−0.0229270			−0.0114635	−	0.999934i	$-0.503649\pi$
$859$	−4.25915e8			−0.0229270			−0.0114635	+	0.999934i	$0.503649\pi$
$860$		−	3.74675e7i		−	0.00200868i
$861$		0			0
$862$	5.45854e9			0.290269
$863$			1.84502e10i			0.977156i	0.872520	+	0.488578i	$0.162484\pi$
$863$			1.84502e10i			0.977156i	−0.872520	+	0.488578i	$0.837516\pi$
$864$		0			0
$865$			3.97105e8i			0.0208617i
$866$			1.99674e10i			1.04474i
$867$		0			0
$868$	1.16876e10			0.606608
$869$		−	7.13131e9i		−	0.368638i
$870$		0			0
$871$	5.00668e9	+	2.71905e9i	0.256736	+	0.139429i
$872$	−1.20943e10			−0.617693
$873$		0			0
$874$	−1.04438e10			−0.529137
$875$	−1.65798e9			−0.0836662
$876$		0			0
$877$		−	1.81523e10i		−	0.908725i	−0.890817	−	0.454363i	$-0.849867\pi$
$877$		−	1.81523e10i		−	0.908725i	0.890817	−	0.454363i	$-0.150133\pi$
$878$		−	1.21423e10i		−	0.605436i
$879$		0			0
$880$	−3.17948e7			−0.00157277
$881$	2.80509e10			1.38207			0.691037	−	0.722819i	$-0.257154\pi$
$881$	2.80509e10			1.38207			0.691037	+	0.722819i	$0.257154\pi$
$882$		0			0
$883$	−2.12689e9			−0.103964			−0.0519819	−	0.998648i	$-0.516554\pi$
$883$	−2.12689e9			−0.103964			−0.0519819	+	0.998648i	$0.516554\pi$
$884$	6.78744e9	−	1.24979e10i	0.330463	−	0.608492i
$885$		0			0
$886$			2.05226e10i			0.991323i
$887$	−2.61618e10			−1.25874			−0.629368	−	0.777107i	$-0.716686\pi$
$887$	−2.61618e10			−1.25874			−0.629368	+	0.777107i	$0.716686\pi$
$888$		0			0
$889$		−	4.92965e10i		−	2.35321i
$890$		−	5.45514e8i		−	0.0259382i
$891$		0			0
$892$			1.80054e10i			0.849425i
$893$	−1.55966e9			−0.0732909
$894$		0			0
$895$		−	8.89248e8i		−	0.0414612i
$896$	−3.36523e9			−0.156292
$897$		0			0
$898$	−1.77963e10			−0.820091
$899$		−	1.81429e10i		−	0.832814i
$900$		0			0
$901$	−1.53940e10			−0.701156
$902$		−	5.47400e9i		−	0.248360i
$903$		0			0
$904$		−	5.31141e9i		−	0.239122i
$905$			1.03095e9i			0.0462345i
$906$		0			0
$907$	−1.57121e10			−0.699213			−0.349607	−	0.936897i	$-0.613685\pi$
$907$	−1.57121e10			−0.699213			−0.349607	+	0.936897i	$0.613685\pi$
$908$			1.83780e10i			0.814700i
$909$		0			0
$910$	−5.91081e8	−	3.21007e8i	−0.0260017	−	0.0141211i
$911$	3.17054e10			1.38937			0.694687	−	0.719312i	$-0.255543\pi$
$911$	3.17054e10			1.38937			0.694687	+	0.719312i	$0.255543\pi$
$912$		0			0
$913$	−7.76106e8			−0.0337500
$914$	−1.26576e10			−0.548328
$915$		0			0
$916$			1.13727e10i			0.488909i
$917$		−	2.61529e10i		−	1.12002i
$918$		0			0
$919$	−7.15392e9			−0.304047			−0.152023	−	0.988377i	$-0.548579\pi$
$919$	−7.15392e9			−0.304047			−0.152023	+	0.988377i	$0.548579\pi$
$920$	3.19781e8			0.0135393
$921$		0			0
$922$	1.06634e10			0.448059
$923$	1.28931e10	+	7.00206e9i	0.539699	+	0.293103i
$924$		0			0
$925$		−	3.14511e10i		−	1.30659i
$926$	−1.62617e10			−0.673017
$927$		0			0
$928$			5.22389e9i			0.214574i
$929$		−	3.59995e10i		−	1.47313i	−0.676365	−	0.736566i	$-0.736446\pi$
$929$		−	3.59995e10i		−	1.47313i	0.676365	−	0.736566i	$-0.263554\pi$
$930$		0			0
$931$		−	2.42141e10i		−	0.983433i
$932$	−8.83526e9			−0.357490
$933$		0			0
$934$			2.48540e8i			0.00998119i
$935$	−2.17760e8			−0.00871238
$936$		0			0
$937$	−1.38764e10			−0.551047			−0.275523	−	0.961294i	$-0.588851\pi$
$937$	−1.38764e10			−0.551047			−0.275523	+	0.961294i	$0.588851\pi$
$938$		−	9.23310e9i		−	0.365290i
$939$		0			0
$940$	4.77556e7			0.00187532
$941$			1.24319e9i			0.0486379i	0.999704	+	0.0243190i	$0.00774173\pi$
$941$			1.24319e9i			0.0486379i	−0.999704	+	0.0243190i	$0.992258\pi$
$942$		0			0
$943$			5.50556e10i			2.13801i
$944$			6.94252e9i			0.268606i
$945$		0			0
$946$	8.30940e8			0.0319118
$947$		−	1.08695e10i		−	0.415896i	−0.978140	−	0.207948i	$-0.933321\pi$
$947$		−	1.08695e10i		−	0.415896i	0.978140	−	0.207948i	$-0.0666785\pi$
$948$		0			0
$949$	−1.04523e10	−	5.67647e9i	−0.396989	−	0.215599i
$950$	8.63611e9			0.326803
$951$		0			0
$952$	−2.30481e10			−0.865778
$953$	4.97595e9			0.186230			0.0931152	−	0.995655i	$-0.470318\pi$
$953$	4.97595e9			0.186230			0.0931152	+	0.995655i	$0.470318\pi$
$954$		0			0
$955$			1.47717e8i			0.00548804i
$956$			4.08563e9i			0.151237i
$957$		0			0
$958$	2.36496e10			0.869051
$959$	−5.17736e10			−1.89558
$960$		0			0
$961$	1.45610e10			0.529247
$962$	1.21821e10	−	2.24314e10i	0.441175	−	0.812350i
$963$		0			0
$964$			1.13801e10i			0.409143i
$965$	1.78412e7			0.000639116
$966$		0			0
$967$			1.95659e10i			0.695838i	0.937525	+	0.347919i	$0.113112\pi$
$967$			1.95659e10i			0.695838i	−0.937525	+	0.347919i	$0.886888\pi$
$968$			9.27230e9i			0.328567i
$969$		0			0
$970$			3.25019e8i			0.0114343i
$971$	6.28877e9			0.220444			0.110222	−	0.993907i	$-0.464844\pi$
$971$	6.28877e9			0.220444			0.110222	+	0.993907i	$0.464844\pi$
$972$		0			0
$973$		−	6.05574e10i		−	2.10752i
$974$	2.51485e9			0.0872078
$975$		0			0
$976$	5.47178e9			0.188389
$977$			1.75153e9i			0.0600878i	0.999549	+	0.0300439i	$0.00956470\pi$
$977$			1.75153e9i			0.0600878i	−0.999549	+	0.0300439i	$0.990435\pi$
$978$		0			0
$979$	1.20982e10			0.412079
$980$			7.41417e8i			0.0251635i
$981$		0			0
$982$		−	1.39440e10i		−	0.469890i
$983$			3.03137e10i			1.01789i	0.860799	+	0.508946i	$0.169965\pi$
$983$			3.03137e10i			1.01789i	−0.860799	+	0.508946i	$0.830035\pi$
$984$		0			0
$985$	4.82939e8			0.0161015
$986$			3.57780e10i			1.18863i
$987$		0			0
$988$	6.15940e9	+	3.34508e9i	0.203184	+	0.110346i
$989$	−8.35730e9			−0.274713
$990$		0			0
$991$	4.08606e10			1.33367			0.666833	−	0.745207i	$-0.267649\pi$
$991$	4.08606e10			1.33367			0.666833	+	0.745207i	$0.267649\pi$
$992$	3.72917e9			0.121289
$993$		0			0
$994$		−	2.37769e10i		−	0.767897i
$995$			7.31517e8i			0.0235420i
$996$		0			0
$997$	5.95647e10			1.90351			0.951757	−	0.306853i	$-0.0992759\pi$
$997$	5.95647e10			1.90351			0.951757	+	0.306853i	$0.0992759\pi$
$998$	2.78277e10			0.886176
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	234.8.b.c.181.3		10
3.2	odd	2		26.8.b.a.25.8	yes	10
12.11	even	2		208.8.f.c.129.5		10
13.12	even	2	inner	234.8.b.c.181.8		10
39.5	even	4		338.8.a.l.1.3		5
39.8	even	4		338.8.a.k.1.3		5
39.38	odd	2		26.8.b.a.25.3	✓	10
156.155	even	2		208.8.f.c.129.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.b.a.25.3	✓	10	39.38	odd	2
26.8.b.a.25.8	yes	10	3.2	odd	2
208.8.f.c.129.5		10	12.11	even	2
208.8.f.c.129.6		10	156.155	even	2
234.8.b.c.181.3		10	1.1	even	1	trivial
234.8.b.c.181.8		10	13.12	even	2	inner
338.8.a.k.1.3		5	39.8	even	4
338.8.a.l.1.3		5	39.5	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)

\(f(q)\)	\(=\)	\(q-8.00000i q^{2} -64.0000 q^{4} +6.61448i q^{5} +1604.67i q^{7} +512.000i q^{8} +52.9158 q^{10} +1173.55i q^{11} +(-3780.46 + 6961.08i) q^{13} +12837.3 q^{14} +4096.00 q^{16} +28053.1 q^{17} +13825.5i q^{19} -423.327i q^{20} +9388.37 q^{22} -94424.9 q^{23} +78081.2 q^{25} +(55688.6 + 30243.7i) q^{26} -102699. i q^{28} -159421. q^{29} +113805. i q^{31} -32768.0i q^{32} -224425. i q^{34} -10614.0 q^{35} -402800. i q^{37} +110604. q^{38} -3386.61 q^{40} -583062. i q^{41} +88507.4 q^{43} -75107.0i q^{44} +755399. i q^{46} +112810. i q^{47} -1.75141e6 q^{49} -624650. i q^{50} +(241949. - 445509. i) q^{52} -548744. q^{53} -7762.40 q^{55} -821589. q^{56} +1.27536e6i q^{58} +1.69495e6i q^{59} +1.33588e6 q^{61} +910442. q^{62} -262144. q^{64} +(-46043.9 - 25005.8i) q^{65} -719239. i q^{67} -1.79540e6 q^{68} +84912.2i q^{70} -1.85217e6i q^{71} +1.50153e6i q^{73} -3.22240e6 q^{74} -884833. i q^{76} -1.88315e6 q^{77} -6.07672e6 q^{79} +27092.9i q^{80} -4.66450e6 q^{82} +661334. i q^{83} +185557. i q^{85} -708059. i q^{86} -600856. q^{88} -1.03091e7i q^{89} +(-1.11702e7 - 6.06637e6i) q^{91} +6.04319e6 q^{92} +902482. q^{94} -91448.6 q^{95} +6.14220e6i q^{97} +1.40113e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 640 q^{4} - 1136 q^{10} + 3432 q^{13} - 3792 q^{14} + 40960 q^{16} + 6918 q^{17} + 17280 q^{22} - 94164 q^{23} - 330788 q^{25} - 118896 q^{26} + 131304 q^{29} - 873450 q^{35} - 602976 q^{38} + 72704 q^{40}+ \cdots - 16547484 q^{95}+O(q^{100}) \) 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

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