LMFDB - Embedded newform 342.10.b.b.341.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,10,Mod(341,342)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("342.341");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	342.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$176.142255968$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			341.5
Character	$\chi$	$=$	342.341
Dual form			342.10.b.b.341.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.0000 q^{2} +256.000 q^{4} -1660.24i q^{5} +7882.01 q^{7} +4096.00 q^{8} +O(q^{10})$	\(q+16.0000 q^{2} +256.000 q^{4} -1660.24i q^{5} +7882.01 q^{7} +4096.00 q^{8} -26563.8i q^{10} -69549.0i q^{11} +59636.3i q^{13} +126112. q^{14} +65536.0 q^{16} -418288. i q^{17} +(330883. + 461741. i) q^{19} -425020. i q^{20} -1.11278e6i q^{22} -1.33436e6i q^{23} -803256. q^{25} +954181. i q^{26} +2.01779e6 q^{28} +3.33733e6 q^{29} -4.85874e6i q^{31} +1.04858e6 q^{32} -6.69261e6i q^{34} -1.30860e7i q^{35} +1.49308e7i q^{37} +(5.29412e6 + 7.38785e6i) q^{38} -6.80032e6i q^{40} +1.95404e7 q^{41} -2.80093e7 q^{43} -1.78045e7i q^{44} -2.13497e7i q^{46} -5.25184e7i q^{47} +2.17725e7 q^{49} -1.28521e7 q^{50} +1.52669e7i q^{52} +2.65773e7 q^{53} -1.15468e8 q^{55} +3.22847e7 q^{56} +5.33972e7 q^{58} -1.71504e8 q^{59} +8.93009e7 q^{61} -7.77399e7i q^{62} +1.67772e7 q^{64} +9.90104e7 q^{65} +2.80368e8i q^{67} -1.07082e8i q^{68} -2.09376e8i q^{70} +3.74217e8 q^{71} +2.68390e8 q^{73} +2.38893e8i q^{74} +(8.47060e7 + 1.18206e8i) q^{76} -5.48186e8i q^{77} -1.93042e8i q^{79} -1.08805e8i q^{80} +3.12646e8 q^{82} +7.23386e8i q^{83} -6.94457e8 q^{85} -4.48149e8 q^{86} -2.84873e8i q^{88} -8.12207e8 q^{89} +4.70054e8i q^{91} -3.41595e8i q^{92} -8.40295e8i q^{94} +(7.66598e8 - 5.49343e8i) q^{95} +1.33158e8i q^{97} +3.48360e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 480 q^{2} + 7680 q^{4} + 1596 q^{7} + 122880 q^{8}+O(q^{10}) $ 30 * q + 480 * q^2 + 7680 * q^4 + 1596 * q^7 + 122880 * q^8	$ 30 q + 480 q^{2} + 7680 q^{4} + 1596 q^{7} + 122880 q^{8} + 25536 q^{14} + 1966080 q^{16} - 323202 q^{19} - 5473830 q^{25} + 408576 q^{28} - 8475660 q^{29} + 31457280 q^{32} - 5171232 q^{38} + 20345076 q^{41} - 60729780 q^{43} + 214984890 q^{49} - 87581280 q^{50} - 181795212 q^{53} - 316864944 q^{55} + 6537216 q^{56} - 135610560 q^{58} + 197198784 q^{59} + 53410728 q^{61} + 503316480 q^{64} + 640935936 q^{65} - 335924712 q^{71} + 136407840 q^{73} - 82739712 q^{76} + 325521216 q^{82} + 269215776 q^{85} - 971676480 q^{86} - 1186853004 q^{89} + 506121804 q^{95} + 3439758240 q^{98}+O(q^{100}) $ 30 * q + 480 * q^2 + 7680 * q^4 + 1596 * q^7 + 122880 * q^8 + 25536 * q^14 + 1966080 * q^16 - 323202 * q^19 - 5473830 * q^25 + 408576 * q^28 - 8475660 * q^29 + 31457280 * q^32 - 5171232 * q^38 + 20345076 * q^41 - 60729780 * q^43 + 214984890 * q^49 - 87581280 * q^50 - 181795212 * q^53 - 316864944 * q^55 + 6537216 * q^56 - 135610560 * q^58 + 197198784 * q^59 + 53410728 * q^61 + 503316480 * q^64 + 640935936 * q^65 - 335924712 * q^71 + 136407840 * q^73 - 82739712 * q^76 + 325521216 * q^82 + 269215776 * q^85 - 971676480 * q^86 - 1186853004 * q^89 + 506121804 * q^95 + 3439758240 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$-1$	$-1$

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 $

$2$	16.0000			0.707107
$2$	16.0000			0.707107
$3$		0			0
$3$		0			0
$4$	256.000			0.500000
$5$		−	1660.24i		−	1.18797i	−0.804477	−	0.593984i	$-0.797554\pi$
$5$		−	1660.24i		−	1.18797i	0.804477	−	0.593984i	$-0.202446\pi$
$6$		0			0
$7$	7882.01			1.24078			0.620392	−	0.784292i	$-0.286974\pi$
$7$	7882.01			1.24078			0.620392	+	0.784292i	$0.286974\pi$
$8$	4096.00			0.353553
$9$		0			0
$10$		−	26563.8i		−	0.840020i
$11$		−	69549.0i		−	1.43227i	−0.697964	−	0.716133i	$-0.745910\pi$
$11$		−	69549.0i		−	1.43227i	0.697964	−	0.716133i	$-0.254090\pi$
$12$		0			0
$13$			59636.3i			0.579116i	0.957160	+	0.289558i	$0.0935084\pi$
$13$			59636.3i			0.579116i	−0.957160	+	0.289558i	$0.906492\pi$
$14$	126112.			0.877366
$15$		0			0
$16$	65536.0			0.250000
$17$		−	418288.i		−	1.21466i	−0.794449	−	0.607331i	$-0.792240\pi$
$17$		−	418288.i		−	1.21466i	0.794449	−	0.607331i	$-0.207760\pi$
$18$		0			0
$19$	330883.	+	461741.i	0.582482	+	0.812843i
$19$	330883.	+	461741.i	0.582482	+	0.812843i
$20$		−	425020.i		−	0.593984i
$21$		0			0
$22$		−	1.11278e6i		−	1.01277i
$23$		−	1.33436e6i		−	0.994252i	−0.867678	−	0.497126i	$-0.834389\pi$
$23$		−	1.33436e6i		−	0.994252i	0.867678	−	0.497126i	$-0.165611\pi$
$24$		0			0
$25$	−803256.			−0.411267
$26$			954181.i			0.409497i
$27$		0			0
$28$	2.01779e6			0.620392
$29$	3.33733e6			0.876209			0.438104	−	0.898924i	$-0.355650\pi$
$29$	3.33733e6			0.876209			0.438104	+	0.898924i	$0.355650\pi$
$30$		0			0
$31$		−	4.85874e6i		−	0.944922i	−0.881351	−	0.472461i	$-0.843366\pi$
$31$		−	4.85874e6i		−	0.944922i	0.881351	−	0.472461i	$-0.156634\pi$
$32$	1.04858e6			0.176777
$33$		0			0
$34$		−	6.69261e6i		−	0.858896i
$35$		−	1.30860e7i		−	1.47401i
$36$		0			0
$37$			1.49308e7i			1.30971i	0.755754	+	0.654856i	$0.227271\pi$
$37$			1.49308e7i			1.30971i	−0.755754	+	0.654856i	$0.772729\pi$
$38$	5.29412e6	+	7.38785e6i	0.411877	+	0.574767i
$39$		0			0
$40$		−	6.80032e6i		−	0.420010i
$41$	1.95404e7			1.07996			0.539978	−	0.841679i	$-0.318433\pi$
$41$	1.95404e7			1.07996			0.539978	+	0.841679i	$0.318433\pi$
$42$		0			0
$43$	−2.80093e7			−1.24938			−0.624690	−	0.780873i	$-0.714775\pi$
$43$	−2.80093e7			−1.24938			−0.624690	+	0.780873i	$0.714775\pi$
$44$		−	1.78045e7i		−	0.716133i
$45$		0			0
$46$		−	2.13497e7i		−	0.703042i
$47$		−	5.25184e7i		−	1.56990i	−0.619560	−	0.784949i	$-0.712689\pi$
$47$		−	5.25184e7i		−	1.56990i	0.619560	−	0.784949i	$-0.287311\pi$
$48$		0			0
$49$	2.17725e7			0.539542
$50$	−1.28521e7			−0.290810
$51$		0			0
$52$			1.52669e7i			0.289558i
$53$	2.65773e7			0.462668			0.231334	−	0.972874i	$-0.425691\pi$
$53$	2.65773e7			0.462668			0.231334	+	0.972874i	$0.425691\pi$
$54$		0			0
$55$	−1.15468e8			−1.70149
$56$	3.22847e7			0.438683
$57$		0			0
$58$	5.33972e7			0.619573
$59$	−1.71504e8			−1.84264			−0.921318	−	0.388809i	$-0.872887\pi$
$59$	−1.71504e8			−1.84264			−0.921318	+	0.388809i	$0.872887\pi$
$60$		0			0
$61$	8.93009e7			0.825794			0.412897	−	0.910778i	$-0.364517\pi$
$61$	8.93009e7			0.825794			0.412897	+	0.910778i	$0.364517\pi$
$62$		−	7.77399e7i		−	0.668161i
$63$		0			0
$64$	1.67772e7			0.125000
$65$	9.90104e7			0.687972
$66$		0			0
$67$			2.80368e8i			1.69977i	0.526965	+	0.849887i	$0.323330\pi$
$67$			2.80368e8i			1.69977i	−0.526965	+	0.849887i	$0.676670\pi$
$68$		−	1.07082e8i		−	0.607331i
$69$		0			0
$70$		−	2.09376e8i		−	1.04228i
$71$	3.74217e8			1.74768			0.873839	−	0.486216i	$-0.161623\pi$
$71$	3.74217e8			1.74768			0.873839	+	0.486216i	$0.161623\pi$
$72$		0			0
$73$	2.68390e8			1.10615			0.553073	−	0.833133i	$-0.313455\pi$
$73$	2.68390e8			1.10615			0.553073	+	0.833133i	$0.313455\pi$
$74$			2.38893e8i			0.926106i
$75$		0			0
$76$	8.47060e7	+	1.18206e8i	0.291241	+	0.406422i
$77$		−	5.48186e8i		−	1.77713i
$78$		0			0
$79$		−	1.93042e8i		−	0.557610i	−0.960348	−	0.278805i	$-0.910062\pi$
$79$		−	1.93042e8i		−	0.557610i	0.960348	−	0.278805i	$-0.0899384\pi$
$80$		−	1.08805e8i		−	0.296992i
$81$		0			0
$82$	3.12646e8			0.763644
$83$			7.23386e8i			1.67309i	0.547900	+	0.836544i	$0.315428\pi$
$83$			7.23386e8i			1.67309i	−0.547900	+	0.836544i	$0.684572\pi$
$84$		0			0
$85$	−6.94457e8			−1.44298
$86$	−4.48149e8			−0.883446
$87$		0			0
$88$		−	2.84873e8i		−	0.506383i
$89$	−8.12207e8			−1.37218			−0.686091	−	0.727515i	$-0.740675\pi$
$89$	−8.12207e8			−1.37218			−0.686091	+	0.727515i	$0.740675\pi$
$90$		0			0
$91$			4.70054e8i			0.718558i
$92$		−	3.41595e8i		−	0.497126i
$93$		0			0
$94$		−	8.40295e8i		−	1.11009i
$95$	7.66598e8	−	5.49343e8i	0.965632	−	0.691970i
$96$		0			0
$97$			1.33158e8i			0.152720i	0.997080	+	0.0763599i	$0.0243298\pi$
$97$			1.33158e8i			0.152720i	−0.997080	+	0.0763599i	$0.975670\pi$
$98$	3.48360e8			0.381514
$99$		0			0
$100$	−2.05634e8			−0.205634
$101$			1.55508e9i			1.48698i	0.668745	+	0.743492i	$0.266832\pi$
$101$			1.55508e9i			1.48698i	−0.668745	+	0.743492i	$0.733168\pi$
$102$		0			0
$103$		−	1.32756e9i		−	1.16222i	−0.813825	−	0.581110i	$-0.802619\pi$
$103$		−	1.32756e9i		−	1.16222i	0.813825	−	0.581110i	$-0.197381\pi$
$104$			2.44270e8i			0.204749i
$105$		0			0
$106$	4.25237e8			0.327156
$107$	−2.23954e9			−1.65170			−0.825852	−	0.563886i	$-0.809306\pi$
$107$	−2.23954e9			−1.65170			−0.825852	+	0.563886i	$0.809306\pi$
$108$		0			0
$109$		−	2.23109e9i		−	1.51390i	−0.653472	−	0.756950i	$-0.726688\pi$
$109$		−	2.23109e9i		−	1.51390i	0.653472	−	0.756950i	$-0.273312\pi$
$110$	−1.84748e9			−1.20313
$111$		0			0
$112$	5.16555e8			0.310196
$113$	−2.89472e9			−1.67014			−0.835072	−	0.550140i	$-0.814574\pi$
$113$	−2.89472e9			−1.67014			−0.835072	+	0.550140i	$0.814574\pi$
$114$		0			0
$115$	−2.21535e9			−1.18114
$116$	8.54355e8			0.438104
$117$		0			0
$118$	−2.74406e9			−1.30294
$119$		−	3.29695e9i		−	1.50713i
$120$		0			0
$121$	−2.47912e9			−1.05139
$122$	1.42881e9			0.583924
$123$		0			0
$124$		−	1.24384e9i		−	0.472461i
$125$		−	1.90905e9i		−	0.699396i
$126$		0			0
$127$			2.49140e9i			0.849820i	0.905236	+	0.424910i	$0.139694\pi$
$127$			2.49140e9i			0.849820i	−0.905236	+	0.424910i	$0.860306\pi$
$128$	2.68435e8			0.0883883
$129$		0			0
$130$	1.58417e9			0.486469
$131$			9.80653e7i			0.0290934i	0.999894	+	0.0145467i	$0.00463052\pi$
$131$			9.80653e7i			0.0290934i	−0.999894	+	0.0145467i	$0.995369\pi$
$132$		0			0
$133$	2.60802e9	+	3.63944e9i	0.722734	+	1.00856i
$134$			4.48588e9i			1.20192i
$135$		0			0
$136$		−	1.71331e9i		−	0.429448i
$137$		−	5.71246e9i		−	1.38542i	−0.721217	−	0.692709i	$-0.756417\pi$
$137$		−	5.71246e9i		−	1.38542i	0.721217	−	0.692709i	$-0.243583\pi$
$138$		0			0
$139$	5.65004e9			1.28376			0.641882	−	0.766803i	$-0.278154\pi$
$139$	5.65004e9			1.28376			0.641882	+	0.766803i	$0.278154\pi$
$140$		−	3.35001e9i		−	0.737005i
$141$		0			0
$142$	5.98748e9			1.23579
$143$	4.14765e9			0.829449
$144$		0			0
$145$		−	5.54074e9i		−	1.04091i
$146$	4.29423e9			0.782164
$147$		0			0
$148$			3.82229e9i			0.654856i
$149$			2.86877e9i			0.476824i	0.971164	+	0.238412i	$0.0766268\pi$
$149$			2.86877e9i			0.476824i	−0.971164	+	0.238412i	$0.923373\pi$
$150$		0			0
$151$		−	3.64369e9i		−	0.570355i	−0.958475	−	0.285178i	$-0.907947\pi$
$151$		−	3.64369e9i		−	0.570355i	0.958475	−	0.285178i	$-0.0920526\pi$
$152$	1.35530e9	+	1.89129e9i	0.205939	+	0.287384i
$153$		0			0
$154$		−	8.77097e9i		−	1.25662i
$155$	−8.06665e9			−1.12254
$156$		0			0
$157$	−1.42916e9			−0.187729			−0.0938645	−	0.995585i	$-0.529922\pi$
$157$	−1.42916e9			−0.187729			−0.0938645	+	0.995585i	$0.529922\pi$
$158$		−	3.08868e9i		−	0.394290i
$159$		0			0
$160$		−	1.74088e9i		−	0.210005i
$161$		−	1.05174e10i		−	1.23365i
$162$		0			0
$163$	6.81464e9			0.756134			0.378067	−	0.925778i	$-0.376589\pi$
$163$	6.81464e9			0.756134			0.378067	+	0.925778i	$0.376589\pi$
$164$	5.00234e9			0.539978
$165$		0			0
$166$			1.15742e10i			1.18305i
$167$	−3.86855e8			−0.0384879			−0.0192440	−	0.999815i	$-0.506126\pi$
$167$	−3.86855e8			−0.0384879			−0.0192440	+	0.999815i	$0.506126\pi$
$168$		0			0
$169$	7.04801e9			0.664624
$170$	−1.11113e10			−1.02034
$171$		0			0
$172$	−7.17039e9			−0.624690
$173$	−1.35644e10			−1.15131			−0.575655	−	0.817693i	$-0.695253\pi$
$173$	−1.35644e10			−1.15131			−0.575655	+	0.817693i	$0.695253\pi$
$174$		0			0
$175$	−6.33127e9			−0.510293
$176$		−	4.55796e9i		−	0.358067i
$177$		0			0
$178$	−1.29953e10			−0.970280
$179$	−8.93717e9			−0.650671			−0.325336	−	0.945599i	$-0.605477\pi$
$179$	−8.93717e9			−0.650671			−0.325336	+	0.945599i	$0.605477\pi$
$180$		0			0
$181$		−	1.29285e10i		−	0.895356i	−0.894195	−	0.447678i	$-0.852251\pi$
$181$		−	1.29285e10i		−	0.895356i	0.894195	−	0.447678i	$-0.147749\pi$
$182$			7.52087e9i			0.508097i
$183$		0			0
$184$		−	5.46552e9i		−	0.351521i
$185$	2.47887e10			1.55590
$186$		0			0
$187$	−2.90915e10			−1.73972
$188$		−	1.34447e10i		−	0.784949i
$189$		0			0
$190$	1.22656e10	−	8.78949e9i	0.682805	−	0.489297i
$191$		−	2.32910e10i		−	1.26630i	−0.774028	−	0.633152i	$-0.781761\pi$
$191$		−	2.32910e10i		−	1.26630i	0.774028	−	0.633152i	$-0.218239\pi$
$192$		0			0
$193$			9.59216e9i			0.497632i	0.968551	+	0.248816i	$0.0800415\pi$
$193$			9.59216e9i			0.497632i	−0.968551	+	0.248816i	$0.919959\pi$
$194$			2.13053e9i			0.107989i
$195$		0			0
$196$	5.57376e9			0.269771
$197$		−	1.06003e10i		−	0.501442i	−0.968059	−	0.250721i	$-0.919332\pi$
$197$		−	1.06003e10i		−	0.501442i	0.968059	−	0.250721i	$-0.0806676\pi$
$198$		0			0
$199$	−2.78787e10			−1.26018			−0.630091	−	0.776521i	$-0.716983\pi$
$199$	−2.78787e10			−1.26018			−0.630091	+	0.776521i	$0.716983\pi$
$200$	−3.29014e9			−0.145405
$201$		0			0
$202$			2.48813e10i			1.05146i
$203$	2.63048e10			1.08718
$204$		0			0
$205$		−	3.24416e10i		−	1.28295i
$206$		−	2.12410e10i		−	0.821813i
$207$		0			0
$208$			3.90833e9i			0.144779i
$209$	3.21136e10	−	2.30126e10i	1.16421	−	0.834270i
$210$		0			0
$211$		−	4.00084e10i		−	1.38957i	−0.719217	−	0.694785i	$-0.755499\pi$
$211$		−	4.00084e10i		−	1.38957i	0.719217	−	0.694785i	$-0.244501\pi$
$212$	6.80379e9			0.231334
$213$		0			0
$214$	−3.58327e10			−1.16793
$215$			4.65021e10i			1.48422i
$216$		0			0
$217$		−	3.82967e10i		−	1.17244i
$218$		−	3.56974e10i		−	1.07049i
$219$		0			0
$220$	−2.95597e10			−0.850743
$221$	2.49452e10			0.703431
$222$		0			0
$223$		−	1.88860e10i		−	0.511410i	−0.966755	−	0.255705i	$-0.917692\pi$
$223$		−	1.88860e10i		−	0.511410i	0.966755	−	0.255705i	$-0.0823075\pi$
$224$	8.26489e9			0.219342
$225$		0			0
$226$	−4.63156e10			−1.18097
$227$	6.47925e10			1.61960			0.809801	−	0.586704i	$-0.199575\pi$
$227$	6.47925e10			1.61960			0.809801	+	0.586704i	$0.199575\pi$
$228$		0			0
$229$	−1.01202e10			−0.243182			−0.121591	−	0.992580i	$-0.538800\pi$
$229$	−1.01202e10			−0.243182			−0.121591	+	0.992580i	$0.538800\pi$
$230$	−3.54455e10			−0.835192
$231$		0			0
$232$	1.36697e10			0.309787
$233$			5.95354e10i			1.32335i	0.749792	+	0.661674i	$0.230154\pi$
$233$			5.95354e10i			1.32335i	−0.749792	+	0.661674i	$0.769846\pi$
$234$		0			0
$235$	−8.71930e10			−1.86499
$236$	−4.39050e10			−0.921318
$237$		0			0
$238$		−	5.27512e10i		−	1.06570i
$239$			1.97579e10i			0.391697i	0.980634	+	0.195849i	$0.0627461\pi$
$239$			1.97579e10i			0.391697i	−0.980634	+	0.195849i	$0.937254\pi$
$240$		0			0
$241$		−	2.81016e9i		−	0.0536604i	−0.999640	−	0.0268302i	$-0.991459\pi$
$241$		−	2.81016e9i		−	0.0536604i	0.999640	−	0.0268302i	$-0.00854135\pi$
$242$	−3.96658e10			−0.743443
$243$		0			0
$244$	2.28610e10			0.412897
$245$		−	3.61474e10i		−	0.640959i
$246$		0			0
$247$	−2.75365e10	+	1.97326e10i	−0.470731	+	0.337325i
$248$		−	1.99014e10i		−	0.334081i
$249$		0			0
$250$		−	3.05448e10i		−	0.494547i
$251$			1.68553e10i			0.268043i	0.990978	+	0.134022i	$0.0427891\pi$
$251$			1.68553e10i			0.268043i	−0.990978	+	0.134022i	$0.957211\pi$
$252$		0			0
$253$	−9.28031e10			−1.42403
$254$			3.98624e10i			0.600914i
$255$		0			0
$256$	4.29497e9			0.0625000
$257$	1.02697e11			1.46845			0.734226	−	0.678905i	$-0.237545\pi$
$257$	1.02697e11			1.46845			0.734226	+	0.678905i	$0.237545\pi$
$258$		0			0
$259$			1.17685e11i			1.62507i
$260$	2.53467e10			0.343986
$261$		0			0
$262$			1.56904e9i			0.0205721i
$263$		−	1.09865e11i		−	1.41599i	−0.706218	−	0.707995i	$-0.749600\pi$
$263$		−	1.09865e11i		−	1.41599i	0.706218	−	0.707995i	$-0.250400\pi$
$264$		0			0
$265$		−	4.41246e10i		−	0.549635i
$266$	4.17283e10	+	5.82311e10i	0.511050	+	0.713161i
$267$		0			0
$268$			7.17741e10i			0.849887i
$269$	−1.33471e9			−0.0155418			−0.00777091	−	0.999970i	$-0.502474\pi$
$269$	−1.33471e9			−0.0155418			−0.00777091	+	0.999970i	$0.502474\pi$
$270$		0			0
$271$	3.99503e10			0.449943			0.224972	−	0.974365i	$-0.427771\pi$
$271$	3.99503e10			0.449943			0.224972	+	0.974365i	$0.427771\pi$
$272$		−	2.74129e10i		−	0.303666i
$273$		0			0
$274$		−	9.13994e10i		−	0.979638i
$275$			5.58657e10i			0.589044i
$276$		0			0
$277$	5.61430e10			0.572976			0.286488	−	0.958084i	$-0.407512\pi$
$277$	5.61430e10			0.572976			0.286488	+	0.958084i	$0.407512\pi$
$278$	9.04007e10			0.907758
$279$		0			0
$280$		−	5.36002e10i		−	0.521141i
$281$	−1.76242e11			−1.68629			−0.843143	−	0.537690i	$-0.819297\pi$
$281$	−1.76242e11			−1.68629			−0.843143	+	0.537690i	$0.819297\pi$
$282$		0			0
$283$	−1.53547e11			−1.42299			−0.711496	−	0.702690i	$-0.751982\pi$
$283$	−1.53547e11			−1.42299			−0.711496	+	0.702690i	$0.751982\pi$
$284$	9.57996e10			0.873839
$285$		0			0
$286$	6.63624e10			0.586509
$287$	1.54018e11			1.33999
$288$		0			0
$289$	−5.63772e10			−0.475404
$290$		−	8.86519e10i		−	0.736033i
$291$		0			0
$292$	6.87077e10			0.553073
$293$	1.20554e10			0.0955601			0.0477801	−	0.998858i	$-0.484785\pi$
$293$	1.20554e10			0.0955601			0.0477801	+	0.998858i	$0.484785\pi$
$294$		0			0
$295$			2.84737e11i			2.18899i
$296$			6.11566e10i			0.463053i
$297$		0			0
$298$			4.59004e10i			0.337165i
$299$	7.95761e10			0.575788
$300$		0			0
$301$	−2.20770e11			−1.55021
$302$		−	5.82991e10i		−	0.403302i
$303$		0			0
$304$	2.16847e10	+	3.02606e10i	0.145621	+	0.203211i
$305$		−	1.48260e11i		−	0.981016i
$306$		0			0
$307$		−	8.17799e10i		−	0.525441i	−0.964872	−	0.262720i	$-0.915380\pi$
$307$		−	8.17799e10i		−	0.525441i	0.964872	−	0.262720i	$-0.0846197\pi$
$308$		−	1.40336e11i		−	0.888566i
$309$		0			0
$310$	−1.29066e11			−0.793754
$311$		−	1.47749e11i		−	0.895578i	−0.894139	−	0.447789i	$-0.852212\pi$
$311$		−	1.47749e11i		−	0.895578i	0.894139	−	0.447789i	$-0.147788\pi$
$312$		0			0
$313$	1.36795e11			0.805603			0.402801	−	0.915287i	$-0.368037\pi$
$313$	1.36795e11			0.805603			0.402801	+	0.915287i	$0.368037\pi$
$314$	−2.28665e10			−0.132744
$315$		0			0
$316$		−	4.94189e10i		−	0.278805i
$317$	−6.96586e8			−0.00387443			−0.00193722	−	0.999998i	$-0.500617\pi$
$317$	−6.96586e8			−0.00387443			−0.00193722	+	0.999998i	$0.500617\pi$
$318$		0			0
$319$		−	2.32108e11i		−	1.25496i
$320$		−	2.78541e10i		−	0.148496i
$321$		0			0
$322$		−	1.68279e11i		−	0.872323i
$323$	1.93141e11	−	1.38404e11i	0.987330	−	0.707519i
$324$		0			0
$325$		−	4.79033e10i		−	0.238172i
$326$	1.09034e11			0.534668
$327$		0			0
$328$	8.00374e10			0.381822
$329$		−	4.13951e11i		−	1.94790i
$330$		0			0
$331$		−	2.23903e11i		−	1.02526i	−0.858610	−	0.512629i	$-0.828672\pi$
$331$		−	2.23903e11i		−	1.02526i	0.858610	−	0.512629i	$-0.171328\pi$
$332$			1.85187e11i			0.836544i
$333$		0			0
$334$	−6.18969e9			−0.0272151
$335$	4.65476e11			2.01928
$336$		0			0
$337$			1.88420e10i			0.0795779i	0.999208	+	0.0397889i	$0.0126686\pi$
$337$			1.88420e10i			0.0795779i	−0.999208	+	0.0397889i	$0.987331\pi$
$338$	1.12768e11			0.469960
$339$		0			0
$340$	−1.77781e11			−0.721490
$341$	−3.37921e11			−1.35338
$342$		0			0
$343$	−1.46457e11			−0.571328
$344$	−1.14726e11			−0.441723
$345$		0			0
$346$	−2.17030e11			−0.814099
$347$			4.13924e11i			1.53263i	0.642464	+	0.766316i	$0.277912\pi$
$347$			4.13924e11i			1.53263i	−0.642464	+	0.766316i	$0.722088\pi$
$348$		0			0
$349$	−5.01939e11			−1.81108			−0.905538	−	0.424265i	$-0.860532\pi$
$349$	−5.01939e11			−1.81108			−0.905538	+	0.424265i	$0.860532\pi$
$350$	−1.01300e11			−0.360832
$351$		0			0
$352$		−	7.29274e10i		−	0.253191i
$353$			1.80801e11i			0.619746i	0.950778	+	0.309873i	$0.100287\pi$
$353$			1.80801e11i			0.619746i	−0.950778	+	0.309873i	$0.899713\pi$
$354$		0			0
$355$		−	6.21289e11i		−	2.07618i
$356$	−2.07925e11			−0.686091
$357$		0			0
$358$	−1.42995e11			−0.460094
$359$			3.64654e11i			1.15866i	0.815093	+	0.579330i	$0.196686\pi$
$359$			3.64654e11i			1.15866i	−0.815093	+	0.579330i	$0.803314\pi$
$360$		0			0
$361$	−1.03721e11	+	3.05564e11i	−0.321429	+	0.946934i
$362$		−	2.06857e11i		−	0.633112i
$363$		0			0
$364$			1.20334e11i			0.359279i
$365$		−	4.45590e11i		−	1.31407i
$366$		0			0
$367$	3.12885e11			0.900299			0.450150	−	0.892953i	$-0.351371\pi$
$367$	3.12885e11			0.900299			0.450150	+	0.892953i	$0.351371\pi$
$368$		−	8.74484e10i		−	0.248563i
$369$		0			0
$370$	3.96619e11			1.10018
$371$	2.09483e11			0.574071
$372$		0			0
$373$		−	1.82387e11i		−	0.487869i	−0.969792	−	0.243935i	$-0.921562\pi$
$373$		−	1.82387e11i		−	0.487869i	0.969792	−	0.243935i	$-0.0784382\pi$
$374$	−4.65464e11			−1.23017
$375$		0			0
$376$		−	2.15116e11i		−	0.555043i
$377$			1.99026e11i			0.507427i
$378$		0			0
$379$			5.01754e11i			1.24915i	0.780965	+	0.624574i	$0.214728\pi$
$379$			5.01754e11i			1.24915i	−0.780965	+	0.624574i	$0.785272\pi$
$380$	1.96249e11	−	1.40632e11i	0.482816	−	0.345985i
$381$		0			0
$382$		−	3.72656e11i		−	0.895412i
$383$	−1.89262e11			−0.449438			−0.224719	−	0.974424i	$-0.572146\pi$
$383$	−1.89262e11			−0.449438			−0.224719	+	0.974424i	$0.572146\pi$
$384$		0			0
$385$	−9.10118e11			−2.11118
$386$			1.53475e11i			0.351879i
$387$		0			0
$388$			3.40885e10i			0.0763599i
$389$		−	1.67228e11i		−	0.370284i	−0.982712	−	0.185142i	$-0.940725\pi$
$389$		−	1.67228e11i		−	0.370284i	0.982712	−	0.185142i	$-0.0592746\pi$
$390$		0			0
$391$	−5.58146e11			−1.20768
$392$	8.91801e10			0.190757
$393$		0			0
$394$		−	1.69605e11i		−	0.354573i
$395$	−3.20496e11			−0.662423
$396$		0			0
$397$	1.72557e11			0.348638			0.174319	−	0.984689i	$-0.444228\pi$
$397$	1.72557e11			0.348638			0.174319	+	0.984689i	$0.444228\pi$
$398$	−4.46059e11			−0.891084
$399$		0			0
$400$	−5.26422e10			−0.102817
$401$	−5.95715e11			−1.15051			−0.575253	−	0.817975i	$-0.695096\pi$
$401$	−5.95715e11			−1.15051			−0.575253	+	0.817975i	$0.695096\pi$
$402$		0			0
$403$	2.89758e11			0.547220
$404$			3.98100e11i			0.743492i
$405$		0			0
$406$	4.20877e11			0.768756
$407$	1.03842e12			1.87586
$408$		0			0
$409$			3.08996e11i			0.546007i	0.962013	+	0.273003i	$0.0880170\pi$
$409$			3.08996e11i			0.546007i	−0.962013	+	0.273003i	$0.911983\pi$
$410$		−	5.19066e11i		−	0.907184i
$411$		0			0
$412$		−	3.39857e11i		−	0.581110i
$413$	−1.35179e12			−2.28631
$414$		0			0
$415$	1.20099e12			1.98757
$416$			6.25332e10i			0.102374i
$417$		0			0
$418$	5.13817e11	−	3.68201e11i	0.823219	−	0.589918i
$419$			5.19086e11i			0.822765i	0.911463	+	0.411382i	$0.134954\pi$
$419$			5.19086e11i			0.822765i	−0.911463	+	0.411382i	$0.865046\pi$
$420$		0			0
$421$			6.17149e11i			0.957460i	0.877962	+	0.478730i	$0.158903\pi$
$421$			6.17149e11i			0.957460i	−0.877962	+	0.478730i	$0.841097\pi$
$422$		−	6.40135e11i		−	0.982575i
$423$		0			0
$424$	1.08861e11			0.163578
$425$			3.35993e11i			0.499551i
$426$		0			0
$427$	7.03870e11			1.02463
$428$	−5.73323e11			−0.825852
$429$		0			0
$430$			7.44033e11i			1.04950i
$431$	7.78680e11			1.08695			0.543477	−	0.839424i	$-0.317107\pi$
$431$	7.78680e11			1.08695			0.543477	+	0.839424i	$0.317107\pi$
$432$		0			0
$433$			4.64020e11i			0.634368i	0.948364	+	0.317184i	$0.102737\pi$
$433$			4.64020e11i			0.634368i	−0.948364	+	0.317184i	$0.897263\pi$
$434$		−	6.12746e11i		−	0.829043i
$435$		0			0
$436$		−	5.71159e11i		−	0.756950i
$437$	6.16126e11	−	4.41515e11i	0.808171	−	0.579134i
$438$		0			0
$439$		−	1.21810e12i		−	1.56528i	−0.622473	−	0.782641i	$-0.713872\pi$
$439$		−	1.21810e12i		−	1.56528i	0.622473	−	0.782641i	$-0.286128\pi$
$440$	−4.72956e11			−0.601566
$441$		0			0
$442$	3.99123e11			0.497401
$443$			7.60818e11i			0.938564i	0.883048	+	0.469282i	$0.155487\pi$
$443$			7.60818e11i			0.938564i	−0.883048	+	0.469282i	$0.844513\pi$
$444$		0			0
$445$			1.34846e12i			1.63011i
$446$		−	3.02177e11i		−	0.361621i
$447$		0			0
$448$	1.32238e11			0.155098
$449$	2.43453e11			0.282687			0.141344	−	0.989961i	$-0.454858\pi$
$449$	2.43453e11			0.282687			0.141344	+	0.989961i	$0.454858\pi$
$450$		0			0
$451$		−	1.35901e12i		−	1.54678i
$452$	−7.41049e11			−0.835072
$453$		0			0
$454$	1.03668e12			1.14523
$455$	7.80401e11			0.853623
$456$		0			0
$457$	1.74520e12			1.87164			0.935820	−	0.352478i	$-0.114661\pi$
$457$	1.74520e12			1.87164			0.935820	+	0.352478i	$0.114661\pi$
$458$	−1.61924e11			−0.171956
$459$		0			0
$460$	−5.67128e11			−0.590570
$461$		−	6.56734e11i		−	0.677229i	−0.940925	−	0.338614i	$-0.890042\pi$
$461$		−	6.56734e11i		−	0.677229i	0.940925	−	0.338614i	$-0.109958\pi$
$462$		0			0
$463$	−1.17383e12			−1.18711			−0.593555	−	0.804794i	$-0.702276\pi$
$463$	−1.17383e12			−1.18711			−0.593555	+	0.804794i	$0.702276\pi$
$464$	2.18715e11			0.219052
$465$		0			0
$466$			9.52567e11i			0.935748i
$467$			1.90738e11i			0.185572i	0.995686	+	0.0927860i	$0.0295772\pi$
$467$			1.90738e11i			0.185572i	−0.995686	+	0.0927860i	$0.970423\pi$
$468$		0			0
$469$			2.20986e12i			2.10905i
$470$	−1.39509e12			−1.31875
$471$		0			0
$472$	−7.02480e11			−0.651470
$473$			1.94802e12i			1.78945i
$474$		0			0
$475$	−2.65784e11	−	3.70896e11i	−0.239556	−	0.334296i
$476$		−	8.44020e11i		−	0.753566i
$477$		0			0
$478$			3.16127e11i			0.276972i
$479$			2.28587e11i			0.198400i	0.995067	+	0.0992002i	$0.0316284\pi$
$479$			2.28587e11i			0.198400i	−0.995067	+	0.0992002i	$0.968372\pi$
$480$		0			0
$481$	−8.90419e11			−0.758475
$482$		−	4.49625e10i		−	0.0379437i
$483$		0			0
$484$	−6.34653e11			−0.525693
$485$	2.21074e11			0.181426
$486$		0			0
$487$			1.17738e12i			0.948501i	0.880390	+	0.474250i	$0.157281\pi$
$487$			1.17738e12i			0.948501i	−0.880390	+	0.474250i	$0.842719\pi$
$488$	3.65776e11			0.291962
$489$		0			0
$490$		−	5.78359e11i		−	0.453226i
$491$		−	5.53397e10i		−	0.0429705i	−0.999769	−	0.0214852i	$-0.993161\pi$
$491$		−	5.53397e10i		−	0.0429705i	0.999769	−	0.0214852i	$-0.00683949\pi$
$492$		0			0
$493$		−	1.39596e12i		−	1.06430i
$494$	−4.40584e11	+	3.15722e11i	−0.332857	+	0.238525i
$495$		0			0
$496$		−	3.18422e11i		−	0.236231i
$497$	2.94958e12			2.16849
$498$		0			0
$499$	1.76482e12			1.27423			0.637114	−	0.770770i	$-0.280128\pi$
$499$	1.76482e12			1.27423			0.637114	+	0.770770i	$0.280128\pi$
$500$		−	4.88717e11i		−	0.349698i
$501$		0			0
$502$			2.69685e11i			0.189535i
$503$			2.44143e12i			1.70054i	0.526344	+	0.850271i	$0.323562\pi$
$503$			2.44143e12i			1.70054i	−0.526344	+	0.850271i	$0.676438\pi$
$504$		0			0
$505$	2.58180e12			1.76649
$506$	−1.48485e12			−1.00694
$507$		0			0
$508$			6.37799e11i			0.424910i
$509$	6.02187e11			0.397650			0.198825	−	0.980035i	$-0.436287\pi$
$509$	6.02187e11			0.397650			0.198825	+	0.980035i	$0.436287\pi$
$510$		0			0
$511$	2.11545e12			1.37249
$512$	6.87195e10			0.0441942
$513$		0			0
$514$	1.64316e12			1.03835
$515$	−2.20407e12			−1.38068
$516$		0			0
$517$	−3.65260e12			−2.24851
$518$			1.88296e12i			1.14910i
$519$		0			0
$520$	4.05546e11			0.243235
$521$	1.58804e12			0.944262			0.472131	−	0.881528i	$-0.343485\pi$
$521$	1.58804e12			0.944262			0.472131	+	0.881528i	$0.343485\pi$
$522$		0			0
$523$			3.22023e12i			1.88204i	0.338349	+	0.941021i	$0.390132\pi$
$523$			3.22023e12i			1.88204i	−0.338349	+	0.941021i	$0.609868\pi$
$524$			2.51047e10i			0.0145467i
$525$		0			0
$526$		−	1.75785e12i		−	1.00126i
$527$	−2.03235e12			−1.14776
$528$		0			0
$529$	2.06460e10			0.0114627
$530$		−	7.05994e11i		−	0.388651i
$531$		0			0
$532$	6.67653e11	+	9.31698e11i	0.361367	+	0.504281i
$533$			1.16532e12i			0.625420i
$534$		0			0
$535$			3.71817e12i			1.96217i
$536$			1.14839e12i			0.600961i
$537$		0			0
$538$	−2.13554e10			−0.0109897
$539$		−	1.51425e12i		−	0.772768i
$540$		0			0
$541$	3.21924e12			1.61572			0.807858	−	0.589377i	$-0.200627\pi$
$541$	3.21924e12			1.61572			0.807858	+	0.589377i	$0.200627\pi$
$542$	6.39204e11			0.318158
$543$		0			0
$544$		−	4.38607e11i		−	0.214724i
$545$	−3.70413e12			−1.79847
$546$		0			0
$547$			2.05525e12i			0.981571i	0.871280	+	0.490785i	$0.163290\pi$
$547$			2.05525e12i			0.981571i	−0.871280	+	0.490785i	$0.836710\pi$
$548$		−	1.46239e12i		−	0.692709i
$549$		0			0
$550$			8.93851e11i			0.416517i
$551$	1.10426e12	+	1.54098e12i	0.510376	+	0.712220i
$552$		0			0
$553$		−	1.52156e12i		−	0.691874i
$554$	8.98288e11			0.405155
$555$		0			0
$556$	1.44641e12			0.641882
$557$			1.20957e12i			0.532454i	0.963910	+	0.266227i	$0.0857771\pi$
$557$			1.20957e12i			0.532454i	−0.963910	+	0.266227i	$0.914223\pi$
$558$		0			0
$559$		−	1.67037e12i		−	0.723537i
$560$		−	8.57604e11i		−	0.368503i
$561$		0			0
$562$	−2.81987e12			−1.19238
$563$	−1.19326e11			−0.0500552			−0.0250276	−	0.999687i	$-0.507967\pi$
$563$	−1.19326e11			−0.0500552			−0.0250276	+	0.999687i	$0.507967\pi$
$564$		0			0
$565$			4.80592e12i			1.98408i
$566$	−2.45675e12			−1.00621
$567$		0			0
$568$	1.53279e12			0.617897
$569$	3.45382e12			1.38132			0.690661	−	0.723179i	$-0.257320\pi$
$569$	3.45382e12			1.38132			0.690661	+	0.723179i	$0.257320\pi$
$570$		0			0
$571$	−3.02014e12			−1.18895			−0.594476	−	0.804113i	$-0.702641\pi$
$571$	−3.02014e12			−1.18895			−0.594476	+	0.804113i	$0.702641\pi$
$572$	1.06180e12			0.414724
$573$		0			0
$574$	2.46428e12			0.947516
$575$			1.07183e12i			0.408903i
$576$		0			0
$577$	−4.79592e11			−0.180128			−0.0900638	−	0.995936i	$-0.528707\pi$
$577$	−4.79592e11			−0.180128			−0.0900638	+	0.995936i	$0.528707\pi$
$578$	−9.02035e11			−0.336162
$579$		0			0
$580$		−	1.41843e12i		−	0.520454i
$581$			5.70174e12i			2.07594i
$582$		0			0
$583$		−	1.84843e12i		−	0.662664i
$584$	1.09932e12			0.391082
$585$		0			0
$586$	1.92886e11			0.0675712
$587$			1.43217e12i			0.497877i	0.968519	+	0.248939i	$0.0800818\pi$
$587$			1.43217e12i			0.497877i	−0.968519	+	0.248939i	$0.919918\pi$
$588$		0			0
$589$	2.24348e12	−	1.60767e12i	0.768074	−	0.550401i
$590$			4.55579e12i			1.54785i
$591$		0			0
$592$			9.78506e11i			0.327428i
$593$			1.16296e12i			0.386204i	0.981179	+	0.193102i	$0.0618548\pi$
$593$			1.16296e12i			0.386204i	−0.981179	+	0.193102i	$0.938145\pi$
$594$		0			0
$595$	−5.47372e12			−1.79042
$596$			7.34406e11i			0.238412i
$597$		0			0
$598$	1.27322e12			0.407143
$599$	−3.35772e12			−1.06567			−0.532837	−	0.846218i	$-0.678874\pi$
$599$	−3.35772e12			−1.06567			−0.532837	+	0.846218i	$0.678874\pi$
$600$		0			0
$601$			1.62837e12i			0.509117i	0.967057	+	0.254558i	$0.0819301\pi$
$601$			1.62837e12i			0.509117i	−0.967057	+	0.254558i	$0.918070\pi$
$602$	−3.53232e12			−1.09616
$603$		0			0
$604$		−	9.32785e11i		−	0.285178i
$605$			4.11591e12i			1.24901i
$606$		0			0
$607$			3.21302e12i			0.960649i	0.877091	+	0.480324i	$0.159481\pi$
$607$			3.21302e12i			0.960649i	−0.877091	+	0.480324i	$0.840519\pi$
$608$	3.46956e11	+	4.84170e11i	0.102969	+	0.143692i
$609$		0			0
$610$		−	2.37217e12i		−	0.693683i
$611$	3.13201e12			0.909154
$612$		0			0
$613$	−3.53747e12			−1.01186			−0.505931	−	0.862574i	$-0.668851\pi$
$613$	−3.53747e12			−1.01186			−0.505931	+	0.862574i	$0.668851\pi$
$614$		−	1.30848e12i		−	0.371543i
$615$		0			0
$616$		−	2.24537e12i		−	0.628311i
$617$		−	1.29645e12i		−	0.360141i	−0.983654	−	0.180070i	$-0.942367\pi$
$617$		−	1.29645e12i		−	0.360141i	0.983654	−	0.180070i	$-0.0576325\pi$
$618$		0			0
$619$	4.11611e12			1.12688			0.563441	−	0.826156i	$-0.309477\pi$
$619$	4.11611e12			1.12688			0.563441	+	0.826156i	$0.309477\pi$
$620$	−2.06506e12			−0.561269
$621$		0			0
$622$		−	2.36399e12i		−	0.633269i
$623$	−6.40183e12			−1.70258
$624$		0			0
$625$	−4.73834e12			−1.24213
$626$	2.18872e12			0.569647
$627$		0			0
$628$	−3.65864e11			−0.0938645
$629$	6.24538e12			1.59086
$630$		0			0
$631$	1.38547e12			0.347909			0.173954	−	0.984754i	$-0.444345\pi$
$631$	1.38547e12			0.347909			0.173954	+	0.984754i	$0.444345\pi$
$632$		−	7.90702e11i		−	0.197145i
$633$		0			0
$634$	−1.11454e10			−0.00273964
$635$	4.13631e12			1.00956
$636$		0			0
$637$			1.29843e12i			0.312458i
$638$		−	3.71372e12i		−	0.887394i
$639$		0			0
$640$		−	4.45666e11i		−	0.105003i
$641$	−1.36709e12			−0.319843			−0.159922	−	0.987130i	$-0.551124\pi$
$641$	−1.36709e12			−0.319843			−0.159922	+	0.987130i	$0.551124\pi$
$642$		0			0
$643$	1.68944e12			0.389757			0.194878	−	0.980827i	$-0.437569\pi$
$643$	1.68944e12			0.389757			0.194878	+	0.980827i	$0.437569\pi$
$644$		−	2.69246e12i		−	0.616826i
$645$		0			0
$646$	3.09025e12	−	2.21447e12i	0.698148	−	0.500292i
$647$		−	5.59422e12i		−	1.25508i	−0.778585	−	0.627539i	$-0.784062\pi$
$647$		−	5.59422e12i		−	1.25508i	0.778585	−	0.627539i	$-0.215938\pi$
$648$		0			0
$649$			1.19279e13i			2.63915i
$650$		−	7.66452e11i		−	0.168413i
$651$		0			0
$652$	1.74455e12			0.378067
$653$			2.07507e11i			0.0446605i	0.999751	+	0.0223302i	$0.00710853\pi$
$653$			2.07507e11i			0.0446605i	−0.999751	+	0.0223302i	$0.992891\pi$
$654$		0			0
$655$	1.62811e11			0.0345620
$656$	1.28060e12			0.269989
$657$		0			0
$658$		−	6.62321e12i		−	1.37738i
$659$	8.49361e12			1.75432			0.877158	−	0.480202i	$-0.159437\pi$
$659$	8.49361e12			1.75432			0.877158	+	0.480202i	$0.159437\pi$
$660$		0			0
$661$			5.50201e12i			1.12102i	0.828146	+	0.560512i	$0.189396\pi$
$661$			5.50201e12i			1.12102i	−0.828146	+	0.560512i	$0.810604\pi$
$662$		−	3.58244e12i		−	0.724967i
$663$		0			0
$664$			2.96299e12i			0.591526i
$665$	6.04233e12	−	4.32993e12i	1.19814	−	0.858585i
$666$		0			0
$667$		−	4.45318e12i		−	0.871172i
$668$	−9.90350e10			−0.0192440
$669$		0			0
$670$	7.44762e12			1.42784
$671$		−	6.21079e12i		−	1.18276i
$672$		0			0
$673$		−	1.51048e12i		−	0.283822i	−0.989879	−	0.141911i	$-0.954675\pi$
$673$		−	1.51048e12i		−	0.283822i	0.989879	−	0.141911i	$-0.0453247\pi$
$674$			3.01472e11i			0.0562701i
$675$		0			0
$676$	1.80429e12			0.332312
$677$	−4.07516e12			−0.745582			−0.372791	−	0.927915i	$-0.621599\pi$
$677$	−4.07516e12			−0.745582			−0.372791	+	0.927915i	$0.621599\pi$
$678$		0			0
$679$			1.04955e12i			0.189492i
$680$	−2.84450e12			−0.510170
$681$		0			0
$682$	−5.40673e12			−0.956985
$683$	4.71303e12			0.828718			0.414359	−	0.910113i	$-0.364006\pi$
$683$	4.71303e12			0.828718			0.414359	+	0.910113i	$0.364006\pi$
$684$		0			0
$685$	−9.48403e12			−1.64583
$686$	−2.34331e12			−0.403990
$687$		0			0
$688$	−1.83562e12			−0.312345
$689$			1.58497e12i			0.267939i
$690$		0			0
$691$	9.18986e12			1.53341			0.766704	−	0.642001i	$-0.221896\pi$
$691$	9.18986e12			1.53341			0.766704	+	0.642001i	$0.221896\pi$
$692$	−3.47248e12			−0.575655
$693$		0			0
$694$			6.62278e12i			1.08373i
$695$		−	9.38040e12i		−	1.52507i
$696$		0			0
$697$		−	8.17351e12i		−	1.31178i
$698$	−8.03103e12			−1.28062
$699$		0			0
$700$	−1.62081e12			−0.255147
$701$			1.97947e12i			0.309612i	0.987945	+	0.154806i	$0.0494752\pi$
$701$			1.97947e12i			0.309612i	−0.987945	+	0.154806i	$0.950525\pi$
$702$		0			0
$703$	−6.89416e12	+	4.94035e12i	−1.06459	+	0.762884i
$704$		−	1.16684e12i		−	0.179033i
$705$		0			0
$706$			2.89281e12i			0.438227i
$707$			1.22571e13i			1.84502i
$708$		0			0
$709$	7.16055e12			1.06424			0.532118	−	0.846670i	$-0.321396\pi$
$709$	7.16055e12			1.06424			0.532118	+	0.846670i	$0.321396\pi$
$710$		−	9.94062e12i		−	1.46808i
$711$		0			0
$712$	−3.32680e12			−0.485140
$713$	−6.48329e12			−0.939491
$714$		0			0
$715$		−	6.88607e12i		−	0.985359i
$716$	−2.28792e12			−0.325336
$717$		0			0
$718$			5.83447e12i			0.819297i
$719$		−	4.61355e11i		−	0.0643807i	−0.999482	−	0.0321903i	$-0.989752\pi$
$719$		−	4.61355e11i		−	0.0643807i	0.999482	−	0.0321903i	$-0.0102483\pi$
$720$		0			0
$721$		−	1.04639e13i		−	1.44206i
$722$	−1.65954e12	+	4.88902e12i	−0.227284	+	0.669583i
$723$		0			0
$724$		−	3.30971e12i		−	0.447678i
$725$	−2.68073e12			−0.360356
$726$		0			0
$727$	−6.23234e12			−0.827459			−0.413730	−	0.910400i	$-0.635774\pi$
$727$	−6.23234e12			−0.827459			−0.413730	+	0.910400i	$0.635774\pi$
$728$			1.92534e12i			0.254049i
$729$		0			0
$730$		−	7.12944e12i		−	0.929185i
$731$			1.17160e13i			1.51758i
$732$		0			0
$733$	5.73454e12			0.733720			0.366860	−	0.930276i	$-0.380433\pi$
$733$	5.73454e12			0.733720			0.366860	+	0.930276i	$0.380433\pi$
$734$	5.00615e12			0.636608
$735$		0			0
$736$		−	1.39917e12i		−	0.175761i
$737$	1.94993e13			2.43453
$738$		0			0
$739$	7.57526e12			0.934325			0.467162	−	0.884172i	$-0.345276\pi$
$739$	7.57526e12			0.934325			0.467162	+	0.884172i	$0.345276\pi$
$740$	6.34590e12			0.777948
$741$		0			0
$742$	3.35172e12			0.405929
$743$	−1.65333e12			−0.199026			−0.0995131	−	0.995036i	$-0.531729\pi$
$743$	−1.65333e12			−0.199026			−0.0995131	+	0.995036i	$0.531729\pi$
$744$		0			0
$745$	4.76284e12			0.566451
$746$		−	2.91819e12i		−	0.344976i
$747$		0			0
$748$	−7.44743e12			−0.869860
$749$	−1.76521e13			−2.04941
$750$		0			0
$751$			8.06900e12i			0.925635i	0.886454	+	0.462818i	$0.153161\pi$
$751$			8.06900e12i			0.925635i	−0.886454	+	0.462818i	$0.846839\pi$
$752$		−	3.44185e12i		−	0.392474i
$753$		0			0
$754$			3.18441e12i			0.358805i
$755$	−6.04939e12			−0.677564
$756$		0			0
$757$	−2.36342e12			−0.261583			−0.130792	−	0.991410i	$-0.541752\pi$
$757$	−2.36342e12			−0.261583			−0.130792	+	0.991410i	$0.541752\pi$
$758$			8.02806e12i			0.883281i
$759$		0			0
$760$	3.13999e12	−	2.25011e12i	0.341402	−	0.244648i
$761$		−	4.25228e12i		−	0.459612i	−0.973236	−	0.229806i	$-0.926191\pi$
$761$		−	4.25228e12i		−	0.459612i	0.973236	−	0.229806i	$-0.0738092\pi$
$762$		0			0
$763$		−	1.75855e13i		−	1.87842i
$764$		−	5.96249e12i		−	0.633152i
$765$		0			0
$766$	−3.02820e12			−0.317800
$767$		−	1.02279e13i		−	1.06710i
$768$		0			0
$769$	1.22186e12			0.125995			0.0629974	−	0.998014i	$-0.479934\pi$
$769$	1.22186e12			0.125995			0.0629974	+	0.998014i	$0.479934\pi$
$770$	−1.45619e13			−1.49283
$771$		0			0
$772$			2.45559e12i			0.248816i
$773$	6.61847e12			0.666730			0.333365	−	0.942798i	$-0.391816\pi$
$773$	6.61847e12			0.666730			0.333365	+	0.942798i	$0.391816\pi$
$774$		0			0
$775$			3.90281e12i			0.388616i
$776$			5.45416e11i			0.0539946i
$777$		0			0
$778$		−	2.67564e12i		−	0.261831i
$779$	6.46558e12	+	9.02259e12i	0.629055	+	0.877835i
$780$		0			0
$781$		−	2.60264e13i		−	2.50314i
$782$	−8.93033e12			−0.853959
$783$		0			0
$784$	1.42688e12			0.134886
$785$			2.37274e12i			0.223016i
$786$		0			0
$787$		−	1.20720e13i		−	1.12174i	−0.827904	−	0.560870i	$-0.810467\pi$
$787$		−	1.20720e13i		−	1.12174i	0.827904	−	0.560870i	$-0.189533\pi$
$788$		−	2.71368e12i		−	0.250721i
$789$		0			0
$790$	−5.12793e12			−0.468404
$791$	−2.28162e13			−2.07229
$792$		0			0
$793$			5.32558e12i			0.478231i
$794$	2.76091e12			0.246524
$795$		0			0
$796$	−7.13694e12			−0.630091
$797$	1.62817e13			1.42935			0.714674	−	0.699458i	$-0.246575\pi$
$797$	1.62817e13			1.42935			0.714674	+	0.699458i	$0.246575\pi$
$798$		0			0
$799$	−2.19678e13			−1.90690
$800$	−8.42275e11			−0.0727025
$801$		0			0
$802$	−9.53144e12			−0.813531
$803$		−	1.86662e13i		−	1.58430i
$804$		0			0
$805$	−1.74614e13			−1.46554
$806$	4.63612e12			0.386943
$807$		0			0
$808$			6.36960e12i			0.525728i
$809$		−	4.79226e12i		−	0.393343i	−0.980469	−	0.196672i	$-0.936987\pi$
$809$		−	4.79226e12i		−	0.393343i	0.980469	−	0.196672i	$-0.0630133\pi$
$810$		0			0
$811$			4.32332e12i			0.350932i	0.984485	+	0.175466i	$0.0561433\pi$
$811$			4.32332e12i			0.350932i	−0.984485	+	0.175466i	$0.943857\pi$
$812$	6.73404e12			0.543592
$813$		0			0
$814$	1.66148e13			1.32643
$815$		−	1.13139e13i		−	0.898263i
$816$		0			0
$817$	−9.26780e12	−	1.29330e13i	−0.727742	−	1.01555i
$818$			4.94394e12i			0.386085i
$819$		0			0
$820$		−	8.30506e12i		−	0.641476i
$821$		−	4.38404e11i		−	0.0336768i	−0.999858	−	0.0168384i	$-0.994640\pi$
$821$		−	4.38404e11i		−	0.0336768i	0.999858	−	0.0168384i	$-0.00536008\pi$
$822$		0			0
$823$	3.74188e12			0.284309			0.142154	−	0.989844i	$-0.454597\pi$
$823$	3.74188e12			0.284309			0.142154	+	0.989844i	$0.454597\pi$
$824$		−	5.43770e12i		−	0.410907i
$825$		0			0
$826$	−2.16287e13			−1.61667
$827$	3.35108e12			0.249121			0.124560	−	0.992212i	$-0.460248\pi$
$827$	3.35108e12			0.249121			0.124560	+	0.992212i	$0.460248\pi$
$828$		0			0
$829$		−	1.01458e13i		−	0.746091i	−0.927813	−	0.373045i	$-0.878314\pi$
$829$		−	1.01458e13i		−	0.746091i	0.927813	−	0.373045i	$-0.121686\pi$
$830$	1.92159e13			1.40543
$831$		0			0
$832$			1.00053e12i			0.0723896i
$833$		−	9.10717e12i		−	0.655362i
$834$		0			0
$835$			6.42271e11i			0.0457224i
$836$	8.22108e12	−	5.89121e12i	0.582104	−	0.417135i
$837$		0			0
$838$			8.30537e12i			0.581783i
$839$	−2.51805e13			−1.75443			−0.877214	−	0.480099i	$-0.840601\pi$
$839$	−2.51805e13			−1.75443			−0.877214	+	0.480099i	$0.840601\pi$
$840$		0			0
$841$	−3.36941e12			−0.232258
$842$			9.87439e12i			0.677027i
$843$		0			0
$844$		−	1.02422e13i		−	0.694785i
$845$		−	1.17013e13i		−	0.789552i
$846$		0			0
$847$	−1.95404e13			−1.30454
$848$	1.74177e12			0.115667
$849$		0			0
$850$			5.37588e12i			0.353236i
$851$	1.99230e13			1.30218
$852$		0			0
$853$	−2.28453e12			−0.147750			−0.0738749	−	0.997268i	$-0.523537\pi$
$853$	−2.28453e12			−0.147750			−0.0738749	+	0.997268i	$0.523537\pi$
$854$	1.12619e13			0.724523
$855$		0			0
$856$	−9.17317e12			−0.583966
$857$	3.65516e12			0.231469			0.115735	−	0.993280i	$-0.463078\pi$
$857$	3.65516e12			0.231469			0.115735	+	0.993280i	$0.463078\pi$
$858$		0			0
$859$	−9.59061e12			−0.601004			−0.300502	−	0.953781i	$-0.597154\pi$
$859$	−9.59061e12			−0.601004			−0.300502	+	0.953781i	$0.597154\pi$
$860$			1.19045e13i			0.742112i
$861$		0			0
$862$	1.24589e13			0.768593
$863$	1.94662e13			1.19463			0.597314	−	0.802008i	$-0.296235\pi$
$863$	1.94662e13			1.19463			0.597314	+	0.802008i	$0.296235\pi$
$864$		0			0
$865$			2.25200e13i			1.36772i
$866$			7.42432e12i			0.448566i
$867$		0			0
$868$		−	9.80394e12i		−	0.586222i
$869$	−1.34259e13			−0.798647
$870$		0			0
$871$	−1.67201e13			−0.984367
$872$		−	9.13854e12i		−	0.535245i
$873$		0			0
$874$	9.85802e12	−	7.06425e12i	0.571463	−	0.409510i
$875$		−	1.50472e13i		−	0.867798i
$876$		0			0
$877$		−	1.86041e13i		−	1.06197i	−0.847382	−	0.530983i	$-0.821823\pi$
$877$		−	1.86041e13i		−	1.06197i	0.847382	−	0.530983i	$-0.178177\pi$
$878$		−	1.94896e13i		−	1.10682i
$879$		0			0
$880$	−7.56729e12			−0.425372
$881$		−	3.34962e12i		−	0.187328i	−0.995604	−	0.0936642i	$-0.970142\pi$
$881$		−	3.34962e12i		−	0.187328i	0.995604	−	0.0936642i	$-0.0298580\pi$
$882$		0			0
$883$	−5.59278e12			−0.309603			−0.154801	−	0.987946i	$-0.549474\pi$
$883$	−5.59278e12			−0.309603			−0.154801	+	0.987946i	$0.549474\pi$
$884$	6.38597e12			0.351715
$885$		0			0
$886$			1.21731e13i			0.663665i
$887$	−3.80386e12			−0.206333			−0.103167	−	0.994664i	$-0.532897\pi$
$887$	−3.80386e12			−0.206333			−0.103167	+	0.994664i	$0.532897\pi$
$888$		0			0
$889$			1.96373e13i			1.05444i
$890$			2.15753e13i			1.15266i
$891$		0			0
$892$		−	4.83482e12i		−	0.255705i
$893$	2.42499e13	−	1.73774e13i	1.27608	−	0.914438i
$894$		0			0
$895$			1.48378e13i			0.772976i
$896$	2.11581e12			0.109671
$897$		0			0
$898$	3.89524e12			0.199890
$899$		−	1.62152e13i		−	0.827949i
$900$		0			0
$901$		−	1.11170e13i		−	0.561986i
$902$		−	2.17442e13i		−	1.09374i
$903$		0			0
$904$	−1.18568e13			−0.590485
$905$	−2.14644e13			−1.06365
$906$		0			0
$907$		−	1.55383e13i		−	0.762378i	−0.924497	−	0.381189i	$-0.875515\pi$
$907$		−	1.55383e13i		−	0.762378i	0.924497	−	0.381189i	$-0.124485\pi$
$908$	1.65869e13			0.809801
$909$		0			0
$910$	1.24864e13			0.603603
$911$	1.14140e13			0.549040			0.274520	−	0.961581i	$-0.411481\pi$
$911$	1.14140e13			0.549040			0.274520	+	0.961581i	$0.411481\pi$
$912$		0			0
$913$	5.03108e13			2.39631
$914$	2.79232e13			1.32345
$915$		0			0
$916$	−2.59078e12			−0.121591
$917$			7.72951e11i			0.0360986i
$918$		0			0
$919$	3.76752e13			1.74235			0.871177	−	0.490969i	$-0.163357\pi$
$919$	3.76752e13			1.74235			0.871177	+	0.490969i	$0.163357\pi$
$920$	−9.07405e12			−0.417596
$921$		0			0
$922$		−	1.05077e13i		−	0.478873i
$923$			2.23169e13i			1.01211i
$924$		0			0
$925$		−	1.19933e13i		−	0.538641i
$926$	−1.87813e13			−0.839413
$927$		0			0
$928$	3.49944e12			0.154893
$929$			1.15781e13i			0.509998i	0.966941	+	0.254999i	$0.0820751\pi$
$929$			1.15781e13i			0.509998i	−0.966941	+	0.254999i	$0.917925\pi$
$930$		0			0
$931$	7.20414e12	+	1.00532e13i	0.314274	+	0.438563i
$932$			1.52411e13i			0.661674i
$933$		0			0
$934$			3.05182e12i			0.131219i
$935$			4.82988e13i			2.06673i
$936$		0			0
$937$	−3.06806e13			−1.30027			−0.650137	−	0.759817i	$-0.725289\pi$
$937$	−3.06806e13			−1.30027			−0.650137	+	0.759817i	$0.725289\pi$
$938$			3.53578e13i			1.49132i
$939$		0			0
$940$	−2.23214e13			−0.932494
$941$	4.50497e13			1.87300			0.936501	−	0.350664i	$-0.114044\pi$
$941$	4.50497e13			1.87300			0.936501	+	0.350664i	$0.114044\pi$
$942$		0			0
$943$		−	2.60738e13i		−	1.07375i
$944$	−1.12397e13			−0.460659
$945$		0			0
$946$			3.11683e13i			1.26533i
$947$		−	1.75244e13i		−	0.708057i	−0.935235	−	0.354028i	$-0.884812\pi$
$947$		−	1.75244e13i		−	0.708057i	0.935235	−	0.354028i	$-0.115188\pi$
$948$		0			0
$949$			1.60058e13i			0.640588i
$950$	−4.25254e12	−	5.93434e12i	−0.169392	−	0.236383i
$951$		0			0
$952$		−	1.35043e13i		−	0.532852i
$953$	−4.14502e13			−1.62783			−0.813914	−	0.580985i	$-0.802668\pi$
$953$	−4.14502e13			−1.62783			−0.813914	+	0.580985i	$0.802668\pi$
$954$		0			0
$955$	−3.86685e13			−1.50433
$956$			5.05803e12i			0.195849i
$957$		0			0
$958$			3.65740e12i			0.140290i
$959$		−	4.50257e13i		−	1.71900i
$960$		0			0
$961$	2.83225e12			0.107122
$962$	−1.42467e13			−0.536323
$963$		0			0
$964$		−	7.19401e11i		−	0.0268302i
$965$	1.59252e13			0.591171
$966$		0			0
$967$	−2.41152e12			−0.0886894			−0.0443447	−	0.999016i	$-0.514120\pi$
$967$	−2.41152e12			−0.0886894			−0.0443447	+	0.999016i	$0.514120\pi$
$968$	−1.01545e13			−0.371721
$969$		0			0
$970$	3.53719e12			0.128288
$971$	4.24449e13			1.53228			0.766142	−	0.642672i	$-0.222174\pi$
$971$	4.24449e13			1.53228			0.766142	+	0.642672i	$0.222174\pi$
$972$		0			0
$973$	4.45337e13			1.59287
$974$			1.88381e13i			0.670691i
$975$		0			0
$976$	5.85242e12			0.206448
$977$	−5.57612e13			−1.95797			−0.978985	−	0.203930i	$-0.934628\pi$
$977$	−5.57612e13			−1.95797			−0.978985	+	0.203930i	$0.934628\pi$
$978$		0			0
$979$			5.64882e13i			1.96533i
$980$		−	9.25375e12i		−	0.320480i
$981$		0			0
$982$		−	8.85435e11i		−	0.0303847i
$983$	−4.69352e13			−1.60327			−0.801637	−	0.597811i	$-0.796037\pi$
$983$	−4.69352e13			−1.60327			−0.801637	+	0.597811i	$0.796037\pi$
$984$		0			0
$985$	−1.75990e13			−0.595697
$986$		−	2.23354e13i		−	0.752572i
$987$		0			0
$988$	−7.04935e12	+	5.05155e12i	−0.235365	+	0.168663i
$989$			3.73744e13i			1.24220i
$990$		0			0
$991$			3.49544e13i			1.15125i	0.817714	+	0.575625i	$0.195241\pi$
$991$			3.49544e13i			1.15125i	−0.817714	+	0.575625i	$0.804759\pi$
$992$		−	5.09476e12i		−	0.167040i
$993$		0			0
$994$	4.71934e13			1.53335
$995$			4.62852e13i			1.49706i
$996$		0			0
$997$	3.38127e13			1.08381			0.541904	−	0.840441i	$-0.317704\pi$
$997$	3.38127e13			1.08381			0.541904	+	0.840441i	$0.317704\pi$
$998$	2.82371e13			0.901015
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.10.b.b.341.5	yes	30
3.2	odd	2		342.10.b.a.341.26	yes	30
19.18	odd	2		342.10.b.a.341.5	✓	30
57.56	even	2	inner	342.10.b.b.341.26	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.10.b.a.341.5	✓	30	19.18	odd	2
342.10.b.a.341.26	yes	30	3.2	odd	2
342.10.b.b.341.5	yes	30	1.1	even	1	trivial
342.10.b.b.341.26	yes	30	57.56	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	342.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+16.0000 q^{2} +256.000 q^{4} -1660.24i q^{5} +7882.01 q^{7} +4096.00 q^{8} +O(q^{10})\)	\(q+16.0000 q^{2} +256.000 q^{4} -1660.24i q^{5} +7882.01 q^{7} +4096.00 q^{8} -26563.8i q^{10} -69549.0i q^{11} +59636.3i q^{13} +126112. q^{14} +65536.0 q^{16} -418288. i q^{17} +(330883. + 461741. i) q^{19} -425020. i q^{20} -1.11278e6i q^{22} -1.33436e6i q^{23} -803256. q^{25} +954181. i q^{26} +2.01779e6 q^{28} +3.33733e6 q^{29} -4.85874e6i q^{31} +1.04858e6 q^{32} -6.69261e6i q^{34} -1.30860e7i q^{35} +1.49308e7i q^{37} +(5.29412e6 + 7.38785e6i) q^{38} -6.80032e6i q^{40} +1.95404e7 q^{41} -2.80093e7 q^{43} -1.78045e7i q^{44} -2.13497e7i q^{46} -5.25184e7i q^{47} +2.17725e7 q^{49} -1.28521e7 q^{50} +1.52669e7i q^{52} +2.65773e7 q^{53} -1.15468e8 q^{55} +3.22847e7 q^{56} +5.33972e7 q^{58} -1.71504e8 q^{59} +8.93009e7 q^{61} -7.77399e7i q^{62} +1.67772e7 q^{64} +9.90104e7 q^{65} +2.80368e8i q^{67} -1.07082e8i q^{68} -2.09376e8i q^{70} +3.74217e8 q^{71} +2.68390e8 q^{73} +2.38893e8i q^{74} +(8.47060e7 + 1.18206e8i) q^{76} -5.48186e8i q^{77} -1.93042e8i q^{79} -1.08805e8i q^{80} +3.12646e8 q^{82} +7.23386e8i q^{83} -6.94457e8 q^{85} -4.48149e8 q^{86} -2.84873e8i q^{88} -8.12207e8 q^{89} +4.70054e8i q^{91} -3.41595e8i q^{92} -8.40295e8i q^{94} +(7.66598e8 - 5.49343e8i) q^{95} +1.33158e8i q^{97} +3.48360e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 480 q^{2} + 7680 q^{4} + 1596 q^{7} + 122880 q^{8}+O(q^{10}) \) 30 * q + 480 * q^2 + 7680 * q^4 + 1596 * q^7 + 122880 * q^8	\( 30 q + 480 q^{2} + 7680 q^{4} + 1596 q^{7} + 122880 q^{8} + 25536 q^{14} + 1966080 q^{16} - 323202 q^{19} - 5473830 q^{25} + 408576 q^{28} - 8475660 q^{29} + 31457280 q^{32} - 5171232 q^{38} + 20345076 q^{41} - 60729780 q^{43} + 214984890 q^{49} - 87581280 q^{50} - 181795212 q^{53} - 316864944 q^{55} + 6537216 q^{56} - 135610560 q^{58} + 197198784 q^{59} + 53410728 q^{61} + 503316480 q^{64} + 640935936 q^{65} - 335924712 q^{71} + 136407840 q^{73} - 82739712 q^{76} + 325521216 q^{82} + 269215776 q^{85} - 971676480 q^{86} - 1186853004 q^{89} + 506121804 q^{95} + 3439758240 q^{98}+O(q^{100}) \) 30 * q + 480 * q^2 + 7680 * q^4 + 1596 * q^7 + 122880 * q^8 + 25536 * q^14 + 1966080 * q^16 - 323202 * q^19 - 5473830 * q^25 + 408576 * q^28 - 8475660 * q^29 + 31457280 * q^32 - 5171232 * q^38 + 20345076 * q^41 - 60729780 * q^43 + 214984890 * q^49 - 87581280 * q^50 - 181795212 * q^53 - 316864944 * q^55 + 6537216 * q^56 - 135610560 * q^58 + 197198784 * q^59 + 53410728 * q^61 + 503316480 * q^64 + 640935936 * q^65 - 335924712 * q^71 + 136407840 * q^73 - 82739712 * q^76 + 325521216 * q^82 + 269215776 * q^85 - 971676480 * q^86 - 1186853004 * q^89 + 506121804 * q^95 + 3439758240 * q^98

Introduction

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