LMFDB - Embedded newform 342.9.d.a.37.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,9,Mod(37,342)]

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([0, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("342.37");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	342.d (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:	$139.323484641$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 $ x^12 + 46118x^10 + 738386961x^8 + 5214446299656x^6 + 17370647163698184x^4 + 24830681474333400768*x^2 + 9218779084612644462864
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.4
Root			$61.5968i$ of defining polynomial
Character	$\chi$	$=$	342.37
Dual form			342.9.d.a.37.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.3137i q^{2} -128.000 q^{4} +154.845 q^{5} +1585.07 q^{7} +1448.15i q^{8} +O(q^{10})$	\(q-11.3137i q^{2} -128.000 q^{4} +154.845 q^{5} +1585.07 q^{7} +1448.15i q^{8} -1751.87i q^{10} +25549.2 q^{11} +10369.3i q^{13} -17933.0i q^{14} +16384.0 q^{16} +132967. q^{17} +(13145.5 + 129656. i) q^{19} -19820.1 q^{20} -289057. i q^{22} +334889. q^{23} -366648. q^{25} +117316. q^{26} -202889. q^{28} +605325. i q^{29} +141494. i q^{31} -185364. i q^{32} -1.50435e6i q^{34} +245440. q^{35} +2.45048e6i q^{37} +(1.46689e6 - 148724. i) q^{38} +224239. i q^{40} -2.38955e6i q^{41} +1.53889e6 q^{43} -3.27030e6 q^{44} -3.78883e6i q^{46} -5.01339e6 q^{47} -3.25235e6 q^{49} +4.14815e6i q^{50} -1.32728e6i q^{52} +1.18301e7i q^{53} +3.95616e6 q^{55} +2.29543e6i q^{56} +6.84847e6 q^{58} -7.32369e6i q^{59} -1.12872e7 q^{61} +1.60082e6 q^{62} -2.09715e6 q^{64} +1.60564e6i q^{65} +1.87052e7i q^{67} -1.70197e7 q^{68} -2.77684e6i q^{70} +2.17638e7i q^{71} -8.89170e6 q^{73} +2.77241e7 q^{74} +(-1.68262e6 - 1.65960e7i) q^{76} +4.04974e7 q^{77} -5.26495e7i q^{79} +2.53698e6 q^{80} -2.70346e7 q^{82} -3.76587e7 q^{83} +2.05892e7 q^{85} -1.74106e7i q^{86} +3.69992e7i q^{88} +1.06654e8i q^{89} +1.64362e7i q^{91} -4.28658e7 q^{92} +5.67200e7i q^{94} +(2.03551e6 + 2.00766e7i) q^{95} +2.45621e7i q^{97} +3.67961e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7}+O(q^{10}) $ 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7	$ 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7} + 12546 q^{11} + 196608 q^{16} - 270810 q^{17} + 41512 q^{19} + 71424 q^{20} + 823956 q^{23} + 865538 q^{25} + 431616 q^{26} + 694016 q^{28} + 1194378 q^{35} - 998784 q^{38} + 7586646 q^{43} - 1605888 q^{44} + 20260530 q^{47} - 19498842 q^{49} - 14858554 q^{55} - 5506560 q^{58} - 41363266 q^{61} - 32266752 q^{62} - 25165824 q^{64} + 34663680 q^{68} + 87906498 q^{73} + 2149632 q^{74} - 5313536 q^{76} + 78817962 q^{77} - 9142272 q^{80} - 49609728 q^{82} + 55944960 q^{83} + 25440254 q^{85} - 105466368 q^{92} - 81396774 q^{95}+O(q^{100}) $ 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7 + 12546 * q^11 + 196608 * q^16 - 270810 * q^17 + 41512 * q^19 + 71424 * q^20 + 823956 * q^23 + 865538 * q^25 + 431616 * q^26 + 694016 * q^28 + 1194378 * q^35 - 998784 * q^38 + 7586646 * q^43 - 1605888 * q^44 + 20260530 * q^47 - 19498842 * q^49 - 14858554 * q^55 - 5506560 * q^58 - 41363266 * q^61 - 32266752 * q^62 - 25165824 * q^64 + 34663680 * q^68 + 87906498 * q^73 + 2149632 * q^74 - 5313536 * q^76 + 78817962 * q^77 - 9142272 * q^80 - 49609728 * q^82 + 55944960 * q^83 + 25440254 * q^85 - 105466368 * q^92 - 81396774 * q^95

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 $

$2$		−	11.3137i		−	0.707107i
$2$		−	11.3137i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−128.000			−0.500000
$5$	154.845			0.247752			0.123876	−	0.992298i	$-0.460468\pi$
$5$	154.845			0.247752			0.123876	+	0.992298i	$0.460468\pi$
$6$		0			0
$7$	1585.07			0.660172			0.330086	−	0.943951i	$-0.392922\pi$
$7$	1585.07			0.660172			0.330086	+	0.943951i	$0.392922\pi$
$8$			1448.15i			0.353553i
$9$		0			0
$10$		−	1751.87i		−	0.175187i
$11$	25549.2			1.74505			0.872524	−	0.488572i	$-0.162482\pi$
$11$	25549.2			1.74505			0.872524	+	0.488572i	$0.162482\pi$
$12$		0			0
$13$			10369.3i			0.363060i	0.983385	+	0.181530i	$0.0581049\pi$
$13$			10369.3i			0.363060i	−0.983385	+	0.181530i	$0.941895\pi$
$14$		−	17933.0i		−	0.466812i
$15$		0			0
$16$	16384.0			0.250000
$17$	132967.			1.59202			0.796008	−	0.605286i	$-0.206941\pi$
$17$	132967.			1.59202			0.796008	+	0.605286i	$0.206941\pi$
$18$		0			0
$19$	13145.5	+	129656.i	0.100870	+	0.994900i
$19$	13145.5	+	129656.i	0.100870	+	0.994900i
$20$	−19820.1			−0.123876
$21$		0			0
$22$		−	289057.i		−	1.23393i
$23$	334889.			1.19671			0.598355	−	0.801231i	$-0.295821\pi$
$23$	334889.			1.19671			0.598355	+	0.801231i	$0.295821\pi$
$24$		0			0
$25$	−366648.			−0.938619
$26$	117316.			0.256722
$27$		0			0
$28$	−202889.			−0.330086
$29$			605325.i			0.855848i	0.903815	+	0.427924i	$0.140755\pi$
$29$			605325.i			0.855848i	−0.903815	+	0.427924i	$0.859245\pi$
$30$		0			0
$31$			141494.i			0.153211i	0.997061	+	0.0766055i	$0.0244082\pi$
$31$			141494.i			0.153211i	−0.997061	+	0.0766055i	$0.975592\pi$
$32$		−	185364.i		−	0.176777i
$33$		0			0
$34$		−	1.50435e6i		−	1.12573i
$35$	245440.			0.163559
$36$		0			0
$37$			2.45048e6i			1.30751i	0.756706	+	0.653755i	$0.226807\pi$
$37$			2.45048e6i			1.30751i	−0.756706	+	0.653755i	$0.773193\pi$
$38$	1.46689e6	−	148724.i	0.703500	−	0.0713260i
$39$		0			0
$40$			224239.i			0.0875934i
$41$		−	2.38955e6i		−	0.845629i	−0.906216	−	0.422815i	$-0.861042\pi$
$41$		−	2.38955e6i		−	0.845629i	0.906216	−	0.422815i	$-0.138958\pi$
$42$		0			0
$43$	1.53889e6			0.450126			0.225063	−	0.974344i	$-0.427741\pi$
$43$	1.53889e6			0.450126			0.225063	+	0.974344i	$0.427741\pi$
$44$	−3.27030e6			−0.872524
$45$		0			0
$46$		−	3.78883e6i		−	0.846202i
$47$	−5.01339e6			−1.02740			−0.513701	−	0.857970i	$-0.671726\pi$
$47$	−5.01339e6			−1.02740			−0.513701	+	0.857970i	$0.671726\pi$
$48$		0			0
$49$	−3.25235e6			−0.564173
$50$			4.14815e6i			0.663704i
$51$		0			0
$52$		−	1.32728e6i		−	0.181530i
$53$			1.18301e7i			1.49929i	0.661840	+	0.749645i	$0.269776\pi$
$53$			1.18301e7i			1.49929i	−0.661840	+	0.749645i	$0.730224\pi$
$54$		0			0
$55$	3.95616e6			0.432338
$56$			2.29543e6i			0.233406i
$57$		0			0
$58$	6.84847e6			0.605176
$59$		−	7.32369e6i		−	0.604396i	−0.953245	−	0.302198i	$-0.902280\pi$
$59$		−	7.32369e6i		−	0.604396i	0.953245	−	0.302198i	$-0.0977204\pi$
$60$		0			0
$61$	−1.12872e7			−0.815203			−0.407602	−	0.913160i	$-0.633635\pi$
$61$	−1.12872e7			−0.815203			−0.407602	+	0.913160i	$0.633635\pi$
$62$	1.60082e6			0.108337
$63$		0			0
$64$	−2.09715e6			−0.125000
$65$			1.60564e6i			0.0899486i
$66$		0			0
$67$			1.87052e7i			0.928245i	0.885771	+	0.464122i	$0.153630\pi$
$67$			1.87052e7i			0.928245i	−0.885771	+	0.464122i	$0.846370\pi$
$68$	−1.70197e7			−0.796008
$69$		0			0
$70$		−	2.77684e6i		−	0.115653i
$71$			2.17638e7i			0.856450i	0.903672	+	0.428225i	$0.140861\pi$
$71$			2.17638e7i			0.856450i	−0.903672	+	0.428225i	$0.859139\pi$
$72$		0			0
$73$	−8.89170e6			−0.313107			−0.156554	−	0.987669i	$-0.550038\pi$
$73$	−8.89170e6			−0.313107			−0.156554	+	0.987669i	$0.550038\pi$
$74$	2.77241e7			0.924549
$75$		0			0
$76$	−1.68262e6	−	1.65960e7i	−0.0504351	−	0.497450i
$77$	4.04974e7			1.15203
$78$		0			0
$79$		−	5.26495e7i		−	1.35172i	−0.737031	−	0.675859i	$-0.763773\pi$
$79$		−	5.26495e7i		−	1.35172i	0.737031	−	0.675859i	$-0.236227\pi$
$80$	2.53698e6			0.0619379
$81$		0			0
$82$	−2.70346e7			−0.597950
$83$	−3.76587e7			−0.793511			−0.396756	−	0.917924i	$-0.629864\pi$
$83$	−3.76587e7			−0.793511			−0.396756	+	0.917924i	$0.629864\pi$
$84$		0			0
$85$	2.05892e7			0.394424
$86$		−	1.74106e7i		−	0.318287i
$87$		0			0
$88$			3.69992e7i			0.616967i
$89$			1.06654e8i			1.69988i	0.526883	+	0.849938i	$0.323360\pi$
$89$			1.06654e8i			1.69988i	−0.526883	+	0.849938i	$0.676640\pi$
$90$		0			0
$91$			1.64362e7i			0.239682i
$92$	−4.28658e7			−0.598355
$93$		0			0
$94$			5.67200e7i			0.726482i
$95$	2.03551e6	+	2.00766e7i	0.0249907	+	0.246488i
$96$		0			0
$97$			2.45621e7i			0.277446i	0.990331	+	0.138723i	$0.0442998\pi$
$97$			2.45621e7i			0.277446i	−0.990331	+	0.138723i	$0.955700\pi$
$98$			3.67961e7i			0.398931i
$99$		0			0
$100$	4.69310e7			0.469310
$101$	−4.35309e7			−0.418323			−0.209162	−	0.977881i	$-0.567073\pi$
$101$	−4.35309e7			−0.418323			−0.209162	+	0.977881i	$0.567073\pi$
$102$		0			0
$103$		−	2.50911e7i		−	0.222931i	−0.993768	−	0.111465i	$-0.964446\pi$
$103$		−	2.50911e7i		−	0.222931i	0.993768	−	0.111465i	$-0.0355544\pi$
$104$	−1.50164e7			−0.128361
$105$		0			0
$106$	1.33843e8			1.06016
$107$		−	2.68160e7i		−	0.204578i	−0.994755	−	0.102289i	$-0.967383\pi$
$107$		−	2.68160e7i		−	0.204578i	0.994755	−	0.102289i	$-0.0326167\pi$
$108$		0			0
$109$		−	1.10533e8i		−	0.783041i	−0.920169	−	0.391521i	$-0.871949\pi$
$109$		−	1.10533e8i		−	0.783041i	0.920169	−	0.391521i	$-0.128051\pi$
$110$		−	4.47589e7i		−	0.305709i
$111$		0			0
$112$	2.59698e7			0.165043
$113$			1.51426e8i			0.928723i	0.885646	+	0.464361i	$0.153716\pi$
$113$			1.51426e8i			0.928723i	−0.885646	+	0.464361i	$0.846284\pi$
$114$		0			0
$115$	5.18557e7			0.296487
$116$		−	7.74816e7i		−	0.427924i
$117$		0			0
$118$	−8.28581e7			−0.427373
$119$	2.10762e8			1.05100
$120$		0			0
$121$	4.38405e8			2.04519
$122$			1.27700e8i			0.576436i
$123$		0			0
$124$		−	1.81112e7i		−	0.0766055i
$125$	−1.17260e8			−0.480296
$126$		0			0
$127$			4.59385e8i			1.76588i	0.469482	+	0.882942i	$0.344441\pi$
$127$			4.59385e8i			1.76588i	−0.469482	+	0.882942i	$0.655559\pi$
$128$			2.37266e7i			0.0883883i
$129$		0			0
$130$	1.81657e7			0.0636033
$131$	3.08184e8			1.04646			0.523232	−	0.852190i	$-0.324726\pi$
$131$	3.08184e8			1.04646			0.523232	+	0.852190i	$0.324726\pi$
$132$		0			0
$133$	2.08366e7	+	2.05515e8i	0.0665916	+	0.656805i
$134$	2.11625e8			0.656368
$135$		0			0
$136$			1.92556e8i			0.562863i
$137$	4.53360e8			1.28695			0.643473	−	0.765468i	$-0.277493\pi$
$137$	4.53360e8			1.28695			0.643473	+	0.765468i	$0.277493\pi$
$138$		0			0
$139$	−3.86640e8			−1.03573			−0.517866	−	0.855462i	$-0.673274\pi$
$139$	−3.86640e8			−1.03573			−0.517866	+	0.855462i	$0.673274\pi$
$140$	−3.14163e7			−0.0817793
$141$		0			0
$142$	2.46230e8			0.605602
$143$			2.64929e8i			0.633556i
$144$		0			0
$145$			9.37314e7i			0.212038i
$146$			1.00598e8i			0.221400i
$147$		0			0
$148$		−	3.13662e8i		−	0.653755i
$149$	−5.42202e8			−1.10006			−0.550029	−	0.835145i	$-0.685383\pi$
$149$	−5.42202e8			−1.10006			−0.550029	+	0.835145i	$0.685383\pi$
$150$		0			0
$151$		−	1.75171e7i		−	0.0336942i	−0.999858	−	0.0168471i	$-0.994637\pi$
$151$		−	1.75171e7i		−	0.0336942i	0.999858	−	0.0168471i	$-0.00536285\pi$
$152$	−1.87762e8	+	1.90367e7i	−0.351750	+	0.0356630i
$153$		0			0
$154$		−	4.58176e8i		−	0.814609i
$155$			2.19095e7i			0.0379583i
$156$		0			0
$157$	−6.98783e8			−1.15012			−0.575061	−	0.818111i	$-0.695022\pi$
$157$	−6.98783e8			−1.15012			−0.575061	+	0.818111i	$0.695022\pi$
$158$	−5.95662e8			−0.955809
$159$		0			0
$160$		−	2.87026e7i		−	0.0437967i
$161$	5.30823e8			0.790035
$162$		0			0
$163$	3.03342e8			0.429716			0.214858	−	0.976645i	$-0.431071\pi$
$163$	3.03342e8			0.429716			0.214858	+	0.976645i	$0.431071\pi$
$164$			3.05862e8i			0.422815i
$165$		0			0
$166$			4.26060e8i			0.561097i
$167$		−	1.38262e9i		−	1.77761i	−0.458286	−	0.888805i	$-0.651536\pi$
$167$		−	1.38262e9i		−	1.77761i	0.458286	−	0.888805i	$-0.348464\pi$
$168$		0			0
$169$	7.08207e8			0.868188
$170$		−	2.32940e8i		−	0.278900i
$171$		0			0
$172$	−1.96978e8			−0.225063
$173$			6.31095e8i			0.704547i	0.935897	+	0.352274i	$0.114591\pi$
$173$			6.31095e8i			0.704547i	−0.935897	+	0.352274i	$0.885409\pi$
$174$		0			0
$175$	−5.81164e8			−0.619650
$176$	4.18599e8			0.436262
$177$		0			0
$178$	1.20665e9			1.20199
$179$		−	1.63061e9i		−	1.58832i	−0.607711	−	0.794158i	$-0.707912\pi$
$179$		−	1.63061e9i		−	1.58832i	0.607711	−	0.794158i	$-0.292088\pi$
$180$		0			0
$181$			8.11994e8i			0.756551i	0.925693	+	0.378276i	$0.123483\pi$
$181$			8.11994e8i			0.756551i	−0.925693	+	0.378276i	$0.876517\pi$
$182$	1.85954e8			0.169481
$183$		0			0
$184$			4.84971e8i			0.423101i
$185$			3.79444e8i			0.323938i
$186$		0			0
$187$	3.39720e9			2.77814
$188$	6.41714e8			0.513701
$189$		0			0
$190$	2.27141e8	−	2.30292e7i	0.174293	−	0.0176711i
$191$	1.14327e9			0.859041			0.429521	−	0.903057i	$-0.358683\pi$
$191$	1.14327e9			0.859041			0.429521	+	0.903057i	$0.358683\pi$
$192$		0			0
$193$		−	1.21153e9i		−	0.873185i	−0.899659	−	0.436592i	$-0.856185\pi$
$193$		−	1.21153e9i		−	0.873185i	0.899659	−	0.436592i	$-0.143815\pi$
$194$	2.77889e8			0.196184
$195$		0			0
$196$	4.16300e8			0.282087
$197$	−4.99652e8			−0.331744			−0.165872	−	0.986147i	$-0.553044\pi$
$197$	−4.99652e8			−0.331744			−0.165872	+	0.986147i	$0.553044\pi$
$198$		0			0
$199$	2.04764e9			1.30570			0.652848	−	0.757489i	$-0.273574\pi$
$199$	2.04764e9			1.30570			0.652848	+	0.757489i	$0.273574\pi$
$200$		−	5.30963e8i		−	0.331852i
$201$		0			0
$202$			4.92496e8i			0.295799i
$203$			9.59484e8i			0.565007i
$204$		0			0
$205$		−	3.70009e8i		−	0.209506i
$206$	−2.83873e8			−0.157636
$207$		0			0
$208$			1.69891e8i			0.0907649i
$209$	3.35857e8	+	3.31262e9i	0.176023	+	1.73615i
$210$		0			0
$211$		−	6.24079e8i		−	0.314854i	−0.987531	−	0.157427i	$-0.949680\pi$
$211$		−	6.24079e8i		−	0.314854i	0.987531	−	0.157427i	$-0.0503199\pi$
$212$		−	1.51426e9i		−	0.749645i
$213$		0			0
$214$	−3.03389e8			−0.144659
$215$	2.38289e8			0.111520
$216$		0			0
$217$			2.24277e8i			0.101146i
$218$	−1.25053e9			−0.553694
$219$		0			0
$220$	−5.06389e8			−0.216169
$221$			1.37878e9i			0.577997i
$222$		0			0
$223$		−	2.29270e9i		−	0.927102i	−0.886070	−	0.463551i	$-0.846575\pi$
$223$		−	2.29270e9i		−	0.927102i	0.886070	−	0.463551i	$-0.153425\pi$
$224$		−	2.93815e8i		−	0.116703i
$225$		0			0
$226$	1.71319e9			0.656706
$227$		−	3.90434e9i		−	1.47043i	−0.677835	−	0.735214i	$-0.737082\pi$
$227$		−	3.90434e9i		−	1.47043i	0.677835	−	0.735214i	$-0.262918\pi$
$228$		0			0
$229$	−3.02177e8			−0.109880			−0.0549401	−	0.998490i	$-0.517497\pi$
$229$	−3.02177e8			−0.109880			−0.0549401	+	0.998490i	$0.517497\pi$
$230$		−	5.86681e8i		−	0.209648i
$231$		0			0
$232$	−8.76604e8			−0.302588
$233$	4.59616e9			1.55945			0.779726	−	0.626121i	$-0.215359\pi$
$233$	4.59616e9			1.55945			0.779726	+	0.626121i	$0.215359\pi$
$234$		0			0
$235$	−7.76297e8			−0.254540
$236$			9.37432e8i			0.302198i
$237$		0			0
$238$		−	2.38450e9i		−	0.743172i
$239$	1.85947e9			0.569897			0.284949	−	0.958543i	$-0.408023\pi$
$239$	1.85947e9			0.569897			0.284949	+	0.958543i	$0.408023\pi$
$240$		0			0
$241$			4.35931e8i			0.129226i	0.997910	+	0.0646130i	$0.0205813\pi$
$241$			4.35931e8i			0.129226i	−0.997910	+	0.0646130i	$0.979419\pi$
$242$		−	4.95998e9i		−	1.44617i
$243$		0			0
$244$	1.44476e9			0.407602
$245$	−5.03609e8			−0.139775
$246$		0			0
$247$	−1.34445e9	+	1.36310e8i	−0.361208	+	0.0366219i
$248$	−2.04905e8			−0.0541683
$249$		0			0
$250$			1.32664e9i			0.339621i
$251$	−1.73172e9			−0.436297			−0.218148	−	0.975916i	$-0.570002\pi$
$251$	−1.73172e9			−0.436297			−0.218148	+	0.975916i	$0.570002\pi$
$252$		0			0
$253$	8.55615e9			2.08832
$254$	5.19735e9			1.24867
$255$		0			0
$256$	2.68435e8			0.0625000
$257$			7.49270e9i			1.71754i	0.512364	+	0.858768i	$0.328770\pi$
$257$			7.49270e9i			1.71754i	−0.512364	+	0.858768i	$0.671230\pi$
$258$		0			0
$259$			3.88419e9i			0.863181i
$260$		−	2.05522e8i		−	0.0449743i
$261$		0			0
$262$		−	3.48670e9i		−	0.739962i
$263$	4.66900e9			0.975890			0.487945	−	0.872874i	$-0.337747\pi$
$263$	4.66900e9			0.975890			0.487945	+	0.872874i	$0.337747\pi$
$264$		0			0
$265$			1.83183e9i			0.371452i
$266$	2.32513e9	−	2.35739e8i	0.464431	−	0.0470874i
$267$		0			0
$268$		−	2.39426e9i		−	0.464122i
$269$			2.06571e9i			0.394512i	0.980352	+	0.197256i	$0.0632030\pi$
$269$			2.06571e9i			0.394512i	−0.980352	+	0.197256i	$0.936797\pi$
$270$		0			0
$271$	1.24856e9			0.231490			0.115745	−	0.993279i	$-0.463074\pi$
$271$	1.24856e9			0.231490			0.115745	+	0.993279i	$0.463074\pi$
$272$	2.17853e9			0.398004
$273$		0			0
$274$		−	5.12918e9i		−	0.910009i
$275$	−9.36758e9			−1.63793
$276$		0			0
$277$	−1.05391e8			−0.0179013			−0.00895067	−	0.999960i	$-0.502849\pi$
$277$	−1.05391e8			−0.0179013			−0.00895067	+	0.999960i	$0.502849\pi$
$278$			4.37433e9i			0.732374i
$279$		0			0
$280$			3.55435e8i			0.0578267i
$281$		−	9.25009e9i		−	1.48361i	−0.670614	−	0.741807i	$-0.733969\pi$
$281$		−	9.25009e9i		−	1.48361i	0.670614	−	0.741807i	$-0.266031\pi$
$282$		0			0
$283$	7.95944e9			1.24090			0.620450	−	0.784246i	$-0.286950\pi$
$283$	7.95944e9			1.24090			0.620450	+	0.784246i	$0.286950\pi$
$284$		−	2.78577e9i		−	0.428225i
$285$		0			0
$286$	2.99733e9			0.447992
$287$		−	3.78760e9i		−	0.558261i
$288$		0			0
$289$	1.07044e10			1.53451
$290$	1.06045e9			0.149933
$291$		0			0
$292$	1.13814e9			0.156554
$293$		−	1.46284e9i		−	0.198484i	−0.995063	−	0.0992420i	$-0.968358\pi$
$293$		−	1.46284e9i		−	0.198484i	0.995063	−	0.0992420i	$-0.0316418\pi$
$294$		0			0
$295$		−	1.13403e9i		−	0.149740i
$296$	−3.54868e9			−0.462274
$297$		0			0
$298$			6.13431e9i			0.777859i
$299$			3.47258e9i			0.434477i
$300$		0			0
$301$	2.43926e9			0.297161
$302$	−1.98184e8			−0.0238254
$303$		0			0
$304$	2.15376e8	+	2.12429e9i	0.0252175	+	0.248725i
$305$	−1.74776e9			−0.201968
$306$		0			0
$307$			1.59981e9i			0.180101i	0.995937	+	0.0900504i	$0.0287028\pi$
$307$			1.59981e9i			0.180101i	−0.995937	+	0.0900504i	$0.971297\pi$
$308$	−5.18367e9			−0.576015
$309$		0			0
$310$	2.47878e8			0.0268405
$311$	1.36371e10			1.45775			0.728873	−	0.684649i	$-0.240045\pi$
$311$	1.36371e10			1.45775			0.728873	+	0.684649i	$0.240045\pi$
$312$		0			0
$313$	−6.84472e9			−0.713146			−0.356573	−	0.934268i	$-0.616055\pi$
$313$	−6.84472e9			−0.713146			−0.356573	+	0.934268i	$0.616055\pi$
$314$			7.90583e9i			0.813259i
$315$		0			0
$316$			6.73914e9i			0.675859i
$317$		−	4.91136e9i		−	0.486368i	−0.969980	−	0.243184i	$-0.921808\pi$
$317$		−	4.91136e9i		−	0.486368i	0.969980	−	0.243184i	$-0.0781919\pi$
$318$		0			0
$319$			1.54656e10i			1.49349i
$320$	−3.24733e8			−0.0309689
$321$		0			0
$322$		−	6.00557e9i		−	0.558639i
$323$	1.74791e9	+	1.72400e10i	0.160587	+	1.58390i
$324$		0			0
$325$		−	3.80190e9i		−	0.340775i
$326$		−	3.43192e9i		−	0.303855i
$327$		0			0
$328$	3.46043e9			0.298975
$329$	−7.94658e9			−0.678261
$330$		0			0
$331$			1.69481e10i			1.41192i	0.708252	+	0.705960i	$0.249484\pi$
$331$			1.69481e10i			1.41192i	−0.708252	+	0.705960i	$0.750516\pi$
$332$	4.82031e9			0.396756
$333$		0			0
$334$	−1.56425e10			−1.25696
$335$			2.89640e9i			0.229974i
$336$		0			0
$337$		−	2.45355e10i		−	1.90229i	−0.308750	−	0.951143i	$-0.599911\pi$
$337$		−	2.45355e10i		−	1.90229i	0.308750	−	0.951143i	$-0.400089\pi$
$338$		−	8.01245e9i		−	0.613901i
$339$		0			0
$340$	−2.63542e9			−0.197212
$341$			3.61505e9i			0.267360i
$342$		0			0
$343$	−1.42928e10			−1.03262
$344$			2.22855e9i			0.159144i
$345$		0			0
$346$	7.14002e9			0.498190
$347$	−1.97537e10			−1.36248			−0.681239	−	0.732061i	$-0.738559\pi$
$347$	−1.97537e10			−1.36248			−0.681239	+	0.732061i	$0.738559\pi$
$348$		0			0
$349$	−7.25657e9			−0.489136			−0.244568	−	0.969632i	$-0.578646\pi$
$349$	−7.25657e9			−0.489136			−0.244568	+	0.969632i	$0.578646\pi$
$350$			6.57512e9i			0.438159i
$351$		0			0
$352$		−	4.73590e9i		−	0.308484i
$353$	1.86135e10			1.19875			0.599375	−	0.800468i	$-0.295416\pi$
$353$	1.86135e10			1.19875			0.599375	+	0.800468i	$0.295416\pi$
$354$		0			0
$355$			3.37002e9i			0.212187i
$356$		−	1.36517e10i		−	0.849938i
$357$		0			0
$358$	−1.84482e10			−1.12311
$359$	−1.90002e10			−1.14388			−0.571940	−	0.820296i	$-0.693809\pi$
$359$	−1.90002e10			−1.14388			−0.571940	+	0.820296i	$0.693809\pi$
$360$		0			0
$361$	−1.66380e10	+	3.40879e9i	−0.979650	+	0.200711i
$362$	9.18666e9			0.534963
$363$		0			0
$364$		−	2.10383e9i		−	0.119841i
$365$	−1.37683e9			−0.0775729
$366$		0			0
$367$	−1.69812e10			−0.936061			−0.468030	−	0.883712i	$-0.655036\pi$
$367$	−1.69812e10			−0.936061			−0.468030	+	0.883712i	$0.655036\pi$
$368$	5.48682e9			0.299178
$369$		0			0
$370$	4.29292e9			0.229058
$371$			1.87516e10i			0.989789i
$372$		0			0
$373$			3.35760e10i			1.73458i	0.497805	+	0.867289i	$0.334140\pi$
$373$			3.35760e10i			1.73458i	−0.497805	+	0.867289i	$0.665860\pi$
$374$		−	3.84349e10i		−	1.96444i
$375$		0			0
$376$		−	7.26016e9i		−	0.363241i
$377$	−6.27683e9			−0.310724
$378$		0			0
$379$			3.39145e9i			0.164372i	0.996617	+	0.0821860i	$0.0261902\pi$
$379$			3.39145e9i			0.164372i	−0.996617	+	0.0821860i	$0.973810\pi$
$380$	−2.60545e8	−	2.56980e9i	−0.0124954	−	0.123244i
$381$		0			0
$382$		−	1.29346e10i		−	0.607434i
$383$			4.19332e10i			1.94878i	0.224864	+	0.974390i	$0.427806\pi$
$383$			4.19332e10i			1.94878i	−0.224864	+	0.974390i	$0.572194\pi$
$384$		0			0
$385$	6.27081e9			0.285417
$386$	−1.37069e10			−0.617435
$387$		0			0
$388$		−	3.14395e9i		−	0.138723i
$389$	1.91799e10			0.837620			0.418810	−	0.908074i	$-0.362447\pi$
$389$	1.91799e10			0.837620			0.418810	+	0.908074i	$0.362447\pi$
$390$		0			0
$391$	4.45291e10			1.90518
$392$		−	4.70990e9i		−	0.199465i
$393$		0			0
$394$			5.65291e9i			0.234578i
$395$		−	8.15250e9i		−	0.334890i
$396$		0			0
$397$	−3.73055e10			−1.50180			−0.750899	−	0.660417i	$-0.770379\pi$
$397$	−3.73055e10			−1.50180			−0.750899	+	0.660417i	$0.770379\pi$
$398$		−	2.31664e10i		−	0.923266i
$399$		0			0
$400$	−6.00716e9			−0.234655
$401$			1.19174e10i			0.460895i	0.973085	+	0.230448i	$0.0740191\pi$
$401$			1.19174e10i			0.460895i	−0.973085	+	0.230448i	$0.925981\pi$
$402$		0			0
$403$	−1.46720e9			−0.0556247
$404$	5.57195e9			0.209162
$405$		0			0
$406$	1.08553e10			0.399520
$407$			6.26080e10i			2.28167i
$408$		0			0
$409$		−	1.57301e10i		−	0.562133i	−0.959688	−	0.281066i	$-0.909312\pi$
$409$		−	1.57301e10i		−	0.562133i	0.959688	−	0.281066i	$-0.0906881\pi$
$410$	−4.18617e9			−0.148143
$411$		0			0
$412$			3.21166e9i			0.111465i
$413$		−	1.16086e10i		−	0.399005i
$414$		0			0
$415$	−5.83125e9			−0.196594
$416$	1.92210e9			0.0641805
$417$		0			0
$418$	3.74780e10	−	3.79979e9i	1.22764	−	0.124467i
$419$	−3.53760e10			−1.14776			−0.573882	−	0.818938i	$-0.694563\pi$
$419$	−3.53760e10			−1.14776			−0.573882	+	0.818938i	$0.694563\pi$
$420$		0			0
$421$			2.22556e10i			0.708454i	0.935159	+	0.354227i	$0.115256\pi$
$421$			2.22556e10i			0.708454i	−0.935159	+	0.354227i	$0.884744\pi$
$422$	−7.06065e9			−0.222636
$423$		0			0
$424$	−1.71319e10			−0.530079
$425$	−4.87520e10			−1.49430
$426$		0			0
$427$	−1.78910e10			−0.538174
$428$			3.43245e9i			0.102289i
$429$		0			0
$430$		−	2.69594e9i		−	0.0788562i
$431$		−	6.16218e10i		−	1.78577i	−0.450285	−	0.892885i	$-0.648678\pi$
$431$		−	6.16218e10i		−	1.78577i	0.450285	−	0.892885i	$-0.351322\pi$
$432$		0			0
$433$		−	4.72412e9i		−	0.134391i	−0.997740	−	0.0671954i	$-0.978595\pi$
$433$		−	4.72412e9i		−	0.134391i	0.997740	−	0.0671954i	$-0.0214051\pi$
$434$	2.53741e9			0.0715207
$435$		0			0
$436$			1.41482e10i			0.391521i
$437$	4.40228e9	+	4.34204e10i	0.120712	+	1.19061i
$438$		0			0
$439$		−	4.36060e10i		−	1.17405i	−0.809568	−	0.587027i	$-0.800298\pi$
$439$		−	4.36060e10i		−	1.17405i	0.809568	−	0.587027i	$-0.199702\pi$
$440$			5.72914e9i			0.152855i
$441$		0			0
$442$	1.55991e10			0.408706
$443$	4.41307e10			1.14585			0.572923	−	0.819610i	$-0.305810\pi$
$443$	4.41307e10			1.14585			0.572923	+	0.819610i	$0.305810\pi$
$444$		0			0
$445$			1.65148e10i			0.421147i
$446$	−2.59389e10			−0.655560
$447$		0			0
$448$	−3.32414e9			−0.0825215
$449$		−	2.32868e10i		−	0.572961i	−0.958086	−	0.286481i	$-0.907515\pi$
$449$		−	2.32868e10i		−	0.572961i	0.958086	−	0.286481i	$-0.0924854\pi$
$450$		0			0
$451$		−	6.10511e10i		−	1.47566i
$452$		−	1.93825e10i		−	0.464361i
$453$		0			0
$454$	−4.41725e10			−1.03975
$455$			2.54505e9i			0.0593815i
$456$		0			0
$457$	−4.60367e10			−1.05545			−0.527727	−	0.849414i	$-0.676956\pi$
$457$	−4.60367e10			−1.05545			−0.527727	+	0.849414i	$0.676956\pi$
$458$			3.41875e9i			0.0776971i
$459$		0			0
$460$	−6.63754e9			−0.148243
$461$	−3.61112e10			−0.799537			−0.399769	−	0.916616i	$-0.630909\pi$
$461$	−3.61112e10			−0.799537			−0.399769	+	0.916616i	$0.630909\pi$
$462$		0			0
$463$	3.50543e9			0.0762812			0.0381406	−	0.999272i	$-0.487857\pi$
$463$	3.50543e9			0.0762812			0.0381406	+	0.999272i	$0.487857\pi$
$464$			9.91764e9i			0.213962i
$465$		0			0
$466$		−	5.19997e10i		−	1.10270i
$467$	−4.73893e9			−0.0996352			−0.0498176	−	0.998758i	$-0.515864\pi$
$467$	−4.73893e9			−0.0996352			−0.0498176	+	0.998758i	$0.515864\pi$
$468$		0			0
$469$			2.96490e10i			0.612801i
$470$			8.78280e9i			0.179987i
$471$		0			0
$472$	1.06058e10			0.213686
$473$	3.93175e10			0.785492
$474$		0			0
$475$	−4.81977e9	−	4.75382e10i	−0.0946787	−	0.933832i
$476$	−2.69775e10			−0.525502
$477$		0			0
$478$		−	2.10375e10i		−	0.402978i
$479$	7.54648e9			0.143352			0.0716758	−	0.997428i	$-0.477165\pi$
$479$	7.54648e9			0.143352			0.0716758	+	0.997428i	$0.477165\pi$
$480$		0			0
$481$	−2.54099e10			−0.474704
$482$	4.93200e9			0.0913765
$483$		0			0
$484$	−5.61158e10			−1.02259
$485$			3.80332e9i			0.0687378i
$486$		0			0
$487$			8.92286e10i			1.58631i	0.609020	+	0.793155i	$0.291563\pi$
$487$			8.92286e10i			1.58631i	−0.609020	+	0.793155i	$0.708437\pi$
$488$		−	1.63456e10i		−	0.288218i
$489$		0			0
$490$			5.69768e9i			0.0988357i
$491$	3.26865e10			0.562397			0.281198	−	0.959650i	$-0.409268\pi$
$491$	3.26865e10			0.562397			0.281198	+	0.959650i	$0.409268\pi$
$492$		0			0
$493$			8.04881e10i			1.36252i
$494$	1.54217e9	+	1.52107e10i	0.0258956	+	0.255413i
$495$		0			0
$496$			2.31823e9i			0.0383027i
$497$			3.44973e10i			0.565404i
$498$		0			0
$499$	−9.51579e10			−1.53477			−0.767384	−	0.641188i	$-0.778442\pi$
$499$	−9.51579e10			−1.53477			−0.767384	+	0.641188i	$0.778442\pi$
$500$	1.50092e10			0.240148
$501$		0			0
$502$			1.95921e10i			0.308508i
$503$	−1.90098e10			−0.296966			−0.148483	−	0.988915i	$-0.547439\pi$
$503$	−1.90098e10			−0.296966			−0.148483	+	0.988915i	$0.547439\pi$
$504$		0			0
$505$	−6.74053e9			−0.103640
$506$		−	9.68018e10i		−	1.47666i
$507$		0			0
$508$		−	5.88013e10i		−	0.882942i
$509$		−	1.06082e11i		−	1.58041i	−0.612845	−	0.790203i	$-0.709975\pi$
$509$		−	1.06082e11i		−	1.58041i	0.612845	−	0.790203i	$-0.290025\pi$
$510$		0			0
$511$	−1.40940e10			−0.206705
$512$		−	3.03700e9i		−	0.0441942i
$513$		0			0
$514$	8.47702e10			1.21448
$515$		−	3.88522e9i		−	0.0552315i
$516$		0			0
$517$	−1.28088e11			−1.79286
$518$	4.39446e10			0.610361
$519$		0			0
$520$	−2.32521e9			−0.0318016
$521$			5.55106e10i			0.753399i	0.926335	+	0.376700i	$0.122941\pi$
$521$			5.55106e10i			0.753399i	−0.926335	+	0.376700i	$0.877059\pi$
$522$		0			0
$523$		−	1.05842e11i		−	1.41465i	−0.706888	−	0.707326i	$-0.749902\pi$
$523$		−	1.05842e11i		−	1.41465i	0.706888	−	0.707326i	$-0.250098\pi$
$524$	−3.94475e10			−0.523232
$525$		0			0
$526$		−	5.28237e10i		−	0.690059i
$527$			1.88139e10i			0.243914i
$528$		0			0
$529$	3.38395e10			0.432116
$530$	2.07248e10			0.262656
$531$		0			0
$532$	−2.66708e9	−	2.63059e10i	−0.0332958	−	0.328402i
$533$	2.47780e10			0.307014
$534$		0			0
$535$		−	4.15232e9i		−	0.0506846i
$536$	−2.70880e10			−0.328184
$537$		0			0
$538$	2.33708e10			0.278962
$539$	−8.30950e10			−0.984509
$540$		0			0
$541$	6.69196e10			0.781204			0.390602	−	0.920560i	$-0.372267\pi$
$541$	6.69196e10			0.781204			0.390602	+	0.920560i	$0.372267\pi$
$542$		−	1.41259e10i		−	0.163688i
$543$		0			0
$544$		−	2.46472e10i		−	0.281431i
$545$		−	1.71154e10i		−	0.194000i
$546$		0			0
$547$		−	9.76775e9i		−	0.109105i	−0.998511	−	0.0545526i	$-0.982627\pi$
$547$		−	9.76775e9i		−	0.109105i	0.998511	−	0.0545526i	$-0.0173732\pi$
$548$	−5.80300e10			−0.643473
$549$		0			0
$550$			1.05982e11i			1.15819i
$551$	−7.84842e10	+	7.95730e9i	−0.851483	+	0.0863295i
$552$		0			0
$553$		−	8.34533e10i		−	0.892366i
$554$			1.19237e9i			0.0126582i
$555$		0			0
$556$	4.94899e10			0.517866
$557$	−1.59399e10			−0.165602			−0.0828010	−	0.996566i	$-0.526387\pi$
$557$	−1.59399e10			−0.165602			−0.0828010	+	0.996566i	$0.526387\pi$
$558$		0			0
$559$			1.59573e10i			0.163423i
$560$	4.02129e9			0.0408896
$561$		0			0
$562$	−1.04653e11			−1.04907
$563$		−	1.84760e10i		−	0.183897i	−0.995764	−	0.0919483i	$-0.970691\pi$
$563$		−	1.84760e10i		−	0.183897i	0.995764	−	0.0919483i	$-0.0293094\pi$
$564$		0			0
$565$			2.34475e10i			0.230093i
$566$		−	9.00508e10i		−	0.877449i
$567$		0			0
$568$	−3.15174e10			−0.302801
$569$			5.39118e10i			0.514322i	0.966369	+	0.257161i	$0.0827870\pi$
$569$			5.39118e10i			0.514322i	−0.966369	+	0.257161i	$0.917213\pi$
$570$		0			0
$571$	5.13331e9			0.0482896			0.0241448	−	0.999708i	$-0.492314\pi$
$571$	5.13331e9			0.0482896			0.0241448	+	0.999708i	$0.492314\pi$
$572$		−	3.39109e10i		−	0.316778i
$573$		0			0
$574$	−4.28519e10			−0.394750
$575$	−1.22786e11			−1.12326
$576$		0			0
$577$	−4.51019e9			−0.0406903			−0.0203452	−	0.999793i	$-0.506477\pi$
$577$	−4.51019e9			−0.0406903			−0.0203452	+	0.999793i	$0.506477\pi$
$578$		−	1.21107e11i		−	1.08507i
$579$		0			0
$580$		−	1.19976e10i		−	0.106019i
$581$	−5.96918e10			−0.523854
$582$		0			0
$583$			3.02251e11i			2.61633i
$584$		−	1.28766e10i		−	0.110700i
$585$		0			0
$586$	−1.65501e10			−0.140349
$587$	5.32145e10			0.448206			0.224103	−	0.974565i	$-0.428055\pi$
$587$	5.32145e10			0.448206			0.224103	+	0.974565i	$0.428055\pi$
$588$		0			0
$589$	−1.83455e10	+	1.86000e9i	−0.152430	+	0.0154544i
$590$	−1.28301e10			−0.105882
$591$		0			0
$592$			4.01487e10i			0.326877i
$593$	2.81590e10			0.227719			0.113859	−	0.993497i	$-0.463679\pi$
$593$	2.81590e10			0.227719			0.113859	+	0.993497i	$0.463679\pi$
$594$		0			0
$595$	3.26354e10			0.260388
$596$	6.94018e10			0.550029
$597$		0			0
$598$	3.92877e10			0.307222
$599$			1.53599e10i			0.119311i	0.998219	+	0.0596554i	$0.0190002\pi$
$599$			1.53599e10i			0.119311i	−0.998219	+	0.0596554i	$0.981000\pi$
$600$		0			0
$601$		−	1.08743e11i		−	0.833494i	−0.909023	−	0.416747i	$-0.863170\pi$
$601$		−	1.08743e11i		−	0.833494i	0.909023	−	0.416747i	$-0.136830\pi$
$602$		−	2.75970e10i		−	0.210124i
$603$		0			0
$604$			2.24219e9i			0.0168471i
$605$	6.78846e10			0.506699
$606$		0			0
$607$			1.12346e11i			0.827565i	0.910376	+	0.413783i	$0.135793\pi$
$607$			1.12346e11i			0.827565i	−0.910376	+	0.413783i	$0.864207\pi$
$608$	2.40336e10	−	2.43670e9i	0.175875	−	0.0178315i
$609$		0			0
$610$			1.97736e10i			0.142813i
$611$		−	5.19856e10i		−	0.373008i
$612$		0			0
$613$	−2.12496e10			−0.150490			−0.0752451	−	0.997165i	$-0.523974\pi$
$613$	−2.12496e10			−0.150490			−0.0752451	+	0.997165i	$0.523974\pi$
$614$	1.80998e10			0.127350
$615$		0			0
$616$			5.86465e10i			0.407304i
$617$	2.05896e11			1.42072			0.710359	−	0.703840i	$-0.248533\pi$
$617$	2.05896e11			1.42072			0.710359	+	0.703840i	$0.248533\pi$
$618$		0			0
$619$	1.59468e9			0.0108620			0.00543100	−	0.999985i	$-0.498271\pi$
$619$	1.59468e9			0.0108620			0.00543100	+	0.999985i	$0.498271\pi$
$620$		−	2.80442e9i		−	0.0189791i
$621$		0			0
$622$		−	1.54287e11i		−	1.03078i
$623$			1.69054e11i			1.12221i
$624$		0			0
$625$	1.25065e11			0.819625
$626$			7.74391e10i			0.504270i
$627$		0			0
$628$	8.94442e10			0.575061
$629$			3.25833e11i			2.08158i
$630$		0			0
$631$	2.84477e11			1.79444			0.897222	−	0.441580i	$-0.145582\pi$
$631$	2.84477e11			1.79444			0.897222	+	0.441580i	$0.145582\pi$
$632$	7.62447e10			0.477905
$633$		0			0
$634$	−5.55657e10			−0.343914
$635$			7.11334e10i			0.437501i
$636$		0			0
$637$		−	3.37247e10i		−	0.204829i
$638$	1.74973e11			1.05606
$639$		0			0
$640$			3.67393e9i			0.0218984i
$641$			5.30174e10i			0.314041i	0.987595	+	0.157021i	$0.0501889\pi$
$641$			5.30174e10i			0.314041i	−0.987595	+	0.157021i	$0.949811\pi$
$642$		0			0
$643$	5.79754e10			0.339156			0.169578	−	0.985517i	$-0.445759\pi$
$643$	5.79754e10			0.339156			0.169578	+	0.985517i	$0.445759\pi$
$644$	−6.79453e10			−0.395017
$645$		0			0
$646$	1.95048e11	−	1.97754e10i	1.11998	−	0.113552i
$647$	2.75434e11			1.57181			0.785904	−	0.618348i	$-0.212198\pi$
$647$	2.75434e11			1.57181			0.785904	+	0.618348i	$0.212198\pi$
$648$		0			0
$649$		−	1.87115e11i		−	1.05470i
$650$	−4.30136e10			−0.240964
$651$		0			0
$652$	−3.88278e10			−0.214858
$653$	2.69428e11			1.48180			0.740900	−	0.671615i	$-0.234399\pi$
$653$	2.69428e11			1.48180			0.740900	+	0.671615i	$0.234399\pi$
$654$		0			0
$655$	4.77206e10			0.259263
$656$		−	3.91503e10i		−	0.211407i
$657$		0			0
$658$			8.99053e10i			0.479603i
$659$		−	8.80932e10i		−	0.467090i	−0.972346	−	0.233545i	$-0.924967\pi$
$659$		−	8.80932e10i		−	0.467090i	0.972346	−	0.233545i	$-0.0750326\pi$
$660$		0			0
$661$			2.53825e10i			0.132962i	0.997788	+	0.0664811i	$0.0211772\pi$
$661$			2.53825e10i			0.132962i	−0.997788	+	0.0664811i	$0.978823\pi$
$662$	1.91746e11			0.998378
$663$		0			0
$664$		−	5.45356e10i		−	0.280549i
$665$	3.22643e9	+	3.18229e10i	0.0164982	+	0.162724i
$666$		0			0
$667$			2.02716e11i			1.02420i
$668$			1.76975e11i			0.888805i
$669$		0			0
$670$	3.27690e10			0.162616
$671$	−2.88379e11			−1.42257
$672$		0			0
$673$			1.21341e11i			0.591490i	0.955267	+	0.295745i	$0.0955679\pi$
$673$			1.21341e11i			0.591490i	−0.955267	+	0.295745i	$0.904432\pi$
$674$	−2.77588e11			−1.34512
$675$		0			0
$676$	−9.06505e10			−0.434094
$677$		−	3.38640e11i		−	1.61207i	−0.591870	−	0.806034i	$-0.701610\pi$
$677$		−	3.38640e11i		−	1.61207i	0.591870	−	0.806034i	$-0.298390\pi$
$678$		0			0
$679$			3.89328e10i			0.183162i
$680$			2.98164e10i			0.139450i
$681$		0			0
$682$	4.08996e10			0.189052
$683$		−	1.36488e11i		−	0.627208i	−0.949554	−	0.313604i	$-0.898464\pi$
$683$		−	1.36488e11i		−	0.627208i	0.949554	−	0.313604i	$-0.101536\pi$
$684$		0			0
$685$	7.02004e10			0.318843
$686$			1.61705e11i			0.730175i
$687$		0			0
$688$	2.52132e10			0.112532
$689$	−1.22671e11			−0.544332
$690$		0			0
$691$	3.85706e11			1.69178			0.845890	−	0.533358i	$-0.179070\pi$
$691$	3.85706e11			1.69178			0.845890	+	0.533358i	$0.179070\pi$
$692$		−	8.07801e10i		−	0.352274i
$693$		0			0
$694$			2.23487e11i			0.963418i
$695$	−5.98692e10			−0.256604
$696$		0			0
$697$		−	3.17730e11i		−	1.34626i
$698$			8.20988e10i			0.345872i
$699$		0			0
$700$	7.43890e10			0.309825
$701$	−4.23246e11			−1.75275			−0.876377	−	0.481626i	$-0.840046\pi$
$701$	−4.23246e11			−1.75275			−0.876377	+	0.481626i	$0.840046\pi$
$702$		0			0
$703$	−3.17721e11	+	3.22128e10i	−1.30084	+	0.131889i
$704$	−5.35806e10			−0.218131
$705$		0			0
$706$		−	2.10587e11i		−	0.847644i
$707$	−6.89996e10			−0.276165
$708$		0			0
$709$	4.73862e10			0.187529			0.0937643	−	0.995594i	$-0.470110\pi$
$709$	4.73862e10			0.187529			0.0937643	+	0.995594i	$0.470110\pi$
$710$	3.81274e10			0.150039
$711$		0			0
$712$	−1.54451e11			−0.600997
$713$			4.73846e10i			0.183349i
$714$		0			0
$715$			4.10229e10i			0.156965i
$716$			2.08718e11i			0.794158i
$717$		0			0
$718$			2.14963e11i			0.808845i
$719$	1.26152e11			0.472039			0.236020	−	0.971748i	$-0.424157\pi$
$719$	1.26152e11			0.472039			0.236020	+	0.971748i	$0.424157\pi$
$720$		0			0
$721$		−	3.97711e10i		−	0.147173i
$722$	3.85661e10	+	1.88237e11i	0.141924	+	0.692717i
$723$		0			0
$724$		−	1.03935e11i		−	0.378276i
$725$		−	2.21941e11i		−	0.803315i
$726$		0			0
$727$	8.49520e10			0.304114			0.152057	−	0.988372i	$-0.451410\pi$
$727$	8.49520e10			0.304114			0.152057	+	0.988372i	$0.451410\pi$
$728$	−2.38021e10			−0.0847403
$729$		0			0
$730$			1.55771e10i			0.0548523i
$731$	2.04622e11			0.716608
$732$		0			0
$733$	5.23738e11			1.81426			0.907128	−	0.420855i	$-0.138270\pi$
$733$	5.23738e11			1.81426			0.907128	+	0.420855i	$0.138270\pi$
$734$			1.92120e11i			0.661895i
$735$		0			0
$736$		−	6.20762e10i		−	0.211551i
$737$			4.77903e11i			1.61983i
$738$		0			0
$739$	3.44499e11			1.15508			0.577538	−	0.816364i	$-0.304014\pi$
$739$	3.44499e11			1.15508			0.577538	+	0.816364i	$0.304014\pi$
$740$		−	4.85689e10i		−	0.161969i
$741$		0			0
$742$	2.12150e11			0.699887
$743$		−	4.53922e10i		−	0.148945i	−0.997223	−	0.0744726i	$-0.976273\pi$
$743$		−	4.53922e10i		−	0.148945i	0.997223	−	0.0744726i	$-0.0237273\pi$
$744$		0			0
$745$	−8.39571e10			−0.272541
$746$	3.79869e11			1.22653
$747$		0			0
$748$	−4.34842e11			−1.38907
$749$		−	4.25054e10i		−	0.135057i
$750$		0			0
$751$			8.96270e10i			0.281760i	0.990027	+	0.140880i	$0.0449931\pi$
$751$			8.96270e10i			0.281760i	−0.990027	+	0.140880i	$0.955007\pi$
$752$	−8.21394e10			−0.256850
$753$		0			0
$754$			7.10142e10i			0.219715i
$755$		−	2.71243e9i		−	0.00834779i
$756$		0			0
$757$	−5.94573e11			−1.81060			−0.905298	−	0.424778i	$-0.860352\pi$
$757$	−5.94573e11			−1.81060			−0.905298	+	0.424778i	$0.860352\pi$
$758$	3.83698e10			0.116229
$759$		0			0
$760$	−2.90740e10	+	2.94774e9i	−0.0871466	+	0.00883556i
$761$	4.29954e11			1.28199			0.640993	−	0.767547i	$-0.278523\pi$
$761$	4.29954e11			1.28199			0.640993	+	0.767547i	$0.278523\pi$
$762$		0			0
$763$		−	1.75202e11i		−	0.516942i
$764$	−1.46338e11			−0.429521
$765$		0			0
$766$	4.74420e11			1.37800
$767$	7.59419e10			0.219432
$768$		0			0
$769$	3.94606e11			1.12839			0.564194	−	0.825642i	$-0.309187\pi$
$769$	3.94606e11			1.12839			0.564194	+	0.825642i	$0.309187\pi$
$770$		−	7.09461e10i		−	0.201821i
$771$		0			0
$772$			1.55076e11i			0.436592i
$773$		−	3.34209e10i		−	0.0936053i	−0.998904	−	0.0468026i	$-0.985097\pi$
$773$		−	3.34209e10i		−	0.0936053i	0.998904	−	0.0468026i	$-0.0149032\pi$
$774$		0			0
$775$		−	5.18783e10i		−	0.143807i
$776$	−3.55698e10			−0.0980921
$777$		0			0
$778$		−	2.16995e11i		−	0.592287i
$779$	3.09820e11	−	3.14118e10i	0.841316	−	0.0852988i
$780$		0			0
$781$			5.56049e11i			1.49455i
$782$		−	5.03789e11i		−	1.34717i
$783$		0			0
$784$	−5.32865e10			−0.141043
$785$	−1.08203e11			−0.284944
$786$		0			0
$787$		−	5.01627e11i		−	1.30762i	−0.756658	−	0.653811i	$-0.773169\pi$
$787$		−	5.01627e11i		−	1.30762i	0.756658	−	0.653811i	$-0.226831\pi$
$788$	6.39554e10			0.165872
$789$		0			0
$790$	−9.22350e10			−0.236803
$791$			2.40021e11i			0.613116i
$792$		0			0
$793$		−	1.17041e11i		−	0.295967i
$794$			4.22064e11i			1.06193i
$795$		0			0
$796$	−2.62098e11			−0.652848
$797$		−	3.80292e11i		−	0.942505i	−0.881998	−	0.471253i	$-0.843802\pi$
$797$		−	3.80292e11i		−	0.942505i	0.881998	−	0.471253i	$-0.156198\pi$
$798$		0			0
$799$	−6.66614e11			−1.63564
$800$			6.79633e10i			0.165926i
$801$		0			0
$802$	1.34829e11			0.325902
$803$	−2.27176e11			−0.546387
$804$		0			0
$805$	8.21951e10			0.195732
$806$			1.65994e10i			0.0393326i
$807$		0			0
$808$		−	6.30395e10i		−	0.147900i
$809$	2.33585e11			0.545319			0.272660	−	0.962111i	$-0.412097\pi$
$809$	2.33585e11			0.545319			0.272660	+	0.962111i	$0.412097\pi$
$810$		0			0
$811$		−	2.37656e10i		−	0.0549371i	−0.999623	−	0.0274685i	$-0.991255\pi$
$811$		−	2.37656e10i		−	0.0549371i	0.999623	−	0.0274685i	$-0.00874461\pi$
$812$		−	1.22814e11i		−	0.282503i
$813$		0			0
$814$	7.08329e11			1.61338
$815$	4.69709e10			0.106463
$816$		0			0
$817$	2.02295e10	+	1.99527e11i	0.0454043	+	0.447831i
$818$	−1.77966e11			−0.397488
$819$		0			0
$820$			4.73611e10i			0.104753i
$821$	5.15004e11			1.13354			0.566771	−	0.823875i	$-0.308192\pi$
$821$	5.15004e11			1.13354			0.566771	+	0.823875i	$0.308192\pi$
$822$		0			0
$823$	2.28392e11			0.497831			0.248916	−	0.968525i	$-0.419926\pi$
$823$	2.28392e11			0.497831			0.248916	+	0.968525i	$0.419926\pi$
$824$	3.63357e10			0.0788179
$825$		0			0
$826$	−1.31336e11			−0.282139
$827$			6.97217e10i			0.149055i	0.997219	+	0.0745274i	$0.0237448\pi$
$827$			6.97217e10i			0.149055i	−0.997219	+	0.0745274i	$0.976255\pi$
$828$		0			0
$829$		−	4.08026e11i		−	0.863913i	−0.901894	−	0.431956i	$-0.857823\pi$
$829$		−	4.08026e11i		−	0.863913i	0.901894	−	0.431956i	$-0.142177\pi$
$830$			6.59731e10i			0.139013i
$831$		0			0
$832$		−	2.17461e10i		−	0.0453825i
$833$	−4.32454e11			−0.898173
$834$		0			0
$835$		−	2.14091e11i		−	0.440406i
$836$	−4.29898e10	−	4.24015e11i	−0.0880116	−	0.868073i
$837$		0			0
$838$			4.00234e11i			0.811591i
$839$		−	9.01495e11i		−	1.81935i	−0.415325	−	0.909673i	$-0.636332\pi$
$839$		−	9.01495e11i		−	1.81935i	0.415325	−	0.909673i	$-0.363668\pi$
$840$		0			0
$841$	1.33828e11			0.267524
$842$	2.51794e11			0.500953
$843$		0			0
$844$			7.98821e10i			0.157427i
$845$	1.09662e11			0.215095
$846$		0			0
$847$	6.94903e11			1.35018
$848$			1.93825e11i			0.374823i
$849$		0			0
$850$			5.51566e11i			1.05663i
$851$			8.20639e11i			1.56471i
$852$		0			0
$853$	−5.99808e10			−0.113296			−0.0566482	−	0.998394i	$-0.518041\pi$
$853$	−5.99808e10			−0.113296			−0.0566482	+	0.998394i	$0.518041\pi$
$854$			2.02413e11i			0.380547i
$855$		0			0
$856$	3.88338e10			0.0723293
$857$			2.70130e10i			0.0500783i	0.999686	+	0.0250391i	$0.00797104\pi$
$857$			2.70130e10i			0.0500783i	−0.999686	+	0.0250391i	$0.992029\pi$
$858$		0			0
$859$	−7.98513e10			−0.146659			−0.0733296	−	0.997308i	$-0.523363\pi$
$859$	−7.98513e10			−0.146659			−0.0733296	+	0.997308i	$0.523363\pi$
$860$	−3.05010e10			−0.0557598
$861$		0			0
$862$	−6.97171e11			−1.26273
$863$		−	1.02525e12i		−	1.84836i	−0.381954	−	0.924181i	$-0.624749\pi$
$863$		−	1.02525e12i		−	1.84836i	0.381954	−	0.924181i	$-0.375251\pi$
$864$		0			0
$865$			9.77217e10i			0.174553i
$866$	−5.34473e10			−0.0950286
$867$		0			0
$868$		−	2.87075e10i		−	0.0505728i
$869$		−	1.34516e12i		−	2.35881i
$870$		0			0
$871$	−1.93960e11			−0.337008
$872$	1.60068e11			0.276847
$873$		0			0
$874$	4.91246e11	−	4.98061e10i	0.841886	−	0.0853565i
$875$	−1.85865e11			−0.317078
$876$		0			0
$877$		−	9.55040e11i		−	1.61445i	−0.590247	−	0.807223i	$-0.700970\pi$
$877$		−	9.55040e11i		−	1.61445i	0.590247	−	0.807223i	$-0.299030\pi$
$878$	−4.93345e11			−0.830181
$879$		0			0
$880$	6.48178e10			0.108085
$881$	−1.13499e12			−1.88403			−0.942014	−	0.335573i	$-0.891070\pi$
$881$	−1.13499e12			−1.88403			−0.942014	+	0.335573i	$0.891070\pi$
$882$		0			0
$883$	2.78656e10			0.0458379			0.0229190	−	0.999737i	$-0.492704\pi$
$883$	2.78656e10			0.0458379			0.0229190	+	0.999737i	$0.492704\pi$
$884$		−	1.76484e11i		−	0.288998i
$885$		0			0
$886$		−	4.99282e11i		−	0.810235i
$887$			6.71858e11i			1.08538i	0.839932	+	0.542691i	$0.182595\pi$
$887$			6.71858e11i			1.08538i	−0.839932	+	0.542691i	$0.817405\pi$
$888$		0			0
$889$			7.28159e11i			1.16579i
$890$	1.86844e11			0.297796
$891$		0			0
$892$			2.93465e11i			0.463551i
$893$	−6.59035e10	−	6.50018e11i	−0.103634	−	1.02216i
$894$		0			0
$895$		−	2.52491e11i		−	0.393508i
$896$			3.76083e10i			0.0583515i
$897$		0			0
$898$	−2.63461e11			−0.405145
$899$	−8.56496e10			−0.131125
$900$		0			0
$901$			1.57301e12i			2.38689i
$902$	−6.90714e11			−1.04345
$903$		0			0
$904$	−2.19288e11			−0.328353
$905$			1.25733e11i			0.187437i
$906$		0			0
$907$		−	6.08911e11i		−	0.899757i	−0.893090	−	0.449878i	$-0.851467\pi$
$907$		−	6.08911e11i		−	0.899757i	0.893090	−	0.449878i	$-0.148533\pi$
$908$			4.99755e11i			0.735214i
$909$		0			0
$910$	2.87940e10			0.0419891
$911$			7.07571e10i			0.102730i	0.998680	+	0.0513649i	$0.0163572\pi$
$911$			7.07571e10i			0.102730i	−0.998680	+	0.0513649i	$0.983643\pi$
$912$		0			0
$913$	−9.62151e11			−1.38471
$914$			5.20846e11i			0.746319i
$915$		0			0
$916$	3.86787e10			0.0549401
$917$	4.88493e11			0.690846
$918$		0			0
$919$	−2.81891e11			−0.395202			−0.197601	−	0.980283i	$-0.563315\pi$
$919$	−2.81891e11			−0.395202			−0.197601	+	0.980283i	$0.563315\pi$
$920$			7.50951e10i			0.104824i
$921$		0			0
$922$			4.08552e11i			0.565358i
$923$	−2.25677e11			−0.310943
$924$		0			0
$925$		−	8.98465e11i		−	1.22725i
$926$		−	3.96594e10i		−	0.0539389i
$927$		0			0
$928$	1.12205e11			0.151294
$929$	1.08860e12			1.46152			0.730762	−	0.682632i	$-0.239165\pi$
$929$	1.08860e12			1.46152			0.730762	+	0.682632i	$0.239165\pi$
$930$		0			0
$931$	−4.27537e10	−	4.21687e11i	−0.0569082	−	0.561296i
$932$	−5.88309e11			−0.779726
$933$		0			0
$934$			5.36149e10i			0.0704527i
$935$	5.26038e11			0.688289
$936$		0			0
$937$	−8.13586e11			−1.05547			−0.527734	−	0.849410i	$-0.676958\pi$
$937$	−8.13586e11			−1.05547			−0.527734	+	0.849410i	$0.676958\pi$
$938$	3.35441e11			0.433316
$939$		0			0
$940$	9.93660e10			0.127270
$941$		−	1.38430e12i		−	1.76552i	−0.469827	−	0.882759i	$-0.655684\pi$
$941$		−	1.38430e12i		−	1.76552i	0.469827	−	0.882759i	$-0.344316\pi$
$942$		0			0
$943$		−	8.00232e11i		−	1.01197i
$944$		−	1.19991e11i		−	0.151099i
$945$		0			0
$946$		−	4.44827e11i		−	0.555427i
$947$	−5.32762e11			−0.662420			−0.331210	−	0.943557i	$-0.607457\pi$
$947$	−5.32762e11			−0.662420			−0.331210	+	0.943557i	$0.607457\pi$
$948$		0			0
$949$		−	9.22011e10i		−	0.113677i
$950$	−5.37834e11	+	5.45295e10i	−0.660319	+	0.0669479i
$951$		0			0
$952$			3.05216e11i			0.371586i
$953$			5.73373e11i			0.695129i	0.937656	+	0.347564i	$0.112991\pi$
$953$			5.73373e11i			0.695129i	−0.937656	+	0.347564i	$0.887009\pi$
$954$		0			0
$955$	1.77029e11			0.212829
$956$	−2.38012e11			−0.284949
$957$		0			0
$958$		−	8.53787e10i		−	0.101365i
$959$	7.18608e11			0.849606
$960$		0			0
$961$	8.32871e11			0.976526
$962$			2.87480e11i			0.335666i
$963$		0			0
$964$		−	5.57992e10i		−	0.0646130i
$965$		−	1.87600e11i		−	0.216333i
$966$		0			0
$967$	−1.36489e12			−1.56096			−0.780479	−	0.625182i	$-0.785025\pi$
$967$	−1.36489e12			−1.56096			−0.780479	+	0.625182i	$0.785025\pi$
$968$			6.34878e11i			0.723084i
$969$		0			0
$970$	4.30296e10			0.0486050
$971$		−	1.57922e12i		−	1.77650i	−0.459361	−	0.888250i	$-0.651922\pi$
$971$		−	1.57922e12i		−	1.77650i	0.459361	−	0.888250i	$-0.348078\pi$
$972$		0			0
$973$	−6.12853e11			−0.683762
$974$	1.00951e12			1.12169
$975$		0			0
$976$	−1.84929e11			−0.203801
$977$		−	5.51758e11i		−	0.605578i	−0.953058	−	0.302789i	$-0.902082\pi$
$977$		−	5.51758e11i		−	0.605578i	0.953058	−	0.302789i	$-0.0979178\pi$
$978$		0			0
$979$			2.72493e12i			2.96636i
$980$	6.44619e10			0.0698874
$981$		0			0
$982$		−	3.69806e11i		−	0.397675i
$983$			8.60977e11i			0.922099i	0.887374	+	0.461050i	$0.152527\pi$
$983$			8.60977e11i			0.922099i	−0.887374	+	0.461050i	$0.847473\pi$
$984$		0			0
$985$	−7.73684e10			−0.0821900
$986$	9.10619e11			0.963450
$987$		0			0
$988$	1.72090e11	−	1.74477e10i	0.180604	−	0.0183109i
$989$	5.15358e11			0.538671
$990$		0			0
$991$		−	1.27354e11i		−	0.132044i	−0.997818	−	0.0660220i	$-0.978969\pi$
$991$		−	1.27354e11i		−	0.132044i	0.997818	−	0.0660220i	$-0.0210308\pi$
$992$	2.62278e10			0.0270841
$993$		0			0
$994$	3.90292e11			0.399801
$995$	3.17067e11			0.323488
$996$		0			0
$997$	−1.40891e12			−1.42595			−0.712975	−	0.701190i	$-0.752653\pi$
$997$	−1.40891e12			−1.42595			−0.712975	+	0.701190i	$0.752653\pi$
$998$			1.07659e12i			1.08525i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.9.d.a.37.4		12
3.2	odd	2		38.9.b.a.37.9	yes	12
12.11	even	2		304.9.e.e.113.9		12
19.18	odd	2	inner	342.9.d.a.37.10		12
57.56	even	2		38.9.b.a.37.4	✓	12
228.227	odd	2		304.9.e.e.113.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.b.a.37.4	✓	12	57.56	even	2
38.9.b.a.37.9	yes	12	3.2	odd	2
304.9.e.e.113.4		12	228.227	odd	2
304.9.e.e.113.9		12	12.11	even	2
342.9.d.a.37.4		12	1.1	even	1	trivial
342.9.d.a.37.10		12	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	342.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-11.3137i q^{2} -128.000 q^{4} +154.845 q^{5} +1585.07 q^{7} +1448.15i q^{8} +O(q^{10})\)	\(q-11.3137i q^{2} -128.000 q^{4} +154.845 q^{5} +1585.07 q^{7} +1448.15i q^{8} -1751.87i q^{10} +25549.2 q^{11} +10369.3i q^{13} -17933.0i q^{14} +16384.0 q^{16} +132967. q^{17} +(13145.5 + 129656. i) q^{19} -19820.1 q^{20} -289057. i q^{22} +334889. q^{23} -366648. q^{25} +117316. q^{26} -202889. q^{28} +605325. i q^{29} +141494. i q^{31} -185364. i q^{32} -1.50435e6i q^{34} +245440. q^{35} +2.45048e6i q^{37} +(1.46689e6 - 148724. i) q^{38} +224239. i q^{40} -2.38955e6i q^{41} +1.53889e6 q^{43} -3.27030e6 q^{44} -3.78883e6i q^{46} -5.01339e6 q^{47} -3.25235e6 q^{49} +4.14815e6i q^{50} -1.32728e6i q^{52} +1.18301e7i q^{53} +3.95616e6 q^{55} +2.29543e6i q^{56} +6.84847e6 q^{58} -7.32369e6i q^{59} -1.12872e7 q^{61} +1.60082e6 q^{62} -2.09715e6 q^{64} +1.60564e6i q^{65} +1.87052e7i q^{67} -1.70197e7 q^{68} -2.77684e6i q^{70} +2.17638e7i q^{71} -8.89170e6 q^{73} +2.77241e7 q^{74} +(-1.68262e6 - 1.65960e7i) q^{76} +4.04974e7 q^{77} -5.26495e7i q^{79} +2.53698e6 q^{80} -2.70346e7 q^{82} -3.76587e7 q^{83} +2.05892e7 q^{85} -1.74106e7i q^{86} +3.69992e7i q^{88} +1.06654e8i q^{89} +1.64362e7i q^{91} -4.28658e7 q^{92} +5.67200e7i q^{94} +(2.03551e6 + 2.00766e7i) q^{95} +2.45621e7i q^{97} +3.67961e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7}+O(q^{10}) \) 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7	\( 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7} + 12546 q^{11} + 196608 q^{16} - 270810 q^{17} + 41512 q^{19} + 71424 q^{20} + 823956 q^{23} + 865538 q^{25} + 431616 q^{26} + 694016 q^{28} + 1194378 q^{35} - 998784 q^{38} + 7586646 q^{43} - 1605888 q^{44} + 20260530 q^{47} - 19498842 q^{49} - 14858554 q^{55} - 5506560 q^{58} - 41363266 q^{61} - 32266752 q^{62} - 25165824 q^{64} + 34663680 q^{68} + 87906498 q^{73} + 2149632 q^{74} - 5313536 q^{76} + 78817962 q^{77} - 9142272 q^{80} - 49609728 q^{82} + 55944960 q^{83} + 25440254 q^{85} - 105466368 q^{92} - 81396774 q^{95}+O(q^{100}) \) 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7 + 12546 * q^11 + 196608 * q^16 - 270810 * q^17 + 41512 * q^19 + 71424 * q^20 + 823956 * q^23 + 865538 * q^25 + 431616 * q^26 + 694016 * q^28 + 1194378 * q^35 - 998784 * q^38 + 7586646 * q^43 - 1605888 * q^44 + 20260530 * q^47 - 19498842 * q^49 - 14858554 * q^55 - 5506560 * q^58 - 41363266 * q^61 - 32266752 * q^62 - 25165824 * q^64 + 34663680 * q^68 + 87906498 * q^73 + 2149632 * q^74 - 5313536 * q^76 + 78817962 * q^77 - 9142272 * q^80 - 49609728 * q^82 + 55944960 * q^83 + 25440254 * q^85 - 105466368 * q^92 - 81396774 * q^95

Introduction

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