LMFDB - Embedded newform 1369.2.b.c.1368.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1369,2,Mod(1368,1369)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1369, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1369.1368");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1369 = 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1369.b (of order $2$, degree $1$, not 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:	$10.9315200367$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 37)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1368.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1369.1368
Dual form			1369.2.b.c.1368.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} +3.00000 q^{3} -2.00000 q^{4} +2.00000i q^{5} -6.00000i q^{6} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} +3.00000 q^{3} -2.00000 q^{4} +2.00000i q^{5} -6.00000i q^{6} -1.00000 q^{7} +6.00000 q^{9} +4.00000 q^{10} +5.00000 q^{11} -6.00000 q^{12} +2.00000i q^{13} +2.00000i q^{14} +6.00000i q^{15} -4.00000 q^{16} -12.0000i q^{18} -4.00000i q^{20} -3.00000 q^{21} -10.0000i q^{22} -2.00000i q^{23} +1.00000 q^{25} +4.00000 q^{26} +9.00000 q^{27} +2.00000 q^{28} +6.00000i q^{29} +12.0000 q^{30} -4.00000i q^{31} +8.00000i q^{32} +15.0000 q^{33} -2.00000i q^{35} -12.0000 q^{36} +6.00000i q^{39} +9.00000 q^{41} +6.00000i q^{42} -2.00000i q^{43} -10.0000 q^{44} +12.0000i q^{45} -4.00000 q^{46} -9.00000 q^{47} -12.0000 q^{48} -6.00000 q^{49} -2.00000i q^{50} -4.00000i q^{52} +1.00000 q^{53} -18.0000i q^{54} +10.0000i q^{55} +12.0000 q^{58} -8.00000i q^{59} -12.0000i q^{60} -8.00000i q^{61} -8.00000 q^{62} -6.00000 q^{63} +8.00000 q^{64} -4.00000 q^{65} -30.0000i q^{66} -8.00000 q^{67} -6.00000i q^{69} -4.00000 q^{70} +9.00000 q^{71} +1.00000 q^{73} +3.00000 q^{75} -5.00000 q^{77} +12.0000 q^{78} -4.00000i q^{79} -8.00000i q^{80} +9.00000 q^{81} -18.0000i q^{82} -15.0000 q^{83} +6.00000 q^{84} -4.00000 q^{86} +18.0000i q^{87} +4.00000i q^{89} +24.0000 q^{90} -2.00000i q^{91} +4.00000i q^{92} -12.0000i q^{93} +18.0000i q^{94} +24.0000i q^{96} -4.00000i q^{97} +12.0000i q^{98} +30.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 4 q^{4} - 2 q^{7} + 12 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 4 * q^4 - 2 * q^7 + 12 * q^9	$ 2 q + 6 q^{3} - 4 q^{4} - 2 q^{7} + 12 q^{9} + 8 q^{10} + 10 q^{11} - 12 q^{12} - 8 q^{16} - 6 q^{21} + 2 q^{25} + 8 q^{26} + 18 q^{27} + 4 q^{28} + 24 q^{30} + 30 q^{33} - 24 q^{36} + 18 q^{41} - 20 q^{44} - 8 q^{46} - 18 q^{47} - 24 q^{48} - 12 q^{49} + 2 q^{53} + 24 q^{58} - 16 q^{62} - 12 q^{63} + 16 q^{64} - 8 q^{65} - 16 q^{67} - 8 q^{70} + 18 q^{71} + 2 q^{73} + 6 q^{75} - 10 q^{77} + 24 q^{78} + 18 q^{81} - 30 q^{83} + 12 q^{84} - 8 q^{86} + 48 q^{90} + 60 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 4 * q^4 - 2 * q^7 + 12 * q^9 + 8 * q^10 + 10 * q^11 - 12 * q^12 - 8 * q^16 - 6 * q^21 + 2 * q^25 + 8 * q^26 + 18 * q^27 + 4 * q^28 + 24 * q^30 + 30 * q^33 - 24 * q^36 + 18 * q^41 - 20 * q^44 - 8 * q^46 - 18 * q^47 - 24 * q^48 - 12 * q^49 + 2 * q^53 + 24 * q^58 - 16 * q^62 - 12 * q^63 + 16 * q^64 - 8 * q^65 - 16 * q^67 - 8 * q^70 + 18 * q^71 + 2 * q^73 + 6 * q^75 - 10 * q^77 + 24 * q^78 + 18 * q^81 - 30 * q^83 + 12 * q^84 - 8 * q^86 + 48 * q^90 + 60 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1369\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 2.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 2.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	−2.00000	−1.00000
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	− 6.00000i	− 2.44949i
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	4.00000	1.26491
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	−6.00000	−1.73205
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	2.00000i	0.534522i
$15$	6.00000i	1.54919i
$16$	−4.00000	−1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 12.0000i	− 2.82843i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	− 4.00000i	− 0.894427i
$21$	−3.00000	−0.654654
$22$	− 10.0000i	− 2.13201i
$23$	− 2.00000i	− 0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$	− 2.00000i	− 0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	9.00000	1.73205
$28$	2.00000	0.377964
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	12.0000	2.19089
$31$	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$	− 4.00000i	− 0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	8.00000i	1.41421i
$33$	15.0000	2.61116
$34$	0	0
$35$	− 2.00000i	− 0.338062i
$36$	−12.0000	−2.00000
$37$	0	0
$37$	0	0
$38$	0	0
$39$	6.00000i	0.960769i
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	6.00000i	0.925820i
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	−10.0000	−1.50756
$45$	12.0000i	1.78885i
$46$	−4.00000	−0.589768
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	−12.0000	−1.73205
$49$	−6.00000	−0.857143
$50$	− 2.00000i	− 0.282843i
$51$	0	0
$52$	− 4.00000i	− 0.554700i
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	− 18.0000i	− 2.44949i
$55$	10.0000i	1.34840i
$56$	0	0
$57$	0	0
$58$	12.0000	1.57568
$59$	− 8.00000i	− 1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$	− 8.00000i	− 1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$	− 12.0000i	− 1.54919i
$61$	− 8.00000i	− 1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$	− 8.00000i	− 1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$	−8.00000	−1.01600
$63$	−6.00000	−0.755929
$64$	8.00000	1.00000
$65$	−4.00000	−0.496139
$66$	− 30.0000i	− 3.69274i
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	− 6.00000i	− 0.722315i
$70$	−4.00000	−0.478091
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	−5.00000	−0.569803
$78$	12.0000	1.35873
$79$	− 4.00000i	− 0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$	− 4.00000i	− 0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$	− 8.00000i	− 0.894427i
$81$	9.00000	1.00000
$82$	− 18.0000i	− 1.98777i
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	6.00000	0.654654
$85$	0	0
$86$	−4.00000	−0.431331
$87$	18.0000i	1.92980i
$88$	0	0
$89$	4.00000i	0.423999i	0.977270	+	0.212000i	$0.0679975\pi$
$89$	4.00000i	0.423999i	−0.977270	+	0.212000i	$0.932002\pi$
$90$	24.0000	2.52982
$91$	− 2.00000i	− 0.209657i
$92$	4.00000i	0.417029i
$93$	− 12.0000i	− 1.24434i
$94$	18.0000i	1.85656i
$95$	0	0
$96$	24.0000i	2.44949i
$97$	− 4.00000i	− 0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$	− 4.00000i	− 0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$	12.0000i	1.21218i
$99$	30.0000	3.01511
$100$	−2.00000	−0.200000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	18.0000i	1.77359i	0.462160	+	0.886796i	$0.347074\pi$
$103$	18.0000i	1.77359i	−0.462160	+	0.886796i	$0.652926\pi$
$104$	0	0
$105$	− 6.00000i	− 0.585540i
$106$	− 2.00000i	− 0.194257i
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−18.0000	−1.73205
$109$	16.0000i	1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$	16.0000i	1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$	20.0000	1.90693
$111$	0	0
$112$	4.00000	0.377964
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	− 12.0000i	− 1.11417i
$117$	12.0000i	1.10940i
$118$	−16.0000	−1.47292
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	−16.0000	−1.44857
$123$	27.0000	2.43451
$124$	8.00000i	0.718421i
$125$	12.0000i	1.07331i
$126$	12.0000i	1.06904i
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	− 6.00000i	− 0.528271i
$130$	8.00000i	0.701646i
$131$	− 12.0000i	− 1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$	− 12.0000i	− 1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	−30.0000	−2.61116
$133$	0	0
$134$	16.0000i	1.38219i
$135$	18.0000i	1.54919i
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−12.0000	−1.02151
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	4.00000i	0.338062i
$141$	−27.0000	−2.27381
$142$	− 18.0000i	− 1.51053i
$143$	10.0000i	0.836242i
$144$	−24.0000	−2.00000
$145$	−12.0000	−0.996546
$146$	− 2.00000i	− 0.165521i
$147$	−18.0000	−1.48461
$148$	0	0
$149$	−5.00000	−0.409616	−0.204808	−	0.978802i	$-0.565657\pi$
$149$	−5.00000	−0.409616	−0.204808	+	0.978802i	$0.565657\pi$
$150$	− 6.00000i	− 0.489898i
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	0	0
$154$	10.0000i	0.805823i
$155$	8.00000	0.642575
$156$	− 12.0000i	− 0.960769i
$157$	23.0000	1.83560	0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000	1.83560	0.917800	+	0.397043i	$0.129964\pi$
$158$	−8.00000	−0.636446
$159$	3.00000	0.237915
$160$	−16.0000	−1.26491
$161$	2.00000i	0.157622i
$162$	− 18.0000i	− 1.41421i
$163$	− 18.0000i	− 1.40987i	−0.709273	−	0.704934i	$-0.750976\pi$
$163$	− 18.0000i	− 1.40987i	0.709273	−	0.704934i	$-0.249024\pi$
$164$	−18.0000	−1.40556
$165$	30.0000i	2.33550i
$166$	30.0000i	2.32845i
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	36.0000	2.72915
$175$	−1.00000	−0.0755929
$176$	−20.0000	−1.50756
$177$	− 24.0000i	− 1.80395i
$178$	8.00000	0.599625
$179$	18.0000i	1.34538i	0.739923	+	0.672692i	$0.234862\pi$
$179$	18.0000i	1.34538i	−0.739923	+	0.672692i	$0.765138\pi$
$180$	− 24.0000i	− 1.78885i
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	−4.00000	−0.296500
$183$	− 24.0000i	− 1.77413i
$184$	0	0
$185$	0	0
$186$	−24.0000	−1.75977
$187$	0	0
$188$	18.0000	1.31278
$189$	−9.00000	−0.654654
$190$	0	0
$191$	4.00000i	0.289430i	0.989473	+	0.144715i	$0.0462265\pi$
$191$	4.00000i	0.289430i	−0.989473	+	0.144715i	$0.953773\pi$
$192$	24.0000	1.73205
$193$	26.0000i	1.87152i	0.352636	+	0.935760i	$0.385285\pi$
$193$	26.0000i	1.87152i	−0.352636	+	0.935760i	$0.614715\pi$
$194$	−8.00000	−0.574367
$195$	−12.0000	−0.859338
$196$	12.0000	0.857143
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	− 60.0000i	− 4.26401i
$199$	2.00000i	0.141776i	0.997484	+	0.0708881i	$0.0225833\pi$
$199$	2.00000i	0.141776i	−0.997484	+	0.0708881i	$0.977417\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	6.00000i	0.422159i
$203$	− 6.00000i	− 0.421117i
$204$	0	0
$205$	18.0000i	1.25717i
$206$	36.0000	2.50824
$207$	− 12.0000i	− 0.834058i
$208$	− 8.00000i	− 0.554700i
$209$	0	0
$210$	−12.0000	−0.828079
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	−2.00000	−0.137361
$213$	27.0000	1.85001
$214$	24.0000i	1.64061i
$215$	4.00000	0.272798
$216$	0	0
$217$	4.00000i	0.271538i
$218$	32.0000	2.16731
$219$	3.00000	0.202721
$220$	− 20.0000i	− 1.34840i
$221$	0	0
$222$	0	0
$223$	−17.0000	−1.13840	−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000	−1.13840	−0.569202	+	0.822198i	$0.692748\pi$
$224$	− 8.00000i	− 0.534522i
$225$	6.00000	0.400000
$226$	−36.0000	−2.39468
$227$	16.0000i	1.06196i	0.847385	+	0.530979i	$0.178176\pi$
$227$	16.0000i	1.06196i	−0.847385	+	0.530979i	$0.821824\pi$
$228$	0	0
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	− 8.00000i	− 0.527504i
$231$	−15.0000	−0.986928
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	24.0000	1.56893
$235$	− 18.0000i	− 1.17419i
$236$	16.0000i	1.04151i
$237$	− 12.0000i	− 0.779484i
$238$	0	0
$239$	6.00000i	0.388108i	0.980991	+	0.194054i	$0.0621637\pi$
$239$	6.00000i	0.388108i	−0.980991	+	0.194054i	$0.937836\pi$
$240$	− 24.0000i	− 1.54919i
$241$	− 14.0000i	− 0.901819i	−0.892570	−	0.450910i	$-0.851100\pi$
$241$	− 14.0000i	− 0.901819i	0.892570	−	0.450910i	$-0.148900\pi$
$242$	− 28.0000i	− 1.79991i
$243$	0	0
$244$	16.0000i	1.02430i
$245$	− 12.0000i	− 0.766652i
$246$	− 54.0000i	− 3.44291i
$247$	0	0
$248$	0	0
$249$	−45.0000	−2.85176
$250$	24.0000	1.51789
$251$	− 2.00000i	− 0.126239i	−0.998006	−	0.0631194i	$-0.979895\pi$
$251$	− 2.00000i	− 0.126239i	0.998006	−	0.0631194i	$-0.0201049\pi$
$252$	12.0000	0.755929
$253$	− 10.0000i	− 0.628695i
$254$	− 2.00000i	− 0.125491i
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−12.0000	−0.747087
$259$	0	0
$260$	8.00000	0.496139
$261$	36.0000i	2.22834i
$262$	−24.0000	−1.48272
$263$	−19.0000	−1.17159	−0.585795	−	0.810459i	$-0.699218\pi$
$263$	−19.0000	−1.17159	−0.585795	+	0.810459i	$0.699218\pi$
$264$	0	0
$265$	2.00000i	0.122859i
$266$	0	0
$267$	12.0000i	0.734388i
$268$	16.0000	0.977356
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	36.0000	2.19089
$271$	−31.0000	−1.88312	−0.941558	−	0.336851i	$-0.890638\pi$
$271$	−31.0000	−1.88312	−0.941558	+	0.336851i	$0.890638\pi$
$272$	0	0
$273$	− 6.00000i	− 0.363137i
$274$	12.0000i	0.724947i
$275$	5.00000	0.301511
$276$	12.0000i	0.722315i
$277$	12.0000i	0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$	12.0000i	0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$	8.00000i	0.479808i
$279$	− 24.0000i	− 1.43684i
$280$	0	0
$281$	− 12.0000i	− 0.715860i	−0.933748	−	0.357930i	$-0.883483\pi$
$281$	− 12.0000i	− 0.715860i	0.933748	−	0.357930i	$-0.116517\pi$
$282$	54.0000i	3.21565i
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−18.0000	−1.06810
$285$	0	0
$286$	20.0000	1.18262
$287$	−9.00000	−0.531253
$288$	48.0000i	2.82843i
$289$	17.0000	1.00000
$290$	24.0000i	1.40933i
$291$	− 12.0000i	− 0.703452i
$292$	−2.00000	−0.117041
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	36.0000i	2.09956i
$295$	16.0000	0.931556
$296$	0	0
$297$	45.0000	2.61116
$298$	10.0000i	0.579284i
$299$	4.00000	0.231326
$300$	−6.00000	−0.346410
$301$	2.00000i	0.115278i
$302$	32.0000i	1.84139i
$303$	−9.00000	−0.517036
$304$	0	0
$305$	16.0000	0.916157
$306$	0	0
$307$	17.0000	0.970241	0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000	0.970241	0.485121	+	0.874447i	$0.338776\pi$
$308$	10.0000	0.569803
$309$	54.0000i	3.07195i
$310$	− 16.0000i	− 0.908739i
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	− 22.0000i	− 1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	− 22.0000i	− 1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$	− 46.0000i	− 2.59593i
$315$	− 12.0000i	− 0.676123i
$316$	8.00000i	0.450035i
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	− 6.00000i	− 0.336463i
$319$	30.0000i	1.67968i
$320$	16.0000i	0.894427i
$321$	−36.0000	−2.00932
$322$	4.00000	0.222911
$323$	0	0
$324$	−18.0000	−1.00000
$325$	2.00000i	0.110940i
$326$	−36.0000	−1.99386
$327$	48.0000i	2.65441i
$328$	0	0
$329$	9.00000	0.496186
$330$	60.0000	3.30289
$331$	2.00000i	0.109930i	0.998488	+	0.0549650i	$0.0175047\pi$
$331$	2.00000i	0.109930i	−0.998488	+	0.0549650i	$0.982495\pi$
$332$	30.0000	1.64646
$333$	0	0
$334$	24.0000	1.31322
$335$	− 16.0000i	− 0.874173i
$336$	12.0000	0.654654
$337$	25.0000	1.36184	0.680918	−	0.732359i	$-0.261581\pi$
$337$	25.0000	1.36184	0.680918	+	0.732359i	$0.261581\pi$
$338$	− 18.0000i	− 0.979071i
$339$	− 54.0000i	− 2.93288i
$340$	0	0
$341$	− 20.0000i	− 1.08306i
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	12.0000	0.646058
$346$	18.0000i	0.967686i
$347$	− 10.0000i	− 0.536828i	−0.963304	−	0.268414i	$-0.913500\pi$
$347$	− 10.0000i	− 0.536828i	0.963304	−	0.268414i	$-0.0864995\pi$
$348$	− 36.0000i	− 1.92980i
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	2.00000i	0.106904i
$351$	18.0000i	0.960769i
$352$	40.0000i	2.13201i
$353$	8.00000i	0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$	8.00000i	0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$	−48.0000	−2.55117
$355$	18.0000i	0.955341i
$356$	− 8.00000i	− 0.423999i
$357$	0	0
$358$	36.0000	1.90266
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	− 10.0000i	− 0.525588i
$363$	42.0000	2.20443
$364$	4.00000i	0.209657i
$365$	2.00000i	0.104685i
$366$	−48.0000	−2.50900
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	8.00000i	0.417029i
$369$	54.0000	2.81113
$370$	0	0
$371$	−1.00000	−0.0519174
$372$	24.0000i	1.24434i
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	0	0
$375$	36.0000i	1.85903i
$376$	0	0
$377$	−12.0000	−0.618031
$378$	18.0000i	0.925820i
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	0	0
$381$	3.00000	0.153695
$382$	8.00000	0.409316
$383$	− 20.0000i	− 1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$	− 20.0000i	− 1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	0	0
$385$	− 10.0000i	− 0.509647i
$386$	52.0000	2.64673
$387$	− 12.0000i	− 0.609994i
$388$	8.00000i	0.406138i
$389$	− 4.00000i	− 0.202808i	−0.994845	−	0.101404i	$-0.967667\pi$
$389$	− 4.00000i	− 0.202808i	0.994845	−	0.101404i	$-0.0323335\pi$
$390$	24.0000i	1.21529i
$391$	0	0
$392$	0	0
$393$	− 36.0000i	− 1.81596i
$394$	− 6.00000i	− 0.302276i
$395$	8.00000	0.402524
$396$	−60.0000	−3.01511
$397$	5.00000	0.250943	0.125471	−	0.992097i	$-0.459956\pi$
$397$	5.00000	0.250943	0.125471	+	0.992097i	$0.459956\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	−4.00000	−0.200000
$401$	18.0000i	0.898877i	0.893311	+	0.449439i	$0.148376\pi$
$401$	18.0000i	0.898877i	−0.893311	+	0.449439i	$0.851624\pi$
$402$	48.0000i	2.39402i
$403$	8.00000	0.398508
$404$	6.00000	0.298511
$405$	18.0000i	0.894427i
$406$	−12.0000	−0.595550
$407$	0	0
$408$	0	0
$409$	20.0000i	0.988936i	0.869196	+	0.494468i	$0.164637\pi$
$409$	20.0000i	0.988936i	−0.869196	+	0.494468i	$0.835363\pi$
$410$	36.0000	1.77791
$411$	−18.0000	−0.887875
$412$	− 36.0000i	− 1.77359i
$413$	8.00000i	0.393654i
$414$	−24.0000	−1.17954
$415$	− 30.0000i	− 1.47264i
$416$	−16.0000	−0.784465
$417$	−12.0000	−0.587643
$418$	0	0
$419$	7.00000	0.341972	0.170986	−	0.985273i	$-0.445305\pi$
$419$	7.00000	0.341972	0.170986	+	0.985273i	$0.445305\pi$
$420$	12.0000i	0.585540i
$421$	− 24.0000i	− 1.16969i	−0.811146	−	0.584844i	$-0.801156\pi$
$421$	− 24.0000i	− 1.16969i	0.811146	−	0.584844i	$-0.198844\pi$
$422$	26.0000i	1.26566i
$423$	−54.0000	−2.62557
$424$	0	0
$425$	0	0
$426$	− 54.0000i	− 2.61631i
$427$	8.00000i	0.387147i
$428$	24.0000	1.16008
$429$	30.0000i	1.44841i
$430$	− 8.00000i	− 0.385794i
$431$	− 30.0000i	− 1.44505i	−0.691345	−	0.722525i	$-0.742982\pi$
$431$	− 30.0000i	− 1.44505i	0.691345	−	0.722525i	$-0.257018\pi$
$432$	−36.0000	−1.73205
$433$	9.00000	0.432512	0.216256	−	0.976337i	$-0.430615\pi$
$433$	9.00000	0.432512	0.216256	+	0.976337i	$0.430615\pi$
$434$	8.00000	0.384012
$435$	−36.0000	−1.72607
$436$	− 32.0000i	− 1.53252i
$437$	0	0
$438$	− 6.00000i	− 0.286691i
$439$	28.0000i	1.33637i	0.743996	+	0.668184i	$0.232928\pi$
$439$	28.0000i	1.33637i	−0.743996	+	0.668184i	$0.767072\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	−1.00000	−0.0475114	−0.0237557	−	0.999718i	$-0.507562\pi$
$443$	−1.00000	−0.0475114	−0.0237557	+	0.999718i	$0.507562\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	34.0000i	1.60995i
$447$	−15.0000	−0.709476
$448$	−8.00000	−0.377964
$449$	− 36.0000i	− 1.69895i	−0.527633	−	0.849473i	$-0.676920\pi$
$449$	− 36.0000i	− 1.69895i	0.527633	−	0.849473i	$-0.323080\pi$
$450$	− 12.0000i	− 0.565685i
$451$	45.0000	2.11897
$452$	36.0000i	1.69330i
$453$	−48.0000	−2.25524
$454$	32.0000	1.50183
$455$	4.00000	0.187523
$456$	0	0
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	− 14.0000i	− 0.654177i
$459$	0	0
$460$	−8.00000	−0.373002
$461$	− 30.0000i	− 1.39724i	−0.715493	−	0.698620i	$-0.753798\pi$
$461$	− 30.0000i	− 1.39724i	0.715493	−	0.698620i	$-0.246202\pi$
$462$	30.0000i	1.39573i
$463$	22.0000i	1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$	22.0000i	1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$	− 24.0000i	− 1.11417i
$465$	24.0000	1.11297
$466$	12.0000i	0.555889i
$467$	2.00000i	0.0925490i	0.998929	+	0.0462745i	$0.0147349\pi$
$467$	2.00000i	0.0925490i	−0.998929	+	0.0462745i	$0.985265\pi$
$468$	− 24.0000i	− 1.10940i
$469$	8.00000	0.369406
$470$	−36.0000	−1.66056
$471$	69.0000	3.17935
$472$	0	0
$473$	− 10.0000i	− 0.459800i
$474$	−24.0000	−1.10236
$475$	0	0
$476$	0	0
$477$	6.00000	0.274721
$478$	12.0000	0.548867
$479$	− 14.0000i	− 0.639676i	−0.947472	−	0.319838i	$-0.896371\pi$
$479$	− 14.0000i	− 0.639676i	0.947472	−	0.319838i	$-0.103629\pi$
$480$	−48.0000	−2.19089
$481$	0	0
$482$	−28.0000	−1.27537
$483$	6.00000i	0.273009i
$484$	−28.0000	−1.27273
$485$	8.00000	0.363261
$486$	0	0
$487$	24.0000i	1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$	24.0000i	1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$	0	0
$489$	− 54.0000i	− 2.44196i
$490$	−24.0000	−1.08421
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	−54.0000	−2.43451
$493$	0	0
$494$	0	0
$495$	60.0000i	2.69680i
$496$	16.0000i	0.718421i
$497$	−9.00000	−0.403705
$498$	90.0000i	4.03300i
$499$	12.0000i	0.537194i	0.963253	+	0.268597i	$0.0865599\pi$
$499$	12.0000i	0.537194i	−0.963253	+	0.268597i	$0.913440\pi$
$500$	− 24.0000i	− 1.07331i
$501$	36.0000i	1.60836i
$502$	−4.00000	−0.178529
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	0	0
$505$	− 6.00000i	− 0.266996i
$506$	−20.0000	−0.889108
$507$	27.0000	1.19911
$508$	−2.00000	−0.0887357
$509$	31.0000	1.37405	0.687025	−	0.726633i	$-0.258916\pi$
$509$	31.0000	1.37405	0.687025	+	0.726633i	$0.258916\pi$
$510$	0	0
$511$	−1.00000	−0.0442374
$512$	− 32.0000i	− 1.41421i
$513$	0	0
$514$	0	0
$515$	−36.0000	−1.58635
$516$	12.0000i	0.528271i
$517$	−45.0000	−1.97910
$518$	0	0
$519$	−27.0000	−1.18517
$520$	0	0
$521$	33.0000	1.44576	0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$522$	72.0000	3.15135
$523$	22.0000i	0.961993i	0.876723	+	0.480996i	$0.159725\pi$
$523$	22.0000i	0.961993i	−0.876723	+	0.480996i	$0.840275\pi$
$524$	24.0000i	1.04844i
$525$	−3.00000	−0.130931
$526$	38.0000i	1.65688i
$527$	0	0
$528$	−60.0000	−2.61116
$529$	19.0000	0.826087
$530$	4.00000	0.173749
$531$	− 48.0000i	− 2.08302i
$532$	0	0
$533$	18.0000i	0.779667i
$534$	24.0000	1.03858
$535$	− 24.0000i	− 1.03761i
$536$	0	0
$537$	54.0000i	2.33027i
$538$	12.0000i	0.517357i
$539$	−30.0000	−1.29219
$540$	− 36.0000i	− 1.54919i
$541$	− 20.0000i	− 0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	− 20.0000i	− 0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$	62.0000i	2.66313i
$543$	15.0000	0.643712
$544$	0	0
$545$	−32.0000	−1.37073
$546$	−12.0000	−0.513553
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	12.0000	0.512615
$549$	− 48.0000i	− 2.04859i
$550$	− 10.0000i	− 0.426401i
$551$	0	0
$552$	0	0
$553$	4.00000i	0.170097i
$554$	24.0000	1.01966
$555$	0	0
$556$	8.00000	0.339276
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	−48.0000	−2.03200
$559$	4.00000	0.169182
$560$	8.00000i	0.338062i
$561$	0	0
$562$	−24.0000	−1.01238
$563$	30.0000i	1.26435i	0.774826	+	0.632175i	$0.217837\pi$
$563$	30.0000i	1.26435i	−0.774826	+	0.632175i	$0.782163\pi$
$564$	54.0000	2.27381
$565$	36.0000	1.51453
$566$	8.00000	0.336265
$567$	−9.00000	−0.377964
$568$	0	0
$569$	− 24.0000i	− 1.00613i	−0.864248	−	0.503066i	$-0.832205\pi$
$569$	− 24.0000i	− 1.00613i	0.864248	−	0.503066i	$-0.167795\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	− 20.0000i	− 0.836242i
$573$	12.0000i	0.501307i
$574$	18.0000i	0.751305i
$575$	− 2.00000i	− 0.0834058i
$576$	48.0000	2.00000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 34.0000i	− 1.41421i
$579$	78.0000i	3.24157i
$580$	24.0000	0.996546
$581$	15.0000	0.622305
$582$	−24.0000	−0.994832
$583$	5.00000	0.207079
$584$	0	0
$585$	−24.0000	−0.992278
$586$	4.00000i	0.165238i
$587$	− 32.0000i	− 1.32078i	−0.750922	−	0.660391i	$-0.770391\pi$
$587$	− 32.0000i	− 1.32078i	0.750922	−	0.660391i	$-0.229609\pi$
$588$	36.0000	1.48461
$589$	0	0
$590$	− 32.0000i	− 1.31742i
$591$	9.00000	0.370211
$592$	0	0
$593$	−5.00000	−0.205325	−0.102663	−	0.994716i	$-0.532736\pi$
$593$	−5.00000	−0.205325	−0.102663	+	0.994716i	$0.532736\pi$
$594$	− 90.0000i	− 3.69274i
$595$	0	0
$596$	10.0000	0.409616
$597$	6.00000i	0.245564i
$598$	− 8.00000i	− 0.327144i
$599$	1.00000	0.0408589	0.0204294	−	0.999791i	$-0.493497\pi$
$599$	1.00000	0.0408589	0.0204294	+	0.999791i	$0.493497\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	4.00000	0.163028
$603$	−48.0000	−1.95471
$604$	32.0000	1.30206
$605$	28.0000i	1.13836i
$606$	18.0000i	0.731200i
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	0	0
$609$	− 18.0000i	− 0.729397i
$610$	− 32.0000i	− 1.29564i
$611$	− 18.0000i	− 0.728202i
$612$	0	0
$613$	−15.0000	−0.605844	−0.302922	−	0.953015i	$-0.597962\pi$
$613$	−15.0000	−0.605844	−0.302922	+	0.953015i	$0.597962\pi$
$614$	− 34.0000i	− 1.37213i
$615$	54.0000i	2.17749i
$616$	0	0
$617$	−17.0000	−0.684394	−0.342197	−	0.939628i	$-0.611171\pi$
$617$	−17.0000	−0.684394	−0.342197	+	0.939628i	$0.611171\pi$
$618$	108.000	4.34440
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	−16.0000	−0.642575
$621$	− 18.0000i	− 0.722315i
$622$	0	0
$623$	− 4.00000i	− 0.160257i
$624$	− 24.0000i	− 0.960769i
$625$	−19.0000	−0.760000
$626$	−44.0000	−1.75859
$627$	0	0
$628$	−46.0000	−1.83560
$629$	0	0
$630$	−24.0000	−0.956183
$631$	− 28.0000i	− 1.11466i	−0.830290	−	0.557331i	$-0.811825\pi$
$631$	− 28.0000i	− 1.11466i	0.830290	−	0.557331i	$-0.188175\pi$
$632$	0	0
$633$	−39.0000	−1.55011
$634$	44.0000i	1.74746i
$635$	2.00000i	0.0793676i
$636$	−6.00000	−0.237915
$637$	− 12.0000i	− 0.475457i
$638$	60.0000	2.37542
$639$	54.0000	2.13621
$640$	0	0
$641$	−1.00000	−0.0394976	−0.0197488	−	0.999805i	$-0.506287\pi$
$641$	−1.00000	−0.0394976	−0.0197488	+	0.999805i	$0.506287\pi$
$642$	72.0000i	2.84161i
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	− 4.00000i	− 0.157622i
$645$	12.0000	0.472500
$646$	0	0
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	0	0
$649$	− 40.0000i	− 1.57014i
$650$	4.00000	0.156893
$651$	12.0000i	0.470317i
$652$	36.0000i	1.40987i
$653$	− 24.0000i	− 0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$	− 24.0000i	− 0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$	96.0000	3.75390
$655$	24.0000	0.937758
$656$	−36.0000	−1.40556
$657$	6.00000	0.234082
$658$	− 18.0000i	− 0.701713i
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	− 60.0000i	− 2.33550i
$661$	− 28.0000i	− 1.08907i	−0.838737	−	0.544537i	$-0.816705\pi$
$661$	− 28.0000i	− 1.08907i	0.838737	−	0.544537i	$-0.183295\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0000	0.464642
$668$	− 24.0000i	− 0.928588i
$669$	−51.0000	−1.97177
$670$	−32.0000	−1.23627
$671$	− 40.0000i	− 1.54418i
$672$	− 24.0000i	− 0.925820i
$673$	27.0000	1.04077	0.520387	−	0.853931i	$-0.325788\pi$
$673$	27.0000	1.04077	0.520387	+	0.853931i	$0.325788\pi$
$674$	− 50.0000i	− 1.92593i
$675$	9.00000	0.346410
$676$	−18.0000	−0.692308
$677$	11.0000	0.422764	0.211382	−	0.977403i	$-0.432204\pi$
$677$	11.0000	0.422764	0.211382	+	0.977403i	$0.432204\pi$
$678$	−108.000	−4.14772
$679$	4.00000i	0.153506i
$680$	0	0
$681$	48.0000i	1.83936i
$682$	−40.0000	−1.53168
$683$	− 18.0000i	− 0.688751i	−0.938832	−	0.344375i	$-0.888091\pi$
$683$	− 18.0000i	− 0.688751i	0.938832	−	0.344375i	$-0.111909\pi$
$684$	0	0
$685$	− 12.0000i	− 0.458496i
$686$	− 26.0000i	− 0.992685i
$687$	21.0000	0.801200
$688$	8.00000i	0.304997i
$689$	2.00000i	0.0761939i
$690$	− 24.0000i	− 0.913664i
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	18.0000	0.684257
$693$	−30.0000	−1.13961
$694$	−20.0000	−0.759190
$695$	− 8.00000i	− 0.303457i
$696$	0	0
$697$	0	0
$698$	− 12.0000i	− 0.454207i
$699$	−18.0000	−0.680823
$700$	2.00000	0.0755929
$701$	12.0000i	0.453234i	0.973984	+	0.226617i	$0.0727665\pi$
$701$	12.0000i	0.453234i	−0.973984	+	0.226617i	$0.927233\pi$
$702$	36.0000	1.35873
$703$	0	0
$704$	40.0000	1.50756
$705$	− 54.0000i	− 2.03376i
$706$	16.0000	0.602168
$707$	3.00000	0.112827
$708$	48.0000i	1.80395i
$709$	− 40.0000i	− 1.50223i	−0.660171	−	0.751116i	$-0.729516\pi$
$709$	− 40.0000i	− 1.50223i	0.660171	−	0.751116i	$-0.270484\pi$
$710$	36.0000	1.35106
$711$	− 24.0000i	− 0.900070i
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−20.0000	−0.747958
$716$	− 36.0000i	− 1.34538i
$717$	18.0000i	0.672222i
$718$	30.0000i	1.11959i
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	− 48.0000i	− 1.78885i
$721$	− 18.0000i	− 0.670355i
$722$	− 38.0000i	− 1.41421i
$723$	− 42.0000i	− 1.56200i
$724$	−10.0000	−0.371647
$725$	6.00000i	0.222834i
$726$	− 84.0000i	− 3.11753i
$727$	16.0000i	0.593407i	0.954970	+	0.296704i	$0.0958873\pi$
$727$	16.0000i	0.593407i	−0.954970	+	0.296704i	$0.904113\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	4.00000	0.148047
$731$	0	0
$732$	48.0000i	1.77413i
$733$	−7.00000	−0.258551	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−7.00000	−0.258551	−0.129275	+	0.991609i	$0.541265\pi$
$734$	− 16.0000i	− 0.590571i
$735$	− 36.0000i	− 1.32788i
$736$	16.0000	0.589768
$737$	−40.0000	−1.47342
$738$	− 108.000i	− 3.97553i
$739$	9.00000	0.331070	0.165535	−	0.986204i	$-0.447065\pi$
$739$	9.00000	0.331070	0.165535	+	0.986204i	$0.447065\pi$
$740$	0	0
$741$	0	0
$742$	2.00000i	0.0734223i
$743$	−21.0000	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$744$	0	0
$745$	− 10.0000i	− 0.366372i
$746$	− 38.0000i	− 1.39128i
$747$	−90.0000	−3.29293
$748$	0	0
$749$	12.0000	0.438470
$750$	72.0000	2.62907
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	36.0000	1.31278
$753$	− 6.00000i	− 0.218652i
$754$	24.0000i	0.874028i
$755$	− 32.0000i	− 1.16460i
$756$	18.0000	0.654654
$757$	50.0000i	1.81728i	0.417579	+	0.908640i	$0.362879\pi$
$757$	50.0000i	1.81728i	−0.417579	+	0.908640i	$0.637121\pi$
$758$	− 30.0000i	− 1.08965i
$759$	− 30.0000i	− 1.08893i
$760$	0	0
$761$	35.0000	1.26875	0.634375	−	0.773026i	$-0.281258\pi$
$761$	35.0000	1.26875	0.634375	+	0.773026i	$0.281258\pi$
$762$	− 6.00000i	− 0.217357i
$763$	− 16.0000i	− 0.579239i
$764$	− 8.00000i	− 0.289430i
$765$	0	0
$766$	−40.0000	−1.44526
$767$	16.0000	0.577727
$768$	48.0000	1.73205
$769$	26.0000i	0.937584i	0.883309	+	0.468792i	$0.155311\pi$
$769$	26.0000i	0.937584i	−0.883309	+	0.468792i	$0.844689\pi$
$770$	−20.0000	−0.720750
$771$	0	0
$772$	− 52.0000i	− 1.87152i
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	−24.0000	−0.862662
$775$	− 4.00000i	− 0.143684i
$776$	0	0
$777$	0	0
$778$	−8.00000	−0.286814
$779$	0	0
$780$	24.0000	0.859338
$781$	45.0000	1.61023
$782$	0	0
$783$	54.0000i	1.92980i
$784$	24.0000	0.857143
$785$	46.0000i	1.64181i
$786$	−72.0000	−2.56815
$787$	−5.00000	−0.178231	−0.0891154	−	0.996021i	$-0.528404\pi$
$787$	−5.00000	−0.178231	−0.0891154	+	0.996021i	$0.528404\pi$
$788$	−6.00000	−0.213741
$789$	−57.0000	−2.02925
$790$	− 16.0000i	− 0.569254i
$791$	18.0000i	0.640006i
$792$	0	0
$793$	16.0000	0.568177
$794$	− 10.0000i	− 0.354887i
$795$	6.00000i	0.212798i
$796$	− 4.00000i	− 0.141776i
$797$	52.0000i	1.84193i	0.389640	+	0.920967i	$0.372599\pi$
$797$	52.0000i	1.84193i	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	0	0
$800$	8.00000i	0.282843i
$801$	24.0000i	0.847998i
$802$	36.0000	1.27120
$803$	5.00000	0.176446
$804$	48.0000	1.69283
$805$	−4.00000	−0.140981
$806$	− 16.0000i	− 0.563576i
$807$	−18.0000	−0.633630
$808$	0	0
$809$	2.00000i	0.0703163i	0.999382	+	0.0351581i	$0.0111935\pi$
$809$	2.00000i	0.0703163i	−0.999382	+	0.0351581i	$0.988807\pi$
$810$	36.0000	1.26491
$811$	47.0000	1.65039	0.825197	−	0.564846i	$-0.191064\pi$
$811$	47.0000	1.65039	0.825197	+	0.564846i	$0.191064\pi$
$812$	12.0000i	0.421117i
$813$	−93.0000	−3.26165
$814$	0	0
$815$	36.0000	1.26102
$816$	0	0
$817$	0	0
$818$	40.0000	1.39857
$819$	− 12.0000i	− 0.419314i
$820$	− 36.0000i	− 1.25717i
$821$	−47.0000	−1.64031	−0.820156	−	0.572140i	$-0.806113\pi$
$821$	−47.0000	−1.64031	−0.820156	+	0.572140i	$0.806113\pi$
$822$	36.0000i	1.25564i
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	15.0000	0.522233
$826$	16.0000	0.556711
$827$	− 22.0000i	− 0.765015i	−0.923952	−	0.382507i	$-0.875061\pi$
$827$	− 22.0000i	− 0.765015i	0.923952	−	0.382507i	$-0.124939\pi$
$828$	24.0000i	0.834058i
$829$	− 4.00000i	− 0.138926i	−0.997585	−	0.0694629i	$-0.977871\pi$
$829$	− 4.00000i	− 0.138926i	0.997585	−	0.0694629i	$-0.0221285\pi$
$830$	−60.0000	−2.08263
$831$	36.0000i	1.24883i
$832$	16.0000i	0.554700i
$833$	0	0
$834$	24.0000i	0.831052i
$835$	−24.0000	−0.830554
$836$	0	0
$837$	− 36.0000i	− 1.24434i
$838$	− 14.0000i	− 0.483622i
$839$	−44.0000	−1.51905	−0.759524	−	0.650479i	$-0.774568\pi$
$839$	−44.0000	−1.51905	−0.759524	+	0.650479i	$0.774568\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	−48.0000	−1.65419
$843$	− 36.0000i	− 1.23991i
$844$	26.0000	0.894957
$845$	18.0000i	0.619219i
$846$	108.000i	3.71312i
$847$	−14.0000	−0.481046
$848$	−4.00000	−0.137361
$849$	12.0000i	0.411839i
$850$	0	0
$851$	0	0
$852$	−54.0000	−1.85001
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	0	0
$857$	48.0000i	1.63965i	0.572615	+	0.819824i	$0.305929\pi$
$857$	48.0000i	1.63965i	−0.572615	+	0.819824i	$0.694071\pi$
$858$	60.0000	2.04837
$859$	20.0000i	0.682391i	0.939992	+	0.341196i	$0.110832\pi$
$859$	20.0000i	0.682391i	−0.939992	+	0.341196i	$0.889168\pi$
$860$	−8.00000	−0.272798
$861$	−27.0000	−0.920158
$862$	−60.0000	−2.04361
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	72.0000i	2.44949i
$865$	− 18.0000i	− 0.612018i
$866$	− 18.0000i	− 0.611665i
$867$	51.0000	1.73205
$868$	− 8.00000i	− 0.271538i
$869$	− 20.0000i	− 0.678454i
$870$	72.0000i	2.44103i
$871$	− 16.0000i	− 0.542139i
$872$	0	0
$873$	− 24.0000i	− 0.812277i
$874$	0	0
$875$	− 12.0000i	− 0.405674i
$876$	−6.00000	−0.202721
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	56.0000	1.88991
$879$	−6.00000	−0.202375
$880$	− 40.0000i	− 1.34840i
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	72.0000i	2.42437i
$883$	48.0000i	1.61533i	0.589643	+	0.807664i	$0.299269\pi$
$883$	48.0000i	1.61533i	−0.589643	+	0.807664i	$0.700731\pi$
$884$	0	0
$885$	48.0000	1.61350
$886$	2.00000i	0.0671913i
$887$	−25.0000	−0.839418	−0.419709	−	0.907659i	$-0.637868\pi$
$887$	−25.0000	−0.839418	−0.419709	+	0.907659i	$0.637868\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	16.0000i	0.536321i
$891$	45.0000	1.50756
$892$	34.0000	1.13840
$893$	0	0
$894$	30.0000i	1.00335i
$895$	−36.0000	−1.20335
$896$	0	0
$897$	12.0000	0.400668
$898$	−72.0000	−2.40267
$899$	24.0000	0.800445
$900$	−12.0000	−0.400000
$901$	0	0
$902$	− 90.0000i	− 2.99667i
$903$	6.00000i	0.199667i
$904$	0	0
$905$	10.0000i	0.332411i
$906$	96.0000i	3.18939i
$907$	− 52.0000i	− 1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$	− 52.0000i	− 1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$	− 32.0000i	− 1.06196i
$909$	−18.0000	−0.597022
$910$	− 8.00000i	− 0.265197i
$911$	− 26.0000i	− 0.861418i	−0.902491	−	0.430709i	$-0.858263\pi$
$911$	− 26.0000i	− 0.861418i	0.902491	−	0.430709i	$-0.141737\pi$
$912$	0	0
$913$	−75.0000	−2.48214
$914$	−36.0000	−1.19077
$915$	48.0000	1.58683
$916$	−14.0000	−0.462573
$917$	12.0000i	0.396275i
$918$	0	0
$919$	− 58.0000i	− 1.91324i	−0.291333	−	0.956622i	$-0.594099\pi$
$919$	− 58.0000i	− 1.91324i	0.291333	−	0.956622i	$-0.405901\pi$
$920$	0	0
$921$	51.0000	1.68051
$922$	−60.0000	−1.97599
$923$	18.0000i	0.592477i
$924$	30.0000	0.986928
$925$	0	0
$926$	44.0000	1.44593
$927$	108.000i	3.54719i
$928$	−48.0000	−1.57568
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	− 48.0000i	− 1.57398i
$931$	0	0
$932$	12.0000	0.393073
$933$	0	0
$934$	4.00000	0.130884
$935$	0	0
$936$	0	0
$937$	37.0000	1.20874	0.604369	−	0.796705i	$-0.293425\pi$
$937$	37.0000	1.20874	0.604369	+	0.796705i	$0.293425\pi$
$938$	− 16.0000i	− 0.522419i
$939$	− 66.0000i	− 2.15383i
$940$	36.0000i	1.17419i
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	− 138.000i	− 4.49628i
$943$	− 18.0000i	− 0.586161i
$944$	32.0000i	1.04151i
$945$	− 18.0000i	− 0.585540i
$946$	−20.0000	−0.650256
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	24.0000i	0.779484i
$949$	2.00000i	0.0649227i
$950$	0	0
$951$	−66.0000	−2.14020
$952$	0	0
$953$	−61.0000	−1.97598	−0.987992	−	0.154506i	$-0.950622\pi$
$953$	−61.0000	−1.97598	−0.987992	+	0.154506i	$0.950622\pi$
$954$	− 12.0000i	− 0.388514i
$955$	−8.00000	−0.258874
$956$	− 12.0000i	− 0.388108i
$957$	90.0000i	2.90929i
$958$	−28.0000	−0.904639
$959$	6.00000	0.193750
$960$	48.0000i	1.54919i
$961$	15.0000	0.483871
$962$	0	0
$963$	−72.0000	−2.32017
$964$	28.0000i	0.901819i
$965$	−52.0000	−1.67394
$966$	12.0000	0.386094
$967$	14.0000i	0.450210i	0.974335	+	0.225105i	$0.0722725\pi$
$967$	14.0000i	0.450210i	−0.974335	+	0.225105i	$0.927728\pi$
$968$	0	0
$969$	0	0
$970$	− 16.0000i	− 0.513729i
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	48.0000	1.53802
$975$	6.00000i	0.192154i
$976$	32.0000i	1.02430i
$977$	28.0000i	0.895799i	0.894084	+	0.447900i	$0.147828\pi$
$977$	28.0000i	0.895799i	−0.894084	+	0.447900i	$0.852172\pi$
$978$	−108.000	−3.45346
$979$	20.0000i	0.639203i
$980$	24.0000i	0.766652i
$981$	96.0000i	3.06504i
$982$	56.0000i	1.78703i
$983$	−9.00000	−0.287055	−0.143528	−	0.989646i	$-0.545845\pi$
$983$	−9.00000	−0.287055	−0.143528	+	0.989646i	$0.545845\pi$
$984$	0	0
$985$	6.00000i	0.191176i
$986$	0	0
$987$	27.0000	0.859419
$988$	0	0
$989$	−4.00000	−0.127193
$990$	120.000	3.81385
$991$	− 18.0000i	− 0.571789i	−0.958261	−	0.285894i	$-0.907709\pi$
$991$	− 18.0000i	− 0.571789i	0.958261	−	0.285894i	$-0.0922907\pi$
$992$	32.0000	1.01600
$993$	6.00000i	0.190404i
$994$	18.0000i	0.570925i
$995$	−4.00000	−0.126809
$996$	90.0000	2.85176
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	24.0000	0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1369.2.b.c.1368.1		2
37.6	odd	4		37.2.a.a.1.1	✓	1
37.31	odd	4		1369.2.a.e.1.1		1
37.36	even	2	inner	1369.2.b.c.1368.2		2
111.80	even	4		333.2.a.d.1.1		1
148.43	even	4		592.2.a.e.1.1		1
185.43	even	4		925.2.b.b.149.2		2
185.117	even	4		925.2.b.b.149.1		2
185.154	odd	4		925.2.a.e.1.1		1
259.6	even	4		1813.2.a.a.1.1		1
296.43	even	4		2368.2.a.b.1.1		1
296.117	odd	4		2368.2.a.q.1.1		1
407.43	even	4		4477.2.a.b.1.1		1
444.191	odd	4		5328.2.a.r.1.1		1
481.376	odd	4		6253.2.a.c.1.1		1
555.524	even	4		8325.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.2.a.a.1.1	✓	1	37.6	odd	4
333.2.a.d.1.1		1	111.80	even	4
592.2.a.e.1.1		1	148.43	even	4
925.2.a.e.1.1		1	185.154	odd	4
925.2.b.b.149.1		2	185.117	even	4
925.2.b.b.149.2		2	185.43	even	4
1369.2.a.e.1.1		1	37.31	odd	4
1369.2.b.c.1368.1		2	1.1	even	1	trivial
1369.2.b.c.1368.2		2	37.36	even	2	inner
1813.2.a.a.1.1		1	259.6	even	4
2368.2.a.b.1.1		1	296.43	even	4
2368.2.a.q.1.1		1	296.117	odd	4
4477.2.a.b.1.1		1	407.43	even	4
5328.2.a.r.1.1		1	444.191	odd	4
6253.2.a.c.1.1		1	481.376	odd	4
8325.2.a.e.1.1		1	555.524	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1369 = 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1369.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +3.00000 q^{3} -2.00000 q^{4} +2.00000i q^{5} -6.00000i q^{6} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} +3.00000 q^{3} -2.00000 q^{4} +2.00000i q^{5} -6.00000i q^{6} -1.00000 q^{7} +6.00000 q^{9} +4.00000 q^{10} +5.00000 q^{11} -6.00000 q^{12} +2.00000i q^{13} +2.00000i q^{14} +6.00000i q^{15} -4.00000 q^{16} -12.0000i q^{18} -4.00000i q^{20} -3.00000 q^{21} -10.0000i q^{22} -2.00000i q^{23} +1.00000 q^{25} +4.00000 q^{26} +9.00000 q^{27} +2.00000 q^{28} +6.00000i q^{29} +12.0000 q^{30} -4.00000i q^{31} +8.00000i q^{32} +15.0000 q^{33} -2.00000i q^{35} -12.0000 q^{36} +6.00000i q^{39} +9.00000 q^{41} +6.00000i q^{42} -2.00000i q^{43} -10.0000 q^{44} +12.0000i q^{45} -4.00000 q^{46} -9.00000 q^{47} -12.0000 q^{48} -6.00000 q^{49} -2.00000i q^{50} -4.00000i q^{52} +1.00000 q^{53} -18.0000i q^{54} +10.0000i q^{55} +12.0000 q^{58} -8.00000i q^{59} -12.0000i q^{60} -8.00000i q^{61} -8.00000 q^{62} -6.00000 q^{63} +8.00000 q^{64} -4.00000 q^{65} -30.0000i q^{66} -8.00000 q^{67} -6.00000i q^{69} -4.00000 q^{70} +9.00000 q^{71} +1.00000 q^{73} +3.00000 q^{75} -5.00000 q^{77} +12.0000 q^{78} -4.00000i q^{79} -8.00000i q^{80} +9.00000 q^{81} -18.0000i q^{82} -15.0000 q^{83} +6.00000 q^{84} -4.00000 q^{86} +18.0000i q^{87} +4.00000i q^{89} +24.0000 q^{90} -2.00000i q^{91} +4.00000i q^{92} -12.0000i q^{93} +18.0000i q^{94} +24.0000i q^{96} -4.00000i q^{97} +12.0000i q^{98} +30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 4 q^{4} - 2 q^{7} + 12 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 4 * q^4 - 2 * q^7 + 12 * q^9	\( 2 q + 6 q^{3} - 4 q^{4} - 2 q^{7} + 12 q^{9} + 8 q^{10} + 10 q^{11} - 12 q^{12} - 8 q^{16} - 6 q^{21} + 2 q^{25} + 8 q^{26} + 18 q^{27} + 4 q^{28} + 24 q^{30} + 30 q^{33} - 24 q^{36} + 18 q^{41} - 20 q^{44} - 8 q^{46} - 18 q^{47} - 24 q^{48} - 12 q^{49} + 2 q^{53} + 24 q^{58} - 16 q^{62} - 12 q^{63} + 16 q^{64} - 8 q^{65} - 16 q^{67} - 8 q^{70} + 18 q^{71} + 2 q^{73} + 6 q^{75} - 10 q^{77} + 24 q^{78} + 18 q^{81} - 30 q^{83} + 12 q^{84} - 8 q^{86} + 48 q^{90} + 60 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 4 * q^4 - 2 * q^7 + 12 * q^9 + 8 * q^10 + 10 * q^11 - 12 * q^12 - 8 * q^16 - 6 * q^21 + 2 * q^25 + 8 * q^26 + 18 * q^27 + 4 * q^28 + 24 * q^30 + 30 * q^33 - 24 * q^36 + 18 * q^41 - 20 * q^44 - 8 * q^46 - 18 * q^47 - 24 * q^48 - 12 * q^49 + 2 * q^53 + 24 * q^58 - 16 * q^62 - 12 * q^63 + 16 * q^64 - 8 * q^65 - 16 * q^67 - 8 * q^70 + 18 * q^71 + 2 * q^73 + 6 * q^75 - 10 * q^77 + 24 * q^78 + 18 * q^81 - 30 * q^83 + 12 * q^84 - 8 * q^86 + 48 * q^90 + 60 * q^99

Introduction

L-functions

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