LMFDB - Embedded newform 1734.2.b.d.577.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1734,2,Mod(577,1734)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1734, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1734.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1734 = 2 \cdot 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1734.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:	$13.8460597105$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 102)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1734.577
Dual form			1734.2.b.d.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +4.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +4.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} +1.00000 q^{8} -1.00000 q^{9} +4.00000i q^{10} +1.00000i q^{12} -6.00000 q^{13} -2.00000i q^{14} -4.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.00000 q^{19} +4.00000i q^{20} +2.00000 q^{21} +6.00000i q^{23} +1.00000i q^{24} -11.0000 q^{25} -6.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} +4.00000i q^{29} -4.00000 q^{30} +6.00000i q^{31} +1.00000 q^{32} +8.00000 q^{35} -1.00000 q^{36} +4.00000i q^{37} -4.00000 q^{38} -6.00000i q^{39} +4.00000i q^{40} -10.0000i q^{41} +2.00000 q^{42} +4.00000 q^{43} -4.00000i q^{45} +6.00000i q^{46} +4.00000 q^{47} +1.00000i q^{48} +3.00000 q^{49} -11.0000 q^{50} -6.00000 q^{52} +2.00000 q^{53} -1.00000i q^{54} -2.00000i q^{56} -4.00000i q^{57} +4.00000i q^{58} -12.0000 q^{59} -4.00000 q^{60} -4.00000i q^{61} +6.00000i q^{62} +2.00000i q^{63} +1.00000 q^{64} -24.0000i q^{65} -12.0000 q^{67} -6.00000 q^{69} +8.00000 q^{70} +6.00000i q^{71} -1.00000 q^{72} -2.00000i q^{73} +4.00000i q^{74} -11.0000i q^{75} -4.00000 q^{76} -6.00000i q^{78} +10.0000i q^{79} +4.00000i q^{80} +1.00000 q^{81} -10.0000i q^{82} +12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} -4.00000 q^{87} -2.00000 q^{89} -4.00000i q^{90} +12.0000i q^{91} +6.00000i q^{92} -6.00000 q^{93} +4.00000 q^{94} -16.0000i q^{95} +1.00000i q^{96} -6.00000i q^{97} +3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 12 q^{13} - 8 q^{15} + 2 q^{16} - 2 q^{18} - 8 q^{19} + 4 q^{21} - 22 q^{25} - 12 q^{26} - 8 q^{30} + 2 q^{32} + 16 q^{35} - 2 q^{36} - 8 q^{38} + 4 q^{42} + 8 q^{43} + 8 q^{47} + 6 q^{49} - 22 q^{50} - 12 q^{52} + 4 q^{53} - 24 q^{59} - 8 q^{60} + 2 q^{64} - 24 q^{67} - 12 q^{69} + 16 q^{70} - 2 q^{72} - 8 q^{76} + 2 q^{81} + 24 q^{83} + 4 q^{84} + 8 q^{86} - 8 q^{87} - 4 q^{89} - 12 q^{93} + 8 q^{94} + 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 12 * q^13 - 8 * q^15 + 2 * q^16 - 2 * q^18 - 8 * q^19 + 4 * q^21 - 22 * q^25 - 12 * q^26 - 8 * q^30 + 2 * q^32 + 16 * q^35 - 2 * q^36 - 8 * q^38 + 4 * q^42 + 8 * q^43 + 8 * q^47 + 6 * q^49 - 22 * q^50 - 12 * q^52 + 4 * q^53 - 24 * q^59 - 8 * q^60 + 2 * q^64 - 24 * q^67 - 12 * q^69 + 16 * q^70 - 2 * q^72 - 8 * q^76 + 2 * q^81 + 24 * q^83 + 4 * q^84 + 8 * q^86 - 8 * q^87 - 4 * q^89 - 12 * q^93 + 8 * q^94 + 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times$.

$n$	$1157$	$1159$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	1.00000	0.500000
$5$	4.00000i	1.78885i	0.447214	+	0.894427i	$0.352416\pi$
$5$	4.00000i	1.78885i	−0.447214	+	0.894427i	$0.647584\pi$
$6$	1.00000i	0.408248i
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	1.00000	0.353553
$9$	−1.00000	−0.333333
$10$	4.00000i	1.26491i
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.00000i	0.288675i
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	− 2.00000i	− 0.534522i
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	4.00000i	0.894427i
$21$	2.00000	0.436436
$22$	0	0
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	1.00000i	0.204124i
$25$	−11.0000	−2.20000
$26$	−6.00000	−1.17670
$27$	− 1.00000i	− 0.192450i
$28$	− 2.00000i	− 0.377964i
$29$	4.00000i	0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$	4.00000i	0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$	−4.00000	−0.730297
$31$	6.00000i	1.07763i	0.842424	+	0.538816i	$0.181128\pi$
$31$	6.00000i	1.07763i	−0.842424	+	0.538816i	$0.818872\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	8.00000	1.35225
$36$	−1.00000	−0.166667
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	−4.00000	−0.648886
$39$	− 6.00000i	− 0.960769i
$40$	4.00000i	0.632456i
$41$	− 10.0000i	− 1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	− 10.0000i	− 1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$	2.00000	0.308607
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	− 4.00000i	− 0.596285i
$46$	6.00000i	0.884652i
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	−11.0000	−1.55563
$51$	0	0
$52$	−6.00000	−0.832050
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	− 1.00000i	− 0.136083i
$55$	0	0
$56$	− 2.00000i	− 0.267261i
$57$	− 4.00000i	− 0.529813i
$58$	4.00000i	0.525226i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−4.00000	−0.516398
$61$	− 4.00000i	− 0.512148i	−0.966657	−	0.256074i	$-0.917571\pi$
$61$	− 4.00000i	− 0.512148i	0.966657	−	0.256074i	$-0.0824290\pi$
$62$	6.00000i	0.762001i
$63$	2.00000i	0.251976i
$64$	1.00000	0.125000
$65$	− 24.0000i	− 2.97683i
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	8.00000	0.956183
$71$	6.00000i	0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$	6.00000i	0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$	−1.00000	−0.117851
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	4.00000i	0.464991i
$75$	− 11.0000i	− 1.27017i
$76$	−4.00000	−0.458831
$77$	0	0
$78$	− 6.00000i	− 0.679366i
$79$	10.0000i	1.12509i	0.826767	+	0.562544i	$0.190177\pi$
$79$	10.0000i	1.12509i	−0.826767	+	0.562544i	$0.809823\pi$
$80$	4.00000i	0.447214i
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	4.00000	0.431331
$87$	−4.00000	−0.428845
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	− 4.00000i	− 0.421637i
$91$	12.0000i	1.25794i
$92$	6.00000i	0.625543i
$93$	−6.00000	−0.622171
$94$	4.00000	0.412568
$95$	− 16.0000i	− 1.64157i
$96$	1.00000i	0.102062i
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	−11.0000	−1.10000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−6.00000	−0.588348
$105$	8.00000i	0.780720i
$106$	2.00000	0.194257
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	16.0000i	1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$	16.0000i	1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	− 2.00000i	− 0.188982i
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	− 4.00000i	− 0.374634i
$115$	−24.0000	−2.23801
$116$	4.00000i	0.371391i
$117$	6.00000	0.554700
$118$	−12.0000	−1.10469
$119$	0	0
$120$	−4.00000	−0.365148
$121$	11.0000	1.00000
$122$	− 4.00000i	− 0.362143i
$123$	10.0000	0.901670
$124$	6.00000i	0.538816i
$125$	− 24.0000i	− 2.14663i
$126$	2.00000i	0.178174i
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	4.00000i	0.352180i
$130$	− 24.0000i	− 2.10494i
$131$	16.0000i	1.39793i	0.715158	+	0.698963i	$0.246355\pi$
$131$	16.0000i	1.39793i	−0.715158	+	0.698963i	$0.753645\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	−12.0000	−1.03664
$135$	4.00000	0.344265
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−6.00000	−0.510754
$139$	− 8.00000i	− 0.678551i	−0.940687	−	0.339276i	$-0.889818\pi$
$139$	− 8.00000i	− 0.678551i	0.940687	−	0.339276i	$-0.110182\pi$
$140$	8.00000	0.676123
$141$	4.00000i	0.336861i
$142$	6.00000i	0.503509i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	−16.0000	−1.32873
$146$	− 2.00000i	− 0.165521i
$147$	3.00000i	0.247436i
$148$	4.00000i	0.328798i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	− 11.0000i	− 0.898146i
$151$	24.0000	1.95309	0.976546	−	0.215308i	$-0.0690756\pi$
$151$	24.0000	1.95309	0.976546	+	0.215308i	$0.0690756\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	0	0
$155$	−24.0000	−1.92773
$156$	− 6.00000i	− 0.480384i
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	10.0000i	0.795557i
$159$	2.00000i	0.158610i
$160$	4.00000i	0.316228i
$161$	12.0000	0.945732
$162$	1.00000	0.0785674
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	− 10.0000i	− 0.780869i
$165$	0	0
$166$	12.0000	0.931381
$167$	2.00000i	0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$	2.00000i	0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$	2.00000	0.154303
$169$	23.0000	1.76923
$170$	0	0
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	4.00000i	0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	4.00000i	0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	−4.00000	−0.303239
$175$	22.0000i	1.66304i
$176$	0	0
$177$	− 12.0000i	− 0.901975i
$178$	−2.00000	−0.149906
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	− 4.00000i	− 0.298142i
$181$	− 20.0000i	− 1.48659i	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	− 20.0000i	− 1.48659i	0.668965	−	0.743294i	$-0.266738\pi$
$182$	12.0000i	0.889499i
$183$	4.00000	0.295689
$184$	6.00000i	0.442326i
$185$	−16.0000	−1.17634
$186$	−6.00000	−0.439941
$187$	0	0
$188$	4.00000	0.291730
$189$	−2.00000	−0.145479
$190$	− 16.0000i	− 1.16076i
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	1.00000i	0.0721688i
$193$	6.00000i	0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$	6.00000i	0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	− 6.00000i	− 0.430775i
$195$	24.0000	1.71868
$196$	3.00000	0.214286
$197$	8.00000i	0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$	8.00000i	0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$	0	0
$199$	− 14.0000i	− 0.992434i	−0.868199	−	0.496217i	$-0.834722\pi$
$199$	− 14.0000i	− 0.992434i	0.868199	−	0.496217i	$-0.165278\pi$
$200$	−11.0000	−0.777817
$201$	− 12.0000i	− 0.846415i
$202$	14.0000	0.985037
$203$	8.00000	0.561490
$204$	0	0
$205$	40.0000	2.79372
$206$	4.00000	0.278693
$207$	− 6.00000i	− 0.417029i
$208$	−6.00000	−0.416025
$209$	0	0
$210$	8.00000i	0.552052i
$211$	8.00000i	0.550743i	0.961338	+	0.275371i	$0.0888008\pi$
$211$	8.00000i	0.550743i	−0.961338	+	0.275371i	$0.911199\pi$
$212$	2.00000	0.137361
$213$	−6.00000	−0.411113
$214$	0	0
$215$	16.0000i	1.09119i
$216$	− 1.00000i	− 0.0680414i
$217$	12.0000	0.814613
$218$	16.0000i	1.08366i
$219$	2.00000	0.135147
$220$	0	0
$221$	0	0
$222$	−4.00000	−0.268462
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	− 2.00000i	− 0.133631i
$225$	11.0000	0.733333
$226$	2.00000i	0.133038i
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	− 4.00000i	− 0.264906i
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−24.0000	−1.58251
$231$	0	0
$232$	4.00000i	0.262613i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	6.00000	0.392232
$235$	16.0000i	1.04372i
$236$	−12.0000	−0.781133
$237$	−10.0000	−0.649570
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	−4.00000	−0.258199
$241$	18.0000i	1.15948i	0.814801	+	0.579741i	$0.196846\pi$
$241$	18.0000i	1.15948i	−0.814801	+	0.579741i	$0.803154\pi$
$242$	11.0000	0.707107
$243$	1.00000i	0.0641500i
$244$	− 4.00000i	− 0.256074i
$245$	12.0000i	0.766652i
$246$	10.0000	0.637577
$247$	24.0000	1.52708
$248$	6.00000i	0.381000i
$249$	12.0000i	0.760469i
$250$	− 24.0000i	− 1.51789i
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	2.00000i	0.125988i
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	4.00000i	0.249029i
$259$	8.00000	0.497096
$260$	− 24.0000i	− 1.48842i
$261$	− 4.00000i	− 0.247594i
$262$	16.0000i	0.988483i
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	8.00000i	0.491436i
$266$	8.00000i	0.490511i
$267$	− 2.00000i	− 0.122398i
$268$	−12.0000	−0.733017
$269$	12.0000i	0.731653i	0.930683	+	0.365826i	$0.119214\pi$
$269$	12.0000i	0.731653i	−0.930683	+	0.365826i	$0.880786\pi$
$270$	4.00000	0.243432
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	−12.0000	−0.726273
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−6.00000	−0.361158
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	− 8.00000i	− 0.479808i
$279$	− 6.00000i	− 0.359211i
$280$	8.00000	0.478091
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	4.00000i	0.238197i
$283$	32.0000i	1.90220i	0.308879	+	0.951101i	$0.400046\pi$
$283$	32.0000i	1.90220i	−0.308879	+	0.951101i	$0.599954\pi$
$284$	6.00000i	0.356034i
$285$	16.0000	0.947758
$286$	0	0
$287$	−20.0000	−1.18056
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	−16.0000	−0.939552
$291$	6.00000	0.351726
$292$	− 2.00000i	− 0.117041i
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	3.00000i	0.174964i
$295$	− 48.0000i	− 2.79467i
$296$	4.00000i	0.232495i
$297$	0	0
$298$	−6.00000	−0.347571
$299$	− 36.0000i	− 2.08193i
$300$	− 11.0000i	− 0.635085i
$301$	− 8.00000i	− 0.461112i
$302$	24.0000	1.38104
$303$	14.0000i	0.804279i
$304$	−4.00000	−0.229416
$305$	16.0000	0.916157
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	4.00000i	0.227552i
$310$	−24.0000	−1.36311
$311$	− 30.0000i	− 1.70114i	−0.525859	−	0.850572i	$-0.676256\pi$
$311$	− 30.0000i	− 1.70114i	0.525859	−	0.850572i	$-0.323744\pi$
$312$	− 6.00000i	− 0.339683i
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	6.00000	0.338600
$315$	−8.00000	−0.450749
$316$	10.0000i	0.562544i
$317$	− 16.0000i	− 0.898650i	−0.893368	−	0.449325i	$-0.851665\pi$
$317$	− 16.0000i	− 0.898650i	0.893368	−	0.449325i	$-0.148335\pi$
$318$	2.00000i	0.112154i
$319$	0	0
$320$	4.00000i	0.223607i
$321$	0	0
$322$	12.0000	0.668734
$323$	0	0
$324$	1.00000	0.0555556
$325$	66.0000	3.66102
$326$	12.0000i	0.664619i
$327$	−16.0000	−0.884802
$328$	− 10.0000i	− 0.552158i
$329$	− 8.00000i	− 0.441054i
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	12.0000	0.658586
$333$	− 4.00000i	− 0.219199i
$334$	2.00000i	0.109435i
$335$	− 48.0000i	− 2.62252i
$336$	2.00000	0.109109
$337$	6.00000i	0.326841i	0.986557	+	0.163420i	$0.0522527\pi$
$337$	6.00000i	0.326841i	−0.986557	+	0.163420i	$0.947747\pi$
$338$	23.0000	1.25104
$339$	−2.00000	−0.108625
$340$	0	0
$341$	0	0
$342$	4.00000	0.216295
$343$	− 20.0000i	− 1.07990i
$344$	4.00000	0.215666
$345$	− 24.0000i	− 1.29212i
$346$	4.00000i	0.215041i
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	−4.00000	−0.214423
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	22.0000i	1.17595i
$351$	6.00000i	0.320256i
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	− 12.0000i	− 0.637793i
$355$	−24.0000	−1.27379
$356$	−2.00000	−0.106000
$357$	0	0
$358$	12.0000	0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	− 4.00000i	− 0.210819i
$361$	−3.00000	−0.157895
$362$	− 20.0000i	− 1.05118i
$363$	11.0000i	0.577350i
$364$	12.0000i	0.628971i
$365$	8.00000	0.418739
$366$	4.00000	0.209083
$367$	10.0000i	0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$	10.0000i	0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$	6.00000i	0.312772i
$369$	10.0000i	0.520579i
$370$	−16.0000	−0.831800
$371$	− 4.00000i	− 0.207670i
$372$	−6.00000	−0.311086
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	4.00000	0.206284
$377$	− 24.0000i	− 1.23606i
$378$	−2.00000	−0.102869
$379$	4.00000i	0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$	4.00000i	0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	− 16.0000i	− 0.820783i
$381$	− 8.00000i	− 0.409852i
$382$	−4.00000	−0.204658
$383$	−28.0000	−1.43073	−0.715367	−	0.698749i	$-0.753740\pi$
$383$	−28.0000	−1.43073	−0.715367	+	0.698749i	$0.753740\pi$
$384$	1.00000i	0.0510310i
$385$	0	0
$386$	6.00000i	0.305392i
$387$	−4.00000	−0.203331
$388$	− 6.00000i	− 0.304604i
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	24.0000	1.21529
$391$	0	0
$392$	3.00000	0.151523
$393$	−16.0000	−0.807093
$394$	8.00000i	0.403034i
$395$	−40.0000	−2.01262
$396$	0	0
$397$	20.0000i	1.00377i	0.864934	+	0.501886i	$0.167360\pi$
$397$	20.0000i	1.00377i	−0.864934	+	0.501886i	$0.832640\pi$
$398$	− 14.0000i	− 0.701757i
$399$	−8.00000	−0.400501
$400$	−11.0000	−0.550000
$401$	30.0000i	1.49813i	0.662497	+	0.749064i	$0.269497\pi$
$401$	30.0000i	1.49813i	−0.662497	+	0.749064i	$0.730503\pi$
$402$	− 12.0000i	− 0.598506i
$403$	− 36.0000i	− 1.79329i
$404$	14.0000	0.696526
$405$	4.00000i	0.198762i
$406$	8.00000	0.397033
$407$	0	0
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	40.0000	1.97546
$411$	− 6.00000i	− 0.295958i
$412$	4.00000	0.197066
$413$	24.0000i	1.18096i
$414$	− 6.00000i	− 0.294884i
$415$	48.0000i	2.35623i
$416$	−6.00000	−0.294174
$417$	8.00000	0.391762
$418$	0	0
$419$	− 12.0000i	− 0.586238i	−0.956076	−	0.293119i	$-0.905307\pi$
$419$	− 12.0000i	− 0.586238i	0.956076	−	0.293119i	$-0.0946933\pi$
$420$	8.00000i	0.390360i
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	8.00000i	0.389434i
$423$	−4.00000	−0.194487
$424$	2.00000	0.0971286
$425$	0	0
$426$	−6.00000	−0.290701
$427$	−8.00000	−0.387147
$428$	0	0
$429$	0	0
$430$	16.0000i	0.771589i
$431$	14.0000i	0.674356i	0.941441	+	0.337178i	$0.109472\pi$
$431$	14.0000i	0.674356i	−0.941441	+	0.337178i	$0.890528\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	12.0000	0.576018
$435$	− 16.0000i	− 0.767141i
$436$	16.0000i	0.766261i
$437$	− 24.0000i	− 1.14808i
$438$	2.00000	0.0955637
$439$	10.0000i	0.477274i	0.971109	+	0.238637i	$0.0767006\pi$
$439$	10.0000i	0.477274i	−0.971109	+	0.238637i	$0.923299\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−4.00000	−0.189832
$445$	− 8.00000i	− 0.379236i
$446$	4.00000	0.189405
$447$	− 6.00000i	− 0.283790i
$448$	− 2.00000i	− 0.0944911i
$449$	− 26.0000i	− 1.22702i	−0.789689	−	0.613508i	$-0.789758\pi$
$449$	− 26.0000i	− 1.22702i	0.789689	−	0.613508i	$-0.210242\pi$
$450$	11.0000	0.518545
$451$	0	0
$452$	2.00000i	0.0940721i
$453$	24.0000i	1.12762i
$454$	4.00000i	0.187729i
$455$	−48.0000	−2.25027
$456$	− 4.00000i	− 0.187317i
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	2.00000	0.0934539
$459$	0	0
$460$	−24.0000	−1.11901
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	4.00000i	0.185695i
$465$	− 24.0000i	− 1.11297i
$466$	6.00000i	0.277945i
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	6.00000	0.277350
$469$	24.0000i	1.10822i
$470$	16.0000i	0.738025i
$471$	6.00000i	0.276465i
$472$	−12.0000	−0.552345
$473$	0	0
$474$	−10.0000	−0.459315
$475$	44.0000	2.01886
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	−20.0000	−0.914779
$479$	− 10.0000i	− 0.456912i	−0.973554	−	0.228456i	$-0.926632\pi$
$479$	− 10.0000i	− 0.456912i	0.973554	−	0.228456i	$-0.0733677\pi$
$480$	−4.00000	−0.182574
$481$	− 24.0000i	− 1.09431i
$482$	18.0000i	0.819878i
$483$	12.0000i	0.546019i
$484$	11.0000	0.500000
$485$	24.0000	1.08978
$486$	1.00000i	0.0453609i
$487$	− 38.0000i	− 1.72194i	−0.508652	−	0.860972i	$-0.669856\pi$
$487$	− 38.0000i	− 1.72194i	0.508652	−	0.860972i	$-0.330144\pi$
$488$	− 4.00000i	− 0.181071i
$489$	−12.0000	−0.542659
$490$	12.0000i	0.542105i
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	10.0000	0.450835
$493$	0	0
$494$	24.0000	1.07981
$495$	0	0
$496$	6.00000i	0.269408i
$497$	12.0000	0.538274
$498$	12.0000i	0.537733i
$499$	32.0000i	1.43252i	0.697835	+	0.716258i	$0.254147\pi$
$499$	32.0000i	1.43252i	−0.697835	+	0.716258i	$0.745853\pi$
$500$	− 24.0000i	− 1.07331i
$501$	−2.00000	−0.0893534
$502$	12.0000	0.535586
$503$	− 26.0000i	− 1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$503$	− 26.0000i	− 1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$504$	2.00000i	0.0890871i
$505$	56.0000i	2.49197i
$506$	0	0
$507$	23.0000i	1.02147i
$508$	−8.00000	−0.354943
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	1.00000	0.0441942
$513$	4.00000i	0.176604i
$514$	−18.0000	−0.793946
$515$	16.0000i	0.705044i
$516$	4.00000i	0.176090i
$517$	0	0
$518$	8.00000	0.351500
$519$	−4.00000	−0.175581
$520$	− 24.0000i	− 1.05247i
$521$	− 10.0000i	− 0.438108i	−0.975713	−	0.219054i	$-0.929703\pi$
$521$	− 10.0000i	− 0.438108i	0.975713	−	0.219054i	$-0.0702971\pi$
$522$	− 4.00000i	− 0.175075i
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	16.0000i	0.698963i
$525$	−22.0000	−0.960159
$526$	12.0000	0.523225
$527$	0	0
$528$	0	0
$529$	−13.0000	−0.565217
$530$	8.00000i	0.347498i
$531$	12.0000	0.520756
$532$	8.00000i	0.346844i
$533$	60.0000i	2.59889i
$534$	− 2.00000i	− 0.0865485i
$535$	0	0
$536$	−12.0000	−0.518321
$537$	12.0000i	0.517838i
$538$	12.0000i	0.517357i
$539$	0	0
$540$	4.00000	0.172133
$541$	12.0000i	0.515920i	0.966156	+	0.257960i	$0.0830503\pi$
$541$	12.0000i	0.515920i	−0.966156	+	0.257960i	$0.916950\pi$
$542$	−16.0000	−0.687259
$543$	20.0000	0.858282
$544$	0	0
$545$	−64.0000	−2.74146
$546$	−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	$0.945291\pi$
$548$	−6.00000	−0.256307
$549$	4.00000i	0.170716i
$550$	0	0
$551$	− 16.0000i	− 0.681623i
$552$	−6.00000	−0.255377
$553$	20.0000	0.850487
$554$	8.00000i	0.339887i
$555$	− 16.0000i	− 0.679162i
$556$	− 8.00000i	− 0.339276i
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	− 6.00000i	− 0.254000i
$559$	−24.0000	−1.01509
$560$	8.00000	0.338062
$561$	0	0
$562$	18.0000	0.759284
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	4.00000i	0.168430i
$565$	−8.00000	−0.336563
$566$	32.0000i	1.34506i
$567$	− 2.00000i	− 0.0839921i
$568$	6.00000i	0.251754i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	16.0000	0.670166
$571$	44.0000i	1.84134i	0.390339	+	0.920671i	$0.372358\pi$
$571$	44.0000i	1.84134i	−0.390339	+	0.920671i	$0.627642\pi$
$572$	0	0
$573$	− 4.00000i	− 0.167102i
$574$	−20.0000	−0.834784
$575$	− 66.0000i	− 2.75239i
$576$	−1.00000	−0.0416667
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	−16.0000	−0.664364
$581$	− 24.0000i	− 0.995688i
$582$	6.00000	0.248708
$583$	0	0
$584$	− 2.00000i	− 0.0827606i
$585$	24.0000i	0.992278i
$586$	2.00000	0.0826192
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	3.00000i	0.123718i
$589$	− 24.0000i	− 0.988903i
$590$	− 48.0000i	− 1.97613i
$591$	−8.00000	−0.329076
$592$	4.00000i	0.164399i
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	14.0000	0.572982
$598$	− 36.0000i	− 1.47215i
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	− 11.0000i	− 0.449073i
$601$	− 38.0000i	− 1.55005i	−0.631929	−	0.775026i	$-0.717737\pi$
$601$	− 38.0000i	− 1.55005i	0.631929	−	0.775026i	$-0.282263\pi$
$602$	− 8.00000i	− 0.326056i
$603$	12.0000	0.488678
$604$	24.0000	0.976546
$605$	44.0000i	1.78885i
$606$	14.0000i	0.568711i
$607$	38.0000i	1.54237i	0.636610	+	0.771186i	$0.280336\pi$
$607$	38.0000i	1.54237i	−0.636610	+	0.771186i	$0.719664\pi$
$608$	−4.00000	−0.162221
$609$	8.00000i	0.324176i
$610$	16.0000	0.647821
$611$	−24.0000	−0.970936
$612$	0	0
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	−12.0000	−0.484281
$615$	40.0000i	1.61296i
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	4.00000i	0.160904i
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	−24.0000	−0.963863
$621$	6.00000	0.240772
$622$	− 30.0000i	− 1.20289i
$623$	4.00000i	0.160257i
$624$	− 6.00000i	− 0.240192i
$625$	41.0000	1.64000
$626$	− 26.0000i	− 1.03917i
$627$	0	0
$628$	6.00000	0.239426
$629$	0	0
$630$	−8.00000	−0.318728
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	10.0000i	0.397779i
$633$	−8.00000	−0.317971
$634$	− 16.0000i	− 0.635441i
$635$	− 32.0000i	− 1.26988i
$636$	2.00000i	0.0793052i
$637$	−18.0000	−0.713186
$638$	0	0
$639$	− 6.00000i	− 0.237356i
$640$	4.00000i	0.158114i
$641$	2.00000i	0.0789953i	0.999220	+	0.0394976i	$0.0125758\pi$
$641$	2.00000i	0.0789953i	−0.999220	+	0.0394976i	$0.987424\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	12.0000	0.472866
$645$	−16.0000	−0.629999
$646$	0	0
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	66.0000	2.58873
$651$	12.0000i	0.470317i
$652$	12.0000i	0.469956i
$653$	20.0000i	0.782660i	0.920250	+	0.391330i	$0.127985\pi$
$653$	20.0000i	0.782660i	−0.920250	+	0.391330i	$0.872015\pi$
$654$	−16.0000	−0.625650
$655$	−64.0000	−2.50069
$656$	− 10.0000i	− 0.390434i
$657$	2.00000i	0.0780274i
$658$	− 8.00000i	− 0.311872i
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	12.0000	0.465690
$665$	−32.0000	−1.24091
$666$	− 4.00000i	− 0.154997i
$667$	−24.0000	−0.929284
$668$	2.00000i	0.0773823i
$669$	4.00000i	0.154649i
$670$	− 48.0000i	− 1.85440i
$671$	0	0
$672$	2.00000	0.0771517
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	6.00000i	0.231111i
$675$	11.0000i	0.423390i
$676$	23.0000	0.884615
$677$	− 8.00000i	− 0.307465i	−0.988113	−	0.153732i	$-0.950871\pi$
$677$	− 8.00000i	− 0.307465i	0.988113	−	0.153732i	$-0.0491294\pi$
$678$	−2.00000	−0.0768095
$679$	−12.0000	−0.460518
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	4.00000	0.152944
$685$	− 24.0000i	− 0.916993i
$686$	− 20.0000i	− 0.763604i
$687$	2.00000i	0.0763048i
$688$	4.00000	0.152499
$689$	−12.0000	−0.457164
$690$	− 24.0000i	− 0.913664i
$691$	16.0000i	0.608669i	0.952565	+	0.304334i	$0.0984340\pi$
$691$	16.0000i	0.608669i	−0.952565	+	0.304334i	$0.901566\pi$
$692$	4.00000i	0.152057i
$693$	0	0
$694$	− 4.00000i	− 0.151838i
$695$	32.0000	1.21383
$696$	−4.00000	−0.151620
$697$	0	0
$698$	30.0000	1.13552
$699$	−6.00000	−0.226941
$700$	22.0000i	0.831522i
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	6.00000i	0.226455i
$703$	− 16.0000i	− 0.603451i
$704$	0	0
$705$	−16.0000	−0.602595
$706$	14.0000	0.526897
$707$	− 28.0000i	− 1.05305i
$708$	− 12.0000i	− 0.450988i
$709$	16.0000i	0.600893i	0.953799	+	0.300446i	$0.0971356\pi$
$709$	16.0000i	0.600893i	−0.953799	+	0.300446i	$0.902864\pi$
$710$	−24.0000	−0.900704
$711$	− 10.0000i	− 0.375029i
$712$	−2.00000	−0.0749532
$713$	−36.0000	−1.34821
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	− 20.0000i	− 0.746914i
$718$	0	0
$719$	− 42.0000i	− 1.56634i	−0.621810	−	0.783168i	$-0.713603\pi$
$719$	− 42.0000i	− 1.56634i	0.621810	−	0.783168i	$-0.286397\pi$
$720$	− 4.00000i	− 0.149071i
$721$	− 8.00000i	− 0.297936i
$722$	−3.00000	−0.111648
$723$	−18.0000	−0.669427
$724$	− 20.0000i	− 0.743294i
$725$	− 44.0000i	− 1.63412i
$726$	11.0000i	0.408248i
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	12.0000i	0.444750i
$729$	−1.00000	−0.0370370
$730$	8.00000	0.296093
$731$	0	0
$732$	4.00000	0.147844
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	10.0000i	0.369107i
$735$	−12.0000	−0.442627
$736$	6.00000i	0.221163i
$737$	0	0
$738$	10.0000i	0.368105i
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	−16.0000	−0.588172
$741$	24.0000i	0.881662i
$742$	− 4.00000i	− 0.146845i
$743$	− 38.0000i	− 1.39408i	−0.717030	−	0.697042i	$-0.754499\pi$
$743$	− 38.0000i	− 1.39408i	0.717030	−	0.697042i	$-0.245501\pi$
$744$	−6.00000	−0.219971
$745$	− 24.0000i	− 0.879292i
$746$	−14.0000	−0.512576
$747$	−12.0000	−0.439057
$748$	0	0
$749$	0	0
$750$	24.0000	0.876356
$751$	34.0000i	1.24068i	0.784334	+	0.620339i	$0.213005\pi$
$751$	34.0000i	1.24068i	−0.784334	+	0.620339i	$0.786995\pi$
$752$	4.00000	0.145865
$753$	12.0000i	0.437304i
$754$	− 24.0000i	− 0.874028i
$755$	96.0000i	3.49380i
$756$	−2.00000	−0.0727393
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	4.00000i	0.145287i
$759$	0	0
$760$	− 16.0000i	− 0.580381i
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	− 8.00000i	− 0.289809i
$763$	32.0000	1.15848
$764$	−4.00000	−0.144715
$765$	0	0
$766$	−28.0000	−1.01168
$767$	72.0000	2.59977
$768$	1.00000i	0.0360844i
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	− 18.0000i	− 0.648254i
$772$	6.00000i	0.215945i
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−4.00000	−0.143777
$775$	− 66.0000i	− 2.37079i
$776$	− 6.00000i	− 0.215387i
$777$	8.00000i	0.286998i
$778$	14.0000	0.501924
$779$	40.0000i	1.43315i
$780$	24.0000	0.859338
$781$	0	0
$782$	0	0
$783$	4.00000	0.142948
$784$	3.00000	0.107143
$785$	24.0000i	0.856597i
$786$	−16.0000	−0.570701
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	8.00000i	0.284988i
$789$	12.0000i	0.427211i
$790$	−40.0000	−1.42314
$791$	4.00000	0.142224
$792$	0	0
$793$	24.0000i	0.852265i
$794$	20.0000i	0.709773i
$795$	−8.00000	−0.283731
$796$	− 14.0000i	− 0.496217i
$797$	−46.0000	−1.62940	−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000	−1.62940	−0.814702	+	0.579880i	$0.803099\pi$
$798$	−8.00000	−0.283197
$799$	0	0
$800$	−11.0000	−0.388909
$801$	2.00000	0.0706665
$802$	30.0000i	1.05934i
$803$	0	0
$804$	− 12.0000i	− 0.423207i
$805$	48.0000i	1.69178i
$806$	− 36.0000i	− 1.26805i
$807$	−12.0000	−0.422420
$808$	14.0000	0.492518
$809$	− 26.0000i	− 0.914111i	−0.889438	−	0.457056i	$-0.848904\pi$
$809$	− 26.0000i	− 0.914111i	0.889438	−	0.457056i	$-0.151096\pi$
$810$	4.00000i	0.140546i
$811$	12.0000i	0.421377i	0.977553	+	0.210688i	$0.0675706\pi$
$811$	12.0000i	0.421377i	−0.977553	+	0.210688i	$0.932429\pi$
$812$	8.00000	0.280745
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	−48.0000	−1.68137
$816$	0	0
$817$	−16.0000	−0.559769
$818$	−26.0000	−0.909069
$819$	− 12.0000i	− 0.419314i
$820$	40.0000	1.39686
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	− 6.00000i	− 0.209274i
$823$	18.0000i	0.627441i	0.949515	+	0.313720i	$0.101575\pi$
$823$	18.0000i	0.627441i	−0.949515	+	0.313720i	$0.898425\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	24.0000i	0.835067i
$827$	16.0000i	0.556375i	0.960527	+	0.278187i	$0.0897336\pi$
$827$	16.0000i	0.556375i	−0.960527	+	0.278187i	$0.910266\pi$
$828$	− 6.00000i	− 0.208514i
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	48.0000i	1.66610i
$831$	−8.00000	−0.277517
$832$	−6.00000	−0.208013
$833$	0	0
$834$	8.00000	0.277017
$835$	−8.00000	−0.276851
$836$	0	0
$837$	6.00000	0.207390
$838$	− 12.0000i	− 0.414533i
$839$	− 30.0000i	− 1.03572i	−0.855467	−	0.517858i	$-0.826730\pi$
$839$	− 30.0000i	− 1.03572i	0.855467	−	0.517858i	$-0.173270\pi$
$840$	8.00000i	0.276026i
$841$	13.0000	0.448276
$842$	34.0000	1.17172
$843$	18.0000i	0.619953i
$844$	8.00000i	0.275371i
$845$	92.0000i	3.16490i
$846$	−4.00000	−0.137523
$847$	− 22.0000i	− 0.755929i
$848$	2.00000	0.0686803
$849$	−32.0000	−1.09824
$850$	0	0
$851$	−24.0000	−0.822709
$852$	−6.00000	−0.205557
$853$	16.0000i	0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$	16.0000i	0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$	−8.00000	−0.273754
$855$	16.0000i	0.547188i
$856$	0	0
$857$	− 10.0000i	− 0.341593i	−0.985306	−	0.170797i	$-0.945366\pi$
$857$	− 10.0000i	− 0.341593i	0.985306	−	0.170797i	$-0.0546341\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	16.0000i	0.545595i
$861$	− 20.0000i	− 0.681598i
$862$	14.0000i	0.476842i
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	−16.0000	−0.544016
$866$	18.0000	0.611665
$867$	0	0
$868$	12.0000	0.407307
$869$	0	0
$870$	− 16.0000i	− 0.542451i
$871$	72.0000	2.43963
$872$	16.0000i	0.541828i
$873$	6.00000i	0.203069i
$874$	− 24.0000i	− 0.811812i
$875$	−48.0000	−1.62270
$876$	2.00000	0.0675737
$877$	− 24.0000i	− 0.810422i	−0.914223	−	0.405211i	$-0.867198\pi$
$877$	− 24.0000i	− 0.810422i	0.914223	−	0.405211i	$-0.132802\pi$
$878$	10.0000i	0.337484i
$879$	2.00000i	0.0674583i
$880$	0	0
$881$	− 50.0000i	− 1.68454i	−0.539054	−	0.842271i	$-0.681218\pi$
$881$	− 50.0000i	− 1.68454i	0.539054	−	0.842271i	$-0.318782\pi$
$882$	−3.00000	−0.101015
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	48.0000	1.61350
$886$	12.0000	0.403148
$887$	− 18.0000i	− 0.604381i	−0.953248	−	0.302190i	$-0.902282\pi$
$887$	− 18.0000i	− 0.604381i	0.953248	−	0.302190i	$-0.0977178\pi$
$888$	−4.00000	−0.134231
$889$	16.0000i	0.536623i
$890$	− 8.00000i	− 0.268161i
$891$	0	0
$892$	4.00000	0.133930
$893$	−16.0000	−0.535420
$894$	− 6.00000i	− 0.200670i
$895$	48.0000i	1.60446i
$896$	− 2.00000i	− 0.0668153i
$897$	36.0000	1.20201
$898$	− 26.0000i	− 0.867631i
$899$	−24.0000	−0.800445
$900$	11.0000	0.366667
$901$	0	0
$902$	0	0
$903$	8.00000	0.266223
$904$	2.00000i	0.0665190i
$905$	80.0000	2.65929
$906$	24.0000i	0.797347i
$907$	24.0000i	0.796907i	0.917189	+	0.398453i	$0.130453\pi$
$907$	24.0000i	0.796907i	−0.917189	+	0.398453i	$0.869547\pi$
$908$	4.00000i	0.132745i
$909$	−14.0000	−0.464351
$910$	−48.0000	−1.59118
$911$	26.0000i	0.861418i	0.902491	+	0.430709i	$0.141737\pi$
$911$	26.0000i	0.861418i	−0.902491	+	0.430709i	$0.858263\pi$
$912$	− 4.00000i	− 0.132453i
$913$	0	0
$914$	−22.0000	−0.727695
$915$	16.0000i	0.528944i
$916$	2.00000	0.0660819
$917$	32.0000	1.05673
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−24.0000	−0.791257
$921$	− 12.0000i	− 0.395413i
$922$	10.0000	0.329332
$923$	− 36.0000i	− 1.18495i
$924$	0	0
$925$	− 44.0000i	− 1.44671i
$926$	−4.00000	−0.131448
$927$	−4.00000	−0.131377
$928$	4.00000i	0.131306i
$929$	− 34.0000i	− 1.11550i	−0.830008	−	0.557752i	$-0.811664\pi$
$929$	− 34.0000i	− 1.11550i	0.830008	−	0.557752i	$-0.188336\pi$
$930$	− 24.0000i	− 0.786991i
$931$	−12.0000	−0.393284
$932$	6.00000i	0.196537i
$933$	30.0000	0.982156
$934$	36.0000	1.17796
$935$	0	0
$936$	6.00000	0.196116
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	24.0000i	0.783628i
$939$	26.0000	0.848478
$940$	16.0000i	0.521862i
$941$	48.0000i	1.56476i	0.622804	+	0.782378i	$0.285993\pi$
$941$	48.0000i	1.56476i	−0.622804	+	0.782378i	$0.714007\pi$
$942$	6.00000i	0.195491i
$943$	60.0000	1.95387
$944$	−12.0000	−0.390567
$945$	− 8.00000i	− 0.260240i
$946$	0	0
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	−10.0000	−0.324785
$949$	12.0000i	0.389536i
$950$	44.0000	1.42755
$951$	16.0000	0.518836
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	−2.00000	−0.0647524
$955$	− 16.0000i	− 0.517748i
$956$	−20.0000	−0.646846
$957$	0	0
$958$	− 10.0000i	− 0.323085i
$959$	12.0000i	0.387500i
$960$	−4.00000	−0.129099
$961$	−5.00000	−0.161290
$962$	− 24.0000i	− 0.773791i
$963$	0	0
$964$	18.0000i	0.579741i
$965$	−24.0000	−0.772587
$966$	12.0000i	0.386094i
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	24.0000	0.770594
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	1.00000i	0.0320750i
$973$	−16.0000	−0.512936
$974$	− 38.0000i	− 1.21760i
$975$	66.0000i	2.11369i
$976$	− 4.00000i	− 0.128037i
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	−12.0000	−0.383718
$979$	0	0
$980$	12.0000i	0.383326i
$981$	− 16.0000i	− 0.510841i
$982$	20.0000	0.638226
$983$	38.0000i	1.21201i	0.795460	+	0.606006i	$0.207229\pi$
$983$	38.0000i	1.21201i	−0.795460	+	0.606006i	$0.792771\pi$
$984$	10.0000	0.318788
$985$	−32.0000	−1.01960
$986$	0	0
$987$	8.00000	0.254643
$988$	24.0000	0.763542
$989$	24.0000i	0.763156i
$990$	0	0
$991$	34.0000i	1.08005i	0.841650	+	0.540023i	$0.181584\pi$
$991$	34.0000i	1.08005i	−0.841650	+	0.540023i	$0.818416\pi$
$992$	6.00000i	0.190500i
$993$	− 20.0000i	− 0.634681i
$994$	12.0000	0.380617
$995$	56.0000	1.77532
$996$	12.0000i	0.380235i
$997$	− 20.0000i	− 0.633406i	−0.948525	−	0.316703i	$-0.897424\pi$
$997$	− 20.0000i	− 0.633406i	0.948525	−	0.316703i	$-0.102576\pi$
$998$	32.0000i	1.01294i
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1734.2.b.d.577.2		2
17.2	even	8		1734.2.f.g.829.2		4
17.4	even	4		1734.2.a.h.1.1		1
17.8	even	8		1734.2.f.g.1483.2		4
17.9	even	8		1734.2.f.g.1483.1		4
17.13	even	4		102.2.a.a.1.1	✓	1
17.15	even	8		1734.2.f.g.829.1		4
17.16	even	2	inner	1734.2.b.d.577.1		2
51.38	odd	4		5202.2.a.g.1.1		1
51.47	odd	4		306.2.a.d.1.1		1
68.47	odd	4		816.2.a.h.1.1		1
85.13	odd	4		2550.2.d.q.2449.2		2
85.47	odd	4		2550.2.d.q.2449.1		2
85.64	even	4		2550.2.a.be.1.1		1
119.13	odd	4		4998.2.a.x.1.1		1
136.13	even	4		3264.2.a.bf.1.1		1
136.115	odd	4		3264.2.a.p.1.1		1
204.47	even	4		2448.2.a.t.1.1		1
255.149	odd	4		7650.2.a.z.1.1		1
408.149	odd	4		9792.2.a.a.1.1		1
408.251	even	4		9792.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.a.a.1.1	✓	1	17.13	even	4
306.2.a.d.1.1		1	51.47	odd	4
816.2.a.h.1.1		1	68.47	odd	4
1734.2.a.h.1.1		1	17.4	even	4
1734.2.b.d.577.1		2	17.16	even	2	inner
1734.2.b.d.577.2		2	1.1	even	1	trivial
1734.2.f.g.829.1		4	17.15	even	8
1734.2.f.g.829.2		4	17.2	even	8
1734.2.f.g.1483.1		4	17.9	even	8
1734.2.f.g.1483.2		4	17.8	even	8
2448.2.a.t.1.1		1	204.47	even	4
2550.2.a.be.1.1		1	85.64	even	4
2550.2.d.q.2449.1		2	85.47	odd	4
2550.2.d.q.2449.2		2	85.13	odd	4
3264.2.a.p.1.1		1	136.115	odd	4
3264.2.a.bf.1.1		1	136.13	even	4
4998.2.a.x.1.1		1	119.13	odd	4
5202.2.a.g.1.1		1	51.38	odd	4
7650.2.a.z.1.1		1	255.149	odd	4
9792.2.a.a.1.1		1	408.149	odd	4
9792.2.a.b.1.1		1	408.251	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:	\( N \)	\(=\)	\( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1734.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +4.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +4.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} +1.00000 q^{8} -1.00000 q^{9} +4.00000i q^{10} +1.00000i q^{12} -6.00000 q^{13} -2.00000i q^{14} -4.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.00000 q^{19} +4.00000i q^{20} +2.00000 q^{21} +6.00000i q^{23} +1.00000i q^{24} -11.0000 q^{25} -6.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} +4.00000i q^{29} -4.00000 q^{30} +6.00000i q^{31} +1.00000 q^{32} +8.00000 q^{35} -1.00000 q^{36} +4.00000i q^{37} -4.00000 q^{38} -6.00000i q^{39} +4.00000i q^{40} -10.0000i q^{41} +2.00000 q^{42} +4.00000 q^{43} -4.00000i q^{45} +6.00000i q^{46} +4.00000 q^{47} +1.00000i q^{48} +3.00000 q^{49} -11.0000 q^{50} -6.00000 q^{52} +2.00000 q^{53} -1.00000i q^{54} -2.00000i q^{56} -4.00000i q^{57} +4.00000i q^{58} -12.0000 q^{59} -4.00000 q^{60} -4.00000i q^{61} +6.00000i q^{62} +2.00000i q^{63} +1.00000 q^{64} -24.0000i q^{65} -12.0000 q^{67} -6.00000 q^{69} +8.00000 q^{70} +6.00000i q^{71} -1.00000 q^{72} -2.00000i q^{73} +4.00000i q^{74} -11.0000i q^{75} -4.00000 q^{76} -6.00000i q^{78} +10.0000i q^{79} +4.00000i q^{80} +1.00000 q^{81} -10.0000i q^{82} +12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} -4.00000 q^{87} -2.00000 q^{89} -4.00000i q^{90} +12.0000i q^{91} +6.00000i q^{92} -6.00000 q^{93} +4.00000 q^{94} -16.0000i q^{95} +1.00000i q^{96} -6.00000i q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 12 q^{13} - 8 q^{15} + 2 q^{16} - 2 q^{18} - 8 q^{19} + 4 q^{21} - 22 q^{25} - 12 q^{26} - 8 q^{30} + 2 q^{32} + 16 q^{35} - 2 q^{36} - 8 q^{38} + 4 q^{42} + 8 q^{43} + 8 q^{47} + 6 q^{49} - 22 q^{50} - 12 q^{52} + 4 q^{53} - 24 q^{59} - 8 q^{60} + 2 q^{64} - 24 q^{67} - 12 q^{69} + 16 q^{70} - 2 q^{72} - 8 q^{76} + 2 q^{81} + 24 q^{83} + 4 q^{84} + 8 q^{86} - 8 q^{87} - 4 q^{89} - 12 q^{93} + 8 q^{94} + 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 12 * q^13 - 8 * q^15 + 2 * q^16 - 2 * q^18 - 8 * q^19 + 4 * q^21 - 22 * q^25 - 12 * q^26 - 8 * q^30 + 2 * q^32 + 16 * q^35 - 2 * q^36 - 8 * q^38 + 4 * q^42 + 8 * q^43 + 8 * q^47 + 6 * q^49 - 22 * q^50 - 12 * q^52 + 4 * q^53 - 24 * q^59 - 8 * q^60 + 2 * q^64 - 24 * q^67 - 12 * q^69 + 16 * q^70 - 2 * q^72 - 8 * q^76 + 2 * q^81 + 24 * q^83 + 4 * q^84 + 8 * q^86 - 8 * q^87 - 4 * q^89 - 12 * q^93 + 8 * q^94 + 6 * q^98

Introduction

L-functions

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