LMFDB - Embedded newform 75.2.b.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,2,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, 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("75.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	75.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:	$0.598878015160$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.2.b.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	\(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -2.00000i q^{12} -1.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} -2.00000i q^{18} +5.00000 q^{19} +3.00000 q^{21} +4.00000i q^{22} -6.00000i q^{23} +2.00000 q^{26} -1.00000i q^{27} +6.00000i q^{28} -10.0000 q^{29} -3.00000 q^{31} -8.00000i q^{32} +2.00000i q^{33} -4.00000 q^{34} +2.00000 q^{36} +2.00000i q^{37} +10.0000i q^{38} +1.00000 q^{39} -8.00000 q^{41} +6.00000i q^{42} -1.00000i q^{43} -4.00000 q^{44} +12.0000 q^{46} +2.00000i q^{47} -4.00000i q^{48} -2.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} +4.00000i q^{53} +2.00000 q^{54} +5.00000i q^{57} -20.0000i q^{58} +10.0000 q^{59} +7.00000 q^{61} -6.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} -4.00000 q^{66} -3.00000i q^{67} -4.00000i q^{68} +6.00000 q^{69} -8.00000 q^{71} +14.0000i q^{73} -4.00000 q^{74} -10.0000 q^{76} -6.00000i q^{77} +2.00000i q^{78} +1.00000 q^{81} -16.0000i q^{82} -6.00000i q^{83} -6.00000 q^{84} +2.00000 q^{86} -10.0000i q^{87} -3.00000 q^{91} +12.0000i q^{92} -3.00000i q^{93} -4.00000 q^{94} +8.00000 q^{96} +17.0000i q^{97} -4.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9	$ 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9} + 4 q^{11} + 12 q^{14} - 8 q^{16} + 10 q^{19} + 6 q^{21} + 4 q^{26} - 20 q^{29} - 6 q^{31} - 8 q^{34} + 4 q^{36} + 2 q^{39} - 16 q^{41} - 8 q^{44} + 24 q^{46} - 4 q^{49} - 4 q^{51} + 4 q^{54} + 20 q^{59} + 14 q^{61} + 16 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} - 8 q^{74} - 20 q^{76} + 2 q^{81} - 12 q^{84} + 4 q^{86} - 6 q^{91} - 8 q^{94} + 16 q^{96} - 4 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9 + 4 * q^11 + 12 * q^14 - 8 * q^16 + 10 * q^19 + 6 * q^21 + 4 * q^26 - 20 * q^29 - 6 * q^31 - 8 * q^34 + 4 * q^36 + 2 * q^39 - 16 * q^41 - 8 * q^44 + 24 * q^46 - 4 * q^49 - 4 * q^51 + 4 * q^54 + 20 * q^59 + 14 * q^61 + 16 * q^64 - 8 * q^66 + 12 * q^69 - 16 * q^71 - 8 * q^74 - 20 * q^76 + 2 * q^81 - 12 * q^84 + 4 * q^86 - 6 * q^91 - 8 * q^94 + 16 * q^96 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$	2.00000i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	− 2.00000i	− 0.577350i
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	6.00000	1.60357
$15$	0	0
$16$	−4.00000	−1.00000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	− 2.00000i	− 0.471405i
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	4.00000i	0.852803i
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	− 1.00000i	− 0.192450i
$28$	6.00000i	1.13389i
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	− 8.00000i	− 1.41421i
$33$	2.00000i	0.348155i
$34$	−4.00000	−0.685994
$35$	0	0
$36$	2.00000	0.333333
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	10.0000i	1.62221i
$39$	1.00000	0.160128
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	6.00000i	0.925820i
$43$	− 1.00000i	− 0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	− 1.00000i	− 0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	12.0000	1.76930
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	− 4.00000i	− 0.577350i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−2.00000	−0.280056
$52$	2.00000i	0.277350i
$53$	4.00000i	0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$	4.00000i	0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$	2.00000	0.272166
$55$	0	0
$56$	0	0
$57$	5.00000i	0.662266i
$58$	− 20.0000i	− 2.62613i
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	− 6.00000i	− 0.762001i
$63$	3.00000i	0.377964i
$64$	8.00000	1.00000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	− 4.00000i	− 0.485071i
$69$	6.00000	0.722315
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−10.0000	−1.14708
$77$	− 6.00000i	− 0.683763i
$78$	2.00000i	0.226455i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 16.0000i	− 1.76690i
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	−6.00000	−0.654654
$85$	0	0
$86$	2.00000	0.215666
$87$	− 10.0000i	− 1.07211i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	12.0000i	1.25109i
$93$	− 3.00000i	− 0.311086i
$94$	−4.00000	−0.412568
$95$	0	0
$96$	8.00000	0.816497
$97$	17.0000i	1.72609i	0.505128	+	0.863044i	$0.331445\pi$
$97$	17.0000i	1.72609i	−0.505128	+	0.863044i	$0.668555\pi$
$98$	− 4.00000i	− 0.404061i
$99$	−2.00000	−0.201008
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	− 4.00000i	− 0.396059i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	0	0
$105$	0	0
$106$	−8.00000	−0.777029
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	2.00000i	0.192450i
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	12.0000i	1.13389i
$113$	4.00000i	0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$	4.00000i	0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	−10.0000	−0.936586
$115$	0	0
$116$	20.0000	1.85695
$117$	1.00000i	0.0924500i
$118$	20.0000i	1.84115i
$119$	6.00000	0.550019
$120$	0	0
$121$	−7.00000	−0.636364
$122$	14.0000i	1.26750i
$123$	− 8.00000i	− 0.721336i
$124$	6.00000	0.538816
$125$	0	0
$126$	−6.00000	−0.534522
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	0	0
$129$	1.00000	0.0880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	− 4.00000i	− 0.348155i
$133$	− 15.0000i	− 1.30066i
$134$	6.00000	0.518321
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	12.0000i	1.02151i
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	− 16.0000i	− 1.34269i
$143$	− 2.00000i	− 0.167248i
$144$	4.00000	0.333333
$145$	0	0
$146$	−28.0000	−2.31730
$147$	− 2.00000i	− 0.164957i
$148$	− 4.00000i	− 0.328798i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	12.0000	0.966988
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 13.0000i	− 1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$	− 13.0000i	− 1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	−18.0000	−1.41860
$162$	2.00000i	0.157135i
$163$	− 11.0000i	− 0.861586i	−0.902451	−	0.430793i	$-0.858234\pi$
$163$	− 11.0000i	− 0.861586i	0.902451	−	0.430793i	$-0.141766\pi$
$164$	16.0000	1.24939
$165$	0	0
$166$	12.0000	0.931381
$167$	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
$169$	12.0000	0.923077
$170$	0	0
$171$	−5.00000	−0.382360
$172$	2.00000i	0.152499i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	20.0000	1.51620
$175$	0	0
$176$	−8.00000	−0.603023
$177$	10.0000i	0.751646i
$178$	0	0
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	− 6.00000i	− 0.444750i
$183$	7.00000i	0.517455i
$184$	0	0
$185$	0	0
$186$	6.00000	0.439941
$187$	4.00000i	0.292509i
$188$	− 4.00000i	− 0.291730i
$189$	−3.00000	−0.218218
$190$	0	0
$191$	22.0000	1.59186	0.795932	−	0.605386i	$-0.206981\pi$
$191$	22.0000	1.59186	0.795932	+	0.605386i	$0.206981\pi$
$192$	8.00000i	0.577350i
$193$	− 11.0000i	− 0.791797i	−0.918294	−	0.395899i	$-0.870433\pi$
$193$	− 11.0000i	− 0.791797i	0.918294	−	0.395899i	$-0.129567\pi$
$194$	−34.0000	−2.44106
$195$	0	0
$196$	4.00000	0.285714
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	− 4.00000i	− 0.284268i
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	24.0000i	1.68863i
$203$	30.0000i	2.10559i
$204$	4.00000	0.280056
$205$	0	0
$206$	−8.00000	−0.557386
$207$	6.00000i	0.417029i
$208$	4.00000i	0.277350i
$209$	10.0000	0.691714
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	− 8.00000i	− 0.549442i
$213$	− 8.00000i	− 0.548151i
$214$	−24.0000	−1.64061
$215$	0	0
$216$	0	0
$217$	9.00000i	0.610960i
$218$	− 10.0000i	− 0.677285i
$219$	−14.0000	−0.946032
$220$	0	0
$221$	2.00000	0.134535
$222$	− 4.00000i	− 0.268462i
$223$	19.0000i	1.27233i	0.771551	+	0.636167i	$0.219481\pi$
$223$	19.0000i	1.27233i	−0.771551	+	0.636167i	$0.780519\pi$
$224$	−24.0000	−1.60357
$225$	0	0
$226$	−8.00000	−0.532152
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	− 10.0000i	− 0.662266i
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−20.0000	−1.30189
$237$	0	0
$238$	12.0000i	0.777844i
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	− 14.0000i	− 0.899954i
$243$	1.00000i	0.0641500i
$244$	−14.0000	−0.896258
$245$	0	0
$246$	16.0000	1.02012
$247$	− 5.00000i	− 0.318142i
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	− 6.00000i	− 0.377964i
$253$	− 12.0000i	− 0.754434i
$254$	16.0000	1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	2.00000i	0.124515i
$259$	6.00000	0.372822
$260$	0	0
$261$	10.0000	0.618984
$262$	24.0000i	1.48272i
$263$	− 16.0000i	− 0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$	− 16.0000i	− 0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$	0	0
$265$	0	0
$266$	30.0000	1.83942
$267$	0	0
$268$	6.00000i	0.366508i
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 8.00000i	− 0.485071i
$273$	− 3.00000i	− 0.181568i
$274$	36.0000	2.17484
$275$	0	0
$276$	−12.0000	−0.722315
$277$	− 3.00000i	− 0.180253i	−0.995930	−	0.0901263i	$-0.971273\pi$
$277$	− 3.00000i	− 0.180253i	0.995930	−	0.0901263i	$-0.0287271\pi$
$278$	− 40.0000i	− 2.39904i
$279$	3.00000	0.179605
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	− 4.00000i	− 0.238197i
$283$	9.00000i	0.534994i	0.963559	+	0.267497i	$0.0861966\pi$
$283$	9.00000i	0.534994i	−0.963559	+	0.267497i	$0.913803\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	4.00000	0.236525
$287$	24.0000i	1.41668i
$288$	8.00000i	0.471405i
$289$	13.0000	0.764706
$290$	0	0
$291$	−17.0000	−0.996558
$292$	− 28.0000i	− 1.63858i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	4.00000	0.233285
$295$	0	0
$296$	0	0
$297$	− 2.00000i	− 0.116052i
$298$	− 20.0000i	− 1.15857i
$299$	−6.00000	−0.346989
$300$	0	0
$301$	−3.00000	−0.172917
$302$	14.0000i	0.805609i
$303$	12.0000i	0.689382i
$304$	−20.0000	−1.14708
$305$	0	0
$306$	4.00000	0.228665
$307$	7.00000i	0.399511i	0.979846	+	0.199756i	$0.0640148\pi$
$307$	7.00000i	0.399511i	−0.979846	+	0.199756i	$0.935985\pi$
$308$	12.0000i	0.683763i
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	− 11.0000i	− 0.621757i	−0.950450	−	0.310878i	$-0.899377\pi$
$313$	− 11.0000i	− 0.621757i	0.950450	−	0.310878i	$-0.100623\pi$
$314$	26.0000	1.46726
$315$	0	0
$316$	0	0
$317$	− 8.00000i	− 0.449325i	−0.974437	−	0.224662i	$-0.927872\pi$
$317$	− 8.00000i	− 0.449325i	0.974437	−	0.224662i	$-0.0721279\pi$
$318$	− 8.00000i	− 0.448618i
$319$	−20.0000	−1.11979
$320$	0	0
$321$	−12.0000	−0.669775
$322$	− 36.0000i	− 2.00620i
$323$	10.0000i	0.556415i
$324$	−2.00000	−0.111111
$325$	0	0
$326$	22.0000	1.21847
$327$	− 5.00000i	− 0.276501i
$328$	0	0
$329$	6.00000	0.330791
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	12.0000i	0.658586i
$333$	− 2.00000i	− 0.109599i
$334$	−24.0000	−1.31322
$335$	0	0
$336$	−12.0000	−0.654654
$337$	− 23.0000i	− 1.25289i	−0.779466	−	0.626445i	$-0.784509\pi$
$337$	− 23.0000i	− 1.25289i	0.779466	−	0.626445i	$-0.215491\pi$
$338$	24.0000i	1.30543i
$339$	−4.00000	−0.217250
$340$	0	0
$341$	−6.00000	−0.324918
$342$	− 10.0000i	− 0.540738i
$343$	− 15.0000i	− 0.809924i
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	2.00000i	0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$	2.00000i	0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$	20.0000i	1.07211i
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	− 16.0000i	− 0.852803i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	−20.0000	−1.06299
$355$	0	0
$356$	0	0
$357$	6.00000i	0.317554i
$358$	20.0000i	1.05703i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	34.0000i	1.78700i
$363$	− 7.00000i	− 0.367405i
$364$	6.00000	0.314485
$365$	0	0
$366$	−14.0000	−0.731792
$367$	27.0000i	1.40939i	0.709511	+	0.704694i	$0.248916\pi$
$367$	27.0000i	1.40939i	−0.709511	+	0.704694i	$0.751084\pi$
$368$	24.0000i	1.25109i
$369$	8.00000	0.416463
$370$	0	0
$371$	12.0000	0.623009
$372$	6.00000i	0.311086i
$373$	29.0000i	1.50156i	0.660551	+	0.750782i	$0.270323\pi$
$373$	29.0000i	1.50156i	−0.660551	+	0.750782i	$0.729677\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	0	0
$377$	10.0000i	0.515026i
$378$	− 6.00000i	− 0.308607i
$379$	−25.0000	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	44.0000i	2.25124i
$383$	− 36.0000i	− 1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$	− 36.0000i	− 1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$	0	0
$385$	0	0
$386$	22.0000	1.11977
$387$	1.00000i	0.0508329i
$388$	− 34.0000i	− 1.72609i
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	12.0000i	0.605320i
$394$	36.0000	1.81365
$395$	0	0
$396$	4.00000	0.201008
$397$	7.00000i	0.351320i	0.984451	+	0.175660i	$0.0562059\pi$
$397$	7.00000i	0.351320i	−0.984451	+	0.175660i	$0.943794\pi$
$398$	10.0000i	0.501255i
$399$	15.0000	0.750939
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	6.00000i	0.299253i
$403$	3.00000i	0.149441i
$404$	−24.0000	−1.19404
$405$	0	0
$406$	−60.0000	−2.97775
$407$	4.00000i	0.198273i
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	− 8.00000i	− 0.394132i
$413$	− 30.0000i	− 1.47620i
$414$	−12.0000	−0.589768
$415$	0	0
$416$	−8.00000	−0.392232
$417$	− 20.0000i	− 0.979404i
$418$	20.0000i	0.978232i
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	− 26.0000i	− 1.26566i
$423$	− 2.00000i	− 0.0972433i
$424$	0	0
$425$	0	0
$426$	16.0000	0.775203
$427$	− 21.0000i	− 1.01626i
$428$	− 24.0000i	− 1.16008i
$429$	2.00000	0.0965609
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	4.00000i	0.192450i
$433$	29.0000i	1.39365i	0.717241	+	0.696826i	$0.245405\pi$
$433$	29.0000i	1.39365i	−0.717241	+	0.696826i	$0.754595\pi$
$434$	−18.0000	−0.864028
$435$	0	0
$436$	10.0000	0.478913
$437$	− 30.0000i	− 1.43509i
$438$	− 28.0000i	− 1.33789i
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	4.00000i	0.190261i
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−38.0000	−1.79935
$447$	− 10.0000i	− 0.472984i
$448$	− 24.0000i	− 1.13389i
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	− 8.00000i	− 0.376288i
$453$	7.00000i	0.328889i
$454$	16.0000	0.750917
$455$	0	0
$456$	0	0
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	30.0000i	1.40181i
$459$	2.00000	0.0933520
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	12.0000i	0.558291i
$463$	24.0000i	1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$	24.0000i	1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	40.0000	1.85695
$465$	0	0
$466$	−48.0000	−2.22356
$467$	− 38.0000i	− 1.75843i	−0.476425	−	0.879215i	$-0.658068\pi$
$467$	− 38.0000i	− 1.75843i	0.476425	−	0.879215i	$-0.341932\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	−9.00000	−0.415581
$470$	0	0
$471$	13.0000	0.599008
$472$	0	0
$473$	− 2.00000i	− 0.0919601i
$474$	0	0
$475$	0	0
$476$	−12.0000	−0.550019
$477$	− 4.00000i	− 0.183147i
$478$	− 40.0000i	− 1.82956i
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	− 46.0000i	− 2.09524i
$483$	− 18.0000i	− 0.819028i
$484$	14.0000	0.636364
$485$	0	0
$486$	−2.00000	−0.0907218
$487$	− 13.0000i	− 0.589086i	−0.955638	−	0.294543i	$-0.904833\pi$
$487$	− 13.0000i	− 0.589086i	0.955638	−	0.294543i	$-0.0951675\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	16.0000i	0.721336i
$493$	− 20.0000i	− 0.900755i
$494$	10.0000	0.449921
$495$	0	0
$496$	12.0000	0.538816
$497$	24.0000i	1.07655i
$498$	12.0000i	0.537733i
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	24.0000i	1.07117i
$503$	− 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
$505$	0	0
$506$	24.0000	1.06693
$507$	12.0000i	0.532939i
$508$	16.0000i	0.709885i
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	42.0000	1.85797
$512$	32.0000i	1.41421i
$513$	− 5.00000i	− 0.220755i
$514$	−24.0000	−1.05859
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	4.00000i	0.175920i
$518$	12.0000i	0.527250i
$519$	6.00000	0.263371
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	20.0000i	0.875376i
$523$	− 31.0000i	− 1.35554i	−0.735276	−	0.677768i	$-0.762948\pi$
$523$	− 31.0000i	− 1.35554i	0.735276	−	0.677768i	$-0.237052\pi$
$524$	−24.0000	−1.04844
$525$	0	0
$526$	32.0000	1.39527
$527$	− 6.00000i	− 0.261364i
$528$	− 8.00000i	− 0.348155i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−10.0000	−0.433963
$532$	30.0000i	1.30066i
$533$	8.00000i	0.346518i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.0000i	0.431532i
$538$	20.0000i	0.862261i
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	− 16.0000i	− 0.687259i
$543$	17.0000i	0.729540i
$544$	16.0000	0.685994
$545$	0	0
$546$	6.00000	0.256776
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	36.0000i	1.53784i
$549$	−7.00000	−0.298753
$550$	0	0
$551$	−50.0000	−2.13007
$552$	0	0
$553$	0	0
$554$	6.00000	0.254916
$555$	0	0
$556$	40.0000	1.69638
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	6.00000i	0.254000i
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	−4.00000	−0.168880
$562$	− 36.0000i	− 1.51857i
$563$	− 6.00000i	− 0.252870i	−0.991975	−	0.126435i	$-0.959647\pi$
$563$	− 6.00000i	− 0.252870i	0.991975	−	0.126435i	$-0.0403535\pi$
$564$	4.00000	0.168430
$565$	0	0
$566$	−18.0000	−0.756596
$567$	− 3.00000i	− 0.125988i
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−13.0000	−0.544033	−0.272017	−	0.962293i	$-0.587691\pi$
$571$	−13.0000	−0.544033	−0.272017	+	0.962293i	$0.587691\pi$
$572$	4.00000i	0.167248i
$573$	22.0000i	0.919063i
$574$	−48.0000	−2.00348
$575$	0	0
$576$	−8.00000	−0.333333
$577$	− 13.0000i	− 0.541197i	−0.962692	−	0.270599i	$-0.912778\pi$
$577$	− 13.0000i	− 0.541197i	0.962692	−	0.270599i	$-0.0872216\pi$
$578$	26.0000i	1.08146i
$579$	11.0000	0.457144
$580$	0	0
$581$	−18.0000	−0.746766
$582$	− 34.0000i	− 1.40935i
$583$	8.00000i	0.331326i
$584$	0	0
$585$	0	0
$586$	12.0000	0.495715
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	4.00000i	0.164957i
$589$	−15.0000	−0.618064
$590$	0	0
$591$	18.0000	0.740421
$592$	− 8.00000i	− 0.328798i
$593$	− 16.0000i	− 0.657041i	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	− 16.0000i	− 0.657041i	0.944497	−	0.328521i	$-0.106550\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	20.0000	0.819232
$597$	5.00000i	0.204636i
$598$	− 12.0000i	− 0.490716i
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	− 6.00000i	− 0.244542i
$603$	3.00000i	0.122169i
$604$	−14.0000	−0.569652
$605$	0	0
$606$	−24.0000	−0.974933
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	− 40.0000i	− 1.62221i
$609$	−30.0000	−1.21566
$610$	0	0
$611$	2.00000	0.0809113
$612$	4.00000i	0.161690i
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	−14.0000	−0.564994
$615$	0	0
$616$	0	0
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	− 8.00000i	− 0.321807i
$619$	25.0000	1.00483	0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000	1.00483	0.502417	+	0.864625i	$0.332444\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	− 36.0000i	− 1.44347i
$623$	0	0
$624$	−4.00000	−0.160128
$625$	0	0
$626$	22.0000	0.879297
$627$	10.0000i	0.399362i
$628$	26.0000i	1.03751i
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−23.0000	−0.915616	−0.457808	−	0.889051i	$-0.651365\pi$
$631$	−23.0000	−0.915616	−0.457808	+	0.889051i	$0.651365\pi$
$632$	0	0
$633$	− 13.0000i	− 0.516704i
$634$	16.0000	0.635441
$635$	0	0
$636$	8.00000	0.317221
$637$	2.00000i	0.0792429i
$638$	− 40.0000i	− 1.58362i
$639$	8.00000	0.316475
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	− 24.0000i	− 0.947204i
$643$	− 36.0000i	− 1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$643$	− 36.0000i	− 1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$644$	36.0000	1.41860
$645$	0	0
$646$	−20.0000	−0.786889
$647$	− 28.0000i	− 1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$	− 28.0000i	− 1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	−9.00000	−0.352738
$652$	22.0000i	0.861586i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	32.0000	1.24939
$657$	− 14.0000i	− 0.546192i
$658$	12.0000i	0.467809i
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	24.0000i	0.932786i
$663$	2.00000i	0.0776736i
$664$	0	0
$665$	0	0
$666$	4.00000	0.154997
$667$	60.0000i	2.32321i
$668$	− 24.0000i	− 0.928588i
$669$	−19.0000	−0.734582
$670$	0	0
$671$	14.0000	0.540464
$672$	− 24.0000i	− 0.925820i
$673$	− 6.00000i	− 0.231283i	−0.993291	−	0.115642i	$-0.963108\pi$
$673$	− 6.00000i	− 0.231283i	0.993291	−	0.115642i	$-0.0368924\pi$
$674$	46.0000	1.77185
$675$	0	0
$676$	−24.0000	−0.923077
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	− 8.00000i	− 0.307238i
$679$	51.0000	1.95720
$680$	0	0
$681$	8.00000	0.306561
$682$	− 12.0000i	− 0.459504i
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	10.0000	0.382360
$685$	0	0
$686$	30.0000	1.14541
$687$	15.0000i	0.572286i
$688$	4.00000i	0.152499i
$689$	4.00000	0.152388
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	12.0000i	0.456172i
$693$	6.00000i	0.227921i
$694$	−4.00000	−0.151838
$695$	0	0
$696$	0	0
$697$	− 16.0000i	− 0.606043i
$698$	− 20.0000i	− 0.757011i
$699$	−24.0000	−0.907763
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	10.0000i	0.377157i
$704$	16.0000	0.603023
$705$	0	0
$706$	12.0000	0.451626
$707$	− 36.0000i	− 1.35392i
$708$	− 20.0000i	− 0.751646i
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	18.0000i	0.674105i
$714$	−12.0000	−0.449089
$715$	0	0
$716$	−20.0000	−0.747435
$717$	− 20.0000i	− 0.746914i
$718$	0	0
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	12.0000i	0.446594i
$723$	− 23.0000i	− 0.855379i
$724$	−34.0000	−1.26360
$725$	0	0
$726$	14.0000	0.519589
$727$	− 43.0000i	− 1.59478i	−0.603463	−	0.797391i	$-0.706213\pi$
$727$	− 43.0000i	− 1.59478i	0.603463	−	0.797391i	$-0.293787\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	2.00000	0.0739727
$732$	− 14.0000i	− 0.517455i
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	−54.0000	−1.99318
$735$	0	0
$736$	−48.0000	−1.76930
$737$	− 6.00000i	− 0.221013i
$738$	16.0000i	0.588968i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	5.00000	0.183680
$742$	24.0000i	0.881068i
$743$	4.00000i	0.146746i	0.997305	+	0.0733729i	$0.0233763\pi$
$743$	4.00000i	0.146746i	−0.997305	+	0.0733729i	$0.976624\pi$
$744$	0	0
$745$	0	0
$746$	−58.0000	−2.12353
$747$	6.00000i	0.219529i
$748$	− 8.00000i	− 0.292509i
$749$	36.0000	1.31541
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	− 8.00000i	− 0.291730i
$753$	12.0000i	0.437304i
$754$	−20.0000	−0.728357
$755$	0	0
$756$	6.00000	0.218218
$757$	− 23.0000i	− 0.835949i	−0.908459	−	0.417975i	$-0.862740\pi$
$757$	− 23.0000i	− 0.835949i	0.908459	−	0.417975i	$-0.137260\pi$
$758$	− 50.0000i	− 1.81608i
$759$	12.0000	0.435572
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	16.0000i	0.579619i
$763$	15.0000i	0.543036i
$764$	−44.0000	−1.59186
$765$	0	0
$766$	72.0000	2.60147
$767$	− 10.0000i	− 0.361079i
$768$	16.0000i	0.577350i
$769$	−35.0000	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	22.0000i	0.791797i
$773$	24.0000i	0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$	24.0000i	0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	0	0
$777$	6.00000i	0.215249i
$778$	0	0
$779$	−40.0000	−1.43315
$780$	0	0
$781$	−16.0000	−0.572525
$782$	24.0000i	0.858238i
$783$	10.0000i	0.357371i
$784$	8.00000	0.285714
$785$	0	0
$786$	−24.0000	−0.856052
$787$	7.00000i	0.249523i	0.992187	+	0.124762i	$0.0398166\pi$
$787$	7.00000i	0.249523i	−0.992187	+	0.124762i	$0.960183\pi$
$788$	36.0000i	1.28245i
$789$	16.0000	0.569615
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	− 7.00000i	− 0.248577i
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−10.0000	−0.354441
$797$	52.0000i	1.84193i	0.389640	+	0.920967i	$0.372599\pi$
$797$	52.0000i	1.84193i	−0.389640	+	0.920967i	$0.627401\pi$
$798$	30.0000i	1.06199i
$799$	−4.00000	−0.141510
$800$	0	0
$801$	0	0
$802$	24.0000i	0.847469i
$803$	28.0000i	0.988099i
$804$	−6.00000	−0.211604
$805$	0	0
$806$	−6.00000	−0.211341
$807$	10.0000i	0.352017i
$808$	0	0
$809$	20.0000	0.703163	0.351581	−	0.936157i	$-0.385644\pi$
$809$	20.0000	0.703163	0.351581	+	0.936157i	$0.385644\pi$
$810$	0	0
$811$	27.0000	0.948098	0.474049	−	0.880498i	$-0.342792\pi$
$811$	27.0000	0.948098	0.474049	+	0.880498i	$0.342792\pi$
$812$	− 60.0000i	− 2.10559i
$813$	− 8.00000i	− 0.280572i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	8.00000	0.280056
$817$	− 5.00000i	− 0.174928i
$818$	− 10.0000i	− 0.349642i
$819$	3.00000	0.104828
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	36.0000i	1.25564i
$823$	− 41.0000i	− 1.42917i	−0.699549	−	0.714585i	$-0.746616\pi$
$823$	− 41.0000i	− 1.42917i	0.699549	−	0.714585i	$-0.253384\pi$
$824$	0	0
$825$	0	0
$826$	60.0000	2.08767
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	− 12.0000i	− 0.417029i
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	3.00000	0.104069
$832$	− 8.00000i	− 0.277350i
$833$	− 4.00000i	− 0.138592i
$834$	40.0000	1.38509
$835$	0	0
$836$	−20.0000	−0.691714
$837$	3.00000i	0.103695i
$838$	40.0000i	1.38178i
$839$	−10.0000	−0.345238	−0.172619	−	0.984989i	$-0.555223\pi$
$839$	−10.0000	−0.345238	−0.172619	+	0.984989i	$0.555223\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	44.0000i	1.51634i
$843$	− 18.0000i	− 0.619953i
$844$	26.0000	0.894957
$845$	0	0
$846$	4.00000	0.137523
$847$	21.0000i	0.721569i
$848$	− 16.0000i	− 0.549442i
$849$	−9.00000	−0.308879
$850$	0	0
$851$	12.0000	0.411355
$852$	16.0000i	0.548151i
$853$	− 51.0000i	− 1.74621i	−0.487535	−	0.873103i	$-0.662104\pi$
$853$	− 51.0000i	− 1.74621i	0.487535	−	0.873103i	$-0.337896\pi$
$854$	42.0000	1.43721
$855$	0	0
$856$	0	0
$857$	− 28.0000i	− 0.956462i	−0.878234	−	0.478231i	$-0.841278\pi$
$857$	− 28.0000i	− 0.956462i	0.878234	−	0.478231i	$-0.158722\pi$
$858$	4.00000i	0.136558i
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	− 36.0000i	− 1.22616i
$863$	− 16.0000i	− 0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$	− 16.0000i	− 0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	−8.00000	−0.272166
$865$	0	0
$866$	−58.0000	−1.97092
$867$	13.0000i	0.441503i
$868$	− 18.0000i	− 0.610960i
$869$	0	0
$870$	0	0
$871$	−3.00000	−0.101651
$872$	0	0
$873$	− 17.0000i	− 0.575363i
$874$	60.0000	2.02953
$875$	0	0
$876$	28.0000	0.946032
$877$	27.0000i	0.911725i	0.890050	+	0.455863i	$0.150669\pi$
$877$	27.0000i	0.911725i	−0.890050	+	0.455863i	$0.849331\pi$
$878$	70.0000i	2.36239i
$879$	6.00000	0.202375
$880$	0	0
$881$	32.0000	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$882$	4.00000i	0.134687i
$883$	− 41.0000i	− 1.37976i	−0.723924	−	0.689880i	$-0.757663\pi$
$883$	− 41.0000i	− 1.37976i	0.723924	−	0.689880i	$-0.242337\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−48.0000	−1.61259
$887$	− 18.0000i	− 0.604381i	−0.953248	−	0.302190i	$-0.902282\pi$
$887$	− 18.0000i	− 0.604381i	0.953248	−	0.302190i	$-0.0977178\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	2.00000	0.0670025
$892$	− 38.0000i	− 1.27233i
$893$	10.0000i	0.334637i
$894$	20.0000	0.668900
$895$	0	0
$896$	0	0
$897$	− 6.00000i	− 0.200334i
$898$	− 40.0000i	− 1.33482i
$899$	30.0000	1.00056
$900$	0	0
$901$	−8.00000	−0.266519
$902$	− 32.0000i	− 1.06548i
$903$	− 3.00000i	− 0.0998337i
$904$	0	0
$905$	0	0
$906$	−14.0000	−0.465119
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	16.0000i	0.530979i
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−58.0000	−1.92163	−0.960813	−	0.277198i	$-0.910594\pi$
$911$	−58.0000	−1.92163	−0.960813	+	0.277198i	$0.910594\pi$
$912$	− 20.0000i	− 0.662266i
$913$	− 12.0000i	− 0.397142i
$914$	−44.0000	−1.45539
$915$	0	0
$916$	−30.0000	−0.991228
$917$	− 36.0000i	− 1.18882i
$918$	4.00000i	0.132020i
$919$	−55.0000	−1.81428	−0.907141	−	0.420826i	$-0.861740\pi$
$919$	−55.0000	−1.81428	−0.907141	+	0.420826i	$0.861740\pi$
$920$	0	0
$921$	−7.00000	−0.230658
$922$	24.0000i	0.790398i
$923$	8.00000i	0.263323i
$924$	−12.0000	−0.394771
$925$	0	0
$926$	−48.0000	−1.57738
$927$	− 4.00000i	− 0.131377i
$928$	80.0000i	2.62613i
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	−10.0000	−0.327737
$932$	− 48.0000i	− 1.57229i
$933$	− 18.0000i	− 0.589294i
$934$	76.0000	2.48680
$935$	0	0
$936$	0	0
$937$	− 33.0000i	− 1.07806i	−0.842286	−	0.539032i	$-0.818790\pi$
$937$	− 33.0000i	− 1.07806i	0.842286	−	0.539032i	$-0.181210\pi$
$938$	− 18.0000i	− 0.587721i
$939$	11.0000	0.358971
$940$	0	0
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	26.0000i	0.847126i
$943$	48.0000i	1.56310i
$944$	−40.0000	−1.30189
$945$	0	0
$946$	4.00000	0.130051
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	14.0000	0.454459
$950$	0	0
$951$	8.00000	0.259418
$952$	0	0
$953$	− 56.0000i	− 1.81402i	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	− 56.0000i	− 1.81402i	0.421111	−	0.907009i	$-0.361640\pi$
$954$	8.00000	0.259010
$955$	0	0
$956$	40.0000	1.29369
$957$	− 20.0000i	− 0.646508i
$958$	− 60.0000i	− 1.93851i
$959$	−54.0000	−1.74375
$960$	0	0
$961$	−22.0000	−0.709677
$962$	4.00000i	0.128965i
$963$	− 12.0000i	− 0.386695i
$964$	46.0000	1.48156
$965$	0	0
$966$	36.0000	1.15828
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	0	0
$969$	−10.0000	−0.321246
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	− 2.00000i	− 0.0641500i
$973$	60.0000i	1.92351i
$974$	26.0000	0.833094
$975$	0	0
$976$	−28.0000	−0.896258
$977$	2.00000i	0.0639857i	0.999488	+	0.0319928i	$0.0101854\pi$
$977$	2.00000i	0.0639857i	−0.999488	+	0.0319928i	$0.989815\pi$
$978$	22.0000i	0.703482i
$979$	0	0
$980$	0	0
$981$	5.00000	0.159638
$982$	− 16.0000i	− 0.510581i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	0	0
$985$	0	0
$986$	40.0000	1.27386
$987$	6.00000i	0.190982i
$988$	10.0000i	0.318142i
$989$	−6.00000	−0.190789
$990$	0	0
$991$	17.0000	0.540023	0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000	0.540023	0.270011	+	0.962857i	$0.412973\pi$
$992$	24.0000i	0.762001i
$993$	12.0000i	0.380808i
$994$	−48.0000	−1.52247
$995$	0	0
$996$	−12.0000	−0.380235
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	10.0000i	0.316544i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.2.b.a.49.2		2
3.2	odd	2		225.2.b.a.199.1		2
4.3	odd	2		1200.2.f.d.49.1		2
5.2	odd	4		75.2.a.a.1.1	✓	1
5.3	odd	4		75.2.a.c.1.1	yes	1
5.4	even	2	inner	75.2.b.a.49.1		2
8.3	odd	2		4800.2.f.y.3649.2		2
8.5	even	2		4800.2.f.l.3649.1		2
12.11	even	2		3600.2.f.p.2449.2		2
15.2	even	4		225.2.a.e.1.1		1
15.8	even	4		225.2.a.a.1.1		1
15.14	odd	2		225.2.b.a.199.2		2
20.3	even	4		1200.2.a.p.1.1		1
20.7	even	4		1200.2.a.c.1.1		1
20.19	odd	2		1200.2.f.d.49.2		2
35.13	even	4		3675.2.a.q.1.1		1
35.27	even	4		3675.2.a.b.1.1		1
40.3	even	4		4800.2.a.be.1.1		1
40.13	odd	4		4800.2.a.bq.1.1		1
40.19	odd	2		4800.2.f.y.3649.1		2
40.27	even	4		4800.2.a.br.1.1		1
40.29	even	2		4800.2.f.l.3649.2		2
40.37	odd	4		4800.2.a.bb.1.1		1
55.32	even	4		9075.2.a.s.1.1		1
55.43	even	4		9075.2.a.a.1.1		1
60.23	odd	4		3600.2.a.bk.1.1		1
60.47	odd	4		3600.2.a.j.1.1		1
60.59	even	2		3600.2.f.p.2449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.a.a.1.1	✓	1	5.2	odd	4
75.2.a.c.1.1	yes	1	5.3	odd	4
75.2.b.a.49.1		2	5.4	even	2	inner
75.2.b.a.49.2		2	1.1	even	1	trivial
225.2.a.a.1.1		1	15.8	even	4
225.2.a.e.1.1		1	15.2	even	4
225.2.b.a.199.1		2	3.2	odd	2
225.2.b.a.199.2		2	15.14	odd	2
1200.2.a.c.1.1		1	20.7	even	4
1200.2.a.p.1.1		1	20.3	even	4
1200.2.f.d.49.1		2	4.3	odd	2
1200.2.f.d.49.2		2	20.19	odd	2
3600.2.a.j.1.1		1	60.47	odd	4
3600.2.a.bk.1.1		1	60.23	odd	4
3600.2.f.p.2449.1		2	60.59	even	2
3600.2.f.p.2449.2		2	12.11	even	2
3675.2.a.b.1.1		1	35.27	even	4
3675.2.a.q.1.1		1	35.13	even	4
4800.2.a.bb.1.1		1	40.37	odd	4
4800.2.a.be.1.1		1	40.3	even	4
4800.2.a.bq.1.1		1	40.13	odd	4
4800.2.a.br.1.1		1	40.27	even	4
4800.2.f.l.3649.1		2	8.5	even	2
4800.2.f.l.3649.2		2	40.29	even	2
4800.2.f.y.3649.1		2	40.19	odd	2
4800.2.f.y.3649.2		2	8.3	odd	2
9075.2.a.a.1.1		1	55.43	even	4
9075.2.a.s.1.1		1	55.32	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -2.00000i q^{12} -1.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} -2.00000i q^{18} +5.00000 q^{19} +3.00000 q^{21} +4.00000i q^{22} -6.00000i q^{23} +2.00000 q^{26} -1.00000i q^{27} +6.00000i q^{28} -10.0000 q^{29} -3.00000 q^{31} -8.00000i q^{32} +2.00000i q^{33} -4.00000 q^{34} +2.00000 q^{36} +2.00000i q^{37} +10.0000i q^{38} +1.00000 q^{39} -8.00000 q^{41} +6.00000i q^{42} -1.00000i q^{43} -4.00000 q^{44} +12.0000 q^{46} +2.00000i q^{47} -4.00000i q^{48} -2.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} +4.00000i q^{53} +2.00000 q^{54} +5.00000i q^{57} -20.0000i q^{58} +10.0000 q^{59} +7.00000 q^{61} -6.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} -4.00000 q^{66} -3.00000i q^{67} -4.00000i q^{68} +6.00000 q^{69} -8.00000 q^{71} +14.0000i q^{73} -4.00000 q^{74} -10.0000 q^{76} -6.00000i q^{77} +2.00000i q^{78} +1.00000 q^{81} -16.0000i q^{82} -6.00000i q^{83} -6.00000 q^{84} +2.00000 q^{86} -10.0000i q^{87} -3.00000 q^{91} +12.0000i q^{92} -3.00000i q^{93} -4.00000 q^{94} +8.00000 q^{96} +17.0000i q^{97} -4.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9} + 4 q^{11} + 12 q^{14} - 8 q^{16} + 10 q^{19} + 6 q^{21} + 4 q^{26} - 20 q^{29} - 6 q^{31} - 8 q^{34} + 4 q^{36} + 2 q^{39} - 16 q^{41} - 8 q^{44} + 24 q^{46} - 4 q^{49} - 4 q^{51} + 4 q^{54} + 20 q^{59} + 14 q^{61} + 16 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} - 8 q^{74} - 20 q^{76} + 2 q^{81} - 12 q^{84} + 4 q^{86} - 6 q^{91} - 8 q^{94} + 16 q^{96} - 4 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9 + 4 * q^11 + 12 * q^14 - 8 * q^16 + 10 * q^19 + 6 * q^21 + 4 * q^26 - 20 * q^29 - 6 * q^31 - 8 * q^34 + 4 * q^36 + 2 * q^39 - 16 * q^41 - 8 * q^44 + 24 * q^46 - 4 * q^49 - 4 * q^51 + 4 * q^54 + 20 * q^59 + 14 * q^61 + 16 * q^64 - 8 * q^66 + 12 * q^69 - 16 * q^71 - 8 * q^74 - 20 * q^76 + 2 * q^81 - 12 * q^84 + 4 * q^86 - 6 * q^91 - 8 * q^94 + 16 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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