LMFDB - Embedded newform 1521.2.b.b.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1351,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.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:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.b.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{4} -2.00000i q^{5} +4.00000i q^{7} -3.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} +1.00000 q^{4} -2.00000i q^{5} +4.00000i q^{7} -3.00000i q^{8} -2.00000 q^{10} +4.00000i q^{11} +4.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} -2.00000i q^{20} +4.00000 q^{22} +1.00000 q^{25} +4.00000i q^{28} +10.0000 q^{29} +4.00000i q^{31} -5.00000i q^{32} -2.00000i q^{34} +8.00000 q^{35} +2.00000i q^{37} -6.00000 q^{40} -6.00000i q^{41} +12.0000 q^{43} +4.00000i q^{44} -9.00000 q^{49} -1.00000i q^{50} -6.00000 q^{53} +8.00000 q^{55} +12.0000 q^{56} -10.0000i q^{58} +12.0000i q^{59} -2.00000 q^{61} +4.00000 q^{62} -7.00000 q^{64} -8.00000i q^{67} +2.00000 q^{68} -8.00000i q^{70} -2.00000i q^{73} +2.00000 q^{74} -16.0000 q^{77} +8.00000 q^{79} +2.00000i q^{80} -6.00000 q^{82} -4.00000i q^{83} -4.00000i q^{85} -12.0000i q^{86} +12.0000 q^{88} -2.00000i q^{89} +10.0000i q^{97} +9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} - 4 q^{10} + 8 q^{14} - 2 q^{16} + 4 q^{17} + 8 q^{22} + 2 q^{25} + 20 q^{29} + 16 q^{35} - 12 q^{40} + 24 q^{43} - 18 q^{49} - 12 q^{53} + 16 q^{55} + 24 q^{56} - 4 q^{61} + 8 q^{62} - 14 q^{64} + 4 q^{68} + 4 q^{74} - 32 q^{77} + 16 q^{79} - 12 q^{82} + 24 q^{88}+O(q^{100}) $ 2 * q + 2 * q^4 - 4 * q^10 + 8 * q^14 - 2 * q^16 + 4 * q^17 + 8 * q^22 + 2 * q^25 + 20 * q^29 + 16 * q^35 - 12 * q^40 + 24 * q^43 - 18 * q^49 - 12 * q^53 + 16 * q^55 + 24 * q^56 - 4 * q^61 + 8 * q^62 - 14 * q^64 + 4 * q^68 + 4 * q^74 - 32 * q^77 + 16 * q^79 - 12 * q^82 + 24 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	− 2.00000i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 2.00000i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 3.00000i	− 1.06066i
$9$	0	0
$10$	−2.00000	−0.632456
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	4.00000	1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	− 2.00000i	− 0.447214i
$21$	0	0
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	4.00000i	0.755929i
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	− 5.00000i	− 0.883883i
$33$	0	0
$34$	− 2.00000i	− 0.342997i
$35$	8.00000	1.35225
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	−6.00000	−0.948683
$41$	− 6.00000i	− 0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	− 6.00000i	− 0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$	0	0
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	4.00000i	0.603023i
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	12.0000	1.60357
$57$	0	0
$58$	− 10.0000i	− 1.31306i
$59$	12.0000i	1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$	12.0000i	1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	− 8.00000i	− 0.956183i
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	0	0
$77$	−16.0000	−1.82337
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	2.00000i	0.223607i
$81$	0	0
$82$	−6.00000	−0.662589
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	− 4.00000i	− 0.433861i
$86$	− 12.0000i	− 1.29399i
$87$	0	0
$88$	12.0000	1.27920
$89$	− 2.00000i	− 0.212000i	−0.994366	−	0.106000i	$-0.966196\pi$
$89$	− 2.00000i	− 0.212000i	0.994366	−	0.106000i	$-0.0338043\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	9.00000i	0.909137i
$99$	0	0
$100$	1.00000	0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	6.00000i	0.582772i
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	− 8.00000i	− 0.762770i
$111$	0	0
$112$	− 4.00000i	− 0.377964i
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	10.0000	0.928477
$117$	0	0
$118$	12.0000	1.10469
$119$	8.00000i	0.733359i
$120$	0	0
$121$	−5.00000	−0.454545
$122$	2.00000i	0.181071i
$123$	0	0
$124$	4.00000i	0.359211i
$125$	− 12.0000i	− 1.07331i
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	− 3.00000i	− 0.265165i
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	− 6.00000i	− 0.514496i
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	8.00000	0.676123
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 20.0000i	− 1.66091i
$146$	−2.00000	−0.165521
$147$	0	0
$148$	2.00000i	0.164399i
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	− 4.00000i	− 0.325515i	−0.986666	−	0.162758i	$-0.947961\pi$
$151$	− 4.00000i	− 0.325515i	0.986666	−	0.162758i	$-0.0520389\pi$
$152$	0	0
$153$	0	0
$154$	16.0000i	1.28932i
$155$	8.00000	0.642575
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	− 8.00000i	− 0.636446i
$159$	0	0
$160$	−10.0000	−0.790569
$161$	0	0
$162$	0	0
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	− 6.00000i	− 0.468521i
$165$	0	0
$166$	−4.00000	−0.310460
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	0	0
$170$	−4.00000	−0.306786
$171$	0	0
$172$	12.0000	0.914991
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	4.00000i	0.302372i
$176$	− 4.00000i	− 0.301511i
$177$	0	0
$178$	−2.00000	−0.149906
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	− 18.0000i	− 1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	− 18.0000i	− 1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−9.00000	−0.642857
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	− 3.00000i	− 0.212132i
$201$	0	0
$202$	18.0000i	1.26648i
$203$	40.0000i	2.80745i
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	12.0000i	0.820303i
$215$	− 24.0000i	− 1.63679i
$216$	0	0
$217$	−16.0000	−1.08615
$218$	−2.00000	−0.135457
$219$	0	0
$220$	8.00000	0.539360
$221$	0	0
$222$	0	0
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	20.0000	1.33631
$225$	0	0
$226$	− 6.00000i	− 0.399114i
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	0	0
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	0	0
$231$	0	0
$232$	− 30.0000i	− 1.96960i
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	0	0
$236$	12.0000i	0.781133i
$237$	0	0
$238$	8.00000	0.518563
$239$	24.0000i	1.55243i	0.630468	+	0.776215i	$0.282863\pi$
$239$	24.0000i	1.55243i	−0.630468	+	0.776215i	$0.717137\pi$
$240$	0	0
$241$	− 10.0000i	− 0.644157i	−0.946713	−	0.322078i	$-0.895619\pi$
$241$	− 10.0000i	− 0.644157i	0.946713	−	0.322078i	$-0.104381\pi$
$242$	5.00000i	0.321412i
$243$	0	0
$244$	−2.00000	−0.128037
$245$	18.0000i	1.14998i
$246$	0	0
$247$	0	0
$248$	12.0000	0.762001
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	− 16.0000i	− 1.00393i
$255$	0	0
$256$	−17.0000	−1.06250
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	4.00000i	0.247121i
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	12.0000i	0.737154i
$266$	0	0
$267$	0	0
$268$	− 8.00000i	− 0.488678i
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	12.0000i	0.728948i	0.931214	+	0.364474i	$0.118751\pi$
$271$	12.0000i	0.728948i	−0.931214	+	0.364474i	$0.881249\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	6.00000	0.362473
$275$	4.00000i	0.241209i
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	− 12.0000i	− 0.719712i
$279$	0	0
$280$	− 24.0000i	− 1.43427i
$281$	− 10.0000i	− 0.596550i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	− 10.0000i	− 0.596550i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−20.0000	−1.17444
$291$	0	0
$292$	− 2.00000i	− 0.117041i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	6.00000	0.348743
$297$	0	0
$298$	6.00000	0.347571
$299$	0	0
$300$	0	0
$301$	48.0000i	2.76667i
$302$	−4.00000	−0.230174
$303$	0	0
$304$	0	0
$305$	4.00000i	0.229039i
$306$	0	0
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	−16.0000	−0.911685
$309$	0	0
$310$	− 8.00000i	− 0.454369i
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	18.0000i	1.01580i
$315$	0	0
$316$	8.00000	0.450035
$317$	− 26.0000i	− 1.46031i	−0.683284	−	0.730153i	$-0.739449\pi$
$317$	− 26.0000i	− 1.46031i	0.683284	−	0.730153i	$-0.260551\pi$
$318$	0	0
$319$	40.0000i	2.23957i
$320$	14.0000i	0.782624i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−8.00000	−0.443079
$327$	0	0
$328$	−18.0000	−0.993884
$329$	0	0
$330$	0	0
$331$	− 16.0000i	− 0.879440i	−0.898135	−	0.439720i	$-0.855078\pi$
$331$	− 16.0000i	− 0.879440i	0.898135	−	0.439720i	$-0.144922\pi$
$332$	− 4.00000i	− 0.219529i
$333$	0	0
$334$	−8.00000	−0.437741
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	0	0
$340$	− 4.00000i	− 0.216930i
$341$	−16.0000	−0.866449
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	− 36.0000i	− 1.94099i
$345$	0	0
$346$	− 6.00000i	− 0.322562i
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	26.0000i	1.39175i	0.718164	+	0.695874i	$0.244983\pi$
$349$	26.0000i	1.39175i	−0.718164	+	0.695874i	$0.755017\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	20.0000	1.06600
$353$	2.00000i	0.106449i	0.998583	+	0.0532246i	$0.0169499\pi$
$353$	2.00000i	0.106449i	−0.998583	+	0.0532246i	$0.983050\pi$
$354$	0	0
$355$	0	0
$356$	− 2.00000i	− 0.106000i
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	24.0000i	1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$	24.0000i	1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	− 10.0000i	− 0.525588i
$363$	0	0
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	0	0
$370$	− 4.00000i	− 0.207950i
$371$	− 24.0000i	− 1.24602i
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 24.0000i	− 1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$379$	− 24.0000i	− 1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$380$	0	0
$381$	0	0
$382$	8.00000i	0.409316i
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	0	0
$385$	32.0000i	1.63087i
$386$	−18.0000	−0.916176
$387$	0	0
$388$	10.0000i	0.507673i
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	0	0
$392$	27.0000i	1.36371i
$393$	0	0
$394$	−18.0000	−0.906827
$395$	− 16.0000i	− 0.805047i
$396$	0	0
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	22.0000i	1.09863i	0.835616	+	0.549314i	$0.185111\pi$
$401$	22.0000i	1.09863i	−0.835616	+	0.549314i	$0.814889\pi$
$402$	0	0
$403$	0	0
$404$	−18.0000	−0.895533
$405$	0	0
$406$	40.0000	1.98517
$407$	−8.00000	−0.396545
$408$	0	0
$409$	34.0000i	1.68119i	0.541663	+	0.840596i	$0.317795\pi$
$409$	34.0000i	1.68119i	−0.541663	+	0.840596i	$0.682205\pi$
$410$	12.0000i	0.592638i
$411$	0	0
$412$	0	0
$413$	−48.0000	−2.36193
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	20.0000i	0.973585i
$423$	0	0
$424$	18.0000i	0.874157i
$425$	2.00000	0.0970143
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−24.0000	−1.15738
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	16.0000i	0.768025i
$435$	0	0
$436$	− 2.00000i	− 0.0957826i
$437$	0	0
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	− 24.0000i	− 1.14416i
$441$	0	0
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	4.00000	0.189405
$447$	0	0
$448$	− 28.0000i	− 1.32288i
$449$	22.0000i	1.03824i	0.854700	+	0.519122i	$0.173741\pi$
$449$	22.0000i	1.03824i	−0.854700	+	0.519122i	$0.826259\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	6.00000	0.282216
$453$	0	0
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	0	0
$461$	38.0000i	1.76984i	0.465746	+	0.884918i	$0.345786\pi$
$461$	38.0000i	1.76984i	−0.465746	+	0.884918i	$0.654214\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	14.0000i	0.648537i
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	0	0
$472$	36.0000	1.65703
$473$	48.0000i	2.20704i
$474$	0	0
$475$	0	0
$476$	8.00000i	0.366679i
$477$	0	0
$478$	24.0000	1.09773
$479$	− 24.0000i	− 1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$	− 24.0000i	− 1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$	0	0
$481$	0	0
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−5.00000	−0.227273
$485$	20.0000	0.908153
$486$	0	0
$487$	12.0000i	0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$	12.0000i	0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	6.00000i	0.271607i
$489$	0	0
$490$	18.0000	0.813157
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	0	0
$496$	− 4.00000i	− 0.179605i
$497$	0	0
$498$	0	0
$499$	− 24.0000i	− 1.07439i	−0.843459	−	0.537194i	$-0.819484\pi$
$499$	− 24.0000i	− 1.07439i	0.843459	−	0.537194i	$-0.180516\pi$
$500$	− 12.0000i	− 0.536656i
$501$	0	0
$502$	12.0000i	0.535586i
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	36.0000i	1.60198i
$506$	0	0
$507$	0	0
$508$	16.0000	0.709885
$509$	− 10.0000i	− 0.443242i	−0.975133	−	0.221621i	$-0.928865\pi$
$509$	− 10.0000i	− 0.443242i	0.975133	−	0.221621i	$-0.0711348\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	11.0000i	0.486136i
$513$	0	0
$514$	− 26.0000i	− 1.14681i
$515$	0	0
$516$	0	0
$517$	0	0
$518$	8.00000i	0.351500i
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	24.0000i	1.04645i
$527$	8.00000i	0.348485i
$528$	0	0
$529$	−23.0000	−1.00000
$530$	12.0000	0.521247
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	24.0000i	1.03761i
$536$	−24.0000	−1.03664
$537$	0	0
$538$	22.0000i	0.948487i
$539$	− 36.0000i	− 1.55063i
$540$	0	0
$541$	− 30.0000i	− 1.28980i	−0.764267	−	0.644900i	$-0.776899\pi$
$541$	− 30.0000i	− 1.28980i	0.764267	−	0.644900i	$-0.223101\pi$
$542$	12.0000	0.515444
$543$	0	0
$544$	− 10.0000i	− 0.428746i
$545$	−4.00000	−0.171341
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	6.00000i	0.256307i
$549$	0	0
$550$	4.00000	0.170561
$551$	0	0
$552$	0	0
$553$	32.0000i	1.36078i
$554$	− 10.0000i	− 0.424859i
$555$	0	0
$556$	12.0000	0.508913
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	0	0
$560$	−8.00000	−0.338062
$561$	0	0
$562$	−10.0000	−0.421825
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	− 12.0000i	− 0.504844i
$566$	12.0000i	0.504398i
$567$	0	0
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0	0
$574$	− 24.0000i	− 1.00174i
$575$	0	0
$576$	0	0
$577$	− 46.0000i	− 1.91501i	−0.288425	−	0.957503i	$-0.593132\pi$
$577$	− 46.0000i	− 1.91501i	0.288425	−	0.957503i	$-0.406868\pi$
$578$	13.0000i	0.540729i
$579$	0	0
$580$	− 20.0000i	− 0.830455i
$581$	16.0000	0.663792
$582$	0	0
$583$	− 24.0000i	− 0.993978i
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−6.00000	−0.247858
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	0	0
$590$	− 24.0000i	− 0.988064i
$591$	0	0
$592$	− 2.00000i	− 0.0821995i
$593$	− 26.0000i	− 1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$	− 26.0000i	− 1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	6.00000i	0.245770i
$597$	0	0
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	48.0000	1.95633
$603$	0	0
$604$	− 4.00000i	− 0.162758i
$605$	10.0000i	0.406558i
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	0	0
$610$	4.00000	0.161955
$611$	0	0
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	48.0000i	1.93398i
$617$	− 22.0000i	− 0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$	− 22.0000i	− 0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$	0	0
$619$	− 24.0000i	− 0.964641i	−0.875995	−	0.482321i	$-0.839794\pi$
$619$	− 24.0000i	− 0.964641i	0.875995	−	0.482321i	$-0.160206\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	0	0
$623$	8.00000	0.320513
$624$	0	0
$625$	−19.0000	−0.760000
$626$	6.00000i	0.239808i
$627$	0	0
$628$	−18.0000	−0.718278
$629$	4.00000i	0.159490i
$630$	0	0
$631$	− 20.0000i	− 0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$	− 20.0000i	− 0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$	− 24.0000i	− 0.954669i
$633$	0	0
$634$	−26.0000	−1.03259
$635$	− 32.0000i	− 1.26988i
$636$	0	0
$637$	0	0
$638$	40.0000	1.58362
$639$	0	0
$640$	−6.00000	−0.237171
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	− 40.0000i	− 1.57745i	−0.614749	−	0.788723i	$-0.710743\pi$
$643$	− 40.0000i	− 1.57745i	0.614749	−	0.788723i	$-0.289257\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	0	0
$651$	0	0
$652$	− 8.00000i	− 0.313304i
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	8.00000i	0.312586i
$656$	6.00000i	0.234261i
$657$	0	0
$658$	0	0
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	− 30.0000i	− 1.16686i	−0.812162	−	0.583432i	$-0.801709\pi$
$661$	− 30.0000i	− 1.16686i	0.812162	−	0.583432i	$-0.198291\pi$
$662$	−16.0000	−0.621858
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 8.00000i	− 0.309529i
$669$	0	0
$670$	16.0000i	0.618134i
$671$	− 8.00000i	− 0.308837i
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	18.0000i	0.693334i
$675$	0	0
$676$	0	0
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	0	0
$679$	−40.0000	−1.53506
$680$	−12.0000	−0.460179
$681$	0	0
$682$	16.0000i	0.612672i
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	−8.00000	−0.305441
$687$	0	0
$688$	−12.0000	−0.457496
$689$	0	0
$690$	0	0
$691$	− 24.0000i	− 0.913003i	−0.889723	−	0.456502i	$-0.849102\pi$
$691$	− 24.0000i	− 0.913003i	0.889723	−	0.456502i	$-0.150898\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	− 12.0000i	− 0.455514i
$695$	− 24.0000i	− 0.910372i
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	26.0000	0.984115
$699$	0	0
$700$	4.00000i	0.151186i
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	0	0
$703$	0	0
$704$	− 28.0000i	− 1.05529i
$705$	0	0
$706$	2.00000	0.0752710
$707$	− 72.0000i	− 2.70784i
$708$	0	0
$709$	26.0000i	0.976450i	0.872718	+	0.488225i	$0.162356\pi$
$709$	26.0000i	0.976450i	−0.872718	+	0.488225i	$0.837644\pi$
$710$	0	0
$711$	0	0
$712$	−6.00000	−0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	24.0000	0.895672
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	0	0
$722$	− 19.0000i	− 0.707107i
$723$	0	0
$724$	10.0000	0.371647
$725$	10.0000	0.371391
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	0	0
$730$	4.00000i	0.148047i
$731$	24.0000	0.887672
$732$	0	0
$733$	30.0000i	1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$	30.0000i	1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$	− 16.0000i	− 0.590571i
$735$	0	0
$736$	0	0
$737$	32.0000	1.17874
$738$	0	0
$739$	− 32.0000i	− 1.17714i	−0.808447	−	0.588570i	$-0.799691\pi$
$739$	− 32.0000i	− 1.17714i	0.808447	−	0.588570i	$-0.200309\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	−24.0000	−0.881068
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	26.0000i	0.951928i
$747$	0	0
$748$	8.00000i	0.292509i
$749$	− 48.0000i	− 1.75388i
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	−24.0000	−0.871719
$759$	0	0
$760$	0	0
$761$	− 10.0000i	− 0.362500i	−0.983437	−	0.181250i	$-0.941986\pi$
$761$	− 10.0000i	− 0.362500i	0.983437	−	0.181250i	$-0.0580143\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	−8.00000	−0.289430
$765$	0	0
$766$	16.0000	0.578103
$767$	0	0
$768$	0	0
$769$	− 30.0000i	− 1.08183i	−0.841078	−	0.540914i	$-0.818079\pi$
$769$	− 30.0000i	− 1.08183i	0.841078	−	0.540914i	$-0.181921\pi$
$770$	32.0000	1.15320
$771$	0	0
$772$	− 18.0000i	− 0.647834i
$773$	− 10.0000i	− 0.359675i	−0.983696	−	0.179838i	$-0.942443\pi$
$773$	− 10.0000i	− 0.359675i	0.983696	−	0.179838i	$-0.0575572\pi$
$774$	0	0
$775$	4.00000i	0.143684i
$776$	30.0000	1.07694
$777$	0	0
$778$	− 22.0000i	− 0.788738i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	9.00000	0.321429
$785$	36.0000i	1.28490i
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	− 18.0000i	− 0.641223i
$789$	0	0
$790$	−16.0000	−0.569254
$791$	24.0000i	0.853342i
$792$	0	0
$793$	0	0
$794$	−38.0000	−1.34857
$795$	0	0
$796$	−8.00000	−0.283552
$797$	46.0000	1.62940	0.814702	−	0.579880i	$-0.196901\pi$
$797$	46.0000	1.62940	0.814702	+	0.579880i	$0.196901\pi$
$798$	0	0
$799$	0	0
$800$	− 5.00000i	− 0.176777i
$801$	0	0
$802$	22.0000	0.776847
$803$	8.00000	0.282314
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	54.0000i	1.89971i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	8.00000i	0.280918i	0.990086	+	0.140459i	$0.0448578\pi$
$811$	8.00000i	0.280918i	−0.990086	+	0.140459i	$0.955142\pi$
$812$	40.0000i	1.40372i
$813$	0	0
$814$	8.00000i	0.280400i
$815$	−16.0000	−0.560456
$816$	0	0
$817$	0	0
$818$	34.0000	1.18878
$819$	0	0
$820$	−12.0000	−0.419058
$821$	22.0000i	0.767805i	0.923374	+	0.383903i	$0.125420\pi$
$821$	22.0000i	0.767805i	−0.923374	+	0.383903i	$0.874580\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	48.0000i	1.67013i
$827$	4.00000i	0.139094i	0.997579	+	0.0695468i	$0.0221553\pi$
$827$	4.00000i	0.139094i	−0.997579	+	0.0695468i	$0.977845\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	8.00000i	0.277684i
$831$	0	0
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	−16.0000	−0.553703
$836$	0	0
$837$	0	0
$838$	4.00000i	0.138178i
$839$	48.0000i	1.65714i	0.559883	+	0.828572i	$0.310846\pi$
$839$	48.0000i	1.65714i	−0.559883	+	0.828572i	$0.689154\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−10.0000	−0.344623
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	− 20.0000i	− 0.687208i
$848$	6.00000	0.206041
$849$	0	0
$850$	− 2.00000i	− 0.0685994i
$851$	0	0
$852$	0	0
$853$	− 30.0000i	− 1.02718i	−0.858036	−	0.513590i	$-0.828315\pi$
$853$	− 30.0000i	− 1.02718i	0.858036	−	0.513590i	$-0.171685\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	36.0000i	1.23045i
$857$	−46.0000	−1.57133	−0.785665	−	0.618652i	$-0.787679\pi$
$857$	−46.0000	−1.57133	−0.785665	+	0.618652i	$0.787679\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	− 24.0000i	− 0.818393i
$861$	0	0
$862$	0	0
$863$	− 16.0000i	− 0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$	− 16.0000i	− 0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	0	0
$865$	− 12.0000i	− 0.408012i
$866$	34.0000i	1.15537i
$867$	0	0
$868$	−16.0000	−0.543075
$869$	32.0000i	1.08553i
$870$	0	0
$871$	0	0
$872$	−6.00000	−0.203186
$873$	0	0
$874$	0	0
$875$	48.0000	1.62270
$876$	0	0
$877$	− 10.0000i	− 0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$	− 10.0000i	− 0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$	32.0000i	1.07995i
$879$	0	0
$880$	−8.00000	−0.269680
$881$	58.0000	1.95407	0.977035	−	0.213080i	$-0.0683494\pi$
$881$	58.0000	1.95407	0.977035	+	0.213080i	$0.0683494\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	0	0
$886$	− 4.00000i	− 0.134383i
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	64.0000i	2.14649i
$890$	4.00000i	0.134080i
$891$	0	0
$892$	4.00000i	0.133930i
$893$	0	0
$894$	0	0
$895$	− 8.00000i	− 0.267411i
$896$	12.0000	0.400892
$897$	0	0
$898$	22.0000	0.734150
$899$	40.0000i	1.33407i
$900$	0	0
$901$	−12.0000	−0.399778
$902$	− 24.0000i	− 0.799113i
$903$	0	0
$904$	− 18.0000i	− 0.598671i
$905$	− 20.0000i	− 0.664822i
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	20.0000i	0.663723i
$909$	0	0
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	2.00000	0.0661541
$915$	0	0
$916$	10.0000i	0.330409i
$917$	− 16.0000i	− 0.528367i
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	0	0
$922$	38.0000	1.25146
$923$	0	0
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	−4.00000	−0.131448
$927$	0	0
$928$	− 50.0000i	− 1.64133i
$929$	− 30.0000i	− 0.984268i	−0.870519	−	0.492134i	$-0.836217\pi$
$929$	− 30.0000i	− 0.984268i	0.870519	−	0.492134i	$-0.163783\pi$
$930$	0	0
$931$	0	0
$932$	−14.0000	−0.458585
$933$	0	0
$934$	− 12.0000i	− 0.392652i
$935$	16.0000	0.523256
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	− 32.0000i	− 1.04484i
$939$	0	0
$940$	0	0
$941$	14.0000i	0.456387i	0.973616	+	0.228193i	$0.0732819\pi$
$941$	14.0000i	0.456387i	−0.973616	+	0.228193i	$0.926718\pi$
$942$	0	0
$943$	0	0
$944$	− 12.0000i	− 0.390567i
$945$	0	0
$946$	48.0000	1.56061
$947$	− 60.0000i	− 1.94974i	−0.222779	−	0.974869i	$-0.571513\pi$
$947$	− 60.0000i	− 1.94974i	0.222779	−	0.974869i	$-0.428487\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	24.0000	0.777844
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	16.0000i	0.517748i
$956$	24.0000i	0.776215i
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−24.0000	−0.775000
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	− 10.0000i	− 0.322078i
$965$	−36.0000	−1.15888
$966$	0	0
$967$	− 52.0000i	− 1.67221i	−0.548572	−	0.836104i	$-0.684828\pi$
$967$	− 52.0000i	− 1.67221i	0.548572	−	0.836104i	$-0.315172\pi$
$968$	15.0000i	0.482118i
$969$	0	0
$970$	− 20.0000i	− 0.642161i
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	0	0
$973$	48.0000i	1.53881i
$974$	12.0000	0.384505
$975$	0	0
$976$	2.00000	0.0640184
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	18.0000i	0.574989i
$981$	0	0
$982$	12.0000i	0.382935i
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	− 20.0000i	− 0.636930i
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	20.0000	0.635001
$993$	0	0
$994$	0	0
$995$	16.0000i	0.507234i
$996$	0	0
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	−24.0000	−0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.b.1351.1		2
3.2	odd	2		507.2.b.a.337.2		2
13.5	odd	4		117.2.a.a.1.1		1
13.8	odd	4		1521.2.a.e.1.1		1
13.12	even	2	inner	1521.2.b.b.1351.2		2
39.2	even	12		507.2.e.a.22.1		2
39.5	even	4		39.2.a.a.1.1	✓	1
39.8	even	4		507.2.a.a.1.1		1
39.11	even	12		507.2.e.b.22.1		2
39.17	odd	6		507.2.j.e.361.1		4
39.20	even	12		507.2.e.b.484.1		2
39.23	odd	6		507.2.j.e.316.2		4
39.29	odd	6		507.2.j.e.316.1		4
39.32	even	12		507.2.e.a.484.1		2
39.35	odd	6		507.2.j.e.361.2		4
39.38	odd	2		507.2.b.a.337.1		2
52.31	even	4		1872.2.a.h.1.1		1
65.18	even	4		2925.2.c.e.2224.2		2
65.44	odd	4		2925.2.a.p.1.1		1
65.57	even	4		2925.2.c.e.2224.1		2
91.83	even	4		5733.2.a.e.1.1		1
104.5	odd	4		7488.2.a.bl.1.1		1
104.83	even	4		7488.2.a.by.1.1		1
117.5	even	12		1053.2.e.b.703.1		2
117.31	odd	12		1053.2.e.d.703.1		2
117.70	odd	12		1053.2.e.d.352.1		2
117.83	even	12		1053.2.e.b.352.1		2
156.47	odd	4		8112.2.a.s.1.1		1
156.83	odd	4		624.2.a.i.1.1		1
195.44	even	4		975.2.a.f.1.1		1
195.83	odd	4		975.2.c.f.274.1		2
195.122	odd	4		975.2.c.f.274.2		2
273.83	odd	4		1911.2.a.f.1.1		1
312.5	even	4		2496.2.a.q.1.1		1
312.83	odd	4		2496.2.a.e.1.1		1
429.395	odd	4		4719.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.a.1.1	✓	1	39.5	even	4
117.2.a.a.1.1		1	13.5	odd	4
507.2.a.a.1.1		1	39.8	even	4
507.2.b.a.337.1		2	39.38	odd	2
507.2.b.a.337.2		2	3.2	odd	2
507.2.e.a.22.1		2	39.2	even	12
507.2.e.a.484.1		2	39.32	even	12
507.2.e.b.22.1		2	39.11	even	12
507.2.e.b.484.1		2	39.20	even	12
507.2.j.e.316.1		4	39.29	odd	6
507.2.j.e.316.2		4	39.23	odd	6
507.2.j.e.361.1		4	39.17	odd	6
507.2.j.e.361.2		4	39.35	odd	6
624.2.a.i.1.1		1	156.83	odd	4
975.2.a.f.1.1		1	195.44	even	4
975.2.c.f.274.1		2	195.83	odd	4
975.2.c.f.274.2		2	195.122	odd	4
1053.2.e.b.352.1		2	117.83	even	12
1053.2.e.b.703.1		2	117.5	even	12
1053.2.e.d.352.1		2	117.70	odd	12
1053.2.e.d.703.1		2	117.31	odd	12
1521.2.a.e.1.1		1	13.8	odd	4
1521.2.b.b.1351.1		2	1.1	even	1	trivial
1521.2.b.b.1351.2		2	13.12	even	2	inner
1872.2.a.h.1.1		1	52.31	even	4
1911.2.a.f.1.1		1	273.83	odd	4
2496.2.a.e.1.1		1	312.83	odd	4
2496.2.a.q.1.1		1	312.5	even	4
2925.2.a.p.1.1		1	65.44	odd	4
2925.2.c.e.2224.1		2	65.57	even	4
2925.2.c.e.2224.2		2	65.18	even	4
4719.2.a.c.1.1		1	429.395	odd	4
5733.2.a.e.1.1		1	91.83	even	4
7488.2.a.bl.1.1		1	104.5	odd	4
7488.2.a.by.1.1		1	104.83	even	4
8112.2.a.s.1.1		1	156.47	odd	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{4} -2.00000i q^{5} +4.00000i q^{7} -3.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{4} -2.00000i q^{5} +4.00000i q^{7} -3.00000i q^{8} -2.00000 q^{10} +4.00000i q^{11} +4.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} -2.00000i q^{20} +4.00000 q^{22} +1.00000 q^{25} +4.00000i q^{28} +10.0000 q^{29} +4.00000i q^{31} -5.00000i q^{32} -2.00000i q^{34} +8.00000 q^{35} +2.00000i q^{37} -6.00000 q^{40} -6.00000i q^{41} +12.0000 q^{43} +4.00000i q^{44} -9.00000 q^{49} -1.00000i q^{50} -6.00000 q^{53} +8.00000 q^{55} +12.0000 q^{56} -10.0000i q^{58} +12.0000i q^{59} -2.00000 q^{61} +4.00000 q^{62} -7.00000 q^{64} -8.00000i q^{67} +2.00000 q^{68} -8.00000i q^{70} -2.00000i q^{73} +2.00000 q^{74} -16.0000 q^{77} +8.00000 q^{79} +2.00000i q^{80} -6.00000 q^{82} -4.00000i q^{83} -4.00000i q^{85} -12.0000i q^{86} +12.0000 q^{88} -2.00000i q^{89} +10.0000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} - 4 q^{10} + 8 q^{14} - 2 q^{16} + 4 q^{17} + 8 q^{22} + 2 q^{25} + 20 q^{29} + 16 q^{35} - 12 q^{40} + 24 q^{43} - 18 q^{49} - 12 q^{53} + 16 q^{55} + 24 q^{56} - 4 q^{61} + 8 q^{62} - 14 q^{64} + 4 q^{68} + 4 q^{74} - 32 q^{77} + 16 q^{79} - 12 q^{82} + 24 q^{88}+O(q^{100}) \) 2 * q + 2 * q^4 - 4 * q^10 + 8 * q^14 - 2 * q^16 + 4 * q^17 + 8 * q^22 + 2 * q^25 + 20 * q^29 + 16 * q^35 - 12 * q^40 + 24 * q^43 - 18 * q^49 - 12 * q^53 + 16 * q^55 + 24 * q^56 - 4 * q^61 + 8 * q^62 - 14 * q^64 + 4 * q^68 + 4 * q^74 - 32 * q^77 + 16 * q^79 - 12 * q^82 + 24 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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