LMFDB - Embedded newform 1690.2.d.d.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1690,2,Mod(1351,1690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1690, 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("1690.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.d (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:	$13.4947179416$
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 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1690.1351
Dual form			1690.2.d.d.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{8} -3.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{8} -3.00000 q^{9} -1.00000 q^{10} +1.00000 q^{16} -2.00000 q^{17} -3.00000i q^{18} -8.00000i q^{19} -1.00000i q^{20} +4.00000 q^{23} -1.00000 q^{25} -2.00000 q^{29} -4.00000i q^{31} +1.00000i q^{32} -2.00000i q^{34} +3.00000 q^{36} -6.00000i q^{37} +8.00000 q^{38} +1.00000 q^{40} +10.0000i q^{41} -3.00000i q^{45} +4.00000i q^{46} -8.00000i q^{47} +7.00000 q^{49} -1.00000i q^{50} +6.00000 q^{53} -2.00000i q^{58} -8.00000i q^{59} -2.00000 q^{61} +4.00000 q^{62} -1.00000 q^{64} +4.00000i q^{67} +2.00000 q^{68} -12.0000i q^{71} +3.00000i q^{72} -10.0000i q^{73} +6.00000 q^{74} +8.00000i q^{76} -8.00000 q^{79} +1.00000i q^{80} +9.00000 q^{81} -10.0000 q^{82} +12.0000i q^{83} -2.00000i q^{85} -10.0000i q^{89} +3.00000 q^{90} -4.00000 q^{92} +8.00000 q^{94} +8.00000 q^{95} -14.0000i q^{97} +7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 6 * q^9	$ 2 q - 2 q^{4} - 6 q^{9} - 2 q^{10} + 2 q^{16} - 4 q^{17} + 8 q^{23} - 2 q^{25} - 4 q^{29} + 6 q^{36} + 16 q^{38} + 2 q^{40} + 14 q^{49} + 12 q^{53} - 4 q^{61} + 8 q^{62} - 2 q^{64} + 4 q^{68} + 12 q^{74} - 16 q^{79} + 18 q^{81} - 20 q^{82} + 6 q^{90} - 8 q^{92} + 16 q^{94} + 16 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 6 * q^9 - 2 * q^10 + 2 * q^16 - 4 * q^17 + 8 * q^23 - 2 * q^25 - 4 * q^29 + 6 * q^36 + 16 * q^38 + 2 * q^40 + 14 * q^49 + 12 * q^53 - 4 * q^61 + 8 * q^62 - 2 * q^64 + 4 * q^68 + 12 * q^74 - 16 * q^79 + 18 * q^81 - 20 * q^82 + 6 * q^90 - 8 * q^92 + 16 * q^94 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$-1$	$1$

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
$2$	1.00000i	0.707107i
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−3.00000	−1.00000
$10$	−1.00000	−0.316228
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	− 3.00000i	− 0.707107i
$19$	− 8.00000i	− 1.83533i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	− 8.00000i	− 1.83533i	0.397360	−	0.917663i	$-0.369927\pi$
$20$	− 1.00000i	− 0.223607i
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$	− 4.00000i	− 0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 2.00000i	− 0.342997i
$35$	0	0
$36$	3.00000	0.500000
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	1.00000	0.158114
$41$	10.0000i	1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$	10.0000i	1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	− 3.00000i	− 0.447214i
$46$	4.00000i	0.589768i
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	− 2.00000i	− 0.262613i
$59$	− 8.00000i	− 1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$	− 8.00000i	− 1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$	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
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	0	0
$71$	− 12.0000i	− 1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$	− 12.0000i	− 1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$	3.00000i	0.353553i
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	8.00000i	0.917663i
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000i	0.111803i
$81$	9.00000	1.00000
$82$	−10.0000	−1.10432
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	− 2.00000i	− 0.216930i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 10.0000i	− 1.06000i	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	− 10.0000i	− 1.06000i	0.847998	−	0.529999i	$-0.177808\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	−4.00000	−0.417029
$93$	0	0
$94$	8.00000	0.825137
$95$	8.00000	0.820783
$96$	0	0
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	7.00000i	0.707107i
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	6.00000i	0.582772i
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	4.00000i	0.373002i
$116$	2.00000	0.185695
$117$	0	0
$118$	8.00000	0.736460
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	4.00000i	0.359211i
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	2.00000i	0.171499i
$137$	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
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	0	0
$144$	−3.00000	−0.250000
$145$	− 2.00000i	− 0.166091i
$146$	10.0000	0.827606
$147$	0	0
$148$	6.00000i	0.493197i
$149$	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
$151$	20.0000i	1.62758i	0.581161	+	0.813788i	$0.302599\pi$
$151$	20.0000i	1.62758i	−0.581161	+	0.813788i	$0.697401\pi$
$152$	−8.00000	−0.648886
$153$	6.00000	0.485071
$154$	0	0
$155$	4.00000	0.321288
$156$	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
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	9.00000i	0.707107i
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	− 10.0000i	− 0.780869i
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	0	0
$169$	0	0
$170$	2.00000	0.153393
$171$	24.0000i	1.83533i
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	10.0000	0.749532
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	3.00000i	0.223607i
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	0	0
$184$	− 4.00000i	− 0.294884i
$185$	6.00000	0.441129
$186$	0	0
$187$	0	0
$188$	8.00000i	0.583460i
$189$	0	0
$190$	8.00000i	0.580381i
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	− 18.0000i	− 1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	− 18.0000i	− 1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 26.0000i	− 1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$	− 26.0000i	− 1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	1.00000i	0.0707107i
$201$	0	0
$202$	− 6.00000i	− 0.422159i
$203$	0	0
$204$	0	0
$205$	−10.0000	−0.698430
$206$	4.00000i	0.278693i
$207$	−12.0000	−0.834058
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	2.00000	0.135457
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	− 14.0000i	− 0.931266i
$227$	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	$0.892813\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	2.00000i	0.131306i
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	8.00000i	0.520756i
$237$	0	0
$238$	0	0
$239$	20.0000i	1.29369i	0.762620	+	0.646846i	$0.223912\pi$
$239$	20.0000i	1.29369i	−0.762620	+	0.646846i	$0.776088\pi$
$240$	0	0
$241$	− 2.00000i	− 0.128831i	−0.997923	−	0.0644157i	$-0.979482\pi$
$241$	− 2.00000i	− 0.128831i	0.997923	−	0.0644157i	$-0.0205183\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	2.00000	0.128037
$245$	7.00000i	0.447214i
$246$	0	0
$247$	0	0
$248$	−4.00000	−0.254000
$249$	0	0
$250$	1.00000	0.0632456
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	0	0
$253$	0	0
$254$	20.0000i	1.25491i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	4.00000i	0.247121i
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	6.00000i	0.368577i
$266$	0	0
$267$	0	0
$268$	− 4.00000i	− 0.244339i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	28.0000i	1.70088i	0.526073	+	0.850439i	$0.323664\pi$
$271$	28.0000i	1.70088i	−0.526073	+	0.850439i	$0.676336\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	4.00000i	0.239904i
$279$	12.0000i	0.718421i
$280$	0	0
$281$	22.0000i	1.31241i	0.754583	+	0.656205i	$0.227839\pi$
$281$	22.0000i	1.31241i	−0.754583	+	0.656205i	$0.772161\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	12.0000i	0.712069i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 3.00000i	− 0.176777i
$289$	−13.0000	−0.764706
$290$	2.00000	0.117444
$291$	0	0
$292$	10.0000i	0.585206i
$293$	− 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
$295$	8.00000	0.465778
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−6.00000	−0.347571
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−20.0000	−1.15087
$303$	0	0
$304$	− 8.00000i	− 0.458831i
$305$	− 2.00000i	− 0.114520i
$306$	6.00000i	0.342997i
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	0	0
$310$	4.00000i	0.227185i
$311$	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
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	0	0
$319$	0	0
$320$	− 1.00000i	− 0.0559017i
$321$	0	0
$322$	0	0
$323$	16.0000i	0.890264i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	20.0000	1.10770
$327$	0	0
$328$	10.0000	0.552158
$329$	0	0
$330$	0	0
$331$	− 8.00000i	− 0.439720i	−0.975531	−	0.219860i	$-0.929440\pi$
$331$	− 8.00000i	− 0.439720i	0.975531	−	0.219860i	$-0.0705600\pi$
$332$	− 12.0000i	− 0.658586i
$333$	18.0000i	0.986394i
$334$	16.0000	0.875481
$335$	−4.00000	−0.218543
$336$	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
$340$	2.00000i	0.108465i
$341$	0	0
$342$	−24.0000	−1.29777
$343$	0	0
$344$	0	0
$345$	0	0
$346$	18.0000i	0.967686i
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	0	0
$349$	− 14.0000i	− 0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$	− 14.0000i	− 0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	10.0000i	0.529999i
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	− 4.00000i	− 0.211112i	−0.994413	−	0.105556i	$-0.966338\pi$
$359$	− 4.00000i	− 0.211112i	0.994413	−	0.105556i	$-0.0336622\pi$
$360$	−3.00000	−0.158114
$361$	−45.0000	−2.36842
$362$	− 22.0000i	− 1.15629i
$363$	0	0
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	4.00000	0.208514
$369$	− 30.0000i	− 1.56174i
$370$	6.00000i	0.311925i
$371$	0	0
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	0	0
$376$	−8.00000	−0.412568
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	0	0
$383$	8.00000i	0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$	8.00000i	0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$	0	0
$385$	0	0
$386$	18.0000	0.916176
$387$	0	0
$388$	14.0000i	0.710742i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	− 7.00000i	− 0.353553i
$393$	0	0
$394$	26.0000	1.30986
$395$	− 8.00000i	− 0.402524i
$396$	0	0
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	− 24.0000i	− 1.20301i
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	30.0000i	1.49813i	0.662497	+	0.749064i	$0.269497\pi$
$401$	30.0000i	1.49813i	−0.662497	+	0.749064i	$0.730503\pi$
$402$	0	0
$403$	0	0
$404$	6.00000	0.298511
$405$	9.00000i	0.447214i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	26.0000i	1.28562i	0.766027	+	0.642809i	$0.222231\pi$
$409$	26.0000i	1.28562i	−0.766027	+	0.642809i	$0.777769\pi$
$410$	− 10.0000i	− 0.493865i
$411$	0	0
$412$	−4.00000	−0.197066
$413$	0	0
$414$	− 12.0000i	− 0.589768i
$415$	−12.0000	−0.589057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	− 26.0000i	− 1.26716i	−0.773676	−	0.633581i	$-0.781584\pi$
$421$	− 26.0000i	− 1.26716i	0.773676	−	0.633581i	$-0.218416\pi$
$422$	4.00000i	0.194717i
$423$	24.0000i	1.16692i
$424$	− 6.00000i	− 0.291386i
$425$	2.00000	0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000i	0.578020i	0.957326	+	0.289010i	$0.0933260\pi$
$431$	12.0000i	0.578020i	−0.957326	+	0.289010i	$0.906674\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	0	0
$436$	2.00000i	0.0957826i
$437$	− 32.0000i	− 1.53077i
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	−21.0000	−1.00000
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	8.00000	0.378811
$447$	0	0
$448$	0	0
$449$	− 18.0000i	− 0.849473i	−0.905317	−	0.424736i	$-0.860367\pi$
$449$	− 18.0000i	− 0.849473i	0.905317	−	0.424736i	$-0.139633\pi$
$450$	3.00000i	0.141421i
$451$	0	0
$452$	14.0000	0.658505
$453$	0	0
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	− 4.00000i	− 0.186501i
$461$	14.0000i	0.652045i	0.945362	+	0.326023i	$0.105709\pi$
$461$	14.0000i	0.652045i	−0.945362	+	0.326023i	$0.894291\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	6.00000i	0.277945i
$467$	16.0000	0.740392	0.370196	−	0.928954i	$-0.379291\pi$
$467$	16.0000	0.740392	0.370196	+	0.928954i	$0.379291\pi$
$468$	0	0
$469$	0	0
$470$	8.00000i	0.369012i
$471$	0	0
$472$	−8.00000	−0.368230
$473$	0	0
$474$	0	0
$475$	8.00000i	0.367065i
$476$	0	0
$477$	−18.0000	−0.824163
$478$	−20.0000	−0.914779
$479$	4.00000i	0.182765i	0.995816	+	0.0913823i	$0.0291285\pi$
$479$	4.00000i	0.182765i	−0.995816	+	0.0913823i	$0.970871\pi$
$480$	0	0
$481$	0	0
$482$	2.00000	0.0910975
$483$	0	0
$484$	−11.0000	−0.500000
$485$	14.0000	0.635707
$486$	0	0
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	2.00000i	0.0905357i
$489$	0	0
$490$	−7.00000	−0.316228
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	− 4.00000i	− 0.179605i
$497$	0	0
$498$	0	0
$499$	8.00000i	0.358129i	0.983837	+	0.179065i	$0.0573071\pi$
$499$	8.00000i	0.358129i	−0.983837	+	0.179065i	$0.942693\pi$
$500$	1.00000i	0.0447214i
$501$	0	0
$502$	28.0000i	1.24970i
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	0	0
$505$	− 6.00000i	− 0.266996i
$506$	0	0
$507$	0	0
$508$	−20.0000	−0.887357
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 2.00000i	− 0.0882162i
$515$	4.00000i	0.176261i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	6.00000i	0.262613i
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	− 28.0000i	− 1.22086i
$527$	8.00000i	0.348485i
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−6.00000	−0.260623
$531$	24.0000i	1.04151i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	− 18.0000i	− 0.776035i
$539$	0	0
$540$	0	0
$541$	2.00000i	0.0859867i	0.999075	+	0.0429934i	$0.0136894\pi$
$541$	2.00000i	0.0859867i	−0.999075	+	0.0429934i	$0.986311\pi$
$542$	−28.0000	−1.20270
$543$	0	0
$544$	− 2.00000i	− 0.0857493i
$545$	2.00000	0.0856706
$546$	0	0
$547$	−40.0000	−1.71028	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000	−1.71028	−0.855138	+	0.518400i	$0.826528\pi$
$548$	− 6.00000i	− 0.256307i
$549$	6.00000	0.256074
$550$	0	0
$551$	16.0000i	0.681623i
$552$	0	0
$553$	0	0
$554$	− 22.0000i	− 0.934690i
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 14.0000i	− 0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$	− 14.0000i	− 0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$	−12.0000	−0.508001
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−22.0000	−0.928014
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	− 14.0000i	− 0.588984i
$566$	− 16.0000i	− 0.672530i
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.00000	−0.166812
$576$	3.00000	0.125000
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 13.0000i	− 0.540729i
$579$	0	0
$580$	2.00000i	0.0830455i
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−10.0000	−0.413803
$585$	0	0
$586$	6.00000	0.247858
$587$	− 20.0000i	− 0.825488i	−0.910847	−	0.412744i	$-0.864570\pi$
$587$	− 20.0000i	− 0.825488i	0.910847	−	0.412744i	$-0.135430\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	8.00000i	0.329355i
$591$	0	0
$592$	− 6.00000i	− 0.246598i
$593$	− 2.00000i	− 0.0821302i	−0.999156	−	0.0410651i	$-0.986925\pi$
$593$	− 2.00000i	− 0.0821302i	0.999156	−	0.0410651i	$-0.0130751\pi$
$594$	0	0
$595$	0	0
$596$	− 6.00000i	− 0.245770i
$597$	0	0
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	− 20.0000i	− 0.813788i
$605$	11.0000i	0.447214i
$606$	0	0
$607$	36.0000	1.46119	0.730597	−	0.682808i	$-0.239242\pi$
$607$	36.0000	1.46119	0.730597	+	0.682808i	$0.239242\pi$
$608$	8.00000	0.324443
$609$	0	0
$610$	2.00000	0.0809776
$611$	0	0
$612$	−6.00000	−0.242536
$613$	38.0000i	1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$	38.0000i	1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	0	0
$617$	26.0000i	1.04672i	0.852111	+	0.523360i	$0.175322\pi$
$617$	26.0000i	1.04672i	−0.852111	+	0.523360i	$0.824678\pi$
$618$	0	0
$619$	24.0000i	0.964641i	0.875995	+	0.482321i	$0.160206\pi$
$619$	24.0000i	0.964641i	−0.875995	+	0.482321i	$0.839794\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 6.00000i	− 0.239808i
$627$	0	0
$628$	18.0000	0.718278
$629$	12.0000i	0.478471i
$630$	0	0
$631$	− 12.0000i	− 0.477712i	−0.971055	−	0.238856i	$-0.923228\pi$
$631$	− 12.0000i	− 0.477712i	0.971055	−	0.238856i	$-0.0767725\pi$
$632$	8.00000i	0.318223i
$633$	0	0
$634$	2.00000	0.0794301
$635$	20.0000i	0.793676i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	36.0000i	1.42414i
$640$	1.00000	0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	0	0
$646$	−16.0000	−0.629512
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	− 9.00000i	− 0.353553i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	20.0000i	0.783260i
$653$	46.0000	1.80012	0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000	1.80012	0.900060	+	0.435767i	$0.143523\pi$
$654$	0	0
$655$	4.00000i	0.156293i
$656$	10.0000i	0.390434i
$657$	30.0000i	1.17041i
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	− 22.0000i	− 0.855701i	−0.903850	−	0.427850i	$-0.859271\pi$
$661$	− 22.0000i	− 0.855701i	0.903850	−	0.427850i	$-0.140729\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	−18.0000	−0.697486
$667$	−8.00000	−0.309761
$668$	16.0000i	0.619059i
$669$	0	0
$670$	− 4.00000i	− 0.154533i
$671$	0	0
$672$	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	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	0	0
$680$	−2.00000	−0.0766965
$681$	0	0
$682$	0	0
$683$	− 28.0000i	− 1.07139i	−0.844411	−	0.535695i	$-0.820050\pi$
$683$	− 28.0000i	− 1.07139i	0.844411	−	0.535695i	$-0.179950\pi$
$684$	− 24.0000i	− 0.917663i
$685$	−6.00000	−0.229248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	32.0000i	1.21734i	0.793424	+	0.608669i	$0.208296\pi$
$691$	32.0000i	1.21734i	−0.793424	+	0.608669i	$0.791704\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	32.0000i	1.21470i
$695$	4.00000i	0.151729i
$696$	0	0
$697$	− 20.0000i	− 0.757554i
$698$	14.0000	0.529908
$699$	0	0
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	−48.0000	−1.81035
$704$	0	0
$705$	0	0
$706$	30.0000	1.12906
$707$	0	0
$708$	0	0
$709$	10.0000i	0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$	10.0000i	0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$	12.0000i	0.450352i
$711$	24.0000	0.900070
$712$	−10.0000	−0.374766
$713$	− 16.0000i	− 0.599205i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	4.00000	0.149279
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	− 3.00000i	− 0.111803i
$721$	0	0
$722$	− 45.0000i	− 1.67473i
$723$	0	0
$724$	22.0000	0.817624
$725$	2.00000	0.0742781
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	10.0000i	0.370117i
$731$	0	0
$732$	0	0
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	− 12.0000i	− 0.442928i
$735$	0	0
$736$	4.00000i	0.147442i
$737$	0	0
$738$	30.0000	1.10432
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	0	0
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	6.00000i	0.219676i
$747$	− 36.0000i	− 1.31717i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	− 8.00000i	− 0.291730i
$753$	0	0
$754$	0	0
$755$	−20.0000	−0.727875
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	0	0
$760$	− 8.00000i	− 0.290191i
$761$	6.00000i	0.217500i	0.994069	+	0.108750i	$0.0346848\pi$
$761$	6.00000i	0.217500i	−0.994069	+	0.108750i	$0.965315\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	6.00000i	0.216930i
$766$	−8.00000	−0.289052
$767$	0	0
$768$	0	0
$769$	− 14.0000i	− 0.504853i	−0.967616	−	0.252426i	$-0.918771\pi$
$769$	− 14.0000i	− 0.504853i	0.967616	−	0.252426i	$-0.0812286\pi$
$770$	0	0
$771$	0	0
$772$	18.0000i	0.647834i
$773$	− 26.0000i	− 0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$	− 26.0000i	− 0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$	0	0
$775$	4.00000i	0.143684i
$776$	−14.0000	−0.502571
$777$	0	0
$778$	− 6.00000i	− 0.215110i
$779$	80.0000	2.86630
$780$	0	0
$781$	0	0
$782$	− 8.00000i	− 0.286079i
$783$	0	0
$784$	7.00000	0.250000
$785$	− 18.0000i	− 0.642448i
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	26.0000i	0.926212i
$789$	0	0
$790$	8.00000	0.284627
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−18.0000	−0.638796
$795$	0	0
$796$	24.0000	0.850657
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	16.0000i	0.566039i
$800$	− 1.00000i	− 0.0353553i
$801$	30.0000i	1.06000i
$802$	−30.0000	−1.05934
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	6.00000i	0.211079i
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	−9.00000	−0.316228
$811$	24.0000i	0.842754i	0.906886	+	0.421377i	$0.138453\pi$
$811$	24.0000i	0.842754i	−0.906886	+	0.421377i	$0.861547\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	0	0
$818$	−26.0000	−0.909069
$819$	0	0
$820$	10.0000	0.349215
$821$	− 42.0000i	− 1.46581i	−0.680331	−	0.732905i	$-0.738164\pi$
$821$	− 42.0000i	− 1.46581i	0.680331	−	0.732905i	$-0.261836\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	− 4.00000i	− 0.139347i
$825$	0	0
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	12.0000	0.417029
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	− 12.0000i	− 0.416526i
$831$	0	0
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	0	0
$838$	− 36.0000i	− 1.24360i
$839$	− 4.00000i	− 0.138095i	−0.997613	−	0.0690477i	$-0.978004\pi$
$839$	− 4.00000i	− 0.138095i	0.997613	−	0.0690477i	$-0.0219961\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	26.0000	0.896019
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−24.0000	−0.825137
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	2.00000i	0.0685994i
$851$	− 24.0000i	− 0.822709i
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	−24.0000	−0.820783
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	0	0
$862$	−12.0000	−0.408722
$863$	40.0000i	1.36162i	0.732462	+	0.680808i	$0.238371\pi$
$863$	40.0000i	1.36162i	−0.732462	+	0.680808i	$0.761629\pi$
$864$	0	0
$865$	18.0000i	0.612018i
$866$	− 34.0000i	− 1.15537i
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	42.0000i	1.42148i
$874$	32.0000	1.08242
$875$	0	0
$876$	0	0
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	24.0000i	0.809961i
$879$	0	0
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	− 21.0000i	− 0.707107i
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	0	0
$886$	24.0000i	0.806296i
$887$	44.0000	1.47738	0.738688	−	0.674048i	$-0.235446\pi$
$887$	44.0000	1.47738	0.738688	+	0.674048i	$0.235446\pi$
$888$	0	0
$889$	0	0
$890$	10.0000i	0.335201i
$891$	0	0
$892$	8.00000i	0.267860i
$893$	−64.0000	−2.14168
$894$	0	0
$895$	− 4.00000i	− 0.133705i
$896$	0	0
$897$	0	0
$898$	18.0000	0.600668
$899$	8.00000i	0.266815i
$900$	−3.00000	−0.100000
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	14.0000i	0.465633i
$905$	− 22.0000i	− 0.731305i
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	− 20.0000i	− 0.663723i
$909$	18.0000	0.597022
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	0	0
$914$	38.0000	1.25693
$915$	0	0
$916$	− 10.0000i	− 0.330409i
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	4.00000	0.131876
$921$	0	0
$922$	−14.0000	−0.461065
$923$	0	0
$924$	0	0
$925$	6.00000i	0.197279i
$926$	−16.0000	−0.525793
$927$	−12.0000	−0.394132
$928$	− 2.00000i	− 0.0656532i
$929$	− 14.0000i	− 0.459325i	−0.973270	−	0.229663i	$-0.926238\pi$
$929$	− 14.0000i	− 0.459325i	0.973270	−	0.229663i	$-0.0737623\pi$
$930$	0	0
$931$	− 56.0000i	− 1.83533i
$932$	−6.00000	−0.196537
$933$	0	0
$934$	16.0000i	0.523536i
$935$	0	0
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	0	0
$940$	−8.00000	−0.260931
$941$	30.0000i	0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$	30.0000i	0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$	0	0
$943$	40.0000i	1.30258i
$944$	− 8.00000i	− 0.260378i
$945$	0	0
$946$	0	0
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	0	0
$950$	−8.00000	−0.259554
$951$	0	0
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	− 18.0000i	− 0.582772i
$955$	0	0
$956$	− 20.0000i	− 0.646846i
$957$	0	0
$958$	−4.00000	−0.129234
$959$	0	0
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	2.00000i	0.0644157i
$965$	18.0000	0.579441
$966$	0	0
$967$	24.0000i	0.771788i	0.922543	+	0.385894i	$0.126107\pi$
$967$	24.0000i	0.771788i	−0.922543	+	0.385894i	$0.873893\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	14.0000i	0.449513i
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	0	0
$974$	8.00000	0.256337
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	0	0
$980$	− 7.00000i	− 0.223607i
$981$	6.00000i	0.191565i
$982$	− 20.0000i	− 0.638226i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	26.0000	0.828429
$986$	4.00000i	0.127386i
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	0	0
$995$	− 24.0000i	− 0.760851i
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	−8.00000	−0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.d.d.1351.2		2
13.2	odd	12		1690.2.e.c.191.1		2
13.3	even	3		1690.2.l.f.1161.1		4
13.4	even	6		1690.2.l.f.361.1		4
13.5	odd	4		130.2.a.b.1.1	✓	1
13.6	odd	12		1690.2.e.c.991.1		2
13.7	odd	12		1690.2.e.i.991.1		2
13.8	odd	4		1690.2.a.b.1.1		1
13.9	even	3		1690.2.l.f.361.2		4
13.10	even	6		1690.2.l.f.1161.2		4
13.11	odd	12		1690.2.e.i.191.1		2
13.12	even	2	inner	1690.2.d.d.1351.1		2
39.5	even	4		1170.2.a.b.1.1		1
52.31	even	4		1040.2.a.e.1.1		1
65.18	even	4		650.2.b.e.599.1		2
65.34	odd	4		8450.2.a.r.1.1		1
65.44	odd	4		650.2.a.d.1.1		1
65.57	even	4		650.2.b.e.599.2		2
91.83	even	4		6370.2.a.r.1.1		1
104.5	odd	4		4160.2.a.i.1.1		1
104.83	even	4		4160.2.a.h.1.1		1
156.83	odd	4		9360.2.a.l.1.1		1
195.44	even	4		5850.2.a.bq.1.1		1
195.83	odd	4		5850.2.e.q.5149.2		2
195.122	odd	4		5850.2.e.q.5149.1		2
260.239	even	4		5200.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.a.b.1.1	✓	1	13.5	odd	4
650.2.a.d.1.1		1	65.44	odd	4
650.2.b.e.599.1		2	65.18	even	4
650.2.b.e.599.2		2	65.57	even	4
1040.2.a.e.1.1		1	52.31	even	4
1170.2.a.b.1.1		1	39.5	even	4
1690.2.a.b.1.1		1	13.8	odd	4
1690.2.d.d.1351.1		2	13.12	even	2	inner
1690.2.d.d.1351.2		2	1.1	even	1	trivial
1690.2.e.c.191.1		2	13.2	odd	12
1690.2.e.c.991.1		2	13.6	odd	12
1690.2.e.i.191.1		2	13.11	odd	12
1690.2.e.i.991.1		2	13.7	odd	12
1690.2.l.f.361.1		4	13.4	even	6
1690.2.l.f.361.2		4	13.9	even	3
1690.2.l.f.1161.1		4	13.3	even	3
1690.2.l.f.1161.2		4	13.10	even	6
4160.2.a.h.1.1		1	104.83	even	4
4160.2.a.i.1.1		1	104.5	odd	4
5200.2.a.r.1.1		1	260.239	even	4
5850.2.a.bq.1.1		1	195.44	even	4
5850.2.e.q.5149.1		2	195.122	odd	4
5850.2.e.q.5149.2		2	195.83	odd	4
6370.2.a.r.1.1		1	91.83	even	4
8450.2.a.r.1.1		1	65.34	odd	4
9360.2.a.l.1.1		1	156.83	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:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{8} -3.00000 q^{9} -1.00000 q^{10} +1.00000 q^{16} -2.00000 q^{17} -3.00000i q^{18} -8.00000i q^{19} -1.00000i q^{20} +4.00000 q^{23} -1.00000 q^{25} -2.00000 q^{29} -4.00000i q^{31} +1.00000i q^{32} -2.00000i q^{34} +3.00000 q^{36} -6.00000i q^{37} +8.00000 q^{38} +1.00000 q^{40} +10.0000i q^{41} -3.00000i q^{45} +4.00000i q^{46} -8.00000i q^{47} +7.00000 q^{49} -1.00000i q^{50} +6.00000 q^{53} -2.00000i q^{58} -8.00000i q^{59} -2.00000 q^{61} +4.00000 q^{62} -1.00000 q^{64} +4.00000i q^{67} +2.00000 q^{68} -12.0000i q^{71} +3.00000i q^{72} -10.0000i q^{73} +6.00000 q^{74} +8.00000i q^{76} -8.00000 q^{79} +1.00000i q^{80} +9.00000 q^{81} -10.0000 q^{82} +12.0000i q^{83} -2.00000i q^{85} -10.0000i q^{89} +3.00000 q^{90} -4.00000 q^{92} +8.00000 q^{94} +8.00000 q^{95} -14.0000i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 6 * q^9	\( 2 q - 2 q^{4} - 6 q^{9} - 2 q^{10} + 2 q^{16} - 4 q^{17} + 8 q^{23} - 2 q^{25} - 4 q^{29} + 6 q^{36} + 16 q^{38} + 2 q^{40} + 14 q^{49} + 12 q^{53} - 4 q^{61} + 8 q^{62} - 2 q^{64} + 4 q^{68} + 12 q^{74} - 16 q^{79} + 18 q^{81} - 20 q^{82} + 6 q^{90} - 8 q^{92} + 16 q^{94} + 16 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 6 * q^9 - 2 * q^10 + 2 * q^16 - 4 * q^17 + 8 * q^23 - 2 * q^25 - 4 * q^29 + 6 * q^36 + 16 * q^38 + 2 * q^40 + 14 * q^49 + 12 * q^53 - 4 * q^61 + 8 * q^62 - 2 * q^64 + 4 * q^68 + 12 * q^74 - 16 * q^79 + 18 * q^81 - 20 * q^82 + 6 * q^90 - 8 * q^92 + 16 * q^94 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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