LMFDB - Embedded newform 1521.2.b.h.1351.3

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:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.h.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{2} -0.438447 q^{4} +3.56155i q^{5} -0.561553i q^{7} +2.43845i q^{8} +O(q^{10})$	$q+1.56155i q^{2} -0.438447 q^{4} +3.56155i q^{5} -0.561553i q^{7} +2.43845i q^{8} -5.56155 q^{10} -2.00000i q^{11} +0.876894 q^{14} -4.68466 q^{16} -1.56155 q^{17} +7.12311i q^{19} -1.56155i q^{20} +3.12311 q^{22} +2.00000 q^{23} -7.68466 q^{25} +0.246211i q^{28} -6.68466 q^{29} +2.56155i q^{31} -2.43845i q^{32} -2.43845i q^{34} +2.00000 q^{35} -7.56155i q^{37} -11.1231 q^{38} -8.68466 q^{40} +1.56155i q^{41} -4.56155 q^{43} +0.876894i q^{44} +3.12311i q^{46} +8.24621i q^{47} +6.68466 q^{49} -12.0000i q^{50} +0.684658 q^{53} +7.12311 q^{55} +1.36932 q^{56} -10.4384i q^{58} -2.87689i q^{59} +3.87689 q^{61} -4.00000 q^{62} -5.56155 q^{64} +4.56155i q^{67} +0.684658 q^{68} +3.12311i q^{70} -14.0000i q^{71} +10.1231i q^{73} +11.8078 q^{74} -3.12311i q^{76} -1.12311 q^{77} +5.43845 q^{79} -16.6847i q^{80} -2.43845 q^{82} +0.876894i q^{83} -5.56155i q^{85} -7.12311i q^{86} +4.87689 q^{88} +4.87689i q^{89} -0.876894 q^{92} -12.8769 q^{94} -25.3693 q^{95} -8.56155i q^{97} +10.4384i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4}+O(q^{10}) $ 4 * q - 10 * q^4	$ 4 q - 10 q^{4} - 14 q^{10} + 20 q^{14} + 6 q^{16} + 2 q^{17} - 4 q^{22} + 8 q^{23} - 6 q^{25} - 2 q^{29} + 8 q^{35} - 28 q^{38} - 10 q^{40} - 10 q^{43} + 2 q^{49} - 22 q^{53} + 12 q^{55} - 44 q^{56} + 32 q^{61} - 16 q^{62} - 14 q^{64} - 22 q^{68} + 6 q^{74} + 12 q^{77} + 30 q^{79} - 18 q^{82} + 36 q^{88} - 20 q^{92} - 68 q^{94} - 52 q^{95}+O(q^{100}) $ 4 * q - 10 * q^4 - 14 * q^10 + 20 * q^14 + 6 * q^16 + 2 * q^17 - 4 * q^22 + 8 * q^23 - 6 * q^25 - 2 * q^29 + 8 * q^35 - 28 * q^38 - 10 * q^40 - 10 * q^43 + 2 * q^49 - 22 * q^53 + 12 * q^55 - 44 * q^56 + 32 * q^61 - 16 * q^62 - 14 * q^64 - 22 * q^68 + 6 * q^74 + 12 * q^77 + 30 * q^79 - 18 * q^82 + 36 * q^88 - 20 * q^92 - 68 * q^94 - 52 * q^95

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 $

$2$	1.56155i	1.10418i	0.833783	+	0.552092i	$0.186170\pi$
$2$	1.56155i	1.10418i	−0.833783	+	0.552092i	$0.813830\pi$
$3$	0	0
$3$	0	0
$4$	−0.438447	−0.219224
$5$	3.56155i	1.59277i	0.604787	+	0.796387i	$0.293258\pi$
$5$	3.56155i	1.59277i	−0.604787	+	0.796387i	$0.706742\pi$
$6$	0	0
$7$	− 0.561553i	− 0.212247i	−0.994353	−	0.106124i	$-0.966156\pi$
$7$	− 0.561553i	− 0.212247i	0.994353	−	0.106124i	$-0.0338439\pi$
$8$	2.43845i	0.862121i
$9$	0	0
$10$	−5.56155	−1.75872
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0.876894	0.234360
$15$	0	0
$16$	−4.68466	−1.17116
$17$	−1.56155	−0.378732	−0.189366	−	0.981907i	$-0.560643\pi$
$17$	−1.56155	−0.378732	−0.189366	+	0.981907i	$0.560643\pi$
$18$	0	0
$19$	7.12311i	1.63415i	0.576530	+	0.817076i	$0.304407\pi$
$19$	7.12311i	1.63415i	−0.576530	+	0.817076i	$0.695593\pi$
$20$	− 1.56155i	− 0.349174i
$21$	0	0
$22$	3.12311	0.665848
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	−7.68466	−1.53693
$26$	0	0
$27$	0	0
$28$	0.246211i	0.0465296i
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	2.56155i	0.460068i	0.973183	+	0.230034i	$0.0738838\pi$
$31$	2.56155i	0.460068i	−0.973183	+	0.230034i	$0.926116\pi$
$32$	− 2.43845i	− 0.431061i
$33$	0	0
$34$	− 2.43845i	− 0.418190i
$35$	2.00000	0.338062
$36$	0	0
$37$	− 7.56155i	− 1.24311i	−0.783370	−	0.621556i	$-0.786501\pi$
$37$	− 7.56155i	− 1.24311i	0.783370	−	0.621556i	$-0.213499\pi$
$38$	−11.1231	−1.80441
$39$	0	0
$40$	−8.68466	−1.37317
$41$	1.56155i	0.243874i	0.992538	+	0.121937i	$0.0389105\pi$
$41$	1.56155i	0.243874i	−0.992538	+	0.121937i	$0.961089\pi$
$42$	0	0
$43$	−4.56155	−0.695630	−0.347815	−	0.937563i	$-0.613076\pi$
$43$	−4.56155	−0.695630	−0.347815	+	0.937563i	$0.613076\pi$
$44$	0.876894i	0.132197i
$45$	0	0
$46$	3.12311i	0.460477i
$47$	8.24621i	1.20283i	0.798935	+	0.601417i	$0.205397\pi$
$47$	8.24621i	1.20283i	−0.798935	+	0.601417i	$0.794603\pi$
$48$	0	0
$49$	6.68466	0.954951
$50$	− 12.0000i	− 1.69706i
$51$	0	0
$52$	0	0
$53$	0.684658	0.0940451	0.0470225	−	0.998894i	$-0.485027\pi$
$53$	0.684658	0.0940451	0.0470225	+	0.998894i	$0.485027\pi$
$54$	0	0
$55$	7.12311	0.960479
$56$	1.36932	0.182983
$57$	0	0
$58$	− 10.4384i	− 1.37064i
$59$	− 2.87689i	− 0.374540i	−0.982309	−	0.187270i	$-0.940036\pi$
$59$	− 2.87689i	− 0.374540i	0.982309	−	0.187270i	$-0.0599639\pi$
$60$	0	0
$61$	3.87689	0.496385	0.248193	−	0.968711i	$-0.420163\pi$
$61$	3.87689	0.496385	0.248193	+	0.968711i	$0.420163\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	−5.56155	−0.695194
$65$	0	0
$66$	0	0
$67$	4.56155i	0.557282i	0.960395	+	0.278641i	$0.0898840\pi$
$67$	4.56155i	0.557282i	−0.960395	+	0.278641i	$0.910116\pi$
$68$	0.684658	0.0830270
$69$	0	0
$70$	3.12311i	0.373283i
$71$	− 14.0000i	− 1.66149i	−0.556650	−	0.830747i	$-0.687914\pi$
$71$	− 14.0000i	− 1.66149i	0.556650	−	0.830747i	$-0.312086\pi$
$72$	0	0
$73$	10.1231i	1.18482i	0.805637	+	0.592410i	$0.201823\pi$
$73$	10.1231i	1.18482i	−0.805637	+	0.592410i	$0.798177\pi$
$74$	11.8078	1.37262
$75$	0	0
$76$	− 3.12311i	− 0.358245i
$77$	−1.12311	−0.127990
$78$	0	0
$79$	5.43845	0.611873	0.305937	−	0.952052i	$-0.401030\pi$
$79$	5.43845	0.611873	0.305937	+	0.952052i	$0.401030\pi$
$80$	− 16.6847i	− 1.86540i
$81$	0	0
$82$	−2.43845	−0.269281
$83$	0.876894i	0.0962517i	0.998841	+	0.0481258i	$0.0153248\pi$
$83$	0.876894i	0.0962517i	−0.998841	+	0.0481258i	$0.984675\pi$
$84$	0	0
$85$	− 5.56155i	− 0.603235i
$86$	− 7.12311i	− 0.768104i
$87$	0	0
$88$	4.87689	0.519879
$89$	4.87689i	0.516950i	0.966018	+	0.258475i	$0.0832199\pi$
$89$	4.87689i	0.516950i	−0.966018	+	0.258475i	$0.916780\pi$
$90$	0	0
$91$	0	0
$92$	−0.876894	−0.0914226
$93$	0	0
$94$	−12.8769	−1.32815
$95$	−25.3693	−2.60284
$96$	0	0
$97$	− 8.56155i	− 0.869294i	−0.900601	−	0.434647i	$-0.856873\pi$
$97$	− 8.56155i	− 0.869294i	0.900601	−	0.434647i	$-0.143127\pi$
$98$	10.4384i	1.05444i
$99$	0	0
$100$	3.36932	0.336932
$101$	−7.56155	−0.752403	−0.376201	−	0.926538i	$-0.622770\pi$
$101$	−7.56155	−0.752403	−0.376201	+	0.926538i	$0.622770\pi$
$102$	0	0
$103$	3.43845	0.338800	0.169400	−	0.985547i	$-0.445817\pi$
$103$	3.43845	0.338800	0.169400	+	0.985547i	$0.445817\pi$
$104$	0	0
$105$	0	0
$106$	1.06913i	0.103843i
$107$	8.24621	0.797191	0.398596	−	0.917127i	$-0.369498\pi$
$107$	8.24621	0.797191	0.398596	+	0.917127i	$0.369498\pi$
$108$	0	0
$109$	− 2.80776i	− 0.268935i	−0.990918	−	0.134468i	$-0.957068\pi$
$109$	− 2.80776i	− 0.268935i	0.990918	−	0.134468i	$-0.0429324\pi$
$110$	11.1231i	1.06055i
$111$	0	0
$112$	2.63068i	0.248576i
$113$	−5.80776	−0.546348	−0.273174	−	0.961965i	$-0.588074\pi$
$113$	−5.80776	−0.546348	−0.273174	+	0.961965i	$0.588074\pi$
$114$	0	0
$115$	7.12311i	0.664233i
$116$	2.93087	0.272124
$117$	0	0
$118$	4.49242	0.413561
$119$	0.876894i	0.0803848i
$120$	0	0
$121$	7.00000	0.636364
$122$	6.05398i	0.548101i
$123$	0	0
$124$	− 1.12311i	− 0.100858i
$125$	− 9.56155i	− 0.855211i
$126$	0	0
$127$	−5.43845	−0.482584	−0.241292	−	0.970453i	$-0.577571\pi$
$127$	−5.43845	−0.482584	−0.241292	+	0.970453i	$0.577571\pi$
$128$	− 13.5616i	− 1.19868i
$129$	0	0
$130$	0	0
$131$	−7.36932	−0.643860	−0.321930	−	0.946763i	$-0.604332\pi$
$131$	−7.36932	−0.643860	−0.321930	+	0.946763i	$0.604332\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−7.12311	−0.615343
$135$	0	0
$136$	− 3.80776i	− 0.326513i
$137$	− 5.56155i	− 0.475156i	−0.971369	−	0.237578i	$-0.923647\pi$
$137$	− 5.56155i	− 0.475156i	0.971369	−	0.237578i	$-0.0763535\pi$
$138$	0	0
$139$	−17.9309	−1.52088	−0.760438	−	0.649410i	$-0.775016\pi$
$139$	−17.9309	−1.52088	−0.760438	+	0.649410i	$0.775016\pi$
$140$	−0.876894	−0.0741111
$141$	0	0
$142$	21.8617	1.83460
$143$	0	0
$144$	0	0
$145$	− 23.8078i	− 1.97713i
$146$	−15.8078	−1.30826
$147$	0	0
$148$	3.31534i	0.272519i
$149$	2.43845i	0.199765i	0.994999	+	0.0998827i	$0.0318468\pi$
$149$	2.43845i	0.199765i	−0.994999	+	0.0998827i	$0.968153\pi$
$150$	0	0
$151$	9.36932i	0.762464i	0.924479	+	0.381232i	$0.124500\pi$
$151$	9.36932i	0.762464i	−0.924479	+	0.381232i	$0.875500\pi$
$152$	−17.3693	−1.40884
$153$	0	0
$154$	− 1.75379i	− 0.141324i
$155$	−9.12311	−0.732785
$156$	0	0
$157$	20.3693	1.62565	0.812824	−	0.582509i	$-0.197929\pi$
$157$	20.3693	1.62565	0.812824	+	0.582509i	$0.197929\pi$
$158$	8.49242i	0.675621i
$159$	0	0
$160$	8.68466	0.686583
$161$	− 1.12311i	− 0.0885131i
$162$	0	0
$163$	4.80776i	0.376573i	0.982114	+	0.188287i	$0.0602934\pi$
$163$	4.80776i	0.376573i	−0.982114	+	0.188287i	$0.939707\pi$
$164$	− 0.684658i	− 0.0534628i
$165$	0	0
$166$	−1.36932	−0.106280
$167$	10.2462i	0.792876i	0.918062	+	0.396438i	$0.129754\pi$
$167$	10.2462i	0.792876i	−0.918062	+	0.396438i	$0.870246\pi$
$168$	0	0
$169$	0	0
$170$	8.68466	0.666083
$171$	0	0
$172$	2.00000	0.152499
$173$	20.2462	1.53929	0.769645	−	0.638471i	$-0.220433\pi$
$173$	20.2462	1.53929	0.769645	+	0.638471i	$0.220433\pi$
$174$	0	0
$175$	4.31534i	0.326209i
$176$	9.36932i	0.706239i
$177$	0	0
$178$	−7.61553	−0.570808
$179$	−4.87689	−0.364516	−0.182258	−	0.983251i	$-0.558341\pi$
$179$	−4.87689	−0.364516	−0.182258	+	0.983251i	$0.558341\pi$
$180$	0	0
$181$	2.68466	0.199549	0.0997745	−	0.995010i	$-0.468188\pi$
$181$	2.68466	0.199549	0.0997745	+	0.995010i	$0.468188\pi$
$182$	0	0
$183$	0	0
$184$	4.87689i	0.359529i
$185$	26.9309	1.98000
$186$	0	0
$187$	3.12311i	0.228384i
$188$	− 3.61553i	− 0.263689i
$189$	0	0
$190$	− 39.6155i	− 2.87401i
$191$	9.12311	0.660125	0.330062	−	0.943959i	$-0.392930\pi$
$191$	9.12311	0.660125	0.330062	+	0.943959i	$0.392930\pi$
$192$	0	0
$193$	13.4924i	0.971206i	0.874180	+	0.485603i	$0.161400\pi$
$193$	13.4924i	0.971206i	−0.874180	+	0.485603i	$0.838600\pi$
$194$	13.3693	0.959861
$195$	0	0
$196$	−2.93087	−0.209348
$197$	− 13.3693i	− 0.952524i	−0.879303	−	0.476262i	$-0.841991\pi$
$197$	− 13.3693i	− 0.952524i	0.879303	−	0.476262i	$-0.158009\pi$
$198$	0	0
$199$	−22.1771	−1.57209	−0.786046	−	0.618168i	$-0.787875\pi$
$199$	−22.1771	−1.57209	−0.786046	+	0.618168i	$0.787875\pi$
$200$	− 18.7386i	− 1.32502i
$201$	0	0
$202$	− 11.8078i	− 0.830791i
$203$	3.75379i	0.263464i
$204$	0	0
$205$	−5.56155	−0.388436
$206$	5.36932i	0.374098i
$207$	0	0
$208$	0	0
$209$	14.2462	0.985431
$210$	0	0
$211$	19.6847	1.35515	0.677574	−	0.735455i	$-0.263031\pi$
$211$	19.6847	1.35515	0.677574	+	0.735455i	$0.263031\pi$
$212$	−0.300187	−0.0206169
$213$	0	0
$214$	12.8769i	0.880246i
$215$	− 16.2462i	− 1.10798i
$216$	0	0
$217$	1.43845	0.0976482
$218$	4.38447	0.296954
$219$	0	0
$220$	−3.12311	−0.210560
$221$	0	0
$222$	0	0
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	−1.36932	−0.0914913
$225$	0	0
$226$	− 9.06913i	− 0.603270i
$227$	− 7.12311i	− 0.472777i	−0.971659	−	0.236389i	$-0.924036\pi$
$227$	− 7.12311i	− 0.472777i	0.971659	−	0.236389i	$-0.0759638\pi$
$228$	0	0
$229$	16.2462i	1.07358i	0.843716	+	0.536790i	$0.180363\pi$
$229$	16.2462i	1.07358i	−0.843716	+	0.536790i	$0.819637\pi$
$230$	−11.1231	−0.733436
$231$	0	0
$232$	− 16.3002i	− 1.07016i
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	−29.3693	−1.91584
$236$	1.26137i	0.0821079i
$237$	0	0
$238$	−1.36932	−0.0887596
$239$	25.3693i	1.64100i	0.571643	+	0.820502i	$0.306306\pi$
$239$	25.3693i	1.64100i	−0.571643	+	0.820502i	$0.693694\pi$
$240$	0	0
$241$	17.8078i	1.14710i	0.819171	+	0.573549i	$0.194434\pi$
$241$	17.8078i	1.14710i	−0.819171	+	0.573549i	$0.805566\pi$
$242$	10.9309i	0.702663i
$243$	0	0
$244$	−1.69981	−0.108819
$245$	23.8078i	1.52102i
$246$	0	0
$247$	0	0
$248$	−6.24621	−0.396635
$249$	0	0
$250$	14.9309	0.944311
$251$	18.7386	1.18277	0.591386	−	0.806389i	$-0.298581\pi$
$251$	18.7386	1.18277	0.591386	+	0.806389i	$0.298581\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	− 8.49242i	− 0.532862i
$255$	0	0
$256$	10.0540	0.628373
$257$	29.1771	1.82002	0.910008	−	0.414590i	$-0.136075\pi$
$257$	29.1771	1.82002	0.910008	+	0.414590i	$0.136075\pi$
$258$	0	0
$259$	−4.24621	−0.263847
$260$	0	0
$261$	0	0
$262$	− 11.5076i	− 0.710941i
$263$	−9.36932	−0.577737	−0.288868	−	0.957369i	$-0.593279\pi$
$263$	−9.36932	−0.577737	−0.288868	+	0.957369i	$0.593279\pi$
$264$	0	0
$265$	2.43845i	0.149793i
$266$	6.24621i	0.382980i
$267$	0	0
$268$	− 2.00000i	− 0.122169i
$269$	21.3693	1.30291	0.651455	−	0.758687i	$-0.274159\pi$
$269$	21.3693	1.30291	0.651455	+	0.758687i	$0.274159\pi$
$270$	0	0
$271$	29.9309i	1.81817i	0.416610	+	0.909085i	$0.363218\pi$
$271$	29.9309i	1.81817i	−0.416610	+	0.909085i	$0.636782\pi$
$272$	7.31534	0.443558
$273$	0	0
$274$	8.68466	0.524659
$275$	15.3693i	0.926805i
$276$	0	0
$277$	−5.31534	−0.319368	−0.159684	−	0.987168i	$-0.551048\pi$
$277$	−5.31534	−0.319368	−0.159684	+	0.987168i	$0.551048\pi$
$278$	− 28.0000i	− 1.67933i
$279$	0	0
$280$	4.87689i	0.291450i
$281$	17.8078i	1.06232i	0.847271	+	0.531161i	$0.178244\pi$
$281$	17.8078i	1.06232i	−0.847271	+	0.531161i	$0.821756\pi$
$282$	0	0
$283$	13.6847	0.813469	0.406734	−	0.913547i	$-0.366667\pi$
$283$	13.6847	0.813469	0.406734	+	0.913547i	$0.366667\pi$
$284$	6.13826i	0.364239i
$285$	0	0
$286$	0	0
$287$	0.876894	0.0517614
$288$	0	0
$289$	−14.5616	−0.856562
$290$	37.1771	2.18311
$291$	0	0
$292$	− 4.43845i	− 0.259740i
$293$	20.4384i	1.19403i	0.802231	+	0.597013i	$0.203646\pi$
$293$	20.4384i	1.19403i	−0.802231	+	0.597013i	$0.796354\pi$
$294$	0	0
$295$	10.2462	0.596557
$296$	18.4384	1.07171
$297$	0	0
$298$	−3.80776	−0.220578
$299$	0	0
$300$	0	0
$301$	2.56155i	0.147645i
$302$	−14.6307	−0.841901
$303$	0	0
$304$	− 33.3693i	− 1.91386i
$305$	13.8078i	0.790630i
$306$	0	0
$307$	− 30.8078i	− 1.75829i	−0.476553	−	0.879146i	$-0.658114\pi$
$307$	− 30.8078i	− 1.75829i	0.476553	−	0.879146i	$-0.341886\pi$
$308$	0.492423	0.0280584
$309$	0	0
$310$	− 14.2462i	− 0.809130i
$311$	−19.1231	−1.08437	−0.542186	−	0.840259i	$-0.682403\pi$
$311$	−19.1231	−1.08437	−0.542186	+	0.840259i	$0.682403\pi$
$312$	0	0
$313$	−13.6847	−0.773503	−0.386751	−	0.922184i	$-0.626403\pi$
$313$	−13.6847	−0.773503	−0.386751	+	0.922184i	$0.626403\pi$
$314$	31.8078i	1.79502i
$315$	0	0
$316$	−2.38447	−0.134137
$317$	− 14.0540i	− 0.789350i	−0.918821	−	0.394675i	$-0.870857\pi$
$317$	− 14.0540i	− 0.789350i	0.918821	−	0.394675i	$-0.129143\pi$
$318$	0	0
$319$	13.3693i	0.748538i
$320$	− 19.8078i	− 1.10729i
$321$	0	0
$322$	1.75379	0.0977348
$323$	− 11.1231i	− 0.618906i
$324$	0	0
$325$	0	0
$326$	−7.50758	−0.415806
$327$	0	0
$328$	−3.80776	−0.210249
$329$	4.63068	0.255298
$330$	0	0
$331$	− 3.19224i	− 0.175461i	−0.996144	−	0.0877306i	$-0.972039\pi$
$331$	− 3.19224i	− 0.175461i	0.996144	−	0.0877306i	$-0.0279615\pi$
$332$	− 0.384472i	− 0.0211006i
$333$	0	0
$334$	−16.0000	−0.875481
$335$	−16.2462	−0.887625
$336$	0	0
$337$	6.12311	0.333547	0.166773	−	0.985995i	$-0.446665\pi$
$337$	6.12311	0.333547	0.166773	+	0.985995i	$0.446665\pi$
$338$	0	0
$339$	0	0
$340$	2.43845i	0.132243i
$341$	5.12311	0.277432
$342$	0	0
$343$	− 7.68466i	− 0.414933i
$344$	− 11.1231i	− 0.599718i
$345$	0	0
$346$	31.6155i	1.69966i
$347$	−27.6155	−1.48248	−0.741240	−	0.671241i	$-0.765762\pi$
$347$	−27.6155	−1.48248	−0.741240	+	0.671241i	$0.765762\pi$
$348$	0	0
$349$	− 6.80776i	− 0.364411i	−0.983260	−	0.182206i	$-0.941676\pi$
$349$	− 6.80776i	− 0.364411i	0.983260	−	0.182206i	$-0.0583237\pi$
$350$	−6.73863	−0.360195
$351$	0	0
$352$	−4.87689	−0.259939
$353$	− 5.31534i	− 0.282907i	−0.989945	−	0.141454i	$-0.954822\pi$
$353$	− 5.31534i	− 0.282907i	0.989945	−	0.141454i	$-0.0451776\pi$
$354$	0	0
$355$	49.8617	2.64639
$356$	− 2.13826i	− 0.113328i
$357$	0	0
$358$	− 7.61553i	− 0.402493i
$359$	− 9.36932i	− 0.494494i	−0.968953	−	0.247247i	$-0.920474\pi$
$359$	− 9.36932i	− 0.494494i	0.968953	−	0.247247i	$-0.0795258\pi$
$360$	0	0
$361$	−31.7386	−1.67045
$362$	4.19224i	0.220339i
$363$	0	0
$364$	0	0
$365$	−36.0540	−1.88715
$366$	0	0
$367$	−17.0540	−0.890210	−0.445105	−	0.895478i	$-0.646834\pi$
$367$	−17.0540	−0.890210	−0.445105	+	0.895478i	$0.646834\pi$
$368$	−9.36932	−0.488409
$369$	0	0
$370$	42.0540i	2.18628i
$371$	− 0.384472i	− 0.0199608i
$372$	0	0
$373$	28.3693	1.46891	0.734454	−	0.678659i	$-0.237438\pi$
$373$	28.3693	1.46891	0.734454	+	0.678659i	$0.237438\pi$
$374$	−4.87689	−0.252178
$375$	0	0
$376$	−20.1080	−1.03699
$377$	0	0
$378$	0	0
$379$	23.6847i	1.21660i	0.793708	+	0.608300i	$0.208148\pi$
$379$	23.6847i	1.21660i	−0.793708	+	0.608300i	$0.791852\pi$
$380$	11.1231	0.570603
$381$	0	0
$382$	14.2462i	0.728900i
$383$	22.7386i	1.16189i	0.813943	+	0.580945i	$0.197317\pi$
$383$	22.7386i	1.16189i	−0.813943	+	0.580945i	$0.802683\pi$
$384$	0	0
$385$	− 4.00000i	− 0.203859i
$386$	−21.0691	−1.07239
$387$	0	0
$388$	3.75379i	0.190570i
$389$	34.0540	1.72661	0.863303	−	0.504687i	$-0.168392\pi$
$389$	34.0540	1.72661	0.863303	+	0.504687i	$0.168392\pi$
$390$	0	0
$391$	−3.12311	−0.157942
$392$	16.3002i	0.823284i
$393$	0	0
$394$	20.8769	1.05176
$395$	19.3693i	0.974576i
$396$	0	0
$397$	− 25.0540i	− 1.25742i	−0.777639	−	0.628711i	$-0.783583\pi$
$397$	− 25.0540i	− 1.25742i	0.777639	−	0.628711i	$-0.216417\pi$
$398$	− 34.6307i	− 1.73588i
$399$	0	0
$400$	36.0000	1.80000
$401$	14.4384i	0.721022i	0.932755	+	0.360511i	$0.117398\pi$
$401$	14.4384i	0.721022i	−0.932755	+	0.360511i	$0.882602\pi$
$402$	0	0
$403$	0	0
$404$	3.31534	0.164944
$405$	0	0
$406$	−5.86174	−0.290913
$407$	−15.1231	−0.749625
$408$	0	0
$409$	6.36932i	0.314942i	0.987524	+	0.157471i	$0.0503342\pi$
$409$	6.36932i	0.314942i	−0.987524	+	0.157471i	$0.949666\pi$
$410$	− 8.68466i	− 0.428905i
$411$	0	0
$412$	−1.50758	−0.0742730
$413$	−1.61553	−0.0794949
$414$	0	0
$415$	−3.12311	−0.153307
$416$	0	0
$417$	0	0
$418$	22.2462i	1.08810i
$419$	34.2462	1.67304	0.836518	−	0.547939i	$-0.184587\pi$
$419$	34.2462	1.67304	0.836518	+	0.547939i	$0.184587\pi$
$420$	0	0
$421$	31.2462i	1.52285i	0.648255	+	0.761424i	$0.275499\pi$
$421$	31.2462i	1.52285i	−0.648255	+	0.761424i	$0.724501\pi$
$422$	30.7386i	1.49633i
$423$	0	0
$424$	1.66950i	0.0810783i
$425$	12.0000	0.582086
$426$	0	0
$427$	− 2.17708i	− 0.105356i
$428$	−3.61553	−0.174763
$429$	0	0
$430$	25.3693	1.22342
$431$	11.1231i	0.535781i	0.963449	+	0.267891i	$0.0863266\pi$
$431$	11.1231i	0.535781i	−0.963449	+	0.267891i	$0.913673\pi$
$432$	0	0
$433$	−8.75379	−0.420680	−0.210340	−	0.977628i	$-0.567457\pi$
$433$	−8.75379	−0.420680	−0.210340	+	0.977628i	$0.567457\pi$
$434$	2.24621i	0.107822i
$435$	0	0
$436$	1.23106i	0.0589569i
$437$	14.2462i	0.681489i
$438$	0	0
$439$	13.6847	0.653133	0.326567	−	0.945174i	$-0.394108\pi$
$439$	13.6847	0.653133	0.326567	+	0.945174i	$0.394108\pi$
$440$	17.3693i	0.828050i
$441$	0	0
$442$	0	0
$443$	34.7386	1.65048	0.825241	−	0.564781i	$-0.191039\pi$
$443$	34.7386	1.65048	0.825241	+	0.564781i	$0.191039\pi$
$444$	0	0
$445$	−17.3693	−0.823385
$446$	−12.4924	−0.591533
$447$	0	0
$448$	3.12311i	0.147553i
$449$	− 8.24621i	− 0.389163i	−0.980886	−	0.194581i	$-0.937665\pi$
$449$	− 8.24621i	− 0.389163i	0.980886	−	0.194581i	$-0.0623348\pi$
$450$	0	0
$451$	3.12311	0.147061
$452$	2.54640	0.119772
$453$	0	0
$454$	11.1231	0.522033
$455$	0	0
$456$	0	0
$457$	12.6155i	0.590130i	0.955477	+	0.295065i	$0.0953412\pi$
$457$	12.6155i	0.590130i	−0.955477	+	0.295065i	$0.904659\pi$
$458$	−25.3693	−1.18543
$459$	0	0
$460$	− 3.12311i	− 0.145616i
$461$	16.1922i	0.754148i	0.926183	+	0.377074i	$0.123070\pi$
$461$	16.1922i	0.754148i	−0.926183	+	0.377074i	$0.876930\pi$
$462$	0	0
$463$	14.3153i	0.665290i	0.943052	+	0.332645i	$0.107941\pi$
$463$	14.3153i	0.665290i	−0.943052	+	0.332645i	$0.892059\pi$
$464$	31.3153	1.45378
$465$	0	0
$466$	40.6004i	1.88078i
$467$	−26.0000	−1.20314	−0.601568	−	0.798821i	$-0.705457\pi$
$467$	−26.0000	−1.20314	−0.601568	+	0.798821i	$0.705457\pi$
$468$	0	0
$469$	2.56155	0.118282
$470$	− 45.8617i	− 2.11544i
$471$	0	0
$472$	7.01515	0.322899
$473$	9.12311i	0.419481i
$474$	0	0
$475$	− 54.7386i	− 2.51158i
$476$	− 0.384472i	− 0.0176222i
$477$	0	0
$478$	−39.6155	−1.81197
$479$	10.2462i	0.468161i	0.972217	+	0.234081i	$0.0752080\pi$
$479$	10.2462i	0.468161i	−0.972217	+	0.234081i	$0.924792\pi$
$480$	0	0
$481$	0	0
$482$	−27.8078	−1.26661
$483$	0	0
$484$	−3.06913	−0.139506
$485$	30.4924	1.38459
$486$	0	0
$487$	7.12311i	0.322779i	0.986891	+	0.161389i	$0.0515975\pi$
$487$	7.12311i	0.322779i	−0.986891	+	0.161389i	$0.948403\pi$
$488$	9.45360i	0.427944i
$489$	0	0
$490$	−37.1771	−1.67949
$491$	−36.2462	−1.63577	−0.817884	−	0.575383i	$-0.804853\pi$
$491$	−36.2462	−1.63577	−0.817884	+	0.575383i	$0.804853\pi$
$492$	0	0
$493$	10.4384	0.470124
$494$	0	0
$495$	0	0
$496$	− 12.0000i	− 0.538816i
$497$	−7.86174	−0.352647
$498$	0	0
$499$	4.49242i	0.201108i	0.994932	+	0.100554i	$0.0320616\pi$
$499$	4.49242i	0.201108i	−0.994932	+	0.100554i	$0.967938\pi$
$500$	4.19224i	0.187482i
$501$	0	0
$502$	29.2614i	1.30600i
$503$	28.2462	1.25944	0.629718	−	0.776824i	$-0.283170\pi$
$503$	28.2462	1.25944	0.629718	+	0.776824i	$0.283170\pi$
$504$	0	0
$505$	− 26.9309i	− 1.19841i
$506$	6.24621	0.277678
$507$	0	0
$508$	2.38447	0.105794
$509$	13.8078i	0.612018i	0.952029	+	0.306009i	$0.0989938\pi$
$509$	13.8078i	0.612018i	−0.952029	+	0.306009i	$0.901006\pi$
$510$	0	0
$511$	5.68466	0.251474
$512$	− 11.4233i	− 0.504843i
$513$	0	0
$514$	45.5616i	2.00963i
$515$	12.2462i	0.539633i
$516$	0	0
$517$	16.4924	0.725336
$518$	− 6.63068i	− 0.291335i
$519$	0	0
$520$	0	0
$521$	9.06913	0.397326	0.198663	−	0.980068i	$-0.436340\pi$
$521$	9.06913	0.397326	0.198663	+	0.980068i	$0.436340\pi$
$522$	0	0
$523$	33.8617	1.48067	0.740335	−	0.672238i	$-0.234667\pi$
$523$	33.8617	1.48067	0.740335	+	0.672238i	$0.234667\pi$
$524$	3.23106	0.141149
$525$	0	0
$526$	− 14.6307i	− 0.637928i
$527$	− 4.00000i	− 0.174243i
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−3.80776	−0.165399
$531$	0	0
$532$	−1.75379	−0.0760364
$533$	0	0
$534$	0	0
$535$	29.3693i	1.26975i
$536$	−11.1231	−0.480445
$537$	0	0
$538$	33.3693i	1.43865i
$539$	− 13.3693i	− 0.575857i
$540$	0	0
$541$	− 19.7386i	− 0.848630i	−0.905515	−	0.424315i	$-0.860515\pi$
$541$	− 19.7386i	− 0.848630i	0.905515	−	0.424315i	$-0.139485\pi$
$542$	−46.7386	−2.00760
$543$	0	0
$544$	3.80776i	0.163257i
$545$	10.0000	0.428353
$546$	0	0
$547$	3.93087	0.168072	0.0840359	−	0.996463i	$-0.473219\pi$
$547$	3.93087	0.168072	0.0840359	+	0.996463i	$0.473219\pi$
$548$	2.43845i	0.104165i
$549$	0	0
$550$	−24.0000	−1.02336
$551$	− 47.6155i	− 2.02849i
$552$	0	0
$553$	− 3.05398i	− 0.129868i
$554$	− 8.30019i	− 0.352641i
$555$	0	0
$556$	7.86174	0.333412
$557$	− 42.9309i	− 1.81904i	−0.415661	−	0.909520i	$-0.636450\pi$
$557$	− 42.9309i	− 1.81904i	0.415661	−	0.909520i	$-0.363550\pi$
$558$	0	0
$559$	0	0
$560$	−9.36932	−0.395926
$561$	0	0
$562$	−27.8078	−1.17300
$563$	−23.3693	−0.984899	−0.492450	−	0.870341i	$-0.663898\pi$
$563$	−23.3693	−0.984899	−0.492450	+	0.870341i	$0.663898\pi$
$564$	0	0
$565$	− 20.6847i	− 0.870210i
$566$	21.3693i	0.898219i
$567$	0	0
$568$	34.1383	1.43241
$569$	−8.73863	−0.366343	−0.183171	−	0.983081i	$-0.558636\pi$
$569$	−8.73863	−0.366343	−0.183171	+	0.983081i	$0.558636\pi$
$570$	0	0
$571$	5.36932	0.224699	0.112349	−	0.993669i	$-0.464162\pi$
$571$	5.36932	0.224699	0.112349	+	0.993669i	$0.464162\pi$
$572$	0	0
$573$	0	0
$574$	1.36932i	0.0571542i
$575$	−15.3693	−0.640945
$576$	0	0
$577$	− 17.3153i	− 0.720847i	−0.932789	−	0.360424i	$-0.882632\pi$
$577$	− 17.3153i	− 0.720847i	0.932789	−	0.360424i	$-0.117368\pi$
$578$	− 22.7386i	− 0.945802i
$579$	0	0
$580$	10.4384i	0.433433i
$581$	0.492423	0.0204291
$582$	0	0
$583$	− 1.36932i	− 0.0567113i
$584$	−24.6847	−1.02146
$585$	0	0
$586$	−31.9157	−1.31843
$587$	− 39.3693i	− 1.62495i	−0.582999	−	0.812473i	$-0.698121\pi$
$587$	− 39.3693i	− 1.62495i	0.582999	−	0.812473i	$-0.301879\pi$
$588$	0	0
$589$	−18.2462	−0.751822
$590$	16.0000i	0.658710i
$591$	0	0
$592$	35.4233i	1.45589i
$593$	− 17.4233i	− 0.715489i	−0.933820	−	0.357744i	$-0.883546\pi$
$593$	− 17.4233i	− 0.715489i	0.933820	−	0.357744i	$-0.116454\pi$
$594$	0	0
$595$	−3.12311	−0.128035
$596$	− 1.06913i	− 0.0437933i
$597$	0	0
$598$	0	0
$599$	41.6155	1.70036	0.850182	−	0.526489i	$-0.176492\pi$
$599$	41.6155	1.70036	0.850182	+	0.526489i	$0.176492\pi$
$600$	0	0
$601$	−7.06913	−0.288356	−0.144178	−	0.989552i	$-0.546054\pi$
$601$	−7.06913	−0.288356	−0.144178	+	0.989552i	$0.546054\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	− 4.10795i	− 0.167150i
$605$	24.9309i	1.01358i
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	17.3693	0.704419
$609$	0	0
$610$	−21.5616	−0.873002
$611$	0	0
$612$	0	0
$613$	− 34.8617i	− 1.40805i	−0.710174	−	0.704026i	$-0.751384\pi$
$613$	− 34.8617i	− 1.40805i	0.710174	−	0.704026i	$-0.248616\pi$
$614$	48.1080	1.94148
$615$	0	0
$616$	− 2.73863i	− 0.110343i
$617$	− 9.80776i	− 0.394846i	−0.980318	−	0.197423i	$-0.936743\pi$
$617$	− 9.80776i	− 0.394846i	0.980318	−	0.197423i	$-0.0632572\pi$
$618$	0	0
$619$	− 29.3002i	− 1.17767i	−0.808252	−	0.588837i	$-0.799586\pi$
$619$	− 29.3002i	− 1.17767i	0.808252	−	0.588837i	$-0.200414\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	− 29.8617i	− 1.19735i
$623$	2.73863	0.109721
$624$	0	0
$625$	−4.36932	−0.174773
$626$	− 21.3693i	− 0.854090i
$627$	0	0
$628$	−8.93087	−0.356380
$629$	11.8078i	0.470806i
$630$	0	0
$631$	18.5616i	0.738924i	0.929246	+	0.369462i	$0.120458\pi$
$631$	18.5616i	0.738924i	−0.929246	+	0.369462i	$0.879542\pi$
$632$	13.2614i	0.527509i
$633$	0	0
$634$	21.9460	0.871588
$635$	− 19.3693i	− 0.768648i
$636$	0	0
$637$	0	0
$638$	−20.8769	−0.826524
$639$	0	0
$640$	48.3002	1.90923
$641$	−19.1771	−0.757449	−0.378725	−	0.925509i	$-0.623637\pi$
$641$	−19.1771	−0.757449	−0.378725	+	0.925509i	$0.623637\pi$
$642$	0	0
$643$	− 31.5464i	− 1.24407i	−0.782990	−	0.622034i	$-0.786306\pi$
$643$	− 31.5464i	− 1.24407i	0.782990	−	0.622034i	$-0.213694\pi$
$644$	0.492423i	0.0194042i
$645$	0	0
$646$	17.3693	0.683387
$647$	−6.38447	−0.250999	−0.125500	−	0.992094i	$-0.540053\pi$
$647$	−6.38447	−0.250999	−0.125500	+	0.992094i	$0.540053\pi$
$648$	0	0
$649$	−5.75379	−0.225856
$650$	0	0
$651$	0	0
$652$	− 2.10795i	− 0.0825537i
$653$	−23.1231	−0.904877	−0.452439	−	0.891796i	$-0.649446\pi$
$653$	−23.1231	−0.904877	−0.452439	+	0.891796i	$0.649446\pi$
$654$	0	0
$655$	− 26.2462i	− 1.02552i
$656$	− 7.31534i	− 0.285616i
$657$	0	0
$658$	7.23106i	0.281896i
$659$	2.24621	0.0875000	0.0437500	−	0.999043i	$-0.486070\pi$
$659$	2.24621	0.0875000	0.0437500	+	0.999043i	$0.486070\pi$
$660$	0	0
$661$	− 5.63068i	− 0.219008i	−0.993986	−	0.109504i	$-0.965074\pi$
$661$	− 5.63068i	− 0.219008i	0.993986	−	0.109504i	$-0.0349263\pi$
$662$	4.98485	0.193742
$663$	0	0
$664$	−2.13826	−0.0829806
$665$	14.2462i	0.552444i
$666$	0	0
$667$	−13.3693	−0.517662
$668$	− 4.49242i	− 0.173817i
$669$	0	0
$670$	− 25.3693i	− 0.980102i
$671$	− 7.75379i	− 0.299332i
$672$	0	0
$673$	23.2462	0.896076	0.448038	−	0.894015i	$-0.352123\pi$
$673$	23.2462	0.896076	0.448038	+	0.894015i	$0.352123\pi$
$674$	9.56155i	0.368297i
$675$	0	0
$676$	0	0
$677$	15.6155	0.600153	0.300077	−	0.953915i	$-0.402988\pi$
$677$	15.6155	0.600153	0.300077	+	0.953915i	$0.402988\pi$
$678$	0	0
$679$	−4.80776	−0.184505
$680$	13.5616	0.520062
$681$	0	0
$682$	8.00000i	0.306336i
$683$	− 38.1080i	− 1.45816i	−0.684428	−	0.729080i	$-0.739948\pi$
$683$	− 38.1080i	− 1.45816i	0.684428	−	0.729080i	$-0.260052\pi$
$684$	0	0
$685$	19.8078	0.756816
$686$	12.0000	0.458162
$687$	0	0
$688$	21.3693	0.814698
$689$	0	0
$690$	0	0
$691$	− 51.3002i	− 1.95155i	−0.218774	−	0.975776i	$-0.570206\pi$
$691$	− 51.3002i	− 1.95155i	0.218774	−	0.975776i	$-0.429794\pi$
$692$	−8.87689	−0.337449
$693$	0	0
$694$	− 43.1231i	− 1.63693i
$695$	− 63.8617i	− 2.42241i
$696$	0	0
$697$	− 2.43845i	− 0.0923628i
$698$	10.6307	0.402377
$699$	0	0
$700$	− 1.89205i	− 0.0715127i
$701$	−5.36932	−0.202796	−0.101398	−	0.994846i	$-0.532332\pi$
$701$	−5.36932	−0.202796	−0.101398	+	0.994846i	$0.532332\pi$
$702$	0	0
$703$	53.8617	2.03143
$704$	11.1231i	0.419218i
$705$	0	0
$706$	8.30019	0.312382
$707$	4.24621i	0.159695i
$708$	0	0
$709$	7.49242i	0.281384i	0.990053	+	0.140692i	$0.0449327\pi$
$709$	7.49242i	0.281384i	−0.990053	+	0.140692i	$0.955067\pi$
$710$	77.8617i	2.92210i
$711$	0	0
$712$	−11.8920	−0.445673
$713$	5.12311i	0.191862i
$714$	0	0
$715$	0	0
$716$	2.13826	0.0799106
$717$	0	0
$718$	14.6307	0.546012
$719$	−23.3693	−0.871528	−0.435764	−	0.900061i	$-0.643522\pi$
$719$	−23.3693	−0.871528	−0.435764	+	0.900061i	$0.643522\pi$
$720$	0	0
$721$	− 1.93087i	− 0.0719093i
$722$	− 49.5616i	− 1.84449i
$723$	0	0
$724$	−1.17708	−0.0437459
$725$	51.3693	1.90781
$726$	0	0
$727$	38.6695	1.43417	0.717086	−	0.696984i	$-0.245475\pi$
$727$	38.6695	1.43417	0.717086	+	0.696984i	$0.245475\pi$
$728$	0	0
$729$	0	0
$730$	− 56.3002i	− 2.08376i
$731$	7.12311	0.263458
$732$	0	0
$733$	20.5076i	0.757465i	0.925506	+	0.378732i	$0.123640\pi$
$733$	20.5076i	0.757465i	−0.925506	+	0.378732i	$0.876360\pi$
$734$	− 26.6307i	− 0.982956i
$735$	0	0
$736$	− 4.87689i	− 0.179765i
$737$	9.12311	0.336054
$738$	0	0
$739$	− 10.2462i	− 0.376913i	−0.982082	−	0.188456i	$-0.939652\pi$
$739$	− 10.2462i	− 0.376913i	0.982082	−	0.188456i	$-0.0603484\pi$
$740$	−11.8078	−0.434062
$741$	0	0
$742$	0.600373	0.0220404
$743$	12.6307i	0.463375i	0.972790	+	0.231687i	$0.0744247\pi$
$743$	12.6307i	0.463375i	−0.972790	+	0.231687i	$0.925575\pi$
$744$	0	0
$745$	−8.68466	−0.318181
$746$	44.3002i	1.62195i
$747$	0	0
$748$	− 1.36932i	− 0.0500672i
$749$	− 4.63068i	− 0.169201i
$750$	0	0
$751$	−44.1080	−1.60952	−0.804761	−	0.593599i	$-0.797707\pi$
$751$	−44.1080	−1.60952	−0.804761	+	0.593599i	$0.797707\pi$
$752$	− 38.6307i	− 1.40872i
$753$	0	0
$754$	0	0
$755$	−33.3693	−1.21443
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	−36.9848	−1.34335
$759$	0	0
$760$	− 61.8617i	− 2.24396i
$761$	9.36932i	0.339637i	0.985475	+	0.169819i	$0.0543182\pi$
$761$	9.36932i	0.339637i	−0.985475	+	0.169819i	$0.945682\pi$
$762$	0	0
$763$	−1.57671	−0.0570807
$764$	−4.00000	−0.144715
$765$	0	0
$766$	−35.5076	−1.28294
$767$	0	0
$768$	0	0
$769$	− 18.0000i	− 0.649097i	−0.945869	−	0.324548i	$-0.894788\pi$
$769$	− 18.0000i	− 0.649097i	0.945869	−	0.324548i	$-0.105212\pi$
$770$	6.24621	0.225098
$771$	0	0
$772$	− 5.91571i	− 0.212911i
$773$	24.2462i	0.872076i	0.899928	+	0.436038i	$0.143619\pi$
$773$	24.2462i	0.872076i	−0.899928	+	0.436038i	$0.856381\pi$
$774$	0	0
$775$	− 19.6847i	− 0.707094i
$776$	20.8769	0.749437
$777$	0	0
$778$	53.1771i	1.90649i
$779$	−11.1231	−0.398527
$780$	0	0
$781$	−28.0000	−1.00192
$782$	− 4.87689i	− 0.174397i
$783$	0	0
$784$	−31.3153	−1.11841
$785$	72.5464i	2.58929i
$786$	0	0
$787$	− 44.1771i	− 1.57474i	−0.616479	−	0.787371i	$-0.711441\pi$
$787$	− 44.1771i	− 1.57474i	0.616479	−	0.787371i	$-0.288559\pi$
$788$	5.86174i	0.208816i
$789$	0	0
$790$	−30.2462	−1.07611
$791$	3.26137i	0.115961i
$792$	0	0
$793$	0	0
$794$	39.1231	1.38843
$795$	0	0
$796$	9.72348	0.344640
$797$	−0.384472	−0.0136187	−0.00680935	−	0.999977i	$-0.502167\pi$
$797$	−0.384472	−0.0136187	−0.00680935	+	0.999977i	$0.502167\pi$
$798$	0	0
$799$	− 12.8769i	− 0.455552i
$800$	18.7386i	0.662511i
$801$	0	0
$802$	−22.5464	−0.796141
$803$	20.2462	0.714473
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	0	0
$808$	− 18.4384i	− 0.648662i
$809$	16.3002	0.573084	0.286542	−	0.958068i	$-0.407494\pi$
$809$	16.3002	0.573084	0.286542	+	0.958068i	$0.407494\pi$
$810$	0	0
$811$	2.56155i	0.0899483i	0.998988	+	0.0449741i	$0.0143205\pi$
$811$	2.56155i	0.0899483i	−0.998988	+	0.0449741i	$0.985679\pi$
$812$	− 1.64584i	− 0.0577576i
$813$	0	0
$814$	− 23.6155i	− 0.827724i
$815$	−17.1231	−0.599796
$816$	0	0
$817$	− 32.4924i	− 1.13677i
$818$	−9.94602	−0.347755
$819$	0	0
$820$	2.43845	0.0851543
$821$	− 6.49242i	− 0.226587i	−0.993562	−	0.113294i	$-0.963860\pi$
$821$	− 6.49242i	− 0.226587i	0.993562	−	0.113294i	$-0.0361401\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	8.38447i	0.292087i
$825$	0	0
$826$	− 2.52273i	− 0.0877771i
$827$	− 14.7386i	− 0.512513i	−0.966609	−	0.256256i	$-0.917511\pi$
$827$	− 14.7386i	− 0.512513i	0.966609	−	0.256256i	$-0.0824891\pi$
$828$	0	0
$829$	−13.4924	−0.468611	−0.234306	−	0.972163i	$-0.575282\pi$
$829$	−13.4924	−0.468611	−0.234306	+	0.972163i	$0.575282\pi$
$830$	− 4.87689i	− 0.169279i
$831$	0	0
$832$	0	0
$833$	−10.4384	−0.361671
$834$	0	0
$835$	−36.4924	−1.26287
$836$	−6.24621	−0.216030
$837$	0	0
$838$	53.4773i	1.84734i
$839$	21.6155i	0.746251i	0.927781	+	0.373125i	$0.121714\pi$
$839$	21.6155i	0.746251i	−0.927781	+	0.373125i	$0.878286\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	−48.7926	−1.68150
$843$	0	0
$844$	−8.63068	−0.297080
$845$	0	0
$846$	0	0
$847$	− 3.93087i	− 0.135066i
$848$	−3.20739	−0.110142
$849$	0	0
$850$	18.7386i	0.642730i
$851$	− 15.1231i	− 0.518413i
$852$	0	0
$853$	− 2.12311i	− 0.0726938i	−0.999339	−	0.0363469i	$-0.988428\pi$
$853$	− 2.12311i	− 0.0726938i	0.999339	−	0.0363469i	$-0.0115721\pi$
$854$	3.39963	0.116333
$855$	0	0
$856$	20.1080i	0.687276i
$857$	35.5616	1.21476	0.607380	−	0.794412i	$-0.292221\pi$
$857$	35.5616	1.21476	0.607380	+	0.794412i	$0.292221\pi$
$858$	0	0
$859$	24.5616	0.838029	0.419015	−	0.907979i	$-0.362376\pi$
$859$	24.5616	0.838029	0.419015	+	0.907979i	$0.362376\pi$
$860$	7.12311i	0.242896i
$861$	0	0
$862$	−17.3693	−0.591601
$863$	− 30.4924i	− 1.03797i	−0.854782	−	0.518987i	$-0.826309\pi$
$863$	− 30.4924i	− 1.03797i	0.854782	−	0.518987i	$-0.173691\pi$
$864$	0	0
$865$	72.1080i	2.45174i
$866$	− 13.6695i	− 0.464509i
$867$	0	0
$868$	−0.630683	−0.0214068
$869$	− 10.8769i	− 0.368973i
$870$	0	0
$871$	0	0
$872$	6.84658	0.231855
$873$	0	0
$874$	−22.2462	−0.752489
$875$	−5.36932	−0.181516
$876$	0	0
$877$	23.5616i	0.795617i	0.917468	+	0.397809i	$0.130229\pi$
$877$	23.5616i	0.795617i	−0.917468	+	0.397809i	$0.869771\pi$
$878$	21.3693i	0.721180i
$879$	0	0
$880$	−33.3693	−1.12488
$881$	−9.06913	−0.305547	−0.152773	−	0.988261i	$-0.548820\pi$
$881$	−9.06913	−0.305547	−0.152773	+	0.988261i	$0.548820\pi$
$882$	0	0
$883$	−8.80776	−0.296405	−0.148202	−	0.988957i	$-0.547349\pi$
$883$	−8.80776	−0.296405	−0.148202	+	0.988957i	$0.547349\pi$
$884$	0	0
$885$	0	0
$886$	54.2462i	1.82244i
$887$	−24.6307	−0.827017	−0.413509	−	0.910500i	$-0.635697\pi$
$887$	−24.6307	−0.827017	−0.413509	+	0.910500i	$0.635697\pi$
$888$	0	0
$889$	3.05398i	0.102427i
$890$	− 27.1231i	− 0.909169i
$891$	0	0
$892$	− 3.50758i	− 0.117442i
$893$	−58.7386	−1.96561
$894$	0	0
$895$	− 17.3693i	− 0.580592i
$896$	−7.61553	−0.254417
$897$	0	0
$898$	12.8769	0.429708
$899$	− 17.1231i	− 0.571088i
$900$	0	0
$901$	−1.06913	−0.0356179
$902$	4.87689i	0.162383i
$903$	0	0
$904$	− 14.1619i	− 0.471019i
$905$	9.56155i	0.317837i
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	3.12311i	0.103644i
$909$	0	0
$910$	0	0
$911$	−38.7386	−1.28347	−0.641734	−	0.766927i	$-0.721785\pi$
$911$	−38.7386	−1.28347	−0.641734	+	0.766927i	$0.721785\pi$
$912$	0	0
$913$	1.75379	0.0580419
$914$	−19.6998	−0.651612
$915$	0	0
$916$	− 7.12311i	− 0.235354i
$917$	4.13826i	0.136657i
$918$	0	0
$919$	11.5076	0.379600	0.189800	−	0.981823i	$-0.439216\pi$
$919$	11.5076	0.379600	0.189800	+	0.981823i	$0.439216\pi$
$920$	−17.3693	−0.572649
$921$	0	0
$922$	−25.2850	−0.832718
$923$	0	0
$924$	0	0
$925$	58.1080i	1.91058i
$926$	−22.3542	−0.734603
$927$	0	0
$928$	16.3002i	0.535080i
$929$	7.80776i	0.256164i	0.991764	+	0.128082i	$0.0408821\pi$
$929$	7.80776i	0.256164i	−0.991764	+	0.128082i	$0.959118\pi$
$930$	0	0
$931$	47.6155i	1.56054i
$932$	−11.3996	−0.373407
$933$	0	0
$934$	− 40.6004i	− 1.32848i
$935$	−11.1231	−0.363764
$936$	0	0
$937$	−7.56155	−0.247025	−0.123513	−	0.992343i	$-0.539416\pi$
$937$	−7.56155	−0.247025	−0.123513	+	0.992343i	$0.539416\pi$
$938$	4.00000i	0.130605i
$939$	0	0
$940$	12.8769	0.419998
$941$	− 30.4924i	− 0.994025i	−0.867744	−	0.497012i	$-0.834430\pi$
$941$	− 30.4924i	− 0.994025i	0.867744	−	0.497012i	$-0.165570\pi$
$942$	0	0
$943$	3.12311i	0.101702i
$944$	13.4773i	0.438648i
$945$	0	0
$946$	−14.2462	−0.463184
$947$	− 38.7386i	− 1.25884i	−0.777067	−	0.629418i	$-0.783293\pi$
$947$	− 38.7386i	− 1.25884i	0.777067	−	0.629418i	$-0.216707\pi$
$948$	0	0
$949$	0	0
$950$	85.4773	2.77325
$951$	0	0
$952$	−2.13826	−0.0693014
$953$	30.9848	1.00370	0.501849	−	0.864955i	$-0.332653\pi$
$953$	30.9848	1.00370	0.501849	+	0.864955i	$0.332653\pi$
$954$	0	0
$955$	32.4924i	1.05143i
$956$	− 11.1231i	− 0.359747i
$957$	0	0
$958$	−16.0000	−0.516937
$959$	−3.12311	−0.100850
$960$	0	0
$961$	24.4384	0.788337
$962$	0	0
$963$	0	0
$964$	− 7.80776i	− 0.251471i
$965$	−48.0540	−1.54691
$966$	0	0
$967$	− 0.876894i	− 0.0281990i	−0.999901	−	0.0140995i	$-0.995512\pi$
$967$	− 0.876894i	− 0.0281990i	0.999901	−	0.0140995i	$-0.00448816\pi$
$968$	17.0691i	0.548623i
$969$	0	0
$970$	47.6155i	1.52884i
$971$	12.9848	0.416704	0.208352	−	0.978054i	$-0.433190\pi$
$971$	12.9848	0.416704	0.208352	+	0.978054i	$0.433190\pi$
$972$	0	0
$973$	10.0691i	0.322801i
$974$	−11.1231	−0.356407
$975$	0	0
$976$	−18.1619	−0.581349
$977$	− 61.1771i	− 1.95723i	−0.205705	−	0.978614i	$-0.565949\pi$
$977$	− 61.1771i	− 1.95723i	0.205705	−	0.978614i	$-0.434051\pi$
$978$	0	0
$979$	9.75379	0.311732
$980$	− 10.4384i	− 0.333444i
$981$	0	0
$982$	− 56.6004i	− 1.80619i
$983$	13.6155i	0.434268i	0.976142	+	0.217134i	$0.0696708\pi$
$983$	13.6155i	0.434268i	−0.976142	+	0.217134i	$0.930329\pi$
$984$	0	0
$985$	47.6155	1.51716
$986$	16.3002i	0.519104i
$987$	0	0
$988$	0	0
$989$	−9.12311	−0.290098
$990$	0	0
$991$	−50.3542	−1.59955	−0.799776	−	0.600298i	$-0.795049\pi$
$991$	−50.3542	−1.59955	−0.799776	+	0.600298i	$0.795049\pi$
$992$	6.24621	0.198317
$993$	0	0
$994$	− 12.2765i	− 0.389388i
$995$	− 78.9848i	− 2.50399i
$996$	0	0
$997$	−20.6155	−0.652900	−0.326450	−	0.945214i	$-0.605853\pi$
$997$	−20.6155	−0.652900	−0.326450	+	0.945214i	$0.605853\pi$
$998$	−7.01515	−0.222061
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.h.1351.3		4
3.2	odd	2		507.2.b.d.337.2		4
13.2	odd	12		117.2.g.c.100.1		4
13.5	odd	4		1521.2.a.g.1.2		2
13.6	odd	12		117.2.g.c.55.1		4
13.8	odd	4		1521.2.a.m.1.1		2
13.12	even	2	inner	1521.2.b.h.1351.2		4
39.2	even	12		39.2.e.b.22.2	yes	4
39.5	even	4		507.2.a.g.1.1		2
39.8	even	4		507.2.a.d.1.2		2
39.11	even	12		507.2.e.g.22.1		4
39.17	odd	6		507.2.j.g.361.3		8
39.20	even	12		507.2.e.g.484.1		4
39.23	odd	6		507.2.j.g.316.2		8
39.29	odd	6		507.2.j.g.316.3		8
39.32	even	12		39.2.e.b.16.2	✓	4
39.35	odd	6		507.2.j.g.361.2		8
39.38	odd	2		507.2.b.d.337.3		4
52.15	even	12		1872.2.t.r.1153.2		4
52.19	even	12		1872.2.t.r.289.2		4
156.47	odd	4		8112.2.a.bo.1.2		2
156.71	odd	12		624.2.q.h.289.1		4
156.83	odd	4		8112.2.a.bk.1.1		2
156.119	odd	12		624.2.q.h.529.1		4
195.2	odd	12		975.2.bb.i.724.3		8
195.32	odd	12		975.2.bb.i.874.2		8
195.119	even	12		975.2.i.k.451.1		4
195.149	even	12		975.2.i.k.601.1		4
195.158	odd	12		975.2.bb.i.724.2		8
195.188	odd	12		975.2.bb.i.874.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.b.16.2	✓	4	39.32	even	12
39.2.e.b.22.2	yes	4	39.2	even	12
117.2.g.c.55.1		4	13.6	odd	12
117.2.g.c.100.1		4	13.2	odd	12
507.2.a.d.1.2		2	39.8	even	4
507.2.a.g.1.1		2	39.5	even	4
507.2.b.d.337.2		4	3.2	odd	2
507.2.b.d.337.3		4	39.38	odd	2
507.2.e.g.22.1		4	39.11	even	12
507.2.e.g.484.1		4	39.20	even	12
507.2.j.g.316.2		8	39.23	odd	6
507.2.j.g.316.3		8	39.29	odd	6
507.2.j.g.361.2		8	39.35	odd	6
507.2.j.g.361.3		8	39.17	odd	6
624.2.q.h.289.1		4	156.71	odd	12
624.2.q.h.529.1		4	156.119	odd	12
975.2.i.k.451.1		4	195.119	even	12
975.2.i.k.601.1		4	195.149	even	12
975.2.bb.i.724.2		8	195.158	odd	12
975.2.bb.i.724.3		8	195.2	odd	12
975.2.bb.i.874.2		8	195.32	odd	12
975.2.bb.i.874.3		8	195.188	odd	12
1521.2.a.g.1.2		2	13.5	odd	4
1521.2.a.m.1.1		2	13.8	odd	4
1521.2.b.h.1351.2		4	13.12	even	2	inner
1521.2.b.h.1351.3		4	1.1	even	1	trivial
1872.2.t.r.289.2		4	52.19	even	12
1872.2.t.r.1153.2		4	52.15	even	12
8112.2.a.bk.1.1		2	156.83	odd	4
8112.2.a.bo.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.56155i q^{2} -0.438447 q^{4} +3.56155i q^{5} -0.561553i q^{7} +2.43845i q^{8} +O(q^{10})\)	\(q+1.56155i q^{2} -0.438447 q^{4} +3.56155i q^{5} -0.561553i q^{7} +2.43845i q^{8} -5.56155 q^{10} -2.00000i q^{11} +0.876894 q^{14} -4.68466 q^{16} -1.56155 q^{17} +7.12311i q^{19} -1.56155i q^{20} +3.12311 q^{22} +2.00000 q^{23} -7.68466 q^{25} +0.246211i q^{28} -6.68466 q^{29} +2.56155i q^{31} -2.43845i q^{32} -2.43845i q^{34} +2.00000 q^{35} -7.56155i q^{37} -11.1231 q^{38} -8.68466 q^{40} +1.56155i q^{41} -4.56155 q^{43} +0.876894i q^{44} +3.12311i q^{46} +8.24621i q^{47} +6.68466 q^{49} -12.0000i q^{50} +0.684658 q^{53} +7.12311 q^{55} +1.36932 q^{56} -10.4384i q^{58} -2.87689i q^{59} +3.87689 q^{61} -4.00000 q^{62} -5.56155 q^{64} +4.56155i q^{67} +0.684658 q^{68} +3.12311i q^{70} -14.0000i q^{71} +10.1231i q^{73} +11.8078 q^{74} -3.12311i q^{76} -1.12311 q^{77} +5.43845 q^{79} -16.6847i q^{80} -2.43845 q^{82} +0.876894i q^{83} -5.56155i q^{85} -7.12311i q^{86} +4.87689 q^{88} +4.87689i q^{89} -0.876894 q^{92} -12.8769 q^{94} -25.3693 q^{95} -8.56155i q^{97} +10.4384i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4}+O(q^{10}) \) 4 * q - 10 * q^4	\( 4 q - 10 q^{4} - 14 q^{10} + 20 q^{14} + 6 q^{16} + 2 q^{17} - 4 q^{22} + 8 q^{23} - 6 q^{25} - 2 q^{29} + 8 q^{35} - 28 q^{38} - 10 q^{40} - 10 q^{43} + 2 q^{49} - 22 q^{53} + 12 q^{55} - 44 q^{56} + 32 q^{61} - 16 q^{62} - 14 q^{64} - 22 q^{68} + 6 q^{74} + 12 q^{77} + 30 q^{79} - 18 q^{82} + 36 q^{88} - 20 q^{92} - 68 q^{94} - 52 q^{95}+O(q^{100}) \) 4 * q - 10 * q^4 - 14 * q^10 + 20 * q^14 + 6 * q^16 + 2 * q^17 - 4 * q^22 + 8 * q^23 - 6 * q^25 - 2 * q^29 + 8 * q^35 - 28 * q^38 - 10 * q^40 - 10 * q^43 + 2 * q^49 - 22 * q^53 + 12 * q^55 - 44 * q^56 + 32 * q^61 - 16 * q^62 - 14 * q^64 - 22 * q^68 + 6 * q^74 + 12 * q^77 + 30 * q^79 - 18 * q^82 + 36 * q^88 - 20 * q^92 - 68 * q^94 - 52 * q^95

Introduction

L-functions

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Varieties

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Database

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