LMFDB - Embedded newform 1425.2.c.a.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(799,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.c (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:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.a.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	\(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} -3.00000 q^{11} -2.00000i q^{12} -6.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} -3.00000i q^{17} -2.00000i q^{18} +1.00000 q^{19} +3.00000 q^{21} -6.00000i q^{22} +4.00000i q^{23} +12.0000 q^{26} -1.00000i q^{27} +6.00000i q^{28} +10.0000 q^{29} +2.00000 q^{31} -8.00000i q^{32} -3.00000i q^{33} +6.00000 q^{34} +2.00000 q^{36} -8.00000i q^{37} +2.00000i q^{38} +6.00000 q^{39} -8.00000 q^{41} +6.00000i q^{42} -1.00000i q^{43} +6.00000 q^{44} -8.00000 q^{46} -3.00000i q^{47} -4.00000i q^{48} -2.00000 q^{49} +3.00000 q^{51} +12.0000i q^{52} -6.00000i q^{53} +2.00000 q^{54} +1.00000i q^{57} +20.0000i q^{58} +7.00000 q^{61} +4.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} +6.00000 q^{66} -8.00000i q^{67} +6.00000i q^{68} -4.00000 q^{69} +12.0000 q^{71} -11.0000i q^{73} +16.0000 q^{74} -2.00000 q^{76} +9.00000i q^{77} +12.0000i q^{78} +1.00000 q^{81} -16.0000i q^{82} +4.00000i q^{83} -6.00000 q^{84} +2.00000 q^{86} +10.0000i q^{87} -10.0000 q^{89} -18.0000 q^{91} -8.00000i q^{92} +2.00000i q^{93} +6.00000 q^{94} +8.00000 q^{96} +2.00000i q^{97} -4.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9	$ 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9} - 6 q^{11} + 12 q^{14} - 8 q^{16} + 2 q^{19} + 6 q^{21} + 24 q^{26} + 20 q^{29} + 4 q^{31} + 12 q^{34} + 4 q^{36} + 12 q^{39} - 16 q^{41} + 12 q^{44} - 16 q^{46} - 4 q^{49} + 6 q^{51} + 4 q^{54} + 14 q^{61} + 16 q^{64} + 12 q^{66} - 8 q^{69} + 24 q^{71} + 32 q^{74} - 4 q^{76} + 2 q^{81} - 12 q^{84} + 4 q^{86} - 20 q^{89} - 36 q^{91} + 12 q^{94} + 16 q^{96} + 6 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9 - 6 * q^11 + 12 * q^14 - 8 * q^16 + 2 * q^19 + 6 * q^21 + 24 * q^26 + 20 * q^29 + 4 * q^31 + 12 * q^34 + 4 * q^36 + 12 * q^39 - 16 * q^41 + 12 * q^44 - 16 * q^46 - 4 * q^49 + 6 * q^51 + 4 * q^54 + 14 * q^61 + 16 * q^64 + 12 * q^66 - 8 * q^69 + 24 * q^71 + 32 * q^74 - 4 * q^76 + 2 * q^81 - 12 * q^84 + 4 * q^86 - 20 * q^89 - 36 * q^91 + 12 * q^94 + 16 * q^96 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$.

$n$	$476$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$	2.00000i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	− 2.00000i	− 0.577350i
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	6.00000	1.60357
$15$	0	0
$16$	−4.00000	−1.00000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	− 2.00000i	− 0.471405i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.00000	0.654654
$22$	− 6.00000i	− 1.27920i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	12.0000	2.35339
$27$	− 1.00000i	− 0.192450i
$28$	6.00000i	1.13389i
$29$	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
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	− 8.00000i	− 1.41421i
$33$	− 3.00000i	− 0.522233i
$34$	6.00000	1.02899
$35$	0	0
$36$	2.00000	0.333333
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	2.00000i	0.324443i
$39$	6.00000	0.960769
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	6.00000i	0.925820i
$43$	− 1.00000i	− 0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	− 1.00000i	− 0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	−8.00000	−1.17954
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	− 4.00000i	− 0.577350i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	3.00000	0.420084
$52$	12.0000i	1.66410i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	2.00000	0.272166
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	20.0000i	2.62613i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	4.00000i	0.508001i
$63$	3.00000i	0.377964i
$64$	8.00000	1.00000
$65$	0	0
$66$	6.00000	0.738549
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	6.00000i	0.727607i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	− 11.0000i	− 1.28745i	−0.765256	−	0.643726i	$-0.777388\pi$
$73$	− 11.0000i	− 1.28745i	0.765256	−	0.643726i	$-0.222612\pi$
$74$	16.0000	1.85996
$75$	0	0
$76$	−2.00000	−0.229416
$77$	9.00000i	1.02565i
$78$	12.0000i	1.35873i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 16.0000i	− 1.76690i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	−6.00000	−0.654654
$85$	0	0
$86$	2.00000	0.215666
$87$	10.0000i	1.07211i
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−18.0000	−1.88691
$92$	− 8.00000i	− 0.834058i
$93$	2.00000i	0.207390i
$94$	6.00000	0.618853
$95$	0	0
$96$	8.00000	0.816497
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 4.00000i	− 0.404061i
$99$	3.00000	0.301511
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	6.00000i	0.594089i
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	0	0
$105$	0	0
$106$	12.0000	1.16554
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	2.00000i	0.192450i
$109$	−20.0000	−1.91565	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000	−1.91565	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	12.0000i	1.13389i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	−20.0000	−1.85695
$117$	6.00000i	0.554700i
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	−2.00000	−0.181818
$122$	14.0000i	1.26750i
$123$	− 8.00000i	− 0.721336i
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−6.00000	−0.534522
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	0	0
$129$	1.00000	0.0880451
$130$	0	0
$131$	−13.0000	−1.13582	−0.567908	−	0.823092i	$-0.692247\pi$
$131$	−13.0000	−1.13582	−0.567908	+	0.823092i	$0.692247\pi$
$132$	6.00000i	0.522233i
$133$	− 3.00000i	− 0.260133i
$134$	16.0000	1.38219
$135$	0	0
$136$	0	0
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	− 8.00000i	− 0.681005i
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	24.0000i	2.01404i
$143$	18.0000i	1.50524i
$144$	4.00000	0.333333
$145$	0	0
$146$	22.0000	1.82073
$147$	− 2.00000i	− 0.164957i
$148$	16.0000i	1.31519i
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	3.00000i	0.242536i
$154$	−18.0000	−1.45048
$155$	0	0
$156$	−12.0000	−0.960769
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	12.0000	0.945732
$162$	2.00000i	0.157135i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	16.0000	1.24939
$165$	0	0
$166$	−8.00000	−0.620920
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	2.00000i	0.152499i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	−20.0000	−1.51620
$175$	0	0
$176$	12.0000	0.904534
$177$	0	0
$178$	− 20.0000i	− 1.49906i
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 36.0000i	− 2.66850i
$183$	7.00000i	0.517455i
$184$	0	0
$185$	0	0
$186$	−4.00000	−0.293294
$187$	9.00000i	0.658145i
$188$	6.00000i	0.437595i
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	8.00000i	0.577350i
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	4.00000	0.285714
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	6.00000i	0.426401i
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	4.00000i	0.281439i
$203$	− 30.0000i	− 2.10559i
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−28.0000	−1.95085
$207$	− 4.00000i	− 0.278019i
$208$	24.0000i	1.66410i
$209$	−3.00000	−0.207514
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	12.0000i	0.824163i
$213$	12.0000i	0.822226i
$214$	−4.00000	−0.273434
$215$	0	0
$216$	0	0
$217$	− 6.00000i	− 0.407307i
$218$	− 40.0000i	− 2.70914i
$219$	11.0000	0.743311
$220$	0	0
$221$	−18.0000	−1.21081
$222$	16.0000i	1.07385i
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	−24.0000	−1.60357
$225$	0	0
$226$	12.0000	0.798228
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	− 2.00000i	− 0.132453i
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	0	0
$231$	−9.00000	−0.592157
$232$	0	0
$233$	− 11.0000i	− 0.720634i	−0.932830	−	0.360317i	$-0.882669\pi$
$233$	− 11.0000i	− 0.720634i	0.932830	−	0.360317i	$-0.117331\pi$
$234$	−12.0000	−0.784465
$235$	0	0
$236$	0	0
$237$	0	0
$238$	− 18.0000i	− 1.16677i
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	− 4.00000i	− 0.257130i
$243$	1.00000i	0.0641500i
$244$	−14.0000	−0.896258
$245$	0	0
$246$	16.0000	1.02012
$247$	− 6.00000i	− 0.381771i
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	27.0000	1.70422	0.852112	−	0.523359i	$-0.175321\pi$
$251$	27.0000	1.70422	0.852112	+	0.523359i	$0.175321\pi$
$252$	− 6.00000i	− 0.377964i
$253$	− 12.0000i	− 0.754434i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	16.0000	1.00000
$257$	− 8.00000i	− 0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$	− 8.00000i	− 0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$	2.00000i	0.124515i
$259$	−24.0000	−1.49129
$260$	0	0
$261$	−10.0000	−0.618984
$262$	− 26.0000i	− 1.60629i
$263$	− 21.0000i	− 1.29492i	−0.762101	−	0.647458i	$-0.775832\pi$
$263$	− 21.0000i	− 1.29492i	0.762101	−	0.647458i	$-0.224168\pi$
$264$	0	0
$265$	0	0
$266$	6.00000	0.367884
$267$	− 10.0000i	− 0.611990i
$268$	16.0000i	0.977356i
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	12.0000i	0.727607i
$273$	− 18.0000i	− 1.08941i
$274$	6.00000	0.362473
$275$	0	0
$276$	8.00000	0.481543
$277$	− 13.0000i	− 0.781094i	−0.920583	−	0.390547i	$-0.872286\pi$
$277$	− 13.0000i	− 0.781094i	0.920583	−	0.390547i	$-0.127714\pi$
$278$	10.0000i	0.599760i
$279$	−2.00000	−0.119737
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	6.00000i	0.357295i
$283$	19.0000i	1.12943i	0.825285	+	0.564716i	$0.191014\pi$
$283$	19.0000i	1.12943i	−0.825285	+	0.564716i	$0.808986\pi$
$284$	−24.0000	−1.42414
$285$	0	0
$286$	−36.0000	−2.12872
$287$	24.0000i	1.41668i
$288$	8.00000i	0.471405i
$289$	8.00000	0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	22.0000i	1.28745i
$293$	4.00000i	0.233682i	0.993151	+	0.116841i	$0.0372769\pi$
$293$	4.00000i	0.233682i	−0.993151	+	0.116841i	$0.962723\pi$
$294$	4.00000	0.233285
$295$	0	0
$296$	0	0
$297$	3.00000i	0.174078i
$298$	− 30.0000i	− 1.73785i
$299$	24.0000	1.38796
$300$	0	0
$301$	−3.00000	−0.172917
$302$	− 16.0000i	− 0.920697i
$303$	2.00000i	0.114897i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−6.00000	−0.342997
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	− 18.0000i	− 1.02565i
$309$	−14.0000	−0.796432
$310$	0	0
$311$	7.00000	0.396934	0.198467	−	0.980108i	$-0.436404\pi$
$311$	7.00000	0.396934	0.198467	+	0.980108i	$0.436404\pi$
$312$	0	0
$313$	14.0000i	0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$	14.0000i	0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	0	0
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	12.0000i	0.672927i
$319$	−30.0000	−1.67968
$320$	0	0
$321$	−2.00000	−0.111629
$322$	24.0000i	1.33747i
$323$	− 3.00000i	− 0.166924i
$324$	−2.00000	−0.111111
$325$	0	0
$326$	32.0000	1.77232
$327$	− 20.0000i	− 1.10600i
$328$	0	0
$329$	−9.00000	−0.496186
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	− 8.00000i	− 0.439057i
$333$	8.00000i	0.438397i
$334$	36.0000	1.96983
$335$	0	0
$336$	−12.0000	−0.654654
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	− 46.0000i	− 2.50207i
$339$	6.00000	0.325875
$340$	0	0
$341$	−6.00000	−0.324918
$342$	− 2.00000i	− 0.108148i
$343$	− 15.0000i	− 0.809924i
$344$	0	0
$345$	0	0
$346$	−28.0000	−1.50529
$347$	− 3.00000i	− 0.161048i	−0.996753	−	0.0805242i	$-0.974341\pi$
$347$	− 3.00000i	− 0.161048i	0.996753	−	0.0805242i	$-0.0256594\pi$
$348$	− 20.0000i	− 1.07211i
$349$	−25.0000	−1.33822	−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000	−1.33822	−0.669110	+	0.743164i	$0.733324\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	24.0000i	1.27920i
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	0	0
$355$	0	0
$356$	20.0000	1.06000
$357$	− 9.00000i	− 0.476331i
$358$	20.0000i	1.05703i
$359$	−25.0000	−1.31945	−0.659725	−	0.751507i	$-0.729327\pi$
$359$	−25.0000	−1.31945	−0.659725	+	0.751507i	$0.729327\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	4.00000i	0.210235i
$363$	− 2.00000i	− 0.104973i
$364$	36.0000	1.88691
$365$	0	0
$366$	−14.0000	−0.731792
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	− 16.0000i	− 0.834058i
$369$	8.00000	0.416463
$370$	0	0
$371$	−18.0000	−0.934513
$372$	− 4.00000i	− 0.207390i
$373$	− 16.0000i	− 0.828449i	−0.910175	−	0.414224i	$-0.864053\pi$
$373$	− 16.0000i	− 0.828449i	0.910175	−	0.414224i	$-0.135947\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	0	0
$377$	− 60.0000i	− 3.09016i
$378$	− 6.00000i	− 0.308607i
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	− 6.00000i	− 0.306987i
$383$	14.0000i	0.715367i	0.933843	+	0.357683i	$0.116433\pi$
$383$	14.0000i	0.715367i	−0.933843	+	0.357683i	$0.883567\pi$
$384$	0	0
$385$	0	0
$386$	−8.00000	−0.407189
$387$	1.00000i	0.0508329i
$388$	− 4.00000i	− 0.203069i
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	− 13.0000i	− 0.655763i
$394$	−4.00000	−0.201517
$395$	0	0
$396$	−6.00000	−0.301511
$397$	7.00000i	0.351320i	0.984451	+	0.175660i	$0.0562059\pi$
$397$	7.00000i	0.351320i	−0.984451	+	0.175660i	$0.943794\pi$
$398$	10.0000i	0.501255i
$399$	3.00000	0.150188
$400$	0	0
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	16.0000i	0.798007i
$403$	− 12.0000i	− 0.597763i
$404$	−4.00000	−0.199007
$405$	0	0
$406$	60.0000	2.97775
$407$	24.0000i	1.18964i
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	− 28.0000i	− 1.37946i
$413$	0	0
$414$	8.00000	0.393179
$415$	0	0
$416$	−48.0000	−2.35339
$417$	5.00000i	0.244851i
$418$	− 6.00000i	− 0.293470i
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	− 56.0000i	− 2.72604i
$423$	3.00000i	0.145865i
$424$	0	0
$425$	0	0
$426$	−24.0000	−1.16280
$427$	− 21.0000i	− 1.01626i
$428$	− 4.00000i	− 0.193347i
$429$	−18.0000	−0.869048
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	4.00000i	0.192450i
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	12.0000	0.576018
$435$	0	0
$436$	40.0000	1.91565
$437$	4.00000i	0.191346i
$438$	22.0000i	1.05120i
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	− 36.0000i	− 1.71235i
$443$	39.0000i	1.85295i	0.376361	+	0.926473i	$0.377175\pi$
$443$	39.0000i	1.85295i	−0.376361	+	0.926473i	$0.622825\pi$
$444$	−16.0000	−0.759326
$445$	0	0
$446$	−8.00000	−0.378811
$447$	− 15.0000i	− 0.709476i
$448$	− 24.0000i	− 1.13389i
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	12.0000i	0.564433i
$453$	− 8.00000i	− 0.375873i
$454$	36.0000	1.68956
$455$	0	0
$456$	0	0
$457$	− 3.00000i	− 0.140334i	−0.997535	−	0.0701670i	$-0.977647\pi$
$457$	− 3.00000i	− 0.140334i	0.997535	−	0.0701670i	$-0.0223532\pi$
$458$	30.0000i	1.40181i
$459$	−3.00000	−0.140028
$460$	0	0
$461$	−33.0000	−1.53696	−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000	−1.53696	−0.768482	+	0.639872i	$0.778987\pi$
$462$	− 18.0000i	− 0.837436i
$463$	− 31.0000i	− 1.44069i	−0.693615	−	0.720346i	$-0.743983\pi$
$463$	− 31.0000i	− 1.44069i	0.693615	−	0.720346i	$-0.256017\pi$
$464$	−40.0000	−1.85695
$465$	0	0
$466$	22.0000	1.01913
$467$	17.0000i	0.786666i	0.919396	+	0.393333i	$0.128678\pi$
$467$	17.0000i	0.786666i	−0.919396	+	0.393333i	$0.871322\pi$
$468$	− 12.0000i	− 0.554700i
$469$	−24.0000	−1.10822
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	3.00000i	0.137940i
$474$	0	0
$475$	0	0
$476$	18.0000	0.825029
$477$	6.00000i	0.274721i
$478$	30.0000i	1.37217i
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	0	0
$481$	−48.0000	−2.18861
$482$	24.0000i	1.09317i
$483$	12.0000i	0.546019i
$484$	4.00000	0.181818
$485$	0	0
$486$	−2.00000	−0.0907218
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	16.0000i	0.721336i
$493$	− 30.0000i	− 1.35113i
$494$	12.0000	0.539906
$495$	0	0
$496$	−8.00000	−0.359211
$497$	− 36.0000i	− 1.61482i
$498$	− 8.00000i	− 0.358489i
$499$	35.0000	1.56682	0.783408	−	0.621508i	$-0.213480\pi$
$499$	35.0000	1.56682	0.783408	+	0.621508i	$0.213480\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	54.0000i	2.41014i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	0	0
$505$	0	0
$506$	24.0000	1.06693
$507$	− 23.0000i	− 1.02147i
$508$	− 4.00000i	− 0.177471i
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	−33.0000	−1.45983
$512$	32.0000i	1.41421i
$513$	− 1.00000i	− 0.0441511i
$514$	16.0000	0.705730
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	9.00000i	0.395820i
$518$	− 48.0000i	− 2.10900i
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	− 20.0000i	− 0.875376i
$523$	14.0000i	0.612177i	0.952003	+	0.306089i	$0.0990204\pi$
$523$	14.0000i	0.612177i	−0.952003	+	0.306089i	$0.900980\pi$
$524$	26.0000	1.13582
$525$	0	0
$526$	42.0000	1.83129
$527$	− 6.00000i	− 0.261364i
$528$	12.0000i	0.522233i
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	6.00000i	0.260133i
$533$	48.0000i	2.07911i
$534$	20.0000	0.865485
$535$	0	0
$536$	0	0
$537$	10.0000i	0.431532i
$538$	60.0000i	2.58678i
$539$	6.00000	0.258438
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	24.0000i	1.03089i
$543$	2.00000i	0.0858282i
$544$	−24.0000	−1.02899
$545$	0	0
$546$	36.0000	1.54066
$547$	2.00000i	0.0855138i	0.999086	+	0.0427569i	$0.0136141\pi$
$547$	2.00000i	0.0855138i	−0.999086	+	0.0427569i	$0.986386\pi$
$548$	6.00000i	0.256307i
$549$	−7.00000	−0.298753
$550$	0	0
$551$	10.0000	0.426014
$552$	0	0
$553$	0	0
$554$	26.0000	1.10463
$555$	0	0
$556$	−10.0000	−0.424094
$557$	− 3.00000i	− 0.127114i	−0.997978	−	0.0635570i	$-0.979756\pi$
$557$	− 3.00000i	− 0.127114i	0.997978	−	0.0635570i	$-0.0202445\pi$
$558$	− 4.00000i	− 0.169334i
$559$	−6.00000	−0.253773
$560$	0	0
$561$	−9.00000	−0.379980
$562$	4.00000i	0.168730i
$563$	44.0000i	1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$	44.0000i	1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−38.0000	−1.59726
$567$	− 3.00000i	− 0.125988i
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	− 36.0000i	− 1.50524i
$573$	− 3.00000i	− 0.125327i
$574$	−48.0000	−2.00348
$575$	0	0
$576$	−8.00000	−0.333333
$577$	− 3.00000i	− 0.124892i	−0.998048	−	0.0624458i	$-0.980110\pi$
$577$	− 3.00000i	− 0.124892i	0.998048	−	0.0624458i	$-0.0198901\pi$
$578$	16.0000i	0.665512i
$579$	−4.00000	−0.166234
$580$	0	0
$581$	12.0000	0.497844
$582$	− 4.00000i	− 0.165805i
$583$	18.0000i	0.745484i
$584$	0	0
$585$	0	0
$586$	−8.00000	−0.330477
$587$	37.0000i	1.52715i	0.645717	+	0.763577i	$0.276559\pi$
$587$	37.0000i	1.52715i	−0.645717	+	0.763577i	$0.723441\pi$
$588$	4.00000i	0.164957i
$589$	2.00000	0.0824086
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	32.0000i	1.31519i
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	−6.00000	−0.246183
$595$	0	0
$596$	30.0000	1.22885
$597$	5.00000i	0.204636i
$598$	48.0000i	1.96287i
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	− 6.00000i	− 0.244542i
$603$	8.00000i	0.325785i
$604$	16.0000	0.651031
$605$	0	0
$606$	−4.00000	−0.162489
$607$	− 18.0000i	− 0.730597i	−0.930890	−	0.365299i	$-0.880967\pi$
$607$	− 18.0000i	− 0.730597i	0.930890	−	0.365299i	$-0.119033\pi$
$608$	− 8.00000i	− 0.324443i
$609$	30.0000	1.21566
$610$	0	0
$611$	−18.0000	−0.728202
$612$	− 6.00000i	− 0.242536i
$613$	9.00000i	0.363507i	0.983344	+	0.181753i	$0.0581772\pi$
$613$	9.00000i	0.363507i	−0.983344	+	0.181753i	$0.941823\pi$
$614$	−24.0000	−0.968561
$615$	0	0
$616$	0	0
$617$	− 23.0000i	− 0.925945i	−0.886373	−	0.462973i	$-0.846783\pi$
$617$	− 23.0000i	− 0.925945i	0.886373	−	0.462973i	$-0.153217\pi$
$618$	− 28.0000i	− 1.12633i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	14.0000i	0.561349i
$623$	30.0000i	1.20192i
$624$	−24.0000	−0.960769
$625$	0	0
$626$	−28.0000	−1.11911
$627$	− 3.00000i	− 0.119808i
$628$	− 4.00000i	− 0.159617i
$629$	−24.0000	−0.956943
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	− 28.0000i	− 1.11290i
$634$	−24.0000	−0.953162
$635$	0	0
$636$	−12.0000	−0.475831
$637$	12.0000i	0.475457i
$638$	− 60.0000i	− 2.37542i
$639$	−12.0000	−0.474713
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	− 4.00000i	− 0.157867i
$643$	− 1.00000i	− 0.0394362i	−0.999806	−	0.0197181i	$-0.993723\pi$
$643$	− 1.00000i	− 0.0394362i	0.999806	−	0.0197181i	$-0.00627687\pi$
$644$	−24.0000	−0.945732
$645$	0	0
$646$	6.00000	0.236067
$647$	27.0000i	1.06148i	0.847535	+	0.530740i	$0.178086\pi$
$647$	27.0000i	1.06148i	−0.847535	+	0.530740i	$0.821914\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	6.00000	0.235159
$652$	32.0000i	1.25322i
$653$	− 1.00000i	− 0.0391330i	−0.999809	−	0.0195665i	$-0.993771\pi$
$653$	− 1.00000i	− 0.0391330i	0.999809	−	0.0195665i	$-0.00622861\pi$
$654$	40.0000	1.56412
$655$	0	0
$656$	32.0000	1.24939
$657$	11.0000i	0.429151i
$658$	− 18.0000i	− 0.701713i
$659$	−10.0000	−0.389545	−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000	−0.389545	−0.194772	+	0.980848i	$0.562397\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	24.0000i	0.932786i
$663$	− 18.0000i	− 0.699062i
$664$	0	0
$665$	0	0
$666$	−16.0000	−0.619987
$667$	40.0000i	1.54881i
$668$	36.0000i	1.39288i
$669$	−4.00000	−0.154649
$670$	0	0
$671$	−21.0000	−0.810696
$672$	− 24.0000i	− 0.925820i
$673$	− 16.0000i	− 0.616755i	−0.951264	−	0.308377i	$-0.900214\pi$
$673$	− 16.0000i	− 0.616755i	0.951264	−	0.308377i	$-0.0997859\pi$
$674$	−44.0000	−1.69482
$675$	0	0
$676$	46.0000	1.76923
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	12.0000i	0.460857i
$679$	6.00000	0.230259
$680$	0	0
$681$	18.0000	0.689761
$682$	− 12.0000i	− 0.459504i
$683$	− 6.00000i	− 0.229584i	−0.993390	−	0.114792i	$-0.963380\pi$
$683$	− 6.00000i	− 0.229584i	0.993390	−	0.114792i	$-0.0366201\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	30.0000	1.14541
$687$	15.0000i	0.572286i
$688$	4.00000i	0.152499i
$689$	−36.0000	−1.37149
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	− 28.0000i	− 1.06440i
$693$	− 9.00000i	− 0.341882i
$694$	6.00000	0.227757
$695$	0	0
$696$	0	0
$697$	24.0000i	0.909065i
$698$	− 50.0000i	− 1.89253i
$699$	11.0000	0.416058
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	− 12.0000i	− 0.452911i
$703$	− 8.00000i	− 0.301726i
$704$	−24.0000	−0.904534
$705$	0	0
$706$	−28.0000	−1.05379
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.00000i	0.299602i
$714$	18.0000	0.673633
$715$	0	0
$716$	−20.0000	−0.747435
$717$	15.0000i	0.560185i
$718$	− 50.0000i	− 1.86598i
$719$	35.0000	1.30528	0.652640	−	0.757668i	$-0.273661\pi$
$719$	35.0000	1.30528	0.652640	+	0.757668i	$0.273661\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	2.00000i	0.0744323i
$723$	12.0000i	0.446285i
$724$	−4.00000	−0.148659
$725$	0	0
$726$	4.00000	0.148454
$727$	7.00000i	0.259616i	0.991539	+	0.129808i	$0.0414360\pi$
$727$	7.00000i	0.259616i	−0.991539	+	0.129808i	$0.958564\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−3.00000	−0.110959
$732$	− 14.0000i	− 0.517455i
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	32.0000	1.17954
$737$	24.0000i	0.884051i
$738$	16.0000i	0.588968i
$739$	45.0000	1.65535	0.827676	−	0.561206i	$-0.189663\pi$
$739$	45.0000	1.65535	0.827676	+	0.561206i	$0.189663\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	− 36.0000i	− 1.32160i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	32.0000	1.17160
$747$	− 4.00000i	− 0.146352i
$748$	− 18.0000i	− 0.658145i
$749$	6.00000	0.219235
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	12.0000i	0.437595i
$753$	27.0000i	0.983935i
$754$	120.000	4.37014
$755$	0	0
$756$	6.00000	0.218218
$757$	− 23.0000i	− 0.835949i	−0.908459	−	0.417975i	$-0.862740\pi$
$757$	− 23.0000i	− 0.835949i	0.908459	−	0.417975i	$-0.137260\pi$
$758$	60.0000i	2.17930i
$759$	12.0000	0.435572
$760$	0	0
$761$	−13.0000	−0.471250	−0.235625	−	0.971844i	$-0.575714\pi$
$761$	−13.0000	−0.471250	−0.235625	+	0.971844i	$0.575714\pi$
$762$	− 4.00000i	− 0.144905i
$763$	60.0000i	2.17215i
$764$	6.00000	0.217072
$765$	0	0
$766$	−28.0000	−1.01168
$767$	0	0
$768$	16.0000i	0.577350i
$769$	45.0000	1.62274	0.811371	−	0.584532i	$-0.198722\pi$
$769$	45.0000	1.62274	0.811371	+	0.584532i	$0.198722\pi$
$770$	0	0
$771$	8.00000	0.288113
$772$	− 8.00000i	− 0.287926i
$773$	− 36.0000i	− 1.29483i	−0.762138	−	0.647415i	$-0.775850\pi$
$773$	− 36.0000i	− 1.29483i	0.762138	−	0.647415i	$-0.224150\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	0	0
$777$	− 24.0000i	− 0.860995i
$778$	30.0000i	1.07555i
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−36.0000	−1.28818
$782$	24.0000i	0.858238i
$783$	− 10.0000i	− 0.357371i
$784$	8.00000	0.285714
$785$	0	0
$786$	26.0000	0.927389
$787$	− 8.00000i	− 0.285169i	−0.989783	−	0.142585i	$-0.954459\pi$
$787$	− 8.00000i	− 0.285169i	0.989783	−	0.142585i	$-0.0455413\pi$
$788$	− 4.00000i	− 0.142494i
$789$	21.0000	0.747620
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	− 42.0000i	− 1.49146i
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−10.0000	−0.354441
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	6.00000i	0.212398i
$799$	−9.00000	−0.318397
$800$	0	0
$801$	10.0000	0.353333
$802$	− 56.0000i	− 1.97743i
$803$	33.0000i	1.16454i
$804$	−16.0000	−0.564276
$805$	0	0
$806$	24.0000	0.845364
$807$	30.0000i	1.05605i
$808$	0	0
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	60.0000i	2.10559i
$813$	12.0000i	0.420858i
$814$	−48.0000	−1.68240
$815$	0	0
$816$	−12.0000	−0.420084
$817$	− 1.00000i	− 0.0349856i
$818$	− 20.0000i	− 0.699284i
$819$	18.0000	0.628971
$820$	0	0
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	6.00000i	0.209274i
$823$	19.0000i	0.662298i	0.943578	+	0.331149i	$0.107436\pi$
$823$	19.0000i	0.662298i	−0.943578	+	0.331149i	$0.892564\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	52.0000i	1.80822i	0.427303	+	0.904109i	$0.359464\pi$
$827$	52.0000i	1.80822i	−0.427303	+	0.904109i	$0.640536\pi$
$828$	8.00000i	0.278019i
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	13.0000	0.450965
$832$	− 48.0000i	− 1.66410i
$833$	6.00000i	0.207888i
$834$	−10.0000	−0.346272
$835$	0	0
$836$	6.00000	0.207514
$837$	− 2.00000i	− 0.0691301i
$838$	− 40.0000i	− 1.38178i
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	4.00000i	0.137849i
$843$	2.00000i	0.0688837i
$844$	56.0000	1.92760
$845$	0	0
$846$	−6.00000	−0.206284
$847$	6.00000i	0.206162i
$848$	24.0000i	0.824163i
$849$	−19.0000	−0.652078
$850$	0	0
$851$	32.0000	1.09695
$852$	− 24.0000i	− 0.822226i
$853$	34.0000i	1.16414i	0.813139	+	0.582069i	$0.197757\pi$
$853$	34.0000i	1.16414i	−0.813139	+	0.582069i	$0.802243\pi$
$854$	42.0000	1.43721
$855$	0	0
$856$	0	0
$857$	− 48.0000i	− 1.63965i	−0.572615	−	0.819824i	$-0.694071\pi$
$857$	− 48.0000i	− 1.63965i	0.572615	−	0.819824i	$-0.305929\pi$
$858$	− 36.0000i	− 1.22902i
$859$	−35.0000	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$859$	−35.0000	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	− 36.0000i	− 1.22616i
$863$	44.0000i	1.49778i	0.662696	+	0.748889i	$0.269412\pi$
$863$	44.0000i	1.49778i	−0.662696	+	0.748889i	$0.730588\pi$
$864$	−8.00000	−0.272166
$865$	0	0
$866$	52.0000	1.76703
$867$	8.00000i	0.271694i
$868$	12.0000i	0.407307i
$869$	0	0
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	− 2.00000i	− 0.0676897i
$874$	−8.00000	−0.270604
$875$	0	0
$876$	−22.0000	−0.743311
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	− 20.0000i	− 0.674967i
$879$	−4.00000	−0.134917
$880$	0	0
$881$	7.00000	0.235836	0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000	0.235836	0.117918	+	0.993023i	$0.462378\pi$
$882$	4.00000i	0.134687i
$883$	− 21.0000i	− 0.706706i	−0.935490	−	0.353353i	$-0.885041\pi$
$883$	− 21.0000i	− 0.706706i	0.935490	−	0.353353i	$-0.114959\pi$
$884$	36.0000	1.21081
$885$	0	0
$886$	−78.0000	−2.62046
$887$	52.0000i	1.74599i	0.487730	+	0.872995i	$0.337825\pi$
$887$	52.0000i	1.74599i	−0.487730	+	0.872995i	$0.662175\pi$
$888$	0	0
$889$	6.00000	0.201234
$890$	0	0
$891$	−3.00000	−0.100504
$892$	− 8.00000i	− 0.267860i
$893$	− 3.00000i	− 0.100391i
$894$	30.0000	1.00335
$895$	0	0
$896$	0	0
$897$	24.0000i	0.801337i
$898$	40.0000i	1.33482i
$899$	20.0000	0.667037
$900$	0	0
$901$	−18.0000	−0.599667
$902$	48.0000i	1.59823i
$903$	− 3.00000i	− 0.0998337i
$904$	0	0
$905$	0	0
$906$	16.0000	0.531564
$907$	2.00000i	0.0664089i	0.999449	+	0.0332045i	$0.0105712\pi$
$907$	2.00000i	0.0664089i	−0.999449	+	0.0332045i	$0.989429\pi$
$908$	36.0000i	1.19470i
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	2.00000	0.0662630	0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000	0.0662630	0.0331315	+	0.999451i	$0.489452\pi$
$912$	− 4.00000i	− 0.132453i
$913$	− 12.0000i	− 0.397142i
$914$	6.00000	0.198462
$915$	0	0
$916$	−30.0000	−0.991228
$917$	39.0000i	1.28789i
$918$	− 6.00000i	− 0.198030i
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	− 66.0000i	− 2.17359i
$923$	− 72.0000i	− 2.36991i
$924$	18.0000	0.592157
$925$	0	0
$926$	62.0000	2.03745
$927$	− 14.0000i	− 0.459820i
$928$	− 80.0000i	− 2.62613i
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	22.0000i	0.720634i
$933$	7.00000i	0.229170i
$934$	−34.0000	−1.11251
$935$	0	0
$936$	0	0
$937$	− 53.0000i	− 1.73143i	−0.500533	−	0.865717i	$-0.666863\pi$
$937$	− 53.0000i	− 1.73143i	0.500533	−	0.865717i	$-0.333137\pi$
$938$	− 48.0000i	− 1.56726i
$939$	−14.0000	−0.456873
$940$	0	0
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	− 4.00000i	− 0.130327i
$943$	− 32.0000i	− 1.04206i
$944$	0	0
$945$	0	0
$946$	−6.00000	−0.195077
$947$	− 28.0000i	− 0.909878i	−0.890523	−	0.454939i	$-0.849661\pi$
$947$	− 28.0000i	− 0.909878i	0.890523	−	0.454939i	$-0.150339\pi$
$948$	0	0
$949$	−66.0000	−2.14245
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	− 16.0000i	− 0.518291i	−0.965838	−	0.259145i	$-0.916559\pi$
$953$	− 16.0000i	− 0.518291i	0.965838	−	0.259145i	$-0.0834409\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	−30.0000	−0.970269
$957$	− 30.0000i	− 0.969762i
$958$	80.0000i	2.58468i
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 96.0000i	− 3.09516i
$963$	− 2.00000i	− 0.0644491i
$964$	−24.0000	−0.772988
$965$	0	0
$966$	−24.0000	−0.772187
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	0	0
$969$	3.00000	0.0963739
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	− 2.00000i	− 0.0641500i
$973$	− 15.0000i	− 0.480878i
$974$	16.0000	0.512673
$975$	0	0
$976$	−28.0000	−0.896258
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	32.0000i	1.02325i
$979$	30.0000	0.958804
$980$	0	0
$981$	20.0000	0.638551
$982$	− 16.0000i	− 0.510581i
$983$	4.00000i	0.127580i	0.997963	+	0.0637901i	$0.0203188\pi$
$983$	4.00000i	0.127580i	−0.997963	+	0.0637901i	$0.979681\pi$
$984$	0	0
$985$	0	0
$986$	60.0000	1.91079
$987$	− 9.00000i	− 0.286473i
$988$	12.0000i	0.381771i
$989$	4.00000	0.127193
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	− 16.0000i	− 0.508001i
$993$	12.0000i	0.380808i
$994$	72.0000	2.28370
$995$	0	0
$996$	8.00000	0.253490
$997$	7.00000i	0.221692i	0.993838	+	0.110846i	$0.0353561\pi$
$997$	7.00000i	0.221692i	−0.993838	+	0.110846i	$0.964644\pi$
$998$	70.0000i	2.21581i
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.a.799.2		2
5.2	odd	4		57.2.a.b.1.1	✓	1
5.3	odd	4		1425.2.a.i.1.1		1
5.4	even	2	inner	1425.2.c.a.799.1		2
15.2	even	4		171.2.a.c.1.1		1
15.8	even	4		4275.2.a.a.1.1		1
20.7	even	4		912.2.a.d.1.1		1
35.27	even	4		2793.2.a.a.1.1		1
40.27	even	4		3648.2.a.y.1.1		1
40.37	odd	4		3648.2.a.h.1.1		1
55.32	even	4		6897.2.a.g.1.1		1
60.47	odd	4		2736.2.a.h.1.1		1
65.12	odd	4		9633.2.a.p.1.1		1
95.37	even	4		1083.2.a.d.1.1		1
105.62	odd	4		8379.2.a.q.1.1		1
285.227	odd	4		3249.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.a.b.1.1	✓	1	5.2	odd	4
171.2.a.c.1.1		1	15.2	even	4
912.2.a.d.1.1		1	20.7	even	4
1083.2.a.d.1.1		1	95.37	even	4
1425.2.a.i.1.1		1	5.3	odd	4
1425.2.c.a.799.1		2	5.4	even	2	inner
1425.2.c.a.799.2		2	1.1	even	1	trivial
2736.2.a.h.1.1		1	60.47	odd	4
2793.2.a.a.1.1		1	35.27	even	4
3249.2.a.a.1.1		1	285.227	odd	4
3648.2.a.h.1.1		1	40.37	odd	4
3648.2.a.y.1.1		1	40.27	even	4
4275.2.a.a.1.1		1	15.8	even	4
6897.2.a.g.1.1		1	55.32	even	4
8379.2.a.q.1.1		1	105.62	odd	4
9633.2.a.p.1.1		1	65.12	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:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} -3.00000 q^{11} -2.00000i q^{12} -6.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} -3.00000i q^{17} -2.00000i q^{18} +1.00000 q^{19} +3.00000 q^{21} -6.00000i q^{22} +4.00000i q^{23} +12.0000 q^{26} -1.00000i q^{27} +6.00000i q^{28} +10.0000 q^{29} +2.00000 q^{31} -8.00000i q^{32} -3.00000i q^{33} +6.00000 q^{34} +2.00000 q^{36} -8.00000i q^{37} +2.00000i q^{38} +6.00000 q^{39} -8.00000 q^{41} +6.00000i q^{42} -1.00000i q^{43} +6.00000 q^{44} -8.00000 q^{46} -3.00000i q^{47} -4.00000i q^{48} -2.00000 q^{49} +3.00000 q^{51} +12.0000i q^{52} -6.00000i q^{53} +2.00000 q^{54} +1.00000i q^{57} +20.0000i q^{58} +7.00000 q^{61} +4.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} +6.00000 q^{66} -8.00000i q^{67} +6.00000i q^{68} -4.00000 q^{69} +12.0000 q^{71} -11.0000i q^{73} +16.0000 q^{74} -2.00000 q^{76} +9.00000i q^{77} +12.0000i q^{78} +1.00000 q^{81} -16.0000i q^{82} +4.00000i q^{83} -6.00000 q^{84} +2.00000 q^{86} +10.0000i q^{87} -10.0000 q^{89} -18.0000 q^{91} -8.00000i q^{92} +2.00000i q^{93} +6.00000 q^{94} +8.00000 q^{96} +2.00000i q^{97} -4.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q - 4 q^{4} - 4 q^{6} - 2 q^{9} - 6 q^{11} + 12 q^{14} - 8 q^{16} + 2 q^{19} + 6 q^{21} + 24 q^{26} + 20 q^{29} + 4 q^{31} + 12 q^{34} + 4 q^{36} + 12 q^{39} - 16 q^{41} + 12 q^{44} - 16 q^{46} - 4 q^{49} + 6 q^{51} + 4 q^{54} + 14 q^{61} + 16 q^{64} + 12 q^{66} - 8 q^{69} + 24 q^{71} + 32 q^{74} - 4 q^{76} + 2 q^{81} - 12 q^{84} + 4 q^{86} - 20 q^{89} - 36 q^{91} + 12 q^{94} + 16 q^{96} + 6 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9 - 6 * q^11 + 12 * q^14 - 8 * q^16 + 2 * q^19 + 6 * q^21 + 24 * q^26 + 20 * q^29 + 4 * q^31 + 12 * q^34 + 4 * q^36 + 12 * q^39 - 16 * q^41 + 12 * q^44 - 16 * q^46 - 4 * q^49 + 6 * q^51 + 4 * q^54 + 14 * q^61 + 16 * q^64 + 12 * q^66 - 8 * q^69 + 24 * q^71 + 32 * q^74 - 4 * q^76 + 2 * q^81 - 12 * q^84 + 4 * q^86 - 20 * q^89 - 36 * q^91 + 12 * q^94 + 16 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

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