LMFDB - Embedded newform 1425.2.c.d.799.1

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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} +2.00000 q^{21} +2.00000i q^{22} -4.00000i q^{23} -3.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -4.00000 q^{29} -5.00000i q^{32} +2.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} -1.00000i q^{38} -4.00000 q^{39} -2.00000i q^{42} -10.0000i q^{43} -2.00000 q^{44} -4.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} -4.00000i q^{52} -2.00000i q^{53} +1.00000 q^{54} +6.00000 q^{56} -1.00000i q^{57} +4.00000i q^{58} -4.00000 q^{59} +2.00000 q^{61} -2.00000i q^{63} -7.00000 q^{64} +2.00000 q^{66} +16.0000i q^{67} -2.00000i q^{68} -4.00000 q^{69} +3.00000i q^{72} -2.00000i q^{73} +1.00000 q^{76} -4.00000i q^{77} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +2.00000 q^{84} -10.0000 q^{86} +4.00000i q^{87} +6.00000i q^{88} +8.00000 q^{91} -4.00000i q^{92} -12.0000 q^{94} -5.00000 q^{96} +16.0000i q^{97} -3.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{11} + 4 q^{14} - 2 q^{16} + 2 q^{19} + 4 q^{21} - 6 q^{24} - 8 q^{26} - 8 q^{29} - 4 q^{34} - 2 q^{36} - 8 q^{39} - 4 q^{44} - 8 q^{46} + 6 q^{49} - 4 q^{51} + 2 q^{54} + 12 q^{56} - 8 q^{59} + 4 q^{61} - 14 q^{64} + 4 q^{66} - 8 q^{69} + 2 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{84} - 20 q^{86} + 16 q^{91} - 24 q^{94} - 10 q^{96} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 4 * q^11 + 4 * q^14 - 2 * q^16 + 2 * q^19 + 4 * q^21 - 6 * q^24 - 8 * q^26 - 8 * q^29 - 4 * q^34 - 2 * q^36 - 8 * q^39 - 4 * q^44 - 8 * q^46 + 6 * q^49 - 4 * q^51 + 2 * q^54 + 12 * q^56 - 8 * q^59 + 4 * q^61 - 14 * q^64 + 4 * q^66 - 8 * q^69 + 2 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^84 - 20 * q^86 + 16 * q^91 - 24 * q^94 - 10 * q^96 + 4 * 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 $

$2$	− 1.00000i	− 0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 3.00000i	− 1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	1.00000i	0.235702i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	2.00000i	0.426401i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	−4.00000	−0.784465
$27$	1.00000i	0.192450i
$28$	2.00000i	0.377964i
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 5.00000i	− 0.883883i
$33$	2.00000i	0.348155i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	− 1.00000i	− 0.162221i
$39$	−4.00000	−0.640513
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	− 2.00000i	− 0.308607i
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	− 4.00000i	− 0.554700i
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	6.00000	0.801784
$57$	− 1.00000i	− 0.132453i
$58$	4.00000i	0.525226i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	2.00000	0.246183
$67$	16.0000i	1.95471i	0.211604	+	0.977356i	$0.432131\pi$
$67$	16.0000i	1.95471i	−0.211604	+	0.977356i	$0.567869\pi$
$68$	− 2.00000i	− 0.242536i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000i	0.353553i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	0	0
$76$	1.00000	0.114708
$77$	− 4.00000i	− 0.455842i
$78$	4.00000i	0.452911i
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−10.0000	−1.07833
$87$	4.00000i	0.428845i
$88$	6.00000i	0.639602i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	− 4.00000i	− 0.417029i
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	−5.00000	−0.510310
$97$	16.0000i	1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$	16.0000i	1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$	− 3.00000i	− 0.303046i
$99$	2.00000	0.201008
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	2.00000i	0.198030i
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	−12.0000	−1.17670
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	1.00000i	0.0962250i
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	− 2.00000i	− 0.188982i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−4.00000	−0.371391
$117$	4.00000i	0.369800i
$118$	4.00000i	0.368230i
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	− 3.00000i	− 0.265165i
$129$	−10.0000	−0.880451
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	2.00000i	0.174078i
$133$	2.00000i	0.173422i
$134$	16.0000	1.38219
$135$	0	0
$136$	−6.00000	−0.514496
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	4.00000i	0.340503i
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	0	0
$143$	8.00000i	0.668994i
$144$	1.00000	0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	24.0000	1.95309	0.976546	−	0.215308i	$-0.0690756\pi$
$151$	24.0000	1.95309	0.976546	+	0.215308i	$0.0690756\pi$
$152$	− 3.00000i	− 0.243332i
$153$	2.00000i	0.161690i
$154$	−4.00000	−0.322329
$155$	0	0
$156$	−4.00000	−0.320256
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	− 8.00000i	− 0.636446i
$159$	−2.00000	−0.158610
$160$	0	0
$161$	8.00000	0.630488
$162$	− 1.00000i	− 0.0785674i
$163$	− 6.00000i	− 0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$	− 6.00000i	− 0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$	0	0
$165$	0	0
$166$	−12.0000	−0.931381
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	− 6.00000i	− 0.462910i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 10.0000i	− 0.762493i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	2.00000	0.150756
$177$	4.00000i	0.300658i
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	− 8.00000i	− 0.592999i
$183$	− 2.00000i	− 0.147844i
$184$	−12.0000	−0.884652
$185$	0	0
$186$	0	0
$187$	4.00000i	0.292509i
$188$	− 12.0000i	− 0.875190i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	7.00000i	0.505181i
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	3.00000	0.214286
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	− 2.00000i	− 0.142134i
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	− 14.0000i	− 0.985037i
$203$	− 8.00000i	− 0.561490i
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−8.00000	−0.557386
$207$	4.00000i	0.278019i
$208$	4.00000i	0.277350i
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	− 2.00000i	− 0.137361i
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	3.00000	0.204124
$217$	0	0
$218$	10.0000i	0.677285i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	24.0000i	1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$	24.0000i	1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$	10.0000	0.668153
$225$	0	0
$226$	14.0000	0.931266
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	12.0000i	0.787839i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−4.00000	−0.260378
$237$	− 8.00000i	− 0.519656i
$238$	− 4.00000i	− 0.259281i
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	7.00000i	0.449977i
$243$	− 1.00000i	− 0.0641500i
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	22.0000	1.38863	0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000	1.38863	0.694314	+	0.719672i	$0.255708\pi$
$252$	− 2.00000i	− 0.125988i
$253$	8.00000i	0.502956i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 22.0000i	− 1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$	− 22.0000i	− 1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$	10.0000i	0.622573i
$259$	0	0
$260$	0	0
$261$	4.00000	0.247594
$262$	14.0000i	0.864923i
$263$	− 16.0000i	− 0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$	− 16.0000i	− 0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	2.00000	0.122628
$267$	0	0
$268$	16.0000i	0.977356i
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	2.00000i	0.121268i
$273$	− 8.00000i	− 0.484182i
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−4.00000	−0.240772
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	8.00000i	0.479808i
$279$	0	0
$280$	0	0
$281$	28.0000	1.67034	0.835170	−	0.549992i	$-0.185369\pi$
$281$	28.0000	1.67034	0.835170	+	0.549992i	$0.185369\pi$
$282$	12.0000i	0.714590i
$283$	26.0000i	1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$283$	26.0000i	1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$284$	0	0
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	5.00000i	0.294628i
$289$	13.0000	0.764706
$290$	0	0
$291$	16.0000	0.937937
$292$	− 2.00000i	− 0.117041i
$293$	26.0000i	1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$	26.0000i	1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	0	0
$297$	− 2.00000i	− 0.116052i
$298$	− 18.0000i	− 1.04271i
$299$	−16.0000	−0.925304
$300$	0	0
$301$	20.0000	1.15278
$302$	− 24.0000i	− 1.38104i
$303$	− 14.0000i	− 0.804279i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	2.00000	0.114332
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 4.00000i	− 0.227921i
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	12.0000i	0.679366i
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	8.00000	0.450035
$317$	− 14.0000i	− 0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$	− 14.0000i	− 0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$	2.00000i	0.112154i
$319$	8.00000	0.447914
$320$	0	0
$321$	12.0000	0.669775
$322$	− 8.00000i	− 0.445823i
$323$	− 2.00000i	− 0.111283i
$324$	1.00000	0.0555556
$325$	0	0
$326$	−6.00000	−0.332309
$327$	10.0000i	0.553001i
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0	0
$334$	8.00000	0.437741
$335$	0	0
$336$	−2.00000	−0.109109
$337$	− 16.0000i	− 0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$	− 16.0000i	− 0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$	3.00000i	0.163178i
$339$	14.0000	0.760376
$340$	0	0
$341$	0	0
$342$	1.00000i	0.0540738i
$343$	20.0000i	1.07990i
$344$	−30.0000	−1.61749
$345$	0	0
$346$	18.0000	0.967686
$347$	28.0000i	1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$	28.0000i	1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$	4.00000i	0.214423i
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	10.0000i	0.533002i
$353$	− 2.00000i	− 0.106449i	−0.998583	−	0.0532246i	$-0.983050\pi$
$353$	− 2.00000i	− 0.106449i	0.998583	−	0.0532246i	$-0.0169499\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	0	0
$357$	− 4.00000i	− 0.211702i
$358$	− 12.0000i	− 0.634220i
$359$	2.00000	0.105556	0.0527780	−	0.998606i	$-0.483192\pi$
$359$	2.00000	0.105556	0.0527780	+	0.998606i	$0.483192\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 10.0000i	− 0.525588i
$363$	7.00000i	0.367405i
$364$	8.00000	0.419314
$365$	0	0
$366$	−2.00000	−0.104542
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	4.00000i	0.208514i
$369$	0	0
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−36.0000	−1.85656
$377$	16.0000i	0.824042i
$378$	2.00000i	0.102869i
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	18.0000i	0.920960i
$383$	8.00000i	0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$	8.00000i	0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	4.00000	0.203595
$387$	10.0000i	0.508329i
$388$	16.0000i	0.812277i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	− 9.00000i	− 0.454569i
$393$	14.0000i	0.706207i
$394$	2.00000	0.100759
$395$	0	0
$396$	2.00000	0.100504
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	12.0000i	0.601506i
$399$	2.00000	0.100125
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	− 16.0000i	− 0.798007i
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	−8.00000	−0.397033
$407$	0	0
$408$	6.00000i	0.297044i
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	− 8.00000i	− 0.394132i
$413$	− 8.00000i	− 0.393654i
$414$	4.00000	0.196589
$415$	0	0
$416$	−20.0000	−0.980581
$417$	8.00000i	0.391762i
$418$	2.00000i	0.0978232i
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	12.0000i	0.584151i
$423$	12.0000i	0.583460i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	12.0000i	0.580042i
$429$	8.00000	0.386244
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 36.0000i	− 1.73005i	−0.501729	−	0.865025i	$-0.667303\pi$
$433$	− 36.0000i	− 1.73005i	0.501729	−	0.865025i	$-0.332697\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 4.00000i	− 0.191346i
$438$	2.00000i	0.0955637i
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	8.00000i	0.380521i
$443$	16.0000i	0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$	16.0000i	0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$	0	0
$445$	0	0
$446$	24.0000	1.13643
$447$	− 18.0000i	− 0.851371i
$448$	− 14.0000i	− 0.661438i
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	14.0000i	0.658505i
$453$	− 24.0000i	− 1.12762i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−3.00000	−0.140488
$457$	14.0000i	0.654892i	0.944870	+	0.327446i	$0.106188\pi$
$457$	14.0000i	0.654892i	−0.944870	+	0.327446i	$0.893812\pi$
$458$	− 14.0000i	− 0.654177i
$459$	2.00000	0.0933520
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	4.00000i	0.186097i
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	−18.0000	−0.833834
$467$	− 16.0000i	− 0.740392i	−0.928954	−	0.370196i	$-0.879291\pi$
$467$	− 16.0000i	− 0.740392i	0.928954	−	0.370196i	$-0.120709\pi$
$468$	4.00000i	0.184900i
$469$	−32.0000	−1.47762
$470$	0	0
$471$	18.0000	0.829396
$472$	12.0000i	0.552345i
$473$	20.0000i	0.919601i
$474$	−8.00000	−0.367452
$475$	0	0
$476$	4.00000	0.183340
$477$	2.00000i	0.0915737i
$478$	− 6.00000i	− 0.274434i
$479$	−38.0000	−1.73626	−0.868132	−	0.496333i	$-0.834679\pi$
$479$	−38.0000	−1.73626	−0.868132	+	0.496333i	$0.834679\pi$
$480$	0	0
$481$	0	0
$482$	− 26.0000i	− 1.18427i
$483$	− 8.00000i	− 0.364013i
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 6.00000i	− 0.271607i
$489$	−6.00000	−0.271329
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	8.00000i	0.360302i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	0	0
$497$	0	0
$498$	12.0000i	0.537733i
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	− 22.0000i	− 0.981908i
$503$	− 8.00000i	− 0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$	− 8.00000i	− 0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	8.00000	0.355643
$507$	3.00000i	0.133235i
$508$	− 4.00000i	− 0.177471i
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	11.0000i	0.486136i
$513$	1.00000i	0.0441511i
$514$	−22.0000	−0.970378
$515$	0	0
$516$	−10.0000	−0.440225
$517$	24.0000i	1.05552i
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	− 4.00000i	− 0.175075i
$523$	− 28.0000i	− 1.22435i	−0.790721	−	0.612177i	$-0.790294\pi$
$523$	− 28.0000i	− 1.22435i	0.790721	−	0.612177i	$-0.209706\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	− 2.00000i	− 0.0870388i
$529$	7.00000	0.304348
$530$	0	0
$531$	4.00000	0.173585
$532$	2.00000i	0.0867110i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	48.0000	2.07328
$537$	− 12.0000i	− 0.517838i
$538$	4.00000i	0.172452i
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	− 12.0000i	− 0.515444i
$543$	− 10.0000i	− 0.429141i
$544$	−10.0000	−0.428746
$545$	0	0
$546$	−8.00000	−0.342368
$547$	− 16.0000i	− 0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$	− 16.0000i	− 0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−4.00000	−0.170406
$552$	12.0000i	0.510754i
$553$	16.0000i	0.680389i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−8.00000	−0.339276
$557$	38.0000i	1.61011i	0.593199	+	0.805056i	$0.297865\pi$
$557$	38.0000i	1.61011i	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	4.00000	0.168880
$562$	− 28.0000i	− 1.18111i
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	26.0000	1.09286
$567$	2.00000i	0.0839921i
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	8.00000i	0.334497i
$573$	18.0000i	0.751961i
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	− 42.0000i	− 1.74848i	−0.485491	−	0.874241i	$-0.661359\pi$
$577$	− 42.0000i	− 1.74848i	0.485491	−	0.874241i	$-0.338641\pi$
$578$	− 13.0000i	− 0.540729i
$579$	4.00000	0.166234
$580$	0	0
$581$	24.0000	0.995688
$582$	− 16.0000i	− 0.663221i
$583$	4.00000i	0.165663i
$584$	−6.00000	−0.248282
$585$	0	0
$586$	26.0000	1.07405
$587$	32.0000i	1.32078i	0.750922	+	0.660391i	$0.229609\pi$
$587$	32.0000i	1.32078i	−0.750922	+	0.660391i	$0.770391\pi$
$588$	− 3.00000i	− 0.123718i
$589$	0	0
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	18.0000	0.737309
$597$	12.0000i	0.491127i
$598$	16.0000i	0.654289i
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	− 20.0000i	− 0.815139i
$603$	− 16.0000i	− 0.651570i
$604$	24.0000	0.976546
$605$	0	0
$606$	−14.0000	−0.568711
$607$	16.0000i	0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$	16.0000i	0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$	− 5.00000i	− 0.202777i
$609$	−8.00000	−0.324176
$610$	0	0
$611$	−48.0000	−1.94187
$612$	2.00000i	0.0808452i
$613$	− 18.0000i	− 0.727013i	−0.931592	−	0.363507i	$-0.881579\pi$
$613$	− 18.0000i	− 0.727013i	0.931592	−	0.363507i	$-0.118421\pi$
$614$	0	0
$615$	0	0
$616$	−12.0000	−0.483494
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	8.00000i	0.321807i
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	18.0000i	0.721734i
$623$	0	0
$624$	4.00000	0.160128
$625$	0	0
$626$	−14.0000	−0.559553
$627$	2.00000i	0.0798723i
$628$	18.0000i	0.718278i
$629$	0	0
$630$	0	0
$631$	48.0000	1.91085	0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000	1.91085	0.955425	+	0.295234i	$0.0953977\pi$
$632$	− 24.0000i	− 0.954669i
$633$	12.0000i	0.476957i
$634$	−14.0000	−0.556011
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	− 12.0000i	− 0.475457i
$638$	− 8.00000i	− 0.316723i
$639$	0	0
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	− 12.0000i	− 0.473602i
$643$	26.0000i	1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$	26.0000i	1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−2.00000	−0.0786889
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	− 3.00000i	− 0.117851i
$649$	8.00000	0.314027
$650$	0	0
$651$	0	0
$652$	− 6.00000i	− 0.234978i
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	0	0
$657$	2.00000i	0.0780274i
$658$	− 24.0000i	− 0.935617i
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	28.0000i	1.08825i
$663$	8.00000i	0.310694i
$664$	−36.0000	−1.39707
$665$	0	0
$666$	0	0
$667$	16.0000i	0.619522i
$668$	8.00000i	0.309529i
$669$	24.0000	0.927894
$670$	0	0
$671$	−4.00000	−0.154418
$672$	− 10.0000i	− 0.385758i
$673$	24.0000i	0.925132i	0.886585	+	0.462566i	$0.153071\pi$
$673$	24.0000i	0.925132i	−0.886585	+	0.462566i	$0.846929\pi$
$674$	−16.0000	−0.616297
$675$	0	0
$676$	−3.00000	−0.115385
$677$	50.0000i	1.92166i	0.277145	+	0.960828i	$0.410612\pi$
$677$	50.0000i	1.92166i	−0.277145	+	0.960828i	$0.589388\pi$
$678$	− 14.0000i	− 0.537667i
$679$	−32.0000	−1.22805
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	20.0000	0.763604
$687$	− 14.0000i	− 0.534133i
$688$	10.0000i	0.381246i
$689$	−8.00000	−0.304776
$690$	0	0
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	18.0000i	0.684257i
$693$	4.00000i	0.151947i
$694$	28.0000	1.06287
$695$	0	0
$696$	12.0000	0.454859
$697$	0	0
$698$	− 18.0000i	− 0.681310i
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	− 4.00000i	− 0.150970i
$703$	0	0
$704$	14.0000	0.527645
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	28.0000i	1.05305i
$708$	4.00000i	0.150329i
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	−4.00000	−0.149696
$715$	0	0
$716$	12.0000	0.448461
$717$	− 6.00000i	− 0.224074i
$718$	− 2.00000i	− 0.0746393i
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	− 1.00000i	− 0.0372161i
$723$	− 26.0000i	− 0.966950i
$724$	10.0000	0.371647
$725$	0	0
$726$	7.00000	0.259794
$727$	14.0000i	0.519231i	0.965712	+	0.259616i	$0.0835959\pi$
$727$	14.0000i	0.519231i	−0.965712	+	0.259616i	$0.916404\pi$
$728$	− 24.0000i	− 0.889499i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−20.0000	−0.739727
$732$	− 2.00000i	− 0.0739221i
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	−20.0000	−0.737210
$737$	− 32.0000i	− 1.17874i
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	− 4.00000i	− 0.146845i
$743$	− 8.00000i	− 0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$	− 8.00000i	− 0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$	0	0
$745$	0	0
$746$	4.00000	0.146450
$747$	12.0000i	0.439057i
$748$	4.00000i	0.146254i
$749$	−24.0000	−0.876941
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	12.0000i	0.437595i
$753$	− 22.0000i	− 0.801725i
$754$	16.0000	0.582686
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	− 20.0000i	− 0.726433i
$759$	8.00000	0.290382
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	4.00000i	0.144905i
$763$	− 20.0000i	− 0.724049i
$764$	−18.0000	−0.651217
$765$	0	0
$766$	8.00000	0.289052
$767$	16.0000i	0.577727i
$768$	17.0000i	0.613435i
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	4.00000i	0.143963i
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	10.0000	0.359443
$775$	0	0
$776$	48.0000	1.72310
$777$	0	0
$778$	− 30.0000i	− 1.07555i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	8.00000i	0.286079i
$783$	− 4.00000i	− 0.142948i
$784$	−3.00000	−0.107143
$785$	0	0
$786$	14.0000	0.499363
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	2.00000i	0.0712470i
$789$	−16.0000	−0.569615
$790$	0	0
$791$	−28.0000	−0.995565
$792$	− 6.00000i	− 0.213201i
$793$	− 8.00000i	− 0.284088i
$794$	2.00000	0.0709773
$795$	0	0
$796$	−12.0000	−0.425329
$797$	14.0000i	0.495905i	0.968772	+	0.247953i	$0.0797578\pi$
$797$	14.0000i	0.495905i	−0.968772	+	0.247953i	$0.920242\pi$
$798$	− 2.00000i	− 0.0707992i
$799$	−24.0000	−0.849059
$800$	0	0
$801$	0	0
$802$	− 12.0000i	− 0.423735i
$803$	4.00000i	0.141157i
$804$	16.0000	0.564276
$805$	0	0
$806$	0	0
$807$	4.00000i	0.140807i
$808$	− 42.0000i	− 1.47755i
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	− 8.00000i	− 0.280745i
$813$	− 12.0000i	− 0.420858i
$814$	0	0
$815$	0	0
$816$	2.00000	0.0700140
$817$	− 10.0000i	− 0.349856i
$818$	− 26.0000i	− 0.909069i
$819$	−8.00000	−0.279543
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	6.00000i	0.209274i
$823$	− 14.0000i	− 0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$	− 14.0000i	− 0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	−8.00000	−0.278356
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	4.00000i	0.139010i
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	28.0000i	0.970725i
$833$	− 6.00000i	− 0.207888i
$834$	8.00000	0.277017
$835$	0	0
$836$	−2.00000	−0.0691714
$837$	0	0
$838$	− 26.0000i	− 0.898155i
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	34.0000i	1.17172i
$843$	− 28.0000i	− 0.964371i
$844$	−12.0000	−0.413057
$845$	0	0
$846$	12.0000	0.412568
$847$	− 14.0000i	− 0.481046i
$848$	2.00000i	0.0686803i
$849$	26.0000	0.892318
$850$	0	0
$851$	0	0
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	36.0000	1.23045
$857$	− 10.0000i	− 0.341593i	−0.985306	−	0.170797i	$-0.945366\pi$
$857$	− 10.0000i	− 0.341593i	0.985306	−	0.170797i	$-0.0546341\pi$
$858$	− 8.00000i	− 0.273115i
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	0	0
$862$	4.00000i	0.136241i
$863$	40.0000i	1.36162i	0.732462	+	0.680808i	$0.238371\pi$
$863$	40.0000i	1.36162i	−0.732462	+	0.680808i	$0.761629\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	−36.0000	−1.22333
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	64.0000	2.16856
$872$	30.0000i	1.01593i
$873$	− 16.0000i	− 0.541518i
$874$	−4.00000	−0.135302
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	− 12.0000i	− 0.405211i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	− 12.0000i	− 0.405211i	0.979260	−	0.202606i	$-0.0649409\pi$
$878$	− 40.0000i	− 1.34993i
$879$	26.0000	0.876958
$880$	0	0
$881$	−10.0000	−0.336909	−0.168454	−	0.985709i	$-0.553878\pi$
$881$	−10.0000	−0.336909	−0.168454	+	0.985709i	$0.553878\pi$
$882$	3.00000i	0.101015i
$883$	− 46.0000i	− 1.54802i	−0.633171	−	0.774012i	$-0.718247\pi$
$883$	− 46.0000i	− 1.54802i	0.633171	−	0.774012i	$-0.281753\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	16.0000	0.537531
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	24.0000i	0.803579i
$893$	− 12.0000i	− 0.401565i
$894$	−18.0000	−0.602010
$895$	0	0
$896$	6.00000	0.200446
$897$	16.0000i	0.534224i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	− 20.0000i	− 0.665558i
$904$	42.0000	1.39690
$905$	0	0
$906$	−24.0000	−0.797347
$907$	− 52.0000i	− 1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$	− 52.0000i	− 1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$	− 12.0000i	− 0.398234i
$909$	−14.0000	−0.464351
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	1.00000i	0.0331133i
$913$	24.0000i	0.794284i
$914$	14.0000	0.463079
$915$	0	0
$916$	14.0000	0.462573
$917$	− 28.0000i	− 0.924641i
$918$	− 2.00000i	− 0.0660098i
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	− 6.00000i	− 0.197599i
$923$	0	0
$924$	−4.00000	−0.131590
$925$	0	0
$926$	−22.0000	−0.722965
$927$	8.00000i	0.262754i
$928$	20.0000i	0.656532i
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	− 18.0000i	− 0.589610i
$933$	18.0000i	0.589294i
$934$	−16.0000	−0.523536
$935$	0	0
$936$	12.0000	0.392232
$937$	42.0000i	1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$	42.0000i	1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$	32.0000i	1.04484i
$939$	−14.0000	−0.456873
$940$	0	0
$941$	−60.0000	−1.95594	−0.977972	−	0.208736i	$-0.933065\pi$
$941$	−60.0000	−1.95594	−0.977972	+	0.208736i	$0.933065\pi$
$942$	− 18.0000i	− 0.586472i
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	20.0000	0.650256
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	− 8.00000i	− 0.259828i
$949$	−8.00000	−0.259691
$950$	0	0
$951$	−14.0000	−0.453981
$952$	− 12.0000i	− 0.388922i
$953$	46.0000i	1.49009i	0.667016	+	0.745043i	$0.267571\pi$
$953$	46.0000i	1.49009i	−0.667016	+	0.745043i	$0.732429\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	6.00000	0.194054
$957$	− 8.00000i	− 0.258603i
$958$	38.0000i	1.22772i
$959$	12.0000	0.387500
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	26.0000	0.837404
$965$	0	0
$966$	−8.00000	−0.257396
$967$	− 34.0000i	− 1.09337i	−0.837340	−	0.546683i	$-0.815890\pi$
$967$	− 34.0000i	− 1.09337i	0.837340	−	0.546683i	$-0.184110\pi$
$968$	21.0000i	0.674966i
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 16.0000i	− 0.512936i
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	6.00000i	0.191859i
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	6.00000i	0.191468i
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	0	0
$985$	0	0
$986$	8.00000	0.254772
$987$	− 24.0000i	− 0.763928i
$988$	− 4.00000i	− 0.127257i
$989$	−40.0000	−1.27193
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	28.0000i	0.888553i
$994$	0	0
$995$	0	0
$996$	−12.0000	−0.380235
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	32.0000i	1.01294i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.d.799.1		2
5.2	odd	4		285.2.a.b.1.1	✓	1
5.3	odd	4		1425.2.a.d.1.1		1
5.4	even	2	inner	1425.2.c.d.799.2		2
15.2	even	4		855.2.a.b.1.1		1
15.8	even	4		4275.2.a.o.1.1		1
20.7	even	4		4560.2.a.v.1.1		1
95.37	even	4		5415.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.b.1.1	✓	1	5.2	odd	4
855.2.a.b.1.1		1	15.2	even	4
1425.2.a.d.1.1		1	5.3	odd	4
1425.2.c.d.799.1		2	1.1	even	1	trivial
1425.2.c.d.799.2		2	5.4	even	2	inner
4275.2.a.o.1.1		1	15.8	even	4
4560.2.a.v.1.1		1	20.7	even	4
5415.2.a.c.1.1		1	95.37	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} +2.00000 q^{21} +2.00000i q^{22} -4.00000i q^{23} -3.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -4.00000 q^{29} -5.00000i q^{32} +2.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} -1.00000i q^{38} -4.00000 q^{39} -2.00000i q^{42} -10.0000i q^{43} -2.00000 q^{44} -4.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} -4.00000i q^{52} -2.00000i q^{53} +1.00000 q^{54} +6.00000 q^{56} -1.00000i q^{57} +4.00000i q^{58} -4.00000 q^{59} +2.00000 q^{61} -2.00000i q^{63} -7.00000 q^{64} +2.00000 q^{66} +16.0000i q^{67} -2.00000i q^{68} -4.00000 q^{69} +3.00000i q^{72} -2.00000i q^{73} +1.00000 q^{76} -4.00000i q^{77} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +2.00000 q^{84} -10.0000 q^{86} +4.00000i q^{87} +6.00000i q^{88} +8.00000 q^{91} -4.00000i q^{92} -12.0000 q^{94} -5.00000 q^{96} +16.0000i q^{97} -3.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{11} + 4 q^{14} - 2 q^{16} + 2 q^{19} + 4 q^{21} - 6 q^{24} - 8 q^{26} - 8 q^{29} - 4 q^{34} - 2 q^{36} - 8 q^{39} - 4 q^{44} - 8 q^{46} + 6 q^{49} - 4 q^{51} + 2 q^{54} + 12 q^{56} - 8 q^{59} + 4 q^{61} - 14 q^{64} + 4 q^{66} - 8 q^{69} + 2 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{84} - 20 q^{86} + 16 q^{91} - 24 q^{94} - 10 q^{96} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 4 * q^11 + 4 * q^14 - 2 * q^16 + 2 * q^19 + 4 * q^21 - 6 * q^24 - 8 * q^26 - 8 * q^29 - 4 * q^34 - 2 * q^36 - 8 * q^39 - 4 * q^44 - 8 * q^46 + 6 * q^49 - 4 * q^51 + 2 * q^54 + 12 * q^56 - 8 * q^59 + 4 * q^61 - 14 * q^64 + 4 * q^66 - 8 * q^69 + 2 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^84 - 20 * q^86 + 16 * q^91 - 24 * q^94 - 10 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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