LMFDB - Embedded newform 1425.2.c.c.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.c.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} -1.00000i q^{12} -2.00000 q^{14} -1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} -2.00000 q^{21} +6.00000i q^{22} +8.00000i q^{23} -3.00000 q^{24} +1.00000i q^{27} -2.00000i q^{28} -4.00000 q^{29} -5.00000i q^{32} +6.00000i q^{33} -6.00000 q^{34} -1.00000 q^{36} +4.00000i q^{37} +1.00000i q^{38} +2.00000i q^{42} +2.00000i q^{43} -6.00000 q^{44} +8.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} -2.00000i q^{53} +1.00000 q^{54} -6.00000 q^{56} +1.00000i q^{57} +4.00000i q^{58} -12.0000 q^{59} +2.00000 q^{61} +2.00000i q^{63} -7.00000 q^{64} +6.00000 q^{66} -8.00000i q^{67} -6.00000i q^{68} +8.00000 q^{69} +16.0000 q^{71} +3.00000i q^{72} -14.0000i q^{73} +4.00000 q^{74} -1.00000 q^{76} +12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -2.00000 q^{84} +2.00000 q^{86} +4.00000i q^{87} +18.0000i q^{88} +8.00000i q^{92} -8.00000 q^{94} -5.00000 q^{96} -12.0000i q^{97} -3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 12 q^{11} - 4 q^{14} - 2 q^{16} - 2 q^{19} - 4 q^{21} - 6 q^{24} - 8 q^{29} - 12 q^{34} - 2 q^{36} - 12 q^{44} + 16 q^{46} + 6 q^{49} - 12 q^{51} + 2 q^{54} - 12 q^{56} - 24 q^{59} + 4 q^{61} - 14 q^{64} + 12 q^{66} + 16 q^{69} + 32 q^{71} + 8 q^{74} - 2 q^{76} - 16 q^{79} + 2 q^{81} - 4 q^{84} + 4 q^{86} - 16 q^{94} - 10 q^{96} + 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 12 * q^11 - 4 * q^14 - 2 * q^16 - 2 * q^19 - 4 * q^21 - 6 * q^24 - 8 * q^29 - 12 * q^34 - 2 * q^36 - 12 * q^44 + 16 * q^46 + 6 * q^49 - 12 * q^51 + 2 * q^54 - 12 * q^56 - 24 * q^59 + 4 * q^61 - 14 * q^64 + 12 * q^66 + 16 * q^69 + 32 * q^71 + 8 * q^74 - 2 * q^76 - 16 * q^79 + 2 * q^81 - 4 * q^84 + 4 * q^86 - 16 * q^94 - 10 * q^96 + 12 * 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	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 3.00000i	− 1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	− 1.00000i	− 0.288675i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	1.00000i	0.235702i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.00000	−0.436436
$22$	6.00000i	1.27920i
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	0	0
$27$	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
$33$	6.00000i	1.04447i
$34$	−6.00000	−1.02899
$35$	0	0
$36$	−1.00000	−0.166667
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	1.00000i	0.162221i
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	2.00000i	0.308607i
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	8.00000	1.17954
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	− 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
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	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
$66$	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
$69$	8.00000	0.963087
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	3.00000i	0.353553i
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−1.00000	−0.114708
$77$	12.0000i	1.36753i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	2.00000	0.215666
$87$	4.00000i	0.428845i
$88$	18.0000i	1.91881i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	8.00000i	0.834058i
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−5.00000	−0.510310
$97$	− 12.0000i	− 1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$	− 12.0000i	− 1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$	− 3.00000i	− 0.303046i
$99$	6.00000	0.603023
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	6.00000i	0.594089i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	1.00000i	0.0962250i
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	2.00000i	0.188982i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	−4.00000	−0.371391
$117$	0	0
$118$	12.0000i	1.10469i
$119$	−12.0000	−1.10004
$120$	0	0
$121$	25.0000	2.27273
$122$	− 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	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	− 3.00000i	− 0.265165i
$129$	2.00000	0.176090
$130$	0	0
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	6.00000i	0.522233i
$133$	2.00000i	0.173422i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−18.0000	−1.54349
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	− 8.00000i	− 0.681005i
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	− 16.0000i	− 1.34269i
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	− 3.00000i	− 0.247436i
$148$	4.00000i	0.328798i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	3.00000i	0.243332i
$153$	6.00000i	0.485071i
$154$	12.0000	0.966988
$155$	0	0
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	8.00000i	0.636446i
$159$	−2.00000	−0.158610
$160$	0	0
$161$	16.0000	1.26098
$162$	− 1.00000i	− 0.0785674i
$163$	22.0000i	1.72317i	0.507611	+	0.861586i	$0.330529\pi$
$163$	22.0000i	1.72317i	−0.507611	+	0.861586i	$0.669471\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	6.00000i	0.462910i
$169$	13.0000	1.00000
$170$	0	0
$171$	1.00000	0.0764719
$172$	2.00000i	0.152499i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	6.00000	0.452267
$177$	12.0000i	0.901975i
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	− 2.00000i	− 0.147844i
$184$	24.0000	1.76930
$185$	0	0
$186$	0	0
$187$	36.0000i	2.63258i
$188$	− 8.00000i	− 0.583460i
$189$	2.00000	0.145479
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	7.00000i	0.505181i
$193$	− 24.0000i	− 1.72756i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	− 24.0000i	− 1.72756i	0.503871	−	0.863779i	$-0.331909\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	3.00000	0.214286
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	− 6.00000i	− 0.426401i
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	18.0000i	1.26648i
$203$	8.00000i	0.561490i
$204$	−6.00000	−0.420084
$205$	0	0
$206$	8.00000	0.557386
$207$	− 8.00000i	− 0.556038i
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	− 2.00000i	− 0.137361i
$213$	− 16.0000i	− 1.09630i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	3.00000	0.204124
$217$	0	0
$218$	− 6.00000i	− 0.406371i
$219$	−14.0000	−0.946032
$220$	0	0
$221$	0	0
$222$	− 4.00000i	− 0.268462i
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	−10.0000	−0.668153
$225$	0	0
$226$	6.00000	0.399114
$227$	− 28.0000i	− 1.85843i	−0.369546	−	0.929213i	$-0.620487\pi$
$227$	− 28.0000i	− 1.85843i	0.369546	−	0.929213i	$-0.379513\pi$
$228$	1.00000i	0.0662266i
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	12.0000i	0.787839i
$233$	− 22.0000i	− 1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	− 22.0000i	− 1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$	0	0
$235$	0	0
$236$	−12.0000	−0.781133
$237$	8.00000i	0.519656i
$238$	12.0000i	0.777844i
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	− 25.0000i	− 1.60706i
$243$	− 1.00000i	− 0.0641500i
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	2.00000i	0.125988i
$253$	− 48.0000i	− 3.01773i
$254$	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	$-0.240706\pi$
$258$	− 2.00000i	− 0.124515i
$259$	8.00000	0.497096
$260$	0	0
$261$	4.00000	0.247594
$262$	2.00000i	0.123560i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	18.0000	1.10782
$265$	0	0
$266$	2.00000	0.122628
$267$	0	0
$268$	− 8.00000i	− 0.488678i
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	6.00000i	0.363803i
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	8.00000	0.481543
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	− 16.0000i	− 0.959616i
$279$	0	0
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	8.00000i	0.476393i
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	0	0
$287$	0	0
$288$	5.00000i	0.294628i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−12.0000	−0.703452
$292$	− 14.0000i	− 0.819288i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	12.0000	0.697486
$297$	− 6.00000i	− 0.348155i
$298$	6.00000i	0.347571i
$299$	0	0
$300$	0	0
$301$	4.00000	0.230556
$302$	16.0000i	0.920697i
$303$	18.0000i	1.03407i
$304$	1.00000	0.0573539
$305$	0	0
$306$	6.00000	0.342997
$307$	− 8.00000i	− 0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$	− 8.00000i	− 0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$	12.0000i	0.683763i
$309$	8.00000	0.455104
$310$	0	0
$311$	34.0000	1.92796	0.963982	−	0.265969i	$-0.0856919\pi$
$311$	34.0000	1.92796	0.963982	+	0.265969i	$0.0856919\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	2.00000i	0.112154i
$319$	24.0000	1.34374
$320$	0	0
$321$	−12.0000	−0.669775
$322$	− 16.0000i	− 0.891645i
$323$	6.00000i	0.333849i
$324$	1.00000	0.0555556
$325$	0	0
$326$	22.0000	1.21847
$327$	− 6.00000i	− 0.331801i
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	0	0
$333$	− 4.00000i	− 0.219199i
$334$	16.0000	0.875481
$335$	0	0
$336$	2.00000	0.109109
$337$	− 12.0000i	− 0.653682i	−0.945079	−	0.326841i	$-0.894016\pi$
$337$	− 12.0000i	− 0.653682i	0.945079	−	0.326841i	$-0.105984\pi$
$338$	− 13.0000i	− 0.707107i
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	− 1.00000i	− 0.0540738i
$343$	− 20.0000i	− 1.07990i
$344$	6.00000	0.323498
$345$	0	0
$346$	−6.00000	−0.322562
$347$	8.00000i	0.429463i	0.976673	+	0.214731i	$0.0688876\pi$
$347$	8.00000i	0.429463i	−0.976673	+	0.214731i	$0.931112\pi$
$348$	4.00000i	0.214423i
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	0	0
$352$	30.0000i	1.59901i
$353$	34.0000i	1.80964i	0.425797	+	0.904819i	$0.359994\pi$
$353$	34.0000i	1.80964i	−0.425797	+	0.904819i	$0.640006\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	0	0
$357$	12.0000i	0.635107i
$358$	− 20.0000i	− 1.05703i
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 10.0000i	− 0.525588i
$363$	− 25.0000i	− 1.31216i
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	− 14.0000i	− 0.730794i	−0.930852	−	0.365397i	$-0.880933\pi$
$367$	− 14.0000i	− 0.730794i	0.930852	−	0.365397i	$-0.119067\pi$
$368$	− 8.00000i	− 0.417029i
$369$	0	0
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	36.0000	1.86152
$375$	0	0
$376$	−24.0000	−1.23771
$377$	0	0
$378$	− 2.00000i	− 0.102869i
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	− 10.0000i	− 0.511645i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−24.0000	−1.22157
$387$	− 2.00000i	− 0.101666i
$388$	− 12.0000i	− 0.609208i
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	− 9.00000i	− 0.454569i
$393$	2.00000i	0.100887i
$394$	6.00000	0.302276
$395$	0	0
$396$	6.00000	0.301511
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	− 20.0000i	− 1.00251i
$399$	2.00000	0.100125
$400$	0	0
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	8.00000i	0.399004i
$403$	0	0
$404$	−18.0000	−0.895533
$405$	0	0
$406$	8.00000	0.397033
$407$	− 24.0000i	− 1.18964i
$408$	18.0000i	0.891133i
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	8.00000i	0.394132i
$413$	24.0000i	1.18096i
$414$	−8.00000	−0.393179
$415$	0	0
$416$	0	0
$417$	− 16.0000i	− 0.783523i
$418$	− 6.00000i	− 0.293470i
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	20.0000i	0.973585i
$423$	8.00000i	0.388973i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−16.0000	−0.775203
$427$	− 4.00000i	− 0.193574i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	16.0000i	0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$	16.0000i	0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$	0	0
$435$	0	0
$436$	6.00000	0.287348
$437$	− 8.00000i	− 0.382692i
$438$	14.0000i	0.668946i
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−8.00000	−0.378811
$447$	6.00000i	0.283790i
$448$	14.0000i	0.661438i
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	0	0
$451$	0	0
$452$	6.00000i	0.282216i
$453$	16.0000i	0.751746i
$454$	−28.0000	−1.31411
$455$	0	0
$456$	3.00000	0.140488
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	2.00000i	0.0934539i
$459$	6.00000	0.280056
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	− 12.0000i	− 0.558291i
$463$	14.0000i	0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$	14.0000i	0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	−22.0000	−1.01913
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	36.0000i	1.65703i
$473$	− 12.0000i	− 0.551761i
$474$	8.00000	0.367452
$475$	0	0
$476$	−12.0000	−0.550019
$477$	2.00000i	0.0915737i
$478$	6.00000i	0.274434i
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	0	0
$482$	− 18.0000i	− 0.819878i
$483$	− 16.0000i	− 0.728025i
$484$	25.0000	1.13636
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	− 6.00000i	− 0.271607i
$489$	22.0000	0.994874
$490$	0	0
$491$	−26.0000	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$491$	−26.0000	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$492$	0	0
$493$	24.0000i	1.08091i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 32.0000i	− 1.43540i
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	6.00000i	0.267793i
$503$	36.0000i	1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$	36.0000i	1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	6.00000	0.267261
$505$	0	0
$506$	−48.0000	−2.13386
$507$	− 13.0000i	− 0.577350i
$508$	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
$511$	−28.0000	−1.23865
$512$	11.0000i	0.486136i
$513$	− 1.00000i	− 0.0441511i
$514$	−22.0000	−0.970378
$515$	0	0
$516$	2.00000	0.0880451
$517$	48.0000i	2.11104i
$518$	− 8.00000i	− 0.351500i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	− 4.00000i	− 0.175075i
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−2.00000	−0.0873704
$525$	0	0
$526$	12.0000	0.523225
$527$	0	0
$528$	− 6.00000i	− 0.261116i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	12.0000	0.520756
$532$	2.00000i	0.0867110i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−24.0000	−1.03664
$537$	− 20.0000i	− 0.863064i
$538$	12.0000i	0.517357i
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	20.0000i	0.859074i
$543$	− 10.0000i	− 0.429141i
$544$	−30.0000	−1.28624
$545$	0	0
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	− 10.0000i	− 0.427179i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	4.00000	0.170406
$552$	− 24.0000i	− 1.02151i
$553$	16.0000i	0.680389i
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	16.0000	0.678551
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	36.0000	1.51992
$562$	4.00000i	0.168730i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	14.0000	0.588464
$567$	− 2.00000i	− 0.0839921i
$568$	− 48.0000i	− 2.01404i
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	0	0
$573$	− 10.0000i	− 0.417756i
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	19.0000i	0.790296i
$579$	−24.0000	−0.997406
$580$	0	0
$581$	0	0
$582$	12.0000i	0.497416i
$583$	12.0000i	0.496989i
$584$	−42.0000	−1.73797
$585$	0	0
$586$	−30.0000	−1.23929
$587$	− 20.0000i	− 0.825488i	−0.910847	−	0.412744i	$-0.864570\pi$
$587$	− 20.0000i	− 0.825488i	0.910847	−	0.412744i	$-0.135430\pi$
$588$	− 3.00000i	− 0.123718i
$589$	0	0
$590$	0	0
$591$	6.00000	0.246807
$592$	− 4.00000i	− 0.164399i
$593$	10.0000i	0.410651i	0.978694	+	0.205325i	$0.0658253\pi$
$593$	10.0000i	0.410651i	−0.978694	+	0.205325i	$0.934175\pi$
$594$	−6.00000	−0.246183
$595$	0	0
$596$	−6.00000	−0.245770
$597$	− 20.0000i	− 0.818546i
$598$	0	0
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	− 4.00000i	− 0.163028i
$603$	8.00000i	0.325785i
$604$	−16.0000	−0.651031
$605$	0	0
$606$	18.0000	0.731200
$607$	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
$611$	0	0
$612$	6.00000i	0.242536i
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	36.0000	1.45048
$617$	34.0000i	1.36879i	0.729112	+	0.684394i	$0.239933\pi$
$617$	34.0000i	1.36879i	−0.729112	+	0.684394i	$0.760067\pi$
$618$	− 8.00000i	− 0.321807i
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	− 34.0000i	− 1.36328i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−10.0000	−0.399680
$627$	− 6.00000i	− 0.239617i
$628$	− 2.00000i	− 0.0798087i
$629$	24.0000	0.956943
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	24.0000i	0.954669i
$633$	20.0000i	0.794929i
$634$	−6.00000	−0.238290
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	0	0
$638$	− 24.0000i	− 0.950169i
$639$	−16.0000	−0.632950
$640$	0	0
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	12.0000i	0.473602i
$643$	− 2.00000i	− 0.0788723i	−0.999222	−	0.0394362i	$-0.987444\pi$
$643$	− 2.00000i	− 0.0788723i	0.999222	−	0.0394362i	$-0.0125562\pi$
$644$	16.0000	0.630488
$645$	0	0
$646$	6.00000	0.236067
$647$	20.0000i	0.786281i	0.919478	+	0.393141i	$0.128611\pi$
$647$	20.0000i	0.786281i	−0.919478	+	0.393141i	$0.871389\pi$
$648$	− 3.00000i	− 0.117851i
$649$	72.0000	2.82625
$650$	0	0
$651$	0	0
$652$	22.0000i	0.861586i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	−6.00000	−0.234619
$655$	0	0
$656$	0	0
$657$	14.0000i	0.546192i
$658$	16.0000i	0.623745i
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	− 28.0000i	− 1.08825i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−4.00000	−0.154997
$667$	− 32.0000i	− 1.23904i
$668$	16.0000i	0.619059i
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−12.0000	−0.463255
$672$	10.0000i	0.385758i
$673$	− 36.0000i	− 1.38770i	−0.720121	−	0.693849i	$-0.755914\pi$
$673$	− 36.0000i	− 1.38770i	0.720121	−	0.693849i	$-0.244086\pi$
$674$	−12.0000	−0.462223
$675$	0	0
$676$	13.0000	0.500000
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	− 6.00000i	− 0.230429i
$679$	−24.0000	−0.921035
$680$	0	0
$681$	−28.0000	−1.07296
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	−20.0000	−0.763604
$687$	2.00000i	0.0763048i
$688$	− 2.00000i	− 0.0762493i
$689$	0	0
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	− 6.00000i	− 0.228086i
$693$	− 12.0000i	− 0.455842i
$694$	8.00000	0.303676
$695$	0	0
$696$	12.0000	0.454859
$697$	0	0
$698$	− 2.00000i	− 0.0757011i
$699$	−22.0000	−0.832116
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	42.0000	1.58293
$705$	0	0
$706$	34.0000	1.27961
$707$	36.0000i	1.35392i
$708$	12.0000i	0.450988i
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	12.0000	0.449089
$715$	0	0
$716$	20.0000	0.747435
$717$	6.00000i	0.224074i
$718$	10.0000i	0.373197i
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	− 1.00000i	− 0.0372161i
$723$	− 18.0000i	− 0.669427i
$724$	10.0000	0.371647
$725$	0	0
$726$	−25.0000	−0.927837
$727$	26.0000i	0.964287i	0.876092	+	0.482143i	$0.160142\pi$
$727$	26.0000i	0.964287i	−0.876092	+	0.482143i	$0.839858\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	− 2.00000i	− 0.0739221i
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	−14.0000	−0.516749
$735$	0	0
$736$	40.0000	1.47442
$737$	48.0000i	1.76810i
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	4.00000i	0.146845i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	36.0000i	1.31629i
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	8.00000i	0.291730i
$753$	6.00000i	0.218652i
$754$	0	0
$755$	0	0
$756$	2.00000	0.0727393
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	20.0000i	0.726433i
$759$	−48.0000	−1.74229
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	− 4.00000i	− 0.144905i
$763$	− 12.0000i	− 0.434429i
$764$	10.0000	0.361787
$765$	0	0
$766$	−8.00000	−0.289052
$767$	0	0
$768$	17.0000i	0.613435i
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	− 24.0000i	− 0.863779i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	−36.0000	−1.29232
$777$	− 8.00000i	− 0.286998i
$778$	− 14.0000i	− 0.501924i
$779$	0	0
$780$	0	0
$781$	−96.0000	−3.43515
$782$	− 48.0000i	− 1.71648i
$783$	− 4.00000i	− 0.142948i
$784$	−3.00000	−0.107143
$785$	0	0
$786$	2.00000	0.0713376
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	6.00000i	0.213741i
$789$	12.0000	0.427211
$790$	0	0
$791$	12.0000	0.426671
$792$	− 18.0000i	− 0.639602i
$793$	0	0
$794$	6.00000	0.212932
$795$	0	0
$796$	20.0000	0.708881
$797$	− 26.0000i	− 0.920967i	−0.887668	−	0.460484i	$-0.847676\pi$
$797$	− 26.0000i	− 0.920967i	0.887668	−	0.460484i	$-0.152324\pi$
$798$	− 2.00000i	− 0.0707992i
$799$	−48.0000	−1.69812
$800$	0	0
$801$	0	0
$802$	− 4.00000i	− 0.141245i
$803$	84.0000i	2.96430i
$804$	−8.00000	−0.282138
$805$	0	0
$806$	0	0
$807$	12.0000i	0.422420i
$808$	54.0000i	1.89971i
$809$	−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
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	8.00000i	0.280745i
$813$	20.0000i	0.701431i
$814$	−24.0000	−0.841200
$815$	0	0
$816$	6.00000	0.210042
$817$	− 2.00000i	− 0.0699711i
$818$	− 10.0000i	− 0.349642i
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	10.0000i	0.348790i
$823$	14.0000i	0.488009i	0.969774	+	0.244005i	$0.0784612\pi$
$823$	14.0000i	0.488009i	−0.969774	+	0.244005i	$0.921539\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	24.0000	0.835067
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	− 8.00000i	− 0.278019i
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	− 18.0000i	− 0.623663i
$834$	−16.0000	−0.554035
$835$	0	0
$836$	6.00000	0.207514
$837$	0	0
$838$	− 30.0000i	− 1.03633i
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	− 6.00000i	− 0.206774i
$843$	4.00000i	0.137767i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	8.00000	0.275046
$847$	− 50.0000i	− 1.71802i
$848$	2.00000i	0.0686803i
$849$	14.0000	0.480479
$850$	0	0
$851$	−32.0000	−1.09695
$852$	− 16.0000i	− 0.548151i
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−36.0000	−1.23045
$857$	46.0000i	1.57133i	0.618652	+	0.785665i	$0.287679\pi$
$857$	46.0000i	1.57133i	−0.618652	+	0.785665i	$0.712321\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	0	0
$862$	− 36.0000i	− 1.22616i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	16.0000	0.543702
$867$	19.0000i	0.645274i
$868$	0	0
$869$	48.0000	1.62829
$870$	0	0
$871$	0	0
$872$	− 18.0000i	− 0.609557i
$873$	12.0000i	0.406138i
$874$	−8.00000	−0.270604
$875$	0	0
$876$	−14.0000	−0.473016
$877$	− 24.0000i	− 0.810422i	−0.914223	−	0.405211i	$-0.867198\pi$
$877$	− 24.0000i	− 0.810422i	0.914223	−	0.405211i	$-0.132802\pi$
$878$	24.0000i	0.809961i
$879$	−30.0000	−1.01187
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	3.00000i	0.101015i
$883$	− 2.00000i	− 0.0673054i	−0.999434	−	0.0336527i	$-0.989286\pi$
$883$	− 2.00000i	− 0.0673054i	0.999434	−	0.0336527i	$-0.0107140\pi$
$884$	0	0
$885$	0	0
$886$	12.0000	0.403148
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	− 12.0000i	− 0.402694i
$889$	8.00000	0.268311
$890$	0	0
$891$	−6.00000	−0.201008
$892$	− 8.00000i	− 0.267860i
$893$	8.00000i	0.267710i
$894$	6.00000	0.200670
$895$	0	0
$896$	−6.00000	−0.200446
$897$	0	0
$898$	− 8.00000i	− 0.266963i
$899$	0	0
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	− 4.00000i	− 0.133112i
$904$	18.0000	0.598671
$905$	0	0
$906$	16.0000	0.531564
$907$	− 52.0000i	− 1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$	− 52.0000i	− 1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$	− 28.0000i	− 0.929213i
$909$	18.0000	0.597022
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	0	0
$914$	−6.00000	−0.198462
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	4.00000i	0.132092i
$918$	− 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
$921$	−8.00000	−0.263609
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	12.0000	0.394771
$925$	0	0
$926$	14.0000	0.460069
$927$	− 8.00000i	− 0.262754i
$928$	20.0000i	0.656532i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	− 22.0000i	− 0.720634i
$933$	− 34.0000i	− 1.11311i
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	30.0000i	0.980057i	0.871706	+	0.490029i	$0.163014\pi$
$937$	30.0000i	0.980057i	−0.871706	+	0.490029i	$0.836986\pi$
$938$	16.0000i	0.522419i
$939$	−10.0000	−0.326338
$940$	0	0
$941$	44.0000	1.43436	0.717180	−	0.696888i	$-0.245433\pi$
$941$	44.0000	1.43436	0.717180	+	0.696888i	$0.245433\pi$
$942$	2.00000i	0.0651635i
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	−12.0000	−0.390154
$947$	48.0000i	1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$	48.0000i	1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$	8.00000i	0.259828i
$949$	0	0
$950$	0	0
$951$	−6.00000	−0.194563
$952$	36.0000i	1.16677i
$953$	30.0000i	0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$	30.0000i	0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−6.00000	−0.194054
$957$	− 24.0000i	− 0.775810i
$958$	18.0000i	0.581554i
$959$	−20.0000	−0.645834
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	12.0000i	0.386695i
$964$	18.0000	0.579741
$965$	0	0
$966$	−16.0000	−0.514792
$967$	− 14.0000i	− 0.450210i	−0.974335	−	0.225105i	$-0.927728\pi$
$967$	− 14.0000i	− 0.450210i	0.974335	−	0.225105i	$-0.0722725\pi$
$968$	− 75.0000i	− 2.41059i
$969$	6.00000	0.192748
$970$	0	0
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 32.0000i	− 1.02587i
$974$	8.00000	0.256337
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	− 22.0000i	− 0.703482i
$979$	0	0
$980$	0	0
$981$	−6.00000	−0.191565
$982$	26.0000i	0.829693i
$983$	− 8.00000i	− 0.255160i	−0.991828	−	0.127580i	$-0.959279\pi$
$983$	− 8.00000i	− 0.255160i	0.991828	−	0.127580i	$-0.0407210\pi$
$984$	0	0
$985$	0	0
$986$	24.0000	0.764316
$987$	16.0000i	0.509286i
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	− 28.0000i	− 0.888553i
$994$	−32.0000	−1.01498
$995$	0	0
$996$	0	0
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	− 24.0000i	− 0.759707i
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.c.799.1		2
5.2	odd	4		1425.2.a.g.1.1		1
5.3	odd	4		285.2.a.a.1.1	✓	1
5.4	even	2	inner	1425.2.c.c.799.2		2
15.2	even	4		4275.2.a.h.1.1		1
15.8	even	4		855.2.a.c.1.1		1
20.3	even	4		4560.2.a.h.1.1		1
95.18	even	4		5415.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.a.1.1	✓	1	5.3	odd	4
855.2.a.c.1.1		1	15.8	even	4
1425.2.a.g.1.1		1	5.2	odd	4
1425.2.c.c.799.1		2	1.1	even	1	trivial
1425.2.c.c.799.2		2	5.4	even	2	inner
4275.2.a.h.1.1		1	15.2	even	4
4560.2.a.h.1.1		1	20.3	even	4
5415.2.a.h.1.1		1	95.18	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} -1.00000i q^{12} -2.00000 q^{14} -1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} -2.00000 q^{21} +6.00000i q^{22} +8.00000i q^{23} -3.00000 q^{24} +1.00000i q^{27} -2.00000i q^{28} -4.00000 q^{29} -5.00000i q^{32} +6.00000i q^{33} -6.00000 q^{34} -1.00000 q^{36} +4.00000i q^{37} +1.00000i q^{38} +2.00000i q^{42} +2.00000i q^{43} -6.00000 q^{44} +8.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} -2.00000i q^{53} +1.00000 q^{54} -6.00000 q^{56} +1.00000i q^{57} +4.00000i q^{58} -12.0000 q^{59} +2.00000 q^{61} +2.00000i q^{63} -7.00000 q^{64} +6.00000 q^{66} -8.00000i q^{67} -6.00000i q^{68} +8.00000 q^{69} +16.0000 q^{71} +3.00000i q^{72} -14.0000i q^{73} +4.00000 q^{74} -1.00000 q^{76} +12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -2.00000 q^{84} +2.00000 q^{86} +4.00000i q^{87} +18.0000i q^{88} +8.00000i q^{92} -8.00000 q^{94} -5.00000 q^{96} -12.0000i q^{97} -3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 12 q^{11} - 4 q^{14} - 2 q^{16} - 2 q^{19} - 4 q^{21} - 6 q^{24} - 8 q^{29} - 12 q^{34} - 2 q^{36} - 12 q^{44} + 16 q^{46} + 6 q^{49} - 12 q^{51} + 2 q^{54} - 12 q^{56} - 24 q^{59} + 4 q^{61} - 14 q^{64} + 12 q^{66} + 16 q^{69} + 32 q^{71} + 8 q^{74} - 2 q^{76} - 16 q^{79} + 2 q^{81} - 4 q^{84} + 4 q^{86} - 16 q^{94} - 10 q^{96} + 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 12 * q^11 - 4 * q^14 - 2 * q^16 - 2 * q^19 - 4 * q^21 - 6 * q^24 - 8 * q^29 - 12 * q^34 - 2 * q^36 - 12 * q^44 + 16 * q^46 + 6 * q^49 - 12 * q^51 + 2 * q^54 - 12 * q^56 - 24 * q^59 + 4 * q^61 - 14 * q^64 + 12 * q^66 + 16 * q^69 + 32 * q^71 + 8 * q^74 - 2 * q^76 - 16 * q^79 + 2 * q^81 - 4 * q^84 + 4 * q^86 - 16 * q^94 - 10 * q^96 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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