LMFDB - Embedded newform 1425.2.c.f.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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} -4.00000 q^{21} +4.00000i q^{22} +4.00000i q^{23} -3.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +2.00000 q^{29} +5.00000i q^{32} +4.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} -6.00000i q^{37} +1.00000i q^{38} +2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} -8.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} +14.0000i q^{53} +1.00000 q^{54} -12.0000 q^{56} +1.00000i q^{57} +2.00000i q^{58} -4.00000 q^{59} +14.0000 q^{61} -4.00000i q^{63} -7.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -3.00000i q^{72} +14.0000i q^{73} +6.00000 q^{74} +1.00000 q^{76} +16.0000i q^{77} +2.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -4.00000 q^{84} +8.00000 q^{86} +2.00000i q^{87} +12.0000i q^{88} +6.00000 q^{89} +8.00000 q^{91} +4.00000i q^{92} +12.0000 q^{94} -5.00000 q^{96} -10.0000i q^{97} -9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{11} - 8 q^{14} - 2 q^{16} + 2 q^{19} - 8 q^{21} - 6 q^{24} + 4 q^{26} + 4 q^{29} - 4 q^{34} - 2 q^{36} + 4 q^{39} - 12 q^{41} + 8 q^{44} - 8 q^{46} - 18 q^{49} - 4 q^{51} + 2 q^{54} - 24 q^{56} - 8 q^{59} + 28 q^{61} - 14 q^{64} - 8 q^{66} - 8 q^{69} + 12 q^{74} + 2 q^{76} - 32 q^{79} + 2 q^{81} - 8 q^{84} + 16 q^{86} + 12 q^{89} + 16 q^{91} + 24 q^{94} - 10 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 - 8 * q^14 - 2 * q^16 + 2 * q^19 - 8 * q^21 - 6 * q^24 + 4 * q^26 + 4 * q^29 - 4 * q^34 - 2 * q^36 + 4 * q^39 - 12 * q^41 + 8 * q^44 - 8 * q^46 - 18 * q^49 - 4 * q^51 + 2 * q^54 - 24 * q^56 - 8 * q^59 + 28 * q^61 - 14 * q^64 - 8 * q^66 - 8 * q^69 + 12 * q^74 + 2 * q^76 - 32 * q^79 + 2 * q^81 - 8 * q^84 + 16 * q^86 + 12 * q^89 + 16 * q^91 + 24 * q^94 - 10 * q^96 - 8 * 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	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	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
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	3.00000i	1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000i	0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	− 1.00000i	− 0.235702i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.00000	−0.872872
$22$	4.00000i	0.852803i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	2.00000	0.392232
$27$	− 1.00000i	− 0.192450i
$28$	4.00000i	0.755929i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000i	0.883883i
$33$	4.00000i	0.696311i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	1.00000i	0.162221i
$39$	2.00000	0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 4.00000i	− 0.617213i
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	4.00000	0.603023
$45$	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
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	− 2.00000i	− 0.277350i
$53$	14.0000i	1.92305i	0.274721	+	0.961524i	$0.411414\pi$
$53$	14.0000i	1.92305i	−0.274721	+	0.961524i	$0.588586\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−12.0000	−1.60357
$57$	1.00000i	0.132453i
$58$	2.00000i	0.262613i
$59$	−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
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	− 4.00000i	− 0.503953i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	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
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	1.00000	0.114708
$77$	16.0000i	1.82337i
$78$	2.00000i	0.226455i
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	8.00000	0.862662
$87$	2.00000i	0.214423i
$88$	12.0000i	1.27920i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	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
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	− 9.00000i	− 0.909137i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	− 2.00000i	− 0.198030i
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−14.0000	−1.35980
$107$	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
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	− 4.00000i	− 0.377964i
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	2.00000	0.185695
$117$	2.00000i	0.184900i
$118$	− 4.00000i	− 0.368230i
$119$	−8.00000	−0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	14.0000i	1.26750i
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	0	0
$126$	4.00000	0.356348
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	3.00000i	0.265165i
$129$	8.00000	0.704361
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	4.00000i	0.348155i
$133$	4.00000i	0.346844i
$134$	4.00000	0.345547
$135$	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
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	− 9.00000i	− 0.742307i
$148$	− 6.00000i	− 0.493197i
$149$	−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
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	3.00000i	0.243332i
$153$	− 2.00000i	− 0.161690i
$154$	−16.0000	−1.28932
$155$	0	0
$156$	2.00000	0.160128
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	− 16.0000i	− 1.27289i
$159$	−14.0000	−1.11027
$160$	0	0
$161$	−16.0000	−1.26098
$162$	1.00000i	0.0785674i
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	− 12.0000i	− 0.925820i
$169$	9.00000	0.692308
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 8.00000i	− 0.609994i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	− 4.00000i	− 0.300658i
$178$	6.00000i	0.449719i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	8.00000i	0.592999i
$183$	14.0000i	1.03491i
$184$	−12.0000	−0.884652
$185$	0	0
$186$	0	0
$187$	8.00000i	0.585018i
$188$	− 12.0000i	− 0.875190i
$189$	4.00000	0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	− 7.00000i	− 0.505181i
$193$	− 22.0000i	− 1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	− 22.0000i	− 1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−9.00000	−0.642857
$197$	22.0000i	1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$	22.0000i	1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$	− 4.00000i	− 0.284268i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	− 10.0000i	− 0.703598i
$203$	8.00000i	0.561490i
$204$	−2.00000	−0.140028
$205$	0	0
$206$	16.0000	1.11477
$207$	− 4.00000i	− 0.278019i
$208$	2.00000i	0.138675i
$209$	4.00000	0.276686
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	14.0000i	0.961524i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	3.00000	0.204124
$217$	0	0
$218$	2.00000i	0.135457i
$219$	−14.0000	−0.946032
$220$	0	0
$221$	4.00000	0.269069
$222$	6.00000i	0.402694i
$223$	24.0000i	1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$	24.0000i	1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$	−20.0000	−1.33631
$225$	0	0
$226$	−10.0000	−0.665190
$227$	− 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
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	−16.0000	−1.05272
$232$	6.00000i	0.393919i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−4.00000	−0.260378
$237$	− 16.0000i	− 1.03931i
$238$	− 8.00000i	− 0.518563i
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	5.00000i	0.321412i
$243$	1.00000i	0.0641500i
$244$	14.0000	0.896258
$245$	0	0
$246$	6.00000	0.382546
$247$	− 2.00000i	− 0.127257i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	− 4.00000i	− 0.251976i
$253$	16.0000i	1.00591i
$254$	8.00000	0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	8.00000i	0.498058i
$259$	24.0000	1.49129
$260$	0	0
$261$	−2.00000	−0.123797
$262$	4.00000i	0.247121i
$263$	4.00000i	0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$	4.00000i	0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	−4.00000	−0.245256
$267$	6.00000i	0.367194i
$268$	− 4.00000i	− 0.244339i
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	− 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	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	4.00000i	0.239904i
$279$	0	0
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	12.0000i	0.714590i
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	0	0
$285$	0	0
$286$	8.00000	0.473050
$287$	− 24.0000i	− 1.41668i
$288$	− 5.00000i	− 0.294628i
$289$	13.0000	0.764706
$290$	0	0
$291$	10.0000	0.586210
$292$	14.0000i	0.819288i
$293$	− 2.00000i	− 0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$	− 2.00000i	− 0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	18.0000	1.04623
$297$	− 4.00000i	− 0.232104i
$298$	− 6.00000i	− 0.347571i
$299$	8.00000	0.462652
$300$	0	0
$301$	32.0000	1.84445
$302$	0	0
$303$	− 10.0000i	− 0.574485i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	2.00000	0.114332
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	16.0000i	0.911685i
$309$	16.0000	0.910208
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	6.00000i	0.339683i
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−16.0000	−0.900070
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	− 14.0000i	− 0.785081i
$319$	8.00000	0.447914
$320$	0	0
$321$	−12.0000	−0.669775
$322$	− 16.0000i	− 0.891645i
$323$	2.00000i	0.111283i
$324$	1.00000	0.0555556
$325$	0	0
$326$	0	0
$327$	2.00000i	0.110600i
$328$	− 18.0000i	− 0.993884i
$329$	48.0000	2.64633
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	8.00000	0.437741
$335$	0	0
$336$	4.00000	0.218218
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	9.00000i	0.489535i
$339$	−10.0000	−0.543125
$340$	0	0
$341$	0	0
$342$	− 1.00000i	− 0.0540738i
$343$	− 8.00000i	− 0.431959i
$344$	24.0000	1.29399
$345$	0	0
$346$	−6.00000	−0.322562
$347$	32.0000i	1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$	32.0000i	1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$	2.00000i	0.107211i
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	20.0000i	1.06600i
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	6.00000	0.317999
$357$	− 8.00000i	− 0.423405i
$358$	− 12.0000i	− 0.634220i
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	22.0000i	1.15629i
$363$	5.00000i	0.262432i
$364$	8.00000	0.419314
$365$	0	0
$366$	−14.0000	−0.731792
$367$	− 20.0000i	− 1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$	− 20.0000i	− 1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$	− 4.00000i	− 0.208514i
$369$	6.00000	0.312348
$370$	0	0
$371$	−56.0000	−2.90738
$372$	0	0
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	36.0000	1.85656
$377$	− 4.00000i	− 0.206010i
$378$	4.00000i	0.205738i
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	− 32.0000i	− 1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$	− 32.0000i	− 1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	22.0000	1.11977
$387$	8.00000i	0.406663i
$388$	− 10.0000i	− 0.507673i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	− 27.0000i	− 1.36371i
$393$	4.00000i	0.201773i
$394$	−22.0000	−1.10834
$395$	0	0
$396$	−4.00000	−0.201008
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	4.00000i	0.199502i
$403$	0	0
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−8.00000	−0.397033
$407$	− 24.0000i	− 1.18964i
$408$	− 6.00000i	− 0.297044i
$409$	−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
$411$	6.00000	0.295958
$412$	− 16.0000i	− 0.788263i
$413$	− 16.0000i	− 0.787309i
$414$	4.00000	0.196589
$415$	0	0
$416$	10.0000	0.490290
$417$	4.00000i	0.195881i
$418$	4.00000i	0.195646i
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	12.0000i	0.584151i
$423$	12.0000i	0.583460i
$424$	−42.0000	−2.03970
$425$	0	0
$426$	0	0
$427$	56.0000i	2.71003i
$428$	12.0000i	0.580042i
$429$	8.00000	0.386244
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	1.00000i	0.0481125i
$433$	− 6.00000i	− 0.288342i	−0.989553	−	0.144171i	$-0.953949\pi$
$433$	− 6.00000i	− 0.288342i	0.989553	−	0.144171i	$-0.0460515\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	4.00000i	0.191346i
$438$	− 14.0000i	− 0.668946i
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	4.00000i	0.190261i
$443$	− 40.0000i	− 1.90046i	−0.311553	−	0.950229i	$-0.600849\pi$
$443$	− 40.0000i	− 1.90046i	0.311553	−	0.950229i	$-0.399151\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	−24.0000	−1.13643
$447$	− 6.00000i	− 0.283790i
$448$	− 28.0000i	− 1.32288i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	10.0000i	0.470360i
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	−3.00000	−0.140488
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	26.0000i	1.21490i
$459$	2.00000	0.0933520
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	− 16.0000i	− 0.744387i
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	18.0000	0.833834
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	2.00000i	0.0924500i
$469$	16.0000	0.738811
$470$	0	0
$471$	18.0000	0.829396
$472$	− 12.0000i	− 0.552345i
$473$	− 32.0000i	− 1.47136i
$474$	16.0000	0.734904
$475$	0	0
$476$	−8.00000	−0.366679
$477$	− 14.0000i	− 0.641016i
$478$	24.0000i	1.09773i
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 22.0000i	− 1.00207i
$483$	− 16.0000i	− 0.728025i
$484$	5.00000	0.227273
$485$	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	$0.941983\pi$
$488$	42.0000i	1.90125i
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	− 6.00000i	− 0.270501i
$493$	4.00000i	0.180151i
$494$	2.00000	0.0899843
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	28.0000i	1.24970i
$503$	20.0000i	0.891756i	0.895094	+	0.445878i	$0.147108\pi$
$503$	20.0000i	0.891756i	−0.895094	+	0.445878i	$0.852892\pi$
$504$	12.0000	0.534522
$505$	0	0
$506$	−16.0000	−0.711287
$507$	9.00000i	0.399704i
$508$	− 8.00000i	− 0.354943i
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	− 11.0000i	− 0.486136i
$513$	− 1.00000i	− 0.0441511i
$514$	2.00000	0.0882162
$515$	0	0
$516$	8.00000	0.352180
$517$	− 48.0000i	− 2.11104i
$518$	24.0000i	1.05450i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−4.00000	−0.174408
$527$	0	0
$528$	− 4.00000i	− 0.174078i
$529$	7.00000	0.304348
$530$	0	0
$531$	4.00000	0.173585
$532$	4.00000i	0.173422i
$533$	12.0000i	0.519778i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	12.0000	0.518321
$537$	− 12.0000i	− 0.517838i
$538$	26.0000i	1.12094i
$539$	−36.0000	−1.55063
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	24.0000i	1.03089i
$543$	22.0000i	0.944110i
$544$	−10.0000	−0.428746
$545$	0	0
$546$	−8.00000	−0.342368
$547$	4.00000i	0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$	4.00000i	0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−14.0000	−0.597505
$550$	0	0
$551$	2.00000	0.0852029
$552$	− 12.0000i	− 0.510754i
$553$	− 64.0000i	− 2.72156i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	4.00000	0.169638
$557$	− 26.0000i	− 1.10166i	−0.834619	−	0.550828i	$-0.814312\pi$
$557$	− 26.0000i	− 1.10166i	0.834619	−	0.550828i	$-0.185688\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	−8.00000	−0.337760
$562$	10.0000i	0.421825i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	−16.0000	−0.672530
$567$	4.00000i	0.167984i
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	− 8.00000i	− 0.334497i
$573$	0	0
$574$	24.0000	1.00174
$575$	0	0
$576$	7.00000	0.291667
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	13.0000i	0.540729i
$579$	22.0000	0.914289
$580$	0	0
$581$	0	0
$582$	10.0000i	0.414513i
$583$	56.0000i	2.31928i
$584$	−42.0000	−1.73797
$585$	0	0
$586$	2.00000	0.0826192
$587$	16.0000i	0.660391i	0.943913	+	0.330195i	$0.107115\pi$
$587$	16.0000i	0.660391i	−0.943913	+	0.330195i	$0.892885\pi$
$588$	− 9.00000i	− 0.371154i
$589$	0	0
$590$	0	0
$591$	−22.0000	−0.904959
$592$	6.00000i	0.246598i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	8.00000i	0.327144i
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	32.0000i	1.30422i
$603$	4.00000i	0.162893i
$604$	0	0
$605$	0	0
$606$	10.0000	0.406222
$607$	− 16.0000i	− 0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$	− 16.0000i	− 0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$	5.00000i	0.202777i
$609$	−8.00000	−0.324176
$610$	0	0
$611$	−24.0000	−0.970936
$612$	− 2.00000i	− 0.0808452i
$613$	42.0000i	1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$	42.0000i	1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	−48.0000	−1.93398
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	16.0000i	0.643614i
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	24.0000i	0.961540i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	4.00000i	0.159745i
$628$	− 18.0000i	− 0.718278i
$629$	12.0000	0.478471
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	− 48.0000i	− 1.90934i
$633$	12.0000i	0.476957i
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	−14.0000	−0.555136
$637$	18.0000i	0.713186i
$638$	8.00000i	0.316723i
$639$	0	0
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	− 12.0000i	− 0.473602i
$643$	40.0000i	1.57745i	0.614749	+	0.788723i	$0.289257\pi$
$643$	40.0000i	1.57745i	−0.614749	+	0.788723i	$0.710743\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	−2.00000	−0.0786889
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	3.00000i	0.117851i
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	6.00000	0.234261
$657$	− 14.0000i	− 0.546192i
$658$	48.0000i	1.87123i
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	− 28.0000i	− 1.08825i
$663$	4.00000i	0.155347i
$664$	0	0
$665$	0	0
$666$	−6.00000	−0.232495
$667$	8.00000i	0.309761i
$668$	− 8.00000i	− 0.309529i
$669$	−24.0000	−0.927894
$670$	0	0
$671$	56.0000	2.16186
$672$	− 20.0000i	− 0.771517i
$673$	− 30.0000i	− 1.15642i	−0.815890	−	0.578208i	$-0.803752\pi$
$673$	− 30.0000i	− 1.15642i	0.815890	−	0.578208i	$-0.196248\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	9.00000	0.346154
$677$	− 14.0000i	− 0.538064i	−0.963131	−	0.269032i	$-0.913296\pi$
$677$	− 14.0000i	− 0.538064i	0.963131	−	0.269032i	$-0.0867037\pi$
$678$	− 10.0000i	− 0.384048i
$679$	40.0000	1.53506
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	8.00000	0.305441
$687$	26.0000i	0.991962i
$688$	8.00000i	0.304997i
$689$	28.0000	1.06672
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	6.00000i	0.228086i
$693$	− 16.0000i	− 0.607790i
$694$	−32.0000	−1.21470
$695$	0	0
$696$	−6.00000	−0.227429
$697$	− 12.0000i	− 0.454532i
$698$	− 30.0000i	− 1.13552i
$699$	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	$0.721929\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 6.00000i	− 0.226294i
$704$	−28.0000	−1.05529
$705$	0	0
$706$	−14.0000	−0.526897
$707$	− 40.0000i	− 1.50435i
$708$	− 4.00000i	− 0.150329i
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	18.0000i	0.674579i
$713$	0	0
$714$	8.00000	0.299392
$715$	0	0
$716$	−12.0000	−0.448461
$717$	24.0000i	0.896296i
$718$	8.00000i	0.298557i
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	64.0000	2.38348
$722$	1.00000i	0.0372161i
$723$	− 22.0000i	− 0.818189i
$724$	22.0000	0.817624
$725$	0	0
$726$	−5.00000	−0.185567
$727$	4.00000i	0.148352i	0.997245	+	0.0741759i	$0.0236326\pi$
$727$	4.00000i	0.148352i	−0.997245	+	0.0741759i	$0.976367\pi$
$728$	24.0000i	0.889499i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	14.0000i	0.517455i
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	20.0000	0.738213
$735$	0	0
$736$	−20.0000	−0.737210
$737$	− 16.0000i	− 0.589368i
$738$	6.00000i	0.220863i
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	− 56.0000i	− 2.05582i
$743$	− 40.0000i	− 1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$	− 40.0000i	− 1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$	0	0
$745$	0	0
$746$	34.0000	1.24483
$747$	0	0
$748$	8.00000i	0.292509i
$749$	−48.0000	−1.75388
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	12.0000i	0.437595i
$753$	28.0000i	1.02038i
$754$	4.00000	0.145671
$755$	0	0
$756$	4.00000	0.145479
$757$	− 50.0000i	− 1.81728i	−0.417579	−	0.908640i	$-0.637121\pi$
$757$	− 50.0000i	− 1.81728i	0.417579	−	0.908640i	$-0.362879\pi$
$758$	− 4.00000i	− 0.145287i
$759$	−16.0000	−0.580763
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	8.00000i	0.289809i
$763$	8.00000i	0.289619i
$764$	0	0
$765$	0	0
$766$	32.0000	1.15621
$767$	8.00000i	0.288863i
$768$	− 17.0000i	− 0.613435i
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	− 22.0000i	− 0.791797i
$773$	− 26.0000i	− 0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$	− 26.0000i	− 0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	30.0000	1.07694
$777$	24.0000i	0.860995i
$778$	− 6.00000i	− 0.215110i
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	− 8.00000i	− 0.286079i
$783$	− 2.00000i	− 0.0714742i
$784$	9.00000	0.321429
$785$	0	0
$786$	−4.00000	−0.142675
$787$	− 44.0000i	− 1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$	− 44.0000i	− 1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$	22.0000i	0.783718i
$789$	−4.00000	−0.142404
$790$	0	0
$791$	−40.0000	−1.42224
$792$	− 12.0000i	− 0.426401i
$793$	− 28.0000i	− 0.994309i
$794$	2.00000	0.0709773
$795$	0	0
$796$	0	0
$797$	− 38.0000i	− 1.34603i	−0.739629	−	0.673015i	$-0.764999\pi$
$797$	− 38.0000i	− 1.34603i	0.739629	−	0.673015i	$-0.235001\pi$
$798$	− 4.00000i	− 0.141598i
$799$	24.0000	0.849059
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 6.00000i	− 0.211867i
$803$	56.0000i	1.97620i
$804$	4.00000	0.141069
$805$	0	0
$806$	0	0
$807$	26.0000i	0.915243i
$808$	− 30.0000i	− 1.05540i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	8.00000i	0.280745i
$813$	24.0000i	0.841717i
$814$	24.0000	0.841200
$815$	0	0
$816$	2.00000	0.0700140
$817$	− 8.00000i	− 0.279885i
$818$	− 10.0000i	− 0.349642i
$819$	−8.00000	−0.279543
$820$	0	0
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	6.00000i	0.209274i
$823$	20.0000i	0.697156i	0.937280	+	0.348578i	$0.113335\pi$
$823$	20.0000i	0.697156i	−0.937280	+	0.348578i	$0.886665\pi$
$824$	48.0000	1.67216
$825$	0	0
$826$	16.0000	0.556711
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	− 4.00000i	− 0.139010i
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	14.0000i	0.485363i
$833$	− 18.0000i	− 0.623663i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	4.00000	0.138343
$837$	0	0
$838$	− 28.0000i	− 0.967244i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	14.0000i	0.482472i
$843$	10.0000i	0.344418i
$844$	12.0000	0.413057
$845$	0	0
$846$	−12.0000	−0.412568
$847$	20.0000i	0.687208i
$848$	− 14.0000i	− 0.480762i
$849$	−16.0000	−0.549119
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	2.00000i	0.0684787i	0.999414	+	0.0342393i	$0.0109009\pi$
$853$	2.00000i	0.0684787i	−0.999414	+	0.0342393i	$0.989099\pi$
$854$	−56.0000	−1.91628
$855$	0	0
$856$	−36.0000	−1.23045
$857$	− 26.0000i	− 0.888143i	−0.895991	−	0.444072i	$-0.853534\pi$
$857$	− 26.0000i	− 0.888143i	0.895991	−	0.444072i	$-0.146466\pi$
$858$	8.00000i	0.273115i
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	32.0000i	1.08992i
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	6.00000	0.203888
$867$	13.0000i	0.441503i
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	−8.00000	−0.271070
$872$	6.00000i	0.203186i
$873$	10.0000i	0.338449i
$874$	−4.00000	−0.135302
$875$	0	0
$876$	−14.0000	−0.473016
$877$	42.0000i	1.41824i	0.705088	+	0.709120i	$0.250907\pi$
$877$	42.0000i	1.41824i	−0.705088	+	0.709120i	$0.749093\pi$
$878$	− 8.00000i	− 0.269987i
$879$	2.00000	0.0674583
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	9.00000i	0.303046i
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	40.0000	1.34383
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	18.0000i	0.604040i
$889$	32.0000	1.07325
$890$	0	0
$891$	4.00000	0.134005
$892$	24.0000i	0.803579i
$893$	− 12.0000i	− 0.401565i
$894$	6.00000	0.200670
$895$	0	0
$896$	−12.0000	−0.400892
$897$	8.00000i	0.267112i
$898$	6.00000i	0.200223i
$899$	0	0
$900$	0	0
$901$	−28.0000	−0.932815
$902$	− 24.0000i	− 0.799113i
$903$	32.0000i	1.06489i
$904$	−30.0000	−0.997785
$905$	0	0
$906$	0	0
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	− 12.0000i	− 0.398234i
$909$	10.0000	0.331679
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	26.0000	0.859064
$917$	16.0000i	0.528367i
$918$	2.00000i	0.0660098i
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	30.0000i	0.987997i
$923$	0	0
$924$	−16.0000	−0.526361
$925$	0	0
$926$	−4.00000	−0.131448
$927$	16.0000i	0.525509i
$928$	10.0000i	0.328266i
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	− 18.0000i	− 0.589610i
$933$	0	0
$934$	8.00000	0.261768
$935$	0	0
$936$	−6.00000	−0.196116
$937$	18.0000i	0.588034i	0.955800	+	0.294017i	$0.0949923\pi$
$937$	18.0000i	0.588034i	−0.955800	+	0.294017i	$0.905008\pi$
$938$	16.0000i	0.522419i
$939$	10.0000	0.326338
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	18.0000i	0.586472i
$943$	− 24.0000i	− 0.781548i
$944$	4.00000	0.130189
$945$	0	0
$946$	32.0000	1.04041
$947$	− 40.0000i	− 1.29983i	−0.760009	−	0.649913i	$-0.774805\pi$
$947$	− 40.0000i	− 1.29983i	0.760009	−	0.649913i	$-0.225195\pi$
$948$	− 16.0000i	− 0.519656i
$949$	28.0000	0.908918
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	− 24.0000i	− 0.777844i
$953$	26.0000i	0.842223i	0.907009	+	0.421111i	$0.138360\pi$
$953$	26.0000i	0.842223i	−0.907009	+	0.421111i	$0.861640\pi$
$954$	14.0000	0.453267
$955$	0	0
$956$	24.0000	0.776215
$957$	8.00000i	0.258603i
$958$	16.0000i	0.516937i
$959$	24.0000	0.775000
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 12.0000i	− 0.386896i
$963$	− 12.0000i	− 0.386695i
$964$	−22.0000	−0.708572
$965$	0	0
$966$	16.0000	0.514792
$967$	− 20.0000i	− 0.643157i	−0.946883	−	0.321578i	$-0.895787\pi$
$967$	− 20.0000i	− 0.643157i	0.946883	−	0.321578i	$-0.104213\pi$
$968$	15.0000i	0.482118i
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	1.00000i	0.0320750i
$973$	16.0000i	0.512936i
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−14.0000	−0.448129
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	12.0000i	0.382935i
$983$	56.0000i	1.78612i	0.449935	+	0.893061i	$0.351447\pi$
$983$	56.0000i	1.78612i	−0.449935	+	0.893061i	$0.648553\pi$
$984$	18.0000	0.573819
$985$	0	0
$986$	−4.00000	−0.127386
$987$	48.0000i	1.52786i
$988$	− 2.00000i	− 0.0636285i
$989$	32.0000	1.01754
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	− 28.0000i	− 0.888553i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	− 20.0000i	− 0.633089i
$999$	−6.00000	−0.189832

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} -4.00000 q^{21} +4.00000i q^{22} +4.00000i q^{23} -3.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +2.00000 q^{29} +5.00000i q^{32} +4.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} -6.00000i q^{37} +1.00000i q^{38} +2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} -8.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} +14.0000i q^{53} +1.00000 q^{54} -12.0000 q^{56} +1.00000i q^{57} +2.00000i q^{58} -4.00000 q^{59} +14.0000 q^{61} -4.00000i q^{63} -7.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -3.00000i q^{72} +14.0000i q^{73} +6.00000 q^{74} +1.00000 q^{76} +16.0000i q^{77} +2.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -4.00000 q^{84} +8.00000 q^{86} +2.00000i q^{87} +12.0000i q^{88} +6.00000 q^{89} +8.00000 q^{91} +4.00000i q^{92} +12.0000 q^{94} -5.00000 q^{96} -10.0000i q^{97} -9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 8 q^{14} - 2 q^{16} + 2 q^{19} - 8 q^{21} - 6 q^{24} + 4 q^{26} + 4 q^{29} - 4 q^{34} - 2 q^{36} + 4 q^{39} - 12 q^{41} + 8 q^{44} - 8 q^{46} - 18 q^{49} - 4 q^{51} + 2 q^{54} - 24 q^{56} - 8 q^{59} + 28 q^{61} - 14 q^{64} - 8 q^{66} - 8 q^{69} + 12 q^{74} + 2 q^{76} - 32 q^{79} + 2 q^{81} - 8 q^{84} + 16 q^{86} + 12 q^{89} + 16 q^{91} + 24 q^{94} - 10 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 - 8 * q^14 - 2 * q^16 + 2 * q^19 - 8 * q^21 - 6 * q^24 + 4 * q^26 + 4 * q^29 - 4 * q^34 - 2 * q^36 + 4 * q^39 - 12 * q^41 + 8 * q^44 - 8 * q^46 - 18 * q^49 - 4 * q^51 + 2 * q^54 - 24 * q^56 - 8 * q^59 + 28 * q^61 - 14 * q^64 - 8 * q^66 - 8 * q^69 + 12 * q^74 + 2 * q^76 - 32 * q^79 + 2 * q^81 - 8 * q^84 + 16 * q^86 + 12 * q^89 + 16 * q^91 + 24 * q^94 - 10 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more