LMFDB - Embedded newform 2366.2.d.d.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2366,2,Mod(337,2366)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2366, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2366.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2366.d (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:	$18.8926051182$
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 182)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2366.337
Dual form			2366.2.d.d.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} -3.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} -3.00000 q^{9} -2.00000 q^{10} +4.00000i q^{11} -1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +3.00000i q^{18} +2.00000i q^{20} +4.00000 q^{22} -8.00000 q^{23} +1.00000 q^{25} +1.00000i q^{28} -10.0000 q^{29} +8.00000i q^{31} -1.00000i q^{32} -6.00000i q^{34} -2.00000 q^{35} +3.00000 q^{36} +6.00000i q^{37} +2.00000 q^{40} +6.00000i q^{41} -4.00000 q^{43} -4.00000i q^{44} +6.00000i q^{45} +8.00000i q^{46} -8.00000i q^{47} -1.00000 q^{49} -1.00000i q^{50} +6.00000 q^{53} +8.00000 q^{55} +1.00000 q^{56} +10.0000i q^{58} +8.00000i q^{59} +10.0000 q^{61} +8.00000 q^{62} +3.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} -6.00000 q^{68} +2.00000i q^{70} +8.00000i q^{71} -3.00000i q^{72} +2.00000i q^{73} +6.00000 q^{74} +4.00000 q^{77} +8.00000 q^{79} -2.00000i q^{80} +9.00000 q^{81} +6.00000 q^{82} -12.0000i q^{85} +4.00000i q^{86} -4.00000 q^{88} +18.0000i q^{89} +6.00000 q^{90} +8.00000 q^{92} -8.00000 q^{94} -2.00000i q^{97} +1.00000i q^{98} -12.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 6 * q^9	$ 2 q - 2 q^{4} - 6 q^{9} - 4 q^{10} - 2 q^{14} + 2 q^{16} + 12 q^{17} + 8 q^{22} - 16 q^{23} + 2 q^{25} - 20 q^{29} - 4 q^{35} + 6 q^{36} + 4 q^{40} - 8 q^{43} - 2 q^{49} + 12 q^{53} + 16 q^{55} + 2 q^{56} + 20 q^{61} + 16 q^{62} - 2 q^{64} - 12 q^{68} + 12 q^{74} + 8 q^{77} + 16 q^{79} + 18 q^{81} + 12 q^{82} - 8 q^{88} + 12 q^{90} + 16 q^{92} - 16 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 6 * q^9 - 4 * q^10 - 2 * q^14 + 2 * q^16 + 12 * q^17 + 8 * q^22 - 16 * q^23 + 2 * q^25 - 20 * q^29 - 4 * q^35 + 6 * q^36 + 4 * q^40 - 8 * q^43 - 2 * q^49 + 12 * q^53 + 16 * q^55 + 2 * q^56 + 20 * q^61 + 16 * q^62 - 2 * q^64 - 12 * q^68 + 12 * q^74 + 8 * q^77 + 16 * q^79 + 18 * q^81 + 12 * q^82 - 8 * q^88 + 12 * q^90 + 16 * q^92 - 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times$.

$n$	$339$	$2199$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	− 2.00000i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 2.00000i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	1.00000i	0.353553i
$9$	−3.00000	−1.00000
$10$	−2.00000	−0.632456
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	3.00000i	0.707107i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	2.00000i	0.447214i
$21$	0	0
$22$	4.00000	0.852803
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	1.00000i	0.188982i
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	8.00000i	1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$	8.00000i	1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 6.00000i	− 1.02899i
$35$	−2.00000	−0.338062
$36$	3.00000	0.500000
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0	0
$40$	2.00000	0.316228
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	− 4.00000i	− 0.603023i
$45$	6.00000i	0.894427i
$46$	8.00000i	1.17954i
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	1.00000	0.133631
$57$	0	0
$58$	10.0000i	1.31306i
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	8.00000	1.01600
$63$	3.00000i	0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	2.00000i	0.239046i
$71$	8.00000i	0.949425i	0.880141	+	0.474713i	$0.157448\pi$
$71$	8.00000i	0.949425i	−0.880141	+	0.474713i	$0.842552\pi$
$72$	− 3.00000i	− 0.353553i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	− 2.00000i	− 0.223607i
$81$	9.00000	1.00000
$82$	6.00000	0.662589
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 12.0000i	− 1.30158i
$86$	4.00000i	0.431331i
$87$	0	0
$88$	−4.00000	−0.426401
$89$	18.0000i	1.90800i	0.299813	+	0.953998i	$0.403076\pi$
$89$	18.0000i	1.90800i	−0.299813	+	0.953998i	$0.596924\pi$
$90$	6.00000	0.632456
$91$	0	0
$92$	8.00000	0.834058
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	1.00000i	0.101015i
$99$	− 12.0000i	− 1.20605i
$100$	−1.00000	−0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	0	0
$106$	− 6.00000i	− 0.582772i
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	2.00000i	0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	2.00000i	0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	− 8.00000i	− 0.762770i
$111$	0	0
$112$	− 1.00000i	− 0.0944911i
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	16.0000i	1.49201i
$116$	10.0000	0.928477
$117$	0	0
$118$	8.00000	0.736460
$119$	− 6.00000i	− 0.550019i
$120$	0	0
$121$	−5.00000	−0.454545
$122$	− 10.0000i	− 0.905357i
$123$	0	0
$124$	− 8.00000i	− 0.718421i
$125$	− 12.0000i	− 1.07331i
$126$	3.00000	0.267261
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000i	0.514496i
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	8.00000	0.671345
$143$	0	0
$144$	−3.00000	−0.250000
$145$	20.0000i	1.66091i
$146$	2.00000	0.165521
$147$	0	0
$148$	− 6.00000i	− 0.493197i
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−18.0000	−1.45521
$154$	− 4.00000i	− 0.322329i
$155$	16.0000	1.28515
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	− 8.00000i	− 0.636446i
$159$	0	0
$160$	−2.00000	−0.158114
$161$	8.00000i	0.630488i
$162$	− 9.00000i	− 0.707107i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	− 6.00000i	− 0.468521i
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	0	0
$170$	−12.0000	−0.920358
$171$	0	0
$172$	4.00000	0.304997
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	− 1.00000i	− 0.0755929i
$176$	4.00000i	0.301511i
$177$	0	0
$178$	18.0000	1.34916
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	− 6.00000i	− 0.447214i
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	− 8.00000i	− 0.589768i
$185$	12.0000	0.882258
$186$	0	0
$187$	24.0000i	1.75505i
$188$	8.00000i	0.583460i
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	−12.0000	−0.852803
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	1.00000i	0.0707107i
$201$	0	0
$202$	− 14.0000i	− 0.985037i
$203$	10.0000i	0.701862i
$204$	0	0
$205$	12.0000	0.838116
$206$	16.0000i	1.11477i
$207$	24.0000	1.66812
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	4.00000i	0.273434i
$215$	8.00000i	0.545595i
$216$	0	0
$217$	8.00000	0.543075
$218$	2.00000	0.135457
$219$	0	0
$220$	−8.00000	−0.539360
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−1.00000	−0.0668153
$225$	−3.00000	−0.200000
$226$	− 2.00000i	− 0.133038i
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 6.00000i	− 0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$	− 6.00000i	− 0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$	16.0000	1.05501
$231$	0	0
$232$	− 10.0000i	− 0.656532i
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	− 8.00000i	− 0.520756i
$237$	0	0
$238$	−6.00000	−0.388922
$239$	24.0000i	1.55243i	0.630468	+	0.776215i	$0.282863\pi$
$239$	24.0000i	1.55243i	−0.630468	+	0.776215i	$0.717137\pi$
$240$	0	0
$241$	18.0000i	1.15948i	0.814801	+	0.579741i	$0.196846\pi$
$241$	18.0000i	1.15948i	−0.814801	+	0.579741i	$0.803154\pi$
$242$	5.00000i	0.321412i
$243$	0	0
$244$	−10.0000	−0.640184
$245$	2.00000i	0.127775i
$246$	0	0
$247$	0	0
$248$	−8.00000	−0.508001
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	− 3.00000i	− 0.188982i
$253$	− 32.0000i	− 2.01182i
$254$	16.0000i	1.00393i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	30.0000	1.85695
$262$	− 8.00000i	− 0.494242i
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	− 12.0000i	− 0.737154i
$266$	0	0
$267$	0	0
$268$	4.00000i	0.244339i
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	− 32.0000i	− 1.94386i	−0.235267	−	0.971931i	$-0.575596\pi$
$271$	− 32.0000i	− 1.94386i	0.235267	−	0.971931i	$-0.424404\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	4.00000i	0.241209i
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	8.00000i	0.479808i
$279$	− 24.0000i	− 1.43684i
$280$	− 2.00000i	− 0.119523i
$281$	10.0000i	0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$	10.0000i	0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	− 8.00000i	− 0.474713i
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	3.00000i	0.176777i
$289$	19.0000	1.11765
$290$	20.0000	1.17444
$291$	0	0
$292$	− 2.00000i	− 0.117041i
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	−6.00000	−0.348743
$297$	0	0
$298$	18.0000	1.04271
$299$	0	0
$300$	0	0
$301$	4.00000i	0.230556i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 20.0000i	− 1.14520i
$306$	18.0000i	1.02899i
$307$	32.0000i	1.82634i	0.407583	+	0.913168i	$0.366372\pi$
$307$	32.0000i	1.82634i	−0.407583	+	0.913168i	$0.633628\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	− 16.0000i	− 0.908739i
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	− 2.00000i	− 0.112867i
$315$	6.00000	0.338062
$316$	−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	$-0.0538913\pi$
$318$	0	0
$319$	− 40.0000i	− 2.23957i
$320$	2.00000i	0.111803i
$321$	0	0
$322$	8.00000	0.445823
$323$	0	0
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−8.00000	−0.441054
$330$	0	0
$331$	28.0000i	1.53902i	0.638635	+	0.769510i	$0.279499\pi$
$331$	28.0000i	1.53902i	−0.638635	+	0.769510i	$0.720501\pi$
$332$	0	0
$333$	− 18.0000i	− 0.986394i
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	0	0
$340$	12.0000i	0.650791i
$341$	−32.0000	−1.73290
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	− 4.00000i	− 0.215666i
$345$	0	0
$346$	2.00000i	0.107521i
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	− 14.0000i	− 0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$	− 14.0000i	− 0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	4.00000	0.213201
$353$	− 10.0000i	− 0.532246i	−0.963939	−	0.266123i	$-0.914257\pi$
$353$	− 10.0000i	− 0.532246i	0.963939	−	0.266123i	$-0.0857428\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	− 18.0000i	− 0.953998i
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−6.00000	−0.316228
$361$	19.0000	1.00000
$362$	− 6.00000i	− 0.315353i
$363$	0	0
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−8.00000	−0.417029
$369$	− 18.0000i	− 0.937043i
$370$	− 12.0000i	− 0.623850i
$371$	− 6.00000i	− 0.311504i
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	0	0
$379$	− 4.00000i	− 0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$	− 4.00000i	− 0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$	0	0
$381$	0	0
$382$	8.00000i	0.409316i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	− 8.00000i	− 0.407718i
$386$	2.00000	0.101797
$387$	12.0000	0.609994
$388$	2.00000i	0.101535i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	18.0000	0.906827
$395$	− 16.0000i	− 0.805047i
$396$	12.0000i	0.603023i
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	1.00000	0.0500000
$401$	18.0000i	0.898877i	0.893311	+	0.449439i	$0.148376\pi$
$401$	18.0000i	0.898877i	−0.893311	+	0.449439i	$0.851624\pi$
$402$	0	0
$403$	0	0
$404$	−14.0000	−0.696526
$405$	− 18.0000i	− 0.894427i
$406$	10.0000	0.496292
$407$	−24.0000	−1.18964
$408$	0	0
$409$	38.0000i	1.87898i	0.342578	+	0.939490i	$0.388700\pi$
$409$	38.0000i	1.87898i	−0.342578	+	0.939490i	$0.611300\pi$
$410$	− 12.0000i	− 0.592638i
$411$	0	0
$412$	16.0000	0.788263
$413$	8.00000	0.393654
$414$	− 24.0000i	− 1.17954i
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	2.00000i	0.0974740i	0.998812	+	0.0487370i	$0.0155196\pi$
$421$	2.00000i	0.0974740i	−0.998812	+	0.0487370i	$0.984480\pi$
$422$	− 20.0000i	− 0.973585i
$423$	24.0000i	1.16692i
$424$	6.00000i	0.291386i
$425$	6.00000	0.291043
$426$	0	0
$427$	− 10.0000i	− 0.483934i
$428$	4.00000	0.193347
$429$	0	0
$430$	8.00000	0.385794
$431$	24.0000i	1.15604i	0.816023	+	0.578020i	$0.196174\pi$
$431$	24.0000i	1.15604i	−0.816023	+	0.578020i	$0.803826\pi$
$432$	0	0
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	− 8.00000i	− 0.384012i
$435$	0	0
$436$	− 2.00000i	− 0.0957826i
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	8.00000i	0.381385i
$441$	3.00000	0.142857
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	36.0000	1.70656
$446$	0	0
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	− 14.0000i	− 0.660701i	−0.943858	−	0.330350i	$-0.892833\pi$
$449$	− 14.0000i	− 0.660701i	0.943858	−	0.330350i	$-0.107167\pi$
$450$	3.00000i	0.141421i
$451$	−24.0000	−1.13012
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	− 16.0000i	− 0.746004i
$461$	− 18.0000i	− 0.838344i	−0.907907	−	0.419172i	$-0.862320\pi$
$461$	− 18.0000i	− 0.838344i	0.907907	−	0.419172i	$-0.137680\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	10.0000i	0.463241i
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	16.0000i	0.738025i
$471$	0	0
$472$	−8.00000	−0.368230
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	0	0
$476$	6.00000i	0.275010i
$477$	−18.0000	−0.824163
$478$	24.0000	1.09773
$479$	24.0000i	1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$	24.0000i	1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$	0	0
$481$	0	0
$482$	18.0000	0.819878
$483$	0	0
$484$	5.00000	0.227273
$485$	−4.00000	−0.181631
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	10.0000i	0.452679i
$489$	0	0
$490$	2.00000	0.0903508
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	−60.0000	−2.70226
$494$	0	0
$495$	−24.0000	−1.07872
$496$	8.00000i	0.359211i
$497$	8.00000	0.358849
$498$	0	0
$499$	− 4.00000i	− 0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$	− 4.00000i	− 0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	12.0000i	0.536656i
$501$	0	0
$502$	24.0000i	1.07117i
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	−3.00000	−0.133631
$505$	− 28.0000i	− 1.24598i
$506$	−32.0000	−1.42257
$507$	0	0
$508$	16.0000	0.709885
$509$	14.0000i	0.620539i	0.950649	+	0.310270i	$0.100419\pi$
$509$	14.0000i	0.620539i	−0.950649	+	0.310270i	$0.899581\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	18.0000i	0.793946i
$515$	32.0000i	1.41009i
$516$	0	0
$517$	32.0000	1.40736
$518$	− 6.00000i	− 0.263625i
$519$	0	0
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	− 30.0000i	− 1.31306i
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	0	0
$527$	48.0000i	2.09091i
$528$	0	0
$529$	41.0000	1.78261
$530$	−12.0000	−0.521247
$531$	− 24.0000i	− 1.04151i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	8.00000i	0.345870i
$536$	4.00000	0.172774
$537$	0	0
$538$	30.0000i	1.29339i
$539$	− 4.00000i	− 0.172292i
$540$	0	0
$541$	− 10.0000i	− 0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$	− 10.0000i	− 0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$	−32.0000	−1.37452
$543$	0	0
$544$	− 6.00000i	− 0.257248i
$545$	4.00000	0.171341
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	6.00000i	0.256307i
$549$	−30.0000	−1.28037
$550$	4.00000	0.170561
$551$	0	0
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	6.00000i	0.254916i
$555$	0	0
$556$	8.00000	0.339276
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	−24.0000	−1.01600
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	10.0000	0.421825
$563$	16.0000	0.674320	0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$564$	0	0
$565$	− 4.00000i	− 0.168281i
$566$	16.0000i	0.672530i
$567$	− 9.00000i	− 0.377964i
$568$	−8.00000	−0.335673
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	0	0
$574$	− 6.00000i	− 0.250435i
$575$	−8.00000	−0.333623
$576$	3.00000	0.125000
$577$	− 42.0000i	− 1.74848i	−0.485491	−	0.874241i	$-0.661359\pi$
$577$	− 42.0000i	− 1.74848i	0.485491	−	0.874241i	$-0.338641\pi$
$578$	− 19.0000i	− 0.790296i
$579$	0	0
$580$	− 20.0000i	− 0.830455i
$581$	0	0
$582$	0	0
$583$	24.0000i	0.993978i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−14.0000	−0.578335
$587$	16.0000i	0.660391i	0.943913	+	0.330195i	$0.107115\pi$
$587$	16.0000i	0.660391i	−0.943913	+	0.330195i	$0.892885\pi$
$588$	0	0
$589$	0	0
$590$	− 16.0000i	− 0.658710i
$591$	0	0
$592$	6.00000i	0.246598i
$593$	− 22.0000i	− 0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$	− 22.0000i	− 0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	− 18.0000i	− 0.737309i
$597$	0	0
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	4.00000	0.163028
$603$	12.0000i	0.488678i
$604$	0	0
$605$	10.0000i	0.406558i
$606$	0	0
$607$	−48.0000	−1.94826	−0.974130	−	0.225989i	$-0.927439\pi$
$607$	−48.0000	−1.94826	−0.974130	+	0.225989i	$0.927439\pi$
$608$	0	0
$609$	0	0
$610$	−20.0000	−0.809776
$611$	0	0
$612$	18.0000	0.727607
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	4.00000i	0.161165i
$617$	− 10.0000i	− 0.402585i	−0.979531	−	0.201292i	$-0.935486\pi$
$617$	− 10.0000i	− 0.402585i	0.979531	−	0.201292i	$-0.0645141\pi$
$618$	0	0
$619$	40.0000i	1.60774i	0.594808	+	0.803868i	$0.297228\pi$
$619$	40.0000i	1.60774i	−0.594808	+	0.803868i	$0.702772\pi$
$620$	−16.0000	−0.642575
$621$	0	0
$622$	24.0000i	0.962312i
$623$	18.0000	0.721155
$624$	0	0
$625$	−19.0000	−0.760000
$626$	− 2.00000i	− 0.0799361i
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	36.0000i	1.43541i
$630$	− 6.00000i	− 0.239046i
$631$	− 8.00000i	− 0.318475i	−0.987240	−	0.159237i	$-0.949096\pi$
$631$	− 8.00000i	− 0.318475i	0.987240	−	0.159237i	$-0.0509036\pi$
$632$	8.00000i	0.318223i
$633$	0	0
$634$	−6.00000	−0.238290
$635$	32.0000i	1.26988i
$636$	0	0
$637$	0	0
$638$	−40.0000	−1.58362
$639$	− 24.0000i	− 0.949425i
$640$	2.00000	0.0790569
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	− 8.00000i	− 0.315489i	−0.987480	−	0.157745i	$-0.949578\pi$
$643$	− 8.00000i	− 0.315489i	0.987480	−	0.157745i	$-0.0504223\pi$
$644$	− 8.00000i	− 0.315244i
$645$	0	0
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	9.00000i	0.353553i
$649$	−32.0000	−1.25611
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	− 16.0000i	− 0.625172i
$656$	6.00000i	0.234261i
$657$	− 6.00000i	− 0.234082i
$658$	8.00000i	0.311872i
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	2.00000i	0.0777910i	0.999243	+	0.0388955i	$0.0123839\pi$
$661$	2.00000i	0.0777910i	−0.999243	+	0.0388955i	$0.987616\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−18.0000	−0.697486
$667$	80.0000	3.09761
$668$	0	0
$669$	0	0
$670$	8.00000i	0.309067i
$671$	40.0000i	1.54418i
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	2.00000i	0.0770371i
$675$	0	0
$676$	0	0
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	12.0000	0.460179
$681$	0	0
$682$	32.0000i	1.22534i
$683$	28.0000i	1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$	28.0000i	1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	1.00000	0.0381802
$687$	0	0
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	− 16.0000i	− 0.608669i	−0.952565	−	0.304334i	$-0.901566\pi$
$691$	− 16.0000i	− 0.608669i	0.952565	−	0.304334i	$-0.0984340\pi$
$692$	2.00000	0.0760286
$693$	−12.0000	−0.455842
$694$	12.0000i	0.455514i
$695$	16.0000i	0.606915i
$696$	0	0
$697$	36.0000i	1.36360i
$698$	−14.0000	−0.529908
$699$	0	0
$700$	1.00000i	0.0377964i
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	0	0
$704$	− 4.00000i	− 0.150756i
$705$	0	0
$706$	−10.0000	−0.376355
$707$	− 14.0000i	− 0.526524i
$708$	0	0
$709$	− 10.0000i	− 0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$	− 10.0000i	− 0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$	− 16.0000i	− 0.600469i
$711$	−24.0000	−0.900070
$712$	−18.0000	−0.674579
$713$	− 64.0000i	− 2.39682i
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	6.00000i	0.223607i
$721$	16.0000i	0.595871i
$722$	− 19.0000i	− 0.707107i
$723$	0	0
$724$	−6.00000	−0.222988
$725$	−10.0000	−0.371391
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	− 4.00000i	− 0.148047i
$731$	−24.0000	−0.887672
$732$	0	0
$733$	− 26.0000i	− 0.960332i	−0.877178	−	0.480166i	$-0.840576\pi$
$733$	− 26.0000i	− 0.960332i	0.877178	−	0.480166i	$-0.159424\pi$
$734$	16.0000i	0.590571i
$735$	0	0
$736$	8.00000i	0.294884i
$737$	16.0000	0.589368
$738$	−18.0000	−0.662589
$739$	− 36.0000i	− 1.32428i	−0.749380	−	0.662141i	$-0.769648\pi$
$739$	− 36.0000i	− 1.32428i	0.749380	−	0.662141i	$-0.230352\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−6.00000	−0.220267
$743$	− 16.0000i	− 0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$	− 16.0000i	− 0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$	0	0
$745$	36.0000	1.31894
$746$	10.0000i	0.366126i
$747$	0	0
$748$	− 24.0000i	− 0.877527i
$749$	4.00000i	0.146157i
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	− 8.00000i	− 0.291730i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	0	0
$761$	− 6.00000i	− 0.217500i	−0.994069	−	0.108750i	$-0.965315\pi$
$761$	− 6.00000i	− 0.217500i	0.994069	−	0.108750i	$-0.0346848\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	8.00000	0.289430
$765$	36.0000i	1.30158i
$766$	−24.0000	−0.867155
$767$	0	0
$768$	0	0
$769$	30.0000i	1.08183i	0.841078	+	0.540914i	$0.181921\pi$
$769$	30.0000i	1.08183i	−0.841078	+	0.540914i	$0.818079\pi$
$770$	−8.00000	−0.288300
$771$	0	0
$772$	− 2.00000i	− 0.0719816i
$773$	46.0000i	1.65451i	0.561830	+	0.827253i	$0.310097\pi$
$773$	46.0000i	1.65451i	−0.561830	+	0.827253i	$0.689903\pi$
$774$	− 12.0000i	− 0.431331i
$775$	8.00000i	0.287368i
$776$	2.00000	0.0717958
$777$	0	0
$778$	30.0000i	1.07555i
$779$	0	0
$780$	0	0
$781$	−32.0000	−1.14505
$782$	48.0000i	1.71648i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	− 4.00000i	− 0.142766i
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	− 18.0000i	− 0.641223i
$789$	0	0
$790$	−16.0000	−0.569254
$791$	− 2.00000i	− 0.0711118i
$792$	12.0000	0.426401
$793$	0	0
$794$	2.00000	0.0709773
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	− 48.0000i	− 1.69812i
$800$	− 1.00000i	− 0.0353553i
$801$	− 54.0000i	− 1.90800i
$802$	18.0000	0.635602
$803$	−8.00000	−0.282314
$804$	0	0
$805$	16.0000	0.563926
$806$	0	0
$807$	0	0
$808$	14.0000i	0.492518i
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−18.0000	−0.632456
$811$	− 8.00000i	− 0.280918i	−0.990086	−	0.140459i	$-0.955142\pi$
$811$	− 8.00000i	− 0.280918i	0.990086	−	0.140459i	$-0.0448578\pi$
$812$	− 10.0000i	− 0.350931i
$813$	0	0
$814$	24.0000i	0.841200i
$815$	−8.00000	−0.280228
$816$	0	0
$817$	0	0
$818$	38.0000	1.32864
$819$	0	0
$820$	−12.0000	−0.419058
$821$	− 14.0000i	− 0.488603i	−0.969699	−	0.244302i	$-0.921441\pi$
$821$	− 14.0000i	− 0.488603i	0.969699	−	0.244302i	$-0.0785587\pi$
$822$	0	0
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	− 16.0000i	− 0.557386i
$825$	0	0
$826$	− 8.00000i	− 0.278356i
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	−24.0000	−0.834058
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	32.0000i	1.10476i	0.833592	+	0.552381i	$0.186281\pi$
$839$	32.0000i	1.10476i	−0.833592	+	0.552381i	$0.813719\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	2.00000	0.0689246
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	24.0000	0.825137
$847$	5.00000i	0.171802i
$848$	6.00000	0.206041
$849$	0	0
$850$	− 6.00000i	− 0.205798i
$851$	− 48.0000i	− 1.64542i
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	− 4.00000i	− 0.136717i
$857$	−34.0000	−1.16142	−0.580709	−	0.814111i	$-0.697225\pi$
$857$	−34.0000	−1.16142	−0.580709	+	0.814111i	$0.697225\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	− 8.00000i	− 0.272798i
$861$	0	0
$862$	24.0000	0.817443
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	0	0
$865$	4.00000i	0.136004i
$866$	− 38.0000i	− 1.29129i
$867$	0	0
$868$	−8.00000	−0.271538
$869$	32.0000i	1.08553i
$870$	0	0
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	6.00000i	0.203069i
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	− 30.0000i	− 1.01303i	−0.862232	−	0.506514i	$-0.830934\pi$
$877$	− 30.0000i	− 1.01303i	0.862232	−	0.506514i	$-0.169066\pi$
$878$	8.00000i	0.269987i
$879$	0	0
$880$	8.00000	0.269680
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	− 3.00000i	− 0.101015i
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	0	0
$886$	36.0000i	1.20944i
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	0	0
$889$	16.0000i	0.536623i
$890$	− 36.0000i	− 1.20672i
$891$	36.0000i	1.20605i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 24.0000i	− 0.802232i
$896$	1.00000	0.0334077
$897$	0	0
$898$	−14.0000	−0.467186
$899$	− 80.0000i	− 2.66815i
$900$	3.00000	0.100000
$901$	36.0000	1.19933
$902$	24.0000i	0.799113i
$903$	0	0
$904$	2.00000i	0.0665190i
$905$	− 12.0000i	− 0.398893i
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	0	0
$909$	−42.0000	−1.39305
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	6.00000i	0.198246i
$917$	− 8.00000i	− 0.264183i
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	−16.0000	−0.527504
$921$	0	0
$922$	−18.0000	−0.592798
$923$	0	0
$924$	0	0
$925$	6.00000i	0.197279i
$926$	16.0000	0.525793
$927$	48.0000	1.57653
$928$	10.0000i	0.328266i
$929$	30.0000i	0.984268i	0.870519	+	0.492134i	$0.163783\pi$
$929$	30.0000i	0.984268i	−0.870519	+	0.492134i	$0.836217\pi$
$930$	0	0
$931$	0	0
$932$	10.0000	0.327561
$933$	0	0
$934$	8.00000i	0.261768i
$935$	48.0000	1.56977
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	4.00000i	0.130605i
$939$	0	0
$940$	16.0000	0.521862
$941$	− 10.0000i	− 0.325991i	−0.986627	−	0.162995i	$-0.947884\pi$
$941$	− 10.0000i	− 0.325991i	0.986627	−	0.162995i	$-0.0521156\pi$
$942$	0	0
$943$	− 48.0000i	− 1.56310i
$944$	8.00000i	0.260378i
$945$	0	0
$946$	−16.0000	−0.520205
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	6.00000	0.194461
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	18.0000i	0.582772i
$955$	16.0000i	0.517748i
$956$	− 24.0000i	− 0.776215i
$957$	0	0
$958$	24.0000	0.775405
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−33.0000	−1.06452
$962$	0	0
$963$	12.0000	0.386695
$964$	− 18.0000i	− 0.579741i
$965$	4.00000	0.128765
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	− 5.00000i	− 0.160706i
$969$	0	0
$970$	4.00000i	0.128432i
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	8.00000i	0.256468i
$974$	16.0000	0.512673
$975$	0	0
$976$	10.0000	0.320092
$977$	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
$979$	−72.0000	−2.30113
$980$	− 2.00000i	− 0.0638877i
$981$	− 6.00000i	− 0.191565i
$982$	28.0000i	0.893516i
$983$	32.0000i	1.02064i	0.859984	+	0.510321i	$0.170473\pi$
$983$	32.0000i	1.02064i	−0.859984	+	0.510321i	$0.829527\pi$
$984$	0	0
$985$	36.0000	1.14706
$986$	60.0000i	1.91079i
$987$	0	0
$988$	0	0
$989$	32.0000	1.01754
$990$	24.0000i	0.762770i
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	− 8.00000i	− 0.253745i
$995$	− 32.0000i	− 1.01447i
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.d.d.337.1		2
13.5	odd	4		2366.2.a.d.1.1		1
13.8	odd	4		182.2.a.c.1.1	✓	1
13.12	even	2	inner	2366.2.d.d.337.2		2
39.8	even	4		1638.2.a.c.1.1		1
52.47	even	4		1456.2.a.i.1.1		1
65.34	odd	4		4550.2.a.g.1.1		1
91.34	even	4		1274.2.a.l.1.1		1
91.47	even	12		1274.2.f.g.1145.1		2
91.60	odd	12		1274.2.f.f.79.1		2
91.73	even	12		1274.2.f.g.79.1		2
91.86	odd	12		1274.2.f.f.1145.1		2
104.21	odd	4		5824.2.a.l.1.1		1
104.99	even	4		5824.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.a.c.1.1	✓	1	13.8	odd	4
1274.2.a.l.1.1		1	91.34	even	4
1274.2.f.f.79.1		2	91.60	odd	12
1274.2.f.f.1145.1		2	91.86	odd	12
1274.2.f.g.79.1		2	91.73	even	12
1274.2.f.g.1145.1		2	91.47	even	12
1456.2.a.i.1.1		1	52.47	even	4
1638.2.a.c.1.1		1	39.8	even	4
2366.2.a.d.1.1		1	13.5	odd	4
2366.2.d.d.337.1		2	1.1	even	1	trivial
2366.2.d.d.337.2		2	13.12	even	2	inner
4550.2.a.g.1.1		1	65.34	odd	4
5824.2.a.l.1.1		1	104.21	odd	4
5824.2.a.m.1.1		1	104.99	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:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2366.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} -3.00000 q^{9} -2.00000 q^{10} +4.00000i q^{11} -1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +3.00000i q^{18} +2.00000i q^{20} +4.00000 q^{22} -8.00000 q^{23} +1.00000 q^{25} +1.00000i q^{28} -10.0000 q^{29} +8.00000i q^{31} -1.00000i q^{32} -6.00000i q^{34} -2.00000 q^{35} +3.00000 q^{36} +6.00000i q^{37} +2.00000 q^{40} +6.00000i q^{41} -4.00000 q^{43} -4.00000i q^{44} +6.00000i q^{45} +8.00000i q^{46} -8.00000i q^{47} -1.00000 q^{49} -1.00000i q^{50} +6.00000 q^{53} +8.00000 q^{55} +1.00000 q^{56} +10.0000i q^{58} +8.00000i q^{59} +10.0000 q^{61} +8.00000 q^{62} +3.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} -6.00000 q^{68} +2.00000i q^{70} +8.00000i q^{71} -3.00000i q^{72} +2.00000i q^{73} +6.00000 q^{74} +4.00000 q^{77} +8.00000 q^{79} -2.00000i q^{80} +9.00000 q^{81} +6.00000 q^{82} -12.0000i q^{85} +4.00000i q^{86} -4.00000 q^{88} +18.0000i q^{89} +6.00000 q^{90} +8.00000 q^{92} -8.00000 q^{94} -2.00000i q^{97} +1.00000i q^{98} -12.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 6 * q^9	\( 2 q - 2 q^{4} - 6 q^{9} - 4 q^{10} - 2 q^{14} + 2 q^{16} + 12 q^{17} + 8 q^{22} - 16 q^{23} + 2 q^{25} - 20 q^{29} - 4 q^{35} + 6 q^{36} + 4 q^{40} - 8 q^{43} - 2 q^{49} + 12 q^{53} + 16 q^{55} + 2 q^{56} + 20 q^{61} + 16 q^{62} - 2 q^{64} - 12 q^{68} + 12 q^{74} + 8 q^{77} + 16 q^{79} + 18 q^{81} + 12 q^{82} - 8 q^{88} + 12 q^{90} + 16 q^{92} - 16 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 6 * q^9 - 4 * q^10 - 2 * q^14 + 2 * q^16 + 12 * q^17 + 8 * q^22 - 16 * q^23 + 2 * q^25 - 20 * q^29 - 4 * q^35 + 6 * q^36 + 4 * q^40 - 8 * q^43 - 2 * q^49 + 12 * q^53 + 16 * q^55 + 2 * q^56 + 20 * q^61 + 16 * q^62 - 2 * q^64 - 12 * q^68 + 12 * q^74 + 8 * q^77 + 16 * q^79 + 18 * q^81 + 12 * q^82 - 8 * q^88 + 12 * q^90 + 16 * q^92 - 16 * q^94

Introduction

L-functions

Modular forms

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Database

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