LMFDB - Embedded newform 2325.2.c.d.1024.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2325,2,Mod(1024,2325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2325, 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("2325.1024");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2325 = 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2325.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:	$18.5652184699$
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 465)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1024.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2325.1024
Dual form			2325.2.c.d.1024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +2.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +8.00000 q^{19} +2.00000 q^{21} -4.00000i q^{22} +8.00000i q^{23} -3.00000 q^{24} -1.00000i q^{27} -2.00000i q^{28} +1.00000 q^{31} +5.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} +8.00000i q^{37} +8.00000i q^{38} -6.00000 q^{41} +2.00000i q^{42} -4.00000 q^{44} -8.00000 q^{46} +4.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} +1.00000 q^{54} +6.00000 q^{56} +8.00000i q^{57} -10.0000 q^{59} -14.0000 q^{61} +1.00000i q^{62} +2.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} +2.00000i q^{67} +2.00000i q^{68} -8.00000 q^{69} +6.00000 q^{71} -3.00000i q^{72} +16.0000i q^{73} -8.00000 q^{74} +8.00000 q^{76} +8.00000i q^{77} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} +2.00000 q^{84} -12.0000i q^{88} -4.00000 q^{89} +8.00000i q^{92} +1.00000i q^{93} -4.00000 q^{94} -5.00000 q^{96} +6.00000i q^{97} +3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} + 4 q^{14} - 2 q^{16} + 16 q^{19} + 4 q^{21} - 6 q^{24} + 2 q^{31} - 4 q^{34} - 2 q^{36} - 12 q^{41} - 8 q^{44} - 16 q^{46} + 6 q^{49} - 4 q^{51} + 2 q^{54}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^11 + 4 * q^14 - 2 * q^16 + 16 * q^19 + 4 * q^21 - 6 * q^24 + 2 * q^31 - 4 * q^34 - 2 * q^36 - 12 * q^41 - 8 * q^44 - 16 * q^46 + 6 * q^49 - 4 * q^51 + 2 * q^54 + 12 * q^56 - 20 * q^59 - 28 * q^61 - 14 * q^64 + 8 * q^66 - 16 * q^69 + 12 * q^71 - 16 * q^74 + 16 * q^76 + 2 * q^81 + 4 * q^84 - 8 * q^89 - 8 * q^94 - 10 * q^96 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2325\mathbb{Z}\right)^\times$.

$n$	$652$	$776$	$1801$
$\chi(n)$	$-1$	$1$	$1$

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
$7$	− 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
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	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
$17$	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
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	− 4.00000i	− 0.852803i
$23$	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
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.00000i	0.883883i
$33$	− 4.00000i	− 0.696311i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	8.00000i	1.29777i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	2.00000i	0.308607i
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−8.00000	−1.17954
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	6.00000	0.801784
$57$	8.00000i	1.05963i
$58$	0	0
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	1.00000i	0.127000i
$63$	2.00000i	0.251976i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	2.00000i	0.242536i
$69$	−8.00000	−0.963087
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	− 3.00000i	− 0.353553i
$73$	16.0000i	1.87266i	0.351123	+	0.936329i	$0.385800\pi$
$73$	16.0000i	1.87266i	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	8.00000	0.917663
$77$	8.00000i	0.911685i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	0	0
$87$	0	0
$88$	− 12.0000i	− 1.27920i
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	0	0
$92$	8.00000i	0.834058i
$93$	1.00000i	0.103695i
$94$	−4.00000	−0.412568
$95$	0	0
$96$	−5.00000	−0.510310
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	3.00000i	0.303046i
$99$	4.00000	0.402015
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	− 2.00000i	− 0.198030i
$103$	− 2.00000i	− 0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$	− 2.00000i	− 0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	0	0
$105$	0	0
$106$	6.00000	0.582772
$107$	16.0000i	1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$	16.0000i	1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	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	$0.908930\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	0	0
$117$	0	0
$118$	− 10.0000i	− 0.920575i
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	− 14.0000i	− 1.26750i
$123$	− 6.00000i	− 0.541002i
$124$	1.00000	0.0898027
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	3.00000i	0.265165i
$129$	0	0
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	− 4.00000i	− 0.348155i
$133$	− 16.0000i	− 1.38738i
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−6.00000	−0.514496
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	− 8.00000i	− 0.681005i
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	6.00000i	0.503509i
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−16.0000	−1.32417
$147$	3.00000i	0.247436i
$148$	8.00000i	0.657596i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	24.0000i	1.94666i
$153$	− 2.00000i	− 0.161690i
$154$	−8.00000	−0.644658
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	16.0000	1.26098
$162$	1.00000i	0.0785674i
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	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
$171$	−8.00000	−0.611775
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	− 10.0000i	− 0.751646i
$178$	− 4.00000i	− 0.299813i
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	− 14.0000i	− 1.03491i
$184$	−24.0000	−1.76930
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	− 8.00000i	− 0.585018i
$188$	4.00000i	0.291730i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	− 7.00000i	− 0.505181i
$193$	− 6.00000i	− 0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	− 6.00000i	− 0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	3.00000	0.214286
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	4.00000i	0.284268i
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	14.0000i	0.985037i
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	2.00000	0.139347
$207$	− 8.00000i	− 0.556038i
$208$	0	0
$209$	−32.0000	−2.21349
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	− 6.00000i	− 0.412082i
$213$	6.00000i	0.411113i
$214$	−16.0000	−1.09374
$215$	0	0
$216$	3.00000	0.204124
$217$	− 2.00000i	− 0.135769i
$218$	14.0000i	0.948200i
$219$	−16.0000	−1.08118
$220$	0	0
$221$	0	0
$222$	− 8.00000i	− 0.536925i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	10.0000	0.668153
$225$	0	0
$226$	−6.00000	−0.399114
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	8.00000i	0.529813i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	−10.0000	−0.650945
$237$	0	0
$238$	4.00000i	0.259281i
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	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
$247$	0	0
$248$	3.00000i	0.190500i
$249$	4.00000	0.253490
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	2.00000i	0.125988i
$253$	− 32.0000i	− 2.01182i
$254$	4.00000	0.250982
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	0	0
$262$	− 14.0000i	− 0.864923i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	12.0000	0.738549
$265$	0	0
$266$	16.0000	0.981023
$267$	− 4.00000i	− 0.244796i
$268$	2.00000i	0.122169i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	18.0000	1.08742
$275$	0	0
$276$	−8.00000	−0.481543
$277$	− 28.0000i	− 1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	− 28.0000i	− 1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$	12.0000i	0.719712i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	− 4.00000i	− 0.238197i
$283$	− 10.0000i	− 0.594438i	−0.954809	−	0.297219i	$-0.903941\pi$
$283$	− 10.0000i	− 0.594438i	0.954809	−	0.297219i	$-0.0960592\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	− 5.00000i	− 0.294628i
$289$	13.0000	0.764706
$290$	0	0
$291$	−6.00000	−0.351726
$292$	16.0000i	0.936329i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−24.0000	−1.39497
$297$	4.00000i	0.232104i
$298$	18.0000i	1.04271i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 24.0000i	− 1.38104i
$303$	14.0000i	0.804279i
$304$	−8.00000	−0.458831
$305$	0	0
$306$	2.00000	0.114332
$307$	22.0000i	1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$	22.0000i	1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$	8.00000i	0.455842i
$309$	2.00000	0.113776
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	8.00000i	0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$	8.00000i	0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	6.00000i	0.336463i
$319$	0	0
$320$	0	0
$321$	−16.0000	−0.893033
$322$	16.0000i	0.891645i
$323$	16.0000i	0.890264i
$324$	1.00000	0.0555556
$325$	0	0
$326$	2.00000	0.110770
$327$	14.0000i	0.774202i
$328$	− 18.0000i	− 0.993884i
$329$	8.00000	0.441054
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	− 4.00000i	− 0.219529i
$333$	− 8.00000i	− 0.438397i
$334$	−16.0000	−0.875481
$335$	0	0
$336$	−2.00000	−0.109109
$337$	28.0000i	1.52526i	0.646837	+	0.762629i	$0.276092\pi$
$337$	28.0000i	1.52526i	−0.646837	+	0.762629i	$0.723908\pi$
$338$	13.0000i	0.707107i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−4.00000	−0.216612
$342$	− 8.00000i	− 0.432590i
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	6.00000	0.322562
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	− 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	$0.878476\pi$
$354$	10.0000	0.531494
$355$	0	0
$356$	−4.00000	−0.212000
$357$	4.00000i	0.211702i
$358$	4.00000i	0.211407i
$359$	−26.0000	−1.37223	−0.686114	−	0.727494i	$-0.740685\pi$
$359$	−26.0000	−1.37223	−0.686114	+	0.727494i	$0.740685\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	− 6.00000i	− 0.315353i
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	14.0000	0.731792
$367$	− 32.0000i	− 1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$	− 32.0000i	− 1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$	− 8.00000i	− 0.417029i
$369$	6.00000	0.312348
$370$	0	0
$371$	−12.0000	−0.623009
$372$	1.00000i	0.0518476i
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	− 2.00000i	− 0.102869i
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	− 14.0000i	− 0.716302i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	6.00000	0.305392
$387$	0	0
$388$	6.00000i	0.304604i
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	9.00000i	0.454569i
$393$	− 14.0000i	− 0.706207i
$394$	−2.00000	−0.100759
$395$	0	0
$396$	4.00000	0.201008
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	8.00000i	0.401004i
$399$	16.0000	0.801002
$400$	0	0
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	− 2.00000i	− 0.0997509i
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	0	0
$407$	− 32.0000i	− 1.58618i
$408$	− 6.00000i	− 0.297044i
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	− 2.00000i	− 0.0985329i
$413$	20.0000i	0.984136i
$414$	8.00000	0.393179
$415$	0	0
$416$	0	0
$417$	12.0000i	0.587643i
$418$	− 32.0000i	− 1.56517i
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	8.00000i	0.389434i
$423$	− 4.00000i	− 0.194487i
$424$	18.0000	0.874157
$425$	0	0
$426$	−6.00000	−0.290701
$427$	28.0000i	1.35501i
$428$	16.0000i	0.773389i
$429$	0	0
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	1.00000i	0.0481125i
$433$	− 24.0000i	− 1.15337i	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	− 24.0000i	− 1.15337i	0.816968	−	0.576683i	$-0.195653\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	14.0000	0.670478
$437$	64.0000i	3.06154i
$438$	− 16.0000i	− 0.764510i
$439$	−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
$443$	20.0000i	0.950229i	0.879924	+	0.475114i	$0.157593\pi$
$443$	20.0000i	0.950229i	−0.879924	+	0.475114i	$0.842407\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	16.0000	0.757622
$447$	18.0000i	0.851371i
$448$	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
$451$	24.0000	1.13012
$452$	6.00000i	0.282216i
$453$	− 24.0000i	− 1.12762i
$454$	20.0000	0.938647
$455$	0	0
$456$	−24.0000	−1.12390
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	− 6.00000i	− 0.280362i
$459$	2.00000	0.0933520
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	− 8.00000i	− 0.372194i
$463$	− 20.0000i	− 0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$	− 20.0000i	− 0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$	0	0
$465$	0	0
$466$	6.00000	0.277945
$467$	40.0000i	1.85098i	0.378773	+	0.925490i	$0.376346\pi$
$467$	40.0000i	1.85098i	−0.378773	+	0.925490i	$0.623654\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	− 30.0000i	− 1.38086i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	4.00000	0.183340
$477$	6.00000i	0.274721i
$478$	20.0000i	0.914779i
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	0	0
$482$	− 22.0000i	− 1.00207i
$483$	16.0000i	0.728025i
$484$	5.00000	0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 4.00000i	− 0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$	− 4.00000i	− 0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$	− 42.0000i	− 1.90125i
$489$	2.00000	0.0904431
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	− 6.00000i	− 0.270501i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	− 12.0000i	− 0.538274i
$498$	4.00000i	0.179244i
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	20.0000i	0.892644i
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	32.0000	1.42257
$507$	13.0000i	0.577350i
$508$	− 4.00000i	− 0.177471i
$509$	28.0000	1.24108	0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	0	0
$511$	32.0000	1.41560
$512$	− 11.0000i	− 0.486136i
$513$	− 8.00000i	− 0.353209i
$514$	18.0000	0.793946
$515$	0	0
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	16.0000i	0.703000i
$519$	6.00000	0.263371
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	8.00000i	0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	8.00000i	0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	−24.0000	−1.04645
$527$	2.00000i	0.0871214i
$528$	4.00000i	0.174078i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	10.0000	0.433963
$532$	− 16.0000i	− 0.693688i
$533$	0	0
$534$	4.00000	0.173097
$535$	0	0
$536$	−6.00000	−0.259161
$537$	4.00000i	0.172613i
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	8.00000i	0.343629i
$543$	− 6.00000i	− 0.257485i
$544$	−10.0000	−0.428746
$545$	0	0
$546$	0	0
$547$	− 22.0000i	− 0.940652i	−0.882493	−	0.470326i	$-0.844136\pi$
$547$	− 22.0000i	− 0.940652i	0.882493	−	0.470326i	$-0.155864\pi$
$548$	− 18.0000i	− 0.768922i
$549$	14.0000	0.597505
$550$	0	0
$551$	0	0
$552$	− 24.0000i	− 1.02151i
$553$	0	0
$554$	28.0000	1.18961
$555$	0	0
$556$	12.0000	0.508913
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	0	0
$560$	0	0
$561$	8.00000	0.337760
$562$	30.0000i	1.26547i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	10.0000	0.420331
$567$	− 2.00000i	− 0.0839921i
$568$	18.0000i	0.755263i
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	− 14.0000i	− 0.584858i
$574$	−12.0000	−0.500870
$575$	0	0
$576$	7.00000	0.291667
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	13.0000i	0.540729i
$579$	6.00000	0.249351
$580$	0	0
$581$	−8.00000	−0.331896
$582$	− 6.00000i	− 0.248708i
$583$	24.0000i	0.993978i
$584$	−48.0000	−1.98625
$585$	0	0
$586$	26.0000	1.07405
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	3.00000i	0.123718i
$589$	8.00000	0.329634
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	− 8.00000i	− 0.328798i
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	18.0000	0.737309
$597$	8.00000i	0.327418i
$598$	0	0
$599$	14.0000	0.572024	0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000	0.572024	0.286012	+	0.958226i	$0.407670\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	− 2.00000i	− 0.0814463i
$604$	−24.0000	−0.976546
$605$	0	0
$606$	−14.0000	−0.568711
$607$	− 46.0000i	− 1.86708i	−0.358470	−	0.933541i	$-0.616702\pi$
$607$	− 46.0000i	− 1.86708i	0.358470	−	0.933541i	$-0.383298\pi$
$608$	40.0000i	1.62221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	− 2.00000i	− 0.0808452i
$613$	44.0000i	1.77714i	0.458738	+	0.888572i	$0.348302\pi$
$613$	44.0000i	1.77714i	−0.458738	+	0.888572i	$0.651698\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	−24.0000	−0.966988
$617$	− 38.0000i	− 1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	− 38.0000i	− 1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$	2.00000i	0.0804518i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	18.0000i	0.721734i
$623$	8.00000i	0.320513i
$624$	0	0
$625$	0	0
$626$	−8.00000	−0.319744
$627$	− 32.0000i	− 1.27796i
$628$	2.00000i	0.0798087i
$629$	−16.0000	−0.637962
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	8.00000i	0.317971i
$634$	6.00000	0.238290
$635$	0	0
$636$	6.00000	0.237915
$637$	0	0
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	− 16.0000i	− 0.631470i
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	16.0000	0.630488
$645$	0	0
$646$	−16.0000	−0.629512
$647$	− 16.0000i	− 0.629025i	−0.949253	−	0.314512i	$-0.898159\pi$
$647$	− 16.0000i	− 0.629025i	0.949253	−	0.314512i	$-0.101841\pi$
$648$	3.00000i	0.117851i
$649$	40.0000	1.57014
$650$	0	0
$651$	2.00000	0.0783862
$652$	− 2.00000i	− 0.0783260i
$653$	− 10.0000i	− 0.391330i	−0.980671	−	0.195665i	$-0.937313\pi$
$653$	− 10.0000i	− 0.391330i	0.980671	−	0.195665i	$-0.0626866\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	6.00000	0.234261
$657$	− 16.0000i	− 0.624219i
$658$	8.00000i	0.311872i
$659$	42.0000	1.63609	0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000	1.63609	0.818044	+	0.575156i	$0.195059\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	− 4.00000i	− 0.155464i
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	16.0000i	0.619059i
$669$	16.0000	0.618596
$670$	0	0
$671$	56.0000	2.16186
$672$	10.0000i	0.385758i
$673$	44.0000i	1.69608i	0.529936	+	0.848038i	$0.322216\pi$
$673$	44.0000i	1.69608i	−0.529936	+	0.848038i	$0.677784\pi$
$674$	−28.0000	−1.07852
$675$	0	0
$676$	13.0000	0.500000
$677$	− 26.0000i	− 0.999261i	−0.866239	−	0.499631i	$-0.833469\pi$
$677$	− 26.0000i	− 0.999261i	0.866239	−	0.499631i	$-0.166531\pi$
$678$	− 6.00000i	− 0.230429i
$679$	12.0000	0.460518
$680$	0	0
$681$	20.0000	0.766402
$682$	− 4.00000i	− 0.153168i
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	20.0000	0.763604
$687$	− 6.00000i	− 0.228914i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	− 6.00000i	− 0.228086i
$693$	− 8.00000i	− 0.303895i
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	14.0000i	0.529908i
$699$	6.00000	0.226941
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	64.0000i	2.41381i
$704$	28.0000	1.05529
$705$	0	0
$706$	−14.0000	−0.526897
$707$	− 28.0000i	− 1.05305i
$708$	− 10.0000i	− 0.375823i
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	0	0
$711$	0	0
$712$	− 12.0000i	− 0.449719i
$713$	8.00000i	0.299602i
$714$	−4.00000	−0.149696
$715$	0	0
$716$	4.00000	0.149487
$717$	20.0000i	0.746914i
$718$	− 26.0000i	− 0.970311i
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	45.0000i	1.67473i
$723$	− 22.0000i	− 0.818189i
$724$	−6.00000	−0.222988
$725$	0	0
$726$	−5.00000	−0.185567
$727$	30.0000i	1.11264i	0.830969	+	0.556319i	$0.187787\pi$
$727$	30.0000i	1.11264i	−0.830969	+	0.556319i	$0.812213\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	− 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	$0.840576\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	−40.0000	−1.47442
$737$	− 8.00000i	− 0.294684i
$738$	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
$741$	0	0
$742$	− 12.0000i	− 0.440534i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	−3.00000	−0.109985
$745$	0	0
$746$	−22.0000	−0.805477
$747$	4.00000i	0.146352i
$748$	− 8.00000i	− 0.292509i
$749$	32.0000	1.16925
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	− 4.00000i	− 0.145865i
$753$	20.0000i	0.728841i
$754$	0	0
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	− 8.00000i	− 0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$	− 8.00000i	− 0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$	− 16.0000i	− 0.581146i
$759$	32.0000	1.16153
$760$	0	0
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	4.00000i	0.144905i
$763$	− 28.0000i	− 1.01367i
$764$	−14.0000	−0.506502
$765$	0	0
$766$	0	0
$767$	0	0
$768$	− 17.0000i	− 0.613435i
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	− 6.00000i	− 0.215945i
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	0	0
$775$	0	0
$776$	−18.0000	−0.646162
$777$	16.0000i	0.573997i
$778$	− 16.0000i	− 0.573628i
$779$	−48.0000	−1.71978
$780$	0	0
$781$	−24.0000	−0.858788
$782$	− 16.0000i	− 0.572159i
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	14.0000	0.499363
$787$	52.0000i	1.85360i	0.375555	+	0.926800i	$0.377452\pi$
$787$	52.0000i	1.85360i	−0.375555	+	0.926800i	$0.622548\pi$
$788$	2.00000i	0.0712470i
$789$	−24.0000	−0.854423
$790$	0	0
$791$	12.0000	0.426671
$792$	12.0000i	0.426401i
$793$	0	0
$794$	−14.0000	−0.496841
$795$	0	0
$796$	8.00000	0.283552
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	16.0000i	0.566394i
$799$	−8.00000	−0.283020
$800$	0	0
$801$	4.00000	0.141333
$802$	16.0000i	0.564980i
$803$	− 64.0000i	− 2.25851i
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	0	0
$807$	0	0
$808$	42.0000i	1.47755i
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	32.0000	1.12160
$815$	0	0
$816$	2.00000	0.0700140
$817$	0	0
$818$	18.0000i	0.629355i
$819$	0	0
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	18.0000i	0.627822i
$823$	44.0000i	1.53374i	0.641800	+	0.766872i	$0.278188\pi$
$823$	44.0000i	1.53374i	−0.641800	+	0.766872i	$0.721812\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	−20.0000	−0.695889
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	− 8.00000i	− 0.278019i
$829$	22.0000	0.764092	0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000	0.764092	0.382046	+	0.924143i	$0.375220\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	0	0
$833$	6.00000i	0.207888i
$834$	−12.0000	−0.415526
$835$	0	0
$836$	−32.0000	−1.10674
$837$	− 1.00000i	− 0.0345651i
$838$	− 14.0000i	− 0.483622i
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	18.0000i	0.620321i
$843$	30.0000i	1.03325i
$844$	8.00000	0.275371
$845$	0	0
$846$	4.00000	0.137523
$847$	− 10.0000i	− 0.343604i
$848$	6.00000i	0.206041i
$849$	10.0000	0.343199
$850$	0	0
$851$	−64.0000	−2.19389
$852$	6.00000i	0.205557i
$853$	2.00000i	0.0684787i	0.999414	+	0.0342393i	$0.0109009\pi$
$853$	2.00000i	0.0684787i	−0.999414	+	0.0342393i	$0.989099\pi$
$854$	−28.0000	−0.958140
$855$	0	0
$856$	−48.0000	−1.64061
$857$	− 6.00000i	− 0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$	− 6.00000i	− 0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	2.00000i	0.0681203i
$863$	− 8.00000i	− 0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$	− 8.00000i	− 0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	24.0000	0.815553
$867$	13.0000i	0.441503i
$868$	− 2.00000i	− 0.0678844i
$869$	0	0
$870$	0	0
$871$	0	0
$872$	42.0000i	1.42230i
$873$	− 6.00000i	− 0.203069i
$874$	−64.0000	−2.16483
$875$	0	0
$876$	−16.0000	−0.540590
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	− 24.0000i	− 0.809961i
$879$	26.0000	0.876958
$880$	0	0
$881$	−28.0000	−0.943344	−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000	−0.943344	−0.471672	+	0.881774i	$0.656349\pi$
$882$	− 3.00000i	− 0.101015i
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	0	0
$885$	0	0
$886$	−20.0000	−0.671913
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	− 24.0000i	− 0.805387i
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−4.00000	−0.134005
$892$	− 16.0000i	− 0.535720i
$893$	32.0000i	1.07084i
$894$	−18.0000	−0.602010
$895$	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
$902$	24.0000i	0.799113i
$903$	0	0
$904$	−18.0000	−0.598671
$905$	0	0
$906$	24.0000	0.797347
$907$	− 18.0000i	− 0.597680i	−0.954303	−	0.298840i	$-0.903400\pi$
$907$	− 18.0000i	− 0.597680i	0.954303	−	0.298840i	$-0.0965997\pi$
$908$	− 20.0000i	− 0.663723i
$909$	−14.0000	−0.464351
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	− 8.00000i	− 0.264906i
$913$	16.0000i	0.529523i
$914$	8.00000	0.264616
$915$	0	0
$916$	−6.00000	−0.198246
$917$	28.0000i	0.924641i
$918$	2.00000i	0.0660098i
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	−22.0000	−0.724925
$922$	12.0000i	0.395199i
$923$	0	0
$924$	−8.00000	−0.263181
$925$	0	0
$926$	20.0000	0.657241
$927$	2.00000i	0.0656886i
$928$	0	0
$929$	44.0000	1.44359	0.721797	−	0.692105i	$-0.243317\pi$
$929$	44.0000	1.44359	0.721797	+	0.692105i	$0.243317\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	− 6.00000i	− 0.196537i
$933$	18.0000i	0.589294i
$934$	−40.0000	−1.30884
$935$	0	0
$936$	0	0
$937$	6.00000i	0.196011i	0.995186	+	0.0980057i	$0.0312463\pi$
$937$	6.00000i	0.196011i	−0.995186	+	0.0980057i	$0.968754\pi$
$938$	4.00000i	0.130605i
$939$	−8.00000	−0.261070
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	− 2.00000i	− 0.0651635i
$943$	− 48.0000i	− 1.56310i
$944$	10.0000	0.325472
$945$	0	0
$946$	0	0
$947$	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
$951$	6.00000	0.194563
$952$	12.0000i	0.388922i
$953$	− 30.0000i	− 0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$	− 30.0000i	− 0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	20.0000	0.646846
$957$	0	0
$958$	10.0000i	0.323085i
$959$	−36.0000	−1.16250
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	− 16.0000i	− 0.515593i
$964$	−22.0000	−0.708572
$965$	0	0
$966$	−16.0000	−0.514792
$967$	24.0000i	0.771788i	0.922543	+	0.385894i	$0.126107\pi$
$967$	24.0000i	0.771788i	−0.922543	+	0.385894i	$0.873893\pi$
$968$	15.0000i	0.482118i
$969$	−16.0000	−0.513994
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	1.00000i	0.0320750i
$973$	− 24.0000i	− 0.769405i
$974$	4.00000	0.128168
$975$	0	0
$976$	14.0000	0.448129
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	2.00000i	0.0639529i
$979$	16.0000	0.511362
$980$	0	0
$981$	−14.0000	−0.446986
$982$	− 24.0000i	− 0.765871i
$983$	8.00000i	0.255160i	0.991828	+	0.127580i	$0.0407210\pi$
$983$	8.00000i	0.255160i	−0.991828	+	0.127580i	$0.959279\pi$
$984$	18.0000	0.573819
$985$	0	0
$986$	0	0
$987$	8.00000i	0.254643i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	5.00000i	0.158750i
$993$	− 4.00000i	− 0.126936i
$994$	12.0000	0.380617
$995$	0	0
$996$	4.00000	0.126745
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	20.0000i	0.633089i
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.c.d.1024.2		2
5.2	odd	4		2325.2.a.d.1.1		1
5.3	odd	4		465.2.a.b.1.1	✓	1
5.4	even	2	inner	2325.2.c.d.1024.1		2
15.2	even	4		6975.2.a.q.1.1		1
15.8	even	4		1395.2.a.a.1.1		1
20.3	even	4		7440.2.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.b.1.1	✓	1	5.3	odd	4
1395.2.a.a.1.1		1	15.8	even	4
2325.2.a.d.1.1		1	5.2	odd	4
2325.2.c.d.1024.1		2	5.4	even	2	inner
2325.2.c.d.1024.2		2	1.1	even	1	trivial
6975.2.a.q.1.1		1	15.2	even	4
7440.2.a.ba.1.1		1	20.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2325.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +2.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +8.00000 q^{19} +2.00000 q^{21} -4.00000i q^{22} +8.00000i q^{23} -3.00000 q^{24} -1.00000i q^{27} -2.00000i q^{28} +1.00000 q^{31} +5.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} +8.00000i q^{37} +8.00000i q^{38} -6.00000 q^{41} +2.00000i q^{42} -4.00000 q^{44} -8.00000 q^{46} +4.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} +1.00000 q^{54} +6.00000 q^{56} +8.00000i q^{57} -10.0000 q^{59} -14.0000 q^{61} +1.00000i q^{62} +2.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} +2.00000i q^{67} +2.00000i q^{68} -8.00000 q^{69} +6.00000 q^{71} -3.00000i q^{72} +16.0000i q^{73} -8.00000 q^{74} +8.00000 q^{76} +8.00000i q^{77} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} +2.00000 q^{84} -12.0000i q^{88} -4.00000 q^{89} +8.00000i q^{92} +1.00000i q^{93} -4.00000 q^{94} -5.00000 q^{96} +6.00000i q^{97} +3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} + 4 q^{14} - 2 q^{16} + 16 q^{19} + 4 q^{21} - 6 q^{24} + 2 q^{31} - 4 q^{34} - 2 q^{36} - 12 q^{41} - 8 q^{44} - 16 q^{46} + 6 q^{49} - 4 q^{51} + 2 q^{54}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^11 + 4 * q^14 - 2 * q^16 + 16 * q^19 + 4 * q^21 - 6 * q^24 + 2 * q^31 - 4 * q^34 - 2 * q^36 - 12 * q^41 - 8 * q^44 - 16 * q^46 + 6 * q^49 - 4 * q^51 + 2 * q^54 + 12 * q^56 - 20 * q^59 - 28 * q^61 - 14 * q^64 + 8 * q^66 - 16 * q^69 + 12 * q^71 - 16 * q^74 + 16 * q^76 + 2 * q^81 + 4 * q^84 - 8 * q^89 - 8 * q^94 - 10 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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