LMFDB - Embedded newform 975.2.c.e.274.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(274,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, 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("975.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.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:	$7.78541419707$
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 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	975.274
Dual form			975.2.c.e.274.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} -1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} -8.00000i q^{23} -3.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} -8.00000 q^{31} -5.00000i q^{32} -4.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} -1.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} -1.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -4.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -7.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} -8.00000 q^{69} +3.00000i q^{72} +6.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} +1.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +4.00000i q^{83} +4.00000 q^{86} -2.00000i q^{87} -12.0000i q^{88} -10.0000 q^{89} -8.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} -5.00000 q^{96} +18.0000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{16} + 8 q^{19} - 6 q^{24} - 2 q^{26} + 4 q^{29} - 16 q^{31} + 4 q^{34} - 2 q^{36} - 2 q^{39} - 12 q^{41} + 8 q^{44} - 16 q^{46} + 14 q^{49} + 4 q^{51} + 2 q^{54} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{66} - 16 q^{69} + 12 q^{74} + 8 q^{76} - 32 q^{79} + 2 q^{81} + 8 q^{86} - 20 q^{89} - 16 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 - 2 * q^16 + 8 * q^19 - 6 * q^24 - 2 * q^26 + 4 * q^29 - 16 * q^31 + 4 * q^34 - 2 * q^36 - 2 * q^39 - 12 * q^41 + 8 * q^44 - 16 * q^46 + 14 * q^49 + 4 * q^51 + 2 * q^54 + 24 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^66 - 16 * q^69 + 12 * q^74 + 8 * q^76 - 32 * q^79 + 2 * q^81 + 8 * q^86 - 20 * q^89 - 16 * q^94 - 10 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 1.00000i	− 0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 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
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	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
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000i	0.192450i
$28$	0	0
$29$	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
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	− 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	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	− 4.00000i	− 0.648886i
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−8.00000	−1.17954
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	2.00000	0.280056
$52$	− 1.00000i	− 0.138675i
$53$	− 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
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	− 2.00000i	− 0.262613i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.00000i	1.01600i
$63$	0	0
$64$	−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
$69$	−8.00000	−0.963087
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000i	0.353553i
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	1.00000i	0.113228i
$79$	−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
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	− 2.00000i	− 0.214423i
$88$	− 12.0000i	− 1.27920i
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	− 8.00000i	− 0.834058i
$93$	8.00000i	0.829561i
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−5.00000	−0.510310
$97$	18.0000i	1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$	18.0000i	1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	− 7.00000i	− 0.707107i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	− 2.00000i	− 0.198030i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−6.00000	−0.582772
$107$	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
$112$	0	0
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	2.00000	0.185695
$117$	1.00000i	0.0924500i
$118$	− 12.0000i	− 1.10469i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000i	0.181071i
$123$	6.00000i	0.541002i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	− 3.00000i	− 0.265165i
$129$	4.00000	0.352180
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	− 4.00000i	− 0.348155i
$133$	0	0
$134$	−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	$-0.0825059\pi$
$138$	8.00000i	0.681005i
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	− 4.00000i	− 0.334497i
$144$	1.00000	0.0833333
$145$	0	0
$146$	6.00000	0.496564
$147$	− 7.00000i	− 0.577350i
$148$	6.00000i	0.493197i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	− 12.0000i	− 0.973329i
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	16.0000i	1.27289i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	−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	$0.787514\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−4.00000	−0.305888
$172$	4.00000i	0.304997i
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	− 12.0000i	− 0.901975i
$178$	10.0000i	0.749532i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	2.00000i	0.147844i
$184$	−24.0000	−1.76930
$185$	0	0
$186$	8.00000	0.586588
$187$	8.00000i	0.585018i
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	7.00000i	0.505181i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	18.0000	1.29232
$195$	0	0
$196$	7.00000	0.500000
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	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
$201$	−4.00000	−0.282138
$202$	− 6.00000i	− 0.422159i
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	8.00000	0.557386
$207$	8.00000i	0.556038i
$208$	1.00000i	0.0693375i
$209$	16.0000	1.10674
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	3.00000	0.204124
$217$	0	0
$218$	− 2.00000i	− 0.135457i
$219$	6.00000	0.405442
$220$	0	0
$221$	2.00000	0.134535
$222$	− 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	$0.702926\pi$
$224$	0	0
$225$	0	0
$226$	−2.00000	−0.133038
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 26.0000i	− 1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	− 26.0000i	− 1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	12.0000	0.781133
$237$	16.0000i	1.03931i
$238$	0	0
$239$	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
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	− 5.00000i	− 0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	6.00000	0.382546
$247$	− 4.00000i	− 0.254514i
$248$	24.0000i	1.52400i
$249$	4.00000	0.253490
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	− 4.00000i	− 0.249029i
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	12.0000i	0.741362i
$263$	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
$266$	0	0
$267$	10.0000i	0.611990i
$268$	− 4.00000i	− 0.244339i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−8.00000	−0.481543
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 20.0000i	− 1.19952i
$279$	8.00000	0.478947
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	8.00000i	0.476393i
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	5.00000i	0.294628i
$289$	13.0000	0.764706
$290$	0	0
$291$	18.0000	1.05518
$292$	6.00000i	0.351123i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	18.0000	1.04623
$297$	4.00000i	0.232104i
$298$	− 10.0000i	− 0.579284i
$299$	−8.00000	−0.462652
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 6.00000i	− 0.344691i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−2.00000	−0.114332
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	3.00000i	0.169842i
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−16.0000	−0.900070
$317$	30.0000i	1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$	30.0000i	1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	6.00000i	0.336463i
$319$	8.00000	0.447914
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	8.00000i	0.445132i
$324$	1.00000	0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	− 2.00000i	− 0.110600i
$328$	18.0000i	0.993884i
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	4.00000i	0.219529i
$333$	− 6.00000i	− 0.328798i
$334$	16.0000	0.875481
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	1.00000i	0.0543928i
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−32.0000	−1.73290
$342$	4.00000i	0.216295i
$343$	0	0
$344$	12.0000	0.646997
$345$	0	0
$346$	2.00000	0.107521
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	− 2.00000i	− 0.107211i
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	− 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	$0.878476\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 22.0000i	− 1.15629i
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	8.00000i	0.417029i
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	8.00000i	0.414781i
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−24.0000	−1.23771
$377$	− 2.00000i	− 0.103005i
$378$	0	0
$379$	−36.0000	−1.84920	−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	− 16.0000i	− 0.818631i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	14.0000	0.712581
$387$	− 4.00000i	− 0.203331i
$388$	18.0000i	0.913812i
$389$	−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
$391$	16.0000	0.809155
$392$	− 21.0000i	− 1.06066i
$393$	12.0000i	0.605320i
$394$	6.00000	0.302276
$395$	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
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	4.00000i	0.199502i
$403$	8.00000i	0.398508i
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	24.0000i	1.18964i
$408$	− 6.00000i	− 0.297044i
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	8.00000i	0.394132i
$413$	0	0
$414$	8.00000	0.393179
$415$	0	0
$416$	−5.00000	−0.245145
$417$	− 20.0000i	− 0.979404i
$418$	− 16.0000i	− 0.782586i
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	− 20.0000i	− 0.973585i
$423$	8.00000i	0.388973i
$424$	−18.0000	−0.874157
$425$	0	0
$426$	0	0
$427$	0	0
$428$	12.0000i	0.580042i
$429$	−4.00000	−0.193122
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 18.0000i	− 0.865025i	−0.901628	−	0.432512i	$-0.857627\pi$
$433$	− 18.0000i	− 0.865025i	0.901628	−	0.432512i	$-0.142373\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 32.0000i	− 1.53077i
$438$	− 6.00000i	− 0.286691i
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 2.00000i	− 0.0951303i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	24.0000	1.13643
$447$	− 10.0000i	− 0.472984i
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	− 2.00000i	− 0.0940721i
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	−12.0000	−0.561951
$457$	26.0000i	1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$	26.0000i	1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$	22.0000i	1.02799i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−26.0000	−1.20443
$467$	4.00000i	0.185098i	0.995708	+	0.0925490i	$0.0295015\pi$
$467$	4.00000i	0.185098i	−0.995708	+	0.0925490i	$0.970499\pi$
$468$	1.00000i	0.0462250i
$469$	0	0
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	− 36.0000i	− 1.65703i
$473$	16.0000i	0.735681i
$474$	16.0000	0.734904
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	− 24.0000i	− 1.09773i
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	14.0000i	0.637683i
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	6.00000i	0.271607i
$489$	20.0000	0.904431
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	6.00000i	0.270501i
$493$	4.00000i	0.180151i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	− 4.00000i	− 0.179244i
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	4.00000i	0.178529i
$503$	− 24.0000i	− 1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$	− 24.0000i	− 1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	0	0
$505$	0	0
$506$	−32.0000	−1.42257
$507$	1.00000i	0.0444116i
$508$	− 16.0000i	− 0.709885i
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	0	0
$512$	11.0000i	0.486136i
$513$	4.00000i	0.176604i
$514$	18.0000	0.793946
$515$	0	0
$516$	4.00000	0.176090
$517$	− 32.0000i	− 1.40736i
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	2.00000i	0.0875376i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	− 16.0000i	− 0.696971i
$528$	4.00000i	0.174078i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	6.00000i	0.259889i
$534$	10.0000	0.432742
$535$	0	0
$536$	−12.0000	−0.518321
$537$	− 12.0000i	− 0.517838i
$538$	14.0000i	0.603583i
$539$	28.0000	1.20605
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	8.00000i	0.343629i
$543$	− 22.0000i	− 0.944110i
$544$	10.0000	0.428746
$545$	0	0
$546$	0	0
$547$	36.0000i	1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$	36.0000i	1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$	− 6.00000i	− 0.256307i
$549$	2.00000	0.0853579
$550$	0	0
$551$	8.00000	0.340811
$552$	24.0000i	1.02151i
$553$	0	0
$554$	−10.0000	−0.424859
$555$	0	0
$556$	20.0000	0.848189
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	− 8.00000i	− 0.338667i
$559$	4.00000	0.169182
$560$	0	0
$561$	8.00000	0.337760
$562$	22.0000i	0.928014i
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	20.0000	0.840663
$567$	0	0
$568$	0	0
$569$	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
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	− 4.00000i	− 0.167248i
$573$	− 16.0000i	− 0.668410i
$574$	0	0
$575$	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
$579$	14.0000	0.581820
$580$	0	0
$581$	0	0
$582$	− 18.0000i	− 0.746124i
$583$	− 24.0000i	− 0.993978i
$584$	18.0000	0.744845
$585$	0	0
$586$	−6.00000	−0.247858
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	− 7.00000i	− 0.288675i
$589$	−32.0000	−1.31854
$590$	0	0
$591$	6.00000	0.246807
$592$	− 6.00000i	− 0.246598i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	10.0000	0.409616
$597$	− 8.00000i	− 0.327418i
$598$	8.00000i	0.327144i
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	0	0
$606$	−6.00000	−0.243733
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	− 20.0000i	− 0.811107i
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	− 2.00000i	− 0.0808452i
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	− 38.0000i	− 1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	− 38.0000i	− 1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$	− 8.00000i	− 0.321807i
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	− 24.0000i	− 0.962312i
$623$	0	0
$624$	1.00000	0.0400320
$625$	0	0
$626$	−26.0000	−1.03917
$627$	− 16.0000i	− 0.638978i
$628$	− 2.00000i	− 0.0798087i
$629$	−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
$633$	− 20.0000i	− 0.794929i
$634$	30.0000	1.19145
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 7.00000i	− 0.277350i
$638$	− 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
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	0	0
$645$	0	0
$646$	8.00000	0.314756
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	− 3.00000i	− 0.117851i
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	20.0000i	0.783260i
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	6.00000	0.234261
$657$	− 6.00000i	− 0.234082i
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	− 20.0000i	− 0.777322i
$663$	− 2.00000i	− 0.0776736i
$664$	12.0000	0.465690
$665$	0	0
$666$	−6.00000	−0.232495
$667$	− 16.0000i	− 0.619522i
$668$	16.0000i	0.619059i
$669$	24.0000	0.927894
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	− 10.0000i	− 0.384331i	−0.981363	−	0.192166i	$-0.938449\pi$
$677$	− 10.0000i	− 0.384331i	0.981363	−	0.192166i	$-0.0615511\pi$
$678$	2.00000i	0.0768095i
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	32.0000i	1.22534i
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	0	0
$687$	22.0000i	0.839352i
$688$	− 4.00000i	− 0.152499i
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−36.0000	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$692$	2.00000i	0.0760286i
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	−6.00000	−0.227429
$697$	− 12.0000i	− 0.454532i
$698$	14.0000i	0.529908i
$699$	−26.0000	−0.983410
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	− 1.00000i	− 0.0377426i
$703$	24.0000i	0.905177i
$704$	−28.0000	−1.05529
$705$	0	0
$706$	14.0000	0.526897
$707$	0	0
$708$	− 12.0000i	− 0.450988i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	30.0000i	1.12430i
$713$	64.0000i	2.39682i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	− 24.0000i	− 0.896296i
$718$	16.0000i	0.597115i
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	0	0
$722$	3.00000i	0.111648i
$723$	14.0000i	0.520666i
$724$	22.0000	0.817624
$725$	0	0
$726$	−5.00000	−0.185567
$727$	− 24.0000i	− 0.890111i	−0.895503	−	0.445055i	$-0.853184\pi$
$727$	− 24.0000i	− 0.890111i	0.895503	−	0.445055i	$-0.146816\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	2.00000i	0.0739221i
$733$	− 30.0000i	− 1.10808i	−0.832492	−	0.554038i	$-0.813086\pi$
$733$	− 30.0000i	− 1.10808i	0.832492	−	0.554038i	$-0.186914\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−40.0000	−1.47442
$737$	− 16.0000i	− 0.589368i
$738$	− 6.00000i	− 0.220863i
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	24.0000	0.879883
$745$	0	0
$746$	−22.0000	−0.805477
$747$	− 4.00000i	− 0.146352i
$748$	8.00000i	0.292509i
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	8.00000i	0.291730i
$753$	4.00000i	0.145768i
$754$	−2.00000	−0.0728357
$755$	0	0
$756$	0	0
$757$	− 42.0000i	− 1.52652i	−0.646094	−	0.763258i	$-0.723599\pi$
$757$	− 42.0000i	− 1.52652i	0.646094	−	0.763258i	$-0.276401\pi$
$758$	36.0000i	1.30758i
$759$	−32.0000	−1.16153
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	16.0000i	0.579619i
$763$	0	0
$764$	16.0000	0.578860
$765$	0	0
$766$	−24.0000	−0.867155
$767$	− 12.0000i	− 0.433295i
$768$	17.0000i	0.613435i
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	14.0000i	0.503871i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	54.0000	1.93849
$777$	0	0
$778$	6.00000i	0.215110i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	− 16.0000i	− 0.572159i
$783$	2.00000i	0.0714742i
$784$	−7.00000	−0.250000
$785$	0	0
$786$	12.0000	0.428026
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	6.00000i	0.213741i
$789$	24.0000	0.854423
$790$	0	0
$791$	0	0
$792$	12.0000i	0.426401i
$793$	2.00000i	0.0710221i
$794$	14.0000	0.496841
$795$	0	0
$796$	8.00000	0.283552
$797$	46.0000i	1.62940i	0.579880	+	0.814702i	$0.303099\pi$
$797$	46.0000i	1.62940i	−0.579880	+	0.814702i	$0.696901\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	10.0000	0.353333
$802$	30.0000i	1.05934i
$803$	24.0000i	0.846942i
$804$	−4.00000	−0.141069
$805$	0	0
$806$	8.00000	0.281788
$807$	14.0000i	0.492823i
$808$	− 18.0000i	− 0.633238i
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	24.0000	0.841200
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	16.0000i	0.559769i
$818$	− 38.0000i	− 1.32864i
$819$	0	0
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	6.00000i	0.209274i
$823$	24.0000i	0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$	24.0000i	0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	0	0
$827$	− 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
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	7.00000i	0.242681i
$833$	14.0000i	0.485071i
$834$	−20.0000	−0.692543
$835$	0	0
$836$	16.0000	0.553372
$837$	− 8.00000i	− 0.276520i
$838$	20.0000i	0.690889i
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	10.0000i	0.344623i
$843$	22.0000i	0.757720i
$844$	20.0000	0.688428
$845$	0	0
$846$	8.00000	0.275046
$847$	0	0
$848$	6.00000i	0.206041i
$849$	20.0000	0.686398
$850$	0	0
$851$	48.0000	1.64542
$852$	0	0
$853$	42.0000i	1.43805i	0.694983	+	0.719026i	$0.255412\pi$
$853$	42.0000i	1.43805i	−0.694983	+	0.719026i	$0.744588\pi$
$854$	0	0
$855$	0	0
$856$	36.0000	1.23045
$857$	10.0000i	0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$	10.0000i	0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$	4.00000i	0.136558i
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	− 8.00000i	− 0.272481i
$863$	40.0000i	1.36162i	0.732462	+	0.680808i	$0.238371\pi$
$863$	40.0000i	1.36162i	−0.732462	+	0.680808i	$0.761629\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	−18.0000	−0.611665
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	−4.00000	−0.135535
$872$	− 6.00000i	− 0.203186i
$873$	− 18.0000i	− 0.609208i
$874$	−32.0000	−1.08242
$875$	0	0
$876$	6.00000	0.202721
$877$	− 50.0000i	− 1.68838i	−0.536044	−	0.844190i	$-0.680082\pi$
$877$	− 50.0000i	− 1.68838i	0.536044	−	0.844190i	$-0.319918\pi$
$878$	− 24.0000i	− 0.809961i
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	7.00000i	0.235702i
$883$	− 36.0000i	− 1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$	− 36.0000i	− 1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	4.00000	0.134383
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	− 18.0000i	− 0.604040i
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	24.0000i	0.803579i
$893$	− 32.0000i	− 1.07084i
$894$	−10.0000	−0.334450
$895$	0	0
$896$	0	0
$897$	8.00000i	0.267112i
$898$	18.0000i	0.600668i
$899$	−16.0000	−0.533630
$900$	0	0
$901$	12.0000	0.399778
$902$	24.0000i	0.799113i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	12.0000i	0.398234i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	4.00000i	0.132453i
$913$	16.0000i	0.529523i
$914$	26.0000	0.860004
$915$	0	0
$916$	−22.0000	−0.726900
$917$	0	0
$918$	2.00000i	0.0660098i
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	18.0000i	0.592798i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	− 8.00000i	− 0.262754i
$928$	− 10.0000i	− 0.328266i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	− 26.0000i	− 0.851658i
$933$	− 24.0000i	− 0.785725i
$934$	4.00000	0.130884
$935$	0	0
$936$	3.00000	0.0980581
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	0	0
$939$	−26.0000	−0.848478
$940$	0	0
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	2.00000i	0.0651635i
$943$	48.0000i	1.56310i
$944$	−12.0000	−0.390567
$945$	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	$0.937538\pi$
$948$	16.0000i	0.519656i
$949$	6.00000	0.194768
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	− 10.0000i	− 0.323932i	−0.986796	−	0.161966i	$-0.948217\pi$
$953$	− 10.0000i	− 0.323932i	0.986796	−	0.161966i	$-0.0517835\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	− 8.00000i	− 0.258603i
$958$	8.00000i	0.258468i
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	− 6.00000i	− 0.193448i
$963$	− 12.0000i	− 0.386695i
$964$	−14.0000	−0.450910
$965$	0	0
$966$	0	0
$967$	− 16.0000i	− 0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$	− 16.0000i	− 0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$	− 15.0000i	− 0.482118i
$969$	8.00000	0.256997
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	− 20.0000i	− 0.639529i
$979$	−40.0000	−1.27841
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	20.0000i	0.638226i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	18.0000	0.573819
$985$	0	0
$986$	4.00000	0.127386
$987$	0	0
$988$	− 4.00000i	− 0.127257i
$989$	32.0000	1.01754
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	40.0000i	1.27000i
$993$	− 20.0000i	− 0.634681i
$994$	0	0
$995$	0	0
$996$	4.00000	0.126745
$997$	38.0000i	1.20347i	0.798695	+	0.601736i	$0.205524\pi$
$997$	38.0000i	1.20347i	−0.798695	+	0.601736i	$0.794476\pi$
$998$	28.0000i	0.886325i
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.c.e.274.1		2
3.2	odd	2		2925.2.c.f.2224.2		2
5.2	odd	4		975.2.a.i.1.1		1
5.3	odd	4		195.2.a.a.1.1	✓	1
5.4	even	2	inner	975.2.c.e.274.2		2
15.2	even	4		2925.2.a.d.1.1		1
15.8	even	4		585.2.a.g.1.1		1
15.14	odd	2		2925.2.c.f.2224.1		2
20.3	even	4		3120.2.a.k.1.1		1
35.13	even	4		9555.2.a.b.1.1		1
60.23	odd	4		9360.2.a.o.1.1		1
65.38	odd	4		2535.2.a.k.1.1		1
195.38	even	4		7605.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.a.1.1	✓	1	5.3	odd	4
585.2.a.g.1.1		1	15.8	even	4
975.2.a.i.1.1		1	5.2	odd	4
975.2.c.e.274.1		2	1.1	even	1	trivial
975.2.c.e.274.2		2	5.4	even	2	inner
2535.2.a.k.1.1		1	65.38	odd	4
2925.2.a.d.1.1		1	15.2	even	4
2925.2.c.f.2224.1		2	15.14	odd	2
2925.2.c.f.2224.2		2	3.2	odd	2
3120.2.a.k.1.1		1	20.3	even	4
7605.2.a.h.1.1		1	195.38	even	4
9360.2.a.o.1.1		1	60.23	odd	4
9555.2.a.b.1.1		1	35.13	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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} -1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} -8.00000i q^{23} -3.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} -8.00000 q^{31} -5.00000i q^{32} -4.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} -1.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} -1.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -4.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -7.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} -8.00000 q^{69} +3.00000i q^{72} +6.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} +1.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +4.00000i q^{83} +4.00000 q^{86} -2.00000i q^{87} -12.0000i q^{88} -10.0000 q^{89} -8.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} -5.00000 q^{96} +18.0000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 2 q^{16} + 8 q^{19} - 6 q^{24} - 2 q^{26} + 4 q^{29} - 16 q^{31} + 4 q^{34} - 2 q^{36} - 2 q^{39} - 12 q^{41} + 8 q^{44} - 16 q^{46} + 14 q^{49} + 4 q^{51} + 2 q^{54} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{66} - 16 q^{69} + 12 q^{74} + 8 q^{76} - 32 q^{79} + 2 q^{81} + 8 q^{86} - 20 q^{89} - 16 q^{94} - 10 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 - 2 * q^16 + 8 * q^19 - 6 * q^24 - 2 * q^26 + 4 * q^29 - 16 * q^31 + 4 * q^34 - 2 * q^36 - 2 * q^39 - 12 * q^41 + 8 * q^44 - 16 * q^46 + 14 * q^49 + 4 * q^51 + 2 * q^54 + 24 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^66 - 16 * q^69 + 12 * q^74 + 8 * q^76 - 32 * q^79 + 2 * q^81 + 8 * q^86 - 20 * q^89 - 16 * q^94 - 10 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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