LMFDB - Embedded newform 1050.2.g.i.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,2,Mod(799,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, 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("1050.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.g (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:	$8.38429221223$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1050.799
Dual form			1050.2.g.i.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} +1.00000i q^{12} +1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -1.00000i q^{18} +1.00000 q^{21} +2.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} +5.00000 q^{29} +7.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} -3.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +1.00000 q^{39} +7.00000 q^{41} +1.00000i q^{42} +11.0000i q^{43} -2.00000 q^{44} -1.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} +3.00000 q^{51} -1.00000i q^{52} +1.00000i q^{53} -1.00000 q^{54} +1.00000 q^{56} +5.00000i q^{58} +5.00000 q^{59} -3.00000 q^{61} +7.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -12.0000i q^{67} -3.00000i q^{68} +1.00000 q^{69} +12.0000 q^{71} +1.00000i q^{72} +6.00000i q^{73} +2.00000 q^{74} +2.00000i q^{77} +1.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} +7.00000i q^{82} +11.0000i q^{83} -1.00000 q^{84} -11.0000 q^{86} -5.00000i q^{87} -2.00000i q^{88} +10.0000 q^{89} -1.00000 q^{91} -1.00000i q^{92} -7.00000i q^{93} -8.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -1.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{21} - 2 q^{24} - 2 q^{26} + 10 q^{29} + 14 q^{31} - 6 q^{34} + 2 q^{36} + 2 q^{39} + 14 q^{41} - 4 q^{44} - 2 q^{46} - 2 q^{49} + 6 q^{51} - 2 q^{54} + 2 q^{56} + 10 q^{59} - 6 q^{61} - 2 q^{64} + 4 q^{66} + 2 q^{69} + 24 q^{71} + 4 q^{74} - 20 q^{79} + 2 q^{81} - 2 q^{84} - 22 q^{86} + 20 q^{89} - 2 q^{91} - 16 q^{94} + 2 q^{96} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 4 * q^11 - 2 * q^14 + 2 * q^16 + 2 * q^21 - 2 * q^24 - 2 * q^26 + 10 * q^29 + 14 * q^31 - 6 * q^34 + 2 * q^36 + 2 * q^39 + 14 * q^41 - 4 * q^44 - 2 * q^46 - 2 * q^49 + 6 * q^51 - 2 * q^54 + 2 * q^56 + 10 * q^59 - 6 * q^61 - 2 * q^64 + 4 * q^66 + 2 * q^69 + 24 * q^71 + 4 * q^74 - 20 * q^79 + 2 * q^81 - 2 * q^84 - 22 * q^86 + 20 * q^89 - 2 * q^91 - 16 * q^94 + 2 * q^96 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$1$	$1$

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
$2$	1.00000i	0.707107i
$3$	− 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
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000i	0.288675i
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	− 1.00000i	− 0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	2.00000i	0.426401i
$23$	1.00000i	0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$	1.00000i	0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000i	0.192450i
$28$	− 1.00000i	− 0.188982i
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	1.00000i	0.176777i
$33$	− 2.00000i	− 0.348155i
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	1.00000i	0.154303i
$43$	11.0000i	1.67748i	0.544529	+	0.838742i	$0.316708\pi$
$43$	11.0000i	1.67748i	−0.544529	+	0.838742i	$0.683292\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−1.00000	−0.147442
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	− 1.00000i	− 0.138675i
$53$	1.00000i	0.137361i	0.997639	+	0.0686803i	$0.0218788\pi$
$53$	1.00000i	0.137361i	−0.997639	+	0.0686803i	$0.978121\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	5.00000i	0.656532i
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	7.00000i	0.889001i
$63$	− 1.00000i	− 0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	− 3.00000i	− 0.363803i
$69$	1.00000	0.120386
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	1.00000i	0.117851i
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	1.00000i	0.113228i
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.00000i	0.773021i
$83$	11.0000i	1.20741i	0.797209	+	0.603703i	$0.206309\pi$
$83$	11.0000i	1.20741i	−0.797209	+	0.603703i	$0.793691\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−11.0000	−1.18616
$87$	− 5.00000i	− 0.536056i
$88$	− 2.00000i	− 0.213201i
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	− 1.00000i	− 0.104257i
$93$	− 7.00000i	− 0.725866i
$94$	−8.00000	−0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	− 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
$99$	−2.00000	−0.201008
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	3.00000i	0.297044i
$103$	11.0000i	1.08386i	0.840423	+	0.541931i	$0.182307\pi$
$103$	11.0000i	1.08386i	−0.840423	+	0.541931i	$0.817693\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	− 2.00000i	− 0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$	− 2.00000i	− 0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	1.00000i	0.0944911i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	−5.00000	−0.464238
$117$	− 1.00000i	− 0.0924500i
$118$	5.00000i	0.460287i
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 3.00000i	− 0.271607i
$123$	− 7.00000i	− 0.631169i
$124$	−7.00000	−0.628619
$125$	0	0
$126$	1.00000	0.0890871
$127$	− 22.0000i	− 1.95218i	−0.217357	−	0.976092i	$-0.569744\pi$
$127$	− 22.0000i	− 1.95218i	0.217357	−	0.976092i	$-0.430256\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	11.0000	0.968496
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	2.00000i	0.174078i
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	3.00000	0.257248
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	1.00000i	0.0851257i
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	12.0000i	1.00702i
$143$	2.00000i	0.167248i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	1.00000i	0.0824786i
$148$	2.00000i	0.164399i
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	− 3.00000i	− 0.242536i
$154$	−2.00000	−0.161165
$155$	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	$-0.0254309\pi$
$158$	− 10.0000i	− 0.795557i
$159$	1.00000	0.0793052
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	1.00000i	0.0785674i
$163$	1.00000i	0.0783260i	0.999233	+	0.0391630i	$0.0124692\pi$
$163$	1.00000i	0.0783260i	−0.999233	+	0.0391630i	$0.987531\pi$
$164$	−7.00000	−0.546608
$165$	0	0
$166$	−11.0000	−0.853766
$167$	18.0000i	1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$	18.0000i	1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$	− 1.00000i	− 0.0771517i
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	− 11.0000i	− 0.838742i
$173$	− 24.0000i	− 1.82469i	−0.409426	−	0.912343i	$-0.634271\pi$
$173$	− 24.0000i	− 1.82469i	0.409426	−	0.912343i	$-0.365729\pi$
$174$	5.00000	0.379049
$175$	0	0
$176$	2.00000	0.150756
$177$	− 5.00000i	− 0.375823i
$178$	10.0000i	0.749532i
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	− 1.00000i	− 0.0741249i
$183$	3.00000i	0.221766i
$184$	1.00000	0.0737210
$185$	0	0
$186$	7.00000	0.513265
$187$	6.00000i	0.438763i
$188$	− 8.00000i	− 0.583460i
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	1.00000i	0.0721688i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 17.0000i	− 1.21120i	−0.795769	−	0.605600i	$-0.792933\pi$
$197$	− 17.0000i	− 1.21120i	0.795769	−	0.605600i	$-0.207067\pi$
$198$	− 2.00000i	− 0.142134i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	12.0000i	0.844317i
$203$	5.00000i	0.350931i
$204$	−3.00000	−0.210042
$205$	0	0
$206$	−11.0000	−0.766406
$207$	− 1.00000i	− 0.0695048i
$208$	1.00000i	0.0693375i
$209$	0	0
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	− 1.00000i	− 0.0686803i
$213$	− 12.0000i	− 0.822226i
$214$	2.00000	0.136717
$215$	0	0
$216$	1.00000	0.0680414
$217$	7.00000i	0.475191i
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	−3.00000	−0.201802
$222$	− 2.00000i	− 0.134231i
$223$	− 9.00000i	− 0.602685i	−0.953516	−	0.301342i	$-0.902565\pi$
$223$	− 9.00000i	− 0.602685i	0.953516	−	0.301342i	$-0.0974347\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−6.00000	−0.399114
$227$	− 17.0000i	− 1.12833i	−0.825662	−	0.564165i	$-0.809198\pi$
$227$	− 17.0000i	− 1.12833i	0.825662	−	0.564165i	$-0.190802\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	− 5.00000i	− 0.328266i
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−5.00000	−0.325472
$237$	10.0000i	0.649570i
$238$	− 3.00000i	− 0.194461i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	− 7.00000i	− 0.449977i
$243$	− 1.00000i	− 0.0641500i
$244$	3.00000	0.192055
$245$	0	0
$246$	7.00000	0.446304
$247$	0	0
$248$	− 7.00000i	− 0.444500i
$249$	11.0000	0.697097
$250$	0	0
$251$	7.00000	0.441836	0.220918	−	0.975292i	$-0.429095\pi$
$251$	7.00000	0.441836	0.220918	+	0.975292i	$0.429095\pi$
$252$	1.00000i	0.0629941i
$253$	2.00000i	0.125739i
$254$	22.0000	1.38040
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.0000i	1.43470i	0.696713	+	0.717350i	$0.254645\pi$
$257$	23.0000i	1.43470i	−0.696713	+	0.717350i	$0.745355\pi$
$258$	11.0000i	0.684830i
$259$	2.00000	0.124274
$260$	0	0
$261$	−5.00000	−0.309492
$262$	− 8.00000i	− 0.494242i
$263$	21.0000i	1.29492i	0.762101	+	0.647458i	$0.224168\pi$
$263$	21.0000i	1.29492i	−0.762101	+	0.647458i	$0.775832\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	12.0000i	0.733017i
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	3.00000i	0.181902i
$273$	1.00000i	0.0605228i
$274$	12.0000	0.724947
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	− 20.0000i	− 1.19952i
$279$	−7.00000	−0.419079
$280$	0	0
$281$	32.0000	1.90896	0.954480	−	0.298275i	$-0.0964112\pi$
$281$	32.0000	1.90896	0.954480	+	0.298275i	$0.0964112\pi$
$282$	8.00000i	0.476393i
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−2.00000	−0.118262
$287$	7.00000i	0.413197i
$288$	− 1.00000i	− 0.0589256i
$289$	8.00000	0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	− 6.00000i	− 0.351123i
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	−2.00000	−0.116248
$297$	2.00000i	0.116052i
$298$	− 15.0000i	− 0.868927i
$299$	−1.00000	−0.0578315
$300$	0	0
$301$	−11.0000	−0.634029
$302$	− 18.0000i	− 1.03578i
$303$	− 12.0000i	− 0.689382i
$304$	0	0
$305$	0	0
$306$	3.00000	0.171499
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	− 2.00000i	− 0.113961i
$309$	11.0000	0.625768
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	− 1.00000i	− 0.0566139i
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	10.0000	0.562544
$317$	3.00000i	0.168497i	0.996445	+	0.0842484i	$0.0268489\pi$
$317$	3.00000i	0.168497i	−0.996445	+	0.0842484i	$0.973151\pi$
$318$	1.00000i	0.0560772i
$319$	10.0000	0.559893
$320$	0	0
$321$	−2.00000	−0.111629
$322$	− 1.00000i	− 0.0557278i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−1.00000	−0.0553849
$327$	0	0
$328$	− 7.00000i	− 0.386510i
$329$	−8.00000	−0.441054
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	− 11.0000i	− 0.603703i
$333$	2.00000i	0.109599i
$334$	−18.0000	−0.984916
$335$	0	0
$336$	1.00000	0.0545545
$337$	− 27.0000i	− 1.47078i	−0.677642	−	0.735392i	$-0.736998\pi$
$337$	− 27.0000i	− 1.47078i	0.677642	−	0.735392i	$-0.263002\pi$
$338$	12.0000i	0.652714i
$339$	6.00000	0.325875
$340$	0	0
$341$	14.0000	0.758143
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	11.0000	0.593080
$345$	0	0
$346$	24.0000	1.29025
$347$	− 2.00000i	− 0.107366i	−0.998558	−	0.0536828i	$-0.982904\pi$
$347$	− 2.00000i	− 0.107366i	0.998558	−	0.0536828i	$-0.0170960\pi$
$348$	5.00000i	0.268028i
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	2.00000i	0.106600i
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	5.00000	0.265747
$355$	0	0
$356$	−10.0000	−0.529999
$357$	3.00000i	0.158777i
$358$	10.0000i	0.528516i
$359$	−25.0000	−1.31945	−0.659725	−	0.751507i	$-0.729327\pi$
$359$	−25.0000	−1.31945	−0.659725	+	0.751507i	$0.729327\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	− 18.0000i	− 0.946059i
$363$	7.00000i	0.367405i
$364$	1.00000	0.0524142
$365$	0	0
$366$	−3.00000	−0.156813
$367$	23.0000i	1.20059i	0.799779	+	0.600295i	$0.204950\pi$
$367$	23.0000i	1.20059i	−0.799779	+	0.600295i	$0.795050\pi$
$368$	1.00000i	0.0521286i
$369$	−7.00000	−0.364405
$370$	0	0
$371$	−1.00000	−0.0519174
$372$	7.00000i	0.362933i
$373$	− 24.0000i	− 1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$	− 24.0000i	− 1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	8.00000	0.412568
$377$	5.00000i	0.257513i
$378$	− 1.00000i	− 0.0514344i
$379$	25.0000	1.28416	0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000	1.28416	0.642082	+	0.766636i	$0.278071\pi$
$380$	0	0
$381$	−22.0000	−1.12709
$382$	− 3.00000i	− 0.153493i
$383$	− 4.00000i	− 0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$	− 4.00000i	− 0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	− 11.0000i	− 0.559161i
$388$	2.00000i	0.101535i
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	1.00000i	0.0505076i
$393$	8.00000i	0.403547i
$394$	17.0000	0.856448
$395$	0	0
$396$	2.00000	0.100504
$397$	13.0000i	0.652451i	0.945292	+	0.326226i	$0.105777\pi$
$397$	13.0000i	0.652451i	−0.945292	+	0.326226i	$0.894223\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	− 12.0000i	− 0.598506i
$403$	7.00000i	0.348695i
$404$	−12.0000	−0.597022
$405$	0	0
$406$	−5.00000	−0.248146
$407$	− 4.00000i	− 0.198273i
$408$	− 3.00000i	− 0.148522i
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	− 11.0000i	− 0.541931i
$413$	5.00000i	0.246034i
$414$	1.00000	0.0491473
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	20.0000i	0.979404i
$418$	0	0
$419$	−25.0000	−1.22133	−0.610665	−	0.791889i	$-0.709098\pi$
$419$	−25.0000	−1.22133	−0.610665	+	0.791889i	$0.709098\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	− 13.0000i	− 0.632830i
$423$	− 8.00000i	− 0.388973i
$424$	1.00000	0.0485643
$425$	0	0
$426$	12.0000	0.581402
$427$	− 3.00000i	− 0.145180i
$428$	2.00000i	0.0966736i
$429$	2.00000	0.0965609
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	1.00000i	0.0481125i
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	−7.00000	−0.336011
$435$	0	0
$436$	0	0
$437$	0	0
$438$	6.00000i	0.286691i
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	− 3.00000i	− 0.142695i
$443$	16.0000i	0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$	16.0000i	0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	9.00000	0.426162
$447$	15.0000i	0.709476i
$448$	− 1.00000i	− 0.0472456i
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	14.0000	0.659234
$452$	− 6.00000i	− 0.282216i
$453$	18.0000i	0.845714i
$454$	17.0000	0.797850
$455$	0	0
$456$	0	0
$457$	33.0000i	1.54367i	0.635820	+	0.771837i	$0.280662\pi$
$457$	33.0000i	1.54367i	−0.635820	+	0.771837i	$0.719338\pi$
$458$	10.0000i	0.467269i
$459$	−3.00000	−0.140028
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	2.00000i	0.0930484i
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	24.0000	1.11178
$467$	33.0000i	1.52706i	0.645774	+	0.763529i	$0.276535\pi$
$467$	33.0000i	1.52706i	−0.645774	+	0.763529i	$0.723465\pi$
$468$	1.00000i	0.0462250i
$469$	12.0000	0.554109
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	− 5.00000i	− 0.230144i
$473$	22.0000i	1.01156i
$474$	−10.0000	−0.459315
$475$	0	0
$476$	3.00000	0.137505
$477$	− 1.00000i	− 0.0457869i
$478$	0	0
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	22.0000i	1.00207i
$483$	1.00000i	0.0455016i
$484$	7.00000	0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	3.00000i	0.135804i
$489$	1.00000	0.0452216
$490$	0	0
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	7.00000i	0.315584i
$493$	15.0000i	0.675566i
$494$	0	0
$495$	0	0
$496$	7.00000	0.314309
$497$	12.0000i	0.538274i
$498$	11.0000i	0.492922i
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	7.00000i	0.312425i
$503$	6.00000i	0.267527i	0.991013	+	0.133763i	$0.0427062\pi$
$503$	6.00000i	0.267527i	−0.991013	+	0.133763i	$0.957294\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	−2.00000	−0.0889108
$507$	− 12.0000i	− 0.532939i
$508$	22.0000i	0.976092i
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−23.0000	−1.01449
$515$	0	0
$516$	−11.0000	−0.484248
$517$	16.0000i	0.703679i
$518$	2.00000i	0.0878750i
$519$	−24.0000	−1.05348
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	− 5.00000i	− 0.218844i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	−21.0000	−0.915644
$527$	21.0000i	0.914774i
$528$	− 2.00000i	− 0.0870388i
$529$	22.0000	0.956522
$530$	0	0
$531$	−5.00000	−0.216982
$532$	0	0
$533$	7.00000i	0.303204i
$534$	10.0000	0.432742
$535$	0	0
$536$	−12.0000	−0.518321
$537$	− 10.0000i	− 0.431532i
$538$	10.0000i	0.431131i
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	− 28.0000i	− 1.20270i
$543$	18.0000i	0.772454i
$544$	−3.00000	−0.128624
$545$	0	0
$546$	−1.00000	−0.0427960
$547$	− 37.0000i	− 1.58201i	−0.611812	−	0.791003i	$-0.709559\pi$
$547$	− 37.0000i	− 1.58201i	0.611812	−	0.791003i	$-0.290441\pi$
$548$	12.0000i	0.512615i
$549$	3.00000	0.128037
$550$	0	0
$551$	0	0
$552$	− 1.00000i	− 0.0425628i
$553$	− 10.0000i	− 0.425243i
$554$	2.00000	0.0849719
$555$	0	0
$556$	20.0000	0.848189
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	− 7.00000i	− 0.296334i
$559$	−11.0000	−0.465250
$560$	0	0
$561$	6.00000	0.253320
$562$	32.0000i	1.34984i
$563$	31.0000i	1.30649i	0.757145	+	0.653247i	$0.226594\pi$
$563$	31.0000i	1.30649i	−0.757145	+	0.653247i	$0.773406\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−16.0000	−0.672530
$567$	1.00000i	0.0419961i
$568$	− 12.0000i	− 0.503509i
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	3.00000i	0.125327i
$574$	−7.00000	−0.292174
$575$	0	0
$576$	1.00000	0.0416667
$577$	38.0000i	1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$	38.0000i	1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$	8.00000i	0.332756i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−11.0000	−0.456357
$582$	− 2.00000i	− 0.0829027i
$583$	2.00000i	0.0828315i
$584$	6.00000	0.248282
$585$	0	0
$586$	24.0000	0.991431
$587$	− 27.0000i	− 1.11441i	−0.830375	−	0.557205i	$-0.811874\pi$
$587$	− 27.0000i	− 1.11441i	0.830375	−	0.557205i	$-0.188126\pi$
$588$	− 1.00000i	− 0.0412393i
$589$	0	0
$590$	0	0
$591$	−17.0000	−0.699287
$592$	− 2.00000i	− 0.0821995i
$593$	− 14.0000i	− 0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$	− 14.0000i	− 0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	15.0000	0.614424
$597$	0	0
$598$	− 1.00000i	− 0.0408930i
$599$	−5.00000	−0.204294	−0.102147	−	0.994769i	$-0.532571\pi$
$599$	−5.00000	−0.204294	−0.102147	+	0.994769i	$0.532571\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	− 11.0000i	− 0.448327i
$603$	12.0000i	0.488678i
$604$	18.0000	0.732410
$605$	0	0
$606$	12.0000	0.487467
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	0	0
$609$	5.00000	0.202610
$610$	0	0
$611$	−8.00000	−0.323645
$612$	3.00000i	0.121268i
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	2.00000	0.0805823
$617$	38.0000i	1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$	38.0000i	1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$	11.0000i	0.442485i
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	− 18.0000i	− 0.721734i
$623$	10.0000i	0.400642i
$624$	1.00000	0.0400320
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	2.00000i	0.0798087i
$629$	6.00000	0.239236
$630$	0	0
$631$	−38.0000	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$631$	−38.0000	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$632$	10.0000i	0.397779i
$633$	13.0000i	0.516704i
$634$	−3.00000	−0.119145
$635$	0	0
$636$	−1.00000	−0.0396526
$637$	− 1.00000i	− 0.0396214i
$638$	10.0000i	0.395904i
$639$	−12.0000	−0.474713
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	− 2.00000i	− 0.0789337i
$643$	− 34.0000i	− 1.34083i	−0.741987	−	0.670415i	$-0.766116\pi$
$643$	− 34.0000i	− 1.34083i	0.741987	−	0.670415i	$-0.233884\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	0	0
$647$	− 22.0000i	− 0.864909i	−0.901656	−	0.432455i	$-0.857648\pi$
$647$	− 22.0000i	− 0.864909i	0.901656	−	0.432455i	$-0.142352\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	10.0000	0.392534
$650$	0	0
$651$	7.00000	0.274352
$652$	− 1.00000i	− 0.0391630i
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	0	0
$655$	0	0
$656$	7.00000	0.273304
$657$	− 6.00000i	− 0.234082i
$658$	− 8.00000i	− 0.311872i
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	17.0000i	0.660724i
$663$	3.00000i	0.116510i
$664$	11.0000	0.426883
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	5.00000i	0.193601i
$668$	− 18.0000i	− 0.696441i
$669$	−9.00000	−0.347960
$670$	0	0
$671$	−6.00000	−0.231627
$672$	1.00000i	0.0385758i
$673$	− 29.0000i	− 1.11787i	−0.829212	−	0.558934i	$-0.811211\pi$
$673$	− 29.0000i	− 1.11787i	0.829212	−	0.558934i	$-0.188789\pi$
$674$	27.0000	1.04000
$675$	0	0
$676$	−12.0000	−0.461538
$677$	− 32.0000i	− 1.22986i	−0.788582	−	0.614930i	$-0.789184\pi$
$677$	− 32.0000i	− 1.22986i	0.788582	−	0.614930i	$-0.210816\pi$
$678$	6.00000i	0.230429i
$679$	2.00000	0.0767530
$680$	0	0
$681$	−17.0000	−0.651441
$682$	14.0000i	0.536088i
$683$	46.0000i	1.76014i	0.474843	+	0.880071i	$0.342505\pi$
$683$	46.0000i	1.76014i	−0.474843	+	0.880071i	$0.657495\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	− 10.0000i	− 0.381524i
$688$	11.0000i	0.419371i
$689$	−1.00000	−0.0380970
$690$	0	0
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	24.0000i	0.912343i
$693$	− 2.00000i	− 0.0759737i
$694$	2.00000	0.0759190
$695$	0	0
$696$	−5.00000	−0.189525
$697$	21.0000i	0.795432i
$698$	− 15.0000i	− 0.567758i
$699$	−24.0000	−0.907763
$700$	0	0
$701$	27.0000	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$702$	− 1.00000i	− 0.0377426i
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−6.00000	−0.225813
$707$	12.0000i	0.451306i
$708$	5.00000i	0.187912i
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	− 10.0000i	− 0.374766i
$713$	7.00000i	0.262152i
$714$	−3.00000	−0.112272
$715$	0	0
$716$	−10.0000	−0.373718
$717$	0	0
$718$	− 25.0000i	− 0.932992i
$719$	10.0000	0.372937	0.186469	−	0.982461i	$-0.440296\pi$
$719$	10.0000	0.372937	0.186469	+	0.982461i	$0.440296\pi$
$720$	0	0
$721$	−11.0000	−0.409661
$722$	− 19.0000i	− 0.707107i
$723$	− 22.0000i	− 0.818189i
$724$	18.0000	0.668965
$725$	0	0
$726$	−7.00000	−0.259794
$727$	13.0000i	0.482143i	0.970507	+	0.241072i	$0.0774989\pi$
$727$	13.0000i	0.482143i	−0.970507	+	0.241072i	$0.922501\pi$
$728$	1.00000i	0.0370625i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−33.0000	−1.22055
$732$	− 3.00000i	− 0.110883i
$733$	11.0000i	0.406294i	0.979148	+	0.203147i	$0.0651170\pi$
$733$	11.0000i	0.406294i	−0.979148	+	0.203147i	$0.934883\pi$
$734$	−23.0000	−0.848945
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	− 24.0000i	− 0.884051i
$738$	− 7.00000i	− 0.257674i
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	0	0
$741$	0	0
$742$	− 1.00000i	− 0.0367112i
$743$	− 29.0000i	− 1.06391i	−0.846774	−	0.531953i	$-0.821458\pi$
$743$	− 29.0000i	− 1.06391i	0.846774	−	0.531953i	$-0.178542\pi$
$744$	−7.00000	−0.256632
$745$	0	0
$746$	24.0000	0.878702
$747$	− 11.0000i	− 0.402469i
$748$	− 6.00000i	− 0.219382i
$749$	2.00000	0.0730784
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	8.00000i	0.291730i
$753$	− 7.00000i	− 0.255094i
$754$	−5.00000	−0.182089
$755$	0	0
$756$	1.00000	0.0363696
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	25.0000i	0.908041i
$759$	2.00000	0.0725954
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	− 22.0000i	− 0.796976i
$763$	0	0
$764$	3.00000	0.108536
$765$	0	0
$766$	4.00000	0.144526
$767$	5.00000i	0.180540i
$768$	− 1.00000i	− 0.0360844i
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	23.0000	0.828325
$772$	14.0000i	0.503871i
$773$	− 24.0000i	− 0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$	− 24.0000i	− 0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$	11.0000	0.395387
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	− 2.00000i	− 0.0717496i
$778$	− 10.0000i	− 0.358517i
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	− 3.00000i	− 0.107280i
$783$	5.00000i	0.178685i
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	−8.00000	−0.285351
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	17.0000i	0.605600i
$789$	21.0000	0.747620
$790$	0	0
$791$	−6.00000	−0.213335
$792$	2.00000i	0.0710669i
$793$	− 3.00000i	− 0.106533i
$794$	−13.0000	−0.461353
$795$	0	0
$796$	0	0
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	−10.0000	−0.353333
$802$	− 8.00000i	− 0.282490i
$803$	12.0000i	0.423471i
$804$	12.0000	0.423207
$805$	0	0
$806$	−7.00000	−0.246564
$807$	− 10.0000i	− 0.352017i
$808$	− 12.0000i	− 0.422159i
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−18.0000	−0.632065	−0.316033	−	0.948748i	$-0.602351\pi$
$811$	−18.0000	−0.632065	−0.316033	+	0.948748i	$0.602351\pi$
$812$	− 5.00000i	− 0.175466i
$813$	28.0000i	0.982003i
$814$	4.00000	0.140200
$815$	0	0
$816$	3.00000	0.105021
$817$	0	0
$818$	10.0000i	0.349642i
$819$	1.00000	0.0349428
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	− 12.0000i	− 0.418548i
$823$	− 54.0000i	− 1.88232i	−0.337959	−	0.941161i	$-0.609737\pi$
$823$	− 54.0000i	− 1.88232i	0.337959	−	0.941161i	$-0.390263\pi$
$824$	11.0000	0.383203
$825$	0	0
$826$	−5.00000	−0.173972
$827$	− 2.00000i	− 0.0695468i	−0.999395	−	0.0347734i	$-0.988929\pi$
$827$	− 2.00000i	− 0.0695468i	0.999395	−	0.0347734i	$-0.0110710\pi$
$828$	1.00000i	0.0347524i
$829$	5.00000	0.173657	0.0868286	−	0.996223i	$-0.472327\pi$
$829$	5.00000	0.173657	0.0868286	+	0.996223i	$0.472327\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	− 1.00000i	− 0.0346688i
$833$	− 3.00000i	− 0.103944i
$834$	−20.0000	−0.692543
$835$	0	0
$836$	0	0
$837$	7.00000i	0.241955i
$838$	− 25.0000i	− 0.863611i
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	22.0000i	0.758170i
$843$	− 32.0000i	− 1.10214i
$844$	13.0000	0.447478
$845$	0	0
$846$	8.00000	0.275046
$847$	− 7.00000i	− 0.240523i
$848$	1.00000i	0.0343401i
$849$	16.0000	0.549119
$850$	0	0
$851$	2.00000	0.0685591
$852$	12.0000i	0.411113i
$853$	− 29.0000i	− 0.992941i	−0.868054	−	0.496471i	$-0.834629\pi$
$853$	− 29.0000i	− 0.992941i	0.868054	−	0.496471i	$-0.165371\pi$
$854$	3.00000	0.102658
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	− 22.0000i	− 0.751506i	−0.926720	−	0.375753i	$-0.877384\pi$
$857$	− 22.0000i	− 0.751506i	0.926720	−	0.375753i	$-0.122616\pi$
$858$	2.00000i	0.0682789i
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	7.00000	0.238559
$862$	− 3.00000i	− 0.102180i
$863$	16.0000i	0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$	16.0000i	0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	14.0000	0.475739
$867$	− 8.00000i	− 0.271694i
$868$	− 7.00000i	− 0.237595i
$869$	−20.0000	−0.678454
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	−6.00000	−0.202721
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	− 35.0000i	− 1.18119i
$879$	−24.0000	−0.809500
$880$	0	0
$881$	7.00000	0.235836	0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000	0.235836	0.117918	+	0.993023i	$0.462378\pi$
$882$	1.00000i	0.0336718i
$883$	− 39.0000i	− 1.31245i	−0.754563	−	0.656227i	$-0.772151\pi$
$883$	− 39.0000i	− 1.31245i	0.754563	−	0.656227i	$-0.227849\pi$
$884$	3.00000	0.100901
$885$	0	0
$886$	−16.0000	−0.537531
$887$	58.0000i	1.94745i	0.227728	+	0.973725i	$0.426870\pi$
$887$	58.0000i	1.94745i	−0.227728	+	0.973725i	$0.573130\pi$
$888$	2.00000i	0.0671156i
$889$	22.0000	0.737856
$890$	0	0
$891$	2.00000	0.0670025
$892$	9.00000i	0.301342i
$893$	0	0
$894$	−15.0000	−0.501675
$895$	0	0
$896$	1.00000	0.0334077
$897$	1.00000i	0.0333890i
$898$	− 20.0000i	− 0.667409i
$899$	35.0000	1.16732
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	14.0000i	0.466149i
$903$	11.0000i	0.366057i
$904$	6.00000	0.199557
$905$	0	0
$906$	−18.0000	−0.598010
$907$	− 17.0000i	− 0.564476i	−0.959344	−	0.282238i	$-0.908923\pi$
$907$	− 17.0000i	− 0.564476i	0.959344	−	0.282238i	$-0.0910767\pi$
$908$	17.0000i	0.564165i
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−53.0000	−1.75597	−0.877984	−	0.478690i	$-0.841112\pi$
$911$	−53.0000	−1.75597	−0.877984	+	0.478690i	$0.841112\pi$
$912$	0	0
$913$	22.0000i	0.728094i
$914$	−33.0000	−1.09154
$915$	0	0
$916$	−10.0000	−0.330409
$917$	− 8.00000i	− 0.264183i
$918$	− 3.00000i	− 0.0990148i
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	12.0000i	0.395199i
$923$	12.0000i	0.394985i
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	4.00000	0.131448
$927$	− 11.0000i	− 0.361287i
$928$	5.00000i	0.164133i
$929$	45.0000	1.47640	0.738201	−	0.674581i	$-0.235676\pi$
$929$	45.0000	1.47640	0.738201	+	0.674581i	$0.235676\pi$
$930$	0	0
$931$	0	0
$932$	24.0000i	0.786146i
$933$	18.0000i	0.589294i
$934$	−33.0000	−1.07979
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	− 42.0000i	− 1.37208i	−0.727564	−	0.686040i	$-0.759347\pi$
$937$	− 42.0000i	− 1.37208i	0.727564	−	0.686040i	$-0.240653\pi$
$938$	12.0000i	0.391814i
$939$	6.00000	0.195803
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	− 2.00000i	− 0.0651635i
$943$	7.00000i	0.227951i
$944$	5.00000	0.162736
$945$	0	0
$946$	−22.0000	−0.715282
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	− 10.0000i	− 0.324785i
$949$	−6.00000	−0.194768
$950$	0	0
$951$	3.00000	0.0972817
$952$	3.00000i	0.0972306i
$953$	16.0000i	0.518291i	0.965838	+	0.259145i	$0.0834409\pi$
$953$	16.0000i	0.518291i	−0.965838	+	0.259145i	$0.916559\pi$
$954$	1.00000	0.0323762
$955$	0	0
$956$	0	0
$957$	− 10.0000i	− 0.323254i
$958$	20.0000i	0.646171i
$959$	12.0000	0.387500
$960$	0	0
$961$	18.0000	0.580645
$962$	2.00000i	0.0644826i
$963$	2.00000i	0.0644491i
$964$	−22.0000	−0.708572
$965$	0	0
$966$	−1.00000	−0.0321745
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	7.00000i	0.224989i
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000i	0.0320750i
$973$	− 20.0000i	− 0.641171i
$974$	2.00000	0.0640841
$975$	0	0
$976$	−3.00000	−0.0960277
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	1.00000i	0.0319765i
$979$	20.0000	0.639203
$980$	0	0
$981$	0	0
$982$	2.00000i	0.0638226i
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	−7.00000	−0.223152
$985$	0	0
$986$	−15.0000	−0.477697
$987$	8.00000i	0.254643i
$988$	0	0
$989$	−11.0000	−0.349780
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	7.00000i	0.222250i
$993$	− 17.0000i	− 0.539479i
$994$	−12.0000	−0.380617
$995$	0	0
$996$	−11.0000	−0.348548
$997$	58.0000i	1.83688i	0.395562	+	0.918439i	$0.370550\pi$
$997$	58.0000i	1.83688i	−0.395562	+	0.918439i	$0.629450\pi$
$998$	− 25.0000i	− 0.791361i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.g.i.799.2		2
3.2	odd	2		3150.2.g.h.2899.1		2
5.2	odd	4		1050.2.a.b.1.1	✓	1
5.3	odd	4		1050.2.a.r.1.1	yes	1
5.4	even	2	inner	1050.2.g.i.799.1		2
15.2	even	4		3150.2.a.y.1.1		1
15.8	even	4		3150.2.a.n.1.1		1
15.14	odd	2		3150.2.g.h.2899.2		2
20.3	even	4		8400.2.a.d.1.1		1
20.7	even	4		8400.2.a.ck.1.1		1
35.13	even	4		7350.2.a.cb.1.1		1
35.27	even	4		7350.2.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.2.a.b.1.1	✓	1	5.2	odd	4
1050.2.a.r.1.1	yes	1	5.3	odd	4
1050.2.g.i.799.1		2	5.4	even	2	inner
1050.2.g.i.799.2		2	1.1	even	1	trivial
3150.2.a.n.1.1		1	15.8	even	4
3150.2.a.y.1.1		1	15.2	even	4
3150.2.g.h.2899.1		2	3.2	odd	2
3150.2.g.h.2899.2		2	15.14	odd	2
7350.2.a.bi.1.1		1	35.27	even	4
7350.2.a.cb.1.1		1	35.13	even	4
8400.2.a.d.1.1		1	20.3	even	4
8400.2.a.ck.1.1		1	20.7	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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} +1.00000i q^{12} +1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -1.00000i q^{18} +1.00000 q^{21} +2.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} +5.00000 q^{29} +7.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} -3.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +1.00000 q^{39} +7.00000 q^{41} +1.00000i q^{42} +11.0000i q^{43} -2.00000 q^{44} -1.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} +3.00000 q^{51} -1.00000i q^{52} +1.00000i q^{53} -1.00000 q^{54} +1.00000 q^{56} +5.00000i q^{58} +5.00000 q^{59} -3.00000 q^{61} +7.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -12.0000i q^{67} -3.00000i q^{68} +1.00000 q^{69} +12.0000 q^{71} +1.00000i q^{72} +6.00000i q^{73} +2.00000 q^{74} +2.00000i q^{77} +1.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} +7.00000i q^{82} +11.0000i q^{83} -1.00000 q^{84} -11.0000 q^{86} -5.00000i q^{87} -2.00000i q^{88} +10.0000 q^{89} -1.00000 q^{91} -1.00000i q^{92} -7.00000i q^{93} -8.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -1.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 4 q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{21} - 2 q^{24} - 2 q^{26} + 10 q^{29} + 14 q^{31} - 6 q^{34} + 2 q^{36} + 2 q^{39} + 14 q^{41} - 4 q^{44} - 2 q^{46} - 2 q^{49} + 6 q^{51} - 2 q^{54} + 2 q^{56} + 10 q^{59} - 6 q^{61} - 2 q^{64} + 4 q^{66} + 2 q^{69} + 24 q^{71} + 4 q^{74} - 20 q^{79} + 2 q^{81} - 2 q^{84} - 22 q^{86} + 20 q^{89} - 2 q^{91} - 16 q^{94} + 2 q^{96} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 4 * q^11 - 2 * q^14 + 2 * q^16 + 2 * q^21 - 2 * q^24 - 2 * q^26 + 10 * q^29 + 14 * q^31 - 6 * q^34 + 2 * q^36 + 2 * q^39 + 14 * q^41 - 4 * q^44 - 2 * q^46 - 2 * q^49 + 6 * q^51 - 2 * q^54 + 2 * q^56 + 10 * q^59 - 6 * q^61 - 2 * q^64 + 4 * q^66 + 2 * q^69 + 24 * q^71 + 4 * q^74 - 20 * q^79 + 2 * q^81 - 2 * q^84 - 22 * q^86 + 20 * q^89 - 2 * q^91 - 16 * q^94 + 2 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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