LMFDB - Embedded newform 8046.2.a.t.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.a (trivial)

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:	yes
Analytic conductor:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + \cdots - 4260 $ x^16 - 4x^15 - 46x^14 + 192x^13 + 752x^12 - 3378x^11 - 5277x^10 + 27132x^9 + 15173x^8 - 102466x^7 - 16612x^6 + 186120x^5 + 117x^4 - 156722x^3 + 10545x^2 + 48580*x - 4260
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$3.39614$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.39614 q^{5} +0.912273 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.39614 q^{5} +0.912273 q^{7} +1.00000 q^{8} +3.39614 q^{10} +4.59744 q^{11} -0.963808 q^{13} +0.912273 q^{14} +1.00000 q^{16} +2.36865 q^{17} -6.97756 q^{19} +3.39614 q^{20} +4.59744 q^{22} -4.67453 q^{23} +6.53378 q^{25} -0.963808 q^{26} +0.912273 q^{28} +4.38966 q^{29} -8.13917 q^{31} +1.00000 q^{32} +2.36865 q^{34} +3.09821 q^{35} +2.94086 q^{37} -6.97756 q^{38} +3.39614 q^{40} -6.42896 q^{41} +9.90304 q^{43} +4.59744 q^{44} -4.67453 q^{46} +13.6140 q^{47} -6.16776 q^{49} +6.53378 q^{50} -0.963808 q^{52} +10.7060 q^{53} +15.6136 q^{55} +0.912273 q^{56} +4.38966 q^{58} +14.5609 q^{59} -7.33496 q^{61} -8.13917 q^{62} +1.00000 q^{64} -3.27323 q^{65} +0.214533 q^{67} +2.36865 q^{68} +3.09821 q^{70} +7.73903 q^{71} +8.97504 q^{73} +2.94086 q^{74} -6.97756 q^{76} +4.19412 q^{77} +0.227439 q^{79} +3.39614 q^{80} -6.42896 q^{82} +8.95630 q^{83} +8.04427 q^{85} +9.90304 q^{86} +4.59744 q^{88} +0.839753 q^{89} -0.879256 q^{91} -4.67453 q^{92} +13.6140 q^{94} -23.6968 q^{95} +4.12389 q^{97} -6.16776 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.39614	1.51880	0.759400	−	0.650624i	$-0.225492\pi$
$5$	3.39614	1.51880	0.759400	+	0.650624i	$0.225492\pi$
$6$	0	0
$7$	0.912273	0.344807	0.172403	−	0.985026i	$-0.444847\pi$
$7$	0.912273	0.344807	0.172403	+	0.985026i	$0.444847\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.39614	1.07395
$11$	4.59744	1.38618	0.693090	−	0.720851i	$-0.256249\pi$
$11$	4.59744	1.38618	0.693090	+	0.720851i	$0.256249\pi$
$12$	0	0
$13$	−0.963808	−0.267312	−0.133656	−	0.991028i	$-0.542672\pi$
$13$	−0.963808	−0.267312	−0.133656	+	0.991028i	$0.542672\pi$
$14$	0.912273	0.243815
$15$	0	0
$16$	1.00000	0.250000
$17$	2.36865	0.574482	0.287241	−	0.957858i	$-0.407262\pi$
$17$	2.36865	0.574482	0.287241	+	0.957858i	$0.407262\pi$
$18$	0	0
$19$	−6.97756	−1.60076	−0.800381	−	0.599492i	$-0.795369\pi$
$19$	−6.97756	−1.60076	−0.800381	+	0.599492i	$0.795369\pi$
$20$	3.39614	0.759400
$21$	0	0
$22$	4.59744	0.980177
$23$	−4.67453	−0.974707	−0.487353	−	0.873205i	$-0.662038\pi$
$23$	−4.67453	−0.974707	−0.487353	+	0.873205i	$0.662038\pi$
$24$	0	0
$25$	6.53378	1.30676
$26$	−0.963808	−0.189018
$27$	0	0
$28$	0.912273	0.172403
$29$	4.38966	0.815140	0.407570	−	0.913174i	$-0.366376\pi$
$29$	4.38966	0.815140	0.407570	+	0.913174i	$0.366376\pi$
$30$	0	0
$31$	−8.13917	−1.46184	−0.730919	−	0.682464i	$-0.760908\pi$
$31$	−8.13917	−1.46184	−0.730919	+	0.682464i	$0.760908\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.36865	0.406220
$35$	3.09821	0.523693
$36$	0	0
$37$	2.94086	0.483474	0.241737	−	0.970342i	$-0.422283\pi$
$37$	2.94086	0.483474	0.241737	+	0.970342i	$0.422283\pi$
$38$	−6.97756	−1.13191
$39$	0	0
$40$	3.39614	0.536977
$41$	−6.42896	−1.00403	−0.502017	−	0.864858i	$-0.667409\pi$
$41$	−6.42896	−1.00403	−0.502017	+	0.864858i	$0.667409\pi$
$42$	0	0
$43$	9.90304	1.51020	0.755100	−	0.655610i	$-0.227588\pi$
$43$	9.90304	1.51020	0.755100	+	0.655610i	$0.227588\pi$
$44$	4.59744	0.693090
$45$	0	0
$46$	−4.67453	−0.689222
$47$	13.6140	1.98580	0.992901	−	0.118940i	$-0.0379496\pi$
$47$	13.6140	1.98580	0.992901	+	0.118940i	$0.0379496\pi$
$48$	0	0
$49$	−6.16776	−0.881108
$50$	6.53378	0.924016
$51$	0	0
$52$	−0.963808	−0.133656
$53$	10.7060	1.47059	0.735294	−	0.677749i	$-0.237044\pi$
$53$	10.7060	1.47059	0.735294	+	0.677749i	$0.237044\pi$
$54$	0	0
$55$	15.6136	2.10533
$56$	0.912273	0.121908
$57$	0	0
$58$	4.38966	0.576391
$59$	14.5609	1.89566	0.947832	−	0.318769i	$-0.103269\pi$
$59$	14.5609	1.89566	0.947832	+	0.318769i	$0.103269\pi$
$60$	0	0
$61$	−7.33496	−0.939146	−0.469573	−	0.882894i	$-0.655592\pi$
$61$	−7.33496	−0.939146	−0.469573	+	0.882894i	$0.655592\pi$
$62$	−8.13917	−1.03368
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.27323	−0.405994
$66$	0	0
$67$	0.214533	0.0262094	0.0131047	−	0.999914i	$-0.495829\pi$
$67$	0.214533	0.0262094	0.0131047	+	0.999914i	$0.495829\pi$
$68$	2.36865	0.287241
$69$	0	0
$70$	3.09821	0.370307
$71$	7.73903	0.918454	0.459227	−	0.888319i	$-0.348127\pi$
$71$	7.73903	0.918454	0.459227	+	0.888319i	$0.348127\pi$
$72$	0	0
$73$	8.97504	1.05045	0.525225	−	0.850964i	$-0.323981\pi$
$73$	8.97504	1.05045	0.525225	+	0.850964i	$0.323981\pi$
$74$	2.94086	0.341868
$75$	0	0
$76$	−6.97756	−0.800381
$77$	4.19412	0.477964
$78$	0	0
$79$	0.227439	0.0255888	0.0127944	−	0.999918i	$-0.495927\pi$
$79$	0.227439	0.0255888	0.0127944	+	0.999918i	$0.495927\pi$
$80$	3.39614	0.379700
$81$	0	0
$82$	−6.42896	−0.709959
$83$	8.95630	0.983082	0.491541	−	0.870854i	$-0.336434\pi$
$83$	8.95630	0.983082	0.491541	+	0.870854i	$0.336434\pi$
$84$	0	0
$85$	8.04427	0.872524
$86$	9.90304	1.06787
$87$	0	0
$88$	4.59744	0.490089
$89$	0.839753	0.0890137	0.0445068	−	0.999009i	$-0.485828\pi$
$89$	0.839753	0.0890137	0.0445068	+	0.999009i	$0.485828\pi$
$90$	0	0
$91$	−0.879256	−0.0921711
$92$	−4.67453	−0.487353
$93$	0	0
$94$	13.6140	1.40417
$95$	−23.6968	−2.43124
$96$	0	0
$97$	4.12389	0.418717	0.209359	−	0.977839i	$-0.432862\pi$
$97$	4.12389	0.418717	0.209359	+	0.977839i	$0.432862\pi$
$98$	−6.16776	−0.623038
$99$	0	0
$100$	6.53378	0.653378
$101$	3.08911	0.307378	0.153689	−	0.988119i	$-0.450885\pi$
$101$	3.08911	0.307378	0.153689	+	0.988119i	$0.450885\pi$
$102$	0	0
$103$	3.75753	0.370240	0.185120	−	0.982716i	$-0.440733\pi$
$103$	3.75753	0.370240	0.185120	+	0.982716i	$0.440733\pi$
$104$	−0.963808	−0.0945091
$105$	0	0
$106$	10.7060	1.03986
$107$	7.20320	0.696360	0.348180	−	0.937428i	$-0.386800\pi$
$107$	7.20320	0.696360	0.348180	+	0.937428i	$0.386800\pi$
$108$	0	0
$109$	−19.2222	−1.84115	−0.920576	−	0.390564i	$-0.872280\pi$
$109$	−19.2222	−1.84115	−0.920576	+	0.390564i	$0.872280\pi$
$110$	15.6136	1.48869
$111$	0	0
$112$	0.912273	0.0862017
$113$	−16.3490	−1.53798	−0.768991	−	0.639259i	$-0.779241\pi$
$113$	−16.3490	−1.53798	−0.768991	+	0.639259i	$0.779241\pi$
$114$	0	0
$115$	−15.8754	−1.48039
$116$	4.38966	0.407570
$117$	0	0
$118$	14.5609	1.34044
$119$	2.16086	0.198085
$120$	0	0
$121$	10.1365	0.921496
$122$	−7.33496	−0.664076
$123$	0	0
$124$	−8.13917	−0.730919
$125$	5.20892	0.465900
$126$	0	0
$127$	5.90509	0.523992	0.261996	−	0.965069i	$-0.415619\pi$
$127$	5.90509	0.523992	0.261996	+	0.965069i	$0.415619\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.27323	−0.287081
$131$	−12.7765	−1.11629	−0.558146	−	0.829743i	$-0.688487\pi$
$131$	−12.7765	−1.11629	−0.558146	+	0.829743i	$0.688487\pi$
$132$	0	0
$133$	−6.36544	−0.551953
$134$	0.214533	0.0185328
$135$	0	0
$136$	2.36865	0.203110
$137$	−5.55451	−0.474554	−0.237277	−	0.971442i	$-0.576255\pi$
$137$	−5.55451	−0.474554	−0.237277	+	0.971442i	$0.576255\pi$
$138$	0	0
$139$	−6.76257	−0.573594	−0.286797	−	0.957991i	$-0.592590\pi$
$139$	−6.76257	−0.573594	−0.286797	+	0.957991i	$0.592590\pi$
$140$	3.09821	0.261846
$141$	0	0
$142$	7.73903	0.649445
$143$	−4.43105	−0.370543
$144$	0	0
$145$	14.9079	1.23804
$146$	8.97504	0.742780
$147$	0	0
$148$	2.94086	0.241737
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	5.68406	0.462562	0.231281	−	0.972887i	$-0.425708\pi$
$151$	5.68406	0.462562	0.231281	+	0.972887i	$0.425708\pi$
$152$	−6.97756	−0.565955
$153$	0	0
$154$	4.19412	0.337972
$155$	−27.6418	−2.22024
$156$	0	0
$157$	11.8575	0.946328	0.473164	−	0.880974i	$-0.343112\pi$
$157$	11.8575	0.946328	0.473164	+	0.880974i	$0.343112\pi$
$158$	0.227439	0.0180940
$159$	0	0
$160$	3.39614	0.268489
$161$	−4.26445	−0.336086
$162$	0	0
$163$	−21.0414	−1.64809	−0.824046	−	0.566522i	$-0.808288\pi$
$163$	−21.0414	−1.64809	−0.824046	+	0.566522i	$0.808288\pi$
$164$	−6.42896	−0.502017
$165$	0	0
$166$	8.95630	0.695144
$167$	16.6717	1.29010	0.645048	−	0.764142i	$-0.276838\pi$
$167$	16.6717	1.29010	0.645048	+	0.764142i	$0.276838\pi$
$168$	0	0
$169$	−12.0711	−0.928544
$170$	8.04427	0.616967
$171$	0	0
$172$	9.90304	0.755100
$173$	−13.5269	−1.02843	−0.514214	−	0.857662i	$-0.671916\pi$
$173$	−13.5269	−1.02843	−0.514214	+	0.857662i	$0.671916\pi$
$174$	0	0
$175$	5.96059	0.450578
$176$	4.59744	0.346545
$177$	0	0
$178$	0.839753	0.0629422
$179$	13.4805	1.00758	0.503789	−	0.863827i	$-0.331939\pi$
$179$	13.4805	1.00758	0.503789	+	0.863827i	$0.331939\pi$
$180$	0	0
$181$	−16.1720	−1.20206	−0.601029	−	0.799227i	$-0.705242\pi$
$181$	−16.1720	−1.20206	−0.601029	+	0.799227i	$0.705242\pi$
$182$	−0.879256	−0.0651748
$183$	0	0
$184$	−4.67453	−0.344611
$185$	9.98757	0.734301
$186$	0	0
$187$	10.8897	0.796336
$188$	13.6140	0.992901
$189$	0	0
$190$	−23.6968	−1.71914
$191$	−16.3983	−1.18654	−0.593268	−	0.805005i	$-0.702163\pi$
$191$	−16.3983	−1.18654	−0.593268	+	0.805005i	$0.702163\pi$
$192$	0	0
$193$	17.1555	1.23488	0.617441	−	0.786617i	$-0.288170\pi$
$193$	17.1555	1.23488	0.617441	+	0.786617i	$0.288170\pi$
$194$	4.12389	0.296078
$195$	0	0
$196$	−6.16776	−0.440554
$197$	−8.71946	−0.621236	−0.310618	−	0.950535i	$-0.600536\pi$
$197$	−8.71946	−0.621236	−0.310618	+	0.950535i	$0.600536\pi$
$198$	0	0
$199$	−2.19412	−0.155537	−0.0777686	−	0.996971i	$-0.524780\pi$
$199$	−2.19412	−0.155537	−0.0777686	+	0.996971i	$0.524780\pi$
$200$	6.53378	0.462008
$201$	0	0
$202$	3.08911	0.217349
$203$	4.00457	0.281066
$204$	0	0
$205$	−21.8336	−1.52493
$206$	3.75753	0.261799
$207$	0	0
$208$	−0.963808	−0.0668280
$209$	−32.0789	−2.21894
$210$	0	0
$211$	9.96103	0.685745	0.342873	−	0.939382i	$-0.388600\pi$
$211$	9.96103	0.685745	0.342873	+	0.939382i	$0.388600\pi$
$212$	10.7060	0.735294
$213$	0	0
$214$	7.20320	0.492401
$215$	33.6321	2.29369
$216$	0	0
$217$	−7.42515	−0.504052
$218$	−19.2222	−1.30189
$219$	0	0
$220$	15.6136	1.05267
$221$	−2.28292	−0.153566
$222$	0	0
$223$	12.6879	0.849642	0.424821	−	0.905277i	$-0.360337\pi$
$223$	12.6879	0.849642	0.424821	+	0.905277i	$0.360337\pi$
$224$	0.912273	0.0609538
$225$	0	0
$226$	−16.3490	−1.08752
$227$	17.9320	1.19019	0.595094	−	0.803656i	$-0.297115\pi$
$227$	17.9320	1.19019	0.595094	+	0.803656i	$0.297115\pi$
$228$	0	0
$229$	−18.4803	−1.22121	−0.610607	−	0.791934i	$-0.709074\pi$
$229$	−18.4803	−1.22121	−0.610607	+	0.791934i	$0.709074\pi$
$230$	−15.8754	−1.04679
$231$	0	0
$232$	4.38966	0.288195
$233$	13.1951	0.864442	0.432221	−	0.901768i	$-0.357730\pi$
$233$	13.1951	0.864442	0.432221	+	0.901768i	$0.357730\pi$
$234$	0	0
$235$	46.2350	3.01604
$236$	14.5609	0.947832
$237$	0	0
$238$	2.16086	0.140067
$239$	8.68447	0.561751	0.280876	−	0.959744i	$-0.409375\pi$
$239$	8.68447	0.561751	0.280876	+	0.959744i	$0.409375\pi$
$240$	0	0
$241$	−7.95788	−0.512612	−0.256306	−	0.966596i	$-0.582505\pi$
$241$	−7.95788	−0.512612	−0.256306	+	0.966596i	$0.582505\pi$
$242$	10.1365	0.651596
$243$	0	0
$244$	−7.33496	−0.469573
$245$	−20.9466	−1.33823
$246$	0	0
$247$	6.72502	0.427903
$248$	−8.13917	−0.516838
$249$	0	0
$250$	5.20892	0.329441
$251$	4.62567	0.291970	0.145985	−	0.989287i	$-0.453365\pi$
$251$	4.62567	0.291970	0.145985	+	0.989287i	$0.453365\pi$
$252$	0	0
$253$	−21.4909	−1.35112
$254$	5.90509	0.370518
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.4005	−1.45968	−0.729842	−	0.683616i	$-0.760406\pi$
$257$	−23.4005	−1.45968	−0.729842	+	0.683616i	$0.760406\pi$
$258$	0	0
$259$	2.68287	0.166705
$260$	−3.27323	−0.202997
$261$	0	0
$262$	−12.7765	−0.789338
$263$	−17.6010	−1.08533	−0.542663	−	0.839951i	$-0.682584\pi$
$263$	−17.6010	−1.08533	−0.542663	+	0.839951i	$0.682584\pi$
$264$	0	0
$265$	36.3592	2.23353
$266$	−6.36544	−0.390290
$267$	0	0
$268$	0.214533	0.0131047
$269$	−17.0522	−1.03969	−0.519845	−	0.854260i	$-0.674010\pi$
$269$	−17.0522	−1.03969	−0.519845	+	0.854260i	$0.674010\pi$
$270$	0	0
$271$	−6.24709	−0.379484	−0.189742	−	0.981834i	$-0.560765\pi$
$271$	−6.24709	−0.379484	−0.189742	+	0.981834i	$0.560765\pi$
$272$	2.36865	0.143620
$273$	0	0
$274$	−5.55451	−0.335560
$275$	30.0386	1.81140
$276$	0	0
$277$	12.6372	0.759297	0.379649	−	0.925131i	$-0.376045\pi$
$277$	12.6372	0.759297	0.379649	+	0.925131i	$0.376045\pi$
$278$	−6.76257	−0.405592
$279$	0	0
$280$	3.09821	0.185153
$281$	18.7016	1.11564	0.557821	−	0.829961i	$-0.311638\pi$
$281$	18.7016	1.11564	0.557821	+	0.829961i	$0.311638\pi$
$282$	0	0
$283$	2.51237	0.149345	0.0746723	−	0.997208i	$-0.476209\pi$
$283$	2.51237	0.149345	0.0746723	+	0.997208i	$0.476209\pi$
$284$	7.73903	0.459227
$285$	0	0
$286$	−4.43105	−0.262013
$287$	−5.86497	−0.346198
$288$	0	0
$289$	−11.3895	−0.669970
$290$	14.9079	0.875423
$291$	0	0
$292$	8.97504	0.525225
$293$	−10.4842	−0.612491	−0.306245	−	0.951953i	$-0.599073\pi$
$293$	−10.4842	−0.612491	−0.306245	+	0.951953i	$0.599073\pi$
$294$	0	0
$295$	49.4508	2.87914
$296$	2.94086	0.170934
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	4.50535	0.260551
$300$	0	0
$301$	9.03428	0.520727
$302$	5.68406	0.327081
$303$	0	0
$304$	−6.97756	−0.400190
$305$	−24.9106	−1.42637
$306$	0	0
$307$	−25.3922	−1.44921	−0.724604	−	0.689166i	$-0.757977\pi$
$307$	−25.3922	−1.44921	−0.724604	+	0.689166i	$0.757977\pi$
$308$	4.19412	0.238982
$309$	0	0
$310$	−27.6418	−1.56995
$311$	−20.5725	−1.16656	−0.583281	−	0.812271i	$-0.698231\pi$
$311$	−20.5725	−1.16656	−0.583281	+	0.812271i	$0.698231\pi$
$312$	0	0
$313$	−17.4072	−0.983914	−0.491957	−	0.870619i	$-0.663718\pi$
$313$	−17.4072	−0.983914	−0.491957	+	0.870619i	$0.663718\pi$
$314$	11.8575	0.669155
$315$	0	0
$316$	0.227439	0.0127944
$317$	24.9335	1.40041	0.700203	−	0.713943i	$-0.253093\pi$
$317$	24.9335	1.40041	0.700203	+	0.713943i	$0.253093\pi$
$318$	0	0
$319$	20.1812	1.12993
$320$	3.39614	0.189850
$321$	0	0
$322$	−4.26445	−0.237648
$323$	−16.5274	−0.919608
$324$	0	0
$325$	−6.29730	−0.349312
$326$	−21.0414	−1.16538
$327$	0	0
$328$	−6.42896	−0.354980
$329$	12.4197	0.684719
$330$	0	0
$331$	−14.2995	−0.785970	−0.392985	−	0.919545i	$-0.628558\pi$
$331$	−14.2995	−0.785970	−0.392985	+	0.919545i	$0.628558\pi$
$332$	8.95630	0.491541
$333$	0	0
$334$	16.6717	0.912235
$335$	0.728585	0.0398068
$336$	0	0
$337$	−9.64909	−0.525620	−0.262810	−	0.964848i	$-0.584649\pi$
$337$	−9.64909	−0.525620	−0.262810	+	0.964848i	$0.584649\pi$
$338$	−12.0711	−0.656580
$339$	0	0
$340$	8.04427	0.436262
$341$	−37.4194	−2.02637
$342$	0	0
$343$	−12.0126	−0.648619
$344$	9.90304	0.533936
$345$	0	0
$346$	−13.5269	−0.727209
$347$	−13.2470	−0.711135	−0.355568	−	0.934651i	$-0.615712\pi$
$347$	−13.2470	−0.711135	−0.355568	+	0.934651i	$0.615712\pi$
$348$	0	0
$349$	−21.9951	−1.17737	−0.588687	−	0.808361i	$-0.700355\pi$
$349$	−21.9951	−1.17737	−0.588687	+	0.808361i	$0.700355\pi$
$350$	5.96059	0.318607
$351$	0	0
$352$	4.59744	0.245044
$353$	17.6746	0.940723	0.470362	−	0.882474i	$-0.344123\pi$
$353$	17.6746	0.940723	0.470362	+	0.882474i	$0.344123\pi$
$354$	0	0
$355$	26.2828	1.39495
$356$	0.839753	0.0445068
$357$	0	0
$358$	13.4805	0.712465
$359$	2.04933	0.108160	0.0540798	−	0.998537i	$-0.482777\pi$
$359$	2.04933	0.108160	0.0540798	+	0.998537i	$0.482777\pi$
$360$	0	0
$361$	29.6863	1.56244
$362$	−16.1720	−0.849984
$363$	0	0
$364$	−0.879256	−0.0460855
$365$	30.4805	1.59542
$366$	0	0
$367$	−15.4421	−0.806072	−0.403036	−	0.915184i	$-0.632045\pi$
$367$	−15.4421	−0.806072	−0.403036	+	0.915184i	$0.632045\pi$
$368$	−4.67453	−0.243677
$369$	0	0
$370$	9.98757	0.519229
$371$	9.76683	0.507069
$372$	0	0
$373$	10.8157	0.560015	0.280008	−	0.959998i	$-0.409663\pi$
$373$	10.8157	0.560015	0.280008	+	0.959998i	$0.409663\pi$
$374$	10.8897	0.563094
$375$	0	0
$376$	13.6140	0.702087
$377$	−4.23079	−0.217897
$378$	0	0
$379$	24.0452	1.23512	0.617560	−	0.786523i	$-0.288121\pi$
$379$	24.0452	1.23512	0.617560	+	0.786523i	$0.288121\pi$
$380$	−23.6968	−1.21562
$381$	0	0
$382$	−16.3983	−0.839008
$383$	−12.6666	−0.647233	−0.323617	−	0.946188i	$-0.604899\pi$
$383$	−12.6666	−0.647233	−0.323617	+	0.946188i	$0.604899\pi$
$384$	0	0
$385$	14.2438	0.725933
$386$	17.1555	0.873193
$387$	0	0
$388$	4.12389	0.209359
$389$	−28.9377	−1.46720	−0.733599	−	0.679582i	$-0.762161\pi$
$389$	−28.9377	−1.46720	−0.733599	+	0.679582i	$0.762161\pi$
$390$	0	0
$391$	−11.0723	−0.559951
$392$	−6.16776	−0.311519
$393$	0	0
$394$	−8.71946	−0.439280
$395$	0.772413	0.0388643
$396$	0	0
$397$	17.9644	0.901609	0.450805	−	0.892623i	$-0.351137\pi$
$397$	17.9644	0.901609	0.450805	+	0.892623i	$0.351137\pi$
$398$	−2.19412	−0.109981
$399$	0	0
$400$	6.53378	0.326689
$401$	−25.6776	−1.28228	−0.641140	−	0.767424i	$-0.721538\pi$
$401$	−25.6776	−1.28228	−0.641140	+	0.767424i	$0.721538\pi$
$402$	0	0
$403$	7.84460	0.390767
$404$	3.08911	0.153689
$405$	0	0
$406$	4.00457	0.198744
$407$	13.5204	0.670182
$408$	0	0
$409$	5.89928	0.291701	0.145850	−	0.989307i	$-0.453408\pi$
$409$	5.89928	0.291701	0.145850	+	0.989307i	$0.453408\pi$
$410$	−21.8336	−1.07829
$411$	0	0
$412$	3.75753	0.185120
$413$	13.2835	0.653638
$414$	0	0
$415$	30.4169	1.49311
$416$	−0.963808	−0.0472546
$417$	0	0
$418$	−32.0789	−1.56903
$419$	−33.1992	−1.62189	−0.810944	−	0.585123i	$-0.801046\pi$
$419$	−33.1992	−1.62189	−0.810944	+	0.585123i	$0.801046\pi$
$420$	0	0
$421$	34.8428	1.69813	0.849067	−	0.528286i	$-0.177165\pi$
$421$	34.8428	1.69813	0.849067	+	0.528286i	$0.177165\pi$
$422$	9.96103	0.484895
$423$	0	0
$424$	10.7060	0.519931
$425$	15.4762	0.750707
$426$	0	0
$427$	−6.69149	−0.323824
$428$	7.20320	0.348180
$429$	0	0
$430$	33.6321	1.62188
$431$	−10.8867	−0.524395	−0.262197	−	0.965014i	$-0.584447\pi$
$431$	−10.8867	−0.524395	−0.262197	+	0.965014i	$0.584447\pi$
$432$	0	0
$433$	0.0200743	0.000964710 0	0.000482355	−	1.00000i	$-0.499846\pi$
$433$	0.0200743	0.000964710 0	0.000482355		1.00000i	$0.499846\pi$
$434$	−7.42515	−0.356419
$435$	0	0
$436$	−19.2222	−0.920576
$437$	32.6168	1.56027
$438$	0	0
$439$	−15.6811	−0.748419	−0.374209	−	0.927344i	$-0.622086\pi$
$439$	−15.6811	−0.748419	−0.374209	+	0.927344i	$0.622086\pi$
$440$	15.6136	0.744347
$441$	0	0
$442$	−2.28292	−0.108588
$443$	−28.7022	−1.36368	−0.681842	−	0.731500i	$-0.738821\pi$
$443$	−28.7022	−1.36368	−0.681842	+	0.731500i	$0.738821\pi$
$444$	0	0
$445$	2.85192	0.135194
$446$	12.6879	0.600788
$447$	0	0
$448$	0.912273	0.0431009
$449$	23.0043	1.08564	0.542819	−	0.839850i	$-0.317357\pi$
$449$	23.0043	1.08564	0.542819	+	0.839850i	$0.317357\pi$
$450$	0	0
$451$	−29.5567	−1.39177
$452$	−16.3490	−0.768991
$453$	0	0
$454$	17.9320	0.841590
$455$	−2.98608	−0.139989
$456$	0	0
$457$	3.93133	0.183900	0.0919500	−	0.995764i	$-0.470690\pi$
$457$	3.93133	0.183900	0.0919500	+	0.995764i	$0.470690\pi$
$458$	−18.4803	−0.863528
$459$	0	0
$460$	−15.8754	−0.740193
$461$	−19.5919	−0.912485	−0.456243	−	0.889855i	$-0.650805\pi$
$461$	−19.5919	−0.912485	−0.456243	+	0.889855i	$0.650805\pi$
$462$	0	0
$463$	35.9914	1.67266	0.836332	−	0.548223i	$-0.184696\pi$
$463$	35.9914	1.67266	0.836332	+	0.548223i	$0.184696\pi$
$464$	4.38966	0.203785
$465$	0	0
$466$	13.1951	0.611253
$467$	−27.8580	−1.28912	−0.644558	−	0.764555i	$-0.722959\pi$
$467$	−27.8580	−1.28912	−0.644558	+	0.764555i	$0.722959\pi$
$468$	0	0
$469$	0.195713	0.00903718
$470$	46.2350	2.13266
$471$	0	0
$472$	14.5609	0.670219
$473$	45.5286	2.09341
$474$	0	0
$475$	−45.5898	−2.09180
$476$	2.16086	0.0990427
$477$	0	0
$478$	8.68447	0.397218
$479$	38.1640	1.74376	0.871878	−	0.489722i	$-0.162902\pi$
$479$	38.1640	1.74376	0.871878	+	0.489722i	$0.162902\pi$
$480$	0	0
$481$	−2.83442	−0.129239
$482$	−7.95788	−0.362471
$483$	0	0
$484$	10.1365	0.460748
$485$	14.0053	0.635948
$486$	0	0
$487$	31.5130	1.42799	0.713996	−	0.700150i	$-0.246884\pi$
$487$	31.5130	1.42799	0.713996	+	0.700150i	$0.246884\pi$
$488$	−7.33496	−0.332038
$489$	0	0
$490$	−20.9466	−0.946270
$491$	−7.14758	−0.322566	−0.161283	−	0.986908i	$-0.551563\pi$
$491$	−7.14758	−0.322566	−0.161283	+	0.986908i	$0.551563\pi$
$492$	0	0
$493$	10.3976	0.468283
$494$	6.72502	0.302573
$495$	0	0
$496$	−8.13917	−0.365460
$497$	7.06011	0.316689
$498$	0	0
$499$	19.2315	0.860921	0.430461	−	0.902609i	$-0.358351\pi$
$499$	19.2315	0.860921	0.430461	+	0.902609i	$0.358351\pi$
$500$	5.20892	0.232950
$501$	0	0
$502$	4.62567	0.206454
$503$	37.8569	1.68795	0.843977	−	0.536379i	$-0.180208\pi$
$503$	37.8569	1.68795	0.843977	+	0.536379i	$0.180208\pi$
$504$	0	0
$505$	10.4911	0.466846
$506$	−21.4909	−0.955386
$507$	0	0
$508$	5.90509	0.261996
$509$	7.23711	0.320779	0.160390	−	0.987054i	$-0.448725\pi$
$509$	7.23711	0.320779	0.160390	+	0.987054i	$0.448725\pi$
$510$	0	0
$511$	8.18769	0.362202
$512$	1.00000	0.0441942
$513$	0	0
$514$	−23.4005	−1.03215
$515$	12.7611	0.562321
$516$	0	0
$517$	62.5894	2.75268
$518$	2.68287	0.117878
$519$	0	0
$520$	−3.27323	−0.143541
$521$	−7.38243	−0.323430	−0.161715	−	0.986837i	$-0.551703\pi$
$521$	−7.38243	−0.323430	−0.161715	+	0.986837i	$0.551703\pi$
$522$	0	0
$523$	−39.7237	−1.73700	−0.868499	−	0.495691i	$-0.834915\pi$
$523$	−39.7237	−1.73700	−0.868499	+	0.495691i	$0.834915\pi$
$524$	−12.7765	−0.558146
$525$	0	0
$526$	−17.6010	−0.767441
$527$	−19.2789	−0.839800
$528$	0	0
$529$	−1.14877	−0.0499467
$530$	36.3592	1.57934
$531$	0	0
$532$	−6.36544	−0.275977
$533$	6.19628	0.268391
$534$	0	0
$535$	24.4631	1.05763
$536$	0.214533	0.00926642
$537$	0	0
$538$	−17.0522	−0.735172
$539$	−28.3559	−1.22137
$540$	0	0
$541$	−23.7991	−1.02320	−0.511601	−	0.859223i	$-0.670947\pi$
$541$	−23.7991	−1.02320	−0.511601	+	0.859223i	$0.670947\pi$
$542$	−6.24709	−0.268336
$543$	0	0
$544$	2.36865	0.101555
$545$	−65.2813	−2.79634
$546$	0	0
$547$	44.5572	1.90513	0.952563	−	0.304340i	$-0.0984359\pi$
$547$	44.5572	1.90513	0.952563	+	0.304340i	$0.0984359\pi$
$548$	−5.55451	−0.237277
$549$	0	0
$550$	30.0386	1.28085
$551$	−30.6291	−1.30484
$552$	0	0
$553$	0.207486	0.00882321
$554$	12.6372	0.536904
$555$	0	0
$556$	−6.76257	−0.286797
$557$	−7.01993	−0.297444	−0.148722	−	0.988879i	$-0.547516\pi$
$557$	−7.01993	−0.297444	−0.148722	+	0.988879i	$0.547516\pi$
$558$	0	0
$559$	−9.54463	−0.403695
$560$	3.09821	0.130923
$561$	0	0
$562$	18.7016	0.788878
$563$	−44.4898	−1.87502	−0.937510	−	0.347959i	$-0.886875\pi$
$563$	−44.4898	−1.87502	−0.937510	+	0.347959i	$0.886875\pi$
$564$	0	0
$565$	−55.5234	−2.33589
$566$	2.51237	0.105603
$567$	0	0
$568$	7.73903	0.324722
$569$	13.4309	0.563054	0.281527	−	0.959553i	$-0.409159\pi$
$569$	13.4309	0.563054	0.281527	+	0.959553i	$0.409159\pi$
$570$	0	0
$571$	−47.0895	−1.97063	−0.985316	−	0.170739i	$-0.945384\pi$
$571$	−47.0895	−1.97063	−0.985316	+	0.170739i	$0.945384\pi$
$572$	−4.43105	−0.185271
$573$	0	0
$574$	−5.86497	−0.244799
$575$	−30.5423	−1.27370
$576$	0	0
$577$	18.0342	0.750773	0.375387	−	0.926868i	$-0.377510\pi$
$577$	18.0342	0.750773	0.375387	+	0.926868i	$0.377510\pi$
$578$	−11.3895	−0.473741
$579$	0	0
$580$	14.9079	0.619018
$581$	8.17060	0.338973
$582$	0	0
$583$	49.2204	2.03850
$584$	8.97504	0.371390
$585$	0	0
$586$	−10.4842	−0.433096
$587$	−6.62444	−0.273420	−0.136710	−	0.990611i	$-0.543653\pi$
$587$	−6.62444	−0.273420	−0.136710	+	0.990611i	$0.543653\pi$
$588$	0	0
$589$	56.7915	2.34005
$590$	49.4508	2.03586
$591$	0	0
$592$	2.94086	0.120869
$593$	−45.7690	−1.87951	−0.939754	−	0.341850i	$-0.888946\pi$
$593$	−45.7690	−1.87951	−0.939754	+	0.341850i	$0.888946\pi$
$594$	0	0
$595$	7.33857	0.300852
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	4.50535	0.184237
$599$	−27.2518	−1.11348	−0.556739	−	0.830687i	$-0.687948\pi$
$599$	−27.2518	−1.11348	−0.556739	+	0.830687i	$0.687948\pi$
$600$	0	0
$601$	−22.0350	−0.898828	−0.449414	−	0.893324i	$-0.648367\pi$
$601$	−22.0350	−0.898828	−0.449414	+	0.893324i	$0.648367\pi$
$602$	9.03428	0.368210
$603$	0	0
$604$	5.68406	0.231281
$605$	34.4248	1.39957
$606$	0	0
$607$	−37.9642	−1.54092	−0.770459	−	0.637489i	$-0.779973\pi$
$607$	−37.9642	−1.54092	−0.770459	+	0.637489i	$0.779973\pi$
$608$	−6.97756	−0.282977
$609$	0	0
$610$	−24.9106	−1.00860
$611$	−13.1213	−0.530829
$612$	0	0
$613$	11.2644	0.454963	0.227482	−	0.973782i	$-0.426951\pi$
$613$	11.2644	0.454963	0.227482	+	0.973782i	$0.426951\pi$
$614$	−25.3922	−1.02474
$615$	0	0
$616$	4.19412	0.168986
$617$	−27.2540	−1.09720	−0.548602	−	0.836084i	$-0.684840\pi$
$617$	−27.2540	−1.09720	−0.548602	+	0.836084i	$0.684840\pi$
$618$	0	0
$619$	26.7052	1.07337	0.536686	−	0.843782i	$-0.319676\pi$
$619$	26.7052	1.07337	0.536686	+	0.843782i	$0.319676\pi$
$620$	−27.6418	−1.11012
$621$	0	0
$622$	−20.5725	−0.824883
$623$	0.766085	0.0306925
$624$	0	0
$625$	−14.9786	−0.599146
$626$	−17.4072	−0.695733
$627$	0	0
$628$	11.8575	0.473164
$629$	6.96586	0.277747
$630$	0	0
$631$	18.7909	0.748055	0.374027	−	0.927418i	$-0.377977\pi$
$631$	18.7909	0.748055	0.374027	+	0.927418i	$0.377977\pi$
$632$	0.227439	0.00904702
$633$	0	0
$634$	24.9335	0.990237
$635$	20.0545	0.795839
$636$	0	0
$637$	5.94453	0.235531
$638$	20.1812	0.798982
$639$	0	0
$640$	3.39614	0.134244
$641$	31.3972	1.24012	0.620058	−	0.784556i	$-0.287109\pi$
$641$	31.3972	1.24012	0.620058	+	0.784556i	$0.287109\pi$
$642$	0	0
$643$	40.7071	1.60533	0.802666	−	0.596429i	$-0.203414\pi$
$643$	40.7071	1.60533	0.802666	+	0.596429i	$0.203414\pi$
$644$	−4.26445	−0.168043
$645$	0	0
$646$	−16.5274	−0.650261
$647$	6.96062	0.273650	0.136825	−	0.990595i	$-0.456310\pi$
$647$	6.96062	0.273650	0.136825	+	0.990595i	$0.456310\pi$
$648$	0	0
$649$	66.9428	2.62773
$650$	−6.29730	−0.247001
$651$	0	0
$652$	−21.0414	−0.824046
$653$	−0.396941	−0.0155335	−0.00776675	−	0.999970i	$-0.502472\pi$
$653$	−0.396941	−0.0155335	−0.00776675	+	0.999970i	$0.502472\pi$
$654$	0	0
$655$	−43.3910	−1.69542
$656$	−6.42896	−0.251009
$657$	0	0
$658$	12.4197	0.484169
$659$	−3.24559	−0.126430	−0.0632150	−	0.998000i	$-0.520135\pi$
$659$	−3.24559	−0.126430	−0.0632150	+	0.998000i	$0.520135\pi$
$660$	0	0
$661$	−41.1498	−1.60054	−0.800271	−	0.599639i	$-0.795311\pi$
$661$	−41.1498	−1.60054	−0.800271	+	0.599639i	$0.795311\pi$
$662$	−14.2995	−0.555765
$663$	0	0
$664$	8.95630	0.347572
$665$	−21.6179	−0.838307
$666$	0	0
$667$	−20.5196	−0.794522
$668$	16.6717	0.645048
$669$	0	0
$670$	0.728585	0.0281477
$671$	−33.7220	−1.30182
$672$	0	0
$673$	−17.4214	−0.671544	−0.335772	−	0.941943i	$-0.608997\pi$
$673$	−17.4214	−0.671544	−0.335772	+	0.941943i	$0.608997\pi$
$674$	−9.64909	−0.371669
$675$	0	0
$676$	−12.0711	−0.464272
$677$	−15.0144	−0.577050	−0.288525	−	0.957472i	$-0.593165\pi$
$677$	−15.0144	−0.577050	−0.288525	+	0.957472i	$0.593165\pi$
$678$	0	0
$679$	3.76211	0.144377
$680$	8.04427	0.308484
$681$	0	0
$682$	−37.4194	−1.43286
$683$	−4.05135	−0.155020	−0.0775102	−	0.996992i	$-0.524697\pi$
$683$	−4.05135	−0.155020	−0.0775102	+	0.996992i	$0.524697\pi$
$684$	0	0
$685$	−18.8639	−0.720753
$686$	−12.0126	−0.458643
$687$	0	0
$688$	9.90304	0.377550
$689$	−10.3186	−0.393106
$690$	0	0
$691$	10.3139	0.392360	0.196180	−	0.980568i	$-0.437146\pi$
$691$	10.3139	0.392360	0.196180	+	0.980568i	$0.437146\pi$
$692$	−13.5269	−0.514214
$693$	0	0
$694$	−13.2470	−0.502849
$695$	−22.9666	−0.871174
$696$	0	0
$697$	−15.2279	−0.576800
$698$	−21.9951	−0.832529
$699$	0	0
$700$	5.96059	0.225289
$701$	−11.5985	−0.438070	−0.219035	−	0.975717i	$-0.570291\pi$
$701$	−11.5985	−0.438070	−0.219035	+	0.975717i	$0.570291\pi$
$702$	0	0
$703$	−20.5200	−0.773927
$704$	4.59744	0.173273
$705$	0	0
$706$	17.6746	0.665192
$707$	2.81811	0.105986
$708$	0	0
$709$	−25.7687	−0.967763	−0.483882	−	0.875133i	$-0.660774\pi$
$709$	−25.7687	−0.967763	−0.483882	+	0.875133i	$0.660774\pi$
$710$	26.2828	0.986377
$711$	0	0
$712$	0.839753	0.0314711
$713$	38.0468	1.42486
$714$	0	0
$715$	−15.0485	−0.562781
$716$	13.4805	0.503789
$717$	0	0
$718$	2.04933	0.0764804
$719$	50.6612	1.88934	0.944672	−	0.328016i	$-0.106380\pi$
$719$	50.6612	1.88934	0.944672	+	0.328016i	$0.106380\pi$
$720$	0	0
$721$	3.42789	0.127661
$722$	29.6863	1.10481
$723$	0	0
$724$	−16.1720	−0.601029
$725$	28.6811	1.06519
$726$	0	0
$727$	−23.6334	−0.876513	−0.438257	−	0.898850i	$-0.644404\pi$
$727$	−23.6334	−0.876513	−0.438257	+	0.898850i	$0.644404\pi$
$728$	−0.879256	−0.0325874
$729$	0	0
$730$	30.4805	1.12813
$731$	23.4568	0.867582
$732$	0	0
$733$	−8.37008	−0.309156	−0.154578	−	0.987981i	$-0.549402\pi$
$733$	−8.37008	−0.309156	−0.154578	+	0.987981i	$0.549402\pi$
$734$	−15.4421	−0.569979
$735$	0	0
$736$	−4.67453	−0.172305
$737$	0.986303	0.0363309
$738$	0	0
$739$	26.7623	0.984466	0.492233	−	0.870463i	$-0.336181\pi$
$739$	26.7623	0.984466	0.492233	+	0.870463i	$0.336181\pi$
$740$	9.98757	0.367150
$741$	0	0
$742$	9.76683	0.358552
$743$	33.5755	1.23176	0.615882	−	0.787838i	$-0.288800\pi$
$743$	33.5755	1.23176	0.615882	+	0.787838i	$0.288800\pi$
$744$	0	0
$745$	−3.39614	−0.124425
$746$	10.8157	0.395990
$747$	0	0
$748$	10.8897	0.398168
$749$	6.57129	0.240110
$750$	0	0
$751$	−2.13765	−0.0780041	−0.0390021	−	0.999239i	$-0.512418\pi$
$751$	−2.13765	−0.0780041	−0.0390021	+	0.999239i	$0.512418\pi$
$752$	13.6140	0.496451
$753$	0	0
$754$	−4.23079	−0.154076
$755$	19.3039	0.702539
$756$	0	0
$757$	−30.1580	−1.09611	−0.548056	−	0.836442i	$-0.684632\pi$
$757$	−30.1580	−1.09611	−0.548056	+	0.836442i	$0.684632\pi$
$758$	24.0452	0.873362
$759$	0	0
$760$	−23.6968	−0.859572
$761$	−10.8269	−0.392476	−0.196238	−	0.980556i	$-0.562872\pi$
$761$	−10.8269	−0.392476	−0.196238	+	0.980556i	$0.562872\pi$
$762$	0	0
$763$	−17.5359	−0.634842
$764$	−16.3983	−0.593268
$765$	0	0
$766$	−12.6666	−0.457663
$767$	−14.0339	−0.506734
$768$	0	0
$769$	35.6939	1.28716	0.643578	−	0.765381i	$-0.277449\pi$
$769$	35.6939	1.28716	0.643578	+	0.765381i	$0.277449\pi$
$770$	14.2438	0.513312
$771$	0	0
$772$	17.1555	0.617441
$773$	17.1708	0.617593	0.308796	−	0.951128i	$-0.400074\pi$
$773$	17.1708	0.617593	0.308796	+	0.951128i	$0.400074\pi$
$774$	0	0
$775$	−53.1795	−1.91027
$776$	4.12389	0.148039
$777$	0	0
$778$	−28.9377	−1.03747
$779$	44.8584	1.60722
$780$	0	0
$781$	35.5797	1.27314
$782$	−11.0723	−0.395945
$783$	0	0
$784$	−6.16776	−0.220277
$785$	40.2696	1.43728
$786$	0	0
$787$	−26.2361	−0.935215	−0.467608	−	0.883936i	$-0.654884\pi$
$787$	−26.2361	−0.935215	−0.467608	+	0.883936i	$0.654884\pi$
$788$	−8.71946	−0.310618
$789$	0	0
$790$	0.772413	0.0274812
$791$	−14.9147	−0.530307
$792$	0	0
$793$	7.06949	0.251045
$794$	17.9644	0.637534
$795$	0	0
$796$	−2.19412	−0.0777686
$797$	15.2369	0.539717	0.269859	−	0.962900i	$-0.413023\pi$
$797$	15.2369	0.539717	0.269859	+	0.962900i	$0.413023\pi$
$798$	0	0
$799$	32.2467	1.14081
$800$	6.53378	0.231004
$801$	0	0
$802$	−25.6776	−0.906709
$803$	41.2622	1.45611
$804$	0	0
$805$	−14.4827	−0.510447
$806$	7.84460	0.276314
$807$	0	0
$808$	3.08911	0.108675
$809$	35.4155	1.24514	0.622572	−	0.782563i	$-0.286088\pi$
$809$	35.4155	1.24514	0.622572	+	0.782563i	$0.286088\pi$
$810$	0	0
$811$	−4.56797	−0.160403	−0.0802016	−	0.996779i	$-0.525556\pi$
$811$	−4.56797	−0.160403	−0.0802016	+	0.996779i	$0.525556\pi$
$812$	4.00457	0.140533
$813$	0	0
$814$	13.5204	0.473891
$815$	−71.4597	−2.50312
$816$	0	0
$817$	−69.0990	−2.41747
$818$	5.89928	0.206264
$819$	0	0
$820$	−21.8336	−0.762464
$821$	33.9967	1.18649	0.593247	−	0.805020i	$-0.297846\pi$
$821$	33.9967	1.18649	0.593247	+	0.805020i	$0.297846\pi$
$822$	0	0
$823$	−37.0749	−1.29235	−0.646174	−	0.763190i	$-0.723632\pi$
$823$	−37.0749	−1.29235	−0.646174	+	0.763190i	$0.723632\pi$
$824$	3.75753	0.130900
$825$	0	0
$826$	13.2835	0.462192
$827$	−45.8541	−1.59450	−0.797251	−	0.603648i	$-0.793713\pi$
$827$	−45.8541	−1.59450	−0.797251	+	0.603648i	$0.793713\pi$
$828$	0	0
$829$	33.1417	1.15106	0.575529	−	0.817781i	$-0.304796\pi$
$829$	33.1417	1.15106	0.575529	+	0.817781i	$0.304796\pi$
$830$	30.4169	1.05579
$831$	0	0
$832$	−0.963808	−0.0334140
$833$	−14.6093	−0.506181
$834$	0	0
$835$	56.6195	1.95940
$836$	−32.0789	−1.10947
$837$	0	0
$838$	−33.1992	−1.14685
$839$	−29.6050	−1.02208	−0.511040	−	0.859557i	$-0.670739\pi$
$839$	−29.6050	−1.02208	−0.511040	+	0.859557i	$0.670739\pi$
$840$	0	0
$841$	−9.73086	−0.335547
$842$	34.8428	1.20076
$843$	0	0
$844$	9.96103	0.342873
$845$	−40.9951	−1.41027
$846$	0	0
$847$	9.24721	0.317738
$848$	10.7060	0.367647
$849$	0	0
$850$	15.4762	0.530830
$851$	−13.7471	−0.471246
$852$	0	0
$853$	1.78603	0.0611526	0.0305763	−	0.999532i	$-0.490266\pi$
$853$	1.78603	0.0611526	0.0305763	+	0.999532i	$0.490266\pi$
$854$	−6.69149	−0.228978
$855$	0	0
$856$	7.20320	0.246200
$857$	24.0648	0.822037	0.411018	−	0.911627i	$-0.365173\pi$
$857$	24.0648	0.822037	0.411018	+	0.911627i	$0.365173\pi$
$858$	0	0
$859$	37.2049	1.26942	0.634708	−	0.772752i	$-0.281120\pi$
$859$	37.2049	1.26942	0.634708	+	0.772752i	$0.281120\pi$
$860$	33.6321	1.14685
$861$	0	0
$862$	−10.8867	−0.370803
$863$	22.7766	0.775324	0.387662	−	0.921802i	$-0.373283\pi$
$863$	22.7766	0.775324	0.387662	+	0.921802i	$0.373283\pi$
$864$	0	0
$865$	−45.9391	−1.56198
$866$	0.0200743	0.000682153 0
$867$	0	0
$868$	−7.42515	−0.252026
$869$	1.04564	0.0354707
$870$	0	0
$871$	−0.206769	−0.00700609
$872$	−19.2222	−0.650945
$873$	0	0
$874$	32.6168	1.10328
$875$	4.75196	0.160646
$876$	0	0
$877$	11.7698	0.397437	0.198718	−	0.980057i	$-0.436322\pi$
$877$	11.7698	0.397437	0.198718	+	0.980057i	$0.436322\pi$
$878$	−15.6811	−0.529212
$879$	0	0
$880$	15.6136	0.526333
$881$	−3.90861	−0.131684	−0.0658422	−	0.997830i	$-0.520973\pi$
$881$	−3.90861	−0.131684	−0.0658422	+	0.997830i	$0.520973\pi$
$882$	0	0
$883$	−37.1332	−1.24963	−0.624815	−	0.780773i	$-0.714826\pi$
$883$	−37.1332	−1.24963	−0.624815	+	0.780773i	$0.714826\pi$
$884$	−2.28292	−0.0767830
$885$	0	0
$886$	−28.7022	−0.964270
$887$	−19.1315	−0.642373	−0.321186	−	0.947016i	$-0.604082\pi$
$887$	−19.1315	−0.642373	−0.321186	+	0.947016i	$0.604082\pi$
$888$	0	0
$889$	5.38705	0.180676
$890$	2.85192	0.0955966
$891$	0	0
$892$	12.6879	0.424821
$893$	−94.9923	−3.17880
$894$	0	0
$895$	45.7816	1.53031
$896$	0.912273	0.0304769
$897$	0	0
$898$	23.0043	0.767662
$899$	−35.7282	−1.19160
$900$	0	0
$901$	25.3589	0.844826
$902$	−29.5567	−0.984132
$903$	0	0
$904$	−16.3490	−0.543759
$905$	−54.9225	−1.82569
$906$	0	0
$907$	−13.6191	−0.452216	−0.226108	−	0.974102i	$-0.572600\pi$
$907$	−13.6191	−0.452216	−0.226108	+	0.974102i	$0.572600\pi$
$908$	17.9320	0.595094
$909$	0	0
$910$	−2.98608	−0.0989875
$911$	−39.4206	−1.30606	−0.653031	−	0.757331i	$-0.726503\pi$
$911$	−39.4206	−1.30606	−0.653031	+	0.757331i	$0.726503\pi$
$912$	0	0
$913$	41.1761	1.36273
$914$	3.93133	0.130037
$915$	0	0
$916$	−18.4803	−0.610607
$917$	−11.6557	−0.384905
$918$	0	0
$919$	−41.1162	−1.35630	−0.678149	−	0.734925i	$-0.737217\pi$
$919$	−41.1162	−1.35630	−0.678149	+	0.734925i	$0.737217\pi$
$920$	−15.8754	−0.523395
$921$	0	0
$922$	−19.5919	−0.645225
$923$	−7.45894	−0.245514
$924$	0	0
$925$	19.2149	0.631783
$926$	35.9914	1.18275
$927$	0	0
$928$	4.38966	0.144098
$929$	−14.8252	−0.486398	−0.243199	−	0.969976i	$-0.578197\pi$
$929$	−14.8252	−0.486398	−0.243199	+	0.969976i	$0.578197\pi$
$930$	0	0
$931$	43.0359	1.41044
$932$	13.1951	0.432221
$933$	0	0
$934$	−27.8580	−0.911543
$935$	36.9830	1.20947
$936$	0	0
$937$	9.47773	0.309624	0.154812	−	0.987944i	$-0.450523\pi$
$937$	9.47773	0.309624	0.154812	+	0.987944i	$0.450523\pi$
$938$	0.195713	0.00639025
$939$	0	0
$940$	46.2350	1.50802
$941$	27.5270	0.897355	0.448678	−	0.893694i	$-0.351895\pi$
$941$	27.5270	0.897355	0.448678	+	0.893694i	$0.351895\pi$
$942$	0	0
$943$	30.0523	0.978639
$944$	14.5609	0.473916
$945$	0	0
$946$	45.5286	1.48026
$947$	32.1785	1.04566	0.522831	−	0.852437i	$-0.324876\pi$
$947$	32.1785	1.04566	0.522831	+	0.852437i	$0.324876\pi$
$948$	0	0
$949$	−8.65022	−0.280798
$950$	−45.5898	−1.47913
$951$	0	0
$952$	2.16086	0.0700337
$953$	−48.6945	−1.57737	−0.788685	−	0.614798i	$-0.789238\pi$
$953$	−48.6945	−1.57737	−0.788685	+	0.614798i	$0.789238\pi$
$954$	0	0
$955$	−55.6908	−1.80211
$956$	8.68447	0.280876
$957$	0	0
$958$	38.1640	1.23302
$959$	−5.06723	−0.163629
$960$	0	0
$961$	35.2461	1.13697
$962$	−2.83442	−0.0913855
$963$	0	0
$964$	−7.95788	−0.256306
$965$	58.2626	1.87554
$966$	0	0
$967$	26.5683	0.854379	0.427190	−	0.904162i	$-0.359504\pi$
$967$	26.5683	0.854379	0.427190	+	0.904162i	$0.359504\pi$
$968$	10.1365	0.325798
$969$	0	0
$970$	14.0053	0.449683
$971$	5.62665	0.180568	0.0902838	−	0.995916i	$-0.471223\pi$
$971$	5.62665	0.180568	0.0902838	+	0.995916i	$0.471223\pi$
$972$	0	0
$973$	−6.16931	−0.197779
$974$	31.5130	1.00974
$975$	0	0
$976$	−7.33496	−0.234786
$977$	−45.5987	−1.45883	−0.729415	−	0.684071i	$-0.760208\pi$
$977$	−45.5987	−1.45883	−0.729415	+	0.684071i	$0.760208\pi$
$978$	0	0
$979$	3.86072	0.123389
$980$	−20.9466	−0.669114
$981$	0	0
$982$	−7.14758	−0.228089
$983$	24.8469	0.792492	0.396246	−	0.918144i	$-0.370313\pi$
$983$	24.8469	0.792492	0.396246	+	0.918144i	$0.370313\pi$
$984$	0	0
$985$	−29.6125	−0.943533
$986$	10.3976	0.331126
$987$	0	0
$988$	6.72502	0.213951
$989$	−46.2920	−1.47200
$990$	0	0
$991$	−16.1678	−0.513589	−0.256794	−	0.966466i	$-0.582666\pi$
$991$	−16.1678	−0.513589	−0.256794	+	0.966466i	$0.582666\pi$
$992$	−8.13917	−0.258419
$993$	0	0
$994$	7.06011	0.223933
$995$	−7.45155	−0.236230
$996$	0	0
$997$	−30.6322	−0.970133	−0.485066	−	0.874477i	$-0.661205\pi$
$997$	−30.6322	−0.970133	−0.485066	+	0.874477i	$0.661205\pi$
$998$	19.2315	0.608763
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.t.1.15	yes	16
3.2	odd	2		8046.2.a.s.1.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.2	✓	16	3.2	odd	2
8046.2.a.t.1.15	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.39614 q^{5} +0.912273 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.39614 q^{5} +0.912273 q^{7} +1.00000 q^{8} +3.39614 q^{10} +4.59744 q^{11} -0.963808 q^{13} +0.912273 q^{14} +1.00000 q^{16} +2.36865 q^{17} -6.97756 q^{19} +3.39614 q^{20} +4.59744 q^{22} -4.67453 q^{23} +6.53378 q^{25} -0.963808 q^{26} +0.912273 q^{28} +4.38966 q^{29} -8.13917 q^{31} +1.00000 q^{32} +2.36865 q^{34} +3.09821 q^{35} +2.94086 q^{37} -6.97756 q^{38} +3.39614 q^{40} -6.42896 q^{41} +9.90304 q^{43} +4.59744 q^{44} -4.67453 q^{46} +13.6140 q^{47} -6.16776 q^{49} +6.53378 q^{50} -0.963808 q^{52} +10.7060 q^{53} +15.6136 q^{55} +0.912273 q^{56} +4.38966 q^{58} +14.5609 q^{59} -7.33496 q^{61} -8.13917 q^{62} +1.00000 q^{64} -3.27323 q^{65} +0.214533 q^{67} +2.36865 q^{68} +3.09821 q^{70} +7.73903 q^{71} +8.97504 q^{73} +2.94086 q^{74} -6.97756 q^{76} +4.19412 q^{77} +0.227439 q^{79} +3.39614 q^{80} -6.42896 q^{82} +8.95630 q^{83} +8.04427 q^{85} +9.90304 q^{86} +4.59744 q^{88} +0.839753 q^{89} -0.879256 q^{91} -4.67453 q^{92} +13.6140 q^{94} -23.6968 q^{95} +4.12389 q^{97} -6.16776 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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