LMFDB - Embedded newform 8464.2.a.ch.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8464,2,Mod(1,8464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.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:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.15594$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.15594 q^{3} +0.0614634 q^{5} +2.55213 q^{7} +6.95994 q^{9} +O(q^{10})$	$q-3.15594 q^{3} +0.0614634 q^{5} +2.55213 q^{7} +6.95994 q^{9} +4.48694 q^{11} -1.07385 q^{13} -0.193975 q^{15} -3.56531 q^{17} +5.69304 q^{19} -8.05438 q^{21} -4.99622 q^{25} -12.4973 q^{27} +5.05833 q^{29} +9.56833 q^{31} -14.1605 q^{33} +0.156863 q^{35} +6.40424 q^{37} +3.38899 q^{39} -0.418182 q^{41} -1.35064 q^{43} +0.427782 q^{45} +9.28734 q^{47} -0.486609 q^{49} +11.2519 q^{51} +2.78586 q^{53} +0.275783 q^{55} -17.9669 q^{57} +13.6650 q^{59} +12.3123 q^{61} +17.7627 q^{63} -0.0660022 q^{65} -6.99968 q^{67} +0.204215 q^{71} +0.687611 q^{73} +15.7678 q^{75} +11.4513 q^{77} +11.1290 q^{79} +18.5610 q^{81} +6.94009 q^{83} -0.219136 q^{85} -15.9638 q^{87} -6.92138 q^{89} -2.74060 q^{91} -30.1971 q^{93} +0.349914 q^{95} -6.90492 q^{97} +31.2288 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−3.15594	−1.82208	−0.911041	−	0.412317i	$-0.864720\pi$
$3$	−3.15594	−1.82208	−0.911041	+	0.412317i	$0.864720\pi$
$4$	0	0
$5$	0.0614634	0.0274873	0.0137436	−	0.999906i	$-0.495625\pi$
$5$	0.0614634	0.0274873	0.0137436	+	0.999906i	$0.495625\pi$
$6$	0	0
$7$	2.55213	0.964616	0.482308	−	0.876002i	$-0.339799\pi$
$7$	2.55213	0.964616	0.482308	+	0.876002i	$0.339799\pi$
$8$	0	0
$9$	6.95994	2.31998
$10$	0	0
$11$	4.48694	1.35286	0.676431	−	0.736506i	$-0.263525\pi$
$11$	4.48694	1.35286	0.676431	+	0.736506i	$0.263525\pi$
$12$	0	0
$13$	−1.07385	−0.297831	−0.148916	−	0.988850i	$-0.547578\pi$
$13$	−1.07385	−0.297831	−0.148916	+	0.988850i	$0.547578\pi$
$14$	0	0
$15$	−0.193975	−0.0500841
$16$	0	0
$17$	−3.56531	−0.864716	−0.432358	−	0.901702i	$-0.642318\pi$
$17$	−3.56531	−0.864716	−0.432358	+	0.901702i	$0.642318\pi$
$18$	0	0
$19$	5.69304	1.30607	0.653037	−	0.757326i	$-0.273495\pi$
$19$	5.69304	1.30607	0.653037	+	0.757326i	$0.273495\pi$
$20$	0	0
$21$	−8.05438	−1.75761
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.99622	−0.999244
$26$	0	0
$27$	−12.4973	−2.40511
$28$	0	0
$29$	5.05833	0.939308	0.469654	−	0.882851i	$-0.344379\pi$
$29$	5.05833	0.939308	0.469654	+	0.882851i	$0.344379\pi$
$30$	0	0
$31$	9.56833	1.71852	0.859262	−	0.511536i	$-0.170923\pi$
$31$	9.56833	1.71852	0.859262	+	0.511536i	$0.170923\pi$
$32$	0	0
$33$	−14.1605	−2.46503
$34$	0	0
$35$	0.156863	0.0265147
$36$	0	0
$37$	6.40424	1.05285	0.526425	−	0.850222i	$-0.323532\pi$
$37$	6.40424	1.05285	0.526425	+	0.850222i	$0.323532\pi$
$38$	0	0
$39$	3.38899	0.542673
$40$	0	0
$41$	−0.418182	−0.0653091	−0.0326546	−	0.999467i	$-0.510396\pi$
$41$	−0.418182	−0.0653091	−0.0326546	+	0.999467i	$0.510396\pi$
$42$	0	0
$43$	−1.35064	−0.205970	−0.102985	−	0.994683i	$-0.532839\pi$
$43$	−1.35064	−0.205970	−0.102985	+	0.994683i	$0.532839\pi$
$44$	0	0
$45$	0.427782	0.0637699
$46$	0	0
$47$	9.28734	1.35470	0.677349	−	0.735662i	$-0.263129\pi$
$47$	9.28734	1.35470	0.677349	+	0.735662i	$0.263129\pi$
$48$	0	0
$49$	−0.486609	−0.0695156
$50$	0	0
$51$	11.2519	1.57558
$52$	0	0
$53$	2.78586	0.382668	0.191334	−	0.981525i	$-0.438719\pi$
$53$	2.78586	0.382668	0.191334	+	0.981525i	$0.438719\pi$
$54$	0	0
$55$	0.275783	0.0371865
$56$	0	0
$57$	−17.9669	−2.37977
$58$	0	0
$59$	13.6650	1.77903	0.889515	−	0.456906i	$-0.151042\pi$
$59$	13.6650	1.77903	0.889515	+	0.456906i	$0.151042\pi$
$60$	0	0
$61$	12.3123	1.57642	0.788212	−	0.615404i	$-0.211007\pi$
$61$	12.3123	1.57642	0.788212	+	0.615404i	$0.211007\pi$
$62$	0	0
$63$	17.7627	2.23789
$64$	0	0
$65$	−0.0660022	−0.00818657
$66$	0	0
$67$	−6.99968	−0.855147	−0.427574	−	0.903981i	$-0.640632\pi$
$67$	−6.99968	−0.855147	−0.427574	+	0.903981i	$0.640632\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.204215	0.0242359	0.0121180	−	0.999927i	$-0.496143\pi$
$71$	0.204215	0.0242359	0.0121180	+	0.999927i	$0.496143\pi$
$72$	0	0
$73$	0.687611	0.0804787	0.0402394	−	0.999190i	$-0.487188\pi$
$73$	0.687611	0.0804787	0.0402394	+	0.999190i	$0.487188\pi$
$74$	0	0
$75$	15.7678	1.82070
$76$	0	0
$77$	11.4513	1.30499
$78$	0	0
$79$	11.1290	1.25211	0.626054	−	0.779780i	$-0.284669\pi$
$79$	11.1290	1.25211	0.626054	+	0.779780i	$0.284669\pi$
$80$	0	0
$81$	18.5610	2.06233
$82$	0	0
$83$	6.94009	0.761774	0.380887	−	0.924622i	$-0.375619\pi$
$83$	6.94009	0.761774	0.380887	+	0.924622i	$0.375619\pi$
$84$	0	0
$85$	−0.219136	−0.0237687
$86$	0	0
$87$	−15.9638	−1.71150
$88$	0	0
$89$	−6.92138	−0.733665	−0.366832	−	0.930287i	$-0.619558\pi$
$89$	−6.92138	−0.733665	−0.366832	+	0.930287i	$0.619558\pi$
$90$	0	0
$91$	−2.74060	−0.287293
$92$	0	0
$93$	−30.1971	−3.13129
$94$	0	0
$95$	0.349914	0.0359004
$96$	0	0
$97$	−6.90492	−0.701089	−0.350544	−	0.936546i	$-0.614003\pi$
$97$	−6.90492	−0.701089	−0.350544	+	0.936546i	$0.614003\pi$
$98$	0	0
$99$	31.2288	3.13862
$100$	0	0
$101$	3.68746	0.366916	0.183458	−	0.983028i	$-0.441271\pi$
$101$	3.68746	0.366916	0.183458	+	0.983028i	$0.441271\pi$
$102$	0	0
$103$	−12.6281	−1.24428	−0.622141	−	0.782906i	$-0.713737\pi$
$103$	−12.6281	−1.24428	−0.622141	+	0.782906i	$0.713737\pi$
$104$	0	0
$105$	−0.495050	−0.0483119
$106$	0	0
$107$	−17.7879	−1.71962	−0.859810	−	0.510614i	$-0.829418\pi$
$107$	−17.7879	−1.71962	−0.859810	+	0.510614i	$0.829418\pi$
$108$	0	0
$109$	−0.738667	−0.0707515	−0.0353757	−	0.999374i	$-0.511263\pi$
$109$	−0.738667	−0.0707515	−0.0353757	+	0.999374i	$0.511263\pi$
$110$	0	0
$111$	−20.2114	−1.91838
$112$	0	0
$113$	−4.16404	−0.391720	−0.195860	−	0.980632i	$-0.562750\pi$
$113$	−4.16404	−0.391720	−0.195860	+	0.980632i	$0.562750\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−7.47390	−0.690963
$118$	0	0
$119$	−9.09916	−0.834119
$120$	0	0
$121$	9.13262	0.830238
$122$	0	0
$123$	1.31976	0.118999
$124$	0	0
$125$	−0.614402	−0.0549538
$126$	0	0
$127$	−2.71578	−0.240986	−0.120493	−	0.992714i	$-0.538448\pi$
$127$	−2.71578	−0.240986	−0.120493	+	0.992714i	$0.538448\pi$
$128$	0	0
$129$	4.26253	0.375295
$130$	0	0
$131$	−9.53661	−0.833218	−0.416609	−	0.909086i	$-0.636782\pi$
$131$	−9.53661	−0.833218	−0.416609	+	0.909086i	$0.636782\pi$
$132$	0	0
$133$	14.5294	1.25986
$134$	0	0
$135$	−0.768128	−0.0661100
$136$	0	0
$137$	−7.21517	−0.616433	−0.308217	−	0.951316i	$-0.599732\pi$
$137$	−7.21517	−0.616433	−0.308217	+	0.951316i	$0.599732\pi$
$138$	0	0
$139$	3.94176	0.334336	0.167168	−	0.985928i	$-0.446538\pi$
$139$	3.94176	0.334336	0.167168	+	0.985928i	$0.446538\pi$
$140$	0	0
$141$	−29.3103	−2.46837
$142$	0	0
$143$	−4.81828	−0.402925
$144$	0	0
$145$	0.310902	0.0258190
$146$	0	0
$147$	1.53571	0.126663
$148$	0	0
$149$	5.87072	0.480948	0.240474	−	0.970656i	$-0.422697\pi$
$149$	5.87072	0.480948	0.240474	+	0.970656i	$0.422697\pi$
$150$	0	0
$151$	−16.4430	−1.33811	−0.669056	−	0.743212i	$-0.733302\pi$
$151$	−16.4430	−1.33811	−0.669056	+	0.743212i	$0.733302\pi$
$152$	0	0
$153$	−24.8144	−2.00612
$154$	0	0
$155$	0.588102	0.0472375
$156$	0	0
$157$	−2.21056	−0.176422	−0.0882111	−	0.996102i	$-0.528115\pi$
$157$	−2.21056	−0.176422	−0.0882111	+	0.996102i	$0.528115\pi$
$158$	0	0
$159$	−8.79201	−0.697252
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−4.63965	−0.363405	−0.181703	−	0.983354i	$-0.558161\pi$
$163$	−4.63965	−0.363405	−0.181703	+	0.983354i	$0.558161\pi$
$164$	0	0
$165$	−0.870353	−0.0677569
$166$	0	0
$167$	15.2566	1.18059	0.590297	−	0.807186i	$-0.299011\pi$
$167$	15.2566	1.18059	0.590297	+	0.807186i	$0.299011\pi$
$168$	0	0
$169$	−11.8469	−0.911297
$170$	0	0
$171$	39.6232	3.03006
$172$	0	0
$173$	13.2591	1.00807	0.504034	−	0.863684i	$-0.331848\pi$
$173$	13.2591	1.00807	0.504034	+	0.863684i	$0.331848\pi$
$174$	0	0
$175$	−12.7510	−0.963887
$176$	0	0
$177$	−43.1259	−3.24154
$178$	0	0
$179$	11.5898	0.866262	0.433131	−	0.901331i	$-0.357409\pi$
$179$	11.5898	0.866262	0.433131	+	0.901331i	$0.357409\pi$
$180$	0	0
$181$	−13.6119	−1.01176	−0.505881	−	0.862603i	$-0.668833\pi$
$181$	−13.6119	−1.01176	−0.505881	+	0.862603i	$0.668833\pi$
$182$	0	0
$183$	−38.8567	−2.87237
$184$	0	0
$185$	0.393626	0.0289400
$186$	0	0
$187$	−15.9973	−1.16984
$188$	0	0
$189$	−31.8949	−2.32001
$190$	0	0
$191$	−6.91622	−0.500440	−0.250220	−	0.968189i	$-0.580503\pi$
$191$	−6.91622	−0.500440	−0.250220	+	0.968189i	$0.580503\pi$
$192$	0	0
$193$	22.3029	1.60540	0.802699	−	0.596384i	$-0.203396\pi$
$193$	22.3029	1.60540	0.802699	+	0.596384i	$0.203396\pi$
$194$	0	0
$195$	0.208299	0.0149166
$196$	0	0
$197$	−3.21889	−0.229336	−0.114668	−	0.993404i	$-0.536580\pi$
$197$	−3.21889	−0.229336	−0.114668	+	0.993404i	$0.536580\pi$
$198$	0	0
$199$	13.8048	0.978597	0.489299	−	0.872116i	$-0.337253\pi$
$199$	13.8048	0.978597	0.489299	+	0.872116i	$0.337253\pi$
$200$	0	0
$201$	22.0906	1.55815
$202$	0	0
$203$	12.9095	0.906072
$204$	0	0
$205$	−0.0257029	−0.00179517
$206$	0	0
$207$	0	0
$208$	0	0
$209$	25.5443	1.76694
$210$	0	0
$211$	22.2456	1.53145	0.765725	−	0.643168i	$-0.222380\pi$
$211$	22.2456	1.53145	0.765725	+	0.643168i	$0.222380\pi$
$212$	0	0
$213$	−0.644491	−0.0441598
$214$	0	0
$215$	−0.0830148	−0.00566156
$216$	0	0
$217$	24.4197	1.65772
$218$	0	0
$219$	−2.17006	−0.146639
$220$	0	0
$221$	3.82860	0.257539
$222$	0	0
$223$	−9.20223	−0.616227	−0.308113	−	0.951350i	$-0.599698\pi$
$223$	−9.20223	−0.616227	−0.308113	+	0.951350i	$0.599698\pi$
$224$	0	0
$225$	−34.7734	−2.31823
$226$	0	0
$227$	11.7349	0.778872	0.389436	−	0.921054i	$-0.372670\pi$
$227$	11.7349	0.778872	0.389436	+	0.921054i	$0.372670\pi$
$228$	0	0
$229$	−12.6272	−0.834428	−0.417214	−	0.908808i	$-0.636993\pi$
$229$	−12.6272	−0.834428	−0.417214	+	0.908808i	$0.636993\pi$
$230$	0	0
$231$	−36.1395	−2.37780
$232$	0	0
$233$	0.0603108	0.00395109	0.00197555	−	0.999998i	$-0.499371\pi$
$233$	0.0603108	0.00395109	0.00197555	+	0.999998i	$0.499371\pi$
$234$	0	0
$235$	0.570832	0.0372370
$236$	0	0
$237$	−35.1224	−2.28144
$238$	0	0
$239$	−19.6516	−1.27115	−0.635577	−	0.772038i	$-0.719238\pi$
$239$	−19.6516	−1.27115	−0.635577	+	0.772038i	$0.719238\pi$
$240$	0	0
$241$	−19.7454	−1.27191	−0.635957	−	0.771725i	$-0.719394\pi$
$241$	−19.7454	−1.27191	−0.635957	+	0.771725i	$0.719394\pi$
$242$	0	0
$243$	−21.0852	−1.35262
$244$	0	0
$245$	−0.0299087	−0.00191079
$246$	0	0
$247$	−6.11345	−0.388989
$248$	0	0
$249$	−21.9025	−1.38801
$250$	0	0
$251$	−1.94939	−0.123044	−0.0615222	−	0.998106i	$-0.519595\pi$
$251$	−1.94939	−0.123044	−0.0615222	+	0.998106i	$0.519595\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.691581	0.0433085
$256$	0	0
$257$	−7.94586	−0.495649	−0.247825	−	0.968805i	$-0.579716\pi$
$257$	−7.94586	−0.495649	−0.247825	+	0.968805i	$0.579716\pi$
$258$	0	0
$259$	16.3445	1.01560
$260$	0	0
$261$	35.2057	2.17918
$262$	0	0
$263$	−11.3326	−0.698796	−0.349398	−	0.936974i	$-0.613614\pi$
$263$	−11.3326	−0.698796	−0.349398	+	0.936974i	$0.613614\pi$
$264$	0	0
$265$	0.171229	0.0105185
$266$	0	0
$267$	21.8434	1.33680
$268$	0	0
$269$	−12.2118	−0.744565	−0.372282	−	0.928119i	$-0.621425\pi$
$269$	−12.2118	−0.744565	−0.372282	+	0.928119i	$0.621425\pi$
$270$	0	0
$271$	−2.19932	−0.133599	−0.0667995	−	0.997766i	$-0.521279\pi$
$271$	−2.19932	−0.133599	−0.0667995	+	0.997766i	$0.521279\pi$
$272$	0	0
$273$	8.64916	0.523471
$274$	0	0
$275$	−22.4177	−1.35184
$276$	0	0
$277$	4.15114	0.249418	0.124709	−	0.992193i	$-0.460200\pi$
$277$	4.15114	0.249418	0.124709	+	0.992193i	$0.460200\pi$
$278$	0	0
$279$	66.5950	3.98694
$280$	0	0
$281$	21.5378	1.28484	0.642418	−	0.766354i	$-0.277931\pi$
$281$	21.5378	1.28484	0.642418	+	0.766354i	$0.277931\pi$
$282$	0	0
$283$	5.35460	0.318298	0.159149	−	0.987255i	$-0.449125\pi$
$283$	5.35460	0.318298	0.159149	+	0.987255i	$0.449125\pi$
$284$	0	0
$285$	−1.10431	−0.0654135
$286$	0	0
$287$	−1.06726	−0.0629982
$288$	0	0
$289$	−4.28854	−0.252267
$290$	0	0
$291$	21.7915	1.27744
$292$	0	0
$293$	9.88688	0.577598	0.288799	−	0.957390i	$-0.406744\pi$
$293$	9.88688	0.577598	0.288799	+	0.957390i	$0.406744\pi$
$294$	0	0
$295$	0.839897	0.0489007
$296$	0	0
$297$	−56.0747	−3.25379
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−3.44701	−0.198682
$302$	0	0
$303$	−11.6374	−0.668550
$304$	0	0
$305$	0.756754	0.0433316
$306$	0	0
$307$	−17.6966	−1.01000	−0.505000	−	0.863119i	$-0.668507\pi$
$307$	−17.6966	−1.01000	−0.505000	+	0.863119i	$0.668507\pi$
$308$	0	0
$309$	39.8534	2.26718
$310$	0	0
$311$	12.7652	0.723846	0.361923	−	0.932208i	$-0.382120\pi$
$311$	12.7652	0.723846	0.361923	+	0.932208i	$0.382120\pi$
$312$	0	0
$313$	22.4298	1.26781	0.633905	−	0.773411i	$-0.281451\pi$
$313$	22.4298	1.26781	0.633905	+	0.773411i	$0.281451\pi$
$314$	0	0
$315$	1.09176	0.0615135
$316$	0	0
$317$	29.4822	1.65589	0.827943	−	0.560812i	$-0.189511\pi$
$317$	29.4822	1.65589	0.827943	+	0.560812i	$0.189511\pi$
$318$	0	0
$319$	22.6964	1.27076
$320$	0	0
$321$	56.1375	3.13329
$322$	0	0
$323$	−20.2975	−1.12938
$324$	0	0
$325$	5.36517	0.297606
$326$	0	0
$327$	2.33119	0.128915
$328$	0	0
$329$	23.7026	1.30676
$330$	0	0
$331$	−7.47656	−0.410949	−0.205474	−	0.978662i	$-0.565874\pi$
$331$	−7.47656	−0.410949	−0.205474	+	0.978662i	$0.565874\pi$
$332$	0	0
$333$	44.5731	2.44259
$334$	0	0
$335$	−0.430224	−0.0235057
$336$	0	0
$337$	−27.2937	−1.48678	−0.743392	−	0.668856i	$-0.766784\pi$
$337$	−27.2937	−1.48678	−0.743392	+	0.668856i	$0.766784\pi$
$338$	0	0
$339$	13.1415	0.713746
$340$	0	0
$341$	42.9325	2.32493
$342$	0	0
$343$	−19.1068	−1.03167
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.6798	−0.680687	−0.340344	−	0.940301i	$-0.610543\pi$
$347$	−12.6798	−0.680687	−0.340344	+	0.940301i	$0.610543\pi$
$348$	0	0
$349$	−14.1692	−0.758460	−0.379230	−	0.925302i	$-0.623811\pi$
$349$	−14.1692	−0.758460	−0.379230	+	0.925302i	$0.623811\pi$
$350$	0	0
$351$	13.4202	0.716317
$352$	0	0
$353$	−21.1891	−1.12778	−0.563890	−	0.825850i	$-0.690696\pi$
$353$	−21.1891	−1.12778	−0.563890	+	0.825850i	$0.690696\pi$
$354$	0	0
$355$	0.0125518	0.000666179 0
$356$	0	0
$357$	28.7164	1.51983
$358$	0	0
$359$	4.09248	0.215993	0.107996	−	0.994151i	$-0.465557\pi$
$359$	4.09248	0.215993	0.107996	+	0.994151i	$0.465557\pi$
$360$	0	0
$361$	13.4107	0.705828
$362$	0	0
$363$	−28.8220	−1.51276
$364$	0	0
$365$	0.0422629	0.00221214
$366$	0	0
$367$	28.9755	1.51251	0.756254	−	0.654278i	$-0.227028\pi$
$367$	28.9755	1.51251	0.756254	+	0.654278i	$0.227028\pi$
$368$	0	0
$369$	−2.91052	−0.151516
$370$	0	0
$371$	7.10990	0.369128
$372$	0	0
$373$	−21.6140	−1.11913	−0.559564	−	0.828787i	$-0.689031\pi$
$373$	−21.6140	−1.11913	−0.559564	+	0.828787i	$0.689031\pi$
$374$	0	0
$375$	1.93901	0.100130
$376$	0	0
$377$	−5.43187	−0.279755
$378$	0	0
$379$	−0.482250	−0.0247715	−0.0123858	−	0.999923i	$-0.503943\pi$
$379$	−0.482250	−0.0247715	−0.0123858	+	0.999923i	$0.503943\pi$
$380$	0	0
$381$	8.57083	0.439097
$382$	0	0
$383$	−0.845058	−0.0431804	−0.0215902	−	0.999767i	$-0.506873\pi$
$383$	−0.845058	−0.0431804	−0.0215902	+	0.999767i	$0.506873\pi$
$384$	0	0
$385$	0.703834	0.0358707
$386$	0	0
$387$	−9.40036	−0.477847
$388$	0	0
$389$	38.8992	1.97227	0.986135	−	0.165945i	$-0.0530675\pi$
$389$	38.8992	1.97227	0.986135	+	0.165945i	$0.0530675\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	30.0970	1.51819
$394$	0	0
$395$	0.684025	0.0344171
$396$	0	0
$397$	−12.7389	−0.639347	−0.319673	−	0.947528i	$-0.603573\pi$
$397$	−12.7389	−0.639347	−0.319673	+	0.947528i	$0.603573\pi$
$398$	0	0
$399$	−45.8539	−2.29557
$400$	0	0
$401$	−19.3386	−0.965721	−0.482861	−	0.875697i	$-0.660402\pi$
$401$	−19.3386	−0.965721	−0.482861	+	0.875697i	$0.660402\pi$
$402$	0	0
$403$	−10.2749	−0.511830
$404$	0	0
$405$	1.14082	0.0566878
$406$	0	0
$407$	28.7354	1.42436
$408$	0	0
$409$	−18.3643	−0.908057	−0.454029	−	0.890987i	$-0.650014\pi$
$409$	−18.3643	−0.908057	−0.454029	+	0.890987i	$0.650014\pi$
$410$	0	0
$411$	22.7706	1.12319
$412$	0	0
$413$	34.8749	1.71608
$414$	0	0
$415$	0.426562	0.0209391
$416$	0	0
$417$	−12.4399	−0.609187
$418$	0	0
$419$	12.2462	0.598265	0.299133	−	0.954212i	$-0.403303\pi$
$419$	12.2462	0.598265	0.299133	+	0.954212i	$0.403303\pi$
$420$	0	0
$421$	36.8858	1.79770	0.898852	−	0.438253i	$-0.144403\pi$
$421$	36.8858	1.79770	0.898852	+	0.438253i	$0.144403\pi$
$422$	0	0
$423$	64.6394	3.14287
$424$	0	0
$425$	17.8131	0.864062
$426$	0	0
$427$	31.4225	1.52064
$428$	0	0
$429$	15.2062	0.734162
$430$	0	0
$431$	−29.0920	−1.40131	−0.700655	−	0.713500i	$-0.747109\pi$
$431$	−29.0920	−1.40131	−0.700655	+	0.713500i	$0.747109\pi$
$432$	0	0
$433$	1.88716	0.0906909	0.0453455	−	0.998971i	$-0.485561\pi$
$433$	1.88716	0.0906909	0.0453455	+	0.998971i	$0.485561\pi$
$434$	0	0
$435$	−0.981188	−0.0470444
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−2.85598	−0.136309	−0.0681544	−	0.997675i	$-0.521711\pi$
$439$	−2.85598	−0.136309	−0.0681544	+	0.997675i	$0.521711\pi$
$440$	0	0
$441$	−3.38677	−0.161275
$442$	0	0
$443$	8.27777	0.393289	0.196644	−	0.980475i	$-0.436996\pi$
$443$	8.27777	0.393289	0.196644	+	0.980475i	$0.436996\pi$
$444$	0	0
$445$	−0.425412	−0.0201664
$446$	0	0
$447$	−18.5276	−0.876326
$448$	0	0
$449$	−8.08237	−0.381430	−0.190715	−	0.981645i	$-0.561081\pi$
$449$	−8.08237	−0.381430	−0.190715	+	0.981645i	$0.561081\pi$
$450$	0	0
$451$	−1.87636	−0.0883543
$452$	0	0
$453$	51.8931	2.43815
$454$	0	0
$455$	−0.168447	−0.00789690
$456$	0	0
$457$	9.74478	0.455842	0.227921	−	0.973680i	$-0.426807\pi$
$457$	9.74478	0.455842	0.227921	+	0.973680i	$0.426807\pi$
$458$	0	0
$459$	44.5569	2.07974
$460$	0	0
$461$	−30.5887	−1.42466	−0.712329	−	0.701845i	$-0.752360\pi$
$461$	−30.5887	−1.42466	−0.712329	+	0.701845i	$0.752360\pi$
$462$	0	0
$463$	−21.0807	−0.979704	−0.489852	−	0.871806i	$-0.662949\pi$
$463$	−21.0807	−0.979704	−0.489852	+	0.871806i	$0.662949\pi$
$464$	0	0
$465$	−1.85601	−0.0860706
$466$	0	0
$467$	−11.1858	−0.517617	−0.258808	−	0.965929i	$-0.583330\pi$
$467$	−11.1858	−0.517617	−0.258808	+	0.965929i	$0.583330\pi$
$468$	0	0
$469$	−17.8641	−0.824889
$470$	0	0
$471$	6.97640	0.321456
$472$	0	0
$473$	−6.06023	−0.278650
$474$	0	0
$475$	−28.4437	−1.30509
$476$	0	0
$477$	19.3895	0.887782
$478$	0	0
$479$	29.3302	1.34013	0.670066	−	0.742302i	$-0.266266\pi$
$479$	29.3302	1.34013	0.670066	+	0.742302i	$0.266266\pi$
$480$	0	0
$481$	−6.87716	−0.313572
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.424400	−0.0192710
$486$	0	0
$487$	40.8306	1.85021	0.925106	−	0.379709i	$-0.123976\pi$
$487$	40.8306	1.85021	0.925106	+	0.379709i	$0.123976\pi$
$488$	0	0
$489$	14.6424	0.662154
$490$	0	0
$491$	−5.04617	−0.227731	−0.113865	−	0.993496i	$-0.536323\pi$
$491$	−5.04617	−0.227731	−0.113865	+	0.993496i	$0.536323\pi$
$492$	0	0
$493$	−18.0345	−0.812234
$494$	0	0
$495$	1.91943	0.0862720
$496$	0	0
$497$	0.521185	0.0233784
$498$	0	0
$499$	5.45513	0.244205	0.122103	−	0.992517i	$-0.461036\pi$
$499$	5.45513	0.244205	0.122103	+	0.992517i	$0.461036\pi$
$500$	0	0
$501$	−48.1490	−2.15114
$502$	0	0
$503$	9.40356	0.419284	0.209642	−	0.977778i	$-0.432770\pi$
$503$	9.40356	0.419284	0.209642	+	0.977778i	$0.432770\pi$
$504$	0	0
$505$	0.226644	0.0100855
$506$	0	0
$507$	37.3879	1.66046
$508$	0	0
$509$	4.34798	0.192721	0.0963604	−	0.995347i	$-0.469280\pi$
$509$	4.34798	0.192721	0.0963604	+	0.995347i	$0.469280\pi$
$510$	0	0
$511$	1.75487	0.0776311
$512$	0	0
$513$	−71.1478	−3.14125
$514$	0	0
$515$	−0.776165	−0.0342019
$516$	0	0
$517$	41.6717	1.83272
$518$	0	0
$519$	−41.8448	−1.83678
$520$	0	0
$521$	38.8959	1.70406	0.852031	−	0.523492i	$-0.175371\pi$
$521$	38.8959	1.70406	0.852031	+	0.523492i	$0.175371\pi$
$522$	0	0
$523$	27.8512	1.21785	0.608924	−	0.793228i	$-0.291601\pi$
$523$	27.8512	1.21785	0.608924	+	0.793228i	$0.291601\pi$
$524$	0	0
$525$	40.2415	1.75628
$526$	0	0
$527$	−34.1141	−1.48603
$528$	0	0
$529$	0	0
$530$	0	0
$531$	95.1075	4.12732
$532$	0	0
$533$	0.449063	0.0194511
$534$	0	0
$535$	−1.09330	−0.0472677
$536$	0	0
$537$	−36.5766	−1.57840
$538$	0	0
$539$	−2.18339	−0.0940451
$540$	0	0
$541$	16.0839	0.691501	0.345750	−	0.938327i	$-0.387624\pi$
$541$	16.0839	0.691501	0.345750	+	0.938327i	$0.387624\pi$
$542$	0	0
$543$	42.9582	1.84351
$544$	0	0
$545$	−0.0454010	−0.00194477
$546$	0	0
$547$	−32.5952	−1.39367	−0.696835	−	0.717232i	$-0.745409\pi$
$547$	−32.5952	−1.39367	−0.696835	+	0.717232i	$0.745409\pi$
$548$	0	0
$549$	85.6926	3.65727
$550$	0	0
$551$	28.7973	1.22681
$552$	0	0
$553$	28.4027	1.20780
$554$	0	0
$555$	−1.24226	−0.0527310
$556$	0	0
$557$	−11.1050	−0.470536	−0.235268	−	0.971931i	$-0.575597\pi$
$557$	−11.1050	−0.470536	−0.235268	+	0.971931i	$0.575597\pi$
$558$	0	0
$559$	1.45038	0.0613444
$560$	0	0
$561$	50.4866	2.13155
$562$	0	0
$563$	40.5744	1.71001	0.855004	−	0.518622i	$-0.173555\pi$
$563$	40.5744	1.71001	0.855004	+	0.518622i	$0.173555\pi$
$564$	0	0
$565$	−0.255936	−0.0107673
$566$	0	0
$567$	47.3701	1.98936
$568$	0	0
$569$	−15.7764	−0.661379	−0.330690	−	0.943740i	$-0.607281\pi$
$569$	−15.7764	−0.661379	−0.330690	+	0.943740i	$0.607281\pi$
$570$	0	0
$571$	13.8376	0.579086	0.289543	−	0.957165i	$-0.406497\pi$
$571$	13.8376	0.579086	0.289543	+	0.957165i	$0.406497\pi$
$572$	0	0
$573$	21.8272	0.911842
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.35982	0.139871	0.0699355	−	0.997552i	$-0.477721\pi$
$577$	3.35982	0.139871	0.0699355	+	0.997552i	$0.477721\pi$
$578$	0	0
$579$	−70.3866	−2.92517
$580$	0	0
$581$	17.7120	0.734819
$582$	0	0
$583$	12.5000	0.517697
$584$	0	0
$585$	−0.459372	−0.0189927
$586$	0	0
$587$	21.0390	0.868372	0.434186	−	0.900823i	$-0.357036\pi$
$587$	21.0390	0.868372	0.434186	+	0.900823i	$0.357036\pi$
$588$	0	0
$589$	54.4729	2.24452
$590$	0	0
$591$	10.1586	0.417869
$592$	0	0
$593$	6.73248	0.276470	0.138235	−	0.990399i	$-0.455857\pi$
$593$	6.73248	0.276470	0.138235	+	0.990399i	$0.455857\pi$
$594$	0	0
$595$	−0.559266	−0.0229277
$596$	0	0
$597$	−43.5671	−1.78308
$598$	0	0
$599$	−3.17780	−0.129842	−0.0649208	−	0.997890i	$-0.520679\pi$
$599$	−3.17780	−0.129842	−0.0649208	+	0.997890i	$0.520679\pi$
$600$	0	0
$601$	21.8661	0.891937	0.445969	−	0.895049i	$-0.352859\pi$
$601$	21.8661	0.891937	0.445969	+	0.895049i	$0.352859\pi$
$602$	0	0
$603$	−48.7174	−1.98392
$604$	0	0
$605$	0.561322	0.0228210
$606$	0	0
$607$	−10.1898	−0.413593	−0.206797	−	0.978384i	$-0.566304\pi$
$607$	−10.1898	−0.413593	−0.206797	+	0.978384i	$0.566304\pi$
$608$	0	0
$609$	−40.7417	−1.65094
$610$	0	0
$611$	−9.97317	−0.403471
$612$	0	0
$613$	22.3679	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$613$	22.3679	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$614$	0	0
$615$	0.0811168	0.00327095
$616$	0	0
$617$	−24.3081	−0.978608	−0.489304	−	0.872113i	$-0.662749\pi$
$617$	−24.3081	−0.978608	−0.489304	+	0.872113i	$0.662749\pi$
$618$	0	0
$619$	24.3008	0.976730	0.488365	−	0.872639i	$-0.337593\pi$
$619$	24.3008	0.976730	0.488365	+	0.872639i	$0.337593\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17.6643	−0.707705
$624$	0	0
$625$	24.9433	0.997734
$626$	0	0
$627$	−80.6163	−3.21951
$628$	0	0
$629$	−22.8331	−0.910416
$630$	0	0
$631$	20.2216	0.805008	0.402504	−	0.915418i	$-0.368140\pi$
$631$	20.2216	0.805008	0.402504	+	0.915418i	$0.368140\pi$
$632$	0	0
$633$	−70.2057	−2.79043
$634$	0	0
$635$	−0.166921	−0.00662406
$636$	0	0
$637$	0.522543	0.0207039
$638$	0	0
$639$	1.42133	0.0562268
$640$	0	0
$641$	2.50833	0.0990732	0.0495366	−	0.998772i	$-0.484226\pi$
$641$	2.50833	0.0990732	0.0495366	+	0.998772i	$0.484226\pi$
$642$	0	0
$643$	33.7473	1.33086	0.665432	−	0.746458i	$-0.268247\pi$
$643$	33.7473	1.33086	0.665432	+	0.746458i	$0.268247\pi$
$644$	0	0
$645$	0.261990	0.0103158
$646$	0	0
$647$	−44.3620	−1.74405	−0.872025	−	0.489462i	$-0.837193\pi$
$647$	−44.3620	−1.74405	−0.872025	+	0.489462i	$0.837193\pi$
$648$	0	0
$649$	61.3140	2.40678
$650$	0	0
$651$	−77.0670	−3.02049
$652$	0	0
$653$	20.1758	0.789539	0.394770	−	0.918780i	$-0.370824\pi$
$653$	20.1758	0.789539	0.394770	+	0.918780i	$0.370824\pi$
$654$	0	0
$655$	−0.586153	−0.0229029
$656$	0	0
$657$	4.78573	0.186709
$658$	0	0
$659$	9.16287	0.356935	0.178467	−	0.983946i	$-0.442886\pi$
$659$	9.16287	0.356935	0.178467	+	0.983946i	$0.442886\pi$
$660$	0	0
$661$	−43.4576	−1.69030	−0.845152	−	0.534525i	$-0.820490\pi$
$661$	−43.4576	−1.69030	−0.845152	+	0.534525i	$0.820490\pi$
$662$	0	0
$663$	−12.0828	−0.469258
$664$	0	0
$665$	0.893027	0.0346301
$666$	0	0
$667$	0	0
$668$	0	0
$669$	29.0417	1.12282
$670$	0	0
$671$	55.2444	2.13268
$672$	0	0
$673$	−27.7365	−1.06916	−0.534581	−	0.845117i	$-0.679531\pi$
$673$	−27.7365	−1.06916	−0.534581	+	0.845117i	$0.679531\pi$
$674$	0	0
$675$	62.4394	2.40329
$676$	0	0
$677$	10.9026	0.419021	0.209511	−	0.977806i	$-0.432813\pi$
$677$	10.9026	0.419021	0.209511	+	0.977806i	$0.432813\pi$
$678$	0	0
$679$	−17.6223	−0.676281
$680$	0	0
$681$	−37.0346	−1.41917
$682$	0	0
$683$	9.76175	0.373523	0.186762	−	0.982405i	$-0.440201\pi$
$683$	9.76175	0.373523	0.186762	+	0.982405i	$0.440201\pi$
$684$	0	0
$685$	−0.443469	−0.0169441
$686$	0	0
$687$	39.8506	1.52039
$688$	0	0
$689$	−2.99159	−0.113970
$690$	0	0
$691$	−5.10327	−0.194138	−0.0970688	−	0.995278i	$-0.530947\pi$
$691$	−5.10327	−0.194138	−0.0970688	+	0.995278i	$0.530947\pi$
$692$	0	0
$693$	79.7002	3.02756
$694$	0	0
$695$	0.242274	0.00918997
$696$	0	0
$697$	1.49095	0.0564738
$698$	0	0
$699$	−0.190337	−0.00719921
$700$	0	0
$701$	−19.3106	−0.729349	−0.364675	−	0.931135i	$-0.618820\pi$
$701$	−19.3106	−0.729349	−0.364675	+	0.931135i	$0.618820\pi$
$702$	0	0
$703$	36.4596	1.37510
$704$	0	0
$705$	−1.80151	−0.0678488
$706$	0	0
$707$	9.41088	0.353933
$708$	0	0
$709$	−45.5785	−1.71174	−0.855868	−	0.517194i	$-0.826977\pi$
$709$	−45.5785	−1.71174	−0.855868	+	0.517194i	$0.826977\pi$
$710$	0	0
$711$	77.4571	2.90487
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−0.296148	−0.0110753
$716$	0	0
$717$	62.0191	2.31614
$718$	0	0
$719$	−15.8594	−0.591457	−0.295729	−	0.955272i	$-0.595562\pi$
$719$	−15.8594	−0.591457	−0.295729	+	0.955272i	$0.595562\pi$
$720$	0	0
$721$	−32.2285	−1.20025
$722$	0	0
$723$	62.3153	2.31753
$724$	0	0
$725$	−25.2725	−0.938598
$726$	0	0
$727$	41.3267	1.53272	0.766361	−	0.642410i	$-0.222065\pi$
$727$	41.3267	1.53272	0.766361	+	0.642410i	$0.222065\pi$
$728$	0	0
$729$	10.8608	0.402253
$730$	0	0
$731$	4.81545	0.178106
$732$	0	0
$733$	12.4809	0.460992	0.230496	−	0.973073i	$-0.425965\pi$
$733$	12.4809	0.460992	0.230496	+	0.973073i	$0.425965\pi$
$734$	0	0
$735$	0.0943899	0.00348162
$736$	0	0
$737$	−31.4071	−1.15690
$738$	0	0
$739$	−11.4138	−0.419863	−0.209931	−	0.977716i	$-0.567324\pi$
$739$	−11.4138	−0.419863	−0.209931	+	0.977716i	$0.567324\pi$
$740$	0	0
$741$	19.2937	0.708770
$742$	0	0
$743$	36.8550	1.35208	0.676039	−	0.736866i	$-0.263695\pi$
$743$	36.8550	1.35208	0.676039	+	0.736866i	$0.263695\pi$
$744$	0	0
$745$	0.360835	0.0132200
$746$	0	0
$747$	48.3026	1.76730
$748$	0	0
$749$	−45.3971	−1.65877
$750$	0	0
$751$	−51.8952	−1.89368	−0.946842	−	0.321700i	$-0.895746\pi$
$751$	−51.8952	−1.89368	−0.946842	+	0.321700i	$0.895746\pi$
$752$	0	0
$753$	6.15215	0.224197
$754$	0	0
$755$	−1.01064	−0.0367811
$756$	0	0
$757$	−17.5670	−0.638482	−0.319241	−	0.947674i	$-0.603428\pi$
$757$	−17.5670	−0.638482	−0.319241	+	0.947674i	$0.603428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.5806	−1.25355	−0.626773	−	0.779202i	$-0.715624\pi$
$761$	−34.5806	−1.25355	−0.626773	+	0.779202i	$0.715624\pi$
$762$	0	0
$763$	−1.88518	−0.0682480
$764$	0	0
$765$	−1.52518	−0.0551429
$766$	0	0
$767$	−14.6741	−0.529851
$768$	0	0
$769$	14.9775	0.540102	0.270051	−	0.962846i	$-0.412959\pi$
$769$	14.9775	0.540102	0.270051	+	0.962846i	$0.412959\pi$
$770$	0	0
$771$	25.0766	0.903113
$772$	0	0
$773$	16.9927	0.611186	0.305593	−	0.952162i	$-0.401145\pi$
$773$	16.9927	0.611186	0.305593	+	0.952162i	$0.401145\pi$
$774$	0	0
$775$	−47.8055	−1.71722
$776$	0	0
$777$	−51.5821	−1.85050
$778$	0	0
$779$	−2.38073	−0.0852985
$780$	0	0
$781$	0.916302	0.0327879
$782$	0	0
$783$	−63.2156	−2.25914
$784$	0	0
$785$	−0.135869	−0.00484937
$786$	0	0
$787$	−2.67492	−0.0953507	−0.0476753	−	0.998863i	$-0.515181\pi$
$787$	−2.67492	−0.0953507	−0.0476753	+	0.998863i	$0.515181\pi$
$788$	0	0
$789$	35.7649	1.27326
$790$	0	0
$791$	−10.6272	−0.377860
$792$	0	0
$793$	−13.2215	−0.469508
$794$	0	0
$795$	−0.540387	−0.0191656
$796$	0	0
$797$	−23.3267	−0.826274	−0.413137	−	0.910669i	$-0.635567\pi$
$797$	−23.3267	−0.826274	−0.413137	+	0.910669i	$0.635567\pi$
$798$	0	0
$799$	−33.1123	−1.17143
$800$	0	0
$801$	−48.1724	−1.70209
$802$	0	0
$803$	3.08527	0.108877
$804$	0	0
$805$	0	0
$806$	0	0
$807$	38.5396	1.35666
$808$	0	0
$809$	−7.84869	−0.275945	−0.137973	−	0.990436i	$-0.544059\pi$
$809$	−7.84869	−0.275945	−0.137973	+	0.990436i	$0.544059\pi$
$810$	0	0
$811$	−3.32512	−0.116761	−0.0583804	−	0.998294i	$-0.518594\pi$
$811$	−3.32512	−0.116761	−0.0583804	+	0.998294i	$0.518594\pi$
$812$	0	0
$813$	6.94091	0.243428
$814$	0	0
$815$	−0.285169	−0.00998903
$816$	0	0
$817$	−7.68924	−0.269012
$818$	0	0
$819$	−19.0744	−0.666514
$820$	0	0
$821$	1.48363	0.0517790	0.0258895	−	0.999665i	$-0.491758\pi$
$821$	1.48363	0.0517790	0.0258895	+	0.999665i	$0.491758\pi$
$822$	0	0
$823$	31.8540	1.11036	0.555180	−	0.831730i	$-0.312649\pi$
$823$	31.8540	1.11036	0.555180	+	0.831730i	$0.312649\pi$
$824$	0	0
$825$	70.7490	2.46316
$826$	0	0
$827$	−9.87348	−0.343335	−0.171667	−	0.985155i	$-0.554915\pi$
$827$	−9.87348	−0.343335	−0.171667	+	0.985155i	$0.554915\pi$
$828$	0	0
$829$	−46.1733	−1.60366	−0.801832	−	0.597549i	$-0.796141\pi$
$829$	−46.1733	−1.60366	−0.801832	+	0.597549i	$0.796141\pi$
$830$	0	0
$831$	−13.1007	−0.454459
$832$	0	0
$833$	1.73491	0.0601112
$834$	0	0
$835$	0.937724	0.0324513
$836$	0	0
$837$	−119.579	−4.13324
$838$	0	0
$839$	31.9927	1.10451	0.552256	−	0.833675i	$-0.313767\pi$
$839$	31.9927	1.10451	0.552256	+	0.833675i	$0.313767\pi$
$840$	0	0
$841$	−3.41331	−0.117700
$842$	0	0
$843$	−67.9719	−2.34108
$844$	0	0
$845$	−0.728148	−0.0250491
$846$	0	0
$847$	23.3077	0.800861
$848$	0	0
$849$	−16.8988	−0.579965
$850$	0	0
$851$	0	0
$852$	0	0
$853$	44.2267	1.51429	0.757147	−	0.653245i	$-0.226593\pi$
$853$	44.2267	1.51429	0.757147	+	0.653245i	$0.226593\pi$
$854$	0	0
$855$	2.43538	0.0832882
$856$	0	0
$857$	48.7698	1.66595	0.832973	−	0.553314i	$-0.186637\pi$
$857$	48.7698	1.66595	0.832973	+	0.553314i	$0.186637\pi$
$858$	0	0
$859$	18.4558	0.629704	0.314852	−	0.949141i	$-0.398045\pi$
$859$	18.4558	0.629704	0.314852	+	0.949141i	$0.398045\pi$
$860$	0	0
$861$	3.36820	0.114788
$862$	0	0
$863$	15.2526	0.519205	0.259602	−	0.965716i	$-0.416409\pi$
$863$	15.2526	0.519205	0.259602	+	0.965716i	$0.416409\pi$
$864$	0	0
$865$	0.814948	0.0277091
$866$	0	0
$867$	13.5344	0.459651
$868$	0	0
$869$	49.9351	1.69393
$870$	0	0
$871$	7.51658	0.254690
$872$	0	0
$873$	−48.0579	−1.62651
$874$	0	0
$875$	−1.56804	−0.0530093
$876$	0	0
$877$	−30.0377	−1.01430	−0.507150	−	0.861858i	$-0.669301\pi$
$877$	−30.0377	−1.01430	−0.507150	+	0.861858i	$0.669301\pi$
$878$	0	0
$879$	−31.2024	−1.05243
$880$	0	0
$881$	11.5828	0.390236	0.195118	−	0.980780i	$-0.437491\pi$
$881$	11.5828	0.390236	0.195118	+	0.980780i	$0.437491\pi$
$882$	0	0
$883$	33.0425	1.11197	0.555984	−	0.831193i	$-0.312342\pi$
$883$	33.0425	1.11197	0.555984	+	0.831193i	$0.312342\pi$
$884$	0	0
$885$	−2.65066	−0.0891011
$886$	0	0
$887$	39.0943	1.31266	0.656329	−	0.754474i	$-0.272108\pi$
$887$	39.0943	1.31266	0.656329	+	0.754474i	$0.272108\pi$
$888$	0	0
$889$	−6.93103	−0.232459
$890$	0	0
$891$	83.2819	2.79005
$892$	0	0
$893$	52.8732	1.76934
$894$	0	0
$895$	0.712348	0.0238112
$896$	0	0
$897$	0	0
$898$	0	0
$899$	48.3998	1.61422
$900$	0	0
$901$	−9.93248	−0.330899
$902$	0	0
$903$	10.8785	0.362015
$904$	0	0
$905$	−0.836632	−0.0278106
$906$	0	0
$907$	−35.1920	−1.16853	−0.584265	−	0.811563i	$-0.698617\pi$
$907$	−35.1920	−1.16853	−0.584265	+	0.811563i	$0.698617\pi$
$908$	0	0
$909$	25.6645	0.851237
$910$	0	0
$911$	−12.1435	−0.402331	−0.201166	−	0.979557i	$-0.564473\pi$
$911$	−12.1435	−0.402331	−0.201166	+	0.979557i	$0.564473\pi$
$912$	0	0
$913$	31.1397	1.03058
$914$	0	0
$915$	−2.38827	−0.0789537
$916$	0	0
$917$	−24.3387	−0.803735
$918$	0	0
$919$	−26.6760	−0.879958	−0.439979	−	0.898008i	$-0.645014\pi$
$919$	−26.6760	−0.879958	−0.439979	+	0.898008i	$0.645014\pi$
$920$	0	0
$921$	55.8494	1.84030
$922$	0	0
$923$	−0.219296	−0.00721821
$924$	0	0
$925$	−31.9970	−1.05205
$926$	0	0
$927$	−87.8907	−2.88671
$928$	0	0
$929$	14.5079	0.475990	0.237995	−	0.971266i	$-0.423510\pi$
$929$	14.5079	0.475990	0.237995	+	0.971266i	$0.423510\pi$
$930$	0	0
$931$	−2.77029	−0.0907925
$932$	0	0
$933$	−40.2860	−1.31891
$934$	0	0
$935$	−0.983252	−0.0321558
$936$	0	0
$937$	−49.7050	−1.62379	−0.811896	−	0.583802i	$-0.801565\pi$
$937$	−49.7050	−1.62379	−0.811896	+	0.583802i	$0.801565\pi$
$938$	0	0
$939$	−70.7872	−2.31005
$940$	0	0
$941$	55.4652	1.80811	0.904057	−	0.427412i	$-0.140575\pi$
$941$	55.4652	1.80811	0.904057	+	0.427412i	$0.140575\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1.96037	−0.0637708
$946$	0	0
$947$	−48.2258	−1.56713	−0.783564	−	0.621311i	$-0.786600\pi$
$947$	−48.2258	−1.56713	−0.783564	+	0.621311i	$0.786600\pi$
$948$	0	0
$949$	−0.738388	−0.0239691
$950$	0	0
$951$	−93.0440	−3.01716
$952$	0	0
$953$	55.5037	1.79794	0.898970	−	0.438010i	$-0.144317\pi$
$953$	55.5037	1.79794	0.898970	+	0.438010i	$0.144317\pi$
$954$	0	0
$955$	−0.425094	−0.0137557
$956$	0	0
$957$	−71.6285	−2.31542
$958$	0	0
$959$	−18.4141	−0.594621
$960$	0	0
$961$	60.5530	1.95332
$962$	0	0
$963$	−123.803	−3.98949
$964$	0	0
$965$	1.37081	0.0441280
$966$	0	0
$967$	−12.7805	−0.410995	−0.205497	−	0.978658i	$-0.565881\pi$
$967$	−12.7805	−0.410995	−0.205497	+	0.978658i	$0.565881\pi$
$968$	0	0
$969$	64.0576	2.05783
$970$	0	0
$971$	−22.0337	−0.707094	−0.353547	−	0.935417i	$-0.615025\pi$
$971$	−22.0337	−0.707094	−0.353547	+	0.935417i	$0.615025\pi$
$972$	0	0
$973$	10.0599	0.322505
$974$	0	0
$975$	−16.9321	−0.542263
$976$	0	0
$977$	39.4490	1.26209	0.631043	−	0.775748i	$-0.282627\pi$
$977$	39.4490	1.26209	0.631043	+	0.775748i	$0.282627\pi$
$978$	0	0
$979$	−31.0558	−0.992548
$980$	0	0
$981$	−5.14108	−0.164142
$982$	0	0
$983$	−17.7019	−0.564603	−0.282302	−	0.959326i	$-0.591098\pi$
$983$	−17.7019	−0.564603	−0.282302	+	0.959326i	$0.591098\pi$
$984$	0	0
$985$	−0.197844	−0.00630383
$986$	0	0
$987$	−74.8038	−2.38103
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−10.3498	−0.328771	−0.164385	−	0.986396i	$-0.552564\pi$
$991$	−10.3498	−0.328771	−0.164385	+	0.986396i	$0.552564\pi$
$992$	0	0
$993$	23.5956	0.748782
$994$	0	0
$995$	0.848491	0.0268990
$996$	0	0
$997$	−37.4240	−1.18523	−0.592615	−	0.805486i	$-0.701904\pi$
$997$	−37.4240	−1.18523	−0.592615	+	0.805486i	$0.701904\pi$
$998$	0	0
$999$	−80.0358	−2.53222

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.ch.1.1		15
4.3	odd	2		4232.2.a.ba.1.15		15
23.11	odd	22		368.2.m.e.305.3		30
23.21	odd	22		368.2.m.e.257.3		30
23.22	odd	2		8464.2.a.cg.1.1		15
92.11	even	22		184.2.i.b.121.1	yes	30
92.67	even	22		184.2.i.b.73.1	✓	30
92.91	even	2		4232.2.a.bb.1.15		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.73.1	✓	30	92.67	even	22
184.2.i.b.121.1	yes	30	92.11	even	22
368.2.m.e.257.3		30	23.21	odd	22
368.2.m.e.305.3		30	23.11	odd	22
4232.2.a.ba.1.15		15	4.3	odd	2
4232.2.a.bb.1.15		15	92.91	even	2
8464.2.a.cg.1.1		15	23.22	odd	2
8464.2.a.ch.1.1		15	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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q-3.15594 q^{3} +0.0614634 q^{5} +2.55213 q^{7} +6.95994 q^{9} +O(q^{10})\)	\(q-3.15594 q^{3} +0.0614634 q^{5} +2.55213 q^{7} +6.95994 q^{9} +4.48694 q^{11} -1.07385 q^{13} -0.193975 q^{15} -3.56531 q^{17} +5.69304 q^{19} -8.05438 q^{21} -4.99622 q^{25} -12.4973 q^{27} +5.05833 q^{29} +9.56833 q^{31} -14.1605 q^{33} +0.156863 q^{35} +6.40424 q^{37} +3.38899 q^{39} -0.418182 q^{41} -1.35064 q^{43} +0.427782 q^{45} +9.28734 q^{47} -0.486609 q^{49} +11.2519 q^{51} +2.78586 q^{53} +0.275783 q^{55} -17.9669 q^{57} +13.6650 q^{59} +12.3123 q^{61} +17.7627 q^{63} -0.0660022 q^{65} -6.99968 q^{67} +0.204215 q^{71} +0.687611 q^{73} +15.7678 q^{75} +11.4513 q^{77} +11.1290 q^{79} +18.5610 q^{81} +6.94009 q^{83} -0.219136 q^{85} -15.9638 q^{87} -6.92138 q^{89} -2.74060 q^{91} -30.1971 q^{93} +0.349914 q^{95} -6.90492 q^{97} +31.2288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * q^99

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