LMFDB - Embedded newform 3864.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3864,2,Mod(1,3864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3864.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:	$30.8541953410$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.39605.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 9x - 2 $ x^4 - 2x^3 - 8x^2 + 9*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.318315$ of defining polynomial
Character	$\chi$	$=$	3864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.681685 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.681685 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.71255 q^{11} +3.66749 q^{13} -0.681685 q^{15} +2.21699 q^{17} +4.34918 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.53531 q^{25} +1.00000 q^{27} -4.63226 q^{29} +8.21699 q^{31} -3.71255 q^{33} +0.681685 q^{35} +3.93391 q^{37} +3.66749 q^{39} -5.99563 q^{41} +5.16305 q^{43} -0.681685 q^{45} +6.41527 q^{47} +1.00000 q^{49} +2.21699 q^{51} -3.60140 q^{53} +2.53079 q^{55} +4.34918 q^{57} +6.83258 q^{59} -4.71939 q^{61} -1.00000 q^{63} -2.50007 q^{65} -2.96477 q^{67} -1.00000 q^{69} +3.12003 q^{71} +10.2787 q^{73} -4.53531 q^{75} +3.71255 q^{77} -11.6137 q^{79} +1.00000 q^{81} +11.5520 q^{83} -1.51129 q^{85} -4.63226 q^{87} +1.88885 q^{89} -3.66749 q^{91} +8.21699 q^{93} -2.96477 q^{95} +7.33062 q^{97} -3.71255 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{13} - 2 q^{15} - 10 q^{17} + 4 q^{19} - 4 q^{21} - 4 q^{23} + 4 q^{27} + 11 q^{29} + 14 q^{31} + 2 q^{35} + 13 q^{37} + 2 q^{39} + 7 q^{41} + 12 q^{43} - 2 q^{45} + 15 q^{47} + 4 q^{49} - 10 q^{51} + q^{53} + 31 q^{55} + 4 q^{57} + 5 q^{59} + 3 q^{61} - 4 q^{63} + 25 q^{65} + 5 q^{67} - 4 q^{69} + 5 q^{71} - 6 q^{73} + 26 q^{79} + 4 q^{81} + 2 q^{83} + 5 q^{85} + 11 q^{87} + 7 q^{89} - 2 q^{91} + 14 q^{93} + 5 q^{95} - 27 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^13 - 2 * q^15 - 10 * q^17 + 4 * q^19 - 4 * q^21 - 4 * q^23 + 4 * q^27 + 11 * q^29 + 14 * q^31 + 2 * q^35 + 13 * q^37 + 2 * q^39 + 7 * q^41 + 12 * q^43 - 2 * q^45 + 15 * q^47 + 4 * q^49 - 10 * q^51 + q^53 + 31 * q^55 + 4 * q^57 + 5 * q^59 + 3 * q^61 - 4 * q^63 + 25 * q^65 + 5 * q^67 - 4 * q^69 + 5 * q^71 - 6 * q^73 + 26 * q^79 + 4 * q^81 + 2 * q^83 + 5 * q^85 + 11 * q^87 + 7 * q^89 - 2 * q^91 + 14 * q^93 + 5 * q^95 - 27 * q^97

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.681685	−0.304859	−0.152429	−	0.988314i	$-0.548710\pi$
$5$	−0.681685	−0.304859	−0.152429	+	0.988314i	$0.548710\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.71255	−1.11938	−0.559688	−	0.828704i	$-0.689079\pi$
$11$	−3.71255	−1.11938	−0.559688	+	0.828704i	$0.689079\pi$
$12$	0	0
$13$	3.66749	1.01718	0.508590	−	0.861009i	$-0.330167\pi$
$13$	3.66749	1.01718	0.508590	+	0.861009i	$0.330167\pi$
$14$	0	0
$15$	−0.681685	−0.176010
$16$	0	0
$17$	2.21699	0.537699	0.268850	−	0.963182i	$-0.413357\pi$
$17$	2.21699	0.537699	0.268850	+	0.963182i	$0.413357\pi$
$18$	0	0
$19$	4.34918	0.997770	0.498885	−	0.866668i	$-0.333743\pi$
$19$	4.34918	0.997770	0.498885	+	0.866668i	$0.333743\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.53531	−0.907061
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.63226	−0.860189	−0.430095	−	0.902784i	$-0.641520\pi$
$29$	−4.63226	−0.860189	−0.430095	+	0.902784i	$0.641520\pi$
$30$	0	0
$31$	8.21699	1.47582	0.737908	−	0.674902i	$-0.235814\pi$
$31$	8.21699	1.47582	0.737908	+	0.674902i	$0.235814\pi$
$32$	0	0
$33$	−3.71255	−0.646272
$34$	0	0
$35$	0.681685	0.115226
$36$	0	0
$37$	3.93391	0.646730	0.323365	−	0.946274i	$-0.395186\pi$
$37$	3.93391	0.646730	0.323365	+	0.946274i	$0.395186\pi$
$38$	0	0
$39$	3.66749	0.587269
$40$	0	0
$41$	−5.99563	−0.936360	−0.468180	−	0.883633i	$-0.655090\pi$
$41$	−5.99563	−0.936360	−0.468180	+	0.883633i	$0.655090\pi$
$42$	0	0
$43$	5.16305	0.787358	0.393679	−	0.919248i	$-0.371202\pi$
$43$	5.16305	0.787358	0.393679	+	0.919248i	$0.371202\pi$
$44$	0	0
$45$	−0.681685	−0.101620
$46$	0	0
$47$	6.41527	0.935763	0.467882	−	0.883791i	$-0.345017\pi$
$47$	6.41527	0.935763	0.467882	+	0.883791i	$0.345017\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.21699	0.310441
$52$	0	0
$53$	−3.60140	−0.494690	−0.247345	−	0.968927i	$-0.579558\pi$
$53$	−3.60140	−0.494690	−0.247345	+	0.968927i	$0.579558\pi$
$54$	0	0
$55$	2.53079	0.341251
$56$	0	0
$57$	4.34918	0.576063
$58$	0	0
$59$	6.83258	0.889526	0.444763	−	0.895648i	$-0.353288\pi$
$59$	6.83258	0.889526	0.444763	+	0.895648i	$0.353288\pi$
$60$	0	0
$61$	−4.71939	−0.604256	−0.302128	−	0.953267i	$-0.597697\pi$
$61$	−4.71939	−0.604256	−0.302128	+	0.953267i	$0.597697\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−2.50007	−0.310096
$66$	0	0
$67$	−2.96477	−0.362204	−0.181102	−	0.983464i	$-0.557966\pi$
$67$	−2.96477	−0.362204	−0.181102	+	0.983464i	$0.557966\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	3.12003	0.370280	0.185140	−	0.982712i	$-0.440726\pi$
$71$	3.12003	0.370280	0.185140	+	0.982712i	$0.440726\pi$
$72$	0	0
$73$	10.2787	1.20303	0.601516	−	0.798860i	$-0.294563\pi$
$73$	10.2787	1.20303	0.601516	+	0.798860i	$0.294563\pi$
$74$	0	0
$75$	−4.53531	−0.523692
$76$	0	0
$77$	3.71255	0.423084
$78$	0	0
$79$	−11.6137	−1.30664	−0.653322	−	0.757080i	$-0.726625\pi$
$79$	−11.6137	−1.30664	−0.653322	+	0.757080i	$0.726625\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.5520	1.26799	0.633997	−	0.773335i	$-0.281413\pi$
$83$	11.5520	1.26799	0.633997	+	0.773335i	$0.281413\pi$
$84$	0	0
$85$	−1.51129	−0.163922
$86$	0	0
$87$	−4.63226	−0.496631
$88$	0	0
$89$	1.88885	0.200218	0.100109	−	0.994976i	$-0.468081\pi$
$89$	1.88885	0.200218	0.100109	+	0.994976i	$0.468081\pi$
$90$	0	0
$91$	−3.66749	−0.384458
$92$	0	0
$93$	8.21699	0.852062
$94$	0	0
$95$	−2.96477	−0.304179
$96$	0	0
$97$	7.33062	0.744311	0.372156	−	0.928170i	$-0.378619\pi$
$97$	7.33062	0.744311	0.372156	+	0.928170i	$0.378619\pi$
$98$	0	0
$99$	−3.71255	−0.373125
$100$	0	0
$101$	4.59251	0.456972	0.228486	−	0.973547i	$-0.426622\pi$
$101$	4.59251	0.456972	0.228486	+	0.973547i	$0.426622\pi$
$102$	0	0
$103$	14.3448	1.41344	0.706718	−	0.707495i	$-0.250175\pi$
$103$	14.3448	1.41344	0.706718	+	0.707495i	$0.250175\pi$
$104$	0	0
$105$	0.681685	0.0665256
$106$	0	0
$107$	1.44613	0.139803	0.0699015	−	0.997554i	$-0.477731\pi$
$107$	1.44613	0.139803	0.0699015	+	0.997554i	$0.477731\pi$
$108$	0	0
$109$	9.11567	0.873122	0.436561	−	0.899675i	$-0.356196\pi$
$109$	9.11567	0.873122	0.436561	+	0.899675i	$0.356196\pi$
$110$	0	0
$111$	3.93391	0.373390
$112$	0	0
$113$	12.8051	1.20461	0.602303	−	0.798268i	$-0.294250\pi$
$113$	12.8051	1.20461	0.602303	+	0.798268i	$0.294250\pi$
$114$	0	0
$115$	0.681685	0.0635675
$116$	0	0
$117$	3.66749	0.339060
$118$	0	0
$119$	−2.21699	−0.203231
$120$	0	0
$121$	2.78301	0.253001
$122$	0	0
$123$	−5.99563	−0.540608
$124$	0	0
$125$	6.50007	0.581384
$126$	0	0
$127$	17.5069	1.55349	0.776744	−	0.629816i	$-0.216870\pi$
$127$	17.5069	1.55349	0.776744	+	0.629816i	$0.216870\pi$
$128$	0	0
$129$	5.16305	0.454581
$130$	0	0
$131$	2.08480	0.182150	0.0910751	−	0.995844i	$-0.470970\pi$
$131$	2.08480	0.182150	0.0910751	+	0.995844i	$0.470970\pi$
$132$	0	0
$133$	−4.34918	−0.377122
$134$	0	0
$135$	−0.681685	−0.0586701
$136$	0	0
$137$	−14.9163	−1.27438	−0.637192	−	0.770705i	$-0.719904\pi$
$137$	−14.9163	−1.27438	−0.637192	+	0.770705i	$0.719904\pi$
$138$	0	0
$139$	5.45050	0.462306	0.231153	−	0.972917i	$-0.425750\pi$
$139$	5.45050	0.462306	0.231153	+	0.972917i	$0.425750\pi$
$140$	0	0
$141$	6.41527	0.540263
$142$	0	0
$143$	−13.6157	−1.13861
$144$	0	0
$145$	3.15774	0.262236
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	7.09900	0.581572	0.290786	−	0.956788i	$-0.406083\pi$
$149$	7.09900	0.581572	0.290786	+	0.956788i	$0.406083\pi$
$150$	0	0
$151$	9.29197	0.756170	0.378085	−	0.925771i	$-0.376583\pi$
$151$	9.29197	0.756170	0.378085	+	0.925771i	$0.376583\pi$
$152$	0	0
$153$	2.21699	0.179233
$154$	0	0
$155$	−5.60140	−0.449915
$156$	0	0
$157$	−1.33499	−0.106543	−0.0532717	−	0.998580i	$-0.516965\pi$
$157$	−1.33499	−0.106543	−0.0532717	+	0.998580i	$0.516965\pi$
$158$	0	0
$159$	−3.60140	−0.285610
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	7.49760	0.587257	0.293629	−	0.955920i	$-0.405137\pi$
$163$	7.49760	0.587257	0.293629	+	0.955920i	$0.405137\pi$
$164$	0	0
$165$	2.53079	0.197022
$166$	0	0
$167$	4.34918	0.336549	0.168275	−	0.985740i	$-0.446180\pi$
$167$	4.34918	0.336549	0.168275	+	0.985740i	$0.446180\pi$
$168$	0	0
$169$	0.450502	0.0346540
$170$	0	0
$171$	4.34918	0.332590
$172$	0	0
$173$	23.7459	1.80537	0.902683	−	0.430306i	$-0.141594\pi$
$173$	23.7459	1.80537	0.902683	+	0.430306i	$0.141594\pi$
$174$	0	0
$175$	4.53531	0.342837
$176$	0	0
$177$	6.83258	0.513568
$178$	0	0
$179$	−6.28512	−0.469772	−0.234886	−	0.972023i	$-0.575472\pi$
$179$	−6.28512	−0.469772	−0.234886	+	0.972023i	$0.575472\pi$
$180$	0	0
$181$	−1.97692	−0.146943	−0.0734717	−	0.997297i	$-0.523408\pi$
$181$	−1.97692	−0.146943	−0.0734717	+	0.997297i	$0.523408\pi$
$182$	0	0
$183$	−4.71939	−0.348868
$184$	0	0
$185$	−2.68168	−0.197161
$186$	0	0
$187$	−8.23068	−0.601887
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−10.6279	−0.769007	−0.384504	−	0.923123i	$-0.625627\pi$
$191$	−10.6279	−0.769007	−0.384504	+	0.923123i	$0.625627\pi$
$192$	0	0
$193$	−16.9531	−1.22031	−0.610154	−	0.792283i	$-0.708892\pi$
$193$	−16.9531	−1.22031	−0.610154	+	0.792283i	$0.708892\pi$
$194$	0	0
$195$	−2.50007	−0.179034
$196$	0	0
$197$	−6.45596	−0.459968	−0.229984	−	0.973194i	$-0.573867\pi$
$197$	−6.45596	−0.459968	−0.229984	+	0.973194i	$0.573867\pi$
$198$	0	0
$199$	15.0256	1.06513	0.532567	−	0.846388i	$-0.321228\pi$
$199$	15.0256	1.06513	0.532567	+	0.846388i	$0.321228\pi$
$200$	0	0
$201$	−2.96477	−0.209119
$202$	0	0
$203$	4.63226	0.325121
$204$	0	0
$205$	4.08713	0.285458
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−16.1465	−1.11688
$210$	0	0
$211$	−1.68416	−0.115943	−0.0579713	−	0.998318i	$-0.518463\pi$
$211$	−1.68416	−0.115943	−0.0579713	+	0.998318i	$0.518463\pi$
$212$	0	0
$213$	3.12003	0.213781
$214$	0	0
$215$	−3.51957	−0.240033
$216$	0	0
$217$	−8.21699	−0.557806
$218$	0	0
$219$	10.2787	0.694571
$220$	0	0
$221$	8.13080	0.546937
$222$	0	0
$223$	−0.742472	−0.0497196	−0.0248598	−	0.999691i	$-0.507914\pi$
$223$	−0.742472	−0.0497196	−0.0248598	+	0.999691i	$0.507914\pi$
$224$	0	0
$225$	−4.53531	−0.302354
$226$	0	0
$227$	−20.0500	−1.33077	−0.665383	−	0.746502i	$-0.731732\pi$
$227$	−20.0500	−1.33077	−0.665383	+	0.746502i	$0.731732\pi$
$228$	0	0
$229$	−11.7710	−0.777850	−0.388925	−	0.921269i	$-0.627153\pi$
$229$	−11.7710	−0.777850	−0.388925	+	0.921269i	$0.627153\pi$
$230$	0	0
$231$	3.71255	0.244268
$232$	0	0
$233$	17.0882	1.11949	0.559743	−	0.828666i	$-0.310900\pi$
$233$	17.0882	1.11949	0.559743	+	0.828666i	$0.310900\pi$
$234$	0	0
$235$	−4.37319	−0.285276
$236$	0	0
$237$	−11.6137	−0.754391
$238$	0	0
$239$	24.5838	1.59019	0.795096	−	0.606483i	$-0.207420\pi$
$239$	24.5838	1.59019	0.795096	+	0.606483i	$0.207420\pi$
$240$	0	0
$241$	−12.3963	−0.798514	−0.399257	−	0.916839i	$-0.630732\pi$
$241$	−12.3963	−0.798514	−0.399257	+	0.916839i	$0.630732\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.681685	−0.0435513
$246$	0	0
$247$	15.9506	1.01491
$248$	0	0
$249$	11.5520	0.732077
$250$	0	0
$251$	6.40653	0.404377	0.202188	−	0.979347i	$-0.435195\pi$
$251$	6.40653	0.404377	0.202188	+	0.979347i	$0.435195\pi$
$252$	0	0
$253$	3.71255	0.233406
$254$	0	0
$255$	−1.51129	−0.0946406
$256$	0	0
$257$	−29.3262	−1.82932	−0.914661	−	0.404223i	$-0.867542\pi$
$257$	−29.3262	−1.82932	−0.914661	+	0.404223i	$0.867542\pi$
$258$	0	0
$259$	−3.93391	−0.244441
$260$	0	0
$261$	−4.63226	−0.286730
$262$	0	0
$263$	−21.0905	−1.30050	−0.650249	−	0.759721i	$-0.725336\pi$
$263$	−21.0905	−1.30050	−0.650249	+	0.759721i	$0.725336\pi$
$264$	0	0
$265$	2.45502	0.150811
$266$	0	0
$267$	1.88885	0.115596
$268$	0	0
$269$	15.7302	0.959085	0.479542	−	0.877519i	$-0.340803\pi$
$269$	15.7302	0.959085	0.479542	+	0.877519i	$0.340803\pi$
$270$	0	0
$271$	21.0055	1.27599	0.637995	−	0.770040i	$-0.279764\pi$
$271$	21.0055	1.27599	0.637995	+	0.770040i	$0.279764\pi$
$272$	0	0
$273$	−3.66749	−0.221967
$274$	0	0
$275$	16.8375	1.01534
$276$	0	0
$277$	−0.643976	−0.0386928	−0.0193464	−	0.999813i	$-0.506159\pi$
$277$	−0.643976	−0.0386928	−0.0193464	+	0.999813i	$0.506159\pi$
$278$	0	0
$279$	8.21699	0.491938
$280$	0	0
$281$	−21.8404	−1.30289	−0.651444	−	0.758697i	$-0.725836\pi$
$281$	−21.8404	−1.30289	−0.651444	+	0.758697i	$0.725836\pi$
$282$	0	0
$283$	−6.89431	−0.409824	−0.204912	−	0.978780i	$-0.565691\pi$
$283$	−6.89431	−0.409824	−0.204912	+	0.978780i	$0.565691\pi$
$284$	0	0
$285$	−2.96477	−0.175618
$286$	0	0
$287$	5.99563	0.353911
$288$	0	0
$289$	−12.0850	−0.710880
$290$	0	0
$291$	7.33062	0.429728
$292$	0	0
$293$	−5.30164	−0.309725	−0.154863	−	0.987936i	$-0.549494\pi$
$293$	−5.30164	−0.309725	−0.154863	+	0.987936i	$0.549494\pi$
$294$	0	0
$295$	−4.65767	−0.271180
$296$	0	0
$297$	−3.71255	−0.215424
$298$	0	0
$299$	−3.66749	−0.212097
$300$	0	0
$301$	−5.16305	−0.297593
$302$	0	0
$303$	4.59251	0.263833
$304$	0	0
$305$	3.21714	0.184213
$306$	0	0
$307$	7.87218	0.449289	0.224645	−	0.974441i	$-0.427878\pi$
$307$	7.87218	0.449289	0.224645	+	0.974441i	$0.427878\pi$
$308$	0	0
$309$	14.3448	0.816048
$310$	0	0
$311$	−10.7390	−0.608955	−0.304478	−	0.952519i	$-0.598482\pi$
$311$	−10.7390	−0.608955	−0.304478	+	0.952519i	$0.598482\pi$
$312$	0	0
$313$	−14.1038	−0.797194	−0.398597	−	0.917126i	$-0.630503\pi$
$313$	−14.1038	−0.797194	−0.398597	+	0.917126i	$0.630503\pi$
$314$	0	0
$315$	0.681685	0.0384086
$316$	0	0
$317$	17.3800	0.976160	0.488080	−	0.872799i	$-0.337697\pi$
$317$	17.3800	0.976160	0.488080	+	0.872799i	$0.337697\pi$
$318$	0	0
$319$	17.1975	0.962875
$320$	0	0
$321$	1.44613	0.0807153
$322$	0	0
$323$	9.64209	0.536500
$324$	0	0
$325$	−16.6332	−0.922644
$326$	0	0
$327$	9.11567	0.504097
$328$	0	0
$329$	−6.41527	−0.353685
$330$	0	0
$331$	27.4251	1.50742	0.753710	−	0.657207i	$-0.228262\pi$
$331$	27.4251	1.50742	0.753710	+	0.657207i	$0.228262\pi$
$332$	0	0
$333$	3.93391	0.215577
$334$	0	0
$335$	2.02104	0.110421
$336$	0	0
$337$	24.8803	1.35531	0.677657	−	0.735378i	$-0.262995\pi$
$337$	24.8803	1.35531	0.677657	+	0.735378i	$0.262995\pi$
$338$	0	0
$339$	12.8051	0.695479
$340$	0	0
$341$	−30.5060	−1.65199
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.681685	0.0367007
$346$	0	0
$347$	−5.84380	−0.313711	−0.156856	−	0.987622i	$-0.550136\pi$
$347$	−5.84380	−0.313711	−0.156856	+	0.987622i	$0.550136\pi$
$348$	0	0
$349$	−28.5750	−1.52959	−0.764793	−	0.644276i	$-0.777159\pi$
$349$	−28.5750	−1.52959	−0.764793	+	0.644276i	$0.777159\pi$
$350$	0	0
$351$	3.66749	0.195756
$352$	0	0
$353$	−1.57039	−0.0835833	−0.0417916	−	0.999126i	$-0.513307\pi$
$353$	−1.57039	−0.0835833	−0.0417916	+	0.999126i	$0.513307\pi$
$354$	0	0
$355$	−2.12688	−0.112883
$356$	0	0
$357$	−2.21699	−0.117336
$358$	0	0
$359$	17.1441	0.904828	0.452414	−	0.891808i	$-0.350563\pi$
$359$	17.1441	0.904828	0.452414	+	0.891808i	$0.350563\pi$
$360$	0	0
$361$	−0.0846542	−0.00445549
$362$	0	0
$363$	2.78301	0.146070
$364$	0	0
$365$	−7.00685	−0.366755
$366$	0	0
$367$	6.56850	0.342873	0.171436	−	0.985195i	$-0.445159\pi$
$367$	6.56850	0.342873	0.171436	+	0.985195i	$0.445159\pi$
$368$	0	0
$369$	−5.99563	−0.312120
$370$	0	0
$371$	3.60140	0.186975
$372$	0	0
$373$	14.8449	0.768639	0.384319	−	0.923200i	$-0.374436\pi$
$373$	14.8449	0.768639	0.384319	+	0.923200i	$0.374436\pi$
$374$	0	0
$375$	6.50007	0.335662
$376$	0	0
$377$	−16.9888	−0.874967
$378$	0	0
$379$	−9.81606	−0.504217	−0.252109	−	0.967699i	$-0.581124\pi$
$379$	−9.81606	−0.504217	−0.252109	+	0.967699i	$0.581124\pi$
$380$	0	0
$381$	17.5069	0.896907
$382$	0	0
$383$	−21.4198	−1.09450	−0.547250	−	0.836969i	$-0.684325\pi$
$383$	−21.4198	−1.09450	−0.547250	+	0.836969i	$0.684325\pi$
$384$	0	0
$385$	−2.53079	−0.128981
$386$	0	0
$387$	5.16305	0.262453
$388$	0	0
$389$	4.34044	0.220069	0.110035	−	0.993928i	$-0.464904\pi$
$389$	4.34044	0.220069	0.110035	+	0.993928i	$0.464904\pi$
$390$	0	0
$391$	−2.21699	−0.112118
$392$	0	0
$393$	2.08480	0.105164
$394$	0	0
$395$	7.91689	0.398342
$396$	0	0
$397$	32.8551	1.64895	0.824476	−	0.565897i	$-0.191470\pi$
$397$	32.8551	1.64895	0.824476	+	0.565897i	$0.191470\pi$
$398$	0	0
$399$	−4.34918	−0.217731
$400$	0	0
$401$	12.9153	0.644962	0.322481	−	0.946576i	$-0.395483\pi$
$401$	12.9153	0.644962	0.322481	+	0.946576i	$0.395483\pi$
$402$	0	0
$403$	30.1358	1.50117
$404$	0	0
$405$	−0.681685	−0.0338732
$406$	0	0
$407$	−14.6048	−0.723934
$408$	0	0
$409$	−33.0301	−1.63323	−0.816616	−	0.577182i	$-0.804152\pi$
$409$	−33.0301	−1.63323	−0.816616	+	0.577182i	$0.804152\pi$
$410$	0	0
$411$	−14.9163	−0.735766
$412$	0	0
$413$	−6.83258	−0.336209
$414$	0	0
$415$	−7.87481	−0.386559
$416$	0	0
$417$	5.45050	0.266912
$418$	0	0
$419$	−0.522051	−0.0255039	−0.0127519	−	0.999919i	$-0.504059\pi$
$419$	−0.522051	−0.0255039	−0.0127519	+	0.999919i	$0.504059\pi$
$420$	0	0
$421$	−18.9888	−0.925457	−0.462728	−	0.886500i	$-0.653129\pi$
$421$	−18.9888	−0.925457	−0.462728	+	0.886500i	$0.653129\pi$
$422$	0	0
$423$	6.41527	0.311921
$424$	0	0
$425$	−10.0547	−0.487726
$426$	0	0
$427$	4.71939	0.228387
$428$	0	0
$429$	−13.6157	−0.657374
$430$	0	0
$431$	−34.9978	−1.68579	−0.842893	−	0.538081i	$-0.819149\pi$
$431$	−34.9978	−1.68579	−0.842893	+	0.538081i	$0.819149\pi$
$432$	0	0
$433$	8.71707	0.418915	0.209458	−	0.977818i	$-0.432830\pi$
$433$	8.71707	0.418915	0.209458	+	0.977818i	$0.432830\pi$
$434$	0	0
$435$	3.15774	0.151402
$436$	0	0
$437$	−4.34918	−0.208049
$438$	0	0
$439$	19.3316	0.922645	0.461322	−	0.887233i	$-0.347375\pi$
$439$	19.3316	0.922645	0.461322	+	0.887233i	$0.347375\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−12.7504	−0.605790	−0.302895	−	0.953024i	$-0.597953\pi$
$443$	−12.7504	−0.605790	−0.302895	+	0.953024i	$0.597953\pi$
$444$	0	0
$445$	−1.28760	−0.0610382
$446$	0	0
$447$	7.09900	0.335771
$448$	0	0
$449$	−12.9080	−0.609166	−0.304583	−	0.952486i	$-0.598517\pi$
$449$	−12.9080	−0.609166	−0.304583	+	0.952486i	$0.598517\pi$
$450$	0	0
$451$	22.2591	1.04814
$452$	0	0
$453$	9.29197	0.436575
$454$	0	0
$455$	2.50007	0.117205
$456$	0	0
$457$	−14.4516	−0.676017	−0.338008	−	0.941143i	$-0.609753\pi$
$457$	−14.4516	−0.676017	−0.338008	+	0.941143i	$0.609753\pi$
$458$	0	0
$459$	2.21699	0.103480
$460$	0	0
$461$	−31.5309	−1.46854	−0.734271	−	0.678856i	$-0.762476\pi$
$461$	−31.5309	−1.46854	−0.734271	+	0.678856i	$0.762476\pi$
$462$	0	0
$463$	−13.9672	−0.649113	−0.324557	−	0.945866i	$-0.605215\pi$
$463$	−13.9672	−0.649113	−0.324557	+	0.945866i	$0.605215\pi$
$464$	0	0
$465$	−5.60140	−0.259759
$466$	0	0
$467$	−8.37662	−0.387624	−0.193812	−	0.981039i	$-0.562085\pi$
$467$	−8.37662	−0.387624	−0.193812	+	0.981039i	$0.562085\pi$
$468$	0	0
$469$	2.96477	0.136900
$470$	0	0
$471$	−1.33499	−0.0615129
$472$	0	0
$473$	−19.1681	−0.881349
$474$	0	0
$475$	−19.7248	−0.905038
$476$	0	0
$477$	−3.60140	−0.164897
$478$	0	0
$479$	6.17957	0.282352	0.141176	−	0.989985i	$-0.454912\pi$
$479$	6.17957	0.282352	0.141176	+	0.989985i	$0.454912\pi$
$480$	0	0
$481$	14.4276	0.657841
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−4.99717	−0.226910
$486$	0	0
$487$	5.84567	0.264893	0.132446	−	0.991190i	$-0.457717\pi$
$487$	5.84567	0.264893	0.132446	+	0.991190i	$0.457717\pi$
$488$	0	0
$489$	7.49760	0.339053
$490$	0	0
$491$	−30.0877	−1.35784	−0.678920	−	0.734212i	$-0.737552\pi$
$491$	−30.0877	−1.35784	−0.678920	+	0.734212i	$0.737552\pi$
$492$	0	0
$493$	−10.2697	−0.462523
$494$	0	0
$495$	2.53079	0.113750
$496$	0	0
$497$	−3.12003	−0.139953
$498$	0	0
$499$	15.3086	0.685309	0.342654	−	0.939462i	$-0.388674\pi$
$499$	15.3086	0.685309	0.342654	+	0.939462i	$0.388674\pi$
$500$	0	0
$501$	4.34918	0.194307
$502$	0	0
$503$	15.0402	0.670609	0.335304	−	0.942110i	$-0.391161\pi$
$503$	15.0402	0.670609	0.335304	+	0.942110i	$0.391161\pi$
$504$	0	0
$505$	−3.13065	−0.139312
$506$	0	0
$507$	0.450502	0.0200075
$508$	0	0
$509$	30.4870	1.35131	0.675656	−	0.737217i	$-0.263861\pi$
$509$	30.4870	1.35131	0.675656	+	0.737217i	$0.263861\pi$
$510$	0	0
$511$	−10.2787	−0.454704
$512$	0	0
$513$	4.34918	0.192021
$514$	0	0
$515$	−9.77864	−0.430898
$516$	0	0
$517$	−23.8170	−1.04747
$518$	0	0
$519$	23.7459	1.04233
$520$	0	0
$521$	−10.1500	−0.444678	−0.222339	−	0.974969i	$-0.571369\pi$
$521$	−10.1500	−0.444678	−0.222339	+	0.974969i	$0.571369\pi$
$522$	0	0
$523$	−2.26437	−0.0990142	−0.0495071	−	0.998774i	$-0.515765\pi$
$523$	−2.26437	−0.0990142	−0.0495071	+	0.998774i	$0.515765\pi$
$524$	0	0
$525$	4.53531	0.197937
$526$	0	0
$527$	18.2170	0.793545
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.83258	0.296509
$532$	0	0
$533$	−21.9889	−0.952446
$534$	0	0
$535$	−0.985808	−0.0426202
$536$	0	0
$537$	−6.28512	−0.271223
$538$	0	0
$539$	−3.71255	−0.159911
$540$	0	0
$541$	31.7512	1.36509	0.682545	−	0.730843i	$-0.260873\pi$
$541$	31.7512	1.36509	0.682545	+	0.730843i	$0.260873\pi$
$542$	0	0
$543$	−1.97692	−0.0848378
$544$	0	0
$545$	−6.21401	−0.266179
$546$	0	0
$547$	25.7710	1.10189	0.550944	−	0.834542i	$-0.314268\pi$
$547$	25.7710	1.10189	0.550944	+	0.834542i	$0.314268\pi$
$548$	0	0
$549$	−4.71939	−0.201419
$550$	0	0
$551$	−20.1465	−0.858271
$552$	0	0
$553$	11.6137	0.493865
$554$	0	0
$555$	−2.68168	−0.113831
$556$	0	0
$557$	−1.65613	−0.0701724	−0.0350862	−	0.999384i	$-0.511171\pi$
$557$	−1.65613	−0.0701724	−0.0350862	+	0.999384i	$0.511171\pi$
$558$	0	0
$559$	18.9354	0.800884
$560$	0	0
$561$	−8.23068	−0.347500
$562$	0	0
$563$	−8.00233	−0.337258	−0.168629	−	0.985680i	$-0.553934\pi$
$563$	−8.00233	−0.337258	−0.168629	+	0.985680i	$0.553934\pi$
$564$	0	0
$565$	−8.72907	−0.367235
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−6.58080	−0.275881	−0.137941	−	0.990440i	$-0.544048\pi$
$569$	−6.58080	−0.275881	−0.137941	+	0.990440i	$0.544048\pi$
$570$	0	0
$571$	−34.9820	−1.46395	−0.731976	−	0.681330i	$-0.761402\pi$
$571$	−34.9820	−1.46395	−0.731976	+	0.681330i	$0.761402\pi$
$572$	0	0
$573$	−10.6279	−0.443987
$574$	0	0
$575$	4.53531	0.189135
$576$	0	0
$577$	11.2558	0.468585	0.234292	−	0.972166i	$-0.424723\pi$
$577$	11.2558	0.468585	0.234292	+	0.972166i	$0.424723\pi$
$578$	0	0
$579$	−16.9531	−0.704545
$580$	0	0
$581$	−11.5520	−0.479257
$582$	0	0
$583$	13.3704	0.553744
$584$	0	0
$585$	−2.50007	−0.103365
$586$	0	0
$587$	−16.3805	−0.676095	−0.338047	−	0.941129i	$-0.609766\pi$
$587$	−16.3805	−0.676095	−0.338047	+	0.941129i	$0.609766\pi$
$588$	0	0
$589$	35.7372	1.47252
$590$	0	0
$591$	−6.45596	−0.265563
$592$	0	0
$593$	−11.2920	−0.463706	−0.231853	−	0.972751i	$-0.574479\pi$
$593$	−11.2920	−0.463706	−0.231853	+	0.972751i	$0.574479\pi$
$594$	0	0
$595$	1.51129	0.0619568
$596$	0	0
$597$	15.0256	0.614955
$598$	0	0
$599$	14.0456	0.573889	0.286945	−	0.957947i	$-0.407360\pi$
$599$	14.0456	0.573889	0.286945	+	0.957947i	$0.407360\pi$
$600$	0	0
$601$	6.71596	0.273950	0.136975	−	0.990575i	$-0.456262\pi$
$601$	6.71596	0.273950	0.136975	+	0.990575i	$0.456262\pi$
$602$	0	0
$603$	−2.96477	−0.120735
$604$	0	0
$605$	−1.89714	−0.0771295
$606$	0	0
$607$	41.5453	1.68627	0.843135	−	0.537701i	$-0.180707\pi$
$607$	41.5453	1.68627	0.843135	+	0.537701i	$0.180707\pi$
$608$	0	0
$609$	4.63226	0.187709
$610$	0	0
$611$	23.5280	0.951839
$612$	0	0
$613$	−9.99376	−0.403644	−0.201822	−	0.979422i	$-0.564686\pi$
$613$	−9.99376	−0.403644	−0.201822	+	0.979422i	$0.564686\pi$
$614$	0	0
$615$	4.08713	0.164809
$616$	0	0
$617$	−15.6157	−0.628666	−0.314333	−	0.949313i	$-0.601781\pi$
$617$	−15.6157	−0.628666	−0.314333	+	0.949313i	$0.601781\pi$
$618$	0	0
$619$	4.23272	0.170127	0.0850637	−	0.996376i	$-0.472891\pi$
$619$	4.23272	0.170127	0.0850637	+	0.996376i	$0.472891\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−1.88885	−0.0756752
$624$	0	0
$625$	18.2455	0.729821
$626$	0	0
$627$	−16.1465	−0.644830
$628$	0	0
$629$	8.72143	0.347746
$630$	0	0
$631$	−42.1468	−1.67784	−0.838919	−	0.544256i	$-0.816812\pi$
$631$	−42.1468	−1.67784	−0.838919	+	0.544256i	$0.816812\pi$
$632$	0	0
$633$	−1.68416	−0.0669395
$634$	0	0
$635$	−11.9342	−0.473594
$636$	0	0
$637$	3.66749	0.145311
$638$	0	0
$639$	3.12003	0.123427
$640$	0	0
$641$	−9.66749	−0.381843	−0.190922	−	0.981605i	$-0.561148\pi$
$641$	−9.66749	−0.381843	−0.190922	+	0.981605i	$0.561148\pi$
$642$	0	0
$643$	30.8753	1.21760	0.608802	−	0.793322i	$-0.291651\pi$
$643$	30.8753	1.21760	0.608802	+	0.793322i	$0.291651\pi$
$644$	0	0
$645$	−3.51957	−0.138583
$646$	0	0
$647$	−11.1106	−0.436805	−0.218402	−	0.975859i	$-0.570085\pi$
$647$	−11.1106	−0.436805	−0.218402	+	0.975859i	$0.570085\pi$
$648$	0	0
$649$	−25.3663	−0.995714
$650$	0	0
$651$	−8.21699	−0.322049
$652$	0	0
$653$	40.6203	1.58959	0.794797	−	0.606876i	$-0.207577\pi$
$653$	40.6203	1.58959	0.794797	+	0.606876i	$0.207577\pi$
$654$	0	0
$655$	−1.42118	−0.0555301
$656$	0	0
$657$	10.2787	0.401011
$658$	0	0
$659$	7.13764	0.278043	0.139022	−	0.990289i	$-0.455604\pi$
$659$	7.13764	0.278043	0.139022	+	0.990289i	$0.455604\pi$
$660$	0	0
$661$	−10.5980	−0.412214	−0.206107	−	0.978529i	$-0.566079\pi$
$661$	−10.5980	−0.412214	−0.206107	+	0.978529i	$0.566079\pi$
$662$	0	0
$663$	8.13080	0.315774
$664$	0	0
$665$	2.96477	0.114969
$666$	0	0
$667$	4.63226	0.179362
$668$	0	0
$669$	−0.742472	−0.0287056
$670$	0	0
$671$	17.5210	0.676390
$672$	0	0
$673$	−32.8352	−1.26570	−0.632852	−	0.774273i	$-0.718116\pi$
$673$	−32.8352	−1.26570	−0.632852	+	0.774273i	$0.718116\pi$
$674$	0	0
$675$	−4.53531	−0.174564
$676$	0	0
$677$	26.2542	1.00903	0.504516	−	0.863402i	$-0.331671\pi$
$677$	26.2542	1.00903	0.504516	+	0.863402i	$0.331671\pi$
$678$	0	0
$679$	−7.33062	−0.281323
$680$	0	0
$681$	−20.0500	−0.768318
$682$	0	0
$683$	−12.8203	−0.490554	−0.245277	−	0.969453i	$-0.578879\pi$
$683$	−12.8203	−0.490554	−0.245277	+	0.969453i	$0.578879\pi$
$684$	0	0
$685$	10.1682	0.388507
$686$	0	0
$687$	−11.7710	−0.449092
$688$	0	0
$689$	−13.2081	−0.503189
$690$	0	0
$691$	−47.2581	−1.79778	−0.898892	−	0.438171i	$-0.855626\pi$
$691$	−47.2581	−1.79778	−0.898892	+	0.438171i	$0.855626\pi$
$692$	0	0
$693$	3.71255	0.141028
$694$	0	0
$695$	−3.71553	−0.140938
$696$	0	0
$697$	−13.2923	−0.503480
$698$	0	0
$699$	17.0882	0.646336
$700$	0	0
$701$	14.3948	0.543685	0.271842	−	0.962342i	$-0.412367\pi$
$701$	14.3948	0.543685	0.271842	+	0.962342i	$0.412367\pi$
$702$	0	0
$703$	17.1093	0.645288
$704$	0	0
$705$	−4.37319	−0.164704
$706$	0	0
$707$	−4.59251	−0.172719
$708$	0	0
$709$	−3.40406	−0.127842	−0.0639210	−	0.997955i	$-0.520361\pi$
$709$	−3.40406	−0.127842	−0.0639210	+	0.997955i	$0.520361\pi$
$710$	0	0
$711$	−11.6137	−0.435548
$712$	0	0
$713$	−8.21699	−0.307729
$714$	0	0
$715$	9.28165	0.347114
$716$	0	0
$717$	24.5838	0.918098
$718$	0	0
$719$	38.6326	1.44075	0.720376	−	0.693584i	$-0.243969\pi$
$719$	38.6326	1.44075	0.720376	+	0.693584i	$0.243969\pi$
$720$	0	0
$721$	−14.3448	−0.534229
$722$	0	0
$723$	−12.3963	−0.461022
$724$	0	0
$725$	21.0087	0.780244
$726$	0	0
$727$	17.7830	0.659535	0.329768	−	0.944062i	$-0.393030\pi$
$727$	17.7830	0.659535	0.329768	+	0.944062i	$0.393030\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.4464	0.423362
$732$	0	0
$733$	1.92050	0.0709354	0.0354677	−	0.999371i	$-0.488708\pi$
$733$	1.92050	0.0709354	0.0354677	+	0.999371i	$0.488708\pi$
$734$	0	0
$735$	−0.681685	−0.0251443
$736$	0	0
$737$	11.0068	0.405442
$738$	0	0
$739$	−8.58925	−0.315961	−0.157980	−	0.987442i	$-0.550498\pi$
$739$	−8.58925	−0.315961	−0.157980	+	0.987442i	$0.550498\pi$
$740$	0	0
$741$	15.9506	0.585959
$742$	0	0
$743$	−10.5308	−0.386337	−0.193169	−	0.981166i	$-0.561876\pi$
$743$	−10.5308	−0.386337	−0.193169	+	0.981166i	$0.561876\pi$
$744$	0	0
$745$	−4.83928	−0.177297
$746$	0	0
$747$	11.5520	0.422665
$748$	0	0
$749$	−1.44613	−0.0528406
$750$	0	0
$751$	−30.8659	−1.12631	−0.563157	−	0.826350i	$-0.690413\pi$
$751$	−30.8659	−1.12631	−0.563157	+	0.826350i	$0.690413\pi$
$752$	0	0
$753$	6.40653	0.233467
$754$	0	0
$755$	−6.33420	−0.230525
$756$	0	0
$757$	−6.50567	−0.236453	−0.118226	−	0.992987i	$-0.537721\pi$
$757$	−6.50567	−0.236453	−0.118226	+	0.992987i	$0.537721\pi$
$758$	0	0
$759$	3.71255	0.134757
$760$	0	0
$761$	−32.7554	−1.18738	−0.593692	−	0.804693i	$-0.702330\pi$
$761$	−32.7554	−1.18738	−0.593692	+	0.804693i	$0.702330\pi$
$762$	0	0
$763$	−9.11567	−0.330009
$764$	0	0
$765$	−1.51129	−0.0546408
$766$	0	0
$767$	25.0584	0.904808
$768$	0	0
$769$	19.5298	0.704264	0.352132	−	0.935950i	$-0.385457\pi$
$769$	19.5298	0.704264	0.352132	+	0.935950i	$0.385457\pi$
$770$	0	0
$771$	−29.3262	−1.05616
$772$	0	0
$773$	−46.4062	−1.66911	−0.834557	−	0.550921i	$-0.814276\pi$
$773$	−46.4062	−1.66911	−0.834557	+	0.550921i	$0.814276\pi$
$774$	0	0
$775$	−37.2666	−1.33865
$776$	0	0
$777$	−3.93391	−0.141128
$778$	0	0
$779$	−26.0761	−0.934272
$780$	0	0
$781$	−11.5833	−0.414482
$782$	0	0
$783$	−4.63226	−0.165544
$784$	0	0
$785$	0.910039	0.0324807
$786$	0	0
$787$	30.4937	1.08698	0.543491	−	0.839415i	$-0.317102\pi$
$787$	30.4937	1.08698	0.543491	+	0.839415i	$0.317102\pi$
$788$	0	0
$789$	−21.0905	−0.750843
$790$	0	0
$791$	−12.8051	−0.455298
$792$	0	0
$793$	−17.3083	−0.614637
$794$	0	0
$795$	2.45502	0.0870706
$796$	0	0
$797$	−38.9621	−1.38011	−0.690054	−	0.723758i	$-0.742413\pi$
$797$	−38.9621	−1.38011	−0.690054	+	0.723758i	$0.742413\pi$
$798$	0	0
$799$	14.2226	0.503159
$800$	0	0
$801$	1.88885	0.0667393
$802$	0	0
$803$	−38.1602	−1.34664
$804$	0	0
$805$	−0.681685	−0.0240262
$806$	0	0
$807$	15.7302	0.553728
$808$	0	0
$809$	6.82728	0.240034	0.120017	−	0.992772i	$-0.461705\pi$
$809$	6.82728	0.240034	0.120017	+	0.992772i	$0.461705\pi$
$810$	0	0
$811$	48.1986	1.69248	0.846241	−	0.532801i	$-0.178861\pi$
$811$	48.1986	1.69248	0.846241	+	0.532801i	$0.178861\pi$
$812$	0	0
$813$	21.0055	0.736693
$814$	0	0
$815$	−5.11100	−0.179030
$816$	0	0
$817$	22.4550	0.785602
$818$	0	0
$819$	−3.66749	−0.128153
$820$	0	0
$821$	56.3174	1.96549	0.982745	−	0.184967i	$-0.0592178\pi$
$821$	56.3174	1.96549	0.982745	+	0.184967i	$0.0592178\pi$
$822$	0	0
$823$	7.86484	0.274151	0.137075	−	0.990561i	$-0.456230\pi$
$823$	7.86484	0.274151	0.137075	+	0.990561i	$0.456230\pi$
$824$	0	0
$825$	16.8375	0.586208
$826$	0	0
$827$	−17.8131	−0.619422	−0.309711	−	0.950831i	$-0.600232\pi$
$827$	−17.8131	−0.619422	−0.309711	+	0.950831i	$0.600232\pi$
$828$	0	0
$829$	−53.7175	−1.86569	−0.932843	−	0.360283i	$-0.882680\pi$
$829$	−53.7175	−1.86569	−0.932843	+	0.360283i	$0.882680\pi$
$830$	0	0
$831$	−0.643976	−0.0223393
$832$	0	0
$833$	2.21699	0.0768142
$834$	0	0
$835$	−2.96477	−0.102600
$836$	0	0
$837$	8.21699	0.284021
$838$	0	0
$839$	3.38033	0.116702	0.0583510	−	0.998296i	$-0.481416\pi$
$839$	3.38033	0.116702	0.0583510	+	0.998296i	$0.481416\pi$
$840$	0	0
$841$	−7.54215	−0.260074
$842$	0	0
$843$	−21.8404	−0.752222
$844$	0	0
$845$	−0.307100	−0.0105646
$846$	0	0
$847$	−2.78301	−0.0956253
$848$	0	0
$849$	−6.89431	−0.236612
$850$	0	0
$851$	−3.93391	−0.134853
$852$	0	0
$853$	28.9072	0.989764	0.494882	−	0.868960i	$-0.335211\pi$
$853$	28.9072	0.989764	0.494882	+	0.868960i	$0.335211\pi$
$854$	0	0
$855$	−2.96477	−0.101393
$856$	0	0
$857$	−19.4329	−0.663815	−0.331908	−	0.943312i	$-0.607692\pi$
$857$	−19.4329	−0.663815	−0.331908	+	0.943312i	$0.607692\pi$
$858$	0	0
$859$	−12.8159	−0.437273	−0.218637	−	0.975806i	$-0.570161\pi$
$859$	−12.8159	−0.437273	−0.218637	+	0.975806i	$0.570161\pi$
$860$	0	0
$861$	5.99563	0.204331
$862$	0	0
$863$	−3.25957	−0.110957	−0.0554785	−	0.998460i	$-0.517668\pi$
$863$	−3.25957	−0.110957	−0.0554785	+	0.998460i	$0.517668\pi$
$864$	0	0
$865$	−16.1872	−0.550382
$866$	0	0
$867$	−12.0850	−0.410427
$868$	0	0
$869$	43.1164	1.46262
$870$	0	0
$871$	−10.8733	−0.368427
$872$	0	0
$873$	7.33062	0.248104
$874$	0	0
$875$	−6.50007	−0.219743
$876$	0	0
$877$	−13.9913	−0.472451	−0.236226	−	0.971698i	$-0.575910\pi$
$877$	−13.9913	−0.472451	−0.236226	+	0.971698i	$0.575910\pi$
$878$	0	0
$879$	−5.30164	−0.178820
$880$	0	0
$881$	−0.174264	−0.00587111	−0.00293555	−	0.999996i	$-0.500934\pi$
$881$	−0.174264	−0.00587111	−0.00293555	+	0.999996i	$0.500934\pi$
$882$	0	0
$883$	51.5852	1.73598	0.867989	−	0.496583i	$-0.165412\pi$
$883$	51.5852	1.73598	0.867989	+	0.496583i	$0.165412\pi$
$884$	0	0
$885$	−4.65767	−0.156566
$886$	0	0
$887$	−42.4004	−1.42367	−0.711833	−	0.702348i	$-0.752135\pi$
$887$	−42.4004	−1.42367	−0.711833	+	0.702348i	$0.752135\pi$
$888$	0	0
$889$	−17.5069	−0.587163
$890$	0	0
$891$	−3.71255	−0.124375
$892$	0	0
$893$	27.9012	0.933676
$894$	0	0
$895$	4.28447	0.143214
$896$	0	0
$897$	−3.66749	−0.122454
$898$	0	0
$899$	−38.0633	−1.26948
$900$	0	0
$901$	−7.98427	−0.265995
$902$	0	0
$903$	−5.16305	−0.171816
$904$	0	0
$905$	1.34764	0.0447970
$906$	0	0
$907$	10.1863	0.338230	0.169115	−	0.985596i	$-0.445909\pi$
$907$	10.1863	0.338230	0.169115	+	0.985596i	$0.445909\pi$
$908$	0	0
$909$	4.59251	0.152324
$910$	0	0
$911$	29.4173	0.974639	0.487319	−	0.873224i	$-0.337975\pi$
$911$	29.4173	0.974639	0.487319	+	0.873224i	$0.337975\pi$
$912$	0	0
$913$	−42.8873	−1.41936
$914$	0	0
$915$	3.21714	0.106355
$916$	0	0
$917$	−2.08480	−0.0688463
$918$	0	0
$919$	−42.7597	−1.41051	−0.705257	−	0.708952i	$-0.749168\pi$
$919$	−42.7597	−1.41051	−0.705257	+	0.708952i	$0.749168\pi$
$920$	0	0
$921$	7.87218	0.259397
$922$	0	0
$923$	11.4427	0.376641
$924$	0	0
$925$	−17.8415	−0.586624
$926$	0	0
$927$	14.3448	0.471145
$928$	0	0
$929$	7.12936	0.233907	0.116953	−	0.993137i	$-0.462687\pi$
$929$	7.12936	0.233907	0.116953	+	0.993137i	$0.462687\pi$
$930$	0	0
$931$	4.34918	0.142539
$932$	0	0
$933$	−10.7390	−0.351580
$934$	0	0
$935$	5.61073	0.183491
$936$	0	0
$937$	−51.4494	−1.68078	−0.840389	−	0.541983i	$-0.817674\pi$
$937$	−51.4494	−1.68078	−0.840389	+	0.541983i	$0.817674\pi$
$938$	0	0
$939$	−14.1038	−0.460260
$940$	0	0
$941$	−21.0999	−0.687838	−0.343919	−	0.938999i	$-0.611755\pi$
$941$	−21.0999	−0.687838	−0.343919	+	0.938999i	$0.611755\pi$
$942$	0	0
$943$	5.99563	0.195245
$944$	0	0
$945$	0.681685	0.0221752
$946$	0	0
$947$	6.17333	0.200606	0.100303	−	0.994957i	$-0.468019\pi$
$947$	6.17333	0.200606	0.100303	+	0.994957i	$0.468019\pi$
$948$	0	0
$949$	37.6971	1.22370
$950$	0	0
$951$	17.3800	0.563586
$952$	0	0
$953$	−32.1838	−1.04254	−0.521268	−	0.853393i	$-0.674541\pi$
$953$	−32.1838	−1.04254	−0.521268	+	0.853393i	$0.674541\pi$
$954$	0	0
$955$	7.24488	0.234439
$956$	0	0
$957$	17.1975	0.555916
$958$	0	0
$959$	14.9163	0.481672
$960$	0	0
$961$	36.5189	1.17803
$962$	0	0
$963$	1.44613	0.0466010
$964$	0	0
$965$	11.5566	0.372021
$966$	0	0
$967$	−8.21822	−0.264280	−0.132140	−	0.991231i	$-0.542185\pi$
$967$	−8.21822	−0.264280	−0.132140	+	0.991231i	$0.542185\pi$
$968$	0	0
$969$	9.64209	0.309748
$970$	0	0
$971$	0.611810	0.0196339	0.00981697	−	0.999952i	$-0.496875\pi$
$971$	0.611810	0.0196339	0.00981697	+	0.999952i	$0.496875\pi$
$972$	0	0
$973$	−5.45050	−0.174735
$974$	0	0
$975$	−16.6332	−0.532689
$976$	0	0
$977$	−53.8950	−1.72426	−0.862128	−	0.506691i	$-0.830868\pi$
$977$	−53.8950	−1.72426	−0.862128	+	0.506691i	$0.830868\pi$
$978$	0	0
$979$	−7.01245	−0.224119
$980$	0	0
$981$	9.11567	0.291041
$982$	0	0
$983$	10.9487	0.349209	0.174604	−	0.984639i	$-0.444135\pi$
$983$	10.9487	0.349209	0.174604	+	0.984639i	$0.444135\pi$
$984$	0	0
$985$	4.40093	0.140225
$986$	0	0
$987$	−6.41527	−0.204200
$988$	0	0
$989$	−5.16305	−0.164175
$990$	0	0
$991$	9.22384	0.293005	0.146502	−	0.989210i	$-0.453198\pi$
$991$	9.22384	0.293005	0.146502	+	0.989210i	$0.453198\pi$
$992$	0	0
$993$	27.4251	0.870309
$994$	0	0
$995$	−10.2427	−0.324715
$996$	0	0
$997$	−33.8410	−1.07175	−0.535877	−	0.844296i	$-0.680019\pi$
$997$	−33.8410	−1.07175	−0.535877	+	0.844296i	$0.680019\pi$
$998$	0	0
$999$	3.93391	0.124463

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3864.2.a.t.1.2	✓	4
4.3	odd	2		7728.2.a.bz.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.t.1.2	✓	4	1.1	even	1	trivial
7728.2.a.bz.1.2		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.681685 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.681685 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.71255 q^{11} +3.66749 q^{13} -0.681685 q^{15} +2.21699 q^{17} +4.34918 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.53531 q^{25} +1.00000 q^{27} -4.63226 q^{29} +8.21699 q^{31} -3.71255 q^{33} +0.681685 q^{35} +3.93391 q^{37} +3.66749 q^{39} -5.99563 q^{41} +5.16305 q^{43} -0.681685 q^{45} +6.41527 q^{47} +1.00000 q^{49} +2.21699 q^{51} -3.60140 q^{53} +2.53079 q^{55} +4.34918 q^{57} +6.83258 q^{59} -4.71939 q^{61} -1.00000 q^{63} -2.50007 q^{65} -2.96477 q^{67} -1.00000 q^{69} +3.12003 q^{71} +10.2787 q^{73} -4.53531 q^{75} +3.71255 q^{77} -11.6137 q^{79} +1.00000 q^{81} +11.5520 q^{83} -1.51129 q^{85} -4.63226 q^{87} +1.88885 q^{89} -3.66749 q^{91} +8.21699 q^{93} -2.96477 q^{95} +7.33062 q^{97} -3.71255 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{13} - 2 q^{15} - 10 q^{17} + 4 q^{19} - 4 q^{21} - 4 q^{23} + 4 q^{27} + 11 q^{29} + 14 q^{31} + 2 q^{35} + 13 q^{37} + 2 q^{39} + 7 q^{41} + 12 q^{43} - 2 q^{45} + 15 q^{47} + 4 q^{49} - 10 q^{51} + q^{53} + 31 q^{55} + 4 q^{57} + 5 q^{59} + 3 q^{61} - 4 q^{63} + 25 q^{65} + 5 q^{67} - 4 q^{69} + 5 q^{71} - 6 q^{73} + 26 q^{79} + 4 q^{81} + 2 q^{83} + 5 q^{85} + 11 q^{87} + 7 q^{89} - 2 q^{91} + 14 q^{93} + 5 q^{95} - 27 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^13 - 2 * q^15 - 10 * q^17 + 4 * q^19 - 4 * q^21 - 4 * q^23 + 4 * q^27 + 11 * q^29 + 14 * q^31 + 2 * q^35 + 13 * q^37 + 2 * q^39 + 7 * q^41 + 12 * q^43 - 2 * q^45 + 15 * q^47 + 4 * q^49 - 10 * q^51 + q^53 + 31 * q^55 + 4 * q^57 + 5 * q^59 + 3 * q^61 - 4 * q^63 + 25 * q^65 + 5 * q^67 - 4 * q^69 + 5 * q^71 - 6 * q^73 + 26 * q^79 + 4 * q^81 + 2 * q^83 + 5 * q^85 + 11 * q^87 + 7 * q^89 - 2 * q^91 + 14 * q^93 + 5 * q^95 - 27 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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