LMFDB - Embedded newform 6128.2.a.m.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6128,2,Mod(1,6128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6128, 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("6128.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6128 = 2^{4} \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6128.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:	$48.9323263586$
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} - 5 x^{15} - 19 x^{14} + 120 x^{13} + 105 x^{12} - 1092 x^{11} - 28 x^{10} + 4827 x^{9} + \cdots + 293 $ x^16 - 5x^15 - 19x^14 + 120x^13 + 105x^12 - 1092x^11 - 28x^10 + 4827x^9 - 1227x^8 - 11282x^7 + 3117x^6 + 14029x^5 - 2361x^4 - 8334x^3 + 120x^2 + 1715*x + 293
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1532)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$2.38050$ of defining polynomial
Character	$\chi$	$=$	6128.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.38050 q^{3} +4.04378 q^{5} +2.78993 q^{7} +2.66676 q^{9} -4.46988 q^{11} -4.31680 q^{13} +9.62621 q^{15} -2.15383 q^{17} +8.54424 q^{19} +6.64141 q^{21} +6.75020 q^{23} +11.3522 q^{25} -0.793271 q^{27} +2.42636 q^{29} -4.50352 q^{31} -10.6405 q^{33} +11.2819 q^{35} +5.66652 q^{37} -10.2761 q^{39} -3.29859 q^{41} +4.92655 q^{43} +10.7838 q^{45} +9.35379 q^{47} +0.783691 q^{49} -5.12718 q^{51} -12.5578 q^{53} -18.0752 q^{55} +20.3395 q^{57} -10.0714 q^{59} +5.19987 q^{61} +7.44007 q^{63} -17.4562 q^{65} -5.00463 q^{67} +16.0688 q^{69} +10.2827 q^{71} -5.99128 q^{73} +27.0238 q^{75} -12.4706 q^{77} +14.5735 q^{79} -9.88867 q^{81} +4.74526 q^{83} -8.70961 q^{85} +5.77595 q^{87} +7.15537 q^{89} -12.0435 q^{91} -10.7206 q^{93} +34.5511 q^{95} +8.19376 q^{97} -11.9201 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 5 q^{3} - 3 q^{5} + 18 q^{7} + 15 q^{9} + 6 q^{11} - 14 q^{13} + 18 q^{15} - q^{17} + 18 q^{19} - 12 q^{21} + 20 q^{23} + 11 q^{25} + 20 q^{27} - 13 q^{29} + 19 q^{31} - 12 q^{33} + 22 q^{35} - 23 q^{37}+ \cdots + 16 q^{99}+O(q^{100}) $ 16 * q + 5 * q^3 - 3 * q^5 + 18 * q^7 + 15 * q^9 + 6 * q^11 - 14 * q^13 + 18 * q^15 - q^17 + 18 * q^19 - 12 * q^21 + 20 * q^23 + 11 * q^25 + 20 * q^27 - 13 * q^29 + 19 * q^31 - 12 * q^33 + 22 * q^35 - 23 * q^37 + 28 * q^39 + 53 * q^43 - 21 * q^45 + 27 * q^47 + 26 * q^49 + 26 * q^51 - 17 * q^53 + 36 * q^55 + 3 * q^57 + 22 * q^59 - 7 * q^61 + 56 * q^63 - 8 * q^65 + 65 * q^67 - 10 * q^69 + 28 * q^71 - 12 * q^73 + 16 * q^75 - 19 * q^77 + 41 * q^79 + 4 * q^81 + 27 * q^83 - 56 * q^85 + 55 * q^87 - 2 * q^89 + 19 * q^91 - 35 * q^93 + 40 * q^95 + 5 * q^97 + 16 * q^99

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
$3$	2.38050	1.37438	0.687190	−	0.726478i	$-0.258844\pi$
$3$	2.38050	1.37438	0.687190	+	0.726478i	$0.258844\pi$
$4$	0	0
$5$	4.04378	1.80843	0.904217	−	0.427072i	$-0.140455\pi$
$5$	4.04378	1.80843	0.904217	+	0.427072i	$0.140455\pi$
$6$	0	0
$7$	2.78993	1.05449	0.527247	−	0.849712i	$-0.323224\pi$
$7$	2.78993	1.05449	0.527247	+	0.849712i	$0.323224\pi$
$8$	0	0
$9$	2.66676	0.888921
$10$	0	0
$11$	−4.46988	−1.34772	−0.673859	−	0.738860i	$-0.735365\pi$
$11$	−4.46988	−1.34772	−0.673859	+	0.738860i	$0.735365\pi$
$12$	0	0
$13$	−4.31680	−1.19726	−0.598632	−	0.801024i	$-0.704289\pi$
$13$	−4.31680	−1.19726	−0.598632	+	0.801024i	$0.704289\pi$
$14$	0	0
$15$	9.62621	2.48548
$16$	0	0
$17$	−2.15383	−0.522380	−0.261190	−	0.965287i	$-0.584115\pi$
$17$	−2.15383	−0.522380	−0.261190	+	0.965287i	$0.584115\pi$
$18$	0	0
$19$	8.54424	1.96018	0.980091	−	0.198547i	$-0.0636222\pi$
$19$	8.54424	1.96018	0.980091	+	0.198547i	$0.0636222\pi$
$20$	0	0
$21$	6.64141	1.44927
$22$	0	0
$23$	6.75020	1.40751	0.703756	−	0.710441i	$-0.251505\pi$
$23$	6.75020	1.40751	0.703756	+	0.710441i	$0.251505\pi$
$24$	0	0
$25$	11.3522	2.27044
$26$	0	0
$27$	−0.793271	−0.152665
$28$	0	0
$29$	2.42636	0.450564	0.225282	−	0.974294i	$-0.427670\pi$
$29$	2.42636	0.450564	0.225282	+	0.974294i	$0.427670\pi$
$30$	0	0
$31$	−4.50352	−0.808856	−0.404428	−	0.914570i	$-0.632529\pi$
$31$	−4.50352	−0.808856	−0.404428	+	0.914570i	$0.632529\pi$
$32$	0	0
$33$	−10.6405	−1.85228
$34$	0	0
$35$	11.2819	1.90698
$36$	0	0
$37$	5.66652	0.931571	0.465785	−	0.884898i	$-0.345772\pi$
$37$	5.66652	0.931571	0.465785	+	0.884898i	$0.345772\pi$
$38$	0	0
$39$	−10.2761	−1.64550
$40$	0	0
$41$	−3.29859	−0.515154	−0.257577	−	0.966258i	$-0.582924\pi$
$41$	−3.29859	−0.515154	−0.257577	+	0.966258i	$0.582924\pi$
$42$	0	0
$43$	4.92655	0.751291	0.375646	−	0.926763i	$-0.377421\pi$
$43$	4.92655	0.751291	0.375646	+	0.926763i	$0.377421\pi$
$44$	0	0
$45$	10.7838	1.60756
$46$	0	0
$47$	9.35379	1.36439	0.682195	−	0.731170i	$-0.261025\pi$
$47$	9.35379	1.36439	0.682195	+	0.731170i	$0.261025\pi$
$48$	0	0
$49$	0.783691	0.111956
$50$	0	0
$51$	−5.12718	−0.717948
$52$	0	0
$53$	−12.5578	−1.72494	−0.862472	−	0.506104i	$-0.831085\pi$
$53$	−12.5578	−1.72494	−0.862472	+	0.506104i	$0.831085\pi$
$54$	0	0
$55$	−18.0752	−2.43726
$56$	0	0
$57$	20.3395	2.69404
$58$	0	0
$59$	−10.0714	−1.31119	−0.655595	−	0.755113i	$-0.727582\pi$
$59$	−10.0714	−1.31119	−0.655595	+	0.755113i	$0.727582\pi$
$60$	0	0
$61$	5.19987	0.665775	0.332887	−	0.942967i	$-0.391977\pi$
$61$	5.19987	0.665775	0.332887	+	0.942967i	$0.391977\pi$
$62$	0	0
$63$	7.44007	0.937361
$64$	0	0
$65$	−17.4562	−2.16517
$66$	0	0
$67$	−5.00463	−0.611413	−0.305707	−	0.952126i	$-0.598893\pi$
$67$	−5.00463	−0.611413	−0.305707	+	0.952126i	$0.598893\pi$
$68$	0	0
$69$	16.0688	1.93446
$70$	0	0
$71$	10.2827	1.22033	0.610166	−	0.792274i	$-0.291103\pi$
$71$	10.2827	1.22033	0.610166	+	0.792274i	$0.291103\pi$
$72$	0	0
$73$	−5.99128	−0.701226	−0.350613	−	0.936520i	$-0.614027\pi$
$73$	−5.99128	−0.701226	−0.350613	+	0.936520i	$0.614027\pi$
$74$	0	0
$75$	27.0238	3.12044
$76$	0	0
$77$	−12.4706	−1.42116
$78$	0	0
$79$	14.5735	1.63964	0.819821	−	0.572620i	$-0.194073\pi$
$79$	14.5735	1.63964	0.819821	+	0.572620i	$0.194073\pi$
$80$	0	0
$81$	−9.88867	−1.09874
$82$	0	0
$83$	4.74526	0.520860	0.260430	−	0.965493i	$-0.416136\pi$
$83$	4.74526	0.520860	0.260430	+	0.965493i	$0.416136\pi$
$84$	0	0
$85$	−8.70961	−0.944690
$86$	0	0
$87$	5.77595	0.619247
$88$	0	0
$89$	7.15537	0.758467	0.379234	−	0.925301i	$-0.376188\pi$
$89$	7.15537	0.758467	0.379234	+	0.925301i	$0.376188\pi$
$90$	0	0
$91$	−12.0435	−1.26251
$92$	0	0
$93$	−10.7206	−1.11168
$94$	0	0
$95$	34.5511	3.54486
$96$	0	0
$97$	8.19376	0.831950	0.415975	−	0.909376i	$-0.363440\pi$
$97$	8.19376	0.831950	0.415975	+	0.909376i	$0.363440\pi$
$98$	0	0
$99$	−11.9201	−1.19802
$100$	0	0
$101$	−1.88995	−0.188057	−0.0940285	−	0.995570i	$-0.529974\pi$
$101$	−1.88995	−0.188057	−0.0940285	+	0.995570i	$0.529974\pi$
$102$	0	0
$103$	−7.22991	−0.712384	−0.356192	−	0.934413i	$-0.615925\pi$
$103$	−7.22991	−0.712384	−0.356192	+	0.934413i	$0.615925\pi$
$104$	0	0
$105$	26.8564	2.62092
$106$	0	0
$107$	−3.76456	−0.363934	−0.181967	−	0.983305i	$-0.558246\pi$
$107$	−3.76456	−0.363934	−0.181967	+	0.983305i	$0.558246\pi$
$108$	0	0
$109$	−19.2306	−1.84195	−0.920977	−	0.389618i	$-0.872607\pi$
$109$	−19.2306	−1.84195	−0.920977	+	0.389618i	$0.872607\pi$
$110$	0	0
$111$	13.4891	1.28033
$112$	0	0
$113$	−10.8173	−1.01761	−0.508803	−	0.860883i	$-0.669912\pi$
$113$	−10.8173	−1.01761	−0.508803	+	0.860883i	$0.669912\pi$
$114$	0	0
$115$	27.2963	2.54540
$116$	0	0
$117$	−11.5119	−1.06427
$118$	0	0
$119$	−6.00902	−0.550846
$120$	0	0
$121$	8.97981	0.816346
$122$	0	0
$123$	−7.85229	−0.708017
$124$	0	0
$125$	25.6869	2.29750
$126$	0	0
$127$	19.6650	1.74499	0.872493	−	0.488627i	$-0.162502\pi$
$127$	19.6650	1.74499	0.872493	+	0.488627i	$0.162502\pi$
$128$	0	0
$129$	11.7276	1.03256
$130$	0	0
$131$	13.2426	1.15701	0.578504	−	0.815680i	$-0.303637\pi$
$131$	13.2426	1.15701	0.578504	+	0.815680i	$0.303637\pi$
$132$	0	0
$133$	23.8378	2.06700
$134$	0	0
$135$	−3.20782	−0.276085
$136$	0	0
$137$	−16.8269	−1.43762	−0.718810	−	0.695207i	$-0.755313\pi$
$137$	−16.8269	−1.43762	−0.718810	+	0.695207i	$0.755313\pi$
$138$	0	0
$139$	−17.5665	−1.48997	−0.744986	−	0.667080i	$-0.767544\pi$
$139$	−17.5665	−1.48997	−0.744986	+	0.667080i	$0.767544\pi$
$140$	0	0
$141$	22.2667	1.87519
$142$	0	0
$143$	19.2956	1.61358
$144$	0	0
$145$	9.81169	0.814816
$146$	0	0
$147$	1.86557	0.153870
$148$	0	0
$149$	2.66547	0.218364	0.109182	−	0.994022i	$-0.465177\pi$
$149$	2.66547	0.218364	0.109182	+	0.994022i	$0.465177\pi$
$150$	0	0
$151$	17.3820	1.41453	0.707265	−	0.706949i	$-0.249929\pi$
$151$	17.3820	1.41453	0.707265	+	0.706949i	$0.249929\pi$
$152$	0	0
$153$	−5.74374	−0.464354
$154$	0	0
$155$	−18.2113	−1.46276
$156$	0	0
$157$	16.7488	1.33670	0.668351	−	0.743846i	$-0.267001\pi$
$157$	16.7488	1.33670	0.668351	+	0.743846i	$0.267001\pi$
$158$	0	0
$159$	−29.8938	−2.37073
$160$	0	0
$161$	18.8325	1.48421
$162$	0	0
$163$	13.3462	1.04535	0.522677	−	0.852531i	$-0.324933\pi$
$163$	13.3462	1.04535	0.522677	+	0.852531i	$0.324933\pi$
$164$	0	0
$165$	−43.0280	−3.34972
$166$	0	0
$167$	−19.1412	−1.48119	−0.740593	−	0.671953i	$-0.765456\pi$
$167$	−19.1412	−1.48119	−0.740593	+	0.671953i	$0.765456\pi$
$168$	0	0
$169$	5.63474	0.433442
$170$	0	0
$171$	22.7855	1.74245
$172$	0	0
$173$	−14.8116	−1.12610	−0.563052	−	0.826422i	$-0.690373\pi$
$173$	−14.8116	−1.12610	−0.563052	+	0.826422i	$0.690373\pi$
$174$	0	0
$175$	31.6718	2.39416
$176$	0	0
$177$	−23.9750	−1.80207
$178$	0	0
$179$	−8.71207	−0.651171	−0.325585	−	0.945513i	$-0.605561\pi$
$179$	−8.71207	−0.651171	−0.325585	+	0.945513i	$0.605561\pi$
$180$	0	0
$181$	−18.5073	−1.37563	−0.687817	−	0.725884i	$-0.741431\pi$
$181$	−18.5073	−1.37563	−0.687817	+	0.725884i	$0.741431\pi$
$182$	0	0
$183$	12.3783	0.915028
$184$	0	0
$185$	22.9142	1.68469
$186$	0	0
$187$	9.62734	0.704021
$188$	0	0
$189$	−2.21317	−0.160984
$190$	0	0
$191$	−4.87908	−0.353038	−0.176519	−	0.984297i	$-0.556484\pi$
$191$	−4.87908	−0.353038	−0.176519	+	0.984297i	$0.556484\pi$
$192$	0	0
$193$	16.4546	1.18443	0.592213	−	0.805781i	$-0.298254\pi$
$193$	16.4546	1.18443	0.592213	+	0.805781i	$0.298254\pi$
$194$	0	0
$195$	−41.5544	−2.97577
$196$	0	0
$197$	−1.18437	−0.0843826	−0.0421913	−	0.999110i	$-0.513434\pi$
$197$	−1.18437	−0.0843826	−0.0421913	+	0.999110i	$0.513434\pi$
$198$	0	0
$199$	9.04190	0.640963	0.320482	−	0.947255i	$-0.396155\pi$
$199$	9.04190	0.640963	0.320482	+	0.947255i	$0.396155\pi$
$200$	0	0
$201$	−11.9135	−0.840314
$202$	0	0
$203$	6.76937	0.475117
$204$	0	0
$205$	−13.3388	−0.931622
$206$	0	0
$207$	18.0012	1.25117
$208$	0	0
$209$	−38.1917	−2.64178
$210$	0	0
$211$	7.82219	0.538502	0.269251	−	0.963070i	$-0.413224\pi$
$211$	7.82219	0.538502	0.269251	+	0.963070i	$0.413224\pi$
$212$	0	0
$213$	24.4779	1.67720
$214$	0	0
$215$	19.9219	1.35866
$216$	0	0
$217$	−12.5645	−0.852933
$218$	0	0
$219$	−14.2622	−0.963751
$220$	0	0
$221$	9.29763	0.625426
$222$	0	0
$223$	−27.0520	−1.81153	−0.905767	−	0.423776i	$-0.860704\pi$
$223$	−27.0520	−1.81153	−0.905767	+	0.423776i	$0.860704\pi$
$224$	0	0
$225$	30.2736	2.01824
$226$	0	0
$227$	−0.984948	−0.0653733	−0.0326867	−	0.999466i	$-0.510406\pi$
$227$	−0.984948	−0.0653733	−0.0326867	+	0.999466i	$0.510406\pi$
$228$	0	0
$229$	−3.00397	−0.198508	−0.0992540	−	0.995062i	$-0.531646\pi$
$229$	−3.00397	−0.198508	−0.0992540	+	0.995062i	$0.531646\pi$
$230$	0	0
$231$	−29.6863	−1.95321
$232$	0	0
$233$	−1.63612	−0.107186	−0.0535930	−	0.998563i	$-0.517067\pi$
$233$	−1.63612	−0.107186	−0.0535930	+	0.998563i	$0.517067\pi$
$234$	0	0
$235$	37.8247	2.46741
$236$	0	0
$237$	34.6920	2.25349
$238$	0	0
$239$	−29.4506	−1.90500	−0.952500	−	0.304540i	$-0.901497\pi$
$239$	−29.4506	−1.90500	−0.952500	+	0.304540i	$0.901497\pi$
$240$	0	0
$241$	10.8878	0.701345	0.350672	−	0.936498i	$-0.385953\pi$
$241$	10.8878	0.701345	0.350672	+	0.936498i	$0.385953\pi$
$242$	0	0
$243$	−21.1601	−1.35742
$244$	0	0
$245$	3.16908	0.202465
$246$	0	0
$247$	−36.8837	−2.34686
$248$	0	0
$249$	11.2961	0.715860
$250$	0	0
$251$	7.70049	0.486050	0.243025	−	0.970020i	$-0.421860\pi$
$251$	7.70049	0.486050	0.243025	+	0.970020i	$0.421860\pi$
$252$	0	0
$253$	−30.1725	−1.89693
$254$	0	0
$255$	−20.7332	−1.29836
$256$	0	0
$257$	2.39675	0.149505	0.0747525	−	0.997202i	$-0.476183\pi$
$257$	2.39675	0.149505	0.0747525	+	0.997202i	$0.476183\pi$
$258$	0	0
$259$	15.8092	0.982335
$260$	0	0
$261$	6.47053	0.400516
$262$	0	0
$263$	−1.27804	−0.0788074	−0.0394037	−	0.999223i	$-0.512546\pi$
$263$	−1.27804	−0.0788074	−0.0394037	+	0.999223i	$0.512546\pi$
$264$	0	0
$265$	−50.7810	−3.11945
$266$	0	0
$267$	17.0333	1.04242
$268$	0	0
$269$	11.0291	0.672458	0.336229	−	0.941780i	$-0.390848\pi$
$269$	11.0291	0.672458	0.336229	+	0.941780i	$0.390848\pi$
$270$	0	0
$271$	−20.8958	−1.26933	−0.634666	−	0.772787i	$-0.718862\pi$
$271$	−20.8958	−1.26933	−0.634666	+	0.772787i	$0.718862\pi$
$272$	0	0
$273$	−28.6696	−1.73516
$274$	0	0
$275$	−50.7429	−3.05991
$276$	0	0
$277$	−19.1837	−1.15264	−0.576318	−	0.817225i	$-0.695511\pi$
$277$	−19.1837	−1.15264	−0.576318	+	0.817225i	$0.695511\pi$
$278$	0	0
$279$	−12.0098	−0.719009
$280$	0	0
$281$	−8.84861	−0.527864	−0.263932	−	0.964541i	$-0.585019\pi$
$281$	−8.84861	−0.527864	−0.263932	+	0.964541i	$0.585019\pi$
$282$	0	0
$283$	−17.9862	−1.06917	−0.534584	−	0.845115i	$-0.679532\pi$
$283$	−17.9862	−1.06917	−0.534584	+	0.845115i	$0.679532\pi$
$284$	0	0
$285$	82.2486	4.87199
$286$	0	0
$287$	−9.20284	−0.543226
$288$	0	0
$289$	−12.3610	−0.727120
$290$	0	0
$291$	19.5052	1.14342
$292$	0	0
$293$	−3.80150	−0.222086	−0.111043	−	0.993816i	$-0.535419\pi$
$293$	−3.80150	−0.222086	−0.111043	+	0.993816i	$0.535419\pi$
$294$	0	0
$295$	−40.7267	−2.37120
$296$	0	0
$297$	3.54583	0.205750
$298$	0	0
$299$	−29.1392	−1.68516
$300$	0	0
$301$	13.7447	0.792232
$302$	0	0
$303$	−4.49902	−0.258462
$304$	0	0
$305$	21.0271	1.20401
$306$	0	0
$307$	1.99827	0.114047	0.0570236	−	0.998373i	$-0.481839\pi$
$307$	1.99827	0.114047	0.0570236	+	0.998373i	$0.481839\pi$
$308$	0	0
$309$	−17.2108	−0.979086
$310$	0	0
$311$	−18.5645	−1.05270	−0.526348	−	0.850269i	$-0.676439\pi$
$311$	−18.5645	−1.05270	−0.526348	+	0.850269i	$0.676439\pi$
$312$	0	0
$313$	−8.26229	−0.467012	−0.233506	−	0.972355i	$-0.575020\pi$
$313$	−8.26229	−0.467012	−0.233506	+	0.972355i	$0.575020\pi$
$314$	0	0
$315$	30.0860	1.69516
$316$	0	0
$317$	20.5266	1.15289	0.576443	−	0.817137i	$-0.304440\pi$
$317$	20.5266	1.15289	0.576443	+	0.817137i	$0.304440\pi$
$318$	0	0
$319$	−10.8455	−0.607234
$320$	0	0
$321$	−8.96153	−0.500184
$322$	0	0
$323$	−18.4028	−1.02396
$324$	0	0
$325$	−49.0051	−2.71831
$326$	0	0
$327$	−45.7783	−2.53154
$328$	0	0
$329$	26.0964	1.43874
$330$	0	0
$331$	−32.4095	−1.78139	−0.890694	−	0.454604i	$-0.849781\pi$
$331$	−32.4095	−1.78139	−0.890694	+	0.454604i	$0.849781\pi$
$332$	0	0
$333$	15.1113	0.828093
$334$	0	0
$335$	−20.2376	−1.10570
$336$	0	0
$337$	4.34808	0.236855	0.118427	−	0.992963i	$-0.462215\pi$
$337$	4.34808	0.236855	0.118427	+	0.992963i	$0.462215\pi$
$338$	0	0
$339$	−25.7505	−1.39858
$340$	0	0
$341$	20.1302	1.09011
$342$	0	0
$343$	−17.3430	−0.936436
$344$	0	0
$345$	64.9788	3.49834
$346$	0	0
$347$	28.4894	1.52939	0.764696	−	0.644391i	$-0.222889\pi$
$347$	28.4894	1.52939	0.764696	+	0.644391i	$0.222889\pi$
$348$	0	0
$349$	−2.91987	−0.156297	−0.0781485	−	0.996942i	$-0.524901\pi$
$349$	−2.91987	−0.156297	−0.0781485	+	0.996942i	$0.524901\pi$
$350$	0	0
$351$	3.42439	0.182780
$352$	0	0
$353$	−2.59932	−0.138348	−0.0691740	−	0.997605i	$-0.522036\pi$
$353$	−2.59932	−0.138348	−0.0691740	+	0.997605i	$0.522036\pi$
$354$	0	0
$355$	41.5810	2.20689
$356$	0	0
$357$	−14.3044	−0.757071
$358$	0	0
$359$	−5.95711	−0.314404	−0.157202	−	0.987566i	$-0.550247\pi$
$359$	−5.95711	−0.314404	−0.157202	+	0.987566i	$0.550247\pi$
$360$	0	0
$361$	54.0040	2.84232
$362$	0	0
$363$	21.3764	1.12197
$364$	0	0
$365$	−24.2274	−1.26812
$366$	0	0
$367$	−14.5124	−0.757542	−0.378771	−	0.925490i	$-0.623653\pi$
$367$	−14.5124	−0.757542	−0.378771	+	0.925490i	$0.623653\pi$
$368$	0	0
$369$	−8.79657	−0.457931
$370$	0	0
$371$	−35.0353	−1.81894
$372$	0	0
$373$	7.97191	0.412770	0.206385	−	0.978471i	$-0.433830\pi$
$373$	7.97191	0.412770	0.206385	+	0.978471i	$0.433830\pi$
$374$	0	0
$375$	61.1475	3.15764
$376$	0	0
$377$	−10.4741	−0.539444
$378$	0	0
$379$	20.9066	1.07390	0.536949	−	0.843615i	$-0.319577\pi$
$379$	20.9066	1.07390	0.536949	+	0.843615i	$0.319577\pi$
$380$	0	0
$381$	46.8124	2.39827
$382$	0	0
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	0	0
$385$	−50.4285	−2.57008
$386$	0	0
$387$	13.1379	0.667839
$388$	0	0
$389$	−26.9030	−1.36404	−0.682019	−	0.731334i	$-0.738898\pi$
$389$	−26.9030	−1.36404	−0.682019	+	0.731334i	$0.738898\pi$
$390$	0	0
$391$	−14.5387	−0.735256
$392$	0	0
$393$	31.5239	1.59017
$394$	0	0
$395$	58.9319	2.96519
$396$	0	0
$397$	−17.3967	−0.873116	−0.436558	−	0.899676i	$-0.643803\pi$
$397$	−17.3967	−0.873116	−0.436558	+	0.899676i	$0.643803\pi$
$398$	0	0
$399$	56.7458	2.84084
$400$	0	0
$401$	12.7831	0.638359	0.319180	−	0.947694i	$-0.396593\pi$
$401$	12.7831	0.638359	0.319180	+	0.947694i	$0.396593\pi$
$402$	0	0
$403$	19.4408	0.968415
$404$	0	0
$405$	−39.9876	−1.98700
$406$	0	0
$407$	−25.3287	−1.25550
$408$	0	0
$409$	13.0575	0.645652	0.322826	−	0.946458i	$-0.395367\pi$
$409$	13.0575	0.645652	0.322826	+	0.946458i	$0.395367\pi$
$410$	0	0
$411$	−40.0564	−1.97584
$412$	0	0
$413$	−28.0986	−1.38264
$414$	0	0
$415$	19.1888	0.941942
$416$	0	0
$417$	−41.8170	−2.04779
$418$	0	0
$419$	20.7601	1.01420	0.507099	−	0.861888i	$-0.330718\pi$
$419$	20.7601	1.01420	0.507099	+	0.861888i	$0.330718\pi$
$420$	0	0
$421$	−2.49579	−0.121637	−0.0608187	−	0.998149i	$-0.519371\pi$
$421$	−2.49579	−0.121637	−0.0608187	+	0.998149i	$0.519371\pi$
$422$	0	0
$423$	24.9443	1.21284
$424$	0	0
$425$	−24.4506	−1.18603
$426$	0	0
$427$	14.5072	0.702055
$428$	0	0
$429$	45.9330	2.21767
$430$	0	0
$431$	5.76858	0.277863	0.138931	−	0.990302i	$-0.455633\pi$
$431$	5.76858	0.277863	0.138931	+	0.990302i	$0.455633\pi$
$432$	0	0
$433$	7.12656	0.342481	0.171240	−	0.985229i	$-0.445223\pi$
$433$	7.12656	0.342481	0.171240	+	0.985229i	$0.445223\pi$
$434$	0	0
$435$	23.3567	1.11987
$436$	0	0
$437$	57.6753	2.75898
$438$	0	0
$439$	−4.09215	−0.195308	−0.0976538	−	0.995220i	$-0.531134\pi$
$439$	−4.09215	−0.195308	−0.0976538	+	0.995220i	$0.531134\pi$
$440$	0	0
$441$	2.08992	0.0995199
$442$	0	0
$443$	13.2905	0.631451	0.315726	−	0.948851i	$-0.397752\pi$
$443$	13.2905	0.631451	0.315726	+	0.948851i	$0.397752\pi$
$444$	0	0
$445$	28.9347	1.37164
$446$	0	0
$447$	6.34514	0.300115
$448$	0	0
$449$	30.3153	1.43067	0.715333	−	0.698784i	$-0.246275\pi$
$449$	30.3153	1.43067	0.715333	+	0.698784i	$0.246275\pi$
$450$	0	0
$451$	14.7443	0.694283
$452$	0	0
$453$	41.3779	1.94410
$454$	0	0
$455$	−48.7015	−2.28316
$456$	0	0
$457$	3.35955	0.157153	0.0785765	−	0.996908i	$-0.474963\pi$
$457$	3.35955	0.157153	0.0785765	+	0.996908i	$0.474963\pi$
$458$	0	0
$459$	1.70857	0.0797492
$460$	0	0
$461$	−28.0322	−1.30559	−0.652795	−	0.757534i	$-0.726404\pi$
$461$	−28.0322	−1.30559	−0.652795	+	0.757534i	$0.726404\pi$
$462$	0	0
$463$	25.6607	1.19255	0.596277	−	0.802779i	$-0.296646\pi$
$463$	25.6607	1.19255	0.596277	+	0.802779i	$0.296646\pi$
$464$	0	0
$465$	−43.3518	−2.01039
$466$	0	0
$467$	−1.23076	−0.0569526	−0.0284763	−	0.999594i	$-0.509066\pi$
$467$	−1.23076	−0.0569526	−0.0284763	+	0.999594i	$0.509066\pi$
$468$	0	0
$469$	−13.9626	−0.644731
$470$	0	0
$471$	39.8705	1.83714
$472$	0	0
$473$	−22.0211	−1.01253
$474$	0	0
$475$	96.9958	4.45047
$476$	0	0
$477$	−33.4886	−1.53334
$478$	0	0
$479$	11.6237	0.531100	0.265550	−	0.964097i	$-0.414447\pi$
$479$	11.6237	0.531100	0.265550	+	0.964097i	$0.414447\pi$
$480$	0	0
$481$	−24.4612	−1.11534
$482$	0	0
$483$	44.8308	2.03987
$484$	0	0
$485$	33.1338	1.50453
$486$	0	0
$487$	−1.67659	−0.0759737	−0.0379869	−	0.999278i	$-0.512095\pi$
$487$	−1.67659	−0.0759737	−0.0379869	+	0.999278i	$0.512095\pi$
$488$	0	0
$489$	31.7706	1.43671
$490$	0	0
$491$	4.35654	0.196608	0.0983040	−	0.995156i	$-0.468658\pi$
$491$	4.35654	0.196608	0.0983040	+	0.995156i	$0.468658\pi$
$492$	0	0
$493$	−5.22596	−0.235366
$494$	0	0
$495$	−48.2023	−2.16653
$496$	0	0
$497$	28.6880	1.28683
$498$	0	0
$499$	0.650046	0.0291001	0.0145500	−	0.999894i	$-0.495368\pi$
$499$	0.650046	0.0291001	0.0145500	+	0.999894i	$0.495368\pi$
$500$	0	0
$501$	−45.5654	−2.03571
$502$	0	0
$503$	19.3179	0.861344	0.430672	−	0.902508i	$-0.358277\pi$
$503$	19.3179	0.861344	0.430672	+	0.902508i	$0.358277\pi$
$504$	0	0
$505$	−7.64254	−0.340089
$506$	0	0
$507$	13.4135	0.595713
$508$	0	0
$509$	−22.3434	−0.990355	−0.495177	−	0.868792i	$-0.664897\pi$
$509$	−22.3434	−0.990355	−0.495177	+	0.868792i	$0.664897\pi$
$510$	0	0
$511$	−16.7152	−0.739438
$512$	0	0
$513$	−6.77790	−0.299252
$514$	0	0
$515$	−29.2362	−1.28830
$516$	0	0
$517$	−41.8103	−1.83882
$518$	0	0
$519$	−35.2589	−1.54769
$520$	0	0
$521$	−31.6603	−1.38706	−0.693532	−	0.720426i	$-0.743946\pi$
$521$	−31.6603	−1.38706	−0.693532	+	0.720426i	$0.743946\pi$
$522$	0	0
$523$	−39.5563	−1.72968	−0.864839	−	0.502049i	$-0.832580\pi$
$523$	−39.5563	−1.72968	−0.864839	+	0.502049i	$0.832580\pi$
$524$	0	0
$525$	75.3945	3.29049
$526$	0	0
$527$	9.69980	0.422530
$528$	0	0
$529$	22.5651	0.981093
$530$	0	0
$531$	−26.8581	−1.16554
$532$	0	0
$533$	14.2394	0.616775
$534$	0	0
$535$	−15.2231	−0.658151
$536$	0	0
$537$	−20.7391	−0.894956
$538$	0	0
$539$	−3.50300	−0.150885
$540$	0	0
$541$	17.6660	0.759521	0.379760	−	0.925085i	$-0.376006\pi$
$541$	17.6660	0.759521	0.379760	+	0.925085i	$0.376006\pi$
$542$	0	0
$543$	−44.0565	−1.89064
$544$	0	0
$545$	−77.7642	−3.33105
$546$	0	0
$547$	12.0615	0.515713	0.257856	−	0.966183i	$-0.416984\pi$
$547$	12.0615	0.515713	0.257856	+	0.966183i	$0.416984\pi$
$548$	0	0
$549$	13.8668	0.591821
$550$	0	0
$551$	20.7314	0.883188
$552$	0	0
$553$	40.6589	1.72899
$554$	0	0
$555$	54.5472	2.31540
$556$	0	0
$557$	14.3543	0.608209	0.304105	−	0.952639i	$-0.401643\pi$
$557$	14.3543	0.608209	0.304105	+	0.952639i	$0.401643\pi$
$558$	0	0
$559$	−21.2669	−0.899494
$560$	0	0
$561$	22.9178	0.967592
$562$	0	0
$563$	−26.1327	−1.10136	−0.550682	−	0.834715i	$-0.685632\pi$
$563$	−26.1327	−1.10136	−0.550682	+	0.834715i	$0.685632\pi$
$564$	0	0
$565$	−43.7428	−1.84027
$566$	0	0
$567$	−27.5887	−1.15861
$568$	0	0
$569$	−25.4846	−1.06837	−0.534186	−	0.845367i	$-0.679382\pi$
$569$	−25.4846	−1.06837	−0.534186	+	0.845367i	$0.679382\pi$
$570$	0	0
$571$	−5.48714	−0.229630	−0.114815	−	0.993387i	$-0.536627\pi$
$571$	−5.48714	−0.229630	−0.114815	+	0.993387i	$0.536627\pi$
$572$	0	0
$573$	−11.6146	−0.485209
$574$	0	0
$575$	76.6295	3.19567
$576$	0	0
$577$	16.7915	0.699038	0.349519	−	0.936929i	$-0.386345\pi$
$577$	16.7915	0.699038	0.349519	+	0.936929i	$0.386345\pi$
$578$	0	0
$579$	39.1701	1.62785
$580$	0	0
$581$	13.2389	0.549244
$582$	0	0
$583$	56.1318	2.32474
$584$	0	0
$585$	−46.5515	−1.92467
$586$	0	0
$587$	9.38292	0.387274	0.193637	−	0.981073i	$-0.437972\pi$
$587$	9.38292	0.387274	0.193637	+	0.981073i	$0.437972\pi$
$588$	0	0
$589$	−38.4792	−1.58551
$590$	0	0
$591$	−2.81938	−0.115974
$592$	0	0
$593$	−5.11281	−0.209958	−0.104979	−	0.994474i	$-0.533478\pi$
$593$	−5.11281	−0.209958	−0.104979	+	0.994474i	$0.533478\pi$
$594$	0	0
$595$	−24.2992	−0.996169
$596$	0	0
$597$	21.5242	0.880927
$598$	0	0
$599$	−41.0792	−1.67845	−0.839226	−	0.543782i	$-0.816992\pi$
$599$	−41.0792	−1.67845	−0.839226	+	0.543782i	$0.816992\pi$
$600$	0	0
$601$	−2.58762	−0.105551	−0.0527756	−	0.998606i	$-0.516807\pi$
$601$	−2.58762	−0.105551	−0.0527756	+	0.998606i	$0.516807\pi$
$602$	0	0
$603$	−13.3462	−0.543498
$604$	0	0
$605$	36.3124	1.47631
$606$	0	0
$607$	−18.6836	−0.758344	−0.379172	−	0.925326i	$-0.623791\pi$
$607$	−18.6836	−0.758344	−0.379172	+	0.925326i	$0.623791\pi$
$608$	0	0
$609$	16.1145	0.652991
$610$	0	0
$611$	−40.3784	−1.63354
$612$	0	0
$613$	12.1431	0.490456	0.245228	−	0.969465i	$-0.421137\pi$
$613$	12.1431	0.490456	0.245228	+	0.969465i	$0.421137\pi$
$614$	0	0
$615$	−31.7530	−1.28040
$616$	0	0
$617$	−4.43471	−0.178535	−0.0892673	−	0.996008i	$-0.528453\pi$
$617$	−4.43471	−0.178535	−0.0892673	+	0.996008i	$0.528453\pi$
$618$	0	0
$619$	4.16552	0.167426	0.0837132	−	0.996490i	$-0.473322\pi$
$619$	4.16552	0.167426	0.0837132	+	0.996490i	$0.473322\pi$
$620$	0	0
$621$	−5.35474	−0.214878
$622$	0	0
$623$	19.9629	0.799798
$624$	0	0
$625$	47.1112	1.88445
$626$	0	0
$627$	−90.9152	−3.63080
$628$	0	0
$629$	−12.2047	−0.486634
$630$	0	0
$631$	−1.02429	−0.0407762	−0.0203881	−	0.999792i	$-0.506490\pi$
$631$	−1.02429	−0.0407762	−0.0203881	+	0.999792i	$0.506490\pi$
$632$	0	0
$633$	18.6207	0.740106
$634$	0	0
$635$	79.5210	3.15569
$636$	0	0
$637$	−3.38304	−0.134041
$638$	0	0
$639$	27.4215	1.08478
$640$	0	0
$641$	24.1346	0.953260	0.476630	−	0.879104i	$-0.341858\pi$
$641$	24.1346	0.953260	0.476630	+	0.879104i	$0.341858\pi$
$642$	0	0
$643$	−8.95871	−0.353297	−0.176649	−	0.984274i	$-0.556526\pi$
$643$	−8.95871	−0.353297	−0.176649	+	0.984274i	$0.556526\pi$
$644$	0	0
$645$	47.4240	1.86732
$646$	0	0
$647$	−25.2632	−0.993199	−0.496600	−	0.867980i	$-0.665418\pi$
$647$	−25.2632	−0.993199	−0.496600	+	0.867980i	$0.665418\pi$
$648$	0	0
$649$	45.0181	1.76711
$650$	0	0
$651$	−29.9097	−1.17225
$652$	0	0
$653$	−13.2596	−0.518890	−0.259445	−	0.965758i	$-0.583540\pi$
$653$	−13.2596	−0.518890	−0.259445	+	0.965758i	$0.583540\pi$
$654$	0	0
$655$	53.5500	2.09237
$656$	0	0
$657$	−15.9773	−0.623334
$658$	0	0
$659$	−13.0807	−0.509553	−0.254776	−	0.967000i	$-0.582002\pi$
$659$	−13.0807	−0.509553	−0.254776	+	0.967000i	$0.582002\pi$
$660$	0	0
$661$	−32.5761	−1.26706	−0.633531	−	0.773717i	$-0.718395\pi$
$661$	−32.5761	−1.26706	−0.633531	+	0.773717i	$0.718395\pi$
$662$	0	0
$663$	22.1330	0.859574
$664$	0	0
$665$	96.3949	3.73803
$666$	0	0
$667$	16.3784	0.634175
$668$	0	0
$669$	−64.3971	−2.48974
$670$	0	0
$671$	−23.2428	−0.897277
$672$	0	0
$673$	−5.54244	−0.213645	−0.106823	−	0.994278i	$-0.534068\pi$
$673$	−5.54244	−0.213645	−0.106823	+	0.994278i	$0.534068\pi$
$674$	0	0
$675$	−9.00536	−0.346617
$676$	0	0
$677$	8.88977	0.341661	0.170831	−	0.985300i	$-0.445355\pi$
$677$	8.88977	0.341661	0.170831	+	0.985300i	$0.445355\pi$
$678$	0	0
$679$	22.8600	0.877286
$680$	0	0
$681$	−2.34467	−0.0898478
$682$	0	0
$683$	9.44506	0.361405	0.180703	−	0.983538i	$-0.442163\pi$
$683$	9.44506	0.361405	0.180703	+	0.983538i	$0.442163\pi$
$684$	0	0
$685$	−68.0444	−2.59984
$686$	0	0
$687$	−7.15094	−0.272826
$688$	0	0
$689$	54.2094	2.06521
$690$	0	0
$691$	−48.3900	−1.84084	−0.920422	−	0.390927i	$-0.872154\pi$
$691$	−48.3900	−1.84084	−0.920422	+	0.390927i	$0.872154\pi$
$692$	0	0
$693$	−33.2562	−1.26330
$694$	0	0
$695$	−71.0351	−2.69452
$696$	0	0
$697$	7.10460	0.269106
$698$	0	0
$699$	−3.89479	−0.147314
$700$	0	0
$701$	37.9341	1.43275	0.716376	−	0.697714i	$-0.245799\pi$
$701$	37.9341	1.43275	0.716376	+	0.697714i	$0.245799\pi$
$702$	0	0
$703$	48.4161	1.82605
$704$	0	0
$705$	90.0416	3.39116
$706$	0	0
$707$	−5.27282	−0.198305
$708$	0	0
$709$	−25.7664	−0.967677	−0.483839	−	0.875157i	$-0.660758\pi$
$709$	−25.7664	−0.967677	−0.483839	+	0.875157i	$0.660758\pi$
$710$	0	0
$711$	38.8639	1.45751
$712$	0	0
$713$	−30.3996	−1.13848
$714$	0	0
$715$	78.0271	2.91805
$716$	0	0
$717$	−70.1070	−2.61819
$718$	0	0
$719$	−38.0635	−1.41953	−0.709765	−	0.704438i	$-0.751199\pi$
$719$	−38.0635	−1.41953	−0.709765	+	0.704438i	$0.751199\pi$
$720$	0	0
$721$	−20.1709	−0.751204
$722$	0	0
$723$	25.9184	0.963914
$724$	0	0
$725$	27.5445	1.02298
$726$	0	0
$727$	−23.1162	−0.857332	−0.428666	−	0.903463i	$-0.641016\pi$
$727$	−23.1162	−0.857332	−0.428666	+	0.903463i	$0.641016\pi$
$728$	0	0
$729$	−20.7056	−0.766873
$730$	0	0
$731$	−10.6109	−0.392459
$732$	0	0
$733$	−18.1098	−0.668900	−0.334450	−	0.942414i	$-0.608550\pi$
$733$	−18.1098	−0.668900	−0.334450	+	0.942414i	$0.608550\pi$
$734$	0	0
$735$	7.54397	0.278264
$736$	0	0
$737$	22.3701	0.824013
$738$	0	0
$739$	18.6316	0.685374	0.342687	−	0.939450i	$-0.388663\pi$
$739$	18.6316	0.685374	0.342687	+	0.939450i	$0.388663\pi$
$740$	0	0
$741$	−87.8016	−3.22547
$742$	0	0
$743$	10.4833	0.384596	0.192298	−	0.981337i	$-0.438406\pi$
$743$	10.4833	0.384596	0.192298	+	0.981337i	$0.438406\pi$
$744$	0	0
$745$	10.7786	0.394897
$746$	0	0
$747$	12.6545	0.463003
$748$	0	0
$749$	−10.5029	−0.383766
$750$	0	0
$751$	−0.549296	−0.0200441	−0.0100221	−	0.999950i	$-0.503190\pi$
$751$	−0.549296	−0.0200441	−0.0100221	+	0.999950i	$0.503190\pi$
$752$	0	0
$753$	18.3310	0.668018
$754$	0	0
$755$	70.2892	2.55809
$756$	0	0
$757$	11.3745	0.413415	0.206708	−	0.978403i	$-0.433725\pi$
$757$	11.3745	0.413415	0.206708	+	0.978403i	$0.433725\pi$
$758$	0	0
$759$	−71.8256	−2.60711
$760$	0	0
$761$	0.184189	0.00667684	0.00333842	−	0.999994i	$-0.498937\pi$
$761$	0.184189	0.00667684	0.00333842	+	0.999994i	$0.498937\pi$
$762$	0	0
$763$	−53.6518	−1.94233
$764$	0	0
$765$	−23.2265	−0.839754
$766$	0	0
$767$	43.4763	1.56984
$768$	0	0
$769$	−7.87222	−0.283879	−0.141940	−	0.989875i	$-0.545334\pi$
$769$	−7.87222	−0.283879	−0.141940	+	0.989875i	$0.545334\pi$
$770$	0	0
$771$	5.70545	0.205477
$772$	0	0
$773$	−16.8402	−0.605701	−0.302850	−	0.953038i	$-0.597938\pi$
$773$	−16.8402	−0.605701	−0.302850	+	0.953038i	$0.597938\pi$
$774$	0	0
$775$	−51.1248	−1.83646
$776$	0	0
$777$	37.6337	1.35010
$778$	0	0
$779$	−28.1840	−1.00980
$780$	0	0
$781$	−45.9624	−1.64466
$782$	0	0
$783$	−1.92476	−0.0687855
$784$	0	0
$785$	67.7286	2.41734
$786$	0	0
$787$	20.4008	0.727212	0.363606	−	0.931553i	$-0.381545\pi$
$787$	20.4008	0.727212	0.363606	+	0.931553i	$0.381545\pi$
$788$	0	0
$789$	−3.04237	−0.108311
$790$	0	0
$791$	−30.1794	−1.07306
$792$	0	0
$793$	−22.4468	−0.797108
$794$	0	0
$795$	−120.884	−4.28731
$796$	0	0
$797$	18.6999	0.662386	0.331193	−	0.943563i	$-0.392549\pi$
$797$	18.6999	0.662386	0.331193	+	0.943563i	$0.392549\pi$
$798$	0	0
$799$	−20.1464	−0.712730
$800$	0	0
$801$	19.0817	0.674217
$802$	0	0
$803$	26.7803	0.945056
$804$	0	0
$805$	76.1548	2.68410
$806$	0	0
$807$	26.2548	0.924213
$808$	0	0
$809$	−13.3498	−0.469353	−0.234676	−	0.972074i	$-0.575403\pi$
$809$	−13.3498	−0.469353	−0.234676	+	0.972074i	$0.575403\pi$
$810$	0	0
$811$	34.4575	1.20997	0.604983	−	0.796238i	$-0.293180\pi$
$811$	34.4575	1.20997	0.604983	+	0.796238i	$0.293180\pi$
$812$	0	0
$813$	−49.7425	−1.74454
$814$	0	0
$815$	53.9691	1.89046
$816$	0	0
$817$	42.0936	1.47267
$818$	0	0
$819$	−32.1173	−1.12227
$820$	0	0
$821$	46.7732	1.63240	0.816198	−	0.577773i	$-0.196078\pi$
$821$	46.7732	1.63240	0.816198	+	0.577773i	$0.196078\pi$
$822$	0	0
$823$	46.6188	1.62503	0.812515	−	0.582940i	$-0.198098\pi$
$823$	46.6188	1.62503	0.812515	+	0.582940i	$0.198098\pi$
$824$	0	0
$825$	−120.793	−4.20548
$826$	0	0
$827$	−16.8462	−0.585799	−0.292899	−	0.956143i	$-0.594620\pi$
$827$	−16.8462	−0.585799	−0.292899	+	0.956143i	$0.594620\pi$
$828$	0	0
$829$	22.6075	0.785192	0.392596	−	0.919711i	$-0.371577\pi$
$829$	22.6075	0.785192	0.392596	+	0.919711i	$0.371577\pi$
$830$	0	0
$831$	−45.6667	−1.58416
$832$	0	0
$833$	−1.68793	−0.0584835
$834$	0	0
$835$	−77.4027	−2.67863
$836$	0	0
$837$	3.57251	0.123484
$838$	0	0
$839$	28.6006	0.987402	0.493701	−	0.869632i	$-0.335644\pi$
$839$	28.6006	0.987402	0.493701	+	0.869632i	$0.335644\pi$
$840$	0	0
$841$	−23.1128	−0.796992
$842$	0	0
$843$	−21.0641	−0.725486
$844$	0	0
$845$	22.7857	0.783851
$846$	0	0
$847$	25.0530	0.860831
$848$	0	0
$849$	−42.8161	−1.46944
$850$	0	0
$851$	38.2501	1.31120
$852$	0	0
$853$	−12.7677	−0.437157	−0.218578	−	0.975819i	$-0.570142\pi$
$853$	−12.7677	−0.437157	−0.218578	+	0.975819i	$0.570142\pi$
$854$	0	0
$855$	92.1394	3.15110
$856$	0	0
$857$	−9.77950	−0.334061	−0.167031	−	0.985952i	$-0.553418\pi$
$857$	−9.77950	−0.334061	−0.167031	+	0.985952i	$0.553418\pi$
$858$	0	0
$859$	18.7336	0.639181	0.319591	−	0.947556i	$-0.396455\pi$
$859$	18.7336	0.639181	0.319591	+	0.947556i	$0.396455\pi$
$860$	0	0
$861$	−21.9073	−0.746599
$862$	0	0
$863$	−51.9818	−1.76948	−0.884741	−	0.466083i	$-0.845665\pi$
$863$	−51.9818	−1.76948	−0.884741	+	0.466083i	$0.845665\pi$
$864$	0	0
$865$	−59.8948	−2.03649
$866$	0	0
$867$	−29.4254	−0.999339
$868$	0	0
$869$	−65.1416	−2.20978
$870$	0	0
$871$	21.6040	0.732023
$872$	0	0
$873$	21.8508	0.739538
$874$	0	0
$875$	71.6645	2.42270
$876$	0	0
$877$	6.96991	0.235357	0.117679	−	0.993052i	$-0.462455\pi$
$877$	6.96991	0.235357	0.117679	+	0.993052i	$0.462455\pi$
$878$	0	0
$879$	−9.04946	−0.305231
$880$	0	0
$881$	−9.71126	−0.327181	−0.163590	−	0.986528i	$-0.552307\pi$
$881$	−9.71126	−0.327181	−0.163590	+	0.986528i	$0.552307\pi$
$882$	0	0
$883$	31.2027	1.05006	0.525028	−	0.851085i	$-0.324055\pi$
$883$	31.2027	1.05006	0.525028	+	0.851085i	$0.324055\pi$
$884$	0	0
$885$	−96.9498	−3.25893
$886$	0	0
$887$	9.79707	0.328953	0.164477	−	0.986381i	$-0.447406\pi$
$887$	9.79707	0.328953	0.164477	+	0.986381i	$0.447406\pi$
$888$	0	0
$889$	54.8639	1.84008
$890$	0	0
$891$	44.2011	1.48079
$892$	0	0
$893$	79.9210	2.67446
$894$	0	0
$895$	−35.2297	−1.17760
$896$	0	0
$897$	−69.3658	−2.31606
$898$	0	0
$899$	−10.9272	−0.364442
$900$	0	0
$901$	27.0473	0.901076
$902$	0	0
$903$	32.7192	1.08883
$904$	0	0
$905$	−74.8393	−2.48774
$906$	0	0
$907$	−58.2183	−1.93311	−0.966553	−	0.256466i	$-0.917442\pi$
$907$	−58.2183	−1.93311	−0.966553	+	0.256466i	$0.917442\pi$
$908$	0	0
$909$	−5.04004	−0.167168
$910$	0	0
$911$	43.7554	1.44968	0.724841	−	0.688916i	$-0.241913\pi$
$911$	43.7554	1.44968	0.724841	+	0.688916i	$0.241913\pi$
$912$	0	0
$913$	−21.2107	−0.701973
$914$	0	0
$915$	50.0550	1.65477
$916$	0	0
$917$	36.9458	1.22006
$918$	0	0
$919$	29.5558	0.974956	0.487478	−	0.873135i	$-0.337917\pi$
$919$	29.5558	0.974956	0.487478	+	0.873135i	$0.337917\pi$
$920$	0	0
$921$	4.75687	0.156744
$922$	0	0
$923$	−44.3883	−1.46106
$924$	0	0
$925$	64.3274	2.11507
$926$	0	0
$927$	−19.2804	−0.633253
$928$	0	0
$929$	−5.71408	−0.187473	−0.0937364	−	0.995597i	$-0.529881\pi$
$929$	−5.71408	−0.187473	−0.0937364	+	0.995597i	$0.529881\pi$
$930$	0	0
$931$	6.69604	0.219454
$932$	0	0
$933$	−44.1927	−1.44680
$934$	0	0
$935$	38.9309	1.27318
$936$	0	0
$937$	34.3809	1.12318	0.561588	−	0.827417i	$-0.310191\pi$
$937$	34.3809	1.12318	0.561588	+	0.827417i	$0.310191\pi$
$938$	0	0
$939$	−19.6683	−0.641852
$940$	0	0
$941$	−7.35681	−0.239825	−0.119913	−	0.992784i	$-0.538261\pi$
$941$	−7.35681	−0.239825	−0.119913	+	0.992784i	$0.538261\pi$
$942$	0	0
$943$	−22.2662	−0.725086
$944$	0	0
$945$	−8.94958	−0.291130
$946$	0	0
$947$	26.4564	0.859717	0.429859	−	0.902896i	$-0.358563\pi$
$947$	26.4564	0.859717	0.429859	+	0.902896i	$0.358563\pi$
$948$	0	0
$949$	25.8631	0.839553
$950$	0	0
$951$	48.8634	1.58450
$952$	0	0
$953$	31.7894	1.02976	0.514879	−	0.857263i	$-0.327837\pi$
$953$	31.7894	1.02976	0.514879	+	0.857263i	$0.327837\pi$
$954$	0	0
$955$	−19.7300	−0.638446
$956$	0	0
$957$	−25.8178	−0.834570
$958$	0	0
$959$	−46.9458	−1.51596
$960$	0	0
$961$	−10.7183	−0.345752
$962$	0	0
$963$	−10.0392	−0.323509
$964$	0	0
$965$	66.5387	2.14196
$966$	0	0
$967$	6.40932	0.206110	0.103055	−	0.994676i	$-0.467138\pi$
$967$	6.40932	0.206110	0.103055	+	0.994676i	$0.467138\pi$
$968$	0	0
$969$	−43.8078	−1.40731
$970$	0	0
$971$	17.4388	0.559639	0.279820	−	0.960053i	$-0.409725\pi$
$971$	17.4388	0.559639	0.279820	+	0.960053i	$0.409725\pi$
$972$	0	0
$973$	−49.0093	−1.57116
$974$	0	0
$975$	−116.656	−3.73600
$976$	0	0
$977$	−24.7820	−0.792845	−0.396423	−	0.918068i	$-0.629748\pi$
$977$	−24.7820	−0.792845	−0.396423	+	0.918068i	$0.629748\pi$
$978$	0	0
$979$	−31.9836	−1.02220
$980$	0	0
$981$	−51.2833	−1.63735
$982$	0	0
$983$	−32.3085	−1.03048	−0.515241	−	0.857045i	$-0.672298\pi$
$983$	−32.3085	−1.03048	−0.515241	+	0.857045i	$0.672298\pi$
$984$	0	0
$985$	−4.78932	−0.152600
$986$	0	0
$987$	62.1224	1.97738
$988$	0	0
$989$	33.2552	1.05745
$990$	0	0
$991$	62.7375	1.99292	0.996461	−	0.0840547i	$-0.0267870\pi$
$991$	62.7375	1.99292	0.996461	+	0.0840547i	$0.0267870\pi$
$992$	0	0
$993$	−77.1507	−2.44830
$994$	0	0
$995$	36.5635	1.15914
$996$	0	0
$997$	5.89546	0.186711	0.0933556	−	0.995633i	$-0.470241\pi$
$997$	5.89546	0.186711	0.0933556	+	0.995633i	$0.470241\pi$
$998$	0	0
$999$	−4.49509	−0.142218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6128.2.a.m.1.14		16
4.3	odd	2		1532.2.a.c.1.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1532.2.a.c.1.3	✓	16	4.3	odd	2
6128.2.a.m.1.14		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:	\( N \)	\(=\)	\( 6128 = 2^{4} \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6128.a (trivial)

\(f(q)\)	\(=\)	\(q+2.38050 q^{3} +4.04378 q^{5} +2.78993 q^{7} +2.66676 q^{9} -4.46988 q^{11} -4.31680 q^{13} +9.62621 q^{15} -2.15383 q^{17} +8.54424 q^{19} +6.64141 q^{21} +6.75020 q^{23} +11.3522 q^{25} -0.793271 q^{27} +2.42636 q^{29} -4.50352 q^{31} -10.6405 q^{33} +11.2819 q^{35} +5.66652 q^{37} -10.2761 q^{39} -3.29859 q^{41} +4.92655 q^{43} +10.7838 q^{45} +9.35379 q^{47} +0.783691 q^{49} -5.12718 q^{51} -12.5578 q^{53} -18.0752 q^{55} +20.3395 q^{57} -10.0714 q^{59} +5.19987 q^{61} +7.44007 q^{63} -17.4562 q^{65} -5.00463 q^{67} +16.0688 q^{69} +10.2827 q^{71} -5.99128 q^{73} +27.0238 q^{75} -12.4706 q^{77} +14.5735 q^{79} -9.88867 q^{81} +4.74526 q^{83} -8.70961 q^{85} +5.77595 q^{87} +7.15537 q^{89} -12.0435 q^{91} -10.7206 q^{93} +34.5511 q^{95} +8.19376 q^{97} -11.9201 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 5 q^{3} - 3 q^{5} + 18 q^{7} + 15 q^{9} + 6 q^{11} - 14 q^{13} + 18 q^{15} - q^{17} + 18 q^{19} - 12 q^{21} + 20 q^{23} + 11 q^{25} + 20 q^{27} - 13 q^{29} + 19 q^{31} - 12 q^{33} + 22 q^{35} - 23 q^{37}+ \cdots + 16 q^{99}+O(q^{100}) \) 16 * q + 5 * q^3 - 3 * q^5 + 18 * q^7 + 15 * q^9 + 6 * q^11 - 14 * q^13 + 18 * q^15 - q^17 + 18 * q^19 - 12 * q^21 + 20 * q^23 + 11 * q^25 + 20 * q^27 - 13 * q^29 + 19 * q^31 - 12 * q^33 + 22 * q^35 - 23 * q^37 + 28 * q^39 + 53 * q^43 - 21 * q^45 + 27 * q^47 + 26 * q^49 + 26 * q^51 - 17 * q^53 + 36 * q^55 + 3 * q^57 + 22 * q^59 - 7 * q^61 + 56 * q^63 - 8 * q^65 + 65 * q^67 - 10 * q^69 + 28 * q^71 - 12 * q^73 + 16 * q^75 - 19 * q^77 + 41 * q^79 + 4 * q^81 + 27 * q^83 - 56 * q^85 + 55 * q^87 - 2 * q^89 + 19 * q^91 - 35 * q^93 + 40 * q^95 + 5 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more