LMFDB - Embedded newform 6024.2.a.p.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 29 x^{12} + 210 x^{11} + 280 x^{10} - 2796 x^{9} - 863 x^{8} + 17652 x^{7} + \cdots - 3364 $ x^14 - 6x^13 - 29x^12 + 210x^11 + 280x^10 - 2796x^9 - 863x^8 + 17652x^7 - 1300x^6 - 53458x^5 + 10892x^4 + 67298x^3 - 13761x^2 - 22140*x - 3364
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.10854$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.10854 q^{5} +5.12441 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.10854 q^{5} +5.12441 q^{7} +1.00000 q^{9} +3.13169 q^{11} +1.17805 q^{13} -1.10854 q^{15} -3.12492 q^{17} +6.71377 q^{19} -5.12441 q^{21} +3.74627 q^{23} -3.77114 q^{25} -1.00000 q^{27} +7.57935 q^{29} -4.19883 q^{31} -3.13169 q^{33} +5.68061 q^{35} +1.50489 q^{37} -1.17805 q^{39} +5.64677 q^{41} -0.425038 q^{43} +1.10854 q^{45} +3.62439 q^{47} +19.2596 q^{49} +3.12492 q^{51} -5.71629 q^{53} +3.47160 q^{55} -6.71377 q^{57} +6.10581 q^{59} -2.49695 q^{61} +5.12441 q^{63} +1.30592 q^{65} -6.96777 q^{67} -3.74627 q^{69} +4.75660 q^{71} -7.61057 q^{73} +3.77114 q^{75} +16.0481 q^{77} +2.31864 q^{79} +1.00000 q^{81} -6.48483 q^{83} -3.46409 q^{85} -7.57935 q^{87} +5.40840 q^{89} +6.03682 q^{91} +4.19883 q^{93} +7.44247 q^{95} -12.2878 q^{97} +3.13169 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.10854	0.495753	0.247877	−	0.968792i	$-0.420267\pi$
$5$	1.10854	0.495753	0.247877	+	0.968792i	$0.420267\pi$
$6$	0	0
$7$	5.12441	1.93685	0.968423	−	0.249312i	$-0.0802046\pi$
$7$	5.12441	1.93685	0.968423	+	0.249312i	$0.0802046\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.13169	0.944240	0.472120	−	0.881534i	$-0.343489\pi$
$11$	3.13169	0.944240	0.472120	+	0.881534i	$0.343489\pi$
$12$	0	0
$13$	1.17805	0.326733	0.163366	−	0.986565i	$-0.447765\pi$
$13$	1.17805	0.326733	0.163366	+	0.986565i	$0.447765\pi$
$14$	0	0
$15$	−1.10854	−0.286223
$16$	0	0
$17$	−3.12492	−0.757903	−0.378952	−	0.925416i	$-0.623715\pi$
$17$	−3.12492	−0.757903	−0.378952	+	0.925416i	$0.623715\pi$
$18$	0	0
$19$	6.71377	1.54024	0.770122	−	0.637897i	$-0.220195\pi$
$19$	6.71377	1.54024	0.770122	+	0.637897i	$0.220195\pi$
$20$	0	0
$21$	−5.12441	−1.11824
$22$	0	0
$23$	3.74627	0.781151	0.390576	−	0.920571i	$-0.372276\pi$
$23$	3.74627	0.781151	0.390576	+	0.920571i	$0.372276\pi$
$24$	0	0
$25$	−3.77114	−0.754229
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.57935	1.40745	0.703725	−	0.710473i	$-0.251519\pi$
$29$	7.57935	1.40745	0.703725	+	0.710473i	$0.251519\pi$
$30$	0	0
$31$	−4.19883	−0.754132	−0.377066	−	0.926186i	$-0.623067\pi$
$31$	−4.19883	−0.754132	−0.377066	+	0.926186i	$0.623067\pi$
$32$	0	0
$33$	−3.13169	−0.545157
$34$	0	0
$35$	5.68061	0.960198
$36$	0	0
$37$	1.50489	0.247402	0.123701	−	0.992320i	$-0.460524\pi$
$37$	1.50489	0.247402	0.123701	+	0.992320i	$0.460524\pi$
$38$	0	0
$39$	−1.17805	−0.188639
$40$	0	0
$41$	5.64677	0.881877	0.440939	−	0.897537i	$-0.354646\pi$
$41$	5.64677	0.881877	0.440939	+	0.897537i	$0.354646\pi$
$42$	0	0
$43$	−0.425038	−0.0648177	−0.0324089	−	0.999475i	$-0.510318\pi$
$43$	−0.425038	−0.0648177	−0.0324089	+	0.999475i	$0.510318\pi$
$44$	0	0
$45$	1.10854	0.165251
$46$	0	0
$47$	3.62439	0.528672	0.264336	−	0.964431i	$-0.414847\pi$
$47$	3.62439	0.528672	0.264336	+	0.964431i	$0.414847\pi$
$48$	0	0
$49$	19.2596	2.75137
$50$	0	0
$51$	3.12492	0.437576
$52$	0	0
$53$	−5.71629	−0.785192	−0.392596	−	0.919711i	$-0.628423\pi$
$53$	−5.71629	−0.785192	−0.392596	+	0.919711i	$0.628423\pi$
$54$	0	0
$55$	3.47160	0.468110
$56$	0	0
$57$	−6.71377	−0.889260
$58$	0	0
$59$	6.10581	0.794909	0.397455	−	0.917622i	$-0.369894\pi$
$59$	6.10581	0.794909	0.397455	+	0.917622i	$0.369894\pi$
$60$	0	0
$61$	−2.49695	−0.319702	−0.159851	−	0.987141i	$-0.551101\pi$
$61$	−2.49695	−0.319702	−0.159851	+	0.987141i	$0.551101\pi$
$62$	0	0
$63$	5.12441	0.645615
$64$	0	0
$65$	1.30592	0.161979
$66$	0	0
$67$	−6.96777	−0.851248	−0.425624	−	0.904900i	$-0.639945\pi$
$67$	−6.96777	−0.851248	−0.425624	+	0.904900i	$0.639945\pi$
$68$	0	0
$69$	−3.74627	−0.450998
$70$	0	0
$71$	4.75660	0.564504	0.282252	−	0.959340i	$-0.408919\pi$
$71$	4.75660	0.564504	0.282252	+	0.959340i	$0.408919\pi$
$72$	0	0
$73$	−7.61057	−0.890750	−0.445375	−	0.895344i	$-0.646930\pi$
$73$	−7.61057	−0.890750	−0.445375	+	0.895344i	$0.646930\pi$
$74$	0	0
$75$	3.77114	0.435454
$76$	0	0
$77$	16.0481	1.82885
$78$	0	0
$79$	2.31864	0.260867	0.130433	−	0.991457i	$-0.458363\pi$
$79$	2.31864	0.260867	0.130433	+	0.991457i	$0.458363\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.48483	−0.711803	−0.355901	−	0.934524i	$-0.615826\pi$
$83$	−6.48483	−0.711803	−0.355901	+	0.934524i	$0.615826\pi$
$84$	0	0
$85$	−3.46409	−0.375733
$86$	0	0
$87$	−7.57935	−0.812591
$88$	0	0
$89$	5.40840	0.573289	0.286645	−	0.958037i	$-0.407460\pi$
$89$	5.40840	0.573289	0.286645	+	0.958037i	$0.407460\pi$
$90$	0	0
$91$	6.03682	0.632831
$92$	0	0
$93$	4.19883	0.435398
$94$	0	0
$95$	7.44247	0.763581
$96$	0	0
$97$	−12.2878	−1.24764	−0.623818	−	0.781570i	$-0.714419\pi$
$97$	−12.2878	−1.24764	−0.623818	+	0.781570i	$0.714419\pi$
$98$	0	0
$99$	3.13169	0.314747
$100$	0	0
$101$	7.44333	0.740639	0.370319	−	0.928905i	$-0.379248\pi$
$101$	7.44333	0.740639	0.370319	+	0.928905i	$0.379248\pi$
$102$	0	0
$103$	−3.50497	−0.345355	−0.172677	−	0.984978i	$-0.555242\pi$
$103$	−3.50497	−0.345355	−0.172677	+	0.984978i	$0.555242\pi$
$104$	0	0
$105$	−5.68061	−0.554371
$106$	0	0
$107$	−7.89790	−0.763518	−0.381759	−	0.924262i	$-0.624682\pi$
$107$	−7.89790	−0.763518	−0.381759	+	0.924262i	$0.624682\pi$
$108$	0	0
$109$	−6.47842	−0.620520	−0.310260	−	0.950652i	$-0.600416\pi$
$109$	−6.47842	−0.620520	−0.310260	+	0.950652i	$0.600416\pi$
$110$	0	0
$111$	−1.50489	−0.142837
$112$	0	0
$113$	1.63266	0.153587	0.0767937	−	0.997047i	$-0.475532\pi$
$113$	1.63266	0.153587	0.0767937	+	0.997047i	$0.475532\pi$
$114$	0	0
$115$	4.15288	0.387258
$116$	0	0
$117$	1.17805	0.108911
$118$	0	0
$119$	−16.0134	−1.46794
$120$	0	0
$121$	−1.19253	−0.108411
$122$	0	0
$123$	−5.64677	−0.509152
$124$	0	0
$125$	−9.72315	−0.869665
$126$	0	0
$127$	−18.4528	−1.63742	−0.818711	−	0.574206i	$-0.805311\pi$
$127$	−18.4528	−1.63742	−0.818711	+	0.574206i	$0.805311\pi$
$128$	0	0
$129$	0.425038	0.0374225
$130$	0	0
$131$	−1.40002	−0.122320	−0.0611602	−	0.998128i	$-0.519480\pi$
$131$	−1.40002	−0.122320	−0.0611602	+	0.998128i	$0.519480\pi$
$132$	0	0
$133$	34.4041	2.98322
$134$	0	0
$135$	−1.10854	−0.0954078
$136$	0	0
$137$	12.7549	1.08973	0.544863	−	0.838525i	$-0.316582\pi$
$137$	12.7549	1.08973	0.544863	+	0.838525i	$0.316582\pi$
$138$	0	0
$139$	−19.7874	−1.67835	−0.839174	−	0.543862i	$-0.816961\pi$
$139$	−19.7874	−1.67835	−0.839174	+	0.543862i	$0.816961\pi$
$140$	0	0
$141$	−3.62439	−0.305229
$142$	0	0
$143$	3.68929	0.308514
$144$	0	0
$145$	8.40200	0.697748
$146$	0	0
$147$	−19.2596	−1.58851
$148$	0	0
$149$	−1.56531	−0.128236	−0.0641178	−	0.997942i	$-0.520423\pi$
$149$	−1.56531	−0.128236	−0.0641178	+	0.997942i	$0.520423\pi$
$150$	0	0
$151$	−4.63521	−0.377208	−0.188604	−	0.982053i	$-0.560396\pi$
$151$	−4.63521	−0.377208	−0.188604	+	0.982053i	$0.560396\pi$
$152$	0	0
$153$	−3.12492	−0.252634
$154$	0	0
$155$	−4.65457	−0.373864
$156$	0	0
$157$	−3.03684	−0.242366	−0.121183	−	0.992630i	$-0.538669\pi$
$157$	−3.03684	−0.242366	−0.121183	+	0.992630i	$0.538669\pi$
$158$	0	0
$159$	5.71629	0.453331
$160$	0	0
$161$	19.1974	1.51297
$162$	0	0
$163$	−16.8906	−1.32298	−0.661489	−	0.749955i	$-0.730075\pi$
$163$	−16.8906	−1.32298	−0.661489	+	0.749955i	$0.730075\pi$
$164$	0	0
$165$	−3.47160	−0.270263
$166$	0	0
$167$	1.92673	0.149095	0.0745474	−	0.997217i	$-0.476249\pi$
$167$	1.92673	0.149095	0.0745474	+	0.997217i	$0.476249\pi$
$168$	0	0
$169$	−11.6122	−0.893246
$170$	0	0
$171$	6.71377	0.513415
$172$	0	0
$173$	−10.7273	−0.815582	−0.407791	−	0.913075i	$-0.633701\pi$
$173$	−10.7273	−0.815582	−0.407791	+	0.913075i	$0.633701\pi$
$174$	0	0
$175$	−19.3249	−1.46082
$176$	0	0
$177$	−6.10581	−0.458941
$178$	0	0
$179$	22.2830	1.66551	0.832756	−	0.553641i	$-0.186762\pi$
$179$	22.2830	1.66551	0.832756	+	0.553641i	$0.186762\pi$
$180$	0	0
$181$	19.7844	1.47056	0.735282	−	0.677762i	$-0.237050\pi$
$181$	19.7844	1.47056	0.735282	+	0.677762i	$0.237050\pi$
$182$	0	0
$183$	2.49695	0.184580
$184$	0	0
$185$	1.66822	0.122650
$186$	0	0
$187$	−9.78626	−0.715643
$188$	0	0
$189$	−5.12441	−0.372746
$190$	0	0
$191$	−8.67724	−0.627863	−0.313932	−	0.949446i	$-0.601646\pi$
$191$	−8.67724	−0.627863	−0.313932	+	0.949446i	$0.601646\pi$
$192$	0	0
$193$	25.3448	1.82436	0.912178	−	0.409794i	$-0.134399\pi$
$193$	25.3448	1.82436	0.912178	+	0.409794i	$0.134399\pi$
$194$	0	0
$195$	−1.30592	−0.0935186
$196$	0	0
$197$	−14.5018	−1.03321	−0.516605	−	0.856224i	$-0.672805\pi$
$197$	−14.5018	−1.03321	−0.516605	+	0.856224i	$0.672805\pi$
$198$	0	0
$199$	8.69113	0.616098	0.308049	−	0.951371i	$-0.400324\pi$
$199$	8.69113	0.616098	0.308049	+	0.951371i	$0.400324\pi$
$200$	0	0
$201$	6.96777	0.491468
$202$	0	0
$203$	38.8397	2.72601
$204$	0	0
$205$	6.25966	0.437194
$206$	0	0
$207$	3.74627	0.260384
$208$	0	0
$209$	21.0254	1.45436
$210$	0	0
$211$	−11.7112	−0.806230	−0.403115	−	0.915149i	$-0.632072\pi$
$211$	−11.7112	−0.806230	−0.403115	+	0.915149i	$0.632072\pi$
$212$	0	0
$213$	−4.75660	−0.325917
$214$	0	0
$215$	−0.471171	−0.0321336
$216$	0	0
$217$	−21.5165	−1.46064
$218$	0	0
$219$	7.61057	0.514275
$220$	0	0
$221$	−3.68131	−0.247632
$222$	0	0
$223$	−15.1061	−1.01158	−0.505788	−	0.862658i	$-0.668798\pi$
$223$	−15.1061	−1.01158	−0.505788	+	0.862658i	$0.668798\pi$
$224$	0	0
$225$	−3.77114	−0.251410
$226$	0	0
$227$	−18.2674	−1.21245	−0.606226	−	0.795293i	$-0.707317\pi$
$227$	−18.2674	−1.21245	−0.606226	+	0.795293i	$0.707317\pi$
$228$	0	0
$229$	−0.509444	−0.0336650	−0.0168325	−	0.999858i	$-0.505358\pi$
$229$	−0.509444	−0.0336650	−0.0168325	+	0.999858i	$0.505358\pi$
$230$	0	0
$231$	−16.0481	−1.05589
$232$	0	0
$233$	18.2335	1.19451	0.597257	−	0.802050i	$-0.296257\pi$
$233$	18.2335	1.19451	0.597257	+	0.802050i	$0.296257\pi$
$234$	0	0
$235$	4.01778	0.262091
$236$	0	0
$237$	−2.31864	−0.150612
$238$	0	0
$239$	3.10314	0.200726	0.100363	−	0.994951i	$-0.468000\pi$
$239$	3.10314	0.200726	0.100363	+	0.994951i	$0.468000\pi$
$240$	0	0
$241$	27.2116	1.75286	0.876428	−	0.481534i	$-0.159920\pi$
$241$	27.2116	1.75286	0.876428	+	0.481534i	$0.159920\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	21.3500	1.36400
$246$	0	0
$247$	7.90916	0.503248
$248$	0	0
$249$	6.48483	0.410960
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	11.7321	0.737594
$254$	0	0
$255$	3.46409	0.216930
$256$	0	0
$257$	0.0186019	0.00116036	0.000580178	−	1.00000i	$-0.499815\pi$
$257$	0.0186019	0.00116036	0.000580178		1.00000i	$0.499815\pi$
$258$	0	0
$259$	7.71166	0.479179
$260$	0	0
$261$	7.57935	0.469150
$262$	0	0
$263$	−26.3022	−1.62186	−0.810931	−	0.585142i	$-0.801039\pi$
$263$	−26.3022	−1.62186	−0.810931	+	0.585142i	$0.801039\pi$
$264$	0	0
$265$	−6.33672	−0.389262
$266$	0	0
$267$	−5.40840	−0.330989
$268$	0	0
$269$	−26.5437	−1.61840	−0.809200	−	0.587534i	$-0.800099\pi$
$269$	−26.5437	−1.61840	−0.809200	+	0.587534i	$0.800099\pi$
$270$	0	0
$271$	−26.0934	−1.58506	−0.792532	−	0.609831i	$-0.791237\pi$
$271$	−26.0934	−1.58506	−0.792532	+	0.609831i	$0.791237\pi$
$272$	0	0
$273$	−6.03682	−0.365365
$274$	0	0
$275$	−11.8100	−0.712172
$276$	0	0
$277$	3.66862	0.220426	0.110213	−	0.993908i	$-0.464847\pi$
$277$	3.66862	0.220426	0.110213	+	0.993908i	$0.464847\pi$
$278$	0	0
$279$	−4.19883	−0.251377
$280$	0	0
$281$	−20.0496	−1.19606	−0.598031	−	0.801473i	$-0.704050\pi$
$281$	−20.0496	−1.19606	−0.598031	+	0.801473i	$0.704050\pi$
$282$	0	0
$283$	24.6173	1.46335	0.731673	−	0.681656i	$-0.238740\pi$
$283$	24.6173	1.46335	0.731673	+	0.681656i	$0.238740\pi$
$284$	0	0
$285$	−7.44247	−0.440854
$286$	0	0
$287$	28.9364	1.70806
$288$	0	0
$289$	−7.23490	−0.425582
$290$	0	0
$291$	12.2878	0.720323
$292$	0	0
$293$	13.6451	0.797157	0.398578	−	0.917134i	$-0.369504\pi$
$293$	13.6451	0.797157	0.398578	+	0.917134i	$0.369504\pi$
$294$	0	0
$295$	6.76853	0.394079
$296$	0	0
$297$	−3.13169	−0.181719
$298$	0	0
$299$	4.41330	0.255228
$300$	0	0
$301$	−2.17807	−0.125542
$302$	0	0
$303$	−7.44333	−0.427608
$304$	0	0
$305$	−2.76797	−0.158494
$306$	0	0
$307$	8.15379	0.465361	0.232681	−	0.972553i	$-0.425250\pi$
$307$	8.15379	0.465361	0.232681	+	0.972553i	$0.425250\pi$
$308$	0	0
$309$	3.50497	0.199391
$310$	0	0
$311$	6.49478	0.368285	0.184143	−	0.982900i	$-0.441049\pi$
$311$	6.49478	0.368285	0.184143	+	0.982900i	$0.441049\pi$
$312$	0	0
$313$	22.9767	1.29872	0.649360	−	0.760481i	$-0.275037\pi$
$313$	22.9767	1.29872	0.649360	+	0.760481i	$0.275037\pi$
$314$	0	0
$315$	5.68061	0.320066
$316$	0	0
$317$	5.91146	0.332020	0.166010	−	0.986124i	$-0.446912\pi$
$317$	5.91146	0.332020	0.166010	+	0.986124i	$0.446912\pi$
$318$	0	0
$319$	23.7362	1.32897
$320$	0	0
$321$	7.89790	0.440818
$322$	0	0
$323$	−20.9800	−1.16736
$324$	0	0
$325$	−4.44260	−0.246431
$326$	0	0
$327$	6.47842	0.358258
$328$	0	0
$329$	18.5729	1.02396
$330$	0	0
$331$	25.3247	1.39197	0.695986	−	0.718055i	$-0.254967\pi$
$331$	25.3247	1.39197	0.695986	+	0.718055i	$0.254967\pi$
$332$	0	0
$333$	1.50489	0.0824672
$334$	0	0
$335$	−7.72404	−0.422009
$336$	0	0
$337$	1.03526	0.0563940	0.0281970	−	0.999602i	$-0.491023\pi$
$337$	1.03526	0.0563940	0.0281970	+	0.999602i	$0.491023\pi$
$338$	0	0
$339$	−1.63266	−0.0886738
$340$	0	0
$341$	−13.1494	−0.712082
$342$	0	0
$343$	62.8233	3.39214
$344$	0	0
$345$	−4.15288	−0.223584
$346$	0	0
$347$	−22.9108	−1.22992	−0.614959	−	0.788559i	$-0.710828\pi$
$347$	−22.9108	−1.22992	−0.614959	+	0.788559i	$0.710828\pi$
$348$	0	0
$349$	−14.5720	−0.780019	−0.390009	−	0.920811i	$-0.627528\pi$
$349$	−14.5720	−0.780019	−0.390009	+	0.920811i	$0.627528\pi$
$350$	0	0
$351$	−1.17805	−0.0628797
$352$	0	0
$353$	35.3217	1.87998	0.939992	−	0.341197i	$-0.110832\pi$
$353$	35.3217	1.87998	0.939992	+	0.341197i	$0.110832\pi$
$354$	0	0
$355$	5.27287	0.279855
$356$	0	0
$357$	16.0134	0.847517
$358$	0	0
$359$	20.2726	1.06995	0.534974	−	0.844868i	$-0.320321\pi$
$359$	20.2726	1.06995	0.534974	+	0.844868i	$0.320321\pi$
$360$	0	0
$361$	26.0747	1.37235
$362$	0	0
$363$	1.19253	0.0625913
$364$	0	0
$365$	−8.43661	−0.441593
$366$	0	0
$367$	−13.9183	−0.726530	−0.363265	−	0.931686i	$-0.618338\pi$
$367$	−13.9183	−0.726530	−0.363265	+	0.931686i	$0.618338\pi$
$368$	0	0
$369$	5.64677	0.293959
$370$	0	0
$371$	−29.2926	−1.52080
$372$	0	0
$373$	−26.3659	−1.36518	−0.682589	−	0.730803i	$-0.739146\pi$
$373$	−26.3659	−1.36518	−0.682589	+	0.730803i	$0.739146\pi$
$374$	0	0
$375$	9.72315	0.502101
$376$	0	0
$377$	8.92886	0.459860
$378$	0	0
$379$	4.13397	0.212348	0.106174	−	0.994348i	$-0.466140\pi$
$379$	4.13397	0.212348	0.106174	+	0.994348i	$0.466140\pi$
$380$	0	0
$381$	18.4528	0.945366
$382$	0	0
$383$	15.1462	0.773937	0.386968	−	0.922093i	$-0.373522\pi$
$383$	15.1462	0.773937	0.386968	+	0.922093i	$0.373522\pi$
$384$	0	0
$385$	17.7899	0.906657
$386$	0	0
$387$	−0.425038	−0.0216059
$388$	0	0
$389$	8.03915	0.407601	0.203801	−	0.979012i	$-0.434671\pi$
$389$	8.03915	0.407601	0.203801	+	0.979012i	$0.434671\pi$
$390$	0	0
$391$	−11.7068	−0.592037
$392$	0	0
$393$	1.40002	0.0706217
$394$	0	0
$395$	2.57030	0.129326
$396$	0	0
$397$	−7.10082	−0.356380	−0.178190	−	0.983996i	$-0.557024\pi$
$397$	−7.10082	−0.356380	−0.178190	+	0.983996i	$0.557024\pi$
$398$	0	0
$399$	−34.4041	−1.72236
$400$	0	0
$401$	−5.90708	−0.294985	−0.147493	−	0.989063i	$-0.547120\pi$
$401$	−5.90708	−0.294985	−0.147493	+	0.989063i	$0.547120\pi$
$402$	0	0
$403$	−4.94644	−0.246400
$404$	0	0
$405$	1.10854	0.0550837
$406$	0	0
$407$	4.71283	0.233607
$408$	0	0
$409$	8.86681	0.438436	0.219218	−	0.975676i	$-0.429649\pi$
$409$	8.86681	0.438436	0.219218	+	0.975676i	$0.429649\pi$
$410$	0	0
$411$	−12.7549	−0.629154
$412$	0	0
$413$	31.2887	1.53962
$414$	0	0
$415$	−7.18869	−0.352879
$416$	0	0
$417$	19.7874	0.968995
$418$	0	0
$419$	15.5039	0.757417	0.378708	−	0.925516i	$-0.376368\pi$
$419$	15.5039	0.757417	0.378708	+	0.925516i	$0.376368\pi$
$420$	0	0
$421$	8.79379	0.428583	0.214292	−	0.976770i	$-0.431256\pi$
$421$	8.79379	0.428583	0.214292	+	0.976770i	$0.431256\pi$
$422$	0	0
$423$	3.62439	0.176224
$424$	0	0
$425$	11.7845	0.571632
$426$	0	0
$427$	−12.7954	−0.619214
$428$	0	0
$429$	−3.68929	−0.178121
$430$	0	0
$431$	38.4849	1.85375	0.926877	−	0.375365i	$-0.122483\pi$
$431$	38.4849	1.85375	0.926877	+	0.375365i	$0.122483\pi$
$432$	0	0
$433$	6.40134	0.307629	0.153814	−	0.988100i	$-0.450844\pi$
$433$	6.40134	0.307629	0.153814	+	0.988100i	$0.450844\pi$
$434$	0	0
$435$	−8.40200	−0.402845
$436$	0	0
$437$	25.1516	1.20316
$438$	0	0
$439$	13.6527	0.651610	0.325805	−	0.945437i	$-0.394365\pi$
$439$	13.6527	0.651610	0.325805	+	0.945437i	$0.394365\pi$
$440$	0	0
$441$	19.2596	0.917124
$442$	0	0
$443$	−24.7866	−1.17765	−0.588823	−	0.808262i	$-0.700409\pi$
$443$	−24.7866	−1.17765	−0.588823	+	0.808262i	$0.700409\pi$
$444$	0	0
$445$	5.99542	0.284210
$446$	0	0
$447$	1.56531	0.0740368
$448$	0	0
$449$	38.4879	1.81636	0.908179	−	0.418582i	$-0.137473\pi$
$449$	38.4879	1.81636	0.908179	+	0.418582i	$0.137473\pi$
$450$	0	0
$451$	17.6839	0.832704
$452$	0	0
$453$	4.63521	0.217781
$454$	0	0
$455$	6.69205	0.313728
$456$	0	0
$457$	12.2821	0.574535	0.287267	−	0.957850i	$-0.407253\pi$
$457$	12.2821	0.574535	0.287267	+	0.957850i	$0.407253\pi$
$458$	0	0
$459$	3.12492	0.145859
$460$	0	0
$461$	−9.99614	−0.465567	−0.232783	−	0.972529i	$-0.574783\pi$
$461$	−9.99614	−0.465567	−0.232783	+	0.972529i	$0.574783\pi$
$462$	0	0
$463$	−10.2007	−0.474067	−0.237033	−	0.971502i	$-0.576175\pi$
$463$	−10.2007	−0.474067	−0.237033	+	0.971502i	$0.576175\pi$
$464$	0	0
$465$	4.65457	0.215850
$466$	0	0
$467$	−28.1446	−1.30238	−0.651188	−	0.758916i	$-0.725729\pi$
$467$	−28.1446	−1.30238	−0.651188	+	0.758916i	$0.725729\pi$
$468$	0	0
$469$	−35.7057	−1.64874
$470$	0	0
$471$	3.03684	0.139930
$472$	0	0
$473$	−1.33109	−0.0612035
$474$	0	0
$475$	−25.3186	−1.16170
$476$	0	0
$477$	−5.71629	−0.261731
$478$	0	0
$479$	9.65608	0.441197	0.220599	−	0.975365i	$-0.429199\pi$
$479$	9.65608	0.441197	0.220599	+	0.975365i	$0.429199\pi$
$480$	0	0
$481$	1.77283	0.0808342
$482$	0	0
$483$	−19.1974	−0.873513
$484$	0	0
$485$	−13.6215	−0.618520
$486$	0	0
$487$	17.9700	0.814298	0.407149	−	0.913362i	$-0.366523\pi$
$487$	17.9700	0.814298	0.407149	+	0.913362i	$0.366523\pi$
$488$	0	0
$489$	16.8906	0.763822
$490$	0	0
$491$	−2.77830	−0.125383	−0.0626915	−	0.998033i	$-0.519968\pi$
$491$	−2.77830	−0.125383	−0.0626915	+	0.998033i	$0.519968\pi$
$492$	0	0
$493$	−23.6848	−1.06671
$494$	0	0
$495$	3.47160	0.156037
$496$	0	0
$497$	24.3748	1.09336
$498$	0	0
$499$	−38.0253	−1.70224	−0.851122	−	0.524968i	$-0.824077\pi$
$499$	−38.0253	−1.70224	−0.851122	+	0.524968i	$0.824077\pi$
$500$	0	0
$501$	−1.92673	−0.0860800
$502$	0	0
$503$	−12.9154	−0.575869	−0.287934	−	0.957650i	$-0.592968\pi$
$503$	−12.9154	−0.575869	−0.287934	+	0.957650i	$0.592968\pi$
$504$	0	0
$505$	8.25121	0.367174
$506$	0	0
$507$	11.6122	0.515716
$508$	0	0
$509$	19.4836	0.863594	0.431797	−	0.901971i	$-0.357880\pi$
$509$	19.4836	0.863594	0.431797	+	0.901971i	$0.357880\pi$
$510$	0	0
$511$	−38.9997	−1.72525
$512$	0	0
$513$	−6.71377	−0.296420
$514$	0	0
$515$	−3.88539	−0.171211
$516$	0	0
$517$	11.3505	0.499193
$518$	0	0
$519$	10.7273	0.470877
$520$	0	0
$521$	8.80593	0.385795	0.192897	−	0.981219i	$-0.438212\pi$
$521$	8.80593	0.385795	0.192897	+	0.981219i	$0.438212\pi$
$522$	0	0
$523$	−28.4089	−1.24223	−0.621117	−	0.783718i	$-0.713321\pi$
$523$	−28.4089	−1.24223	−0.621117	+	0.783718i	$0.713321\pi$
$524$	0	0
$525$	19.3249	0.843407
$526$	0	0
$527$	13.1210	0.571559
$528$	0	0
$529$	−8.96547	−0.389803
$530$	0	0
$531$	6.10581	0.264970
$532$	0	0
$533$	6.65219	0.288138
$534$	0	0
$535$	−8.75512	−0.378517
$536$	0	0
$537$	−22.2830	−0.961583
$538$	0	0
$539$	60.3151	2.59796
$540$	0	0
$541$	0.265791	0.0114273	0.00571363	−	0.999984i	$-0.498181\pi$
$541$	0.265791	0.0114273	0.00571363	+	0.999984i	$0.498181\pi$
$542$	0	0
$543$	−19.7844	−0.849030
$544$	0	0
$545$	−7.18158	−0.307625
$546$	0	0
$547$	9.30268	0.397754	0.198877	−	0.980024i	$-0.436271\pi$
$547$	9.30268	0.397754	0.198877	+	0.980024i	$0.436271\pi$
$548$	0	0
$549$	−2.49695	−0.106567
$550$	0	0
$551$	50.8860	2.16782
$552$	0	0
$553$	11.8816	0.505259
$554$	0	0
$555$	−1.66822	−0.0708122
$556$	0	0
$557$	−8.57197	−0.363206	−0.181603	−	0.983372i	$-0.558129\pi$
$557$	−8.57197	−0.363206	−0.181603	+	0.983372i	$0.558129\pi$
$558$	0	0
$559$	−0.500717	−0.0211781
$560$	0	0
$561$	9.78626	0.413176
$562$	0	0
$563$	26.5796	1.12020	0.560098	−	0.828426i	$-0.310763\pi$
$563$	26.5796	1.12020	0.560098	+	0.828426i	$0.310763\pi$
$564$	0	0
$565$	1.80986	0.0761415
$566$	0	0
$567$	5.12441	0.215205
$568$	0	0
$569$	−4.31072	−0.180715	−0.0903574	−	0.995909i	$-0.528801\pi$
$569$	−4.31072	−0.180715	−0.0903574	+	0.995909i	$0.528801\pi$
$570$	0	0
$571$	18.9581	0.793373	0.396687	−	0.917954i	$-0.370160\pi$
$571$	18.9581	0.793373	0.396687	+	0.917954i	$0.370160\pi$
$572$	0	0
$573$	8.67724	0.362497
$574$	0	0
$575$	−14.1277	−0.589166
$576$	0	0
$577$	23.1217	0.962568	0.481284	−	0.876565i	$-0.340171\pi$
$577$	23.1217	0.962568	0.481284	+	0.876565i	$0.340171\pi$
$578$	0	0
$579$	−25.3448	−1.05329
$580$	0	0
$581$	−33.2310	−1.37865
$582$	0	0
$583$	−17.9016	−0.741410
$584$	0	0
$585$	1.30592	0.0539930
$586$	0	0
$587$	3.64881	0.150602	0.0753012	−	0.997161i	$-0.476008\pi$
$587$	3.64881	0.150602	0.0753012	+	0.997161i	$0.476008\pi$
$588$	0	0
$589$	−28.1900	−1.16155
$590$	0	0
$591$	14.5018	0.596525
$592$	0	0
$593$	10.5805	0.434491	0.217245	−	0.976117i	$-0.430293\pi$
$593$	10.5805	0.434491	0.217245	+	0.976117i	$0.430293\pi$
$594$	0	0
$595$	−17.7514	−0.727738
$596$	0	0
$597$	−8.69113	−0.355704
$598$	0	0
$599$	−27.6803	−1.13098	−0.565492	−	0.824754i	$-0.691314\pi$
$599$	−27.6803	−1.13098	−0.565492	+	0.824754i	$0.691314\pi$
$600$	0	0
$601$	−35.9967	−1.46834	−0.734169	−	0.678967i	$-0.762428\pi$
$601$	−35.9967	−1.46834	−0.734169	+	0.678967i	$0.762428\pi$
$602$	0	0
$603$	−6.96777	−0.283749
$604$	0	0
$605$	−1.32196	−0.0537453
$606$	0	0
$607$	13.2223	0.536676	0.268338	−	0.963325i	$-0.413526\pi$
$607$	13.2223	0.536676	0.268338	+	0.963325i	$0.413526\pi$
$608$	0	0
$609$	−38.8397	−1.57386
$610$	0	0
$611$	4.26972	0.172734
$612$	0	0
$613$	33.3580	1.34732	0.673658	−	0.739043i	$-0.264722\pi$
$613$	33.3580	1.34732	0.673658	+	0.739043i	$0.264722\pi$
$614$	0	0
$615$	−6.25966	−0.252414
$616$	0	0
$617$	13.0040	0.523521	0.261761	−	0.965133i	$-0.415697\pi$
$617$	13.0040	0.523521	0.261761	+	0.965133i	$0.415697\pi$
$618$	0	0
$619$	23.4182	0.941258	0.470629	−	0.882331i	$-0.344027\pi$
$619$	23.4182	0.941258	0.470629	+	0.882331i	$0.344027\pi$
$620$	0	0
$621$	−3.74627	−0.150333
$622$	0	0
$623$	27.7149	1.11037
$624$	0	0
$625$	8.07723	0.323089
$626$	0	0
$627$	−21.0254	−0.839675
$628$	0	0
$629$	−4.70264	−0.187507
$630$	0	0
$631$	45.6114	1.81576	0.907880	−	0.419231i	$-0.137700\pi$
$631$	45.6114	1.81576	0.907880	+	0.419231i	$0.137700\pi$
$632$	0	0
$633$	11.7112	0.465477
$634$	0	0
$635$	−20.4556	−0.811757
$636$	0	0
$637$	22.6888	0.898964
$638$	0	0
$639$	4.75660	0.188168
$640$	0	0
$641$	−32.4839	−1.28304	−0.641518	−	0.767108i	$-0.721695\pi$
$641$	−32.4839	−1.28304	−0.641518	+	0.767108i	$0.721695\pi$
$642$	0	0
$643$	−43.7770	−1.72640	−0.863198	−	0.504865i	$-0.831542\pi$
$643$	−43.7770	−1.72640	−0.863198	+	0.504865i	$0.831542\pi$
$644$	0	0
$645$	0.471171	0.0185524
$646$	0	0
$647$	24.7704	0.973825	0.486912	−	0.873451i	$-0.338123\pi$
$647$	24.7704	0.973825	0.486912	+	0.873451i	$0.338123\pi$
$648$	0	0
$649$	19.1215	0.750585
$650$	0	0
$651$	21.5165	0.843300
$652$	0	0
$653$	41.5652	1.62657	0.813286	−	0.581864i	$-0.197677\pi$
$653$	41.5652	1.62657	0.813286	+	0.581864i	$0.197677\pi$
$654$	0	0
$655$	−1.55198	−0.0606407
$656$	0	0
$657$	−7.61057	−0.296917
$658$	0	0
$659$	−40.6596	−1.58387	−0.791937	−	0.610603i	$-0.790927\pi$
$659$	−40.6596	−1.58387	−0.791937	+	0.610603i	$0.790927\pi$
$660$	0	0
$661$	−37.1079	−1.44333	−0.721665	−	0.692243i	$-0.756623\pi$
$661$	−37.1079	−1.44333	−0.721665	+	0.692243i	$0.756623\pi$
$662$	0	0
$663$	3.68131	0.142970
$664$	0	0
$665$	38.1383	1.47894
$666$	0	0
$667$	28.3943	1.09943
$668$	0	0
$669$	15.1061	0.584034
$670$	0	0
$671$	−7.81969	−0.301876
$672$	0	0
$673$	−15.8649	−0.611547	−0.305774	−	0.952104i	$-0.598915\pi$
$673$	−15.8649	−0.611547	−0.305774	+	0.952104i	$0.598915\pi$
$674$	0	0
$675$	3.77114	0.145151
$676$	0	0
$677$	50.8738	1.95524	0.977619	−	0.210382i	$-0.0674708\pi$
$677$	50.8738	1.95524	0.977619	+	0.210382i	$0.0674708\pi$
$678$	0	0
$679$	−62.9677	−2.41648
$680$	0	0
$681$	18.2674	0.700009
$682$	0	0
$683$	−0.603495	−0.0230921	−0.0115461	−	0.999933i	$-0.503675\pi$
$683$	−0.603495	−0.0230921	−0.0115461	+	0.999933i	$0.503675\pi$
$684$	0	0
$685$	14.1393	0.540236
$686$	0	0
$687$	0.509444	0.0194365
$688$	0	0
$689$	−6.73408	−0.256548
$690$	0	0
$691$	−45.4942	−1.73068	−0.865341	−	0.501183i	$-0.832898\pi$
$691$	−45.4942	−1.73068	−0.865341	+	0.501183i	$0.832898\pi$
$692$	0	0
$693$	16.0481	0.609616
$694$	0	0
$695$	−21.9351	−0.832047
$696$	0	0
$697$	−17.6457	−0.668378
$698$	0	0
$699$	−18.2335	−0.689653
$700$	0	0
$701$	−36.8381	−1.39136	−0.695678	−	0.718354i	$-0.744896\pi$
$701$	−36.8381	−1.39136	−0.695678	+	0.718354i	$0.744896\pi$
$702$	0	0
$703$	10.1035	0.381059
$704$	0	0
$705$	−4.01778	−0.151318
$706$	0	0
$707$	38.1427	1.43450
$708$	0	0
$709$	−29.2026	−1.09673	−0.548363	−	0.836241i	$-0.684749\pi$
$709$	−29.2026	−1.09673	−0.548363	+	0.836241i	$0.684749\pi$
$710$	0	0
$711$	2.31864	0.0869556
$712$	0	0
$713$	−15.7299	−0.589091
$714$	0	0
$715$	4.08972	0.152947
$716$	0	0
$717$	−3.10314	−0.115889
$718$	0	0
$719$	−17.7930	−0.663565	−0.331783	−	0.943356i	$-0.607650\pi$
$719$	−17.7930	−0.663565	−0.331783	+	0.943356i	$0.607650\pi$
$720$	0	0
$721$	−17.9609	−0.668899
$722$	0	0
$723$	−27.2116	−1.01201
$724$	0	0
$725$	−28.5828	−1.06154
$726$	0	0
$727$	19.2004	0.712104	0.356052	−	0.934466i	$-0.384123\pi$
$727$	19.2004	0.712104	0.356052	+	0.934466i	$0.384123\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.32821	0.0491256
$732$	0	0
$733$	5.43177	0.200627	0.100314	−	0.994956i	$-0.468015\pi$
$733$	5.43177	0.200627	0.100314	+	0.994956i	$0.468015\pi$
$734$	0	0
$735$	−21.3500	−0.787507
$736$	0	0
$737$	−21.8209	−0.803783
$738$	0	0
$739$	10.8190	0.397984	0.198992	−	0.980001i	$-0.436233\pi$
$739$	10.8190	0.397984	0.198992	+	0.980001i	$0.436233\pi$
$740$	0	0
$741$	−7.90916	−0.290550
$742$	0	0
$743$	3.51588	0.128985	0.0644926	−	0.997918i	$-0.479457\pi$
$743$	3.51588	0.128985	0.0644926	+	0.997918i	$0.479457\pi$
$744$	0	0
$745$	−1.73521	−0.0635732
$746$	0	0
$747$	−6.48483	−0.237268
$748$	0	0
$749$	−40.4721	−1.47882
$750$	0	0
$751$	−19.6330	−0.716420	−0.358210	−	0.933641i	$-0.616613\pi$
$751$	−19.6330	−0.716420	−0.358210	+	0.933641i	$0.616613\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−5.13831	−0.187002
$756$	0	0
$757$	−16.1154	−0.585722	−0.292861	−	0.956155i	$-0.594607\pi$
$757$	−16.1154	−0.585722	−0.292861	+	0.956155i	$0.594607\pi$
$758$	0	0
$759$	−11.7321	−0.425850
$760$	0	0
$761$	25.8647	0.937595	0.468798	−	0.883306i	$-0.344687\pi$
$761$	25.8647	0.937595	0.468798	+	0.883306i	$0.344687\pi$
$762$	0	0
$763$	−33.1981	−1.20185
$764$	0	0
$765$	−3.46409	−0.125244
$766$	0	0
$767$	7.19296	0.259723
$768$	0	0
$769$	−4.79053	−0.172751	−0.0863754	−	0.996263i	$-0.527528\pi$
$769$	−4.79053	−0.172751	−0.0863754	+	0.996263i	$0.527528\pi$
$770$	0	0
$771$	−0.0186019	−0.000669932 0
$772$	0	0
$773$	−40.2333	−1.44709	−0.723545	−	0.690277i	$-0.757488\pi$
$773$	−40.2333	−1.44709	−0.723545	+	0.690277i	$0.757488\pi$
$774$	0	0
$775$	15.8344	0.568788
$776$	0	0
$777$	−7.71166	−0.276654
$778$	0	0
$779$	37.9111	1.35831
$780$	0	0
$781$	14.8962	0.533027
$782$	0	0
$783$	−7.57935	−0.270864
$784$	0	0
$785$	−3.36645	−0.120154
$786$	0	0
$787$	30.6272	1.09174	0.545870	−	0.837870i	$-0.316199\pi$
$787$	30.6272	1.09174	0.545870	+	0.837870i	$0.316199\pi$
$788$	0	0
$789$	26.3022	0.936382
$790$	0	0
$791$	8.36641	0.297475
$792$	0	0
$793$	−2.94154	−0.104457
$794$	0	0
$795$	6.33672	0.224740
$796$	0	0
$797$	−44.7217	−1.58412	−0.792062	−	0.610440i	$-0.790993\pi$
$797$	−44.7217	−1.58412	−0.792062	+	0.610440i	$0.790993\pi$
$798$	0	0
$799$	−11.3259	−0.400682
$800$	0	0
$801$	5.40840	0.191096
$802$	0	0
$803$	−23.8340	−0.841082
$804$	0	0
$805$	21.2811	0.750060
$806$	0	0
$807$	26.5437	0.934383
$808$	0	0
$809$	24.7809	0.871250	0.435625	−	0.900128i	$-0.356527\pi$
$809$	24.7809	0.871250	0.435625	+	0.900128i	$0.356527\pi$
$810$	0	0
$811$	54.2137	1.90370	0.951849	−	0.306566i	$-0.0991799\pi$
$811$	54.2137	1.90370	0.951849	+	0.306566i	$0.0991799\pi$
$812$	0	0
$813$	26.0934	0.915137
$814$	0	0
$815$	−18.7239	−0.655871
$816$	0	0
$817$	−2.85361	−0.0998351
$818$	0	0
$819$	6.03682	0.210944
$820$	0	0
$821$	−2.50638	−0.0874733	−0.0437367	−	0.999043i	$-0.513926\pi$
$821$	−2.50638	−0.0874733	−0.0437367	+	0.999043i	$0.513926\pi$
$822$	0	0
$823$	30.1562	1.05118	0.525590	−	0.850738i	$-0.323845\pi$
$823$	30.1562	1.05118	0.525590	+	0.850738i	$0.323845\pi$
$824$	0	0
$825$	11.8100	0.411173
$826$	0	0
$827$	11.3341	0.394126	0.197063	−	0.980391i	$-0.436860\pi$
$827$	11.3341	0.394126	0.197063	+	0.980391i	$0.436860\pi$
$828$	0	0
$829$	−1.59043	−0.0552380	−0.0276190	−	0.999619i	$-0.508793\pi$
$829$	−1.59043	−0.0552380	−0.0276190	+	0.999619i	$0.508793\pi$
$830$	0	0
$831$	−3.66862	−0.127263
$832$	0	0
$833$	−60.1847	−2.08528
$834$	0	0
$835$	2.13585	0.0739143
$836$	0	0
$837$	4.19883	0.145133
$838$	0	0
$839$	−33.8915	−1.17007	−0.585033	−	0.811009i	$-0.698918\pi$
$839$	−33.8915	−1.17007	−0.585033	+	0.811009i	$0.698918\pi$
$840$	0	0
$841$	28.4465	0.980915
$842$	0	0
$843$	20.0496	0.690546
$844$	0	0
$845$	−12.8726	−0.442830
$846$	0	0
$847$	−6.11099	−0.209976
$848$	0	0
$849$	−24.6173	−0.844863
$850$	0	0
$851$	5.63771	0.193258
$852$	0	0
$853$	−7.01551	−0.240207	−0.120103	−	0.992761i	$-0.538323\pi$
$853$	−7.01551	−0.240207	−0.120103	+	0.992761i	$0.538323\pi$
$854$	0	0
$855$	7.44247	0.254527
$856$	0	0
$857$	−23.2807	−0.795254	−0.397627	−	0.917547i	$-0.630166\pi$
$857$	−23.2807	−0.795254	−0.397627	+	0.917547i	$0.630166\pi$
$858$	0	0
$859$	3.99330	0.136250	0.0681249	−	0.997677i	$-0.478298\pi$
$859$	3.99330	0.136250	0.0681249	+	0.997677i	$0.478298\pi$
$860$	0	0
$861$	−28.9364	−0.986149
$862$	0	0
$863$	−50.2347	−1.71001	−0.855004	−	0.518621i	$-0.826446\pi$
$863$	−50.2347	−1.71001	−0.855004	+	0.518621i	$0.826446\pi$
$864$	0	0
$865$	−11.8916	−0.404328
$866$	0	0
$867$	7.23490	0.245710
$868$	0	0
$869$	7.26125	0.246321
$870$	0	0
$871$	−8.20839	−0.278131
$872$	0	0
$873$	−12.2878	−0.415878
$874$	0	0
$875$	−49.8254	−1.68441
$876$	0	0
$877$	−38.6509	−1.30515	−0.652575	−	0.757725i	$-0.726311\pi$
$877$	−38.6509	−1.30515	−0.652575	+	0.757725i	$0.726311\pi$
$878$	0	0
$879$	−13.6451	−0.460239
$880$	0	0
$881$	26.6998	0.899540	0.449770	−	0.893144i	$-0.351506\pi$
$881$	26.6998	0.899540	0.449770	+	0.893144i	$0.351506\pi$
$882$	0	0
$883$	36.2222	1.21897	0.609486	−	0.792797i	$-0.291376\pi$
$883$	36.2222	1.21897	0.609486	+	0.792797i	$0.291376\pi$
$884$	0	0
$885$	−6.76853	−0.227522
$886$	0	0
$887$	−8.61474	−0.289255	−0.144627	−	0.989486i	$-0.546198\pi$
$887$	−8.61474	−0.289255	−0.144627	+	0.989486i	$0.546198\pi$
$888$	0	0
$889$	−94.5598	−3.17143
$890$	0	0
$891$	3.13169	0.104916
$892$	0	0
$893$	24.3333	0.814284
$894$	0	0
$895$	24.7016	0.825683
$896$	0	0
$897$	−4.41330	−0.147356
$898$	0	0
$899$	−31.8244	−1.06140
$900$	0	0
$901$	17.8629	0.595100
$902$	0	0
$903$	2.17807	0.0724817
$904$	0	0
$905$	21.9318	0.729037
$906$	0	0
$907$	2.33836	0.0776440	0.0388220	−	0.999246i	$-0.487639\pi$
$907$	2.33836	0.0776440	0.0388220	+	0.999246i	$0.487639\pi$
$908$	0	0
$909$	7.44333	0.246880
$910$	0	0
$911$	11.9046	0.394417	0.197209	−	0.980362i	$-0.436812\pi$
$911$	11.9046	0.394417	0.197209	+	0.980362i	$0.436812\pi$
$912$	0	0
$913$	−20.3085	−0.672112
$914$	0	0
$915$	2.76797	0.0915063
$916$	0	0
$917$	−7.17428	−0.236916
$918$	0	0
$919$	−43.9153	−1.44863	−0.724316	−	0.689469i	$-0.757844\pi$
$919$	−43.9153	−1.44863	−0.724316	+	0.689469i	$0.757844\pi$
$920$	0	0
$921$	−8.15379	−0.268677
$922$	0	0
$923$	5.60352	0.184442
$924$	0	0
$925$	−5.67514	−0.186597
$926$	0	0
$927$	−3.50497	−0.115118
$928$	0	0
$929$	10.6535	0.349529	0.174765	−	0.984610i	$-0.444084\pi$
$929$	10.6535	0.349529	0.174765	+	0.984610i	$0.444084\pi$
$930$	0	0
$931$	129.305	4.23779
$932$	0	0
$933$	−6.49478	−0.212630
$934$	0	0
$935$	−10.8485	−0.354782
$936$	0	0
$937$	34.1412	1.11534	0.557672	−	0.830061i	$-0.311695\pi$
$937$	34.1412	1.11534	0.557672	+	0.830061i	$0.311695\pi$
$938$	0	0
$939$	−22.9767	−0.749816
$940$	0	0
$941$	−14.6382	−0.477193	−0.238597	−	0.971119i	$-0.576687\pi$
$941$	−14.6382	−0.477193	−0.238597	+	0.971119i	$0.576687\pi$
$942$	0	0
$943$	21.1543	0.688879
$944$	0	0
$945$	−5.68061	−0.184790
$946$	0	0
$947$	−37.6214	−1.22253	−0.611266	−	0.791425i	$-0.709339\pi$
$947$	−37.6214	−1.22253	−0.611266	+	0.791425i	$0.709339\pi$
$948$	0	0
$949$	−8.96565	−0.291037
$950$	0	0
$951$	−5.91146	−0.191692
$952$	0	0
$953$	39.0346	1.26445	0.632227	−	0.774783i	$-0.282141\pi$
$953$	39.0346	1.26445	0.632227	+	0.774783i	$0.282141\pi$
$954$	0	0
$955$	−9.61906	−0.311265
$956$	0	0
$957$	−23.7362	−0.767281
$958$	0	0
$959$	65.3615	2.11063
$960$	0	0
$961$	−13.3698	−0.431285
$962$	0	0
$963$	−7.89790	−0.254506
$964$	0	0
$965$	28.0956	0.904431
$966$	0	0
$967$	−8.40670	−0.270341	−0.135171	−	0.990822i	$-0.543158\pi$
$967$	−8.40670	−0.270341	−0.135171	+	0.990822i	$0.543158\pi$
$968$	0	0
$969$	20.9800	0.673973
$970$	0	0
$971$	20.9439	0.672123	0.336061	−	0.941840i	$-0.390905\pi$
$971$	20.9439	0.672123	0.336061	+	0.941840i	$0.390905\pi$
$972$	0	0
$973$	−101.399	−3.25070
$974$	0	0
$975$	4.44260	0.142277
$976$	0	0
$977$	−18.3747	−0.587860	−0.293930	−	0.955827i	$-0.594963\pi$
$977$	−18.3747	−0.587860	−0.293930	+	0.955827i	$0.594963\pi$
$978$	0	0
$979$	16.9374	0.541322
$980$	0	0
$981$	−6.47842	−0.206840
$982$	0	0
$983$	48.0392	1.53221	0.766106	−	0.642714i	$-0.222192\pi$
$983$	48.0392	1.53221	0.766106	+	0.642714i	$0.222192\pi$
$984$	0	0
$985$	−16.0758	−0.512218
$986$	0	0
$987$	−18.5729	−0.591181
$988$	0	0
$989$	−1.59231	−0.0506324
$990$	0	0
$991$	−20.8608	−0.662664	−0.331332	−	0.943514i	$-0.607498\pi$
$991$	−20.8608	−0.662664	−0.331332	+	0.943514i	$0.607498\pi$
$992$	0	0
$993$	−25.3247	−0.803656
$994$	0	0
$995$	9.63445	0.305433
$996$	0	0
$997$	−49.2337	−1.55925	−0.779623	−	0.626249i	$-0.784589\pi$
$997$	−49.2337	−1.55925	−0.779623	+	0.626249i	$0.784589\pi$
$998$	0	0
$999$	−1.50489	−0.0476125

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.p.1.8	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.p.1.8	✓	14	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.10854 q^{5} +5.12441 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.10854 q^{5} +5.12441 q^{7} +1.00000 q^{9} +3.13169 q^{11} +1.17805 q^{13} -1.10854 q^{15} -3.12492 q^{17} +6.71377 q^{19} -5.12441 q^{21} +3.74627 q^{23} -3.77114 q^{25} -1.00000 q^{27} +7.57935 q^{29} -4.19883 q^{31} -3.13169 q^{33} +5.68061 q^{35} +1.50489 q^{37} -1.17805 q^{39} +5.64677 q^{41} -0.425038 q^{43} +1.10854 q^{45} +3.62439 q^{47} +19.2596 q^{49} +3.12492 q^{51} -5.71629 q^{53} +3.47160 q^{55} -6.71377 q^{57} +6.10581 q^{59} -2.49695 q^{61} +5.12441 q^{63} +1.30592 q^{65} -6.96777 q^{67} -3.74627 q^{69} +4.75660 q^{71} -7.61057 q^{73} +3.77114 q^{75} +16.0481 q^{77} +2.31864 q^{79} +1.00000 q^{81} -6.48483 q^{83} -3.46409 q^{85} -7.57935 q^{87} +5.40840 q^{89} +6.03682 q^{91} +4.19883 q^{93} +7.44247 q^{95} -12.2878 q^{97} +3.13169 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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