LMFDB - Embedded newform 6024.2.a.p.1.5

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			1.5
Root			$-1.55864$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.55864 q^{5} -1.20577 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.55864 q^{5} -1.20577 q^{7} +1.00000 q^{9} -0.221005 q^{11} +4.83624 q^{13} +1.55864 q^{15} +6.20555 q^{17} -3.00737 q^{19} +1.20577 q^{21} -1.45466 q^{23} -2.57064 q^{25} -1.00000 q^{27} +1.15068 q^{29} -5.70285 q^{31} +0.221005 q^{33} +1.87936 q^{35} +8.16576 q^{37} -4.83624 q^{39} -0.660432 q^{41} +7.32166 q^{43} -1.55864 q^{45} +3.54575 q^{47} -5.54612 q^{49} -6.20555 q^{51} +2.53764 q^{53} +0.344468 q^{55} +3.00737 q^{57} -12.4392 q^{59} -8.51976 q^{61} -1.20577 q^{63} -7.53795 q^{65} +2.96050 q^{67} +1.45466 q^{69} -10.2911 q^{71} +10.6084 q^{73} +2.57064 q^{75} +0.266481 q^{77} +1.81922 q^{79} +1.00000 q^{81} -1.70577 q^{83} -9.67221 q^{85} -1.15068 q^{87} +11.8623 q^{89} -5.83138 q^{91} +5.70285 q^{93} +4.68741 q^{95} -6.16322 q^{97} -0.221005 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.55864	−0.697045	−0.348522	−	0.937300i	$-0.613316\pi$
$5$	−1.55864	−0.697045	−0.348522	+	0.937300i	$0.613316\pi$
$6$	0	0
$7$	−1.20577	−0.455737	−0.227869	−	0.973692i	$-0.573176\pi$
$7$	−1.20577	−0.455737	−0.227869	+	0.973692i	$0.573176\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.221005	−0.0666356	−0.0333178	−	0.999445i	$-0.510607\pi$
$11$	−0.221005	−0.0666356	−0.0333178	+	0.999445i	$0.510607\pi$
$12$	0	0
$13$	4.83624	1.34133	0.670665	−	0.741760i	$-0.266009\pi$
$13$	4.83624	1.34133	0.670665	+	0.741760i	$0.266009\pi$
$14$	0	0
$15$	1.55864	0.402439
$16$	0	0
$17$	6.20555	1.50507	0.752533	−	0.658554i	$-0.228832\pi$
$17$	6.20555	1.50507	0.752533	+	0.658554i	$0.228832\pi$
$18$	0	0
$19$	−3.00737	−0.689939	−0.344970	−	0.938614i	$-0.612111\pi$
$19$	−3.00737	−0.689939	−0.344970	+	0.938614i	$0.612111\pi$
$20$	0	0
$21$	1.20577	0.263120
$22$	0	0
$23$	−1.45466	−0.303317	−0.151658	−	0.988433i	$-0.548461\pi$
$23$	−1.45466	−0.303317	−0.151658	+	0.988433i	$0.548461\pi$
$24$	0	0
$25$	−2.57064	−0.514129
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.15068	0.213675	0.106838	−	0.994276i	$-0.465927\pi$
$29$	1.15068	0.213675	0.106838	+	0.994276i	$0.465927\pi$
$30$	0	0
$31$	−5.70285	−1.02426	−0.512131	−	0.858907i	$-0.671144\pi$
$31$	−5.70285	−1.02426	−0.512131	+	0.858907i	$0.671144\pi$
$32$	0	0
$33$	0.221005	0.0384721
$34$	0	0
$35$	1.87936	0.317669
$36$	0	0
$37$	8.16576	1.34244	0.671221	−	0.741257i	$-0.265770\pi$
$37$	8.16576	1.34244	0.671221	+	0.741257i	$0.265770\pi$
$38$	0	0
$39$	−4.83624	−0.774417
$40$	0	0
$41$	−0.660432	−0.103142	−0.0515711	−	0.998669i	$-0.516423\pi$
$41$	−0.660432	−0.103142	−0.0515711	+	0.998669i	$0.516423\pi$
$42$	0	0
$43$	7.32166	1.11654	0.558271	−	0.829658i	$-0.311465\pi$
$43$	7.32166	1.11654	0.558271	+	0.829658i	$0.311465\pi$
$44$	0	0
$45$	−1.55864	−0.232348
$46$	0	0
$47$	3.54575	0.517201	0.258600	−	0.965984i	$-0.416739\pi$
$47$	3.54575	0.517201	0.258600	+	0.965984i	$0.416739\pi$
$48$	0	0
$49$	−5.54612	−0.792304
$50$	0	0
$51$	−6.20555	−0.868951
$52$	0	0
$53$	2.53764	0.348571	0.174286	−	0.984695i	$-0.444238\pi$
$53$	2.53764	0.348571	0.174286	+	0.984695i	$0.444238\pi$
$54$	0	0
$55$	0.344468	0.0464480
$56$	0	0
$57$	3.00737	0.398336
$58$	0	0
$59$	−12.4392	−1.61945	−0.809725	−	0.586810i	$-0.800384\pi$
$59$	−12.4392	−1.61945	−0.809725	+	0.586810i	$0.800384\pi$
$60$	0	0
$61$	−8.51976	−1.09084	−0.545422	−	0.838162i	$-0.683631\pi$
$61$	−8.51976	−1.09084	−0.545422	+	0.838162i	$0.683631\pi$
$62$	0	0
$63$	−1.20577	−0.151912
$64$	0	0
$65$	−7.53795	−0.934967
$66$	0	0
$67$	2.96050	0.361682	0.180841	−	0.983512i	$-0.442118\pi$
$67$	2.96050	0.361682	0.180841	+	0.983512i	$0.442118\pi$
$68$	0	0
$69$	1.45466	0.175120
$70$	0	0
$71$	−10.2911	−1.22133	−0.610664	−	0.791890i	$-0.709097\pi$
$71$	−10.2911	−1.22133	−0.610664	+	0.791890i	$0.709097\pi$
$72$	0	0
$73$	10.6084	1.24162	0.620810	−	0.783961i	$-0.286804\pi$
$73$	10.6084	1.24162	0.620810	+	0.783961i	$0.286804\pi$
$74$	0	0
$75$	2.57064	0.296832
$76$	0	0
$77$	0.266481	0.0303683
$78$	0	0
$79$	1.81922	0.204678	0.102339	−	0.994750i	$-0.467367\pi$
$79$	1.81922	0.204678	0.102339	+	0.994750i	$0.467367\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.70577	−0.187232	−0.0936162	−	0.995608i	$-0.529843\pi$
$83$	−1.70577	−0.187232	−0.0936162	+	0.995608i	$0.529843\pi$
$84$	0	0
$85$	−9.67221	−1.04910
$86$	0	0
$87$	−1.15068	−0.123365
$88$	0	0
$89$	11.8623	1.25740	0.628701	−	0.777647i	$-0.283587\pi$
$89$	11.8623	1.25740	0.628701	+	0.777647i	$0.283587\pi$
$90$	0	0
$91$	−5.83138	−0.611294
$92$	0	0
$93$	5.70285	0.591358
$94$	0	0
$95$	4.68741	0.480918
$96$	0	0
$97$	−6.16322	−0.625780	−0.312890	−	0.949789i	$-0.601297\pi$
$97$	−6.16322	−0.625780	−0.312890	+	0.949789i	$0.601297\pi$
$98$	0	0
$99$	−0.221005	−0.0222119
$100$	0	0
$101$	6.72468	0.669131	0.334565	−	0.942373i	$-0.391411\pi$
$101$	6.72468	0.669131	0.334565	+	0.942373i	$0.391411\pi$
$102$	0	0
$103$	13.8894	1.36856	0.684281	−	0.729218i	$-0.260116\pi$
$103$	13.8894	1.36856	0.684281	+	0.729218i	$0.260116\pi$
$104$	0	0
$105$	−1.87936	−0.183406
$106$	0	0
$107$	6.94730	0.671621	0.335811	−	0.941930i	$-0.390990\pi$
$107$	6.94730	0.671621	0.335811	+	0.941930i	$0.390990\pi$
$108$	0	0
$109$	−13.4503	−1.28830	−0.644152	−	0.764897i	$-0.722790\pi$
$109$	−13.4503	−1.28830	−0.644152	+	0.764897i	$0.722790\pi$
$110$	0	0
$111$	−8.16576	−0.775060
$112$	0	0
$113$	−10.6447	−1.00137	−0.500685	−	0.865630i	$-0.666919\pi$
$113$	−10.6447	−1.00137	−0.500685	+	0.865630i	$0.666919\pi$
$114$	0	0
$115$	2.26728	0.211425
$116$	0	0
$117$	4.83624	0.447110
$118$	0	0
$119$	−7.48245	−0.685915
$120$	0	0
$121$	−10.9512	−0.995560
$122$	0	0
$123$	0.660432	0.0595492
$124$	0	0
$125$	11.7999	1.05542
$126$	0	0
$127$	−5.48527	−0.486739	−0.243369	−	0.969934i	$-0.578253\pi$
$127$	−5.48527	−0.486739	−0.243369	+	0.969934i	$0.578253\pi$
$128$	0	0
$129$	−7.32166	−0.644636
$130$	0	0
$131$	2.95751	0.258399	0.129200	−	0.991619i	$-0.458759\pi$
$131$	2.95751	0.258399	0.129200	+	0.991619i	$0.458759\pi$
$132$	0	0
$133$	3.62619	0.314431
$134$	0	0
$135$	1.55864	0.134146
$136$	0	0
$137$	17.8018	1.52091	0.760456	−	0.649389i	$-0.224975\pi$
$137$	17.8018	1.52091	0.760456	+	0.649389i	$0.224975\pi$
$138$	0	0
$139$	−13.0603	−1.10776	−0.553878	−	0.832598i	$-0.686853\pi$
$139$	−13.0603	−1.10776	−0.553878	+	0.832598i	$0.686853\pi$
$140$	0	0
$141$	−3.54575	−0.298606
$142$	0	0
$143$	−1.06883	−0.0893804
$144$	0	0
$145$	−1.79349	−0.148941
$146$	0	0
$147$	5.54612	0.457437
$148$	0	0
$149$	−11.1633	−0.914537	−0.457269	−	0.889329i	$-0.651172\pi$
$149$	−11.1633	−0.914537	−0.457269	+	0.889329i	$0.651172\pi$
$150$	0	0
$151$	5.09133	0.414326	0.207163	−	0.978306i	$-0.433577\pi$
$151$	5.09133	0.414326	0.207163	+	0.978306i	$0.433577\pi$
$152$	0	0
$153$	6.20555	0.501689
$154$	0	0
$155$	8.88868	0.713956
$156$	0	0
$157$	−18.4053	−1.46890	−0.734450	−	0.678663i	$-0.762560\pi$
$157$	−18.4053	−1.46890	−0.734450	+	0.678663i	$0.762560\pi$
$158$	0	0
$159$	−2.53764	−0.201248
$160$	0	0
$161$	1.75398	0.138233
$162$	0	0
$163$	−9.19102	−0.719896	−0.359948	−	0.932972i	$-0.617206\pi$
$163$	−9.19102	−0.719896	−0.359948	+	0.932972i	$0.617206\pi$
$164$	0	0
$165$	−0.344468	−0.0268168
$166$	0	0
$167$	14.1064	1.09158	0.545792	−	0.837921i	$-0.316229\pi$
$167$	14.1064	1.09158	0.545792	+	0.837921i	$0.316229\pi$
$168$	0	0
$169$	10.3892	0.799167
$170$	0	0
$171$	−3.00737	−0.229980
$172$	0	0
$173$	19.1605	1.45675	0.728373	−	0.685180i	$-0.240277\pi$
$173$	19.1605	1.45675	0.728373	+	0.685180i	$0.240277\pi$
$174$	0	0
$175$	3.09960	0.234308
$176$	0	0
$177$	12.4392	0.934990
$178$	0	0
$179$	10.6209	0.793847	0.396924	−	0.917852i	$-0.370078\pi$
$179$	10.6209	0.793847	0.396924	+	0.917852i	$0.370078\pi$
$180$	0	0
$181$	−6.14553	−0.456794	−0.228397	−	0.973568i	$-0.573348\pi$
$181$	−6.14553	−0.456794	−0.228397	+	0.973568i	$0.573348\pi$
$182$	0	0
$183$	8.51976	0.629799
$184$	0	0
$185$	−12.7275	−0.935742
$186$	0	0
$187$	−1.37146	−0.100291
$188$	0	0
$189$	1.20577	0.0877067
$190$	0	0
$191$	14.2392	1.03031	0.515155	−	0.857097i	$-0.327734\pi$
$191$	14.2392	1.03031	0.515155	+	0.857097i	$0.327734\pi$
$192$	0	0
$193$	17.6605	1.27123	0.635617	−	0.772005i	$-0.280746\pi$
$193$	17.6605	1.27123	0.635617	+	0.772005i	$0.280746\pi$
$194$	0	0
$195$	7.53795	0.539804
$196$	0	0
$197$	14.6848	1.04625	0.523125	−	0.852256i	$-0.324766\pi$
$197$	14.6848	1.04625	0.523125	+	0.852256i	$0.324766\pi$
$198$	0	0
$199$	21.7002	1.53829	0.769143	−	0.639077i	$-0.220683\pi$
$199$	21.7002	1.53829	0.769143	+	0.639077i	$0.220683\pi$
$200$	0	0
$201$	−2.96050	−0.208817
$202$	0	0
$203$	−1.38745	−0.0973798
$204$	0	0
$205$	1.02938	0.0718947
$206$	0	0
$207$	−1.45466	−0.101106
$208$	0	0
$209$	0.664646	0.0459745
$210$	0	0
$211$	11.5282	0.793633	0.396817	−	0.917898i	$-0.370115\pi$
$211$	11.5282	0.793633	0.396817	+	0.917898i	$0.370115\pi$
$212$	0	0
$213$	10.2911	0.705134
$214$	0	0
$215$	−11.4118	−0.778280
$216$	0	0
$217$	6.87631	0.466794
$218$	0	0
$219$	−10.6084	−0.716850
$220$	0	0
$221$	30.0115	2.01879
$222$	0	0
$223$	16.1662	1.08257	0.541284	−	0.840840i	$-0.317938\pi$
$223$	16.1662	1.08257	0.541284	+	0.840840i	$0.317938\pi$
$224$	0	0
$225$	−2.57064	−0.171376
$226$	0	0
$227$	10.5109	0.697633	0.348817	−	0.937191i	$-0.386584\pi$
$227$	10.5109	0.697633	0.348817	+	0.937191i	$0.386584\pi$
$228$	0	0
$229$	4.75418	0.314165	0.157082	−	0.987585i	$-0.449791\pi$
$229$	4.75418	0.314165	0.157082	+	0.987585i	$0.449791\pi$
$230$	0	0
$231$	−0.266481	−0.0175332
$232$	0	0
$233$	16.1767	1.05977	0.529886	−	0.848069i	$-0.322235\pi$
$233$	16.1767	1.05977	0.529886	+	0.848069i	$0.322235\pi$
$234$	0	0
$235$	−5.52654	−0.360512
$236$	0	0
$237$	−1.81922	−0.118171
$238$	0	0
$239$	19.6314	1.26985	0.634926	−	0.772573i	$-0.281030\pi$
$239$	19.6314	1.26985	0.634926	+	0.772573i	$0.281030\pi$
$240$	0	0
$241$	26.1045	1.68154	0.840770	−	0.541393i	$-0.182103\pi$
$241$	26.1045	1.68154	0.840770	+	0.541393i	$0.182103\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	8.64441	0.552271
$246$	0	0
$247$	−14.5444	−0.925436
$248$	0	0
$249$	1.70577	0.108099
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	0.321487	0.0202117
$254$	0	0
$255$	9.67221	0.605697
$256$	0	0
$257$	1.42058	0.0886134	0.0443067	−	0.999018i	$-0.485892\pi$
$257$	1.42058	0.0886134	0.0443067	+	0.999018i	$0.485892\pi$
$258$	0	0
$259$	−9.84601	−0.611801
$260$	0	0
$261$	1.15068	0.0712251
$262$	0	0
$263$	−13.9650	−0.861117	−0.430559	−	0.902563i	$-0.641683\pi$
$263$	−13.9650	−0.861117	−0.430559	+	0.902563i	$0.641683\pi$
$264$	0	0
$265$	−3.95526	−0.242970
$266$	0	0
$267$	−11.8623	−0.725962
$268$	0	0
$269$	11.7559	0.716770	0.358385	−	0.933574i	$-0.383328\pi$
$269$	11.7559	0.716770	0.358385	+	0.933574i	$0.383328\pi$
$270$	0	0
$271$	−1.96657	−0.119460	−0.0597302	−	0.998215i	$-0.519024\pi$
$271$	−1.96657	−0.119460	−0.0597302	+	0.998215i	$0.519024\pi$
$272$	0	0
$273$	5.83138	0.352931
$274$	0	0
$275$	0.568126	0.0342593
$276$	0	0
$277$	3.87722	0.232959	0.116480	−	0.993193i	$-0.462839\pi$
$277$	3.87722	0.232959	0.116480	+	0.993193i	$0.462839\pi$
$278$	0	0
$279$	−5.70285	−0.341421
$280$	0	0
$281$	5.96162	0.355641	0.177820	−	0.984063i	$-0.443095\pi$
$281$	5.96162	0.355641	0.177820	+	0.984063i	$0.443095\pi$
$282$	0	0
$283$	14.5355	0.864046	0.432023	−	0.901863i	$-0.357800\pi$
$283$	14.5355	0.864046	0.432023	+	0.901863i	$0.357800\pi$
$284$	0	0
$285$	−4.68741	−0.277658
$286$	0	0
$287$	0.796328	0.0470058
$288$	0	0
$289$	21.5088	1.26523
$290$	0	0
$291$	6.16322	0.361294
$292$	0	0
$293$	−19.2222	−1.12297	−0.561486	−	0.827486i	$-0.689770\pi$
$293$	−19.2222	−1.12297	−0.561486	+	0.827486i	$0.689770\pi$
$294$	0	0
$295$	19.3883	1.12883
$296$	0	0
$297$	0.221005	0.0128240
$298$	0	0
$299$	−7.03506	−0.406848
$300$	0	0
$301$	−8.82822	−0.508850
$302$	0	0
$303$	−6.72468	−0.386323
$304$	0	0
$305$	13.2792	0.760367
$306$	0	0
$307$	−19.7100	−1.12491	−0.562456	−	0.826827i	$-0.690143\pi$
$307$	−19.7100	−1.12491	−0.562456	+	0.826827i	$0.690143\pi$
$308$	0	0
$309$	−13.8894	−0.790140
$310$	0	0
$311$	18.4123	1.04407	0.522033	−	0.852925i	$-0.325174\pi$
$311$	18.4123	1.04407	0.522033	+	0.852925i	$0.325174\pi$
$312$	0	0
$313$	−16.5056	−0.932954	−0.466477	−	0.884533i	$-0.654477\pi$
$313$	−16.5056	−0.932954	−0.466477	+	0.884533i	$0.654477\pi$
$314$	0	0
$315$	1.87936	0.105890
$316$	0	0
$317$	17.6295	0.990174	0.495087	−	0.868844i	$-0.335136\pi$
$317$	17.6295	0.990174	0.495087	+	0.868844i	$0.335136\pi$
$318$	0	0
$319$	−0.254306	−0.0142384
$320$	0	0
$321$	−6.94730	−0.387761
$322$	0	0
$323$	−18.6624	−1.03840
$324$	0	0
$325$	−12.4322	−0.689617
$326$	0	0
$327$	13.4503	0.743803
$328$	0	0
$329$	−4.27535	−0.235708
$330$	0	0
$331$	−20.0335	−1.10114	−0.550569	−	0.834789i	$-0.685589\pi$
$331$	−20.0335	−1.10114	−0.550569	+	0.834789i	$0.685589\pi$
$332$	0	0
$333$	8.16576	0.447481
$334$	0	0
$335$	−4.61435	−0.252109
$336$	0	0
$337$	22.4444	1.22262	0.611311	−	0.791390i	$-0.290642\pi$
$337$	22.4444	1.22262	0.611311	+	0.791390i	$0.290642\pi$
$338$	0	0
$339$	10.6447	0.578141
$340$	0	0
$341$	1.26036	0.0682523
$342$	0	0
$343$	15.1277	0.816820
$344$	0	0
$345$	−2.26728	−0.122066
$346$	0	0
$347$	−29.4688	−1.58197	−0.790984	−	0.611836i	$-0.790431\pi$
$347$	−29.4688	−1.58197	−0.790984	+	0.611836i	$0.790431\pi$
$348$	0	0
$349$	14.8204	0.793319	0.396660	−	0.917966i	$-0.370169\pi$
$349$	14.8204	0.793319	0.396660	+	0.917966i	$0.370169\pi$
$350$	0	0
$351$	−4.83624	−0.258139
$352$	0	0
$353$	8.53307	0.454170	0.227085	−	0.973875i	$-0.427081\pi$
$353$	8.53307	0.454170	0.227085	+	0.973875i	$0.427081\pi$
$354$	0	0
$355$	16.0401	0.851320
$356$	0	0
$357$	7.48245	0.396013
$358$	0	0
$359$	4.20953	0.222170	0.111085	−	0.993811i	$-0.464567\pi$
$359$	4.20953	0.222170	0.111085	+	0.993811i	$0.464567\pi$
$360$	0	0
$361$	−9.95570	−0.523984
$362$	0	0
$363$	10.9512	0.574787
$364$	0	0
$365$	−16.5347	−0.865464
$366$	0	0
$367$	15.1719	0.791969	0.395985	−	0.918257i	$-0.370403\pi$
$367$	15.1719	0.791969	0.395985	+	0.918257i	$0.370403\pi$
$368$	0	0
$369$	−0.660432	−0.0343807
$370$	0	0
$371$	−3.05980	−0.158857
$372$	0	0
$373$	−24.1894	−1.25248	−0.626241	−	0.779630i	$-0.715407\pi$
$373$	−24.1894	−1.25248	−0.626241	+	0.779630i	$0.715407\pi$
$374$	0	0
$375$	−11.7999	−0.609344
$376$	0	0
$377$	5.56494	0.286609
$378$	0	0
$379$	−9.40290	−0.482995	−0.241497	−	0.970402i	$-0.577638\pi$
$379$	−9.40290	−0.482995	−0.241497	+	0.970402i	$0.577638\pi$
$380$	0	0
$381$	5.48527	0.281019
$382$	0	0
$383$	−10.7756	−0.550605	−0.275303	−	0.961358i	$-0.588778\pi$
$383$	−10.7756	−0.550605	−0.275303	+	0.961358i	$0.588778\pi$
$384$	0	0
$385$	−0.415348	−0.0211681
$386$	0	0
$387$	7.32166	0.372181
$388$	0	0
$389$	21.5844	1.09437	0.547186	−	0.837011i	$-0.315699\pi$
$389$	21.5844	1.09437	0.547186	+	0.837011i	$0.315699\pi$
$390$	0	0
$391$	−9.02694	−0.456512
$392$	0	0
$393$	−2.95751	−0.149187
$394$	0	0
$395$	−2.83551	−0.142670
$396$	0	0
$397$	−13.3154	−0.668279	−0.334140	−	0.942524i	$-0.608446\pi$
$397$	−13.3154	−0.668279	−0.334140	+	0.942524i	$0.608446\pi$
$398$	0	0
$399$	−3.62619	−0.181537
$400$	0	0
$401$	32.5015	1.62305	0.811524	−	0.584320i	$-0.198639\pi$
$401$	32.5015	1.62305	0.811524	+	0.584320i	$0.198639\pi$
$402$	0	0
$403$	−27.5803	−1.37387
$404$	0	0
$405$	−1.55864	−0.0774494
$406$	0	0
$407$	−1.80468	−0.0894545
$408$	0	0
$409$	−1.80764	−0.0893820	−0.0446910	−	0.999001i	$-0.514230\pi$
$409$	−1.80764	−0.0893820	−0.0446910	+	0.999001i	$0.514230\pi$
$410$	0	0
$411$	−17.8018	−0.878099
$412$	0	0
$413$	14.9988	0.738044
$414$	0	0
$415$	2.65868	0.130509
$416$	0	0
$417$	13.0603	0.639564
$418$	0	0
$419$	14.1590	0.691712	0.345856	−	0.938288i	$-0.387589\pi$
$419$	14.1590	0.691712	0.345856	+	0.938288i	$0.387589\pi$
$420$	0	0
$421$	14.0419	0.684359	0.342179	−	0.939635i	$-0.388835\pi$
$421$	14.0419	0.684359	0.342179	+	0.939635i	$0.388835\pi$
$422$	0	0
$423$	3.54575	0.172400
$424$	0	0
$425$	−15.9523	−0.773798
$426$	0	0
$427$	10.2729	0.497138
$428$	0	0
$429$	1.06883	0.0516038
$430$	0	0
$431$	8.93472	0.430370	0.215185	−	0.976573i	$-0.430965\pi$
$431$	8.93472	0.430370	0.215185	+	0.976573i	$0.430965\pi$
$432$	0	0
$433$	10.5495	0.506975	0.253487	−	0.967339i	$-0.418422\pi$
$433$	10.5495	0.506975	0.253487	+	0.967339i	$0.418422\pi$
$434$	0	0
$435$	1.79349	0.0859912
$436$	0	0
$437$	4.37469	0.209270
$438$	0	0
$439$	−24.3993	−1.16452	−0.582258	−	0.813004i	$-0.697831\pi$
$439$	−24.3993	−1.16452	−0.582258	+	0.813004i	$0.697831\pi$
$440$	0	0
$441$	−5.54612	−0.264101
$442$	0	0
$443$	−24.6492	−1.17112	−0.585560	−	0.810629i	$-0.699125\pi$
$443$	−24.6492	−1.17112	−0.585560	+	0.810629i	$0.699125\pi$
$444$	0	0
$445$	−18.4891	−0.876465
$446$	0	0
$447$	11.1633	0.528008
$448$	0	0
$449$	8.09838	0.382186	0.191093	−	0.981572i	$-0.438797\pi$
$449$	8.09838	0.382186	0.191093	+	0.981572i	$0.438797\pi$
$450$	0	0
$451$	0.145959	0.00687295
$452$	0	0
$453$	−5.09133	−0.239211
$454$	0	0
$455$	9.08901	0.426099
$456$	0	0
$457$	27.6383	1.29287	0.646433	−	0.762970i	$-0.276260\pi$
$457$	27.6383	1.29287	0.646433	+	0.762970i	$0.276260\pi$
$458$	0	0
$459$	−6.20555	−0.289650
$460$	0	0
$461$	36.0197	1.67760	0.838802	−	0.544437i	$-0.183257\pi$
$461$	36.0197	1.67760	0.838802	+	0.544437i	$0.183257\pi$
$462$	0	0
$463$	6.70699	0.311700	0.155850	−	0.987781i	$-0.450188\pi$
$463$	6.70699	0.311700	0.155850	+	0.987781i	$0.450188\pi$
$464$	0	0
$465$	−8.88868	−0.412203
$466$	0	0
$467$	−27.6408	−1.27907	−0.639533	−	0.768764i	$-0.720872\pi$
$467$	−27.6408	−1.27907	−0.639533	+	0.768764i	$0.720872\pi$
$468$	0	0
$469$	−3.56967	−0.164832
$470$	0	0
$471$	18.4053	0.848070
$472$	0	0
$473$	−1.61813	−0.0744015
$474$	0	0
$475$	7.73089	0.354718
$476$	0	0
$477$	2.53764	0.116190
$478$	0	0
$479$	13.6070	0.621720	0.310860	−	0.950456i	$-0.399383\pi$
$479$	13.6070	0.621720	0.310860	+	0.950456i	$0.399383\pi$
$480$	0	0
$481$	39.4915	1.80066
$482$	0	0
$483$	−1.75398	−0.0798087
$484$	0	0
$485$	9.60623	0.436197
$486$	0	0
$487$	2.42133	0.109721	0.0548604	−	0.998494i	$-0.482529\pi$
$487$	2.42133	0.109721	0.0548604	+	0.998494i	$0.482529\pi$
$488$	0	0
$489$	9.19102	0.415632
$490$	0	0
$491$	−7.75251	−0.349866	−0.174933	−	0.984580i	$-0.555971\pi$
$491$	−7.75251	−0.349866	−0.174933	+	0.984580i	$0.555971\pi$
$492$	0	0
$493$	7.14058	0.321596
$494$	0	0
$495$	0.344468	0.0154827
$496$	0	0
$497$	12.4087	0.556604
$498$	0	0
$499$	−6.40173	−0.286581	−0.143290	−	0.989681i	$-0.545768\pi$
$499$	−6.40173	−0.286581	−0.143290	+	0.989681i	$0.545768\pi$
$500$	0	0
$501$	−14.1064	−0.630226
$502$	0	0
$503$	21.8282	0.973273	0.486637	−	0.873604i	$-0.338224\pi$
$503$	21.8282	0.973273	0.486637	+	0.873604i	$0.338224\pi$
$504$	0	0
$505$	−10.4813	−0.466414
$506$	0	0
$507$	−10.3892	−0.461400
$508$	0	0
$509$	−10.9985	−0.487500	−0.243750	−	0.969838i	$-0.578378\pi$
$509$	−10.9985	−0.487500	−0.243750	+	0.969838i	$0.578378\pi$
$510$	0	0
$511$	−12.7913	−0.565853
$512$	0	0
$513$	3.00737	0.132779
$514$	0	0
$515$	−21.6486	−0.953949
$516$	0	0
$517$	−0.783630	−0.0344640
$518$	0	0
$519$	−19.1605	−0.841053
$520$	0	0
$521$	−22.6218	−0.991078	−0.495539	−	0.868586i	$-0.665029\pi$
$521$	−22.6218	−0.991078	−0.495539	+	0.868586i	$0.665029\pi$
$522$	0	0
$523$	−32.3714	−1.41550	−0.707752	−	0.706461i	$-0.750290\pi$
$523$	−32.3714	−1.41550	−0.707752	+	0.706461i	$0.750290\pi$
$524$	0	0
$525$	−3.09960	−0.135278
$526$	0	0
$527$	−35.3893	−1.54158
$528$	0	0
$529$	−20.8840	−0.907999
$530$	0	0
$531$	−12.4392	−0.539817
$532$	0	0
$533$	−3.19401	−0.138348
$534$	0	0
$535$	−10.8283	−0.468150
$536$	0	0
$537$	−10.6209	−0.458328
$538$	0	0
$539$	1.22572	0.0527956
$540$	0	0
$541$	15.4887	0.665912	0.332956	−	0.942942i	$-0.391954\pi$
$541$	15.4887	0.665912	0.332956	+	0.942942i	$0.391954\pi$
$542$	0	0
$543$	6.14553	0.263730
$544$	0	0
$545$	20.9642	0.898006
$546$	0	0
$547$	17.5549	0.750593	0.375297	−	0.926905i	$-0.377541\pi$
$547$	17.5549	0.750593	0.375297	+	0.926905i	$0.377541\pi$
$548$	0	0
$549$	−8.51976	−0.363615
$550$	0	0
$551$	−3.46052	−0.147423
$552$	0	0
$553$	−2.19356	−0.0932796
$554$	0	0
$555$	12.7275	0.540251
$556$	0	0
$557$	41.5375	1.76000	0.879999	−	0.474975i	$-0.157543\pi$
$557$	41.5375	1.76000	0.879999	+	0.474975i	$0.157543\pi$
$558$	0	0
$559$	35.4093	1.49765
$560$	0	0
$561$	1.37146	0.0579031
$562$	0	0
$563$	31.3490	1.32120	0.660602	−	0.750736i	$-0.270301\pi$
$563$	31.3490	1.32120	0.660602	+	0.750736i	$0.270301\pi$
$564$	0	0
$565$	16.5913	0.698000
$566$	0	0
$567$	−1.20577	−0.0506375
$568$	0	0
$569$	−3.51852	−0.147504	−0.0737521	−	0.997277i	$-0.523497\pi$
$569$	−3.51852	−0.147504	−0.0737521	+	0.997277i	$0.523497\pi$
$570$	0	0
$571$	43.9252	1.83821	0.919106	−	0.394011i	$-0.128913\pi$
$571$	43.9252	1.83821	0.919106	+	0.394011i	$0.128913\pi$
$572$	0	0
$573$	−14.2392	−0.594850
$574$	0	0
$575$	3.73940	0.155944
$576$	0	0
$577$	−2.15907	−0.0898834	−0.0449417	−	0.998990i	$-0.514310\pi$
$577$	−2.15907	−0.0898834	−0.0449417	+	0.998990i	$0.514310\pi$
$578$	0	0
$579$	−17.6605	−0.733947
$580$	0	0
$581$	2.05676	0.0853288
$582$	0	0
$583$	−0.560832	−0.0232273
$584$	0	0
$585$	−7.53795	−0.311656
$586$	0	0
$587$	−14.9079	−0.615314	−0.307657	−	0.951497i	$-0.599545\pi$
$587$	−14.9079	−0.615314	−0.307657	+	0.951497i	$0.599545\pi$
$588$	0	0
$589$	17.1506	0.706678
$590$	0	0
$591$	−14.6848	−0.604053
$592$	0	0
$593$	−13.1392	−0.539563	−0.269782	−	0.962922i	$-0.586951\pi$
$593$	−13.1392	−0.539563	−0.269782	+	0.962922i	$0.586951\pi$
$594$	0	0
$595$	11.6624	0.478113
$596$	0	0
$597$	−21.7002	−0.888130
$598$	0	0
$599$	−28.8798	−1.18000	−0.589998	−	0.807404i	$-0.700872\pi$
$599$	−28.8798	−1.18000	−0.589998	+	0.807404i	$0.700872\pi$
$600$	0	0
$601$	−5.56359	−0.226944	−0.113472	−	0.993541i	$-0.536197\pi$
$601$	−5.56359	−0.226944	−0.113472	+	0.993541i	$0.536197\pi$
$602$	0	0
$603$	2.96050	0.120561
$604$	0	0
$605$	17.0689	0.693950
$606$	0	0
$607$	−1.08895	−0.0441990	−0.0220995	−	0.999756i	$-0.507035\pi$
$607$	−1.08895	−0.0441990	−0.0220995	+	0.999756i	$0.507035\pi$
$608$	0	0
$609$	1.38745	0.0562223
$610$	0	0
$611$	17.1481	0.693737
$612$	0	0
$613$	26.1309	1.05542	0.527709	−	0.849425i	$-0.323051\pi$
$613$	26.1309	1.05542	0.527709	+	0.849425i	$0.323051\pi$
$614$	0	0
$615$	−1.02938	−0.0415084
$616$	0	0
$617$	−30.3868	−1.22333	−0.611664	−	0.791118i	$-0.709499\pi$
$617$	−30.3868	−1.22333	−0.611664	+	0.791118i	$0.709499\pi$
$618$	0	0
$619$	15.9348	0.640473	0.320236	−	0.947338i	$-0.396238\pi$
$619$	15.9348	0.640473	0.320236	+	0.947338i	$0.396238\pi$
$620$	0	0
$621$	1.45466	0.0583733
$622$	0	0
$623$	−14.3032	−0.573045
$624$	0	0
$625$	−5.53857	−0.221543
$626$	0	0
$627$	−0.664646	−0.0265434
$628$	0	0
$629$	50.6730	2.02047
$630$	0	0
$631$	−40.9848	−1.63158	−0.815789	−	0.578349i	$-0.803697\pi$
$631$	−40.9848	−1.63158	−0.815789	+	0.578349i	$0.803697\pi$
$632$	0	0
$633$	−11.5282	−0.458204
$634$	0	0
$635$	8.54955	0.339279
$636$	0	0
$637$	−26.8224	−1.06274
$638$	0	0
$639$	−10.2911	−0.407109
$640$	0	0
$641$	37.5480	1.48306	0.741528	−	0.670922i	$-0.234101\pi$
$641$	37.5480	1.48306	0.741528	+	0.670922i	$0.234101\pi$
$642$	0	0
$643$	29.4843	1.16275	0.581374	−	0.813637i	$-0.302516\pi$
$643$	29.4843	1.16275	0.581374	+	0.813637i	$0.302516\pi$
$644$	0	0
$645$	11.4118	0.449340
$646$	0	0
$647$	35.6241	1.40053	0.700264	−	0.713884i	$-0.253066\pi$
$647$	35.6241	1.40053	0.700264	+	0.713884i	$0.253066\pi$
$648$	0	0
$649$	2.74914	0.107913
$650$	0	0
$651$	−6.87631	−0.269504
$652$	0	0
$653$	19.8545	0.776965	0.388483	−	0.921456i	$-0.372999\pi$
$653$	19.8545	0.776965	0.388483	+	0.921456i	$0.372999\pi$
$654$	0	0
$655$	−4.60970	−0.180116
$656$	0	0
$657$	10.6084	0.413873
$658$	0	0
$659$	4.36864	0.170178	0.0850890	−	0.996373i	$-0.472883\pi$
$659$	4.36864	0.170178	0.0850890	+	0.996373i	$0.472883\pi$
$660$	0	0
$661$	3.63700	0.141463	0.0707315	−	0.997495i	$-0.477467\pi$
$661$	3.63700	0.141463	0.0707315	+	0.997495i	$0.477467\pi$
$662$	0	0
$663$	−30.0115	−1.16555
$664$	0	0
$665$	−5.65193	−0.219172
$666$	0	0
$667$	−1.67384	−0.0648113
$668$	0	0
$669$	−16.1662	−0.625021
$670$	0	0
$671$	1.88291	0.0726891
$672$	0	0
$673$	−11.0663	−0.426576	−0.213288	−	0.976989i	$-0.568417\pi$
$673$	−11.0663	−0.426576	−0.213288	+	0.976989i	$0.568417\pi$
$674$	0	0
$675$	2.57064	0.0989441
$676$	0	0
$677$	10.8256	0.416062	0.208031	−	0.978122i	$-0.433295\pi$
$677$	10.8256	0.416062	0.208031	+	0.978122i	$0.433295\pi$
$678$	0	0
$679$	7.43141	0.285191
$680$	0	0
$681$	−10.5109	−0.402779
$682$	0	0
$683$	−34.4525	−1.31829	−0.659144	−	0.752017i	$-0.729081\pi$
$683$	−34.4525	−1.31829	−0.659144	+	0.752017i	$0.729081\pi$
$684$	0	0
$685$	−27.7466	−1.06014
$686$	0	0
$687$	−4.75418	−0.181383
$688$	0	0
$689$	12.2726	0.467550
$690$	0	0
$691$	−17.0003	−0.646721	−0.323361	−	0.946276i	$-0.604813\pi$
$691$	−17.0003	−0.646721	−0.323361	+	0.946276i	$0.604813\pi$
$692$	0	0
$693$	0.266481	0.0101228
$694$	0	0
$695$	20.3562	0.772156
$696$	0	0
$697$	−4.09835	−0.155236
$698$	0	0
$699$	−16.1767	−0.611859
$700$	0	0
$701$	27.6425	1.04404	0.522022	−	0.852932i	$-0.325178\pi$
$701$	27.6425	1.04404	0.522022	+	0.852932i	$0.325178\pi$
$702$	0	0
$703$	−24.5575	−0.926204
$704$	0	0
$705$	5.52654	0.208142
$706$	0	0
$707$	−8.10840	−0.304948
$708$	0	0
$709$	−40.3156	−1.51408	−0.757042	−	0.653366i	$-0.773356\pi$
$709$	−40.3156	−1.51408	−0.757042	+	0.653366i	$0.773356\pi$
$710$	0	0
$711$	1.81922	0.0682262
$712$	0	0
$713$	8.29568	0.310676
$714$	0	0
$715$	1.66593	0.0623021
$716$	0	0
$717$	−19.6314	−0.733150
$718$	0	0
$719$	17.0139	0.634510	0.317255	−	0.948340i	$-0.397239\pi$
$719$	17.0139	0.634510	0.317255	+	0.948340i	$0.397239\pi$
$720$	0	0
$721$	−16.7474	−0.623705
$722$	0	0
$723$	−26.1045	−0.970837
$724$	0	0
$725$	−2.95798	−0.109857
$726$	0	0
$727$	13.0904	0.485494	0.242747	−	0.970090i	$-0.421951\pi$
$727$	13.0904	0.485494	0.242747	+	0.970090i	$0.421951\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	45.4349	1.68047
$732$	0	0
$733$	−23.6276	−0.872706	−0.436353	−	0.899776i	$-0.643730\pi$
$733$	−23.6276	−0.872706	−0.436353	+	0.899776i	$0.643730\pi$
$734$	0	0
$735$	−8.64441	−0.318854
$736$	0	0
$737$	−0.654286	−0.0241009
$738$	0	0
$739$	−41.0534	−1.51018	−0.755088	−	0.655624i	$-0.772406\pi$
$739$	−41.0534	−1.51018	−0.755088	+	0.655624i	$0.772406\pi$
$740$	0	0
$741$	14.5444	0.534301
$742$	0	0
$743$	50.7131	1.86048	0.930241	−	0.366949i	$-0.119598\pi$
$743$	50.7131	1.86048	0.930241	+	0.366949i	$0.119598\pi$
$744$	0	0
$745$	17.3996	0.637473
$746$	0	0
$747$	−1.70577	−0.0624108
$748$	0	0
$749$	−8.37683	−0.306083
$750$	0	0
$751$	−36.0171	−1.31428	−0.657142	−	0.753767i	$-0.728235\pi$
$751$	−36.0171	−1.31428	−0.657142	+	0.753767i	$0.728235\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−7.93554	−0.288804
$756$	0	0
$757$	−37.9888	−1.38073	−0.690364	−	0.723462i	$-0.742550\pi$
$757$	−37.9888	−1.38073	−0.690364	+	0.723462i	$0.742550\pi$
$758$	0	0
$759$	−0.321487	−0.0116692
$760$	0	0
$761$	−38.6247	−1.40014	−0.700072	−	0.714072i	$-0.746849\pi$
$761$	−38.6247	−1.40014	−0.700072	+	0.714072i	$0.746849\pi$
$762$	0	0
$763$	16.2179	0.587128
$764$	0	0
$765$	−9.67221	−0.349700
$766$	0	0
$767$	−60.1590	−2.17222
$768$	0	0
$769$	34.5995	1.24769	0.623844	−	0.781549i	$-0.285570\pi$
$769$	34.5995	1.24769	0.623844	+	0.781549i	$0.285570\pi$
$770$	0	0
$771$	−1.42058	−0.0511610
$772$	0	0
$773$	−43.5293	−1.56564	−0.782819	−	0.622249i	$-0.786219\pi$
$773$	−43.5293	−1.56564	−0.782819	+	0.622249i	$0.786219\pi$
$774$	0	0
$775$	14.6600	0.526603
$776$	0	0
$777$	9.84601	0.353224
$778$	0	0
$779$	1.98617	0.0711618
$780$	0	0
$781$	2.27439	0.0813839
$782$	0	0
$783$	−1.15068	−0.0411218
$784$	0	0
$785$	28.6872	1.02389
$786$	0	0
$787$	26.6348	0.949429	0.474715	−	0.880140i	$-0.342551\pi$
$787$	26.6348	0.949429	0.474715	+	0.880140i	$0.342551\pi$
$788$	0	0
$789$	13.9650	0.497166
$790$	0	0
$791$	12.8350	0.456362
$792$	0	0
$793$	−41.2036	−1.46318
$794$	0	0
$795$	3.95526	0.140279
$796$	0	0
$797$	−0.324424	−0.0114917	−0.00574584	−	0.999983i	$-0.501829\pi$
$797$	−0.324424	−0.0114917	−0.00574584	+	0.999983i	$0.501829\pi$
$798$	0	0
$799$	22.0033	0.778422
$800$	0	0
$801$	11.8623	0.419134
$802$	0	0
$803$	−2.34451	−0.0827361
$804$	0	0
$805$	−2.73382	−0.0963544
$806$	0	0
$807$	−11.7559	−0.413827
$808$	0	0
$809$	11.2545	0.395687	0.197843	−	0.980234i	$-0.436606\pi$
$809$	11.2545	0.395687	0.197843	+	0.980234i	$0.436606\pi$
$810$	0	0
$811$	−18.3184	−0.643246	−0.321623	−	0.946868i	$-0.604228\pi$
$811$	−18.3184	−0.643246	−0.321623	+	0.946868i	$0.604228\pi$
$812$	0	0
$813$	1.96657	0.0689705
$814$	0	0
$815$	14.3255	0.501800
$816$	0	0
$817$	−22.0190	−0.770346
$818$	0	0
$819$	−5.83138	−0.203765
$820$	0	0
$821$	31.3325	1.09351	0.546755	−	0.837293i	$-0.315863\pi$
$821$	31.3325	1.09351	0.546755	+	0.837293i	$0.315863\pi$
$822$	0	0
$823$	15.4731	0.539357	0.269679	−	0.962950i	$-0.413083\pi$
$823$	15.4731	0.539357	0.269679	+	0.962950i	$0.413083\pi$
$824$	0	0
$825$	−0.568126	−0.0197796
$826$	0	0
$827$	47.6558	1.65716	0.828578	−	0.559874i	$-0.189150\pi$
$827$	47.6558	1.65716	0.828578	+	0.559874i	$0.189150\pi$
$828$	0	0
$829$	42.9523	1.49179	0.745897	−	0.666061i	$-0.232021\pi$
$829$	42.9523	1.49179	0.745897	+	0.666061i	$0.232021\pi$
$830$	0	0
$831$	−3.87722	−0.134499
$832$	0	0
$833$	−34.4167	−1.19247
$834$	0	0
$835$	−21.9867	−0.760883
$836$	0	0
$837$	5.70285	0.197119
$838$	0	0
$839$	42.0115	1.45040	0.725198	−	0.688540i	$-0.241748\pi$
$839$	42.0115	1.45040	0.725198	+	0.688540i	$0.241748\pi$
$840$	0	0
$841$	−27.6759	−0.954343
$842$	0	0
$843$	−5.96162	−0.205329
$844$	0	0
$845$	−16.1930	−0.557055
$846$	0	0
$847$	13.2045	0.453714
$848$	0	0
$849$	−14.5355	−0.498857
$850$	0	0
$851$	−11.8784	−0.407185
$852$	0	0
$853$	36.6489	1.25484	0.627418	−	0.778683i	$-0.284112\pi$
$853$	36.6489	1.25484	0.627418	+	0.778683i	$0.284112\pi$
$854$	0	0
$855$	4.68741	0.160306
$856$	0	0
$857$	0.991945	0.0338842	0.0169421	−	0.999856i	$-0.494607\pi$
$857$	0.991945	0.0338842	0.0169421	+	0.999856i	$0.494607\pi$
$858$	0	0
$859$	−20.0171	−0.682974	−0.341487	−	0.939887i	$-0.610931\pi$
$859$	−20.0171	−0.682974	−0.341487	+	0.939887i	$0.610931\pi$
$860$	0	0
$861$	−0.796328	−0.0271388
$862$	0	0
$863$	38.8712	1.32319	0.661596	−	0.749860i	$-0.269879\pi$
$863$	38.8712	1.32319	0.661596	+	0.749860i	$0.269879\pi$
$864$	0	0
$865$	−29.8643	−1.01542
$866$	0	0
$867$	−21.5088	−0.730478
$868$	0	0
$869$	−0.402058	−0.0136389
$870$	0	0
$871$	14.3177	0.485135
$872$	0	0
$873$	−6.16322	−0.208593
$874$	0	0
$875$	−14.2279	−0.480992
$876$	0	0
$877$	−44.6650	−1.50823	−0.754115	−	0.656743i	$-0.771934\pi$
$877$	−44.6650	−1.50823	−0.754115	+	0.656743i	$0.771934\pi$
$878$	0	0
$879$	19.2222	0.648349
$880$	0	0
$881$	−32.0463	−1.07967	−0.539834	−	0.841771i	$-0.681513\pi$
$881$	−32.0463	−1.07967	−0.539834	+	0.841771i	$0.681513\pi$
$882$	0	0
$883$	−30.6849	−1.03263	−0.516315	−	0.856399i	$-0.672696\pi$
$883$	−30.6849	−1.03263	−0.516315	+	0.856399i	$0.672696\pi$
$884$	0	0
$885$	−19.3883	−0.651730
$886$	0	0
$887$	17.3523	0.582633	0.291316	−	0.956627i	$-0.405907\pi$
$887$	17.3523	0.582633	0.291316	+	0.956627i	$0.405907\pi$
$888$	0	0
$889$	6.61396	0.221825
$890$	0	0
$891$	−0.221005	−0.00740396
$892$	0	0
$893$	−10.6634	−0.356837
$894$	0	0
$895$	−16.5542	−0.553347
$896$	0	0
$897$	7.03506	0.234894
$898$	0	0
$899$	−6.56213	−0.218859
$900$	0	0
$901$	15.7474	0.524623
$902$	0	0
$903$	8.82822	0.293785
$904$	0	0
$905$	9.57867	0.318406
$906$	0	0
$907$	−49.1213	−1.63105	−0.815524	−	0.578724i	$-0.803551\pi$
$907$	−49.1213	−1.63105	−0.815524	+	0.578724i	$0.803551\pi$
$908$	0	0
$909$	6.72468	0.223044
$910$	0	0
$911$	22.1009	0.732236	0.366118	−	0.930568i	$-0.380687\pi$
$911$	22.1009	0.732236	0.366118	+	0.930568i	$0.380687\pi$
$912$	0	0
$913$	0.376984	0.0124764
$914$	0	0
$915$	−13.2792	−0.438998
$916$	0	0
$917$	−3.56607	−0.117762
$918$	0	0
$919$	−45.8415	−1.51217	−0.756087	−	0.654472i	$-0.772891\pi$
$919$	−45.8415	−1.51217	−0.756087	+	0.654472i	$0.772891\pi$
$920$	0	0
$921$	19.7100	0.649468
$922$	0	0
$923$	−49.7701	−1.63820
$924$	0	0
$925$	−20.9913	−0.690188
$926$	0	0
$927$	13.8894	0.456188
$928$	0	0
$929$	43.4814	1.42658	0.713290	−	0.700869i	$-0.247204\pi$
$929$	43.4814	1.42658	0.713290	+	0.700869i	$0.247204\pi$
$930$	0	0
$931$	16.6793	0.546641
$932$	0	0
$933$	−18.4123	−0.602792
$934$	0	0
$935$	2.13761	0.0699073
$936$	0	0
$937$	−32.3659	−1.05735	−0.528673	−	0.848825i	$-0.677310\pi$
$937$	−32.3659	−1.05735	−0.528673	+	0.848825i	$0.677310\pi$
$938$	0	0
$939$	16.5056	0.538641
$940$	0	0
$941$	−9.70056	−0.316229	−0.158115	−	0.987421i	$-0.550542\pi$
$941$	−9.70056	−0.316229	−0.158115	+	0.987421i	$0.550542\pi$
$942$	0	0
$943$	0.960702	0.0312848
$944$	0	0
$945$	−1.87936	−0.0611355
$946$	0	0
$947$	6.75553	0.219525	0.109763	−	0.993958i	$-0.464991\pi$
$947$	6.75553	0.219525	0.109763	+	0.993958i	$0.464991\pi$
$948$	0	0
$949$	51.3048	1.66542
$950$	0	0
$951$	−17.6295	−0.571677
$952$	0	0
$953$	−46.8770	−1.51849	−0.759247	−	0.650802i	$-0.774433\pi$
$953$	−46.8770	−1.51849	−0.759247	+	0.650802i	$0.774433\pi$
$954$	0	0
$955$	−22.1937	−0.718172
$956$	0	0
$957$	0.254306	0.00822054
$958$	0	0
$959$	−21.4649	−0.693136
$960$	0	0
$961$	1.52248	0.0491122
$962$	0	0
$963$	6.94730	0.223874
$964$	0	0
$965$	−27.5264	−0.886106
$966$	0	0
$967$	27.3962	0.881003	0.440502	−	0.897752i	$-0.354801\pi$
$967$	27.3962	0.881003	0.440502	+	0.897752i	$0.354801\pi$
$968$	0	0
$969$	18.6624	0.599523
$970$	0	0
$971$	−53.9378	−1.73094	−0.865472	−	0.500957i	$-0.832982\pi$
$971$	−53.9378	−1.73094	−0.865472	+	0.500957i	$0.832982\pi$
$972$	0	0
$973$	15.7476	0.504846
$974$	0	0
$975$	12.4322	0.398150
$976$	0	0
$977$	46.5088	1.48795	0.743974	−	0.668208i	$-0.232938\pi$
$977$	46.5088	1.48795	0.743974	+	0.668208i	$0.232938\pi$
$978$	0	0
$979$	−2.62163	−0.0837878
$980$	0	0
$981$	−13.4503	−0.429435
$982$	0	0
$983$	−54.4551	−1.73685	−0.868424	−	0.495822i	$-0.834867\pi$
$983$	−54.4551	−1.73685	−0.868424	+	0.495822i	$0.834867\pi$
$984$	0	0
$985$	−22.8883	−0.729283
$986$	0	0
$987$	4.27535	0.136086
$988$	0	0
$989$	−10.6505	−0.338666
$990$	0	0
$991$	2.21224	0.0702740	0.0351370	−	0.999383i	$-0.488813\pi$
$991$	2.21224	0.0702740	0.0351370	+	0.999383i	$0.488813\pi$
$992$	0	0
$993$	20.0335	0.635743
$994$	0	0
$995$	−33.8228	−1.07225
$996$	0	0
$997$	51.1813	1.62093	0.810464	−	0.585789i	$-0.199215\pi$
$997$	51.1813	1.62093	0.810464	+	0.585789i	$0.199215\pi$
$998$	0	0
$999$	−8.16576	−0.258353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.p.1.5	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.p.1.5	✓	14	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.55864 q^{5} -1.20577 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.55864 q^{5} -1.20577 q^{7} +1.00000 q^{9} -0.221005 q^{11} +4.83624 q^{13} +1.55864 q^{15} +6.20555 q^{17} -3.00737 q^{19} +1.20577 q^{21} -1.45466 q^{23} -2.57064 q^{25} -1.00000 q^{27} +1.15068 q^{29} -5.70285 q^{31} +0.221005 q^{33} +1.87936 q^{35} +8.16576 q^{37} -4.83624 q^{39} -0.660432 q^{41} +7.32166 q^{43} -1.55864 q^{45} +3.54575 q^{47} -5.54612 q^{49} -6.20555 q^{51} +2.53764 q^{53} +0.344468 q^{55} +3.00737 q^{57} -12.4392 q^{59} -8.51976 q^{61} -1.20577 q^{63} -7.53795 q^{65} +2.96050 q^{67} +1.45466 q^{69} -10.2911 q^{71} +10.6084 q^{73} +2.57064 q^{75} +0.266481 q^{77} +1.81922 q^{79} +1.00000 q^{81} -1.70577 q^{83} -9.67221 q^{85} -1.15068 q^{87} +11.8623 q^{89} -5.83138 q^{91} +5.70285 q^{93} +4.68741 q^{95} -6.16322 q^{97} -0.221005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) \) 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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