LMFDB - Embedded newform 6024.2.a.p.1.10

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			1.10
Root			$2.29106$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.29106 q^{5} -4.25127 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.29106 q^{5} -4.25127 q^{7} +1.00000 q^{9} -6.06576 q^{11} +2.44568 q^{13} -2.29106 q^{15} +5.39502 q^{17} -1.43938 q^{19} +4.25127 q^{21} +1.68306 q^{23} +0.248959 q^{25} -1.00000 q^{27} -6.04771 q^{29} +5.89508 q^{31} +6.06576 q^{33} -9.73992 q^{35} -5.59860 q^{37} -2.44568 q^{39} +6.81731 q^{41} -1.64680 q^{43} +2.29106 q^{45} +3.10083 q^{47} +11.0733 q^{49} -5.39502 q^{51} -9.76969 q^{53} -13.8970 q^{55} +1.43938 q^{57} -6.28536 q^{59} +2.30692 q^{61} -4.25127 q^{63} +5.60320 q^{65} -7.68981 q^{67} -1.68306 q^{69} +6.54067 q^{71} -5.35117 q^{73} -0.248959 q^{75} +25.7872 q^{77} -7.76709 q^{79} +1.00000 q^{81} +7.64073 q^{83} +12.3603 q^{85} +6.04771 q^{87} -14.7267 q^{89} -10.3972 q^{91} -5.89508 q^{93} -3.29771 q^{95} +15.4903 q^{97} -6.06576 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.29106	1.02459	0.512297	−	0.858809i	$-0.328795\pi$
$5$	2.29106	1.02459	0.512297	+	0.858809i	$0.328795\pi$
$6$	0	0
$7$	−4.25127	−1.60683	−0.803415	−	0.595419i	$-0.796986\pi$
$7$	−4.25127	−1.60683	−0.803415	+	0.595419i	$0.796986\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.06576	−1.82890	−0.914448	−	0.404703i	$-0.867375\pi$
$11$	−6.06576	−1.82890	−0.914448	+	0.404703i	$0.867375\pi$
$12$	0	0
$13$	2.44568	0.678309	0.339155	−	0.940731i	$-0.389859\pi$
$13$	2.44568	0.678309	0.339155	+	0.940731i	$0.389859\pi$
$14$	0	0
$15$	−2.29106	−0.591549
$16$	0	0
$17$	5.39502	1.30848	0.654242	−	0.756285i	$-0.272988\pi$
$17$	5.39502	1.30848	0.654242	+	0.756285i	$0.272988\pi$
$18$	0	0
$19$	−1.43938	−0.330217	−0.165108	−	0.986275i	$-0.552797\pi$
$19$	−1.43938	−0.330217	−0.165108	+	0.986275i	$0.552797\pi$
$20$	0	0
$21$	4.25127	0.927704
$22$	0	0
$23$	1.68306	0.350942	0.175471	−	0.984485i	$-0.443855\pi$
$23$	1.68306	0.350942	0.175471	+	0.984485i	$0.443855\pi$
$24$	0	0
$25$	0.248959	0.0497917
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.04771	−1.12303	−0.561516	−	0.827466i	$-0.689782\pi$
$29$	−6.04771	−1.12303	−0.561516	+	0.827466i	$0.689782\pi$
$30$	0	0
$31$	5.89508	1.05879	0.529394	−	0.848376i	$-0.322419\pi$
$31$	5.89508	1.05879	0.529394	+	0.848376i	$0.322419\pi$
$32$	0	0
$33$	6.06576	1.05591
$34$	0	0
$35$	−9.73992	−1.64635
$36$	0	0
$37$	−5.59860	−0.920405	−0.460202	−	0.887814i	$-0.652223\pi$
$37$	−5.59860	−0.920405	−0.460202	+	0.887814i	$0.652223\pi$
$38$	0	0
$39$	−2.44568	−0.391622
$40$	0	0
$41$	6.81731	1.06468	0.532342	−	0.846529i	$-0.321312\pi$
$41$	6.81731	1.06468	0.532342	+	0.846529i	$0.321312\pi$
$42$	0	0
$43$	−1.64680	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$43$	−1.64680	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$44$	0	0
$45$	2.29106	0.341531
$46$	0	0
$47$	3.10083	0.452302	0.226151	−	0.974092i	$-0.427386\pi$
$47$	3.10083	0.452302	0.226151	+	0.974092i	$0.427386\pi$
$48$	0	0
$49$	11.0733	1.58190
$50$	0	0
$51$	−5.39502	−0.755454
$52$	0	0
$53$	−9.76969	−1.34197	−0.670985	−	0.741471i	$-0.734129\pi$
$53$	−9.76969	−1.34197	−0.670985	+	0.741471i	$0.734129\pi$
$54$	0	0
$55$	−13.8970	−1.87388
$56$	0	0
$57$	1.43938	0.190651
$58$	0	0
$59$	−6.28536	−0.818284	−0.409142	−	0.912471i	$-0.634172\pi$
$59$	−6.28536	−0.818284	−0.409142	+	0.912471i	$0.634172\pi$
$60$	0	0
$61$	2.30692	0.295371	0.147686	−	0.989034i	$-0.452818\pi$
$61$	2.30692	0.295371	0.147686	+	0.989034i	$0.452818\pi$
$62$	0	0
$63$	−4.25127	−0.535610
$64$	0	0
$65$	5.60320	0.694991
$66$	0	0
$67$	−7.68981	−0.939460	−0.469730	−	0.882810i	$-0.655649\pi$
$67$	−7.68981	−0.939460	−0.469730	+	0.882810i	$0.655649\pi$
$68$	0	0
$69$	−1.68306	−0.202616
$70$	0	0
$71$	6.54067	0.776234	0.388117	−	0.921610i	$-0.373126\pi$
$71$	6.54067	0.776234	0.388117	+	0.921610i	$0.373126\pi$
$72$	0	0
$73$	−5.35117	−0.626307	−0.313153	−	0.949703i	$-0.601385\pi$
$73$	−5.35117	−0.626307	−0.313153	+	0.949703i	$0.601385\pi$
$74$	0	0
$75$	−0.248959	−0.0287473
$76$	0	0
$77$	25.7872	2.93873
$78$	0	0
$79$	−7.76709	−0.873866	−0.436933	−	0.899494i	$-0.643935\pi$
$79$	−7.76709	−0.873866	−0.436933	+	0.899494i	$0.643935\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.64073	0.838679	0.419340	−	0.907829i	$-0.362262\pi$
$83$	7.64073	0.838679	0.419340	+	0.907829i	$0.362262\pi$
$84$	0	0
$85$	12.3603	1.34066
$86$	0	0
$87$	6.04771	0.648382
$88$	0	0
$89$	−14.7267	−1.56103	−0.780514	−	0.625138i	$-0.785043\pi$
$89$	−14.7267	−1.56103	−0.780514	+	0.625138i	$0.785043\pi$
$90$	0	0
$91$	−10.3972	−1.08993
$92$	0	0
$93$	−5.89508	−0.611291
$94$	0	0
$95$	−3.29771	−0.338338
$96$	0	0
$97$	15.4903	1.57280	0.786399	−	0.617719i	$-0.211943\pi$
$97$	15.4903	1.57280	0.786399	+	0.617719i	$0.211943\pi$
$98$	0	0
$99$	−6.06576	−0.609632
$100$	0	0
$101$	4.37977	0.435803	0.217902	−	0.975971i	$-0.430079\pi$
$101$	4.37977	0.435803	0.217902	+	0.975971i	$0.430079\pi$
$102$	0	0
$103$	14.7536	1.45371	0.726857	−	0.686789i	$-0.240980\pi$
$103$	14.7536	1.45371	0.726857	+	0.686789i	$0.240980\pi$
$104$	0	0
$105$	9.73992	0.950519
$106$	0	0
$107$	5.15496	0.498349	0.249174	−	0.968459i	$-0.419841\pi$
$107$	5.15496	0.498349	0.249174	+	0.968459i	$0.419841\pi$
$108$	0	0
$109$	5.98502	0.573261	0.286631	−	0.958041i	$-0.407465\pi$
$109$	5.98502	0.573261	0.286631	+	0.958041i	$0.407465\pi$
$110$	0	0
$111$	5.59860	0.531396
$112$	0	0
$113$	−5.23282	−0.492262	−0.246131	−	0.969237i	$-0.579159\pi$
$113$	−5.23282	−0.492262	−0.246131	+	0.969237i	$0.579159\pi$
$114$	0	0
$115$	3.85599	0.359573
$116$	0	0
$117$	2.44568	0.226103
$118$	0	0
$119$	−22.9357	−2.10251
$120$	0	0
$121$	25.7935	2.34486
$122$	0	0
$123$	−6.81731	−0.614696
$124$	0	0
$125$	−10.8849	−0.973577
$126$	0	0
$127$	15.6526	1.38894	0.694471	−	0.719520i	$-0.255638\pi$
$127$	15.6526	1.38894	0.694471	+	0.719520i	$0.255638\pi$
$128$	0	0
$129$	1.64680	0.144993
$130$	0	0
$131$	5.07410	0.443326	0.221663	−	0.975123i	$-0.428851\pi$
$131$	5.07410	0.443326	0.221663	+	0.975123i	$0.428851\pi$
$132$	0	0
$133$	6.11920	0.530602
$134$	0	0
$135$	−2.29106	−0.197183
$136$	0	0
$137$	21.1750	1.80910	0.904551	−	0.426366i	$-0.140206\pi$
$137$	21.1750	1.80910	0.904551	+	0.426366i	$0.140206\pi$
$138$	0	0
$139$	0.786899	0.0667439	0.0333719	−	0.999443i	$-0.489375\pi$
$139$	0.786899	0.0667439	0.0333719	+	0.999443i	$0.489375\pi$
$140$	0	0
$141$	−3.10083	−0.261137
$142$	0	0
$143$	−14.8349	−1.24056
$144$	0	0
$145$	−13.8557	−1.15065
$146$	0	0
$147$	−11.0733	−0.913312
$148$	0	0
$149$	5.16729	0.423321	0.211661	−	0.977343i	$-0.432113\pi$
$149$	5.16729	0.423321	0.211661	+	0.977343i	$0.432113\pi$
$150$	0	0
$151$	17.0211	1.38516	0.692580	−	0.721341i	$-0.256474\pi$
$151$	17.0211	1.38516	0.692580	+	0.721341i	$0.256474\pi$
$152$	0	0
$153$	5.39502	0.436161
$154$	0	0
$155$	13.5060	1.08483
$156$	0	0
$157$	8.38665	0.669328	0.334664	−	0.942338i	$-0.391377\pi$
$157$	8.38665	0.669328	0.334664	+	0.942338i	$0.391377\pi$
$158$	0	0
$159$	9.76969	0.774787
$160$	0	0
$161$	−7.15514	−0.563904
$162$	0	0
$163$	13.5743	1.06322	0.531612	−	0.846988i	$-0.321587\pi$
$163$	13.5743	1.06322	0.531612	+	0.846988i	$0.321587\pi$
$164$	0	0
$165$	13.8970	1.08188
$166$	0	0
$167$	−2.20529	−0.170650	−0.0853251	−	0.996353i	$-0.527193\pi$
$167$	−2.20529	−0.170650	−0.0853251	+	0.996353i	$0.527193\pi$
$168$	0	0
$169$	−7.01865	−0.539896
$170$	0	0
$171$	−1.43938	−0.110072
$172$	0	0
$173$	−16.0222	−1.21814	−0.609072	−	0.793115i	$-0.708458\pi$
$173$	−16.0222	−1.21814	−0.609072	+	0.793115i	$0.708458\pi$
$174$	0	0
$175$	−1.05839	−0.0800068
$176$	0	0
$177$	6.28536	0.472436
$178$	0	0
$179$	26.1636	1.95556	0.977778	−	0.209642i	$-0.0672300\pi$
$179$	26.1636	1.95556	0.977778	+	0.209642i	$0.0672300\pi$
$180$	0	0
$181$	−12.4832	−0.927866	−0.463933	−	0.885870i	$-0.653562\pi$
$181$	−12.4832	−0.927866	−0.463933	+	0.885870i	$0.653562\pi$
$182$	0	0
$183$	−2.30692	−0.170533
$184$	0	0
$185$	−12.8267	−0.943041
$186$	0	0
$187$	−32.7249	−2.39308
$188$	0	0
$189$	4.25127	0.309235
$190$	0	0
$191$	7.21810	0.522283	0.261142	−	0.965300i	$-0.415901\pi$
$191$	7.21810	0.522283	0.261142	+	0.965300i	$0.415901\pi$
$192$	0	0
$193$	17.3651	1.24997	0.624985	−	0.780637i	$-0.285105\pi$
$193$	17.3651	1.24997	0.624985	+	0.780637i	$0.285105\pi$
$194$	0	0
$195$	−5.60320	−0.401253
$196$	0	0
$197$	0.113977	0.00812053	0.00406026	−	0.999992i	$-0.498708\pi$
$197$	0.113977	0.00812053	0.00406026	+	0.999992i	$0.498708\pi$
$198$	0	0
$199$	23.1617	1.64189	0.820946	−	0.571005i	$-0.193446\pi$
$199$	23.1617	1.64189	0.820946	+	0.571005i	$0.193446\pi$
$200$	0	0
$201$	7.68981	0.542398
$202$	0	0
$203$	25.7105	1.80452
$204$	0	0
$205$	15.6189	1.09087
$206$	0	0
$207$	1.68306	0.116981
$208$	0	0
$209$	8.73095	0.603932
$210$	0	0
$211$	12.4403	0.856427	0.428213	−	0.903678i	$-0.359143\pi$
$211$	12.4403	0.856427	0.428213	+	0.903678i	$0.359143\pi$
$212$	0	0
$213$	−6.54067	−0.448159
$214$	0	0
$215$	−3.77292	−0.257311
$216$	0	0
$217$	−25.0616	−1.70129
$218$	0	0
$219$	5.35117	0.361598
$220$	0	0
$221$	13.1945	0.887557
$222$	0	0
$223$	7.49486	0.501893	0.250946	−	0.968001i	$-0.419258\pi$
$223$	7.49486	0.501893	0.250946	+	0.968001i	$0.419258\pi$
$224$	0	0
$225$	0.248959	0.0165972
$226$	0	0
$227$	12.1984	0.809637	0.404819	−	0.914397i	$-0.367335\pi$
$227$	12.1984	0.809637	0.404819	+	0.914397i	$0.367335\pi$
$228$	0	0
$229$	−25.3318	−1.67398	−0.836988	−	0.547222i	$-0.815685\pi$
$229$	−25.3318	−1.67398	−0.836988	+	0.547222i	$0.815685\pi$
$230$	0	0
$231$	−25.7872	−1.69667
$232$	0	0
$233$	−2.63111	−0.172370	−0.0861849	−	0.996279i	$-0.527468\pi$
$233$	−2.63111	−0.172370	−0.0861849	+	0.996279i	$0.527468\pi$
$234$	0	0
$235$	7.10418	0.463426
$236$	0	0
$237$	7.76709	0.504527
$238$	0	0
$239$	3.35385	0.216943	0.108471	−	0.994100i	$-0.465404\pi$
$239$	3.35385	0.216943	0.108471	+	0.994100i	$0.465404\pi$
$240$	0	0
$241$	−9.40253	−0.605670	−0.302835	−	0.953043i	$-0.597933\pi$
$241$	−9.40253	−0.605670	−0.302835	+	0.953043i	$0.597933\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	25.3697	1.62081
$246$	0	0
$247$	−3.52027	−0.223989
$248$	0	0
$249$	−7.64073	−0.484212
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−10.2090	−0.641836
$254$	0	0
$255$	−12.3603	−0.774033
$256$	0	0
$257$	3.96337	0.247228	0.123614	−	0.992330i	$-0.460551\pi$
$257$	3.96337	0.247228	0.123614	+	0.992330i	$0.460551\pi$
$258$	0	0
$259$	23.8012	1.47893
$260$	0	0
$261$	−6.04771	−0.374344
$262$	0	0
$263$	−1.41988	−0.0875537	−0.0437769	−	0.999041i	$-0.513939\pi$
$263$	−1.41988	−0.0875537	−0.0437769	+	0.999041i	$0.513939\pi$
$264$	0	0
$265$	−22.3830	−1.37497
$266$	0	0
$267$	14.7267	0.901260
$268$	0	0
$269$	18.6879	1.13942	0.569710	−	0.821846i	$-0.307055\pi$
$269$	18.6879	1.13942	0.569710	+	0.821846i	$0.307055\pi$
$270$	0	0
$271$	−9.90443	−0.601651	−0.300826	−	0.953679i	$-0.597262\pi$
$271$	−9.90443	−0.601651	−0.300826	+	0.953679i	$0.597262\pi$
$272$	0	0
$273$	10.3972	0.629270
$274$	0	0
$275$	−1.51012	−0.0910639
$276$	0	0
$277$	−22.4974	−1.35174	−0.675869	−	0.737021i	$-0.736232\pi$
$277$	−22.4974	−1.35174	−0.675869	+	0.737021i	$0.736232\pi$
$278$	0	0
$279$	5.89508	0.352929
$280$	0	0
$281$	−16.8243	−1.00365	−0.501826	−	0.864969i	$-0.667338\pi$
$281$	−16.8243	−1.00365	−0.501826	+	0.864969i	$0.667338\pi$
$282$	0	0
$283$	−6.70725	−0.398705	−0.199352	−	0.979928i	$-0.563884\pi$
$283$	−6.70725	−0.398705	−0.199352	+	0.979928i	$0.563884\pi$
$284$	0	0
$285$	3.29771	0.195340
$286$	0	0
$287$	−28.9822	−1.71077
$288$	0	0
$289$	12.1062	0.712132
$290$	0	0
$291$	−15.4903	−0.908055
$292$	0	0
$293$	4.17652	0.243995	0.121998	−	0.992530i	$-0.461070\pi$
$293$	4.17652	0.243995	0.121998	+	0.992530i	$0.461070\pi$
$294$	0	0
$295$	−14.4001	−0.838408
$296$	0	0
$297$	6.06576	0.351971
$298$	0	0
$299$	4.11622	0.238047
$300$	0	0
$301$	7.00099	0.403530
$302$	0	0
$303$	−4.37977	−0.251611
$304$	0	0
$305$	5.28530	0.302635
$306$	0	0
$307$	−8.49929	−0.485080	−0.242540	−	0.970141i	$-0.577981\pi$
$307$	−8.49929	−0.485080	−0.242540	+	0.970141i	$0.577981\pi$
$308$	0	0
$309$	−14.7536	−0.839303
$310$	0	0
$311$	−27.4466	−1.55636	−0.778178	−	0.628044i	$-0.783856\pi$
$311$	−27.4466	−1.55636	−0.778178	+	0.628044i	$0.783856\pi$
$312$	0	0
$313$	14.1521	0.799926	0.399963	−	0.916531i	$-0.369023\pi$
$313$	14.1521	0.799926	0.399963	+	0.916531i	$0.369023\pi$
$314$	0	0
$315$	−9.73992	−0.548783
$316$	0	0
$317$	23.4619	1.31775	0.658875	−	0.752252i	$-0.271033\pi$
$317$	23.4619	1.31775	0.658875	+	0.752252i	$0.271033\pi$
$318$	0	0
$319$	36.6840	2.05391
$320$	0	0
$321$	−5.15496	−0.287722
$322$	0	0
$323$	−7.76549	−0.432084
$324$	0	0
$325$	0.608873	0.0337742
$326$	0	0
$327$	−5.98502	−0.330973
$328$	0	0
$329$	−13.1825	−0.726773
$330$	0	0
$331$	11.7597	0.646373	0.323187	−	0.946335i	$-0.395246\pi$
$331$	11.7597	0.646373	0.323187	+	0.946335i	$0.395246\pi$
$332$	0	0
$333$	−5.59860	−0.306802
$334$	0	0
$335$	−17.6178	−0.962565
$336$	0	0
$337$	−27.7689	−1.51267	−0.756335	−	0.654184i	$-0.773012\pi$
$337$	−27.7689	−1.51267	−0.756335	+	0.654184i	$0.773012\pi$
$338$	0	0
$339$	5.23282	0.284208
$340$	0	0
$341$	−35.7582	−1.93641
$342$	0	0
$343$	−17.3168	−0.935020
$344$	0	0
$345$	−3.85599	−0.207599
$346$	0	0
$347$	33.0478	1.77410	0.887050	−	0.461674i	$-0.152751\pi$
$347$	33.0478	1.77410	0.887050	+	0.461674i	$0.152751\pi$
$348$	0	0
$349$	19.6185	1.05016	0.525078	−	0.851054i	$-0.324036\pi$
$349$	19.6185	1.05016	0.525078	+	0.851054i	$0.324036\pi$
$350$	0	0
$351$	−2.44568	−0.130541
$352$	0	0
$353$	14.2503	0.758469	0.379235	−	0.925300i	$-0.376187\pi$
$353$	14.2503	0.758469	0.379235	+	0.925300i	$0.376187\pi$
$354$	0	0
$355$	14.9851	0.795325
$356$	0	0
$357$	22.9357	1.21389
$358$	0	0
$359$	−19.3779	−1.02273	−0.511364	−	0.859364i	$-0.670860\pi$
$359$	−19.3779	−1.02273	−0.511364	+	0.859364i	$0.670860\pi$
$360$	0	0
$361$	−16.9282	−0.890957
$362$	0	0
$363$	−25.7935	−1.35381
$364$	0	0
$365$	−12.2598	−0.641710
$366$	0	0
$367$	−6.73168	−0.351391	−0.175696	−	0.984445i	$-0.556217\pi$
$367$	−6.73168	−0.351391	−0.175696	+	0.984445i	$0.556217\pi$
$368$	0	0
$369$	6.81731	0.354895
$370$	0	0
$371$	41.5336	2.15632
$372$	0	0
$373$	1.75568	0.0909056	0.0454528	−	0.998966i	$-0.485527\pi$
$373$	1.75568	0.0909056	0.0454528	+	0.998966i	$0.485527\pi$
$374$	0	0
$375$	10.8849	0.562095
$376$	0	0
$377$	−14.7908	−0.761763
$378$	0	0
$379$	6.75152	0.346802	0.173401	−	0.984851i	$-0.444524\pi$
$379$	6.75152	0.346802	0.173401	+	0.984851i	$0.444524\pi$
$380$	0	0
$381$	−15.6526	−0.801906
$382$	0	0
$383$	−8.06453	−0.412078	−0.206039	−	0.978544i	$-0.566057\pi$
$383$	−8.06453	−0.412078	−0.206039	+	0.978544i	$0.566057\pi$
$384$	0	0
$385$	59.0801	3.01100
$386$	0	0
$387$	−1.64680	−0.0837115
$388$	0	0
$389$	33.3487	1.69085	0.845423	−	0.534097i	$-0.179348\pi$
$389$	33.3487	1.69085	0.845423	+	0.534097i	$0.179348\pi$
$390$	0	0
$391$	9.08013	0.459202
$392$	0	0
$393$	−5.07410	−0.255955
$394$	0	0
$395$	−17.7949	−0.895357
$396$	0	0
$397$	28.7404	1.44244	0.721220	−	0.692707i	$-0.243582\pi$
$397$	28.7404	1.44244	0.721220	+	0.692707i	$0.243582\pi$
$398$	0	0
$399$	−6.11920	−0.306343
$400$	0	0
$401$	1.46487	0.0731519	0.0365760	−	0.999331i	$-0.488355\pi$
$401$	1.46487	0.0731519	0.0365760	+	0.999331i	$0.488355\pi$
$402$	0	0
$403$	14.4175	0.718185
$404$	0	0
$405$	2.29106	0.113844
$406$	0	0
$407$	33.9598	1.68332
$408$	0	0
$409$	10.7695	0.532517	0.266259	−	0.963902i	$-0.414212\pi$
$409$	10.7695	0.532517	0.266259	+	0.963902i	$0.414212\pi$
$410$	0	0
$411$	−21.1750	−1.04449
$412$	0	0
$413$	26.7208	1.31484
$414$	0	0
$415$	17.5054	0.859305
$416$	0	0
$417$	−0.786899	−0.0385346
$418$	0	0
$419$	7.46007	0.364448	0.182224	−	0.983257i	$-0.441670\pi$
$419$	7.46007	0.364448	0.182224	+	0.983257i	$0.441670\pi$
$420$	0	0
$421$	19.2546	0.938411	0.469206	−	0.883089i	$-0.344540\pi$
$421$	19.2546	0.938411	0.469206	+	0.883089i	$0.344540\pi$
$422$	0	0
$423$	3.10083	0.150767
$424$	0	0
$425$	1.34314	0.0651517
$426$	0	0
$427$	−9.80736	−0.474611
$428$	0	0
$429$	14.8349	0.716236
$430$	0	0
$431$	36.3501	1.75092	0.875462	−	0.483287i	$-0.160557\pi$
$431$	36.3501	1.75092	0.875462	+	0.483287i	$0.160557\pi$
$432$	0	0
$433$	−22.6136	−1.08674	−0.543369	−	0.839494i	$-0.682852\pi$
$433$	−22.6136	−1.08674	−0.543369	+	0.839494i	$0.682852\pi$
$434$	0	0
$435$	13.8557	0.664328
$436$	0	0
$437$	−2.42256	−0.115887
$438$	0	0
$439$	−3.54403	−0.169147	−0.0845736	−	0.996417i	$-0.526953\pi$
$439$	−3.54403	−0.169147	−0.0845736	+	0.996417i	$0.526953\pi$
$440$	0	0
$441$	11.0733	0.527301
$442$	0	0
$443$	−2.87974	−0.136821	−0.0684103	−	0.997657i	$-0.521793\pi$
$443$	−2.87974	−0.136821	−0.0684103	+	0.997657i	$0.521793\pi$
$444$	0	0
$445$	−33.7398	−1.59942
$446$	0	0
$447$	−5.16729	−0.244405
$448$	0	0
$449$	41.4182	1.95464	0.977322	−	0.211757i	$-0.0679186\pi$
$449$	41.4182	1.95464	0.977322	+	0.211757i	$0.0679186\pi$
$450$	0	0
$451$	−41.3522	−1.94720
$452$	0	0
$453$	−17.0211	−0.799723
$454$	0	0
$455$	−23.8207	−1.11673
$456$	0	0
$457$	17.2177	0.805410	0.402705	−	0.915330i	$-0.368070\pi$
$457$	17.2177	0.805410	0.402705	+	0.915330i	$0.368070\pi$
$458$	0	0
$459$	−5.39502	−0.251818
$460$	0	0
$461$	−9.86371	−0.459399	−0.229699	−	0.973262i	$-0.573774\pi$
$461$	−9.86371	−0.459399	−0.229699	+	0.973262i	$0.573774\pi$
$462$	0	0
$463$	−36.3939	−1.69137	−0.845683	−	0.533685i	$-0.820807\pi$
$463$	−36.3939	−1.69137	−0.845683	+	0.533685i	$0.820807\pi$
$464$	0	0
$465$	−13.5060	−0.626325
$466$	0	0
$467$	−23.1881	−1.07302	−0.536509	−	0.843895i	$-0.680257\pi$
$467$	−23.1881	−1.07302	−0.536509	+	0.843895i	$0.680257\pi$
$468$	0	0
$469$	32.6915	1.50955
$470$	0	0
$471$	−8.38665	−0.386437
$472$	0	0
$473$	9.98909	0.459299
$474$	0	0
$475$	−0.358346	−0.0164421
$476$	0	0
$477$	−9.76969	−0.447324
$478$	0	0
$479$	34.8259	1.59123	0.795617	−	0.605800i	$-0.207147\pi$
$479$	34.8259	1.59123	0.795617	+	0.605800i	$0.207147\pi$
$480$	0	0
$481$	−13.6924	−0.624319
$482$	0	0
$483$	7.15514	0.325570
$484$	0	0
$485$	35.4891	1.61148
$486$	0	0
$487$	−9.86373	−0.446968	−0.223484	−	0.974708i	$-0.571743\pi$
$487$	−9.86373	−0.446968	−0.223484	+	0.974708i	$0.571743\pi$
$488$	0	0
$489$	−13.5743	−0.613852
$490$	0	0
$491$	−13.7221	−0.619271	−0.309636	−	0.950855i	$-0.600207\pi$
$491$	−13.7221	−0.619271	−0.309636	+	0.950855i	$0.600207\pi$
$492$	0	0
$493$	−32.6275	−1.46947
$494$	0	0
$495$	−13.8970	−0.624625
$496$	0	0
$497$	−27.8062	−1.24728
$498$	0	0
$499$	−1.27119	−0.0569064	−0.0284532	−	0.999595i	$-0.509058\pi$
$499$	−1.27119	−0.0569064	−0.0284532	+	0.999595i	$0.509058\pi$
$500$	0	0
$501$	2.20529	0.0985249
$502$	0	0
$503$	4.88664	0.217884	0.108942	−	0.994048i	$-0.465254\pi$
$503$	4.88664	0.217884	0.108942	+	0.994048i	$0.465254\pi$
$504$	0	0
$505$	10.0343	0.446521
$506$	0	0
$507$	7.01865	0.311709
$508$	0	0
$509$	26.5264	1.17576	0.587881	−	0.808947i	$-0.299962\pi$
$509$	26.5264	1.17576	0.587881	+	0.808947i	$0.299962\pi$
$510$	0	0
$511$	22.7493	1.00637
$512$	0	0
$513$	1.43938	0.0635503
$514$	0	0
$515$	33.8014	1.48947
$516$	0	0
$517$	−18.8089	−0.827214
$518$	0	0
$519$	16.0222	0.703296
$520$	0	0
$521$	−6.93340	−0.303758	−0.151879	−	0.988399i	$-0.548532\pi$
$521$	−6.93340	−0.303758	−0.151879	+	0.988399i	$0.548532\pi$
$522$	0	0
$523$	5.33400	0.233239	0.116620	−	0.993177i	$-0.462794\pi$
$523$	5.33400	0.233239	0.116620	+	0.993177i	$0.462794\pi$
$524$	0	0
$525$	1.05839	0.0461920
$526$	0	0
$527$	31.8041	1.38541
$528$	0	0
$529$	−20.1673	−0.876840
$530$	0	0
$531$	−6.28536	−0.272761
$532$	0	0
$533$	16.6729	0.722186
$534$	0	0
$535$	11.8103	0.510605
$536$	0	0
$537$	−26.1636	−1.12904
$538$	0	0
$539$	−67.1682	−2.89314
$540$	0	0
$541$	20.6908	0.889566	0.444783	−	0.895638i	$-0.353281\pi$
$541$	20.6908	0.889566	0.444783	+	0.895638i	$0.353281\pi$
$542$	0	0
$543$	12.4832	0.535704
$544$	0	0
$545$	13.7120	0.587360
$546$	0	0
$547$	−7.91342	−0.338353	−0.169177	−	0.985586i	$-0.554111\pi$
$547$	−7.91342	−0.338353	−0.169177	+	0.985586i	$0.554111\pi$
$548$	0	0
$549$	2.30692	0.0984571
$550$	0	0
$551$	8.70496	0.370844
$552$	0	0
$553$	33.0200	1.40415
$554$	0	0
$555$	12.8267	0.544465
$556$	0	0
$557$	31.9645	1.35438	0.677190	−	0.735808i	$-0.263198\pi$
$557$	31.9645	1.35438	0.677190	+	0.735808i	$0.263198\pi$
$558$	0	0
$559$	−4.02754	−0.170347
$560$	0	0
$561$	32.7249	1.38165
$562$	0	0
$563$	−19.5446	−0.823709	−0.411854	−	0.911250i	$-0.635119\pi$
$563$	−19.5446	−0.823709	−0.411854	+	0.911250i	$0.635119\pi$
$564$	0	0
$565$	−11.9887	−0.504369
$566$	0	0
$567$	−4.25127	−0.178537
$568$	0	0
$569$	27.8854	1.16902	0.584509	−	0.811388i	$-0.301287\pi$
$569$	27.8854	1.16902	0.584509	+	0.811388i	$0.301287\pi$
$570$	0	0
$571$	8.16300	0.341611	0.170805	−	0.985305i	$-0.445363\pi$
$571$	8.16300	0.341611	0.170805	+	0.985305i	$0.445363\pi$
$572$	0	0
$573$	−7.21810	−0.301540
$574$	0	0
$575$	0.419012	0.0174740
$576$	0	0
$577$	−44.4408	−1.85009	−0.925047	−	0.379852i	$-0.875975\pi$
$577$	−44.4408	−1.85009	−0.925047	+	0.379852i	$0.875975\pi$
$578$	0	0
$579$	−17.3651	−0.721671
$580$	0	0
$581$	−32.4828	−1.34762
$582$	0	0
$583$	59.2606	2.45433
$584$	0	0
$585$	5.60320	0.231664
$586$	0	0
$587$	20.4418	0.843725	0.421863	−	0.906660i	$-0.361376\pi$
$587$	20.4418	0.843725	0.421863	+	0.906660i	$0.361376\pi$
$588$	0	0
$589$	−8.48527	−0.349629
$590$	0	0
$591$	−0.113977	−0.00468839
$592$	0	0
$593$	−4.91479	−0.201826	−0.100913	−	0.994895i	$-0.532176\pi$
$593$	−4.91479	−0.201826	−0.100913	+	0.994895i	$0.532176\pi$
$594$	0	0
$595$	−52.5471	−2.15422
$596$	0	0
$597$	−23.1617	−0.947947
$598$	0	0
$599$	−22.3478	−0.913108	−0.456554	−	0.889696i	$-0.650917\pi$
$599$	−22.3478	−0.913108	−0.456554	+	0.889696i	$0.650917\pi$
$600$	0	0
$601$	29.1536	1.18920	0.594600	−	0.804021i	$-0.297310\pi$
$601$	29.1536	1.18920	0.594600	+	0.804021i	$0.297310\pi$
$602$	0	0
$603$	−7.68981	−0.313153
$604$	0	0
$605$	59.0944	2.40253
$606$	0	0
$607$	14.5615	0.591031	0.295516	−	0.955338i	$-0.404509\pi$
$607$	14.5615	0.591031	0.295516	+	0.955338i	$0.404509\pi$
$608$	0	0
$609$	−25.7105	−1.04184
$610$	0	0
$611$	7.58363	0.306801
$612$	0	0
$613$	−25.9536	−1.04825	−0.524127	−	0.851640i	$-0.675608\pi$
$613$	−25.9536	−1.04825	−0.524127	+	0.851640i	$0.675608\pi$
$614$	0	0
$615$	−15.6189	−0.629813
$616$	0	0
$617$	44.6151	1.79614	0.898068	−	0.439856i	$-0.144970\pi$
$617$	44.6151	1.79614	0.898068	+	0.439856i	$0.144970\pi$
$618$	0	0
$619$	19.6845	0.791187	0.395593	−	0.918426i	$-0.370539\pi$
$619$	19.6845	0.791187	0.395593	+	0.918426i	$0.370539\pi$
$620$	0	0
$621$	−1.68306	−0.0675388
$622$	0	0
$623$	62.6073	2.50831
$624$	0	0
$625$	−26.1828	−1.04731
$626$	0	0
$627$	−8.73095	−0.348680
$628$	0	0
$629$	−30.2046	−1.20434
$630$	0	0
$631$	19.5194	0.777057	0.388528	−	0.921437i	$-0.372984\pi$
$631$	19.5194	0.777057	0.388528	+	0.921437i	$0.372984\pi$
$632$	0	0
$633$	−12.4403	−0.494458
$634$	0	0
$635$	35.8610	1.42310
$636$	0	0
$637$	27.0818	1.07302
$638$	0	0
$639$	6.54067	0.258745
$640$	0	0
$641$	13.8787	0.548177	0.274088	−	0.961705i	$-0.411624\pi$
$641$	13.8787	0.548177	0.274088	+	0.961705i	$0.411624\pi$
$642$	0	0
$643$	15.9289	0.628173	0.314087	−	0.949394i	$-0.398302\pi$
$643$	15.9289	0.628173	0.314087	+	0.949394i	$0.398302\pi$
$644$	0	0
$645$	3.77292	0.148558
$646$	0	0
$647$	37.7897	1.48567	0.742833	−	0.669477i	$-0.233482\pi$
$647$	37.7897	1.48567	0.742833	+	0.669477i	$0.233482\pi$
$648$	0	0
$649$	38.1255	1.49656
$650$	0	0
$651$	25.0616	0.982241
$652$	0	0
$653$	12.1193	0.474264	0.237132	−	0.971477i	$-0.423793\pi$
$653$	12.1193	0.474264	0.237132	+	0.971477i	$0.423793\pi$
$654$	0	0
$655$	11.6251	0.454229
$656$	0	0
$657$	−5.35117	−0.208769
$658$	0	0
$659$	−35.9331	−1.39975	−0.699877	−	0.714264i	$-0.746762\pi$
$659$	−35.9331	−1.39975	−0.699877	+	0.714264i	$0.746762\pi$
$660$	0	0
$661$	−5.12314	−0.199267	−0.0996335	−	0.995024i	$-0.531767\pi$
$661$	−5.12314	−0.199267	−0.0996335	+	0.995024i	$0.531767\pi$
$662$	0	0
$663$	−13.1945	−0.512431
$664$	0	0
$665$	14.0195	0.543652
$666$	0	0
$667$	−10.1786	−0.394118
$668$	0	0
$669$	−7.49486	−0.289768
$670$	0	0
$671$	−13.9933	−0.540203
$672$	0	0
$673$	−10.7217	−0.413290	−0.206645	−	0.978416i	$-0.566254\pi$
$673$	−10.7217	−0.413290	−0.206645	+	0.978416i	$0.566254\pi$
$674$	0	0
$675$	−0.248959	−0.00958242
$676$	0	0
$677$	−3.24557	−0.124737	−0.0623687	−	0.998053i	$-0.519865\pi$
$677$	−3.24557	−0.124737	−0.0623687	+	0.998053i	$0.519865\pi$
$678$	0	0
$679$	−65.8533	−2.52722
$680$	0	0
$681$	−12.1984	−0.467444
$682$	0	0
$683$	−14.3704	−0.549867	−0.274933	−	0.961463i	$-0.588656\pi$
$683$	−14.3704	−0.549867	−0.274933	+	0.961463i	$0.588656\pi$
$684$	0	0
$685$	48.5132	1.85359
$686$	0	0
$687$	25.3318	0.966470
$688$	0	0
$689$	−23.8935	−0.910271
$690$	0	0
$691$	−10.8913	−0.414325	−0.207163	−	0.978306i	$-0.566423\pi$
$691$	−10.8913	−0.414325	−0.207163	+	0.978306i	$0.566423\pi$
$692$	0	0
$693$	25.7872	0.979575
$694$	0	0
$695$	1.80283	0.0683853
$696$	0	0
$697$	36.7795	1.39312
$698$	0	0
$699$	2.63111	0.0995178
$700$	0	0
$701$	−38.2380	−1.44423	−0.722115	−	0.691773i	$-0.756830\pi$
$701$	−38.2380	−1.44423	−0.722115	+	0.691773i	$0.756830\pi$
$702$	0	0
$703$	8.05853	0.303933
$704$	0	0
$705$	−7.10418	−0.267559
$706$	0	0
$707$	−18.6196	−0.700262
$708$	0	0
$709$	−0.951508	−0.0357346	−0.0178673	−	0.999840i	$-0.505688\pi$
$709$	−0.951508	−0.0357346	−0.0178673	+	0.999840i	$0.505688\pi$
$710$	0	0
$711$	−7.76709	−0.291289
$712$	0	0
$713$	9.92175	0.371573
$714$	0	0
$715$	−33.9877	−1.27107
$716$	0	0
$717$	−3.35385	−0.125252
$718$	0	0
$719$	10.2357	0.381728	0.190864	−	0.981616i	$-0.438871\pi$
$719$	10.2357	0.381728	0.190864	+	0.981616i	$0.438871\pi$
$720$	0	0
$721$	−62.7215	−2.33587
$722$	0	0
$723$	9.40253	0.349684
$724$	0	0
$725$	−1.50563	−0.0559177
$726$	0	0
$727$	−30.7464	−1.14032	−0.570161	−	0.821533i	$-0.693119\pi$
$727$	−30.7464	−1.14032	−0.570161	+	0.821533i	$0.693119\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.88451	−0.328606
$732$	0	0
$733$	−42.3484	−1.56417	−0.782086	−	0.623170i	$-0.785844\pi$
$733$	−42.3484	−1.56417	−0.782086	+	0.623170i	$0.785844\pi$
$734$	0	0
$735$	−25.3697	−0.935774
$736$	0	0
$737$	46.6446	1.71818
$738$	0	0
$739$	−28.8704	−1.06201	−0.531007	−	0.847367i	$-0.678186\pi$
$739$	−28.8704	−1.06201	−0.531007	+	0.847367i	$0.678186\pi$
$740$	0	0
$741$	3.52027	0.129320
$742$	0	0
$743$	9.55657	0.350596	0.175298	−	0.984515i	$-0.443911\pi$
$743$	9.55657	0.350596	0.175298	+	0.984515i	$0.443911\pi$
$744$	0	0
$745$	11.8386	0.433732
$746$	0	0
$747$	7.64073	0.279560
$748$	0	0
$749$	−21.9152	−0.800762
$750$	0	0
$751$	21.7905	0.795146	0.397573	−	0.917571i	$-0.369853\pi$
$751$	21.7905	0.795146	0.397573	+	0.917571i	$0.369853\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	38.9965	1.41923
$756$	0	0
$757$	6.25682	0.227408	0.113704	−	0.993515i	$-0.463728\pi$
$757$	6.25682	0.227408	0.113704	+	0.993515i	$0.463728\pi$
$758$	0	0
$759$	10.2090	0.370564
$760$	0	0
$761$	28.4544	1.03147	0.515736	−	0.856747i	$-0.327518\pi$
$761$	28.4544	1.03147	0.515736	+	0.856747i	$0.327518\pi$
$762$	0	0
$763$	−25.4440	−0.921133
$764$	0	0
$765$	12.3603	0.446888
$766$	0	0
$767$	−15.3720	−0.555050
$768$	0	0
$769$	−32.5864	−1.17510	−0.587548	−	0.809189i	$-0.699907\pi$
$769$	−32.5864	−1.17510	−0.587548	+	0.809189i	$0.699907\pi$
$770$	0	0
$771$	−3.96337	−0.142737
$772$	0	0
$773$	−30.1254	−1.08353	−0.541767	−	0.840529i	$-0.682244\pi$
$773$	−30.1254	−1.08353	−0.541767	+	0.840529i	$0.682244\pi$
$774$	0	0
$775$	1.46763	0.0527189
$776$	0	0
$777$	−23.8012	−0.853863
$778$	0	0
$779$	−9.81271	−0.351577
$780$	0	0
$781$	−39.6741	−1.41965
$782$	0	0
$783$	6.04771	0.216127
$784$	0	0
$785$	19.2143	0.685789
$786$	0	0
$787$	18.1112	0.645595	0.322797	−	0.946468i	$-0.395377\pi$
$787$	18.1112	0.645595	0.322797	+	0.946468i	$0.395377\pi$
$788$	0	0
$789$	1.41988	0.0505492
$790$	0	0
$791$	22.2462	0.790982
$792$	0	0
$793$	5.64199	0.200353
$794$	0	0
$795$	22.3830	0.793842
$796$	0	0
$797$	−26.2081	−0.928340	−0.464170	−	0.885746i	$-0.653647\pi$
$797$	−26.2081	−0.928340	−0.464170	+	0.885746i	$0.653647\pi$
$798$	0	0
$799$	16.7290	0.591830
$800$	0	0
$801$	−14.7267	−0.520343
$802$	0	0
$803$	32.4589	1.14545
$804$	0	0
$805$	−16.3928	−0.577772
$806$	0	0
$807$	−18.6879	−0.657845
$808$	0	0
$809$	29.3122	1.03056	0.515282	−	0.857021i	$-0.327687\pi$
$809$	29.3122	1.03056	0.515282	+	0.857021i	$0.327687\pi$
$810$	0	0
$811$	−15.4488	−0.542481	−0.271240	−	0.962512i	$-0.587434\pi$
$811$	−15.4488	−0.542481	−0.271240	+	0.962512i	$0.587434\pi$
$812$	0	0
$813$	9.90443	0.347363
$814$	0	0
$815$	31.0996	1.08937
$816$	0	0
$817$	2.37037	0.0829288
$818$	0	0
$819$	−10.3972	−0.363309
$820$	0	0
$821$	−28.4656	−0.993455	−0.496727	−	0.867907i	$-0.665465\pi$
$821$	−28.4656	−0.993455	−0.496727	+	0.867907i	$0.665465\pi$
$822$	0	0
$823$	−12.2897	−0.428393	−0.214197	−	0.976791i	$-0.568713\pi$
$823$	−12.2897	−0.428393	−0.214197	+	0.976791i	$0.568713\pi$
$824$	0	0
$825$	1.51012	0.0525758
$826$	0	0
$827$	24.9088	0.866166	0.433083	−	0.901354i	$-0.357426\pi$
$827$	24.9088	0.866166	0.433083	+	0.901354i	$0.357426\pi$
$828$	0	0
$829$	49.7766	1.72881	0.864407	−	0.502793i	$-0.167694\pi$
$829$	49.7766	1.72881	0.864407	+	0.502793i	$0.167694\pi$
$830$	0	0
$831$	22.4974	0.780427
$832$	0	0
$833$	59.7408	2.06990
$834$	0	0
$835$	−5.05245	−0.174847
$836$	0	0
$837$	−5.89508	−0.203764
$838$	0	0
$839$	−33.2852	−1.14913	−0.574567	−	0.818458i	$-0.694829\pi$
$839$	−33.2852	−1.14913	−0.574567	+	0.818458i	$0.694829\pi$
$840$	0	0
$841$	7.57478	0.261199
$842$	0	0
$843$	16.8243	0.579458
$844$	0	0
$845$	−16.0802	−0.553174
$846$	0	0
$847$	−109.655	−3.76780
$848$	0	0
$849$	6.70725	0.230192
$850$	0	0
$851$	−9.42277	−0.323008
$852$	0	0
$853$	−22.0186	−0.753903	−0.376951	−	0.926233i	$-0.623028\pi$
$853$	−22.0186	−0.753903	−0.376951	+	0.926233i	$0.623028\pi$
$854$	0	0
$855$	−3.29771	−0.112779
$856$	0	0
$857$	53.7397	1.83571	0.917857	−	0.396912i	$-0.129918\pi$
$857$	53.7397	1.83571	0.917857	+	0.396912i	$0.129918\pi$
$858$	0	0
$859$	−53.3048	−1.81874	−0.909369	−	0.415991i	$-0.863435\pi$
$859$	−53.3048	−1.81874	−0.909369	+	0.415991i	$0.863435\pi$
$860$	0	0
$861$	28.9822	0.987712
$862$	0	0
$863$	−47.9022	−1.63061	−0.815305	−	0.579031i	$-0.803431\pi$
$863$	−47.9022	−1.63061	−0.815305	+	0.579031i	$0.803431\pi$
$864$	0	0
$865$	−36.7078	−1.24810
$866$	0	0
$867$	−12.1062	−0.411149
$868$	0	0
$869$	47.1133	1.59821
$870$	0	0
$871$	−18.8068	−0.637245
$872$	0	0
$873$	15.4903	0.524266
$874$	0	0
$875$	46.2748	1.56437
$876$	0	0
$877$	−16.4393	−0.555115	−0.277558	−	0.960709i	$-0.589525\pi$
$877$	−16.4393	−0.555115	−0.277558	+	0.960709i	$0.589525\pi$
$878$	0	0
$879$	−4.17652	−0.140871
$880$	0	0
$881$	11.4489	0.385725	0.192862	−	0.981226i	$-0.438223\pi$
$881$	11.4489	0.385725	0.192862	+	0.981226i	$0.438223\pi$
$882$	0	0
$883$	−46.2396	−1.55609	−0.778043	−	0.628211i	$-0.783788\pi$
$883$	−46.2396	−1.55609	−0.778043	+	0.628211i	$0.783788\pi$
$884$	0	0
$885$	14.4001	0.484055
$886$	0	0
$887$	30.3875	1.02031	0.510156	−	0.860082i	$-0.329588\pi$
$887$	30.3875	1.02031	0.510156	+	0.860082i	$0.329588\pi$
$888$	0	0
$889$	−66.5434	−2.23180
$890$	0	0
$891$	−6.06576	−0.203211
$892$	0	0
$893$	−4.46327	−0.149358
$894$	0	0
$895$	59.9423	2.00365
$896$	0	0
$897$	−4.11622	−0.137436
$898$	0	0
$899$	−35.6517	−1.18905
$900$	0	0
$901$	−52.7077	−1.75595
$902$	0	0
$903$	−7.00099	−0.232978
$904$	0	0
$905$	−28.5997	−0.950686
$906$	0	0
$907$	−11.7640	−0.390619	−0.195309	−	0.980742i	$-0.562571\pi$
$907$	−11.7640	−0.390619	−0.195309	+	0.980742i	$0.562571\pi$
$908$	0	0
$909$	4.37977	0.145268
$910$	0	0
$911$	−45.7756	−1.51661	−0.758307	−	0.651897i	$-0.773973\pi$
$911$	−45.7756	−1.51661	−0.758307	+	0.651897i	$0.773973\pi$
$912$	0	0
$913$	−46.3469	−1.53386
$914$	0	0
$915$	−5.28530	−0.174727
$916$	0	0
$917$	−21.5714	−0.712350
$918$	0	0
$919$	43.1367	1.42295	0.711475	−	0.702712i	$-0.248028\pi$
$919$	43.1367	1.42295	0.711475	+	0.702712i	$0.248028\pi$
$920$	0	0
$921$	8.49929	0.280061
$922$	0	0
$923$	15.9964	0.526527
$924$	0	0
$925$	−1.39382	−0.0458285
$926$	0	0
$927$	14.7536	0.484572
$928$	0	0
$929$	26.4226	0.866896	0.433448	−	0.901179i	$-0.357297\pi$
$929$	26.4226	0.866896	0.433448	+	0.901179i	$0.357297\pi$
$930$	0	0
$931$	−15.9387	−0.522371
$932$	0	0
$933$	27.4466	0.898562
$934$	0	0
$935$	−74.9748	−2.45194
$936$	0	0
$937$	−8.34169	−0.272511	−0.136256	−	0.990674i	$-0.543507\pi$
$937$	−8.34169	−0.272511	−0.136256	+	0.990674i	$0.543507\pi$
$938$	0	0
$939$	−14.1521	−0.461838
$940$	0	0
$941$	27.3962	0.893091	0.446546	−	0.894761i	$-0.352654\pi$
$941$	27.3962	0.893091	0.446546	+	0.894761i	$0.352654\pi$
$942$	0	0
$943$	11.4739	0.373642
$944$	0	0
$945$	9.73992	0.316840
$946$	0	0
$947$	18.7481	0.609232	0.304616	−	0.952475i	$-0.401472\pi$
$947$	18.7481	0.609232	0.304616	+	0.952475i	$0.401472\pi$
$948$	0	0
$949$	−13.0872	−0.424830
$950$	0	0
$951$	−23.4619	−0.760803
$952$	0	0
$953$	7.29484	0.236303	0.118151	−	0.992996i	$-0.462303\pi$
$953$	7.29484	0.236303	0.118151	+	0.992996i	$0.462303\pi$
$954$	0	0
$955$	16.5371	0.535128
$956$	0	0
$957$	−36.6840	−1.18582
$958$	0	0
$959$	−90.0207	−2.90692
$960$	0	0
$961$	3.75195	0.121031
$962$	0	0
$963$	5.15496	0.166116
$964$	0	0
$965$	39.7846	1.28071
$966$	0	0
$967$	−52.3466	−1.68335	−0.841677	−	0.539981i	$-0.818431\pi$
$967$	−52.3466	−1.68335	−0.841677	+	0.539981i	$0.818431\pi$
$968$	0	0
$969$	7.76549	0.249464
$970$	0	0
$971$	40.7026	1.30621	0.653104	−	0.757268i	$-0.273466\pi$
$971$	40.7026	1.30621	0.653104	+	0.757268i	$0.273466\pi$
$972$	0	0
$973$	−3.34532	−0.107246
$974$	0	0
$975$	−0.608873	−0.0194995
$976$	0	0
$977$	31.8794	1.01991	0.509956	−	0.860201i	$-0.329662\pi$
$977$	31.8794	1.01991	0.509956	+	0.860201i	$0.329662\pi$
$978$	0	0
$979$	89.3287	2.85496
$980$	0	0
$981$	5.98502	0.191087
$982$	0	0
$983$	20.6129	0.657449	0.328725	−	0.944426i	$-0.393381\pi$
$983$	20.6129	0.657449	0.328725	+	0.944426i	$0.393381\pi$
$984$	0	0
$985$	0.261128	0.00832024
$986$	0	0
$987$	13.1825	0.419603
$988$	0	0
$989$	−2.77166	−0.0881335
$990$	0	0
$991$	−33.1988	−1.05460	−0.527298	−	0.849680i	$-0.676795\pi$
$991$	−33.1988	−1.05460	−0.527298	+	0.849680i	$0.676795\pi$
$992$	0	0
$993$	−11.7597	−0.373184
$994$	0	0
$995$	53.0650	1.68227
$996$	0	0
$997$	−5.58278	−0.176808	−0.0884042	−	0.996085i	$-0.528177\pi$
$997$	−5.58278	−0.176808	−0.0884042	+	0.996085i	$0.528177\pi$
$998$	0	0
$999$	5.59860	0.177132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.p.1.10	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.p.1.10	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.29106 q^{5} -4.25127 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.29106 q^{5} -4.25127 q^{7} +1.00000 q^{9} -6.06576 q^{11} +2.44568 q^{13} -2.29106 q^{15} +5.39502 q^{17} -1.43938 q^{19} +4.25127 q^{21} +1.68306 q^{23} +0.248959 q^{25} -1.00000 q^{27} -6.04771 q^{29} +5.89508 q^{31} +6.06576 q^{33} -9.73992 q^{35} -5.59860 q^{37} -2.44568 q^{39} +6.81731 q^{41} -1.64680 q^{43} +2.29106 q^{45} +3.10083 q^{47} +11.0733 q^{49} -5.39502 q^{51} -9.76969 q^{53} -13.8970 q^{55} +1.43938 q^{57} -6.28536 q^{59} +2.30692 q^{61} -4.25127 q^{63} +5.60320 q^{65} -7.68981 q^{67} -1.68306 q^{69} +6.54067 q^{71} -5.35117 q^{73} -0.248959 q^{75} +25.7872 q^{77} -7.76709 q^{79} +1.00000 q^{81} +7.64073 q^{83} +12.3603 q^{85} +6.04771 q^{87} -14.7267 q^{89} -10.3972 q^{91} -5.89508 q^{93} -3.29771 q^{95} +15.4903 q^{97} -6.06576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) \) 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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