LMFDB - Embedded newform 6024.2.a.n.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:	$1$
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} - 7 x^{13} - 22 x^{12} + 214 x^{11} + 91 x^{10} - 2481 x^{9} + 1285 x^{8} + 13253 x^{7} + \cdots - 5832 $ x^14 - 7x^13 - 22x^12 + 214x^11 + 91x^10 - 2481x^9 + 1285x^8 + 13253x^7 - 14287x^6 - 29907x^5 + 49572x^4 + 10044x^3 - 53784x^2 + 32076*x - 5832
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.68008$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.680080 q^{5} -1.01908 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.680080 q^{5} -1.01908 q^{7} +1.00000 q^{9} -1.55965 q^{11} -3.74701 q^{13} -0.680080 q^{15} +4.50402 q^{17} +2.35885 q^{19} +1.01908 q^{21} +0.503609 q^{23} -4.53749 q^{25} -1.00000 q^{27} -1.62556 q^{29} +4.16977 q^{31} +1.55965 q^{33} -0.693057 q^{35} +7.25353 q^{37} +3.74701 q^{39} -7.86444 q^{41} +1.59953 q^{43} +0.680080 q^{45} +9.14846 q^{47} -5.96147 q^{49} -4.50402 q^{51} -8.24789 q^{53} -1.06069 q^{55} -2.35885 q^{57} +7.41283 q^{59} -10.3450 q^{61} -1.01908 q^{63} -2.54826 q^{65} +15.7452 q^{67} -0.503609 q^{69} +5.08491 q^{71} -2.61870 q^{73} +4.53749 q^{75} +1.58941 q^{77} -11.0665 q^{79} +1.00000 q^{81} -6.86134 q^{83} +3.06310 q^{85} +1.62556 q^{87} +3.33660 q^{89} +3.81850 q^{91} -4.16977 q^{93} +1.60421 q^{95} -13.3115 q^{97} -1.55965 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9} + 10 q^{11} - 9 q^{13} + 7 q^{15} - 22 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 23 q^{25} - 14 q^{27} - 12 q^{29} - 13 q^{31} - 10 q^{33} + 23 q^{35} + 5 q^{37} + 9 q^{39} - 52 q^{41} + 16 q^{43} - 7 q^{45} + q^{47} + 9 q^{49} + 22 q^{51} - 13 q^{53} - 12 q^{55} + 6 q^{57} + 12 q^{59} - 20 q^{61} + q^{63} - 40 q^{65} + 21 q^{67} - 3 q^{69} - 5 q^{71} - 14 q^{73} - 23 q^{75} - 14 q^{77} - 23 q^{79} + 14 q^{81} + 25 q^{83} - 11 q^{85} + 12 q^{87} - 79 q^{89} + 6 q^{91} + 13 q^{93} + 3 q^{95} - 17 q^{97} + 10 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9 + 10 * q^11 - 9 * q^13 + 7 * q^15 - 22 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 23 * q^25 - 14 * q^27 - 12 * q^29 - 13 * q^31 - 10 * q^33 + 23 * q^35 + 5 * q^37 + 9 * q^39 - 52 * q^41 + 16 * q^43 - 7 * q^45 + q^47 + 9 * q^49 + 22 * q^51 - 13 * q^53 - 12 * q^55 + 6 * q^57 + 12 * q^59 - 20 * q^61 + q^63 - 40 * q^65 + 21 * q^67 - 3 * q^69 - 5 * q^71 - 14 * q^73 - 23 * q^75 - 14 * q^77 - 23 * q^79 + 14 * q^81 + 25 * q^83 - 11 * q^85 + 12 * q^87 - 79 * q^89 + 6 * q^91 + 13 * q^93 + 3 * q^95 - 17 * q^97 + 10 * 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
$5$	0.680080	0.304141	0.152071	−	0.988370i	$-0.451406\pi$
$5$	0.680080	0.304141	0.152071	+	0.988370i	$0.451406\pi$
$6$	0	0
$7$	−1.01908	−0.385176	−0.192588	−	0.981280i	$-0.561688\pi$
$7$	−1.01908	−0.385176	−0.192588	+	0.981280i	$0.561688\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.55965	−0.470253	−0.235126	−	0.971965i	$-0.575550\pi$
$11$	−1.55965	−0.470253	−0.235126	+	0.971965i	$0.575550\pi$
$12$	0	0
$13$	−3.74701	−1.03923	−0.519616	−	0.854400i	$-0.673925\pi$
$13$	−3.74701	−1.03923	−0.519616	+	0.854400i	$0.673925\pi$
$14$	0	0
$15$	−0.680080	−0.175596
$16$	0	0
$17$	4.50402	1.09239	0.546193	−	0.837659i	$-0.316077\pi$
$17$	4.50402	1.09239	0.546193	+	0.837659i	$0.316077\pi$
$18$	0	0
$19$	2.35885	0.541158	0.270579	−	0.962698i	$-0.412785\pi$
$19$	2.35885	0.541158	0.270579	+	0.962698i	$0.412785\pi$
$20$	0	0
$21$	1.01908	0.222382
$22$	0	0
$23$	0.503609	0.105010	0.0525048	−	0.998621i	$-0.483280\pi$
$23$	0.503609	0.105010	0.0525048	+	0.998621i	$0.483280\pi$
$24$	0	0
$25$	−4.53749	−0.907498
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.62556	−0.301858	−0.150929	−	0.988545i	$-0.548227\pi$
$29$	−1.62556	−0.301858	−0.150929	+	0.988545i	$0.548227\pi$
$30$	0	0
$31$	4.16977	0.748913	0.374457	−	0.927244i	$-0.377829\pi$
$31$	4.16977	0.748913	0.374457	+	0.927244i	$0.377829\pi$
$32$	0	0
$33$	1.55965	0.271501
$34$	0	0
$35$	−0.693057	−0.117148
$36$	0	0
$37$	7.25353	1.19247	0.596236	−	0.802809i	$-0.296662\pi$
$37$	7.25353	1.19247	0.596236	+	0.802809i	$0.296662\pi$
$38$	0	0
$39$	3.74701	0.600001
$40$	0	0
$41$	−7.86444	−1.22822	−0.614110	−	0.789220i	$-0.710485\pi$
$41$	−7.86444	−1.22822	−0.614110	+	0.789220i	$0.710485\pi$
$42$	0	0
$43$	1.59953	0.243926	0.121963	−	0.992535i	$-0.461081\pi$
$43$	1.59953	0.243926	0.121963	+	0.992535i	$0.461081\pi$
$44$	0	0
$45$	0.680080	0.101380
$46$	0	0
$47$	9.14846	1.33444	0.667220	−	0.744861i	$-0.267484\pi$
$47$	9.14846	1.33444	0.667220	+	0.744861i	$0.267484\pi$
$48$	0	0
$49$	−5.96147	−0.851639
$50$	0	0
$51$	−4.50402	−0.630689
$52$	0	0
$53$	−8.24789	−1.13294	−0.566468	−	0.824084i	$-0.691690\pi$
$53$	−8.24789	−1.13294	−0.566468	+	0.824084i	$0.691690\pi$
$54$	0	0
$55$	−1.06069	−0.143023
$56$	0	0
$57$	−2.35885	−0.312438
$58$	0	0
$59$	7.41283	0.965068	0.482534	−	0.875877i	$-0.339716\pi$
$59$	7.41283	0.965068	0.482534	+	0.875877i	$0.339716\pi$
$60$	0	0
$61$	−10.3450	−1.32454	−0.662270	−	0.749265i	$-0.730407\pi$
$61$	−10.3450	−1.32454	−0.662270	+	0.749265i	$0.730407\pi$
$62$	0	0
$63$	−1.01908	−0.128392
$64$	0	0
$65$	−2.54826	−0.316073
$66$	0	0
$67$	15.7452	1.92359	0.961793	−	0.273779i	$-0.0882736\pi$
$67$	15.7452	1.92359	0.961793	+	0.273779i	$0.0882736\pi$
$68$	0	0
$69$	−0.503609	−0.0606274
$70$	0	0
$71$	5.08491	0.603468	0.301734	−	0.953392i	$-0.402435\pi$
$71$	5.08491	0.603468	0.301734	+	0.953392i	$0.402435\pi$
$72$	0	0
$73$	−2.61870	−0.306496	−0.153248	−	0.988188i	$-0.548973\pi$
$73$	−2.61870	−0.306496	−0.153248	+	0.988188i	$0.548973\pi$
$74$	0	0
$75$	4.53749	0.523944
$76$	0	0
$77$	1.58941	0.181130
$78$	0	0
$79$	−11.0665	−1.24507	−0.622537	−	0.782591i	$-0.713898\pi$
$79$	−11.0665	−1.24507	−0.622537	+	0.782591i	$0.713898\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.86134	−0.753130	−0.376565	−	0.926390i	$-0.622895\pi$
$83$	−6.86134	−0.753130	−0.376565	+	0.926390i	$0.622895\pi$
$84$	0	0
$85$	3.06310	0.332239
$86$	0	0
$87$	1.62556	0.174278
$88$	0	0
$89$	3.33660	0.353679	0.176840	−	0.984240i	$-0.443413\pi$
$89$	3.33660	0.353679	0.176840	+	0.984240i	$0.443413\pi$
$90$	0	0
$91$	3.81850	0.400288
$92$	0	0
$93$	−4.16977	−0.432385
$94$	0	0
$95$	1.60421	0.164589
$96$	0	0
$97$	−13.3115	−1.35157	−0.675787	−	0.737097i	$-0.736196\pi$
$97$	−13.3115	−1.35157	−0.675787	+	0.737097i	$0.736196\pi$
$98$	0	0
$99$	−1.55965	−0.156751
$100$	0	0
$101$	17.7156	1.76277	0.881386	−	0.472396i	$-0.156611\pi$
$101$	17.7156	1.76277	0.881386	+	0.472396i	$0.156611\pi$
$102$	0	0
$103$	−17.2489	−1.69959	−0.849793	−	0.527116i	$-0.823273\pi$
$103$	−17.2489	−1.69959	−0.849793	+	0.527116i	$0.823273\pi$
$104$	0	0
$105$	0.693057	0.0676354
$106$	0	0
$107$	−9.79844	−0.947251	−0.473625	−	0.880726i	$-0.657055\pi$
$107$	−9.79844	−0.947251	−0.473625	+	0.880726i	$0.657055\pi$
$108$	0	0
$109$	−0.786029	−0.0752879	−0.0376440	−	0.999291i	$-0.511985\pi$
$109$	−0.786029	−0.0752879	−0.0376440	+	0.999291i	$0.511985\pi$
$110$	0	0
$111$	−7.25353	−0.688475
$112$	0	0
$113$	−2.61653	−0.246142	−0.123071	−	0.992398i	$-0.539274\pi$
$113$	−2.61653	−0.246142	−0.123071	+	0.992398i	$0.539274\pi$
$114$	0	0
$115$	0.342494	0.0319378
$116$	0	0
$117$	−3.74701	−0.346411
$118$	0	0
$119$	−4.58996	−0.420761
$120$	0	0
$121$	−8.56749	−0.778862
$122$	0	0
$123$	7.86444	0.709113
$124$	0	0
$125$	−6.48626	−0.580149
$126$	0	0
$127$	−1.36351	−0.120992	−0.0604959	−	0.998168i	$-0.519268\pi$
$127$	−1.36351	−0.120992	−0.0604959	+	0.998168i	$0.519268\pi$
$128$	0	0
$129$	−1.59953	−0.140831
$130$	0	0
$131$	−15.0256	−1.31280	−0.656398	−	0.754415i	$-0.727921\pi$
$131$	−15.0256	−1.31280	−0.656398	+	0.754415i	$0.727921\pi$
$132$	0	0
$133$	−2.40386	−0.208441
$134$	0	0
$135$	−0.680080	−0.0585320
$136$	0	0
$137$	−7.71407	−0.659057	−0.329529	−	0.944146i	$-0.606890\pi$
$137$	−7.71407	−0.659057	−0.329529	+	0.944146i	$0.606890\pi$
$138$	0	0
$139$	17.7822	1.50826	0.754132	−	0.656723i	$-0.228058\pi$
$139$	17.7822	1.50826	0.754132	+	0.656723i	$0.228058\pi$
$140$	0	0
$141$	−9.14846	−0.770439
$142$	0	0
$143$	5.84402	0.488702
$144$	0	0
$145$	−1.10551	−0.0918075
$146$	0	0
$147$	5.96147	0.491694
$148$	0	0
$149$	12.5608	1.02902	0.514511	−	0.857483i	$-0.327973\pi$
$149$	12.5608	1.02902	0.514511	+	0.857483i	$0.327973\pi$
$150$	0	0
$151$	−0.355240	−0.0289090	−0.0144545	−	0.999896i	$-0.504601\pi$
$151$	−0.355240	−0.0289090	−0.0144545	+	0.999896i	$0.504601\pi$
$152$	0	0
$153$	4.50402	0.364129
$154$	0	0
$155$	2.83578	0.227775
$156$	0	0
$157$	2.73936	0.218625	0.109312	−	0.994007i	$-0.465135\pi$
$157$	2.73936	0.218625	0.109312	+	0.994007i	$0.465135\pi$
$158$	0	0
$159$	8.24789	0.654101
$160$	0	0
$161$	−0.513218	−0.0404473
$162$	0	0
$163$	−0.888539	−0.0695958	−0.0347979	−	0.999394i	$-0.511079\pi$
$163$	−0.888539	−0.0695958	−0.0347979	+	0.999394i	$0.511079\pi$
$164$	0	0
$165$	1.06069	0.0825745
$166$	0	0
$167$	−6.40514	−0.495644	−0.247822	−	0.968806i	$-0.579715\pi$
$167$	−6.40514	−0.495644	−0.247822	+	0.968806i	$0.579715\pi$
$168$	0	0
$169$	1.04005	0.0800037
$170$	0	0
$171$	2.35885	0.180386
$172$	0	0
$173$	−8.39208	−0.638038	−0.319019	−	0.947748i	$-0.603353\pi$
$173$	−8.39208	−0.638038	−0.319019	+	0.947748i	$0.603353\pi$
$174$	0	0
$175$	4.62407	0.349547
$176$	0	0
$177$	−7.41283	−0.557183
$178$	0	0
$179$	6.29521	0.470526	0.235263	−	0.971932i	$-0.424405\pi$
$179$	6.29521	0.470526	0.235263	+	0.971932i	$0.424405\pi$
$180$	0	0
$181$	−20.1784	−1.49985	−0.749923	−	0.661525i	$-0.769910\pi$
$181$	−20.1784	−1.49985	−0.749923	+	0.661525i	$0.769910\pi$
$182$	0	0
$183$	10.3450	0.764724
$184$	0	0
$185$	4.93298	0.362680
$186$	0	0
$187$	−7.02471	−0.513698
$188$	0	0
$189$	1.01908	0.0741272
$190$	0	0
$191$	−25.2081	−1.82400	−0.911999	−	0.410193i	$-0.865461\pi$
$191$	−25.2081	−1.82400	−0.911999	+	0.410193i	$0.865461\pi$
$192$	0	0
$193$	−0.797302	−0.0573910	−0.0286955	−	0.999588i	$-0.509135\pi$
$193$	−0.797302	−0.0573910	−0.0286955	+	0.999588i	$0.509135\pi$
$194$	0	0
$195$	2.54826	0.182485
$196$	0	0
$197$	−15.6086	−1.11206	−0.556032	−	0.831161i	$-0.687677\pi$
$197$	−15.6086	−1.11206	−0.556032	+	0.831161i	$0.687677\pi$
$198$	0	0
$199$	15.6916	1.11235	0.556176	−	0.831065i	$-0.312268\pi$
$199$	15.6916	1.11235	0.556176	+	0.831065i	$0.312268\pi$
$200$	0	0
$201$	−15.7452	−1.11058
$202$	0	0
$203$	1.65657	0.116269
$204$	0	0
$205$	−5.34845	−0.373552
$206$	0	0
$207$	0.503609	0.0350032
$208$	0	0
$209$	−3.67899	−0.254481
$210$	0	0
$211$	12.8880	0.887247	0.443624	−	0.896213i	$-0.353693\pi$
$211$	12.8880	0.887247	0.443624	+	0.896213i	$0.353693\pi$
$212$	0	0
$213$	−5.08491	−0.348412
$214$	0	0
$215$	1.08781	0.0741881
$216$	0	0
$217$	−4.24934	−0.288464
$218$	0	0
$219$	2.61870	0.176955
$220$	0	0
$221$	−16.8766	−1.13524
$222$	0	0
$223$	−5.99483	−0.401443	−0.200722	−	0.979648i	$-0.564329\pi$
$223$	−5.99483	−0.401443	−0.200722	+	0.979648i	$0.564329\pi$
$224$	0	0
$225$	−4.53749	−0.302499
$226$	0	0
$227$	0.239494	0.0158958	0.00794790	−	0.999968i	$-0.497470\pi$
$227$	0.239494	0.0158958	0.00794790	+	0.999968i	$0.497470\pi$
$228$	0	0
$229$	23.1460	1.52953	0.764767	−	0.644307i	$-0.222854\pi$
$229$	23.1460	1.52953	0.764767	+	0.644307i	$0.222854\pi$
$230$	0	0
$231$	−1.58941	−0.104576
$232$	0	0
$233$	−23.9559	−1.56940	−0.784700	−	0.619876i	$-0.787183\pi$
$233$	−23.9559	−1.56940	−0.784700	+	0.619876i	$0.787183\pi$
$234$	0	0
$235$	6.22169	0.405858
$236$	0	0
$237$	11.0665	0.718844
$238$	0	0
$239$	−12.7704	−0.826045	−0.413023	−	0.910721i	$-0.635527\pi$
$239$	−12.7704	−0.826045	−0.413023	+	0.910721i	$0.635527\pi$
$240$	0	0
$241$	8.49660	0.547314	0.273657	−	0.961827i	$-0.411767\pi$
$241$	8.49660	0.547314	0.273657	+	0.961827i	$0.411767\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.05428	−0.259018
$246$	0	0
$247$	−8.83864	−0.562389
$248$	0	0
$249$	6.86134	0.434820
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−0.785454	−0.0493811
$254$	0	0
$255$	−3.06310	−0.191819
$256$	0	0
$257$	22.9505	1.43161	0.715805	−	0.698300i	$-0.246060\pi$
$257$	22.9505	1.43161	0.715805	+	0.698300i	$0.246060\pi$
$258$	0	0
$259$	−7.39193	−0.459312
$260$	0	0
$261$	−1.62556	−0.100619
$262$	0	0
$263$	−11.9641	−0.737736	−0.368868	−	0.929482i	$-0.620255\pi$
$263$	−11.9641	−0.737736	−0.368868	+	0.929482i	$0.620255\pi$
$264$	0	0
$265$	−5.60923	−0.344572
$266$	0	0
$267$	−3.33660	−0.204197
$268$	0	0
$269$	−24.5763	−1.49845	−0.749223	−	0.662318i	$-0.769573\pi$
$269$	−24.5763	−1.49845	−0.749223	+	0.662318i	$0.769573\pi$
$270$	0	0
$271$	0.835452	0.0507501	0.0253751	−	0.999678i	$-0.491922\pi$
$271$	0.835452	0.0507501	0.0253751	+	0.999678i	$0.491922\pi$
$272$	0	0
$273$	−3.81850	−0.231106
$274$	0	0
$275$	7.07691	0.426754
$276$	0	0
$277$	−5.96777	−0.358568	−0.179284	−	0.983797i	$-0.557378\pi$
$277$	−5.96777	−0.358568	−0.179284	+	0.983797i	$0.557378\pi$
$278$	0	0
$279$	4.16977	0.249638
$280$	0	0
$281$	15.3083	0.913217	0.456609	−	0.889668i	$-0.349064\pi$
$281$	15.3083	0.913217	0.456609	+	0.889668i	$0.349064\pi$
$282$	0	0
$283$	−11.2926	−0.671276	−0.335638	−	0.941991i	$-0.608952\pi$
$283$	−11.2926	−0.671276	−0.335638	+	0.941991i	$0.608952\pi$
$284$	0	0
$285$	−1.60421	−0.0950252
$286$	0	0
$287$	8.01451	0.473081
$288$	0	0
$289$	3.28622	0.193307
$290$	0	0
$291$	13.3115	0.780332
$292$	0	0
$293$	−10.6118	−0.619950	−0.309975	−	0.950745i	$-0.600321\pi$
$293$	−10.6118	−0.619950	−0.309975	+	0.950745i	$0.600321\pi$
$294$	0	0
$295$	5.04132	0.293517
$296$	0	0
$297$	1.55965	0.0905002
$298$	0	0
$299$	−1.88702	−0.109129
$300$	0	0
$301$	−1.63005	−0.0939547
$302$	0	0
$303$	−17.7156	−1.01774
$304$	0	0
$305$	−7.03542	−0.402847
$306$	0	0
$307$	−3.21240	−0.183341	−0.0916707	−	0.995789i	$-0.529221\pi$
$307$	−3.21240	−0.183341	−0.0916707	+	0.995789i	$0.529221\pi$
$308$	0	0
$309$	17.2489	0.981257
$310$	0	0
$311$	−25.9511	−1.47155	−0.735777	−	0.677224i	$-0.763183\pi$
$311$	−25.9511	−1.47155	−0.735777	+	0.677224i	$0.763183\pi$
$312$	0	0
$313$	19.6345	1.10981	0.554903	−	0.831915i	$-0.312756\pi$
$313$	19.6345	1.10981	0.554903	+	0.831915i	$0.312756\pi$
$314$	0	0
$315$	−0.693057	−0.0390493
$316$	0	0
$317$	−10.0929	−0.566874	−0.283437	−	0.958991i	$-0.591475\pi$
$317$	−10.0929	−0.566874	−0.283437	+	0.958991i	$0.591475\pi$
$318$	0	0
$319$	2.53530	0.141950
$320$	0	0
$321$	9.79844	0.546895
$322$	0	0
$323$	10.6243	0.591154
$324$	0	0
$325$	17.0020	0.943101
$326$	0	0
$327$	0.786029	0.0434675
$328$	0	0
$329$	−9.32302	−0.513995
$330$	0	0
$331$	−4.38425	−0.240980	−0.120490	−	0.992715i	$-0.538447\pi$
$331$	−4.38425	−0.240980	−0.120490	+	0.992715i	$0.538447\pi$
$332$	0	0
$333$	7.25353	0.397491
$334$	0	0
$335$	10.7080	0.585041
$336$	0	0
$337$	18.5751	1.01185	0.505924	−	0.862578i	$-0.331152\pi$
$337$	18.5751	1.01185	0.505924	+	0.862578i	$0.331152\pi$
$338$	0	0
$339$	2.61653	0.142110
$340$	0	0
$341$	−6.50339	−0.352179
$342$	0	0
$343$	13.2088	0.713208
$344$	0	0
$345$	−0.342494	−0.0184393
$346$	0	0
$347$	27.7739	1.49098	0.745490	−	0.666517i	$-0.232216\pi$
$347$	27.7739	1.49098	0.745490	+	0.666517i	$0.232216\pi$
$348$	0	0
$349$	−36.7486	−1.96711	−0.983554	−	0.180616i	$-0.942191\pi$
$349$	−36.7486	−1.96711	−0.983554	+	0.180616i	$0.942191\pi$
$350$	0	0
$351$	3.74701	0.200000
$352$	0	0
$353$	−19.9622	−1.06248	−0.531241	−	0.847221i	$-0.678274\pi$
$353$	−19.9622	−1.06248	−0.531241	+	0.847221i	$0.678274\pi$
$354$	0	0
$355$	3.45815	0.183539
$356$	0	0
$357$	4.58996	0.242927
$358$	0	0
$359$	15.3151	0.808301	0.404150	−	0.914693i	$-0.367567\pi$
$359$	15.3151	0.808301	0.404150	+	0.914693i	$0.367567\pi$
$360$	0	0
$361$	−13.4358	−0.707148
$362$	0	0
$363$	8.56749	0.449676
$364$	0	0
$365$	−1.78093	−0.0932180
$366$	0	0
$367$	−6.13759	−0.320379	−0.160190	−	0.987086i	$-0.551211\pi$
$367$	−6.13759	−0.320379	−0.160190	+	0.987086i	$0.551211\pi$
$368$	0	0
$369$	−7.86444	−0.409407
$370$	0	0
$371$	8.40527	0.436380
$372$	0	0
$373$	28.6736	1.48466	0.742332	−	0.670032i	$-0.233720\pi$
$373$	28.6736	1.48466	0.742332	+	0.670032i	$0.233720\pi$
$374$	0	0
$375$	6.48626	0.334949
$376$	0	0
$377$	6.09097	0.313701
$378$	0	0
$379$	−10.0800	−0.517776	−0.258888	−	0.965907i	$-0.583356\pi$
$379$	−10.0800	−0.517776	−0.258888	+	0.965907i	$0.583356\pi$
$380$	0	0
$381$	1.36351	0.0698547
$382$	0	0
$383$	−19.9819	−1.02103	−0.510515	−	0.859869i	$-0.670545\pi$
$383$	−19.9819	−1.02103	−0.510515	+	0.859869i	$0.670545\pi$
$384$	0	0
$385$	1.08093	0.0550892
$386$	0	0
$387$	1.59953	0.0813088
$388$	0	0
$389$	−35.5529	−1.80260	−0.901301	−	0.433194i	$-0.857387\pi$
$389$	−35.5529	−1.80260	−0.901301	+	0.433194i	$0.857387\pi$
$390$	0	0
$391$	2.26827	0.114711
$392$	0	0
$393$	15.0256	0.757943
$394$	0	0
$395$	−7.52608	−0.378678
$396$	0	0
$397$	−24.9442	−1.25191	−0.625957	−	0.779857i	$-0.715292\pi$
$397$	−24.9442	−1.25191	−0.625957	+	0.779857i	$0.715292\pi$
$398$	0	0
$399$	2.40386	0.120344
$400$	0	0
$401$	−34.5521	−1.72545	−0.862724	−	0.505674i	$-0.831244\pi$
$401$	−34.5521	−1.72545	−0.862724	+	0.505674i	$0.831244\pi$
$402$	0	0
$403$	−15.6242	−0.778295
$404$	0	0
$405$	0.680080	0.0337935
$406$	0	0
$407$	−11.3130	−0.560764
$408$	0	0
$409$	15.9863	0.790469	0.395235	−	0.918580i	$-0.370663\pi$
$409$	15.9863	0.790469	0.395235	+	0.918580i	$0.370663\pi$
$410$	0	0
$411$	7.71407	0.380507
$412$	0	0
$413$	−7.55428	−0.371722
$414$	0	0
$415$	−4.66626	−0.229058
$416$	0	0
$417$	−17.7822	−0.870796
$418$	0	0
$419$	8.63660	0.421926	0.210963	−	0.977494i	$-0.432340\pi$
$419$	8.63660	0.421926	0.210963	+	0.977494i	$0.432340\pi$
$420$	0	0
$421$	33.3142	1.62364	0.811818	−	0.583911i	$-0.198478\pi$
$421$	33.3142	1.62364	0.811818	+	0.583911i	$0.198478\pi$
$422$	0	0
$423$	9.14846	0.444813
$424$	0	0
$425$	−20.4370	−0.991338
$426$	0	0
$427$	10.5424	0.510182
$428$	0	0
$429$	−5.84402	−0.282152
$430$	0	0
$431$	−13.1360	−0.632739	−0.316369	−	0.948636i	$-0.602464\pi$
$431$	−13.1360	−0.632739	−0.316369	+	0.948636i	$0.602464\pi$
$432$	0	0
$433$	−36.9514	−1.77577	−0.887886	−	0.460064i	$-0.847826\pi$
$433$	−36.9514	−1.77577	−0.887886	+	0.460064i	$0.847826\pi$
$434$	0	0
$435$	1.10551	0.0530051
$436$	0	0
$437$	1.18794	0.0568269
$438$	0	0
$439$	−28.9378	−1.38113	−0.690563	−	0.723273i	$-0.742637\pi$
$439$	−28.9378	−1.38113	−0.690563	+	0.723273i	$0.742637\pi$
$440$	0	0
$441$	−5.96147	−0.283880
$442$	0	0
$443$	−35.8535	−1.70345	−0.851725	−	0.523989i	$-0.824443\pi$
$443$	−35.8535	−1.70345	−0.851725	+	0.523989i	$0.824443\pi$
$444$	0	0
$445$	2.26916	0.107568
$446$	0	0
$447$	−12.5608	−0.594107
$448$	0	0
$449$	−14.9555	−0.705794	−0.352897	−	0.935662i	$-0.614803\pi$
$449$	−14.9555	−0.705794	−0.352897	+	0.935662i	$0.614803\pi$
$450$	0	0
$451$	12.2658	0.577574
$452$	0	0
$453$	0.355240	0.0166906
$454$	0	0
$455$	2.59689	0.121744
$456$	0	0
$457$	35.1905	1.64614	0.823071	−	0.567938i	$-0.192259\pi$
$457$	35.1905	1.64614	0.823071	+	0.567938i	$0.192259\pi$
$458$	0	0
$459$	−4.50402	−0.210230
$460$	0	0
$461$	4.25208	0.198039	0.0990197	−	0.995085i	$-0.468429\pi$
$461$	4.25208	0.198039	0.0990197	+	0.995085i	$0.468429\pi$
$462$	0	0
$463$	−32.3609	−1.50394	−0.751969	−	0.659199i	$-0.770895\pi$
$463$	−32.3609	−1.50394	−0.751969	+	0.659199i	$0.770895\pi$
$464$	0	0
$465$	−2.83578	−0.131506
$466$	0	0
$467$	14.7895	0.684375	0.342188	−	0.939632i	$-0.388832\pi$
$467$	14.7895	0.684375	0.342188	+	0.939632i	$0.388832\pi$
$468$	0	0
$469$	−16.0457	−0.740920
$470$	0	0
$471$	−2.73936	−0.126223
$472$	0	0
$473$	−2.49471	−0.114707
$474$	0	0
$475$	−10.7033	−0.491100
$476$	0	0
$477$	−8.24789	−0.377645
$478$	0	0
$479$	−2.18597	−0.0998793	−0.0499397	−	0.998752i	$-0.515903\pi$
$479$	−2.18597	−0.0998793	−0.0499397	+	0.998752i	$0.515903\pi$
$480$	0	0
$481$	−27.1790	−1.23926
$482$	0	0
$483$	0.513218	0.0233522
$484$	0	0
$485$	−9.05286	−0.411069
$486$	0	0
$487$	−22.8041	−1.03335	−0.516677	−	0.856180i	$-0.672831\pi$
$487$	−22.8041	−1.03335	−0.516677	+	0.856180i	$0.672831\pi$
$488$	0	0
$489$	0.888539	0.0401811
$490$	0	0
$491$	33.7016	1.52093	0.760466	−	0.649378i	$-0.224971\pi$
$491$	33.7016	1.52093	0.760466	+	0.649378i	$0.224971\pi$
$492$	0	0
$493$	−7.32155	−0.329746
$494$	0	0
$495$	−1.06069	−0.0476744
$496$	0	0
$497$	−5.18194	−0.232442
$498$	0	0
$499$	−23.3858	−1.04689	−0.523446	−	0.852059i	$-0.675354\pi$
$499$	−23.3858	−1.04689	−0.523446	+	0.852059i	$0.675354\pi$
$500$	0	0
$501$	6.40514	0.286160
$502$	0	0
$503$	31.9223	1.42335	0.711673	−	0.702511i	$-0.247938\pi$
$503$	31.9223	1.42335	0.711673	+	0.702511i	$0.247938\pi$
$504$	0	0
$505$	12.0481	0.536132
$506$	0	0
$507$	−1.04005	−0.0461902
$508$	0	0
$509$	−0.452800	−0.0200700	−0.0100350	−	0.999950i	$-0.503194\pi$
$509$	−0.452800	−0.0200700	−0.0100350	+	0.999950i	$0.503194\pi$
$510$	0	0
$511$	2.66867	0.118055
$512$	0	0
$513$	−2.35885	−0.104146
$514$	0	0
$515$	−11.7306	−0.516914
$516$	0	0
$517$	−14.2684	−0.627524
$518$	0	0
$519$	8.39208	0.368371
$520$	0	0
$521$	−29.8910	−1.30955	−0.654774	−	0.755824i	$-0.727236\pi$
$521$	−29.8910	−1.30955	−0.654774	+	0.755824i	$0.727236\pi$
$522$	0	0
$523$	−19.8076	−0.866127	−0.433063	−	0.901363i	$-0.642567\pi$
$523$	−19.8076	−0.866127	−0.433063	+	0.901363i	$0.642567\pi$
$524$	0	0
$525$	−4.62407	−0.201811
$526$	0	0
$527$	18.7807	0.818102
$528$	0	0
$529$	−22.7464	−0.988973
$530$	0	0
$531$	7.41283	0.321689
$532$	0	0
$533$	29.4681	1.27641
$534$	0	0
$535$	−6.66372	−0.288098
$536$	0	0
$537$	−6.29521	−0.271658
$538$	0	0
$539$	9.29783	0.400486
$540$	0	0
$541$	−18.7988	−0.808223	−0.404112	−	0.914710i	$-0.632419\pi$
$541$	−18.7988	−0.808223	−0.404112	+	0.914710i	$0.632419\pi$
$542$	0	0
$543$	20.1784	0.865937
$544$	0	0
$545$	−0.534563	−0.0228982
$546$	0	0
$547$	29.0916	1.24387	0.621934	−	0.783069i	$-0.286347\pi$
$547$	29.0916	1.24387	0.621934	+	0.783069i	$0.286347\pi$
$548$	0	0
$549$	−10.3450	−0.441513
$550$	0	0
$551$	−3.83445	−0.163353
$552$	0	0
$553$	11.2776	0.479573
$554$	0	0
$555$	−4.93298	−0.209393
$556$	0	0
$557$	−2.40656	−0.101969	−0.0509847	−	0.998699i	$-0.516236\pi$
$557$	−2.40656	−0.101969	−0.0509847	+	0.998699i	$0.516236\pi$
$558$	0	0
$559$	−5.99346	−0.253496
$560$	0	0
$561$	7.02471	0.296583
$562$	0	0
$563$	−36.2766	−1.52888	−0.764438	−	0.644697i	$-0.776983\pi$
$563$	−36.2766	−1.52888	−0.764438	+	0.644697i	$0.776983\pi$
$564$	0	0
$565$	−1.77945	−0.0748620
$566$	0	0
$567$	−1.01908	−0.0427974
$568$	0	0
$569$	−28.8028	−1.20748	−0.603738	−	0.797183i	$-0.706323\pi$
$569$	−28.8028	−1.20748	−0.603738	+	0.797183i	$0.706323\pi$
$570$	0	0
$571$	−20.1250	−0.842207	−0.421103	−	0.907013i	$-0.638357\pi$
$571$	−20.1250	−0.842207	−0.421103	+	0.907013i	$0.638357\pi$
$572$	0	0
$573$	25.2081	1.05309
$574$	0	0
$575$	−2.28512	−0.0952961
$576$	0	0
$577$	23.5392	0.979951	0.489975	−	0.871736i	$-0.337006\pi$
$577$	23.5392	0.979951	0.489975	+	0.871736i	$0.337006\pi$
$578$	0	0
$579$	0.797302	0.0331347
$580$	0	0
$581$	6.99226	0.290088
$582$	0	0
$583$	12.8638	0.532766
$584$	0	0
$585$	−2.54826	−0.105358
$586$	0	0
$587$	15.2592	0.629816	0.314908	−	0.949122i	$-0.398026\pi$
$587$	15.2592	0.629816	0.314908	+	0.949122i	$0.398026\pi$
$588$	0	0
$589$	9.83589	0.405281
$590$	0	0
$591$	15.6086	0.642051
$592$	0	0
$593$	14.0429	0.576674	0.288337	−	0.957529i	$-0.406898\pi$
$593$	14.0429	0.576674	0.288337	+	0.957529i	$0.406898\pi$
$594$	0	0
$595$	−3.12154	−0.127971
$596$	0	0
$597$	−15.6916	−0.642216
$598$	0	0
$599$	−13.6050	−0.555885	−0.277942	−	0.960598i	$-0.589652\pi$
$599$	−13.6050	−0.555885	−0.277942	+	0.960598i	$0.589652\pi$
$600$	0	0
$601$	3.56430	0.145391	0.0726954	−	0.997354i	$-0.476840\pi$
$601$	3.56430	0.145391	0.0726954	+	0.997354i	$0.476840\pi$
$602$	0	0
$603$	15.7452	0.641195
$604$	0	0
$605$	−5.82658	−0.236884
$606$	0	0
$607$	35.2032	1.42885	0.714427	−	0.699710i	$-0.246688\pi$
$607$	35.2032	1.42885	0.714427	+	0.699710i	$0.246688\pi$
$608$	0	0
$609$	−1.65657	−0.0671278
$610$	0	0
$611$	−34.2793	−1.38679
$612$	0	0
$613$	31.3860	1.26767	0.633835	−	0.773469i	$-0.281480\pi$
$613$	31.3860	1.26767	0.633835	+	0.773469i	$0.281480\pi$
$614$	0	0
$615$	5.34845	0.215670
$616$	0	0
$617$	−15.7110	−0.632503	−0.316251	−	0.948675i	$-0.602424\pi$
$617$	−15.7110	−0.632503	−0.316251	+	0.948675i	$0.602424\pi$
$618$	0	0
$619$	−35.8045	−1.43910	−0.719552	−	0.694439i	$-0.755653\pi$
$619$	−35.8045	−1.43910	−0.719552	+	0.694439i	$0.755653\pi$
$620$	0	0
$621$	−0.503609	−0.0202091
$622$	0	0
$623$	−3.40027	−0.136229
$624$	0	0
$625$	18.2763	0.731051
$626$	0	0
$627$	3.67899	0.146925
$628$	0	0
$629$	32.6701	1.30264
$630$	0	0
$631$	3.27460	0.130360	0.0651798	−	0.997874i	$-0.479238\pi$
$631$	3.27460	0.130360	0.0651798	+	0.997874i	$0.479238\pi$
$632$	0	0
$633$	−12.8880	−0.512252
$634$	0	0
$635$	−0.927295	−0.0367986
$636$	0	0
$637$	22.3377	0.885051
$638$	0	0
$639$	5.08491	0.201156
$640$	0	0
$641$	−24.8999	−0.983489	−0.491744	−	0.870740i	$-0.663641\pi$
$641$	−24.8999	−0.983489	−0.491744	+	0.870740i	$0.663641\pi$
$642$	0	0
$643$	15.1235	0.596413	0.298206	−	0.954501i	$-0.403612\pi$
$643$	15.1235	0.596413	0.298206	+	0.954501i	$0.403612\pi$
$644$	0	0
$645$	−1.08781	−0.0428325
$646$	0	0
$647$	12.0625	0.474225	0.237113	−	0.971482i	$-0.423799\pi$
$647$	12.0625	0.474225	0.237113	+	0.971482i	$0.423799\pi$
$648$	0	0
$649$	−11.5614	−0.453826
$650$	0	0
$651$	4.24934	0.166545
$652$	0	0
$653$	21.9257	0.858020	0.429010	−	0.903300i	$-0.358862\pi$
$653$	21.9257	0.858020	0.429010	+	0.903300i	$0.358862\pi$
$654$	0	0
$655$	−10.2186	−0.399275
$656$	0	0
$657$	−2.61870	−0.102165
$658$	0	0
$659$	−42.9219	−1.67200	−0.835999	−	0.548731i	$-0.815111\pi$
$659$	−42.9219	−1.67200	−0.835999	+	0.548731i	$0.815111\pi$
$660$	0	0
$661$	28.8242	1.12113	0.560565	−	0.828110i	$-0.310584\pi$
$661$	28.8242	1.12113	0.560565	+	0.828110i	$0.310584\pi$
$662$	0	0
$663$	16.8766	0.655433
$664$	0	0
$665$	−1.63482	−0.0633956
$666$	0	0
$667$	−0.818645	−0.0316981
$668$	0	0
$669$	5.99483	0.231773
$670$	0	0
$671$	16.1346	0.622869
$672$	0	0
$673$	12.2137	0.470805	0.235403	−	0.971898i	$-0.424359\pi$
$673$	12.2137	0.470805	0.235403	+	0.971898i	$0.424359\pi$
$674$	0	0
$675$	4.53749	0.174648
$676$	0	0
$677$	−38.4569	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$677$	−38.4569	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$678$	0	0
$679$	13.5655	0.520595
$680$	0	0
$681$	−0.239494	−0.00917744
$682$	0	0
$683$	−6.09916	−0.233378	−0.116689	−	0.993169i	$-0.537228\pi$
$683$	−6.09916	−0.233378	−0.116689	+	0.993169i	$0.537228\pi$
$684$	0	0
$685$	−5.24619	−0.200446
$686$	0	0
$687$	−23.1460	−0.883076
$688$	0	0
$689$	30.9049	1.17738
$690$	0	0
$691$	40.8412	1.55367	0.776836	−	0.629703i	$-0.216824\pi$
$691$	40.8412	1.55367	0.776836	+	0.629703i	$0.216824\pi$
$692$	0	0
$693$	1.58941	0.0603768
$694$	0	0
$695$	12.0933	0.458725
$696$	0	0
$697$	−35.4216	−1.34169
$698$	0	0
$699$	23.9559	0.906094
$700$	0	0
$701$	−28.2097	−1.06547	−0.532733	−	0.846283i	$-0.678835\pi$
$701$	−28.2097	−1.06547	−0.532733	+	0.846283i	$0.678835\pi$
$702$	0	0
$703$	17.1100	0.645317
$704$	0	0
$705$	−6.22169	−0.234322
$706$	0	0
$707$	−18.0537	−0.678979
$708$	0	0
$709$	−32.1710	−1.20821	−0.604103	−	0.796906i	$-0.706469\pi$
$709$	−32.1710	−1.20821	−0.604103	+	0.796906i	$0.706469\pi$
$710$	0	0
$711$	−11.0665	−0.415025
$712$	0	0
$713$	2.09993	0.0786431
$714$	0	0
$715$	3.97441	0.148634
$716$	0	0
$717$	12.7704	0.476918
$718$	0	0
$719$	22.2038	0.828063	0.414031	−	0.910263i	$-0.364120\pi$
$719$	22.2038	0.828063	0.414031	+	0.910263i	$0.364120\pi$
$720$	0	0
$721$	17.5780	0.654641
$722$	0	0
$723$	−8.49660	−0.315992
$724$	0	0
$725$	7.37595	0.273936
$726$	0	0
$727$	11.9024	0.441436	0.220718	−	0.975338i	$-0.429160\pi$
$727$	11.9024	0.441436	0.220718	+	0.975338i	$0.429160\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.20433	0.266462
$732$	0	0
$733$	50.6835	1.87204	0.936019	−	0.351948i	$-0.114481\pi$
$733$	50.6835	1.87204	0.936019	+	0.351948i	$0.114481\pi$
$734$	0	0
$735$	4.05428	0.149544
$736$	0	0
$737$	−24.5571	−0.904571
$738$	0	0
$739$	16.0446	0.590211	0.295105	−	0.955465i	$-0.404645\pi$
$739$	16.0446	0.590211	0.295105	+	0.955465i	$0.404645\pi$
$740$	0	0
$741$	8.83864	0.324696
$742$	0	0
$743$	31.8770	1.16945	0.584727	−	0.811230i	$-0.301202\pi$
$743$	31.8770	1.16945	0.584727	+	0.811230i	$0.301202\pi$
$744$	0	0
$745$	8.54237	0.312968
$746$	0	0
$747$	−6.86134	−0.251043
$748$	0	0
$749$	9.98540	0.364859
$750$	0	0
$751$	−2.11597	−0.0772127	−0.0386064	−	0.999254i	$-0.512292\pi$
$751$	−2.11597	−0.0772127	−0.0386064	+	0.999254i	$0.512292\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	−0.241592	−0.00879242
$756$	0	0
$757$	34.5007	1.25395	0.626974	−	0.779040i	$-0.284293\pi$
$757$	34.5007	1.25395	0.626974	+	0.779040i	$0.284293\pi$
$758$	0	0
$759$	0.785454	0.0285102
$760$	0	0
$761$	−13.0080	−0.471539	−0.235769	−	0.971809i	$-0.575761\pi$
$761$	−13.0080	−0.471539	−0.235769	+	0.971809i	$0.575761\pi$
$762$	0	0
$763$	0.801028	0.0289991
$764$	0	0
$765$	3.06310	0.110746
$766$	0	0
$767$	−27.7759	−1.00293
$768$	0	0
$769$	41.4591	1.49505	0.747526	−	0.664233i	$-0.231242\pi$
$769$	41.4591	1.49505	0.747526	+	0.664233i	$0.231242\pi$
$770$	0	0
$771$	−22.9505	−0.826540
$772$	0	0
$773$	11.6399	0.418657	0.209328	−	0.977845i	$-0.432872\pi$
$773$	11.6399	0.418657	0.209328	+	0.977845i	$0.432872\pi$
$774$	0	0
$775$	−18.9203	−0.679637
$776$	0	0
$777$	7.39193	0.265184
$778$	0	0
$779$	−18.5511	−0.664661
$780$	0	0
$781$	−7.93069	−0.283783
$782$	0	0
$783$	1.62556	0.0580927
$784$	0	0
$785$	1.86299	0.0664928
$786$	0	0
$787$	37.7329	1.34503	0.672517	−	0.740082i	$-0.265213\pi$
$787$	37.7329	1.34503	0.672517	+	0.740082i	$0.265213\pi$
$788$	0	0
$789$	11.9641	0.425932
$790$	0	0
$791$	2.66645	0.0948082
$792$	0	0
$793$	38.7627	1.37650
$794$	0	0
$795$	5.60923	0.198939
$796$	0	0
$797$	−11.2859	−0.399768	−0.199884	−	0.979820i	$-0.564057\pi$
$797$	−11.2859	−0.399768	−0.199884	+	0.979820i	$0.564057\pi$
$798$	0	0
$799$	41.2049	1.45772
$800$	0	0
$801$	3.33660	0.117893
$802$	0	0
$803$	4.08426	0.144130
$804$	0	0
$805$	−0.349030	−0.0123017
$806$	0	0
$807$	24.5763	0.865128
$808$	0	0
$809$	−23.0021	−0.808710	−0.404355	−	0.914602i	$-0.632504\pi$
$809$	−23.0021	−0.808710	−0.404355	+	0.914602i	$0.632504\pi$
$810$	0	0
$811$	−4.71627	−0.165611	−0.0828053	−	0.996566i	$-0.526388\pi$
$811$	−4.71627	−0.165611	−0.0828053	+	0.996566i	$0.526388\pi$
$812$	0	0
$813$	−0.835452	−0.0293006
$814$	0	0
$815$	−0.604278	−0.0211669
$816$	0	0
$817$	3.77307	0.132003
$818$	0	0
$819$	3.81850	0.133429
$820$	0	0
$821$	20.1364	0.702765	0.351383	−	0.936232i	$-0.385712\pi$
$821$	20.1364	0.702765	0.351383	+	0.936232i	$0.385712\pi$
$822$	0	0
$823$	20.3697	0.710043	0.355021	−	0.934858i	$-0.384474\pi$
$823$	20.3697	0.710043	0.355021	+	0.934858i	$0.384474\pi$
$824$	0	0
$825$	−7.07691	−0.246386
$826$	0	0
$827$	29.1385	1.01325	0.506623	−	0.862168i	$-0.330894\pi$
$827$	29.1385	1.01325	0.506623	+	0.862168i	$0.330894\pi$
$828$	0	0
$829$	−45.2022	−1.56994	−0.784968	−	0.619536i	$-0.787321\pi$
$829$	−45.2022	−1.56994	−0.784968	+	0.619536i	$0.787321\pi$
$830$	0	0
$831$	5.96777	0.207020
$832$	0	0
$833$	−26.8506	−0.930319
$834$	0	0
$835$	−4.35601	−0.150746
$836$	0	0
$837$	−4.16977	−0.144128
$838$	0	0
$839$	12.3205	0.425350	0.212675	−	0.977123i	$-0.431782\pi$
$839$	12.3205	0.425350	0.212675	+	0.977123i	$0.431782\pi$
$840$	0	0
$841$	−26.3576	−0.908882
$842$	0	0
$843$	−15.3083	−0.527246
$844$	0	0
$845$	0.707316	0.0243324
$846$	0	0
$847$	8.73096	0.299999
$848$	0	0
$849$	11.2926	0.387562
$850$	0	0
$851$	3.65294	0.125221
$852$	0	0
$853$	−14.9328	−0.511289	−0.255644	−	0.966771i	$-0.582288\pi$
$853$	−14.9328	−0.511289	−0.255644	+	0.966771i	$0.582288\pi$
$854$	0	0
$855$	1.60421	0.0548628
$856$	0	0
$857$	11.5629	0.394982	0.197491	−	0.980305i	$-0.436721\pi$
$857$	11.5629	0.394982	0.197491	+	0.980305i	$0.436721\pi$
$858$	0	0
$859$	18.1512	0.619310	0.309655	−	0.950849i	$-0.399786\pi$
$859$	18.1512	0.619310	0.309655	+	0.950849i	$0.399786\pi$
$860$	0	0
$861$	−8.01451	−0.273134
$862$	0	0
$863$	−8.96887	−0.305304	−0.152652	−	0.988280i	$-0.548781\pi$
$863$	−8.96887	−0.305304	−0.152652	+	0.988280i	$0.548781\pi$
$864$	0	0
$865$	−5.70729	−0.194054
$866$	0	0
$867$	−3.28622	−0.111606
$868$	0	0
$869$	17.2598	0.585499
$870$	0	0
$871$	−58.9974	−1.99905
$872$	0	0
$873$	−13.3115	−0.450525
$874$	0	0
$875$	6.61002	0.223460
$876$	0	0
$877$	25.5065	0.861295	0.430647	−	0.902520i	$-0.358285\pi$
$877$	25.5065	0.861295	0.430647	+	0.902520i	$0.358285\pi$
$878$	0	0
$879$	10.6118	0.357928
$880$	0	0
$881$	−56.3169	−1.89737	−0.948683	−	0.316230i	$-0.897583\pi$
$881$	−56.3169	−1.89737	−0.948683	+	0.316230i	$0.897583\pi$
$882$	0	0
$883$	24.4824	0.823899	0.411950	−	0.911207i	$-0.364848\pi$
$883$	24.4824	0.823899	0.411950	+	0.911207i	$0.364848\pi$
$884$	0	0
$885$	−5.04132	−0.169462
$886$	0	0
$887$	9.34990	0.313939	0.156969	−	0.987603i	$-0.449828\pi$
$887$	9.34990	0.313939	0.156969	+	0.987603i	$0.449828\pi$
$888$	0	0
$889$	1.38953	0.0466032
$890$	0	0
$891$	−1.55965	−0.0522503
$892$	0	0
$893$	21.5799	0.722143
$894$	0	0
$895$	4.28125	0.143106
$896$	0	0
$897$	1.88702	0.0630059
$898$	0	0
$899$	−6.77820	−0.226066
$900$	0	0
$901$	−37.1487	−1.23760
$902$	0	0
$903$	1.63005	0.0542448
$904$	0	0
$905$	−13.7229	−0.456165
$906$	0	0
$907$	54.4069	1.80655	0.903276	−	0.429060i	$-0.141155\pi$
$907$	54.4069	1.80655	0.903276	+	0.429060i	$0.141155\pi$
$908$	0	0
$909$	17.7156	0.587591
$910$	0	0
$911$	−29.1279	−0.965051	−0.482526	−	0.875882i	$-0.660280\pi$
$911$	−29.1279	−0.965051	−0.482526	+	0.875882i	$0.660280\pi$
$912$	0	0
$913$	10.7013	0.354161
$914$	0	0
$915$	7.03542	0.232584
$916$	0	0
$917$	15.3123	0.505658
$918$	0	0
$919$	−50.8890	−1.67867	−0.839337	−	0.543612i	$-0.817056\pi$
$919$	−50.8890	−1.67867	−0.839337	+	0.543612i	$0.817056\pi$
$920$	0	0
$921$	3.21240	0.105852
$922$	0	0
$923$	−19.0532	−0.627143
$924$	0	0
$925$	−32.9128	−1.08217
$926$	0	0
$927$	−17.2489	−0.566529
$928$	0	0
$929$	46.6535	1.53065	0.765326	−	0.643643i	$-0.222578\pi$
$929$	46.6535	1.53065	0.765326	+	0.643643i	$0.222578\pi$
$930$	0	0
$931$	−14.0622	−0.460872
$932$	0	0
$933$	25.9511	0.849602
$934$	0	0
$935$	−4.77737	−0.156237
$936$	0	0
$937$	−19.9674	−0.652305	−0.326153	−	0.945317i	$-0.605752\pi$
$937$	−19.9674	−0.652305	−0.326153	+	0.945317i	$0.605752\pi$
$938$	0	0
$939$	−19.6345	−0.640746
$940$	0	0
$941$	−56.8712	−1.85395	−0.926974	−	0.375125i	$-0.877600\pi$
$941$	−56.8712	−1.85395	−0.926974	+	0.375125i	$0.877600\pi$
$942$	0	0
$943$	−3.96060	−0.128975
$944$	0	0
$945$	0.693057	0.0225451
$946$	0	0
$947$	−53.3893	−1.73492	−0.867460	−	0.497507i	$-0.834249\pi$
$947$	−53.3893	−1.73492	−0.867460	+	0.497507i	$0.834249\pi$
$948$	0	0
$949$	9.81229	0.318520
$950$	0	0
$951$	10.0929	0.327285
$952$	0	0
$953$	48.1320	1.55915	0.779575	−	0.626309i	$-0.215435\pi$
$953$	48.1320	1.55915	0.779575	+	0.626309i	$0.215435\pi$
$954$	0	0
$955$	−17.1436	−0.554753
$956$	0	0
$957$	−2.53530	−0.0819547
$958$	0	0
$959$	7.86126	0.253853
$960$	0	0
$961$	−13.6130	−0.439129
$962$	0	0
$963$	−9.79844	−0.315750
$964$	0	0
$965$	−0.542229	−0.0174550
$966$	0	0
$967$	−6.58165	−0.211652	−0.105826	−	0.994385i	$-0.533749\pi$
$967$	−6.58165	−0.211652	−0.105826	+	0.994385i	$0.533749\pi$
$968$	0	0
$969$	−10.6243	−0.341303
$970$	0	0
$971$	16.2125	0.520283	0.260142	−	0.965570i	$-0.416231\pi$
$971$	16.2125	0.520283	0.260142	+	0.965570i	$0.416231\pi$
$972$	0	0
$973$	−18.1215	−0.580947
$974$	0	0
$975$	−17.0020	−0.544500
$976$	0	0
$977$	−9.10953	−0.291440	−0.145720	−	0.989326i	$-0.546550\pi$
$977$	−9.10953	−0.291440	−0.145720	+	0.989326i	$0.546550\pi$
$978$	0	0
$979$	−5.20394	−0.166319
$980$	0	0
$981$	−0.786029	−0.0250960
$982$	0	0
$983$	29.2998	0.934519	0.467260	−	0.884120i	$-0.345241\pi$
$983$	29.2998	0.934519	0.467260	+	0.884120i	$0.345241\pi$
$984$	0	0
$985$	−10.6151	−0.338225
$986$	0	0
$987$	9.32302	0.296755
$988$	0	0
$989$	0.805539	0.0256146
$990$	0	0
$991$	14.2679	0.453235	0.226617	−	0.973984i	$-0.427233\pi$
$991$	14.2679	0.453235	0.226617	+	0.973984i	$0.427233\pi$
$992$	0	0
$993$	4.38425	0.139130
$994$	0	0
$995$	10.6716	0.338312
$996$	0	0
$997$	37.8132	1.19756	0.598778	−	0.800915i	$-0.295653\pi$
$997$	37.8132	1.19756	0.598778	+	0.800915i	$0.295653\pi$
$998$	0	0
$999$	−7.25353	−0.229492

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.n.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.680080 q^{5} -1.01908 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.680080 q^{5} -1.01908 q^{7} +1.00000 q^{9} -1.55965 q^{11} -3.74701 q^{13} -0.680080 q^{15} +4.50402 q^{17} +2.35885 q^{19} +1.01908 q^{21} +0.503609 q^{23} -4.53749 q^{25} -1.00000 q^{27} -1.62556 q^{29} +4.16977 q^{31} +1.55965 q^{33} -0.693057 q^{35} +7.25353 q^{37} +3.74701 q^{39} -7.86444 q^{41} +1.59953 q^{43} +0.680080 q^{45} +9.14846 q^{47} -5.96147 q^{49} -4.50402 q^{51} -8.24789 q^{53} -1.06069 q^{55} -2.35885 q^{57} +7.41283 q^{59} -10.3450 q^{61} -1.01908 q^{63} -2.54826 q^{65} +15.7452 q^{67} -0.503609 q^{69} +5.08491 q^{71} -2.61870 q^{73} +4.53749 q^{75} +1.58941 q^{77} -11.0665 q^{79} +1.00000 q^{81} -6.86134 q^{83} +3.06310 q^{85} +1.62556 q^{87} +3.33660 q^{89} +3.81850 q^{91} -4.16977 q^{93} +1.60421 q^{95} -13.3115 q^{97} -1.55965 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9} + 10 q^{11} - 9 q^{13} + 7 q^{15} - 22 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 23 q^{25} - 14 q^{27} - 12 q^{29} - 13 q^{31} - 10 q^{33} + 23 q^{35} + 5 q^{37} + 9 q^{39} - 52 q^{41} + 16 q^{43} - 7 q^{45} + q^{47} + 9 q^{49} + 22 q^{51} - 13 q^{53} - 12 q^{55} + 6 q^{57} + 12 q^{59} - 20 q^{61} + q^{63} - 40 q^{65} + 21 q^{67} - 3 q^{69} - 5 q^{71} - 14 q^{73} - 23 q^{75} - 14 q^{77} - 23 q^{79} + 14 q^{81} + 25 q^{83} - 11 q^{85} + 12 q^{87} - 79 q^{89} + 6 q^{91} + 13 q^{93} + 3 q^{95} - 17 q^{97} + 10 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9 + 10 * q^11 - 9 * q^13 + 7 * q^15 - 22 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 23 * q^25 - 14 * q^27 - 12 * q^29 - 13 * q^31 - 10 * q^33 + 23 * q^35 + 5 * q^37 + 9 * q^39 - 52 * q^41 + 16 * q^43 - 7 * q^45 + q^47 + 9 * q^49 + 22 * q^51 - 13 * q^53 - 12 * q^55 + 6 * q^57 + 12 * q^59 - 20 * q^61 + q^63 - 40 * q^65 + 21 * q^67 - 3 * q^69 - 5 * q^71 - 14 * q^73 - 23 * q^75 - 14 * q^77 - 23 * q^79 + 14 * q^81 + 25 * q^83 - 11 * q^85 + 12 * q^87 - 79 * q^89 + 6 * q^91 + 13 * q^93 + 3 * q^95 - 17 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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