LMFDB - Embedded newform 6024.2.a.r.1.19

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 9 x^{19} - 31 x^{18} + 471 x^{17} - 82 x^{16} - 9476 x^{15} + 12881 x^{14} + 94079 x^{13} + \cdots + 24832 $ x^20 - 9x^19 - 31x^18 + 471x^17 - 82x^16 - 9476x^15 + 12881x^14 + 94079x^13 - 181272x^12 - 504914x^11 + 1108012x^10 + 1591114x^9 - 3399127x^8 - 3224089x^7 + 5089722x^6 + 4287384x^5 - 2840216x^4 - 2820864x^3 - 208576x^2 + 204992*x + 24832
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Root			$4.35600$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +4.35600 q^{5} -4.92117 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.35600 q^{5} -4.92117 q^{7} +1.00000 q^{9} +1.24997 q^{11} +5.88291 q^{13} +4.35600 q^{15} -2.94010 q^{17} -3.68083 q^{19} -4.92117 q^{21} -0.211167 q^{23} +13.9747 q^{25} +1.00000 q^{27} +5.01806 q^{29} +2.92456 q^{31} +1.24997 q^{33} -21.4366 q^{35} -4.27512 q^{37} +5.88291 q^{39} -2.80912 q^{41} -1.34096 q^{43} +4.35600 q^{45} +8.33504 q^{47} +17.2179 q^{49} -2.94010 q^{51} +5.50512 q^{53} +5.44488 q^{55} -3.68083 q^{57} +9.06367 q^{59} -10.0314 q^{61} -4.92117 q^{63} +25.6260 q^{65} +4.09535 q^{67} -0.211167 q^{69} +3.07818 q^{71} +11.8342 q^{73} +13.9747 q^{75} -6.15133 q^{77} -16.2304 q^{79} +1.00000 q^{81} +8.87346 q^{83} -12.8071 q^{85} +5.01806 q^{87} -10.6447 q^{89} -28.9508 q^{91} +2.92456 q^{93} -16.0337 q^{95} +2.62685 q^{97} +1.24997 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * 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
$5$	4.35600	1.94806	0.974031	−	0.226413i	$-0.0726999\pi$
$5$	4.35600	1.94806	0.974031	+	0.226413i	$0.0726999\pi$
$6$	0	0
$7$	−4.92117	−1.86003	−0.930013	−	0.367527i	$-0.880204\pi$
$7$	−4.92117	−1.86003	−0.930013	+	0.367527i	$0.880204\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.24997	0.376881	0.188441	−	0.982085i	$-0.439657\pi$
$11$	1.24997	0.376881	0.188441	+	0.982085i	$0.439657\pi$
$12$	0	0
$13$	5.88291	1.63163	0.815814	−	0.578315i	$-0.196290\pi$
$13$	5.88291	1.63163	0.815814	+	0.578315i	$0.196290\pi$
$14$	0	0
$15$	4.35600	1.12471
$16$	0	0
$17$	−2.94010	−0.713078	−0.356539	−	0.934280i	$-0.616043\pi$
$17$	−2.94010	−0.713078	−0.356539	+	0.934280i	$0.616043\pi$
$18$	0	0
$19$	−3.68083	−0.844441	−0.422220	−	0.906493i	$-0.638749\pi$
$19$	−3.68083	−0.844441	−0.422220	+	0.906493i	$0.638749\pi$
$20$	0	0
$21$	−4.92117	−1.07389
$22$	0	0
$23$	−0.211167	−0.0440313	−0.0220157	−	0.999758i	$-0.507008\pi$
$23$	−0.211167	−0.0440313	−0.0220157	+	0.999758i	$0.507008\pi$
$24$	0	0
$25$	13.9747	2.79495
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.01806	0.931830	0.465915	−	0.884829i	$-0.345725\pi$
$29$	5.01806	0.931830	0.465915	+	0.884829i	$0.345725\pi$
$30$	0	0
$31$	2.92456	0.525267	0.262633	−	0.964896i	$-0.415409\pi$
$31$	2.92456	0.525267	0.262633	+	0.964896i	$0.415409\pi$
$32$	0	0
$33$	1.24997	0.217592
$34$	0	0
$35$	−21.4366	−3.62345
$36$	0	0
$37$	−4.27512	−0.702826	−0.351413	−	0.936221i	$-0.614299\pi$
$37$	−4.27512	−0.702826	−0.351413	+	0.936221i	$0.614299\pi$
$38$	0	0
$39$	5.88291	0.942020
$40$	0	0
$41$	−2.80912	−0.438711	−0.219356	−	0.975645i	$-0.570396\pi$
$41$	−2.80912	−0.438711	−0.219356	+	0.975645i	$0.570396\pi$
$42$	0	0
$43$	−1.34096	−0.204494	−0.102247	−	0.994759i	$-0.532603\pi$
$43$	−1.34096	−0.204494	−0.102247	+	0.994759i	$0.532603\pi$
$44$	0	0
$45$	4.35600	0.649354
$46$	0	0
$47$	8.33504	1.21579	0.607896	−	0.794017i	$-0.292014\pi$
$47$	8.33504	1.21579	0.607896	+	0.794017i	$0.292014\pi$
$48$	0	0
$49$	17.2179	2.45970
$50$	0	0
$51$	−2.94010	−0.411696
$52$	0	0
$53$	5.50512	0.756186	0.378093	−	0.925768i	$-0.376580\pi$
$53$	5.50512	0.756186	0.378093	+	0.925768i	$0.376580\pi$
$54$	0	0
$55$	5.44488	0.734188
$56$	0	0
$57$	−3.68083	−0.487538
$58$	0	0
$59$	9.06367	1.17999	0.589995	−	0.807407i	$-0.299130\pi$
$59$	9.06367	1.17999	0.589995	+	0.807407i	$0.299130\pi$
$60$	0	0
$61$	−10.0314	−1.28439	−0.642197	−	0.766540i	$-0.721977\pi$
$61$	−10.0314	−1.28439	−0.642197	+	0.766540i	$0.721977\pi$
$62$	0	0
$63$	−4.92117	−0.620009
$64$	0	0
$65$	25.6260	3.17851
$66$	0	0
$67$	4.09535	0.500327	0.250164	−	0.968204i	$-0.419516\pi$
$67$	4.09535	0.500327	0.250164	+	0.968204i	$0.419516\pi$
$68$	0	0
$69$	−0.211167	−0.0254215
$70$	0	0
$71$	3.07818	0.365312	0.182656	−	0.983177i	$-0.441530\pi$
$71$	3.07818	0.365312	0.182656	+	0.983177i	$0.441530\pi$
$72$	0	0
$73$	11.8342	1.38509	0.692544	−	0.721375i	$-0.256490\pi$
$73$	11.8342	1.38509	0.692544	+	0.721375i	$0.256490\pi$
$74$	0	0
$75$	13.9747	1.61366
$76$	0	0
$77$	−6.15133	−0.701009
$78$	0	0
$79$	−16.2304	−1.82606	−0.913030	−	0.407893i	$-0.866263\pi$
$79$	−16.2304	−1.82606	−0.913030	+	0.407893i	$0.866263\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.87346	0.973989	0.486994	−	0.873405i	$-0.338093\pi$
$83$	8.87346	0.973989	0.486994	+	0.873405i	$0.338093\pi$
$84$	0	0
$85$	−12.8071	−1.38912
$86$	0	0
$87$	5.01806	0.537992
$88$	0	0
$89$	−10.6447	−1.12834	−0.564169	−	0.825660i	$-0.690803\pi$
$89$	−10.6447	−1.12834	−0.564169	+	0.825660i	$0.690803\pi$
$90$	0	0
$91$	−28.9508	−3.03487
$92$	0	0
$93$	2.92456	0.303263
$94$	0	0
$95$	−16.0337	−1.64502
$96$	0	0
$97$	2.62685	0.266717	0.133358	−	0.991068i	$-0.457424\pi$
$97$	2.62685	0.266717	0.133358	+	0.991068i	$0.457424\pi$
$98$	0	0
$99$	1.24997	0.125627
$100$	0	0
$101$	−1.32221	−0.131565	−0.0657826	−	0.997834i	$-0.520954\pi$
$101$	−1.32221	−0.131565	−0.0657826	+	0.997834i	$0.520954\pi$
$102$	0	0
$103$	−10.4394	−1.02863	−0.514313	−	0.857603i	$-0.671953\pi$
$103$	−10.4394	−1.02863	−0.514313	+	0.857603i	$0.671953\pi$
$104$	0	0
$105$	−21.4366	−2.09200
$106$	0	0
$107$	6.84402	0.661637	0.330818	−	0.943694i	$-0.392675\pi$
$107$	6.84402	0.661637	0.330818	+	0.943694i	$0.392675\pi$
$108$	0	0
$109$	5.70996	0.546915	0.273458	−	0.961884i	$-0.411833\pi$
$109$	5.70996	0.546915	0.273458	+	0.961884i	$0.411833\pi$
$110$	0	0
$111$	−4.27512	−0.405777
$112$	0	0
$113$	16.1165	1.51612	0.758058	−	0.652187i	$-0.226148\pi$
$113$	16.1165	1.51612	0.758058	+	0.652187i	$0.226148\pi$
$114$	0	0
$115$	−0.919843	−0.0857758
$116$	0	0
$117$	5.88291	0.543876
$118$	0	0
$119$	14.4687	1.32634
$120$	0	0
$121$	−9.43757	−0.857961
$122$	0	0
$123$	−2.80912	−0.253290
$124$	0	0
$125$	39.0940	3.49667
$126$	0	0
$127$	16.4604	1.46063	0.730313	−	0.683112i	$-0.239374\pi$
$127$	16.4604	1.46063	0.730313	+	0.683112i	$0.239374\pi$
$128$	0	0
$129$	−1.34096	−0.118065
$130$	0	0
$131$	10.2248	0.893349	0.446674	−	0.894697i	$-0.352608\pi$
$131$	10.2248	0.893349	0.446674	+	0.894697i	$0.352608\pi$
$132$	0	0
$133$	18.1140	1.57068
$134$	0	0
$135$	4.35600	0.374905
$136$	0	0
$137$	0.168777	0.0144196	0.00720978	−	0.999974i	$-0.497705\pi$
$137$	0.168777	0.0144196	0.00720978	+	0.999974i	$0.497705\pi$
$138$	0	0
$139$	−19.0858	−1.61883	−0.809416	−	0.587235i	$-0.800216\pi$
$139$	−19.0858	−1.61883	−0.809416	+	0.587235i	$0.800216\pi$
$140$	0	0
$141$	8.33504	0.701937
$142$	0	0
$143$	7.35349	0.614929
$144$	0	0
$145$	21.8587	1.81526
$146$	0	0
$147$	17.2179	1.42011
$148$	0	0
$149$	−4.16772	−0.341433	−0.170716	−	0.985320i	$-0.554608\pi$
$149$	−4.16772	−0.341433	−0.170716	+	0.985320i	$0.554608\pi$
$150$	0	0
$151$	−11.1077	−0.903932	−0.451966	−	0.892035i	$-0.649277\pi$
$151$	−11.1077	−0.903932	−0.451966	+	0.892035i	$0.649277\pi$
$152$	0	0
$153$	−2.94010	−0.237693
$154$	0	0
$155$	12.7394	1.02325
$156$	0	0
$157$	−12.7249	−1.01556	−0.507778	−	0.861488i	$-0.669533\pi$
$157$	−12.7249	−1.01556	−0.507778	+	0.861488i	$0.669533\pi$
$158$	0	0
$159$	5.50512	0.436584
$160$	0	0
$161$	1.03919	0.0818994
$162$	0	0
$163$	7.33904	0.574838	0.287419	−	0.957805i	$-0.407203\pi$
$163$	7.33904	0.574838	0.287419	+	0.957805i	$0.407203\pi$
$164$	0	0
$165$	5.44488	0.423884
$166$	0	0
$167$	−3.15770	−0.244350	−0.122175	−	0.992509i	$-0.538987\pi$
$167$	−3.15770	−0.244350	−0.122175	+	0.992509i	$0.538987\pi$
$168$	0	0
$169$	21.6087	1.66221
$170$	0	0
$171$	−3.68083	−0.281480
$172$	0	0
$173$	19.7487	1.50147	0.750734	−	0.660604i	$-0.229700\pi$
$173$	19.7487	1.50147	0.750734	+	0.660604i	$0.229700\pi$
$174$	0	0
$175$	−68.7720	−5.19868
$176$	0	0
$177$	9.06367	0.681267
$178$	0	0
$179$	19.1557	1.43176	0.715880	−	0.698223i	$-0.246026\pi$
$179$	19.1557	1.43176	0.715880	+	0.698223i	$0.246026\pi$
$180$	0	0
$181$	7.06594	0.525207	0.262603	−	0.964904i	$-0.415419\pi$
$181$	7.06594	0.525207	0.262603	+	0.964904i	$0.415419\pi$
$182$	0	0
$183$	−10.0314	−0.741545
$184$	0	0
$185$	−18.6224	−1.36915
$186$	0	0
$187$	−3.67504	−0.268746
$188$	0	0
$189$	−4.92117	−0.357962
$190$	0	0
$191$	2.28936	0.165652	0.0828260	−	0.996564i	$-0.473605\pi$
$191$	2.28936	0.165652	0.0828260	+	0.996564i	$0.473605\pi$
$192$	0	0
$193$	−5.77635	−0.415791	−0.207895	−	0.978151i	$-0.566661\pi$
$193$	−5.77635	−0.415791	−0.207895	+	0.978151i	$0.566661\pi$
$194$	0	0
$195$	25.6260	1.83511
$196$	0	0
$197$	−6.86517	−0.489123	−0.244561	−	0.969634i	$-0.578644\pi$
$197$	−6.86517	−0.489123	−0.244561	+	0.969634i	$0.578644\pi$
$198$	0	0
$199$	11.0860	0.785864	0.392932	−	0.919567i	$-0.371461\pi$
$199$	11.0860	0.785864	0.392932	+	0.919567i	$0.371461\pi$
$200$	0	0
$201$	4.09535	0.288864
$202$	0	0
$203$	−24.6947	−1.73323
$204$	0	0
$205$	−12.2365	−0.854637
$206$	0	0
$207$	−0.211167	−0.0146771
$208$	0	0
$209$	−4.60094	−0.318254
$210$	0	0
$211$	19.1539	1.31861	0.659303	−	0.751877i	$-0.270851\pi$
$211$	19.1539	1.31861	0.659303	+	0.751877i	$0.270851\pi$
$212$	0	0
$213$	3.07818	0.210913
$214$	0	0
$215$	−5.84122	−0.398368
$216$	0	0
$217$	−14.3922	−0.977009
$218$	0	0
$219$	11.8342	0.799681
$220$	0	0
$221$	−17.2963	−1.16348
$222$	0	0
$223$	−15.3644	−1.02888	−0.514439	−	0.857527i	$-0.672000\pi$
$223$	−15.3644	−1.02888	−0.514439	+	0.857527i	$0.672000\pi$
$224$	0	0
$225$	13.9747	0.931650
$226$	0	0
$227$	−13.7669	−0.913742	−0.456871	−	0.889533i	$-0.651030\pi$
$227$	−13.7669	−0.913742	−0.456871	+	0.889533i	$0.651030\pi$
$228$	0	0
$229$	26.1119	1.72552	0.862761	−	0.505613i	$-0.168734\pi$
$229$	26.1119	1.72552	0.862761	+	0.505613i	$0.168734\pi$
$230$	0	0
$231$	−6.15133	−0.404727
$232$	0	0
$233$	8.17364	0.535473	0.267737	−	0.963492i	$-0.413724\pi$
$233$	8.17364	0.535473	0.267737	+	0.963492i	$0.413724\pi$
$234$	0	0
$235$	36.3075	2.36844
$236$	0	0
$237$	−16.2304	−1.05428
$238$	0	0
$239$	−12.2266	−0.790874	−0.395437	−	0.918493i	$-0.629407\pi$
$239$	−12.2266	−0.790874	−0.395437	+	0.918493i	$0.629407\pi$
$240$	0	0
$241$	−8.83509	−0.569118	−0.284559	−	0.958659i	$-0.591847\pi$
$241$	−8.83509	−0.569118	−0.284559	+	0.958659i	$0.591847\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	75.0010	4.79164
$246$	0	0
$247$	−21.6540	−1.37781
$248$	0	0
$249$	8.87346	0.562333
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−0.263953	−0.0165946
$254$	0	0
$255$	−12.8071	−0.802009
$256$	0	0
$257$	−25.8055	−1.60970	−0.804850	−	0.593478i	$-0.797754\pi$
$257$	−25.8055	−1.60970	−0.804850	+	0.593478i	$0.797754\pi$
$258$	0	0
$259$	21.0386	1.30727
$260$	0	0
$261$	5.01806	0.310610
$262$	0	0
$263$	3.59173	0.221476	0.110738	−	0.993850i	$-0.464679\pi$
$263$	3.59173	0.221476	0.110738	+	0.993850i	$0.464679\pi$
$264$	0	0
$265$	23.9803	1.47310
$266$	0	0
$267$	−10.6447	−0.651446
$268$	0	0
$269$	12.4895	0.761497	0.380748	−	0.924679i	$-0.375666\pi$
$269$	12.4895	0.761497	0.380748	+	0.924679i	$0.375666\pi$
$270$	0	0
$271$	16.5999	1.00837	0.504187	−	0.863595i	$-0.331792\pi$
$271$	16.5999	1.00837	0.504187	+	0.863595i	$0.331792\pi$
$272$	0	0
$273$	−28.9508	−1.75218
$274$	0	0
$275$	17.4681	1.05336
$276$	0	0
$277$	−8.04888	−0.483610	−0.241805	−	0.970325i	$-0.577739\pi$
$277$	−8.04888	−0.483610	−0.241805	+	0.970325i	$0.577739\pi$
$278$	0	0
$279$	2.92456	0.175089
$280$	0	0
$281$	−19.1102	−1.14002	−0.570010	−	0.821638i	$-0.693060\pi$
$281$	−19.1102	−1.14002	−0.570010	+	0.821638i	$0.693060\pi$
$282$	0	0
$283$	−24.0081	−1.42713	−0.713565	−	0.700589i	$-0.752921\pi$
$283$	−24.0081	−1.42713	−0.713565	+	0.700589i	$0.752921\pi$
$284$	0	0
$285$	−16.0337	−0.949755
$286$	0	0
$287$	13.8242	0.816014
$288$	0	0
$289$	−8.35584	−0.491520
$290$	0	0
$291$	2.62685	0.153989
$292$	0	0
$293$	−24.9548	−1.45787	−0.728937	−	0.684580i	$-0.759985\pi$
$293$	−24.9548	−1.45787	−0.728937	+	0.684580i	$0.759985\pi$
$294$	0	0
$295$	39.4814	2.29869
$296$	0	0
$297$	1.24997	0.0725308
$298$	0	0
$299$	−1.24228	−0.0718427
$300$	0	0
$301$	6.59908	0.380365
$302$	0	0
$303$	−1.32221	−0.0759591
$304$	0	0
$305$	−43.6969	−2.50208
$306$	0	0
$307$	18.5038	1.05607	0.528034	−	0.849223i	$-0.322929\pi$
$307$	18.5038	1.05607	0.528034	+	0.849223i	$0.322929\pi$
$308$	0	0
$309$	−10.4394	−0.593877
$310$	0	0
$311$	−2.07695	−0.117773	−0.0588866	−	0.998265i	$-0.518755\pi$
$311$	−2.07695	−0.117773	−0.0588866	+	0.998265i	$0.518755\pi$
$312$	0	0
$313$	−11.6912	−0.660825	−0.330413	−	0.943837i	$-0.607188\pi$
$313$	−11.6912	−0.660825	−0.330413	+	0.943837i	$0.607188\pi$
$314$	0	0
$315$	−21.4366	−1.20782
$316$	0	0
$317$	21.6443	1.21566	0.607832	−	0.794066i	$-0.292039\pi$
$317$	21.6443	1.21566	0.607832	+	0.794066i	$0.292039\pi$
$318$	0	0
$319$	6.27244	0.351189
$320$	0	0
$321$	6.84402	0.381996
$322$	0	0
$323$	10.8220	0.602152
$324$	0	0
$325$	82.2122	4.56031
$326$	0	0
$327$	5.70996	0.315762
$328$	0	0
$329$	−41.0181	−2.26140
$330$	0	0
$331$	2.60418	0.143138	0.0715692	−	0.997436i	$-0.477199\pi$
$331$	2.60418	0.143138	0.0715692	+	0.997436i	$0.477199\pi$
$332$	0	0
$333$	−4.27512	−0.234275
$334$	0	0
$335$	17.8394	0.974669
$336$	0	0
$337$	−25.6067	−1.39489	−0.697444	−	0.716639i	$-0.745679\pi$
$337$	−25.6067	−1.39489	−0.697444	+	0.716639i	$0.745679\pi$
$338$	0	0
$339$	16.1165	0.875330
$340$	0	0
$341$	3.65562	0.197963
$342$	0	0
$343$	−50.2838	−2.71507
$344$	0	0
$345$	−0.919843	−0.0495227
$346$	0	0
$347$	−7.38191	−0.396282	−0.198141	−	0.980174i	$-0.563490\pi$
$347$	−7.38191	−0.396282	−0.198141	+	0.980174i	$0.563490\pi$
$348$	0	0
$349$	20.8489	1.11601	0.558007	−	0.829836i	$-0.311566\pi$
$349$	20.8489	1.11601	0.558007	+	0.829836i	$0.311566\pi$
$350$	0	0
$351$	5.88291	0.314007
$352$	0	0
$353$	−3.08924	−0.164424	−0.0822119	−	0.996615i	$-0.526198\pi$
$353$	−3.08924	−0.164424	−0.0822119	+	0.996615i	$0.526198\pi$
$354$	0	0
$355$	13.4085	0.711651
$356$	0	0
$357$	14.4687	0.765765
$358$	0	0
$359$	20.8667	1.10130	0.550651	−	0.834736i	$-0.314380\pi$
$359$	20.8667	1.10130	0.550651	+	0.834736i	$0.314380\pi$
$360$	0	0
$361$	−5.45148	−0.286920
$362$	0	0
$363$	−9.43757	−0.495344
$364$	0	0
$365$	51.5498	2.69824
$366$	0	0
$367$	−10.7509	−0.561192	−0.280596	−	0.959826i	$-0.590532\pi$
$367$	−10.7509	−0.561192	−0.280596	+	0.959826i	$0.590532\pi$
$368$	0	0
$369$	−2.80912	−0.146237
$370$	0	0
$371$	−27.0916	−1.40653
$372$	0	0
$373$	−25.8239	−1.33711	−0.668557	−	0.743661i	$-0.733088\pi$
$373$	−25.8239	−1.33711	−0.668557	+	0.743661i	$0.733088\pi$
$374$	0	0
$375$	39.0940	2.01880
$376$	0	0
$377$	29.5208	1.52040
$378$	0	0
$379$	11.6536	0.598603	0.299301	−	0.954159i	$-0.403246\pi$
$379$	11.6536	0.598603	0.299301	+	0.954159i	$0.403246\pi$
$380$	0	0
$381$	16.4604	0.843293
$382$	0	0
$383$	33.0297	1.68774	0.843869	−	0.536549i	$-0.180272\pi$
$383$	33.0297	1.68774	0.843869	+	0.536549i	$0.180272\pi$
$384$	0	0
$385$	−26.7952	−1.36561
$386$	0	0
$387$	−1.34096	−0.0681648
$388$	0	0
$389$	7.72824	0.391837	0.195919	−	0.980620i	$-0.437231\pi$
$389$	7.72824	0.391837	0.195919	+	0.980620i	$0.437231\pi$
$390$	0	0
$391$	0.620850	0.0313978
$392$	0	0
$393$	10.2248	0.515775
$394$	0	0
$395$	−70.6995	−3.55728
$396$	0	0
$397$	−37.5093	−1.88254	−0.941268	−	0.337659i	$-0.890365\pi$
$397$	−37.5093	−1.88254	−0.941268	+	0.337659i	$0.890365\pi$
$398$	0	0
$399$	18.1140	0.906833
$400$	0	0
$401$	−33.3532	−1.66558	−0.832790	−	0.553589i	$-0.813258\pi$
$401$	−33.3532	−1.66558	−0.832790	+	0.553589i	$0.813258\pi$
$402$	0	0
$403$	17.2049	0.857039
$404$	0	0
$405$	4.35600	0.216451
$406$	0	0
$407$	−5.34379	−0.264882
$408$	0	0
$409$	−2.19049	−0.108313	−0.0541565	−	0.998532i	$-0.517247\pi$
$409$	−2.19049	−0.108313	−0.0541565	+	0.998532i	$0.517247\pi$
$410$	0	0
$411$	0.168777	0.00832514
$412$	0	0
$413$	−44.6038	−2.19481
$414$	0	0
$415$	38.6528	1.89739
$416$	0	0
$417$	−19.0858	−0.934633
$418$	0	0
$419$	21.3702	1.04400	0.522002	−	0.852944i	$-0.325185\pi$
$419$	21.3702	1.04400	0.522002	+	0.852944i	$0.325185\pi$
$420$	0	0
$421$	3.11948	0.152034	0.0760170	−	0.997107i	$-0.475780\pi$
$421$	3.11948	0.152034	0.0760170	+	0.997107i	$0.475780\pi$
$422$	0	0
$423$	8.33504	0.405264
$424$	0	0
$425$	−41.0871	−1.99302
$426$	0	0
$427$	49.3663	2.38900
$428$	0	0
$429$	7.35349	0.355030
$430$	0	0
$431$	−30.6506	−1.47639	−0.738195	−	0.674588i	$-0.764321\pi$
$431$	−30.6506	−1.47639	−0.738195	+	0.674588i	$0.764321\pi$
$432$	0	0
$433$	−28.6111	−1.37496	−0.687480	−	0.726203i	$-0.741283\pi$
$433$	−28.6111	−1.37496	−0.687480	+	0.726203i	$0.741283\pi$
$434$	0	0
$435$	21.8587	1.04804
$436$	0	0
$437$	0.777269	0.0371818
$438$	0	0
$439$	36.3700	1.73584	0.867922	−	0.496701i	$-0.165455\pi$
$439$	36.3700	1.73584	0.867922	+	0.496701i	$0.165455\pi$
$440$	0	0
$441$	17.2179	0.819898
$442$	0	0
$443$	−27.7305	−1.31752	−0.658759	−	0.752354i	$-0.728918\pi$
$443$	−27.7305	−1.31752	−0.658759	+	0.752354i	$0.728918\pi$
$444$	0	0
$445$	−46.3684	−2.19807
$446$	0	0
$447$	−4.16772	−0.197126
$448$	0	0
$449$	−5.34630	−0.252307	−0.126154	−	0.992011i	$-0.540263\pi$
$449$	−5.34630	−0.252307	−0.126154	+	0.992011i	$0.540263\pi$
$450$	0	0
$451$	−3.51133	−0.165342
$452$	0	0
$453$	−11.1077	−0.521886
$454$	0	0
$455$	−126.110	−5.91211
$456$	0	0
$457$	−6.48450	−0.303332	−0.151666	−	0.988432i	$-0.548464\pi$
$457$	−6.48450	−0.303332	−0.151666	+	0.988432i	$0.548464\pi$
$458$	0	0
$459$	−2.94010	−0.137232
$460$	0	0
$461$	−2.57371	−0.119870	−0.0599348	−	0.998202i	$-0.519089\pi$
$461$	−2.57371	−0.119870	−0.0599348	+	0.998202i	$0.519089\pi$
$462$	0	0
$463$	−6.62080	−0.307695	−0.153847	−	0.988095i	$-0.549166\pi$
$463$	−6.62080	−0.307695	−0.153847	+	0.988095i	$0.549166\pi$
$464$	0	0
$465$	12.7394	0.590775
$466$	0	0
$467$	16.3421	0.756224	0.378112	−	0.925760i	$-0.376573\pi$
$467$	16.3421	0.756224	0.378112	+	0.925760i	$0.376573\pi$
$468$	0	0
$469$	−20.1539	−0.930621
$470$	0	0
$471$	−12.7249	−0.586331
$472$	0	0
$473$	−1.67616	−0.0770700
$474$	0	0
$475$	−51.4387	−2.36017
$476$	0	0
$477$	5.50512	0.252062
$478$	0	0
$479$	−20.3204	−0.928463	−0.464232	−	0.885714i	$-0.653669\pi$
$479$	−20.3204	−0.928463	−0.464232	+	0.885714i	$0.653669\pi$
$480$	0	0
$481$	−25.1502	−1.14675
$482$	0	0
$483$	1.03919	0.0472846
$484$	0	0
$485$	11.4426	0.519581
$486$	0	0
$487$	−6.86437	−0.311054	−0.155527	−	0.987832i	$-0.549708\pi$
$487$	−6.86437	−0.311054	−0.155527	+	0.987832i	$0.549708\pi$
$488$	0	0
$489$	7.33904	0.331883
$490$	0	0
$491$	7.12006	0.321324	0.160662	−	0.987009i	$-0.448637\pi$
$491$	7.12006	0.321324	0.160662	+	0.987009i	$0.448637\pi$
$492$	0	0
$493$	−14.7536	−0.664468
$494$	0	0
$495$	5.44488	0.244729
$496$	0	0
$497$	−15.1482	−0.679490
$498$	0	0
$499$	−36.1334	−1.61755	−0.808777	−	0.588115i	$-0.799870\pi$
$499$	−36.1334	−1.61755	−0.808777	+	0.588115i	$0.799870\pi$
$500$	0	0
$501$	−3.15770	−0.141076
$502$	0	0
$503$	19.0397	0.848940	0.424470	−	0.905442i	$-0.360460\pi$
$503$	19.0397	0.848940	0.424470	+	0.905442i	$0.360460\pi$
$504$	0	0
$505$	−5.75956	−0.256297
$506$	0	0
$507$	21.6087	0.959675
$508$	0	0
$509$	37.4609	1.66043	0.830214	−	0.557445i	$-0.188218\pi$
$509$	37.4609	1.66043	0.830214	+	0.557445i	$0.188218\pi$
$510$	0	0
$511$	−58.2381	−2.57630
$512$	0	0
$513$	−3.68083	−0.162513
$514$	0	0
$515$	−45.4741	−2.00383
$516$	0	0
$517$	10.4186	0.458209
$518$	0	0
$519$	19.7487	0.866873
$520$	0	0
$521$	−0.104215	−0.00456572	−0.00228286	−	0.999997i	$-0.500727\pi$
$521$	−0.104215	−0.00456572	−0.00228286	+	0.999997i	$0.500727\pi$
$522$	0	0
$523$	−23.9009	−1.04511	−0.522557	−	0.852604i	$-0.675022\pi$
$523$	−23.9009	−1.04511	−0.522557	+	0.852604i	$0.675022\pi$
$524$	0	0
$525$	−68.7720	−3.00146
$526$	0	0
$527$	−8.59849	−0.374556
$528$	0	0
$529$	−22.9554	−0.998061
$530$	0	0
$531$	9.06367	0.393330
$532$	0	0
$533$	−16.5258	−0.715813
$534$	0	0
$535$	29.8126	1.28891
$536$	0	0
$537$	19.1557	0.826627
$538$	0	0
$539$	21.5219	0.927013
$540$	0	0
$541$	3.84475	0.165299	0.0826493	−	0.996579i	$-0.473662\pi$
$541$	3.84475	0.165299	0.0826493	+	0.996579i	$0.473662\pi$
$542$	0	0
$543$	7.06594	0.303228
$544$	0	0
$545$	24.8726	1.06543
$546$	0	0
$547$	42.5252	1.81824	0.909122	−	0.416530i	$-0.136754\pi$
$547$	42.5252	1.81824	0.909122	+	0.416530i	$0.136754\pi$
$548$	0	0
$549$	−10.0314	−0.428131
$550$	0	0
$551$	−18.4706	−0.786876
$552$	0	0
$553$	79.8723	3.39652
$554$	0	0
$555$	−18.6224	−0.790479
$556$	0	0
$557$	12.8538	0.544635	0.272317	−	0.962207i	$-0.412210\pi$
$557$	12.8538	0.544635	0.272317	+	0.962207i	$0.412210\pi$
$558$	0	0
$559$	−7.88875	−0.333658
$560$	0	0
$561$	−3.67504	−0.155160
$562$	0	0
$563$	−41.3079	−1.74092	−0.870459	−	0.492241i	$-0.836178\pi$
$563$	−41.3079	−1.74092	−0.870459	+	0.492241i	$0.836178\pi$
$564$	0	0
$565$	70.2036	2.95349
$566$	0	0
$567$	−4.92117	−0.206670
$568$	0	0
$569$	−29.1483	−1.22196	−0.610980	−	0.791646i	$-0.709224\pi$
$569$	−29.1483	−1.22196	−0.610980	+	0.791646i	$0.709224\pi$
$570$	0	0
$571$	42.1639	1.76450	0.882251	−	0.470779i	$-0.156027\pi$
$571$	42.1639	1.76450	0.882251	+	0.470779i	$0.156027\pi$
$572$	0	0
$573$	2.28936	0.0956393
$574$	0	0
$575$	−2.95100	−0.123065
$576$	0	0
$577$	−0.591605	−0.0246288	−0.0123144	−	0.999924i	$-0.503920\pi$
$577$	−0.591605	−0.0246288	−0.0123144	+	0.999924i	$0.503920\pi$
$578$	0	0
$579$	−5.77635	−0.240057
$580$	0	0
$581$	−43.6678	−1.81164
$582$	0	0
$583$	6.88125	0.284992
$584$	0	0
$585$	25.6260	1.05950
$586$	0	0
$587$	−25.2165	−1.04080	−0.520398	−	0.853924i	$-0.674216\pi$
$587$	−25.2165	−1.04080	−0.520398	+	0.853924i	$0.674216\pi$
$588$	0	0
$589$	−10.7648	−0.443556
$590$	0	0
$591$	−6.86517	−0.282395
$592$	0	0
$593$	40.9802	1.68285	0.841427	−	0.540371i	$-0.181716\pi$
$593$	40.9802	1.68285	0.841427	+	0.540371i	$0.181716\pi$
$594$	0	0
$595$	63.0257	2.58380
$596$	0	0
$597$	11.0860	0.453719
$598$	0	0
$599$	−11.8358	−0.483596	−0.241798	−	0.970327i	$-0.577737\pi$
$599$	−11.8358	−0.483596	−0.241798	+	0.970327i	$0.577737\pi$
$600$	0	0
$601$	35.5743	1.45111	0.725553	−	0.688166i	$-0.241584\pi$
$601$	35.5743	1.45111	0.725553	+	0.688166i	$0.241584\pi$
$602$	0	0
$603$	4.09535	0.166776
$604$	0	0
$605$	−41.1100	−1.67136
$606$	0	0
$607$	20.4568	0.830316	0.415158	−	0.909749i	$-0.363726\pi$
$607$	20.4568	0.830316	0.415158	+	0.909749i	$0.363726\pi$
$608$	0	0
$609$	−24.6947	−1.00068
$610$	0	0
$611$	49.0344	1.98372
$612$	0	0
$613$	0.0660007	0.00266574	0.00133287	−	0.999999i	$-0.499576\pi$
$613$	0.0660007	0.00266574	0.00133287	+	0.999999i	$0.499576\pi$
$614$	0	0
$615$	−12.2365	−0.493425
$616$	0	0
$617$	−39.4308	−1.58742	−0.793712	−	0.608294i	$-0.791854\pi$
$617$	−39.4308	−1.58742	−0.793712	+	0.608294i	$0.791854\pi$
$618$	0	0
$619$	5.33752	0.214533	0.107267	−	0.994230i	$-0.465790\pi$
$619$	5.33752	0.214533	0.107267	+	0.994230i	$0.465790\pi$
$620$	0	0
$621$	−0.211167	−0.00847383
$622$	0	0
$623$	52.3844	2.09874
$624$	0	0
$625$	100.420	4.01679
$626$	0	0
$627$	−4.60094	−0.183744
$628$	0	0
$629$	12.5693	0.501170
$630$	0	0
$631$	−1.18202	−0.0470556	−0.0235278	−	0.999723i	$-0.507490\pi$
$631$	−1.18202	−0.0470556	−0.0235278	+	0.999723i	$0.507490\pi$
$632$	0	0
$633$	19.1539	0.761298
$634$	0	0
$635$	71.7016	2.84539
$636$	0	0
$637$	101.291	4.01330
$638$	0	0
$639$	3.07818	0.121771
$640$	0	0
$641$	34.9053	1.37868	0.689339	−	0.724439i	$-0.257901\pi$
$641$	34.9053	1.37868	0.689339	+	0.724439i	$0.257901\pi$
$642$	0	0
$643$	26.3194	1.03793	0.518967	−	0.854794i	$-0.326317\pi$
$643$	26.3194	1.03793	0.518967	+	0.854794i	$0.326317\pi$
$644$	0	0
$645$	−5.84122	−0.229998
$646$	0	0
$647$	−21.0201	−0.826387	−0.413193	−	0.910643i	$-0.635587\pi$
$647$	−21.0201	−0.826387	−0.413193	+	0.910643i	$0.635587\pi$
$648$	0	0
$649$	11.3293	0.444716
$650$	0	0
$651$	−14.3922	−0.564076
$652$	0	0
$653$	−37.2269	−1.45680	−0.728401	−	0.685151i	$-0.759736\pi$
$653$	−37.2269	−1.45680	−0.728401	+	0.685151i	$0.759736\pi$
$654$	0	0
$655$	44.5394	1.74030
$656$	0	0
$657$	11.8342	0.461696
$658$	0	0
$659$	27.9195	1.08759	0.543794	−	0.839219i	$-0.316987\pi$
$659$	27.9195	1.08759	0.543794	+	0.839219i	$0.316987\pi$
$660$	0	0
$661$	46.3672	1.80348	0.901738	−	0.432284i	$-0.142292\pi$
$661$	46.3672	1.80348	0.901738	+	0.432284i	$0.142292\pi$
$662$	0	0
$663$	−17.2963	−0.671734
$664$	0	0
$665$	78.9045	3.05979
$666$	0	0
$667$	−1.05965	−0.0410297
$668$	0	0
$669$	−15.3644	−0.594023
$670$	0	0
$671$	−12.5390	−0.484064
$672$	0	0
$673$	−19.6448	−0.757251	−0.378625	−	0.925550i	$-0.623603\pi$
$673$	−19.6448	−0.757251	−0.378625	+	0.925550i	$0.623603\pi$
$674$	0	0
$675$	13.9747	0.537888
$676$	0	0
$677$	33.3262	1.28083	0.640415	−	0.768029i	$-0.278762\pi$
$677$	33.3262	1.28083	0.640415	+	0.768029i	$0.278762\pi$
$678$	0	0
$679$	−12.9272	−0.496100
$680$	0	0
$681$	−13.7669	−0.527549
$682$	0	0
$683$	−40.2051	−1.53841	−0.769204	−	0.639004i	$-0.779347\pi$
$683$	−40.2051	−1.53841	−0.769204	+	0.639004i	$0.779347\pi$
$684$	0	0
$685$	0.735191	0.0280902
$686$	0	0
$687$	26.1119	0.996230
$688$	0	0
$689$	32.3861	1.23381
$690$	0	0
$691$	−28.3747	−1.07942	−0.539712	−	0.841850i	$-0.681467\pi$
$691$	−28.3747	−1.07942	−0.539712	+	0.841850i	$0.681467\pi$
$692$	0	0
$693$	−6.15133	−0.233670
$694$	0	0
$695$	−83.1376	−3.15359
$696$	0	0
$697$	8.25909	0.312835
$698$	0	0
$699$	8.17364	0.309155
$700$	0	0
$701$	50.9161	1.92307	0.961536	−	0.274678i	$-0.0885711\pi$
$701$	50.9161	1.92307	0.961536	+	0.274678i	$0.0885711\pi$
$702$	0	0
$703$	15.7360	0.593495
$704$	0	0
$705$	36.3075	1.36742
$706$	0	0
$707$	6.50683	0.244714
$708$	0	0
$709$	17.4452	0.655170	0.327585	−	0.944822i	$-0.393765\pi$
$709$	17.4452	0.655170	0.327585	+	0.944822i	$0.393765\pi$
$710$	0	0
$711$	−16.2304	−0.608686
$712$	0	0
$713$	−0.617570	−0.0231282
$714$	0	0
$715$	32.0318	1.19792
$716$	0	0
$717$	−12.2266	−0.456611
$718$	0	0
$719$	−15.8495	−0.591088	−0.295544	−	0.955329i	$-0.595501\pi$
$719$	−15.8495	−0.591088	−0.295544	+	0.955329i	$0.595501\pi$
$720$	0	0
$721$	51.3740	1.91327
$722$	0	0
$723$	−8.83509	−0.328580
$724$	0	0
$725$	70.1261	2.60442
$726$	0	0
$727$	33.8241	1.25447	0.627234	−	0.778831i	$-0.284187\pi$
$727$	33.8241	1.25447	0.627234	+	0.778831i	$0.284187\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.94255	0.145820
$732$	0	0
$733$	−12.2394	−0.452073	−0.226036	−	0.974119i	$-0.572577\pi$
$733$	−12.2394	−0.452073	−0.226036	+	0.974119i	$0.572577\pi$
$734$	0	0
$735$	75.0010	2.76645
$736$	0	0
$737$	5.11908	0.188564
$738$	0	0
$739$	−25.0078	−0.919925	−0.459962	−	0.887938i	$-0.652137\pi$
$739$	−25.0078	−0.919925	−0.459962	+	0.887938i	$0.652137\pi$
$740$	0	0
$741$	−21.6540	−0.795480
$742$	0	0
$743$	−27.7456	−1.01789	−0.508944	−	0.860800i	$-0.669964\pi$
$743$	−27.7456	−1.01789	−0.508944	+	0.860800i	$0.669964\pi$
$744$	0	0
$745$	−18.1546	−0.665133
$746$	0	0
$747$	8.87346	0.324663
$748$	0	0
$749$	−33.6806	−1.23066
$750$	0	0
$751$	−41.3433	−1.50864	−0.754319	−	0.656508i	$-0.772033\pi$
$751$	−41.3433	−1.50864	−0.754319	+	0.656508i	$0.772033\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−48.3852	−1.76092
$756$	0	0
$757$	27.6997	1.00676	0.503381	−	0.864064i	$-0.332089\pi$
$757$	27.6997	1.00676	0.503381	+	0.864064i	$0.332089\pi$
$758$	0	0
$759$	−0.263953	−0.00958088
$760$	0	0
$761$	−45.5382	−1.65076	−0.825379	−	0.564579i	$-0.809038\pi$
$761$	−45.5382	−1.65076	−0.825379	+	0.564579i	$0.809038\pi$
$762$	0	0
$763$	−28.0997	−1.01728
$764$	0	0
$765$	−12.8071	−0.463040
$766$	0	0
$767$	53.3208	1.92530
$768$	0	0
$769$	−13.1931	−0.475756	−0.237878	−	0.971295i	$-0.576452\pi$
$769$	−13.1931	−0.475756	−0.237878	+	0.971295i	$0.576452\pi$
$770$	0	0
$771$	−25.8055	−0.929361
$772$	0	0
$773$	−28.1143	−1.01120	−0.505600	−	0.862768i	$-0.668729\pi$
$773$	−28.1143	−1.01120	−0.505600	+	0.862768i	$0.668729\pi$
$774$	0	0
$775$	40.8700	1.46809
$776$	0	0
$777$	21.0386	0.754755
$778$	0	0
$779$	10.3399	0.370466
$780$	0	0
$781$	3.84764	0.137679
$782$	0	0
$783$	5.01806	0.179331
$784$	0	0
$785$	−55.4296	−1.97837
$786$	0	0
$787$	29.0548	1.03569	0.517846	−	0.855474i	$-0.326734\pi$
$787$	29.0548	1.03569	0.517846	+	0.855474i	$0.326734\pi$
$788$	0	0
$789$	3.59173	0.127869
$790$	0	0
$791$	−79.3121	−2.82001
$792$	0	0
$793$	−59.0141	−2.09565
$794$	0	0
$795$	23.9803	0.850494
$796$	0	0
$797$	−37.6316	−1.33298	−0.666489	−	0.745515i	$-0.732204\pi$
$797$	−37.6316	−1.33298	−0.666489	+	0.745515i	$0.732204\pi$
$798$	0	0
$799$	−24.5058	−0.866954
$800$	0	0
$801$	−10.6447	−0.376112
$802$	0	0
$803$	14.7924	0.522014
$804$	0	0
$805$	4.52670	0.159545
$806$	0	0
$807$	12.4895	0.439650
$808$	0	0
$809$	40.4662	1.42272	0.711358	−	0.702830i	$-0.248080\pi$
$809$	40.4662	1.42272	0.711358	+	0.702830i	$0.248080\pi$
$810$	0	0
$811$	−53.0724	−1.86362	−0.931811	−	0.362943i	$-0.881772\pi$
$811$	−53.0724	−1.86362	−0.931811	+	0.362943i	$0.881772\pi$
$812$	0	0
$813$	16.5999	0.582185
$814$	0	0
$815$	31.9689	1.11982
$816$	0	0
$817$	4.93584	0.172683
$818$	0	0
$819$	−28.9508	−1.01162
$820$	0	0
$821$	−7.59255	−0.264982	−0.132491	−	0.991184i	$-0.542298\pi$
$821$	−7.59255	−0.264982	−0.132491	+	0.991184i	$0.542298\pi$
$822$	0	0
$823$	1.17368	0.0409120	0.0204560	−	0.999791i	$-0.493488\pi$
$823$	1.17368	0.0409120	0.0204560	+	0.999791i	$0.493488\pi$
$824$	0	0
$825$	17.4681	0.608160
$826$	0	0
$827$	9.61335	0.334289	0.167144	−	0.985932i	$-0.446545\pi$
$827$	9.61335	0.334289	0.167144	+	0.985932i	$0.446545\pi$
$828$	0	0
$829$	31.7642	1.10322	0.551609	−	0.834103i	$-0.314014\pi$
$829$	31.7642	1.10322	0.551609	+	0.834103i	$0.314014\pi$
$830$	0	0
$831$	−8.04888	−0.279213
$832$	0	0
$833$	−50.6222	−1.75395
$834$	0	0
$835$	−13.7549	−0.476009
$836$	0	0
$837$	2.92456	0.101088
$838$	0	0
$839$	41.7763	1.44228	0.721140	−	0.692790i	$-0.243619\pi$
$839$	41.7763	1.44228	0.721140	+	0.692790i	$0.243619\pi$
$840$	0	0
$841$	−3.81908	−0.131692
$842$	0	0
$843$	−19.1102	−0.658190
$844$	0	0
$845$	94.1275	3.23808
$846$	0	0
$847$	46.4438	1.59583
$848$	0	0
$849$	−24.0081	−0.823954
$850$	0	0
$851$	0.902764	0.0309463
$852$	0	0
$853$	17.6633	0.604779	0.302389	−	0.953184i	$-0.402216\pi$
$853$	17.6633	0.604779	0.302389	+	0.953184i	$0.402216\pi$
$854$	0	0
$855$	−16.0337	−0.548341
$856$	0	0
$857$	−34.8030	−1.18885	−0.594425	−	0.804151i	$-0.702620\pi$
$857$	−34.8030	−1.18885	−0.594425	+	0.804151i	$0.702620\pi$
$858$	0	0
$859$	1.26007	0.0429929	0.0214964	−	0.999769i	$-0.493157\pi$
$859$	1.26007	0.0429929	0.0214964	+	0.999769i	$0.493157\pi$
$860$	0	0
$861$	13.8242	0.471126
$862$	0	0
$863$	−15.7124	−0.534856	−0.267428	−	0.963578i	$-0.586174\pi$
$863$	−15.7124	−0.534856	−0.267428	+	0.963578i	$0.586174\pi$
$864$	0	0
$865$	86.0255	2.92496
$866$	0	0
$867$	−8.35584	−0.283779
$868$	0	0
$869$	−20.2875	−0.688207
$870$	0	0
$871$	24.0926	0.816347
$872$	0	0
$873$	2.62685	0.0889056
$874$	0	0
$875$	−192.388	−6.50390
$876$	0	0
$877$	−53.8996	−1.82006	−0.910030	−	0.414543i	$-0.863941\pi$
$877$	−53.8996	−1.82006	−0.910030	+	0.414543i	$0.863941\pi$
$878$	0	0
$879$	−24.9548	−0.841704
$880$	0	0
$881$	−3.73223	−0.125742	−0.0628709	−	0.998022i	$-0.520026\pi$
$881$	−3.73223	−0.125742	−0.0628709	+	0.998022i	$0.520026\pi$
$882$	0	0
$883$	−24.3727	−0.820207	−0.410103	−	0.912039i	$-0.634507\pi$
$883$	−24.3727	−0.820207	−0.410103	+	0.912039i	$0.634507\pi$
$884$	0	0
$885$	39.4814	1.32715
$886$	0	0
$887$	7.46111	0.250520	0.125260	−	0.992124i	$-0.460024\pi$
$887$	7.46111	0.250520	0.125260	+	0.992124i	$0.460024\pi$
$888$	0	0
$889$	−81.0045	−2.71680
$890$	0	0
$891$	1.24997	0.0418757
$892$	0	0
$893$	−30.6799	−1.02666
$894$	0	0
$895$	83.4420	2.78916
$896$	0	0
$897$	−1.24228	−0.0414784
$898$	0	0
$899$	14.6756	0.489459
$900$	0	0
$901$	−16.1856	−0.539220
$902$	0	0
$903$	6.59908	0.219604
$904$	0	0
$905$	30.7792	1.02314
$906$	0	0
$907$	2.42755	0.0806055	0.0403028	−	0.999188i	$-0.487168\pi$
$907$	2.42755	0.0806055	0.0403028	+	0.999188i	$0.487168\pi$
$908$	0	0
$909$	−1.32221	−0.0438550
$910$	0	0
$911$	−16.2233	−0.537502	−0.268751	−	0.963210i	$-0.586611\pi$
$911$	−16.2233	−0.537502	−0.268751	+	0.963210i	$0.586611\pi$
$912$	0	0
$913$	11.0916	0.367078
$914$	0	0
$915$	−43.6969	−1.44458
$916$	0	0
$917$	−50.3182	−1.66165
$918$	0	0
$919$	−18.0037	−0.593887	−0.296943	−	0.954895i	$-0.595967\pi$
$919$	−18.0037	−0.593887	−0.296943	+	0.954895i	$0.595967\pi$
$920$	0	0
$921$	18.5038	0.609722
$922$	0	0
$923$	18.1087	0.596054
$924$	0	0
$925$	−59.7438	−1.96436
$926$	0	0
$927$	−10.4394	−0.342875
$928$	0	0
$929$	38.3525	1.25831	0.629153	−	0.777281i	$-0.283402\pi$
$929$	38.3525	1.25831	0.629153	+	0.777281i	$0.283402\pi$
$930$	0	0
$931$	−63.3761	−2.07707
$932$	0	0
$933$	−2.07695	−0.0679964
$934$	0	0
$935$	−16.0085	−0.523533
$936$	0	0
$937$	58.7250	1.91846	0.959232	−	0.282621i	$-0.0912039\pi$
$937$	58.7250	1.91846	0.959232	+	0.282621i	$0.0912039\pi$
$938$	0	0
$939$	−11.6912	−0.381528
$940$	0	0
$941$	−13.2238	−0.431085	−0.215543	−	0.976494i	$-0.569152\pi$
$941$	−13.2238	−0.431085	−0.215543	+	0.976494i	$0.569152\pi$
$942$	0	0
$943$	0.593193	0.0193170
$944$	0	0
$945$	−21.4366	−0.697333
$946$	0	0
$947$	−30.4452	−0.989335	−0.494668	−	0.869082i	$-0.664710\pi$
$947$	−30.4452	−0.989335	−0.494668	+	0.869082i	$0.664710\pi$
$948$	0	0
$949$	69.6196	2.25995
$950$	0	0
$951$	21.6443	0.701864
$952$	0	0
$953$	−16.3961	−0.531122	−0.265561	−	0.964094i	$-0.585557\pi$
$953$	−16.3961	−0.531122	−0.265561	+	0.964094i	$0.585557\pi$
$954$	0	0
$955$	9.97244	0.322701
$956$	0	0
$957$	6.27244	0.202759
$958$	0	0
$959$	−0.830578	−0.0268208
$960$	0	0
$961$	−22.4469	−0.724095
$962$	0	0
$963$	6.84402	0.220546
$964$	0	0
$965$	−25.1618	−0.809986
$966$	0	0
$967$	−58.9524	−1.89578	−0.947891	−	0.318596i	$-0.896789\pi$
$967$	−58.9524	−1.89578	−0.947891	+	0.318596i	$0.896789\pi$
$968$	0	0
$969$	10.8220	0.347653
$970$	0	0
$971$	−22.7235	−0.729232	−0.364616	−	0.931158i	$-0.618800\pi$
$971$	−22.7235	−0.729232	−0.364616	+	0.931158i	$0.618800\pi$
$972$	0	0
$973$	93.9241	3.01107
$974$	0	0
$975$	82.2122	2.63290
$976$	0	0
$977$	−28.7579	−0.920048	−0.460024	−	0.887907i	$-0.652159\pi$
$977$	−28.7579	−0.920048	−0.460024	+	0.887907i	$0.652159\pi$
$978$	0	0
$979$	−13.3056	−0.425249
$980$	0	0
$981$	5.70996	0.182305
$982$	0	0
$983$	−18.8833	−0.602285	−0.301142	−	0.953579i	$-0.597368\pi$
$983$	−18.8833	−0.602285	−0.301142	+	0.953579i	$0.597368\pi$
$984$	0	0
$985$	−29.9047	−0.952842
$986$	0	0
$987$	−41.0181	−1.30562
$988$	0	0
$989$	0.283166	0.00900415
$990$	0	0
$991$	59.8200	1.90025	0.950123	−	0.311875i	$-0.100957\pi$
$991$	59.8200	1.90025	0.950123	+	0.311875i	$0.100957\pi$
$992$	0	0
$993$	2.60418	0.0826410
$994$	0	0
$995$	48.2905	1.53091
$996$	0	0
$997$	−26.7247	−0.846379	−0.423189	−	0.906041i	$-0.639090\pi$
$997$	−26.7247	−0.846379	−0.423189	+	0.906041i	$0.639090\pi$
$998$	0	0
$999$	−4.27512	−0.135259

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.r.1.19	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.r.1.19	✓	20	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.35600 q^{5} -4.92117 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.35600 q^{5} -4.92117 q^{7} +1.00000 q^{9} +1.24997 q^{11} +5.88291 q^{13} +4.35600 q^{15} -2.94010 q^{17} -3.68083 q^{19} -4.92117 q^{21} -0.211167 q^{23} +13.9747 q^{25} +1.00000 q^{27} +5.01806 q^{29} +2.92456 q^{31} +1.24997 q^{33} -21.4366 q^{35} -4.27512 q^{37} +5.88291 q^{39} -2.80912 q^{41} -1.34096 q^{43} +4.35600 q^{45} +8.33504 q^{47} +17.2179 q^{49} -2.94010 q^{51} +5.50512 q^{53} +5.44488 q^{55} -3.68083 q^{57} +9.06367 q^{59} -10.0314 q^{61} -4.92117 q^{63} +25.6260 q^{65} +4.09535 q^{67} -0.211167 q^{69} +3.07818 q^{71} +11.8342 q^{73} +13.9747 q^{75} -6.15133 q^{77} -16.2304 q^{79} +1.00000 q^{81} +8.87346 q^{83} -12.8071 q^{85} +5.01806 q^{87} -10.6447 q^{89} -28.9508 q^{91} +2.92456 q^{93} -16.0337 q^{95} +2.62685 q^{97} +1.24997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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