LMFDB - Embedded newform 6024.2.a.r.1.6

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			1.6
Root			$-1.23330$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.23330 q^{5} +3.99613 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.23330 q^{5} +3.99613 q^{7} +1.00000 q^{9} +3.68256 q^{11} +0.129260 q^{13} -1.23330 q^{15} +7.05618 q^{17} +1.10359 q^{19} +3.99613 q^{21} -0.579729 q^{23} -3.47898 q^{25} +1.00000 q^{27} +0.608569 q^{29} +4.86643 q^{31} +3.68256 q^{33} -4.92842 q^{35} -6.53898 q^{37} +0.129260 q^{39} -7.63937 q^{41} +12.3032 q^{43} -1.23330 q^{45} +3.24983 q^{47} +8.96905 q^{49} +7.05618 q^{51} +6.66028 q^{53} -4.54169 q^{55} +1.10359 q^{57} +2.03503 q^{59} +8.67400 q^{61} +3.99613 q^{63} -0.159416 q^{65} -10.0829 q^{67} -0.579729 q^{69} -4.73157 q^{71} -14.6873 q^{73} -3.47898 q^{75} +14.7160 q^{77} +11.1271 q^{79} +1.00000 q^{81} +4.61373 q^{83} -8.70238 q^{85} +0.608569 q^{87} -15.3708 q^{89} +0.516540 q^{91} +4.86643 q^{93} -1.36105 q^{95} +5.79346 q^{97} +3.68256 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.23330	−0.551548	−0.275774	−	0.961223i	$-0.588934\pi$
$5$	−1.23330	−0.551548	−0.275774	+	0.961223i	$0.588934\pi$
$6$	0	0
$7$	3.99613	1.51039	0.755197	−	0.655497i	$-0.227541\pi$
$7$	3.99613	1.51039	0.755197	+	0.655497i	$0.227541\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.68256	1.11033	0.555166	−	0.831739i	$-0.312655\pi$
$11$	3.68256	1.11033	0.555166	+	0.831739i	$0.312655\pi$
$12$	0	0
$13$	0.129260	0.0358503	0.0179252	−	0.999839i	$-0.494294\pi$
$13$	0.129260	0.0358503	0.0179252	+	0.999839i	$0.494294\pi$
$14$	0	0
$15$	−1.23330	−0.318436
$16$	0	0
$17$	7.05618	1.71138	0.855688	−	0.517492i	$-0.173134\pi$
$17$	7.05618	1.71138	0.855688	+	0.517492i	$0.173134\pi$
$18$	0	0
$19$	1.10359	0.253180	0.126590	−	0.991955i	$-0.459597\pi$
$19$	1.10359	0.253180	0.126590	+	0.991955i	$0.459597\pi$
$20$	0	0
$21$	3.99613	0.872027
$22$	0	0
$23$	−0.579729	−0.120882	−0.0604409	−	0.998172i	$-0.519251\pi$
$23$	−0.579729	−0.120882	−0.0604409	+	0.998172i	$0.519251\pi$
$24$	0	0
$25$	−3.47898	−0.695795
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.608569	0.113008	0.0565042	−	0.998402i	$-0.482005\pi$
$29$	0.608569	0.113008	0.0565042	+	0.998402i	$0.482005\pi$
$30$	0	0
$31$	4.86643	0.874037	0.437019	−	0.899452i	$-0.356034\pi$
$31$	4.86643	0.874037	0.437019	+	0.899452i	$0.356034\pi$
$32$	0	0
$33$	3.68256	0.641051
$34$	0	0
$35$	−4.92842	−0.833055
$36$	0	0
$37$	−6.53898	−1.07500	−0.537501	−	0.843263i	$-0.680632\pi$
$37$	−6.53898	−1.07500	−0.537501	+	0.843263i	$0.680632\pi$
$38$	0	0
$39$	0.129260	0.0206982
$40$	0	0
$41$	−7.63937	−1.19307	−0.596534	−	0.802587i	$-0.703456\pi$
$41$	−7.63937	−1.19307	−0.596534	+	0.802587i	$0.703456\pi$
$42$	0	0
$43$	12.3032	1.87622	0.938112	−	0.346332i	$-0.112573\pi$
$43$	12.3032	1.87622	0.938112	+	0.346332i	$0.112573\pi$
$44$	0	0
$45$	−1.23330	−0.183849
$46$	0	0
$47$	3.24983	0.474037	0.237019	−	0.971505i	$-0.423830\pi$
$47$	3.24983	0.474037	0.237019	+	0.971505i	$0.423830\pi$
$48$	0	0
$49$	8.96905	1.28129
$50$	0	0
$51$	7.05618	0.988063
$52$	0	0
$53$	6.66028	0.914860	0.457430	−	0.889246i	$-0.348770\pi$
$53$	6.66028	0.914860	0.457430	+	0.889246i	$0.348770\pi$
$54$	0	0
$55$	−4.54169	−0.612401
$56$	0	0
$57$	1.10359	0.146174
$58$	0	0
$59$	2.03503	0.264938	0.132469	−	0.991187i	$-0.457710\pi$
$59$	2.03503	0.264938	0.132469	+	0.991187i	$0.457710\pi$
$60$	0	0
$61$	8.67400	1.11059	0.555296	−	0.831653i	$-0.312605\pi$
$61$	8.67400	1.11059	0.555296	+	0.831653i	$0.312605\pi$
$62$	0	0
$63$	3.99613	0.503465
$64$	0	0
$65$	−0.159416	−0.0197732
$66$	0	0
$67$	−10.0829	−1.23183	−0.615913	−	0.787814i	$-0.711213\pi$
$67$	−10.0829	−1.23183	−0.615913	+	0.787814i	$0.711213\pi$
$68$	0	0
$69$	−0.579729	−0.0697911
$70$	0	0
$71$	−4.73157	−0.561534	−0.280767	−	0.959776i	$-0.590589\pi$
$71$	−4.73157	−0.561534	−0.280767	+	0.959776i	$0.590589\pi$
$72$	0	0
$73$	−14.6873	−1.71902	−0.859508	−	0.511122i	$-0.829230\pi$
$73$	−14.6873	−1.71902	−0.859508	+	0.511122i	$0.829230\pi$
$74$	0	0
$75$	−3.47898	−0.401717
$76$	0	0
$77$	14.7160	1.67704
$78$	0	0
$79$	11.1271	1.25190	0.625950	−	0.779863i	$-0.284711\pi$
$79$	11.1271	1.25190	0.625950	+	0.779863i	$0.284711\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.61373	0.506423	0.253211	−	0.967411i	$-0.418513\pi$
$83$	4.61373	0.506423	0.253211	+	0.967411i	$0.418513\pi$
$84$	0	0
$85$	−8.70238	−0.943906
$86$	0	0
$87$	0.608569	0.0652454
$88$	0	0
$89$	−15.3708	−1.62930	−0.814650	−	0.579953i	$-0.803071\pi$
$89$	−15.3708	−1.62930	−0.814650	+	0.579953i	$0.803071\pi$
$90$	0	0
$91$	0.516540	0.0541482
$92$	0	0
$93$	4.86643	0.504626
$94$	0	0
$95$	−1.36105	−0.139641
$96$	0	0
$97$	5.79346	0.588237	0.294118	−	0.955769i	$-0.404974\pi$
$97$	5.79346	0.588237	0.294118	+	0.955769i	$0.404974\pi$
$98$	0	0
$99$	3.68256	0.370111
$100$	0	0
$101$	−11.3877	−1.13312	−0.566562	−	0.824019i	$-0.691727\pi$
$101$	−11.3877	−1.13312	−0.566562	+	0.824019i	$0.691727\pi$
$102$	0	0
$103$	2.87697	0.283476	0.141738	−	0.989904i	$-0.454731\pi$
$103$	2.87697	0.283476	0.141738	+	0.989904i	$0.454731\pi$
$104$	0	0
$105$	−4.92842	−0.480964
$106$	0	0
$107$	−18.4093	−1.77970	−0.889849	−	0.456255i	$-0.849191\pi$
$107$	−18.4093	−1.77970	−0.889849	+	0.456255i	$0.849191\pi$
$108$	0	0
$109$	−8.56936	−0.820796	−0.410398	−	0.911906i	$-0.634610\pi$
$109$	−8.56936	−0.820796	−0.410398	+	0.911906i	$0.634610\pi$
$110$	0	0
$111$	−6.53898	−0.620653
$112$	0	0
$113$	−10.3686	−0.975393	−0.487697	−	0.873013i	$-0.662163\pi$
$113$	−10.3686	−0.975393	−0.487697	+	0.873013i	$0.662163\pi$
$114$	0	0
$115$	0.714979	0.0666721
$116$	0	0
$117$	0.129260	0.0119501
$118$	0	0
$119$	28.1974	2.58485
$120$	0	0
$121$	2.56122	0.232838
$122$	0	0
$123$	−7.63937	−0.688819
$124$	0	0
$125$	10.4571	0.935312
$126$	0	0
$127$	18.5723	1.64802	0.824012	−	0.566573i	$-0.191731\pi$
$127$	18.5723	1.64802	0.824012	+	0.566573i	$0.191731\pi$
$128$	0	0
$129$	12.3032	1.08324
$130$	0	0
$131$	−9.83655	−0.859424	−0.429712	−	0.902966i	$-0.641385\pi$
$131$	−9.83655	−0.859424	−0.429712	+	0.902966i	$0.641385\pi$
$132$	0	0
$133$	4.41008	0.382402
$134$	0	0
$135$	−1.23330	−0.106145
$136$	0	0
$137$	−4.46093	−0.381123	−0.190561	−	0.981675i	$-0.561031\pi$
$137$	−4.46093	−0.381123	−0.190561	+	0.981675i	$0.561031\pi$
$138$	0	0
$139$	−18.2429	−1.54734	−0.773671	−	0.633587i	$-0.781582\pi$
$139$	−18.2429	−1.54734	−0.773671	+	0.633587i	$0.781582\pi$
$140$	0	0
$141$	3.24983	0.273685
$142$	0	0
$143$	0.476008	0.0398058
$144$	0	0
$145$	−0.750547	−0.0623295
$146$	0	0
$147$	8.96905	0.739754
$148$	0	0
$149$	6.55914	0.537346	0.268673	−	0.963232i	$-0.413415\pi$
$149$	6.55914	0.537346	0.268673	+	0.963232i	$0.413415\pi$
$150$	0	0
$151$	14.2652	1.16089	0.580443	−	0.814301i	$-0.302879\pi$
$151$	14.2652	1.16089	0.580443	+	0.814301i	$0.302879\pi$
$152$	0	0
$153$	7.05618	0.570459
$154$	0	0
$155$	−6.00177	−0.482073
$156$	0	0
$157$	−5.01253	−0.400043	−0.200022	−	0.979791i	$-0.564101\pi$
$157$	−5.01253	−0.400043	−0.200022	+	0.979791i	$0.564101\pi$
$158$	0	0
$159$	6.66028	0.528194
$160$	0	0
$161$	−2.31667	−0.182579
$162$	0	0
$163$	19.2360	1.50668	0.753340	−	0.657631i	$-0.228442\pi$
$163$	19.2360	1.50668	0.753340	+	0.657631i	$0.228442\pi$
$164$	0	0
$165$	−4.54169	−0.353570
$166$	0	0
$167$	10.6277	0.822394	0.411197	−	0.911547i	$-0.365111\pi$
$167$	10.6277	0.822394	0.411197	+	0.911547i	$0.365111\pi$
$168$	0	0
$169$	−12.9833	−0.998715
$170$	0	0
$171$	1.10359	0.0843934
$172$	0	0
$173$	−20.9386	−1.59193	−0.795967	−	0.605340i	$-0.793037\pi$
$173$	−20.9386	−1.59193	−0.795967	+	0.605340i	$0.793037\pi$
$174$	0	0
$175$	−13.9024	−1.05093
$176$	0	0
$177$	2.03503	0.152962
$178$	0	0
$179$	12.3410	0.922412	0.461206	−	0.887293i	$-0.347417\pi$
$179$	12.3410	0.922412	0.461206	+	0.887293i	$0.347417\pi$
$180$	0	0
$181$	12.9716	0.964171	0.482085	−	0.876124i	$-0.339879\pi$
$181$	12.9716	0.964171	0.482085	+	0.876124i	$0.339879\pi$
$182$	0	0
$183$	8.67400	0.641201
$184$	0	0
$185$	8.06452	0.592915
$186$	0	0
$187$	25.9848	1.90020
$188$	0	0
$189$	3.99613	0.290676
$190$	0	0
$191$	−2.73391	−0.197819	−0.0989095	−	0.995096i	$-0.531535\pi$
$191$	−2.73391	−0.197819	−0.0989095	+	0.995096i	$0.531535\pi$
$192$	0	0
$193$	21.9507	1.58005	0.790024	−	0.613076i	$-0.210068\pi$
$193$	21.9507	1.58005	0.790024	+	0.613076i	$0.210068\pi$
$194$	0	0
$195$	−0.159416	−0.0114160
$196$	0	0
$197$	−26.4946	−1.88766	−0.943831	−	0.330427i	$-0.892807\pi$
$197$	−26.4946	−1.88766	−0.943831	+	0.330427i	$0.892807\pi$
$198$	0	0
$199$	0.248014	0.0175812	0.00879061	−	0.999961i	$-0.497202\pi$
$199$	0.248014	0.0175812	0.00879061	+	0.999961i	$0.497202\pi$
$200$	0	0
$201$	−10.0829	−0.711195
$202$	0	0
$203$	2.43192	0.170687
$204$	0	0
$205$	9.42162	0.658035
$206$	0	0
$207$	−0.579729	−0.0402939
$208$	0	0
$209$	4.06402	0.281114
$210$	0	0
$211$	−2.74587	−0.189033	−0.0945166	−	0.995523i	$-0.530131\pi$
$211$	−2.74587	−0.189033	−0.0945166	+	0.995523i	$0.530131\pi$
$212$	0	0
$213$	−4.73157	−0.324202
$214$	0	0
$215$	−15.1735	−1.03483
$216$	0	0
$217$	19.4469	1.32014
$218$	0	0
$219$	−14.6873	−0.992474
$220$	0	0
$221$	0.912084	0.0613534
$222$	0	0
$223$	21.5503	1.44311	0.721556	−	0.692356i	$-0.243427\pi$
$223$	21.5503	1.44311	0.721556	+	0.692356i	$0.243427\pi$
$224$	0	0
$225$	−3.47898	−0.231932
$226$	0	0
$227$	3.60697	0.239403	0.119702	−	0.992810i	$-0.461806\pi$
$227$	3.60697	0.239403	0.119702	+	0.992810i	$0.461806\pi$
$228$	0	0
$229$	6.03746	0.398967	0.199483	−	0.979901i	$-0.436074\pi$
$229$	6.03746	0.398967	0.199483	+	0.979901i	$0.436074\pi$
$230$	0	0
$231$	14.7160	0.968240
$232$	0	0
$233$	−1.00138	−0.0656025	−0.0328012	−	0.999462i	$-0.510443\pi$
$233$	−1.00138	−0.0656025	−0.0328012	+	0.999462i	$0.510443\pi$
$234$	0	0
$235$	−4.00802	−0.261454
$236$	0	0
$237$	11.1271	0.722785
$238$	0	0
$239$	−16.4870	−1.06645	−0.533227	−	0.845972i	$-0.679021\pi$
$239$	−16.4870	−1.06645	−0.533227	+	0.845972i	$0.679021\pi$
$240$	0	0
$241$	21.8627	1.40830	0.704151	−	0.710050i	$-0.251328\pi$
$241$	21.8627	1.40830	0.704151	+	0.710050i	$0.251328\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−11.0615	−0.706694
$246$	0	0
$247$	0.142650	0.00907660
$248$	0	0
$249$	4.61373	0.292383
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−2.13488	−0.134219
$254$	0	0
$255$	−8.70238	−0.544964
$256$	0	0
$257$	13.5102	0.842742	0.421371	−	0.906888i	$-0.361549\pi$
$257$	13.5102	0.842742	0.421371	+	0.906888i	$0.361549\pi$
$258$	0	0
$259$	−26.1306	−1.62368
$260$	0	0
$261$	0.608569	0.0376695
$262$	0	0
$263$	−27.1545	−1.67442	−0.837208	−	0.546885i	$-0.815814\pi$
$263$	−27.1545	−1.67442	−0.837208	+	0.546885i	$0.815814\pi$
$264$	0	0
$265$	−8.21411	−0.504589
$266$	0	0
$267$	−15.3708	−0.940677
$268$	0	0
$269$	0.449890	0.0274303	0.0137151	−	0.999906i	$-0.495634\pi$
$269$	0.449890	0.0274303	0.0137151	+	0.999906i	$0.495634\pi$
$270$	0	0
$271$	20.1140	1.22184	0.610921	−	0.791692i	$-0.290799\pi$
$271$	20.1140	1.22184	0.610921	+	0.791692i	$0.290799\pi$
$272$	0	0
$273$	0.516540	0.0312625
$274$	0	0
$275$	−12.8115	−0.772564
$276$	0	0
$277$	24.2845	1.45911	0.729556	−	0.683921i	$-0.239727\pi$
$277$	24.2845	1.45911	0.729556	+	0.683921i	$0.239727\pi$
$278$	0	0
$279$	4.86643	0.291346
$280$	0	0
$281$	18.4155	1.09858	0.549289	−	0.835632i	$-0.314898\pi$
$281$	18.4155	1.09858	0.549289	+	0.835632i	$0.314898\pi$
$282$	0	0
$283$	−28.7710	−1.71026	−0.855129	−	0.518415i	$-0.826522\pi$
$283$	−28.7710	−1.71026	−0.855129	+	0.518415i	$0.826522\pi$
$284$	0	0
$285$	−1.36105	−0.0806218
$286$	0	0
$287$	−30.5279	−1.80201
$288$	0	0
$289$	32.7897	1.92881
$290$	0	0
$291$	5.79346	0.339619
$292$	0	0
$293$	−19.8248	−1.15818	−0.579090	−	0.815264i	$-0.696592\pi$
$293$	−19.8248	−1.15818	−0.579090	+	0.815264i	$0.696592\pi$
$294$	0	0
$295$	−2.50980	−0.146126
$296$	0	0
$297$	3.68256	0.213684
$298$	0	0
$299$	−0.0749359	−0.00433365
$300$	0	0
$301$	49.1653	2.83384
$302$	0	0
$303$	−11.3877	−0.654209
$304$	0	0
$305$	−10.6976	−0.612545
$306$	0	0
$307$	17.1076	0.976381	0.488191	−	0.872737i	$-0.337657\pi$
$307$	17.1076	0.976381	0.488191	+	0.872737i	$0.337657\pi$
$308$	0	0
$309$	2.87697	0.163665
$310$	0	0
$311$	15.5291	0.880573	0.440286	−	0.897857i	$-0.354877\pi$
$311$	15.5291	0.880573	0.440286	+	0.897857i	$0.354877\pi$
$312$	0	0
$313$	−3.60440	−0.203733	−0.101867	−	0.994798i	$-0.532481\pi$
$313$	−3.60440	−0.203733	−0.101867	+	0.994798i	$0.532481\pi$
$314$	0	0
$315$	−4.92842	−0.277685
$316$	0	0
$317$	−1.07834	−0.0605655	−0.0302828	−	0.999541i	$-0.509641\pi$
$317$	−1.07834	−0.0605655	−0.0302828	+	0.999541i	$0.509641\pi$
$318$	0	0
$319$	2.24109	0.125477
$320$	0	0
$321$	−18.4093	−1.02751
$322$	0	0
$323$	7.78711	0.433287
$324$	0	0
$325$	−0.449693	−0.0249445
$326$	0	0
$327$	−8.56936	−0.473887
$328$	0	0
$329$	12.9868	0.715983
$330$	0	0
$331$	−22.5156	−1.23757	−0.618783	−	0.785562i	$-0.712374\pi$
$331$	−22.5156	−1.23757	−0.618783	+	0.785562i	$0.712374\pi$
$332$	0	0
$333$	−6.53898	−0.358334
$334$	0	0
$335$	12.4353	0.679411
$336$	0	0
$337$	29.0928	1.58479	0.792393	−	0.610011i	$-0.208835\pi$
$337$	29.0928	1.58479	0.792393	+	0.610011i	$0.208835\pi$
$338$	0	0
$339$	−10.3686	−0.563143
$340$	0	0
$341$	17.9209	0.970472
$342$	0	0
$343$	7.86856	0.424862
$344$	0	0
$345$	0.714979	0.0384932
$346$	0	0
$347$	29.8622	1.60308	0.801542	−	0.597938i	$-0.204013\pi$
$347$	29.8622	1.60308	0.801542	+	0.597938i	$0.204013\pi$
$348$	0	0
$349$	28.0809	1.50314	0.751568	−	0.659656i	$-0.229298\pi$
$349$	28.0809	1.50314	0.751568	+	0.659656i	$0.229298\pi$
$350$	0	0
$351$	0.129260	0.00689940
$352$	0	0
$353$	27.1208	1.44349	0.721746	−	0.692158i	$-0.243340\pi$
$353$	27.1208	1.44349	0.721746	+	0.692158i	$0.243340\pi$
$354$	0	0
$355$	5.83544	0.309713
$356$	0	0
$357$	28.1974	1.49237
$358$	0	0
$359$	15.8956	0.838940	0.419470	−	0.907769i	$-0.362216\pi$
$359$	15.8956	0.838940	0.419470	+	0.907769i	$0.362216\pi$
$360$	0	0
$361$	−17.7821	−0.935900
$362$	0	0
$363$	2.56122	0.134429
$364$	0	0
$365$	18.1138	0.948120
$366$	0	0
$367$	−28.7374	−1.50008	−0.750041	−	0.661392i	$-0.769966\pi$
$367$	−28.7374	−1.50008	−0.750041	+	0.661392i	$0.769966\pi$
$368$	0	0
$369$	−7.63937	−0.397690
$370$	0	0
$371$	26.6153	1.38180
$372$	0	0
$373$	9.02079	0.467079	0.233539	−	0.972347i	$-0.424969\pi$
$373$	9.02079	0.467079	0.233539	+	0.972347i	$0.424969\pi$
$374$	0	0
$375$	10.4571	0.540003
$376$	0	0
$377$	0.0786637	0.00405139
$378$	0	0
$379$	−27.8802	−1.43211	−0.716056	−	0.698043i	$-0.754055\pi$
$379$	−27.8802	−1.43211	−0.716056	+	0.698043i	$0.754055\pi$
$380$	0	0
$381$	18.5723	0.951487
$382$	0	0
$383$	19.4453	0.993609	0.496805	−	0.867862i	$-0.334507\pi$
$383$	19.4453	0.993609	0.496805	+	0.867862i	$0.334507\pi$
$384$	0	0
$385$	−18.1492	−0.924968
$386$	0	0
$387$	12.3032	0.625408
$388$	0	0
$389$	−13.9146	−0.705499	−0.352749	−	0.935718i	$-0.614753\pi$
$389$	−13.9146	−0.705499	−0.352749	+	0.935718i	$0.614753\pi$
$390$	0	0
$391$	−4.09067	−0.206874
$392$	0	0
$393$	−9.83655	−0.496188
$394$	0	0
$395$	−13.7231	−0.690483
$396$	0	0
$397$	−5.37573	−0.269800	−0.134900	−	0.990859i	$-0.543071\pi$
$397$	−5.37573	−0.269800	−0.134900	+	0.990859i	$0.543071\pi$
$398$	0	0
$399$	4.41008	0.220780
$400$	0	0
$401$	25.1264	1.25475	0.627377	−	0.778716i	$-0.284128\pi$
$401$	25.1264	1.25475	0.627377	+	0.778716i	$0.284128\pi$
$402$	0	0
$403$	0.629036	0.0313345
$404$	0	0
$405$	−1.23330	−0.0612831
$406$	0	0
$407$	−24.0802	−1.19361
$408$	0	0
$409$	−29.3884	−1.45316	−0.726582	−	0.687079i	$-0.758893\pi$
$409$	−29.3884	−1.45316	−0.726582	+	0.687079i	$0.758893\pi$
$410$	0	0
$411$	−4.46093	−0.220041
$412$	0	0
$413$	8.13223	0.400161
$414$	0	0
$415$	−5.69011	−0.279316
$416$	0	0
$417$	−18.2429	−0.893359
$418$	0	0
$419$	−15.1457	−0.739918	−0.369959	−	0.929048i	$-0.620628\pi$
$419$	−15.1457	−0.739918	−0.369959	+	0.929048i	$0.620628\pi$
$420$	0	0
$421$	1.27650	0.0622130	0.0311065	−	0.999516i	$-0.490097\pi$
$421$	1.27650	0.0622130	0.0311065	+	0.999516i	$0.490097\pi$
$422$	0	0
$423$	3.24983	0.158012
$424$	0	0
$425$	−24.5483	−1.19077
$426$	0	0
$427$	34.6624	1.67743
$428$	0	0
$429$	0.476008	0.0229819
$430$	0	0
$431$	−23.2812	−1.12142	−0.560709	−	0.828013i	$-0.689471\pi$
$431$	−23.2812	−1.12142	−0.560709	+	0.828013i	$0.689471\pi$
$432$	0	0
$433$	10.6263	0.510668	0.255334	−	0.966853i	$-0.417815\pi$
$433$	10.6263	0.510668	0.255334	+	0.966853i	$0.417815\pi$
$434$	0	0
$435$	−0.750547	−0.0359860
$436$	0	0
$437$	−0.639781	−0.0306049
$438$	0	0
$439$	1.09694	0.0523540	0.0261770	−	0.999657i	$-0.491667\pi$
$439$	1.09694	0.0523540	0.0261770	+	0.999657i	$0.491667\pi$
$440$	0	0
$441$	8.96905	0.427097
$442$	0	0
$443$	−27.0766	−1.28645	−0.643225	−	0.765678i	$-0.722404\pi$
$443$	−27.0766	−1.28645	−0.643225	+	0.765678i	$0.722404\pi$
$444$	0	0
$445$	18.9568	0.898637
$446$	0	0
$447$	6.55914	0.310237
$448$	0	0
$449$	−4.34739	−0.205166	−0.102583	−	0.994724i	$-0.532711\pi$
$449$	−4.34739	−0.205166	−0.102583	+	0.994724i	$0.532711\pi$
$450$	0	0
$451$	−28.1324	−1.32470
$452$	0	0
$453$	14.2652	0.670238
$454$	0	0
$455$	−0.637049	−0.0298653
$456$	0	0
$457$	27.5672	1.28954	0.644770	−	0.764377i	$-0.276953\pi$
$457$	27.5672	1.28954	0.644770	+	0.764377i	$0.276953\pi$
$458$	0	0
$459$	7.05618	0.329354
$460$	0	0
$461$	8.01862	0.373464	0.186732	−	0.982411i	$-0.440210\pi$
$461$	8.01862	0.373464	0.186732	+	0.982411i	$0.440210\pi$
$462$	0	0
$463$	−25.3025	−1.17591	−0.587953	−	0.808895i	$-0.700066\pi$
$463$	−25.3025	−1.17591	−0.587953	+	0.808895i	$0.700066\pi$
$464$	0	0
$465$	−6.00177	−0.278325
$466$	0	0
$467$	−20.7524	−0.960308	−0.480154	−	0.877184i	$-0.659419\pi$
$467$	−20.7524	−0.960308	−0.480154	+	0.877184i	$0.659419\pi$
$468$	0	0
$469$	−40.2927	−1.86054
$470$	0	0
$471$	−5.01253	−0.230965
$472$	0	0
$473$	45.3073	2.08323
$474$	0	0
$475$	−3.83935	−0.176162
$476$	0	0
$477$	6.66028	0.304953
$478$	0	0
$479$	30.8383	1.40904	0.704520	−	0.709684i	$-0.251162\pi$
$479$	30.8383	1.40904	0.704520	+	0.709684i	$0.251162\pi$
$480$	0	0
$481$	−0.845231	−0.0385392
$482$	0	0
$483$	−2.31667	−0.105412
$484$	0	0
$485$	−7.14506	−0.324441
$486$	0	0
$487$	37.4930	1.69897	0.849485	−	0.527613i	$-0.176913\pi$
$487$	37.4930	1.69897	0.849485	+	0.527613i	$0.176913\pi$
$488$	0	0
$489$	19.2360	0.869882
$490$	0	0
$491$	9.96286	0.449617	0.224809	−	0.974403i	$-0.427824\pi$
$491$	9.96286	0.449617	0.224809	+	0.974403i	$0.427824\pi$
$492$	0	0
$493$	4.29417	0.193400
$494$	0	0
$495$	−4.54169	−0.204134
$496$	0	0
$497$	−18.9080	−0.848138
$498$	0	0
$499$	−15.0397	−0.673269	−0.336634	−	0.941635i	$-0.609289\pi$
$499$	−15.0397	−0.673269	−0.336634	+	0.941635i	$0.609289\pi$
$500$	0	0
$501$	10.6277	0.474809
$502$	0	0
$503$	38.3823	1.71138	0.855691	−	0.517487i	$-0.173132\pi$
$503$	38.3823	1.71138	0.855691	+	0.517487i	$0.173132\pi$
$504$	0	0
$505$	14.0445	0.624972
$506$	0	0
$507$	−12.9833	−0.576608
$508$	0	0
$509$	−42.2942	−1.87466	−0.937329	−	0.348445i	$-0.886710\pi$
$509$	−42.2942	−1.87466	−0.937329	+	0.348445i	$0.886710\pi$
$510$	0	0
$511$	−58.6923	−2.59639
$512$	0	0
$513$	1.10359	0.0487246
$514$	0	0
$515$	−3.54816	−0.156351
$516$	0	0
$517$	11.9677	0.526339
$518$	0	0
$519$	−20.9386	−0.919103
$520$	0	0
$521$	−27.4448	−1.20238	−0.601190	−	0.799106i	$-0.705306\pi$
$521$	−27.4448	−1.20238	−0.601190	+	0.799106i	$0.705306\pi$
$522$	0	0
$523$	22.2859	0.974496	0.487248	−	0.873264i	$-0.338001\pi$
$523$	22.2859	0.974496	0.487248	+	0.873264i	$0.338001\pi$
$524$	0	0
$525$	−13.9024	−0.606752
$526$	0	0
$527$	34.3384	1.49581
$528$	0	0
$529$	−22.6639	−0.985388
$530$	0	0
$531$	2.03503	0.0883126
$532$	0	0
$533$	−0.987466	−0.0427719
$534$	0	0
$535$	22.7042	0.981589
$536$	0	0
$537$	12.3410	0.532555
$538$	0	0
$539$	33.0290	1.42266
$540$	0	0
$541$	2.14159	0.0920742	0.0460371	−	0.998940i	$-0.485341\pi$
$541$	2.14159	0.0920742	0.0460371	+	0.998940i	$0.485341\pi$
$542$	0	0
$543$	12.9716	0.556664
$544$	0	0
$545$	10.5686	0.452708
$546$	0	0
$547$	−9.28527	−0.397009	−0.198505	−	0.980100i	$-0.563608\pi$
$547$	−9.28527	−0.397009	−0.198505	+	0.980100i	$0.563608\pi$
$548$	0	0
$549$	8.67400	0.370197
$550$	0	0
$551$	0.671609	0.0286115
$552$	0	0
$553$	44.4654	1.89086
$554$	0	0
$555$	8.06452	0.342320
$556$	0	0
$557$	−3.23040	−0.136876	−0.0684381	−	0.997655i	$-0.521802\pi$
$557$	−3.23040	−0.136876	−0.0684381	+	0.997655i	$0.521802\pi$
$558$	0	0
$559$	1.59032	0.0672633
$560$	0	0
$561$	25.9848	1.09708
$562$	0	0
$563$	−34.2785	−1.44467	−0.722333	−	0.691545i	$-0.756930\pi$
$563$	−34.2785	−1.44467	−0.722333	+	0.691545i	$0.756930\pi$
$564$	0	0
$565$	12.7875	0.537976
$566$	0	0
$567$	3.99613	0.167822
$568$	0	0
$569$	12.5602	0.526550	0.263275	−	0.964721i	$-0.415197\pi$
$569$	12.5602	0.526550	0.263275	+	0.964721i	$0.415197\pi$
$570$	0	0
$571$	−11.7407	−0.491334	−0.245667	−	0.969354i	$-0.579007\pi$
$571$	−11.7407	−0.491334	−0.245667	+	0.969354i	$0.579007\pi$
$572$	0	0
$573$	−2.73391	−0.114211
$574$	0	0
$575$	2.01686	0.0841090
$576$	0	0
$577$	−36.1892	−1.50658	−0.753288	−	0.657691i	$-0.771533\pi$
$577$	−36.1892	−1.50658	−0.753288	+	0.657691i	$0.771533\pi$
$578$	0	0
$579$	21.9507	0.912241
$580$	0	0
$581$	18.4371	0.764898
$582$	0	0
$583$	24.5268	1.01580
$584$	0	0
$585$	−0.159416	−0.00659106
$586$	0	0
$587$	−30.2066	−1.24676	−0.623379	−	0.781920i	$-0.714241\pi$
$587$	−30.2066	−1.24676	−0.623379	+	0.781920i	$0.714241\pi$
$588$	0	0
$589$	5.37054	0.221289
$590$	0	0
$591$	−26.4946	−1.08984
$592$	0	0
$593$	5.73381	0.235460	0.117730	−	0.993046i	$-0.462438\pi$
$593$	5.73381	0.235460	0.117730	+	0.993046i	$0.462438\pi$
$594$	0	0
$595$	−34.7758	−1.42567
$596$	0	0
$597$	0.248014	0.0101505
$598$	0	0
$599$	−37.3273	−1.52515	−0.762576	−	0.646898i	$-0.776066\pi$
$599$	−37.3273	−1.52515	−0.762576	+	0.646898i	$0.776066\pi$
$600$	0	0
$601$	20.2305	0.825218	0.412609	−	0.910908i	$-0.364618\pi$
$601$	20.2305	0.825218	0.412609	+	0.910908i	$0.364618\pi$
$602$	0	0
$603$	−10.0829	−0.410609
$604$	0	0
$605$	−3.15874	−0.128421
$606$	0	0
$607$	8.26415	0.335431	0.167716	−	0.985835i	$-0.446361\pi$
$607$	8.26415	0.335431	0.167716	+	0.985835i	$0.446361\pi$
$608$	0	0
$609$	2.43192	0.0985463
$610$	0	0
$611$	0.420074	0.0169944
$612$	0	0
$613$	−28.4930	−1.15082	−0.575410	−	0.817865i	$-0.695157\pi$
$613$	−28.4930	−1.15082	−0.575410	+	0.817865i	$0.695157\pi$
$614$	0	0
$615$	9.42162	0.379916
$616$	0	0
$617$	19.1837	0.772305	0.386153	−	0.922435i	$-0.373804\pi$
$617$	19.1837	0.772305	0.386153	+	0.922435i	$0.373804\pi$
$618$	0	0
$619$	−16.0863	−0.646562	−0.323281	−	0.946303i	$-0.604786\pi$
$619$	−16.0863	−0.646562	−0.323281	+	0.946303i	$0.604786\pi$
$620$	0	0
$621$	−0.579729	−0.0232637
$622$	0	0
$623$	−61.4236	−2.46089
$624$	0	0
$625$	4.49814	0.179926
$626$	0	0
$627$	4.06402	0.162301
$628$	0	0
$629$	−46.1403	−1.83973
$630$	0	0
$631$	33.4360	1.33107	0.665533	−	0.746368i	$-0.268204\pi$
$631$	33.4360	1.33107	0.665533	+	0.746368i	$0.268204\pi$
$632$	0	0
$633$	−2.74587	−0.109138
$634$	0	0
$635$	−22.9052	−0.908964
$636$	0	0
$637$	1.15934	0.0459348
$638$	0	0
$639$	−4.73157	−0.187178
$640$	0	0
$641$	28.0550	1.10811	0.554054	−	0.832481i	$-0.313080\pi$
$641$	28.0550	1.10811	0.554054	+	0.832481i	$0.313080\pi$
$642$	0	0
$643$	−23.1356	−0.912378	−0.456189	−	0.889883i	$-0.650786\pi$
$643$	−23.1356	−0.912378	−0.456189	+	0.889883i	$0.650786\pi$
$644$	0	0
$645$	−15.1735	−0.597458
$646$	0	0
$647$	−27.6112	−1.08551	−0.542755	−	0.839891i	$-0.682619\pi$
$647$	−27.6112	−1.08551	−0.542755	+	0.839891i	$0.682619\pi$
$648$	0	0
$649$	7.49410	0.294169
$650$	0	0
$651$	19.4469	0.762184
$652$	0	0
$653$	−2.29513	−0.0898153	−0.0449076	−	0.998991i	$-0.514299\pi$
$653$	−2.29513	−0.0898153	−0.0449076	+	0.998991i	$0.514299\pi$
$654$	0	0
$655$	12.1314	0.474013
$656$	0	0
$657$	−14.6873	−0.573005
$658$	0	0
$659$	18.2994	0.712842	0.356421	−	0.934325i	$-0.383997\pi$
$659$	18.2994	0.712842	0.356421	+	0.934325i	$0.383997\pi$
$660$	0	0
$661$	−6.71033	−0.261001	−0.130501	−	0.991448i	$-0.541658\pi$
$661$	−6.71033	−0.261001	−0.130501	+	0.991448i	$0.541658\pi$
$662$	0	0
$663$	0.912084	0.0354224
$664$	0	0
$665$	−5.43894	−0.210913
$666$	0	0
$667$	−0.352805	−0.0136607
$668$	0	0
$669$	21.5503	0.833181
$670$	0	0
$671$	31.9425	1.23313
$672$	0	0
$673$	−40.1096	−1.54611	−0.773056	−	0.634338i	$-0.781273\pi$
$673$	−40.1096	−1.54611	−0.773056	+	0.634338i	$0.781273\pi$
$674$	0	0
$675$	−3.47898	−0.133906
$676$	0	0
$677$	24.8462	0.954919	0.477459	−	0.878654i	$-0.341558\pi$
$677$	24.8462	0.954919	0.477459	+	0.878654i	$0.341558\pi$
$678$	0	0
$679$	23.1514	0.888470
$680$	0	0
$681$	3.60697	0.138220
$682$	0	0
$683$	−6.03929	−0.231087	−0.115543	−	0.993302i	$-0.536861\pi$
$683$	−6.03929	−0.231087	−0.115543	+	0.993302i	$0.536861\pi$
$684$	0	0
$685$	5.50165	0.210207
$686$	0	0
$687$	6.03746	0.230344
$688$	0	0
$689$	0.860909	0.0327980
$690$	0	0
$691$	−44.5726	−1.69562	−0.847812	−	0.530297i	$-0.822080\pi$
$691$	−44.5726	−1.69562	−0.847812	+	0.530297i	$0.822080\pi$
$692$	0	0
$693$	14.7160	0.559013
$694$	0	0
$695$	22.4989	0.853434
$696$	0	0
$697$	−53.9048	−2.04179
$698$	0	0
$699$	−1.00138	−0.0378756
$700$	0	0
$701$	35.7303	1.34951	0.674757	−	0.738040i	$-0.264248\pi$
$701$	35.7303	1.34951	0.674757	+	0.738040i	$0.264248\pi$
$702$	0	0
$703$	−7.21634	−0.272169
$704$	0	0
$705$	−4.00802	−0.150951
$706$	0	0
$707$	−45.5069	−1.71146
$708$	0	0
$709$	−36.9676	−1.38835	−0.694174	−	0.719807i	$-0.744230\pi$
$709$	−36.9676	−1.38835	−0.694174	+	0.719807i	$0.744230\pi$
$710$	0	0
$711$	11.1271	0.417300
$712$	0	0
$713$	−2.82121	−0.105655
$714$	0	0
$715$	−0.587060	−0.0219548
$716$	0	0
$717$	−16.4870	−0.615717
$718$	0	0
$719$	−29.9770	−1.11795	−0.558976	−	0.829184i	$-0.688806\pi$
$719$	−29.9770	−1.11795	−0.558976	+	0.829184i	$0.688806\pi$
$720$	0	0
$721$	11.4967	0.428161
$722$	0	0
$723$	21.8627	0.813084
$724$	0	0
$725$	−2.11720	−0.0786307
$726$	0	0
$727$	−8.46899	−0.314097	−0.157049	−	0.987591i	$-0.550198\pi$
$727$	−8.46899	−0.314097	−0.157049	+	0.987591i	$0.550198\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	86.8138	3.21092
$732$	0	0
$733$	23.7697	0.877952	0.438976	−	0.898499i	$-0.355341\pi$
$733$	23.7697	0.877952	0.438976	+	0.898499i	$0.355341\pi$
$734$	0	0
$735$	−11.0615	−0.408010
$736$	0	0
$737$	−37.1309	−1.36774
$738$	0	0
$739$	−43.5247	−1.60108	−0.800541	−	0.599279i	$-0.795454\pi$
$739$	−43.5247	−1.60108	−0.800541	+	0.599279i	$0.795454\pi$
$740$	0	0
$741$	0.142650	0.00524038
$742$	0	0
$743$	−7.17549	−0.263243	−0.131622	−	0.991300i	$-0.542018\pi$
$743$	−7.17549	−0.263243	−0.131622	+	0.991300i	$0.542018\pi$
$744$	0	0
$745$	−8.08938	−0.296372
$746$	0	0
$747$	4.61373	0.168808
$748$	0	0
$749$	−73.5661	−2.68805
$750$	0	0
$751$	35.5128	1.29588	0.647939	−	0.761692i	$-0.275631\pi$
$751$	35.5128	1.29588	0.647939	+	0.761692i	$0.275631\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−17.5933	−0.640284
$756$	0	0
$757$	−27.6178	−1.00379	−0.501893	−	0.864930i	$-0.667363\pi$
$757$	−27.6178	−1.00379	−0.501893	+	0.864930i	$0.667363\pi$
$758$	0	0
$759$	−2.13488	−0.0774914
$760$	0	0
$761$	−5.76020	−0.208807	−0.104403	−	0.994535i	$-0.533293\pi$
$761$	−5.76020	−0.208807	−0.104403	+	0.994535i	$0.533293\pi$
$762$	0	0
$763$	−34.2443	−1.23973
$764$	0	0
$765$	−8.70238	−0.314635
$766$	0	0
$767$	0.263048	0.00949811
$768$	0	0
$769$	5.87954	0.212021	0.106011	−	0.994365i	$-0.466192\pi$
$769$	5.87954	0.212021	0.106011	+	0.994365i	$0.466192\pi$
$770$	0	0
$771$	13.5102	0.486557
$772$	0	0
$773$	9.87239	0.355085	0.177543	−	0.984113i	$-0.443185\pi$
$773$	9.87239	0.355085	0.177543	+	0.984113i	$0.443185\pi$
$774$	0	0
$775$	−16.9302	−0.608151
$776$	0	0
$777$	−26.1306	−0.937431
$778$	0	0
$779$	−8.43071	−0.302062
$780$	0	0
$781$	−17.4243	−0.623489
$782$	0	0
$783$	0.608569	0.0217485
$784$	0	0
$785$	6.18194	0.220643
$786$	0	0
$787$	−24.2595	−0.864757	−0.432379	−	0.901692i	$-0.642326\pi$
$787$	−24.2595	−0.864757	−0.432379	+	0.901692i	$0.642326\pi$
$788$	0	0
$789$	−27.1545	−0.966724
$790$	0	0
$791$	−41.4341	−1.47323
$792$	0	0
$793$	1.12120	0.0398151
$794$	0	0
$795$	−8.21411	−0.291324
$796$	0	0
$797$	18.4337	0.652954	0.326477	−	0.945205i	$-0.394138\pi$
$797$	18.4337	0.652954	0.326477	+	0.945205i	$0.394138\pi$
$798$	0	0
$799$	22.9314	0.811256
$800$	0	0
$801$	−15.3708	−0.543100
$802$	0	0
$803$	−54.0867	−1.90868
$804$	0	0
$805$	2.85715	0.100701
$806$	0	0
$807$	0.449890	0.0158369
$808$	0	0
$809$	−45.1828	−1.58854	−0.794272	−	0.607562i	$-0.792148\pi$
$809$	−45.1828	−1.58854	−0.794272	+	0.607562i	$0.792148\pi$
$810$	0	0
$811$	15.0087	0.527027	0.263513	−	0.964656i	$-0.415119\pi$
$811$	15.0087	0.527027	0.263513	+	0.964656i	$0.415119\pi$
$812$	0	0
$813$	20.1140	0.705430
$814$	0	0
$815$	−23.7237	−0.831006
$816$	0	0
$817$	13.5777	0.475023
$818$	0	0
$819$	0.516540	0.0180494
$820$	0	0
$821$	−0.264672	−0.00923711	−0.00461855	−	0.999989i	$-0.501470\pi$
$821$	−0.264672	−0.00923711	−0.00461855	+	0.999989i	$0.501470\pi$
$822$	0	0
$823$	−0.988268	−0.0344489	−0.0172244	−	0.999852i	$-0.505483\pi$
$823$	−0.988268	−0.0344489	−0.0172244	+	0.999852i	$0.505483\pi$
$824$	0	0
$825$	−12.8115	−0.446040
$826$	0	0
$827$	26.9570	0.937386	0.468693	−	0.883361i	$-0.344725\pi$
$827$	26.9570	0.937386	0.468693	+	0.883361i	$0.344725\pi$
$828$	0	0
$829$	−22.1995	−0.771022	−0.385511	−	0.922703i	$-0.625975\pi$
$829$	−22.1995	−0.771022	−0.385511	+	0.922703i	$0.625975\pi$
$830$	0	0
$831$	24.2845	0.842419
$832$	0	0
$833$	63.2872	2.19277
$834$	0	0
$835$	−13.1071	−0.453589
$836$	0	0
$837$	4.86643	0.168209
$838$	0	0
$839$	52.0343	1.79642	0.898211	−	0.439564i	$-0.144867\pi$
$839$	52.0343	1.79642	0.898211	+	0.439564i	$0.144867\pi$
$840$	0	0
$841$	−28.6296	−0.987229
$842$	0	0
$843$	18.4155	0.634265
$844$	0	0
$845$	16.0123	0.550839
$846$	0	0
$847$	10.2350	0.351677
$848$	0	0
$849$	−28.7710	−0.987418
$850$	0	0
$851$	3.79084	0.129948
$852$	0	0
$853$	25.7928	0.883129	0.441565	−	0.897230i	$-0.354424\pi$
$853$	25.7928	0.883129	0.441565	+	0.897230i	$0.354424\pi$
$854$	0	0
$855$	−1.36105	−0.0465470
$856$	0	0
$857$	−29.8116	−1.01835	−0.509173	−	0.860664i	$-0.670049\pi$
$857$	−29.8116	−1.01835	−0.509173	+	0.860664i	$0.670049\pi$
$858$	0	0
$859$	12.1693	0.415211	0.207605	−	0.978213i	$-0.433433\pi$
$859$	12.1693	0.415211	0.207605	+	0.978213i	$0.433433\pi$
$860$	0	0
$861$	−30.5279	−1.04039
$862$	0	0
$863$	14.2098	0.483706	0.241853	−	0.970313i	$-0.422245\pi$
$863$	14.2098	0.483706	0.241853	+	0.970313i	$0.422245\pi$
$864$	0	0
$865$	25.8236	0.878028
$866$	0	0
$867$	32.7897	1.11360
$868$	0	0
$869$	40.9763	1.39002
$870$	0	0
$871$	−1.30332	−0.0441614
$872$	0	0
$873$	5.79346	0.196079
$874$	0	0
$875$	41.7879	1.41269
$876$	0	0
$877$	54.1400	1.82818	0.914088	−	0.405515i	$-0.132908\pi$
$877$	54.1400	1.82818	0.914088	+	0.405515i	$0.132908\pi$
$878$	0	0
$879$	−19.8248	−0.668675
$880$	0	0
$881$	−23.9111	−0.805585	−0.402793	−	0.915291i	$-0.631960\pi$
$881$	−23.9111	−0.805585	−0.402793	+	0.915291i	$0.631960\pi$
$882$	0	0
$883$	−55.5158	−1.86826	−0.934128	−	0.356938i	$-0.883821\pi$
$883$	−55.5158	−1.86826	−0.934128	+	0.356938i	$0.883821\pi$
$884$	0	0
$885$	−2.50980	−0.0843659
$886$	0	0
$887$	−52.0644	−1.74815	−0.874076	−	0.485789i	$-0.838532\pi$
$887$	−52.0644	−1.74815	−0.874076	+	0.485789i	$0.838532\pi$
$888$	0	0
$889$	74.2172	2.48917
$890$	0	0
$891$	3.68256	0.123370
$892$	0	0
$893$	3.58648	0.120017
$894$	0	0
$895$	−15.2202	−0.508754
$896$	0	0
$897$	−0.0749359	−0.00250204
$898$	0	0
$899$	2.96156	0.0987736
$900$	0	0
$901$	46.9961	1.56567
$902$	0	0
$903$	49.1653	1.63612
$904$	0	0
$905$	−15.9978	−0.531786
$906$	0	0
$907$	9.75489	0.323906	0.161953	−	0.986798i	$-0.448221\pi$
$907$	9.75489	0.323906	0.161953	+	0.986798i	$0.448221\pi$
$908$	0	0
$909$	−11.3877	−0.377708
$910$	0	0
$911$	−17.7664	−0.588627	−0.294313	−	0.955709i	$-0.595091\pi$
$911$	−17.7664	−0.588627	−0.294313	+	0.955709i	$0.595091\pi$
$912$	0	0
$913$	16.9903	0.562298
$914$	0	0
$915$	−10.6976	−0.353653
$916$	0	0
$917$	−39.3081	−1.29807
$918$	0	0
$919$	−28.0339	−0.924752	−0.462376	−	0.886684i	$-0.653003\pi$
$919$	−28.0339	−0.924752	−0.462376	+	0.886684i	$0.653003\pi$
$920$	0	0
$921$	17.1076	0.563714
$922$	0	0
$923$	−0.611604	−0.0201312
$924$	0	0
$925$	22.7490	0.747981
$926$	0	0
$927$	2.87697	0.0944920
$928$	0	0
$929$	26.7165	0.876540	0.438270	−	0.898843i	$-0.355591\pi$
$929$	26.7165	0.876540	0.438270	+	0.898843i	$0.355591\pi$
$930$	0	0
$931$	9.89813	0.324398
$932$	0	0
$933$	15.5291	0.508399
$934$	0	0
$935$	−32.0470	−1.04805
$936$	0	0
$937$	20.4803	0.669061	0.334531	−	0.942385i	$-0.391422\pi$
$937$	20.4803	0.669061	0.334531	+	0.942385i	$0.391422\pi$
$938$	0	0
$939$	−3.60440	−0.117625
$940$	0	0
$941$	−37.4966	−1.22235	−0.611177	−	0.791494i	$-0.709304\pi$
$941$	−37.4966	−1.22235	−0.611177	+	0.791494i	$0.709304\pi$
$942$	0	0
$943$	4.42876	0.144220
$944$	0	0
$945$	−4.92842	−0.160321
$946$	0	0
$947$	23.1643	0.752739	0.376369	−	0.926470i	$-0.377172\pi$
$947$	23.1643	0.752739	0.376369	+	0.926470i	$0.377172\pi$
$948$	0	0
$949$	−1.89848	−0.0616273
$950$	0	0
$951$	−1.07834	−0.0349675
$952$	0	0
$953$	37.6329	1.21905	0.609524	−	0.792768i	$-0.291361\pi$
$953$	37.6329	1.21905	0.609524	+	0.792768i	$0.291361\pi$
$954$	0	0
$955$	3.37173	0.109107
$956$	0	0
$957$	2.24109	0.0724441
$958$	0	0
$959$	−17.8264	−0.575646
$960$	0	0
$961$	−7.31782	−0.236059
$962$	0	0
$963$	−18.4093	−0.593233
$964$	0	0
$965$	−27.0718	−0.871472
$966$	0	0
$967$	13.2230	0.425223	0.212612	−	0.977137i	$-0.431803\pi$
$967$	13.2230	0.425223	0.212612	+	0.977137i	$0.431803\pi$
$968$	0	0
$969$	7.78711	0.250158
$970$	0	0
$971$	37.9680	1.21845	0.609226	−	0.792996i	$-0.291480\pi$
$971$	37.9680	1.21845	0.609226	+	0.792996i	$0.291480\pi$
$972$	0	0
$973$	−72.9010	−2.33710
$974$	0	0
$975$	−0.449693	−0.0144017
$976$	0	0
$977$	−34.6988	−1.11011	−0.555056	−	0.831813i	$-0.687303\pi$
$977$	−34.6988	−1.11011	−0.555056	+	0.831813i	$0.687303\pi$
$978$	0	0
$979$	−56.6038	−1.80906
$980$	0	0
$981$	−8.56936	−0.273599
$982$	0	0
$983$	−38.7819	−1.23695	−0.618475	−	0.785804i	$-0.712249\pi$
$983$	−38.7819	−1.23695	−0.618475	+	0.785804i	$0.712249\pi$
$984$	0	0
$985$	32.6758	1.04114
$986$	0	0
$987$	12.9868	0.413373
$988$	0	0
$989$	−7.13253	−0.226801
$990$	0	0
$991$	−43.2538	−1.37400	−0.687001	−	0.726656i	$-0.741073\pi$
$991$	−43.2538	−1.37400	−0.687001	+	0.726656i	$0.741073\pi$
$992$	0	0
$993$	−22.5156	−0.714510
$994$	0	0
$995$	−0.305875	−0.00969688
$996$	0	0
$997$	58.6283	1.85678	0.928388	−	0.371612i	$-0.121195\pi$
$997$	58.6283	1.85678	0.928388	+	0.371612i	$0.121195\pi$
$998$	0	0
$999$	−6.53898	−0.206884

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.r.1.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.r.1.6	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.23330 q^{5} +3.99613 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.23330 q^{5} +3.99613 q^{7} +1.00000 q^{9} +3.68256 q^{11} +0.129260 q^{13} -1.23330 q^{15} +7.05618 q^{17} +1.10359 q^{19} +3.99613 q^{21} -0.579729 q^{23} -3.47898 q^{25} +1.00000 q^{27} +0.608569 q^{29} +4.86643 q^{31} +3.68256 q^{33} -4.92842 q^{35} -6.53898 q^{37} +0.129260 q^{39} -7.63937 q^{41} +12.3032 q^{43} -1.23330 q^{45} +3.24983 q^{47} +8.96905 q^{49} +7.05618 q^{51} +6.66028 q^{53} -4.54169 q^{55} +1.10359 q^{57} +2.03503 q^{59} +8.67400 q^{61} +3.99613 q^{63} -0.159416 q^{65} -10.0829 q^{67} -0.579729 q^{69} -4.73157 q^{71} -14.6873 q^{73} -3.47898 q^{75} +14.7160 q^{77} +11.1271 q^{79} +1.00000 q^{81} +4.61373 q^{83} -8.70238 q^{85} +0.608569 q^{87} -15.3708 q^{89} +0.516540 q^{91} +4.86643 q^{93} -1.36105 q^{95} +5.79346 q^{97} +3.68256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 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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