LMFDB - Embedded newform 6039.2.a.p.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53628 q^{2} +0.360147 q^{4} +2.16725 q^{5} +2.96367 q^{7} +2.51927 q^{8} +O(q^{10})$	\(q-1.53628 q^{2} +0.360147 q^{4} +2.16725 q^{5} +2.96367 q^{7} +2.51927 q^{8} -3.32950 q^{10} -1.00000 q^{11} +1.11824 q^{13} -4.55302 q^{14} -4.59059 q^{16} +6.92345 q^{17} +6.93391 q^{19} +0.780531 q^{20} +1.53628 q^{22} -0.602257 q^{23} -0.303015 q^{25} -1.71793 q^{26} +1.06736 q^{28} +4.99192 q^{29} -1.25237 q^{31} +2.01388 q^{32} -10.6363 q^{34} +6.42302 q^{35} -3.94654 q^{37} -10.6524 q^{38} +5.45989 q^{40} -1.36557 q^{41} +11.4226 q^{43} -0.360147 q^{44} +0.925234 q^{46} -3.48067 q^{47} +1.78333 q^{49} +0.465515 q^{50} +0.402733 q^{52} +9.97119 q^{53} -2.16725 q^{55} +7.46627 q^{56} -7.66897 q^{58} +4.08773 q^{59} +1.00000 q^{61} +1.92398 q^{62} +6.08730 q^{64} +2.42352 q^{65} -5.56579 q^{67} +2.49346 q^{68} -9.86753 q^{70} +6.01977 q^{71} +4.24844 q^{73} +6.06298 q^{74} +2.49723 q^{76} -2.96367 q^{77} -15.1185 q^{79} -9.94897 q^{80} +2.09790 q^{82} +10.9464 q^{83} +15.0049 q^{85} -17.5482 q^{86} -2.51927 q^{88} -3.58975 q^{89} +3.31410 q^{91} -0.216901 q^{92} +5.34728 q^{94} +15.0275 q^{95} -15.8550 q^{97} -2.73968 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * q^98

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 $

$2$	−1.53628	−1.08631	−0.543156	−	0.839632i	$-0.682771\pi$
$2$	−1.53628	−1.08631	−0.543156	+	0.839632i	$0.682771\pi$
$3$	0	0
$3$	0	0
$4$	0.360147	0.180074
$5$	2.16725	0.969225	0.484612	−	0.874729i	$-0.338961\pi$
$5$	2.16725	0.969225	0.484612	+	0.874729i	$0.338961\pi$
$6$	0	0
$7$	2.96367	1.12016	0.560081	−	0.828438i	$-0.310770\pi$
$7$	2.96367	1.12016	0.560081	+	0.828438i	$0.310770\pi$
$8$	2.51927	0.890696
$9$	0	0
$10$	−3.32950	−1.05288
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.11824	0.310145	0.155073	−	0.987903i	$-0.450439\pi$
$13$	1.11824	0.310145	0.155073	+	0.987903i	$0.450439\pi$
$14$	−4.55302	−1.21684
$15$	0	0
$16$	−4.59059	−1.14765
$17$	6.92345	1.67918	0.839592	−	0.543218i	$-0.182794\pi$
$17$	6.92345	1.67918	0.839592	+	0.543218i	$0.182794\pi$
$18$	0	0
$19$	6.93391	1.59075	0.795374	−	0.606118i	$-0.207274\pi$
$19$	6.93391	1.59075	0.795374	+	0.606118i	$0.207274\pi$
$20$	0.780531	0.174532
$21$	0	0
$22$	1.53628	0.327535
$23$	−0.602257	−0.125579	−0.0627896	−	0.998027i	$-0.520000\pi$
$23$	−0.602257	−0.125579	−0.0627896	+	0.998027i	$0.520000\pi$
$24$	0	0
$25$	−0.303015	−0.0606030
$26$	−1.71793	−0.336914
$27$	0	0
$28$	1.06736	0.201712
$29$	4.99192	0.926976	0.463488	−	0.886103i	$-0.346598\pi$
$29$	4.99192	0.926976	0.463488	+	0.886103i	$0.346598\pi$
$30$	0	0
$31$	−1.25237	−0.224932	−0.112466	−	0.993656i	$-0.535875\pi$
$31$	−1.25237	−0.224932	−0.112466	+	0.993656i	$0.535875\pi$
$32$	2.01388	0.356007
$33$	0	0
$34$	−10.6363	−1.82412
$35$	6.42302	1.08569
$36$	0	0
$37$	−3.94654	−0.648807	−0.324404	−	0.945919i	$-0.605164\pi$
$37$	−3.94654	−0.648807	−0.324404	+	0.945919i	$0.605164\pi$
$38$	−10.6524	−1.72805
$39$	0	0
$40$	5.45989	0.863285
$41$	−1.36557	−0.213266	−0.106633	−	0.994298i	$-0.534007\pi$
$41$	−1.36557	−0.213266	−0.106633	+	0.994298i	$0.534007\pi$
$42$	0	0
$43$	11.4226	1.74193	0.870964	−	0.491348i	$-0.163495\pi$
$43$	11.4226	1.74193	0.870964	+	0.491348i	$0.163495\pi$
$44$	−0.360147	−0.0542943
$45$	0	0
$46$	0.925234	0.136418
$47$	−3.48067	−0.507708	−0.253854	−	0.967242i	$-0.581698\pi$
$47$	−3.48067	−0.507708	−0.253854	+	0.967242i	$0.581698\pi$
$48$	0	0
$49$	1.78333	0.254761
$50$	0.465515	0.0658338
$51$	0	0
$52$	0.402733	0.0558490
$53$	9.97119	1.36965	0.684824	−	0.728708i	$-0.259879\pi$
$53$	9.97119	1.36965	0.684824	+	0.728708i	$0.259879\pi$
$54$	0	0
$55$	−2.16725	−0.292232
$56$	7.46627	0.997723
$57$	0	0
$58$	−7.66897	−1.00698
$59$	4.08773	0.532177	0.266088	−	0.963949i	$-0.414269\pi$
$59$	4.08773	0.532177	0.266088	+	0.963949i	$0.414269\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	1.92398	0.244346
$63$	0	0
$64$	6.08730	0.760912
$65$	2.42352	0.300600
$66$	0	0
$67$	−5.56579	−0.679970	−0.339985	−	0.940431i	$-0.610422\pi$
$67$	−5.56579	−0.679970	−0.339985	+	0.940431i	$0.610422\pi$
$68$	2.49346	0.302377
$69$	0	0
$70$	−9.86753	−1.17940
$71$	6.01977	0.714415	0.357208	−	0.934025i	$-0.383729\pi$
$71$	6.01977	0.714415	0.357208	+	0.934025i	$0.383729\pi$
$72$	0	0
$73$	4.24844	0.497242	0.248621	−	0.968601i	$-0.420023\pi$
$73$	4.24844	0.497242	0.248621	+	0.968601i	$0.420023\pi$
$74$	6.06298	0.704807
$75$	0	0
$76$	2.49723	0.286452
$77$	−2.96367	−0.337741
$78$	0	0
$79$	−15.1185	−1.70096	−0.850482	−	0.526005i	$-0.823689\pi$
$79$	−15.1185	−1.70096	−0.850482	+	0.526005i	$0.823689\pi$
$80$	−9.94897	−1.11233
$81$	0	0
$82$	2.09790	0.231674
$83$	10.9464	1.20152	0.600761	−	0.799429i	$-0.294864\pi$
$83$	10.9464	1.20152	0.600761	+	0.799429i	$0.294864\pi$
$84$	0	0
$85$	15.0049	1.62751
$86$	−17.5482	−1.89228
$87$	0	0
$88$	−2.51927	−0.268555
$89$	−3.58975	−0.380513	−0.190256	−	0.981734i	$-0.560932\pi$
$89$	−3.58975	−0.380513	−0.190256	+	0.981734i	$0.560932\pi$
$90$	0	0
$91$	3.31410	0.347412
$92$	−0.216901	−0.0226135
$93$	0	0
$94$	5.34728	0.551530
$95$	15.0275	1.54179
$96$	0	0
$97$	−15.8550	−1.60983	−0.804914	−	0.593391i	$-0.797789\pi$
$97$	−15.8550	−1.60983	−0.804914	+	0.593391i	$0.797789\pi$
$98$	−2.73968	−0.276750
$99$	0	0
$100$	−0.109130	−0.0109130
$101$	1.66067	0.165243	0.0826216	−	0.996581i	$-0.473671\pi$
$101$	1.66067	0.165243	0.0826216	+	0.996581i	$0.473671\pi$
$102$	0	0
$103$	−11.6642	−1.14930	−0.574652	−	0.818398i	$-0.694862\pi$
$103$	−11.6642	−1.14930	−0.574652	+	0.818398i	$0.694862\pi$
$104$	2.81716	0.276245
$105$	0	0
$106$	−15.3185	−1.48787
$107$	5.00509	0.483860	0.241930	−	0.970294i	$-0.422220\pi$
$107$	5.00509	0.483860	0.241930	+	0.970294i	$0.422220\pi$
$108$	0	0
$109$	11.0556	1.05894	0.529469	−	0.848329i	$-0.322391\pi$
$109$	11.0556	1.05894	0.529469	+	0.848329i	$0.322391\pi$
$110$	3.32950	0.317455
$111$	0	0
$112$	−13.6050	−1.28555
$113$	−0.0342434	−0.00322135	−0.00161067	−	0.999999i	$-0.500513\pi$
$113$	−0.0342434	−0.00322135	−0.00161067	+	0.999999i	$0.500513\pi$
$114$	0	0
$115$	−1.30524	−0.121715
$116$	1.79783	0.166924
$117$	0	0
$118$	−6.27988	−0.578110
$119$	20.5188	1.88096
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.53628	−0.139088
$123$	0	0
$124$	−0.451037	−0.0405043
$125$	−11.4930	−1.02796
$126$	0	0
$127$	12.8091	1.13662	0.568312	−	0.822813i	$-0.307597\pi$
$127$	12.8091	1.13662	0.568312	+	0.822813i	$0.307597\pi$
$128$	−13.3795	−1.18260
$129$	0	0
$130$	−3.72319	−0.326546
$131$	2.28901	0.199992	0.0999960	−	0.994988i	$-0.468117\pi$
$131$	2.28901	0.199992	0.0999960	+	0.994988i	$0.468117\pi$
$132$	0	0
$133$	20.5498	1.78190
$134$	8.55060	0.738659
$135$	0	0
$136$	17.4420	1.49564
$137$	12.8333	1.09643	0.548213	−	0.836339i	$-0.315308\pi$
$137$	12.8333	1.09643	0.548213	+	0.836339i	$0.315308\pi$
$138$	0	0
$139$	−14.3877	−1.22035	−0.610174	−	0.792268i	$-0.708900\pi$
$139$	−14.3877	−1.22035	−0.610174	+	0.792268i	$0.708900\pi$
$140$	2.31323	0.195504
$141$	0	0
$142$	−9.24803	−0.776078
$143$	−1.11824	−0.0935123
$144$	0	0
$145$	10.8187	0.898448
$146$	−6.52678	−0.540160
$147$	0	0
$148$	−1.42134	−0.116833
$149$	−1.61267	−0.132115	−0.0660577	−	0.997816i	$-0.521042\pi$
$149$	−1.61267	−0.132115	−0.0660577	+	0.997816i	$0.521042\pi$
$150$	0	0
$151$	−1.44516	−0.117605	−0.0588025	−	0.998270i	$-0.518728\pi$
$151$	−1.44516	−0.117605	−0.0588025	+	0.998270i	$0.518728\pi$
$152$	17.4684	1.41687
$153$	0	0
$154$	4.55302	0.366892
$155$	−2.71420	−0.218010
$156$	0	0
$157$	2.08168	0.166136	0.0830681	−	0.996544i	$-0.473528\pi$
$157$	2.08168	0.166136	0.0830681	+	0.996544i	$0.473528\pi$
$158$	23.2262	1.84778
$159$	0	0
$160$	4.36459	0.345051
$161$	−1.78489	−0.140669
$162$	0	0
$163$	−13.8859	−1.08763	−0.543816	−	0.839205i	$-0.683021\pi$
$163$	−13.8859	−1.08763	−0.543816	+	0.839205i	$0.683021\pi$
$164$	−0.491807	−0.0384037
$165$	0	0
$166$	−16.8167	−1.30523
$167$	−4.64625	−0.359538	−0.179769	−	0.983709i	$-0.557535\pi$
$167$	−4.64625	−0.359538	−0.179769	+	0.983709i	$0.557535\pi$
$168$	0	0
$169$	−11.7495	−0.903810
$170$	−23.0516	−1.76798
$171$	0	0
$172$	4.11381	0.313675
$173$	18.2979	1.39116	0.695580	−	0.718448i	$-0.255147\pi$
$173$	18.2979	1.39116	0.695580	+	0.718448i	$0.255147\pi$
$174$	0	0
$175$	−0.898036	−0.0678852
$176$	4.59059	0.346029
$177$	0	0
$178$	5.51485	0.413355
$179$	1.14913	0.0858900	0.0429450	−	0.999077i	$-0.486326\pi$
$179$	1.14913	0.0858900	0.0429450	+	0.999077i	$0.486326\pi$
$180$	0	0
$181$	−14.3074	−1.06346	−0.531731	−	0.846913i	$-0.678458\pi$
$181$	−14.3074	−1.06346	−0.531731	+	0.846913i	$0.678458\pi$
$182$	−5.09138	−0.377398
$183$	0	0
$184$	−1.51725	−0.111853
$185$	−8.55315	−0.628840
$186$	0	0
$187$	−6.92345	−0.506293
$188$	−1.25356	−0.0914250
$189$	0	0
$190$	−23.0865	−1.67487
$191$	−1.63500	−0.118305	−0.0591524	−	0.998249i	$-0.518840\pi$
$191$	−1.63500	−0.118305	−0.0591524	+	0.998249i	$0.518840\pi$
$192$	0	0
$193$	23.1138	1.66377	0.831883	−	0.554951i	$-0.187263\pi$
$193$	23.1138	1.66377	0.831883	+	0.554951i	$0.187263\pi$
$194$	24.3576	1.74878
$195$	0	0
$196$	0.642261	0.0458758
$197$	−11.9309	−0.850039	−0.425020	−	0.905184i	$-0.639733\pi$
$197$	−11.9309	−0.850039	−0.425020	+	0.905184i	$0.639733\pi$
$198$	0	0
$199$	−3.55582	−0.252065	−0.126033	−	0.992026i	$-0.540224\pi$
$199$	−3.55582	−0.252065	−0.126033	+	0.992026i	$0.540224\pi$
$200$	−0.763376	−0.0539789
$201$	0	0
$202$	−2.55125	−0.179506
$203$	14.7944	1.03836
$204$	0	0
$205$	−2.95954	−0.206703
$206$	17.9194	1.24850
$207$	0	0
$208$	−5.13340	−0.355937
$209$	−6.93391	−0.479629
$210$	0	0
$211$	−13.1085	−0.902425	−0.451212	−	0.892417i	$-0.649008\pi$
$211$	−13.1085	−0.902425	−0.451212	+	0.892417i	$0.649008\pi$
$212$	3.59110	0.246638
$213$	0	0
$214$	−7.68920	−0.525623
$215$	24.7556	1.68832
$216$	0	0
$217$	−3.71160	−0.251960
$218$	−16.9845	−1.15034
$219$	0	0
$220$	−0.780531	−0.0526234
$221$	7.74211	0.520791
$222$	0	0
$223$	−23.4695	−1.57163	−0.785816	−	0.618461i	$-0.787757\pi$
$223$	−23.4695	−1.57163	−0.785816	+	0.618461i	$0.787757\pi$
$224$	5.96847	0.398785
$225$	0	0
$226$	0.0526073	0.00349939
$227$	29.3689	1.94928	0.974640	−	0.223777i	$-0.0718386\pi$
$227$	29.3689	1.94928	0.974640	+	0.223777i	$0.0718386\pi$
$228$	0	0
$229$	−28.9810	−1.91512	−0.957559	−	0.288239i	$-0.906930\pi$
$229$	−28.9810	−1.91512	−0.957559	+	0.288239i	$0.906930\pi$
$230$	2.00522	0.132220
$231$	0	0
$232$	12.5760	0.825653
$233$	−17.1527	−1.12371	−0.561855	−	0.827236i	$-0.689912\pi$
$233$	−17.1527	−1.12371	−0.561855	+	0.827236i	$0.689912\pi$
$234$	0	0
$235$	−7.54350	−0.492084
$236$	1.47218	0.0958310
$237$	0	0
$238$	−31.5226	−2.04331
$239$	−24.6961	−1.59746	−0.798729	−	0.601690i	$-0.794494\pi$
$239$	−24.6961	−1.59746	−0.798729	+	0.601690i	$0.794494\pi$
$240$	0	0
$241$	−16.6705	−1.07384	−0.536919	−	0.843634i	$-0.680412\pi$
$241$	−16.6705	−1.07384	−0.536919	+	0.843634i	$0.680412\pi$
$242$	−1.53628	−0.0987556
$243$	0	0
$244$	0.360147	0.0230561
$245$	3.86492	0.246921
$246$	0	0
$247$	7.75381	0.493363
$248$	−3.15505	−0.200346
$249$	0	0
$250$	17.6564	1.11669
$251$	22.5316	1.42218	0.711091	−	0.703100i	$-0.248201\pi$
$251$	22.5316	1.42218	0.711091	+	0.703100i	$0.248201\pi$
$252$	0	0
$253$	0.602257	0.0378636
$254$	−19.6783	−1.23473
$255$	0	0
$256$	8.38008	0.523755
$257$	−14.2743	−0.890409	−0.445205	−	0.895429i	$-0.646869\pi$
$257$	−14.2743	−0.890409	−0.445205	+	0.895429i	$0.646869\pi$
$258$	0	0
$259$	−11.6962	−0.726769
$260$	0.872824	0.0541302
$261$	0	0
$262$	−3.51656	−0.217254
$263$	−11.6036	−0.715510	−0.357755	−	0.933816i	$-0.616458\pi$
$263$	−11.6036	−0.715510	−0.357755	+	0.933816i	$0.616458\pi$
$264$	0	0
$265$	21.6101	1.32750
$266$	−31.5702	−1.93569
$267$	0	0
$268$	−2.00451	−0.122445
$269$	−26.6831	−1.62690	−0.813449	−	0.581636i	$-0.802413\pi$
$269$	−26.6831	−1.62690	−0.813449	+	0.581636i	$0.802413\pi$
$270$	0	0
$271$	−19.5820	−1.18952	−0.594761	−	0.803903i	$-0.702753\pi$
$271$	−19.5820	−1.18952	−0.594761	+	0.803903i	$0.702753\pi$
$272$	−31.7827	−1.92711
$273$	0	0
$274$	−19.7156	−1.19106
$275$	0.303015	0.0182725
$276$	0	0
$277$	5.54773	0.333331	0.166665	−	0.986013i	$-0.446700\pi$
$277$	5.54773	0.333331	0.166665	+	0.986013i	$0.446700\pi$
$278$	22.1035	1.32568
$279$	0	0
$280$	16.1813	0.967018
$281$	−4.91656	−0.293297	−0.146649	−	0.989189i	$-0.546849\pi$
$281$	−4.91656	−0.293297	−0.146649	+	0.989189i	$0.546849\pi$
$282$	0	0
$283$	17.1732	1.02084	0.510420	−	0.859925i	$-0.329490\pi$
$283$	17.1732	1.02084	0.510420	+	0.859925i	$0.329490\pi$
$284$	2.16800	0.128647
$285$	0	0
$286$	1.71793	0.101583
$287$	−4.04710	−0.238893
$288$	0	0
$289$	30.9342	1.81966
$290$	−16.6206	−0.975995
$291$	0	0
$292$	1.53007	0.0895403
$293$	16.5723	0.968164	0.484082	−	0.875023i	$-0.339154\pi$
$293$	16.5723	0.968164	0.484082	+	0.875023i	$0.339154\pi$
$294$	0	0
$295$	8.85914	0.515799
$296$	−9.94239	−0.577890
$297$	0	0
$298$	2.47751	0.143519
$299$	−0.673470	−0.0389478
$300$	0	0
$301$	33.8527	1.95124
$302$	2.22016	0.127756
$303$	0	0
$304$	−31.8307	−1.82562
$305$	2.16725	0.124097
$306$	0	0
$307$	−8.88162	−0.506901	−0.253451	−	0.967348i	$-0.581566\pi$
$307$	−8.88162	−0.506901	−0.253451	+	0.967348i	$0.581566\pi$
$308$	−1.06736	−0.0608183
$309$	0	0
$310$	4.16976	0.236826
$311$	−33.1230	−1.87823	−0.939117	−	0.343597i	$-0.888355\pi$
$311$	−33.1230	−1.87823	−0.939117	+	0.343597i	$0.888355\pi$
$312$	0	0
$313$	−0.858895	−0.0485476	−0.0242738	−	0.999705i	$-0.507727\pi$
$313$	−0.858895	−0.0485476	−0.0242738	+	0.999705i	$0.507727\pi$
$314$	−3.19804	−0.180476
$315$	0	0
$316$	−5.44489	−0.306299
$317$	18.1664	1.02033	0.510165	−	0.860077i	$-0.329584\pi$
$317$	18.1664	1.02033	0.510165	+	0.860077i	$0.329584\pi$
$318$	0	0
$319$	−4.99192	−0.279494
$320$	13.1927	0.737495
$321$	0	0
$322$	2.74209	0.152810
$323$	48.0066	2.67116
$324$	0	0
$325$	−0.338845	−0.0187957
$326$	21.3327	1.18151
$327$	0	0
$328$	−3.44024	−0.189956
$329$	−10.3156	−0.568715
$330$	0	0
$331$	2.83812	0.155997	0.0779987	−	0.996953i	$-0.475147\pi$
$331$	2.83812	0.155997	0.0779987	+	0.996953i	$0.475147\pi$
$332$	3.94231	0.216363
$333$	0	0
$334$	7.13793	0.390570
$335$	−12.0625	−0.659043
$336$	0	0
$337$	−6.25080	−0.340503	−0.170251	−	0.985401i	$-0.554458\pi$
$337$	−6.25080	−0.340503	−0.170251	+	0.985401i	$0.554458\pi$
$338$	18.0505	0.981820
$339$	0	0
$340$	5.40397	0.293071
$341$	1.25237	0.0678195
$342$	0	0
$343$	−15.4605	−0.834788
$344$	28.7765	1.55153
$345$	0	0
$346$	−28.1106	−1.51123
$347$	−0.687843	−0.0369254	−0.0184627	−	0.999830i	$-0.505877\pi$
$347$	−0.687843	−0.0369254	−0.0184627	+	0.999830i	$0.505877\pi$
$348$	0	0
$349$	−5.99137	−0.320711	−0.160355	−	0.987059i	$-0.551264\pi$
$349$	−5.99137	−0.320711	−0.160355	+	0.987059i	$0.551264\pi$
$350$	1.37963	0.0737445
$351$	0	0
$352$	−2.01388	−0.107340
$353$	−21.5599	−1.14752	−0.573758	−	0.819025i	$-0.694515\pi$
$353$	−21.5599	−1.14752	−0.573758	+	0.819025i	$0.694515\pi$
$354$	0	0
$355$	13.0464	0.692429
$356$	−1.29284	−0.0685203
$357$	0	0
$358$	−1.76538	−0.0933034
$359$	−9.54787	−0.503917	−0.251959	−	0.967738i	$-0.581075\pi$
$359$	−9.54787	−0.503917	−0.251959	+	0.967738i	$0.581075\pi$
$360$	0	0
$361$	29.0792	1.53048
$362$	21.9802	1.15525
$363$	0	0
$364$	1.19357	0.0625599
$365$	9.20745	0.481940
$366$	0	0
$367$	18.2671	0.953537	0.476769	−	0.879029i	$-0.341808\pi$
$367$	18.2671	0.953537	0.476769	+	0.879029i	$0.341808\pi$
$368$	2.76471	0.144121
$369$	0	0
$370$	13.1400	0.683116
$371$	29.5513	1.53423
$372$	0	0
$373$	−21.4678	−1.11156	−0.555781	−	0.831329i	$-0.687581\pi$
$373$	−21.4678	−1.11156	−0.555781	+	0.831329i	$0.687581\pi$
$374$	10.6363	0.549992
$375$	0	0
$376$	−8.76875	−0.452214
$377$	5.58218	0.287497
$378$	0	0
$379$	−22.3560	−1.14835	−0.574176	−	0.818732i	$-0.694677\pi$
$379$	−22.3560	−1.14835	−0.574176	+	0.818732i	$0.694677\pi$
$380$	5.41213	0.277636
$381$	0	0
$382$	2.51182	0.128516
$383$	31.2050	1.59450	0.797252	−	0.603647i	$-0.206286\pi$
$383$	31.2050	1.59450	0.797252	+	0.603647i	$0.206286\pi$
$384$	0	0
$385$	−6.42302	−0.327347
$386$	−35.5092	−1.80737
$387$	0	0
$388$	−5.71013	−0.289888
$389$	31.3867	1.59137	0.795684	−	0.605712i	$-0.207112\pi$
$389$	31.3867	1.59137	0.795684	+	0.605712i	$0.207112\pi$
$390$	0	0
$391$	−4.16970	−0.210871
$392$	4.49268	0.226915
$393$	0	0
$394$	18.3291	0.923408
$395$	−32.7656	−1.64862
$396$	0	0
$397$	22.1105	1.10969	0.554847	−	0.831952i	$-0.312777\pi$
$397$	22.1105	1.10969	0.554847	+	0.831952i	$0.312777\pi$
$398$	5.46272	0.273821
$399$	0	0
$400$	1.39102	0.0695509
$401$	33.4242	1.66912	0.834562	−	0.550915i	$-0.185721\pi$
$401$	33.4242	1.66912	0.834562	+	0.550915i	$0.185721\pi$
$402$	0	0
$403$	−1.40045	−0.0697615
$404$	0.598087	0.0297559
$405$	0	0
$406$	−22.7283	−1.12799
$407$	3.94654	0.195623
$408$	0	0
$409$	9.67708	0.478501	0.239250	−	0.970958i	$-0.423098\pi$
$409$	9.67708	0.478501	0.239250	+	0.970958i	$0.423098\pi$
$410$	4.54667	0.224544
$411$	0	0
$412$	−4.20082	−0.206959
$413$	12.1147	0.596123
$414$	0	0
$415$	23.7236	1.16455
$416$	2.25201	0.110414
$417$	0	0
$418$	10.6524	0.521027
$419$	−31.9454	−1.56063	−0.780317	−	0.625384i	$-0.784942\pi$
$419$	−31.9454	−1.56063	−0.780317	+	0.625384i	$0.784942\pi$
$420$	0	0
$421$	30.9318	1.50753	0.753763	−	0.657146i	$-0.228237\pi$
$421$	30.9318	1.50753	0.753763	+	0.657146i	$0.228237\pi$
$422$	20.1382	0.980315
$423$	0	0
$424$	25.1201	1.21994
$425$	−2.09791	−0.101764
$426$	0	0
$427$	2.96367	0.143422
$428$	1.80257	0.0871305
$429$	0	0
$430$	−38.0315	−1.83404
$431$	−0.341449	−0.0164470	−0.00822351	−	0.999966i	$-0.502618\pi$
$431$	−0.341449	−0.0164470	−0.00822351	+	0.999966i	$0.502618\pi$
$432$	0	0
$433$	2.92341	0.140490	0.0702450	−	0.997530i	$-0.477622\pi$
$433$	2.92341	0.140490	0.0702450	+	0.997530i	$0.477622\pi$
$434$	5.70205	0.273707
$435$	0	0
$436$	3.98166	0.190687
$437$	−4.17600	−0.199765
$438$	0	0
$439$	32.8788	1.56922	0.784609	−	0.619991i	$-0.212864\pi$
$439$	32.8788	1.56922	0.784609	+	0.619991i	$0.212864\pi$
$440$	−5.45989	−0.260290
$441$	0	0
$442$	−11.8940	−0.565741
$443$	23.6725	1.12471	0.562356	−	0.826895i	$-0.309895\pi$
$443$	23.6725	1.12471	0.562356	+	0.826895i	$0.309895\pi$
$444$	0	0
$445$	−7.77989	−0.368802
$446$	36.0556	1.70728
$447$	0	0
$448$	18.0407	0.852344
$449$	23.9338	1.12950	0.564752	−	0.825261i	$-0.308972\pi$
$449$	23.9338	1.12950	0.564752	+	0.825261i	$0.308972\pi$
$450$	0	0
$451$	1.36557	0.0643022
$452$	−0.0123327	−0.000580080 0
$453$	0	0
$454$	−45.1187	−2.11753
$455$	7.18250	0.336721
$456$	0	0
$457$	32.7798	1.53337	0.766687	−	0.642022i	$-0.221904\pi$
$457$	32.7798	1.53337	0.766687	+	0.642022i	$0.221904\pi$
$458$	44.5228	2.08041
$459$	0	0
$460$	−0.470080	−0.0219176
$461$	−27.3848	−1.27544	−0.637718	−	0.770270i	$-0.720121\pi$
$461$	−27.3848	−1.27544	−0.637718	+	0.770270i	$0.720121\pi$
$462$	0	0
$463$	3.92457	0.182390	0.0911952	−	0.995833i	$-0.470931\pi$
$463$	3.92457	0.182390	0.0911952	+	0.995833i	$0.470931\pi$
$464$	−22.9158	−1.06384
$465$	0	0
$466$	26.3513	1.22070
$467$	−10.0855	−0.466701	−0.233351	−	0.972393i	$-0.574969\pi$
$467$	−10.0855	−0.466701	−0.233351	+	0.972393i	$0.574969\pi$
$468$	0	0
$469$	−16.4952	−0.761675
$470$	11.5889	0.534556
$471$	0	0
$472$	10.2981	0.474007
$473$	−11.4226	−0.525211
$474$	0	0
$475$	−2.10108	−0.0964042
$476$	7.38980	0.338711
$477$	0	0
$478$	37.9401	1.73534
$479$	−3.63175	−0.165939	−0.0829695	−	0.996552i	$-0.526440\pi$
$479$	−3.63175	−0.165939	−0.0829695	+	0.996552i	$0.526440\pi$
$480$	0	0
$481$	−4.41319	−0.201224
$482$	25.6104	1.16652
$483$	0	0
$484$	0.360147	0.0163703
$485$	−34.3617	−1.56029
$486$	0	0
$487$	−6.40952	−0.290443	−0.145222	−	0.989399i	$-0.546390\pi$
$487$	−6.40952	−0.290443	−0.145222	+	0.989399i	$0.546390\pi$
$488$	2.51927	0.114042
$489$	0	0
$490$	−5.93759	−0.268233
$491$	15.8126	0.713613	0.356807	−	0.934178i	$-0.383865\pi$
$491$	15.8126	0.713613	0.356807	+	0.934178i	$0.383865\pi$
$492$	0	0
$493$	34.5613	1.55656
$494$	−11.9120	−0.535946
$495$	0	0
$496$	5.74910	0.258142
$497$	17.8406	0.800260
$498$	0	0
$499$	−4.57896	−0.204982	−0.102491	−	0.994734i	$-0.532681\pi$
$499$	−4.57896	−0.204982	−0.102491	+	0.994734i	$0.532681\pi$
$500$	−4.13917	−0.185109
$501$	0	0
$502$	−34.6148	−1.54493
$503$	−9.30226	−0.414767	−0.207384	−	0.978260i	$-0.566495\pi$
$503$	−9.30226	−0.414767	−0.207384	+	0.978260i	$0.566495\pi$
$504$	0	0
$505$	3.59910	0.160158
$506$	−0.925234	−0.0411317
$507$	0	0
$508$	4.61316	0.204676
$509$	33.7782	1.49719	0.748596	−	0.663026i	$-0.230728\pi$
$509$	33.7782	1.49719	0.748596	+	0.663026i	$0.230728\pi$
$510$	0	0
$511$	12.5910	0.556992
$512$	13.8849	0.613634
$513$	0	0
$514$	21.9294	0.967262
$515$	−25.2792	−1.11393
$516$	0	0
$517$	3.48067	0.153080
$518$	17.9687	0.789497
$519$	0	0
$520$	6.10549	0.267743
$521$	−27.0350	−1.18442	−0.592212	−	0.805782i	$-0.701745\pi$
$521$	−27.0350	−1.18442	−0.592212	+	0.805782i	$0.701745\pi$
$522$	0	0
$523$	37.3914	1.63501	0.817507	−	0.575919i	$-0.195356\pi$
$523$	37.3914	1.63501	0.817507	+	0.575919i	$0.195356\pi$
$524$	0.824382	0.0360133
$525$	0	0
$526$	17.8264	0.777267
$527$	−8.67071	−0.377702
$528$	0	0
$529$	−22.6373	−0.984230
$530$	−33.1991	−1.44208
$531$	0	0
$532$	7.40096	0.320872
$533$	−1.52704	−0.0661435
$534$	0	0
$535$	10.8473	0.468969
$536$	−14.0217	−0.605646
$537$	0	0
$538$	40.9927	1.76732
$539$	−1.78333	−0.0768133
$540$	0	0
$541$	37.7680	1.62377	0.811886	−	0.583815i	$-0.198441\pi$
$541$	37.7680	1.62377	0.811886	+	0.583815i	$0.198441\pi$
$542$	30.0834	1.29219
$543$	0	0
$544$	13.9430	0.597801
$545$	23.9604	1.02635
$546$	0	0
$547$	−6.32281	−0.270344	−0.135172	−	0.990822i	$-0.543159\pi$
$547$	−6.32281	−0.270344	−0.135172	+	0.990822i	$0.543159\pi$
$548$	4.62190	0.197438
$549$	0	0
$550$	−0.465515	−0.0198496
$551$	34.6135	1.47459
$552$	0	0
$553$	−44.8062	−1.90535
$554$	−8.52286	−0.362101
$555$	0	0
$556$	−5.18169	−0.219752
$557$	−33.1235	−1.40349	−0.701744	−	0.712430i	$-0.747595\pi$
$557$	−33.1235	−1.40349	−0.701744	+	0.712430i	$0.747595\pi$
$558$	0	0
$559$	12.7732	0.540250
$560$	−29.4854	−1.24599
$561$	0	0
$562$	7.55320	0.318612
$563$	−9.13968	−0.385192	−0.192596	−	0.981278i	$-0.561691\pi$
$563$	−9.13968	−0.385192	−0.192596	+	0.981278i	$0.561691\pi$
$564$	0	0
$565$	−0.0742140	−0.00312221
$566$	−26.3828	−1.10895
$567$	0	0
$568$	15.1654	0.636326
$569$	43.1897	1.81061	0.905304	−	0.424764i	$-0.139643\pi$
$569$	43.1897	1.81061	0.905304	+	0.424764i	$0.139643\pi$
$570$	0	0
$571$	7.63115	0.319353	0.159677	−	0.987169i	$-0.448955\pi$
$571$	7.63115	0.319353	0.159677	+	0.987169i	$0.448955\pi$
$572$	−0.402733	−0.0168391
$573$	0	0
$574$	6.21747	0.259512
$575$	0.182493	0.00761049
$576$	0	0
$577$	40.0501	1.66731	0.833654	−	0.552287i	$-0.186245\pi$
$577$	40.0501	1.66731	0.833654	+	0.552287i	$0.186245\pi$
$578$	−47.5235	−1.97672
$579$	0	0
$580$	3.89634	0.161787
$581$	32.4415	1.34590
$582$	0	0
$583$	−9.97119	−0.412964
$584$	10.7030	0.442892
$585$	0	0
$586$	−25.4596	−1.05173
$587$	−44.4272	−1.83371	−0.916854	−	0.399223i	$-0.869280\pi$
$587$	−44.4272	−1.83371	−0.916854	+	0.399223i	$0.869280\pi$
$588$	0	0
$589$	−8.68381	−0.357810
$590$	−13.6101	−0.560318
$591$	0	0
$592$	18.1169	0.744602
$593$	9.39583	0.385840	0.192920	−	0.981214i	$-0.438204\pi$
$593$	9.39583	0.385840	0.192920	+	0.981214i	$0.438204\pi$
$594$	0	0
$595$	44.4694	1.82307
$596$	−0.580800	−0.0237905
$597$	0	0
$598$	1.03464	0.0423095
$599$	−6.52995	−0.266807	−0.133403	−	0.991062i	$-0.542591\pi$
$599$	−6.52995	−0.266807	−0.133403	+	0.991062i	$0.542591\pi$
$600$	0	0
$601$	−15.5451	−0.634099	−0.317050	−	0.948409i	$-0.602692\pi$
$601$	−15.5451	−0.634099	−0.317050	+	0.948409i	$0.602692\pi$
$602$	−52.0072	−2.11965
$603$	0	0
$604$	−0.520469	−0.0211776
$605$	2.16725	0.0881114
$606$	0	0
$607$	11.8074	0.479249	0.239625	−	0.970866i	$-0.422976\pi$
$607$	11.8074	0.479249	0.239625	+	0.970866i	$0.422976\pi$
$608$	13.9641	0.566318
$609$	0	0
$610$	−3.32950	−0.134808
$611$	−3.89224	−0.157463
$612$	0	0
$613$	10.1209	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$613$	10.1209	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$614$	13.6446	0.550653
$615$	0	0
$616$	−7.46627	−0.300825
$617$	18.1856	0.732124	0.366062	−	0.930591i	$-0.380706\pi$
$617$	18.1856	0.732124	0.366062	+	0.930591i	$0.380706\pi$
$618$	0	0
$619$	37.6887	1.51484	0.757419	−	0.652929i	$-0.226460\pi$
$619$	37.6887	1.51484	0.757419	+	0.652929i	$0.226460\pi$
$620$	−0.977511	−0.0392578
$621$	0	0
$622$	50.8861	2.04035
$623$	−10.6388	−0.426235
$624$	0	0
$625$	−23.3931	−0.935724
$626$	1.31950	0.0527379
$627$	0	0
$628$	0.749712	0.0299168
$629$	−27.3237	−1.08947
$630$	0	0
$631$	2.98191	0.118708	0.0593540	−	0.998237i	$-0.481096\pi$
$631$	2.98191	0.118708	0.0593540	+	0.998237i	$0.481096\pi$
$632$	−38.0875	−1.51504
$633$	0	0
$634$	−27.9087	−1.10840
$635$	27.7606	1.10164
$636$	0	0
$637$	1.99419	0.0790129
$638$	7.66897	0.303617
$639$	0	0
$640$	−28.9968	−1.14620
$641$	−1.83933	−0.0726491	−0.0363246	−	0.999340i	$-0.511565\pi$
$641$	−1.83933	−0.0726491	−0.0363246	+	0.999340i	$0.511565\pi$
$642$	0	0
$643$	15.0684	0.594239	0.297119	−	0.954840i	$-0.403974\pi$
$643$	15.0684	0.594239	0.297119	+	0.954840i	$0.403974\pi$
$644$	−0.642824	−0.0253308
$645$	0	0
$646$	−73.7515	−2.90171
$647$	9.19964	0.361675	0.180838	−	0.983513i	$-0.442119\pi$
$647$	9.19964	0.361675	0.180838	+	0.983513i	$0.442119\pi$
$648$	0	0
$649$	−4.08773	−0.160457
$650$	0.520560	0.0204180
$651$	0	0
$652$	−5.00099	−0.195854
$653$	29.7910	1.16581	0.582906	−	0.812540i	$-0.301916\pi$
$653$	29.7910	1.16581	0.582906	+	0.812540i	$0.301916\pi$
$654$	0	0
$655$	4.96087	0.193837
$656$	6.26878	0.244755
$657$	0	0
$658$	15.8476	0.617802
$659$	14.7506	0.574601	0.287301	−	0.957840i	$-0.407242\pi$
$659$	14.7506	0.574601	0.287301	+	0.957840i	$0.407242\pi$
$660$	0	0
$661$	−18.0393	−0.701648	−0.350824	−	0.936442i	$-0.614098\pi$
$661$	−18.0393	−0.701648	−0.350824	+	0.936442i	$0.614098\pi$
$662$	−4.36014	−0.169462
$663$	0	0
$664$	27.5769	1.07019
$665$	44.5366	1.72706
$666$	0	0
$667$	−3.00642	−0.116409
$668$	−1.67334	−0.0647433
$669$	0	0
$670$	18.5313	0.715927
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	6.65379	0.256485	0.128242	−	0.991743i	$-0.459066\pi$
$673$	6.65379	0.256485	0.128242	+	0.991743i	$0.459066\pi$
$674$	9.60296	0.369892
$675$	0	0
$676$	−4.23156	−0.162752
$677$	−23.5492	−0.905070	−0.452535	−	0.891747i	$-0.649480\pi$
$677$	−23.5492	−0.905070	−0.452535	+	0.891747i	$0.649480\pi$
$678$	0	0
$679$	−46.9889	−1.80327
$680$	37.8013	1.44961
$681$	0	0
$682$	−1.92398	−0.0736731
$683$	38.8826	1.48780	0.743901	−	0.668290i	$-0.232973\pi$
$683$	38.8826	1.48780	0.743901	+	0.668290i	$0.232973\pi$
$684$	0	0
$685$	27.8131	1.06268
$686$	23.7516	0.906840
$687$	0	0
$688$	−52.4364	−1.99912
$689$	11.1502	0.424790
$690$	0	0
$691$	8.93629	0.339953	0.169976	−	0.985448i	$-0.445631\pi$
$691$	8.93629	0.339953	0.169976	+	0.985448i	$0.445631\pi$
$692$	6.58993	0.250512
$693$	0	0
$694$	1.05672	0.0401125
$695$	−31.1817	−1.18279
$696$	0	0
$697$	−9.45447	−0.358113
$698$	9.20441	0.348392
$699$	0	0
$700$	−0.323425	−0.0122243
$701$	−29.3901	−1.11005	−0.555025	−	0.831834i	$-0.687291\pi$
$701$	−29.3901	−1.11005	−0.555025	+	0.831834i	$0.687291\pi$
$702$	0	0
$703$	−27.3650	−1.03209
$704$	−6.08730	−0.229424
$705$	0	0
$706$	33.1219	1.24656
$707$	4.92168	0.185099
$708$	0	0
$709$	28.4876	1.06987	0.534936	−	0.844892i	$-0.320336\pi$
$709$	28.4876	1.06987	0.534936	+	0.844892i	$0.320336\pi$
$710$	−20.0428	−0.752194
$711$	0	0
$712$	−9.04354	−0.338921
$713$	0.754247	0.0282468
$714$	0	0
$715$	−2.42352	−0.0906344
$716$	0.413856	0.0154665
$717$	0	0
$718$	14.6682	0.547412
$719$	−44.8105	−1.67115	−0.835574	−	0.549378i	$-0.814865\pi$
$719$	−44.8105	−1.67115	−0.835574	+	0.549378i	$0.814865\pi$
$720$	0	0
$721$	−34.5687	−1.28740
$722$	−44.6736	−1.66258
$723$	0	0
$724$	−5.15278	−0.191502
$725$	−1.51263	−0.0561775
$726$	0	0
$727$	−37.3792	−1.38632	−0.693158	−	0.720785i	$-0.743781\pi$
$727$	−37.3792	−1.38632	−0.693158	+	0.720785i	$0.743781\pi$
$728$	8.34912	0.309439
$729$	0	0
$730$	−14.1452	−0.523537
$731$	79.0837	2.92502
$732$	0	0
$733$	19.9355	0.736334	0.368167	−	0.929760i	$-0.379985\pi$
$733$	19.9355	0.736334	0.368167	+	0.929760i	$0.379985\pi$
$734$	−28.0634	−1.03584
$735$	0	0
$736$	−1.21287	−0.0447071
$737$	5.56579	0.205019
$738$	0	0
$739$	25.9077	0.953029	0.476514	−	0.879167i	$-0.341900\pi$
$739$	25.9077	0.953029	0.476514	+	0.879167i	$0.341900\pi$
$740$	−3.08039	−0.113238
$741$	0	0
$742$	−45.3990	−1.66665
$743$	−53.6023	−1.96648	−0.983239	−	0.182324i	$-0.941638\pi$
$743$	−53.6023	−1.96648	−0.983239	+	0.182324i	$0.941638\pi$
$744$	0	0
$745$	−3.49507	−0.128050
$746$	32.9805	1.20750
$747$	0	0
$748$	−2.49346	−0.0911700
$749$	14.8334	0.542001
$750$	0	0
$751$	−49.9819	−1.82387	−0.911933	−	0.410339i	$-0.865410\pi$
$751$	−49.9819	−1.82387	−0.911933	+	0.410339i	$0.865410\pi$
$752$	15.9783	0.582670
$753$	0	0
$754$	−8.57578	−0.312311
$755$	−3.13202	−0.113986
$756$	0	0
$757$	−15.9641	−0.580225	−0.290113	−	0.956993i	$-0.593693\pi$
$757$	−15.9641	−0.580225	−0.290113	+	0.956993i	$0.593693\pi$
$758$	34.3451	1.24747
$759$	0	0
$760$	37.8584	1.37327
$761$	−23.6666	−0.857912	−0.428956	−	0.903325i	$-0.641118\pi$
$761$	−23.6666	−0.857912	−0.428956	+	0.903325i	$0.641118\pi$
$762$	0	0
$763$	32.7653	1.18618
$764$	−0.588843	−0.0213036
$765$	0	0
$766$	−47.9396	−1.73213
$767$	4.57107	0.165052
$768$	0	0
$769$	2.05401	0.0740696	0.0370348	−	0.999314i	$-0.488209\pi$
$769$	2.05401	0.0740696	0.0370348	+	0.999314i	$0.488209\pi$
$770$	9.86753	0.355601
$771$	0	0
$772$	8.32437	0.299601
$773$	−40.4222	−1.45389	−0.726944	−	0.686697i	$-0.759060\pi$
$773$	−40.4222	−1.45389	−0.726944	+	0.686697i	$0.759060\pi$
$774$	0	0
$775$	0.379486	0.0136316
$776$	−39.9429	−1.43387
$777$	0	0
$778$	−48.2186	−1.72872
$779$	−9.46875	−0.339253
$780$	0	0
$781$	−6.01977	−0.215404
$782$	6.40581	0.229071
$783$	0	0
$784$	−8.18652	−0.292376
$785$	4.51153	0.161023
$786$	0	0
$787$	16.3124	0.581473	0.290736	−	0.956803i	$-0.406100\pi$
$787$	16.3124	0.581473	0.290736	+	0.956803i	$0.406100\pi$
$788$	−4.29687	−0.153070
$789$	0	0
$790$	50.3370	1.79091
$791$	−0.101486	−0.00360843
$792$	0	0
$793$	1.11824	0.0397100
$794$	−33.9678	−1.20547
$795$	0	0
$796$	−1.28062	−0.0453903
$797$	−4.40506	−0.156035	−0.0780176	−	0.996952i	$-0.524859\pi$
$797$	−4.40506	−0.156035	−0.0780176	+	0.996952i	$0.524859\pi$
$798$	0	0
$799$	−24.0983	−0.852536
$800$	−0.610236	−0.0215751
$801$	0	0
$802$	−51.3488	−1.81319
$803$	−4.24844	−0.149924
$804$	0	0
$805$	−3.86831	−0.136340
$806$	2.15148	0.0757828
$807$	0	0
$808$	4.18368	0.147181
$809$	14.1528	0.497586	0.248793	−	0.968557i	$-0.419966\pi$
$809$	14.1528	0.497586	0.248793	+	0.968557i	$0.419966\pi$
$810$	0	0
$811$	49.7568	1.74720	0.873598	−	0.486648i	$-0.161781\pi$
$811$	49.7568	1.74720	0.873598	+	0.486648i	$0.161781\pi$
$812$	5.32816	0.186982
$813$	0	0
$814$	−6.06298	−0.212507
$815$	−30.0944	−1.05416
$816$	0	0
$817$	79.2032	2.77097
$818$	−14.8667	−0.519801
$819$	0	0
$820$	−1.06587	−0.0372218
$821$	4.12520	0.143971	0.0719853	−	0.997406i	$-0.477067\pi$
$821$	4.12520	0.143971	0.0719853	+	0.997406i	$0.477067\pi$
$822$	0	0
$823$	−2.13671	−0.0744809	−0.0372405	−	0.999306i	$-0.511857\pi$
$823$	−2.13671	−0.0744809	−0.0372405	+	0.999306i	$0.511857\pi$
$824$	−29.3851	−1.02368
$825$	0	0
$826$	−18.6115	−0.647576
$827$	−15.4530	−0.537353	−0.268676	−	0.963231i	$-0.586586\pi$
$827$	−15.4530	−0.537353	−0.268676	+	0.963231i	$0.586586\pi$
$828$	0	0
$829$	17.3229	0.601651	0.300825	−	0.953679i	$-0.402738\pi$
$829$	17.3229	0.601651	0.300825	+	0.953679i	$0.402738\pi$
$830$	−36.4460	−1.26506
$831$	0	0
$832$	6.80709	0.235993
$833$	12.3468	0.427790
$834$	0	0
$835$	−10.0696	−0.348473
$836$	−2.49723	−0.0863686
$837$	0	0
$838$	49.0770	1.69534
$839$	23.7300	0.819250	0.409625	−	0.912254i	$-0.365660\pi$
$839$	23.7300	0.819250	0.409625	+	0.912254i	$0.365660\pi$
$840$	0	0
$841$	−4.08076	−0.140716
$842$	−47.5199	−1.63764
$843$	0	0
$844$	−4.72098	−0.162503
$845$	−25.4642	−0.875995
$846$	0	0
$847$	2.96367	0.101833
$848$	−45.7736	−1.57187
$849$	0	0
$850$	3.22297	0.110547
$851$	2.37683	0.0814767
$852$	0	0
$853$	−5.60379	−0.191870	−0.0959350	−	0.995388i	$-0.530584\pi$
$853$	−5.60379	−0.191870	−0.0959350	+	0.995388i	$0.530584\pi$
$854$	−4.55302	−0.155801
$855$	0	0
$856$	12.6092	0.430972
$857$	20.2530	0.691829	0.345915	−	0.938266i	$-0.387569\pi$
$857$	20.2530	0.691829	0.345915	+	0.938266i	$0.387569\pi$
$858$	0	0
$859$	−22.5525	−0.769483	−0.384741	−	0.923024i	$-0.625709\pi$
$859$	−22.5525	−0.769483	−0.384741	+	0.923024i	$0.625709\pi$
$860$	8.91567	0.304022
$861$	0	0
$862$	0.524560	0.0178666
$863$	12.8191	0.436366	0.218183	−	0.975908i	$-0.429987\pi$
$863$	12.8191	0.436366	0.218183	+	0.975908i	$0.429987\pi$
$864$	0	0
$865$	39.6561	1.34835
$866$	−4.49117	−0.152616
$867$	0	0
$868$	−1.33672	−0.0453714
$869$	15.1185	0.512860
$870$	0	0
$871$	−6.22391	−0.210889
$872$	27.8521	0.943192
$873$	0	0
$874$	6.41549	0.217007
$875$	−34.0614	−1.15148
$876$	0	0
$877$	3.58171	0.120946	0.0604729	−	0.998170i	$-0.480739\pi$
$877$	3.58171	0.120946	0.0604729	+	0.998170i	$0.480739\pi$
$878$	−50.5109	−1.70466
$879$	0	0
$880$	9.94897	0.335380
$881$	36.6210	1.23379	0.616896	−	0.787045i	$-0.288390\pi$
$881$	36.6210	1.23379	0.616896	+	0.787045i	$0.288390\pi$
$882$	0	0
$883$	−39.9528	−1.34452	−0.672260	−	0.740315i	$-0.734676\pi$
$883$	−39.9528	−1.34452	−0.672260	+	0.740315i	$0.734676\pi$
$884$	2.78830	0.0937807
$885$	0	0
$886$	−36.3675	−1.22179
$887$	−35.7841	−1.20151	−0.600756	−	0.799433i	$-0.705134\pi$
$887$	−35.7841	−1.20151	−0.600756	+	0.799433i	$0.705134\pi$
$888$	0	0
$889$	37.9619	1.27320
$890$	11.9521	0.400634
$891$	0	0
$892$	−8.45247	−0.283010
$893$	−24.1347	−0.807637
$894$	0	0
$895$	2.49046	0.0832468
$896$	−39.6525	−1.32470
$897$	0	0
$898$	−36.7689	−1.22699
$899$	−6.25171	−0.208506
$900$	0	0
$901$	69.0350	2.29989
$902$	−2.09790	−0.0698523
$903$	0	0
$904$	−0.0862682	−0.00286924
$905$	−31.0078	−1.03073
$906$	0	0
$907$	17.5669	0.583301	0.291650	−	0.956525i	$-0.405796\pi$
$907$	17.5669	0.583301	0.291650	+	0.956525i	$0.405796\pi$
$908$	10.5771	0.351014
$909$	0	0
$910$	−11.0343	−0.365784
$911$	48.1419	1.59501	0.797506	−	0.603312i	$-0.206152\pi$
$911$	48.1419	1.59501	0.797506	+	0.603312i	$0.206152\pi$
$912$	0	0
$913$	−10.9464	−0.362273
$914$	−50.3588	−1.66572
$915$	0	0
$916$	−10.4374	−0.344862
$917$	6.78387	0.224023
$918$	0	0
$919$	9.75158	0.321675	0.160838	−	0.986981i	$-0.448580\pi$
$919$	9.75158	0.321675	0.160838	+	0.986981i	$0.448580\pi$
$920$	−3.28826	−0.108411
$921$	0	0
$922$	42.0706	1.38552
$923$	6.73157	0.221572
$924$	0	0
$925$	1.19586	0.0393197
$926$	−6.02923	−0.198133
$927$	0	0
$928$	10.0531	0.330010
$929$	19.2963	0.633090	0.316545	−	0.948577i	$-0.397477\pi$
$929$	19.2963	0.633090	0.316545	+	0.948577i	$0.397477\pi$
$930$	0	0
$931$	12.3654	0.405261
$932$	−6.17750	−0.202351
$933$	0	0
$934$	15.4941	0.506983
$935$	−15.0049	−0.490712
$936$	0	0
$937$	38.7671	1.26647	0.633234	−	0.773961i	$-0.281727\pi$
$937$	38.7671	1.26647	0.633234	+	0.773961i	$0.281727\pi$
$938$	25.3411	0.827417
$939$	0	0
$940$	−2.71677	−0.0886113
$941$	11.5036	0.375008	0.187504	−	0.982264i	$-0.439960\pi$
$941$	11.5036	0.375008	0.187504	+	0.982264i	$0.439960\pi$
$942$	0	0
$943$	0.822425	0.0267818
$944$	−18.7651	−0.610751
$945$	0	0
$946$	17.5482	0.570543
$947$	−36.2501	−1.17797	−0.588985	−	0.808144i	$-0.700472\pi$
$947$	−36.2501	−1.17797	−0.588985	+	0.808144i	$0.700472\pi$
$948$	0	0
$949$	4.75079	0.154217
$950$	3.22784	0.104725
$951$	0	0
$952$	51.6924	1.67536
$953$	17.5362	0.568053	0.284026	−	0.958817i	$-0.408330\pi$
$953$	17.5362	0.568053	0.284026	+	0.958817i	$0.408330\pi$
$954$	0	0
$955$	−3.54347	−0.114664
$956$	−8.89424	−0.287660
$957$	0	0
$958$	5.57938	0.180262
$959$	38.0338	1.22817
$960$	0	0
$961$	−29.4316	−0.949406
$962$	6.77989	0.218592
$963$	0	0
$964$	−6.00382	−0.193370
$965$	50.0934	1.61256
$966$	0	0
$967$	−14.1752	−0.455844	−0.227922	−	0.973679i	$-0.573193\pi$
$967$	−14.1752	−0.455844	−0.227922	+	0.973679i	$0.573193\pi$
$968$	2.51927	0.0809723
$969$	0	0
$970$	52.7891	1.69496
$971$	29.6419	0.951255	0.475627	−	0.879647i	$-0.342221\pi$
$971$	29.6419	0.951255	0.475627	+	0.879647i	$0.342221\pi$
$972$	0	0
$973$	−42.6403	−1.36699
$974$	9.84681	0.315512
$975$	0	0
$976$	−4.59059	−0.146941
$977$	−9.22781	−0.295224	−0.147612	−	0.989045i	$-0.547159\pi$
$977$	−9.22781	−0.295224	−0.147612	+	0.989045i	$0.547159\pi$
$978$	0	0
$979$	3.58975	0.114729
$980$	1.39194	0.0444639
$981$	0	0
$982$	−24.2926	−0.775207
$983$	−39.2361	−1.25144	−0.625718	−	0.780049i	$-0.715194\pi$
$983$	−39.2361	−1.25144	−0.625718	+	0.780049i	$0.715194\pi$
$984$	0	0
$985$	−25.8572	−0.823879
$986$	−53.0957	−1.69091
$987$	0	0
$988$	2.79251	0.0888417
$989$	−6.87933	−0.218750
$990$	0	0
$991$	−22.0776	−0.701317	−0.350659	−	0.936503i	$-0.614042\pi$
$991$	−22.0776	−0.701317	−0.350659	+	0.936503i	$0.614042\pi$
$992$	−2.52212	−0.0800773
$993$	0	0
$994$	−27.4081	−0.869332
$995$	−7.70636	−0.244308
$996$	0	0
$997$	47.8055	1.51401	0.757007	−	0.653407i	$-0.226661\pi$
$997$	47.8055	1.51401	0.757007	+	0.653407i	$0.226661\pi$
$998$	7.03455	0.222675
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.p.1.6	yes	25
3.2	odd	2		6039.2.a.m.1.20	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.20	✓	25	3.2	odd	2
6039.2.a.p.1.6	yes	25	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53628 q^{2} +0.360147 q^{4} +2.16725 q^{5} +2.96367 q^{7} +2.51927 q^{8} +O(q^{10})\)	\(q-1.53628 q^{2} +0.360147 q^{4} +2.16725 q^{5} +2.96367 q^{7} +2.51927 q^{8} -3.32950 q^{10} -1.00000 q^{11} +1.11824 q^{13} -4.55302 q^{14} -4.59059 q^{16} +6.92345 q^{17} +6.93391 q^{19} +0.780531 q^{20} +1.53628 q^{22} -0.602257 q^{23} -0.303015 q^{25} -1.71793 q^{26} +1.06736 q^{28} +4.99192 q^{29} -1.25237 q^{31} +2.01388 q^{32} -10.6363 q^{34} +6.42302 q^{35} -3.94654 q^{37} -10.6524 q^{38} +5.45989 q^{40} -1.36557 q^{41} +11.4226 q^{43} -0.360147 q^{44} +0.925234 q^{46} -3.48067 q^{47} +1.78333 q^{49} +0.465515 q^{50} +0.402733 q^{52} +9.97119 q^{53} -2.16725 q^{55} +7.46627 q^{56} -7.66897 q^{58} +4.08773 q^{59} +1.00000 q^{61} +1.92398 q^{62} +6.08730 q^{64} +2.42352 q^{65} -5.56579 q^{67} +2.49346 q^{68} -9.86753 q^{70} +6.01977 q^{71} +4.24844 q^{73} +6.06298 q^{74} +2.49723 q^{76} -2.96367 q^{77} -15.1185 q^{79} -9.94897 q^{80} +2.09790 q^{82} +10.9464 q^{83} +15.0049 q^{85} -17.5482 q^{86} -2.51927 q^{88} -3.58975 q^{89} +3.31410 q^{91} -0.216901 q^{92} +5.34728 q^{94} +15.0275 q^{95} -15.8550 q^{97} -2.73968 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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