LMFDB - Embedded newform 6039.2.a.k.1.9

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:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.686089$ of defining polynomial
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)

$f(q)$	$=$	$q-0.686089 q^{2} -1.52928 q^{4} -3.77631 q^{5} +4.48234 q^{7} +2.42140 q^{8} +O(q^{10})$	\(q-0.686089 q^{2} -1.52928 q^{4} -3.77631 q^{5} +4.48234 q^{7} +2.42140 q^{8} +2.59089 q^{10} -1.00000 q^{11} -0.769699 q^{13} -3.07528 q^{14} +1.39726 q^{16} +3.80128 q^{17} +6.11093 q^{19} +5.77505 q^{20} +0.686089 q^{22} +3.48519 q^{23} +9.26054 q^{25} +0.528082 q^{26} -6.85475 q^{28} -4.50868 q^{29} +1.54601 q^{31} -5.80145 q^{32} -2.60802 q^{34} -16.9267 q^{35} +10.3247 q^{37} -4.19264 q^{38} -9.14398 q^{40} +1.01077 q^{41} -3.73505 q^{43} +1.52928 q^{44} -2.39115 q^{46} +8.78295 q^{47} +13.0913 q^{49} -6.35356 q^{50} +1.17709 q^{52} -3.35846 q^{53} +3.77631 q^{55} +10.8535 q^{56} +3.09336 q^{58} -9.89674 q^{59} -1.00000 q^{61} -1.06070 q^{62} +1.18579 q^{64} +2.90662 q^{65} -7.87874 q^{67} -5.81322 q^{68} +11.6132 q^{70} +13.3078 q^{71} +7.03634 q^{73} -7.08369 q^{74} -9.34533 q^{76} -4.48234 q^{77} -16.1138 q^{79} -5.27651 q^{80} -0.693478 q^{82} -8.42331 q^{83} -14.3548 q^{85} +2.56258 q^{86} -2.42140 q^{88} -2.88714 q^{89} -3.45005 q^{91} -5.32984 q^{92} -6.02589 q^{94} -23.0768 q^{95} +13.9532 q^{97} -8.98183 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * 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 $

$2$	−0.686089	−0.485138	−0.242569	−	0.970134i	$-0.577990\pi$
$2$	−0.686089	−0.485138	−0.242569	+	0.970134i	$0.577990\pi$
$3$	0	0
$3$	0	0
$4$	−1.52928	−0.764641
$5$	−3.77631	−1.68882	−0.844409	−	0.535698i	$-0.820048\pi$
$5$	−3.77631	−1.68882	−0.844409	+	0.535698i	$0.820048\pi$
$6$	0	0
$7$	4.48234	1.69416	0.847082	−	0.531462i	$-0.178357\pi$
$7$	4.48234	1.69416	0.847082	+	0.531462i	$0.178357\pi$
$8$	2.42140	0.856095
$9$	0	0
$10$	2.59089	0.819311
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.769699	−0.213476	−0.106738	−	0.994287i	$-0.534041\pi$
$13$	−0.769699	−0.213476	−0.106738	+	0.994287i	$0.534041\pi$
$14$	−3.07528	−0.821904
$15$	0	0
$16$	1.39726	0.349316
$17$	3.80128	0.921945	0.460972	−	0.887414i	$-0.347501\pi$
$17$	3.80128	0.921945	0.460972	+	0.887414i	$0.347501\pi$
$18$	0	0
$19$	6.11093	1.40194	0.700971	−	0.713189i	$-0.252750\pi$
$19$	6.11093	1.40194	0.700971	+	0.713189i	$0.252750\pi$
$20$	5.77505	1.29134
$21$	0	0
$22$	0.686089	0.146275
$23$	3.48519	0.726713	0.363357	−	0.931650i	$-0.381631\pi$
$23$	3.48519	0.726713	0.363357	+	0.931650i	$0.381631\pi$
$24$	0	0
$25$	9.26054	1.85211
$26$	0.528082	0.103565
$27$	0	0
$28$	−6.85475	−1.29543
$29$	−4.50868	−0.837241	−0.418621	−	0.908161i	$-0.637486\pi$
$29$	−4.50868	−0.837241	−0.418621	+	0.908161i	$0.637486\pi$
$30$	0	0
$31$	1.54601	0.277671	0.138835	−	0.990315i	$-0.455664\pi$
$31$	1.54601	0.277671	0.138835	+	0.990315i	$0.455664\pi$
$32$	−5.80145	−1.02556
$33$	0	0
$34$	−2.60802	−0.447271
$35$	−16.9267	−2.86114
$36$	0	0
$37$	10.3247	1.69737	0.848687	−	0.528895i	$-0.177393\pi$
$37$	10.3247	1.69737	0.848687	+	0.528895i	$0.177393\pi$
$38$	−4.19264	−0.680136
$39$	0	0
$40$	−9.14398	−1.44579
$41$	1.01077	0.157856	0.0789278	−	0.996880i	$-0.474850\pi$
$41$	1.01077	0.157856	0.0789278	+	0.996880i	$0.474850\pi$
$42$	0	0
$43$	−3.73505	−0.569590	−0.284795	−	0.958588i	$-0.591925\pi$
$43$	−3.73505	−0.569590	−0.284795	+	0.958588i	$0.591925\pi$
$44$	1.52928	0.230548
$45$	0	0
$46$	−2.39115	−0.352557
$47$	8.78295	1.28112	0.640562	−	0.767906i	$-0.278701\pi$
$47$	8.78295	1.28112	0.640562	+	0.767906i	$0.278701\pi$
$48$	0	0
$49$	13.0913	1.87019
$50$	−6.35356	−0.898529
$51$	0	0
$52$	1.17709	0.163232
$53$	−3.35846	−0.461320	−0.230660	−	0.973034i	$-0.574088\pi$
$53$	−3.35846	−0.461320	−0.230660	+	0.973034i	$0.574088\pi$
$54$	0	0
$55$	3.77631	0.509198
$56$	10.8535	1.45037
$57$	0	0
$58$	3.09336	0.406178
$59$	−9.89674	−1.28845	−0.644223	−	0.764838i	$-0.722819\pi$
$59$	−9.89674	−1.28845	−0.644223	+	0.764838i	$0.722819\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−1.06070	−0.134709
$63$	0	0
$64$	1.18579	0.148223
$65$	2.90662	0.360522
$66$	0	0
$67$	−7.87874	−0.962542	−0.481271	−	0.876572i	$-0.659825\pi$
$67$	−7.87874	−0.962542	−0.481271	+	0.876572i	$0.659825\pi$
$68$	−5.81322	−0.704957
$69$	0	0
$70$	11.6132	1.38805
$71$	13.3078	1.57934	0.789671	−	0.613530i	$-0.210251\pi$
$71$	13.3078	1.57934	0.789671	+	0.613530i	$0.210251\pi$
$72$	0	0
$73$	7.03634	0.823541	0.411771	−	0.911288i	$-0.364911\pi$
$73$	7.03634	0.823541	0.411771	+	0.911288i	$0.364911\pi$
$74$	−7.08369	−0.823462
$75$	0	0
$76$	−9.34533	−1.07198
$77$	−4.48234	−0.510810
$78$	0	0
$79$	−16.1138	−1.81295	−0.906474	−	0.422261i	$-0.861237\pi$
$79$	−16.1138	−1.81295	−0.906474	+	0.422261i	$0.861237\pi$
$80$	−5.27651	−0.589931
$81$	0	0
$82$	−0.693478	−0.0765819
$83$	−8.42331	−0.924579	−0.462289	−	0.886729i	$-0.652972\pi$
$83$	−8.42331	−0.924579	−0.462289	+	0.886729i	$0.652972\pi$
$84$	0	0
$85$	−14.3548	−1.55700
$86$	2.56258	0.276330
$87$	0	0
$88$	−2.42140	−0.258122
$89$	−2.88714	−0.306036	−0.153018	−	0.988223i	$-0.548899\pi$
$89$	−2.88714	−0.306036	−0.153018	+	0.988223i	$0.548899\pi$
$90$	0	0
$91$	−3.45005	−0.361664
$92$	−5.32984	−0.555675
$93$	0	0
$94$	−6.02589	−0.621523
$95$	−23.0768	−2.36763
$96$	0	0
$97$	13.9532	1.41673	0.708367	−	0.705844i	$-0.249432\pi$
$97$	13.9532	1.41673	0.708367	+	0.705844i	$0.249432\pi$
$98$	−8.98183	−0.907302
$99$	0	0
$100$	−14.1620	−1.41620
$101$	−1.01604	−0.101100	−0.0505498	−	0.998722i	$-0.516097\pi$
$101$	−1.01604	−0.101100	−0.0505498	+	0.998722i	$0.516097\pi$
$102$	0	0
$103$	−12.6258	−1.24406	−0.622028	−	0.782995i	$-0.713691\pi$
$103$	−12.6258	−1.24406	−0.622028	+	0.782995i	$0.713691\pi$
$104$	−1.86375	−0.182756
$105$	0	0
$106$	2.30420	0.223804
$107$	12.7922	1.23667	0.618334	−	0.785916i	$-0.287808\pi$
$107$	12.7922	1.23667	0.618334	+	0.785916i	$0.287808\pi$
$108$	0	0
$109$	−18.3052	−1.75332	−0.876661	−	0.481109i	$-0.840234\pi$
$109$	−18.3052	−1.75332	−0.876661	+	0.481109i	$0.840234\pi$
$110$	−2.59089	−0.247032
$111$	0	0
$112$	6.26301	0.591799
$113$	13.6164	1.28092	0.640461	−	0.767990i	$-0.278743\pi$
$113$	13.6164	1.28092	0.640461	+	0.767990i	$0.278743\pi$
$114$	0	0
$115$	−13.1612	−1.22729
$116$	6.89504	0.640189
$117$	0	0
$118$	6.79005	0.625075
$119$	17.0386	1.56193
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.686089	0.0621156
$123$	0	0
$124$	−2.36428	−0.212318
$125$	−16.0892	−1.43906
$126$	0	0
$127$	−8.22802	−0.730119	−0.365059	−	0.930984i	$-0.618951\pi$
$127$	−8.22802	−0.730119	−0.365059	+	0.930984i	$0.618951\pi$
$128$	10.7894	0.953653
$129$	0	0
$130$	−1.99420	−0.174903
$131$	15.6721	1.36928	0.684639	−	0.728882i	$-0.259960\pi$
$131$	15.6721	1.36928	0.684639	+	0.728882i	$0.259960\pi$
$132$	0	0
$133$	27.3912	2.37512
$134$	5.40552	0.466966
$135$	0	0
$136$	9.20442	0.789272
$137$	−3.16502	−0.270406	−0.135203	−	0.990818i	$-0.543169\pi$
$137$	−3.16502	−0.270406	−0.135203	+	0.990818i	$0.543169\pi$
$138$	0	0
$139$	3.56579	0.302446	0.151223	−	0.988500i	$-0.451679\pi$
$139$	3.56579	0.302446	0.151223	+	0.988500i	$0.451679\pi$
$140$	25.8857	2.18774
$141$	0	0
$142$	−9.13032	−0.766200
$143$	0.769699	0.0643655
$144$	0	0
$145$	17.0262	1.41395
$146$	−4.82756	−0.399531
$147$	0	0
$148$	−15.7894	−1.29788
$149$	−13.3424	−1.09305	−0.546524	−	0.837443i	$-0.684049\pi$
$149$	−13.3424	−1.09305	−0.546524	+	0.837443i	$0.684049\pi$
$150$	0	0
$151$	22.1019	1.79863	0.899315	−	0.437302i	$-0.144066\pi$
$151$	22.1019	1.79863	0.899315	+	0.437302i	$0.144066\pi$
$152$	14.7970	1.20020
$153$	0	0
$154$	3.07528	0.247813
$155$	−5.83820	−0.468936
$156$	0	0
$157$	−9.85879	−0.786817	−0.393409	−	0.919364i	$-0.628704\pi$
$157$	−9.85879	−0.786817	−0.393409	+	0.919364i	$0.628704\pi$
$158$	11.0555	0.879531
$159$	0	0
$160$	21.9081	1.73199
$161$	15.6218	1.23117
$162$	0	0
$163$	−11.6798	−0.914830	−0.457415	−	0.889253i	$-0.651225\pi$
$163$	−11.6798	−0.914830	−0.457415	+	0.889253i	$0.651225\pi$
$164$	−1.54575	−0.120703
$165$	0	0
$166$	5.77914	0.448549
$167$	−2.77827	−0.214989	−0.107494	−	0.994206i	$-0.534283\pi$
$167$	−2.77827	−0.214989	−0.107494	+	0.994206i	$0.534283\pi$
$168$	0	0
$169$	−12.4076	−0.954428
$170$	9.84868	0.755360
$171$	0	0
$172$	5.71194	0.435531
$173$	12.5194	0.951835	0.475918	−	0.879490i	$-0.342116\pi$
$173$	12.5194	0.951835	0.475918	+	0.879490i	$0.342116\pi$
$174$	0	0
$175$	41.5089	3.13778
$176$	−1.39726	−0.105323
$177$	0	0
$178$	1.98084	0.148470
$179$	−11.8950	−0.889076	−0.444538	−	0.895760i	$-0.646632\pi$
$179$	−11.8950	−0.889076	−0.444538	+	0.895760i	$0.646632\pi$
$180$	0	0
$181$	−2.77705	−0.206416	−0.103208	−	0.994660i	$-0.532911\pi$
$181$	−2.77705	−0.206416	−0.103208	+	0.994660i	$0.532911\pi$
$182$	2.36704	0.175457
$183$	0	0
$184$	8.43906	0.622136
$185$	−38.9894	−2.86656
$186$	0	0
$187$	−3.80128	−0.277977
$188$	−13.4316	−0.979600
$189$	0	0
$190$	15.8327	1.14863
$191$	10.4918	0.759157	0.379578	−	0.925160i	$-0.376069\pi$
$191$	10.4918	0.759157	0.379578	+	0.925160i	$0.376069\pi$
$192$	0	0
$193$	16.1834	1.16491	0.582454	−	0.812864i	$-0.302093\pi$
$193$	16.1834	1.16491	0.582454	+	0.812864i	$0.302093\pi$
$194$	−9.57315	−0.687312
$195$	0	0
$196$	−20.0203	−1.43002
$197$	−5.10016	−0.363372	−0.181686	−	0.983357i	$-0.558155\pi$
$197$	−5.10016	−0.363372	−0.181686	+	0.983357i	$0.558155\pi$
$198$	0	0
$199$	3.56997	0.253069	0.126534	−	0.991962i	$-0.459615\pi$
$199$	3.56997	0.253069	0.126534	+	0.991962i	$0.459615\pi$
$200$	22.4235	1.58558
$201$	0	0
$202$	0.697093	0.0490473
$203$	−20.2094	−1.41842
$204$	0	0
$205$	−3.81698	−0.266590
$206$	8.66242	0.603540
$207$	0	0
$208$	−1.07547	−0.0745706
$209$	−6.11093	−0.422702
$210$	0	0
$211$	−15.7416	−1.08370	−0.541848	−	0.840476i	$-0.682275\pi$
$211$	−15.7416	−1.08370	−0.541848	+	0.840476i	$0.682275\pi$
$212$	5.13603	0.352744
$213$	0	0
$214$	−8.77659	−0.599955
$215$	14.1047	0.961934
$216$	0	0
$217$	6.92972	0.470420
$218$	12.5590	0.850604
$219$	0	0
$220$	−5.77505	−0.389354
$221$	−2.92584	−0.196813
$222$	0	0
$223$	15.6928	1.05087	0.525435	−	0.850834i	$-0.323903\pi$
$223$	15.6928	1.05087	0.525435	+	0.850834i	$0.323903\pi$
$224$	−26.0041	−1.73747
$225$	0	0
$226$	−9.34206	−0.621425
$227$	4.05098	0.268873	0.134436	−	0.990922i	$-0.457078\pi$
$227$	4.05098	0.268873	0.134436	+	0.990922i	$0.457078\pi$
$228$	0	0
$229$	7.07401	0.467463	0.233732	−	0.972301i	$-0.424906\pi$
$229$	7.07401	0.467463	0.233732	+	0.972301i	$0.424906\pi$
$230$	9.02975	0.595404
$231$	0	0
$232$	−10.9173	−0.716758
$233$	5.99348	0.392646	0.196323	−	0.980539i	$-0.437100\pi$
$233$	5.99348	0.392646	0.196323	+	0.980539i	$0.437100\pi$
$234$	0	0
$235$	−33.1672	−2.16359
$236$	15.1349	0.985198
$237$	0	0
$238$	−11.6900	−0.757750
$239$	6.10444	0.394864	0.197432	−	0.980317i	$-0.436740\pi$
$239$	6.10444	0.394864	0.197432	+	0.980317i	$0.436740\pi$
$240$	0	0
$241$	3.66176	0.235875	0.117937	−	0.993021i	$-0.462372\pi$
$241$	3.66176	0.235875	0.117937	+	0.993021i	$0.462372\pi$
$242$	−0.686089	−0.0441035
$243$	0	0
$244$	1.52928	0.0979022
$245$	−49.4370	−3.15842
$246$	0	0
$247$	−4.70357	−0.299281
$248$	3.74350	0.237713
$249$	0	0
$250$	11.0386	0.698142
$251$	28.4222	1.79399	0.896996	−	0.442039i	$-0.145745\pi$
$251$	28.4222	1.79399	0.896996	+	0.442039i	$0.145745\pi$
$252$	0	0
$253$	−3.48519	−0.219112
$254$	5.64516	0.354209
$255$	0	0
$256$	−9.77403	−0.610877
$257$	−14.7040	−0.917210	−0.458605	−	0.888640i	$-0.651651\pi$
$257$	−14.7040	−0.917210	−0.458605	+	0.888640i	$0.651651\pi$
$258$	0	0
$259$	46.2789	2.87563
$260$	−4.44505	−0.275670
$261$	0	0
$262$	−10.7525	−0.664289
$263$	13.3143	0.820996	0.410498	−	0.911862i	$-0.365355\pi$
$263$	13.3143	0.820996	0.410498	+	0.911862i	$0.365355\pi$
$264$	0	0
$265$	12.6826	0.779085
$266$	−18.7928	−1.15226
$267$	0	0
$268$	12.0488	0.735999
$269$	23.1975	1.41438	0.707190	−	0.707024i	$-0.249963\pi$
$269$	23.1975	1.41438	0.707190	+	0.707024i	$0.249963\pi$
$270$	0	0
$271$	−21.4301	−1.30178	−0.650892	−	0.759171i	$-0.725605\pi$
$271$	−21.4301	−1.30178	−0.650892	+	0.759171i	$0.725605\pi$
$272$	5.31139	0.322050
$273$	0	0
$274$	2.17149	0.131184
$275$	−9.26054	−0.558432
$276$	0	0
$277$	11.5072	0.691402	0.345701	−	0.938345i	$-0.387641\pi$
$277$	11.5072	0.691402	0.345701	+	0.938345i	$0.387641\pi$
$278$	−2.44645	−0.146728
$279$	0	0
$280$	−40.9864	−2.44940
$281$	−9.24018	−0.551223	−0.275611	−	0.961269i	$-0.588880\pi$
$281$	−9.24018	−0.551223	−0.275611	+	0.961269i	$0.588880\pi$
$282$	0	0
$283$	25.8120	1.53437	0.767184	−	0.641428i	$-0.221658\pi$
$283$	25.8120	1.53437	0.767184	+	0.641428i	$0.221658\pi$
$284$	−20.3513	−1.20763
$285$	0	0
$286$	−0.528082	−0.0312262
$287$	4.53061	0.267433
$288$	0	0
$289$	−2.55030	−0.150018
$290$	−11.6815	−0.685961
$291$	0	0
$292$	−10.7605	−0.629713
$293$	−12.6913	−0.741436	−0.370718	−	0.928746i	$-0.620888\pi$
$293$	−12.6913	−0.741436	−0.370718	+	0.928746i	$0.620888\pi$
$294$	0	0
$295$	37.3732	2.17595
$296$	25.0003	1.45311
$297$	0	0
$298$	9.15405	0.530280
$299$	−2.68255	−0.155136
$300$	0	0
$301$	−16.7417	−0.964978
$302$	−15.1639	−0.872585
$303$	0	0
$304$	8.53858	0.489721
$305$	3.77631	0.216231
$306$	0	0
$307$	−17.9853	−1.02647	−0.513237	−	0.858247i	$-0.671554\pi$
$307$	−17.9853	−1.02647	−0.513237	+	0.858247i	$0.671554\pi$
$308$	6.85475	0.390586
$309$	0	0
$310$	4.00553	0.227499
$311$	−20.5838	−1.16720	−0.583600	−	0.812041i	$-0.698356\pi$
$311$	−20.5838	−1.16720	−0.583600	+	0.812041i	$0.698356\pi$
$312$	0	0
$313$	−20.1778	−1.14052	−0.570259	−	0.821465i	$-0.693157\pi$
$313$	−20.1778	−1.14052	−0.570259	+	0.821465i	$0.693157\pi$
$314$	6.76401	0.381715
$315$	0	0
$316$	24.6426	1.38625
$317$	15.3821	0.863948	0.431974	−	0.901886i	$-0.357817\pi$
$317$	15.3821	0.863948	0.431974	+	0.901886i	$0.357817\pi$
$318$	0	0
$319$	4.50868	0.252438
$320$	−4.47790	−0.250322
$321$	0	0
$322$	−10.7180	−0.597289
$323$	23.2293	1.29251
$324$	0	0
$325$	−7.12783	−0.395381
$326$	8.01336	0.443819
$327$	0	0
$328$	2.44748	0.135139
$329$	39.3681	2.17044
$330$	0	0
$331$	17.0595	0.937674	0.468837	−	0.883285i	$-0.344673\pi$
$331$	17.0595	0.937674	0.468837	+	0.883285i	$0.344673\pi$
$332$	12.8816	0.706970
$333$	0	0
$334$	1.90614	0.104299
$335$	29.7526	1.62556
$336$	0	0
$337$	−4.88401	−0.266049	−0.133025	−	0.991113i	$-0.542469\pi$
$337$	−4.88401	−0.266049	−0.133025	+	0.991113i	$0.542469\pi$
$338$	8.51270	0.463030
$339$	0	0
$340$	21.9525	1.19054
$341$	−1.54601	−0.0837209
$342$	0	0
$343$	27.3035	1.47425
$344$	−9.04406	−0.487623
$345$	0	0
$346$	−8.58945	−0.461772
$347$	18.7220	1.00505	0.502526	−	0.864562i	$-0.332404\pi$
$347$	18.7220	1.00505	0.502526	+	0.864562i	$0.332404\pi$
$348$	0	0
$349$	−4.28971	−0.229623	−0.114811	−	0.993387i	$-0.536626\pi$
$349$	−4.28971	−0.229623	−0.114811	+	0.993387i	$0.536626\pi$
$350$	−28.4788	−1.52226
$351$	0	0
$352$	5.80145	0.309218
$353$	30.8366	1.64127	0.820634	−	0.571454i	$-0.193620\pi$
$353$	30.8366	1.64127	0.820634	+	0.571454i	$0.193620\pi$
$354$	0	0
$355$	−50.2543	−2.66722
$356$	4.41525	0.234008
$357$	0	0
$358$	8.16105	0.431325
$359$	−17.4702	−0.922040	−0.461020	−	0.887390i	$-0.652516\pi$
$359$	−17.4702	−0.922040	−0.461020	+	0.887390i	$0.652516\pi$
$360$	0	0
$361$	18.3434	0.965444
$362$	1.90530	0.100140
$363$	0	0
$364$	5.27610	0.276543
$365$	−26.5714	−1.39081
$366$	0	0
$367$	32.6736	1.70555	0.852774	−	0.522279i	$-0.174918\pi$
$367$	32.6736	1.70555	0.852774	+	0.522279i	$0.174918\pi$
$368$	4.86974	0.253853
$369$	0	0
$370$	26.7502	1.39068
$371$	−15.0537	−0.781551
$372$	0	0
$373$	15.0278	0.778109	0.389055	−	0.921215i	$-0.372802\pi$
$373$	15.0278	0.778109	0.389055	+	0.921215i	$0.372802\pi$
$374$	2.60802	0.134857
$375$	0	0
$376$	21.2671	1.09676
$377$	3.47033	0.178731
$378$	0	0
$379$	19.3890	0.995947	0.497973	−	0.867192i	$-0.334078\pi$
$379$	19.3890	0.995947	0.497973	+	0.867192i	$0.334078\pi$
$380$	35.2909	1.81038
$381$	0	0
$382$	−7.19828	−0.368296
$383$	29.4718	1.50594	0.752970	−	0.658055i	$-0.228621\pi$
$383$	29.4718	1.50594	0.752970	+	0.658055i	$0.228621\pi$
$384$	0	0
$385$	16.9267	0.862665
$386$	−11.1033	−0.565142
$387$	0	0
$388$	−21.3384	−1.08329
$389$	12.3906	0.628229	0.314114	−	0.949385i	$-0.398292\pi$
$389$	12.3906	0.628229	0.314114	+	0.949385i	$0.398292\pi$
$390$	0	0
$391$	13.2482	0.669990
$392$	31.6994	1.60106
$393$	0	0
$394$	3.49917	0.176286
$395$	60.8509	3.06174
$396$	0	0
$397$	−4.89557	−0.245701	−0.122851	−	0.992425i	$-0.539204\pi$
$397$	−4.89557	−0.245701	−0.122851	+	0.992425i	$0.539204\pi$
$398$	−2.44932	−0.122773
$399$	0	0
$400$	12.9394	0.646971
$401$	−0.312953	−0.0156281	−0.00781407	−	0.999969i	$-0.502487\pi$
$401$	−0.312953	−0.0156281	−0.00781407	+	0.999969i	$0.502487\pi$
$402$	0	0
$403$	−1.18996	−0.0592761
$404$	1.55381	0.0773048
$405$	0	0
$406$	13.8655	0.688132
$407$	−10.3247	−0.511778
$408$	0	0
$409$	2.56948	0.127053	0.0635264	−	0.997980i	$-0.479765\pi$
$409$	2.56948	0.127053	0.0635264	+	0.997980i	$0.479765\pi$
$410$	2.61879	0.129333
$411$	0	0
$412$	19.3084	0.951256
$413$	−44.3605	−2.18284
$414$	0	0
$415$	31.8091	1.56145
$416$	4.46537	0.218933
$417$	0	0
$418$	4.19264	0.205069
$419$	10.7998	0.527604	0.263802	−	0.964577i	$-0.415023\pi$
$419$	10.7998	0.527604	0.263802	+	0.964577i	$0.415023\pi$
$420$	0	0
$421$	25.5815	1.24677	0.623383	−	0.781917i	$-0.285758\pi$
$421$	25.5815	1.24677	0.623383	+	0.781917i	$0.285758\pi$
$422$	10.8001	0.525743
$423$	0	0
$424$	−8.13218	−0.394934
$425$	35.2019	1.70754
$426$	0	0
$427$	−4.48234	−0.216915
$428$	−19.5629	−0.945606
$429$	0	0
$430$	−9.67710	−0.466671
$431$	−24.2472	−1.16795	−0.583973	−	0.811773i	$-0.698503\pi$
$431$	−24.2472	−1.16795	−0.583973	+	0.811773i	$0.698503\pi$
$432$	0	0
$433$	−5.72897	−0.275317	−0.137658	−	0.990480i	$-0.543958\pi$
$433$	−5.72897	−0.275317	−0.137658	+	0.990480i	$0.543958\pi$
$434$	−4.75440	−0.228219
$435$	0	0
$436$	27.9938	1.34066
$437$	21.2978	1.01881
$438$	0	0
$439$	3.68355	0.175806	0.0879031	−	0.996129i	$-0.471983\pi$
$439$	3.68355	0.175806	0.0879031	+	0.996129i	$0.471983\pi$
$440$	9.14398	0.435922
$441$	0	0
$442$	2.00739	0.0954816
$443$	38.8838	1.84743	0.923713	−	0.383086i	$-0.125139\pi$
$443$	38.8838	1.84743	0.923713	+	0.383086i	$0.125139\pi$
$444$	0	0
$445$	10.9027	0.516840
$446$	−10.7667	−0.509817
$447$	0	0
$448$	5.31510	0.251115
$449$	−10.7057	−0.505234	−0.252617	−	0.967566i	$-0.581291\pi$
$449$	−10.7057	−0.505234	−0.252617	+	0.967566i	$0.581291\pi$
$450$	0	0
$451$	−1.01077	−0.0475953
$452$	−20.8233	−0.979446
$453$	0	0
$454$	−2.77933	−0.130441
$455$	13.0285	0.610784
$456$	0	0
$457$	35.1021	1.64201	0.821005	−	0.570922i	$-0.193414\pi$
$457$	35.1021	1.64201	0.821005	+	0.570922i	$0.193414\pi$
$458$	−4.85340	−0.226785
$459$	0	0
$460$	20.1272	0.938434
$461$	−7.65121	−0.356352	−0.178176	−	0.983999i	$-0.557020\pi$
$461$	−7.65121	−0.356352	−0.178176	+	0.983999i	$0.557020\pi$
$462$	0	0
$463$	−32.3318	−1.50259	−0.751293	−	0.659968i	$-0.770570\pi$
$463$	−32.3318	−1.50259	−0.751293	+	0.659968i	$0.770570\pi$
$464$	−6.29982	−0.292462
$465$	0	0
$466$	−4.11206	−0.190488
$467$	−1.52940	−0.0707723	−0.0353862	−	0.999374i	$-0.511266\pi$
$467$	−1.52940	−0.0707723	−0.0353862	+	0.999374i	$0.511266\pi$
$468$	0	0
$469$	−35.3152	−1.63070
$470$	22.7556	1.04964
$471$	0	0
$472$	−23.9640	−1.10303
$473$	3.73505	0.171738
$474$	0	0
$475$	56.5905	2.59655
$476$	−26.0568	−1.19431
$477$	0	0
$478$	−4.18819	−0.191563
$479$	−34.2341	−1.56419	−0.782097	−	0.623157i	$-0.785850\pi$
$479$	−34.2341	−1.56419	−0.782097	+	0.623157i	$0.785850\pi$
$480$	0	0
$481$	−7.94693	−0.362349
$482$	−2.51230	−0.114432
$483$	0	0
$484$	−1.52928	−0.0695128
$485$	−52.6917	−2.39261
$486$	0	0
$487$	35.9633	1.62965	0.814827	−	0.579704i	$-0.196832\pi$
$487$	35.9633	1.62965	0.814827	+	0.579704i	$0.196832\pi$
$488$	−2.42140	−0.109612
$489$	0	0
$490$	33.9182	1.53227
$491$	−23.3680	−1.05459	−0.527293	−	0.849684i	$-0.676793\pi$
$491$	−23.3680	−1.05459	−0.527293	+	0.849684i	$0.676793\pi$
$492$	0	0
$493$	−17.1387	−0.771890
$494$	3.22707	0.145193
$495$	0	0
$496$	2.16018	0.0969948
$497$	59.6499	2.67567
$498$	0	0
$499$	0.906030	0.0405595	0.0202797	−	0.999794i	$-0.493544\pi$
$499$	0.906030	0.0405595	0.0202797	+	0.999794i	$0.493544\pi$
$500$	24.6048	1.10036
$501$	0	0
$502$	−19.5002	−0.870334
$503$	15.4553	0.689116	0.344558	−	0.938765i	$-0.388029\pi$
$503$	15.4553	0.689116	0.344558	+	0.938765i	$0.388029\pi$
$504$	0	0
$505$	3.83688	0.170739
$506$	2.39115	0.106300
$507$	0	0
$508$	12.5830	0.558278
$509$	−35.6136	−1.57855	−0.789273	−	0.614043i	$-0.789542\pi$
$509$	−35.6136	−1.57855	−0.789273	+	0.614043i	$0.789542\pi$
$510$	0	0
$511$	31.5392	1.39521
$512$	−14.8728	−0.657293
$513$	0	0
$514$	10.0883	0.444974
$515$	47.6789	2.10099
$516$	0	0
$517$	−8.78295	−0.386274
$518$	−31.7515	−1.39508
$519$	0	0
$520$	7.03811	0.308641
$521$	16.6516	0.729521	0.364761	−	0.931101i	$-0.381151\pi$
$521$	16.6516	0.729521	0.364761	+	0.931101i	$0.381151\pi$
$522$	0	0
$523$	−0.581619	−0.0254324	−0.0127162	−	0.999919i	$-0.504048\pi$
$523$	−0.581619	−0.0254324	−0.0127162	+	0.999919i	$0.504048\pi$
$524$	−23.9671	−1.04701
$525$	0	0
$526$	−9.13481	−0.398297
$527$	5.87679	0.255997
$528$	0	0
$529$	−10.8534	−0.471888
$530$	−8.70139	−0.377964
$531$	0	0
$532$	−41.8889	−1.81611
$533$	−0.777988	−0.0336984
$534$	0	0
$535$	−48.3073	−2.08851
$536$	−19.0776	−0.824027
$537$	0	0
$538$	−15.9156	−0.686170
$539$	−13.0913	−0.563884
$540$	0	0
$541$	28.5442	1.22721	0.613607	−	0.789612i	$-0.289718\pi$
$541$	28.5442	1.22721	0.613607	+	0.789612i	$0.289718\pi$
$542$	14.7029	0.631545
$543$	0	0
$544$	−22.0529	−0.945511
$545$	69.1262	2.96104
$546$	0	0
$547$	−25.6176	−1.09533	−0.547664	−	0.836698i	$-0.684483\pi$
$547$	−25.6176	−1.09533	−0.547664	+	0.836698i	$0.684483\pi$
$548$	4.84021	0.206763
$549$	0	0
$550$	6.35356	0.270917
$551$	−27.5522	−1.17376
$552$	0	0
$553$	−72.2277	−3.07143
$554$	−7.89499	−0.335426
$555$	0	0
$556$	−5.45309	−0.231263
$557$	10.2215	0.433100	0.216550	−	0.976272i	$-0.430520\pi$
$557$	10.2215	0.433100	0.216550	+	0.976272i	$0.430520\pi$
$558$	0	0
$559$	2.87486	0.121594
$560$	−23.6511	−0.999441
$561$	0	0
$562$	6.33959	0.267419
$563$	−1.25185	−0.0527590	−0.0263795	−	0.999652i	$-0.508398\pi$
$563$	−1.25185	−0.0527590	−0.0263795	+	0.999652i	$0.508398\pi$
$564$	0	0
$565$	−51.4198	−2.16325
$566$	−17.7094	−0.744380
$567$	0	0
$568$	32.2235	1.35207
$569$	−27.7083	−1.16159	−0.580796	−	0.814049i	$-0.697258\pi$
$569$	−27.7083	−1.16159	−0.580796	+	0.814049i	$0.697258\pi$
$570$	0	0
$571$	38.4361	1.60850	0.804251	−	0.594290i	$-0.202567\pi$
$571$	38.4361	1.60850	0.804251	+	0.594290i	$0.202567\pi$
$572$	−1.17709	−0.0492164
$573$	0	0
$574$	−3.10840	−0.129742
$575$	32.2748	1.34595
$576$	0	0
$577$	−40.3318	−1.67904	−0.839518	−	0.543332i	$-0.817163\pi$
$577$	−40.3318	−1.67904	−0.839518	+	0.543332i	$0.817163\pi$
$578$	1.74973	0.0727793
$579$	0	0
$580$	−26.0378	−1.08116
$581$	−37.7561	−1.56639
$582$	0	0
$583$	3.35846	0.139093
$584$	17.0378	0.705029
$585$	0	0
$586$	8.70739	0.359699
$587$	−21.4539	−0.885499	−0.442750	−	0.896645i	$-0.645997\pi$
$587$	−21.4539	−0.885499	−0.442750	+	0.896645i	$0.645997\pi$
$588$	0	0
$589$	9.44752	0.389278
$590$	−25.6414	−1.05564
$591$	0	0
$592$	14.4264	0.592920
$593$	4.78901	0.196661	0.0983305	−	0.995154i	$-0.468650\pi$
$593$	4.78901	0.196661	0.0983305	+	0.995154i	$0.468650\pi$
$594$	0	0
$595$	−64.3431	−2.63781
$596$	20.4042	0.835790
$597$	0	0
$598$	1.84047	0.0752624
$599$	24.9946	1.02125	0.510626	−	0.859803i	$-0.329414\pi$
$599$	24.9946	1.02125	0.510626	+	0.859803i	$0.329414\pi$
$600$	0	0
$601$	11.5085	0.469440	0.234720	−	0.972063i	$-0.424583\pi$
$601$	11.5085	0.469440	0.234720	+	0.972063i	$0.424583\pi$
$602$	11.4863	0.468148
$603$	0	0
$604$	−33.8001	−1.37531
$605$	−3.77631	−0.153529
$606$	0	0
$607$	48.6573	1.97494	0.987470	−	0.157807i	$-0.0504424\pi$
$607$	48.6573	1.97494	0.987470	+	0.157807i	$0.0504424\pi$
$608$	−35.4523	−1.43778
$609$	0	0
$610$	−2.59089	−0.104902
$611$	−6.76023	−0.273490
$612$	0	0
$613$	21.1808	0.855484	0.427742	−	0.903901i	$-0.359309\pi$
$613$	21.1808	0.855484	0.427742	+	0.903901i	$0.359309\pi$
$614$	12.3395	0.497982
$615$	0	0
$616$	−10.8535	−0.437302
$617$	10.2709	0.413489	0.206745	−	0.978395i	$-0.433713\pi$
$617$	10.2709	0.413489	0.206745	+	0.978395i	$0.433713\pi$
$618$	0	0
$619$	7.70550	0.309710	0.154855	−	0.987937i	$-0.450509\pi$
$619$	7.70550	0.309710	0.154855	+	0.987937i	$0.450509\pi$
$620$	8.92825	0.358567
$621$	0	0
$622$	14.1223	0.566253
$623$	−12.9411	−0.518476
$624$	0	0
$625$	14.4550	0.578199
$626$	13.8438	0.553309
$627$	0	0
$628$	15.0769	0.601633
$629$	39.2471	1.56489
$630$	0	0
$631$	−16.0146	−0.637532	−0.318766	−	0.947833i	$-0.603268\pi$
$631$	−16.0146	−0.637532	−0.318766	+	0.947833i	$0.603268\pi$
$632$	−39.0181	−1.55206
$633$	0	0
$634$	−10.5535	−0.419134
$635$	31.0716	1.23304
$636$	0	0
$637$	−10.0764	−0.399241
$638$	−3.09336	−0.122467
$639$	0	0
$640$	−40.7440	−1.61055
$641$	−24.9229	−0.984396	−0.492198	−	0.870483i	$-0.663806\pi$
$641$	−24.9229	−0.984396	−0.492198	+	0.870483i	$0.663806\pi$
$642$	0	0
$643$	11.8125	0.465838	0.232919	−	0.972496i	$-0.425172\pi$
$643$	11.8125	0.465838	0.232919	+	0.972496i	$0.425172\pi$
$644$	−23.8902	−0.941404
$645$	0	0
$646$	−15.9374	−0.627048
$647$	37.8707	1.48885	0.744426	−	0.667706i	$-0.232723\pi$
$647$	37.8707	1.48885	0.744426	+	0.667706i	$0.232723\pi$
$648$	0	0
$649$	9.89674	0.388481
$650$	4.89033	0.191815
$651$	0	0
$652$	17.8616	0.699516
$653$	33.5190	1.31170	0.655850	−	0.754891i	$-0.272310\pi$
$653$	33.5190	1.31170	0.655850	+	0.754891i	$0.272310\pi$
$654$	0	0
$655$	−59.1828	−2.31246
$656$	1.41231	0.0551415
$657$	0	0
$658$	−27.0101	−1.05296
$659$	−27.9658	−1.08939	−0.544696	−	0.838633i	$-0.683355\pi$
$659$	−27.9658	−1.08939	−0.544696	+	0.838633i	$0.683355\pi$
$660$	0	0
$661$	−18.1170	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$661$	−18.1170	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$662$	−11.7043	−0.454902
$663$	0	0
$664$	−20.3962	−0.791527
$665$	−103.438	−4.01115
$666$	0	0
$667$	−15.7136	−0.608434
$668$	4.24875	0.164389
$669$	0	0
$670$	−20.4129	−0.788621
$671$	1.00000	0.0386046
$672$	0	0
$673$	−1.95653	−0.0754186	−0.0377093	−	0.999289i	$-0.512006\pi$
$673$	−1.95653	−0.0754186	−0.0377093	+	0.999289i	$0.512006\pi$
$674$	3.35087	0.129071
$675$	0	0
$676$	18.9747	0.729794
$677$	20.6599	0.794025	0.397013	−	0.917813i	$-0.370047\pi$
$677$	20.6599	0.794025	0.397013	+	0.917813i	$0.370047\pi$
$678$	0	0
$679$	62.5430	2.40018
$680$	−34.7588	−1.33294
$681$	0	0
$682$	1.06070	0.0406162
$683$	12.0060	0.459397	0.229699	−	0.973262i	$-0.426226\pi$
$683$	12.0060	0.459397	0.229699	+	0.973262i	$0.426226\pi$
$684$	0	0
$685$	11.9521	0.456666
$686$	−18.7326	−0.715214
$687$	0	0
$688$	−5.21885	−0.198967
$689$	2.58500	0.0984807
$690$	0	0
$691$	−29.6509	−1.12797	−0.563986	−	0.825784i	$-0.690733\pi$
$691$	−29.6509	−1.12797	−0.563986	+	0.825784i	$0.690733\pi$
$692$	−19.1457	−0.727812
$693$	0	0
$694$	−12.8450	−0.487589
$695$	−13.4655	−0.510777
$696$	0	0
$697$	3.84221	0.145534
$698$	2.94313	0.111399
$699$	0	0
$700$	−63.4788	−2.39927
$701$	2.43910	0.0921235	0.0460618	−	0.998939i	$-0.485333\pi$
$701$	2.43910	0.0921235	0.0460618	+	0.998939i	$0.485333\pi$
$702$	0	0
$703$	63.0937	2.37962
$704$	−1.18579	−0.0446910
$705$	0	0
$706$	−21.1567	−0.796242
$707$	−4.55422	−0.171279
$708$	0	0
$709$	−36.3219	−1.36410	−0.682049	−	0.731307i	$-0.738911\pi$
$709$	−36.3219	−1.36410	−0.682049	+	0.731307i	$0.738911\pi$
$710$	34.4790	1.29397
$711$	0	0
$712$	−6.99093	−0.261996
$713$	5.38813	0.201787
$714$	0	0
$715$	−2.90662	−0.108702
$716$	18.1908	0.679823
$717$	0	0
$718$	11.9861	0.447317
$719$	−3.36651	−0.125550	−0.0627748	−	0.998028i	$-0.519995\pi$
$719$	−3.36651	−0.125550	−0.0627748	+	0.998028i	$0.519995\pi$
$720$	0	0
$721$	−56.5931	−2.10764
$722$	−12.5852	−0.468374
$723$	0	0
$724$	4.24688	0.157834
$725$	−41.7528	−1.55066
$726$	0	0
$727$	8.51027	0.315629	0.157814	−	0.987469i	$-0.449555\pi$
$727$	8.51027	0.315629	0.157814	+	0.987469i	$0.449555\pi$
$728$	−8.35396	−0.309618
$729$	0	0
$730$	18.2304	0.674736
$731$	−14.1980	−0.525130
$732$	0	0
$733$	−14.2509	−0.526368	−0.263184	−	0.964746i	$-0.584773\pi$
$733$	−14.2509	−0.526368	−0.263184	+	0.964746i	$0.584773\pi$
$734$	−22.4170	−0.827427
$735$	0	0
$736$	−20.2192	−0.745289
$737$	7.87874	0.290217
$738$	0	0
$739$	4.37346	0.160880	0.0804402	−	0.996759i	$-0.474367\pi$
$739$	4.37346	0.160880	0.0804402	+	0.996759i	$0.474367\pi$
$740$	59.6258	2.19189
$741$	0	0
$742$	10.3282	0.379161
$743$	4.92865	0.180815	0.0904074	−	0.995905i	$-0.471183\pi$
$743$	4.92865	0.180815	0.0904074	+	0.995905i	$0.471183\pi$
$744$	0	0
$745$	50.3849	1.84596
$746$	−10.3104	−0.377491
$747$	0	0
$748$	5.81322	0.212552
$749$	57.3389	2.09512
$750$	0	0
$751$	−35.7468	−1.30442	−0.652210	−	0.758038i	$-0.726158\pi$
$751$	−35.7468	−1.30442	−0.652210	+	0.758038i	$0.726158\pi$
$752$	12.2721	0.447517
$753$	0	0
$754$	−2.38095	−0.0867093
$755$	−83.4638	−3.03756
$756$	0	0
$757$	18.0088	0.654540	0.327270	−	0.944931i	$-0.393871\pi$
$757$	18.0088	0.654540	0.327270	+	0.944931i	$0.393871\pi$
$758$	−13.3026	−0.483172
$759$	0	0
$760$	−55.8782	−2.02691
$761$	−49.4533	−1.79268	−0.896341	−	0.443366i	$-0.853784\pi$
$761$	−49.4533	−1.79268	−0.896341	+	0.443366i	$0.853784\pi$
$762$	0	0
$763$	−82.0501	−2.97041
$764$	−16.0448	−0.580482
$765$	0	0
$766$	−20.2203	−0.730589
$767$	7.61751	0.275052
$768$	0	0
$769$	36.3593	1.31115	0.655575	−	0.755130i	$-0.272426\pi$
$769$	36.3593	1.31115	0.655575	+	0.755130i	$0.272426\pi$
$770$	−11.6132	−0.418512
$771$	0	0
$772$	−24.7490	−0.890736
$773$	−16.1097	−0.579427	−0.289714	−	0.957113i	$-0.593560\pi$
$773$	−16.1097	−0.579427	−0.289714	+	0.957113i	$0.593560\pi$
$774$	0	0
$775$	14.3168	0.514276
$776$	33.7863	1.21286
$777$	0	0
$778$	−8.50107	−0.304778
$779$	6.17674	0.221305
$780$	0	0
$781$	−13.3078	−0.476190
$782$	−9.08944	−0.325038
$783$	0	0
$784$	18.2921	0.653288
$785$	37.2299	1.32879
$786$	0	0
$787$	15.6275	0.557061	0.278530	−	0.960427i	$-0.410153\pi$
$787$	15.6275	0.557061	0.278530	+	0.960427i	$0.410153\pi$
$788$	7.79959	0.277849
$789$	0	0
$790$	−41.7492	−1.48537
$791$	61.0333	2.17009
$792$	0	0
$793$	0.769699	0.0273328
$794$	3.35880	0.119199
$795$	0	0
$796$	−5.45949	−0.193507
$797$	37.0889	1.31376	0.656879	−	0.753996i	$-0.271876\pi$
$797$	37.0889	1.31376	0.656879	+	0.753996i	$0.271876\pi$
$798$	0	0
$799$	33.3864	1.18113
$800$	−53.7246	−1.89945
$801$	0	0
$802$	0.214714	0.00758181
$803$	−7.03634	−0.248307
$804$	0	0
$805$	−58.9929	−2.07923
$806$	0.816418	0.0287571
$807$	0	0
$808$	−2.46024	−0.0865508
$809$	−24.3691	−0.856771	−0.428386	−	0.903596i	$-0.640918\pi$
$809$	−24.3691	−0.856771	−0.428386	+	0.903596i	$0.640918\pi$
$810$	0	0
$811$	39.6765	1.39323	0.696614	−	0.717446i	$-0.254689\pi$
$811$	39.6765	1.39323	0.696614	+	0.717446i	$0.254689\pi$
$812$	30.9059	1.08458
$813$	0	0
$814$	7.08369	0.248283
$815$	44.1064	1.54498
$816$	0	0
$817$	−22.8246	−0.798532
$818$	−1.76289	−0.0616382
$819$	0	0
$820$	5.83724	0.203845
$821$	−25.3389	−0.884333	−0.442167	−	0.896933i	$-0.645790\pi$
$821$	−25.3389	−0.884333	−0.442167	+	0.896933i	$0.645790\pi$
$822$	0	0
$823$	−10.4823	−0.365389	−0.182695	−	0.983170i	$-0.558482\pi$
$823$	−10.4823	−0.365389	−0.182695	+	0.983170i	$0.558482\pi$
$824$	−30.5721	−1.06503
$825$	0	0
$826$	30.4353	1.05898
$827$	29.9510	1.04150	0.520749	−	0.853710i	$-0.325653\pi$
$827$	29.9510	1.04150	0.520749	+	0.853710i	$0.325653\pi$
$828$	0	0
$829$	−28.6181	−0.993946	−0.496973	−	0.867766i	$-0.665555\pi$
$829$	−28.6181	−0.993946	−0.496973	+	0.867766i	$0.665555\pi$
$830$	−21.8239	−0.757517
$831$	0	0
$832$	−0.912699	−0.0316421
$833$	49.7638	1.72421
$834$	0	0
$835$	10.4916	0.363077
$836$	9.34533	0.323215
$837$	0	0
$838$	−7.40962	−0.255961
$839$	−36.0462	−1.24445	−0.622227	−	0.782837i	$-0.713772\pi$
$839$	−36.0462	−1.24445	−0.622227	+	0.782837i	$0.713772\pi$
$840$	0	0
$841$	−8.67179	−0.299027
$842$	−17.5512	−0.604854
$843$	0	0
$844$	24.0733	0.828638
$845$	46.8549	1.61186
$846$	0	0
$847$	4.48234	0.154015
$848$	−4.69265	−0.161146
$849$	0	0
$850$	−24.1516	−0.828394
$851$	35.9837	1.23350
$852$	0	0
$853$	−42.5109	−1.45554	−0.727772	−	0.685819i	$-0.759444\pi$
$853$	−42.5109	−1.45554	−0.727772	+	0.685819i	$0.759444\pi$
$854$	3.07528	0.105234
$855$	0	0
$856$	30.9750	1.05871
$857$	−8.59466	−0.293588	−0.146794	−	0.989167i	$-0.546895\pi$
$857$	−8.59466	−0.293588	−0.146794	+	0.989167i	$0.546895\pi$
$858$	0	0
$859$	−9.56524	−0.326362	−0.163181	−	0.986596i	$-0.552175\pi$
$859$	−9.56524	−0.326362	−0.163181	+	0.986596i	$0.552175\pi$
$860$	−21.5701	−0.735534
$861$	0	0
$862$	16.6357	0.566616
$863$	−40.9448	−1.39378	−0.696888	−	0.717180i	$-0.745432\pi$
$863$	−40.9448	−1.39378	−0.696888	+	0.717180i	$0.745432\pi$
$864$	0	0
$865$	−47.2773	−1.60748
$866$	3.93058	0.133567
$867$	0	0
$868$	−10.5975	−0.359702
$869$	16.1138	0.546625
$870$	0	0
$871$	6.06426	0.205480
$872$	−44.3243	−1.50101
$873$	0	0
$874$	−14.6122	−0.494264
$875$	−72.1170	−2.43800
$876$	0	0
$877$	35.9784	1.21490	0.607451	−	0.794357i	$-0.292192\pi$
$877$	35.9784	1.21490	0.607451	+	0.794357i	$0.292192\pi$
$878$	−2.52725	−0.0852904
$879$	0	0
$880$	5.27651	0.177871
$881$	−20.3508	−0.685636	−0.342818	−	0.939402i	$-0.611381\pi$
$881$	−20.3508	−0.685636	−0.342818	+	0.939402i	$0.611381\pi$
$882$	0	0
$883$	29.2431	0.984108	0.492054	−	0.870565i	$-0.336246\pi$
$883$	29.2431	0.984108	0.492054	+	0.870565i	$0.336246\pi$
$884$	4.47443	0.150491
$885$	0	0
$886$	−26.6778	−0.896257
$887$	7.26531	0.243945	0.121973	−	0.992533i	$-0.461078\pi$
$887$	7.26531	0.243945	0.121973	+	0.992533i	$0.461078\pi$
$888$	0	0
$889$	−36.8808	−1.23694
$890$	−7.48026	−0.250739
$891$	0	0
$892$	−23.9987	−0.803538
$893$	53.6720	1.79606
$894$	0	0
$895$	44.9193	1.50149
$896$	48.3615	1.61564
$897$	0	0
$898$	7.34509	0.245109
$899$	−6.97044	−0.232477
$900$	0	0
$901$	−12.7664	−0.425311
$902$	0.693478	0.0230903
$903$	0	0
$904$	32.9708	1.09659
$905$	10.4870	0.348599
$906$	0	0
$907$	24.6557	0.818680	0.409340	−	0.912382i	$-0.365759\pi$
$907$	24.6557	0.818680	0.409340	+	0.912382i	$0.365759\pi$
$908$	−6.19509	−0.205591
$909$	0	0
$910$	−8.93870	−0.296315
$911$	−9.27932	−0.307437	−0.153719	−	0.988115i	$-0.549125\pi$
$911$	−9.27932	−0.307437	−0.153719	+	0.988115i	$0.549125\pi$
$912$	0	0
$913$	8.42331	0.278771
$914$	−24.0832	−0.796602
$915$	0	0
$916$	−10.8181	−0.357442
$917$	70.2477	2.31978
$918$	0	0
$919$	−0.138091	−0.00455519	−0.00227759	−	0.999997i	$-0.500725\pi$
$919$	−0.138091	−0.00455519	−0.00227759	+	0.999997i	$0.500725\pi$
$920$	−31.8685	−1.05067
$921$	0	0
$922$	5.24942	0.172880
$923$	−10.2430	−0.337152
$924$	0	0
$925$	95.6126	3.14372
$926$	22.1825	0.728963
$927$	0	0
$928$	26.1569	0.858642
$929$	4.82781	0.158395	0.0791977	−	0.996859i	$-0.474764\pi$
$929$	4.82781	0.158395	0.0791977	+	0.996859i	$0.474764\pi$
$930$	0	0
$931$	80.0002	2.62190
$932$	−9.16571	−0.300233
$933$	0	0
$934$	1.04931	0.0343344
$935$	14.3548	0.469453
$936$	0	0
$937$	−16.4702	−0.538058	−0.269029	−	0.963132i	$-0.586703\pi$
$937$	−16.4702	−0.538058	−0.269029	+	0.963132i	$0.586703\pi$
$938$	24.2294	0.791117
$939$	0	0
$940$	50.7219	1.65437
$941$	−54.5905	−1.77960	−0.889800	−	0.456350i	$-0.849157\pi$
$941$	−54.5905	−1.77960	−0.889800	+	0.456350i	$0.849157\pi$
$942$	0	0
$943$	3.52273	0.114716
$944$	−13.8284	−0.450075
$945$	0	0
$946$	−2.56258	−0.0833166
$947$	−3.35352	−0.108975	−0.0544874	−	0.998514i	$-0.517352\pi$
$947$	−3.35352	−0.108975	−0.0544874	+	0.998514i	$0.517352\pi$
$948$	0	0
$949$	−5.41586	−0.175806
$950$	−38.8262	−1.25969
$951$	0	0
$952$	41.2573	1.33716
$953$	−19.0563	−0.617293	−0.308647	−	0.951177i	$-0.599876\pi$
$953$	−19.0563	−0.617293	−0.308647	+	0.951177i	$0.599876\pi$
$954$	0	0
$955$	−39.6202	−1.28208
$956$	−9.33541	−0.301929
$957$	0	0
$958$	23.4876	0.758851
$959$	−14.1867	−0.458112
$960$	0	0
$961$	−28.6099	−0.922899
$962$	5.45231	0.175789
$963$	0	0
$964$	−5.59987	−0.180360
$965$	−61.1137	−1.96732
$966$	0	0
$967$	−25.4500	−0.818415	−0.409208	−	0.912441i	$-0.634195\pi$
$967$	−25.4500	−0.818415	−0.409208	+	0.912441i	$0.634195\pi$
$968$	2.42140	0.0778268
$969$	0	0
$970$	36.1512	1.16075
$971$	29.6230	0.950648	0.475324	−	0.879811i	$-0.342331\pi$
$971$	29.6230	0.950648	0.475324	+	0.879811i	$0.342331\pi$
$972$	0	0
$973$	15.9831	0.512393
$974$	−24.6741	−0.790608
$975$	0	0
$976$	−1.39726	−0.0447253
$977$	−7.18850	−0.229980	−0.114990	−	0.993367i	$-0.536684\pi$
$977$	−7.18850	−0.229980	−0.114990	+	0.993367i	$0.536684\pi$
$978$	0	0
$979$	2.88714	0.0922734
$980$	75.6031	2.41505
$981$	0	0
$982$	16.0326	0.511620
$983$	39.9478	1.27414	0.637068	−	0.770807i	$-0.280147\pi$
$983$	39.9478	1.27414	0.637068	+	0.770807i	$0.280147\pi$
$984$	0	0
$985$	19.2598	0.613669
$986$	11.7587	0.374474
$987$	0	0
$988$	7.19309	0.228843
$989$	−13.0174	−0.413928
$990$	0	0
$991$	11.8963	0.377897	0.188949	−	0.981987i	$-0.439492\pi$
$991$	11.8963	0.377897	0.188949	+	0.981987i	$0.439492\pi$
$992$	−8.96908	−0.284768
$993$	0	0
$994$	−40.9252	−1.29807
$995$	−13.4813	−0.427387
$996$	0	0
$997$	−6.22191	−0.197050	−0.0985249	−	0.995135i	$-0.531412\pi$
$997$	−6.22191	−0.197050	−0.0985249	+	0.995135i	$0.531412\pi$
$998$	−0.621618	−0.0196770
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.9		19
3.2	odd	2		671.2.a.c.1.11	✓	19
33.32	even	2		7381.2.a.i.1.9		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.11	✓	19	3.2	odd	2
6039.2.a.k.1.9		19	1.1	even	1	trivial
7381.2.a.i.1.9		19	33.32	even	2

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)

\(f(q)\)	\(=\)	\(q-0.686089 q^{2} -1.52928 q^{4} -3.77631 q^{5} +4.48234 q^{7} +2.42140 q^{8} +O(q^{10})\)	\(q-0.686089 q^{2} -1.52928 q^{4} -3.77631 q^{5} +4.48234 q^{7} +2.42140 q^{8} +2.59089 q^{10} -1.00000 q^{11} -0.769699 q^{13} -3.07528 q^{14} +1.39726 q^{16} +3.80128 q^{17} +6.11093 q^{19} +5.77505 q^{20} +0.686089 q^{22} +3.48519 q^{23} +9.26054 q^{25} +0.528082 q^{26} -6.85475 q^{28} -4.50868 q^{29} +1.54601 q^{31} -5.80145 q^{32} -2.60802 q^{34} -16.9267 q^{35} +10.3247 q^{37} -4.19264 q^{38} -9.14398 q^{40} +1.01077 q^{41} -3.73505 q^{43} +1.52928 q^{44} -2.39115 q^{46} +8.78295 q^{47} +13.0913 q^{49} -6.35356 q^{50} +1.17709 q^{52} -3.35846 q^{53} +3.77631 q^{55} +10.8535 q^{56} +3.09336 q^{58} -9.89674 q^{59} -1.00000 q^{61} -1.06070 q^{62} +1.18579 q^{64} +2.90662 q^{65} -7.87874 q^{67} -5.81322 q^{68} +11.6132 q^{70} +13.3078 q^{71} +7.03634 q^{73} -7.08369 q^{74} -9.34533 q^{76} -4.48234 q^{77} -16.1138 q^{79} -5.27651 q^{80} -0.693478 q^{82} -8.42331 q^{83} -14.3548 q^{85} +2.56258 q^{86} -2.42140 q^{88} -2.88714 q^{89} -3.45005 q^{91} -5.32984 q^{92} -6.02589 q^{94} -23.0768 q^{95} +13.9532 q^{97} -8.98183 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more