LMFDB - Embedded newform 6039.2.a.j.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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} + \cdots - 1 $ x^14 - x^13 - 21x^12 + 20x^11 + 167x^10 - 148x^9 - 627x^8 + 497x^7 + 1123x^6 - 745x^5 - 802x^4 + 386x^3 + 74x^2 - 15x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.231279$ 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)

$f(q)$	$=$	$q-0.231279 q^{2} -1.94651 q^{4} -3.70694 q^{5} -0.911108 q^{7} +0.912746 q^{8} +O(q^{10})$	\(q-0.231279 q^{2} -1.94651 q^{4} -3.70694 q^{5} -0.911108 q^{7} +0.912746 q^{8} +0.857338 q^{10} +1.00000 q^{11} -1.28993 q^{13} +0.210720 q^{14} +3.68192 q^{16} +7.36730 q^{17} +5.91624 q^{19} +7.21559 q^{20} -0.231279 q^{22} -6.40611 q^{23} +8.74139 q^{25} +0.298334 q^{26} +1.77348 q^{28} -9.58456 q^{29} -8.65847 q^{31} -2.67705 q^{32} -1.70390 q^{34} +3.37742 q^{35} -8.74406 q^{37} -1.36830 q^{38} -3.38349 q^{40} -6.54649 q^{41} +9.59278 q^{43} -1.94651 q^{44} +1.48160 q^{46} -5.82694 q^{47} -6.16988 q^{49} -2.02170 q^{50} +2.51086 q^{52} +5.01034 q^{53} -3.70694 q^{55} -0.831610 q^{56} +2.21671 q^{58} -1.55369 q^{59} +1.00000 q^{61} +2.00252 q^{62} -6.74470 q^{64} +4.78168 q^{65} +12.7209 q^{67} -14.3405 q^{68} -0.781128 q^{70} +0.506871 q^{71} -3.71011 q^{73} +2.02232 q^{74} -11.5160 q^{76} -0.911108 q^{77} -5.28303 q^{79} -13.6486 q^{80} +1.51407 q^{82} -7.43641 q^{83} -27.3101 q^{85} -2.21861 q^{86} +0.912746 q^{88} -12.7615 q^{89} +1.17526 q^{91} +12.4695 q^{92} +1.34765 q^{94} -21.9311 q^{95} +19.0085 q^{97} +1.42697 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * 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 $

$2$	−0.231279	−0.163539	−0.0817696	−	0.996651i	$-0.526057\pi$
$2$	−0.231279	−0.163539	−0.0817696	+	0.996651i	$0.526057\pi$
$3$	0	0
$3$	0	0
$4$	−1.94651	−0.973255
$5$	−3.70694	−1.65779	−0.828896	−	0.559402i	$-0.811031\pi$
$5$	−3.70694	−1.65779	−0.828896	+	0.559402i	$0.811031\pi$
$6$	0	0
$7$	−0.911108	−0.344366	−0.172183	−	0.985065i	$-0.555082\pi$
$7$	−0.911108	−0.344366	−0.172183	+	0.985065i	$0.555082\pi$
$8$	0.912746	0.322705
$9$	0	0
$10$	0.857338	0.271114
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.28993	−0.357761	−0.178881	−	0.983871i	$-0.557248\pi$
$13$	−1.28993	−0.357761	−0.178881	+	0.983871i	$0.557248\pi$
$14$	0.210720	0.0563174
$15$	0	0
$16$	3.68192	0.920480
$17$	7.36730	1.78683	0.893416	−	0.449230i	$-0.148301\pi$
$17$	7.36730	1.78683	0.893416	+	0.449230i	$0.148301\pi$
$18$	0	0
$19$	5.91624	1.35728	0.678640	−	0.734472i	$-0.262570\pi$
$19$	5.91624	1.35728	0.678640	+	0.734472i	$0.262570\pi$
$20$	7.21559	1.61346
$21$	0	0
$22$	−0.231279	−0.0493089
$23$	−6.40611	−1.33577	−0.667883	−	0.744267i	$-0.732799\pi$
$23$	−6.40611	−1.33577	−0.667883	+	0.744267i	$0.732799\pi$
$24$	0	0
$25$	8.74139	1.74828
$26$	0.298334	0.0585080
$27$	0	0
$28$	1.77348	0.335156
$29$	−9.58456	−1.77981	−0.889904	−	0.456147i	$-0.849229\pi$
$29$	−9.58456	−1.77981	−0.889904	+	0.456147i	$0.849229\pi$
$30$	0	0
$31$	−8.65847	−1.55511	−0.777553	−	0.628817i	$-0.783539\pi$
$31$	−8.65847	−1.55511	−0.777553	+	0.628817i	$0.783539\pi$
$32$	−2.67705	−0.473239
$33$	0	0
$34$	−1.70390	−0.292217
$35$	3.37742	0.570888
$36$	0	0
$37$	−8.74406	−1.43751	−0.718757	−	0.695261i	$-0.755289\pi$
$37$	−8.74406	−1.43751	−0.718757	+	0.695261i	$0.755289\pi$
$38$	−1.36830	−0.221968
$39$	0	0
$40$	−3.38349	−0.534977
$41$	−6.54649	−1.02239	−0.511195	−	0.859465i	$-0.670797\pi$
$41$	−6.54649	−1.02239	−0.511195	+	0.859465i	$0.670797\pi$
$42$	0	0
$43$	9.59278	1.46289	0.731443	−	0.681903i	$-0.238847\pi$
$43$	9.59278	1.46289	0.731443	+	0.681903i	$0.238847\pi$
$44$	−1.94651	−0.293447
$45$	0	0
$46$	1.48160	0.218450
$47$	−5.82694	−0.849946	−0.424973	−	0.905206i	$-0.639716\pi$
$47$	−5.82694	−0.849946	−0.424973	+	0.905206i	$0.639716\pi$
$48$	0	0
$49$	−6.16988	−0.881412
$50$	−2.02170	−0.285912
$51$	0	0
$52$	2.51086	0.348193
$53$	5.01034	0.688223	0.344111	−	0.938929i	$-0.388180\pi$
$53$	5.01034	0.688223	0.344111	+	0.938929i	$0.388180\pi$
$54$	0	0
$55$	−3.70694	−0.499843
$56$	−0.831610	−0.111129
$57$	0	0
$58$	2.21671	0.291069
$59$	−1.55369	−0.202273	−0.101137	−	0.994873i	$-0.532248\pi$
$59$	−1.55369	−0.202273	−0.101137	+	0.994873i	$0.532248\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	2.00252	0.254321
$63$	0	0
$64$	−6.74470	−0.843087
$65$	4.78168	0.593094
$66$	0	0
$67$	12.7209	1.55410	0.777052	−	0.629436i	$-0.216714\pi$
$67$	12.7209	1.55410	0.777052	+	0.629436i	$0.216714\pi$
$68$	−14.3405	−1.73904
$69$	0	0
$70$	−0.781128	−0.0933626
$71$	0.506871	0.0601545	0.0300773	−	0.999548i	$-0.490425\pi$
$71$	0.506871	0.0601545	0.0300773	+	0.999548i	$0.490425\pi$
$72$	0	0
$73$	−3.71011	−0.434236	−0.217118	−	0.976145i	$-0.569666\pi$
$73$	−3.71011	−0.434236	−0.217118	+	0.976145i	$0.569666\pi$
$74$	2.02232	0.235090
$75$	0	0
$76$	−11.5160	−1.32098
$77$	−0.911108	−0.103830
$78$	0	0
$79$	−5.28303	−0.594387	−0.297194	−	0.954817i	$-0.596051\pi$
$79$	−5.28303	−0.594387	−0.297194	+	0.954817i	$0.596051\pi$
$80$	−13.6486	−1.52597
$81$	0	0
$82$	1.51407	0.167201
$83$	−7.43641	−0.816252	−0.408126	−	0.912926i	$-0.633818\pi$
$83$	−7.43641	−0.816252	−0.408126	+	0.912926i	$0.633818\pi$
$84$	0	0
$85$	−27.3101	−2.96220
$86$	−2.21861	−0.239239
$87$	0	0
$88$	0.912746	0.0972991
$89$	−12.7615	−1.35272	−0.676360	−	0.736571i	$-0.736444\pi$
$89$	−12.7615	−1.35272	−0.676360	+	0.736571i	$0.736444\pi$
$90$	0	0
$91$	1.17526	0.123201
$92$	12.4695	1.30004
$93$	0	0
$94$	1.34765	0.139000
$95$	−21.9311	−2.25009
$96$	0	0
$97$	19.0085	1.93002	0.965012	−	0.262207i	$-0.0844504\pi$
$97$	19.0085	1.93002	0.965012	+	0.262207i	$0.0844504\pi$
$98$	1.42697	0.144145
$99$	0	0
$100$	−17.0152	−1.70152
$101$	5.60028	0.557249	0.278624	−	0.960400i	$-0.410122\pi$
$101$	5.60028	0.557249	0.278624	+	0.960400i	$0.410122\pi$
$102$	0	0
$103$	16.0507	1.58153	0.790763	−	0.612122i	$-0.209684\pi$
$103$	16.0507	1.58153	0.790763	+	0.612122i	$0.209684\pi$
$104$	−1.17738	−0.115451
$105$	0	0
$106$	−1.15879	−0.112551
$107$	−16.1333	−1.55966	−0.779830	−	0.625991i	$-0.784695\pi$
$107$	−16.1333	−1.55966	−0.779830	+	0.625991i	$0.784695\pi$
$108$	0	0
$109$	−15.2826	−1.46381	−0.731905	−	0.681407i	$-0.761369\pi$
$109$	−15.2826	−1.46381	−0.731905	+	0.681407i	$0.761369\pi$
$110$	0.857338	0.0817440
$111$	0	0
$112$	−3.35463	−0.316982
$113$	−13.4720	−1.26734	−0.633668	−	0.773605i	$-0.718451\pi$
$113$	−13.4720	−1.26734	−0.633668	+	0.773605i	$0.718451\pi$
$114$	0	0
$115$	23.7470	2.21442
$116$	18.6564	1.73221
$117$	0	0
$118$	0.359337	0.0330796
$119$	−6.71240	−0.615325
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.231279	−0.0209391
$123$	0	0
$124$	16.8538	1.51352
$125$	−13.8691	−1.24049
$126$	0	0
$127$	14.9118	1.32321	0.661605	−	0.749852i	$-0.269875\pi$
$127$	14.9118	1.32321	0.661605	+	0.749852i	$0.269875\pi$
$128$	6.91400	0.611117
$129$	0	0
$130$	−1.10590	−0.0969942
$131$	−1.52367	−0.133123	−0.0665617	−	0.997782i	$-0.521203\pi$
$131$	−1.52367	−0.133123	−0.0665617	+	0.997782i	$0.521203\pi$
$132$	0	0
$133$	−5.39033	−0.467401
$134$	−2.94208	−0.254157
$135$	0	0
$136$	6.72448	0.576619
$137$	−3.70337	−0.316400	−0.158200	−	0.987407i	$-0.550569\pi$
$137$	−3.70337	−0.316400	−0.158200	+	0.987407i	$0.550569\pi$
$138$	0	0
$139$	6.10661	0.517956	0.258978	−	0.965883i	$-0.416614\pi$
$139$	6.10661	0.517956	0.258978	+	0.965883i	$0.416614\pi$
$140$	−6.57418	−0.555620
$141$	0	0
$142$	−0.117229	−0.00983762
$143$	−1.28993	−0.107869
$144$	0	0
$145$	35.5294	2.95055
$146$	0.858073	0.0710146
$147$	0	0
$148$	17.0204	1.39907
$149$	13.5246	1.10797	0.553987	−	0.832525i	$-0.313106\pi$
$149$	13.5246	1.10797	0.553987	+	0.832525i	$0.313106\pi$
$150$	0	0
$151$	−1.09346	−0.0889849	−0.0444924	−	0.999010i	$-0.514167\pi$
$151$	−1.09346	−0.0889849	−0.0444924	+	0.999010i	$0.514167\pi$
$152$	5.40003	0.438000
$153$	0	0
$154$	0.210720	0.0169803
$155$	32.0964	2.57804
$156$	0	0
$157$	14.4608	1.15410	0.577049	−	0.816709i	$-0.304204\pi$
$157$	14.4608	1.15410	0.577049	+	0.816709i	$0.304204\pi$
$158$	1.22186	0.0972057
$159$	0	0
$160$	9.92364	0.784533
$161$	5.83665	0.459993
$162$	0	0
$163$	1.40734	0.110231	0.0551155	−	0.998480i	$-0.482447\pi$
$163$	1.40734	0.110231	0.0551155	+	0.998480i	$0.482447\pi$
$164$	12.7428	0.995046
$165$	0	0
$166$	1.71989	0.133489
$167$	14.0105	1.08416	0.542082	−	0.840326i	$-0.317636\pi$
$167$	14.0105	1.08416	0.542082	+	0.840326i	$0.317636\pi$
$168$	0	0
$169$	−11.3361	−0.872007
$170$	6.31627	0.484436
$171$	0	0
$172$	−18.6724	−1.42376
$173$	−2.67772	−0.203584	−0.101792	−	0.994806i	$-0.532458\pi$
$173$	−2.67772	−0.203584	−0.101792	+	0.994806i	$0.532458\pi$
$174$	0	0
$175$	−7.96434	−0.602048
$176$	3.68192	0.277535
$177$	0	0
$178$	2.95148	0.221223
$179$	−2.14059	−0.159995	−0.0799977	−	0.996795i	$-0.525491\pi$
$179$	−2.14059	−0.159995	−0.0799977	+	0.996795i	$0.525491\pi$
$180$	0	0
$181$	10.4522	0.776903	0.388451	−	0.921469i	$-0.373010\pi$
$181$	10.4522	0.776903	0.388451	+	0.921469i	$0.373010\pi$
$182$	−0.271814	−0.0201482
$183$	0	0
$184$	−5.84715	−0.431058
$185$	32.4137	2.38310
$186$	0	0
$187$	7.36730	0.538750
$188$	11.3422	0.827214
$189$	0	0
$190$	5.07222	0.367978
$191$	−6.90631	−0.499723	−0.249862	−	0.968282i	$-0.580385\pi$
$191$	−6.90631	−0.499723	−0.249862	+	0.968282i	$0.580385\pi$
$192$	0	0
$193$	13.5215	0.973300	0.486650	−	0.873597i	$-0.338219\pi$
$193$	13.5215	0.973300	0.486650	+	0.873597i	$0.338219\pi$
$194$	−4.39628	−0.315634
$195$	0	0
$196$	12.0097	0.857838
$197$	12.6509	0.901342	0.450671	−	0.892690i	$-0.351185\pi$
$197$	12.6509	0.901342	0.450671	+	0.892690i	$0.351185\pi$
$198$	0	0
$199$	3.63188	0.257457	0.128728	−	0.991680i	$-0.458910\pi$
$199$	3.63188	0.257457	0.128728	+	0.991680i	$0.458910\pi$
$200$	7.97867	0.564177
$201$	0	0
$202$	−1.29523	−0.0911320
$203$	8.73257	0.612906
$204$	0	0
$205$	24.2674	1.69491
$206$	−3.71220	−0.258642
$207$	0	0
$208$	−4.74941	−0.329312
$209$	5.91624	0.409235
$210$	0	0
$211$	−18.3470	−1.26306	−0.631530	−	0.775352i	$-0.717573\pi$
$211$	−18.3470	−1.26306	−0.631530	+	0.775352i	$0.717573\pi$
$212$	−9.75267	−0.669816
$213$	0	0
$214$	3.73129	0.255066
$215$	−35.5598	−2.42516
$216$	0	0
$217$	7.88880	0.535526
$218$	3.53456	0.239390
$219$	0	0
$220$	7.21559	0.486475
$221$	−9.50328	−0.639260
$222$	0	0
$223$	6.99961	0.468729	0.234364	−	0.972149i	$-0.424699\pi$
$223$	6.99961	0.468729	0.234364	+	0.972149i	$0.424699\pi$
$224$	2.43908	0.162968
$225$	0	0
$226$	3.11579	0.207259
$227$	−6.85426	−0.454933	−0.227467	−	0.973786i	$-0.573044\pi$
$227$	−6.85426	−0.454933	−0.227467	+	0.973786i	$0.573044\pi$
$228$	0	0
$229$	−2.15050	−0.142109	−0.0710544	−	0.997472i	$-0.522636\pi$
$229$	−2.15050	−0.142109	−0.0710544	+	0.997472i	$0.522636\pi$
$230$	−5.49220	−0.362145
$231$	0	0
$232$	−8.74827	−0.574352
$233$	0.0878610	0.00575597	0.00287798	−	0.999996i	$-0.499084\pi$
$233$	0.0878610	0.00575597	0.00287798	+	0.999996i	$0.499084\pi$
$234$	0	0
$235$	21.6001	1.40903
$236$	3.02428	0.196864
$237$	0	0
$238$	1.55244	0.100630
$239$	1.34617	0.0870763	0.0435382	−	0.999052i	$-0.486137\pi$
$239$	1.34617	0.0870763	0.0435382	+	0.999052i	$0.486137\pi$
$240$	0	0
$241$	4.54006	0.292451	0.146225	−	0.989251i	$-0.453288\pi$
$241$	4.54006	0.292451	0.146225	+	0.989251i	$0.453288\pi$
$242$	−0.231279	−0.0148672
$243$	0	0
$244$	−1.94651	−0.124613
$245$	22.8714	1.46120
$246$	0	0
$247$	−7.63152	−0.485582
$248$	−7.90298	−0.501840
$249$	0	0
$250$	3.20763	0.202869
$251$	5.83748	0.368458	0.184229	−	0.982883i	$-0.441021\pi$
$251$	5.83748	0.368458	0.184229	+	0.982883i	$0.441021\pi$
$252$	0	0
$253$	−6.40611	−0.402748
$254$	−3.44880	−0.216397
$255$	0	0
$256$	11.8903	0.743145
$257$	−19.2484	−1.20068	−0.600341	−	0.799744i	$-0.704968\pi$
$257$	−19.2484	−1.20068	−0.600341	+	0.799744i	$0.704968\pi$
$258$	0	0
$259$	7.96678	0.495032
$260$	−9.30759	−0.577232
$261$	0	0
$262$	0.352393	0.0217709
$263$	12.4662	0.768698	0.384349	−	0.923188i	$-0.374426\pi$
$263$	12.4662	0.768698	0.384349	+	0.923188i	$0.374426\pi$
$264$	0	0
$265$	−18.5730	−1.14093
$266$	1.24667	0.0764384
$267$	0	0
$268$	−24.7614	−1.51254
$269$	10.9441	0.667276	0.333638	−	0.942701i	$-0.391724\pi$
$269$	10.9441	0.667276	0.333638	+	0.942701i	$0.391724\pi$
$270$	0	0
$271$	−17.9716	−1.09169	−0.545847	−	0.837885i	$-0.683792\pi$
$271$	−17.9716	−1.09169	−0.545847	+	0.837885i	$0.683792\pi$
$272$	27.1258	1.64474
$273$	0	0
$274$	0.856514	0.0517439
$275$	8.74139	0.527125
$276$	0	0
$277$	−5.74306	−0.345067	−0.172534	−	0.985004i	$-0.555195\pi$
$277$	−5.74306	−0.345067	−0.172534	+	0.985004i	$0.555195\pi$
$278$	−1.41233	−0.0847061
$279$	0	0
$280$	3.08273	0.184228
$281$	31.3802	1.87199	0.935993	−	0.352019i	$-0.114505\pi$
$281$	31.3802	1.87199	0.935993	+	0.352019i	$0.114505\pi$
$282$	0	0
$283$	−22.4497	−1.33450	−0.667248	−	0.744835i	$-0.732528\pi$
$283$	−22.4497	−1.33450	−0.667248	+	0.744835i	$0.732528\pi$
$284$	−0.986629	−0.0585457
$285$	0	0
$286$	0.298334	0.0176408
$287$	5.96456	0.352077
$288$	0	0
$289$	37.2771	2.19277
$290$	−8.21721	−0.482531
$291$	0	0
$292$	7.22177	0.422622
$293$	15.8824	0.927861	0.463931	−	0.885872i	$-0.346439\pi$
$293$	15.8824	0.927861	0.463931	+	0.885872i	$0.346439\pi$
$294$	0	0
$295$	5.75944	0.335327
$296$	−7.98111	−0.463893
$297$	0	0
$298$	−3.12795	−0.181197
$299$	8.26341	0.477885
$300$	0	0
$301$	−8.74006	−0.503769
$302$	0.252896	0.0145525
$303$	0	0
$304$	21.7831	1.24935
$305$	−3.70694	−0.212259
$306$	0	0
$307$	13.2053	0.753665	0.376832	−	0.926281i	$-0.377013\pi$
$307$	13.2053	0.753665	0.376832	+	0.926281i	$0.377013\pi$
$308$	1.77348	0.101053
$309$	0	0
$310$	−7.42323	−0.421611
$311$	−7.86121	−0.445768	−0.222884	−	0.974845i	$-0.571547\pi$
$311$	−7.86121	−0.445768	−0.222884	+	0.974845i	$0.571547\pi$
$312$	0	0
$313$	−17.6385	−0.996988	−0.498494	−	0.866893i	$-0.666113\pi$
$313$	−17.6385	−0.996988	−0.498494	+	0.866893i	$0.666113\pi$
$314$	−3.34449	−0.188740
$315$	0	0
$316$	10.2835	0.578490
$317$	−10.5734	−0.593859	−0.296930	−	0.954899i	$-0.595963\pi$
$317$	−10.5734	−0.593859	−0.296930	+	0.954899i	$0.595963\pi$
$318$	0	0
$319$	−9.58456	−0.536632
$320$	25.0022	1.39766
$321$	0	0
$322$	−1.34990	−0.0752268
$323$	43.5867	2.42523
$324$	0	0
$325$	−11.2758	−0.625466
$326$	−0.325488	−0.0180271
$327$	0	0
$328$	−5.97528	−0.329930
$329$	5.30897	0.292693
$330$	0	0
$331$	−25.7221	−1.41381	−0.706906	−	0.707307i	$-0.749910\pi$
$331$	−25.7221	−1.41381	−0.706906	+	0.707307i	$0.749910\pi$
$332$	14.4750	0.794421
$333$	0	0
$334$	−3.24034	−0.177303
$335$	−47.1556	−2.57638
$336$	0	0
$337$	−22.1471	−1.20643	−0.603216	−	0.797578i	$-0.706114\pi$
$337$	−22.1471	−1.20643	−0.603216	+	0.797578i	$0.706114\pi$
$338$	2.62180	0.142607
$339$	0	0
$340$	53.1594	2.88297
$341$	−8.65847	−0.468882
$342$	0	0
$343$	11.9992	0.647895
$344$	8.75578	0.472080
$345$	0	0
$346$	0.619303	0.0332939
$347$	−16.8724	−0.905757	−0.452879	−	0.891572i	$-0.649603\pi$
$347$	−16.8724	−0.905757	−0.452879	+	0.891572i	$0.649603\pi$
$348$	0	0
$349$	−36.7014	−1.96458	−0.982291	−	0.187362i	$-0.940006\pi$
$349$	−36.7014	−1.96458	−0.982291	+	0.187362i	$0.940006\pi$
$350$	1.84199	0.0984584
$351$	0	0
$352$	−2.67705	−0.142687
$353$	−0.505931	−0.0269280	−0.0134640	−	0.999909i	$-0.504286\pi$
$353$	−0.505931	−0.0269280	−0.0134640	+	0.999909i	$0.504286\pi$
$354$	0	0
$355$	−1.87894	−0.0997237
$356$	24.8405	1.31654
$357$	0	0
$358$	0.495075	0.0261655
$359$	−6.62258	−0.349527	−0.174763	−	0.984610i	$-0.555916\pi$
$359$	−6.62258	−0.349527	−0.174763	+	0.984610i	$0.555916\pi$
$360$	0	0
$361$	16.0019	0.842206
$362$	−2.41737	−0.127054
$363$	0	0
$364$	−2.28766	−0.119906
$365$	13.7532	0.719873
$366$	0	0
$367$	19.2989	1.00739	0.503696	−	0.863881i	$-0.331973\pi$
$367$	19.2989	1.00739	0.503696	+	0.863881i	$0.331973\pi$
$368$	−23.5868	−1.22955
$369$	0	0
$370$	−7.49662	−0.389731
$371$	−4.56496	−0.237001
$372$	0	0
$373$	−33.3340	−1.72597	−0.862985	−	0.505229i	$-0.831408\pi$
$373$	−33.3340	−1.72597	−0.862985	+	0.505229i	$0.831408\pi$
$374$	−1.70390	−0.0881068
$375$	0	0
$376$	−5.31852	−0.274282
$377$	12.3634	0.636747
$378$	0	0
$379$	38.6940	1.98757	0.993787	−	0.111294i	$-0.0354997\pi$
$379$	38.6940	1.98757	0.993787	+	0.111294i	$0.0354997\pi$
$380$	42.6892	2.18991
$381$	0	0
$382$	1.59729	0.0817244
$383$	9.35020	0.477773	0.238886	−	0.971048i	$-0.423218\pi$
$383$	9.35020	0.477773	0.238886	+	0.971048i	$0.423218\pi$
$384$	0	0
$385$	3.37742	0.172129
$386$	−3.12725	−0.159173
$387$	0	0
$388$	−37.0003	−1.87840
$389$	−22.9294	−1.16257	−0.581283	−	0.813701i	$-0.697449\pi$
$389$	−22.9294	−1.16257	−0.581283	+	0.813701i	$0.697449\pi$
$390$	0	0
$391$	−47.1957	−2.38679
$392$	−5.63154	−0.284436
$393$	0	0
$394$	−2.92590	−0.147405
$395$	19.5839	0.985371
$396$	0	0
$397$	17.2747	0.866993	0.433496	−	0.901155i	$-0.357280\pi$
$397$	17.2747	0.866993	0.433496	+	0.901155i	$0.357280\pi$
$398$	−0.839978	−0.0421043
$399$	0	0
$400$	32.1851	1.60925
$401$	31.6046	1.57826	0.789129	−	0.614228i	$-0.210532\pi$
$401$	31.6046	1.57826	0.789129	+	0.614228i	$0.210532\pi$
$402$	0	0
$403$	11.1688	0.556357
$404$	−10.9010	−0.542345
$405$	0	0
$406$	−2.01966	−0.100234
$407$	−8.74406	−0.433427
$408$	0	0
$409$	20.8224	1.02960	0.514800	−	0.857310i	$-0.327866\pi$
$409$	20.8224	1.02960	0.514800	+	0.857310i	$0.327866\pi$
$410$	−5.61255	−0.277184
$411$	0	0
$412$	−31.2429	−1.53923
$413$	1.41558	0.0696561
$414$	0	0
$415$	27.5663	1.35318
$416$	3.45319	0.169307
$417$	0	0
$418$	−1.36830	−0.0669260
$419$	12.4302	0.607256	0.303628	−	0.952791i	$-0.401802\pi$
$419$	12.4302	0.607256	0.303628	+	0.952791i	$0.401802\pi$
$420$	0	0
$421$	−9.53980	−0.464942	−0.232471	−	0.972603i	$-0.574681\pi$
$421$	−9.53980	−0.464942	−0.232471	+	0.972603i	$0.574681\pi$
$422$	4.24328	0.206560
$423$	0	0
$424$	4.57317	0.222093
$425$	64.4004	3.12388
$426$	0	0
$427$	−0.911108	−0.0440916
$428$	31.4035	1.51795
$429$	0	0
$430$	8.22426	0.396609
$431$	12.5341	0.603745	0.301872	−	0.953348i	$-0.402388\pi$
$431$	12.5341	0.603745	0.301872	+	0.953348i	$0.402388\pi$
$432$	0	0
$433$	−18.0140	−0.865696	−0.432848	−	0.901467i	$-0.642491\pi$
$433$	−18.0140	−0.865696	−0.432848	+	0.901467i	$0.642491\pi$
$434$	−1.82452	−0.0875796
$435$	0	0
$436$	29.7478	1.42466
$437$	−37.9001	−1.81301
$438$	0	0
$439$	−4.17247	−0.199141	−0.0995707	−	0.995030i	$-0.531747\pi$
$439$	−4.17247	−0.199141	−0.0995707	+	0.995030i	$0.531747\pi$
$440$	−3.38349	−0.161302
$441$	0	0
$442$	2.19791	0.104544
$443$	31.4449	1.49399	0.746995	−	0.664830i	$-0.231496\pi$
$443$	31.4449	1.49399	0.746995	+	0.664830i	$0.231496\pi$
$444$	0	0
$445$	47.3062	2.24253
$446$	−1.61887	−0.0766556
$447$	0	0
$448$	6.14514	0.290331
$449$	−13.4654	−0.635471	−0.317735	−	0.948179i	$-0.602922\pi$
$449$	−13.4654	−0.635471	−0.317735	+	0.948179i	$0.602922\pi$
$450$	0	0
$451$	−6.54649	−0.308262
$452$	26.2233	1.23344
$453$	0	0
$454$	1.58525	0.0743994
$455$	−4.35663	−0.204242
$456$	0	0
$457$	16.0392	0.750280	0.375140	−	0.926968i	$-0.377595\pi$
$457$	16.0392	0.750280	0.375140	+	0.926968i	$0.377595\pi$
$458$	0.497365	0.0232404
$459$	0	0
$460$	−46.2238	−2.15520
$461$	2.07719	0.0967444	0.0483722	−	0.998829i	$-0.484597\pi$
$461$	2.07719	0.0967444	0.0483722	+	0.998829i	$0.484597\pi$
$462$	0	0
$463$	22.3955	1.04081	0.520404	−	0.853920i	$-0.325781\pi$
$463$	22.3955	1.04081	0.520404	+	0.853920i	$0.325781\pi$
$464$	−35.2896	−1.63828
$465$	0	0
$466$	−0.0203204	−0.000941327 0
$467$	−7.84551	−0.363047	−0.181523	−	0.983387i	$-0.558103\pi$
$467$	−7.84551	−0.363047	−0.181523	+	0.983387i	$0.558103\pi$
$468$	0	0
$469$	−11.5901	−0.535181
$470$	−4.99566	−0.230432
$471$	0	0
$472$	−1.41813	−0.0652745
$473$	9.59278	0.441077
$474$	0	0
$475$	51.7162	2.37290
$476$	13.0658	0.598868
$477$	0	0
$478$	−0.311341	−0.0142404
$479$	23.9318	1.09347	0.546736	−	0.837305i	$-0.315870\pi$
$479$	23.9318	1.09347	0.546736	+	0.837305i	$0.315870\pi$
$480$	0	0
$481$	11.2792	0.514287
$482$	−1.05002	−0.0478272
$483$	0	0
$484$	−1.94651	−0.0884777
$485$	−70.4634	−3.19958
$486$	0	0
$487$	25.8238	1.17019	0.585094	−	0.810965i	$-0.301058\pi$
$487$	25.8238	1.17019	0.585094	+	0.810965i	$0.301058\pi$
$488$	0.912746	0.0413181
$489$	0	0
$490$	−5.28968	−0.238963
$491$	31.3212	1.41351	0.706754	−	0.707459i	$-0.250159\pi$
$491$	31.3212	1.41351	0.706754	+	0.707459i	$0.250159\pi$
$492$	0	0
$493$	−70.6123	−3.18022
$494$	1.76501	0.0794117
$495$	0	0
$496$	−31.8798	−1.43144
$497$	−0.461814	−0.0207152
$498$	0	0
$499$	24.5909	1.10084	0.550420	−	0.834888i	$-0.314467\pi$
$499$	24.5909	1.10084	0.550420	+	0.834888i	$0.314467\pi$
$500$	26.9963	1.20731
$501$	0	0
$502$	−1.35009	−0.0602574
$503$	2.31095	0.103040	0.0515200	−	0.998672i	$-0.483593\pi$
$503$	2.31095	0.103040	0.0515200	+	0.998672i	$0.483593\pi$
$504$	0	0
$505$	−20.7599	−0.923803
$506$	1.48160	0.0658652
$507$	0	0
$508$	−29.0260	−1.28782
$509$	29.9915	1.32935	0.664676	−	0.747132i	$-0.268570\pi$
$509$	29.9915	1.32935	0.664676	+	0.747132i	$0.268570\pi$
$510$	0	0
$511$	3.38031	0.149536
$512$	−16.5780	−0.732650
$513$	0	0
$514$	4.45176	0.196359
$515$	−59.4991	−2.62184
$516$	0	0
$517$	−5.82694	−0.256268
$518$	−1.84255	−0.0809571
$519$	0	0
$520$	4.36446	0.191394
$521$	17.7422	0.777300	0.388650	−	0.921385i	$-0.372942\pi$
$521$	17.7422	0.777300	0.388650	+	0.921385i	$0.372942\pi$
$522$	0	0
$523$	−16.2858	−0.712129	−0.356064	−	0.934461i	$-0.615882\pi$
$523$	−16.2858	−0.712129	−0.356064	+	0.934461i	$0.615882\pi$
$524$	2.96583	0.129563
$525$	0	0
$526$	−2.88317	−0.125712
$527$	−63.7895	−2.77872
$528$	0	0
$529$	18.0382	0.784269
$530$	4.29555	0.186587
$531$	0	0
$532$	10.4923	0.454901
$533$	8.44449	0.365772
$534$	0	0
$535$	59.8050	2.58559
$536$	11.6110	0.501517
$537$	0	0
$538$	−2.53115	−0.109126
$539$	−6.16988	−0.265756
$540$	0	0
$541$	−4.81097	−0.206840	−0.103420	−	0.994638i	$-0.532979\pi$
$541$	−4.81097	−0.206840	−0.103420	+	0.994638i	$0.532979\pi$
$542$	4.15645	0.178535
$543$	0	0
$544$	−19.7226	−0.845599
$545$	56.6517	2.42669
$546$	0	0
$547$	12.5709	0.537491	0.268745	−	0.963211i	$-0.413391\pi$
$547$	12.5709	0.537491	0.268745	+	0.963211i	$0.413391\pi$
$548$	7.20865	0.307938
$549$	0	0
$550$	−2.02170	−0.0862057
$551$	−56.7046	−2.41570
$552$	0	0
$553$	4.81341	0.204687
$554$	1.32825	0.0564320
$555$	0	0
$556$	−11.8866	−0.504103
$557$	31.8497	1.34951	0.674757	−	0.738040i	$-0.264248\pi$
$557$	31.8497	1.34951	0.674757	+	0.738040i	$0.264248\pi$
$558$	0	0
$559$	−12.3740	−0.523364
$560$	12.4354	0.525491
$561$	0	0
$562$	−7.25759	−0.306143
$563$	−37.6429	−1.58646	−0.793229	−	0.608923i	$-0.791602\pi$
$563$	−37.6429	−1.58646	−0.793229	+	0.608923i	$0.791602\pi$
$564$	0	0
$565$	49.9397	2.10098
$566$	5.19216	0.218243
$567$	0	0
$568$	0.462645	0.0194121
$569$	3.93199	0.164838	0.0824189	−	0.996598i	$-0.473735\pi$
$569$	3.93199	0.164838	0.0824189	+	0.996598i	$0.473735\pi$
$570$	0	0
$571$	18.8819	0.790185	0.395092	−	0.918641i	$-0.370713\pi$
$571$	18.8819	0.790185	0.395092	+	0.918641i	$0.370713\pi$
$572$	2.51086	0.104984
$573$	0	0
$574$	−1.37948	−0.0575783
$575$	−55.9982	−2.33529
$576$	0	0
$577$	34.1619	1.42218	0.711088	−	0.703103i	$-0.248203\pi$
$577$	34.1619	1.42218	0.711088	+	0.703103i	$0.248203\pi$
$578$	−8.62143	−0.358604
$579$	0	0
$580$	−69.1583	−2.87164
$581$	6.77537	0.281090
$582$	0	0
$583$	5.01034	0.207507
$584$	−3.38639	−0.140130
$585$	0	0
$586$	−3.67328	−0.151742
$587$	4.92186	0.203147	0.101573	−	0.994828i	$-0.467612\pi$
$587$	4.92186	0.203147	0.101573	+	0.994828i	$0.467612\pi$
$588$	0	0
$589$	−51.2256	−2.11071
$590$	−1.33204	−0.0548392
$591$	0	0
$592$	−32.1949	−1.32320
$593$	−38.0886	−1.56411	−0.782056	−	0.623208i	$-0.785829\pi$
$593$	−38.0886	−1.56411	−0.782056	+	0.623208i	$0.785829\pi$
$594$	0	0
$595$	24.8825	1.02008
$596$	−26.3257	−1.07834
$597$	0	0
$598$	−1.91116	−0.0781530
$599$	35.5566	1.45280	0.726401	−	0.687271i	$-0.241191\pi$
$599$	35.5566	1.45280	0.726401	+	0.687271i	$0.241191\pi$
$600$	0	0
$601$	−3.34841	−0.136584	−0.0682922	−	0.997665i	$-0.521755\pi$
$601$	−3.34841	−0.136584	−0.0682922	+	0.997665i	$0.521755\pi$
$602$	2.02140	0.0823859
$603$	0	0
$604$	2.12844	0.0866050
$605$	−3.70694	−0.150708
$606$	0	0
$607$	−24.1878	−0.981753	−0.490877	−	0.871229i	$-0.663323\pi$
$607$	−24.1878	−0.981753	−0.490877	+	0.871229i	$0.663323\pi$
$608$	−15.8380	−0.642318
$609$	0	0
$610$	0.857338	0.0347126
$611$	7.51633	0.304078
$612$	0	0
$613$	−19.1329	−0.772769	−0.386384	−	0.922338i	$-0.626276\pi$
$613$	−19.1329	−0.772769	−0.386384	+	0.922338i	$0.626276\pi$
$614$	−3.05411	−0.123254
$615$	0	0
$616$	−0.831610	−0.0335065
$617$	−22.2994	−0.897741	−0.448870	−	0.893597i	$-0.648173\pi$
$617$	−22.2994	−0.897741	−0.448870	+	0.893597i	$0.648173\pi$
$618$	0	0
$619$	31.9197	1.28296	0.641480	−	0.767140i	$-0.278321\pi$
$619$	31.9197	1.28296	0.641480	+	0.767140i	$0.278321\pi$
$620$	−62.4759	−2.50909
$621$	0	0
$622$	1.81814	0.0729006
$623$	11.6271	0.465831
$624$	0	0
$625$	7.70489	0.308196
$626$	4.07943	0.163047
$627$	0	0
$628$	−28.1481	−1.12323
$629$	−64.4201	−2.56860
$630$	0	0
$631$	0.258718	0.0102994	0.00514971	−	0.999987i	$-0.498361\pi$
$631$	0.258718	0.0102994	0.00514971	+	0.999987i	$0.498361\pi$
$632$	−4.82207	−0.191812
$633$	0	0
$634$	2.44540	0.0971193
$635$	−55.2772	−2.19361
$636$	0	0
$637$	7.95870	0.315335
$638$	2.21671	0.0877605
$639$	0	0
$640$	−25.6298	−1.01311
$641$	26.1212	1.03173	0.515863	−	0.856671i	$-0.327471\pi$
$641$	26.1212	1.03173	0.515863	+	0.856671i	$0.327471\pi$
$642$	0	0
$643$	7.45790	0.294111	0.147056	−	0.989128i	$-0.453020\pi$
$643$	7.45790	0.294111	0.147056	+	0.989128i	$0.453020\pi$
$644$	−11.3611	−0.447690
$645$	0	0
$646$	−10.0807	−0.396620
$647$	44.3581	1.74390	0.871949	−	0.489597i	$-0.162856\pi$
$647$	44.3581	1.74390	0.871949	+	0.489597i	$0.162856\pi$
$648$	0	0
$649$	−1.55369	−0.0609877
$650$	2.60785	0.102288
$651$	0	0
$652$	−2.73939	−0.107283
$653$	6.14521	0.240481	0.120240	−	0.992745i	$-0.461633\pi$
$653$	6.14521	0.240481	0.120240	+	0.992745i	$0.461633\pi$
$654$	0	0
$655$	5.64814	0.220691
$656$	−24.1036	−0.941089
$657$	0	0
$658$	−1.22785	−0.0478668
$659$	−27.8267	−1.08398	−0.541988	−	0.840387i	$-0.682328\pi$
$659$	−27.8267	−1.08398	−0.541988	+	0.840387i	$0.682328\pi$
$660$	0	0
$661$	41.3110	1.60681	0.803406	−	0.595431i	$-0.203019\pi$
$661$	41.3110	1.60681	0.803406	+	0.595431i	$0.203019\pi$
$662$	5.94898	0.231214
$663$	0	0
$664$	−6.78756	−0.263408
$665$	19.9816	0.774854
$666$	0	0
$667$	61.3997	2.37741
$668$	−27.2716	−1.05517
$669$	0	0
$670$	10.9061	0.421340
$671$	1.00000	0.0386046
$672$	0	0
$673$	−19.1769	−0.739214	−0.369607	−	0.929188i	$-0.620508\pi$
$673$	−19.1769	−0.739214	−0.369607	+	0.929188i	$0.620508\pi$
$674$	5.12218	0.197299
$675$	0	0
$676$	22.0658	0.848685
$677$	6.49815	0.249744	0.124872	−	0.992173i	$-0.460148\pi$
$677$	6.49815	0.249744	0.124872	+	0.992173i	$0.460148\pi$
$678$	0	0
$679$	−17.3188	−0.664635
$680$	−24.9272	−0.955915
$681$	0	0
$682$	2.00252	0.0766806
$683$	30.5994	1.17085	0.585426	−	0.810726i	$-0.300927\pi$
$683$	30.5994	1.17085	0.585426	+	0.810726i	$0.300927\pi$
$684$	0	0
$685$	13.7282	0.524526
$686$	−2.77516	−0.105956
$687$	0	0
$688$	35.3199	1.34656
$689$	−6.46297	−0.246220
$690$	0	0
$691$	21.3756	0.813166	0.406583	−	0.913614i	$-0.366720\pi$
$691$	21.3756	0.813166	0.406583	+	0.913614i	$0.366720\pi$
$692$	5.21222	0.198139
$693$	0	0
$694$	3.90224	0.148127
$695$	−22.6368	−0.858663
$696$	0	0
$697$	−48.2299	−1.82684
$698$	8.48828	0.321286
$699$	0	0
$700$	15.5027	0.585946
$701$	13.7805	0.520483	0.260241	−	0.965544i	$-0.416198\pi$
$701$	13.7805	0.520483	0.260241	+	0.965544i	$0.416198\pi$
$702$	0	0
$703$	−51.7320	−1.95111
$704$	−6.74470	−0.254200
$705$	0	0
$706$	0.117012	0.00440379
$707$	−5.10246	−0.191898
$708$	0	0
$709$	19.3933	0.728332	0.364166	−	0.931334i	$-0.381354\pi$
$709$	19.3933	0.728332	0.364166	+	0.931334i	$0.381354\pi$
$710$	0.434560	0.0163087
$711$	0	0
$712$	−11.6480	−0.436529
$713$	55.4671	2.07726
$714$	0	0
$715$	4.78168	0.178825
$716$	4.16669	0.155716
$717$	0	0
$718$	1.53167	0.0571613
$719$	−6.37432	−0.237722	−0.118861	−	0.992911i	$-0.537924\pi$
$719$	−6.37432	−0.237722	−0.118861	+	0.992911i	$0.537924\pi$
$720$	0	0
$721$	−14.6240	−0.544624
$722$	−3.70091	−0.137734
$723$	0	0
$724$	−20.3452	−0.756124
$725$	−83.7824	−3.11160
$726$	0	0
$727$	−21.3988	−0.793637	−0.396819	−	0.917897i	$-0.629886\pi$
$727$	−21.3988	−0.793637	−0.396819	+	0.917897i	$0.629886\pi$
$728$	1.07272	0.0397575
$729$	0	0
$730$	−3.18082	−0.117728
$731$	70.6729	2.61393
$732$	0	0
$733$	23.1610	0.855470	0.427735	−	0.903904i	$-0.359312\pi$
$733$	23.1610	0.855470	0.427735	+	0.903904i	$0.359312\pi$
$734$	−4.46343	−0.164748
$735$	0	0
$736$	17.1494	0.632137
$737$	12.7209	0.468580
$738$	0	0
$739$	−31.1840	−1.14712	−0.573561	−	0.819163i	$-0.694439\pi$
$739$	−31.1840	−1.14712	−0.573561	+	0.819163i	$0.694439\pi$
$740$	−63.0936	−2.31937
$741$	0	0
$742$	1.05578	0.0387589
$743$	11.7663	0.431663	0.215831	−	0.976431i	$-0.430754\pi$
$743$	11.7663	0.431663	0.215831	+	0.976431i	$0.430754\pi$
$744$	0	0
$745$	−50.1347	−1.83679
$746$	7.70947	0.282264
$747$	0	0
$748$	−14.3405	−0.524341
$749$	14.6991	0.537095
$750$	0	0
$751$	−0.685367	−0.0250094	−0.0125047	−	0.999922i	$-0.503980\pi$
$751$	−0.685367	−0.0250094	−0.0125047	+	0.999922i	$0.503980\pi$
$752$	−21.4543	−0.782359
$753$	0	0
$754$	−2.85940	−0.104133
$755$	4.05341	0.147519
$756$	0	0
$757$	−32.9419	−1.19730	−0.598648	−	0.801013i	$-0.704295\pi$
$757$	−32.9419	−1.19730	−0.598648	+	0.801013i	$0.704295\pi$
$758$	−8.94911	−0.325046
$759$	0	0
$760$	−20.0176	−0.726114
$761$	−24.4080	−0.884791	−0.442396	−	0.896820i	$-0.645871\pi$
$761$	−24.4080	−0.884791	−0.442396	+	0.896820i	$0.645871\pi$
$762$	0	0
$763$	13.9241	0.504087
$764$	13.4432	0.486358
$765$	0	0
$766$	−2.16251	−0.0781346
$767$	2.00415	0.0723656
$768$	0	0
$769$	−0.979264	−0.0353132	−0.0176566	−	0.999844i	$-0.505621\pi$
$769$	−0.979264	−0.0353132	−0.0176566	+	0.999844i	$0.505621\pi$
$770$	−0.781128	−0.0281499
$771$	0	0
$772$	−26.3198	−0.947269
$773$	47.3585	1.70337	0.851683	−	0.524057i	$-0.175582\pi$
$773$	47.3585	1.70337	0.851683	+	0.524057i	$0.175582\pi$
$774$	0	0
$775$	−75.6870	−2.71876
$776$	17.3500	0.622827
$777$	0	0
$778$	5.30310	0.190125
$779$	−38.7306	−1.38767
$780$	0	0
$781$	0.506871	0.0181373
$782$	10.9154	0.390334
$783$	0	0
$784$	−22.7170	−0.811322
$785$	−53.6053	−1.91326
$786$	0	0
$787$	−21.9000	−0.780651	−0.390326	−	0.920677i	$-0.627638\pi$
$787$	−21.9000	−0.780651	−0.390326	+	0.920677i	$0.627638\pi$
$788$	−24.6252	−0.877235
$789$	0	0
$790$	−4.52934	−0.161147
$791$	12.2744	0.436428
$792$	0	0
$793$	−1.28993	−0.0458067
$794$	−3.99528	−0.141787
$795$	0	0
$796$	−7.06948	−0.250571
$797$	−16.0797	−0.569571	−0.284785	−	0.958591i	$-0.591922\pi$
$797$	−16.0797	−0.569571	−0.284785	+	0.958591i	$0.591922\pi$
$798$	0	0
$799$	−42.9288	−1.51871
$800$	−23.4011	−0.827353
$801$	0	0
$802$	−7.30949	−0.258107
$803$	−3.71011	−0.130927
$804$	0	0
$805$	−21.6361	−0.762573
$806$	−2.58311	−0.0909862
$807$	0	0
$808$	5.11164	0.179827
$809$	11.8133	0.415333	0.207667	−	0.978200i	$-0.433413\pi$
$809$	11.8133	0.415333	0.207667	+	0.978200i	$0.433413\pi$
$810$	0	0
$811$	29.4126	1.03281	0.516407	−	0.856343i	$-0.327269\pi$
$811$	29.4126	1.03281	0.516407	+	0.856343i	$0.327269\pi$
$812$	−16.9980	−0.596514
$813$	0	0
$814$	2.02232	0.0708823
$815$	−5.21691	−0.182740
$816$	0	0
$817$	56.7532	1.98554
$818$	−4.81579	−0.168380
$819$	0	0
$820$	−47.2368	−1.64958
$821$	30.6948	1.07125	0.535627	−	0.844454i	$-0.320075\pi$
$821$	30.6948	1.07125	0.535627	+	0.844454i	$0.320075\pi$
$822$	0	0
$823$	31.6432	1.10301	0.551505	−	0.834171i	$-0.314054\pi$
$823$	31.6432	1.10301	0.551505	+	0.834171i	$0.314054\pi$
$824$	14.6503	0.510366
$825$	0	0
$826$	−0.327394	−0.0113915
$827$	13.0222	0.452825	0.226413	−	0.974031i	$-0.427300\pi$
$827$	13.0222	0.452825	0.226413	+	0.974031i	$0.427300\pi$
$828$	0	0
$829$	47.5998	1.65321	0.826604	−	0.562783i	$-0.190269\pi$
$829$	47.5998	1.65321	0.826604	+	0.562783i	$0.190269\pi$
$830$	−6.37552	−0.221298
$831$	0	0
$832$	8.70017	0.301624
$833$	−45.4554	−1.57494
$834$	0	0
$835$	−51.9360	−1.79732
$836$	−11.5160	−0.398290
$837$	0	0
$838$	−2.87485	−0.0993102
$839$	2.77869	0.0959309	0.0479655	−	0.998849i	$-0.484726\pi$
$839$	2.77869	0.0959309	0.0479655	+	0.998849i	$0.484726\pi$
$840$	0	0
$841$	62.8638	2.16772
$842$	2.20636	0.0760362
$843$	0	0
$844$	35.7126	1.22928
$845$	42.0222	1.44561
$846$	0	0
$847$	−0.911108	−0.0313060
$848$	18.4477	0.633495
$849$	0	0
$850$	−14.8945	−0.510877
$851$	56.0154	1.92018
$852$	0	0
$853$	19.4760	0.666845	0.333423	−	0.942777i	$-0.391796\pi$
$853$	19.4760	0.666845	0.333423	+	0.942777i	$0.391796\pi$
$854$	0.210720	0.00721071
$855$	0	0
$856$	−14.7256	−0.503310
$857$	34.8220	1.18950	0.594749	−	0.803912i	$-0.297252\pi$
$857$	34.8220	1.18950	0.594749	+	0.803912i	$0.297252\pi$
$858$	0	0
$859$	−8.18830	−0.279381	−0.139691	−	0.990195i	$-0.544611\pi$
$859$	−8.18830	−0.279381	−0.139691	+	0.990195i	$0.544611\pi$
$860$	69.2176	2.36030
$861$	0	0
$862$	−2.89887	−0.0987359
$863$	−34.4697	−1.17336	−0.586681	−	0.809818i	$-0.699566\pi$
$863$	−34.4697	−1.17336	−0.586681	+	0.809818i	$0.699566\pi$
$864$	0	0
$865$	9.92616	0.337500
$866$	4.16626	0.141575
$867$	0	0
$868$	−15.3556	−0.521204
$869$	−5.28303	−0.179215
$870$	0	0
$871$	−16.4090	−0.555999
$872$	−13.9492	−0.472378
$873$	0	0
$874$	8.76551	0.296498
$875$	12.6362	0.427182
$876$	0	0
$877$	22.6206	0.763845	0.381922	−	0.924194i	$-0.375262\pi$
$877$	22.6206	0.763845	0.381922	+	0.924194i	$0.375262\pi$
$878$	0.965007	0.0325674
$879$	0	0
$880$	−13.6486	−0.460096
$881$	−20.0360	−0.675029	−0.337514	−	0.941320i	$-0.609586\pi$
$881$	−20.0360	−0.675029	−0.337514	+	0.941320i	$0.609586\pi$
$882$	0	0
$883$	−16.1716	−0.544217	−0.272108	−	0.962267i	$-0.587721\pi$
$883$	−16.1716	−0.544217	−0.272108	+	0.962267i	$0.587721\pi$
$884$	18.4982	0.622163
$885$	0	0
$886$	−7.27255	−0.244326
$887$	14.1009	0.473460	0.236730	−	0.971575i	$-0.423924\pi$
$887$	14.1009	0.473460	0.236730	+	0.971575i	$0.423924\pi$
$888$	0	0
$889$	−13.5863	−0.455669
$890$	−10.9410	−0.366742
$891$	0	0
$892$	−13.6248	−0.456193
$893$	−34.4736	−1.15361
$894$	0	0
$895$	7.93504	0.265239
$896$	−6.29940	−0.210448
$897$	0	0
$898$	3.11427	0.103924
$899$	82.9876	2.76779
$900$	0	0
$901$	36.9127	1.22974
$902$	1.51407	0.0504129
$903$	0	0
$904$	−12.2965	−0.408975
$905$	−38.7455	−1.28794
$906$	0	0
$907$	−36.0807	−1.19804	−0.599019	−	0.800735i	$-0.704443\pi$
$907$	−36.0807	−1.19804	−0.599019	+	0.800735i	$0.704443\pi$
$908$	13.3419	0.442766
$909$	0	0
$910$	1.00760	0.0334015
$911$	20.4544	0.677684	0.338842	−	0.940843i	$-0.389965\pi$
$911$	20.4544	0.677684	0.338842	+	0.940843i	$0.389965\pi$
$912$	0	0
$913$	−7.43641	−0.246109
$914$	−3.70953	−0.122700
$915$	0	0
$916$	4.18596	0.138308
$917$	1.38823	0.0458432
$918$	0	0
$919$	43.7171	1.44209	0.721046	−	0.692887i	$-0.243662\pi$
$919$	43.7171	1.44209	0.721046	+	0.692887i	$0.243662\pi$
$920$	21.6750	0.714604
$921$	0	0
$922$	−0.480411	−0.0158215
$923$	−0.653827	−0.0215210
$924$	0	0
$925$	−76.4352	−2.51317
$926$	−5.17962	−0.170213
$927$	0	0
$928$	25.6583	0.842275
$929$	34.9056	1.14521	0.572607	−	0.819830i	$-0.305932\pi$
$929$	34.9056	1.14521	0.572607	+	0.819830i	$0.305932\pi$
$930$	0	0
$931$	−36.5025	−1.19632
$932$	−0.171022	−0.00560202
$933$	0	0
$934$	1.81451	0.0593724
$935$	−27.3101	−0.893136
$936$	0	0
$937$	27.5062	0.898589	0.449294	−	0.893384i	$-0.351675\pi$
$937$	27.5062	0.898589	0.449294	+	0.893384i	$0.351675\pi$
$938$	2.68055	0.0875232
$939$	0	0
$940$	−42.0448	−1.37135
$941$	−35.8790	−1.16962	−0.584811	−	0.811170i	$-0.698831\pi$
$941$	−35.8790	−1.16962	−0.584811	+	0.811170i	$0.698831\pi$
$942$	0	0
$943$	41.9375	1.36567
$944$	−5.72057	−0.186189
$945$	0	0
$946$	−2.21861	−0.0721333
$947$	11.3161	0.367724	0.183862	−	0.982952i	$-0.441140\pi$
$947$	11.3161	0.367724	0.183862	+	0.982952i	$0.441140\pi$
$948$	0	0
$949$	4.78578	0.155353
$950$	−11.9609	−0.388062
$951$	0	0
$952$	−6.12672	−0.198568
$953$	2.13994	0.0693195	0.0346598	−	0.999399i	$-0.488965\pi$
$953$	2.13994	0.0693195	0.0346598	+	0.999399i	$0.488965\pi$
$954$	0	0
$955$	25.6013	0.828438
$956$	−2.62033	−0.0847475
$957$	0	0
$958$	−5.53493	−0.178826
$959$	3.37417	0.108958
$960$	0	0
$961$	43.9690	1.41836
$962$	−2.60865	−0.0841062
$963$	0	0
$964$	−8.83727	−0.284629
$965$	−50.1234	−1.61353
$966$	0	0
$967$	28.0975	0.903555	0.451778	−	0.892131i	$-0.350790\pi$
$967$	28.0975	0.903555	0.451778	+	0.892131i	$0.350790\pi$
$968$	0.912746	0.0293368
$969$	0	0
$970$	16.2967	0.523257
$971$	50.8194	1.63087	0.815436	−	0.578847i	$-0.196497\pi$
$971$	50.8194	1.63087	0.815436	+	0.578847i	$0.196497\pi$
$972$	0	0
$973$	−5.56378	−0.178366
$974$	−5.97251	−0.191372
$975$	0	0
$976$	3.68192	0.117855
$977$	27.5896	0.882670	0.441335	−	0.897342i	$-0.354505\pi$
$977$	27.5896	0.882670	0.441335	+	0.897342i	$0.354505\pi$
$978$	0	0
$979$	−12.7615	−0.407860
$980$	−44.5193	−1.42212
$981$	0	0
$982$	−7.24396	−0.231164
$983$	−28.6548	−0.913945	−0.456973	−	0.889481i	$-0.651066\pi$
$983$	−28.6548	−0.913945	−0.456973	+	0.889481i	$0.651066\pi$
$984$	0	0
$985$	−46.8962	−1.49424
$986$	16.3312	0.520091
$987$	0	0
$988$	14.8548	0.472595
$989$	−61.4524	−1.95407
$990$	0	0
$991$	19.1844	0.609412	0.304706	−	0.952447i	$-0.401442\pi$
$991$	19.1844	0.609412	0.304706	+	0.952447i	$0.401442\pi$
$992$	23.1791	0.735937
$993$	0	0
$994$	0.106808	0.00338775
$995$	−13.4631	−0.426810
$996$	0	0
$997$	−4.14208	−0.131181	−0.0655905	−	0.997847i	$-0.520893\pi$
$997$	−4.14208	−0.131181	−0.0655905	+	0.997847i	$0.520893\pi$
$998$	−5.68737	−0.180031
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.j.1.6		14
3.2	odd	2		2013.2.a.h.1.9	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.h.1.9	✓	14	3.2	odd	2
6039.2.a.j.1.6		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-0.231279 q^{2} -1.94651 q^{4} -3.70694 q^{5} -0.911108 q^{7} +0.912746 q^{8} +O(q^{10})\)	\(q-0.231279 q^{2} -1.94651 q^{4} -3.70694 q^{5} -0.911108 q^{7} +0.912746 q^{8} +0.857338 q^{10} +1.00000 q^{11} -1.28993 q^{13} +0.210720 q^{14} +3.68192 q^{16} +7.36730 q^{17} +5.91624 q^{19} +7.21559 q^{20} -0.231279 q^{22} -6.40611 q^{23} +8.74139 q^{25} +0.298334 q^{26} +1.77348 q^{28} -9.58456 q^{29} -8.65847 q^{31} -2.67705 q^{32} -1.70390 q^{34} +3.37742 q^{35} -8.74406 q^{37} -1.36830 q^{38} -3.38349 q^{40} -6.54649 q^{41} +9.59278 q^{43} -1.94651 q^{44} +1.48160 q^{46} -5.82694 q^{47} -6.16988 q^{49} -2.02170 q^{50} +2.51086 q^{52} +5.01034 q^{53} -3.70694 q^{55} -0.831610 q^{56} +2.21671 q^{58} -1.55369 q^{59} +1.00000 q^{61} +2.00252 q^{62} -6.74470 q^{64} +4.78168 q^{65} +12.7209 q^{67} -14.3405 q^{68} -0.781128 q^{70} +0.506871 q^{71} -3.71011 q^{73} +2.02232 q^{74} -11.5160 q^{76} -0.911108 q^{77} -5.28303 q^{79} -13.6486 q^{80} +1.51407 q^{82} -7.43641 q^{83} -27.3101 q^{85} -2.21861 q^{86} +0.912746 q^{88} -12.7615 q^{89} +1.17526 q^{91} +12.4695 q^{92} +1.34765 q^{94} -21.9311 q^{95} +19.0085 q^{97} +1.42697 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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