LMFDB - Embedded newform 6039.2.a.j.1.7

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			1.7
Root			$-0.0561655$ 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)

$f(q)$	$=$	$q-0.0561655 q^{2} -1.99685 q^{4} +2.87016 q^{5} +3.53988 q^{7} +0.224485 q^{8} +O(q^{10})$	\(q-0.0561655 q^{2} -1.99685 q^{4} +2.87016 q^{5} +3.53988 q^{7} +0.224485 q^{8} -0.161204 q^{10} +1.00000 q^{11} +1.66374 q^{13} -0.198819 q^{14} +3.98108 q^{16} +6.51228 q^{17} -4.27158 q^{19} -5.73126 q^{20} -0.0561655 q^{22} +8.02418 q^{23} +3.23780 q^{25} -0.0934449 q^{26} -7.06858 q^{28} -3.26557 q^{29} +6.97737 q^{31} -0.672570 q^{32} -0.365766 q^{34} +10.1600 q^{35} +0.826353 q^{37} +0.239916 q^{38} +0.644307 q^{40} +11.9366 q^{41} -2.21299 q^{43} -1.99685 q^{44} -0.450683 q^{46} -10.4001 q^{47} +5.53072 q^{49} -0.181853 q^{50} -3.32224 q^{52} -8.11158 q^{53} +2.87016 q^{55} +0.794649 q^{56} +0.183413 q^{58} -2.00484 q^{59} +1.00000 q^{61} -0.391888 q^{62} -7.92439 q^{64} +4.77520 q^{65} +10.0237 q^{67} -13.0040 q^{68} -0.570642 q^{70} +2.13305 q^{71} +2.54693 q^{73} -0.0464125 q^{74} +8.52969 q^{76} +3.53988 q^{77} -6.81997 q^{79} +11.4263 q^{80} -0.670428 q^{82} -0.562991 q^{83} +18.6913 q^{85} +0.124294 q^{86} +0.224485 q^{88} -7.23252 q^{89} +5.88944 q^{91} -16.0231 q^{92} +0.584128 q^{94} -12.2601 q^{95} +11.1816 q^{97} -0.310636 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.0561655	−0.0397150	−0.0198575	−	0.999803i	$-0.506321\pi$
$2$	−0.0561655	−0.0397150	−0.0198575	+	0.999803i	$0.506321\pi$
$3$	0	0
$3$	0	0
$4$	−1.99685	−0.998423
$5$	2.87016	1.28357	0.641786	−	0.766883i	$-0.278194\pi$
$5$	2.87016	1.28357	0.641786	+	0.766883i	$0.278194\pi$
$6$	0	0
$7$	3.53988	1.33795	0.668974	−	0.743286i	$-0.266734\pi$
$7$	3.53988	1.33795	0.668974	+	0.743286i	$0.266734\pi$
$8$	0.224485	0.0793674
$9$	0	0
$10$	−0.161204	−0.0509771
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.66374	0.461439	0.230719	−	0.973020i	$-0.425892\pi$
$13$	1.66374	0.461439	0.230719	+	0.973020i	$0.425892\pi$
$14$	−0.198819	−0.0531366
$15$	0	0
$16$	3.98108	0.995271
$17$	6.51228	1.57946	0.789730	−	0.613455i	$-0.210221\pi$
$17$	6.51228	1.57946	0.789730	+	0.613455i	$0.210221\pi$
$18$	0	0
$19$	−4.27158	−0.979968	−0.489984	−	0.871731i	$-0.662997\pi$
$19$	−4.27158	−0.979968	−0.489984	+	0.871731i	$0.662997\pi$
$20$	−5.73126	−1.28155
$21$	0	0
$22$	−0.0561655	−0.0119745
$23$	8.02418	1.67316	0.836579	−	0.547846i	$-0.184552\pi$
$23$	8.02418	1.67316	0.836579	+	0.547846i	$0.184552\pi$
$24$	0	0
$25$	3.23780	0.647559
$26$	−0.0934449	−0.0183261
$27$	0	0
$28$	−7.06858	−1.33584
$29$	−3.26557	−0.606402	−0.303201	−	0.952927i	$-0.598055\pi$
$29$	−3.26557	−0.606402	−0.303201	+	0.952927i	$0.598055\pi$
$30$	0	0
$31$	6.97737	1.25317	0.626587	−	0.779352i	$-0.284451\pi$
$31$	6.97737	1.25317	0.626587	+	0.779352i	$0.284451\pi$
$32$	−0.672570	−0.118895
$33$	0	0
$34$	−0.365766	−0.0627283
$35$	10.1600	1.71735
$36$	0	0
$37$	0.826353	0.135852	0.0679258	−	0.997690i	$-0.478362\pi$
$37$	0.826353	0.135852	0.0679258	+	0.997690i	$0.478362\pi$
$38$	0.239916	0.0389194
$39$	0	0
$40$	0.644307	0.101874
$41$	11.9366	1.86419	0.932095	−	0.362214i	$-0.117979\pi$
$41$	11.9366	1.86419	0.932095	+	0.362214i	$0.117979\pi$
$42$	0	0
$43$	−2.21299	−0.337477	−0.168739	−	0.985661i	$-0.553969\pi$
$43$	−2.21299	−0.337477	−0.168739	+	0.985661i	$0.553969\pi$
$44$	−1.99685	−0.301036
$45$	0	0
$46$	−0.450683	−0.0664495
$47$	−10.4001	−1.51701	−0.758506	−	0.651666i	$-0.774070\pi$
$47$	−10.4001	−1.51701	−0.758506	+	0.651666i	$0.774070\pi$
$48$	0	0
$49$	5.53072	0.790103
$50$	−0.181853	−0.0257178
$51$	0	0
$52$	−3.32224	−0.460711
$53$	−8.11158	−1.11421	−0.557106	−	0.830442i	$-0.688088\pi$
$53$	−8.11158	−1.11421	−0.557106	+	0.830442i	$0.688088\pi$
$54$	0	0
$55$	2.87016	0.387012
$56$	0.794649	0.106189
$57$	0	0
$58$	0.183413	0.0240833
$59$	−2.00484	−0.261007	−0.130504	−	0.991448i	$-0.541659\pi$
$59$	−2.00484	−0.261007	−0.130504	+	0.991448i	$0.541659\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−0.391888	−0.0497698
$63$	0	0
$64$	−7.92439	−0.990549
$65$	4.77520	0.592291
$66$	0	0
$67$	10.0237	1.22459	0.612293	−	0.790631i	$-0.290247\pi$
$67$	10.0237	1.22459	0.612293	+	0.790631i	$0.290247\pi$
$68$	−13.0040	−1.57697
$69$	0	0
$70$	−0.570642	−0.0682047
$71$	2.13305	0.253146	0.126573	−	0.991957i	$-0.459602\pi$
$71$	2.13305	0.253146	0.126573	+	0.991957i	$0.459602\pi$
$72$	0	0
$73$	2.54693	0.298096	0.149048	−	0.988830i	$-0.452379\pi$
$73$	2.54693	0.298096	0.149048	+	0.988830i	$0.452379\pi$
$74$	−0.0464125	−0.00539535
$75$	0	0
$76$	8.52969	0.978422
$77$	3.53988	0.403406
$78$	0	0
$79$	−6.81997	−0.767306	−0.383653	−	0.923477i	$-0.625334\pi$
$79$	−6.81997	−0.767306	−0.383653	+	0.923477i	$0.625334\pi$
$80$	11.4263	1.27750
$81$	0	0
$82$	−0.670428	−0.0740364
$83$	−0.562991	−0.0617963	−0.0308982	−	0.999523i	$-0.509837\pi$
$83$	−0.562991	−0.0617963	−0.0308982	+	0.999523i	$0.509837\pi$
$84$	0	0
$85$	18.6913	2.02735
$86$	0.124294	0.0134029
$87$	0	0
$88$	0.224485	0.0239302
$89$	−7.23252	−0.766646	−0.383323	−	0.923614i	$-0.625220\pi$
$89$	−7.23252	−0.766646	−0.383323	+	0.923614i	$0.625220\pi$
$90$	0	0
$91$	5.88944	0.617381
$92$	−16.0231	−1.67052
$93$	0	0
$94$	0.584128	0.0602482
$95$	−12.2601	−1.25786
$96$	0	0
$97$	11.1816	1.13532	0.567660	−	0.823263i	$-0.307849\pi$
$97$	11.1816	1.13532	0.567660	+	0.823263i	$0.307849\pi$
$98$	−0.310636	−0.0313789
$99$	0	0
$100$	−6.46538	−0.646538
$101$	−10.8938	−1.08398	−0.541989	−	0.840386i	$-0.682329\pi$
$101$	−10.8938	−1.08398	−0.541989	+	0.840386i	$0.682329\pi$
$102$	0	0
$103$	−1.11625	−0.109988	−0.0549938	−	0.998487i	$-0.517514\pi$
$103$	−1.11625	−0.109988	−0.0549938	+	0.998487i	$0.517514\pi$
$104$	0.373485	0.0366232
$105$	0	0
$106$	0.455591	0.0442509
$107$	2.80022	0.270707	0.135354	−	0.990797i	$-0.456783\pi$
$107$	2.80022	0.270707	0.135354	+	0.990797i	$0.456783\pi$
$108$	0	0
$109$	10.6923	1.02414	0.512069	−	0.858944i	$-0.328879\pi$
$109$	10.6923	1.02414	0.512069	+	0.858944i	$0.328879\pi$
$110$	−0.161204	−0.0153702
$111$	0	0
$112$	14.0925	1.33162
$113$	−13.9443	−1.31177	−0.655885	−	0.754861i	$-0.727704\pi$
$113$	−13.9443	−1.31177	−0.655885	+	0.754861i	$0.727704\pi$
$114$	0	0
$115$	23.0307	2.14762
$116$	6.52085	0.605446
$117$	0	0
$118$	0.112603	0.0103659
$119$	23.0527	2.11323
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.0561655	−0.00508499
$123$	0	0
$124$	−13.9327	−1.25120
$125$	−5.05780	−0.452383
$126$	0	0
$127$	−15.3492	−1.36202	−0.681012	−	0.732272i	$-0.738460\pi$
$127$	−15.3492	−1.36202	−0.681012	+	0.732272i	$0.738460\pi$
$128$	1.79022	0.158234
$129$	0	0
$130$	−0.268202	−0.0235228
$131$	−17.1289	−1.49656	−0.748280	−	0.663383i	$-0.769120\pi$
$131$	−17.1289	−1.49656	−0.748280	+	0.663383i	$0.769120\pi$
$132$	0	0
$133$	−15.1209	−1.31115
$134$	−0.562985	−0.0486345
$135$	0	0
$136$	1.46191	0.125358
$137$	1.15929	0.0990451	0.0495225	−	0.998773i	$-0.484230\pi$
$137$	1.15929	0.0990451	0.0495225	+	0.998773i	$0.484230\pi$
$138$	0	0
$139$	7.66907	0.650482	0.325241	−	0.945631i	$-0.394555\pi$
$139$	7.66907	0.650482	0.325241	+	0.945631i	$0.394555\pi$
$140$	−20.2879	−1.71464
$141$	0	0
$142$	−0.119804	−0.0100537
$143$	1.66374	0.139129
$144$	0	0
$145$	−9.37271	−0.778361
$146$	−0.143050	−0.0118389
$147$	0	0
$148$	−1.65010	−0.135637
$149$	1.19140	0.0976031	0.0488015	−	0.998808i	$-0.484460\pi$
$149$	1.19140	0.0976031	0.0488015	+	0.998808i	$0.484460\pi$
$150$	0	0
$151$	11.2781	0.917798	0.458899	−	0.888488i	$-0.348244\pi$
$151$	11.2781	0.917798	0.458899	+	0.888488i	$0.348244\pi$
$152$	−0.958906	−0.0777775
$153$	0	0
$154$	−0.198819	−0.0160213
$155$	20.0262	1.60854
$156$	0	0
$157$	17.0614	1.36165	0.680825	−	0.732446i	$-0.261621\pi$
$157$	17.0614	1.36165	0.680825	+	0.732446i	$0.261621\pi$
$158$	0.383047	0.0304736
$159$	0	0
$160$	−1.93038	−0.152610
$161$	28.4046	2.23860
$162$	0	0
$163$	−18.8321	−1.47504	−0.737521	−	0.675325i	$-0.764004\pi$
$163$	−18.8321	−1.47504	−0.737521	+	0.675325i	$0.764004\pi$
$164$	−23.8356	−1.86125
$165$	0	0
$166$	0.0316207	0.00245424
$167$	−20.7776	−1.60782	−0.803911	−	0.594749i	$-0.797251\pi$
$167$	−20.7776	−1.60782	−0.803911	+	0.594749i	$0.797251\pi$
$168$	0	0
$169$	−10.2320	−0.787074
$170$	−1.04980	−0.0805163
$171$	0	0
$172$	4.41899	0.336945
$173$	−8.58089	−0.652393	−0.326197	−	0.945302i	$-0.605767\pi$
$173$	−8.58089	−0.652393	−0.326197	+	0.945302i	$0.605767\pi$
$174$	0	0
$175$	11.4614	0.866400
$176$	3.98108	0.300085
$177$	0	0
$178$	0.406218	0.0304474
$179$	−1.62970	−0.121810	−0.0609049	−	0.998144i	$-0.519399\pi$
$179$	−1.62970	−0.121810	−0.0609049	+	0.998144i	$0.519399\pi$
$180$	0	0
$181$	−6.80075	−0.505496	−0.252748	−	0.967532i	$-0.581334\pi$
$181$	−6.80075	−0.505496	−0.252748	+	0.967532i	$0.581334\pi$
$182$	−0.330783	−0.0245193
$183$	0	0
$184$	1.80131	0.132794
$185$	2.37176	0.174375
$186$	0	0
$187$	6.51228	0.476225
$188$	20.7674	1.51462
$189$	0	0
$190$	0.688595	0.0499559
$191$	−21.1056	−1.52714	−0.763572	−	0.645723i	$-0.776556\pi$
$191$	−21.1056	−1.52714	−0.763572	+	0.645723i	$0.776556\pi$
$192$	0	0
$193$	−15.3449	−1.10455	−0.552276	−	0.833661i	$-0.686241\pi$
$193$	−15.3449	−1.10455	−0.552276	+	0.833661i	$0.686241\pi$
$194$	−0.628021	−0.0450893
$195$	0	0
$196$	−11.0440	−0.788856
$197$	−8.01939	−0.571358	−0.285679	−	0.958325i	$-0.592219\pi$
$197$	−8.01939	−0.571358	−0.285679	+	0.958325i	$0.592219\pi$
$198$	0	0
$199$	18.9435	1.34287	0.671435	−	0.741063i	$-0.265678\pi$
$199$	18.9435	1.34287	0.671435	+	0.741063i	$0.265678\pi$
$200$	0.726836	0.0513951
$201$	0	0
$202$	0.611859	0.0430502
$203$	−11.5597	−0.811334
$204$	0	0
$205$	34.2600	2.39282
$206$	0.0626949	0.00436816
$207$	0	0
$208$	6.62349	0.459257
$209$	−4.27158	−0.295471
$210$	0	0
$211$	−14.2888	−0.983682	−0.491841	−	0.870685i	$-0.663676\pi$
$211$	−14.2888	−0.983682	−0.491841	+	0.870685i	$0.663676\pi$
$212$	16.1976	1.11245
$213$	0	0
$214$	−0.157276	−0.0107511
$215$	−6.35162	−0.433177
$216$	0	0
$217$	24.6990	1.67668
$218$	−0.600540	−0.0406737
$219$	0	0
$220$	−5.73126	−0.386401
$221$	10.8348	0.728824
$222$	0	0
$223$	−2.84596	−0.190580	−0.0952899	−	0.995450i	$-0.530378\pi$
$223$	−2.84596	−0.190580	−0.0952899	+	0.995450i	$0.530378\pi$
$224$	−2.38081	−0.159075
$225$	0	0
$226$	0.783189	0.0520970
$227$	19.4833	1.29315	0.646576	−	0.762849i	$-0.276200\pi$
$227$	19.4833	1.29315	0.646576	+	0.762849i	$0.276200\pi$
$228$	0	0
$229$	−11.7161	−0.774219	−0.387109	−	0.922034i	$-0.626526\pi$
$229$	−11.7161	−0.774219	−0.387109	+	0.922034i	$0.626526\pi$
$230$	−1.29353	−0.0852928
$231$	0	0
$232$	−0.733072	−0.0481286
$233$	27.8004	1.82126	0.910632	−	0.413219i	$-0.135596\pi$
$233$	27.8004	1.82126	0.910632	+	0.413219i	$0.135596\pi$
$234$	0	0
$235$	−29.8499	−1.94720
$236$	4.00335	0.260596
$237$	0	0
$238$	−1.29476	−0.0839271
$239$	−1.27569	−0.0825176	−0.0412588	−	0.999148i	$-0.513137\pi$
$239$	−1.27569	−0.0825176	−0.0412588	+	0.999148i	$0.513137\pi$
$240$	0	0
$241$	14.9945	0.965878	0.482939	−	0.875654i	$-0.339569\pi$
$241$	14.9945	0.965878	0.482939	+	0.875654i	$0.339569\pi$
$242$	−0.0561655	−0.00361046
$243$	0	0
$244$	−1.99685	−0.127835
$245$	15.8740	1.01415
$246$	0	0
$247$	−7.10681	−0.452195
$248$	1.56632	0.0994611
$249$	0	0
$250$	0.284074	0.0179664
$251$	25.6541	1.61927	0.809637	−	0.586931i	$-0.199664\pi$
$251$	25.6541	1.61927	0.809637	+	0.586931i	$0.199664\pi$
$252$	0	0
$253$	8.02418	0.504476
$254$	0.862098	0.0540929
$255$	0	0
$256$	15.7482	0.984264
$257$	23.5239	1.46738	0.733690	−	0.679485i	$-0.237797\pi$
$257$	23.5239	1.46738	0.733690	+	0.679485i	$0.237797\pi$
$258$	0	0
$259$	2.92519	0.181762
$260$	−9.53533	−0.591356
$261$	0	0
$262$	0.962055	0.0594360
$263$	7.55266	0.465717	0.232859	−	0.972511i	$-0.425192\pi$
$263$	7.55266	0.465717	0.232859	+	0.972511i	$0.425192\pi$
$264$	0	0
$265$	−23.2815	−1.43017
$266$	0.849271	0.0520722
$267$	0	0
$268$	−20.0157	−1.22266
$269$	−25.3390	−1.54495	−0.772474	−	0.635046i	$-0.780981\pi$
$269$	−25.3390	−1.54495	−0.772474	+	0.635046i	$0.780981\pi$
$270$	0	0
$271$	−24.0494	−1.46090	−0.730449	−	0.682967i	$-0.760689\pi$
$271$	−24.0494	−1.46090	−0.730449	+	0.682967i	$0.760689\pi$
$272$	25.9259	1.57199
$273$	0	0
$274$	−0.0651123	−0.00393358
$275$	3.23780	0.195246
$276$	0	0
$277$	25.9274	1.55782	0.778912	−	0.627133i	$-0.215772\pi$
$277$	25.9274	1.55782	0.778912	+	0.627133i	$0.215772\pi$
$278$	−0.430738	−0.0258339
$279$	0	0
$280$	2.28077	0.136302
$281$	16.8801	1.00698	0.503491	−	0.864000i	$-0.332049\pi$
$281$	16.8801	1.00698	0.503491	+	0.864000i	$0.332049\pi$
$282$	0	0
$283$	14.2351	0.846191	0.423095	−	0.906085i	$-0.360944\pi$
$283$	14.2351	0.846191	0.423095	+	0.906085i	$0.360944\pi$
$284$	−4.25937	−0.252747
$285$	0	0
$286$	−0.0934449	−0.00552552
$287$	42.2542	2.49419
$288$	0	0
$289$	25.4098	1.49469
$290$	0.526423	0.0309126
$291$	0	0
$292$	−5.08583	−0.297626
$293$	−25.1762	−1.47081	−0.735406	−	0.677627i	$-0.763008\pi$
$293$	−25.1762	−1.47081	−0.735406	+	0.677627i	$0.763008\pi$
$294$	0	0
$295$	−5.75419	−0.335022
$296$	0.185504	0.0107822
$297$	0	0
$298$	−0.0669155	−0.00387631
$299$	13.3502	0.772060
$300$	0	0
$301$	−7.83369	−0.451527
$302$	−0.633440	−0.0364504
$303$	0	0
$304$	−17.0055	−0.975333
$305$	2.87016	0.164345
$306$	0	0
$307$	20.2369	1.15498	0.577490	−	0.816398i	$-0.304032\pi$
$307$	20.2369	1.15498	0.577490	+	0.816398i	$0.304032\pi$
$308$	−7.06858	−0.402770
$309$	0	0
$310$	−1.12478	−0.0638832
$311$	−20.5089	−1.16295	−0.581477	−	0.813563i	$-0.697525\pi$
$311$	−20.5089	−1.16295	−0.581477	+	0.813563i	$0.697525\pi$
$312$	0	0
$313$	−10.5734	−0.597644	−0.298822	−	0.954309i	$-0.596594\pi$
$313$	−10.5734	−0.597644	−0.298822	+	0.954309i	$0.596594\pi$
$314$	−0.958264	−0.0540780
$315$	0	0
$316$	13.6184	0.766096
$317$	−20.3836	−1.14486	−0.572429	−	0.819955i	$-0.693999\pi$
$317$	−20.3836	−1.14486	−0.572429	+	0.819955i	$0.693999\pi$
$318$	0	0
$319$	−3.26557	−0.182837
$320$	−22.7442	−1.27144
$321$	0	0
$322$	−1.59536	−0.0889059
$323$	−27.8177	−1.54782
$324$	0	0
$325$	5.38686	0.298809
$326$	1.05771	0.0585813
$327$	0	0
$328$	2.67960	0.147956
$329$	−36.8151	−2.02968
$330$	0	0
$331$	11.9595	0.657353	0.328676	−	0.944443i	$-0.393397\pi$
$331$	11.9595	0.657353	0.328676	+	0.944443i	$0.393397\pi$
$332$	1.12421	0.0616989
$333$	0	0
$334$	1.16699	0.0638547
$335$	28.7695	1.57185
$336$	0	0
$337$	−17.6438	−0.961117	−0.480559	−	0.876963i	$-0.659566\pi$
$337$	−17.6438	−0.961117	−0.480559	+	0.876963i	$0.659566\pi$
$338$	0.574684	0.0312587
$339$	0	0
$340$	−37.3236	−2.02415
$341$	6.97737	0.377846
$342$	0	0
$343$	−5.20108	−0.280832
$344$	−0.496782	−0.0267847
$345$	0	0
$346$	0.481950	0.0259098
$347$	−0.648489	−0.0348127	−0.0174063	−	0.999848i	$-0.505541\pi$
$347$	−0.648489	−0.0348127	−0.0174063	+	0.999848i	$0.505541\pi$
$348$	0	0
$349$	−23.3381	−1.24926	−0.624631	−	0.780920i	$-0.714751\pi$
$349$	−23.3381	−1.24926	−0.624631	+	0.780920i	$0.714751\pi$
$350$	−0.643735	−0.0344091
$351$	0	0
$352$	−0.672570	−0.0358481
$353$	35.7398	1.90224	0.951119	−	0.308825i	$-0.0999356\pi$
$353$	35.7398	1.90224	0.951119	+	0.308825i	$0.0999356\pi$
$354$	0	0
$355$	6.12219	0.324932
$356$	14.4422	0.765437
$357$	0	0
$358$	0.0915332	0.00483768
$359$	−10.9472	−0.577770	−0.288885	−	0.957364i	$-0.593285\pi$
$359$	−10.9472	−0.577770	−0.288885	+	0.957364i	$0.593285\pi$
$360$	0	0
$361$	−0.753598	−0.0396631
$362$	0.381968	0.0200758
$363$	0	0
$364$	−11.7603	−0.616407
$365$	7.31009	0.382628
$366$	0	0
$367$	17.0313	0.889025	0.444513	−	0.895773i	$-0.353377\pi$
$367$	17.0313	0.889025	0.444513	+	0.895773i	$0.353377\pi$
$368$	31.9449	1.66525
$369$	0	0
$370$	−0.133211	−0.00692532
$371$	−28.7140	−1.49076
$372$	0	0
$373$	26.7579	1.38547	0.692735	−	0.721192i	$-0.256406\pi$
$373$	26.7579	1.38547	0.692735	+	0.721192i	$0.256406\pi$
$374$	−0.365766	−0.0189133
$375$	0	0
$376$	−2.33467	−0.120401
$377$	−5.43307	−0.279818
$378$	0	0
$379$	7.77600	0.399426	0.199713	−	0.979854i	$-0.435999\pi$
$379$	7.77600	0.399426	0.199713	+	0.979854i	$0.435999\pi$
$380$	24.4815	1.25588
$381$	0	0
$382$	1.18540	0.0606506
$383$	28.3894	1.45063	0.725316	−	0.688416i	$-0.241694\pi$
$383$	28.3894	1.45063	0.725316	+	0.688416i	$0.241694\pi$
$384$	0	0
$385$	10.1600	0.517801
$386$	0.861857	0.0438673
$387$	0	0
$388$	−22.3279	−1.13353
$389$	−32.7737	−1.66169	−0.830846	−	0.556503i	$-0.812143\pi$
$389$	−32.7737	−1.66169	−0.830846	+	0.556503i	$0.812143\pi$
$390$	0	0
$391$	52.2557	2.64269
$392$	1.24156	0.0627084
$393$	0	0
$394$	0.450413	0.0226915
$395$	−19.5744	−0.984893
$396$	0	0
$397$	23.4981	1.17934	0.589668	−	0.807646i	$-0.299259\pi$
$397$	23.4981	1.17934	0.589668	+	0.807646i	$0.299259\pi$
$398$	−1.06397	−0.0533321
$399$	0	0
$400$	12.8899	0.644497
$401$	5.67692	0.283492	0.141746	−	0.989903i	$-0.454728\pi$
$401$	5.67692	0.283492	0.141746	+	0.989903i	$0.454728\pi$
$402$	0	0
$403$	11.6086	0.578263
$404$	21.7533	1.08227
$405$	0	0
$406$	0.649258	0.0322221
$407$	0.826353	0.0409608
$408$	0	0
$409$	15.8544	0.783950	0.391975	−	0.919976i	$-0.371792\pi$
$409$	15.8544	0.783950	0.391975	+	0.919976i	$0.371792\pi$
$410$	−1.92423	−0.0950311
$411$	0	0
$412$	2.22898	0.109814
$413$	−7.09687	−0.349214
$414$	0	0
$415$	−1.61587	−0.0793201
$416$	−1.11898	−0.0548626
$417$	0	0
$418$	0.239916	0.0117347
$419$	−9.40019	−0.459229	−0.229615	−	0.973282i	$-0.573747\pi$
$419$	−9.40019	−0.459229	−0.229615	+	0.973282i	$0.573747\pi$
$420$	0	0
$421$	19.3416	0.942650	0.471325	−	0.881960i	$-0.343776\pi$
$421$	19.3416	0.942650	0.471325	+	0.881960i	$0.343776\pi$
$422$	0.802538	0.0390669
$423$	0	0
$424$	−1.82093	−0.0884321
$425$	21.0854	1.02279
$426$	0	0
$427$	3.53988	0.171307
$428$	−5.59160	−0.270280
$429$	0	0
$430$	0.356742	0.0172036
$431$	−22.0619	−1.06268	−0.531342	−	0.847158i	$-0.678312\pi$
$431$	−22.0619	−1.06268	−0.531342	+	0.847158i	$0.678312\pi$
$432$	0	0
$433$	16.4389	0.790005	0.395002	−	0.918680i	$-0.370744\pi$
$433$	16.4389	0.790005	0.395002	+	0.918680i	$0.370744\pi$
$434$	−1.38723	−0.0665894
$435$	0	0
$436$	−21.3509	−1.02252
$437$	−34.2760	−1.63964
$438$	0	0
$439$	29.1315	1.39037	0.695185	−	0.718831i	$-0.255322\pi$
$439$	29.1315	1.39037	0.695185	+	0.718831i	$0.255322\pi$
$440$	0.644307	0.0307161
$441$	0	0
$442$	−0.608540	−0.0289453
$443$	39.8835	1.89492	0.947461	−	0.319871i	$-0.103640\pi$
$443$	39.8835	1.89492	0.947461	+	0.319871i	$0.103640\pi$
$444$	0	0
$445$	−20.7585	−0.984046
$446$	0.159845	0.00756888
$447$	0	0
$448$	−28.0514	−1.32530
$449$	6.39145	0.301631	0.150816	−	0.988562i	$-0.451810\pi$
$449$	6.39145	0.301631	0.150816	+	0.988562i	$0.451810\pi$
$450$	0	0
$451$	11.9366	0.562074
$452$	27.8446	1.30970
$453$	0	0
$454$	−1.09429	−0.0513576
$455$	16.9036	0.792453
$456$	0	0
$457$	−6.55699	−0.306723	−0.153362	−	0.988170i	$-0.549010\pi$
$457$	−6.55699	−0.306723	−0.153362	+	0.988170i	$0.549010\pi$
$458$	0.658038	0.0307481
$459$	0	0
$460$	−45.9887	−2.14423
$461$	7.90935	0.368375	0.184188	−	0.982891i	$-0.441035\pi$
$461$	7.90935	0.368375	0.184188	+	0.982891i	$0.441035\pi$
$462$	0	0
$463$	29.9602	1.39237	0.696184	−	0.717864i	$-0.254880\pi$
$463$	29.9602	1.39237	0.696184	+	0.717864i	$0.254880\pi$
$464$	−13.0005	−0.603534
$465$	0	0
$466$	−1.56142	−0.0723315
$467$	−11.4395	−0.529355	−0.264678	−	0.964337i	$-0.585266\pi$
$467$	−11.4395	−0.529355	−0.264678	+	0.964337i	$0.585266\pi$
$468$	0	0
$469$	35.4826	1.63843
$470$	1.67654	0.0773329
$471$	0	0
$472$	−0.450055	−0.0207155
$473$	−2.21299	−0.101753
$474$	0	0
$475$	−13.8305	−0.634587
$476$	−46.0326	−2.10990
$477$	0	0
$478$	0.0716499	0.00327719
$479$	−6.05533	−0.276675	−0.138338	−	0.990385i	$-0.544176\pi$
$479$	−6.05533	−0.276675	−0.138338	+	0.990385i	$0.544176\pi$
$480$	0	0
$481$	1.37484	0.0626872
$482$	−0.842172	−0.0383599
$483$	0	0
$484$	−1.99685	−0.0907657
$485$	32.0930	1.45727
$486$	0	0
$487$	5.74022	0.260114	0.130057	−	0.991507i	$-0.458484\pi$
$487$	5.74022	0.260114	0.130057	+	0.991507i	$0.458484\pi$
$488$	0.224485	0.0101620
$489$	0	0
$490$	−0.891573	−0.0402772
$491$	−5.78303	−0.260984	−0.130492	−	0.991449i	$-0.541656\pi$
$491$	−5.78303	−0.260984	−0.130492	+	0.991449i	$0.541656\pi$
$492$	0	0
$493$	−21.2663	−0.957788
$494$	0.399158	0.0179590
$495$	0	0
$496$	27.7775	1.24725
$497$	7.55073	0.338697
$498$	0	0
$499$	−9.90735	−0.443514	−0.221757	−	0.975102i	$-0.571179\pi$
$499$	−9.90735	−0.443514	−0.221757	+	0.975102i	$0.571179\pi$
$500$	10.0996	0.451670
$501$	0	0
$502$	−1.44088	−0.0643095
$503$	−33.8356	−1.50865	−0.754327	−	0.656499i	$-0.772037\pi$
$503$	−33.8356	−1.50865	−0.754327	+	0.656499i	$0.772037\pi$
$504$	0	0
$505$	−31.2670	−1.39136
$506$	−0.450683	−0.0200353
$507$	0	0
$508$	30.6501	1.35988
$509$	18.3516	0.813422	0.406711	−	0.913557i	$-0.366676\pi$
$509$	18.3516	0.813422	0.406711	+	0.913557i	$0.366676\pi$
$510$	0	0
$511$	9.01582	0.398836
$512$	−4.46494	−0.197324
$513$	0	0
$514$	−1.32123	−0.0582770
$515$	−3.20382	−0.141177
$516$	0	0
$517$	−10.4001	−0.457396
$518$	−0.164295	−0.00721869
$519$	0	0
$520$	1.07196	0.0470086
$521$	−14.9139	−0.653389	−0.326694	−	0.945130i	$-0.605935\pi$
$521$	−14.9139	−0.653389	−0.326694	+	0.945130i	$0.605935\pi$
$522$	0	0
$523$	12.6492	0.553112	0.276556	−	0.960998i	$-0.410807\pi$
$523$	12.6492	0.553112	0.276556	+	0.960998i	$0.410807\pi$
$524$	34.2038	1.49420
$525$	0	0
$526$	−0.424199	−0.0184960
$527$	45.4386	1.97934
$528$	0	0
$529$	41.3875	1.79946
$530$	1.30762	0.0567993
$531$	0	0
$532$	30.1940	1.30908
$533$	19.8595	0.860210
$534$	0	0
$535$	8.03706	0.347472
$536$	2.25016	0.0971923
$537$	0	0
$538$	1.42318	0.0613577
$539$	5.53072	0.238225
$540$	0	0
$541$	16.0561	0.690305	0.345153	−	0.938547i	$-0.387827\pi$
$541$	16.0561	0.690305	0.345153	+	0.938547i	$0.387827\pi$
$542$	1.35075	0.0580196
$543$	0	0
$544$	−4.37996	−0.187789
$545$	30.6886	1.31456
$546$	0	0
$547$	7.91664	0.338491	0.169246	−	0.985574i	$-0.445867\pi$
$547$	7.91664	0.338491	0.169246	+	0.985574i	$0.445867\pi$
$548$	−2.31493	−0.0988889
$549$	0	0
$550$	−0.181853	−0.00775422
$551$	13.9492	0.594254
$552$	0	0
$553$	−24.1418	−1.02662
$554$	−1.45622	−0.0618691
$555$	0	0
$556$	−15.3140	−0.649456
$557$	30.0118	1.27164	0.635821	−	0.771837i	$-0.280662\pi$
$557$	30.0118	1.27164	0.635821	+	0.771837i	$0.280662\pi$
$558$	0	0
$559$	−3.68184	−0.155725
$560$	40.4478	1.70923
$561$	0	0
$562$	−0.948080	−0.0399923
$563$	0.239936	0.0101121	0.00505604	−	0.999987i	$-0.498391\pi$
$563$	0.239936	0.0101121	0.00505604	+	0.999987i	$0.498391\pi$
$564$	0	0
$565$	−40.0223	−1.68375
$566$	−0.799524	−0.0336065
$567$	0	0
$568$	0.478838	0.0200916
$569$	7.76917	0.325701	0.162850	−	0.986651i	$-0.447931\pi$
$569$	7.76917	0.325701	0.162850	+	0.986651i	$0.447931\pi$
$570$	0	0
$571$	−46.4165	−1.94247	−0.971235	−	0.238124i	$-0.923468\pi$
$571$	−46.4165	−1.94247	−0.971235	+	0.238124i	$0.923468\pi$
$572$	−3.32224	−0.138910
$573$	0	0
$574$	−2.37323	−0.0990567
$575$	25.9807	1.08347
$576$	0	0
$577$	41.0776	1.71008	0.855042	−	0.518559i	$-0.173531\pi$
$577$	41.0776	1.71008	0.855042	+	0.518559i	$0.173531\pi$
$578$	−1.42715	−0.0593618
$579$	0	0
$580$	18.7159	0.777133
$581$	−1.99292	−0.0826802
$582$	0	0
$583$	−8.11158	−0.335947
$584$	0.571748	0.0236591
$585$	0	0
$586$	1.41404	0.0584133
$587$	−32.5698	−1.34430	−0.672150	−	0.740415i	$-0.734629\pi$
$587$	−32.5698	−1.34430	−0.672150	+	0.740415i	$0.734629\pi$
$588$	0	0
$589$	−29.8044	−1.22807
$590$	0.323187	0.0133054
$591$	0	0
$592$	3.28978	0.135209
$593$	37.9151	1.55699	0.778493	−	0.627653i	$-0.215984\pi$
$593$	37.9151	1.55699	0.778493	+	0.627653i	$0.215984\pi$
$594$	0	0
$595$	66.1647	2.71249
$596$	−2.37904	−0.0974491
$597$	0	0
$598$	−0.749820	−0.0306624
$599$	6.61867	0.270432	0.135216	−	0.990816i	$-0.456827\pi$
$599$	6.61867	0.270432	0.135216	+	0.990816i	$0.456827\pi$
$600$	0	0
$601$	28.1603	1.14868	0.574341	−	0.818616i	$-0.305258\pi$
$601$	28.1603	1.14868	0.574341	+	0.818616i	$0.305258\pi$
$602$	0.439984	0.0179324
$603$	0	0
$604$	−22.5206	−0.916351
$605$	2.87016	0.116688
$606$	0	0
$607$	−43.2487	−1.75541	−0.877705	−	0.479201i	$-0.840927\pi$
$607$	−43.2487	−1.75541	−0.877705	+	0.479201i	$0.840927\pi$
$608$	2.87293	0.116513
$609$	0	0
$610$	−0.161204	−0.00652695
$611$	−17.3031	−0.700009
$612$	0	0
$613$	−28.6880	−1.15870	−0.579349	−	0.815079i	$-0.696693\pi$
$613$	−28.6880	−1.15870	−0.579349	+	0.815079i	$0.696693\pi$
$614$	−1.13661	−0.0458700
$615$	0	0
$616$	0.794649	0.0320173
$617$	0.710736	0.0286132	0.0143066	−	0.999898i	$-0.495446\pi$
$617$	0.710736	0.0286132	0.0143066	+	0.999898i	$0.495446\pi$
$618$	0	0
$619$	−47.7932	−1.92097	−0.960486	−	0.278329i	$-0.910220\pi$
$619$	−47.7932	−1.92097	−0.960486	+	0.278329i	$0.910220\pi$
$620$	−39.9891	−1.60600
$621$	0	0
$622$	1.15189	0.0461867
$623$	−25.6022	−1.02573
$624$	0	0
$625$	−30.7057	−1.22823
$626$	0.593861	0.0237355
$627$	0	0
$628$	−34.0690	−1.35950
$629$	5.38144	0.214572
$630$	0	0
$631$	−20.5702	−0.818889	−0.409444	−	0.912335i	$-0.634277\pi$
$631$	−20.5702	−0.818889	−0.409444	+	0.912335i	$0.634277\pi$
$632$	−1.53098	−0.0608991
$633$	0	0
$634$	1.14486	0.0454680
$635$	−44.0547	−1.74826
$636$	0	0
$637$	9.20169	0.364584
$638$	0.183413	0.00726138
$639$	0	0
$640$	5.13820	0.203105
$641$	13.0911	0.517069	0.258535	−	0.966002i	$-0.416760\pi$
$641$	13.0911	0.517069	0.258535	+	0.966002i	$0.416760\pi$
$642$	0	0
$643$	23.2585	0.917226	0.458613	−	0.888636i	$-0.348346\pi$
$643$	23.2585	0.917226	0.458613	+	0.888636i	$0.348346\pi$
$644$	−56.7196	−2.23507
$645$	0	0
$646$	1.56240	0.0614717
$647$	−30.4427	−1.19683	−0.598413	−	0.801188i	$-0.704202\pi$
$647$	−30.4427	−1.19683	−0.598413	+	0.801188i	$0.704202\pi$
$648$	0	0
$649$	−2.00484	−0.0786967
$650$	−0.302556	−0.0118672
$651$	0	0
$652$	37.6047	1.47271
$653$	28.5204	1.11609	0.558045	−	0.829811i	$-0.311552\pi$
$653$	28.5204	1.11609	0.558045	+	0.829811i	$0.311552\pi$
$654$	0	0
$655$	−49.1627	−1.92094
$656$	47.5207	1.85537
$657$	0	0
$658$	2.06774	0.0806089
$659$	−9.78396	−0.381129	−0.190564	−	0.981675i	$-0.561032\pi$
$659$	−9.78396	−0.381129	−0.190564	+	0.981675i	$0.561032\pi$
$660$	0	0
$661$	−26.0822	−1.01448	−0.507239	−	0.861805i	$-0.669334\pi$
$661$	−26.0822	−1.01448	−0.507239	+	0.861805i	$0.669334\pi$
$662$	−0.671711	−0.0261068
$663$	0	0
$664$	−0.126383	−0.00490462
$665$	−43.3992	−1.68295
$666$	0	0
$667$	−26.2036	−1.01461
$668$	41.4897	1.60529
$669$	0	0
$670$	−1.61586	−0.0624259
$671$	1.00000	0.0386046
$672$	0	0
$673$	−6.37530	−0.245750	−0.122875	−	0.992422i	$-0.539211\pi$
$673$	−6.37530	−0.245750	−0.122875	+	0.992422i	$0.539211\pi$
$674$	0.990972	0.0381708
$675$	0	0
$676$	20.4316	0.785833
$677$	22.2919	0.856747	0.428374	−	0.903602i	$-0.359087\pi$
$677$	22.2919	0.856747	0.428374	+	0.903602i	$0.359087\pi$
$678$	0	0
$679$	39.5815	1.51900
$680$	4.19591	0.160906
$681$	0	0
$682$	−0.391888	−0.0150062
$683$	−24.9233	−0.953662	−0.476831	−	0.878995i	$-0.658215\pi$
$683$	−24.9233	−0.953662	−0.476831	+	0.878995i	$0.658215\pi$
$684$	0	0
$685$	3.32735	0.127132
$686$	0.292121	0.0111532
$687$	0	0
$688$	−8.81008	−0.335881
$689$	−13.4956	−0.514141
$690$	0	0
$691$	−0.490891	−0.0186744	−0.00933718	−	0.999956i	$-0.502972\pi$
$691$	−0.490891	−0.0186744	−0.00933718	+	0.999956i	$0.502972\pi$
$692$	17.1347	0.651364
$693$	0	0
$694$	0.0364227	0.00138259
$695$	22.0114	0.834941
$696$	0	0
$697$	77.7347	2.94441
$698$	1.31080	0.0496145
$699$	0	0
$700$	−22.8866	−0.865033
$701$	12.9149	0.487791	0.243895	−	0.969802i	$-0.421575\pi$
$701$	12.9149	0.487791	0.243895	+	0.969802i	$0.421575\pi$
$702$	0	0
$703$	−3.52983	−0.133130
$704$	−7.92439	−0.298662
$705$	0	0
$706$	−2.00735	−0.0755474
$707$	−38.5628	−1.45031
$708$	0	0
$709$	−21.6649	−0.813642	−0.406821	−	0.913508i	$-0.633363\pi$
$709$	−21.6649	−0.813642	−0.406821	+	0.913508i	$0.633363\pi$
$710$	−0.343856	−0.0129047
$711$	0	0
$712$	−1.62359	−0.0608467
$713$	55.9877	2.09676
$714$	0	0
$715$	4.77520	0.178582
$716$	3.25427	0.121618
$717$	0	0
$718$	0.614854	0.0229462
$719$	−3.05414	−0.113900	−0.0569502	−	0.998377i	$-0.518138\pi$
$719$	−3.05414	−0.113900	−0.0569502	+	0.998377i	$0.518138\pi$
$720$	0	0
$721$	−3.95140	−0.147158
$722$	0.0423262	0.00157522
$723$	0	0
$724$	13.5800	0.504698
$725$	−10.5733	−0.392681
$726$	0	0
$727$	−11.0808	−0.410964	−0.205482	−	0.978661i	$-0.565876\pi$
$727$	−11.0808	−0.410964	−0.205482	+	0.978661i	$0.565876\pi$
$728$	1.32209	0.0489999
$729$	0	0
$730$	−0.410575	−0.0151961
$731$	−14.4116	−0.533032
$732$	0	0
$733$	−10.7441	−0.396843	−0.198421	−	0.980117i	$-0.563581\pi$
$733$	−10.7441	−0.396843	−0.198421	+	0.980117i	$0.563581\pi$
$734$	−0.956571	−0.0353077
$735$	0	0
$736$	−5.39682	−0.198929
$737$	10.0237	0.369227
$738$	0	0
$739$	−25.1340	−0.924569	−0.462284	−	0.886732i	$-0.652970\pi$
$739$	−25.1340	−0.924569	−0.462284	+	0.886732i	$0.652970\pi$
$740$	−4.73604	−0.174100
$741$	0	0
$742$	1.61274	0.0592054
$743$	25.0852	0.920285	0.460143	−	0.887845i	$-0.347798\pi$
$743$	25.0852	0.920285	0.460143	+	0.887845i	$0.347798\pi$
$744$	0	0
$745$	3.41950	0.125281
$746$	−1.50287	−0.0550240
$747$	0	0
$748$	−13.0040	−0.475474
$749$	9.91242	0.362192
$750$	0	0
$751$	40.8602	1.49101	0.745505	−	0.666500i	$-0.232209\pi$
$751$	40.8602	1.49101	0.745505	+	0.666500i	$0.232209\pi$
$752$	−41.4037	−1.50984
$753$	0	0
$754$	0.305151	0.0111130
$755$	32.3699	1.17806
$756$	0	0
$757$	−23.1061	−0.839807	−0.419903	−	0.907569i	$-0.637936\pi$
$757$	−23.1061	−0.839807	−0.419903	+	0.907569i	$0.637936\pi$
$758$	−0.436743	−0.0158632
$759$	0	0
$760$	−2.75221	−0.0998331
$761$	24.9673	0.905066	0.452533	−	0.891748i	$-0.350521\pi$
$761$	24.9673	0.905066	0.452533	+	0.891748i	$0.350521\pi$
$762$	0	0
$763$	37.8495	1.37024
$764$	42.1445	1.52474
$765$	0	0
$766$	−1.59451	−0.0576119
$767$	−3.33553	−0.120439
$768$	0	0
$769$	−39.7456	−1.43326	−0.716632	−	0.697452i	$-0.754317\pi$
$769$	−39.7456	−1.43326	−0.716632	+	0.697452i	$0.754317\pi$
$770$	−0.570642	−0.0205645
$771$	0	0
$772$	30.6415	1.10281
$773$	−47.6555	−1.71405	−0.857025	−	0.515276i	$-0.827690\pi$
$773$	−47.6555	−1.71405	−0.857025	+	0.515276i	$0.827690\pi$
$774$	0	0
$775$	22.5913	0.811504
$776$	2.51010	0.0901074
$777$	0	0
$778$	1.84075	0.0659941
$779$	−50.9883	−1.82685
$780$	0	0
$781$	2.13305	0.0763265
$782$	−2.93497	−0.104954
$783$	0	0
$784$	22.0182	0.786366
$785$	48.9690	1.74778
$786$	0	0
$787$	25.6637	0.914811	0.457405	−	0.889258i	$-0.348779\pi$
$787$	25.6637	0.914811	0.457405	+	0.889258i	$0.348779\pi$
$788$	16.0135	0.570457
$789$	0	0
$790$	1.09940	0.0391151
$791$	−49.3611	−1.75508
$792$	0	0
$793$	1.66374	0.0590812
$794$	−1.31978	−0.0468374
$795$	0	0
$796$	−37.8273	−1.34075
$797$	−8.44244	−0.299047	−0.149523	−	0.988758i	$-0.547774\pi$
$797$	−8.44244	−0.299047	−0.149523	+	0.988758i	$0.547774\pi$
$798$	0	0
$799$	−67.7284	−2.39606
$800$	−2.17764	−0.0769913
$801$	0	0
$802$	−0.318848	−0.0112589
$803$	2.54693	0.0898793
$804$	0	0
$805$	81.5257	2.87340
$806$	−0.652000	−0.0229657
$807$	0	0
$808$	−2.44550	−0.0860325
$809$	−14.1868	−0.498780	−0.249390	−	0.968403i	$-0.580230\pi$
$809$	−14.1868	−0.498780	−0.249390	+	0.968403i	$0.580230\pi$
$810$	0	0
$811$	−2.52616	−0.0887054	−0.0443527	−	0.999016i	$-0.514123\pi$
$811$	−2.52616	−0.0887054	−0.0443527	+	0.999016i	$0.514123\pi$
$812$	23.0830	0.810054
$813$	0	0
$814$	−0.0464125	−0.00162676
$815$	−54.0510	−1.89332
$816$	0	0
$817$	9.45295	0.330717
$818$	−0.890471	−0.0311346
$819$	0	0
$820$	−68.4120	−2.38905
$821$	−14.9783	−0.522746	−0.261373	−	0.965238i	$-0.584175\pi$
$821$	−14.9783	−0.522746	−0.261373	+	0.965238i	$0.584175\pi$
$822$	0	0
$823$	1.14435	0.0398894	0.0199447	−	0.999801i	$-0.493651\pi$
$823$	1.14435	0.0398894	0.0199447	+	0.999801i	$0.493651\pi$
$824$	−0.250582	−0.00872944
$825$	0	0
$826$	0.398599	0.0138690
$827$	−40.5209	−1.40905	−0.704525	−	0.709679i	$-0.748840\pi$
$827$	−40.5209	−1.40905	−0.704525	+	0.709679i	$0.748840\pi$
$828$	0	0
$829$	−41.9288	−1.45625	−0.728124	−	0.685446i	$-0.759607\pi$
$829$	−41.9288	−1.45625	−0.728124	+	0.685446i	$0.759607\pi$
$830$	0.0907564	0.00315020
$831$	0	0
$832$	−13.1841	−0.457078
$833$	36.0176	1.24794
$834$	0	0
$835$	−59.6351	−2.06376
$836$	8.52969	0.295005
$837$	0	0
$838$	0.527967	0.0182383
$839$	17.9705	0.620412	0.310206	−	0.950669i	$-0.399602\pi$
$839$	17.9705	0.620412	0.310206	+	0.950669i	$0.399602\pi$
$840$	0	0
$841$	−18.3360	−0.632277
$842$	−1.08633	−0.0374374
$843$	0	0
$844$	28.5325	0.982130
$845$	−29.3673	−1.01027
$846$	0	0
$847$	3.53988	0.121632
$848$	−32.2929	−1.10894
$849$	0	0
$850$	−1.18427	−0.0406203
$851$	6.63081	0.227301
$852$	0	0
$853$	−29.2979	−1.00314	−0.501571	−	0.865117i	$-0.667244\pi$
$853$	−29.2979	−1.00314	−0.501571	+	0.865117i	$0.667244\pi$
$854$	−0.198819	−0.00680345
$855$	0	0
$856$	0.628606	0.0214853
$857$	−13.4529	−0.459543	−0.229772	−	0.973245i	$-0.573798\pi$
$857$	−13.4529	−0.459543	−0.229772	+	0.973245i	$0.573798\pi$
$858$	0	0
$859$	11.4298	0.389981	0.194990	−	0.980805i	$-0.437532\pi$
$859$	11.4298	0.389981	0.194990	+	0.980805i	$0.437532\pi$
$860$	12.6832	0.432493
$861$	0	0
$862$	1.23912	0.0422045
$863$	−22.5690	−0.768259	−0.384129	−	0.923279i	$-0.625498\pi$
$863$	−22.5690	−0.768259	−0.384129	+	0.923279i	$0.625498\pi$
$864$	0	0
$865$	−24.6285	−0.837394
$866$	−0.923301	−0.0313751
$867$	0	0
$868$	−49.3202	−1.67404
$869$	−6.81997	−0.231352
$870$	0	0
$871$	16.6768	0.565072
$872$	2.40027	0.0812833
$873$	0	0
$874$	1.92513	0.0651184
$875$	−17.9040	−0.605265
$876$	0	0
$877$	42.1546	1.42346	0.711729	−	0.702454i	$-0.247912\pi$
$877$	42.1546	1.42346	0.711729	+	0.702454i	$0.247912\pi$
$878$	−1.63619	−0.0552186
$879$	0	0
$880$	11.4263	0.385181
$881$	−4.79100	−0.161413	−0.0807065	−	0.996738i	$-0.525718\pi$
$881$	−4.79100	−0.161413	−0.0807065	+	0.996738i	$0.525718\pi$
$882$	0	0
$883$	35.8055	1.20495	0.602476	−	0.798137i	$-0.294181\pi$
$883$	35.8055	1.20495	0.602476	+	0.798137i	$0.294181\pi$
$884$	−21.6353	−0.727675
$885$	0	0
$886$	−2.24008	−0.0752569
$887$	−17.9056	−0.601210	−0.300605	−	0.953749i	$-0.597189\pi$
$887$	−17.9056	−0.601210	−0.300605	+	0.953749i	$0.597189\pi$
$888$	0	0
$889$	−54.3344	−1.82232
$890$	1.16591	0.0390814
$891$	0	0
$892$	5.68295	0.190279
$893$	44.4249	1.48662
$894$	0	0
$895$	−4.67751	−0.156352
$896$	6.33714	0.211709
$897$	0	0
$898$	−0.358979	−0.0119793
$899$	−22.7851	−0.759927
$900$	0	0
$901$	−52.8249	−1.75985
$902$	−0.670428	−0.0223228
$903$	0	0
$904$	−3.13029	−0.104112
$905$	−19.5192	−0.648841
$906$	0	0
$907$	16.6708	0.553544	0.276772	−	0.960936i	$-0.410735\pi$
$907$	16.6708	0.553544	0.276772	+	0.960936i	$0.410735\pi$
$908$	−38.9052	−1.29111
$909$	0	0
$910$	−0.949400	−0.0314723
$911$	36.4248	1.20681	0.603403	−	0.797436i	$-0.293811\pi$
$911$	36.4248	1.20681	0.603403	+	0.797436i	$0.293811\pi$
$912$	0	0
$913$	−0.562991	−0.0186323
$914$	0.368277	0.0121815
$915$	0	0
$916$	23.3952	0.772998
$917$	−60.6342	−2.00232
$918$	0	0
$919$	4.21597	0.139072	0.0695359	−	0.997579i	$-0.477848\pi$
$919$	4.21597	0.139072	0.0695359	+	0.997579i	$0.477848\pi$
$920$	5.17004	0.170451
$921$	0	0
$922$	−0.444233	−0.0146300
$923$	3.54884	0.116812
$924$	0	0
$925$	2.67556	0.0879719
$926$	−1.68273	−0.0552979
$927$	0	0
$928$	2.19633	0.0720979
$929$	−28.4286	−0.932713	−0.466356	−	0.884597i	$-0.654434\pi$
$929$	−28.4286	−0.932713	−0.466356	+	0.884597i	$0.654434\pi$
$930$	0	0
$931$	−23.6249	−0.774275
$932$	−55.5131	−1.81839
$933$	0	0
$934$	0.642503	0.0210234
$935$	18.6913	0.611270
$936$	0	0
$937$	−20.5654	−0.671843	−0.335921	−	0.941890i	$-0.609048\pi$
$937$	−20.5654	−0.671843	−0.335921	+	0.941890i	$0.609048\pi$
$938$	−1.99290	−0.0650704
$939$	0	0
$940$	59.6057	1.94412
$941$	7.87253	0.256637	0.128318	−	0.991733i	$-0.459042\pi$
$941$	7.87253	0.256637	0.128318	+	0.991733i	$0.459042\pi$
$942$	0	0
$943$	95.7818	3.11908
$944$	−7.98142	−0.259773
$945$	0	0
$946$	0.124294	0.00404113
$947$	−51.6536	−1.67852	−0.839258	−	0.543734i	$-0.817010\pi$
$947$	−51.6536	−1.67852	−0.839258	+	0.543734i	$0.817010\pi$
$948$	0	0
$949$	4.23744	0.137553
$950$	0.776798	0.0252026
$951$	0	0
$952$	5.17498	0.167722
$953$	50.9942	1.65186	0.825932	−	0.563769i	$-0.190649\pi$
$953$	50.9942	1.65186	0.825932	+	0.563769i	$0.190649\pi$
$954$	0	0
$955$	−60.5762	−1.96020
$956$	2.54736	0.0823875
$957$	0	0
$958$	0.340101	0.0109882
$959$	4.10375	0.132517
$960$	0	0
$961$	17.6838	0.570444
$962$	−0.0772185	−0.00248962
$963$	0	0
$964$	−29.9416	−0.964355
$965$	−44.0424	−1.41777
$966$	0	0
$967$	31.7512	1.02105	0.510524	−	0.859863i	$-0.329451\pi$
$967$	31.7512	1.02105	0.510524	+	0.859863i	$0.329451\pi$
$968$	0.224485	0.00721522
$969$	0	0
$970$	−1.80252	−0.0578754
$971$	−27.2104	−0.873222	−0.436611	−	0.899650i	$-0.643821\pi$
$971$	−27.2104	−0.873222	−0.436611	+	0.899650i	$0.643821\pi$
$972$	0	0
$973$	27.1476	0.870311
$974$	−0.322402	−0.0103304
$975$	0	0
$976$	3.98108	0.127431
$977$	−49.0474	−1.56916	−0.784582	−	0.620025i	$-0.787123\pi$
$977$	−49.0474	−1.56916	−0.784582	+	0.620025i	$0.787123\pi$
$978$	0	0
$979$	−7.23252	−0.231152
$980$	−31.6980	−1.01255
$981$	0	0
$982$	0.324807	0.0103650
$983$	−29.9573	−0.955490	−0.477745	−	0.878499i	$-0.658546\pi$
$983$	−29.9573	−0.955490	−0.477745	+	0.878499i	$0.658546\pi$
$984$	0	0
$985$	−23.0169	−0.733380
$986$	1.19444	0.0380386
$987$	0	0
$988$	14.1912	0.451482
$989$	−17.7574	−0.564653
$990$	0	0
$991$	−45.8678	−1.45704	−0.728519	−	0.685026i	$-0.759791\pi$
$991$	−45.8678	−1.45704	−0.728519	+	0.685026i	$0.759791\pi$
$992$	−4.69277	−0.148996
$993$	0	0
$994$	−0.424091	−0.0134513
$995$	54.3709	1.72367
$996$	0	0
$997$	−17.7103	−0.560892	−0.280446	−	0.959870i	$-0.590482\pi$
$997$	−17.7103	−0.560892	−0.280446	+	0.959870i	$0.590482\pi$
$998$	0.556452	0.0176142
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.j.1.7		14
3.2	odd	2		2013.2.a.h.1.8	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.h.1.8	✓	14	3.2	odd	2
6039.2.a.j.1.7		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)

\(f(q)\)	\(=\)	\(q-0.0561655 q^{2} -1.99685 q^{4} +2.87016 q^{5} +3.53988 q^{7} +0.224485 q^{8} +O(q^{10})\)	\(q-0.0561655 q^{2} -1.99685 q^{4} +2.87016 q^{5} +3.53988 q^{7} +0.224485 q^{8} -0.161204 q^{10} +1.00000 q^{11} +1.66374 q^{13} -0.198819 q^{14} +3.98108 q^{16} +6.51228 q^{17} -4.27158 q^{19} -5.73126 q^{20} -0.0561655 q^{22} +8.02418 q^{23} +3.23780 q^{25} -0.0934449 q^{26} -7.06858 q^{28} -3.26557 q^{29} +6.97737 q^{31} -0.672570 q^{32} -0.365766 q^{34} +10.1600 q^{35} +0.826353 q^{37} +0.239916 q^{38} +0.644307 q^{40} +11.9366 q^{41} -2.21299 q^{43} -1.99685 q^{44} -0.450683 q^{46} -10.4001 q^{47} +5.53072 q^{49} -0.181853 q^{50} -3.32224 q^{52} -8.11158 q^{53} +2.87016 q^{55} +0.794649 q^{56} +0.183413 q^{58} -2.00484 q^{59} +1.00000 q^{61} -0.391888 q^{62} -7.92439 q^{64} +4.77520 q^{65} +10.0237 q^{67} -13.0040 q^{68} -0.570642 q^{70} +2.13305 q^{71} +2.54693 q^{73} -0.0464125 q^{74} +8.52969 q^{76} +3.53988 q^{77} -6.81997 q^{79} +11.4263 q^{80} -0.670428 q^{82} -0.562991 q^{83} +18.6913 q^{85} +0.124294 q^{86} +0.224485 q^{88} -7.23252 q^{89} +5.88944 q^{91} -16.0231 q^{92} +0.584128 q^{94} -12.2601 q^{95} +11.1816 q^{97} -0.310636 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) \) 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

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