LMFDB - Embedded newform 6039.2.a.e.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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} + \cdots + 1 $ x^12 - x^11 - 16x^10 + 13x^9 + 93x^8 - 59x^7 - 238x^6 + 108x^5 + 257x^4 - 71x^3 - 93x^2 + 13x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.0557908$ 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)

$f(q)$	$=$	$q+0.0557908 q^{2} -1.99689 q^{4} +3.85264 q^{5} -0.441454 q^{7} -0.222990 q^{8} +O(q^{10})$	\(q+0.0557908 q^{2} -1.99689 q^{4} +3.85264 q^{5} -0.441454 q^{7} -0.222990 q^{8} +0.214942 q^{10} +1.00000 q^{11} +0.753369 q^{13} -0.0246291 q^{14} +3.98133 q^{16} -4.62965 q^{17} +1.94492 q^{19} -7.69330 q^{20} +0.0557908 q^{22} -4.57610 q^{23} +9.84287 q^{25} +0.0420311 q^{26} +0.881533 q^{28} -6.43394 q^{29} -8.50703 q^{31} +0.668101 q^{32} -0.258292 q^{34} -1.70076 q^{35} -2.46764 q^{37} +0.108509 q^{38} -0.859100 q^{40} -0.878330 q^{41} -12.5343 q^{43} -1.99689 q^{44} -0.255304 q^{46} +6.11430 q^{47} -6.80512 q^{49} +0.549142 q^{50} -1.50439 q^{52} -8.63319 q^{53} +3.85264 q^{55} +0.0984396 q^{56} -0.358955 q^{58} +11.9080 q^{59} -1.00000 q^{61} -0.474614 q^{62} -7.92539 q^{64} +2.90246 q^{65} +5.71519 q^{67} +9.24490 q^{68} -0.0948870 q^{70} -10.4969 q^{71} -2.39728 q^{73} -0.137672 q^{74} -3.88379 q^{76} -0.441454 q^{77} +14.6611 q^{79} +15.3387 q^{80} -0.0490028 q^{82} -8.84232 q^{83} -17.8364 q^{85} -0.699301 q^{86} -0.222990 q^{88} -4.49383 q^{89} -0.332577 q^{91} +9.13796 q^{92} +0.341122 q^{94} +7.49309 q^{95} +4.95529 q^{97} -0.379663 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) $ 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8	$ 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) $ 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8 - 8 * q^10 + 12 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 20 * q^19 + 9 * q^20 - q^22 + 9 * q^23 + 3 * q^25 + 18 * q^26 - 31 * q^28 - 18 * q^29 - 21 * q^31 - 18 * q^32 - 12 * q^34 + 4 * q^35 - 18 * q^37 + 2 * q^38 - 26 * q^40 - 15 * q^41 - 33 * q^43 + 9 * q^44 - 28 * q^46 + 20 * q^47 + 15 * q^49 + 2 * q^50 - 27 * q^52 + 3 * q^55 + 8 * q^56 - 11 * q^58 + 21 * q^59 - 12 * q^61 + 9 * q^62 - 12 * q^64 - 17 * q^65 - 34 * q^67 + 16 * q^68 - 36 * q^70 + 5 * q^71 - 2 * q^73 - 6 * q^74 - 27 * q^76 - 9 * q^77 - 31 * q^79 + 60 * q^80 - 12 * q^82 + 32 * q^83 - 40 * q^85 - 18 * q^86 - 6 * q^88 - 27 * q^89 - 45 * q^91 + 78 * q^92 - 13 * q^94 - 37 * q^95 - 19 * q^97 - 4 * 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 $

$2$	0.0557908	0.0394501	0.0197250	−	0.999805i	$-0.493721\pi$
$2$	0.0557908	0.0394501	0.0197250	+	0.999805i	$0.493721\pi$
$3$	0	0
$3$	0	0
$4$	−1.99689	−0.998444
$5$	3.85264	1.72295	0.861477	−	0.507796i	$-0.169540\pi$
$5$	3.85264	1.72295	0.861477	+	0.507796i	$0.169540\pi$
$6$	0	0
$7$	−0.441454	−0.166854	−0.0834269	−	0.996514i	$-0.526587\pi$
$7$	−0.441454	−0.166854	−0.0834269	+	0.996514i	$0.526587\pi$
$8$	−0.222990	−0.0788387
$9$	0	0
$10$	0.214942	0.0679707
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.753369	0.208947	0.104473	−	0.994528i	$-0.466684\pi$
$13$	0.753369	0.208947	0.104473	+	0.994528i	$0.466684\pi$
$14$	−0.0246291	−0.00658240
$15$	0	0
$16$	3.98133	0.995333
$17$	−4.62965	−1.12286	−0.561428	−	0.827526i	$-0.689748\pi$
$17$	−4.62965	−1.12286	−0.561428	+	0.827526i	$0.689748\pi$
$18$	0	0
$19$	1.94492	0.446196	0.223098	−	0.974796i	$-0.428383\pi$
$19$	1.94492	0.446196	0.223098	+	0.974796i	$0.428383\pi$
$20$	−7.69330	−1.72027
$21$	0	0
$22$	0.0557908	0.0118946
$23$	−4.57610	−0.954183	−0.477091	−	0.878854i	$-0.658309\pi$
$23$	−4.57610	−0.954183	−0.477091	+	0.878854i	$0.658309\pi$
$24$	0	0
$25$	9.84287	1.96857
$26$	0.0420311	0.00824297
$27$	0	0
$28$	0.881533	0.166594
$29$	−6.43394	−1.19475	−0.597377	−	0.801961i	$-0.703790\pi$
$29$	−6.43394	−1.19475	−0.597377	+	0.801961i	$0.703790\pi$
$30$	0	0
$31$	−8.50703	−1.52791	−0.763954	−	0.645271i	$-0.776745\pi$
$31$	−8.50703	−1.52791	−0.763954	+	0.645271i	$0.776745\pi$
$32$	0.668101	0.118105
$33$	0	0
$34$	−0.258292	−0.0442968
$35$	−1.70076	−0.287482
$36$	0	0
$37$	−2.46764	−0.405678	−0.202839	−	0.979212i	$-0.565017\pi$
$37$	−2.46764	−0.405678	−0.202839	+	0.979212i	$0.565017\pi$
$38$	0.108509	0.0176025
$39$	0	0
$40$	−0.859100	−0.135836
$41$	−0.878330	−0.137172	−0.0685861	−	0.997645i	$-0.521849\pi$
$41$	−0.878330	−0.137172	−0.0685861	+	0.997645i	$0.521849\pi$
$42$	0	0
$43$	−12.5343	−1.91147	−0.955734	−	0.294232i	$-0.904936\pi$
$43$	−12.5343	−1.91147	−0.955734	+	0.294232i	$0.904936\pi$
$44$	−1.99689	−0.301042
$45$	0	0
$46$	−0.255304	−0.0376426
$47$	6.11430	0.891862	0.445931	−	0.895067i	$-0.352873\pi$
$47$	6.11430	0.891862	0.445931	+	0.895067i	$0.352873\pi$
$48$	0	0
$49$	−6.80512	−0.972160
$50$	0.549142	0.0776604
$51$	0	0
$52$	−1.50439	−0.208622
$53$	−8.63319	−1.18586	−0.592930	−	0.805254i	$-0.702029\pi$
$53$	−8.63319	−1.18586	−0.592930	+	0.805254i	$0.702029\pi$
$54$	0	0
$55$	3.85264	0.519490
$56$	0.0984396	0.0131545
$57$	0	0
$58$	−0.358955	−0.0471331
$59$	11.9080	1.55029	0.775143	−	0.631785i	$-0.217678\pi$
$59$	11.9080	1.55029	0.775143	+	0.631785i	$0.217678\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−0.474614	−0.0602761
$63$	0	0
$64$	−7.92539	−0.990674
$65$	2.90246	0.360006
$66$	0	0
$67$	5.71519	0.698222	0.349111	−	0.937081i	$-0.386484\pi$
$67$	5.71519	0.698222	0.349111	+	0.937081i	$0.386484\pi$
$68$	9.24490	1.12111
$69$	0	0
$70$	−0.0948870	−0.0113412
$71$	−10.4969	−1.24575	−0.622876	−	0.782321i	$-0.714036\pi$
$71$	−10.4969	−1.24575	−0.622876	+	0.782321i	$0.714036\pi$
$72$	0	0
$73$	−2.39728	−0.280580	−0.140290	−	0.990110i	$-0.544804\pi$
$73$	−2.39728	−0.280580	−0.140290	+	0.990110i	$0.544804\pi$
$74$	−0.137672	−0.0160040
$75$	0	0
$76$	−3.88379	−0.445501
$77$	−0.441454	−0.0503083
$78$	0	0
$79$	14.6611	1.64950	0.824750	−	0.565497i	$-0.191316\pi$
$79$	14.6611	1.64950	0.824750	+	0.565497i	$0.191316\pi$
$80$	15.3387	1.71491
$81$	0	0
$82$	−0.0490028	−0.00541145
$83$	−8.84232	−0.970571	−0.485285	−	0.874356i	$-0.661284\pi$
$83$	−8.84232	−0.970571	−0.485285	+	0.874356i	$0.661284\pi$
$84$	0	0
$85$	−17.8364	−1.93463
$86$	−0.699301	−0.0754075
$87$	0	0
$88$	−0.222990	−0.0237708
$89$	−4.49383	−0.476345	−0.238173	−	0.971223i	$-0.576548\pi$
$89$	−4.49383	−0.476345	−0.238173	+	0.971223i	$0.576548\pi$
$90$	0	0
$91$	−0.332577	−0.0348636
$92$	9.13796	0.952698
$93$	0	0
$94$	0.341122	0.0351840
$95$	7.49309	0.768775
$96$	0	0
$97$	4.95529	0.503133	0.251567	−	0.967840i	$-0.419054\pi$
$97$	4.95529	0.503133	0.251567	+	0.967840i	$0.419054\pi$
$98$	−0.379663	−0.0383518
$99$	0	0
$100$	−19.6551	−1.96551
$101$	−5.46677	−0.543964	−0.271982	−	0.962302i	$-0.587679\pi$
$101$	−5.46677	−0.543964	−0.271982	+	0.962302i	$0.587679\pi$
$102$	0	0
$103$	0.558769	0.0550571	0.0275286	−	0.999621i	$-0.491236\pi$
$103$	0.558769	0.0550571	0.0275286	+	0.999621i	$0.491236\pi$
$104$	−0.167993	−0.0164731
$105$	0	0
$106$	−0.481653	−0.0467822
$107$	1.19369	0.115399	0.0576993	−	0.998334i	$-0.481624\pi$
$107$	1.19369	0.115399	0.0576993	+	0.998334i	$0.481624\pi$
$108$	0	0
$109$	2.00887	0.192415	0.0962073	−	0.995361i	$-0.469329\pi$
$109$	2.00887	0.192415	0.0962073	+	0.995361i	$0.469329\pi$
$110$	0.214942	0.0204939
$111$	0	0
$112$	−1.75757	−0.166075
$113$	7.98832	0.751478	0.375739	−	0.926726i	$-0.377389\pi$
$113$	7.98832	0.751478	0.375739	+	0.926726i	$0.377389\pi$
$114$	0	0
$115$	−17.6301	−1.64401
$116$	12.8479	1.19289
$117$	0	0
$118$	0.664356	0.0611589
$119$	2.04378	0.187353
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.0557908	−0.00505106
$123$	0	0
$124$	16.9876	1.52553
$125$	18.6578	1.66881
$126$	0	0
$127$	−9.70800	−0.861445	−0.430723	−	0.902484i	$-0.641741\pi$
$127$	−9.70800	−0.861445	−0.430723	+	0.902484i	$0.641741\pi$
$128$	−1.77837	−0.157187
$129$	0	0
$130$	0.161931	0.0142023
$131$	1.94083	0.169571	0.0847855	−	0.996399i	$-0.472980\pi$
$131$	1.94083	0.169571	0.0847855	+	0.996399i	$0.472980\pi$
$132$	0	0
$133$	−0.858593	−0.0744495
$134$	0.318855	0.0275449
$135$	0	0
$136$	1.03237	0.0885246
$137$	−11.3790	−0.972172	−0.486086	−	0.873911i	$-0.661576\pi$
$137$	−11.3790	−0.972172	−0.486086	+	0.873911i	$0.661576\pi$
$138$	0	0
$139$	−16.9073	−1.43406	−0.717030	−	0.697043i	$-0.754499\pi$
$139$	−16.9073	−1.43406	−0.717030	+	0.697043i	$0.754499\pi$
$140$	3.39623	0.287034
$141$	0	0
$142$	−0.585630	−0.0491450
$143$	0.753369	0.0629999
$144$	0	0
$145$	−24.7877	−2.05851
$146$	−0.133746	−0.0110689
$147$	0	0
$148$	4.92760	0.405047
$149$	7.08406	0.580348	0.290174	−	0.956974i	$-0.406287\pi$
$149$	7.08406	0.580348	0.290174	+	0.956974i	$0.406287\pi$
$150$	0	0
$151$	−22.6679	−1.84469	−0.922343	−	0.386372i	$-0.873728\pi$
$151$	−22.6679	−1.84469	−0.922343	+	0.386372i	$0.873728\pi$
$152$	−0.433698	−0.0351775
$153$	0	0
$154$	−0.0246291	−0.00198467
$155$	−32.7746	−2.63252
$156$	0	0
$157$	−15.6861	−1.25189	−0.625943	−	0.779869i	$-0.715286\pi$
$157$	−15.6861	−1.25189	−0.625943	+	0.779869i	$0.715286\pi$
$158$	0.817954	0.0650729
$159$	0	0
$160$	2.57396	0.203489
$161$	2.02014	0.159209
$162$	0	0
$163$	17.9933	1.40934	0.704671	−	0.709534i	$-0.251094\pi$
$163$	17.9933	1.40934	0.704671	+	0.709534i	$0.251094\pi$
$164$	1.75393	0.136959
$165$	0	0
$166$	−0.493320	−0.0382891
$167$	−6.97705	−0.539900	−0.269950	−	0.962874i	$-0.587007\pi$
$167$	−6.97705	−0.539900	−0.269950	+	0.962874i	$0.587007\pi$
$168$	0	0
$169$	−12.4324	−0.956341
$170$	−0.995108	−0.0763213
$171$	0	0
$172$	25.0297	1.90849
$173$	19.3710	1.47275	0.736375	−	0.676574i	$-0.236536\pi$
$173$	19.3710	1.47275	0.736375	+	0.676574i	$0.236536\pi$
$174$	0	0
$175$	−4.34517	−0.328464
$176$	3.98133	0.300104
$177$	0	0
$178$	−0.250715	−0.0187918
$179$	18.1478	1.35643	0.678216	−	0.734862i	$-0.262753\pi$
$179$	18.1478	1.35643	0.678216	+	0.734862i	$0.262753\pi$
$180$	0	0
$181$	−5.42045	−0.402899	−0.201449	−	0.979499i	$-0.564565\pi$
$181$	−5.42045	−0.402899	−0.201449	+	0.979499i	$0.564565\pi$
$182$	−0.0185548	−0.00137537
$183$	0	0
$184$	1.02042	0.0752266
$185$	−9.50695	−0.698965
$186$	0	0
$187$	−4.62965	−0.338554
$188$	−12.2096	−0.890474
$189$	0	0
$190$	0.418046	0.0303282
$191$	11.9976	0.868119	0.434059	−	0.900884i	$-0.357081\pi$
$191$	11.9976	0.868119	0.434059	+	0.900884i	$0.357081\pi$
$192$	0	0
$193$	7.08933	0.510301	0.255151	−	0.966901i	$-0.417875\pi$
$193$	7.08933	0.510301	0.255151	+	0.966901i	$0.417875\pi$
$194$	0.276460	0.0198486
$195$	0	0
$196$	13.5891	0.970647
$197$	9.90659	0.705815	0.352908	−	0.935658i	$-0.385193\pi$
$197$	9.90659	0.705815	0.352908	+	0.935658i	$0.385193\pi$
$198$	0	0
$199$	10.8852	0.771633	0.385817	−	0.922575i	$-0.373920\pi$
$199$	10.8852	0.771633	0.385817	+	0.922575i	$0.373920\pi$
$200$	−2.19486	−0.155200
$201$	0	0
$202$	−0.304995	−0.0214594
$203$	2.84029	0.199349
$204$	0	0
$205$	−3.38389	−0.236341
$206$	0.0311742	0.00217201
$207$	0	0
$208$	2.99941	0.207972
$209$	1.94492	0.134533
$210$	0	0
$211$	−0.0357265	−0.00245951	−0.00122976	−	0.999999i	$-0.500391\pi$
$211$	−0.0357265	−0.00245951	−0.00122976	+	0.999999i	$0.500391\pi$
$212$	17.2395	1.18401
$213$	0	0
$214$	0.0665971	0.00455248
$215$	−48.2903	−3.29337
$216$	0	0
$217$	3.75546	0.254937
$218$	0.112076	0.00759077
$219$	0	0
$220$	−7.69330	−0.518682
$221$	−3.48784	−0.234617
$222$	0	0
$223$	3.59788	0.240932	0.120466	−	0.992717i	$-0.461561\pi$
$223$	3.59788	0.240932	0.120466	+	0.992717i	$0.461561\pi$
$224$	−0.294936	−0.0197062
$225$	0	0
$226$	0.445675	0.0296458
$227$	29.0427	1.92763	0.963815	−	0.266572i	$-0.0858909\pi$
$227$	29.0427	1.92763	0.963815	+	0.266572i	$0.0858909\pi$
$228$	0	0
$229$	1.33227	0.0880388	0.0440194	−	0.999031i	$-0.485984\pi$
$229$	1.33227	0.0880388	0.0440194	+	0.999031i	$0.485984\pi$
$230$	−0.983597	−0.0648565
$231$	0	0
$232$	1.43470	0.0941929
$233$	−13.8414	−0.906777	−0.453389	−	0.891313i	$-0.649785\pi$
$233$	−13.8414	−0.906777	−0.453389	+	0.891313i	$0.649785\pi$
$234$	0	0
$235$	23.5562	1.53664
$236$	−23.7789	−1.54787
$237$	0	0
$238$	0.114024	0.00739108
$239$	−13.3411	−0.862963	−0.431481	−	0.902122i	$-0.642009\pi$
$239$	−13.3411	−0.862963	−0.431481	+	0.902122i	$0.642009\pi$
$240$	0	0
$241$	−22.8238	−1.47021	−0.735104	−	0.677954i	$-0.762867\pi$
$241$	−22.8238	−1.47021	−0.735104	+	0.677954i	$0.762867\pi$
$242$	0.0557908	0.00358637
$243$	0	0
$244$	1.99689	0.127838
$245$	−26.2177	−1.67499
$246$	0	0
$247$	1.46524	0.0932312
$248$	1.89698	0.120458
$249$	0	0
$250$	1.04094	0.0658346
$251$	−9.93136	−0.626862	−0.313431	−	0.949611i	$-0.601478\pi$
$251$	−9.93136	−0.626862	−0.313431	+	0.949611i	$0.601478\pi$
$252$	0	0
$253$	−4.57610	−0.287697
$254$	−0.541617	−0.0339841
$255$	0	0
$256$	15.7516	0.984473
$257$	−0.987613	−0.0616056	−0.0308028	−	0.999525i	$-0.509806\pi$
$257$	−0.987613	−0.0616056	−0.0308028	+	0.999525i	$0.509806\pi$
$258$	0	0
$259$	1.08935	0.0676889
$260$	−5.79589	−0.359446
$261$	0	0
$262$	0.108280	0.00668959
$263$	22.4248	1.38277	0.691387	−	0.722485i	$-0.257000\pi$
$263$	22.4248	1.38277	0.691387	+	0.722485i	$0.257000\pi$
$264$	0	0
$265$	−33.2606	−2.04318
$266$	−0.0479016	−0.00293704
$267$	0	0
$268$	−11.4126	−0.697135
$269$	−8.28612	−0.505214	−0.252607	−	0.967569i	$-0.581288\pi$
$269$	−8.28612	−0.505214	−0.252607	+	0.967569i	$0.581288\pi$
$270$	0	0
$271$	9.45223	0.574182	0.287091	−	0.957903i	$-0.407312\pi$
$271$	9.45223	0.574182	0.287091	+	0.957903i	$0.407312\pi$
$272$	−18.4322	−1.11762
$273$	0	0
$274$	−0.634843	−0.0383523
$275$	9.84287	0.593547
$276$	0	0
$277$	−16.8025	−1.00957	−0.504783	−	0.863246i	$-0.668428\pi$
$277$	−16.8025	−1.00957	−0.504783	+	0.863246i	$0.668428\pi$
$278$	−0.943273	−0.0565737
$279$	0	0
$280$	0.379253	0.0226647
$281$	−13.4681	−0.803438	−0.401719	−	0.915763i	$-0.631587\pi$
$281$	−13.4681	−0.803438	−0.401719	+	0.915763i	$0.631587\pi$
$282$	0	0
$283$	−4.06678	−0.241745	−0.120872	−	0.992668i	$-0.538569\pi$
$283$	−4.06678	−0.241745	−0.120872	+	0.992668i	$0.538569\pi$
$284$	20.9611	1.24381
$285$	0	0
$286$	0.0420311	0.00248535
$287$	0.387742	0.0228877
$288$	0	0
$289$	4.43370	0.260806
$290$	−1.38293	−0.0812082
$291$	0	0
$292$	4.78710	0.280144
$293$	12.9649	0.757415	0.378708	−	0.925516i	$-0.376369\pi$
$293$	12.9649	0.757415	0.378708	+	0.925516i	$0.376369\pi$
$294$	0	0
$295$	45.8772	2.67107
$296$	0.550259	0.0319831
$297$	0	0
$298$	0.395225	0.0228948
$299$	−3.44749	−0.199374
$300$	0	0
$301$	5.53333	0.318936
$302$	−1.26466	−0.0727730
$303$	0	0
$304$	7.74339	0.444114
$305$	−3.85264	−0.220602
$306$	0	0
$307$	−21.2017	−1.21005	−0.605023	−	0.796208i	$-0.706836\pi$
$307$	−21.2017	−1.21005	−0.605023	+	0.796208i	$0.706836\pi$
$308$	0.881533	0.0502300
$309$	0	0
$310$	−1.82852	−0.103853
$311$	−21.3927	−1.21307	−0.606534	−	0.795057i	$-0.707441\pi$
$311$	−21.3927	−1.21307	−0.606534	+	0.795057i	$0.707441\pi$
$312$	0	0
$313$	19.3646	1.09455	0.547276	−	0.836952i	$-0.315665\pi$
$313$	19.3646	1.09455	0.547276	+	0.836952i	$0.315665\pi$
$314$	−0.875139	−0.0493870
$315$	0	0
$316$	−29.2765	−1.64693
$317$	−14.6901	−0.825078	−0.412539	−	0.910940i	$-0.635358\pi$
$317$	−14.6901	−0.825078	−0.412539	+	0.910940i	$0.635358\pi$
$318$	0	0
$319$	−6.43394	−0.360232
$320$	−30.5337	−1.70689
$321$	0	0
$322$	0.112705	0.00628081
$323$	−9.00432	−0.501014
$324$	0	0
$325$	7.41531	0.411327
$326$	1.00386	0.0555987
$327$	0	0
$328$	0.195859	0.0108145
$329$	−2.69918	−0.148811
$330$	0	0
$331$	21.7285	1.19431	0.597153	−	0.802127i	$-0.296298\pi$
$331$	21.7285	1.19431	0.597153	+	0.802127i	$0.296298\pi$
$332$	17.6571	0.969060
$333$	0	0
$334$	−0.389255	−0.0212991
$335$	22.0186	1.20300
$336$	0	0
$337$	−32.9088	−1.79266	−0.896328	−	0.443392i	$-0.853775\pi$
$337$	−32.9088	−1.79266	−0.896328	+	0.443392i	$0.853775\pi$
$338$	−0.693616	−0.0377277
$339$	0	0
$340$	35.6173	1.93162
$341$	−8.50703	−0.460682
$342$	0	0
$343$	6.09432	0.329062
$344$	2.79503	0.150698
$345$	0	0
$346$	1.08072	0.0581001
$347$	−4.71626	−0.253182	−0.126591	−	0.991955i	$-0.540404\pi$
$347$	−4.71626	−0.253182	−0.126591	+	0.991955i	$0.540404\pi$
$348$	0	0
$349$	−27.5317	−1.47374	−0.736869	−	0.676036i	$-0.763697\pi$
$349$	−27.5317	−1.47374	−0.736869	+	0.676036i	$0.763697\pi$
$350$	−0.242421	−0.0129579
$351$	0	0
$352$	0.668101	0.0356099
$353$	−0.567527	−0.0302064	−0.0151032	−	0.999886i	$-0.504808\pi$
$353$	−0.567527	−0.0302064	−0.0151032	+	0.999886i	$0.504808\pi$
$354$	0	0
$355$	−40.4408	−2.14637
$356$	8.97367	0.475604
$357$	0	0
$358$	1.01248	0.0535114
$359$	13.9804	0.737855	0.368928	−	0.929458i	$-0.379725\pi$
$359$	13.9804	0.737855	0.368928	+	0.929458i	$0.379725\pi$
$360$	0	0
$361$	−15.2173	−0.800909
$362$	−0.302411	−0.0158944
$363$	0	0
$364$	0.664120	0.0348093
$365$	−9.23587	−0.483427
$366$	0	0
$367$	−12.9960	−0.678385	−0.339192	−	0.940717i	$-0.610154\pi$
$367$	−12.9960	−0.678385	−0.339192	+	0.940717i	$0.610154\pi$
$368$	−18.2190	−0.949730
$369$	0	0
$370$	−0.530401	−0.0275742
$371$	3.81115	0.197865
$372$	0	0
$373$	−1.13883	−0.0589662	−0.0294831	−	0.999565i	$-0.509386\pi$
$373$	−1.13883	−0.0589662	−0.0294831	+	0.999565i	$0.509386\pi$
$374$	−0.258292	−0.0133560
$375$	0	0
$376$	−1.36343	−0.0703133
$377$	−4.84713	−0.249640
$378$	0	0
$379$	−32.8553	−1.68766	−0.843831	−	0.536610i	$-0.819705\pi$
$379$	−32.8553	−1.68766	−0.843831	+	0.536610i	$0.819705\pi$
$380$	−14.9629	−0.767579
$381$	0	0
$382$	0.669358	0.0342473
$383$	−7.90734	−0.404046	−0.202023	−	0.979381i	$-0.564752\pi$
$383$	−7.90734	−0.404046	−0.202023	+	0.979381i	$0.564752\pi$
$384$	0	0
$385$	−1.70076	−0.0866790
$386$	0.395520	0.0201314
$387$	0	0
$388$	−9.89515	−0.502350
$389$	−1.25985	−0.0638769	−0.0319384	−	0.999490i	$-0.510168\pi$
$389$	−1.25985	−0.0638769	−0.0319384	+	0.999490i	$0.510168\pi$
$390$	0	0
$391$	21.1858	1.07141
$392$	1.51747	0.0766439
$393$	0	0
$394$	0.552697	0.0278445
$395$	56.4839	2.84202
$396$	0	0
$397$	−12.6957	−0.637181	−0.318590	−	0.947893i	$-0.603209\pi$
$397$	−12.6957	−0.637181	−0.318590	+	0.947893i	$0.603209\pi$
$398$	0.607296	0.0304410
$399$	0	0
$400$	39.1877	1.95939
$401$	19.4169	0.969634	0.484817	−	0.874616i	$-0.338886\pi$
$401$	19.4169	0.969634	0.484817	+	0.874616i	$0.338886\pi$
$402$	0	0
$403$	−6.40893	−0.319252
$404$	10.9165	0.543117
$405$	0	0
$406$	0.158462	0.00786434
$407$	−2.46764	−0.122316
$408$	0	0
$409$	−23.7774	−1.17572	−0.587859	−	0.808963i	$-0.700029\pi$
$409$	−23.7774	−1.17572	−0.587859	+	0.808963i	$0.700029\pi$
$410$	−0.188790	−0.00932368
$411$	0	0
$412$	−1.11580	−0.0549714
$413$	−5.25682	−0.258671
$414$	0	0
$415$	−34.0663	−1.67225
$416$	0.503327	0.0246776
$417$	0	0
$418$	0.108509	0.00530734
$419$	35.1183	1.71564	0.857820	−	0.513950i	$-0.171818\pi$
$419$	35.1183	1.71564	0.857820	+	0.513950i	$0.171818\pi$
$420$	0	0
$421$	37.7862	1.84159	0.920794	−	0.390050i	$-0.127542\pi$
$421$	37.7862	1.84159	0.920794	+	0.390050i	$0.127542\pi$
$422$	−0.00199321	−9.70279e−5 0
$423$	0	0
$424$	1.92511	0.0934917
$425$	−45.5691	−2.21043
$426$	0	0
$427$	0.441454	0.0213634
$428$	−2.38367	−0.115219
$429$	0	0
$430$	−2.69416	−0.129924
$431$	13.8987	0.669477	0.334739	−	0.942311i	$-0.391352\pi$
$431$	13.8987	0.669477	0.334739	+	0.942311i	$0.391352\pi$
$432$	0	0
$433$	27.3091	1.31239	0.656197	−	0.754590i	$-0.272164\pi$
$433$	27.3091	1.31239	0.656197	+	0.754590i	$0.272164\pi$
$434$	0.209520	0.0100573
$435$	0	0
$436$	−4.01148	−0.192115
$437$	−8.90016	−0.425752
$438$	0	0
$439$	20.5170	0.979225	0.489612	−	0.871940i	$-0.337138\pi$
$439$	20.5170	0.979225	0.489612	+	0.871940i	$0.337138\pi$
$440$	−0.859100	−0.0409560
$441$	0	0
$442$	−0.194589	−0.00925567
$443$	23.6464	1.12347	0.561737	−	0.827316i	$-0.310133\pi$
$443$	23.6464	1.12347	0.561737	+	0.827316i	$0.310133\pi$
$444$	0	0
$445$	−17.3131	−0.820721
$446$	0.200729	0.00950478
$447$	0	0
$448$	3.49870	0.165298
$449$	6.85563	0.323537	0.161769	−	0.986829i	$-0.448280\pi$
$449$	6.85563	0.323537	0.161769	+	0.986829i	$0.448280\pi$
$450$	0	0
$451$	−0.878330	−0.0413590
$452$	−15.9518	−0.750308
$453$	0	0
$454$	1.62031	0.0760451
$455$	−1.28130	−0.0600684
$456$	0	0
$457$	−5.51224	−0.257852	−0.128926	−	0.991654i	$-0.541153\pi$
$457$	−5.51224	−0.257852	−0.128926	+	0.991654i	$0.541153\pi$
$458$	0.0743284	0.00347314
$459$	0	0
$460$	35.2053	1.64146
$461$	−36.7463	−1.71144	−0.855722	−	0.517435i	$-0.826887\pi$
$461$	−36.7463	−1.71144	−0.855722	+	0.517435i	$0.826887\pi$
$462$	0	0
$463$	15.5303	0.721753	0.360876	−	0.932614i	$-0.382478\pi$
$463$	15.5303	0.721753	0.360876	+	0.932614i	$0.382478\pi$
$464$	−25.6157	−1.18918
$465$	0	0
$466$	−0.772221	−0.0357724
$467$	4.59549	0.212654	0.106327	−	0.994331i	$-0.466091\pi$
$467$	4.59549	0.212654	0.106327	+	0.994331i	$0.466091\pi$
$468$	0	0
$469$	−2.52299	−0.116501
$470$	1.31422	0.0606205
$471$	0	0
$472$	−2.65536	−0.122223
$473$	−12.5343	−0.576329
$474$	0	0
$475$	19.1436	0.878369
$476$	−4.08120	−0.187061
$477$	0	0
$478$	−0.744310	−0.0340439
$479$	−2.29275	−0.104758	−0.0523792	−	0.998627i	$-0.516680\pi$
$479$	−2.29275	−0.104758	−0.0523792	+	0.998627i	$0.516680\pi$
$480$	0	0
$481$	−1.85904	−0.0847651
$482$	−1.27336	−0.0579998
$483$	0	0
$484$	−1.99689	−0.0907676
$485$	19.0910	0.866876
$486$	0	0
$487$	−28.6856	−1.29987	−0.649933	−	0.759991i	$-0.725203\pi$
$487$	−28.6856	−1.29987	−0.649933	+	0.759991i	$0.725203\pi$
$488$	0.222990	0.0100943
$489$	0	0
$490$	−1.46271	−0.0660784
$491$	−28.5345	−1.28774	−0.643871	−	0.765134i	$-0.722673\pi$
$491$	−28.5345	−1.28774	−0.643871	+	0.765134i	$0.722673\pi$
$492$	0	0
$493$	29.7869	1.34154
$494$	0.0817472	0.00367798
$495$	0	0
$496$	−33.8693	−1.52078
$497$	4.63389	0.207858
$498$	0	0
$499$	−36.5523	−1.63631	−0.818153	−	0.575000i	$-0.805002\pi$
$499$	−36.5523	−1.63631	−0.818153	+	0.575000i	$0.805002\pi$
$500$	−37.2576	−1.66621
$501$	0	0
$502$	−0.554079	−0.0247297
$503$	−4.78902	−0.213532	−0.106766	−	0.994284i	$-0.534050\pi$
$503$	−4.78902	−0.213532	−0.106766	+	0.994284i	$0.534050\pi$
$504$	0	0
$505$	−21.0615	−0.937225
$506$	−0.255304	−0.0113497
$507$	0	0
$508$	19.3858	0.860105
$509$	7.52291	0.333447	0.166724	−	0.986004i	$-0.446681\pi$
$509$	7.52291	0.333447	0.166724	+	0.986004i	$0.446681\pi$
$510$	0	0
$511$	1.05829	0.0468159
$512$	4.43553	0.196024
$513$	0	0
$514$	−0.0550997	−0.00243035
$515$	2.15274	0.0948610
$516$	0	0
$517$	6.11430	0.268907
$518$	0.0607757	0.00267033
$519$	0	0
$520$	−0.647219	−0.0283824
$521$	−5.38247	−0.235810	−0.117905	−	0.993025i	$-0.537618\pi$
$521$	−5.38247	−0.235810	−0.117905	+	0.993025i	$0.537618\pi$
$522$	0	0
$523$	25.6396	1.12114	0.560571	−	0.828106i	$-0.310582\pi$
$523$	25.6396	1.12114	0.560571	+	0.828106i	$0.310582\pi$
$524$	−3.87562	−0.169307
$525$	0	0
$526$	1.25110	0.0545505
$527$	39.3846	1.71562
$528$	0	0
$529$	−2.05931	−0.0895351
$530$	−1.85564	−0.0806037
$531$	0	0
$532$	1.71451	0.0743336
$533$	−0.661707	−0.0286617
$534$	0	0
$535$	4.59887	0.198827
$536$	−1.27443	−0.0550469
$537$	0	0
$538$	−0.462289	−0.0199307
$539$	−6.80512	−0.293117
$540$	0	0
$541$	24.6924	1.06161	0.530805	−	0.847494i	$-0.321890\pi$
$541$	24.6924	1.06161	0.530805	+	0.847494i	$0.321890\pi$
$542$	0.527347	0.0226515
$543$	0	0
$544$	−3.09308	−0.132615
$545$	7.73945	0.331522
$546$	0	0
$547$	−34.5139	−1.47571	−0.737854	−	0.674960i	$-0.764161\pi$
$547$	−34.5139	−1.47571	−0.737854	+	0.674960i	$0.764161\pi$
$548$	22.7225	0.970659
$549$	0	0
$550$	0.549142	0.0234155
$551$	−12.5135	−0.533094
$552$	0	0
$553$	−6.47219	−0.275226
$554$	−0.937428	−0.0398275
$555$	0	0
$556$	33.7620	1.43183
$557$	7.56468	0.320526	0.160263	−	0.987074i	$-0.448766\pi$
$557$	7.56468	0.320526	0.160263	+	0.987074i	$0.448766\pi$
$558$	0	0
$559$	−9.44297	−0.399395
$560$	−6.77131	−0.286140
$561$	0	0
$562$	−0.751395	−0.0316957
$563$	13.9614	0.588405	0.294202	−	0.955743i	$-0.404946\pi$
$563$	13.9614	0.588405	0.294202	+	0.955743i	$0.404946\pi$
$564$	0	0
$565$	30.7761	1.29476
$566$	−0.226889	−0.00953685
$567$	0	0
$568$	2.34070	0.0982135
$569$	−38.9567	−1.63315	−0.816575	−	0.577239i	$-0.804130\pi$
$569$	−38.9567	−1.63315	−0.816575	+	0.577239i	$0.804130\pi$
$570$	0	0
$571$	−25.6502	−1.07343	−0.536713	−	0.843765i	$-0.680334\pi$
$571$	−25.6502	−1.07343	−0.536713	+	0.843765i	$0.680334\pi$
$572$	−1.50439	−0.0629018
$573$	0	0
$574$	0.0216325	0.000902921 0
$575$	−45.0419	−1.87838
$576$	0	0
$577$	−17.8624	−0.743620	−0.371810	−	0.928309i	$-0.621263\pi$
$577$	−17.8624	−0.743620	−0.371810	+	0.928309i	$0.621263\pi$
$578$	0.247360	0.0102888
$579$	0	0
$580$	49.4982	2.05530
$581$	3.90347	0.161943
$582$	0	0
$583$	−8.63319	−0.357550
$584$	0.534569	0.0221206
$585$	0	0
$586$	0.723320	0.0298801
$587$	6.62058	0.273261	0.136630	−	0.990622i	$-0.456373\pi$
$587$	6.62058	0.273261	0.136630	+	0.990622i	$0.456373\pi$
$588$	0	0
$589$	−16.5455	−0.681746
$590$	2.55953	0.105374
$591$	0	0
$592$	−9.82451	−0.403785
$593$	21.1203	0.867306	0.433653	−	0.901080i	$-0.357224\pi$
$593$	21.1203	0.867306	0.433653	+	0.901080i	$0.357224\pi$
$594$	0	0
$595$	7.87395	0.322801
$596$	−14.1461	−0.579445
$597$	0	0
$598$	−0.192338	−0.00786530
$599$	−43.8941	−1.79346	−0.896731	−	0.442575i	$-0.854065\pi$
$599$	−43.8941	−1.79346	−0.896731	+	0.442575i	$0.854065\pi$
$600$	0	0
$601$	5.93734	0.242189	0.121095	−	0.992641i	$-0.461360\pi$
$601$	5.93734	0.242189	0.121095	+	0.992641i	$0.461360\pi$
$602$	0.308709	0.0125820
$603$	0	0
$604$	45.2652	1.84182
$605$	3.85264	0.156632
$606$	0	0
$607$	−8.99069	−0.364921	−0.182460	−	0.983213i	$-0.558406\pi$
$607$	−8.99069	−0.364921	−0.182460	+	0.983213i	$0.558406\pi$
$608$	1.29941	0.0526978
$609$	0	0
$610$	−0.214942	−0.00870276
$611$	4.60632	0.186352
$612$	0	0
$613$	−4.64652	−0.187671	−0.0938357	−	0.995588i	$-0.529913\pi$
$613$	−4.64652	−0.187671	−0.0938357	+	0.995588i	$0.529913\pi$
$614$	−1.18286	−0.0477364
$615$	0	0
$616$	0.0984396	0.00396625
$617$	−33.2502	−1.33860	−0.669301	−	0.742992i	$-0.733406\pi$
$617$	−33.2502	−1.33860	−0.669301	+	0.742992i	$0.733406\pi$
$618$	0	0
$619$	−35.1882	−1.41433	−0.707166	−	0.707048i	$-0.750027\pi$
$619$	−35.1882	−1.41433	−0.707166	+	0.707048i	$0.750027\pi$
$620$	65.4471	2.62842
$621$	0	0
$622$	−1.19352	−0.0478556
$623$	1.98382	0.0794800
$624$	0	0
$625$	22.6677	0.906709
$626$	1.08037	0.0431802
$627$	0	0
$628$	31.3233	1.24994
$629$	11.4243	0.455518
$630$	0	0
$631$	12.4067	0.493904	0.246952	−	0.969028i	$-0.420571\pi$
$631$	12.4067	0.493904	0.246952	+	0.969028i	$0.420571\pi$
$632$	−3.26927	−0.130045
$633$	0	0
$634$	−0.819572	−0.0325494
$635$	−37.4015	−1.48423
$636$	0	0
$637$	−5.12676	−0.203130
$638$	−0.358955	−0.0142112
$639$	0	0
$640$	−6.85141	−0.270826
$641$	−16.8996	−0.667493	−0.333746	−	0.942663i	$-0.608313\pi$
$641$	−16.8996	−0.667493	−0.333746	+	0.942663i	$0.608313\pi$
$642$	0	0
$643$	−27.9036	−1.10041	−0.550205	−	0.835030i	$-0.685450\pi$
$643$	−27.9036	−1.10041	−0.550205	+	0.835030i	$0.685450\pi$
$644$	−4.03399	−0.158961
$645$	0	0
$646$	−0.502358	−0.0197650
$647$	8.15768	0.320712	0.160356	−	0.987059i	$-0.448736\pi$
$647$	8.15768	0.320712	0.160356	+	0.987059i	$0.448736\pi$
$648$	0	0
$649$	11.9080	0.467429
$650$	0.413706	0.0162269
$651$	0	0
$652$	−35.9306	−1.40715
$653$	17.0492	0.667187	0.333594	−	0.942717i	$-0.391739\pi$
$653$	17.0492	0.667187	0.333594	+	0.942717i	$0.391739\pi$
$654$	0	0
$655$	7.47732	0.292163
$656$	−3.49693	−0.136532
$657$	0	0
$658$	−0.150590	−0.00587059
$659$	−40.3761	−1.57283	−0.786414	−	0.617700i	$-0.788065\pi$
$659$	−40.3761	−1.57283	−0.786414	+	0.617700i	$0.788065\pi$
$660$	0	0
$661$	−15.5753	−0.605807	−0.302904	−	0.953021i	$-0.597956\pi$
$661$	−15.5753	−0.605807	−0.302904	+	0.953021i	$0.597956\pi$
$662$	1.21225	0.0471155
$663$	0	0
$664$	1.97175	0.0765186
$665$	−3.30785	−0.128273
$666$	0	0
$667$	29.4424	1.14001
$668$	13.9324	0.539060
$669$	0	0
$670$	1.22844	0.0474586
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−12.6784	−0.488715	−0.244358	−	0.969685i	$-0.578577\pi$
$673$	−12.6784	−0.488715	−0.244358	+	0.969685i	$0.578577\pi$
$674$	−1.83601	−0.0707204
$675$	0	0
$676$	24.8262	0.954853
$677$	−42.2085	−1.62221	−0.811103	−	0.584904i	$-0.801132\pi$
$677$	−42.2085	−1.62221	−0.811103	+	0.584904i	$0.801132\pi$
$678$	0	0
$679$	−2.18753	−0.0839497
$680$	3.97734	0.152524
$681$	0	0
$682$	−0.474614	−0.0181739
$683$	29.5196	1.12953	0.564767	−	0.825250i	$-0.308966\pi$
$683$	29.5196	1.12953	0.564767	+	0.825250i	$0.308966\pi$
$684$	0	0
$685$	−43.8392	−1.67501
$686$	0.340007	0.0129815
$687$	0	0
$688$	−49.9034	−1.90255
$689$	−6.50397	−0.247782
$690$	0	0
$691$	−4.12885	−0.157069	−0.0785344	−	0.996911i	$-0.525024\pi$
$691$	−4.12885	−0.157069	−0.0785344	+	0.996911i	$0.525024\pi$
$692$	−38.6817	−1.47046
$693$	0	0
$694$	−0.263124	−0.00998804
$695$	−65.1378	−2.47082
$696$	0	0
$697$	4.06637	0.154025
$698$	−1.53602	−0.0581391
$699$	0	0
$700$	8.67682	0.327953
$701$	31.8152	1.20164	0.600822	−	0.799383i	$-0.294840\pi$
$701$	31.8152	1.20164	0.600822	+	0.799383i	$0.294840\pi$
$702$	0	0
$703$	−4.79937	−0.181012
$704$	−7.92539	−0.298700
$705$	0	0
$706$	−0.0316628	−0.00119165
$707$	2.41333	0.0907624
$708$	0	0
$709$	5.23265	0.196516	0.0982582	−	0.995161i	$-0.468673\pi$
$709$	5.23265	0.196516	0.0982582	+	0.995161i	$0.468673\pi$
$710$	−2.25622	−0.0846746
$711$	0	0
$712$	1.00208	0.0375545
$713$	38.9290	1.45790
$714$	0	0
$715$	2.90246	0.108546
$716$	−36.2392	−1.35432
$717$	0	0
$718$	0.779976	0.0291084
$719$	27.8327	1.03799	0.518993	−	0.854778i	$-0.326307\pi$
$719$	27.8327	1.03799	0.518993	+	0.854778i	$0.326307\pi$
$720$	0	0
$721$	−0.246671	−0.00918649
$722$	−0.848984	−0.0315959
$723$	0	0
$724$	10.8240	0.402272
$725$	−63.3285	−2.35196
$726$	0	0
$727$	40.1787	1.49015	0.745073	−	0.666983i	$-0.232415\pi$
$727$	40.1787	1.49015	0.745073	+	0.666983i	$0.232415\pi$
$728$	0.0741613	0.00274860
$729$	0	0
$730$	−0.515277	−0.0190712
$731$	58.0296	2.14630
$732$	0	0
$733$	51.8028	1.91338	0.956690	−	0.291110i	$-0.0940245\pi$
$733$	51.8028	1.91338	0.956690	+	0.291110i	$0.0940245\pi$
$734$	−0.725056	−0.0267623
$735$	0	0
$736$	−3.05730	−0.112693
$737$	5.71519	0.210522
$738$	0	0
$739$	−30.1172	−1.10788	−0.553939	−	0.832557i	$-0.686876\pi$
$739$	−30.1172	−1.10788	−0.553939	+	0.832557i	$0.686876\pi$
$740$	18.9843	0.697877
$741$	0	0
$742$	0.212627	0.00780580
$743$	39.9540	1.46577	0.732885	−	0.680352i	$-0.238173\pi$
$743$	39.9540	1.46577	0.732885	+	0.680352i	$0.238173\pi$
$744$	0	0
$745$	27.2923	0.999914
$746$	−0.0635361	−0.00232622
$747$	0	0
$748$	9.24490	0.338027
$749$	−0.526960	−0.0192547
$750$	0	0
$751$	19.7225	0.719684	0.359842	−	0.933013i	$-0.382831\pi$
$751$	19.7225	0.719684	0.359842	+	0.933013i	$0.382831\pi$
$752$	24.3431	0.887700
$753$	0	0
$754$	−0.270426	−0.00984832
$755$	−87.3313	−3.17831
$756$	0	0
$757$	16.1399	0.586613	0.293307	−	0.956018i	$-0.405244\pi$
$757$	16.1399	0.586613	0.293307	+	0.956018i	$0.405244\pi$
$758$	−1.83302	−0.0665783
$759$	0	0
$760$	−1.67088	−0.0606093
$761$	7.34329	0.266194	0.133097	−	0.991103i	$-0.457508\pi$
$761$	7.34329	0.266194	0.133097	+	0.991103i	$0.457508\pi$
$762$	0	0
$763$	−0.886822	−0.0321051
$764$	−23.9579	−0.866768
$765$	0	0
$766$	−0.441157	−0.0159397
$767$	8.97110	0.323928
$768$	0	0
$769$	47.8086	1.72402	0.862012	−	0.506889i	$-0.169204\pi$
$769$	47.8086	1.72402	0.862012	+	0.506889i	$0.169204\pi$
$770$	−0.0948870	−0.00341949
$771$	0	0
$772$	−14.1566	−0.509507
$773$	16.8234	0.605095	0.302548	−	0.953134i	$-0.402163\pi$
$773$	16.8234	0.605095	0.302548	+	0.953134i	$0.402163\pi$
$774$	0	0
$775$	−83.7336	−3.00780
$776$	−1.10498	−0.0396664
$777$	0	0
$778$	−0.0702880	−0.00251995
$779$	−1.70828	−0.0612056
$780$	0	0
$781$	−10.4969	−0.375608
$782$	1.18197	0.0422672
$783$	0	0
$784$	−27.0934	−0.967623
$785$	−60.4329	−2.15694
$786$	0	0
$787$	6.44538	0.229753	0.114876	−	0.993380i	$-0.463353\pi$
$787$	6.44538	0.229753	0.114876	+	0.993380i	$0.463353\pi$
$788$	−19.7823	−0.704717
$789$	0	0
$790$	3.15129	0.112118
$791$	−3.52647	−0.125387
$792$	0	0
$793$	−0.753369	−0.0267529
$794$	−0.708305	−0.0251368
$795$	0	0
$796$	−21.7366	−0.770432
$797$	39.1866	1.38806	0.694030	−	0.719946i	$-0.255833\pi$
$797$	39.1866	1.38806	0.694030	+	0.719946i	$0.255833\pi$
$798$	0	0
$799$	−28.3071	−1.00143
$800$	6.57603	0.232498
$801$	0	0
$802$	1.08328	0.0382521
$803$	−2.39728	−0.0845982
$804$	0	0
$805$	7.78287	0.274310
$806$	−0.357560	−0.0125945
$807$	0	0
$808$	1.21903	0.0428854
$809$	19.1481	0.673211	0.336606	−	0.941646i	$-0.390721\pi$
$809$	19.1481	0.673211	0.336606	+	0.941646i	$0.390721\pi$
$810$	0	0
$811$	−20.9180	−0.734529	−0.367265	−	0.930117i	$-0.619706\pi$
$811$	−20.9180	−0.734529	−0.367265	+	0.930117i	$0.619706\pi$
$812$	−5.67174	−0.199039
$813$	0	0
$814$	−0.137672	−0.00482539
$815$	69.3217	2.42823
$816$	0	0
$817$	−24.3783	−0.852889
$818$	−1.32656	−0.0463822
$819$	0	0
$820$	6.75726	0.235974
$821$	−3.44283	−0.120156	−0.0600779	−	0.998194i	$-0.519135\pi$
$821$	−3.44283	−0.120156	−0.0600779	+	0.998194i	$0.519135\pi$
$822$	0	0
$823$	48.7180	1.69820	0.849102	−	0.528229i	$-0.177144\pi$
$823$	48.7180	1.69820	0.849102	+	0.528229i	$0.177144\pi$
$824$	−0.124600	−0.00434064
$825$	0	0
$826$	−0.293282	−0.0102046
$827$	47.1026	1.63792	0.818958	−	0.573853i	$-0.194552\pi$
$827$	47.1026	1.63792	0.818958	+	0.573853i	$0.194552\pi$
$828$	0	0
$829$	27.8153	0.966065	0.483033	−	0.875602i	$-0.339535\pi$
$829$	27.8153	0.966065	0.483033	+	0.875602i	$0.339535\pi$
$830$	−1.90059	−0.0659703
$831$	0	0
$832$	−5.97074	−0.206998
$833$	31.5053	1.09160
$834$	0	0
$835$	−26.8801	−0.930224
$836$	−3.88379	−0.134324
$837$	0	0
$838$	1.95928	0.0676821
$839$	52.0679	1.79758	0.898791	−	0.438377i	$-0.144446\pi$
$839$	52.0679	1.79758	0.898791	+	0.438377i	$0.144446\pi$
$840$	0	0
$841$	12.3956	0.427436
$842$	2.10812	0.0726507
$843$	0	0
$844$	0.0713417	0.00245568
$845$	−47.8978	−1.64773
$846$	0	0
$847$	−0.441454	−0.0151685
$848$	−34.3716	−1.18033
$849$	0	0
$850$	−2.54234	−0.0872014
$851$	11.2922	0.387091
$852$	0	0
$853$	−57.5053	−1.96894	−0.984472	−	0.175543i	$-0.943832\pi$
$853$	−57.5053	−1.96894	−0.984472	+	0.175543i	$0.943832\pi$
$854$	0.0246291	0.000842789 0
$855$	0	0
$856$	−0.266181	−0.00909788
$857$	−47.1919	−1.61204	−0.806022	−	0.591886i	$-0.798383\pi$
$857$	−47.1919	−1.61204	−0.806022	+	0.591886i	$0.798383\pi$
$858$	0	0
$859$	−43.5042	−1.48435	−0.742173	−	0.670208i	$-0.766205\pi$
$859$	−43.5042	−1.48435	−0.742173	+	0.670208i	$0.766205\pi$
$860$	96.4303	3.28825
$861$	0	0
$862$	0.775421	0.0264109
$863$	10.3453	0.352157	0.176079	−	0.984376i	$-0.443659\pi$
$863$	10.3453	0.352157	0.176079	+	0.984376i	$0.443659\pi$
$864$	0	0
$865$	74.6295	2.53748
$866$	1.52360	0.0517740
$867$	0	0
$868$	−7.49923	−0.254541
$869$	14.6611	0.497343
$870$	0	0
$871$	4.30565	0.145891
$872$	−0.447957	−0.0151697
$873$	0	0
$874$	−0.496547	−0.0167960
$875$	−8.23658	−0.278447
$876$	0	0
$877$	−3.50085	−0.118215	−0.0591077	−	0.998252i	$-0.518826\pi$
$877$	−3.50085	−0.118215	−0.0591077	+	0.998252i	$0.518826\pi$
$878$	1.14466	0.0386305
$879$	0	0
$880$	15.3387	0.517066
$881$	49.7484	1.67607	0.838034	−	0.545618i	$-0.183705\pi$
$881$	49.7484	1.67607	0.838034	+	0.545618i	$0.183705\pi$
$882$	0	0
$883$	24.8510	0.836303	0.418151	−	0.908377i	$-0.362678\pi$
$883$	24.8510	0.836303	0.418151	+	0.908377i	$0.362678\pi$
$884$	6.96482	0.234252
$885$	0	0
$886$	1.31925	0.0443212
$887$	38.2709	1.28501	0.642506	−	0.766281i	$-0.277895\pi$
$887$	38.2709	1.28501	0.642506	+	0.766281i	$0.277895\pi$
$888$	0	0
$889$	4.28563	0.143735
$890$	−0.965914	−0.0323775
$891$	0	0
$892$	−7.18457	−0.240557
$893$	11.8918	0.397945
$894$	0	0
$895$	69.9172	2.33707
$896$	0.785067	0.0262272
$897$	0	0
$898$	0.382481	0.0127636
$899$	54.7338	1.82547
$900$	0	0
$901$	39.9687	1.33155
$902$	−0.0490028	−0.00163161
$903$	0	0
$904$	−1.78131	−0.0592456
$905$	−20.8831	−0.694177
$906$	0	0
$907$	38.5601	1.28037	0.640183	−	0.768223i	$-0.278859\pi$
$907$	38.5601	1.28037	0.640183	+	0.768223i	$0.278859\pi$
$908$	−57.9950	−1.92463
$909$	0	0
$910$	−0.0714849	−0.00236970
$911$	−19.2862	−0.638979	−0.319489	−	0.947590i	$-0.603511\pi$
$911$	−19.2862	−0.638979	−0.319489	+	0.947590i	$0.603511\pi$
$912$	0	0
$913$	−8.84232	−0.292638
$914$	−0.307532	−0.0101723
$915$	0	0
$916$	−2.66039	−0.0879018
$917$	−0.856786	−0.0282936
$918$	0	0
$919$	58.5184	1.93035	0.965173	−	0.261614i	$-0.0842547\pi$
$919$	58.5184	1.93035	0.965173	+	0.261614i	$0.0842547\pi$
$920$	3.93133	0.129612
$921$	0	0
$922$	−2.05010	−0.0675166
$923$	−7.90803	−0.260296
$924$	0	0
$925$	−24.2887	−0.798607
$926$	0.866447	0.0284732
$927$	0	0
$928$	−4.29853	−0.141106
$929$	−51.0136	−1.67370	−0.836850	−	0.547432i	$-0.815606\pi$
$929$	−51.0136	−1.67370	−0.836850	+	0.547432i	$0.815606\pi$
$930$	0	0
$931$	−13.2354	−0.433774
$932$	27.6396	0.905366
$933$	0	0
$934$	0.256386	0.00838920
$935$	−17.8364	−0.583313
$936$	0	0
$937$	−47.0136	−1.53587	−0.767934	−	0.640529i	$-0.778715\pi$
$937$	−47.0136	−1.53587	−0.767934	+	0.640529i	$0.778715\pi$
$938$	−0.140760	−0.00459597
$939$	0	0
$940$	−47.0391	−1.53425
$941$	8.99834	0.293338	0.146669	−	0.989186i	$-0.453145\pi$
$941$	8.99834	0.293338	0.146669	+	0.989186i	$0.453145\pi$
$942$	0	0
$943$	4.01933	0.130887
$944$	47.4096	1.54305
$945$	0	0
$946$	−0.699301	−0.0227362
$947$	33.9862	1.10440	0.552202	−	0.833710i	$-0.313788\pi$
$947$	33.9862	1.10440	0.552202	+	0.833710i	$0.313788\pi$
$948$	0	0
$949$	−1.80604	−0.0586264
$950$	1.06804	0.0346517
$951$	0	0
$952$	−0.455741	−0.0147707
$953$	−34.8498	−1.12890	−0.564448	−	0.825468i	$-0.690911\pi$
$953$	−34.8498	−1.12890	−0.564448	+	0.825468i	$0.690911\pi$
$954$	0	0
$955$	46.2226	1.49573
$956$	26.6406	0.861620
$957$	0	0
$958$	−0.127914	−0.00413272
$959$	5.02329	0.162211
$960$	0	0
$961$	41.3696	1.33450
$962$	−0.103718	−0.00334399
$963$	0	0
$964$	45.5765	1.46792
$965$	27.3127	0.879226
$966$	0	0
$967$	19.8748	0.639131	0.319566	−	0.947564i	$-0.396463\pi$
$967$	19.8748	0.639131	0.319566	+	0.947564i	$0.396463\pi$
$968$	−0.222990	−0.00716716
$969$	0	0
$970$	1.06510	0.0341983
$971$	15.1136	0.485018	0.242509	−	0.970149i	$-0.422030\pi$
$971$	15.1136	0.485018	0.242509	+	0.970149i	$0.422030\pi$
$972$	0	0
$973$	7.46379	0.239278
$974$	−1.60039	−0.0512798
$975$	0	0
$976$	−3.98133	−0.127439
$977$	23.5244	0.752612	0.376306	−	0.926495i	$-0.377194\pi$
$977$	23.5244	0.752612	0.376306	+	0.926495i	$0.377194\pi$
$978$	0	0
$979$	−4.49383	−0.143623
$980$	52.3538	1.67238
$981$	0	0
$982$	−1.59196	−0.0508015
$983$	40.9132	1.30493	0.652464	−	0.757820i	$-0.273735\pi$
$983$	40.9132	1.30493	0.652464	+	0.757820i	$0.273735\pi$
$984$	0	0
$985$	38.1666	1.21609
$986$	1.66184	0.0529237
$987$	0	0
$988$	−2.92593	−0.0930861
$989$	57.3584	1.82389
$990$	0	0
$991$	−18.8787	−0.599703	−0.299851	−	0.953986i	$-0.596937\pi$
$991$	−18.8787	−0.599703	−0.299851	+	0.953986i	$0.596937\pi$
$992$	−5.68356	−0.180453
$993$	0	0
$994$	0.258529	0.00820003
$995$	41.9369	1.32949
$996$	0	0
$997$	−42.2297	−1.33743	−0.668715	−	0.743519i	$-0.733155\pi$
$997$	−42.2297	−1.33743	−0.668715	+	0.743519i	$0.733155\pi$
$998$	−2.03928	−0.0645524
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.e.1.7		12
3.2	odd	2		2013.2.a.d.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.d.1.6	✓	12	3.2	odd	2
6039.2.a.e.1.7		12	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)

\(f(q)\)	\(=\)	\(q+0.0557908 q^{2} -1.99689 q^{4} +3.85264 q^{5} -0.441454 q^{7} -0.222990 q^{8} +O(q^{10})\)	\(q+0.0557908 q^{2} -1.99689 q^{4} +3.85264 q^{5} -0.441454 q^{7} -0.222990 q^{8} +0.214942 q^{10} +1.00000 q^{11} +0.753369 q^{13} -0.0246291 q^{14} +3.98133 q^{16} -4.62965 q^{17} +1.94492 q^{19} -7.69330 q^{20} +0.0557908 q^{22} -4.57610 q^{23} +9.84287 q^{25} +0.0420311 q^{26} +0.881533 q^{28} -6.43394 q^{29} -8.50703 q^{31} +0.668101 q^{32} -0.258292 q^{34} -1.70076 q^{35} -2.46764 q^{37} +0.108509 q^{38} -0.859100 q^{40} -0.878330 q^{41} -12.5343 q^{43} -1.99689 q^{44} -0.255304 q^{46} +6.11430 q^{47} -6.80512 q^{49} +0.549142 q^{50} -1.50439 q^{52} -8.63319 q^{53} +3.85264 q^{55} +0.0984396 q^{56} -0.358955 q^{58} +11.9080 q^{59} -1.00000 q^{61} -0.474614 q^{62} -7.92539 q^{64} +2.90246 q^{65} +5.71519 q^{67} +9.24490 q^{68} -0.0948870 q^{70} -10.4969 q^{71} -2.39728 q^{73} -0.137672 q^{74} -3.88379 q^{76} -0.441454 q^{77} +14.6611 q^{79} +15.3387 q^{80} -0.0490028 q^{82} -8.84232 q^{83} -17.8364 q^{85} -0.699301 q^{86} -0.222990 q^{88} -4.49383 q^{89} -0.332577 q^{91} +9.13796 q^{92} +0.341122 q^{94} +7.49309 q^{95} +4.95529 q^{97} -0.379663 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) \) 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) \) 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8 - 8 * q^10 + 12 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 20 * q^19 + 9 * q^20 - q^22 + 9 * q^23 + 3 * q^25 + 18 * q^26 - 31 * q^28 - 18 * q^29 - 21 * q^31 - 18 * q^32 - 12 * q^34 + 4 * q^35 - 18 * q^37 + 2 * q^38 - 26 * q^40 - 15 * q^41 - 33 * q^43 + 9 * q^44 - 28 * q^46 + 20 * q^47 + 15 * q^49 + 2 * q^50 - 27 * q^52 + 3 * q^55 + 8 * q^56 - 11 * q^58 + 21 * q^59 - 12 * q^61 + 9 * q^62 - 12 * q^64 - 17 * q^65 - 34 * q^67 + 16 * q^68 - 36 * q^70 + 5 * q^71 - 2 * q^73 - 6 * q^74 - 27 * q^76 - 9 * q^77 - 31 * q^79 + 60 * q^80 - 12 * q^82 + 32 * q^83 - 40 * q^85 - 18 * q^86 - 6 * q^88 - 27 * q^89 - 45 * q^91 + 78 * q^92 - 13 * q^94 - 37 * q^95 - 19 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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