LMFDB - Embedded newform 6039.2.a.e.1.4

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			1.4
Root			$1.32082$ 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)

$f(q)$	$=$	$q-1.32082 q^{2} -0.255442 q^{4} -0.125081 q^{5} +1.14683 q^{7} +2.97903 q^{8} +O(q^{10})$	\(q-1.32082 q^{2} -0.255442 q^{4} -0.125081 q^{5} +1.14683 q^{7} +2.97903 q^{8} +0.165209 q^{10} +1.00000 q^{11} -2.50418 q^{13} -1.51475 q^{14} -3.42387 q^{16} -3.85395 q^{17} +6.27754 q^{19} +0.0319509 q^{20} -1.32082 q^{22} +3.82413 q^{23} -4.98435 q^{25} +3.30756 q^{26} -0.292949 q^{28} +1.92089 q^{29} -3.36232 q^{31} -1.43575 q^{32} +5.09036 q^{34} -0.143447 q^{35} -4.87160 q^{37} -8.29148 q^{38} -0.372619 q^{40} +12.3084 q^{41} -5.75302 q^{43} -0.255442 q^{44} -5.05098 q^{46} -4.24272 q^{47} -5.68478 q^{49} +6.58342 q^{50} +0.639672 q^{52} +6.57191 q^{53} -0.125081 q^{55} +3.41644 q^{56} -2.53715 q^{58} -10.4740 q^{59} -1.00000 q^{61} +4.44100 q^{62} +8.74410 q^{64} +0.313225 q^{65} -1.28591 q^{67} +0.984460 q^{68} +0.189467 q^{70} +5.24754 q^{71} -8.55185 q^{73} +6.43449 q^{74} -1.60355 q^{76} +1.14683 q^{77} -12.1425 q^{79} +0.428260 q^{80} -16.2572 q^{82} +9.44925 q^{83} +0.482055 q^{85} +7.59869 q^{86} +2.97903 q^{88} -10.8900 q^{89} -2.87187 q^{91} -0.976843 q^{92} +5.60386 q^{94} -0.785200 q^{95} +12.7500 q^{97} +7.50855 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.32082	−0.933959	−0.466979	−	0.884268i	$-0.654658\pi$
$2$	−1.32082	−0.933959	−0.466979	+	0.884268i	$0.654658\pi$
$3$	0	0
$3$	0	0
$4$	−0.255442	−0.127721
$5$	−0.125081	−0.0559378	−0.0279689	−	0.999609i	$-0.508904\pi$
$5$	−0.125081	−0.0559378	−0.0279689	+	0.999609i	$0.508904\pi$
$6$	0	0
$7$	1.14683	0.433462	0.216731	−	0.976231i	$-0.430461\pi$
$7$	1.14683	0.433462	0.216731	+	0.976231i	$0.430461\pi$
$8$	2.97903	1.05324
$9$	0	0
$10$	0.165209	0.0522436
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.50418	−0.694534	−0.347267	−	0.937766i	$-0.612890\pi$
$13$	−2.50418	−0.694534	−0.347267	+	0.937766i	$0.612890\pi$
$14$	−1.51475	−0.404835
$15$	0	0
$16$	−3.42387	−0.855966
$17$	−3.85395	−0.934719	−0.467360	−	0.884067i	$-0.654795\pi$
$17$	−3.85395	−0.934719	−0.467360	+	0.884067i	$0.654795\pi$
$18$	0	0
$19$	6.27754	1.44017	0.720083	−	0.693888i	$-0.244104\pi$
$19$	6.27754	1.44017	0.720083	+	0.693888i	$0.244104\pi$
$20$	0.0319509	0.00714444
$21$	0	0
$22$	−1.32082	−0.281599
$23$	3.82413	0.797386	0.398693	−	0.917084i	$-0.369464\pi$
$23$	3.82413	0.797386	0.398693	+	0.917084i	$0.369464\pi$
$24$	0	0
$25$	−4.98435	−0.996871
$26$	3.30756	0.648666
$27$	0	0
$28$	−0.292949	−0.0553621
$29$	1.92089	0.356701	0.178350	−	0.983967i	$-0.442924\pi$
$29$	1.92089	0.356701	0.178350	+	0.983967i	$0.442924\pi$
$30$	0	0
$31$	−3.36232	−0.603890	−0.301945	−	0.953325i	$-0.597636\pi$
$31$	−3.36232	−0.603890	−0.301945	+	0.953325i	$0.597636\pi$
$32$	−1.43575	−0.253808
$33$	0	0
$34$	5.09036	0.872989
$35$	−0.143447	−0.0242469
$36$	0	0
$37$	−4.87160	−0.800886	−0.400443	−	0.916322i	$-0.631144\pi$
$37$	−4.87160	−0.800886	−0.400443	+	0.916322i	$0.631144\pi$
$38$	−8.29148	−1.34506
$39$	0	0
$40$	−0.372619	−0.0589162
$41$	12.3084	1.92225	0.961127	−	0.276105i	$-0.0890438\pi$
$41$	12.3084	1.92225	0.961127	+	0.276105i	$0.0890438\pi$
$42$	0	0
$43$	−5.75302	−0.877327	−0.438664	−	0.898651i	$-0.644548\pi$
$43$	−5.75302	−0.877327	−0.438664	+	0.898651i	$0.644548\pi$
$44$	−0.255442	−0.0385093
$45$	0	0
$46$	−5.05098	−0.744726
$47$	−4.24272	−0.618864	−0.309432	−	0.950922i	$-0.600139\pi$
$47$	−4.24272	−0.618864	−0.309432	+	0.950922i	$0.600139\pi$
$48$	0	0
$49$	−5.68478	−0.812111
$50$	6.58342	0.931036
$51$	0	0
$52$	0.639672	0.0887066
$53$	6.57191	0.902721	0.451360	−	0.892342i	$-0.350939\pi$
$53$	6.57191	0.902721	0.451360	+	0.892342i	$0.350939\pi$
$54$	0	0
$55$	−0.125081	−0.0168659
$56$	3.41644	0.456541
$57$	0	0
$58$	−2.53715	−0.333144
$59$	−10.4740	−1.36360	−0.681799	−	0.731539i	$-0.738802\pi$
$59$	−10.4740	−1.36360	−0.681799	+	0.731539i	$0.738802\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	4.44100	0.564008
$63$	0	0
$64$	8.74410	1.09301
$65$	0.313225	0.0388507
$66$	0	0
$67$	−1.28591	−0.157100	−0.0785498	−	0.996910i	$-0.525029\pi$
$67$	−1.28591	−0.157100	−0.0785498	+	0.996910i	$0.525029\pi$
$68$	0.984460	0.119383
$69$	0	0
$70$	0.189467	0.0226456
$71$	5.24754	0.622768	0.311384	−	0.950284i	$-0.399207\pi$
$71$	5.24754	0.622768	0.311384	+	0.950284i	$0.399207\pi$
$72$	0	0
$73$	−8.55185	−1.00092	−0.500459	−	0.865760i	$-0.666835\pi$
$73$	−8.55185	−1.00092	−0.500459	+	0.865760i	$0.666835\pi$
$74$	6.43449	0.747995
$75$	0	0
$76$	−1.60355	−0.183940
$77$	1.14683	0.130694
$78$	0	0
$79$	−12.1425	−1.36613	−0.683067	−	0.730356i	$-0.739354\pi$
$79$	−12.1425	−1.36613	−0.683067	+	0.730356i	$0.739354\pi$
$80$	0.428260	0.0478809
$81$	0	0
$82$	−16.2572	−1.79531
$83$	9.44925	1.03719	0.518595	−	0.855020i	$-0.326455\pi$
$83$	9.44925	1.03719	0.518595	+	0.855020i	$0.326455\pi$
$84$	0	0
$85$	0.482055	0.0522862
$86$	7.59869	0.819387
$87$	0	0
$88$	2.97903	0.317565
$89$	−10.8900	−1.15434	−0.577171	−	0.816624i	$-0.695843\pi$
$89$	−10.8900	−1.15434	−0.577171	+	0.816624i	$0.695843\pi$
$90$	0	0
$91$	−2.87187	−0.301054
$92$	−0.976843	−0.101843
$93$	0	0
$94$	5.60386	0.577994
$95$	−0.785200	−0.0805598
$96$	0	0
$97$	12.7500	1.29456	0.647282	−	0.762251i	$-0.275906\pi$
$97$	12.7500	1.29456	0.647282	+	0.762251i	$0.275906\pi$
$98$	7.50855	0.758478
$99$	0	0
$100$	1.27321	0.127321
$101$	−6.02286	−0.599297	−0.299648	−	0.954050i	$-0.596869\pi$
$101$	−6.02286	−0.599297	−0.299648	+	0.954050i	$0.596869\pi$
$102$	0	0
$103$	−12.4091	−1.22271	−0.611354	−	0.791357i	$-0.709375\pi$
$103$	−12.4091	−1.22271	−0.611354	+	0.791357i	$0.709375\pi$
$104$	−7.46001	−0.731514
$105$	0	0
$106$	−8.68029	−0.843104
$107$	8.84453	0.855033	0.427517	−	0.904007i	$-0.359389\pi$
$107$	8.84453	0.855033	0.427517	+	0.904007i	$0.359389\pi$
$108$	0	0
$109$	4.51481	0.432441	0.216220	−	0.976345i	$-0.430627\pi$
$109$	4.51481	0.432441	0.216220	+	0.976345i	$0.430627\pi$
$110$	0.165209	0.0157520
$111$	0	0
$112$	−3.92660	−0.371028
$113$	9.02713	0.849201	0.424601	−	0.905381i	$-0.360415\pi$
$113$	9.02713	0.849201	0.424601	+	0.905381i	$0.360415\pi$
$114$	0	0
$115$	−0.478325	−0.0446041
$116$	−0.490677	−0.0455582
$117$	0	0
$118$	13.8342	1.27354
$119$	−4.41983	−0.405165
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.32082	0.119581
$123$	0	0
$124$	0.858877	0.0771294
$125$	1.24885	0.111701
$126$	0	0
$127$	21.5594	1.91309	0.956545	−	0.291585i	$-0.0941825\pi$
$127$	21.5594	1.91309	0.956545	+	0.291585i	$0.0941825\pi$
$128$	−8.67785	−0.767021
$129$	0	0
$130$	−0.413712	−0.0362850
$131$	−4.29632	−0.375371	−0.187685	−	0.982229i	$-0.560099\pi$
$131$	−4.29632	−0.375371	−0.187685	+	0.982229i	$0.560099\pi$
$132$	0	0
$133$	7.19928	0.624257
$134$	1.69846	0.146724
$135$	0	0
$136$	−11.4810	−0.984488
$137$	−10.8595	−0.927793	−0.463897	−	0.885889i	$-0.653549\pi$
$137$	−10.8595	−0.927793	−0.463897	+	0.885889i	$0.653549\pi$
$138$	0	0
$139$	−17.1720	−1.45651	−0.728256	−	0.685305i	$-0.759669\pi$
$139$	−17.1720	−1.45651	−0.728256	+	0.685305i	$0.759669\pi$
$140$	0.0366423	0.00309684
$141$	0	0
$142$	−6.93104	−0.581640
$143$	−2.50418	−0.209410
$144$	0	0
$145$	−0.240267	−0.0199531
$146$	11.2954	0.934816
$147$	0	0
$148$	1.24441	0.102290
$149$	−1.82618	−0.149606	−0.0748031	−	0.997198i	$-0.523833\pi$
$149$	−1.82618	−0.149606	−0.0748031	+	0.997198i	$0.523833\pi$
$150$	0	0
$151$	−23.2017	−1.88813	−0.944065	−	0.329760i	$-0.893032\pi$
$151$	−23.2017	−1.88813	−0.944065	+	0.329760i	$0.893032\pi$
$152$	18.7010	1.51685
$153$	0	0
$154$	−1.51475	−0.122062
$155$	0.420561	0.0337803
$156$	0	0
$157$	2.99686	0.239175	0.119588	−	0.992824i	$-0.461843\pi$
$157$	2.99686	0.239175	0.119588	+	0.992824i	$0.461843\pi$
$158$	16.0380	1.27591
$159$	0	0
$160$	0.179585	0.0141974
$161$	4.38563	0.345636
$162$	0	0
$163$	−19.2741	−1.50967	−0.754834	−	0.655916i	$-0.772283\pi$
$163$	−19.2741	−1.50967	−0.754834	+	0.655916i	$0.772283\pi$
$164$	−3.14409	−0.245512
$165$	0	0
$166$	−12.4807	−0.968693
$167$	20.0450	1.55112	0.775562	−	0.631271i	$-0.217466\pi$
$167$	20.0450	1.55112	0.775562	+	0.631271i	$0.217466\pi$
$168$	0	0
$169$	−6.72909	−0.517622
$170$	−0.636706	−0.0488331
$171$	0	0
$172$	1.46956	0.112053
$173$	−5.28474	−0.401791	−0.200896	−	0.979613i	$-0.564385\pi$
$173$	−5.28474	−0.401791	−0.200896	+	0.979613i	$0.564385\pi$
$174$	0	0
$175$	−5.71621	−0.432105
$176$	−3.42387	−0.258084
$177$	0	0
$178$	14.3837	1.07811
$179$	12.8122	0.957632	0.478816	−	0.877915i	$-0.341066\pi$
$179$	12.8122	0.957632	0.478816	+	0.877915i	$0.341066\pi$
$180$	0	0
$181$	13.1985	0.981035	0.490518	−	0.871431i	$-0.336808\pi$
$181$	13.1985	0.981035	0.490518	+	0.871431i	$0.336808\pi$
$182$	3.79322	0.281172
$183$	0	0
$184$	11.3922	0.839843
$185$	0.609344	0.0447998
$186$	0	0
$187$	−3.85395	−0.281828
$188$	1.08377	0.0790420
$189$	0	0
$190$	1.03711	0.0752395
$191$	−5.50669	−0.398450	−0.199225	−	0.979954i	$-0.563842\pi$
$191$	−5.50669	−0.398450	−0.199225	+	0.979954i	$0.563842\pi$
$192$	0	0
$193$	26.3631	1.89766	0.948828	−	0.315793i	$-0.102271\pi$
$193$	26.3631	1.89766	0.948828	+	0.315793i	$0.102271\pi$
$194$	−16.8404	−1.20907
$195$	0	0
$196$	1.45213	0.103724
$197$	−16.8218	−1.19851	−0.599253	−	0.800560i	$-0.704536\pi$
$197$	−16.8218	−1.19851	−0.599253	+	0.800560i	$0.704536\pi$
$198$	0	0
$199$	2.71037	0.192133	0.0960666	−	0.995375i	$-0.469374\pi$
$199$	2.71037	0.192133	0.0960666	+	0.995375i	$0.469374\pi$
$200$	−14.8485	−1.04995
$201$	0	0
$202$	7.95509	0.559719
$203$	2.20294	0.154616
$204$	0	0
$205$	−1.53955	−0.107527
$206$	16.3902	1.14196
$207$	0	0
$208$	8.57397	0.594498
$209$	6.27754	0.434227
$210$	0	0
$211$	1.89225	0.130268	0.0651341	−	0.997877i	$-0.479252\pi$
$211$	1.89225	0.130268	0.0651341	+	0.997877i	$0.479252\pi$
$212$	−1.67874	−0.115296
$213$	0	0
$214$	−11.6820	−0.798566
$215$	0.719592	0.0490758
$216$	0	0
$217$	−3.85601	−0.261763
$218$	−5.96324	−0.403882
$219$	0	0
$220$	0.0319509	0.00215413
$221$	9.65097	0.649194
$222$	0	0
$223$	21.6038	1.44670	0.723349	−	0.690482i	$-0.242602\pi$
$223$	21.6038	1.44670	0.723349	+	0.690482i	$0.242602\pi$
$224$	−1.64657	−0.110016
$225$	0	0
$226$	−11.9232	−0.793119
$227$	11.0851	0.735742	0.367871	−	0.929877i	$-0.380087\pi$
$227$	11.0851	0.735742	0.367871	+	0.929877i	$0.380087\pi$
$228$	0	0
$229$	7.87823	0.520608	0.260304	−	0.965527i	$-0.416177\pi$
$229$	7.87823	0.520608	0.260304	+	0.965527i	$0.416177\pi$
$230$	0.631780	0.0416583
$231$	0	0
$232$	5.72239	0.375693
$233$	15.9084	1.04219	0.521097	−	0.853497i	$-0.325523\pi$
$233$	15.9084	1.04219	0.521097	+	0.853497i	$0.325523\pi$
$234$	0	0
$235$	0.530683	0.0346179
$236$	2.67550	0.174160
$237$	0	0
$238$	5.83778	0.378407
$239$	−0.635463	−0.0411047	−0.0205523	−	0.999789i	$-0.506542\pi$
$239$	−0.635463	−0.0411047	−0.0205523	+	0.999789i	$0.506542\pi$
$240$	0	0
$241$	21.8933	1.41027	0.705136	−	0.709072i	$-0.250886\pi$
$241$	21.8933	1.41027	0.705136	+	0.709072i	$0.250886\pi$
$242$	−1.32082	−0.0849053
$243$	0	0
$244$	0.255442	0.0163530
$245$	0.711056	0.0454277
$246$	0	0
$247$	−15.7201	−1.00024
$248$	−10.0164	−0.636044
$249$	0	0
$250$	−1.64950	−0.104324
$251$	−14.6001	−0.921547	−0.460774	−	0.887518i	$-0.652428\pi$
$251$	−14.6001	−0.921547	−0.460774	+	0.887518i	$0.652428\pi$
$252$	0	0
$253$	3.82413	0.240421
$254$	−28.4761	−1.78675
$255$	0	0
$256$	−6.02634	−0.376646
$257$	2.67965	0.167152	0.0835760	−	0.996501i	$-0.473366\pi$
$257$	2.67965	0.167152	0.0835760	+	0.996501i	$0.473366\pi$
$258$	0	0
$259$	−5.58690	−0.347153
$260$	−0.0800107	−0.00496205
$261$	0	0
$262$	5.67465	0.350581
$263$	−27.9847	−1.72561	−0.862805	−	0.505536i	$-0.831295\pi$
$263$	−27.9847	−1.72561	−0.862805	+	0.505536i	$0.831295\pi$
$264$	0	0
$265$	−0.822019	−0.0504962
$266$	−9.50893	−0.583030
$267$	0	0
$268$	0.328477	0.0200649
$269$	−12.9481	−0.789457	−0.394729	−	0.918798i	$-0.629161\pi$
$269$	−12.9481	−0.789457	−0.394729	+	0.918798i	$0.629161\pi$
$270$	0	0
$271$	10.2416	0.622134	0.311067	−	0.950388i	$-0.399314\pi$
$271$	10.2416	0.622134	0.311067	+	0.950388i	$0.399314\pi$
$272$	13.1954	0.800088
$273$	0	0
$274$	14.3435	0.866521
$275$	−4.98435	−0.300568
$276$	0	0
$277$	−20.9312	−1.25764	−0.628818	−	0.777552i	$-0.716461\pi$
$277$	−20.9312	−1.25764	−0.628818	+	0.777552i	$0.716461\pi$
$278$	22.6811	1.36032
$279$	0	0
$280$	−0.427331	−0.0255379
$281$	−19.8232	−1.18255	−0.591275	−	0.806470i	$-0.701375\pi$
$281$	−19.8232	−1.18255	−0.591275	+	0.806470i	$0.701375\pi$
$282$	0	0
$283$	−17.2247	−1.02390	−0.511952	−	0.859014i	$-0.671078\pi$
$283$	−17.2247	−1.02390	−0.511952	+	0.859014i	$0.671078\pi$
$284$	−1.34044	−0.0795405
$285$	0	0
$286$	3.30756	0.195580
$287$	14.1157	0.833224
$288$	0	0
$289$	−2.14710	−0.126300
$290$	0.317348	0.0186353
$291$	0	0
$292$	2.18450	0.127838
$293$	11.1269	0.650039	0.325020	−	0.945707i	$-0.394629\pi$
$293$	11.1269	0.650039	0.325020	+	0.945707i	$0.394629\pi$
$294$	0	0
$295$	1.31010	0.0762767
$296$	−14.5126	−0.843529
$297$	0	0
$298$	2.41204	0.139726
$299$	−9.57630	−0.553812
$300$	0	0
$301$	−6.59774	−0.380288
$302$	30.6452	1.76344
$303$	0	0
$304$	−21.4935	−1.23273
$305$	0.125081	0.00716211
$306$	0	0
$307$	−10.3242	−0.589234	−0.294617	−	0.955615i	$-0.595192\pi$
$307$	−10.3242	−0.589234	−0.294617	+	0.955615i	$0.595192\pi$
$308$	−0.292949	−0.0166923
$309$	0	0
$310$	−0.555484	−0.0315494
$311$	−21.2489	−1.20491	−0.602457	−	0.798152i	$-0.705811\pi$
$311$	−21.2489	−1.20491	−0.602457	+	0.798152i	$0.705811\pi$
$312$	0	0
$313$	−16.8625	−0.953125	−0.476563	−	0.879141i	$-0.658117\pi$
$313$	−16.8625	−0.953125	−0.476563	+	0.879141i	$0.658117\pi$
$314$	−3.95830	−0.223380
$315$	0	0
$316$	3.10169	0.174484
$317$	2.45125	0.137676	0.0688379	−	0.997628i	$-0.478071\pi$
$317$	2.45125	0.137676	0.0688379	+	0.997628i	$0.478071\pi$
$318$	0	0
$319$	1.92089	0.107549
$320$	−1.09372	−0.0611407
$321$	0	0
$322$	−5.79262	−0.322810
$323$	−24.1933	−1.34615
$324$	0	0
$325$	12.4817	0.692361
$326$	25.4576	1.40997
$327$	0	0
$328$	36.6672	2.02461
$329$	−4.86568	−0.268254
$330$	0	0
$331$	−0.781991	−0.0429821	−0.0214911	−	0.999769i	$-0.506841\pi$
$331$	−0.781991	−0.0429821	−0.0214911	+	0.999769i	$0.506841\pi$
$332$	−2.41374	−0.132471
$333$	0	0
$334$	−26.4757	−1.44869
$335$	0.160843	0.00878781
$336$	0	0
$337$	−8.40420	−0.457806	−0.228903	−	0.973449i	$-0.573514\pi$
$337$	−8.40420	−0.457806	−0.228903	+	0.973449i	$0.573514\pi$
$338$	8.88790	0.483438
$339$	0	0
$340$	−0.123137	−0.00667804
$341$	−3.36232	−0.182080
$342$	0	0
$343$	−14.5473	−0.785480
$344$	−17.1384	−0.924040
$345$	0	0
$346$	6.98017	0.375257
$347$	−27.0984	−1.45472	−0.727360	−	0.686256i	$-0.759253\pi$
$347$	−27.0984	−1.45472	−0.727360	+	0.686256i	$0.759253\pi$
$348$	0	0
$349$	−12.4898	−0.668564	−0.334282	−	0.942473i	$-0.608494\pi$
$349$	−12.4898	−0.668564	−0.334282	+	0.942473i	$0.608494\pi$
$350$	7.55007	0.403568
$351$	0	0
$352$	−1.43575	−0.0765259
$353$	−9.28788	−0.494344	−0.247172	−	0.968972i	$-0.579501\pi$
$353$	−9.28788	−0.494344	−0.247172	+	0.968972i	$0.579501\pi$
$354$	0	0
$355$	−0.656366	−0.0348363
$356$	2.78177	0.147434
$357$	0	0
$358$	−16.9226	−0.894389
$359$	−30.8759	−1.62957	−0.814784	−	0.579764i	$-0.803145\pi$
$359$	−30.8759	−1.62957	−0.814784	+	0.579764i	$0.803145\pi$
$360$	0	0
$361$	20.4075	1.07408
$362$	−17.4328	−0.916246
$363$	0	0
$364$	0.733596	0.0384509
$365$	1.06967	0.0559892
$366$	0	0
$367$	1.47818	0.0771603	0.0385802	−	0.999256i	$-0.487717\pi$
$367$	1.47818	0.0771603	0.0385802	+	0.999256i	$0.487717\pi$
$368$	−13.0933	−0.682536
$369$	0	0
$370$	−0.804831	−0.0418412
$371$	7.53687	0.391295
$372$	0	0
$373$	24.5719	1.27228	0.636142	−	0.771572i	$-0.280529\pi$
$373$	24.5719	1.27228	0.636142	+	0.771572i	$0.280529\pi$
$374$	5.09036	0.263216
$375$	0	0
$376$	−12.6392	−0.651816
$377$	−4.81026	−0.247741
$378$	0	0
$379$	−2.91173	−0.149565	−0.0747826	−	0.997200i	$-0.523826\pi$
$379$	−2.91173	−0.149565	−0.0747826	+	0.997200i	$0.523826\pi$
$380$	0.200573	0.0102892
$381$	0	0
$382$	7.27333	0.372136
$383$	−37.0515	−1.89324	−0.946622	−	0.322346i	$-0.895528\pi$
$383$	−37.0515	−1.89324	−0.946622	+	0.322346i	$0.895528\pi$
$384$	0	0
$385$	−0.143447	−0.00731071
$386$	−34.8208	−1.77233
$387$	0	0
$388$	−3.25688	−0.165343
$389$	−14.5321	−0.736806	−0.368403	−	0.929666i	$-0.620095\pi$
$389$	−14.5321	−0.736806	−0.368403	+	0.929666i	$0.620095\pi$
$390$	0	0
$391$	−14.7380	−0.745332
$392$	−16.9351	−0.855352
$393$	0	0
$394$	22.2186	1.11936
$395$	1.51879	0.0764185
$396$	0	0
$397$	−23.3346	−1.17113	−0.585564	−	0.810626i	$-0.699127\pi$
$397$	−23.3346	−1.17113	−0.585564	+	0.810626i	$0.699127\pi$
$398$	−3.57991	−0.179445
$399$	0	0
$400$	17.0658	0.853288
$401$	−0.540093	−0.0269709	−0.0134855	−	0.999909i	$-0.504293\pi$
$401$	−0.540093	−0.0269709	−0.0134855	+	0.999909i	$0.504293\pi$
$402$	0	0
$403$	8.41984	0.419422
$404$	1.53849	0.0765428
$405$	0	0
$406$	−2.90968	−0.144405
$407$	−4.87160	−0.241476
$408$	0	0
$409$	−0.914514	−0.0452198	−0.0226099	−	0.999744i	$-0.507198\pi$
$409$	−0.914514	−0.0452198	−0.0226099	+	0.999744i	$0.507198\pi$
$410$	2.03346	0.100426
$411$	0	0
$412$	3.16981	0.156165
$413$	−12.0119	−0.591068
$414$	0	0
$415$	−1.18192	−0.0580182
$416$	3.59538	0.176278
$417$	0	0
$418$	−8.29148	−0.405550
$419$	4.99943	0.244238	0.122119	−	0.992515i	$-0.461031\pi$
$419$	4.99943	0.244238	0.122119	+	0.992515i	$0.461031\pi$
$420$	0	0
$421$	−1.76551	−0.0860457	−0.0430228	−	0.999074i	$-0.513699\pi$
$421$	−1.76551	−0.0860457	−0.0430228	+	0.999074i	$0.513699\pi$
$422$	−2.49932	−0.121665
$423$	0	0
$424$	19.5779	0.950786
$425$	19.2094	0.931794
$426$	0	0
$427$	−1.14683	−0.0554991
$428$	−2.25926	−0.109206
$429$	0	0
$430$	−0.950450	−0.0458347
$431$	2.49144	0.120009	0.0600043	−	0.998198i	$-0.480889\pi$
$431$	2.49144	0.120009	0.0600043	+	0.998198i	$0.480889\pi$
$432$	0	0
$433$	−8.69076	−0.417651	−0.208826	−	0.977953i	$-0.566964\pi$
$433$	−8.69076	−0.417651	−0.208826	+	0.977953i	$0.566964\pi$
$434$	5.09308	0.244476
$435$	0	0
$436$	−1.15327	−0.0552317
$437$	24.0061	1.14837
$438$	0	0
$439$	−3.39355	−0.161965	−0.0809826	−	0.996716i	$-0.525806\pi$
$439$	−3.39355	−0.161965	−0.0809826	+	0.996716i	$0.525806\pi$
$440$	−0.372619	−0.0177639
$441$	0	0
$442$	−12.7472	−0.606321
$443$	11.6791	0.554889	0.277444	−	0.960742i	$-0.410513\pi$
$443$	11.6791	0.554889	0.277444	+	0.960742i	$0.410513\pi$
$444$	0	0
$445$	1.36213	0.0645713
$446$	−28.5347	−1.35116
$447$	0	0
$448$	10.0280	0.473779
$449$	−21.0387	−0.992877	−0.496438	−	0.868072i	$-0.665359\pi$
$449$	−21.0387	−0.992877	−0.496438	+	0.868072i	$0.665359\pi$
$450$	0	0
$451$	12.3084	0.579582
$452$	−2.30591	−0.108461
$453$	0	0
$454$	−14.6414	−0.687153
$455$	0.359216	0.0168403
$456$	0	0
$457$	−19.6944	−0.921263	−0.460632	−	0.887591i	$-0.652377\pi$
$457$	−19.6944	−0.921263	−0.460632	+	0.887591i	$0.652377\pi$
$458$	−10.4057	−0.486227
$459$	0	0
$460$	0.122184	0.00569687
$461$	−6.46808	−0.301249	−0.150624	−	0.988591i	$-0.548128\pi$
$461$	−6.46808	−0.301249	−0.150624	+	0.988591i	$0.548128\pi$
$462$	0	0
$463$	−27.2380	−1.26586	−0.632929	−	0.774210i	$-0.718147\pi$
$463$	−27.2380	−1.26586	−0.632929	+	0.774210i	$0.718147\pi$
$464$	−6.57688	−0.305324
$465$	0	0
$466$	−21.0121	−0.973367
$467$	3.62126	0.167572	0.0837860	−	0.996484i	$-0.473299\pi$
$467$	3.62126	0.167572	0.0837860	+	0.996484i	$0.473299\pi$
$468$	0	0
$469$	−1.47473	−0.0680966
$470$	−0.700935	−0.0323317
$471$	0	0
$472$	−31.2023	−1.43620
$473$	−5.75302	−0.264524
$474$	0	0
$475$	−31.2895	−1.43566
$476$	1.12901	0.0517480
$477$	0	0
$478$	0.839330	0.0383901
$479$	23.3655	1.06760	0.533798	−	0.845612i	$-0.320764\pi$
$479$	23.3655	1.06760	0.533798	+	0.845612i	$0.320764\pi$
$480$	0	0
$481$	12.1994	0.556243
$482$	−28.9171	−1.31714
$483$	0	0
$484$	−0.255442	−0.0116110
$485$	−1.59478	−0.0724151
$486$	0	0
$487$	36.6938	1.66275	0.831377	−	0.555709i	$-0.187553\pi$
$487$	36.6938	1.66275	0.831377	+	0.555709i	$0.187553\pi$
$488$	−2.97903	−0.134854
$489$	0	0
$490$	−0.939176	−0.0424276
$491$	3.77646	0.170429	0.0852146	−	0.996363i	$-0.472842\pi$
$491$	3.77646	0.170429	0.0852146	+	0.996363i	$0.472842\pi$
$492$	0	0
$493$	−7.40302	−0.333415
$494$	20.7634	0.934187
$495$	0	0
$496$	11.5121	0.516909
$497$	6.01804	0.269946
$498$	0	0
$499$	−27.0496	−1.21091	−0.605454	−	0.795880i	$-0.707008\pi$
$499$	−27.0496	−1.21091	−0.605454	+	0.795880i	$0.707008\pi$
$500$	−0.319009	−0.0142665
$501$	0	0
$502$	19.2840	0.860687
$503$	−9.82735	−0.438180	−0.219090	−	0.975705i	$-0.570309\pi$
$503$	−9.82735	−0.438180	−0.219090	+	0.975705i	$0.570309\pi$
$504$	0	0
$505$	0.753344	0.0335234
$506$	−5.05098	−0.224543
$507$	0	0
$508$	−5.50718	−0.244342
$509$	−22.3951	−0.992647	−0.496323	−	0.868138i	$-0.665317\pi$
$509$	−22.3951	−0.992647	−0.496323	+	0.868138i	$0.665317\pi$
$510$	0	0
$511$	−9.80753	−0.433859
$512$	25.3154	1.11879
$513$	0	0
$514$	−3.53933	−0.156113
$515$	1.55214	0.0683956
$516$	0	0
$517$	−4.24272	−0.186595
$518$	7.37928	0.324227
$519$	0	0
$520$	0.933104	0.0409193
$521$	36.5278	1.60031	0.800156	−	0.599792i	$-0.204750\pi$
$521$	36.5278	1.60031	0.800156	+	0.599792i	$0.204750\pi$
$522$	0	0
$523$	−36.5507	−1.59825	−0.799125	−	0.601164i	$-0.794704\pi$
$523$	−36.5507	−1.59825	−0.799125	+	0.601164i	$0.794704\pi$
$524$	1.09746	0.0479427
$525$	0	0
$526$	36.9627	1.61165
$527$	12.9582	0.564467
$528$	0	0
$529$	−8.37603	−0.364175
$530$	1.08574	0.0471614
$531$	0	0
$532$	−1.83900	−0.0797307
$533$	−30.8225	−1.33507
$534$	0	0
$535$	−1.10628	−0.0478287
$536$	−3.83077	−0.165464
$537$	0	0
$538$	17.1020	0.737321
$539$	−5.68478	−0.244861
$540$	0	0
$541$	6.27049	0.269590	0.134795	−	0.990874i	$-0.456962\pi$
$541$	6.27049	0.269590	0.134795	+	0.990874i	$0.456962\pi$
$542$	−13.5273	−0.581048
$543$	0	0
$544$	5.53331	0.237239
$545$	−0.564716	−0.0241898
$546$	0	0
$547$	−41.7002	−1.78297	−0.891486	−	0.453048i	$-0.850337\pi$
$547$	−41.7002	−1.78297	−0.891486	+	0.453048i	$0.850337\pi$
$548$	2.77398	0.118499
$549$	0	0
$550$	6.58342	0.280718
$551$	12.0585	0.513709
$552$	0	0
$553$	−13.9254	−0.592166
$554$	27.6463	1.17458
$555$	0	0
$556$	4.38646	0.186027
$557$	12.6770	0.537142	0.268571	−	0.963260i	$-0.413449\pi$
$557$	12.6770	0.537142	0.268571	+	0.963260i	$0.413449\pi$
$558$	0	0
$559$	14.4066	0.609334
$560$	0.491142	0.0207545
$561$	0	0
$562$	26.1828	1.10445
$563$	27.2535	1.14860	0.574298	−	0.818646i	$-0.305275\pi$
$563$	27.2535	1.14860	0.574298	+	0.818646i	$0.305275\pi$
$564$	0	0
$565$	−1.12912	−0.0475025
$566$	22.7507	0.956285
$567$	0	0
$568$	15.6325	0.655927
$569$	−38.4443	−1.61167	−0.805835	−	0.592140i	$-0.798283\pi$
$569$	−38.4443	−1.61167	−0.805835	+	0.592140i	$0.798283\pi$
$570$	0	0
$571$	−43.7668	−1.83158	−0.915792	−	0.401654i	$-0.868435\pi$
$571$	−43.7668	−1.83158	−0.915792	+	0.401654i	$0.868435\pi$
$572$	0.639672	0.0267460
$573$	0	0
$574$	−18.6443	−0.778196
$575$	−19.0608	−0.794891
$576$	0	0
$577$	43.8337	1.82482	0.912411	−	0.409276i	$-0.134219\pi$
$577$	43.8337	1.82482	0.912411	+	0.409276i	$0.134219\pi$
$578$	2.83593	0.117959
$579$	0	0
$580$	0.0613742	0.00254843
$581$	10.8367	0.449582
$582$	0	0
$583$	6.57191	0.272181
$584$	−25.4762	−1.05421
$585$	0	0
$586$	−14.6966	−0.607110
$587$	32.6296	1.34677	0.673383	−	0.739294i	$-0.264840\pi$
$587$	32.6296	1.34677	0.673383	+	0.739294i	$0.264840\pi$
$588$	0	0
$589$	−21.1071	−0.869702
$590$	−1.73040	−0.0712393
$591$	0	0
$592$	16.6797	0.685532
$593$	−15.3437	−0.630090	−0.315045	−	0.949077i	$-0.602020\pi$
$593$	−15.3437	−0.630090	−0.315045	+	0.949077i	$0.602020\pi$
$594$	0	0
$595$	0.552835	0.0226640
$596$	0.466482	0.0191078
$597$	0	0
$598$	12.6485	0.517237
$599$	36.4818	1.49061	0.745303	−	0.666726i	$-0.232305\pi$
$599$	36.4818	1.49061	0.745303	+	0.666726i	$0.232305\pi$
$600$	0	0
$601$	−13.4313	−0.547874	−0.273937	−	0.961748i	$-0.588326\pi$
$601$	−13.4313	−0.547874	−0.273937	+	0.961748i	$0.588326\pi$
$602$	8.71441	0.355173
$603$	0	0
$604$	5.92670	0.241154
$605$	−0.125081	−0.00508526
$606$	0	0
$607$	−46.9219	−1.90450	−0.952250	−	0.305320i	$-0.901237\pi$
$607$	−46.9219	−1.90450	−0.952250	+	0.305320i	$0.901237\pi$
$608$	−9.01300	−0.365525
$609$	0	0
$610$	−0.165209	−0.00668911
$611$	10.6245	0.429822
$612$	0	0
$613$	30.0763	1.21477	0.607385	−	0.794408i	$-0.292219\pi$
$613$	30.0763	1.21477	0.607385	+	0.794408i	$0.292219\pi$
$614$	13.6364	0.550320
$615$	0	0
$616$	3.41644	0.137652
$617$	33.6749	1.35570	0.677849	−	0.735201i	$-0.262912\pi$
$617$	33.6749	1.35570	0.677849	+	0.735201i	$0.262912\pi$
$618$	0	0
$619$	−31.5312	−1.26735	−0.633673	−	0.773601i	$-0.718454\pi$
$619$	−31.5312	−1.26735	−0.633673	+	0.773601i	$0.718454\pi$
$620$	−0.107429	−0.00431445
$621$	0	0
$622$	28.0659	1.12534
$623$	−12.4890	−0.500362
$624$	0	0
$625$	24.7656	0.990623
$626$	22.2723	0.890179
$627$	0	0
$628$	−0.765524	−0.0305477
$629$	18.7749	0.748604
$630$	0	0
$631$	45.7220	1.82016	0.910082	−	0.414428i	$-0.136018\pi$
$631$	45.7220	1.82016	0.910082	+	0.414428i	$0.136018\pi$
$632$	−36.1727	−1.43887
$633$	0	0
$634$	−3.23765	−0.128584
$635$	−2.69667	−0.107014
$636$	0	0
$637$	14.2357	0.564039
$638$	−2.53715	−0.100447
$639$	0	0
$640$	1.08543	0.0429055
$641$	7.94352	0.313750	0.156875	−	0.987618i	$-0.449858\pi$
$641$	7.94352	0.313750	0.156875	+	0.987618i	$0.449858\pi$
$642$	0	0
$643$	−23.5096	−0.927130	−0.463565	−	0.886063i	$-0.653430\pi$
$643$	−23.5096	−0.927130	−0.463565	+	0.886063i	$0.653430\pi$
$644$	−1.12027	−0.0441450
$645$	0	0
$646$	31.9549	1.25725
$647$	−35.2670	−1.38649	−0.693244	−	0.720703i	$-0.743819\pi$
$647$	−35.2670	−1.38649	−0.693244	+	0.720703i	$0.743819\pi$
$648$	0	0
$649$	−10.4740	−0.411140
$650$	−16.4861	−0.646636
$651$	0	0
$652$	4.92343	0.192816
$653$	22.4739	0.879474	0.439737	−	0.898127i	$-0.355072\pi$
$653$	22.4739	0.879474	0.439737	+	0.898127i	$0.355072\pi$
$654$	0	0
$655$	0.537387	0.0209974
$656$	−42.1424	−1.64539
$657$	0	0
$658$	6.42668	0.250538
$659$	14.6896	0.572225	0.286112	−	0.958196i	$-0.407637\pi$
$659$	14.6896	0.572225	0.286112	+	0.958196i	$0.407637\pi$
$660$	0	0
$661$	−35.6070	−1.38495	−0.692477	−	0.721440i	$-0.743480\pi$
$661$	−35.6070	−1.38495	−0.692477	+	0.721440i	$0.743480\pi$
$662$	1.03287	0.0401435
$663$	0	0
$664$	28.1496	1.09242
$665$	−0.900492	−0.0349196
$666$	0	0
$667$	7.34574	0.284428
$668$	−5.12032	−0.198111
$669$	0	0
$670$	−0.212445	−0.00820745
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	21.9541	0.846270	0.423135	−	0.906067i	$-0.360930\pi$
$673$	21.9541	0.846270	0.423135	+	0.906067i	$0.360930\pi$
$674$	11.1004	0.427572
$675$	0	0
$676$	1.71889	0.0661113
$677$	−34.0087	−1.30706	−0.653531	−	0.756900i	$-0.726713\pi$
$677$	−34.0087	−1.30706	−0.653531	+	0.756900i	$0.726713\pi$
$678$	0	0
$679$	14.6221	0.561144
$680$	1.43605	0.0550701
$681$	0	0
$682$	4.44100	0.170055
$683$	−19.3276	−0.739552	−0.369776	−	0.929121i	$-0.620566\pi$
$683$	−19.3276	−0.739552	−0.369776	+	0.929121i	$0.620566\pi$
$684$	0	0
$685$	1.35832	0.0518987
$686$	19.2143	0.733606
$687$	0	0
$688$	19.6976	0.750962
$689$	−16.4572	−0.626970
$690$	0	0
$691$	−18.9410	−0.720548	−0.360274	−	0.932847i	$-0.617317\pi$
$691$	−18.9410	−0.720548	−0.360274	+	0.932847i	$0.617317\pi$
$692$	1.34994	0.0513172
$693$	0	0
$694$	35.7921	1.35865
$695$	2.14789	0.0814741
$696$	0	0
$697$	−47.4360	−1.79677
$698$	16.4967	0.624411
$699$	0	0
$700$	1.46016	0.0551889
$701$	38.8008	1.46549	0.732744	−	0.680505i	$-0.238239\pi$
$701$	38.8008	1.46549	0.732744	+	0.680505i	$0.238239\pi$
$702$	0	0
$703$	−30.5817	−1.15341
$704$	8.74410	0.329556
$705$	0	0
$706$	12.2676	0.461697
$707$	−6.90720	−0.259772
$708$	0	0
$709$	−22.3779	−0.840420	−0.420210	−	0.907427i	$-0.638044\pi$
$709$	−22.3779	−0.840420	−0.420210	+	0.907427i	$0.638044\pi$
$710$	0.866939	0.0325357
$711$	0	0
$712$	−32.4417	−1.21580
$713$	−12.8579	−0.481533
$714$	0	0
$715$	0.313225	0.0117139
$716$	−3.27278	−0.122310
$717$	0	0
$718$	40.7814	1.52195
$719$	10.6672	0.397819	0.198910	−	0.980018i	$-0.436260\pi$
$719$	10.6672	0.397819	0.198910	+	0.980018i	$0.436260\pi$
$720$	0	0
$721$	−14.2312	−0.529997
$722$	−26.9546	−1.00315
$723$	0	0
$724$	−3.37145	−0.125299
$725$	−9.57441	−0.355585
$726$	0	0
$727$	−22.7163	−0.842499	−0.421250	−	0.906945i	$-0.638408\pi$
$727$	−22.7163	−0.842499	−0.421250	+	0.906945i	$0.638408\pi$
$728$	−8.55538	−0.317083
$729$	0	0
$730$	−1.41284	−0.0522916
$731$	22.1718	0.820054
$732$	0	0
$733$	−21.4233	−0.791287	−0.395644	−	0.918404i	$-0.629478\pi$
$733$	−21.4233	−0.791287	−0.395644	+	0.918404i	$0.629478\pi$
$734$	−1.95240	−0.0720645
$735$	0	0
$736$	−5.49050	−0.202383
$737$	−1.28591	−0.0473673
$738$	0	0
$739$	33.6543	1.23799	0.618996	−	0.785394i	$-0.287540\pi$
$739$	33.6543	1.23799	0.618996	+	0.785394i	$0.287540\pi$
$740$	−0.155652	−0.00572188
$741$	0	0
$742$	−9.95482	−0.365453
$743$	−49.9515	−1.83254	−0.916271	−	0.400558i	$-0.868816\pi$
$743$	−49.9515	−1.83254	−0.916271	+	0.400558i	$0.868816\pi$
$744$	0	0
$745$	0.228419	0.00836864
$746$	−32.4550	−1.18826
$747$	0	0
$748$	0.984460	0.0359954
$749$	10.1432	0.370624
$750$	0	0
$751$	16.5995	0.605723	0.302862	−	0.953035i	$-0.402058\pi$
$751$	16.5995	0.605723	0.302862	+	0.953035i	$0.402058\pi$
$752$	14.5265	0.529727
$753$	0	0
$754$	6.35347	0.231380
$755$	2.90209	0.105618
$756$	0	0
$757$	7.43752	0.270321	0.135161	−	0.990824i	$-0.456845\pi$
$757$	7.43752	0.270321	0.135161	+	0.990824i	$0.456845\pi$
$758$	3.84586	0.139688
$759$	0	0
$760$	−2.33913	−0.0848492
$761$	−6.45053	−0.233831	−0.116916	−	0.993142i	$-0.537301\pi$
$761$	−6.45053	−0.233831	−0.116916	+	0.993142i	$0.537301\pi$
$762$	0	0
$763$	5.17773	0.187446
$764$	1.40664	0.0508904
$765$	0	0
$766$	48.9383	1.76821
$767$	26.2288	0.947066
$768$	0	0
$769$	−31.7200	−1.14385	−0.571926	−	0.820305i	$-0.693803\pi$
$769$	−31.7200	−1.14385	−0.571926	+	0.820305i	$0.693803\pi$
$770$	0.189467	0.00682791
$771$	0	0
$772$	−6.73424	−0.242370
$773$	20.6384	0.742310	0.371155	−	0.928571i	$-0.378962\pi$
$773$	20.6384	0.742310	0.371155	+	0.928571i	$0.378962\pi$
$774$	0	0
$775$	16.7590	0.602000
$776$	37.9825	1.36349
$777$	0	0
$778$	19.1942	0.688146
$779$	77.2667	2.76837
$780$	0	0
$781$	5.24754	0.187772
$782$	19.4662	0.696109
$783$	0	0
$784$	19.4639	0.695140
$785$	−0.374850	−0.0133790
$786$	0	0
$787$	44.1526	1.57387	0.786936	−	0.617035i	$-0.211666\pi$
$787$	44.1526	1.57387	0.786936	+	0.617035i	$0.211666\pi$
$788$	4.29700	0.153074
$789$	0	0
$790$	−2.00604	−0.0713718
$791$	10.3526	0.368096
$792$	0	0
$793$	2.50418	0.0889260
$794$	30.8207	1.09379
$795$	0	0
$796$	−0.692343	−0.0245394
$797$	−34.6903	−1.22879	−0.614396	−	0.788998i	$-0.710600\pi$
$797$	−34.6903	−1.22879	−0.614396	+	0.788998i	$0.710600\pi$
$798$	0	0
$799$	16.3512	0.578464
$800$	7.15630	0.253013
$801$	0	0
$802$	0.713364	0.0251897
$803$	−8.55185	−0.301788
$804$	0	0
$805$	−0.548558	−0.0193341
$806$	−11.1211	−0.391723
$807$	0	0
$808$	−17.9423	−0.631206
$809$	−51.8848	−1.82417	−0.912087	−	0.409997i	$-0.865530\pi$
$809$	−51.8848	−1.82417	−0.912087	+	0.409997i	$0.865530\pi$
$810$	0	0
$811$	22.2643	0.781806	0.390903	−	0.920432i	$-0.372163\pi$
$811$	22.2643	0.781806	0.390903	+	0.920432i	$0.372163\pi$
$812$	−0.562723	−0.0197477
$813$	0	0
$814$	6.43449	0.225529
$815$	2.41083	0.0844475
$816$	0	0
$817$	−36.1148	−1.26350
$818$	1.20791	0.0422334
$819$	0	0
$820$	0.393265	0.0137334
$821$	−22.1713	−0.773785	−0.386892	−	0.922125i	$-0.626452\pi$
$821$	−22.1713	−0.773785	−0.386892	+	0.922125i	$0.626452\pi$
$822$	0	0
$823$	−21.2506	−0.740748	−0.370374	−	0.928883i	$-0.620770\pi$
$823$	−21.2506	−0.740748	−0.370374	+	0.928883i	$0.620770\pi$
$824$	−36.9671	−1.28781
$825$	0	0
$826$	15.8655	0.552033
$827$	−35.1976	−1.22394	−0.611970	−	0.790881i	$-0.709623\pi$
$827$	−35.1976	−1.22394	−0.611970	+	0.790881i	$0.709623\pi$
$828$	0	0
$829$	−26.1779	−0.909196	−0.454598	−	0.890697i	$-0.650217\pi$
$829$	−26.1779	−0.909196	−0.454598	+	0.890697i	$0.650217\pi$
$830$	1.56110	0.0541866
$831$	0	0
$832$	−21.8968	−0.759134
$833$	21.9088	0.759096
$834$	0	0
$835$	−2.50724	−0.0867666
$836$	−1.60355	−0.0554598
$837$	0	0
$838$	−6.60334	−0.228108
$839$	1.51914	0.0524467	0.0262233	−	0.999656i	$-0.491652\pi$
$839$	1.51914	0.0524467	0.0262233	+	0.999656i	$0.491652\pi$
$840$	0	0
$841$	−25.3102	−0.872765
$842$	2.33192	0.0803631
$843$	0	0
$844$	−0.483361	−0.0166380
$845$	0.841680	0.0289547
$846$	0	0
$847$	1.14683	0.0394056
$848$	−22.5013	−0.772699
$849$	0	0
$850$	−25.3721	−0.870258
$851$	−18.6296	−0.638616
$852$	0	0
$853$	10.5213	0.360244	0.180122	−	0.983644i	$-0.442351\pi$
$853$	10.5213	0.360244	0.180122	+	0.983644i	$0.442351\pi$
$854$	1.51475	0.0518338
$855$	0	0
$856$	26.3481	0.900559
$857$	40.5631	1.38561	0.692804	−	0.721126i	$-0.256375\pi$
$857$	40.5631	1.38561	0.692804	+	0.721126i	$0.256375\pi$
$858$	0	0
$859$	13.0459	0.445122	0.222561	−	0.974919i	$-0.428558\pi$
$859$	13.0459	0.445122	0.222561	+	0.974919i	$0.428558\pi$
$860$	−0.183814	−0.00626801
$861$	0	0
$862$	−3.29074	−0.112083
$863$	40.3377	1.37311	0.686556	−	0.727077i	$-0.259122\pi$
$863$	40.3377	1.37311	0.686556	+	0.727077i	$0.259122\pi$
$864$	0	0
$865$	0.661019	0.0224753
$866$	11.4789	0.390069
$867$	0	0
$868$	0.984987	0.0334326
$869$	−12.1425	−0.411905
$870$	0	0
$871$	3.22016	0.109111
$872$	13.4497	0.455466
$873$	0	0
$874$	−31.7077	−1.07253
$875$	1.43222	0.0484179
$876$	0	0
$877$	−23.0218	−0.777392	−0.388696	−	0.921366i	$-0.627074\pi$
$877$	−23.0218	−0.777392	−0.388696	+	0.921366i	$0.627074\pi$
$878$	4.48225	0.151269
$879$	0	0
$880$	0.428260	0.0144366
$881$	−50.1735	−1.69039	−0.845195	−	0.534459i	$-0.820516\pi$
$881$	−50.1735	−1.69039	−0.845195	+	0.534459i	$0.820516\pi$
$882$	0	0
$883$	−38.1075	−1.28242	−0.641209	−	0.767366i	$-0.721567\pi$
$883$	−38.1075	−1.28242	−0.641209	+	0.767366i	$0.721567\pi$
$884$	−2.46526	−0.0829157
$885$	0	0
$886$	−15.4259	−0.518243
$887$	−31.0229	−1.04165	−0.520823	−	0.853665i	$-0.674375\pi$
$887$	−31.0229	−1.04165	−0.520823	+	0.853665i	$0.674375\pi$
$888$	0	0
$889$	24.7250	0.829251
$890$	−1.79913	−0.0603070
$891$	0	0
$892$	−5.51852	−0.184774
$893$	−26.6338	−0.891268
$894$	0	0
$895$	−1.60256	−0.0535678
$896$	−9.95203	−0.332474
$897$	0	0
$898$	27.7882	0.927306
$899$	−6.45865	−0.215408
$900$	0	0
$901$	−25.3278	−0.843790
$902$	−16.2572	−0.541305
$903$	0	0
$904$	26.8921	0.894417
$905$	−1.65088	−0.0548770
$906$	0	0
$907$	−3.45224	−0.114630	−0.0573148	−	0.998356i	$-0.518254\pi$
$907$	−3.45224	−0.114630	−0.0573148	+	0.998356i	$0.518254\pi$
$908$	−2.83159	−0.0939697
$909$	0	0
$910$	−0.474458	−0.0157281
$911$	−26.5820	−0.880701	−0.440351	−	0.897826i	$-0.645146\pi$
$911$	−26.5820	−0.880701	−0.440351	+	0.897826i	$0.645146\pi$
$912$	0	0
$913$	9.44925	0.312725
$914$	26.0127	0.860422
$915$	0	0
$916$	−2.01243	−0.0664926
$917$	−4.92715	−0.162709
$918$	0	0
$919$	−16.8704	−0.556503	−0.278252	−	0.960508i	$-0.589755\pi$
$919$	−16.8704	−0.556503	−0.278252	+	0.960508i	$0.589755\pi$
$920$	−1.42494	−0.0469790
$921$	0	0
$922$	8.54316	0.281354
$923$	−13.1408	−0.432534
$924$	0	0
$925$	24.2818	0.798380
$926$	35.9764	1.18226
$927$	0	0
$928$	−2.75793	−0.0905334
$929$	−4.20582	−0.137989	−0.0689943	−	0.997617i	$-0.521979\pi$
$929$	−4.20582	−0.137989	−0.0689943	+	0.997617i	$0.521979\pi$
$930$	0	0
$931$	−35.6864	−1.16958
$932$	−4.06367	−0.133110
$933$	0	0
$934$	−4.78302	−0.156505
$935$	0.482055	0.0157649
$936$	0	0
$937$	30.6465	1.00118	0.500589	−	0.865685i	$-0.333117\pi$
$937$	30.6465	1.00118	0.500589	+	0.865685i	$0.333117\pi$
$938$	1.94785	0.0635994
$939$	0	0
$940$	−0.135559	−0.00442144
$941$	−12.7265	−0.414873	−0.207437	−	0.978248i	$-0.566512\pi$
$941$	−12.7265	−0.414873	−0.207437	+	0.978248i	$0.566512\pi$
$942$	0	0
$943$	47.0691	1.53278
$944$	35.8616	1.16719
$945$	0	0
$946$	7.59869	0.247055
$947$	9.96361	0.323774	0.161887	−	0.986809i	$-0.448242\pi$
$947$	9.96361	0.323774	0.161887	+	0.986809i	$0.448242\pi$
$948$	0	0
$949$	21.4154	0.695172
$950$	41.3277	1.34085
$951$	0	0
$952$	−13.1668	−0.426738
$953$	−18.5859	−0.602056	−0.301028	−	0.953615i	$-0.597330\pi$
$953$	−18.5859	−0.602056	−0.301028	+	0.953615i	$0.597330\pi$
$954$	0	0
$955$	0.688781	0.0222884
$956$	0.162324	0.00524993
$957$	0	0
$958$	−30.8615	−0.997090
$959$	−12.4541	−0.402163
$960$	0	0
$961$	−19.6948	−0.635317
$962$	−16.1131	−0.519508
$963$	0	0
$964$	−5.59247	−0.180121
$965$	−3.29751	−0.106151
$966$	0	0
$967$	−37.1048	−1.19321	−0.596605	−	0.802535i	$-0.703484\pi$
$967$	−37.1048	−1.19321	−0.596605	+	0.802535i	$0.703484\pi$
$968$	2.97903	0.0957495
$969$	0	0
$970$	2.10641	0.0676327
$971$	10.2379	0.328549	0.164275	−	0.986415i	$-0.447472\pi$
$971$	10.2379	0.328549	0.164275	+	0.986415i	$0.447472\pi$
$972$	0	0
$973$	−19.6934	−0.631342
$974$	−48.4658	−1.55294
$975$	0	0
$976$	3.42387	0.109595
$977$	−21.0939	−0.674854	−0.337427	−	0.941352i	$-0.609557\pi$
$977$	−21.0939	−0.674854	−0.337427	+	0.941352i	$0.609557\pi$
$978$	0	0
$979$	−10.8900	−0.348047
$980$	−0.181634	−0.00580208
$981$	0	0
$982$	−4.98801	−0.159174
$983$	35.5793	1.13480	0.567401	−	0.823442i	$-0.307949\pi$
$983$	35.5793	1.13480	0.567401	+	0.823442i	$0.307949\pi$
$984$	0	0
$985$	2.10409	0.0670418
$986$	9.77803	0.311396
$987$	0	0
$988$	4.01557	0.127752
$989$	−22.0003	−0.699568
$990$	0	0
$991$	29.0555	0.922977	0.461489	−	0.887146i	$-0.347316\pi$
$991$	29.0555	0.922977	0.461489	+	0.887146i	$0.347316\pi$
$992$	4.82745	0.153272
$993$	0	0
$994$	−7.94873	−0.252118
$995$	−0.339016	−0.0107475
$996$	0	0
$997$	33.8463	1.07192	0.535962	−	0.844242i	$-0.319949\pi$
$997$	33.8463	1.07192	0.535962	+	0.844242i	$0.319949\pi$
$998$	35.7276	1.13094
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.e.1.4		12
3.2	odd	2		2013.2.a.d.1.9	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.d.1.9	✓	12	3.2	odd	2
6039.2.a.e.1.4		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)

\(f(q)\)	\(=\)	\(q-1.32082 q^{2} -0.255442 q^{4} -0.125081 q^{5} +1.14683 q^{7} +2.97903 q^{8} +O(q^{10})\)	\(q-1.32082 q^{2} -0.255442 q^{4} -0.125081 q^{5} +1.14683 q^{7} +2.97903 q^{8} +0.165209 q^{10} +1.00000 q^{11} -2.50418 q^{13} -1.51475 q^{14} -3.42387 q^{16} -3.85395 q^{17} +6.27754 q^{19} +0.0319509 q^{20} -1.32082 q^{22} +3.82413 q^{23} -4.98435 q^{25} +3.30756 q^{26} -0.292949 q^{28} +1.92089 q^{29} -3.36232 q^{31} -1.43575 q^{32} +5.09036 q^{34} -0.143447 q^{35} -4.87160 q^{37} -8.29148 q^{38} -0.372619 q^{40} +12.3084 q^{41} -5.75302 q^{43} -0.255442 q^{44} -5.05098 q^{46} -4.24272 q^{47} -5.68478 q^{49} +6.58342 q^{50} +0.639672 q^{52} +6.57191 q^{53} -0.125081 q^{55} +3.41644 q^{56} -2.53715 q^{58} -10.4740 q^{59} -1.00000 q^{61} +4.44100 q^{62} +8.74410 q^{64} +0.313225 q^{65} -1.28591 q^{67} +0.984460 q^{68} +0.189467 q^{70} +5.24754 q^{71} -8.55185 q^{73} +6.43449 q^{74} -1.60355 q^{76} +1.14683 q^{77} -12.1425 q^{79} +0.428260 q^{80} -16.2572 q^{82} +9.44925 q^{83} +0.482055 q^{85} +7.59869 q^{86} +2.97903 q^{88} -10.8900 q^{89} -2.87187 q^{91} -0.976843 q^{92} +5.60386 q^{94} -0.785200 q^{95} +12.7500 q^{97} +7.50855 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) \) 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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