LMFDB - Embedded newform 6016.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6016,2,Mod(1,6016)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6016, 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("6016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6016 = 2^{7} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6016.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:	$48.0380018560$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} + \cdots - 832 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 163x^9 - 840x^8 - 438x^7 + 3358x^6 + 5x^5 - 6230x^4 + 1488x^3 + 4720x^2 - 1440x - 832
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.10994$ of defining polynomial
Character	$\chi$	$=$	6016.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.10994 q^{3} +0.407310 q^{5} -3.05265 q^{7} +6.67175 q^{9} +O(q^{10})$	$q-3.10994 q^{3} +0.407310 q^{5} -3.05265 q^{7} +6.67175 q^{9} -2.16670 q^{11} +0.550265 q^{13} -1.26671 q^{15} -3.34079 q^{17} +6.78215 q^{19} +9.49357 q^{21} -5.22765 q^{23} -4.83410 q^{25} -11.4189 q^{27} -3.25095 q^{29} -8.03298 q^{31} +6.73833 q^{33} -1.24337 q^{35} +9.82204 q^{37} -1.71129 q^{39} +8.04458 q^{41} +3.91084 q^{43} +2.71747 q^{45} +1.00000 q^{47} +2.31866 q^{49} +10.3897 q^{51} -6.41908 q^{53} -0.882520 q^{55} -21.0921 q^{57} +2.73544 q^{59} -4.63323 q^{61} -20.3665 q^{63} +0.224128 q^{65} -11.2924 q^{67} +16.2577 q^{69} -5.75360 q^{71} -6.94230 q^{73} +15.0338 q^{75} +6.61419 q^{77} -15.0416 q^{79} +15.4970 q^{81} -9.73633 q^{83} -1.36074 q^{85} +10.1103 q^{87} +9.09741 q^{89} -1.67976 q^{91} +24.9821 q^{93} +2.76244 q^{95} -15.1387 q^{97} -14.4557 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9	$ 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9} + 10 q^{11} + 4 q^{13} - 14 q^{15} + 10 q^{17} + 8 q^{19} + 10 q^{21} - 18 q^{23} + 23 q^{25} + 16 q^{27} + 14 q^{29} - 4 q^{31} + 14 q^{33} + 14 q^{35} + 16 q^{37} - 12 q^{39} + 10 q^{41} + 12 q^{43} + 10 q^{45} + 13 q^{47} + 9 q^{49} + 22 q^{51} + 26 q^{53} + 2 q^{55} + 20 q^{57} + 30 q^{59} + 18 q^{61} - 12 q^{63} - 4 q^{65} + 4 q^{67} + 2 q^{69} - 36 q^{71} + 10 q^{73} + 38 q^{75} + 42 q^{77} + 21 q^{81} + 12 q^{83} + 4 q^{85} - 6 q^{87} + 50 q^{89} - 4 q^{91} + 52 q^{93} - 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) $ 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9 + 10 * q^11 + 4 * q^13 - 14 * q^15 + 10 * q^17 + 8 * q^19 + 10 * q^21 - 18 * q^23 + 23 * q^25 + 16 * q^27 + 14 * q^29 - 4 * q^31 + 14 * q^33 + 14 * q^35 + 16 * q^37 - 12 * q^39 + 10 * q^41 + 12 * q^43 + 10 * q^45 + 13 * q^47 + 9 * q^49 + 22 * q^51 + 26 * q^53 + 2 * q^55 + 20 * q^57 + 30 * q^59 + 18 * q^61 - 12 * q^63 - 4 * q^65 + 4 * q^67 + 2 * q^69 - 36 * q^71 + 10 * q^73 + 38 * q^75 + 42 * q^77 + 21 * q^81 + 12 * q^83 + 4 * q^85 - 6 * q^87 + 50 * q^89 - 4 * q^91 + 52 * q^93 - 8 * q^95 - 10 * q^97 + 34 * q^99

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	0
$2$	0	0
$3$	−3.10994	−1.79553	−0.897763	−	0.440478i	$-0.854809\pi$
$3$	−3.10994	−1.79553	−0.897763	+	0.440478i	$0.854809\pi$
$4$	0	0
$5$	0.407310	0.182155	0.0910773	−	0.995844i	$-0.470969\pi$
$5$	0.407310	0.182155	0.0910773	+	0.995844i	$0.470969\pi$
$6$	0	0
$7$	−3.05265	−1.15379	−0.576896	−	0.816817i	$-0.695736\pi$
$7$	−3.05265	−1.15379	−0.576896	+	0.816817i	$0.695736\pi$
$8$	0	0
$9$	6.67175	2.22392
$10$	0	0
$11$	−2.16670	−0.653286	−0.326643	−	0.945148i	$-0.605917\pi$
$11$	−2.16670	−0.653286	−0.326643	+	0.945148i	$0.605917\pi$
$12$	0	0
$13$	0.550265	0.152616	0.0763080	−	0.997084i	$-0.475687\pi$
$13$	0.550265	0.152616	0.0763080	+	0.997084i	$0.475687\pi$
$14$	0	0
$15$	−1.26671	−0.327063
$16$	0	0
$17$	−3.34079	−0.810261	−0.405131	−	0.914259i	$-0.632774\pi$
$17$	−3.34079	−0.810261	−0.405131	+	0.914259i	$0.632774\pi$
$18$	0	0
$19$	6.78215	1.55593	0.777966	−	0.628306i	$-0.216252\pi$
$19$	6.78215	1.55593	0.777966	+	0.628306i	$0.216252\pi$
$20$	0	0
$21$	9.49357	2.07167
$22$	0	0
$23$	−5.22765	−1.09004	−0.545020	−	0.838423i	$-0.683478\pi$
$23$	−5.22765	−1.09004	−0.545020	+	0.838423i	$0.683478\pi$
$24$	0	0
$25$	−4.83410	−0.966820
$26$	0	0
$27$	−11.4189	−2.19758
$28$	0	0
$29$	−3.25095	−0.603685	−0.301843	−	0.953358i	$-0.597602\pi$
$29$	−3.25095	−0.603685	−0.301843	+	0.953358i	$0.597602\pi$
$30$	0	0
$31$	−8.03298	−1.44277	−0.721383	−	0.692536i	$-0.756493\pi$
$31$	−8.03298	−1.44277	−0.721383	+	0.692536i	$0.756493\pi$
$32$	0	0
$33$	6.73833	1.17299
$34$	0	0
$35$	−1.24337	−0.210169
$36$	0	0
$37$	9.82204	1.61473	0.807366	−	0.590050i	$-0.200892\pi$
$37$	9.82204	1.61473	0.807366	+	0.590050i	$0.200892\pi$
$38$	0	0
$39$	−1.71129	−0.274026
$40$	0	0
$41$	8.04458	1.25635	0.628176	−	0.778071i	$-0.283802\pi$
$41$	8.04458	1.25635	0.628176	+	0.778071i	$0.283802\pi$
$42$	0	0
$43$	3.91084	0.596398	0.298199	−	0.954504i	$-0.403614\pi$
$43$	3.91084	0.596398	0.298199	+	0.954504i	$0.403614\pi$
$44$	0	0
$45$	2.71747	0.405097
$46$	0	0
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	2.31866	0.331238
$50$	0	0
$51$	10.3897	1.45485
$52$	0	0
$53$	−6.41908	−0.881728	−0.440864	−	0.897574i	$-0.645328\pi$
$53$	−6.41908	−0.881728	−0.440864	+	0.897574i	$0.645328\pi$
$54$	0	0
$55$	−0.882520	−0.118999
$56$	0	0
$57$	−21.0921	−2.79372
$58$	0	0
$59$	2.73544	0.356124	0.178062	−	0.984019i	$-0.443017\pi$
$59$	2.73544	0.356124	0.178062	+	0.984019i	$0.443017\pi$
$60$	0	0
$61$	−4.63323	−0.593224	−0.296612	−	0.954998i	$-0.595857\pi$
$61$	−4.63323	−0.593224	−0.296612	+	0.954998i	$0.595857\pi$
$62$	0	0
$63$	−20.3665	−2.56594
$64$	0	0
$65$	0.224128	0.0277997
$66$	0	0
$67$	−11.2924	−1.37958	−0.689791	−	0.724009i	$-0.742297\pi$
$67$	−11.2924	−1.37958	−0.689791	+	0.724009i	$0.742297\pi$
$68$	0	0
$69$	16.2577	1.95720
$70$	0	0
$71$	−5.75360	−0.682827	−0.341414	−	0.939913i	$-0.610906\pi$
$71$	−5.75360	−0.682827	−0.341414	+	0.939913i	$0.610906\pi$
$72$	0	0
$73$	−6.94230	−0.812535	−0.406267	−	0.913754i	$-0.633170\pi$
$73$	−6.94230	−0.812535	−0.406267	+	0.913754i	$0.633170\pi$
$74$	0	0
$75$	15.0338	1.73595
$76$	0	0
$77$	6.61419	0.753757
$78$	0	0
$79$	−15.0416	−1.69231	−0.846155	−	0.532937i	$-0.821088\pi$
$79$	−15.0416	−1.69231	−0.846155	+	0.532937i	$0.821088\pi$
$80$	0	0
$81$	15.4970	1.72189
$82$	0	0
$83$	−9.73633	−1.06870	−0.534350	−	0.845263i	$-0.679444\pi$
$83$	−9.73633	−1.06870	−0.534350	+	0.845263i	$0.679444\pi$
$84$	0	0
$85$	−1.36074	−0.147593
$86$	0	0
$87$	10.1103	1.08393
$88$	0	0
$89$	9.09741	0.964324	0.482162	−	0.876082i	$-0.339852\pi$
$89$	9.09741	0.964324	0.482162	+	0.876082i	$0.339852\pi$
$90$	0	0
$91$	−1.67976	−0.176087
$92$	0	0
$93$	24.9821	2.59052
$94$	0	0
$95$	2.76244	0.283420
$96$	0	0
$97$	−15.1387	−1.53710	−0.768552	−	0.639787i	$-0.779022\pi$
$97$	−15.1387	−1.53710	−0.768552	+	0.639787i	$0.779022\pi$
$98$	0	0
$99$	−14.4557	−1.45285
$100$	0	0
$101$	3.77611	0.375737	0.187869	−	0.982194i	$-0.439842\pi$
$101$	3.77611	0.375737	0.187869	+	0.982194i	$0.439842\pi$
$102$	0	0
$103$	4.04840	0.398901	0.199451	−	0.979908i	$-0.436084\pi$
$103$	4.04840	0.398901	0.199451	+	0.979908i	$0.436084\pi$
$104$	0	0
$105$	3.86682	0.377363
$106$	0	0
$107$	14.0615	1.35937	0.679687	−	0.733503i	$-0.262116\pi$
$107$	14.0615	1.35937	0.679687	+	0.733503i	$0.262116\pi$
$108$	0	0
$109$	9.62922	0.922312	0.461156	−	0.887319i	$-0.347435\pi$
$109$	9.62922	0.922312	0.461156	+	0.887319i	$0.347435\pi$
$110$	0	0
$111$	−30.5460	−2.89930
$112$	0	0
$113$	−6.99339	−0.657883	−0.328941	−	0.944350i	$-0.606692\pi$
$113$	−6.99339	−0.657883	−0.328941	+	0.944350i	$0.606692\pi$
$114$	0	0
$115$	−2.12927	−0.198556
$116$	0	0
$117$	3.67123	0.339405
$118$	0	0
$119$	10.1983	0.934874
$120$	0	0
$121$	−6.30539	−0.573218
$122$	0	0
$123$	−25.0182	−2.25582
$124$	0	0
$125$	−4.00553	−0.358265
$126$	0	0
$127$	−0.458497	−0.0406851	−0.0203425	−	0.999793i	$-0.506476\pi$
$127$	−0.458497	−0.0406851	−0.0203425	+	0.999793i	$0.506476\pi$
$128$	0	0
$129$	−12.1625	−1.07085
$130$	0	0
$131$	−5.47957	−0.478752	−0.239376	−	0.970927i	$-0.576943\pi$
$131$	−5.47957	−0.478752	−0.239376	+	0.970927i	$0.576943\pi$
$132$	0	0
$133$	−20.7035	−1.79522
$134$	0	0
$135$	−4.65105	−0.400298
$136$	0	0
$137$	11.6905	0.998790	0.499395	−	0.866374i	$-0.333556\pi$
$137$	11.6905	0.998790	0.499395	+	0.866374i	$0.333556\pi$
$138$	0	0
$139$	−9.69114	−0.821992	−0.410996	−	0.911637i	$-0.634819\pi$
$139$	−9.69114	−0.821992	−0.410996	+	0.911637i	$0.634819\pi$
$140$	0	0
$141$	−3.10994	−0.261905
$142$	0	0
$143$	−1.19226	−0.0997018
$144$	0	0
$145$	−1.32414	−0.109964
$146$	0	0
$147$	−7.21091	−0.594746
$148$	0	0
$149$	−9.94488	−0.814716	−0.407358	−	0.913269i	$-0.633550\pi$
$149$	−9.94488	−0.814716	−0.407358	+	0.913269i	$0.633550\pi$
$150$	0	0
$151$	7.83567	0.637657	0.318829	−	0.947812i	$-0.396711\pi$
$151$	7.83567	0.637657	0.318829	+	0.947812i	$0.396711\pi$
$152$	0	0
$153$	−22.2889	−1.80195
$154$	0	0
$155$	−3.27191	−0.262806
$156$	0	0
$157$	−17.9714	−1.43427	−0.717135	−	0.696934i	$-0.754547\pi$
$157$	−17.9714	−1.43427	−0.717135	+	0.696934i	$0.754547\pi$
$158$	0	0
$159$	19.9630	1.58317
$160$	0	0
$161$	15.9582	1.25768
$162$	0	0
$163$	−9.23108	−0.723034	−0.361517	−	0.932366i	$-0.617741\pi$
$163$	−9.23108	−0.723034	−0.361517	+	0.932366i	$0.617741\pi$
$164$	0	0
$165$	2.74459	0.213666
$166$	0	0
$167$	16.7235	1.29411	0.647053	−	0.762445i	$-0.276001\pi$
$167$	16.7235	1.29411	0.647053	+	0.762445i	$0.276001\pi$
$168$	0	0
$169$	−12.6972	−0.976708
$170$	0	0
$171$	45.2488	3.46026
$172$	0	0
$173$	0.659422	0.0501349	0.0250675	−	0.999686i	$-0.492020\pi$
$173$	0.659422	0.0501349	0.0250675	+	0.999686i	$0.492020\pi$
$174$	0	0
$175$	14.7568	1.11551
$176$	0	0
$177$	−8.50707	−0.639430
$178$	0	0
$179$	0.422467	0.0315767	0.0157883	−	0.999875i	$-0.494974\pi$
$179$	0.422467	0.0315767	0.0157883	+	0.999875i	$0.494974\pi$
$180$	0	0
$181$	15.4430	1.14787	0.573935	−	0.818901i	$-0.305416\pi$
$181$	15.4430	1.14787	0.573935	+	0.818901i	$0.305416\pi$
$182$	0	0
$183$	14.4091	1.06515
$184$	0	0
$185$	4.00061	0.294131
$186$	0	0
$187$	7.23851	0.529332
$188$	0	0
$189$	34.8580	2.53555
$190$	0	0
$191$	6.20080	0.448674	0.224337	−	0.974512i	$-0.427978\pi$
$191$	6.20080	0.448674	0.224337	+	0.974512i	$0.427978\pi$
$192$	0	0
$193$	−0.914661	−0.0658387	−0.0329194	−	0.999458i	$-0.510480\pi$
$193$	−0.914661	−0.0658387	−0.0329194	+	0.999458i	$0.510480\pi$
$194$	0	0
$195$	−0.697026	−0.0499151
$196$	0	0
$197$	−27.3854	−1.95113	−0.975565	−	0.219712i	$-0.929488\pi$
$197$	−27.3854	−1.95113	−0.975565	+	0.219712i	$0.929488\pi$
$198$	0	0
$199$	16.5462	1.17293	0.586464	−	0.809975i	$-0.300519\pi$
$199$	16.5462	1.17293	0.586464	+	0.809975i	$0.300519\pi$
$200$	0	0
$201$	35.1186	2.47707
$202$	0	0
$203$	9.92399	0.696528
$204$	0	0
$205$	3.27664	0.228850
$206$	0	0
$207$	−34.8776	−2.42416
$208$	0	0
$209$	−14.6949	−1.01647
$210$	0	0
$211$	23.9892	1.65149	0.825743	−	0.564046i	$-0.190756\pi$
$211$	23.9892	1.65149	0.825743	+	0.564046i	$0.190756\pi$
$212$	0	0
$213$	17.8934	1.22603
$214$	0	0
$215$	1.59292	0.108637
$216$	0	0
$217$	24.5219	1.66465
$218$	0	0
$219$	21.5902	1.45893
$220$	0	0
$221$	−1.83832	−0.123659
$222$	0	0
$223$	−15.0889	−1.01043	−0.505215	−	0.862993i	$-0.668587\pi$
$223$	−15.0889	−1.01043	−0.505215	+	0.862993i	$0.668587\pi$
$224$	0	0
$225$	−32.2519	−2.15013
$226$	0	0
$227$	19.4047	1.28794	0.643968	−	0.765053i	$-0.277287\pi$
$227$	19.4047	1.28794	0.643968	+	0.765053i	$0.277287\pi$
$228$	0	0
$229$	5.07540	0.335392	0.167696	−	0.985839i	$-0.446367\pi$
$229$	5.07540	0.335392	0.167696	+	0.985839i	$0.446367\pi$
$230$	0	0
$231$	−20.5697	−1.35339
$232$	0	0
$233$	28.1567	1.84461	0.922304	−	0.386465i	$-0.126304\pi$
$233$	28.1567	1.84461	0.922304	+	0.386465i	$0.126304\pi$
$234$	0	0
$235$	0.407310	0.0265700
$236$	0	0
$237$	46.7785	3.03859
$238$	0	0
$239$	−7.86942	−0.509030	−0.254515	−	0.967069i	$-0.581916\pi$
$239$	−7.86942	−0.509030	−0.254515	+	0.967069i	$0.581916\pi$
$240$	0	0
$241$	15.8599	1.02162	0.510812	−	0.859693i	$-0.329345\pi$
$241$	15.8599	1.02162	0.510812	+	0.859693i	$0.329345\pi$
$242$	0	0
$243$	−13.9380	−0.894122
$244$	0	0
$245$	0.944415	0.0603365
$246$	0	0
$247$	3.73198	0.237460
$248$	0	0
$249$	30.2794	1.91888
$250$	0	0
$251$	26.3744	1.66474	0.832369	−	0.554222i	$-0.186984\pi$
$251$	26.3744	1.66474	0.832369	+	0.554222i	$0.186984\pi$
$252$	0	0
$253$	11.3268	0.712108
$254$	0	0
$255$	4.23182	0.265007
$256$	0	0
$257$	−21.0078	−1.31043	−0.655217	−	0.755441i	$-0.727423\pi$
$257$	−21.0078	−1.31043	−0.655217	+	0.755441i	$0.727423\pi$
$258$	0	0
$259$	−29.9832	−1.86307
$260$	0	0
$261$	−21.6895	−1.34255
$262$	0	0
$263$	−4.00628	−0.247038	−0.123519	−	0.992342i	$-0.539418\pi$
$263$	−4.00628	−0.247038	−0.123519	+	0.992342i	$0.539418\pi$
$264$	0	0
$265$	−2.61455	−0.160611
$266$	0	0
$267$	−28.2924	−1.73147
$268$	0	0
$269$	−7.17613	−0.437536	−0.218768	−	0.975777i	$-0.570204\pi$
$269$	−7.17613	−0.437536	−0.218768	+	0.975777i	$0.570204\pi$
$270$	0	0
$271$	18.5170	1.12483	0.562414	−	0.826856i	$-0.309873\pi$
$271$	18.5170	1.12483	0.562414	+	0.826856i	$0.309873\pi$
$272$	0	0
$273$	5.22397	0.316169
$274$	0	0
$275$	10.4741	0.631610
$276$	0	0
$277$	10.5411	0.633354	0.316677	−	0.948533i	$-0.397433\pi$
$277$	10.5411	0.633354	0.316677	+	0.948533i	$0.397433\pi$
$278$	0	0
$279$	−53.5940	−3.20859
$280$	0	0
$281$	7.60641	0.453760	0.226880	−	0.973923i	$-0.427147\pi$
$281$	7.60641	0.453760	0.226880	+	0.973923i	$0.427147\pi$
$282$	0	0
$283$	12.8983	0.766724	0.383362	−	0.923598i	$-0.374766\pi$
$283$	12.8983	0.766724	0.383362	+	0.923598i	$0.374766\pi$
$284$	0	0
$285$	−8.59102	−0.508888
$286$	0	0
$287$	−24.5573	−1.44957
$288$	0	0
$289$	−5.83910	−0.343476
$290$	0	0
$291$	47.0806	2.75991
$292$	0	0
$293$	−1.98565	−0.116003	−0.0580015	−	0.998316i	$-0.518473\pi$
$293$	−1.98565	−0.116003	−0.0580015	+	0.998316i	$0.518473\pi$
$294$	0	0
$295$	1.11417	0.0648696
$296$	0	0
$297$	24.7415	1.43564
$298$	0	0
$299$	−2.87659	−0.166357
$300$	0	0
$301$	−11.9384	−0.688119
$302$	0	0
$303$	−11.7435	−0.674646
$304$	0	0
$305$	−1.88716	−0.108058
$306$	0	0
$307$	−7.55506	−0.431190	−0.215595	−	0.976483i	$-0.569169\pi$
$307$	−7.55506	−0.431190	−0.215595	+	0.976483i	$0.569169\pi$
$308$	0	0
$309$	−12.5903	−0.716238
$310$	0	0
$311$	−21.5882	−1.22415	−0.612077	−	0.790799i	$-0.709666\pi$
$311$	−21.5882	−1.22415	−0.612077	+	0.790799i	$0.709666\pi$
$312$	0	0
$313$	−14.6067	−0.825622	−0.412811	−	0.910817i	$-0.635453\pi$
$313$	−14.6067	−0.825622	−0.412811	+	0.910817i	$0.635453\pi$
$314$	0	0
$315$	−8.29548	−0.467397
$316$	0	0
$317$	8.34814	0.468878	0.234439	−	0.972131i	$-0.424675\pi$
$317$	8.34814	0.468878	0.234439	+	0.972131i	$0.424675\pi$
$318$	0	0
$319$	7.04384	0.394379
$320$	0	0
$321$	−43.7304	−2.44079
$322$	0	0
$323$	−22.6578	−1.26071
$324$	0	0
$325$	−2.66003	−0.147552
$326$	0	0
$327$	−29.9463	−1.65604
$328$	0	0
$329$	−3.05265	−0.168298
$330$	0	0
$331$	31.8095	1.74841	0.874205	−	0.485558i	$-0.161383\pi$
$331$	31.8095	1.74841	0.874205	+	0.485558i	$0.161383\pi$
$332$	0	0
$333$	65.5302	3.59103
$334$	0	0
$335$	−4.59949	−0.251297
$336$	0	0
$337$	−6.01424	−0.327616	−0.163808	−	0.986492i	$-0.552378\pi$
$337$	−6.01424	−0.327616	−0.163808	+	0.986492i	$0.552378\pi$
$338$	0	0
$339$	21.7490	1.18125
$340$	0	0
$341$	17.4051	0.942539
$342$	0	0
$343$	14.2905	0.771613
$344$	0	0
$345$	6.62192	0.356512
$346$	0	0
$347$	12.8677	0.690775	0.345387	−	0.938460i	$-0.387747\pi$
$347$	12.8677	0.690775	0.345387	+	0.938460i	$0.387747\pi$
$348$	0	0
$349$	−10.9705	−0.587237	−0.293619	−	0.955923i	$-0.594860\pi$
$349$	−10.9705	−0.587237	−0.293619	+	0.955923i	$0.594860\pi$
$350$	0	0
$351$	−6.28344	−0.335385
$352$	0	0
$353$	24.9810	1.32961	0.664803	−	0.747019i	$-0.268516\pi$
$353$	24.9810	1.32961	0.664803	+	0.747019i	$0.268516\pi$
$354$	0	0
$355$	−2.34350	−0.124380
$356$	0	0
$357$	−31.7160	−1.67859
$358$	0	0
$359$	23.2115	1.22506	0.612528	−	0.790449i	$-0.290153\pi$
$359$	23.2115	1.22506	0.612528	+	0.790449i	$0.290153\pi$
$360$	0	0
$361$	26.9976	1.42092
$362$	0	0
$363$	19.6094	1.02923
$364$	0	0
$365$	−2.82767	−0.148007
$366$	0	0
$367$	−18.5333	−0.967430	−0.483715	−	0.875226i	$-0.660713\pi$
$367$	−18.5333	−0.967430	−0.483715	+	0.875226i	$0.660713\pi$
$368$	0	0
$369$	53.6714	2.79402
$370$	0	0
$371$	19.5952	1.01733
$372$	0	0
$373$	−32.7310	−1.69475	−0.847373	−	0.530998i	$-0.821817\pi$
$373$	−32.7310	−1.69475	−0.847373	+	0.530998i	$0.821817\pi$
$374$	0	0
$375$	12.4570	0.643275
$376$	0	0
$377$	−1.78888	−0.0921320
$378$	0	0
$379$	18.8297	0.967215	0.483608	−	0.875285i	$-0.339326\pi$
$379$	18.8297	0.967215	0.483608	+	0.875285i	$0.339326\pi$
$380$	0	0
$381$	1.42590	0.0730511
$382$	0	0
$383$	−37.4769	−1.91498	−0.957489	−	0.288470i	$-0.906853\pi$
$383$	−37.4769	−1.91498	−0.957489	+	0.288470i	$0.906853\pi$
$384$	0	0
$385$	2.69402	0.137300
$386$	0	0
$387$	26.0922	1.32634
$388$	0	0
$389$	9.88039	0.500956	0.250478	−	0.968122i	$-0.419412\pi$
$389$	9.88039	0.500956	0.250478	+	0.968122i	$0.419412\pi$
$390$	0	0
$391$	17.4645	0.883217
$392$	0	0
$393$	17.0411	0.859612
$394$	0	0
$395$	−6.12659	−0.308262
$396$	0	0
$397$	−8.26429	−0.414773	−0.207387	−	0.978259i	$-0.566496\pi$
$397$	−8.26429	−0.414773	−0.207387	+	0.978259i	$0.566496\pi$
$398$	0	0
$399$	64.3868	3.22337
$400$	0	0
$401$	28.0473	1.40061	0.700307	−	0.713841i	$-0.253046\pi$
$401$	28.0473	1.40061	0.700307	+	0.713841i	$0.253046\pi$
$402$	0	0
$403$	−4.42026	−0.220189
$404$	0	0
$405$	6.31208	0.313650
$406$	0	0
$407$	−21.2814	−1.05488
$408$	0	0
$409$	5.98600	0.295989	0.147994	−	0.988988i	$-0.452718\pi$
$409$	5.98600	0.295989	0.147994	+	0.988988i	$0.452718\pi$
$410$	0	0
$411$	−36.3569	−1.79335
$412$	0	0
$413$	−8.35034	−0.410893
$414$	0	0
$415$	−3.96570	−0.194669
$416$	0	0
$417$	30.1389	1.47591
$418$	0	0
$419$	−3.61266	−0.176490	−0.0882449	−	0.996099i	$-0.528126\pi$
$419$	−3.61266	−0.176490	−0.0882449	+	0.996099i	$0.528126\pi$
$420$	0	0
$421$	37.4594	1.82566	0.912829	−	0.408341i	$-0.133893\pi$
$421$	37.4594	1.82566	0.912829	+	0.408341i	$0.133893\pi$
$422$	0	0
$423$	6.67175	0.324392
$424$	0	0
$425$	16.1497	0.783377
$426$	0	0
$427$	14.1436	0.684458
$428$	0	0
$429$	3.70786	0.179017
$430$	0	0
$431$	6.49150	0.312685	0.156342	−	0.987703i	$-0.450030\pi$
$431$	6.49150	0.312685	0.156342	+	0.987703i	$0.450030\pi$
$432$	0	0
$433$	14.1159	0.678369	0.339184	−	0.940720i	$-0.389849\pi$
$433$	14.1159	0.678369	0.339184	+	0.940720i	$0.389849\pi$
$434$	0	0
$435$	4.11801	0.197443
$436$	0	0
$437$	−35.4547	−1.69603
$438$	0	0
$439$	−23.0935	−1.10219	−0.551097	−	0.834441i	$-0.685791\pi$
$439$	−23.0935	−1.10219	−0.551097	+	0.834441i	$0.685791\pi$
$440$	0	0
$441$	15.4695	0.736645
$442$	0	0
$443$	4.41824	0.209917	0.104958	−	0.994477i	$-0.466529\pi$
$443$	4.41824	0.209917	0.104958	+	0.994477i	$0.466529\pi$
$444$	0	0
$445$	3.70547	0.175656
$446$	0	0
$447$	30.9280	1.46284
$448$	0	0
$449$	34.6386	1.63470	0.817348	−	0.576144i	$-0.195443\pi$
$449$	34.6386	1.63470	0.817348	+	0.576144i	$0.195443\pi$
$450$	0	0
$451$	−17.4302	−0.820758
$452$	0	0
$453$	−24.3685	−1.14493
$454$	0	0
$455$	−0.684185	−0.0320751
$456$	0	0
$457$	22.1776	1.03742	0.518712	−	0.854949i	$-0.326412\pi$
$457$	22.1776	1.03742	0.518712	+	0.854949i	$0.326412\pi$
$458$	0	0
$459$	38.1483	1.78061
$460$	0	0
$461$	22.1451	1.03140	0.515699	−	0.856770i	$-0.327532\pi$
$461$	22.1451	1.03140	0.515699	+	0.856770i	$0.327532\pi$
$462$	0	0
$463$	33.6623	1.56442	0.782210	−	0.623015i	$-0.214092\pi$
$463$	33.6623	1.56442	0.782210	+	0.623015i	$0.214092\pi$
$464$	0	0
$465$	10.1755	0.471876
$466$	0	0
$467$	27.6298	1.27855	0.639276	−	0.768977i	$-0.279234\pi$
$467$	27.6298	1.27855	0.639276	+	0.768977i	$0.279234\pi$
$468$	0	0
$469$	34.4716	1.59175
$470$	0	0
$471$	55.8899	2.57527
$472$	0	0
$473$	−8.47364	−0.389618
$474$	0	0
$475$	−32.7856	−1.50431
$476$	0	0
$477$	−42.8265	−1.96089
$478$	0	0
$479$	−19.8209	−0.905640	−0.452820	−	0.891602i	$-0.649582\pi$
$479$	−19.8209	−0.905640	−0.452820	+	0.891602i	$0.649582\pi$
$480$	0	0
$481$	5.40472	0.246434
$482$	0	0
$483$	−49.6290	−2.25820
$484$	0	0
$485$	−6.16615	−0.279990
$486$	0	0
$487$	4.49345	0.203618	0.101809	−	0.994804i	$-0.467537\pi$
$487$	4.49345	0.203618	0.101809	+	0.994804i	$0.467537\pi$
$488$	0	0
$489$	28.7081	1.29823
$490$	0	0
$491$	10.6023	0.478475	0.239238	−	0.970961i	$-0.423103\pi$
$491$	10.6023	0.478475	0.239238	+	0.970961i	$0.423103\pi$
$492$	0	0
$493$	10.8607	0.489143
$494$	0	0
$495$	−5.88795	−0.264644
$496$	0	0
$497$	17.5637	0.787841
$498$	0	0
$499$	−19.4794	−0.872016	−0.436008	−	0.899943i	$-0.643608\pi$
$499$	−19.4794	−0.872016	−0.436008	+	0.899943i	$0.643608\pi$
$500$	0	0
$501$	−52.0092	−2.32360
$502$	0	0
$503$	7.10841	0.316948	0.158474	−	0.987363i	$-0.449343\pi$
$503$	7.10841	0.316948	0.158474	+	0.987363i	$0.449343\pi$
$504$	0	0
$505$	1.53805	0.0684422
$506$	0	0
$507$	39.4876	1.75371
$508$	0	0
$509$	4.46106	0.197733	0.0988666	−	0.995101i	$-0.468478\pi$
$509$	4.46106	0.197733	0.0988666	+	0.995101i	$0.468478\pi$
$510$	0	0
$511$	21.1924	0.937497
$512$	0	0
$513$	−77.4449	−3.41928
$514$	0	0
$515$	1.64896	0.0726617
$516$	0	0
$517$	−2.16670	−0.0952915
$518$	0	0
$519$	−2.05077	−0.0900186
$520$	0	0
$521$	−29.6999	−1.30118	−0.650589	−	0.759430i	$-0.725478\pi$
$521$	−29.6999	−1.30118	−0.650589	+	0.759430i	$0.725478\pi$
$522$	0	0
$523$	39.9686	1.74771	0.873853	−	0.486191i	$-0.161614\pi$
$523$	39.9686	1.74771	0.873853	+	0.486191i	$0.161614\pi$
$524$	0	0
$525$	−45.8928	−2.00293
$526$	0	0
$527$	26.8365	1.16902
$528$	0	0
$529$	4.32829	0.188186
$530$	0	0
$531$	18.2502	0.791990
$532$	0	0
$533$	4.42665	0.191739
$534$	0	0
$535$	5.72738	0.247616
$536$	0	0
$537$	−1.31385	−0.0566968
$538$	0	0
$539$	−5.02386	−0.216393
$540$	0	0
$541$	−13.3928	−0.575801	−0.287901	−	0.957660i	$-0.592957\pi$
$541$	−13.3928	−0.575801	−0.287901	+	0.957660i	$0.592957\pi$
$542$	0	0
$543$	−48.0269	−2.06103
$544$	0	0
$545$	3.92208	0.168003
$546$	0	0
$547$	28.7590	1.22965	0.614824	−	0.788665i	$-0.289227\pi$
$547$	28.7590	1.22965	0.614824	+	0.788665i	$0.289227\pi$
$548$	0	0
$549$	−30.9117	−1.31928
$550$	0	0
$551$	−22.0484	−0.939294
$552$	0	0
$553$	45.9167	1.95258
$554$	0	0
$555$	−12.4417	−0.528120
$556$	0	0
$557$	19.6279	0.831659	0.415830	−	0.909442i	$-0.363491\pi$
$557$	19.6279	0.831659	0.415830	+	0.909442i	$0.363491\pi$
$558$	0	0
$559$	2.15200	0.0910198
$560$	0	0
$561$	−22.5114	−0.950430
$562$	0	0
$563$	−29.6491	−1.24956	−0.624780	−	0.780801i	$-0.714812\pi$
$563$	−29.6491	−1.24956	−0.624780	+	0.780801i	$0.714812\pi$
$564$	0	0
$565$	−2.84848	−0.119836
$566$	0	0
$567$	−47.3069	−1.98670
$568$	0	0
$569$	44.2613	1.85553	0.927765	−	0.373164i	$-0.121727\pi$
$569$	44.2613	1.85553	0.927765	+	0.373164i	$0.121727\pi$
$570$	0	0
$571$	28.6496	1.19895	0.599475	−	0.800393i	$-0.295376\pi$
$571$	28.6496	1.19895	0.599475	+	0.800393i	$0.295376\pi$
$572$	0	0
$573$	−19.2841	−0.805607
$574$	0	0
$575$	25.2710	1.05387
$576$	0	0
$577$	5.20852	0.216833	0.108417	−	0.994106i	$-0.465422\pi$
$577$	5.20852	0.216833	0.108417	+	0.994106i	$0.465422\pi$
$578$	0	0
$579$	2.84454	0.118215
$580$	0	0
$581$	29.7216	1.23306
$582$	0	0
$583$	13.9082	0.576020
$584$	0	0
$585$	1.49533	0.0618242
$586$	0	0
$587$	6.64376	0.274217	0.137109	−	0.990556i	$-0.456219\pi$
$587$	6.64376	0.274217	0.137109	+	0.990556i	$0.456219\pi$
$588$	0	0
$589$	−54.4809	−2.24485
$590$	0	0
$591$	85.1671	3.50331
$592$	0	0
$593$	−0.225118	−0.00924447	−0.00462223	−	0.999989i	$-0.501471\pi$
$593$	−0.225118	−0.00924447	−0.00462223	+	0.999989i	$0.501471\pi$
$594$	0	0
$595$	4.15386	0.170291
$596$	0	0
$597$	−51.4577	−2.10603
$598$	0	0
$599$	17.4269	0.712045	0.356023	−	0.934477i	$-0.384133\pi$
$599$	17.4269	0.712045	0.356023	+	0.934477i	$0.384133\pi$
$600$	0	0
$601$	−34.2190	−1.39582	−0.697910	−	0.716185i	$-0.745887\pi$
$601$	−34.2190	−1.39582	−0.697910	+	0.716185i	$0.745887\pi$
$602$	0	0
$603$	−75.3398	−3.06807
$604$	0	0
$605$	−2.56825	−0.104414
$606$	0	0
$607$	29.8314	1.21082	0.605410	−	0.795914i	$-0.293009\pi$
$607$	29.8314	1.21082	0.605410	+	0.795914i	$0.293009\pi$
$608$	0	0
$609$	−30.8631	−1.25063
$610$	0	0
$611$	0.550265	0.0222613
$612$	0	0
$613$	−39.6737	−1.60241	−0.801203	−	0.598393i	$-0.795806\pi$
$613$	−39.6737	−1.60241	−0.801203	+	0.598393i	$0.795806\pi$
$614$	0	0
$615$	−10.1902	−0.410907
$616$	0	0
$617$	−25.4005	−1.02259	−0.511294	−	0.859406i	$-0.670834\pi$
$617$	−25.4005	−1.02259	−0.511294	+	0.859406i	$0.670834\pi$
$618$	0	0
$619$	7.43419	0.298805	0.149403	−	0.988776i	$-0.452265\pi$
$619$	7.43419	0.298805	0.149403	+	0.988776i	$0.452265\pi$
$620$	0	0
$621$	59.6942	2.39544
$622$	0	0
$623$	−27.7712	−1.11263
$624$	0	0
$625$	22.5390	0.901560
$626$	0	0
$627$	45.7004	1.82510
$628$	0	0
$629$	−32.8134	−1.30836
$630$	0	0
$631$	−32.0345	−1.27527	−0.637636	−	0.770338i	$-0.720088\pi$
$631$	−32.0345	−1.27527	−0.637636	+	0.770338i	$0.720088\pi$
$632$	0	0
$633$	−74.6051	−2.96529
$634$	0	0
$635$	−0.186751	−0.00741097
$636$	0	0
$637$	1.27588	0.0505521
$638$	0	0
$639$	−38.3866	−1.51855
$640$	0	0
$641$	17.2147	0.679938	0.339969	−	0.940437i	$-0.389583\pi$
$641$	17.2147	0.679938	0.339969	+	0.940437i	$0.389583\pi$
$642$	0	0
$643$	−14.1099	−0.556439	−0.278219	−	0.960518i	$-0.589744\pi$
$643$	−14.1099	−0.556439	−0.278219	+	0.960518i	$0.589744\pi$
$644$	0	0
$645$	−4.95391	−0.195060
$646$	0	0
$647$	−16.4499	−0.646714	−0.323357	−	0.946277i	$-0.604811\pi$
$647$	−16.4499	−0.646714	−0.323357	+	0.946277i	$0.604811\pi$
$648$	0	0
$649$	−5.92689	−0.232651
$650$	0	0
$651$	−76.2616	−2.98893
$652$	0	0
$653$	27.6510	1.08207	0.541035	−	0.841000i	$-0.318033\pi$
$653$	27.6510	1.08207	0.541035	+	0.841000i	$0.318033\pi$
$654$	0	0
$655$	−2.23188	−0.0872068
$656$	0	0
$657$	−46.3173	−1.80701
$658$	0	0
$659$	7.95658	0.309945	0.154972	−	0.987919i	$-0.450471\pi$
$659$	7.95658	0.309945	0.154972	+	0.987919i	$0.450471\pi$
$660$	0	0
$661$	45.3923	1.76555	0.882777	−	0.469792i	$-0.155671\pi$
$661$	45.3923	1.76555	0.882777	+	0.469792i	$0.155671\pi$
$662$	0	0
$663$	5.71707	0.222033
$664$	0	0
$665$	−8.43275	−0.327008
$666$	0	0
$667$	16.9948	0.658041
$668$	0	0
$669$	46.9258	1.81426
$670$	0	0
$671$	10.0388	0.387545
$672$	0	0
$673$	−8.24632	−0.317872	−0.158936	−	0.987289i	$-0.550806\pi$
$673$	−8.24632	−0.317872	−0.158936	+	0.987289i	$0.550806\pi$
$674$	0	0
$675$	55.2003	2.12466
$676$	0	0
$677$	−9.09676	−0.349617	−0.174808	−	0.984602i	$-0.555931\pi$
$677$	−9.09676	−0.349617	−0.174808	+	0.984602i	$0.555931\pi$
$678$	0	0
$679$	46.2132	1.77350
$680$	0	0
$681$	−60.3475	−2.31252
$682$	0	0
$683$	−26.9031	−1.02942	−0.514710	−	0.857364i	$-0.672100\pi$
$683$	−26.9031	−1.02942	−0.514710	+	0.857364i	$0.672100\pi$
$684$	0	0
$685$	4.76167	0.181934
$686$	0	0
$687$	−15.7842	−0.602205
$688$	0	0
$689$	−3.53219	−0.134566
$690$	0	0
$691$	−13.1874	−0.501673	−0.250836	−	0.968029i	$-0.580706\pi$
$691$	−13.1874	−0.501673	−0.250836	+	0.968029i	$0.580706\pi$
$692$	0	0
$693$	44.1282	1.67629
$694$	0	0
$695$	−3.94730	−0.149730
$696$	0	0
$697$	−26.8753	−1.01797
$698$	0	0
$699$	−87.5658	−3.31204
$700$	0	0
$701$	6.53505	0.246825	0.123413	−	0.992355i	$-0.460616\pi$
$701$	6.53505	0.246825	0.123413	+	0.992355i	$0.460616\pi$
$702$	0	0
$703$	66.6145	2.51241
$704$	0	0
$705$	−1.26671	−0.0477071
$706$	0	0
$707$	−11.5271	−0.433523
$708$	0	0
$709$	−43.6864	−1.64068	−0.820339	−	0.571877i	$-0.806215\pi$
$709$	−43.6864	−1.64068	−0.820339	+	0.571877i	$0.806215\pi$
$710$	0	0
$711$	−100.354	−3.76356
$712$	0	0
$713$	41.9936	1.57267
$714$	0	0
$715$	−0.485620	−0.0181611
$716$	0	0
$717$	24.4734	0.913977
$718$	0	0
$719$	−5.47228	−0.204082	−0.102041	−	0.994780i	$-0.532537\pi$
$719$	−5.47228	−0.204082	−0.102041	+	0.994780i	$0.532537\pi$
$720$	0	0
$721$	−12.3584	−0.460249
$722$	0	0
$723$	−49.3233	−1.83435
$724$	0	0
$725$	15.7154	0.583655
$726$	0	0
$727$	−4.64274	−0.172190	−0.0860948	−	0.996287i	$-0.527439\pi$
$727$	−4.64274	−0.172190	−0.0860948	+	0.996287i	$0.527439\pi$
$728$	0	0
$729$	−3.14464	−0.116468
$730$	0	0
$731$	−13.0653	−0.483238
$732$	0	0
$733$	14.4222	0.532696	0.266348	−	0.963877i	$-0.414183\pi$
$733$	14.4222	0.532696	0.266348	+	0.963877i	$0.414183\pi$
$734$	0	0
$735$	−2.93708	−0.108336
$736$	0	0
$737$	24.4672	0.901261
$738$	0	0
$739$	36.1648	1.33034	0.665172	−	0.746690i	$-0.268358\pi$
$739$	36.1648	1.33034	0.665172	+	0.746690i	$0.268358\pi$
$740$	0	0
$741$	−11.6062	−0.426366
$742$	0	0
$743$	45.9267	1.68489	0.842444	−	0.538784i	$-0.181116\pi$
$743$	45.9267	1.68489	0.842444	+	0.538784i	$0.181116\pi$
$744$	0	0
$745$	−4.05065	−0.148404
$746$	0	0
$747$	−64.9583	−2.37670
$748$	0	0
$749$	−42.9247	−1.56844
$750$	0	0
$751$	−30.5643	−1.11531	−0.557653	−	0.830074i	$-0.688298\pi$
$751$	−30.5643	−1.11531	−0.557653	+	0.830074i	$0.688298\pi$
$752$	0	0
$753$	−82.0229	−2.98908
$754$	0	0
$755$	3.19154	0.116152
$756$	0	0
$757$	−1.46143	−0.0531164	−0.0265582	−	0.999647i	$-0.508455\pi$
$757$	−1.46143	−0.0531164	−0.0265582	+	0.999647i	$0.508455\pi$
$758$	0	0
$759$	−35.2256	−1.27861
$760$	0	0
$761$	−26.1612	−0.948343	−0.474171	−	0.880433i	$-0.657252\pi$
$761$	−26.1612	−0.948343	−0.474171	+	0.880433i	$0.657252\pi$
$762$	0	0
$763$	−29.3946	−1.06416
$764$	0	0
$765$	−9.07851	−0.328234
$766$	0	0
$767$	1.50522	0.0543502
$768$	0	0
$769$	−34.7026	−1.25141	−0.625704	−	0.780061i	$-0.715188\pi$
$769$	−34.7026	−1.25141	−0.625704	+	0.780061i	$0.715188\pi$
$770$	0	0
$771$	65.3332	2.35292
$772$	0	0
$773$	35.1723	1.26506	0.632530	−	0.774536i	$-0.282017\pi$
$773$	35.1723	1.26506	0.632530	+	0.774536i	$0.282017\pi$
$774$	0	0
$775$	38.8322	1.39489
$776$	0	0
$777$	93.2462	3.34519
$778$	0	0
$779$	54.5596	1.95480
$780$	0	0
$781$	12.4664	0.446081
$782$	0	0
$783$	37.1223	1.32664
$784$	0	0
$785$	−7.31991	−0.261259
$786$	0	0
$787$	1.57031	0.0559756	0.0279878	−	0.999608i	$-0.491090\pi$
$787$	1.57031	0.0559756	0.0279878	+	0.999608i	$0.491090\pi$
$788$	0	0
$789$	12.4593	0.443563
$790$	0	0
$791$	21.3484	0.759060
$792$	0	0
$793$	−2.54950	−0.0905354
$794$	0	0
$795$	8.13111	0.288381
$796$	0	0
$797$	−11.3579	−0.402318	−0.201159	−	0.979559i	$-0.564471\pi$
$797$	−11.3579	−0.402318	−0.201159	+	0.979559i	$0.564471\pi$
$798$	0	0
$799$	−3.34079	−0.118189
$800$	0	0
$801$	60.6957	2.14458
$802$	0	0
$803$	15.0419	0.530818
$804$	0	0
$805$	6.49992	0.229092
$806$	0	0
$807$	22.3174	0.785608
$808$	0	0
$809$	35.4551	1.24653	0.623267	−	0.782009i	$-0.285805\pi$
$809$	35.4551	1.24653	0.623267	+	0.782009i	$0.285805\pi$
$810$	0	0
$811$	−1.60023	−0.0561917	−0.0280959	−	0.999605i	$-0.508944\pi$
$811$	−1.60023	−0.0561917	−0.0280959	+	0.999605i	$0.508944\pi$
$812$	0	0
$813$	−57.5868	−2.01966
$814$	0	0
$815$	−3.75991	−0.131704
$816$	0	0
$817$	26.5239	0.927954
$818$	0	0
$819$	−11.2070	−0.391603
$820$	0	0
$821$	−9.29436	−0.324375	−0.162188	−	0.986760i	$-0.551855\pi$
$821$	−9.29436	−0.324375	−0.162188	+	0.986760i	$0.551855\pi$
$822$	0	0
$823$	−17.5978	−0.613421	−0.306711	−	0.951803i	$-0.599228\pi$
$823$	−17.5978	−0.613421	−0.306711	+	0.951803i	$0.599228\pi$
$824$	0	0
$825$	−32.5737	−1.13407
$826$	0	0
$827$	−49.0737	−1.70646	−0.853230	−	0.521535i	$-0.825359\pi$
$827$	−49.0737	−1.70646	−0.853230	+	0.521535i	$0.825359\pi$
$828$	0	0
$829$	−7.75210	−0.269241	−0.134621	−	0.990897i	$-0.542982\pi$
$829$	−7.75210	−0.269241	−0.134621	+	0.990897i	$0.542982\pi$
$830$	0	0
$831$	−32.7823	−1.13720
$832$	0	0
$833$	−7.74618	−0.268389
$834$	0	0
$835$	6.81166	0.235727
$836$	0	0
$837$	91.7281	3.17059
$838$	0	0
$839$	39.9387	1.37884	0.689419	−	0.724363i	$-0.257866\pi$
$839$	39.9387	1.37884	0.689419	+	0.724363i	$0.257866\pi$
$840$	0	0
$841$	−18.4314	−0.635564
$842$	0	0
$843$	−23.6555	−0.814739
$844$	0	0
$845$	−5.17170	−0.177912
$846$	0	0
$847$	19.2481	0.661374
$848$	0	0
$849$	−40.1130	−1.37667
$850$	0	0
$851$	−51.3461	−1.76012
$852$	0	0
$853$	−12.4661	−0.426832	−0.213416	−	0.976961i	$-0.568459\pi$
$853$	−12.4661	−0.426832	−0.213416	+	0.976961i	$0.568459\pi$
$854$	0	0
$855$	18.4303	0.630303
$856$	0	0
$857$	15.2044	0.519373	0.259686	−	0.965693i	$-0.416381\pi$
$857$	15.2044	0.519373	0.259686	+	0.965693i	$0.416381\pi$
$858$	0	0
$859$	41.5280	1.41692	0.708459	−	0.705752i	$-0.249391\pi$
$859$	41.5280	1.41692	0.708459	+	0.705752i	$0.249391\pi$
$860$	0	0
$861$	76.3718	2.60274
$862$	0	0
$863$	8.60696	0.292984	0.146492	−	0.989212i	$-0.453202\pi$
$863$	8.60696	0.292984	0.146492	+	0.989212i	$0.453202\pi$
$864$	0	0
$865$	0.268589	0.00913231
$866$	0	0
$867$	18.1593	0.616721
$868$	0	0
$869$	32.5907	1.10556
$870$	0	0
$871$	−6.21378	−0.210546
$872$	0	0
$873$	−101.002	−3.41839
$874$	0	0
$875$	12.2275	0.413364
$876$	0	0
$877$	1.78883	0.0604044	0.0302022	−	0.999544i	$-0.490385\pi$
$877$	1.78883	0.0604044	0.0302022	+	0.999544i	$0.490385\pi$
$878$	0	0
$879$	6.17526	0.208286
$880$	0	0
$881$	−5.30452	−0.178714	−0.0893570	−	0.996000i	$-0.528481\pi$
$881$	−5.30452	−0.178714	−0.0893570	+	0.996000i	$0.528481\pi$
$882$	0	0
$883$	−4.44640	−0.149633	−0.0748167	−	0.997197i	$-0.523837\pi$
$883$	−4.44640	−0.149633	−0.0748167	+	0.997197i	$0.523837\pi$
$884$	0	0
$885$	−3.46501	−0.116475
$886$	0	0
$887$	32.9646	1.10684	0.553421	−	0.832902i	$-0.313322\pi$
$887$	32.9646	1.10684	0.553421	+	0.832902i	$0.313322\pi$
$888$	0	0
$889$	1.39963	0.0469421
$890$	0	0
$891$	−33.5774	−1.12489
$892$	0	0
$893$	6.78215	0.226956
$894$	0	0
$895$	0.172075	0.00575183
$896$	0	0
$897$	8.94603	0.298699
$898$	0	0
$899$	26.1148	0.870977
$900$	0	0
$901$	21.4448	0.714430
$902$	0	0
$903$	37.1278	1.23554
$904$	0	0
$905$	6.29009	0.209090
$906$	0	0
$907$	−39.7603	−1.32022	−0.660110	−	0.751169i	$-0.729490\pi$
$907$	−39.7603	−1.32022	−0.660110	+	0.751169i	$0.729490\pi$
$908$	0	0
$909$	25.1933	0.835608
$910$	0	0
$911$	−23.2653	−0.770813	−0.385407	−	0.922747i	$-0.625939\pi$
$911$	−23.2653	−0.770813	−0.385407	+	0.922747i	$0.625939\pi$
$912$	0	0
$913$	21.0957	0.698167
$914$	0	0
$915$	5.86896	0.194022
$916$	0	0
$917$	16.7272	0.552380
$918$	0	0
$919$	−11.7907	−0.388939	−0.194469	−	0.980909i	$-0.562298\pi$
$919$	−11.7907	−0.388939	−0.194469	+	0.980909i	$0.562298\pi$
$920$	0	0
$921$	23.4958	0.774213
$922$	0	0
$923$	−3.16600	−0.104210
$924$	0	0
$925$	−47.4807	−1.56116
$926$	0	0
$927$	27.0099	0.887123
$928$	0	0
$929$	20.8905	0.685397	0.342698	−	0.939446i	$-0.388659\pi$
$929$	20.8905	0.685397	0.342698	+	0.939446i	$0.388659\pi$
$930$	0	0
$931$	15.7255	0.515383
$932$	0	0
$933$	67.1380	2.19800
$934$	0	0
$935$	2.94832	0.0964203
$936$	0	0
$937$	32.9342	1.07591	0.537957	−	0.842972i	$-0.319196\pi$
$937$	32.9342	1.07591	0.537957	+	0.842972i	$0.319196\pi$
$938$	0	0
$939$	45.4262	1.48243
$940$	0	0
$941$	−45.1275	−1.47111	−0.735556	−	0.677463i	$-0.763079\pi$
$941$	−45.1275	−1.47111	−0.735556	+	0.677463i	$0.763079\pi$
$942$	0	0
$943$	−42.0542	−1.36947
$944$	0	0
$945$	14.1980	0.461861
$946$	0	0
$947$	−30.4571	−0.989721	−0.494861	−	0.868972i	$-0.664781\pi$
$947$	−30.4571	−0.989721	−0.494861	+	0.868972i	$0.664781\pi$
$948$	0	0
$949$	−3.82010	−0.124006
$950$	0	0
$951$	−25.9622	−0.841883
$952$	0	0
$953$	8.75634	0.283646	0.141823	−	0.989892i	$-0.454704\pi$
$953$	8.75634	0.283646	0.141823	+	0.989892i	$0.454704\pi$
$954$	0	0
$955$	2.52565	0.0817281
$956$	0	0
$957$	−21.9059	−0.708118
$958$	0	0
$959$	−35.6871	−1.15240
$960$	0	0
$961$	33.5288	1.08157
$962$	0	0
$963$	93.8146	3.02313
$964$	0	0
$965$	−0.372550	−0.0119928
$966$	0	0
$967$	−47.9641	−1.54242	−0.771210	−	0.636581i	$-0.780348\pi$
$967$	−47.9641	−1.54242	−0.771210	+	0.636581i	$0.780348\pi$
$968$	0	0
$969$	70.4644	2.26364
$970$	0	0
$971$	−56.6897	−1.81926	−0.909629	−	0.415422i	$-0.863634\pi$
$971$	−56.6897	−1.81926	−0.909629	+	0.415422i	$0.863634\pi$
$972$	0	0
$973$	29.5836	0.948408
$974$	0	0
$975$	8.27255	0.264934
$976$	0	0
$977$	−9.49520	−0.303778	−0.151889	−	0.988398i	$-0.548536\pi$
$977$	−9.49520	−0.303778	−0.151889	+	0.988398i	$0.548536\pi$
$978$	0	0
$979$	−19.7114	−0.629979
$980$	0	0
$981$	64.2437	2.05114
$982$	0	0
$983$	2.08823	0.0666042	0.0333021	−	0.999445i	$-0.489398\pi$
$983$	2.08823	0.0666042	0.0333021	+	0.999445i	$0.489398\pi$
$984$	0	0
$985$	−11.1543	−0.355407
$986$	0	0
$987$	9.49357	0.302184
$988$	0	0
$989$	−20.4445	−0.650097
$990$	0	0
$991$	−1.54116	−0.0489567	−0.0244784	−	0.999700i	$-0.507792\pi$
$991$	−1.54116	−0.0489567	−0.0244784	+	0.999700i	$0.507792\pi$
$992$	0	0
$993$	−98.9258	−3.13932
$994$	0	0
$995$	6.73943	0.213654
$996$	0	0
$997$	−37.2343	−1.17922	−0.589611	−	0.807687i	$-0.700719\pi$
$997$	−37.2343	−1.17922	−0.589611	+	0.807687i	$0.700719\pi$
$998$	0	0
$999$	−112.157	−3.54850

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6016.2.a.p.1.1	yes	13
4.3	odd	2		6016.2.a.n.1.13	yes	13
8.3	odd	2		6016.2.a.o.1.1	yes	13
8.5	even	2		6016.2.a.m.1.13	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6016.2.a.m.1.13	✓	13	8.5	even	2
6016.2.a.n.1.13	yes	13	4.3	odd	2
6016.2.a.o.1.1	yes	13	8.3	odd	2
6016.2.a.p.1.1	yes	13	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:	\( N \)	\(=\)	\( 6016 = 2^{7} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6016.a (trivial)

\(f(q)\)	\(=\)	\(q-3.10994 q^{3} +0.407310 q^{5} -3.05265 q^{7} +6.67175 q^{9} +O(q^{10})\)	\(q-3.10994 q^{3} +0.407310 q^{5} -3.05265 q^{7} +6.67175 q^{9} -2.16670 q^{11} +0.550265 q^{13} -1.26671 q^{15} -3.34079 q^{17} +6.78215 q^{19} +9.49357 q^{21} -5.22765 q^{23} -4.83410 q^{25} -11.4189 q^{27} -3.25095 q^{29} -8.03298 q^{31} +6.73833 q^{33} -1.24337 q^{35} +9.82204 q^{37} -1.71129 q^{39} +8.04458 q^{41} +3.91084 q^{43} +2.71747 q^{45} +1.00000 q^{47} +2.31866 q^{49} +10.3897 q^{51} -6.41908 q^{53} -0.882520 q^{55} -21.0921 q^{57} +2.73544 q^{59} -4.63323 q^{61} -20.3665 q^{63} +0.224128 q^{65} -11.2924 q^{67} +16.2577 q^{69} -5.75360 q^{71} -6.94230 q^{73} +15.0338 q^{75} +6.61419 q^{77} -15.0416 q^{79} +15.4970 q^{81} -9.73633 q^{83} -1.36074 q^{85} +10.1103 q^{87} +9.09741 q^{89} -1.67976 q^{91} +24.9821 q^{93} +2.76244 q^{95} -15.1387 q^{97} -14.4557 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9	\( 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9} + 10 q^{11} + 4 q^{13} - 14 q^{15} + 10 q^{17} + 8 q^{19} + 10 q^{21} - 18 q^{23} + 23 q^{25} + 16 q^{27} + 14 q^{29} - 4 q^{31} + 14 q^{33} + 14 q^{35} + 16 q^{37} - 12 q^{39} + 10 q^{41} + 12 q^{43} + 10 q^{45} + 13 q^{47} + 9 q^{49} + 22 q^{51} + 26 q^{53} + 2 q^{55} + 20 q^{57} + 30 q^{59} + 18 q^{61} - 12 q^{63} - 4 q^{65} + 4 q^{67} + 2 q^{69} - 36 q^{71} + 10 q^{73} + 38 q^{75} + 42 q^{77} + 21 q^{81} + 12 q^{83} + 4 q^{85} - 6 q^{87} + 50 q^{89} - 4 q^{91} + 52 q^{93} - 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) \) 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9 + 10 * q^11 + 4 * q^13 - 14 * q^15 + 10 * q^17 + 8 * q^19 + 10 * q^21 - 18 * q^23 + 23 * q^25 + 16 * q^27 + 14 * q^29 - 4 * q^31 + 14 * q^33 + 14 * q^35 + 16 * q^37 - 12 * q^39 + 10 * q^41 + 12 * q^43 + 10 * q^45 + 13 * q^47 + 9 * q^49 + 22 * q^51 + 26 * q^53 + 2 * q^55 + 20 * q^57 + 30 * q^59 + 18 * q^61 - 12 * q^63 - 4 * q^65 + 4 * q^67 + 2 * q^69 - 36 * q^71 + 10 * q^73 + 38 * q^75 + 42 * q^77 + 21 * q^81 + 12 * q^83 + 4 * q^85 - 6 * q^87 + 50 * q^89 - 4 * q^91 + 52 * q^93 - 8 * q^95 - 10 * q^97 + 34 * q^99

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