LMFDB - Embedded newform 6036.2.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 27 x^{13} + 106 x^{12} + 266 x^{11} - 1004 x^{10} - 1105 x^{9} + 4076 x^{8} + \cdots - 44 $ x^15 - 4x^14 - 27x^13 + 106x^12 + 266x^11 - 1004x^10 - 1105x^9 + 4076x^8 + 1501x^7 - 7100x^6 - 134x^5 + 5356x^4 - 1041x^3 - 1381x^2 + 543x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.04745$ of defining polynomial
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.04745 q^{5} +3.89716 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.04745 q^{5} +3.89716 q^{7} +1.00000 q^{9} -4.70738 q^{11} -0.326886 q^{13} -2.04745 q^{15} -5.01396 q^{17} +1.82435 q^{19} +3.89716 q^{21} -1.20909 q^{23} -0.807938 q^{25} +1.00000 q^{27} +2.55087 q^{29} +7.52387 q^{31} -4.70738 q^{33} -7.97926 q^{35} +9.54315 q^{37} -0.326886 q^{39} -8.47431 q^{41} -4.76687 q^{43} -2.04745 q^{45} -4.57901 q^{47} +8.18789 q^{49} -5.01396 q^{51} -1.27828 q^{53} +9.63813 q^{55} +1.82435 q^{57} -7.43663 q^{59} -0.825352 q^{61} +3.89716 q^{63} +0.669283 q^{65} -4.57998 q^{67} -1.20909 q^{69} +5.48910 q^{71} -13.0048 q^{73} -0.807938 q^{75} -18.3454 q^{77} -13.2651 q^{79} +1.00000 q^{81} +3.86744 q^{83} +10.2659 q^{85} +2.55087 q^{87} -4.91231 q^{89} -1.27393 q^{91} +7.52387 q^{93} -3.73527 q^{95} -9.32029 q^{97} -4.70738 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.04745	−0.915649	−0.457824	−	0.889043i	$-0.651371\pi$
$5$	−2.04745	−0.915649	−0.457824	+	0.889043i	$0.651371\pi$
$6$	0	0
$7$	3.89716	1.47299	0.736495	−	0.676443i	$-0.236480\pi$
$7$	3.89716	1.47299	0.736495	+	0.676443i	$0.236480\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.70738	−1.41933	−0.709664	−	0.704541i	$-0.751153\pi$
$11$	−4.70738	−1.41933	−0.709664	+	0.704541i	$0.751153\pi$
$12$	0	0
$13$	−0.326886	−0.0906618	−0.0453309	−	0.998972i	$-0.514434\pi$
$13$	−0.326886	−0.0906618	−0.0453309	+	0.998972i	$0.514434\pi$
$14$	0	0
$15$	−2.04745	−0.528650
$16$	0	0
$17$	−5.01396	−1.21606	−0.608032	−	0.793912i	$-0.708041\pi$
$17$	−5.01396	−1.21606	−0.608032	+	0.793912i	$0.708041\pi$
$18$	0	0
$19$	1.82435	0.418535	0.209267	−	0.977858i	$-0.432892\pi$
$19$	1.82435	0.418535	0.209267	+	0.977858i	$0.432892\pi$
$20$	0	0
$21$	3.89716	0.850431
$22$	0	0
$23$	−1.20909	−0.252112	−0.126056	−	0.992023i	$-0.540232\pi$
$23$	−1.20909	−0.252112	−0.126056	+	0.992023i	$0.540232\pi$
$24$	0	0
$25$	−0.807938	−0.161588
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.55087	0.473685	0.236843	−	0.971548i	$-0.423887\pi$
$29$	2.55087	0.473685	0.236843	+	0.971548i	$0.423887\pi$
$30$	0	0
$31$	7.52387	1.35133	0.675663	−	0.737210i	$-0.263857\pi$
$31$	7.52387	1.35133	0.675663	+	0.737210i	$0.263857\pi$
$32$	0	0
$33$	−4.70738	−0.819449
$34$	0	0
$35$	−7.97926	−1.34874
$36$	0	0
$37$	9.54315	1.56888	0.784442	−	0.620202i	$-0.212949\pi$
$37$	9.54315	1.56888	0.784442	+	0.620202i	$0.212949\pi$
$38$	0	0
$39$	−0.326886	−0.0523436
$40$	0	0
$41$	−8.47431	−1.32346	−0.661732	−	0.749740i	$-0.730178\pi$
$41$	−8.47431	−1.32346	−0.661732	+	0.749740i	$0.730178\pi$
$42$	0	0
$43$	−4.76687	−0.726941	−0.363470	−	0.931606i	$-0.618408\pi$
$43$	−4.76687	−0.726941	−0.363470	+	0.931606i	$0.618408\pi$
$44$	0	0
$45$	−2.04745	−0.305216
$46$	0	0
$47$	−4.57901	−0.667917	−0.333959	−	0.942588i	$-0.608385\pi$
$47$	−4.57901	−0.667917	−0.333959	+	0.942588i	$0.608385\pi$
$48$	0	0
$49$	8.18789	1.16970
$50$	0	0
$51$	−5.01396	−0.702095
$52$	0	0
$53$	−1.27828	−0.175586	−0.0877928	−	0.996139i	$-0.527981\pi$
$53$	−1.27828	−0.175586	−0.0877928	+	0.996139i	$0.527981\pi$
$54$	0	0
$55$	9.63813	1.29960
$56$	0	0
$57$	1.82435	0.241641
$58$	0	0
$59$	−7.43663	−0.968167	−0.484083	−	0.875022i	$-0.660847\pi$
$59$	−7.43663	−0.968167	−0.484083	+	0.875022i	$0.660847\pi$
$60$	0	0
$61$	−0.825352	−0.105675	−0.0528377	−	0.998603i	$-0.516827\pi$
$61$	−0.825352	−0.105675	−0.0528377	+	0.998603i	$0.516827\pi$
$62$	0	0
$63$	3.89716	0.490997
$64$	0	0
$65$	0.669283	0.0830144
$66$	0	0
$67$	−4.57998	−0.559533	−0.279767	−	0.960068i	$-0.590257\pi$
$67$	−4.57998	−0.559533	−0.279767	+	0.960068i	$0.590257\pi$
$68$	0	0
$69$	−1.20909	−0.145557
$70$	0	0
$71$	5.48910	0.651437	0.325718	−	0.945467i	$-0.394394\pi$
$71$	5.48910	0.651437	0.325718	+	0.945467i	$0.394394\pi$
$72$	0	0
$73$	−13.0048	−1.52209	−0.761046	−	0.648698i	$-0.775314\pi$
$73$	−13.0048	−1.52209	−0.761046	+	0.648698i	$0.775314\pi$
$74$	0	0
$75$	−0.807938	−0.0932927
$76$	0	0
$77$	−18.3454	−2.09065
$78$	0	0
$79$	−13.2651	−1.49244	−0.746222	−	0.665697i	$-0.768134\pi$
$79$	−13.2651	−1.49244	−0.746222	+	0.665697i	$0.768134\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.86744	0.424507	0.212254	−	0.977215i	$-0.431920\pi$
$83$	3.86744	0.424507	0.212254	+	0.977215i	$0.431920\pi$
$84$	0	0
$85$	10.2659	1.11349
$86$	0	0
$87$	2.55087	0.273482
$88$	0	0
$89$	−4.91231	−0.520704	−0.260352	−	0.965514i	$-0.583839\pi$
$89$	−4.91231	−0.520704	−0.260352	+	0.965514i	$0.583839\pi$
$90$	0	0
$91$	−1.27393	−0.133544
$92$	0	0
$93$	7.52387	0.780189
$94$	0	0
$95$	−3.73527	−0.383231
$96$	0	0
$97$	−9.32029	−0.946332	−0.473166	−	0.880973i	$-0.656889\pi$
$97$	−9.32029	−0.946332	−0.473166	+	0.880973i	$0.656889\pi$
$98$	0	0
$99$	−4.70738	−0.473109
$100$	0	0
$101$	2.59179	0.257893	0.128946	−	0.991652i	$-0.458841\pi$
$101$	2.59179	0.257893	0.128946	+	0.991652i	$0.458841\pi$
$102$	0	0
$103$	−10.1898	−1.00403	−0.502015	−	0.864859i	$-0.667408\pi$
$103$	−10.1898	−1.00403	−0.502015	+	0.864859i	$0.667408\pi$
$104$	0	0
$105$	−7.97926	−0.778696
$106$	0	0
$107$	2.11685	0.204644	0.102322	−	0.994751i	$-0.467373\pi$
$107$	2.11685	0.204644	0.102322	+	0.994751i	$0.467373\pi$
$108$	0	0
$109$	−6.58991	−0.631198	−0.315599	−	0.948893i	$-0.602206\pi$
$109$	−6.58991	−0.631198	−0.315599	+	0.948893i	$0.602206\pi$
$110$	0	0
$111$	9.54315	0.905796
$112$	0	0
$113$	0.987955	0.0929390	0.0464695	−	0.998920i	$-0.485203\pi$
$113$	0.987955	0.0929390	0.0464695	+	0.998920i	$0.485203\pi$
$114$	0	0
$115$	2.47554	0.230846
$116$	0	0
$117$	−0.326886	−0.0302206
$118$	0	0
$119$	−19.5402	−1.79125
$120$	0	0
$121$	11.1594	1.01449
$122$	0	0
$123$	−8.47431	−0.764102
$124$	0	0
$125$	11.8915	1.06361
$126$	0	0
$127$	−0.523002	−0.0464089	−0.0232045	−	0.999731i	$-0.507387\pi$
$127$	−0.523002	−0.0464089	−0.0232045	+	0.999731i	$0.507387\pi$
$128$	0	0
$129$	−4.76687	−0.419699
$130$	0	0
$131$	−11.8830	−1.03823	−0.519113	−	0.854706i	$-0.673737\pi$
$131$	−11.8830	−1.03823	−0.519113	+	0.854706i	$0.673737\pi$
$132$	0	0
$133$	7.10980	0.616498
$134$	0	0
$135$	−2.04745	−0.176217
$136$	0	0
$137$	−6.31909	−0.539876	−0.269938	−	0.962878i	$-0.587003\pi$
$137$	−6.31909	−0.539876	−0.269938	+	0.962878i	$0.587003\pi$
$138$	0	0
$139$	0.794226	0.0673654	0.0336827	−	0.999433i	$-0.489276\pi$
$139$	0.794226	0.0673654	0.0336827	+	0.999433i	$0.489276\pi$
$140$	0	0
$141$	−4.57901	−0.385622
$142$	0	0
$143$	1.53877	0.128679
$144$	0	0
$145$	−5.22279	−0.433729
$146$	0	0
$147$	8.18789	0.675326
$148$	0	0
$149$	−8.51557	−0.697623	−0.348811	−	0.937193i	$-0.613415\pi$
$149$	−8.51557	−0.697623	−0.348811	+	0.937193i	$0.613415\pi$
$150$	0	0
$151$	−5.19256	−0.422565	−0.211282	−	0.977425i	$-0.567764\pi$
$151$	−5.19256	−0.422565	−0.211282	+	0.977425i	$0.567764\pi$
$152$	0	0
$153$	−5.01396	−0.405355
$154$	0	0
$155$	−15.4048	−1.23734
$156$	0	0
$157$	22.7694	1.81720	0.908598	−	0.417672i	$-0.137154\pi$
$157$	22.7694	1.81720	0.908598	+	0.417672i	$0.137154\pi$
$158$	0	0
$159$	−1.27828	−0.101374
$160$	0	0
$161$	−4.71200	−0.371358
$162$	0	0
$163$	−19.9768	−1.56471	−0.782353	−	0.622835i	$-0.785981\pi$
$163$	−19.9768	−1.56471	−0.782353	+	0.622835i	$0.785981\pi$
$164$	0	0
$165$	9.63813	0.750327
$166$	0	0
$167$	−4.06406	−0.314486	−0.157243	−	0.987560i	$-0.550261\pi$
$167$	−4.06406	−0.314486	−0.157243	+	0.987560i	$0.550261\pi$
$168$	0	0
$169$	−12.8931	−0.991780
$170$	0	0
$171$	1.82435	0.139512
$172$	0	0
$173$	14.4948	1.10202	0.551009	−	0.834499i	$-0.314243\pi$
$173$	14.4948	1.10202	0.551009	+	0.834499i	$0.314243\pi$
$174$	0	0
$175$	−3.14867	−0.238017
$176$	0	0
$177$	−7.43663	−0.558971
$178$	0	0
$179$	14.8963	1.11340	0.556700	−	0.830714i	$-0.312067\pi$
$179$	14.8963	1.11340	0.556700	+	0.830714i	$0.312067\pi$
$180$	0	0
$181$	0.0958839	0.00712699	0.00356350	−	0.999994i	$-0.498866\pi$
$181$	0.0958839	0.00712699	0.00356350	+	0.999994i	$0.498866\pi$
$182$	0	0
$183$	−0.825352	−0.0610118
$184$	0	0
$185$	−19.5391	−1.43655
$186$	0	0
$187$	23.6026	1.72599
$188$	0	0
$189$	3.89716	0.283477
$190$	0	0
$191$	−12.2133	−0.883721	−0.441860	−	0.897084i	$-0.645681\pi$
$191$	−12.2133	−0.883721	−0.441860	+	0.897084i	$0.645681\pi$
$192$	0	0
$193$	26.3900	1.89960	0.949798	−	0.312863i	$-0.101288\pi$
$193$	26.3900	1.89960	0.949798	+	0.312863i	$0.101288\pi$
$194$	0	0
$195$	0.669283	0.0479284
$196$	0	0
$197$	0.655343	0.0466913	0.0233456	−	0.999727i	$-0.492568\pi$
$197$	0.655343	0.0466913	0.0233456	+	0.999727i	$0.492568\pi$
$198$	0	0
$199$	−7.47514	−0.529899	−0.264949	−	0.964262i	$-0.585355\pi$
$199$	−7.47514	−0.529899	−0.264949	+	0.964262i	$0.585355\pi$
$200$	0	0
$201$	−4.57998	−0.323047
$202$	0	0
$203$	9.94118	0.697734
$204$	0	0
$205$	17.3507	1.21183
$206$	0	0
$207$	−1.20909	−0.0840372
$208$	0	0
$209$	−8.58791	−0.594038
$210$	0	0
$211$	−5.77812	−0.397782	−0.198891	−	0.980022i	$-0.563734\pi$
$211$	−5.77812	−0.397782	−0.198891	+	0.980022i	$0.563734\pi$
$212$	0	0
$213$	5.48910	0.376107
$214$	0	0
$215$	9.75994	0.665622
$216$	0	0
$217$	29.3218	1.99049
$218$	0	0
$219$	−13.0048	−0.878780
$220$	0	0
$221$	1.63899	0.110251
$222$	0	0
$223$	23.7158	1.58813	0.794064	−	0.607835i	$-0.207962\pi$
$223$	23.7158	1.58813	0.794064	+	0.607835i	$0.207962\pi$
$224$	0	0
$225$	−0.807938	−0.0538626
$226$	0	0
$227$	−19.1401	−1.27038	−0.635188	−	0.772358i	$-0.719077\pi$
$227$	−19.1401	−1.27038	−0.635188	+	0.772358i	$0.719077\pi$
$228$	0	0
$229$	−17.0306	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$229$	−17.0306	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$230$	0	0
$231$	−18.3454	−1.20704
$232$	0	0
$233$	0.00287566	0.000188391 0	9.41955e−5	−	1.00000i	$-0.499970\pi$
$233$	0.00287566	0.000188391 0	9.41955e−5		1.00000i	$0.499970\pi$
$234$	0	0
$235$	9.37530	0.611577
$236$	0	0
$237$	−13.2651	−0.861663
$238$	0	0
$239$	16.7797	1.08539	0.542693	−	0.839931i	$-0.317405\pi$
$239$	16.7797	1.08539	0.542693	+	0.839931i	$0.317405\pi$
$240$	0	0
$241$	2.01472	0.129780	0.0648898	−	0.997892i	$-0.479330\pi$
$241$	2.01472	0.129780	0.0648898	+	0.997892i	$0.479330\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−16.7643	−1.07103
$246$	0	0
$247$	−0.596355	−0.0379451
$248$	0	0
$249$	3.86744	0.245089
$250$	0	0
$251$	−3.46557	−0.218745	−0.109372	−	0.994001i	$-0.534884\pi$
$251$	−3.46557	−0.218745	−0.109372	+	0.994001i	$0.534884\pi$
$252$	0	0
$253$	5.69162	0.357829
$254$	0	0
$255$	10.2659	0.642873
$256$	0	0
$257$	−1.64966	−0.102903	−0.0514514	−	0.998676i	$-0.516385\pi$
$257$	−1.64966	−0.102903	−0.0514514	+	0.998676i	$0.516385\pi$
$258$	0	0
$259$	37.1912	2.31095
$260$	0	0
$261$	2.55087	0.157895
$262$	0	0
$263$	11.1195	0.685660	0.342830	−	0.939397i	$-0.388614\pi$
$263$	11.1195	0.685660	0.342830	+	0.939397i	$0.388614\pi$
$264$	0	0
$265$	2.61722	0.160775
$266$	0	0
$267$	−4.91231	−0.300629
$268$	0	0
$269$	−0.751890	−0.0458435	−0.0229218	−	0.999737i	$-0.507297\pi$
$269$	−0.751890	−0.0458435	−0.0229218	+	0.999737i	$0.507297\pi$
$270$	0	0
$271$	−18.2087	−1.10610	−0.553051	−	0.833148i	$-0.686536\pi$
$271$	−18.2087	−1.10610	−0.553051	+	0.833148i	$0.686536\pi$
$272$	0	0
$273$	−1.27393	−0.0771016
$274$	0	0
$275$	3.80327	0.229346
$276$	0	0
$277$	−25.2198	−1.51531	−0.757654	−	0.652656i	$-0.773655\pi$
$277$	−25.2198	−1.51531	−0.757654	+	0.652656i	$0.773655\pi$
$278$	0	0
$279$	7.52387	0.450442
$280$	0	0
$281$	2.79663	0.166833	0.0834166	−	0.996515i	$-0.473417\pi$
$281$	2.79663	0.166833	0.0834166	+	0.996515i	$0.473417\pi$
$282$	0	0
$283$	−13.4378	−0.798792	−0.399396	−	0.916779i	$-0.630780\pi$
$283$	−13.4378	−0.798792	−0.399396	+	0.916779i	$0.630780\pi$
$284$	0	0
$285$	−3.73527	−0.221258
$286$	0	0
$287$	−33.0258	−1.94945
$288$	0	0
$289$	8.13983	0.478814
$290$	0	0
$291$	−9.32029	−0.546365
$292$	0	0
$293$	23.3446	1.36381	0.681903	−	0.731443i	$-0.261153\pi$
$293$	23.3446	1.36381	0.681903	+	0.731443i	$0.261153\pi$
$294$	0	0
$295$	15.2261	0.886501
$296$	0	0
$297$	−4.70738	−0.273150
$298$	0	0
$299$	0.395233	0.0228569
$300$	0	0
$301$	−18.5773	−1.07078
$302$	0	0
$303$	2.59179	0.148894
$304$	0	0
$305$	1.68987	0.0967616
$306$	0	0
$307$	−33.2791	−1.89934	−0.949670	−	0.313253i	$-0.898581\pi$
$307$	−33.2791	−1.89934	−0.949670	+	0.313253i	$0.898581\pi$
$308$	0	0
$309$	−10.1898	−0.579677
$310$	0	0
$311$	14.4700	0.820519	0.410260	−	0.911969i	$-0.365438\pi$
$311$	14.4700	0.820519	0.410260	+	0.911969i	$0.365438\pi$
$312$	0	0
$313$	−22.2012	−1.25488	−0.627442	−	0.778663i	$-0.715898\pi$
$313$	−22.2012	−1.25488	−0.627442	+	0.778663i	$0.715898\pi$
$314$	0	0
$315$	−7.97926	−0.449580
$316$	0	0
$317$	−6.95963	−0.390892	−0.195446	−	0.980714i	$-0.562615\pi$
$317$	−6.95963	−0.390892	−0.195446	+	0.980714i	$0.562615\pi$
$318$	0	0
$319$	−12.0079	−0.672315
$320$	0	0
$321$	2.11685	0.118151
$322$	0	0
$323$	−9.14723	−0.508966
$324$	0	0
$325$	0.264104	0.0146498
$326$	0	0
$327$	−6.58991	−0.364423
$328$	0	0
$329$	−17.8452	−0.983835
$330$	0	0
$331$	−13.9114	−0.764640	−0.382320	−	0.924030i	$-0.624875\pi$
$331$	−13.9114	−0.764640	−0.382320	+	0.924030i	$0.624875\pi$
$332$	0	0
$333$	9.54315	0.522961
$334$	0	0
$335$	9.37729	0.512336
$336$	0	0
$337$	1.97814	0.107756	0.0538782	−	0.998548i	$-0.482842\pi$
$337$	1.97814	0.107756	0.0538782	+	0.998548i	$0.482842\pi$
$338$	0	0
$339$	0.987955	0.0536584
$340$	0	0
$341$	−35.4177	−1.91797
$342$	0	0
$343$	4.62940	0.249964
$344$	0	0
$345$	2.47554	0.133279
$346$	0	0
$347$	−9.00457	−0.483391	−0.241695	−	0.970352i	$-0.577703\pi$
$347$	−9.00457	−0.483391	−0.241695	+	0.970352i	$0.577703\pi$
$348$	0	0
$349$	0.401688	0.0215019	0.0107509	−	0.999942i	$-0.496578\pi$
$349$	0.401688	0.0215019	0.0107509	+	0.999942i	$0.496578\pi$
$350$	0	0
$351$	−0.326886	−0.0174479
$352$	0	0
$353$	−9.46991	−0.504032	−0.252016	−	0.967723i	$-0.581094\pi$
$353$	−9.46991	−0.504032	−0.252016	+	0.967723i	$0.581094\pi$
$354$	0	0
$355$	−11.2387	−0.596487
$356$	0	0
$357$	−19.5402	−1.03418
$358$	0	0
$359$	23.7941	1.25581	0.627903	−	0.778292i	$-0.283914\pi$
$359$	23.7941	1.25581	0.627903	+	0.778292i	$0.283914\pi$
$360$	0	0
$361$	−15.6717	−0.824828
$362$	0	0
$363$	11.1594	0.585716
$364$	0	0
$365$	26.6266	1.39370
$366$	0	0
$367$	−13.3921	−0.699061	−0.349531	−	0.936925i	$-0.613659\pi$
$367$	−13.3921	−0.699061	−0.349531	+	0.936925i	$0.613659\pi$
$368$	0	0
$369$	−8.47431	−0.441155
$370$	0	0
$371$	−4.98167	−0.258636
$372$	0	0
$373$	−1.99893	−0.103501	−0.0517503	−	0.998660i	$-0.516480\pi$
$373$	−1.99893	−0.103501	−0.0517503	+	0.998660i	$0.516480\pi$
$374$	0	0
$375$	11.8915	0.614073
$376$	0	0
$377$	−0.833845	−0.0429452
$378$	0	0
$379$	−13.9518	−0.716654	−0.358327	−	0.933596i	$-0.616653\pi$
$379$	−13.9518	−0.716654	−0.358327	+	0.933596i	$0.616653\pi$
$380$	0	0
$381$	−0.523002	−0.0267942
$382$	0	0
$383$	−27.3853	−1.39932	−0.699661	−	0.714475i	$-0.746666\pi$
$383$	−27.3853	−1.39932	−0.699661	+	0.714475i	$0.746666\pi$
$384$	0	0
$385$	37.5614	1.91430
$386$	0	0
$387$	−4.76687	−0.242314
$388$	0	0
$389$	4.95165	0.251059	0.125529	−	0.992090i	$-0.459937\pi$
$389$	4.95165	0.251059	0.125529	+	0.992090i	$0.459937\pi$
$390$	0	0
$391$	6.06231	0.306584
$392$	0	0
$393$	−11.8830	−0.599420
$394$	0	0
$395$	27.1597	1.36656
$396$	0	0
$397$	5.20465	0.261214	0.130607	−	0.991434i	$-0.458307\pi$
$397$	5.20465	0.261214	0.130607	+	0.991434i	$0.458307\pi$
$398$	0	0
$399$	7.10980	0.355935
$400$	0	0
$401$	25.0630	1.25158	0.625792	−	0.779990i	$-0.284776\pi$
$401$	25.0630	1.25158	0.625792	+	0.779990i	$0.284776\pi$
$402$	0	0
$403$	−2.45945	−0.122514
$404$	0	0
$405$	−2.04745	−0.101739
$406$	0	0
$407$	−44.9232	−2.22676
$408$	0	0
$409$	−21.9726	−1.08648	−0.543239	−	0.839578i	$-0.682802\pi$
$409$	−21.9726	−1.08648	−0.543239	+	0.839578i	$0.682802\pi$
$410$	0	0
$411$	−6.31909	−0.311698
$412$	0	0
$413$	−28.9818	−1.42610
$414$	0	0
$415$	−7.91841	−0.388699
$416$	0	0
$417$	0.794226	0.0388934
$418$	0	0
$419$	27.1619	1.32694	0.663472	−	0.748201i	$-0.269082\pi$
$419$	27.1619	1.32694	0.663472	+	0.748201i	$0.269082\pi$
$420$	0	0
$421$	23.0600	1.12388	0.561938	−	0.827179i	$-0.310056\pi$
$421$	23.0600	1.12388	0.561938	+	0.827179i	$0.310056\pi$
$422$	0	0
$423$	−4.57901	−0.222639
$424$	0	0
$425$	4.05097	0.196501
$426$	0	0
$427$	−3.21653	−0.155659
$428$	0	0
$429$	1.53877	0.0742927
$430$	0	0
$431$	16.0875	0.774906	0.387453	−	0.921890i	$-0.373355\pi$
$431$	16.0875	0.774906	0.387453	+	0.921890i	$0.373355\pi$
$432$	0	0
$433$	−0.424933	−0.0204210	−0.0102105	−	0.999948i	$-0.503250\pi$
$433$	−0.424933	−0.0204210	−0.0102105	+	0.999948i	$0.503250\pi$
$434$	0	0
$435$	−5.22279	−0.250414
$436$	0	0
$437$	−2.20580	−0.105518
$438$	0	0
$439$	−18.7546	−0.895110	−0.447555	−	0.894257i	$-0.647705\pi$
$439$	−18.7546	−0.895110	−0.447555	+	0.894257i	$0.647705\pi$
$440$	0	0
$441$	8.18789	0.389900
$442$	0	0
$443$	−2.60334	−0.123689	−0.0618443	−	0.998086i	$-0.519698\pi$
$443$	−2.60334	−0.123689	−0.0618443	+	0.998086i	$0.519698\pi$
$444$	0	0
$445$	10.0577	0.476782
$446$	0	0
$447$	−8.51557	−0.402773
$448$	0	0
$449$	−2.69027	−0.126962	−0.0634808	−	0.997983i	$-0.520220\pi$
$449$	−2.69027	−0.126962	−0.0634808	+	0.997983i	$0.520220\pi$
$450$	0	0
$451$	39.8917	1.87843
$452$	0	0
$453$	−5.19256	−0.243968
$454$	0	0
$455$	2.60831	0.122279
$456$	0	0
$457$	3.37173	0.157723	0.0788613	−	0.996886i	$-0.474872\pi$
$457$	3.37173	0.157723	0.0788613	+	0.996886i	$0.474872\pi$
$458$	0	0
$459$	−5.01396	−0.234032
$460$	0	0
$461$	17.8684	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$461$	17.8684	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$462$	0	0
$463$	−29.6764	−1.37918	−0.689589	−	0.724201i	$-0.742209\pi$
$463$	−29.6764	−1.37918	−0.689589	+	0.724201i	$0.742209\pi$
$464$	0	0
$465$	−15.4048	−0.714379
$466$	0	0
$467$	−24.8093	−1.14804	−0.574018	−	0.818843i	$-0.694616\pi$
$467$	−24.8093	−1.14804	−0.574018	+	0.818843i	$0.694616\pi$
$468$	0	0
$469$	−17.8489	−0.824187
$470$	0	0
$471$	22.7694	1.04916
$472$	0	0
$473$	22.4394	1.03177
$474$	0	0
$475$	−1.47396	−0.0676301
$476$	0	0
$477$	−1.27828	−0.0585285
$478$	0	0
$479$	−20.2970	−0.927394	−0.463697	−	0.885994i	$-0.653477\pi$
$479$	−20.2970	−0.927394	−0.463697	+	0.885994i	$0.653477\pi$
$480$	0	0
$481$	−3.11952	−0.142238
$482$	0	0
$483$	−4.71200	−0.214404
$484$	0	0
$485$	19.0828	0.866507
$486$	0	0
$487$	−1.05720	−0.0479061	−0.0239530	−	0.999713i	$-0.507625\pi$
$487$	−1.05720	−0.0479061	−0.0239530	+	0.999713i	$0.507625\pi$
$488$	0	0
$489$	−19.9768	−0.903383
$490$	0	0
$491$	−11.2245	−0.506556	−0.253278	−	0.967393i	$-0.581509\pi$
$491$	−11.2245	−0.506556	−0.253278	+	0.967393i	$0.581509\pi$
$492$	0	0
$493$	−12.7900	−0.576032
$494$	0	0
$495$	9.63813	0.433202
$496$	0	0
$497$	21.3919	0.959559
$498$	0	0
$499$	6.38326	0.285754	0.142877	−	0.989740i	$-0.454365\pi$
$499$	6.38326	0.285754	0.142877	+	0.989740i	$0.454365\pi$
$500$	0	0
$501$	−4.06406	−0.181569
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−5.30656	−0.236139
$506$	0	0
$507$	−12.8931	−0.572605
$508$	0	0
$509$	−20.9545	−0.928794	−0.464397	−	0.885627i	$-0.653729\pi$
$509$	−20.9545	−0.928794	−0.464397	+	0.885627i	$0.653729\pi$
$510$	0	0
$511$	−50.6817	−2.24202
$512$	0	0
$513$	1.82435	0.0805471
$514$	0	0
$515$	20.8631	0.919338
$516$	0	0
$517$	21.5551	0.947993
$518$	0	0
$519$	14.4948	0.636250
$520$	0	0
$521$	−14.5474	−0.637331	−0.318666	−	0.947867i	$-0.603235\pi$
$521$	−14.5474	−0.637331	−0.318666	+	0.947867i	$0.603235\pi$
$522$	0	0
$523$	37.6429	1.64601	0.823004	−	0.568036i	$-0.192297\pi$
$523$	37.6429	1.64601	0.823004	+	0.568036i	$0.192297\pi$
$524$	0	0
$525$	−3.14867	−0.137419
$526$	0	0
$527$	−37.7244	−1.64330
$528$	0	0
$529$	−21.5381	−0.936440
$530$	0	0
$531$	−7.43663	−0.322722
$532$	0	0
$533$	2.77013	0.119988
$534$	0	0
$535$	−4.33415	−0.187382
$536$	0	0
$537$	14.8963	0.642822
$538$	0	0
$539$	−38.5435	−1.66018
$540$	0	0
$541$	0.131363	0.00564775	0.00282387	−	0.999996i	$-0.499101\pi$
$541$	0.131363	0.00564775	0.00282387	+	0.999996i	$0.499101\pi$
$542$	0	0
$543$	0.0958839	0.00411477
$544$	0	0
$545$	13.4925	0.577956
$546$	0	0
$547$	14.9179	0.637843	0.318922	−	0.947781i	$-0.396679\pi$
$547$	14.9179	0.637843	0.318922	+	0.947781i	$0.396679\pi$
$548$	0	0
$549$	−0.825352	−0.0352252
$550$	0	0
$551$	4.65369	0.198254
$552$	0	0
$553$	−51.6964	−2.19836
$554$	0	0
$555$	−19.5391	−0.829390
$556$	0	0
$557$	−5.52271	−0.234005	−0.117002	−	0.993132i	$-0.537329\pi$
$557$	−5.52271	−0.234005	−0.117002	+	0.993132i	$0.537329\pi$
$558$	0	0
$559$	1.55822	0.0659057
$560$	0	0
$561$	23.6026	0.996503
$562$	0	0
$563$	18.3716	0.774271	0.387135	−	0.922023i	$-0.373465\pi$
$563$	18.3716	0.774271	0.387135	+	0.922023i	$0.373465\pi$
$564$	0	0
$565$	−2.02279	−0.0850995
$566$	0	0
$567$	3.89716	0.163666
$568$	0	0
$569$	21.1494	0.886627	0.443313	−	0.896367i	$-0.353803\pi$
$569$	21.1494	0.886627	0.443313	+	0.896367i	$0.353803\pi$
$570$	0	0
$571$	44.9516	1.88117	0.940583	−	0.339564i	$-0.110279\pi$
$571$	44.9516	1.88117	0.940583	+	0.339564i	$0.110279\pi$
$572$	0	0
$573$	−12.2133	−0.510216
$574$	0	0
$575$	0.976866	0.0407381
$576$	0	0
$577$	8.14898	0.339247	0.169623	−	0.985509i	$-0.445745\pi$
$577$	8.14898	0.339247	0.169623	+	0.985509i	$0.445745\pi$
$578$	0	0
$579$	26.3900	1.09673
$580$	0	0
$581$	15.0721	0.625295
$582$	0	0
$583$	6.01735	0.249213
$584$	0	0
$585$	0.669283	0.0276715
$586$	0	0
$587$	34.6714	1.43104	0.715521	−	0.698592i	$-0.246190\pi$
$587$	34.6714	1.43104	0.715521	+	0.698592i	$0.246190\pi$
$588$	0	0
$589$	13.7262	0.565578
$590$	0	0
$591$	0.655343	0.0269572
$592$	0	0
$593$	17.4584	0.716930	0.358465	−	0.933543i	$-0.383300\pi$
$593$	17.4584	0.716930	0.358465	+	0.933543i	$0.383300\pi$
$594$	0	0
$595$	40.0077	1.64016
$596$	0	0
$597$	−7.47514	−0.305937
$598$	0	0
$599$	−33.2755	−1.35960	−0.679801	−	0.733397i	$-0.737934\pi$
$599$	−33.2755	−1.35960	−0.679801	+	0.733397i	$0.737934\pi$
$600$	0	0
$601$	−21.3864	−0.872368	−0.436184	−	0.899857i	$-0.643670\pi$
$601$	−21.3864	−0.872368	−0.436184	+	0.899857i	$0.643670\pi$
$602$	0	0
$603$	−4.57998	−0.186511
$604$	0	0
$605$	−22.8483	−0.928916
$606$	0	0
$607$	4.03385	0.163729	0.0818645	−	0.996643i	$-0.473913\pi$
$607$	4.03385	0.163729	0.0818645	+	0.996643i	$0.473913\pi$
$608$	0	0
$609$	9.94118	0.402837
$610$	0	0
$611$	1.49681	0.0605546
$612$	0	0
$613$	18.8434	0.761079	0.380539	−	0.924765i	$-0.375738\pi$
$613$	18.8434	0.761079	0.380539	+	0.924765i	$0.375738\pi$
$614$	0	0
$615$	17.3507	0.699649
$616$	0	0
$617$	21.9673	0.884372	0.442186	−	0.896923i	$-0.354203\pi$
$617$	21.9673	0.884372	0.442186	+	0.896923i	$0.354203\pi$
$618$	0	0
$619$	44.2152	1.77716	0.888579	−	0.458723i	$-0.151693\pi$
$619$	44.2152	1.77716	0.888579	+	0.458723i	$0.151693\pi$
$620$	0	0
$621$	−1.20909	−0.0485189
$622$	0	0
$623$	−19.1441	−0.766992
$624$	0	0
$625$	−20.3075	−0.812302
$626$	0	0
$627$	−8.58791	−0.342968
$628$	0	0
$629$	−47.8490	−1.90786
$630$	0	0
$631$	−10.3264	−0.411087	−0.205543	−	0.978648i	$-0.565896\pi$
$631$	−10.3264	−0.411087	−0.205543	+	0.978648i	$0.565896\pi$
$632$	0	0
$633$	−5.77812	−0.229660
$634$	0	0
$635$	1.07082	0.0424942
$636$	0	0
$637$	−2.67651	−0.106047
$638$	0	0
$639$	5.48910	0.217146
$640$	0	0
$641$	−17.9443	−0.708759	−0.354379	−	0.935102i	$-0.615308\pi$
$641$	−17.9443	−0.708759	−0.354379	+	0.935102i	$0.615308\pi$
$642$	0	0
$643$	−8.52846	−0.336330	−0.168165	−	0.985759i	$-0.553784\pi$
$643$	−8.52846	−0.336330	−0.168165	+	0.985759i	$0.553784\pi$
$644$	0	0
$645$	9.75994	0.384297
$646$	0	0
$647$	13.9515	0.548490	0.274245	−	0.961660i	$-0.411572\pi$
$647$	13.9515	0.548490	0.274245	+	0.961660i	$0.411572\pi$
$648$	0	0
$649$	35.0070	1.37415
$650$	0	0
$651$	29.3218	1.14921
$652$	0	0
$653$	36.8740	1.44299	0.721496	−	0.692418i	$-0.243455\pi$
$653$	36.8740	1.44299	0.721496	+	0.692418i	$0.243455\pi$
$654$	0	0
$655$	24.3299	0.950650
$656$	0	0
$657$	−13.0048	−0.507364
$658$	0	0
$659$	47.4745	1.84935	0.924673	−	0.380763i	$-0.124339\pi$
$659$	47.4745	1.84935	0.924673	+	0.380763i	$0.124339\pi$
$660$	0	0
$661$	22.2703	0.866213	0.433107	−	0.901343i	$-0.357417\pi$
$661$	22.2703	0.866213	0.433107	+	0.901343i	$0.357417\pi$
$662$	0	0
$663$	1.63899	0.0636532
$664$	0	0
$665$	−14.5570	−0.564495
$666$	0	0
$667$	−3.08422	−0.119422
$668$	0	0
$669$	23.7158	0.916906
$670$	0	0
$671$	3.88524	0.149988
$672$	0	0
$673$	−3.63814	−0.140240	−0.0701201	−	0.997539i	$-0.522338\pi$
$673$	−3.63814	−0.140240	−0.0701201	+	0.997539i	$0.522338\pi$
$674$	0	0
$675$	−0.807938	−0.0310976
$676$	0	0
$677$	−11.8147	−0.454075	−0.227037	−	0.973886i	$-0.572904\pi$
$677$	−11.8147	−0.454075	−0.227037	+	0.973886i	$0.572904\pi$
$678$	0	0
$679$	−36.3227	−1.39394
$680$	0	0
$681$	−19.1401	−0.733451
$682$	0	0
$683$	8.99351	0.344127	0.172064	−	0.985086i	$-0.444957\pi$
$683$	8.99351	0.344127	0.172064	+	0.985086i	$0.444957\pi$
$684$	0	0
$685$	12.9380	0.494337
$686$	0	0
$687$	−17.0306	−0.649759
$688$	0	0
$689$	0.417852	0.0159189
$690$	0	0
$691$	33.2929	1.26652	0.633260	−	0.773939i	$-0.281716\pi$
$691$	33.2929	1.26652	0.633260	+	0.773939i	$0.281716\pi$
$692$	0	0
$693$	−18.3454	−0.696885
$694$	0	0
$695$	−1.62614	−0.0616830
$696$	0	0
$697$	42.4899	1.60942
$698$	0	0
$699$	0.00287566	0.000108768 0
$700$	0	0
$701$	−18.6421	−0.704103	−0.352051	−	0.935981i	$-0.614516\pi$
$701$	−18.6421	−0.704103	−0.352051	+	0.935981i	$0.614516\pi$
$702$	0	0
$703$	17.4101	0.656633
$704$	0	0
$705$	9.37530	0.353094
$706$	0	0
$707$	10.1006	0.379873
$708$	0	0
$709$	−14.0309	−0.526943	−0.263471	−	0.964667i	$-0.584867\pi$
$709$	−14.0309	−0.526943	−0.263471	+	0.964667i	$0.584867\pi$
$710$	0	0
$711$	−13.2651	−0.497482
$712$	0	0
$713$	−9.09700	−0.340685
$714$	0	0
$715$	−3.15057	−0.117825
$716$	0	0
$717$	16.7797	0.626648
$718$	0	0
$719$	−41.6576	−1.55357	−0.776784	−	0.629767i	$-0.783150\pi$
$719$	−41.6576	−1.55357	−0.776784	+	0.629767i	$0.783150\pi$
$720$	0	0
$721$	−39.7113	−1.47892
$722$	0	0
$723$	2.01472	0.0749283
$724$	0	0
$725$	−2.06095	−0.0765417
$726$	0	0
$727$	34.2156	1.26898	0.634492	−	0.772929i	$-0.281209\pi$
$727$	34.2156	1.26898	0.634492	+	0.772929i	$0.281209\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	23.9009	0.884007
$732$	0	0
$733$	16.0439	0.592595	0.296297	−	0.955096i	$-0.404248\pi$
$733$	16.0439	0.592595	0.296297	+	0.955096i	$0.404248\pi$
$734$	0	0
$735$	−16.7643	−0.618361
$736$	0	0
$737$	21.5597	0.794161
$738$	0	0
$739$	30.2862	1.11410	0.557048	−	0.830480i	$-0.311934\pi$
$739$	30.2862	1.11410	0.557048	+	0.830480i	$0.311934\pi$
$740$	0	0
$741$	−0.596355	−0.0219076
$742$	0	0
$743$	23.4411	0.859971	0.429985	−	0.902836i	$-0.358519\pi$
$743$	23.4411	0.859971	0.429985	+	0.902836i	$0.358519\pi$
$744$	0	0
$745$	17.4352	0.638777
$746$	0	0
$747$	3.86744	0.141502
$748$	0	0
$749$	8.24971	0.301438
$750$	0	0
$751$	24.5229	0.894854	0.447427	−	0.894321i	$-0.352340\pi$
$751$	24.5229	0.894854	0.447427	+	0.894321i	$0.352340\pi$
$752$	0	0
$753$	−3.46557	−0.126292
$754$	0	0
$755$	10.6315	0.386921
$756$	0	0
$757$	−18.1214	−0.658634	−0.329317	−	0.944219i	$-0.606819\pi$
$757$	−18.1214	−0.658634	−0.329317	+	0.944219i	$0.606819\pi$
$758$	0	0
$759$	5.69162	0.206593
$760$	0	0
$761$	13.7164	0.497217	0.248609	−	0.968604i	$-0.420027\pi$
$761$	13.7164	0.497217	0.248609	+	0.968604i	$0.420027\pi$
$762$	0	0
$763$	−25.6819	−0.929749
$764$	0	0
$765$	10.2659	0.371163
$766$	0	0
$767$	2.43093	0.0877758
$768$	0	0
$769$	−8.23898	−0.297105	−0.148553	−	0.988905i	$-0.547461\pi$
$769$	−8.23898	−0.297105	−0.148553	+	0.988905i	$0.547461\pi$
$770$	0	0
$771$	−1.64966	−0.0594109
$772$	0	0
$773$	−47.1354	−1.69534	−0.847671	−	0.530523i	$-0.821996\pi$
$773$	−47.1354	−1.69534	−0.847671	+	0.530523i	$0.821996\pi$
$774$	0	0
$775$	−6.07882	−0.218358
$776$	0	0
$777$	37.1912	1.33423
$778$	0	0
$779$	−15.4601	−0.553916
$780$	0	0
$781$	−25.8393	−0.924602
$782$	0	0
$783$	2.55087	0.0911608
$784$	0	0
$785$	−46.6193	−1.66391
$786$	0	0
$787$	−26.4429	−0.942589	−0.471295	−	0.881976i	$-0.656213\pi$
$787$	−26.4429	−0.942589	−0.471295	+	0.881976i	$0.656213\pi$
$788$	0	0
$789$	11.1195	0.395866
$790$	0	0
$791$	3.85022	0.136898
$792$	0	0
$793$	0.269796	0.00958073
$794$	0	0
$795$	2.61722	0.0928233
$796$	0	0
$797$	−15.0538	−0.533231	−0.266616	−	0.963803i	$-0.585905\pi$
$797$	−15.0538	−0.533231	−0.266616	+	0.963803i	$0.585905\pi$
$798$	0	0
$799$	22.9590	0.812231
$800$	0	0
$801$	−4.91231	−0.173568
$802$	0	0
$803$	61.2183	2.16035
$804$	0	0
$805$	9.64760	0.340033
$806$	0	0
$807$	−0.751890	−0.0264678
$808$	0	0
$809$	43.4681	1.52826	0.764129	−	0.645064i	$-0.223169\pi$
$809$	43.4681	1.52826	0.764129	+	0.645064i	$0.223169\pi$
$810$	0	0
$811$	23.5643	0.827453	0.413727	−	0.910401i	$-0.364227\pi$
$811$	23.5643	0.827453	0.413727	+	0.910401i	$0.364227\pi$
$812$	0	0
$813$	−18.2087	−0.638608
$814$	0	0
$815$	40.9016	1.43272
$816$	0	0
$817$	−8.69644	−0.304250
$818$	0	0
$819$	−1.27393	−0.0445146
$820$	0	0
$821$	32.1403	1.12170	0.560852	−	0.827916i	$-0.310474\pi$
$821$	32.1403	1.12170	0.560852	+	0.827916i	$0.310474\pi$
$822$	0	0
$823$	26.4882	0.923320	0.461660	−	0.887057i	$-0.347254\pi$
$823$	26.4882	0.923320	0.461660	+	0.887057i	$0.347254\pi$
$824$	0	0
$825$	3.80327	0.132413
$826$	0	0
$827$	47.0849	1.63730	0.818651	−	0.574292i	$-0.194722\pi$
$827$	47.0849	1.63730	0.818651	+	0.574292i	$0.194722\pi$
$828$	0	0
$829$	16.1417	0.560626	0.280313	−	0.959909i	$-0.409562\pi$
$829$	16.1417	0.560626	0.280313	+	0.959909i	$0.409562\pi$
$830$	0	0
$831$	−25.2198	−0.874863
$832$	0	0
$833$	−41.0538	−1.42243
$834$	0	0
$835$	8.32096	0.287959
$836$	0	0
$837$	7.52387	0.260063
$838$	0	0
$839$	8.36319	0.288729	0.144365	−	0.989525i	$-0.453886\pi$
$839$	8.36319	0.288729	0.144365	+	0.989525i	$0.453886\pi$
$840$	0	0
$841$	−22.4930	−0.775622
$842$	0	0
$843$	2.79663	0.0963211
$844$	0	0
$845$	26.3981	0.908122
$846$	0	0
$847$	43.4899	1.49433
$848$	0	0
$849$	−13.4378	−0.461183
$850$	0	0
$851$	−11.5385	−0.395534
$852$	0	0
$853$	8.87310	0.303809	0.151905	−	0.988395i	$-0.451459\pi$
$853$	8.87310	0.303809	0.151905	+	0.988395i	$0.451459\pi$
$854$	0	0
$855$	−3.73527	−0.127744
$856$	0	0
$857$	−53.7147	−1.83486	−0.917429	−	0.397899i	$-0.869739\pi$
$857$	−53.7147	−1.83486	−0.917429	+	0.397899i	$0.869739\pi$
$858$	0	0
$859$	48.4400	1.65275	0.826376	−	0.563119i	$-0.190399\pi$
$859$	48.4400	1.65275	0.826376	+	0.563119i	$0.190399\pi$
$860$	0	0
$861$	−33.0258	−1.12551
$862$	0	0
$863$	−23.7291	−0.807747	−0.403874	−	0.914815i	$-0.632336\pi$
$863$	−23.7291	−0.807747	−0.403874	+	0.914815i	$0.632336\pi$
$864$	0	0
$865$	−29.6774	−1.00906
$866$	0	0
$867$	8.13983	0.276443
$868$	0	0
$869$	62.4440	2.11827
$870$	0	0
$871$	1.49713	0.0507283
$872$	0	0
$873$	−9.32029	−0.315444
$874$	0	0
$875$	46.3430	1.56668
$876$	0	0
$877$	−10.5283	−0.355514	−0.177757	−	0.984074i	$-0.556884\pi$
$877$	−10.5283	−0.355514	−0.177757	+	0.984074i	$0.556884\pi$
$878$	0	0
$879$	23.3446	0.787394
$880$	0	0
$881$	−31.9326	−1.07584	−0.537918	−	0.842997i	$-0.680789\pi$
$881$	−31.9326	−1.07584	−0.537918	+	0.842997i	$0.680789\pi$
$882$	0	0
$883$	−38.5893	−1.29863	−0.649316	−	0.760519i	$-0.724945\pi$
$883$	−38.5893	−1.29863	−0.649316	+	0.760519i	$0.724945\pi$
$884$	0	0
$885$	15.2261	0.511821
$886$	0	0
$887$	47.1741	1.58395	0.791976	−	0.610552i	$-0.209053\pi$
$887$	47.1741	1.58395	0.791976	+	0.610552i	$0.209053\pi$
$888$	0	0
$889$	−2.03822	−0.0683598
$890$	0	0
$891$	−4.70738	−0.157703
$892$	0	0
$893$	−8.35372	−0.279547
$894$	0	0
$895$	−30.4994	−1.01948
$896$	0	0
$897$	0.395233	0.0131964
$898$	0	0
$899$	19.1924	0.640104
$900$	0	0
$901$	6.40926	0.213523
$902$	0	0
$903$	−18.5773	−0.618213
$904$	0	0
$905$	−0.196318	−0.00652582
$906$	0	0
$907$	2.31330	0.0768120	0.0384060	−	0.999262i	$-0.487772\pi$
$907$	2.31330	0.0768120	0.0384060	+	0.999262i	$0.487772\pi$
$908$	0	0
$909$	2.59179	0.0859642
$910$	0	0
$911$	−42.4575	−1.40668	−0.703340	−	0.710854i	$-0.748309\pi$
$911$	−42.4575	−1.40668	−0.703340	+	0.710854i	$0.748309\pi$
$912$	0	0
$913$	−18.2055	−0.602514
$914$	0	0
$915$	1.68987	0.0558653
$916$	0	0
$917$	−46.3101	−1.52930
$918$	0	0
$919$	−6.19226	−0.204264	−0.102132	−	0.994771i	$-0.532566\pi$
$919$	−6.19226	−0.204264	−0.102132	+	0.994771i	$0.532566\pi$
$920$	0	0
$921$	−33.2791	−1.09658
$922$	0	0
$923$	−1.79431	−0.0590604
$924$	0	0
$925$	−7.71028	−0.253512
$926$	0	0
$927$	−10.1898	−0.334676
$928$	0	0
$929$	−36.0820	−1.18381	−0.591906	−	0.806007i	$-0.701624\pi$
$929$	−36.0820	−1.18381	−0.591906	+	0.806007i	$0.701624\pi$
$930$	0	0
$931$	14.9376	0.489560
$932$	0	0
$933$	14.4700	0.473727
$934$	0	0
$935$	−48.3252	−1.58040
$936$	0	0
$937$	26.8658	0.877668	0.438834	−	0.898568i	$-0.355392\pi$
$937$	26.8658	0.877668	0.438834	+	0.898568i	$0.355392\pi$
$938$	0	0
$939$	−22.2012	−0.724508
$940$	0	0
$941$	26.4271	0.861500	0.430750	−	0.902471i	$-0.358249\pi$
$941$	26.4271	0.861500	0.430750	+	0.902471i	$0.358249\pi$
$942$	0	0
$943$	10.2462	0.333661
$944$	0	0
$945$	−7.97926	−0.259565
$946$	0	0
$947$	−35.4368	−1.15154	−0.575770	−	0.817612i	$-0.695298\pi$
$947$	−35.4368	−1.15154	−0.575770	+	0.817612i	$0.695298\pi$
$948$	0	0
$949$	4.25107	0.137996
$950$	0	0
$951$	−6.95963	−0.225682
$952$	0	0
$953$	13.3955	0.433922	0.216961	−	0.976180i	$-0.430386\pi$
$953$	13.3955	0.433922	0.216961	+	0.976180i	$0.430386\pi$
$954$	0	0
$955$	25.0061	0.809177
$956$	0	0
$957$	−12.0079	−0.388161
$958$	0	0
$959$	−24.6265	−0.795232
$960$	0	0
$961$	25.6086	0.826084
$962$	0	0
$963$	2.11685	0.0682145
$964$	0	0
$965$	−54.0323	−1.73936
$966$	0	0
$967$	13.6998	0.440557	0.220278	−	0.975437i	$-0.429303\pi$
$967$	13.6998	0.440557	0.220278	+	0.975437i	$0.429303\pi$
$968$	0	0
$969$	−9.14723	−0.293851
$970$	0	0
$971$	20.8360	0.668658	0.334329	−	0.942457i	$-0.391490\pi$
$971$	20.8360	0.668658	0.334329	+	0.942457i	$0.391490\pi$
$972$	0	0
$973$	3.09523	0.0992285
$974$	0	0
$975$	0.264104	0.00845809
$976$	0	0
$977$	21.2444	0.679668	0.339834	−	0.940485i	$-0.389629\pi$
$977$	21.2444	0.679668	0.339834	+	0.940485i	$0.389629\pi$
$978$	0	0
$979$	23.1241	0.739049
$980$	0	0
$981$	−6.58991	−0.210399
$982$	0	0
$983$	22.2204	0.708720	0.354360	−	0.935109i	$-0.384699\pi$
$983$	22.2204	0.708720	0.354360	+	0.935109i	$0.384699\pi$
$984$	0	0
$985$	−1.34178	−0.0427528
$986$	0	0
$987$	−17.8452	−0.568017
$988$	0	0
$989$	5.76355	0.183270
$990$	0	0
$991$	12.9681	0.411944	0.205972	−	0.978558i	$-0.433964\pi$
$991$	12.9681	0.411944	0.205972	+	0.978558i	$0.433964\pi$
$992$	0	0
$993$	−13.9114	−0.441465
$994$	0	0
$995$	15.3050	0.485201
$996$	0	0
$997$	38.7803	1.22818	0.614092	−	0.789235i	$-0.289522\pi$
$997$	38.7803	1.22818	0.614092	+	0.789235i	$0.289522\pi$
$998$	0	0
$999$	9.54315	0.301932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.4	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.4	✓	15	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.04745 q^{5} +3.89716 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.04745 q^{5} +3.89716 q^{7} +1.00000 q^{9} -4.70738 q^{11} -0.326886 q^{13} -2.04745 q^{15} -5.01396 q^{17} +1.82435 q^{19} +3.89716 q^{21} -1.20909 q^{23} -0.807938 q^{25} +1.00000 q^{27} +2.55087 q^{29} +7.52387 q^{31} -4.70738 q^{33} -7.97926 q^{35} +9.54315 q^{37} -0.326886 q^{39} -8.47431 q^{41} -4.76687 q^{43} -2.04745 q^{45} -4.57901 q^{47} +8.18789 q^{49} -5.01396 q^{51} -1.27828 q^{53} +9.63813 q^{55} +1.82435 q^{57} -7.43663 q^{59} -0.825352 q^{61} +3.89716 q^{63} +0.669283 q^{65} -4.57998 q^{67} -1.20909 q^{69} +5.48910 q^{71} -13.0048 q^{73} -0.807938 q^{75} -18.3454 q^{77} -13.2651 q^{79} +1.00000 q^{81} +3.86744 q^{83} +10.2659 q^{85} +2.55087 q^{87} -4.91231 q^{89} -1.27393 q^{91} +7.52387 q^{93} -3.73527 q^{95} -9.32029 q^{97} -4.70738 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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