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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [36,16,Mod(13,36)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("36.13"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 36.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(51.3696618360\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 13.11
Character \(\chi\) \(=\) 36.13
Dual form 36.16.e.a.25.11

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2055.12 - 3182.04i) q^{3} +(14056.2 - 24346.0i) q^{5} +(1.43872e6 + 2.49194e6i) q^{7} +(-5.90186e6 - 1.30790e7i) q^{9} +(4.20164e7 + 7.27745e7i) q^{11} +(-1.43726e7 + 2.48941e7i) q^{13} +(-4.85828e7 - 9.47612e7i) q^{15} -1.95001e9 q^{17} -6.48866e9 q^{19} +(1.08862e10 + 5.43165e8i) q^{21} +(-9.53728e9 + 1.65191e10i) q^{23} +(1.48636e10 + 2.57446e10i) q^{25} +(-5.37468e10 - 8.09886e9i) q^{27} +(-8.83490e10 - 1.53025e11i) q^{29} +(-1.03273e11 + 1.78874e11i) q^{31} +(3.17920e11 + 1.58626e10i) q^{33} +8.08918e10 q^{35} +7.52014e11 q^{37} +(4.96765e10 + 9.68945e10i) q^{39} +(-1.83425e11 + 3.17701e11i) q^{41} +(-8.74863e11 - 1.51531e12i) q^{43} +(-4.01378e11 - 4.01532e10i) q^{45} +(2.08116e12 + 3.60468e12i) q^{47} +(-1.76608e12 + 3.05893e12i) q^{49} +(-4.00750e12 + 6.20500e12i) q^{51} -4.46975e11 q^{53} +2.36236e12 q^{55} +(-1.33350e13 + 2.06472e13i) q^{57} +(-1.01835e13 + 1.76384e13i) q^{59} +(1.12839e13 + 1.95443e13i) q^{61} +(2.41009e13 - 3.35241e13i) q^{63} +(4.04047e11 + 6.99830e11i) q^{65} +(-4.04151e13 + 7.00010e13i) q^{67} +(3.29641e13 + 6.42967e13i) q^{69} +4.45541e13 q^{71} +8.53807e11 q^{73} +(1.12467e14 + 5.61151e12i) q^{75} +(-1.20900e14 + 2.09405e14i) q^{77} +(4.07554e13 + 7.05904e13i) q^{79} +(-1.36227e14 + 1.54380e14i) q^{81} +(-4.11445e13 - 7.12643e13i) q^{83} +(-2.74096e13 + 4.74748e13i) q^{85} +(-6.68499e14 - 3.33546e13i) q^{87} +4.30863e14 q^{89} -8.27128e13 q^{91} +(3.56946e14 + 6.96226e14i) q^{93} +(-9.12057e13 + 1.57973e14i) q^{95} +(-1.05369e14 - 1.82504e14i) q^{97} +(7.03840e14 - 9.79036e14i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 1911 q^{3} + 159846 q^{5} + 803706 q^{7} - 9247029 q^{9} + 85327287 q^{11} - 41509092 q^{13} + 1011307500 q^{15} - 4181504694 q^{17} - 4752917850 q^{19} - 38851541184 q^{21} - 2587803774 q^{23} - 90297075903 q^{25}+ \cdots + 18\!\cdots\!18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2055.12 − 3182.04i 0.542535 − 0.840033i
\(4\) 0 0
\(5\) 14056.2 − 24346.0i 0.0804622 − 0.139365i −0.822986 − 0.568061i \(-0.807694\pi\)
0.903449 + 0.428697i \(0.141027\pi\)
\(6\) 0 0
\(7\) 1.43872e6 + 2.49194e6i 0.660302 + 1.14368i 0.980536 + 0.196338i \(0.0629049\pi\)
−0.320235 + 0.947338i \(0.603762\pi\)
\(8\) 0 0
\(9\) −5.90186e6 − 1.30790e7i −0.411311 − 0.911495i
\(10\) 0 0
\(11\) 4.20164e7 + 7.27745e7i 0.650090 + 1.12599i 0.983101 + 0.183066i \(0.0586022\pi\)
−0.333010 + 0.942923i \(0.608064\pi\)
\(12\) 0 0
\(13\) −1.43726e7 + 2.48941e7i −0.0635272 + 0.110032i −0.896040 − 0.443974i \(-0.853568\pi\)
0.832513 + 0.554006i \(0.186902\pi\)
\(14\) 0 0
\(15\) −4.85828e7 − 9.47612e7i −0.0734173 − 0.143201i
\(16\) 0 0
\(17\) −1.95001e9 −1.15257 −0.576287 − 0.817247i \(-0.695499\pi\)
−0.576287 + 0.817247i \(0.695499\pi\)
\(18\) 0 0
\(19\) −6.48866e9 −1.66534 −0.832671 − 0.553768i \(-0.813189\pi\)
−0.832671 + 0.553768i \(0.813189\pi\)
\(20\) 0 0
\(21\) 1.08862e10 + 5.43165e8i 1.31896 + 0.0658093i
\(22\) 0 0
\(23\) −9.53728e9 + 1.65191e10i −0.584071 + 1.01164i 0.410919 + 0.911672i \(0.365208\pi\)
−0.994991 + 0.0999695i \(0.968125\pi\)
\(24\) 0 0
\(25\) 1.48636e10 + 2.57446e10i 0.487052 + 0.843598i
\(26\) 0 0
\(27\) −5.37468e10 − 8.09886e9i −0.988837 − 0.149003i
\(28\) 0 0
\(29\) −8.83490e10 − 1.53025e11i −0.951079 − 1.64732i −0.743097 − 0.669184i \(-0.766644\pi\)
−0.207982 − 0.978133i \(-0.566690\pi\)
\(30\) 0 0
\(31\) −1.03273e11 + 1.78874e11i −0.674176 + 1.16771i 0.302533 + 0.953139i \(0.402168\pi\)
−0.976709 + 0.214569i \(0.931165\pi\)
\(32\) 0 0
\(33\) 3.17920e11 + 1.58626e10i 1.29856 + 0.0647916i
\(34\) 0 0
\(35\) 8.08918e10 0.212517
\(36\) 0 0
\(37\) 7.52014e11 1.30230 0.651152 − 0.758947i \(-0.274286\pi\)
0.651152 + 0.758947i \(0.274286\pi\)
\(38\) 0 0
\(39\) 4.96765e10 + 9.68945e10i 0.0579651 + 0.113061i
\(40\) 0 0
\(41\) −1.83425e11 + 3.17701e11i −0.147089 + 0.254765i −0.930150 − 0.367179i \(-0.880324\pi\)
0.783062 + 0.621944i \(0.213657\pi\)
\(42\) 0 0
\(43\) −8.74863e11 − 1.51531e12i −0.490825 − 0.850134i 0.509119 − 0.860696i \(-0.329971\pi\)
−0.999944 + 0.0105623i \(0.996638\pi\)
\(44\) 0 0
\(45\) −4.01378e11 − 4.01532e10i −0.160125 − 0.0160187i
\(46\) 0 0
\(47\) 2.08116e12 + 3.60468e12i 0.599201 + 1.03785i 0.992939 + 0.118624i \(0.0378483\pi\)
−0.393738 + 0.919223i \(0.628818\pi\)
\(48\) 0 0
\(49\) −1.76608e12 + 3.05893e12i −0.371996 + 0.644317i
\(50\) 0 0
\(51\) −4.00750e12 + 6.20500e12i −0.625312 + 0.968201i
\(52\) 0 0
\(53\) −4.46975e11 −0.0522654 −0.0261327 − 0.999658i \(-0.508319\pi\)
−0.0261327 + 0.999658i \(0.508319\pi\)
\(54\) 0 0
\(55\) 2.36236e12 0.209231
\(56\) 0 0
\(57\) −1.33350e13 + 2.06472e13i −0.903506 + 1.39894i
\(58\) 0 0
\(59\) −1.01835e13 + 1.76384e13i −0.532731 + 0.922717i 0.466538 + 0.884501i \(0.345501\pi\)
−0.999269 + 0.0382163i \(0.987832\pi\)
\(60\) 0 0
\(61\) 1.12839e13 + 1.95443e13i 0.459711 + 0.796243i 0.998945 − 0.0459127i \(-0.0146196\pi\)
−0.539234 + 0.842156i \(0.681286\pi\)
\(62\) 0 0
\(63\) 2.41009e13 − 3.35241e13i 0.770866 − 1.07227i
\(64\) 0 0
\(65\) 4.04047e11 + 6.99830e11i 0.0102231 + 0.0177069i
\(66\) 0 0
\(67\) −4.04151e13 + 7.00010e13i −0.814671 + 1.41105i 0.0948927 + 0.995488i \(0.469749\pi\)
−0.909564 + 0.415564i \(0.863584\pi\)
\(68\) 0 0
\(69\) 3.29641e13 + 6.42967e13i 0.532933 + 1.03949i
\(70\) 0 0
\(71\) 4.45541e13 0.581367 0.290683 − 0.956819i \(-0.406117\pi\)
0.290683 + 0.956819i \(0.406117\pi\)
\(72\) 0 0
\(73\) 8.53807e11 0.00904562 0.00452281 − 0.999990i \(-0.498560\pi\)
0.00452281 + 0.999990i \(0.498560\pi\)
\(74\) 0 0
\(75\) 1.12467e14 + 5.61151e12i 0.972893 + 0.0485423i
\(76\) 0 0
\(77\) −1.20900e14 + 2.09405e14i −0.858511 + 1.48698i
\(78\) 0 0
\(79\) 4.07554e13 + 7.05904e13i 0.238771 + 0.413564i 0.960362 − 0.278756i \(-0.0899221\pi\)
−0.721591 + 0.692320i \(0.756589\pi\)
\(80\) 0 0
\(81\) −1.36227e14 + 1.54380e14i −0.661646 + 0.749816i
\(82\) 0 0
\(83\) −4.11445e13 − 7.12643e13i −0.166428 − 0.288261i 0.770734 − 0.637158i \(-0.219890\pi\)
−0.937161 + 0.348896i \(0.886557\pi\)
\(84\) 0 0
\(85\) −2.74096e13 + 4.74748e13i −0.0927386 + 0.160628i
\(86\) 0 0
\(87\) −6.68499e14 − 3.33546e13i −1.89979 − 0.0947898i
\(88\) 0 0
\(89\) 4.30863e14 1.03256 0.516279 − 0.856420i \(-0.327317\pi\)
0.516279 + 0.856420i \(0.327317\pi\)
\(90\) 0 0
\(91\) −8.27128e13 −0.167789
\(92\) 0 0
\(93\) 3.56946e14 + 6.96226e14i 0.615149 + 1.19985i
\(94\) 0 0
\(95\) −9.12057e13 + 1.57973e14i −0.133997 + 0.232090i
\(96\) 0 0
\(97\) −1.05369e14 − 1.82504e14i −0.132411 − 0.229343i 0.792194 − 0.610269i \(-0.208939\pi\)
−0.924606 + 0.380926i \(0.875605\pi\)
\(98\) 0 0
\(99\) 7.03840e14 − 9.79036e14i 0.758944 − 1.05569i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 36.16.e.a.13.11 ✓ 30
3.2 odd 2 108.16.e.a.37.8 30
9.2 odd 6 108.16.e.a.73.8 30
9.7 even 3 inner 36.16.e.a.25.11 yes 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.16.e.a.13.11 ✓ 30 1.1 even 1 trivial
36.16.e.a.25.11 yes 30 9.7 even 3 inner
108.16.e.a.37.8 30 3.2 odd 2
108.16.e.a.73.8 30 9.2 odd 6