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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [936,2,Mod(217,936)] 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("936.217"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.t (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,4,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.27870912.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.3
Root \(-1.08870 - 1.88569i\) of defining polynomial
Character \(\chi\) \(=\) 936.289
Dual form 936.2.t.i.217.3

$q$-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)
 
\(f(q)\) \(=\) \(q+3.17741 q^{5} +(1.08870 - 1.88569i) q^{7} +(-1.45926 - 2.52751i) q^{11} +(-1.21815 - 3.39354i) q^{13} +(3.04797 - 5.27923i) q^{17} +(-1.45926 + 2.52751i) q^{19} +(-0.281852 - 0.488181i) q^{23} +5.09593 q^{25} +(-1.32982 - 2.30331i) q^{29} -7.09593 q^{31} +(3.45926 - 5.99162i) q^{35} +(3.76611 + 6.52310i) q^{37} +(1.30685 + 2.26354i) q^{41} +(-5.00723 + 8.67277i) q^{43} +3.43630 q^{47} +(1.12944 + 1.95625i) q^{49} +7.17741 q^{53} +(-4.63667 - 8.03095i) q^{55} +(7.27334 - 12.5978i) q^{59} +(-4.39556 + 7.61333i) q^{61} +(-3.87056 - 10.7827i) q^{65} +(-5.52500 - 9.56958i) q^{67} +(-3.71815 + 6.44002i) q^{71} +13.1919 q^{73} -6.35482 q^{77} +9.96853 q^{79} +13.7911 q^{83} +(9.68464 - 16.7743i) q^{85} +(2.00000 + 3.46410i) q^{89} +(-7.72538 - 1.39751i) q^{91} +(-4.63667 + 8.03095i) q^{95} +(-0.629444 + 1.09023i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{5} - q^{7} - q^{13} + 2 q^{17} - 8 q^{23} - 2 q^{25} + 2 q^{29} - 10 q^{31} + 12 q^{35} - 6 q^{41} - 5 q^{43} + 8 q^{47} + 8 q^{49} + 28 q^{53} - 4 q^{55} - 4 q^{59} - 5 q^{61} - 22 q^{65}+ \cdots - 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 3.17741 1.42098 0.710490 − 0.703707i \(-0.248473\pi\)
0.710490 + 0.703707i \(0.248473\pi\)
\(6\) 0 0
\(7\) 1.08870 − 1.88569i 0.411492 − 0.712725i −0.583561 − 0.812069i \(-0.698341\pi\)
0.995053 + 0.0993444i \(0.0316746\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.45926 − 2.52751i −0.439984 − 0.762074i 0.557704 − 0.830040i \(-0.311682\pi\)
−0.997688 + 0.0679657i \(0.978349\pi\)
\(12\) 0 0
\(13\) −1.21815 − 3.39354i −0.337854 − 0.941199i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.04797 − 5.27923i 0.739240 − 1.28040i −0.213597 − 0.976922i \(-0.568518\pi\)
0.952838 − 0.303480i \(-0.0981486\pi\)
\(18\) 0 0
\(19\) −1.45926 + 2.52751i −0.334778 + 0.579852i −0.983442 − 0.181223i \(-0.941994\pi\)
0.648665 + 0.761074i \(0.275328\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.281852 − 0.488181i −0.0587701 − 0.101793i 0.835143 − 0.550032i \(-0.185385\pi\)
−0.893914 + 0.448239i \(0.852051\pi\)
\(24\) 0 0
\(25\) 5.09593 1.01919
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.32982 − 2.30331i −0.246941 − 0.427714i 0.715735 − 0.698372i \(-0.246092\pi\)
−0.962676 + 0.270658i \(0.912759\pi\)
\(30\) 0 0
\(31\) −7.09593 −1.27447 −0.637234 − 0.770671i \(-0.719921\pi\)
−0.637234 + 0.770671i \(0.719921\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.45926 − 5.99162i 0.584722 − 1.01277i
\(36\) 0 0
\(37\) 3.76611 + 6.52310i 0.619145 + 1.07239i 0.989642 + 0.143557i \(0.0458542\pi\)
−0.370497 + 0.928834i \(0.620812\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.30685 + 2.26354i 0.204096 + 0.353505i 0.949844 − 0.312723i \(-0.101241\pi\)
−0.745748 + 0.666228i \(0.767908\pi\)
\(42\) 0 0
\(43\) −5.00723 + 8.67277i −0.763595 + 1.32259i 0.177391 + 0.984140i \(0.443234\pi\)
−0.940986 + 0.338445i \(0.890099\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.43630 0.501235 0.250618 − 0.968086i \(-0.419366\pi\)
0.250618 + 0.968086i \(0.419366\pi\)
\(48\) 0 0
\(49\) 1.12944 + 1.95625i 0.161349 + 0.279465i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.17741 0.985893 0.492947 − 0.870060i \(-0.335920\pi\)
0.492947 + 0.870060i \(0.335920\pi\)
\(54\) 0 0
\(55\) −4.63667 − 8.03095i −0.625209 − 1.08289i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.27334 − 12.5978i 0.946908 − 1.64009i 0.195024 − 0.980798i \(-0.437521\pi\)
0.751884 − 0.659295i \(-0.229145\pi\)
\(60\) 0 0
\(61\) −4.39556 + 7.61333i −0.562794 + 0.974787i 0.434458 + 0.900692i \(0.356940\pi\)
−0.997251 + 0.0740948i \(0.976393\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.87056 − 10.7827i −0.480083 − 1.33743i
\(66\) 0 0
\(67\) −5.52500 − 9.56958i −0.674986 − 1.16911i −0.976473 − 0.215640i \(-0.930816\pi\)
0.301487 − 0.953470i \(-0.402517\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.71815 + 6.44002i −0.441263 + 0.764290i −0.997783 − 0.0665439i \(-0.978803\pi\)
0.556520 + 0.830834i \(0.312136\pi\)
\(72\) 0 0
\(73\) 13.1919 1.54399 0.771995 − 0.635628i \(-0.219259\pi\)
0.771995 + 0.635628i \(0.219259\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.35482 −0.724199
\(78\) 0 0
\(79\) 9.96853 1.12155 0.560773 − 0.827969i \(-0.310504\pi\)
0.560773 + 0.827969i \(0.310504\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.7911 1.51377 0.756886 − 0.653547i \(-0.226720\pi\)
0.756886 + 0.653547i \(0.226720\pi\)
\(84\) 0 0
\(85\) 9.68464 − 16.7743i 1.05045 − 1.81943i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 + 3.46410i 0.212000 + 0.367194i 0.952340 − 0.305038i \(-0.0986691\pi\)
−0.740341 + 0.672232i \(0.765336\pi\)
\(90\) 0 0
\(91\) −7.72538 − 1.39751i −0.809839 − 0.146499i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.63667 + 8.03095i −0.475712 + 0.823958i
\(96\) 0 0
\(97\) −0.629444 + 1.09023i −0.0639103 + 0.110696i −0.896210 − 0.443630i \(-0.853690\pi\)
0.832300 + 0.554326i \(0.187024\pi\)
\(98\) 0 0
\(99\) 0 0
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 936.2.t.i.289.3 yes 6
3.2 odd 2 936.2.t.g.289.1 yes 6
4.3 odd 2 1872.2.t.v.289.3 6
12.11 even 2 1872.2.t.t.289.1 6
13.9 even 3 inner 936.2.t.i.217.3 yes 6
39.35 odd 6 936.2.t.g.217.1 ✓ 6
52.35 odd 6 1872.2.t.v.1153.3 6
156.35 even 6 1872.2.t.t.1153.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
936.2.t.g.217.1 ✓ 6 39.35 odd 6
936.2.t.g.289.1 yes 6 3.2 odd 2
936.2.t.i.217.3 yes 6 13.9 even 3 inner
936.2.t.i.289.3 yes 6 1.1 even 1 trivial
1872.2.t.t.289.1 6 12.11 even 2
1872.2.t.t.1153.1 6 156.35 even 6
1872.2.t.v.289.3 6 4.3 odd 2
1872.2.t.v.1153.3 6 52.35 odd 6