Show commands: Magma / Pari/GP / SageMath

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 = [2,0,0,0,-6,0,4] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 217.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 936.217
Dual form 936.2.t.b.289.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{5} +(2.00000 + 3.46410i) q^{7} +(2.00000 - 3.46410i) q^{11} +(-2.50000 - 2.59808i) q^{13} +(1.50000 + 2.59808i) q^{17} +(2.00000 + 3.46410i) q^{19} +(-4.00000 + 6.92820i) q^{23} +4.00000 q^{25} +(-2.50000 + 4.33013i) q^{29} -8.00000 q^{31} +(-6.00000 - 10.3923i) q^{35} +(-3.50000 + 6.06218i) q^{37} +(-4.50000 + 7.79423i) q^{41} +(-4.00000 - 6.92820i) q^{43} +4.00000 q^{47} +(-4.50000 + 7.79423i) q^{49} +5.00000 q^{53} +(-6.00000 + 10.3923i) q^{55} +(2.00000 + 3.46410i) q^{59} +(2.50000 + 4.33013i) q^{61} +(7.50000 + 7.79423i) q^{65} +(-4.00000 + 6.92820i) q^{67} +(-2.00000 - 3.46410i) q^{71} +11.0000 q^{73} +16.0000 q^{77} -4.00000 q^{79} +(-4.50000 - 7.79423i) q^{85} +(-3.00000 + 5.19615i) q^{89} +(4.00000 - 13.8564i) q^{91} +(-6.00000 - 10.3923i) q^{95} +(7.00000 + 12.1244i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{5} + 4 q^{7} + 4 q^{11} - 5 q^{13} + 3 q^{17} + 4 q^{19} - 8 q^{23} + 8 q^{25} - 5 q^{29} - 16 q^{31} - 12 q^{35} - 7 q^{37} - 9 q^{41} - 8 q^{43} + 8 q^{47} - 9 q^{49} + 10 q^{53} - 12 q^{55}+ \cdots + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

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
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 2.00000 + 3.46410i 0.755929 + 1.30931i 0.944911 + 0.327327i \(0.106148\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 − 3.46410i 0.603023 − 1.04447i −0.389338 − 0.921095i \(-0.627296\pi\)
0.992361 − 0.123371i \(-0.0393705\pi\)
\(12\) 0 0
\(13\) −2.50000 − 2.59808i −0.693375 − 0.720577i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 − 0.150675i \(-0.0481447\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(18\) 0 0
\(19\) 2.00000 + 3.46410i 0.458831 + 0.794719i 0.998899 − 0.0469020i \(-0.0149348\pi\)
−0.540068 + 0.841621i \(0.681602\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 + 6.92820i −0.834058 + 1.44463i 0.0607377 + 0.998154i \(0.480655\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.50000 + 4.33013i −0.464238 + 0.804084i −0.999167 − 0.0408130i \(-0.987005\pi\)
0.534928 + 0.844897i \(0.320339\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 − 10.3923i −1.01419 − 1.75662i
\(36\) 0 0
\(37\) −3.50000 + 6.06218i −0.575396 + 0.996616i 0.420602 + 0.907245i \(0.361819\pi\)
−0.995998 + 0.0893706i \(0.971514\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.50000 + 7.79423i −0.702782 + 1.21725i 0.264704 + 0.964330i \(0.414726\pi\)
−0.967486 + 0.252924i \(0.918608\pi\)
\(42\) 0 0
\(43\) −4.00000 − 6.92820i −0.609994 − 1.05654i −0.991241 − 0.132068i \(-0.957838\pi\)
0.381246 − 0.924473i \(-0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) −4.50000 + 7.79423i −0.642857 + 1.11346i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.00000 0.686803 0.343401 − 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) 0 0
\(55\) −6.00000 + 10.3923i −0.809040 + 1.40130i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.00000 + 3.46410i 0.260378 + 0.450988i 0.966342 − 0.257260i \(-0.0828195\pi\)
−0.705965 + 0.708247i \(0.749486\pi\)
\(60\) 0 0
\(61\) 2.50000 + 4.33013i 0.320092 + 0.554416i 0.980507 − 0.196485i \(-0.0629528\pi\)
−0.660415 + 0.750901i \(0.729619\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.50000 + 7.79423i 0.930261 + 0.966755i
\(66\) 0 0
\(67\) −4.00000 + 6.92820i −0.488678 + 0.846415i −0.999915 − 0.0130248i \(-0.995854\pi\)
0.511237 + 0.859440i \(0.329187\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 − 3.46410i −0.237356 − 0.411113i 0.722599 − 0.691268i \(-0.242948\pi\)
−0.959955 + 0.280155i \(0.909614\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 − 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −4.50000 − 7.79423i −0.488094 − 0.845403i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 + 5.19615i −0.317999 + 0.550791i −0.980071 − 0.198650i \(-0.936344\pi\)
0.662071 + 0.749441i \(0.269678\pi\)
\(90\) 0 0
\(91\) 4.00000 − 13.8564i 0.419314 − 1.45255i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.00000 − 10.3923i −0.615587 − 1.06623i
\(96\) 0 0
\(97\) 7.00000 + 12.1244i 0.710742 + 1.23104i 0.964579 + 0.263795i \(0.0849741\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 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.b.217.1 2
3.2 odd 2 312.2.q.c.217.1 ✓ 2
4.3 odd 2 1872.2.t.a.1153.1 2
12.11 even 2 624.2.q.e.529.1 2
13.3 even 3 inner 936.2.t.b.289.1 2
39.17 odd 6 4056.2.a.b.1.1 1
39.20 even 12 4056.2.c.b.337.2 2
39.29 odd 6 312.2.q.c.289.1 yes 2
39.32 even 12 4056.2.c.b.337.1 2
39.35 odd 6 4056.2.a.j.1.1 1
52.3 odd 6 1872.2.t.a.289.1 2
156.35 even 6 8112.2.a.bh.1.1 1
156.95 even 6 8112.2.a.r.1.1 1
156.107 even 6 624.2.q.e.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.q.c.217.1 ✓ 2 3.2 odd 2
312.2.q.c.289.1 yes 2 39.29 odd 6
624.2.q.e.289.1 2 156.107 even 6
624.2.q.e.529.1 2 12.11 even 2
936.2.t.b.217.1 2 1.1 even 1 trivial
936.2.t.b.289.1 2 13.3 even 3 inner
1872.2.t.a.289.1 2 52.3 odd 6
1872.2.t.a.1153.1 2 4.3 odd 2
4056.2.a.b.1.1 1 39.17 odd 6
4056.2.a.j.1.1 1 39.35 odd 6
4056.2.c.b.337.1 2 39.32 even 12
4056.2.c.b.337.2 2 39.20 even 12
8112.2.a.r.1.1 1 156.95 even 6
8112.2.a.bh.1.1 1 156.35 even 6