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Properties

Label 936.2.bi.b
Level $936$
Weight $2$
Character orbit 936.bi
Analytic conductor $7.474$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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(361,936)] 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("936.361"); 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, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.bi (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,-6] 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.195105024.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{5} + \beta_{3} - 1) q^{5} + (\beta_{3} + \beta_1 - 1) q^{7} + ( - \beta_{7} - \beta_{6} - \beta_{5} - 1) q^{11} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots + 1) q^{13} + (\beta_{5} + 2 \beta_{2} + 1) q^{17}+ \cdots + ( - 4 \beta_{5} + 2 \beta_{4} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{7} - 6 q^{11} + 6 q^{13} - 6 q^{19} - 2 q^{23} - 20 q^{25} + 8 q^{29} - 16 q^{35} - 24 q^{37} - 12 q^{41} + 6 q^{43} + 2 q^{49} - 20 q^{53} + 16 q^{55} - 18 q^{59} - 4 q^{61} - 14 q^{65} - 42 q^{67}+ \cdots + 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - \nu^{6} + 2\nu^{5} + 10\nu^{4} + 10\nu^{3} + 42\nu^{2} - 45\nu - 27 ) / 108 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 2\nu^{5} - 10\nu^{4} - 10\nu^{3} + 66\nu^{2} + 45\nu - 81 ) / 108 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{6} + 2\nu^{5} + 10\nu^{4} + 10\nu^{3} - 66\nu^{2} + 171\nu - 27 ) / 108 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 2\nu^{6} - 10\nu^{5} - 2\nu^{4} + 22\nu^{3} - 18\nu^{2} - 9\nu + 54 ) / 54 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 8\nu^{6} - 4\nu^{5} - 26\nu^{4} + 52\nu^{3} + 18\nu^{2} - 153\nu + 162 ) / 108 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\nu^{7} - 37\nu^{6} + 26\nu^{5} + 64\nu^{4} - 158\nu^{3} - 24\nu^{2} + 450\nu - 729 ) / 108 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} - 49\nu^{6} + 14\nu^{5} + 130\nu^{4} - 218\nu^{3} - 150\nu^{2} + 801\nu - 891 ) / 108 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} + 5\beta_{5} + \beta_{4} + 3\beta_{3} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{4} - \beta_{2} + 4\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{7} + 3\beta_{6} - 5\beta_{5} - 7\beta_{4} + 8\beta_{3} - 5\beta_{2} + 4\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{7} - 4\beta_{6} - 28\beta_{5} - 2\beta_{4} + 4\beta_{3} - 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -20\beta_{7} + 16\beta_{6} - 56\beta_{5} + 20\beta_{4} + 7\beta_{3} - 9\beta_{2} + 4\beta _1 + 3 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(1 + \beta_{5}\) \(1\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−1.58726 − 0.693255i
1.30512 − 1.13871i
0.560908 + 1.63871i
1.72124 + 0.193255i
1.72124 − 0.193255i
0.560908 − 1.63871i
1.30512 + 1.13871i
−1.58726 + 0.693255i
0 0 0 − 3.44247i 0 −2.74922 − 1.58726i 0 0 0
361.2 0 0 0 1.12182i 0 −2.26053 − 1.30512i 0 0 0
361.3 0 0 0 2.61023i 0 −0.971521 − 0.560908i 0 0 0
361.4 0 0 0 3.17452i 0 2.98127 + 1.72124i 0 0 0
433.1 0 0 0 − 3.17452i 0 2.98127 − 1.72124i 0 0 0
433.2 0 0 0 − 2.61023i 0 −0.971521 + 0.560908i 0 0 0
433.3 0 0 0 − 1.12182i 0 −2.26053 + 1.30512i 0 0 0
433.4 0 0 0 3.44247i 0 −2.74922 + 1.58726i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.bi.b 8
3.b odd 2 1 104.2.o.a ✓ 8
4.b odd 2 1 1872.2.by.n 8
12.b even 2 1 208.2.w.c 8
13.e even 6 1 inner 936.2.bi.b 8
24.f even 2 1 832.2.w.i 8
24.h odd 2 1 832.2.w.g 8
39.d odd 2 1 1352.2.o.f 8
39.f even 4 1 1352.2.i.k 8
39.f even 4 1 1352.2.i.l 8
39.h odd 6 1 104.2.o.a ✓ 8
39.h odd 6 1 1352.2.f.f 8
39.i odd 6 1 1352.2.f.f 8
39.i odd 6 1 1352.2.o.f 8
39.k even 12 1 1352.2.a.k 4
39.k even 12 1 1352.2.a.l 4
39.k even 12 1 1352.2.i.k 8
39.k even 12 1 1352.2.i.l 8
52.i odd 6 1 1872.2.by.n 8
156.p even 6 1 2704.2.f.q 8
156.r even 6 1 208.2.w.c 8
156.r even 6 1 2704.2.f.q 8
156.v odd 12 1 2704.2.a.bd 4
156.v odd 12 1 2704.2.a.be 4
312.ba even 6 1 832.2.w.i 8
312.bg odd 6 1 832.2.w.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.o.a ✓ 8 3.b odd 2 1
104.2.o.a ✓ 8 39.h odd 6 1
208.2.w.c 8 12.b even 2 1
208.2.w.c 8 156.r even 6 1
832.2.w.g 8 24.h odd 2 1
832.2.w.g 8 312.bg odd 6 1
832.2.w.i 8 24.f even 2 1
832.2.w.i 8 312.ba even 6 1
936.2.bi.b 8 1.a even 1 1 trivial
936.2.bi.b 8 13.e even 6 1 inner
1352.2.a.k 4 39.k even 12 1
1352.2.a.l 4 39.k even 12 1
1352.2.f.f 8 39.h odd 6 1
1352.2.f.f 8 39.i odd 6 1
1352.2.i.k 8 39.f even 4 1
1352.2.i.k 8 39.k even 12 1
1352.2.i.l 8 39.f even 4 1
1352.2.i.l 8 39.k even 12 1
1352.2.o.f 8 39.d odd 2 1
1352.2.o.f 8 39.i odd 6 1
1872.2.by.n 8 4.b odd 2 1
1872.2.by.n 8 52.i odd 6 1
2704.2.a.bd 4 156.v odd 12 1
2704.2.a.be 4 156.v odd 12 1
2704.2.f.q 8 156.p even 6 1
2704.2.f.q 8 156.r even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 30T_{5}^{6} + 305T_{5}^{4} + 1152T_{5}^{2} + 1024 \) acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 30 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{8} - 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 34 T^{6} + \cdots + 9409 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 80656 \) Copy content Toggle raw display
$23$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{8} - 8 T^{7} + \cdots + 3721 \) Copy content Toggle raw display
$31$ \( T^{8} + 192 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{8} + 24 T^{7} + \cdots + 89401 \) Copy content Toggle raw display
$41$ \( (T^{4} + 6 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 6 T^{7} + \cdots + 692224 \) Copy content Toggle raw display
$47$ \( T^{8} + 192 T^{6} + \cdots + 135424 \) Copy content Toggle raw display
$53$ \( (T^{4} + 10 T^{3} + \cdots - 1724)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 18 T^{7} + \cdots + 43264 \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{7} + \cdots + 896809 \) Copy content Toggle raw display
$67$ \( T^{8} + 42 T^{7} + \cdots + 55696 \) Copy content Toggle raw display
$71$ \( T^{8} - 54 T^{7} + \cdots + 1430416 \) Copy content Toggle raw display
$73$ \( T^{8} + 198 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots + 3328)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 248 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$89$ \( T^{8} - 18 T^{7} + \cdots + 186267904 \) Copy content Toggle raw display
$97$ \( T^{8} - 30 T^{7} + \cdots + 55830784 \) Copy content Toggle raw display
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