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Properties

Label 2366.2.a.bh
Level $2366$
Weight $2$
Character orbit 2366.a
Self dual yes
Analytic conductor $18.893$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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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] = [2366,2,Mod(1,2366)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2366, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2366.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,6,2,6,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.285686784.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 12x^{3} + 21x^{2} + 2x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + ( - \beta_{5} + \beta_1 - 1) q^{5} + \beta_1 q^{6} + q^{7} + q^{8} + (\beta_{5} - \beta_{4} + \beta_{2} + \cdots + 1) q^{9} + ( - \beta_{5} + \beta_1 - 1) q^{10}+ \cdots + (2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{7} + 6 q^{8} + 6 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} + 6 q^{14} + 14 q^{15} + 6 q^{16} + 4 q^{17} + 6 q^{18} - 4 q^{19} - 2 q^{20} + 2 q^{21}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 10x^{4} + 12x^{3} + 21x^{2} + 2x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} + 7\nu^{3} - 26\nu^{2} - 9\nu + 1 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} + 3\nu^{4} + 24\nu^{3} - 17\nu^{2} - 68\nu - 13 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{5} - 7\nu^{4} - 26\nu^{3} + 43\nu^{2} + 37\nu - 3 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} - 11\nu^{4} - 33\nu^{3} + 74\nu^{2} + 41\nu - 24 ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} + 8\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} - 5\beta_{4} + \beta_{3} + 11\beta_{2} + 13\beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 13\beta_{4} + 11\beta_{3} + 20\beta_{2} + 73\beta _1 + 42 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.55629
−0.865515
−0.466545
0.252878
2.29079
3.34469
1.00000 −2.55629 1.00000 −3.48754 −2.55629 1.00000 1.00000 3.53463 −3.48754
1.2 1.00000 −0.865515 1.00000 −3.71131 −0.865515 1.00000 1.00000 −2.25088 −3.71131
1.3 1.00000 −0.466545 1.00000 3.38938 −0.466545 1.00000 1.00000 −2.78234 3.38938
1.4 1.00000 0.252878 1.00000 1.14776 0.252878 1.00000 1.00000 −2.93605 1.14776
1.5 1.00000 2.29079 1.00000 −0.901839 2.29079 1.00000 1.00000 2.24770 −0.901839
1.6 1.00000 3.34469 1.00000 1.56356 3.34469 1.00000 1.00000 8.18694 1.56356
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(13\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2366.2.a.bh 6
13.b even 2 1 2366.2.a.bf 6
13.d odd 4 2 2366.2.d.r 12
13.f odd 12 2 182.2.m.b ✓ 12
39.k even 12 2 1638.2.bj.g 12
52.l even 12 2 1456.2.cc.d 12
91.w even 12 2 1274.2.v.d 12
91.x odd 12 2 1274.2.o.d 12
91.ba even 12 2 1274.2.o.e 12
91.bc even 12 2 1274.2.m.c 12
91.bd odd 12 2 1274.2.v.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.m.b ✓ 12 13.f odd 12 2
1274.2.m.c 12 91.bc even 12 2
1274.2.o.d 12 91.x odd 12 2
1274.2.o.e 12 91.ba even 12 2
1274.2.v.d 12 91.w even 12 2
1274.2.v.e 12 91.bd odd 12 2
1456.2.cc.d 12 52.l even 12 2
1638.2.bj.g 12 39.k even 12 2
2366.2.a.bf 6 13.b even 2 1
2366.2.a.bh 6 1.a even 1 1 trivial
2366.2.d.r 12 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\):

\( T_{3}^{6} - 2T_{3}^{5} - 10T_{3}^{4} + 12T_{3}^{3} + 21T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{6} + 2T_{5}^{5} - 19T_{5}^{4} - 24T_{5}^{3} + 93T_{5}^{2} + 10T_{5} - 71 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + \cdots - 71 \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 2 T^{5} + \cdots - 704 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} + \cdots - 176 \) Copy content Toggle raw display
$19$ \( T^{6} + 4 T^{5} + \cdots - 19232 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + \cdots + 3142 \) Copy content Toggle raw display
$29$ \( T^{6} - 10 T^{5} + \cdots - 368 \) Copy content Toggle raw display
$31$ \( T^{6} + 2 T^{5} + \cdots - 32 \) Copy content Toggle raw display
$37$ \( T^{6} - 155 T^{4} + \cdots - 32 \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{5} + \cdots - 32 \) Copy content Toggle raw display
$43$ \( T^{6} - 26 T^{5} + \cdots + 2944 \) Copy content Toggle raw display
$47$ \( T^{6} + 8 T^{5} + \cdots + 5632 \) Copy content Toggle raw display
$53$ \( T^{6} - 18 T^{5} + \cdots + 44928 \) Copy content Toggle raw display
$59$ \( T^{6} - 2 T^{5} + \cdots - 2 \) Copy content Toggle raw display
$61$ \( T^{6} - 28 T^{5} + \cdots - 283487 \) Copy content Toggle raw display
$67$ \( T^{6} - 6 T^{5} + \cdots - 32 \) Copy content Toggle raw display
$71$ \( T^{6} - 4 T^{5} + \cdots - 27392 \) Copy content Toggle raw display
$73$ \( T^{6} + 22 T^{5} + \cdots + 52048 \) Copy content Toggle raw display
$79$ \( T^{6} - 22 T^{5} + \cdots + 8032 \) Copy content Toggle raw display
$83$ \( T^{6} + 10 T^{5} + \cdots + 29656 \) Copy content Toggle raw display
$89$ \( T^{6} + 4 T^{5} + \cdots - 101504 \) Copy content Toggle raw display
$97$ \( T^{6} + 12 T^{5} + \cdots - 80000 \) Copy content Toggle raw display
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