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Properties

Label 8464.2.a.cc
Level $8464$
Weight $2$
Character orbit 8464.a
Self dual yes
Analytic conductor $67.585$
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] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-2,0,0,0,0,0,18,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(67.5853802708\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.299900807424.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 14x^{6} + 30x^{5} + 37x^{4} - 88x^{3} + 24x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 4232)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_1 q^{5} + (\beta_{5} + \beta_{3}) q^{7} + ( - \beta_{4} + 2) q^{9} + (\beta_{7} + \beta_{3} - \beta_1) q^{11} - \beta_{4} q^{13} + (\beta_{7} - 2 \beta_{5} + \cdots - \beta_1) q^{15}+ \cdots + (3 \beta_{7} - 2 \beta_{5} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 18 q^{9} + 2 q^{13} + 34 q^{25} + 10 q^{27} + 22 q^{29} - 28 q^{31} + 16 q^{35} + 8 q^{39} + 6 q^{41} - 12 q^{47} + 8 q^{49} - 32 q^{55} + 58 q^{59} - 60 q^{71} + 68 q^{73} - 54 q^{75} + 8 q^{77}+ \cdots - 52 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 14x^{6} + 30x^{5} + 37x^{4} - 88x^{3} + 24x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + \nu^{5} - 13\nu^{4} - 13\nu^{3} + 34\nu^{2} + 46\nu - 12 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 14\nu^{4} + 28\nu^{3} + 39\nu^{2} - 62\nu - 14 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - 13\nu^{5} + 43\nu^{4} + 18\nu^{3} - 118\nu^{2} + 68\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 2\nu^{6} + 44\nu^{5} - 34\nu^{4} - 143\nu^{3} + 104\nu^{2} + 74\nu - 28 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} - 41\nu^{5} + 75\nu^{4} + 100\nu^{3} - 208\nu^{2} + 28\nu + 28 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 8\nu^{6} + 40\nu^{5} - 118\nu^{4} - 79\nu^{3} + 338\nu^{2} - 90\nu - 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 8\nu^{6} + 40\nu^{5} - 118\nu^{4} - 79\nu^{3} + 346\nu^{2} - 98\nu - 88 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 17 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} + 9\beta_{6} + 11\beta_{5} + 9\beta_{4} - 6\beta_{3} + 7\beta_{2} + 5\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 27\beta_{7} - 39\beta_{6} - 3\beta_{5} + 5\beta_{4} - 12\beta_{3} + 9\beta_{2} + 17\beta _1 + 133 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -91\beta_{7} + 95\beta_{6} + 117\beta_{5} + 87\beta_{4} - 78\beta_{3} + 53\beta_{2} + 39\beta _1 - 41 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 269\beta_{7} - 429\beta_{6} - 93\beta_{5} + 15\beta_{4} - 156\beta_{3} + 75\beta_{2} + 199\beta _1 + 1207 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -953\beta_{7} + 1041\beta_{6} + 1223\beta_{5} + 849\beta_{4} - 826\beta_{3} + 451\beta_{2} + 333\beta _1 - 775 ) / 4 \) 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
−0.368185
1.77499
−1.88765
0.778627
2.13112
3.04100
−0.196290
−3.27361
0 −3.06033 0 −2.99022 0 −4.86472 0 6.36561 0
1.2 0 −3.06033 0 2.99022 0 4.86472 0 6.36561 0
1.3 0 −1.36075 0 −4.25299 0 −0.454719 0 −1.14835 0
1.4 0 −1.36075 0 4.25299 0 0.454719 0 −1.14835 0
1.5 0 0.308608 0 −1.98578 0 2.43259 0 −2.90476 0
1.6 0 0.308608 0 1.98578 0 −2.43259 0 −2.90476 0
1.7 0 3.11247 0 −2.45507 0 1.48669 0 6.68750 0
1.8 0 3.11247 0 2.45507 0 −1.48669 0 6.68750 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(23\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8464.2.a.cc 8
4.b odd 2 1 4232.2.a.w ✓ 8
23.b odd 2 1 inner 8464.2.a.cc 8
92.b even 2 1 4232.2.a.w ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4232.2.a.w ✓ 8 4.b odd 2 1
4232.2.a.w ✓ 8 92.b even 2 1
8464.2.a.cc 8 1.a even 1 1 trivial
8464.2.a.cc 8 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8464))\):

\( T_{3}^{4} + T_{3}^{3} - 10T_{3}^{2} - 10T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{8} - 37T_{5}^{6} + 455T_{5}^{4} - 2255T_{5}^{2} + 3844 \) Copy content Toggle raw display
\( T_{7}^{8} - 32T_{7}^{6} + 212T_{7}^{4} - 352T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{4} - T_{13}^{3} - 37T_{13}^{2} + 25T_{13} + 316 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} - 10 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 37 T^{6} + \cdots + 3844 \) Copy content Toggle raw display
$7$ \( T^{8} - 32 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{8} - 64 T^{6} + \cdots + 3136 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} - 37 T^{2} + \cdots + 316)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 59 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{8} - 47 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} - 11 T^{3} + \cdots - 896)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 14 T^{3} + \cdots - 2144)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 116 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( (T^{4} - 3 T^{3} + \cdots + 558)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 227 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( (T^{4} + 6 T^{3} + \cdots + 904)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 257 T^{6} + \cdots + 11723776 \) Copy content Toggle raw display
$59$ \( (T^{4} - 29 T^{3} + \cdots - 2996)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 121 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$67$ \( T^{8} - 43 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( (T^{4} + 30 T^{3} + \cdots - 12968)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 34 T^{3} + \cdots + 2647)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 256 T^{6} + \cdots + 802816 \) Copy content Toggle raw display
$83$ \( T^{8} - 59 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( T^{8} - 193 T^{6} + \cdots + 71824 \) Copy content Toggle raw display
$97$ \( T^{8} - 441 T^{6} + \cdots + 24324624 \) Copy content Toggle raw display
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