Properties

Label 784.2.x.i
Level $784$
Weight $2$
Character orbit 784.x
Analytic conductor $6.260$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} - \beta_{2} - 1) q^{4} + (\beta_{7} + 2 \beta_{5} - \beta_{3} - \beta_1) q^{8} + ( - 3 \beta_{5} - 3 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} - \beta_{2} - 1) q^{4} + (\beta_{7} + 2 \beta_{5} - \beta_{3} - \beta_1) q^{8} + ( - 3 \beta_{5} - 3 \beta_{3}) q^{9} + ( - 2 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3) q^{11} + (\beta_{4} + 3 \beta_{2}) q^{16} + (3 \beta_{6} - 3 \beta_{4} - 3 \beta_{2} + 3) q^{18} + ( - 3 \beta_{7} + \beta_{6} + 4 \beta_{5} + 3 \beta_{3} + 3 \beta_1 + 5) q^{22} + (2 \beta_{4} + 4 \beta_{2}) q^{23} - 5 \beta_{3} q^{25} + (4 \beta_{7} + 4 \beta_{6} - \beta_{5} - 4 \beta_{3} - 4 \beta_1 + 1) q^{29} + (\beta_{7} + 5 \beta_{3}) q^{32} + ( - 3 \beta_{7} + 6 \beta_{5} + 3 \beta_{3} + 3 \beta_1) q^{36} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{37} + ( - 2 \beta_{7} + 2 \beta_{6} - 5 \beta_{5} + 2 \beta_{3} + 2 \beta_1 - 5) q^{43} + (2 \beta_{5} + 7 \beta_{4} + 2 \beta_{3} + \beta_{2} + 5 \beta_1) q^{44} + (2 \beta_{7} + 6 \beta_{3}) q^{46} + (5 \beta_{6} + 5) q^{50} + ( - 4 \beta_{7} + 4 \beta_{6} - 7 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 7) q^{53} + (8 \beta_{5} - 5 \beta_{4} + 8 \beta_{3} + 3 \beta_{2} + \beta_1) q^{58} + ( - 5 \beta_{6} - 7) q^{64} + ( - 6 \beta_{7} - 6 \beta_{6} + 5 \beta_{4} + \beta_{3} + 6 \beta_{2} - 5) q^{67} - 16 \beta_{5} q^{71} + (9 \beta_{4} + 3 \beta_{2}) q^{72} + ( - \beta_{7} + 5 \beta_{6} + 3 \beta_{4} + 9 \beta_{3} - 5 \beta_{2} - 3) q^{74} + ( - 6 \beta_{5} - 6 \beta_{3} - 12 \beta_1) q^{79} + ( - 9 \beta_{4} + 9) q^{81} + (4 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} - 7 \beta_{2} - 5 \beta_1) q^{86} + (7 \beta_{7} + 3 \beta_{6} + 7 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 7) q^{88} + ( - 6 \beta_{6} - 10) q^{92} + ( - 6 \beta_{7} - 6 \beta_{6} - 3 \beta_{5} + 6 \beta_{3} + 6 \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{4} + 8 q^{11} - 2 q^{16} + 6 q^{18} + 36 q^{22} - 8 q^{29} - 12 q^{37} - 48 q^{43} + 26 q^{44} + 20 q^{50} + 20 q^{53} - 26 q^{58} - 36 q^{64} - 8 q^{67} + 30 q^{72} - 22 q^{74} + 36 q^{81} + 2 q^{86} - 34 q^{88} - 56 q^{92} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 5\nu^{4} - 5\nu^{2} - 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 5\nu^{5} - 5\nu^{3} - 12\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 36 ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 7 ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 3\nu^{5} + 5\nu^{3} + 12\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{5} - \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{6} - 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10\beta_{5} - 10\beta_{3} - 7\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(\beta_{5}\) \(1\) \(-\beta_{4}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−0.228425 1.39564i
1.09445 + 0.895644i
−1.09445 + 0.895644i
0.228425 1.39564i
−1.09445 0.895644i
0.228425 + 1.39564i
−0.228425 + 1.39564i
1.09445 0.895644i
−0.228425 1.39564i 0 −1.89564 + 0.637600i 0 0 0 1.32288 + 2.50000i 2.59808 1.50000i 0
165.2 1.09445 + 0.895644i 0 0.395644 + 1.96048i 0 0 0 −1.32288 + 2.50000i 2.59808 1.50000i 0
373.1 −1.09445 + 0.895644i 0 0.395644 1.96048i 0 0 0 1.32288 + 2.50000i −2.59808 1.50000i 0
373.2 0.228425 1.39564i 0 −1.89564 0.637600i 0 0 0 −1.32288 + 2.50000i −2.59808 1.50000i 0
557.1 −1.09445 0.895644i 0 0.395644 + 1.96048i 0 0 0 1.32288 2.50000i −2.59808 + 1.50000i 0
557.2 0.228425 + 1.39564i 0 −1.89564 + 0.637600i 0 0 0 −1.32288 2.50000i −2.59808 + 1.50000i 0
765.1 −0.228425 + 1.39564i 0 −1.89564 0.637600i 0 0 0 1.32288 2.50000i 2.59808 + 1.50000i 0
765.2 1.09445 0.895644i 0 0.395644 1.96048i 0 0 0 −1.32288 2.50000i 2.59808 + 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 765.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
7.d odd 6 1 inner
16.e even 4 1 inner
112.l odd 4 1 inner
112.w even 12 1 inner
112.x odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.x.i 8
7.b odd 2 1 CM 784.2.x.i 8
7.c even 3 1 784.2.m.f 4
7.c even 3 1 inner 784.2.x.i 8
7.d odd 6 1 784.2.m.f 4
7.d odd 6 1 inner 784.2.x.i 8
16.e even 4 1 inner 784.2.x.i 8
112.l odd 4 1 inner 784.2.x.i 8
112.w even 12 1 784.2.m.f 4
112.w even 12 1 inner 784.2.x.i 8
112.x odd 12 1 784.2.m.f 4
112.x odd 12 1 inner 784.2.x.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.m.f 4 7.c even 3 1
784.2.m.f 4 7.d odd 6 1
784.2.m.f 4 112.w even 12 1
784.2.m.f 4 112.x odd 12 1
784.2.x.i 8 1.a even 1 1 trivial
784.2.x.i 8 7.b odd 2 1 CM
784.2.x.i 8 7.c even 3 1 inner
784.2.x.i 8 7.d odd 6 1 inner
784.2.x.i 8 16.e even 4 1 inner
784.2.x.i 8 112.l odd 4 1 inner
784.2.x.i 8 112.w even 12 1 inner
784.2.x.i 8 112.x odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{6} + 5 T^{4} + 12 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 28 T^{2} + 784)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + 8 T^{2} - 216 T + 2916)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 2085136 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 24 T^{3} + 288 T^{2} + 1392 T + 3364)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 20 T^{7} + 200 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 193877776 \) Copy content Toggle raw display
$71$ \( (T^{2} + 256)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 252 T^{2} + 63504)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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