Properties

Label 1216.2.b.d
Level $1216$
Weight $2$
Character orbit 1216.b
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} -4 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} -4 q^{9} -\beta_{7} q^{13} -\beta_{4} q^{15} -3 q^{17} + ( \beta_{1} + \beta_{5} ) q^{19} + \beta_{7} q^{21} + 3 \beta_{2} q^{23} -7 q^{25} -\beta_{1} q^{27} -3 \beta_{7} q^{29} + \beta_{5} q^{35} -4 \beta_{7} q^{37} + 7 \beta_{2} q^{39} -\beta_{6} q^{41} + 3 \beta_{5} q^{43} -4 \beta_{3} q^{45} + 6 q^{49} -3 \beta_{1} q^{51} + 3 \beta_{7} q^{53} + ( -7 - \beta_{6} ) q^{57} + 3 \beta_{1} q^{59} -2 \beta_{3} q^{61} -4 \beta_{2} q^{63} + \beta_{6} q^{65} -5 \beta_{1} q^{67} + 3 \beta_{7} q^{69} + \beta_{4} q^{71} + 7 q^{73} -7 \beta_{1} q^{75} -\beta_{4} q^{79} -5 q^{81} -5 \beta_{5} q^{83} -3 \beta_{3} q^{85} + 21 \beta_{2} q^{87} + \beta_{6} q^{89} + \beta_{1} q^{91} + ( -12 \beta_{2} - \beta_{4} ) q^{95} -\beta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 32q^{9} + O(q^{10}) \) \( 8q - 32q^{9} - 24q^{17} - 56q^{25} + 48q^{49} - 56q^{57} + 56q^{73} - 40q^{81} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{6} - 9 \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu \)\()/40\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 26 \)\()/5\)
\(\beta_{4}\)\(=\)\( -\nu^{6} - 3 \nu^{4} - \nu^{2} - 6 \)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} - \nu^{3} - 8 \nu \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{7} + 25 \nu^{5} + 55 \nu^{3} + 184 \nu \)\()/20\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} + 5 \nu^{5} + 11 \nu^{3} + 16 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{2}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + 3 \beta_{3} + 2 \beta_{1} - 6\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 5 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{4} - \beta_{3} + 6 \beta_{1} - 2\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{7} + \beta_{6} + 11 \beta_{5} - 22 \beta_{2}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{1} - 9\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(14 \beta_{7} - 7 \beta_{6} - 13 \beta_{5} - 26 \beta_{2}\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
1.09445 0.895644i
−1.09445 + 0.895644i
0.228425 + 1.39564i
−0.228425 1.39564i
−0.228425 + 1.39564i
0.228425 1.39564i
−1.09445 0.895644i
1.09445 + 0.895644i
0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.2 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.3 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.4 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.5 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.6 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.7 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.8 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner
152.b even 2 1 inner
152.g odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.b.d 8
4.b odd 2 1 inner 1216.2.b.d 8
8.b even 2 1 inner 1216.2.b.d 8
8.d odd 2 1 inner 1216.2.b.d 8
19.b odd 2 1 inner 1216.2.b.d 8
76.d even 2 1 inner 1216.2.b.d 8
152.b even 2 1 inner 1216.2.b.d 8
152.g odd 2 1 inner 1216.2.b.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.b.d 8 1.a even 1 1 trivial
1216.2.b.d 8 4.b odd 2 1 inner
1216.2.b.d 8 8.b even 2 1 inner
1216.2.b.d 8 8.d odd 2 1 inner
1216.2.b.d 8 19.b odd 2 1 inner
1216.2.b.d 8 76.d even 2 1 inner
1216.2.b.d 8 152.b even 2 1 inner
1216.2.b.d 8 152.g 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 \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{2} + 7 \)
\( T_{5}^{2} + 12 \)
\( T_{13}^{2} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 7 + T^{2} )^{4} \)
$5$ \( ( 12 + T^{2} )^{4} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( T^{8} \)
$13$ \( ( -7 + T^{2} )^{4} \)
$17$ \( ( 3 + T )^{8} \)
$19$ \( ( 361 - 10 T^{2} + T^{4} )^{2} \)
$23$ \( ( 9 + T^{2} )^{4} \)
$29$ \( ( -63 + T^{2} )^{4} \)
$31$ \( T^{8} \)
$37$ \( ( -112 + T^{2} )^{4} \)
$41$ \( ( 84 + T^{2} )^{4} \)
$43$ \( ( -108 + T^{2} )^{4} \)
$47$ \( T^{8} \)
$53$ \( ( -63 + T^{2} )^{4} \)
$59$ \( ( 63 + T^{2} )^{4} \)
$61$ \( ( 48 + T^{2} )^{4} \)
$67$ \( ( 175 + T^{2} )^{4} \)
$71$ \( ( -84 + T^{2} )^{4} \)
$73$ \( ( -7 + T )^{8} \)
$79$ \( ( -84 + T^{2} )^{4} \)
$83$ \( ( -300 + T^{2} )^{4} \)
$89$ \( ( 84 + T^{2} )^{4} \)
$97$ \( ( 84 + T^{2} )^{4} \)
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