Properties

Label 1216.2.b.e
Level $1216$
Weight $2$
Character orbit 1216.b
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM discriminant -19
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: 8.0.2702336256.1
Defining polynomial: \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + \beta_{6} q^{7} + 3 q^{9} +O(q^{10})\) \( q + \beta_{3} q^{5} + \beta_{6} q^{7} + 3 q^{9} + ( -2 \beta_{1} + \beta_{5} ) q^{11} + ( -3 - \beta_{4} ) q^{17} + ( \beta_{1} + 2 \beta_{5} ) q^{19} + \beta_{7} q^{23} -\beta_{4} q^{25} + ( 4 \beta_{1} - 3 \beta_{5} ) q^{35} + ( 2 \beta_{1} + 3 \beta_{5} ) q^{43} + 3 \beta_{3} q^{45} + ( \beta_{6} + 2 \beta_{7} ) q^{47} + ( -8 - 3 \beta_{4} ) q^{49} + ( 3 \beta_{6} - \beta_{7} ) q^{55} + ( 2 \beta_{2} - \beta_{3} ) q^{61} + 3 \beta_{6} q^{63} + ( 7 - 3 \beta_{4} ) q^{73} + ( -\beta_{2} + 10 \beta_{3} ) q^{77} + 9 q^{81} + ( 2 \beta_{1} + 4 \beta_{5} ) q^{83} + ( \beta_{2} - 6 \beta_{3} ) q^{85} + ( \beta_{6} - 2 \beta_{7} ) q^{95} + ( -6 \beta_{1} + 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 28q^{17} - 4q^{25} - 76q^{49} + 44q^{73} + 72q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + 16 \nu^{5} + 44 \nu^{3} + 175 \nu \)\()/500\)
\(\beta_{2}\)\(=\)\((\)\( -19 \nu^{6} - 196 \nu^{4} - 1764 \nu^{2} - 4975 \)\()/700\)
\(\beta_{3}\)\(=\)\((\)\( 17 \nu^{6} + 28 \nu^{4} + 252 \nu^{2} + 325 \)\()/700\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 9 \nu^{4} + 31 \nu^{2} + 125 \)\()/25\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 4 \nu^{5} + 36 \nu^{3} + 45 \nu \)\()/100\)
\(\beta_{6}\)\(=\)\((\)\( -23 \nu^{7} - 182 \nu^{5} - 938 \nu^{3} - 5525 \nu \)\()/1750\)
\(\beta_{7}\)\(=\)\((\)\( 18 \nu^{7} + 112 \nu^{5} + 308 \nu^{3} + 1250 \nu \)\()/875\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - 4 \beta_{6} - 4 \beta_{5} - 4 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(-4 \beta_{4} + \beta_{3} - 5 \beta_{2} - 16\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{7} + 16 \beta_{5} + 8 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(36 \beta_{4} - 47 \beta_{3} + 11 \beta_{2} - 80\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(67 \beta_{7} + 44 \beta_{6} - 44 \beta_{5} + 180 \beta_{1}\)\()/8\)
\(\nu^{6}\)\(=\)\(49 \beta_{3} + 7 \beta_{2} + 27\)
\(\nu^{7}\)\(=\)\((\)\(281 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} - 1116 \beta_{1}\)\()/8\)

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
−0.656712 + 2.13746i
0.656712 2.13746i
1.52274 + 1.63746i
−1.52274 1.63746i
−1.52274 + 1.63746i
1.52274 1.63746i
0.656712 + 2.13746i
−0.656712 2.13746i
0 0 0 3.04547i 0 5.27492i 0 3.00000 0
607.2 0 0 0 3.04547i 0 5.27492i 0 3.00000 0
607.3 0 0 0 1.31342i 0 2.27492i 0 3.00000 0
607.4 0 0 0 1.31342i 0 2.27492i 0 3.00000 0
607.5 0 0 0 1.31342i 0 2.27492i 0 3.00000 0
607.6 0 0 0 1.31342i 0 2.27492i 0 3.00000 0
607.7 0 0 0 3.04547i 0 5.27492i 0 3.00000 0
607.8 0 0 0 3.04547i 0 5.27492i 0 3.00000 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
8.d 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.e 8
4.b odd 2 1 inner 1216.2.b.e 8
8.b even 2 1 inner 1216.2.b.e 8
8.d odd 2 1 inner 1216.2.b.e 8
19.b odd 2 1 CM 1216.2.b.e 8
76.d even 2 1 inner 1216.2.b.e 8
152.b even 2 1 inner 1216.2.b.e 8
152.g odd 2 1 inner 1216.2.b.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.b.e 8 1.a even 1 1 trivial
1216.2.b.e 8 4.b odd 2 1 inner
1216.2.b.e 8 8.b even 2 1 inner
1216.2.b.e 8 8.d odd 2 1 inner
1216.2.b.e 8 19.b odd 2 1 CM
1216.2.b.e 8 76.d even 2 1 inner
1216.2.b.e 8 152.b even 2 1 inner
1216.2.b.e 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 \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3} \)
\( T_{5}^{4} + 11 T_{5}^{2} + 16 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 16 + 11 T^{2} + T^{4} )^{2} \)
$7$ \( ( 144 + 33 T^{2} + T^{4} )^{2} \)
$11$ \( ( 196 - 47 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( ( -2 + 7 T + T^{2} )^{4} \)
$19$ \( ( -19 + T^{2} )^{4} \)
$23$ \( ( 16 + T^{2} )^{4} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( ( 1764 - 87 T^{2} + T^{4} )^{2} \)
$47$ \( ( 784 + 113 T^{2} + T^{4} )^{2} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 26896 + 347 T^{2} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( ( -98 - 11 T + T^{2} )^{4} \)
$79$ \( T^{8} \)
$83$ \( ( -76 + T^{2} )^{4} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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