Properties

Label 1216.2.c.h
Level $1216$
Weight $2$
Character orbit 1216.c
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.303595776.1
Defining polynomial: \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{1} q^{5} + ( -\beta_{3} + 2 \beta_{6} ) q^{7} + ( -5 + \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + \beta_{1} q^{5} + ( -\beta_{3} + 2 \beta_{6} ) q^{7} + ( -5 + \beta_{7} ) q^{9} + ( -\beta_{2} - \beta_{4} ) q^{11} + ( 3 \beta_{1} - \beta_{5} ) q^{13} + ( 2 \beta_{3} + 2 \beta_{6} ) q^{15} + 5 q^{17} + \beta_{4} q^{19} + ( -\beta_{1} - 5 \beta_{5} ) q^{21} + ( -\beta_{3} + \beta_{6} ) q^{23} + ( 2 + \beta_{7} ) q^{25} + ( -3 \beta_{2} + 8 \beta_{4} ) q^{27} + ( \beta_{1} - 3 \beta_{5} ) q^{29} + 2 \beta_{3} q^{31} + 8 q^{33} + ( -\beta_{2} + 5 \beta_{4} ) q^{35} + 4 \beta_{1} q^{37} + ( 7 \beta_{3} + 3 \beta_{6} ) q^{39} + ( -\beta_{2} - 7 \beta_{4} ) q^{43} + ( -7 \beta_{1} - 2 \beta_{5} ) q^{45} + ( 4 \beta_{3} - 3 \beta_{6} ) q^{47} + 4 q^{49} + 5 \beta_{2} q^{51} + ( -\beta_{1} + 5 \beta_{5} ) q^{53} + ( -2 \beta_{3} - \beta_{6} ) q^{55} -\beta_{7} q^{57} + ( -\beta_{2} - 2 \beta_{4} ) q^{59} + ( -3 \beta_{1} - 4 \beta_{5} ) q^{61} -11 \beta_{6} q^{63} + ( -8 + 2 \beta_{7} ) q^{65} + ( \beta_{2} + 6 \beta_{4} ) q^{67} + ( \beta_{1} - 3 \beta_{5} ) q^{69} + ( 2 \beta_{3} - 4 \beta_{6} ) q^{71} + ( -5 - 4 \beta_{7} ) q^{73} + ( \beta_{2} + 8 \beta_{4} ) q^{75} + ( -\beta_{1} + 6 \beta_{5} ) q^{77} + ( -6 \beta_{3} + 4 \beta_{6} ) q^{79} + ( 9 - 8 \beta_{7} ) q^{81} -8 \beta_{4} q^{83} + 5 \beta_{1} q^{85} + ( 5 \beta_{3} - 7 \beta_{6} ) q^{87} + ( -2 + 4 \beta_{7} ) q^{89} + ( -\beta_{2} + 16 \beta_{4} ) q^{91} + ( -6 \beta_{1} + 2 \beta_{5} ) q^{93} -\beta_{6} q^{95} + ( 4 + 2 \beta_{7} ) q^{97} + ( 5 \beta_{2} - 3 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 44q^{9} + O(q^{10}) \) \( 8q - 44q^{9} + 40q^{17} + 12q^{25} + 64q^{33} + 32q^{49} + 4q^{57} - 72q^{65} - 24q^{73} + 104q^{81} - 32q^{89} + 24q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{6} + 4 \nu^{4} + 20 \nu^{2} + 27 \)\()/36\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu \)\()/72\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} + 2 \nu^{3} - 9 \nu \)\()/54\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{7} + 16 \nu^{5} + 8 \nu^{3} + 81 \nu \)\()/216\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{6} + 16 \nu^{4} + 80 \nu^{2} + 153 \)\()/72\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 5 \nu^{5} + 16 \nu^{3} + 18 \nu \)\()/27\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{4} + 7 \nu^{2} + 18 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} + \beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + 2 \beta_{5} + \beta_{1} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} - 4 \beta_{4} - \beta_{3}\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{7} - 6 \beta_{5} + 5 \beta_{1} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{6} + 15 \beta_{4} + 16 \beta_{3} - \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(8 \beta_{5} - 16 \beta_{1} - 5\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{6} + 35 \beta_{4} - 48 \beta_{3} - 13 \beta_{2}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
609.1
1.26217 1.18614i
−1.26217 1.18614i
−0.396143 1.68614i
0.396143 1.68614i
0.396143 + 1.68614i
−0.396143 + 1.68614i
−1.26217 + 1.18614i
1.26217 + 1.18614i
0 3.37228i 0 2.52434i 0 −3.31662 0 −8.37228 0
609.2 0 3.37228i 0 2.52434i 0 3.31662 0 −8.37228 0
609.3 0 2.37228i 0 0.792287i 0 3.31662 0 −2.62772 0
609.4 0 2.37228i 0 0.792287i 0 −3.31662 0 −2.62772 0
609.5 0 2.37228i 0 0.792287i 0 −3.31662 0 −2.62772 0
609.6 0 2.37228i 0 0.792287i 0 3.31662 0 −2.62772 0
609.7 0 3.37228i 0 2.52434i 0 3.31662 0 −8.37228 0
609.8 0 3.37228i 0 2.52434i 0 −3.31662 0 −8.37228 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 609.8
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.c.h 8
4.b odd 2 1 inner 1216.2.c.h 8
8.b even 2 1 inner 1216.2.c.h 8
8.d odd 2 1 inner 1216.2.c.h 8
16.e even 4 1 4864.2.a.bi 4
16.e even 4 1 4864.2.a.bl 4
16.f odd 4 1 4864.2.a.bi 4
16.f odd 4 1 4864.2.a.bl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.h 8 1.a even 1 1 trivial
1216.2.c.h 8 4.b odd 2 1 inner
1216.2.c.h 8 8.b even 2 1 inner
1216.2.c.h 8 8.d odd 2 1 inner
4864.2.a.bi 4 16.e even 4 1
4864.2.a.bi 4 16.f odd 4 1
4864.2.a.bl 4 16.e even 4 1
4864.2.a.bl 4 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{4} + 17 T_{3}^{2} + 64 \)
\( T_{5}^{4} + 7 T_{5}^{2} + 4 \)
\( T_{7}^{2} - 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 64 + 17 T^{2} + T^{4} )^{2} \)
$5$ \( ( 4 + 7 T^{2} + T^{4} )^{2} \)
$7$ \( ( -11 + T^{2} )^{4} \)
$11$ \( ( 64 + 17 T^{2} + T^{4} )^{2} \)
$13$ \( ( 576 + 51 T^{2} + T^{4} )^{2} \)
$17$ \( ( -5 + T )^{8} \)
$19$ \( ( 1 + T^{2} )^{4} \)
$23$ \( ( 4 - 7 T^{2} + T^{4} )^{2} \)
$29$ \( ( 256 + 43 T^{2} + T^{4} )^{2} \)
$31$ \( ( -12 + T^{2} )^{4} \)
$37$ \( ( 1024 + 112 T^{2} + T^{4} )^{2} \)
$41$ \( T^{8} \)
$43$ \( ( 1156 + 101 T^{2} + T^{4} )^{2} \)
$47$ \( ( 36 - 87 T^{2} + T^{4} )^{2} \)
$53$ \( ( 3364 + 127 T^{2} + T^{4} )^{2} \)
$59$ \( ( 36 + 21 T^{2} + T^{4} )^{2} \)
$61$ \( ( 4356 + 231 T^{2} + T^{4} )^{2} \)
$67$ \( ( 484 + 77 T^{2} + T^{4} )^{2} \)
$71$ \( ( -44 + T^{2} )^{4} \)
$73$ \( ( -123 + 6 T + T^{2} )^{4} \)
$79$ \( ( 16 - 184 T^{2} + T^{4} )^{2} \)
$83$ \( ( 64 + T^{2} )^{4} \)
$89$ \( ( -116 + 8 T + T^{2} )^{4} \)
$97$ \( ( -24 - 6 T + T^{2} )^{4} \)
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