Properties

Label 1216.2.c.f
Level $1216$
Weight $2$
Character orbit 1216.c
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \beta_{1} q^{3} -\beta_{3} q^{5} + \beta_{2} q^{7} - q^{9} +O(q^{10})\) \( q -2 \beta_{1} q^{3} -\beta_{3} q^{5} + \beta_{2} q^{7} - q^{9} -5 \beta_{1} q^{11} -2 \beta_{2} q^{15} + 5 q^{17} + \beta_{1} q^{19} -2 \beta_{3} q^{21} + 2 \beta_{2} q^{23} -6 q^{25} -4 \beta_{1} q^{27} + 2 \beta_{3} q^{29} -10 q^{33} -11 \beta_{1} q^{35} + 2 \beta_{3} q^{37} -6 q^{41} + \beta_{1} q^{43} + \beta_{3} q^{45} + 3 \beta_{2} q^{47} + 4 q^{49} -10 \beta_{1} q^{51} + 4 \beta_{3} q^{53} -5 \beta_{2} q^{55} + 2 q^{57} + 6 \beta_{1} q^{59} + 3 \beta_{3} q^{61} -\beta_{2} q^{63} -8 \beta_{1} q^{67} -4 \beta_{3} q^{69} -2 \beta_{2} q^{71} -9 q^{73} + 12 \beta_{1} q^{75} -5 \beta_{3} q^{77} -4 \beta_{2} q^{79} -11 q^{81} + 4 \beta_{1} q^{83} -5 \beta_{3} q^{85} + 4 \beta_{2} q^{87} -4 q^{89} + \beta_{2} q^{95} -12 q^{97} + 5 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 20q^{17} - 24q^{25} - 40q^{33} - 24q^{41} + 16q^{49} + 8q^{57} - 36q^{73} - 44q^{81} - 16q^{89} - 48q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 2 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 8 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 4 \beta_{1}\)

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
1.65831 + 0.500000i
−1.65831 + 0.500000i
−1.65831 0.500000i
1.65831 0.500000i
0 2.00000i 0 3.31662i 0 3.31662 0 −1.00000 0
609.2 0 2.00000i 0 3.31662i 0 −3.31662 0 −1.00000 0
609.3 0 2.00000i 0 3.31662i 0 −3.31662 0 −1.00000 0
609.4 0 2.00000i 0 3.31662i 0 3.31662 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
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.f 4
4.b odd 2 1 inner 1216.2.c.f 4
8.b even 2 1 inner 1216.2.c.f 4
8.d odd 2 1 inner 1216.2.c.f 4
16.e even 4 1 4864.2.a.r 2
16.e even 4 1 4864.2.a.y 2
16.f odd 4 1 4864.2.a.r 2
16.f odd 4 1 4864.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.f 4 1.a even 1 1 trivial
1216.2.c.f 4 4.b odd 2 1 inner
1216.2.c.f 4 8.b even 2 1 inner
1216.2.c.f 4 8.d odd 2 1 inner
4864.2.a.r 2 16.e even 4 1
4864.2.a.r 2 16.f odd 4 1
4864.2.a.y 2 16.e even 4 1
4864.2.a.y 2 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 \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 4 + T^{2} )^{2} \)
$5$ \( ( 11 + T^{2} )^{2} \)
$7$ \( ( -11 + T^{2} )^{2} \)
$11$ \( ( 25 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( -5 + T )^{4} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( ( -44 + T^{2} )^{2} \)
$29$ \( ( 44 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 44 + T^{2} )^{2} \)
$41$ \( ( 6 + T )^{4} \)
$43$ \( ( 1 + T^{2} )^{2} \)
$47$ \( ( -99 + T^{2} )^{2} \)
$53$ \( ( 176 + T^{2} )^{2} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( ( 99 + T^{2} )^{2} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( -44 + T^{2} )^{2} \)
$73$ \( ( 9 + T )^{4} \)
$79$ \( ( -176 + T^{2} )^{2} \)
$83$ \( ( 16 + T^{2} )^{2} \)
$89$ \( ( 4 + T )^{4} \)
$97$ \( ( 12 + T )^{4} \)
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