Properties

Label 1216.2.c.e
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{12}^{3} q^{3} + ( 1 - 2 \zeta_{12}^{2} ) q^{5} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} - q^{9} +O(q^{10})\) \( q + 2 \zeta_{12}^{3} q^{3} + ( 1 - 2 \zeta_{12}^{2} ) q^{5} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} - q^{9} + 3 \zeta_{12}^{3} q^{11} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{15} -3 q^{17} + \zeta_{12}^{3} q^{19} + ( -2 + 4 \zeta_{12}^{2} ) q^{21} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{23} + 2 q^{25} + 4 \zeta_{12}^{3} q^{27} + ( 2 - 4 \zeta_{12}^{2} ) q^{29} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{31} -6 q^{33} -3 \zeta_{12}^{3} q^{35} + ( -6 + 12 \zeta_{12}^{2} ) q^{37} + 6 q^{41} + \zeta_{12}^{3} q^{43} + ( -1 + 2 \zeta_{12}^{2} ) q^{45} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{47} -4 q^{49} -6 \zeta_{12}^{3} q^{51} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{55} -2 q^{57} + 6 \zeta_{12}^{3} q^{59} + ( -3 + 6 \zeta_{12}^{2} ) q^{61} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{63} + 4 \zeta_{12}^{3} q^{67} + ( -4 + 8 \zeta_{12}^{2} ) q^{69} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} - q^{73} + 4 \zeta_{12}^{3} q^{75} + ( -3 + 6 \zeta_{12}^{2} ) q^{77} -11 q^{81} -12 \zeta_{12}^{3} q^{83} + ( -3 + 6 \zeta_{12}^{2} ) q^{85} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{87} + ( -8 + 16 \zeta_{12}^{2} ) q^{93} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{95} + 4 q^{97} -3 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} - 12q^{17} + 8q^{25} - 24q^{33} + 24q^{41} - 16q^{49} - 8q^{57} - 4q^{73} - 44q^{81} + 16q^{97} + O(q^{100}) \)

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
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 2.00000i 0 1.73205i 0 −1.73205 0 −1.00000 0
609.2 0 2.00000i 0 1.73205i 0 1.73205 0 −1.00000 0
609.3 0 2.00000i 0 1.73205i 0 1.73205 0 −1.00000 0
609.4 0 2.00000i 0 1.73205i 0 −1.73205 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.e 4
4.b odd 2 1 inner 1216.2.c.e 4
8.b even 2 1 inner 1216.2.c.e 4
8.d odd 2 1 inner 1216.2.c.e 4
16.e even 4 1 4864.2.a.q 2
16.e even 4 1 4864.2.a.z 2
16.f odd 4 1 4864.2.a.q 2
16.f odd 4 1 4864.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.e 4 1.a even 1 1 trivial
1216.2.c.e 4 4.b odd 2 1 inner
1216.2.c.e 4 8.b even 2 1 inner
1216.2.c.e 4 8.d odd 2 1 inner
4864.2.a.q 2 16.e even 4 1
4864.2.a.q 2 16.f odd 4 1
4864.2.a.z 2 16.e even 4 1
4864.2.a.z 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 \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 4 + T^{2} )^{2} \)
$5$ \( ( 3 + T^{2} )^{2} \)
$7$ \( ( -3 + T^{2} )^{2} \)
$11$ \( ( 9 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( 3 + T )^{4} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( ( -12 + T^{2} )^{2} \)
$29$ \( ( 12 + T^{2} )^{2} \)
$31$ \( ( -48 + T^{2} )^{2} \)
$37$ \( ( 108 + T^{2} )^{2} \)
$41$ \( ( -6 + T )^{4} \)
$43$ \( ( 1 + T^{2} )^{2} \)
$47$ \( ( -27 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( ( 27 + T^{2} )^{2} \)
$67$ \( ( 16 + T^{2} )^{2} \)
$71$ \( ( -12 + T^{2} )^{2} \)
$73$ \( ( 1 + T )^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 144 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( -4 + T )^{4} \)
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