Properties

Label 1134.2.e.s
Level $1134$
Weight $2$
Character orbit 1134.e
Analytic conductor $9.055$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + q^{8} + ( \beta_{2} + \beta_{3} ) q^{11} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{14} + q^{16} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( \beta_{2} + \beta_{3} ) q^{22} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{23} + ( 5 + 5 \beta_{1} ) q^{25} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{26} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{28} + ( 2 \beta_{2} - \beta_{3} ) q^{29} + ( 5 - \beta_{2} + 2 \beta_{3} ) q^{31} + q^{32} + ( 4 + 4 \beta_{1} ) q^{37} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{38} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{43} + ( \beta_{2} + \beta_{3} ) q^{44} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{46} + ( 9 + \beta_{2} - 2 \beta_{3} ) q^{47} + ( -5 \beta_{1} + 2 \beta_{3} ) q^{49} + ( 5 + 5 \beta_{1} ) q^{50} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{52} + ( -2 \beta_{2} + \beta_{3} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{56} + ( 2 \beta_{2} - \beta_{3} ) q^{58} + ( 2 + \beta_{2} - 2 \beta_{3} ) q^{61} + ( 5 - \beta_{2} + 2 \beta_{3} ) q^{62} + q^{64} + ( -4 - \beta_{2} + 2 \beta_{3} ) q^{67} + ( -3 - \beta_{2} + 2 \beta_{3} ) q^{71} -7 \beta_{1} q^{73} + ( 4 + 4 \beta_{1} ) q^{74} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{76} + ( -12 - 6 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{77} + ( 5 + \beta_{2} - 2 \beta_{3} ) q^{79} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 12 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{83} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{86} + ( \beta_{2} + \beta_{3} ) q^{88} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -12 - 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{91} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{92} + ( 9 + \beta_{2} - 2 \beta_{3} ) q^{94} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -5 \beta_{1} + 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{4} - 2q^{7} + 4q^{8} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{4} - 2q^{7} + 4q^{8} - 4q^{13} - 2q^{14} + 4q^{16} - 4q^{19} - 6q^{23} + 10q^{25} - 4q^{26} - 2q^{28} + 20q^{31} + 4q^{32} + 8q^{37} - 4q^{38} - 6q^{41} - 4q^{43} - 6q^{46} + 36q^{47} + 10q^{49} + 10q^{50} - 4q^{52} - 2q^{56} + 8q^{61} + 20q^{62} + 4q^{64} - 16q^{67} - 12q^{71} + 14q^{73} + 8q^{74} - 4q^{76} - 36q^{77} + 20q^{79} - 6q^{82} - 24q^{83} - 4q^{86} - 6q^{89} - 40q^{91} - 6q^{92} + 36q^{94} + 8q^{97} + 10q^{98} + O(q^{100}) \)

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1 - \beta_{1}\) \(-1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
1.00000 0 1.00000 0 0 −2.62132 + 0.358719i 1.00000 0 0
865.2 1.00000 0 1.00000 0 0 1.62132 2.09077i 1.00000 0 0
919.1 1.00000 0 1.00000 0 0 −2.62132 0.358719i 1.00000 0 0
919.2 1.00000 0 1.00000 0 0 1.62132 + 2.09077i 1.00000 0 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
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.e.s 4
3.b odd 2 1 1134.2.e.r 4
7.c even 3 1 1134.2.h.r 4
9.c even 3 1 1134.2.g.i 4
9.c even 3 1 1134.2.h.r 4
9.d odd 6 1 1134.2.g.j yes 4
9.d odd 6 1 1134.2.h.s 4
21.h odd 6 1 1134.2.h.s 4
63.g even 3 1 1134.2.g.i 4
63.h even 3 1 inner 1134.2.e.s 4
63.h even 3 1 7938.2.a.bq 2
63.i even 6 1 7938.2.a.bj 2
63.j odd 6 1 1134.2.e.r 4
63.j odd 6 1 7938.2.a.bk 2
63.n odd 6 1 1134.2.g.j yes 4
63.t odd 6 1 7938.2.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.e.r 4 3.b odd 2 1
1134.2.e.r 4 63.j odd 6 1
1134.2.e.s 4 1.a even 1 1 trivial
1134.2.e.s 4 63.h even 3 1 inner
1134.2.g.i 4 9.c even 3 1
1134.2.g.i 4 63.g even 3 1
1134.2.g.j yes 4 9.d odd 6 1
1134.2.g.j yes 4 63.n odd 6 1
1134.2.h.r 4 7.c even 3 1
1134.2.h.r 4 9.c even 3 1
1134.2.h.s 4 9.d odd 6 1
1134.2.h.s 4 21.h odd 6 1
7938.2.a.bj 2 63.i even 6 1
7938.2.a.bk 2 63.j odd 6 1
7938.2.a.bp 2 63.t odd 6 1
7938.2.a.bq 2 63.h even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1134, [\chi])\):

\( T_{5} \)
\( T_{11}^{4} + 18 T_{11}^{2} + 324 \)
\( T_{17} \)
\( T_{23}^{4} + 6 T_{23}^{3} + 45 T_{23}^{2} - 54 T_{23} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( 324 + 18 T^{2} + T^{4} \)
$13$ \( 196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( 196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4} \)
$23$ \( 81 - 54 T + 45 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( 324 + 18 T^{2} + T^{4} \)
$31$ \( ( 7 - 10 T + T^{2} )^{2} \)
$37$ \( ( 16 - 4 T + T^{2} )^{2} \)
$41$ \( 3969 - 378 T + 99 T^{2} + 6 T^{3} + T^{4} \)
$43$ \( 4624 - 272 T + 84 T^{2} + 4 T^{3} + T^{4} \)
$47$ \( ( 63 - 18 T + T^{2} )^{2} \)
$53$ \( 324 + 18 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -14 - 4 T + T^{2} )^{2} \)
$67$ \( ( -2 + 8 T + T^{2} )^{2} \)
$71$ \( ( -9 + 6 T + T^{2} )^{2} \)
$73$ \( ( 49 - 7 T + T^{2} )^{2} \)
$79$ \( ( 7 - 10 T + T^{2} )^{2} \)
$83$ \( 15876 + 3024 T + 450 T^{2} + 24 T^{3} + T^{4} \)
$89$ \( 3969 - 378 T + 99 T^{2} + 6 T^{3} + T^{4} \)
$97$ \( 3136 + 448 T + 120 T^{2} - 8 T^{3} + T^{4} \)
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