Properties

Label 1161.1.i.b
Level $1161$
Weight $1$
Character orbit 1161.i
Analytic conductor $0.579$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1161 = 3^{3} \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1161.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.579414479667\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \( 1 \)
Twist minimal: yes
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.49923.1

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} - q^{4} -\beta_{1} q^{5} +O(q^{10})\) \( q -\beta_{3} q^{2} - q^{4} -\beta_{1} q^{5} + ( -2 + 2 \beta_{2} ) q^{10} -\beta_{2} q^{13} - q^{16} + ( -1 + \beta_{2} ) q^{19} + \beta_{1} q^{20} -\beta_{1} q^{23} + \beta_{2} q^{25} + ( -\beta_{1} + \beta_{3} ) q^{26} + ( 1 - \beta_{2} ) q^{31} + \beta_{3} q^{32} + \beta_{1} q^{38} + \beta_{3} q^{41} - q^{43} + ( -2 + 2 \beta_{2} ) q^{46} -\beta_{3} q^{47} + ( 1 - \beta_{2} ) q^{49} + ( \beta_{1} - \beta_{3} ) q^{50} + \beta_{2} q^{52} + \beta_{1} q^{53} + \beta_{2} q^{61} -\beta_{1} q^{62} + q^{64} + \beta_{3} q^{65} + ( \beta_{1} - \beta_{3} ) q^{71} -2 \beta_{2} q^{73} + ( 1 - \beta_{2} ) q^{76} -\beta_{2} q^{79} + \beta_{1} q^{80} + 2 q^{82} + \beta_{3} q^{86} -\beta_{1} q^{89} + \beta_{1} q^{92} -2 q^{94} + ( \beta_{1} - \beta_{3} ) q^{95} + q^{97} -\beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{10} - 2q^{13} - 4q^{16} - 2q^{19} + 2q^{25} + 2q^{31} - 4q^{43} - 4q^{46} + 2q^{49} + 2q^{52} + 2q^{61} + 4q^{64} - 4q^{73} + 2q^{76} - 2q^{79} + 8q^{82} - 8q^{94} + 4q^{97} + O(q^{100}) \)

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1161\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(947\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
350.1
1.22474 + 0.707107i
−1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
1.41421i 0 −1.00000 −1.22474 0.707107i 0 0 0 0 −1.00000 + 1.73205i
350.2 1.41421i 0 −1.00000 1.22474 + 0.707107i 0 0 0 0 −1.00000 + 1.73205i
1025.1 1.41421i 0 −1.00000 1.22474 0.707107i 0 0 0 0 −1.00000 1.73205i
1025.2 1.41421i 0 −1.00000 −1.22474 + 0.707107i 0 0 0 0 −1.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
43.c even 3 1 inner
129.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1161.1.i.b 4
3.b odd 2 1 inner 1161.1.i.b 4
9.c even 3 1 3483.1.o.b 4
9.c even 3 1 3483.1.p.b 4
9.d odd 6 1 3483.1.o.b 4
9.d odd 6 1 3483.1.p.b 4
43.c even 3 1 inner 1161.1.i.b 4
129.f odd 6 1 inner 1161.1.i.b 4
387.e even 3 1 3483.1.o.b 4
387.g even 3 1 3483.1.p.b 4
387.o odd 6 1 3483.1.p.b 4
387.p odd 6 1 3483.1.o.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.1.i.b 4 1.a even 1 1 trivial
1161.1.i.b 4 3.b odd 2 1 inner
1161.1.i.b 4 43.c even 3 1 inner
1161.1.i.b 4 129.f odd 6 1 inner
3483.1.o.b 4 9.c even 3 1
3483.1.o.b 4 9.d odd 6 1
3483.1.o.b 4 387.e even 3 1
3483.1.o.b 4 387.p odd 6 1
3483.1.p.b 4 9.c even 3 1
3483.1.p.b 4 9.d odd 6 1
3483.1.p.b 4 387.g even 3 1
3483.1.p.b 4 387.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1161, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ 1
$5$ \( 1 - T^{4} + T^{8} \)
$7$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$11$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$13$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$17$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
$19$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$23$ \( 1 - T^{4} + T^{8} \)
$29$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
$31$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$37$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$41$ \( ( 1 + T^{4} )^{2} \)
$43$ \( ( 1 + T )^{4} \)
$47$ \( ( 1 + T^{4} )^{2} \)
$53$ \( 1 - T^{4} + T^{8} \)
$59$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$61$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$67$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$71$ \( 1 - T^{4} + T^{8} \)
$73$ \( ( 1 + T + T^{2} )^{4} \)
$79$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$83$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
$89$ \( 1 - T^{4} + T^{8} \)
$97$ \( ( 1 - T + T^{2} )^{4} \)
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