Properties

Label 1161.1.i.c
Level $1161$
Weight $1$
Character orbit 1161.i
Analytic conductor $0.579$
Analytic rank $0$
Dimension $8$
Projective image $A_{5}$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
Defining polynomial: \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(A_{5}\)
Projective field Galois closure of 5.1.149769.1

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( -\beta_{3} - \beta_{6} - \beta_{7} ) q^{5} + ( -\beta_{2} - \beta_{4} ) q^{7} + \beta_{7} q^{8} +O(q^{10})\) \( q + \beta_{7} q^{2} + ( -\beta_{3} - \beta_{6} - \beta_{7} ) q^{5} + ( -\beta_{2} - \beta_{4} ) q^{7} + \beta_{7} q^{8} + ( 1 - \beta_{4} - \beta_{5} ) q^{10} + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{11} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{13} + \beta_{1} q^{14} - q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} + \beta_{4} q^{19} + ( -1 - \beta_{2} ) q^{22} + \beta_{6} q^{23} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{25} + ( -\beta_{1} - \beta_{3} ) q^{26} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{34} + \beta_{7} q^{35} -\beta_{6} q^{38} + ( 1 - \beta_{4} - \beta_{5} ) q^{40} + ( -\beta_{1} + \beta_{6} ) q^{41} + ( -1 + \beta_{5} ) q^{43} + \beta_{4} q^{46} -\beta_{4} q^{49} + ( -\beta_{1} - \beta_{3} ) q^{50} -\beta_{6} q^{53} + ( 2 - \beta_{4} - 2 \beta_{5} ) q^{55} + \beta_{1} q^{56} + ( \beta_{1} - \beta_{6} ) q^{59} -\beta_{5} q^{61} - q^{64} + ( \beta_{1} - \beta_{6} - 2 \beta_{7} ) q^{65} + ( -1 + \beta_{5} ) q^{67} - q^{70} + \beta_{3} q^{71} + \beta_{3} q^{77} + ( -\beta_{2} - \beta_{4} - \beta_{5} ) q^{79} + ( \beta_{3} + \beta_{6} + \beta_{7} ) q^{80} -\beta_{2} q^{82} + ( -\beta_{3} - \beta_{7} ) q^{83} + ( -2 - \beta_{2} ) q^{85} + ( -\beta_{3} - \beta_{7} ) q^{86} + ( -1 - \beta_{2} ) q^{88} + ( 1 - \beta_{5} ) q^{91} + \beta_{3} q^{95} + ( 1 + \beta_{2} ) q^{97} + \beta_{6} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{7} + O(q^{10}) \) \( 8q + 2q^{7} + 2q^{10} + 2q^{13} - 8q^{16} + 2q^{19} - 4q^{22} + 2q^{25} + 2q^{34} + 2q^{40} - 4q^{43} + 2q^{46} - 2q^{49} + 6q^{55} - 4q^{61} - 8q^{64} - 4q^{67} - 8q^{70} - 2q^{79} + 4q^{82} - 12q^{85} - 4q^{88} + 4q^{91} + 4q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 5 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{6} + 8 \nu^{4} - 16 \nu^{2} + 1 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{6} + 8 \nu^{4} - 24 \nu^{2} + 9 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{5} - 24 \nu^{3} + 9 \nu \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{5} - 20 \nu^{3} + \nu \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{7} - 2 \beta_{6} + 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-2 \beta_{5} + 3 \beta_{4} + 3 \beta_{2}\)
\(\nu^{5}\)\(=\)\(3 \beta_{7} - 5 \beta_{6} + 3 \beta_{3}\)
\(\nu^{6}\)\(=\)\(8 \beta_{2} - 5\)
\(\nu^{7}\)\(=\)\(8 \beta_{3} - 13 \beta_{1}\)

Character values

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

\(n\) \(433\) \(947\)
\(\chi(n)\) \(-\beta_{5}\) \(-1\)

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
1.40126 0.809017i
−0.535233 + 0.309017i
0.535233 0.309017i
−1.40126 + 0.809017i
0.535233 + 0.309017i
−1.40126 0.809017i
1.40126 + 0.809017i
−0.535233 0.309017i
1.00000i 0 0 −0.535233 0.309017i 0 0.809017 + 1.40126i 1.00000i 0 −0.309017 + 0.535233i
350.2 1.00000i 0 0 1.40126 + 0.809017i 0 −0.309017 0.535233i 1.00000i 0 0.809017 1.40126i
350.3 1.00000i 0 0 −1.40126 0.809017i 0 −0.309017 0.535233i 1.00000i 0 0.809017 1.40126i
350.4 1.00000i 0 0 0.535233 + 0.309017i 0 0.809017 + 1.40126i 1.00000i 0 −0.309017 + 0.535233i
1025.1 1.00000i 0 0 −1.40126 + 0.809017i 0 −0.309017 + 0.535233i 1.00000i 0 0.809017 + 1.40126i
1025.2 1.00000i 0 0 0.535233 0.309017i 0 0.809017 1.40126i 1.00000i 0 −0.309017 0.535233i
1025.3 1.00000i 0 0 −0.535233 + 0.309017i 0 0.809017 1.40126i 1.00000i 0 −0.309017 0.535233i
1025.4 1.00000i 0 0 1.40126 0.809017i 0 −0.309017 + 0.535233i 1.00000i 0 0.809017 + 1.40126i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1025.4
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.c 8
3.b odd 2 1 inner 1161.1.i.c 8
9.c even 3 1 3483.1.o.c 8
9.c even 3 1 3483.1.p.c 8
9.d odd 6 1 3483.1.o.c 8
9.d odd 6 1 3483.1.p.c 8
43.c even 3 1 inner 1161.1.i.c 8
129.f odd 6 1 inner 1161.1.i.c 8
387.e even 3 1 3483.1.o.c 8
387.g even 3 1 3483.1.p.c 8
387.o odd 6 1 3483.1.p.c 8
387.p odd 6 1 3483.1.o.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.1.i.c 8 1.a even 1 1 trivial
1161.1.i.c 8 3.b odd 2 1 inner
1161.1.i.c 8 43.c even 3 1 inner
1161.1.i.c 8 129.f odd 6 1 inner
3483.1.o.c 8 9.c even 3 1
3483.1.o.c 8 9.d odd 6 1
3483.1.o.c 8 387.e even 3 1
3483.1.o.c 8 387.p odd 6 1
3483.1.p.c 8 9.c even 3 1
3483.1.p.c 8 9.d odd 6 1
3483.1.p.c 8 387.g even 3 1
3483.1.p.c 8 387.o odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ 1
$5$ \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
$7$ \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
$11$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$13$ \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
$17$ \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
$19$ \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
$23$ \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
$29$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$31$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$37$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$41$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$43$ \( ( 1 + T + T^{2} )^{4} \)
$47$ \( ( 1 - T )^{8}( 1 + T )^{8} \)
$53$ \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
$59$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$61$ \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
$67$ \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
$71$ \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
$73$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$79$ \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
$83$ \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
$89$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$97$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
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