Properties

Label 1232.1.cd.a
Level $1232$
Weight $1$
Character orbit 1232.cd
Analytic conductor $0.615$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1232.cd (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.717409.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{10}^{3} q^{7} + \zeta_{10}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{10}^{3} q^{7} + \zeta_{10}^{4} q^{9} + \zeta_{10} q^{11} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{23} + \zeta_{10}^{2} q^{25} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{29} + ( 1 - \zeta_{10} ) q^{37} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{43} -\zeta_{10} q^{49} + ( 1 - \zeta_{10}^{3} ) q^{53} -\zeta_{10}^{2} q^{63} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{67} + ( -1 - \zeta_{10}^{2} ) q^{71} + \zeta_{10}^{4} q^{77} + ( -1 + \zeta_{10}^{3} ) q^{79} -\zeta_{10}^{3} q^{81} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{7} - q^{9} + O(q^{10}) \) \( 4 q + q^{7} - q^{9} + q^{11} + 2 q^{23} - q^{25} - 2 q^{29} + 3 q^{37} + 2 q^{43} - q^{49} + 3 q^{53} + q^{63} + 2 q^{67} - 3 q^{71} - q^{77} - 3 q^{79} - q^{81} - 4 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0 0 0 0 0 0.809017 + 0.587785i 0 0.309017 0.951057i 0
433.1 0 0 0 0 0 −0.309017 0.951057i 0 −0.809017 0.587785i 0
993.1 0 0 0 0 0 −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0
1105.1 0 0 0 0 0 0.809017 0.587785i 0 0.309017 + 0.951057i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
11.c even 5 1 inner
77.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.1.cd.a 4
4.b odd 2 1 77.1.j.a 4
7.b odd 2 1 CM 1232.1.cd.a 4
11.c even 5 1 inner 1232.1.cd.a 4
12.b even 2 1 693.1.br.a 4
20.d odd 2 1 1925.1.bn.a 4
20.e even 4 2 1925.1.cb.a 8
28.d even 2 1 77.1.j.a 4
28.f even 6 2 539.1.u.a 8
28.g odd 6 2 539.1.u.a 8
44.c even 2 1 847.1.j.b 4
44.g even 10 1 847.1.d.b 2
44.g even 10 2 847.1.j.a 4
44.g even 10 1 847.1.j.b 4
44.h odd 10 1 77.1.j.a 4
44.h odd 10 1 847.1.d.a 2
44.h odd 10 2 847.1.j.c 4
77.j odd 10 1 inner 1232.1.cd.a 4
84.h odd 2 1 693.1.br.a 4
132.o even 10 1 693.1.br.a 4
140.c even 2 1 1925.1.bn.a 4
140.j odd 4 2 1925.1.cb.a 8
220.n odd 10 1 1925.1.bn.a 4
220.v even 20 2 1925.1.cb.a 8
308.g odd 2 1 847.1.j.b 4
308.s odd 10 1 847.1.d.b 2
308.s odd 10 2 847.1.j.a 4
308.s odd 10 1 847.1.j.b 4
308.t even 10 1 77.1.j.a 4
308.t even 10 1 847.1.d.a 2
308.t even 10 2 847.1.j.c 4
308.bb odd 30 2 539.1.u.a 8
308.be even 30 2 539.1.u.a 8
924.bk odd 10 1 693.1.br.a 4
1540.bw even 10 1 1925.1.bn.a 4
1540.ct odd 20 2 1925.1.cb.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.1.j.a 4 4.b odd 2 1
77.1.j.a 4 28.d even 2 1
77.1.j.a 4 44.h odd 10 1
77.1.j.a 4 308.t even 10 1
539.1.u.a 8 28.f even 6 2
539.1.u.a 8 28.g odd 6 2
539.1.u.a 8 308.bb odd 30 2
539.1.u.a 8 308.be even 30 2
693.1.br.a 4 12.b even 2 1
693.1.br.a 4 84.h odd 2 1
693.1.br.a 4 132.o even 10 1
693.1.br.a 4 924.bk odd 10 1
847.1.d.a 2 44.h odd 10 1
847.1.d.a 2 308.t even 10 1
847.1.d.b 2 44.g even 10 1
847.1.d.b 2 308.s odd 10 1
847.1.j.a 4 44.g even 10 2
847.1.j.a 4 308.s odd 10 2
847.1.j.b 4 44.c even 2 1
847.1.j.b 4 44.g even 10 1
847.1.j.b 4 308.g odd 2 1
847.1.j.b 4 308.s odd 10 1
847.1.j.c 4 44.h odd 10 2
847.1.j.c 4 308.t even 10 2
1232.1.cd.a 4 1.a even 1 1 trivial
1232.1.cd.a 4 7.b odd 2 1 CM
1232.1.cd.a 4 11.c even 5 1 inner
1232.1.cd.a 4 77.j odd 10 1 inner
1925.1.bn.a 4 20.d odd 2 1
1925.1.bn.a 4 140.c even 2 1
1925.1.bn.a 4 220.n odd 10 1
1925.1.bn.a 4 1540.bw even 10 1
1925.1.cb.a 8 20.e even 4 2
1925.1.cb.a 8 140.j odd 4 2
1925.1.cb.a 8 220.v even 20 2
1925.1.cb.a 8 1540.ct odd 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1232, [\chi])\).

Hecke characteristic polynomials

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