Properties

Label 1232.2.q.l
Level $1232$
Weight $2$
Character orbit 1232.q
Analytic conductor $9.838$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 308)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} + ( -1 + \beta_{4} ) q^{11} + ( -2 \beta_{1} - \beta_{3} ) q^{13} + ( 5 + 2 \beta_{1} - 2 \beta_{3} ) q^{15} + ( 2 - 2 \beta_{4} + \beta_{5} ) q^{17} + ( \beta_{1} + 4 \beta_{4} - \beta_{5} ) q^{19} + ( 1 + \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{21} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{23} + ( -4 - \beta_{2} + 4 \beta_{4} - 2 \beta_{5} ) q^{25} + ( -7 - \beta_{1} + 2 \beta_{3} ) q^{27} + ( 1 + 3 \beta_{1} + \beta_{3} ) q^{29} + ( 7 + \beta_{2} - 7 \beta_{4} + 3 \beta_{5} ) q^{31} + ( -\beta_{2} - \beta_{3} ) q^{33} + ( 1 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{35} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{37} + ( 6 + 2 \beta_{2} - 6 \beta_{4} + 3 \beta_{5} ) q^{39} + ( 6 + 2 \beta_{1} - \beta_{3} ) q^{41} + ( -3 - 2 \beta_{1} + 3 \beta_{3} ) q^{43} + ( 6 + 6 \beta_{2} - 6 \beta_{4} + 3 \beta_{5} ) q^{45} + ( -\beta_{1} + 4 \beta_{4} + \beta_{5} ) q^{47} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{49} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{51} + ( -2 + \beta_{2} + 2 \beta_{4} + 4 \beta_{5} ) q^{53} + ( \beta_{1} + \beta_{3} ) q^{55} + ( -1 - \beta_{1} - 4 \beta_{3} ) q^{57} + ( 3 - 3 \beta_{2} - 3 \beta_{4} ) q^{59} + ( 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{61} + ( -8 - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{63} + ( 2 \beta_{1} + 13 \beta_{4} - 2 \beta_{5} ) q^{65} + ( 2 - 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} ) q^{67} + ( -1 - \beta_{1} + \beta_{3} ) q^{69} + ( 4 - \beta_{1} ) q^{71} + ( -7 - \beta_{2} + 7 \beta_{4} ) q^{73} + ( -\beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{75} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{77} + ( -5 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{79} + ( -4 - 5 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{81} + ( -2 + \beta_{1} - 4 \beta_{3} ) q^{83} + ( -4 - 2 \beta_{1} - 3 \beta_{3} ) q^{85} + ( -7 - \beta_{2} + 7 \beta_{4} - 4 \beta_{5} ) q^{87} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{89} + ( 6 + \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{91} + ( 2 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{93} + ( 4 - 5 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} ) q^{95} + ( 10 + 6 \beta_{1} - \beta_{3} ) q^{97} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + 2 q^{5} - 4 q^{7} - 4 q^{9} + O(q^{10}) \) \( 6 q + q^{3} + 2 q^{5} - 4 q^{7} - 4 q^{9} - 3 q^{11} + 6 q^{13} + 30 q^{15} + 5 q^{17} + 11 q^{19} - 10 q^{21} - 4 q^{23} - 11 q^{25} - 44 q^{27} - 2 q^{29} + 19 q^{31} + q^{33} + 17 q^{35} + 8 q^{37} + 17 q^{39} + 34 q^{41} - 20 q^{43} + 21 q^{45} + 13 q^{47} - 12 q^{49} - 9 q^{53} - 4 q^{55} + 4 q^{57} + 6 q^{59} - 4 q^{61} - 50 q^{63} + 37 q^{65} + 8 q^{67} - 6 q^{69} + 26 q^{71} - 22 q^{73} + 13 q^{75} + 2 q^{77} + 5 q^{79} - 19 q^{81} - 6 q^{83} - 14 q^{85} - 18 q^{87} + 3 q^{89} + 31 q^{91} - 14 q^{93} + 3 q^{95} + 50 q^{97} + 8 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 2 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47\)\()/3\)

Character values

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

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.500000 + 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 0.224437i
0.500000 2.05195i
0.500000 + 1.41036i
0 −0.794182 1.37556i 0 −1.64400 + 2.84748i 0 −2.64400 + 0.0963576i 0 0.238550 0.413181i 0
177.2 0 −0.296790 0.514055i 0 0.933463 1.61680i 0 −0.0665372 2.64491i 0 1.32383 2.29294i 0
177.3 0 1.59097 + 2.75564i 0 1.71053 2.96273i 0 0.710533 + 2.54856i 0 −3.56238 + 6.17023i 0
529.1 0 −0.794182 + 1.37556i 0 −1.64400 2.84748i 0 −2.64400 0.0963576i 0 0.238550 + 0.413181i 0
529.2 0 −0.296790 + 0.514055i 0 0.933463 + 1.61680i 0 −0.0665372 + 2.64491i 0 1.32383 + 2.29294i 0
529.3 0 1.59097 2.75564i 0 1.71053 + 2.96273i 0 0.710533 2.54856i 0 −3.56238 6.17023i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.q.l 6
4.b odd 2 1 308.2.i.a 6
7.c even 3 1 inner 1232.2.q.l 6
7.c even 3 1 8624.2.a.ci 3
7.d odd 6 1 8624.2.a.cn 3
12.b even 2 1 2772.2.s.f 6
28.d even 2 1 2156.2.i.l 6
28.f even 6 1 2156.2.a.h 3
28.f even 6 1 2156.2.i.l 6
28.g odd 6 1 308.2.i.a 6
28.g odd 6 1 2156.2.a.i 3
84.n even 6 1 2772.2.s.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.i.a 6 4.b odd 2 1
308.2.i.a 6 28.g odd 6 1
1232.2.q.l 6 1.a even 1 1 trivial
1232.2.q.l 6 7.c even 3 1 inner
2156.2.a.h 3 28.f even 6 1
2156.2.a.i 3 28.g odd 6 1
2156.2.i.l 6 28.d even 2 1
2156.2.i.l 6 28.f even 6 1
2772.2.s.f 6 12.b even 2 1
2772.2.s.f 6 84.n even 6 1
8624.2.a.ci 3 7.c even 3 1
8624.2.a.cn 3 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1232, [\chi])\):

\( T_{3}^{6} - T_{3}^{5} + 7 T_{3}^{4} + 12 T_{3}^{3} + 33 T_{3}^{2} + 18 T_{3} + 9 \)
\( T_{13}^{3} - 3 T_{13}^{2} - 24 T_{13} + 79 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 9 + 18 T + 33 T^{2} + 12 T^{3} + 7 T^{4} - T^{5} + T^{6} \)
$5$ \( 441 - 231 T + 163 T^{2} - 20 T^{3} + 15 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( 343 + 196 T + 98 T^{2} + 55 T^{3} + 14 T^{4} + 4 T^{5} + T^{6} \)
$11$ \( ( 1 + T + T^{2} )^{3} \)
$13$ \( ( 79 - 24 T - 3 T^{2} + T^{3} )^{2} \)
$17$ \( 9 + 12 T + 31 T^{2} - 26 T^{3} + 21 T^{4} - 5 T^{5} + T^{6} \)
$19$ \( 1089 - 1188 T + 933 T^{2} - 330 T^{3} + 85 T^{4} - 11 T^{5} + T^{6} \)
$23$ \( 9 - 3 T + 13 T^{2} + 10 T^{3} + 15 T^{4} + 4 T^{5} + T^{6} \)
$29$ \( ( -129 - 50 T + T^{2} + T^{3} )^{2} \)
$31$ \( 9409 + 7760 T + 8243 T^{2} - 1714 T^{3} + 281 T^{4} - 19 T^{5} + T^{6} \)
$37$ \( 64 - 224 T + 848 T^{2} + 208 T^{3} + 92 T^{4} - 8 T^{5} + T^{6} \)
$41$ \( ( -33 + 76 T - 17 T^{2} + T^{3} )^{2} \)
$43$ \( ( -73 - 31 T + 10 T^{2} + T^{3} )^{2} \)
$47$ \( 3969 - 3276 T + 1885 T^{2} - 550 T^{3} + 117 T^{4} - 13 T^{5} + T^{6} \)
$53$ \( 81 - 378 T + 1683 T^{2} - 396 T^{3} + 123 T^{4} + 9 T^{5} + T^{6} \)
$59$ \( 59049 - 10935 T + 3483 T^{2} - 216 T^{3} + 81 T^{4} - 6 T^{5} + T^{6} \)
$61$ \( 222784 + 47200 T + 11888 T^{2} + 544 T^{3} + 116 T^{4} + 4 T^{5} + T^{6} \)
$67$ \( 5184 + 4320 T + 3024 T^{2} + 624 T^{3} + 124 T^{4} - 8 T^{5} + T^{6} \)
$71$ \( ( -63 + 52 T - 13 T^{2} + T^{3} )^{2} \)
$73$ \( 124609 + 54715 T + 16259 T^{2} + 2704 T^{3} + 329 T^{4} + 22 T^{5} + T^{6} \)
$79$ \( 91809 + 39996 T + 15909 T^{2} + 1266 T^{3} + 157 T^{4} - 5 T^{5} + T^{6} \)
$83$ \( ( -477 - 96 T + 3 T^{2} + T^{3} )^{2} \)
$89$ \( 59049 + 26244 T + 10935 T^{2} + 810 T^{3} + 117 T^{4} - 3 T^{5} + T^{6} \)
$97$ \( ( 1171 + 56 T - 25 T^{2} + T^{3} )^{2} \)
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