Properties

Label 1232.2.q.k
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: 6.0.1783323.2
Defining polynomial: \(x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( -\beta_{3} + \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( -\beta_{3} + \beta_{5} ) q^{9} + ( -1 + \beta_{4} ) q^{11} + ( -4 - \beta_{2} ) q^{13} + ( 3 + 2 \beta_{2} ) q^{15} + ( 1 - \beta_{4} + \beta_{5} ) q^{17} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{19} + ( \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{21} + ( -\beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{23} + ( -2 + 3 \beta_{1} + 2 \beta_{4} ) q^{25} + ( 2 \beta_{2} + \beta_{3} ) q^{27} + ( -2 + 3 \beta_{2} - \beta_{3} ) q^{29} + ( -2 + 3 \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{31} + ( \beta_{1} + \beta_{2} ) q^{33} + ( -1 - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{35} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{37} + ( -3 + 4 \beta_{1} + 3 \beta_{4} - \beta_{5} ) q^{39} + ( -2 - \beta_{2} + 4 \beta_{3} ) q^{41} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{43} + ( 3 - 3 \beta_{4} - \beta_{5} ) q^{45} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{47} + ( -3 - 3 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{51} + ( -6 + \beta_{1} + 6 \beta_{4} - 2 \beta_{5} ) q^{53} + ( -1 - \beta_{2} - \beta_{3} ) q^{55} + ( 6 - 2 \beta_{2} + \beta_{3} ) q^{57} + ( 3 - \beta_{1} - 3 \beta_{4} + 6 \beta_{5} ) q^{59} + ( -2 \beta_{3} + 8 \beta_{4} + 2 \beta_{5} ) q^{61} + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{63} + ( -6 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - 7 \beta_{4} + 4 \beta_{5} ) q^{65} + ( -6 + 2 \beta_{1} + 6 \beta_{4} ) q^{67} + ( 3 \beta_{2} + \beta_{3} ) q^{69} + ( -1 + 4 \beta_{2} + 3 \beta_{3} ) q^{71} + ( 5 + 5 \beta_{1} - 5 \beta_{4} ) q^{73} + ( 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} - 3 \beta_{5} ) q^{75} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{77} + ( -3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{79} + ( 6 - \beta_{1} - 6 \beta_{4} + 4 \beta_{5} ) q^{81} + ( 3 - 2 \beta_{2} + \beta_{3} ) q^{83} + ( 4 + \beta_{2} ) q^{85} + ( 9 + 3 \beta_{1} - 9 \beta_{4} + 4 \beta_{5} ) q^{87} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{89} + ( 7 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} ) q^{91} + ( \beta_{1} + \beta_{2} + 4 \beta_{3} + 9 \beta_{4} - 4 \beta_{5} ) q^{93} + ( -6 + \beta_{1} + 6 \beta_{4} + 4 \beta_{5} ) q^{95} + ( 4 + 3 \beta_{2} ) q^{97} + \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{3} + 2q^{5} - 2q^{7} + O(q^{10}) \) \( 6q - q^{3} + 2q^{5} - 2q^{7} - 3q^{11} - 22q^{13} + 14q^{15} + 3q^{17} - 11q^{19} + 10q^{21} + 12q^{23} - 3q^{25} - 4q^{27} - 18q^{29} - 3q^{31} - q^{33} - 9q^{35} + 4q^{37} - 5q^{39} - 10q^{41} - 4q^{43} + 9q^{45} - 3q^{47} - 24q^{49} + 2q^{51} - 17q^{53} - 4q^{55} + 40q^{57} + 8q^{59} + 24q^{61} - 12q^{63} - 15q^{65} - 16q^{67} - 6q^{69} - 14q^{71} + 20q^{73} + 25q^{75} - 2q^{77} + 3q^{79} + 17q^{81} + 22q^{83} + 22q^{85} + 30q^{87} - q^{89} + 15q^{91} + 26q^{93} - 17q^{95} + 18q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 5 \nu^{4} + 25 \nu^{3} - 19 \nu^{2} + 12 \nu - 60 \)\()/83\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{5} - 20 \nu^{4} + 17 \nu^{3} - 76 \nu^{2} + 48 \nu - 240 \)\()/83\)
\(\beta_{4}\)\(=\)\((\)\( -20 \nu^{5} + 17 \nu^{4} - 85 \nu^{3} - 35 \nu^{2} - 323 \nu + 204 \)\()/249\)
\(\beta_{5}\)\(=\)\((\)\( -16 \nu^{5} - 3 \nu^{4} - 68 \nu^{3} - 28 \nu^{2} - 275 \nu - 36 \)\()/83\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} - \beta_{3}\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + 4 \beta_{2}\)
\(\nu^{4}\)\(=\)\(-5 \beta_{5} + 12 \beta_{4} - \beta_{1} - 12\)
\(\nu^{5}\)\(=\)\(-6 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} - 17 \beta_{2} - 17 \beta_{1}\)

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
1.09935 + 1.90412i
0.356769 + 0.617942i
−0.956115 1.65604i
1.09935 1.90412i
0.356769 0.617942i
−0.956115 + 1.65604i
0 −1.09935 1.90412i 0 0.317776 0.550404i 0 −0.317776 + 2.62660i 0 −0.917122 + 1.58850i 0
177.2 0 −0.356769 0.617942i 0 −1.10220 + 1.90907i 0 1.10220 2.40523i 0 1.24543 2.15715i 0
177.3 0 0.956115 + 1.65604i 0 1.78442 3.09071i 0 −1.78442 1.95341i 0 −0.328310 + 0.568650i 0
529.1 0 −1.09935 + 1.90412i 0 0.317776 + 0.550404i 0 −0.317776 2.62660i 0 −0.917122 1.58850i 0
529.2 0 −0.356769 + 0.617942i 0 −1.10220 1.90907i 0 1.10220 + 2.40523i 0 1.24543 + 2.15715i 0
529.3 0 0.956115 1.65604i 0 1.78442 + 3.09071i 0 −1.78442 + 1.95341i 0 −0.328310 0.568650i 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.k 6
4.b odd 2 1 77.2.e.b 6
7.c even 3 1 inner 1232.2.q.k 6
7.c even 3 1 8624.2.a.cl 3
7.d odd 6 1 8624.2.a.ck 3
12.b even 2 1 693.2.i.g 6
28.d even 2 1 539.2.e.l 6
28.f even 6 1 539.2.a.i 3
28.f even 6 1 539.2.e.l 6
28.g odd 6 1 77.2.e.b 6
28.g odd 6 1 539.2.a.h 3
44.c even 2 1 847.2.e.d 6
44.g even 10 4 847.2.n.d 24
44.h odd 10 4 847.2.n.e 24
84.j odd 6 1 4851.2.a.bn 3
84.n even 6 1 693.2.i.g 6
84.n even 6 1 4851.2.a.bo 3
308.m odd 6 1 5929.2.a.w 3
308.n even 6 1 847.2.e.d 6
308.n even 6 1 5929.2.a.v 3
308.bb odd 30 4 847.2.n.e 24
308.bc even 30 4 847.2.n.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.b 6 4.b odd 2 1
77.2.e.b 6 28.g odd 6 1
539.2.a.h 3 28.g odd 6 1
539.2.a.i 3 28.f even 6 1
539.2.e.l 6 28.d even 2 1
539.2.e.l 6 28.f even 6 1
693.2.i.g 6 12.b even 2 1
693.2.i.g 6 84.n even 6 1
847.2.e.d 6 44.c even 2 1
847.2.e.d 6 308.n even 6 1
847.2.n.d 24 44.g even 10 4
847.2.n.d 24 308.bc even 30 4
847.2.n.e 24 44.h odd 10 4
847.2.n.e 24 308.bb odd 30 4
1232.2.q.k 6 1.a even 1 1 trivial
1232.2.q.k 6 7.c even 3 1 inner
4851.2.a.bn 3 84.j odd 6 1
4851.2.a.bo 3 84.n even 6 1
5929.2.a.v 3 308.n even 6 1
5929.2.a.w 3 308.m odd 6 1
8624.2.a.ck 3 7.d odd 6 1
8624.2.a.cl 3 7.c even 3 1

Hecke kernels

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

\( T_{3}^{6} + T_{3}^{5} + 5 T_{3}^{4} + 2 T_{3}^{3} + 19 T_{3}^{2} + 12 T_{3} + 9 \)
\( T_{13}^{3} + 11 T_{13}^{2} + 36 T_{13} + 35 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 9 + 12 T + 19 T^{2} + 2 T^{3} + 5 T^{4} + T^{5} + T^{6} \)
$5$ \( 25 - 35 T + 59 T^{2} + 4 T^{3} + 11 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( 343 + 98 T + 98 T^{2} + 23 T^{3} + 14 T^{4} + 2 T^{5} + T^{6} \)
$11$ \( ( 1 + T + T^{2} )^{3} \)
$13$ \( ( 35 + 36 T + 11 T^{2} + T^{3} )^{2} \)
$17$ \( 49 - 14 T + 25 T^{2} - 8 T^{3} + 11 T^{4} - 3 T^{5} + T^{6} \)
$19$ \( 3249 - 1140 T + 1027 T^{2} + 334 T^{3} + 101 T^{4} + 11 T^{5} + T^{6} \)
$23$ \( 2209 - 2021 T + 1285 T^{2} - 422 T^{3} + 101 T^{4} - 12 T^{5} + T^{6} \)
$29$ \( ( -53 - 20 T + 9 T^{2} + T^{3} )^{2} \)
$31$ \( 11449 + 4708 T + 2257 T^{2} + 82 T^{3} + 53 T^{4} + 3 T^{5} + T^{6} \)
$37$ \( 23104 - 5472 T + 1904 T^{2} - 160 T^{3} + 52 T^{4} - 4 T^{5} + T^{6} \)
$41$ \( ( -109 - 80 T + 5 T^{2} + T^{3} )^{2} \)
$43$ \( ( -41 - 25 T + 2 T^{2} + T^{3} )^{2} \)
$47$ \( 49 + 14 T + 25 T^{2} + 8 T^{3} + 11 T^{4} + 3 T^{5} + T^{6} \)
$53$ \( 441 + 1554 T + 5119 T^{2} + 1216 T^{3} + 215 T^{4} + 17 T^{5} + T^{6} \)
$59$ \( 1750329 - 207711 T + 35233 T^{2} - 1390 T^{3} + 221 T^{4} - 8 T^{5} + T^{6} \)
$61$ \( 141376 - 64672 T + 20560 T^{2} - 3376 T^{3} + 404 T^{4} - 24 T^{5} + T^{6} \)
$67$ \( 5184 + 4896 T + 3472 T^{2} + 944 T^{3} + 188 T^{4} + 16 T^{5} + T^{6} \)
$71$ \( ( -419 - 86 T + 7 T^{2} + T^{3} )^{2} \)
$73$ \( 390625 + 15625 T + 13125 T^{2} - 1750 T^{3} + 375 T^{4} - 20 T^{5} + T^{6} \)
$79$ \( 19881 - 5358 T + 1867 T^{2} - 168 T^{3} + 47 T^{4} - 3 T^{5} + T^{6} \)
$83$ \( ( 3 + 16 T - 11 T^{2} + T^{3} )^{2} \)
$89$ \( 9 + 24 T + 67 T^{2} - 2 T^{3} + 9 T^{4} + T^{5} + T^{6} \)
$97$ \( ( 47 - 12 T - 9 T^{2} + T^{3} )^{2} \)
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