Properties

Label 912.2.q.l
Level $912$
Weight $2$
Character orbit 912.q
Analytic conductor $7.282$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 57)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{3} - \beta_1) q^{7} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{3} - \beta_1) q^{7} - \beta_{2} q^{9} + ( - \beta_{4} - \beta_{3} - \beta_1) q^{11} + ( - \beta_{5} + \beta_{4}) q^{13} + (\beta_{2} + \beta_1) q^{15} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{19} - \beta_{3} q^{21} + ( - \beta_{5} + \beta_{4} + 4 \beta_{2} - \beta_1) q^{23} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{25} - q^{27} + ( - 2 \beta_{2} - 2 \beta_1) q^{29} + (\beta_{4} + \beta_{3} + \beta_1 - 5) q^{31} + ( - \beta_{5} - \beta_{3}) q^{33} + ( - \beta_{5} - \beta_{3} - 6 \beta_{2} + 6) q^{35} - q^{37} + \beta_{4} q^{39} + (2 \beta_{3} - 2 \beta_{2} + 2) q^{41} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{43} + (\beta_{3} + \beta_1 + 1) q^{45} - 6 \beta_{2} q^{47} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{49} + (3 \beta_{2} + 3 \beta_1) q^{53} + (\beta_{5} - 2 \beta_{3} - 3 \beta_{2} + 3) q^{55} + (\beta_{4} + \beta_{3} - 1) q^{57} + ( - \beta_{5} - \beta_{3}) q^{59} + ( - \beta_{5} + \beta_{4} - 4 \beta_{2} + 2 \beta_1) q^{61} + \beta_1 q^{63} + ( - 2 \beta_{4} + \beta_{3} + \beta_1 + 3) q^{65} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - \beta_1) q^{67} + (\beta_{4} - \beta_{3} - \beta_1 + 4) q^{69} + (2 \beta_{5} + 2 \beta_{3} + 6 \beta_{2} - 6) q^{71} + ( - 2 \beta_{3} + 7 \beta_{2} - 7) q^{73} + (\beta_{4} - 2) q^{75} + ( - 3 \beta_{3} - 3 \beta_1 + 3) q^{77} + (\beta_{5} - \beta_{3} - 3 \beta_{2} + 3) q^{79} + (\beta_{2} - 1) q^{81} + ( - 2 \beta_{4} + 2) q^{83} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{87} + (5 \beta_{2} - \beta_1) q^{89} + (\beta_{5} - \beta_{4} + 3 \beta_{2} + \beta_1) q^{91} + (\beta_{5} + \beta_{3} + 5 \beta_{2} - 5) q^{93} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 2) q^{95} + (3 \beta_{5} - 2 \beta_{3} + \beta_{2} - 1) q^{97} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 2 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 2 q^{5} + 2 q^{7} - 3 q^{9} + q^{13} + 2 q^{15} - 4 q^{19} + q^{21} + 14 q^{23} - 5 q^{25} - 6 q^{27} - 4 q^{29} - 30 q^{31} + 18 q^{35} - 6 q^{37} + 2 q^{39} + 4 q^{41} - 3 q^{43} + 4 q^{45} - 18 q^{47} - 4 q^{49} + 6 q^{53} + 12 q^{55} - 5 q^{57} - 13 q^{61} - q^{63} + 12 q^{65} + 9 q^{67} + 28 q^{69} - 18 q^{71} - 19 q^{73} - 10 q^{75} + 24 q^{77} + 11 q^{79} - 3 q^{81} + 8 q^{83} - 8 q^{87} + 16 q^{89} + 7 q^{91} - 15 q^{93} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} - 4\nu^{4} + \nu^{3} - 9\nu^{2} + 21\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} + 8\nu^{4} - 2\nu^{3} - 9\nu^{2} + 12\nu - 18 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - 2\nu^{4} + 14\nu^{3} + 18\nu^{2} + 24\nu + 45 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\nu^{5} + 5\nu^{4} - 8\nu^{3} + 36\nu^{2} - 6\nu - 153 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{3} + 3\beta_{2} + 4\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} + 3\beta_{2} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 7\beta_{4} + 2\beta_{3} - 24\beta_{2} + 4\beta _1 + 9 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} + \beta_{4} + 8\beta_{3} - 6\beta_{2} - 2\beta _1 + 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{5} - 2\beta_{4} + 2\beta_{3} - 33\beta_{2} + 22\beta _1 + 81 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.403374 + 1.68443i
1.71903 0.211943i
−1.62241 0.606458i
0.403374 1.68443i
1.71903 + 0.211943i
−1.62241 + 0.606458i
0 0.500000 0.866025i 0 −1.66044 + 2.87597i 0 −2.32088 0 −0.500000 0.866025i 0
49.2 0 0.500000 0.866025i 0 −0.675970 + 1.17081i 0 −0.351939 0 −0.500000 0.866025i 0
49.3 0 0.500000 0.866025i 0 1.33641 2.31473i 0 3.67282 0 −0.500000 0.866025i 0
577.1 0 0.500000 + 0.866025i 0 −1.66044 2.87597i 0 −2.32088 0 −0.500000 + 0.866025i 0
577.2 0 0.500000 + 0.866025i 0 −0.675970 1.17081i 0 −0.351939 0 −0.500000 + 0.866025i 0
577.3 0 0.500000 + 0.866025i 0 1.33641 + 2.31473i 0 3.67282 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.q.l 6
3.b odd 2 1 2736.2.s.z 6
4.b odd 2 1 57.2.e.b 6
12.b even 2 1 171.2.f.b 6
19.c even 3 1 inner 912.2.q.l 6
57.h odd 6 1 2736.2.s.z 6
76.f even 6 1 1083.2.a.o 3
76.g odd 6 1 57.2.e.b 6
76.g odd 6 1 1083.2.a.l 3
228.m even 6 1 171.2.f.b 6
228.m even 6 1 3249.2.a.y 3
228.n odd 6 1 3249.2.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.b 6 4.b odd 2 1
57.2.e.b 6 76.g odd 6 1
171.2.f.b 6 12.b even 2 1
171.2.f.b 6 228.m even 6 1
912.2.q.l 6 1.a even 1 1 trivial
912.2.q.l 6 19.c even 3 1 inner
1083.2.a.l 3 76.g odd 6 1
1083.2.a.o 3 76.f even 6 1
2736.2.s.z 6 3.b odd 2 1
2736.2.s.z 6 57.h odd 6 1
3249.2.a.t 3 228.n odd 6 1
3249.2.a.y 3 228.m even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{6} + 2T_{5}^{5} + 12T_{5}^{4} + 8T_{5}^{3} + 88T_{5}^{2} + 96T_{5} + 144 \) Copy content Toggle raw display
\( T_{7}^{3} - T_{7}^{2} - 9T_{7} - 3 \) Copy content Toggle raw display
\( T_{13}^{6} - T_{13}^{5} + 22T_{13}^{4} + 27T_{13}^{3} + 438T_{13}^{2} + 63T_{13} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + 12 T^{4} + 8 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( (T^{3} - T^{2} - 9 T - 3)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 24 T + 36)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - T^{5} + 22 T^{4} + 27 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 4 T^{5} + 17 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 14 T^{5} + 168 T^{4} + \cdots + 24336 \) Copy content Toggle raw display
$29$ \( T^{6} + 4 T^{5} + 48 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$31$ \( (T^{3} + 15 T^{2} + 51 T - 31)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} - 4 T^{5} + 48 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + 30 T^{4} - 89 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 36)^{3} \) Copy content Toggle raw display
$53$ \( T^{6} - 6 T^{5} + 108 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$59$ \( T^{6} + 24 T^{4} - 72 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$61$ \( T^{6} + 13 T^{5} + 158 T^{4} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{6} - 9 T^{5} + 162 T^{4} + \cdots + 292681 \) Copy content Toggle raw display
$71$ \( T^{6} + 18 T^{5} + 312 T^{4} + \cdots + 419904 \) Copy content Toggle raw display
$73$ \( T^{6} + 19 T^{5} + 278 T^{4} + \cdots + 961 \) Copy content Toggle raw display
$79$ \( T^{6} - 11 T^{5} + 118 T^{4} + \cdots + 29241 \) Copy content Toggle raw display
$83$ \( (T^{3} - 4 T^{2} - 80 T + 192)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 16 T^{5} + 180 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$97$ \( T^{6} - 2 T^{5} + 272 T^{4} + \cdots + 2096704 \) Copy content Toggle raw display
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