Properties

Label 912.3.be.e
Level $912$
Weight $3$
Character orbit 912.be
Analytic conductor $24.850$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.be (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 2) q^{3} + (\beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - \beta_1 - 2) q^{7} + ( - 3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 2) q^{3} + (\beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - \beta_1 - 2) q^{7} + ( - 3 \beta_{2} + 3) q^{9} + (5 \beta_{3} + 5 \beta_1 + 5) q^{11} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 2) q^{13} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{15} + (5 \beta_{5} - 5 \beta_{4} - \beta_{2} - 6 \beta_1) q^{17} + ( - 5 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 3) q^{19} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{21} + ( - 5 \beta_{5} + 5 \beta_{3} - 4 \beta_{2} + 4) q^{23} + (5 \beta_{5} - 8 \beta_{3} + 6 \beta_{2} - 6) q^{25} + ( - 6 \beta_{2} + 3) q^{27} + (5 \beta_{5} - 10 \beta_{4} - 4 \beta_{3} + 13 \beta_{2} - 8 \beta_1 + 13) q^{29} + (2 \beta_{5} - \beta_{4} - 7 \beta_{3} + 2 \beta_{2} + 7 \beta_1 - 1) q^{31} + (10 \beta_{3} - 5 \beta_{2} + 5 \beta_1 + 10) q^{33} + ( - 4 \beta_{5} + 4 \beta_{4} - 13 \beta_{2} - 3 \beta_1) q^{35} + ( - 4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 1) q^{37} + ( - 3 \beta_{4} - 6 \beta_{3} - 6 \beta_1 + 6) q^{39} + ( - 20 \beta_{3} - 2 \beta_{2} - 10 \beta_1 + 4) q^{41} + (11 \beta_{5} - 11 \beta_{4} + 11 \beta_{2} + \beta_1) q^{43} + ( - 3 \beta_{4} + 3 \beta_{3} + 3 \beta_1 + 6) q^{45} + (2 \beta_{5} - 12 \beta_{3} + 20 \beta_{2} - 20) q^{47} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 39) q^{49} + (5 \beta_{5} - 10 \beta_{4} - 6 \beta_{3} - \beta_{2} - 12 \beta_1 - 1) q^{51} + (8 \beta_{5} - 16 \beta_{4} + 7 \beta_{3} + 3 \beta_{2} + 14 \beta_1 + 3) q^{53} + (15 \beta_{5} - 15 \beta_{4} + 55 \beta_{2} + 10 \beta_1) q^{55} + ( - 8 \beta_{5} + 7 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} - \beta_1 + 1) q^{57} + (\beta_{5} + \beta_{4} + 14 \beta_{3} - 18 \beta_{2} + 7 \beta_1 + 36) q^{59} + ( - 11 \beta_{5} - 10 \beta_{2} + 10) q^{61} + ( - 3 \beta_{3} + 6 \beta_{2} - 6) q^{63} + ( - 2 \beta_{5} + \beta_{4} - 5 \beta_{3} + 8 \beta_{2} + 5 \beta_1 - 4) q^{65} + (6 \beta_{5} - 12 \beta_{4} - 23 \beta_{3} - 28 \beta_{2} - 46 \beta_1 - 28) q^{67} + ( - 10 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 8 \beta_{2} - 5 \beta_1 + 4) q^{69} + (3 \beta_{5} + 3 \beta_{4} - 8 \beta_{3} + 25 \beta_{2} - 4 \beta_1 - 50) q^{71} + ( - 2 \beta_{5} + 2 \beta_{4} + 19 \beta_{2} + 8 \beta_1) q^{73} + (10 \beta_{5} - 5 \beta_{4} - 8 \beta_{3} + 12 \beta_{2} + 8 \beta_1 - 6) q^{75} + (5 \beta_{4} - 5 \beta_{3} - 5 \beta_1 - 40) q^{77} + ( - 7 \beta_{5} - 7 \beta_{4} - 38 \beta_{3} + 21 \beta_{2} - 19 \beta_1 - 42) q^{79} - 9 \beta_{2} q^{81} + (11 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 13) q^{83} + ( - 8 \beta_{5} - 18 \beta_{3} + 34 \beta_{2} - 34) q^{85} + ( - 15 \beta_{4} - 12 \beta_{3} - 12 \beta_1 + 39) q^{87} + ( - 9 \beta_{3} + 21 \beta_{2} - 18 \beta_1 + 21) q^{89} + ( - \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - 6 \beta_1 + 5) q^{91} + (3 \beta_{5} - 3 \beta_{4} + 3 \beta_{2} + 21 \beta_1) q^{93} + ( - 10 \beta_{5} + 4 \beta_{4} + 28 \beta_{3} - 67 \beta_{2} + 13 \beta_1 + 82) q^{95} + (\beta_{5} + \beta_{4} + 48 \beta_{3} - 9 \beta_{2} + 24 \beta_1 + 18) q^{97} + (15 \beta_{3} - 15 \beta_{2} + 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{3} + 4 q^{5} - 10 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 9 q^{3} + 4 q^{5} - 10 q^{7} + 9 q^{9} + 20 q^{11} + 21 q^{13} + 12 q^{15} - 2 q^{17} + 10 q^{19} - 15 q^{21} + 2 q^{23} - 5 q^{25} + 114 q^{29} + 30 q^{33} - 32 q^{35} + 42 q^{39} + 48 q^{41} + 21 q^{43} + 24 q^{45} - 46 q^{47} - 240 q^{49} - 6 q^{51} - 18 q^{53} + 140 q^{55} - 3 q^{57} + 144 q^{59} + 19 q^{61} - 15 q^{63} - 201 q^{67} - 204 q^{71} + 51 q^{73} - 220 q^{77} - 153 q^{79} - 27 q^{81} - 52 q^{83} - 92 q^{85} + 228 q^{87} + 216 q^{89} + 57 q^{91} - 15 q^{93} + 248 q^{95} + 12 q^{97} + 30 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
1.71903 0.211943i
−1.62241 0.606458i
0.403374 + 1.68443i
1.71903 + 0.211943i
−1.62241 + 0.606458i
0.403374 1.68443i
0 1.50000 0.866025i 0 −1.41016 2.44247i 0 −2.35194 0 1.50000 2.59808i 0
145.2 0 1.50000 0.866025i 0 −0.764419 1.32401i 0 1.67282 0 1.50000 2.59808i 0
145.3 0 1.50000 0.866025i 0 4.17458 + 7.23058i 0 −4.32088 0 1.50000 2.59808i 0
673.1 0 1.50000 + 0.866025i 0 −1.41016 + 2.44247i 0 −2.35194 0 1.50000 + 2.59808i 0
673.2 0 1.50000 + 0.866025i 0 −0.764419 + 1.32401i 0 1.67282 0 1.50000 + 2.59808i 0
673.3 0 1.50000 + 0.866025i 0 4.17458 7.23058i 0 −4.32088 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.be.e 6
4.b odd 2 1 228.3.l.d 6
12.b even 2 1 684.3.y.f 6
19.d odd 6 1 inner 912.3.be.e 6
76.f even 6 1 228.3.l.d 6
228.n odd 6 1 684.3.y.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.d 6 4.b odd 2 1
228.3.l.d 6 76.f even 6 1
684.3.y.f 6 12.b even 2 1
684.3.y.f 6 228.n odd 6 1
912.3.be.e 6 1.a even 1 1 trivial
912.3.be.e 6 19.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{6} - 4T_{5}^{5} + 48T_{5}^{4} + 200T_{5}^{3} + 880T_{5}^{2} + 1152T_{5} + 1296 \) Copy content Toggle raw display
\( T_{7}^{3} + 5T_{7}^{2} - T_{7} - 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 4 T^{5} + 48 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( (T^{3} + 5 T^{2} - T - 17)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 10 T^{2} - 200 T + 1500)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 21 T^{5} + 60 T^{4} + \cdots + 70227 \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} + 672 T^{4} + \cdots + 10810944 \) Copy content Toggle raw display
$19$ \( T^{6} - 10 T^{5} + 551 T^{4} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{6} - 2 T^{5} + 936 T^{4} + \cdots + 96118416 \) Copy content Toggle raw display
$29$ \( T^{6} - 114 T^{5} + \cdots + 2525784768 \) Copy content Toggle raw display
$31$ \( T^{6} + 3177 T^{4} + \cdots + 21531123 \) Copy content Toggle raw display
$37$ \( T^{6} + 1113 T^{4} + 265851 T^{2} + \cdots + 41067 \) Copy content Toggle raw display
$41$ \( T^{6} - 48 T^{5} - 1776 T^{4} + \cdots + 42052608 \) Copy content Toggle raw display
$43$ \( T^{6} - 21 T^{5} + \cdots + 547045321 \) Copy content Toggle raw display
$47$ \( T^{6} + 46 T^{5} + \cdots + 340623936 \) Copy content Toggle raw display
$53$ \( T^{6} + 18 T^{5} + \cdots + 42880911408 \) Copy content Toggle raw display
$59$ \( T^{6} - 144 T^{5} + \cdots + 999406512 \) Copy content Toggle raw display
$61$ \( T^{6} - 19 T^{5} + \cdots + 870427009 \) Copy content Toggle raw display
$67$ \( T^{6} + 201 T^{5} + \cdots + 1309260316923 \) Copy content Toggle raw display
$71$ \( T^{6} + 204 T^{5} + \cdots + 794333952 \) Copy content Toggle raw display
$73$ \( T^{6} - 51 T^{5} + \cdots + 102556129 \) Copy content Toggle raw display
$79$ \( T^{6} + 153 T^{5} + \cdots + 372052109763 \) Copy content Toggle raw display
$83$ \( (T^{3} + 26 T^{2} - 2540 T - 19368)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 216 T^{5} + 18468 T^{4} + \cdots + 314928 \) Copy content Toggle raw display
$97$ \( T^{6} - 12 T^{5} + \cdots + 319704890112 \) Copy content Toggle raw display
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