Properties

Label 912.3.be.f
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: 6.0.92607408.1
Defining polynomial: \( x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 57)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{3} + (\beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{5} + (\beta_{5} - \beta_{4} - \beta_1 + 4) q^{7} + 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{3} + (\beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{5} + (\beta_{5} - \beta_{4} - \beta_1 + 4) q^{7} + 3 \beta_{3} q^{9} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1 + 5) q^{11} + (\beta_{5} - 3 \beta_{4} - \beta_{2} - 3 \beta_1) q^{13} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 4) q^{15} + ( - \beta_{5} + 3 \beta_{4} - 11 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 11) q^{17} + (2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} - 2) q^{19} + (2 \beta_{5} - \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{21} + (3 \beta_{4} - 20 \beta_{3} - 3 \beta_1) q^{23} + (\beta_{5} - \beta_{4} - 6 \beta_{3} + \beta_{2} + \beta_1) q^{25} + (6 \beta_{3} - 3) q^{27} + (\beta_{5} + 3 \beta_{4} + 11 \beta_{3} - \beta_{2} + 3 \beta_1 - 22) q^{29} + ( - \beta_{5} - \beta_{4} - 30 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 15) q^{31} + (3 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 5) q^{33} + (2 \beta_{5} - 5 \beta_{4} - 11 \beta_{3} - 7 \beta_{2} - 7 \beta_1 + 11) q^{35} + (4 \beta_{5} + 4 \beta_{4} - 18 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 9) q^{37} + (4 \beta_{5} - 4 \beta_{4} + \beta_{2} - 5 \beta_1) q^{39} + (2 \beta_{4} + 10 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 10) q^{41} + ( - \beta_{5} + 2 \beta_{4} + 37 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 37) q^{43} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{2} + 6) q^{45} + (2 \beta_{5} + 2 \beta_{4} + 16 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{47} + (12 \beta_{5} - 12 \beta_{4} - 3 \beta_{2} - 9 \beta_1 - 3) q^{49} + ( - 5 \beta_{5} + 7 \beta_{4} - 11 \beta_{3} + 5 \beta_{2} + 7 \beta_1 + 22) q^{51} + ( - 9 \beta_{5} + 13 \beta_{3} + 9 \beta_{2} - 26) q^{53} + (9 \beta_{5} - 7 \beta_{4} + 41 \beta_{3} - 16 \beta_{2} - 16 \beta_1 - 41) q^{55} + (7 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} - 7) q^{57} + ( - 5 \beta_{5} + 6 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{59} + ( - 5 \beta_{5} + 5 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} - 5 \beta_1) q^{61} + (3 \beta_{5} + 12 \beta_{3} + 3 \beta_{2}) q^{63} + ( - 11 \beta_{5} - 11 \beta_{4} + 8 \beta_{3} - 15 \beta_{2} - 4 \beta_1 - 4) q^{65} + (11 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} - 11 \beta_{2} - 2 \beta_1 + 16) q^{67} + (3 \beta_{5} + 3 \beta_{4} - 40 \beta_{3} - 3 \beta_{2} - 6 \beta_1 + 20) q^{69} + (17 \beta_{5} - 13 \beta_{4} - 5 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 5) q^{71} + (6 \beta_{5} - 14 \beta_{4} + 17 \beta_{3} - 20 \beta_{2} - 20 \beta_1 - 17) q^{73} + ( - 12 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 6) q^{75} + (3 \beta_{5} - 3 \beta_{4} + \beta_{2} - 4 \beta_1 + 44) q^{77} + (5 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + 11 \beta_{2} - 11 \beta_1 - 3) q^{79} + (9 \beta_{3} - 9) q^{81} + ( - 24 \beta_{5} + 24 \beta_{4} + 9 \beta_{2} + 15 \beta_1 + 47) q^{83} + (14 \beta_{5} + 20 \beta_{4} + 14 \beta_{3} + 14 \beta_{2} - 20 \beta_1) q^{85} + ( - 2 \beta_{5} + 2 \beta_{4} - 5 \beta_{2} + 7 \beta_1 - 33) q^{87} + (3 \beta_{5} + 12 \beta_{4} + 7 \beta_{3} - 3 \beta_{2} + 12 \beta_1 - 14) q^{89} + (16 \beta_{5} - 19 \beta_{4} - 49 \beta_{3} - 16 \beta_{2} - 19 \beta_1 + 98) q^{91} + (\beta_{5} - 4 \beta_{4} - 45 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 45) q^{93} + (16 \beta_{5} - 15 \beta_{4} + 75 \beta_{3} - 9 \beta_{2} - 21 \beta_1 - 49) q^{95} + (7 \beta_{5} - 7 \beta_{4} - 23 \beta_{3} - 23) q^{97} + (3 \beta_{5} + 6 \beta_{4} + 15 \beta_{3} + 3 \beta_{2} - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{3} + 4 q^{5} + 22 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 9 q^{3} + 4 q^{5} + 22 q^{7} + 9 q^{9} + 36 q^{11} - 3 q^{13} + 12 q^{15} + 38 q^{17} + 10 q^{19} + 33 q^{21} - 54 q^{23} - 21 q^{25} - 102 q^{29} + 54 q^{33} + 24 q^{35} - 6 q^{39} + 96 q^{41} - 107 q^{43} + 24 q^{45} + 50 q^{47} - 48 q^{49} + 114 q^{51} - 90 q^{53} - 148 q^{55} - 3 q^{57} + 27 q^{61} + 33 q^{63} + 39 q^{67} - 84 q^{71} - 77 q^{73} + 260 q^{77} - 9 q^{79} - 27 q^{81} + 348 q^{83} + 68 q^{85} - 204 q^{87} - 72 q^{89} + 393 q^{91} + 129 q^{93} - 104 q^{95} - 228 q^{97} + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 26\nu^{3} + 34\nu^{2} - 56\nu + 34 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 14\nu^{4} - 13\nu^{3} + 109\nu^{2} + 41\nu + 63 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 9\nu^{4} + 33\nu^{3} - 121\nu^{2} + 271\nu - 98 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - 8\beta_{2} - 6\beta _1 - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} - \beta_{3} - 3\beta_{2} - 11\beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{5} - 3\beta_{4} - 15\beta_{3} + 61\beta_{2} + 29\beta _1 + 196 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\) \(1\)

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
0.500000 + 2.93068i
0.500000 2.69511i
0.500000 + 0.630453i
0.500000 2.93068i
0.500000 + 2.69511i
0.500000 0.630453i
0 1.50000 0.866025i 0 −3.20750 5.55555i 0 2.26281 0 1.50000 2.59808i 0
145.2 0 1.50000 0.866025i 0 2.32722 + 4.03087i 0 10.6817 0 1.50000 2.59808i 0
145.3 0 1.50000 0.866025i 0 2.88028 + 4.98878i 0 −1.94451 0 1.50000 2.59808i 0
673.1 0 1.50000 + 0.866025i 0 −3.20750 + 5.55555i 0 2.26281 0 1.50000 + 2.59808i 0
673.2 0 1.50000 + 0.866025i 0 2.32722 4.03087i 0 10.6817 0 1.50000 + 2.59808i 0
673.3 0 1.50000 + 0.866025i 0 2.88028 4.98878i 0 −1.94451 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.f 6
4.b odd 2 1 57.3.g.b 6
12.b even 2 1 171.3.p.c 6
19.d odd 6 1 inner 912.3.be.f 6
76.f even 6 1 57.3.g.b 6
228.n odd 6 1 171.3.p.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.g.b 6 4.b odd 2 1
57.3.g.b 6 76.f even 6 1
171.3.p.c 6 12.b even 2 1
171.3.p.c 6 228.n odd 6 1
912.3.be.f 6 1.a even 1 1 trivial
912.3.be.f 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 \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{6} - 4T_{5}^{5} + 56T_{5}^{4} - 184T_{5}^{3} + 2288T_{5}^{2} - 6880T_{5} + 29584 \) Copy content Toggle raw display
\( T_{7}^{3} - 11T_{7}^{2} - T_{7} + 47 \) 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
$5$ \( T^{6} - 4 T^{5} + 56 T^{4} + \cdots + 29584 \) Copy content Toggle raw display
$7$ \( (T^{3} - 11 T^{2} - T + 47)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 18 T^{2} - 96 T + 1084)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} - 340 T^{4} + \cdots + 2883 \) Copy content Toggle raw display
$17$ \( T^{6} - 38 T^{5} + 1408 T^{4} + \cdots + 52186176 \) Copy content Toggle raw display
$19$ \( T^{6} - 10 T^{5} - 249 T^{4} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{6} + 54 T^{5} + 2352 T^{4} + \cdots + 21049744 \) Copy content Toggle raw display
$29$ \( T^{6} + 102 T^{5} + \cdots + 487228608 \) Copy content Toggle raw display
$31$ \( T^{6} + 2345 T^{4} + \cdots + 112326483 \) Copy content Toggle raw display
$37$ \( T^{6} + 4713 T^{4} + \cdots + 3652564347 \) Copy content Toggle raw display
$41$ \( T^{6} - 96 T^{5} + 3824 T^{4} + \cdots + 1354752 \) Copy content Toggle raw display
$43$ \( T^{6} + 107 T^{5} + \cdots + 1477402969 \) Copy content Toggle raw display
$47$ \( T^{6} - 50 T^{5} + 1976 T^{4} + \cdots + 5798464 \) Copy content Toggle raw display
$53$ \( T^{6} + 90 T^{5} + \cdots + 6327041328 \) Copy content Toggle raw display
$59$ \( T^{6} - 1048 T^{4} + \cdots + 25509168 \) Copy content Toggle raw display
$61$ \( T^{6} - 27 T^{5} + \cdots + 1215986641 \) Copy content Toggle raw display
$67$ \( T^{6} - 39 T^{5} + \cdots + 11787475467 \) Copy content Toggle raw display
$71$ \( T^{6} + 84 T^{5} + \cdots + 149905029888 \) Copy content Toggle raw display
$73$ \( T^{6} + 77 T^{5} + \cdots + 434693631969 \) Copy content Toggle raw display
$79$ \( T^{6} + 9 T^{5} + \cdots + 32898206883 \) Copy content Toggle raw display
$83$ \( (T^{3} - 174 T^{2} - 4140 T + 1174072)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 72 T^{5} + \cdots + 591631797168 \) Copy content Toggle raw display
$97$ \( T^{6} + 228 T^{5} + \cdots + 68659968 \) Copy content Toggle raw display
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