Properties

Label 912.3.be.d
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: 6.0.6967728.1
Defining polynomial: \( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 2) q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_{5} + \beta_{4} - \beta_1 - 4) q^{7} + (3 \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 2) q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_{5} + \beta_{4} - \beta_1 - 4) q^{7} + (3 \beta_{3} + 3) q^{9} + (3 \beta_1 - 1) q^{11} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{13} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{15} + ( - 2 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 2) q^{17} + ( - 2 \beta_{5} - 16 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{19} + ( - 3 \beta_{5} + 5 \beta_{3} + \beta_{2} + 2 \beta_1 + 7) q^{21} + (4 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 11) q^{23} + (4 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 7) q^{25} + ( - 6 \beta_{3} - 3) q^{27} + ( - 4 \beta_{4} - 8 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 8) q^{29} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 2) q^{31} + (\beta_{3} - 3 \beta_{2} - 6 \beta_1 + 2) q^{33} + ( - 5 \beta_{3} + 9 \beta_{2}) q^{35} + (9 \beta_{5} - 9 \beta_{4} - 24 \beta_{3} - 3) q^{37} + (\beta_{5} + \beta_{4} - 6 \beta_1 + 7) q^{39} + (8 \beta_{5} + 2 \beta_{3} - 6 \beta_{2} - 12 \beta_1 + 12) q^{41} + (4 \beta_{3} - \beta_{2}) q^{43} + (3 \beta_1 - 3) q^{45} + ( - 12 \beta_{5} + 6 \beta_{4} + 16 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 4) q^{47} + ( - 4 \beta_{5} - 4 \beta_{4} + 6 \beta_1 + 14) q^{49} + ( - 6 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 4) q^{51} + (12 \beta_{4} - 17 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 17) q^{53} + (6 \beta_{5} - 12 \beta_{4} - 64 \beta_{3} - 2 \beta_{2} + 6) q^{55} + (4 \beta_{5} - 2 \beta_{4} + 16 \beta_{3} - 5 \beta_{2} - 4 \beta_1 - 12) q^{57} + (14 \beta_{5} - 21 \beta_{3} - 5 \beta_{2} - 10 \beta_1 - 28) q^{59} + ( - 10 \beta_{5} + 5 \beta_{4} - 38 \beta_{3} - 12 \beta_{2} - 12 \beta_1 - 48) q^{61} + (6 \beta_{5} - 3 \beta_{4} - 15 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 9) q^{63} + (10 \beta_{5} - 10 \beta_{4} - 74 \beta_{3} - 2 \beta_{2} - \beta_1 - 27) q^{65} + (8 \beta_{4} - 4 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{67} + ( - 6 \beta_{5} + 6 \beta_{4} - 14 \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 13) q^{69} + (6 \beta_{5} + 40 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 86) q^{71} + ( - 10 \beta_{5} + 20 \beta_{4} - 31 \beta_{3} + 20 \beta_{2} - 10) q^{73} + ( - 6 \beta_{5} + 6 \beta_{4} - 6 \beta_{3} - 9) q^{75} + (2 \beta_{5} + 2 \beta_{4} - 29 \beta_1 + 7) q^{77} + (7 \beta_{5} + 11 \beta_{3} + \beta_{2} + 2 \beta_1 + 29) q^{79} + 9 \beta_{3} q^{81} + ( - 2 \beta_{5} - 2 \beta_{4} + 42) q^{83} + ( - 8 \beta_{5} + 4 \beta_{4} + 38 \beta_{3} + 14 \beta_{2} + 14 \beta_1 + 30) q^{85} + (4 \beta_{5} + 4 \beta_{4} - 6 \beta_1 - 20) q^{87} + ( - 71 \beta_{3} - \beta_{2} + \beta_1 + 71) q^{89} + ( - 2 \beta_{4} + 4 \beta_{3} + 15 \beta_{2} - 15 \beta_1 - 4) q^{91} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 9 \beta_{2} + 1) q^{93} + ( - 2 \beta_{5} - 4 \beta_{4} + 9 \beta_{3} - 19 \beta_{2} - 12 \beta_1 + 42) q^{95} + (34 \beta_{5} - 10 \beta_{3} - 8 \beta_{2} - 16 \beta_1 + 14) q^{97} + ( - 3 \beta_{3} + 9 \beta_{2} + 9 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} - 2 q^{5} - 26 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 9 q^{3} - 2 q^{5} - 26 q^{7} + 9 q^{9} - 15 q^{13} + 6 q^{15} - 10 q^{17} + 46 q^{19} + 39 q^{21} + 24 q^{23} + 15 q^{25} + 66 q^{29} + 6 q^{35} + 30 q^{39} + 24 q^{41} - 11 q^{43} - 12 q^{45} + 26 q^{47} + 96 q^{49} + 30 q^{51} + 180 q^{53} + 176 q^{55} - 141 q^{57} - 162 q^{59} - 141 q^{61} - 39 q^{63} + 63 q^{67} + 372 q^{71} + 103 q^{73} - 16 q^{77} + 123 q^{79} - 27 q^{81} + 252 q^{83} + 116 q^{85} - 132 q^{87} + 642 q^{89} - 87 q^{91} - 21 q^{93} + 214 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} - 16\nu^{4} + 128\nu^{3} - 100\nu^{2} + 14\nu + 281 ) / 393 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -56\nu^{5} + 55\nu^{4} - 440\nu^{3} - 344\nu^{2} - 1964\nu - 8 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -56\nu^{5} + 55\nu^{4} - 440\nu^{3} - 344\nu^{2} - 2750\nu - 8 ) / 393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 196\nu^{5} - 127\nu^{4} + 1540\nu^{3} + 1466\nu^{2} + 10018\nu + 1469 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -210\nu^{5} + 239\nu^{4} - 1650\nu^{3} - 766\nu^{2} - 10116\nu + 2459 ) / 393 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} - \beta_{4} - 11\beta_{3} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} + 7\beta _1 - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{5} + 8\beta_{4} + 46\beta_{3} - 3\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -28\beta_{5} + 14\beta_{4} + 193\beta_{3} - 55\beta_{2} - 55\beta _1 + 165 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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
1.56632 + 2.71294i
0.0702177 + 0.121621i
−1.13654 1.96854i
1.56632 2.71294i
0.0702177 0.121621i
−1.13654 + 1.96854i
0 −1.50000 + 0.866025i 0 −3.13264 5.42589i 0 −8.36156 0 1.50000 2.59808i 0
145.2 0 −1.50000 + 0.866025i 0 −0.140435 0.243241i 0 5.24143 0 1.50000 2.59808i 0
145.3 0 −1.50000 + 0.866025i 0 2.27307 + 3.93708i 0 −9.87987 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 −3.13264 + 5.42589i 0 −8.36156 0 1.50000 + 2.59808i 0
673.2 0 −1.50000 0.866025i 0 −0.140435 + 0.243241i 0 5.24143 0 1.50000 + 2.59808i 0
673.3 0 −1.50000 0.866025i 0 2.27307 3.93708i 0 −9.87987 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.d 6
4.b odd 2 1 57.3.g.a 6
12.b even 2 1 171.3.p.e 6
19.d odd 6 1 inner 912.3.be.d 6
76.f even 6 1 57.3.g.a 6
228.n odd 6 1 171.3.p.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.g.a 6 4.b odd 2 1
57.3.g.a 6 76.f even 6 1
171.3.p.e 6 12.b even 2 1
171.3.p.e 6 228.n odd 6 1
912.3.be.d 6 1.a even 1 1 trivial
912.3.be.d 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 \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{6} + 2T_{5}^{5} + 32T_{5}^{4} - 40T_{5}^{3} + 800T_{5}^{2} + 224T_{5} + 64 \) Copy content Toggle raw display
\( T_{7}^{3} + 13T_{7}^{2} - 13T_{7} - 433 \) 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
$5$ \( T^{6} + 2 T^{5} + 32 T^{4} - 40 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{3} + 13 T^{2} - 13 T - 433)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 264 T + 304)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 15 T^{5} - 172 T^{4} + \cdots + 3518667 \) Copy content Toggle raw display
$17$ \( T^{6} + 10 T^{5} + 472 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$19$ \( T^{6} - 46 T^{5} + 1383 T^{4} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{6} - 24 T^{5} + 696 T^{4} + \cdots + 1517824 \) Copy content Toggle raw display
$29$ \( T^{6} - 66 T^{5} + 1520 T^{4} + \cdots + 19293888 \) Copy content Toggle raw display
$31$ \( T^{6} + 2033 T^{4} + \cdots + 124768803 \) Copy content Toggle raw display
$37$ \( T^{6} + 5913 T^{4} + \cdots + 1689765867 \) Copy content Toggle raw display
$41$ \( T^{6} - 24 T^{5} + \cdots + 1907539968 \) Copy content Toggle raw display
$43$ \( T^{6} + 11 T^{5} + 110 T^{4} + \cdots + 2209 \) Copy content Toggle raw display
$47$ \( T^{6} - 26 T^{5} + \cdots + 2266521664 \) Copy content Toggle raw display
$53$ \( T^{6} - 180 T^{5} + 11016 T^{4} + \cdots + 2495232 \) Copy content Toggle raw display
$59$ \( T^{6} + 162 T^{5} + \cdots + 48350430912 \) Copy content Toggle raw display
$61$ \( T^{6} + 141 T^{5} + \cdots + 558907255201 \) Copy content Toggle raw display
$67$ \( T^{6} - 63 T^{5} + \cdots + 2349816507 \) Copy content Toggle raw display
$71$ \( T^{6} - 372 T^{5} + \cdots + 100019515392 \) Copy content Toggle raw display
$73$ \( T^{6} - 103 T^{5} + \cdots + 214090364601 \) Copy content Toggle raw display
$79$ \( T^{6} - 123 T^{5} + \cdots + 2549342403 \) Copy content Toggle raw display
$83$ \( (T^{3} - 126 T^{2} + 4908 T - 57704)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 642 T^{5} + \cdots + 3516172210368 \) Copy content Toggle raw display
$97$ \( T^{6} + 12 T^{5} + \cdots + 3000768049152 \) Copy content Toggle raw display
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