Properties

Label 912.2.bn.m
Level $912$
Weight $2$
Character orbit 912.bn
Analytic conductor $7.282$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3} + \beta_1) q^{3} - \beta_{5} q^{5} + (\beta_{3} - 2 \beta_1 + 2) q^{7} + ( - 2 \beta_{6} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{3} + \beta_1) q^{3} - \beta_{5} q^{5} + (\beta_{3} - 2 \beta_1 + 2) q^{7} + ( - 2 \beta_{6} + \beta_{2} - 1) q^{9} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4}) q^{11} + (\beta_{2} - 2 \beta_1 + 1) q^{13} + ( - \beta_{7} + \beta_{4} + 2 \beta_1) q^{15} + ( - 4 \beta_{7} + 2 \beta_{4}) q^{17} + ( - 3 \beta_{3} - 2 \beta_1) q^{19} + (2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{21} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4}) q^{23} + (3 \beta_{2} - 3) q^{25} + ( - \beta_{7} + 5 \beta_{3}) q^{27} + (2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 2 \beta_{4}) q^{29} + ( - \beta_{3} + 4 \beta_{2} - 2) q^{31} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 4) q^{33} + (2 \beta_{7} - 2 \beta_{5} - \beta_{4}) q^{35} + (6 \beta_{3} + 2 \beta_{2} - 1) q^{37} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_1 - 2) q^{39} + 4 \beta_{4} q^{41} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{43} + ( - \beta_{6} + \beta_{5} + 4) q^{45} + (2 \beta_{7} - 4 \beta_{6} + 2 \beta_{4}) q^{47} + (4 \beta_{3} - 8 \beta_1) q^{49} + (2 \beta_{6} - 4 \beta_{5} + 4 \beta_{2} + 4) q^{51} + (6 \beta_{7} - \beta_{6} + 2 \beta_{5} - 6 \beta_{4}) q^{53} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{55} + (2 \beta_{6} - 5 \beta_{5} - 3 \beta_{2} - 2) q^{57} + (8 \beta_{6} - 4 \beta_{5} + \beta_{4}) q^{59} + ( - 4 \beta_{3} - 7 \beta_{2} + 2 \beta_1 + 7) q^{61} + (2 \beta_{7} - 4 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{63} + (2 \beta_{7} - \beta_{6} - \beta_{5}) q^{65} + \beta_1 q^{67} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + 4 \beta_{3} + 4 \beta_{2} - 2) q^{69} + ( - 4 \beta_{6} + 2 \beta_{5} - 6 \beta_{4}) q^{71} + 3 \beta_{2} q^{73} + (3 \beta_{7} + 3 \beta_{3}) q^{75} + (\beta_{6} - \beta_{5}) q^{77} + (7 \beta_{3} + 2 \beta_{2} - 7 \beta_1 - 4) q^{79} + (4 \beta_{5} + 7 \beta_{2}) q^{81} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{4}) q^{83} + ( - 8 \beta_{3} + 4 \beta_1) q^{85} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} + 8 \beta_1 - 4) q^{87} + (3 \beta_{6} - 6 \beta_{5}) q^{89} + (4 \beta_{2} - 7 \beta_1 + 4) q^{91} + (4 \beta_{7} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{93} + (2 \beta_{7} + 3 \beta_{4}) q^{95} + (4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{97} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - 8 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} - 4 q^{9} + 12 q^{13} - 12 q^{21} - 12 q^{25} + 24 q^{33} - 16 q^{39} + 16 q^{43} + 32 q^{45} + 48 q^{51} - 16 q^{55} - 28 q^{57} + 28 q^{61} - 8 q^{63} + 12 q^{73} - 24 q^{79} + 28 q^{81} - 32 q^{87} + 48 q^{91} - 4 q^{93} + 24 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0 −1.57313 + 0.724745i 0 1.22474 + 0.707107i 0 3.73205 0 1.94949 2.28024i 0
65.2 0 −0.158919 1.72474i 0 −1.22474 0.707107i 0 3.73205 0 −2.94949 + 0.548188i 0
65.3 0 0.158919 + 1.72474i 0 −1.22474 0.707107i 0 0.267949 0 −2.94949 + 0.548188i 0
65.4 0 1.57313 0.724745i 0 1.22474 + 0.707107i 0 0.267949 0 1.94949 2.28024i 0
449.1 0 −1.57313 0.724745i 0 1.22474 0.707107i 0 3.73205 0 1.94949 + 2.28024i 0
449.2 0 −0.158919 + 1.72474i 0 −1.22474 + 0.707107i 0 3.73205 0 −2.94949 0.548188i 0
449.3 0 0.158919 1.72474i 0 −1.22474 + 0.707107i 0 0.267949 0 −2.94949 0.548188i 0
449.4 0 1.57313 + 0.724745i 0 1.22474 0.707107i 0 0.267949 0 1.94949 + 2.28024i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bn.m 8
3.b odd 2 1 inner 912.2.bn.m 8
4.b odd 2 1 57.2.f.a 8
12.b even 2 1 57.2.f.a 8
19.d odd 6 1 inner 912.2.bn.m 8
57.f even 6 1 inner 912.2.bn.m 8
76.f even 6 1 57.2.f.a 8
76.f even 6 1 1083.2.d.b 8
76.g odd 6 1 1083.2.d.b 8
228.m even 6 1 1083.2.d.b 8
228.n odd 6 1 57.2.f.a 8
228.n odd 6 1 1083.2.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.f.a 8 4.b odd 2 1
57.2.f.a 8 12.b even 2 1
57.2.f.a 8 76.f even 6 1
57.2.f.a 8 228.n odd 6 1
912.2.bn.m 8 1.a even 1 1 trivial
912.2.bn.m 8 3.b odd 2 1 inner
912.2.bn.m 8 19.d odd 6 1 inner
912.2.bn.m 8 57.f even 6 1 inner
1083.2.d.b 8 76.f even 6 1
1083.2.d.b 8 76.g odd 6 1
1083.2.d.b 8 228.m even 6 1
1083.2.d.b 8 228.n odd 6 1

Hecke kernels

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

\( T_{5}^{4} - 2T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 1 \) Copy content Toggle raw display
\( T_{17}^{4} - 24T_{17}^{2} + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{6} - 5 T^{4} + 18 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{3} + 11 T^{2} + 6 T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 26 T^{2} + 361)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 28 T^{6} + 780 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{8} + 64 T^{6} + 3840 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$31$ \( (T^{4} + 26 T^{2} + 121)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 78 T^{2} + 1089)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + 51 T^{2} - 104 T + 169)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 112 T^{6} + 12480 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{8} + 156 T^{6} + \cdots + 18974736 \) Copy content Toggle raw display
$59$ \( T^{8} + 196 T^{6} + \cdots + 78074896 \) Copy content Toggle raw display
$61$ \( (T^{4} - 14 T^{3} + 159 T^{2} - 518 T + 1369)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 192 T^{6} + 34560 T^{4} + \cdots + 5308416 \) Copy content Toggle raw display
$73$ \( (T^{2} - 3 T + 9)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 12 T^{3} + 11 T^{2} - 444 T + 1369)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 64 T^{2} + 256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 54 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 12 T^{3} + 44 T^{2} + 48 T + 16)^{2} \) Copy content Toggle raw display
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