Properties

Label 912.2.bn.k
Level $912$
Weight $2$
Character orbit 912.bn
Analytic conductor $7.282$
Analytic rank $0$
Dimension $4$
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.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{3} ) q^{3} + ( -2 + \beta_{1} - \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{3} ) q^{3} + ( -2 + \beta_{1} - \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{9} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{11} + ( 2 - \beta_{2} ) q^{13} + ( 4 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{15} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{17} + ( -3 - 2 \beta_{2} ) q^{19} + ( -5 - \beta_{1} + 2 \beta_{2} ) q^{21} + ( -3 + 2 \beta_{2} + \beta_{3} ) q^{23} + ( 3 - 6 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{25} + ( 5 - 2 \beta_{1} + 2 \beta_{3} ) q^{27} + ( 1 - 2 \beta_{1} - 8 \beta_{2} + \beta_{3} ) q^{29} + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{31} + ( 2 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{33} + ( -4 + 4 \beta_{1} ) q^{35} + ( -2 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{37} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{39} + ( -3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{41} + ( -3 - 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{43} + ( -6 - 3 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{45} + ( 5 - 2 \beta_{2} + \beta_{3} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{49} + ( 3 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{51} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{53} + ( 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{55} + ( 2 - 2 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{57} + ( -4 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 2 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -5 + 2 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{63} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{65} + ( 2 - \beta_{2} ) q^{67} + ( 1 + 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{69} + ( 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{71} + ( -9 + 2 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} ) q^{73} + ( -12 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{75} + ( -8 + 4 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} ) q^{77} + ( -3 + 6 \beta_{1} + 3 \beta_{2} ) q^{79} + ( -\beta_{2} + 5 \beta_{3} ) q^{81} + ( -2 - 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -1 + 2 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{85} + ( 5 - 8 \beta_{1} - 10 \beta_{2} + \beta_{3} ) q^{87} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{89} + ( -1 - \beta_{2} - 3 \beta_{3} ) q^{91} + ( -3 + 3 \beta_{1} - 6 \beta_{2} ) q^{93} + ( 4 - 5 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -6 + 9 \beta_{1} + 3 \beta_{2} ) q^{97} + ( 6 - 6 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} - 9 q^{5} - 2 q^{7} + 5 q^{9} + O(q^{10}) \) \( 4 q + q^{3} - 9 q^{5} - 2 q^{7} + 5 q^{9} + 6 q^{13} + 11 q^{15} + 15 q^{17} - 16 q^{19} - 17 q^{21} - 9 q^{23} + 9 q^{25} + 16 q^{27} - 15 q^{29} - 8 q^{33} - 12 q^{35} + 3 q^{39} - 3 q^{41} - 8 q^{43} - 17 q^{45} + 15 q^{47} + 6 q^{49} + 11 q^{51} + 9 q^{53} + 2 q^{55} - q^{57} - 9 q^{59} + 8 q^{61} - 19 q^{63} - 18 q^{65} + 6 q^{67} + q^{69} + 3 q^{71} - 16 q^{73} - 54 q^{75} - 7 q^{81} - 17 q^{85} - 9 q^{87} + 9 q^{89} - 3 q^{91} - 21 q^{93} + 27 q^{95} - 9 q^{97} - 4 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 2 \beta_{1} + 3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
0 −1.18614 + 1.26217i 0 −3.68614 2.12819i 0 2.37228 0 −0.186141 2.99422i 0
65.2 0 1.68614 0.396143i 0 −0.813859 0.469882i 0 −3.37228 0 2.68614 1.33591i 0
449.1 0 −1.18614 1.26217i 0 −3.68614 + 2.12819i 0 2.37228 0 −0.186141 + 2.99422i 0
449.2 0 1.68614 + 0.396143i 0 −0.813859 + 0.469882i 0 −3.37228 0 2.68614 + 1.33591i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.k 4
3.b odd 2 1 912.2.bn.l 4
4.b odd 2 1 228.2.p.d yes 4
12.b even 2 1 228.2.p.c 4
19.d odd 6 1 912.2.bn.l 4
57.f even 6 1 inner 912.2.bn.k 4
76.f even 6 1 228.2.p.c 4
228.n odd 6 1 228.2.p.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.p.c 4 12.b even 2 1
228.2.p.c 4 76.f even 6 1
228.2.p.d yes 4 4.b odd 2 1
228.2.p.d yes 4 228.n odd 6 1
912.2.bn.k 4 1.a even 1 1 trivial
912.2.bn.k 4 57.f even 6 1 inner
912.2.bn.l 4 3.b odd 2 1
912.2.bn.l 4 19.d odd 6 1

Hecke kernels

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

\( T_{5}^{4} + 9 T_{5}^{3} + 31 T_{5}^{2} + 36 T_{5} + 16 \)
\( T_{7}^{2} + T_{7} - 8 \)
\( T_{17}^{4} - 15 T_{17}^{3} + 91 T_{17}^{2} - 240 T_{17} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 - 3 T - 2 T^{2} - T^{3} + T^{4} \)
$5$ \( 16 + 36 T + 31 T^{2} + 9 T^{3} + T^{4} \)
$7$ \( ( -8 + T + T^{2} )^{2} \)
$11$ \( 64 + 28 T^{2} + T^{4} \)
$13$ \( ( 3 - 3 T + T^{2} )^{2} \)
$17$ \( 256 - 240 T + 91 T^{2} - 15 T^{3} + T^{4} \)
$19$ \( ( 19 + 8 T + T^{2} )^{2} \)
$23$ \( 16 + 36 T + 31 T^{2} + 9 T^{3} + T^{4} \)
$29$ \( 2304 + 720 T + 177 T^{2} + 15 T^{3} + T^{4} \)
$31$ \( 324 + 63 T^{2} + T^{4} \)
$37$ \( 576 + 51 T^{2} + T^{4} \)
$41$ \( 5184 - 216 T + 81 T^{2} + 3 T^{3} + T^{4} \)
$43$ \( 289 - 136 T + 81 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( 256 - 240 T + 91 T^{2} - 15 T^{3} + T^{4} \)
$53$ \( 144 - 108 T + 69 T^{2} - 9 T^{3} + T^{4} \)
$59$ \( 144 + 108 T + 69 T^{2} + 9 T^{3} + T^{4} \)
$61$ \( 289 + 136 T + 81 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( ( 3 - 3 T + T^{2} )^{2} \)
$71$ \( 5184 + 216 T + 81 T^{2} - 3 T^{3} + T^{4} \)
$73$ \( 961 + 496 T + 225 T^{2} + 16 T^{3} + T^{4} \)
$79$ \( 9801 - 99 T^{2} + T^{4} \)
$83$ \( 256 + 76 T^{2} + T^{4} \)
$89$ \( 144 - 108 T + 69 T^{2} - 9 T^{3} + T^{4} \)
$97$ \( 46656 - 1944 T - 189 T^{2} + 9 T^{3} + T^{4} \)
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