Properties

Label 912.2.bb.b
Level $912$
Weight $2$
Character orbit 912.bb
Analytic conductor $7.282$
Analytic rank $0$
Dimension $2$
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.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( -1 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( -1 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -2 + 4 \zeta_{6} ) q^{11} + ( -6 + 3 \zeta_{6} ) q^{13} + ( -6 + 6 \zeta_{6} ) q^{17} + ( 3 + 2 \zeta_{6} ) q^{19} + ( 1 + \zeta_{6} ) q^{21} + ( 4 - 2 \zeta_{6} ) q^{23} + 5 \zeta_{6} q^{25} - q^{27} + ( -4 + 2 \zeta_{6} ) q^{29} - q^{31} + ( 2 + 2 \zeta_{6} ) q^{33} + ( 5 - 10 \zeta_{6} ) q^{37} + ( -3 + 6 \zeta_{6} ) q^{39} + ( -4 - 4 \zeta_{6} ) q^{41} + ( 3 + 3 \zeta_{6} ) q^{43} + ( 12 - 6 \zeta_{6} ) q^{47} + 4 q^{49} + 6 \zeta_{6} q^{51} + ( -12 + 6 \zeta_{6} ) q^{53} + ( 5 - 3 \zeta_{6} ) q^{57} + ( -12 + 12 \zeta_{6} ) q^{59} + 5 \zeta_{6} q^{61} + ( 2 - \zeta_{6} ) q^{63} + 13 \zeta_{6} q^{67} + ( 2 - 4 \zeta_{6} ) q^{69} + ( 5 - 5 \zeta_{6} ) q^{73} + 5 q^{75} -6 q^{77} + ( -1 + \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + ( 10 - 20 \zeta_{6} ) q^{83} + ( -2 + 4 \zeta_{6} ) q^{87} -9 \zeta_{6} q^{91} + ( -1 + \zeta_{6} ) q^{93} + ( 8 + 8 \zeta_{6} ) q^{97} + ( 4 - 2 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - q^{9} - 9q^{13} - 6q^{17} + 8q^{19} + 3q^{21} + 6q^{23} + 5q^{25} - 2q^{27} - 6q^{29} - 2q^{31} + 6q^{33} - 12q^{41} + 9q^{43} + 18q^{47} + 8q^{49} + 6q^{51} - 18q^{53} + 7q^{57} - 12q^{59} + 5q^{61} + 3q^{63} + 13q^{67} + 5q^{73} + 10q^{75} - 12q^{77} - q^{79} - q^{81} - 9q^{91} - q^{93} + 24q^{97} + 6q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(\zeta_{6}\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 1.73205i 0 −0.500000 + 0.866025i 0
559.1 0 0.500000 0.866025i 0 0 0 1.73205i 0 −0.500000 0.866025i 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
76.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.bb.b yes 2
3.b odd 2 1 2736.2.bm.b 2
4.b odd 2 1 912.2.bb.a 2
12.b even 2 1 2736.2.bm.a 2
19.d odd 6 1 912.2.bb.a 2
57.f even 6 1 2736.2.bm.a 2
76.f even 6 1 inner 912.2.bb.b yes 2
228.n odd 6 1 2736.2.bm.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bb.a 2 4.b odd 2 1
912.2.bb.a 2 19.d odd 6 1
912.2.bb.b yes 2 1.a even 1 1 trivial
912.2.bb.b yes 2 76.f even 6 1 inner
2736.2.bm.a 2 12.b even 2 1
2736.2.bm.a 2 57.f even 6 1
2736.2.bm.b 2 3.b odd 2 1
2736.2.bm.b 2 228.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 \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5} \)
\( T_{23}^{2} - 6 T_{23} + 12 \)
\( T_{31} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 3 + T^{2} \)
$11$ \( 12 + T^{2} \)
$13$ \( 27 + 9 T + T^{2} \)
$17$ \( 36 + 6 T + T^{2} \)
$19$ \( 19 - 8 T + T^{2} \)
$23$ \( 12 - 6 T + T^{2} \)
$29$ \( 12 + 6 T + T^{2} \)
$31$ \( ( 1 + T )^{2} \)
$37$ \( 75 + T^{2} \)
$41$ \( 48 + 12 T + T^{2} \)
$43$ \( 27 - 9 T + T^{2} \)
$47$ \( 108 - 18 T + T^{2} \)
$53$ \( 108 + 18 T + T^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 25 - 5 T + T^{2} \)
$67$ \( 169 - 13 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 25 - 5 T + T^{2} \)
$79$ \( 1 + T + T^{2} \)
$83$ \( 300 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 192 - 24 T + T^{2} \)
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