Properties

Label 912.2.bn.a
Level $912$
Weight $2$
Character orbit 912.bn
Analytic conductor $7.282$
Analytic rank $0$
Dimension $2$
CM discriminant -3
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: \(2\)
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: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{3} - q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{3} - q^{7} + 3 \zeta_{6} q^{9} + ( 2 - \zeta_{6} ) q^{13} + ( 5 - 2 \zeta_{6} ) q^{19} + ( 1 + \zeta_{6} ) q^{21} -5 \zeta_{6} q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -1 + 2 \zeta_{6} ) q^{31} + ( 7 - 14 \zeta_{6} ) q^{37} -3 q^{39} + ( 5 - 5 \zeta_{6} ) q^{43} -6 q^{49} + ( -7 - \zeta_{6} ) q^{57} -13 \zeta_{6} q^{61} -3 \zeta_{6} q^{63} + ( 18 - 9 \zeta_{6} ) q^{67} + ( 7 - 7 \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} + ( -7 - 7 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -2 + \zeta_{6} ) q^{91} + ( 3 - 3 \zeta_{6} ) q^{93} + ( 8 + 8 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 2 q^{7} + 3 q^{9} + O(q^{10}) \) \( 2 q - 3 q^{3} - 2 q^{7} + 3 q^{9} + 3 q^{13} + 8 q^{19} + 3 q^{21} - 5 q^{25} - 6 q^{39} + 5 q^{43} - 12 q^{49} - 15 q^{57} - 13 q^{61} - 3 q^{63} + 27 q^{67} + 7 q^{73} - 21 q^{79} - 9 q^{81} - 3 q^{91} + 3 q^{93} + 24 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.50000 0.866025i 0 0 0 −1.00000 0 1.50000 + 2.59808i 0
449.1 0 −1.50000 + 0.866025i 0 0 0 −1.00000 0 1.50000 2.59808i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
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.a 2
3.b odd 2 1 CM 912.2.bn.a 2
4.b odd 2 1 228.2.p.b 2
12.b even 2 1 228.2.p.b 2
19.d odd 6 1 inner 912.2.bn.a 2
57.f even 6 1 inner 912.2.bn.a 2
76.f even 6 1 228.2.p.b 2
228.n odd 6 1 228.2.p.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.p.b 2 4.b odd 2 1
228.2.p.b 2 12.b even 2 1
228.2.p.b 2 76.f even 6 1
228.2.p.b 2 228.n odd 6 1
912.2.bn.a 2 1.a even 1 1 trivial
912.2.bn.a 2 3.b odd 2 1 CM
912.2.bn.a 2 19.d odd 6 1 inner
912.2.bn.a 2 57.f even 6 1 inner

Hecke kernels

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

\( T_{5} \)
\( T_{7} + 1 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 3 - 3 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 19 - 8 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 3 + T^{2} \)
$37$ \( 147 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 25 - 5 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 169 + 13 T + T^{2} \)
$67$ \( 243 - 27 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 49 - 7 T + T^{2} \)
$79$ \( 147 + 21 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 192 - 24 T + T^{2} \)
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