Properties

Label 912.2.f.c
Level $912$
Weight $2$
Character orbit 912.f
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - 4 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - 4 q^{7} - 3 q^{9} - 4 \beta q^{13} + ( - \beta + 4) q^{19} - 4 \beta q^{21} + 5 q^{25} - 3 \beta q^{27} - 6 \beta q^{31} + 4 \beta q^{37} + 12 q^{39} - 8 q^{43} + 9 q^{49} + (4 \beta + 3) q^{57} - 14 q^{61} + 12 q^{63} - 2 \beta q^{67} - 10 q^{73} + 5 \beta q^{75} - 10 \beta q^{79} + 9 q^{81} + 16 \beta q^{91} + 18 q^{93} - 8 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{7} - 6 q^{9} + 8 q^{19} + 10 q^{25} + 24 q^{39} - 16 q^{43} + 18 q^{49} + 6 q^{57} - 28 q^{61} + 24 q^{63} - 20 q^{73} + 18 q^{81} + 36 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.73205i 0 0 0 −4.00000 0 −3.00000 0
113.2 0 1.73205i 0 0 0 −4.00000 0 −3.00000 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.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.f.c 2
3.b odd 2 1 CM 912.2.f.c 2
4.b odd 2 1 228.2.d.a 2
12.b even 2 1 228.2.d.a 2
19.b odd 2 1 inner 912.2.f.c 2
57.d even 2 1 inner 912.2.f.c 2
76.d even 2 1 228.2.d.a 2
228.b odd 2 1 228.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.d.a 2 4.b odd 2 1
228.2.d.a 2 12.b even 2 1
228.2.d.a 2 76.d even 2 1
228.2.d.a 2 228.b odd 2 1
912.2.f.c 2 1.a even 1 1 trivial
912.2.f.c 2 3.b odd 2 1 CM
912.2.f.c 2 19.b odd 2 1 inner
912.2.f.c 2 57.d even 2 1 inner

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 48 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 108 \) Copy content Toggle raw display
$37$ \( T^{2} + 48 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 12 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 300 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 192 \) Copy content Toggle raw display
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