Properties

Label 912.2.f.e
Level $912$
Weight $2$
Character orbit 912.f
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.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: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{3} + ( - 4 \zeta_{6} + 2) q^{5} - q^{7} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} + 1) q^{3} + ( - 4 \zeta_{6} + 2) q^{5} - q^{7} + 3 \zeta_{6} q^{9} + ( - 4 \zeta_{6} + 2) q^{11} + (2 \zeta_{6} - 1) q^{13} + ( - 6 \zeta_{6} + 6) q^{15} + ( - 2 \zeta_{6} + 1) q^{17} + (2 \zeta_{6} + 3) q^{19} + ( - \zeta_{6} - 1) q^{21} + ( - 6 \zeta_{6} + 3) q^{23} - 7 q^{25} + (6 \zeta_{6} - 3) q^{27} + 9 q^{29} + ( - 12 \zeta_{6} + 6) q^{31} + ( - 6 \zeta_{6} + 6) q^{33} + (4 \zeta_{6} - 2) q^{35} + ( - 8 \zeta_{6} + 4) q^{37} + (3 \zeta_{6} - 3) q^{39} - 2 q^{43} + ( - 6 \zeta_{6} + 12) q^{45} + (4 \zeta_{6} - 2) q^{47} - 6 q^{49} + ( - 3 \zeta_{6} + 3) q^{51} - 9 q^{53} - 12 q^{55} + (7 \zeta_{6} + 1) q^{57} + 3 q^{59} - 8 q^{61} - 3 \zeta_{6} q^{63} + 6 q^{65} + (10 \zeta_{6} - 5) q^{67} + ( - 9 \zeta_{6} + 9) q^{69} + 12 q^{71} + 11 q^{73} + ( - 7 \zeta_{6} - 7) q^{75} + (4 \zeta_{6} - 2) q^{77} + (8 \zeta_{6} - 4) q^{79} + (9 \zeta_{6} - 9) q^{81} + (12 \zeta_{6} - 6) q^{83} - 6 q^{85} + (9 \zeta_{6} + 9) q^{87} + 6 q^{89} + ( - 2 \zeta_{6} + 1) q^{91} + ( - 18 \zeta_{6} + 18) q^{93} + ( - 16 \zeta_{6} + 14) q^{95} + (16 \zeta_{6} - 8) q^{97} + ( - 6 \zeta_{6} + 12) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{15} + 8 q^{19} - 3 q^{21} - 14 q^{25} + 18 q^{29} + 6 q^{33} - 3 q^{39} - 4 q^{43} + 18 q^{45} - 12 q^{49} + 3 q^{51} - 18 q^{53} - 24 q^{55} + 9 q^{57} + 6 q^{59} - 16 q^{61} - 3 q^{63} + 12 q^{65} + 9 q^{69} + 24 q^{71} + 22 q^{73} - 21 q^{75} - 9 q^{81} - 12 q^{85} + 27 q^{87} + 12 q^{89} + 18 q^{93} + 12 q^{95} + 18 q^{99}+O(q^{100}) \) 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 1.50000 0.866025i 0 3.46410i 0 −1.00000 0 1.50000 2.59808i 0
113.2 0 1.50000 + 0.866025i 0 3.46410i 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
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.e 2
3.b odd 2 1 912.2.f.a 2
4.b odd 2 1 114.2.b.c yes 2
12.b even 2 1 114.2.b.b 2
19.b odd 2 1 912.2.f.a 2
57.d even 2 1 inner 912.2.f.e 2
76.d even 2 1 114.2.b.b 2
228.b odd 2 1 114.2.b.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.b.b 2 12.b even 2 1
114.2.b.b 2 76.d even 2 1
114.2.b.c yes 2 4.b odd 2 1
114.2.b.c yes 2 228.b odd 2 1
912.2.f.a 2 3.b odd 2 1
912.2.f.a 2 19.b odd 2 1
912.2.f.e 2 1.a even 1 1 trivial
912.2.f.e 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 \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{29} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 3 \) Copy content Toggle raw display
$17$ \( T^{2} + 3 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 27 \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) 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
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T - 3)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 75 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T - 11)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 48 \) Copy content Toggle raw display
$83$ \( T^{2} + 108 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 192 \) Copy content Toggle raw display
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