Properties

Label 912.2.f.b
Level $912$
Weight $2$
Character orbit 912.f
Analytic conductor $7.282$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{3} + \beta q^{5} + 4 q^{7} + ( -1 - 2 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{3} + \beta q^{5} + 4 q^{7} + ( -1 - 2 \beta ) q^{9} -4 \beta q^{11} -3 \beta q^{13} + ( -2 - \beta ) q^{15} -2 \beta q^{17} + ( -1 - 3 \beta ) q^{19} + ( -4 + 4 \beta ) q^{21} -\beta q^{23} + 3 q^{25} + ( 5 + \beta ) q^{27} -6 q^{29} + 3 \beta q^{31} + ( 8 + 4 \beta ) q^{33} + 4 \beta q^{35} -3 \beta q^{37} + ( 6 + 3 \beta ) q^{39} -2 q^{43} + ( 4 - \beta ) q^{45} -\beta q^{47} + 9 q^{49} + ( 4 + 2 \beta ) q^{51} + 6 q^{53} + 8 q^{55} + ( 7 + 2 \beta ) q^{57} -12 q^{59} + 2 q^{61} + ( -4 - 8 \beta ) q^{63} + 6 q^{65} + ( 2 + \beta ) q^{69} + 12 q^{71} -4 q^{73} + ( -3 + 3 \beta ) q^{75} -16 \beta q^{77} + 3 \beta q^{79} + ( -7 + 4 \beta ) q^{81} + 2 \beta q^{83} + 4 q^{85} + ( 6 - 6 \beta ) q^{87} + 6 q^{89} -12 \beta q^{91} + ( -6 - 3 \beta ) q^{93} + ( 6 - \beta ) q^{95} + 6 \beta q^{97} + ( -16 + 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 8 q^{7} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 8 q^{7} - 2 q^{9} - 4 q^{15} - 2 q^{19} - 8 q^{21} + 6 q^{25} + 10 q^{27} - 12 q^{29} + 16 q^{33} + 12 q^{39} - 4 q^{43} + 8 q^{45} + 18 q^{49} + 8 q^{51} + 12 q^{53} + 16 q^{55} + 14 q^{57} - 24 q^{59} + 4 q^{61} - 8 q^{63} + 12 q^{65} + 4 q^{69} + 24 q^{71} - 8 q^{73} - 6 q^{75} - 14 q^{81} + 8 q^{85} + 12 q^{87} + 12 q^{89} - 12 q^{93} + 12 q^{95} - 32 q^{99} + O(q^{100}) \)

Character values

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

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

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
1.41421i
1.41421i
0 −1.00000 1.41421i 0 1.41421i 0 4.00000 0 −1.00000 + 2.82843i 0
113.2 0 −1.00000 + 1.41421i 0 1.41421i 0 4.00000 0 −1.00000 2.82843i 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.b 2
3.b odd 2 1 912.2.f.d 2
4.b odd 2 1 114.2.b.d yes 2
12.b even 2 1 114.2.b.a 2
19.b odd 2 1 912.2.f.d 2
57.d even 2 1 inner 912.2.f.b 2
76.d even 2 1 114.2.b.a 2
228.b odd 2 1 114.2.b.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.b.a 2 12.b even 2 1
114.2.b.a 2 76.d even 2 1
114.2.b.d yes 2 4.b odd 2 1
114.2.b.d yes 2 228.b odd 2 1
912.2.f.b 2 1.a even 1 1 trivial
912.2.f.b 2 57.d even 2 1 inner
912.2.f.d 2 3.b odd 2 1
912.2.f.d 2 19.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 2 \)
\( T_{7} - 4 \)
\( T_{29} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + 2 T + T^{2} \)
$5$ \( 2 + T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( 32 + T^{2} \)
$13$ \( 18 + T^{2} \)
$17$ \( 8 + T^{2} \)
$19$ \( 19 + 2 T + T^{2} \)
$23$ \( 2 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( 18 + T^{2} \)
$37$ \( 18 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 2 + T )^{2} \)
$47$ \( 2 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( ( 4 + T )^{2} \)
$79$ \( 18 + T^{2} \)
$83$ \( 8 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 72 + T^{2} \)
show more
show less