Properties

Label 912.2.f.f
Level $912$
Weight $2$
Character orbit 912.f
Analytic conductor $7.282$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
Defining polynomial: \( x^{4} + 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{3} - \beta_{3} q^{5} - q^{7} + ( - \beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{3} - \beta_{3} q^{5} - q^{7} + ( - \beta_{3} - 2) q^{9} - \beta_{3} q^{11} + ( - \beta_{2} - 2 \beta_1) q^{13} + (2 \beta_{2} - \beta_1) q^{15} + 3 \beta_{3} q^{17} + ( - \beta_{2} - 2 \beta_1 - 3) q^{19} + (\beta_{2} + \beta_1) q^{21} + 2 \beta_{3} q^{23} + (4 \beta_{2} + \beta_1) q^{27} + 4 \beta_{2} q^{29} + ( - \beta_{2} - 2 \beta_1) q^{31} + (2 \beta_{2} - \beta_1) q^{33} + \beta_{3} q^{35} + ( - 3 \beta_{2} - 6 \beta_1) q^{37} + ( - \beta_{3} - 5) q^{39} - 7 \beta_{2} q^{41} + 5 q^{43} + (2 \beta_{3} - 5) q^{45} + \beta_{3} q^{47} - 6 q^{49} + ( - 6 \beta_{2} + 3 \beta_1) q^{51} + 3 \beta_{2} q^{53} - 5 q^{55} + ( - \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 5) q^{57} - \beta_{2} q^{59} - q^{61} + (\beta_{3} + 2) q^{63} + 5 \beta_{2} q^{65} + (2 \beta_{2} + 4 \beta_1) q^{67} + ( - 4 \beta_{2} + 2 \beta_1) q^{69} + 3 \beta_{2} q^{71} - 3 q^{73} + \beta_{3} q^{77} + ( - 4 \beta_{2} - 8 \beta_1) q^{79} + (4 \beta_{3} - 1) q^{81} - 4 \beta_{3} q^{83} + 15 q^{85} + (4 \beta_{3} - 4) q^{87} - 9 \beta_{2} q^{89} + (\beta_{2} + 2 \beta_1) q^{91} + ( - \beta_{3} - 5) q^{93} + (3 \beta_{3} + 5 \beta_{2}) q^{95} + ( - \beta_{2} - 2 \beta_1) q^{97} + (2 \beta_{3} - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} - 8 q^{9} - 12 q^{19} - 20 q^{39} + 20 q^{43} - 20 q^{45} - 24 q^{49} - 20 q^{55} - 20 q^{57} - 4 q^{61} + 8 q^{63} - 12 q^{73} - 4 q^{81} + 60 q^{85} - 16 q^{87} - 20 q^{93} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} - \beta_1 \) 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
−0.707107 + 1.58114i
−0.707107 1.58114i
0.707107 + 1.58114i
0.707107 1.58114i
0 −0.707107 1.58114i 0 2.23607i 0 −1.00000 0 −2.00000 + 2.23607i 0
113.2 0 −0.707107 + 1.58114i 0 2.23607i 0 −1.00000 0 −2.00000 2.23607i 0
113.3 0 0.707107 1.58114i 0 2.23607i 0 −1.00000 0 −2.00000 2.23607i 0
113.4 0 0.707107 + 1.58114i 0 2.23607i 0 −1.00000 0 −2.00000 + 2.23607i 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 inner
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.f 4
3.b odd 2 1 inner 912.2.f.f 4
4.b odd 2 1 57.2.d.a 4
12.b even 2 1 57.2.d.a 4
19.b odd 2 1 inner 912.2.f.f 4
57.d even 2 1 inner 912.2.f.f 4
76.d even 2 1 57.2.d.a 4
228.b odd 2 1 57.2.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.d.a 4 4.b odd 2 1
57.2.d.a 4 12.b even 2 1
57.2.d.a 4 76.d even 2 1
57.2.d.a 4 228.b odd 2 1
912.2.f.f 4 1.a even 1 1 trivial
912.2.f.f 4 3.b odd 2 1 inner
912.2.f.f 4 19.b odd 2 1 inner
912.2.f.f 4 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 \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{2} + 5 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{29}^{2} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 90)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$43$ \( (T - 5)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$73$ \( (T + 3)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 160)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
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