Properties

Label 3344.2.a.r
Level $3344$
Weight $2$
Character orbit 3344.a
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) 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 1672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_1 q^{5} + ( - \beta_1 - 1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_1 q^{5} + ( - \beta_1 - 1) q^{7} + \beta_{2} q^{9} + q^{11} + ( - 2 \beta_{2} + \beta_1 - 1) q^{13} + ( - \beta_{2} - 3) q^{15} + 2 \beta_1 q^{17} + q^{19} + (\beta_{2} + \beta_1 + 3) q^{21} + (\beta_{2} - \beta_1 - 2) q^{23} + (\beta_{2} - 2) q^{25} + (2 \beta_1 - 1) q^{27} + (\beta_{2} - 2 \beta_1 + 4) q^{29} + (3 \beta_{2} - 2 \beta_1 - 1) q^{31} - \beta_1 q^{33} + ( - \beta_{2} - \beta_1 - 3) q^{35} + (\beta_{2} + \beta_1 + 4) q^{37} + ( - \beta_{2} + 3 \beta_1 - 1) q^{39} + ( - 4 \beta_{2} + 3 \beta_1 - 3) q^{41} + ( - 3 \beta_{2} - 4) q^{43} + (\beta_1 + 1) q^{45} + (2 \beta_{2} + 2 \beta_1 + 2) q^{47} + (\beta_{2} + 2 \beta_1 - 3) q^{49} + ( - 2 \beta_{2} - 6) q^{51} + ( - 2 \beta_{2} - 2) q^{53} + \beta_1 q^{55} - \beta_1 q^{57} + (3 \beta_{2} - \beta_1 - 4) q^{59} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{61} + ( - \beta_{2} - \beta_1 - 1) q^{63} + (\beta_{2} - 3 \beta_1 + 1) q^{65} + ( - \beta_{2} + 6 \beta_1 - 3) q^{67} + (\beta_{2} + \beta_1 + 2) q^{69} + ( - \beta_1 - 6) q^{71} + ( - 2 \beta_1 + 2) q^{73} + (\beta_1 - 1) q^{75} + ( - \beta_1 - 1) q^{77} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{79} + ( - 5 \beta_{2} + \beta_1 - 6) q^{81} + ( - 3 \beta_{2} + 4 \beta_1 - 8) q^{83} + (2 \beta_{2} + 6) q^{85} + (2 \beta_{2} - 5 \beta_1 + 5) q^{87} + ( - 5 \beta_{2} - 3 \beta_1 - 4) q^{89} + (\beta_{2} + 2 \beta_1) q^{91} + (2 \beta_{2} - 2 \beta_1 + 3) q^{93} + \beta_1 q^{95} + (3 \beta_{2} - \beta_1 - 4) q^{97} + \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{7} - q^{9} + 3 q^{11} - q^{13} - 8 q^{15} + 3 q^{19} + 8 q^{21} - 7 q^{23} - 7 q^{25} - 3 q^{27} + 11 q^{29} - 6 q^{31} - 8 q^{35} + 11 q^{37} - 2 q^{39} - 5 q^{41} - 9 q^{43} + 3 q^{45} + 4 q^{47} - 10 q^{49} - 16 q^{51} - 4 q^{53} - 15 q^{59} - 4 q^{61} - 2 q^{63} + 2 q^{65} - 8 q^{67} + 5 q^{69} - 18 q^{71} + 6 q^{73} - 3 q^{75} - 3 q^{77} - 4 q^{79} - 13 q^{81} - 21 q^{83} + 16 q^{85} + 13 q^{87} - 7 q^{89} - q^{91} + 7 q^{93} - 15 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.11491
−0.254102
−1.86081
0 −2.11491 0 2.11491 0 −3.11491 0 1.47283 0
1.2 0 0.254102 0 −0.254102 0 −0.745898 0 −2.93543 0
1.3 0 1.86081 0 −1.86081 0 0.860806 0 0.462598 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3344.2.a.r 3
4.b odd 2 1 1672.2.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1672.2.a.e 3 4.b odd 2 1
3344.2.a.r 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):

\( T_{3}^{3} - 4T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 4T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 4T + 1 \) Copy content Toggle raw display
$5$ \( T^{3} - 4T - 1 \) Copy content Toggle raw display
$7$ \( T^{3} + 3T^{2} - T - 2 \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + T^{2} - 19 T - 32 \) Copy content Toggle raw display
$17$ \( T^{3} - 16T - 8 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 7 T^{2} + 10 T - 4 \) Copy content Toggle raw display
$29$ \( T^{3} - 11 T^{2} + 25 T - 16 \) Copy content Toggle raw display
$31$ \( T^{3} + 6 T^{2} - 34 T - 31 \) Copy content Toggle raw display
$37$ \( T^{3} - 11 T^{2} + 28 T - 16 \) Copy content Toggle raw display
$41$ \( T^{3} + 5 T^{2} - 77 T - 212 \) Copy content Toggle raw display
$43$ \( T^{3} + 9 T^{2} - 21 T - 218 \) Copy content Toggle raw display
$47$ \( T^{3} - 4 T^{2} - 44 T - 32 \) Copy content Toggle raw display
$53$ \( T^{3} + 4 T^{2} - 16 T - 56 \) Copy content Toggle raw display
$59$ \( T^{3} + 15 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$61$ \( T^{3} + 4 T^{2} - 20 T - 16 \) Copy content Toggle raw display
$67$ \( T^{3} + 8 T^{2} - 110 T - 191 \) Copy content Toggle raw display
$71$ \( T^{3} + 18 T^{2} + 104 T + 193 \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} - 4 T + 32 \) Copy content Toggle raw display
$79$ \( T^{3} + 4 T^{2} - 56 T + 104 \) Copy content Toggle raw display
$83$ \( T^{3} + 21 T^{2} + 71 T + 14 \) Copy content Toggle raw display
$89$ \( T^{3} + 7 T^{2} - 198 T - 148 \) Copy content Toggle raw display
$97$ \( T^{3} + 15 T^{2} + 32 T + 16 \) Copy content Toggle raw display
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