Properties

Label 3344.2.a.j
Level $3344$
Weight $2$
Character orbit 3344.a
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} - \beta q^{5} + (\beta - 1) q^{7} + \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} - \beta q^{5} + (\beta - 1) q^{7} + \beta q^{9} + q^{11} + ( - \beta - 3) q^{13} + (\beta + 3) q^{15} + 2 \beta q^{17} + q^{19} - 3 q^{21} + ( - 2 \beta + 2) q^{23} + (\beta - 2) q^{25} + (2 \beta - 3) q^{27} + (\beta - 8) q^{29} + (3 \beta + 1) q^{31} - \beta q^{33} - 3 q^{35} + 8 q^{37} + (4 \beta + 3) q^{39} + ( - \beta - 5) q^{41} + \beta q^{43} + ( - \beta - 3) q^{45} + (4 \beta + 2) q^{47} + ( - \beta - 3) q^{49} + ( - 2 \beta - 6) q^{51} + (2 \beta + 2) q^{53} - \beta q^{55} - \beta q^{57} + 4 q^{59} - 2 q^{61} + 3 q^{63} + (4 \beta + 3) q^{65} + ( - 5 \beta - 1) q^{67} + 6 q^{69} + (3 \beta + 6) q^{71} + (2 \beta - 14) q^{73} + (\beta - 3) q^{75} + (\beta - 1) q^{77} + (6 \beta - 2) q^{79} + ( - 2 \beta - 6) q^{81} + ( - 3 \beta - 4) q^{83} + ( - 2 \beta - 6) q^{85} + (7 \beta - 3) q^{87} + 6 \beta q^{89} - 3 \beta q^{91} + ( - 4 \beta - 9) q^{93} - \beta q^{95} + (8 \beta - 4) q^{97} + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - q^{5} - q^{7} + q^{9} + 2 q^{11} - 7 q^{13} + 7 q^{15} + 2 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} - 3 q^{25} - 4 q^{27} - 15 q^{29} + 5 q^{31} - q^{33} - 6 q^{35} + 16 q^{37} + 10 q^{39} - 11 q^{41} + q^{43} - 7 q^{45} + 8 q^{47} - 7 q^{49} - 14 q^{51} + 6 q^{53} - q^{55} - q^{57} + 8 q^{59} - 4 q^{61} + 6 q^{63} + 10 q^{65} - 7 q^{67} + 12 q^{69} + 15 q^{71} - 26 q^{73} - 5 q^{75} - q^{77} + 2 q^{79} - 14 q^{81} - 11 q^{83} - 14 q^{85} + q^{87} + 6 q^{89} - 3 q^{91} - 22 q^{93} - q^{95} + q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
0 −2.30278 0 −2.30278 0 1.30278 0 2.30278 0
1.2 0 1.30278 0 1.30278 0 −2.30278 0 −1.30278 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.j 2
4.b odd 2 1 1672.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1672.2.a.d 2 4.b odd 2 1
3344.2.a.j 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$5$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$7$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 9 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$29$ \( T^{2} + 15T + 53 \) Copy content Toggle raw display
$31$ \( T^{2} - 5T - 23 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 11T + 27 \) Copy content Toggle raw display
$43$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 4 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 7T - 69 \) Copy content Toggle raw display
$71$ \( T^{2} - 15T + 27 \) Copy content Toggle raw display
$73$ \( T^{2} + 26T + 156 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 116 \) Copy content Toggle raw display
$83$ \( T^{2} + 11T + 1 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 108 \) Copy content Toggle raw display
$97$ \( T^{2} - 208 \) Copy content Toggle raw display
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