Properties

Label 3344.2.a.p
Level $3344$
Weight $2$
Character orbit 3344.a
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) 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 418)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_1 + 2) q^{5} + \beta_{2} q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_1 + 2) q^{5} + \beta_{2} q^{7} + (\beta_{2} + 1) q^{9} + q^{11} + (\beta_{2} + 2) q^{13} + (\beta_{2} - 2 \beta_1 + 4) q^{15} + (2 \beta_{2} + 2) q^{17} + q^{19} + ( - \beta_{2} - \beta_1) q^{21} + 2 \beta_{2} q^{23} + (\beta_{2} - 4 \beta_1 + 3) q^{25} + ( - \beta_{2} + \beta_1) q^{27} + ( - 2 \beta_{2} - \beta_1 + 2) q^{29} + ( - \beta_{2} + 2 \beta_1) q^{31} - \beta_1 q^{33} + (\beta_{2} - \beta_1) q^{35} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{37} + ( - \beta_{2} - 3 \beta_1) q^{39} + ( - \beta_{2} + 2 \beta_1 - 2) q^{41} + ( - 2 \beta_{2} + 3 \beta_1) q^{43} + (\beta_{2} - 2 \beta_1 + 2) q^{45} + (2 \beta_{2} + 2 \beta_1) q^{47} + ( - 2 \beta_{2} + \beta_1 - 3) q^{49} + ( - 2 \beta_{2} - 4 \beta_1) q^{51} + (2 \beta_1 - 6) q^{53} + ( - \beta_1 + 2) q^{55} - \beta_1 q^{57} + ( - 4 \beta_{2} - 4 \beta_1 + 4) q^{59} + 2 q^{61} + ( - \beta_{2} + \beta_1 + 4) q^{63} + (\beta_{2} - 3 \beta_1 + 4) q^{65} + \beta_{2} q^{67} + ( - 2 \beta_{2} - 2 \beta_1) q^{69} + ( - 2 \beta_{2} + \beta_1) q^{71} + (4 \beta_{2} + 2 \beta_1 + 2) q^{73} + (3 \beta_{2} - 4 \beta_1 + 16) q^{75} + \beta_{2} q^{77} + ( - 2 \beta_{2} - 4 \beta_1) q^{79} + ( - 3 \beta_{2} + \beta_1 - 7) q^{81} + (2 \beta_{2} - 5 \beta_1) q^{83} + (2 \beta_{2} - 4 \beta_1 + 4) q^{85} + (3 \beta_{2} + 4) q^{87} + ( - 2 \beta_{2} + 8 \beta_1 + 2) q^{89} + (\beta_1 + 4) q^{91} + ( - \beta_{2} + \beta_1 - 8) q^{93} + ( - \beta_1 + 2) q^{95} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{97} + (\beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} + 5 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} + 5 q^{5} - q^{7} + 2 q^{9} + 3 q^{11} + 5 q^{13} + 9 q^{15} + 4 q^{17} + 3 q^{19} - 2 q^{23} + 4 q^{25} + 2 q^{27} + 7 q^{29} + 3 q^{31} - q^{33} - 2 q^{35} - 14 q^{37} - 2 q^{39} - 3 q^{41} + 5 q^{43} + 3 q^{45} - 6 q^{49} - 2 q^{51} - 16 q^{53} + 5 q^{55} - q^{57} + 12 q^{59} + 6 q^{61} + 14 q^{63} + 8 q^{65} - q^{67} + 3 q^{71} + 4 q^{73} + 41 q^{75} - q^{77} - 2 q^{79} - 17 q^{81} - 7 q^{83} + 6 q^{85} + 9 q^{87} + 16 q^{89} + 13 q^{91} - 22 q^{93} + 5 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
2.39138
0.772866
−2.16425
0 −2.39138 0 −0.391382 0 1.71871 0 2.71871 0
1.2 0 −0.772866 0 1.22713 0 −3.40268 0 −2.40268 0
1.3 0 2.16425 0 4.16425 0 0.683969 0 1.68397 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.p 3
4.b odd 2 1 418.2.a.h 3
12.b even 2 1 3762.2.a.bd 3
44.c even 2 1 4598.2.a.bm 3
76.d even 2 1 7942.2.a.bc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.h 3 4.b odd 2 1
3344.2.a.p 3 1.a even 1 1 trivial
3762.2.a.bd 3 12.b even 2 1
4598.2.a.bm 3 44.c even 2 1
7942.2.a.bc 3 76.d even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 5T - 4 \) Copy content Toggle raw display
$5$ \( T^{3} - 5 T^{2} + 3 T + 2 \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 7T + 4 \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 5T^{2} + T + 14 \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 24 T + 88 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 2 T^{2} - 28 T + 32 \) Copy content Toggle raw display
$29$ \( T^{3} - 7 T^{2} - 19 T + 86 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} - 25 T + 76 \) Copy content Toggle raw display
$37$ \( T^{3} + 14 T^{2} + 16 T - 128 \) Copy content Toggle raw display
$41$ \( T^{3} + 3 T^{2} - 25 T + 22 \) Copy content Toggle raw display
$43$ \( T^{3} - 5 T^{2} - 67 T + 268 \) Copy content Toggle raw display
$47$ \( T^{3} - 52T - 128 \) Copy content Toggle raw display
$53$ \( T^{3} + 16 T^{2} + 64 T + 56 \) Copy content Toggle raw display
$59$ \( T^{3} - 12 T^{2} - 160 T + 1792 \) Copy content Toggle raw display
$61$ \( (T - 2)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} + T^{2} - 7T + 4 \) Copy content Toggle raw display
$71$ \( T^{3} - 3 T^{2} - 31 T - 28 \) Copy content Toggle raw display
$73$ \( T^{3} - 4 T^{2} - 136 T + 56 \) Copy content Toggle raw display
$79$ \( T^{3} + 2 T^{2} - 116 T + 352 \) Copy content Toggle raw display
$83$ \( T^{3} + 7 T^{2} - 143 T - 1108 \) Copy content Toggle raw display
$89$ \( T^{3} - 16 T^{2} - 280 T + 4424 \) Copy content Toggle raw display
$97$ \( T^{3} + 14 T^{2} + 16 T - 128 \) Copy content Toggle raw display
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