Properties

Label 3344.2.a.t
Level $3344$
Weight $2$
Character orbit 3344.a
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.246832.1
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) 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 209)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{2} - \beta_1 - 1) q^{5} + (\beta_{2} + \beta_1 - 2) q^{7} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{2} - \beta_1 - 1) q^{5} + (\beta_{2} + \beta_1 - 2) q^{7} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{9} - q^{11} + (\beta_{4} - \beta_1 + 1) q^{13} + (2 \beta_{4} + 3 \beta_{3} + \beta_{2} - 2) q^{15} + (2 \beta_{3} + 2 \beta_1 - 2) q^{17} + q^{19} + (2 \beta_{3} - \beta_{2} + 2) q^{21} + ( - 3 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{23} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{25} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{27} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{29} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{31} + \beta_{3} q^{33} + (\beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{35} + ( - 3 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{37} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{39} + (\beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{41} + (\beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 4) q^{43} + (4 \beta_{3} + 3 \beta_{2} + \beta_1 - 8) q^{45} + (4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2) q^{47} + (\beta_{4} + 3 \beta_{3} + 2 \beta_1 - 1) q^{49} + (2 \beta_{3} + 2 \beta_{2} - 6) q^{51} + (4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{53} + ( - \beta_{2} + \beta_1 + 1) q^{55} - \beta_{3} q^{57} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_1 - 3) q^{59} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{61} + (\beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 - 4) q^{63} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{65} + (5 \beta_{3} - \beta_{2} + 4 \beta_1 - 4) q^{67} + (5 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{69} + (2 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{71} + ( - 2 \beta_{4} + 2 \beta_{2} - 6 \beta_1 + 2) q^{73} + ( - 5 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} - \beta_1 + 7) q^{75} + ( - \beta_{2} - \beta_1 + 2) q^{77} + ( - 2 \beta_1 - 8) q^{79} + (\beta_{4} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{81} + ( - \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 6) q^{83} + ( - 4 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{85} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 6) q^{87} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 5) q^{89} + ( - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{91} + (4 \beta_{4} + 6 \beta_{3} + 4 \beta_{2} - 9) q^{93} + (\beta_{2} - \beta_1 - 1) q^{95} + ( - 3 \beta_{4} + \beta_{3} - 5 \beta_{2} + \beta_1 + 5) q^{97} + (\beta_{4} + \beta_{3} + 2 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39} + 2 q^{41} - 20 q^{43} - 28 q^{45} + 20 q^{47} + 3 q^{49} - 24 q^{51} - 14 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{67} - 5 q^{69} - 23 q^{71} + 18 q^{75} + 6 q^{77} - 44 q^{79} + q^{81} + 14 q^{83} - 12 q^{85} - 28 q^{87} - 27 q^{89} - 24 q^{91} - 27 q^{93} - 5 q^{95} + 15 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 7\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 3\beta_{3} + 8\beta_{2} + 10\beta _1 + 6 \) 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
−1.15351
0.245526
2.71457
−1.51908
1.71250
0 −2.26452 0 0.637602 0 −2.66942 0 2.12805 0
1.2 0 −2.15766 0 −3.43077 0 −3.93972 0 1.65548 0
1.3 0 −0.121872 0 −1.06025 0 3.36889 0 −2.98515 0
1.4 0 0.563416 0 2.34577 0 −1.69239 0 −2.68256 0
1.5 0 2.98063 0 −3.49235 0 −1.06736 0 5.88418 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.t 5
4.b odd 2 1 209.2.a.c 5
12.b even 2 1 1881.2.a.k 5
20.d odd 2 1 5225.2.a.h 5
44.c even 2 1 2299.2.a.n 5
76.d even 2 1 3971.2.a.h 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.a.c 5 4.b odd 2 1
1881.2.a.k 5 12.b even 2 1
2299.2.a.n 5 44.c even 2 1
3344.2.a.t 5 1.a even 1 1 trivial
3971.2.a.h 5 76.d even 2 1
5225.2.a.h 5 20.d odd 2 1

Hecke kernels

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

\( T_{3}^{5} + T_{3}^{4} - 9T_{3}^{3} - 11T_{3}^{2} + 7T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{5} + 5T_{5}^{4} - 3T_{5}^{3} - 33T_{5}^{2} - 9T_{5} + 19 \) Copy content Toggle raw display
\( T_{7}^{5} + 6T_{7}^{4} - T_{7}^{3} - 62T_{7}^{2} - 119T_{7} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + T^{4} - 9 T^{3} - 11 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{5} + 5 T^{4} - 3 T^{3} - 33 T^{2} + \cdots + 19 \) Copy content Toggle raw display
$7$ \( T^{5} + 6 T^{4} - T^{3} - 62 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( (T + 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - 4 T^{4} - 9 T^{3} + 26 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( T^{5} + 4 T^{4} - 32 T^{3} - 64 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$19$ \( (T - 1)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} + 3 T^{4} - 76 T^{3} - 388 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$29$ \( T^{5} - 10 T^{4} - 37 T^{3} + \cdots + 490 \) Copy content Toggle raw display
$31$ \( T^{5} + 11 T^{4} - 3 T^{3} - 193 T^{2} + \cdots + 757 \) Copy content Toggle raw display
$37$ \( T^{5} - T^{4} - 80 T^{3} + 104 T^{2} + \cdots - 3088 \) Copy content Toggle raw display
$41$ \( T^{5} - 2 T^{4} - 189 T^{3} + \cdots - 4112 \) Copy content Toggle raw display
$43$ \( T^{5} + 20 T^{4} + 23 T^{3} + \cdots - 11266 \) Copy content Toggle raw display
$47$ \( T^{5} - 20 T^{4} + 28 T^{3} + \cdots - 13184 \) Copy content Toggle raw display
$53$ \( T^{5} + 14 T^{4} - 88 T^{3} + \cdots + 30304 \) Copy content Toggle raw display
$59$ \( T^{5} + 3 T^{4} - 164 T^{3} + \cdots + 2000 \) Copy content Toggle raw display
$61$ \( T^{5} + 10 T^{4} - 24 T^{3} + \cdots - 736 \) Copy content Toggle raw display
$67$ \( T^{5} + 9 T^{4} - 195 T^{3} + \cdots - 17689 \) Copy content Toggle raw display
$71$ \( T^{5} + 23 T^{4} - 17 T^{3} + \cdots - 19081 \) Copy content Toggle raw display
$73$ \( T^{5} - 340 T^{3} - 1168 T^{2} + \cdots + 155392 \) Copy content Toggle raw display
$79$ \( T^{5} + 44 T^{4} + 748 T^{3} + \cdots + 36800 \) Copy content Toggle raw display
$83$ \( T^{5} - 14 T^{4} - 69 T^{3} + \cdots + 3908 \) Copy content Toggle raw display
$89$ \( T^{5} + 27 T^{4} + 268 T^{3} + \cdots + 320 \) Copy content Toggle raw display
$97$ \( T^{5} - 15 T^{4} - 124 T^{3} + \cdots - 37456 \) Copy content Toggle raw display
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