Properties

Label 3344.2.a.u
Level $3344$
Weight $2$
Character orbit 3344.a
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.3979184.1
Defining polynomial: \( x^{5} - 2x^{4} - 9x^{3} + 18x^{2} + 5x - 12 \) 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_1 + 1) q^{5} + (\beta_1 - 2) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_1 + 1) q^{5} + (\beta_1 - 2) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{9} + q^{11} + ( - \beta_{2} + \beta_1) q^{13} + (\beta_{4} + \beta_{3}) q^{15} + 2 q^{17} - q^{19} + ( - 2 \beta_{4} + \beta_{3}) q^{21} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{23} + (\beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{25} + (\beta_{4} - \beta_{2} + \beta_1 - 2) q^{27} + (\beta_{3} + \beta_{2} - \beta_1 + 4) q^{29} + (\beta_{4} - 3 \beta_{3}) q^{31} + \beta_{4} q^{33} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{35} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{37} + (\beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{39} + (2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 6) q^{41} + (\beta_{4} + \beta_{3} - \beta_{2} - 1) q^{43} + ( - 2 \beta_{2} - \beta_1 + 2) q^{45} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2) q^{47} + (\beta_{4} - \beta_{3} + \beta_{2} - 4 \beta_1 + 2) q^{49} + 2 \beta_{4} q^{51} + 2 q^{53} + (\beta_1 + 1) q^{55} - \beta_{4} q^{57} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{59} + ( - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{61} + (3 \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{63} + ( - \beta_{3} - 2 \beta_{2} + 4) q^{65} + ( - 3 \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{67} + (\beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{69} + ( - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{71} + (2 \beta_{4} + 2 \beta_{3} + 2) q^{73} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{75} + (\beta_1 - 2) q^{77} + (2 \beta_1 - 2) q^{79} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 2) q^{81} + (\beta_{4} - 3 \beta_{3} - \beta_{2} - 1) q^{83} + (2 \beta_1 + 2) q^{85} + (4 \beta_{4} - 4 \beta_{3} + 3 \beta_1) q^{87} + (\beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_1 + 8) q^{89} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{91} + ( - 4 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 1) q^{93} + ( - \beta_1 - 1) q^{95} + (\beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{97} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9} + 5 q^{11} + 4 q^{13} - q^{15} + 10 q^{17} - 5 q^{19} + 2 q^{21} - 5 q^{23} + 6 q^{25} - 7 q^{27} + 16 q^{29} - q^{31} - q^{33} + 10 q^{35} - q^{37} + 4 q^{39} + 28 q^{41} - 4 q^{43} + 12 q^{45} + 4 q^{47} - q^{49} - 2 q^{51} + 10 q^{53} + 7 q^{55} + q^{57} + q^{59} - 16 q^{61} - 12 q^{63} + 24 q^{65} + 9 q^{67} - 7 q^{69} - 7 q^{71} + 8 q^{73} + 8 q^{75} - 8 q^{77} - 6 q^{79} + 5 q^{81} - 4 q^{83} + 14 q^{85} + 2 q^{87} + 31 q^{89} + 12 q^{91} - 7 q^{93} - 7 q^{95} + 13 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 9x^{3} + 18x^{2} + 5x - 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} - 7\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 10\nu^{2} + \nu + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - \nu^{3} - 9\nu^{2} + 8\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{4} - 8\beta_{3} + 10\beta_{2} - \beta _1 + 38 \) 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
−0.828542
1.85873
2.94407
−2.89005
0.915789
0 −2.88332 0 0.171458 0 −2.82854 0 5.31352 0
1.2 0 −2.35480 0 2.85873 0 −0.141265 0 2.54511 0
1.3 0 0.576583 0 3.94407 0 0.944071 0 −2.66755 0
1.4 0 0.804734 0 −1.89005 0 −4.89005 0 −2.35240 0
1.5 0 2.85680 0 1.91579 0 −1.08421 0 5.16133 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.u 5
4.b odd 2 1 1672.2.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1672.2.a.g 5 4.b odd 2 1
3344.2.a.u 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):

\( T_{3}^{5} + T_{3}^{4} - 11T_{3}^{3} - 7T_{3}^{2} + 23T_{3} - 9 \) Copy content Toggle raw display
\( T_{5}^{5} - 7T_{5}^{4} + 9T_{5}^{3} + 23T_{5}^{2} - 45T_{5} + 7 \) Copy content Toggle raw display
\( T_{7}^{5} + 8T_{7}^{4} + 15T_{7}^{3} - 4T_{7}^{2} - 15T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + T^{4} - 11 T^{3} - 7 T^{2} + \cdots - 9 \) Copy content Toggle raw display
$5$ \( T^{5} - 7 T^{4} + 9 T^{3} + 23 T^{2} + \cdots + 7 \) Copy content Toggle raw display
$7$ \( T^{5} + 8 T^{4} + 15 T^{3} - 4 T^{2} + \cdots - 2 \) Copy content Toggle raw display
$11$ \( (T - 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - 4 T^{4} - 29 T^{3} + 48 T^{2} + \cdots + 296 \) Copy content Toggle raw display
$17$ \( (T - 2)^{5} \) Copy content Toggle raw display
$19$ \( (T + 1)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} + 5 T^{4} - 32 T^{3} - 20 T^{2} + \cdots - 144 \) Copy content Toggle raw display
$29$ \( T^{5} - 16 T^{4} + 37 T^{3} + \cdots + 2224 \) Copy content Toggle raw display
$31$ \( T^{5} + T^{4} - 161 T^{3} + \cdots + 18999 \) Copy content Toggle raw display
$37$ \( T^{5} + T^{4} - 40 T^{3} - 80 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$41$ \( T^{5} - 28 T^{4} + 231 T^{3} + \cdots + 12148 \) Copy content Toggle raw display
$43$ \( T^{5} + 4 T^{4} - 53 T^{3} - 416 T^{2} + \cdots - 874 \) Copy content Toggle raw display
$47$ \( T^{5} - 4 T^{4} - 116 T^{3} + \cdots + 3712 \) Copy content Toggle raw display
$53$ \( (T - 2)^{5} \) Copy content Toggle raw display
$59$ \( T^{5} - T^{4} - 84 T^{3} - 100 T^{2} + \cdots + 3856 \) Copy content Toggle raw display
$61$ \( T^{5} + 16 T^{4} - 192 T^{3} + \cdots + 107968 \) Copy content Toggle raw display
$67$ \( T^{5} - 9 T^{4} - 97 T^{3} + 133 T^{2} + \cdots - 983 \) Copy content Toggle raw display
$71$ \( T^{5} + 7 T^{4} - 363 T^{3} + \cdots + 222813 \) Copy content Toggle raw display
$73$ \( T^{5} - 8 T^{4} - 140 T^{3} + \cdots - 1024 \) Copy content Toggle raw display
$79$ \( T^{5} + 6 T^{4} - 28 T^{3} - 88 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$83$ \( T^{5} + 4 T^{4} - 229 T^{3} + \cdots + 17222 \) Copy content Toggle raw display
$89$ \( T^{5} - 31 T^{4} + 232 T^{3} + \cdots + 52224 \) Copy content Toggle raw display
$97$ \( T^{5} - 13 T^{4} - 280 T^{3} + \cdots - 120176 \) Copy content Toggle raw display
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