Properties

Label 3344.2.a.x
Level $3344$
Weight $2$
Character orbit 3344.a
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 836)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{5} q^{5} - \beta_{4} q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{5} q^{5} - \beta_{4} q^{7} + (\beta_{2} + 2) q^{9} - q^{11} + \beta_{3} q^{13} + ( - \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{15} + ( - \beta_{5} - \beta_{4} - 2 \beta_1 + 3) q^{17} - q^{19} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{21} + (\beta_{5} + \beta_{2} + 2 \beta_1) q^{23} + (\beta_{2} + 2 \beta_1 + 4) q^{25} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 3) q^{27} + (\beta_{5} - \beta_{2} + \beta_1) q^{29} + ( - \beta_{5} - \beta_{3} + \beta_{2} + 4) q^{31} - \beta_1 q^{33} + ( - \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 4) q^{35} + (2 \beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 + 3) q^{37} + ( - 2 \beta_{5} + \beta_{4} - \beta_{2} - 2) q^{39} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{41} + (\beta_{2} + 2 \beta_1 - 2) q^{43} + ( - \beta_{5} + 2 \beta_{4} + 4 \beta_1 - 3) q^{45} + ( - 2 \beta_{3} - 2) q^{47} + ( - \beta_{2} + 2 \beta_1 + 7) q^{49} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 7) q^{51} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{53} + \beta_{5} q^{55} - \beta_1 q^{57} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 3) q^{59} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3) q^{61} + (3 \beta_{5} - \beta_{2} - 2 \beta_1 + 1) q^{63} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_1 + 1) q^{65} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{67} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{69} + ( - 3 \beta_1 + 6) q^{71} + (2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 4) q^{73} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 7) q^{75} + \beta_{4} q^{77} + (3 \beta_{5} + \beta_{4} + 2 \beta_{2} + 4 \beta_1 - 1) q^{79} + (\beta_{5} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{81} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 3) q^{83} + ( - \beta_{5} - \beta_{4} - 4 \beta_{2} + 1) q^{85} + (2 \beta_{5} - \beta_{4} - 2 \beta_1 + 6) q^{87} + ( - 2 \beta_{5} + \beta_{4} - \beta_{2} - 3) q^{89} + ( - 4 \beta_{5} - \beta_{4} - \beta_{2} - 7 \beta_1 - 1) q^{91} + (2 \beta_{2} + 6 \beta_1 + 1) q^{93} + \beta_{5} q^{95} + ( - \beta_{4} + \beta_{2} - 7) q^{97} + ( - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + \nu^{4} - 12\nu^{3} - 5\nu^{2} + 28\nu - 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} + 15\nu^{3} - 25\nu^{2} - 43\nu + 48 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{5} + \nu^{4} + 24\nu^{3} - 20\nu^{2} - 50\nu + 42 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{3} + 7\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 2\beta_{3} + 10\beta_{2} - 2\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{5} + 12\beta_{4} - 11\beta_{3} - 5\beta_{2} + 58\beta _1 - 46 \) 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
−3.25580
−1.64714
0.658537
1.24102
2.18868
2.81471
0 −3.25580 0 −2.84405 0 1.37411 0 7.60023 0
1.2 0 −1.64714 0 1.84900 0 −3.60453 0 −0.286920 0
1.3 0 0.658537 0 −2.39807 0 −4.45908 0 −2.56633 0
1.4 0 1.24102 0 2.83235 0 4.46564 0 −1.45986 0
1.5 0 2.18868 0 −3.62873 0 4.31127 0 1.79031 0
1.6 0 2.81471 0 4.18951 0 −4.08740 0 4.92257 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.x 6
4.b odd 2 1 836.2.a.d 6
12.b even 2 1 7524.2.a.r 6
44.c even 2 1 9196.2.a.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
836.2.a.d 6 4.b odd 2 1
3344.2.a.x 6 1.a even 1 1 trivial
7524.2.a.r 6 12.b even 2 1
9196.2.a.k 6 44.c even 2 1

Hecke kernels

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

\( T_{3}^{6} - 2T_{3}^{5} - 12T_{3}^{4} + 28T_{3}^{3} + 16T_{3}^{2} - 60T_{3} + 27 \) Copy content Toggle raw display
\( T_{5}^{6} - 28T_{5}^{4} - 6T_{5}^{3} + 228T_{5}^{2} + 48T_{5} - 543 \) Copy content Toggle raw display
\( T_{7}^{6} + 2T_{7}^{5} - 43T_{7}^{4} - 78T_{7}^{3} + 547T_{7}^{2} + 760T_{7} - 1738 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} - 12 T^{4} + 28 T^{3} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( T^{6} - 28 T^{4} - 6 T^{3} + 228 T^{2} + \cdots - 543 \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} - 43 T^{4} + \cdots - 1738 \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} - 49 T^{4} + \cdots + 2462 \) Copy content Toggle raw display
$17$ \( T^{6} - 12 T^{5} - 18 T^{4} + \cdots - 2592 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 2 T^{5} - 75 T^{4} + \cdots - 1584 \) Copy content Toggle raw display
$29$ \( T^{6} - 4 T^{5} - 69 T^{4} + \cdots - 2682 \) Copy content Toggle raw display
$31$ \( T^{6} - 20 T^{5} + 88 T^{4} + \cdots - 7593 \) Copy content Toggle raw display
$37$ \( T^{6} - 22 T^{5} + 67 T^{4} + \cdots + 91824 \) Copy content Toggle raw display
$41$ \( T^{6} + 2 T^{5} - 159 T^{4} + \cdots + 24912 \) Copy content Toggle raw display
$43$ \( T^{6} + 10 T^{5} - 25 T^{4} + \cdots + 4900 \) Copy content Toggle raw display
$47$ \( T^{6} + 16 T^{5} - 96 T^{4} + \cdots + 328896 \) Copy content Toggle raw display
$53$ \( T^{6} - 12 T^{5} - 96 T^{4} + \cdots - 31104 \) Copy content Toggle raw display
$59$ \( T^{6} - 14 T^{5} - 81 T^{4} + \cdots + 31536 \) Copy content Toggle raw display
$61$ \( T^{6} - 12 T^{5} - 214 T^{4} + \cdots + 109728 \) Copy content Toggle raw display
$67$ \( T^{6} - 12 T^{5} - 184 T^{4} + \cdots + 253887 \) Copy content Toggle raw display
$71$ \( T^{6} - 30 T^{5} + 252 T^{4} + \cdots + 2187 \) Copy content Toggle raw display
$73$ \( T^{6} - 24 T^{5} - 52 T^{4} + \cdots - 217728 \) Copy content Toggle raw display
$79$ \( T^{6} - 370 T^{4} + 1184 T^{3} + \cdots + 428832 \) Copy content Toggle raw display
$83$ \( T^{6} - 14 T^{5} - 135 T^{4} + \cdots - 177642 \) Copy content Toggle raw display
$89$ \( T^{6} + 14 T^{5} - 173 T^{4} + \cdots + 19776 \) Copy content Toggle raw display
$97$ \( T^{6} + 46 T^{5} + 791 T^{4} + \cdots + 752 \) Copy content Toggle raw display
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