Properties

Label 3344.2.a.ba
Level $3344$
Weight $2$
Character orbit 3344.a
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{4} q^{5} + (\beta_{5} - \beta_{2} - 2) q^{7} + ( - \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{4} q^{5} + (\beta_{5} - \beta_{2} - 2) q^{7} + ( - \beta_{6} + 2) q^{9} + q^{11} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{13} + ( - \beta_{5} + \beta_{4} - \beta_1 - 1) q^{15} + (\beta_{6} + \beta_{5} - \beta_{3}) q^{17} - q^{19} + (\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 3) q^{21} + ( - 2 \beta_{5} + \beta_{3} + \beta_{2} - 1) q^{23} + (\beta_{6} + 2 \beta_{3}) q^{25} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{27} + (\beta_{3} - 3) q^{29} + ( - \beta_{5} + \beta_{4} + \beta_1 - 3) q^{31} + \beta_{2} q^{33} + (\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{35} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{2} + 1) q^{37} + (2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 5) q^{39} + (\beta_{6} + \beta_{4} - \beta_{2} - \beta_1 - 2) q^{41} + (2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{43} + ( - \beta_{6} - \beta_{3} - \beta_{2}) q^{45} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{47} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} + 4) q^{49} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 4) q^{51} + (2 \beta_{4} + 2 \beta_{3}) q^{53} - \beta_{4} q^{55} - \beta_{2} q^{57} + ( - \beta_{6} - \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{59} + (\beta_{6} + \beta_{5} - \beta_{3} - 2 \beta_1 + 2) q^{61} + (\beta_{6} + \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 1) q^{63} + ( - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{65} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 1) q^{67} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 1) q^{69} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{71} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{2}) q^{73} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 4) q^{75} + (\beta_{5} - \beta_{2} - 2) q^{77} + (\beta_{6} + \beta_{5} + \beta_{3} - 4 \beta_1 - 8) q^{79} + ( - 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 2) q^{81} + ( - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{83} + (\beta_{6} + \beta_{5} - \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{85} + (\beta_{5} - 3 \beta_{2} - 2) q^{87} + (\beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + \beta_{2} - 1) q^{89} + ( - 3 \beta_{6} - \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 1) q^{91} + (\beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - 5 \beta_{2} - 2) q^{93} + \beta_{4} q^{95} + (\beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 4 \beta_1 - 3) q^{97} + ( - \beta_{6} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 7\nu^{2} - 2\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 9\nu^{2} + 2\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 9\nu^{3} + 14\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 10\nu^{4} + 23\nu^{2} - 4\nu - 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 12\nu^{4} + 4\nu^{3} - 41\nu^{2} - 20\nu + 34 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{3} + 9\beta_{2} + 2\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{6} + 9\beta_{5} + 2\beta_{4} + 9\beta_{3} + 31\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{5} + 47\beta_{3} + 67\beta_{2} + 24\beta _1 + 158 \) 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
1.19313
−2.03821
−1.45416
2.61330
0.456669
2.78328
−2.55401
0 −3.16232 0 1.08235 0 1.30958 0 7.00029 0
1.2 0 −1.87275 0 −3.24760 0 −1.92338 0 0.507178 0
1.3 0 −1.71116 0 2.59296 0 2.00933 0 −0.0719365 0
1.4 0 −1.19599 0 4.07680 0 −3.61829 0 −1.56960 0
1.5 0 0.835165 0 0.221953 0 −4.69915 0 −2.30250 0
1.6 0 2.10880 0 −2.97131 0 1.34513 0 1.44705 0
1.7 0 2.99825 0 0.244850 0 −4.42321 0 5.98952 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.ba 7
4.b odd 2 1 209.2.a.d 7
12.b even 2 1 1881.2.a.p 7
20.d odd 2 1 5225.2.a.n 7
44.c even 2 1 2299.2.a.q 7
76.d even 2 1 3971.2.a.i 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.a.d 7 4.b odd 2 1
1881.2.a.p 7 12.b even 2 1
2299.2.a.q 7 44.c even 2 1
3344.2.a.ba 7 1.a even 1 1 trivial
3971.2.a.i 7 76.d even 2 1
5225.2.a.n 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):

\( T_{3}^{7} + 2T_{3}^{6} - 14T_{3}^{5} - 28T_{3}^{4} + 46T_{3}^{3} + 100T_{3}^{2} - 17T_{3} - 64 \) Copy content Toggle raw display
\( T_{5}^{7} - 2T_{5}^{6} - 20T_{5}^{5} + 34T_{5}^{4} + 88T_{5}^{3} - 156T_{5}^{2} + 57T_{5} - 6 \) Copy content Toggle raw display
\( T_{7}^{7} + 10T_{7}^{6} + 17T_{7}^{5} - 86T_{7}^{4} - 185T_{7}^{3} + 316T_{7}^{2} + 394T_{7} - 512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 2 T^{6} - 14 T^{5} - 28 T^{4} + \cdots - 64 \) Copy content Toggle raw display
$5$ \( T^{7} - 2 T^{6} - 20 T^{5} + 34 T^{4} + \cdots - 6 \) Copy content Toggle raw display
$7$ \( T^{7} + 10 T^{6} + 17 T^{5} + \cdots - 512 \) Copy content Toggle raw display
$11$ \( (T - 1)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} + 4 T^{6} - 51 T^{5} + \cdots - 5716 \) Copy content Toggle raw display
$17$ \( T^{7} - 2 T^{6} - 70 T^{5} + \cdots - 17088 \) Copy content Toggle raw display
$19$ \( (T + 1)^{7} \) Copy content Toggle raw display
$23$ \( T^{7} + 10 T^{6} - 51 T^{5} + \cdots - 1920 \) Copy content Toggle raw display
$29$ \( T^{7} + 18 T^{6} + 117 T^{5} + \cdots - 276 \) Copy content Toggle raw display
$31$ \( T^{7} + 24 T^{6} + 214 T^{5} + 904 T^{4} + \cdots - 4 \) Copy content Toggle raw display
$37$ \( T^{7} - 121 T^{5} - 194 T^{4} + \cdots - 8992 \) Copy content Toggle raw display
$41$ \( T^{7} + 12 T^{6} - 5 T^{5} + \cdots - 1824 \) Copy content Toggle raw display
$43$ \( T^{7} + 2 T^{6} - 89 T^{5} + \cdots - 4976 \) Copy content Toggle raw display
$47$ \( T^{7} + 8 T^{6} - 152 T^{5} + \cdots - 79872 \) Copy content Toggle raw display
$53$ \( T^{7} - 2 T^{6} - 160 T^{5} + \cdots + 768 \) Copy content Toggle raw display
$59$ \( T^{7} - 10 T^{6} - 345 T^{5} + \cdots + 6552192 \) Copy content Toggle raw display
$61$ \( T^{7} - 14 T^{6} - 34 T^{5} + \cdots - 36544 \) Copy content Toggle raw display
$67$ \( T^{7} + 8 T^{6} - 170 T^{5} + \cdots - 13544 \) Copy content Toggle raw display
$71$ \( T^{7} + 10 T^{6} - 134 T^{5} + \cdots - 39756 \) Copy content Toggle raw display
$73$ \( T^{7} + 6 T^{6} - 220 T^{5} + \cdots + 67328 \) Copy content Toggle raw display
$79$ \( T^{7} + 52 T^{6} + 970 T^{5} + \cdots + 203264 \) Copy content Toggle raw display
$83$ \( T^{7} - 10 T^{6} - 219 T^{5} + \cdots - 576936 \) Copy content Toggle raw display
$89$ \( T^{7} - 401 T^{5} - 698 T^{4} + \cdots - 8199552 \) Copy content Toggle raw display
$97$ \( T^{7} + 24 T^{6} - 189 T^{5} + \cdots - 17393056 \) Copy content Toggle raw display
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