Properties

Label 3344.2.a.z
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: 6.6.106392688.1
Defining polynomial: \( x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + (\beta_{5} - 1) q^{5} + \beta_{5} q^{7} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + (\beta_{5} - 1) q^{5} + \beta_{5} q^{7} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{9} + q^{11} + ( - 2 \beta_{4} + \beta_{3} - \beta_1) q^{13} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{15} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{17} - q^{19} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{21} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{23} + ( - 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 2) q^{25} + (2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2) q^{27} + (\beta_{5} + \beta_{4} + \beta_{3} - 1) q^{29} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{31} + ( - \beta_1 + 1) q^{33} + ( - \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 6) q^{35} + (\beta_{5} + 3 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{37} + (\beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{39} + (4 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{41} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{43} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{45} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 2) q^{47} + ( - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{49} + (4 \beta_{4} - 3 \beta_{3} - \beta_{2} + 5) q^{51} + (3 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{53} + (\beta_{5} - 1) q^{55} + (\beta_1 - 1) q^{57} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 5) q^{59} + (2 \beta_{4} - 2 \beta_{3} + 2) q^{61} + (\beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{63} + (\beta_{5} - 3 \beta_{4} - \beta_1 + 1) q^{65} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 4) q^{67} + ( - 3 \beta_{5} - 3 \beta_{4} - \beta_{2} - \beta_1 + 1) q^{69} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{71} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{73} + ( - 2 \beta_{5} + 3 \beta_{3} + 4 \beta_{2} + \beta_1 + 6) q^{75} + \beta_{5} q^{77} + ( - 2 \beta_{5} - 4 \beta_1 + 6) q^{79} + (\beta_{5} + \beta_{4} - 3 \beta_{2} - 3 \beta_1 - 1) q^{81} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{83} + ( - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2}) q^{85} + ( - \beta_{5} - 2 \beta_{3} - 2 \beta_{2}) q^{87} + (3 \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_1 - 4) q^{89} + (\beta_{5} - 5 \beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{91} + (4 \beta_{5} - \beta_{3} - \beta_{2} - 4) q^{93} + ( - \beta_{5} + 1) q^{95} + ( - \beta_{5} - \beta_{4} + 5 \beta_{3} + \beta_1) q^{97} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} - q^{17} - 6 q^{19} + 5 q^{21} + 4 q^{23} + 5 q^{25} + 16 q^{27} - 7 q^{29} + q^{31} + 4 q^{33} + 32 q^{35} + 5 q^{37} + 13 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 7 q^{49} + 35 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 18 q^{59} + 16 q^{61} + 6 q^{63} + 10 q^{65} + 18 q^{67} - 2 q^{69} + 7 q^{71} - 21 q^{73} + 23 q^{75} + 3 q^{77} + 22 q^{79} - 10 q^{81} + 16 q^{83} + 3 q^{87} - 13 q^{89} + 7 q^{91} - 9 q^{93} + 3 q^{95} - 15 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 10\nu^{2} + 11\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 12\nu^{2} + 9\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} + 5\nu^{3} - 24\nu^{2} - 5\nu + 22 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 8\beta_{4} - 7\beta_{3} + \beta_{2} + 10\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 13\beta_{4} - 9\beta_{3} + 9\beta_{2} + 41\beta _1 + 20 \) 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
2.80207
2.50058
0.972165
−0.870457
−1.25613
−2.14822
0 −1.80207 0 0.702208 0 1.70221 0 0.247456 0
1.2 0 −1.50058 0 −2.88388 0 −1.88388 0 −0.748274 0
1.3 0 0.0278351 0 −0.122408 0 0.877592 0 −2.99923 0
1.4 0 1.87046 0 2.83301 0 3.83301 0 0.498611 0
1.5 0 2.25613 0 −4.20604 0 −3.20604 0 2.09011 0
1.6 0 3.14822 0 0.677107 0 1.67711 0 6.91132 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.z 6
4.b odd 2 1 1672.2.a.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1672.2.a.i 6 4.b odd 2 1
3344.2.a.z 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):

\( T_{3}^{6} - 4T_{3}^{5} - 4T_{3}^{4} + 24T_{3}^{3} + 2T_{3}^{2} - 36T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} + 3T_{5}^{5} - 13T_{5}^{4} - 23T_{5}^{3} + 41T_{5}^{2} - 11T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{6} - 3T_{7}^{5} - 13T_{7}^{4} + 39T_{7}^{3} + 17T_{7}^{2} - 101T_{7} + 58 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} - 4 T^{4} + 24 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} - 13 T^{4} - 23 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} - 13 T^{4} + 39 T^{3} + \cdots + 58 \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} - 45 T^{4} - 115 T^{3} + \cdots - 212 \) Copy content Toggle raw display
$17$ \( T^{6} + T^{5} - 82 T^{4} - 136 T^{3} + \cdots - 3232 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 4 T^{5} - 77 T^{4} + \cdots - 2096 \) Copy content Toggle raw display
$29$ \( T^{6} + 7 T^{5} - 27 T^{4} - 133 T^{3} + \cdots - 844 \) Copy content Toggle raw display
$31$ \( T^{6} - T^{5} - 71 T^{4} + 277 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$37$ \( T^{6} - 5 T^{5} - 130 T^{4} + \cdots + 13856 \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} - 185 T^{4} + \cdots - 151432 \) Copy content Toggle raw display
$43$ \( T^{6} - 16 T^{5} + 11 T^{4} + \cdots - 152 \) Copy content Toggle raw display
$47$ \( T^{6} - 4 T^{5} - 84 T^{4} + \cdots - 8192 \) Copy content Toggle raw display
$53$ \( T^{6} + 11 T^{5} - 90 T^{4} + \cdots - 40736 \) Copy content Toggle raw display
$59$ \( T^{6} - 18 T^{5} - 67 T^{4} + \cdots - 6736 \) Copy content Toggle raw display
$61$ \( T^{6} - 16 T^{5} + 48 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$67$ \( T^{6} - 18 T^{5} + 24 T^{4} + \cdots + 13891 \) Copy content Toggle raw display
$71$ \( T^{6} - 7 T^{5} - 141 T^{4} + 1249 T^{3} + \cdots - 34 \) Copy content Toggle raw display
$73$ \( T^{6} + 21 T^{5} - 12 T^{4} + \cdots + 99328 \) Copy content Toggle raw display
$79$ \( T^{6} - 22 T^{5} - 12 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( T^{6} - 16 T^{5} - 125 T^{4} + \cdots + 122336 \) Copy content Toggle raw display
$89$ \( T^{6} + 13 T^{5} - 144 T^{4} + \cdots + 38912 \) Copy content Toggle raw display
$97$ \( T^{6} + 15 T^{5} - 248 T^{4} + \cdots + 424064 \) Copy content Toggle raw display
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