Properties

Label 3344.2.a.w
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: 6.6.744786576.1
Defining polynomial: \( x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{5} + 1) q^{5} + (\beta_{2} - \beta_1 + 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{5} + 1) q^{5} + (\beta_{2} - \beta_1 + 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 3) q^{9} + q^{11} + ( - \beta_{5} - \beta_{3} + 1) q^{13} + ( - \beta_{4} - \beta_{2} - \beta_1 - 2) q^{15} + 2 \beta_{4} q^{17} + q^{19} + ( - 3 \beta_{4} - \beta_{2} + 2) q^{21} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{23} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 3) q^{25} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - 3 \beta_1 - 1) q^{27} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{29} + (\beta_{4} + \beta_{2} + \beta_1) q^{31} - \beta_1 q^{33} + (2 \beta_{5} - \beta_{4} + \beta_{3} + 1) q^{35} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{37} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 3) q^{39} + ( - \beta_{5} + \beta_{3} + 1) q^{41} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{43} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 + 9) q^{45} + ( - 2 \beta_{5} + 2 \beta_{4}) q^{47} + ( - 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{49} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{2} + 4 \beta_1 - 4) q^{51} + (2 \beta_{5} - 2 \beta_1 + 4) q^{53} + (\beta_{5} + 1) q^{55} - \beta_1 q^{57} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1 - 1) q^{59} + ( - 2 \beta_{5} - 2 \beta_1 + 4) q^{61} + (4 \beta_{5} - \beta_{4} + \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 7) q^{63} + (4 \beta_{5} - \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 6) q^{65} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{67} + ( - \beta_{5} + 3 \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1 - 7) q^{69} + ( - 2 \beta_{5} - 2 \beta_{2} + \beta_1) q^{71} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 4) q^{73} + ( - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 7) q^{75} + (\beta_{2} - \beta_1 + 1) q^{77} + ( - 2 \beta_{4} + 2 \beta_1 + 2) q^{79} + (2 \beta_{5} - 3 \beta_{4} + \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 12) q^{81} + (\beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{83} + (2 \beta_{4} + 2 \beta_1 + 2) q^{85} + ( - 4 \beta_{4} - 5 \beta_{2} - \beta_1 + 3) q^{87} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{89} + ( - 3 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{91} + ( - 3 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 12) q^{93} + (\beta_{5} + 1) q^{95} + ( - 3 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 5) q^{97} + (\beta_{5} - \beta_{4} + \beta_{3} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} + 6 q^{11} + 8 q^{13} - 9 q^{15} - 2 q^{17} + 6 q^{19} + 18 q^{21} - 5 q^{23} + 13 q^{25} - 7 q^{27} + 10 q^{29} - 3 q^{31} - q^{33} + 4 q^{35} + 7 q^{37} + 10 q^{39} + 6 q^{41} - 16 q^{43} + 42 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{53} + 5 q^{55} - q^{57} - 15 q^{59} + 24 q^{61} + 20 q^{63} - 28 q^{65} - 25 q^{67} - 33 q^{69} + 9 q^{71} - 26 q^{73} - 28 q^{75} + 2 q^{77} + 16 q^{79} + 58 q^{81} + 2 q^{83} + 12 q^{85} + 36 q^{87} + 7 q^{89} - 8 q^{91} - 55 q^{93} + 5 q^{95} + 37 q^{97} + 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 24\nu^{4} + \nu^{3} - 253\nu^{2} - 48\nu + 137 ) / 97 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{5} + \nu^{4} + 93\nu^{3} + 42\nu^{2} - 487\nu - 354 ) / 97 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{5} - 2\nu^{4} - 89\nu^{3} + 13\nu^{2} + 101\nu + 29 ) / 97 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\nu^{5} - 3\nu^{4} - 182\nu^{3} + 68\nu^{2} + 588\nu - 199 ) / 97 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{3} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + 2\beta_{4} + \beta_{3} + 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{5} - 12\beta_{4} + 10\beta_{3} + 4\beta_{2} - 2\beta _1 + 57 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{5} + 33\beta_{4} + 12\beta_{3} + \beta_{2} + 87\beta _1 + 12 \) 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
3.27915
2.82559
0.837309
−0.566270
−2.05861
−3.31717
0 −3.27915 0 3.53478 0 2.34459 0 7.75283 0
1.2 0 −2.82559 0 −0.343563 0 −4.77460 0 4.98394 0
1.3 0 −0.837309 0 3.44987 0 −0.535976 0 −2.29891 0
1.4 0 0.566270 0 −3.92909 0 2.44546 0 −2.67934 0
1.5 0 2.05861 0 0.680036 0 −1.59123 0 1.23787 0
1.6 0 3.31717 0 1.60797 0 4.11176 0 8.00361 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.w 6
4.b odd 2 1 836.2.a.e 6
12.b even 2 1 7524.2.a.q 6
44.c even 2 1 9196.2.a.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
836.2.a.e 6 4.b odd 2 1
3344.2.a.w 6 1.a even 1 1 trivial
7524.2.a.q 6 12.b even 2 1
9196.2.a.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):

\( T_{3}^{6} + T_{3}^{5} - 17T_{3}^{4} - 13T_{3}^{3} + 69T_{3}^{2} + 21T_{3} - 30 \) Copy content Toggle raw display
\( T_{5}^{6} - 5T_{5}^{5} - 9T_{5}^{4} + 77T_{5}^{3} - 99T_{5}^{2} + 9T_{5} + 18 \) Copy content Toggle raw display
\( T_{7}^{6} - 2T_{7}^{5} - 25T_{7}^{4} + 58T_{7}^{3} + 81T_{7}^{2} - 156T_{7} - 96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} - 17 T^{4} - 13 T^{3} + \cdots - 30 \) Copy content Toggle raw display
$5$ \( T^{6} - 5 T^{5} - 9 T^{4} + 77 T^{3} + \cdots + 18 \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} - 25 T^{4} + 58 T^{3} + \cdots - 96 \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 8 T^{5} - 23 T^{4} + 246 T^{3} + \cdots - 40 \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} - 72 T^{4} - 48 T^{3} + \cdots - 256 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 5 T^{5} - 88 T^{4} + \cdots - 7232 \) Copy content Toggle raw display
$29$ \( T^{6} - 10 T^{5} - 91 T^{4} + \cdots - 11424 \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} - 79 T^{4} - 387 T^{3} + \cdots - 142 \) Copy content Toggle raw display
$37$ \( T^{6} - 7 T^{5} - 110 T^{4} + \cdots - 5856 \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{5} - 43 T^{4} + 464 T^{3} + \cdots - 896 \) Copy content Toggle raw display
$43$ \( T^{6} + 16 T^{5} - 49 T^{4} + \cdots + 39880 \) Copy content Toggle raw display
$47$ \( T^{6} - 124 T^{4} - 176 T^{3} + \cdots + 10752 \) Copy content Toggle raw display
$53$ \( T^{6} - 20 T^{5} + 52 T^{4} + \cdots + 32960 \) Copy content Toggle raw display
$59$ \( T^{6} + 15 T^{5} - 106 T^{4} + \cdots - 28064 \) Copy content Toggle raw display
$61$ \( T^{6} - 24 T^{5} + 60 T^{4} + \cdots + 25152 \) Copy content Toggle raw display
$67$ \( T^{6} + 25 T^{5} + 77 T^{4} + \cdots - 56022 \) Copy content Toggle raw display
$71$ \( T^{6} - 9 T^{5} - 157 T^{4} + \cdots - 82154 \) Copy content Toggle raw display
$73$ \( T^{6} + 26 T^{5} + 20 T^{4} + \cdots + 60672 \) Copy content Toggle raw display
$79$ \( T^{6} - 16 T^{5} - 20 T^{4} + \cdots - 4096 \) Copy content Toggle raw display
$83$ \( T^{6} - 2 T^{5} - 77 T^{4} + \cdots - 7968 \) Copy content Toggle raw display
$89$ \( T^{6} - 7 T^{5} - 226 T^{4} + \cdots + 1152 \) Copy content Toggle raw display
$97$ \( T^{6} - 37 T^{5} + 334 T^{4} + \cdots + 405664 \) Copy content Toggle raw display
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