Properties

Label 3344.2.a.y
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: 6.6.57500224.1
Defining polynomial: \( x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 18x + 2 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{4} - 1) q^{5} + ( - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{2} - 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{4} - 1) q^{5} + ( - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{2} - 2 \beta_1 + 1) q^{9} - q^{11} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{13} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_1 - 1) q^{15} + ( - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{17} + q^{19} + (\beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{21} + (2 \beta_{5} - \beta_{4} + \beta_{2} + 2) q^{23} + (\beta_{2} - 2 \beta_1 + 1) q^{25} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{27} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{29} + ( - 3 \beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{31} + (\beta_1 - 1) q^{33} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 2) q^{35} + ( - \beta_{5} + 4 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{37} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 3 \beta_1 + 2) q^{39} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2) q^{41} + ( - 2 \beta_{5} - \beta_{2} - 2 \beta_1 + 5) q^{43} + (2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{45} + ( - 2 \beta_{4} + 4 \beta_1 + 2) q^{47} + (2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + \beta_{2} + 4) q^{49} + ( - \beta_{5} + 3 \beta_{4} + \beta_{2} + \beta_1 + 3) q^{51} + ( - 2 \beta_{4} + 2 \beta_{2} - 2) q^{53} + (\beta_{4} + 1) q^{55} + ( - \beta_1 + 1) q^{57} + (\beta_{5} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{59} + ( - \beta_{5} + \beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{61} + (2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 9) q^{63} + (2 \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 7) q^{65} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 7) q^{67} + (\beta_{5} - 4 \beta_{4} - 3 \beta_1 + 1) q^{69} + ( - 2 \beta_{5} - \beta_1 + 3) q^{71} + ( - 2 \beta_{5} + 2 \beta_{3} - 6 \beta_1 + 4) q^{73} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 6) q^{75} + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{77} + (\beta_{5} - \beta_{4} - \beta_{2} + 5 \beta_1 - 3) q^{79} + (\beta_{4} - 2 \beta_{3} - \beta_{2} + 2) q^{81} + (\beta_{5} + \beta_{4} - 2 \beta_{2} - 3 \beta_1 + 6) q^{83} + (\beta_{5} + \beta_{4} + \beta_{2} + 5 \beta_1 - 3) q^{85} + (\beta_{5} + 3 \beta_{2} - 3 \beta_1 + 5) q^{87} + (3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_1 + 1) q^{89} + (5 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 12) q^{91} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 3) q^{93} + ( - \beta_{4} - 1) q^{95} + ( - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{97} + ( - \beta_{2} + 2 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 18x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 6\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - \nu^{4} - 7\nu^{3} + 3\nu^{2} + 10\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} + 6\beta_{2} + 2\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + \beta_{4} + 9\beta_{3} + 10\beta_{2} + 20\beta _1 + 14 \) 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
2.72058
1.64887
1.12938
0.121186
−1.67361
−1.94641
0 −1.72058 0 −2.22719 0 1.96772 0 −0.0396114 0
1.2 0 −0.648873 0 1.55597 0 3.00900 0 −2.57896 0
1.3 0 −0.129383 0 −1.42012 0 −5.10098 0 −2.98326 0
1.4 0 0.878814 0 −1.66503 0 −1.34977 0 −2.22769 0
1.5 0 2.67361 0 3.02460 0 0.969288 0 4.14820 0
1.6 0 2.94641 0 −3.26823 0 4.50474 0 5.68132 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.y 6
4.b odd 2 1 1672.2.a.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1672.2.a.h 6 4.b odd 2 1
3344.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):

\( T_{3}^{6} - 4T_{3}^{5} - 2T_{3}^{4} + 16T_{3}^{3} - 8T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{6} + 4T_{5}^{5} - 8T_{5}^{4} - 46T_{5}^{3} - 20T_{5}^{2} + 88T_{5} + 81 \) Copy content Toggle raw display
\( T_{7}^{6} - 4T_{7}^{5} - 23T_{7}^{4} + 116T_{7}^{3} - 65T_{7}^{2} - 206T_{7} + 178 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} - 2 T^{4} + 16 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 4 T^{5} - 8 T^{4} - 46 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{6} - 4 T^{5} - 23 T^{4} + 116 T^{3} + \cdots + 178 \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} - 67 T^{4} + 90 T^{3} + \cdots - 962 \) Copy content Toggle raw display
$17$ \( T^{6} + 10 T^{5} - 6 T^{4} + \cdots + 1184 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 14 T^{5} + 5 T^{4} + \cdots - 4304 \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} - 65 T^{4} + \cdots + 11038 \) Copy content Toggle raw display
$31$ \( T^{6} + 4 T^{5} - 156 T^{4} + \cdots + 82899 \) Copy content Toggle raw display
$37$ \( T^{6} + 4 T^{5} - 195 T^{4} + \cdots - 48848 \) Copy content Toggle raw display
$41$ \( T^{6} + 8 T^{5} - 97 T^{4} + \cdots - 4160 \) Copy content Toggle raw display
$43$ \( T^{6} - 26 T^{5} + 163 T^{4} + \cdots + 61972 \) Copy content Toggle raw display
$47$ \( T^{6} - 24 T^{5} + 64 T^{4} + \cdots - 32448 \) Copy content Toggle raw display
$53$ \( T^{6} + 8 T^{5} - 72 T^{4} + \cdots - 18176 \) Copy content Toggle raw display
$59$ \( T^{6} - 4 T^{5} - 111 T^{4} + \cdots - 5200 \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} - 262 T^{4} + \cdots - 408608 \) Copy content Toggle raw display
$67$ \( T^{6} - 44 T^{5} + 660 T^{4} + \cdots - 96461 \) Copy content Toggle raw display
$71$ \( T^{6} - 16 T^{5} + 42 T^{4} + \cdots + 2319 \) Copy content Toggle raw display
$73$ \( T^{6} - 12 T^{5} - 276 T^{4} + \cdots - 470656 \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} - 186 T^{4} + \cdots - 13152 \) Copy content Toggle raw display
$83$ \( T^{6} - 28 T^{5} + 117 T^{4} + \cdots + 516266 \) Copy content Toggle raw display
$89$ \( T^{6} - 239 T^{4} - 704 T^{3} + \cdots + 69376 \) Copy content Toggle raw display
$97$ \( T^{6} - 4 T^{5} - 259 T^{4} + \cdots + 144880 \) Copy content Toggle raw display
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