Properties

Label 3344.2.a.q
Level $3344$
Weight $2$
Character orbit 3344.a
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 6 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)

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
2.66908
−0.523976
−2.14510
0 −2.66908 0 −4.12398 0 4.21417 0 4.12398 0
1.2 0 0.523976 0 2.72545 0 4.67750 0 −2.72545 0
1.3 0 2.14510 0 −1.60147 0 −2.89167 0 1.60147 0
\(n\): e.g. 2-40 or 990-1000
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.q 3
4.b odd 2 1 418.2.a.g 3
12.b even 2 1 3762.2.a.bg 3
44.c even 2 1 4598.2.a.bo 3
76.d even 2 1 7942.2.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.g 3 4.b odd 2 1
3344.2.a.q 3 1.a even 1 1 trivial
3762.2.a.bg 3 12.b even 2 1
4598.2.a.bo 3 44.c even 2 1
7942.2.a.bi 3 76.d even 2 1

Hecke kernels

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

\( T_{3}^{3} - 6 T_{3} + 3 \)
\( T_{5}^{3} + 3 T_{5}^{2} - 9 T_{5} - 18 \)
\( T_{7}^{3} - 6 T_{7}^{2} - 6 T_{7} + 57 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 3 - 6 T + T^{3} \)
$5$ \( -18 - 9 T + 3 T^{2} + T^{3} \)
$7$ \( 57 - 6 T - 6 T^{2} + T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( -29 - 18 T + T^{3} \)
$17$ \( -4 + 18 T + 9 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -76 + 66 T - 15 T^{2} + T^{3} \)
$29$ \( 147 - 24 T - 6 T^{2} + T^{3} \)
$31$ \( 134 - 51 T - 3 T^{2} + T^{3} \)
$37$ \( -96 - 24 T + 6 T^{2} + T^{3} \)
$41$ \( -122 - 45 T + 9 T^{2} + T^{3} \)
$43$ \( 184 + 93 T - 21 T^{2} + T^{3} \)
$47$ \( 672 - 60 T - 12 T^{2} + T^{3} \)
$53$ \( -452 - 90 T + 9 T^{2} + T^{3} \)
$59$ \( 64 - 24 T - 3 T^{2} + T^{3} \)
$61$ \( -192 + 156 T - 24 T^{2} + T^{3} \)
$67$ \( -123 - 96 T + T^{3} \)
$71$ \( -1306 + 3 T + 21 T^{2} + T^{3} \)
$73$ \( -12 - 6 T + 3 T^{2} + T^{3} \)
$79$ \( 944 - 132 T - 6 T^{2} + T^{3} \)
$83$ \( -1104 + 345 T - 33 T^{2} + T^{3} \)
$89$ \( 144 - 36 T - 6 T^{2} + T^{3} \)
$97$ \( 256 - 96 T - 12 T^{2} + T^{3} \)
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