Properties

Label 3344.2.a.k
Level $3344$
Weight $2$
Character orbit 3344.a
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) 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 418)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + 2 q^{5} + (\beta - 2) q^{7} + (\beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + 2 q^{5} + (\beta - 2) q^{7} + (\beta + 1) q^{9} - q^{11} + (\beta - 2) q^{13} - 2 \beta q^{15} + ( - \beta + 2) q^{17} + q^{19} + (\beta - 4) q^{21} + (\beta - 4) q^{23} - q^{25} + (\beta - 4) q^{27} + ( - 3 \beta + 2) q^{29} - 2 q^{31} + \beta q^{33} + (2 \beta - 4) q^{35} - 2 \beta q^{37} + (\beta - 4) q^{39} + 2 q^{41} + (2 \beta + 2) q^{45} - 8 q^{47} + ( - 3 \beta + 1) q^{49} + ( - \beta + 4) q^{51} + 5 \beta q^{53} - 2 q^{55} - \beta q^{57} + 3 \beta q^{59} + (4 \beta - 4) q^{61} + 2 q^{63} + (2 \beta - 4) q^{65} + 3 \beta q^{67} + (3 \beta - 4) q^{69} - 6 q^{71} + ( - 3 \beta - 2) q^{73} + \beta q^{75} + ( - \beta + 2) q^{77} + 4 q^{79} - 7 q^{81} + ( - 4 \beta - 4) q^{83} + ( - 2 \beta + 4) q^{85} + (\beta + 12) q^{87} + (2 \beta - 6) q^{89} + ( - 3 \beta + 8) q^{91} + 2 \beta q^{93} + 2 q^{95} + ( - 2 \beta - 2) q^{97} + ( - \beta - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{11} - 3 q^{13} - 2 q^{15} + 3 q^{17} + 2 q^{19} - 7 q^{21} - 7 q^{23} - 2 q^{25} - 7 q^{27} + q^{29} - 4 q^{31} + q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 4 q^{41} + 6 q^{45} - 16 q^{47} - q^{49} + 7 q^{51} + 5 q^{53} - 4 q^{55} - q^{57} + 3 q^{59} - 4 q^{61} + 4 q^{63} - 6 q^{65} + 3 q^{67} - 5 q^{69} - 12 q^{71} - 7 q^{73} + q^{75} + 3 q^{77} + 8 q^{79} - 14 q^{81} - 12 q^{83} + 6 q^{85} + 25 q^{87} - 10 q^{89} + 13 q^{91} + 2 q^{93} + 4 q^{95} - 6 q^{97} - 3 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
0 −2.56155 0 2.00000 0 0.561553 0 3.56155 0
1.2 0 1.56155 0 2.00000 0 −3.56155 0 −0.561553 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.k 2
4.b odd 2 1 418.2.a.e 2
12.b even 2 1 3762.2.a.y 2
44.c even 2 1 4598.2.a.bj 2
76.d even 2 1 7942.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.e 2 4.b odd 2 1
3344.2.a.k 2 1.a even 1 1 trivial
3762.2.a.y 2 12.b even 2 1
4598.2.a.bj 2 44.c even 2 1
7942.2.a.x 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):

\( T_{3}^{2} + T_{3} - 4 \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 3T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 5T - 100 \) Copy content Toggle raw display
$59$ \( T^{2} - 3T - 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T - 36 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 7T - 26 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12T - 32 \) Copy content Toggle raw display
$89$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
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