Properties

Label 3344.2.a.n
Level $3344$
Weight $2$
Character orbit 3344.a
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) 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 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} - q^{5} + (\beta + 2) q^{7} + 2 \beta q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} - q^{5} + (\beta + 2) q^{7} + 2 \beta q^{9} + q^{11} + (3 \beta - 2) q^{13} + ( - \beta - 1) q^{15} + (\beta + 2) q^{17} + q^{19} + (3 \beta + 4) q^{21} + 3 q^{23} - 4 q^{25} + ( - \beta + 1) q^{27} + ( - 3 \beta - 2) q^{29} + (\beta + 5) q^{31} + (\beta + 1) q^{33} + ( - \beta - 2) q^{35} + (5 \beta + 3) q^{37} + (\beta + 4) q^{39} + ( - 4 \beta + 4) q^{41} + (4 \beta - 6) q^{43} - 2 \beta q^{45} + ( - 2 \beta - 6) q^{47} + (4 \beta - 1) q^{49} + (3 \beta + 4) q^{51} + ( - 6 \beta + 4) q^{53} - q^{55} + (\beta + 1) q^{57} + ( - \beta + 3) q^{59} + ( - 5 \beta - 4) q^{61} + (4 \beta + 4) q^{63} + ( - 3 \beta + 2) q^{65} + (\beta + 9) q^{67} + (3 \beta + 3) q^{69} + (\beta + 11) q^{71} + ( - 6 \beta + 4) q^{73} + ( - 4 \beta - 4) q^{75} + (\beta + 2) q^{77} + ( - \beta + 16) q^{79} + ( - 6 \beta - 1) q^{81} + ( - \beta - 2) q^{83} + ( - \beta - 2) q^{85} + ( - 5 \beta - 8) q^{87} + (7 \beta - 5) q^{89} + (4 \beta + 2) q^{91} + (6 \beta + 7) q^{93} - q^{95} + (\beta + 1) q^{97} + 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{19} + 8 q^{21} + 6 q^{23} - 8 q^{25} + 2 q^{27} - 4 q^{29} + 10 q^{31} + 2 q^{33} - 4 q^{35} + 6 q^{37} + 8 q^{39} + 8 q^{41} - 12 q^{43} - 12 q^{47} - 2 q^{49} + 8 q^{51} + 8 q^{53} - 2 q^{55} + 2 q^{57} + 6 q^{59} - 8 q^{61} + 8 q^{63} + 4 q^{65} + 18 q^{67} + 6 q^{69} + 22 q^{71} + 8 q^{73} - 8 q^{75} + 4 q^{77} + 32 q^{79} - 2 q^{81} - 4 q^{83} - 4 q^{85} - 16 q^{87} - 10 q^{89} + 4 q^{91} + 14 q^{93} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 −0.414214 0 −1.00000 0 0.585786 0 −2.82843 0
1.2 0 2.41421 0 −1.00000 0 3.41421 0 2.82843 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.n 2
4.b odd 2 1 209.2.a.b 2
12.b even 2 1 1881.2.a.d 2
20.d odd 2 1 5225.2.a.f 2
44.c even 2 1 2299.2.a.f 2
76.d even 2 1 3971.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.a.b 2 4.b odd 2 1
1881.2.a.d 2 12.b even 2 1
2299.2.a.f 2 44.c even 2 1
3344.2.a.n 2 1.a even 1 1 trivial
3971.2.a.d 2 76.d even 2 1
5225.2.a.f 2 20.d odd 2 1

Hecke kernels

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

\( T_{3}^{2} - 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( (T - 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 23 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T - 41 \) Copy content Toggle raw display
$41$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 34 \) Copy content Toggle raw display
$67$ \( T^{2} - 18T + 79 \) Copy content Toggle raw display
$71$ \( T^{2} - 22T + 119 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$79$ \( T^{2} - 32T + 254 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$89$ \( T^{2} + 10T - 73 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
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