Properties

Label 3440.2.a.j
Level 3440
Weight 2
Character orbit 3440.a
Self dual yes
Analytic conductor 27.469
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3440 = 2^{4} \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3440.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.4685382953\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 430)
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.

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

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 −1.41421 0 1.00000 0 −1.00000 0 −1.00000 0
1.2 0 1.41421 0 1.00000 0 −1.00000 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 3440.2.a.j 2
4.b odd 2 1 430.2.a.g 2
12.b even 2 1 3870.2.a.bc 2
20.d odd 2 1 2150.2.a.v 2
20.e even 4 2 2150.2.b.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.a.g 2 4.b odd 2 1
2150.2.a.v 2 20.d odd 2 1
2150.2.b.o 4 20.e even 4 2
3440.2.a.j 2 1.a even 1 1 trivial
3870.2.a.bc 2 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(43\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3440))\):

\( T_{3}^{2} - 2 \)
\( T_{7} + 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 4 T^{2} + 9 T^{4} \)
$5$ \( ( 1 - T )^{2} \)
$7$ \( ( 1 + T + 7 T^{2} )^{2} \)
$11$ \( 1 + 4 T + 24 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ \( 1 - 2 T + 19 T^{2} - 26 T^{3} + 169 T^{4} \)
$17$ \( 1 + 32 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )^{2} \)
$23$ \( 1 + 4 T + 92 T^{3} + 529 T^{4} \)
$29$ \( 1 + 6 T + 49 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( 1 + 2 T + 61 T^{2} + 62 T^{3} + 961 T^{4} \)
$37$ \( 1 + 4 T + 60 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 2 T + 75 T^{2} - 82 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( 1 + 44 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 74 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 14 T + 121 T^{2} + 854 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 6 T + 125 T^{2} + 402 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 4 T + 48 T^{2} + 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 10 T + 153 T^{2} + 730 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 14 T + 205 T^{2} - 1106 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 4 T + 138 T^{2} - 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 4 T + 132 T^{2} + 356 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 4 T - 44 T^{2} + 388 T^{3} + 9409 T^{4} \)
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