Properties

Label 3136.2.a.bk
Level $3136$
Weight $2$
Character orbit 3136.a
Self dual yes
Analytic conductor $25.041$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.0410860739\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 392)
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 \beta q^{5} - q^{9} +O(q^{10})\) \( q + \beta q^{3} + 2 \beta q^{5} - q^{9} -6 q^{11} -4 \beta q^{13} + 4 q^{15} -\beta q^{17} -3 \beta q^{19} + 4 q^{23} + 3 q^{25} -4 \beta q^{27} + 6 q^{29} -2 \beta q^{31} -6 \beta q^{33} -2 q^{37} -8 q^{39} + \beta q^{41} -10 q^{43} -2 \beta q^{45} + 2 \beta q^{47} -2 q^{51} + 2 q^{53} -12 \beta q^{55} -6 q^{57} + \beta q^{59} -6 \beta q^{61} -16 q^{65} -4 q^{67} + 4 \beta q^{69} -12 q^{71} + 7 \beta q^{73} + 3 \beta q^{75} -4 q^{79} -5 q^{81} + \beta q^{83} -4 q^{85} + 6 \beta q^{87} + 3 \beta q^{89} -4 q^{93} -12 q^{95} -9 \beta q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 12q^{11} + 8q^{15} + 8q^{23} + 6q^{25} + 12q^{29} - 4q^{37} - 16q^{39} - 20q^{43} - 4q^{51} + 4q^{53} - 12q^{57} - 32q^{65} - 8q^{67} - 24q^{71} - 8q^{79} - 10q^{81} - 8q^{85} - 8q^{93} - 24q^{95} + 12q^{99} + O(q^{100}) \)

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 −2.82843 0 0 0 −1.00000 0
1.2 0 1.41421 0 2.82843 0 0 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.a.bk 2
4.b odd 2 1 3136.2.a.bp 2
7.b odd 2 1 inner 3136.2.a.bk 2
8.b even 2 1 392.2.a.g 2
8.d odd 2 1 784.2.a.k 2
24.f even 2 1 7056.2.a.ct 2
24.h odd 2 1 3528.2.a.be 2
28.d even 2 1 3136.2.a.bp 2
40.f even 2 1 9800.2.a.bv 2
56.e even 2 1 784.2.a.k 2
56.h odd 2 1 392.2.a.g 2
56.j odd 6 2 392.2.i.h 4
56.k odd 6 2 784.2.i.n 4
56.m even 6 2 784.2.i.n 4
56.p even 6 2 392.2.i.h 4
168.e odd 2 1 7056.2.a.ct 2
168.i even 2 1 3528.2.a.be 2
168.s odd 6 2 3528.2.s.bj 4
168.ba even 6 2 3528.2.s.bj 4
280.c odd 2 1 9800.2.a.bv 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.g 2 8.b even 2 1
392.2.a.g 2 56.h odd 2 1
392.2.i.h 4 56.j odd 6 2
392.2.i.h 4 56.p even 6 2
784.2.a.k 2 8.d odd 2 1
784.2.a.k 2 56.e even 2 1
784.2.i.n 4 56.k odd 6 2
784.2.i.n 4 56.m even 6 2
3136.2.a.bk 2 1.a even 1 1 trivial
3136.2.a.bk 2 7.b odd 2 1 inner
3136.2.a.bp 2 4.b odd 2 1
3136.2.a.bp 2 28.d even 2 1
3528.2.a.be 2 24.h odd 2 1
3528.2.a.be 2 168.i even 2 1
3528.2.s.bj 4 168.s odd 6 2
3528.2.s.bj 4 168.ba even 6 2
7056.2.a.ct 2 24.f even 2 1
7056.2.a.ct 2 168.e odd 2 1
9800.2.a.bv 2 40.f even 2 1
9800.2.a.bv 2 280.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3136))\):

\( T_{3}^{2} - 2 \)
\( T_{5}^{2} - 8 \)
\( T_{11} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 4 T^{2} + 9 T^{4} \)
$5$ \( 1 + 2 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( ( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 6 T^{2} + 169 T^{4} \)
$17$ \( 1 + 32 T^{2} + 289 T^{4} \)
$19$ \( 1 + 20 T^{2} + 361 T^{4} \)
$23$ \( ( 1 - 4 T + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 54 T^{2} + 961 T^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 80 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 + 10 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 86 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 2 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 116 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 50 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 48 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 164 T^{2} + 6889 T^{4} \)
$89$ \( 1 + 160 T^{2} + 7921 T^{4} \)
$97$ \( 1 + 32 T^{2} + 9409 T^{4} \)
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