Properties

Label 3136.2.a.bl
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 1568)
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.

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

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 0 0 0 0 −1.00000 0
1.2 0 1.41421 0 0 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.bl 2
4.b odd 2 1 3136.2.a.bo 2
7.b odd 2 1 inner 3136.2.a.bl 2
8.b even 2 1 1568.2.a.r yes 2
8.d odd 2 1 1568.2.a.q 2
28.d even 2 1 3136.2.a.bo 2
56.e even 2 1 1568.2.a.q 2
56.h odd 2 1 1568.2.a.r yes 2
56.j odd 6 2 1568.2.i.q 4
56.k odd 6 2 1568.2.i.r 4
56.m even 6 2 1568.2.i.r 4
56.p even 6 2 1568.2.i.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.q 2 8.d odd 2 1
1568.2.a.q 2 56.e even 2 1
1568.2.a.r yes 2 8.b even 2 1
1568.2.a.r yes 2 56.h odd 2 1
1568.2.i.q 4 56.j odd 6 2
1568.2.i.q 4 56.p even 6 2
1568.2.i.r 4 56.k odd 6 2
1568.2.i.r 4 56.m even 6 2
3136.2.a.bl 2 1.a even 1 1 trivial
3136.2.a.bl 2 7.b odd 2 1 inner
3136.2.a.bo 2 4.b odd 2 1
3136.2.a.bo 2 28.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3136))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( -8 + T^{2} \)
$17$ \( -18 + T^{2} \)
$19$ \( -18 + T^{2} \)
$23$ \( ( -8 + T )^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( -72 + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( -18 + T^{2} \)
$43$ \( ( 6 + T )^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -162 + T^{2} \)
$61$ \( -32 + T^{2} \)
$67$ \( ( 12 + T )^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( -2 + T^{2} \)
$79$ \( ( -12 + T )^{2} \)
$83$ \( -98 + T^{2} \)
$89$ \( -18 + T^{2} \)
$97$ \( -338 + T^{2} \)
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