Properties

Label 3136.2.a.bu
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) 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 224)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + q^{5} - 3 \beta q^{11} + \beta q^{15} - 5 q^{17} + \beta q^{19} - \beta q^{23} - 4 q^{25} - 3 \beta q^{27} - 8 q^{29} + 5 \beta q^{31} - 9 q^{33} + 5 q^{37} - 4 q^{41} - 4 \beta q^{43} - 5 \beta q^{47} - 5 \beta q^{51} + q^{53} - 3 \beta q^{55} + 3 q^{57} + \beta q^{59} + 11 q^{61} + 7 \beta q^{67} - 3 q^{69} - 8 \beta q^{71} - 15 q^{73} - 4 \beta q^{75} - \beta q^{79} - 9 q^{81} + 4 \beta q^{83} - 5 q^{85} - 8 \beta q^{87} - 7 q^{89} + 15 q^{93} + \beta q^{95} - 12 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 10 q^{17} - 8 q^{25} - 16 q^{29} - 18 q^{33} + 10 q^{37} - 8 q^{41} + 2 q^{53} + 6 q^{57} + 22 q^{61} - 6 q^{69} - 30 q^{73} - 18 q^{81} - 10 q^{85} - 14 q^{89} + 30 q^{93} - 24 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
0 −1.73205 0 1.00000 0 0 0 0 0
1.2 0 1.73205 0 1.00000 0 0 0 0 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
4.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.bu 2
4.b odd 2 1 inner 3136.2.a.bu 2
7.b odd 2 1 3136.2.a.bh 2
7.d odd 6 2 448.2.i.i 4
8.b even 2 1 1568.2.a.n 2
8.d odd 2 1 1568.2.a.n 2
28.d even 2 1 3136.2.a.bh 2
28.f even 6 2 448.2.i.i 4
56.e even 2 1 1568.2.a.s 2
56.h odd 2 1 1568.2.a.s 2
56.j odd 6 2 224.2.i.b 4
56.k odd 6 2 1568.2.i.u 4
56.m even 6 2 224.2.i.b 4
56.p even 6 2 1568.2.i.u 4
168.ba even 6 2 2016.2.s.r 4
168.be odd 6 2 2016.2.s.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.b 4 56.j odd 6 2
224.2.i.b 4 56.m even 6 2
448.2.i.i 4 7.d odd 6 2
448.2.i.i 4 28.f even 6 2
1568.2.a.n 2 8.b even 2 1
1568.2.a.n 2 8.d odd 2 1
1568.2.a.s 2 56.e even 2 1
1568.2.a.s 2 56.h odd 2 1
1568.2.i.u 4 56.k odd 6 2
1568.2.i.u 4 56.p even 6 2
2016.2.s.r 4 168.ba even 6 2
2016.2.s.r 4 168.be odd 6 2
3136.2.a.bh 2 7.b odd 2 1
3136.2.a.bh 2 28.d even 2 1
3136.2.a.bu 2 1.a even 1 1 trivial
3136.2.a.bu 2 4.b odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} - 3 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 27 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 5)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 3 \) Copy content Toggle raw display
$23$ \( T^{2} - 3 \) Copy content Toggle raw display
$29$ \( (T + 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 75 \) Copy content Toggle raw display
$37$ \( (T - 5)^{2} \) Copy content Toggle raw display
$41$ \( (T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 48 \) Copy content Toggle raw display
$47$ \( T^{2} - 75 \) Copy content Toggle raw display
$53$ \( (T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 3 \) Copy content Toggle raw display
$61$ \( (T - 11)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 147 \) Copy content Toggle raw display
$71$ \( T^{2} - 192 \) Copy content Toggle raw display
$73$ \( (T + 15)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 3 \) Copy content Toggle raw display
$83$ \( T^{2} - 48 \) Copy content Toggle raw display
$89$ \( (T + 7)^{2} \) Copy content Toggle raw display
$97$ \( (T + 12)^{2} \) Copy content Toggle raw display
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