Properties

Label 3136.2.a.be
Level $3136$
Weight $2$
Character orbit 3136.a
Self dual yes
Analytic conductor $25.041$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
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 224)
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.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{3} + ( - 2 \beta + 1) q^{5} - 2 \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{3} + ( - 2 \beta + 1) q^{5} - 2 \beta q^{9} + ( - \beta - 1) q^{11} + 2 \beta q^{13} + (3 \beta - 5) q^{15} + ( - 2 \beta + 3) q^{17} + ( - \beta - 5) q^{19} + (3 \beta + 1) q^{23} + ( - 4 \beta + 4) q^{25} + ( - \beta - 1) q^{27} + 2 \beta q^{29} + ( - \beta + 7) q^{31} - q^{33} + ( - 4 \beta - 3) q^{37} + ( - 2 \beta + 4) q^{39} + (2 \beta + 4) q^{41} + (4 \beta + 4) q^{43} + ( - 2 \beta + 8) q^{45} + (\beta + 9) q^{47} + (5 \beta - 7) q^{51} + q^{53} + (\beta + 3) q^{55} + ( - 4 \beta + 3) q^{57} + ( - 7 \beta - 1) q^{59} + (4 \beta + 3) q^{61} + (2 \beta - 8) q^{65} + (3 \beta - 7) q^{67} + ( - 2 \beta + 5) q^{69} + ( - 4 \beta - 8) q^{71} + (4 \beta + 9) q^{73} + (8 \beta - 12) q^{75} + ( - 5 \beta + 1) q^{79} + (6 \beta - 1) q^{81} + (8 \beta - 4) q^{83} + ( - 8 \beta + 11) q^{85} + ( - 2 \beta + 4) q^{87} + 9 q^{89} + (8 \beta - 9) q^{93} + (9 \beta - 1) q^{95} + ( - 2 \beta + 4) q^{97} + (2 \beta + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{11} - 10 q^{15} + 6 q^{17} - 10 q^{19} + 2 q^{23} + 8 q^{25} - 2 q^{27} + 14 q^{31} - 2 q^{33} - 6 q^{37} + 8 q^{39} + 8 q^{41} + 8 q^{43} + 16 q^{45} + 18 q^{47} - 14 q^{51} + 2 q^{53} + 6 q^{55} + 6 q^{57} - 2 q^{59} + 6 q^{61} - 16 q^{65} - 14 q^{67} + 10 q^{69} - 16 q^{71} + 18 q^{73} - 24 q^{75} + 2 q^{79} - 2 q^{81} - 8 q^{83} + 22 q^{85} + 8 q^{87} + 18 q^{89} - 18 q^{93} - 2 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) \) 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 −2.41421 0 3.82843 0 0 0 2.82843 0
1.2 0 0.414214 0 −1.82843 0 0 0 −2.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3136.2.a.be 2
4.b odd 2 1 3136.2.a.bx 2
7.b odd 2 1 3136.2.a.bw 2
7.d odd 6 2 448.2.i.g 4
8.b even 2 1 1568.2.a.u 2
8.d odd 2 1 1568.2.a.j 2
28.d even 2 1 3136.2.a.bd 2
28.f even 6 2 448.2.i.j 4
56.e even 2 1 1568.2.a.w 2
56.h odd 2 1 1568.2.a.l 2
56.j odd 6 2 224.2.i.d yes 4
56.k odd 6 2 1568.2.i.x 4
56.m even 6 2 224.2.i.a 4
56.p even 6 2 1568.2.i.o 4
168.ba even 6 2 2016.2.s.s 4
168.be odd 6 2 2016.2.s.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.a 4 56.m even 6 2
224.2.i.d yes 4 56.j odd 6 2
448.2.i.g 4 7.d odd 6 2
448.2.i.j 4 28.f even 6 2
1568.2.a.j 2 8.d odd 2 1
1568.2.a.l 2 56.h odd 2 1
1568.2.a.u 2 8.b even 2 1
1568.2.a.w 2 56.e even 2 1
1568.2.i.o 4 56.p even 6 2
1568.2.i.x 4 56.k odd 6 2
2016.2.s.q 4 168.be odd 6 2
2016.2.s.s 4 168.ba even 6 2
3136.2.a.bd 2 28.d even 2 1
3136.2.a.be 2 1.a even 1 1 trivial
3136.2.a.bw 2 7.b odd 2 1
3136.2.a.bx 2 4.b odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 2T_{5} - 7 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 1 \) 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
$5$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 8 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 10T + 23 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 17 \) Copy content Toggle raw display
$29$ \( T^{2} - 8 \) Copy content Toggle raw display
$31$ \( T^{2} - 14T + 47 \) Copy content Toggle raw display
$37$ \( T^{2} + 6T - 23 \) Copy content Toggle raw display
$41$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 18T + 79 \) Copy content Toggle raw display
$53$ \( (T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 2T - 97 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 23 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 31 \) Copy content Toggle raw display
$71$ \( T^{2} + 16T + 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 18T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 49 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 112 \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
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