Properties

Label 3136.2.a.by
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{3} + ( 2 - 2 \beta ) q^{5} + ( 1 + 4 \beta ) q^{9} +O(q^{10})\) \( q + 2 \beta q^{3} + ( 2 - 2 \beta ) q^{5} + ( 1 + 4 \beta ) q^{9} + ( -4 + 4 \beta ) q^{11} + ( 2 + 2 \beta ) q^{13} -4 q^{15} + ( -2 + 4 \beta ) q^{17} -2 \beta q^{19} -4 q^{23} + ( 3 - 4 \beta ) q^{25} + ( 8 + 4 \beta ) q^{27} + ( 2 - 4 \beta ) q^{29} + 4 \beta q^{31} + 8 q^{33} + ( 2 - 4 \beta ) q^{37} + ( 4 + 8 \beta ) q^{39} + ( 6 - 4 \beta ) q^{41} + ( 4 - 4 \beta ) q^{43} + ( -6 - 2 \beta ) q^{45} + ( 8 - 4 \beta ) q^{47} + ( 8 + 4 \beta ) q^{51} + 10 q^{53} + ( -16 + 8 \beta ) q^{55} + ( -4 - 4 \beta ) q^{57} + ( 8 - 2 \beta ) q^{59} + ( 10 - 2 \beta ) q^{61} -4 \beta q^{65} + 4 q^{67} -8 \beta q^{69} -8 \beta q^{71} + ( -2 - 8 \beta ) q^{73} + ( -8 - 2 \beta ) q^{75} + ( -8 + 8 \beta ) q^{79} + ( 5 + 12 \beta ) q^{81} + ( 8 - 2 \beta ) q^{83} + ( -12 + 4 \beta ) q^{85} + ( -8 - 4 \beta ) q^{87} + 6 q^{89} + ( 8 + 8 \beta ) q^{93} + 4 q^{95} + ( -10 + 4 \beta ) q^{97} + ( 12 + 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} + 6q^{9} - 4q^{11} + 6q^{13} - 8q^{15} - 2q^{19} - 8q^{23} + 2q^{25} + 20q^{27} + 4q^{31} + 16q^{33} + 16q^{39} + 8q^{41} + 4q^{43} - 14q^{45} + 12q^{47} + 20q^{51} + 20q^{53} - 24q^{55} - 12q^{57} + 14q^{59} + 18q^{61} - 4q^{65} + 8q^{67} - 8q^{69} - 8q^{71} - 12q^{73} - 18q^{75} - 8q^{79} + 22q^{81} + 14q^{83} - 20q^{85} - 20q^{87} + 12q^{89} + 24q^{93} + 8q^{95} - 16q^{97} + 28q^{99} + O(q^{100}) \)

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
−0.618034
1.61803
0 −1.23607 0 3.23607 0 0 0 −1.47214 0
1.2 0 3.23607 0 −1.23607 0 0 0 7.47214 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.by 2
4.b odd 2 1 3136.2.a.bf 2
7.b odd 2 1 448.2.a.i 2
8.b even 2 1 1568.2.a.k 2
8.d odd 2 1 1568.2.a.v 2
21.c even 2 1 4032.2.a.bv 2
28.d even 2 1 448.2.a.j 2
56.e even 2 1 224.2.a.c 2
56.h odd 2 1 224.2.a.d yes 2
56.j odd 6 2 1568.2.i.m 4
56.k odd 6 2 1568.2.i.n 4
56.m even 6 2 1568.2.i.v 4
56.p even 6 2 1568.2.i.w 4
84.h odd 2 1 4032.2.a.bw 2
112.j even 4 2 1792.2.b.k 4
112.l odd 4 2 1792.2.b.m 4
168.e odd 2 1 2016.2.a.r 2
168.i even 2 1 2016.2.a.o 2
280.c odd 2 1 5600.2.a.z 2
280.n even 2 1 5600.2.a.bk 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 56.e even 2 1
224.2.a.d yes 2 56.h odd 2 1
448.2.a.i 2 7.b odd 2 1
448.2.a.j 2 28.d even 2 1
1568.2.a.k 2 8.b even 2 1
1568.2.a.v 2 8.d odd 2 1
1568.2.i.m 4 56.j odd 6 2
1568.2.i.n 4 56.k odd 6 2
1568.2.i.v 4 56.m even 6 2
1568.2.i.w 4 56.p even 6 2
1792.2.b.k 4 112.j even 4 2
1792.2.b.m 4 112.l odd 4 2
2016.2.a.o 2 168.i even 2 1
2016.2.a.r 2 168.e odd 2 1
3136.2.a.bf 2 4.b odd 2 1
3136.2.a.by 2 1.a even 1 1 trivial
4032.2.a.bv 2 21.c even 2 1
4032.2.a.bw 2 84.h odd 2 1
5600.2.a.z 2 280.c odd 2 1
5600.2.a.bk 2 280.n even 2 1

Hecke kernels

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

\( T_{3}^{2} - 2 T_{3} - 4 \)
\( T_{5}^{2} - 2 T_{5} - 4 \)
\( T_{11}^{2} + 4 T_{11} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 - 2 T + T^{2} \)
$5$ \( -4 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -16 + 4 T + T^{2} \)
$13$ \( 4 - 6 T + T^{2} \)
$17$ \( -20 + T^{2} \)
$19$ \( -4 + 2 T + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( -16 - 4 T + T^{2} \)
$37$ \( -20 + T^{2} \)
$41$ \( -4 - 8 T + T^{2} \)
$43$ \( -16 - 4 T + T^{2} \)
$47$ \( 16 - 12 T + T^{2} \)
$53$ \( ( -10 + T )^{2} \)
$59$ \( 44 - 14 T + T^{2} \)
$61$ \( 76 - 18 T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( -64 + 8 T + T^{2} \)
$73$ \( -44 + 12 T + T^{2} \)
$79$ \( -64 + 8 T + T^{2} \)
$83$ \( 44 - 14 T + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 44 + 16 T + T^{2} \)
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