Properties

Label 656.2.a.i
Level $656$
Weight $2$
Character orbit 656.a
Self dual yes
Analytic conductor $5.238$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 656 = 2^{4} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 656.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.23818637260\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Defining polynomial: \(x^{4} - 10 x^{2} - 6 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} ) q^{3} + ( 2 - \beta_{2} + \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + ( 3 - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} ) q^{3} + ( 2 - \beta_{2} + \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + ( 3 - \beta_{2} - \beta_{3} ) q^{9} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + 2 \beta_{1} q^{13} + ( 1 + 2 \beta_{1} - 3 \beta_{3} ) q^{15} + 2 \beta_{3} q^{17} + ( -2 + \beta_{1} + \beta_{2} ) q^{19} + ( -2 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{21} + ( 2 + 2 \beta_{2} ) q^{23} + ( 3 - 2 \beta_{1} ) q^{25} + ( 5 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{27} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -8 - 2 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{33} + ( 7 - 4 \beta_{1} + \beta_{3} ) q^{35} + ( 4 - \beta_{2} - \beta_{3} ) q^{37} + ( 6 - 2 \beta_{1} ) q^{39} - q^{41} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{43} + ( 6 + 2 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{45} + ( 2 + 3 \beta_{1} - \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{49} + ( -4 - 2 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{51} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -1 - 4 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{55} + ( -4 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{57} + ( -2 - 2 \beta_{2} ) q^{59} + ( 6 + 2 \beta_{1} ) q^{61} + ( 6 + \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{63} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( -7 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{67} + ( -6 + 2 \beta_{3} ) q^{69} + ( 4 - \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{71} + ( 4 - \beta_{2} + 3 \beta_{3} ) q^{73} + ( -6 + 5 \beta_{1} - 3 \beta_{2} ) q^{75} + ( 2 - 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{77} + ( 6 + \beta_{1} - 3 \beta_{2} ) q^{79} + ( 7 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{81} + ( 2 + 4 \beta_{1} - 2 \beta_{3} ) q^{83} + ( 4 - 4 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -12 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{87} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{89} + ( -12 + 2 \beta_{1} - 6 \beta_{3} ) q^{91} + ( -2 - 4 \beta_{2} + 6 \beta_{3} ) q^{93} + ( -7 + 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{95} + ( 2 - 4 \beta_{3} ) q^{97} + ( -16 - 7 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 4q^{5} + 12q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 4q^{5} + 12q^{9} - 4q^{11} + 10q^{15} - 4q^{17} - 6q^{19} + 12q^{23} + 12q^{25} + 10q^{27} - 4q^{29} + 8q^{31} - 20q^{33} + 26q^{35} + 16q^{37} + 24q^{39} - 4q^{41} - 4q^{43} + 4q^{45} + 6q^{47} + 16q^{49} + 4q^{51} - 16q^{53} + 2q^{55} + 4q^{57} - 12q^{59} + 24q^{61} + 10q^{63} + 4q^{65} - 28q^{67} - 28q^{69} + 2q^{71} + 8q^{73} - 30q^{75} + 8q^{77} + 18q^{79} + 28q^{81} + 12q^{83} + 32q^{85} - 44q^{87} + 4q^{89} - 36q^{91} - 28q^{93} - 14q^{95} + 16q^{97} - 58q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 10 x^{2} - 6 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 7 \nu - 3 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 3 \nu^{2} + 7 \nu - 12 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} + 7 \beta_{1} + 3\)

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
−1.55466
−2.46810
3.31526
0.707500
0 −2.92968 0 −3.33307 0 −2.77840 0 5.58303 0
1.2 0 −2.21551 0 3.59669 0 5.06479 0 1.90849 0
1.3 0 −0.0950939 0 1.17025 0 −3.14501 0 −2.99096 0
1.4 0 3.24028 0 2.56613 0 0.858626 0 7.49944 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(41\) \(1\)

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 656.2.a.i 4
3.b odd 2 1 5904.2.a.bp 4
4.b odd 2 1 164.2.a.a 4
8.b even 2 1 2624.2.a.y 4
8.d odd 2 1 2624.2.a.v 4
12.b even 2 1 1476.2.a.g 4
20.d odd 2 1 4100.2.a.c 4
20.e even 4 2 4100.2.d.c 8
28.d even 2 1 8036.2.a.i 4
164.d odd 2 1 6724.2.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.a.a 4 4.b odd 2 1
656.2.a.i 4 1.a even 1 1 trivial
1476.2.a.g 4 12.b even 2 1
2624.2.a.v 4 8.d odd 2 1
2624.2.a.y 4 8.b even 2 1
4100.2.a.c 4 20.d odd 2 1
4100.2.d.c 8 20.e even 4 2
5904.2.a.bp 4 3.b odd 2 1
6724.2.a.c 4 164.d odd 2 1
8036.2.a.i 4 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2 T_{3}^{3} - 10 T_{3}^{2} - 22 T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(656))\).