Properties

Label 980.2.a.j
Level $980$
Weight $2$
Character orbit 980.a
Self dual yes
Analytic conductor $7.825$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.82533939809\)
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: yes
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.

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

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 −2.41421 0 1.00000 0 0 0 2.82843 0
1.2 0 0.414214 0 1.00000 0 0 0 −2.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

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 980.2.a.j 2
3.b odd 2 1 8820.2.a.bg 2
4.b odd 2 1 3920.2.a.bx 2
5.b even 2 1 4900.2.a.z 2
5.c odd 4 2 4900.2.e.q 4
7.b odd 2 1 980.2.a.k yes 2
7.c even 3 2 980.2.i.l 4
7.d odd 6 2 980.2.i.k 4
21.c even 2 1 8820.2.a.bl 2
28.d even 2 1 3920.2.a.bo 2
35.c odd 2 1 4900.2.a.x 2
35.f even 4 2 4900.2.e.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 1.a even 1 1 trivial
980.2.a.k yes 2 7.b odd 2 1
980.2.i.k 4 7.d odd 6 2
980.2.i.l 4 7.c even 3 2
3920.2.a.bo 2 28.d even 2 1
3920.2.a.bx 2 4.b odd 2 1
4900.2.a.x 2 35.c odd 2 1
4900.2.a.z 2 5.b even 2 1
4900.2.e.q 4 5.c odd 4 2
4900.2.e.r 4 35.f even 4 2
8820.2.a.bg 2 3.b odd 2 1
8820.2.a.bl 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(980))\):

\( T_{3}^{2} + 2 T_{3} - 1 \)
\( T_{11}^{2} + 2 T_{11} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -1 + 2 T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -7 + 2 T + T^{2} \)
$13$ \( 23 + 10 T + T^{2} \)
$17$ \( 23 + 10 T + T^{2} \)
$19$ \( -28 - 4 T + T^{2} \)
$23$ \( 2 - 4 T + T^{2} \)
$29$ \( -31 - 2 T + T^{2} \)
$31$ \( 34 + 12 T + T^{2} \)
$37$ \( 2 + 4 T + T^{2} \)
$41$ \( 2 + 4 T + T^{2} \)
$43$ \( 4 - 12 T + T^{2} \)
$47$ \( -97 + 2 T + T^{2} \)
$53$ \( 46 + 16 T + T^{2} \)
$59$ \( 2 + 4 T + T^{2} \)
$61$ \( 56 + 16 T + T^{2} \)
$67$ \( -34 + 8 T + T^{2} \)
$71$ \( -68 + 4 T + T^{2} \)
$73$ \( 56 + 16 T + T^{2} \)
$79$ \( -199 + 2 T + T^{2} \)
$83$ \( ( -8 + T )^{2} \)
$89$ \( -288 + T^{2} \)
$97$ \( -153 + 6 T + T^{2} \)
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