Properties

Label 980.2.i.k
Level $980$
Weight $2$
Character orbit 980.i
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} -\beta_{2} q^{5} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} -\beta_{2} q^{5} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{11} + ( 5 + \beta_{3} ) q^{13} + ( -1 - \beta_{3} ) q^{15} + ( -5 - \beta_{1} - 5 \beta_{2} ) q^{17} + ( 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{19} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{25} + ( 1 - \beta_{3} ) q^{27} + ( 1 - 4 \beta_{3} ) q^{29} + ( -6 + \beta_{1} - 6 \beta_{2} ) q^{31} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{33} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{37} + ( -7 + 6 \beta_{1} - 7 \beta_{2} ) q^{39} + ( 2 + \beta_{3} ) q^{41} + ( 6 - 4 \beta_{3} ) q^{43} -2 \beta_{1} q^{45} + ( -7 \beta_{1} + \beta_{2} - 7 \beta_{3} ) q^{47} + ( -4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{51} + ( 8 - 3 \beta_{1} + 8 \beta_{2} ) q^{53} + ( 1 - 2 \beta_{3} ) q^{55} + ( -10 - 6 \beta_{3} ) q^{57} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{59} + ( -2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{61} + ( \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{65} + ( 4 + 5 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 4 + 3 \beta_{3} ) q^{69} + ( -2 - 6 \beta_{3} ) q^{71} + ( -8 + 2 \beta_{1} - 8 \beta_{2} ) q^{73} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{75} + ( 10 \beta_{1} - \beta_{2} + 10 \beta_{3} ) q^{79} + ( 1 - 6 \beta_{1} + \beta_{2} ) q^{81} -8 q^{83} + ( -5 + \beta_{3} ) q^{85} + ( 7 - 3 \beta_{1} + 7 \beta_{2} ) q^{87} + ( 12 \beta_{1} + 12 \beta_{3} ) q^{89} + ( -7 \beta_{1} + 8 \beta_{2} - 7 \beta_{3} ) q^{93} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{95} + ( 3 - 9 \beta_{3} ) q^{97} + ( 8 - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{5} + O(q^{10}) \) \( 4 q - 2 q^{3} + 2 q^{5} + 2 q^{11} + 20 q^{13} - 4 q^{15} - 10 q^{17} + 4 q^{19} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 4 q^{29} - 12 q^{31} - 6 q^{33} + 4 q^{37} - 14 q^{39} + 8 q^{41} + 24 q^{43} - 2 q^{47} - 6 q^{51} + 16 q^{53} + 4 q^{55} - 40 q^{57} - 4 q^{59} - 16 q^{61} + 10 q^{65} + 8 q^{67} + 16 q^{69} - 8 q^{71} - 16 q^{73} - 2 q^{75} + 2 q^{79} + 2 q^{81} - 32 q^{83} - 20 q^{85} + 14 q^{87} - 16 q^{93} - 4 q^{95} + 12 q^{97} + 32 q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 −1.20711 + 2.09077i 0 0.500000 + 0.866025i 0 0 0 −1.41421 2.44949i 0
361.2 0 0.207107 0.358719i 0 0.500000 + 0.866025i 0 0 0 1.41421 + 2.44949i 0
961.1 0 −1.20711 2.09077i 0 0.500000 0.866025i 0 0 0 −1.41421 + 2.44949i 0
961.2 0 0.207107 + 0.358719i 0 0.500000 0.866025i 0 0 0 1.41421 2.44949i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.i.k 4
7.b odd 2 1 980.2.i.l 4
7.c even 3 1 980.2.a.k yes 2
7.c even 3 1 inner 980.2.i.k 4
7.d odd 6 1 980.2.a.j 2
7.d odd 6 1 980.2.i.l 4
21.g even 6 1 8820.2.a.bg 2
21.h odd 6 1 8820.2.a.bl 2
28.f even 6 1 3920.2.a.bx 2
28.g odd 6 1 3920.2.a.bo 2
35.i odd 6 1 4900.2.a.z 2
35.j even 6 1 4900.2.a.x 2
35.k even 12 2 4900.2.e.q 4
35.l odd 12 2 4900.2.e.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 7.d odd 6 1
980.2.a.k yes 2 7.c even 3 1
980.2.i.k 4 1.a even 1 1 trivial
980.2.i.k 4 7.c even 3 1 inner
980.2.i.l 4 7.b odd 2 1
980.2.i.l 4 7.d odd 6 1
3920.2.a.bo 2 28.g odd 6 1
3920.2.a.bx 2 28.f even 6 1
4900.2.a.x 2 35.j even 6 1
4900.2.a.z 2 35.i odd 6 1
4900.2.e.q 4 35.k even 12 2
4900.2.e.r 4 35.l odd 12 2
8820.2.a.bg 2 21.g even 6 1
8820.2.a.bl 2 21.h odd 6 1

Hecke kernels

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

\( T_{3}^{4} + 2 T_{3}^{3} + 5 T_{3}^{2} - 2 T_{3} + 1 \)
\( T_{11}^{4} - 2 T_{11}^{3} + 11 T_{11}^{2} + 14 T_{11} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 49 + 14 T + 11 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( ( 23 - 10 T + T^{2} )^{2} \)
$17$ \( 529 + 230 T + 77 T^{2} + 10 T^{3} + T^{4} \)
$19$ \( 784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( 4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4} \)
$29$ \( ( -31 - 2 T + T^{2} )^{2} \)
$31$ \( 1156 + 408 T + 110 T^{2} + 12 T^{3} + T^{4} \)
$37$ \( 4 - 8 T + 14 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( ( 2 - 4 T + T^{2} )^{2} \)
$43$ \( ( 4 - 12 T + T^{2} )^{2} \)
$47$ \( 9409 - 194 T + 101 T^{2} + 2 T^{3} + T^{4} \)
$53$ \( 2116 - 736 T + 210 T^{2} - 16 T^{3} + T^{4} \)
$59$ \( 4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4} \)
$61$ \( 3136 + 896 T + 200 T^{2} + 16 T^{3} + T^{4} \)
$67$ \( 1156 + 272 T + 98 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( ( -68 + 4 T + T^{2} )^{2} \)
$73$ \( 3136 + 896 T + 200 T^{2} + 16 T^{3} + T^{4} \)
$79$ \( 39601 + 398 T + 203 T^{2} - 2 T^{3} + T^{4} \)
$83$ \( ( 8 + T )^{4} \)
$89$ \( 82944 + 288 T^{2} + T^{4} \)
$97$ \( ( -153 - 6 T + T^{2} )^{2} \)
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