Properties

Label 980.2.e.c
Level $980$
Weight $2$
Character orbit 980.e
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.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} + 16 \nu - 25 \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 4 \nu^{2} + 4 \nu + 15 \)\()/10\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu + 5 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} - 5 \beta_{2} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(-4 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 7\)

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
−1.63746 + 1.52274i
2.13746 0.656712i
−1.63746 1.52274i
2.13746 + 0.656712i
0 1.73205i 0 −2.13746 + 0.656712i 0 0 0 0 0
589.2 0 1.73205i 0 1.63746 1.52274i 0 0 0 0 0
589.3 0 1.73205i 0 −2.13746 0.656712i 0 0 0 0 0
589.4 0 1.73205i 0 1.63746 + 1.52274i 0 0 0 0 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.e.c 4
5.b even 2 1 inner 980.2.e.c 4
5.c odd 4 2 4900.2.a.bf 4
7.b odd 2 1 980.2.e.f 4
7.c even 3 1 980.2.q.b 4
7.c even 3 1 980.2.q.g 4
7.d odd 6 1 140.2.q.a 4
7.d odd 6 1 140.2.q.b yes 4
21.g even 6 1 1260.2.bm.a 4
21.g even 6 1 1260.2.bm.b 4
28.f even 6 1 560.2.bw.a 4
28.f even 6 1 560.2.bw.e 4
35.c odd 2 1 980.2.e.f 4
35.f even 4 2 4900.2.a.be 4
35.i odd 6 1 140.2.q.a 4
35.i odd 6 1 140.2.q.b yes 4
35.j even 6 1 980.2.q.b 4
35.j even 6 1 980.2.q.g 4
35.k even 12 4 700.2.i.f 8
105.p even 6 1 1260.2.bm.a 4
105.p even 6 1 1260.2.bm.b 4
140.s even 6 1 560.2.bw.a 4
140.s even 6 1 560.2.bw.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 7.d odd 6 1
140.2.q.a 4 35.i odd 6 1
140.2.q.b yes 4 7.d odd 6 1
140.2.q.b yes 4 35.i odd 6 1
560.2.bw.a 4 28.f even 6 1
560.2.bw.a 4 140.s even 6 1
560.2.bw.e 4 28.f even 6 1
560.2.bw.e 4 140.s even 6 1
700.2.i.f 8 35.k even 12 4
980.2.e.c 4 1.a even 1 1 trivial
980.2.e.c 4 5.b even 2 1 inner
980.2.e.f 4 7.b odd 2 1
980.2.e.f 4 35.c odd 2 1
980.2.q.b 4 7.c even 3 1
980.2.q.b 4 35.j even 6 1
980.2.q.g 4 7.c even 3 1
980.2.q.g 4 35.j even 6 1
1260.2.bm.a 4 21.g even 6 1
1260.2.bm.a 4 105.p even 6 1
1260.2.bm.b 4 21.g even 6 1
1260.2.bm.b 4 105.p even 6 1
4900.2.a.be 4 35.f even 4 2
4900.2.a.bf 4 5.c odd 4 2

Hecke kernels

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

\( T_{3}^{2} + 3 \)
\( T_{11}^{2} + 3 T_{11} - 12 \)
\( T_{19}^{2} + T_{19} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( 25 + 5 T - 4 T^{2} + T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -12 + 3 T + T^{2} )^{2} \)
$13$ \( 256 + 44 T^{2} + T^{4} \)
$17$ \( 4 + 23 T^{2} + T^{4} \)
$19$ \( ( -14 + T + T^{2} )^{2} \)
$23$ \( 49 + 62 T^{2} + T^{4} \)
$29$ \( ( -14 + T + T^{2} )^{2} \)
$31$ \( ( -14 - T + T^{2} )^{2} \)
$37$ \( 3136 + 131 T^{2} + T^{4} \)
$41$ \( ( 42 - 15 T + T^{2} )^{2} \)
$43$ \( 196 + 47 T^{2} + T^{4} \)
$47$ \( 196 + 47 T^{2} + T^{4} \)
$53$ \( 1764 + 87 T^{2} + T^{4} \)
$59$ \( ( -14 - T + T^{2} )^{2} \)
$61$ \( ( -21 - 12 T + T^{2} )^{2} \)
$67$ \( 2401 + 206 T^{2} + T^{4} \)
$71$ \( ( -48 - 6 T + T^{2} )^{2} \)
$73$ \( 196 + 47 T^{2} + T^{4} \)
$79$ \( ( -2 + 7 T + T^{2} )^{2} \)
$83$ \( 1764 + 87 T^{2} + T^{4} \)
$89$ \( ( 7 + T )^{4} \)
$97$ \( ( 48 + T^{2} )^{2} \)
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