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: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_1 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_1 q^{5} + ( - \beta_{3} - \beta_1 - 2) q^{11} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{13} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{15} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{17} + (\beta_{3} + \beta_1) q^{19} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{3} - 2 \beta_{2} + 2) q^{25} - 3 \beta_{2} q^{27} + ( - \beta_{3} - \beta_1 - 1) q^{29} + ( - \beta_{3} - \beta_1) q^{31} + (3 \beta_{3} - 3 \beta_1) q^{33} + (\beta_{3} + 4 \beta_{2} - \beta_1) q^{37} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{39} + ( - \beta_{3} - \beta_1 + 7) q^{41} + (\beta_{3} - 3 \beta_{2} - \beta_1) q^{43} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{47} + ( - \beta_{3} - \beta_1 + 4) q^{51} + ( - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{53} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 7) q^{55} + ( - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{57} + ( - \beta_{3} - \beta_1) q^{59} + (2 \beta_{3} + 2 \beta_1 + 7) q^{61} + (2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 - 4) q^{65} + ( - 4 \beta_{3} + 5 \beta_{2} + 4 \beta_1) q^{67} + ( - 2 \beta_{3} - 2 \beta_1 - 7) q^{69} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{71} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{73} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 - 7) q^{75} + ( - \beta_{3} - \beta_1 - 4) q^{79} - 9 q^{81} + (3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{83} + (3 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 1) q^{85} + (3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{87} - 7 q^{89} + (3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{93} + ( - \beta_{3} - 2 \beta_{2} + 7) q^{95} - 4 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{5} - 6 q^{11} - 3 q^{15} - 2 q^{19} + 9 q^{25} - 2 q^{29} + 2 q^{31} + 12 q^{39} + 30 q^{41} + 18 q^{51} - 27 q^{55} + 2 q^{59} + 24 q^{61} - 16 q^{65} - 24 q^{69} + 12 q^{71} - 27 q^{75} - 14 q^{79} - 36 q^{81} - 5 q^{85} - 28 q^{89} + 29 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} + 16\nu - 25 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - 4\nu^{2} + 4\nu + 15 ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu + 5 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} - 5\beta_{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{3} + 2\beta_{2} + 4\beta _1 + 7 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{2} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 12 \) Copy content Toggle raw display
\( T_{19}^{2} + T_{19} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

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