Properties

Label 980.2.e.a
Level $980$
Weight $2$
Character orbit 980.e
Analytic conductor $7.825$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{5} + 3 q^{9} + 2 \beta q^{13} - 2 \beta q^{17} + 4 q^{19} - 4 \beta q^{23} + (2 \beta - 3) q^{25} - 2 q^{29} + 8 q^{31} - 4 \beta q^{37} - 6 q^{41} - 4 \beta q^{43} + ( - 3 \beta - 3) q^{45} - 4 \beta q^{47} - 4 q^{59} + 6 q^{61} + ( - 2 \beta + 8) q^{65} + 4 \beta q^{67} + 12 q^{71} - 2 \beta q^{73} + 4 q^{79} + 9 q^{81} + (2 \beta - 8) q^{85} - 10 q^{89} + ( - 4 \beta - 4) q^{95} + 6 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 6 q^{9} + 8 q^{19} - 6 q^{25} - 4 q^{29} + 16 q^{31} - 12 q^{41} - 6 q^{45} - 8 q^{59} + 12 q^{61} + 16 q^{65} + 24 q^{71} + 8 q^{79} + 18 q^{81} - 16 q^{85} - 20 q^{89} - 8 q^{95}+O(q^{100}) \) 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
1.00000i
1.00000i
0 0 0 −1.00000 2.00000i 0 0 0 3.00000 0
589.2 0 0 0 −1.00000 + 2.00000i 0 0 0 3.00000 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.a 2
5.b even 2 1 inner 980.2.e.a 2
5.c odd 4 1 4900.2.a.l 1
5.c odd 4 1 4900.2.a.m 1
7.b odd 2 1 140.2.e.b 2
7.c even 3 2 980.2.q.e 4
7.d odd 6 2 980.2.q.d 4
21.c even 2 1 1260.2.k.b 2
28.d even 2 1 560.2.g.c 2
35.c odd 2 1 140.2.e.b 2
35.f even 4 1 700.2.a.f 1
35.f even 4 1 700.2.a.h 1
35.i odd 6 2 980.2.q.d 4
35.j even 6 2 980.2.q.e 4
56.e even 2 1 2240.2.g.c 2
56.h odd 2 1 2240.2.g.d 2
84.h odd 2 1 5040.2.t.g 2
105.g even 2 1 1260.2.k.b 2
105.k odd 4 1 6300.2.a.g 1
105.k odd 4 1 6300.2.a.y 1
140.c even 2 1 560.2.g.c 2
140.j odd 4 1 2800.2.a.o 1
140.j odd 4 1 2800.2.a.s 1
280.c odd 2 1 2240.2.g.d 2
280.n even 2 1 2240.2.g.c 2
420.o odd 2 1 5040.2.t.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.b 2 7.b odd 2 1
140.2.e.b 2 35.c odd 2 1
560.2.g.c 2 28.d even 2 1
560.2.g.c 2 140.c even 2 1
700.2.a.f 1 35.f even 4 1
700.2.a.h 1 35.f even 4 1
980.2.e.a 2 1.a even 1 1 trivial
980.2.e.a 2 5.b even 2 1 inner
980.2.q.d 4 7.d odd 6 2
980.2.q.d 4 35.i odd 6 2
980.2.q.e 4 7.c even 3 2
980.2.q.e 4 35.j even 6 2
1260.2.k.b 2 21.c even 2 1
1260.2.k.b 2 105.g even 2 1
2240.2.g.c 2 56.e even 2 1
2240.2.g.c 2 280.n even 2 1
2240.2.g.d 2 56.h odd 2 1
2240.2.g.d 2 280.c odd 2 1
2800.2.a.o 1 140.j odd 4 1
2800.2.a.s 1 140.j odd 4 1
4900.2.a.l 1 5.c odd 4 1
4900.2.a.m 1 5.c odd 4 1
5040.2.t.g 2 84.h odd 2 1
5040.2.t.g 2 420.o odd 2 1
6300.2.a.g 1 105.k odd 4 1
6300.2.a.y 1 105.k odd 4 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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