Properties

Label 980.2.e.b
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: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} + ( 2 + i ) q^{5} -6 q^{9} +O(q^{10})\) \( q + 3 i q^{3} + ( 2 + i ) q^{5} -6 q^{9} + 3 q^{11} + i q^{13} + ( -3 + 6 i ) q^{15} + 5 i q^{17} -8 q^{19} -2 i q^{23} + ( 3 + 4 i ) q^{25} -9 i q^{27} + q^{29} + 2 q^{31} + 9 i q^{33} + 10 i q^{37} -3 q^{39} + 6 q^{41} + 4 i q^{43} + ( -12 - 6 i ) q^{45} -11 i q^{47} -15 q^{51} -6 i q^{53} + ( 6 + 3 i ) q^{55} -24 i q^{57} -10 q^{59} + ( -1 + 2 i ) q^{65} -10 i q^{67} + 6 q^{69} -10 i q^{73} + ( -12 + 9 i ) q^{75} + 7 q^{79} + 9 q^{81} + 12 i q^{83} + ( -5 + 10 i ) q^{85} + 3 i q^{87} + 8 q^{89} + 6 i q^{93} + ( -16 - 8 i ) q^{95} -3 i q^{97} -18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 12 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{5} - 12 q^{9} + 6 q^{11} - 6 q^{15} - 16 q^{19} + 6 q^{25} + 2 q^{29} + 4 q^{31} - 6 q^{39} + 12 q^{41} - 24 q^{45} - 30 q^{51} + 12 q^{55} - 20 q^{59} - 2 q^{65} + 12 q^{69} - 24 q^{75} + 14 q^{79} + 18 q^{81} - 10 q^{85} + 16 q^{89} - 32 q^{95} - 36 q^{99} + O(q^{100}) \)

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 3.00000i 0 2.00000 1.00000i 0 0 0 −6.00000 0
589.2 0 3.00000i 0 2.00000 + 1.00000i 0 0 0 −6.00000 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.b 2
5.b even 2 1 inner 980.2.e.b 2
5.c odd 4 1 4900.2.a.b 1
5.c odd 4 1 4900.2.a.w 1
7.b odd 2 1 140.2.e.a 2
7.c even 3 2 980.2.q.c 4
7.d odd 6 2 980.2.q.f 4
21.c even 2 1 1260.2.k.c 2
28.d even 2 1 560.2.g.a 2
35.c odd 2 1 140.2.e.a 2
35.f even 4 1 700.2.a.a 1
35.f even 4 1 700.2.a.j 1
35.i odd 6 2 980.2.q.f 4
35.j even 6 2 980.2.q.c 4
56.e even 2 1 2240.2.g.f 2
56.h odd 2 1 2240.2.g.e 2
84.h odd 2 1 5040.2.t.s 2
105.g even 2 1 1260.2.k.c 2
105.k odd 4 1 6300.2.a.c 1
105.k odd 4 1 6300.2.a.t 1
140.c even 2 1 560.2.g.a 2
140.j odd 4 1 2800.2.a.a 1
140.j odd 4 1 2800.2.a.bf 1
280.c odd 2 1 2240.2.g.e 2
280.n even 2 1 2240.2.g.f 2
420.o odd 2 1 5040.2.t.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 7.b odd 2 1
140.2.e.a 2 35.c odd 2 1
560.2.g.a 2 28.d even 2 1
560.2.g.a 2 140.c even 2 1
700.2.a.a 1 35.f even 4 1
700.2.a.j 1 35.f even 4 1
980.2.e.b 2 1.a even 1 1 trivial
980.2.e.b 2 5.b even 2 1 inner
980.2.q.c 4 7.c even 3 2
980.2.q.c 4 35.j even 6 2
980.2.q.f 4 7.d odd 6 2
980.2.q.f 4 35.i odd 6 2
1260.2.k.c 2 21.c even 2 1
1260.2.k.c 2 105.g even 2 1
2240.2.g.e 2 56.h odd 2 1
2240.2.g.e 2 280.c odd 2 1
2240.2.g.f 2 56.e even 2 1
2240.2.g.f 2 280.n even 2 1
2800.2.a.a 1 140.j odd 4 1
2800.2.a.bf 1 140.j odd 4 1
4900.2.a.b 1 5.c odd 4 1
4900.2.a.w 1 5.c odd 4 1
5040.2.t.s 2 84.h odd 2 1
5040.2.t.s 2 420.o odd 2 1
6300.2.a.c 1 105.k odd 4 1
6300.2.a.t 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 \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{2} + 9 \)
\( T_{11} - 3 \)
\( T_{19} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 25 + T^{2} \)
$19$ \( ( 8 + T )^{2} \)
$23$ \( 4 + T^{2} \)
$29$ \( ( -1 + T )^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 121 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 10 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( 100 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( -7 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -8 + T )^{2} \)
$97$ \( 9 + T^{2} \)
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