Properties

Label 980.2.k.c
Level $980$
Weight $2$
Character orbit 980.k
Analytic conductor $7.825$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.k (of order \(4\), degree \(2\), 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + i ) q^{2} + 2 i q^{4} + ( 1 + 2 i ) q^{5} + ( -2 + 2 i ) q^{8} -3 i q^{9} +O(q^{10})\) \( q + ( 1 + i ) q^{2} + 2 i q^{4} + ( 1 + 2 i ) q^{5} + ( -2 + 2 i ) q^{8} -3 i q^{9} + ( -1 + 3 i ) q^{10} + ( -5 + 5 i ) q^{13} -4 q^{16} + ( 5 + 5 i ) q^{17} + ( 3 - 3 i ) q^{18} + ( -4 + 2 i ) q^{20} + ( -3 + 4 i ) q^{25} -10 q^{26} -4 i q^{29} + ( -4 - 4 i ) q^{32} + 10 i q^{34} + 6 q^{36} + ( 7 + 7 i ) q^{37} + ( -6 - 2 i ) q^{40} -10 q^{41} + ( 6 - 3 i ) q^{45} + ( -7 + i ) q^{50} + ( -10 - 10 i ) q^{52} + ( 9 - 9 i ) q^{53} + ( 4 - 4 i ) q^{58} + 10 q^{61} -8 i q^{64} + ( -15 - 5 i ) q^{65} + ( -10 + 10 i ) q^{68} + ( 6 + 6 i ) q^{72} + ( 5 - 5 i ) q^{73} + 14 i q^{74} + ( -4 - 8 i ) q^{80} -9 q^{81} + ( -10 - 10 i ) q^{82} + ( -5 + 15 i ) q^{85} + 10 i q^{89} + ( 9 + 3 i ) q^{90} + ( -5 - 5 i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8} - 2 q^{10} - 10 q^{13} - 8 q^{16} + 10 q^{17} + 6 q^{18} - 8 q^{20} - 6 q^{25} - 20 q^{26} - 8 q^{32} + 12 q^{36} + 14 q^{37} - 12 q^{40} - 20 q^{41} + 12 q^{45} - 14 q^{50} - 20 q^{52} + 18 q^{53} + 8 q^{58} + 20 q^{61} - 30 q^{65} - 20 q^{68} + 12 q^{72} + 10 q^{73} - 8 q^{80} - 18 q^{81} - 20 q^{82} - 10 q^{85} + 18 q^{90} - 10 q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
687.1
1.00000i
1.00000i
1.00000 + 1.00000i 0 2.00000i 1.00000 + 2.00000i 0 0 −2.00000 + 2.00000i 3.00000i −1.00000 + 3.00000i
883.1 1.00000 1.00000i 0 2.00000i 1.00000 2.00000i 0 0 −2.00000 2.00000i 3.00000i −1.00000 3.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.k.c yes 2
4.b odd 2 1 CM 980.2.k.c yes 2
5.c odd 4 1 inner 980.2.k.c yes 2
7.b odd 2 1 980.2.k.b 2
7.c even 3 2 980.2.x.a 4
7.d odd 6 2 980.2.x.b 4
20.e even 4 1 inner 980.2.k.c yes 2
28.d even 2 1 980.2.k.b 2
28.f even 6 2 980.2.x.b 4
28.g odd 6 2 980.2.x.a 4
35.f even 4 1 980.2.k.b 2
35.k even 12 2 980.2.x.b 4
35.l odd 12 2 980.2.x.a 4
140.j odd 4 1 980.2.k.b 2
140.w even 12 2 980.2.x.a 4
140.x odd 12 2 980.2.x.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.k.b 2 7.b odd 2 1
980.2.k.b 2 28.d even 2 1
980.2.k.b 2 35.f even 4 1
980.2.k.b 2 140.j odd 4 1
980.2.k.c yes 2 1.a even 1 1 trivial
980.2.k.c yes 2 4.b odd 2 1 CM
980.2.k.c yes 2 5.c odd 4 1 inner
980.2.k.c yes 2 20.e even 4 1 inner
980.2.x.a 4 7.c even 3 2
980.2.x.a 4 28.g odd 6 2
980.2.x.a 4 35.l odd 12 2
980.2.x.a 4 140.w even 12 2
980.2.x.b 4 7.d odd 6 2
980.2.x.b 4 28.f even 6 2
980.2.x.b 4 35.k even 12 2
980.2.x.b 4 140.x odd 12 2

Hecke kernels

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

\( T_{3} \)
\( T_{13}^{2} + 10 T_{13} + 50 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 - 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 50 + 10 T + T^{2} \)
$17$ \( 50 - 10 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 16 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 98 - 14 T + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 162 - 18 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 50 - 10 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 100 + T^{2} \)
$97$ \( 50 + 10 T + T^{2} \)
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