Properties

Label 980.2.q.d
Level $980$
Weight $2$
Character orbit 980.q
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
CM no
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.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} -3 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} -3 \zeta_{12}^{2} q^{9} -4 \zeta_{12}^{3} q^{13} -4 \zeta_{12} q^{17} + 4 \zeta_{12}^{2} q^{19} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{23} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{25} -2 q^{29} + ( 8 - 8 \zeta_{12}^{2} ) q^{31} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{37} + 6 q^{41} -8 \zeta_{12}^{3} q^{43} + ( -3 - 6 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{45} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{47} + ( -4 + 4 \zeta_{12}^{2} ) q^{59} + 6 \zeta_{12}^{2} q^{61} + ( -4 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{65} -8 \zeta_{12} q^{67} + 12 q^{71} -4 \zeta_{12} q^{73} -4 \zeta_{12}^{2} q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( -8 + 4 \zeta_{12}^{3} ) q^{85} -10 \zeta_{12}^{2} q^{89} + ( 4 + 8 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{95} -12 \zeta_{12}^{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} - 6q^{9} + O(q^{10}) \) \( 4q - 2q^{5} - 6q^{9} + 8q^{19} + 6q^{25} - 8q^{29} + 16q^{31} + 24q^{41} - 6q^{45} - 8q^{59} + 12q^{61} - 16q^{65} + 48q^{71} - 8q^{79} - 18q^{81} - 32q^{85} - 20q^{89} + 8q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
569.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 0 0 −2.23205 0.133975i 0 0 0 −1.50000 + 2.59808i 0
569.2 0 0 0 1.23205 + 1.86603i 0 0 0 −1.50000 + 2.59808i 0
949.1 0 0 0 −2.23205 + 0.133975i 0 0 0 −1.50000 2.59808i 0
949.2 0 0 0 1.23205 1.86603i 0 0 0 −1.50000 2.59808i 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
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.q.d 4
5.b even 2 1 inner 980.2.q.d 4
7.b odd 2 1 980.2.q.e 4
7.c even 3 1 140.2.e.b 2
7.c even 3 1 inner 980.2.q.d 4
7.d odd 6 1 980.2.e.a 2
7.d odd 6 1 980.2.q.e 4
21.h odd 6 1 1260.2.k.b 2
28.g odd 6 1 560.2.g.c 2
35.c odd 2 1 980.2.q.e 4
35.i odd 6 1 980.2.e.a 2
35.i odd 6 1 980.2.q.e 4
35.j even 6 1 140.2.e.b 2
35.j even 6 1 inner 980.2.q.d 4
35.k even 12 1 4900.2.a.l 1
35.k even 12 1 4900.2.a.m 1
35.l odd 12 1 700.2.a.f 1
35.l odd 12 1 700.2.a.h 1
56.k odd 6 1 2240.2.g.c 2
56.p even 6 1 2240.2.g.d 2
84.n even 6 1 5040.2.t.g 2
105.o odd 6 1 1260.2.k.b 2
105.x even 12 1 6300.2.a.g 1
105.x even 12 1 6300.2.a.y 1
140.p odd 6 1 560.2.g.c 2
140.w even 12 1 2800.2.a.o 1
140.w even 12 1 2800.2.a.s 1
280.bf even 6 1 2240.2.g.d 2
280.bi odd 6 1 2240.2.g.c 2
420.ba even 6 1 5040.2.t.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.b 2 7.c even 3 1
140.2.e.b 2 35.j even 6 1
560.2.g.c 2 28.g odd 6 1
560.2.g.c 2 140.p odd 6 1
700.2.a.f 1 35.l odd 12 1
700.2.a.h 1 35.l odd 12 1
980.2.e.a 2 7.d odd 6 1
980.2.e.a 2 35.i odd 6 1
980.2.q.d 4 1.a even 1 1 trivial
980.2.q.d 4 5.b even 2 1 inner
980.2.q.d 4 7.c even 3 1 inner
980.2.q.d 4 35.j even 6 1 inner
980.2.q.e 4 7.b odd 2 1
980.2.q.e 4 7.d odd 6 1
980.2.q.e 4 35.c odd 2 1
980.2.q.e 4 35.i odd 6 1
1260.2.k.b 2 21.h odd 6 1
1260.2.k.b 2 105.o odd 6 1
2240.2.g.c 2 56.k odd 6 1
2240.2.g.c 2 280.bi odd 6 1
2240.2.g.d 2 56.p even 6 1
2240.2.g.d 2 280.bf even 6 1
2800.2.a.o 1 140.w even 12 1
2800.2.a.s 1 140.w even 12 1
4900.2.a.l 1 35.k even 12 1
4900.2.a.m 1 35.k even 12 1
5040.2.t.g 2 84.n even 6 1
5040.2.t.g 2 420.ba even 6 1
6300.2.a.g 1 105.x even 12 1
6300.2.a.y 1 105.x even 12 1

Hecke kernels

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

\( T_{3} \)
\( T_{11} \)
\( T_{19}^{2} - 4 T_{19} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 + 10 T - T^{2} + 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 16 + T^{2} )^{2} \)
$17$ \( 256 - 16 T^{2} + T^{4} \)
$19$ \( ( 16 - 4 T + T^{2} )^{2} \)
$23$ \( 4096 - 64 T^{2} + T^{4} \)
$29$ \( ( 2 + T )^{4} \)
$31$ \( ( 64 - 8 T + T^{2} )^{2} \)
$37$ \( 4096 - 64 T^{2} + T^{4} \)
$41$ \( ( -6 + T )^{4} \)
$43$ \( ( 64 + T^{2} )^{2} \)
$47$ \( 4096 - 64 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 16 + 4 T + T^{2} )^{2} \)
$61$ \( ( 36 - 6 T + T^{2} )^{2} \)
$67$ \( 4096 - 64 T^{2} + T^{4} \)
$71$ \( ( -12 + T )^{4} \)
$73$ \( 256 - 16 T^{2} + T^{4} \)
$79$ \( ( 16 + 4 T + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( ( 100 + 10 T + T^{2} )^{2} \)
$97$ \( ( 144 + T^{2} )^{2} \)
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