Properties

Label 980.2.q.f
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + 3 \zeta_{12} q^{3} + ( -\zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + 6 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + 3 \zeta_{12} q^{3} + ( -\zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + 6 \zeta_{12}^{2} q^{9} + ( -3 + 3 \zeta_{12}^{2} ) q^{11} -\zeta_{12}^{3} q^{13} + ( -3 + 6 \zeta_{12}^{3} ) q^{15} + 5 \zeta_{12} q^{17} -8 \zeta_{12}^{2} q^{19} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{23} + ( -3 - 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} + 9 \zeta_{12}^{3} q^{27} + q^{29} + ( 2 - 2 \zeta_{12}^{2} ) q^{31} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{33} + ( 10 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{37} + ( 3 - 3 \zeta_{12}^{2} ) q^{39} -6 q^{41} + 4 \zeta_{12}^{3} q^{43} + ( -12 - 6 \zeta_{12} + 12 \zeta_{12}^{2} ) q^{45} + ( 11 \zeta_{12} - 11 \zeta_{12}^{3} ) q^{47} + 15 \zeta_{12}^{2} q^{51} + 6 \zeta_{12} q^{53} + ( -6 - 3 \zeta_{12}^{3} ) q^{55} -24 \zeta_{12}^{3} q^{57} + ( -10 + 10 \zeta_{12}^{2} ) q^{59} + ( 2 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} + 10 \zeta_{12} q^{67} -6 q^{69} -10 \zeta_{12} q^{73} + ( -9 \zeta_{12} - 12 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{75} -7 \zeta_{12}^{2} q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} -12 \zeta_{12}^{3} q^{83} + ( -5 + 10 \zeta_{12}^{3} ) q^{85} + 3 \zeta_{12} q^{87} + 8 \zeta_{12}^{2} q^{89} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{93} + ( 16 + 8 \zeta_{12} - 16 \zeta_{12}^{2} ) q^{95} + 3 \zeta_{12}^{3} q^{97} -18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} + 12q^{9} + O(q^{10}) \) \( 4q + 4q^{5} + 12q^{9} - 6q^{11} - 12q^{15} - 16q^{19} - 6q^{25} + 4q^{29} + 4q^{31} + 6q^{39} - 24q^{41} - 24q^{45} + 30q^{51} - 24q^{55} - 20q^{59} + 2q^{65} - 24q^{69} - 24q^{75} - 14q^{79} - 18q^{81} - 20q^{85} + 16q^{89} + 32q^{95} - 72q^{99} + O(q^{100}) \)

Character values

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

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

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 −2.59808 + 1.50000i 0 1.86603 1.23205i 0 0 0 3.00000 5.19615i 0
569.2 0 2.59808 1.50000i 0 0.133975 2.23205i 0 0 0 3.00000 5.19615i 0
949.1 0 −2.59808 1.50000i 0 1.86603 + 1.23205i 0 0 0 3.00000 + 5.19615i 0
949.2 0 2.59808 + 1.50000i 0 0.133975 + 2.23205i 0 0 0 3.00000 + 5.19615i 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.f 4
5.b even 2 1 inner 980.2.q.f 4
7.b odd 2 1 980.2.q.c 4
7.c even 3 1 140.2.e.a 2
7.c even 3 1 inner 980.2.q.f 4
7.d odd 6 1 980.2.e.b 2
7.d odd 6 1 980.2.q.c 4
21.h odd 6 1 1260.2.k.c 2
28.g odd 6 1 560.2.g.a 2
35.c odd 2 1 980.2.q.c 4
35.i odd 6 1 980.2.e.b 2
35.i odd 6 1 980.2.q.c 4
35.j even 6 1 140.2.e.a 2
35.j even 6 1 inner 980.2.q.f 4
35.k even 12 1 4900.2.a.b 1
35.k even 12 1 4900.2.a.w 1
35.l odd 12 1 700.2.a.a 1
35.l odd 12 1 700.2.a.j 1
56.k odd 6 1 2240.2.g.f 2
56.p even 6 1 2240.2.g.e 2
84.n even 6 1 5040.2.t.s 2
105.o odd 6 1 1260.2.k.c 2
105.x even 12 1 6300.2.a.c 1
105.x even 12 1 6300.2.a.t 1
140.p odd 6 1 560.2.g.a 2
140.w even 12 1 2800.2.a.a 1
140.w even 12 1 2800.2.a.bf 1
280.bf even 6 1 2240.2.g.e 2
280.bi odd 6 1 2240.2.g.f 2
420.ba even 6 1 5040.2.t.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 7.c even 3 1
140.2.e.a 2 35.j even 6 1
560.2.g.a 2 28.g odd 6 1
560.2.g.a 2 140.p odd 6 1
700.2.a.a 1 35.l odd 12 1
700.2.a.j 1 35.l odd 12 1
980.2.e.b 2 7.d odd 6 1
980.2.e.b 2 35.i odd 6 1
980.2.q.c 4 7.b odd 2 1
980.2.q.c 4 7.d odd 6 1
980.2.q.c 4 35.c odd 2 1
980.2.q.c 4 35.i odd 6 1
980.2.q.f 4 1.a even 1 1 trivial
980.2.q.f 4 5.b even 2 1 inner
980.2.q.f 4 7.c even 3 1 inner
980.2.q.f 4 35.j even 6 1 inner
1260.2.k.c 2 21.h odd 6 1
1260.2.k.c 2 105.o odd 6 1
2240.2.g.e 2 56.p even 6 1
2240.2.g.e 2 280.bf even 6 1
2240.2.g.f 2 56.k odd 6 1
2240.2.g.f 2 280.bi odd 6 1
2800.2.a.a 1 140.w even 12 1
2800.2.a.bf 1 140.w even 12 1
4900.2.a.b 1 35.k even 12 1
4900.2.a.w 1 35.k even 12 1
5040.2.t.s 2 84.n even 6 1
5040.2.t.s 2 420.ba even 6 1
6300.2.a.c 1 105.x even 12 1
6300.2.a.t 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 \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{4} - 9 T_{3}^{2} + 81 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{19}^{2} + 8 T_{19} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 9 T^{2} + T^{4} \)
$5$ \( 25 - 20 T + 11 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 9 + 3 T + T^{2} )^{2} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( 625 - 25 T^{2} + T^{4} \)
$19$ \( ( 64 + 8 T + T^{2} )^{2} \)
$23$ \( 16 - 4 T^{2} + T^{4} \)
$29$ \( ( -1 + T )^{4} \)
$31$ \( ( 4 - 2 T + T^{2} )^{2} \)
$37$ \( 10000 - 100 T^{2} + T^{4} \)
$41$ \( ( 6 + T )^{4} \)
$43$ \( ( 16 + T^{2} )^{2} \)
$47$ \( 14641 - 121 T^{2} + T^{4} \)
$53$ \( 1296 - 36 T^{2} + T^{4} \)
$59$ \( ( 100 + 10 T + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( 10000 - 100 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 10000 - 100 T^{2} + T^{4} \)
$79$ \( ( 49 + 7 T + T^{2} )^{2} \)
$83$ \( ( 144 + T^{2} )^{2} \)
$89$ \( ( 64 - 8 T + T^{2} )^{2} \)
$97$ \( ( 9 + T^{2} )^{2} \)
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