Properties

Label 980.2.q.e
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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} - \beta_1) q^{5} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2} - \beta_1) q^{5} - 3 \beta_{2} q^{9} + 2 \beta_{3} q^{13} + 2 \beta_1 q^{17} - 4 \beta_{2} q^{19} + (4 \beta_{3} - 4 \beta_1) q^{23} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{25} - 2 q^{29} + (8 \beta_{2} - 8) q^{31} + (4 \beta_{3} - 4 \beta_1) q^{37} - 6 q^{41} - 4 \beta_{3} q^{43} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{45} + (4 \beta_{3} - 4 \beta_1) q^{47} + ( - 4 \beta_{2} + 4) q^{59} - 6 \beta_{2} q^{61} + (2 \beta_{3} - 8 \beta_{2} - 2 \beta_1) q^{65} - 4 \beta_1 q^{67} + 12 q^{71} + 2 \beta_1 q^{73} - 4 \beta_{2} q^{79} + (9 \beta_{2} - 9) q^{81} + (2 \beta_{3} - 8) q^{85} + 10 \beta_{2} q^{89} + ( - 4 \beta_{2} + 4 \beta_1 + 4) q^{95} + 6 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 6 q^{9} - 8 q^{19} + 6 q^{25} - 8 q^{29} - 16 q^{31} - 24 q^{41} + 6 q^{45} + 8 q^{59} - 12 q^{61} - 16 q^{65} + 48 q^{71} - 8 q^{79} - 18 q^{81} - 32 q^{85} + 20 q^{89} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\) \(1\)

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 −1.23205 1.86603i 0 0 0 −1.50000 + 2.59808i 0
569.2 0 0 0 2.23205 + 0.133975i 0 0 0 −1.50000 + 2.59808i 0
949.1 0 0 0 −1.23205 + 1.86603i 0 0 0 −1.50000 2.59808i 0
949.2 0 0 0 2.23205 0.133975i 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.e 4
5.b even 2 1 inner 980.2.q.e 4
7.b odd 2 1 980.2.q.d 4
7.c even 3 1 980.2.e.a 2
7.c even 3 1 inner 980.2.q.e 4
7.d odd 6 1 140.2.e.b 2
7.d odd 6 1 980.2.q.d 4
21.g even 6 1 1260.2.k.b 2
28.f even 6 1 560.2.g.c 2
35.c odd 2 1 980.2.q.d 4
35.i odd 6 1 140.2.e.b 2
35.i odd 6 1 980.2.q.d 4
35.j even 6 1 980.2.e.a 2
35.j even 6 1 inner 980.2.q.e 4
35.k even 12 1 700.2.a.f 1
35.k even 12 1 700.2.a.h 1
35.l odd 12 1 4900.2.a.l 1
35.l odd 12 1 4900.2.a.m 1
56.j odd 6 1 2240.2.g.d 2
56.m even 6 1 2240.2.g.c 2
84.j odd 6 1 5040.2.t.g 2
105.p even 6 1 1260.2.k.b 2
105.w odd 12 1 6300.2.a.g 1
105.w odd 12 1 6300.2.a.y 1
140.s even 6 1 560.2.g.c 2
140.x odd 12 1 2800.2.a.o 1
140.x odd 12 1 2800.2.a.s 1
280.ba even 6 1 2240.2.g.c 2
280.bk odd 6 1 2240.2.g.d 2
420.be odd 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.d odd 6 1
140.2.e.b 2 35.i odd 6 1
560.2.g.c 2 28.f even 6 1
560.2.g.c 2 140.s even 6 1
700.2.a.f 1 35.k even 12 1
700.2.a.h 1 35.k even 12 1
980.2.e.a 2 7.c even 3 1
980.2.e.a 2 35.j even 6 1
980.2.q.d 4 7.b odd 2 1
980.2.q.d 4 7.d odd 6 1
980.2.q.d 4 35.c odd 2 1
980.2.q.d 4 35.i odd 6 1
980.2.q.e 4 1.a even 1 1 trivial
980.2.q.e 4 5.b even 2 1 inner
980.2.q.e 4 7.c even 3 1 inner
980.2.q.e 4 35.j even 6 1 inner
1260.2.k.b 2 21.g even 6 1
1260.2.k.b 2 105.p even 6 1
2240.2.g.c 2 56.m even 6 1
2240.2.g.c 2 280.ba even 6 1
2240.2.g.d 2 56.j odd 6 1
2240.2.g.d 2 280.bk odd 6 1
2800.2.a.o 1 140.x odd 12 1
2800.2.a.s 1 140.x odd 12 1
4900.2.a.l 1 35.l odd 12 1
4900.2.a.m 1 35.l odd 12 1
5040.2.t.g 2 84.j odd 6 1
5040.2.t.g 2 420.be odd 6 1
6300.2.a.g 1 105.w odd 12 1
6300.2.a.y 1 105.w odd 12 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{19}^{2} + 4T_{19} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

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