Properties

Label 980.2.x.c
Level $980$
Weight $2$
Character orbit 980.x
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{2} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{5} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + 3 \zeta_{12} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{2} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{5} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + 3 \zeta_{12} q^{9} + (\zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12}) q^{10} + ( - \zeta_{12}^{3} + 1) q^{13} + ( - 4 \zeta_{12}^{2} + 4) q^{16} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{17} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{18} + (4 \zeta_{12}^{3} + 2) q^{20} + (4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12}) q^{25} + ( - 2 \zeta_{12}^{2} + 2) q^{26} + 4 \zeta_{12}^{3} q^{29} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12}) q^{32} + 6 \zeta_{12}^{3} q^{34} + 6 q^{36} + ( - 7 \zeta_{12}^{2} + 7 \zeta_{12} + 7) q^{37} + (2 \zeta_{12}^{2} + 6 \zeta_{12} - 2) q^{40} + 8 q^{41} + (6 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 6 \zeta_{12}) q^{45} + (\zeta_{12}^{3} - 7) q^{50} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{52} + (9 \zeta_{12}^{3} - 9 \zeta_{12}^{2} - 9 \zeta_{12}) q^{53} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{58} + ( - 12 \zeta_{12}^{2} + 12) q^{61} - 8 \zeta_{12}^{3} q^{64} + (\zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{65} + (6 \zeta_{12}^{2} + 6 \zeta_{12} - 6) q^{68} + ( - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{72} + (11 \zeta_{12}^{3} - 11 \zeta_{12}^{2} - 11 \zeta_{12}) q^{73} + ( - 14 \zeta_{12}^{3} + 14 \zeta_{12}) q^{74} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 4 \zeta_{12}) q^{80} + 9 \zeta_{12}^{2} q^{81} + ( - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 8) q^{82} + ( - 3 \zeta_{12}^{3} - 9) q^{85} - 16 \zeta_{12} q^{89} + (9 \zeta_{12}^{3} - 3) q^{90} + ( - 13 \zeta_{12}^{3} - 13) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 4 q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 4 q^{5} + 8 q^{8} + 6 q^{10} + 4 q^{13} + 8 q^{16} + 6 q^{17} + 6 q^{18} + 8 q^{20} - 6 q^{25} + 4 q^{26} - 8 q^{32} + 24 q^{36} + 14 q^{37} - 4 q^{40} + 32 q^{41} + 6 q^{45} - 28 q^{50} - 4 q^{52} - 18 q^{53} - 8 q^{58} + 24 q^{61} - 2 q^{65} - 12 q^{68} + 12 q^{72} - 22 q^{73} + 16 q^{80} + 18 q^{81} + 16 q^{82} - 36 q^{85} - 12 q^{90} - 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(\zeta_{12}^{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.366025 + 1.36603i 0 −1.73205 1.00000i −1.86603 1.23205i 0 0 2.00000 2.00000i −2.59808 + 1.50000i 2.36603 2.09808i
263.1 1.36603 + 0.366025i 0 1.73205 + 1.00000i −0.133975 2.23205i 0 0 2.00000 + 2.00000i 2.59808 1.50000i 0.633975 3.09808i
667.1 1.36603 0.366025i 0 1.73205 1.00000i −0.133975 + 2.23205i 0 0 2.00000 2.00000i 2.59808 + 1.50000i 0.633975 + 3.09808i
863.1 −0.366025 1.36603i 0 −1.73205 + 1.00000i −1.86603 + 1.23205i 0 0 2.00000 + 2.00000i −2.59808 1.50000i 2.36603 + 2.09808i
\(n\): 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
7.c even 3 1 inner
20.e even 4 1 inner
28.g odd 6 1 inner
35.l odd 12 1 inner
140.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.x.c 4
4.b odd 2 1 CM 980.2.x.c 4
5.c odd 4 1 inner 980.2.x.c 4
7.b odd 2 1 980.2.x.d 4
7.c even 3 1 980.2.k.a 2
7.c even 3 1 inner 980.2.x.c 4
7.d odd 6 1 20.2.e.a 2
7.d odd 6 1 980.2.x.d 4
20.e even 4 1 inner 980.2.x.c 4
21.g even 6 1 180.2.k.c 2
28.d even 2 1 980.2.x.d 4
28.f even 6 1 20.2.e.a 2
28.f even 6 1 980.2.x.d 4
28.g odd 6 1 980.2.k.a 2
28.g odd 6 1 inner 980.2.x.c 4
35.f even 4 1 980.2.x.d 4
35.i odd 6 1 100.2.e.b 2
35.k even 12 1 20.2.e.a 2
35.k even 12 1 100.2.e.b 2
35.k even 12 1 980.2.x.d 4
35.l odd 12 1 980.2.k.a 2
35.l odd 12 1 inner 980.2.x.c 4
56.j odd 6 1 320.2.n.e 2
56.m even 6 1 320.2.n.e 2
84.j odd 6 1 180.2.k.c 2
105.p even 6 1 900.2.k.c 2
105.w odd 12 1 180.2.k.c 2
105.w odd 12 1 900.2.k.c 2
112.v even 12 1 1280.2.o.g 2
112.v even 12 1 1280.2.o.j 2
112.x odd 12 1 1280.2.o.g 2
112.x odd 12 1 1280.2.o.j 2
140.j odd 4 1 980.2.x.d 4
140.s even 6 1 100.2.e.b 2
140.w even 12 1 980.2.k.a 2
140.w even 12 1 inner 980.2.x.c 4
140.x odd 12 1 20.2.e.a 2
140.x odd 12 1 100.2.e.b 2
140.x odd 12 1 980.2.x.d 4
280.ba even 6 1 1600.2.n.h 2
280.bk odd 6 1 1600.2.n.h 2
280.bp odd 12 1 320.2.n.e 2
280.bp odd 12 1 1600.2.n.h 2
280.bv even 12 1 320.2.n.e 2
280.bv even 12 1 1600.2.n.h 2
420.be odd 6 1 900.2.k.c 2
420.br even 12 1 180.2.k.c 2
420.br even 12 1 900.2.k.c 2
560.ce odd 12 1 1280.2.o.j 2
560.ch even 12 1 1280.2.o.g 2
560.cz even 12 1 1280.2.o.j 2
560.da odd 12 1 1280.2.o.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.e.a 2 7.d odd 6 1
20.2.e.a 2 28.f even 6 1
20.2.e.a 2 35.k even 12 1
20.2.e.a 2 140.x odd 12 1
100.2.e.b 2 35.i odd 6 1
100.2.e.b 2 35.k even 12 1
100.2.e.b 2 140.s even 6 1
100.2.e.b 2 140.x odd 12 1
180.2.k.c 2 21.g even 6 1
180.2.k.c 2 84.j odd 6 1
180.2.k.c 2 105.w odd 12 1
180.2.k.c 2 420.br even 12 1
320.2.n.e 2 56.j odd 6 1
320.2.n.e 2 56.m even 6 1
320.2.n.e 2 280.bp odd 12 1
320.2.n.e 2 280.bv even 12 1
900.2.k.c 2 105.p even 6 1
900.2.k.c 2 105.w odd 12 1
900.2.k.c 2 420.be odd 6 1
900.2.k.c 2 420.br even 12 1
980.2.k.a 2 7.c even 3 1
980.2.k.a 2 28.g odd 6 1
980.2.k.a 2 35.l odd 12 1
980.2.k.a 2 140.w even 12 1
980.2.x.c 4 1.a even 1 1 trivial
980.2.x.c 4 4.b odd 2 1 CM
980.2.x.c 4 5.c odd 4 1 inner
980.2.x.c 4 7.c even 3 1 inner
980.2.x.c 4 20.e even 4 1 inner
980.2.x.c 4 28.g odd 6 1 inner
980.2.x.c 4 35.l odd 12 1 inner
980.2.x.c 4 140.w even 12 1 inner
980.2.x.d 4 7.b odd 2 1
980.2.x.d 4 7.d odd 6 1
980.2.x.d 4 28.d even 2 1
980.2.x.d 4 28.f even 6 1
980.2.x.d 4 35.f even 4 1
980.2.x.d 4 35.k even 12 1
980.2.x.d 4 140.j odd 4 1
980.2.x.d 4 140.x odd 12 1
1280.2.o.g 2 112.v even 12 1
1280.2.o.g 2 112.x odd 12 1
1280.2.o.g 2 560.ch even 12 1
1280.2.o.g 2 560.da odd 12 1
1280.2.o.j 2 112.v even 12 1
1280.2.o.j 2 112.x odd 12 1
1280.2.o.j 2 560.ce odd 12 1
1280.2.o.j 2 560.cz even 12 1
1600.2.n.h 2 280.ba even 6 1
1600.2.n.h 2 280.bk odd 6 1
1600.2.n.h 2 280.bp odd 12 1
1600.2.n.h 2 280.bv even 12 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 11 T^{2} + 20 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 14 T^{3} + 98 T^{2} + \cdots + 9604 \) Copy content Toggle raw display
$41$ \( (T - 8)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 26244 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T + 144)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 22 T^{3} + 242 T^{2} + \cdots + 58564 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 256 T^{2} + 65536 \) Copy content Toggle raw display
$97$ \( (T^{2} + 26 T + 338)^{2} \) Copy content Toggle raw display
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