Properties

Label 980.2.i.b
Level $980$
Weight $2$
Character orbit 980.i
Analytic conductor $7.825$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 + 3 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} -6 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -3 + 3 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} -6 \zeta_{6} q^{9} + ( 5 - 5 \zeta_{6} ) q^{11} -3 q^{13} -3 q^{15} + ( 1 - \zeta_{6} ) q^{17} -6 \zeta_{6} q^{19} -6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 9 q^{27} -9 q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} + 15 \zeta_{6} q^{33} -2 \zeta_{6} q^{37} + ( 9 - 9 \zeta_{6} ) q^{39} -4 q^{41} + 10 q^{43} + ( 6 - 6 \zeta_{6} ) q^{45} + \zeta_{6} q^{47} + 3 \zeta_{6} q^{51} + ( -4 + 4 \zeta_{6} ) q^{53} + 5 q^{55} + 18 q^{57} + ( 8 - 8 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} -3 \zeta_{6} q^{65} + ( -12 + 12 \zeta_{6} ) q^{67} + 18 q^{69} + 8 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} -3 \zeta_{6} q^{75} -13 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -4 q^{83} + q^{85} + ( 27 - 27 \zeta_{6} ) q^{87} -4 \zeta_{6} q^{89} + 12 \zeta_{6} q^{93} + ( 6 - 6 \zeta_{6} ) q^{95} -13 q^{97} -30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} + q^{5} - 6q^{9} + O(q^{10}) \) \( 2q - 3q^{3} + q^{5} - 6q^{9} + 5q^{11} - 6q^{13} - 6q^{15} + q^{17} - 6q^{19} - 6q^{23} - q^{25} + 18q^{27} - 18q^{29} + 4q^{31} + 15q^{33} - 2q^{37} + 9q^{39} - 8q^{41} + 20q^{43} + 6q^{45} + q^{47} + 3q^{51} - 4q^{53} + 10q^{55} + 36q^{57} + 8q^{59} + 8q^{61} - 3q^{65} - 12q^{67} + 36q^{69} + 16q^{71} - 2q^{73} - 3q^{75} - 13q^{79} - 9q^{81} - 8q^{83} + 2q^{85} + 27q^{87} - 4q^{89} + 12q^{93} + 6q^{95} - 26q^{97} - 60q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.50000 + 2.59808i 0 0.500000 + 0.866025i 0 0 0 −3.00000 5.19615i 0
961.1 0 −1.50000 2.59808i 0 0.500000 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.i.b 2
7.b odd 2 1 980.2.i.j 2
7.c even 3 1 140.2.a.b 1
7.c even 3 1 inner 980.2.i.b 2
7.d odd 6 1 980.2.a.b 1
7.d odd 6 1 980.2.i.j 2
21.g even 6 1 8820.2.a.n 1
21.h odd 6 1 1260.2.a.h 1
28.f even 6 1 3920.2.a.bl 1
28.g odd 6 1 560.2.a.a 1
35.i odd 6 1 4900.2.a.u 1
35.j even 6 1 700.2.a.b 1
35.k even 12 2 4900.2.e.a 2
35.l odd 12 2 700.2.e.a 2
56.k odd 6 1 2240.2.a.bb 1
56.p even 6 1 2240.2.a.c 1
84.n even 6 1 5040.2.a.bd 1
105.o odd 6 1 6300.2.a.bf 1
105.x even 12 2 6300.2.k.p 2
140.p odd 6 1 2800.2.a.be 1
140.w even 12 2 2800.2.g.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.b 1 7.c even 3 1
560.2.a.a 1 28.g odd 6 1
700.2.a.b 1 35.j even 6 1
700.2.e.a 2 35.l odd 12 2
980.2.a.b 1 7.d odd 6 1
980.2.i.b 2 1.a even 1 1 trivial
980.2.i.b 2 7.c even 3 1 inner
980.2.i.j 2 7.b odd 2 1
980.2.i.j 2 7.d odd 6 1
1260.2.a.h 1 21.h odd 6 1
2240.2.a.c 1 56.p even 6 1
2240.2.a.bb 1 56.k odd 6 1
2800.2.a.be 1 140.p odd 6 1
2800.2.g.c 2 140.w even 12 2
3920.2.a.bl 1 28.f even 6 1
4900.2.a.u 1 35.i odd 6 1
4900.2.e.a 2 35.k even 12 2
5040.2.a.bd 1 84.n even 6 1
6300.2.a.bf 1 105.o odd 6 1
6300.2.k.p 2 105.x even 12 2
8820.2.a.n 1 21.g even 6 1

Hecke kernels

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

\( T_{3}^{2} + 3 T_{3} + 9 \)
\( T_{11}^{2} - 5 T_{11} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + 3 T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 25 - 5 T + T^{2} \)
$13$ \( ( 3 + T )^{2} \)
$17$ \( 1 - T + T^{2} \)
$19$ \( 36 + 6 T + T^{2} \)
$23$ \( 36 + 6 T + T^{2} \)
$29$ \( ( 9 + T )^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( ( 4 + T )^{2} \)
$43$ \( ( -10 + T )^{2} \)
$47$ \( 1 - T + T^{2} \)
$53$ \( 16 + 4 T + T^{2} \)
$59$ \( 64 - 8 T + T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( 144 + 12 T + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 4 + 2 T + T^{2} \)
$79$ \( 169 + 13 T + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( 16 + 4 T + T^{2} \)
$97$ \( ( 13 + T )^{2} \)
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