Properties

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

Related objects

Downloads

Learn more

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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} + q^{13} + q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} + 2 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 5 q^{27} -9 q^{29} + ( 8 - 8 \zeta_{6} ) q^{31} + 3 \zeta_{6} q^{33} + 10 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{39} + 2 q^{43} + ( -2 + 2 \zeta_{6} ) q^{45} -3 \zeta_{6} q^{47} + 3 \zeta_{6} q^{51} -3 q^{55} + 2 q^{57} + ( 12 - 12 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} + \zeta_{6} q^{65} + ( -8 + 8 \zeta_{6} ) q^{67} + 6 q^{69} + ( 14 - 14 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} -5 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} -3 q^{85} + ( -9 + 9 \zeta_{6} ) q^{87} + 12 \zeta_{6} q^{89} -8 \zeta_{6} q^{93} + ( -2 + 2 \zeta_{6} ) q^{95} -17 q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + q^{3} + q^{5} + 2q^{9} - 3q^{11} + 2q^{13} + 2q^{15} - 3q^{17} + 2q^{19} + 6q^{23} - q^{25} + 10q^{27} - 18q^{29} + 8q^{31} + 3q^{33} + 10q^{37} + q^{39} + 4q^{43} - 2q^{45} - 3q^{47} + 3q^{51} - 6q^{55} + 4q^{57} + 12q^{59} + 8q^{61} + q^{65} - 8q^{67} + 12q^{69} + 14q^{73} + q^{75} - 5q^{79} - q^{81} + 24q^{83} - 6q^{85} - 9q^{87} + 12q^{89} - 8q^{93} - 2q^{95} - 34q^{97} - 12q^{99} + O(q^{100}) \)

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 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 0 1.00000 + 1.73205i 0
961.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 0 1.00000 1.73205i 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.h 2
7.b odd 2 1 980.2.i.d 2
7.c even 3 1 980.2.a.c 1
7.c even 3 1 inner 980.2.i.h 2
7.d odd 6 1 140.2.a.a 1
7.d odd 6 1 980.2.i.d 2
21.g even 6 1 1260.2.a.c 1
21.h odd 6 1 8820.2.a.r 1
28.f even 6 1 560.2.a.c 1
28.g odd 6 1 3920.2.a.u 1
35.i odd 6 1 700.2.a.d 1
35.j even 6 1 4900.2.a.p 1
35.k even 12 2 700.2.e.c 2
35.l odd 12 2 4900.2.e.l 2
56.j odd 6 1 2240.2.a.g 1
56.m even 6 1 2240.2.a.r 1
84.j odd 6 1 5040.2.a.h 1
105.p even 6 1 6300.2.a.d 1
105.w odd 12 2 6300.2.k.c 2
140.s even 6 1 2800.2.a.y 1
140.x odd 12 2 2800.2.g.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.a 1 7.d odd 6 1
560.2.a.c 1 28.f even 6 1
700.2.a.d 1 35.i odd 6 1
700.2.e.c 2 35.k even 12 2
980.2.a.c 1 7.c even 3 1
980.2.i.d 2 7.b odd 2 1
980.2.i.d 2 7.d odd 6 1
980.2.i.h 2 1.a even 1 1 trivial
980.2.i.h 2 7.c even 3 1 inner
1260.2.a.c 1 21.g even 6 1
2240.2.a.g 1 56.j odd 6 1
2240.2.a.r 1 56.m even 6 1
2800.2.a.y 1 140.s even 6 1
2800.2.g.j 2 140.x odd 12 2
3920.2.a.u 1 28.g odd 6 1
4900.2.a.p 1 35.j even 6 1
4900.2.e.l 2 35.l odd 12 2
5040.2.a.h 1 84.j odd 6 1
6300.2.a.d 1 105.p even 6 1
6300.2.k.c 2 105.w odd 12 2
8820.2.a.r 1 21.h odd 6 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 9 + 3 T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( 9 + 3 T + T^{2} \)
$19$ \( 4 - 2 T + T^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( ( 9 + T )^{2} \)
$31$ \( 64 - 8 T + T^{2} \)
$37$ \( 100 - 10 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 9 + 3 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 144 - 12 T + T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( 64 + 8 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 196 - 14 T + T^{2} \)
$79$ \( 25 + 5 T + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( 144 - 12 T + T^{2} \)
$97$ \( ( 17 + T )^{2} \)
show more
show less