Properties

Label 980.2.i.e
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{11} -5 q^{13} - q^{15} + ( -1 + \zeta_{6} ) q^{17} + 6 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -5 q^{27} + 3 q^{29} + ( -2 + 2 \zeta_{6} ) q^{31} + \zeta_{6} q^{33} -8 \zeta_{6} q^{37} + ( 5 - 5 \zeta_{6} ) q^{39} -10 q^{41} -2 q^{43} + ( -2 + 2 \zeta_{6} ) q^{45} + 7 \zeta_{6} q^{47} -\zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{53} + q^{55} -6 q^{57} + ( -14 + 14 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} -5 \zeta_{6} q^{65} + ( -14 + 14 \zeta_{6} ) q^{67} -4 q^{69} + ( 10 - 10 \zeta_{6} ) q^{73} -\zeta_{6} q^{75} + 11 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -4 q^{83} - q^{85} + ( -3 + 3 \zeta_{6} ) q^{87} -4 \zeta_{6} q^{89} -2 \zeta_{6} q^{93} + ( -6 + 6 \zeta_{6} ) q^{95} -3 q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - q^{3} + q^{5} + 2q^{9} + q^{11} - 10q^{13} - 2q^{15} - q^{17} + 6q^{19} + 4q^{23} - q^{25} - 10q^{27} + 6q^{29} - 2q^{31} + q^{33} - 8q^{37} + 5q^{39} - 20q^{41} - 4q^{43} - 2q^{45} + 7q^{47} - q^{51} + 2q^{53} + 2q^{55} - 12q^{57} - 14q^{59} + 8q^{61} - 5q^{65} - 14q^{67} - 8q^{69} + 10q^{73} - q^{75} + 11q^{79} - q^{81} - 8q^{83} - 2q^{85} - 3q^{87} - 4q^{89} - 2q^{93} - 6q^{95} - 6q^{97} + 4q^{99} + O(q^{100}) \)

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.e 2
7.b odd 2 1 980.2.i.g 2
7.c even 3 1 980.2.a.f yes 1
7.c even 3 1 inner 980.2.i.e 2
7.d odd 6 1 980.2.a.d 1
7.d odd 6 1 980.2.i.g 2
21.g even 6 1 8820.2.a.i 1
21.h odd 6 1 8820.2.a.v 1
28.f even 6 1 3920.2.a.z 1
28.g odd 6 1 3920.2.a.n 1
35.i odd 6 1 4900.2.a.o 1
35.j even 6 1 4900.2.a.h 1
35.k even 12 2 4900.2.e.j 2
35.l odd 12 2 4900.2.e.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.d 1 7.d odd 6 1
980.2.a.f yes 1 7.c even 3 1
980.2.i.e 2 1.a even 1 1 trivial
980.2.i.e 2 7.c even 3 1 inner
980.2.i.g 2 7.b odd 2 1
980.2.i.g 2 7.d odd 6 1
3920.2.a.n 1 28.g odd 6 1
3920.2.a.z 1 28.f even 6 1
4900.2.a.h 1 35.j even 6 1
4900.2.a.o 1 35.i odd 6 1
4900.2.e.j 2 35.k even 12 2
4900.2.e.k 2 35.l odd 12 2
8820.2.a.i 1 21.g even 6 1
8820.2.a.v 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 \(S_{2}^{\mathrm{new}}(980, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( 5 + T )^{2} \)
$17$ \( 1 + T + T^{2} \)
$19$ \( 36 - 6 T + T^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( 4 + 2 T + T^{2} \)
$37$ \( 64 + 8 T + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( ( 2 + T )^{2} \)
$47$ \( 49 - 7 T + T^{2} \)
$53$ \( 4 - 2 T + T^{2} \)
$59$ \( 196 + 14 T + T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( 196 + 14 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 121 - 11 T + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( 16 + 4 T + T^{2} \)
$97$ \( ( 3 + T )^{2} \)
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