Properties

Label 980.2.o.a
Level $980$
Weight $2$
Character orbit 980.o
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
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.o (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: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} -2 \zeta_{12} q^{4} + \zeta_{12} q^{5} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( -2 + 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} -2 \zeta_{12} q^{4} + \zeta_{12} q^{5} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( -2 + 2 \zeta_{12}^{3} ) q^{8} + ( 1 - \zeta_{12}^{3} ) q^{10} + ( -2 - 2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{11} + ( -2 + 4 \zeta_{12}^{2} ) q^{12} + ( -2 + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{13} + ( 1 - 2 \zeta_{12}^{2} ) q^{15} + 4 \zeta_{12}^{2} q^{16} + ( 4 - 3 \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{17} + ( -6 + 6 \zeta_{12}^{2} ) q^{19} -2 \zeta_{12}^{2} q^{20} + ( -1 + \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{22} + ( -2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{23} + ( 4 + 2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{24} + \zeta_{12}^{2} q^{25} + ( 4 + 5 \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 1 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{29} + ( -2 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{30} + 6 \zeta_{12}^{2} q^{31} + ( 4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{32} + ( 2 + 3 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{33} + ( -5 + 5 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{34} + ( 6 - 2 \zeta_{12} - 6 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{37} + ( 6 + 6 \zeta_{12}^{3} ) q^{38} + ( 6 + 6 \zeta_{12} - 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{39} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{40} + ( -2 + 4 \zeta_{12}^{2} ) q^{41} -2 \zeta_{12}^{3} q^{43} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{44} + ( -4 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{46} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{47} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{48} + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{50} + ( 3 - 6 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{51} + ( 6 + 4 \zeta_{12} - 6 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{52} -2 \zeta_{12}^{2} q^{53} + ( -6 + 3 \zeta_{12} + 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{54} + ( -2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{55} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{57} + ( -8 + 5 \zeta_{12} + 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{58} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{59} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{60} + ( 2 - 6 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{61} + ( 6 + 6 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{62} -8 \zeta_{12}^{3} q^{64} + ( -3 - 2 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{65} + ( 5 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{66} + ( -4 + 2 \zeta_{12}^{2} ) q^{67} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{68} + ( -2 + 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{69} + ( -2 + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{71} + ( -8 + 6 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{73} + ( -8 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{74} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{75} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{76} + ( 3 - 3 \zeta_{12} - 9 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{78} + ( -5 + 6 \zeta_{12} - 5 \zeta_{12}^{2} ) q^{79} + 4 \zeta_{12}^{3} q^{80} + 9 \zeta_{12}^{2} q^{81} + ( 4 + 2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{82} + ( 12 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{83} + ( -3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{85} + ( -2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{86} + ( -\zeta_{12} + 12 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{87} + ( 4 + 2 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{88} + ( -2 - 6 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{89} + ( 4 \zeta_{12} + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{92} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{93} + ( 1 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{94} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{95} + ( 4 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{96} + ( -6 + 12 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 6q^{6} - 8q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 6q^{6} - 8q^{8} + 4q^{10} - 6q^{11} + 8q^{16} + 12q^{17} - 12q^{19} - 4q^{20} + 2q^{22} - 12q^{23} + 12q^{24} + 2q^{25} + 18q^{26} + 4q^{29} - 6q^{30} + 12q^{31} + 8q^{32} + 12q^{33} - 18q^{34} + 12q^{37} + 24q^{38} + 18q^{39} - 4q^{40} + 16q^{44} - 8q^{46} + 2q^{50} + 18q^{51} + 12q^{52} - 4q^{53} - 18q^{54} - 8q^{55} - 26q^{58} + 12q^{61} + 12q^{62} - 6q^{65} + 12q^{66} - 12q^{67} + 24q^{68} - 24q^{73} - 24q^{74} - 6q^{78} - 30q^{79} + 18q^{81} + 12q^{82} + 48q^{83} - 12q^{85} - 4q^{86} + 24q^{87} + 4q^{88} - 12q^{89} + 8q^{92} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−1.36603 + 0.366025i 0.866025 1.50000i 1.73205 1.00000i −0.866025 + 0.500000i −0.633975 + 2.36603i 0 −2.00000 + 2.00000i 0 1.00000 1.00000i
31.2 0.366025 + 1.36603i −0.866025 + 1.50000i −1.73205 + 1.00000i 0.866025 0.500000i −2.36603 0.633975i 0 −2.00000 2.00000i 0 1.00000 + 1.00000i
411.1 −1.36603 0.366025i 0.866025 + 1.50000i 1.73205 + 1.00000i −0.866025 0.500000i −0.633975 2.36603i 0 −2.00000 2.00000i 0 1.00000 + 1.00000i
411.2 0.366025 1.36603i −0.866025 1.50000i −1.73205 1.00000i 0.866025 + 0.500000i −2.36603 + 0.633975i 0 −2.00000 + 2.00000i 0 1.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f 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.o.a 4
4.b odd 2 1 980.2.o.d 4
7.b odd 2 1 980.2.o.b 4
7.c even 3 1 140.2.g.a 4
7.c even 3 1 980.2.o.c 4
7.d odd 6 1 140.2.g.b yes 4
7.d odd 6 1 980.2.o.d 4
21.g even 6 1 1260.2.c.b 4
21.h odd 6 1 1260.2.c.a 4
28.d even 2 1 980.2.o.c 4
28.f even 6 1 140.2.g.a 4
28.f even 6 1 inner 980.2.o.a 4
28.g odd 6 1 140.2.g.b yes 4
28.g odd 6 1 980.2.o.b 4
35.i odd 6 1 700.2.g.g 4
35.j even 6 1 700.2.g.f 4
35.k even 12 1 700.2.c.c 4
35.k even 12 1 700.2.c.f 4
35.l odd 12 1 700.2.c.b 4
35.l odd 12 1 700.2.c.e 4
56.j odd 6 1 2240.2.k.b 4
56.k odd 6 1 2240.2.k.b 4
56.m even 6 1 2240.2.k.a 4
56.p even 6 1 2240.2.k.a 4
84.j odd 6 1 1260.2.c.a 4
84.n even 6 1 1260.2.c.b 4
140.p odd 6 1 700.2.g.g 4
140.s even 6 1 700.2.g.f 4
140.w even 12 1 700.2.c.c 4
140.w even 12 1 700.2.c.f 4
140.x odd 12 1 700.2.c.b 4
140.x odd 12 1 700.2.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a 4 7.c even 3 1
140.2.g.a 4 28.f even 6 1
140.2.g.b yes 4 7.d odd 6 1
140.2.g.b yes 4 28.g odd 6 1
700.2.c.b 4 35.l odd 12 1
700.2.c.b 4 140.x odd 12 1
700.2.c.c 4 35.k even 12 1
700.2.c.c 4 140.w even 12 1
700.2.c.e 4 35.l odd 12 1
700.2.c.e 4 140.x odd 12 1
700.2.c.f 4 35.k even 12 1
700.2.c.f 4 140.w even 12 1
700.2.g.f 4 35.j even 6 1
700.2.g.f 4 140.s even 6 1
700.2.g.g 4 35.i odd 6 1
700.2.g.g 4 140.p odd 6 1
980.2.o.a 4 1.a even 1 1 trivial
980.2.o.a 4 28.f even 6 1 inner
980.2.o.b 4 7.b odd 2 1
980.2.o.b 4 28.g odd 6 1
980.2.o.c 4 7.c even 3 1
980.2.o.c 4 28.d even 2 1
980.2.o.d 4 4.b odd 2 1
980.2.o.d 4 7.d odd 6 1
1260.2.c.a 4 21.h odd 6 1
1260.2.c.a 4 84.j odd 6 1
1260.2.c.b 4 21.g even 6 1
1260.2.c.b 4 84.n even 6 1
2240.2.k.a 4 56.m even 6 1
2240.2.k.a 4 56.p even 6 1
2240.2.k.b 4 56.j odd 6 1
2240.2.k.b 4 56.k odd 6 1

Hecke kernels

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

\( T_{3}^{4} + 3 T_{3}^{2} + 9 \)
\( T_{11}^{4} + 6 T_{11}^{3} + 11 T_{11}^{2} - 6 T_{11} + 1 \)
\( T_{17}^{4} - 12 T_{17}^{3} + 51 T_{17}^{2} - 36 T_{17} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( 1 - T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 1 - 6 T + 11 T^{2} + 6 T^{3} + T^{4} \)
$13$ \( 9 + 42 T^{2} + T^{4} \)
$17$ \( 9 - 36 T + 51 T^{2} - 12 T^{3} + T^{4} \)
$19$ \( ( 36 + 6 T + T^{2} )^{2} \)
$23$ \( 64 + 96 T + 56 T^{2} + 12 T^{3} + T^{4} \)
$29$ \( ( -47 - 2 T + T^{2} )^{2} \)
$31$ \( ( 36 - 6 T + T^{2} )^{2} \)
$37$ \( 576 - 288 T + 120 T^{2} - 12 T^{3} + T^{4} \)
$41$ \( ( 12 + T^{2} )^{2} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( 9 + 3 T^{2} + T^{4} \)
$53$ \( ( 4 + 2 T + T^{2} )^{2} \)
$59$ \( 144 + 12 T^{2} + T^{4} \)
$61$ \( 576 + 288 T + 24 T^{2} - 12 T^{3} + T^{4} \)
$67$ \( ( 12 + 6 T + T^{2} )^{2} \)
$71$ \( 16 + 56 T^{2} + T^{4} \)
$73$ \( 144 + 288 T + 204 T^{2} + 24 T^{3} + T^{4} \)
$79$ \( 1521 + 1170 T + 339 T^{2} + 30 T^{3} + T^{4} \)
$83$ \( ( 132 - 24 T + T^{2} )^{2} \)
$89$ \( 576 - 288 T + 24 T^{2} + 12 T^{3} + T^{4} \)
$97$ \( 9801 + 234 T^{2} + T^{4} \)
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