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Properties

Label 980.2.o.d
Level $980$
Weight $2$
Character orbit 980.o
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [980,2,Mod(31,980)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("980.31"); S:= CuspForms(chi, 2); N := Newforms(S);
 
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

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} + 1) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12}^{3} q^{4} - \zeta_{12} q^{5} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \cdots + 2) q^{6} + (2 \zeta_{12}^{3} - 2) q^{8} + \cdots + (3 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 6 q^{6} - 8 q^{8} + 2 q^{10} + 6 q^{11} + 12 q^{12} - 16 q^{16} + 12 q^{17} + 12 q^{19} + 4 q^{20} + 2 q^{22} + 12 q^{23} + 12 q^{24} + 2 q^{25} + 12 q^{26} + 4 q^{29} - 12 q^{31} - 16 q^{32}+ \cdots - 24 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
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
1.00000 − 1.00000i −0.866025 + 1.50000i − 2.00000i −0.866025 + 0.500000i 0.633975 + 2.36603i 0 −2.00000 − 2.00000i 0 −0.366025 + 1.36603i
31.2 1.00000 + 1.00000i 0.866025 − 1.50000i 2.00000i 0.866025 − 0.500000i 2.36603 − 0.633975i 0 −2.00000 + 2.00000i 0 1.36603 + 0.366025i
411.1 1.00000 − 1.00000i 0.866025 + 1.50000i − 2.00000i 0.866025 + 0.500000i 2.36603 + 0.633975i 0 −2.00000 − 2.00000i 0 1.36603 − 0.366025i
411.2 1.00000 + 1.00000i −0.866025 − 1.50000i 2.00000i −0.866025 − 0.500000i 0.633975 − 2.36603i 0 −2.00000 + 2.00000i 0 −0.366025 − 1.36603i
\(n\): 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.d 4
4.b odd 2 1 980.2.o.a 4
7.b odd 2 1 980.2.o.c 4
7.c even 3 1 140.2.g.b yes 4
7.c even 3 1 980.2.o.b 4
7.d odd 6 1 140.2.g.a ✓ 4
7.d odd 6 1 980.2.o.a 4
21.g even 6 1 1260.2.c.a 4
21.h odd 6 1 1260.2.c.b 4
28.d even 2 1 980.2.o.b 4
28.f even 6 1 140.2.g.b yes 4
28.f even 6 1 inner 980.2.o.d 4
28.g odd 6 1 140.2.g.a ✓ 4
28.g odd 6 1 980.2.o.c 4
35.i odd 6 1 700.2.g.f 4
35.j even 6 1 700.2.g.g 4
35.k even 12 1 700.2.c.b 4
35.k even 12 1 700.2.c.e 4
35.l odd 12 1 700.2.c.c 4
35.l odd 12 1 700.2.c.f 4
56.j odd 6 1 2240.2.k.a 4
56.k odd 6 1 2240.2.k.a 4
56.m even 6 1 2240.2.k.b 4
56.p even 6 1 2240.2.k.b 4
84.j odd 6 1 1260.2.c.b 4
84.n even 6 1 1260.2.c.a 4
140.p odd 6 1 700.2.g.f 4
140.s even 6 1 700.2.g.g 4
140.w even 12 1 700.2.c.b 4
140.w even 12 1 700.2.c.e 4
140.x odd 12 1 700.2.c.c 4
140.x odd 12 1 700.2.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a ✓ 4 7.d odd 6 1
140.2.g.a ✓ 4 28.g odd 6 1
140.2.g.b yes 4 7.c even 3 1
140.2.g.b yes 4 28.f even 6 1
700.2.c.b 4 35.k even 12 1
700.2.c.b 4 140.w even 12 1
700.2.c.c 4 35.l odd 12 1
700.2.c.c 4 140.x odd 12 1
700.2.c.e 4 35.k even 12 1
700.2.c.e 4 140.w even 12 1
700.2.c.f 4 35.l odd 12 1
700.2.c.f 4 140.x odd 12 1
700.2.g.f 4 35.i odd 6 1
700.2.g.f 4 140.p odd 6 1
700.2.g.g 4 35.j even 6 1
700.2.g.g 4 140.s even 6 1
980.2.o.a 4 4.b odd 2 1
980.2.o.a 4 7.d odd 6 1
980.2.o.b 4 7.c even 3 1
980.2.o.b 4 28.d even 2 1
980.2.o.c 4 7.b odd 2 1
980.2.o.c 4 28.g odd 6 1
980.2.o.d 4 1.a even 1 1 trivial
980.2.o.d 4 28.f even 6 1 inner
1260.2.c.a 4 21.g even 6 1
1260.2.c.a 4 84.n even 6 1
1260.2.c.b 4 21.h odd 6 1
1260.2.c.b 4 84.j odd 6 1
2240.2.k.a 4 56.j odd 6 1
2240.2.k.a 4 56.k odd 6 1
2240.2.k.b 4 56.m even 6 1
2240.2.k.b 4 56.p even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

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