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Properties

Label 980.2.o.f
Level $980$
Weight $2$
Character orbit 980.o
Analytic conductor $7.825$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(31,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 16 q^{9} + 30 q^{12} - 14 q^{16} - 8 q^{22} - 36 q^{24} + 16 q^{25} - 30 q^{26} - 40 q^{29} + 2 q^{32} + 60 q^{36} + 8 q^{37} + 60 q^{38} - 18 q^{44} - 12 q^{45} + 2 q^{46} + 4 q^{50} + 36 q^{52} - 8 q^{53} - 12 q^{54} + 48 q^{57} + 2 q^{58} + 14 q^{60} - 24 q^{61} + 4 q^{64} + 4 q^{65} - 24 q^{66} - 60 q^{68} + 4 q^{72} + 72 q^{73} + 38 q^{74} + 120 q^{78} - 36 q^{81} - 42 q^{82} + 28 q^{86} + 4 q^{88} + 60 q^{89} - 4 q^{92} - 8 q^{93} - 18 q^{94} + 60 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.39328 + 0.242400i 0.406021 − 0.703249i 1.88248 − 0.675465i −0.866025 + 0.500000i −0.395235 + 1.07825i 0 −2.45910 + 1.39743i 1.17029 + 2.02701i 1.08542 − 0.906567i
31.2 −1.27092 + 0.620297i 0.331177 − 0.573616i 1.23046 − 1.57669i 0.866025 − 0.500000i −0.0650866 + 0.934447i 0 −0.585797 + 2.76710i 1.28064 + 2.21814i −0.790498 + 1.17265i
31.3 −1.26796 − 0.626319i 1.49907 − 2.59647i 1.21545 + 1.58830i 0.866025 − 0.500000i −3.52698 + 2.35332i 0 −0.546365 − 2.77516i −2.99443 − 5.18651i −1.41125 + 0.0915727i
31.4 −0.950668 + 1.04701i −1.36859 + 2.37047i −0.192463 − 1.99072i 0.866025 − 0.500000i −1.18083 − 3.68646i 0 2.26727 + 1.69100i −2.24609 − 3.89033i −0.299797 + 1.38207i
31.5 −0.894275 − 1.09557i −0.895374 + 1.55083i −0.400544 + 1.95948i −0.866025 + 0.500000i 2.49976 − 0.405928i 0 2.50494 − 1.31349i −0.103389 − 0.179074i 1.32225 + 0.501653i
31.6 −0.501653 − 1.32225i 0.895374 − 1.55083i −1.49669 + 1.32662i −0.866025 + 0.500000i −2.49976 − 0.405928i 0 2.50494 + 1.31349i −0.103389 − 0.179074i 1.09557 + 0.894275i
31.7 −0.397222 + 1.35728i −0.556469 + 0.963833i −1.68443 − 1.07828i −0.866025 + 0.500000i −1.08715 − 1.13814i 0 2.13263 − 1.85793i 0.880685 + 1.52539i −0.334637 − 1.37405i
31.8 0.0915727 − 1.41125i −1.49907 + 2.59647i −1.98323 − 0.258463i 0.866025 − 0.500000i 3.52698 + 2.35332i 0 −0.546365 + 2.77516i −2.99443 − 5.18651i −0.626319 − 1.26796i
31.9 0.288532 + 1.38447i 0.450639 − 0.780530i −1.83350 + 0.798926i 0.866025 − 0.500000i 1.21064 + 0.398687i 0 −1.63511 − 2.30790i 1.09385 + 1.89460i 0.942109 + 1.05472i
31.10 0.569639 + 1.29442i 1.51353 − 2.62152i −1.35102 + 1.47470i −0.866025 + 0.500000i 4.25550 + 0.465823i 0 −2.67847 − 0.908739i −3.08156 − 5.33743i −1.14053 − 0.836177i
31.11 0.836177 + 1.14053i −1.51353 + 2.62152i −0.601615 + 1.90737i −0.866025 + 0.500000i −4.25550 + 0.465823i 0 −2.67847 + 0.908739i −3.08156 − 5.33743i −1.29442 − 0.569639i
31.12 0.906567 − 1.08542i −0.406021 + 0.703249i −0.356272 − 1.96801i −0.866025 + 0.500000i 0.395235 + 1.07825i 0 −2.45910 − 1.39743i 1.17029 + 2.02701i −0.242400 + 1.39328i
31.13 1.05472 + 0.942109i −0.450639 + 0.780530i 0.224860 + 1.98732i 0.866025 − 0.500000i −1.21064 + 0.398687i 0 −1.63511 + 2.30790i 1.09385 + 1.89460i 1.38447 + 0.288532i
31.14 1.17265 − 0.790498i −0.331177 + 0.573616i 0.750225 − 1.85396i 0.866025 − 0.500000i 0.0650866 + 0.934447i 0 −0.585797 − 2.76710i 1.28064 + 2.21814i 0.620297 − 1.27092i
31.15 1.37405 + 0.334637i 0.556469 − 0.963833i 1.77604 + 0.919616i −0.866025 + 0.500000i 1.08715 − 1.13814i 0 2.13263 + 1.85793i 0.880685 + 1.52539i −1.35728 + 0.397222i
31.16 1.38207 − 0.299797i 1.36859 − 2.37047i 1.82024 − 0.828682i 0.866025 − 0.500000i 1.18083 − 3.68646i 0 2.26727 − 1.69100i −2.24609 − 3.89033i 1.04701 − 0.950668i
411.1 −1.39328 − 0.242400i 0.406021 + 0.703249i 1.88248 + 0.675465i −0.866025 − 0.500000i −0.395235 − 1.07825i 0 −2.45910 − 1.39743i 1.17029 − 2.02701i 1.08542 + 0.906567i
411.2 −1.27092 − 0.620297i 0.331177 + 0.573616i 1.23046 + 1.57669i 0.866025 + 0.500000i −0.0650866 − 0.934447i 0 −0.585797 − 2.76710i 1.28064 − 2.21814i −0.790498 − 1.17265i
411.3 −1.26796 + 0.626319i 1.49907 + 2.59647i 1.21545 − 1.58830i 0.866025 + 0.500000i −3.52698 − 2.35332i 0 −0.546365 + 2.77516i −2.99443 + 5.18651i −1.41125 − 0.0915727i
411.4 −0.950668 − 1.04701i −1.36859 − 2.37047i −0.192463 + 1.99072i 0.866025 + 0.500000i −1.18083 + 3.68646i 0 2.26727 − 1.69100i −2.24609 + 3.89033i −0.299797 − 1.38207i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
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.f 32
4.b odd 2 1 inner 980.2.o.f 32
7.b odd 2 1 140.2.o.a ✓ 32
7.c even 3 1 140.2.o.a ✓ 32
7.c even 3 1 980.2.g.a 32
7.d odd 6 1 980.2.g.a 32
7.d odd 6 1 inner 980.2.o.f 32
28.d even 2 1 140.2.o.a ✓ 32
28.f even 6 1 980.2.g.a 32
28.f even 6 1 inner 980.2.o.f 32
28.g odd 6 1 140.2.o.a ✓ 32
28.g odd 6 1 980.2.g.a 32
35.c odd 2 1 700.2.p.c 32
35.f even 4 1 700.2.t.c 32
35.f even 4 1 700.2.t.d 32
35.j even 6 1 700.2.p.c 32
35.l odd 12 1 700.2.t.c 32
35.l odd 12 1 700.2.t.d 32
140.c even 2 1 700.2.p.c 32
140.j odd 4 1 700.2.t.c 32
140.j odd 4 1 700.2.t.d 32
140.p odd 6 1 700.2.p.c 32
140.w even 12 1 700.2.t.c 32
140.w even 12 1 700.2.t.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.o.a ✓ 32 7.b odd 2 1
140.2.o.a ✓ 32 7.c even 3 1
140.2.o.a ✓ 32 28.d even 2 1
140.2.o.a ✓ 32 28.g odd 6 1
700.2.p.c 32 35.c odd 2 1
700.2.p.c 32 35.j even 6 1
700.2.p.c 32 140.c even 2 1
700.2.p.c 32 140.p odd 6 1
700.2.t.c 32 35.f even 4 1
700.2.t.c 32 35.l odd 12 1
700.2.t.c 32 140.j odd 4 1
700.2.t.c 32 140.w even 12 1
700.2.t.d 32 35.f even 4 1
700.2.t.d 32 35.l odd 12 1
700.2.t.d 32 140.j odd 4 1
700.2.t.d 32 140.w even 12 1
980.2.g.a 32 7.c even 3 1
980.2.g.a 32 7.d odd 6 1
980.2.g.a 32 28.f even 6 1
980.2.g.a 32 28.g odd 6 1
980.2.o.f 32 1.a even 1 1 trivial
980.2.o.f 32 4.b odd 2 1 inner
980.2.o.f 32 7.d odd 6 1 inner
980.2.o.f 32 28.f even 6 1 inner

Hecke kernels

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

\( T_{3}^{32} + 32 T_{3}^{30} + 629 T_{3}^{28} + 7904 T_{3}^{26} + 73006 T_{3}^{24} + 483232 T_{3}^{22} + \cdots + 331776 \) Copy content Toggle raw display
\( T_{11}^{32} - 78 T_{11}^{30} + 3683 T_{11}^{28} - 113038 T_{11}^{26} + 2566385 T_{11}^{24} + \cdots + 268435456 \) Copy content Toggle raw display
\( T_{17}^{16} - 64 T_{17}^{14} + 2988 T_{17}^{12} - 672 T_{17}^{11} - 61696 T_{17}^{10} + 39360 T_{17}^{9} + \cdots + 9437184 \) Copy content Toggle raw display