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Properties

Label 980.2.bg.b
Level $980$
Weight $2$
Character orbit 980.bg
Analytic conductor $7.825$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(81,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.bg (of order \(21\), degree \(12\), 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: \(84\)
Relative dimension: \(7\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 11 q^{3} + 7 q^{5} + 14 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 11 q^{3} + 7 q^{5} + 14 q^{7} + 8 q^{9} + 8 q^{11} + q^{15} - 22 q^{17} + 9 q^{19} + 14 q^{21} - 12 q^{23} + 7 q^{25} + 13 q^{27} + 22 q^{29} + 14 q^{31} + 77 q^{33} - q^{37} - 31 q^{39} - 28 q^{41} - 6 q^{43} - 6 q^{45} + 58 q^{47} + 28 q^{49} + 14 q^{51} + 34 q^{53} - 9 q^{55} + 15 q^{57} + 4 q^{59} - 28 q^{61} + 77 q^{63} - 14 q^{65} - 7 q^{67} - 18 q^{69} + 16 q^{71} - 56 q^{73} - 11 q^{75} - 42 q^{77} + 44 q^{79} - 8 q^{81} - 20 q^{83} + 23 q^{85} - 102 q^{87} - 71 q^{89} - 63 q^{91} + 3 q^{93} - 26 q^{95} + 84 q^{97} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 −2.79102 − 0.860916i 0 −0.733052 + 0.680173i 0 2.38129 + 1.15302i 0 4.56990 + 3.11571i 0
81.2 0 −2.42278 − 0.747329i 0 −0.733052 + 0.680173i 0 1.12256 − 2.39580i 0 2.83264 + 1.93126i 0
81.3 0 −1.10756 − 0.341638i 0 −0.733052 + 0.680173i 0 −2.50718 − 0.845018i 0 −1.36874 − 0.933190i 0
81.4 0 −0.781149 − 0.240953i 0 −0.733052 + 0.680173i 0 −0.949851 + 2.46937i 0 −1.92658 − 1.31352i 0
81.5 0 0.678408 + 0.209261i 0 −0.733052 + 0.680173i 0 1.71966 − 2.01066i 0 −2.06227 − 1.40603i 0
81.6 0 1.51514 + 0.467360i 0 −0.733052 + 0.680173i 0 −0.906055 − 2.48577i 0 −0.401485 − 0.273728i 0
81.7 0 2.24168 + 0.691467i 0 −0.733052 + 0.680173i 0 1.83316 + 1.90775i 0 2.06830 + 1.41014i 0
121.1 0 −2.79102 + 0.860916i 0 −0.733052 − 0.680173i 0 2.38129 − 1.15302i 0 4.56990 − 3.11571i 0
121.2 0 −2.42278 + 0.747329i 0 −0.733052 − 0.680173i 0 1.12256 + 2.39580i 0 2.83264 − 1.93126i 0
121.3 0 −1.10756 + 0.341638i 0 −0.733052 − 0.680173i 0 −2.50718 + 0.845018i 0 −1.36874 + 0.933190i 0
121.4 0 −0.781149 + 0.240953i 0 −0.733052 − 0.680173i 0 −0.949851 − 2.46937i 0 −1.92658 + 1.31352i 0
121.5 0 0.678408 − 0.209261i 0 −0.733052 − 0.680173i 0 1.71966 + 2.01066i 0 −2.06227 + 1.40603i 0
121.6 0 1.51514 − 0.467360i 0 −0.733052 − 0.680173i 0 −0.906055 + 2.48577i 0 −0.401485 + 0.273728i 0
121.7 0 2.24168 − 0.691467i 0 −0.733052 − 0.680173i 0 1.83316 − 1.90775i 0 2.06830 − 1.41014i 0
221.1 0 −0.851118 − 2.16861i 0 −0.988831 − 0.149042i 0 1.49497 + 2.18290i 0 −1.77933 + 1.65098i 0
221.2 0 −0.833959 − 2.12489i 0 −0.988831 − 0.149042i 0 −2.05763 + 1.66318i 0 −1.62053 + 1.50363i 0
221.3 0 −0.626027 − 1.59509i 0 −0.988831 − 0.149042i 0 −0.596041 − 2.57774i 0 0.0467499 − 0.0433775i 0
221.4 0 −0.192499 − 0.490479i 0 −0.988831 − 0.149042i 0 −2.61364 + 0.410949i 0 1.99564 − 1.85169i 0
221.5 0 0.0742142 + 0.189095i 0 −0.988831 − 0.149042i 0 2.62986 − 0.289537i 0 2.16891 − 2.01245i 0
221.6 0 0.499361 + 1.27235i 0 −0.988831 − 0.149042i 0 0.188317 + 2.63904i 0 0.829639 − 0.769793i 0
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.bg.b ✓ 84
49.g even 21 1 inner 980.2.bg.b ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.bg.b ✓ 84 1.a even 1 1 trivial
980.2.bg.b ✓ 84 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{84} + 11 T_{3}^{83} + 46 T_{3}^{82} + 56 T_{3}^{81} - 295 T_{3}^{80} - 1589 T_{3}^{79} + \cdots + 9406489 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\). Copy content Toggle raw display