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Properties

Label 980.2.bg.c
Level $980$
Weight $2$
Character orbit 980.bg
Analytic conductor $7.825$
Analytic rank $0$
Dimension $120$
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: \(120\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - q^{3} - 10 q^{5} - 6 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - q^{3} - 10 q^{5} - 6 q^{7} + 11 q^{9} + 3 q^{11} + 6 q^{13} - 2 q^{15} + 6 q^{17} + 6 q^{19} - 14 q^{21} - 27 q^{23} + 10 q^{25} - 37 q^{27} - 10 q^{29} + 6 q^{31} - 59 q^{33} + 7 q^{35} + 9 q^{37} + 14 q^{39} + 28 q^{41} + 15 q^{43} - 11 q^{45} + 105 q^{47} + 2 q^{49} + 17 q^{51} + 33 q^{53} - q^{55} - 8 q^{57} + 15 q^{59} - 15 q^{61} - 95 q^{63} + 24 q^{65} - 27 q^{67} + 2 q^{69} - 44 q^{71} - 17 q^{73} - q^{75} - 25 q^{77} - 57 q^{79} - 38 q^{81} - 17 q^{83} - 16 q^{85} - 116 q^{87} - 67 q^{89} + 3 q^{91} + 35 q^{93} + 8 q^{95} - 122 q^{97} - 36 q^{99}+O(q^{100}) \) 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 −3.10646 − 0.958216i 0 0.733052 − 0.680173i 0 −2.18647 − 1.48975i 0 6.25320 + 4.26336i 0
81.2 0 −2.38897 − 0.736899i 0 0.733052 − 0.680173i 0 0.256638 + 2.63327i 0 2.68543 + 1.83090i 0
81.3 0 −2.05906 − 0.635134i 0 0.733052 − 0.680173i 0 2.64454 + 0.0799134i 0 1.35760 + 0.925593i 0
81.4 0 −0.706567 − 0.217947i 0 0.733052 − 0.680173i 0 −0.0636519 − 2.64499i 0 −2.02698 − 1.38197i 0
81.5 0 −0.538993 − 0.166257i 0 0.733052 − 0.680173i 0 −1.45039 + 2.21278i 0 −2.21584 − 1.51074i 0
81.6 0 0.491334 + 0.151556i 0 0.733052 − 0.680173i 0 −2.25857 − 1.37800i 0 −2.26028 − 1.54103i 0
81.7 0 0.632858 + 0.195211i 0 0.733052 − 0.680173i 0 2.63745 − 0.209474i 0 −2.11631 − 1.44288i 0
81.8 0 1.32696 + 0.409312i 0 0.733052 − 0.680173i 0 −2.35184 − 1.21196i 0 −0.885434 − 0.603679i 0
81.9 0 2.46809 + 0.761305i 0 0.733052 − 0.680173i 0 2.42533 − 1.05726i 0 3.03316 + 2.06798i 0
81.10 0 2.92523 + 0.902314i 0 0.733052 − 0.680173i 0 −1.67749 + 2.04598i 0 5.26408 + 3.58899i 0
121.1 0 −3.10646 + 0.958216i 0 0.733052 + 0.680173i 0 −2.18647 + 1.48975i 0 6.25320 − 4.26336i 0
121.2 0 −2.38897 + 0.736899i 0 0.733052 + 0.680173i 0 0.256638 − 2.63327i 0 2.68543 − 1.83090i 0
121.3 0 −2.05906 + 0.635134i 0 0.733052 + 0.680173i 0 2.64454 − 0.0799134i 0 1.35760 − 0.925593i 0
121.4 0 −0.706567 + 0.217947i 0 0.733052 + 0.680173i 0 −0.0636519 + 2.64499i 0 −2.02698 + 1.38197i 0
121.5 0 −0.538993 + 0.166257i 0 0.733052 + 0.680173i 0 −1.45039 − 2.21278i 0 −2.21584 + 1.51074i 0
121.6 0 0.491334 − 0.151556i 0 0.733052 + 0.680173i 0 −2.25857 + 1.37800i 0 −2.26028 + 1.54103i 0
121.7 0 0.632858 − 0.195211i 0 0.733052 + 0.680173i 0 2.63745 + 0.209474i 0 −2.11631 + 1.44288i 0
121.8 0 1.32696 − 0.409312i 0 0.733052 + 0.680173i 0 −2.35184 + 1.21196i 0 −0.885434 + 0.603679i 0
121.9 0 2.46809 − 0.761305i 0 0.733052 + 0.680173i 0 2.42533 + 1.05726i 0 3.03316 − 2.06798i 0
121.10 0 2.92523 − 0.902314i 0 0.733052 + 0.680173i 0 −1.67749 − 2.04598i 0 5.26408 − 3.58899i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.10
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.c ✓ 120
49.g even 21 1 inner 980.2.bg.c ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.bg.c ✓ 120 1.a even 1 1 trivial
980.2.bg.c ✓ 120 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{120} + T_{3}^{119} - 20 T_{3}^{118} - 12 T_{3}^{117} + 148 T_{3}^{116} + 11 T_{3}^{115} + \cdots + 19\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\). Copy content Toggle raw display