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Properties

Label 770.2.bh.a
Level $770$
Weight $2$
Character orbit 770.bh
Analytic conductor $6.148$
Analytic rank $0$
Dimension $144$
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] = [770,2,Mod(57,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.bh (of order \(20\), degree \(8\), 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: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 4 q^{3} - 12 q^{11} + 16 q^{12} + 4 q^{15} + 36 q^{16} - 8 q^{20} + 12 q^{22} + 16 q^{23} + 8 q^{25} + 36 q^{26} + 64 q^{27} - 8 q^{31} - 56 q^{33} + 44 q^{36} - 52 q^{37} + 44 q^{38} - 4 q^{42} - 16 q^{45} - 40 q^{46} - 12 q^{47} - 4 q^{48} - 80 q^{50} + 20 q^{51} - 40 q^{52} + 32 q^{53} - 40 q^{55} - 144 q^{56} - 80 q^{57} - 56 q^{58} - 52 q^{60} - 40 q^{62} - 64 q^{66} - 112 q^{67} + 12 q^{70} - 48 q^{71} - 168 q^{75} + 12 q^{77} - 48 q^{78} + 176 q^{81} + 20 q^{82} + 80 q^{83} + 40 q^{85} + 88 q^{86} - 28 q^{88} + 36 q^{91} + 4 q^{92} + 40 q^{93} - 40 q^{95} - 20 q^{96} - 100 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1 −0.453990 − 0.891007i −0.483212 + 3.05088i −0.587785 + 0.809017i 2.13876 + 0.652474i 2.93773 − 0.954526i −0.987688 + 0.156434i 0.987688 + 0.156434i −6.22121 − 2.02139i −0.389616 − 2.20186i
57.2 −0.453990 − 0.891007i −0.351319 + 2.21814i −0.587785 + 0.809017i −0.189096 − 2.22806i 2.13588 − 0.693988i −0.987688 + 0.156434i 0.987688 + 0.156434i −1.94356 − 0.631502i −1.89937 + 1.18000i
57.3 −0.453990 − 0.891007i −0.240707 + 1.51977i −0.587785 + 0.809017i −2.07494 + 0.833451i 1.46340 − 0.475488i −0.987688 + 0.156434i 0.987688 + 0.156434i 0.601419 + 0.195413i 1.68461 + 1.47040i
57.4 −0.453990 − 0.891007i −0.175073 + 1.10537i −0.587785 + 0.809017i 2.22966 + 0.169132i 1.06437 − 0.345835i −0.987688 + 0.156434i 0.987688 + 0.156434i 1.66199 + 0.540012i −0.861547 − 2.06343i
57.5 −0.453990 − 0.891007i 0.0591474 − 0.373442i −0.587785 + 0.809017i 0.994482 + 2.00275i −0.359591 + 0.116838i −0.987688 + 0.156434i 0.987688 + 0.156434i 2.71721 + 0.882875i 1.33298 − 1.79532i
57.6 −0.453990 − 0.891007i 0.116139 − 0.733273i −0.587785 + 0.809017i 1.01259 − 1.99366i −0.706077 + 0.229418i −0.987688 + 0.156434i 0.987688 + 0.156434i 2.32897 + 0.756728i −2.23607 + 0.00287657i
57.7 −0.453990 − 0.891007i 0.275311 − 1.73824i −0.587785 + 0.809017i −0.895448 + 2.04894i −1.67378 + 0.543843i −0.987688 + 0.156434i 0.987688 + 0.156434i −0.0925285 − 0.0300643i 2.23215 − 0.132351i
57.8 −0.453990 − 0.891007i 0.357321 − 2.25604i −0.587785 + 0.809017i 0.551484 − 2.16699i −2.17236 + 0.705843i −0.987688 + 0.156434i 0.987688 + 0.156434i −2.10885 − 0.685207i −2.18117 + 0.492419i
57.9 −0.453990 − 0.891007i 0.479725 − 3.02887i −0.587785 + 0.809017i −2.22865 + 0.181956i −2.91653 + 0.947638i −0.987688 + 0.156434i 0.987688 + 0.156434i −6.09073 − 1.97900i 1.17391 + 1.90314i
57.10 0.453990 + 0.891007i −0.496458 + 3.13451i −0.587785 + 0.809017i 1.97486 + 1.04877i −3.01826 + 0.980692i 0.987688 − 0.156434i −0.987688 − 0.156434i −6.72554 − 2.18526i −0.0378880 + 2.23575i
57.11 0.453990 + 0.891007i −0.224568 + 1.41787i −0.587785 + 0.809017i −0.518606 − 2.17510i −1.36528 + 0.443607i 0.987688 − 0.156434i −0.987688 − 0.156434i 0.893248 + 0.290234i 1.70258 − 1.44955i
57.12 0.453990 + 0.891007i −0.103513 + 0.653554i −0.587785 + 0.809017i 2.09571 + 0.779736i −0.629315 + 0.204477i 0.987688 − 0.156434i −0.987688 − 0.156434i 2.43675 + 0.791748i 0.256684 + 2.22129i
57.13 0.453990 + 0.891007i −0.0913462 + 0.576737i −0.587785 + 0.809017i −0.555103 + 2.16607i −0.555347 + 0.180443i 0.987688 − 0.156434i −0.987688 − 0.156434i 2.52889 + 0.821685i −2.18199 + 0.488775i
57.14 0.453990 + 0.891007i −0.0789261 + 0.498320i −0.587785 + 0.809017i −2.16205 − 0.570545i −0.479838 + 0.155909i 0.987688 − 0.156434i −0.987688 − 0.156434i 2.61108 + 0.848390i −0.473193 − 2.18543i
57.15 0.453990 + 0.891007i 0.160962 − 1.01627i −0.587785 + 0.809017i 1.76753 − 1.36961i 0.978579 − 0.317960i 0.987688 − 0.156434i −0.987688 − 0.156434i 1.84627 + 0.599890i 2.02278 + 0.953087i
57.16 0.453990 + 0.891007i 0.271189 − 1.71222i −0.587785 + 0.809017i −2.18245 + 0.486716i 1.64871 − 0.535700i 0.987688 − 0.156434i −0.987688 − 0.156434i −0.00497708 − 0.00161715i −1.42448 − 1.72362i
57.17 0.453990 + 0.891007i 0.428429 − 2.70499i −0.587785 + 0.809017i 1.85778 + 1.24445i 2.60467 − 0.846309i 0.987688 − 0.156434i −0.987688 − 0.156434i −4.28027 − 1.39074i −0.265400 + 2.22026i
57.18 0.453990 + 0.891007i 0.539364 − 3.40541i −0.587785 + 0.809017i −0.738825 − 2.11048i 3.27911 − 1.06545i 0.987688 − 0.156434i −0.987688 − 0.156434i −8.45274 − 2.74646i 1.54503 − 1.61644i
127.1 −0.891007 − 0.453990i −3.05088 + 0.483212i 0.587785 + 0.809017i −1.34678 − 1.78499i 2.93773 + 0.954526i 0.156434 − 0.987688i −0.156434 − 0.987688i 6.22121 − 2.02139i 0.389616 + 2.20186i
127.2 −0.891007 − 0.453990i −2.21814 + 0.351319i 0.587785 + 0.809017i −1.15664 + 1.91368i 2.13588 + 0.693988i 0.156434 − 0.987688i −0.156434 − 0.987688i 1.94356 − 0.631502i 1.89937 − 1.18000i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.bh.a ✓ 144
5.c odd 4 1 inner 770.2.bh.a ✓ 144
11.d odd 10 1 inner 770.2.bh.a ✓ 144
55.l even 20 1 inner 770.2.bh.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.bh.a ✓ 144 1.a even 1 1 trivial
770.2.bh.a ✓ 144 5.c odd 4 1 inner
770.2.bh.a ✓ 144 11.d odd 10 1 inner
770.2.bh.a ✓ 144 55.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{144} - 4 T_{3}^{143} + 8 T_{3}^{142} - 24 T_{3}^{141} - 241 T_{3}^{140} + 980 T_{3}^{139} + \cdots + 11\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display