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Properties

Label 770.2.bi.a
Level $770$
Weight $2$
Character orbit 770.bi
Analytic conductor $6.148$
Analytic rank $0$
Dimension $384$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [770,2,Mod(27,770)]
 
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, 10, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.bi (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: \(384\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 384 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 384 q - 4 q^{7} + 24 q^{15} + 96 q^{16} + 16 q^{22} + 16 q^{23} - 40 q^{25} + 4 q^{28} + 16 q^{30} - 32 q^{35} + 80 q^{36} - 64 q^{37} + 44 q^{42} - 80 q^{43} - 16 q^{46} - 64 q^{50} - 136 q^{51} + 80 q^{53} + 8 q^{56} - 112 q^{57} - 96 q^{58} + 68 q^{63} - 272 q^{65} + 160 q^{67} - 8 q^{70} + 16 q^{71} + 96 q^{77} + 24 q^{81} - 80 q^{85} - 16 q^{88} - 84 q^{91} + 16 q^{92} + 88 q^{93} - 184 q^{95} - 32 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −0.453990 + 0.891007i −0.454148 − 2.86738i −0.587785 − 0.809017i −1.32097 + 1.80417i 2.76103 + 0.897114i −2.60332 − 0.471965i 0.987688 − 0.156434i −5.16244 + 1.67738i −1.00782 − 1.99607i
27.2 −0.453990 + 0.891007i −0.445742 − 2.81431i −0.587785 − 0.809017i 2.06857 + 0.849127i 2.70993 + 0.880509i 1.00237 − 2.44852i 0.987688 − 0.156434i −4.86846 + 1.58186i −1.69569 + 1.45761i
27.3 −0.453990 + 0.891007i −0.442024 − 2.79083i −0.587785 − 0.809017i 0.523489 − 2.17393i 2.68732 + 0.873164i −2.62728 + 0.312118i 0.987688 − 0.156434i −4.74017 + 1.54018i 1.69932 + 1.45337i
27.4 −0.453990 + 0.891007i −0.376983 − 2.38018i −0.587785 − 0.809017i 0.805590 + 2.08591i 2.29190 + 0.744683i 0.945612 + 2.47100i 0.987688 − 0.156434i −2.66995 + 0.867519i −2.22429 − 0.229198i
27.5 −0.453990 + 0.891007i −0.370844 − 2.34142i −0.587785 − 0.809017i −1.72342 − 1.42472i 2.25458 + 0.732557i 1.14583 + 2.38476i 0.987688 − 0.156434i −2.49155 + 0.809553i 2.05185 − 0.888763i
27.6 −0.453990 + 0.891007i −0.355652 − 2.24550i −0.587785 − 0.809017i −1.77729 + 1.35692i 2.16222 + 0.702547i 2.50760 − 0.843757i 0.987688 − 0.156434i −2.06261 + 0.670181i −0.402154 − 2.19961i
27.7 −0.453990 + 0.891007i −0.275125 − 1.73707i −0.587785 − 0.809017i 2.06245 − 0.863892i 1.67264 + 0.543475i −0.479663 + 2.60191i 0.987688 − 0.156434i −0.0885490 + 0.0287713i −0.166598 + 2.22985i
27.8 −0.453990 + 0.891007i −0.205427 − 1.29702i −0.587785 − 0.809017i −0.486212 − 2.18257i 1.24891 + 0.405797i −0.970889 − 2.46117i 0.987688 − 0.156434i 1.21312 − 0.394165i 2.16542 + 0.557647i
27.9 −0.453990 + 0.891007i −0.160862 − 1.01564i −0.587785 − 0.809017i 2.16890 − 0.543949i 0.977976 + 0.317764i 2.64207 − 0.139461i 0.987688 − 0.156434i 1.84751 − 0.600293i −0.499997 + 2.17945i
27.10 −0.453990 + 0.891007i −0.0917129 − 0.579052i −0.587785 − 0.809017i −2.22927 − 0.174197i 0.557576 + 0.181167i −1.87792 + 1.86371i 0.987688 − 0.156434i 2.52628 − 0.820838i 1.16728 − 1.90721i
27.11 −0.453990 + 0.891007i −0.0293280 − 0.185170i −0.587785 − 0.809017i −0.0774713 + 2.23473i 0.178302 + 0.0579339i 2.16357 + 1.52281i 0.987688 − 0.156434i 2.81974 − 0.916190i −1.95598 − 1.08357i
27.12 −0.453990 + 0.891007i −0.00505097 − 0.0318906i −0.587785 − 0.809017i 2.20552 + 0.368341i 0.0307078 + 0.00997756i −2.58959 + 0.542252i 0.987688 − 0.156434i 2.85218 − 0.926729i −1.32948 + 1.79791i
27.13 −0.453990 + 0.891007i 0.00505097 + 0.0318906i −0.587785 − 0.809017i −2.20552 − 0.368341i −0.0307078 − 0.00997756i −2.29528 − 1.31594i 0.987688 − 0.156434i 2.85218 − 0.926729i 1.32948 − 1.79791i
27.14 −0.453990 + 0.891007i 0.0293280 + 0.185170i −0.587785 − 0.809017i 0.0774713 − 2.23473i −0.178302 − 0.0579339i 2.52825 − 0.779702i 0.987688 − 0.156434i 2.81974 − 0.916190i 1.95598 + 1.08357i
27.15 −0.453990 + 0.891007i 0.0917129 + 0.579052i −0.587785 − 0.809017i 2.22927 + 0.174197i −0.557576 − 0.181167i −1.21009 − 2.35280i 0.987688 − 0.156434i 2.52628 − 0.820838i −1.16728 + 1.90721i
27.16 −0.453990 + 0.891007i 0.160862 + 1.01564i −0.587785 − 0.809017i −2.16890 + 0.543949i −0.977976 − 0.317764i 2.46967 + 0.949081i 0.987688 − 0.156434i 1.84751 − 0.600293i 0.499997 − 2.17945i
27.17 −0.453990 + 0.891007i 0.205427 + 1.29702i −0.587785 − 0.809017i 0.486212 + 2.18257i −1.24891 − 0.405797i −1.68391 + 2.04069i 0.987688 − 0.156434i 1.21312 − 0.394165i −2.16542 − 0.557647i
27.18 −0.453990 + 0.891007i 0.275125 + 1.73707i −0.587785 − 0.809017i −2.06245 + 0.863892i −1.67264 − 0.543475i 0.347847 − 2.62279i 0.987688 − 0.156434i −0.0885490 + 0.0287713i 0.166598 − 2.22985i
27.19 −0.453990 + 0.891007i 0.355652 + 2.24550i −0.587785 − 0.809017i 1.77729 − 1.35692i −2.16222 − 0.702547i 2.12414 + 1.57735i 0.987688 − 0.156434i −2.06261 + 0.670181i 0.402154 + 2.19961i
27.20 −0.453990 + 0.891007i 0.370844 + 2.34142i −0.587785 − 0.809017i 1.72342 + 1.42472i −2.25458 − 0.732557i 1.82668 − 1.91396i 0.987688 − 0.156434i −2.49155 + 0.809553i −2.05185 + 0.888763i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
11.c even 5 1 inner
35.f even 4 1 inner
55.k odd 20 1 inner
77.j odd 10 1 inner
385.bk 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.bi.a ✓ 384
5.c odd 4 1 inner 770.2.bi.a ✓ 384
7.b odd 2 1 inner 770.2.bi.a ✓ 384
11.c even 5 1 inner 770.2.bi.a ✓ 384
35.f even 4 1 inner 770.2.bi.a ✓ 384
55.k odd 20 1 inner 770.2.bi.a ✓ 384
77.j odd 10 1 inner 770.2.bi.a ✓ 384
385.bk even 20 1 inner 770.2.bi.a ✓ 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.bi.a ✓ 384 1.a even 1 1 trivial
770.2.bi.a ✓ 384 5.c odd 4 1 inner
770.2.bi.a ✓ 384 7.b odd 2 1 inner
770.2.bi.a ✓ 384 11.c even 5 1 inner
770.2.bi.a ✓ 384 35.f even 4 1 inner
770.2.bi.a ✓ 384 55.k odd 20 1 inner
770.2.bi.a ✓ 384 77.j odd 10 1 inner
770.2.bi.a ✓ 384 385.bk even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(770, [\chi])\).