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Properties

Label 760.2.bi.a
Level $760$
Weight $2$
Character orbit 760.bi
Analytic conductor $6.069$
Analytic rank $0$
Dimension $160$
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] = [760,2,Mod(331,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("760.331");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.bi (of order \(6\), degree \(2\), 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: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(80\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 2 q^{4} + 2 q^{6} + 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 2 q^{4} + 2 q^{6} + 80 q^{9} - 6 q^{10} + 2 q^{16} - 8 q^{19} + 12 q^{24} + 80 q^{25} - 4 q^{26} + 20 q^{28} - 16 q^{30} + 30 q^{32} - 6 q^{34} + 36 q^{36} - 10 q^{38} - 24 q^{41} + 2 q^{42} + 24 q^{44} - 18 q^{48} - 128 q^{49} + 24 q^{51} - 78 q^{52} - 38 q^{54} + 8 q^{57} + 4 q^{58} - 6 q^{60} - 32 q^{62} - 32 q^{64} - 12 q^{66} - 60 q^{68} - 36 q^{70} + 84 q^{72} - 16 q^{73} - 28 q^{74} - 32 q^{76} - 120 q^{78} - 72 q^{81} - 38 q^{82} - 96 q^{86} + 30 q^{90} - 80 q^{92} - 160 q^{96} - 6 q^{98} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
331.1 −1.41404 + 0.0220988i −2.73453 − 1.57878i 1.99902 − 0.0624972i −0.866025 − 0.500000i 3.90163 + 2.17203i 3.09677i −2.82532 + 0.132550i 3.48511 + 6.03639i 1.23564 + 0.687882i
331.2 −1.40887 − 0.122873i 1.43910 + 0.830862i 1.96980 + 0.346222i 0.866025 + 0.500000i −1.92540 − 1.34740i 2.51816i −2.73265 − 0.729816i −0.119336 − 0.206696i −1.15868 − 0.810844i
331.3 −1.40339 + 0.174657i 0.913769 + 0.527565i 1.93899 − 0.490224i 0.866025 + 0.500000i −1.37451 − 0.580781i 0.412371i −2.63553 + 1.02663i −0.943351 − 1.63393i −1.30270 − 0.550436i
331.4 −1.38804 + 0.270828i −0.829078 − 0.478669i 1.85330 − 0.751840i −0.866025 − 0.500000i 1.28043 + 0.439873i − 0.728770i −2.36884 + 1.54551i −1.04175 − 1.80437i 1.33749 + 0.459476i
331.5 −1.38749 − 0.273628i 2.26443 + 1.30737i 1.85026 + 0.759312i 0.866025 + 0.500000i −2.78415 − 2.43358i − 4.97950i −2.35944 − 1.55982i 1.91844 + 3.32284i −1.06479 − 0.930714i
331.6 −1.38307 + 0.295179i 2.62009 + 1.51271i 1.82574 − 0.816502i −0.866025 − 0.500000i −4.07028 − 1.31878i 0.260581i −2.28410 + 1.66820i 3.07660 + 5.32882i 1.34536 + 0.435901i
331.7 −1.36218 + 0.380082i −2.18371 − 1.26077i 1.71107 − 1.03548i 0.866025 + 0.500000i 3.45381 + 0.887403i − 2.14168i −1.93722 + 2.06086i 1.67907 + 2.90824i −1.36972 − 0.351929i
331.8 −1.34866 − 0.425568i −0.163524 − 0.0944106i 1.63778 + 1.14789i −0.866025 − 0.500000i 0.180361 + 0.196919i − 0.178380i −1.72031 − 2.24511i −1.48217 − 2.56720i 0.955193 + 1.04288i
331.9 −1.34297 − 0.443221i 1.45450 + 0.839755i 1.60711 + 1.19046i −0.866025 − 0.500000i −1.58114 − 1.77243i − 1.39716i −1.63066 − 2.31105i −0.0896240 − 0.155233i 0.941431 + 1.05532i
331.10 −1.30520 + 0.544476i −1.53915 − 0.888628i 1.40709 − 1.42130i 0.866025 + 0.500000i 2.49273 + 0.321808i 3.73845i −1.06267 + 2.62121i 0.0793181 + 0.137383i −1.40257 − 0.181070i
331.11 −1.28633 − 0.587661i −0.617577 − 0.356558i 1.30931 + 1.51186i 0.866025 + 0.500000i 0.584875 + 0.821580i − 4.66584i −0.795747 − 2.71418i −1.24573 − 2.15767i −0.820167 − 1.15210i
331.12 −1.27386 − 0.614225i −2.15052 − 1.24160i 1.24546 + 1.56488i −0.866025 − 0.500000i 1.97685 + 2.90254i − 2.51323i −0.625353 − 2.75843i 1.58316 + 2.74211i 0.796085 + 1.16887i
331.13 −1.22816 + 0.701150i −0.404372 − 0.233465i 1.01678 − 1.72225i −0.866025 − 0.500000i 0.660330 + 0.00320734i 2.98667i −0.0412128 + 2.82813i −1.39099 − 2.40926i 1.41420 + 0.00686901i
331.14 −1.22040 + 0.714585i 2.30025 + 1.32805i 0.978738 − 1.74415i 0.866025 + 0.500000i −3.75622 + 0.0229750i 2.43751i 0.0518969 + 2.82795i 2.02743 + 3.51161i −1.41419 + 0.00864991i
331.15 −1.16887 − 0.796085i −2.15052 − 1.24160i 0.732496 + 1.86103i 0.866025 + 0.500000i 1.52525 + 3.16327i 2.51323i 0.625353 − 2.75843i 1.58316 + 2.74211i −0.614225 − 1.27386i
331.16 −1.15210 − 0.820167i −0.617577 − 0.356558i 0.654653 + 1.88982i −0.866025 − 0.500000i 0.419071 + 0.917306i 4.66584i 0.795747 − 2.71418i −1.24573 − 2.15767i 0.587661 + 1.28633i
331.17 −1.12431 + 0.857862i −0.0181753 − 0.0104935i 0.528146 − 1.92901i 0.866025 + 0.500000i 0.0294366 − 0.00379393i − 2.74805i 1.06102 + 2.62188i −1.49978 − 2.59769i −1.40261 + 0.180775i
331.18 −1.05532 − 0.941431i 1.45450 + 0.839755i 0.227414 + 1.98703i 0.866025 + 0.500000i −0.744394 − 2.25552i 1.39716i 1.63066 − 2.31105i −0.0896240 − 0.155233i −0.443221 − 1.34297i
331.19 −1.04288 − 0.955193i −0.163524 − 0.0944106i 0.175214 + 1.99231i 0.866025 + 0.500000i 0.0803562 + 0.254656i 0.178380i 1.72031 − 2.24511i −1.48217 − 2.56720i −0.425568 − 1.34866i
331.20 −1.04115 + 0.957082i −1.65698 − 0.956659i 0.167989 − 1.99293i −0.866025 − 0.500000i 2.64077 − 0.589841i − 2.69731i 1.73250 + 2.23572i 0.330392 + 0.572255i 1.38020 − 0.308282i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.d odd 6 1 inner
152.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.bi.a ✓ 160
8.d odd 2 1 inner 760.2.bi.a ✓ 160
19.d odd 6 1 inner 760.2.bi.a ✓ 160
152.o even 6 1 inner 760.2.bi.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.bi.a ✓ 160 1.a even 1 1 trivial
760.2.bi.a ✓ 160 8.d odd 2 1 inner
760.2.bi.a ✓ 160 19.d odd 6 1 inner
760.2.bi.a ✓ 160 152.o even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(760, [\chi])\).