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Properties

Label 760.2.bf.b
Level $760$
Weight $2$
Character orbit 760.bf
Analytic conductor $6.069$
Analytic rank $0$
Dimension $224$
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] = [760,2,Mod(179,760)]
 
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, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.179");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.bf (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: \(224\)
Relative dimension: \(112\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 12 q^{4} + 4 q^{6} - 124 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q - 12 q^{4} + 4 q^{6} - 124 q^{9} + 6 q^{10} - 8 q^{11} + 24 q^{14} + 12 q^{16} - 4 q^{19} - 24 q^{20} + 18 q^{24} + 18 q^{25} - 116 q^{26} - 40 q^{30} + 18 q^{34} - 16 q^{35} + 16 q^{36} + 48 q^{40} - 24 q^{41} - 38 q^{44} + 136 q^{49} - 36 q^{51} - 4 q^{54} - 12 q^{59} + 42 q^{60} + 60 q^{64} + 22 q^{66} + 12 q^{70} - 52 q^{74} + 6 q^{76} - 22 q^{80} - 64 q^{81} + 108 q^{86} - 96 q^{89} - 30 q^{90} + 28 q^{96} + 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1 −1.41420 − 0.00511652i −0.00963822 + 0.0166939i 1.99995 + 0.0144716i −2.01865 + 0.961794i 0.0137158 − 0.0235592i −2.41934 −2.82826 − 0.0306986i 1.49981 + 2.59775i 2.85971 − 1.34985i
179.2 −1.41395 + 0.0274370i −1.24620 + 2.15849i 1.99849 − 0.0775888i 2.09156 + 0.790814i 1.70284 − 3.08618i 1.76849 −2.82364 + 0.164539i −1.60605 − 2.78176i −2.97905 − 1.06078i
179.3 −1.41273 − 0.0647840i 0.518942 − 0.898833i 1.99161 + 0.183045i −1.63275 − 1.52778i −0.791354 + 1.23619i 4.87002 −2.80174 − 0.387617i 0.961399 + 1.66519i 2.20766 + 2.26412i
179.4 −1.40965 − 0.113516i −1.61208 + 2.79220i 1.97423 + 0.320035i −0.892456 − 2.05025i 2.58943 − 3.75303i −4.76466 −2.74664 − 0.675243i −3.69759 − 6.40442i 1.02532 + 2.99144i
179.5 −1.40963 − 0.113745i 1.31769 − 2.28231i 1.97412 + 0.320677i −0.101608 + 2.23376i −2.11706 + 3.06734i −1.79986 −2.74631 − 0.676584i −1.97263 − 3.41670i 0.397309 − 3.13722i
179.6 −1.40101 − 0.192821i −0.625953 + 1.08418i 1.92564 + 0.540287i 1.49993 − 1.65838i 1.08602 − 1.39825i 1.44752 −2.59366 − 1.12825i 0.716364 + 1.24078i −2.42118 + 2.03418i
179.7 −1.38995 − 0.260850i 1.10728 − 1.91786i 1.86391 + 0.725136i 0.129919 − 2.23229i −2.03933 + 2.37689i −4.03476 −2.40159 − 1.49410i −0.952125 − 1.64913i −0.762874 + 3.06888i
179.8 −1.38469 − 0.287464i 0.962897 − 1.66779i 1.83473 + 0.796095i 1.82653 + 1.28988i −1.81274 + 2.03257i 3.29436 −2.31168 − 1.62976i −0.354342 − 0.613739i −2.15839 − 2.31114i
179.9 −1.37125 + 0.345926i 0.961755 − 1.66581i 1.76067 − 0.948704i 2.20104 − 0.394246i −0.742564 + 2.61694i −0.442628 −2.08614 + 1.90997i −0.349947 − 0.606126i −2.88180 + 1.30201i
179.10 −1.35308 + 0.411300i −0.834091 + 1.44469i 1.66166 − 1.11305i −1.44764 + 1.70422i 0.534394 − 2.29784i 2.81929 −1.79057 + 2.18949i 0.108583 + 0.188071i 1.25783 − 2.90136i
179.11 −1.34938 + 0.423299i 1.43085 − 2.47830i 1.64164 − 1.14238i −2.18977 + 0.452655i −0.881691 + 3.94984i 1.88254 −1.73162 + 2.23640i −2.59466 − 4.49408i 2.76322 − 1.53773i
179.12 −1.33453 − 0.468009i −1.54679 + 2.67911i 1.56193 + 1.24914i −0.886639 + 2.05277i 3.31808 − 2.85144i 1.31469 −1.49984 − 2.39802i −3.28509 − 5.68994i 2.14396 − 2.32453i
179.13 −1.31968 + 0.508378i −0.289899 + 0.502120i 1.48310 − 1.34179i 2.22714 + 0.199572i 0.127307 − 0.810015i −3.60789 −1.27508 + 2.52471i 1.33192 + 2.30695i −3.04057 + 0.868860i
179.14 −1.30811 + 0.537457i 0.183448 − 0.317741i 1.42228 − 1.40610i −0.803072 − 2.08688i −0.0691971 + 0.514233i −2.23274 −1.10478 + 2.60374i 1.43269 + 2.48150i 2.17211 + 2.29825i
179.15 −1.29251 − 0.573957i −0.775159 + 1.34262i 1.34115 + 1.48369i −1.81668 − 1.30371i 1.77250 − 1.29043i −0.589386 −0.881872 − 2.68743i 0.298257 + 0.516595i 1.59980 + 2.72775i
179.16 −1.27528 + 0.611288i −0.946271 + 1.63899i 1.25265 − 1.55912i −2.23011 + 0.163062i 0.204862 − 2.66861i −1.64866 −0.644407 + 2.75404i −0.290858 − 0.503781i 2.74433 − 1.57119i
179.17 −1.26678 + 0.628712i −1.29905 + 2.25003i 1.20944 − 1.59287i −0.878485 − 2.05627i 0.230992 − 3.66701i 3.78564 −0.530634 + 2.77821i −1.87509 − 3.24775i 2.40565 + 2.05252i
179.18 −1.25569 − 0.650563i −0.617466 + 1.06948i 1.15354 + 1.63382i 0.782848 + 2.09455i 1.47111 − 0.941242i −2.31345 −0.385587 − 2.80202i 0.737473 + 1.27734i 0.379620 − 3.13941i
179.19 −1.25442 + 0.653014i 1.43377 − 2.48337i 1.14714 − 1.63831i 0.875328 − 2.05762i −0.176880 + 4.05146i 1.17999 −0.369162 + 2.80423i −2.61140 − 4.52308i 0.245626 + 3.15272i
179.20 −1.20622 + 0.738270i −0.0958018 + 0.165934i 0.909915 − 1.78103i 1.38848 + 1.75275i −0.00694613 − 0.270879i 2.48009 0.217325 + 2.82007i 1.48164 + 2.56628i −2.96881 − 1.08912i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.112
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
19.d odd 6 1 inner
40.e odd 2 1 inner
95.h odd 6 1 inner
152.o even 6 1 inner
760.bf 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.bf.b ✓ 224
5.b even 2 1 inner 760.2.bf.b ✓ 224
8.d odd 2 1 inner 760.2.bf.b ✓ 224
19.d odd 6 1 inner 760.2.bf.b ✓ 224
40.e odd 2 1 inner 760.2.bf.b ✓ 224
95.h odd 6 1 inner 760.2.bf.b ✓ 224
152.o even 6 1 inner 760.2.bf.b ✓ 224
760.bf even 6 1 inner 760.2.bf.b ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.bf.b ✓ 224 1.a even 1 1 trivial
760.2.bf.b ✓ 224 5.b even 2 1 inner
760.2.bf.b ✓ 224 8.d odd 2 1 inner
760.2.bf.b ✓ 224 19.d odd 6 1 inner
760.2.bf.b ✓ 224 40.e odd 2 1 inner
760.2.bf.b ✓ 224 95.h odd 6 1 inner
760.2.bf.b ✓ 224 152.o even 6 1 inner
760.2.bf.b ✓ 224 760.bf even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{112} + 115 T_{3}^{110} + 7012 T_{3}^{108} + 295041 T_{3}^{106} + 9519883 T_{3}^{104} + \cdots + 7651198566400 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display