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Properties

Label 760.2.bx.b
Level $760$
Weight $2$
Character orbit 760.bx
Analytic conductor $6.069$
Analytic rank $0$
Dimension $672$
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(59,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 9, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.bx (of order \(18\), degree \(6\), 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: \(672\)
Relative dimension: \(112\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 672 q - 6 q^{4} - 6 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 672 q - 6 q^{4} - 6 q^{6} - 24 q^{9} - 15 q^{10} - 12 q^{11} + 30 q^{14} - 6 q^{16} - 24 q^{19} + 18 q^{20} - 36 q^{24} - 12 q^{25} - 30 q^{26} - 24 q^{30} - 36 q^{34} - 42 q^{35} + 30 q^{36} + 6 q^{40} - 12 q^{41} - 12 q^{44} - 18 q^{46} - 204 q^{49} - 36 q^{50} - 12 q^{54} - 24 q^{59} - 24 q^{60} - 126 q^{64} + 162 q^{65} + 24 q^{66} - 21 q^{70} + 102 q^{74} - 48 q^{76} + 45 q^{80} + 12 q^{81} - 126 q^{84} - 126 q^{86} - 192 q^{89} + 66 q^{90} - 36 q^{91} + 24 q^{96} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −1.41417 − 0.0110864i 1.80042 + 1.51073i 1.99975 + 0.0313560i −2.07319 − 0.837783i −2.52935 − 2.15639i 1.95459 + 3.38544i −2.82764 − 0.0665128i 0.438259 + 2.48549i 2.92256 + 1.20775i
59.2 −1.41340 − 0.0478337i 2.48477 + 2.08497i 1.99542 + 0.135217i 0.395721 + 2.20077i −3.41225 − 3.06576i 0.519457 + 0.899726i −2.81387 − 0.286564i 1.30604 + 7.40692i −0.454043 − 3.12951i
59.3 −1.41065 + 0.100336i −1.90237 − 1.59627i 1.97987 − 0.283077i 1.84844 + 1.25829i 2.84374 + 2.06091i −0.702797 − 1.21728i −2.76449 + 0.597973i 0.549960 + 3.11898i −2.73375 − 1.58954i
59.4 −1.41004 + 0.108601i −0.279501 − 0.234530i 1.97641 − 0.306262i −1.13073 − 1.92911i 0.419578 + 0.300341i −1.50960 − 2.61470i −2.75355 + 0.646480i −0.497828 − 2.82332i 1.80387 + 2.59732i
59.5 −1.40591 + 0.153005i 0.0588901 + 0.0494147i 1.95318 − 0.430223i −0.594638 + 2.15555i −0.0903551 − 0.0604622i 1.13840 + 1.97177i −2.68017 + 0.903703i −0.519918 − 2.94860i 0.506198 − 3.12150i
59.6 −1.40455 − 0.165057i −1.53543 − 1.28838i 1.94551 + 0.463661i 1.58297 − 1.57930i 1.94393 + 2.06302i −0.305304 − 0.528803i −2.65604 − 0.972354i 0.176678 + 1.00199i −2.48403 + 1.95693i
59.7 −1.37630 − 0.325281i 1.53543 + 1.28838i 1.78838 + 0.895367i 2.22778 + 0.192303i −1.69412 − 2.27263i −0.305304 − 0.528803i −2.17010 − 1.81402i 0.176678 + 1.00199i −3.00354 − 0.989322i
59.8 −1.35207 + 0.414632i 1.24080 + 1.04116i 1.65616 − 1.12122i 2.21671 − 0.293610i −2.10934 − 0.893236i −2.39079 − 4.14097i −1.77435 + 2.20266i −0.0653614 − 0.370683i −2.87539 + 1.31610i
59.9 −1.34453 − 0.438464i −2.48477 − 2.08497i 1.61550 + 1.17905i −1.11149 − 1.94026i 2.42665 + 3.89278i 0.519457 + 0.899726i −1.65511 − 2.29360i 1.30604 + 7.40692i 0.643695 + 3.09607i
59.10 −1.33268 − 0.473257i −1.80042 − 1.51073i 1.55206 + 1.26140i −1.04964 + 1.97440i 1.68442 + 2.86538i 1.95459 + 3.38544i −1.47142 − 2.41556i 0.438259 + 2.48549i 2.33323 − 2.13449i
59.11 −1.32951 + 0.482089i −1.20583 − 1.01181i 1.53518 − 1.28188i 1.46165 + 1.69222i 2.09095 + 0.763895i −0.194154 − 0.336285i −1.42305 + 2.44437i −0.0906791 − 0.514267i −2.75907 − 1.54517i
59.12 −1.30254 + 0.550817i 1.02727 + 0.861978i 1.39320 − 1.43492i −0.171671 − 2.22947i −1.81284 − 0.556922i 0.503971 + 0.872903i −1.02432 + 2.63643i −0.208677 − 1.18346i 1.45164 + 2.80940i
59.13 −1.29596 + 0.566126i 1.22081 + 1.02438i 1.35900 − 1.46735i −1.87433 + 1.21938i −2.16205 − 0.636422i −0.296032 − 0.512742i −0.930504 + 2.67099i −0.0799219 − 0.453260i 1.73873 − 2.64137i
59.14 −1.29126 − 0.576755i 1.90237 + 1.59627i 1.33471 + 1.48948i 0.607174 − 2.15205i −1.53579 − 3.15841i −0.702797 − 1.21728i −0.864387 − 2.69311i 0.549960 + 3.11898i −2.02523 + 2.42867i
59.15 −1.28786 − 0.584312i 0.279501 + 0.234530i 1.31716 + 1.50502i 0.373820 + 2.20460i −0.222920 − 0.465357i −1.50960 − 2.61470i −0.816909 − 2.70789i −0.497828 − 2.82332i 0.806747 − 3.05764i
59.16 −1.26879 − 0.624628i −0.0588901 − 0.0494147i 1.21968 + 1.58505i −1.84108 − 1.26902i 0.0438537 + 0.0994815i 1.13840 + 1.97177i −0.557457 − 2.77295i −0.519918 − 2.94860i 1.54329 + 2.76012i
59.17 −1.26853 + 0.625167i 0.558467 + 0.468610i 1.21833 − 1.58608i 1.98709 + 1.02541i −1.00139 − 0.245310i 1.80004 + 3.11776i −0.553923 + 2.77366i −0.428654 − 2.43102i −3.16174 − 0.0585007i
59.18 −1.26391 + 0.634448i −1.62628 − 1.36461i 1.19495 − 1.60377i −0.493348 − 2.18096i 2.92124 + 0.692957i 1.64330 + 2.84627i −0.492803 + 2.78517i 0.261675 + 1.48403i 2.00726 + 2.44355i
59.19 −1.24887 + 0.663571i −2.08482 − 1.74937i 1.11935 − 1.65743i −2.23315 + 0.114140i 3.76450 + 0.801311i −1.01479 − 1.75767i −0.298101 + 2.81267i 0.765228 + 4.33982i 2.71318 − 1.62440i
59.20 −1.13031 + 0.849946i 2.52318 + 2.11720i 0.555182 − 1.92140i 1.83816 − 1.27326i −4.65148 − 0.248515i 1.25779 + 2.17856i 1.00556 + 2.64364i 1.36296 + 7.72975i −0.995479 + 3.00150i
See next 80 embeddings (of 672 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.112
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.f odd 18 1 inner
40.e odd 2 1 inner
95.o odd 18 1 inner
152.v even 18 1 inner
760.bx even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.bx.b ✓ 672
5.b even 2 1 inner 760.2.bx.b ✓ 672
8.d odd 2 1 inner 760.2.bx.b ✓ 672
19.f odd 18 1 inner 760.2.bx.b ✓ 672
40.e odd 2 1 inner 760.2.bx.b ✓ 672
95.o odd 18 1 inner 760.2.bx.b ✓ 672
152.v even 18 1 inner 760.2.bx.b ✓ 672
760.bx even 18 1 inner 760.2.bx.b ✓ 672
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.bx.b ✓ 672 1.a even 1 1 trivial
760.2.bx.b ✓ 672 5.b even 2 1 inner
760.2.bx.b ✓ 672 8.d odd 2 1 inner
760.2.bx.b ✓ 672 19.f odd 18 1 inner
760.2.bx.b ✓ 672 40.e odd 2 1 inner
760.2.bx.b ✓ 672 95.o odd 18 1 inner
760.2.bx.b ✓ 672 152.v even 18 1 inner
760.2.bx.b ✓ 672 760.bx even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{336} + 6 T_{3}^{334} + 3 T_{3}^{332} + 6367 T_{3}^{330} + 37035 T_{3}^{328} + \cdots + 98\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display