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Properties

Label 230.4.l.a
Level $230$
Weight $4$
Character orbit 230.l
Analytic conductor $13.570$
Analytic rank $0$
Dimension $720$
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] = [230,4,Mod(7,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([11, 38]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.l (of order \(44\), degree \(20\), 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: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(720\)
Relative dimension: \(36\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 720 q + 16 q^{3} + 16 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 720 q + 16 q^{3} + 16 q^{6} + 64 q^{12} - 192 q^{13} + 1152 q^{16} + 144 q^{18} + 1660 q^{23} - 880 q^{25} - 304 q^{26} - 728 q^{27} - 352 q^{28} - 608 q^{31} - 3872 q^{33} - 688 q^{35} - 2816 q^{36} - 2376 q^{37} + 2208 q^{41} + 256 q^{46} + 3640 q^{47} - 256 q^{48} - 272 q^{50} - 768 q^{52} + 2360 q^{55} - 3872 q^{56} - 7656 q^{57} + 2112 q^{61} + 1376 q^{62} + 10912 q^{66} + 1104 q^{70} + 22752 q^{71} - 128 q^{72} + 1296 q^{73} - 5396 q^{75} + 752 q^{77} - 6000 q^{78} - 3928 q^{81} - 80 q^{82} + 3060 q^{85} - 10472 q^{86} - 6760 q^{87} - 1104 q^{92} + 5288 q^{93} + 9392 q^{95} - 256 q^{96} + 25344 q^{97} + 14352 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.99490 + 0.142678i −1.77498 + 8.15947i 3.95929 − 0.569259i −0.0181895 + 11.1803i 2.37674 − 16.5306i 13.3136 − 24.3820i −7.81717 + 1.70052i −38.8664 − 17.7497i −1.55890 − 22.3063i
7.2 −1.99490 + 0.142678i −1.74558 + 8.02432i 3.95929 − 0.569259i 10.9645 + 2.18620i 2.33737 − 16.2568i −7.93704 + 14.5356i −7.81717 + 1.70052i −36.7825 − 16.7980i −22.1851 − 2.79686i
7.3 −1.99490 + 0.142678i −1.69513 + 7.79240i 3.95929 − 0.569259i −4.22141 − 10.3528i 2.26982 − 15.7870i −6.18513 + 11.3272i −7.81717 + 1.70052i −33.2880 − 15.2021i 9.89842 + 20.0505i
7.4 −1.99490 + 0.142678i −1.37958 + 6.34183i 3.95929 − 0.569259i 0.935085 − 11.1412i 1.84729 − 12.8482i 6.29353 − 11.5257i −7.81717 + 1.70052i −13.7555 − 6.28194i −0.275801 + 22.3590i
7.5 −1.99490 + 0.142678i −0.982064 + 4.51448i 3.95929 − 0.569259i −10.4680 + 3.92695i 1.31501 − 9.14607i 9.44192 − 17.2916i −7.81717 + 1.70052i 5.14402 + 2.34919i 20.3224 − 9.32744i
7.6 −1.99490 + 0.142678i −0.773121 + 3.55398i 3.95929 − 0.569259i 8.49866 + 7.26448i 1.03523 − 7.20016i 5.42233 − 9.93026i −7.81717 + 1.70052i 12.5270 + 5.72089i −17.9905 − 13.2794i
7.7 −1.99490 + 0.142678i −0.655343 + 3.01257i 3.95929 − 0.569259i −3.24111 + 10.7002i 0.877520 − 6.10328i −11.6907 + 21.4100i −7.81717 + 1.70052i 15.9140 + 7.26768i 4.93902 − 21.8084i
7.8 −1.99490 + 0.142678i −0.293810 + 1.35062i 3.95929 − 0.569259i −11.1728 + 0.409416i 0.393418 − 2.73628i −14.3881 + 26.3498i −7.81717 + 1.70052i 22.8222 + 10.4226i 22.2303 − 2.41087i
7.9 −1.99490 + 0.142678i −0.143842 + 0.661230i 3.95929 − 0.569259i 10.6128 − 3.51689i 0.192607 − 1.33961i −7.16158 + 13.1154i −7.81717 + 1.70052i 24.1435 + 11.0260i −20.6697 + 8.53008i
7.10 −1.99490 + 0.142678i 0.273744 − 1.25838i 3.95929 − 0.569259i −8.63728 − 7.09911i −0.366550 + 2.54941i 9.01505 − 16.5098i −7.81717 + 1.70052i 23.0515 + 10.5273i 18.2434 + 12.9297i
7.11 −1.99490 + 0.142678i 0.610767 − 2.80765i 3.95929 − 0.569259i −4.39809 − 10.2789i −0.817831 + 5.68814i −6.68930 + 12.2505i −7.81717 + 1.70052i 17.0502 + 7.78656i 10.2404 + 19.8780i
7.12 −1.99490 + 0.142678i 0.777625 − 3.57469i 3.95929 − 0.569259i 5.59520 + 9.67955i −1.04126 + 7.24211i −1.72659 + 3.16201i −7.81717 + 1.70052i 12.3864 + 5.65667i −12.5429 − 18.5115i
7.13 −1.99490 + 0.142678i 1.03021 − 4.73580i 3.95929 − 0.569259i −7.52636 + 8.26764i −1.37948 + 9.59447i 3.77900 − 6.92072i −7.81717 + 1.70052i 3.19355 + 1.45845i 13.8348 − 17.5670i
7.14 −1.99490 + 0.142678i 1.08789 − 5.00093i 3.95929 − 0.569259i 3.50493 − 10.6168i −1.45670 + 10.1316i −3.80039 + 6.95990i −7.81717 + 1.70052i 0.734227 + 0.335310i −5.47721 + 21.6795i
7.15 −1.99490 + 0.142678i 1.32678 − 6.09912i 3.95929 − 0.569259i 9.97777 + 5.04421i −1.77659 + 12.3565i 14.8793 − 27.2494i −7.81717 + 1.70052i −10.8788 − 4.96819i −20.6244 − 8.63909i
7.16 −1.99490 + 0.142678i 1.70555 − 7.84030i 3.95929 − 0.569259i −0.684672 + 11.1594i −2.28377 + 15.8840i −9.42724 + 17.2647i −7.81717 + 1.70052i −34.0014 − 15.5279i −0.226343 − 22.3595i
7.17 −1.99490 + 0.142678i 2.00431 − 9.21365i 3.95929 − 0.569259i −10.6008 − 3.55292i −2.68381 + 18.6663i 13.0324 − 23.8670i −7.81717 + 1.70052i −56.3140 − 25.7178i 21.6545 + 5.57523i
7.18 −1.99490 + 0.142678i 2.05873 − 9.46383i 3.95929 − 0.569259i 9.45713 − 5.96345i −2.75668 + 19.1732i −4.94222 + 9.05100i −7.81717 + 1.70052i −60.7656 − 27.7507i −18.0152 + 13.2458i
7.19 1.99490 − 0.142678i −2.09073 + 9.61093i 3.95929 − 0.569259i −11.0180 − 1.89851i −2.79953 + 19.4712i −1.98561 + 3.63638i 7.81717 − 1.70052i −63.4388 − 28.9715i −22.2507 − 2.21531i
7.20 1.99490 − 0.142678i −1.86789 + 8.58657i 3.95929 − 0.569259i 7.63461 + 8.16778i −2.50115 + 17.3959i −14.6956 + 26.9129i 7.81717 − 1.70052i −45.6801 − 20.8614i 16.3957 + 15.2046i
See next 80 embeddings (of 720 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.d odd 22 1 inner
115.l even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.4.l.a ✓ 720
5.c odd 4 1 inner 230.4.l.a ✓ 720
23.d odd 22 1 inner 230.4.l.a ✓ 720
115.l even 44 1 inner 230.4.l.a ✓ 720
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.l.a ✓ 720 1.a even 1 1 trivial
230.4.l.a ✓ 720 5.c odd 4 1 inner
230.4.l.a ✓ 720 23.d odd 22 1 inner
230.4.l.a ✓ 720 115.l even 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(230, [\chi])\).