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Properties

Label 385.2.bb.a
Level $385$
Weight $2$
Character orbit 385.bb
Analytic conductor $3.074$
Analytic rank $0$
Dimension $144$
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] = [385,2,Mod(64,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.bb (of order \(10\), degree \(4\), 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: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(36\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 36 q^{4} + 2 q^{5} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 36 q^{4} + 2 q^{5} + 26 q^{9} - 16 q^{10} - 2 q^{11} + 2 q^{15} - 84 q^{16} - 16 q^{19} + 22 q^{20} - 36 q^{21} + 32 q^{24} - 6 q^{25} - 92 q^{26} - 20 q^{29} - 20 q^{30} - 4 q^{31} + 16 q^{34} + 12 q^{36} - 36 q^{39} - 46 q^{40} - 52 q^{44} - 40 q^{45} + 92 q^{46} + 36 q^{49} - 38 q^{50} + 4 q^{51} - 128 q^{54} + 34 q^{55} - 24 q^{56} - 40 q^{59} + 12 q^{60} + 32 q^{61} + 132 q^{64} + 28 q^{65} + 52 q^{66} - 44 q^{69} + 4 q^{70} - 28 q^{71} + 16 q^{74} - 30 q^{75} + 104 q^{76} - 52 q^{79} + 8 q^{80} - 82 q^{81} - 48 q^{84} - 40 q^{85} + 72 q^{86} - 64 q^{89} - 292 q^{90} + 18 q^{91} - 12 q^{94} - 56 q^{95} + 216 q^{96} + 216 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −2.52691 − 0.821042i 0.640755 + 0.881923i 4.09311 + 2.97382i −0.889856 − 2.05138i −0.895031 − 2.75462i −0.587785 + 0.809017i −4.77786 − 6.57616i 0.559829 − 1.72298i 0.564314 + 5.91425i
64.2 −2.41507 − 0.784704i −1.17435 − 1.61635i 3.59877 + 2.61466i 0.476024 − 2.18481i 1.56777 + 4.82511i 0.587785 − 0.809017i −3.65436 − 5.02980i −0.306442 + 0.943130i −2.86406 + 4.90294i
64.3 −2.40348 − 0.780938i 1.66984 + 2.29834i 3.54881 + 2.57836i −0.646153 + 2.14067i −2.21857 − 6.82805i −0.587785 + 0.809017i −3.54509 − 4.87940i −1.56694 + 4.82255i 3.22475 − 4.64046i
64.4 −2.32863 − 0.756619i 0.231363 + 0.318443i 3.23203 + 2.34821i 2.03462 + 0.927532i −0.297818 − 0.916591i 0.587785 − 0.809017i −2.87116 − 3.95182i 0.879174 − 2.70582i −4.03610 − 3.69932i
64.5 −2.20581 − 0.716711i −0.168497 − 0.231916i 2.73389 + 1.98628i −1.74499 + 1.39821i 0.205455 + 0.632326i 0.587785 − 0.809017i −1.88031 − 2.58802i 0.901657 − 2.77502i 4.85123 − 1.83354i
64.6 −2.04743 − 0.665250i −1.31491 − 1.80982i 2.13137 + 1.54853i 2.15255 − 0.605421i 1.48821 + 4.58023i −0.587785 + 0.809017i −0.802912 − 1.10511i −0.619413 + 1.90636i −4.80995 − 0.192427i
64.7 −1.77973 − 0.578270i −1.05796 − 1.45616i 1.21502 + 0.882760i 0.149921 + 2.23104i 1.04084 + 3.20336i −0.587785 + 0.809017i 0.547942 + 0.754177i −0.0740683 + 0.227959i 1.02332 − 4.05734i
64.8 −1.72675 − 0.561057i 0.927009 + 1.27592i 1.04886 + 0.762043i 1.55493 − 1.60692i −0.884855 − 2.72330i −0.587785 + 0.809017i 0.750807 + 1.03340i 0.158429 − 0.487594i −3.58656 + 1.90234i
64.9 −1.66980 − 0.542550i 1.71974 + 2.36702i 0.875826 + 0.636325i −0.941331 − 2.02827i −1.58739 − 4.88550i 0.587785 − 0.809017i 0.946768 + 1.30311i −1.71823 + 5.28818i 0.471391 + 3.89752i
64.10 −1.39894 − 0.454544i −1.89456 − 2.60764i 0.132396 + 0.0961917i −2.20536 − 0.369286i 1.46509 + 4.50910i 0.587785 − 0.809017i 1.58770 + 2.18528i −2.28337 + 7.02748i 2.91732 + 1.51904i
64.11 −1.31407 − 0.426966i −0.112101 − 0.154293i −0.0735615 − 0.0534456i −1.34982 − 1.78269i 0.0814296 + 0.250615i 0.587785 − 0.809017i 1.69812 + 2.33726i 0.915811 − 2.81858i 1.01261 + 2.91890i
64.12 −1.26927 − 0.412412i 0.0399448 + 0.0549793i −0.177063 − 0.128644i −2.16602 + 0.555284i −0.0280267 − 0.0862574i −0.587785 + 0.809017i 1.74060 + 2.39572i 0.925624 − 2.84878i 2.97828 + 0.188487i
64.13 −1.13778 − 0.369687i −1.56650 − 2.15610i −0.460160 − 0.334326i 1.15417 + 1.91517i 0.985249 + 3.03229i 0.587785 − 0.809017i 1.80634 + 2.48621i −1.26781 + 3.90191i −0.605177 − 2.60573i
64.14 −0.686010 − 0.222898i 1.74439 + 2.40095i −1.19711 − 0.869749i −2.21915 + 0.274521i −0.661502 − 2.03589i −0.587785 + 0.809017i 1.47532 + 2.03060i −1.79459 + 5.52319i 1.58355 + 0.306321i
64.15 −0.375456 − 0.121993i −0.833244 − 1.14686i −1.49195 − 1.08396i −1.09898 − 1.94737i 0.172937 + 0.532246i −0.587785 + 0.809017i 0.892014 + 1.22775i 0.306054 − 0.941936i 0.175055 + 0.865219i
64.16 −0.315289 − 0.102444i 1.64785 + 2.26806i −1.52912 − 1.11097i 2.23607 0.000113848i −0.287199 − 0.883908i 0.587785 − 0.809017i 0.758023 + 1.04333i −1.50167 + 4.62167i −0.704997 − 0.229107i
64.17 −0.270411 − 0.0878620i 0.439014 + 0.604251i −1.55263 − 1.12805i −1.50445 + 1.65428i −0.0656237 − 0.201969i 0.587785 − 0.809017i 0.654983 + 0.901507i 0.754665 − 2.32262i 0.552168 − 0.315151i
64.18 −0.240760 − 0.0782276i 0.608344 + 0.837314i −1.56619 − 1.13790i 1.63717 + 1.52305i −0.0809637 − 0.249181i −0.587785 + 0.809017i 0.585655 + 0.806085i 0.596039 − 1.83442i −0.275019 − 0.494761i
64.19 0.240760 + 0.0782276i −0.608344 − 0.837314i −1.56619 − 1.13790i −0.429267 + 2.19448i −0.0809637 − 0.249181i 0.587785 − 0.809017i −0.585655 − 0.806085i 0.596039 − 1.83442i −0.275019 + 0.494761i
64.20 0.270411 + 0.0878620i −0.439014 − 0.604251i −1.55263 − 1.12805i 2.18948 + 0.454045i −0.0656237 − 0.201969i −0.587785 + 0.809017i −0.654983 − 0.901507i 0.754665 − 2.32262i 0.552168 + 0.315151i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.bb.a ✓ 144
5.b even 2 1 inner 385.2.bb.a ✓ 144
11.c even 5 1 inner 385.2.bb.a ✓ 144
55.j even 10 1 inner 385.2.bb.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.bb.a ✓ 144 1.a even 1 1 trivial
385.2.bb.a ✓ 144 5.b even 2 1 inner
385.2.bb.a ✓ 144 11.c even 5 1 inner
385.2.bb.a ✓ 144 55.j even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(385, [\chi])\).