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Properties

Label 465.2.ba.a
Level $465$
Weight $2$
Character orbit 465.ba
Analytic conductor $3.713$
Analytic rank $0$
Dimension $128$
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] = [465,2,Mod(4,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.ba (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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 36 q^{4} - 20 q^{6} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 36 q^{4} - 20 q^{6} + 32 q^{9} + 2 q^{10} + 4 q^{14} - 4 q^{15} - 40 q^{16} - 4 q^{19} + 36 q^{20} + 20 q^{21} - 20 q^{24} + 12 q^{25} + 24 q^{26} - 28 q^{29} - 8 q^{30} - 12 q^{31} - 44 q^{34} + 124 q^{36} - 4 q^{39} - 22 q^{40} + 36 q^{41} - 80 q^{44} + 46 q^{46} + 64 q^{49} + 68 q^{50} - 8 q^{51} - 10 q^{54} + 12 q^{55} - 432 q^{56} + 88 q^{59} + 12 q^{60} - 120 q^{61} + 40 q^{64} - 4 q^{65} + 12 q^{66} - 24 q^{69} - 74 q^{70} + 44 q^{71} + 132 q^{74} - 12 q^{75} - 14 q^{76} - 132 q^{79} + 70 q^{80} - 32 q^{81} + 8 q^{84} + 48 q^{85} - 104 q^{86} + 52 q^{89} - 2 q^{90} - 32 q^{91} - 204 q^{94} + 34 q^{95} - 20 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.49874 + 0.811888i 0.951057 + 0.309017i 3.96648 − 2.88182i 2.23400 − 0.0962096i −2.62733 2.20823 + 3.03937i −4.48287 + 6.17014i 0.809017 + 0.587785i −5.50406 + 2.05416i
4.2 −2.36868 + 0.769631i −0.951057 − 0.309017i 3.40028 − 2.47045i −0.0416760 − 2.23568i 2.49058 1.08798 + 1.49748i −3.22499 + 4.43882i 0.809017 + 0.587785i 1.81936 + 5.26353i
4.3 −2.34433 + 0.761718i 0.951057 + 0.309017i 3.29762 − 2.39586i −2.05846 + 0.873354i −2.46497 1.42844 + 1.96608i −3.00798 + 4.14013i 0.809017 + 0.587785i 4.16045 − 3.61539i
4.4 −2.30094 + 0.747620i −0.951057 − 0.309017i 3.11734 − 2.26488i −1.05352 + 1.97233i 2.41935 −0.528967 − 0.728060i −2.63543 + 3.62735i 0.809017 + 0.587785i 0.949523 − 5.32585i
4.5 −2.28150 + 0.741303i 0.951057 + 0.309017i 3.03766 − 2.20699i 1.23533 + 1.86386i −2.39891 −2.71113 − 3.73155i −2.47429 + 3.40556i 0.809017 + 0.587785i −4.20008 − 3.33663i
4.6 −1.82273 + 0.592242i 0.951057 + 0.309017i 1.35357 − 0.983429i 1.06683 − 1.96517i −1.91654 −0.295926 − 0.407308i 0.368247 − 0.506848i 0.809017 + 0.587785i −0.780683 + 4.21380i
4.7 −1.75841 + 0.571340i −0.951057 − 0.309017i 1.14753 − 0.833726i 1.89346 + 1.18945i 1.84890 2.64500 + 3.64052i 0.632036 − 0.869922i 0.809017 + 0.587785i −4.00906 − 1.00972i
4.8 −1.71190 + 0.556229i 0.951057 + 0.309017i 1.00316 − 0.728840i −1.35623 − 1.77782i −1.79999 −2.23429 − 3.07523i 0.804112 − 1.10677i 0.809017 + 0.587785i 3.31060 + 2.28907i
4.9 −1.36240 + 0.442671i −0.951057 − 0.309017i 0.0421447 − 0.0306199i 2.03530 − 0.926033i 1.43251 −1.00919 − 1.38903i 1.64016 − 2.25748i 0.809017 + 0.587785i −2.36297 + 2.16260i
4.10 −1.28040 + 0.416026i 0.951057 + 0.309017i −0.151696 + 0.110214i −0.613177 + 2.15035i −1.34629 1.52844 + 2.10372i 1.73104 − 2.38257i 0.809017 + 0.587785i −0.109493 − 3.00840i
4.11 −1.03892 + 0.337565i 0.951057 + 0.309017i −0.652633 + 0.474165i −1.88420 + 1.20407i −1.09238 −1.49943 − 2.06379i 1.80214 − 2.48044i 0.809017 + 0.587785i 1.55108 − 1.88697i
4.12 −0.932426 + 0.302963i −0.951057 − 0.309017i −0.840403 + 0.610589i 0.949797 + 2.02432i 0.980410 −2.34180 − 3.22321i 1.75117 − 2.41028i 0.809017 + 0.587785i −1.49891 − 1.59978i
4.13 −0.869491 + 0.282515i −0.951057 − 0.309017i −0.941833 + 0.684282i −0.569561 − 2.16231i 0.914237 0.633796 + 0.872346i 1.70035 − 2.34033i 0.809017 + 0.587785i 1.10611 + 1.71920i
4.14 −0.598879 + 0.194588i 0.951057 + 0.309017i −1.29724 + 0.942501i 1.97230 + 1.05357i −0.629699 0.877498 + 1.20777i 1.33375 − 1.83575i 0.809017 + 0.587785i −1.38618 − 0.247178i
4.15 −0.464076 + 0.150787i −0.951057 − 0.309017i −1.42540 + 1.03562i −2.22182 + 0.252038i 0.487958 −0.802309 − 1.10428i 1.07897 − 1.48507i 0.809017 + 0.587785i 0.993088 − 0.451987i
4.16 −0.0800113 + 0.0259972i −0.951057 − 0.309017i −1.61231 + 1.17141i −1.58838 + 1.57386i 0.0841288 1.24192 + 1.70936i 0.197449 − 0.271765i 0.809017 + 0.587785i 0.0861727 − 0.167220i
4.17 0.0800113 − 0.0259972i 0.951057 + 0.309017i −1.61231 + 1.17141i −1.58838 − 1.57386i 0.0841288 −1.24192 − 1.70936i −0.197449 + 0.271765i 0.809017 + 0.587785i −0.168005 − 0.0846329i
4.18 0.464076 − 0.150787i 0.951057 + 0.309017i −1.42540 + 1.03562i −2.22182 − 0.252038i 0.487958 0.802309 + 1.10428i −1.07897 + 1.48507i 0.809017 + 0.587785i −1.06910 + 0.218057i
4.19 0.598879 − 0.194588i −0.951057 − 0.309017i −1.29724 + 0.942501i 1.97230 − 1.05357i −0.629699 −0.877498 − 1.20777i −1.33375 + 1.83575i 0.809017 + 0.587785i 0.976159 − 1.01475i
4.20 0.869491 − 0.282515i 0.951057 + 0.309017i −0.941833 + 0.684282i −0.569561 + 2.16231i 0.914237 −0.633796 − 0.872346i −1.70035 + 2.34033i 0.809017 + 0.587785i 0.115657 + 2.04102i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.d even 5 1 inner
155.n even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.ba.a ✓ 128
5.b even 2 1 inner 465.2.ba.a ✓ 128
31.d even 5 1 inner 465.2.ba.a ✓ 128
155.n even 10 1 inner 465.2.ba.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.ba.a ✓ 128 1.a even 1 1 trivial
465.2.ba.a ✓ 128 5.b even 2 1 inner
465.2.ba.a ✓ 128 31.d even 5 1 inner
465.2.ba.a ✓ 128 155.n even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(465, [\chi])\).