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Properties

Label 462.2.ba.a
Level $462$
Weight $2$
Character orbit 462.ba
Analytic conductor $3.689$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [462,2,Mod(19,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 25, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.ba (of order \(30\), degree \(8\), 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: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{4} - 22 q^{5} - 16 q^{6} + 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{4} - 22 q^{5} - 16 q^{6} + 4 q^{7} - 8 q^{9} + 2 q^{10} + 4 q^{11} + 2 q^{14} - 6 q^{15} + 8 q^{16} + 30 q^{17} + 10 q^{19} + 20 q^{20} - 4 q^{21} - 2 q^{22} + 4 q^{23} - 8 q^{24} - 12 q^{26} - 10 q^{28} - 20 q^{29} + 18 q^{30} - 16 q^{31} - 14 q^{33} + 42 q^{35} - 16 q^{36} - 14 q^{37} + 12 q^{38} + 18 q^{39} + 18 q^{40} - 28 q^{41} - 6 q^{42} + 6 q^{44} - 12 q^{45} - 42 q^{46} + 24 q^{47} + 116 q^{49} + 26 q^{51} + 32 q^{54} - 14 q^{55} - 4 q^{56} + 20 q^{58} + 30 q^{59} + 2 q^{60} - 32 q^{61} - 8 q^{62} + 4 q^{63} + 16 q^{64} + 12 q^{65} + 4 q^{66} + 16 q^{67} - 30 q^{68} - 20 q^{70} - 24 q^{71} - 64 q^{73} + 4 q^{74} + 12 q^{75} - 48 q^{77} - 60 q^{79} - 18 q^{80} + 8 q^{81} - 68 q^{82} + 8 q^{83} + 2 q^{84} - 80 q^{85} - 18 q^{86} + 10 q^{87} - 8 q^{88} - 24 q^{89} + 4 q^{90} - 172 q^{91} + 8 q^{92} - 104 q^{93} - 6 q^{94} - 118 q^{95} + 8 q^{96} + 120 q^{97} + 40 q^{98} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 − 0.207912i −0.993461 − 2.23135i −0.809017 − 0.587785i 2.46197 − 0.968858i −0.951057 + 0.309017i 0.104528 + 0.994522i 1.22126 + 2.11528i
19.2 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 − 0.207912i −0.716911 − 1.61021i −0.809017 − 0.587785i 1.87226 + 1.86939i −0.951057 + 0.309017i 0.104528 + 0.994522i 0.881296 + 1.52645i
19.3 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 − 0.207912i −0.281694 − 0.632696i −0.809017 − 0.587785i −2.57173 − 0.621464i −0.951057 + 0.309017i 0.104528 + 0.994522i 0.346286 + 0.599785i
19.4 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 − 0.207912i 1.49050 + 3.34771i −0.809017 − 0.587785i −2.49930 − 0.868052i −0.951057 + 0.309017i 0.104528 + 0.994522i −1.83226 − 3.17357i
19.5 0.994522 − 0.104528i −0.743145 − 0.669131i 0.978148 − 0.207912i −1.50643 − 3.38350i −0.809017 − 0.587785i 2.60972 − 0.435184i 0.951057 − 0.309017i 0.104528 + 0.994522i −1.85185 − 3.20750i
19.6 0.994522 − 0.104528i −0.743145 − 0.669131i 0.978148 − 0.207912i −0.0779848 − 0.175157i −0.809017 − 0.587785i −1.57120 − 2.12869i 0.951057 − 0.309017i 0.104528 + 0.994522i −0.0958664 − 0.166046i
19.7 0.994522 − 0.104528i −0.743145 − 0.669131i 0.978148 − 0.207912i 0.688924 + 1.54735i −0.809017 − 0.587785i 0.640384 + 2.56708i 0.951057 − 0.309017i 0.104528 + 0.994522i 0.846892 + 1.46686i
19.8 0.994522 − 0.104528i −0.743145 − 0.669131i 0.978148 − 0.207912i 1.48256 + 3.32989i −0.809017 − 0.587785i 1.16475 − 2.37557i 0.951057 − 0.309017i 0.104528 + 0.994522i 1.82251 + 3.15668i
61.1 −0.207912 + 0.978148i −0.994522 − 0.104528i −0.913545 − 0.406737i −2.80028 + 2.52138i 0.309017 − 0.951057i −2.64322 + 0.115723i 0.587785 − 0.809017i 0.978148 + 0.207912i −1.88407 − 3.26331i
61.2 −0.207912 + 0.978148i −0.994522 − 0.104528i −0.913545 − 0.406737i −1.32335 + 1.19155i 0.309017 − 0.951057i 2.59833 − 0.498689i 0.587785 − 0.809017i 0.978148 + 0.207912i −0.890371 − 1.54217i
61.3 −0.207912 + 0.978148i −0.994522 − 0.104528i −0.913545 − 0.406737i −0.624935 + 0.562694i 0.309017 − 0.951057i 1.46102 − 2.20577i 0.587785 − 0.809017i 0.978148 + 0.207912i −0.420466 − 0.728269i
61.4 −0.207912 + 0.978148i −0.994522 − 0.104528i −0.913545 − 0.406737i 1.09698 − 0.987728i 0.309017 − 0.951057i −2.51719 + 0.814704i 0.587785 − 0.809017i 0.978148 + 0.207912i 0.738068 + 1.27837i
61.5 0.207912 − 0.978148i 0.994522 + 0.104528i −0.913545 − 0.406737i −2.17695 + 1.96014i 0.309017 − 0.951057i −2.29320 + 1.31956i −0.587785 + 0.809017i 0.978148 + 0.207912i 1.46469 + 2.53692i
61.6 0.207912 − 0.978148i 0.994522 + 0.104528i −0.913545 − 0.406737i −2.02809 + 1.82610i 0.309017 − 0.951057i 2.53777 + 0.748146i −0.587785 + 0.809017i 0.978148 + 0.207912i 1.36453 + 2.36343i
61.7 0.207912 − 0.978148i 0.994522 + 0.104528i −0.913545 − 0.406737i −1.26898 + 1.14260i 0.309017 − 0.951057i −0.554537 − 2.58698i −0.587785 + 0.809017i 0.978148 + 0.207912i 0.853791 + 1.47881i
61.8 0.207912 − 0.978148i 0.994522 + 0.104528i −0.913545 − 0.406737i 2.13316 − 1.92070i 0.309017 − 0.951057i 2.34031 + 1.23407i −0.587785 + 0.809017i 0.978148 + 0.207912i −1.43522 − 2.48588i
73.1 −0.994522 − 0.104528i 0.743145 − 0.669131i 0.978148 + 0.207912i −0.993461 + 2.23135i −0.809017 + 0.587785i 2.46197 + 0.968858i −0.951057 − 0.309017i 0.104528 − 0.994522i 1.22126 − 2.11528i
73.2 −0.994522 − 0.104528i 0.743145 − 0.669131i 0.978148 + 0.207912i −0.716911 + 1.61021i −0.809017 + 0.587785i 1.87226 − 1.86939i −0.951057 − 0.309017i 0.104528 − 0.994522i 0.881296 − 1.52645i
73.3 −0.994522 − 0.104528i 0.743145 − 0.669131i 0.978148 + 0.207912i −0.281694 + 0.632696i −0.809017 + 0.587785i −2.57173 + 0.621464i −0.951057 − 0.309017i 0.104528 − 0.994522i 0.346286 − 0.599785i
73.4 −0.994522 − 0.104528i 0.743145 − 0.669131i 0.978148 + 0.207912i 1.49050 − 3.34771i −0.809017 + 0.587785i −2.49930 + 0.868052i −0.951057 − 0.309017i 0.104528 − 0.994522i −1.83226 + 3.17357i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.n even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.ba.a ✓ 64
7.d odd 6 1 462.2.ba.b yes 64
11.d odd 10 1 462.2.ba.b yes 64
77.n even 30 1 inner 462.2.ba.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.ba.a ✓ 64 1.a even 1 1 trivial
462.2.ba.a ✓ 64 77.n even 30 1 inner
462.2.ba.b yes 64 7.d odd 6 1
462.2.ba.b yes 64 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{64} + 22 T_{5}^{63} + 262 T_{5}^{62} + 2248 T_{5}^{61} + 15382 T_{5}^{60} + 87548 T_{5}^{59} + \cdots + 1291129965841 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display