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Properties

Label 462.2.y.a
Level $462$
Weight $2$
Character orbit 462.y
Analytic conductor $3.689$
Analytic rank $0$
Dimension $24$
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] = [462,2,Mod(25,462)]
 
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, 20, 24]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.y (of order \(15\), 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: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 5 q^{5} - 6 q^{6} + 11 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 5 q^{5} - 6 q^{6} + 11 q^{7} + 6 q^{8} + 3 q^{9} + 10 q^{10} + 6 q^{11} + 12 q^{12} - 2 q^{13} - 5 q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{18} + 7 q^{19} - 10 q^{20} - 4 q^{21} + 2 q^{22} + 24 q^{23} + 3 q^{24} + 4 q^{25} + 4 q^{26} + 6 q^{27} + 14 q^{28} - 6 q^{29} - 7 q^{31} + 12 q^{32} - q^{33} - 24 q^{34} + 4 q^{35} - 6 q^{36} - q^{37} + 8 q^{38} - q^{39} + 16 q^{41} - 4 q^{42} + 52 q^{43} - 4 q^{44} - 10 q^{45} - 4 q^{46} + 27 q^{47} + 6 q^{48} - 33 q^{49} - 22 q^{50} - 8 q^{51} - 4 q^{52} + 13 q^{53} - 12 q^{54} + 30 q^{55} + 14 q^{56} - 16 q^{57} - 3 q^{58} + 14 q^{59} - 5 q^{60} + 9 q^{61} - 4 q^{62} + 5 q^{63} - 6 q^{64} - 50 q^{65} - 4 q^{66} - 20 q^{67} - 2 q^{68} - 32 q^{69} - 17 q^{70} - 18 q^{71} - 3 q^{72} + 7 q^{73} + q^{74} + 11 q^{75} - 4 q^{76} - 34 q^{77} - 12 q^{78} + q^{79} + 3 q^{81} - 2 q^{82} - 68 q^{83} - 5 q^{84} - 19 q^{86} - 8 q^{87} - q^{88} - 42 q^{89} + 10 q^{90} - 16 q^{91} + 32 q^{92} + 7 q^{93} + 8 q^{94} + 38 q^{95} + 3 q^{96} - 80 q^{97} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.913545 − 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i −0.269783 + 2.56681i −0.809017 − 0.587785i −0.146873 + 2.64167i −0.309017 − 0.951057i 0.913545 + 0.406737i 1.29048 − 2.23517i
25.2 −0.913545 − 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i −0.126123 + 1.19998i −0.809017 − 0.587785i 0.836823 − 2.50993i −0.309017 − 0.951057i 0.913545 + 0.406737i 0.603294 − 1.04494i
25.3 −0.913545 − 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i 0.0177181 − 0.168577i −0.809017 − 0.587785i 2.46190 + 0.969054i −0.309017 − 0.951057i 0.913545 + 0.406737i −0.0847527 + 0.146796i
37.1 −0.913545 + 0.406737i 0.978148 − 0.207912i 0.669131 − 0.743145i −0.269783 − 2.56681i −0.809017 + 0.587785i −0.146873 − 2.64167i −0.309017 + 0.951057i 0.913545 − 0.406737i 1.29048 + 2.23517i
37.2 −0.913545 + 0.406737i 0.978148 − 0.207912i 0.669131 − 0.743145i −0.126123 − 1.19998i −0.809017 + 0.587785i 0.836823 + 2.50993i −0.309017 + 0.951057i 0.913545 − 0.406737i 0.603294 + 1.04494i
37.3 −0.913545 + 0.406737i 0.978148 − 0.207912i 0.669131 − 0.743145i 0.0177181 + 0.168577i −0.809017 + 0.587785i 2.46190 − 0.969054i −0.309017 + 0.951057i 0.913545 − 0.406737i −0.0847527 − 0.146796i
163.1 −0.669131 − 0.743145i −0.913545 − 0.406737i −0.104528 + 0.994522i −2.39103 − 0.508229i 0.309017 + 0.951057i −0.0798814 + 2.64455i 0.809017 − 0.587785i 0.669131 + 0.743145i 1.22222 + 2.11695i
163.2 −0.669131 − 0.743145i −0.913545 − 0.406737i −0.104528 + 0.994522i −1.47473 − 0.313464i 0.309017 + 0.951057i 2.56873 − 0.633736i 0.809017 − 0.587785i 0.669131 + 0.743145i 0.753838 + 1.30569i
163.3 −0.669131 − 0.743145i −0.913545 − 0.406737i −0.104528 + 0.994522i 2.51399 + 0.534365i 0.309017 + 0.951057i 1.73654 − 1.99611i 0.809017 − 0.587785i 0.669131 + 0.743145i −1.28508 − 2.22582i
235.1 0.104528 − 0.994522i −0.669131 − 0.743145i −0.978148 − 0.207912i −0.154851 − 0.0689440i −0.809017 + 0.587785i −2.56131 + 0.663085i −0.309017 + 0.951057i −0.104528 + 0.994522i −0.0847527 + 0.146796i
235.2 0.104528 − 0.994522i −0.669131 − 0.743145i −0.978148 − 0.207912i 1.10227 + 0.490763i −0.809017 + 0.587785i 0.798293 + 2.52244i −0.309017 + 0.951057i −0.104528 + 0.994522i 0.603294 − 1.04494i
235.3 0.104528 − 0.994522i −0.669131 − 0.743145i −0.978148 − 0.207912i 2.35782 + 1.04977i −0.809017 + 0.587785i −1.43391 − 2.22349i −0.309017 + 0.951057i −0.104528 + 0.994522i 1.29048 − 2.23517i
247.1 0.978148 + 0.207912i 0.104528 − 0.994522i 0.913545 + 0.406737i −1.71977 − 1.91000i 0.309017 − 0.951057i −1.36179 − 2.26838i 0.809017 + 0.587785i −0.978148 − 0.207912i −1.28508 − 2.22582i
247.2 0.978148 + 0.207912i 0.104528 − 0.994522i 0.913545 + 0.406737i 1.00883 + 1.12042i 0.309017 − 0.951057i 0.191063 − 2.63884i 0.809017 + 0.587785i −0.978148 − 0.207912i 0.753838 + 1.30569i
247.3 0.978148 + 0.207912i 0.104528 − 0.994522i 0.913545 + 0.406737i 1.63565 + 1.81658i 0.309017 − 0.951057i 2.49043 + 0.893181i 0.809017 + 0.587785i −0.978148 − 0.207912i 1.22222 + 2.11695i
289.1 0.104528 + 0.994522i −0.669131 + 0.743145i −0.978148 + 0.207912i −0.154851 + 0.0689440i −0.809017 − 0.587785i −2.56131 − 0.663085i −0.309017 − 0.951057i −0.104528 − 0.994522i −0.0847527 − 0.146796i
289.2 0.104528 + 0.994522i −0.669131 + 0.743145i −0.978148 + 0.207912i 1.10227 − 0.490763i −0.809017 − 0.587785i 0.798293 − 2.52244i −0.309017 − 0.951057i −0.104528 − 0.994522i 0.603294 + 1.04494i
289.3 0.104528 + 0.994522i −0.669131 + 0.743145i −0.978148 + 0.207912i 2.35782 − 1.04977i −0.809017 − 0.587785i −1.43391 + 2.22349i −0.309017 − 0.951057i −0.104528 − 0.994522i 1.29048 + 2.23517i
361.1 0.978148 − 0.207912i 0.104528 + 0.994522i 0.913545 − 0.406737i −1.71977 + 1.91000i 0.309017 + 0.951057i −1.36179 + 2.26838i 0.809017 − 0.587785i −0.978148 + 0.207912i −1.28508 + 2.22582i
361.2 0.978148 − 0.207912i 0.104528 + 0.994522i 0.913545 − 0.406737i 1.00883 − 1.12042i 0.309017 + 0.951057i 0.191063 + 2.63884i 0.809017 − 0.587785i −0.978148 + 0.207912i 0.753838 − 1.30569i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.y.a ✓ 24
7.c even 3 1 inner 462.2.y.a ✓ 24
11.c even 5 1 inner 462.2.y.a ✓ 24
77.m even 15 1 inner 462.2.y.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.y.a ✓ 24 1.a even 1 1 trivial
462.2.y.a ✓ 24 7.c even 3 1 inner
462.2.y.a ✓ 24 11.c even 5 1 inner
462.2.y.a ✓ 24 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 5 T_{5}^{23} + 3 T_{5}^{22} + 20 T_{5}^{21} - 20 T_{5}^{20} - 5 T_{5}^{19} - 122 T_{5}^{18} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display