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Properties

Label 462.2.y.c
Level $462$
Weight $2$
Character orbit 462.y
Analytic conductor $3.689$
Analytic rank $0$
Dimension $40$
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: \(40\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + q^{5} + 10 q^{6} + 9 q^{7} + 10 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + q^{5} + 10 q^{6} + 9 q^{7} + 10 q^{8} + 5 q^{9} + 4 q^{10} + 2 q^{11} - 20 q^{12} - 2 q^{13} - 3 q^{14} - 2 q^{15} + 5 q^{16} + 6 q^{17} - 5 q^{18} - q^{19} - 2 q^{20} - 2 q^{21} + 4 q^{22} - 20 q^{23} - 5 q^{24} + 16 q^{25} + 4 q^{26} - 10 q^{27} + 13 q^{28} + 16 q^{29} - q^{30} + 30 q^{31} + 20 q^{32} + 2 q^{33} + 32 q^{34} + 3 q^{35} - 10 q^{36} + 19 q^{37} + 6 q^{38} + q^{39} - q^{40} - 12 q^{41} + 11 q^{42} - 36 q^{43} - 3 q^{44} - 4 q^{45} + 10 q^{46} + 25 q^{47} - 10 q^{48} + 29 q^{49} + 12 q^{50} - 14 q^{51} - 4 q^{52} - 18 q^{53} + 20 q^{54} + 12 q^{55} - 4 q^{56} + 12 q^{57} + 8 q^{58} - 14 q^{59} + q^{60} + 35 q^{61} - 40 q^{62} + 3 q^{63} - 10 q^{64} - 2 q^{65} + 3 q^{66} - 24 q^{67} + 6 q^{68} - 40 q^{69} - 32 q^{70} - 34 q^{71} - 5 q^{72} - 39 q^{73} - 19 q^{74} + 6 q^{75} - 28 q^{76} - 31 q^{77} + 12 q^{78} - 17 q^{79} + q^{80} + 5 q^{81} + 4 q^{82} - 24 q^{83} + 3 q^{84} - 112 q^{85} + 7 q^{86} + 2 q^{87} - 2 q^{88} - 38 q^{89} + 2 q^{90} - 36 q^{91} - 40 q^{92} + 30 q^{93} - 10 q^{94} - 42 q^{95} - 5 q^{96} + 4 q^{97} - 50 q^{98} - 4 q^{99}+O(q^{100}) \) 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 −0.435132 + 4.14000i 0.809017 + 0.587785i 2.64557 − 0.0306495i −0.309017 − 0.951057i 0.913545 + 0.406737i 2.08140 − 3.60509i
25.2 −0.913545 − 0.406737i −0.978148 − 0.207912i 0.669131 + 0.743145i −0.168631 + 1.60442i 0.809017 + 0.587785i −1.86877 − 1.87289i −0.309017 − 0.951057i 0.913545 + 0.406737i 0.806628 − 1.39712i
25.3 −0.913545 − 0.406737i −0.978148 − 0.207912i 0.669131 + 0.743145i −0.0207786 + 0.197695i 0.809017 + 0.587785i −2.07849 + 1.63704i −0.309017 − 0.951057i 0.913545 + 0.406737i 0.0993921 − 0.172152i
25.4 −0.913545 − 0.406737i −0.978148 − 0.207912i 0.669131 + 0.743145i 0.171980 − 1.63628i 0.809017 + 0.587785i 2.59574 + 0.512007i −0.309017 − 0.951057i 0.913545 + 0.406737i −0.822649 + 1.42487i
25.5 −0.913545 − 0.406737i −0.978148 − 0.207912i 0.669131 + 0.743145i 0.348032 − 3.31131i 0.809017 + 0.587785i 1.39315 − 2.24925i −0.309017 − 0.951057i 0.913545 + 0.406737i −1.66477 + 2.88347i
37.1 −0.913545 + 0.406737i −0.978148 + 0.207912i 0.669131 − 0.743145i −0.435132 − 4.14000i 0.809017 − 0.587785i 2.64557 + 0.0306495i −0.309017 + 0.951057i 0.913545 − 0.406737i 2.08140 + 3.60509i
37.2 −0.913545 + 0.406737i −0.978148 + 0.207912i 0.669131 − 0.743145i −0.168631 − 1.60442i 0.809017 − 0.587785i −1.86877 + 1.87289i −0.309017 + 0.951057i 0.913545 − 0.406737i 0.806628 + 1.39712i
37.3 −0.913545 + 0.406737i −0.978148 + 0.207912i 0.669131 − 0.743145i −0.0207786 − 0.197695i 0.809017 − 0.587785i −2.07849 − 1.63704i −0.309017 + 0.951057i 0.913545 − 0.406737i 0.0993921 + 0.172152i
37.4 −0.913545 + 0.406737i −0.978148 + 0.207912i 0.669131 − 0.743145i 0.171980 + 1.63628i 0.809017 − 0.587785i 2.59574 − 0.512007i −0.309017 + 0.951057i 0.913545 − 0.406737i −0.822649 − 1.42487i
37.5 −0.913545 + 0.406737i −0.978148 + 0.207912i 0.669131 − 0.743145i 0.348032 + 3.31131i 0.809017 − 0.587785i 1.39315 + 2.24925i −0.309017 + 0.951057i 0.913545 − 0.406737i −1.66477 − 2.88347i
163.1 −0.669131 − 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i −3.51246 − 0.746597i −0.309017 − 0.951057i 2.57655 + 0.601166i 0.809017 − 0.587785i 0.669131 + 0.743145i 1.79547 + 3.10984i
163.2 −0.669131 − 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i −2.08985 − 0.444211i −0.309017 − 0.951057i −2.27148 + 1.35660i 0.809017 − 0.587785i 0.669131 + 0.743145i 1.06827 + 1.85029i
163.3 −0.669131 − 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i −1.13392 − 0.241022i −0.309017 − 0.951057i −1.18884 − 2.36361i 0.809017 − 0.587785i 0.669131 + 0.743145i 0.579626 + 1.00394i
163.4 −0.669131 − 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i 1.87534 + 0.398616i −0.309017 − 0.951057i 1.79574 + 1.94302i 0.809017 − 0.587785i 0.669131 + 0.743145i −0.958617 − 1.66037i
163.5 −0.669131 − 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i 3.88274 + 0.825303i −0.309017 − 0.951057i 1.63079 − 2.08339i 0.809017 − 0.587785i 0.669131 + 0.743145i −1.98474 − 3.43768i
235.1 0.104528 − 0.994522i 0.669131 + 0.743145i −0.978148 − 0.207912i −3.04169 − 1.35425i 0.809017 − 0.587785i 0.194999 + 2.63856i −0.309017 + 0.951057i −0.104528 + 0.994522i −1.66477 + 2.88347i
235.2 0.104528 − 0.994522i 0.669131 + 0.743145i −0.978148 − 0.207912i −1.50305 − 0.669203i 0.809017 − 0.587785i −2.40095 + 1.11151i −0.309017 + 0.951057i −0.104528 + 0.994522i −0.822649 + 1.42487i
235.3 0.104528 − 0.994522i 0.669131 + 0.743145i −0.978148 − 0.207912i 0.181598 + 0.0808528i 0.809017 − 0.587785i 0.719303 − 2.54610i −0.309017 + 0.951057i −0.104528 + 0.994522i 0.0993921 − 0.172152i
235.4 0.104528 − 0.994522i 0.669131 + 0.743145i −0.978148 − 0.207912i 1.47378 + 0.656170i 0.809017 − 0.587785i 2.61272 + 0.416765i −0.309017 + 0.951057i −0.104528 + 0.994522i 0.806628 − 1.39712i
235.5 0.104528 − 0.994522i 0.669131 + 0.743145i −0.978148 − 0.207912i 3.80291 + 1.69317i 0.809017 − 0.587785i −2.12230 + 1.57983i −0.309017 + 0.951057i −0.104528 + 0.994522i 2.08140 − 3.60509i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.5
Significant digits:
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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.c ✓ 40
7.c even 3 1 inner 462.2.y.c ✓ 40
11.c even 5 1 inner 462.2.y.c ✓ 40
77.m even 15 1 inner 462.2.y.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.y.c ✓ 40 1.a even 1 1 trivial
462.2.y.c ✓ 40 7.c even 3 1 inner
462.2.y.c ✓ 40 11.c even 5 1 inner
462.2.y.c ✓ 40 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} - T_{5}^{39} - 20 T_{5}^{38} - 7 T_{5}^{37} + 120 T_{5}^{36} + 636 T_{5}^{35} + \cdots + 60037250625 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display