↑

Properties

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

Related objects

Downloads

Learn more

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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{4} - 2 q^{5} + 16 q^{6} + 16 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{4} - 2 q^{5} + 16 q^{6} + 16 q^{7} - 8 q^{9} - 2 q^{10} + 4 q^{11} + 2 q^{14} - 6 q^{15} + 8 q^{16} + 30 q^{17} - 10 q^{19} - 20 q^{20} + 4 q^{21} - 2 q^{22} + 4 q^{23} + 8 q^{24} - 12 q^{26} - 20 q^{29} - 18 q^{30} + 34 q^{31} + 8 q^{33} - 2 q^{35} - 16 q^{36} - 14 q^{37} + 12 q^{38} - 18 q^{39} + 12 q^{40} + 28 q^{41} + 4 q^{42} + 6 q^{44} - 12 q^{45} + 42 q^{46} + 24 q^{47} - 44 q^{49} + 14 q^{51} - 32 q^{54} + 14 q^{55} - 4 q^{56} - 10 q^{58} - 30 q^{59} + 2 q^{60} - 28 q^{61} + 8 q^{62} + 16 q^{63} + 16 q^{64} - 12 q^{65} - 4 q^{66} + 16 q^{67} - 30 q^{68} - 30 q^{70} - 24 q^{71} - 116 q^{73} - 44 q^{74} + 12 q^{75} - 32 q^{77} - 18 q^{80} + 8 q^{81} - 28 q^{82} - 8 q^{83} - 2 q^{84} - 80 q^{85} - 18 q^{86} - 10 q^{87} - 14 q^{88} - 24 q^{89} - 4 q^{90} + 48 q^{91} + 8 q^{92} + 76 q^{93} + 6 q^{94} + 98 q^{95} - 8 q^{96} - 120 q^{97} - 40 q^{98} + 8 q^{99}+O(q^{100}) \) 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 −1.44711 − 3.25025i 0.809017 + 0.587785i 1.04897 + 2.42892i −0.951057 + 0.309017i 0.104528 + 0.994522i 1.77892 + 3.08119i
19.2 −0.994522 + 0.104528i −0.743145 − 0.669131i 0.978148 − 0.207912i −1.26939 − 2.85109i 0.809017 + 0.587785i −0.613482 − 2.57364i −0.951057 + 0.309017i 0.104528 + 0.994522i 1.56045 + 2.70278i
19.3 −0.994522 + 0.104528i −0.743145 − 0.669131i 0.978148 − 0.207912i 0.0572849 + 0.128664i 0.809017 + 0.587785i −2.64421 + 0.0903816i −0.951057 + 0.309017i 0.104528 + 0.994522i −0.0704201 − 0.121971i
19.4 −0.994522 + 0.104528i −0.743145 − 0.669131i 0.978148 − 0.207912i 0.949965 + 2.13366i 0.809017 + 0.587785i 2.43658 + 1.03106i −0.951057 + 0.309017i 0.104528 + 0.994522i −1.16779 − 2.02267i
19.5 0.994522 − 0.104528i 0.743145 + 0.669131i 0.978148 − 0.207912i −1.01671 − 2.28357i 0.809017 + 0.587785i −0.151373 − 2.64142i 0.951057 − 0.309017i 0.104528 + 0.994522i −1.24984 − 2.16479i
19.6 0.994522 − 0.104528i 0.743145 + 0.669131i 0.978148 − 0.207912i −0.748568 − 1.68131i 0.809017 + 0.587785i 2.42705 + 1.05329i 0.951057 − 0.309017i 0.104528 + 0.994522i −0.920212 − 1.59385i
19.7 0.994522 − 0.104528i 0.743145 + 0.669131i 0.978148 − 0.207912i 0.316797 + 0.711538i 0.809017 + 0.587785i −0.370234 + 2.61972i 0.951057 − 0.309017i 0.104528 + 0.994522i 0.389438 + 0.674526i
19.8 0.994522 − 0.104528i 0.743145 + 0.669131i 0.978148 − 0.207912i 1.33064 + 2.98866i 0.809017 + 0.587785i 1.90285 − 1.83825i 0.951057 − 0.309017i 0.104528 + 0.994522i 1.63575 + 2.83319i
61.1 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 − 0.406737i −3.24579 + 2.92252i −0.309017 + 0.951057i −0.190680 − 2.63887i 0.587785 − 0.809017i 0.978148 + 0.207912i −2.18382 − 3.78249i
61.2 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 − 0.406737i −0.468951 + 0.422245i −0.309017 + 0.951057i −0.177812 + 2.63977i 0.587785 − 0.809017i 0.978148 + 0.207912i −0.315518 − 0.546493i
61.3 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 − 0.406737i 0.131864 − 0.118731i −0.309017 + 0.951057i −1.17519 − 2.37043i 0.587785 − 0.809017i 0.978148 + 0.207912i 0.0887206 + 0.153669i
61.4 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 − 0.406737i 1.55595 − 1.40099i −0.309017 + 0.951057i 2.62525 − 0.328695i 0.587785 − 0.809017i 0.978148 + 0.207912i 1.04687 + 1.81324i
61.5 0.207912 − 0.978148i −0.994522 − 0.104528i −0.913545 − 0.406737i −1.40387 + 1.26405i −0.309017 + 0.951057i 1.39244 − 2.24969i −0.587785 + 0.809017i 0.978148 + 0.207912i 0.944546 + 1.63600i
61.6 0.207912 − 0.978148i −0.994522 − 0.104528i −0.913545 − 0.406737i −0.716223 + 0.644890i −0.309017 + 0.951057i 2.17783 + 1.50235i −0.587785 + 0.809017i 0.978148 + 0.207912i 0.481886 + 0.834652i
61.7 0.207912 − 0.978148i −0.994522 − 0.104528i −0.913545 − 0.406737i 0.192170 − 0.173031i −0.309017 + 0.951057i −1.35536 + 2.27222i −0.587785 + 0.809017i 0.978148 + 0.207912i −0.129295 − 0.223946i
61.8 0.207912 − 0.978148i −0.994522 − 0.104528i −0.913545 − 0.406737i 2.61658 − 2.35598i −0.309017 + 0.951057i 1.99807 − 1.73428i −0.587785 + 0.809017i 0.978148 + 0.207912i −1.76048 − 3.04924i
73.1 −0.994522 − 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i −1.44711 + 3.25025i 0.809017 − 0.587785i 1.04897 − 2.42892i −0.951057 − 0.309017i 0.104528 − 0.994522i 1.77892 − 3.08119i
73.2 −0.994522 − 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i −1.26939 + 2.85109i 0.809017 − 0.587785i −0.613482 + 2.57364i −0.951057 − 0.309017i 0.104528 − 0.994522i 1.56045 − 2.70278i
73.3 −0.994522 − 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i 0.0572849 − 0.128664i 0.809017 − 0.587785i −2.64421 − 0.0903816i −0.951057 − 0.309017i 0.104528 − 0.994522i −0.0704201 + 0.121971i
73.4 −0.994522 − 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i 0.949965 − 2.13366i 0.809017 − 0.587785i 2.43658 − 1.03106i −0.951057 − 0.309017i 0.104528 − 0.994522i −1.16779 + 2.02267i
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.b yes 64
7.d odd 6 1 462.2.ba.a ✓ 64
11.d odd 10 1 462.2.ba.a ✓ 64
77.n even 30 1 inner 462.2.ba.b yes 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.ba.a ✓ 64 7.d odd 6 1
462.2.ba.a ✓ 64 11.d odd 10 1
462.2.ba.b yes 64 1.a even 1 1 trivial
462.2.ba.b yes 64 77.n even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{64} + 2 T_{5}^{63} + 22 T_{5}^{62} + 8 T_{5}^{61} + 52 T_{5}^{60} - 1232 T_{5}^{59} + \cdots + 1291129965841 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display