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Properties

Label 468.2.cb.h
Level $468$
Weight $2$
Character orbit 468.cb
Analytic conductor $3.737$
Analytic rank $0$
Dimension $24$
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] = [468,2,Mod(19,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.cb (of order \(12\), 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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{8} + 12 q^{10} - 4 q^{11} + 22 q^{14} + 12 q^{16} - 22 q^{19} + 4 q^{20} - 34 q^{22} + 4 q^{26} - 14 q^{28} - 8 q^{29} - 2 q^{31} + 34 q^{32} + 20 q^{34} - 24 q^{35} - 12 q^{37} + 12 q^{38} + 28 q^{40} + 36 q^{41} + 10 q^{43} - 20 q^{44} - 2 q^{46} - 20 q^{47} - 54 q^{49} + 24 q^{50} + 4 q^{52} + 36 q^{53} - 24 q^{55} - 70 q^{56} + 48 q^{58} + 36 q^{59} - 2 q^{61} - 64 q^{62} - 8 q^{64} - 16 q^{65} - 16 q^{67} + 10 q^{68} - 36 q^{71} - 40 q^{73} + 30 q^{74} + 58 q^{76} + 48 q^{80} - 30 q^{82} + 24 q^{83} + 18 q^{85} - 30 q^{86} - 78 q^{88} - 66 q^{89} + 42 q^{91} - 36 q^{92} + 6 q^{94} + 28 q^{95} + 4 q^{97} - 44 q^{98}+O(q^{100}) \) 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 −1.17059 − 0.793543i 0 0.740580 + 1.85783i −2.78626 + 2.78626i 0 −0.826736 − 0.221523i 0.607351 − 2.76245i 0 5.47259 − 1.05056i
19.2 −1.12047 + 0.862866i 0 0.510925 − 1.93364i −0.756294 + 0.756294i 0 −0.936058 − 0.250816i 1.09599 + 2.60745i 0 0.194828 − 1.49999i
19.3 −0.0144886 + 1.41414i 0 −1.99958 − 0.0409778i 2.26006 − 2.26006i 0 −3.65215 − 0.978592i 0.0869195 − 2.82709i 0 3.16330 + 3.22879i
19.4 0.737204 − 1.20687i 0 −0.913059 − 1.77942i −0.218120 + 0.218120i 0 3.40856 + 0.913320i −2.82063 − 0.209852i 0 0.102443 + 0.424042i
19.5 1.15700 + 0.813242i 0 0.677275 + 1.88183i −1.41026 + 1.41026i 0 0.844432 + 0.226265i −0.746782 + 2.72806i 0 −2.77854 + 0.484781i
19.6 1.41136 − 0.0898367i 0 1.98386 − 0.253583i 1.54484 − 1.54484i 0 −1.07009 − 0.286731i 2.77715 − 0.536120i 0 2.04154 − 2.31911i
163.1 −1.41092 − 0.0965326i 0 1.98136 + 0.272399i 1.91901 + 1.91901i 0 0.104554 − 0.390202i −2.76924 − 0.575598i 0 −2.52232 − 2.89281i
163.2 −0.806088 − 1.16199i 0 −0.700446 + 1.87333i 0.922964 + 0.922964i 0 −0.757268 + 2.82616i 2.74142 − 0.696159i 0 0.328486 − 1.81647i
163.3 0.101912 − 1.41054i 0 −1.97923 − 0.287501i −2.43250 − 2.43250i 0 0.522652 − 1.95057i −0.607239 + 2.76247i 0 −3.67903 + 3.18323i
163.4 0.400093 + 1.35644i 0 −1.67985 + 1.08540i −1.23779 − 1.23779i 0 1.04473 − 3.89897i −2.14438 − 1.84435i 0 1.18375 − 2.17421i
163.5 1.31582 + 0.518286i 0 1.46276 + 1.36394i −0.713774 − 0.713774i 0 −0.872392 + 3.25581i 1.21781 + 2.55283i 0 −0.569258 − 1.30914i
163.6 1.39918 − 0.205665i 0 1.91540 − 0.575524i 1.90811 + 1.90811i 0 1.18978 − 4.44032i 2.56163 − 1.19919i 0 3.06221 + 2.27735i
271.1 −1.17059 + 0.793543i 0 0.740580 − 1.85783i −2.78626 − 2.78626i 0 −0.826736 + 0.221523i 0.607351 + 2.76245i 0 5.47259 + 1.05056i
271.2 −1.12047 − 0.862866i 0 0.510925 + 1.93364i −0.756294 − 0.756294i 0 −0.936058 + 0.250816i 1.09599 − 2.60745i 0 0.194828 + 1.49999i
271.3 −0.0144886 − 1.41414i 0 −1.99958 + 0.0409778i 2.26006 + 2.26006i 0 −3.65215 + 0.978592i 0.0869195 + 2.82709i 0 3.16330 − 3.22879i
271.4 0.737204 + 1.20687i 0 −0.913059 + 1.77942i −0.218120 − 0.218120i 0 3.40856 − 0.913320i −2.82063 + 0.209852i 0 0.102443 − 0.424042i
271.5 1.15700 − 0.813242i 0 0.677275 − 1.88183i −1.41026 − 1.41026i 0 0.844432 − 0.226265i −0.746782 − 2.72806i 0 −2.77854 − 0.484781i
271.6 1.41136 + 0.0898367i 0 1.98386 + 0.253583i 1.54484 + 1.54484i 0 −1.07009 + 0.286731i 2.77715 + 0.536120i 0 2.04154 + 2.31911i
379.1 −1.41092 + 0.0965326i 0 1.98136 − 0.272399i 1.91901 − 1.91901i 0 0.104554 + 0.390202i −2.76924 + 0.575598i 0 −2.52232 + 2.89281i
379.2 −0.806088 + 1.16199i 0 −0.700446 − 1.87333i 0.922964 − 0.922964i 0 −0.757268 − 2.82616i 2.74142 + 0.696159i 0 0.328486 + 1.81647i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.cb.h 24
3.b odd 2 1 156.2.w.c ✓ 24
4.b odd 2 1 468.2.cb.g 24
12.b even 2 1 156.2.w.d yes 24
13.f odd 12 1 468.2.cb.g 24
39.k even 12 1 156.2.w.d yes 24
52.l even 12 1 inner 468.2.cb.h 24
156.v odd 12 1 156.2.w.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.w.c ✓ 24 3.b odd 2 1
156.2.w.c ✓ 24 156.v odd 12 1
156.2.w.d yes 24 12.b even 2 1
156.2.w.d yes 24 39.k even 12 1
468.2.cb.g 24 4.b odd 2 1
468.2.cb.g 24 13.f odd 12 1
468.2.cb.h 24 1.a even 1 1 trivial
468.2.cb.h 24 52.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\):

\( T_{5}^{24} + 2 T_{5}^{23} + 2 T_{5}^{22} - 14 T_{5}^{21} + 293 T_{5}^{20} + 388 T_{5}^{19} + \cdots + 1106704 \) Copy content Toggle raw display
\( T_{7}^{24} + 2 T_{7}^{23} + 29 T_{7}^{22} + 112 T_{7}^{21} + 209 T_{7}^{20} + 816 T_{7}^{19} + \cdots + 2560000 \) Copy content Toggle raw display
\( T_{17}^{24} - 81 T_{17}^{22} + 4243 T_{17}^{20} + 2304 T_{17}^{19} - 129234 T_{17}^{18} + \cdots + 2247518464 \) Copy content Toggle raw display