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Properties

Label 468.3.f.b
Level $468$
Weight $3$
Character orbit 468.f
Analytic conductor $12.752$
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] = [468,3,Mod(235,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.235");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 468.f (of order \(2\), degree \(1\), 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: \(12.7520763721\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} + 8 q^{4} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} + 8 q^{4} + 32 q^{8} - 12 q^{10} - 32 q^{14} + 4 q^{16} - 84 q^{20} + 28 q^{22} + 104 q^{25} - 96 q^{28} - 64 q^{29} - 44 q^{32} + 40 q^{34} - 192 q^{37} + 104 q^{38} + 220 q^{40} + 220 q^{44} - 104 q^{46} - 248 q^{49} - 100 q^{50} - 52 q^{52} - 336 q^{53} - 168 q^{56} - 16 q^{58} + 16 q^{61} - 152 q^{62} - 16 q^{64} - 400 q^{68} + 208 q^{70} + 112 q^{73} + 104 q^{74} - 264 q^{76} + 272 q^{77} + 300 q^{80} - 4 q^{82} + 64 q^{85} - 288 q^{86} - 492 q^{88} - 328 q^{92} - 884 q^{94} - 80 q^{97} + 572 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1 −1.96946 − 0.348182i 0 3.75754 + 1.37146i 1.51686 0 10.1057i −6.92280 − 4.00934i 0 −2.98739 − 0.528144i
235.2 −1.96946 + 0.348182i 0 3.75754 − 1.37146i 1.51686 0 − 10.1057i −6.92280 + 4.00934i 0 −2.98739 + 0.528144i
235.3 −1.88011 − 0.682039i 0 3.06965 + 2.56462i −5.46052 0 − 10.4764i −4.02211 − 6.91539i 0 10.2664 + 3.72428i
235.4 −1.88011 + 0.682039i 0 3.06965 − 2.56462i −5.46052 0 10.4764i −4.02211 + 6.91539i 0 10.2664 − 3.72428i
235.5 −1.69411 − 1.06301i 0 1.74001 + 3.60171i −5.51671 0 − 1.01261i 0.880886 − 7.95135i 0 9.34592 + 5.86433i
235.6 −1.69411 + 1.06301i 0 1.74001 − 3.60171i −5.51671 0 1.01261i 0.880886 + 7.95135i 0 9.34592 − 5.86433i
235.7 −1.41412 − 1.41431i 0 −0.000546494 4.00000i 9.54425 0 − 4.66845i 5.65801 − 5.65569i 0 −13.4967 − 13.4985i
235.8 −1.41412 + 1.41431i 0 −0.000546494 − 4.00000i 9.54425 0 4.66845i 5.65801 + 5.65569i 0 −13.4967 + 13.4985i
235.9 −0.969763 − 1.74916i 0 −2.11912 + 3.39254i 4.25416 0 10.6496i 7.98914 + 0.416717i 0 −4.12553 − 7.44120i
235.10 −0.969763 + 1.74916i 0 −2.11912 − 3.39254i 4.25416 0 − 10.6496i 7.98914 − 0.416717i 0 −4.12553 + 7.44120i
235.11 −0.805717 − 1.83052i 0 −2.70164 + 2.94977i −5.48615 0 6.35783i 7.57638 + 2.56874i 0 4.42028 + 10.0425i
235.12 −0.805717 + 1.83052i 0 −2.70164 − 2.94977i −5.48615 0 − 6.35783i 7.57638 − 2.56874i 0 4.42028 − 10.0425i
235.13 −0.337137 − 1.97138i 0 −3.77268 + 1.32925i 3.28562 0 − 11.8734i 3.89237 + 6.98924i 0 −1.10771 − 6.47720i
235.14 −0.337137 + 1.97138i 0 −3.77268 − 1.32925i 3.28562 0 11.8734i 3.89237 − 6.98924i 0 −1.10771 + 6.47720i
235.15 0.253155 − 1.98391i 0 −3.87182 − 1.00448i 5.54446 0 3.35464i −2.97297 + 7.42708i 0 1.40361 − 10.9997i
235.16 0.253155 + 1.98391i 0 −3.87182 + 1.00448i 5.54446 0 − 3.35464i −2.97297 − 7.42708i 0 1.40361 + 10.9997i
235.17 1.25145 − 1.56009i 0 −0.867741 − 3.90474i −6.81265 0 − 11.2335i −7.17767 − 3.53285i 0 −8.52570 + 10.6283i
235.18 1.25145 + 1.56009i 0 −0.867741 + 3.90474i −6.81265 0 11.2335i −7.17767 + 3.53285i 0 −8.52570 − 10.6283i
235.19 1.66943 − 1.10137i 0 1.57397 − 3.67731i 0.251667 0 2.33352i −1.42246 − 7.87252i 0 0.420139 − 0.277178i
235.20 1.66943 + 1.10137i 0 1.57397 + 3.67731i 0.251667 0 − 2.33352i −1.42246 + 7.87252i 0 0.420139 + 0.277178i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.3.f.b 24
3.b odd 2 1 156.3.f.a ✓ 24
4.b odd 2 1 inner 468.3.f.b 24
12.b even 2 1 156.3.f.a ✓ 24
24.f even 2 1 2496.3.k.e 24
24.h odd 2 1 2496.3.k.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.f.a ✓ 24 3.b odd 2 1
156.3.f.a ✓ 24 12.b even 2 1
468.3.f.b 24 1.a even 1 1 trivial
468.3.f.b 24 4.b odd 2 1 inner
2496.3.k.e 24 24.f even 2 1
2496.3.k.e 24 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 176 T_{5}^{10} - 80 T_{5}^{9} + 11344 T_{5}^{8} + 3840 T_{5}^{7} - 349632 T_{5}^{6} + \cdots - 9678848 \) acting on \(S_{3}^{\mathrm{new}}(468, [\chi])\). Copy content Toggle raw display