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Properties

Label 165.3.h.a
Level $165$
Weight $3$
Character orbit 165.h
Analytic conductor $4.496$
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] = [165,3,Mod(109,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.109");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 165.h (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: \(4.49592436194\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4} + 4 q^{5} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4} + 4 q^{5} - 72 q^{9} - 28 q^{11} - 24 q^{14} - 12 q^{15} + 152 q^{16} + 12 q^{20} - 80 q^{26} - 96 q^{31} + 104 q^{34} - 144 q^{36} - 92 q^{44} - 12 q^{45} - 264 q^{49} + 64 q^{55} + 280 q^{56} + 296 q^{59} - 12 q^{60} - 112 q^{64} - 36 q^{66} + 192 q^{69} - 208 q^{70} + 608 q^{71} + 312 q^{75} + 4 q^{80} + 216 q^{81} - 800 q^{86} - 272 q^{89} - 736 q^{91} + 84 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −3.63542 − 1.73205i 9.21627 2.02329 + 4.57234i 6.29673i 4.25629 −18.9633 −3.00000 −7.35550 − 16.6224i
109.2 −3.63542 1.73205i 9.21627 2.02329 − 4.57234i − 6.29673i 4.25629 −18.9633 −3.00000 −7.35550 + 16.6224i
109.3 −3.46254 − 1.73205i 7.98919 −3.32476 − 3.73443i 5.99730i −5.79297 −13.8127 −3.00000 11.5121 + 12.9306i
109.4 −3.46254 1.73205i 7.98919 −3.32476 + 3.73443i − 5.99730i −5.79297 −13.8127 −3.00000 11.5121 − 12.9306i
109.5 −2.63268 − 1.73205i 2.93098 4.42289 − 2.33196i 4.55993i −0.961295 2.81437 −3.00000 −11.6440 + 6.13929i
109.6 −2.63268 1.73205i 2.93098 4.42289 + 2.33196i − 4.55993i −0.961295 2.81437 −3.00000 −11.6440 − 6.13929i
109.7 −1.49337 − 1.73205i −1.76983 −1.35352 − 4.81331i 2.58660i 8.34829 8.61652 −3.00000 2.02132 + 7.18807i
109.8 −1.49337 1.73205i −1.76983 −1.35352 + 4.81331i − 2.58660i 8.34829 8.61652 −3.00000 2.02132 − 7.18807i
109.9 −1.26695 − 1.73205i −2.39485 −4.75221 + 1.55451i 2.19442i 1.85326 8.10193 −3.00000 6.02080 − 1.96948i
109.10 −1.26695 1.73205i −2.39485 −4.75221 − 1.55451i − 2.19442i 1.85326 8.10193 −3.00000 6.02080 + 1.96948i
109.11 −0.168030 − 1.73205i −3.97177 3.98431 + 3.02080i 0.291037i −10.1130 1.33950 −3.00000 −0.669486 − 0.507586i
109.12 −0.168030 1.73205i −3.97177 3.98431 − 3.02080i − 0.291037i −10.1130 1.33950 −3.00000 −0.669486 + 0.507586i
109.13 0.168030 − 1.73205i −3.97177 3.98431 + 3.02080i − 0.291037i 10.1130 −1.33950 −3.00000 0.669486 + 0.507586i
109.14 0.168030 1.73205i −3.97177 3.98431 − 3.02080i 0.291037i 10.1130 −1.33950 −3.00000 0.669486 − 0.507586i
109.15 1.26695 − 1.73205i −2.39485 −4.75221 + 1.55451i − 2.19442i −1.85326 −8.10193 −3.00000 −6.02080 + 1.96948i
109.16 1.26695 1.73205i −2.39485 −4.75221 − 1.55451i 2.19442i −1.85326 −8.10193 −3.00000 −6.02080 − 1.96948i
109.17 1.49337 − 1.73205i −1.76983 −1.35352 − 4.81331i − 2.58660i −8.34829 −8.61652 −3.00000 −2.02132 − 7.18807i
109.18 1.49337 1.73205i −1.76983 −1.35352 + 4.81331i 2.58660i −8.34829 −8.61652 −3.00000 −2.02132 + 7.18807i
109.19 2.63268 − 1.73205i 2.93098 4.42289 − 2.33196i − 4.55993i 0.961295 −2.81437 −3.00000 11.6440 − 6.13929i
109.20 2.63268 1.73205i 2.93098 4.42289 + 2.33196i 4.55993i 0.961295 −2.81437 −3.00000 11.6440 + 6.13929i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.3.h.a ✓ 24
3.b odd 2 1 495.3.h.h 24
5.b even 2 1 inner 165.3.h.a ✓ 24
5.c odd 4 2 825.3.b.e 24
11.b odd 2 1 inner 165.3.h.a ✓ 24
15.d odd 2 1 495.3.h.h 24
33.d even 2 1 495.3.h.h 24
55.d odd 2 1 inner 165.3.h.a ✓ 24
55.e even 4 2 825.3.b.e 24
165.d even 2 1 495.3.h.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.3.h.a ✓ 24 1.a even 1 1 trivial
165.3.h.a ✓ 24 5.b even 2 1 inner
165.3.h.a ✓ 24 11.b odd 2 1 inner
165.3.h.a ✓ 24 55.d odd 2 1 inner
495.3.h.h 24 3.b odd 2 1
495.3.h.h 24 15.d odd 2 1
495.3.h.h 24 33.d even 2 1
495.3.h.h 24 165.d even 2 1
825.3.b.e 24 5.c odd 4 2
825.3.b.e 24 55.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(165, [\chi])\).