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Properties

Label 165.2.p.b
Level $165$
Weight $2$
Character orbit 165.p
Analytic conductor $1.318$
Analytic rank $0$
Dimension $48$
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] = [165,2,Mod(41,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.p (of order \(10\), 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: \(1.31753163335\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{3} - 4 q^{4} - 20 q^{6} + 10 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{3} - 4 q^{4} - 20 q^{6} + 10 q^{7} + 2 q^{9} + 8 q^{12} + 6 q^{15} + 32 q^{16} - 30 q^{18} - 100 q^{19} - 82 q^{22} + 100 q^{24} + 12 q^{25} + 14 q^{27} + 30 q^{28} + 10 q^{30} + 10 q^{31} - 46 q^{33} - 28 q^{34} + 14 q^{36} + 6 q^{37} - 50 q^{40} - 52 q^{42} + 32 q^{45} + 20 q^{46} - 80 q^{48} - 26 q^{49} - 30 q^{51} + 40 q^{52} + 6 q^{55} - 70 q^{57} + 92 q^{58} + 44 q^{60} + 70 q^{61} - 20 q^{63} + 18 q^{64} + 76 q^{66} + 20 q^{67} + 42 q^{69} - 4 q^{70} - 80 q^{72} + 90 q^{73} - 6 q^{75} - 108 q^{78} - 100 q^{79} + 38 q^{81} - 34 q^{82} + 70 q^{84} + 20 q^{85} + 74 q^{88} + 20 q^{90} - 86 q^{91} + 76 q^{93} + 10 q^{94} - 30 q^{96} + 6 q^{97} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −2.09233 + 1.52016i 1.72815 − 0.116121i 1.44890 − 4.45925i −0.587785 + 0.809017i −3.43934 + 2.87004i 0.326781 + 0.106178i 2.14883 + 6.61341i 2.97303 − 0.401348i − 2.58626i
41.2 −1.54632 + 1.12347i −0.455760 − 1.67101i 0.510897 − 1.57238i −0.587785 + 0.809017i 2.58208 + 2.07189i −0.0795268 − 0.0258398i −0.204777 − 0.630238i −2.58457 + 1.52316i − 1.91136i
41.3 −1.45632 + 1.05808i −1.62444 − 0.600999i 0.383300 − 1.17968i 0.587785 − 0.809017i 3.00160 − 0.843534i −0.145486 − 0.0472713i −0.422546 − 1.30046i 2.27760 + 1.95257i 1.80011i
41.4 −1.37228 + 0.997022i 1.42851 + 0.979464i 0.271073 − 0.834277i 0.587785 − 0.809017i −2.93687 + 0.0801565i 3.82974 + 1.24436i −0.588527 − 1.81130i 1.08130 + 2.79836i 1.69623i
41.5 −0.860692 + 0.625329i −1.27571 + 1.17157i −0.268280 + 0.825682i −0.587785 + 0.809017i 0.365379 − 1.80609i −2.18340 − 0.709430i −0.942926 − 2.90203i 0.254870 − 2.98915i − 1.06387i
41.6 −0.131900 + 0.0958309i 0.642850 − 1.60834i −0.609820 + 1.87683i 0.587785 − 0.809017i 0.0693364 + 0.273744i 3.41501 + 1.10960i −0.200186 − 0.616109i −2.17349 − 2.06784i 0.163037i
41.7 0.131900 − 0.0958309i −1.46543 − 0.923314i −0.609820 + 1.87683i −0.587785 + 0.809017i −0.281772 + 0.0186487i 3.41501 + 1.10960i 0.200186 + 0.616109i 1.29498 + 2.70611i 0.163037i
41.8 0.860692 − 0.625329i 1.72070 + 0.197973i −0.268280 + 0.825682i 0.587785 − 0.809017i 1.60479 − 0.905610i −2.18340 − 0.709430i 0.942926 + 2.90203i 2.92161 + 0.681304i − 1.06387i
41.9 1.37228 − 0.997022i −0.579977 + 1.63206i 0.271073 − 0.834277i −0.587785 + 0.809017i 0.831309 + 2.81790i 3.82974 + 1.24436i 0.588527 + 1.81130i −2.32725 − 1.89312i 1.69623i
41.10 1.45632 − 1.05808i 0.960940 − 1.44104i 0.383300 − 1.17968i −0.587785 + 0.809017i −0.125297 − 3.11536i −0.145486 − 0.0472713i 0.422546 + 1.30046i −1.15319 − 2.76950i 1.80011i
41.11 1.54632 − 1.12347i −0.613479 − 1.61977i 0.510897 − 1.57238i 0.587785 − 0.809017i −2.76839 − 1.81546i −0.0795268 − 0.0258398i 0.204777 + 0.630238i −2.24729 + 1.98739i − 1.91136i
41.12 2.09233 − 1.52016i −1.46636 + 0.921840i 1.44890 − 4.45925i 0.587785 − 0.809017i −1.66676 + 4.15790i 0.326781 + 0.106178i −2.14883 − 6.61341i 1.30042 − 2.70350i − 2.58626i
101.1 −0.848632 + 2.61182i −0.139896 + 1.72639i −4.48340 − 3.25738i −0.951057 + 0.309017i −4.39031 − 1.83046i −0.0372155 + 0.0512227i 7.86897 − 5.71714i −2.96086 − 0.483031i − 2.74623i
101.2 −0.664086 + 2.04385i 1.72041 + 0.200478i −2.11826 − 1.53901i 0.951057 − 0.309017i −1.55225 + 3.38312i −1.16392 + 1.60200i 1.07501 − 0.781038i 2.91962 + 0.689809i 2.14903i
101.3 −0.491121 + 1.51151i −1.07720 − 1.35634i −0.425442 − 0.309102i −0.951057 + 0.309017i 2.57916 − 0.962071i −1.78891 + 2.46222i −1.89539 + 1.37708i −0.679300 + 2.92208i − 1.58930i
101.4 −0.212587 + 0.654275i 1.00646 − 1.40962i 1.23515 + 0.897390i 0.951057 − 0.309017i 0.708322 + 0.958169i −0.891460 + 1.22699i −1.96284 + 1.42608i −0.974079 − 2.83746i 0.687946i
101.5 −0.140144 + 0.431318i 0.180419 + 1.72263i 1.45164 + 1.05468i 0.951057 − 0.309017i −0.768285 − 0.163597i 2.72178 − 3.74621i −1.39214 + 1.01145i −2.93490 + 0.621591i 0.453514i
101.6 −0.0403345 + 0.124137i −0.644015 + 1.60787i 1.60425 + 1.16556i −0.951057 + 0.309017i −0.173620 − 0.144799i −1.50339 + 2.06924i −0.420589 + 0.305576i −2.17049 − 2.07099i − 0.130525i
101.7 0.0403345 − 0.124137i −1.72819 + 0.115636i 1.60425 + 1.16556i 0.951057 − 0.309017i −0.0553509 + 0.219196i −1.50339 + 2.06924i 0.420589 − 0.305576i 2.97326 − 0.399681i − 0.130525i
101.8 0.140144 − 0.431318i −1.58256 − 0.703911i 1.45164 + 1.05468i −0.951057 + 0.309017i −0.525396 + 0.583940i 2.72178 − 3.74621i 1.39214 − 1.01145i 2.00902 + 2.22797i 0.453514i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.p.b ✓ 48
3.b odd 2 1 inner 165.2.p.b ✓ 48
5.b even 2 1 825.2.bi.e 48
5.c odd 4 1 825.2.bs.g 48
5.c odd 4 1 825.2.bs.h 48
11.d odd 10 1 inner 165.2.p.b ✓ 48
15.d odd 2 1 825.2.bi.e 48
15.e even 4 1 825.2.bs.g 48
15.e even 4 1 825.2.bs.h 48
33.f even 10 1 inner 165.2.p.b ✓ 48
55.h odd 10 1 825.2.bi.e 48
55.l even 20 1 825.2.bs.g 48
55.l even 20 1 825.2.bs.h 48
165.r even 10 1 825.2.bi.e 48
165.u odd 20 1 825.2.bs.g 48
165.u odd 20 1 825.2.bs.h 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.p.b ✓ 48 1.a even 1 1 trivial
165.2.p.b ✓ 48 3.b odd 2 1 inner
165.2.p.b ✓ 48 11.d odd 10 1 inner
165.2.p.b ✓ 48 33.f even 10 1 inner
825.2.bi.e 48 5.b even 2 1
825.2.bi.e 48 15.d odd 2 1
825.2.bi.e 48 55.h odd 10 1
825.2.bi.e 48 165.r even 10 1
825.2.bs.g 48 5.c odd 4 1
825.2.bs.g 48 15.e even 4 1
825.2.bs.g 48 55.l even 20 1
825.2.bs.g 48 165.u odd 20 1
825.2.bs.h 48 5.c odd 4 1
825.2.bs.h 48 15.e even 4 1
825.2.bs.h 48 55.l even 20 1
825.2.bs.h 48 165.u odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 14 T_{2}^{46} + 135 T_{2}^{44} + 1083 T_{2}^{42} + 8563 T_{2}^{40} + 45065 T_{2}^{38} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(165, [\chi])\). Copy content Toggle raw display