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Properties

Label 165.6.f.a
Level $165$
Weight $6$
Character orbit 165.f
Analytic conductor $26.463$
Analytic rank $0$
Dimension $80$
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] = [165,6,Mod(131,165)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.131");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(26.4633302691\)
Analytic rank: \(0\)
Dimension: \(80\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 44 q^{3} + 1280 q^{4} - 352 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 44 q^{3} + 1280 q^{4} - 352 q^{9} + 2112 q^{12} + 1100 q^{15} + 13792 q^{16} - 9892 q^{22} - 50000 q^{25} - 10780 q^{27} + 2112 q^{31} + 6316 q^{33} + 69560 q^{34} - 17268 q^{36} - 7456 q^{37} - 100712 q^{42} + 61352 q^{48} - 233408 q^{49} - 15800 q^{55} - 93728 q^{58} + 62700 q^{60} + 212400 q^{64} + 203724 q^{66} + 182072 q^{67} - 122584 q^{69} + 6600 q^{70} - 27500 q^{75} - 489128 q^{78} + 194872 q^{81} - 237544 q^{82} - 641716 q^{88} + 168272 q^{91} + 433336 q^{93} + 949008 q^{97} + 328952 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1 −10.9622 −2.76318 − 15.3416i 88.1691 25.0000i 30.2904 + 168.177i 90.0333i −615.734 −227.730 + 84.7831i − 274.054i
131.2 −10.9622 −2.76318 + 15.3416i 88.1691 − 25.0000i 30.2904 − 168.177i − 90.0333i −615.734 −227.730 − 84.7831i 274.054i
131.3 −10.5656 14.6766 + 5.25334i 79.6325 − 25.0000i −155.067 − 55.5049i 67.9017i −503.268 187.805 + 154.202i 264.141i
131.4 −10.5656 14.6766 − 5.25334i 79.6325 25.0000i −155.067 + 55.5049i − 67.9017i −503.268 187.805 − 154.202i − 264.141i
131.5 −9.99646 −15.1349 + 3.73284i 67.9292 25.0000i 151.296 − 37.3152i − 75.9052i −359.165 215.132 − 112.993i − 249.911i
131.6 −9.99646 −15.1349 − 3.73284i 67.9292 − 25.0000i 151.296 + 37.3152i 75.9052i −359.165 215.132 + 112.993i 249.911i
131.7 −9.78187 5.46056 − 14.6008i 63.6850 − 25.0000i −53.4145 + 142.823i 214.687i −309.938 −183.364 − 159.457i 244.547i
131.8 −9.78187 5.46056 + 14.6008i 63.6850 25.0000i −53.4145 − 142.823i − 214.687i −309.938 −183.364 + 159.457i − 244.547i
131.9 −9.43613 12.0254 + 9.91910i 57.0405 25.0000i −113.474 − 93.5979i 156.233i −236.286 46.2229 + 238.563i − 235.903i
131.10 −9.43613 12.0254 − 9.91910i 57.0405 − 25.0000i −113.474 + 93.5979i − 156.233i −236.286 46.2229 − 238.563i 235.903i
131.11 −8.20151 −3.15142 − 15.2666i 35.2648 25.0000i 25.8464 + 125.209i − 179.318i −26.7763 −223.137 + 96.2227i − 205.038i
131.12 −8.20151 −3.15142 + 15.2666i 35.2648 − 25.0000i 25.8464 − 125.209i 179.318i −26.7763 −223.137 − 96.2227i 205.038i
131.13 −7.77066 −12.8111 − 8.88121i 28.3832 25.0000i 99.5507 + 69.0129i 223.165i 28.1048 85.2482 + 227.556i − 194.267i
131.14 −7.77066 −12.8111 + 8.88121i 28.3832 − 25.0000i 99.5507 − 69.0129i − 223.165i 28.1048 85.2482 − 227.556i 194.267i
131.15 −7.54491 −0.606932 + 15.5766i 24.9257 25.0000i 4.57925 − 117.524i 74.1145i 53.3749 −242.263 − 18.9079i − 188.623i
131.16 −7.54491 −0.606932 − 15.5766i 24.9257 − 25.0000i 4.57925 + 117.524i − 74.1145i 53.3749 −242.263 + 18.9079i 188.623i
131.17 −7.04255 −14.9021 + 4.57456i 17.5974 − 25.0000i 104.949 − 32.2165i 17.4527i 101.431 201.147 − 136.341i 176.064i
131.18 −7.04255 −14.9021 − 4.57456i 17.5974 25.0000i 104.949 + 32.2165i − 17.4527i 101.431 201.147 + 136.341i − 176.064i
131.19 −6.89239 14.9703 + 4.34640i 15.5051 − 25.0000i −103.181 − 29.9571i − 243.497i 113.689 205.218 + 130.134i 172.310i
131.20 −6.89239 14.9703 − 4.34640i 15.5051 25.0000i −103.181 + 29.9571i 243.497i 113.689 205.218 − 130.134i − 172.310i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.80
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.6.f.a ✓ 80
3.b odd 2 1 inner 165.6.f.a ✓ 80
11.b odd 2 1 inner 165.6.f.a ✓ 80
33.d even 2 1 inner 165.6.f.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.f.a ✓ 80 1.a even 1 1 trivial
165.6.f.a ✓ 80 3.b odd 2 1 inner
165.6.f.a ✓ 80 11.b odd 2 1 inner
165.6.f.a ✓ 80 33.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(165, [\chi])\).