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Properties

Label 855.2.bs.g
Level $855$
Weight $2$
Character orbit 855.bs
Analytic conductor $6.827$
Analytic rank $0$
Dimension $42$
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] = [855,2,Mod(226,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.bs (of order \(9\), degree \(6\), 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: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 3 q^{2} + 3 q^{4} + 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 3 q^{2} + 3 q^{4} + 21 q^{8} + 3 q^{10} + 9 q^{13} - 18 q^{14} - 9 q^{16} + 24 q^{17} + 3 q^{19} - 48 q^{20} - 12 q^{22} - 6 q^{23} - 24 q^{26} + 6 q^{28} - 24 q^{29} + 12 q^{31} - 15 q^{32} + 75 q^{34} + 6 q^{35} + 48 q^{37} + 63 q^{38} + 6 q^{40} - 12 q^{41} + 9 q^{43} + 36 q^{44} + 30 q^{46} + 30 q^{47} - 15 q^{49} + 6 q^{50} - 42 q^{52} - 72 q^{53} - 72 q^{56} - 24 q^{58} - 36 q^{59} - 54 q^{61} + 54 q^{62} - 81 q^{64} + 12 q^{65} - 9 q^{67} + 27 q^{68} - 36 q^{70} + 9 q^{73} + 48 q^{74} - 24 q^{76} + 12 q^{77} - 9 q^{80} - 18 q^{82} - 24 q^{85} - 96 q^{86} - 60 q^{88} - 24 q^{89} + 30 q^{91} - 6 q^{92} + 6 q^{95} - 78 q^{97} - 39 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1 −0.475059 + 2.69419i 0 −5.15361 − 1.87576i 0.939693 − 0.342020i 0 2.17622 + 3.76932i 4.76617 − 8.25525i 0 0.475059 + 2.69419i
226.2 −0.391279 + 2.21906i 0 −2.89173 − 1.05250i 0.939693 − 0.342020i 0 −1.46356 − 2.53496i 1.21374 − 2.10227i 0 0.391279 + 2.21906i
226.3 −0.221137 + 1.25413i 0 0.355446 + 0.129372i 0.939693 − 0.342020i 0 −0.739830 − 1.28142i −1.51433 + 2.62289i 0 0.221137 + 1.25413i
226.4 −0.180825 + 1.02551i 0 0.860411 + 0.313164i 0.939693 − 0.342020i 0 1.07128 + 1.85551i −1.51807 + 2.62937i 0 0.180825 + 1.02551i
226.5 0.0565306 − 0.320601i 0 1.77980 + 0.647793i 0.939693 − 0.342020i 0 −1.55579 − 2.69471i 0.633843 − 1.09785i 0 −0.0565306 − 0.320601i
226.6 0.202653 − 1.14930i 0 0.599562 + 0.218223i 0.939693 − 0.342020i 0 2.26060 + 3.91547i 1.53934 − 2.66621i 0 −0.202653 − 1.14930i
226.7 0.335469 − 1.90254i 0 −1.62773 − 0.592445i 0.939693 − 0.342020i 0 −0.982870 − 1.70238i 0.258685 − 0.448056i 0 −0.335469 − 1.90254i
271.1 −2.36807 + 0.861905i 0 3.33276 − 2.79652i −0.766044 − 0.642788i 0 1.61695 + 2.80064i −2.96183 + 5.13003i 0 2.36807 + 0.861905i
271.2 −1.39519 + 0.507808i 0 0.156599 − 0.131402i −0.766044 − 0.642788i 0 0.218101 + 0.377763i 1.33297 − 2.30878i 0 1.39519 + 0.507808i
271.3 −1.15079 + 0.418853i 0 −0.383212 + 0.321553i −0.766044 − 0.642788i 0 −2.28518 − 3.95806i 1.53096 − 2.65169i 0 1.15079 + 0.418853i
271.4 0.324158 − 0.117984i 0 −1.44093 + 1.20908i −0.766044 − 0.642788i 0 −0.760300 − 1.31688i −0.669398 + 1.15943i 0 −0.324158 − 0.117984i
271.5 0.683015 − 0.248597i 0 −1.12738 + 0.945984i −0.766044 − 0.642788i 0 0.631559 + 1.09389i −1.26170 + 2.18533i 0 −0.683015 − 0.248597i
271.6 1.83913 − 0.669389i 0 1.40223 − 1.17661i −0.766044 − 0.642788i 0 2.19746 + 3.80612i −0.165885 + 0.287321i 0 −1.83913 − 0.669389i
271.7 2.50743 − 0.912631i 0 3.92224 − 3.29115i −0.766044 − 0.642788i 0 −1.44494 − 2.50271i 4.16279 − 7.21016i 0 −2.50743 − 0.912631i
541.1 −0.475059 − 2.69419i 0 −5.15361 + 1.87576i 0.939693 + 0.342020i 0 2.17622 − 3.76932i 4.76617 + 8.25525i 0 0.475059 − 2.69419i
541.2 −0.391279 − 2.21906i 0 −2.89173 + 1.05250i 0.939693 + 0.342020i 0 −1.46356 + 2.53496i 1.21374 + 2.10227i 0 0.391279 − 2.21906i
541.3 −0.221137 − 1.25413i 0 0.355446 − 0.129372i 0.939693 + 0.342020i 0 −0.739830 + 1.28142i −1.51433 − 2.62289i 0 0.221137 − 1.25413i
541.4 −0.180825 − 1.02551i 0 0.860411 − 0.313164i 0.939693 + 0.342020i 0 1.07128 − 1.85551i −1.51807 − 2.62937i 0 0.180825 − 1.02551i
541.5 0.0565306 + 0.320601i 0 1.77980 − 0.647793i 0.939693 + 0.342020i 0 −1.55579 + 2.69471i 0.633843 + 1.09785i 0 −0.0565306 + 0.320601i
541.6 0.202653 + 1.14930i 0 0.599562 − 0.218223i 0.939693 + 0.342020i 0 2.26060 − 3.91547i 1.53934 + 2.66621i 0 −0.202653 + 1.14930i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.bs.g ✓ 42
3.b odd 2 1 855.2.bs.h yes 42
19.e even 9 1 inner 855.2.bs.g ✓ 42
57.l odd 18 1 855.2.bs.h yes 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.bs.g ✓ 42 1.a even 1 1 trivial
855.2.bs.g ✓ 42 19.e even 9 1 inner
855.2.bs.h yes 42 3.b odd 2 1
855.2.bs.h yes 42 57.l odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} + 3 T_{2}^{41} + 3 T_{2}^{40} - 15 T_{2}^{39} - 36 T_{2}^{38} + 12 T_{2}^{37} + 568 T_{2}^{36} + \cdots + 46656 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display