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Properties

Label 261.2.o.c
Level $261$
Weight $2$
Character orbit 261.o
Analytic conductor $2.084$
Analytic rank $0$
Dimension $36$
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] = [261,2,Mod(64,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([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("261.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.o (of order \(14\), 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: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 12 q^{4} - 14 q^{13} + 4 q^{16} - 8 q^{22} - 40 q^{25} - 128 q^{28} - 28 q^{31} + 28 q^{37} - 42 q^{40} - 14 q^{43} - 14 q^{49} + 116 q^{52} - 14 q^{55} + 112 q^{58} + 98 q^{64} + 56 q^{67} + 126 q^{73} - 14 q^{76} - 14 q^{79} + 70 q^{82} + 14 q^{85} + 156 q^{88} + 66 q^{91} + 24 q^{94} - 42 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −1.91655 − 1.52840i 0 0.892126 + 3.90866i 1.05454 − 1.32235i 0 −0.213302 + 0.934536i 2.13697 − 4.43747i 0 −4.04215 + 0.922593i
64.2 −1.15245 − 0.919048i 0 0.0384498 + 0.168459i −0.925863 + 1.16099i 0 0.747874 − 3.27665i −1.16861 + 2.42665i 0 2.13402 − 0.487076i
64.3 −0.0991871 − 0.0790990i 0 −0.441460 − 1.93416i 2.33067 − 2.92257i 0 −0.702135 + 3.07625i −0.219293 + 0.455366i 0 −0.462345 + 0.105527i
64.4 0.0991871 + 0.0790990i 0 −0.441460 − 1.93416i −2.33067 + 2.92257i 0 −0.702135 + 3.07625i 0.219293 − 0.455366i 0 −0.462345 + 0.105527i
64.5 1.15245 + 0.919048i 0 0.0384498 + 0.168459i 0.925863 − 1.16099i 0 0.747874 − 3.27665i 1.16861 − 2.42665i 0 2.13402 − 0.487076i
64.6 1.91655 + 1.52840i 0 0.892126 + 3.90866i −1.05454 + 1.32235i 0 −0.213302 + 0.934536i −2.13697 + 4.43747i 0 −4.04215 + 0.922593i
91.1 −2.55860 − 0.583983i 0 4.40345 + 2.12059i −0.652317 + 2.85799i 0 −3.93253 + 1.89381i −5.92459 − 4.72470i 0 3.33803 − 6.93150i
91.2 −1.94866 − 0.444770i 0 1.79753 + 0.865647i 0.935274 − 4.09770i 0 1.35429 − 0.652192i 0.00763820 + 0.00609126i 0 −3.64507 + 7.56907i
91.3 −0.588528 − 0.134328i 0 −1.47362 − 0.709656i −0.403208 + 1.76657i 0 0.375329 − 0.180749i 1.71586 + 1.36836i 0 0.474599 − 0.985514i
91.4 0.588528 + 0.134328i 0 −1.47362 − 0.709656i 0.403208 − 1.76657i 0 0.375329 − 0.180749i −1.71586 − 1.36836i 0 0.474599 − 0.985514i
91.5 1.94866 + 0.444770i 0 1.79753 + 0.865647i −0.935274 + 4.09770i 0 1.35429 − 0.652192i −0.00763820 − 0.00609126i 0 −3.64507 + 7.56907i
91.6 2.55860 + 0.583983i 0 4.40345 + 2.12059i 0.652317 − 2.85799i 0 −3.93253 + 1.89381i 5.92459 + 4.72470i 0 3.33803 − 6.93150i
100.1 −1.12008 + 2.32586i 0 −2.90808 − 3.64662i −1.94007 − 0.934287i 0 2.03725 − 2.55464i 6.70523 − 1.53043i 0 4.34604 − 3.46585i
100.2 −0.548846 + 1.13969i 0 0.249318 + 0.312635i 3.34782 + 1.61222i 0 2.25904 − 2.83274i −2.95963 + 0.675517i 0 −3.67487 + 2.93061i
100.3 −0.492918 + 1.02355i 0 0.442285 + 0.554608i −1.55374 − 0.748244i 0 −1.92582 + 2.41490i −3.00083 + 0.684920i 0 1.53174 − 1.22152i
100.4 0.492918 − 1.02355i 0 0.442285 + 0.554608i 1.55374 + 0.748244i 0 −1.92582 + 2.41490i 3.00083 − 0.684920i 0 1.53174 − 1.22152i
100.5 0.548846 − 1.13969i 0 0.249318 + 0.312635i −3.34782 − 1.61222i 0 2.25904 − 2.83274i 2.95963 − 0.675517i 0 −3.67487 + 2.93061i
100.6 1.12008 − 2.32586i 0 −2.90808 − 3.64662i 1.94007 + 0.934287i 0 2.03725 − 2.55464i −6.70523 + 1.53043i 0 4.34604 − 3.46585i
109.1 −2.55860 + 0.583983i 0 4.40345 − 2.12059i −0.652317 − 2.85799i 0 −3.93253 − 1.89381i −5.92459 + 4.72470i 0 3.33803 + 6.93150i
109.2 −1.94866 + 0.444770i 0 1.79753 − 0.865647i 0.935274 + 4.09770i 0 1.35429 + 0.652192i 0.00763820 − 0.00609126i 0 −3.64507 − 7.56907i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.e even 14 1 inner
87.h odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.2.o.c ✓ 36
3.b odd 2 1 inner 261.2.o.c ✓ 36
29.e even 14 1 inner 261.2.o.c ✓ 36
29.f odd 28 2 7569.2.a.bw 36
87.h odd 14 1 inner 261.2.o.c ✓ 36
87.k even 28 2 7569.2.a.bw 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.o.c ✓ 36 1.a even 1 1 trivial
261.2.o.c ✓ 36 3.b odd 2 1 inner
261.2.o.c ✓ 36 29.e even 14 1 inner
261.2.o.c ✓ 36 87.h odd 14 1 inner
7569.2.a.bw 36 29.f odd 28 2
7569.2.a.bw 36 87.k even 28 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 12 T_{2}^{34} + 80 T_{2}^{32} - 529 T_{2}^{30} + 3970 T_{2}^{28} - 25351 T_{2}^{26} + \cdots + 841 \) acting on \(S_{2}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display