Properties

Label 180.2.q.a
Level $180$
Weight $2$
Character orbit 180.q
Analytic conductor $1.437$
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] = [180,2,Mod(11,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.q (of order \(6\), degree \(2\), 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: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{6} + 4 q^{9} - 22 q^{12} - 30 q^{14} - 16 q^{18} - 4 q^{21} - 28 q^{24} + 24 q^{25} - 12 q^{29} + 16 q^{30} - 44 q^{33} - 6 q^{34} + 42 q^{36} + 60 q^{38} - 6 q^{40} - 60 q^{41} - 18 q^{42} - 8 q^{45} - 12 q^{46} - 12 q^{48} + 24 q^{49} - 18 q^{52} - 32 q^{54} - 42 q^{56} - 12 q^{57} - 18 q^{58} + 14 q^{60} - 48 q^{64} + 16 q^{66} + 48 q^{68} + 36 q^{69} + 60 q^{72} - 24 q^{73} + 84 q^{74} + 6 q^{76} + 48 q^{77} + 38 q^{78} - 36 q^{82} + 50 q^{84} - 54 q^{86} - 18 q^{90} + 60 q^{92} - 32 q^{93} + 18 q^{94} - 18 q^{96} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.39556 0.228922i −1.60382 0.654027i 1.89519 + 0.638951i −0.866025 + 0.500000i 2.08851 + 1.27989i −1.04714 0.604565i −2.49858 1.32555i 2.14450 + 2.09789i 1.32305 0.499529i
11.2 −1.37234 0.341569i 1.63013 0.585384i 1.76666 + 0.937501i 0.866025 0.500000i −2.43705 + 0.246546i 1.44910 + 0.836639i −2.10425 1.89001i 2.31465 1.90850i −1.35927 + 0.390365i
11.3 −1.34552 + 0.435401i 0.531460 1.64850i 1.62085 1.17168i −0.866025 + 0.500000i 0.00266887 + 2.44949i 3.61144 + 2.08506i −1.67074 + 2.28224i −2.43510 1.75222i 0.947554 1.04983i
11.4 −1.29101 + 0.577305i −1.30449 + 1.13944i 1.33344 1.49062i 0.866025 0.500000i 1.02631 2.22411i 2.51585 + 1.45253i −0.860949 + 2.69421i 0.403372 2.97276i −0.829399 + 1.14547i
11.5 −1.18360 + 0.774003i −0.427686 1.67842i 0.801838 1.83223i 0.866025 0.500000i 1.80531 + 1.65555i −4.06955 2.34956i 0.469091 + 2.78926i −2.63417 + 1.43567i −0.638030 + 1.26211i
11.6 −0.981980 1.01770i −1.63013 + 0.585384i −0.0714305 + 1.99872i 0.866025 0.500000i 2.19650 + 1.08415i −1.44910 0.836639i 2.10425 1.89001i 2.31465 1.90850i −1.35927 0.390365i
11.7 −0.962433 + 1.03621i 0.968388 + 1.43605i −0.147445 1.99456i −0.866025 + 0.500000i −2.42005 0.378648i 1.90150 + 1.09783i 2.20868 + 1.76684i −1.12445 + 2.78130i 0.315389 1.37860i
11.8 −0.896034 1.09413i 1.60382 + 0.654027i −0.394247 + 1.96076i −0.866025 + 0.500000i −0.721488 2.34082i 1.04714 + 0.604565i 2.49858 1.32555i 2.14450 + 2.09789i 1.32305 + 0.499529i
11.9 −0.660227 + 1.25064i −1.67907 + 0.425108i −1.12820 1.65141i −0.866025 + 0.500000i 0.576911 2.38058i −1.87435 1.08216i 2.81019 0.320666i 2.63857 1.42758i −0.0535467 1.41320i
11.10 −0.295692 1.38296i −0.531460 + 1.64850i −1.82513 + 0.817857i −0.866025 + 0.500000i 2.43695 + 0.247538i −3.61144 2.08506i 1.67074 + 2.28224i −2.43510 1.75222i 0.947554 + 1.04983i
11.11 −0.236271 + 1.39434i −1.35535 1.07843i −1.88835 0.658884i 0.866025 0.500000i 1.82393 1.63501i 3.33246 + 1.92400i 1.36487 2.47732i 0.673959 + 2.92332i 0.492552 + 1.32567i
11.12 −0.145547 1.40670i 1.30449 1.13944i −1.95763 + 0.409483i 0.866025 0.500000i −1.79271 1.66919i −2.51585 1.45253i 0.860949 + 2.69421i 0.403372 2.97276i −0.829399 1.14547i
11.13 0.0785043 1.41203i 0.427686 + 1.67842i −1.98767 0.221701i 0.866025 0.500000i 2.40356 0.472143i 4.06955 + 2.34956i −0.469091 + 2.78926i −2.63417 + 1.43567i −0.638030 1.26211i
11.14 0.304404 + 1.38106i 1.45681 + 0.936861i −1.81468 + 0.840803i 0.866025 0.500000i −0.850407 + 2.29713i 0.737635 + 0.425874i −1.71360 2.25024i 1.24458 + 2.72965i 0.954154 + 1.04383i
11.15 0.416164 1.35159i −0.968388 1.43605i −1.65361 1.12497i −0.866025 + 0.500000i −2.34396 + 0.711237i −1.90150 1.09783i −2.20868 + 1.76684i −1.12445 + 2.78130i 0.315389 + 1.37860i
11.16 0.538420 + 1.30771i −0.398623 + 1.68556i −1.42021 + 1.40819i −0.866025 + 0.500000i −2.41884 + 0.386254i 0.891819 + 0.514892i −2.60618 1.09902i −2.68220 1.34380i −1.12014 0.863299i
11.17 0.752973 1.19709i 1.67907 0.425108i −0.866065 1.80276i −0.866025 + 0.500000i 0.755401 2.33010i 1.87435 + 1.08216i −2.81019 0.320666i 2.63857 1.42758i −0.0535467 + 1.41320i
11.18 1.04050 + 0.957789i 0.561951 1.63836i 0.165280 + 1.99316i 0.866025 0.500000i 2.15391 1.16648i 0.811773 + 0.468677i −1.73705 + 2.23218i −2.36842 1.84135i 1.37999 + 0.309220i
11.19 1.08940 0.901786i 1.35535 + 1.07843i 0.373566 1.96480i 0.866025 0.500000i 2.44903 0.0473955i −3.33246 1.92400i −1.36487 2.47732i 0.673959 + 2.92332i 0.492552 1.32567i
11.20 1.18642 + 0.769686i 1.70656 + 0.296046i 0.815168 + 1.82634i −0.866025 + 0.500000i 1.79683 + 1.66475i −3.55496 2.05246i −0.438576 + 2.79422i 2.82471 + 1.01044i −1.41231 0.0733593i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.2.q.a 48
3.b odd 2 1 540.2.q.a 48
4.b odd 2 1 inner 180.2.q.a 48
5.b even 2 1 900.2.r.f 48
5.c odd 4 1 900.2.o.b 48
5.c odd 4 1 900.2.o.c 48
9.c even 3 1 540.2.q.a 48
9.c even 3 1 1620.2.e.b 48
9.d odd 6 1 inner 180.2.q.a 48
9.d odd 6 1 1620.2.e.b 48
12.b even 2 1 540.2.q.a 48
20.d odd 2 1 900.2.r.f 48
20.e even 4 1 900.2.o.b 48
20.e even 4 1 900.2.o.c 48
36.f odd 6 1 540.2.q.a 48
36.f odd 6 1 1620.2.e.b 48
36.h even 6 1 inner 180.2.q.a 48
36.h even 6 1 1620.2.e.b 48
45.h odd 6 1 900.2.r.f 48
45.l even 12 1 900.2.o.b 48
45.l even 12 1 900.2.o.c 48
180.n even 6 1 900.2.r.f 48
180.v odd 12 1 900.2.o.b 48
180.v odd 12 1 900.2.o.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.q.a 48 1.a even 1 1 trivial
180.2.q.a 48 4.b odd 2 1 inner
180.2.q.a 48 9.d odd 6 1 inner
180.2.q.a 48 36.h even 6 1 inner
540.2.q.a 48 3.b odd 2 1
540.2.q.a 48 9.c even 3 1
540.2.q.a 48 12.b even 2 1
540.2.q.a 48 36.f odd 6 1
900.2.o.b 48 5.c odd 4 1
900.2.o.b 48 20.e even 4 1
900.2.o.b 48 45.l even 12 1
900.2.o.b 48 180.v odd 12 1
900.2.o.c 48 5.c odd 4 1
900.2.o.c 48 20.e even 4 1
900.2.o.c 48 45.l even 12 1
900.2.o.c 48 180.v odd 12 1
900.2.r.f 48 5.b even 2 1
900.2.r.f 48 20.d odd 2 1
900.2.r.f 48 45.h odd 6 1
900.2.r.f 48 180.n even 6 1
1620.2.e.b 48 9.c even 3 1
1620.2.e.b 48 9.d odd 6 1
1620.2.e.b 48 36.f odd 6 1
1620.2.e.b 48 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(180, [\chi])\).