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Properties

Label 270.2.k.e
Level $270$
Weight $2$
Character orbit 270.k
Analytic conductor $2.156$
Analytic rank $0$
Dimension $24$
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] = [270,2,Mod(31,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.k (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: \(2.15596085457\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{3} + 3 q^{7} + 12 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{3} + 3 q^{7} + 12 q^{8} + 9 q^{9} - 12 q^{10} + 9 q^{13} - 3 q^{14} + 15 q^{17} - 12 q^{18} - 9 q^{19} + 24 q^{21} - 12 q^{23} + 3 q^{24} - 12 q^{26} - 3 q^{27} + 6 q^{28} - 33 q^{29} + 6 q^{30} - 21 q^{31} + 6 q^{33} - 15 q^{34} + 3 q^{35} - 6 q^{36} - 12 q^{37} + 12 q^{38} - 36 q^{39} + 21 q^{41} + 15 q^{42} + 12 q^{43} + 9 q^{44} - 12 q^{45} - 15 q^{46} - 21 q^{47} + 27 q^{49} + 3 q^{51} + 9 q^{52} - 60 q^{53} + 36 q^{54} + 18 q^{55} - 3 q^{56} - 24 q^{57} - 21 q^{58} + 36 q^{59} + 33 q^{61} + 18 q^{62} + 39 q^{63} - 12 q^{64} - 9 q^{66} - 12 q^{67} - 3 q^{68} - 48 q^{69} - 6 q^{70} + 12 q^{71} + 6 q^{74} - 12 q^{76} - 60 q^{77} - 36 q^{78} - 15 q^{79} - 24 q^{80} - 15 q^{81} + 6 q^{82} - 27 q^{83} + 3 q^{84} - 15 q^{85} - 12 q^{86} + 78 q^{87} - 9 q^{88} + 30 q^{89} - 9 q^{90} - 9 q^{91} + 6 q^{92} - 30 q^{93} + 12 q^{94} - 15 q^{95} + 3 q^{97} + 27 q^{98} - 111 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0.939693 + 0.342020i −1.60992 − 0.638862i 0.766044 + 0.642788i −0.173648 + 0.984808i −1.29433 − 1.15096i −3.84295 + 3.22462i 0.500000 + 0.866025i 2.18371 + 2.05704i −0.500000 + 0.866025i
31.2 0.939693 + 0.342020i −0.718131 − 1.57616i 0.766044 + 0.642788i −0.173648 + 0.984808i −0.135743 − 1.72672i 3.34524 − 2.80699i 0.500000 + 0.866025i −1.96858 + 2.26378i −0.500000 + 0.866025i
31.3 0.939693 + 0.342020i 0.137658 + 1.72657i 0.766044 + 0.642788i −0.173648 + 0.984808i −0.461166 + 1.66953i 1.26626 − 1.06252i 0.500000 + 0.866025i −2.96210 + 0.475354i −0.500000 + 0.866025i
31.4 0.939693 + 0.342020i 1.69040 − 0.377572i 0.766044 + 0.642788i −0.173648 + 0.984808i 1.71759 + 0.223348i −0.268547 + 0.225338i 0.500000 + 0.866025i 2.71488 − 1.27649i −0.500000 + 0.866025i
61.1 0.939693 − 0.342020i −1.60992 + 0.638862i 0.766044 − 0.642788i −0.173648 − 0.984808i −1.29433 + 1.15096i −3.84295 − 3.22462i 0.500000 − 0.866025i 2.18371 − 2.05704i −0.500000 − 0.866025i
61.2 0.939693 − 0.342020i −0.718131 + 1.57616i 0.766044 − 0.642788i −0.173648 − 0.984808i −0.135743 + 1.72672i 3.34524 + 2.80699i 0.500000 − 0.866025i −1.96858 − 2.26378i −0.500000 − 0.866025i
61.3 0.939693 − 0.342020i 0.137658 − 1.72657i 0.766044 − 0.642788i −0.173648 − 0.984808i −0.461166 − 1.66953i 1.26626 + 1.06252i 0.500000 − 0.866025i −2.96210 − 0.475354i −0.500000 − 0.866025i
61.4 0.939693 − 0.342020i 1.69040 + 0.377572i 0.766044 − 0.642788i −0.173648 − 0.984808i 1.71759 − 0.223348i −0.268547 − 0.225338i 0.500000 − 0.866025i 2.71488 + 1.27649i −0.500000 − 0.866025i
121.1 −0.173648 − 0.984808i −1.61088 − 0.636438i −0.939693 + 0.342020i −0.766044 − 0.642788i −0.347042 + 1.69693i −1.33944 − 0.487517i 0.500000 + 0.866025i 2.18989 + 2.05046i −0.500000 + 0.866025i
121.2 −0.173648 − 0.984808i −1.29249 + 1.15303i −0.939693 + 0.342020i −0.766044 − 0.642788i 1.35995 + 1.07263i 0.582091 + 0.211864i 0.500000 + 0.866025i 0.341047 − 2.98055i −0.500000 + 0.866025i
121.3 −0.173648 − 0.984808i 0.683937 − 1.59130i −0.939693 + 0.342020i −0.766044 − 0.642788i −1.68589 − 0.397221i −3.30330 − 1.20230i 0.500000 + 0.866025i −2.06446 − 2.17670i −0.500000 + 0.866025i
121.4 −0.173648 − 0.984808i 1.71943 + 0.208681i −0.939693 + 0.342020i −0.766044 − 0.642788i −0.0930655 − 1.72955i 4.56065 + 1.65994i 0.500000 + 0.866025i 2.91290 + 0.717628i −0.500000 + 0.866025i
151.1 −0.766044 − 0.642788i −1.67972 − 0.422540i 0.173648 + 0.984808i 0.939693 + 0.342020i 1.01514 + 1.40339i 0.192018 − 1.08899i 0.500000 − 0.866025i 2.64292 + 1.41950i −0.500000 − 0.866025i
151.2 −0.766044 − 0.642788i −1.08744 + 1.34814i 0.173648 + 0.984808i 0.939693 + 0.342020i 1.69959 − 0.333742i −0.314431 + 1.78323i 0.500000 − 0.866025i −0.634956 − 2.93204i −0.500000 − 0.866025i
151.3 −0.766044 − 0.642788i 1.09919 − 1.33858i 0.173648 + 0.984808i 0.939693 + 0.342020i −1.70245 + 0.318864i −0.158856 + 0.900919i 0.500000 − 0.866025i −0.583569 − 2.94269i −0.500000 − 0.866025i
151.4 −0.766044 − 0.642788i 1.16797 + 1.27900i 0.173648 + 0.984808i 0.939693 + 0.342020i −0.0725903 − 1.73053i 0.781269 − 4.43080i 0.500000 − 0.866025i −0.271692 + 2.98767i −0.500000 − 0.866025i
211.1 −0.766044 + 0.642788i −1.67972 + 0.422540i 0.173648 − 0.984808i 0.939693 − 0.342020i 1.01514 − 1.40339i 0.192018 + 1.08899i 0.500000 + 0.866025i 2.64292 − 1.41950i −0.500000 + 0.866025i
211.2 −0.766044 + 0.642788i −1.08744 − 1.34814i 0.173648 − 0.984808i 0.939693 − 0.342020i 1.69959 + 0.333742i −0.314431 − 1.78323i 0.500000 + 0.866025i −0.634956 + 2.93204i −0.500000 + 0.866025i
211.3 −0.766044 + 0.642788i 1.09919 + 1.33858i 0.173648 − 0.984808i 0.939693 − 0.342020i −1.70245 − 0.318864i −0.158856 − 0.900919i 0.500000 + 0.866025i −0.583569 + 2.94269i −0.500000 + 0.866025i
211.4 −0.766044 + 0.642788i 1.16797 − 1.27900i 0.173648 − 0.984808i 0.939693 − 0.342020i −0.0725903 + 1.73053i 0.781269 + 4.43080i 0.500000 + 0.866025i −0.271692 − 2.98767i −0.500000 + 0.866025i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.2.k.e ✓ 24
3.b odd 2 1 810.2.k.e 24
27.e even 9 1 inner 270.2.k.e ✓ 24
27.e even 9 1 7290.2.a.s 12
27.f odd 18 1 810.2.k.e 24
27.f odd 18 1 7290.2.a.v 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.k.e ✓ 24 1.a even 1 1 trivial
270.2.k.e ✓ 24 27.e even 9 1 inner
810.2.k.e 24 3.b odd 2 1
810.2.k.e 24 27.f odd 18 1
7290.2.a.s 12 27.e even 9 1
7290.2.a.v 12 27.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} - 3 T_{7}^{23} - 9 T_{7}^{22} + 29 T_{7}^{21} + 303 T_{7}^{20} - 1899 T_{7}^{19} + \cdots + 2483776 \) acting on \(S_{2}^{\mathrm{new}}(270, [\chi])\). Copy content Toggle raw display