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Properties

Label 135.2.k.b
Level $135$
Weight $2$
Character orbit 135.k
Analytic conductor $1.078$
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] = [135,2,Mod(16,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.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: \(1.07798042729\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 3 q^{3} - 6 q^{6} - 9 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 3 q^{3} - 6 q^{6} - 9 q^{8} - 3 q^{9} - 3 q^{10} - 6 q^{11} - 39 q^{12} - 3 q^{13} - 27 q^{14} + 6 q^{15} - 12 q^{16} - 12 q^{17} + 6 q^{18} - 24 q^{19} + 12 q^{21} + 33 q^{22} + 18 q^{23} - 3 q^{24} - 18 q^{26} + 39 q^{27} + 60 q^{28} - 21 q^{30} - 12 q^{31} + 36 q^{32} - 33 q^{33} + 51 q^{34} - 12 q^{35} - 36 q^{36} - 24 q^{37} - 24 q^{38} - 30 q^{39} - 9 q^{40} + 33 q^{41} + 78 q^{42} - 24 q^{43} + 12 q^{44} - 12 q^{45} - 30 q^{46} + 27 q^{47} + 141 q^{48} + 18 q^{49} - 39 q^{51} - 84 q^{52} + 36 q^{53} + 30 q^{54} - 24 q^{56} - 81 q^{58} - 27 q^{59} - 6 q^{61} - 18 q^{62} - 6 q^{63} - 63 q^{64} + 6 q^{65} - 30 q^{66} + 12 q^{67} - 3 q^{68} + 51 q^{69} + 9 q^{70} + 12 q^{71} - 153 q^{72} - 39 q^{73} - 72 q^{74} + 138 q^{76} - 54 q^{77} - 48 q^{78} - 18 q^{79} + 78 q^{80} + 45 q^{81} + 48 q^{82} + 57 q^{83} - 51 q^{84} + 9 q^{85} + 18 q^{86} - 39 q^{87} + 126 q^{88} + 9 q^{89} + 54 q^{90} - 69 q^{91} + 24 q^{92} - 18 q^{93} - 33 q^{94} + 3 q^{95} + 153 q^{96} + 21 q^{97} - 15 q^{98} - 9 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.09652 − 1.75919i −1.01713 + 1.40194i 0.953350 + 5.40672i −0.939693 − 0.342020i 4.59871 − 1.14987i 0.438601 − 2.48743i 4.77590 − 8.27211i −0.930888 − 2.85192i 1.36840 + 2.37015i
16.2 −1.13174 − 0.949646i 1.73105 + 0.0588713i 0.0317206 + 0.179896i −0.939693 − 0.342020i −1.90320 − 1.71051i 0.642118 − 3.64163i −1.34245 + 2.32519i 2.99307 + 0.203818i 0.738693 + 1.27945i
16.3 −0.844223 − 0.708387i −1.70750 + 0.290600i −0.136396 − 0.773542i −0.939693 − 0.342020i 1.64737 + 0.964239i −0.518523 + 2.94069i −1.53487 + 2.65848i 2.83110 − 0.992398i 0.551027 + 0.954407i
16.4 0.601643 + 0.504838i −0.953823 − 1.44576i −0.240184 − 1.36215i −0.939693 − 0.342020i 0.156014 − 1.35136i 0.247142 − 1.40161i 1.32855 − 2.30112i −1.18044 + 2.75800i −0.392694 − 0.680167i
16.5 0.786397 + 0.659865i 1.53844 − 0.795743i −0.164299 − 0.931783i −0.939693 − 0.342020i 1.73491 + 0.389393i −0.712481 + 4.04068i 1.51222 − 2.61923i 1.73359 − 2.44840i −0.513284 − 0.889034i
16.6 1.42971 + 1.19967i 0.697646 + 1.58534i 0.257569 + 1.46075i −0.939693 − 0.342020i −0.904448 + 3.10352i 0.603654 − 3.42349i 0.482190 − 0.835177i −2.02658 + 2.21201i −0.933178 − 1.61631i
16.7 2.02078 + 1.69563i −1.72837 + 0.112798i 0.861072 + 4.88338i −0.939693 − 0.342020i −3.68392 − 2.70275i −0.00591834 + 0.0335646i −3.90246 + 6.75926i 2.97455 − 0.389915i −1.31897 − 2.28452i
31.1 −2.42143 − 0.881327i 0.345069 + 1.69733i 3.55448 + 2.98256i 0.173648 − 0.984808i 0.660344 − 4.41408i −2.32834 + 1.95371i −3.40147 − 5.89151i −2.76186 + 1.17139i −1.28841 + 2.23160i
31.2 −2.20793 − 0.803619i 0.127781 − 1.72733i 2.69705 + 2.26309i 0.173648 − 0.984808i −1.67025 + 3.71113i 3.56068 − 2.98776i −1.78659 − 3.09446i −2.96734 − 0.441440i −1.17481 + 2.03484i
31.3 −0.867363 − 0.315694i −1.06092 + 1.36910i −0.879434 − 0.737932i 0.173648 − 0.984808i 1.35242 − 0.852584i 2.85743 − 2.39767i 1.45286 + 2.51642i −0.748897 − 2.90502i −0.461514 + 0.799366i
31.4 0.157838 + 0.0574483i −0.188780 − 1.72173i −1.51048 − 1.26744i 0.173648 − 0.984808i 0.0691141 − 0.282600i −1.41056 + 1.18360i −0.333566 − 0.577753i −2.92872 + 0.650056i 0.0839839 − 0.145464i
31.5 0.281020 + 0.102283i 1.72885 − 0.105208i −1.46358 − 1.22809i 0.173648 − 0.984808i 0.496603 + 0.147266i 1.00072 − 0.839702i −0.584737 − 1.01279i 2.97786 − 0.363779i 0.149528 − 0.258989i
31.6 1.71599 + 0.624569i 0.0386016 + 1.73162i 1.02244 + 0.857933i 0.173648 − 0.984808i −1.01528 + 2.99555i 0.570617 − 0.478805i −0.607452 − 1.05214i −2.99702 + 0.133687i 0.913059 − 1.58146i
31.7 2.40218 + 0.874320i −1.31696 − 1.12500i 3.47392 + 2.91497i 0.173648 − 0.984808i −2.17995 − 3.85389i −1.18637 + 0.995479i 3.24001 + 5.61186i 0.468745 + 2.96315i 1.27817 − 2.21386i
61.1 −2.42143 + 0.881327i 0.345069 − 1.69733i 3.55448 − 2.98256i 0.173648 + 0.984808i 0.660344 + 4.41408i −2.32834 − 1.95371i −3.40147 + 5.89151i −2.76186 − 1.17139i −1.28841 − 2.23160i
61.2 −2.20793 + 0.803619i 0.127781 + 1.72733i 2.69705 − 2.26309i 0.173648 + 0.984808i −1.67025 − 3.71113i 3.56068 + 2.98776i −1.78659 + 3.09446i −2.96734 + 0.441440i −1.17481 − 2.03484i
61.3 −0.867363 + 0.315694i −1.06092 − 1.36910i −0.879434 + 0.737932i 0.173648 + 0.984808i 1.35242 + 0.852584i 2.85743 + 2.39767i 1.45286 − 2.51642i −0.748897 + 2.90502i −0.461514 − 0.799366i
61.4 0.157838 − 0.0574483i −0.188780 + 1.72173i −1.51048 + 1.26744i 0.173648 + 0.984808i 0.0691141 + 0.282600i −1.41056 − 1.18360i −0.333566 + 0.577753i −2.92872 − 0.650056i 0.0839839 + 0.145464i
61.5 0.281020 − 0.102283i 1.72885 + 0.105208i −1.46358 + 1.22809i 0.173648 + 0.984808i 0.496603 − 0.147266i 1.00072 + 0.839702i −0.584737 + 1.01279i 2.97786 + 0.363779i 0.149528 + 0.258989i
61.6 1.71599 − 0.624569i 0.0386016 − 1.73162i 1.02244 − 0.857933i 0.173648 + 0.984808i −1.01528 − 2.99555i 0.570617 + 0.478805i −0.607452 + 1.05214i −2.99702 − 0.133687i 0.913059 + 1.58146i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.7
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 135.2.k.b ✓ 42
3.b odd 2 1 405.2.k.b 42
5.b even 2 1 675.2.l.e 42
5.c odd 4 2 675.2.u.d 84
27.e even 9 1 inner 135.2.k.b ✓ 42
27.e even 9 1 3645.2.a.l 21
27.f odd 18 1 405.2.k.b 42
27.f odd 18 1 3645.2.a.k 21
135.p even 18 1 675.2.l.e 42
135.r odd 36 2 675.2.u.d 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.k.b ✓ 42 1.a even 1 1 trivial
135.2.k.b ✓ 42 27.e even 9 1 inner
405.2.k.b 42 3.b odd 2 1
405.2.k.b 42 27.f odd 18 1
675.2.l.e 42 5.b even 2 1
675.2.l.e 42 135.p even 18 1
675.2.u.d 84 5.c odd 4 2
675.2.u.d 84 135.r odd 36 2
3645.2.a.k 21 27.f odd 18 1
3645.2.a.l 21 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} + 7 T_{2}^{39} + 3 T_{2}^{38} - 36 T_{2}^{37} + 402 T_{2}^{36} + 153 T_{2}^{35} + \cdots + 29241 \) acting on \(S_{2}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display