↑

Properties

Label 135.2.k.a
Level $135$
Weight $2$
Character orbit 135.k
Analytic conductor $1.078$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(30\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{3} - 9 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{3} - 9 q^{8} - 3 q^{9} + 3 q^{10} - 6 q^{11} + 3 q^{12} + 3 q^{13} - 9 q^{14} - 6 q^{15} + 12 q^{16} - 12 q^{17} + 6 q^{18} + 24 q^{19} - 36 q^{21} - 51 q^{22} + 18 q^{23} + 45 q^{24} - 18 q^{26} - 9 q^{27} - 60 q^{28} + 18 q^{29} - 3 q^{30} + 12 q^{31} + 36 q^{32} + 27 q^{33} - 69 q^{34} - 12 q^{35} - 42 q^{36} + 24 q^{37} - 24 q^{38} + 6 q^{39} + 9 q^{40} - 75 q^{41} - 18 q^{42} + 6 q^{43} + 12 q^{44} - 6 q^{45} + 30 q^{46} + 45 q^{47} - 27 q^{48} - 36 q^{49} + 21 q^{51} + 30 q^{52} + 36 q^{53} + 18 q^{54} + 30 q^{56} + 30 q^{57} + 27 q^{58} - 27 q^{59} - 12 q^{61} + 36 q^{62} + 18 q^{63} + 27 q^{64} + 6 q^{65} + 78 q^{66} - 30 q^{67} + 69 q^{68} - 117 q^{69} + 27 q^{70} + 12 q^{71} + 9 q^{72} + 21 q^{73} - 30 q^{76} - 36 q^{77} + 66 q^{78} + 54 q^{79} + 6 q^{80} - 27 q^{81} - 48 q^{82} - 87 q^{83} + 45 q^{84} + 27 q^{85} + 18 q^{86} - 27 q^{87} - 18 q^{88} + 9 q^{89} - 6 q^{90} + 51 q^{91} + 24 q^{92} + 36 q^{93} + 15 q^{94} + 21 q^{95} - 15 q^{96} - 75 q^{97} - 15 q^{98} + 123 q^{99}+O(q^{100}) \) 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 −1.76334 − 1.47962i 1.47207 + 0.912694i 0.572799 + 3.24850i 0.939693 + 0.342020i −1.24532 − 3.78748i −0.745456 + 4.22769i 1.49463 − 2.58877i 1.33398 + 2.68710i −1.15094 − 1.99348i
16.2 −0.531925 − 0.446338i 0.942993 − 1.45285i −0.263570 − 1.49478i 0.939693 + 0.342020i −1.15006 + 0.351912i −0.0506352 + 0.287167i −1.22136 + 2.11545i −1.22153 − 2.74005i −0.347189 − 0.601349i
16.3 0.186537 + 0.156523i −1.71306 + 0.255766i −0.337000 − 1.91122i 0.939693 + 0.342020i −0.359583 − 0.220424i 0.688903 − 3.90696i 0.479795 − 0.831029i 2.86917 − 0.876286i 0.121753 + 0.210883i
16.4 1.25101 + 1.04972i −0.905836 + 1.47630i 0.115814 + 0.656812i 0.939693 + 0.342020i −2.68292 + 0.895989i −0.576430 + 3.26909i 1.08849 − 1.88532i −1.35892 − 2.67457i 0.816539 + 1.41429i
16.5 1.62376 + 1.36249i −0.235856 − 1.71592i 0.432902 + 2.45511i 0.939693 + 0.342020i 1.95495 − 3.10759i −0.0109747 + 0.0622407i −0.522478 + 0.904959i −2.88874 + 0.809420i 1.05983 + 1.83568i
31.1 −2.22676 − 0.810474i −1.72562 − 0.149064i 2.76950 + 2.32388i −0.173648 + 0.984808i 3.72174 + 1.73050i −0.151115 + 0.126801i −1.91389 − 3.31495i 2.95556 + 0.514456i 1.18483 − 2.05219i
31.2 −1.38905 − 0.505573i 1.28409 + 1.16238i 0.141770 + 0.118959i −0.173648 + 0.984808i −1.19600 − 2.26380i 2.40771 − 2.02030i 1.34141 + 2.32340i 0.297765 + 2.98519i 0.739099 − 1.28016i
31.3 −0.282715 − 0.102900i −0.864728 + 1.50075i −1.46275 − 1.22739i −0.173648 + 0.984808i 0.398898 − 0.335304i −3.28585 + 2.75715i 0.588102 + 1.01862i −1.50449 − 2.59548i 0.150430 − 0.260552i
31.4 1.22047 + 0.444217i 1.42893 + 0.978851i −0.239858 − 0.201265i −0.173648 + 0.984808i 1.30916 + 1.82942i −1.16252 + 0.975467i −1.50214 − 2.60178i 1.08370 + 2.79743i −0.649401 + 1.12480i
31.5 1.73836 + 0.632710i 0.550980 − 1.64208i 1.08947 + 0.914177i −0.173648 + 0.984808i 1.99676 − 2.50591i −0.872404 + 0.732034i −0.534435 − 0.925669i −2.39284 − 1.80950i −0.924960 + 1.60208i
61.1 −2.22676 + 0.810474i −1.72562 + 0.149064i 2.76950 − 2.32388i −0.173648 − 0.984808i 3.72174 − 1.73050i −0.151115 − 0.126801i −1.91389 + 3.31495i 2.95556 − 0.514456i 1.18483 + 2.05219i
61.2 −1.38905 + 0.505573i 1.28409 − 1.16238i 0.141770 − 0.118959i −0.173648 − 0.984808i −1.19600 + 2.26380i 2.40771 + 2.02030i 1.34141 − 2.32340i 0.297765 − 2.98519i 0.739099 + 1.28016i
61.3 −0.282715 + 0.102900i −0.864728 − 1.50075i −1.46275 + 1.22739i −0.173648 − 0.984808i 0.398898 + 0.335304i −3.28585 − 2.75715i 0.588102 − 1.01862i −1.50449 + 2.59548i 0.150430 + 0.260552i
61.4 1.22047 − 0.444217i 1.42893 − 0.978851i −0.239858 + 0.201265i −0.173648 − 0.984808i 1.30916 − 1.82942i −1.16252 − 0.975467i −1.50214 + 2.60178i 1.08370 − 2.79743i −0.649401 − 1.12480i
61.5 1.73836 − 0.632710i 0.550980 + 1.64208i 1.08947 − 0.914177i −0.173648 − 0.984808i 1.99676 + 2.50591i −0.872404 − 0.732034i −0.534435 + 0.925669i −2.39284 + 1.80950i −0.924960 − 1.60208i
76.1 −1.76334 + 1.47962i 1.47207 − 0.912694i 0.572799 − 3.24850i 0.939693 − 0.342020i −1.24532 + 3.78748i −0.745456 − 4.22769i 1.49463 + 2.58877i 1.33398 − 2.68710i −1.15094 + 1.99348i
76.2 −0.531925 + 0.446338i 0.942993 + 1.45285i −0.263570 + 1.49478i 0.939693 − 0.342020i −1.15006 − 0.351912i −0.0506352 − 0.287167i −1.22136 − 2.11545i −1.22153 + 2.74005i −0.347189 + 0.601349i
76.3 0.186537 − 0.156523i −1.71306 − 0.255766i −0.337000 + 1.91122i 0.939693 − 0.342020i −0.359583 + 0.220424i 0.688903 + 3.90696i 0.479795 + 0.831029i 2.86917 + 0.876286i 0.121753 − 0.210883i
76.4 1.25101 − 1.04972i −0.905836 − 1.47630i 0.115814 − 0.656812i 0.939693 − 0.342020i −2.68292 − 0.895989i −0.576430 − 3.26909i 1.08849 + 1.88532i −1.35892 + 2.67457i 0.816539 − 1.41429i
76.5 1.62376 − 1.36249i −0.235856 + 1.71592i 0.432902 − 2.45511i 0.939693 − 0.342020i 1.95495 + 3.10759i −0.0109747 − 0.0622407i −0.522478 − 0.904959i −2.88874 − 0.809420i 1.05983 − 1.83568i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.5
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.a ✓ 30
3.b odd 2 1 405.2.k.a 30
5.b even 2 1 675.2.l.d 30
5.c odd 4 2 675.2.u.c 60
27.e even 9 1 inner 135.2.k.a ✓ 30
27.e even 9 1 3645.2.a.h 15
27.f odd 18 1 405.2.k.a 30
27.f odd 18 1 3645.2.a.g 15
135.p even 18 1 675.2.l.d 30
135.r odd 36 2 675.2.u.c 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.k.a ✓ 30 1.a even 1 1 trivial
135.2.k.a ✓ 30 27.e even 9 1 inner
405.2.k.a 30 3.b odd 2 1
405.2.k.a 30 27.f odd 18 1
675.2.l.d 30 5.b even 2 1
675.2.l.d 30 135.p even 18 1
675.2.u.c 60 5.c odd 4 2
675.2.u.c 60 135.r odd 36 2
3645.2.a.g 15 27.f odd 18 1
3645.2.a.h 15 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} + 7 T_{2}^{27} - 3 T_{2}^{26} - 36 T_{2}^{25} + 169 T_{2}^{24} + 93 T_{2}^{23} - 315 T_{2}^{22} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display