Properties

Label 228.2.w.a
Level $228$
Weight $2$
Character orbit 228.w
Analytic conductor $1.821$
Analytic rank $0$
Dimension $60$
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] = [228,2,Mod(67,228)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(228, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 0, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("228.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 228.w (of order \(18\), 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: \(1.82058916609\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{2} - 3 q^{4} + 3 q^{6} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{2} - 3 q^{4} + 3 q^{6} + 3 q^{8} - 6 q^{10} - 6 q^{13} - 9 q^{14} - 27 q^{16} - 18 q^{19} + 30 q^{20} - 6 q^{21} + 12 q^{22} + 18 q^{24} + 30 q^{27} - 18 q^{28} + 12 q^{31} - 63 q^{32} - 39 q^{34} - 3 q^{36} - 48 q^{38} + 33 q^{40} - 12 q^{41} - 9 q^{42} + 18 q^{43} - 9 q^{44} - 69 q^{46} + 6 q^{48} + 30 q^{49} - 27 q^{50} - 3 q^{52} + 12 q^{53} + 3 q^{54} - 42 q^{56} - 36 q^{58} - 3 q^{60} + 27 q^{62} + 12 q^{63} + 21 q^{64} - 36 q^{65} + 24 q^{66} - 42 q^{67} + 93 q^{68} - 72 q^{69} + 132 q^{70} - 24 q^{71} + 18 q^{72} + 18 q^{73} + 27 q^{74} - 60 q^{75} + 66 q^{76} - 72 q^{77} - 33 q^{78} - 12 q^{79} + 6 q^{80} - 9 q^{82} + 69 q^{84} - 60 q^{85} + 123 q^{86} + 36 q^{88} - 12 q^{89} + 15 q^{90} + 84 q^{91} - 99 q^{92} - 12 q^{93} + 24 q^{95} + 12 q^{97} + 117 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.37612 0.326044i −0.766044 + 0.642788i 1.78739 + 0.897349i −1.25586 0.457097i 1.26374 0.634786i −1.18216 0.682521i −2.16708 1.81762i 0.173648 0.984808i 1.57918 + 1.03849i
67.2 −1.33191 + 0.475423i −0.766044 + 0.642788i 1.54795 1.26644i 3.39140 + 1.23437i 0.714703 1.22033i −1.70277 0.983096i −1.45963 + 2.42270i 0.173648 0.984808i −5.10387 0.0317150i
67.3 −0.669968 1.24545i −0.766044 + 0.642788i −1.10229 + 1.66882i −2.74412 0.998777i 1.31378 + 0.523422i 3.79172 + 2.18915i 2.81693 + 0.254784i 0.173648 0.984808i 0.594545 + 4.08681i
67.4 −0.662654 + 1.24936i −0.766044 + 0.642788i −1.12178 1.65578i 1.27204 + 0.462984i −0.295448 1.38301i 3.79268 + 2.18970i 2.81201 0.304291i 0.173648 0.984808i −1.42135 + 1.28243i
67.5 −0.585310 1.28741i −0.766044 + 0.642788i −1.31482 + 1.50706i 0.378363 + 0.137713i 1.27590 + 0.609980i −3.35222 1.93540i 2.70978 + 0.810613i 0.173648 0.984808i −0.0441675 0.567712i
67.6 0.370262 1.36488i −0.766044 + 0.642788i −1.72581 1.01073i 3.29537 + 1.19942i 0.593693 + 1.28356i 2.21173 + 1.27694i −2.01853 + 1.98130i 0.173648 0.984808i 2.85721 4.05370i
67.7 0.541986 + 1.30624i −0.766044 + 0.642788i −1.41250 + 1.41592i −3.72560 1.35601i −1.25482 0.652253i 0.752955 + 0.434719i −2.61509 1.07765i 0.173648 0.984808i −0.247958 5.60144i
67.8 0.866706 1.11751i −0.766044 + 0.642788i −0.497640 1.93710i −2.00728 0.730589i 0.0543836 + 1.41317i −3.36529 1.94295i −2.59603 1.12278i 0.173648 0.984808i −2.55616 + 1.60994i
67.9 1.15291 + 0.819021i −0.766044 + 0.642788i 0.658410 + 1.88852i 1.35681 + 0.493837i −1.40964 + 0.113671i −0.113130 0.0653155i −0.787646 + 2.71654i 0.173648 0.984808i 1.15981 + 1.68060i
67.10 1.36774 0.359580i −0.766044 + 0.642788i 1.74140 0.983622i 0.0388803 + 0.0141513i −0.816613 + 1.15462i 1.39317 + 0.804345i 2.02809 1.97151i 0.173648 0.984808i 0.0582665 + 0.00537462i
79.1 −1.41413 0.0150734i 0.939693 0.342020i 1.99955 + 0.0426317i −0.175493 + 0.995269i −1.33401 + 0.469498i 2.28954 1.32186i −2.82698 0.0904268i 0.766044 0.642788i 0.263172 1.40480i
79.2 −1.13985 + 0.837107i 0.939693 0.342020i 0.598505 1.90835i 0.316000 1.79212i −0.784799 + 1.17647i 0.258690 0.149354i 0.915287 + 2.67624i 0.766044 0.642788i 1.14001 + 2.30727i
79.3 −0.663081 + 1.24913i 0.939693 0.342020i −1.12065 1.65655i −0.379809 + 2.15400i −0.195865 + 1.40058i −3.95398 + 2.28283i 2.81232 0.301406i 0.766044 0.642788i −2.43878 1.90271i
79.4 −0.578676 1.29040i 0.939693 0.342020i −1.33027 + 1.49345i −0.540387 + 3.06469i −0.985121 1.01466i 1.31815 0.761035i 2.69694 + 0.852357i 0.766044 0.642788i 4.26739 1.07615i
79.5 −0.488367 1.32721i 0.939693 0.342020i −1.52300 + 1.29633i 0.683388 3.87569i −0.912849 1.08014i −0.181667 + 0.104885i 2.46429 + 1.38825i 0.766044 0.642788i −5.47761 + 0.985755i
79.6 −0.0863378 + 1.41158i 0.939693 0.342020i −1.98509 0.243745i 0.432248 2.45140i 0.401656 + 1.35598i 1.95860 1.13080i 0.515453 2.78106i 0.766044 0.642788i 3.42301 + 0.821799i
79.7 0.869785 1.11511i 0.939693 0.342020i −0.486948 1.93981i 0.234529 1.33008i 0.435940 1.34535i −1.38264 + 0.798268i −2.58665 1.14422i 0.766044 0.642788i −1.27920 1.41841i
79.8 1.13356 + 0.845604i 0.939693 0.342020i 0.569909 + 1.91708i 0.162908 0.923898i 1.35441 + 0.406908i 2.27351 1.31261i −0.975067 + 2.65504i 0.766044 0.642788i 0.965918 0.909537i
79.9 1.24913 0.663079i 0.939693 0.342020i 1.12065 1.65654i −0.711633 + 4.03587i 0.947012 1.05032i 1.83842 1.06142i 0.301424 2.81232i 0.766044 0.642788i 1.78718 + 5.51320i
79.10 1.38401 + 0.290702i 0.939693 0.342020i 1.83099 + 0.804669i −0.0217506 + 0.123354i 1.39997 0.200190i −3.23383 + 1.86705i 2.30019 + 1.64594i 0.766044 0.642788i −0.0659622 + 0.164400i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.k even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 228.2.w.a 60
3.b odd 2 1 684.2.cf.c 60
4.b odd 2 1 228.2.w.b yes 60
12.b even 2 1 684.2.cf.b 60
19.f odd 18 1 228.2.w.b yes 60
57.j even 18 1 684.2.cf.b 60
76.k even 18 1 inner 228.2.w.a 60
228.u odd 18 1 684.2.cf.c 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.w.a 60 1.a even 1 1 trivial
228.2.w.a 60 76.k even 18 1 inner
228.2.w.b yes 60 4.b odd 2 1
228.2.w.b yes 60 19.f odd 18 1
684.2.cf.b 60 12.b even 2 1
684.2.cf.b 60 57.j even 18 1
684.2.cf.c 60 3.b odd 2 1
684.2.cf.c 60 228.u odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{60} - 120 T_{7}^{58} + 8037 T_{7}^{56} - 234 T_{7}^{55} - 370070 T_{7}^{54} + 29574 T_{7}^{53} + 12934554 T_{7}^{52} - 2059938 T_{7}^{51} - 359272269 T_{7}^{50} + 96327522 T_{7}^{49} + \cdots + 54\!\cdots\!89 \) acting on \(S_{2}^{\mathrm{new}}(228, [\chi])\). Copy content Toggle raw display