Properties

Label 945.2.bt.d
Level $945$
Weight $2$
Character orbit 945.bt
Analytic conductor $7.546$
Analytic rank $0$
Dimension $120$
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] = [945,2,Mod(106,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([10, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bt (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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 9 q^{8} + 3 q^{10} + 60 q^{12} - 3 q^{13} - 6 q^{15} + 12 q^{16} + 36 q^{18} + 6 q^{19} + 9 q^{22} - 6 q^{23} - 12 q^{24} + 54 q^{26} - 6 q^{27} - 138 q^{28} - 18 q^{29} + 21 q^{30} - 51 q^{32} + 30 q^{33} - 21 q^{34} + 60 q^{35} - 90 q^{36} - 54 q^{37} + 15 q^{38} + 15 q^{39} + 9 q^{40} + 45 q^{41} - 3 q^{42} + 9 q^{43} - 48 q^{44} + 12 q^{45} - 24 q^{46} + 3 q^{47} - 42 q^{48} - 6 q^{51} + 15 q^{52} - 36 q^{53} - 90 q^{54} + 24 q^{55} - 9 q^{56} - 48 q^{57} + 81 q^{58} + 12 q^{59} - 15 q^{60} - 24 q^{61} + 27 q^{62} - 15 q^{63} - 123 q^{64} + 6 q^{65} - 21 q^{66} + 60 q^{67} + 105 q^{68} - 30 q^{69} - 42 q^{71} + 159 q^{72} - 6 q^{73} + 21 q^{74} - 84 q^{76} + 9 q^{77} + 45 q^{78} + 30 q^{79} + 174 q^{80} - 24 q^{81} + 36 q^{82} - 42 q^{83} + 3 q^{85} - 75 q^{86} - 6 q^{87} - 195 q^{88} - 75 q^{89} - 39 q^{90} + 18 q^{91} - 39 q^{92} + 18 q^{93} - 69 q^{94} - 3 q^{95} - 24 q^{96} - 57 q^{97} + 3 q^{98} + 87 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1 −0.478635 + 2.71447i −1.68671 0.393722i −5.25988 1.91444i 0.766044 0.642788i 1.87607 4.39007i 0.939693 0.342020i 4.95791 8.58736i 2.68997 + 1.32819i 1.37817 + 2.38707i
106.2 −0.447881 + 2.54006i 0.214551 1.71871i −4.37192 1.59125i 0.766044 0.642788i 4.26954 + 1.31475i 0.939693 0.342020i 3.42073 5.92487i −2.90794 0.737503i 1.28962 + 2.23369i
106.3 −0.429464 + 2.43561i 1.70788 0.288365i −3.86839 1.40798i 0.766044 0.642788i −0.0311277 + 4.28357i 0.939693 0.342020i 2.61743 4.53353i 2.83369 0.984983i 1.23659 + 2.14184i
106.4 −0.395944 + 2.24551i 0.590363 + 1.62833i −3.00615 1.09415i 0.766044 0.642788i −3.89019 + 0.680937i 0.939693 0.342020i 1.36704 2.36778i −2.30294 + 1.92262i 1.14007 + 1.97467i
106.5 −0.325155 + 1.84404i −1.34970 + 1.08550i −1.41539 0.515160i 0.766044 0.642788i −1.56285 2.84186i 0.939693 0.342020i −0.462294 + 0.800716i 0.643383 2.93020i 0.936246 + 1.62163i
106.6 −0.238716 + 1.35382i 1.73190 0.0230899i 0.103533 + 0.0376830i 0.766044 0.642788i −0.382171 + 2.35019i 0.939693 0.342020i −1.45044 + 2.51223i 2.99893 0.0799785i 0.687354 + 1.19053i
106.7 −0.166984 + 0.947014i −0.149438 + 1.72559i 1.01043 + 0.367768i 0.766044 0.642788i −1.60921 0.429667i 0.939693 0.342020i −1.47863 + 2.56106i −2.95534 0.515739i 0.480812 + 0.832790i
106.8 −0.158687 + 0.899960i −0.829508 1.52050i 1.09464 + 0.398416i 0.766044 0.642788i 1.50002 0.505240i 0.939693 0.342020i −1.44611 + 2.50473i −1.62383 + 2.52253i 0.456922 + 0.791411i
106.9 −0.134776 + 0.764355i 0.458197 1.67035i 1.31331 + 0.478006i 0.766044 0.642788i 1.21498 + 0.575349i 0.939693 0.342020i −1.31852 + 2.28374i −2.58011 1.53069i 0.388073 + 0.672163i
106.10 −0.0320960 + 0.182025i −1.00120 + 1.41336i 1.84728 + 0.672356i 0.766044 0.642788i −0.225133 0.227608i 0.939693 0.342020i −0.366509 + 0.634813i −0.995179 2.83013i 0.0924166 + 0.160070i
106.11 0.0446705 0.253339i 0.324260 1.70143i 1.81720 + 0.661407i 0.766044 0.642788i −0.416553 0.158151i 0.939693 0.342020i 0.505982 0.876387i −2.78971 1.10341i −0.128623 0.222782i
106.12 0.0735518 0.417133i 1.58140 + 0.706518i 1.71080 + 0.622679i 0.766044 0.642788i 0.411027 0.607689i 0.939693 0.342020i 0.809139 1.40147i 2.00167 + 2.23458i −0.211784 0.366820i
106.13 0.154907 0.878522i −1.60676 + 0.646788i 1.13158 + 0.411862i 0.766044 0.642788i 0.319319 + 1.51176i 0.939693 0.342020i 1.42919 2.47544i 2.16333 2.07846i −0.446037 0.772559i
106.14 0.200144 1.13507i −1.71494 0.242873i 0.631054 + 0.229685i 0.766044 0.642788i −0.618912 + 1.89797i 0.939693 0.342020i 1.53959 2.66665i 2.88203 + 0.833024i −0.576291 0.998166i
106.15 0.231428 1.31249i 1.14379 1.30067i 0.210302 + 0.0765436i 0.766044 0.642788i −1.44241 1.80223i 0.939693 0.342020i 1.48187 2.56668i −0.383476 2.97539i −0.666371 1.15419i
106.16 0.276984 1.57085i 1.16016 + 1.28609i −0.511473 0.186161i 0.766044 0.642788i 2.34160 1.46622i 0.939693 0.342020i 1.16098 2.01088i −0.308053 + 2.98414i −0.797543 1.38138i
106.17 0.344690 1.95483i −1.23324 1.21619i −1.82317 0.663580i 0.766044 0.642788i −2.80253 + 1.99158i 0.939693 0.342020i 0.0593724 0.102836i 0.0417824 + 2.99971i −0.992494 1.71905i
106.18 0.398691 2.26109i 1.45776 0.935386i −3.07419 1.11891i 0.766044 0.642788i −1.53380 3.66905i 0.939693 0.342020i −1.45964 + 2.52818i 1.25010 2.72713i −1.14799 1.98837i
106.19 0.437486 2.48111i 0.160588 + 1.72459i −4.08512 1.48686i 0.766044 0.642788i 4.34915 + 0.356048i 0.939693 0.342020i −2.95686 + 5.12144i −2.94842 + 0.553898i −1.25969 2.18185i
106.20 0.472138 2.67763i −1.72539 + 0.151715i −5.06738 1.84437i 0.766044 0.642788i −0.408387 + 4.69159i 0.939693 0.342020i −4.61211 + 7.98841i 2.95397 0.523536i −1.35947 2.35466i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.20
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 945.2.bt.d 120
27.e even 9 1 inner 945.2.bt.d 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.bt.d 120 1.a even 1 1 trivial
945.2.bt.d 120 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} - 7 T_{2}^{117} - 3 T_{2}^{116} + 39 T_{2}^{115} + 1004 T_{2}^{114} - 48 T_{2}^{113} - 330 T_{2}^{112} - 5114 T_{2}^{111} - 2004 T_{2}^{110} + 38355 T_{2}^{109} + 635350 T_{2}^{108} - 16425 T_{2}^{107} - 255267 T_{2}^{106} + \cdots + 37654626304 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display