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Properties

Label 405.3.s.a
Level $405$
Weight $3$
Character orbit 405.s
Analytic conductor $11.035$
Analytic rank $0$
Dimension $408$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [405,3,Mod(37,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([28, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.s (of order \(36\), degree \(12\), not 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: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(408\)
Relative dimension: \(34\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 408 q + 12 q^{2} + 12 q^{5} - 12 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 408 q + 12 q^{2} + 12 q^{5} - 12 q^{7} + 6 q^{8} - 6 q^{10} + 60 q^{11} - 12 q^{13} - 24 q^{16} + 6 q^{17} + 300 q^{20} - 12 q^{22} + 156 q^{23} + 6 q^{25} + 48 q^{26} - 24 q^{28} - 24 q^{31} - 72 q^{32} - 156 q^{35} - 6 q^{37} + 252 q^{38} - 108 q^{40} + 384 q^{41} - 12 q^{43} - 12 q^{46} + 210 q^{47} - 276 q^{50} - 60 q^{52} - 516 q^{53} - 24 q^{55} - 912 q^{56} - 12 q^{58} - 312 q^{61} + 6 q^{62} - 420 q^{65} - 480 q^{67} + 540 q^{68} - 12 q^{70} + 12 q^{71} - 6 q^{73} - 216 q^{76} + 876 q^{77} + 1644 q^{80} - 24 q^{82} + 372 q^{83} - 12 q^{85} - 516 q^{86} + 348 q^{88} - 12 q^{91} - 2082 q^{92} - 198 q^{95} + 600 q^{97} + 1032 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −3.80701 − 0.333070i 0 10.4432 + 1.84141i 3.90040 − 3.12840i 0 3.00418 − 4.29041i −24.3786 − 6.53222i 0 −15.8908 + 10.6108i
37.2 −3.75537 − 0.328553i 0 10.0557 + 1.77308i 2.50965 + 4.32454i 0 −0.0930392 + 0.132874i −22.6151 − 6.05970i 0 −8.00382 − 17.0648i
37.3 −3.31284 − 0.289836i 0 6.95165 + 1.22576i −4.31093 + 2.53295i 0 1.07419 − 1.53410i −9.82571 − 2.63279i 0 15.0156 − 7.14180i
37.4 −3.11531 − 0.272554i 0 5.69163 + 1.00359i −0.988365 − 4.90134i 0 −0.485519 + 0.693393i −5.37504 − 1.44024i 0 1.74318 + 15.5386i
37.5 −3.01752 − 0.263999i 0 5.09649 + 0.898648i 3.14357 + 3.88818i 0 −3.95134 + 5.64310i −3.43817 − 0.921254i 0 −8.45932 − 12.5625i
37.6 −3.00625 − 0.263013i 0 5.02913 + 0.886771i −4.79837 − 1.40556i 0 −6.32550 + 9.03376i −3.22596 − 0.864394i 0 14.0554 + 5.48750i
37.7 −2.74079 − 0.239788i 0 3.51520 + 0.619824i 2.64835 − 4.24102i 0 3.37892 − 4.82559i 1.14426 + 0.306603i 0 −8.27551 + 10.9887i
37.8 −2.35308 − 0.205868i 0 1.55539 + 0.274257i −4.75024 + 1.56052i 0 2.05445 − 2.93406i 5.52284 + 1.47984i 0 11.4990 − 2.69411i
37.9 −2.10279 − 0.183971i 0 0.448665 + 0.0791117i 1.60745 + 4.73457i 0 1.45218 − 2.07392i 7.22671 + 1.93639i 0 −2.50911 − 10.2515i
37.10 −1.71130 − 0.149720i 0 −1.03309 − 0.182161i 4.75957 + 1.53184i 0 7.63890 − 10.9095i 8.37788 + 2.24485i 0 −7.91572 − 3.33404i
37.11 −1.51047 − 0.132149i 0 −1.67517 − 0.295378i 1.60774 − 4.73447i 0 −3.38273 + 4.83104i 8.34957 + 2.23726i 0 −3.05409 + 6.93881i
37.12 −1.41431 − 0.123736i 0 −1.95427 − 0.344591i 4.88629 − 1.06028i 0 −4.38775 + 6.26635i 8.20665 + 2.19897i 0 −7.04191 + 0.894960i
37.13 −1.34162 − 0.117377i 0 −2.15306 − 0.379643i −3.14234 − 3.88918i 0 6.77150 − 9.67070i 8.04746 + 2.15631i 0 3.75933 + 5.58663i
37.14 −1.18325 − 0.103521i 0 −2.54988 − 0.449612i −2.11870 + 4.52892i 0 −7.28025 + 10.3973i 7.55976 + 2.02563i 0 2.97578 − 5.13949i
37.15 −0.532015 − 0.0465453i 0 −3.65836 − 0.645067i −1.67973 + 4.70941i 0 1.89461 − 2.70578i 3.97968 + 1.06635i 0 1.11284 − 2.42729i
37.16 −0.0325506 − 0.00284781i 0 −3.93818 − 0.694407i 4.95271 − 0.686078i 0 −2.28630 + 3.26518i 0.252459 + 0.0676462i 0 −0.163168 + 0.00822790i
37.17 0.0277177 + 0.00242499i 0 −3.93847 − 0.694458i −4.64756 − 1.84396i 0 2.12874 − 3.04016i −0.214983 − 0.0576046i 0 −0.124348 − 0.0623806i
37.18 0.0902713 + 0.00789772i 0 −3.93114 − 0.693167i −3.56314 − 3.50771i 0 −5.48605 + 7.83489i −0.699509 − 0.187433i 0 −0.293946 − 0.344787i
37.19 0.553138 + 0.0483933i 0 −3.63561 − 0.641056i 0.321397 − 4.98966i 0 3.24240 − 4.63063i −4.12530 − 1.10537i 0 0.419244 − 2.74442i
37.20 0.723224 + 0.0632739i 0 −3.42018 − 0.603070i −1.21658 + 4.84973i 0 4.26087 − 6.08515i −5.24040 − 1.40416i 0 −1.18672 + 3.43047i
See next 80 embeddings (of 408 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
27.e even 9 1 inner
135.r odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.s.a 408
3.b odd 2 1 135.3.r.a ✓ 408
5.c odd 4 1 inner 405.3.s.a 408
15.e even 4 1 135.3.r.a ✓ 408
27.e even 9 1 inner 405.3.s.a 408
27.f odd 18 1 135.3.r.a ✓ 408
135.q even 36 1 135.3.r.a ✓ 408
135.r odd 36 1 inner 405.3.s.a 408
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.3.r.a ✓ 408 3.b odd 2 1
135.3.r.a ✓ 408 15.e even 4 1
135.3.r.a ✓ 408 27.f odd 18 1
135.3.r.a ✓ 408 135.q even 36 1
405.3.s.a 408 1.a even 1 1 trivial
405.3.s.a 408 5.c odd 4 1 inner
405.3.s.a 408 27.e even 9 1 inner
405.3.s.a 408 135.r odd 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(405, [\chi])\).