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Properties

Label 405.2.r.a
Level $405$
Weight $2$
Character orbit 405.r
Analytic conductor $3.234$
Analytic rank $0$
Dimension $192$
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] = [405,2,Mod(8,405)]
 
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([2, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.r (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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 12 q^{2} + 12 q^{5} - 12 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 12 q^{2} + 12 q^{5} - 12 q^{7} + 18 q^{8} - 6 q^{10} + 36 q^{11} - 12 q^{13} - 24 q^{16} + 18 q^{17} - 36 q^{20} - 12 q^{22} + 36 q^{23} - 30 q^{25} - 24 q^{28} - 24 q^{31} + 48 q^{32} - 36 q^{35} - 6 q^{37} - 12 q^{38} - 36 q^{40} - 24 q^{41} - 12 q^{43} - 12 q^{46} + 6 q^{47} - 36 q^{50} + 12 q^{52} - 24 q^{55} - 180 q^{56} - 12 q^{58} - 60 q^{61} + 18 q^{62} + 84 q^{65} + 24 q^{67} + 60 q^{68} - 12 q^{70} + 36 q^{71} - 6 q^{73} - 72 q^{76} - 132 q^{77} - 24 q^{82} - 48 q^{83} - 12 q^{85} - 12 q^{86} - 48 q^{88} - 12 q^{91} - 258 q^{92} - 18 q^{95} + 24 q^{97} - 324 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −2.05499 + 1.43892i 0 1.46845 − 4.03452i 1.82626 + 1.29027i 0 −1.14217 + 2.44940i 1.48912 + 5.55747i 0 −5.60952 − 0.0236494i
8.2 −1.94894 + 1.36466i 0 1.25203 − 3.43992i −0.278496 − 2.21866i 0 −1.02475 + 2.19758i 1.02263 + 3.81650i 0 3.57049 + 3.94398i
8.3 −1.65044 + 1.15565i 0 0.704386 − 1.93528i 0.268922 + 2.21984i 0 0.618828 − 1.32708i 0.0310202 + 0.115769i 0 −3.00920 − 3.35293i
8.4 −1.23715 + 0.866263i 0 0.0960923 − 0.264011i 2.23168 − 0.140023i 0 0.677826 − 1.45360i −0.671958 − 2.50778i 0 −2.63963 + 2.10645i
8.5 −0.865920 + 0.606324i 0 −0.301851 + 0.829330i −2.13636 + 0.660292i 0 −1.22161 + 2.61975i −0.788655 − 2.94330i 0 1.44956 − 1.86708i
8.6 −0.490069 + 0.343150i 0 −0.561625 + 1.54305i 1.25284 − 1.85213i 0 −0.0636379 + 0.136472i −0.563947 − 2.10468i 0 0.0215816 + 1.33758i
8.7 −0.263290 + 0.184358i 0 −0.648706 + 1.78231i −2.21531 − 0.303968i 0 2.07991 − 4.46037i −0.324162 − 1.20979i 0 0.639309 − 0.328378i
8.8 −0.0928047 + 0.0649826i 0 −0.679650 + 1.86732i −0.584024 + 2.15845i 0 0.344848 − 0.739528i −0.116914 − 0.436329i 0 −0.0860616 − 0.238266i
8.9 0.377486 − 0.264318i 0 −0.611409 + 1.67983i −0.538986 − 2.17014i 0 −1.52168 + 3.26325i 0.451753 + 1.68596i 0 −0.777066 − 0.676732i
8.10 0.807333 − 0.565301i 0 −0.351818 + 0.966613i 2.04189 + 0.911409i 0 −0.275357 + 0.590505i 0.772562 + 2.88324i 0 2.16371 − 0.418474i
8.11 1.08817 − 0.761943i 0 −0.0804885 + 0.221140i −1.23426 − 1.86457i 0 0.827388 − 1.77434i 0.768546 + 2.86825i 0 −2.76377 − 1.08853i
8.12 1.22359 − 0.856769i 0 0.0790861 − 0.217287i −2.10603 + 0.751431i 0 −2.03610 + 4.36642i 0.683816 + 2.55204i 0 −1.93312 + 2.72382i
8.13 1.42277 − 0.996232i 0 0.347747 − 0.955427i 0.893670 + 2.04972i 0 0.429055 − 0.920112i 0.442010 + 1.64960i 0 3.31348 + 2.02597i
8.14 1.66533 − 1.16608i 0 0.729546 − 2.00441i 1.48319 − 1.67336i 0 1.13512 − 2.43427i −0.0700073 − 0.261271i 0 0.518731 − 4.51621i
8.15 2.12047 − 1.48477i 0 1.60780 − 4.41741i 2.09416 − 0.783897i 0 −1.25154 + 2.68394i −1.80956 − 6.75336i 0 3.27669 − 4.77156i
8.16 2.18018 − 1.52658i 0 1.73870 − 4.77703i −1.64880 + 1.51045i 0 1.30062 − 2.78918i −2.12414 − 7.92739i 0 −1.28885 + 5.81007i
17.1 −2.47401 + 1.15365i 0 3.50423 − 4.17618i 1.69173 − 1.46221i 0 −0.0671403 − 0.00587402i −2.43862 + 9.10104i 0 −2.49848 + 5.56917i
17.2 −1.82102 + 0.849158i 0 1.30949 − 1.56058i 1.44997 + 1.70223i 0 3.18459 + 0.278616i −0.0193446 + 0.0721951i 0 −4.08589 − 1.86855i
17.3 −1.81626 + 0.846935i 0 1.29592 − 1.54442i −0.828498 − 2.07692i 0 −2.21198 − 0.193523i −0.00834683 + 0.0311508i 0 3.26378 + 3.07054i
17.4 −1.64734 + 0.768168i 0 0.838077 − 0.998782i −0.861745 + 2.06335i 0 −4.49995 − 0.393694i 0.327512 − 1.22229i 0 −0.165408 − 4.06100i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
27.f odd 18 1 inner
135.q even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.r.a 192
3.b odd 2 1 135.2.q.a ✓ 192
5.c odd 4 1 inner 405.2.r.a 192
15.d odd 2 1 675.2.ba.b 192
15.e even 4 1 135.2.q.a ✓ 192
15.e even 4 1 675.2.ba.b 192
27.e even 9 1 135.2.q.a ✓ 192
27.f odd 18 1 inner 405.2.r.a 192
135.p even 18 1 675.2.ba.b 192
135.q even 36 1 inner 405.2.r.a 192
135.r odd 36 1 135.2.q.a ✓ 192
135.r odd 36 1 675.2.ba.b 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.q.a ✓ 192 3.b odd 2 1
135.2.q.a ✓ 192 15.e even 4 1
135.2.q.a ✓ 192 27.e even 9 1
135.2.q.a ✓ 192 135.r odd 36 1
405.2.r.a 192 1.a even 1 1 trivial
405.2.r.a 192 5.c odd 4 1 inner
405.2.r.a 192 27.f odd 18 1 inner
405.2.r.a 192 135.q even 36 1 inner
675.2.ba.b 192 15.d odd 2 1
675.2.ba.b 192 15.e even 4 1
675.2.ba.b 192 135.p even 18 1
675.2.ba.b 192 135.r odd 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(405, [\chi])\).