↑

Properties

Label 63.9.q.a
Level $63$
Weight $9$
Character orbit 63.q
Analytic conductor $25.665$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,9,Mod(44,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.44");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 63.q (of order \(6\), degree \(2\), 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: \(25.6648524339\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 3072 q^{4} - 1458 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 3072 q^{4} - 1458 q^{7} - 11776 q^{10} + 1404 q^{13} - 514428 q^{16} + 24498 q^{19} - 529792 q^{22} + 1177458 q^{25} + 837284 q^{28} + 2985158 q^{31} - 11986992 q^{34} + 1100006 q^{37} + 24874380 q^{40} - 8895620 q^{43} - 3817512 q^{46} + 38509298 q^{49} + 15703324 q^{52} + 26803072 q^{55} - 51071956 q^{58} + 59934648 q^{61} - 370921960 q^{64} - 73910310 q^{67} + 187962148 q^{70} + 159392822 q^{73} + 39366376 q^{76} + 64994458 q^{79} + 206277244 q^{82} - 196502976 q^{85} - 365637780 q^{88} + 210338518 q^{91} - 71989068 q^{94} + 387792392 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −27.4555 − 15.8514i 0 374.537 + 648.717i 794.041 + 458.440i 0 −2372.24 − 370.515i − 15631.9i 0 −14533.9 − 25173.4i
44.2 −25.7287 − 14.8545i 0 313.312 + 542.673i −5.85354 − 3.37954i 0 2208.77 + 941.340i − 11010.9i 0 100.403 + 173.903i
44.3 −22.2448 − 12.8430i 0 201.886 + 349.677i −794.533 − 458.724i 0 1902.93 − 1464.13i − 3795.69i 0 11782.8 + 20408.4i
44.4 −19.4152 − 11.2094i 0 123.300 + 213.561i 373.345 + 215.551i 0 −2400.66 + 40.3637i 210.749i 0 −4832.37 − 8369.92i
44.5 −16.8268 − 9.71498i 0 60.7615 + 105.242i −114.955 − 66.3692i 0 −394.290 − 2368.40i 2612.88i 0 1289.55 + 2233.57i
44.6 −16.4314 − 9.48667i 0 51.9940 + 90.0562i 417.886 + 241.267i 0 1698.52 + 1697.01i 2884.18i 0 −4577.64 − 7928.71i
44.7 −14.6979 − 8.48582i 0 16.0184 + 27.7448i −331.889 − 191.616i 0 −2399.91 + 72.4085i 3801.02i 0 3252.05 + 5632.71i
44.8 −12.5278 − 7.23295i 0 −23.3690 − 40.4763i −1048.98 − 605.629i 0 308.356 + 2381.12i 4379.37i 0 8760.96 + 15174.4i
44.9 −6.01710 − 3.47398i 0 −103.863 − 179.896i 711.958 + 411.049i 0 2399.13 + 94.7876i 3221.95i 0 −2855.95 − 4946.65i
44.10 −3.74034 − 2.15949i 0 −118.673 − 205.548i 499.994 + 288.672i 0 88.7490 − 2399.36i 2130.75i 0 −1246.77 − 2159.46i
44.11 −0.379964 − 0.219372i 0 −127.904 − 221.536i 328.280 + 189.533i 0 −1403.85 + 1947.82i 224.553i 0 −83.1564 − 144.031i
44.12 0.379964 + 0.219372i 0 −127.904 − 221.536i −328.280 − 189.533i 0 −1403.85 + 1947.82i − 224.553i 0 −83.1564 − 144.031i
44.13 3.74034 + 2.15949i 0 −118.673 − 205.548i −499.994 − 288.672i 0 88.7490 − 2399.36i − 2130.75i 0 −1246.77 − 2159.46i
44.14 6.01710 + 3.47398i 0 −103.863 − 179.896i −711.958 − 411.049i 0 2399.13 + 94.7876i − 3221.95i 0 −2855.95 − 4946.65i
44.15 12.5278 + 7.23295i 0 −23.3690 − 40.4763i 1048.98 + 605.629i 0 308.356 + 2381.12i − 4379.37i 0 8760.96 + 15174.4i
44.16 14.6979 + 8.48582i 0 16.0184 + 27.7448i 331.889 + 191.616i 0 −2399.91 + 72.4085i − 3801.02i 0 3252.05 + 5632.71i
44.17 16.4314 + 9.48667i 0 51.9940 + 90.0562i −417.886 − 241.267i 0 1698.52 + 1697.01i − 2884.18i 0 −4577.64 − 7928.71i
44.18 16.8268 + 9.71498i 0 60.7615 + 105.242i 114.955 + 66.3692i 0 −394.290 − 2368.40i − 2612.88i 0 1289.55 + 2233.57i
44.19 19.4152 + 11.2094i 0 123.300 + 213.561i −373.345 − 215.551i 0 −2400.66 + 40.3637i − 210.749i 0 −4832.37 − 8369.92i
44.20 22.2448 + 12.8430i 0 201.886 + 349.677i 794.533 + 458.724i 0 1902.93 − 1464.13i 3795.69i 0 11782.8 + 20408.4i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.9.q.a ✓ 44
3.b odd 2 1 inner 63.9.q.a ✓ 44
7.c even 3 1 inner 63.9.q.a ✓ 44
21.h odd 6 1 inner 63.9.q.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.9.q.a ✓ 44 1.a even 1 1 trivial
63.9.q.a ✓ 44 3.b odd 2 1 inner
63.9.q.a ✓ 44 7.c even 3 1 inner
63.9.q.a ✓ 44 21.h odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(63, [\chi])\).