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Properties

Label 663.2.cg.a
Level $663$
Weight $2$
Character orbit 663.cg
Analytic conductor $5.294$
Analytic rank $0$
Dimension $320$
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] = [663,2,Mod(94,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([0, 8, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.94");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.cg (of order \(24\), degree \(8\), 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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(40\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q + 96 q^{14} + 112 q^{16} + 16 q^{17} + 16 q^{20} - 40 q^{22} + 8 q^{23} + 48 q^{25} - 16 q^{26} + 24 q^{28} + 8 q^{29} - 32 q^{31} + 120 q^{32} - 8 q^{33} - 64 q^{34} - 16 q^{39} - 8 q^{41} + 24 q^{42} - 16 q^{43} - 8 q^{45} + 16 q^{46} - 32 q^{52} + 16 q^{53} + 40 q^{56} - 64 q^{58} + 32 q^{59} - 48 q^{61} - 40 q^{62} - 120 q^{65} + 32 q^{66} - 32 q^{67} + 32 q^{68} + 72 q^{69} + 64 q^{70} + 128 q^{71} - 112 q^{73} - 32 q^{74} - 32 q^{76} + 32 q^{77} - 8 q^{78} - 96 q^{79} - 64 q^{80} + 40 q^{82} - 40 q^{85} - 256 q^{86} + 24 q^{87} - 64 q^{88} - 32 q^{91} + 32 q^{94} - 32 q^{95} - 192 q^{96} - 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1 −2.69142 − 0.721165i −0.130526 − 0.991445i 4.99163 + 2.88192i 1.31375 − 3.17167i −0.363694 + 2.76253i −3.35998 − 0.442349i −7.41572 − 7.41572i −0.965926 + 0.258819i −5.82315 + 7.58888i
94.2 −2.60948 − 0.699208i 0.130526 + 0.991445i 4.58844 + 2.64914i −0.935443 + 2.25836i 0.352621 − 2.67842i −0.402306 − 0.0529646i −6.30059 − 6.30059i −0.965926 + 0.258819i 4.02008 − 5.23907i
94.3 −2.53256 − 0.678597i −0.130526 − 0.991445i 4.22131 + 2.43718i −1.02580 + 2.47649i −0.342226 + 2.59947i 2.58586 + 0.340434i −5.32894 − 5.32894i −0.965926 + 0.258819i 4.27843 − 5.57576i
94.4 −2.23414 − 0.598636i 0.130526 + 0.991445i 2.90096 + 1.67487i 0.0799704 − 0.193066i 0.301901 − 2.29316i −4.15651 − 0.547215i −2.20751 − 2.20751i −0.965926 + 0.258819i −0.294241 + 0.383463i
94.5 −2.23393 − 0.598580i −0.130526 − 0.991445i 2.90010 + 1.67437i 0.138459 − 0.334269i −0.301873 + 2.29295i 2.81092 + 0.370065i −2.20568 − 2.20568i −0.965926 + 0.258819i −0.509394 + 0.663855i
94.6 −2.12425 − 0.569190i 0.130526 + 0.991445i 2.45640 + 1.41820i 0.417519 − 1.00798i 0.287051 − 2.18037i 0.521180 + 0.0686147i −1.30066 − 1.30066i −0.965926 + 0.258819i −1.46065 + 1.90355i
94.7 −1.90474 − 0.510374i −0.130526 − 0.991445i 1.63551 + 0.944262i −0.169779 + 0.409882i −0.257389 + 1.95506i −4.66158 − 0.613709i 0.155439 + 0.155439i −0.965926 + 0.258819i 0.532577 − 0.694068i
94.8 −1.80243 − 0.482961i −0.130526 − 0.991445i 1.28347 + 0.741011i 0.722299 − 1.74378i −0.243564 + 1.85005i −1.36122 − 0.179208i 0.683458 + 0.683458i −0.965926 + 0.258819i −2.14408 + 2.79422i
94.9 −1.65612 − 0.443757i 0.130526 + 0.991445i 0.813778 + 0.469835i −0.507969 + 1.22635i 0.223793 − 1.69988i 3.24889 + 0.427724i 1.28551 + 1.28551i −0.965926 + 0.258819i 1.38546 − 1.80557i
94.10 −1.47355 − 0.394837i 0.130526 + 0.991445i 0.283410 + 0.163627i 1.59058 − 3.84000i 0.199122 − 1.51248i 3.71230 + 0.488734i 1.80442 + 1.80442i −0.965926 + 0.258819i −3.85998 + 5.03042i
94.11 −1.38032 − 0.369855i 0.130526 + 0.991445i 0.0364331 + 0.0210347i 0.609950 − 1.47255i 0.186523 − 1.41678i −2.94449 − 0.387650i 1.97842 + 1.97842i −0.965926 + 0.258819i −1.38655 + 1.80699i
94.12 −1.03139 − 0.276361i −0.130526 − 0.991445i −0.744656 − 0.429927i −1.34429 + 3.24541i −0.139373 + 1.05864i 1.20540 + 0.158694i 2.15928 + 2.15928i −0.965926 + 0.258819i 2.28340 − 2.97578i
94.13 −0.943188 − 0.252727i 0.130526 + 0.991445i −0.906317 − 0.523263i −1.41269 + 3.41053i 0.127454 − 0.968107i −4.34304 − 0.571773i 2.10351 + 2.10351i −0.965926 + 0.258819i 2.19436 − 2.85975i
94.14 −0.934800 − 0.250479i 0.130526 + 0.991445i −0.920939 − 0.531704i −0.952967 + 2.30067i 0.126320 − 0.959497i −1.26262 − 0.166226i 2.09636 + 2.09636i −0.965926 + 0.258819i 1.46710 − 1.91196i
94.15 −0.881009 − 0.236066i −0.130526 − 0.991445i −1.01160 − 0.584048i 0.0422882 − 0.102093i −0.119051 + 0.904285i −0.241493 − 0.0317931i 2.04324 + 2.04324i −0.965926 + 0.258819i −0.0613569 + 0.0799619i
94.16 −0.835967 − 0.223997i 0.130526 + 0.991445i −1.08338 − 0.625493i 1.04609 − 2.52548i 0.112965 − 0.858052i −0.00109541 0.000144214i 1.98951 + 1.98951i −0.965926 + 0.258819i −1.44019 + 1.87690i
94.17 −0.763944 − 0.204698i −0.130526 − 0.991445i −1.19034 − 0.687244i 0.141231 − 0.340961i −0.103232 + 0.784127i 3.17886 + 0.418505i 1.88717 + 1.88717i −0.965926 + 0.258819i −0.177686 + 0.231565i
94.18 −0.366725 − 0.0982636i 0.130526 + 0.991445i −1.60722 − 0.927929i −0.798146 + 1.92689i 0.0495557 − 0.376413i 4.18476 + 0.550934i 1.03515 + 1.03515i −0.965926 + 0.258819i 0.482043 − 0.628211i
94.19 −0.220821 − 0.0591688i −0.130526 − 0.991445i −1.68679 − 0.973869i 0.220040 − 0.531223i −0.0298397 + 0.226655i −0.897729 − 0.118188i 0.638160 + 0.638160i −0.965926 + 0.258819i −0.0800212 + 0.104286i
94.20 −0.205057 − 0.0549449i −0.130526 − 0.991445i −1.69302 − 0.977466i 1.43968 − 3.47570i −0.0277095 + 0.210475i −1.39625 − 0.183819i 0.593684 + 0.593684i −0.965926 + 0.258819i −0.486190 + 0.633615i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 94.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
17.d even 8 1 inner
221.be even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.cg.a ✓ 320
13.c even 3 1 inner 663.2.cg.a ✓ 320
17.d even 8 1 inner 663.2.cg.a ✓ 320
221.be even 24 1 inner 663.2.cg.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.cg.a ✓ 320 1.a even 1 1 trivial
663.2.cg.a ✓ 320 13.c even 3 1 inner
663.2.cg.a ✓ 320 17.d even 8 1 inner
663.2.cg.a ✓ 320 221.be even 24 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(663, [\chi])\).