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Properties

Label 416.2.bg.a
Level $416$
Weight $2$
Character orbit 416.bg
Analytic conductor $3.322$
Analytic rank $0$
Dimension $216$
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] = [416,2,Mod(77,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.bg (of order \(8\), degree \(4\), 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: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(54\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 8 q^{3} - 8 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 8 q^{3} - 8 q^{4} - 8 q^{9} - 8 q^{10} - 24 q^{12} - 4 q^{13} + 24 q^{14} - 8 q^{16} + 8 q^{22} - 8 q^{23} - 8 q^{25} + 16 q^{26} - 56 q^{27} - 8 q^{29} - 24 q^{30} - 8 q^{35} + 16 q^{36} - 48 q^{38} - 28 q^{39} + 32 q^{40} + 32 q^{42} - 8 q^{43} - 112 q^{48} + 16 q^{51} - 12 q^{52} - 8 q^{53} - 40 q^{55} - 48 q^{56} - 8 q^{61} + 56 q^{62} - 8 q^{64} - 8 q^{65} - 56 q^{66} - 8 q^{69} - 8 q^{74} - 16 q^{75} - 8 q^{77} + 84 q^{78} - 88 q^{82} - 8 q^{87} + 72 q^{88} + 88 q^{90} + 44 q^{91} - 80 q^{92} + 24 q^{94} + 112 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −1.41210 − 0.0772002i −0.397360 + 0.959313i 1.98808 + 0.218030i 1.53059 + 3.69517i 0.635174 − 1.32397i 2.05838 + 2.05838i −2.79055 − 0.461361i 1.35893 + 1.35893i −1.87609 − 5.33613i
77.2 −1.41090 + 0.0967282i 1.21855 − 2.94185i 1.98129 − 0.272948i −1.46867 − 3.54569i −1.43470 + 4.26853i 1.96843 + 1.96843i −2.76900 + 0.576749i −5.04829 − 5.04829i 2.41512 + 4.86055i
77.3 −1.40524 + 0.159029i −0.700567 + 1.69132i 1.94942 − 0.446949i −0.175581 − 0.423890i 0.715499 − 2.48813i −0.506941 − 0.506941i −2.66833 + 0.938086i −0.248445 − 0.248445i 0.314145 + 0.567746i
77.4 −1.39490 + 0.232927i 0.834650 − 2.01502i 1.89149 − 0.649819i 1.24361 + 3.00234i −0.694900 + 3.00517i −2.87336 − 2.87336i −2.48708 + 1.34701i −1.24236 − 1.24236i −2.43404 − 3.89829i
77.5 −1.37997 − 0.309324i −0.935921 + 2.25951i 1.80864 + 0.853715i −1.28599 − 3.10465i 1.99046 − 2.82856i 2.22807 + 2.22807i −2.23179 − 1.73756i −2.10813 − 2.10813i 0.814283 + 4.68210i
77.6 −1.37688 − 0.322781i 0.0930705 − 0.224692i 1.79162 + 0.888865i −0.454046 − 1.09616i −0.200674 + 0.279334i −0.425170 − 0.425170i −2.17995 − 1.80217i 2.07950 + 2.07950i 0.271348 + 1.65585i
77.7 −1.31886 + 0.510506i 0.324061 − 0.782352i 1.47877 − 1.34657i 0.174055 + 0.420207i −0.0279946 + 1.19725i 2.63267 + 2.63267i −1.26285 + 2.53085i 1.61426 + 1.61426i −0.444072 − 0.465337i
77.8 −1.21662 + 0.720993i −0.843536 + 2.03648i 0.960339 − 1.75435i −0.608708 − 1.46955i −0.442020 − 3.08581i −1.01919 − 1.01919i 0.0965056 + 2.82678i −1.31436 − 1.31436i 1.80010 + 1.34901i
77.9 −1.21243 − 0.728017i 0.979432 − 2.36456i 0.939982 + 1.76534i −0.107364 − 0.259200i −2.90893 + 2.15382i −0.641321 − 0.641321i 0.145536 − 2.82468i −2.51053 − 2.51053i −0.0585303 + 0.392425i
77.10 −1.17991 − 0.779620i −0.174001 + 0.420075i 0.784386 + 1.83977i 0.828671 + 2.00059i 0.532805 − 0.359998i −1.24243 − 1.24243i 0.508812 − 2.78229i 1.97513 + 1.97513i 0.581940 − 3.00657i
77.11 −1.11277 − 0.872782i −1.21033 + 2.92200i 0.476504 + 1.94241i 0.0390281 + 0.0942221i 3.89709 − 2.19515i −2.68922 − 2.68922i 1.16506 − 2.57733i −4.95187 − 4.95187i 0.0388061 − 0.138910i
77.12 −1.08935 + 0.901840i −0.465988 + 1.12499i 0.373369 − 1.96484i 1.10333 + 2.66368i −0.506941 − 1.64576i −3.08810 − 3.08810i 1.36524 + 2.47712i 1.07285 + 1.07285i −3.60412 − 1.90665i
77.13 −1.04284 + 0.955242i 0.348689 − 0.841810i 0.175025 − 1.99233i −1.67361 − 4.04044i 0.440506 + 1.21095i −1.63113 − 1.63113i 1.72063 + 2.24487i 1.53426 + 1.53426i 5.60491 + 2.61483i
77.14 −1.01393 − 0.985874i 0.225020 − 0.543247i 0.0561057 + 1.99921i −0.861807 − 2.08059i −0.763728 + 0.328973i 1.89986 + 1.89986i 1.91408 − 2.08237i 1.87684 + 1.87684i −1.17738 + 2.95920i
77.15 −0.988066 + 1.01179i 0.708345 − 1.71010i −0.0474507 − 1.99944i 0.188293 + 0.454578i 1.03037 + 2.40639i 0.461953 + 0.461953i 2.06990 + 1.92757i −0.301353 − 0.301353i −0.645985 − 0.258640i
77.16 −0.918245 − 1.07556i −0.882106 + 2.12959i −0.313652 + 1.97525i 0.820581 + 1.98106i 3.10049 − 1.00673i 2.89688 + 2.89688i 2.41251 − 1.47641i −1.63573 − 1.63573i 1.37725 − 2.70168i
77.17 −0.797764 + 1.16772i −0.541522 + 1.30735i −0.727144 − 1.86313i 0.954453 + 2.30425i −1.09461 − 1.67530i 2.30724 + 2.30724i 2.75571 + 0.637240i 0.705404 + 0.705404i −3.45215 − 0.723717i
77.18 −0.636692 − 1.26278i 0.993510 − 2.39855i −1.18925 + 1.60801i −0.590580 − 1.42579i −3.66140 + 0.272545i −3.67238 − 3.67238i 2.78775 + 0.477957i −2.64464 − 2.64464i −1.42444 + 1.65356i
77.19 −0.608769 − 1.27648i 0.959908 − 2.31742i −1.25880 + 1.55416i 1.15665 + 2.79239i −3.54251 + 0.185471i 1.46396 + 1.46396i 2.75017 + 0.660710i −2.32771 − 2.32771i 2.86030 − 3.17635i
77.20 −0.544463 + 1.30521i 1.26194 − 3.04659i −1.40712 − 1.42127i −0.106710 − 0.257621i 3.28935 + 3.30585i −1.31255 − 1.31255i 2.62118 − 1.06275i −5.56791 − 5.56791i 0.394348 0.000986456i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
32.g even 8 1 inner
416.bg even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.bg.a ✓ 216
13.b even 2 1 inner 416.2.bg.a ✓ 216
32.g even 8 1 inner 416.2.bg.a ✓ 216
416.bg even 8 1 inner 416.2.bg.a ✓ 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.bg.a ✓ 216 1.a even 1 1 trivial
416.2.bg.a ✓ 216 13.b even 2 1 inner
416.2.bg.a ✓ 216 32.g even 8 1 inner
416.2.bg.a ✓ 216 416.bg even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(416, [\chi])\).