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Properties

Label 663.2.bh.a
Level $663$
Weight $2$
Character orbit 663.bh
Analytic conductor $5.294$
Analytic rank $0$
Dimension $176$
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(25,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.bh (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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(44\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 176 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q - 16 q^{10} - 48 q^{14} - 224 q^{16} + 16 q^{17} + 64 q^{22} - 16 q^{23} - 64 q^{25} - 8 q^{26} + 64 q^{29} - 16 q^{39} + 48 q^{42} + 32 q^{43} - 32 q^{52} + 112 q^{53} - 80 q^{56} + 48 q^{61} - 80 q^{62} - 96 q^{65} - 32 q^{66} - 64 q^{68} - 80 q^{69} + 160 q^{74} + 16 q^{77} - 8 q^{78} + 48 q^{79} - 96 q^{82} + 48 q^{87} + 64 q^{88} + 16 q^{90} - 16 q^{91} - 64 q^{94} + 64 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −1.95407 − 1.95407i 0.382683 − 0.923880i 5.63678i −0.865555 + 2.08963i −2.55311 + 1.05753i −1.23904 − 2.99131i 7.10651 − 7.10651i −0.707107 − 0.707107i 5.77464 − 2.39194i
25.2 −1.90518 − 1.90518i −0.382683 + 0.923880i 5.25939i 0.744605 − 1.79764i 2.48923 − 1.03107i 0.699814 + 1.68950i 6.20971 − 6.20971i −0.707107 − 0.707107i −4.84342 + 2.00621i
25.3 −1.78542 − 1.78542i 0.382683 − 0.923880i 4.37544i 1.39840 − 3.37605i −2.33276 + 0.966262i 0.514649 + 1.24247i 4.24115 − 4.24115i −0.707107 − 0.707107i −8.52439 + 3.53092i
25.4 −1.73735 − 1.73735i −0.382683 + 0.923880i 4.03676i −0.261674 + 0.631738i 2.26995 − 0.940246i −1.22387 − 2.95467i 3.53856 − 3.53856i −0.707107 − 0.707107i 1.55217 − 0.642929i
25.5 −1.68987 − 1.68987i 0.382683 − 0.923880i 3.71130i 0.735185 − 1.77489i −2.20792 + 0.914549i −1.32827 − 3.20674i 2.89187 − 2.89187i −0.707107 − 0.707107i −4.24170 + 1.75697i
25.6 −1.59424 − 1.59424i −0.382683 + 0.923880i 3.08320i −1.69636 + 4.09538i 2.08297 − 0.862796i 1.82763 + 4.41228i 1.72688 − 1.72688i −0.707107 − 0.707107i 9.23343 − 3.82461i
25.7 −1.48812 − 1.48812i 0.382683 − 0.923880i 2.42903i −0.433771 + 1.04722i −1.94433 + 0.805367i 0.0435768 + 0.105204i 0.638443 − 0.638443i −0.707107 − 0.707107i 2.20389 − 0.912883i
25.8 −1.39826 − 1.39826i −0.382683 + 0.923880i 1.91028i 1.43534 − 3.46521i 1.82692 − 0.756734i −1.60702 − 3.87968i −0.125454 + 0.125454i −0.707107 − 0.707107i −6.85224 + 2.83829i
25.9 −1.32058 − 1.32058i 0.382683 − 0.923880i 1.48785i 0.166365 − 0.401642i −1.72542 + 0.714691i 1.72917 + 4.17458i −0.676336 + 0.676336i −0.707107 − 0.707107i −0.750097 + 0.310700i
25.10 −1.30955 − 1.30955i −0.382683 + 0.923880i 1.42986i 1.35765 − 3.27765i 1.71102 − 0.708726i 0.755648 + 1.82430i −0.746623 + 0.746623i −0.707107 − 0.707107i −6.07016 + 2.51434i
25.11 −1.25532 − 1.25532i −0.382683 + 0.923880i 1.15167i −0.732053 + 1.76733i 1.64016 − 0.679376i −1.13685 − 2.74460i −1.06493 + 1.06493i −0.707107 − 0.707107i 3.13754 − 1.29961i
25.12 −1.22146 − 1.22146i 0.382683 − 0.923880i 0.983908i −1.53837 + 3.71396i −1.59591 + 0.661047i −0.366025 − 0.883662i −1.24111 + 1.24111i −0.707107 − 0.707107i 6.41549 − 2.65738i
25.13 −1.05113 − 1.05113i −0.382683 + 0.923880i 0.209746i 0.0148646 − 0.0358864i 1.37337 − 0.568867i 1.66699 + 4.02447i −1.88179 + 1.88179i −0.707107 − 0.707107i −0.0533458 + 0.0220966i
25.14 −0.910967 − 0.910967i −0.382683 + 0.923880i − 0.340278i −0.776616 + 1.87492i 1.19024 − 0.493012i −0.588016 − 1.41960i −2.13192 + 2.13192i −0.707107 − 0.707107i 2.41546 − 1.00052i
25.15 −0.764274 − 0.764274i 0.382683 − 0.923880i − 0.831770i 1.32262 − 3.19309i −0.998572 + 0.413622i −0.145543 − 0.351372i −2.16425 + 2.16425i −0.707107 − 0.707107i −3.45124 + 1.42955i
25.16 −0.763198 − 0.763198i 0.382683 − 0.923880i − 0.835058i −0.388194 + 0.937182i −0.997166 + 0.413040i 1.16233 + 2.80612i −2.16371 + 2.16371i −0.707107 − 0.707107i 1.01152 − 0.418987i
25.17 −0.761058 − 0.761058i −0.382683 + 0.923880i − 0.841580i 0.569722 − 1.37543i 0.994371 − 0.411882i 0.168946 + 0.407871i −2.16261 + 2.16261i −0.707107 − 0.707107i −1.48037 + 0.613191i
25.18 −0.646609 − 0.646609i 0.382683 − 0.923880i − 1.16379i −0.874291 + 2.11073i −0.844836 + 0.349942i −1.77452 − 4.28407i −2.04574 + 2.04574i −0.707107 − 0.707107i 1.93014 − 0.799490i
25.19 −0.321174 − 0.321174i 0.382683 − 0.923880i − 1.79369i 0.193991 − 0.468337i −0.419634 + 0.173818i −0.279608 − 0.675033i −1.21844 + 1.21844i −0.707107 − 0.707107i −0.212723 + 0.0881126i
25.20 −0.163086 − 0.163086i −0.382683 + 0.923880i − 1.94681i −0.313750 + 0.757459i 0.213082 − 0.0882614i 0.929804 + 2.24474i −0.643668 + 0.643668i −0.707107 − 0.707107i 0.174699 − 0.0723626i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
17.d even 8 1 inner
221.p even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.bh.a ✓ 176
13.b even 2 1 inner 663.2.bh.a ✓ 176
17.d even 8 1 inner 663.2.bh.a ✓ 176
221.p even 8 1 inner 663.2.bh.a ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.bh.a ✓ 176 1.a even 1 1 trivial
663.2.bh.a ✓ 176 13.b even 2 1 inner
663.2.bh.a ✓ 176 17.d even 8 1 inner
663.2.bh.a ✓ 176 221.p even 8 1 inner

Hecke kernels

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