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Properties

Label 663.2.u.a
Level $663$
Weight $2$
Character orbit 663.u
Analytic conductor $5.294$
Analytic rank $0$
Dimension $80$
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(64,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.u (of order \(4\), 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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 72 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 72 q^{4} + 4 q^{13} + 40 q^{14} + 56 q^{16} + 16 q^{17} - 16 q^{22} - 8 q^{23} - 16 q^{30} - 80 q^{38} - 8 q^{39} - 32 q^{48} + 40 q^{52} + 56 q^{55} + 16 q^{56} + 24 q^{61} + 72 q^{62} - 24 q^{64} - 8 q^{65} - 16 q^{68} + 24 q^{69} + 80 q^{74} + 32 q^{75} - 36 q^{78} + 24 q^{79} - 80 q^{81} - 48 q^{82} - 32 q^{88} - 4 q^{91} - 56 q^{92} - 128 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −2.72455 −0.707107 − 0.707107i 5.42320 0.857114 − 0.857114i 1.92655 + 1.92655i −0.721768 − 0.721768i −9.32669 1.00000i −2.33525 + 2.33525i
64.2 −2.55052 −0.707107 − 0.707107i 4.50513 −1.93171 + 1.93171i 1.80349 + 1.80349i 0.421479 + 0.421479i −6.38937 1.00000i 4.92686 − 4.92686i
64.3 −2.52110 0.707107 + 0.707107i 4.35594 −0.200034 + 0.200034i −1.78269 − 1.78269i 0.721903 + 0.721903i −5.93954 1.00000i 0.504306 − 0.504306i
64.4 −2.37549 0.707107 + 0.707107i 3.64295 −1.63781 + 1.63781i −1.67973 − 1.67973i 1.51543 + 1.51543i −3.90282 1.00000i 3.89060 − 3.89060i
64.5 −2.33800 0.707107 + 0.707107i 3.46624 2.85327 − 2.85327i −1.65322 − 1.65322i −2.09854 − 2.09854i −3.42807 1.00000i −6.67095 + 6.67095i
64.6 −2.12891 −0.707107 − 0.707107i 2.53227 0.941123 − 0.941123i 1.50537 + 1.50537i −3.15870 − 3.15870i −1.13316 1.00000i −2.00357 + 2.00357i
64.7 −1.96748 0.707107 + 0.707107i 1.87099 1.73641 − 1.73641i −1.39122 − 1.39122i −0.0272937 − 0.0272937i 0.253821 1.00000i −3.41636 + 3.41636i
64.8 −1.92637 0.707107 + 0.707107i 1.71091 −2.26304 + 2.26304i −1.36215 − 1.36215i −3.45940 − 3.45940i 0.556895 1.00000i 4.35945 − 4.35945i
64.9 −1.89891 −0.707107 − 0.707107i 1.60585 1.38527 − 1.38527i 1.34273 + 1.34273i 2.95122 + 2.95122i 0.748449 1.00000i −2.63050 + 2.63050i
64.10 −1.79751 −0.707107 − 0.707107i 1.23103 −2.85875 + 2.85875i 1.27103 + 1.27103i −1.50627 − 1.50627i 1.38224 1.00000i 5.13862 − 5.13862i
64.11 −1.42352 −0.707107 − 0.707107i 0.0264098 1.13266 − 1.13266i 1.00658 + 1.00658i −0.0139855 − 0.0139855i 2.80945 1.00000i −1.61236 + 1.61236i
64.12 −1.27232 0.707107 + 0.707107i −0.381213 1.59414 − 1.59414i −0.899663 − 0.899663i 2.18373 + 2.18373i 3.02965 1.00000i −2.02825 + 2.02825i
64.13 −0.994055 −0.707107 − 0.707107i −1.01185 −0.419562 + 0.419562i 0.702903 + 0.702903i −2.59225 − 2.59225i 2.99395 1.00000i 0.417068 − 0.417068i
64.14 −0.933332 0.707107 + 0.707107i −1.12889 −2.33579 + 2.33579i −0.659965 − 0.659965i 3.02690 + 3.02690i 2.92029 1.00000i 2.18007 − 2.18007i
64.15 −0.924037 0.707107 + 0.707107i −1.14616 −0.360185 + 0.360185i −0.653393 − 0.653393i −1.06827 − 1.06827i 2.90716 1.00000i 0.332825 − 0.332825i
64.16 −0.773870 0.707107 + 0.707107i −1.40112 0.968793 − 0.968793i −0.547209 − 0.547209i −2.52172 − 2.52172i 2.63203 1.00000i −0.749720 + 0.749720i
64.17 −0.564398 −0.707107 − 0.707107i −1.68146 −1.45879 + 1.45879i 0.399090 + 0.399090i −0.0958549 − 0.0958549i 2.07781 1.00000i 0.823339 − 0.823339i
64.18 −0.450108 −0.707107 − 0.707107i −1.79740 2.54550 − 2.54550i 0.318275 + 0.318275i −0.230605 − 0.230605i 1.70924 1.00000i −1.14575 + 1.14575i
64.19 −0.408444 −0.707107 − 0.707107i −1.83317 0.402109 − 0.402109i 0.288814 + 0.288814i 2.59513 + 2.59513i 1.56564 1.00000i −0.164239 + 0.164239i
64.20 −0.101762 0.707107 + 0.707107i −1.98964 −1.80634 + 1.80634i −0.0719568 − 0.0719568i −0.455691 − 0.455691i 0.405995 1.00000i 0.183818 − 0.183818i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
17.c even 4 1 inner
221.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.u.a ✓ 80
13.b even 2 1 inner 663.2.u.a ✓ 80
17.c even 4 1 inner 663.2.u.a ✓ 80
221.k even 4 1 inner 663.2.u.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.u.a ✓ 80 1.a even 1 1 trivial
663.2.u.a ✓ 80 13.b even 2 1 inner
663.2.u.a ✓ 80 17.c even 4 1 inner
663.2.u.a ✓ 80 221.k even 4 1 inner

Hecke kernels

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