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Properties

Label 663.2.i.h
Level $663$
Weight $2$
Character orbit 663.i
Analytic conductor $5.294$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [663,2,Mod(256,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.256");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.i (of order \(3\), 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: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 11 q^{3} - 12 q^{4} - 3 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 11 q^{3} - 12 q^{4} - 3 q^{7} - 11 q^{9} + 9 q^{10} - 2 q^{11} + 24 q^{12} - q^{13} + 42 q^{14} - 18 q^{16} + 11 q^{17} + 21 q^{20} + 6 q^{21} - 3 q^{22} - 18 q^{23} + 14 q^{25} - 12 q^{26} + 22 q^{27} - 19 q^{28} - 15 q^{29} + 9 q^{30} - 26 q^{31} - 35 q^{32} - 2 q^{33} - 15 q^{35} - 12 q^{36} + 18 q^{37} + 10 q^{38} + 2 q^{39} + 24 q^{40} - 21 q^{42} + 11 q^{43} + 6 q^{44} + 29 q^{46} + 22 q^{47} - 18 q^{48} - 28 q^{49} + 2 q^{50} - 22 q^{51} + 32 q^{52} + 38 q^{53} - 13 q^{55} - 43 q^{56} - 68 q^{58} + 19 q^{59} - 42 q^{60} - 18 q^{61} - 6 q^{62} - 3 q^{63} + 40 q^{64} + 33 q^{65} + 6 q^{66} - 2 q^{67} + 12 q^{68} - 18 q^{69} - 116 q^{70} - 11 q^{71} + 60 q^{73} - 35 q^{74} - 7 q^{75} + 10 q^{76} + 40 q^{77} + 27 q^{78} - 28 q^{79} + 43 q^{80} - 11 q^{81} + 35 q^{82} + 10 q^{83} - 19 q^{84} - 22 q^{86} - 15 q^{87} + 5 q^{88} - 3 q^{89} - 18 q^{90} + 31 q^{91} + 108 q^{92} + 13 q^{93} - 55 q^{94} - 26 q^{95} + 70 q^{96} - 31 q^{97} + 54 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
256.1 −1.39887 + 2.42291i −0.500000 + 0.866025i −2.91366 − 5.04661i −2.12968 −1.39887 − 2.42291i −2.16160 − 3.74400i 10.7078 −0.500000 − 0.866025i 2.97914 − 5.16002i
256.2 −1.10307 + 1.91057i −0.500000 + 0.866025i −1.43351 − 2.48292i 1.69398 −1.10307 − 1.91057i 0.0628344 + 0.108832i 1.91278 −0.500000 − 0.866025i −1.86857 + 3.23647i
256.3 −0.758996 + 1.31462i −0.500000 + 0.866025i −0.152150 − 0.263532i −3.77954 −0.758996 − 1.31462i 0.250738 + 0.434291i −2.57406 −0.500000 − 0.866025i 2.86866 − 4.96866i
256.4 −0.558776 + 0.967828i −0.500000 + 0.866025i 0.375539 + 0.650453i −0.988234 −0.558776 − 0.967828i −1.64242 − 2.84475i −3.07447 −0.500000 − 0.866025i 0.552201 − 0.956441i
256.5 −0.334354 + 0.579119i −0.500000 + 0.866025i 0.776414 + 1.34479i 2.85507 −0.334354 − 0.579119i −1.01523 − 1.75843i −2.37581 −0.500000 − 0.866025i −0.954607 + 1.65343i
256.6 −0.110719 + 0.191771i −0.500000 + 0.866025i 0.975482 + 1.68959i −1.56526 −0.110719 − 0.191771i 1.37363 + 2.37921i −0.874896 −0.500000 − 0.866025i 0.173304 − 0.300171i
256.7 0.313535 − 0.543059i −0.500000 + 0.866025i 0.803391 + 1.39151i 2.72481 0.313535 + 0.543059i −0.354504 − 0.614019i 2.26171 −0.500000 − 0.866025i 0.854325 − 1.47973i
256.8 0.519476 − 0.899759i −0.500000 + 0.866025i 0.460289 + 0.797244i 1.38648 0.519476 + 0.899759i 1.90205 + 3.29445i 3.03434 −0.500000 − 0.866025i 0.720244 − 1.24750i
256.9 1.02850 − 1.78141i −0.500000 + 0.866025i −1.11561 − 1.93229i 3.23889 1.02850 + 1.78141i −2.47067 − 4.27932i −0.475611 −0.500000 − 0.866025i 3.33118 − 5.76978i
256.10 1.19083 − 2.06258i −0.500000 + 0.866025i −1.83616 − 3.18032i −0.495625 1.19083 + 2.06258i 2.45730 + 4.25616i −3.98291 −0.500000 − 0.866025i −0.590206 + 1.02227i
256.11 1.21244 − 2.10001i −0.500000 + 0.866025i −1.94002 − 3.36022i −2.94090 1.21244 + 2.10001i 0.0978636 + 0.169505i −4.55889 −0.500000 − 0.866025i −3.56567 + 6.17592i
562.1 −1.39887 − 2.42291i −0.500000 − 0.866025i −2.91366 + 5.04661i −2.12968 −1.39887 + 2.42291i −2.16160 + 3.74400i 10.7078 −0.500000 + 0.866025i 2.97914 + 5.16002i
562.2 −1.10307 − 1.91057i −0.500000 − 0.866025i −1.43351 + 2.48292i 1.69398 −1.10307 + 1.91057i 0.0628344 − 0.108832i 1.91278 −0.500000 + 0.866025i −1.86857 − 3.23647i
562.3 −0.758996 − 1.31462i −0.500000 − 0.866025i −0.152150 + 0.263532i −3.77954 −0.758996 + 1.31462i 0.250738 − 0.434291i −2.57406 −0.500000 + 0.866025i 2.86866 + 4.96866i
562.4 −0.558776 − 0.967828i −0.500000 − 0.866025i 0.375539 − 0.650453i −0.988234 −0.558776 + 0.967828i −1.64242 + 2.84475i −3.07447 −0.500000 + 0.866025i 0.552201 + 0.956441i
562.5 −0.334354 − 0.579119i −0.500000 − 0.866025i 0.776414 − 1.34479i 2.85507 −0.334354 + 0.579119i −1.01523 + 1.75843i −2.37581 −0.500000 + 0.866025i −0.954607 − 1.65343i
562.6 −0.110719 − 0.191771i −0.500000 − 0.866025i 0.975482 − 1.68959i −1.56526 −0.110719 + 0.191771i 1.37363 − 2.37921i −0.874896 −0.500000 + 0.866025i 0.173304 + 0.300171i
562.7 0.313535 + 0.543059i −0.500000 − 0.866025i 0.803391 − 1.39151i 2.72481 0.313535 − 0.543059i −0.354504 + 0.614019i 2.26171 −0.500000 + 0.866025i 0.854325 + 1.47973i
562.8 0.519476 + 0.899759i −0.500000 − 0.866025i 0.460289 − 0.797244i 1.38648 0.519476 − 0.899759i 1.90205 − 3.29445i 3.03434 −0.500000 + 0.866025i 0.720244 + 1.24750i
562.9 1.02850 + 1.78141i −0.500000 − 0.866025i −1.11561 + 1.93229i 3.23889 1.02850 − 1.78141i −2.47067 + 4.27932i −0.475611 −0.500000 + 0.866025i 3.33118 + 5.76978i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 256.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.i.h ✓ 22
13.c even 3 1 inner 663.2.i.h ✓ 22
13.c even 3 1 8619.2.a.bl 11
13.e even 6 1 8619.2.a.bk 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.i.h ✓ 22 1.a even 1 1 trivial
663.2.i.h ✓ 22 13.c even 3 1 inner
8619.2.a.bk 11 13.e even 6 1
8619.2.a.bl 11 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\):

\( T_{2}^{22} + 17 T_{2}^{20} + 189 T_{2}^{18} + 7 T_{2}^{17} + 1218 T_{2}^{16} + 195 T_{2}^{15} + \cdots + 144 \) Copy content Toggle raw display
\( T_{5}^{11} - 31 T_{5}^{9} + 3 T_{5}^{8} + 343 T_{5}^{7} - 27 T_{5}^{6} - 1666 T_{5}^{5} - 61 T_{5}^{4} + \cdots - 1074 \) Copy content Toggle raw display