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Properties

Label 396.2.p.b
Level $396$
Weight $2$
Character orbit 396.p
Analytic conductor $3.162$
Analytic rank $0$
Dimension $60$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [396,2,Mod(23,396)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(396, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("396.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 396.p (of order \(6\), 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: \(3.16207592004\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 13 q^{6} + 30 q^{11} - 3 q^{12} - 15 q^{14} + 6 q^{15} - 37 q^{18} - 21 q^{20} - 4 q^{21} - 31 q^{24} + 30 q^{25} + 36 q^{27} - 12 q^{29} - 16 q^{30} + 45 q^{32} - 30 q^{34} + 33 q^{36} + 15 q^{38}+ \cdots - 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.40955 + 0.114728i 1.54758 + 0.777805i 1.97368 − 0.323429i 0.757506 + 0.437346i −2.27064 − 0.918807i 2.63124 − 1.51915i −2.74489 + 0.682325i 1.79004 + 2.40744i −1.11792 − 0.529556i
23.2 −1.33260 − 0.473464i −1.64959 − 0.528078i 1.55166 + 1.26188i −3.39072 − 1.95763i 1.94822 + 1.48474i −4.36027 + 2.51740i −1.47030 − 2.41624i 2.44227 + 1.74222i 3.59162 + 4.21413i
23.3 −1.29075 − 0.577904i −0.272833 − 1.71043i 1.33205 + 1.49186i −0.882040 − 0.509246i −0.636305 + 2.36540i 3.46600 − 2.00109i −0.857193 − 2.69541i −2.85112 + 0.933322i 0.844195 + 1.16704i
23.4 −1.28587 − 0.588681i −1.70003 + 0.331522i 1.30691 + 1.51393i 3.46980 + 2.00329i 2.38117 + 0.574481i 0.592509 − 0.342086i −0.789289 − 2.71607i 2.78019 − 1.12719i −3.28241 − 4.61857i
23.5 −1.25644 + 0.649130i −1.62209 + 0.607300i 1.15726 − 1.63118i 0.645969 + 0.372951i 1.64384 − 1.81598i −1.48984 + 0.860160i −0.395177 + 2.80068i 2.26237 − 1.97019i −1.05371 − 0.0492705i
23.6 −1.24656 + 0.667903i 0.207882 + 1.71953i 1.10781 − 1.66516i −2.99390 − 1.72853i −1.40762 − 2.00465i 0.186759 − 0.107825i −0.268787 + 2.81563i −2.91357 + 0.714919i 4.88655 + 0.155077i
23.7 −1.20995 − 0.732136i 1.73071 − 0.0680655i 0.927953 + 1.77170i 2.51433 + 1.45165i −2.14391 − 1.18476i −3.20405 + 1.84986i 0.174347 − 2.82305i 2.99073 − 0.235604i −1.97940 − 3.59725i
23.8 −1.07088 + 0.923695i −0.486132 − 1.66243i 0.293575 − 1.97834i 3.41974 + 1.97439i 2.05617 + 1.33123i 3.12910 − 1.80659i 1.51299 + 2.38974i −2.52735 + 1.61632i −5.48586 + 1.04446i
23.9 −0.885646 + 1.10256i 1.70451 + 0.307646i −0.431263 − 1.95295i −1.61716 − 0.933665i −1.84879 + 1.60685i −3.83877 + 2.21632i 2.53518 + 1.25413i 2.81071 + 1.04877i 2.46165 − 0.956109i
23.10 −0.839971 − 1.13774i −1.01224 + 1.40548i −0.588899 + 1.91133i −3.41563 − 1.97202i 2.44932 − 0.0288909i 3.07465 − 1.77515i 2.66926 − 0.935452i −0.950728 − 2.84537i 0.625392 + 5.54253i
23.11 −0.504411 − 1.32120i −1.61916 − 0.615083i −1.49114 + 1.33286i 0.524204 + 0.302649i 0.00407397 + 2.44949i 1.76489 − 1.01896i 2.51312 + 1.29778i 2.24335 + 1.99183i 0.135446 − 0.845238i
23.12 −0.485157 + 1.32839i 0.986970 − 1.42334i −1.52924 − 1.28896i −0.335872 − 0.193916i 1.41191 + 2.00163i −0.905329 + 0.522692i 2.45416 − 1.40609i −1.05178 − 2.80958i 0.420546 − 0.352089i
23.13 −0.430236 − 1.34718i 1.71291 − 0.256761i −1.62979 + 1.15921i 1.16537 + 0.672829i −1.08286 − 2.19714i 2.14636 − 1.23920i 2.26286 + 1.69689i 2.86815 − 0.879620i 0.405037 − 1.85944i
23.14 −0.325009 + 1.37636i −1.19384 − 1.25489i −1.78874 − 0.894661i −2.13707 − 1.23384i 2.11519 − 1.23530i 1.06895 − 0.617157i 1.81273 − 2.17118i −0.149504 + 2.99627i 2.39277 − 2.54037i
23.15 0.0730390 − 1.41233i −0.0943311 + 1.72948i −1.98933 − 0.206310i 1.90456 + 1.09960i 2.43570 + 0.259546i −1.35594 + 0.782849i −0.436675 + 2.79452i −2.98220 − 0.326288i 1.69209 − 2.60954i
23.16 0.212544 + 1.39815i −1.51418 + 0.840981i −1.90965 + 0.594337i −1.75916 − 1.01565i −1.49765 − 1.93831i −0.400003 + 0.230942i −1.23686 − 2.54366i 1.58550 − 2.54680i 1.04613 − 2.67544i
23.17 0.270212 − 1.38816i −0.650105 − 1.60542i −1.85397 − 0.750195i −0.596873 − 0.344605i −2.40424 + 0.468646i −2.15220 + 1.24257i −1.54235 + 2.37089i −2.15473 + 2.08738i −0.639648 + 0.735438i
23.18 0.331663 + 1.37477i 0.841231 + 1.51404i −1.78000 + 0.911922i 0.809368 + 0.467289i −1.80246 + 1.65865i 1.27147 − 0.734083i −1.84405 − 2.14464i −1.58466 + 2.54732i −0.373979 + 1.26768i
23.19 0.396259 + 1.35756i −0.579495 − 1.63223i −1.68596 + 1.07589i 2.58308 + 1.49134i 1.98623 − 1.43349i −4.06629 + 2.34767i −2.12867 − 1.86246i −2.32837 + 1.89174i −1.00102 + 4.09765i
23.20 0.695917 + 1.23114i 1.28011 − 1.16676i −1.03140 + 1.71354i 0.447034 + 0.258095i 2.32729 + 0.764017i 3.33992 − 1.92830i −2.82737 − 0.0773133i 0.277340 − 2.98715i −0.00665214 + 0.729974i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.30
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 396.2.p.b yes 60
4.b odd 2 1 396.2.p.a ✓ 60
9.d odd 6 1 396.2.p.a ✓ 60
36.h even 6 1 inner 396.2.p.b yes 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
396.2.p.a ✓ 60 4.b odd 2 1
396.2.p.a ✓ 60 9.d odd 6 1
396.2.p.b yes 60 1.a even 1 1 trivial
396.2.p.b yes 60 36.h even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{60} - 120 T_{7}^{58} + 8109 T_{7}^{56} - 2352 T_{7}^{55} - 375352 T_{7}^{54} + \cdots + 96\!\cdots\!04 \) acting on \(S_{2}^{\mathrm{new}}(396, [\chi])\). Copy content Toggle raw display