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Properties

Label 456.2.bk.a
Level $456$
Weight $2$
Character orbit 456.bk
Analytic conductor $3.641$
Analytic rank $0$
Dimension $240$
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] = [456,2,Mod(61,456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(456, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("456.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.bk (of order \(18\), degree \(6\), 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: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(40\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 6 q^{4} + 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 6 q^{4} + 6 q^{6} - 24 q^{10} + 18 q^{14} - 6 q^{16} + 60 q^{20} - 36 q^{28} - 72 q^{31} - 90 q^{32} + 66 q^{34} - 6 q^{36} + 72 q^{38} - 114 q^{40} - 60 q^{44} + 30 q^{46} + 72 q^{47} + 24 q^{48} - 120 q^{49} + 18 q^{50} + 6 q^{52} - 6 q^{54} + 72 q^{58} - 12 q^{60} + 12 q^{62} + 30 q^{64} - 48 q^{66} - 78 q^{68} - 108 q^{70} - 36 q^{72} - 24 q^{73} - 96 q^{74} - 48 q^{76} - 96 q^{78} - 114 q^{80} - 66 q^{82} - 36 q^{84} - 24 q^{86} - 24 q^{88} + 6 q^{90} - 108 q^{94} + 120 q^{95} - 12 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −1.41080 + 0.0982454i 0.984808 − 0.173648i 1.98070 − 0.277209i −2.00499 − 2.38946i −1.37230 + 0.341735i −0.650971 + 1.12751i −2.76712 + 0.585679i 0.939693 − 0.342020i 3.06339 + 3.17406i
61.2 −1.40084 + 0.194048i −0.984808 + 0.173648i 1.92469 − 0.543658i 1.54992 + 1.84712i 1.34586 − 0.434352i −2.50148 + 4.33269i −2.59068 + 1.13506i 0.939693 − 0.342020i −2.52961 − 2.28676i
61.3 −1.39993 − 0.200508i −0.984808 + 0.173648i 1.91959 + 0.561394i 1.13283 + 1.35006i 1.41348 − 0.0456327i 1.93035 − 3.34346i −2.57473 − 1.17081i 0.939693 − 0.342020i −1.31518 − 2.11712i
61.4 −1.38719 − 0.275126i −0.984808 + 0.173648i 1.84861 + 0.763306i −1.78078 − 2.12225i 1.41389 + 0.0300627i −0.439462 + 0.761171i −2.35438 − 1.56745i 0.939693 − 0.342020i 1.88640 + 3.43391i
61.5 −1.31763 + 0.513660i 0.984808 − 0.173648i 1.47231 − 1.35363i 0.782380 + 0.932405i −1.20842 + 0.734660i −1.79007 + 3.10050i −1.24466 + 2.53985i 0.939693 − 0.342020i −1.50983 − 0.826689i
61.6 −1.28710 + 0.585973i 0.984808 − 0.173648i 1.31327 − 1.50842i 2.76639 + 3.29686i −1.16580 + 0.800574i 2.20445 − 3.81822i −0.806426 + 2.71103i 0.939693 − 0.342020i −5.49250 − 2.62237i
61.7 −1.23950 − 0.680912i 0.984808 − 0.173648i 1.07272 + 1.68798i 1.78078 + 2.12225i −1.33891 − 0.455330i −0.439462 + 0.761171i −0.180268 − 2.82268i 0.939693 − 0.342020i −0.762210 − 3.84308i
61.8 −1.20129 − 0.746258i 0.984808 − 0.173648i 0.886199 + 1.79294i −1.13283 − 1.35006i −1.31263 − 0.526318i 1.93035 − 3.34346i 0.273416 − 2.81518i 0.939693 − 0.342020i 0.353370 + 2.46719i
61.9 −1.10998 + 0.876323i −0.984808 + 0.173648i 0.464117 − 1.94540i −0.290862 − 0.346635i 0.940946 − 1.05576i 0.982047 − 1.70096i 1.18964 + 2.56608i 0.939693 − 0.342020i 0.626615 + 0.129870i
61.10 −1.01758 − 0.982103i −0.984808 + 0.173648i 0.0709471 + 1.99874i 2.00499 + 2.38946i 1.17266 + 0.790481i −0.650971 + 1.12751i 1.89078 − 2.10356i 0.939693 − 0.342020i 0.306449 − 4.40058i
61.11 −0.999573 + 1.00043i 0.984808 − 0.173648i −0.00170955 − 2.00000i −0.280818 − 0.334666i −0.810664 + 1.15880i −0.191923 + 0.332421i 2.00256 + 1.99743i 0.939693 − 0.342020i 0.615507 + 0.0535848i
61.12 −0.948372 − 1.04909i 0.984808 − 0.173648i −0.201180 + 1.98986i −1.54992 − 1.84712i −1.11614 − 0.868469i −2.50148 + 4.33269i 2.27833 − 1.67607i 0.939693 − 0.342020i −0.467897 + 3.37776i
61.13 −0.783185 + 1.17755i −0.984808 + 0.173648i −0.773242 − 1.84448i −2.37243 − 2.82735i 0.566808 − 1.29566i −1.53699 + 2.66215i 2.77755 + 0.534038i 0.939693 − 0.342020i 5.18739 − 0.579311i
61.14 −0.679191 − 1.24044i −0.984808 + 0.173648i −1.07740 + 1.68500i −0.782380 − 0.932405i 0.884273 + 1.10366i −1.79007 + 3.10050i 2.82190 + 0.192022i 0.939693 − 0.342020i −0.625210 + 1.60378i
61.15 −0.609322 − 1.27622i −0.984808 + 0.173648i −1.25745 + 1.55525i −2.76639 − 3.29686i 0.821678 + 1.15102i 2.20445 − 3.81822i 2.75103 + 0.657129i 0.939693 − 0.342020i −2.52188 + 5.53936i
61.16 −0.292311 + 1.38367i 0.984808 − 0.173648i −1.82911 − 0.808927i 0.464877 + 0.554019i −0.0475977 + 1.41341i 1.23281 − 2.13529i 1.65396 − 2.29443i 0.939693 − 0.342020i −0.902471 + 0.481293i
61.17 −0.287006 − 1.38478i 0.984808 − 0.173648i −1.83526 + 0.794882i 0.290862 + 0.346635i −0.523111 − 1.31391i 0.982047 − 1.70096i 1.62747 + 2.31330i 0.939693 − 0.342020i 0.396536 − 0.502267i
61.18 −0.276939 + 1.38683i −0.984808 + 0.173648i −1.84661 − 0.768136i 1.58746 + 1.89186i 0.0319107 − 1.41385i 1.26268 − 2.18703i 1.57667 − 2.34821i 0.939693 − 0.342020i −3.06333 + 1.67761i
61.19 −0.172542 + 1.40365i 0.984808 − 0.173648i −1.94046 − 0.484378i −1.72128 − 2.05134i 0.0738198 + 1.41229i −1.30350 + 2.25773i 1.01471 − 2.64015i 0.939693 − 0.342020i 3.17635 − 2.06213i
61.20 −0.122655 − 1.40888i −0.984808 + 0.173648i −1.96991 + 0.345613i 0.280818 + 0.334666i 0.365442 + 1.36618i −0.191923 + 0.332421i 0.728547 + 2.73299i 0.939693 − 0.342020i 0.437062 − 0.436688i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
19.e even 9 1 inner
152.t even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.bk.a ✓ 240
8.b even 2 1 inner 456.2.bk.a ✓ 240
19.e even 9 1 inner 456.2.bk.a ✓ 240
152.t even 18 1 inner 456.2.bk.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bk.a ✓ 240 1.a even 1 1 trivial
456.2.bk.a ✓ 240 8.b even 2 1 inner
456.2.bk.a ✓ 240 19.e even 9 1 inner
456.2.bk.a ✓ 240 152.t even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(456, [\chi])\).