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Properties

Label 459.2.y.a
Level $459$
Weight $2$
Character orbit 459.y
Analytic conductor $3.665$
Analytic rank $0$
Dimension $256$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [459,2,Mod(44,459)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("459.44"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(459, base_ring=CyclotomicField(48)) chi = DirichletCharacter(H, H._module([40, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 459 = 3^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 459.y (of order \(48\), degree \(16\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.66513345278\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(16\) over \(\Q(\zeta_{48})\)
Twist minimal: no (minimal twist has level 153)
Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q + 24 q^{2} - 8 q^{4} + 24 q^{5} - 8 q^{7} - 32 q^{10} + 24 q^{11} - 8 q^{13} + 24 q^{14} - 32 q^{19} + 24 q^{20} - 8 q^{22} + 24 q^{23} - 8 q^{25} - 32 q^{28} + 24 q^{29} - 8 q^{31} + 24 q^{32} - 56 q^{34}+ \cdots - 8 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −0.343100 − 2.60610i 0 −4.74220 + 1.27067i −1.80716 − 0.613447i 0 1.25167 + 3.68730i 2.92671 + 7.06570i 0 −0.978671 + 4.92011i
44.2 −0.322211 − 2.44744i 0 −3.95427 + 1.05954i 1.16211 + 0.394485i 0 −0.737380 − 2.17225i 1.97792 + 4.77513i 0 0.591030 − 2.97131i
44.3 −0.286947 − 2.17958i 0 −2.73637 + 0.733209i 2.65023 + 0.899631i 0 0.480424 + 1.41528i 0.700707 + 1.69166i 0 1.20034 − 6.03453i
44.4 −0.208175 − 1.58125i 0 −0.525156 + 0.140715i −2.62395 − 0.890712i 0 −0.361760 − 1.06571i −0.888848 − 2.14587i 0 −0.862195 + 4.33455i
44.5 −0.202568 − 1.53866i 0 −0.394575 + 0.105726i −0.711743 − 0.241604i 0 0.882571 + 2.59997i −0.945194 − 2.28190i 0 −0.227569 + 1.14407i
44.6 −0.163117 − 1.23899i 0 0.423355 − 0.113438i −0.805077 − 0.273287i 0 −1.30408 − 3.84170i −1.16607 − 2.81515i 0 −0.207279 + 1.04206i
44.7 −0.148889 − 1.13093i 0 0.675027 − 0.180873i 2.45283 + 0.832623i 0 0.510405 + 1.50361i −1.17810 − 2.84418i 0 0.576435 − 2.89793i
44.8 0.0177816 + 0.135065i 0 1.91393 − 0.512835i 0.0115553 + 0.00392250i 0 −0.410532 − 1.20939i 0.207565 + 0.501106i 0 −0.000324320 0.00163047i
44.9 0.0252039 + 0.191443i 0 1.89584 − 0.507988i 2.71539 + 0.921751i 0 −0.832539 − 2.45258i 0.292821 + 0.706933i 0 −0.108024 + 0.543074i
44.10 0.0707100 + 0.537096i 0 1.64838 − 0.441682i 0.339563 + 0.115266i 0 1.52158 + 4.48243i 0.768405 + 1.85509i 0 −0.0378984 + 0.190528i
44.11 0.116095 + 0.881827i 0 1.16771 − 0.312887i −3.62509 − 1.23055i 0 0.512374 + 1.50940i 1.09222 + 2.63686i 0 0.664279 − 3.33956i
44.12 0.134617 + 1.02252i 0 0.904431 − 0.242342i 2.75111 + 0.933877i 0 0.386058 + 1.13729i 1.15890 + 2.79784i 0 −0.584559 + 2.93878i
44.13 0.223909 + 1.70076i 0 −0.910594 + 0.243993i −2.45157 − 0.832196i 0 −1.60093 − 4.71620i 0.694074 + 1.67564i 0 0.866436 − 4.35587i
44.14 0.254538 + 1.93341i 0 −1.74143 + 0.466616i −0.119506 − 0.0405669i 0 0.279445 + 0.823220i 0.147116 + 0.355169i 0 0.0480136 − 0.241380i
44.15 0.306288 + 2.32649i 0 −3.38688 + 0.907511i 4.08435 + 1.38645i 0 −0.552259 − 1.62690i −1.35269 − 3.26568i 0 −1.97457 + 9.92685i
44.16 0.352832 + 2.68002i 0 −5.12617 + 1.37355i −0.805821 − 0.273539i 0 −0.655568 − 1.93124i −3.42093 − 8.25886i 0 0.448772 − 2.25613i
62.1 −1.45899 − 1.90139i 0 −0.969004 + 3.61637i 0.502973 + 0.0329666i 0 0.0485612 + 0.740900i 3.86148 − 1.59948i 0 −0.671150 − 1.00445i
62.2 −1.34092 − 1.74752i 0 −0.738117 + 2.75469i 1.77127 + 0.116095i 0 −0.295171 − 4.50344i 1.73356 − 0.718065i 0 −2.17224 − 3.25099i
62.3 −1.13784 − 1.48286i 0 −0.386558 + 1.44266i 1.43402 + 0.0939907i 0 0.180003 + 2.74631i −0.874557 + 0.362253i 0 −1.49231 − 2.23340i
62.4 −1.06529 − 1.38832i 0 −0.274934 + 1.02607i −2.70285 − 0.177154i 0 0.103700 + 1.58216i −1.51606 + 0.627973i 0 2.63338 + 3.94113i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
17.e odd 16 1 inner
153.s even 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 459.2.y.a 256
3.b odd 2 1 153.2.s.a ✓ 256
9.c even 3 1 153.2.s.a ✓ 256
9.d odd 6 1 inner 459.2.y.a 256
17.e odd 16 1 inner 459.2.y.a 256
51.i even 16 1 153.2.s.a ✓ 256
153.s even 48 1 inner 459.2.y.a 256
153.t odd 48 1 153.2.s.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.s.a ✓ 256 3.b odd 2 1
153.2.s.a ✓ 256 9.c even 3 1
153.2.s.a ✓ 256 51.i even 16 1
153.2.s.a ✓ 256 153.t odd 48 1
459.2.y.a 256 1.a even 1 1 trivial
459.2.y.a 256 9.d odd 6 1 inner
459.2.y.a 256 17.e odd 16 1 inner
459.2.y.a 256 153.s even 48 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(459, [\chi])\).