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Properties

Label 504.3.l.h
Level $504$
Weight $3$
Character orbit 504.l
Analytic conductor $13.733$
Analytic rank $0$
Dimension $32$
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] = [504,3,Mod(181,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("504.181"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 504.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,2,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7330053238\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} - 6 q^{4} + 2 q^{8} - 38 q^{14} - 46 q^{16} + 16 q^{22} - 64 q^{23} + 160 q^{25} - 122 q^{28} + 122 q^{32} + 216 q^{44} - 260 q^{46} - 16 q^{49} - 122 q^{50} - 98 q^{56} - 160 q^{58} - 174 q^{64}+ \cdots + 2 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 −1.91858 − 0.564843i 0 3.36191 + 2.16739i −0.227906 0 2.25697 + 6.62617i −5.22585 − 6.05727i 0 0.437256 + 0.128731i
181.2 −1.91858 − 0.564843i 0 3.36191 + 2.16739i 0.227906 0 2.25697 − 6.62617i −5.22585 − 6.05727i 0 −0.437256 − 0.128731i
181.3 −1.91858 + 0.564843i 0 3.36191 − 2.16739i −0.227906 0 2.25697 − 6.62617i −5.22585 + 6.05727i 0 0.437256 − 0.128731i
181.4 −1.91858 + 0.564843i 0 3.36191 − 2.16739i 0.227906 0 2.25697 + 6.62617i −5.22585 + 6.05727i 0 −0.437256 + 0.128731i
181.5 −1.44694 − 1.38071i 0 0.187253 + 3.99561i −5.24467 0 −6.31936 + 3.01094i 5.24586 − 6.03995i 0 7.58871 + 7.24140i
181.6 −1.44694 − 1.38071i 0 0.187253 + 3.99561i 5.24467 0 −6.31936 − 3.01094i 5.24586 − 6.03995i 0 −7.58871 − 7.24140i
181.7 −1.44694 + 1.38071i 0 0.187253 − 3.99561i −5.24467 0 −6.31936 − 3.01094i 5.24586 + 6.03995i 0 7.58871 − 7.24140i
181.8 −1.44694 + 1.38071i 0 0.187253 − 3.99561i 5.24467 0 −6.31936 + 3.01094i 5.24586 + 6.03995i 0 −7.58871 + 7.24140i
181.9 −1.19641 − 1.60268i 0 −1.13719 + 3.83494i −9.09019 0 3.83254 − 5.85761i 7.50675 − 2.76563i 0 10.8756 + 14.5687i
181.10 −1.19641 − 1.60268i 0 −1.13719 + 3.83494i 9.09019 0 3.83254 + 5.85761i 7.50675 − 2.76563i 0 −10.8756 − 14.5687i
181.11 −1.19641 + 1.60268i 0 −1.13719 − 3.83494i −9.09019 0 3.83254 + 5.85761i 7.50675 + 2.76563i 0 10.8756 − 14.5687i
181.12 −1.19641 + 1.60268i 0 −1.13719 − 3.83494i 9.09019 0 3.83254 − 5.85761i 7.50675 + 2.76563i 0 −10.8756 + 14.5687i
181.13 −0.172794 − 1.99252i 0 −3.94028 + 0.688593i −6.16756 0 6.35406 + 2.93700i 2.05289 + 7.73212i 0 1.06572 + 12.2890i
181.14 −0.172794 − 1.99252i 0 −3.94028 + 0.688593i 6.16756 0 6.35406 − 2.93700i 2.05289 + 7.73212i 0 −1.06572 − 12.2890i
181.15 −0.172794 + 1.99252i 0 −3.94028 − 0.688593i −6.16756 0 6.35406 − 2.93700i 2.05289 − 7.73212i 0 1.06572 − 12.2890i
181.16 −0.172794 + 1.99252i 0 −3.94028 − 0.688593i 6.16756 0 6.35406 + 2.93700i 2.05289 − 7.73212i 0 −1.06572 + 12.2890i
181.17 0.606108 − 1.90595i 0 −3.26527 − 2.31042i −5.53884 0 −5.95316 − 3.68238i −6.38264 + 4.82306i 0 −3.35714 + 10.5567i
181.18 0.606108 − 1.90595i 0 −3.26527 − 2.31042i 5.53884 0 −5.95316 + 3.68238i −6.38264 + 4.82306i 0 3.35714 − 10.5567i
181.19 0.606108 + 1.90595i 0 −3.26527 + 2.31042i −5.53884 0 −5.95316 + 3.68238i −6.38264 − 4.82306i 0 −3.35714 − 10.5567i
181.20 0.606108 + 1.90595i 0 −3.26527 + 2.31042i 5.53884 0 −5.95316 − 3.68238i −6.38264 − 4.82306i 0 3.35714 + 10.5567i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.b even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.3.l.h 32
3.b odd 2 1 168.3.l.a ✓ 32
4.b odd 2 1 2016.3.l.h 32
7.b odd 2 1 inner 504.3.l.h 32
8.b even 2 1 inner 504.3.l.h 32
8.d odd 2 1 2016.3.l.h 32
12.b even 2 1 672.3.l.a 32
21.c even 2 1 168.3.l.a ✓ 32
24.f even 2 1 672.3.l.a 32
24.h odd 2 1 168.3.l.a ✓ 32
28.d even 2 1 2016.3.l.h 32
56.e even 2 1 2016.3.l.h 32
56.h odd 2 1 inner 504.3.l.h 32
84.h odd 2 1 672.3.l.a 32
168.e odd 2 1 672.3.l.a 32
168.i even 2 1 168.3.l.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.l.a ✓ 32 3.b odd 2 1
168.3.l.a ✓ 32 21.c even 2 1
168.3.l.a ✓ 32 24.h odd 2 1
168.3.l.a ✓ 32 168.i even 2 1
504.3.l.h 32 1.a even 1 1 trivial
504.3.l.h 32 7.b odd 2 1 inner
504.3.l.h 32 8.b even 2 1 inner
504.3.l.h 32 56.h odd 2 1 inner
672.3.l.a 32 12.b even 2 1
672.3.l.a 32 24.f even 2 1
672.3.l.a 32 84.h odd 2 1
672.3.l.a 32 168.e odd 2 1
2016.3.l.h 32 4.b odd 2 1
2016.3.l.h 32 8.d odd 2 1
2016.3.l.h 32 28.d even 2 1
2016.3.l.h 32 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(504, [\chi])\):

\( T_{5}^{16} - 240 T_{5}^{14} + 22368 T_{5}^{12} - 1034336 T_{5}^{10} + 24858624 T_{5}^{8} + \cdots + 99680256 \) Copy content Toggle raw display
\( T_{11}^{16} + 1144 T_{11}^{14} + 534672 T_{11}^{12} + 130731744 T_{11}^{10} + 17758113408 T_{11}^{8} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display