↑

Properties

Label 403.2.h.a
Level $403$
Weight $2$
Character orbit 403.h
Analytic conductor $3.218$
Analytic rank $0$
Dimension $30$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [403,2,Mod(118,403)] 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("403.118"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(403, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 6 q^{2} - 2 q^{3} + 22 q^{4} + 7 q^{5} + 6 q^{7} - 24 q^{8} - 13 q^{9} - 3 q^{10} - 3 q^{11} + 12 q^{12} + 15 q^{13} + 3 q^{14} + 4 q^{15} + 22 q^{16} + 4 q^{17} - 6 q^{18} + 5 q^{19} + 10 q^{20}+ \cdots + 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1 −2.75405 1.04794 + 1.81508i 5.58479 1.03612 − 1.79462i −2.88606 − 4.99881i 2.05805 + 3.56464i −9.87267 −0.696336 + 1.20609i −2.85353 + 4.94246i
118.2 −2.54177 −0.358627 − 0.621161i 4.46058 −1.54812 + 2.68142i 0.911547 + 1.57885i −1.09448 − 1.89569i −6.25421 1.24277 − 2.15255i 3.93495 − 6.81554i
118.3 −2.19969 −0.332345 − 0.575638i 2.83864 2.12692 − 3.68393i 0.731057 + 1.26623i −0.0876538 − 0.151821i −1.84476 1.27909 − 2.21546i −4.67856 + 8.10350i
118.4 −1.54088 −0.945700 − 1.63800i 0.374297 0.246770 − 0.427419i 1.45721 + 2.52395i −2.41271 − 4.17893i 2.50501 −0.288697 + 0.500038i −0.380242 + 0.658599i
118.5 −1.20534 −1.24518 − 2.15671i −0.547152 0.667125 − 1.15549i 1.50087 + 2.59957i 1.94340 + 3.36606i 3.07019 −1.60094 + 2.77291i −0.804114 + 1.39277i
118.6 −1.11604 0.521580 + 0.903404i −0.754455 −0.0866186 + 0.150028i −0.582104 − 1.00823i −0.0906774 − 0.157058i 3.07408 0.955908 − 1.65568i 0.0966698 − 0.167437i
118.7 −0.899619 1.46679 + 2.54055i −1.19069 −1.03352 + 1.79011i −1.31955 − 2.28553i 1.98012 + 3.42966i 2.87040 −2.80293 + 4.85482i 0.929773 − 1.61041i
118.8 0.0439466 0.820034 + 1.42034i −1.99807 0.865000 − 1.49822i 0.0360377 + 0.0624192i −2.17574 − 3.76849i −0.175702 0.155089 − 0.268621i 0.0380138 − 0.0658419i
118.9 0.165462 −0.350644 − 0.607334i −1.97262 −1.35983 + 2.35530i −0.0580182 − 0.100491i 0.231883 + 0.401634i −0.657317 1.25410 − 2.17216i −0.225001 + 0.389713i
118.10 0.352077 −1.65894 − 2.87336i −1.87604 1.09161 − 1.89072i −0.584073 − 1.01164i −1.14275 − 1.97930i −1.36466 −4.00415 + 6.93538i 0.384330 − 0.665679i
118.11 1.21548 −0.671133 − 1.16244i −0.522608 1.76937 − 3.06463i −0.815749 − 1.41292i 0.724247 + 1.25443i −3.06618 0.599162 − 1.03778i 2.15063 − 3.72500i
118.12 1.41262 −1.31623 − 2.27977i −0.00449076 −1.72422 + 2.98644i −1.85934 − 3.22047i 2.33584 + 4.04580i −2.83159 −1.96492 + 3.40333i −2.43568 + 4.21871i
118.13 1.58004 1.43561 + 2.48655i 0.496524 1.71834 − 2.97625i 2.26832 + 3.92885i 0.908007 + 1.57271i −2.37555 −2.62196 + 4.54137i 2.71504 − 4.70259i
118.14 2.10012 0.700181 + 1.21275i 2.41051 −0.938510 + 1.62555i 1.47047 + 2.54692i 0.810515 + 1.40385i 0.862110 0.519492 − 0.899787i −1.97098 + 3.41385i
118.15 2.38763 −0.113336 − 0.196303i 3.70079 0.669573 − 1.15973i −0.270604 − 0.468700i −0.988047 − 1.71135i 4.06087 1.47431 − 2.55358i 1.59869 − 2.76902i
222.1 −2.75405 1.04794 − 1.81508i 5.58479 1.03612 + 1.79462i −2.88606 + 4.99881i 2.05805 − 3.56464i −9.87267 −0.696336 − 1.20609i −2.85353 − 4.94246i
222.2 −2.54177 −0.358627 + 0.621161i 4.46058 −1.54812 − 2.68142i 0.911547 − 1.57885i −1.09448 + 1.89569i −6.25421 1.24277 + 2.15255i 3.93495 + 6.81554i
222.3 −2.19969 −0.332345 + 0.575638i 2.83864 2.12692 + 3.68393i 0.731057 − 1.26623i −0.0876538 + 0.151821i −1.84476 1.27909 + 2.21546i −4.67856 − 8.10350i
222.4 −1.54088 −0.945700 + 1.63800i 0.374297 0.246770 + 0.427419i 1.45721 − 2.52395i −2.41271 + 4.17893i 2.50501 −0.288697 − 0.500038i −0.380242 − 0.658599i
222.5 −1.20534 −1.24518 + 2.15671i −0.547152 0.667125 + 1.15549i 1.50087 − 2.59957i 1.94340 − 3.36606i 3.07019 −1.60094 − 2.77291i −0.804114 − 1.39277i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.15
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.h.a ✓ 30
31.c even 3 1 inner 403.2.h.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.h.a ✓ 30 1.a even 1 1 trivial
403.2.h.a ✓ 30 31.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} + 3 T_{2}^{14} - 16 T_{2}^{13} - 49 T_{2}^{12} + 96 T_{2}^{11} + 303 T_{2}^{10} - 268 T_{2}^{9} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\). Copy content Toggle raw display