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Properties

Label 275.3.bk.c
Level $275$
Weight $3$
Character orbit 275.bk
Analytic conductor $7.493$
Analytic rank $0$
Dimension $128$
Inner twists $8$

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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] = [275,3,Mod(82,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([5, 8])) 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("275.82"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 275.bk (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128] 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: \(7.49320726991\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 48 q^{6} + 32 q^{11} + 280 q^{16} - 8 q^{21} + 528 q^{26} - 16 q^{31} + 336 q^{36} - 604 q^{41} - 1152 q^{46} - 1024 q^{51} - 992 q^{56} - 320 q^{61} + 1624 q^{66} + 536 q^{71} + 1776 q^{76} + 3680 q^{81}+ \cdots - 6584 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −1.72172 + 3.37907i 2.24871 − 0.356160i −6.10266 − 8.39959i 0 −2.66816 + 8.21175i 0.264482 + 0.0418898i 23.9070 − 3.78649i −3.62968 + 1.17935i 0
82.2 −1.54871 + 3.03951i 2.39044 − 0.378609i −4.48900 − 6.17857i 0 −2.55131 + 7.85213i 2.23278 + 0.353637i 12.2547 − 1.94095i −2.98864 + 0.971069i 0
82.3 −1.35668 + 2.66263i −5.64189 + 0.893588i −2.89790 − 3.98862i 0 5.27495 − 16.2346i 7.26024 + 1.14991i 2.74554 − 0.434852i 22.4730 − 7.30190i 0
82.4 −1.11368 + 2.18572i −3.11704 + 0.493690i −1.18596 − 1.63233i 0 2.39232 − 7.36280i −1.55118 − 0.245682i −4.80296 + 0.760714i 0.912687 − 0.296550i 0
82.5 −0.807274 + 1.58436i 0.0974353 − 0.0154322i 0.492622 + 0.678036i 0 −0.0542067 + 0.166831i −2.29002 − 0.362704i −8.49706 + 1.34580i −8.55025 + 2.77815i 0
82.6 −0.534746 + 1.04950i 5.70073 − 0.902907i 1.53565 + 2.11364i 0 −2.10084 + 6.46573i −1.94938 − 0.308752i −7.69295 + 1.21844i 23.1236 − 7.51330i 0
82.7 −0.319362 + 0.626784i 4.31726 − 0.683787i 2.06028 + 2.83573i 0 −0.950184 + 2.92437i 13.0084 + 2.06033i −5.21454 + 0.825902i 9.61169 − 3.12303i 0
82.8 −0.166354 + 0.326489i 0.928883 − 0.147121i 2.27222 + 3.12744i 0 −0.106490 + 0.327744i −10.4047 − 1.64794i −2.84673 + 0.450878i −7.71833 + 2.50784i 0
82.9 0.166354 − 0.326489i −0.928883 + 0.147121i 2.27222 + 3.12744i 0 −0.106490 + 0.327744i 10.4047 + 1.64794i 2.84673 − 0.450878i −7.71833 + 2.50784i 0
82.10 0.319362 − 0.626784i −4.31726 + 0.683787i 2.06028 + 2.83573i 0 −0.950184 + 2.92437i −13.0084 − 2.06033i 5.21454 − 0.825902i 9.61169 − 3.12303i 0
82.11 0.534746 − 1.04950i −5.70073 + 0.902907i 1.53565 + 2.11364i 0 −2.10084 + 6.46573i 1.94938 + 0.308752i 7.69295 − 1.21844i 23.1236 − 7.51330i 0
82.12 0.807274 − 1.58436i −0.0974353 + 0.0154322i 0.492622 + 0.678036i 0 −0.0542067 + 0.166831i 2.29002 + 0.362704i 8.49706 − 1.34580i −8.55025 + 2.77815i 0
82.13 1.11368 − 2.18572i 3.11704 − 0.493690i −1.18596 − 1.63233i 0 2.39232 − 7.36280i 1.55118 + 0.245682i 4.80296 − 0.760714i 0.912687 − 0.296550i 0
82.14 1.35668 − 2.66263i 5.64189 − 0.893588i −2.89790 − 3.98862i 0 5.27495 − 16.2346i −7.26024 − 1.14991i −2.74554 + 0.434852i 22.4730 − 7.30190i 0
82.15 1.54871 − 3.03951i −2.39044 + 0.378609i −4.48900 − 6.17857i 0 −2.55131 + 7.85213i −2.23278 − 0.353637i −12.2547 + 1.94095i −2.98864 + 0.971069i 0
82.16 1.72172 − 3.37907i −2.24871 + 0.356160i −6.10266 − 8.39959i 0 −2.66816 + 8.21175i −0.264482 − 0.0418898i −23.9070 + 3.78649i −3.62968 + 1.17935i 0
93.1 −3.37907 − 1.72172i −0.356160 − 2.24871i 6.10266 + 8.39959i 0 −2.66816 + 8.21175i −0.0418898 + 0.264482i −3.78649 − 23.9070i 3.62968 − 1.17935i 0
93.2 −3.03951 − 1.54871i −0.378609 − 2.39044i 4.48900 + 6.17857i 0 −2.55131 + 7.85213i −0.353637 + 2.23278i −1.94095 − 12.2547i 2.98864 − 0.971069i 0
93.3 −2.66263 − 1.35668i 0.893588 + 5.64189i 2.89790 + 3.98862i 0 5.27495 − 16.2346i −1.14991 + 7.26024i −0.434852 − 2.74554i −22.4730 + 7.30190i 0
93.4 −2.18572 − 1.11368i 0.493690 + 3.11704i 1.18596 + 1.63233i 0 2.39232 − 7.36280i 0.245682 − 1.55118i 0.760714 + 4.80296i −0.912687 + 0.296550i 0
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
11.c even 5 1 inner
55.j even 10 1 inner
55.k odd 20 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.3.bk.c ✓ 128
5.b even 2 1 inner 275.3.bk.c ✓ 128
5.c odd 4 2 inner 275.3.bk.c ✓ 128
11.c even 5 1 inner 275.3.bk.c ✓ 128
55.j even 10 1 inner 275.3.bk.c ✓ 128
55.k odd 20 2 inner 275.3.bk.c ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.3.bk.c ✓ 128 1.a even 1 1 trivial
275.3.bk.c ✓ 128 5.b even 2 1 inner
275.3.bk.c ✓ 128 5.c odd 4 2 inner
275.3.bk.c ✓ 128 11.c even 5 1 inner
275.3.bk.c ✓ 128 55.j even 10 1 inner
275.3.bk.c ✓ 128 55.k odd 20 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} - 626 T_{2}^{124} + 223399 T_{2}^{120} - 60936494 T_{2}^{116} + 14980842035 T_{2}^{112} + \cdots + 51\!\cdots\!41 \) acting on \(S_{3}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display