↑

Properties

Label 158.3.h.a
Level $158$
Weight $3$
Character orbit 158.h
Analytic conductor $4.305$
Analytic rank $0$
Dimension $336$
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] = [158,3,Mod(3,158)] 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("158.3"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(158, base_ring=CyclotomicField(78)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 158 = 2 \cdot 79 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 158.h (of order \(78\), degree \(24\), 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: \(4.30518817689\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(14\) over \(\Q(\zeta_{78})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q + 6 q^{3} + 28 q^{4} + 6 q^{7} - 44 q^{9} - 16 q^{10} - 28 q^{11} + 24 q^{13} - 260 q^{15} + 56 q^{16} + 48 q^{18} + 34 q^{19} + 156 q^{21} - 176 q^{22} + 6 q^{23} + 170 q^{25} + 48 q^{26} - 546 q^{27}+ \cdots - 812 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.39590 − 0.226856i −5.59597 − 0.225510i 1.89707 + 0.633336i −7.06611 + 5.30983i 7.76026 + 1.58427i −2.81526 + 4.45198i −2.50445 − 1.31444i 22.2932 + 1.79969i 11.0681 − 5.80901i
3.2 −1.39590 − 0.226856i −3.73807 − 0.150639i 1.89707 + 0.633336i 6.57335 − 4.93955i 5.18380 + 1.05828i −5.84455 + 9.24242i −2.50445 − 1.31444i 4.97965 + 0.401999i −10.2963 + 5.40391i
3.3 −1.39590 − 0.226856i −2.55035 − 0.102776i 1.89707 + 0.633336i 0.219168 − 0.164694i 3.53672 + 0.722027i 1.14253 − 1.80676i −2.50445 − 1.31444i −2.47708 − 0.199970i −0.343298 + 0.180177i
3.4 −1.39590 − 0.226856i −0.186135 − 0.00750097i 1.89707 + 0.633336i −5.56264 + 4.18005i 0.258124 + 0.0526963i 4.53253 − 7.16763i −2.50445 − 1.31444i −8.93623 − 0.721407i 8.71316 − 4.57302i
3.5 −1.39590 − 0.226856i 1.95277 + 0.0786938i 1.89707 + 0.633336i −2.12938 + 1.60013i −2.70801 − 0.552845i −5.96105 + 9.42665i −2.50445 − 1.31444i −5.16371 − 0.416858i 3.33540 − 1.75055i
3.6 −1.39590 − 0.226856i 2.59298 + 0.104494i 1.89707 + 0.633336i 6.35889 − 4.77840i −3.59584 − 0.734095i 2.58422 − 4.08662i −2.50445 − 1.31444i −2.25819 − 0.182300i −9.96039 + 5.22761i
3.7 −1.39590 − 0.226856i 5.79413 + 0.233496i 1.89707 + 0.633336i −2.55373 + 1.91901i −8.03506 − 1.64037i 3.75813 − 5.94301i −2.50445 − 1.31444i 24.5466 + 1.98161i 4.00009 − 2.09941i
3.8 1.39590 + 0.226856i −5.06743 − 0.204211i 1.89707 + 0.633336i 3.03611 − 2.28149i −7.02730 − 1.43463i −5.64621 + 8.92877i 2.50445 + 1.31444i 16.6664 + 1.34545i 4.75568 − 2.49597i
3.9 1.39590 + 0.226856i −4.77860 − 0.192571i 1.89707 + 0.633336i −0.779646 + 0.585866i −6.62676 − 1.35286i 3.94628 − 6.24054i 2.50445 + 1.31444i 13.8271 + 1.11624i −1.22121 + 0.640943i
3.10 1.39590 + 0.226856i −1.21410 − 0.0489267i 1.89707 + 0.633336i −5.58202 + 4.19461i −1.68367 − 0.343723i −4.14179 + 6.54971i 2.50445 + 1.31444i −7.49916 − 0.605395i −8.74351 + 4.58895i
3.11 1.39590 + 0.226856i −0.615776 − 0.0248149i 1.89707 + 0.633336i 5.61999 − 4.22314i −0.853932 − 0.174332i 0.536230 − 0.847981i 2.50445 + 1.31444i −8.59225 − 0.693638i 8.80298 − 4.62016i
3.12 1.39590 + 0.226856i 2.21105 + 0.0891024i 1.89707 + 0.633336i 0.499313 − 0.375209i 3.06620 + 0.625969i 7.32221 − 11.5791i 2.50445 + 1.31444i −4.08999 − 0.330178i 0.782109 − 0.410482i
3.13 1.39590 + 0.226856i 3.36668 + 0.135673i 1.89707 + 0.633336i 4.56546 − 3.43072i 4.66877 + 0.953137i −3.56114 + 5.63150i 2.50445 + 1.31444i 2.34533 + 0.189335i 7.15121 − 3.75324i
3.14 1.39590 + 0.226856i 4.36752 + 0.176005i 1.89707 + 0.633336i −3.19876 + 2.40371i 6.05670 + 1.23648i −1.05902 + 1.67471i 2.50445 + 1.31444i 10.0735 + 0.813215i −5.01044 + 2.62968i
7.1 −0.894413 + 1.09546i −2.76698 + 4.37563i −0.400051 − 1.95958i −2.97215 − 1.26632i −2.31849 − 6.94472i −3.41540 − 0.137636i 2.50445 + 1.31444i −7.63170 − 16.0835i 4.04552 − 2.12325i
7.2 −0.894413 + 1.09546i −2.05516 + 3.24998i −0.400051 − 1.95958i 7.16544 + 3.05291i −1.72205 − 5.15816i 13.0731 + 0.526826i 2.50445 + 1.31444i −2.48042 − 5.22738i −9.75320 + 5.11887i
7.3 −0.894413 + 1.09546i −0.567893 + 0.898052i −0.400051 − 1.95958i 5.48490 + 2.33690i −0.475846 − 1.42533i −10.2604 − 0.413480i 2.50445 + 1.31444i 3.37424 + 7.11106i −7.46573 + 3.91832i
7.4 −0.894413 + 1.09546i −0.472998 + 0.747987i −0.400051 − 1.95958i −0.957780 − 0.408072i −0.396332 − 1.18716i −4.78404 − 0.192790i 2.50445 + 1.31444i 3.52248 + 7.42346i 1.30368 − 0.684222i
7.5 −0.894413 + 1.09546i 0.0394111 − 0.0623236i −0.400051 − 1.95958i −4.75882 − 2.02755i 0.0330231 + 0.0989162i 5.36757 + 0.216306i 2.50445 + 1.31444i 3.85590 + 8.12614i 6.47744 − 3.39962i
7.6 −0.894413 + 1.09546i 2.23257 − 3.53053i −0.400051 − 1.95958i 5.68127 + 2.42056i 1.87070 + 5.60344i 1.03703 + 0.0417909i 2.50445 + 1.31444i −3.62203 − 7.63326i −7.73302 + 4.05860i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 3.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.h odd 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 158.3.h.a ✓ 336
79.h odd 78 1 inner 158.3.h.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
158.3.h.a ✓ 336 1.a even 1 1 trivial
158.3.h.a ✓ 336 79.h odd 78 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(158, [\chi])\).