Properties

Label 232.4.s.a
Level $232$
Weight $4$
Character orbit 232.s
Analytic conductor $13.688$
Analytic rank $0$
Dimension $528$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [232,4,Mod(45,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 7, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.45");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 232.s (of order \(14\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.6884431213\)
Analytic rank: \(0\)
Dimension: \(528\)
Relative dimension: \(88\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 528 q - 3 q^{2} + 5 q^{4} - 51 q^{6} + 18 q^{7} - 24 q^{8} + 746 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 528 q - 3 q^{2} + 5 q^{4} - 51 q^{6} + 18 q^{7} - 24 q^{8} + 746 q^{9} - 21 q^{10} + 12 q^{12} + 81 q^{14} - 118 q^{15} - 155 q^{16} + 28 q^{17} - 914 q^{18} - 287 q^{20} - 472 q^{22} + 594 q^{23} + 17 q^{24} + 2034 q^{25} - 101 q^{26} + 256 q^{28} + 302 q^{30} - 10 q^{31} - 358 q^{32} - 118 q^{33} - 279 q^{34} + 318 q^{36} - 1934 q^{38} - 1618 q^{39} + 323 q^{40} + 212 q^{41} - 2132 q^{42} - 1192 q^{44} + 4652 q^{46} - 10 q^{47} + 544 q^{48} - 3734 q^{49} + 258 q^{50} + 4214 q^{52} - 1778 q^{54} - 510 q^{55} + 2759 q^{56} - 132 q^{57} - 3955 q^{58} + 7050 q^{60} + 2287 q^{62} - 6 q^{63} - 298 q^{64} - 420 q^{65} + 2315 q^{66} - 904 q^{68} + 10686 q^{70} + 498 q^{71} - 693 q^{72} - 1860 q^{73} + 766 q^{74} - 3657 q^{76} + 3125 q^{78} - 1510 q^{79} - 2224 q^{80} - 4718 q^{81} + 2362 q^{82} + 961 q^{84} - 4520 q^{86} - 2738 q^{87} - 630 q^{88} - 1078 q^{89} + 5748 q^{90} - 478 q^{92} - 5610 q^{94} - 510 q^{95} - 256 q^{96} + 772 q^{97} + 1430 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −2.82734 + 0.0784509i 3.89696 + 3.10772i 7.98769 0.443615i −2.39521 + 4.97371i −11.2618 8.48087i 4.79075 6.00741i −22.5491 + 1.88089i −0.479694 2.10168i 6.38189 14.2503i
45.2 −2.82725 0.0816801i 5.31996 + 4.24253i 7.98666 + 0.461860i −1.20408 + 2.50030i −14.6943 12.4292i −2.55349 + 3.20198i −22.5425 1.95814i 4.29489 + 18.8171i 3.60846 6.97062i
45.3 −2.82293 + 0.176241i −2.18494 1.74244i 7.93788 0.995030i 5.84824 12.1440i 6.47504 + 4.53370i −7.03876 + 8.82633i −22.2327 + 4.20788i −4.27016 18.7088i −14.3689 + 35.3124i
45.4 −2.80684 0.348789i −3.62321 2.88941i 7.75669 + 1.95799i 3.00172 6.23314i 9.16197 + 9.37385i 15.7972 19.8091i −21.0889 8.20120i −1.22913 5.38517i −10.5994 + 16.4485i
45.5 −2.76834 0.579913i 0.136331 + 0.108720i 7.32740 + 3.21079i −3.15436 + 6.55010i −0.314361 0.380034i −17.1707 + 21.5314i −18.4228 13.1378i −6.00130 26.2934i 12.5308 16.3036i
45.6 −2.74857 + 0.667358i −6.45303 5.14612i 7.10927 3.66856i −2.02619 + 4.20744i 21.1709 + 9.83798i 9.78624 12.2716i −17.0921 + 14.8277i 9.15096 + 40.0930i 2.76127 12.9166i
45.7 −2.73296 0.728638i −6.28332 5.01078i 6.93817 + 3.98268i 7.75346 16.1002i 13.5210 + 18.2725i −10.1355 + 12.7095i −16.0598 15.9399i 8.36410 + 36.6455i −32.9211 + 38.3518i
45.8 −2.71900 + 0.779112i −4.55540 3.63281i 6.78597 4.23682i −6.30657 + 13.0957i 15.2165 + 6.32847i −20.2050 + 25.3362i −15.1501 + 16.8069i 1.54631 + 6.77482i 6.94456 40.5209i
45.9 −2.65907 + 0.964034i 3.90835 + 3.11681i 6.14128 5.12686i 9.06800 18.8299i −13.3973 4.52001i −12.7745 + 16.0187i −11.3876 + 19.5531i −0.447331 1.95988i −5.95975 + 58.8118i
45.10 −2.62649 + 1.04956i 1.06985 + 0.853177i 5.79685 5.51331i −9.50600 + 19.7394i −3.70541 1.11799i 9.71878 12.1870i −9.43879 + 20.5648i −5.59140 24.4975i 4.24968 61.8224i
45.11 −2.59668 1.12126i −2.56621 2.04649i 5.48553 + 5.82314i −6.45055 + 13.3947i 4.36900 + 8.19149i 10.5690 13.2531i −7.71493 21.2716i −3.61072 15.8196i 31.7690 27.5491i
45.12 −2.57901 1.16135i 7.45001 + 5.94119i 5.30255 + 5.99024i 4.54665 9.44122i −12.3139 23.9744i −7.61140 + 9.54439i −6.71857 21.6070i 14.1969 + 62.2008i −22.6904 + 19.0687i
45.13 −2.53868 + 1.24705i 0.541991 + 0.432224i 4.88976 6.33169i 3.55640 7.38494i −1.91494 0.421388i 17.4431 21.8729i −4.51761 + 22.1719i −5.90113 25.8545i 0.180806 + 23.1830i
45.14 −2.52067 + 1.28304i 7.63258 + 6.08678i 4.70760 6.46827i 4.15605 8.63012i −27.0489 5.54985i 19.9393 25.0030i −3.56724 + 22.3445i 15.1993 + 66.5927i 0.596784 + 27.0861i
45.15 −2.51596 1.29226i −7.45001 5.94119i 4.66012 + 6.50256i −4.54665 + 9.44122i 11.0664 + 24.5752i −7.61140 + 9.54439i −3.32168 22.3823i 14.1969 + 62.2008i 23.6397 17.8783i
45.16 −2.49565 1.33107i 2.56621 + 2.04649i 4.45649 + 6.64377i 6.45055 13.3947i −3.68034 8.52313i 10.5690 13.2531i −2.27848 22.5124i −3.61072 15.8196i −33.9276 + 24.8423i
45.17 −2.45878 + 1.39801i −1.85288 1.47762i 4.09116 6.87477i 0.379073 0.787153i 6.62153 + 1.04281i −7.38060 + 9.25499i −0.448292 + 22.6230i −4.75827 20.8474i 0.168389 + 2.46538i
45.18 −2.30539 + 1.63865i 7.18394 + 5.72900i 2.62963 7.55546i −8.29334 + 17.2213i −25.9496 1.43559i −7.58299 + 9.50877i 6.31845 + 21.7273i 12.7795 + 55.9906i −9.10035 53.2917i
45.19 −2.27365 1.68242i 6.28332 + 5.01078i 2.33894 + 7.65045i −7.75346 + 16.1002i −5.85582 21.9639i −10.1355 + 12.7095i 7.55334 21.3295i 8.36410 + 36.6455i 44.7159 23.5617i
45.20 −2.21092 + 1.76404i −7.04646 5.61936i 1.77633 7.80030i 5.26037 10.9233i 25.4919 0.00627760i 1.30089 1.63127i 9.83273 + 20.3793i 12.0673 + 52.8702i 7.63884 + 33.4300i
See next 80 embeddings (of 528 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
29.d even 7 1 inner
232.s even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.4.s.a 528
8.b even 2 1 inner 232.4.s.a 528
29.d even 7 1 inner 232.4.s.a 528
232.s even 14 1 inner 232.4.s.a 528
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.s.a 528 1.a even 1 1 trivial
232.4.s.a 528 8.b even 2 1 inner
232.4.s.a 528 29.d even 7 1 inner
232.4.s.a 528 232.s even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(232, [\chi])\).