↑

Properties

Label 171.4.u.b
Level $171$
Weight $4$
Character orbit 171.u
Analytic conductor $10.089$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [171,4,Mod(28,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.28");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 171.u (of order \(9\), 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: \(10.0893266110\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{2} - 24 q^{4} + 6 q^{5} + 3 q^{7} + 75 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{2} - 24 q^{4} + 6 q^{5} + 3 q^{7} + 75 q^{8} + 75 q^{10} - 39 q^{11} - 156 q^{13} - 93 q^{14} + 504 q^{16} - 12 q^{17} + 546 q^{19} + 198 q^{20} - 6 q^{22} - 6 q^{23} - 498 q^{25} + 639 q^{26} - 1368 q^{28} + 630 q^{29} - 591 q^{31} - 147 q^{32} - 408 q^{34} - 2001 q^{35} - 72 q^{37} - 2934 q^{38} + 2886 q^{40} + 477 q^{41} + 588 q^{43} + 3423 q^{44} - 1728 q^{46} + 1242 q^{47} - 639 q^{49} + 1788 q^{50} + 2733 q^{52} + 300 q^{53} + 315 q^{55} - 4638 q^{56} - 2820 q^{58} - 2097 q^{59} - 2316 q^{61} + 1320 q^{62} - 1785 q^{64} + 2433 q^{65} + 57 q^{67} + 438 q^{68} - 213 q^{70} + 792 q^{71} + 4068 q^{73} - 4287 q^{74} + 5538 q^{76} - 3786 q^{77} + 1824 q^{79} + 2739 q^{80} + 2205 q^{82} - 1071 q^{83} - 2394 q^{85} + 5256 q^{86} + 1101 q^{88} + 3006 q^{89} - 3285 q^{91} + 1452 q^{92} - 1086 q^{94} + 3078 q^{95} - 2535 q^{97} + 2403 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −0.824938 − 4.67846i 0 −13.6899 + 4.98271i 14.0244 + 5.10445i 0 3.64598 − 6.31503i 15.6022 + 27.0238i 0 12.3117 − 69.8233i
28.2 −0.477474 − 2.70789i 0 0.412858 − 0.150268i −5.11043 − 1.86004i 0 11.7392 − 20.3328i −11.6027 − 20.0964i 0 −2.59670 + 14.7266i
28.3 0.163781 + 0.928850i 0 6.68160 − 2.43190i 3.55727 + 1.29474i 0 −11.7083 + 20.2794i 7.12592 + 12.3424i 0 −0.620006 + 3.51623i
28.4 0.851642 + 4.82990i 0 −15.0851 + 5.49052i −6.94640 − 2.52828i 0 14.5751 − 25.2448i −19.7481 − 34.2048i 0 6.29551 − 35.7036i
55.1 −0.824938 + 4.67846i 0 −13.6899 − 4.98271i 14.0244 − 5.10445i 0 3.64598 + 6.31503i 15.6022 − 27.0238i 0 12.3117 + 69.8233i
55.2 −0.477474 + 2.70789i 0 0.412858 + 0.150268i −5.11043 + 1.86004i 0 11.7392 + 20.3328i −11.6027 + 20.0964i 0 −2.59670 − 14.7266i
55.3 0.163781 − 0.928850i 0 6.68160 + 2.43190i 3.55727 − 1.29474i 0 −11.7083 − 20.2794i 7.12592 − 12.3424i 0 −0.620006 − 3.51623i
55.4 0.851642 − 4.82990i 0 −15.0851 − 5.49052i −6.94640 + 2.52828i 0 14.5751 + 25.2448i −19.7481 + 34.2048i 0 6.29551 + 35.7036i
73.1 −2.98693 − 2.50633i 0 1.25086 + 7.09397i −2.58856 + 14.6804i 0 5.35146 + 9.26900i −1.55302 + 2.68992i 0 44.5258 − 37.3616i
73.2 −1.38101 − 1.15881i 0 −0.824823 − 4.67780i 3.13553 − 17.7825i 0 −14.1277 − 24.4699i −11.4927 + 19.9060i 0 −24.9367 + 20.9244i
73.3 1.74757 + 1.46638i 0 −0.485473 − 2.75325i −1.27381 + 7.22413i 0 13.1019 + 22.6932i 12.3141 − 21.3286i 0 −12.8194 + 10.7568i
73.4 2.26193 + 1.89799i 0 0.124797 + 0.707761i 0.0925462 − 0.524856i 0 −11.8949 − 20.6025i 10.7499 − 18.6194i 0 1.20550 − 1.01154i
82.1 −2.98693 + 2.50633i 0 1.25086 − 7.09397i −2.58856 − 14.6804i 0 5.35146 − 9.26900i −1.55302 − 2.68992i 0 44.5258 + 37.3616i
82.2 −1.38101 + 1.15881i 0 −0.824823 + 4.67780i 3.13553 + 17.7825i 0 −14.1277 + 24.4699i −11.4927 − 19.9060i 0 −24.9367 − 20.9244i
82.3 1.74757 − 1.46638i 0 −0.485473 + 2.75325i −1.27381 − 7.22413i 0 13.1019 − 22.6932i 12.3141 + 21.3286i 0 −12.8194 − 10.7568i
82.4 2.26193 − 1.89799i 0 0.124797 − 0.707761i 0.0925462 + 0.524856i 0 −11.8949 + 20.6025i 10.7499 + 18.6194i 0 1.20550 + 1.01154i
100.1 −2.98397 + 1.08608i 0 1.59618 − 1.33936i 6.63193 + 5.56485i 0 −4.36312 − 7.55715i 9.39359 − 16.2702i 0 −25.8334 − 9.40257i
100.2 −0.579114 + 0.210780i 0 −5.83741 + 4.89817i −9.88078 − 8.29096i 0 6.05695 + 10.4909i 4.81321 − 8.33672i 0 7.46967 + 2.71874i
100.3 2.16911 − 0.789492i 0 −2.04660 + 1.71731i −6.28846 − 5.27664i 0 −1.44586 − 2.50430i −12.3168 + 21.3333i 0 −17.8062 − 6.48094i
100.4 5.03941 − 1.83419i 0 15.9030 − 13.3442i 7.64679 + 6.41642i 0 −9.43070 − 16.3344i 34.2145 − 59.2612i 0 50.3042 + 18.3092i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.4.u.b 24
3.b odd 2 1 19.4.e.a ✓ 24
19.e even 9 1 inner 171.4.u.b 24
57.j even 18 1 361.4.a.m 12
57.l odd 18 1 19.4.e.a ✓ 24
57.l odd 18 1 361.4.a.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.e.a ✓ 24 3.b odd 2 1
19.4.e.a ✓ 24 57.l odd 18 1
171.4.u.b 24 1.a even 1 1 trivial
171.4.u.b 24 19.e even 9 1 inner
361.4.a.m 12 57.j even 18 1
361.4.a.n 12 57.l odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 6 T_{2}^{23} + 30 T_{2}^{22} - 165 T_{2}^{21} + 324 T_{2}^{20} - 507 T_{2}^{19} + \cdots + 4804153344 \) acting on \(S_{4}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display