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Properties

Label 368.3.p.d
Level $368$
Weight $3$
Character orbit 368.p
Analytic conductor $10.027$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [368,3,Mod(17,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 368.p (of order \(22\), degree \(10\), not 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: \(10.0272737285\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(12\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 52 q^{9} - 4 q^{23} + 28 q^{25} + 96 q^{27} + 72 q^{29} - 8 q^{31} - 68 q^{35} + 352 q^{37} - 272 q^{39} - 88 q^{43} - 128 q^{47} - 124 q^{49} + 264 q^{51} - 176 q^{53} + 384 q^{55} - 528 q^{57} + 236 q^{59} + 280 q^{69} - 64 q^{71} + 144 q^{73} + 920 q^{75} - 216 q^{77} + 792 q^{79} - 1308 q^{81} + 528 q^{83} - 84 q^{85} + 488 q^{87} - 132 q^{89} - 552 q^{93} - 400 q^{95} + 396 q^{97} - 1584 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −4.92267 − 1.44543i 0 −0.256283 − 0.398785i 0 2.28953 + 0.329185i 0 14.5721 + 9.36493i 0
17.2 0 −3.94918 − 1.15958i 0 2.91222 + 4.53150i 0 −0.892557 − 0.128330i 0 6.68009 + 4.29303i 0
17.3 0 −3.55562 − 1.04402i 0 −4.97979 − 7.74870i 0 −7.18141 − 1.03253i 0 3.98113 + 2.55852i 0
17.4 0 −2.36121 − 0.693313i 0 2.71504 + 4.22469i 0 −9.50031 − 1.36594i 0 −2.47667 − 1.59166i 0
17.5 0 −1.39991 − 0.411052i 0 −4.80464 − 7.47616i 0 2.52864 + 0.363563i 0 −5.78049 − 3.71489i 0
17.6 0 −0.606518 − 0.178090i 0 −0.924166 − 1.43803i 0 13.0629 + 1.87817i 0 −7.23513 − 4.64974i 0
17.7 0 1.35898 + 0.399032i 0 −0.451980 − 0.703295i 0 −2.77270 − 0.398654i 0 −5.88368 − 3.78121i 0
17.8 0 1.42820 + 0.419357i 0 1.20977 + 1.88243i 0 −5.66991 − 0.815209i 0 −5.70739 − 3.66792i 0
17.9 0 1.53697 + 0.451294i 0 4.84539 + 7.53958i 0 9.14584 + 1.31497i 0 −5.41268 − 3.47852i 0
17.10 0 2.47704 + 0.727325i 0 −1.45163 − 2.25878i 0 −6.76396 − 0.972511i 0 −1.96455 − 1.26254i 0
17.11 0 4.66046 + 1.36844i 0 −2.04753 − 3.18602i 0 1.52914 + 0.219857i 0 12.2760 + 7.88932i 0
17.12 0 5.33345 + 1.56604i 0 3.23359 + 5.03157i 0 4.22475 + 0.607427i 0 18.4219 + 11.8390i 0
33.1 0 −4.55057 − 2.92447i 0 1.14162 + 0.521359i 0 −0.166662 + 0.567598i 0 8.41640 + 18.4294i 0
33.2 0 −3.60932 − 2.31957i 0 −1.93673 − 0.884477i 0 2.15085 − 7.32513i 0 3.90805 + 8.55743i 0
33.3 0 −2.93467 − 1.88600i 0 8.72769 + 3.98580i 0 −1.92467 + 6.55483i 0 1.31657 + 2.88288i 0
33.4 0 −1.47276 − 0.946488i 0 −8.03016 − 3.66725i 0 0.614011 − 2.09113i 0 −2.46554 − 5.39878i 0
33.5 0 −0.933084 − 0.599657i 0 7.14600 + 3.26347i 0 3.46763 − 11.8097i 0 −3.22768 − 7.06763i 0
33.6 0 −0.780462 − 0.501573i 0 1.25882 + 0.574885i 0 −3.04076 + 10.3559i 0 −3.38119 − 7.40377i 0
33.7 0 −0.735026 − 0.472373i 0 −2.37096 − 1.08278i 0 −2.33484 + 7.95173i 0 −3.42161 − 7.49228i 0
33.8 0 0.754411 + 0.484830i 0 −2.49877 − 1.14115i 0 1.65998 − 5.65338i 0 −3.40466 − 7.45517i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.3.p.d 120
4.b odd 2 1 184.3.l.a ✓ 120
23.d odd 22 1 inner 368.3.p.d 120
92.h even 22 1 184.3.l.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.3.l.a ✓ 120 4.b odd 2 1
184.3.l.a ✓ 120 92.h even 22 1
368.3.p.d 120 1.a even 1 1 trivial
368.3.p.d 120 23.d odd 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{120} + 80 T_{3}^{118} - 32 T_{3}^{117} + 4364 T_{3}^{116} - 4064 T_{3}^{115} + \cdots + 14\!\cdots\!21 \) acting on \(S_{3}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display