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Properties

Label 368.3.p.c
Level $368$
Weight $3$
Character orbit 368.p
Analytic conductor $10.027$
Analytic rank $0$
Dimension $40$
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: \(40\)
Relative dimension: \(4\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 92)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{3} - 6 q^{9} - 2 q^{13} - 77 q^{15} + 55 q^{17} - 33 q^{19} + 33 q^{21} + 50 q^{23} - 54 q^{25} + 191 q^{27} + q^{29} + 53 q^{31} - 121 q^{33} + 156 q^{35} - 352 q^{37} + 306 q^{39} + 6 q^{41} + 88 q^{43} + 58 q^{47} + 292 q^{49} - 264 q^{51} + 176 q^{53} - 518 q^{55} + 891 q^{57} - 696 q^{59} + 308 q^{61} - 275 q^{63} + 231 q^{65} - 22 q^{67} - 233 q^{69} + 215 q^{71} - 400 q^{73} + 557 q^{75} - 239 q^{77} + 748 q^{79} - 684 q^{81} + 671 q^{83} - 794 q^{85} - 62 q^{87} - 44 q^{89} - 346 q^{93} - 561 q^{95} + 693 q^{97} - 1353 q^{99}+O(q^{100}) \) 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 −3.29776 − 0.968311i 0 4.37331 + 6.80500i 0 9.96857 + 1.43326i 0 2.36634 + 1.52076i 0
17.2 0 −3.26124 − 0.957586i 0 −1.69914 − 2.64391i 0 −10.8401 − 1.55857i 0 2.14742 + 1.38006i 0
17.3 0 1.77627 + 0.521559i 0 −2.67542 − 4.16303i 0 5.18338 + 0.745258i 0 −4.68818 − 3.01291i 0
17.4 0 4.50964 + 1.32415i 0 3.25659 + 5.06735i 0 −7.52267 − 1.08160i 0 11.0122 + 7.07709i 0
33.1 0 −4.12457 − 2.65070i 0 0.451755 + 0.206310i 0 −0.430959 + 1.46771i 0 6.24713 + 13.6793i 0
33.2 0 −0.368269 − 0.236672i 0 5.43355 + 2.48142i 0 1.13570 − 3.86785i 0 −3.65913 − 8.01237i 0
33.3 0 1.35972 + 0.873837i 0 −3.48403 − 1.59110i 0 −2.68421 + 9.14158i 0 −2.65350 − 5.81035i 0
33.4 0 4.74748 + 3.05102i 0 0.739883 + 0.337893i 0 3.51711 − 11.9782i 0 9.49108 + 20.7826i 0
65.1 0 −3.29776 + 0.968311i 0 4.37331 − 6.80500i 0 9.96857 − 1.43326i 0 2.36634 − 1.52076i 0
65.2 0 −3.26124 + 0.957586i 0 −1.69914 + 2.64391i 0 −10.8401 + 1.55857i 0 2.14742 − 1.38006i 0
65.3 0 1.77627 − 0.521559i 0 −2.67542 + 4.16303i 0 5.18338 − 0.745258i 0 −4.68818 + 3.01291i 0
65.4 0 4.50964 − 1.32415i 0 3.25659 − 5.06735i 0 −7.52267 + 1.08160i 0 11.0122 − 7.07709i 0
97.1 0 −2.65957 + 3.06930i 0 2.44160 + 0.351049i 0 0.284672 − 0.130005i 0 −1.06649 − 7.41762i 0
97.2 0 0.434995 − 0.502011i 0 1.14824 + 0.165091i 0 −11.0573 + 5.04971i 0 1.21804 + 8.47165i 0
97.3 0 0.710283 − 0.819710i 0 −9.64449 − 1.38667i 0 4.98879 − 2.27830i 0 1.11341 + 7.74394i 0
97.4 0 2.05837 − 2.37548i 0 5.87759 + 0.845069i 0 11.1861 − 5.10852i 0 −0.125205 − 0.870819i 0
113.1 0 −0.652910 − 4.54109i 0 −1.15784 + 3.94325i 0 4.50708 + 3.90541i 0 −11.5598 + 3.39426i 0
113.2 0 −0.174403 − 1.21300i 0 2.39977 − 8.17285i 0 −3.00025 − 2.59973i 0 7.19448 − 2.11249i 0
113.3 0 0.0496937 + 0.345627i 0 −1.74128 + 5.93027i 0 −6.80138 − 5.89343i 0 8.51845 − 2.50124i 0
113.4 0 0.591227 + 4.11207i 0 0.143195 − 0.487679i 0 4.01704 + 3.48078i 0 −7.92417 + 2.32675i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
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.c 40
4.b odd 2 1 92.3.f.a ✓ 40
23.d odd 22 1 inner 368.3.p.c 40
92.g odd 22 1 2116.3.d.c 40
92.h even 22 1 92.3.f.a ✓ 40
92.h even 22 1 2116.3.d.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.3.f.a ✓ 40 4.b odd 2 1
92.3.f.a ✓ 40 92.h even 22 1
368.3.p.c 40 1.a even 1 1 trivial
368.3.p.c 40 23.d odd 22 1 inner
2116.3.d.c 40 92.g odd 22 1
2116.3.d.c 40 92.h even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 2 T_{3}^{39} + 23 T_{3}^{38} - 119 T_{3}^{37} + 788 T_{3}^{36} - 779 T_{3}^{35} + \cdots + 55296371439409 \) acting on \(S_{3}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display