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Properties

Label 368.3.p.b
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 46)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} + 24 q^{9} + 4 q^{13} + 154 q^{15} - 110 q^{17} + 66 q^{19} - 66 q^{21} + 8 q^{23} + 264 q^{25} - 286 q^{27} + 70 q^{29} - 82 q^{31} + 242 q^{33} + 276 q^{35} - 352 q^{37} + 204 q^{39} - 108 q^{41} + 88 q^{43} - 116 q^{47} + 412 q^{49} - 264 q^{51} + 176 q^{53} + 76 q^{55} - 198 q^{57} + 300 q^{59} - 616 q^{61} + 550 q^{63} - 462 q^{65} + 44 q^{67} + 106 q^{69} - 430 q^{71} + 368 q^{73} - 418 q^{75} + 646 q^{77} - 704 q^{79} + 636 q^{81} - 814 q^{83} + 736 q^{85} + 100 q^{87} - 44 q^{89} + 140 q^{93} + 594 q^{95} - 990 q^{97} + 1122 q^{99}+O(q^{100}) \) 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 −5.38424 − 1.58096i 0 −2.08772 − 3.24855i 0 5.96665 + 0.857874i 0 18.9193 + 12.1587i 0
17.2 0 −2.54103 − 0.746114i 0 0.680920 + 1.05953i 0 2.11717 + 0.304404i 0 −1.67113 − 1.07397i 0
17.3 0 3.94151 + 1.15733i 0 −3.93626 − 6.12494i 0 −12.3131 − 1.77035i 0 6.62478 + 4.25749i 0
17.4 0 4.52996 + 1.33012i 0 −1.16762 − 1.81685i 0 10.6509 + 1.53137i 0 11.1801 + 7.18499i 0
33.1 0 −3.42514 − 2.20120i 0 −7.87646 − 3.59706i 0 −2.42105 + 8.24534i 0 3.14754 + 6.89216i 0
33.2 0 −2.46501 − 1.58417i 0 3.29959 + 1.50687i 0 −1.07230 + 3.65191i 0 −0.172039 − 0.376712i 0
33.3 0 0.252593 + 0.162332i 0 4.73531 + 2.16254i 0 0.352465 − 1.20038i 0 −3.70128 − 8.10468i 0
33.4 0 2.40885 + 1.54807i 0 −6.44075 − 2.94139i 0 0.0656098 − 0.223446i 0 −0.332712 − 0.728539i 0
65.1 0 −5.38424 + 1.58096i 0 −2.08772 + 3.24855i 0 5.96665 − 0.857874i 0 18.9193 − 12.1587i 0
65.2 0 −2.54103 + 0.746114i 0 0.680920 − 1.05953i 0 2.11717 − 0.304404i 0 −1.67113 + 1.07397i 0
65.3 0 3.94151 − 1.15733i 0 −3.93626 + 6.12494i 0 −12.3131 + 1.77035i 0 6.62478 − 4.25749i 0
65.4 0 4.52996 − 1.33012i 0 −1.16762 + 1.81685i 0 10.6509 − 1.53137i 0 11.1801 − 7.18499i 0
97.1 0 −2.98027 + 3.43941i 0 −6.93065 − 0.996477i 0 −9.13753 + 4.17297i 0 −1.66673 − 11.5924i 0
97.2 0 −1.65855 + 1.91407i 0 −3.34128 − 0.480403i 0 3.29739 − 1.50587i 0 0.367960 + 2.55922i 0
97.3 0 0.658994 − 0.760519i 0 7.10773 + 1.02194i 0 −4.73539 + 2.16258i 0 1.13672 + 7.90604i 0
97.4 0 2.89167 − 3.33717i 0 3.51835 + 0.505863i 0 −0.228956 + 0.104560i 0 −1.49409 − 10.3916i 0
113.1 0 −0.417040 − 2.90057i 0 −1.07755 + 3.66981i 0 −4.93613 − 4.27718i 0 0.396024 − 0.116283i 0
113.2 0 −0.244112 − 1.69783i 0 0.188923 − 0.643413i 0 2.70796 + 2.34646i 0 5.81239 − 1.70667i 0
113.3 0 0.264805 + 1.84176i 0 1.43372 − 4.88280i 0 8.74875 + 7.58083i 0 5.31348 − 1.56018i 0
113.4 0 0.769132 + 5.34943i 0 0.167244 − 0.569580i 0 −3.96555 − 3.43617i 0 −19.3894 + 5.69324i 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.b 40
4.b odd 2 1 46.3.d.a ✓ 40
12.b even 2 1 414.3.l.a 40
23.d odd 22 1 inner 368.3.p.b 40
92.g odd 22 1 1058.3.b.e 40
92.h even 22 1 46.3.d.a ✓ 40
92.h even 22 1 1058.3.b.e 40
276.j odd 22 1 414.3.l.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.3.d.a ✓ 40 4.b odd 2 1
46.3.d.a ✓ 40 92.h even 22 1
368.3.p.b 40 1.a even 1 1 trivial
368.3.p.b 40 23.d odd 22 1 inner
414.3.l.a 40 12.b even 2 1
414.3.l.a 40 276.j odd 22 1
1058.3.b.e 40 92.g odd 22 1
1058.3.b.e 40 92.h even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 4 T_{3}^{39} + 14 T_{3}^{38} + 154 T_{3}^{37} + 395 T_{3}^{36} - 1736 T_{3}^{35} + \cdots + 10\!\cdots\!89 \) acting on \(S_{3}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display