Properties

Label 368.3.p.a
Level $368$
Weight $3$
Character orbit 368.p
Analytic conductor $10.027$
Analytic rank $0$
Dimension $30$
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: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 23)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 11 q^{3} - 11 q^{5} + 11 q^{7} - 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 11 q^{3} - 11 q^{5} + 11 q^{7} - 38 q^{9} + 11 q^{11} - 11 q^{13} - 66 q^{15} + 44 q^{17} - 22 q^{19} + 22 q^{21} - 36 q^{23} - 152 q^{25} + 62 q^{27} - 88 q^{29} + 110 q^{31} - 132 q^{33} - 209 q^{35} + 341 q^{37} - 295 q^{39} + 77 q^{41} - 77 q^{43} + 110 q^{47} - 422 q^{49} + 275 q^{51} - 187 q^{53} + 165 q^{55} - 176 q^{57} + q^{59} + 297 q^{61} - 264 q^{63} + 220 q^{65} - 11 q^{67} - 66 q^{69} + 176 q^{71} - 121 q^{73} - 154 q^{75} + 110 q^{77} - 33 q^{79} + 494 q^{81} + 154 q^{83} + 275 q^{85} - 271 q^{87} + 121 q^{89} + 260 q^{93} + 154 q^{95} + 154 q^{97} + 242 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −0.844537 0.247978i 0 −3.24760 5.05336i 0 3.20136 + 0.460286i 0 −6.91953 4.44691i 0
17.2 0 −0.201951 0.0592983i 0 2.90325 + 4.51755i 0 −7.13192 1.02541i 0 −7.53401 4.84182i 0
17.3 0 2.61464 + 0.767728i 0 3.01510 + 4.69158i 0 2.67922 + 0.385214i 0 −1.32434 0.851101i 0
33.1 0 −2.64748 1.70143i 0 −2.08252 0.951056i 0 2.90589 9.89656i 0 0.375553 + 0.822346i 0
33.2 0 1.87410 + 1.20441i 0 −2.69128 1.22907i 0 1.33225 4.53722i 0 −1.67709 3.67232i 0
33.3 0 3.80315 + 2.44414i 0 6.26010 + 2.85889i 0 −2.54176 + 8.65643i 0 4.75142 + 10.4042i 0
65.1 0 −0.844537 + 0.247978i 0 −3.24760 + 5.05336i 0 3.20136 0.460286i 0 −6.91953 + 4.44691i 0
65.2 0 −0.201951 + 0.0592983i 0 2.90325 4.51755i 0 −7.13192 + 1.02541i 0 −7.53401 + 4.84182i 0
65.3 0 2.61464 0.767728i 0 3.01510 4.69158i 0 2.67922 0.385214i 0 −1.32434 + 0.851101i 0
97.1 0 −2.35762 + 2.72084i 0 5.05070 + 0.726181i 0 8.85488 4.04389i 0 −0.563759 3.92103i 0
97.2 0 0.590042 0.680945i 0 −1.77862 0.255727i 0 −2.68734 + 1.22727i 0 1.16530 + 8.10482i 0
97.3 0 3.16934 3.65762i 0 −5.40865 0.777647i 0 0.889564 0.406250i 0 −2.05259 14.2761i 0
113.1 0 −0.749667 5.21405i 0 1.45779 4.96477i 0 1.27914 + 1.10838i 0 −17.9889 + 5.28201i 0
113.2 0 0.238691 + 1.66013i 0 −2.65558 + 9.04406i 0 5.31379 + 4.60443i 0 5.93637 1.74307i 0
113.3 0 0.365090 + 2.53926i 0 0.682875 2.32566i 0 −6.72814 5.82996i 0 2.32089 0.681474i 0
129.1 0 −2.35762 2.72084i 0 5.05070 0.726181i 0 8.85488 + 4.04389i 0 −0.563759 + 3.92103i 0
129.2 0 0.590042 + 0.680945i 0 −1.77862 + 0.255727i 0 −2.68734 1.22727i 0 1.16530 8.10482i 0
129.3 0 3.16934 + 3.65762i 0 −5.40865 + 0.777647i 0 0.889564 + 0.406250i 0 −2.05259 + 14.2761i 0
145.1 0 −2.64748 + 1.70143i 0 −2.08252 + 0.951056i 0 2.90589 + 9.89656i 0 0.375553 0.822346i 0
145.2 0 1.87410 1.20441i 0 −2.69128 + 1.22907i 0 1.33225 + 4.53722i 0 −1.67709 + 3.67232i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.3
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.a 30
4.b odd 2 1 23.3.d.a 30
12.b even 2 1 207.3.j.a 30
23.d odd 22 1 inner 368.3.p.a 30
92.g odd 22 1 529.3.b.b 30
92.h even 22 1 23.3.d.a 30
92.h even 22 1 529.3.b.b 30
276.j odd 22 1 207.3.j.a 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.3.d.a 30 4.b odd 2 1
23.3.d.a 30 92.h even 22 1
207.3.j.a 30 12.b even 2 1
207.3.j.a 30 276.j odd 22 1
368.3.p.a 30 1.a even 1 1 trivial
368.3.p.a 30 23.d odd 22 1 inner
529.3.b.b 30 92.g odd 22 1
529.3.b.b 30 92.h even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} - 11 T_{3}^{29} + 93 T_{3}^{28} - 600 T_{3}^{27} + 3512 T_{3}^{26} - 16076 T_{3}^{25} + 69780 T_{3}^{24} - 288419 T_{3}^{23} + 1153932 T_{3}^{22} - 3865349 T_{3}^{21} + 13169090 T_{3}^{20} + \cdots + 324612289 \) acting on \(S_{3}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display