Properties

Label 368.2.j.d
Level $368$
Weight $2$
Character orbit 368.j
Analytic conductor $2.938$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [368,2,Mod(93,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.93");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.j (of order \(4\), degree \(2\), 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: \(2.93849479438\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} - 4 q^{3} + 4 q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} - 4 q^{3} + 4 q^{5} + 2 q^{8} - 4 q^{11} + 38 q^{12} - 12 q^{13} - 6 q^{14} + 40 q^{17} - 8 q^{18} + 8 q^{19} - 2 q^{20} + 34 q^{22} - 10 q^{24} - 2 q^{26} - 28 q^{27} + 10 q^{28} - 16 q^{29} - 30 q^{30} - 12 q^{31} + 16 q^{32} - 28 q^{33} + 4 q^{34} + 8 q^{35} - 24 q^{36} - 4 q^{37} + 20 q^{38} + 40 q^{42} - 28 q^{43} + 50 q^{44} + 20 q^{45} - 4 q^{47} + 20 q^{48} - 24 q^{49} + 10 q^{50} - 12 q^{51} - 44 q^{52} + 32 q^{53} + 38 q^{54} - 40 q^{56} - 40 q^{58} + 24 q^{59} + 34 q^{60} - 16 q^{61} - 42 q^{62} - 4 q^{63} + 24 q^{64} + 4 q^{65} - 12 q^{66} - 10 q^{68} - 4 q^{69} + 14 q^{70} + 4 q^{74} + 44 q^{75} + 8 q^{76} - 20 q^{77} + 4 q^{78} + 12 q^{79} - 4 q^{80} - 48 q^{81} - 4 q^{82} - 16 q^{83} - 66 q^{84} + 8 q^{85} - 28 q^{88} - 40 q^{90} - 72 q^{91} + 4 q^{93} - 26 q^{94} - 40 q^{95} - 32 q^{96} - 124 q^{97} + 54 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1 −1.41334 0.0497254i 0.387219 + 0.387219i 1.99505 + 0.140558i 1.36361 1.36361i −0.528018 0.566527i 1.65176i −2.81270 0.297860i 2.70012i −1.99505 + 1.85944i
93.2 −1.27953 + 0.602332i 0.203642 + 0.203642i 1.27439 1.54140i 1.88186 1.88186i −0.383226 0.137906i 4.71330i −0.702189 + 2.73988i 2.91706i −1.27439 + 3.54140i
93.3 −1.20316 + 0.743237i 2.30229 + 2.30229i 0.895196 1.78847i 1.94640 1.94640i −4.48117 1.05888i 3.32801i 0.252192 + 2.81716i 7.60105i −0.895196 + 3.78847i
93.4 −1.11221 0.873496i −1.75354 1.75354i 0.474011 + 1.94302i 0.238712 0.238712i 0.418590 + 3.48200i 4.31299i 1.17002 2.57508i 3.14978i −0.474011 + 0.0569833i
93.5 −0.741286 1.20436i 1.96106 + 1.96106i −0.900989 + 1.78556i −0.463078 + 0.463078i 0.908123 3.81553i 1.33926i 2.81835 0.238491i 4.69148i 0.900989 + 0.214442i
93.6 −0.408999 1.35378i −1.35709 1.35709i −1.66544 + 1.10739i −0.944781 + 0.944781i −1.28215 + 2.39224i 1.03914i 2.18032 + 1.80172i 0.683369i 1.66544 + 0.892612i
93.7 −0.340155 + 1.37270i −1.72800 1.72800i −1.76859 0.933858i 1.71285 1.71285i 2.95981 1.78424i 1.69946i 1.88350 2.11008i 2.97199i 1.76859 + 2.93386i
93.8 0.326476 + 1.37601i −0.248916 0.248916i −1.78683 + 0.898472i 1.04954 1.04954i 0.261246 0.423776i 3.80824i −1.81967 2.16537i 2.87608i 1.78683 + 1.10153i
93.9 0.730764 + 1.21078i −2.15319 2.15319i −0.931969 + 1.76959i 0.480015 0.480015i 1.03356 4.18052i 2.39661i −2.82363 + 0.164741i 6.27249i 0.931969 + 0.230414i
93.10 0.737124 1.20692i −0.383765 0.383765i −0.913296 1.77929i −1.94404 + 1.94404i −0.746054 + 0.180290i 1.91069i −2.82067 0.209289i 2.70545i 0.913296 + 3.77929i
93.11 1.29063 0.578171i 1.73006 + 1.73006i 1.33144 1.49241i −1.86880 + 1.86880i 3.23314 + 1.23259i 1.07441i 0.855521 2.69594i 2.98625i −1.33144 + 3.49241i
93.12 1.41369 0.0386035i −0.959768 0.959768i 1.99702 0.109146i −1.45229 + 1.45229i −1.39386 1.31976i 3.25050i 2.81895 0.231391i 1.15769i −1.99702 + 2.10915i
277.1 −1.41334 + 0.0497254i 0.387219 0.387219i 1.99505 0.140558i 1.36361 + 1.36361i −0.528018 + 0.566527i 1.65176i −2.81270 + 0.297860i 2.70012i −1.99505 1.85944i
277.2 −1.27953 0.602332i 0.203642 0.203642i 1.27439 + 1.54140i 1.88186 + 1.88186i −0.383226 + 0.137906i 4.71330i −0.702189 2.73988i 2.91706i −1.27439 3.54140i
277.3 −1.20316 0.743237i 2.30229 2.30229i 0.895196 + 1.78847i 1.94640 + 1.94640i −4.48117 + 1.05888i 3.32801i 0.252192 2.81716i 7.60105i −0.895196 3.78847i
277.4 −1.11221 + 0.873496i −1.75354 + 1.75354i 0.474011 1.94302i 0.238712 + 0.238712i 0.418590 3.48200i 4.31299i 1.17002 + 2.57508i 3.14978i −0.474011 0.0569833i
277.5 −0.741286 + 1.20436i 1.96106 1.96106i −0.900989 1.78556i −0.463078 0.463078i 0.908123 + 3.81553i 1.33926i 2.81835 + 0.238491i 4.69148i 0.900989 0.214442i
277.6 −0.408999 + 1.35378i −1.35709 + 1.35709i −1.66544 1.10739i −0.944781 0.944781i −1.28215 2.39224i 1.03914i 2.18032 1.80172i 0.683369i 1.66544 0.892612i
277.7 −0.340155 1.37270i −1.72800 + 1.72800i −1.76859 + 0.933858i 1.71285 + 1.71285i 2.95981 + 1.78424i 1.69946i 1.88350 + 2.11008i 2.97199i 1.76859 2.93386i
277.8 0.326476 1.37601i −0.248916 + 0.248916i −1.78683 0.898472i 1.04954 + 1.04954i 0.261246 + 0.423776i 3.80824i −1.81967 + 2.16537i 2.87608i 1.78683 1.10153i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.2.j.d 24
4.b odd 2 1 1472.2.j.d 24
16.e even 4 1 inner 368.2.j.d 24
16.f odd 4 1 1472.2.j.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.2.j.d 24 1.a even 1 1 trivial
368.2.j.d 24 16.e even 4 1 inner
1472.2.j.d 24 4.b odd 2 1
1472.2.j.d 24 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 4 T_{3}^{23} + 8 T_{3}^{22} + 12 T_{3}^{21} + 208 T_{3}^{20} + 828 T_{3}^{19} + 1720 T_{3}^{18} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display