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Properties

Label 186.2.j.a
Level $186$
Weight $2$
Character orbit 186.j
Analytic conductor $1.485$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [186,2,Mod(23,186)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(186, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("186.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 186 = 2 \cdot 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 186.j (of order \(10\), degree \(4\), 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: \(1.48521747760\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{4} - 18 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{4} - 18 q^{7} - 4 q^{9} + 4 q^{10} + 10 q^{15} - 12 q^{16} + 8 q^{18} + 20 q^{19} - 10 q^{22} - 92 q^{25} - 22 q^{28} + 40 q^{31} - 12 q^{33} - 20 q^{34} + 4 q^{36} - 8 q^{39} + 6 q^{40} - 50 q^{43} + 6 q^{45} + 20 q^{46} - 10 q^{48} - 34 q^{49} - 10 q^{51} + 20 q^{55} - 20 q^{58} + 10 q^{60} - 148 q^{63} + 12 q^{64} - 36 q^{66} + 60 q^{67} + 54 q^{69} - 24 q^{70} - 8 q^{72} - 20 q^{73} + 30 q^{75} + 30 q^{76} + 10 q^{78} + 50 q^{79} - 56 q^{81} + 12 q^{82} + 20 q^{84} + 20 q^{85} + 104 q^{87} + 98 q^{90} + 70 q^{91} - 10 q^{93} + 24 q^{94} + 22 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.951057 − 0.309017i −1.58495 − 0.698515i 0.809017 + 0.587785i − 1.37063i 1.29153 + 1.15410i −0.358484 − 0.260454i −0.587785 − 0.809017i 2.02415 + 2.21423i −0.423548 + 1.30355i
23.2 −0.951057 − 0.309017i −1.01081 + 1.40651i 0.809017 + 0.587785i 3.83089i 1.39597 − 1.02531i −3.86758 − 2.80996i −0.587785 − 0.809017i −0.956535 − 2.84342i 1.18381 − 3.64339i
23.3 −0.951057 − 0.309017i −0.845246 + 1.51181i 0.809017 + 0.587785i − 3.62278i 1.27105 − 1.17662i 0.937401 + 0.681062i −0.587785 − 0.809017i −1.57112 − 2.55570i −1.11950 + 3.44547i
23.4 −0.951057 − 0.309017i −0.303810 − 1.70520i 0.809017 + 0.587785i 3.78035i −0.237994 + 1.71562i 2.28063 + 1.65697i −0.587785 − 0.809017i −2.81540 + 1.03611i 1.16819 − 3.59533i
23.5 −0.951057 − 0.309017i 1.24859 − 1.20042i 0.809017 + 0.587785i − 1.64455i −1.55843 + 0.755834i −2.68311 − 1.94939i −0.587785 − 0.809017i 0.117969 − 2.99768i −0.508194 + 1.56406i
23.6 −0.951057 − 0.309017i 1.37819 + 1.04909i 0.809017 + 0.587785i 0.644755i −0.986550 − 1.42363i 0.882122 + 0.640899i −0.587785 − 0.809017i 0.798817 + 2.89169i 0.199240 − 0.613198i
23.7 0.951057 + 0.309017i −1.64449 − 0.543752i 0.809017 + 0.587785i − 3.83089i −1.39597 − 1.02531i −3.86758 − 2.80996i 0.587785 + 0.809017i 2.40867 + 1.78838i 1.18381 − 3.64339i
23.8 0.951057 + 0.309017i −1.57244 − 0.726255i 0.809017 + 0.587785i 3.62278i −1.27105 − 1.17662i 0.937401 + 0.681062i 0.587785 + 0.809017i 1.94511 + 2.28398i −1.11950 + 3.44547i
23.9 0.951057 + 0.309017i −0.871678 + 1.49672i 0.809017 + 0.587785i 1.37063i −1.29153 + 1.15410i −0.358484 − 0.260454i 0.587785 + 0.809017i −1.48036 − 2.60932i −0.423548 + 1.30355i
23.10 0.951057 + 0.309017i 0.498339 − 1.65881i 0.809017 + 0.587785i − 0.644755i 0.986550 − 1.42363i 0.882122 + 0.640899i 0.587785 + 0.809017i −2.50332 − 1.65330i 0.199240 − 0.613198i
23.11 0.951057 + 0.309017i 0.756502 + 1.55811i 0.809017 + 0.587785i − 3.78035i 0.237994 + 1.71562i 2.28063 + 1.65697i 0.587785 + 0.809017i −1.85541 + 2.35743i 1.16819 − 3.59533i
23.12 0.951057 + 0.309017i 1.71572 + 0.237258i 0.809017 + 0.587785i 1.64455i 1.55843 + 0.755834i −2.68311 − 1.94939i 0.587785 + 0.809017i 2.88742 + 0.814139i −0.508194 + 1.56406i
29.1 −0.587785 + 0.809017i −1.39814 − 1.02235i −0.309017 − 0.951057i 1.55836i 1.64890 − 0.530201i 0.227448 + 0.700013i 0.951057 + 0.309017i 0.909609 + 2.85878i −1.26074 − 0.915980i
29.2 −0.587785 + 0.809017i −1.25037 + 1.19857i −0.309017 − 0.951057i − 2.26518i −0.234716 − 1.71607i −1.15341 − 3.54984i 0.951057 + 0.309017i 0.126851 − 2.99732i 1.83257 + 1.33144i
29.3 −0.587785 + 0.809017i 0.305970 + 1.70481i −0.309017 − 0.951057i − 0.974922i −1.55907 − 0.754528i 1.13785 + 3.50196i 0.951057 + 0.309017i −2.81276 + 1.04324i 0.788729 + 0.573045i
29.4 −0.587785 + 0.809017i 0.686333 − 1.59027i −0.309017 − 0.951057i − 0.0948010i 0.883136 + 1.48999i −1.34266 − 4.13227i 0.951057 + 0.309017i −2.05789 − 2.18290i 0.0766956 + 0.0557226i
29.5 −0.587785 + 0.809017i 1.04720 + 1.37963i −0.309017 − 0.951057i 4.23437i −1.73167 + 0.0362735i −0.764711 − 2.35354i 0.951057 + 0.309017i −0.806759 + 2.88949i −3.42568 − 2.48890i
29.6 −0.587785 + 0.809017i 1.72705 − 0.131557i −0.309017 − 0.951057i − 3.07586i −0.908701 + 1.47454i 0.204495 + 0.629370i 0.951057 + 0.309017i 2.96539 − 0.454412i 2.48842 + 1.80794i
29.7 0.587785 − 0.809017i −1.72452 + 0.161322i −0.309017 − 0.951057i 0.0948010i −0.883136 + 1.48999i −1.34266 − 4.13227i −0.951057 − 0.309017i 2.94795 − 0.556408i 0.0766956 + 0.0557226i
29.8 0.587785 − 0.809017i −0.658806 + 1.60187i −0.309017 − 0.951057i 3.07586i 0.908701 + 1.47454i 0.204495 + 0.629370i −0.951057 − 0.309017i −2.13195 − 2.11064i 2.48842 + 1.80794i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.f odd 10 1 inner
93.k even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 186.2.j.a ✓ 48
3.b odd 2 1 inner 186.2.j.a ✓ 48
31.f odd 10 1 inner 186.2.j.a ✓ 48
93.k even 10 1 inner 186.2.j.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
186.2.j.a ✓ 48 1.a even 1 1 trivial
186.2.j.a ✓ 48 3.b odd 2 1 inner
186.2.j.a ✓ 48 31.f odd 10 1 inner
186.2.j.a ✓ 48 93.k even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(186, [\chi])\).