↑

Properties

Label 286.2.g.a
Level $286$
Weight $2$
Character orbit 286.g
Analytic conductor $2.284$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [286,2,Mod(21,286)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(286, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("286.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 286 = 2 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 286.g (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: \(2.28372149781\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{9} + 8 q^{11} + 8 q^{14} - 32 q^{15} - 28 q^{16} - 4 q^{22} + 4 q^{26} - 48 q^{27} + 28 q^{31} + 4 q^{33} + 16 q^{34} - 24 q^{37} - 8 q^{44} + 64 q^{45} - 52 q^{47} + 8 q^{53} + 8 q^{55} + 4 q^{58} - 24 q^{59} - 32 q^{60} - 16 q^{66} + 24 q^{67} - 8 q^{70} - 60 q^{71} + 64 q^{78} - 4 q^{81} + 12 q^{86} + 68 q^{89} - 72 q^{91} + 16 q^{92} + 16 q^{93} - 52 q^{97} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −0.707107 + 0.707107i −3.26605 − 1.00000i 1.97421 + 1.97421i 2.30945 − 2.30945i 0.588595 + 0.588595i 0.707107 + 0.707107i 7.66707 −2.79195
21.2 −0.707107 + 0.707107i −1.68321 − 1.00000i −0.756111 − 0.756111i 1.19021 − 1.19021i 0.145112 + 0.145112i 0.707107 + 0.707107i −0.166795 1.06930
21.3 −0.707107 + 0.707107i −1.10788 − 1.00000i 1.36420 + 1.36420i 0.783387 − 0.783387i −3.51104 − 3.51104i 0.707107 + 0.707107i −1.77261 −1.92928
21.4 −0.707107 + 0.707107i −0.155179 − 1.00000i −2.15078 − 2.15078i 0.109728 − 0.109728i 1.48383 + 1.48383i 0.707107 + 0.707107i −2.97592 3.04166
21.5 −0.707107 + 0.707107i 1.59256 − 1.00000i 1.57203 + 1.57203i −1.12611 + 1.12611i 2.04591 + 2.04591i 0.707107 + 0.707107i −0.463768 −2.22319
21.6 −0.707107 + 0.707107i 2.16685 − 1.00000i −2.66993 − 2.66993i −1.53219 + 1.53219i −1.96943 − 1.96943i 0.707107 + 0.707107i 1.69522 3.77586
21.7 −0.707107 + 0.707107i 2.45292 − 1.00000i 0.666377 + 0.666377i −1.73447 + 1.73447i −0.197191 − 0.197191i 0.707107 + 0.707107i 3.01680 −0.942399
21.8 0.707107 − 0.707107i −3.26605 − 1.00000i 1.97421 + 1.97421i −2.30945 + 2.30945i −0.588595 − 0.588595i −0.707107 − 0.707107i 7.66707 2.79195
21.9 0.707107 − 0.707107i −1.68321 − 1.00000i −0.756111 − 0.756111i −1.19021 + 1.19021i −0.145112 − 0.145112i −0.707107 − 0.707107i −0.166795 −1.06930
21.10 0.707107 − 0.707107i −1.10788 − 1.00000i 1.36420 + 1.36420i −0.783387 + 0.783387i 3.51104 + 3.51104i −0.707107 − 0.707107i −1.77261 1.92928
21.11 0.707107 − 0.707107i −0.155179 − 1.00000i −2.15078 − 2.15078i −0.109728 + 0.109728i −1.48383 − 1.48383i −0.707107 − 0.707107i −2.97592 −3.04166
21.12 0.707107 − 0.707107i 1.59256 − 1.00000i 1.57203 + 1.57203i 1.12611 − 1.12611i −2.04591 − 2.04591i −0.707107 − 0.707107i −0.463768 2.22319
21.13 0.707107 − 0.707107i 2.16685 − 1.00000i −2.66993 − 2.66993i 1.53219 − 1.53219i 1.96943 + 1.96943i −0.707107 − 0.707107i 1.69522 −3.77586
21.14 0.707107 − 0.707107i 2.45292 − 1.00000i 0.666377 + 0.666377i 1.73447 − 1.73447i 0.197191 + 0.197191i −0.707107 − 0.707107i 3.01680 0.942399
109.1 −0.707107 − 0.707107i −3.26605 1.00000i 1.97421 − 1.97421i 2.30945 + 2.30945i 0.588595 − 0.588595i 0.707107 − 0.707107i 7.66707 −2.79195
109.2 −0.707107 − 0.707107i −1.68321 1.00000i −0.756111 + 0.756111i 1.19021 + 1.19021i 0.145112 − 0.145112i 0.707107 − 0.707107i −0.166795 1.06930
109.3 −0.707107 − 0.707107i −1.10788 1.00000i 1.36420 − 1.36420i 0.783387 + 0.783387i −3.51104 + 3.51104i 0.707107 − 0.707107i −1.77261 −1.92928
109.4 −0.707107 − 0.707107i −0.155179 1.00000i −2.15078 + 2.15078i 0.109728 + 0.109728i 1.48383 − 1.48383i 0.707107 − 0.707107i −2.97592 3.04166
109.5 −0.707107 − 0.707107i 1.59256 1.00000i 1.57203 − 1.57203i −1.12611 − 1.12611i 2.04591 − 2.04591i 0.707107 − 0.707107i −0.463768 −2.22319
109.6 −0.707107 − 0.707107i 2.16685 1.00000i −2.66993 + 2.66993i −1.53219 − 1.53219i −1.96943 + 1.96943i 0.707107 − 0.707107i 1.69522 3.77586
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
13.d odd 4 1 inner
143.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 286.2.g.a ✓ 28
11.b odd 2 1 inner 286.2.g.a ✓ 28
13.d odd 4 1 inner 286.2.g.a ✓ 28
143.g even 4 1 inner 286.2.g.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
286.2.g.a ✓ 28 1.a even 1 1 trivial
286.2.g.a ✓ 28 11.b odd 2 1 inner
286.2.g.a ✓ 28 13.d odd 4 1 inner
286.2.g.a ✓ 28 143.g even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(286, [\chi])\).