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Properties

Label 1110.2.o.a
Level $1110$
Weight $2$
Character orbit 1110.o
Analytic conductor $8.863$
Analytic rank $0$
Dimension $36$
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] = [1110,2,Mod(253,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.253");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.o (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: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{2} + 36 q^{4} + 4 q^{5} + 4 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{2} + 36 q^{4} + 4 q^{5} + 4 q^{7} - 36 q^{8} - 4 q^{10} - 8 q^{13} - 4 q^{14} + 36 q^{16} - 4 q^{19} + 4 q^{20} + 4 q^{25} + 8 q^{26} + 4 q^{28} - 36 q^{29} - 4 q^{31} - 36 q^{32} + 4 q^{33} + 28 q^{35} + 28 q^{37} + 4 q^{38} - 4 q^{39} - 4 q^{40} - 32 q^{43} + 16 q^{47} - 4 q^{50} - 8 q^{52} - 8 q^{53} - 8 q^{55} - 4 q^{56} - 8 q^{57} + 36 q^{58} + 4 q^{59} - 4 q^{61} + 4 q^{62} - 4 q^{63} + 36 q^{64} - 52 q^{65} - 4 q^{66} - 16 q^{67} + 8 q^{69} - 28 q^{70} - 8 q^{71} + 4 q^{73} - 28 q^{74} + 16 q^{75} - 4 q^{76} - 8 q^{77} + 4 q^{78} + 12 q^{79} + 4 q^{80} - 36 q^{81} + 8 q^{83} - 8 q^{85} + 32 q^{86} + 24 q^{89} + 56 q^{91} - 8 q^{93} - 16 q^{94} + 20 q^{95} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
253.1 −1.00000 −0.707107 + 0.707107i 1.00000 −2.23554 + 0.0484605i 0.707107 − 0.707107i 2.77693 − 2.77693i −1.00000 − 1.00000i 2.23554 − 0.0484605i
253.2 −1.00000 −0.707107 + 0.707107i 1.00000 −2.22064 − 0.262217i 0.707107 − 0.707107i −2.69527 + 2.69527i −1.00000 − 1.00000i 2.22064 + 0.262217i
253.3 −1.00000 −0.707107 + 0.707107i 1.00000 0.0687789 − 2.23501i 0.707107 − 0.707107i 2.13605 − 2.13605i −1.00000 − 1.00000i −0.0687789 + 2.23501i
253.4 −1.00000 −0.707107 + 0.707107i 1.00000 −0.530977 + 2.17211i 0.707107 − 0.707107i −1.93388 + 1.93388i −1.00000 − 1.00000i 0.530977 − 2.17211i
253.5 −1.00000 −0.707107 + 0.707107i 1.00000 1.63524 + 1.52512i 0.707107 − 0.707107i 1.84299 − 1.84299i −1.00000 − 1.00000i −1.63524 − 1.52512i
253.6 −1.00000 −0.707107 + 0.707107i 1.00000 1.69741 − 1.45560i 0.707107 − 0.707107i −0.764053 + 0.764053i −1.00000 − 1.00000i −1.69741 + 1.45560i
253.7 −1.00000 −0.707107 + 0.707107i 1.00000 −0.866398 + 2.06140i 0.707107 − 0.707107i −0.662100 + 0.662100i −1.00000 − 1.00000i 0.866398 − 2.06140i
253.8 −1.00000 −0.707107 + 0.707107i 1.00000 2.23596 + 0.0221544i 0.707107 − 0.707107i −0.281427 + 0.281427i −1.00000 − 1.00000i −2.23596 − 0.0221544i
253.9 −1.00000 −0.707107 + 0.707107i 1.00000 1.21617 − 1.87641i 0.707107 − 0.707107i −2.24767 + 2.24767i −1.00000 − 1.00000i −1.21617 + 1.87641i
253.10 −1.00000 0.707107 − 0.707107i 1.00000 1.91070 + 1.16157i −0.707107 + 0.707107i 2.90687 − 2.90687i −1.00000 − 1.00000i −1.91070 − 1.16157i
253.11 −1.00000 0.707107 − 0.707107i 1.00000 −0.891783 − 2.05054i −0.707107 + 0.707107i −1.56460 + 1.56460i −1.00000 − 1.00000i 0.891783 + 2.05054i
253.12 −1.00000 0.707107 − 0.707107i 1.00000 0.407420 + 2.19864i −0.707107 + 0.707107i 1.64494 − 1.64494i −1.00000 − 1.00000i −0.407420 − 2.19864i
253.13 −1.00000 0.707107 − 0.707107i 1.00000 −1.65797 − 1.50038i −0.707107 + 0.707107i −1.06172 + 1.06172i −1.00000 − 1.00000i 1.65797 + 1.50038i
253.14 −1.00000 0.707107 − 0.707107i 1.00000 −2.02734 + 0.943349i −0.707107 + 0.707107i −0.516238 + 0.516238i −1.00000 − 1.00000i 2.02734 − 0.943349i
253.15 −1.00000 0.707107 − 0.707107i 1.00000 1.94044 − 1.11117i −0.707107 + 0.707107i 0.636201 − 0.636201i −1.00000 − 1.00000i −1.94044 + 1.11117i
253.16 −1.00000 0.707107 − 0.707107i 1.00000 0.914315 − 2.04060i −0.707107 + 0.707107i 0.867388 − 0.867388i −1.00000 − 1.00000i −0.914315 + 2.04060i
253.17 −1.00000 0.707107 − 0.707107i 1.00000 2.05225 + 0.887844i −0.707107 + 0.707107i −1.84414 + 1.84414i −1.00000 − 1.00000i −2.05225 − 0.887844i
253.18 −1.00000 0.707107 − 0.707107i 1.00000 −1.64803 + 1.51129i −0.707107 + 0.707107i 2.75974 − 2.75974i −1.00000 − 1.00000i 1.64803 − 1.51129i
487.1 −1.00000 −0.707107 − 0.707107i 1.00000 −2.23554 − 0.0484605i 0.707107 + 0.707107i 2.77693 + 2.77693i −1.00000 1.00000i 2.23554 + 0.0484605i
487.2 −1.00000 −0.707107 − 0.707107i 1.00000 −2.22064 + 0.262217i 0.707107 + 0.707107i −2.69527 − 2.69527i −1.00000 1.00000i 2.22064 − 0.262217i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 253.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.o.a yes 36
5.c odd 4 1 1110.2.l.a ✓ 36
37.d odd 4 1 1110.2.l.a ✓ 36
185.k even 4 1 inner 1110.2.o.a yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.l.a ✓ 36 5.c odd 4 1
1110.2.l.a ✓ 36 37.d odd 4 1
1110.2.o.a yes 36 1.a even 1 1 trivial
1110.2.o.a yes 36 185.k even 4 1 inner