↑

Properties

Label 1110.2.u.e
Level $1110$
Weight $2$
Character orbit 1110.u
Analytic conductor $8.863$
Analytic rank $0$
Dimension $40$
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] = [1110,2,Mod(191,1110)]
 
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([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.u (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: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 24 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 24 q^{7} - 8 q^{9} + 40 q^{10} - 8 q^{12} - 16 q^{13} - 40 q^{16} + 8 q^{18} + 16 q^{19} - 24 q^{31} + 24 q^{33} - 8 q^{34} + 16 q^{39} - 20 q^{42} - 32 q^{43} - 8 q^{45} + 72 q^{46} + 96 q^{49} + 20 q^{51} + 16 q^{52} + 24 q^{54} - 16 q^{57} + 40 q^{61} - 24 q^{63} - 44 q^{66} - 24 q^{70} + 8 q^{72} + 8 q^{75} - 16 q^{76} + 48 q^{79} + 24 q^{81} - 56 q^{82} - 8 q^{84} + 12 q^{87} - 8 q^{90} - 64 q^{91} + 12 q^{93} + 24 q^{94} + 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 −0.707107 − 0.707107i −1.73082 + 0.0652635i 1.00000i −0.707107 + 0.707107i 1.27002 + 1.17773i −5.10391 0.707107 − 0.707107i 2.99148 − 0.225919i 1.00000
191.2 −0.707107 − 0.707107i 0.347239 + 1.69689i 1.00000i −0.707107 + 0.707107i 0.954345 − 1.44542i −3.94859 0.707107 − 0.707107i −2.75885 + 1.17845i 1.00000
191.3 −0.707107 − 0.707107i 1.64725 + 0.535321i 1.00000i −0.707107 + 0.707107i −0.786252 − 1.54331i −2.53955 0.707107 − 0.707107i 2.42686 + 1.76361i 1.00000
191.4 −0.707107 − 0.707107i 0.878679 + 1.49262i 1.00000i −0.707107 + 0.707107i 0.434124 − 1.67676i 2.38487 0.707107 − 0.707107i −1.45585 + 2.62307i 1.00000
191.5 −0.707107 − 0.707107i −1.08352 − 1.35129i 1.00000i −0.707107 + 0.707107i −0.189341 + 1.72167i −3.77650 0.707107 − 0.707107i −0.651967 + 2.92830i 1.00000
191.6 −0.707107 − 0.707107i −0.912132 + 1.47242i 1.00000i −0.707107 + 0.707107i 1.68613 − 0.396182i 0.363832 0.707107 − 0.707107i −1.33603 − 2.68608i 1.00000
191.7 −0.707107 − 0.707107i 0.544013 − 1.64440i 1.00000i −0.707107 + 0.707107i −1.54744 + 0.778090i −1.56844 0.707107 − 0.707107i −2.40810 − 1.78915i 1.00000
191.8 −0.707107 − 0.707107i −1.59695 + 0.670623i 1.00000i −0.707107 + 0.707107i 1.60342 + 0.655016i 3.28633 0.707107 − 0.707107i 2.10053 − 2.14191i 1.00000
191.9 −0.707107 − 0.707107i 0.349356 − 1.69645i 1.00000i −0.707107 + 0.707107i −1.44660 + 0.952541i 2.97676 0.707107 − 0.707107i −2.75590 − 1.18533i 1.00000
191.10 −0.707107 − 0.707107i 1.55689 + 0.759006i 1.00000i −0.707107 + 0.707107i −0.564190 − 1.63759i 1.92521 0.707107 − 0.707107i 1.84782 + 2.36338i 1.00000
191.11 0.707107 + 0.707107i 1.08352 − 1.35129i 1.00000i 0.707107 − 0.707107i 1.72167 − 0.189341i −3.77650 −0.707107 + 0.707107i −0.651967 − 2.92830i 1.00000
191.12 0.707107 + 0.707107i −0.347239 + 1.69689i 1.00000i 0.707107 − 0.707107i −1.44542 + 0.954345i −3.94859 −0.707107 + 0.707107i −2.75885 − 1.17845i 1.00000
191.13 0.707107 + 0.707107i −0.544013 − 1.64440i 1.00000i 0.707107 − 0.707107i 0.778090 − 1.54744i −1.56844 −0.707107 + 0.707107i −2.40810 + 1.78915i 1.00000
191.14 0.707107 + 0.707107i −1.55689 + 0.759006i 1.00000i 0.707107 − 0.707107i −1.63759 − 0.564190i 1.92521 −0.707107 + 0.707107i 1.84782 − 2.36338i 1.00000
191.15 0.707107 + 0.707107i 1.73082 + 0.0652635i 1.00000i 0.707107 − 0.707107i 1.17773 + 1.27002i −5.10391 −0.707107 + 0.707107i 2.99148 + 0.225919i 1.00000
191.16 0.707107 + 0.707107i −1.64725 + 0.535321i 1.00000i 0.707107 − 0.707107i −1.54331 − 0.786252i −2.53955 −0.707107 + 0.707107i 2.42686 − 1.76361i 1.00000
191.17 0.707107 + 0.707107i 0.912132 + 1.47242i 1.00000i 0.707107 − 0.707107i −0.396182 + 1.68613i 0.363832 −0.707107 + 0.707107i −1.33603 + 2.68608i 1.00000
191.18 0.707107 + 0.707107i −0.349356 − 1.69645i 1.00000i 0.707107 − 0.707107i 0.952541 − 1.44660i 2.97676 −0.707107 + 0.707107i −2.75590 + 1.18533i 1.00000
191.19 0.707107 + 0.707107i 1.59695 + 0.670623i 1.00000i 0.707107 − 0.707107i 0.655016 + 1.60342i 3.28633 −0.707107 + 0.707107i 2.10053 + 2.14191i 1.00000
191.20 0.707107 + 0.707107i −0.878679 + 1.49262i 1.00000i 0.707107 − 0.707107i −1.67676 + 0.434124i 2.38487 −0.707107 + 0.707107i −1.45585 − 2.62307i 1.00000
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.d odd 4 1 inner
111.g 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.u.e ✓ 40
3.b odd 2 1 inner 1110.2.u.e ✓ 40
37.d odd 4 1 inner 1110.2.u.e ✓ 40
111.g even 4 1 inner 1110.2.u.e ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.u.e ✓ 40 1.a even 1 1 trivial
1110.2.u.e ✓ 40 3.b odd 2 1 inner
1110.2.u.e ✓ 40 37.d odd 4 1 inner
1110.2.u.e ✓ 40 111.g even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 6 T_{7}^{9} - 29 T_{7}^{8} - 184 T_{7}^{7} + 331 T_{7}^{6} + 1998 T_{7}^{5} - 1945 T_{7}^{4} + \cdots - 4954 \) acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\). Copy content Toggle raw display