Properties

Label 1110.2.i.c
Level $1110$
Weight $2$
Character orbit 1110.i
Analytic conductor $8.863$
Analytic rank $0$
Dimension $2$
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(121,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.i (of order \(3\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} - \zeta_{6} q^{5} - q^{6} - 2 \zeta_{6} q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} - \zeta_{6} q^{5} - q^{6} - 2 \zeta_{6} q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} - q^{10} - 3 q^{11} + (\zeta_{6} - 1) q^{12} + \zeta_{6} q^{13} - 2 q^{14} + (\zeta_{6} - 1) q^{15} + (\zeta_{6} - 1) q^{16} + (3 \zeta_{6} - 3) q^{17} + \zeta_{6} q^{18} - 2 \zeta_{6} q^{19} + (\zeta_{6} - 1) q^{20} + (2 \zeta_{6} - 2) q^{21} + (3 \zeta_{6} - 3) q^{22} + \zeta_{6} q^{24} + (\zeta_{6} - 1) q^{25} + q^{26} + q^{27} + (2 \zeta_{6} - 2) q^{28} - 3 q^{29} + \zeta_{6} q^{30} - 4 q^{31} + \zeta_{6} q^{32} + 3 \zeta_{6} q^{33} + 3 \zeta_{6} q^{34} + (2 \zeta_{6} - 2) q^{35} + q^{36} + (3 \zeta_{6} + 4) q^{37} - 2 q^{38} + ( - \zeta_{6} + 1) q^{39} + \zeta_{6} q^{40} + 6 \zeta_{6} q^{41} + 2 \zeta_{6} q^{42} - 4 q^{43} + 3 \zeta_{6} q^{44} + q^{45} - 3 q^{47} + q^{48} + ( - 3 \zeta_{6} + 3) q^{49} + \zeta_{6} q^{50} + 3 q^{51} + ( - \zeta_{6} + 1) q^{52} + (6 \zeta_{6} - 6) q^{53} + ( - \zeta_{6} + 1) q^{54} + 3 \zeta_{6} q^{55} + 2 \zeta_{6} q^{56} + (2 \zeta_{6} - 2) q^{57} + (3 \zeta_{6} - 3) q^{58} + ( - 3 \zeta_{6} + 3) q^{59} + q^{60} - 8 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + 2 q^{63} + q^{64} + ( - \zeta_{6} + 1) q^{65} + 3 q^{66} + 7 \zeta_{6} q^{67} + 3 q^{68} + 2 \zeta_{6} q^{70} + ( - \zeta_{6} + 1) q^{72} - 16 q^{73} + ( - 4 \zeta_{6} + 7) q^{74} + q^{75} + (2 \zeta_{6} - 2) q^{76} + 6 \zeta_{6} q^{77} - \zeta_{6} q^{78} - 8 \zeta_{6} q^{79} + q^{80} - \zeta_{6} q^{81} + 6 q^{82} + (6 \zeta_{6} - 6) q^{83} + 2 q^{84} + 3 q^{85} + (4 \zeta_{6} - 4) q^{86} + 3 \zeta_{6} q^{87} + 3 q^{88} + (6 \zeta_{6} - 6) q^{89} + ( - \zeta_{6} + 1) q^{90} + ( - 2 \zeta_{6} + 2) q^{91} + 4 \zeta_{6} q^{93} + (3 \zeta_{6} - 3) q^{94} + (2 \zeta_{6} - 2) q^{95} + ( - \zeta_{6} + 1) q^{96} + 14 q^{97} - 3 \zeta_{6} q^{98} + ( - 3 \zeta_{6} + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} - q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} - q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} - q^{9} - 2 q^{10} - 6 q^{11} - q^{12} + q^{13} - 4 q^{14} - q^{15} - q^{16} - 3 q^{17} + q^{18} - 2 q^{19} - q^{20} - 2 q^{21} - 3 q^{22} + q^{24} - q^{25} + 2 q^{26} + 2 q^{27} - 2 q^{28} - 6 q^{29} + q^{30} - 8 q^{31} + q^{32} + 3 q^{33} + 3 q^{34} - 2 q^{35} + 2 q^{36} + 11 q^{37} - 4 q^{38} + q^{39} + q^{40} + 6 q^{41} + 2 q^{42} - 8 q^{43} + 3 q^{44} + 2 q^{45} - 6 q^{47} + 2 q^{48} + 3 q^{49} + q^{50} + 6 q^{51} + q^{52} - 6 q^{53} + q^{54} + 3 q^{55} + 2 q^{56} - 2 q^{57} - 3 q^{58} + 3 q^{59} + 2 q^{60} - 8 q^{61} - 4 q^{62} + 4 q^{63} + 2 q^{64} + q^{65} + 6 q^{66} + 7 q^{67} + 6 q^{68} + 2 q^{70} + q^{72} - 32 q^{73} + 10 q^{74} + 2 q^{75} - 2 q^{76} + 6 q^{77} - q^{78} - 8 q^{79} + 2 q^{80} - q^{81} + 12 q^{82} - 6 q^{83} + 4 q^{84} + 6 q^{85} - 4 q^{86} + 3 q^{87} + 6 q^{88} - 6 q^{89} + q^{90} + 2 q^{91} + 4 q^{93} - 3 q^{94} - 2 q^{95} + q^{96} + 28 q^{97} - 3 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.00000 −1.00000 1.73205i −1.00000 −0.500000 + 0.866025i −1.00000
211.1 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.00000 −1.00000 + 1.73205i −1.00000 −0.500000 0.866025i −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.i.c 2
37.c even 3 1 inner 1110.2.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.c 2 1.a even 1 1 trivial
1110.2.i.c 2 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T - 14)^{2} \) Copy content Toggle raw display
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