Properties

Label 1110.2.i.d
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{5} - q^{6} + \zeta_{6} q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{5} - q^{6} + \zeta_{6} q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} + q^{10} + 6 q^{11} + (\zeta_{6} - 1) q^{12} + \zeta_{6} q^{13} + q^{14} + ( - \zeta_{6} + 1) q^{15} + (\zeta_{6} - 1) q^{16} + (6 \zeta_{6} - 6) q^{17} + \zeta_{6} q^{18} + 7 \zeta_{6} q^{19} + ( - \zeta_{6} + 1) q^{20} + ( - \zeta_{6} + 1) q^{21} + ( - 6 \zeta_{6} + 6) q^{22} - 6 q^{23} + \zeta_{6} q^{24} + (\zeta_{6} - 1) q^{25} + q^{26} + q^{27} + ( - \zeta_{6} + 1) q^{28} + 9 q^{29} - \zeta_{6} q^{30} - 4 q^{31} + \zeta_{6} q^{32} - 6 \zeta_{6} q^{33} + 6 \zeta_{6} q^{34} + (\zeta_{6} - 1) q^{35} + q^{36} + ( - 3 \zeta_{6} + 7) q^{37} + 7 q^{38} + ( - \zeta_{6} + 1) q^{39} - \zeta_{6} q^{40} - 6 \zeta_{6} q^{41} - \zeta_{6} q^{42} + 2 q^{43} - 6 \zeta_{6} q^{44} - q^{45} + (6 \zeta_{6} - 6) q^{46} + 12 q^{47} + q^{48} + ( - 6 \zeta_{6} + 6) q^{49} + \zeta_{6} q^{50} + 6 q^{51} + ( - \zeta_{6} + 1) q^{52} + (6 \zeta_{6} - 6) q^{53} + ( - \zeta_{6} + 1) q^{54} + 6 \zeta_{6} q^{55} - \zeta_{6} q^{56} + ( - 7 \zeta_{6} + 7) q^{57} + ( - 9 \zeta_{6} + 9) q^{58} + ( - 6 \zeta_{6} + 6) q^{59} - q^{60} - 8 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} - q^{63} + q^{64} + (\zeta_{6} - 1) q^{65} - 6 q^{66} + 10 \zeta_{6} q^{67} + 6 q^{68} + 6 \zeta_{6} q^{69} + \zeta_{6} q^{70} + 3 \zeta_{6} q^{71} + ( - \zeta_{6} + 1) q^{72} - 10 q^{73} + ( - 7 \zeta_{6} + 4) q^{74} + q^{75} + ( - 7 \zeta_{6} + 7) q^{76} + 6 \zeta_{6} q^{77} - \zeta_{6} q^{78} - 14 \zeta_{6} q^{79} - q^{80} - \zeta_{6} q^{81} - 6 q^{82} + ( - 3 \zeta_{6} + 3) q^{83} - q^{84} - 6 q^{85} + ( - 2 \zeta_{6} + 2) q^{86} - 9 \zeta_{6} q^{87} - 6 q^{88} + (\zeta_{6} - 1) q^{90} + (\zeta_{6} - 1) q^{91} + 6 \zeta_{6} q^{92} + 4 \zeta_{6} q^{93} + ( - 12 \zeta_{6} + 12) q^{94} + (7 \zeta_{6} - 7) q^{95} + ( - \zeta_{6} + 1) q^{96} - 10 q^{97} - 6 \zeta_{6} q^{98} + (6 \zeta_{6} - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} - q^{9} + 2 q^{10} + 12 q^{11} - q^{12} + q^{13} + 2 q^{14} + q^{15} - q^{16} - 6 q^{17} + q^{18} + 7 q^{19} + q^{20} + q^{21} + 6 q^{22} - 12 q^{23} + q^{24} - q^{25} + 2 q^{26} + 2 q^{27} + q^{28} + 18 q^{29} - q^{30} - 8 q^{31} + q^{32} - 6 q^{33} + 6 q^{34} - q^{35} + 2 q^{36} + 11 q^{37} + 14 q^{38} + q^{39} - q^{40} - 6 q^{41} - q^{42} + 4 q^{43} - 6 q^{44} - 2 q^{45} - 6 q^{46} + 24 q^{47} + 2 q^{48} + 6 q^{49} + q^{50} + 12 q^{51} + q^{52} - 6 q^{53} + q^{54} + 6 q^{55} - q^{56} + 7 q^{57} + 9 q^{58} + 6 q^{59} - 2 q^{60} - 8 q^{61} - 4 q^{62} - 2 q^{63} + 2 q^{64} - q^{65} - 12 q^{66} + 10 q^{67} + 12 q^{68} + 6 q^{69} + q^{70} + 3 q^{71} + q^{72} - 20 q^{73} + q^{74} + 2 q^{75} + 7 q^{76} + 6 q^{77} - q^{78} - 14 q^{79} - 2 q^{80} - q^{81} - 12 q^{82} + 3 q^{83} - 2 q^{84} - 12 q^{85} + 2 q^{86} - 9 q^{87} - 12 q^{88} - q^{90} - q^{91} + 6 q^{92} + 4 q^{93} + 12 q^{94} - 7 q^{95} + q^{96} - 20 q^{97} - 6 q^{98} - 6 q^{99}+O(q^{100}) \) 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 0.500000 + 0.866025i −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 0.500000 0.866025i −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.d 2
37.c even 3 1 inner 1110.2.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.d 2 1.a even 1 1 trivial
1110.2.i.d 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} - 6 \) 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
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) 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
$43$ \( (T - 2)^{2} \) Copy content Toggle raw display
$47$ \( (T - 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$83$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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