Properties

Label 1110.2.x.b
Level $1110$
Weight $2$
Character orbit 1110.x
Analytic conductor $8.863$
Analytic rank $0$
Dimension $4$
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(751,1110)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.751");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.x (of order \(6\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{3} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{5} + \zeta_{12}^{3} q^{6} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + (\zeta_{12}^{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{3} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{5} + \zeta_{12}^{3} q^{6} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + (\zeta_{12}^{2} - 1) q^{9} + q^{10} + (\zeta_{12}^{3} - 2 \zeta_{12} + 2) q^{11} + (\zeta_{12}^{2} - 1) q^{12} + ( - \zeta_{12}^{2} + 2) q^{13} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{14} + \zeta_{12} q^{15} + (\zeta_{12}^{2} - 1) q^{16} + (3 \zeta_{12}^{2} + 2 \zeta_{12} + 3) q^{17} + (\zeta_{12}^{3} - \zeta_{12}) q^{18} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{19} + \zeta_{12} q^{20} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 1) q^{21} + ( - \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{22} + 2 \zeta_{12}^{3} q^{23} + (\zeta_{12}^{3} - \zeta_{12}) q^{24} + ( - \zeta_{12}^{2} + 1) q^{25} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{26} - q^{27} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 1) q^{28} + (6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{29} + \zeta_{12}^{2} q^{30} + (8 \zeta_{12}^{2} - 4) q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12}) q^{33} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 3 \zeta_{12}) q^{34} + ( - \zeta_{12}^{2} + \zeta_{12} - 1) q^{35} - q^{36} + (7 \zeta_{12}^{2} - 4) q^{37} + 2 q^{38} + (\zeta_{12}^{2} + 1) q^{39} + \zeta_{12}^{2} q^{40} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{41} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{42} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{43} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12}) q^{44} + \zeta_{12}^{3} q^{45} + (2 \zeta_{12}^{2} - 2) q^{46} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{47} - q^{48} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 2 \zeta_{12} + 3) q^{49} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{50} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{51} + (\zeta_{12}^{2} + 1) q^{52} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{53} - \zeta_{12} q^{54} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{55} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{56} + 2 \zeta_{12} q^{57} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + \zeta_{12} - 6) q^{58} + ( - 2 \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{59} + \zeta_{12}^{3} q^{60} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{61} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{62} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 1) q^{63} - q^{64} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{65} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{66} + ( - 5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 5 \zeta_{12}) q^{67} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{68} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{69} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{70} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{71} - \zeta_{12} q^{72} + (6 \zeta_{12}^{3} - 12 \zeta_{12} - 2) q^{73} + (7 \zeta_{12}^{3} - 4 \zeta_{12}) q^{74} + q^{75} + 2 \zeta_{12} q^{76} + ( - 3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 3 \zeta_{12}) q^{77} + (\zeta_{12}^{3} + \zeta_{12}) q^{78} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} - 8) q^{79} + \zeta_{12}^{3} q^{80} - \zeta_{12}^{2} q^{81} + ( - 3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{82} + ( - 6 \zeta_{12}^{3} - \zeta_{12}^{2} + 3 \zeta_{12} + 1) q^{83} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 1) q^{84} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} + 2) q^{85} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{86} + (6 \zeta_{12}^{3} - \zeta_{12}^{2} - 6 \zeta_{12} + 2) q^{87} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{88} + (4 \zeta_{12}^{2} + 6 \zeta_{12} + 4) q^{89} + (\zeta_{12}^{2} - 1) q^{90} + (\zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{91} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{92} + (4 \zeta_{12}^{2} - 8) q^{93} + ( - \zeta_{12}^{2} - 1) q^{94} + ( - 2 \zeta_{12}^{2} + 2) q^{95} - \zeta_{12} q^{96} + (8 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{97} + ( - 3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 3 \zeta_{12} + 4) q^{98} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{4} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 2 q^{4} + 2 q^{7} - 2 q^{9} + 4 q^{10} + 8 q^{11} - 2 q^{12} + 6 q^{13} - 2 q^{16} + 18 q^{17} - 2 q^{21} - 6 q^{22} + 2 q^{25} - 4 q^{27} - 2 q^{28} + 2 q^{30} + 4 q^{33} + 4 q^{34} - 6 q^{35} - 4 q^{36} - 2 q^{37} + 8 q^{38} + 6 q^{39} + 2 q^{40} - 6 q^{41} + 6 q^{42} + 4 q^{44} - 4 q^{46} - 4 q^{48} + 6 q^{49} + 6 q^{52} + 6 q^{53} - 6 q^{55} + 6 q^{56} - 12 q^{58} - 12 q^{59} - 6 q^{61} - 4 q^{63} - 4 q^{64} - 4 q^{67} + 2 q^{70} - 6 q^{71} - 8 q^{73} + 4 q^{75} + 10 q^{77} - 24 q^{79} - 2 q^{81} + 2 q^{83} - 4 q^{84} + 8 q^{85} + 8 q^{86} + 6 q^{87} + 24 q^{89} - 2 q^{90} + 6 q^{91} - 24 q^{93} - 6 q^{94} + 4 q^{95} + 12 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(\zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i 1.36603 2.36603i 1.00000i −0.500000 0.866025i 1.00000
751.2 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i −0.366025 + 0.633975i 1.00000i −0.500000 0.866025i 1.00000
841.1 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 1.36603 + 2.36603i 1.00000i −0.500000 + 0.866025i 1.00000
841.2 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i −0.366025 0.633975i 1.00000i −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.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.x.b 4
37.e even 6 1 inner 1110.2.x.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.x.b 4 1.a even 1 1 trivial
1110.2.x.b 4 37.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 2T_{7}^{3} + 6T_{7}^{2} + 4T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 18 T^{3} + 131 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$19$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 78T^{2} + 1089 \) Copy content Toggle raw display
$31$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 37)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$43$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 54 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + 51 T^{2} + 36 T + 9 \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} + 87 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$73$ \( (T^{2} + 4 T - 104)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 24 T^{3} + 236 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$83$ \( T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676 \) Copy content Toggle raw display
$89$ \( T^{4} - 24 T^{3} + 204 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$97$ \( T^{4} + 152T^{2} + 2704 \) Copy content Toggle raw display
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