Properties

Label 1110.2.i.j
Level $1110$
Weight $2$
Character orbit 1110.i
Analytic conductor $8.863$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{73})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 19x^{2} + 18x + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{2} + 1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{2} - 1) q^{5} - q^{6} + q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{2} + 1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{2} - 1) q^{5} - q^{6} + q^{8} - \beta_{2} q^{9} + q^{10} + q^{11} + \beta_{2} q^{12} + (\beta_{2} - 1) q^{13} + \beta_{2} q^{15} - \beta_{2} q^{16} + (3 \beta_{2} + \beta_1) q^{17} + (\beta_{2} - 1) q^{18} + (\beta_{3} + \beta_1 - 1) q^{19} - \beta_{2} q^{20} - \beta_{2} q^{22} + (\beta_{3} + 1) q^{23} + ( - \beta_{2} + 1) q^{24} - \beta_{2} q^{25} + q^{26} - q^{27} + (\beta_{3} + 4) q^{29} + ( - \beta_{2} + 1) q^{30} + ( - 2 \beta_{3} + 2) q^{31} + (\beta_{2} - 1) q^{32} + ( - \beta_{2} + 1) q^{33} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{34} + q^{36} + (\beta_{3} - 5 \beta_{2} + 2) q^{37} + ( - \beta_{3} + 1) q^{38} + \beta_{2} q^{39} + (\beta_{2} - 1) q^{40} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{41} + (\beta_{2} - 1) q^{44} + q^{45} + ( - 2 \beta_{2} + \beta_1) q^{46} + 3 q^{47} - q^{48} + 7 \beta_{2} q^{49} + (\beta_{2} - 1) q^{50} + ( - \beta_{3} + 4) q^{51} - \beta_{2} q^{52} + (4 \beta_{2} + 2 \beta_1) q^{53} + \beta_{2} q^{54} + (\beta_{2} - 1) q^{55} + \beta_1 q^{57} + ( - 5 \beta_{2} + \beta_1) q^{58} + (3 \beta_{2} + 2 \beta_1) q^{59} - q^{60} + ( - 10 \beta_{2} + 10) q^{61} - 2 \beta_1 q^{62} + q^{64} - \beta_{2} q^{65} - q^{66} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{67} + (\beta_{3} - 4) q^{68} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{69} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 4) q^{71} - \beta_{2} q^{72} + (2 \beta_{2} + \beta_1 - 5) q^{74} - q^{75} - \beta_1 q^{76} + ( - \beta_{2} + 1) q^{78} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{79} + q^{80} + (\beta_{2} - 1) q^{81} + ( - 2 \beta_{3} - 2) q^{82} - 2 \beta_1 q^{83} + (\beta_{3} - 4) q^{85} + (\beta_{3} - 5 \beta_{2} + \beta_1 + 4) q^{87} + q^{88} + (12 \beta_{2} - \beta_1) q^{89} - \beta_{2} q^{90} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{92} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{93} - 3 \beta_{2} q^{94} - \beta_1 q^{95} + \beta_{2} q^{96} + ( - 2 \beta_{3} - 8) q^{97} + ( - 7 \beta_{2} + 7) q^{98} - \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} - 2 q^{9} + 4 q^{10} + 4 q^{11} + 2 q^{12} - 2 q^{13} + 2 q^{15} - 2 q^{16} + 7 q^{17} - 2 q^{18} - q^{19} - 2 q^{20} - 2 q^{22} + 6 q^{23} + 2 q^{24} - 2 q^{25} + 4 q^{26} - 4 q^{27} + 18 q^{29} + 2 q^{30} + 4 q^{31} - 2 q^{32} + 2 q^{33} + 7 q^{34} + 4 q^{36} + 2 q^{38} + 2 q^{39} - 2 q^{40} + 6 q^{41} - 2 q^{44} + 4 q^{45} - 3 q^{46} + 12 q^{47} - 4 q^{48} + 14 q^{49} - 2 q^{50} + 14 q^{51} - 2 q^{52} + 10 q^{53} + 2 q^{54} - 2 q^{55} + q^{57} - 9 q^{58} + 8 q^{59} - 4 q^{60} + 20 q^{61} - 2 q^{62} + 4 q^{64} - 2 q^{65} - 4 q^{66} + 5 q^{67} - 14 q^{68} + 3 q^{69} + 10 q^{71} - 2 q^{72} - 15 q^{74} - 4 q^{75} - q^{76} + 2 q^{78} + 6 q^{79} + 4 q^{80} - 2 q^{81} - 12 q^{82} - 2 q^{83} - 14 q^{85} + 9 q^{87} + 4 q^{88} + 23 q^{89} - 2 q^{90} - 3 q^{92} + 2 q^{93} - 6 q^{94} - q^{95} + 2 q^{96} - 36 q^{97} + 14 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 19x^{2} + 18x + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 19\nu^{2} - 19\nu + 324 ) / 342 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 37 ) / 19 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 18\beta_{2} + \beta _1 - 19 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 19\beta_{3} - 37 \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−1.88600 + 3.26665i
2.38600 4.13267i
−1.88600 3.26665i
2.38600 + 4.13267i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.00000 0 1.00000 −0.500000 + 0.866025i 1.00000
121.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.00000 0 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 1.00000 −0.500000 0.866025i 1.00000
211.2 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.00000 0 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.j 4
37.c even 3 1 inner 1110.2.i.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.j 4 1.a even 1 1 trivial
1110.2.i.j 4 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 7 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{4} + T^{3} + \cdots + 324 \) Copy content Toggle raw display
$23$ \( (T^{2} - 3 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 9 T + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T - 72)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + T^{2} + 1369 \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T - 3)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 10 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} + \cdots + 3249 \) Copy content Toggle raw display
$61$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 5 T^{3} + \cdots + 24964 \) Copy content Toggle raw display
$71$ \( T^{4} - 10 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$83$ \( T^{4} + 2 T^{3} + \cdots + 5184 \) Copy content Toggle raw display
$89$ \( T^{4} - 23 T^{3} + \cdots + 12996 \) Copy content Toggle raw display
$97$ \( (T^{2} + 18 T + 8)^{2} \) Copy content Toggle raw display
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