Properties

Label 1110.2.i.i
Level $1110$
Weight $2$
Character orbit 1110.i
Analytic conductor $8.863$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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: \(\Q(\sqrt{-3}, \sqrt{145})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 37x^{2} + 36x + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} - \beta_{2} q^{3} - \beta_{2} q^{4} + \beta_{2} q^{5} + q^{6} - \beta_{2} q^{7} + q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} - \beta_{2} q^{3} - \beta_{2} q^{4} + \beta_{2} q^{5} + q^{6} - \beta_{2} q^{7} + q^{8} + (\beta_{2} - 1) q^{9} - q^{10} - 2 q^{11} + (\beta_{2} - 1) q^{12} + \beta_{2} q^{13} + q^{14} + ( - \beta_{2} + 1) q^{15} + (\beta_{2} - 1) q^{16} + ( - 2 \beta_{2} + 2) q^{17} - \beta_{2} q^{18} + \beta_1 q^{19} + ( - \beta_{2} + 1) q^{20} + (\beta_{2} - 1) q^{21} + ( - 2 \beta_{2} + 2) q^{22} - \beta_{2} q^{24} + (\beta_{2} - 1) q^{25} - q^{26} + q^{27} + (\beta_{2} - 1) q^{28} + (\beta_{3} - 1) q^{29} + \beta_{2} q^{30} + ( - \beta_{3} + 4) q^{31} - \beta_{2} q^{32} + 2 \beta_{2} q^{33} + 2 \beta_{2} q^{34} + ( - \beta_{2} + 1) q^{35} + q^{36} + (\beta_1 - 1) q^{37} + (\beta_{3} - 1) q^{38} + ( - \beta_{2} + 1) q^{39} + \beta_{2} q^{40} - \beta_{2} q^{42} + ( - \beta_{3} - 4) q^{43} + 2 \beta_{2} q^{44} - q^{45} + ( - 2 \beta_{3} + 2) q^{47} + q^{48} + ( - 6 \beta_{2} + 6) q^{49} - \beta_{2} q^{50} - 2 q^{51} + ( - \beta_{2} + 1) q^{52} + (\beta_{2} - 1) q^{54} - 2 \beta_{2} q^{55} - \beta_{2} q^{56} + ( - \beta_{3} - \beta_1 + 1) q^{57} + ( - \beta_{3} - \beta_1 + 1) q^{58} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{59} - q^{60} - 10 \beta_{2} q^{61} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{62} + q^{63} + q^{64} + (\beta_{2} - 1) q^{65} - 2 q^{66} + (5 \beta_{2} + \beta_1) q^{67} - 2 q^{68} + \beta_{2} q^{70} + (6 \beta_{2} + \beta_1) q^{71} + (\beta_{2} - 1) q^{72} + ( - \beta_{3} + 4) q^{73} + (\beta_{3} - \beta_{2}) q^{74} + q^{75} + ( - \beta_{3} - \beta_1 + 1) q^{76} + 2 \beta_{2} q^{77} + \beta_{2} q^{78} + (3 \beta_{2} + \beta_1) q^{79} - q^{80} - \beta_{2} q^{81} + ( - \beta_{3} - 10 \beta_{2} + \cdots + 11) q^{83}+ \cdots + ( - 2 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9} - 4 q^{10} - 8 q^{11} - 2 q^{12} + 2 q^{13} + 4 q^{14} + 2 q^{15} - 2 q^{16} + 4 q^{17} - 2 q^{18} + q^{19} + 2 q^{20} - 2 q^{21} + 4 q^{22} - 2 q^{24} - 2 q^{25} - 4 q^{26} + 4 q^{27} - 2 q^{28} - 2 q^{29} + 2 q^{30} + 14 q^{31} - 2 q^{32} + 4 q^{33} + 4 q^{34} + 2 q^{35} + 4 q^{36} - 3 q^{37} - 2 q^{38} + 2 q^{39} + 2 q^{40} - 2 q^{42} - 18 q^{43} + 4 q^{44} - 4 q^{45} + 4 q^{47} + 4 q^{48} + 12 q^{49} - 2 q^{50} - 8 q^{51} + 2 q^{52} - 2 q^{54} - 4 q^{55} - 2 q^{56} + q^{57} + q^{58} - 6 q^{59} - 4 q^{60} - 20 q^{61} - 7 q^{62} + 4 q^{63} + 4 q^{64} - 2 q^{65} - 8 q^{66} + 11 q^{67} - 8 q^{68} + 2 q^{70} + 13 q^{71} - 2 q^{72} + 14 q^{73} + 4 q^{75} + q^{76} + 4 q^{77} + 2 q^{78} + 7 q^{79} - 4 q^{80} - 2 q^{81} + 21 q^{83} + 4 q^{84} + 8 q^{85} + 9 q^{86} + q^{87} - 8 q^{88} + 2 q^{89} + 2 q^{90} + 2 q^{91} - 7 q^{93} - 2 q^{94} - q^{95} - 2 q^{96} - 2 q^{97} + 12 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 37x^{2} + 36x + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 37\nu^{2} - 37\nu + 1296 ) / 1332 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 73 ) / 37 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 36\beta_{2} + \beta _1 - 37 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 37\beta_{3} - 73 \) Copy content Toggle raw display

Character values

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

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

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
−2.76040 4.78115i
3.26040 + 5.64718i
−2.76040 + 4.78115i
3.26040 5.64718i
−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
121.2 −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
211.2 −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.i 4
37.c even 3 1 inner 1110.2.i.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.i 4 1.a even 1 1 trivial
1110.2.i.i 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} + 2 \) 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
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + T - 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 7 T - 24)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 3 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 9 T - 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 2 T - 144)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + \cdots + 18496 \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 11 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$71$ \( T^{4} - 13 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$73$ \( (T^{2} - 7 T - 24)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 7 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$83$ \( T^{4} - 21 T^{3} + \cdots + 5476 \) Copy content Toggle raw display
$89$ \( T^{4} - 2 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
$97$ \( (T^{2} + T - 36)^{2} \) Copy content Toggle raw display
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