Properties

Label 1110.2.d.i
Level $1110$
Weight $2$
Character orbit 1110.d
Analytic conductor $8.863$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1110,2,Mod(889,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.889");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.d (of order \(2\), degree \(1\), 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: \(6\)
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{4} q^{3} - q^{4} + ( - \beta_{3} - \beta_1) q^{5} - q^{6} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} + \beta_{4} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_{4} q^{3} - q^{4} + ( - \beta_{3} - \beta_1) q^{5} - q^{6} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} + \beta_{4} q^{8} - q^{9} + (\beta_{5} + \beta_{2}) q^{10} + (2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{11} + \beta_{4} q^{12} + (\beta_{5} + \beta_{4}) q^{13} + (\beta_{2} + \beta_1 + 1) q^{14} + (\beta_{5} + \beta_{2}) q^{15} + q^{16} + ( - 2 \beta_{5} - 3 \beta_{4} + \cdots + \beta_1) q^{17}+ \cdots + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9} - 10 q^{11} + 6 q^{14} + 6 q^{16} - 20 q^{19} - 2 q^{20} + 6 q^{21} + 6 q^{24} + 10 q^{25} + 4 q^{26} + 34 q^{29} - 2 q^{30} - 2 q^{31} - 14 q^{34} - 20 q^{35} + 6 q^{36} + 4 q^{39} - 18 q^{41} + 10 q^{44} - 2 q^{45} + 20 q^{46} - 4 q^{49} - 4 q^{50} - 14 q^{51} + 6 q^{54} - 42 q^{55} - 6 q^{56} + 8 q^{59} + 6 q^{61} - 6 q^{64} + 8 q^{65} + 10 q^{66} + 20 q^{69} - 2 q^{70} + 4 q^{71} - 6 q^{74} - 4 q^{75} + 20 q^{76} + 24 q^{79} + 2 q^{80} + 6 q^{81} - 6 q^{84} + 4 q^{85} + 6 q^{86} - 12 q^{89} + 12 q^{91} - 8 q^{94} - 28 q^{95} - 6 q^{96} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} - 5\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16 \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
889.1
0.432320 0.432320i
−1.75233 + 1.75233i
1.32001 1.32001i
0.432320 + 0.432320i
−1.75233 1.75233i
1.32001 + 1.32001i
1.00000i 1.00000i −1.00000 −2.19388 + 0.432320i −1.00000 1.86464i 1.00000i −1.00000 0.432320 + 2.19388i
889.2 1.00000i 1.00000i −1.00000 1.38900 1.75233i −1.00000 2.50466i 1.00000i −1.00000 −1.75233 1.38900i
889.3 1.00000i 1.00000i −1.00000 1.80487 + 1.32001i −1.00000 3.64002i 1.00000i −1.00000 1.32001 1.80487i
889.4 1.00000i 1.00000i −1.00000 −2.19388 0.432320i −1.00000 1.86464i 1.00000i −1.00000 0.432320 2.19388i
889.5 1.00000i 1.00000i −1.00000 1.38900 + 1.75233i −1.00000 2.50466i 1.00000i −1.00000 −1.75233 + 1.38900i
889.6 1.00000i 1.00000i −1.00000 1.80487 1.32001i −1.00000 3.64002i 1.00000i −1.00000 1.32001 + 1.80487i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 889.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.d.i 6
3.b odd 2 1 3330.2.d.n 6
5.b even 2 1 inner 1110.2.d.i 6
5.c odd 4 1 5550.2.a.cf 3
5.c odd 4 1 5550.2.a.cg 3
15.d odd 2 1 3330.2.d.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.i 6 1.a even 1 1 trivial
1110.2.d.i 6 5.b even 2 1 inner
3330.2.d.n 6 3.b odd 2 1
3330.2.d.n 6 15.d odd 2 1
5550.2.a.cf 3 5.c odd 4 1
5550.2.a.cg 3 5.c odd 4 1

Hecke kernels

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

\( T_{7}^{6} + 23T_{7}^{4} + 151T_{7}^{2} + 289 \) Copy content Toggle raw display
\( T_{11}^{3} + 5T_{11}^{2} - 11T_{11} - 59 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 23 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( (T^{3} + 5 T^{2} - 11 T - 59)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 14 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 55 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{3} + 10 T^{2} + 25 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 82 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{3} - 17 T^{2} + \cdots + 50)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 24 T + 20)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$41$ \( (T^{3} + 9 T^{2} - 30 T - 34)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 185 T^{4} + \cdots + 59536 \) Copy content Toggle raw display
$47$ \( T^{6} + 44 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$53$ \( T^{6} + 91 T^{4} + \cdots + 121 \) Copy content Toggle raw display
$59$ \( (T^{3} - 4 T^{2} + \cdots + 232)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 3 T^{2} + \cdots + 108)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 312 T^{4} + \cdots + 295936 \) Copy content Toggle raw display
$71$ \( (T^{3} - 2 T^{2} + \cdots - 148)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 54 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$79$ \( (T^{3} - 12 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 206 T^{4} + \cdots + 1936 \) Copy content Toggle raw display
$89$ \( (T^{3} + 6 T^{2} - 45 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 493 T^{4} + \cdots + 1085764 \) Copy content Toggle raw display
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