Properties

Label 1110.2.d.f
Level $1110$
Weight $2$
Character orbit 1110.d
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(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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_1 q^{3} - q^{4} + \beta_{2} q^{5} - q^{6} - 2 \beta_1 q^{7} + \beta_1 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_1 q^{3} - q^{4} + \beta_{2} q^{5} - q^{6} - 2 \beta_1 q^{7} + \beta_1 q^{8} - q^{9} + \beta_{3} q^{10} + ( - \beta_{3} + 3) q^{11} + \beta_1 q^{12} + 2 \beta_{2} q^{13} - 2 q^{14} + \beta_{3} q^{15} + q^{16} + 2 \beta_{2} q^{17} + \beta_1 q^{18} + ( - \beta_{3} + 5) q^{19} - \beta_{2} q^{20} - 2 q^{21} + (\beta_{2} - 3 \beta_1) q^{22} + 4 \beta_1 q^{23} + q^{24} - 5 q^{25} + 2 \beta_{3} q^{26} + \beta_1 q^{27} + 2 \beta_1 q^{28} - 4 q^{29} - \beta_{2} q^{30} + (2 \beta_{3} + 2) q^{31} - \beta_1 q^{32} + (\beta_{2} - 3 \beta_1) q^{33} + 2 \beta_{3} q^{34} + 2 \beta_{3} q^{35} + q^{36} - \beta_1 q^{37} + (\beta_{2} - 5 \beta_1) q^{38} + 2 \beta_{3} q^{39} - \beta_{3} q^{40} + 2 \beta_{3} q^{41} + 2 \beta_1 q^{42} + ( - 2 \beta_{2} + 6 \beta_1) q^{43} + (\beta_{3} - 3) q^{44} - \beta_{2} q^{45} + 4 q^{46} + (3 \beta_{2} - 3 \beta_1) q^{47} - \beta_1 q^{48} + 3 q^{49} + 5 \beta_1 q^{50} + 2 \beta_{3} q^{51} - 2 \beta_{2} q^{52} + ( - 2 \beta_{2} + 8 \beta_1) q^{53} + q^{54} + (3 \beta_{2} - 5 \beta_1) q^{55} + 2 q^{56} + (\beta_{2} - 5 \beta_1) q^{57} + 4 \beta_1 q^{58} + (4 \beta_{3} + 2) q^{59} - \beta_{3} q^{60} + ( - \beta_{3} + 3) q^{61} + ( - 2 \beta_{2} - 2 \beta_1) q^{62} + 2 \beta_1 q^{63} - q^{64} - 10 q^{65} + (\beta_{3} - 3) q^{66} + ( - 2 \beta_{2} + 6 \beta_1) q^{67} - 2 \beta_{2} q^{68} + 4 q^{69} - 2 \beta_{2} q^{70} + (2 \beta_{3} + 2) q^{71} - \beta_1 q^{72} + ( - 2 \beta_{2} + 6 \beta_1) q^{73} - q^{74} + 5 \beta_1 q^{75} + (\beta_{3} - 5) q^{76} + (2 \beta_{2} - 6 \beta_1) q^{77} - 2 \beta_{2} q^{78} + ( - 2 \beta_{3} - 2) q^{79} + \beta_{2} q^{80} + q^{81} - 2 \beta_{2} q^{82} + (6 \beta_{2} - 2 \beta_1) q^{83} + 2 q^{84} - 10 q^{85} + ( - 2 \beta_{3} + 6) q^{86} + 4 \beta_1 q^{87} + ( - \beta_{2} + 3 \beta_1) q^{88} + (2 \beta_{3} + 12) q^{89} - \beta_{3} q^{90} + 4 \beta_{3} q^{91} - 4 \beta_1 q^{92} + ( - 2 \beta_{2} - 2 \beta_1) q^{93} + (3 \beta_{3} - 3) q^{94} + (5 \beta_{2} - 5 \beta_1) q^{95} - q^{96} + ( - 5 \beta_{2} - 3 \beta_1) q^{97} - 3 \beta_1 q^{98} + (\beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 12 q^{11} - 8 q^{14} + 4 q^{16} + 20 q^{19} - 8 q^{21} + 4 q^{24} - 20 q^{25} - 16 q^{29} + 8 q^{31} + 4 q^{36} - 12 q^{44} + 16 q^{46} + 12 q^{49} + 4 q^{54} + 8 q^{56} + 8 q^{59} + 12 q^{61} - 4 q^{64} - 40 q^{65} - 12 q^{66} + 16 q^{69} + 8 q^{71} - 4 q^{74} - 20 q^{76} - 8 q^{79} + 4 q^{81} + 8 q^{84} - 40 q^{85} + 24 q^{86} + 48 q^{89} - 12 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) 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
1.61803i
0.618034i
0.618034i
1.61803i
1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 −2.23607
889.2 1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 2.23607
889.3 1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 2.23607
889.4 1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 −2.23607
\(n\): e.g. 2-40 or 990-1000
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.f 4
3.b odd 2 1 3330.2.d.j 4
5.b even 2 1 inner 1110.2.d.f 4
5.c odd 4 1 5550.2.a.bu 2
5.c odd 4 1 5550.2.a.bz 2
15.d odd 2 1 3330.2.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.f 4 1.a even 1 1 trivial
1110.2.d.f 4 5.b even 2 1 inner
3330.2.d.j 4 3.b odd 2 1
3330.2.d.j 4 15.d odd 2 1
5550.2.a.bu 2 5.c odd 4 1
5550.2.a.bz 2 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 112T^{2} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 1936 \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T - 76)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 112T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 112T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 368 T^{2} + 30976 \) Copy content Toggle raw display
$89$ \( (T^{2} - 24 T + 124)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 268 T^{2} + 13456 \) Copy content Toggle raw display
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