Properties

Label 1110.2.d.h
Level $1110$
Weight $2$
Character orbit 1110.d
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(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: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{2} q^{3} - q^{4} + (\beta_{3} + \beta_{2} + 1) q^{5} + q^{6} + (\beta_{3} + \beta_{2} + \beta_1) q^{7} - \beta_{2} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{2} q^{3} - q^{4} + (\beta_{3} + \beta_{2} + 1) q^{5} + q^{6} + (\beta_{3} + \beta_{2} + \beta_1) q^{7} - \beta_{2} q^{8} - q^{9} + (\beta_{2} - \beta_1 - 1) q^{10} + ( - \beta_{3} + \beta_1 - 1) q^{11} + \beta_{2} q^{12} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{13} + (\beta_{3} - \beta_1 - 1) q^{14} + ( - \beta_{2} + \beta_1 + 1) q^{15} + q^{16} + (\beta_{3} + \beta_{2} + \beta_1) q^{17} - \beta_{2} q^{18} - 3 q^{19} + ( - \beta_{3} - \beta_{2} - 1) q^{20} + ( - \beta_{3} + \beta_1 + 1) q^{21} + (\beta_{3} - \beta_{2} + \beta_1) q^{22} - \beta_{2} q^{23} - q^{24} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{25} + (\beta_{3} - \beta_1 + 3) q^{26} + \beta_{2} q^{27} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{28} + ( - \beta_{3} + \beta_1 - 6) q^{29} + (\beta_{3} + \beta_{2} + 1) q^{30} - 2 q^{31} + \beta_{2} q^{32} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{33} + (\beta_{3} - \beta_1 - 1) q^{34} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{35} + q^{36} - \beta_{2} q^{37} - 3 \beta_{2} q^{38} + ( - \beta_{3} + \beta_1 - 3) q^{39} + ( - \beta_{2} + \beta_1 + 1) q^{40} + (\beta_{3} - \beta_1 + 2) q^{41} + (\beta_{3} + \beta_{2} + \beta_1) q^{42} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{43} + (\beta_{3} - \beta_1 + 1) q^{44} + ( - \beta_{3} - \beta_{2} - 1) q^{45} + q^{46} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{47} - \beta_{2} q^{48} + (2 \beta_{3} - 2 \beta_1) q^{49} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{50} + ( - \beta_{3} + \beta_1 + 1) q^{51} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{52} + 3 \beta_{2} q^{53} - q^{54} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{55} + ( - \beta_{3} + \beta_1 + 1) q^{56} + 3 \beta_{2} q^{57} + (\beta_{3} - 6 \beta_{2} + \beta_1) q^{58} + ( - 3 \beta_{3} + 3 \beta_1 - 4) q^{59} + (\beta_{2} - \beta_1 - 1) q^{60} + (2 \beta_{3} - 2 \beta_1 - 10) q^{61} - 2 \beta_{2} q^{62} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{63} - q^{64} + (2 \beta_{3} - 6 \beta_{2} + 3 \beta_1) q^{65} + ( - \beta_{3} + \beta_1 - 1) q^{66} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{67} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{68} - q^{69} + ( - \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 2) q^{70} + (3 \beta_{3} - 3 \beta_1 + 6) q^{71} + \beta_{2} q^{72} + (4 \beta_{3} - \beta_{2} + 4 \beta_1) q^{73} + q^{74} + (2 \beta_{3} + 2 \beta_1 - 1) q^{75} + 3 q^{76} + 5 \beta_{2} q^{77} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{78} + (3 \beta_{3} - 3 \beta_1 - 6) q^{79} + (\beta_{3} + \beta_{2} + 1) q^{80} + q^{81} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{82} + (\beta_{3} - \beta_{2} + \beta_1) q^{83} + (\beta_{3} - \beta_1 - 1) q^{84} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{85} + (2 \beta_{3} - 2 \beta_1 - 2) q^{86} + ( - \beta_{3} + 6 \beta_{2} - \beta_1) q^{87} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{88} + (3 \beta_{3} - 3 \beta_1 + 7) q^{89} + ( - \beta_{2} + \beta_1 + 1) q^{90} + ( - 2 \beta_{3} + 2 \beta_1 - 3) q^{91} + \beta_{2} q^{92} + 2 \beta_{2} q^{93} + ( - \beta_{3} + \beta_1 + 4) q^{94} + ( - 3 \beta_{3} - 3 \beta_{2} - 3) q^{95} + q^{96} + (3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{97} + ( - 2 \beta_{3} - 2 \beta_1) q^{98} + (\beta_{3} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{9} - 4 q^{10} - 4 q^{11} - 4 q^{14} + 4 q^{15} + 4 q^{16} - 12 q^{19} - 4 q^{20} + 4 q^{21} - 4 q^{24} + 12 q^{26} - 24 q^{29} + 4 q^{30} - 8 q^{31} - 4 q^{34} - 16 q^{35} + 4 q^{36} - 12 q^{39} + 4 q^{40} + 8 q^{41} + 4 q^{44} - 4 q^{45} + 4 q^{46} + 4 q^{50} + 4 q^{51} - 4 q^{54} - 16 q^{55} + 4 q^{56} - 16 q^{59} - 4 q^{60} - 40 q^{61} - 4 q^{64} - 4 q^{66} - 4 q^{69} + 8 q^{70} + 24 q^{71} + 4 q^{74} - 4 q^{75} + 12 q^{76} - 24 q^{79} + 4 q^{80} + 4 q^{81} - 4 q^{84} - 16 q^{85} - 8 q^{86} + 28 q^{89} + 4 q^{90} - 12 q^{91} + 16 q^{94} - 12 q^{95} + 4 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) 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
1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
1.00000i 1.00000i −1.00000 −0.224745 2.22474i 1.00000 3.44949i 1.00000i −1.00000 −2.22474 + 0.224745i
889.2 1.00000i 1.00000i −1.00000 2.22474 + 0.224745i 1.00000 1.44949i 1.00000i −1.00000 0.224745 2.22474i
889.3 1.00000i 1.00000i −1.00000 −0.224745 + 2.22474i 1.00000 3.44949i 1.00000i −1.00000 −2.22474 0.224745i
889.4 1.00000i 1.00000i −1.00000 2.22474 0.224745i 1.00000 1.44949i 1.00000i −1.00000 0.224745 + 2.22474i
\(n\): 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.h 4
3.b odd 2 1 3330.2.d.h 4
5.b even 2 1 inner 1110.2.d.h 4
5.c odd 4 1 5550.2.a.bt 2
5.c odd 4 1 5550.2.a.cc 2
15.d odd 2 1 3330.2.d.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.h 4 1.a even 1 1 trivial
1110.2.d.h 4 5.b even 2 1 inner
3330.2.d.h 4 3.b odd 2 1
3330.2.d.h 4 15.d odd 2 1
5550.2.a.bt 2 5.c odd 4 1
5550.2.a.cc 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{4} + 14T_{7}^{2} + 25 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 5 \) 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
$5$ \( T^{4} - 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 30T^{2} + 9 \) Copy content Toggle raw display
$17$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$19$ \( (T + 3)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 30)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$47$ \( T^{4} + 44T^{2} + 100 \) Copy content Toggle raw display
$53$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 8 T - 38)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 20 T + 76)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T - 18)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 194T^{2} + 9025 \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T - 18)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$89$ \( (T^{2} - 14 T - 5)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 140T^{2} + 1444 \) Copy content Toggle raw display
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