Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1110,2,Mod(739,1110)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.739");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.e (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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - i q^{3} + q^{4} + ( - i + 2) q^{5} + i q^{6} + 3 i q^{7} - q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - i q^{3} + q^{4} + ( - i + 2) q^{5} + i q^{6} + 3 i q^{7} - q^{8} - q^{9} + (i - 2) q^{10} - 3 q^{11} - i q^{12} - 4 q^{13} - 3 i q^{14} + ( - 2 i - 1) q^{15} + q^{16} + 7 q^{17} + q^{18} - 4 i q^{19} + ( - i + 2) q^{20} + 3 q^{21} + 3 q^{22} + 6 q^{23} + i q^{24} + ( - 4 i + 3) q^{25} + 4 q^{26} + i q^{27} + 3 i q^{28} - 9 i q^{29} + (2 i + 1) q^{30} - 5 i q^{31} - q^{32} + 3 i q^{33} - 7 q^{34} + (6 i + 3) q^{35} - q^{36} + ( - i + 6) q^{37} + 4 i q^{38} + 4 i q^{39} + (i - 2) q^{40} + 7 q^{41} - 3 q^{42} + q^{43} - 3 q^{44} + (i - 2) q^{45} - 6 q^{46} + 8 i q^{47} - i q^{48} - 2 q^{49} + (4 i - 3) q^{50} - 7 i q^{51} - 4 q^{52} + i q^{53} - i q^{54} + (3 i - 6) q^{55} - 3 i q^{56} - 4 q^{57} + 9 i q^{58} + 6 i q^{59} + ( - 2 i - 1) q^{60} - 5 i q^{61} + 5 i q^{62} - 3 i q^{63} + q^{64} + (4 i - 8) q^{65} - 3 i q^{66} - 2 i q^{67} + 7 q^{68} - 6 i q^{69} + ( - 6 i - 3) q^{70} + 12 q^{71} + q^{72} - 4 i q^{73} + (i - 6) q^{74} + ( - 3 i - 4) q^{75} - 4 i q^{76} - 9 i q^{77} - 4 i q^{78} - 4 i q^{79} + ( - i + 2) q^{80} + q^{81} - 7 q^{82} - 14 i q^{83} + 3 q^{84} + ( - 7 i + 14) q^{85} - q^{86} - 9 q^{87} + 3 q^{88} + 6 i q^{89} + ( - i + 2) q^{90} - 12 i q^{91} + 6 q^{92} - 5 q^{93} - 8 i q^{94} + ( - 8 i - 4) q^{95} + i q^{96} + 7 q^{97} + 2 q^{98} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8} - 2 q^{9} - 4 q^{10} - 6 q^{11} - 8 q^{13} - 2 q^{15} + 2 q^{16} + 14 q^{17} + 2 q^{18} + 4 q^{20} + 6 q^{21} + 6 q^{22} + 12 q^{23} + 6 q^{25} + 8 q^{26} + 2 q^{30} - 2 q^{32} - 14 q^{34} + 6 q^{35} - 2 q^{36} + 12 q^{37} - 4 q^{40} + 14 q^{41} - 6 q^{42} + 2 q^{43} - 6 q^{44} - 4 q^{45} - 12 q^{46} - 4 q^{49} - 6 q^{50} - 8 q^{52} - 12 q^{55} - 8 q^{57} - 2 q^{60} + 2 q^{64} - 16 q^{65} + 14 q^{68} - 6 q^{70} + 24 q^{71} + 2 q^{72} - 12 q^{74} - 8 q^{75} + 4 q^{80} + 2 q^{81} - 14 q^{82} + 6 q^{84} + 28 q^{85} - 2 q^{86} - 18 q^{87} + 6 q^{88} + 4 q^{90} + 12 q^{92} - 10 q^{93} - 8 q^{95} + 14 q^{97} + 4 q^{98} + 6 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
739.1
1.00000i
1.00000i
−1.00000 1.00000i 1.00000 2.00000 1.00000i 1.00000i 3.00000i −1.00000 −1.00000 −2.00000 + 1.00000i
739.2 −1.00000 1.00000i 1.00000 2.00000 + 1.00000i 1.00000i 3.00000i −1.00000 −1.00000 −2.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.d 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.e.a 2
3.b odd 2 1 3330.2.e.b 2
5.b even 2 1 1110.2.e.b yes 2
15.d odd 2 1 3330.2.e.a 2
37.b even 2 1 1110.2.e.b yes 2
111.d odd 2 1 3330.2.e.a 2
185.d even 2 1 inner 1110.2.e.a 2
555.b odd 2 1 3330.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.e.a 2 1.a even 1 1 trivial
1110.2.e.a 2 185.d even 2 1 inner
1110.2.e.b yes 2 5.b even 2 1
1110.2.e.b yes 2 37.b even 2 1
3330.2.e.a 2 15.d odd 2 1
3330.2.e.a 2 111.d odd 2 1
3330.2.e.b 2 3.b odd 2 1
3330.2.e.b 2 555.b odd 2 1

Hecke kernels

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

\( T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T - 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 81 \) Copy content Toggle raw display
$31$ \( T^{2} + 25 \) Copy content Toggle raw display
$37$ \( T^{2} - 12T + 37 \) Copy content Toggle raw display
$41$ \( (T - 7)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 1 \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 196 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
show more
show less