Properties

Label 110.2.b.a
Level $110$
Weight $2$
Character orbit 110.b
Analytic conductor $0.878$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [110,2,Mod(89,110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(110, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("110.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 110.b (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: \(0.878354422234\)
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: \(\Z[a_1, a_2]\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(101\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 −2.00000 1.00000i −1.00000 3.00000i 1.00000i 2.00000 −1.00000 + 2.00000i
89.2 1.00000i 1.00000i −1.00000 −2.00000 + 1.00000i −1.00000 3.00000i 1.00000i 2.00000 −1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 110.2.b.a 2
3.b odd 2 1 990.2.c.d 2
4.b odd 2 1 880.2.b.a 2
5.b even 2 1 inner 110.2.b.a 2
5.c odd 4 1 550.2.a.e 1
5.c odd 4 1 550.2.a.j 1
11.b odd 2 1 1210.2.b.a 2
15.d odd 2 1 990.2.c.d 2
15.e even 4 1 4950.2.a.q 1
15.e even 4 1 4950.2.a.ba 1
20.d odd 2 1 880.2.b.a 2
20.e even 4 1 4400.2.a.k 1
20.e even 4 1 4400.2.a.s 1
55.d odd 2 1 1210.2.b.a 2
55.e even 4 1 6050.2.a.f 1
55.e even 4 1 6050.2.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.b.a 2 1.a even 1 1 trivial
110.2.b.a 2 5.b even 2 1 inner
550.2.a.e 1 5.c odd 4 1
550.2.a.j 1 5.c odd 4 1
880.2.b.a 2 4.b odd 2 1
880.2.b.a 2 20.d odd 2 1
990.2.c.d 2 3.b odd 2 1
990.2.c.d 2 15.d odd 2 1
1210.2.b.a 2 11.b odd 2 1
1210.2.b.a 2 55.d odd 2 1
4400.2.a.k 1 20.e even 4 1
4400.2.a.s 1 20.e even 4 1
4950.2.a.q 1 15.e even 4 1
4950.2.a.ba 1 15.e even 4 1
6050.2.a.f 1 55.e even 4 1
6050.2.a.bk 1 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(110, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) 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
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T - 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( (T - 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 49 \) Copy content Toggle raw display
$41$ \( (T + 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 81 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 13)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T + 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T - 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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