Properties

Label 1110.2.a.p
Level $1110$
Weight $2$
Character orbit 1110.a
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1110,2,Mod(1,1110)]
 
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, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

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: yes
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.37228
−2.37228
−1.00000 −1.00000 1.00000 1.00000 1.00000 −3.37228 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 2.37228 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(5\) \( -1 \)
\(37\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.a.p 2
3.b odd 2 1 3330.2.a.be 2
4.b odd 2 1 8880.2.a.br 2
5.b even 2 1 5550.2.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.p 2 1.a even 1 1 trivial
3330.2.a.be 2 3.b odd 2 1
5550.2.a.ca 2 5.b even 2 1
8880.2.a.br 2 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1110))\):

\( T_{7}^{2} + T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 6 \) Copy content Toggle raw display
\( T_{13}^{2} + 3T_{13} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$13$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$19$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$23$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} - 5T - 2 \) Copy content Toggle raw display
$59$ \( T^{2} - 18T + 48 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$71$ \( T^{2} - 14T + 16 \) Copy content Toggle raw display
$73$ \( T^{2} - 9T - 54 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 7T + 4 \) Copy content Toggle raw display
$89$ \( T^{2} - 21T + 102 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 128 \) Copy content Toggle raw display
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