Properties

Label 1110.2.a.r
Level $1110$
Weight $2$
Character orbit 1110.a
Self dual yes
Analytic conductor $8.863$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - q^{10} + ( - 2 \beta - 1) q^{11} - q^{12} + (3 \beta - 2) q^{13} - q^{14} + q^{15} + q^{16} - 5 q^{17} + q^{18} + (\beta - 2) q^{19} - q^{20} + q^{21} + ( - 2 \beta - 1) q^{22} + ( - \beta - 4) q^{23} - q^{24} + q^{25} + (3 \beta - 2) q^{26} - q^{27} - q^{28} + ( - 3 \beta + 1) q^{29} + q^{30} + ( - \beta - 1) q^{31} + q^{32} + (2 \beta + 1) q^{33} - 5 q^{34} + q^{35} + q^{36} + q^{37} + (\beta - 2) q^{38} + ( - 3 \beta + 2) q^{39} - q^{40} + (3 \beta + 1) q^{41} + q^{42} + ( - 3 \beta - 5) q^{43} + ( - 2 \beta - 1) q^{44} - q^{45} + ( - \beta - 4) q^{46} - 8 q^{47} - q^{48} - 6 q^{49} + q^{50} + 5 q^{51} + (3 \beta - 2) q^{52} + (4 \beta - 5) q^{53} - q^{54} + (2 \beta + 1) q^{55} - q^{56} + ( - \beta + 2) q^{57} + ( - 3 \beta + 1) q^{58} + (4 \beta - 6) q^{59} + q^{60} + ( - 5 \beta - 1) q^{61} + ( - \beta - 1) q^{62} - q^{63} + q^{64} + ( - 3 \beta + 2) q^{65} + (2 \beta + 1) q^{66} + (2 \beta + 8) q^{67} - 5 q^{68} + (\beta + 4) q^{69} + q^{70} + (2 \beta - 4) q^{71} + q^{72} + ( - \beta - 4) q^{73} + q^{74} - q^{75} + (\beta - 2) q^{76} + (2 \beta + 1) q^{77} + ( - 3 \beta + 2) q^{78} + (4 \beta + 4) q^{79} - q^{80} + q^{81} + (3 \beta + 1) q^{82} + (\beta - 10) q^{83} + q^{84} + 5 q^{85} + ( - 3 \beta - 5) q^{86} + (3 \beta - 1) q^{87} + ( - 2 \beta - 1) q^{88} + (3 \beta - 2) q^{89} - q^{90} + ( - 3 \beta + 2) q^{91} + ( - \beta - 4) q^{92} + (\beta + 1) q^{93} - 8 q^{94} + ( - \beta + 2) q^{95} - q^{96} + ( - 3 \beta + 11) q^{97} - 6 q^{98} + ( - 2 \beta - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} - q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 10 q^{17} + 2 q^{18} - 3 q^{19} - 2 q^{20} + 2 q^{21} - 4 q^{22} - 9 q^{23} - 2 q^{24} + 2 q^{25} - q^{26} - 2 q^{27} - 2 q^{28} - q^{29} + 2 q^{30} - 3 q^{31} + 2 q^{32} + 4 q^{33} - 10 q^{34} + 2 q^{35} + 2 q^{36} + 2 q^{37} - 3 q^{38} + q^{39} - 2 q^{40} + 5 q^{41} + 2 q^{42} - 13 q^{43} - 4 q^{44} - 2 q^{45} - 9 q^{46} - 16 q^{47} - 2 q^{48} - 12 q^{49} + 2 q^{50} + 10 q^{51} - q^{52} - 6 q^{53} - 2 q^{54} + 4 q^{55} - 2 q^{56} + 3 q^{57} - q^{58} - 8 q^{59} + 2 q^{60} - 7 q^{61} - 3 q^{62} - 2 q^{63} + 2 q^{64} + q^{65} + 4 q^{66} + 18 q^{67} - 10 q^{68} + 9 q^{69} + 2 q^{70} - 6 q^{71} + 2 q^{72} - 9 q^{73} + 2 q^{74} - 2 q^{75} - 3 q^{76} + 4 q^{77} + q^{78} + 12 q^{79} - 2 q^{80} + 2 q^{81} + 5 q^{82} - 19 q^{83} + 2 q^{84} + 10 q^{85} - 13 q^{86} + q^{87} - 4 q^{88} - q^{89} - 2 q^{90} + q^{91} - 9 q^{92} + 3 q^{93} - 16 q^{94} + 3 q^{95} - 2 q^{96} + 19 q^{97} - 12 q^{98} - 4 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.00000 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.00000 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.r 2
3.b odd 2 1 3330.2.a.bc 2
4.b odd 2 1 8880.2.a.bp 2
5.b even 2 1 5550.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.r 2 1.a even 1 1 trivial
3330.2.a.bc 2 3.b odd 2 1
5550.2.a.bw 2 5.b even 2 1
8880.2.a.bp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1110))\):

\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 13 \) Copy content Toggle raw display
\( T_{13}^{2} + T_{13} - 38 \) 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
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 13 \) Copy content Toggle raw display
$13$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$17$ \( (T + 5)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$43$ \( T^{2} + 13T + 4 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T - 59 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T - 94 \) Copy content Toggle raw display
$67$ \( T^{2} - 18T + 64 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$73$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 19T + 86 \) Copy content Toggle raw display
$89$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$97$ \( T^{2} - 19T + 52 \) Copy content Toggle raw display
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