Properties

Label 5610.2.a.bs
Level $5610$
Weight $2$
Character orbit 5610.a
Self dual yes
Analytic conductor $44.796$
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] = [5610,2,Mod(1,5610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.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: \(44.7960755339\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + (\beta + 1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + (\beta + 1) q^{7} - q^{8} + q^{9} + q^{10} - q^{11} + q^{12} + ( - \beta - 3) q^{13} + ( - \beta - 1) q^{14} - q^{15} + q^{16} + q^{17} - q^{18} + ( - \beta - 3) q^{19} - q^{20} + (\beta + 1) q^{21} + q^{22} + (\beta - 1) q^{23} - q^{24} + q^{25} + (\beta + 3) q^{26} + q^{27} + (\beta + 1) q^{28} + 6 q^{29} + q^{30} + ( - 2 \beta - 2) q^{31} - q^{32} - q^{33} - q^{34} + ( - \beta - 1) q^{35} + q^{36} + 2 \beta q^{37} + (\beta + 3) q^{38} + ( - \beta - 3) q^{39} + q^{40} + ( - 2 \beta + 2) q^{41} + ( - \beta - 1) q^{42} + ( - \beta - 9) q^{43} - q^{44} - q^{45} + ( - \beta + 1) q^{46} + 12 q^{47} + q^{48} + (2 \beta + 7) q^{49} - q^{50} + q^{51} + ( - \beta - 3) q^{52} + ( - \beta - 5) q^{53} - q^{54} + q^{55} + ( - \beta - 1) q^{56} + ( - \beta - 3) q^{57} - 6 q^{58} + (\beta - 7) q^{59} - q^{60} + 2 q^{61} + (2 \beta + 2) q^{62} + (\beta + 1) q^{63} + q^{64} + (\beta + 3) q^{65} + q^{66} - 10 q^{67} + q^{68} + (\beta - 1) q^{69} + (\beta + 1) q^{70} + ( - \beta - 5) q^{71} - q^{72} + (\beta - 5) q^{73} - 2 \beta q^{74} + q^{75} + ( - \beta - 3) q^{76} + ( - \beta - 1) q^{77} + (\beta + 3) q^{78} + ( - \beta - 3) q^{79} - q^{80} + q^{81} + (2 \beta - 2) q^{82} + (\beta + 1) q^{84} - q^{85} + (\beta + 9) q^{86} + 6 q^{87} + q^{88} + 6 q^{89} + q^{90} + ( - 4 \beta - 16) q^{91} + (\beta - 1) q^{92} + ( - 2 \beta - 2) q^{93} - 12 q^{94} + (\beta + 3) q^{95} - q^{96} + (2 \beta - 6) q^{97} + ( - 2 \beta - 7) q^{98} - q^{99} +O(q^{100}) \) 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 \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 6 q^{19} - 2 q^{20} + 2 q^{21} + 2 q^{22} - 2 q^{23} - 2 q^{24} + 2 q^{25} + 6 q^{26} + 2 q^{27} + 2 q^{28} + 12 q^{29} + 2 q^{30} - 4 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} - 2 q^{35} + 2 q^{36} + 6 q^{38} - 6 q^{39} + 2 q^{40} + 4 q^{41} - 2 q^{42} - 18 q^{43} - 2 q^{44} - 2 q^{45} + 2 q^{46} + 24 q^{47} + 2 q^{48} + 14 q^{49} - 2 q^{50} + 2 q^{51} - 6 q^{52} - 10 q^{53} - 2 q^{54} + 2 q^{55} - 2 q^{56} - 6 q^{57} - 12 q^{58} - 14 q^{59} - 2 q^{60} + 4 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} + 6 q^{65} + 2 q^{66} - 20 q^{67} + 2 q^{68} - 2 q^{69} + 2 q^{70} - 10 q^{71} - 2 q^{72} - 10 q^{73} + 2 q^{75} - 6 q^{76} - 2 q^{77} + 6 q^{78} - 6 q^{79} - 2 q^{80} + 2 q^{81} - 4 q^{82} + 2 q^{84} - 2 q^{85} + 18 q^{86} + 12 q^{87} + 2 q^{88} + 12 q^{89} + 2 q^{90} - 32 q^{91} - 2 q^{92} - 4 q^{93} - 24 q^{94} + 6 q^{95} - 2 q^{96} - 12 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100}) \) 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
−1.30278
2.30278
−1.00000 1.00000 1.00000 −1.00000 −1.00000 −2.60555 −1.00000 1.00000 1.00000
1.2 −1.00000 1.00000 1.00000 −1.00000 −1.00000 4.60555 −1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(5\) \( +1 \)
\(11\) \( +1 \)
\(17\) \( -1 \)

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 5610.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5610.2.a.bs 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7}^{2} - 2T_{7} - 12 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} - 4 \) Copy content Toggle raw display
\( T_{19}^{2} + 6T_{19} - 4 \) Copy content Toggle raw display
\( T_{23}^{2} + 2T_{23} - 12 \) 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
$7$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$37$ \( T^{2} - 52 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$43$ \( T^{2} + 18T + 68 \) Copy content Toggle raw display
$47$ \( (T - 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 12 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 36 \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( (T + 10)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 10T + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 12 \) Copy content Toggle raw display
$79$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 12T - 16 \) Copy content Toggle raw display
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