Properties

Label 5610.2.a.ce
Level $5610$
Weight $2$
Character orbit 5610.a
Self dual yes
Analytic conductor $44.796$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.2777.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. 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} + \beta_1 + 1) q^{13} + \beta_1 q^{14} - q^{15} + q^{16} + q^{17} - q^{18} + ( - \beta_1 - 2) q^{19} + q^{20} + \beta_1 q^{21} + q^{22} + (\beta_{3} + \beta_1 - 1) q^{23} + q^{24} + q^{25} + (\beta_{3} - \beta_1 - 1) q^{26} - q^{27} - \beta_1 q^{28} + ( - 2 \beta_{2} - 2) q^{29} + q^{30} + (3 \beta_{2} - \beta_1) q^{31} - q^{32} + q^{33} - q^{34} - \beta_1 q^{35} + q^{36} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{37} + (\beta_1 + 2) q^{38} + (\beta_{3} - \beta_1 - 1) q^{39} - q^{40} + 2 \beta_1 q^{41} - \beta_1 q^{42} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{43} - q^{44} + q^{45} + ( - \beta_{3} - \beta_1 + 1) q^{46} + (2 \beta_{2} - 2 \beta_1) q^{47} - q^{48} + ( - \beta_{2} + \beta_1 - 3) q^{49} - q^{50} - q^{51} + ( - \beta_{3} + \beta_1 + 1) q^{52} + (\beta_{3} + \beta_{2} + 1) q^{53} + q^{54} - q^{55} + \beta_1 q^{56} + (\beta_1 + 2) q^{57} + (2 \beta_{2} + 2) q^{58} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 7) q^{59} - q^{60} + ( - \beta_{2} + \beta_1 - 2) q^{61} + ( - 3 \beta_{2} + \beta_1) q^{62} - \beta_1 q^{63} + q^{64} + ( - \beta_{3} + \beta_1 + 1) q^{65} - q^{66} + ( - \beta_{2} + 3 \beta_1 + 2) q^{67} + q^{68} + ( - \beta_{3} - \beta_1 + 1) q^{69} + \beta_1 q^{70} + ( - \beta_{3} + \beta_{2} - 5) q^{71} - q^{72} + ( - \beta_{3} + \beta_{2} - 4 \beta_1 + 5) q^{73} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{74} - q^{75} + ( - \beta_1 - 2) q^{76} + \beta_1 q^{77} + ( - \beta_{3} + \beta_1 + 1) q^{78} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{79} + q^{80} + q^{81} - 2 \beta_1 q^{82} + (\beta_{3} + \beta_{2} + \beta_1 - 7) q^{83} + \beta_1 q^{84} + q^{85} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{86} + (2 \beta_{2} + 2) q^{87} + q^{88} + ( - 2 \beta_{3} - 4) q^{89} - q^{90} + ( - \beta_{2} - \beta_1 - 4) q^{91} + (\beta_{3} + \beta_1 - 1) q^{92} + ( - 3 \beta_{2} + \beta_1) q^{93} + ( - 2 \beta_{2} + 2 \beta_1) q^{94} + ( - \beta_1 - 2) q^{95} + q^{96} + (\beta_{2} + 3 \beta_1 + 4) q^{97} + (\beta_{2} - \beta_1 + 3) q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - q^{7} - 4 q^{8} + 4 q^{9} - 4 q^{10} - 4 q^{11} - 4 q^{12} + 3 q^{13} + q^{14} - 4 q^{15} + 4 q^{16} + 4 q^{17} - 4 q^{18} - 9 q^{19} + 4 q^{20} + q^{21} + 4 q^{22} - q^{23} + 4 q^{24} + 4 q^{25} - 3 q^{26} - 4 q^{27} - q^{28} - 8 q^{29} + 4 q^{30} - q^{31} - 4 q^{32} + 4 q^{33} - 4 q^{34} - q^{35} + 4 q^{36} + q^{37} + 9 q^{38} - 3 q^{39} - 4 q^{40} + 2 q^{41} - q^{42} + 4 q^{43} - 4 q^{44} + 4 q^{45} + q^{46} - 2 q^{47} - 4 q^{48} - 11 q^{49} - 4 q^{50} - 4 q^{51} + 3 q^{52} + 6 q^{53} + 4 q^{54} - 4 q^{55} + q^{56} + 9 q^{57} + 8 q^{58} - 24 q^{59} - 4 q^{60} - 7 q^{61} + q^{62} - q^{63} + 4 q^{64} + 3 q^{65} - 4 q^{66} + 11 q^{67} + 4 q^{68} + q^{69} + q^{70} - 22 q^{71} - 4 q^{72} + 14 q^{73} - q^{74} - 4 q^{75} - 9 q^{76} + q^{77} + 3 q^{78} - 12 q^{79} + 4 q^{80} + 4 q^{81} - 2 q^{82} - 25 q^{83} + q^{84} + 4 q^{85} - 4 q^{86} + 8 q^{87} + 4 q^{88} - 20 q^{89} - 4 q^{90} - 17 q^{91} - q^{92} + q^{93} + 2 q^{94} - 9 q^{95} + 4 q^{96} + 19 q^{97} + 11 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} + x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} - 2\nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 2\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} - 2\beta _1 + 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{3} + 2\beta_{2} - 6\beta _1 + 9 ) / 4 \) 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
−0.679643
−1.50848
2.36234
0.825785
−1.00000 −1.00000 1.00000 1.00000 1.00000 −2.94272 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 −1.32584 −1.00000 1.00000 −1.00000
1.3 −1.00000 −1.00000 1.00000 1.00000 1.00000 0.846618 −1.00000 1.00000 −1.00000
1.4 −1.00000 −1.00000 1.00000 1.00000 1.00000 2.42194 −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.ce 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5610.2.a.ce 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7}^{4} + T_{7}^{3} - 8T_{7}^{2} - 4T_{7} + 8 \) Copy content Toggle raw display
\( T_{13}^{4} - 3T_{13}^{3} - 42T_{13}^{2} + 128T_{13} - 88 \) Copy content Toggle raw display
\( T_{19}^{4} + 9T_{19}^{3} + 22T_{19}^{2} + 8T_{19} - 8 \) Copy content Toggle raw display
\( T_{23}^{4} + T_{23}^{3} - 46T_{23}^{2} - 16T_{23} + 488 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} - 8 T^{2} - 4 T + 8 \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 3 T^{3} - 42 T^{2} + 128 T - 88 \) Copy content Toggle raw display
$17$ \( (T - 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 9 T^{3} + 22 T^{2} + 8 T - 8 \) Copy content Toggle raw display
$23$ \( T^{4} + T^{3} - 46 T^{2} - 16 T + 488 \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} - 56 T^{2} - 352 T + 592 \) Copy content Toggle raw display
$31$ \( T^{4} + T^{3} - 164 T^{2} - 64 T + 6464 \) Copy content Toggle raw display
$37$ \( T^{4} - T^{3} - 102 T^{2} - 300 T + 296 \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} - 32 T^{2} + 32 T + 128 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} - 92 T^{2} + 424 T + 512 \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} - 80 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} - 52 T^{2} + 96 T - 16 \) Copy content Toggle raw display
$59$ \( T^{4} + 24 T^{3} + 148 T^{2} + \cdots - 1504 \) Copy content Toggle raw display
$61$ \( T^{4} + 7 T^{3} - 2 T^{2} - 60 T - 8 \) Copy content Toggle raw display
$67$ \( T^{4} - 11 T^{3} - 26 T^{2} + 76 T + 88 \) Copy content Toggle raw display
$71$ \( T^{4} + 22 T^{3} + 132 T^{2} + \cdots - 1648 \) Copy content Toggle raw display
$73$ \( T^{4} - 14 T^{3} - 80 T^{2} + \cdots - 4288 \) Copy content Toggle raw display
$79$ \( T^{4} + 12 T^{3} - 44 T^{2} + \cdots - 656 \) Copy content Toggle raw display
$83$ \( T^{4} + 25 T^{3} + 152 T^{2} + \cdots - 512 \) Copy content Toggle raw display
$89$ \( T^{4} + 20 T^{3} - 720 T + 752 \) Copy content Toggle raw display
$97$ \( T^{4} - 19 T^{3} + 16 T^{2} + \cdots - 3776 \) Copy content Toggle raw display
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