Properties

Label 510.2.a.h
Level $510$
Weight $2$
Character orbit 510.a
Self dual yes
Analytic conductor $4.072$
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] = [510,2,Mod(1,510)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(510, 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("510.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.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: \(4.07237050309\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) 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 \(\beta = 2\sqrt{6}\). 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 \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + \beta q^{7} - q^{8} + q^{9} - q^{10} - q^{12} + ( - \beta + 2) q^{13} - \beta q^{14} - q^{15} + q^{16} - q^{17} - q^{18} + 4 q^{19} + q^{20} - \beta q^{21} - 4 q^{23} + q^{24} + q^{25} + (\beta - 2) q^{26} - q^{27} + \beta q^{28} + 6 q^{29} + q^{30} + 4 q^{31} - q^{32} + q^{34} + \beta q^{35} + q^{36} + 6 q^{37} - 4 q^{38} + (\beta - 2) q^{39} - q^{40} + (\beta + 2) q^{41} + \beta q^{42} + ( - \beta + 4) q^{43} + q^{45} + 4 q^{46} - 2 \beta q^{47} - q^{48} + 17 q^{49} - q^{50} + q^{51} + ( - \beta + 2) q^{52} + (2 \beta + 2) q^{53} + q^{54} - \beta q^{56} - 4 q^{57} - 6 q^{58} - \beta q^{59} - q^{60} + ( - 2 \beta + 2) q^{61} - 4 q^{62} + \beta q^{63} + q^{64} + ( - \beta + 2) q^{65} + ( - \beta - 4) q^{67} - q^{68} + 4 q^{69} - \beta q^{70} + (\beta - 4) q^{71} - q^{72} + (\beta - 6) q^{73} - 6 q^{74} - q^{75} + 4 q^{76} + ( - \beta + 2) q^{78} + (2 \beta + 4) q^{79} + q^{80} + q^{81} + ( - \beta - 2) q^{82} + ( - 2 \beta + 4) q^{83} - \beta q^{84} - q^{85} + (\beta - 4) q^{86} - 6 q^{87} + (2 \beta + 2) q^{89} - q^{90} + (2 \beta - 24) q^{91} - 4 q^{92} - 4 q^{93} + 2 \beta q^{94} + 4 q^{95} + q^{96} + (3 \beta + 2) q^{97} - 17 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{12} + 4 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} + 8 q^{19} + 2 q^{20} - 8 q^{23} + 2 q^{24} + 2 q^{25} - 4 q^{26} - 2 q^{27} + 12 q^{29} + 2 q^{30} + 8 q^{31} - 2 q^{32} + 2 q^{34} + 2 q^{36} + 12 q^{37} - 8 q^{38} - 4 q^{39} - 2 q^{40} + 4 q^{41} + 8 q^{43} + 2 q^{45} + 8 q^{46} - 2 q^{48} + 34 q^{49} - 2 q^{50} + 2 q^{51} + 4 q^{52} + 4 q^{53} + 2 q^{54} - 8 q^{57} - 12 q^{58} - 2 q^{60} + 4 q^{61} - 8 q^{62} + 2 q^{64} + 4 q^{65} - 8 q^{67} - 2 q^{68} + 8 q^{69} - 8 q^{71} - 2 q^{72} - 12 q^{73} - 12 q^{74} - 2 q^{75} + 8 q^{76} + 4 q^{78} + 8 q^{79} + 2 q^{80} + 2 q^{81} - 4 q^{82} + 8 q^{83} - 2 q^{85} - 8 q^{86} - 12 q^{87} + 4 q^{89} - 2 q^{90} - 48 q^{91} - 8 q^{92} - 8 q^{93} + 8 q^{95} + 2 q^{96} + 4 q^{97} - 34 q^{98}+O(q^{100}) \) 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
−2.44949
2.44949
−1.00000 −1.00000 1.00000 1.00000 1.00000 −4.89898 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 4.89898 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 510.2.a.h 2
3.b odd 2 1 1530.2.a.s 2
4.b odd 2 1 4080.2.a.bq 2
5.b even 2 1 2550.2.a.bl 2
5.c odd 4 2 2550.2.d.u 4
15.d odd 2 1 7650.2.a.cu 2
17.b even 2 1 8670.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.h 2 1.a even 1 1 trivial
1530.2.a.s 2 3.b odd 2 1
2550.2.a.bl 2 5.b even 2 1
2550.2.d.u 4 5.c odd 4 2
4080.2.a.bq 2 4.b odd 2 1
7650.2.a.cu 2 15.d odd 2 1
8670.2.a.be 2 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(510))\):

\( T_{7}^{2} - 24 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 20 \) 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
$7$ \( T^{2} - 24 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 20 \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T - 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 20 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 8 \) Copy content Toggle raw display
$47$ \( T^{2} - 96 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 92 \) Copy content Toggle raw display
$59$ \( T^{2} - 24 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 92 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 8 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 8 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 12 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 80 \) Copy content Toggle raw display
$83$ \( T^{2} - 8T - 80 \) Copy content Toggle raw display
$89$ \( T^{2} - 4T - 92 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 212 \) Copy content Toggle raw display
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