Properties

Label 726.2.a.m
Level $726$
Weight $2$
Character orbit 726.a
Self dual yes
Analytic conductor $5.797$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [726,2,Mod(1,726)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(726, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("726.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 726.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: \(5.79713918674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + ( - \beta + 3) q^{5} + q^{6} + \beta q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + ( - \beta + 3) q^{5} + q^{6} + \beta q^{7} + q^{8} + q^{9} + ( - \beta + 3) q^{10} + q^{12} + (2 \beta - 2) q^{13} + \beta q^{14} + ( - \beta + 3) q^{15} + q^{16} - 4 q^{17} + q^{18} + ( - 6 \beta + 2) q^{19} + ( - \beta + 3) q^{20} + \beta q^{21} + (6 \beta - 2) q^{23} + q^{24} + ( - 5 \beta + 5) q^{25} + (2 \beta - 2) q^{26} + q^{27} + \beta q^{28} + (3 \beta - 5) q^{29} + ( - \beta + 3) q^{30} + ( - \beta - 4) q^{31} + q^{32} - 4 q^{34} + (2 \beta - 1) q^{35} + q^{36} + (8 \beta - 4) q^{37} + ( - 6 \beta + 2) q^{38} + (2 \beta - 2) q^{39} + ( - \beta + 3) q^{40} + ( - 6 \beta + 2) q^{41} + \beta q^{42} + (2 \beta + 6) q^{43} + ( - \beta + 3) q^{45} + (6 \beta - 2) q^{46} + 4 q^{47} + q^{48} + (\beta - 6) q^{49} + ( - 5 \beta + 5) q^{50} - 4 q^{51} + (2 \beta - 2) q^{52} + (\beta + 2) q^{53} + q^{54} + \beta q^{56} + ( - 6 \beta + 2) q^{57} + (3 \beta - 5) q^{58} + ( - 7 \beta + 6) q^{59} + ( - \beta + 3) q^{60} - 6 \beta q^{61} + ( - \beta - 4) q^{62} + \beta q^{63} + q^{64} + (6 \beta - 8) q^{65} - 8 \beta q^{67} - 4 q^{68} + (6 \beta - 2) q^{69} + (2 \beta - 1) q^{70} + 6 q^{71} + q^{72} + (9 \beta - 9) q^{73} + (8 \beta - 4) q^{74} + ( - 5 \beta + 5) q^{75} + ( - 6 \beta + 2) q^{76} + (2 \beta - 2) q^{78} + ( - 3 \beta + 5) q^{79} + ( - \beta + 3) q^{80} + q^{81} + ( - 6 \beta + 2) q^{82} + (\beta - 1) q^{83} + \beta q^{84} + (4 \beta - 12) q^{85} + (2 \beta + 6) q^{86} + (3 \beta - 5) q^{87} + (2 \beta + 8) q^{89} + ( - \beta + 3) q^{90} + 2 q^{91} + (6 \beta - 2) q^{92} + ( - \beta - 4) q^{93} + 4 q^{94} + ( - 14 \beta + 12) q^{95} + q^{96} + (\beta - 9) q^{97} + (\beta - 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 5 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 5 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9} + 5 q^{10} + 2 q^{12} - 2 q^{13} + q^{14} + 5 q^{15} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 2 q^{19} + 5 q^{20} + q^{21} + 2 q^{23} + 2 q^{24} + 5 q^{25} - 2 q^{26} + 2 q^{27} + q^{28} - 7 q^{29} + 5 q^{30} - 9 q^{31} + 2 q^{32} - 8 q^{34} + 2 q^{36} - 2 q^{38} - 2 q^{39} + 5 q^{40} - 2 q^{41} + q^{42} + 14 q^{43} + 5 q^{45} + 2 q^{46} + 8 q^{47} + 2 q^{48} - 11 q^{49} + 5 q^{50} - 8 q^{51} - 2 q^{52} + 5 q^{53} + 2 q^{54} + q^{56} - 2 q^{57} - 7 q^{58} + 5 q^{59} + 5 q^{60} - 6 q^{61} - 9 q^{62} + q^{63} + 2 q^{64} - 10 q^{65} - 8 q^{67} - 8 q^{68} + 2 q^{69} + 12 q^{71} + 2 q^{72} - 9 q^{73} + 5 q^{75} - 2 q^{76} - 2 q^{78} + 7 q^{79} + 5 q^{80} + 2 q^{81} - 2 q^{82} - q^{83} + q^{84} - 20 q^{85} + 14 q^{86} - 7 q^{87} + 18 q^{89} + 5 q^{90} + 4 q^{91} + 2 q^{92} - 9 q^{93} + 8 q^{94} + 10 q^{95} + 2 q^{96} - 17 q^{97} - 11 q^{98}+O(q^{100}) \) 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
1.61803
−0.618034
1.00000 1.00000 1.00000 1.38197 1.00000 1.61803 1.00000 1.00000 1.38197
1.2 1.00000 1.00000 1.00000 3.61803 1.00000 −0.618034 1.00000 1.00000 3.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(-1\)

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 726.2.a.m 2
3.b odd 2 1 2178.2.a.o 2
4.b odd 2 1 5808.2.a.by 2
11.b odd 2 1 726.2.a.k 2
11.c even 5 2 726.2.e.a 4
11.c even 5 2 726.2.e.c 4
11.d odd 10 2 66.2.e.b 4
11.d odd 10 2 726.2.e.j 4
33.d even 2 1 2178.2.a.v 2
33.f even 10 2 198.2.f.a 4
44.c even 2 1 5808.2.a.bz 2
44.g even 10 2 528.2.y.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.e.b 4 11.d odd 10 2
198.2.f.a 4 33.f even 10 2
528.2.y.g 4 44.g even 10 2
726.2.a.k 2 11.b odd 2 1
726.2.a.m 2 1.a even 1 1 trivial
726.2.e.a 4 11.c even 5 2
726.2.e.c 4 11.c even 5 2
726.2.e.j 4 11.d odd 10 2
2178.2.a.o 2 3.b odd 2 1
2178.2.a.v 2 33.d even 2 1
5808.2.a.by 2 4.b odd 2 1
5808.2.a.bz 2 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(726))\):

\( T_{5}^{2} - 5T_{5} + 5 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 1 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 4 \) 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
$5$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$29$ \( T^{2} + 7T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$37$ \( T^{2} - 80 \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$43$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
$47$ \( (T - 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$59$ \( T^{2} - 5T - 55 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 9T - 81 \) Copy content Toggle raw display
$79$ \( T^{2} - 7T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$97$ \( T^{2} + 17T + 71 \) Copy content Toggle raw display
show more
show less