Properties

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

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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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + ( - 3 \beta + 1) q^{5} + q^{6} + (\beta - 4) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + ( - 3 \beta + 1) q^{5} + q^{6} + (\beta - 4) q^{7} - q^{8} + q^{9} + (3 \beta - 1) q^{10} - q^{12} + ( - 2 \beta + 2) q^{13} + ( - \beta + 4) q^{14} + (3 \beta - 1) q^{15} + q^{16} - 4 \beta q^{17} - q^{18} + ( - 2 \beta + 2) q^{19} + ( - 3 \beta + 1) q^{20} + ( - \beta + 4) q^{21} + (2 \beta - 2) q^{23} + q^{24} + (3 \beta + 5) q^{25} + (2 \beta - 2) q^{26} - q^{27} + (\beta - 4) q^{28} + (\beta + 1) q^{29} + ( - 3 \beta + 1) q^{30} + ( - \beta + 8) q^{31} - q^{32} + 4 \beta q^{34} + (10 \beta - 7) q^{35} + q^{36} + (4 \beta + 4) q^{37} + (2 \beta - 2) q^{38} + (2 \beta - 2) q^{39} + (3 \beta - 1) q^{40} + ( - 2 \beta + 2) q^{41} + (\beta - 4) q^{42} + ( - 2 \beta + 2) q^{43} + ( - 3 \beta + 1) q^{45} + ( - 2 \beta + 2) q^{46} - 4 \beta q^{47} - q^{48} + ( - 7 \beta + 10) q^{49} + ( - 3 \beta - 5) q^{50} + 4 \beta q^{51} + ( - 2 \beta + 2) q^{52} + (3 \beta - 2) q^{53} + q^{54} + ( - \beta + 4) q^{56} + (2 \beta - 2) q^{57} + ( - \beta - 1) q^{58} + (3 \beta - 2) q^{59} + (3 \beta - 1) q^{60} + (2 \beta + 8) q^{61} + (\beta - 8) q^{62} + (\beta - 4) q^{63} + q^{64} + ( - 2 \beta + 8) q^{65} + 4 q^{67} - 4 \beta q^{68} + ( - 2 \beta + 2) q^{69} + ( - 10 \beta + 7) q^{70} + ( - 4 \beta + 10) q^{71} - q^{72} + ( - 7 \beta + 7) q^{73} + ( - 4 \beta - 4) q^{74} + ( - 3 \beta - 5) q^{75} + ( - 2 \beta + 2) q^{76} + ( - 2 \beta + 2) q^{78} + ( - 7 \beta + 5) q^{79} + ( - 3 \beta + 1) q^{80} + q^{81} + (2 \beta - 2) q^{82} + (7 \beta + 5) q^{83} + ( - \beta + 4) q^{84} + (8 \beta + 12) q^{85} + (2 \beta - 2) q^{86} + ( - \beta - 1) q^{87} + (6 \beta - 8) q^{89} + (3 \beta - 1) q^{90} + (8 \beta - 10) q^{91} + (2 \beta - 2) q^{92} + (\beta - 8) q^{93} + 4 \beta q^{94} + ( - 2 \beta + 8) q^{95} + q^{96} + ( - 7 \beta + 7) q^{97} + (7 \beta - 10) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - q^{5} + 2 q^{6} - 7 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - q^{5} + 2 q^{6} - 7 q^{7} - 2 q^{8} + 2 q^{9} + q^{10} - 2 q^{12} + 2 q^{13} + 7 q^{14} + q^{15} + 2 q^{16} - 4 q^{17} - 2 q^{18} + 2 q^{19} - q^{20} + 7 q^{21} - 2 q^{23} + 2 q^{24} + 13 q^{25} - 2 q^{26} - 2 q^{27} - 7 q^{28} + 3 q^{29} - q^{30} + 15 q^{31} - 2 q^{32} + 4 q^{34} - 4 q^{35} + 2 q^{36} + 12 q^{37} - 2 q^{38} - 2 q^{39} + q^{40} + 2 q^{41} - 7 q^{42} + 2 q^{43} - q^{45} + 2 q^{46} - 4 q^{47} - 2 q^{48} + 13 q^{49} - 13 q^{50} + 4 q^{51} + 2 q^{52} - q^{53} + 2 q^{54} + 7 q^{56} - 2 q^{57} - 3 q^{58} - q^{59} + q^{60} + 18 q^{61} - 15 q^{62} - 7 q^{63} + 2 q^{64} + 14 q^{65} + 8 q^{67} - 4 q^{68} + 2 q^{69} + 4 q^{70} + 16 q^{71} - 2 q^{72} + 7 q^{73} - 12 q^{74} - 13 q^{75} + 2 q^{76} + 2 q^{78} + 3 q^{79} - q^{80} + 2 q^{81} - 2 q^{82} + 17 q^{83} + 7 q^{84} + 32 q^{85} - 2 q^{86} - 3 q^{87} - 10 q^{89} + q^{90} - 12 q^{91} - 2 q^{92} - 15 q^{93} + 4 q^{94} + 14 q^{95} + 2 q^{96} + 7 q^{97} - 13 q^{98}+O(q^{100}) \) 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 −3.85410 1.00000 −2.38197 −1.00000 1.00000 3.85410
1.2 −1.00000 −1.00000 1.00000 2.85410 1.00000 −4.61803 −1.00000 1.00000 −2.85410
\(n\): 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.j 2
3.b odd 2 1 2178.2.a.bb 2
4.b odd 2 1 5808.2.a.cg 2
11.b odd 2 1 726.2.a.l 2
11.c even 5 2 726.2.e.n 4
11.c even 5 2 726.2.e.r 4
11.d odd 10 2 66.2.e.a 4
11.d odd 10 2 726.2.e.f 4
33.d even 2 1 2178.2.a.t 2
33.f even 10 2 198.2.f.c 4
44.c even 2 1 5808.2.a.cb 2
44.g even 10 2 528.2.y.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.e.a 4 11.d odd 10 2
198.2.f.c 4 33.f even 10 2
528.2.y.d 4 44.g even 10 2
726.2.a.j 2 1.a even 1 1 trivial
726.2.a.l 2 11.b odd 2 1
726.2.e.f 4 11.d odd 10 2
726.2.e.n 4 11.c even 5 2
726.2.e.r 4 11.c even 5 2
2178.2.a.t 2 33.d even 2 1
2178.2.a.bb 2 3.b odd 2 1
5808.2.a.cb 2 44.c even 2 1
5808.2.a.cg 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(726))\):

\( T_{5}^{2} + T_{5} - 11 \) Copy content Toggle raw display
\( T_{7}^{2} + 7T_{7} + 11 \) 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
$5$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$7$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} - 15T + 55 \) Copy content Toggle raw display
$37$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$59$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$61$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$73$ \( T^{2} - 7T - 49 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T - 59 \) Copy content Toggle raw display
$83$ \( T^{2} - 17T + 11 \) Copy content Toggle raw display
$89$ \( T^{2} + 10T - 20 \) Copy content Toggle raw display
$97$ \( T^{2} - 7T - 49 \) Copy content Toggle raw display
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