Properties

Label 722.2.a.b
Level $722$
Weight $2$
Character orbit 722.a
Self dual yes
Analytic conductor $5.765$
Analytic rank $0$
Dimension $1$
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] = [722,2,Mod(1,722)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("722.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.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: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - 4 q^{5} - q^{6} + 3 q^{7} - q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - 4 q^{5} - q^{6} + 3 q^{7} - q^{8} - 2 q^{9} + 4 q^{10} + 2 q^{11} + q^{12} + q^{13} - 3 q^{14} - 4 q^{15} + q^{16} + 3 q^{17} + 2 q^{18} - 4 q^{20} + 3 q^{21} - 2 q^{22} - q^{23} - q^{24} + 11 q^{25} - q^{26} - 5 q^{27} + 3 q^{28} + 5 q^{29} + 4 q^{30} + 8 q^{31} - q^{32} + 2 q^{33} - 3 q^{34} - 12 q^{35} - 2 q^{36} + 2 q^{37} + q^{39} + 4 q^{40} + 8 q^{41} - 3 q^{42} + 4 q^{43} + 2 q^{44} + 8 q^{45} + q^{46} + 8 q^{47} + q^{48} + 2 q^{49} - 11 q^{50} + 3 q^{51} + q^{52} + q^{53} + 5 q^{54} - 8 q^{55} - 3 q^{56} - 5 q^{58} - 15 q^{59} - 4 q^{60} + 2 q^{61} - 8 q^{62} - 6 q^{63} + q^{64} - 4 q^{65} - 2 q^{66} - 3 q^{67} + 3 q^{68} - q^{69} + 12 q^{70} - 2 q^{71} + 2 q^{72} + 9 q^{73} - 2 q^{74} + 11 q^{75} + 6 q^{77} - q^{78} + 10 q^{79} - 4 q^{80} + q^{81} - 8 q^{82} - 6 q^{83} + 3 q^{84} - 12 q^{85} - 4 q^{86} + 5 q^{87} - 2 q^{88} - 8 q^{90} + 3 q^{91} - q^{92} + 8 q^{93} - 8 q^{94} - q^{96} + 2 q^{97} - 2 q^{98} - 4 q^{99}+O(q^{100}) \) 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
0
−1.00000 1.00000 1.00000 −4.00000 −1.00000 3.00000 −1.00000 −2.00000 4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(19\) \( -1 \)

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 722.2.a.b 1
3.b odd 2 1 6498.2.a.y 1
4.b odd 2 1 5776.2.a.d 1
19.b odd 2 1 38.2.a.b 1
19.c even 3 2 722.2.c.f 2
19.d odd 6 2 722.2.c.d 2
19.e even 9 6 722.2.e.d 6
19.f odd 18 6 722.2.e.c 6
57.d even 2 1 342.2.a.d 1
76.d even 2 1 304.2.a.d 1
95.d odd 2 1 950.2.a.b 1
95.g even 4 2 950.2.b.c 2
133.c even 2 1 1862.2.a.f 1
152.b even 2 1 1216.2.a.g 1
152.g odd 2 1 1216.2.a.n 1
209.d even 2 1 4598.2.a.a 1
228.b odd 2 1 2736.2.a.w 1
247.d odd 2 1 6422.2.a.b 1
285.b even 2 1 8550.2.a.u 1
380.d even 2 1 7600.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 19.b odd 2 1
304.2.a.d 1 76.d even 2 1
342.2.a.d 1 57.d even 2 1
722.2.a.b 1 1.a even 1 1 trivial
722.2.c.d 2 19.d odd 6 2
722.2.c.f 2 19.c even 3 2
722.2.e.c 6 19.f odd 18 6
722.2.e.d 6 19.e even 9 6
950.2.a.b 1 95.d odd 2 1
950.2.b.c 2 95.g even 4 2
1216.2.a.g 1 152.b even 2 1
1216.2.a.n 1 152.g odd 2 1
1862.2.a.f 1 133.c even 2 1
2736.2.a.w 1 228.b odd 2 1
4598.2.a.a 1 209.d even 2 1
5776.2.a.d 1 4.b odd 2 1
6422.2.a.b 1 247.d odd 2 1
6498.2.a.y 1 3.b odd 2 1
7600.2.a.h 1 380.d even 2 1
8550.2.a.u 1 285.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(722))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} + 4 \) Copy content Toggle raw display
\( T_{7} - 3 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T - 3 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T - 5 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 8 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T - 1 \) Copy content Toggle raw display
$59$ \( T + 15 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 3 \) Copy content Toggle raw display
$71$ \( T + 2 \) Copy content Toggle raw display
$73$ \( T - 9 \) Copy content Toggle raw display
$79$ \( T - 10 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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