Properties

Label 722.2.a.k
Level $722$
Weight $2$
Character orbit 722.a
Self dual yes
Analytic conductor $5.765$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + 2 q^{5} - \beta_1 q^{6} + (2 \beta_{2} + 2) q^{7} - q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + 2 q^{5} - \beta_1 q^{6} + (2 \beta_{2} + 2) q^{7} - q^{8} + (\beta_{2} - 1) q^{9} - 2 q^{10} + ( - \beta_{2} - \beta_1 + 2) q^{11} + \beta_1 q^{12} + (2 \beta_{2} - 2 \beta_1 + 2) q^{13} + ( - 2 \beta_{2} - 2) q^{14} + 2 \beta_1 q^{15} + q^{16} + ( - 4 \beta_{2} + 4 \beta_1 + 1) q^{17} + ( - \beta_{2} + 1) q^{18} + 2 q^{20} + (4 \beta_1 + 2) q^{21} + (\beta_{2} + \beta_1 - 2) q^{22} - 2 \beta_{2} q^{23} - \beta_1 q^{24} - q^{25} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{26} + ( - 3 \beta_1 + 1) q^{27} + (2 \beta_{2} + 2) q^{28} + (2 \beta_{2} - 4 \beta_1 - 4) q^{29} - 2 \beta_1 q^{30} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{31} - q^{32} + ( - \beta_{2} + \beta_1 - 3) q^{33} + (4 \beta_{2} - 4 \beta_1 - 1) q^{34} + (4 \beta_{2} + 4) q^{35} + (\beta_{2} - 1) q^{36} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{37} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{39} - 2 q^{40} + \beta_{2} q^{41} + ( - 4 \beta_1 - 2) q^{42} + ( - 2 \beta_1 + 3) q^{43} + ( - \beta_{2} - \beta_1 + 2) q^{44} + (2 \beta_{2} - 2) q^{45} + 2 \beta_{2} q^{46} + (2 \beta_{2} - 6 \beta_1 - 2) q^{47} + \beta_1 q^{48} + (4 \beta_{2} + 4 \beta_1 + 5) q^{49} + q^{50} + (4 \beta_{2} - 3 \beta_1 + 4) q^{51} + (2 \beta_{2} - 2 \beta_1 + 2) q^{52} + ( - 6 \beta_{2} + 4 \beta_1) q^{53} + (3 \beta_1 - 1) q^{54} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{55} + ( - 2 \beta_{2} - 2) q^{56} + ( - 2 \beta_{2} + 4 \beta_1 + 4) q^{58} + ( - 3 \beta_{2} + \beta_1 + 2) q^{59} + 2 \beta_1 q^{60} + (2 \beta_1 + 6) q^{61} + (4 \beta_{2} - 2 \beta_1 - 2) q^{62} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{63} + q^{64} + (4 \beta_{2} - 4 \beta_1 + 4) q^{65} + (\beta_{2} - \beta_1 + 3) q^{66} + ( - 3 \beta_{2} + 6) q^{67} + ( - 4 \beta_{2} + 4 \beta_1 + 1) q^{68} + ( - 2 \beta_1 - 2) q^{69} + ( - 4 \beta_{2} - 4) q^{70} + (4 \beta_{2} - 2 \beta_1 + 4) q^{71} + ( - \beta_{2} + 1) q^{72} + (3 \beta_{2} - 7 \beta_1 + 4) q^{73} + (2 \beta_{2} + 2 \beta_1 - 2) q^{74} - \beta_1 q^{75} + (4 \beta_{2} - 6 \beta_1 - 2) q^{77} + (2 \beta_{2} - 4 \beta_1 + 2) q^{78} + ( - 2 \beta_1 + 6) q^{79} + 2 q^{80} + ( - 6 \beta_{2} + \beta_1 - 3) q^{81} - \beta_{2} q^{82} + (\beta_{2} - 4 \beta_1 + 2) q^{83} + (4 \beta_1 + 2) q^{84} + ( - 8 \beta_{2} + 8 \beta_1 + 2) q^{85} + (2 \beta_1 - 3) q^{86} + ( - 4 \beta_{2} - 2 \beta_1 - 6) q^{87} + (\beta_{2} + \beta_1 - 2) q^{88} + (4 \beta_{2} + 4 \beta_1 - 3) q^{89} + ( - 2 \beta_{2} + 2) q^{90} + (4 \beta_{2} - 4 \beta_1 + 8) q^{91} - 2 \beta_{2} q^{92} + (2 \beta_{2} - 2 \beta_1) q^{93} + ( - 2 \beta_{2} + 6 \beta_1 + 2) q^{94} - \beta_1 q^{96} + \beta_{2} q^{97} + ( - 4 \beta_{2} - 4 \beta_1 - 5) q^{98} + (4 \beta_{2} - \beta_1 - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{7} - 3 q^{8} - 3 q^{9} - 6 q^{10} + 6 q^{11} + 6 q^{13} - 6 q^{14} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 6 q^{20} + 6 q^{21} - 6 q^{22} - 3 q^{25} - 6 q^{26} + 3 q^{27} + 6 q^{28} - 12 q^{29} + 6 q^{31} - 3 q^{32} - 9 q^{33} - 3 q^{34} + 12 q^{35} - 3 q^{36} + 6 q^{37} - 6 q^{39} - 6 q^{40} - 6 q^{42} + 9 q^{43} + 6 q^{44} - 6 q^{45} - 6 q^{47} + 15 q^{49} + 3 q^{50} + 12 q^{51} + 6 q^{52} - 3 q^{54} + 12 q^{55} - 6 q^{56} + 12 q^{58} + 6 q^{59} + 18 q^{61} - 6 q^{62} + 6 q^{63} + 3 q^{64} + 12 q^{65} + 9 q^{66} + 18 q^{67} + 3 q^{68} - 6 q^{69} - 12 q^{70} + 12 q^{71} + 3 q^{72} + 12 q^{73} - 6 q^{74} - 6 q^{77} + 6 q^{78} + 18 q^{79} + 6 q^{80} - 9 q^{81} + 6 q^{83} + 6 q^{84} + 6 q^{85} - 9 q^{86} - 18 q^{87} - 6 q^{88} - 9 q^{89} + 6 q^{90} + 24 q^{91} + 6 q^{94} - 15 q^{98} - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
−1.53209
−0.347296
1.87939
−1.00000 −1.53209 1.00000 2.00000 1.53209 2.69459 −1.00000 −0.652704 −2.00000
1.2 −1.00000 −0.347296 1.00000 2.00000 0.347296 −1.75877 −1.00000 −2.87939 −2.00000
1.3 −1.00000 1.87939 1.00000 2.00000 −1.87939 5.06418 −1.00000 0.532089 −2.00000
\(n\): 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.k 3
3.b odd 2 1 6498.2.a.bq 3
4.b odd 2 1 5776.2.a.bo 3
19.b odd 2 1 722.2.a.l 3
19.c even 3 2 722.2.c.l 6
19.d odd 6 2 722.2.c.k 6
19.e even 9 2 722.2.e.a 6
19.e even 9 2 722.2.e.k 6
19.e even 9 2 722.2.e.l 6
19.f odd 18 2 38.2.e.a 6
19.f odd 18 2 722.2.e.b 6
19.f odd 18 2 722.2.e.m 6
57.d even 2 1 6498.2.a.bl 3
57.j even 18 2 342.2.u.c 6
76.d even 2 1 5776.2.a.bn 3
76.k even 18 2 304.2.u.c 6
95.o odd 18 2 950.2.l.d 6
95.r even 36 4 950.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.e.a 6 19.f odd 18 2
304.2.u.c 6 76.k even 18 2
342.2.u.c 6 57.j even 18 2
722.2.a.k 3 1.a even 1 1 trivial
722.2.a.l 3 19.b odd 2 1
722.2.c.k 6 19.d odd 6 2
722.2.c.l 6 19.c even 3 2
722.2.e.a 6 19.e even 9 2
722.2.e.b 6 19.f odd 18 2
722.2.e.k 6 19.e even 9 2
722.2.e.l 6 19.e even 9 2
722.2.e.m 6 19.f odd 18 2
950.2.l.d 6 95.o odd 18 2
950.2.u.b 12 95.r even 36 4
5776.2.a.bn 3 76.d even 2 1
5776.2.a.bo 3 4.b odd 2 1
6498.2.a.bl 3 57.d even 2 1
6498.2.a.bq 3 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(722))\):

\( T_{3}^{3} - 3T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} - 6T_{7}^{2} + 24 \) Copy content Toggle raw display
\( T_{13}^{3} - 6T_{13}^{2} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 3T - 1 \) Copy content Toggle raw display
$5$ \( (T - 2)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 6T^{2} + 24 \) Copy content Toggle raw display
$11$ \( T^{3} - 6 T^{2} + 3 T + 19 \) Copy content Toggle raw display
$13$ \( T^{3} - 6T^{2} + 8 \) Copy content Toggle raw display
$17$ \( T^{3} - 3 T^{2} - 45 T + 111 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 12T - 8 \) Copy content Toggle raw display
$29$ \( T^{3} + 12 T^{2} + 12 T - 152 \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 24 T - 8 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 24 T + 136 \) Copy content Toggle raw display
$41$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$43$ \( T^{3} - 9 T^{2} + 15 T + 17 \) Copy content Toggle raw display
$47$ \( T^{3} + 6 T^{2} - 72 T - 296 \) Copy content Toggle raw display
$53$ \( T^{3} - 84T - 136 \) Copy content Toggle raw display
$59$ \( T^{3} - 6 T^{2} - 9 T - 3 \) Copy content Toggle raw display
$61$ \( T^{3} - 18 T^{2} + 96 T - 152 \) Copy content Toggle raw display
$67$ \( T^{3} - 18 T^{2} + 81 T - 81 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} + 12 T + 152 \) Copy content Toggle raw display
$73$ \( T^{3} - 12 T^{2} - 63 T + 57 \) Copy content Toggle raw display
$79$ \( T^{3} - 18 T^{2} + 96 T - 136 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} - 27 T + 51 \) Copy content Toggle raw display
$89$ \( T^{3} + 9 T^{2} - 117 T - 981 \) Copy content Toggle raw display
$97$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
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