Properties

Label 722.2.c.h
Level $722$
Weight $2$
Character orbit 722.c
Analytic conductor $5.765$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [722,2,Mod(429,722)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("722.429");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.c (of order \(3\), degree \(2\), not minimal)

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: no
Analytic conductor: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{2} + 2 \beta_1 q^{3} + \beta_{3} q^{4} + (3 \beta_{3} - \beta_1 + 3) q^{5} + ( - 2 \beta_{2} - 2 \beta_1) q^{6} + ( - 2 \beta_{2} - 2) q^{7} + q^{8} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{2} + 2 \beta_1 q^{3} + \beta_{3} q^{4} + (3 \beta_{3} - \beta_1 + 3) q^{5} + ( - 2 \beta_{2} - 2 \beta_1) q^{6} + ( - 2 \beta_{2} - 2) q^{7} + q^{8} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{9} + ( - 3 \beta_{3} + \beta_{2} + \beta_1) q^{10} + ( - 2 \beta_{2} - 2) q^{11} + 2 \beta_{2} q^{12} + ( - 2 \beta_{3} - \beta_{2} - \beta_1) q^{13} + (2 \beta_{3} - 2 \beta_1 + 2) q^{14} + ( - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{15} + ( - \beta_{3} - 1) q^{16} + (4 \beta_{3} + \beta_1 + 4) q^{17} + ( - 4 \beta_{2} + 1) q^{18} + ( - \beta_{2} - 3) q^{20} + (4 \beta_{3} + 4) q^{21} + (2 \beta_{3} - 2 \beta_1 + 2) q^{22} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{23} + 2 \beta_1 q^{24} + (5 \beta_{3} - 5 \beta_{2} - 5 \beta_1) q^{25} + (\beta_{2} - 2) q^{26} + (4 \beta_{2} - 8) q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{28} + (6 \beta_{3} - 5 \beta_{2} - 5 \beta_1) q^{29} + ( - 4 \beta_{2} - 2) q^{30} + 2 \beta_{2} q^{31} + \beta_{3} q^{32} + (4 \beta_{3} + 4) q^{33} + ( - 4 \beta_{3} - \beta_{2} - \beta_1) q^{34} + ( - 8 \beta_{3} + 6 \beta_1 - 8) q^{35} + ( - \beta_{3} - 4 \beta_1 - 1) q^{36} + (\beta_{2} - 9) q^{37} + ( - 6 \beta_{2} + 2) q^{39} + (3 \beta_{3} - \beta_1 + 3) q^{40} + ( - \beta_{3} - 3 \beta_1 - 1) q^{41} - 4 \beta_{3} q^{42} + (8 \beta_{3} - 2 \beta_1 + 8) q^{43} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{44} + (7 \beta_{2} + 1) q^{45} + (2 \beta_{2} + 4) q^{46} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{47} + ( - 2 \beta_{2} - 2 \beta_1) q^{48} + (4 \beta_{2} + 1) q^{49} + (5 \beta_{2} + 5) q^{50} + (2 \beta_{3} + 10 \beta_{2} + 10 \beta_1) q^{51} + (2 \beta_{3} + \beta_1 + 2) q^{52} + (3 \beta_{3} - 5 \beta_{2} - 5 \beta_1) q^{53} + (8 \beta_{3} + 4 \beta_1 + 8) q^{54} + ( - 8 \beta_{3} + 6 \beta_1 - 8) q^{55} + ( - 2 \beta_{2} - 2) q^{56} + (5 \beta_{2} + 6) q^{58} + (2 \beta_{3} + 4 \beta_1 + 2) q^{59} + (2 \beta_{3} - 4 \beta_1 + 2) q^{60} + (2 \beta_{3} + \beta_{2} + \beta_1) q^{61} + 2 \beta_1 q^{62} + (6 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{63} + q^{64} + 5 q^{65} - 4 \beta_{3} q^{66} + ( - 8 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{67} + (\beta_{2} - 4) q^{68} + (4 \beta_{2} + 4) q^{69} + (8 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{70} + ( - 2 \beta_{3} - 8 \beta_1 - 2) q^{71} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{72} + (5 \beta_{3} - \beta_1 + 5) q^{73} + (9 \beta_{3} + \beta_1 + 9) q^{74} + 10 q^{75} + (4 \beta_{2} + 8) q^{77} + ( - 2 \beta_{3} - 6 \beta_1 - 2) q^{78} + ( - 4 \beta_{3} - 2 \beta_1 - 4) q^{79} + ( - 3 \beta_{3} + \beta_{2} + \beta_1) q^{80} + ( - 5 \beta_{3} - 12 \beta_1 - 5) q^{81} + (\beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{82} + ( - 4 \beta_{2} + 2) q^{83} - 4 q^{84} + (11 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{85} + ( - 8 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{86} + (2 \beta_{2} + 10) q^{87} + ( - 2 \beta_{2} - 2) q^{88} + ( - 7 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{89} + ( - \beta_{3} + 7 \beta_1 - 1) q^{90} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{91} + ( - 4 \beta_{3} + 2 \beta_1 - 4) q^{92} + ( - 4 \beta_{3} - 4 \beta_1 - 4) q^{93} + (4 \beta_{2} + 2) q^{94} + 2 \beta_{2} q^{96} + (8 \beta_{3} - \beta_1 + 8) q^{97} + ( - \beta_{3} + 4 \beta_1 - 1) q^{98} + (6 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 5 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 5 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{8} - 6 q^{9} + 5 q^{10} - 4 q^{11} - 4 q^{12} + 5 q^{13} + 2 q^{14} - 2 q^{16} + 9 q^{17} + 12 q^{18} - 10 q^{20} + 8 q^{21} + 2 q^{22} - 6 q^{23} + 2 q^{24} - 5 q^{25} - 10 q^{26} - 40 q^{27} + 2 q^{28} - 7 q^{29} - 4 q^{31} - 2 q^{32} + 8 q^{33} + 9 q^{34} - 10 q^{35} - 6 q^{36} - 38 q^{37} + 20 q^{39} + 5 q^{40} - 5 q^{41} + 8 q^{42} + 14 q^{43} + 2 q^{44} - 10 q^{45} + 12 q^{46} + 2 q^{48} - 4 q^{49} + 10 q^{50} - 14 q^{51} + 5 q^{52} - q^{53} + 20 q^{54} - 10 q^{55} - 4 q^{56} + 14 q^{58} + 8 q^{59} - 5 q^{61} + 2 q^{62} - 14 q^{63} + 4 q^{64} + 20 q^{65} + 8 q^{66} + 10 q^{67} - 18 q^{68} + 8 q^{69} - 10 q^{70} - 12 q^{71} - 6 q^{72} + 9 q^{73} + 19 q^{74} + 40 q^{75} + 24 q^{77} - 10 q^{78} - 10 q^{79} + 5 q^{80} - 22 q^{81} - 5 q^{82} + 16 q^{83} - 16 q^{84} - 20 q^{85} + 14 q^{86} + 36 q^{87} - 4 q^{88} + 11 q^{89} + 5 q^{90} - 6 q^{92} - 12 q^{93} - 4 q^{96} + 15 q^{97} + 2 q^{98} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/722\mathbb{Z}\right)^\times\).

\(n\) \(363\)
\(\chi(n)\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
429.1
−0.309017 + 0.535233i
0.809017 1.40126i
−0.309017 0.535233i
0.809017 + 1.40126i
−0.500000 + 0.866025i −0.618034 + 1.07047i −0.500000 0.866025i 1.80902 3.13331i −0.618034 1.07047i −3.23607 1.00000 0.736068 + 1.27491i 1.80902 + 3.13331i
429.2 −0.500000 + 0.866025i 1.61803 2.80252i −0.500000 0.866025i 0.690983 1.19682i 1.61803 + 2.80252i 1.23607 1.00000 −3.73607 6.47106i 0.690983 + 1.19682i
653.1 −0.500000 0.866025i −0.618034 1.07047i −0.500000 + 0.866025i 1.80902 + 3.13331i −0.618034 + 1.07047i −3.23607 1.00000 0.736068 1.27491i 1.80902 3.13331i
653.2 −0.500000 0.866025i 1.61803 + 2.80252i −0.500000 + 0.866025i 0.690983 + 1.19682i 1.61803 2.80252i 1.23607 1.00000 −3.73607 + 6.47106i 0.690983 1.19682i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.c.h 4
19.b odd 2 1 722.2.c.i 4
19.c even 3 1 722.2.a.i yes 2
19.c even 3 1 inner 722.2.c.h 4
19.d odd 6 1 722.2.a.h 2
19.d odd 6 1 722.2.c.i 4
19.e even 9 6 722.2.e.p 12
19.f odd 18 6 722.2.e.q 12
57.f even 6 1 6498.2.a.bk 2
57.h odd 6 1 6498.2.a.be 2
76.f even 6 1 5776.2.a.t 2
76.g odd 6 1 5776.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.h 2 19.d odd 6 1
722.2.a.i yes 2 19.c even 3 1
722.2.c.h 4 1.a even 1 1 trivial
722.2.c.h 4 19.c even 3 1 inner
722.2.c.i 4 19.b odd 2 1
722.2.c.i 4 19.d odd 6 1
722.2.e.p 12 19.e even 9 6
722.2.e.q 12 19.f odd 18 6
5776.2.a.t 2 76.f even 6 1
5776.2.a.be 2 76.g odd 6 1
6498.2.a.be 2 57.h odd 6 1
6498.2.a.bk 2 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\):

\( T_{3}^{4} - 2T_{3}^{3} + 8T_{3}^{2} + 8T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{4} - 5T_{5}^{3} + 20T_{5}^{2} - 25T_{5} + 25 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 8 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$5$ \( T^{4} - 5 T^{3} + 20 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 5 T^{3} + 20 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$17$ \( T^{4} - 9 T^{3} + 62 T^{2} - 171 T + 361 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 32 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$29$ \( T^{4} + 7 T^{3} + 68 T^{2} - 133 T + 361 \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 19 T + 89)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 5 T^{3} + 30 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$43$ \( T^{4} - 14 T^{3} + 152 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$47$ \( T^{4} + 20T^{2} + 400 \) Copy content Toggle raw display
$53$ \( T^{4} + T^{3} + 32 T^{2} - 31 T + 961 \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} + 68 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$61$ \( T^{4} + 5 T^{3} + 20 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} + 120 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$71$ \( T^{4} + 12 T^{3} + 188 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$73$ \( T^{4} - 9 T^{3} + 62 T^{2} - 171 T + 361 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} + 80 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$83$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 11 T^{3} + 102 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$97$ \( T^{4} - 15 T^{3} + 170 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
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