Properties

Label 722.2.e.e
Level $722$
Weight $2$
Character orbit 722.e
Analytic conductor $5.765$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $6$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [722,2,Mod(99,722)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("722.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.e (of order \(9\), degree \(6\), 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: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(363\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−0.173648 0.984808i
−0.766044 0.642788i
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.939693 0.342020i −0.173648 0.984808i 0.766044 + 0.642788i 0 −0.173648 + 0.984808i 0.500000 0.866025i −0.500000 0.866025i 1.87939 0.684040i 0
245.1 0.173648 0.984808i −0.766044 0.642788i −0.939693 0.342020i 0 −0.766044 + 0.642788i 0.500000 + 0.866025i −0.500000 + 0.866025i −0.347296 1.96962i 0
389.1 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i 0 −0.766044 0.642788i 0.500000 0.866025i −0.500000 0.866025i −0.347296 + 1.96962i 0
415.1 0.766044 + 0.642788i 0.939693 0.342020i 0.173648 + 0.984808i 0 0.939693 + 0.342020i 0.500000 + 0.866025i −0.500000 + 0.866025i −1.53209 + 1.28558i 0
423.1 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 0.642788i 0 −0.173648 0.984808i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.87939 + 0.684040i 0
595.1 0.766044 0.642788i 0.939693 + 0.342020i 0.173648 0.984808i 0 0.939693 0.342020i 0.500000 0.866025i −0.500000 0.866025i −1.53209 1.28558i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 2 inner
19.e even 9 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.e.e 6
19.b odd 2 1 722.2.e.f 6
19.c even 3 2 inner 722.2.e.e 6
19.d odd 6 2 722.2.e.f 6
19.e even 9 1 722.2.a.e 1
19.e even 9 2 722.2.c.c 2
19.e even 9 3 inner 722.2.e.e 6
19.f odd 18 1 38.2.a.a 1
19.f odd 18 2 722.2.c.e 2
19.f odd 18 3 722.2.e.f 6
57.j even 18 1 342.2.a.e 1
57.l odd 18 1 6498.2.a.f 1
76.k even 18 1 304.2.a.c 1
76.l odd 18 1 5776.2.a.m 1
95.o odd 18 1 950.2.a.d 1
95.r even 36 2 950.2.b.b 2
133.ba even 18 1 1862.2.a.b 1
152.s odd 18 1 1216.2.a.e 1
152.v even 18 1 1216.2.a.m 1
209.p even 18 1 4598.2.a.p 1
228.u odd 18 1 2736.2.a.n 1
247.bl odd 18 1 6422.2.a.h 1
285.bf even 18 1 8550.2.a.m 1
380.bb even 18 1 7600.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.a 1 19.f odd 18 1
304.2.a.c 1 76.k even 18 1
342.2.a.e 1 57.j even 18 1
722.2.a.e 1 19.e even 9 1
722.2.c.c 2 19.e even 9 2
722.2.c.e 2 19.f odd 18 2
722.2.e.e 6 1.a even 1 1 trivial
722.2.e.e 6 19.c even 3 2 inner
722.2.e.e 6 19.e even 9 3 inner
722.2.e.f 6 19.b odd 2 1
722.2.e.f 6 19.d odd 6 2
722.2.e.f 6 19.f odd 18 3
950.2.a.d 1 95.o odd 18 1
950.2.b.b 2 95.r even 36 2
1216.2.a.e 1 152.s odd 18 1
1216.2.a.m 1 152.v even 18 1
1862.2.a.b 1 133.ba even 18 1
2736.2.a.n 1 228.u odd 18 1
4598.2.a.p 1 209.p even 18 1
5776.2.a.m 1 76.l odd 18 1
6422.2.a.h 1 247.bl odd 18 1
6498.2.a.f 1 57.l odd 18 1
7600.2.a.n 1 380.bb even 18 1
8550.2.a.m 1 285.bf even 18 1

Hecke kernels

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

\( T_{3}^{6} - T_{3}^{3} + 1 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display
\( T_{13}^{6} - 125T_{13}^{3} + 15625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 36)^{3} \) Copy content Toggle raw display
$13$ \( T^{6} - 125 T^{3} + 15625 \) Copy content Toggle raw display
$17$ \( T^{6} + 27T^{3} + 729 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 27T^{3} + 729 \) Copy content Toggle raw display
$29$ \( T^{6} - 729 T^{3} + 531441 \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 16)^{3} \) Copy content Toggle raw display
$37$ \( (T + 2)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 512 T^{3} + 262144 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} + 27T^{3} + 729 \) Copy content Toggle raw display
$59$ \( T^{6} - 729 T^{3} + 531441 \) Copy content Toggle raw display
$61$ \( T^{6} - 1000 T^{3} + 1000000 \) Copy content Toggle raw display
$67$ \( T^{6} - 125 T^{3} + 15625 \) Copy content Toggle raw display
$71$ \( T^{6} + 216 T^{3} + 46656 \) Copy content Toggle raw display
$73$ \( T^{6} - 343 T^{3} + 117649 \) Copy content Toggle raw display
$79$ \( T^{6} + 1000 T^{3} + 1000000 \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T + 36)^{3} \) Copy content Toggle raw display
$89$ \( T^{6} + 1728 T^{3} + 2985984 \) Copy content Toggle raw display
$97$ \( T^{6} + 1000 T^{3} + 1000000 \) Copy content Toggle raw display
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