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Properties

Label 722.2.e.o
Level $722$
Weight $2$
Character orbit 722.e
Analytic conductor $5.765$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.186694177220038656.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 343x^{6} + 117649 \) 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)
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 basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + ( - \beta_{7} - \beta_1) q^{3} + \beta_{2} q^{4} + ( - \beta_{10} + \beta_{7} - \beta_{4}) q^{5} - \beta_{5} q^{6} + (\beta_{9} + \beta_{6} + \beta_{3}) q^{7} + (\beta_{6} + 1) q^{8} + 4 \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} + ( - \beta_{7} - \beta_1) q^{3} + \beta_{2} q^{4} + ( - \beta_{10} + \beta_{7} - \beta_{4}) q^{5} - \beta_{5} q^{6} + (\beta_{9} + \beta_{6} + \beta_{3}) q^{7} + (\beta_{6} + 1) q^{8} + 4 \beta_{8} q^{9} + (\beta_{11} - \beta_{8} + \beta_{5}) q^{10} + (2 \beta_{6} + \beta_{3} + 2) q^{11} + ( - \beta_{9} - \beta_{3}) q^{12} + (2 \beta_{8} + 2 \beta_{2}) q^{13} + (\beta_{10} + \beta_{7} + \beta_{4}) q^{14} + (\beta_{11} + 7 \beta_{2}) q^{15} + \beta_{4} q^{16} - 4 q^{18} + (\beta_{9} + 1) q^{20} + ( - 7 \beta_{10} + \beta_1) q^{21} + (\beta_{7} + 2 \beta_{4} + \beta_1) q^{22} + (\beta_{11} + \beta_{2}) q^{23} - \beta_{7} q^{24} + ( - 3 \beta_{8} + 2 \beta_{5} - 3 \beta_{2}) q^{25} + 2 \beta_{6} q^{26} + \beta_{3} q^{27} + (\beta_{11} + \beta_{8} + \beta_{5}) q^{28} + ( - \beta_{11} + \beta_{8} - \beta_{5}) q^{29} + (7 \beta_{6} - \beta_{3} + 7) q^{30} + (\beta_{9} + 3 \beta_{6} + \beta_{3}) q^{31} + (\beta_{8} + \beta_{2}) q^{32} + ( - 7 \beta_{10} - 2 \beta_{7} - 7 \beta_{4}) q^{33} - 6 \beta_{4} q^{35} + 4 \beta_{10} q^{36} + ( - \beta_{9} - 3) q^{37} - 2 \beta_{9} q^{39} + ( - \beta_{10} - \beta_1) q^{40} + (2 \beta_{7} - 5 \beta_{4} + 2 \beta_1) q^{41} + ( - \beta_{11} + 7 \beta_{2}) q^{42} + ( - 6 \beta_{10} - 2 \beta_{7} - 6 \beta_{4}) q^{43} + (2 \beta_{8} + \beta_{5} + 2 \beta_{2}) q^{44} + ( - 4 \beta_{9} + 4 \beta_{6} - 4 \beta_{3}) q^{45} + (\beta_{6} - \beta_{3} + 1) q^{46} + ( - \beta_{11} + 7 \beta_{8} - \beta_{5}) q^{47} + ( - \beta_{11} - \beta_{5}) q^{48} + ( - \beta_{6} - 2 \beta_{3} - 1) q^{49} + (2 \beta_{9} - 3 \beta_{6} + 2 \beta_{3}) q^{50} + (2 \beta_{10} + 2 \beta_{4}) q^{52} + ( - 4 \beta_{11} + 2 \beta_{2}) q^{53} + (\beta_{7} + \beta_1) q^{54} + (5 \beta_{10} - \beta_1) q^{55} + (\beta_{9} - 1) q^{56} + ( - \beta_{9} - 1) q^{58} - 3 \beta_1 q^{59} + ( - \beta_{7} + 7 \beta_{4} - \beta_1) q^{60} + ( - 3 \beta_{11} - 7 \beta_{2}) q^{61} + (3 \beta_{10} + \beta_{7} + 3 \beta_{4}) q^{62} + ( - 4 \beta_{8} - 4 \beta_{5} - 4 \beta_{2}) q^{63} + \beta_{6} q^{64} + (2 \beta_{6} - 2 \beta_{3} + 2) q^{65} + ( - 2 \beta_{11} - 7 \beta_{8} - 2 \beta_{5}) q^{66} + ( - \beta_{11} + 2 \beta_{8} - \beta_{5}) q^{67} + ( - \beta_{9} + 7 \beta_{6} - \beta_{3}) q^{69} + ( - 6 \beta_{8} - 6 \beta_{2}) q^{70} + ( - 8 \beta_{10} + 2 \beta_{7} - 8 \beta_{4}) q^{71} - 4 \beta_{2} q^{72} + ( - 2 \beta_{7} + 7 \beta_{4} - 2 \beta_1) q^{73} + (3 \beta_{10} + \beta_1) q^{74} + (3 \beta_{9} + 14) q^{75} + (3 \beta_{9} - 9) q^{77} + 2 \beta_1 q^{78} + 4 \beta_{4} q^{79} + (\beta_{11} + \beta_{2}) q^{80} + (5 \beta_{10} + 5 \beta_{4}) q^{81} + ( - 5 \beta_{8} + 2 \beta_{5} - 5 \beta_{2}) q^{82} + (3 \beta_{9} + 3 \beta_{3}) q^{83} + (7 \beta_{6} + \beta_{3} + 7) q^{84} + ( - 2 \beta_{11} - 6 \beta_{8} - 2 \beta_{5}) q^{86} + ( - 7 \beta_{6} + \beta_{3} - 7) q^{87} + (\beta_{9} + 2 \beta_{6} + \beta_{3}) q^{88} + (4 \beta_{10} - 4 \beta_{7} + 4 \beta_{4}) q^{90} + (2 \beta_{11} - 2 \beta_{2}) q^{91} + ( - \beta_{7} + \beta_{4} - \beta_1) q^{92} + ( - 7 \beta_{10} + 3 \beta_1) q^{93} + ( - \beta_{9} - 7) q^{94} - \beta_{9} q^{96} + (9 \beta_{10} + 2 \beta_1) q^{97} + ( - 2 \beta_{7} - \beta_{4} - 2 \beta_1) q^{98} + (4 \beta_{11} - 8 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{7} + 6 q^{8} + 12 q^{11} - 48 q^{18} + 12 q^{20} - 12 q^{26} + 42 q^{30} - 18 q^{31} - 36 q^{37} - 24 q^{45} + 6 q^{46} - 6 q^{49} + 18 q^{50} - 12 q^{56} - 12 q^{58} - 6 q^{64} + 12 q^{65} - 42 q^{69} + 168 q^{75} - 108 q^{77} + 42 q^{84} - 42 q^{87} - 12 q^{88} - 84 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 343x^{6} + 117649 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 49 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 49 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 343 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 343 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} ) / 2401 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} ) / 2401 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} ) / 16807 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} ) / 16807 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 49\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 49\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 343\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 343\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2401\beta_{8} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2401\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 16807\beta_{10} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16807\beta_{11} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(363\)
\(\chi(n)\) \(\beta_{2}\)

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
2.48619 + 0.904900i
−2.48619 − 0.904900i
−0.459430 + 2.60556i
0.459430 − 2.60556i
−0.459430 − 2.60556i
0.459430 + 2.60556i
2.02676 + 1.70066i
−2.02676 − 1.70066i
2.48619 − 0.904900i
−2.48619 + 0.904900i
2.02676 − 1.70066i
−2.02676 + 1.70066i
0.939693 + 0.342020i −0.459430 − 2.60556i 0.766044 + 0.642788i −1.26072 + 1.05787i 0.459430 − 2.60556i −1.82288 + 3.15731i 0.500000 + 0.866025i −3.75877 + 1.36808i −1.54650 + 0.562880i
99.2 0.939693 + 0.342020i 0.459430 + 2.60556i 0.766044 + 0.642788i 2.79281 − 2.34344i −0.459430 + 2.60556i 0.822876 − 1.42526i 0.500000 + 0.866025i −3.75877 + 1.36808i 3.42589 − 1.24692i
245.1 −0.173648 + 0.984808i −2.02676 − 1.70066i −0.939693 − 0.342020i 1.54650 − 0.562880i 2.02676 − 1.70066i −1.82288 − 3.15731i 0.500000 − 0.866025i 0.694593 + 3.93923i 0.285782 + 1.62075i
245.2 −0.173648 + 0.984808i 2.02676 + 1.70066i −0.939693 − 0.342020i −3.42589 + 1.24692i −2.02676 + 1.70066i 0.822876 + 1.42526i 0.500000 − 0.866025i 0.694593 + 3.93923i −0.633078 − 3.59036i
389.1 −0.173648 − 0.984808i −2.02676 + 1.70066i −0.939693 + 0.342020i 1.54650 + 0.562880i 2.02676 + 1.70066i −1.82288 + 3.15731i 0.500000 + 0.866025i 0.694593 − 3.93923i 0.285782 − 1.62075i
389.2 −0.173648 − 0.984808i 2.02676 − 1.70066i −0.939693 + 0.342020i −3.42589 − 1.24692i −2.02676 − 1.70066i 0.822876 − 1.42526i 0.500000 + 0.866025i 0.694593 − 3.93923i −0.633078 + 3.59036i
415.1 −0.766044 − 0.642788i −2.48619 + 0.904900i 0.173648 + 0.984808i 0.633078 − 3.59036i 2.48619 + 0.904900i 0.822876 + 1.42526i 0.500000 − 0.866025i 3.06418 − 2.57115i −2.79281 + 2.34344i
415.2 −0.766044 − 0.642788i 2.48619 − 0.904900i 0.173648 + 0.984808i −0.285782 + 1.62075i −2.48619 − 0.904900i −1.82288 − 3.15731i 0.500000 − 0.866025i 3.06418 − 2.57115i 1.26072 − 1.05787i
423.1 0.939693 − 0.342020i −0.459430 + 2.60556i 0.766044 − 0.642788i −1.26072 − 1.05787i 0.459430 + 2.60556i −1.82288 − 3.15731i 0.500000 − 0.866025i −3.75877 − 1.36808i −1.54650 − 0.562880i
423.2 0.939693 − 0.342020i 0.459430 − 2.60556i 0.766044 − 0.642788i 2.79281 + 2.34344i −0.459430 − 2.60556i 0.822876 + 1.42526i 0.500000 − 0.866025i −3.75877 − 1.36808i 3.42589 + 1.24692i
595.1 −0.766044 + 0.642788i −2.48619 − 0.904900i 0.173648 − 0.984808i 0.633078 + 3.59036i 2.48619 − 0.904900i 0.822876 − 1.42526i 0.500000 + 0.866025i 3.06418 + 2.57115i −2.79281 − 2.34344i
595.2 −0.766044 + 0.642788i 2.48619 + 0.904900i 0.173648 − 0.984808i −0.285782 − 1.62075i −2.48619 + 0.904900i −1.82288 + 3.15731i 0.500000 + 0.866025i 3.06418 + 2.57115i 1.26072 + 1.05787i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.2
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.o 12
19.b odd 2 1 722.2.e.n 12
19.c even 3 2 inner 722.2.e.o 12
19.d odd 6 2 722.2.e.n 12
19.e even 9 1 722.2.a.g 2
19.e even 9 2 722.2.c.j 4
19.e even 9 3 inner 722.2.e.o 12
19.f odd 18 2 38.2.c.b ✓ 4
19.f odd 18 1 722.2.a.j 2
19.f odd 18 3 722.2.e.n 12
57.j even 18 2 342.2.g.f 4
57.j even 18 1 6498.2.a.ba 2
57.l odd 18 1 6498.2.a.bg 2
76.k even 18 2 304.2.i.e 4
76.k even 18 1 5776.2.a.ba 2
76.l odd 18 1 5776.2.a.z 2
95.o odd 18 2 950.2.e.k 4
95.r even 36 4 950.2.j.g 8
152.s odd 18 2 1216.2.i.l 4
152.v even 18 2 1216.2.i.k 4
228.u odd 18 2 2736.2.s.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.b ✓ 4 19.f odd 18 2
304.2.i.e 4 76.k even 18 2
342.2.g.f 4 57.j even 18 2
722.2.a.g 2 19.e even 9 1
722.2.a.j 2 19.f odd 18 1
722.2.c.j 4 19.e even 9 2
722.2.e.n 12 19.b odd 2 1
722.2.e.n 12 19.d odd 6 2
722.2.e.n 12 19.f odd 18 3
722.2.e.o 12 1.a even 1 1 trivial
722.2.e.o 12 19.c even 3 2 inner
722.2.e.o 12 19.e even 9 3 inner
950.2.e.k 4 95.o odd 18 2
950.2.j.g 8 95.r even 36 4
1216.2.i.k 4 152.v even 18 2
1216.2.i.l 4 152.s odd 18 2
2736.2.s.v 4 228.u odd 18 2
5776.2.a.z 2 76.l odd 18 1
5776.2.a.ba 2 76.k even 18 1
6498.2.a.ba 2 57.j even 18 1
6498.2.a.bg 2 57.l odd 18 1

Hecke kernels

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

\( T_{3}^{12} + 343T_{3}^{6} + 117649 \) Copy content Toggle raw display
\( T_{5}^{12} + 44T_{5}^{9} + 2152T_{5}^{6} - 9504T_{5}^{3} + 46656 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 10T_{7}^{2} - 12T_{7} + 36 \) Copy content Toggle raw display
\( T_{13}^{6} - 8T_{13}^{3} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 343 T^{6} + 117649 \) Copy content Toggle raw display
$5$ \( T^{12} + 44 T^{9} + \cdots + 46656 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 10 T^{2} + \cdots + 36)^{3} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + 19 T^{2} + \cdots + 9)^{3} \) Copy content Toggle raw display
$13$ \( (T^{6} - 8 T^{3} + 64)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 44 T^{9} + \cdots + 46656 \) Copy content Toggle raw display
$29$ \( T^{12} + 44 T^{9} + \cdots + 46656 \) Copy content Toggle raw display
$31$ \( (T^{4} + 6 T^{3} + 34 T^{2} + \cdots + 4)^{3} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 2)^{6} \) Copy content Toggle raw display
$41$ \( T^{12} - 1090 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$43$ \( T^{12} + 1440 T^{9} + \cdots + 262144 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 5489031744 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1586874322944 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 62523502209 \) Copy content Toggle raw display
$61$ \( T^{12} - 3332 T^{9} + \cdots + 7529536 \) Copy content Toggle raw display
$67$ \( T^{12} + 100 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 2176782336 \) Copy content Toggle raw display
$73$ \( T^{12} + 1862 T^{9} + \cdots + 85766121 \) Copy content Toggle raw display
$79$ \( (T^{6} + 64 T^{3} + 4096)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 63 T^{2} + 3969)^{3} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 22164361129 \) Copy content Toggle raw display
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