Properties

Label 361.2.e.j
Level $361$
Weight $2$
Character orbit 361.e
Analytic conductor $2.883$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $6$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [361,2,Mod(28,361)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(361, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("361.28");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 361.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: \(2.88259951297\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.6053445140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{9} + 17x^{6} + 4x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{9} + 17x^{6} + 4x^{3} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 21 ) / 34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 55\nu ) / 34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} - 55\nu^{2} ) / 34 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + 89\nu^{2} ) / 34 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{9} + 17\nu^{6} - 85\nu^{3} + 1 ) / 34 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{10} - 17\nu^{7} + 68\nu^{4} + 16\nu ) / 17 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{9} + 17\nu^{6} - 68\nu^{3} + 1 ) / 17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -12\nu^{10} + 51\nu^{7} - 221\nu^{4} + 3\nu ) / 34 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -12\nu^{11} + 51\nu^{8} - 221\nu^{5} + 3\nu^{2} ) / 34 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 21\nu^{11} - 85\nu^{8} + 357\nu^{5} + 84\nu^{2} ) / 34 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} - 3\beta_{7} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{11} - 5\beta_{10} + 3\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5\beta_{8} - 8\beta_{6} + 8\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -8\beta_{9} - 13\beta_{7} + 8\beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -13\beta_{11} - 21\beta_{10} - 21\beta_{4} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 34\beta_{2} - 21 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 34\beta_{3} - 55\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -55\beta_{5} - 89\beta_{4} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
0.107320 + 0.608645i
−0.280969 1.59345i
0.473442 + 0.397265i
−1.23949 1.04005i
1.52045 0.553400i
−0.580762 + 0.211380i
1.52045 + 0.553400i
−0.580762 0.211380i
0.473442 0.397265i
−1.23949 + 1.04005i
0.107320 0.608645i
−0.280969 + 1.59345i
−0.107320 0.608645i 2.00553 1.68284i 1.52045 0.553400i 1.16152 + 0.422760i −1.23949 1.04005i −1.50000 + 2.59808i −1.11803 1.93649i 0.669258 3.79555i 0.132655 0.752326i
28.2 0.280969 + 1.59345i 0.292603 0.245523i −0.580762 + 0.211380i −3.04091 1.10680i 0.473442 + 0.397265i −1.50000 + 2.59808i 1.11803 + 1.93649i −0.495610 + 2.81074i 0.909234 5.15652i
54.1 −0.473442 0.397265i −2.46015 + 0.895420i −0.280969 1.59345i −0.214641 + 1.21729i 1.52045 + 0.553400i −1.50000 2.59808i −1.11803 + 1.93649i 2.95241 2.47737i 0.585206 0.491046i
54.2 1.23949 + 1.04005i −0.358931 + 0.130640i 0.107320 + 0.608645i 0.561937 3.18690i −0.580762 0.211380i −1.50000 2.59808i 1.11803 1.93649i −2.18637 + 1.83458i 4.01106 3.36568i
62.1 −1.52045 + 0.553400i 0.0663277 0.376163i 0.473442 0.397265i 2.47897 + 2.08010i 0.107320 + 0.608645i −1.50000 2.59808i 1.11803 1.93649i 2.68198 + 0.976160i −4.92029 1.79084i
62.2 0.580762 0.211380i 0.454617 2.57826i −1.23949 + 1.04005i −0.946883 0.794529i −0.280969 1.59345i −1.50000 2.59808i −1.11803 + 1.93649i −3.62167 1.31818i −0.717861 0.261280i
99.1 −1.52045 0.553400i 0.0663277 + 0.376163i 0.473442 + 0.397265i 2.47897 2.08010i 0.107320 0.608645i −1.50000 + 2.59808i 1.11803 + 1.93649i 2.68198 0.976160i −4.92029 + 1.79084i
99.2 0.580762 + 0.211380i 0.454617 + 2.57826i −1.23949 1.04005i −0.946883 + 0.794529i −0.280969 + 1.59345i −1.50000 + 2.59808i −1.11803 1.93649i −3.62167 + 1.31818i −0.717861 + 0.261280i
234.1 −0.473442 + 0.397265i −2.46015 0.895420i −0.280969 + 1.59345i −0.214641 1.21729i 1.52045 0.553400i −1.50000 + 2.59808i −1.11803 1.93649i 2.95241 + 2.47737i 0.585206 + 0.491046i
234.2 1.23949 1.04005i −0.358931 0.130640i 0.107320 0.608645i 0.561937 + 3.18690i −0.580762 + 0.211380i −1.50000 + 2.59808i 1.11803 + 1.93649i −2.18637 1.83458i 4.01106 + 3.36568i
245.1 −0.107320 + 0.608645i 2.00553 + 1.68284i 1.52045 + 0.553400i 1.16152 0.422760i −1.23949 + 1.04005i −1.50000 2.59808i −1.11803 + 1.93649i 0.669258 + 3.79555i 0.132655 + 0.752326i
245.2 0.280969 1.59345i 0.292603 + 0.245523i −0.580762 0.211380i −3.04091 + 1.10680i 0.473442 0.397265i −1.50000 2.59808i 1.11803 1.93649i −0.495610 2.81074i 0.909234 + 5.15652i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.2
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 361.2.e.j 12
19.b odd 2 1 361.2.e.i 12
19.c even 3 2 inner 361.2.e.j 12
19.d odd 6 2 361.2.e.i 12
19.e even 9 1 361.2.a.f yes 2
19.e even 9 2 361.2.c.d 4
19.e even 9 3 inner 361.2.e.j 12
19.f odd 18 1 361.2.a.c 2
19.f odd 18 2 361.2.c.g 4
19.f odd 18 3 361.2.e.i 12
57.j even 18 1 3249.2.a.o 2
57.l odd 18 1 3249.2.a.i 2
76.k even 18 1 5776.2.a.bg 2
76.l odd 18 1 5776.2.a.s 2
95.o odd 18 1 9025.2.a.s 2
95.p even 18 1 9025.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
361.2.a.c 2 19.f odd 18 1
361.2.a.f yes 2 19.e even 9 1
361.2.c.d 4 19.e even 9 2
361.2.c.g 4 19.f odd 18 2
361.2.e.i 12 19.b odd 2 1
361.2.e.i 12 19.d odd 6 2
361.2.e.i 12 19.f odd 18 3
361.2.e.j 12 1.a even 1 1 trivial
361.2.e.j 12 19.c even 3 2 inner
361.2.e.j 12 19.e even 9 3 inner
3249.2.a.i 2 57.l odd 18 1
3249.2.a.o 2 57.j even 18 1
5776.2.a.s 2 76.l odd 18 1
5776.2.a.bg 2 76.k even 18 1
9025.2.a.n 2 95.p even 18 1
9025.2.a.s 2 95.o odd 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(361, [\chi])\):

\( T_{2}^{12} + 4T_{2}^{9} + 17T_{2}^{6} - 4T_{2}^{3} + 1 \) Copy content Toggle raw display
\( T_{3}^{12} + 18T_{3}^{9} + 323T_{3}^{6} + 18T_{3}^{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{9} + 17 T^{6} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 18 T^{9} + 323 T^{6} + 18 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} + 32 T^{9} + 1088 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( (T^{2} + 3 T + 9)^{6} \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} + 2 T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$13$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 144 T^{9} + 20672 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 598 T^{9} + \cdots + 4750104241 \) Copy content Toggle raw display
$29$ \( T^{12} - 50 T^{9} + 2375 T^{6} + \cdots + 15625 \) Copy content Toggle raw display
$31$ \( (T^{4} - 11 T^{3} + 102 T^{2} - 209 T + 361)^{3} \) Copy content Toggle raw display
$37$ \( (T^{2} - 11 T + 19)^{6} \) Copy content Toggle raw display
$41$ \( (T^{6} - 27 T^{3} + 729)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 322 T^{9} + 103683 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( (T^{6} + 27 T^{3} + 729)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 558 T^{9} + \cdots + 42180533641 \) Copy content Toggle raw display
$59$ \( T^{12} - 3600 T^{9} + 12960125 T^{6} + \cdots + 15625 \) Copy content Toggle raw display
$61$ \( T^{12} - 1264 T^{9} + \cdots + 42180533641 \) Copy content Toggle raw display
$67$ \( (T^{6} - 343 T^{3} + 117649)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 414 T^{9} + 172727 T^{6} + \cdots + 1771561 \) Copy content Toggle raw display
$73$ \( T^{12} + 1208 T^{9} + \cdots + 594823321 \) Copy content Toggle raw display
$79$ \( T^{12} + 5832000 T^{6} + \cdots + 34012224000000 \) Copy content Toggle raw display
$83$ \( (T^{4} + 8 T^{3} + 68 T^{2} - 32 T + 16)^{3} \) Copy content Toggle raw display
$89$ \( T^{12} - 2300 T^{9} + \cdots + 735091890625 \) Copy content Toggle raw display
$97$ \( T^{12} + 3024 T^{9} + \cdots + 941480149401 \) Copy content Toggle raw display
show more
show less