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Properties

Label 361.2.e.l
Level $361$
Weight $2$
Character orbit 361.e
Analytic conductor $2.883$
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] = [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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{11} - 2 \beta_{10} - \beta_{5}) q^{2} - 2 \beta_{5} q^{3} + 3 \beta_{3} q^{4} + ( - \beta_{11} + \beta_{10} + \beta_{4}) q^{5} + ( - 2 \beta_{9} + 4 \beta_{7} + 2 \beta_{3}) q^{6} - 2 \beta_{6} q^{7} + (\beta_{8} + 2 \beta_{6} - 2 \beta_{2} - 1) q^{8} - \beta_{9} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} - 2 \beta_{10} - \beta_{5}) q^{2} - 2 \beta_{5} q^{3} + 3 \beta_{3} q^{4} + ( - \beta_{11} + \beta_{10} + \beta_{4}) q^{5} + ( - 2 \beta_{9} + 4 \beta_{7} + 2 \beta_{3}) q^{6} - 2 \beta_{6} q^{7} + (\beta_{8} + 2 \beta_{6} - 2 \beta_{2} - 1) q^{8} - \beta_{9} q^{9} + ( - 3 \beta_{9} + \beta_{7} - \beta_1) q^{10} + (4 \beta_{8} + 2 \beta_{6} - 2 \beta_{2} - 4) q^{11} + 6 \beta_{8} q^{12} - 3 \beta_{7} q^{13} + ( - 4 \beta_{11} - 2 \beta_{10} - 2 \beta_{4}) q^{14} + ( - 2 \beta_{3} - 2 \beta_1) q^{15} - \beta_{5} q^{16} - \beta_{10} q^{17} + (2 \beta_{2} + 1) q^{18} + (3 \beta_{2} + 3) q^{20} + 4 \beta_{10} q^{21} + (8 \beta_{5} - 6 \beta_{4}) q^{22} + ( - 2 \beta_{3} - 4 \beta_1) q^{23} + (2 \beta_{11} - 4 \beta_{10} - 4 \beta_{4}) q^{24} + ( - 3 \beta_{9} - \beta_{7} + 3 \beta_{3}) q^{25} + ( - 6 \beta_{8} + 3 \beta_{6}) q^{26} + (4 \beta_{8} - 4) q^{27} + (6 \beta_{7} - 6 \beta_1) q^{28} + (4 \beta_{9} - 5 \beta_{7} + 5 \beta_1) q^{29} + ( - 6 \beta_{8} - 2 \beta_{6} + \cdots + 6) q^{30}+ \cdots + (4 \beta_{3} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{7} - 18 q^{11} + 36 q^{12} + 18 q^{20} - 45 q^{26} - 24 q^{27} + 30 q^{30} - 36 q^{31} - 66 q^{37} + 36 q^{39} - 3 q^{45} + 60 q^{46} + 6 q^{49} - 15 q^{50} + 60 q^{56} - 150 q^{58} + 78 q^{64} - 18 q^{65} + 9 q^{68} + 84 q^{75} + 24 q^{77} + 18 q^{83} - 36 q^{84} - 18 q^{87} - 30 q^{88} - 60 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

\(\nu\)\(=\) \( \beta_1 \) 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

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
1.52045 − 0.553400i
−0.580762 + 0.211380i
−0.280969 − 1.59345i
0.107320 + 0.608645i
0.473442 − 0.397265i
−1.23949 + 1.04005i
0.473442 + 0.397265i
−1.23949 − 1.04005i
−0.280969 + 1.59345i
0.107320 − 0.608645i
1.52045 + 0.553400i
−0.580762 − 0.211380i
−0.388289 − 2.20210i −1.53209 + 1.28558i −2.81908 + 1.02606i 0.580762 + 0.211380i 3.42585 + 2.87463i 1.61803 − 2.80252i 1.11803 + 1.93649i 0.173648 − 0.984808i 0.239976 − 1.36097i
28.2 0.388289 + 2.20210i −1.53209 + 1.28558i −2.81908 + 1.02606i −1.52045 − 0.553400i −3.42585 − 2.87463i −0.618034 + 1.07047i −1.11803 − 1.93649i 0.173648 − 0.984808i 0.628265 − 3.56307i
54.1 −1.71293 − 1.43732i 1.87939 − 0.684040i 0.520945 + 2.95442i −0.107320 + 0.608645i −4.20243 − 1.52956i 1.61803 + 2.80252i 1.11803 − 1.93649i 0.766044 − 0.642788i 1.05865 − 0.888311i
54.2 1.71293 + 1.43732i 1.87939 − 0.684040i 0.520945 + 2.95442i 0.280969 − 1.59345i 4.20243 + 1.52956i −0.618034 − 1.07047i −1.11803 + 1.93649i 0.766044 − 0.642788i 2.77157 − 2.32563i
62.1 −2.10122 + 0.764780i −0.347296 + 1.96962i 2.29813 − 1.92836i 1.23949 + 1.04005i −0.776578 − 4.40419i −0.618034 − 1.07047i −1.11803 + 1.93649i −0.939693 − 0.342020i −3.39984 − 1.23744i
62.2 2.10122 − 0.764780i −0.347296 + 1.96962i 2.29813 − 1.92836i −0.473442 − 0.397265i 0.776578 + 4.40419i 1.61803 + 2.80252i 1.11803 − 1.93649i −0.939693 − 0.342020i −1.29862 − 0.472660i
99.1 −2.10122 − 0.764780i −0.347296 − 1.96962i 2.29813 + 1.92836i 1.23949 − 1.04005i −0.776578 + 4.40419i −0.618034 + 1.07047i −1.11803 − 1.93649i −0.939693 + 0.342020i −3.39984 + 1.23744i
99.2 2.10122 + 0.764780i −0.347296 − 1.96962i 2.29813 + 1.92836i −0.473442 + 0.397265i 0.776578 − 4.40419i 1.61803 − 2.80252i 1.11803 + 1.93649i −0.939693 + 0.342020i −1.29862 + 0.472660i
234.1 −1.71293 + 1.43732i 1.87939 + 0.684040i 0.520945 − 2.95442i −0.107320 − 0.608645i −4.20243 + 1.52956i 1.61803 − 2.80252i 1.11803 + 1.93649i 0.766044 + 0.642788i 1.05865 + 0.888311i
234.2 1.71293 − 1.43732i 1.87939 + 0.684040i 0.520945 − 2.95442i 0.280969 + 1.59345i 4.20243 − 1.52956i −0.618034 + 1.07047i −1.11803 − 1.93649i 0.766044 + 0.642788i 2.77157 + 2.32563i
245.1 −0.388289 + 2.20210i −1.53209 − 1.28558i −2.81908 − 1.02606i 0.580762 − 0.211380i 3.42585 − 2.87463i 1.61803 + 2.80252i 1.11803 − 1.93649i 0.173648 + 0.984808i 0.239976 + 1.36097i
245.2 0.388289 − 2.20210i −1.53209 − 1.28558i −2.81908 − 1.02606i −1.52045 + 0.553400i −3.42585 + 2.87463i −0.618034 − 1.07047i −1.11803 + 1.93649i 0.173648 + 0.984808i 0.628265 + 3.56307i
\(n\): 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.l 12
19.b odd 2 1 361.2.e.k 12
19.c even 3 2 inner 361.2.e.l 12
19.d odd 6 2 361.2.e.k 12
19.e even 9 1 361.2.a.d ✓ 2
19.e even 9 2 361.2.c.f 4
19.e even 9 3 inner 361.2.e.l 12
19.f odd 18 1 361.2.a.e yes 2
19.f odd 18 2 361.2.c.e 4
19.f odd 18 3 361.2.e.k 12
57.j even 18 1 3249.2.a.m 2
57.l odd 18 1 3249.2.a.n 2
76.k even 18 1 5776.2.a.r 2
76.l odd 18 1 5776.2.a.bh 2
95.o odd 18 1 9025.2.a.o 2
95.p even 18 1 9025.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
361.2.a.d ✓ 2 19.e even 9 1
361.2.a.e yes 2 19.f odd 18 1
361.2.c.e 4 19.f odd 18 2
361.2.c.f 4 19.e even 9 2
361.2.e.k 12 19.b odd 2 1
361.2.e.k 12 19.d odd 6 2
361.2.e.k 12 19.f odd 18 3
361.2.e.l 12 1.a even 1 1 trivial
361.2.e.l 12 19.c even 3 2 inner
361.2.e.l 12 19.e even 9 3 inner
3249.2.a.m 2 57.j even 18 1
3249.2.a.n 2 57.l odd 18 1
5776.2.a.r 2 76.k even 18 1
5776.2.a.bh 2 76.l odd 18 1
9025.2.a.o 2 95.o odd 18 1
9025.2.a.r 2 95.p even 18 1

Hecke kernels

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

\( T_{2}^{12} + 125T_{2}^{6} + 15625 \) Copy content Toggle raw display
\( T_{3}^{6} - 8T_{3}^{3} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 125 T^{6} + 15625 \) Copy content Toggle raw display
$3$ \( (T^{6} - 8 T^{3} + 64)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} + 8 T^{2} + \cdots + 16)^{3} \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} + 32 T^{2} + \cdots + 16)^{3} \) Copy content Toggle raw display
$13$ \( T^{12} - 108 T^{9} + \cdots + 531441 \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 8000 T^{6} + 64000000 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 36)^{6} \) Copy content Toggle raw display
$37$ \( (T^{2} + 11 T + 19)^{6} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$43$ \( T^{12} + 608 T^{9} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 7256313856 \) Copy content Toggle raw display
$53$ \( T^{12} - 1364 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{12} - 32 T^{9} + \cdots + 4096 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 887503681 \) Copy content Toggle raw display
$67$ \( T^{12} - 608 T^{9} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( T^{12} + 32 T^{9} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 282429536481 \) Copy content Toggle raw display
$79$ \( (T^{6} - 8 T^{3} + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 6 T^{3} + \cdots + 1296)^{3} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 51520374361 \) Copy content Toggle raw display
$97$ \( T^{12} + 1364 T^{9} + \cdots + 1 \) Copy content Toggle raw display
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