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Properties

Label 390.2.be.a
Level $390$
Weight $2$
Character orbit 390.be
Analytic conductor $3.114$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [390,2,Mod(17,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.be (of order \(12\), degree \(4\), 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: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.11007531417600000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{12} - \beta_{11} + \beta_{2}) q^{3} + \beta_{2} q^{4} + ( - \beta_{10} + 2 \beta_{3} + \beta_1) q^{5} + ( - \beta_{15} + \beta_{14} + \cdots - \beta_1) q^{6}+ \cdots + (\beta_{7} + \beta_{5} - \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{12} - \beta_{11} + \beta_{2}) q^{3} + \beta_{2} q^{4} + ( - \beta_{10} + 2 \beta_{3} + \beta_1) q^{5} + ( - \beta_{15} + \beta_{14} + \cdots - \beta_1) q^{6}+ \cdots + ( - 5 \beta_{12} - 2 \beta_{11} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} - 24 q^{7} - 16 q^{10} + 8 q^{12} - 16 q^{13} + 8 q^{16} + 8 q^{22} - 64 q^{25} + 56 q^{27} - 24 q^{28} - 4 q^{30} + 60 q^{33} + 16 q^{36} - 48 q^{37} + 16 q^{40} + 20 q^{42} - 8 q^{43} - 48 q^{46} + 4 q^{48} + 80 q^{51} - 16 q^{52} + 8 q^{55} - 24 q^{58} + 16 q^{61} - 48 q^{63} - 16 q^{66} + 72 q^{67} + 28 q^{75} - 24 q^{76} - 20 q^{78} - 8 q^{81} + 16 q^{82} - 96 q^{85} + 40 q^{87} - 8 q^{88} + 32 q^{90} + 56 q^{91} - 24 q^{93} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{13} + 233\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} + 377\nu^{2} ) / 144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{15} - 305\nu^{3} ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} + 13\nu^{12} - 96\nu^{8} + 624\nu^{4} + 233\nu^{2} - 91 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{14} - 13\nu^{12} + 96\nu^{8} - 624\nu^{4} + 610\nu^{2} + 91 ) / 144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} - 21\nu^{13} + 144\nu^{9} - 1008\nu^{5} + 377\nu^{3} + 147\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{12} + 48\nu^{8} - 336\nu^{4} + 1 ) / 48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} + 7\nu^{13} - 48\nu^{9} + 336\nu^{5} + 329\nu^{3} - 49\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -35\nu^{14} - \nu^{12} + 240\nu^{10} - 1632\nu^{6} + 5\nu^{2} - 233 ) / 144 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 35\nu^{13} - 240\nu^{9} + 1632\nu^{5} - 5\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\nu^{14} - 7\nu^{12} - 120\nu^{10} + 48\nu^{8} + 816\nu^{6} - 312\nu^{4} - 119\nu^{2} + 1 ) / 72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7\nu^{14} - 48\nu^{10} + 330\nu^{6} - \nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 7\nu^{15} - 48\nu^{11} + 330\nu^{7} - \nu^{3} + 18\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -56\nu^{15} + \nu^{13} + 384\nu^{11} - 2640\nu^{7} + 8\nu^{3} + 377\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 89\nu^{15} - 624\nu^{11} + 4272\nu^{7} - 623\nu^{3} ) / 144 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + \beta_{13} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - \beta_{4} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{6} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{11} + 3\beta_{9} - 4\beta_{7} + 3\beta_{5} - 3\beta_{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{14} + 5\beta_{13} - 6\beta_{10} + 5\beta_{8} - 5\beta_{6} + 5\beta_{3} - 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5\beta_{12} - 4\beta_{11} + 4\beta_{9} - 4\beta_{4} + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -16\beta_{15} - 13\beta_{14} + 13\beta_{13} + 16\beta_{3} + 13\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -26\beta_{7} + 21\beta_{5} - 21\beta_{4} - 21\beta_{2} - 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -21\beta_{10} + 17\beta_{8} - 17\beta_{6} + 17\beta_{3} + 21\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 68\beta_{12} - 55\beta_{11} + 55\beta_{9} + 55\beta_{5} - 123\beta_{2} + 55 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -110\beta_{15} - 89\beta_{14} + 89\beta_{13} - 89\beta_{8} - 89\beta_{6} - 89\beta_{3} + 89\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -72\beta_{11} - 72\beta_{9} - 72\beta_{4} - 161 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -233\beta_{14} - 233\beta_{13} + 521\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 377\beta_{5} + 377\beta_{4} - 843\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -305\beta_{8} - 305\beta_{6} - 682\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(-1\) \(-\beta_{12}\) \(1 + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.56290 − 0.418778i
−0.596975 + 0.159959i
−1.56290 + 0.418778i
0.596975 − 0.159959i
1.56290 + 0.418778i
−0.596975 − 0.159959i
−1.56290 − 0.418778i
0.596975 + 0.159959i
−0.159959 − 0.596975i
0.418778 + 1.56290i
0.159959 + 0.596975i
−0.418778 − 1.56290i
−0.159959 + 0.596975i
0.418778 − 1.56290i
0.159959 − 0.596975i
−0.418778 + 1.56290i
−0.965926 + 0.258819i −1.34425 + 1.09224i 0.866025 − 0.500000i 0.707107 − 2.12132i 1.01575 − 1.40294i −0.206150 − 0.0552376i −0.707107 + 0.707107i 0.614017 − 2.93649i −0.133975 + 2.23205i
17.2 −0.965926 + 0.258819i 1.71028 + 0.273784i 0.866025 − 0.500000i 0.707107 − 2.12132i −1.72286 + 0.178197i −4.52590 − 1.21271i −0.707107 + 0.707107i 2.85008 + 0.936492i −0.133975 + 2.23205i
17.3 0.965926 − 0.258819i −1.34425 + 1.09224i 0.866025 − 0.500000i −0.707107 + 2.12132i −1.01575 + 1.40294i −4.52590 − 1.21271i 0.707107 − 0.707107i 0.614017 − 2.93649i −0.133975 + 2.23205i
17.4 0.965926 − 0.258819i 1.71028 + 0.273784i 0.866025 − 0.500000i −0.707107 + 2.12132i 1.72286 − 0.178197i −0.206150 − 0.0552376i 0.707107 − 0.707107i 2.85008 + 0.936492i −0.133975 + 2.23205i
23.1 −0.965926 − 0.258819i −1.34425 − 1.09224i 0.866025 + 0.500000i 0.707107 + 2.12132i 1.01575 + 1.40294i −0.206150 + 0.0552376i −0.707107 − 0.707107i 0.614017 + 2.93649i −0.133975 − 2.23205i
23.2 −0.965926 − 0.258819i 1.71028 − 0.273784i 0.866025 + 0.500000i 0.707107 + 2.12132i −1.72286 − 0.178197i −4.52590 + 1.21271i −0.707107 − 0.707107i 2.85008 − 0.936492i −0.133975 − 2.23205i
23.3 0.965926 + 0.258819i −1.34425 − 1.09224i 0.866025 + 0.500000i −0.707107 − 2.12132i −1.01575 − 1.40294i −4.52590 + 1.21271i 0.707107 + 0.707107i 0.614017 + 2.93649i −0.133975 − 2.23205i
23.4 0.965926 + 0.258819i 1.71028 − 0.273784i 0.866025 + 0.500000i −0.707107 − 2.12132i 1.72286 + 0.178197i −0.206150 + 0.0552376i 0.707107 + 0.707107i 2.85008 − 0.936492i −0.133975 − 2.23205i
173.1 −0.258819 − 0.965926i −1.09224 − 1.34425i −0.866025 + 0.500000i −0.707107 − 2.12132i −1.01575 + 1.40294i −1.21271 + 4.52590i 0.707107 + 0.707107i −0.614017 + 2.93649i −1.86603 + 1.23205i
173.2 −0.258819 − 0.965926i −0.273784 + 1.71028i −0.866025 + 0.500000i −0.707107 − 2.12132i 1.72286 − 0.178197i −0.0552376 + 0.206150i 0.707107 + 0.707107i −2.85008 − 0.936492i −1.86603 + 1.23205i
173.3 0.258819 + 0.965926i −1.09224 − 1.34425i −0.866025 + 0.500000i 0.707107 + 2.12132i 1.01575 − 1.40294i −0.0552376 + 0.206150i −0.707107 − 0.707107i −0.614017 + 2.93649i −1.86603 + 1.23205i
173.4 0.258819 + 0.965926i −0.273784 + 1.71028i −0.866025 + 0.500000i 0.707107 + 2.12132i −1.72286 + 0.178197i −1.21271 + 4.52590i −0.707107 − 0.707107i −2.85008 − 0.936492i −1.86603 + 1.23205i
257.1 −0.258819 + 0.965926i −1.09224 + 1.34425i −0.866025 − 0.500000i −0.707107 + 2.12132i −1.01575 − 1.40294i −1.21271 − 4.52590i 0.707107 − 0.707107i −0.614017 − 2.93649i −1.86603 − 1.23205i
257.2 −0.258819 + 0.965926i −0.273784 − 1.71028i −0.866025 − 0.500000i −0.707107 + 2.12132i 1.72286 + 0.178197i −0.0552376 − 0.206150i 0.707107 − 0.707107i −2.85008 + 0.936492i −1.86603 − 1.23205i
257.3 0.258819 − 0.965926i −1.09224 + 1.34425i −0.866025 − 0.500000i 0.707107 − 2.12132i 1.01575 + 1.40294i −0.0552376 − 0.206150i −0.707107 + 0.707107i −0.614017 − 2.93649i −1.86603 − 1.23205i
257.4 0.258819 − 0.965926i −0.273784 − 1.71028i −0.866025 − 0.500000i 0.707107 − 2.12132i −1.72286 − 0.178197i −1.21271 − 4.52590i −0.707107 + 0.707107i −2.85008 + 0.936492i −1.86603 − 1.23205i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
13.e even 6 1 inner
15.e even 4 1 inner
39.h odd 6 1 inner
65.r odd 12 1 inner
195.bf even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.be.a ✓ 16
3.b odd 2 1 inner 390.2.be.a ✓ 16
5.c odd 4 1 inner 390.2.be.a ✓ 16
13.e even 6 1 inner 390.2.be.a ✓ 16
15.e even 4 1 inner 390.2.be.a ✓ 16
39.h odd 6 1 inner 390.2.be.a ✓ 16
65.r odd 12 1 inner 390.2.be.a ✓ 16
195.bf even 12 1 inner 390.2.be.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.be.a ✓ 16 1.a even 1 1 trivial
390.2.be.a ✓ 16 3.b odd 2 1 inner
390.2.be.a ✓ 16 5.c odd 4 1 inner
390.2.be.a ✓ 16 13.e even 6 1 inner
390.2.be.a ✓ 16 15.e even 4 1 inner
390.2.be.a ✓ 16 39.h odd 6 1 inner
390.2.be.a ✓ 16 65.r odd 12 1 inner
390.2.be.a ✓ 16 195.bf even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 12T_{7}^{7} + 72T_{7}^{6} + 288T_{7}^{5} + 623T_{7}^{4} + 288T_{7}^{3} + 72T_{7}^{2} + 12T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 8 T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 34 T^{6} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 8 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 1928 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{8} + 26 T^{6} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 144 T^{4} + 20736)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 52 T^{6} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 104 T^{2} + 784)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 24 T^{7} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 136 T^{6} + \cdots + 7311616)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 4 T^{7} + \cdots + 11316496)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 12050 T^{4} + 625)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 17618 T^{4} + 28561)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 8 T^{2} + 64)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T + 4)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 18 T^{3} + \cdots + 2916)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 124 T^{6} + \cdots + 11316496)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 2312 T^{4} + 38416)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 132 T^{2} + 36)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 256)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} - 94 T^{6} + \cdots + 83521)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 12 T^{3} + \cdots + 576)^{4} \) Copy content Toggle raw display
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