↑

Properties

Label 1530.2.d.g
Level $1530$
Weight $2$
Character orbit 1530.d
Analytic conductor $12.217$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1530,2,Mod(919,1530)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1530, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1530.919");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1530.d (of order \(2\), degree \(1\), 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: \(12.2171115093\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 170)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - q^{4} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} - \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - q^{4} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} - \beta_{4} q^{8} + (\beta_{5} + \beta_{2}) q^{10} - 2 q^{11} + ( - \beta_{5} - \beta_{2} + \beta_1) q^{13} + (\beta_{2} + \beta_1) q^{14} + q^{16} + \beta_{4} q^{17} + (\beta_{3} - \beta_{2} - \beta_1) q^{19} + ( - \beta_{3} - \beta_1) q^{20} - 2 \beta_{4} q^{22} + (4 \beta_{4} - \beta_{2} + \beta_1) q^{23} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 2) q^{25}+ \cdots + (2 \beta_{5} + \beta_{4} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 2 q^{5} - 12 q^{11} + 6 q^{16} - 2 q^{19} + 2 q^{20} + 10 q^{25} - 2 q^{26} + 34 q^{29} + 6 q^{31} - 6 q^{34} - 20 q^{35} - 32 q^{41} + 12 q^{44} - 24 q^{46} + 2 q^{49} + 4 q^{50} + 4 q^{55} + 18 q^{59} - 18 q^{61} - 6 q^{64} - 12 q^{65} + 4 q^{70} + 2 q^{71} + 16 q^{74} + 2 q^{76} - 8 q^{79} - 2 q^{80} - 8 q^{86} - 18 q^{89} - 24 q^{91} + 30 q^{94} + 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} - 5\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times\).

\(n\) \(307\) \(1261\) \(1361\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
919.1
1.32001 + 1.32001i
−1.75233 − 1.75233i
0.432320 + 0.432320i
1.32001 − 1.32001i
−1.75233 + 1.75233i
0.432320 − 0.432320i
− 1.00000i 0 −1.00000 −1.80487 + 1.32001i 0 2.64002i 1.00000i 0 1.32001 + 1.80487i
919.2 − 1.00000i 0 −1.00000 −1.38900 − 1.75233i 0 − 3.50466i 1.00000i 0 −1.75233 + 1.38900i
919.3 − 1.00000i 0 −1.00000 2.19388 + 0.432320i 0 0.864641i 1.00000i 0 0.432320 − 2.19388i
919.4 1.00000i 0 −1.00000 −1.80487 − 1.32001i 0 − 2.64002i − 1.00000i 0 1.32001 − 1.80487i
919.5 1.00000i 0 −1.00000 −1.38900 + 1.75233i 0 3.50466i − 1.00000i 0 −1.75233 − 1.38900i
919.6 1.00000i 0 −1.00000 2.19388 − 0.432320i 0 − 0.864641i − 1.00000i 0 0.432320 + 2.19388i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 919.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1530.2.d.g 6
3.b odd 2 1 170.2.c.b ✓ 6
5.b even 2 1 inner 1530.2.d.g 6
5.c odd 4 1 7650.2.a.dj 3
5.c odd 4 1 7650.2.a.do 3
12.b even 2 1 1360.2.e.c 6
15.d odd 2 1 170.2.c.b ✓ 6
15.e even 4 1 850.2.a.p 3
15.e even 4 1 850.2.a.q 3
60.h even 2 1 1360.2.e.c 6
60.l odd 4 1 6800.2.a.bk 3
60.l odd 4 1 6800.2.a.bp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.c.b ✓ 6 3.b odd 2 1
170.2.c.b ✓ 6 15.d odd 2 1
850.2.a.p 3 15.e even 4 1
850.2.a.q 3 15.e even 4 1
1360.2.e.c 6 12.b even 2 1
1360.2.e.c 6 60.h even 2 1
1530.2.d.g 6 1.a even 1 1 trivial
1530.2.d.g 6 5.b even 2 1 inner
6800.2.a.bk 3 60.l odd 4 1
6800.2.a.bp 3 60.l odd 4 1
7650.2.a.dj 3 5.c odd 4 1
7650.2.a.do 3 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1530, [\chi])\):

\( T_{7}^{6} + 20T_{7}^{4} + 100T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{29}^{3} - 17T_{29}^{2} + 66T_{29} + 50 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 20 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T + 2)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 17 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$19$ \( (T^{3} + T^{2} - 24 T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 68 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( (T^{3} - 17 T^{2} + \cdots + 50)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 3 T^{2} - 46 T - 74)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 308 T^{4} + \cdots + 1032256 \) Copy content Toggle raw display
$41$ \( (T^{3} + 16 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 72 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( T^{6} + 257 T^{4} + \cdots + 440896 \) Copy content Toggle raw display
$53$ \( T^{6} + 265 T^{4} + \cdots + 583696 \) Copy content Toggle raw display
$59$ \( (T^{3} - 9 T^{2} + \cdots + 160)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 9 T^{2} - 30 T - 34)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 132 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T^{3} - T^{2} - 118 T - 334)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 17 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$79$ \( (T^{3} + 4 T^{2} + \cdots - 160)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 328 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$89$ \( (T^{3} + 9 T^{2} + \cdots - 160)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 89 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
show more
show less