Properties

Label 1530.2.d.h
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: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + ( - \beta_{5} - \beta_{2}) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + ( - \beta_{5} - \beta_{2}) q^{7} + \beta_1 q^{8} + \beta_{4} q^{10} + 4 q^{11} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{13} + (\beta_{4} + \beta_{3}) q^{14} + q^{16} - \beta_1 q^{17} + (2 \beta_{5} - 2 \beta_{2}) q^{19} + \beta_{2} q^{20} - 4 \beta_1 q^{22} + ( - \beta_{5} - \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{25} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{26} + (\beta_{5} + \beta_{2}) q^{28} + ( - \beta_{5} + \beta_{2}) q^{29} + ( - \beta_{4} - \beta_{3} + 2) q^{31} - \beta_1 q^{32} - q^{34} + ( - \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 6) q^{35} + (\beta_{4} - \beta_{3} + 4 \beta_1) q^{37} + (2 \beta_{4} - 2 \beta_{3}) q^{38} - \beta_{4} q^{40} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{41} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4 \beta_1) q^{43} - 4 q^{44} + (\beta_{4} + \beta_{3} + 2) q^{46} + ( - 2 \beta_{5} - 2 \beta_{2}) q^{47} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 5) q^{49} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 2) q^{50} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{52} + ( - 2 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 6 \beta_1) q^{53} - 4 \beta_{2} q^{55} + ( - \beta_{4} - \beta_{3}) q^{56} + ( - \beta_{4} + \beta_{3}) q^{58} + ( - 3 \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - 4) q^{59} + ( - 3 \beta_{4} - 3 \beta_{3} + 4) q^{61} + ( - \beta_{5} - \beta_{2} - 2 \beta_1) q^{62} - q^{64} + ( - \beta_{5} - \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{65} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 8 \beta_1) q^{67} + \beta_1 q^{68} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + 6 \beta_1 - 2) q^{70} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 4) q^{71} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 12 \beta_1) q^{73} + ( - \beta_{5} + \beta_{2} + 4) q^{74} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{76} + ( - 4 \beta_{5} - 4 \beta_{2}) q^{77} + ( - 4 \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} - 6) q^{79} - \beta_{2} q^{80} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2}) q^{82} + (2 \beta_{4} - 2 \beta_{3}) q^{83} + \beta_{4} q^{85} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 4) q^{86} + 4 \beta_1 q^{88} + ( - 2 \beta_{4} - 2 \beta_{3} + 4) q^{89} + (2 \beta_{4} + 2 \beta_{3} + 8) q^{91} + (\beta_{5} + \beta_{2} - 2 \beta_1) q^{92} + (2 \beta_{4} + 2 \beta_{3}) q^{94} + ( - 2 \beta_{5} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 8) q^{95} + (\beta_{5} + 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 12 \beta_1) q^{97} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 5 \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 \( 6 q - 6 q^{4} - 2 q^{5} + 24 q^{11} + 6 q^{16} - 8 q^{19} + 2 q^{20} - 2 q^{25} + 4 q^{26} + 4 q^{29} + 12 q^{31} - 6 q^{34} - 32 q^{35} + 4 q^{41} - 24 q^{44} + 12 q^{46} - 22 q^{49} - 8 q^{50} - 8 q^{55} - 12 q^{59} + 24 q^{61} - 6 q^{64} + 24 q^{65} - 8 q^{70} - 16 q^{71} + 28 q^{74} + 8 q^{76} - 20 q^{79} - 2 q^{80} + 28 q^{86} + 24 q^{89} + 48 q^{91} + 56 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 30\nu^{2} - 32\nu + 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{5} + 3\nu^{4} + 10\nu^{3} - 32\nu^{2} - 74\nu - 3 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -10\nu^{5} + 11\nu^{4} - 17\nu^{3} - 10\nu^{2} - 72\nu - 11 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{5} - 27\nu^{4} + 25\nu^{3} + 12\nu^{2} + 68\nu - 65 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{5} + 37\nu^{4} - 30\nu^{3} - 42\nu^{2} - 54\nu + 55 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{2} - 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} - 3\beta_{3} + 3\beta_{2} - 4\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} - 5\beta_{4} - 5\beta_{3} + \beta_{2} - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{5} - 11\beta_{4} - 5\beta_{3} - 5\beta_{2} + 18\beta _1 - 18 ) / 2 \) Copy content Toggle raw display

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
−0.854638 + 0.854638i
1.45161 1.45161i
0.403032 0.403032i
−0.854638 0.854638i
1.45161 + 1.45161i
0.403032 + 0.403032i
1.00000i 0 −1.00000 −2.17009 0.539189i 0 1.07838i 1.00000i 0 −0.539189 + 2.17009i
919.2 1.00000i 0 −1.00000 −0.311108 + 2.21432i 0 4.42864i 1.00000i 0 2.21432 + 0.311108i
919.3 1.00000i 0 −1.00000 1.48119 1.67513i 0 3.35026i 1.00000i 0 −1.67513 1.48119i
919.4 1.00000i 0 −1.00000 −2.17009 + 0.539189i 0 1.07838i 1.00000i 0 −0.539189 2.17009i
919.5 1.00000i 0 −1.00000 −0.311108 2.21432i 0 4.42864i 1.00000i 0 2.21432 0.311108i
919.6 1.00000i 0 −1.00000 1.48119 + 1.67513i 0 3.35026i 1.00000i 0 −1.67513 + 1.48119i
\(n\): 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.h 6
3.b odd 2 1 1530.2.d.j yes 6
5.b even 2 1 inner 1530.2.d.h 6
5.c odd 4 1 7650.2.a.dk 3
5.c odd 4 1 7650.2.a.dp 3
15.d odd 2 1 1530.2.d.j yes 6
15.e even 4 1 7650.2.a.di 3
15.e even 4 1 7650.2.a.dn 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1530.2.d.h 6 1.a even 1 1 trivial
1530.2.d.h 6 5.b even 2 1 inner
1530.2.d.j yes 6 3.b odd 2 1
1530.2.d.j yes 6 15.d odd 2 1
7650.2.a.di 3 15.e even 4 1
7650.2.a.dk 3 5.c odd 4 1
7650.2.a.dn 3 15.e even 4 1
7650.2.a.dp 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 \(S_{2}^{\mathrm{new}}(1530, [\chi])\):

\( T_{7}^{6} + 32T_{7}^{4} + 256T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{29}^{3} - 2T_{29}^{2} - 12T_{29} + 8 \) 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
$5$ \( T^{6} + 2 T^{5} + 3 T^{4} + 12 T^{3} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T - 4)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 44 T^{4} + 432 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$19$ \( (T^{3} + 4 T^{2} - 48 T - 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 44 T^{4} + 112 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 12 T + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} - 4 T + 40)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 92 T^{4} + 1584 T^{2} + \cdots + 1600 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} - 36 T + 104)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 108 T^{4} + 1712 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$53$ \( T^{6} + 460 T^{4} + 69680 T^{2} + \cdots + 3474496 \) Copy content Toggle raw display
$59$ \( (T^{3} + 6 T^{2} - 100 T - 632)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 12 T^{2} - 96 T + 944)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 204 T^{4} + 9392 T^{2} + \cdots + 53824 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} - 64 T + 80)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 428 T^{4} + 50352 T^{2} + \cdots + 1459264 \) Copy content Toggle raw display
$79$ \( (T^{3} + 10 T^{2} - 164 T - 1432)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 112 T^{4} + 2816 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$89$ \( (T^{3} - 12 T^{2} - 16 T + 320)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 524 T^{4} + 67504 T^{2} + \cdots + 287296 \) Copy content Toggle raw display
show more
show less