Properties

Label 270.3.b.d
Level $270$
Weight $3$
Character orbit 270.b
Analytic conductor $7.357$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [270,3,Mod(269,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.269");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 270.b (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: \(7.35696713773\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.31744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 14x^{2} + 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + 2 q^{4} + (\beta_{2} - \beta_1 + 3) q^{5} + (2 \beta_{3} + \beta_1) q^{7} + 2 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + 2 q^{4} + (\beta_{2} - \beta_1 + 3) q^{5} + (2 \beta_{3} + \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{10} + (3 \beta_{3} - 2 \beta_1) q^{11} - 3 \beta_{3} q^{13} + (3 \beta_{3} + \beta_1) q^{14} + 4 q^{16} + (7 \beta_{2} - 3) q^{17} + ( - 3 \beta_{2} - 3) q^{19} + (2 \beta_{2} - 2 \beta_1 + 6) q^{20} + (\beta_{3} + 5 \beta_1) q^{22} + (14 \beta_{2} + 15) q^{23} + ( - 2 \beta_{3} + 12 \beta_{2} + \cdots - 3) q^{25}+ \cdots + ( - 45 \beta_{2} - 132) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 12 q^{5} + 8 q^{10} + 16 q^{16} - 12 q^{17} - 12 q^{19} + 24 q^{20} + 60 q^{23} - 12 q^{25} - 68 q^{31} + 56 q^{34} + 72 q^{35} - 24 q^{38} + 16 q^{40} + 112 q^{46} - 240 q^{47} - 180 q^{49} + 96 q^{50} + 204 q^{53} - 88 q^{55} - 196 q^{61} - 120 q^{62} + 32 q^{64} - 24 q^{65} - 24 q^{68} + 80 q^{70} - 24 q^{76} - 312 q^{77} + 180 q^{79} + 48 q^{80} + 108 q^{83} + 20 q^{85} + 456 q^{91} + 120 q^{92} + 112 q^{94} - 60 q^{95} - 528 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 14x^{2} + 31 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 7\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 13\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{3} + 13\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
3.35300i
3.35300i
1.66053i
1.66053i
−1.41421 0 2.00000 1.58579 4.74186i 0 0.813575i −2.82843 0 −2.24264 + 6.70601i
269.2 −1.41421 0 2.00000 1.58579 + 4.74186i 0 0.813575i −2.82843 0 −2.24264 6.70601i
269.3 1.41421 0 2.00000 4.41421 2.34834i 0 13.6872i 2.82843 0 6.24264 3.32106i
269.4 1.41421 0 2.00000 4.41421 + 2.34834i 0 13.6872i 2.82843 0 6.24264 + 3.32106i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.3.b.d yes 4
3.b odd 2 1 270.3.b.a 4
4.b odd 2 1 2160.3.c.m 4
5.b even 2 1 270.3.b.a 4
5.c odd 4 2 1350.3.d.o 8
9.c even 3 2 810.3.j.a 8
9.d odd 6 2 810.3.j.f 8
12.b even 2 1 2160.3.c.g 4
15.d odd 2 1 inner 270.3.b.d yes 4
15.e even 4 2 1350.3.d.o 8
20.d odd 2 1 2160.3.c.g 4
45.h odd 6 2 810.3.j.a 8
45.j even 6 2 810.3.j.f 8
60.h even 2 1 2160.3.c.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.a 4 3.b odd 2 1
270.3.b.a 4 5.b even 2 1
270.3.b.d yes 4 1.a even 1 1 trivial
270.3.b.d yes 4 15.d odd 2 1 inner
810.3.j.a 8 9.c even 3 2
810.3.j.a 8 45.h odd 6 2
810.3.j.f 8 9.d odd 6 2
810.3.j.f 8 45.j even 6 2
1350.3.d.o 8 5.c odd 4 2
1350.3.d.o 8 15.e even 4 2
2160.3.c.g 4 12.b even 2 1
2160.3.c.g 4 20.d odd 2 1
2160.3.c.m 4 4.b odd 2 1
2160.3.c.m 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(270, [\chi])\):

\( T_{7}^{4} + 188T_{7}^{2} + 124 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} - 89 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 12 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{4} + 188T^{2} + 124 \) Copy content Toggle raw display
$11$ \( T^{4} + 388 T^{2} + 35836 \) Copy content Toggle raw display
$13$ \( T^{4} + 324 T^{2} + 10044 \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 89)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T - 9)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 30 T - 167)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 2268 T^{2} + 813564 \) Copy content Toggle raw display
$31$ \( (T^{2} + 34 T - 161)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 5408 T^{2} + 2293504 \) Copy content Toggle raw display
$41$ \( T^{4} + 1372 T^{2} + 65596 \) Copy content Toggle raw display
$43$ \( T^{4} + 1724 T^{2} + 65596 \) Copy content Toggle raw display
$47$ \( (T^{2} + 120 T + 3208)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 102 T + 2439)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 3076 T^{2} + 2327356 \) Copy content Toggle raw display
$61$ \( (T^{2} + 98 T + 1249)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 24464 T^{2} + 131041216 \) Copy content Toggle raw display
$71$ \( T^{4} + 15876 T^{2} + 53524476 \) Copy content Toggle raw display
$73$ \( T^{4} + 8036 T^{2} + 35836 \) Copy content Toggle raw display
$79$ \( (T^{2} - 90 T - 5913)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 54 T - 5999)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 23548 T^{2} + 87702844 \) Copy content Toggle raw display
$97$ \( T^{4} + 18464 T^{2} + 84193024 \) Copy content Toggle raw display
show more
show less