Properties

Label 270.2.f.b
Level $270$
Weight $2$
Character orbit 270.f
Analytic conductor $2.156$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [270,2,Mod(53,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.f (of order \(4\), degree \(2\), 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: \(2.15596085457\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{24}^{5} + \zeta_{24}) q^{2} - \zeta_{24}^{6} q^{4} + ( - \zeta_{24}^{7} + 2 \zeta_{24}) q^{5} + (\zeta_{24}^{4} + \zeta_{24}^{2}) q^{7} - \zeta_{24}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{24}^{5} + \zeta_{24}) q^{2} - \zeta_{24}^{6} q^{4} + ( - \zeta_{24}^{7} + 2 \zeta_{24}) q^{5} + (\zeta_{24}^{4} + \zeta_{24}^{2}) q^{7} - \zeta_{24}^{3} q^{8} + ( - 2 \zeta_{24}^{6} + \cdots + 2 \zeta_{24}^{2}) q^{10} + \cdots + (\zeta_{24}^{5} - 5 \zeta_{24}^{3} + \zeta_{24}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} - 4 q^{10} - 8 q^{16} - 20 q^{22} + 16 q^{25} + 4 q^{28} - 16 q^{31} + 24 q^{37} - 8 q^{40} - 24 q^{43} - 16 q^{46} - 36 q^{55} + 8 q^{58} + 48 q^{61} - 8 q^{67} + 20 q^{70} + 68 q^{73} + 32 q^{76} + 40 q^{82} + 40 q^{85} - 20 q^{88} - 48 q^{91} - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(217\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 + 0.258819i
−0.707107 0.707107i 0 1.00000i −2.19067 0.448288i 0 1.36603 1.36603i 0.707107 0.707107i 0 1.23205 + 1.86603i
53.2 −0.707107 0.707107i 0 1.00000i 1.48356 1.67303i 0 −0.366025 + 0.366025i 0.707107 0.707107i 0 −2.23205 + 0.133975i
53.3 0.707107 + 0.707107i 0 1.00000i −1.48356 + 1.67303i 0 −0.366025 + 0.366025i −0.707107 + 0.707107i 0 −2.23205 + 0.133975i
53.4 0.707107 + 0.707107i 0 1.00000i 2.19067 + 0.448288i 0 1.36603 1.36603i −0.707107 + 0.707107i 0 1.23205 + 1.86603i
107.1 −0.707107 + 0.707107i 0 1.00000i −2.19067 + 0.448288i 0 1.36603 + 1.36603i 0.707107 + 0.707107i 0 1.23205 1.86603i
107.2 −0.707107 + 0.707107i 0 1.00000i 1.48356 + 1.67303i 0 −0.366025 0.366025i 0.707107 + 0.707107i 0 −2.23205 0.133975i
107.3 0.707107 0.707107i 0 1.00000i −1.48356 1.67303i 0 −0.366025 0.366025i −0.707107 0.707107i 0 −2.23205 0.133975i
107.4 0.707107 0.707107i 0 1.00000i 2.19067 0.448288i 0 1.36603 + 1.36603i −0.707107 0.707107i 0 1.23205 1.86603i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
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
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.2.f.b 8
3.b odd 2 1 inner 270.2.f.b 8
4.b odd 2 1 2160.2.w.b 8
5.b even 2 1 1350.2.f.a 8
5.c odd 4 1 inner 270.2.f.b 8
5.c odd 4 1 1350.2.f.a 8
9.c even 3 1 810.2.m.d 8
9.c even 3 1 810.2.m.e 8
9.d odd 6 1 810.2.m.d 8
9.d odd 6 1 810.2.m.e 8
12.b even 2 1 2160.2.w.b 8
15.d odd 2 1 1350.2.f.a 8
15.e even 4 1 inner 270.2.f.b 8
15.e even 4 1 1350.2.f.a 8
20.e even 4 1 2160.2.w.b 8
45.k odd 12 1 810.2.m.d 8
45.k odd 12 1 810.2.m.e 8
45.l even 12 1 810.2.m.d 8
45.l even 12 1 810.2.m.e 8
60.l odd 4 1 2160.2.w.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.f.b 8 1.a even 1 1 trivial
270.2.f.b 8 3.b odd 2 1 inner
270.2.f.b 8 5.c odd 4 1 inner
270.2.f.b 8 15.e even 4 1 inner
810.2.m.d 8 9.c even 3 1
810.2.m.d 8 9.d odd 6 1
810.2.m.d 8 45.k odd 12 1
810.2.m.d 8 45.l even 12 1
810.2.m.e 8 9.c even 3 1
810.2.m.e 8 9.d odd 6 1
810.2.m.e 8 45.k odd 12 1
810.2.m.e 8 45.l even 12 1
1350.2.f.a 8 5.b even 2 1
1350.2.f.a 8 5.c odd 4 1
1350.2.f.a 8 15.d odd 2 1
1350.2.f.a 8 15.e even 4 1
2160.2.w.b 8 4.b odd 2 1
2160.2.w.b 8 12.b even 2 1
2160.2.w.b 8 20.e even 4 1
2160.2.w.b 8 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 2T_{7}^{3} + 2T_{7}^{2} + 2T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(270, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 576)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 896T^{4} + 4096 \) Copy content Toggle raw display
$19$ \( (T^{4} + 56 T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 52 T^{2} + 484)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 71)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 12 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 112 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 12 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 2786 T^{4} + 279841 \) Copy content Toggle raw display
$59$ \( (T^{4} - 124 T^{2} + 2116)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T + 24)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{3} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 256 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 34 T^{3} + \cdots + 20449)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 64386 T^{4} + 1185921 \) Copy content Toggle raw display
$89$ \( (T^{4} - 112 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 30 T^{3} + \cdots + 1521)^{2} \) Copy content Toggle raw display
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