Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [270,2,Mod(91,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.e (of order \(3\), degree \(2\), not 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - q^{8} - q^{10} + ( - 6 \zeta_{6} + 6) q^{11} - 2 \zeta_{6} q^{13} - \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} - 4 q^{19} + (\zeta_{6} - 1) q^{20} - 6 \zeta_{6} q^{22} + 9 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 2 q^{26} - q^{28} + ( - 3 \zeta_{6} + 3) q^{29} + 4 \zeta_{6} q^{31} + \zeta_{6} q^{32} - q^{35} + 8 q^{37} + (4 \zeta_{6} - 4) q^{38} + \zeta_{6} q^{40} - 3 \zeta_{6} q^{41} + (8 \zeta_{6} - 8) q^{43} - 6 q^{44} + 9 q^{46} + (3 \zeta_{6} - 3) q^{47} + 6 \zeta_{6} q^{49} + \zeta_{6} q^{50} + (2 \zeta_{6} - 2) q^{52} - 6 q^{53} - 6 q^{55} + (\zeta_{6} - 1) q^{56} - 3 \zeta_{6} q^{58} + 6 \zeta_{6} q^{59} + ( - 13 \zeta_{6} + 13) q^{61} + 4 q^{62} + q^{64} + (2 \zeta_{6} - 2) q^{65} + 13 \zeta_{6} q^{67} + (\zeta_{6} - 1) q^{70} + 6 q^{71} - 4 q^{73} + ( - 8 \zeta_{6} + 8) q^{74} + 4 \zeta_{6} q^{76} - 6 \zeta_{6} q^{77} + ( - 10 \zeta_{6} + 10) q^{79} + q^{80} - 3 q^{82} + (9 \zeta_{6} - 9) q^{83} + 8 \zeta_{6} q^{86} + (6 \zeta_{6} - 6) q^{88} - 9 q^{89} - 2 q^{91} + ( - 9 \zeta_{6} + 9) q^{92} + 3 \zeta_{6} q^{94} + 4 \zeta_{6} q^{95} + (2 \zeta_{6} - 2) q^{97} + 6 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} - q^{5} + q^{7} - 2 q^{8} - 2 q^{10} + 6 q^{11} - 2 q^{13} - q^{14} - q^{16} - 8 q^{19} - q^{20} - 6 q^{22} + 9 q^{23} - q^{25} - 4 q^{26} - 2 q^{28} + 3 q^{29} + 4 q^{31} + q^{32} - 2 q^{35} + 16 q^{37} - 4 q^{38} + q^{40} - 3 q^{41} - 8 q^{43} - 12 q^{44} + 18 q^{46} - 3 q^{47} + 6 q^{49} + q^{50} - 2 q^{52} - 12 q^{53} - 12 q^{55} - q^{56} - 3 q^{58} + 6 q^{59} + 13 q^{61} + 8 q^{62} + 2 q^{64} - 2 q^{65} + 13 q^{67} - q^{70} + 12 q^{71} - 8 q^{73} + 8 q^{74} + 4 q^{76} - 6 q^{77} + 10 q^{79} + 2 q^{80} - 6 q^{82} - 9 q^{83} + 8 q^{86} - 6 q^{88} - 18 q^{89} - 4 q^{91} + 9 q^{92} + 3 q^{94} + 4 q^{95} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 + 0.866025i −1.00000 0 −1.00000
181.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.500000 0.866025i 0 0.500000 0.866025i −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.2.e.b 2
3.b odd 2 1 90.2.e.a 2
4.b odd 2 1 2160.2.q.b 2
5.b even 2 1 1350.2.e.b 2
5.c odd 4 2 1350.2.j.e 4
9.c even 3 1 inner 270.2.e.b 2
9.c even 3 1 810.2.a.b 1
9.d odd 6 1 90.2.e.a 2
9.d odd 6 1 810.2.a.g 1
12.b even 2 1 720.2.q.b 2
15.d odd 2 1 450.2.e.e 2
15.e even 4 2 450.2.j.c 4
36.f odd 6 1 2160.2.q.b 2
36.f odd 6 1 6480.2.a.v 1
36.h even 6 1 720.2.q.b 2
36.h even 6 1 6480.2.a.g 1
45.h odd 6 1 450.2.e.e 2
45.h odd 6 1 4050.2.a.n 1
45.j even 6 1 1350.2.e.b 2
45.j even 6 1 4050.2.a.ba 1
45.k odd 12 2 1350.2.j.e 4
45.k odd 12 2 4050.2.c.a 2
45.l even 12 2 450.2.j.c 4
45.l even 12 2 4050.2.c.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.a 2 3.b odd 2 1
90.2.e.a 2 9.d odd 6 1
270.2.e.b 2 1.a even 1 1 trivial
270.2.e.b 2 9.c even 3 1 inner
450.2.e.e 2 15.d odd 2 1
450.2.e.e 2 45.h odd 6 1
450.2.j.c 4 15.e even 4 2
450.2.j.c 4 45.l even 12 2
720.2.q.b 2 12.b even 2 1
720.2.q.b 2 36.h even 6 1
810.2.a.b 1 9.c even 3 1
810.2.a.g 1 9.d odd 6 1
1350.2.e.b 2 5.b even 2 1
1350.2.e.b 2 45.j even 6 1
1350.2.j.e 4 5.c odd 4 2
1350.2.j.e 4 45.k odd 12 2
2160.2.q.b 2 4.b odd 2 1
2160.2.q.b 2 36.f odd 6 1
4050.2.a.n 1 45.h odd 6 1
4050.2.a.ba 1 45.j even 6 1
4050.2.c.a 2 45.k odd 12 2
4050.2.c.t 2 45.l even 12 2
6480.2.a.g 1 36.h even 6 1
6480.2.a.v 1 36.f odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$83$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$89$ \( (T + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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