Properties

Label 1350.2.e.d
Level $1350$
Weight $2$
Character orbit 1350.e
Analytic conductor $10.780$
Analytic rank $1$
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] = [1350,2,Mod(451,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.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: \(10.7798042729\)
Analytic rank: \(1\)
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 450)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 2) q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 2) q^{7} + q^{8} - 4 \zeta_{6} q^{13} + 2 \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} - 6 q^{17} - 7 q^{19} + 4 q^{26} - 2 q^{28} + (6 \zeta_{6} - 6) q^{29} + 10 \zeta_{6} q^{31} - \zeta_{6} q^{32} + ( - 6 \zeta_{6} + 6) q^{34} - 2 q^{37} + ( - 7 \zeta_{6} + 7) q^{38} + 9 \zeta_{6} q^{41} + (\zeta_{6} - 1) q^{43} + (6 \zeta_{6} - 6) q^{47} + 3 \zeta_{6} q^{49} + (4 \zeta_{6} - 4) q^{52} - 12 q^{53} + ( - 2 \zeta_{6} + 2) q^{56} - 6 \zeta_{6} q^{58} - 9 \zeta_{6} q^{59} + ( - 4 \zeta_{6} + 4) q^{61} - 10 q^{62} + q^{64} - 13 \zeta_{6} q^{67} + 6 \zeta_{6} q^{68} - 6 q^{71} + q^{73} + ( - 2 \zeta_{6} + 2) q^{74} + 7 \zeta_{6} q^{76} + (2 \zeta_{6} - 2) q^{79} - 9 q^{82} + (9 \zeta_{6} - 9) q^{83} - \zeta_{6} q^{86} - 15 q^{89} - 8 q^{91} - 6 \zeta_{6} q^{94} + ( - 17 \zeta_{6} + 17) q^{97} - 3 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + 2 q^{7} + 2 q^{8} - 4 q^{13} + 2 q^{14} - q^{16} - 12 q^{17} - 14 q^{19} + 8 q^{26} - 4 q^{28} - 6 q^{29} + 10 q^{31} - q^{32} + 6 q^{34} - 4 q^{37} + 7 q^{38} + 9 q^{41} - q^{43} - 6 q^{47} + 3 q^{49} - 4 q^{52} - 24 q^{53} + 2 q^{56} - 6 q^{58} - 9 q^{59} + 4 q^{61} - 20 q^{62} + 2 q^{64} - 13 q^{67} + 6 q^{68} - 12 q^{71} + 2 q^{73} + 2 q^{74} + 7 q^{76} - 2 q^{79} - 18 q^{82} - 9 q^{83} - q^{86} - 30 q^{89} - 16 q^{91} - 6 q^{94} + 17 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 1.00000 1.73205i 1.00000 0 0
901.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 1.00000 + 1.73205i 1.00000 0 0
\(n\): 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 1350.2.e.d 2
3.b odd 2 1 450.2.e.f yes 2
5.b even 2 1 1350.2.e.h 2
5.c odd 4 2 1350.2.j.b 4
9.c even 3 1 inner 1350.2.e.d 2
9.c even 3 1 4050.2.a.u 1
9.d odd 6 1 450.2.e.f yes 2
9.d odd 6 1 4050.2.a.d 1
15.d odd 2 1 450.2.e.c 2
15.e even 4 2 450.2.j.b 4
45.h odd 6 1 450.2.e.c 2
45.h odd 6 1 4050.2.a.bg 1
45.j even 6 1 1350.2.e.h 2
45.j even 6 1 4050.2.a.o 1
45.k odd 12 2 1350.2.j.b 4
45.k odd 12 2 4050.2.c.m 2
45.l even 12 2 450.2.j.b 4
45.l even 12 2 4050.2.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.c 2 15.d odd 2 1
450.2.e.c 2 45.h odd 6 1
450.2.e.f yes 2 3.b odd 2 1
450.2.e.f yes 2 9.d odd 6 1
450.2.j.b 4 15.e even 4 2
450.2.j.b 4 45.l even 12 2
1350.2.e.d 2 1.a even 1 1 trivial
1350.2.e.d 2 9.c even 3 1 inner
1350.2.e.h 2 5.b even 2 1
1350.2.e.h 2 45.j even 6 1
1350.2.j.b 4 5.c odd 4 2
1350.2.j.b 4 45.k odd 12 2
4050.2.a.d 1 9.d odd 6 1
4050.2.a.o 1 45.j even 6 1
4050.2.a.u 1 9.c even 3 1
4050.2.a.bg 1 45.h odd 6 1
4050.2.c.h 2 45.l even 12 2
4050.2.c.m 2 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{2} - 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17} + 6 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( (T + 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( (T - 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$83$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$89$ \( (T + 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
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