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Properties

Label 450.2.e.i
Level $450$
Weight $2$
Character orbit 450.e
Analytic conductor $3.593$
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] = [450,2,Mod(151,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, 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("450.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.e (of order \(3\), 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: \(3.59326809096\)
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 18)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + ( - \zeta_{6} + 2) q^{6} + ( - 2 \zeta_{6} + 2) q^{7} - q^{8} + 3 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} + ( - 2 \zeta_{6} + 1) q^{12} + \cdots + 9 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} - q^{4} + 3 q^{6} + 2 q^{7} - 2 q^{8} + 3 q^{9} + 3 q^{11} + 2 q^{13} - 2 q^{14} - q^{16} + 6 q^{17} + 6 q^{18} - 2 q^{19} + 6 q^{21} - 3 q^{22} - 6 q^{23} - 3 q^{24} + 4 q^{26}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0.500000 − 0.866025i 1.50000 + 0.866025i −0.500000 − 0.866025i 0 1.50000 − 0.866025i 1.00000 − 1.73205i −1.00000 1.50000 + 2.59808i 0
301.1 0.500000 + 0.866025i 1.50000 − 0.866025i −0.500000 + 0.866025i 0 1.50000 + 0.866025i 1.00000 + 1.73205i −1.00000 1.50000 − 2.59808i 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 450.2.e.i 2
3.b odd 2 1 1350.2.e.c 2
5.b even 2 1 18.2.c.a ✓ 2
5.c odd 4 2 450.2.j.e 4
9.c even 3 1 inner 450.2.e.i 2
9.c even 3 1 4050.2.a.c 1
9.d odd 6 1 1350.2.e.c 2
9.d odd 6 1 4050.2.a.v 1
15.d odd 2 1 54.2.c.a 2
15.e even 4 2 1350.2.j.a 4
20.d odd 2 1 144.2.i.c 2
35.c odd 2 1 882.2.f.d 2
35.i odd 6 1 882.2.e.g 2
35.i odd 6 1 882.2.h.b 2
35.j even 6 1 882.2.e.i 2
35.j even 6 1 882.2.h.c 2
40.e odd 2 1 576.2.i.a 2
40.f even 2 1 576.2.i.g 2
45.h odd 6 1 54.2.c.a 2
45.h odd 6 1 162.2.a.b 1
45.j even 6 1 18.2.c.a ✓ 2
45.j even 6 1 162.2.a.c 1
45.k odd 12 2 450.2.j.e 4
45.k odd 12 2 4050.2.c.c 2
45.l even 12 2 1350.2.j.a 4
45.l even 12 2 4050.2.c.r 2
60.h even 2 1 432.2.i.b 2
105.g even 2 1 2646.2.f.g 2
105.o odd 6 1 2646.2.e.b 2
105.o odd 6 1 2646.2.h.h 2
105.p even 6 1 2646.2.e.c 2
105.p even 6 1 2646.2.h.i 2
120.i odd 2 1 1728.2.i.e 2
120.m even 2 1 1728.2.i.f 2
180.n even 6 1 432.2.i.b 2
180.n even 6 1 1296.2.a.f 1
180.p odd 6 1 144.2.i.c 2
180.p odd 6 1 1296.2.a.g 1
315.q odd 6 1 882.2.h.b 2
315.r even 6 1 882.2.h.c 2
315.u even 6 1 2646.2.e.c 2
315.v odd 6 1 2646.2.e.b 2
315.z even 6 1 2646.2.f.g 2
315.z even 6 1 7938.2.a.i 1
315.bg odd 6 1 882.2.f.d 2
315.bg odd 6 1 7938.2.a.x 1
315.bn odd 6 1 882.2.e.g 2
315.bo even 6 1 882.2.e.i 2
315.bq even 6 1 2646.2.h.i 2
315.br odd 6 1 2646.2.h.h 2
360.z odd 6 1 576.2.i.a 2
360.z odd 6 1 5184.2.a.o 1
360.bd even 6 1 1728.2.i.f 2
360.bd even 6 1 5184.2.a.p 1
360.bh odd 6 1 1728.2.i.e 2
360.bh odd 6 1 5184.2.a.q 1
360.bk even 6 1 576.2.i.g 2
360.bk even 6 1 5184.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a ✓ 2 5.b even 2 1
18.2.c.a ✓ 2 45.j even 6 1
54.2.c.a 2 15.d odd 2 1
54.2.c.a 2 45.h odd 6 1
144.2.i.c 2 20.d odd 2 1
144.2.i.c 2 180.p odd 6 1
162.2.a.b 1 45.h odd 6 1
162.2.a.c 1 45.j even 6 1
432.2.i.b 2 60.h even 2 1
432.2.i.b 2 180.n even 6 1
450.2.e.i 2 1.a even 1 1 trivial
450.2.e.i 2 9.c even 3 1 inner
450.2.j.e 4 5.c odd 4 2
450.2.j.e 4 45.k odd 12 2
576.2.i.a 2 40.e odd 2 1
576.2.i.a 2 360.z odd 6 1
576.2.i.g 2 40.f even 2 1
576.2.i.g 2 360.bk even 6 1
882.2.e.g 2 35.i odd 6 1
882.2.e.g 2 315.bn odd 6 1
882.2.e.i 2 35.j even 6 1
882.2.e.i 2 315.bo even 6 1
882.2.f.d 2 35.c odd 2 1
882.2.f.d 2 315.bg odd 6 1
882.2.h.b 2 35.i odd 6 1
882.2.h.b 2 315.q odd 6 1
882.2.h.c 2 35.j even 6 1
882.2.h.c 2 315.r even 6 1
1296.2.a.f 1 180.n even 6 1
1296.2.a.g 1 180.p odd 6 1
1350.2.e.c 2 3.b odd 2 1
1350.2.e.c 2 9.d odd 6 1
1350.2.j.a 4 15.e even 4 2
1350.2.j.a 4 45.l even 12 2
1728.2.i.e 2 120.i odd 2 1
1728.2.i.e 2 360.bh odd 6 1
1728.2.i.f 2 120.m even 2 1
1728.2.i.f 2 360.bd even 6 1
2646.2.e.b 2 105.o odd 6 1
2646.2.e.b 2 315.v odd 6 1
2646.2.e.c 2 105.p even 6 1
2646.2.e.c 2 315.u even 6 1
2646.2.f.g 2 105.g even 2 1
2646.2.f.g 2 315.z even 6 1
2646.2.h.h 2 105.o odd 6 1
2646.2.h.h 2 315.br odd 6 1
2646.2.h.i 2 105.p even 6 1
2646.2.h.i 2 315.bq even 6 1
4050.2.a.c 1 9.c even 3 1
4050.2.a.v 1 9.d odd 6 1
4050.2.c.c 2 45.k odd 12 2
4050.2.c.r 2 45.l even 12 2
5184.2.a.o 1 360.z odd 6 1
5184.2.a.p 1 360.bd even 6 1
5184.2.a.q 1 360.bh odd 6 1
5184.2.a.r 1 360.bk even 6 1
7938.2.a.i 1 315.z even 6 1
7938.2.a.x 1 315.bg odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} - 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{17} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( (T - 4)^{2} \) 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
$59$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 11)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
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