Properties

Label 630.2.k.e
Level $630$
Weight $2$
Character orbit 630.k
Analytic conductor $5.031$
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] = [630,2,Mod(361,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.k (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: \(5.03057532734\)
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 70)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + ( - 2 \zeta_{6} - 1) q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + ( - 2 \zeta_{6} - 1) q^{7} - q^{8} + ( - \zeta_{6} + 1) q^{10} + ( - 3 \zeta_{6} + 3) q^{11} + 5 q^{13} + ( - 3 \zeta_{6} + 2) q^{14} - \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + \zeta_{6} q^{19} + q^{20} + 3 q^{22} + 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 5 \zeta_{6} q^{26} + ( - \zeta_{6} + 3) q^{28} + 6 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + ( - \zeta_{6} + 1) q^{32} + 6 q^{34} + (3 \zeta_{6} - 2) q^{35} - 11 \zeta_{6} q^{37} + (\zeta_{6} - 1) q^{38} + \zeta_{6} q^{40} - 3 q^{41} - 10 q^{43} + 3 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{46} + 3 \zeta_{6} q^{47} + (8 \zeta_{6} - 3) q^{49} - q^{50} + (5 \zeta_{6} - 5) q^{52} + ( - 3 \zeta_{6} + 3) q^{53} - 3 q^{55} + (2 \zeta_{6} + 1) q^{56} + 6 \zeta_{6} q^{58} + 4 \zeta_{6} q^{61} + 4 q^{62} + q^{64} - 5 \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} + 6 \zeta_{6} q^{68} + (\zeta_{6} - 3) q^{70} - 12 q^{71} + ( - 4 \zeta_{6} + 4) q^{73} + ( - 11 \zeta_{6} + 11) q^{74} - q^{76} + (3 \zeta_{6} - 9) q^{77} + 10 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} - 3 \zeta_{6} q^{82} + 12 q^{83} - 6 q^{85} - 10 \zeta_{6} q^{86} + (3 \zeta_{6} - 3) q^{88} + 6 \zeta_{6} q^{89} + ( - 10 \zeta_{6} - 5) q^{91} - 3 q^{92} + (3 \zeta_{6} - 3) q^{94} + ( - \zeta_{6} + 1) q^{95} + 14 q^{97} + (5 \zeta_{6} - 8) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - q^{5} - 4 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} - q^{5} - 4 q^{7} - 2 q^{8} + q^{10} + 3 q^{11} + 10 q^{13} + q^{14} - q^{16} + 6 q^{17} + q^{19} + 2 q^{20} + 6 q^{22} + 3 q^{23} - q^{25} + 5 q^{26} + 5 q^{28} + 12 q^{29} + 4 q^{31} + q^{32} + 12 q^{34} - q^{35} - 11 q^{37} - q^{38} + q^{40} - 6 q^{41} - 20 q^{43} + 3 q^{44} - 3 q^{46} + 3 q^{47} + 2 q^{49} - 2 q^{50} - 5 q^{52} + 3 q^{53} - 6 q^{55} + 4 q^{56} + 6 q^{58} + 4 q^{61} + 8 q^{62} + 2 q^{64} - 5 q^{65} + 4 q^{67} + 6 q^{68} - 5 q^{70} - 24 q^{71} + 4 q^{73} + 11 q^{74} - 2 q^{76} - 15 q^{77} + 10 q^{79} - q^{80} - 3 q^{82} + 24 q^{83} - 12 q^{85} - 10 q^{86} - 3 q^{88} + 6 q^{89} - 20 q^{91} - 6 q^{92} - 3 q^{94} + q^{95} + 28 q^{97} - 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 −2.00000 1.73205i −1.00000 0 0.500000 0.866025i
541.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −2.00000 + 1.73205i −1.00000 0 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.k.e 2
3.b odd 2 1 70.2.e.b 2
7.c even 3 1 inner 630.2.k.e 2
7.c even 3 1 4410.2.a.m 1
7.d odd 6 1 4410.2.a.c 1
12.b even 2 1 560.2.q.d 2
15.d odd 2 1 350.2.e.h 2
15.e even 4 2 350.2.j.a 4
21.c even 2 1 490.2.e.a 2
21.g even 6 1 490.2.a.j 1
21.g even 6 1 490.2.e.a 2
21.h odd 6 1 70.2.e.b 2
21.h odd 6 1 490.2.a.g 1
84.j odd 6 1 3920.2.a.g 1
84.n even 6 1 560.2.q.d 2
84.n even 6 1 3920.2.a.be 1
105.o odd 6 1 350.2.e.h 2
105.o odd 6 1 2450.2.a.p 1
105.p even 6 1 2450.2.a.f 1
105.w odd 12 2 2450.2.c.p 2
105.x even 12 2 350.2.j.a 4
105.x even 12 2 2450.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 3.b odd 2 1
70.2.e.b 2 21.h odd 6 1
350.2.e.h 2 15.d odd 2 1
350.2.e.h 2 105.o odd 6 1
350.2.j.a 4 15.e even 4 2
350.2.j.a 4 105.x even 12 2
490.2.a.g 1 21.h odd 6 1
490.2.a.j 1 21.g even 6 1
490.2.e.a 2 21.c even 2 1
490.2.e.a 2 21.g even 6 1
560.2.q.d 2 12.b even 2 1
560.2.q.d 2 84.n even 6 1
630.2.k.e 2 1.a even 1 1 trivial
630.2.k.e 2 7.c even 3 1 inner
2450.2.a.f 1 105.p even 6 1
2450.2.a.p 1 105.o odd 6 1
2450.2.c.f 2 105.x even 12 2
2450.2.c.p 2 105.w odd 12 2
3920.2.a.g 1 84.j odd 6 1
3920.2.a.be 1 84.n even 6 1
4410.2.a.c 1 7.d odd 6 1
4410.2.a.m 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{13} - 5 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 36 \) 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
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( (T + 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T - 14)^{2} \) Copy content Toggle raw display
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