Properties

Label 630.2.i.d
Level $630$
Weight $2$
Character orbit 630.i
Analytic conductor $5.031$
Analytic rank $0$
Dimension $2$
CM no
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(121,630)]
 
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([2, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.121");
 
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.i (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + ( - \zeta_{6} + 2) q^{3} + q^{4} + ( - \zeta_{6} + 1) q^{5} + ( - \zeta_{6} + 2) q^{6} + ( - 2 \zeta_{6} - 1) q^{7} + q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \zeta_{6} + 2) q^{3} + q^{4} + ( - \zeta_{6} + 1) q^{5} + ( - \zeta_{6} + 2) q^{6} + ( - 2 \zeta_{6} - 1) q^{7} + q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + ( - \zeta_{6} + 1) q^{10} + ( - \zeta_{6} + 2) q^{12} + 4 \zeta_{6} q^{13} + ( - 2 \zeta_{6} - 1) q^{14} + ( - 2 \zeta_{6} + 1) q^{15} + q^{16} + ( - 3 \zeta_{6} + 3) q^{18} - 2 \zeta_{6} q^{19} + ( - \zeta_{6} + 1) q^{20} + ( - \zeta_{6} - 4) q^{21} + ( - 3 \zeta_{6} + 3) q^{23} + ( - \zeta_{6} + 2) q^{24} - \zeta_{6} q^{25} + 4 \zeta_{6} q^{26} + ( - 6 \zeta_{6} + 3) q^{27} + ( - 2 \zeta_{6} - 1) q^{28} + (6 \zeta_{6} - 6) q^{29} + ( - 2 \zeta_{6} + 1) q^{30} + 2 q^{31} + q^{32} + (\zeta_{6} - 3) q^{35} + ( - 3 \zeta_{6} + 3) q^{36} - 2 \zeta_{6} q^{37} - 2 \zeta_{6} q^{38} + (4 \zeta_{6} + 4) q^{39} + ( - \zeta_{6} + 1) q^{40} + 6 \zeta_{6} q^{41} + ( - \zeta_{6} - 4) q^{42} + (11 \zeta_{6} - 11) q^{43} - 3 \zeta_{6} q^{45} + ( - 3 \zeta_{6} + 3) q^{46} + 3 q^{47} + ( - \zeta_{6} + 2) q^{48} + (8 \zeta_{6} - 3) q^{49} - \zeta_{6} q^{50} + 4 \zeta_{6} q^{52} + (6 \zeta_{6} - 6) q^{53} + ( - 6 \zeta_{6} + 3) q^{54} + ( - 2 \zeta_{6} - 1) q^{56} + ( - 2 \zeta_{6} - 2) q^{57} + (6 \zeta_{6} - 6) q^{58} + 12 q^{59} + ( - 2 \zeta_{6} + 1) q^{60} - 7 q^{61} + 2 q^{62} + (3 \zeta_{6} - 9) q^{63} + q^{64} + 4 q^{65} - 7 q^{67} + ( - 6 \zeta_{6} + 3) q^{69} + (\zeta_{6} - 3) q^{70} + ( - 3 \zeta_{6} + 3) q^{72} + (2 \zeta_{6} - 2) q^{73} - 2 \zeta_{6} q^{74} + ( - \zeta_{6} - 1) q^{75} - 2 \zeta_{6} q^{76} + (4 \zeta_{6} + 4) q^{78} - 4 q^{79} + ( - \zeta_{6} + 1) q^{80} - 9 \zeta_{6} q^{81} + 6 \zeta_{6} q^{82} + ( - 12 \zeta_{6} + 12) q^{83} + ( - \zeta_{6} - 4) q^{84} + (11 \zeta_{6} - 11) q^{86} + (12 \zeta_{6} - 6) q^{87} + 15 \zeta_{6} q^{89} - 3 \zeta_{6} q^{90} + ( - 12 \zeta_{6} + 8) q^{91} + ( - 3 \zeta_{6} + 3) q^{92} + ( - 2 \zeta_{6} + 4) q^{93} + 3 q^{94} - 2 q^{95} + ( - \zeta_{6} + 2) q^{96} + ( - 10 \zeta_{6} + 10) q^{97} + (8 \zeta_{6} - 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} - 4 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} - 4 q^{7} + 2 q^{8} + 3 q^{9} + q^{10} + 3 q^{12} + 4 q^{13} - 4 q^{14} + 2 q^{16} + 3 q^{18} - 2 q^{19} + q^{20} - 9 q^{21} + 3 q^{23} + 3 q^{24} - q^{25} + 4 q^{26} - 4 q^{28} - 6 q^{29} + 4 q^{31} + 2 q^{32} - 5 q^{35} + 3 q^{36} - 2 q^{37} - 2 q^{38} + 12 q^{39} + q^{40} + 6 q^{41} - 9 q^{42} - 11 q^{43} - 3 q^{45} + 3 q^{46} + 6 q^{47} + 3 q^{48} + 2 q^{49} - q^{50} + 4 q^{52} - 6 q^{53} - 4 q^{56} - 6 q^{57} - 6 q^{58} + 24 q^{59} - 14 q^{61} + 4 q^{62} - 15 q^{63} + 2 q^{64} + 8 q^{65} - 14 q^{67} - 5 q^{70} + 3 q^{72} - 2 q^{73} - 2 q^{74} - 3 q^{75} - 2 q^{76} + 12 q^{78} - 8 q^{79} + q^{80} - 9 q^{81} + 6 q^{82} + 12 q^{83} - 9 q^{84} - 11 q^{86} + 15 q^{89} - 3 q^{90} + 4 q^{91} + 3 q^{92} + 6 q^{93} + 6 q^{94} - 4 q^{95} + 3 q^{96} + 10 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 1.50000 + 0.866025i 1.00000 0.500000 + 0.866025i 1.50000 + 0.866025i −2.00000 + 1.73205i 1.00000 1.50000 + 2.59808i 0.500000 + 0.866025i
151.1 1.00000 1.50000 0.866025i 1.00000 0.500000 0.866025i 1.50000 0.866025i −2.00000 1.73205i 1.00000 1.50000 2.59808i 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
63.h 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.i.d 2
3.b odd 2 1 1890.2.i.a 2
7.c even 3 1 630.2.l.b yes 2
9.c even 3 1 630.2.l.b yes 2
9.d odd 6 1 1890.2.l.d 2
21.h odd 6 1 1890.2.l.d 2
63.h even 3 1 inner 630.2.i.d 2
63.j odd 6 1 1890.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.i.d 2 1.a even 1 1 trivial
630.2.i.d 2 63.h even 3 1 inner
630.2.l.b yes 2 7.c even 3 1
630.2.l.b yes 2 9.c even 3 1
1890.2.i.a 2 3.b odd 2 1
1890.2.i.a 2 63.j odd 6 1
1890.2.l.d 2 9.d odd 6 1
1890.2.l.d 2 21.h odd 6 1

Hecke kernels

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

\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) 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
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$47$ \( (T - 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T + 7)^{2} \) Copy content Toggle raw display
$67$ \( (T + 7)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$89$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$97$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
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