Properties

Label 630.6.a.e
Level $630$
Weight $6$
Character orbit 630.a
Self dual yes
Analytic conductor $101.042$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,6,Mod(1,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 630.a (trivial)

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: yes
Analytic conductor: \(101.041806482\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 16 q^{4} + 25 q^{5} + 49 q^{7} - 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 16 q^{4} + 25 q^{5} + 49 q^{7} - 64 q^{8} - 100 q^{10} + 267 q^{11} - 1087 q^{13} - 196 q^{14} + 256 q^{16} + 513 q^{17} - 802 q^{19} + 400 q^{20} - 1068 q^{22} + 1290 q^{23} + 625 q^{25} + 4348 q^{26} + 784 q^{28} - 1779 q^{29} - 2584 q^{31} - 1024 q^{32} - 2052 q^{34} + 1225 q^{35} + 13862 q^{37} + 3208 q^{38} - 1600 q^{40} + 11904 q^{41} - 598 q^{43} + 4272 q^{44} - 5160 q^{46} + 17019 q^{47} + 2401 q^{49} - 2500 q^{50} - 17392 q^{52} - 27852 q^{53} + 6675 q^{55} - 3136 q^{56} + 7116 q^{58} - 30912 q^{59} - 1780 q^{61} + 10336 q^{62} + 4096 q^{64} - 27175 q^{65} + 25052 q^{67} + 8208 q^{68} - 4900 q^{70} + 51984 q^{71} + 47690 q^{73} - 55448 q^{74} - 12832 q^{76} + 13083 q^{77} - 102121 q^{79} + 6400 q^{80} - 47616 q^{82} + 83676 q^{83} + 12825 q^{85} + 2392 q^{86} - 17088 q^{88} + 32400 q^{89} - 53263 q^{91} + 20640 q^{92} - 68076 q^{94} - 20050 q^{95} - 148645 q^{97} - 9604 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−4.00000 0 16.0000 25.0000 0 49.0000 −64.0000 0 −100.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.6.a.e 1
3.b odd 2 1 70.6.a.f 1
12.b even 2 1 560.6.a.f 1
15.d odd 2 1 350.6.a.d 1
15.e even 4 2 350.6.c.b 2
21.c even 2 1 490.6.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.6.a.f 1 3.b odd 2 1
350.6.a.d 1 15.d odd 2 1
350.6.c.b 2 15.e even 4 2
490.6.a.l 1 21.c even 2 1
560.6.a.f 1 12.b even 2 1
630.6.a.e 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 267 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(630))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 267 \) Copy content Toggle raw display
$13$ \( T + 1087 \) Copy content Toggle raw display
$17$ \( T - 513 \) Copy content Toggle raw display
$19$ \( T + 802 \) Copy content Toggle raw display
$23$ \( T - 1290 \) Copy content Toggle raw display
$29$ \( T + 1779 \) Copy content Toggle raw display
$31$ \( T + 2584 \) Copy content Toggle raw display
$37$ \( T - 13862 \) Copy content Toggle raw display
$41$ \( T - 11904 \) Copy content Toggle raw display
$43$ \( T + 598 \) Copy content Toggle raw display
$47$ \( T - 17019 \) Copy content Toggle raw display
$53$ \( T + 27852 \) Copy content Toggle raw display
$59$ \( T + 30912 \) Copy content Toggle raw display
$61$ \( T + 1780 \) Copy content Toggle raw display
$67$ \( T - 25052 \) Copy content Toggle raw display
$71$ \( T - 51984 \) Copy content Toggle raw display
$73$ \( T - 47690 \) Copy content Toggle raw display
$79$ \( T + 102121 \) Copy content Toggle raw display
$83$ \( T - 83676 \) Copy content Toggle raw display
$89$ \( T - 32400 \) Copy content Toggle raw display
$97$ \( T + 148645 \) Copy content Toggle raw display
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