Properties

Label 1050.6.a.h
Level $1050$
Weight $6$
Character orbit 1050.a
Self dual yes
Analytic conductor $168.403$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,6,Mod(1,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, 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("1050.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1050.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: \(168.403010804\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} - 36 q^{6} - 49 q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} - 36 q^{6} - 49 q^{7} + 64 q^{8} + 81 q^{9} - 154 q^{11} - 144 q^{12} + 404 q^{13} - 196 q^{14} + 256 q^{16} - 2182 q^{17} + 324 q^{18} - 2494 q^{19} + 441 q^{21} - 616 q^{22} + 3472 q^{23} - 576 q^{24} + 1616 q^{26} - 729 q^{27} - 784 q^{28} + 5958 q^{29} - 6410 q^{31} + 1024 q^{32} + 1386 q^{33} - 8728 q^{34} + 1296 q^{36} + 11150 q^{37} - 9976 q^{38} - 3636 q^{39} + 7834 q^{41} + 1764 q^{42} - 16236 q^{43} - 2464 q^{44} + 13888 q^{46} + 2800 q^{47} - 2304 q^{48} + 2401 q^{49} + 19638 q^{51} + 6464 q^{52} + 30924 q^{53} - 2916 q^{54} - 3136 q^{56} + 22446 q^{57} + 23832 q^{58} - 11536 q^{59} - 38834 q^{61} - 25640 q^{62} - 3969 q^{63} + 4096 q^{64} + 5544 q^{66} + 48756 q^{67} - 34912 q^{68} - 31248 q^{69} - 77882 q^{71} + 5184 q^{72} + 47540 q^{73} + 44600 q^{74} - 39904 q^{76} + 7546 q^{77} - 14544 q^{78} - 36480 q^{79} + 6561 q^{81} + 31336 q^{82} - 25716 q^{83} + 7056 q^{84} - 64944 q^{86} - 53622 q^{87} - 9856 q^{88} - 100826 q^{89} - 19796 q^{91} + 55552 q^{92} + 57690 q^{93} + 11200 q^{94} - 9216 q^{96} + 89024 q^{97} + 9604 q^{98} - 12474 q^{99}+O(q^{100}) \) 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 −9.00000 16.0000 0 −36.0000 −49.0000 64.0000 81.0000 0
\(n\): 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 1050.6.a.h 1
5.b even 2 1 210.6.a.e 1
5.c odd 4 2 1050.6.g.f 2
15.d odd 2 1 630.6.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.6.a.e 1 5.b even 2 1
630.6.a.p 1 15.d odd 2 1
1050.6.a.h 1 1.a even 1 1 trivial
1050.6.g.f 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1050))\):

\( T_{11} + 154 \) Copy content Toggle raw display
\( T_{13} - 404 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T + 154 \) Copy content Toggle raw display
$13$ \( T - 404 \) Copy content Toggle raw display
$17$ \( T + 2182 \) Copy content Toggle raw display
$19$ \( T + 2494 \) Copy content Toggle raw display
$23$ \( T - 3472 \) Copy content Toggle raw display
$29$ \( T - 5958 \) Copy content Toggle raw display
$31$ \( T + 6410 \) Copy content Toggle raw display
$37$ \( T - 11150 \) Copy content Toggle raw display
$41$ \( T - 7834 \) Copy content Toggle raw display
$43$ \( T + 16236 \) Copy content Toggle raw display
$47$ \( T - 2800 \) Copy content Toggle raw display
$53$ \( T - 30924 \) Copy content Toggle raw display
$59$ \( T + 11536 \) Copy content Toggle raw display
$61$ \( T + 38834 \) Copy content Toggle raw display
$67$ \( T - 48756 \) Copy content Toggle raw display
$71$ \( T + 77882 \) Copy content Toggle raw display
$73$ \( T - 47540 \) Copy content Toggle raw display
$79$ \( T + 36480 \) Copy content Toggle raw display
$83$ \( T + 25716 \) Copy content Toggle raw display
$89$ \( T + 100826 \) Copy content Toggle raw display
$97$ \( T - 89024 \) Copy content Toggle raw display
show more
show less