Properties

Label 1050.6.a.n
Level $1050$
Weight $6$
Character orbit 1050.a
Self dual yes
Analytic conductor $168.403$
Analytic rank $1$
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] = [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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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 \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + 36 q^{6} - 49 q^{7} + 64 q^{8} + 81 q^{9} + 216 q^{11} + 144 q^{12} - 998 q^{13} - 196 q^{14} + 256 q^{16} - 1302 q^{17} + 324 q^{18} + 884 q^{19} - 441 q^{21} + 864 q^{22} + 2268 q^{23} + 576 q^{24} - 3992 q^{26} + 729 q^{27} - 784 q^{28} - 1482 q^{29} + 8360 q^{31} + 1024 q^{32} + 1944 q^{33} - 5208 q^{34} + 1296 q^{36} + 4714 q^{37} + 3536 q^{38} - 8982 q^{39} - 9786 q^{41} - 1764 q^{42} - 19436 q^{43} + 3456 q^{44} + 9072 q^{46} - 22200 q^{47} + 2304 q^{48} + 2401 q^{49} - 11718 q^{51} - 15968 q^{52} - 26790 q^{53} + 2916 q^{54} - 3136 q^{56} + 7956 q^{57} - 5928 q^{58} + 28092 q^{59} - 38866 q^{61} + 33440 q^{62} - 3969 q^{63} + 4096 q^{64} + 7776 q^{66} - 23948 q^{67} - 20832 q^{68} + 20412 q^{69} - 20628 q^{71} + 5184 q^{72} - 290 q^{73} + 18856 q^{74} + 14144 q^{76} - 10584 q^{77} - 35928 q^{78} - 99544 q^{79} + 6561 q^{81} - 39144 q^{82} - 19308 q^{83} - 7056 q^{84} - 77744 q^{86} - 13338 q^{87} + 13824 q^{88} + 36390 q^{89} + 48902 q^{91} + 36288 q^{92} + 75240 q^{93} - 88800 q^{94} + 9216 q^{96} + 79078 q^{97} + 9604 q^{98} + 17496 q^{99}+O(q^{100}) \) 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.n 1
5.b even 2 1 42.6.a.a 1
5.c odd 4 2 1050.6.g.o 2
15.d odd 2 1 126.6.a.k 1
20.d odd 2 1 336.6.a.j 1
35.c odd 2 1 294.6.a.h 1
35.i odd 6 2 294.6.e.h 2
35.j even 6 2 294.6.e.r 2
60.h even 2 1 1008.6.a.x 1
105.g even 2 1 882.6.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.a 1 5.b even 2 1
126.6.a.k 1 15.d odd 2 1
294.6.a.h 1 35.c odd 2 1
294.6.e.h 2 35.i odd 6 2
294.6.e.r 2 35.j even 6 2
336.6.a.j 1 20.d odd 2 1
882.6.a.o 1 105.g even 2 1
1008.6.a.x 1 60.h even 2 1
1050.6.a.n 1 1.a even 1 1 trivial
1050.6.g.o 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1050))\):

\( T_{11} - 216 \) Copy content Toggle raw display
\( T_{13} + 998 \) 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
$11$ \( T - 216 \) Copy content Toggle raw display
$13$ \( T + 998 \) Copy content Toggle raw display
$17$ \( T + 1302 \) Copy content Toggle raw display
$19$ \( T - 884 \) Copy content Toggle raw display
$23$ \( T - 2268 \) Copy content Toggle raw display
$29$ \( T + 1482 \) Copy content Toggle raw display
$31$ \( T - 8360 \) Copy content Toggle raw display
$37$ \( T - 4714 \) Copy content Toggle raw display
$41$ \( T + 9786 \) Copy content Toggle raw display
$43$ \( T + 19436 \) Copy content Toggle raw display
$47$ \( T + 22200 \) Copy content Toggle raw display
$53$ \( T + 26790 \) Copy content Toggle raw display
$59$ \( T - 28092 \) Copy content Toggle raw display
$61$ \( T + 38866 \) Copy content Toggle raw display
$67$ \( T + 23948 \) Copy content Toggle raw display
$71$ \( T + 20628 \) Copy content Toggle raw display
$73$ \( T + 290 \) Copy content Toggle raw display
$79$ \( T + 99544 \) Copy content Toggle raw display
$83$ \( T + 19308 \) Copy content Toggle raw display
$89$ \( T - 36390 \) Copy content Toggle raw display
$97$ \( T - 79078 \) Copy content Toggle raw display
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