Properties

Label 294.6.a.j
Level $294$
Weight $6$
Character orbit 294.a
Self dual yes
Analytic conductor $47.153$
Analytic rank $1$
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] = [294,6,Mod(1,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("294.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 294.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: \(47.1528430250\)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} + 6 q^{5} - 36 q^{6} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} + 6 q^{5} - 36 q^{6} + 64 q^{8} + 81 q^{9} + 24 q^{10} - 666 q^{11} - 144 q^{12} + 559 q^{13} - 54 q^{15} + 256 q^{16} + 1740 q^{17} + 324 q^{18} - 1157 q^{19} + 96 q^{20} - 2664 q^{22} - 3468 q^{23} - 576 q^{24} - 3089 q^{25} + 2236 q^{26} - 729 q^{27} + 3372 q^{29} - 216 q^{30} - 6293 q^{31} + 1024 q^{32} + 5994 q^{33} + 6960 q^{34} + 1296 q^{36} + 3131 q^{37} - 4628 q^{38} - 5031 q^{39} + 384 q^{40} + 4866 q^{41} - 11407 q^{43} - 10656 q^{44} + 486 q^{45} - 13872 q^{46} - 2310 q^{47} - 2304 q^{48} - 12356 q^{50} - 15660 q^{51} + 8944 q^{52} - 28296 q^{53} - 2916 q^{54} - 3996 q^{55} + 10413 q^{57} + 13488 q^{58} - 20544 q^{59} - 864 q^{60} + 4630 q^{61} - 25172 q^{62} + 4096 q^{64} + 3354 q^{65} + 23976 q^{66} - 18745 q^{67} + 27840 q^{68} + 31212 q^{69} - 38226 q^{71} + 5184 q^{72} - 70589 q^{73} + 12524 q^{74} + 27801 q^{75} - 18512 q^{76} - 20124 q^{78} - 62293 q^{79} + 1536 q^{80} + 6561 q^{81} + 19464 q^{82} - 79818 q^{83} + 10440 q^{85} - 45628 q^{86} - 30348 q^{87} - 42624 q^{88} + 18120 q^{89} + 1944 q^{90} - 55488 q^{92} + 56637 q^{93} - 9240 q^{94} - 6942 q^{95} - 9216 q^{96} - 124754 q^{97} - 53946 q^{99}+O(q^{100}) \) 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 6.00000 −36.0000 0 64.0000 81.0000 24.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 294.6.a.j 1
3.b odd 2 1 882.6.a.e 1
7.b odd 2 1 294.6.a.l 1
7.c even 3 2 294.6.e.e 2
7.d odd 6 2 42.6.e.a 2
21.c even 2 1 882.6.a.f 1
21.g even 6 2 126.6.g.c 2
28.f even 6 2 336.6.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.a 2 7.d odd 6 2
126.6.g.c 2 21.g even 6 2
294.6.a.j 1 1.a even 1 1 trivial
294.6.a.l 1 7.b odd 2 1
294.6.e.e 2 7.c even 3 2
336.6.q.c 2 28.f even 6 2
882.6.a.e 1 3.b odd 2 1
882.6.a.f 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(294))\):

\( T_{5} - 6 \) Copy content Toggle raw display
\( T_{11} + 666 \) 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
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 666 \) Copy content Toggle raw display
$13$ \( T - 559 \) Copy content Toggle raw display
$17$ \( T - 1740 \) Copy content Toggle raw display
$19$ \( T + 1157 \) Copy content Toggle raw display
$23$ \( T + 3468 \) Copy content Toggle raw display
$29$ \( T - 3372 \) Copy content Toggle raw display
$31$ \( T + 6293 \) Copy content Toggle raw display
$37$ \( T - 3131 \) Copy content Toggle raw display
$41$ \( T - 4866 \) Copy content Toggle raw display
$43$ \( T + 11407 \) Copy content Toggle raw display
$47$ \( T + 2310 \) Copy content Toggle raw display
$53$ \( T + 28296 \) Copy content Toggle raw display
$59$ \( T + 20544 \) Copy content Toggle raw display
$61$ \( T - 4630 \) Copy content Toggle raw display
$67$ \( T + 18745 \) Copy content Toggle raw display
$71$ \( T + 38226 \) Copy content Toggle raw display
$73$ \( T + 70589 \) Copy content Toggle raw display
$79$ \( T + 62293 \) Copy content Toggle raw display
$83$ \( T + 79818 \) Copy content Toggle raw display
$89$ \( T - 18120 \) Copy content Toggle raw display
$97$ \( T + 124754 \) Copy content Toggle raw display
show more
show less