Properties

Label 294.6.a.d
Level $294$
Weight $6$
Character orbit 294.a
Self dual yes
Analytic conductor $47.153$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} + 86 q^{5} + 36 q^{6} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} + 86 q^{5} + 36 q^{6} - 64 q^{8} + 81 q^{9} - 344 q^{10} + 34 q^{11} - 144 q^{12} - 3 q^{13} - 774 q^{15} + 256 q^{16} - 1904 q^{17} - 324 q^{18} - 1489 q^{19} + 1376 q^{20} - 136 q^{22} - 224 q^{23} + 576 q^{24} + 4271 q^{25} + 12 q^{26} - 729 q^{27} - 6508 q^{29} + 3096 q^{30} + 1731 q^{31} - 1024 q^{32} - 306 q^{33} + 7616 q^{34} + 1296 q^{36} - 7633 q^{37} + 5956 q^{38} + 27 q^{39} - 5504 q^{40} + 15414 q^{41} + 18491 q^{43} + 544 q^{44} + 6966 q^{45} + 896 q^{46} + 18462 q^{47} - 2304 q^{48} - 17084 q^{50} + 17136 q^{51} - 48 q^{52} - 19956 q^{53} + 2916 q^{54} + 2924 q^{55} + 13401 q^{57} + 26032 q^{58} - 31828 q^{59} - 12384 q^{60} - 57654 q^{61} - 6924 q^{62} + 4096 q^{64} - 258 q^{65} + 1224 q^{66} - 60563 q^{67} - 30464 q^{68} + 2016 q^{69} - 44834 q^{71} - 5184 q^{72} + 20821 q^{73} + 30532 q^{74} - 38439 q^{75} - 23824 q^{76} - 108 q^{78} - 30531 q^{79} + 22016 q^{80} + 6561 q^{81} - 61656 q^{82} + 110602 q^{83} - 163744 q^{85} - 73964 q^{86} + 58572 q^{87} - 2176 q^{88} - 58992 q^{89} - 27864 q^{90} - 3584 q^{92} - 15579 q^{93} - 73848 q^{94} - 128054 q^{95} + 9216 q^{96} - 119846 q^{97} + 2754 q^{99}+O(q^{100}) \) 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 86.0000 36.0000 0 −64.0000 81.0000 −344.000
\(n\): 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.d 1
3.b odd 2 1 882.6.a.m 1
7.b odd 2 1 294.6.a.e 1
7.c even 3 2 42.6.e.b 2
7.d odd 6 2 294.6.e.m 2
21.c even 2 1 882.6.a.y 1
21.h odd 6 2 126.6.g.b 2
28.g odd 6 2 336.6.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.b 2 7.c even 3 2
126.6.g.b 2 21.h odd 6 2
294.6.a.d 1 1.a even 1 1 trivial
294.6.a.e 1 7.b odd 2 1
294.6.e.m 2 7.d odd 6 2
336.6.q.a 2 28.g odd 6 2
882.6.a.m 1 3.b odd 2 1
882.6.a.y 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(294))\):

\( T_{5} - 86 \) Copy content Toggle raw display
\( T_{11} - 34 \) 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
$5$ \( T - 86 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 34 \) Copy content Toggle raw display
$13$ \( T + 3 \) Copy content Toggle raw display
$17$ \( T + 1904 \) Copy content Toggle raw display
$19$ \( T + 1489 \) Copy content Toggle raw display
$23$ \( T + 224 \) Copy content Toggle raw display
$29$ \( T + 6508 \) Copy content Toggle raw display
$31$ \( T - 1731 \) Copy content Toggle raw display
$37$ \( T + 7633 \) Copy content Toggle raw display
$41$ \( T - 15414 \) Copy content Toggle raw display
$43$ \( T - 18491 \) Copy content Toggle raw display
$47$ \( T - 18462 \) Copy content Toggle raw display
$53$ \( T + 19956 \) Copy content Toggle raw display
$59$ \( T + 31828 \) Copy content Toggle raw display
$61$ \( T + 57654 \) Copy content Toggle raw display
$67$ \( T + 60563 \) Copy content Toggle raw display
$71$ \( T + 44834 \) Copy content Toggle raw display
$73$ \( T - 20821 \) Copy content Toggle raw display
$79$ \( T + 30531 \) Copy content Toggle raw display
$83$ \( T - 110602 \) Copy content Toggle raw display
$89$ \( T + 58992 \) Copy content Toggle raw display
$97$ \( T + 119846 \) Copy content Toggle raw display
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