Properties

Label 294.6.a.a
Level $294$
Weight $6$
Character orbit 294.a
Self dual yes
Analytic conductor $47.153$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} - 26 q^{5} + 36 q^{6} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} - 26 q^{5} + 36 q^{6} - 64 q^{8} + 81 q^{9} + 104 q^{10} - 358 q^{11} - 144 q^{12} - 332 q^{13} + 234 q^{15} + 256 q^{16} - 126 q^{17} - 324 q^{18} + 2200 q^{19} - 416 q^{20} + 1432 q^{22} - 2142 q^{23} + 576 q^{24} - 2449 q^{25} + 1328 q^{26} - 729 q^{27} - 3610 q^{29} - 936 q^{30} - 5668 q^{31} - 1024 q^{32} + 3222 q^{33} + 504 q^{34} + 1296 q^{36} - 2922 q^{37} - 8800 q^{38} + 2988 q^{39} + 1664 q^{40} + 2142 q^{41} + 6388 q^{43} - 5728 q^{44} - 2106 q^{45} + 8568 q^{46} + 6520 q^{47} - 2304 q^{48} + 9796 q^{50} + 1134 q^{51} - 5312 q^{52} - 10702 q^{53} + 2916 q^{54} + 9308 q^{55} - 19800 q^{57} + 14440 q^{58} - 42524 q^{59} + 3744 q^{60} + 44840 q^{61} + 22672 q^{62} + 4096 q^{64} + 8632 q^{65} - 12888 q^{66} - 1448 q^{67} - 2016 q^{68} + 19278 q^{69} - 4402 q^{71} - 5184 q^{72} - 20500 q^{73} + 11688 q^{74} + 22041 q^{75} + 35200 q^{76} - 11952 q^{78} + 65236 q^{79} - 6656 q^{80} + 6561 q^{81} - 8568 q^{82} + 102804 q^{83} + 3276 q^{85} - 25552 q^{86} + 32490 q^{87} + 22912 q^{88} + 128006 q^{89} + 8424 q^{90} - 34272 q^{92} + 51012 q^{93} - 26080 q^{94} - 57200 q^{95} + 9216 q^{96} + 113324 q^{97} - 28998 q^{99}+O(q^{100}) \) 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 −26.0000 36.0000 0 −64.0000 81.0000 104.000
\(n\): 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.a 1
3.b odd 2 1 882.6.a.t 1
7.b odd 2 1 294.6.a.g yes 1
7.c even 3 2 294.6.e.q 2
7.d odd 6 2 294.6.e.j 2
21.c even 2 1 882.6.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.6.a.a 1 1.a even 1 1 trivial
294.6.a.g yes 1 7.b odd 2 1
294.6.e.j 2 7.d odd 6 2
294.6.e.q 2 7.c even 3 2
882.6.a.p 1 21.c even 2 1
882.6.a.t 1 3.b odd 2 1

Hecke kernels

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

\( T_{5} + 26 \) Copy content Toggle raw display
\( T_{11} + 358 \) 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
$5$ \( T + 26 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 358 \) Copy content Toggle raw display
$13$ \( T + 332 \) Copy content Toggle raw display
$17$ \( T + 126 \) Copy content Toggle raw display
$19$ \( T - 2200 \) Copy content Toggle raw display
$23$ \( T + 2142 \) Copy content Toggle raw display
$29$ \( T + 3610 \) Copy content Toggle raw display
$31$ \( T + 5668 \) Copy content Toggle raw display
$37$ \( T + 2922 \) Copy content Toggle raw display
$41$ \( T - 2142 \) Copy content Toggle raw display
$43$ \( T - 6388 \) Copy content Toggle raw display
$47$ \( T - 6520 \) Copy content Toggle raw display
$53$ \( T + 10702 \) Copy content Toggle raw display
$59$ \( T + 42524 \) Copy content Toggle raw display
$61$ \( T - 44840 \) Copy content Toggle raw display
$67$ \( T + 1448 \) Copy content Toggle raw display
$71$ \( T + 4402 \) Copy content Toggle raw display
$73$ \( T + 20500 \) Copy content Toggle raw display
$79$ \( T - 65236 \) Copy content Toggle raw display
$83$ \( T - 102804 \) Copy content Toggle raw display
$89$ \( T - 128006 \) Copy content Toggle raw display
$97$ \( T - 113324 \) Copy content Toggle raw display
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