Properties

Label 21.6.a.a
Level $21$
Weight $6$
Character orbit 21.a
Self dual yes
Analytic conductor $3.368$
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] = [21,6,Mod(1,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, 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("21.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.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: \(3.36806021607\)
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)
 
\(f(q)\) \(=\) \( q - 6 q^{2} - 9 q^{3} + 4 q^{4} + 78 q^{5} + 54 q^{6} + 49 q^{7} + 168 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{2} - 9 q^{3} + 4 q^{4} + 78 q^{5} + 54 q^{6} + 49 q^{7} + 168 q^{8} + 81 q^{9} - 468 q^{10} + 444 q^{11} - 36 q^{12} - 442 q^{13} - 294 q^{14} - 702 q^{15} - 1136 q^{16} - 126 q^{17} - 486 q^{18} + 2684 q^{19} + 312 q^{20} - 441 q^{21} - 2664 q^{22} + 4200 q^{23} - 1512 q^{24} + 2959 q^{25} + 2652 q^{26} - 729 q^{27} + 196 q^{28} - 5442 q^{29} + 4212 q^{30} + 80 q^{31} + 1440 q^{32} - 3996 q^{33} + 756 q^{34} + 3822 q^{35} + 324 q^{36} - 5434 q^{37} - 16104 q^{38} + 3978 q^{39} + 13104 q^{40} + 7962 q^{41} + 2646 q^{42} - 11524 q^{43} + 1776 q^{44} + 6318 q^{45} - 25200 q^{46} - 13920 q^{47} + 10224 q^{48} + 2401 q^{49} - 17754 q^{50} + 1134 q^{51} - 1768 q^{52} - 9594 q^{53} + 4374 q^{54} + 34632 q^{55} + 8232 q^{56} - 24156 q^{57} + 32652 q^{58} + 27492 q^{59} - 2808 q^{60} + 49478 q^{61} - 480 q^{62} + 3969 q^{63} + 27712 q^{64} - 34476 q^{65} + 23976 q^{66} - 59356 q^{67} - 504 q^{68} - 37800 q^{69} - 22932 q^{70} + 32040 q^{71} + 13608 q^{72} - 61846 q^{73} + 32604 q^{74} - 26631 q^{75} + 10736 q^{76} + 21756 q^{77} - 23868 q^{78} - 65776 q^{79} - 88608 q^{80} + 6561 q^{81} - 47772 q^{82} + 40188 q^{83} - 1764 q^{84} - 9828 q^{85} + 69144 q^{86} + 48978 q^{87} + 74592 q^{88} - 7974 q^{89} - 37908 q^{90} - 21658 q^{91} + 16800 q^{92} - 720 q^{93} + 83520 q^{94} + 209352 q^{95} - 12960 q^{96} - 143662 q^{97} - 14406 q^{98} + 35964 q^{99}+O(q^{100}) \) 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
−6.00000 −9.00000 4.00000 78.0000 54.0000 49.0000 168.000 81.0000 −468.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 21.6.a.a 1
3.b odd 2 1 63.6.a.d 1
4.b odd 2 1 336.6.a.r 1
5.b even 2 1 525.6.a.d 1
5.c odd 4 2 525.6.d.b 2
7.b odd 2 1 147.6.a.b 1
7.c even 3 2 147.6.e.j 2
7.d odd 6 2 147.6.e.i 2
12.b even 2 1 1008.6.a.c 1
21.c even 2 1 441.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 1.a even 1 1 trivial
63.6.a.d 1 3.b odd 2 1
147.6.a.b 1 7.b odd 2 1
147.6.e.i 2 7.d odd 6 2
147.6.e.j 2 7.c even 3 2
336.6.a.r 1 4.b odd 2 1
441.6.a.j 1 21.c even 2 1
525.6.a.d 1 5.b even 2 1
525.6.d.b 2 5.c odd 4 2
1008.6.a.c 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 6 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T - 78 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 444 \) Copy content Toggle raw display
$13$ \( T + 442 \) Copy content Toggle raw display
$17$ \( T + 126 \) Copy content Toggle raw display
$19$ \( T - 2684 \) Copy content Toggle raw display
$23$ \( T - 4200 \) Copy content Toggle raw display
$29$ \( T + 5442 \) Copy content Toggle raw display
$31$ \( T - 80 \) Copy content Toggle raw display
$37$ \( T + 5434 \) Copy content Toggle raw display
$41$ \( T - 7962 \) Copy content Toggle raw display
$43$ \( T + 11524 \) Copy content Toggle raw display
$47$ \( T + 13920 \) Copy content Toggle raw display
$53$ \( T + 9594 \) Copy content Toggle raw display
$59$ \( T - 27492 \) Copy content Toggle raw display
$61$ \( T - 49478 \) Copy content Toggle raw display
$67$ \( T + 59356 \) Copy content Toggle raw display
$71$ \( T - 32040 \) Copy content Toggle raw display
$73$ \( T + 61846 \) Copy content Toggle raw display
$79$ \( T + 65776 \) Copy content Toggle raw display
$83$ \( T - 40188 \) Copy content Toggle raw display
$89$ \( T + 7974 \) Copy content Toggle raw display
$97$ \( T + 143662 \) Copy content Toggle raw display
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