Properties

Label 21.6.a.d
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)
 
\(f(q)\) \(=\) \( q + 10 q^{2} + 9 q^{3} + 68 q^{4} - 106 q^{5} + 90 q^{6} - 49 q^{7} + 360 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 10 q^{2} + 9 q^{3} + 68 q^{4} - 106 q^{5} + 90 q^{6} - 49 q^{7} + 360 q^{8} + 81 q^{9} - 1060 q^{10} + 92 q^{11} + 612 q^{12} + 670 q^{13} - 490 q^{14} - 954 q^{15} + 1424 q^{16} - 222 q^{17} + 810 q^{18} - 908 q^{19} - 7208 q^{20} - 441 q^{21} + 920 q^{22} - 1176 q^{23} + 3240 q^{24} + 8111 q^{25} + 6700 q^{26} + 729 q^{27} - 3332 q^{28} + 1118 q^{29} - 9540 q^{30} + 3696 q^{31} + 2720 q^{32} + 828 q^{33} - 2220 q^{34} + 5194 q^{35} + 5508 q^{36} + 4182 q^{37} - 9080 q^{38} + 6030 q^{39} - 38160 q^{40} - 6662 q^{41} - 4410 q^{42} - 3700 q^{43} + 6256 q^{44} - 8586 q^{45} - 11760 q^{46} - 7056 q^{47} + 12816 q^{48} + 2401 q^{49} + 81110 q^{50} - 1998 q^{51} + 45560 q^{52} - 37578 q^{53} + 7290 q^{54} - 9752 q^{55} - 17640 q^{56} - 8172 q^{57} + 11180 q^{58} + 32700 q^{59} - 64872 q^{60} - 10802 q^{61} + 36960 q^{62} - 3969 q^{63} - 18368 q^{64} - 71020 q^{65} + 8280 q^{66} + 64996 q^{67} - 15096 q^{68} - 10584 q^{69} + 51940 q^{70} - 61320 q^{71} + 29160 q^{72} + 38922 q^{73} + 41820 q^{74} + 72999 q^{75} - 61744 q^{76} - 4508 q^{77} + 60300 q^{78} - 88096 q^{79} - 150944 q^{80} + 6561 q^{81} - 66620 q^{82} + 71892 q^{83} - 29988 q^{84} + 23532 q^{85} - 37000 q^{86} + 10062 q^{87} + 33120 q^{88} + 111818 q^{89} - 85860 q^{90} - 32830 q^{91} - 79968 q^{92} + 33264 q^{93} - 70560 q^{94} + 96248 q^{95} + 24480 q^{96} - 150846 q^{97} + 24010 q^{98} + 7452 q^{99}+O(q^{100}) \) 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
10.0000 9.00000 68.0000 −106.000 90.0000 −49.0000 360.000 81.0000 −1060.00
\(n\): 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.d 1
3.b odd 2 1 63.6.a.a 1
4.b odd 2 1 336.6.a.a 1
5.b even 2 1 525.6.a.a 1
5.c odd 4 2 525.6.d.a 2
7.b odd 2 1 147.6.a.g 1
7.c even 3 2 147.6.e.a 2
7.d odd 6 2 147.6.e.b 2
12.b even 2 1 1008.6.a.bc 1
21.c even 2 1 441.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.d 1 1.a even 1 1 trivial
63.6.a.a 1 3.b odd 2 1
147.6.a.g 1 7.b odd 2 1
147.6.e.a 2 7.c even 3 2
147.6.e.b 2 7.d odd 6 2
336.6.a.a 1 4.b odd 2 1
441.6.a.b 1 21.c even 2 1
525.6.a.a 1 5.b even 2 1
525.6.d.a 2 5.c odd 4 2
1008.6.a.bc 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 10 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 10 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T + 106 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T - 92 \) Copy content Toggle raw display
$13$ \( T - 670 \) Copy content Toggle raw display
$17$ \( T + 222 \) Copy content Toggle raw display
$19$ \( T + 908 \) Copy content Toggle raw display
$23$ \( T + 1176 \) Copy content Toggle raw display
$29$ \( T - 1118 \) Copy content Toggle raw display
$31$ \( T - 3696 \) Copy content Toggle raw display
$37$ \( T - 4182 \) Copy content Toggle raw display
$41$ \( T + 6662 \) Copy content Toggle raw display
$43$ \( T + 3700 \) Copy content Toggle raw display
$47$ \( T + 7056 \) Copy content Toggle raw display
$53$ \( T + 37578 \) Copy content Toggle raw display
$59$ \( T - 32700 \) Copy content Toggle raw display
$61$ \( T + 10802 \) Copy content Toggle raw display
$67$ \( T - 64996 \) Copy content Toggle raw display
$71$ \( T + 61320 \) Copy content Toggle raw display
$73$ \( T - 38922 \) Copy content Toggle raw display
$79$ \( T + 88096 \) Copy content Toggle raw display
$83$ \( T - 71892 \) Copy content Toggle raw display
$89$ \( T - 111818 \) Copy content Toggle raw display
$97$ \( T + 150846 \) Copy content Toggle raw display
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