Properties

Label 42.6.a.a
Level $42$
Weight $6$
Character orbit 42.a
Self dual yes
Analytic conductor $6.736$
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] = [42,6,Mod(1,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, 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("42.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 42.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: \(6.73612043215\)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} - 54 q^{5} + 36 q^{6} + 49 q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} - 54 q^{5} + 36 q^{6} + 49 q^{7} - 64 q^{8} + 81 q^{9} + 216 q^{10} + 216 q^{11} - 144 q^{12} + 998 q^{13} - 196 q^{14} + 486 q^{15} + 256 q^{16} + 1302 q^{17} - 324 q^{18} + 884 q^{19} - 864 q^{20} - 441 q^{21} - 864 q^{22} - 2268 q^{23} + 576 q^{24} - 209 q^{25} - 3992 q^{26} - 729 q^{27} + 784 q^{28} - 1482 q^{29} - 1944 q^{30} + 8360 q^{31} - 1024 q^{32} - 1944 q^{33} - 5208 q^{34} - 2646 q^{35} + 1296 q^{36} - 4714 q^{37} - 3536 q^{38} - 8982 q^{39} + 3456 q^{40} - 9786 q^{41} + 1764 q^{42} + 19436 q^{43} + 3456 q^{44} - 4374 q^{45} + 9072 q^{46} + 22200 q^{47} - 2304 q^{48} + 2401 q^{49} + 836 q^{50} - 11718 q^{51} + 15968 q^{52} + 26790 q^{53} + 2916 q^{54} - 11664 q^{55} - 3136 q^{56} - 7956 q^{57} + 5928 q^{58} + 28092 q^{59} + 7776 q^{60} - 38866 q^{61} - 33440 q^{62} + 3969 q^{63} + 4096 q^{64} - 53892 q^{65} + 7776 q^{66} + 23948 q^{67} + 20832 q^{68} + 20412 q^{69} + 10584 q^{70} - 20628 q^{71} - 5184 q^{72} + 290 q^{73} + 18856 q^{74} + 1881 q^{75} + 14144 q^{76} + 10584 q^{77} + 35928 q^{78} - 99544 q^{79} - 13824 q^{80} + 6561 q^{81} + 39144 q^{82} + 19308 q^{83} - 7056 q^{84} - 70308 q^{85} - 77744 q^{86} + 13338 q^{87} - 13824 q^{88} + 36390 q^{89} + 17496 q^{90} + 48902 q^{91} - 36288 q^{92} - 75240 q^{93} - 88800 q^{94} - 47736 q^{95} + 9216 q^{96} - 79078 q^{97} - 9604 q^{98} + 17496 q^{99}+O(q^{100}) \) 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 −54.0000 36.0000 49.0000 −64.0000 81.0000 216.000
\(n\): 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 42.6.a.a 1
3.b odd 2 1 126.6.a.k 1
4.b odd 2 1 336.6.a.j 1
5.b even 2 1 1050.6.a.n 1
5.c odd 4 2 1050.6.g.o 2
7.b odd 2 1 294.6.a.h 1
7.c even 3 2 294.6.e.r 2
7.d odd 6 2 294.6.e.h 2
12.b even 2 1 1008.6.a.x 1
21.c even 2 1 882.6.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.a 1 1.a even 1 1 trivial
126.6.a.k 1 3.b odd 2 1
294.6.a.h 1 7.b odd 2 1
294.6.e.h 2 7.d odd 6 2
294.6.e.r 2 7.c even 3 2
336.6.a.j 1 4.b odd 2 1
882.6.a.o 1 21.c even 2 1
1008.6.a.x 1 12.b even 2 1
1050.6.a.n 1 5.b even 2 1
1050.6.g.o 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 54 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(42))\). 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
$5$ \( T + 54 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 216 \) Copy content Toggle raw display
$13$ \( T - 998 \) Copy content Toggle raw display
$17$ \( T - 1302 \) Copy content Toggle raw display
$19$ \( T - 884 \) Copy content Toggle raw display
$23$ \( T + 2268 \) Copy content Toggle raw display
$29$ \( T + 1482 \) Copy content Toggle raw display
$31$ \( T - 8360 \) Copy content Toggle raw display
$37$ \( T + 4714 \) Copy content Toggle raw display
$41$ \( T + 9786 \) Copy content Toggle raw display
$43$ \( T - 19436 \) Copy content Toggle raw display
$47$ \( T - 22200 \) Copy content Toggle raw display
$53$ \( T - 26790 \) Copy content Toggle raw display
$59$ \( T - 28092 \) Copy content Toggle raw display
$61$ \( T + 38866 \) Copy content Toggle raw display
$67$ \( T - 23948 \) Copy content Toggle raw display
$71$ \( T + 20628 \) Copy content Toggle raw display
$73$ \( T - 290 \) Copy content Toggle raw display
$79$ \( T + 99544 \) Copy content Toggle raw display
$83$ \( T - 19308 \) Copy content Toggle raw display
$89$ \( T - 36390 \) Copy content Toggle raw display
$97$ \( T + 79078 \) Copy content Toggle raw display
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