Properties

Label 42.6.a.c
Level $42$
Weight $6$
Character orbit 42.a
Self dual yes
Analytic conductor $6.736$
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] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} - 72 q^{5} - 36 q^{6} + 49 q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} - 72 q^{5} - 36 q^{6} + 49 q^{7} - 64 q^{8} + 81 q^{9} + 288 q^{10} - 414 q^{11} + 144 q^{12} - 1054 q^{13} - 196 q^{14} - 648 q^{15} + 256 q^{16} - 1848 q^{17} - 324 q^{18} + 236 q^{19} - 1152 q^{20} + 441 q^{21} + 1656 q^{22} + 2898 q^{23} - 576 q^{24} + 2059 q^{25} + 4216 q^{26} + 729 q^{27} + 784 q^{28} - 6522 q^{29} + 2592 q^{30} + 6200 q^{31} - 1024 q^{32} - 3726 q^{33} + 7392 q^{34} - 3528 q^{35} + 1296 q^{36} + 9650 q^{37} - 944 q^{38} - 9486 q^{39} + 4608 q^{40} + 8484 q^{41} - 1764 q^{42} - 10804 q^{43} - 6624 q^{44} - 5832 q^{45} - 11592 q^{46} + 60 q^{47} + 2304 q^{48} + 2401 q^{49} - 8236 q^{50} - 16632 q^{51} - 16864 q^{52} + 22506 q^{53} - 2916 q^{54} + 29808 q^{55} - 3136 q^{56} + 2124 q^{57} + 26088 q^{58} - 28176 q^{59} - 10368 q^{60} - 35194 q^{61} - 24800 q^{62} + 3969 q^{63} + 4096 q^{64} + 75888 q^{65} + 14904 q^{66} - 28216 q^{67} - 29568 q^{68} + 26082 q^{69} + 14112 q^{70} - 6642 q^{71} - 5184 q^{72} - 52090 q^{73} - 38600 q^{74} + 18531 q^{75} + 3776 q^{76} - 20286 q^{77} + 37944 q^{78} + 43340 q^{79} - 18432 q^{80} + 6561 q^{81} - 33936 q^{82} + 25716 q^{83} + 7056 q^{84} + 133056 q^{85} + 43216 q^{86} - 58698 q^{87} + 26496 q^{88} + 98724 q^{89} + 23328 q^{90} - 51646 q^{91} + 46368 q^{92} + 55800 q^{93} - 240 q^{94} - 16992 q^{95} - 9216 q^{96} - 148954 q^{97} - 9604 q^{98} - 33534 q^{99}+O(q^{100}) \) 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 −72.0000 −36.0000 49.0000 −64.0000 81.0000 288.000
\(n\): 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.c 1
3.b odd 2 1 126.6.a.l 1
4.b odd 2 1 336.6.a.b 1
5.b even 2 1 1050.6.a.g 1
5.c odd 4 2 1050.6.g.b 2
7.b odd 2 1 294.6.a.c 1
7.c even 3 2 294.6.e.l 2
7.d odd 6 2 294.6.e.n 2
12.b even 2 1 1008.6.a.ba 1
21.c even 2 1 882.6.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.c 1 1.a even 1 1 trivial
126.6.a.l 1 3.b odd 2 1
294.6.a.c 1 7.b odd 2 1
294.6.e.l 2 7.c even 3 2
294.6.e.n 2 7.d odd 6 2
336.6.a.b 1 4.b odd 2 1
882.6.a.n 1 21.c even 2 1
1008.6.a.ba 1 12.b even 2 1
1050.6.a.g 1 5.b even 2 1
1050.6.g.b 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 72 \) 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
$5$ \( T + 72 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T + 414 \) Copy content Toggle raw display
$13$ \( T + 1054 \) Copy content Toggle raw display
$17$ \( T + 1848 \) Copy content Toggle raw display
$19$ \( T - 236 \) Copy content Toggle raw display
$23$ \( T - 2898 \) Copy content Toggle raw display
$29$ \( T + 6522 \) Copy content Toggle raw display
$31$ \( T - 6200 \) Copy content Toggle raw display
$37$ \( T - 9650 \) Copy content Toggle raw display
$41$ \( T - 8484 \) Copy content Toggle raw display
$43$ \( T + 10804 \) Copy content Toggle raw display
$47$ \( T - 60 \) Copy content Toggle raw display
$53$ \( T - 22506 \) Copy content Toggle raw display
$59$ \( T + 28176 \) Copy content Toggle raw display
$61$ \( T + 35194 \) Copy content Toggle raw display
$67$ \( T + 28216 \) Copy content Toggle raw display
$71$ \( T + 6642 \) Copy content Toggle raw display
$73$ \( T + 52090 \) Copy content Toggle raw display
$79$ \( T - 43340 \) Copy content Toggle raw display
$83$ \( T - 25716 \) Copy content Toggle raw display
$89$ \( T - 98724 \) Copy content Toggle raw display
$97$ \( T + 148954 \) Copy content Toggle raw display
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