Properties

Label 42.6.a.f
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + 24 q^{5} + 36 q^{6} + 49 q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + 24 q^{5} + 36 q^{6} + 49 q^{7} + 64 q^{8} + 81 q^{9} + 96 q^{10} + 66 q^{11} + 144 q^{12} + 98 q^{13} + 196 q^{14} + 216 q^{15} + 256 q^{16} - 216 q^{17} + 324 q^{18} - 340 q^{19} + 384 q^{20} + 441 q^{21} + 264 q^{22} - 1038 q^{23} + 576 q^{24} - 2549 q^{25} + 392 q^{26} + 729 q^{27} + 784 q^{28} - 2490 q^{29} + 864 q^{30} - 7048 q^{31} + 1024 q^{32} + 594 q^{33} - 864 q^{34} + 1176 q^{35} + 1296 q^{36} - 12238 q^{37} - 1360 q^{38} + 882 q^{39} + 1536 q^{40} + 6468 q^{41} + 1764 q^{42} - 15412 q^{43} + 1056 q^{44} + 1944 q^{45} - 4152 q^{46} + 20604 q^{47} + 2304 q^{48} + 2401 q^{49} - 10196 q^{50} - 1944 q^{51} + 1568 q^{52} + 32490 q^{53} + 2916 q^{54} + 1584 q^{55} + 3136 q^{56} - 3060 q^{57} - 9960 q^{58} + 34224 q^{59} + 3456 q^{60} + 35654 q^{61} - 28192 q^{62} + 3969 q^{63} + 4096 q^{64} + 2352 q^{65} + 2376 q^{66} + 12680 q^{67} - 3456 q^{68} - 9342 q^{69} + 4704 q^{70} - 42642 q^{71} + 5184 q^{72} + 33734 q^{73} - 48952 q^{74} - 22941 q^{75} - 5440 q^{76} + 3234 q^{77} + 3528 q^{78} - 85108 q^{79} + 6144 q^{80} + 6561 q^{81} + 25872 q^{82} - 106764 q^{83} + 7056 q^{84} - 5184 q^{85} - 61648 q^{86} - 22410 q^{87} + 4224 q^{88} + 34884 q^{89} + 7776 q^{90} + 4802 q^{91} - 16608 q^{92} - 63432 q^{93} + 82416 q^{94} - 8160 q^{95} + 9216 q^{96} + 18662 q^{97} + 9604 q^{98} + 5346 q^{99}+O(q^{100}) \) 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 24.0000 36.0000 49.0000 64.0000 81.0000 96.0000
\(n\): 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.f 1
3.b odd 2 1 126.6.a.b 1
4.b odd 2 1 336.6.a.g 1
5.b even 2 1 1050.6.a.a 1
5.c odd 4 2 1050.6.g.m 2
7.b odd 2 1 294.6.a.i 1
7.c even 3 2 294.6.e.b 2
7.d odd 6 2 294.6.e.f 2
12.b even 2 1 1008.6.a.k 1
21.c even 2 1 882.6.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.f 1 1.a even 1 1 trivial
126.6.a.b 1 3.b odd 2 1
294.6.a.i 1 7.b odd 2 1
294.6.e.b 2 7.c even 3 2
294.6.e.f 2 7.d odd 6 2
336.6.a.g 1 4.b odd 2 1
882.6.a.i 1 21.c even 2 1
1008.6.a.k 1 12.b even 2 1
1050.6.a.a 1 5.b even 2 1
1050.6.g.m 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 24 \) 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
$5$ \( T - 24 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 66 \) Copy content Toggle raw display
$13$ \( T - 98 \) Copy content Toggle raw display
$17$ \( T + 216 \) Copy content Toggle raw display
$19$ \( T + 340 \) Copy content Toggle raw display
$23$ \( T + 1038 \) Copy content Toggle raw display
$29$ \( T + 2490 \) Copy content Toggle raw display
$31$ \( T + 7048 \) Copy content Toggle raw display
$37$ \( T + 12238 \) Copy content Toggle raw display
$41$ \( T - 6468 \) Copy content Toggle raw display
$43$ \( T + 15412 \) Copy content Toggle raw display
$47$ \( T - 20604 \) Copy content Toggle raw display
$53$ \( T - 32490 \) Copy content Toggle raw display
$59$ \( T - 34224 \) Copy content Toggle raw display
$61$ \( T - 35654 \) Copy content Toggle raw display
$67$ \( T - 12680 \) Copy content Toggle raw display
$71$ \( T + 42642 \) Copy content Toggle raw display
$73$ \( T - 33734 \) Copy content Toggle raw display
$79$ \( T + 85108 \) Copy content Toggle raw display
$83$ \( T + 106764 \) Copy content Toggle raw display
$89$ \( T - 34884 \) Copy content Toggle raw display
$97$ \( T - 18662 \) Copy content Toggle raw display
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