Properties

Label 42.8.a.f
Level $42$
Weight $8$
Character orbit 42.a
Self dual yes
Analytic conductor $13.120$
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] = [42,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("42.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(13.1201710703\)
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)
 
\(f(q)\) \(=\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + 470 q^{5} + 216 q^{6} - 343 q^{7} + 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + 470 q^{5} + 216 q^{6} - 343 q^{7} + 512 q^{8} + 729 q^{9} + 3760 q^{10} - 7268 q^{11} + 1728 q^{12} + 11382 q^{13} - 2744 q^{14} + 12690 q^{15} + 4096 q^{16} + 20842 q^{17} + 5832 q^{18} - 13684 q^{19} + 30080 q^{20} - 9261 q^{21} - 58144 q^{22} - 46736 q^{23} + 13824 q^{24} + 142775 q^{25} + 91056 q^{26} + 19683 q^{27} - 21952 q^{28} + 2774 q^{29} + 101520 q^{30} - 11232 q^{31} + 32768 q^{32} - 196236 q^{33} + 166736 q^{34} - 161210 q^{35} + 46656 q^{36} - 239698 q^{37} - 109472 q^{38} + 307314 q^{39} + 240640 q^{40} + 52962 q^{41} - 74088 q^{42} - 874732 q^{43} - 465152 q^{44} + 342630 q^{45} - 373888 q^{46} - 451296 q^{47} + 110592 q^{48} + 117649 q^{49} + 1142200 q^{50} + 562734 q^{51} + 728448 q^{52} - 1943538 q^{53} + 157464 q^{54} - 3415960 q^{55} - 175616 q^{56} - 369468 q^{57} + 22192 q^{58} + 1563908 q^{59} + 812160 q^{60} - 3183690 q^{61} - 89856 q^{62} - 250047 q^{63} + 262144 q^{64} + 5349540 q^{65} - 1569888 q^{66} + 2128684 q^{67} + 1333888 q^{68} - 1261872 q^{69} - 1289680 q^{70} + 5694544 q^{71} + 373248 q^{72} - 2253878 q^{73} - 1917584 q^{74} + 3854925 q^{75} - 875776 q^{76} + 2492924 q^{77} + 2458512 q^{78} + 132912 q^{79} + 1925120 q^{80} + 531441 q^{81} + 423696 q^{82} + 6950524 q^{83} - 592704 q^{84} + 9795740 q^{85} - 6997856 q^{86} + 74898 q^{87} - 3721216 q^{88} - 10626510 q^{89} + 2741040 q^{90} - 3904026 q^{91} - 2991104 q^{92} - 303264 q^{93} - 3610368 q^{94} - 6431480 q^{95} + 884736 q^{96} + 2407090 q^{97} + 941192 q^{98} - 5298372 q^{99}+O(q^{100}) \) 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
8.00000 27.0000 64.0000 470.000 216.000 −343.000 512.000 729.000 3760.00
\(n\): 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 42.8.a.f 1
3.b odd 2 1 126.8.a.a 1
4.b odd 2 1 336.8.a.f 1
7.b odd 2 1 294.8.a.k 1
7.c even 3 2 294.8.e.a 2
7.d odd 6 2 294.8.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.8.a.f 1 1.a even 1 1 trivial
126.8.a.a 1 3.b odd 2 1
294.8.a.k 1 7.b odd 2 1
294.8.e.a 2 7.c even 3 2
294.8.e.f 2 7.d odd 6 2
336.8.a.f 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 470 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(42))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 8 \) Copy content Toggle raw display
$3$ \( T - 27 \) Copy content Toggle raw display
$5$ \( T - 470 \) Copy content Toggle raw display
$7$ \( T + 343 \) Copy content Toggle raw display
$11$ \( T + 7268 \) Copy content Toggle raw display
$13$ \( T - 11382 \) Copy content Toggle raw display
$17$ \( T - 20842 \) Copy content Toggle raw display
$19$ \( T + 13684 \) Copy content Toggle raw display
$23$ \( T + 46736 \) Copy content Toggle raw display
$29$ \( T - 2774 \) Copy content Toggle raw display
$31$ \( T + 11232 \) Copy content Toggle raw display
$37$ \( T + 239698 \) Copy content Toggle raw display
$41$ \( T - 52962 \) Copy content Toggle raw display
$43$ \( T + 874732 \) Copy content Toggle raw display
$47$ \( T + 451296 \) Copy content Toggle raw display
$53$ \( T + 1943538 \) Copy content Toggle raw display
$59$ \( T - 1563908 \) Copy content Toggle raw display
$61$ \( T + 3183690 \) Copy content Toggle raw display
$67$ \( T - 2128684 \) Copy content Toggle raw display
$71$ \( T - 5694544 \) Copy content Toggle raw display
$73$ \( T + 2253878 \) Copy content Toggle raw display
$79$ \( T - 132912 \) Copy content Toggle raw display
$83$ \( T - 6950524 \) Copy content Toggle raw display
$89$ \( T + 10626510 \) Copy content Toggle raw display
$97$ \( T - 2407090 \) Copy content Toggle raw display
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