Properties

Label 70.6.a.c
Level $70$
Weight $6$
Character orbit 70.a
Self dual yes
Analytic conductor $11.227$
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] = [70,6,Mod(1,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, 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("70.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 70.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: \(11.2268673869\)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} - 3 q^{3} + 16 q^{4} - 25 q^{5} + 12 q^{6} + 49 q^{7} - 64 q^{8} - 234 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - 3 q^{3} + 16 q^{4} - 25 q^{5} + 12 q^{6} + 49 q^{7} - 64 q^{8} - 234 q^{9} + 100 q^{10} + 405 q^{11} - 48 q^{12} - 391 q^{13} - 196 q^{14} + 75 q^{15} + 256 q^{16} + 999 q^{17} + 936 q^{18} + 2342 q^{19} - 400 q^{20} - 147 q^{21} - 1620 q^{22} + 2430 q^{23} + 192 q^{24} + 625 q^{25} + 1564 q^{26} + 1431 q^{27} + 784 q^{28} + 8259 q^{29} - 300 q^{30} + 4016 q^{31} - 1024 q^{32} - 1215 q^{33} - 3996 q^{34} - 1225 q^{35} - 3744 q^{36} - 7042 q^{37} - 9368 q^{38} + 1173 q^{39} + 1600 q^{40} + 3336 q^{41} + 588 q^{42} - 23518 q^{43} + 6480 q^{44} + 5850 q^{45} - 9720 q^{46} + 10317 q^{47} - 768 q^{48} + 2401 q^{49} - 2500 q^{50} - 2997 q^{51} - 6256 q^{52} + 3084 q^{53} - 5724 q^{54} - 10125 q^{55} - 3136 q^{56} - 7026 q^{57} - 33036 q^{58} - 18816 q^{59} + 1200 q^{60} + 21668 q^{61} - 16064 q^{62} - 11466 q^{63} + 4096 q^{64} + 9775 q^{65} + 4860 q^{66} + 52124 q^{67} + 15984 q^{68} - 7290 q^{69} + 4900 q^{70} - 28560 q^{71} + 14976 q^{72} - 70342 q^{73} + 28168 q^{74} - 1875 q^{75} + 37472 q^{76} + 19845 q^{77} - 4692 q^{78} + 58823 q^{79} - 6400 q^{80} + 52569 q^{81} - 13344 q^{82} + 756 q^{83} - 2352 q^{84} - 24975 q^{85} + 94072 q^{86} - 24777 q^{87} - 25920 q^{88} + 135384 q^{89} - 23400 q^{90} - 19159 q^{91} + 38880 q^{92} - 12048 q^{93} - 41268 q^{94} - 58550 q^{95} + 3072 q^{96} + 110435 q^{97} - 9604 q^{98} - 94770 q^{99}+O(q^{100}) \) 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 −3.00000 16.0000 −25.0000 12.0000 49.0000 −64.0000 −234.000 100.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(7\) \( -1 \)

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 70.6.a.c 1
3.b odd 2 1 630.6.a.n 1
4.b odd 2 1 560.6.a.d 1
5.b even 2 1 350.6.a.k 1
5.c odd 4 2 350.6.c.e 2
7.b odd 2 1 490.6.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.6.a.c 1 1.a even 1 1 trivial
350.6.a.k 1 5.b even 2 1
350.6.c.e 2 5.c odd 4 2
490.6.a.e 1 7.b odd 2 1
560.6.a.d 1 4.b odd 2 1
630.6.a.n 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 3 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(70))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 25 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 405 \) Copy content Toggle raw display
$13$ \( T + 391 \) Copy content Toggle raw display
$17$ \( T - 999 \) Copy content Toggle raw display
$19$ \( T - 2342 \) Copy content Toggle raw display
$23$ \( T - 2430 \) Copy content Toggle raw display
$29$ \( T - 8259 \) Copy content Toggle raw display
$31$ \( T - 4016 \) Copy content Toggle raw display
$37$ \( T + 7042 \) Copy content Toggle raw display
$41$ \( T - 3336 \) Copy content Toggle raw display
$43$ \( T + 23518 \) Copy content Toggle raw display
$47$ \( T - 10317 \) Copy content Toggle raw display
$53$ \( T - 3084 \) Copy content Toggle raw display
$59$ \( T + 18816 \) Copy content Toggle raw display
$61$ \( T - 21668 \) Copy content Toggle raw display
$67$ \( T - 52124 \) Copy content Toggle raw display
$71$ \( T + 28560 \) Copy content Toggle raw display
$73$ \( T + 70342 \) Copy content Toggle raw display
$79$ \( T - 58823 \) Copy content Toggle raw display
$83$ \( T - 756 \) Copy content Toggle raw display
$89$ \( T - 135384 \) Copy content Toggle raw display
$97$ \( T - 110435 \) Copy content Toggle raw display
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