Properties

Label 70.6.a.b
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} + 25 q^{5} + 36 q^{6} - 49 q^{7} - 64 q^{8} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} + 25 q^{5} + 36 q^{6} - 49 q^{7} - 64 q^{8} - 162 q^{9} - 100 q^{10} - 187 q^{11} - 144 q^{12} + 627 q^{13} + 196 q^{14} - 225 q^{15} + 256 q^{16} + 1813 q^{17} + 648 q^{18} + 258 q^{19} + 400 q^{20} + 441 q^{21} + 748 q^{22} + 2970 q^{23} + 576 q^{24} + 625 q^{25} - 2508 q^{26} + 3645 q^{27} - 784 q^{28} + 1299 q^{29} + 900 q^{30} + 1916 q^{31} - 1024 q^{32} + 1683 q^{33} - 7252 q^{34} - 1225 q^{35} - 2592 q^{36} + 6578 q^{37} - 1032 q^{38} - 5643 q^{39} - 1600 q^{40} + 6676 q^{41} - 1764 q^{42} + 3178 q^{43} - 2992 q^{44} - 4050 q^{45} - 11880 q^{46} - 22001 q^{47} - 2304 q^{48} + 2401 q^{49} - 2500 q^{50} - 16317 q^{51} + 10032 q^{52} + 26168 q^{53} - 14580 q^{54} - 4675 q^{55} + 3136 q^{56} - 2322 q^{57} - 5196 q^{58} + 3932 q^{59} - 3600 q^{60} - 48740 q^{61} - 7664 q^{62} + 7938 q^{63} + 4096 q^{64} + 15675 q^{65} - 6732 q^{66} - 44832 q^{67} + 29008 q^{68} - 26730 q^{69} + 4900 q^{70} + 63736 q^{71} + 10368 q^{72} + 60470 q^{73} - 26312 q^{74} - 5625 q^{75} + 4128 q^{76} + 9163 q^{77} + 22572 q^{78} - 43721 q^{79} + 6400 q^{80} + 6561 q^{81} - 26704 q^{82} + 97276 q^{83} + 7056 q^{84} + 45325 q^{85} - 12712 q^{86} - 11691 q^{87} + 11968 q^{88} + 45560 q^{89} + 16200 q^{90} - 30723 q^{91} + 47520 q^{92} - 17244 q^{93} + 88004 q^{94} + 6450 q^{95} + 9216 q^{96} - 57295 q^{97} - 9604 q^{98} + 30294 q^{99}+O(q^{100}) \) 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 25.0000 36.0000 −49.0000 −64.0000 −162.000 −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.b 1
3.b odd 2 1 630.6.a.i 1
4.b odd 2 1 560.6.a.e 1
5.b even 2 1 350.6.a.l 1
5.c odd 4 2 350.6.c.g 2
7.b odd 2 1 490.6.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.6.a.b 1 1.a even 1 1 trivial
350.6.a.l 1 5.b even 2 1
350.6.c.g 2 5.c odd 4 2
490.6.a.g 1 7.b odd 2 1
560.6.a.e 1 4.b odd 2 1
630.6.a.i 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 9 \) 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
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T + 187 \) Copy content Toggle raw display
$13$ \( T - 627 \) Copy content Toggle raw display
$17$ \( T - 1813 \) Copy content Toggle raw display
$19$ \( T - 258 \) Copy content Toggle raw display
$23$ \( T - 2970 \) Copy content Toggle raw display
$29$ \( T - 1299 \) Copy content Toggle raw display
$31$ \( T - 1916 \) Copy content Toggle raw display
$37$ \( T - 6578 \) Copy content Toggle raw display
$41$ \( T - 6676 \) Copy content Toggle raw display
$43$ \( T - 3178 \) Copy content Toggle raw display
$47$ \( T + 22001 \) Copy content Toggle raw display
$53$ \( T - 26168 \) Copy content Toggle raw display
$59$ \( T - 3932 \) Copy content Toggle raw display
$61$ \( T + 48740 \) Copy content Toggle raw display
$67$ \( T + 44832 \) Copy content Toggle raw display
$71$ \( T - 63736 \) Copy content Toggle raw display
$73$ \( T - 60470 \) Copy content Toggle raw display
$79$ \( T + 43721 \) Copy content Toggle raw display
$83$ \( T - 97276 \) Copy content Toggle raw display
$89$ \( T - 45560 \) Copy content Toggle raw display
$97$ \( T + 57295 \) Copy content Toggle raw display
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