Properties

Label 70.6.a.e
Level $70$
Weight $6$
Character orbit 70.a
Self dual yes
Analytic conductor $11.227$
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] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - 17 q^{3} + 16 q^{4} + 25 q^{5} - 68 q^{6} - 49 q^{7} + 64 q^{8} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 17 q^{3} + 16 q^{4} + 25 q^{5} - 68 q^{6} - 49 q^{7} + 64 q^{8} + 46 q^{9} + 100 q^{10} - 715 q^{11} - 272 q^{12} + 331 q^{13} - 196 q^{14} - 425 q^{15} + 256 q^{16} - 1699 q^{17} + 184 q^{18} - 1718 q^{19} + 400 q^{20} + 833 q^{21} - 2860 q^{22} - 3950 q^{23} - 1088 q^{24} + 625 q^{25} + 1324 q^{26} + 3349 q^{27} - 784 q^{28} + 4579 q^{29} - 1700 q^{30} + 6756 q^{31} + 1024 q^{32} + 12155 q^{33} - 6796 q^{34} - 1225 q^{35} + 736 q^{36} - 16518 q^{37} - 6872 q^{38} - 5627 q^{39} + 1600 q^{40} + 18876 q^{41} + 3332 q^{42} + 2258 q^{43} - 11440 q^{44} + 1150 q^{45} - 15800 q^{46} - 537 q^{47} - 4352 q^{48} + 2401 q^{49} + 2500 q^{50} + 28883 q^{51} + 5296 q^{52} - 10984 q^{53} + 13396 q^{54} - 17875 q^{55} - 3136 q^{56} + 29206 q^{57} + 18316 q^{58} - 25956 q^{59} - 6800 q^{60} + 39188 q^{61} + 27024 q^{62} - 2254 q^{63} + 4096 q^{64} + 8275 q^{65} + 48620 q^{66} + 4416 q^{67} - 27184 q^{68} + 67150 q^{69} - 4900 q^{70} - 31880 q^{71} + 2944 q^{72} - 5018 q^{73} - 66072 q^{74} - 10625 q^{75} - 27488 q^{76} + 35035 q^{77} - 22508 q^{78} - 27977 q^{79} + 6400 q^{80} - 68111 q^{81} + 75504 q^{82} + 37644 q^{83} + 13328 q^{84} - 42475 q^{85} + 9032 q^{86} - 77843 q^{87} - 45760 q^{88} - 17216 q^{89} + 4600 q^{90} - 16219 q^{91} - 63200 q^{92} - 114852 q^{93} - 2148 q^{94} - 42950 q^{95} - 17408 q^{96} - 63175 q^{97} + 9604 q^{98} - 32890 q^{99}+O(q^{100}) \) 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 −17.0000 16.0000 25.0000 −68.0000 −49.0000 64.0000 46.0000 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.e 1
3.b odd 2 1 630.6.a.b 1
4.b odd 2 1 560.6.a.h 1
5.b even 2 1 350.6.a.e 1
5.c odd 4 2 350.6.c.a 2
7.b odd 2 1 490.6.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.6.a.e 1 1.a even 1 1 trivial
350.6.a.e 1 5.b even 2 1
350.6.c.a 2 5.c odd 4 2
490.6.a.m 1 7.b odd 2 1
560.6.a.h 1 4.b odd 2 1
630.6.a.b 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 17 \) 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
$3$ \( T + 17 \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T + 715 \) Copy content Toggle raw display
$13$ \( T - 331 \) Copy content Toggle raw display
$17$ \( T + 1699 \) Copy content Toggle raw display
$19$ \( T + 1718 \) Copy content Toggle raw display
$23$ \( T + 3950 \) Copy content Toggle raw display
$29$ \( T - 4579 \) Copy content Toggle raw display
$31$ \( T - 6756 \) Copy content Toggle raw display
$37$ \( T + 16518 \) Copy content Toggle raw display
$41$ \( T - 18876 \) Copy content Toggle raw display
$43$ \( T - 2258 \) Copy content Toggle raw display
$47$ \( T + 537 \) Copy content Toggle raw display
$53$ \( T + 10984 \) Copy content Toggle raw display
$59$ \( T + 25956 \) Copy content Toggle raw display
$61$ \( T - 39188 \) Copy content Toggle raw display
$67$ \( T - 4416 \) Copy content Toggle raw display
$71$ \( T + 31880 \) Copy content Toggle raw display
$73$ \( T + 5018 \) Copy content Toggle raw display
$79$ \( T + 27977 \) Copy content Toggle raw display
$83$ \( T - 37644 \) Copy content Toggle raw display
$89$ \( T + 17216 \) Copy content Toggle raw display
$97$ \( T + 63175 \) Copy content Toggle raw display
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