Properties

Label 70.4.a.c
Level $70$
Weight $4$
Character orbit 70.a
Self dual yes
Analytic conductor $4.130$
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] = [70,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(4.13013370040\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} - q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} + 7 q^{7} - 8 q^{8} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} + 7 q^{7} - 8 q^{8} - 26 q^{9} + 10 q^{10} - 65 q^{11} - 4 q^{12} + 13 q^{13} - 14 q^{14} + 5 q^{15} + 16 q^{16} - 73 q^{17} + 52 q^{18} - 142 q^{19} - 20 q^{20} - 7 q^{21} + 130 q^{22} + 130 q^{23} + 8 q^{24} + 25 q^{25} - 26 q^{26} + 53 q^{27} + 28 q^{28} + 111 q^{29} - 10 q^{30} + 256 q^{31} - 32 q^{32} + 65 q^{33} + 146 q^{34} - 35 q^{35} - 104 q^{36} - 266 q^{37} + 284 q^{38} - 13 q^{39} + 40 q^{40} - 424 q^{41} + 14 q^{42} + 534 q^{43} - 260 q^{44} + 130 q^{45} - 260 q^{46} - 269 q^{47} - 16 q^{48} + 49 q^{49} - 50 q^{50} + 73 q^{51} + 52 q^{52} - 132 q^{53} - 106 q^{54} + 325 q^{55} - 56 q^{56} + 142 q^{57} - 222 q^{58} - 224 q^{59} + 20 q^{60} - 572 q^{61} - 512 q^{62} - 182 q^{63} + 64 q^{64} - 65 q^{65} - 130 q^{66} - 108 q^{67} - 292 q^{68} - 130 q^{69} + 70 q^{70} + 560 q^{71} + 208 q^{72} + 586 q^{73} + 532 q^{74} - 25 q^{75} - 568 q^{76} - 455 q^{77} + 26 q^{78} + 57 q^{79} - 80 q^{80} + 649 q^{81} + 848 q^{82} + 252 q^{83} - 28 q^{84} + 365 q^{85} - 1068 q^{86} - 111 q^{87} + 520 q^{88} - 184 q^{89} - 260 q^{90} + 91 q^{91} + 520 q^{92} - 256 q^{93} + 538 q^{94} + 710 q^{95} + 32 q^{96} - 605 q^{97} - 98 q^{98} + 1690 q^{99}+O(q^{100}) \) 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
−2.00000 −1.00000 4.00000 −5.00000 2.00000 7.00000 −8.00000 −26.0000 10.0000
\(n\): 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 70.4.a.c 1
3.b odd 2 1 630.4.a.x 1
4.b odd 2 1 560.4.a.i 1
5.b even 2 1 350.4.a.r 1
5.c odd 4 2 350.4.c.h 2
7.b odd 2 1 490.4.a.d 1
7.c even 3 2 490.4.e.o 2
7.d odd 6 2 490.4.e.n 2
8.b even 2 1 2240.4.a.v 1
8.d odd 2 1 2240.4.a.r 1
35.c odd 2 1 2450.4.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.a.c 1 1.a even 1 1 trivial
350.4.a.r 1 5.b even 2 1
350.4.c.h 2 5.c odd 4 2
490.4.a.d 1 7.b odd 2 1
490.4.e.n 2 7.d odd 6 2
490.4.e.o 2 7.c even 3 2
560.4.a.i 1 4.b odd 2 1
630.4.a.x 1 3.b odd 2 1
2240.4.a.r 1 8.d odd 2 1
2240.4.a.v 1 8.b even 2 1
2450.4.a.bc 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(70))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T + 65 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T + 73 \) Copy content Toggle raw display
$19$ \( T + 142 \) Copy content Toggle raw display
$23$ \( T - 130 \) Copy content Toggle raw display
$29$ \( T - 111 \) Copy content Toggle raw display
$31$ \( T - 256 \) Copy content Toggle raw display
$37$ \( T + 266 \) Copy content Toggle raw display
$41$ \( T + 424 \) Copy content Toggle raw display
$43$ \( T - 534 \) Copy content Toggle raw display
$47$ \( T + 269 \) Copy content Toggle raw display
$53$ \( T + 132 \) Copy content Toggle raw display
$59$ \( T + 224 \) Copy content Toggle raw display
$61$ \( T + 572 \) Copy content Toggle raw display
$67$ \( T + 108 \) Copy content Toggle raw display
$71$ \( T - 560 \) Copy content Toggle raw display
$73$ \( T - 586 \) Copy content Toggle raw display
$79$ \( T - 57 \) Copy content Toggle raw display
$83$ \( T - 252 \) Copy content Toggle raw display
$89$ \( T + 184 \) Copy content Toggle raw display
$97$ \( T + 605 \) Copy content Toggle raw display
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