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Properties

Label 70.2.i.b
Level $70$
Weight $2$
Character orbit 70.i
Analytic conductor $0.559$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [70,2,Mod(9,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 70.i (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(0.558952814149\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{2} - \zeta_{12} + 2) q^{5} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{2} - \zeta_{12} + 2) q^{5} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 2 \zeta_{12}^{3} + \cdots + 2 \zeta_{12}) q^{10}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 4 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 4 q^{5} - 6 q^{9} - 2 q^{10} - 6 q^{11} - 10 q^{14} - 2 q^{16} - 10 q^{19} + 8 q^{20} - 6 q^{25} + 10 q^{26} + 16 q^{29} + 4 q^{31} + 8 q^{34} + 10 q^{35} - 12 q^{36} + 2 q^{40} + 12 q^{41} + 6 q^{44} + 12 q^{45} + 14 q^{46} - 4 q^{49} - 16 q^{50} - 24 q^{55} - 8 q^{56} - 8 q^{59} - 12 q^{61} - 4 q^{64} - 10 q^{65} - 4 q^{70} - 24 q^{71} + 2 q^{74} - 20 q^{76} + 28 q^{79} + 4 q^{80} - 18 q^{81} - 8 q^{85} - 4 q^{86} + 4 q^{89} + 12 q^{90} + 40 q^{91} - 14 q^{94} + 20 q^{95} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(57\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.86603 − 1.23205i 0 1.73205 − 2.00000i − 1.00000i −1.50000 + 2.59808i −2.23205 + 0.133975i
9.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.133975 − 2.23205i 0 −1.73205 + 2.00000i 1.00000i −1.50000 + 2.59808i 1.23205 − 1.86603i
39.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.86603 + 1.23205i 0 1.73205 + 2.00000i 1.00000i −1.50000 − 2.59808i −2.23205 − 0.133975i
39.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.133975 + 2.23205i 0 −1.73205 − 2.00000i − 1.00000i −1.50000 − 2.59808i 1.23205 + 1.86603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.2.i.b ✓ 4
3.b odd 2 1 630.2.u.a 4
4.b odd 2 1 560.2.bw.d 4
5.b even 2 1 inner 70.2.i.b ✓ 4
5.c odd 4 1 350.2.e.c 2
5.c odd 4 1 350.2.e.j 2
7.b odd 2 1 490.2.i.a 4
7.c even 3 1 inner 70.2.i.b ✓ 4
7.c even 3 1 490.2.c.a 2
7.d odd 6 1 490.2.c.d 2
7.d odd 6 1 490.2.i.a 4
15.d odd 2 1 630.2.u.a 4
20.d odd 2 1 560.2.bw.d 4
21.h odd 6 1 630.2.u.a 4
28.g odd 6 1 560.2.bw.d 4
35.c odd 2 1 490.2.i.a 4
35.i odd 6 1 490.2.c.d 2
35.i odd 6 1 490.2.i.a 4
35.j even 6 1 inner 70.2.i.b ✓ 4
35.j even 6 1 490.2.c.a 2
35.k even 12 1 2450.2.a.j 1
35.k even 12 1 2450.2.a.bb 1
35.l odd 12 1 350.2.e.c 2
35.l odd 12 1 350.2.e.j 2
35.l odd 12 1 2450.2.a.k 1
35.l odd 12 1 2450.2.a.ba 1
105.o odd 6 1 630.2.u.a 4
140.p odd 6 1 560.2.bw.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.b ✓ 4 1.a even 1 1 trivial
70.2.i.b ✓ 4 5.b even 2 1 inner
70.2.i.b ✓ 4 7.c even 3 1 inner
70.2.i.b ✓ 4 35.j even 6 1 inner
350.2.e.c 2 5.c odd 4 1
350.2.e.c 2 35.l odd 12 1
350.2.e.j 2 5.c odd 4 1
350.2.e.j 2 35.l odd 12 1
490.2.c.a 2 7.c even 3 1
490.2.c.a 2 35.j even 6 1
490.2.c.d 2 7.d odd 6 1
490.2.c.d 2 35.i odd 6 1
490.2.i.a 4 7.b odd 2 1
490.2.i.a 4 7.d odd 6 1
490.2.i.a 4 35.c odd 2 1
490.2.i.a 4 35.i odd 6 1
560.2.bw.d 4 4.b odd 2 1
560.2.bw.d 4 20.d odd 2 1
560.2.bw.d 4 28.g odd 6 1
560.2.bw.d 4 140.p odd 6 1
630.2.u.a 4 3.b odd 2 1
630.2.u.a 4 15.d odd 2 1
630.2.u.a 4 21.h odd 6 1
630.2.u.a 4 105.o odd 6 1
2450.2.a.j 1 35.k even 12 1
2450.2.a.k 1 35.l odd 12 1
2450.2.a.ba 1 35.l odd 12 1
2450.2.a.bb 1 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(70, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$29$ \( (T - 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$41$ \( (T - 3)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$53$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 256 T^{2} + 65536 \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
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