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Properties

Label 70.2.i.a
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{5} - 3 q^{6} + ( - 3 \zeta_{12}^{3} + \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{5} - 3 q^{6} + ( - 3 \zeta_{12}^{3} + \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12}) q^{10} - 3 \zeta_{12} q^{12} - 2 \zeta_{12}^{3} q^{13} + ( - 2 \zeta_{12}^{2} + 3) q^{14} + ( - 3 \zeta_{12}^{3} - 6) q^{15} + (\zeta_{12}^{2} - 1) q^{16} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{17} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{18} + (2 \zeta_{12}^{2} - 2) q^{19} + (2 \zeta_{12}^{3} - 1) q^{20} + (9 \zeta_{12}^{2} - 3) q^{21} + \zeta_{12} q^{23} - 3 \zeta_{12}^{2} q^{24} + (4 \zeta_{12}^{3} + \cdots - 4 \zeta_{12}) q^{25} + \cdots + ( - 5 \zeta_{12}^{3} - 3 \zeta_{12}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{5} - 12 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{5} - 12 q^{6} + 12 q^{9} + 4 q^{10} + 8 q^{14} - 24 q^{15} - 2 q^{16} - 4 q^{19} - 4 q^{20} + 6 q^{21} - 6 q^{24} + 6 q^{25} + 4 q^{26} + 4 q^{29} + 6 q^{30} - 20 q^{31} + 8 q^{34} + 16 q^{35} + 24 q^{36} + 12 q^{39} - 4 q^{40} - 12 q^{41} + 12 q^{45} + 2 q^{46} - 22 q^{49} - 16 q^{50} - 12 q^{51} - 18 q^{54} + 10 q^{56} + 4 q^{59} - 12 q^{60} + 18 q^{61} - 4 q^{64} + 8 q^{65} - 12 q^{69} + 2 q^{70} + 24 q^{71} - 16 q^{74} + 24 q^{75} - 8 q^{76} - 20 q^{79} - 2 q^{80} - 18 q^{81} - 24 q^{84} + 16 q^{85} - 10 q^{86} - 14 q^{89} + 48 q^{90} - 20 q^{91} + 16 q^{94} - 4 q^{95} + 6 q^{96}+O(q^{100}) \) 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 2.59808 − 1.50000i 0.500000 + 0.866025i −2.23205 − 0.133975i −3.00000 −0.866025 + 2.50000i − 1.00000i 3.00000 − 5.19615i 1.86603 + 1.23205i
9.2 0.866025 + 0.500000i −2.59808 + 1.50000i 0.500000 + 0.866025i 1.23205 + 1.86603i −3.00000 0.866025 − 2.50000i 1.00000i 3.00000 − 5.19615i 0.133975 + 2.23205i
39.1 −0.866025 + 0.500000i 2.59808 + 1.50000i 0.500000 − 0.866025i −2.23205 + 0.133975i −3.00000 −0.866025 − 2.50000i 1.00000i 3.00000 + 5.19615i 1.86603 − 1.23205i
39.2 0.866025 − 0.500000i −2.59808 − 1.50000i 0.500000 − 0.866025i 1.23205 − 1.86603i −3.00000 0.866025 + 2.50000i − 1.00000i 3.00000 + 5.19615i 0.133975 − 2.23205i
\(n\): 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.a ✓ 4
3.b odd 2 1 630.2.u.b 4
4.b odd 2 1 560.2.bw.c 4
5.b even 2 1 inner 70.2.i.a ✓ 4
5.c odd 4 1 350.2.e.f 2
5.c odd 4 1 350.2.e.g 2
7.b odd 2 1 490.2.i.b 4
7.c even 3 1 inner 70.2.i.a ✓ 4
7.c even 3 1 490.2.c.c 2
7.d odd 6 1 490.2.c.b 2
7.d odd 6 1 490.2.i.b 4
15.d odd 2 1 630.2.u.b 4
20.d odd 2 1 560.2.bw.c 4
21.h odd 6 1 630.2.u.b 4
28.g odd 6 1 560.2.bw.c 4
35.c odd 2 1 490.2.i.b 4
35.i odd 6 1 490.2.c.b 2
35.i odd 6 1 490.2.i.b 4
35.j even 6 1 inner 70.2.i.a ✓ 4
35.j even 6 1 490.2.c.c 2
35.k even 12 1 2450.2.a.c 1
35.k even 12 1 2450.2.a.bh 1
35.l odd 12 1 350.2.e.f 2
35.l odd 12 1 350.2.e.g 2
35.l odd 12 1 2450.2.a.r 1
35.l odd 12 1 2450.2.a.s 1
105.o odd 6 1 630.2.u.b 4
140.p odd 6 1 560.2.bw.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.a ✓ 4 1.a even 1 1 trivial
70.2.i.a ✓ 4 5.b even 2 1 inner
70.2.i.a ✓ 4 7.c even 3 1 inner
70.2.i.a ✓ 4 35.j even 6 1 inner
350.2.e.f 2 5.c odd 4 1
350.2.e.f 2 35.l odd 12 1
350.2.e.g 2 5.c odd 4 1
350.2.e.g 2 35.l odd 12 1
490.2.c.b 2 7.d odd 6 1
490.2.c.b 2 35.i odd 6 1
490.2.c.c 2 7.c even 3 1
490.2.c.c 2 35.j even 6 1
490.2.i.b 4 7.b odd 2 1
490.2.i.b 4 7.d odd 6 1
490.2.i.b 4 35.c odd 2 1
490.2.i.b 4 35.i odd 6 1
560.2.bw.c 4 4.b odd 2 1
560.2.bw.c 4 20.d odd 2 1
560.2.bw.c 4 28.g odd 6 1
560.2.bw.c 4 140.p odd 6 1
630.2.u.b 4 3.b odd 2 1
630.2.u.b 4 15.d odd 2 1
630.2.u.b 4 21.h odd 6 1
630.2.u.b 4 105.o odd 6 1
2450.2.a.c 1 35.k even 12 1
2450.2.a.r 1 35.l odd 12 1
2450.2.a.s 1 35.l odd 12 1
2450.2.a.bh 1 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 9T_{3}^{2} + 81 \) 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
$3$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$41$ \( (T + 3)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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