Properties

Label 70.2.g.a
Level $70$
Weight $2$
Character orbit 70.g
Analytic conductor $0.559$
Analytic rank $0$
Dimension $8$
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(13,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, base_ring=CyclotomicField(4))
 
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("70.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 70.g (of order \(4\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{16}^{6} q^{2} + ( - \zeta_{16}^{3} + \zeta_{16}) q^{3} - \zeta_{16}^{4} q^{4} + (2 \zeta_{16}^{7} - \zeta_{16}^{3}) q^{5} + ( - \zeta_{16}^{7} - \zeta_{16}) q^{6} + ( - \zeta_{16}^{7} - \zeta_{16}^{6} + \zeta_{16}^{5} + \zeta_{16}^{4} + \zeta_{16}^{3} + \zeta_{16} - 1) q^{7} - \zeta_{16}^{2} q^{8} + (\zeta_{16}^{6} + \zeta_{16}^{4} + \zeta_{16}^{2}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{16}^{6} q^{2} + ( - \zeta_{16}^{3} + \zeta_{16}) q^{3} - \zeta_{16}^{4} q^{4} + (2 \zeta_{16}^{7} - \zeta_{16}^{3}) q^{5} + ( - \zeta_{16}^{7} - \zeta_{16}) q^{6} + ( - \zeta_{16}^{7} - \zeta_{16}^{6} + \zeta_{16}^{5} + \zeta_{16}^{4} + \zeta_{16}^{3} + \zeta_{16} - 1) q^{7} - \zeta_{16}^{2} q^{8} + (\zeta_{16}^{6} + \zeta_{16}^{4} + \zeta_{16}^{2}) q^{9} + (2 \zeta_{16}^{5} - \zeta_{16}) q^{10} + (2 \zeta_{16}^{6} - 2 \zeta_{16}^{2}) q^{11} + (\zeta_{16}^{7} - \zeta_{16}^{5}) q^{12} + ( - 3 \zeta_{16}^{7} - 3 \zeta_{16}^{5} + 2 \zeta_{16}^{3} - 2 \zeta_{16}) q^{13} + ( - \zeta_{16}^{7} + \zeta_{16}^{6} - \zeta_{16}^{5} - \zeta_{16}^{4} + \zeta_{16}^{3} + \zeta_{16}^{2} + \zeta_{16}) q^{14} + (\zeta_{16}^{6} - \zeta_{16}^{4} + 2 \zeta_{16}^{2} - 2) q^{15} - q^{16} + (2 \zeta_{16}^{7} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{3} + 2 \zeta_{16}) q^{17} + (\zeta_{16}^{4} + \zeta_{16}^{2} + 1) q^{18} + (3 \zeta_{16}^{7} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} - 3 \zeta_{16}) q^{19} + (\zeta_{16}^{7} + 2 \zeta_{16}^{3}) q^{20} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}^{3} - 2 \zeta_{16} + 2) q^{21} + (2 \zeta_{16}^{4} - 2) q^{22} + ( - \zeta_{16}^{4} - 2 \zeta_{16}^{2} - 1) q^{23} + (\zeta_{16}^{5} - \zeta_{16}^{3}) q^{24} + ( - 3 \zeta_{16}^{6} + 4 \zeta_{16}^{2}) q^{25} + (2 \zeta_{16}^{7} - 3 \zeta_{16}^{5} - 3 \zeta_{16}^{3} + 2 \zeta_{16}) q^{26} + ( - 3 \zeta_{16}^{7} + 3 \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16}) q^{27} + ( - \zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{4} - \zeta_{16}^{3} - \zeta_{16}^{2} + \zeta_{16} + 1) q^{28} + ( - 2 \zeta_{16}^{6} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{2}) q^{29} + (2 \zeta_{16}^{6} + \zeta_{16}^{4} - \zeta_{16}^{2} + 2) q^{30} + ( - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3}) q^{31} + \zeta_{16}^{6} q^{32} + (2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} + 2 \zeta_{16}) q^{33} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} + 2 \zeta_{16}) q^{34} + ( - 3 \zeta_{16}^{7} + \zeta_{16}^{6} + 2 \zeta_{16}^{5} - 3 \zeta_{16}^{4} - \zeta_{16}^{3} - 3 \zeta_{16}^{2} - \zeta_{16} - 1) q^{35} + ( - \zeta_{16}^{6} + \zeta_{16}^{2} + 1) q^{36} + ( - 2 \zeta_{16}^{6} - 4 \zeta_{16}^{4} + 4) q^{37} + (3 \zeta_{16}^{7} + 3 \zeta_{16}^{5} + 2 \zeta_{16}^{3} - 2 \zeta_{16}) q^{38} + ( - 5 \zeta_{16}^{6} + 4 \zeta_{16}^{4} - 5 \zeta_{16}^{2}) q^{39} + (\zeta_{16}^{5} + 2 \zeta_{16}) q^{40} + (2 \zeta_{16}^{7} + 2 \zeta_{16}) q^{41} + (2 \zeta_{16}^{7} - 2 \zeta_{16}^{6} - 2 \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16}) q^{42} + (4 \zeta_{16}^{4} + 4) q^{43} + (2 \zeta_{16}^{6} + 2 \zeta_{16}^{2}) q^{44} + ( - \zeta_{16}^{7} - 3 \zeta_{16}^{5} - 2 \zeta_{16}^{3} - \zeta_{16}) q^{45} + (\zeta_{16}^{6} - \zeta_{16}^{2} - 2) q^{46} + ( - 2 \zeta_{16}^{3} - 2 \zeta_{16}) q^{47} + (\zeta_{16}^{3} - \zeta_{16}) q^{48} + (2 \zeta_{16}^{7} + 4 \zeta_{16}^{6} - 2 \zeta_{16}^{5} + \zeta_{16}^{4} + 2 \zeta_{16}^{3} + 4 \zeta_{16}^{2} - 2 \zeta_{16}) q^{49} + ( - 3 \zeta_{16}^{4} + 4) q^{50} + ( - 4 \zeta_{16}^{6} + 4 \zeta_{16}^{2} - 4) q^{51} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} - 3 \zeta_{16}^{3} - 3 \zeta_{16}) q^{52} + ( - 4 \zeta_{16}^{4} + 6 \zeta_{16}^{2} - 4) q^{53} + ( - \zeta_{16}^{7} - 3 \zeta_{16}^{5} + 3 \zeta_{16}^{3} + \zeta_{16}) q^{54} + ( - 2 \zeta_{16}^{5} + 6 \zeta_{16}) q^{55} + ( - \zeta_{16}^{7} - \zeta_{16}^{6} - \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16}^{2} - \zeta_{16} - 1) q^{56} + (4 \zeta_{16}^{6} + \zeta_{16}^{4} - 1) q^{57} + ( - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{2} - 2) q^{58} + ( - 5 \zeta_{16}^{5} + 5 \zeta_{16}^{3}) q^{59} + ( - 2 \zeta_{16}^{6} + 2 \zeta_{16}^{4} + \zeta_{16}^{2} - 1) q^{60} + ( - \zeta_{16}^{7} + 6 \zeta_{16}^{5} + 6 \zeta_{16}^{3} - \zeta_{16}) q^{61} + ( - 2 \zeta_{16}^{3} - 2 \zeta_{16}) q^{62} + (3 \zeta_{16}^{7} + 3 \zeta_{16}^{5} + \zeta_{16}^{3} - \zeta_{16}^{2} - \zeta_{16}) q^{63} + \zeta_{16}^{4} q^{64} + (4 \zeta_{16}^{6} + 8 \zeta_{16}^{4} - 7 \zeta_{16}^{2} + 1) q^{65} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{3} - 2 \zeta_{16}) q^{66} + (12 \zeta_{16}^{6} - 2 \zeta_{16}^{4} + 2) q^{67} + ( - 2 \zeta_{16}^{7} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{3} - 2 \zeta_{16}) q^{68} + (\zeta_{16}^{7} + \zeta_{16}^{5} - \zeta_{16}^{3} - \zeta_{16}) q^{69} + (\zeta_{16}^{7} + \zeta_{16}^{6} - 3 \zeta_{16}^{5} + \zeta_{16}^{4} + 2 \zeta_{16}^{3} - 3 \zeta_{16}^{2} - \zeta_{16} - 3) q^{70} + (\zeta_{16}^{6} - \zeta_{16}^{2} - 2) q^{71} + ( - \zeta_{16}^{6} - \zeta_{16}^{4} + 1) q^{72} + ( - 2 \zeta_{16}^{7} - 2 \zeta_{16}^{5} - 4 \zeta_{16}^{3} + 4 \zeta_{16}) q^{73} + ( - 4 \zeta_{16}^{6} - 2 \zeta_{16}^{4} - 4 \zeta_{16}^{2}) q^{74} + ( - 3 \zeta_{16}^{7} - 4 \zeta_{16}^{5} + 4 \zeta_{16}^{3} - 3 \zeta_{16}) q^{75} + (2 \zeta_{16}^{7} + 3 \zeta_{16}^{5} + 3 \zeta_{16}^{3} + 2 \zeta_{16}) q^{76} + ( - 4 \zeta_{16}^{6} + 2 \zeta_{16}^{4} - 4 \zeta_{16}^{3} - 4 \zeta_{16} - 2) q^{77} + ( - 5 \zeta_{16}^{4} + 4 \zeta_{16}^{2} - 5) q^{78} + (5 \zeta_{16}^{6} + 2 \zeta_{16}^{4} + 5 \zeta_{16}^{2}) q^{79} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{3}) q^{80} + (5 \zeta_{16}^{6} - 5 \zeta_{16}^{2} + 3) q^{81} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}^{5}) q^{82} + (5 \zeta_{16}^{7} + 5 \zeta_{16}^{5} - 2 \zeta_{16}^{3} + 2 \zeta_{16}) q^{83} + ( - \zeta_{16}^{7} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{3} + \zeta_{16}) q^{84} + ( - 6 \zeta_{16}^{6} + 2 \zeta_{16}^{4} - 2 \zeta_{16}^{2} - 6) q^{85} + ( - 4 \zeta_{16}^{6} + 4 \zeta_{16}^{2}) q^{86} + ( - 2 \zeta_{16}^{3} - 2 \zeta_{16}) q^{87} + (2 \zeta_{16}^{4} + 2) q^{88} + ( - 10 \zeta_{16}^{7} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} + 10 \zeta_{16}) q^{89} + (\zeta_{16}^{7} - \zeta_{16}^{5} - 3 \zeta_{16}^{3} - 2 \zeta_{16}) q^{90} + (7 \zeta_{16}^{7} - 6 \zeta_{16}^{6} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} + 6 \zeta_{16}^{2} + 7 \zeta_{16} + 2) q^{91} + (2 \zeta_{16}^{6} + \zeta_{16}^{4} - 1) q^{92} + ( - 2 \zeta_{16}^{4} - 2) q^{93} + (2 \zeta_{16}^{7} - 2 \zeta_{16}) q^{94} + ( - 4 \zeta_{16}^{6} - \zeta_{16}^{4} + 7 \zeta_{16}^{2} + 8) q^{95} + (\zeta_{16}^{7} + \zeta_{16}) q^{96} + (6 \zeta_{16}^{7} - 6 \zeta_{16}^{5} - 6 \zeta_{16}^{3} - 6 \zeta_{16}) q^{97} + (2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} + 4 \zeta_{16}^{4} - 2 \zeta_{16}^{3} + \zeta_{16}^{2} + 2 \zeta_{16} + 4) q^{98} + ( - 2 \zeta_{16}^{6} - 4 \zeta_{16}^{4} - 2 \zeta_{16}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} - 16 q^{15} - 8 q^{16} + 8 q^{18} + 16 q^{21} - 16 q^{22} - 8 q^{23} + 8 q^{28} + 16 q^{30} - 8 q^{35} + 8 q^{36} + 32 q^{37} + 32 q^{43} - 16 q^{46} + 32 q^{50} - 32 q^{51} - 32 q^{53} - 8 q^{56} - 8 q^{57} - 16 q^{58} - 8 q^{60} + 8 q^{65} + 16 q^{67} - 24 q^{70} - 16 q^{71} + 8 q^{72} - 16 q^{77} - 40 q^{78} + 24 q^{81} - 48 q^{85} + 16 q^{88} + 16 q^{91} - 8 q^{92} - 16 q^{93} + 64 q^{95} + 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(31\) \(57\)
\(\chi(n)\) \(-1\) \(-\zeta_{16}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−0.382683 + 0.923880i
0.382683 0.923880i
−0.923880 0.382683i
0.923880 + 0.382683i
−0.382683 0.923880i
0.382683 + 0.923880i
−0.923880 + 0.382683i
0.923880 0.382683i
−0.707107 + 0.707107i −1.30656 + 1.30656i 1.00000i −0.158513 + 2.23044i 1.84776i −2.47247 + 0.941740i 0.707107 + 0.707107i 0.414214i −1.46508 1.68925i
13.2 −0.707107 + 0.707107i 1.30656 1.30656i 1.00000i 0.158513 2.23044i 1.84776i −0.941740 + 2.47247i 0.707107 + 0.707107i 0.414214i 1.46508 + 1.68925i
13.3 0.707107 0.707107i −0.541196 + 0.541196i 1.00000i 2.23044 + 0.158513i 0.765367i −2.14065 1.55487i −0.707107 0.707107i 2.41421i 1.68925 1.46508i
13.4 0.707107 0.707107i 0.541196 0.541196i 1.00000i −2.23044 0.158513i 0.765367i 1.55487 + 2.14065i −0.707107 0.707107i 2.41421i −1.68925 + 1.46508i
27.1 −0.707107 0.707107i −1.30656 1.30656i 1.00000i −0.158513 2.23044i 1.84776i −2.47247 0.941740i 0.707107 0.707107i 0.414214i −1.46508 + 1.68925i
27.2 −0.707107 0.707107i 1.30656 + 1.30656i 1.00000i 0.158513 + 2.23044i 1.84776i −0.941740 2.47247i 0.707107 0.707107i 0.414214i 1.46508 1.68925i
27.3 0.707107 + 0.707107i −0.541196 0.541196i 1.00000i 2.23044 0.158513i 0.765367i −2.14065 + 1.55487i −0.707107 + 0.707107i 2.41421i 1.68925 + 1.46508i
27.4 0.707107 + 0.707107i 0.541196 + 0.541196i 1.00000i −2.23044 + 0.158513i 0.765367i 1.55487 2.14065i −0.707107 + 0.707107i 2.41421i −1.68925 1.46508i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.2.g.a 8
3.b odd 2 1 630.2.p.a 8
4.b odd 2 1 560.2.bj.c 8
5.b even 2 1 350.2.g.a 8
5.c odd 4 1 inner 70.2.g.a 8
5.c odd 4 1 350.2.g.a 8
7.b odd 2 1 inner 70.2.g.a 8
7.c even 3 2 490.2.l.a 16
7.d odd 6 2 490.2.l.a 16
15.e even 4 1 630.2.p.a 8
20.e even 4 1 560.2.bj.c 8
21.c even 2 1 630.2.p.a 8
28.d even 2 1 560.2.bj.c 8
35.c odd 2 1 350.2.g.a 8
35.f even 4 1 inner 70.2.g.a 8
35.f even 4 1 350.2.g.a 8
35.k even 12 2 490.2.l.a 16
35.l odd 12 2 490.2.l.a 16
105.k odd 4 1 630.2.p.a 8
140.j odd 4 1 560.2.bj.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.g.a 8 1.a even 1 1 trivial
70.2.g.a 8 5.c odd 4 1 inner
70.2.g.a 8 7.b odd 2 1 inner
70.2.g.a 8 35.f even 4 1 inner
350.2.g.a 8 5.b even 2 1
350.2.g.a 8 5.c odd 4 1
350.2.g.a 8 35.c odd 2 1
350.2.g.a 8 35.f even 4 1
490.2.l.a 16 7.c even 3 2
490.2.l.a 16 7.d odd 6 2
490.2.l.a 16 35.k even 12 2
490.2.l.a 16 35.l odd 12 2
560.2.bj.c 8 4.b odd 2 1
560.2.bj.c 8 20.e even 4 1
560.2.bj.c 8 28.d even 2 1
560.2.bj.c 8 140.j odd 4 1
630.2.p.a 8 3.b odd 2 1
630.2.p.a 8 15.e even 4 1
630.2.p.a 8 21.c even 2 1
630.2.p.a 8 105.k odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(70, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$5$ \( T^{8} - 48T^{4} + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 1548 T^{4} + 334084 \) Copy content Toggle raw display
$17$ \( T^{8} + 768 T^{4} + 16384 \) Copy content Toggle raw display
$19$ \( (T^{4} - 52 T^{2} + 98)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} + 8 T^{2} - 8 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 24 T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 16 T^{2} + 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 16 T^{3} + 128 T^{2} - 448 T + 784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 16 T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T + 32)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 192T^{4} + 1024 \) Copy content Toggle raw display
$53$ \( (T^{4} + 16 T^{3} + 128 T^{2} - 64 T + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 100 T^{2} + 1250)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 148 T^{2} + 4418)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} + 32 T^{2} + 1088 T + 18496)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T + 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 3264 T^{4} + \cdots + 2458624 \) Copy content Toggle raw display
$79$ \( (T^{4} + 108 T^{2} + 2116)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 6732 T^{4} + \cdots + 11303044 \) Copy content Toggle raw display
$89$ \( (T^{4} - 416 T^{2} + 36992)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 62208 T^{4} + \cdots + 107495424 \) Copy content Toggle raw display
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