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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Newspace parameters

Level: \( N \) = \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 70.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.558952814149\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
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

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.

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

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.

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 27.4
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

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( ( 1 - 8 T^{2} + 32 T^{4} - 72 T^{6} + 81 T^{8} )( 1 + 8 T^{2} + 32 T^{4} + 72 T^{6} + 81 T^{8} ) \)
$5$ \( 1 - 48 T^{4} + 625 T^{8} \)
$7$ \( 1 + 8 T + 32 T^{2} + 88 T^{3} + 226 T^{4} + 616 T^{5} + 1568 T^{6} + 2744 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 + 14 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 - 240 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( 1 - 252 T^{4} + 17030 T^{8} - 21047292 T^{12} + 6975757441 T^{16} \)
$19$ \( ( 1 + 24 T^{2} + 288 T^{4} + 8664 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 4 T + 8 T^{2} + 84 T^{3} + 878 T^{4} + 1932 T^{5} + 4232 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 92 T^{2} + 3670 T^{4} - 77372 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 108 T^{2} + 4806 T^{4} - 103788 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 16 T + 128 T^{2} - 1040 T^{3} + 7666 T^{4} - 38480 T^{5} + 175232 T^{6} - 810448 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 148 T^{2} + 8806 T^{4} - 248788 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 8 T + 32 T^{2} - 344 T^{3} + 1849 T^{4} )^{4} \)
$47$ \( 1 + 6020 T^{4} + 17872774 T^{8} + 29375679620 T^{12} + 23811286661761 T^{16} \)
$53$ \( ( 1 + 16 T + 128 T^{2} + 784 T^{3} + 4786 T^{4} + 41552 T^{5} + 359552 T^{6} + 2382032 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 136 T^{2} + 10336 T^{4} + 473416 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 96 T^{2} + 8688 T^{4} - 357216 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 8 T + 32 T^{2} + 552 T^{3} - 8974 T^{4} + 36984 T^{5} + 143648 T^{6} - 2406104 T^{7} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} )^{4} \)
$73$ \( 1 + 1220 T^{4} + 55730374 T^{8} + 34645854020 T^{12} + 806460091894081 T^{16} \)
$79$ \( ( 1 - 208 T^{2} + 22498 T^{4} - 1298128 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( 1 - 4224 T^{4} + 99283874 T^{8} - 200463947904 T^{12} + 2252292232139041 T^{16} \)
$89$ \( ( 1 - 60 T^{2} + 10470 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( 1 - 11900 T^{4} + 108781062 T^{8} - 1053498443900 T^{12} + 7837433594376961 T^{16} \)
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