Properties

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

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Newspace parameters

Level: \( N \) = \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 70.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.558952814149\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(31\) \(57\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 1.00000 1.73205i −0.500000 + 0.866025i 0.500000 + 0.866025i −2.00000 −2.00000 1.73205i 1.00000 −0.500000 0.866025i 0.500000 0.866025i
51.1 −0.500000 + 0.866025i 1.00000 + 1.73205i −0.500000 0.866025i 0.500000 0.866025i −2.00000 −2.00000 + 1.73205i 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.2.e.b 2
3.b odd 2 1 630.2.k.e 2
4.b odd 2 1 560.2.q.d 2
5.b even 2 1 350.2.e.h 2
5.c odd 4 2 350.2.j.a 4
7.b odd 2 1 490.2.e.a 2
7.c even 3 1 inner 70.2.e.b 2
7.c even 3 1 490.2.a.g 1
7.d odd 6 1 490.2.a.j 1
7.d odd 6 1 490.2.e.a 2
21.g even 6 1 4410.2.a.c 1
21.h odd 6 1 630.2.k.e 2
21.h odd 6 1 4410.2.a.m 1
28.f even 6 1 3920.2.a.g 1
28.g odd 6 1 560.2.q.d 2
28.g odd 6 1 3920.2.a.be 1
35.i odd 6 1 2450.2.a.f 1
35.j even 6 1 350.2.e.h 2
35.j even 6 1 2450.2.a.p 1
35.k even 12 2 2450.2.c.p 2
35.l odd 12 2 350.2.j.a 4
35.l odd 12 2 2450.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 1.a even 1 1 trivial
70.2.e.b 2 7.c even 3 1 inner
350.2.e.h 2 5.b even 2 1
350.2.e.h 2 35.j even 6 1
350.2.j.a 4 5.c odd 4 2
350.2.j.a 4 35.l odd 12 2
490.2.a.g 1 7.c even 3 1
490.2.a.j 1 7.d odd 6 1
490.2.e.a 2 7.b odd 2 1
490.2.e.a 2 7.d odd 6 1
560.2.q.d 2 4.b odd 2 1
560.2.q.d 2 28.g odd 6 1
630.2.k.e 2 3.b odd 2 1
630.2.k.e 2 21.h odd 6 1
2450.2.a.f 1 35.i odd 6 1
2450.2.a.p 1 35.j even 6 1
2450.2.c.f 2 35.l odd 12 2
2450.2.c.p 2 35.k even 12 2
3920.2.a.g 1 28.f even 6 1
3920.2.a.be 1 28.g odd 6 1
4410.2.a.c 1 21.g even 6 1
4410.2.a.m 1 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(70, [\chi])\):

\( T_{3}^{2} - 2 T_{3} + 4 \)
\( T_{13} - 5 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \)
$37$ \( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( ( 1 - 3 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 10 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )^{2} \)
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