Properties

Label 70.2.e.d
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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
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.

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

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

Hecke kernels

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

\( T_{3}^{2} - 2 T_{3} + 4 \)
\( T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + T + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \)
$37$ \( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 3 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 2 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{2} \)
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