Properties

Label 70.2.e.c
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.

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

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 −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.00000 −0.500000 2.59808i −1.00000 1.00000 + 1.73205i −0.500000 + 0.866025i
51.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i −1.00000 −0.500000 + 2.59808i −1.00000 1.00000 1.73205i −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.c 2
3.b odd 2 1 630.2.k.b 2
4.b odd 2 1 560.2.q.g 2
5.b even 2 1 350.2.e.e 2
5.c odd 4 2 350.2.j.b 4
7.b odd 2 1 490.2.e.h 2
7.c even 3 1 inner 70.2.e.c 2
7.c even 3 1 490.2.a.c 1
7.d odd 6 1 490.2.a.b 1
7.d odd 6 1 490.2.e.h 2
21.g even 6 1 4410.2.a.bd 1
21.h odd 6 1 630.2.k.b 2
21.h odd 6 1 4410.2.a.bm 1
28.f even 6 1 3920.2.a.bc 1
28.g odd 6 1 560.2.q.g 2
28.g odd 6 1 3920.2.a.p 1
35.i odd 6 1 2450.2.a.bc 1
35.j even 6 1 350.2.e.e 2
35.j even 6 1 2450.2.a.w 1
35.k even 12 2 2450.2.c.l 2
35.l odd 12 2 350.2.j.b 4
35.l odd 12 2 2450.2.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.c 2 1.a even 1 1 trivial
70.2.e.c 2 7.c even 3 1 inner
350.2.e.e 2 5.b even 2 1
350.2.e.e 2 35.j even 6 1
350.2.j.b 4 5.c odd 4 2
350.2.j.b 4 35.l odd 12 2
490.2.a.b 1 7.d odd 6 1
490.2.a.c 1 7.c even 3 1
490.2.e.h 2 7.b odd 2 1
490.2.e.h 2 7.d odd 6 1
560.2.q.g 2 4.b odd 2 1
560.2.q.g 2 28.g odd 6 1
630.2.k.b 2 3.b odd 2 1
630.2.k.b 2 21.h odd 6 1
2450.2.a.w 1 35.j even 6 1
2450.2.a.bc 1 35.i odd 6 1
2450.2.c.g 2 35.l odd 12 2
2450.2.c.l 2 35.k even 12 2
3920.2.a.p 1 28.g odd 6 1
3920.2.a.bc 1 28.f even 6 1
4410.2.a.bd 1 21.g even 6 1
4410.2.a.bm 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 \(S_{2}^{\mathrm{new}}(70, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 7 + T + T^{2} \)
$11$ \( 36 - 6 T + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( 9 - 3 T + T^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( 64 + 8 T + T^{2} \)
$37$ \( 16 - 4 T + T^{2} \)
$41$ \( ( -9 + T )^{2} \)
$43$ \( ( 7 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( 36 - 6 T + T^{2} \)
$61$ \( 25 + 5 T + T^{2} \)
$67$ \( 25 + 5 T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( 256 - 16 T + T^{2} \)
$79$ \( 4 + 2 T + T^{2} \)
$83$ \( ( -3 + T )^{2} \)
$89$ \( 225 - 15 T + T^{2} \)
$97$ \( ( -14 + T )^{2} \)
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