Properties

Label 70.2.i.a
Level $70$
Weight $2$
Character orbit 70.i
Analytic conductor $0.559$
Analytic rank $0$
Dimension $4$
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.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.558952814149\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(31\) \(57\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i 2.59808 1.50000i 0.500000 + 0.866025i −2.23205 0.133975i −3.00000 −0.866025 + 2.50000i 1.00000i 3.00000 5.19615i 1.86603 + 1.23205i
9.2 0.866025 + 0.500000i −2.59808 + 1.50000i 0.500000 + 0.866025i 1.23205 + 1.86603i −3.00000 0.866025 2.50000i 1.00000i 3.00000 5.19615i 0.133975 + 2.23205i
39.1 −0.866025 + 0.500000i 2.59808 + 1.50000i 0.500000 0.866025i −2.23205 + 0.133975i −3.00000 −0.866025 2.50000i 1.00000i 3.00000 + 5.19615i 1.86603 1.23205i
39.2 0.866025 0.500000i −2.59808 1.50000i 0.500000 0.866025i 1.23205 1.86603i −3.00000 0.866025 + 2.50000i 1.00000i 3.00000 + 5.19615i 0.133975 2.23205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.2.i.a 4
3.b odd 2 1 630.2.u.b 4
4.b odd 2 1 560.2.bw.c 4
5.b even 2 1 inner 70.2.i.a 4
5.c odd 4 1 350.2.e.f 2
5.c odd 4 1 350.2.e.g 2
7.b odd 2 1 490.2.i.b 4
7.c even 3 1 inner 70.2.i.a 4
7.c even 3 1 490.2.c.c 2
7.d odd 6 1 490.2.c.b 2
7.d odd 6 1 490.2.i.b 4
15.d odd 2 1 630.2.u.b 4
20.d odd 2 1 560.2.bw.c 4
21.h odd 6 1 630.2.u.b 4
28.g odd 6 1 560.2.bw.c 4
35.c odd 2 1 490.2.i.b 4
35.i odd 6 1 490.2.c.b 2
35.i odd 6 1 490.2.i.b 4
35.j even 6 1 inner 70.2.i.a 4
35.j even 6 1 490.2.c.c 2
35.k even 12 1 2450.2.a.c 1
35.k even 12 1 2450.2.a.bh 1
35.l odd 12 1 350.2.e.f 2
35.l odd 12 1 350.2.e.g 2
35.l odd 12 1 2450.2.a.r 1
35.l odd 12 1 2450.2.a.s 1
105.o odd 6 1 630.2.u.b 4
140.p odd 6 1 560.2.bw.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.a 4 1.a even 1 1 trivial
70.2.i.a 4 5.b even 2 1 inner
70.2.i.a 4 7.c even 3 1 inner
70.2.i.a 4 35.j even 6 1 inner
350.2.e.f 2 5.c odd 4 1
350.2.e.f 2 35.l odd 12 1
350.2.e.g 2 5.c odd 4 1
350.2.e.g 2 35.l odd 12 1
490.2.c.b 2 7.d odd 6 1
490.2.c.b 2 35.i odd 6 1
490.2.c.c 2 7.c even 3 1
490.2.c.c 2 35.j even 6 1
490.2.i.b 4 7.b odd 2 1
490.2.i.b 4 7.d odd 6 1
490.2.i.b 4 35.c odd 2 1
490.2.i.b 4 35.i odd 6 1
560.2.bw.c 4 4.b odd 2 1
560.2.bw.c 4 20.d odd 2 1
560.2.bw.c 4 28.g odd 6 1
560.2.bw.c 4 140.p odd 6 1
630.2.u.b 4 3.b odd 2 1
630.2.u.b 4 15.d odd 2 1
630.2.u.b 4 21.h odd 6 1
630.2.u.b 4 105.o odd 6 1
2450.2.a.c 1 35.k even 12 1
2450.2.a.r 1 35.l odd 12 1
2450.2.a.s 1 35.l odd 12 1
2450.2.a.bh 1 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 9 T_{3}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(70, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 81 - 9 T^{2} + T^{4} \)
$5$ \( 25 + 10 T - T^{2} + 2 T^{3} + T^{4} \)
$7$ \( 49 + 11 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( 16 - 4 T^{2} + T^{4} \)
$19$ \( ( 4 + 2 T + T^{2} )^{2} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( -1 + T )^{4} \)
$31$ \( ( 100 + 10 T + T^{2} )^{2} \)
$37$ \( 4096 - 64 T^{2} + T^{4} \)
$41$ \( ( 3 + T )^{4} \)
$43$ \( ( 25 + T^{2} )^{2} \)
$47$ \( 4096 - 64 T^{2} + T^{4} \)
$53$ \( 1296 - 36 T^{2} + T^{4} \)
$59$ \( ( 4 - 2 T + T^{2} )^{2} \)
$61$ \( ( 81 - 9 T + T^{2} )^{2} \)
$67$ \( 2401 - 49 T^{2} + T^{4} \)
$71$ \( ( -6 + T )^{4} \)
$73$ \( 10000 - 100 T^{2} + T^{4} \)
$79$ \( ( 100 + 10 T + T^{2} )^{2} \)
$83$ \( ( 81 + T^{2} )^{2} \)
$89$ \( ( 49 + 7 T + T^{2} )^{2} \)
$97$ \( T^{4} \)
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