# 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 $4$

# Related objects

## 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: $$x^{4} - x^{2} + 1$$ 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{3}^{4} - 9T_{3}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(70, [\chi])$$.

## Hecke characteristic polynomials

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