# Properties

 Label 1560.1.cs.f Level $1560$ Weight $1$ Character orbit 1560.cs Analytic conductor $0.779$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ CM discriminant -104 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1560.cs (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.778541419707$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{24}^{9} q^{2} + \zeta_{24}^{2} q^{3} - \zeta_{24}^{6} q^{4} - \zeta_{24}^{11} q^{5} - \zeta_{24}^{11} q^{6} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{7} - \zeta_{24}^{3} q^{8} + \zeta_{24}^{4} q^{9} +O(q^{10})$$ q - z^9 * q^2 + z^2 * q^3 - z^6 * q^4 - z^11 * q^5 - z^11 * q^6 + (-z^5 - z) * q^7 - z^3 * q^8 + z^4 * q^9 $$q - \zeta_{24}^{9} q^{2} + \zeta_{24}^{2} q^{3} - \zeta_{24}^{6} q^{4} - \zeta_{24}^{11} q^{5} - \zeta_{24}^{11} q^{6} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{7} - \zeta_{24}^{3} q^{8} + \zeta_{24}^{4} q^{9} - \zeta_{24}^{8} q^{10} - \zeta_{24}^{8} q^{12} + \zeta_{24}^{9} q^{13} + (\zeta_{24}^{10} - \zeta_{24}^{2}) q^{14} + \zeta_{24} q^{15} - q^{16} + ( - \zeta_{24}^{4} + \zeta_{24}^{2}) q^{17} + \zeta_{24} q^{18} - \zeta_{24}^{5} q^{20} + ( - \zeta_{24}^{7} - \zeta_{24}^{3}) q^{21} - \zeta_{24}^{5} q^{24} - \zeta_{24}^{10} q^{25} + \zeta_{24}^{6} q^{26} + \zeta_{24}^{6} q^{27} + (\zeta_{24}^{11} + \zeta_{24}^{7}) q^{28} - \zeta_{24}^{10} q^{30} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{31} + \zeta_{24}^{9} q^{32} + ( - \zeta_{24}^{11} - \zeta_{24}) q^{34} + ( - \zeta_{24}^{4} - 1) q^{35} - \zeta_{24}^{10} q^{36} + \zeta_{24}^{3} q^{37} + \zeta_{24}^{11} q^{39} - \zeta_{24}^{2} q^{40} + ( - \zeta_{24}^{4} - 1) q^{42} + (\zeta_{24}^{10} + \zeta_{24}^{8}) q^{43} + \zeta_{24}^{3} q^{45} + (\zeta_{24}^{11} + \zeta_{24}^{7}) q^{47} - \zeta_{24}^{2} q^{48} + (\zeta_{24}^{10} + \zeta_{24}^{6} + \zeta_{24}^{2}) q^{49} - \zeta_{24}^{7} q^{50} + ( - \zeta_{24}^{6} + \zeta_{24}^{4}) q^{51} + \zeta_{24}^{3} q^{52} + \zeta_{24}^{3} q^{54} + (\zeta_{24}^{8} + \zeta_{24}^{4}) q^{56} - \zeta_{24}^{7} q^{60} + ( - \zeta_{24}^{6} + 1) q^{62} + ( - \zeta_{24}^{9} - \zeta_{24}^{5}) q^{63} + \zeta_{24}^{6} q^{64} + \zeta_{24}^{8} q^{65} + (\zeta_{24}^{10} - \zeta_{24}^{8}) q^{68} + (\zeta_{24}^{9} - \zeta_{24}) q^{70} + ( - \zeta_{24}^{7} - \zeta_{24}^{5}) q^{71} - \zeta_{24}^{7} q^{72} + q^{74} + q^{75} + \zeta_{24}^{8} q^{78} + \zeta_{24}^{11} q^{80} + \zeta_{24}^{8} q^{81} + (\zeta_{24}^{9} - \zeta_{24}) q^{84} + ( - \zeta_{24}^{3} + \zeta_{24}) q^{85} + (\zeta_{24}^{7} + \zeta_{24}^{5}) q^{86} + q^{90} + ( - \zeta_{24}^{10} + \zeta_{24}^{2}) q^{91} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{93} + (\zeta_{24}^{8} + \zeta_{24}^{4}) q^{94} + \zeta_{24}^{11} q^{96} + ( - \zeta_{24}^{11} + \zeta_{24}^{7} + \zeta_{24}^{3}) q^{98} +O(q^{100})$$ q - z^9 * q^2 + z^2 * q^3 - z^6 * q^4 - z^11 * q^5 - z^11 * q^6 + (-z^5 - z) * q^7 - z^3 * q^8 + z^4 * q^9 - z^8 * q^10 - z^8 * q^12 + z^9 * q^13 + (z^10 - z^2) * q^14 + z * q^15 - q^16 + (-z^4 + z^2) * q^17 + z * q^18 - z^5 * q^20 + (-z^7 - z^3) * q^21 - z^5 * q^24 - z^10 * q^25 + z^6 * q^26 + z^6 * q^27 + (z^11 + z^7) * q^28 - z^10 * q^30 + (-z^9 + z^3) * q^31 + z^9 * q^32 + (-z^11 - z) * q^34 + (-z^4 - 1) * q^35 - z^10 * q^36 + z^3 * q^37 + z^11 * q^39 - z^2 * q^40 + (-z^4 - 1) * q^42 + (z^10 + z^8) * q^43 + z^3 * q^45 + (z^11 + z^7) * q^47 - z^2 * q^48 + (z^10 + z^6 + z^2) * q^49 - z^7 * q^50 + (-z^6 + z^4) * q^51 + z^3 * q^52 + z^3 * q^54 + (z^8 + z^4) * q^56 - z^7 * q^60 + (-z^6 + 1) * q^62 + (-z^9 - z^5) * q^63 + z^6 * q^64 + z^8 * q^65 + (z^10 - z^8) * q^68 + (z^9 - z) * q^70 + (-z^7 - z^5) * q^71 - z^7 * q^72 + q^74 + q^75 + z^8 * q^78 + z^11 * q^80 + z^8 * q^81 + (z^9 - z) * q^84 + (-z^3 + z) * q^85 + (z^7 + z^5) * q^86 + q^90 + (-z^10 + z^2) * q^91 + (-z^11 + z^5) * q^93 + (z^8 + z^4) * q^94 + z^11 * q^96 + (-z^11 + z^7 + z^3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{9}+O(q^{10})$$ 8 * q + 4 * q^9 $$8 q + 4 q^{9} + 4 q^{10} + 4 q^{12} - 8 q^{16} - 4 q^{17} - 12 q^{35} - 12 q^{42} - 4 q^{43} + 4 q^{51} + 8 q^{62} - 4 q^{65} + 4 q^{68} + 8 q^{74} + 8 q^{75} - 4 q^{78} - 4 q^{81} + 8 q^{90}+O(q^{100})$$ 8 * q + 4 * q^9 + 4 * q^10 + 4 * q^12 - 8 * q^16 - 4 * q^17 - 12 * q^35 - 12 * q^42 - 4 * q^43 + 4 * q^51 + 8 * q^62 - 4 * q^65 + 4 * q^68 + 8 * q^74 + 8 * q^75 - 4 * q^78 - 4 * q^81 + 8 * q^90

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$$.

 $$n$$ $$391$$ $$521$$ $$781$$ $$937$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$\zeta_{24}^{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
467.1
 0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.965926 − 0.258819i
−0.707107 + 0.707107i −0.866025 + 0.500000i 1.00000i 0.258819 0.965926i 0.258819 0.965926i −1.22474 1.22474i 0.707107 + 0.707107i 0.500000 0.866025i 0.500000 + 0.866025i
467.2 −0.707107 + 0.707107i 0.866025 + 0.500000i 1.00000i −0.965926 + 0.258819i −0.965926 + 0.258819i 1.22474 + 1.22474i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.500000 0.866025i
467.3 0.707107 0.707107i −0.866025 + 0.500000i 1.00000i −0.258819 + 0.965926i −0.258819 + 0.965926i 1.22474 + 1.22474i −0.707107 0.707107i 0.500000 0.866025i 0.500000 + 0.866025i
467.4 0.707107 0.707107i 0.866025 + 0.500000i 1.00000i 0.965926 0.258819i 0.965926 0.258819i −1.22474 1.22474i −0.707107 0.707107i 0.500000 + 0.866025i 0.500000 0.866025i
1403.1 −0.707107 0.707107i −0.866025 0.500000i 1.00000i 0.258819 + 0.965926i 0.258819 + 0.965926i −1.22474 + 1.22474i 0.707107 0.707107i 0.500000 + 0.866025i 0.500000 0.866025i
1403.2 −0.707107 0.707107i 0.866025 0.500000i 1.00000i −0.965926 0.258819i −0.965926 0.258819i 1.22474 1.22474i 0.707107 0.707107i 0.500000 0.866025i 0.500000 + 0.866025i
1403.3 0.707107 + 0.707107i −0.866025 0.500000i 1.00000i −0.258819 0.965926i −0.258819 0.965926i 1.22474 1.22474i −0.707107 + 0.707107i 0.500000 + 0.866025i 0.500000 0.866025i
1403.4 0.707107 + 0.707107i 0.866025 0.500000i 1.00000i 0.965926 + 0.258819i 0.965926 + 0.258819i −1.22474 + 1.22474i −0.707107 + 0.707107i 0.500000 0.866025i 0.500000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1403.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.h odd 2 1 CM by $$\Q(\sqrt{-26})$$
8.d odd 2 1 inner
13.b even 2 1 inner
15.e even 4 1 inner
120.q odd 4 1 inner
195.s even 4 1 inner
1560.cs odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.1.cs.f yes 8
3.b odd 2 1 1560.1.cs.c 8
5.c odd 4 1 1560.1.cs.c 8
8.d odd 2 1 inner 1560.1.cs.f yes 8
13.b even 2 1 inner 1560.1.cs.f yes 8
15.e even 4 1 inner 1560.1.cs.f yes 8
24.f even 2 1 1560.1.cs.c 8
39.d odd 2 1 1560.1.cs.c 8
40.k even 4 1 1560.1.cs.c 8
65.h odd 4 1 1560.1.cs.c 8
104.h odd 2 1 CM 1560.1.cs.f yes 8
120.q odd 4 1 inner 1560.1.cs.f yes 8
195.s even 4 1 inner 1560.1.cs.f yes 8
312.h even 2 1 1560.1.cs.c 8
520.bc even 4 1 1560.1.cs.c 8
1560.cs odd 4 1 inner 1560.1.cs.f yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.1.cs.c 8 3.b odd 2 1
1560.1.cs.c 8 5.c odd 4 1
1560.1.cs.c 8 24.f even 2 1
1560.1.cs.c 8 39.d odd 2 1
1560.1.cs.c 8 40.k even 4 1
1560.1.cs.c 8 65.h odd 4 1
1560.1.cs.c 8 312.h even 2 1
1560.1.cs.c 8 520.bc even 4 1
1560.1.cs.f yes 8 1.a even 1 1 trivial
1560.1.cs.f yes 8 8.d odd 2 1 inner
1560.1.cs.f yes 8 13.b even 2 1 inner
1560.1.cs.f yes 8 15.e even 4 1 inner
1560.1.cs.f yes 8 104.h odd 2 1 CM
1560.1.cs.f yes 8 120.q odd 4 1 inner
1560.1.cs.f yes 8 195.s even 4 1 inner
1560.1.cs.f yes 8 1560.cs odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1560, [\chi])$$:

 $$T_{7}^{4} + 9$$ T7^4 + 9 $$T_{11}$$ T11 $$T_{17}^{4} + 2T_{17}^{3} + 2T_{17}^{2} - 2T_{17} + 1$$ T17^4 + 2*T17^3 + 2*T17^2 - 2*T17 + 1 $$T_{103}$$ T103

## Hecke characteristic polynomials

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