# Properties

 Label 960.1.cl.a Level $960$ Weight $1$ Character orbit 960.cl Analytic conductor $0.479$ Analytic rank $0$ Dimension $16$ Projective image $D_{16}$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 960.cl (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.479102412128$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{16})$$ Coefficient field: $$\Q(\zeta_{32})$$ Defining polynomial: $$x^{16} + 1$$ x^16 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{16}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{32}^{13} q^{2} + \zeta_{32} q^{3} - \zeta_{32}^{10} q^{4} + \zeta_{32}^{11} q^{5} - \zeta_{32}^{14} q^{6} - \zeta_{32}^{7} q^{8} + \zeta_{32}^{2} q^{9} +O(q^{10})$$ q - z^13 * q^2 + z * q^3 - z^10 * q^4 + z^11 * q^5 - z^14 * q^6 - z^7 * q^8 + z^2 * q^9 $$q - \zeta_{32}^{13} q^{2} + \zeta_{32} q^{3} - \zeta_{32}^{10} q^{4} + \zeta_{32}^{11} q^{5} - \zeta_{32}^{14} q^{6} - \zeta_{32}^{7} q^{8} + \zeta_{32}^{2} q^{9} + \zeta_{32}^{8} q^{10} - \zeta_{32}^{11} q^{12} + \zeta_{32}^{12} q^{15} - \zeta_{32}^{4} q^{16} + (\zeta_{32}^{15} - \zeta_{32}^{9}) q^{17} - \zeta_{32}^{15} q^{18} + (\zeta_{32}^{6} - \zeta_{32}^{4}) q^{19} + \zeta_{32}^{5} q^{20} + (\zeta_{32}^{13} + \zeta_{32}^{7}) q^{23} - \zeta_{32}^{8} q^{24} - \zeta_{32}^{6} q^{25} + \zeta_{32}^{3} q^{27} + \zeta_{32}^{9} q^{30} + ( - \zeta_{32}^{14} - \zeta_{32}^{2}) q^{31} - \zeta_{32} q^{32} + (\zeta_{32}^{12} - \zeta_{32}^{6}) q^{34} - \zeta_{32}^{12} q^{36} + (\zeta_{32}^{3} - \zeta_{32}) q^{38} + \zeta_{32}^{2} q^{40} + \zeta_{32}^{13} q^{45} + (\zeta_{32}^{10} + \zeta_{32}^{4}) q^{46} + ( - \zeta_{32}^{5} - \zeta_{32}^{3}) q^{47} - \zeta_{32}^{5} q^{48} - \zeta_{32}^{12} q^{49} - \zeta_{32}^{3} q^{50} + ( - \zeta_{32}^{10} - 1) q^{51} + (\zeta_{32}^{9} + \zeta_{32}^{5}) q^{53} + q^{54} + (\zeta_{32}^{7} - \zeta_{32}^{5}) q^{57} + \zeta_{32}^{6} q^{60} + (\zeta_{32}^{10} - \zeta_{32}^{8}) q^{61} + (\zeta_{32}^{15} - \zeta_{32}^{11}) q^{62} + \zeta_{32}^{14} q^{64} + (\zeta_{32}^{9} - \zeta_{32}^{3}) q^{68} + (\zeta_{32}^{14} + \zeta_{32}^{8}) q^{69} - \zeta_{32}^{9} q^{72} - \zeta_{32}^{7} q^{75} + (\zeta_{32}^{14} + 1) q^{76} + (\zeta_{32}^{8} - 1) q^{79} - \zeta_{32}^{15} q^{80} + \zeta_{32}^{4} q^{81} + (\zeta_{32}^{15} + \zeta_{32}^{11}) q^{83} + ( - \zeta_{32}^{10} + \zeta_{32}^{4}) q^{85} + \zeta_{32}^{10} q^{90} + (\zeta_{32}^{7} + \zeta_{32}) q^{92} + ( - \zeta_{32}^{15} - \zeta_{32}^{3}) q^{93} + ( - \zeta_{32}^{2} - 1) q^{94} + ( - \zeta_{32}^{15} - \zeta_{32}) q^{95} - \zeta_{32}^{2} q^{96} - \zeta_{32}^{9} q^{98} +O(q^{100})$$ q - z^13 * q^2 + z * q^3 - z^10 * q^4 + z^11 * q^5 - z^14 * q^6 - z^7 * q^8 + z^2 * q^9 + z^8 * q^10 - z^11 * q^12 + z^12 * q^15 - z^4 * q^16 + (z^15 - z^9) * q^17 - z^15 * q^18 + (z^6 - z^4) * q^19 + z^5 * q^20 + (z^13 + z^7) * q^23 - z^8 * q^24 - z^6 * q^25 + z^3 * q^27 + z^9 * q^30 + (-z^14 - z^2) * q^31 - z * q^32 + (z^12 - z^6) * q^34 - z^12 * q^36 + (z^3 - z) * q^38 + z^2 * q^40 + z^13 * q^45 + (z^10 + z^4) * q^46 + (-z^5 - z^3) * q^47 - z^5 * q^48 - z^12 * q^49 - z^3 * q^50 + (-z^10 - 1) * q^51 + (z^9 + z^5) * q^53 + q^54 + (z^7 - z^5) * q^57 + z^6 * q^60 + (z^10 - z^8) * q^61 + (z^15 - z^11) * q^62 + z^14 * q^64 + (z^9 - z^3) * q^68 + (z^14 + z^8) * q^69 - z^9 * q^72 - z^7 * q^75 + (z^14 + 1) * q^76 + (z^8 - 1) * q^79 - z^15 * q^80 + z^4 * q^81 + (z^15 + z^11) * q^83 + (-z^10 + z^4) * q^85 + z^10 * q^90 + (z^7 + z) * q^92 + (-z^15 - z^3) * q^93 + (-z^2 - 1) * q^94 + (-z^15 - z) * q^95 - z^2 * q^96 - z^9 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q+O(q^{10})$$ 16 * q $$16 q - 16 q^{51} + 16 q^{54} + 16 q^{76} - 16 q^{79} - 16 q^{94}+O(q^{100})$$ 16 * q - 16 * q^51 + 16 * q^54 + 16 * q^76 - 16 * q^79 - 16 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$577$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-\zeta_{32}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 −0.980785 − 0.195090i 0.980785 + 0.195090i 0.195090 + 0.980785i −0.195090 − 0.980785i 0.831470 − 0.555570i −0.831470 + 0.555570i 0.831470 + 0.555570i −0.831470 − 0.555570i 0.195090 − 0.980785i −0.195090 + 0.980785i −0.980785 + 0.195090i 0.980785 − 0.195090i 0.555570 + 0.831470i −0.555570 − 0.831470i 0.555570 − 0.831470i −0.555570 + 0.831470i
−0.831470 + 0.555570i −0.980785 0.195090i 0.382683 0.923880i 0.555570 0.831470i 0.923880 0.382683i 0 0.195090 + 0.980785i 0.923880 + 0.382683i 1.00000i
29.2 0.831470 0.555570i 0.980785 + 0.195090i 0.382683 0.923880i −0.555570 + 0.831470i 0.923880 0.382683i 0 −0.195090 0.980785i 0.923880 + 0.382683i 1.00000i
149.1 −0.555570 + 0.831470i 0.195090 + 0.980785i −0.382683 0.923880i −0.831470 + 0.555570i −0.923880 0.382683i 0 0.980785 + 0.195090i −0.923880 + 0.382683i 1.00000i
149.2 0.555570 0.831470i −0.195090 0.980785i −0.382683 0.923880i 0.831470 0.555570i −0.923880 0.382683i 0 −0.980785 0.195090i −0.923880 + 0.382683i 1.00000i
269.1 −0.195090 + 0.980785i 0.831470 0.555570i −0.923880 0.382683i 0.980785 0.195090i 0.382683 + 0.923880i 0 0.555570 0.831470i 0.382683 0.923880i 1.00000i
269.2 0.195090 0.980785i −0.831470 + 0.555570i −0.923880 0.382683i −0.980785 + 0.195090i 0.382683 + 0.923880i 0 −0.555570 + 0.831470i 0.382683 0.923880i 1.00000i
389.1 −0.195090 0.980785i 0.831470 + 0.555570i −0.923880 + 0.382683i 0.980785 + 0.195090i 0.382683 0.923880i 0 0.555570 + 0.831470i 0.382683 + 0.923880i 1.00000i
389.2 0.195090 + 0.980785i −0.831470 0.555570i −0.923880 + 0.382683i −0.980785 0.195090i 0.382683 0.923880i 0 −0.555570 0.831470i 0.382683 + 0.923880i 1.00000i
509.1 −0.555570 0.831470i 0.195090 0.980785i −0.382683 + 0.923880i −0.831470 0.555570i −0.923880 + 0.382683i 0 0.980785 0.195090i −0.923880 0.382683i 1.00000i
509.2 0.555570 + 0.831470i −0.195090 + 0.980785i −0.382683 + 0.923880i 0.831470 + 0.555570i −0.923880 + 0.382683i 0 −0.980785 + 0.195090i −0.923880 0.382683i 1.00000i
629.1 −0.831470 0.555570i −0.980785 + 0.195090i 0.382683 + 0.923880i 0.555570 + 0.831470i 0.923880 + 0.382683i 0 0.195090 0.980785i 0.923880 0.382683i 1.00000i
629.2 0.831470 + 0.555570i 0.980785 0.195090i 0.382683 + 0.923880i −0.555570 0.831470i 0.923880 + 0.382683i 0 −0.195090 + 0.980785i 0.923880 0.382683i 1.00000i
749.1 −0.980785 0.195090i 0.555570 + 0.831470i 0.923880 + 0.382683i −0.195090 0.980785i −0.382683 0.923880i 0 −0.831470 0.555570i −0.382683 + 0.923880i 1.00000i
749.2 0.980785 + 0.195090i −0.555570 0.831470i 0.923880 + 0.382683i 0.195090 + 0.980785i −0.382683 0.923880i 0 0.831470 + 0.555570i −0.382683 + 0.923880i 1.00000i
869.1 −0.980785 + 0.195090i 0.555570 0.831470i 0.923880 0.382683i −0.195090 + 0.980785i −0.382683 + 0.923880i 0 −0.831470 + 0.555570i −0.382683 0.923880i 1.00000i
869.2 0.980785 0.195090i −0.555570 + 0.831470i 0.923880 0.382683i 0.195090 0.980785i −0.382683 + 0.923880i 0 0.831470 0.555570i −0.382683 0.923880i 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 869.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
64.i even 16 1 inner
192.q odd 16 1 inner
320.bf even 16 1 inner
960.cl odd 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.1.cl.a 16
3.b odd 2 1 inner 960.1.cl.a 16
4.b odd 2 1 3840.1.cl.a 16
5.b even 2 1 inner 960.1.cl.a 16
12.b even 2 1 3840.1.cl.a 16
15.d odd 2 1 CM 960.1.cl.a 16
20.d odd 2 1 3840.1.cl.a 16
60.h even 2 1 3840.1.cl.a 16
64.i even 16 1 inner 960.1.cl.a 16
64.j odd 16 1 3840.1.cl.a 16
192.q odd 16 1 inner 960.1.cl.a 16
192.s even 16 1 3840.1.cl.a 16
320.bf even 16 1 inner 960.1.cl.a 16
320.bh odd 16 1 3840.1.cl.a 16
960.cl odd 16 1 inner 960.1.cl.a 16
960.cp even 16 1 3840.1.cl.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.1.cl.a 16 1.a even 1 1 trivial
960.1.cl.a 16 3.b odd 2 1 inner
960.1.cl.a 16 5.b even 2 1 inner
960.1.cl.a 16 15.d odd 2 1 CM
960.1.cl.a 16 64.i even 16 1 inner
960.1.cl.a 16 192.q odd 16 1 inner
960.1.cl.a 16 320.bf even 16 1 inner
960.1.cl.a 16 960.cl odd 16 1 inner
3840.1.cl.a 16 4.b odd 2 1
3840.1.cl.a 16 12.b even 2 1
3840.1.cl.a 16 20.d odd 2 1
3840.1.cl.a 16 60.h even 2 1
3840.1.cl.a 16 64.j odd 16 1
3840.1.cl.a 16 192.s even 16 1
3840.1.cl.a 16 320.bh odd 16 1
3840.1.cl.a 16 960.cp even 16 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(960, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 1$$
$3$ $$T^{16} + 1$$
$5$ $$T^{16} + 1$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$T^{16}$$
$17$ $$T^{16} + 24 T^{12} + 148 T^{8} + 176 T^{4} + \cdots + 4$$
$19$ $$(T^{8} - 8 T^{5} + 2 T^{4} + 12 T^{2} + 8 T + 2)^{2}$$
$23$ $$T^{16} + 16 T^{10} + 140 T^{8} + 192 T^{6} + \cdots + 4$$
$29$ $$T^{16}$$
$31$ $$(T^{4} + 4 T^{2} + 2)^{4}$$
$37$ $$T^{16}$$
$41$ $$T^{16}$$
$43$ $$T^{16}$$
$47$ $$T^{16} + 24 T^{12} + 148 T^{8} + 176 T^{4} + \cdots + 4$$
$53$ $$T^{16} - 16 T^{12} + 128 T^{8} + \cdots + 16$$
$59$ $$T^{16}$$
$61$ $$(T^{8} + 4 T^{6} + 6 T^{4} - 8 T^{3} + 4 T^{2} + \cdots + 2)^{2}$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$T^{16}$$
$79$ $$(T^{2} + 2 T + 2)^{8}$$
$83$ $$T^{16} + 16 T^{12} + 128 T^{8} + \cdots + 16$$
$89$ $$T^{16}$$
$97$ $$T^{16}$$