# Properties

 Label 960.3.c.c Level $960$ Weight $3$ Character orbit 960.c Self dual yes Analytic conductor $26.158$ Analytic rank $0$ Dimension $1$ CM discriminant -15 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 960.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.1581053786$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 5 q^{5} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 5 * q^5 + 9 * q^9 $$q + 3 q^{3} - 5 q^{5} + 9 q^{9} - 15 q^{15} - 14 q^{17} + 22 q^{19} + 34 q^{23} + 25 q^{25} + 27 q^{27} + 2 q^{31} - 45 q^{45} - 14 q^{47} + 49 q^{49} - 42 q^{51} + 86 q^{53} + 66 q^{57} + 118 q^{61} + 102 q^{69} + 75 q^{75} + 98 q^{79} + 81 q^{81} - 154 q^{83} + 70 q^{85} + 6 q^{93} - 110 q^{95}+O(q^{100})$$ q + 3 * q^3 - 5 * q^5 + 9 * q^9 - 15 * q^15 - 14 * q^17 + 22 * q^19 + 34 * q^23 + 25 * q^25 + 27 * q^27 + 2 * q^31 - 45 * q^45 - 14 * q^47 + 49 * q^49 - 42 * q^51 + 86 * q^53 + 66 * q^57 + 118 * q^61 + 102 * q^69 + 75 * q^75 + 98 * q^79 + 81 * q^81 - 154 * q^83 + 70 * q^85 + 6 * q^93 - 110 * q^95

## Character values

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

 $$n$$ $$511$$ $$577$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0
0 3.00000 0 −5.00000 0 0 0 9.00000 0
 $$n$$: 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.c.c 1
3.b odd 2 1 960.3.c.b 1
4.b odd 2 1 960.3.c.a 1
5.b even 2 1 960.3.c.b 1
8.b even 2 1 15.3.d.b yes 1
8.d odd 2 1 240.3.c.b 1
12.b even 2 1 960.3.c.d 1
15.d odd 2 1 CM 960.3.c.c 1
20.d odd 2 1 960.3.c.d 1
24.f even 2 1 240.3.c.a 1
24.h odd 2 1 15.3.d.a 1
40.e odd 2 1 240.3.c.a 1
40.f even 2 1 15.3.d.a 1
40.i odd 4 2 75.3.c.d 2
40.k even 4 2 1200.3.l.l 2
60.h even 2 1 960.3.c.a 1
72.j odd 6 2 405.3.h.b 2
72.n even 6 2 405.3.h.a 2
120.i odd 2 1 15.3.d.b yes 1
120.m even 2 1 240.3.c.b 1
120.q odd 4 2 1200.3.l.l 2
120.w even 4 2 75.3.c.d 2
360.bh odd 6 2 405.3.h.a 2
360.bk even 6 2 405.3.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.d.a 1 24.h odd 2 1
15.3.d.a 1 40.f even 2 1
15.3.d.b yes 1 8.b even 2 1
15.3.d.b yes 1 120.i odd 2 1
75.3.c.d 2 40.i odd 4 2
75.3.c.d 2 120.w even 4 2
240.3.c.a 1 24.f even 2 1
240.3.c.a 1 40.e odd 2 1
240.3.c.b 1 8.d odd 2 1
240.3.c.b 1 120.m even 2 1
405.3.h.a 2 72.n even 6 2
405.3.h.a 2 360.bh odd 6 2
405.3.h.b 2 72.j odd 6 2
405.3.h.b 2 360.bk even 6 2
960.3.c.a 1 4.b odd 2 1
960.3.c.a 1 60.h even 2 1
960.3.c.b 1 3.b odd 2 1
960.3.c.b 1 5.b even 2 1
960.3.c.c 1 1.a even 1 1 trivial
960.3.c.c 1 15.d odd 2 1 CM
960.3.c.d 1 12.b even 2 1
960.3.c.d 1 20.d odd 2 1
1200.3.l.l 2 40.k even 4 2
1200.3.l.l 2 120.q odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(960, [\chi])$$:

 $$T_{7}$$ T7 $$T_{17} + 14$$ T17 + 14 $$T_{19} - 22$$ T19 - 22

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 5$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 14$$
$19$ $$T - 22$$
$23$ $$T - 34$$
$29$ $$T$$
$31$ $$T - 2$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T + 14$$
$53$ $$T - 86$$
$59$ $$T$$
$61$ $$T - 118$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T - 98$$
$83$ $$T + 154$$
$89$ $$T$$
$97$ $$T$$