# Properties

 Label 2960.1.gf.a Level $2960$ Weight $1$ Character orbit 2960.gf Analytic conductor $1.477$ Analytic rank $0$ Dimension $12$ Projective image $D_{36}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2960.gf (of order $$36$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.47723243739$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{36})$$ Defining polynomial: $$x^{12} - x^{6} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{36}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{36} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{36}^{2} q^{5} -\zeta_{36}^{17} q^{9} +O(q^{10})$$ $$q -\zeta_{36}^{2} q^{5} -\zeta_{36}^{17} q^{9} -\zeta_{36} q^{13} + ( -\zeta_{36} - \zeta_{36}^{15} ) q^{17} + \zeta_{36}^{4} q^{25} + ( -\zeta_{36}^{7} - \zeta_{36}^{14} ) q^{29} -\zeta_{36}^{10} q^{37} + ( \zeta_{36}^{8} - \zeta_{36}^{12} ) q^{41} -\zeta_{36} q^{45} -\zeta_{36}^{5} q^{49} + ( -\zeta_{36}^{5} - \zeta_{36}^{8} ) q^{53} + ( 1 + \zeta_{36}^{11} ) q^{61} + \zeta_{36}^{3} q^{65} + ( -\zeta_{36}^{3} + \zeta_{36}^{6} ) q^{73} -\zeta_{36}^{16} q^{81} + ( \zeta_{36}^{3} + \zeta_{36}^{17} ) q^{85} + ( -\zeta_{36}^{4} - \zeta_{36}^{9} ) q^{89} + ( -\zeta_{36}^{2} + \zeta_{36}^{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + O(q^{10})$$ $$12 q + 6 q^{41} + 12 q^{61} + 6 q^{73} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2960\mathbb{Z}\right)^\times$$.

 $$n$$ $$741$$ $$1777$$ $$2481$$ $$2591$$ $$\chi(n)$$ $$1$$ $$-\zeta_{36}^{9}$$ $$-\zeta_{36}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
383.1
 0.342020 − 0.939693i −0.984808 + 0.173648i 0.642788 − 0.766044i −0.984808 − 0.173648i 0.342020 + 0.939693i 0.642788 + 0.766044i −0.642788 − 0.766044i −0.342020 − 0.939693i 0.984808 + 0.173648i −0.642788 + 0.766044i 0.984808 − 0.173648i −0.342020 + 0.939693i
0 0 0 0.766044 + 0.642788i 0 0 0 0.342020 + 0.939693i 0
463.1 0 0 0 −0.939693 + 0.342020i 0 0 0 −0.984808 0.173648i 0
607.1 0 0 0 0.173648 + 0.984808i 0 0 0 0.642788 + 0.766044i 0
927.1 0 0 0 −0.939693 0.342020i 0 0 0 −0.984808 + 0.173648i 0
1167.1 0 0 0 0.766044 0.642788i 0 0 0 0.342020 0.939693i 0
1263.1 0 0 0 0.173648 0.984808i 0 0 0 0.642788 0.766044i 0
1327.1 0 0 0 0.173648 0.984808i 0 0 0 −0.642788 + 0.766044i 0
1423.1 0 0 0 0.766044 0.642788i 0 0 0 −0.342020 + 0.939693i 0
1663.1 0 0 0 −0.939693 0.342020i 0 0 0 0.984808 0.173648i 0
1983.1 0 0 0 0.173648 + 0.984808i 0 0 0 −0.642788 0.766044i 0
2127.1 0 0 0 −0.939693 + 0.342020i 0 0 0 0.984808 + 0.173648i 0
2207.1 0 0 0 0.766044 + 0.642788i 0 0 0 −0.342020 0.939693i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2207.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
185.bc even 36 1 inner
740.bw odd 36 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.gf.a 12
4.b odd 2 1 CM 2960.1.gf.a 12
5.c odd 4 1 2960.1.go.a yes 12
20.e even 4 1 2960.1.go.a yes 12
37.i odd 36 1 2960.1.go.a yes 12
148.q even 36 1 2960.1.go.a yes 12
185.bc even 36 1 inner 2960.1.gf.a 12
740.bw odd 36 1 inner 2960.1.gf.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.gf.a 12 1.a even 1 1 trivial
2960.1.gf.a 12 4.b odd 2 1 CM
2960.1.gf.a 12 185.bc even 36 1 inner
2960.1.gf.a 12 740.bw odd 36 1 inner
2960.1.go.a yes 12 5.c odd 4 1
2960.1.go.a yes 12 20.e even 4 1
2960.1.go.a yes 12 37.i odd 36 1
2960.1.go.a yes 12 148.q even 36 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2960, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$( 1 + T^{3} + T^{6} )^{2}$$
$7$ $$T^{12}$$
$11$ $$T^{12}$$
$13$ $$1 - T^{6} + T^{12}$$
$17$ $$9 + 27 T^{2} + 9 T^{4} - 24 T^{6} + 18 T^{8} - 3 T^{10} + T^{12}$$
$19$ $$T^{12}$$
$23$ $$T^{12}$$
$29$ $$1 - 6 T + 45 T^{2} - 110 T^{3} + 105 T^{4} - 36 T^{5} + 2 T^{6} + 6 T^{7} - 9 T^{8} + 2 T^{9} + T^{12}$$
$31$ $$T^{12}$$
$37$ $$( 1 + T^{3} + T^{6} )^{2}$$
$41$ $$( 3 - 9 T + 9 T^{2} - 6 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} )^{2}$$
$43$ $$T^{12}$$
$47$ $$T^{12}$$
$53$ $$1 - 4 T^{3} + 53 T^{6} - 14 T^{9} + T^{12}$$
$59$ $$T^{12}$$
$61$ $$1 - 6 T + 51 T^{2} - 200 T^{3} + 480 T^{4} - 786 T^{5} + 923 T^{6} - 792 T^{7} + 495 T^{8} - 220 T^{9} + 66 T^{10} - 12 T^{11} + T^{12}$$
$67$ $$T^{12}$$
$71$ $$T^{12}$$
$73$ $$( 1 + 2 T + 2 T^{2} - 2 T^{3} + T^{4} )^{3}$$
$79$ $$T^{12}$$
$83$ $$T^{12}$$
$89$ $$1 + 12 T + 39 T^{2} + 14 T^{3} - 12 T^{4} + 12 T^{5} + 23 T^{6} + 15 T^{8} - 2 T^{9} + 6 T^{10} + T^{12}$$
$97$ $$( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} )^{2}$$