# Properties

 Label 2960.1.fm.a Level $2960$ Weight $1$ Character orbit 2960.fm Analytic conductor $1.477$ Analytic rank $0$ Dimension $4$ Projective image $S_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.47723243739$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.0.350464000.5

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} -\zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{5} + q^{6} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12} q^{2} -\zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{5} + q^{6} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{7} + \zeta_{12}^{3} q^{8} - q^{10} + ( 1 + \zeta_{12}^{3} ) q^{11} + \zeta_{12} q^{12} + \zeta_{12}^{5} q^{13} + ( -1 - \zeta_{12}^{3} ) q^{14} + \zeta_{12}^{4} q^{15} + \zeta_{12}^{4} q^{16} -\zeta_{12} q^{20} + ( -\zeta_{12} + \zeta_{12}^{4} ) q^{21} + ( \zeta_{12} + \zeta_{12}^{4} ) q^{22} + ( 1 + \zeta_{12}^{3} ) q^{23} + \zeta_{12}^{2} q^{24} -\zeta_{12}^{4} q^{25} - q^{26} + \zeta_{12}^{3} q^{27} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{28} + \zeta_{12}^{5} q^{30} - q^{31} + \zeta_{12}^{5} q^{32} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{33} + ( \zeta_{12} - \zeta_{12}^{4} ) q^{35} -\zeta_{12}^{5} q^{37} + \zeta_{12}^{4} q^{39} -\zeta_{12}^{2} q^{40} -\zeta_{12}^{5} q^{41} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{42} - q^{43} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{44} + ( \zeta_{12} + \zeta_{12}^{4} ) q^{46} + \zeta_{12}^{3} q^{48} + \zeta_{12} q^{49} -\zeta_{12}^{5} q^{50} -\zeta_{12} q^{52} + \zeta_{12}^{4} q^{53} + \zeta_{12}^{4} q^{54} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{55} + ( -\zeta_{12}^{2} - \zeta_{12}^{5} ) q^{56} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{59} - q^{60} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{61} -\zeta_{12} q^{62} - q^{64} -\zeta_{12}^{4} q^{65} + ( 1 + \zeta_{12}^{3} ) q^{66} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{69} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{70} -2 \zeta_{12}^{5} q^{71} + q^{74} -\zeta_{12}^{3} q^{75} -2 \zeta_{12}^{2} q^{77} + \zeta_{12}^{5} q^{78} -\zeta_{12}^{3} q^{80} + \zeta_{12}^{2} q^{81} + q^{82} + ( -1 - \zeta_{12}^{3} ) q^{84} -\zeta_{12} q^{86} + ( -1 + \zeta_{12}^{3} ) q^{88} + ( \zeta_{12} - \zeta_{12}^{4} ) q^{91} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{92} + \zeta_{12}^{5} q^{93} + \zeta_{12}^{4} q^{96} + \zeta_{12}^{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 4q^{6} - 2q^{7} + O(q^{10})$$ $$4q + 2q^{4} + 4q^{6} - 2q^{7} - 4q^{10} + 4q^{11} - 4q^{14} - 2q^{15} - 2q^{16} - 2q^{21} - 2q^{22} + 4q^{23} + 2q^{24} + 2q^{25} - 4q^{26} + 2q^{28} - 4q^{31} + 2q^{33} + 2q^{35} - 2q^{39} - 2q^{40} - 2q^{42} - 4q^{43} + 2q^{44} - 2q^{46} - 2q^{53} - 2q^{54} - 2q^{55} - 2q^{56} + 2q^{59} - 4q^{60} + 2q^{61} - 4q^{64} + 2q^{65} + 4q^{66} + 2q^{69} + 2q^{70} + 4q^{74} - 4q^{77} + 2q^{81} + 4q^{82} - 4q^{84} - 4q^{88} + 2q^{91} + 2q^{92} - 2q^{96} + 2q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$741$$ $$1777$$ $$2481$$ $$2591$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$\zeta_{12}^{3}$$ $$-\zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1157.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000 −1.36603 0.366025i 1.00000i 0 −1.00000
1453.1 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000 0.366025 1.36603i 1.00000i 0 −1.00000
2357.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000 0.366025 + 1.36603i 1.00000i 0 −1.00000
2653.1 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.866025 0.500000i 1.00000 −1.36603 + 0.366025i 1.00000i 0 −1.00000
 $$n$$: 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
37.c even 3 1 inner
80.i odd 4 1 inner
2960.fm odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.fm.a yes 4
5.c odd 4 1 2960.1.dn.a 4
16.e even 4 1 2960.1.dn.a 4
37.c even 3 1 inner 2960.1.fm.a yes 4
80.i odd 4 1 inner 2960.1.fm.a yes 4
185.s odd 12 1 2960.1.dn.a 4
592.bj even 12 1 2960.1.dn.a 4
2960.fm odd 12 1 inner 2960.1.fm.a yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.dn.a 4 5.c odd 4 1
2960.1.dn.a 4 16.e even 4 1
2960.1.dn.a 4 185.s odd 12 1
2960.1.dn.a 4 592.bj even 12 1
2960.1.fm.a yes 4 1.a even 1 1 trivial
2960.1.fm.a yes 4 37.c even 3 1 inner
2960.1.fm.a yes 4 80.i odd 4 1 inner
2960.1.fm.a yes 4 2960.fm odd 12 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$1 - T^{2} + T^{4}$$
$7$ $$4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$( 2 - 2 T + T^{2} )^{2}$$
$13$ $$1 - T^{2} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 2 - 2 T + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 1 + T )^{4}$$
$37$ $$1 - T^{2} + T^{4}$$
$41$ $$1 - T^{2} + T^{4}$$
$43$ $$( 1 + T )^{4}$$
$47$ $$T^{4}$$
$53$ $$( 1 + T + T^{2} )^{2}$$
$59$ $$4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$61$ $$4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$67$ $$T^{4}$$
$71$ $$16 - 4 T^{2} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$