# Properties

 Label 2520.1.hf.a Level $2520$ Weight $1$ Character orbit 2520.hf Analytic conductor $1.258$ Analytic rank $0$ Dimension $16$ Projective image $D_{24}$ CM discriminant -56 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2520.hf (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.25764383184$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{48})$$ Defining polynomial: $$x^{16} - x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{24}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{48}^{2} q^{2} + \zeta_{48}^{15} q^{3} + \zeta_{48}^{4} q^{4} + \zeta_{48}^{21} q^{5} + \zeta_{48}^{17} q^{6} + \zeta_{48}^{14} q^{7} + \zeta_{48}^{6} q^{8} -\zeta_{48}^{6} q^{9} +O(q^{10})$$ $$q + \zeta_{48}^{2} q^{2} + \zeta_{48}^{15} q^{3} + \zeta_{48}^{4} q^{4} + \zeta_{48}^{21} q^{5} + \zeta_{48}^{17} q^{6} + \zeta_{48}^{14} q^{7} + \zeta_{48}^{6} q^{8} -\zeta_{48}^{6} q^{9} + \zeta_{48}^{23} q^{10} + \zeta_{48}^{19} q^{12} + ( -\zeta_{48} - \zeta_{48}^{19} ) q^{13} + \zeta_{48}^{16} q^{14} -\zeta_{48}^{12} q^{15} + \zeta_{48}^{8} q^{16} -\zeta_{48}^{8} q^{18} + ( \zeta_{48}^{7} + \zeta_{48}^{17} ) q^{19} -\zeta_{48} q^{20} -\zeta_{48}^{5} q^{21} + ( -1 + \zeta_{48}^{20} ) q^{23} + \zeta_{48}^{21} q^{24} -\zeta_{48}^{18} q^{25} + ( -\zeta_{48}^{3} - \zeta_{48}^{21} ) q^{26} -\zeta_{48}^{21} q^{27} + \zeta_{48}^{18} q^{28} -\zeta_{48}^{14} q^{30} + \zeta_{48}^{10} q^{32} -\zeta_{48}^{11} q^{35} -\zeta_{48}^{10} q^{36} + ( \zeta_{48}^{9} + \zeta_{48}^{19} ) q^{38} + ( \zeta_{48}^{10} - \zeta_{48}^{16} ) q^{39} -\zeta_{48}^{3} q^{40} -\zeta_{48}^{7} q^{42} + \zeta_{48}^{3} q^{45} + ( -\zeta_{48}^{2} + \zeta_{48}^{22} ) q^{46} + \zeta_{48}^{23} q^{48} -\zeta_{48}^{4} q^{49} -\zeta_{48}^{20} q^{50} + ( -\zeta_{48}^{5} - \zeta_{48}^{23} ) q^{52} -\zeta_{48}^{23} q^{54} + \zeta_{48}^{20} q^{56} + ( -\zeta_{48}^{8} + \zeta_{48}^{22} ) q^{57} + ( -\zeta_{48}^{13} - \zeta_{48}^{19} ) q^{59} -\zeta_{48}^{16} q^{60} + ( \zeta_{48} + \zeta_{48}^{15} ) q^{61} -\zeta_{48}^{20} q^{63} + \zeta_{48}^{12} q^{64} + ( \zeta_{48}^{16} - \zeta_{48}^{22} ) q^{65} + ( -\zeta_{48}^{11} - \zeta_{48}^{15} ) q^{69} -\zeta_{48}^{13} q^{70} + ( -\zeta_{48}^{2} - \zeta_{48}^{22} ) q^{71} -\zeta_{48}^{12} q^{72} + \zeta_{48}^{9} q^{75} + ( \zeta_{48}^{11} + \zeta_{48}^{21} ) q^{76} + ( \zeta_{48}^{12} - \zeta_{48}^{18} ) q^{78} + ( \zeta_{48}^{6} - \zeta_{48}^{10} ) q^{79} -\zeta_{48}^{5} q^{80} + \zeta_{48}^{12} q^{81} + ( \zeta_{48}^{11} + \zeta_{48}^{17} ) q^{83} -\zeta_{48}^{9} q^{84} + \zeta_{48}^{5} q^{90} + ( \zeta_{48}^{9} - \zeta_{48}^{15} ) q^{91} + ( -1 - \zeta_{48}^{4} ) q^{92} + ( -\zeta_{48}^{4} - \zeta_{48}^{14} ) q^{95} -\zeta_{48} q^{96} -\zeta_{48}^{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 8q^{14} + 8q^{16} - 8q^{18} - 16q^{23} + 8q^{39} - 8q^{57} + 8q^{60} - 8q^{65} - 16q^{92} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$1081$$ $$1261$$ $$2017$$ $$\chi(n)$$ $$-\zeta_{48}^{16}$$ $$1$$ $$-1$$ $$-1$$ $$\zeta_{48}^{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
293.1
 −0.608761 − 0.793353i 0.608761 + 0.793353i 0.793353 − 0.608761i −0.793353 + 0.608761i 0.130526 − 0.991445i −0.130526 + 0.991445i 0.991445 + 0.130526i −0.991445 − 0.130526i 0.130526 + 0.991445i −0.130526 − 0.991445i 0.991445 − 0.130526i −0.991445 + 0.130526i −0.608761 + 0.793353i 0.608761 − 0.793353i 0.793353 + 0.608761i −0.793353 − 0.608761i
−0.258819 + 0.965926i −0.382683 0.923880i −0.866025 0.500000i −0.923880 0.382683i 0.991445 0.130526i 0.965926 + 0.258819i 0.707107 0.707107i −0.707107 + 0.707107i 0.608761 0.793353i
293.2 −0.258819 + 0.965926i 0.382683 + 0.923880i −0.866025 0.500000i 0.923880 + 0.382683i −0.991445 + 0.130526i 0.965926 + 0.258819i 0.707107 0.707107i −0.707107 + 0.707107i −0.608761 + 0.793353i
293.3 0.258819 0.965926i −0.923880 + 0.382683i −0.866025 0.500000i 0.382683 0.923880i 0.130526 + 0.991445i −0.965926 0.258819i −0.707107 + 0.707107i 0.707107 0.707107i −0.793353 0.608761i
293.4 0.258819 0.965926i 0.923880 0.382683i −0.866025 0.500000i −0.382683 + 0.923880i −0.130526 0.991445i −0.965926 0.258819i −0.707107 + 0.707107i 0.707107 0.707107i 0.793353 + 0.608761i
797.1 −0.965926 0.258819i −0.923880 0.382683i 0.866025 + 0.500000i 0.382683 + 0.923880i 0.793353 + 0.608761i 0.258819 0.965926i −0.707107 0.707107i 0.707107 + 0.707107i −0.130526 0.991445i
797.2 −0.965926 0.258819i 0.923880 + 0.382683i 0.866025 + 0.500000i −0.382683 0.923880i −0.793353 0.608761i 0.258819 0.965926i −0.707107 0.707107i 0.707107 + 0.707107i 0.130526 + 0.991445i
797.3 0.965926 + 0.258819i −0.382683 + 0.923880i 0.866025 + 0.500000i −0.923880 + 0.382683i −0.608761 + 0.793353i −0.258819 + 0.965926i 0.707107 + 0.707107i −0.707107 0.707107i −0.991445 + 0.130526i
797.4 0.965926 + 0.258819i 0.382683 0.923880i 0.866025 + 0.500000i 0.923880 0.382683i 0.608761 0.793353i −0.258819 + 0.965926i 0.707107 + 0.707107i −0.707107 0.707107i 0.991445 0.130526i
1973.1 −0.965926 + 0.258819i −0.923880 + 0.382683i 0.866025 0.500000i 0.382683 0.923880i 0.793353 0.608761i 0.258819 + 0.965926i −0.707107 + 0.707107i 0.707107 0.707107i −0.130526 + 0.991445i
1973.2 −0.965926 + 0.258819i 0.923880 0.382683i 0.866025 0.500000i −0.382683 + 0.923880i −0.793353 + 0.608761i 0.258819 + 0.965926i −0.707107 + 0.707107i 0.707107 0.707107i 0.130526 0.991445i
1973.3 0.965926 0.258819i −0.382683 0.923880i 0.866025 0.500000i −0.923880 0.382683i −0.608761 0.793353i −0.258819 0.965926i 0.707107 0.707107i −0.707107 + 0.707107i −0.991445 0.130526i
1973.4 0.965926 0.258819i 0.382683 + 0.923880i 0.866025 0.500000i 0.923880 + 0.382683i 0.608761 + 0.793353i −0.258819 0.965926i 0.707107 0.707107i −0.707107 + 0.707107i 0.991445 + 0.130526i
2477.1 −0.258819 0.965926i −0.382683 + 0.923880i −0.866025 + 0.500000i −0.923880 + 0.382683i 0.991445 + 0.130526i 0.965926 0.258819i 0.707107 + 0.707107i −0.707107 0.707107i 0.608761 + 0.793353i
2477.2 −0.258819 0.965926i 0.382683 0.923880i −0.866025 + 0.500000i 0.923880 0.382683i −0.991445 0.130526i 0.965926 0.258819i 0.707107 + 0.707107i −0.707107 0.707107i −0.608761 0.793353i
2477.3 0.258819 + 0.965926i −0.923880 0.382683i −0.866025 + 0.500000i 0.382683 + 0.923880i 0.130526 0.991445i −0.965926 + 0.258819i −0.707107 0.707107i 0.707107 + 0.707107i −0.793353 + 0.608761i
2477.4 0.258819 + 0.965926i 0.923880 + 0.382683i −0.866025 + 0.500000i −0.382683 0.923880i −0.130526 + 0.991445i −0.965926 + 0.258819i −0.707107 0.707107i 0.707107 + 0.707107i 0.793353 0.608761i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2477.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
7.b odd 2 1 inner
8.b even 2 1 inner
45.l even 12 1 inner
315.cf odd 12 1 inner
360.br even 12 1 inner
2520.hf odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.1.hf.a 16
5.c odd 4 1 2520.1.hf.b yes 16
7.b odd 2 1 inner 2520.1.hf.a 16
8.b even 2 1 inner 2520.1.hf.a 16
9.d odd 6 1 2520.1.hf.b yes 16
35.f even 4 1 2520.1.hf.b yes 16
40.i odd 4 1 2520.1.hf.b yes 16
45.l even 12 1 inner 2520.1.hf.a 16
56.h odd 2 1 CM 2520.1.hf.a 16
63.o even 6 1 2520.1.hf.b yes 16
72.j odd 6 1 2520.1.hf.b yes 16
280.s even 4 1 2520.1.hf.b yes 16
315.cf odd 12 1 inner 2520.1.hf.a 16
360.br even 12 1 inner 2520.1.hf.a 16
504.cc even 6 1 2520.1.hf.b yes 16
2520.hf odd 12 1 inner 2520.1.hf.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.1.hf.a 16 1.a even 1 1 trivial
2520.1.hf.a 16 7.b odd 2 1 inner
2520.1.hf.a 16 8.b even 2 1 inner
2520.1.hf.a 16 45.l even 12 1 inner
2520.1.hf.a 16 56.h odd 2 1 CM
2520.1.hf.a 16 315.cf odd 12 1 inner
2520.1.hf.a 16 360.br even 12 1 inner
2520.1.hf.a 16 2520.hf odd 12 1 inner
2520.1.hf.b yes 16 5.c odd 4 1
2520.1.hf.b yes 16 9.d odd 6 1
2520.1.hf.b yes 16 35.f even 4 1
2520.1.hf.b yes 16 40.i odd 4 1
2520.1.hf.b yes 16 63.o even 6 1
2520.1.hf.b yes 16 72.j odd 6 1
2520.1.hf.b yes 16 280.s even 4 1
2520.1.hf.b yes 16 504.cc even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23}^{4} + 4 T_{23}^{3} + 5 T_{23}^{2} + 2 T_{23} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2520, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{4} + T^{8} )^{2}$$
$3$ $$( 1 + T^{8} )^{2}$$
$5$ $$( 1 + T^{8} )^{2}$$
$7$ $$( 1 - T^{4} + T^{8} )^{2}$$
$11$ $$T^{16}$$
$13$ $$16 - 48 T^{4} + 140 T^{8} - 12 T^{12} + T^{16}$$
$17$ $$T^{16}$$
$19$ $$( 1 + 16 T^{2} + 20 T^{4} + 8 T^{6} + T^{8} )^{2}$$
$23$ $$( 1 + 2 T + 5 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$29$ $$T^{16}$$
$31$ $$T^{16}$$
$37$ $$T^{16}$$
$41$ $$T^{16}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$( 4 + 8 T^{2} + 14 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$61$ $$1 + 16 T^{2} + 236 T^{4} + 304 T^{6} + 271 T^{8} + 128 T^{10} + 44 T^{12} + 8 T^{14} + T^{16}$$
$67$ $$T^{16}$$
$71$ $$( 1 + 4 T^{2} + T^{4} )^{4}$$
$73$ $$T^{16}$$
$79$ $$( 1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8} )^{2}$$
$83$ $$16 - 48 T^{4} + 140 T^{8} - 12 T^{12} + T^{16}$$
$89$ $$T^{16}$$
$97$ $$T^{16}$$