# Properties

 Label 2520.1.h.e Level $2520$ Weight $1$ Character orbit 2520.h Analytic conductor $1.258$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ 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.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.25764383184$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.11200.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{2} - q^{4} + \zeta_{8}^{3} q^{5} + \zeta_{8}^{2} q^{7} -\zeta_{8}^{2} q^{8} +O(q^{10})$$ $$q + \zeta_{8}^{2} q^{2} - q^{4} + \zeta_{8}^{3} q^{5} + \zeta_{8}^{2} q^{7} -\zeta_{8}^{2} q^{8} -\zeta_{8} q^{10} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{13} - q^{14} + q^{16} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{19} -\zeta_{8}^{3} q^{20} -\zeta_{8}^{2} q^{25} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{26} -\zeta_{8}^{2} q^{28} + \zeta_{8}^{2} q^{32} -\zeta_{8} q^{35} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{38} + \zeta_{8} q^{40} - q^{49} + q^{50} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{52} + q^{56} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{59} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{61} - q^{64} + ( -1 - \zeta_{8}^{2} ) q^{65} -\zeta_{8}^{3} q^{70} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{76} + \zeta_{8}^{3} q^{80} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{83} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{91} + ( 1 - \zeta_{8}^{2} ) q^{95} -\zeta_{8}^{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{14} + 4q^{16} - 4q^{49} + 4q^{50} + 4q^{56} - 4q^{64} - 4q^{65} + 4q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$281$$ $$631$$ $$1081$$ $$1261$$ $$2017$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1189.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
1.00000i 0 −1.00000 −0.707107 0.707107i 0 1.00000i 1.00000i 0 −0.707107 + 0.707107i
1189.2 1.00000i 0 −1.00000 0.707107 + 0.707107i 0 1.00000i 1.00000i 0 0.707107 0.707107i
1189.3 1.00000i 0 −1.00000 −0.707107 + 0.707107i 0 1.00000i 1.00000i 0 −0.707107 0.707107i
1189.4 1.00000i 0 −1.00000 0.707107 0.707107i 0 1.00000i 1.00000i 0 0.707107 + 0.707107i
 $$n$$: 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
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
5.b even 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
35.c odd 2 1 inner
40.f even 2 1 inner
280.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.1.h.e 4
3.b odd 2 1 280.1.c.a 4
5.b even 2 1 inner 2520.1.h.e 4
7.b odd 2 1 inner 2520.1.h.e 4
8.b even 2 1 inner 2520.1.h.e 4
12.b even 2 1 1120.1.c.a 4
15.d odd 2 1 280.1.c.a 4
15.e even 4 1 1400.1.m.b 2
15.e even 4 1 1400.1.m.e 2
21.c even 2 1 280.1.c.a 4
21.g even 6 2 1960.1.bk.a 8
21.h odd 6 2 1960.1.bk.a 8
24.f even 2 1 1120.1.c.a 4
24.h odd 2 1 280.1.c.a 4
35.c odd 2 1 inner 2520.1.h.e 4
40.f even 2 1 inner 2520.1.h.e 4
56.h odd 2 1 CM 2520.1.h.e 4
60.h even 2 1 1120.1.c.a 4
84.h odd 2 1 1120.1.c.a 4
105.g even 2 1 280.1.c.a 4
105.k odd 4 1 1400.1.m.b 2
105.k odd 4 1 1400.1.m.e 2
105.o odd 6 2 1960.1.bk.a 8
105.p even 6 2 1960.1.bk.a 8
120.i odd 2 1 280.1.c.a 4
120.m even 2 1 1120.1.c.a 4
120.w even 4 1 1400.1.m.b 2
120.w even 4 1 1400.1.m.e 2
168.e odd 2 1 1120.1.c.a 4
168.i even 2 1 280.1.c.a 4
168.s odd 6 2 1960.1.bk.a 8
168.ba even 6 2 1960.1.bk.a 8
280.c odd 2 1 inner 2520.1.h.e 4
420.o odd 2 1 1120.1.c.a 4
840.b odd 2 1 1120.1.c.a 4
840.u even 2 1 280.1.c.a 4
840.bp odd 4 1 1400.1.m.b 2
840.bp odd 4 1 1400.1.m.e 2
840.cb even 6 2 1960.1.bk.a 8
840.cg odd 6 2 1960.1.bk.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.c.a 4 3.b odd 2 1
280.1.c.a 4 15.d odd 2 1
280.1.c.a 4 21.c even 2 1
280.1.c.a 4 24.h odd 2 1
280.1.c.a 4 105.g even 2 1
280.1.c.a 4 120.i odd 2 1
280.1.c.a 4 168.i even 2 1
280.1.c.a 4 840.u even 2 1
1120.1.c.a 4 12.b even 2 1
1120.1.c.a 4 24.f even 2 1
1120.1.c.a 4 60.h even 2 1
1120.1.c.a 4 84.h odd 2 1
1120.1.c.a 4 120.m even 2 1
1120.1.c.a 4 168.e odd 2 1
1120.1.c.a 4 420.o odd 2 1
1120.1.c.a 4 840.b odd 2 1
1400.1.m.b 2 15.e even 4 1
1400.1.m.b 2 105.k odd 4 1
1400.1.m.b 2 120.w even 4 1
1400.1.m.b 2 840.bp odd 4 1
1400.1.m.e 2 15.e even 4 1
1400.1.m.e 2 105.k odd 4 1
1400.1.m.e 2 120.w even 4 1
1400.1.m.e 2 840.bp odd 4 1
1960.1.bk.a 8 21.g even 6 2
1960.1.bk.a 8 21.h odd 6 2
1960.1.bk.a 8 105.o odd 6 2
1960.1.bk.a 8 105.p even 6 2
1960.1.bk.a 8 168.s odd 6 2
1960.1.bk.a 8 168.ba even 6 2
1960.1.bk.a 8 840.cb even 6 2
1960.1.bk.a 8 840.cg odd 6 2
2520.1.h.e 4 1.a even 1 1 trivial
2520.1.h.e 4 5.b even 2 1 inner
2520.1.h.e 4 7.b odd 2 1 inner
2520.1.h.e 4 8.b even 2 1 inner
2520.1.h.e 4 35.c odd 2 1 inner
2520.1.h.e 4 40.f even 2 1 inner
2520.1.h.e 4 56.h odd 2 1 CM
2520.1.h.e 4 280.c odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2520, [\chi])$$:

 $$T_{13}^{2} + 2$$ $$T_{53}$$ $$T_{59}^{2} - 2$$

## Hecke characteristic polynomials

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