# Properties

 Label 5780.2.c.a Level $5780$ Weight $2$ Character orbit 5780.c Analytic conductor $46.154$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5780 = 2^{2} \cdot 5 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5780.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$46.1535323683$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} + i q^{5} + 2 i q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} + i q^{5} + 2 i q^{7} - q^{9} + 2 q^{13} -2 q^{15} + 4 q^{19} -4 q^{21} + 6 i q^{23} - q^{25} + 4 i q^{27} -6 i q^{29} + 4 i q^{31} -2 q^{35} -2 i q^{37} + 4 i q^{39} + 6 i q^{41} + 10 q^{43} -i q^{45} -6 q^{47} + 3 q^{49} + 6 q^{53} + 8 i q^{57} -12 q^{59} + 2 i q^{61} -2 i q^{63} + 2 i q^{65} + 2 q^{67} -12 q^{69} + 12 i q^{71} -2 i q^{73} -2 i q^{75} + 8 i q^{79} -11 q^{81} -6 q^{83} + 12 q^{87} -6 q^{89} + 4 i q^{91} -8 q^{93} + 4 i q^{95} -2 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 4q^{13} - 4q^{15} + 8q^{19} - 8q^{21} - 2q^{25} - 4q^{35} + 20q^{43} - 12q^{47} + 6q^{49} + 12q^{53} - 24q^{59} + 4q^{67} - 24q^{69} - 22q^{81} - 12q^{83} + 24q^{87} - 12q^{89} - 16q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5780\mathbb{Z}\right)^\times$$.

 $$n$$ $$581$$ $$1157$$ $$2891$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5201.1
 − 1.00000i 1.00000i
0 2.00000i 0 1.00000i 0 2.00000i 0 −1.00000 0
5201.2 0 2.00000i 0 1.00000i 0 2.00000i 0 −1.00000 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
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5780.2.c.a 2
17.b even 2 1 inner 5780.2.c.a 2
17.c even 4 1 20.2.a.a 1
17.c even 4 1 5780.2.a.f 1
51.f odd 4 1 180.2.a.a 1
68.f odd 4 1 80.2.a.b 1
85.f odd 4 1 100.2.c.a 2
85.i odd 4 1 100.2.c.a 2
85.j even 4 1 100.2.a.a 1
119.f odd 4 1 980.2.a.h 1
119.m odd 12 2 980.2.i.c 2
119.n even 12 2 980.2.i.i 2
136.i even 4 1 320.2.a.f 1
136.j odd 4 1 320.2.a.a 1
153.m odd 12 2 1620.2.i.b 2
153.n even 12 2 1620.2.i.h 2
187.f odd 4 1 2420.2.a.a 1
204.l even 4 1 720.2.a.h 1
221.h odd 4 1 3380.2.f.b 2
221.i odd 4 1 3380.2.f.b 2
221.k even 4 1 3380.2.a.c 1
255.i odd 4 1 900.2.a.b 1
255.k even 4 1 900.2.d.c 2
255.r even 4 1 900.2.d.c 2
272.i odd 4 1 1280.2.d.g 2
272.j even 4 1 1280.2.d.c 2
272.s even 4 1 1280.2.d.c 2
272.t odd 4 1 1280.2.d.g 2
323.g odd 4 1 7220.2.a.f 1
340.i even 4 1 400.2.c.b 2
340.n odd 4 1 400.2.a.c 1
340.s even 4 1 400.2.c.b 2
357.l even 4 1 8820.2.a.g 1
408.q even 4 1 2880.2.a.f 1
408.t odd 4 1 2880.2.a.m 1
476.k even 4 1 3920.2.a.h 1
595.l even 4 1 4900.2.e.f 2
595.r even 4 1 4900.2.e.f 2
595.u odd 4 1 4900.2.a.e 1
680.s odd 4 1 1600.2.c.d 2
680.t even 4 1 1600.2.c.e 2
680.bc odd 4 1 1600.2.a.w 1
680.be even 4 1 1600.2.a.c 1
680.bk odd 4 1 1600.2.c.d 2
680.bl even 4 1 1600.2.c.e 2
748.j even 4 1 9680.2.a.ba 1
1020.q odd 4 1 3600.2.f.j 2
1020.ba even 4 1 3600.2.a.be 1
1020.bl odd 4 1 3600.2.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 17.c even 4 1
80.2.a.b 1 68.f odd 4 1
100.2.a.a 1 85.j even 4 1
100.2.c.a 2 85.f odd 4 1
100.2.c.a 2 85.i odd 4 1
180.2.a.a 1 51.f odd 4 1
320.2.a.a 1 136.j odd 4 1
320.2.a.f 1 136.i even 4 1
400.2.a.c 1 340.n odd 4 1
400.2.c.b 2 340.i even 4 1
400.2.c.b 2 340.s even 4 1
720.2.a.h 1 204.l even 4 1
900.2.a.b 1 255.i odd 4 1
900.2.d.c 2 255.k even 4 1
900.2.d.c 2 255.r even 4 1
980.2.a.h 1 119.f odd 4 1
980.2.i.c 2 119.m odd 12 2
980.2.i.i 2 119.n even 12 2
1280.2.d.c 2 272.j even 4 1
1280.2.d.c 2 272.s even 4 1
1280.2.d.g 2 272.i odd 4 1
1280.2.d.g 2 272.t odd 4 1
1600.2.a.c 1 680.be even 4 1
1600.2.a.w 1 680.bc odd 4 1
1600.2.c.d 2 680.s odd 4 1
1600.2.c.d 2 680.bk odd 4 1
1600.2.c.e 2 680.t even 4 1
1600.2.c.e 2 680.bl even 4 1
1620.2.i.b 2 153.m odd 12 2
1620.2.i.h 2 153.n even 12 2
2420.2.a.a 1 187.f odd 4 1
2880.2.a.f 1 408.q even 4 1
2880.2.a.m 1 408.t odd 4 1
3380.2.a.c 1 221.k even 4 1
3380.2.f.b 2 221.h odd 4 1
3380.2.f.b 2 221.i odd 4 1
3600.2.a.be 1 1020.ba even 4 1
3600.2.f.j 2 1020.q odd 4 1
3600.2.f.j 2 1020.bl odd 4 1
3920.2.a.h 1 476.k even 4 1
4900.2.a.e 1 595.u odd 4 1
4900.2.e.f 2 595.l even 4 1
4900.2.e.f 2 595.r even 4 1
5780.2.a.f 1 17.c even 4 1
5780.2.c.a 2 1.a even 1 1 trivial
5780.2.c.a 2 17.b even 2 1 inner
7220.2.a.f 1 323.g odd 4 1
8820.2.a.g 1 357.l even 4 1
9680.2.a.ba 1 748.j even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(5780, [\chi])$$:

 $$T_{3}^{2} + 4$$ $$T_{7}^{2} + 4$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$36 + T^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$36 + T^{2}$$
$43$ $$( -10 + T )^{2}$$
$47$ $$( 6 + T )^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$4 + T^{2}$$
$67$ $$( -2 + T )^{2}$$
$71$ $$144 + T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$64 + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$4 + T^{2}$$