# Properties

 Label 4680.2.l.c Level $4680$ Weight $2$ Character orbit 4680.l Analytic conductor $37.370$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4680.l (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$37.3699881460$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1560) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 i ) q^{5} + 5 i q^{7} +O(q^{10})$$ $$q + ( 1 - 2 i ) q^{5} + 5 i q^{7} -5 q^{11} + i q^{13} -3 i q^{17} + 4 q^{19} -5 i q^{23} + ( -3 - 4 i ) q^{25} -4 q^{29} + ( 10 + 5 i ) q^{35} + 7 i q^{37} -11 q^{41} -12 i q^{43} + 6 i q^{47} -18 q^{49} -i q^{53} + ( -5 + 10 i ) q^{55} + 12 q^{59} -7 q^{61} + ( 2 + i ) q^{65} -4 i q^{67} + 7 q^{71} -14 i q^{73} -25 i q^{77} + 5 q^{79} + 2 i q^{83} + ( -6 - 3 i ) q^{85} -3 q^{89} -5 q^{91} + ( 4 - 8 i ) q^{95} + i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + O(q^{10})$$ $$2 q + 2 q^{5} - 10 q^{11} + 8 q^{19} - 6 q^{25} - 8 q^{29} + 20 q^{35} - 22 q^{41} - 36 q^{49} - 10 q^{55} + 24 q^{59} - 14 q^{61} + 4 q^{65} + 14 q^{71} + 10 q^{79} - 12 q^{85} - 6 q^{89} - 10 q^{91} + 8 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$$.

 $$n$$ $$937$$ $$1081$$ $$2081$$ $$2341$$ $$3511$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2809.1
 1.00000i − 1.00000i
0 0 0 1.00000 2.00000i 0 5.00000i 0 0 0
2809.2 0 0 0 1.00000 + 2.00000i 0 5.00000i 0 0 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4680.2.l.c 2
3.b odd 2 1 1560.2.l.a 2
5.b even 2 1 inner 4680.2.l.c 2
12.b even 2 1 3120.2.l.b 2
15.d odd 2 1 1560.2.l.a 2
15.e even 4 1 7800.2.a.l 1
15.e even 4 1 7800.2.a.m 1
60.h even 2 1 3120.2.l.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.a 2 3.b odd 2 1
1560.2.l.a 2 15.d odd 2 1
3120.2.l.b 2 12.b even 2 1
3120.2.l.b 2 60.h even 2 1
4680.2.l.c 2 1.a even 1 1 trivial
4680.2.l.c 2 5.b even 2 1 inner
7800.2.a.l 1 15.e even 4 1
7800.2.a.m 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4680, [\chi])$$:

 $$T_{7}^{2} + 25$$ $$T_{11} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 - 2 T + T^{2}$$
$7$ $$25 + T^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$25 + T^{2}$$
$29$ $$( 4 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$49 + T^{2}$$
$41$ $$( 11 + T )^{2}$$
$43$ $$144 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$1 + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$( 7 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -7 + T )^{2}$$
$73$ $$196 + T^{2}$$
$79$ $$( -5 + T )^{2}$$
$83$ $$4 + T^{2}$$
$89$ $$( 3 + T )^{2}$$
$97$ $$1 + T^{2}$$