# Properties

 Label 1360.2.e.b Level $1360$ Weight $2$ Character orbit 1360.e Analytic conductor $10.860$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1360 = 2^{4} \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1360.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.8596546749$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 170) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + ( 1 - 2 i ) q^{5} + 2 q^{9} +O(q^{10})$$ $$q + i q^{3} + ( 1 - 2 i ) q^{5} + 2 q^{9} + 6 q^{11} -3 i q^{13} + ( 2 + i ) q^{15} + i q^{17} -7 q^{19} -8 i q^{23} + ( -3 - 4 i ) q^{25} + 5 i q^{27} + 5 q^{29} -5 q^{31} + 6 i q^{33} + 8 i q^{37} + 3 q^{39} -4 i q^{43} + ( 2 - 4 i ) q^{45} -3 i q^{47} + 7 q^{49} - q^{51} -9 i q^{53} + ( 6 - 12 i ) q^{55} -7 i q^{57} + 5 q^{59} -3 q^{61} + ( -6 - 3 i ) q^{65} + 2 i q^{67} + 8 q^{69} + 15 q^{71} + 11 i q^{73} + ( 4 - 3 i ) q^{75} + 8 q^{79} + q^{81} + 4 i q^{83} + ( 2 + i ) q^{85} + 5 i q^{87} + q^{89} -5 i q^{93} + ( -7 + 14 i ) q^{95} -9 i q^{97} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + 4q^{9} + O(q^{10})$$ $$2q + 2q^{5} + 4q^{9} + 12q^{11} + 4q^{15} - 14q^{19} - 6q^{25} + 10q^{29} - 10q^{31} + 6q^{39} + 4q^{45} + 14q^{49} - 2q^{51} + 12q^{55} + 10q^{59} - 6q^{61} - 12q^{65} + 16q^{69} + 30q^{71} + 8q^{75} + 16q^{79} + 2q^{81} + 4q^{85} + 2q^{89} - 14q^{95} + 24q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1360\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$341$$ $$511$$ $$817$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1089.1
 − 1.00000i 1.00000i
0 1.00000i 0 1.00000 + 2.00000i 0 0 0 2.00000 0
1089.2 0 1.00000i 0 1.00000 2.00000i 0 0 0 2.00000 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
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 1360.2.e.b 2
4.b odd 2 1 170.2.c.a 2
5.b even 2 1 inner 1360.2.e.b 2
5.c odd 4 1 6800.2.a.g 1
5.c odd 4 1 6800.2.a.r 1
12.b even 2 1 1530.2.d.b 2
20.d odd 2 1 170.2.c.a 2
20.e even 4 1 850.2.a.d 1
20.e even 4 1 850.2.a.h 1
60.h even 2 1 1530.2.d.b 2
60.l odd 4 1 7650.2.a.s 1
60.l odd 4 1 7650.2.a.cb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.c.a 2 4.b odd 2 1
170.2.c.a 2 20.d odd 2 1
850.2.a.d 1 20.e even 4 1
850.2.a.h 1 20.e even 4 1
1360.2.e.b 2 1.a even 1 1 trivial
1360.2.e.b 2 5.b even 2 1 inner
1530.2.d.b 2 12.b even 2 1
1530.2.d.b 2 60.h even 2 1
6800.2.a.g 1 5.c odd 4 1
6800.2.a.r 1 5.c odd 4 1
7650.2.a.s 1 60.l odd 4 1
7650.2.a.cb 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1360, [\chi])$$:

 $$T_{3}^{2} + 1$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -6 + T )^{2}$$
$13$ $$9 + T^{2}$$
$17$ $$1 + T^{2}$$
$19$ $$( 7 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$( 5 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$9 + T^{2}$$
$53$ $$81 + T^{2}$$
$59$ $$( -5 + T )^{2}$$
$61$ $$( 3 + T )^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( -15 + T )^{2}$$
$73$ $$121 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( -1 + T )^{2}$$
$97$ $$81 + T^{2}$$