# Properties

 Label 1360.2.o.a Level $1360$ Weight $2$ Character orbit 1360.o 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.o (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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( 2 - i ) q^{5} -2 q^{7} -2 q^{9} +O(q^{10})$$ $$q - q^{3} + ( 2 - i ) q^{5} -2 q^{7} -2 q^{9} + i q^{13} + ( -2 + i ) q^{15} + ( 1 + 4 i ) q^{17} + 5 q^{19} + 2 q^{21} + 4 q^{23} + ( 3 - 4 i ) q^{25} + 5 q^{27} -9 i q^{29} -5 i q^{31} + ( -4 + 2 i ) q^{35} + 2 q^{37} -i q^{39} -10 i q^{41} -6 i q^{43} + ( -4 + 2 i ) q^{45} + 7 i q^{47} -3 q^{49} + ( -1 - 4 i ) q^{51} + i q^{53} -5 q^{57} + 5 q^{59} + 5 i q^{61} + 4 q^{63} + ( 1 + 2 i ) q^{65} + 2 i q^{67} -4 q^{69} -5 i q^{71} + 11 q^{73} + ( -3 + 4 i ) q^{75} -16 i q^{79} + q^{81} -6 i q^{83} + ( 6 + 7 i ) q^{85} + 9 i q^{87} + 5 q^{89} -2 i q^{91} + 5 i q^{93} + ( 10 - 5 i ) q^{95} + 7 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 4q^{5} - 4q^{7} - 4q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 4q^{5} - 4q^{7} - 4q^{9} - 4q^{15} + 2q^{17} + 10q^{19} + 4q^{21} + 8q^{23} + 6q^{25} + 10q^{27} - 8q^{35} + 4q^{37} - 8q^{45} - 6q^{49} - 2q^{51} - 10q^{57} + 10q^{59} + 8q^{63} + 2q^{65} - 8q^{69} + 22q^{73} - 6q^{75} + 2q^{81} + 12q^{85} + 10q^{89} + 20q^{95} + 14q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
849.1
 1.00000i − 1.00000i
0 −1.00000 0 2.00000 1.00000i 0 −2.00000 0 −2.00000 0
849.2 0 −1.00000 0 2.00000 + 1.00000i 0 −2.00000 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
85.c 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.o.a 2
4.b odd 2 1 170.2.d.b yes 2
5.b even 2 1 1360.2.o.b 2
12.b even 2 1 1530.2.f.b 2
17.b even 2 1 1360.2.o.b 2
20.d odd 2 1 170.2.d.a 2
20.e even 4 1 850.2.b.c 2
20.e even 4 1 850.2.b.i 2
60.h even 2 1 1530.2.f.e 2
68.d odd 2 1 170.2.d.a 2
85.c even 2 1 inner 1360.2.o.a 2
204.h even 2 1 1530.2.f.e 2
340.d odd 2 1 170.2.d.b yes 2
340.r even 4 1 850.2.b.c 2
340.r even 4 1 850.2.b.i 2
1020.b even 2 1 1530.2.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.d.a 2 20.d odd 2 1
170.2.d.a 2 68.d odd 2 1
170.2.d.b yes 2 4.b odd 2 1
170.2.d.b yes 2 340.d odd 2 1
850.2.b.c 2 20.e even 4 1
850.2.b.c 2 340.r even 4 1
850.2.b.i 2 20.e even 4 1
850.2.b.i 2 340.r even 4 1
1360.2.o.a 2 1.a even 1 1 trivial
1360.2.o.a 2 85.c even 2 1 inner
1360.2.o.b 2 5.b even 2 1
1360.2.o.b 2 17.b even 2 1
1530.2.f.b 2 12.b even 2 1
1530.2.f.b 2 1020.b even 2 1
1530.2.f.e 2 60.h even 2 1
1530.2.f.e 2 204.h even 2 1

## Hecke kernels

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

 $$T_{3} + 1$$ $$T_{7} + 2$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T + 3 T^{2} )^{2}$$
$5$ $$1 - 4 T + 5 T^{2}$$
$7$ $$( 1 + 2 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$1 - 25 T^{2} + 169 T^{4}$$
$17$ $$1 - 2 T + 17 T^{2}$$
$19$ $$( 1 - 5 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 4 T + 23 T^{2} )^{2}$$
$29$ $$1 + 23 T^{2} + 841 T^{4}$$
$31$ $$1 - 37 T^{2} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 8 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} )$$
$43$ $$1 - 50 T^{2} + 1849 T^{4}$$
$47$ $$1 - 45 T^{2} + 2209 T^{4}$$
$53$ $$1 - 105 T^{2} + 2809 T^{4}$$
$59$ $$( 1 - 5 T + 59 T^{2} )^{2}$$
$61$ $$1 - 97 T^{2} + 3721 T^{4}$$
$67$ $$1 - 130 T^{2} + 4489 T^{4}$$
$71$ $$1 - 117 T^{2} + 5041 T^{4}$$
$73$ $$( 1 - 11 T + 73 T^{2} )^{2}$$
$79$ $$1 + 98 T^{2} + 6241 T^{4}$$
$83$ $$1 - 130 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 5 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 7 T + 97 T^{2} )^{2}$$