# Properties

 Label 1350.2.c.b Level $1350$ Weight $2$ Character orbit 1350.c Analytic conductor $10.780$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1350.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 54) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + i q^{7} -i q^{8} +O(q^{10})$$ $$q + i q^{2} - q^{4} + i q^{7} -i q^{8} -3 q^{11} -4 i q^{13} - q^{14} + q^{16} -2 q^{19} -3 i q^{22} -6 i q^{23} + 4 q^{26} -i q^{28} -6 q^{29} + 5 q^{31} + i q^{32} -2 i q^{37} -2 i q^{38} -6 q^{41} -10 i q^{43} + 3 q^{44} + 6 q^{46} -6 i q^{47} + 6 q^{49} + 4 i q^{52} + 9 i q^{53} + q^{56} -6 i q^{58} -12 q^{59} + 8 q^{61} + 5 i q^{62} - q^{64} -14 i q^{67} -7 i q^{73} + 2 q^{74} + 2 q^{76} -3 i q^{77} -8 q^{79} -6 i q^{82} -3 i q^{83} + 10 q^{86} + 3 i q^{88} + 18 q^{89} + 4 q^{91} + 6 i q^{92} + 6 q^{94} + i q^{97} + 6 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + O(q^{10})$$ $$2 q - 2 q^{4} - 6 q^{11} - 2 q^{14} + 2 q^{16} - 4 q^{19} + 8 q^{26} - 12 q^{29} + 10 q^{31} - 12 q^{41} + 6 q^{44} + 12 q^{46} + 12 q^{49} + 2 q^{56} - 24 q^{59} + 16 q^{61} - 2 q^{64} + 4 q^{74} + 4 q^{76} - 16 q^{79} + 20 q^{86} + 36 q^{89} + 8 q^{91} + 12 q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 0 0
649.2 1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 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 1350.2.c.b 2
3.b odd 2 1 1350.2.c.k 2
5.b even 2 1 inner 1350.2.c.b 2
5.c odd 4 1 54.2.a.a 1
5.c odd 4 1 1350.2.a.r 1
15.d odd 2 1 1350.2.c.k 2
15.e even 4 1 54.2.a.b yes 1
15.e even 4 1 1350.2.a.h 1
20.e even 4 1 432.2.a.g 1
35.f even 4 1 2646.2.a.a 1
40.i odd 4 1 1728.2.a.c 1
40.k even 4 1 1728.2.a.d 1
45.k odd 12 2 162.2.c.c 2
45.l even 12 2 162.2.c.b 2
55.e even 4 1 6534.2.a.bc 1
60.l odd 4 1 432.2.a.b 1
65.h odd 4 1 9126.2.a.u 1
105.k odd 4 1 2646.2.a.bd 1
120.q odd 4 1 1728.2.a.z 1
120.w even 4 1 1728.2.a.y 1
165.l odd 4 1 6534.2.a.b 1
180.v odd 12 2 1296.2.i.o 2
180.x even 12 2 1296.2.i.c 2
195.s even 4 1 9126.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 5.c odd 4 1
54.2.a.b yes 1 15.e even 4 1
162.2.c.b 2 45.l even 12 2
162.2.c.c 2 45.k odd 12 2
432.2.a.b 1 60.l odd 4 1
432.2.a.g 1 20.e even 4 1
1296.2.i.c 2 180.x even 12 2
1296.2.i.o 2 180.v odd 12 2
1350.2.a.h 1 15.e even 4 1
1350.2.a.r 1 5.c odd 4 1
1350.2.c.b 2 1.a even 1 1 trivial
1350.2.c.b 2 5.b even 2 1 inner
1350.2.c.k 2 3.b odd 2 1
1350.2.c.k 2 15.d odd 2 1
1728.2.a.c 1 40.i odd 4 1
1728.2.a.d 1 40.k even 4 1
1728.2.a.y 1 120.w even 4 1
1728.2.a.z 1 120.q odd 4 1
2646.2.a.a 1 35.f even 4 1
2646.2.a.bd 1 105.k odd 4 1
6534.2.a.b 1 165.l odd 4 1
6534.2.a.bc 1 55.e even 4 1
9126.2.a.r 1 195.s even 4 1
9126.2.a.u 1 65.h odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{2} + 1$$ $$T_{11} + 3$$ $$T_{13}^{2} + 16$$ $$T_{17}$$ $$T_{29} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$100 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$81 + T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$( -8 + T )^{2}$$
$67$ $$196 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$49 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$9 + T^{2}$$
$89$ $$( -18 + T )^{2}$$
$97$ $$1 + T^{2}$$