# Properties

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

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## 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: $$x^{2} + 1$$ 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 $$q - i q^{2} - q^{4} + i q^{7} + i q^{8} + 3 q^{11} - 4 i q^{13} + q^{14} + q^{16} - 2 q^{19} - 3 i q^{22} + 6 i q^{23} - 4 q^{26} - i q^{28} + 6 q^{29} + 5 q^{31} - i q^{32} - 2 i q^{37} + 2 i q^{38} + 6 q^{41} - 10 i q^{43} - 3 q^{44} + 6 q^{46} + 6 i q^{47} + 6 q^{49} + 4 i q^{52} - 9 i q^{53} - q^{56} - 6 i q^{58} + 12 q^{59} + 8 q^{61} - 5 i q^{62} - q^{64} - 14 i q^{67} - 7 i q^{73} - 2 q^{74} + 2 q^{76} + 3 i q^{77} - 8 q^{79} - 6 i q^{82} + 3 i q^{83} - 10 q^{86} + 3 i q^{88} - 18 q^{89} + 4 q^{91} - 6 i q^{92} + 6 q^{94} + i q^{97} - 6 i q^{98} +O(q^{100})$$ 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 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} + 6 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{19} - 8 q^{26} + 12 q^{29} + 10 q^{31} + 12 q^{41} - 6 q^{44} + 12 q^{46} + 12 q^{49} - 2 q^{56} + 24 q^{59} + 16 q^{61} - 2 q^{64} - 4 q^{74} + 4 q^{76} - 16 q^{79} - 20 q^{86} - 36 q^{89} + 8 q^{91} + 12 q^{94}+O(q^{100})$$ 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

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

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

## Hecke kernels

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

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11} - 3$$ T11 - 3 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{17}$$ T17 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

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