# Properties

 Label 1350.3.d.f Level $1350$ Weight $3$ Character orbit 1350.d Analytic conductor $36.785$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1350.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7848356886$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 2 q^{4} - 5 q^{7} - 2 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 2 * q^4 - 5 * q^7 - 2*b * q^8 $$q + \beta q^{2} - 2 q^{4} - 5 q^{7} - 2 \beta q^{8} - \beta q^{11} + 9 q^{13} - 5 \beta q^{14} + 4 q^{16} - 8 \beta q^{17} - 21 q^{19} + 2 q^{22} + \beta q^{23} + 9 \beta q^{26} + 10 q^{28} + 27 \beta q^{29} + 40 q^{31} + 4 \beta q^{32} + 16 q^{34} + 25 q^{37} - 21 \beta q^{38} - 37 \beta q^{41} + 64 q^{43} + 2 \beta q^{44} - 2 q^{46} - 16 \beta q^{47} - 24 q^{49} - 18 q^{52} + 51 \beta q^{53} + 10 \beta q^{56} - 54 q^{58} + 64 \beta q^{59} - 97 q^{61} + 40 \beta q^{62} - 8 q^{64} + 131 q^{67} + 16 \beta q^{68} + 63 \beta q^{71} - 17 q^{73} + 25 \beta q^{74} + 42 q^{76} + 5 \beta q^{77} + 117 q^{79} + 74 q^{82} - 41 \beta q^{83} + 64 \beta q^{86} - 4 q^{88} + 104 \beta q^{89} - 45 q^{91} - 2 \beta q^{92} + 32 q^{94} + 41 q^{97} - 24 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 2 * q^4 - 5 * q^7 - 2*b * q^8 - b * q^11 + 9 * q^13 - 5*b * q^14 + 4 * q^16 - 8*b * q^17 - 21 * q^19 + 2 * q^22 + b * q^23 + 9*b * q^26 + 10 * q^28 + 27*b * q^29 + 40 * q^31 + 4*b * q^32 + 16 * q^34 + 25 * q^37 - 21*b * q^38 - 37*b * q^41 + 64 * q^43 + 2*b * q^44 - 2 * q^46 - 16*b * q^47 - 24 * q^49 - 18 * q^52 + 51*b * q^53 + 10*b * q^56 - 54 * q^58 + 64*b * q^59 - 97 * q^61 + 40*b * q^62 - 8 * q^64 + 131 * q^67 + 16*b * q^68 + 63*b * q^71 - 17 * q^73 + 25*b * q^74 + 42 * q^76 + 5*b * q^77 + 117 * q^79 + 74 * q^82 - 41*b * q^83 + 64*b * q^86 - 4 * q^88 + 104*b * q^89 - 45 * q^91 - 2*b * q^92 + 32 * q^94 + 41 * q^97 - 24*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 10 q^{7}+O(q^{10})$$ 2 * q - 4 * q^4 - 10 * q^7 $$2 q - 4 q^{4} - 10 q^{7} + 18 q^{13} + 8 q^{16} - 42 q^{19} + 4 q^{22} + 20 q^{28} + 80 q^{31} + 32 q^{34} + 50 q^{37} + 128 q^{43} - 4 q^{46} - 48 q^{49} - 36 q^{52} - 108 q^{58} - 194 q^{61} - 16 q^{64} + 262 q^{67} - 34 q^{73} + 84 q^{76} + 234 q^{79} + 148 q^{82} - 8 q^{88} - 90 q^{91} + 64 q^{94} + 82 q^{97}+O(q^{100})$$ 2 * q - 4 * q^4 - 10 * q^7 + 18 * q^13 + 8 * q^16 - 42 * q^19 + 4 * q^22 + 20 * q^28 + 80 * q^31 + 32 * q^34 + 50 * q^37 + 128 * q^43 - 4 * q^46 - 48 * q^49 - 36 * q^52 - 108 * q^58 - 194 * q^61 - 16 * q^64 + 262 * q^67 - 34 * q^73 + 84 * q^76 + 234 * q^79 + 148 * q^82 - 8 * q^88 - 90 * q^91 + 64 * q^94 + 82 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1
 − 1.41421i 1.41421i
1.41421i 0 −2.00000 0 0 −5.00000 2.82843i 0 0
701.2 1.41421i 0 −2.00000 0 0 −5.00000 2.82843i 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.3.d.f 2
3.b odd 2 1 inner 1350.3.d.f 2
5.b even 2 1 1350.3.d.g 2
5.c odd 4 2 270.3.b.c 4
15.d odd 2 1 1350.3.d.g 2
15.e even 4 2 270.3.b.c 4
20.e even 4 2 2160.3.c.i 4
45.k odd 12 4 810.3.j.e 8
45.l even 12 4 810.3.j.e 8
60.l odd 4 2 2160.3.c.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.c 4 5.c odd 4 2
270.3.b.c 4 15.e even 4 2
810.3.j.e 8 45.k odd 12 4
810.3.j.e 8 45.l even 12 4
1350.3.d.f 2 1.a even 1 1 trivial
1350.3.d.f 2 3.b odd 2 1 inner
1350.3.d.g 2 5.b even 2 1
1350.3.d.g 2 15.d odd 2 1
2160.3.c.i 4 20.e even 4 2
2160.3.c.i 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7} + 5$$ T7 + 5 $$T_{11}^{2} + 2$$ T11^2 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 5)^{2}$$
$11$ $$T^{2} + 2$$
$13$ $$(T - 9)^{2}$$
$17$ $$T^{2} + 128$$
$19$ $$(T + 21)^{2}$$
$23$ $$T^{2} + 2$$
$29$ $$T^{2} + 1458$$
$31$ $$(T - 40)^{2}$$
$37$ $$(T - 25)^{2}$$
$41$ $$T^{2} + 2738$$
$43$ $$(T - 64)^{2}$$
$47$ $$T^{2} + 512$$
$53$ $$T^{2} + 5202$$
$59$ $$T^{2} + 8192$$
$61$ $$(T + 97)^{2}$$
$67$ $$(T - 131)^{2}$$
$71$ $$T^{2} + 7938$$
$73$ $$(T + 17)^{2}$$
$79$ $$(T - 117)^{2}$$
$83$ $$T^{2} + 3362$$
$89$ $$T^{2} + 21632$$
$97$ $$(T - 41)^{2}$$