# Properties

 Label 1350.4.c.s Level $1350$ Weight $4$ Character orbit 1350.c Analytic conductor $79.653$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1350.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.6525785077$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 i q^{2} - 4 q^{4} + 29 i q^{7} + 8 i q^{8} +O(q^{10})$$ q - 2*i * q^2 - 4 * q^4 + 29*i * q^7 + 8*i * q^8 $$q - 2 i q^{2} - 4 q^{4} + 29 i q^{7} + 8 i q^{8} + 57 q^{11} - 20 i q^{13} + 58 q^{14} + 16 q^{16} + 72 i q^{17} + 106 q^{19} - 114 i q^{22} + 174 i q^{23} - 40 q^{26} - 116 i q^{28} - 210 q^{29} + 47 q^{31} - 32 i q^{32} + 144 q^{34} + 2 i q^{37} - 212 i q^{38} + 6 q^{41} - 218 i q^{43} - 228 q^{44} + 348 q^{46} - 474 i q^{47} - 498 q^{49} + 80 i q^{52} + 81 i q^{53} - 232 q^{56} + 420 i q^{58} + 84 q^{59} + 56 q^{61} - 94 i q^{62} - 64 q^{64} - 142 i q^{67} - 288 i q^{68} - 360 q^{71} + 1159 i q^{73} + 4 q^{74} - 424 q^{76} + 1653 i q^{77} + 160 q^{79} - 12 i q^{82} + 735 i q^{83} - 436 q^{86} + 456 i q^{88} - 954 q^{89} + 580 q^{91} - 696 i q^{92} - 948 q^{94} + 191 i q^{97} + 996 i q^{98} +O(q^{100})$$ q - 2*i * q^2 - 4 * q^4 + 29*i * q^7 + 8*i * q^8 + 57 * q^11 - 20*i * q^13 + 58 * q^14 + 16 * q^16 + 72*i * q^17 + 106 * q^19 - 114*i * q^22 + 174*i * q^23 - 40 * q^26 - 116*i * q^28 - 210 * q^29 + 47 * q^31 - 32*i * q^32 + 144 * q^34 + 2*i * q^37 - 212*i * q^38 + 6 * q^41 - 218*i * q^43 - 228 * q^44 + 348 * q^46 - 474*i * q^47 - 498 * q^49 + 80*i * q^52 + 81*i * q^53 - 232 * q^56 + 420*i * q^58 + 84 * q^59 + 56 * q^61 - 94*i * q^62 - 64 * q^64 - 142*i * q^67 - 288*i * q^68 - 360 * q^71 + 1159*i * q^73 + 4 * q^74 - 424 * q^76 + 1653*i * q^77 + 160 * q^79 - 12*i * q^82 + 735*i * q^83 - 436 * q^86 + 456*i * q^88 - 954 * q^89 + 580 * q^91 - 696*i * q^92 - 948 * q^94 + 191*i * q^97 + 996*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} + 114 q^{11} + 116 q^{14} + 32 q^{16} + 212 q^{19} - 80 q^{26} - 420 q^{29} + 94 q^{31} + 288 q^{34} + 12 q^{41} - 456 q^{44} + 696 q^{46} - 996 q^{49} - 464 q^{56} + 168 q^{59} + 112 q^{61} - 128 q^{64} - 720 q^{71} + 8 q^{74} - 848 q^{76} + 320 q^{79} - 872 q^{86} - 1908 q^{89} + 1160 q^{91} - 1896 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 + 114 * q^11 + 116 * q^14 + 32 * q^16 + 212 * q^19 - 80 * q^26 - 420 * q^29 + 94 * q^31 + 288 * q^34 + 12 * q^41 - 456 * q^44 + 696 * q^46 - 996 * q^49 - 464 * q^56 + 168 * q^59 + 112 * q^61 - 128 * q^64 - 720 * q^71 + 8 * q^74 - 848 * q^76 + 320 * q^79 - 872 * q^86 - 1908 * q^89 + 1160 * q^91 - 1896 * 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
2.00000i 0 −4.00000 0 0 29.0000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 29.0000i 8.00000i 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 1350.4.c.s 2
3.b odd 2 1 1350.4.c.b 2
5.b even 2 1 inner 1350.4.c.s 2
5.c odd 4 1 54.4.a.b 1
5.c odd 4 1 1350.4.a.o 1
15.d odd 2 1 1350.4.c.b 2
15.e even 4 1 54.4.a.c yes 1
15.e even 4 1 1350.4.a.a 1
20.e even 4 1 432.4.a.e 1
40.i odd 4 1 1728.4.a.v 1
40.k even 4 1 1728.4.a.u 1
45.k odd 12 2 162.4.c.g 2
45.l even 12 2 162.4.c.b 2
60.l odd 4 1 432.4.a.j 1
120.q odd 4 1 1728.4.a.k 1
120.w even 4 1 1728.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.a.b 1 5.c odd 4 1
54.4.a.c yes 1 15.e even 4 1
162.4.c.b 2 45.l even 12 2
162.4.c.g 2 45.k odd 12 2
432.4.a.e 1 20.e even 4 1
432.4.a.j 1 60.l odd 4 1
1350.4.a.a 1 15.e even 4 1
1350.4.a.o 1 5.c odd 4 1
1350.4.c.b 2 3.b odd 2 1
1350.4.c.b 2 15.d odd 2 1
1350.4.c.s 2 1.a even 1 1 trivial
1350.4.c.s 2 5.b even 2 1 inner
1728.4.a.k 1 120.q odd 4 1
1728.4.a.l 1 120.w even 4 1
1728.4.a.u 1 40.k even 4 1
1728.4.a.v 1 40.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{2} + 841$$ T7^2 + 841 $$T_{11} - 57$$ T11 - 57

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 841$$
$11$ $$(T - 57)^{2}$$
$13$ $$T^{2} + 400$$
$17$ $$T^{2} + 5184$$
$19$ $$(T - 106)^{2}$$
$23$ $$T^{2} + 30276$$
$29$ $$(T + 210)^{2}$$
$31$ $$(T - 47)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 47524$$
$47$ $$T^{2} + 224676$$
$53$ $$T^{2} + 6561$$
$59$ $$(T - 84)^{2}$$
$61$ $$(T - 56)^{2}$$
$67$ $$T^{2} + 20164$$
$71$ $$(T + 360)^{2}$$
$73$ $$T^{2} + 1343281$$
$79$ $$(T - 160)^{2}$$
$83$ $$T^{2} + 540225$$
$89$ $$(T + 954)^{2}$$
$97$ $$T^{2} + 36481$$