# Properties

 Label 1350.4.c.g 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: $$\Z[a_1, \ldots, a_{17}]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 i q^{2} - 4 q^{4} + 8 i q^{7} + 8 i q^{8} +O(q^{10})$$ q - 2*i * q^2 - 4 * q^4 + 8*i * q^7 + 8*i * q^8 $$q - 2 i q^{2} - 4 q^{4} + 8 i q^{7} + 8 i q^{8} - 18 q^{11} - 8 i q^{13} + 16 q^{14} + 16 q^{16} - 15 i q^{17} - 23 q^{19} + 36 i q^{22} + 63 i q^{23} - 16 q^{26} - 32 i q^{28} + 156 q^{29} - 85 q^{31} - 32 i q^{32} - 30 q^{34} + 74 i q^{37} + 46 i q^{38} - 246 q^{41} + 190 i q^{43} + 72 q^{44} + 126 q^{46} - 288 i q^{47} + 279 q^{49} + 32 i q^{52} - 177 i q^{53} - 64 q^{56} - 312 i q^{58} + 792 q^{59} - 907 q^{61} + 170 i q^{62} - 64 q^{64} - 322 i q^{67} + 60 i q^{68} + 270 q^{71} - 254 i q^{73} + 148 q^{74} + 92 q^{76} - 144 i q^{77} + 1123 q^{79} + 492 i q^{82} - 771 i q^{83} + 380 q^{86} - 144 i q^{88} - 198 q^{89} + 64 q^{91} - 252 i q^{92} - 576 q^{94} - 1192 i q^{97} - 558 i q^{98} +O(q^{100})$$ q - 2*i * q^2 - 4 * q^4 + 8*i * q^7 + 8*i * q^8 - 18 * q^11 - 8*i * q^13 + 16 * q^14 + 16 * q^16 - 15*i * q^17 - 23 * q^19 + 36*i * q^22 + 63*i * q^23 - 16 * q^26 - 32*i * q^28 + 156 * q^29 - 85 * q^31 - 32*i * q^32 - 30 * q^34 + 74*i * q^37 + 46*i * q^38 - 246 * q^41 + 190*i * q^43 + 72 * q^44 + 126 * q^46 - 288*i * q^47 + 279 * q^49 + 32*i * q^52 - 177*i * q^53 - 64 * q^56 - 312*i * q^58 + 792 * q^59 - 907 * q^61 + 170*i * q^62 - 64 * q^64 - 322*i * q^67 + 60*i * q^68 + 270 * q^71 - 254*i * q^73 + 148 * q^74 + 92 * q^76 - 144*i * q^77 + 1123 * q^79 + 492*i * q^82 - 771*i * q^83 + 380 * q^86 - 144*i * q^88 - 198 * q^89 + 64 * q^91 - 252*i * q^92 - 576 * q^94 - 1192*i * q^97 - 558*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} - 36 q^{11} + 32 q^{14} + 32 q^{16} - 46 q^{19} - 32 q^{26} + 312 q^{29} - 170 q^{31} - 60 q^{34} - 492 q^{41} + 144 q^{44} + 252 q^{46} + 558 q^{49} - 128 q^{56} + 1584 q^{59} - 1814 q^{61} - 128 q^{64} + 540 q^{71} + 296 q^{74} + 184 q^{76} + 2246 q^{79} + 760 q^{86} - 396 q^{89} + 128 q^{91} - 1152 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 - 36 * q^11 + 32 * q^14 + 32 * q^16 - 46 * q^19 - 32 * q^26 + 312 * q^29 - 170 * q^31 - 60 * q^34 - 492 * q^41 + 144 * q^44 + 252 * q^46 + 558 * q^49 - 128 * q^56 + 1584 * q^59 - 1814 * q^61 - 128 * q^64 + 540 * q^71 + 296 * q^74 + 184 * q^76 + 2246 * q^79 + 760 * q^86 - 396 * q^89 + 128 * q^91 - 1152 * 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 8.00000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 8.00000i 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.g 2
3.b odd 2 1 1350.4.c.n 2
5.b even 2 1 inner 1350.4.c.g 2
5.c odd 4 1 270.4.a.b 1
5.c odd 4 1 1350.4.a.t 1
15.d odd 2 1 1350.4.c.n 2
15.e even 4 1 270.4.a.l yes 1
15.e even 4 1 1350.4.a.f 1
20.e even 4 1 2160.4.a.c 1
45.k odd 12 2 810.4.e.v 2
45.l even 12 2 810.4.e.b 2
60.l odd 4 1 2160.4.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.b 1 5.c odd 4 1
270.4.a.l yes 1 15.e even 4 1
810.4.e.b 2 45.l even 12 2
810.4.e.v 2 45.k odd 12 2
1350.4.a.f 1 15.e even 4 1
1350.4.a.t 1 5.c odd 4 1
1350.4.c.g 2 1.a even 1 1 trivial
1350.4.c.g 2 5.b even 2 1 inner
1350.4.c.n 2 3.b odd 2 1
1350.4.c.n 2 15.d odd 2 1
2160.4.a.c 1 20.e even 4 1
2160.4.a.m 1 60.l odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 64$$ T7^2 + 64 $$T_{11} + 18$$ T11 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 64$$
$11$ $$(T + 18)^{2}$$
$13$ $$T^{2} + 64$$
$17$ $$T^{2} + 225$$
$19$ $$(T + 23)^{2}$$
$23$ $$T^{2} + 3969$$
$29$ $$(T - 156)^{2}$$
$31$ $$(T + 85)^{2}$$
$37$ $$T^{2} + 5476$$
$41$ $$(T + 246)^{2}$$
$43$ $$T^{2} + 36100$$
$47$ $$T^{2} + 82944$$
$53$ $$T^{2} + 31329$$
$59$ $$(T - 792)^{2}$$
$61$ $$(T + 907)^{2}$$
$67$ $$T^{2} + 103684$$
$71$ $$(T - 270)^{2}$$
$73$ $$T^{2} + 64516$$
$79$ $$(T - 1123)^{2}$$
$83$ $$T^{2} + 594441$$
$89$ $$(T + 198)^{2}$$
$97$ $$T^{2} + 1420864$$