# Properties

 Label 1350.4.c.l 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - 4 q^{4} + 22 i q^{7} - 8 i q^{8} +O(q^{10})$$ q + 2*i * q^2 - 4 * q^4 + 22*i * q^7 - 8*i * q^8 $$q + 2 i q^{2} - 4 q^{4} + 22 i q^{7} - 8 i q^{8} + 12 q^{11} + 38 i q^{13} - 44 q^{14} + 16 q^{16} - 105 i q^{17} + 157 q^{19} + 24 i q^{22} + 117 i q^{23} - 76 q^{26} - 88 i q^{28} + 66 q^{29} - 25 q^{31} + 32 i q^{32} + 210 q^{34} - 314 i q^{37} + 314 i q^{38} + 504 q^{41} + 380 i q^{43} - 48 q^{44} - 234 q^{46} - 252 i q^{47} - 141 q^{49} - 152 i q^{52} - 3 i q^{53} + 176 q^{56} + 132 i q^{58} - 318 q^{59} + 293 q^{61} - 50 i q^{62} - 64 q^{64} + 322 i q^{67} + 420 i q^{68} + 120 q^{71} + 44 i q^{73} + 628 q^{74} - 628 q^{76} + 264 i q^{77} - 917 q^{79} + 1008 i q^{82} - 309 i q^{83} - 760 q^{86} - 96 i q^{88} + 1272 q^{89} - 836 q^{91} - 468 i q^{92} + 504 q^{94} - 1328 i q^{97} - 282 i q^{98} +O(q^{100})$$ q + 2*i * q^2 - 4 * q^4 + 22*i * q^7 - 8*i * q^8 + 12 * q^11 + 38*i * q^13 - 44 * q^14 + 16 * q^16 - 105*i * q^17 + 157 * q^19 + 24*i * q^22 + 117*i * q^23 - 76 * q^26 - 88*i * q^28 + 66 * q^29 - 25 * q^31 + 32*i * q^32 + 210 * q^34 - 314*i * q^37 + 314*i * q^38 + 504 * q^41 + 380*i * q^43 - 48 * q^44 - 234 * q^46 - 252*i * q^47 - 141 * q^49 - 152*i * q^52 - 3*i * q^53 + 176 * q^56 + 132*i * q^58 - 318 * q^59 + 293 * q^61 - 50*i * q^62 - 64 * q^64 + 322*i * q^67 + 420*i * q^68 + 120 * q^71 + 44*i * q^73 + 628 * q^74 - 628 * q^76 + 264*i * q^77 - 917 * q^79 + 1008*i * q^82 - 309*i * q^83 - 760 * q^86 - 96*i * q^88 + 1272 * q^89 - 836 * q^91 - 468*i * q^92 + 504 * q^94 - 1328*i * q^97 - 282*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} + 24 q^{11} - 88 q^{14} + 32 q^{16} + 314 q^{19} - 152 q^{26} + 132 q^{29} - 50 q^{31} + 420 q^{34} + 1008 q^{41} - 96 q^{44} - 468 q^{46} - 282 q^{49} + 352 q^{56} - 636 q^{59} + 586 q^{61} - 128 q^{64} + 240 q^{71} + 1256 q^{74} - 1256 q^{76} - 1834 q^{79} - 1520 q^{86} + 2544 q^{89} - 1672 q^{91} + 1008 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 + 24 * q^11 - 88 * q^14 + 32 * q^16 + 314 * q^19 - 152 * q^26 + 132 * q^29 - 50 * q^31 + 420 * q^34 + 1008 * q^41 - 96 * q^44 - 468 * q^46 - 282 * q^49 + 352 * q^56 - 636 * q^59 + 586 * q^61 - 128 * q^64 + 240 * q^71 + 1256 * q^74 - 1256 * q^76 - 1834 * q^79 - 1520 * q^86 + 2544 * q^89 - 1672 * q^91 + 1008 * 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 22.0000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 22.0000i 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.l 2
3.b odd 2 1 1350.4.c.i 2
5.b even 2 1 inner 1350.4.c.l 2
5.c odd 4 1 270.4.a.c 1
5.c odd 4 1 1350.4.a.z 1
15.d odd 2 1 1350.4.c.i 2
15.e even 4 1 270.4.a.g yes 1
15.e even 4 1 1350.4.a.l 1
20.e even 4 1 2160.4.a.r 1
45.k odd 12 2 810.4.e.s 2
45.l even 12 2 810.4.e.k 2
60.l odd 4 1 2160.4.a.i 1

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

## Hecke kernels

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

 $$T_{7}^{2} + 484$$ T7^2 + 484 $$T_{11} - 12$$ T11 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 484$$
$11$ $$(T - 12)^{2}$$
$13$ $$T^{2} + 1444$$
$17$ $$T^{2} + 11025$$
$19$ $$(T - 157)^{2}$$
$23$ $$T^{2} + 13689$$
$29$ $$(T - 66)^{2}$$
$31$ $$(T + 25)^{2}$$
$37$ $$T^{2} + 98596$$
$41$ $$(T - 504)^{2}$$
$43$ $$T^{2} + 144400$$
$47$ $$T^{2} + 63504$$
$53$ $$T^{2} + 9$$
$59$ $$(T + 318)^{2}$$
$61$ $$(T - 293)^{2}$$
$67$ $$T^{2} + 103684$$
$71$ $$(T - 120)^{2}$$
$73$ $$T^{2} + 1936$$
$79$ $$(T + 917)^{2}$$
$83$ $$T^{2} + 95481$$
$89$ $$(T - 1272)^{2}$$
$97$ $$T^{2} + 1763584$$