# Properties

 Label 1350.4.c.d 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - 4 q^{4} + 4 i q^{7} - 8 i q^{8} +O(q^{10})$$ q + 2*i * q^2 - 4 * q^4 + 4*i * q^7 - 8*i * q^8 $$q + 2 i q^{2} - 4 q^{4} + 4 i q^{7} - 8 i q^{8} - 42 q^{11} + 20 i q^{13} - 8 q^{14} + 16 q^{16} + 93 i q^{17} - 59 q^{19} - 84 i q^{22} - 9 i q^{23} - 40 q^{26} - 16 i q^{28} + 120 q^{29} + 47 q^{31} + 32 i q^{32} - 186 q^{34} + 262 i q^{37} - 118 i q^{38} - 126 q^{41} - 178 i q^{43} + 168 q^{44} + 18 q^{46} + 144 i q^{47} + 327 q^{49} - 80 i q^{52} - 741 i q^{53} + 32 q^{56} + 240 i q^{58} - 444 q^{59} + 221 q^{61} + 94 i q^{62} - 64 q^{64} + 538 i q^{67} - 372 i q^{68} - 690 q^{71} - 1126 i q^{73} - 524 q^{74} + 236 q^{76} - 168 i q^{77} - 665 q^{79} - 252 i q^{82} - 75 i q^{83} + 356 q^{86} + 336 i q^{88} - 1086 q^{89} - 80 q^{91} + 36 i q^{92} - 288 q^{94} - 1544 i q^{97} + 654 i q^{98} +O(q^{100})$$ q + 2*i * q^2 - 4 * q^4 + 4*i * q^7 - 8*i * q^8 - 42 * q^11 + 20*i * q^13 - 8 * q^14 + 16 * q^16 + 93*i * q^17 - 59 * q^19 - 84*i * q^22 - 9*i * q^23 - 40 * q^26 - 16*i * q^28 + 120 * q^29 + 47 * q^31 + 32*i * q^32 - 186 * q^34 + 262*i * q^37 - 118*i * q^38 - 126 * q^41 - 178*i * q^43 + 168 * q^44 + 18 * q^46 + 144*i * q^47 + 327 * q^49 - 80*i * q^52 - 741*i * q^53 + 32 * q^56 + 240*i * q^58 - 444 * q^59 + 221 * q^61 + 94*i * q^62 - 64 * q^64 + 538*i * q^67 - 372*i * q^68 - 690 * q^71 - 1126*i * q^73 - 524 * q^74 + 236 * q^76 - 168*i * q^77 - 665 * q^79 - 252*i * q^82 - 75*i * q^83 + 356 * q^86 + 336*i * q^88 - 1086 * q^89 - 80 * q^91 + 36*i * q^92 - 288 * q^94 - 1544*i * q^97 + 654*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} - 84 q^{11} - 16 q^{14} + 32 q^{16} - 118 q^{19} - 80 q^{26} + 240 q^{29} + 94 q^{31} - 372 q^{34} - 252 q^{41} + 336 q^{44} + 36 q^{46} + 654 q^{49} + 64 q^{56} - 888 q^{59} + 442 q^{61} - 128 q^{64} - 1380 q^{71} - 1048 q^{74} + 472 q^{76} - 1330 q^{79} + 712 q^{86} - 2172 q^{89} - 160 q^{91} - 576 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 - 84 * q^11 - 16 * q^14 + 32 * q^16 - 118 * q^19 - 80 * q^26 + 240 * q^29 + 94 * q^31 - 372 * q^34 - 252 * q^41 + 336 * q^44 + 36 * q^46 + 654 * q^49 + 64 * q^56 - 888 * q^59 + 442 * q^61 - 128 * q^64 - 1380 * q^71 - 1048 * q^74 + 472 * q^76 - 1330 * q^79 + 712 * q^86 - 2172 * q^89 - 160 * q^91 - 576 * 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 4.00000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 4.00000i 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.d 2
3.b odd 2 1 1350.4.c.q 2
5.b even 2 1 inner 1350.4.c.d 2
5.c odd 4 1 270.4.a.e 1
5.c odd 4 1 1350.4.a.u 1
15.d odd 2 1 1350.4.c.q 2
15.e even 4 1 270.4.a.i yes 1
15.e even 4 1 1350.4.a.g 1
20.e even 4 1 2160.4.a.o 1
45.k odd 12 2 810.4.e.q 2
45.l even 12 2 810.4.e.h 2
60.l odd 4 1 2160.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.e 1 5.c odd 4 1
270.4.a.i yes 1 15.e even 4 1
810.4.e.h 2 45.l even 12 2
810.4.e.q 2 45.k odd 12 2
1350.4.a.g 1 15.e even 4 1
1350.4.a.u 1 5.c odd 4 1
1350.4.c.d 2 1.a even 1 1 trivial
1350.4.c.d 2 5.b even 2 1 inner
1350.4.c.q 2 3.b odd 2 1
1350.4.c.q 2 15.d odd 2 1
2160.4.a.e 1 60.l odd 4 1
2160.4.a.o 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 $$S_{4}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} + 42$$ T11 + 42

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 42)^{2}$$
$13$ $$T^{2} + 400$$
$17$ $$T^{2} + 8649$$
$19$ $$(T + 59)^{2}$$
$23$ $$T^{2} + 81$$
$29$ $$(T - 120)^{2}$$
$31$ $$(T - 47)^{2}$$
$37$ $$T^{2} + 68644$$
$41$ $$(T + 126)^{2}$$
$43$ $$T^{2} + 31684$$
$47$ $$T^{2} + 20736$$
$53$ $$T^{2} + 549081$$
$59$ $$(T + 444)^{2}$$
$61$ $$(T - 221)^{2}$$
$67$ $$T^{2} + 289444$$
$71$ $$(T + 690)^{2}$$
$73$ $$T^{2} + 1267876$$
$79$ $$(T + 665)^{2}$$
$83$ $$T^{2} + 5625$$
$89$ $$(T + 1086)^{2}$$
$97$ $$T^{2} + 2383936$$