# Properties

 Label 384.4.d.a Level $384$ Weight $4$ Character orbit 384.d Analytic conductor $22.657$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.6567334422$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - 3 i q^{3} + 8 i q^{5} - 12 q^{7} - 9 q^{9} +O(q^{10})$$ q - 3*i * q^3 + 8*i * q^5 - 12 * q^7 - 9 * q^9 $$q - 3 i q^{3} + 8 i q^{5} - 12 q^{7} - 9 q^{9} - 12 i q^{11} + 20 i q^{13} + 24 q^{15} + 62 q^{17} - 108 i q^{19} + 36 i q^{21} + 72 q^{23} + 61 q^{25} + 27 i q^{27} - 128 i q^{29} + 204 q^{31} - 36 q^{33} - 96 i q^{35} + 228 i q^{37} + 60 q^{39} - 22 q^{41} - 204 i q^{43} - 72 i q^{45} + 600 q^{47} - 199 q^{49} - 186 i q^{51} - 256 i q^{53} + 96 q^{55} - 324 q^{57} - 828 i q^{59} - 84 i q^{61} + 108 q^{63} - 160 q^{65} - 348 i q^{67} - 216 i q^{69} - 456 q^{71} + 822 q^{73} - 183 i q^{75} + 144 i q^{77} + 1356 q^{79} + 81 q^{81} - 108 i q^{83} + 496 i q^{85} - 384 q^{87} - 938 q^{89} - 240 i q^{91} - 612 i q^{93} + 864 q^{95} + 1278 q^{97} + 108 i q^{99} +O(q^{100})$$ q - 3*i * q^3 + 8*i * q^5 - 12 * q^7 - 9 * q^9 - 12*i * q^11 + 20*i * q^13 + 24 * q^15 + 62 * q^17 - 108*i * q^19 + 36*i * q^21 + 72 * q^23 + 61 * q^25 + 27*i * q^27 - 128*i * q^29 + 204 * q^31 - 36 * q^33 - 96*i * q^35 + 228*i * q^37 + 60 * q^39 - 22 * q^41 - 204*i * q^43 - 72*i * q^45 + 600 * q^47 - 199 * q^49 - 186*i * q^51 - 256*i * q^53 + 96 * q^55 - 324 * q^57 - 828*i * q^59 - 84*i * q^61 + 108 * q^63 - 160 * q^65 - 348*i * q^67 - 216*i * q^69 - 456 * q^71 + 822 * q^73 - 183*i * q^75 + 144*i * q^77 + 1356 * q^79 + 81 * q^81 - 108*i * q^83 + 496*i * q^85 - 384 * q^87 - 938 * q^89 - 240*i * q^91 - 612*i * q^93 + 864 * q^95 + 1278 * q^97 + 108*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 24 q^{7} - 18 q^{9}+O(q^{10})$$ 2 * q - 24 * q^7 - 18 * q^9 $$2 q - 24 q^{7} - 18 q^{9} + 48 q^{15} + 124 q^{17} + 144 q^{23} + 122 q^{25} + 408 q^{31} - 72 q^{33} + 120 q^{39} - 44 q^{41} + 1200 q^{47} - 398 q^{49} + 192 q^{55} - 648 q^{57} + 216 q^{63} - 320 q^{65} - 912 q^{71} + 1644 q^{73} + 2712 q^{79} + 162 q^{81} - 768 q^{87} - 1876 q^{89} + 1728 q^{95} + 2556 q^{97}+O(q^{100})$$ 2 * q - 24 * q^7 - 18 * q^9 + 48 * q^15 + 124 * q^17 + 144 * q^23 + 122 * q^25 + 408 * q^31 - 72 * q^33 + 120 * q^39 - 44 * q^41 + 1200 * q^47 - 398 * q^49 + 192 * q^55 - 648 * q^57 + 216 * q^63 - 320 * q^65 - 912 * q^71 + 1644 * q^73 + 2712 * q^79 + 162 * q^81 - 768 * q^87 - 1876 * q^89 + 1728 * q^95 + 2556 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 1.00000i − 1.00000i
0 3.00000i 0 8.00000i 0 −12.0000 0 −9.00000 0
193.2 0 3.00000i 0 8.00000i 0 −12.0000 0 −9.00000 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.4.d.a 2
3.b odd 2 1 1152.4.d.b 2
4.b odd 2 1 384.4.d.b yes 2
8.b even 2 1 inner 384.4.d.a 2
8.d odd 2 1 384.4.d.b yes 2
12.b even 2 1 1152.4.d.g 2
16.e even 4 1 768.4.a.a 1
16.e even 4 1 768.4.a.d 1
16.f odd 4 1 768.4.a.b 1
16.f odd 4 1 768.4.a.c 1
24.f even 2 1 1152.4.d.g 2
24.h odd 2 1 1152.4.d.b 2
48.i odd 4 1 2304.4.a.f 1
48.i odd 4 1 2304.4.a.l 1
48.k even 4 1 2304.4.a.e 1
48.k even 4 1 2304.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.a 2 1.a even 1 1 trivial
384.4.d.a 2 8.b even 2 1 inner
384.4.d.b yes 2 4.b odd 2 1
384.4.d.b yes 2 8.d odd 2 1
768.4.a.a 1 16.e even 4 1
768.4.a.b 1 16.f odd 4 1
768.4.a.c 1 16.f odd 4 1
768.4.a.d 1 16.e even 4 1
1152.4.d.b 2 3.b odd 2 1
1152.4.d.b 2 24.h odd 2 1
1152.4.d.g 2 12.b even 2 1
1152.4.d.g 2 24.f even 2 1
2304.4.a.e 1 48.k even 4 1
2304.4.a.f 1 48.i odd 4 1
2304.4.a.k 1 48.k even 4 1
2304.4.a.l 1 48.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 $$S_{4}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{2} + 64$$ T5^2 + 64 $$T_{7} + 12$$ T7 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2} + 64$$
$7$ $$(T + 12)^{2}$$
$11$ $$T^{2} + 144$$
$13$ $$T^{2} + 400$$
$17$ $$(T - 62)^{2}$$
$19$ $$T^{2} + 11664$$
$23$ $$(T - 72)^{2}$$
$29$ $$T^{2} + 16384$$
$31$ $$(T - 204)^{2}$$
$37$ $$T^{2} + 51984$$
$41$ $$(T + 22)^{2}$$
$43$ $$T^{2} + 41616$$
$47$ $$(T - 600)^{2}$$
$53$ $$T^{2} + 65536$$
$59$ $$T^{2} + 685584$$
$61$ $$T^{2} + 7056$$
$67$ $$T^{2} + 121104$$
$71$ $$(T + 456)^{2}$$
$73$ $$(T - 822)^{2}$$
$79$ $$(T - 1356)^{2}$$
$83$ $$T^{2} + 11664$$
$89$ $$(T + 938)^{2}$$
$97$ $$(T - 1278)^{2}$$