# Properties

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

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \beta_1 q^{3} + \beta_{2} q^{5} - 5 \beta_{3} q^{7} - 9 q^{9}+O(q^{10})$$ q + 3*b1 * q^3 + b2 * q^5 - 5*b3 * q^7 - 9 * q^9 $$q + 3 \beta_1 q^{3} + \beta_{2} q^{5} - 5 \beta_{3} q^{7} - 9 q^{9} + 20 \beta_1 q^{11} - 14 \beta_{2} q^{13} - 3 \beta_{3} q^{15} - 34 q^{17} - 52 \beta_1 q^{19} - 15 \beta_{2} q^{21} - 22 \beta_{3} q^{23} + 117 q^{25} - 27 \beta_1 q^{27} - 71 \beta_{2} q^{29} - 39 \beta_{3} q^{31} - 60 q^{33} - 40 \beta_1 q^{35} - 96 \beta_{2} q^{37} + 42 \beta_{3} q^{39} + 26 q^{41} + 252 \beta_1 q^{43} - 9 \beta_{2} q^{45} + 122 \beta_{3} q^{47} - 143 q^{49} - 102 \beta_1 q^{51} - 241 \beta_{2} q^{53} - 20 \beta_{3} q^{55} + 156 q^{57} + 364 \beta_1 q^{59} - 260 \beta_{2} q^{61} + 45 \beta_{3} q^{63} + 112 q^{65} - 628 \beta_1 q^{67} - 66 \beta_{2} q^{69} + 118 \beta_{3} q^{71} - 338 q^{73} + 351 \beta_1 q^{75} - 100 \beta_{2} q^{77} - 279 \beta_{3} q^{79} + 81 q^{81} - 1036 \beta_1 q^{83} - 34 \beta_{2} q^{85} + 213 \beta_{3} q^{87} - 234 q^{89} + 560 \beta_1 q^{91} - 117 \beta_{2} q^{93} + 52 \beta_{3} q^{95} - 178 q^{97} - 180 \beta_1 q^{99}+O(q^{100})$$ q + 3*b1 * q^3 + b2 * q^5 - 5*b3 * q^7 - 9 * q^9 + 20*b1 * q^11 - 14*b2 * q^13 - 3*b3 * q^15 - 34 * q^17 - 52*b1 * q^19 - 15*b2 * q^21 - 22*b3 * q^23 + 117 * q^25 - 27*b1 * q^27 - 71*b2 * q^29 - 39*b3 * q^31 - 60 * q^33 - 40*b1 * q^35 - 96*b2 * q^37 + 42*b3 * q^39 + 26 * q^41 + 252*b1 * q^43 - 9*b2 * q^45 + 122*b3 * q^47 - 143 * q^49 - 102*b1 * q^51 - 241*b2 * q^53 - 20*b3 * q^55 + 156 * q^57 + 364*b1 * q^59 - 260*b2 * q^61 + 45*b3 * q^63 + 112 * q^65 - 628*b1 * q^67 - 66*b2 * q^69 + 118*b3 * q^71 - 338 * q^73 + 351*b1 * q^75 - 100*b2 * q^77 - 279*b3 * q^79 + 81 * q^81 - 1036*b1 * q^83 - 34*b2 * q^85 + 213*b3 * q^87 - 234 * q^89 + 560*b1 * q^91 - 117*b2 * q^93 + 52*b3 * q^95 - 178 * q^97 - 180*b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 36 q^{9}+O(q^{10})$$ 4 * q - 36 * q^9 $$4 q - 36 q^{9} - 136 q^{17} + 468 q^{25} - 240 q^{33} + 104 q^{41} - 572 q^{49} + 624 q^{57} + 448 q^{65} - 1352 q^{73} + 324 q^{81} - 936 q^{89} - 712 q^{97}+O(q^{100})$$ 4 * q - 36 * q^9 - 136 * q^17 + 468 * q^25 - 240 * q^33 + 104 * q^41 - 572 * q^49 + 624 * q^57 + 448 * q^65 - 1352 * q^73 + 324 * q^81 - 936 * q^89 - 712 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$2\zeta_{8}^{3} + 2\zeta_{8}$$ 2*v^3 + 2*v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 4$$ (b3 + b2) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 4$$ (-b3 + b2) / 4

## 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
 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
0 3.00000i 0 2.82843i 0 −14.1421 0 −9.00000 0
193.2 0 3.00000i 0 2.82843i 0 14.1421 0 −9.00000 0
193.3 0 3.00000i 0 2.82843i 0 14.1421 0 −9.00000 0
193.4 0 3.00000i 0 2.82843i 0 −14.1421 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 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.d 4
3.b odd 2 1 1152.4.d.n 4
4.b odd 2 1 inner 384.4.d.d 4
8.b even 2 1 inner 384.4.d.d 4
8.d odd 2 1 inner 384.4.d.d 4
12.b even 2 1 1152.4.d.n 4
16.e even 4 1 768.4.a.h 2
16.e even 4 1 768.4.a.m 2
16.f odd 4 1 768.4.a.h 2
16.f odd 4 1 768.4.a.m 2
24.f even 2 1 1152.4.d.n 4
24.h odd 2 1 1152.4.d.n 4
48.i odd 4 1 2304.4.a.bb 2
48.i odd 4 1 2304.4.a.bh 2
48.k even 4 1 2304.4.a.bb 2
48.k even 4 1 2304.4.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.d 4 1.a even 1 1 trivial
384.4.d.d 4 4.b odd 2 1 inner
384.4.d.d 4 8.b even 2 1 inner
384.4.d.d 4 8.d odd 2 1 inner
768.4.a.h 2 16.e even 4 1
768.4.a.h 2 16.f odd 4 1
768.4.a.m 2 16.e even 4 1
768.4.a.m 2 16.f odd 4 1
1152.4.d.n 4 3.b odd 2 1
1152.4.d.n 4 12.b even 2 1
1152.4.d.n 4 24.f even 2 1
1152.4.d.n 4 24.h odd 2 1
2304.4.a.bb 2 48.i odd 4 1
2304.4.a.bb 2 48.k even 4 1
2304.4.a.bh 2 48.i odd 4 1
2304.4.a.bh 2 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{2} + 8$$ T5^2 + 8 $$T_{7}^{2} - 200$$ T7^2 - 200

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 9)^{2}$$
$5$ $$(T^{2} + 8)^{2}$$
$7$ $$(T^{2} - 200)^{2}$$
$11$ $$(T^{2} + 400)^{2}$$
$13$ $$(T^{2} + 1568)^{2}$$
$17$ $$(T + 34)^{4}$$
$19$ $$(T^{2} + 2704)^{2}$$
$23$ $$(T^{2} - 3872)^{2}$$
$29$ $$(T^{2} + 40328)^{2}$$
$31$ $$(T^{2} - 12168)^{2}$$
$37$ $$(T^{2} + 73728)^{2}$$
$41$ $$(T - 26)^{4}$$
$43$ $$(T^{2} + 63504)^{2}$$
$47$ $$(T^{2} - 119072)^{2}$$
$53$ $$(T^{2} + 464648)^{2}$$
$59$ $$(T^{2} + 132496)^{2}$$
$61$ $$(T^{2} + 540800)^{2}$$
$67$ $$(T^{2} + 394384)^{2}$$
$71$ $$(T^{2} - 111392)^{2}$$
$73$ $$(T + 338)^{4}$$
$79$ $$(T^{2} - 622728)^{2}$$
$83$ $$(T^{2} + 1073296)^{2}$$
$89$ $$(T + 234)^{4}$$
$97$ $$(T + 178)^{4}$$