# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(193,384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$22.6567334422$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + 3 \beta_1 q^{3} + (\beta_{2} - 4 \beta_1) q^{5} + ( - \beta_{3} + 8) q^{7} - 9 q^{9}+O(q^{10})$$ q + 3*b1 * q^3 + (b2 - 4*b1) * q^5 + (-b3 + 8) * q^7 - 9 * q^9 $$q + 3 \beta_1 q^{3} + (\beta_{2} - 4 \beta_1) q^{5} + ( - \beta_{3} + 8) q^{7} - 9 q^{9} + ( - 4 \beta_{2} - 4 \beta_1) q^{11} + ( - 2 \beta_{2} - 36 \beta_1) q^{13} + ( - 3 \beta_{3} + 12) q^{15} + ( - 4 \beta_{3} - 18) q^{17} + (4 \beta_{2} - 68 \beta_1) q^{19} + ( - 3 \beta_{2} + 24 \beta_1) q^{21} + (2 \beta_{3} + 128) q^{23} + (8 \beta_{3} - 99) q^{25} - 27 \beta_1 q^{27} + (9 \beta_{2} - 76 \beta_1) q^{29} + (13 \beta_{3} + 40) q^{31} + (12 \beta_{3} + 12) q^{33} + (12 \beta_{2} - 240 \beta_1) q^{35} + ( - 4 \beta_{2} + 68 \beta_1) q^{37} + (6 \beta_{3} + 108) q^{39} + ( - 20 \beta_{3} + 218) q^{41} + (4 \beta_{2} - 356 \beta_1) q^{43} + ( - 9 \beta_{2} + 36 \beta_1) q^{45} + ( - 14 \beta_{3} + 112) q^{47} + ( - 16 \beta_{3} - 71) q^{49} + ( - 12 \beta_{2} - 54 \beta_1) q^{51} + (15 \beta_{2} + 172 \beta_1) q^{53} + ( - 12 \beta_{3} + 816) q^{55} + ( - 12 \beta_{3} + 204) q^{57} - 324 \beta_1 q^{59} - 324 \beta_1 q^{61} + (9 \beta_{3} - 72) q^{63} + (28 \beta_{3} + 272) q^{65} + ( - 48 \beta_{2} - 228 \beta_1) q^{67} + (6 \beta_{2} + 384 \beta_1) q^{69} + ( - 2 \beta_{3} + 1024) q^{71} + (48 \beta_{3} - 330) q^{73} + (24 \beta_{2} - 297 \beta_1) q^{75} + ( - 28 \beta_{2} + 800 \beta_1) q^{77} + ( - 59 \beta_{3} - 248) q^{79} + 81 q^{81} + ( - 28 \beta_{2} - 388 \beta_1) q^{83} + ( - 2 \beta_{2} - 760 \beta_1) q^{85} + ( - 27 \beta_{3} + 228) q^{87} + (8 \beta_{3} - 266) q^{89} + (20 \beta_{2} + 128 \beta_1) q^{91} + (39 \beta_{2} + 120 \beta_1) q^{93} + (84 \beta_{3} - 1104) q^{95} + ( - 88 \beta_{3} - 610) q^{97} + (36 \beta_{2} + 36 \beta_1) q^{99}+O(q^{100})$$ q + 3*b1 * q^3 + (b2 - 4*b1) * q^5 + (-b3 + 8) * q^7 - 9 * q^9 + (-4*b2 - 4*b1) * q^11 + (-2*b2 - 36*b1) * q^13 + (-3*b3 + 12) * q^15 + (-4*b3 - 18) * q^17 + (4*b2 - 68*b1) * q^19 + (-3*b2 + 24*b1) * q^21 + (2*b3 + 128) * q^23 + (8*b3 - 99) * q^25 - 27*b1 * q^27 + (9*b2 - 76*b1) * q^29 + (13*b3 + 40) * q^31 + (12*b3 + 12) * q^33 + (12*b2 - 240*b1) * q^35 + (-4*b2 + 68*b1) * q^37 + (6*b3 + 108) * q^39 + (-20*b3 + 218) * q^41 + (4*b2 - 356*b1) * q^43 + (-9*b2 + 36*b1) * q^45 + (-14*b3 + 112) * q^47 + (-16*b3 - 71) * q^49 + (-12*b2 - 54*b1) * q^51 + (15*b2 + 172*b1) * q^53 + (-12*b3 + 816) * q^55 + (-12*b3 + 204) * q^57 - 324*b1 * q^59 - 324*b1 * q^61 + (9*b3 - 72) * q^63 + (28*b3 + 272) * q^65 + (-48*b2 - 228*b1) * q^67 + (6*b2 + 384*b1) * q^69 + (-2*b3 + 1024) * q^71 + (48*b3 - 330) * q^73 + (24*b2 - 297*b1) * q^75 + (-28*b2 + 800*b1) * q^77 + (-59*b3 - 248) * q^79 + 81 * q^81 + (-28*b2 - 388*b1) * q^83 + (-2*b2 - 760*b1) * q^85 + (-27*b3 + 228) * q^87 + (8*b3 - 266) * q^89 + (20*b2 + 128*b1) * q^91 + (39*b2 + 120*b1) * q^93 + (84*b3 - 1104) * q^95 + (-88*b3 - 610) * q^97 + (36*b2 + 36*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 32 q^{7} - 36 q^{9}+O(q^{10})$$ 4 * q + 32 * q^7 - 36 * q^9 $$4 q + 32 q^{7} - 36 q^{9} + 48 q^{15} - 72 q^{17} + 512 q^{23} - 396 q^{25} + 160 q^{31} + 48 q^{33} + 432 q^{39} + 872 q^{41} + 448 q^{47} - 284 q^{49} + 3264 q^{55} + 816 q^{57} - 288 q^{63} + 1088 q^{65} + 4096 q^{71} - 1320 q^{73} - 992 q^{79} + 324 q^{81} + 912 q^{87} - 1064 q^{89} - 4416 q^{95} - 2440 q^{97}+O(q^{100})$$ 4 * q + 32 * q^7 - 36 * q^9 + 48 * q^15 - 72 * q^17 + 512 * q^23 - 396 * q^25 + 160 * q^31 + 48 * q^33 + 432 * q^39 + 872 * q^41 + 448 * q^47 - 284 * q^49 + 3264 * q^55 + 816 * q^57 - 288 * q^63 + 1088 * q^65 + 4096 * q^71 - 1320 * q^73 - 992 * q^79 + 324 * q^81 + 912 * q^87 - 1064 * q^89 - 4416 * q^95 - 2440 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 3$$ (v^3 + 4*v) / 3 $$\beta_{2}$$ $$=$$ $$( 4\nu^{3} + 40\nu ) / 3$$ (4*v^3 + 40*v) / 3 $$\beta_{3}$$ $$=$$ $$8\nu^{2} + 28$$ 8*v^2 + 28
 $$\nu$$ $$=$$ $$( \beta_{2} - 4\beta_1 ) / 8$$ (b2 - 4*b1) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 28 ) / 8$$ (b3 - 28) / 8 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 10\beta_1 ) / 2$$ (-b2 + 10*b1) / 2

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 − 1.30278i 2.30278i − 2.30278i 1.30278i
0 3.00000i 0 10.4222i 0 −6.42221 0 −9.00000 0
193.2 0 3.00000i 0 18.4222i 0 22.4222 0 −9.00000 0
193.3 0 3.00000i 0 18.4222i 0 22.4222 0 −9.00000 0
193.4 0 3.00000i 0 10.4222i 0 −6.42221 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.e yes 4
3.b odd 2 1 1152.4.d.o 4
4.b odd 2 1 384.4.d.c 4
8.b even 2 1 inner 384.4.d.e yes 4
8.d odd 2 1 384.4.d.c 4
12.b even 2 1 1152.4.d.i 4
16.e even 4 1 768.4.a.e 2
16.e even 4 1 768.4.a.p 2
16.f odd 4 1 768.4.a.j 2
16.f odd 4 1 768.4.a.k 2
24.f even 2 1 1152.4.d.i 4
24.h odd 2 1 1152.4.d.o 4
48.i odd 4 1 2304.4.a.s 2
48.i odd 4 1 2304.4.a.bp 2
48.k even 4 1 2304.4.a.t 2
48.k even 4 1 2304.4.a.bq 2

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

 $$T_{5}^{4} + 448T_{5}^{2} + 36864$$ T5^4 + 448*T5^2 + 36864 $$T_{7}^{2} - 16T_{7} - 144$$ T7^2 - 16*T7 - 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 9)^{2}$$
$5$ $$T^{4} + 448 T^{2} + 36864$$
$7$ $$(T^{2} - 16 T - 144)^{2}$$
$11$ $$T^{4} + 6688 T^{2} + \cdots + 10969344$$
$13$ $$T^{4} + 4256 T^{2} + 215296$$
$17$ $$(T^{2} + 36 T - 3004)^{2}$$
$19$ $$T^{4} + 15904 T^{2} + \cdots + 1679616$$
$23$ $$(T^{2} - 256 T + 15552)^{2}$$
$29$ $$T^{4} + 45248 T^{2} + \cdots + 122589184$$
$31$ $$(T^{2} - 80 T - 33552)^{2}$$
$37$ $$T^{4} + 15904 T^{2} + \cdots + 1679616$$
$41$ $$(T^{2} - 436 T - 35676)^{2}$$
$43$ $$T^{4} + 260128 T^{2} + \cdots + 15229534464$$
$47$ $$(T^{2} - 224 T - 28224)^{2}$$
$53$ $$T^{4} + 152768 T^{2} + \cdots + 296390656$$
$59$ $$(T^{2} + 104976)^{2}$$
$61$ $$(T^{2} + 104976)^{2}$$
$67$ $$T^{4} + 1062432 T^{2} + \cdots + 182540853504$$
$71$ $$(T^{2} - 2048 T + 1047744)^{2}$$
$73$ $$(T^{2} + 660 T - 370332)^{2}$$
$79$ $$(T^{2} + 496 T - 662544)^{2}$$
$83$ $$T^{4} + 627232 T^{2} + \cdots + 156950784$$
$89$ $$(T^{2} + 532 T + 57444)^{2}$$
$97$ $$(T^{2} + 1220 T - 1238652)^{2}$$