# Properties

 Label 384.4.f.f Level $384$ Weight $4$ Character orbit 384.f 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.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.6567334422$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{26})$$ Defining polynomial: $$x^{4} + 169$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( \beta_{1} + 2 \beta_{2} ) q^{5} + \beta_{3} q^{7} + ( 25 - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( \beta_{1} + 2 \beta_{2} ) q^{5} + \beta_{3} q^{7} + ( 25 - \beta_{3} ) q^{9} -23 \beta_{1} q^{11} -22 \beta_{1} q^{13} + ( 52 - \beta_{3} ) q^{15} -2 \beta_{3} q^{17} + ( -\beta_{1} - 2 \beta_{2} ) q^{19} + ( 27 \beta_{1} + 2 \beta_{2} ) q^{21} + 88 q^{23} -21 q^{25} + ( -27 \beta_{1} + 23 \beta_{2} ) q^{27} + ( 25 \beta_{1} + 50 \beta_{2} ) q^{29} + 21 \beta_{3} q^{31} + ( -46 - 23 \beta_{3} ) q^{33} + 52 \beta_{1} q^{35} + 166 \beta_{1} q^{37} + ( -44 - 22 \beta_{3} ) q^{39} -48 \beta_{3} q^{41} + ( -23 \beta_{1} - 46 \beta_{2} ) q^{43} + ( -27 \beta_{1} + 50 \beta_{2} ) q^{45} -384 q^{47} + 239 q^{49} + ( -54 \beta_{1} - 4 \beta_{2} ) q^{51} + ( 45 \beta_{1} + 90 \beta_{2} ) q^{53} -46 \beta_{3} q^{55} + ( -52 + \beta_{3} ) q^{57} + 315 \beta_{1} q^{59} -118 \beta_{1} q^{61} + ( 104 + 25 \beta_{3} ) q^{63} -44 \beta_{3} q^{65} + ( 5 \beta_{1} + 10 \beta_{2} ) q^{67} + 88 \beta_{2} q^{69} -680 q^{71} + 422 q^{73} -21 \beta_{2} q^{75} + ( 46 \beta_{1} + 92 \beta_{2} ) q^{77} + 73 \beta_{3} q^{79} + ( 521 - 50 \beta_{3} ) q^{81} + 93 \beta_{1} q^{83} -104 \beta_{1} q^{85} + ( 1300 - 25 \beta_{3} ) q^{87} -94 \beta_{3} q^{89} + ( 44 \beta_{1} + 88 \beta_{2} ) q^{91} + ( 567 \beta_{1} + 42 \beta_{2} ) q^{93} -104 q^{95} -1062 q^{97} + ( -621 \beta_{1} - 92 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 100q^{9} + O(q^{10})$$ $$4q + 100q^{9} + 208q^{15} + 352q^{23} - 84q^{25} - 184q^{33} - 176q^{39} - 1536q^{47} + 956q^{49} - 208q^{57} + 416q^{63} - 2720q^{71} + 1688q^{73} + 2084q^{81} + 5200q^{87} - 416q^{95} - 4248q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 169$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2}$$$$/13$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - \nu^{2} + 13 \nu$$$$)/13$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + 26 \nu$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$13 \beta_{1}$$$$/2$$ $$\nu^{3}$$ $$=$$ $$($$$$13 \beta_{3} - 26 \beta_{2} - 13 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 −2.54951 − 2.54951i −2.54951 + 2.54951i 2.54951 + 2.54951i 2.54951 − 2.54951i
0 −5.09902 1.00000i 0 −10.1980 0 10.1980i 0 25.0000 + 10.1980i 0
191.2 0 −5.09902 + 1.00000i 0 −10.1980 0 10.1980i 0 25.0000 10.1980i 0
191.3 0 5.09902 1.00000i 0 10.1980 0 10.1980i 0 25.0000 10.1980i 0
191.4 0 5.09902 + 1.00000i 0 10.1980 0 10.1980i 0 25.0000 + 10.1980i 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
12.b even 2 1 inner
24.f 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.f.f yes 4
3.b odd 2 1 384.4.f.e 4
4.b odd 2 1 384.4.f.e 4
8.b even 2 1 inner 384.4.f.f yes 4
8.d odd 2 1 384.4.f.e 4
12.b even 2 1 inner 384.4.f.f yes 4
16.e even 4 1 768.4.c.d 2
16.e even 4 1 768.4.c.g 2
16.f odd 4 1 768.4.c.e 2
16.f odd 4 1 768.4.c.f 2
24.f even 2 1 inner 384.4.f.f yes 4
24.h odd 2 1 384.4.f.e 4
48.i odd 4 1 768.4.c.e 2
48.i odd 4 1 768.4.c.f 2
48.k even 4 1 768.4.c.d 2
48.k even 4 1 768.4.c.g 2

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

 $$T_{5}^{2} - 104$$ $$T_{23} - 88$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$729 - 50 T^{2} + T^{4}$$
$5$ $$( -104 + T^{2} )^{2}$$
$7$ $$( 104 + T^{2} )^{2}$$
$11$ $$( 2116 + T^{2} )^{2}$$
$13$ $$( 1936 + T^{2} )^{2}$$
$17$ $$( 416 + T^{2} )^{2}$$
$19$ $$( -104 + T^{2} )^{2}$$
$23$ $$( -88 + T )^{4}$$
$29$ $$( -65000 + T^{2} )^{2}$$
$31$ $$( 45864 + T^{2} )^{2}$$
$37$ $$( 110224 + T^{2} )^{2}$$
$41$ $$( 239616 + T^{2} )^{2}$$
$43$ $$( -55016 + T^{2} )^{2}$$
$47$ $$( 384 + T )^{4}$$
$53$ $$( -210600 + T^{2} )^{2}$$
$59$ $$( 396900 + T^{2} )^{2}$$
$61$ $$( 55696 + T^{2} )^{2}$$
$67$ $$( -2600 + T^{2} )^{2}$$
$71$ $$( 680 + T )^{4}$$
$73$ $$( -422 + T )^{4}$$
$79$ $$( 554216 + T^{2} )^{2}$$
$83$ $$( 34596 + T^{2} )^{2}$$
$89$ $$( 918944 + T^{2} )^{2}$$
$97$ $$( 1062 + T )^{4}$$