# Properties

 Label 384.6.d.h Level $384$ Weight $6$ Character orbit 384.d Analytic conductor $61.587$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$61.5873868082$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{61})$$ Defining polynomial: $$x^{4} + 31 x^{2} + 225$$ 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

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.

 $$f(q)$$ $$=$$ $$q -9 \beta_{1} q^{3} + ( -20 \beta_{1} + \beta_{2} ) q^{5} + ( 8 - 5 \beta_{3} ) q^{7} -81 q^{9} +O(q^{10})$$ $$q -9 \beta_{1} q^{3} + ( -20 \beta_{1} + \beta_{2} ) q^{5} + ( 8 - 5 \beta_{3} ) q^{7} -81 q^{9} + ( 172 \beta_{1} - 4 \beta_{2} ) q^{11} + ( 108 \beta_{1} + 14 \beta_{2} ) q^{13} + ( -180 + 9 \beta_{3} ) q^{15} + ( 462 - 52 \beta_{3} ) q^{17} + ( 268 \beta_{1} - 28 \beta_{2} ) q^{19} + ( -72 \beta_{1} + 45 \beta_{2} ) q^{21} + ( 1792 - 22 \beta_{3} ) q^{23} + ( 1749 + 40 \beta_{3} ) q^{25} + 729 \beta_{1} q^{27} + ( 4324 \beta_{1} - 63 \beta_{2} ) q^{29} + ( 1768 - 151 \beta_{3} ) q^{31} + ( 1548 - 36 \beta_{3} ) q^{33} + ( -5040 \beta_{1} + 108 \beta_{2} ) q^{35} + ( 1556 \beta_{1} - 164 \beta_{2} ) q^{37} + ( 972 + 126 \beta_{3} ) q^{39} + ( 1258 + 412 \beta_{3} ) q^{41} + ( -1844 \beta_{1} - 508 \beta_{2} ) q^{43} + ( 1620 \beta_{1} - 81 \beta_{2} ) q^{45} + ( 2288 + 298 \beta_{3} ) q^{47} + ( 7657 - 80 \beta_{3} ) q^{49} + ( -4158 \beta_{1} + 468 \beta_{2} ) q^{51} + ( -484 \beta_{1} - 33 \beta_{2} ) q^{53} + ( 7344 - 252 \beta_{3} ) q^{55} + ( 2412 - 252 \beta_{3} ) q^{57} + ( -13908 \beta_{1} + 768 \beta_{2} ) q^{59} + ( 14220 \beta_{1} + 672 \beta_{2} ) q^{61} + ( -648 + 405 \beta_{3} ) q^{63} + ( -11504 + 172 \beta_{3} ) q^{65} + ( -39252 \beta_{1} + 336 \beta_{2} ) q^{67} + ( -16128 \beta_{1} + 198 \beta_{2} ) q^{69} + ( 37760 + 502 \beta_{3} ) q^{71} + ( 8550 - 1680 \beta_{3} ) q^{73} + ( -15741 \beta_{1} - 360 \beta_{2} ) q^{75} + ( 20896 \beta_{1} - 892 \beta_{2} ) q^{77} + ( 50248 - 751 \beta_{3} ) q^{79} + 6561 q^{81} + ( -35252 \beta_{1} - 2236 \beta_{2} ) q^{83} + ( -59992 \beta_{1} + 1502 \beta_{2} ) q^{85} + ( 38916 - 567 \beta_{3} ) q^{87} + ( -45178 - 88 \beta_{3} ) q^{89} + ( -67456 \beta_{1} - 428 \beta_{2} ) q^{91} + ( -15912 \beta_{1} + 1359 \beta_{2} ) q^{93} + ( 32688 - 828 \beta_{3} ) q^{95} + ( -49954 + 2824 \beta_{3} ) q^{97} + ( -13932 \beta_{1} + 324 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 32q^{7} - 324q^{9} + O(q^{10})$$ $$4q + 32q^{7} - 324q^{9} - 720q^{15} + 1848q^{17} + 7168q^{23} + 6996q^{25} + 7072q^{31} + 6192q^{33} + 3888q^{39} + 5032q^{41} + 9152q^{47} + 30628q^{49} + 29376q^{55} + 9648q^{57} - 2592q^{63} - 46016q^{65} + 151040q^{71} + 34200q^{73} + 200992q^{79} + 26244q^{81} + 155664q^{87} - 180712q^{89} + 130752q^{95} - 199816q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 31 x^{2} + 225$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 16 \nu$$$$)/15$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{3} + 184 \nu$$$$)/15$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{2} + 124$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 4 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 124$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{2} + 23 \beta_{1}$$

## 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
 − 4.40512i 3.40512i − 3.40512i 4.40512i
0 9.00000i 0 51.2410i 0 164.205 0 −81.0000 0
193.2 0 9.00000i 0 11.2410i 0 −148.205 0 −81.0000 0
193.3 0 9.00000i 0 11.2410i 0 −148.205 0 −81.0000 0
193.4 0 9.00000i 0 51.2410i 0 164.205 0 −81.0000 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 384.6.d.h yes 4
4.b odd 2 1 384.6.d.g 4
8.b even 2 1 inner 384.6.d.h yes 4
8.d odd 2 1 384.6.d.g 4
16.e even 4 1 768.6.a.q 2
16.e even 4 1 768.6.a.r 2
16.f odd 4 1 768.6.a.m 2
16.f odd 4 1 768.6.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.6.d.g 4 4.b odd 2 1
384.6.d.g 4 8.d odd 2 1
384.6.d.h yes 4 1.a even 1 1 trivial
384.6.d.h yes 4 8.b even 2 1 inner
768.6.a.m 2 16.f odd 4 1
768.6.a.q 2 16.e even 4 1
768.6.a.r 2 16.e even 4 1
768.6.a.v 2 16.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{4} + 2752 T_{5}^{2} + 331776$$ $$T_{7}^{2} - 16 T_{7} - 24336$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 81 + T^{2} )^{2}$$
$5$ $$331776 + 2752 T^{2} + T^{4}$$
$7$ $$( -24336 - 16 T + T^{2} )^{2}$$
$11$ $$195105024 + 90400 T^{2} + T^{4}$$
$13$ $$32267655424 + 405920 T^{2} + T^{4}$$
$17$ $$( -2425660 - 924 T + T^{2} )^{2}$$
$19$ $$480748089600 + 1674016 T^{2} + T^{4}$$
$23$ $$( 2738880 - 3584 T + T^{2} )^{2}$$
$29$ $$219728206925824 + 45141440 T^{2} + T^{4}$$
$31$ $$( -19127952 - 3536 T + T^{2} )^{2}$$
$37$ $$567838398009600 + 57343264 T^{2} + T^{4}$$
$41$ $$( -164087580 - 2516 T + T^{2} )^{2}$$
$43$ $$61737404508336384 + 510541600 T^{2} + T^{4}$$
$47$ $$( -81437760 - 4576 T + T^{2} )^{2}$$
$53$ $$686591217664 + 2594240 T^{2} + T^{4}$$
$59$ $$146104176222777600 + 1538201376 T^{2} + T^{4}$$
$61$ $$56900178980557056 + 1285908768 T^{2} + T^{4}$$
$67$ $$2046424686977528064 + 3301812000 T^{2} + T^{4}$$
$71$ $$( 1179861696 - 75520 T + T^{2} )^{2}$$
$73$ $$( -2681559900 - 17100 T + T^{2} )^{2}$$
$79$ $$( 1974396528 - 100496 T + T^{2} )^{2}$$
$83$ $$13227767487008043264 + 12244813600 T^{2} + T^{4}$$
$89$ $$( 2033493540 + 90356 T + T^{2} )^{2}$$
$97$ $$( -5288174460 + 99908 T + T^{2} )^{2}$$