# Properties

 Label 384.7.b.b Level $384$ Weight $7$ Character orbit 384.b Analytic conductor $88.341$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$88.3407681100$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}\cdot 3^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{3} + ( 60 - 120 \zeta_{12}^{2} ) q^{5} -108 \zeta_{12}^{3} q^{7} + 243 q^{9} +O(q^{10})$$ $$q + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{3} + ( 60 - 120 \zeta_{12}^{2} ) q^{5} -108 \zeta_{12}^{3} q^{7} + 243 q^{9} + ( 456 \zeta_{12} - 228 \zeta_{12}^{3} ) q^{11} + ( -1272 + 2544 \zeta_{12}^{2} ) q^{13} -1620 \zeta_{12}^{3} q^{15} + 974 q^{17} + ( 1128 \zeta_{12} - 564 \zeta_{12}^{3} ) q^{19} + ( 972 - 1944 \zeta_{12}^{2} ) q^{21} + 20952 \zeta_{12}^{3} q^{23} + 4825 q^{25} + ( 4374 \zeta_{12} - 2187 \zeta_{12}^{3} ) q^{27} + ( 5484 - 10968 \zeta_{12}^{2} ) q^{29} + 15660 \zeta_{12}^{3} q^{31} + 6156 q^{33} + ( -12960 \zeta_{12} + 6480 \zeta_{12}^{3} ) q^{35} + ( -29712 + 59424 \zeta_{12}^{2} ) q^{37} + 34344 \zeta_{12}^{3} q^{39} + 33298 q^{41} + ( 18888 \zeta_{12} - 9444 \zeta_{12}^{3} ) q^{43} + ( 14580 - 29160 \zeta_{12}^{2} ) q^{45} -73224 \zeta_{12}^{3} q^{47} + 105985 q^{49} + ( 17532 \zeta_{12} - 8766 \zeta_{12}^{3} ) q^{51} + ( 95076 - 190152 \zeta_{12}^{2} ) q^{53} -41040 \zeta_{12}^{3} q^{55} + 15228 q^{57} + ( 86664 \zeta_{12} - 43332 \zeta_{12}^{3} ) q^{59} + ( -2784 + 5568 \zeta_{12}^{2} ) q^{61} -26244 \zeta_{12}^{3} q^{63} + 228960 q^{65} + ( 301992 \zeta_{12} - 150996 \zeta_{12}^{3} ) q^{67} + ( -188568 + 377136 \zeta_{12}^{2} ) q^{69} + 165672 \zeta_{12}^{3} q^{71} -113618 q^{73} + ( 86850 \zeta_{12} - 43425 \zeta_{12}^{3} ) q^{75} + ( 24624 - 49248 \zeta_{12}^{2} ) q^{77} -658260 \zeta_{12}^{3} q^{79} + 59049 q^{81} + ( 665640 \zeta_{12} - 332820 \zeta_{12}^{3} ) q^{83} + ( 58440 - 116880 \zeta_{12}^{2} ) q^{85} -148068 \zeta_{12}^{3} q^{87} -464290 q^{89} + ( 274752 \zeta_{12} - 137376 \zeta_{12}^{3} ) q^{91} + ( -140940 + 281880 \zeta_{12}^{2} ) q^{93} -101520 \zeta_{12}^{3} q^{95} + 51694 q^{97} + ( 110808 \zeta_{12} - 55404 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 972q^{9} + O(q^{10})$$ $$4q + 972q^{9} + 3896q^{17} + 19300q^{25} + 24624q^{33} + 133192q^{41} + 423940q^{49} + 60912q^{57} + 915840q^{65} - 454472q^{73} + 236196q^{81} - 1857160q^{89} + 206776q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
0 −15.5885 0 103.923i 0 108.000i 0 243.000 0
319.2 0 −15.5885 0 103.923i 0 108.000i 0 243.000 0
319.3 0 15.5885 0 103.923i 0 108.000i 0 243.000 0
319.4 0 15.5885 0 103.923i 0 108.000i 0 243.000 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 384.7.b.b 4
4.b odd 2 1 inner 384.7.b.b 4
8.b even 2 1 inner 384.7.b.b 4
8.d odd 2 1 inner 384.7.b.b 4
16.e even 4 2 768.7.g.d 4
16.f odd 4 2 768.7.g.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.7.b.b 4 1.a even 1 1 trivial
384.7.b.b 4 4.b odd 2 1 inner
384.7.b.b 4 8.b even 2 1 inner
384.7.b.b 4 8.d odd 2 1 inner
768.7.g.d 4 16.e even 4 2
768.7.g.d 4 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 10800$$ acting on $$S_{7}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -243 + T^{2} )^{2}$$
$5$ $$( 10800 + T^{2} )^{2}$$
$7$ $$( 11664 + T^{2} )^{2}$$
$11$ $$( -155952 + T^{2} )^{2}$$
$13$ $$( 4853952 + T^{2} )^{2}$$
$17$ $$( -974 + T )^{4}$$
$19$ $$( -954288 + T^{2} )^{2}$$
$23$ $$( 438986304 + T^{2} )^{2}$$
$29$ $$( 90222768 + T^{2} )^{2}$$
$31$ $$( 245235600 + T^{2} )^{2}$$
$37$ $$( 2648408832 + T^{2} )^{2}$$
$41$ $$( -33298 + T )^{4}$$
$43$ $$( -267567408 + T^{2} )^{2}$$
$47$ $$( 5361754176 + T^{2} )^{2}$$
$53$ $$( 27118337328 + T^{2} )^{2}$$
$59$ $$( -5632986672 + T^{2} )^{2}$$
$61$ $$( 23251968 + T^{2} )^{2}$$
$67$ $$( -68399376048 + T^{2} )^{2}$$
$71$ $$( 27447211584 + T^{2} )^{2}$$
$73$ $$( 113618 + T )^{4}$$
$79$ $$( 433306227600 + T^{2} )^{2}$$
$83$ $$( -332307457200 + T^{2} )^{2}$$
$89$ $$( 464290 + T )^{4}$$
$97$ $$( -51694 + T )^{4}$$