# Properties

 Label 384.7.b.a 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{3} + 196 \zeta_{12}^{3} q^{5} + ( 156 - 312 \zeta_{12}^{2} ) q^{7} + 243 q^{9} +O(q^{10})$$ $$q + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{3} + 196 \zeta_{12}^{3} q^{5} + ( 156 - 312 \zeta_{12}^{2} ) q^{7} + 243 q^{9} + ( 1992 \zeta_{12} - 996 \zeta_{12}^{3} ) q^{11} + 2880 \zeta_{12}^{3} q^{13} + ( -1764 + 3528 \zeta_{12}^{2} ) q^{15} -2898 q^{17} + ( 11304 \zeta_{12} - 5652 \zeta_{12}^{3} ) q^{19} -4212 \zeta_{12}^{3} q^{21} + ( 936 - 1872 \zeta_{12}^{2} ) q^{23} -22791 q^{25} + ( 4374 \zeta_{12} - 2187 \zeta_{12}^{3} ) q^{27} + 14596 \zeta_{12}^{3} q^{29} + ( 3972 - 7944 \zeta_{12}^{2} ) q^{31} + 26892 q^{33} + ( 61152 \zeta_{12} - 30576 \zeta_{12}^{3} ) q^{35} -12168 \zeta_{12}^{3} q^{37} + ( -25920 + 51840 \zeta_{12}^{2} ) q^{39} + 18738 q^{41} + ( 156168 \zeta_{12} - 78084 \zeta_{12}^{3} ) q^{43} + 47628 \zeta_{12}^{3} q^{45} + ( 87336 - 174672 \zeta_{12}^{2} ) q^{47} + 44641 q^{49} + ( -52164 \zeta_{12} + 26082 \zeta_{12}^{3} ) q^{51} + 144236 \zeta_{12}^{3} q^{53} + ( -195216 + 390432 \zeta_{12}^{2} ) q^{55} + 152604 q^{57} + ( -385272 \zeta_{12} + 192636 \zeta_{12}^{3} ) q^{59} + 333432 \zeta_{12}^{3} q^{61} + ( 37908 - 75816 \zeta_{12}^{2} ) q^{63} -564480 q^{65} + ( -368856 \zeta_{12} + 184428 \zeta_{12}^{3} ) q^{67} -25272 \zeta_{12}^{3} q^{69} + ( -242856 + 485712 \zeta_{12}^{2} ) q^{71} + 628238 q^{73} + ( -410238 \zeta_{12} + 205119 \zeta_{12}^{3} ) q^{75} -466128 \zeta_{12}^{3} q^{77} + ( 168516 - 337032 \zeta_{12}^{2} ) q^{79} + 59049 q^{81} + ( -412248 \zeta_{12} + 206124 \zeta_{12}^{3} ) q^{83} -568008 \zeta_{12}^{3} q^{85} + ( -131364 + 262728 \zeta_{12}^{2} ) q^{87} -1375074 q^{89} + ( 898560 \zeta_{12} - 449280 \zeta_{12}^{3} ) q^{91} -107244 \zeta_{12}^{3} q^{93} + ( -1107792 + 2215584 \zeta_{12}^{2} ) q^{95} + 489710 q^{97} + ( 484056 \zeta_{12} - 242028 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 972q^{9} + O(q^{10})$$ $$4q + 972q^{9} - 11592q^{17} - 91164q^{25} + 107568q^{33} + 74952q^{41} + 178564q^{49} + 610416q^{57} - 2257920q^{65} + 2512952q^{73} + 236196q^{81} - 5500296q^{89} + 1958840q^{97} + O(q^{100})$$

## 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 0 196.000i 0 270.200i 0 243.000 0
319.2 0 −15.5885 0 196.000i 0 270.200i 0 243.000 0
319.3 0 15.5885 0 196.000i 0 270.200i 0 243.000 0
319.4 0 15.5885 0 196.000i 0 270.200i 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.a 4
4.b odd 2 1 inner 384.7.b.a 4
8.b even 2 1 inner 384.7.b.a 4
8.d odd 2 1 inner 384.7.b.a 4
16.e even 4 1 768.7.g.a 2
16.e even 4 1 768.7.g.b 2
16.f odd 4 1 768.7.g.a 2
16.f odd 4 1 768.7.g.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 38416$$ acting on $$S_{7}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -243 + T^{2} )^{2}$$
$5$ $$( 38416 + T^{2} )^{2}$$
$7$ $$( 73008 + T^{2} )^{2}$$
$11$ $$( -2976048 + T^{2} )^{2}$$
$13$ $$( 8294400 + T^{2} )^{2}$$
$17$ $$( 2898 + T )^{4}$$
$19$ $$( -95835312 + T^{2} )^{2}$$
$23$ $$( 2628288 + T^{2} )^{2}$$
$29$ $$( 213043216 + T^{2} )^{2}$$
$31$ $$( 47330352 + T^{2} )^{2}$$
$37$ $$( 148060224 + T^{2} )^{2}$$
$41$ $$( -18738 + T )^{4}$$
$43$ $$( -18291333168 + T^{2} )^{2}$$
$47$ $$( 22882730688 + T^{2} )^{2}$$
$53$ $$( 20804023696 + T^{2} )^{2}$$
$59$ $$( -111325885488 + T^{2} )^{2}$$
$61$ $$( 111176898624 + T^{2} )^{2}$$
$67$ $$( -102041061552 + T^{2} )^{2}$$
$71$ $$( 176937110208 + T^{2} )^{2}$$
$73$ $$( -628238 + T )^{4}$$
$79$ $$( 85192926768 + T^{2} )^{2}$$
$83$ $$( -127461310128 + T^{2} )^{2}$$
$89$ $$( 1375074 + T )^{4}$$
$97$ $$( -489710 + T )^{4}$$