# Properties

 Label 384.7.b.c Level $384$ Weight $7$ Character orbit 384.b Analytic conductor $88.341$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4 x^{7} + 106 x^{6} - 304 x^{5} + 4359 x^{4} - 8216 x^{3} + 73366 x^{2} - 69308 x + 604693$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{32}\cdot 3^{10}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} -\beta_{1} q^{5} + \beta_{5} q^{7} + 243 q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} -\beta_{1} q^{5} + \beta_{5} q^{7} + 243 q^{9} + ( -17 \beta_{4} + \beta_{7} ) q^{11} + \beta_{3} q^{13} + \beta_{6} q^{15} + ( -818 + \beta_{2} ) q^{17} + ( -195 \beta_{4} - 5 \beta_{7} ) q^{19} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{21} + ( 2 \beta_{5} + 4 \beta_{6} ) q^{23} + ( -7079 - 2 \beta_{2} ) q^{25} + 243 \beta_{4} q^{27} + ( 47 \beta_{1} + 14 \beta_{3} ) q^{29} + ( -61 \beta_{5} + 20 \beta_{6} ) q^{31} + ( -4212 - 3 \beta_{2} ) q^{33} + ( -325 \beta_{4} + 33 \beta_{7} ) q^{35} + ( -110 \beta_{1} - 27 \beta_{3} ) q^{37} + ( 81 \beta_{5} - \beta_{6} ) q^{39} + ( -62446 + \beta_{2} ) q^{41} + ( 529 \beta_{4} - 25 \beta_{7} ) q^{43} -243 \beta_{1} q^{45} + ( 198 \beta_{5} + 32 \beta_{6} ) q^{47} + ( -51839 + 12 \beta_{2} ) q^{49} + ( -845 \beta_{4} - 81 \beta_{7} ) q^{51} + ( 949 \beta_{1} - 70 \beta_{3} ) q^{53} + ( -400 \beta_{5} - 76 \beta_{6} ) q^{55} + ( -46980 + 15 \beta_{2} ) q^{57} + ( -10620 \beta_{4} + 32 \beta_{7} ) q^{59} + ( -462 \beta_{1} - 5 \beta_{3} ) q^{61} + 243 \beta_{5} q^{63} + ( -4512 - 31 \beta_{2} ) q^{65} + ( -10568 \beta_{4} + 140 \beta_{7} ) q^{67} + ( -978 \beta_{1} + 6 \beta_{3} ) q^{69} + ( 94 \beta_{5} - 228 \beta_{6} ) q^{71} + ( -245330 - 20 \beta_{2} ) q^{73} + ( -7025 \beta_{4} + 162 \beta_{7} ) q^{75} + ( 3284 \beta_{1} + 124 \beta_{3} ) q^{77} + ( -1525 \beta_{5} - 4 \beta_{6} ) q^{79} + 59049 q^{81} + ( -13683 \beta_{4} - 309 \beta_{7} ) q^{83} + ( -4334 \beta_{1} + 400 \beta_{3} ) q^{85} + ( 1134 \beta_{5} - 61 \beta_{6} ) q^{87} + ( -211874 + 38 \beta_{2} ) q^{89} + ( -56279 \beta_{4} - 357 \beta_{7} ) q^{91} + ( -4677 \beta_{1} - 183 \beta_{3} ) q^{93} + ( 2000 \beta_{5} + 100 \beta_{6} ) q^{95} + ( 954094 - 10 \beta_{2} ) q^{97} + ( -4131 \beta_{4} + 243 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 1944 q^{9} + O(q^{10})$$ $$8 q + 1944 q^{9} - 6544 q^{17} - 56632 q^{25} - 33696 q^{33} - 499568 q^{41} - 414712 q^{49} - 375840 q^{57} - 36096 q^{65} - 1962640 q^{73} + 472392 q^{81} - 1694992 q^{89} + 7632752 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 106 x^{6} - 304 x^{5} + 4359 x^{4} - 8216 x^{3} + 73366 x^{2} - 69308 x + 604693$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{6} - 6 \nu^{5} + 418 \nu^{4} - 826 \nu^{3} + 15758 \nu^{2} - 15346 \nu + 184860$$$$)/959$$ $$\beta_{2}$$ $$=$$ $$($$$$72 \nu^{6} - 216 \nu^{5} + 5184 \nu^{4} - 10008 \nu^{3} + 103680 \nu^{2} - 98712 \nu - 42696$$$$)/137$$ $$\beta_{3}$$ $$=$$ $$($$$$146 \nu^{6} - 438 \nu^{5} + 13252 \nu^{4} - 25774 \nu^{3} + 477116 \nu^{2} - 464302 \nu + 5226282$$$$)/959$$ $$\beta_{4}$$ $$=$$ $$($$$$-3078 \nu^{7} + 10773 \nu^{6} - 265689 \nu^{5} + 637290 \nu^{4} - 9124497 \nu^{3} + 13054842 \nu^{2} - 79324947 \nu + 37507653$$$$)/13442440$$ $$\beta_{5}$$ $$=$$ $$($$$$6066 \nu^{7} - 21231 \nu^{6} + 1938603 \nu^{5} - 4793430 \nu^{4} + 138256659 \nu^{3} - 202602174 \nu^{2} + 4301554449 \nu - 2117169471$$$$)/23524270$$ $$\beta_{6}$$ $$=$$ $$($$$$92502 \nu^{7} - 323757 \nu^{6} + 5437665 \nu^{5} - 12784770 \nu^{4} + 57721113 \nu^{3} - 73958778 \nu^{2} - 1012485933 \nu + 518150979$$$$)/4704854$$ $$\beta_{7}$$ $$=$$ $$($$$$-55910 \nu^{7} + 195685 \nu^{6} - 5832297 \nu^{5} + 14091530 \nu^{4} - 193915041 \nu^{3} + 276878874 \nu^{2} - 1778978627 \nu + 843807893$$$$)/2688488$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + 9 \beta_{5} + 96 \beta_{4} + 432$$$$)/864$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} + 18 \beta_{5} + 192 \beta_{4} + 12 \beta_{3} - 3 \beta_{2} - 120 \beta_{1} - 42336$$$$)/1728$$ $$\nu^{3}$$ $$=$$ $$($$$$162 \beta_{7} - 176 \beta_{6} - 576 \beta_{5} - 30474 \beta_{4} + 36 \beta_{3} - 9 \beta_{2} - 360 \beta_{1} - 127872$$$$)/3456$$ $$\nu^{4}$$ $$=$$ $$($$$$81 \beta_{7} - 89 \beta_{6} - 297 \beta_{5} - 15333 \beta_{4} - 264 \beta_{3} + 54 \beta_{2} + 5664 \beta_{1} + 364176$$$$)/864$$ $$\nu^{5}$$ $$=$$ $$($$$$-12960 \beta_{7} + 2852 \beta_{6} + 2484 \beta_{5} + 1424736 \beta_{4} - 2700 \beta_{3} + 555 \beta_{2} + 57240 \beta_{1} + 3855168$$$$)/3456$$ $$\nu^{6}$$ $$=$$ $$($$$$-19845 \beta_{7} + 4724 \beta_{6} + 5220 \beta_{5} + 2213865 \beta_{4} + 19188 \beta_{3} + 39 \beta_{2} - 581976 \beta_{1} + 7627392$$$$)/1728$$ $$\nu^{7}$$ $$=$$ $$($$$$283311 \beta_{7} + 74398 \beta_{6} + 254286 \beta_{5} - 26602971 \beta_{4} + 71904 \beta_{3} - 840 \beta_{2} - 2137296 \beta_{1} + 19874592$$$$)/1728$$

## 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
 −1.23205 + 6.41320i −1.23205 − 3.79090i −1.23205 + 3.79090i −1.23205 − 6.41320i 2.23205 − 6.41320i 2.23205 + 3.79090i 2.23205 − 3.79090i 2.23205 + 6.41320i
0 −15.5885 0 195.235i 0 277.510i 0 243.000 0
319.2 0 −15.5885 0 85.3891i 0 511.824i 0 243.000 0
319.3 0 −15.5885 0 85.3891i 0 511.824i 0 243.000 0
319.4 0 −15.5885 0 195.235i 0 277.510i 0 243.000 0
319.5 0 15.5885 0 195.235i 0 277.510i 0 243.000 0
319.6 0 15.5885 0 85.3891i 0 511.824i 0 243.000 0
319.7 0 15.5885 0 85.3891i 0 511.824i 0 243.000 0
319.8 0 15.5885 0 195.235i 0 277.510i 0 243.000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 319.8 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.c 8
4.b odd 2 1 inner 384.7.b.c 8
8.b even 2 1 inner 384.7.b.c 8
8.d odd 2 1 inner 384.7.b.c 8
16.e even 4 2 768.7.g.g 8
16.f odd 4 2 768.7.g.g 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 45408 T_{5}^{2} + 277920000$$ acting on $$S_{7}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -243 + T^{2} )^{4}$$
$5$ $$( 277920000 + 45408 T^{2} + T^{4} )^{2}$$
$7$ $$( 20174323968 + 338976 T^{2} + T^{4} )^{2}$$
$11$ $$( 4522189383936 - 4545120 T^{2} + T^{4} )^{2}$$
$13$ $$( 11711515004928 + 9088896 T^{2} + T^{4} )^{2}$$
$17$ $$( -58718780 + 1636 T + T^{2} )^{4}$$
$19$ $$( 2107360836000000 - 128143200 T^{2} + T^{4} )^{2}$$
$23$ $$( 2996687502839808 + 180514944 T^{2} + T^{4} )^{2}$$
$29$ $$( 123449102940268800 + 1869854304 T^{2} + T^{4} )^{2}$$
$31$ $$( 6459874959912599808 + 5276545056 T^{2} + T^{4} )^{2}$$
$37$ $$( 1035562157965836288 + 7121639424 T^{2} + T^{4} )^{2}$$
$41$ $$( 3840115012 + 124892 T + T^{2} )^{4}$$
$43$ $$( 1701874998525913344 - 2889761376 T^{2} + T^{4} )^{2}$$
$47$ $$( 80088648191955505152 + 26657465472 T^{2} + T^{4} )^{2}$$
$53$ $$($$$$91\!\cdots\!00$$$$+ 86629009248 T^{2} + T^{4} )^{2}$$
$59$ $$($$$$63\!\cdots\!00$$$$- 59428064352 T^{2} + T^{4} )^{2}$$
$61$ $$( 19934164088443699200 + 9877596672 T^{2} + T^{4} )^{2}$$
$67$ $$($$$$24\!\cdots\!84$$$$- 140980616544 T^{2} + T^{4} )^{2}$$
$71$ $$($$$$55\!\cdots\!92$$$$+ 569594613888 T^{2} + T^{4} )^{2}$$
$73$ $$( 36431647300 + 490660 T + T^{2} )^{4}$$
$79$ $$($$$$11\!\cdots\!00$$$$+ 790499817504 T^{2} + T^{4} )^{2}$$
$83$ $$($$$$27\!\cdots\!44$$$$- 509657219424 T^{2} + T^{4} )^{2}$$
$89$ $$( -40865541500 + 423748 T + T^{2} )^{4}$$
$97$ $$( 904356570436 - 1908188 T + T^{2} )^{4}$$