# Properties

 Label 96.8.f.b Level 96 Weight 8 Character orbit 96.f Analytic conductor 29.989 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 96.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.9889624465$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{6}, \sqrt{-26})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 24) 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.

 $$f(q)$$ $$=$$ $$q + ( 9 + 9 \beta_{1} ) q^{3} -5 \beta_{2} q^{5} + 31 \beta_{3} q^{7} + ( -2025 + 162 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( 9 + 9 \beta_{1} ) q^{3} -5 \beta_{2} q^{5} + 31 \beta_{3} q^{7} + ( -2025 + 162 \beta_{1} ) q^{9} + 350 \beta_{1} q^{11} -203 \beta_{3} q^{13} + ( -45 \beta_{2} - 90 \beta_{3} ) q^{15} + 5404 \beta_{1} q^{17} -11570 q^{19} + ( -3627 \beta_{2} + 279 \beta_{3} ) q^{21} + 2834 \beta_{2} q^{23} -68525 q^{25} + ( -56133 - 16767 \beta_{1} ) q^{27} + 2795 \beta_{2} q^{29} -1435 \beta_{3} q^{31} + ( -81900 + 3150 \beta_{1} ) q^{33} -29760 \beta_{1} q^{35} -5661 \beta_{3} q^{37} + ( 23751 \beta_{2} - 1827 \beta_{3} ) q^{39} -79160 \beta_{1} q^{41} -495062 q^{43} + ( 10125 \beta_{2} - 1620 \beta_{3} ) q^{45} -57968 \beta_{2} q^{47} -1575113 q^{49} + ( -1264536 + 48636 \beta_{1} ) q^{51} + 29211 \beta_{2} q^{53} -3500 \beta_{3} q^{55} + ( -104130 - 104130 \beta_{1} ) q^{57} + 282506 \beta_{1} q^{59} -35755 \beta_{3} q^{61} + ( -65286 \beta_{2} - 62775 \beta_{3} ) q^{63} + 194880 \beta_{1} q^{65} -1400126 q^{67} + ( 25506 \beta_{2} + 51012 \beta_{3} ) q^{69} + 182154 \beta_{2} q^{71} -2223598 q^{73} + ( -616725 - 616725 \beta_{1} ) q^{75} -141050 \beta_{2} q^{77} + 111901 \beta_{3} q^{79} + ( 3418281 - 656100 \beta_{1} ) q^{81} -590962 \beta_{1} q^{83} -54040 \beta_{3} q^{85} + ( 25155 \beta_{2} + 50310 \beta_{3} ) q^{87} + 1146364 \beta_{1} q^{89} + 15707328 q^{91} + ( 167895 \beta_{2} - 12915 \beta_{3} ) q^{93} + 57850 \beta_{2} q^{95} + 6867926 q^{97} + ( -1474200 - 708750 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 36q^{3} - 8100q^{9} + O(q^{10})$$ $$4q + 36q^{3} - 8100q^{9} - 46280q^{19} - 274100q^{25} - 224532q^{27} - 327600q^{33} - 1980248q^{43} - 6300452q^{49} - 5058144q^{51} - 416520q^{57} - 5600504q^{67} - 8894392q^{73} - 2466900q^{75} + 13673124q^{81} + 62829312q^{91} + 27471704q^{97} - 5896800q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 10 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 18 \nu$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} - 2 \nu$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{2} + 40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 8 \beta_{1}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 40$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{2} - 8 \beta_{1}$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$65$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 1.22474 − 2.54951i −1.22474 − 2.54951i 1.22474 + 2.54951i −1.22474 + 2.54951i
0 9.00000 45.8912i 0 −97.9796 0 1548.76i 0 −2025.00 826.041i 0
47.2 0 9.00000 45.8912i 0 97.9796 0 1548.76i 0 −2025.00 826.041i 0
47.3 0 9.00000 + 45.8912i 0 −97.9796 0 1548.76i 0 −2025.00 + 826.041i 0
47.4 0 9.00000 + 45.8912i 0 97.9796 0 1548.76i 0 −2025.00 + 826.041i 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
3.b odd 2 1 inner
8.d odd 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 96.8.f.b 4
3.b odd 2 1 inner 96.8.f.b 4
4.b odd 2 1 24.8.f.b 4
8.b even 2 1 24.8.f.b 4
8.d odd 2 1 inner 96.8.f.b 4
12.b even 2 1 24.8.f.b 4
24.f even 2 1 inner 96.8.f.b 4
24.h odd 2 1 24.8.f.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.f.b 4 4.b odd 2 1
24.8.f.b 4 8.b even 2 1
24.8.f.b 4 12.b even 2 1
24.8.f.b 4 24.h odd 2 1
96.8.f.b 4 1.a even 1 1 trivial
96.8.f.b 4 3.b odd 2 1 inner
96.8.f.b 4 8.d odd 2 1 inner
96.8.f.b 4 24.f even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 9600$$ acting on $$S_{8}^{\mathrm{new}}(96, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 - 18 T + 2187 T^{2} )^{2}$$
$5$ $$( 1 + 146650 T^{2} + 6103515625 T^{4} )^{2}$$
$7$ $$( 1 + 751570 T^{2} + 678223072849 T^{4} )^{2}$$
$11$ $$( 1 - 35789342 T^{2} + 379749833583241 T^{4} )^{2}$$
$13$ $$( 1 - 22639370 T^{2} + 3937376385699289 T^{4} )^{2}$$
$17$ $$( 1 - 61393730 T^{2} + 168377826559400929 T^{4} )^{2}$$
$19$ $$( 1 + 11570 T + 893871739 T^{2} )^{4}$$
$23$ $$( 1 + 3725533390 T^{2} + 11592836324538749809 T^{4} )^{2}$$
$29$ $$( 1 + 31499935018 T^{2} +$$$$29\!\cdots\!81$$$$T^{4} )^{2}$$
$31$ $$( 1 - 49885402622 T^{2} +$$$$75\!\cdots\!21$$$$T^{4} )^{2}$$
$37$ $$( 1 - 109874639450 T^{2} +$$$$90\!\cdots\!89$$$$T^{4} )^{2}$$
$41$ $$( 1 - 226584602162 T^{2} +$$$$37\!\cdots\!61$$$$T^{4} )^{2}$$
$43$ $$( 1 + 495062 T + 271818611107 T^{2} )^{4}$$
$47$ $$( 1 - 277104744290 T^{2} +$$$$25\!\cdots\!69$$$$T^{4} )^{2}$$
$53$ $$( 1 + 2021761791610 T^{2} +$$$$13\!\cdots\!69$$$$T^{4} )^{2}$$
$59$ $$( 1 - 2902252328702 T^{2} +$$$$61\!\cdots\!61$$$$T^{4} )^{2}$$
$61$ $$( 1 - 3094549289642 T^{2} +$$$$98\!\cdots\!41$$$$T^{4} )^{2}$$
$67$ $$( 1 + 1400126 T + 6060711605323 T^{2} )^{4}$$
$71$ $$( 1 + 5449089705838 T^{2} +$$$$82\!\cdots\!81$$$$T^{4} )^{2}$$
$73$ $$( 1 + 2223598 T + 11047398519097 T^{2} )^{4}$$
$79$ $$( 1 - 7153320805022 T^{2} +$$$$36\!\cdots\!81$$$$T^{4} )^{2}$$
$83$ $$( 1 - 45191963757710 T^{2} +$$$$73\!\cdots\!29$$$$T^{4} )^{2}$$
$89$ $$( 1 - 54294758858162 T^{2} +$$$$19\!\cdots\!41$$$$T^{4} )^{2}$$
$97$ $$( 1 - 6867926 T + 80798284478113 T^{2} )^{4}$$