# Properties

 Label 864.2.f.a Level 864 Weight 2 Character orbit 864.f Analytic conductor 6.899 Analytic rank 0 Dimension 8 CM no Inner twists 4

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## Newspace parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 864.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.23123460096.3 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 216) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} -\beta_{4} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} -\beta_{4} q^{7} -\beta_{6} q^{11} + ( \beta_{4} + \beta_{7} ) q^{13} + ( \beta_{3} - \beta_{6} ) q^{17} + ( -2 - \beta_{2} ) q^{19} + ( \beta_{1} - \beta_{5} ) q^{23} + ( 1 + 2 \beta_{2} ) q^{25} -2 \beta_{5} q^{29} -2 \beta_{7} q^{31} + ( 2 \beta_{3} - \beta_{6} ) q^{35} + ( \beta_{4} - \beta_{7} ) q^{37} -2 \beta_{6} q^{41} -2 q^{43} + ( 3 \beta_{1} - \beta_{5} ) q^{47} + ( -2 - 4 \beta_{2} ) q^{49} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{53} + 2 \beta_{7} q^{55} -\beta_{6} q^{59} + ( -\beta_{4} - \beta_{7} ) q^{61} + ( -\beta_{3} - \beta_{6} ) q^{65} + ( 4 + 3 \beta_{2} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{71} + ( -1 + 4 \beta_{2} ) q^{73} + ( -\beta_{1} - 2 \beta_{5} ) q^{77} + ( 3 \beta_{4} + 2 \beta_{7} ) q^{79} -2 \beta_{3} q^{83} + ( -4 \beta_{4} + 2 \beta_{7} ) q^{85} + ( 3 \beta_{3} + \beta_{6} ) q^{89} + ( 6 - \beta_{2} ) q^{91} + ( -3 \beta_{1} - \beta_{5} ) q^{95} + ( -1 - 6 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 16q^{19} + 8q^{25} - 16q^{43} - 16q^{49} + 32q^{67} - 8q^{73} + 48q^{91} - 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 2 x^{6} + 6 x^{4} + 8 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 2 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} + 2 \nu^{2} + 4$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{5} + 6 \nu^{3} + 16 \nu$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 2 \nu^{4} - 2 \nu^{2} + 4$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{5} - 6 \nu^{3}$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{3} + 4 \nu$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} + 10 \nu^{2} + 8$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{3}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{2} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} - \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{2} - 2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{5} - \beta_{3} + 4 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-\beta_{7} - 2 \beta_{4} + 3 \beta_{2} + 1$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{6} + \beta_{3} - \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
431.1
 −0.563016 + 1.29731i −0.563016 − 1.29731i 1.08766 + 0.903873i 1.08766 − 0.903873i −1.08766 − 0.903873i −1.08766 + 0.903873i 0.563016 − 1.29731i 0.563016 + 1.29731i
0 0 0 −3.07638 0 3.99102i 0 0 0
431.2 0 0 0 −3.07638 0 3.99102i 0 0 0
431.3 0 0 0 −1.59245 0 1.43937i 0 0 0
431.4 0 0 0 −1.59245 0 1.43937i 0 0 0
431.5 0 0 0 1.59245 0 1.43937i 0 0 0
431.6 0 0 0 1.59245 0 1.43937i 0 0 0
431.7 0 0 0 3.07638 0 3.99102i 0 0 0
431.8 0 0 0 3.07638 0 3.99102i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 431.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 864.2.f.a 8
3.b odd 2 1 inner 864.2.f.a 8
4.b odd 2 1 216.2.f.a 8
8.b even 2 1 216.2.f.a 8
8.d odd 2 1 inner 864.2.f.a 8
9.c even 3 2 2592.2.p.f 16
9.d odd 6 2 2592.2.p.f 16
12.b even 2 1 216.2.f.a 8
24.f even 2 1 inner 864.2.f.a 8
24.h odd 2 1 216.2.f.a 8
36.f odd 6 2 648.2.l.f 16
36.h even 6 2 648.2.l.f 16
72.j odd 6 2 648.2.l.f 16
72.l even 6 2 2592.2.p.f 16
72.n even 6 2 648.2.l.f 16
72.p odd 6 2 2592.2.p.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.f.a 8 4.b odd 2 1
216.2.f.a 8 8.b even 2 1
216.2.f.a 8 12.b even 2 1
216.2.f.a 8 24.h odd 2 1
648.2.l.f 16 36.f odd 6 2
648.2.l.f 16 36.h even 6 2
648.2.l.f 16 72.j odd 6 2
648.2.l.f 16 72.n even 6 2
864.2.f.a 8 1.a even 1 1 trivial
864.2.f.a 8 3.b odd 2 1 inner
864.2.f.a 8 8.d odd 2 1 inner
864.2.f.a 8 24.f even 2 1 inner
2592.2.p.f 16 9.c even 3 2
2592.2.p.f 16 9.d odd 6 2
2592.2.p.f 16 72.l even 6 2
2592.2.p.f 16 72.p odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 12 T_{5}^{2} + 24$$ acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$( 1 + 8 T^{2} + 54 T^{4} + 200 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 - 10 T^{2} + 75 T^{4} - 490 T^{6} + 2401 T^{8} )^{2}$$
$11$ $$( 1 - 16 T^{2} + 198 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 22 T^{2} + 267 T^{4} - 3718 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 16 T^{2} + 54 T^{4} - 4624 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 4 T + 39 T^{2} + 76 T^{3} + 361 T^{4} )^{4}$$
$23$ $$( 1 + 56 T^{2} + 1542 T^{4} + 29624 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 20 T^{2} + 1014 T^{4} + 16820 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 28 T^{2} + 1926 T^{4} - 26908 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 94 T^{2} + 4515 T^{4} - 128686 T^{6} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 52 T^{2} + 2310 T^{4} - 87412 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 2 T + 43 T^{2} )^{8}$$
$47$ $$( 1 + 56 T^{2} + 4902 T^{4} + 123704 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 68 T^{2} + 5046 T^{4} + 191012 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 208 T^{2} + 17670 T^{4} - 724048 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 214 T^{2} + 18699 T^{4} - 796294 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - 8 T + 123 T^{2} - 536 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 + 140 T^{2} + 13254 T^{4} + 705740 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 2 T + 99 T^{2} + 146 T^{3} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 130 T^{2} + 13635 T^{4} - 811330 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 - 172 T^{2} + 20406 T^{4} - 1184908 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 80 T^{2} + 16470 T^{4} + 633680 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 2 T + 87 T^{2} + 194 T^{3} + 9409 T^{4} )^{4}$$
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