# Properties

 Label 9.5.b.a Level 9 Weight 5 Character orbit 9.b Analytic conductor 0.930 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 9.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.930329667755$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -2 q^{4} -7 \beta q^{5} -28 q^{7} + 14 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} -2 q^{4} -7 \beta q^{5} -28 q^{7} + 14 \beta q^{8} + 126 q^{10} + 4 \beta q^{11} -112 q^{13} -28 \beta q^{14} -284 q^{16} -21 \beta q^{17} + 560 q^{19} + 14 \beta q^{20} -72 q^{22} + 188 \beta q^{23} -257 q^{25} -112 \beta q^{26} + 56 q^{28} -233 \beta q^{29} -364 q^{31} -60 \beta q^{32} + 378 q^{34} + 196 \beta q^{35} -826 q^{37} + 560 \beta q^{38} + 1764 q^{40} -427 \beta q^{41} + 1736 q^{43} -8 \beta q^{44} -3384 q^{46} -308 \beta q^{47} -1617 q^{49} -257 \beta q^{50} + 224 q^{52} + 423 \beta q^{53} + 504 q^{55} -392 \beta q^{56} + 4194 q^{58} + 1064 \beta q^{59} + 2618 q^{61} -364 \beta q^{62} -3464 q^{64} + 784 \beta q^{65} -3784 q^{67} + 42 \beta q^{68} -3528 q^{70} -2028 \beta q^{71} + 6608 q^{73} -826 \beta q^{74} -1120 q^{76} -112 \beta q^{77} -4276 q^{79} + 1988 \beta q^{80} + 7686 q^{82} + 28 \beta q^{83} -2646 q^{85} + 1736 \beta q^{86} -1008 q^{88} + 1029 \beta q^{89} + 3136 q^{91} -376 \beta q^{92} + 5544 q^{94} -3920 \beta q^{95} -5824 q^{97} -1617 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} - 56q^{7} + O(q^{10})$$ $$2q - 4q^{4} - 56q^{7} + 252q^{10} - 224q^{13} - 568q^{16} + 1120q^{19} - 144q^{22} - 514q^{25} + 112q^{28} - 728q^{31} + 756q^{34} - 1652q^{37} + 3528q^{40} + 3472q^{43} - 6768q^{46} - 3234q^{49} + 448q^{52} + 1008q^{55} + 8388q^{58} + 5236q^{61} - 6928q^{64} - 7568q^{67} - 7056q^{70} + 13216q^{73} - 2240q^{76} - 8552q^{79} + 15372q^{82} - 5292q^{85} - 2016q^{88} + 6272q^{91} + 11088q^{94} - 11648q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/9\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 − 1.41421i 1.41421i
4.24264i 0 −2.00000 29.6985i 0 −28.0000 59.3970i 0 126.000
8.2 4.24264i 0 −2.00000 29.6985i 0 −28.0000 59.3970i 0 126.000
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.5.b.a 2
3.b odd 2 1 inner 9.5.b.a 2
4.b odd 2 1 144.5.e.c 2
5.b even 2 1 225.5.c.a 2
5.c odd 4 2 225.5.d.a 4
7.b odd 2 1 441.5.b.a 2
8.b even 2 1 576.5.e.d 2
8.d odd 2 1 576.5.e.g 2
9.c even 3 2 81.5.d.c 4
9.d odd 6 2 81.5.d.c 4
12.b even 2 1 144.5.e.c 2
15.d odd 2 1 225.5.c.a 2
15.e even 4 2 225.5.d.a 4
21.c even 2 1 441.5.b.a 2
24.f even 2 1 576.5.e.g 2
24.h odd 2 1 576.5.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.5.b.a 2 1.a even 1 1 trivial
9.5.b.a 2 3.b odd 2 1 inner
81.5.d.c 4 9.c even 3 2
81.5.d.c 4 9.d odd 6 2
144.5.e.c 2 4.b odd 2 1
144.5.e.c 2 12.b even 2 1
225.5.c.a 2 5.b even 2 1
225.5.c.a 2 15.d odd 2 1
225.5.d.a 4 5.c odd 4 2
225.5.d.a 4 15.e even 4 2
441.5.b.a 2 7.b odd 2 1
441.5.b.a 2 21.c even 2 1
576.5.e.d 2 8.b even 2 1
576.5.e.d 2 24.h odd 2 1
576.5.e.g 2 8.d odd 2 1
576.5.e.g 2 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(9, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 14 T^{2} + 256 T^{4}$$
$3$ 1
$5$ $$1 - 368 T^{2} + 390625 T^{4}$$
$7$ $$( 1 + 28 T + 2401 T^{2} )^{2}$$
$11$ $$1 - 28994 T^{2} + 214358881 T^{4}$$
$13$ $$( 1 + 112 T + 28561 T^{2} )^{2}$$
$17$ $$1 - 159104 T^{2} + 6975757441 T^{4}$$
$19$ $$( 1 - 560 T + 130321 T^{2} )^{2}$$
$23$ $$1 + 76510 T^{2} + 78310985281 T^{4}$$
$29$ $$1 - 437360 T^{2} + 500246412961 T^{4}$$
$31$ $$( 1 + 364 T + 923521 T^{2} )^{2}$$
$37$ $$( 1 + 826 T + 1874161 T^{2} )^{2}$$
$41$ $$1 - 2369600 T^{2} + 7984925229121 T^{4}$$
$43$ $$( 1 - 1736 T + 3418801 T^{2} )^{2}$$
$47$ $$1 - 8051810 T^{2} + 23811286661761 T^{4}$$
$53$ $$1 - 12560240 T^{2} + 62259690411361 T^{4}$$
$59$ $$1 - 3856994 T^{2} + 146830437604321 T^{4}$$
$61$ $$( 1 - 2618 T + 13845841 T^{2} )^{2}$$
$67$ $$( 1 + 3784 T + 20151121 T^{2} )^{2}$$
$71$ $$1 + 23206750 T^{2} + 645753531245761 T^{4}$$
$73$ $$( 1 - 6608 T + 28398241 T^{2} )^{2}$$
$79$ $$( 1 + 4276 T + 38950081 T^{2} )^{2}$$
$83$ $$1 - 94902530 T^{2} + 2252292232139041 T^{4}$$
$89$ $$1 - 106425344 T^{2} + 3936588805702081 T^{4}$$
$97$ $$( 1 + 5824 T + 88529281 T^{2} )^{2}$$