# Properties

 Label 9.9.b.a Level 9 Weight 9 Character orbit 9.b Analytic conductor 3.666 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.66640749055$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 238 q^{4} + 233 \beta q^{5} + 1652 q^{7} + 494 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} + 238 q^{4} + 233 \beta q^{5} + 1652 q^{7} + 494 \beta q^{8} -4194 q^{10} -5036 \beta q^{11} -26272 q^{13} + 1652 \beta q^{14} + 52036 q^{16} + 3579 \beta q^{17} + 46640 q^{19} + 55454 \beta q^{20} + 90648 q^{22} -77332 \beta q^{23} -586577 q^{25} -26272 \beta q^{26} + 393176 q^{28} -144953 \beta q^{29} -196444 q^{31} + 178500 \beta q^{32} -64422 q^{34} + 384916 \beta q^{35} + 2819414 q^{37} + 46640 \beta q^{38} -2071836 q^{40} -166507 \beta q^{41} -2213464 q^{43} -1198568 \beta q^{44} + 1391976 q^{46} + 384892 \beta q^{47} -3035697 q^{49} -586577 \beta q^{50} -6252736 q^{52} + 1222983 \beta q^{53} + 21120984 q^{55} + 816088 \beta q^{56} + 2609154 q^{58} + 2768264 \beta q^{59} -17405302 q^{61} -196444 \beta q^{62} + 10108216 q^{64} -6121376 \beta q^{65} -14322664 q^{67} + 851802 \beta q^{68} -6928488 q^{70} -3687708 \beta q^{71} -8906992 q^{73} + 2819414 \beta q^{74} + 11100320 q^{76} -8319472 \beta q^{77} + 32758844 q^{79} + 12124388 \beta q^{80} + 2997126 q^{82} + 20055628 \beta q^{83} -15010326 q^{85} -2213464 \beta q^{86} + 44780112 q^{88} -13891371 \beta q^{89} -43401344 q^{91} -18405016 \beta q^{92} -6928056 q^{94} + 10867120 \beta q^{95} -24451744 q^{97} -3035697 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 476q^{4} + 3304q^{7} + O(q^{10})$$ $$2q + 476q^{4} + 3304q^{7} - 8388q^{10} - 52544q^{13} + 104072q^{16} + 93280q^{19} + 181296q^{22} - 1173154q^{25} + 786352q^{28} - 392888q^{31} - 128844q^{34} + 5638828q^{37} - 4143672q^{40} - 4426928q^{43} + 2783952q^{46} - 6071394q^{49} - 12505472q^{52} + 42241968q^{55} + 5218308q^{58} - 34810604q^{61} + 20216432q^{64} - 28645328q^{67} - 13856976q^{70} - 17813984q^{73} + 22200640q^{76} + 65517688q^{79} + 5994252q^{82} - 30020652q^{85} + 89560224q^{88} - 86802688q^{91} - 13856112q^{94} - 48903488q^{97} + O(q^{100})$$

## 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 0 238.000 988.535i 0 1652.00 2095.86i 0 −4194.00
8.2 4.24264i 0 238.000 988.535i 0 1652.00 2095.86i 0 −4194.00
 $$n$$: 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 9.9.b.a 2
3.b odd 2 1 inner 9.9.b.a 2
4.b odd 2 1 144.9.e.a 2
5.b even 2 1 225.9.c.a 2
5.c odd 4 2 225.9.d.a 4
9.c even 3 2 81.9.d.b 4
9.d odd 6 2 81.9.d.b 4
12.b even 2 1 144.9.e.a 2
15.d odd 2 1 225.9.c.a 2
15.e even 4 2 225.9.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.9.b.a 2 1.a even 1 1 trivial
9.9.b.a 2 3.b odd 2 1 inner
81.9.d.b 4 9.c even 3 2
81.9.d.b 4 9.d odd 6 2
144.9.e.a 2 4.b odd 2 1
144.9.e.a 2 12.b even 2 1
225.9.c.a 2 5.b even 2 1
225.9.c.a 2 15.d odd 2 1
225.9.d.a 4 5.c odd 4 2
225.9.d.a 4 15.e even 4 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 494 T^{2} + 65536 T^{4}$$
$3$ 1
$5$ $$1 + 195952 T^{2} + 152587890625 T^{4}$$
$7$ $$( 1 - 1652 T + 5764801 T^{2} )^{2}$$
$11$ $$1 + 27785566 T^{2} + 45949729863572161 T^{4}$$
$13$ $$( 1 + 26272 T + 815730721 T^{2} )^{2}$$
$17$ $$1 - 13720948544 T^{2} + 48661191875666868481 T^{4}$$
$19$ $$( 1 - 46640 T + 16983563041 T^{2} )^{2}$$
$23$ $$1 - 48977682530 T^{2} +$$$$61\!\cdots\!61$$$$T^{4}$$
$29$ $$1 - 622288126160 T^{2} +$$$$25\!\cdots\!21$$$$T^{4}$$
$31$ $$( 1 + 196444 T + 852891037441 T^{2} )^{2}$$
$37$ $$( 1 - 2819414 T + 3512479453921 T^{2} )^{2}$$
$41$ $$1 - 15470807999360 T^{2} +$$$$63\!\cdots\!41$$$$T^{4}$$
$43$ $$( 1 + 2213464 T + 11688200277601 T^{2} )^{2}$$
$47$ $$1 - 44956019993570 T^{2} +$$$$56\!\cdots\!21$$$$T^{4}$$
$53$ $$1 - 97597007293520 T^{2} +$$$$38\!\cdots\!21$$$$T^{4}$$
$59$ $$1 - 155721734882114 T^{2} +$$$$21\!\cdots\!41$$$$T^{4}$$
$61$ $$( 1 + 17405302 T + 191707312997281 T^{2} )^{2}$$
$67$ $$( 1 + 14322664 T + 406067677556641 T^{2} )^{2}$$
$71$ $$1 - 1046721637212770 T^{2} +$$$$41\!\cdots\!21$$$$T^{4}$$
$73$ $$( 1 + 8906992 T + 806460091894081 T^{2} )^{2}$$
$79$ $$( 1 - 32758844 T + 1517108809906561 T^{2} )^{2}$$
$83$ $$1 + 2735523396260830 T^{2} +$$$$50\!\cdots\!81$$$$T^{4}$$
$89$ $$1 - 4399714222730624 T^{2} +$$$$15\!\cdots\!61$$$$T^{4}$$
$97$ $$( 1 + 24451744 T + 7837433594376961 T^{2} )^{2}$$