# Properties

 Label 9.12.a.c Level 9 Weight 12 Character orbit 9.a Self dual yes Analytic conductor 6.915 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$9 = 3^{2}$$ Weight: $$k$$ = $$12$$ Character orbit: $$[\chi]$$ = 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.91508862504$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{70})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{70}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 472 q^{4} + 224 \beta q^{5} + 58100 q^{7} -1576 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} + 472 q^{4} + 224 \beta q^{5} + 58100 q^{7} -1576 \beta q^{8} + 564480 q^{10} -3200 \beta q^{11} + 762650 q^{13} + 58100 \beta q^{14} -4938176 q^{16} -178752 \beta q^{17} -10301704 q^{19} + 105728 \beta q^{20} -8064000 q^{22} + 207104 \beta q^{23} + 77615395 q^{25} + 762650 \beta q^{26} + 27423200 q^{28} -2808800 \beta q^{29} + 106159508 q^{31} -1710528 \beta q^{32} -450455040 q^{34} + 13014400 \beta q^{35} -9574450 q^{37} -10301704 \beta q^{38} -889620480 q^{40} -2161600 \beta q^{41} + 1590697400 q^{43} -1510400 \beta q^{44} + 521902080 q^{46} -28695296 \beta q^{47} + 1398283257 q^{49} + 77615395 \beta q^{50} + 359970800 q^{52} + 20943648 \beta q^{53} -1806336000 q^{55} -91565600 \beta q^{56} -7078176000 q^{58} -115091200 \beta q^{59} -3092621098 q^{61} + 106159508 \beta q^{62} + 5802853888 q^{64} + 170833600 \beta q^{65} -9113820400 q^{67} -84370944 \beta q^{68} + 32796288000 q^{70} + 66739200 \beta q^{71} + 620142950 q^{73} -9574450 \beta q^{74} -4862404288 q^{76} -185920000 \beta q^{77} + 10618486484 q^{79} -1106151424 \beta q^{80} -5447232000 q^{82} + 1195288192 \beta q^{83} -100901928960 q^{85} + 1590697400 \beta q^{86} + 12708864000 q^{88} -1223376000 \beta q^{89} + 44309965000 q^{91} + 97753088 \beta q^{92} -72312145920 q^{94} -2307581696 \beta q^{95} + 131872902350 q^{97} + 1398283257 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 944q^{4} + 116200q^{7} + O(q^{10})$$ $$2q + 944q^{4} + 116200q^{7} + 1128960q^{10} + 1525300q^{13} - 9876352q^{16} - 20603408q^{19} - 16128000q^{22} + 155230790q^{25} + 54846400q^{28} + 212319016q^{31} - 900910080q^{34} - 19148900q^{37} - 1779240960q^{40} + 3181394800q^{43} + 1043804160q^{46} + 2796566514q^{49} + 719941600q^{52} - 3612672000q^{55} - 14156352000q^{58} - 6185242196q^{61} + 11605707776q^{64} - 18227640800q^{67} + 65592576000q^{70} + 1240285900q^{73} - 9724808576q^{76} + 21236972968q^{79} - 10894464000q^{82} - 201803857920q^{85} + 25417728000q^{88} + 88619930000q^{91} - 144624291840q^{94} + 263745804700q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −8.36660 8.36660
−50.1996 0 472.000 −11244.7 0 58100.0 79114.6 0 564480.
1.2 50.1996 0 472.000 11244.7 0 58100.0 −79114.6 0 564480.
 $$n$$: 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 9.12.a.c 2
3.b odd 2 1 inner 9.12.a.c 2
4.b odd 2 1 144.12.a.r 2
5.b even 2 1 225.12.a.j 2
5.c odd 4 2 225.12.b.g 4
9.c even 3 2 81.12.c.g 4
9.d odd 6 2 81.12.c.g 4
12.b even 2 1 144.12.a.r 2
15.d odd 2 1 225.12.a.j 2
15.e even 4 2 225.12.b.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.12.a.c 2 1.a even 1 1 trivial
9.12.a.c 2 3.b odd 2 1 inner
81.12.c.g 4 9.c even 3 2
81.12.c.g 4 9.d odd 6 2
144.12.a.r 2 4.b odd 2 1
144.12.a.r 2 12.b even 2 1
225.12.a.j 2 5.b even 2 1
225.12.a.j 2 15.d odd 2 1
225.12.b.g 4 5.c odd 4 2
225.12.b.g 4 15.e even 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2520$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 1576 T^{2} + 4194304 T^{4}$$
$3$ 
$5$ $$1 - 28787270 T^{2} + 2384185791015625 T^{4}$$
$7$ $$( 1 - 58100 T + 1977326743 T^{2} )^{2}$$
$11$ $$1 + 544818541222 T^{2} +$$$$81\!\cdots\!21$$$$T^{4}$$
$13$ $$( 1 - 762650 T + 1792160394037 T^{2} )^{2}$$
$17$ $$1 - 11975946694814 T^{2} +$$$$11\!\cdots\!89$$$$T^{4}$$
$19$ $$( 1 + 10301704 T + 116490258898219 T^{2} )^{2}$$
$23$ $$1 + 1797531507451534 T^{2} +$$$$90\!\cdots\!29$$$$T^{4}$$
$29$ $$1 + 4519838782611658 T^{2} +$$$$14\!\cdots\!41$$$$T^{4}$$
$31$ $$( 1 - 106159508 T + 25408476896404831 T^{2} )^{2}$$
$37$ $$( 1 + 9574450 T + 177917621779460413 T^{2} )^{2}$$
$41$ $$1 + 1088883326741296882 T^{2} +$$$$30\!\cdots\!81$$$$T^{4}$$
$43$ $$( 1 - 1590697400 T + 929293739471222707 T^{2} )^{2}$$
$47$ $$1 + 2869299998598432286 T^{2} +$$$$61\!\cdots\!09$$$$T^{4}$$
$53$ $$1 + 17432708152043665114 T^{2} +$$$$85\!\cdots\!09$$$$T^{4}$$
$59$ $$1 + 26931896409526885318 T^{2} +$$$$90\!\cdots\!81$$$$T^{4}$$
$61$ $$( 1 + 3092621098 T + 43513917611435838661 T^{2} )^{2}$$
$67$ $$( 1 + 9113820400 T +$$$$12\!\cdots\!83$$$$T^{2} )^{2}$$
$71$ $$1 +$$$$45\!\cdots\!42$$$$T^{2} +$$$$53\!\cdots\!41$$$$T^{4}$$
$73$ $$( 1 - 620142950 T +$$$$31\!\cdots\!77$$$$T^{2} )^{2}$$
$79$ $$( 1 - 10618486484 T +$$$$74\!\cdots\!79$$$$T^{2} )^{2}$$
$83$ $$1 -$$$$10\!\cdots\!46$$$$T^{2} +$$$$16\!\cdots\!89$$$$T^{4}$$
$89$ $$1 +$$$$17\!\cdots\!78$$$$T^{2} +$$$$77\!\cdots\!21$$$$T^{4}$$
$97$ $$( 1 - 131872902350 T +$$$$71\!\cdots\!53$$$$T^{2} )^{2}$$