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

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Newspace parameters

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

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: $$3$$ 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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

There are no other newforms in $$S_{9}^{\mathrm{new}}(9, [\chi])$$.