Properties

 Label 9.9.b.a Level $9$ Weight $9$ Character orbit 9.b Analytic conductor $3.666$ Analytic rank $0$ Dimension $2$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,9,Mod(8,9)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9.8");

S:= CuspForms(chi, 9);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$3.66640749055$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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

$p$ $F_p(T)$
$2$ $$T^{2} + 18$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 977202$$
$7$ $$(T - 1652)^{2}$$
$11$ $$T^{2} + 456503328$$
$13$ $$(T + 26272)^{2}$$
$17$ $$T^{2} + 230566338$$
$19$ $$(T - 46640)^{2}$$
$23$ $$T^{2} + 107644288032$$
$29$ $$T^{2} + 378204699762$$
$31$ $$(T + 196444)^{2}$$
$37$ $$(T - 2819414)^{2}$$
$41$ $$T^{2} + 499042458882$$
$43$ $$(T + 2213464)^{2}$$
$47$ $$T^{2} + 2666553329952$$
$53$ $$T^{2} + 26922373529202$$
$59$ $$T^{2} + 137939140326528$$
$61$ $$(T + 17405302)^{2}$$
$67$ $$(T + 14322664)^{2}$$
$71$ $$T^{2} + 244785425278752$$
$73$ $$(T + 8906992)^{2}$$
$79$ $$(T - 32758844)^{2}$$
$83$ $$T^{2} + 72\!\cdots\!12$$
$89$ $$T^{2} + 34\!\cdots\!38$$
$97$ $$(T + 24451744)^{2}$$