# Properties

 Label 450.9.b.a Level $450$ Weight $9$ Character orbit 450.b Analytic conductor $183.320$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,9,Mod(449,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("450.449");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 450.b (of order $$2$$, degree $$1$$, not 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: $$183.320374528$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 18) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 \beta_{3} q^{2} + 128 q^{4} - 1766 \beta_1 q^{7} + 1024 \beta_{3} q^{8}+O(q^{10})$$ q + 8*b3 * q^2 + 128 * q^4 - 1766*b1 * q^7 + 1024*b3 * q^8 $$q + 8 \beta_{3} q^{2} + 128 q^{4} - 1766 \beta_1 q^{7} + 1024 \beta_{3} q^{8} + 14268 \beta_{2} q^{11} + 20912 \beta_1 q^{13} - 28256 \beta_{2} q^{14} + 16384 q^{16} + 67023 \beta_{3} q^{17} + 36304 q^{19} + 114144 \beta_1 q^{22} - 292476 \beta_{3} q^{23} + 334592 \beta_{2} q^{26} - 226048 \beta_1 q^{28} + 190347 \beta_{2} q^{29} - 471196 q^{31} + 131072 \beta_{3} q^{32} + 1072368 q^{34} - 1503701 \beta_1 q^{37} + 290432 \beta_{3} q^{38} + 1212927 \beta_{2} q^{41} - 1811860 \beta_1 q^{43} + 1826304 \beta_{2} q^{44} - 4679616 q^{46} - 4252980 \beta_{3} q^{47} - 6710223 q^{49} + 2676736 \beta_1 q^{52} - 7266699 \beta_{3} q^{53} - 3616768 \beta_{2} q^{56} + 1522776 \beta_1 q^{58} + 1900776 \beta_{2} q^{59} - 5440630 q^{61} - 3769568 \beta_{3} q^{62} + 2097152 q^{64} - 3060788 \beta_1 q^{67} + 8578944 \beta_{3} q^{68} + 14986476 \beta_{2} q^{71} + 24515576 \beta_1 q^{73} - 24059216 \beta_{2} q^{74} + 4646912 q^{76} + 50394576 \beta_{3} q^{77} - 8357756 q^{79} + 9703416 \beta_1 q^{82} + 36339492 \beta_{3} q^{83} - 28989760 \beta_{2} q^{86} + 14610432 \beta_1 q^{88} + 75898881 \beta_{2} q^{89} + 147722368 q^{91} - 37436928 \beta_{3} q^{92} - 68047680 q^{94} + 10215664 \beta_1 q^{97} - 53681784 \beta_{3} q^{98}+O(q^{100})$$ q + 8*b3 * q^2 + 128 * q^4 - 1766*b1 * q^7 + 1024*b3 * q^8 + 14268*b2 * q^11 + 20912*b1 * q^13 - 28256*b2 * q^14 + 16384 * q^16 + 67023*b3 * q^17 + 36304 * q^19 + 114144*b1 * q^22 - 292476*b3 * q^23 + 334592*b2 * q^26 - 226048*b1 * q^28 + 190347*b2 * q^29 - 471196 * q^31 + 131072*b3 * q^32 + 1072368 * q^34 - 1503701*b1 * q^37 + 290432*b3 * q^38 + 1212927*b2 * q^41 - 1811860*b1 * q^43 + 1826304*b2 * q^44 - 4679616 * q^46 - 4252980*b3 * q^47 - 6710223 * q^49 + 2676736*b1 * q^52 - 7266699*b3 * q^53 - 3616768*b2 * q^56 + 1522776*b1 * q^58 + 1900776*b2 * q^59 - 5440630 * q^61 - 3769568*b3 * q^62 + 2097152 * q^64 - 3060788*b1 * q^67 + 8578944*b3 * q^68 + 14986476*b2 * q^71 + 24515576*b1 * q^73 - 24059216*b2 * q^74 + 4646912 * q^76 + 50394576*b3 * q^77 - 8357756 * q^79 + 9703416*b1 * q^82 + 36339492*b3 * q^83 - 28989760*b2 * q^86 + 14610432*b1 * q^88 + 75898881*b2 * q^89 + 147722368 * q^91 - 37436928*b3 * q^92 - 68047680 * q^94 + 10215664*b1 * q^97 - 53681784*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 512 q^{4}+O(q^{10})$$ 4 * q + 512 * q^4 $$4 q + 512 q^{4} + 65536 q^{16} + 145216 q^{19} - 1884784 q^{31} + 4289472 q^{34} - 18718464 q^{46} - 26840892 q^{49} - 21762520 q^{61} + 8388608 q^{64} + 18587648 q^{76} - 33431024 q^{79} + 590889472 q^{91} - 272190720 q^{94}+O(q^{100})$$ 4 * q + 512 * q^4 + 65536 * q^16 + 145216 * q^19 - 1884784 * q^31 + 4289472 * q^34 - 18718464 * q^46 - 26840892 * q^49 - 21762520 * q^61 + 8388608 * q^64 + 18587648 * q^76 - 33431024 * q^79 + 590889472 * q^91 - 272190720 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}^{2}$$ 2*v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
−11.3137 0 128.000 0 0 3532.00i −1448.15 0 0
449.2 −11.3137 0 128.000 0 0 3532.00i −1448.15 0 0
449.3 11.3137 0 128.000 0 0 3532.00i 1448.15 0 0
449.4 11.3137 0 128.000 0 0 3532.00i 1448.15 0 0
 $$n$$: 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
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.9.b.a 4
3.b odd 2 1 inner 450.9.b.a 4
5.b even 2 1 inner 450.9.b.a 4
5.c odd 4 1 18.9.b.a 2
5.c odd 4 1 450.9.d.b 2
15.d odd 2 1 inner 450.9.b.a 4
15.e even 4 1 18.9.b.a 2
15.e even 4 1 450.9.d.b 2
20.e even 4 1 144.9.e.d 2
45.k odd 12 2 162.9.d.d 4
45.l even 12 2 162.9.d.d 4
60.l odd 4 1 144.9.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.b.a 2 5.c odd 4 1
18.9.b.a 2 15.e even 4 1
144.9.e.d 2 20.e even 4 1
144.9.e.d 2 60.l odd 4 1
162.9.d.d 4 45.k odd 12 2
162.9.d.d 4 45.l even 12 2
450.9.b.a 4 1.a even 1 1 trivial
450.9.b.a 4 3.b odd 2 1 inner
450.9.b.a 4 5.b even 2 1 inner
450.9.b.a 4 15.d odd 2 1 inner
450.9.d.b 2 5.c odd 4 1
450.9.d.b 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 12475024$$ acting on $$S_{9}^{\mathrm{new}}(450, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 128)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 12475024)^{2}$$
$11$ $$(T^{2} + 407151648)^{2}$$
$13$ $$(T^{2} + 1749246976)^{2}$$
$17$ $$(T^{2} - 8984165058)^{2}$$
$19$ $$(T - 36304)^{4}$$
$23$ $$(T^{2} - 171084421152)^{2}$$
$29$ $$(T^{2} + 72463960818)^{2}$$
$31$ $$(T + 471196)^{4}$$
$37$ $$(T^{2} + 9044466789604)^{2}$$
$41$ $$(T^{2} + 2942383814658)^{2}$$
$43$ $$(T^{2} + 13131346638400)^{2}$$
$47$ $$(T^{2} - 36175677760800)^{2}$$
$53$ $$(T^{2} - 105609828713202)^{2}$$
$59$ $$(T^{2} + 7225898804352)^{2}$$
$61$ $$(T + 5440630)^{4}$$
$67$ $$(T^{2} + 37473692723776)^{2}$$
$71$ $$(T^{2} + 449188925797152)^{2}$$
$73$ $$(T^{2} + 24\!\cdots\!04)^{2}$$
$79$ $$(T + 8357756)^{4}$$
$83$ $$(T^{2} - 26\!\cdots\!28)^{2}$$
$89$ $$(T^{2} + 11\!\cdots\!22)^{2}$$
$97$ $$(T^{2} + 417439163843584)^{2}$$