# Properties

 Label 450.9.d.b Level $450$ Weight $9$ Character orbit 450.d Analytic conductor $183.320$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,9,Mod(251,450)]

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, 0]))

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

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

chi := DirichletCharacter("450.251");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 450.d (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: $$183.320374528$$ 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 \beta q^{2} - 128 q^{4} + 3532 q^{7} + 1024 \beta q^{8} +O(q^{10})$$ q - 8*b * q^2 - 128 * q^4 + 3532 * q^7 + 1024*b * q^8 $$q - 8 \beta q^{2} - 128 q^{4} + 3532 q^{7} + 1024 \beta q^{8} - 14268 \beta q^{11} + 41824 q^{13} - 28256 \beta q^{14} + 16384 q^{16} - 67023 \beta q^{17} - 36304 q^{19} - 228288 q^{22} - 292476 \beta q^{23} - 334592 \beta q^{26} - 452096 q^{28} + 190347 \beta q^{29} - 471196 q^{31} - 131072 \beta q^{32} - 1072368 q^{34} + 3007402 q^{37} + 290432 \beta q^{38} - 1212927 \beta q^{41} - 3623720 q^{43} + 1826304 \beta q^{44} - 4679616 q^{46} + 4252980 \beta q^{47} + 6710223 q^{49} - 5353472 q^{52} - 7266699 \beta q^{53} + 3616768 \beta q^{56} + 3045552 q^{58} + 1900776 \beta q^{59} - 5440630 q^{61} + 3769568 \beta q^{62} - 2097152 q^{64} + 6121576 q^{67} + 8578944 \beta q^{68} - 14986476 \beta q^{71} + 49031152 q^{73} - 24059216 \beta q^{74} + 4646912 q^{76} - 50394576 \beta q^{77} + 8357756 q^{79} - 19406832 q^{82} + 36339492 \beta q^{83} + 28989760 \beta q^{86} + 29220864 q^{88} + 75898881 \beta q^{89} + 147722368 q^{91} + 37436928 \beta q^{92} + 68047680 q^{94} - 20431328 q^{97} - 53681784 \beta q^{98} +O(q^{100})$$ q - 8*b * q^2 - 128 * q^4 + 3532 * q^7 + 1024*b * q^8 - 14268*b * q^11 + 41824 * q^13 - 28256*b * q^14 + 16384 * q^16 - 67023*b * q^17 - 36304 * q^19 - 228288 * q^22 - 292476*b * q^23 - 334592*b * q^26 - 452096 * q^28 + 190347*b * q^29 - 471196 * q^31 - 131072*b * q^32 - 1072368 * q^34 + 3007402 * q^37 + 290432*b * q^38 - 1212927*b * q^41 - 3623720 * q^43 + 1826304*b * q^44 - 4679616 * q^46 + 4252980*b * q^47 + 6710223 * q^49 - 5353472 * q^52 - 7266699*b * q^53 + 3616768*b * q^56 + 3045552 * q^58 + 1900776*b * q^59 - 5440630 * q^61 + 3769568*b * q^62 - 2097152 * q^64 + 6121576 * q^67 + 8578944*b * q^68 - 14986476*b * q^71 + 49031152 * q^73 - 24059216*b * q^74 + 4646912 * q^76 - 50394576*b * q^77 + 8357756 * q^79 - 19406832 * q^82 + 36339492*b * q^83 + 28989760*b * q^86 + 29220864 * q^88 + 75898881*b * q^89 + 147722368 * q^91 + 37436928*b * q^92 + 68047680 * q^94 - 20431328 * q^97 - 53681784*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 256 q^{4} + 7064 q^{7}+O(q^{10})$$ 2 * q - 256 * q^4 + 7064 * q^7 $$2 q - 256 q^{4} + 7064 q^{7} + 83648 q^{13} + 32768 q^{16} - 72608 q^{19} - 456576 q^{22} - 904192 q^{28} - 942392 q^{31} - 2144736 q^{34} + 6014804 q^{37} - 7247440 q^{43} - 9359232 q^{46} + 13420446 q^{49} - 10706944 q^{52} + 6091104 q^{58} - 10881260 q^{61} - 4194304 q^{64} + 12243152 q^{67} + 98062304 q^{73} + 9293824 q^{76} + 16715512 q^{79} - 38813664 q^{82} + 58441728 q^{88} + 295444736 q^{91} + 136095360 q^{94} - 40862656 q^{97}+O(q^{100})$$ 2 * q - 256 * q^4 + 7064 * q^7 + 83648 * q^13 + 32768 * q^16 - 72608 * q^19 - 456576 * q^22 - 904192 * q^28 - 942392 * q^31 - 2144736 * q^34 + 6014804 * q^37 - 7247440 * q^43 - 9359232 * q^46 + 13420446 * q^49 - 10706944 * q^52 + 6091104 * q^58 - 10881260 * q^61 - 4194304 * q^64 + 12243152 * q^67 + 98062304 * q^73 + 9293824 * q^76 + 16715512 * q^79 - 38813664 * q^82 + 58441728 * q^88 + 295444736 * q^91 + 136095360 * q^94 - 40862656 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 1.41421i − 1.41421i
11.3137i 0 −128.000 0 0 3532.00 1448.15i 0 0
251.2 11.3137i 0 −128.000 0 0 3532.00 1448.15i 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.9.d.b 2
3.b odd 2 1 inner 450.9.d.b 2
5.b even 2 1 18.9.b.a 2
5.c odd 4 2 450.9.b.a 4
15.d odd 2 1 18.9.b.a 2
15.e even 4 2 450.9.b.a 4
20.d odd 2 1 144.9.e.d 2
45.h odd 6 2 162.9.d.d 4
45.j even 6 2 162.9.d.d 4
60.h even 2 1 144.9.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.b.a 2 5.b even 2 1
18.9.b.a 2 15.d odd 2 1
144.9.e.d 2 20.d odd 2 1
144.9.e.d 2 60.h even 2 1
162.9.d.d 4 45.h odd 6 2
162.9.d.d 4 45.j even 6 2
450.9.b.a 4 5.c odd 4 2
450.9.b.a 4 15.e even 4 2
450.9.d.b 2 1.a even 1 1 trivial
450.9.d.b 2 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 3532$$ acting on $$S_{9}^{\mathrm{new}}(450, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 128$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 3532)^{2}$$
$11$ $$T^{2} + 407151648$$
$13$ $$(T - 41824)^{2}$$
$17$ $$T^{2} + 8984165058$$
$19$ $$(T + 36304)^{2}$$
$23$ $$T^{2} + 171084421152$$
$29$ $$T^{2} + 72463960818$$
$31$ $$(T + 471196)^{2}$$
$37$ $$(T - 3007402)^{2}$$
$41$ $$T^{2} + 2942383814658$$
$43$ $$(T + 3623720)^{2}$$
$47$ $$T^{2} + 36175677760800$$
$53$ $$T^{2} + 105609828713202$$
$59$ $$T^{2} + 7225898804352$$
$61$ $$(T + 5440630)^{2}$$
$67$ $$(T - 6121576)^{2}$$
$71$ $$T^{2} + 449188925797152$$
$73$ $$(T - 49031152)^{2}$$
$79$ $$(T - 8357756)^{2}$$
$83$ $$T^{2} + 26\!\cdots\!28$$
$89$ $$T^{2} + 11\!\cdots\!22$$
$97$ $$(T + 20431328)^{2}$$