# Properties

 Label 450.3.g.a Level $450$ Weight $3$ Character orbit 450.g Analytic conductor $12.262$ 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] = [450,3,Mod(307,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("450.307");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 450.g (of order $$4$$, degree $$2$$, 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: $$12.2616118962$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - i - 1) q^{2} + 2 i q^{4} + ( - 8 i - 8) q^{7} + ( - 2 i + 2) q^{8}+O(q^{10})$$ q + (-i - 1) * q^2 + 2*i * q^4 + (-8*i - 8) * q^7 + (-2*i + 2) * q^8 $$q + ( - i - 1) q^{2} + 2 i q^{4} + ( - 8 i - 8) q^{7} + ( - 2 i + 2) q^{8} - 4 q^{11} + ( - 3 i + 3) q^{13} + 16 i q^{14} - 4 q^{16} + (19 i + 19) q^{17} - 8 i q^{19} + (4 i + 4) q^{22} + (20 i - 20) q^{23} - 6 q^{26} + ( - 16 i + 16) q^{28} + 38 i q^{29} - 44 q^{31} + (4 i + 4) q^{32} - 38 i q^{34} + (3 i + 3) q^{37} + (8 i - 8) q^{38} - 70 q^{41} + (36 i - 36) q^{43} - 8 i q^{44} + 40 q^{46} + 79 i q^{49} + (6 i + 6) q^{52} + ( - 17 i + 17) q^{53} - 32 q^{56} + ( - 38 i + 38) q^{58} + 92 i q^{59} + 72 q^{61} + (44 i + 44) q^{62} - 8 i q^{64} + ( - 44 i - 44) q^{67} + (38 i - 38) q^{68} + 88 q^{71} + (55 i - 55) q^{73} - 6 i q^{74} + 16 q^{76} + (32 i + 32) q^{77} + 12 i q^{79} + (70 i + 70) q^{82} + (24 i - 24) q^{83} + 72 q^{86} + (8 i - 8) q^{88} - 26 i q^{89} - 48 q^{91} + ( - 40 i - 40) q^{92} + (57 i + 57) q^{97} + ( - 79 i + 79) q^{98} +O(q^{100})$$ q + (-i - 1) * q^2 + 2*i * q^4 + (-8*i - 8) * q^7 + (-2*i + 2) * q^8 - 4 * q^11 + (-3*i + 3) * q^13 + 16*i * q^14 - 4 * q^16 + (19*i + 19) * q^17 - 8*i * q^19 + (4*i + 4) * q^22 + (20*i - 20) * q^23 - 6 * q^26 + (-16*i + 16) * q^28 + 38*i * q^29 - 44 * q^31 + (4*i + 4) * q^32 - 38*i * q^34 + (3*i + 3) * q^37 + (8*i - 8) * q^38 - 70 * q^41 + (36*i - 36) * q^43 - 8*i * q^44 + 40 * q^46 + 79*i * q^49 + (6*i + 6) * q^52 + (-17*i + 17) * q^53 - 32 * q^56 + (-38*i + 38) * q^58 + 92*i * q^59 + 72 * q^61 + (44*i + 44) * q^62 - 8*i * q^64 + (-44*i - 44) * q^67 + (38*i - 38) * q^68 + 88 * q^71 + (55*i - 55) * q^73 - 6*i * q^74 + 16 * q^76 + (32*i + 32) * q^77 + 12*i * q^79 + (70*i + 70) * q^82 + (24*i - 24) * q^83 + 72 * q^86 + (8*i - 8) * q^88 - 26*i * q^89 - 48 * q^91 + (-40*i - 40) * q^92 + (57*i + 57) * q^97 + (-79*i + 79) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 16 q^{7} + 4 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 16 * q^7 + 4 * q^8 $$2 q - 2 q^{2} - 16 q^{7} + 4 q^{8} - 8 q^{11} + 6 q^{13} - 8 q^{16} + 38 q^{17} + 8 q^{22} - 40 q^{23} - 12 q^{26} + 32 q^{28} - 88 q^{31} + 8 q^{32} + 6 q^{37} - 16 q^{38} - 140 q^{41} - 72 q^{43} + 80 q^{46} + 12 q^{52} + 34 q^{53} - 64 q^{56} + 76 q^{58} + 144 q^{61} + 88 q^{62} - 88 q^{67} - 76 q^{68} + 176 q^{71} - 110 q^{73} + 32 q^{76} + 64 q^{77} + 140 q^{82} - 48 q^{83} + 144 q^{86} - 16 q^{88} - 96 q^{91} - 80 q^{92} + 114 q^{97} + 158 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 16 * q^7 + 4 * q^8 - 8 * q^11 + 6 * q^13 - 8 * q^16 + 38 * q^17 + 8 * q^22 - 40 * q^23 - 12 * q^26 + 32 * q^28 - 88 * q^31 + 8 * q^32 + 6 * q^37 - 16 * q^38 - 140 * q^41 - 72 * q^43 + 80 * q^46 + 12 * q^52 + 34 * q^53 - 64 * q^56 + 76 * q^58 + 144 * q^61 + 88 * q^62 - 88 * q^67 - 76 * q^68 + 176 * q^71 - 110 * q^73 + 32 * q^76 + 64 * q^77 + 140 * q^82 - 48 * q^83 + 144 * q^86 - 16 * q^88 - 96 * q^91 - 80 * q^92 + 114 * q^97 + 158 * q^98

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 1.00000i − 1.00000i
−1.00000 1.00000i 0 2.00000i 0 0 −8.00000 8.00000i 2.00000 2.00000i 0 0
343.1 −1.00000 + 1.00000i 0 2.00000i 0 0 −8.00000 + 8.00000i 2.00000 + 2.00000i 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.g.a 2
3.b odd 2 1 450.3.g.d 2
5.b even 2 1 90.3.g.c yes 2
5.c odd 4 1 90.3.g.c yes 2
5.c odd 4 1 inner 450.3.g.a 2
15.d odd 2 1 90.3.g.a 2
15.e even 4 1 90.3.g.a 2
15.e even 4 1 450.3.g.d 2
20.d odd 2 1 720.3.bh.d 2
20.e even 4 1 720.3.bh.d 2
60.h even 2 1 720.3.bh.b 2
60.l odd 4 1 720.3.bh.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.g.a 2 15.d odd 2 1
90.3.g.a 2 15.e even 4 1
90.3.g.c yes 2 5.b even 2 1
90.3.g.c yes 2 5.c odd 4 1
450.3.g.a 2 1.a even 1 1 trivial
450.3.g.a 2 5.c odd 4 1 inner
450.3.g.d 2 3.b odd 2 1
450.3.g.d 2 15.e even 4 1
720.3.bh.b 2 60.h even 2 1
720.3.bh.b 2 60.l odd 4 1
720.3.bh.d 2 20.d odd 2 1
720.3.bh.d 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{2} + 16T_{7} + 128$$ T7^2 + 16*T7 + 128 $$T_{11} + 4$$ T11 + 4 $$T_{17}^{2} - 38T_{17} + 722$$ T17^2 - 38*T17 + 722

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16T + 128$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} - 6T + 18$$
$17$ $$T^{2} - 38T + 722$$
$19$ $$T^{2} + 64$$
$23$ $$T^{2} + 40T + 800$$
$29$ $$T^{2} + 1444$$
$31$ $$(T + 44)^{2}$$
$37$ $$T^{2} - 6T + 18$$
$41$ $$(T + 70)^{2}$$
$43$ $$T^{2} + 72T + 2592$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 34T + 578$$
$59$ $$T^{2} + 8464$$
$61$ $$(T - 72)^{2}$$
$67$ $$T^{2} + 88T + 3872$$
$71$ $$(T - 88)^{2}$$
$73$ $$T^{2} + 110T + 6050$$
$79$ $$T^{2} + 144$$
$83$ $$T^{2} + 48T + 1152$$
$89$ $$T^{2} + 676$$
$97$ $$T^{2} - 114T + 6498$$