# Properties

 Label 450.3.g.e Level $450$ Weight $3$ Character orbit 450.g Analytic conductor $12.262$ 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] = [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 50) 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.

 $$f(q)$$ $$=$$ $$q + (i + 1) q^{2} + 2 i q^{4} + ( - 3 i - 3) q^{7} + (2 i - 2) q^{8}+O(q^{10})$$ q + (i + 1) * q^2 + 2*i * q^4 + (-3*i - 3) * q^7 + (2*i - 2) * q^8 $$q + (i + 1) q^{2} + 2 i q^{4} + ( - 3 i - 3) q^{7} + (2 i - 2) q^{8} - 12 q^{11} + (12 i - 12) q^{13} - 6 i q^{14} - 4 q^{16} + ( - 12 i - 12) q^{17} + 20 i q^{19} + ( - 12 i - 12) q^{22} + (3 i - 3) q^{23} - 24 q^{26} + ( - 6 i + 6) q^{28} + 30 i q^{29} - 8 q^{31} + ( - 4 i - 4) q^{32} - 24 i q^{34} + ( - 48 i - 48) q^{37} + (20 i - 20) q^{38} + 48 q^{41} + (27 i - 27) q^{43} - 24 i q^{44} - 6 q^{46} + ( - 27 i - 27) q^{47} - 31 i q^{49} + ( - 24 i - 24) q^{52} + ( - 12 i + 12) q^{53} + 12 q^{56} + (30 i - 30) q^{58} + 60 i q^{59} + 32 q^{61} + ( - 8 i - 8) q^{62} - 8 i q^{64} + ( - 3 i - 3) q^{67} + ( - 24 i + 24) q^{68} + 48 q^{71} + (12 i - 12) q^{73} - 96 i q^{74} - 40 q^{76} + (36 i + 36) q^{77} + 40 i q^{79} + (48 i + 48) q^{82} + (93 i - 93) q^{83} - 54 q^{86} + ( - 24 i + 24) q^{88} - 30 i q^{89} + 72 q^{91} + ( - 6 i - 6) q^{92} - 54 i q^{94} + (12 i + 12) q^{97} + ( - 31 i + 31) q^{98} +O(q^{100})$$ q + (i + 1) * q^2 + 2*i * q^4 + (-3*i - 3) * q^7 + (2*i - 2) * q^8 - 12 * q^11 + (12*i - 12) * q^13 - 6*i * q^14 - 4 * q^16 + (-12*i - 12) * q^17 + 20*i * q^19 + (-12*i - 12) * q^22 + (3*i - 3) * q^23 - 24 * q^26 + (-6*i + 6) * q^28 + 30*i * q^29 - 8 * q^31 + (-4*i - 4) * q^32 - 24*i * q^34 + (-48*i - 48) * q^37 + (20*i - 20) * q^38 + 48 * q^41 + (27*i - 27) * q^43 - 24*i * q^44 - 6 * q^46 + (-27*i - 27) * q^47 - 31*i * q^49 + (-24*i - 24) * q^52 + (-12*i + 12) * q^53 + 12 * q^56 + (30*i - 30) * q^58 + 60*i * q^59 + 32 * q^61 + (-8*i - 8) * q^62 - 8*i * q^64 + (-3*i - 3) * q^67 + (-24*i + 24) * q^68 + 48 * q^71 + (12*i - 12) * q^73 - 96*i * q^74 - 40 * q^76 + (36*i + 36) * q^77 + 40*i * q^79 + (48*i + 48) * q^82 + (93*i - 93) * q^83 - 54 * q^86 + (-24*i + 24) * q^88 - 30*i * q^89 + 72 * q^91 + (-6*i - 6) * q^92 - 54*i * q^94 + (12*i + 12) * q^97 + (-31*i + 31) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 6 q^{7} - 4 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 6 * q^7 - 4 * q^8 $$2 q + 2 q^{2} - 6 q^{7} - 4 q^{8} - 24 q^{11} - 24 q^{13} - 8 q^{16} - 24 q^{17} - 24 q^{22} - 6 q^{23} - 48 q^{26} + 12 q^{28} - 16 q^{31} - 8 q^{32} - 96 q^{37} - 40 q^{38} + 96 q^{41} - 54 q^{43} - 12 q^{46} - 54 q^{47} - 48 q^{52} + 24 q^{53} + 24 q^{56} - 60 q^{58} + 64 q^{61} - 16 q^{62} - 6 q^{67} + 48 q^{68} + 96 q^{71} - 24 q^{73} - 80 q^{76} + 72 q^{77} + 96 q^{82} - 186 q^{83} - 108 q^{86} + 48 q^{88} + 144 q^{91} - 12 q^{92} + 24 q^{97} + 62 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 6 * q^7 - 4 * q^8 - 24 * q^11 - 24 * q^13 - 8 * q^16 - 24 * q^17 - 24 * q^22 - 6 * q^23 - 48 * q^26 + 12 * q^28 - 16 * q^31 - 8 * q^32 - 96 * q^37 - 40 * q^38 + 96 * q^41 - 54 * q^43 - 12 * q^46 - 54 * q^47 - 48 * q^52 + 24 * q^53 + 24 * q^56 - 60 * q^58 + 64 * q^61 - 16 * q^62 - 6 * q^67 + 48 * q^68 + 96 * q^71 - 24 * q^73 - 80 * q^76 + 72 * q^77 + 96 * q^82 - 186 * q^83 - 108 * q^86 + 48 * q^88 + 144 * q^91 - 12 * q^92 + 24 * q^97 + 62 * 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 −3.00000 3.00000i −2.00000 + 2.00000i 0 0
343.1 1.00000 1.00000i 0 2.00000i 0 0 −3.00000 + 3.00000i −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.e 2
3.b odd 2 1 50.3.c.a 2
5.b even 2 1 450.3.g.c 2
5.c odd 4 1 450.3.g.c 2
5.c odd 4 1 inner 450.3.g.e 2
12.b even 2 1 400.3.p.a 2
15.d odd 2 1 50.3.c.b yes 2
15.e even 4 1 50.3.c.a 2
15.e even 4 1 50.3.c.b yes 2
60.h even 2 1 400.3.p.g 2
60.l odd 4 1 400.3.p.a 2
60.l odd 4 1 400.3.p.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.3.c.a 2 3.b odd 2 1
50.3.c.a 2 15.e even 4 1
50.3.c.b yes 2 15.d odd 2 1
50.3.c.b yes 2 15.e even 4 1
400.3.p.a 2 12.b even 2 1
400.3.p.a 2 60.l odd 4 1
400.3.p.g 2 60.h even 2 1
400.3.p.g 2 60.l odd 4 1
450.3.g.c 2 5.b even 2 1
450.3.g.c 2 5.c odd 4 1
450.3.g.e 2 1.a even 1 1 trivial
450.3.g.e 2 5.c odd 4 1 inner

## Hecke kernels

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

 $$T_{7}^{2} + 6T_{7} + 18$$ T7^2 + 6*T7 + 18 $$T_{11} + 12$$ T11 + 12 $$T_{17}^{2} + 24T_{17} + 288$$ T17^2 + 24*T17 + 288

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 6T + 18$$
$11$ $$(T + 12)^{2}$$
$13$ $$T^{2} + 24T + 288$$
$17$ $$T^{2} + 24T + 288$$
$19$ $$T^{2} + 400$$
$23$ $$T^{2} + 6T + 18$$
$29$ $$T^{2} + 900$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 96T + 4608$$
$41$ $$(T - 48)^{2}$$
$43$ $$T^{2} + 54T + 1458$$
$47$ $$T^{2} + 54T + 1458$$
$53$ $$T^{2} - 24T + 288$$
$59$ $$T^{2} + 3600$$
$61$ $$(T - 32)^{2}$$
$67$ $$T^{2} + 6T + 18$$
$71$ $$(T - 48)^{2}$$
$73$ $$T^{2} + 24T + 288$$
$79$ $$T^{2} + 1600$$
$83$ $$T^{2} + 186T + 17298$$
$89$ $$T^{2} + 900$$
$97$ $$T^{2} - 24T + 288$$