# Properties

 Label 450.3.g.c 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 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.c 2
3.b odd 2 1 50.3.c.b yes 2
5.b even 2 1 450.3.g.e 2
5.c odd 4 1 inner 450.3.g.c 2
5.c odd 4 1 450.3.g.e 2
12.b even 2 1 400.3.p.g 2
15.d odd 2 1 50.3.c.a 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.a 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 15.d odd 2 1
50.3.c.a 2 15.e even 4 1
50.3.c.b yes 2 3.b odd 2 1
50.3.c.b yes 2 15.e even 4 1
400.3.p.a 2 60.h even 2 1
400.3.p.a 2 60.l odd 4 1
400.3.p.g 2 12.b even 2 1
400.3.p.g 2 60.l odd 4 1
450.3.g.c 2 1.a even 1 1 trivial
450.3.g.c 2 5.c odd 4 1 inner
450.3.g.e 2 5.b even 2 1
450.3.g.e 2 5.c odd 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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