# Properties

 Label 450.3.g.j Level $450$ Weight $3$ Character orbit 450.g Analytic conductor $12.262$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 30) 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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} + 2 \beta_{2} q^{4} + (4 \beta_{2} + \beta_1 + 4) q^{7} + (2 \beta_{2} - 2) q^{8}+O(q^{10})$$ q + (b2 + 1) * q^2 + 2*b2 * q^4 + (4*b2 + b1 + 4) * q^7 + (2*b2 - 2) * q^8 $$q + (\beta_{2} + 1) q^{2} + 2 \beta_{2} q^{4} + (4 \beta_{2} + \beta_1 + 4) q^{7} + (2 \beta_{2} - 2) q^{8} + (\beta_{3} - \beta_1 + 4) q^{11} + ( - 2 \beta_{3} + 3 \beta_{2} - 3) q^{13} + (\beta_{3} + 8 \beta_{2} + \beta_1) q^{14} - 4 q^{16} + (11 \beta_{2} - \beta_1 + 11) q^{17} + ( - \beta_{3} + 16 \beta_{2} - \beta_1) q^{19} + (4 \beta_{2} - 2 \beta_1 + 4) q^{22} + (3 \beta_{3} + 4 \beta_{2} - 4) q^{23} + ( - 2 \beta_{3} + 2 \beta_1 - 6) q^{26} + (2 \beta_{3} + 8 \beta_{2} - 8) q^{28} + ( - \beta_{3} + 16 \beta_{2} - \beta_1) q^{29} + ( - 2 \beta_{3} + 2 \beta_1 - 20) q^{31} + ( - 4 \beta_{2} - 4) q^{32} + ( - \beta_{3} + 22 \beta_{2} - \beta_1) q^{34} + (27 \beta_{2} + 27) q^{37} + ( - 2 \beta_{3} + 16 \beta_{2} - 16) q^{38} + ( - \beta_{3} + \beta_1 - 8) q^{41} + (5 \beta_{3} + 12 \beta_{2} - 12) q^{43} + ( - 2 \beta_{3} + 8 \beta_{2} - 2 \beta_1) q^{44} + (3 \beta_{3} - 3 \beta_1 - 8) q^{46} + (24 \beta_{2} + 3 \beta_1 + 24) q^{47} + (8 \beta_{3} + 31 \beta_{2} + 8 \beta_1) q^{49} + ( - 6 \beta_{2} + 4 \beta_1 - 6) q^{52} + ( - 9 \beta_{3} - 25 \beta_{2} + 25) q^{53} + (2 \beta_{3} - 2 \beta_1 - 16) q^{56} + ( - 2 \beta_{3} + 16 \beta_{2} - 16) q^{58} - 20 \beta_{2} q^{59} + (4 \beta_{3} - 4 \beta_1 - 24) q^{61} + ( - 20 \beta_{2} + 4 \beta_1 - 20) q^{62} - 8 \beta_{2} q^{64} + (4 \beta_{2} - 9 \beta_1 + 4) q^{67} + ( - 2 \beta_{3} + 22 \beta_{2} - 22) q^{68} + (\beta_{3} - \beta_1 - 16) q^{71} + ( - 2 \beta_{3} - 47 \beta_{2} + 47) q^{73} + 54 \beta_{2} q^{74} + ( - 2 \beta_{3} + 2 \beta_1 - 32) q^{76} + ( - 32 \beta_{2} - 4 \beta_1 - 32) q^{77} + ( - 13 \beta_{3} - 12 \beta_{2} - 13 \beta_1) q^{79} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{82} + (7 \beta_{3} - 48 \beta_{2} + 48) q^{83} + (5 \beta_{3} - 5 \beta_1 - 24) q^{86} + ( - 4 \beta_{3} + 8 \beta_{2} - 8) q^{88} + (3 \beta_{3} - 88 \beta_{2} + 3 \beta_1) q^{89} + ( - 5 \beta_{3} + 5 \beta_1 + 72) q^{91} + ( - 8 \beta_{2} - 6 \beta_1 - 8) q^{92} + (3 \beta_{3} + 48 \beta_{2} + 3 \beta_1) q^{94} + ( - 33 \beta_{2} + 10 \beta_1 - 33) q^{97} + (16 \beta_{3} + 31 \beta_{2} - 31) q^{98}+O(q^{100})$$ q + (b2 + 1) * q^2 + 2*b2 * q^4 + (4*b2 + b1 + 4) * q^7 + (2*b2 - 2) * q^8 + (b3 - b1 + 4) * q^11 + (-2*b3 + 3*b2 - 3) * q^13 + (b3 + 8*b2 + b1) * q^14 - 4 * q^16 + (11*b2 - b1 + 11) * q^17 + (-b3 + 16*b2 - b1) * q^19 + (4*b2 - 2*b1 + 4) * q^22 + (3*b3 + 4*b2 - 4) * q^23 + (-2*b3 + 2*b1 - 6) * q^26 + (2*b3 + 8*b2 - 8) * q^28 + (-b3 + 16*b2 - b1) * q^29 + (-2*b3 + 2*b1 - 20) * q^31 + (-4*b2 - 4) * q^32 + (-b3 + 22*b2 - b1) * q^34 + (27*b2 + 27) * q^37 + (-2*b3 + 16*b2 - 16) * q^38 + (-b3 + b1 - 8) * q^41 + (5*b3 + 12*b2 - 12) * q^43 + (-2*b3 + 8*b2 - 2*b1) * q^44 + (3*b3 - 3*b1 - 8) * q^46 + (24*b2 + 3*b1 + 24) * q^47 + (8*b3 + 31*b2 + 8*b1) * q^49 + (-6*b2 + 4*b1 - 6) * q^52 + (-9*b3 - 25*b2 + 25) * q^53 + (2*b3 - 2*b1 - 16) * q^56 + (-2*b3 + 16*b2 - 16) * q^58 - 20*b2 * q^59 + (4*b3 - 4*b1 - 24) * q^61 + (-20*b2 + 4*b1 - 20) * q^62 - 8*b2 * q^64 + (4*b2 - 9*b1 + 4) * q^67 + (-2*b3 + 22*b2 - 22) * q^68 + (b3 - b1 - 16) * q^71 + (-2*b3 - 47*b2 + 47) * q^73 + 54*b2 * q^74 + (-2*b3 + 2*b1 - 32) * q^76 + (-32*b2 - 4*b1 - 32) * q^77 + (-13*b3 - 12*b2 - 13*b1) * q^79 + (-8*b2 + 2*b1 - 8) * q^82 + (7*b3 - 48*b2 + 48) * q^83 + (5*b3 - 5*b1 - 24) * q^86 + (-4*b3 + 8*b2 - 8) * q^88 + (3*b3 - 88*b2 + 3*b1) * q^89 + (-5*b3 + 5*b1 + 72) * q^91 + (-8*b2 - 6*b1 - 8) * q^92 + (3*b3 + 48*b2 + 3*b1) * q^94 + (-33*b2 + 10*b1 - 33) * q^97 + (16*b3 + 31*b2 - 31) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 16 q^{7} - 8 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 + 16 * q^7 - 8 * q^8 $$4 q + 4 q^{2} + 16 q^{7} - 8 q^{8} + 16 q^{11} - 12 q^{13} - 16 q^{16} + 44 q^{17} + 16 q^{22} - 16 q^{23} - 24 q^{26} - 32 q^{28} - 80 q^{31} - 16 q^{32} + 108 q^{37} - 64 q^{38} - 32 q^{41} - 48 q^{43} - 32 q^{46} + 96 q^{47} - 24 q^{52} + 100 q^{53} - 64 q^{56} - 64 q^{58} - 96 q^{61} - 80 q^{62} + 16 q^{67} - 88 q^{68} - 64 q^{71} + 188 q^{73} - 128 q^{76} - 128 q^{77} - 32 q^{82} + 192 q^{83} - 96 q^{86} - 32 q^{88} + 288 q^{91} - 32 q^{92} - 132 q^{97} - 124 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 + 16 * q^7 - 8 * q^8 + 16 * q^11 - 12 * q^13 - 16 * q^16 + 44 * q^17 + 16 * q^22 - 16 * q^23 - 24 * q^26 - 32 * q^28 - 80 * q^31 - 16 * q^32 + 108 * q^37 - 64 * q^38 - 32 * q^41 - 48 * q^43 - 32 * q^46 + 96 * q^47 - 24 * q^52 + 100 * q^53 - 64 * q^56 - 64 * q^58 - 96 * q^61 - 80 * q^62 + 16 * q^67 - 88 * q^68 - 64 * q^71 + 188 * q^73 - 128 * q^76 - 128 * q^77 - 32 * q^82 + 192 * q^83 - 96 * q^86 - 32 * q^88 + 288 * q^91 - 32 * q^92 - 132 * q^97 - 124 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu$$ 4*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( 4\nu^{3} ) / 3$$ (4*v^3) / 3
 $$\nu$$ $$=$$ $$( \beta_1 ) / 4$$ (b1) / 4 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} ) / 4$$ (3*b3) / 4

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
1.00000 + 1.00000i 0 2.00000i 0 0 −0.898979 0.898979i −2.00000 + 2.00000i 0 0
307.2 1.00000 + 1.00000i 0 2.00000i 0 0 8.89898 + 8.89898i −2.00000 + 2.00000i 0 0
343.1 1.00000 1.00000i 0 2.00000i 0 0 −0.898979 + 0.898979i −2.00000 2.00000i 0 0
343.2 1.00000 1.00000i 0 2.00000i 0 0 8.89898 8.89898i −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.j 4
3.b odd 2 1 150.3.f.b 4
5.b even 2 1 90.3.g.d 4
5.c odd 4 1 90.3.g.d 4
5.c odd 4 1 inner 450.3.g.j 4
12.b even 2 1 1200.3.bg.d 4
15.d odd 2 1 30.3.f.a 4
15.e even 4 1 30.3.f.a 4
15.e even 4 1 150.3.f.b 4
20.d odd 2 1 720.3.bh.i 4
20.e even 4 1 720.3.bh.i 4
60.h even 2 1 240.3.bg.b 4
60.l odd 4 1 240.3.bg.b 4
60.l odd 4 1 1200.3.bg.d 4
120.i odd 2 1 960.3.bg.e 4
120.m even 2 1 960.3.bg.g 4
120.q odd 4 1 960.3.bg.g 4
120.w even 4 1 960.3.bg.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.f.a 4 15.d odd 2 1
30.3.f.a 4 15.e even 4 1
90.3.g.d 4 5.b even 2 1
90.3.g.d 4 5.c odd 4 1
150.3.f.b 4 3.b odd 2 1
150.3.f.b 4 15.e even 4 1
240.3.bg.b 4 60.h even 2 1
240.3.bg.b 4 60.l odd 4 1
450.3.g.j 4 1.a even 1 1 trivial
450.3.g.j 4 5.c odd 4 1 inner
720.3.bh.i 4 20.d odd 2 1
720.3.bh.i 4 20.e even 4 1
960.3.bg.e 4 120.i odd 2 1
960.3.bg.e 4 120.w even 4 1
960.3.bg.g 4 120.m even 2 1
960.3.bg.g 4 120.q odd 4 1
1200.3.bg.d 4 12.b even 2 1
1200.3.bg.d 4 60.l odd 4 1

## Hecke kernels

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

 $$T_{7}^{4} - 16T_{7}^{3} + 128T_{7}^{2} + 256T_{7} + 256$$ T7^4 - 16*T7^3 + 128*T7^2 + 256*T7 + 256 $$T_{11}^{2} - 8T_{11} - 80$$ T11^2 - 8*T11 - 80 $$T_{17}^{4} - 44T_{17}^{3} + 968T_{17}^{2} - 8536T_{17} + 37636$$ T17^4 - 44*T17^3 + 968*T17^2 - 8536*T17 + 37636

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 256$$
$11$ $$(T^{2} - 8 T - 80)^{2}$$
$13$ $$T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 30276$$
$17$ $$T^{4} - 44 T^{3} + 968 T^{2} + \cdots + 37636$$
$19$ $$T^{4} + 704 T^{2} + 25600$$
$23$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 160000$$
$29$ $$T^{4} + 704 T^{2} + 25600$$
$31$ $$(T^{2} + 40 T + 16)^{2}$$
$37$ $$(T^{2} - 54 T + 1458)^{2}$$
$41$ $$(T^{2} + 16 T - 32)^{2}$$
$43$ $$T^{4} + 48 T^{3} + 1152 T^{2} + \cdots + 831744$$
$47$ $$T^{4} - 96 T^{3} + 4608 T^{2} + \cdots + 518400$$
$53$ $$T^{4} - 100 T^{3} + 5000 T^{2} + \cdots + 6959044$$
$59$ $$(T^{2} + 400)^{2}$$
$61$ $$(T^{2} + 48 T - 960)^{2}$$
$67$ $$T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 14868736$$
$71$ $$(T^{2} + 32 T + 160)^{2}$$
$73$ $$T^{4} - 188 T^{3} + \cdots + 17859076$$
$79$ $$T^{4} + 32736 T^{2} + \cdots + 258566400$$
$83$ $$T^{4} - 192 T^{3} + 18432 T^{2} + \cdots + 5089536$$
$89$ $$T^{4} + 17216 T^{2} + \cdots + 47334400$$
$97$ $$T^{4} + 132 T^{3} + 8712 T^{2} + \cdots + 6874884$$