# Properties

 Label 450.3.b.b Level $450$ Weight $3$ Character orbit 450.b Analytic conductor $12.262$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,3,Mod(449,450)]

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, 1]))

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

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

chi := DirichletCharacter("450.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 450.b (of order $$2$$, degree $$1$$, not 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + 2 q^{4} + 2 \beta_1 q^{7} + 2 \beta_{3} q^{8}+O(q^{10})$$ q + b3 * q^2 + 2 * q^4 + 2*b1 * q^7 + 2*b3 * q^8 $$q + \beta_{3} q^{2} + 2 q^{4} + 2 \beta_1 q^{7} + 2 \beta_{3} q^{8} + 12 \beta_{2} q^{11} + 4 \beta_1 q^{13} + 4 \beta_{2} q^{14} + 4 q^{16} - 9 \beta_{3} q^{17} + 16 q^{19} + 12 \beta_1 q^{22} + 12 \beta_{3} q^{23} + 8 \beta_{2} q^{26} + 4 \beta_1 q^{28} - 3 \beta_{2} q^{29} + 44 q^{31} + 4 \beta_{3} q^{32} - 18 q^{34} + 17 \beta_1 q^{37} + 16 \beta_{3} q^{38} + 33 \beta_{2} q^{41} - 20 \beta_1 q^{43} + 24 \beta_{2} q^{44} + 24 q^{46} - 60 \beta_{3} q^{47} + 33 q^{49} + 8 \beta_1 q^{52} - 27 \beta_{3} q^{53} + 8 \beta_{2} q^{56} - 3 \beta_1 q^{58} - 24 \beta_{2} q^{59} + 50 q^{61} + 44 \beta_{3} q^{62} + 8 q^{64} - 4 \beta_1 q^{67} - 18 \beta_{3} q^{68} - 36 \beta_{2} q^{71} - 8 \beta_1 q^{73} + 34 \beta_{2} q^{74} + 32 q^{76} - 48 \beta_{3} q^{77} + 76 q^{79} + 33 \beta_1 q^{82} - 84 \beta_{3} q^{83} - 40 \beta_{2} q^{86} + 24 \beta_1 q^{88} - 9 \beta_{2} q^{89} - 32 q^{91} + 24 \beta_{3} q^{92} - 120 q^{94} - 88 \beta_1 q^{97} + 33 \beta_{3} q^{98}+O(q^{100})$$ q + b3 * q^2 + 2 * q^4 + 2*b1 * q^7 + 2*b3 * q^8 + 12*b2 * q^11 + 4*b1 * q^13 + 4*b2 * q^14 + 4 * q^16 - 9*b3 * q^17 + 16 * q^19 + 12*b1 * q^22 + 12*b3 * q^23 + 8*b2 * q^26 + 4*b1 * q^28 - 3*b2 * q^29 + 44 * q^31 + 4*b3 * q^32 - 18 * q^34 + 17*b1 * q^37 + 16*b3 * q^38 + 33*b2 * q^41 - 20*b1 * q^43 + 24*b2 * q^44 + 24 * q^46 - 60*b3 * q^47 + 33 * q^49 + 8*b1 * q^52 - 27*b3 * q^53 + 8*b2 * q^56 - 3*b1 * q^58 - 24*b2 * q^59 + 50 * q^61 + 44*b3 * q^62 + 8 * q^64 - 4*b1 * q^67 - 18*b3 * q^68 - 36*b2 * q^71 - 8*b1 * q^73 + 34*b2 * q^74 + 32 * q^76 - 48*b3 * q^77 + 76 * q^79 + 33*b1 * q^82 - 84*b3 * q^83 - 40*b2 * q^86 + 24*b1 * q^88 - 9*b2 * q^89 - 32 * q^91 + 24*b3 * q^92 - 120 * q^94 - 88*b1 * q^97 + 33*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4}+O(q^{10})$$ 4 * q + 8 * q^4 $$4 q + 8 q^{4} + 16 q^{16} + 64 q^{19} + 176 q^{31} - 72 q^{34} + 96 q^{46} + 132 q^{49} + 200 q^{61} + 32 q^{64} + 128 q^{76} + 304 q^{79} - 128 q^{91} - 480 q^{94}+O(q^{100})$$ 4 * q + 8 * q^4 + 16 * q^16 + 64 * q^19 + 176 * q^31 - 72 * q^34 + 96 * q^46 + 132 * q^49 + 200 * q^61 + 32 * q^64 + 128 * q^76 + 304 * q^79 - 128 * q^91 - 480 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}^{2}$$ 2*v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
−1.41421 0 2.00000 0 0 4.00000i −2.82843 0 0
449.2 −1.41421 0 2.00000 0 0 4.00000i −2.82843 0 0
449.3 1.41421 0 2.00000 0 0 4.00000i 2.82843 0 0
449.4 1.41421 0 2.00000 0 0 4.00000i 2.82843 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
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.b.b 4
3.b odd 2 1 inner 450.3.b.b 4
4.b odd 2 1 3600.3.c.b 4
5.b even 2 1 inner 450.3.b.b 4
5.c odd 4 1 18.3.b.a 2
5.c odd 4 1 450.3.d.f 2
12.b even 2 1 3600.3.c.b 4
15.d odd 2 1 inner 450.3.b.b 4
15.e even 4 1 18.3.b.a 2
15.e even 4 1 450.3.d.f 2
20.d odd 2 1 3600.3.c.b 4
20.e even 4 1 144.3.e.b 2
20.e even 4 1 3600.3.l.d 2
35.f even 4 1 882.3.b.a 2
35.k even 12 2 882.3.s.d 4
35.l odd 12 2 882.3.s.b 4
40.i odd 4 1 576.3.e.c 2
40.k even 4 1 576.3.e.f 2
45.k odd 12 2 162.3.d.b 4
45.l even 12 2 162.3.d.b 4
55.e even 4 1 2178.3.c.d 2
60.h even 2 1 3600.3.c.b 4
60.l odd 4 1 144.3.e.b 2
60.l odd 4 1 3600.3.l.d 2
65.f even 4 1 3042.3.d.a 4
65.h odd 4 1 3042.3.c.e 2
65.k even 4 1 3042.3.d.a 4
80.i odd 4 1 2304.3.h.f 4
80.j even 4 1 2304.3.h.c 4
80.s even 4 1 2304.3.h.c 4
80.t odd 4 1 2304.3.h.f 4
105.k odd 4 1 882.3.b.a 2
105.w odd 12 2 882.3.s.d 4
105.x even 12 2 882.3.s.b 4
120.q odd 4 1 576.3.e.f 2
120.w even 4 1 576.3.e.c 2
165.l odd 4 1 2178.3.c.d 2
180.v odd 12 2 1296.3.q.f 4
180.x even 12 2 1296.3.q.f 4
195.j odd 4 1 3042.3.d.a 4
195.s even 4 1 3042.3.c.e 2
195.u odd 4 1 3042.3.d.a 4
240.z odd 4 1 2304.3.h.c 4
240.bb even 4 1 2304.3.h.f 4
240.bd odd 4 1 2304.3.h.c 4
240.bf even 4 1 2304.3.h.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 5.c odd 4 1
18.3.b.a 2 15.e even 4 1
144.3.e.b 2 20.e even 4 1
144.3.e.b 2 60.l odd 4 1
162.3.d.b 4 45.k odd 12 2
162.3.d.b 4 45.l even 12 2
450.3.b.b 4 1.a even 1 1 trivial
450.3.b.b 4 3.b odd 2 1 inner
450.3.b.b 4 5.b even 2 1 inner
450.3.b.b 4 15.d odd 2 1 inner
450.3.d.f 2 5.c odd 4 1
450.3.d.f 2 15.e even 4 1
576.3.e.c 2 40.i odd 4 1
576.3.e.c 2 120.w even 4 1
576.3.e.f 2 40.k even 4 1
576.3.e.f 2 120.q odd 4 1
882.3.b.a 2 35.f even 4 1
882.3.b.a 2 105.k odd 4 1
882.3.s.b 4 35.l odd 12 2
882.3.s.b 4 105.x even 12 2
882.3.s.d 4 35.k even 12 2
882.3.s.d 4 105.w odd 12 2
1296.3.q.f 4 180.v odd 12 2
1296.3.q.f 4 180.x even 12 2
2178.3.c.d 2 55.e even 4 1
2178.3.c.d 2 165.l odd 4 1
2304.3.h.c 4 80.j even 4 1
2304.3.h.c 4 80.s even 4 1
2304.3.h.c 4 240.z odd 4 1
2304.3.h.c 4 240.bd odd 4 1
2304.3.h.f 4 80.i odd 4 1
2304.3.h.f 4 80.t odd 4 1
2304.3.h.f 4 240.bb even 4 1
2304.3.h.f 4 240.bf even 4 1
3042.3.c.e 2 65.h odd 4 1
3042.3.c.e 2 195.s even 4 1
3042.3.d.a 4 65.f even 4 1
3042.3.d.a 4 65.k even 4 1
3042.3.d.a 4 195.j odd 4 1
3042.3.d.a 4 195.u odd 4 1
3600.3.c.b 4 4.b odd 2 1
3600.3.c.b 4 12.b even 2 1
3600.3.c.b 4 20.d odd 2 1
3600.3.c.b 4 60.h even 2 1
3600.3.l.d 2 20.e even 4 1
3600.3.l.d 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 16$$ acting on $$S_{3}^{\mathrm{new}}(450, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 16)^{2}$$
$11$ $$(T^{2} + 288)^{2}$$
$13$ $$(T^{2} + 64)^{2}$$
$17$ $$(T^{2} - 162)^{2}$$
$19$ $$(T - 16)^{4}$$
$23$ $$(T^{2} - 288)^{2}$$
$29$ $$(T^{2} + 18)^{2}$$
$31$ $$(T - 44)^{4}$$
$37$ $$(T^{2} + 1156)^{2}$$
$41$ $$(T^{2} + 2178)^{2}$$
$43$ $$(T^{2} + 1600)^{2}$$
$47$ $$(T^{2} - 7200)^{2}$$
$53$ $$(T^{2} - 1458)^{2}$$
$59$ $$(T^{2} + 1152)^{2}$$
$61$ $$(T - 50)^{4}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} + 2592)^{2}$$
$73$ $$(T^{2} + 256)^{2}$$
$79$ $$(T - 76)^{4}$$
$83$ $$(T^{2} - 14112)^{2}$$
$89$ $$(T^{2} + 162)^{2}$$
$97$ $$(T^{2} + 30976)^{2}$$