# Properties

 Label 450.3.g.i Level $450$ Weight $3$ Character orbit 450.g 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(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: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} + 2 \beta_{2} q^{4} + \beta_1 q^{7} + (2 \beta_{2} - 2) q^{8}+O(q^{10})$$ q + (b2 + 1) * q^2 + 2*b2 * q^4 + b1 * q^7 + (2*b2 - 2) * q^8 $$q + (\beta_{2} + 1) q^{2} + 2 \beta_{2} q^{4} + \beta_1 q^{7} + (2 \beta_{2} - 2) q^{8} + ( - 6 \beta_{3} + 6 \beta_1) q^{11} - 9 \beta_{3} q^{13} + (\beta_{3} + \beta_1) q^{14} - 4 q^{16} + (6 \beta_{2} + 6) q^{17} + 17 \beta_{2} q^{19} + 12 \beta_1 q^{22} + (30 \beta_{2} - 30) q^{23} + ( - 9 \beta_{3} + 9 \beta_1) q^{26} + 2 \beta_{3} q^{28} + (18 \beta_{3} + 18 \beta_1) q^{29} + 31 q^{31} + ( - 4 \beta_{2} - 4) q^{32} + 12 \beta_{2} q^{34} - 16 \beta_1 q^{37} + (17 \beta_{2} - 17) q^{38} + ( - 30 \beta_{3} + 30 \beta_1) q^{41} - 17 \beta_{3} q^{43} + (12 \beta_{3} + 12 \beta_1) q^{44} - 60 q^{46} - 46 \beta_{2} q^{49} + 18 \beta_1 q^{52} + (42 \beta_{2} - 42) q^{53} + (2 \beta_{3} - 2 \beta_1) q^{56} + 36 \beta_{3} q^{58} + (12 \beta_{3} + 12 \beta_1) q^{59} + 47 q^{61} + (31 \beta_{2} + 31) q^{62} - 8 \beta_{2} q^{64} - 57 \beta_1 q^{67} + (12 \beta_{2} - 12) q^{68} + ( - 18 \beta_{3} + 18 \beta_1) q^{71} + 40 \beta_{3} q^{73} + ( - 16 \beta_{3} - 16 \beta_1) q^{74} - 34 q^{76} + (18 \beta_{2} + 18) q^{77} - 38 \beta_{2} q^{79} + 60 \beta_1 q^{82} + ( - 24 \beta_{2} + 24) q^{83} + ( - 17 \beta_{3} + 17 \beta_1) q^{86} + 24 \beta_{3} q^{88} + ( - 36 \beta_{3} - 36 \beta_1) q^{89} + 27 q^{91} + ( - 60 \beta_{2} - 60) q^{92} - 79 \beta_1 q^{97} + ( - 46 \beta_{2} + 46) q^{98}+O(q^{100})$$ q + (b2 + 1) * q^2 + 2*b2 * q^4 + b1 * q^7 + (2*b2 - 2) * q^8 + (-6*b3 + 6*b1) * q^11 - 9*b3 * q^13 + (b3 + b1) * q^14 - 4 * q^16 + (6*b2 + 6) * q^17 + 17*b2 * q^19 + 12*b1 * q^22 + (30*b2 - 30) * q^23 + (-9*b3 + 9*b1) * q^26 + 2*b3 * q^28 + (18*b3 + 18*b1) * q^29 + 31 * q^31 + (-4*b2 - 4) * q^32 + 12*b2 * q^34 - 16*b1 * q^37 + (17*b2 - 17) * q^38 + (-30*b3 + 30*b1) * q^41 - 17*b3 * q^43 + (12*b3 + 12*b1) * q^44 - 60 * q^46 - 46*b2 * q^49 + 18*b1 * q^52 + (42*b2 - 42) * q^53 + (2*b3 - 2*b1) * q^56 + 36*b3 * q^58 + (12*b3 + 12*b1) * q^59 + 47 * q^61 + (31*b2 + 31) * q^62 - 8*b2 * q^64 - 57*b1 * q^67 + (12*b2 - 12) * q^68 + (-18*b3 + 18*b1) * q^71 + 40*b3 * q^73 + (-16*b3 - 16*b1) * q^74 - 34 * q^76 + (18*b2 + 18) * q^77 - 38*b2 * q^79 + 60*b1 * q^82 + (-24*b2 + 24) * q^83 + (-17*b3 + 17*b1) * q^86 + 24*b3 * q^88 + (-36*b3 - 36*b1) * q^89 + 27 * q^91 + (-60*b2 - 60) * q^92 - 79*b1 * q^97 + (-46*b2 + 46) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 8 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 - 8 * q^8 $$4 q + 4 q^{2} - 8 q^{8} - 16 q^{16} + 24 q^{17} - 120 q^{23} + 124 q^{31} - 16 q^{32} - 68 q^{38} - 240 q^{46} - 168 q^{53} + 188 q^{61} + 124 q^{62} - 48 q^{68} - 136 q^{76} + 72 q^{77} + 96 q^{83} + 108 q^{91} - 240 q^{92} + 184 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 - 8 * q^8 - 16 * q^16 + 24 * q^17 - 120 * q^23 + 124 * q^31 - 16 * q^32 - 68 * q^38 - 240 * q^46 - 168 * q^53 + 188 * q^61 + 124 * q^62 - 48 * q^68 - 136 * q^76 + 72 * q^77 + 96 * q^83 + 108 * q^91 - 240 * q^92 + 184 * q^98

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

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

## 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 −1.22474 1.22474i −2.00000 + 2.00000i 0 0
307.2 1.00000 + 1.00000i 0 2.00000i 0 0 1.22474 + 1.22474i −2.00000 + 2.00000i 0 0
343.1 1.00000 1.00000i 0 2.00000i 0 0 −1.22474 + 1.22474i −2.00000 2.00000i 0 0
343.2 1.00000 1.00000i 0 2.00000i 0 0 1.22474 1.22474i −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
15.d odd 2 1 inner
15.e even 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.i yes 4
3.b odd 2 1 450.3.g.f 4
5.b even 2 1 450.3.g.f 4
5.c odd 4 1 450.3.g.f 4
5.c odd 4 1 inner 450.3.g.i yes 4
15.d odd 2 1 inner 450.3.g.i yes 4
15.e even 4 1 450.3.g.f 4
15.e even 4 1 inner 450.3.g.i yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.3.g.f 4 3.b odd 2 1
450.3.g.f 4 5.b even 2 1
450.3.g.f 4 5.c odd 4 1
450.3.g.f 4 15.e even 4 1
450.3.g.i yes 4 1.a even 1 1 trivial
450.3.g.i yes 4 5.c odd 4 1 inner
450.3.g.i yes 4 15.d odd 2 1 inner
450.3.g.i yes 4 15.e even 4 1 inner

## Hecke kernels

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

 $$T_{7}^{4} + 9$$ T7^4 + 9 $$T_{11}^{2} - 216$$ T11^2 - 216 $$T_{17}^{2} - 12T_{17} + 72$$ T17^2 - 12*T17 + 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 9$$
$11$ $$(T^{2} - 216)^{2}$$
$13$ $$T^{4} + 59049$$
$17$ $$(T^{2} - 12 T + 72)^{2}$$
$19$ $$(T^{2} + 289)^{2}$$
$23$ $$(T^{2} + 60 T + 1800)^{2}$$
$29$ $$(T^{2} + 1944)^{2}$$
$31$ $$(T - 31)^{4}$$
$37$ $$T^{4} + 589824$$
$41$ $$(T^{2} - 5400)^{2}$$
$43$ $$T^{4} + 751689$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 84 T + 3528)^{2}$$
$59$ $$(T^{2} + 864)^{2}$$
$61$ $$(T - 47)^{4}$$
$67$ $$T^{4} + 95004009$$
$71$ $$(T^{2} - 1944)^{2}$$
$73$ $$T^{4} + 23040000$$
$79$ $$(T^{2} + 1444)^{2}$$
$83$ $$(T^{2} - 48 T + 1152)^{2}$$
$89$ $$(T^{2} + 7776)^{2}$$
$97$ $$T^{4} + 350550729$$