# Properties

 Label 450.3.g.h 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: $$1$$ Twist minimal: no (minimal twist has level 150) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 1) q^{2} - 2 \beta_{2} q^{4} + ( - \beta_{3} + 6 \beta_{2} - 6) q^{7} + ( - 2 \beta_{2} - 2) q^{8}+O(q^{10})$$ q + (-b2 + 1) * q^2 - 2*b2 * q^4 + (-b3 + 6*b2 - 6) * q^7 + (-2*b2 - 2) * q^8 $$q + ( - \beta_{2} + 1) q^{2} - 2 \beta_{2} q^{4} + ( - \beta_{3} + 6 \beta_{2} - 6) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + (6 \beta_{3} - 6 \beta_1 - 6) q^{11} + (12 \beta_{2} - 3 \beta_1 + 12) q^{13} + ( - \beta_{3} + 12 \beta_{2} - \beta_1) q^{14} - 4 q^{16} + (6 \beta_{3} - 6 \beta_{2} + 6) q^{17} + (6 \beta_{3} + 19 \beta_{2} + 6 \beta_1) q^{19} + (12 \beta_{3} + 6 \beta_{2} - 6) q^{22} + (6 \beta_{2} - 18 \beta_1 + 6) q^{23} + (3 \beta_{3} - 3 \beta_1 + 24) q^{26} + (12 \beta_{2} - 2 \beta_1 + 12) q^{28} + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{29} + (18 \beta_{3} - 18 \beta_1 - 5) q^{31} + (4 \beta_{2} - 4) q^{32} + (6 \beta_{3} - 12 \beta_{2} + 6 \beta_1) q^{34} + ( - 20 \beta_{3} - 12 \beta_{2} + 12) q^{37} + (19 \beta_{2} + 12 \beta_1 + 19) q^{38} + ( - 6 \beta_{3} + 6 \beta_1 - 48) q^{41} + (18 \beta_{2} + 5 \beta_1 + 18) q^{43} + (12 \beta_{3} + 12 \beta_{2} + 12 \beta_1) q^{44} + (18 \beta_{3} - 18 \beta_1 + 12) q^{46} + ( - 18 \beta_{3} + 36 \beta_{2} - 36) q^{47} + (12 \beta_{3} - 26 \beta_{2} + 12 \beta_1) q^{49} + (6 \beta_{3} - 24 \beta_{2} + 24) q^{52} + (30 \beta_{2} + 24 \beta_1 + 30) q^{53} + (2 \beta_{3} - 2 \beta_1 + 24) q^{56} + ( - 6 \beta_{2} + 12 \beta_1 - 6) q^{58} - 30 \beta_{2} q^{59} + (24 \beta_{3} - 24 \beta_1 + 11) q^{61} + (36 \beta_{3} + 5 \beta_{2} - 5) q^{62} + 8 \beta_{2} q^{64} + (9 \beta_{3} + 6 \beta_{2} - 6) q^{67} + ( - 12 \beta_{2} + 12 \beta_1 - 12) q^{68} + (6 \beta_{3} - 6 \beta_1 + 24) q^{71} + (12 \beta_{2} - 28 \beta_1 + 12) q^{73} + ( - 20 \beta_{3} - 24 \beta_{2} - 20 \beta_1) q^{74} + ( - 12 \beta_{3} + 12 \beta_1 + 38) q^{76} + ( - 66 \beta_{3} - 18 \beta_{2} + 18) q^{77} + ( - 12 \beta_{3} + 2 \beta_{2} - 12 \beta_1) q^{79} + ( - 12 \beta_{3} + 48 \beta_{2} - 48) q^{82} + ( - 12 \beta_{2} - 42 \beta_1 - 12) q^{83} + ( - 5 \beta_{3} + 5 \beta_1 + 36) q^{86} + (12 \beta_{2} + 24 \beta_1 + 12) q^{88} + (12 \beta_{3} - 12 \beta_{2} + 12 \beta_1) q^{89} + ( - 30 \beta_{3} + 30 \beta_1 - 153) q^{91} + (36 \beta_{3} - 12 \beta_{2} + 12) q^{92} + ( - 18 \beta_{3} + 72 \beta_{2} - 18 \beta_1) q^{94} + ( - 5 \beta_{3} + 48 \beta_{2} - 48) q^{97} + ( - 26 \beta_{2} + 24 \beta_1 - 26) q^{98}+O(q^{100})$$ q + (-b2 + 1) * q^2 - 2*b2 * q^4 + (-b3 + 6*b2 - 6) * q^7 + (-2*b2 - 2) * q^8 + (6*b3 - 6*b1 - 6) * q^11 + (12*b2 - 3*b1 + 12) * q^13 + (-b3 + 12*b2 - b1) * q^14 - 4 * q^16 + (6*b3 - 6*b2 + 6) * q^17 + (6*b3 + 19*b2 + 6*b1) * q^19 + (12*b3 + 6*b2 - 6) * q^22 + (6*b2 - 18*b1 + 6) * q^23 + (3*b3 - 3*b1 + 24) * q^26 + (12*b2 - 2*b1 + 12) * q^28 + (6*b3 - 6*b2 + 6*b1) * q^29 + (18*b3 - 18*b1 - 5) * q^31 + (4*b2 - 4) * q^32 + (6*b3 - 12*b2 + 6*b1) * q^34 + (-20*b3 - 12*b2 + 12) * q^37 + (19*b2 + 12*b1 + 19) * q^38 + (-6*b3 + 6*b1 - 48) * q^41 + (18*b2 + 5*b1 + 18) * q^43 + (12*b3 + 12*b2 + 12*b1) * q^44 + (18*b3 - 18*b1 + 12) * q^46 + (-18*b3 + 36*b2 - 36) * q^47 + (12*b3 - 26*b2 + 12*b1) * q^49 + (6*b3 - 24*b2 + 24) * q^52 + (30*b2 + 24*b1 + 30) * q^53 + (2*b3 - 2*b1 + 24) * q^56 + (-6*b2 + 12*b1 - 6) * q^58 - 30*b2 * q^59 + (24*b3 - 24*b1 + 11) * q^61 + (36*b3 + 5*b2 - 5) * q^62 + 8*b2 * q^64 + (9*b3 + 6*b2 - 6) * q^67 + (-12*b2 + 12*b1 - 12) * q^68 + (6*b3 - 6*b1 + 24) * q^71 + (12*b2 - 28*b1 + 12) * q^73 + (-20*b3 - 24*b2 - 20*b1) * q^74 + (-12*b3 + 12*b1 + 38) * q^76 + (-66*b3 - 18*b2 + 18) * q^77 + (-12*b3 + 2*b2 - 12*b1) * q^79 + (-12*b3 + 48*b2 - 48) * q^82 + (-12*b2 - 42*b1 - 12) * q^83 + (-5*b3 + 5*b1 + 36) * q^86 + (12*b2 + 24*b1 + 12) * q^88 + (12*b3 - 12*b2 + 12*b1) * q^89 + (-30*b3 + 30*b1 - 153) * q^91 + (36*b3 - 12*b2 + 12) * q^92 + (-18*b3 + 72*b2 - 18*b1) * q^94 + (-5*b3 + 48*b2 - 48) * q^97 + (-26*b2 + 24*b1 - 26) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 24 q^{7} - 8 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 - 24 * q^7 - 8 * q^8 $$4 q + 4 q^{2} - 24 q^{7} - 8 q^{8} - 24 q^{11} + 48 q^{13} - 16 q^{16} + 24 q^{17} - 24 q^{22} + 24 q^{23} + 96 q^{26} + 48 q^{28} - 20 q^{31} - 16 q^{32} + 48 q^{37} + 76 q^{38} - 192 q^{41} + 72 q^{43} + 48 q^{46} - 144 q^{47} + 96 q^{52} + 120 q^{53} + 96 q^{56} - 24 q^{58} + 44 q^{61} - 20 q^{62} - 24 q^{67} - 48 q^{68} + 96 q^{71} + 48 q^{73} + 152 q^{76} + 72 q^{77} - 192 q^{82} - 48 q^{83} + 144 q^{86} + 48 q^{88} - 612 q^{91} + 48 q^{92} - 192 q^{97} - 104 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 - 24 * q^7 - 8 * q^8 - 24 * q^11 + 48 * q^13 - 16 * q^16 + 24 * q^17 - 24 * q^22 + 24 * q^23 + 96 * q^26 + 48 * q^28 - 20 * q^31 - 16 * q^32 + 48 * q^37 + 76 * q^38 - 192 * q^41 + 72 * q^43 + 48 * q^46 - 144 * q^47 + 96 * q^52 + 120 * q^53 + 96 * q^56 - 24 * q^58 + 44 * q^61 - 20 * q^62 - 24 * q^67 - 48 * q^68 + 96 * q^71 + 48 * q^73 + 152 * q^76 + 72 * q^77 - 192 * q^82 - 48 * q^83 + 144 * q^86 + 48 * q^88 - 612 * q^91 + 48 * q^92 - 192 * q^97 - 104 * 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 −7.22474 7.22474i −2.00000 + 2.00000i 0 0
307.2 1.00000 + 1.00000i 0 2.00000i 0 0 −4.77526 4.77526i −2.00000 + 2.00000i 0 0
343.1 1.00000 1.00000i 0 2.00000i 0 0 −7.22474 + 7.22474i −2.00000 2.00000i 0 0
343.2 1.00000 1.00000i 0 2.00000i 0 0 −4.77526 + 4.77526i −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.h 4
3.b odd 2 1 150.3.f.a 4
5.b even 2 1 450.3.g.g 4
5.c odd 4 1 450.3.g.g 4
5.c odd 4 1 inner 450.3.g.h 4
12.b even 2 1 1200.3.bg.p 4
15.d odd 2 1 150.3.f.c yes 4
15.e even 4 1 150.3.f.a 4
15.e even 4 1 150.3.f.c yes 4
60.h even 2 1 1200.3.bg.a 4
60.l odd 4 1 1200.3.bg.a 4
60.l odd 4 1 1200.3.bg.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.3.f.a 4 3.b odd 2 1
150.3.f.a 4 15.e even 4 1
150.3.f.c yes 4 15.d odd 2 1
150.3.f.c yes 4 15.e even 4 1
450.3.g.g 4 5.b even 2 1
450.3.g.g 4 5.c odd 4 1
450.3.g.h 4 1.a even 1 1 trivial
450.3.g.h 4 5.c odd 4 1 inner
1200.3.bg.a 4 60.h even 2 1
1200.3.bg.a 4 60.l odd 4 1
1200.3.bg.p 4 12.b even 2 1
1200.3.bg.p 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 $$S_{3}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{4} + 24T_{7}^{3} + 288T_{7}^{2} + 1656T_{7} + 4761$$ T7^4 + 24*T7^3 + 288*T7^2 + 1656*T7 + 4761 $$T_{11}^{2} + 12T_{11} - 180$$ T11^2 + 12*T11 - 180 $$T_{17}^{4} - 24T_{17}^{3} + 288T_{17}^{2} + 864T_{17} + 1296$$ T17^4 - 24*T17^3 + 288*T17^2 + 864*T17 + 1296

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 24 T^{3} + 288 T^{2} + \cdots + 4761$$
$11$ $$(T^{2} + 12 T - 180)^{2}$$
$13$ $$T^{4} - 48 T^{3} + 1152 T^{2} + \cdots + 68121$$
$17$ $$T^{4} - 24 T^{3} + 288 T^{2} + \cdots + 1296$$
$19$ $$T^{4} + 1154 T^{2} + 21025$$
$23$ $$T^{4} - 24 T^{3} + 288 T^{2} + \cdots + 810000$$
$29$ $$T^{4} + 504 T^{2} + 32400$$
$31$ $$(T^{2} + 10 T - 1919)^{2}$$
$37$ $$T^{4} - 48 T^{3} + 1152 T^{2} + \cdots + 831744$$
$41$ $$(T^{2} + 96 T + 2088)^{2}$$
$43$ $$T^{4} - 72 T^{3} + 2592 T^{2} + \cdots + 328329$$
$47$ $$T^{4} + 144 T^{3} + 10368 T^{2} + \cdots + 2624400$$
$53$ $$T^{4} - 120 T^{3} + 7200 T^{2} + \cdots + 5184$$
$59$ $$(T^{2} + 900)^{2}$$
$61$ $$(T^{2} - 22 T - 3335)^{2}$$
$67$ $$T^{4} + 24 T^{3} + 288 T^{2} + \cdots + 29241$$
$71$ $$(T^{2} - 48 T + 360)^{2}$$
$73$ $$T^{4} - 48 T^{3} + 1152 T^{2} + \cdots + 4260096$$
$79$ $$T^{4} + 1736 T^{2} + 739600$$
$83$ $$T^{4} + 48 T^{3} + 1152 T^{2} + \cdots + 25040016$$
$89$ $$T^{4} + 2016 T^{2} + 518400$$
$97$ $$T^{4} + 192 T^{3} + \cdots + 20548089$$