# Properties

 Label 45.3.d.a Level $45$ Weight $3$ Character orbit 45.d Analytic conductor $1.226$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,3,Mod(44,45)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, 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("45.44");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 45.d (of order $$2$$, degree $$1$$, 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: $$1.22616118962$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 8x^{2} + 9$$ x^4 + 8*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + 3 q^{4} + ( - \beta_{3} - \beta_1) q^{5} + \beta_{2} q^{7} - \beta_{3} q^{8}+O(q^{10})$$ q + b3 * q^2 + 3 * q^4 + (-b3 - b1) * q^5 + b2 * q^7 - b3 * q^8 $$q + \beta_{3} q^{2} + 3 q^{4} + ( - \beta_{3} - \beta_1) q^{5} + \beta_{2} q^{7} - \beta_{3} q^{8} + ( - \beta_{2} - 7) q^{10} - \beta_1 q^{11} - \beta_{2} q^{13} + 7 \beta_1 q^{14} - 19 q^{16} + 4 \beta_{3} q^{17} + 20 q^{19} + ( - 3 \beta_{3} - 3 \beta_1) q^{20} - \beta_{2} q^{22} - 2 \beta_{3} q^{23} + (2 \beta_{2} - 11) q^{25} - 7 \beta_1 q^{26} + 3 \beta_{2} q^{28} + 2 \beta_1 q^{29} + 26 q^{31} - 15 \beta_{3} q^{32} + 28 q^{34} + (18 \beta_{3} - 7 \beta_1) q^{35} - 3 \beta_{2} q^{37} + 20 \beta_{3} q^{38} + (\beta_{2} + 7) q^{40} + 13 \beta_1 q^{41} - 2 \beta_{2} q^{43} - 3 \beta_1 q^{44} - 14 q^{46} - 8 \beta_{3} q^{47} - 77 q^{49} + ( - 11 \beta_{3} + 14 \beta_1) q^{50} - 3 \beta_{2} q^{52} - 32 \beta_{3} q^{53} + (\beta_{2} - 18) q^{55} - 7 \beta_1 q^{56} + 2 \beta_{2} q^{58} - 11 \beta_1 q^{59} - 22 q^{61} + 26 \beta_{3} q^{62} - 29 q^{64} + ( - 18 \beta_{3} + 7 \beta_1) q^{65} + 8 \beta_{2} q^{67} + 12 \beta_{3} q^{68} + ( - 7 \beta_{2} + 126) q^{70} + 12 \beta_1 q^{71} - 6 \beta_{2} q^{73} - 21 \beta_1 q^{74} + 60 q^{76} + 18 \beta_{3} q^{77} + 14 q^{79} + (19 \beta_{3} + 19 \beta_1) q^{80} + 13 \beta_{2} q^{82} + 28 \beta_{3} q^{83} + ( - 4 \beta_{2} - 28) q^{85} - 14 \beta_1 q^{86} + \beta_{2} q^{88} - 21 \beta_1 q^{89} + 126 q^{91} - 6 \beta_{3} q^{92} - 56 q^{94} + ( - 20 \beta_{3} - 20 \beta_1) q^{95} - 2 \beta_{2} q^{97} - 77 \beta_{3} q^{98}+O(q^{100})$$ q + b3 * q^2 + 3 * q^4 + (-b3 - b1) * q^5 + b2 * q^7 - b3 * q^8 + (-b2 - 7) * q^10 - b1 * q^11 - b2 * q^13 + 7*b1 * q^14 - 19 * q^16 + 4*b3 * q^17 + 20 * q^19 + (-3*b3 - 3*b1) * q^20 - b2 * q^22 - 2*b3 * q^23 + (2*b2 - 11) * q^25 - 7*b1 * q^26 + 3*b2 * q^28 + 2*b1 * q^29 + 26 * q^31 - 15*b3 * q^32 + 28 * q^34 + (18*b3 - 7*b1) * q^35 - 3*b2 * q^37 + 20*b3 * q^38 + (b2 + 7) * q^40 + 13*b1 * q^41 - 2*b2 * q^43 - 3*b1 * q^44 - 14 * q^46 - 8*b3 * q^47 - 77 * q^49 + (-11*b3 + 14*b1) * q^50 - 3*b2 * q^52 - 32*b3 * q^53 + (b2 - 18) * q^55 - 7*b1 * q^56 + 2*b2 * q^58 - 11*b1 * q^59 - 22 * q^61 + 26*b3 * q^62 - 29 * q^64 + (-18*b3 + 7*b1) * q^65 + 8*b2 * q^67 + 12*b3 * q^68 + (-7*b2 + 126) * q^70 + 12*b1 * q^71 - 6*b2 * q^73 - 21*b1 * q^74 + 60 * q^76 + 18*b3 * q^77 + 14 * q^79 + (19*b3 + 19*b1) * q^80 + 13*b2 * q^82 + 28*b3 * q^83 + (-4*b2 - 28) * q^85 - 14*b1 * q^86 + b2 * q^88 - 21*b1 * q^89 + 126 * q^91 - 6*b3 * q^92 - 56 * q^94 + (-20*b3 - 20*b1) * q^95 - 2*b2 * q^97 - 77*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{4}+O(q^{10})$$ 4 * q + 12 * q^4 $$4 q + 12 q^{4} - 28 q^{10} - 76 q^{16} + 80 q^{19} - 44 q^{25} + 104 q^{31} + 112 q^{34} + 28 q^{40} - 56 q^{46} - 308 q^{49} - 72 q^{55} - 88 q^{61} - 116 q^{64} + 504 q^{70} + 240 q^{76} + 56 q^{79} - 112 q^{85} + 504 q^{91} - 224 q^{94}+O(q^{100})$$ 4 * q + 12 * q^4 - 28 * q^10 - 76 * q^16 + 80 * q^19 - 44 * q^25 + 104 * q^31 + 112 * q^34 + 28 * q^40 - 56 * q^46 - 308 * q^49 - 72 * q^55 - 88 * q^61 - 116 * q^64 + 504 * q^70 + 240 * q^76 + 56 * q^79 - 112 * q^85 + 504 * q^91 - 224 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 5\nu$$ v^3 + 5*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 11\nu$$ v^3 + 11*v $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 6$$ (b2 - b1) / 6 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ b3 - 4 $$\nu^{3}$$ $$=$$ $$( -5\beta_{2} + 11\beta_1 ) / 6$$ (-5*b2 + 11*b1) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
44.1
 − 2.57794i 2.57794i 1.16372i − 1.16372i
−2.64575 0 3.00000 2.64575 4.24264i 0 11.2250i 2.64575 0 −7.00000 + 11.2250i
44.2 −2.64575 0 3.00000 2.64575 + 4.24264i 0 11.2250i 2.64575 0 −7.00000 11.2250i
44.3 2.64575 0 3.00000 −2.64575 4.24264i 0 11.2250i −2.64575 0 −7.00000 11.2250i
44.4 2.64575 0 3.00000 −2.64575 + 4.24264i 0 11.2250i −2.64575 0 −7.00000 + 11.2250i
 $$n$$: 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 45.3.d.a 4
3.b odd 2 1 inner 45.3.d.a 4
4.b odd 2 1 720.3.c.a 4
5.b even 2 1 inner 45.3.d.a 4
5.c odd 4 2 225.3.c.d 4
8.b even 2 1 2880.3.c.b 4
8.d odd 2 1 2880.3.c.g 4
9.c even 3 2 405.3.h.j 8
9.d odd 6 2 405.3.h.j 8
12.b even 2 1 720.3.c.a 4
15.d odd 2 1 inner 45.3.d.a 4
15.e even 4 2 225.3.c.d 4
20.d odd 2 1 720.3.c.a 4
20.e even 4 2 3600.3.l.s 4
24.f even 2 1 2880.3.c.g 4
24.h odd 2 1 2880.3.c.b 4
40.e odd 2 1 2880.3.c.g 4
40.f even 2 1 2880.3.c.b 4
45.h odd 6 2 405.3.h.j 8
45.j even 6 2 405.3.h.j 8
60.h even 2 1 720.3.c.a 4
60.l odd 4 2 3600.3.l.s 4
120.i odd 2 1 2880.3.c.b 4
120.m even 2 1 2880.3.c.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.d.a 4 1.a even 1 1 trivial
45.3.d.a 4 3.b odd 2 1 inner
45.3.d.a 4 5.b even 2 1 inner
45.3.d.a 4 15.d odd 2 1 inner
225.3.c.d 4 5.c odd 4 2
225.3.c.d 4 15.e even 4 2
405.3.h.j 8 9.c even 3 2
405.3.h.j 8 9.d odd 6 2
405.3.h.j 8 45.h odd 6 2
405.3.h.j 8 45.j even 6 2
720.3.c.a 4 4.b odd 2 1
720.3.c.a 4 12.b even 2 1
720.3.c.a 4 20.d odd 2 1
720.3.c.a 4 60.h even 2 1
2880.3.c.b 4 8.b even 2 1
2880.3.c.b 4 24.h odd 2 1
2880.3.c.b 4 40.f even 2 1
2880.3.c.b 4 120.i odd 2 1
2880.3.c.g 4 8.d odd 2 1
2880.3.c.g 4 24.f even 2 1
2880.3.c.g 4 40.e odd 2 1
2880.3.c.g 4 120.m even 2 1
3600.3.l.s 4 20.e even 4 2
3600.3.l.s 4 60.l odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 7)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 22T^{2} + 625$$
$7$ $$(T^{2} + 126)^{2}$$
$11$ $$(T^{2} + 18)^{2}$$
$13$ $$(T^{2} + 126)^{2}$$
$17$ $$(T^{2} - 112)^{2}$$
$19$ $$(T - 20)^{4}$$
$23$ $$(T^{2} - 28)^{2}$$
$29$ $$(T^{2} + 72)^{2}$$
$31$ $$(T - 26)^{4}$$
$37$ $$(T^{2} + 1134)^{2}$$
$41$ $$(T^{2} + 3042)^{2}$$
$43$ $$(T^{2} + 504)^{2}$$
$47$ $$(T^{2} - 448)^{2}$$
$53$ $$(T^{2} - 7168)^{2}$$
$59$ $$(T^{2} + 2178)^{2}$$
$61$ $$(T + 22)^{4}$$
$67$ $$(T^{2} + 8064)^{2}$$
$71$ $$(T^{2} + 2592)^{2}$$
$73$ $$(T^{2} + 4536)^{2}$$
$79$ $$(T - 14)^{4}$$
$83$ $$(T^{2} - 5488)^{2}$$
$89$ $$(T^{2} + 7938)^{2}$$
$97$ $$(T^{2} + 504)^{2}$$