# Properties

 Label 45.3.g.b Level $45$ Weight $3$ Character orbit 45.g Analytic conductor $1.226$ 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] = [45,3,Mod(28,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("45.28");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 45.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: $$1.22616118962$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{2} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{5} + (\beta_{2} - 2 \beta_1 + 1) q^{7} + (\beta_{3} + 3 \beta_{2} - 3) q^{8}+O(q^{10})$$ q + (b2 + b1 + 1) * q^2 + (2*b3 + b2 + 2*b1) * q^4 + (b3 - 3*b2 - 2*b1 + 1) * q^5 + (b2 - 2*b1 + 1) * q^7 + (b3 + 3*b2 - 3) * q^8 $$q + (\beta_{2} + \beta_1 + 1) q^{2} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{5} + (\beta_{2} - 2 \beta_1 + 1) q^{7} + (\beta_{3} + 3 \beta_{2} - 3) q^{8} + ( - 4 \beta_{3} - 8 \beta_{2} - 2 \beta_1 + 1) q^{10} + ( - 3 \beta_{3} + 3 \beta_1 - 4) q^{11} + (2 \beta_{3} + 8 \beta_{2} - 8) q^{13} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{14} + ( - 4 \beta_{3} + 4 \beta_1 - 5) q^{16} + (10 \beta_{2} - 6 \beta_1 + 10) q^{17} + ( - 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{19} + ( - 6 \beta_{3} - 17 \beta_{2} + 7 \beta_1 + 9) q^{20} + (5 \beta_{2} + 2 \beta_1 + 5) q^{22} + (2 \beta_{3} + 14 \beta_{2} - 14) q^{23} + (14 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 4) q^{25} + (10 \beta_{3} - 10 \beta_1 - 22) q^{26} + (2 \beta_{3} - 11 \beta_{2} + 11) q^{28} + (7 \beta_{3} - 18 \beta_{2} + 7 \beta_1) q^{29} + (6 \beta_{3} - 6 \beta_1 - 4) q^{31} + (19 \beta_{2} + 7 \beta_1 + 19) q^{32} + (4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{34} + (5 \beta_{3} + 10 \beta_{2} - 5 \beta_1 + 10) q^{35} + (16 \beta_{2} + 18 \beta_1 + 16) q^{37} + ( - 18 \beta_{3} - 24 \beta_{2} + 24) q^{38} + ( - 8 \beta_{3} + 9 \beta_{2} + 6 \beta_1 + 12) q^{40} + ( - 6 \beta_{3} + 6 \beta_1 + 14) q^{41} + ( - 20 \beta_{3} + 2 \beta_{2} - 2) q^{43} + ( - 5 \beta_{3} + 32 \beta_{2} - 5 \beta_1) q^{44} + (16 \beta_{3} - 16 \beta_1 - 34) q^{46} + ( - 32 \beta_{2} + 10 \beta_1 - 32) q^{47} + ( - 4 \beta_{3} - 35 \beta_{2} - 4 \beta_1) q^{49} + (19 \beta_{3} + 13 \beta_{2} - 8 \beta_1 - 41) q^{50} + ( - 20 \beta_{2} - 34 \beta_1 - 20) q^{52} + (12 \beta_{3} + 14 \beta_{2} - 14) q^{53} + ( - 16 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 31) q^{55} + ( - 5 \beta_{3} + 5 \beta_1) q^{56} + ( - 4 \beta_{3} + 3 \beta_{2} - 3) q^{58} + ( - 31 \beta_{3} - 36 \beta_{2} - 31 \beta_1) q^{59} + ( - 18 \beta_{3} + 18 \beta_1 + 50) q^{61} + ( - 22 \beta_{2} - 16 \beta_1 - 22) q^{62} + (10 \beta_{3} + 79 \beta_{2} + 10 \beta_1) q^{64} + ( - 22 \beta_{3} + 26 \beta_{2} + 14 \beta_1 + 28) q^{65} + ( - 50 \beta_{2} - 4 \beta_1 - 50) q^{67} + (34 \beta_{3} - 26 \beta_{2} + 26) q^{68} + (10 \beta_{3} + 5 \beta_{2} - 15) q^{70} + 68 q^{71} + (48 \beta_{3} - 19 \beta_{2} + 19) q^{73} + (34 \beta_{3} + 86 \beta_{2} + 34 \beta_1) q^{74} + ( - 18 \beta_{3} + 18 \beta_1 + 78) q^{76} + ( - 22 \beta_{2} + 14 \beta_1 - 22) q^{77} + (10 \beta_{3} + 10 \beta_1) q^{79} + ( - 21 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 41) q^{80} + (32 \beta_{2} + 26 \beta_1 + 32) q^{82} + ( - 14 \beta_{3} - 4 \beta_{2} + 4) q^{83} + (8 \beta_{3} + 16 \beta_{2} - 36 \beta_1 + 58) q^{85} + ( - 18 \beta_{3} + 18 \beta_1 + 56) q^{86} + (14 \beta_{3} - 3 \beta_{2} + 3) q^{88} + (36 \beta_{3} + 6 \beta_{2} + 36 \beta_1) q^{89} + ( - 14 \beta_{3} + 14 \beta_1 - 4) q^{91} + ( - 26 \beta_{2} - 58 \beta_1 - 26) q^{92} + ( - 22 \beta_{3} - 34 \beta_{2} - 22 \beta_1) q^{94} + (24 \beta_{3} + 48 \beta_{2} - 18 \beta_1 - 36) q^{95} + ( - 5 \beta_{2} + 16 \beta_1 - 5) q^{97} + ( - 43 \beta_{3} - 47 \beta_{2} + 47) q^{98}+O(q^{100})$$ q + (b2 + b1 + 1) * q^2 + (2*b3 + b2 + 2*b1) * q^4 + (b3 - 3*b2 - 2*b1 + 1) * q^5 + (b2 - 2*b1 + 1) * q^7 + (b3 + 3*b2 - 3) * q^8 + (-4*b3 - 8*b2 - 2*b1 + 1) * q^10 + (-3*b3 + 3*b1 - 4) * q^11 + (2*b3 + 8*b2 - 8) * q^13 + (-b3 - 4*b2 - b1) * q^14 + (-4*b3 + 4*b1 - 5) * q^16 + (10*b2 - 6*b1 + 10) * q^17 + (-6*b3 - 6*b2 - 6*b1) * q^19 + (-6*b3 - 17*b2 + 7*b1 + 9) * q^20 + (5*b2 + 2*b1 + 5) * q^22 + (2*b3 + 14*b2 - 14) * q^23 + (14*b3 + 3*b2 + 2*b1 + 4) * q^25 + (10*b3 - 10*b1 - 22) * q^26 + (2*b3 - 11*b2 + 11) * q^28 + (7*b3 - 18*b2 + 7*b1) * q^29 + (6*b3 - 6*b1 - 4) * q^31 + (19*b2 + 7*b1 + 19) * q^32 + (4*b3 + 2*b2 + 4*b1) * q^34 + (5*b3 + 10*b2 - 5*b1 + 10) * q^35 + (16*b2 + 18*b1 + 16) * q^37 + (-18*b3 - 24*b2 + 24) * q^38 + (-8*b3 + 9*b2 + 6*b1 + 12) * q^40 + (-6*b3 + 6*b1 + 14) * q^41 + (-20*b3 + 2*b2 - 2) * q^43 + (-5*b3 + 32*b2 - 5*b1) * q^44 + (16*b3 - 16*b1 - 34) * q^46 + (-32*b2 + 10*b1 - 32) * q^47 + (-4*b3 - 35*b2 - 4*b1) * q^49 + (19*b3 + 13*b2 - 8*b1 - 41) * q^50 + (-20*b2 - 34*b1 - 20) * q^52 + (12*b3 + 14*b2 - 14) * q^53 + (-16*b3 + 3*b2 + 2*b1 - 31) * q^55 + (-5*b3 + 5*b1) * q^56 + (-4*b3 + 3*b2 - 3) * q^58 + (-31*b3 - 36*b2 - 31*b1) * q^59 + (-18*b3 + 18*b1 + 50) * q^61 + (-22*b2 - 16*b1 - 22) * q^62 + (10*b3 + 79*b2 + 10*b1) * q^64 + (-22*b3 + 26*b2 + 14*b1 + 28) * q^65 + (-50*b2 - 4*b1 - 50) * q^67 + (34*b3 - 26*b2 + 26) * q^68 + (10*b3 + 5*b2 - 15) * q^70 + 68 * q^71 + (48*b3 - 19*b2 + 19) * q^73 + (34*b3 + 86*b2 + 34*b1) * q^74 + (-18*b3 + 18*b1 + 78) * q^76 + (-22*b2 + 14*b1 - 22) * q^77 + (10*b3 + 10*b1) * q^79 + (-21*b3 + 3*b2 + 2*b1 - 41) * q^80 + (32*b2 + 26*b1 + 32) * q^82 + (-14*b3 - 4*b2 + 4) * q^83 + (8*b3 + 16*b2 - 36*b1 + 58) * q^85 + (-18*b3 + 18*b1 + 56) * q^86 + (14*b3 - 3*b2 + 3) * q^88 + (36*b3 + 6*b2 + 36*b1) * q^89 + (-14*b3 + 14*b1 - 4) * q^91 + (-26*b2 - 58*b1 - 26) * q^92 + (-22*b3 - 34*b2 - 22*b1) * q^94 + (24*b3 + 48*b2 - 18*b1 - 36) * q^95 + (-5*b2 + 16*b1 - 5) * q^97 + (-43*b3 - 47*b2 + 47) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 4 q^{5} + 4 q^{7} - 12 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 + 4 * q^5 + 4 * q^7 - 12 * q^8 $$4 q + 4 q^{2} + 4 q^{5} + 4 q^{7} - 12 q^{8} + 4 q^{10} - 16 q^{11} - 32 q^{13} - 20 q^{16} + 40 q^{17} + 36 q^{20} + 20 q^{22} - 56 q^{23} + 16 q^{25} - 88 q^{26} + 44 q^{28} - 16 q^{31} + 76 q^{32} + 40 q^{35} + 64 q^{37} + 96 q^{38} + 48 q^{40} + 56 q^{41} - 8 q^{43} - 136 q^{46} - 128 q^{47} - 164 q^{50} - 80 q^{52} - 56 q^{53} - 124 q^{55} - 12 q^{58} + 200 q^{61} - 88 q^{62} + 112 q^{65} - 200 q^{67} + 104 q^{68} - 60 q^{70} + 272 q^{71} + 76 q^{73} + 312 q^{76} - 88 q^{77} - 164 q^{80} + 128 q^{82} + 16 q^{83} + 232 q^{85} + 224 q^{86} + 12 q^{88} - 16 q^{91} - 104 q^{92} - 144 q^{95} - 20 q^{97} + 188 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 + 4 * q^5 + 4 * q^7 - 12 * q^8 + 4 * q^10 - 16 * q^11 - 32 * q^13 - 20 * q^16 + 40 * q^17 + 36 * q^20 + 20 * q^22 - 56 * q^23 + 16 * q^25 - 88 * q^26 + 44 * q^28 - 16 * q^31 + 76 * q^32 + 40 * q^35 + 64 * q^37 + 96 * q^38 + 48 * q^40 + 56 * q^41 - 8 * q^43 - 136 * q^46 - 128 * q^47 - 164 * q^50 - 80 * q^52 - 56 * q^53 - 124 * q^55 - 12 * q^58 + 200 * q^61 - 88 * q^62 + 112 * q^65 - 200 * q^67 + 104 * q^68 - 60 * q^70 + 272 * q^71 + 76 * q^73 + 312 * q^76 - 88 * q^77 - 164 * q^80 + 128 * q^82 + 16 * q^83 + 232 * q^85 + 224 * q^86 + 12 * q^88 - 16 * q^91 - 104 * q^92 - 144 * q^95 - 20 * q^97 + 188 * 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 $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 −1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
−0.224745 + 0.224745i 0 3.89898i 4.67423 + 1.77526i 0 3.44949 3.44949i −1.77526 1.77526i 0 −1.44949 + 0.651531i
28.2 2.22474 2.22474i 0 5.89898i −2.67423 + 4.22474i 0 −1.44949 + 1.44949i −4.22474 4.22474i 0 3.44949 + 15.3485i
37.1 −0.224745 0.224745i 0 3.89898i 4.67423 1.77526i 0 3.44949 + 3.44949i −1.77526 + 1.77526i 0 −1.44949 0.651531i
37.2 2.22474 + 2.22474i 0 5.89898i −2.67423 4.22474i 0 −1.44949 1.44949i −4.22474 + 4.22474i 0 3.44949 15.3485i
 $$n$$: 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 45.3.g.b 4
3.b odd 2 1 15.3.f.a 4
4.b odd 2 1 720.3.bh.k 4
5.b even 2 1 225.3.g.a 4
5.c odd 4 1 inner 45.3.g.b 4
5.c odd 4 1 225.3.g.a 4
9.c even 3 2 405.3.l.f 8
9.d odd 6 2 405.3.l.h 8
12.b even 2 1 240.3.bg.a 4
15.d odd 2 1 75.3.f.c 4
15.e even 4 1 15.3.f.a 4
15.e even 4 1 75.3.f.c 4
20.e even 4 1 720.3.bh.k 4
24.f even 2 1 960.3.bg.h 4
24.h odd 2 1 960.3.bg.i 4
45.k odd 12 2 405.3.l.f 8
45.l even 12 2 405.3.l.h 8
60.h even 2 1 1200.3.bg.k 4
60.l odd 4 1 240.3.bg.a 4
60.l odd 4 1 1200.3.bg.k 4
120.q odd 4 1 960.3.bg.h 4
120.w even 4 1 960.3.bg.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.f.a 4 3.b odd 2 1
15.3.f.a 4 15.e even 4 1
45.3.g.b 4 1.a even 1 1 trivial
45.3.g.b 4 5.c odd 4 1 inner
75.3.f.c 4 15.d odd 2 1
75.3.f.c 4 15.e even 4 1
225.3.g.a 4 5.b even 2 1
225.3.g.a 4 5.c odd 4 1
240.3.bg.a 4 12.b even 2 1
240.3.bg.a 4 60.l odd 4 1
405.3.l.f 8 9.c even 3 2
405.3.l.f 8 45.k odd 12 2
405.3.l.h 8 9.d odd 6 2
405.3.l.h 8 45.l even 12 2
720.3.bh.k 4 4.b odd 2 1
720.3.bh.k 4 20.e even 4 1
960.3.bg.h 4 24.f even 2 1
960.3.bg.h 4 120.q odd 4 1
960.3.bg.i 4 24.h odd 2 1
960.3.bg.i 4 120.w even 4 1
1200.3.bg.k 4 60.h even 2 1
1200.3.bg.k 4 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4T_{2}^{3} + 8T_{2}^{2} + 4T_{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4 T^{3} + 8 T^{2} + 4 T + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 4 T^{3} - 100 T + 625$$
$7$ $$T^{4} - 4 T^{3} + 8 T^{2} + 40 T + 100$$
$11$ $$(T^{2} + 8 T - 38)^{2}$$
$13$ $$T^{4} + 32 T^{3} + 512 T^{2} + \cdots + 13456$$
$17$ $$T^{4} - 40 T^{3} + 800 T^{2} + \cdots + 8464$$
$19$ $$T^{4} + 504 T^{2} + 32400$$
$23$ $$T^{4} + 56 T^{3} + 1568 T^{2} + \cdots + 144400$$
$29$ $$T^{4} + 1236T^{2} + 900$$
$31$ $$(T^{2} + 8 T - 200)^{2}$$
$37$ $$T^{4} - 64 T^{3} + 2048 T^{2} + \cdots + 211600$$
$41$ $$(T^{2} - 28 T - 20)^{2}$$
$43$ $$T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 1420864$$
$47$ $$T^{4} + 128 T^{3} + 8192 T^{2} + \cdots + 3055504$$
$53$ $$T^{4} + 56 T^{3} + 1568 T^{2} + \cdots + 1600$$
$59$ $$T^{4} + 14124 T^{2} + \cdots + 19980900$$
$61$ $$(T^{2} - 100 T + 556)^{2}$$
$67$ $$T^{4} + 200 T^{3} + \cdots + 24522304$$
$71$ $$(T - 68)^{4}$$
$73$ $$T^{4} - 76 T^{3} + 2888 T^{2} + \cdots + 38316100$$
$79$ $$(T^{2} + 600)^{2}$$
$83$ $$T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 309136$$
$89$ $$T^{4} + 15624 T^{2} + \cdots + 59907600$$
$97$ $$T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 515524$$