# Properties

 Label 45.4.b.b Level $45$ Weight $4$ Character orbit 45.b Analytic conductor $2.655$ Analytic rank $0$ Dimension $4$ Inner twists $2$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,4,Mod(19,45)]

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([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 45.b (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: $$2.65508595026$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 21x^{2} + 100$$ x^4 + 21*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{3} - 5) q^{4} + (\beta_{3} - \beta_{2} - 2) q^{5} + (2 \beta_{2} - 4 \beta_1) q^{7} + (4 \beta_{2} + 5 \beta_1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b3 - 5) * q^4 + (b3 - b2 - 2) * q^5 + (2*b2 - 4*b1) * q^7 + (4*b2 + 5*b1) * q^8 $$q - \beta_1 q^{2} + (\beta_{3} - 5) q^{4} + (\beta_{3} - \beta_{2} - 2) q^{5} + (2 \beta_{2} - 4 \beta_1) q^{7} + (4 \beta_{2} + 5 \beta_1) q^{8} + (\beta_{3} + 4 \beta_{2} + 10 \beta_1 + 3) q^{10} + ( - 2 \beta_{3} + 22) q^{11} + ( - 8 \beta_{2} - 2 \beta_1) q^{13} + (2 \beta_{3} - 58) q^{14} + ( - \beta_{3} + 13) q^{16} + ( - 6 \beta_{2} - 16 \beta_1) q^{17} + ( - 8 \beta_{3} - 24) q^{19} + ( - 6 \beta_{3} - 4 \beta_{2} + \cdots + 102) q^{20}+ \cdots + ( - 48 \beta_{2} - 67 \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b3 - 5) * q^4 + (b3 - b2 - 2) * q^5 + (2*b2 - 4*b1) * q^7 + (4*b2 + 5*b1) * q^8 + (b3 + 4*b2 + 10*b1 + 3) * q^10 + (-2*b3 + 22) * q^11 + (-8*b2 - 2*b1) * q^13 + (2*b3 - 58) * q^14 + (-b3 + 13) * q^16 + (-6*b2 - 16*b1) * q^17 + (-8*b3 - 24) * q^19 + (-6*b3 - 4*b2 + 5*b1 + 102) * q^20 + (-8*b2 - 38*b1) * q^22 + (-6*b2 + 2*b1) * q^23 + (-6*b3 - 14*b2 + 10*b1 + 67) * q^25 + (10*b3 - 2) * q^26 + (24*b2 + 42*b1) * q^28 + (-14*b3 - 152) * q^29 + (4*b3 + 24) * q^31 + (28*b2 + 19*b1) * q^32 + (22*b3 - 190) * q^34 + (10*b3 + 30*b2 + 30*b1 + 70) * q^35 + (12*b2 + 66*b1) * q^37 + (-32*b2 - 40*b1) * q^38 + (7*b3 + 8*b2 - 70*b1 + 101) * q^40 + (-4*b3 + 206) * q^41 + (14*b2 - 82*b1) * q^43 + (30*b3 - 294) * q^44 + (4*b3 + 44) * q^46 + (-38*b2 + 54*b1) * q^47 + (-12*b3 - 29) * q^49 + (4*b3 - 24*b2 - 115*b1 + 172) * q^50 + (-24*b2 + 66*b1) * q^52 + (54*b2 - 28*b1) * q^53 + (24*b3 - 4*b2 - 10*b1 - 228) * q^55 + (-50*b3 + 10) * q^56 + (-56*b2 + 40*b1) * q^58 + (2*b3 - 94) * q^59 + (-32*b3 + 186) * q^61 + (16*b2 + 8*b1) * q^62 + (-55*b3 + 267) * q^64 + (-22*b3 - 48*b2 + 60*b1 - 226) * q^65 + (46*b2 + 106*b1) * q^67 + (40*b2 + 238*b1) * q^68 + (-60*b3 + 40*b2 + 10*b1 + 300) * q^70 + (60*b3 - 12) * q^71 + (-84*b2 - 192*b1) * q^73 + (-78*b3 + 822) * q^74 + (8*b3 - 616) * q^76 + (-24*b2 - 132*b1) * q^77 + (100*b3 - 240) * q^79 + (14*b3 - 4*b2 - 5*b1 - 118) * q^80 + (-16*b2 - 238*b1) * q^82 + (-30*b2 - 238*b1) * q^83 + (-2*b3 + 22*b2 + 190*b1 - 126) * q^85 + (68*b3 - 1108) * q^86 + (56*b2 + 230*b1) * q^88 + (48*b3 + 534) * q^89 + (84*b3 + 444) * q^91 + (-32*b2 + 4*b1) * q^92 + (-16*b3 + 816) * q^94 + (-16*b3 + 96*b2 - 40*b1 - 688) * q^95 + (-4*b2 - 244*b1) * q^97 + (-48*b2 - 67*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 18 q^{4} - 6 q^{5}+O(q^{10})$$ 4 * q - 18 * q^4 - 6 * q^5 $$4 q - 18 q^{4} - 6 q^{5} + 14 q^{10} + 84 q^{11} - 228 q^{14} + 50 q^{16} - 112 q^{19} + 396 q^{20} + 256 q^{25} + 12 q^{26} - 636 q^{29} + 104 q^{31} - 716 q^{34} + 300 q^{35} + 418 q^{40} + 816 q^{41} - 1116 q^{44} + 184 q^{46} - 140 q^{49} + 696 q^{50} - 864 q^{55} - 60 q^{56} - 372 q^{59} + 680 q^{61} + 958 q^{64} - 948 q^{65} + 1080 q^{70} + 72 q^{71} + 3132 q^{74} - 2448 q^{76} - 760 q^{79} - 444 q^{80} - 508 q^{85} - 4296 q^{86} + 2232 q^{89} + 1944 q^{91} + 3232 q^{94} - 2784 q^{95}+O(q^{100})$$ 4 * q - 18 * q^4 - 6 * q^5 + 14 * q^10 + 84 * q^11 - 228 * q^14 + 50 * q^16 - 112 * q^19 + 396 * q^20 + 256 * q^25 + 12 * q^26 - 636 * q^29 + 104 * q^31 - 716 * q^34 + 300 * q^35 + 418 * q^40 + 816 * q^41 - 1116 * q^44 + 184 * q^46 - 140 * q^49 + 696 * q^50 - 864 * q^55 - 60 * q^56 - 372 * q^59 + 680 * q^61 + 958 * q^64 - 948 * q^65 + 1080 * q^70 + 72 * q^71 + 3132 * q^74 - 2448 * q^76 - 760 * q^79 - 444 * q^80 - 508 * q^85 - 4296 * q^86 + 2232 * q^89 + 1944 * q^91 + 3232 * q^94 - 2784 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + \nu ) / 10$$ (v^3 + v) / 10 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 13\nu ) / 2$$ (v^3 + 13*v) / 2 $$\beta_{3}$$ $$=$$ $$3\nu^{2} + 32$$ 3*v^2 + 32
 $$\nu$$ $$=$$ $$( \beta_{2} - 5\beta_1 ) / 6$$ (b2 - 5*b1) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 32 ) / 3$$ (b3 - 32) / 3 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 65\beta_1 ) / 6$$ (-b2 + 65*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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 3.70156i − 2.70156i 2.70156i 3.70156i
4.70156i 0 −14.1047 −11.1047 1.29844i 0 16.2094i 28.7016i 0 −6.10469 + 52.2094i
19.2 1.70156i 0 5.10469 8.10469 + 7.70156i 0 22.2094i 22.2984i 0 13.1047 13.7906i
19.3 1.70156i 0 5.10469 8.10469 7.70156i 0 22.2094i 22.2984i 0 13.1047 + 13.7906i
19.4 4.70156i 0 −14.1047 −11.1047 + 1.29844i 0 16.2094i 28.7016i 0 −6.10469 52.2094i
 $$n$$: 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.4.b.b 4
3.b odd 2 1 15.4.b.a 4
4.b odd 2 1 720.4.f.j 4
5.b even 2 1 inner 45.4.b.b 4
5.c odd 4 1 225.4.a.i 2
5.c odd 4 1 225.4.a.o 2
12.b even 2 1 240.4.f.f 4
15.d odd 2 1 15.4.b.a 4
15.e even 4 1 75.4.a.c 2
15.e even 4 1 75.4.a.f 2
20.d odd 2 1 720.4.f.j 4
24.f even 2 1 960.4.f.p 4
24.h odd 2 1 960.4.f.q 4
60.h even 2 1 240.4.f.f 4
60.l odd 4 1 1200.4.a.bn 2
60.l odd 4 1 1200.4.a.bt 2
120.i odd 2 1 960.4.f.q 4
120.m even 2 1 960.4.f.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 3.b odd 2 1
15.4.b.a 4 15.d odd 2 1
45.4.b.b 4 1.a even 1 1 trivial
45.4.b.b 4 5.b even 2 1 inner
75.4.a.c 2 15.e even 4 1
75.4.a.f 2 15.e even 4 1
225.4.a.i 2 5.c odd 4 1
225.4.a.o 2 5.c odd 4 1
240.4.f.f 4 12.b even 2 1
240.4.f.f 4 60.h even 2 1
720.4.f.j 4 4.b odd 2 1
720.4.f.j 4 20.d odd 2 1
960.4.f.p 4 24.f even 2 1
960.4.f.p 4 120.m even 2 1
960.4.f.q 4 24.h odd 2 1
960.4.f.q 4 120.i odd 2 1
1200.4.a.bn 2 60.l odd 4 1
1200.4.a.bt 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 25T_{2}^{2} + 64$$ acting on $$S_{4}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 25T^{2} + 64$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6 T^{3} + \cdots + 15625$$
$7$ $$T^{4} + 756 T^{2} + 129600$$
$11$ $$(T^{2} - 42 T + 72)^{2}$$
$13$ $$T^{4} + 3780 T^{2} + 1327104$$
$17$ $$T^{4} + 7252 T^{2} + 2483776$$
$19$ $$(T^{2} + 56 T - 5120)^{2}$$
$23$ $$T^{4} + 2464 T^{2} + 6400$$
$29$ $$(T^{2} + 318 T + 7200)^{2}$$
$31$ $$(T^{2} - 52 T - 800)^{2}$$
$37$ $$T^{4} + 106596 T^{2} + 41990400$$
$41$ $$(T^{2} - 408 T + 40140)^{2}$$
$43$ $$T^{4} + \cdots + 8256266496$$
$47$ $$T^{4} + \cdots + 6186766336$$
$53$ $$T^{4} + 218644 T^{2} + 813390400$$
$59$ $$(T^{2} + 186 T + 8280)^{2}$$
$61$ $$(T^{2} - 340 T - 65564)^{2}$$
$67$ $$T^{4} + \cdots + 9419867136$$
$71$ $$(T^{2} - 36 T - 331776)^{2}$$
$73$ $$T^{4} + \cdots + 104976000000$$
$79$ $$(T^{2} + 380 T - 886400)^{2}$$
$83$ $$T^{4} + \cdots + 40558737664$$
$89$ $$(T^{2} - 1116 T + 98820)^{2}$$
$97$ $$T^{4} + \cdots + 196199387136$$
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