# Properties

 Label 45.3.c.a Level $45$ Weight $3$ Character orbit 45.c 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(26,45)]

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, 0]))

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

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

chi := DirichletCharacter("45.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 45.c (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: $$\Q(\sqrt{-2}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 9$$ x^4 - 4*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - \beta_1) q^{2} + ( - 2 \beta_{3} - 3) q^{4} - \beta_{2} q^{5} + (\beta_{3} + 4) q^{7} + (3 \beta_{2} + 9 \beta_1) q^{8}+O(q^{10})$$ q + (-b2 - b1) * q^2 + (-2*b3 - 3) * q^4 - b2 * q^5 + (b3 + 4) * q^7 + (3*b2 + 9*b1) * q^8 $$q + ( - \beta_{2} - \beta_1) q^{2} + ( - 2 \beta_{3} - 3) q^{4} - \beta_{2} q^{5} + (\beta_{3} + 4) q^{7} + (3 \beta_{2} + 9 \beta_1) q^{8} + ( - \beta_{3} - 5) q^{10} + (2 \beta_{2} - 7 \beta_1) q^{11} + (5 \beta_{3} - 6) q^{13} + ( - 6 \beta_{2} - 9 \beta_1) q^{14} + (4 \beta_{3} + 21) q^{16} + (8 \beta_{2} - 4 \beta_1) q^{17} + 2 \beta_{3} q^{19} + (3 \beta_{2} + 10 \beta_1) q^{20} + ( - 5 \beta_{3} - 4) q^{22} + ( - 6 \beta_{2} + 18 \beta_1) q^{23} - 5 q^{25} + ( - 4 \beta_{2} - 19 \beta_1) q^{26} + ( - 11 \beta_{3} - 32) q^{28} + (10 \beta_{2} + 16 \beta_1) q^{29} + ( - 14 \beta_{3} - 14) q^{31} + ( - 17 \beta_{2} - 5 \beta_1) q^{32} + (4 \beta_{3} + 32) q^{34} + ( - 4 \beta_{2} - 5 \beta_1) q^{35} + (9 \beta_{3} + 38) q^{37} + ( - 4 \beta_{2} - 10 \beta_1) q^{38} + (9 \beta_{3} + 15) q^{40} + ( - 8 \beta_{2} + \beta_1) q^{41} + ( - 10 \beta_{3} - 12) q^{43} + (22 \beta_{2} + \beta_1) q^{44} + (12 \beta_{3} + 6) q^{46} + ( - 4 \beta_{2} - 22 \beta_1) q^{47} + (8 \beta_{3} - 23) q^{49} + (5 \beta_{2} + 5 \beta_1) q^{50} + ( - 3 \beta_{3} - 82) q^{52} + (8 \beta_{2} - 22 \beta_1) q^{53} + ( - 7 \beta_{3} + 10) q^{55} + (30 \beta_{2} + 51 \beta_1) q^{56} + (26 \beta_{3} + 82) q^{58} + ( - 22 \beta_{2} + 17 \beta_1) q^{59} + (2 \beta_{3} - 42) q^{61} + (42 \beta_{2} + 84 \beta_1) q^{62} + ( - 6 \beta_{3} - 11) q^{64} + (6 \beta_{2} - 25 \beta_1) q^{65} + ( - 8 \beta_{3} + 52) q^{67} + ( - 8 \beta_{2} - 68 \beta_1) q^{68} + ( - 9 \beta_{3} - 30) q^{70} + ( - 44 \beta_{2} + 4 \beta_1) q^{71} + (2 \beta_{3} + 54) q^{73} + ( - 56 \beta_{2} - 83 \beta_1) q^{74} + ( - 6 \beta_{3} - 40) q^{76} + ( - 6 \beta_{2} - 18 \beta_1) q^{77} + ( - 26 \beta_{3} - 14) q^{79} + ( - 21 \beta_{2} - 20 \beta_1) q^{80} + ( - 7 \beta_{3} - 38) q^{82} + (24 \beta_{2} + 18 \beta_1) q^{83} + ( - 4 \beta_{3} + 40) q^{85} + (32 \beta_{2} + 62 \beta_1) q^{86} + (3 \beta_{3} + 96) q^{88} + (12 \beta_{2} + 57 \beta_1) q^{89} + (14 \beta_{3} + 26) q^{91} + ( - 54 \beta_{2} + 6 \beta_1) q^{92} + ( - 26 \beta_{3} - 64) q^{94} - 10 \beta_1 q^{95} + (18 \beta_{3} - 58) q^{97} + (7 \beta_{2} - 17 \beta_1) q^{98}+O(q^{100})$$ q + (-b2 - b1) * q^2 + (-2*b3 - 3) * q^4 - b2 * q^5 + (b3 + 4) * q^7 + (3*b2 + 9*b1) * q^8 + (-b3 - 5) * q^10 + (2*b2 - 7*b1) * q^11 + (5*b3 - 6) * q^13 + (-6*b2 - 9*b1) * q^14 + (4*b3 + 21) * q^16 + (8*b2 - 4*b1) * q^17 + 2*b3 * q^19 + (3*b2 + 10*b1) * q^20 + (-5*b3 - 4) * q^22 + (-6*b2 + 18*b1) * q^23 - 5 * q^25 + (-4*b2 - 19*b1) * q^26 + (-11*b3 - 32) * q^28 + (10*b2 + 16*b1) * q^29 + (-14*b3 - 14) * q^31 + (-17*b2 - 5*b1) * q^32 + (4*b3 + 32) * q^34 + (-4*b2 - 5*b1) * q^35 + (9*b3 + 38) * q^37 + (-4*b2 - 10*b1) * q^38 + (9*b3 + 15) * q^40 + (-8*b2 + b1) * q^41 + (-10*b3 - 12) * q^43 + (22*b2 + b1) * q^44 + (12*b3 + 6) * q^46 + (-4*b2 - 22*b1) * q^47 + (8*b3 - 23) * q^49 + (5*b2 + 5*b1) * q^50 + (-3*b3 - 82) * q^52 + (8*b2 - 22*b1) * q^53 + (-7*b3 + 10) * q^55 + (30*b2 + 51*b1) * q^56 + (26*b3 + 82) * q^58 + (-22*b2 + 17*b1) * q^59 + (2*b3 - 42) * q^61 + (42*b2 + 84*b1) * q^62 + (-6*b3 - 11) * q^64 + (6*b2 - 25*b1) * q^65 + (-8*b3 + 52) * q^67 + (-8*b2 - 68*b1) * q^68 + (-9*b3 - 30) * q^70 + (-44*b2 + 4*b1) * q^71 + (2*b3 + 54) * q^73 + (-56*b2 - 83*b1) * q^74 + (-6*b3 - 40) * q^76 + (-6*b2 - 18*b1) * q^77 + (-26*b3 - 14) * q^79 + (-21*b2 - 20*b1) * q^80 + (-7*b3 - 38) * q^82 + (24*b2 + 18*b1) * q^83 + (-4*b3 + 40) * q^85 + (32*b2 + 62*b1) * q^86 + (3*b3 + 96) * q^88 + (12*b2 + 57*b1) * q^89 + (14*b3 + 26) * q^91 + (-54*b2 + 6*b1) * q^92 + (-26*b3 - 64) * q^94 - 10*b1 * q^95 + (18*b3 - 58) * q^97 + (7*b2 - 17*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{4} + 16 q^{7}+O(q^{10})$$ 4 * q - 12 * q^4 + 16 * q^7 $$4 q - 12 q^{4} + 16 q^{7} - 20 q^{10} - 24 q^{13} + 84 q^{16} - 16 q^{22} - 20 q^{25} - 128 q^{28} - 56 q^{31} + 128 q^{34} + 152 q^{37} + 60 q^{40} - 48 q^{43} + 24 q^{46} - 92 q^{49} - 328 q^{52} + 40 q^{55} + 328 q^{58} - 168 q^{61} - 44 q^{64} + 208 q^{67} - 120 q^{70} + 216 q^{73} - 160 q^{76} - 56 q^{79} - 152 q^{82} + 160 q^{85} + 384 q^{88} + 104 q^{91} - 256 q^{94} - 232 q^{97}+O(q^{100})$$ 4 * q - 12 * q^4 + 16 * q^7 - 20 * q^10 - 24 * q^13 + 84 * q^16 - 16 * q^22 - 20 * q^25 - 128 * q^28 - 56 * q^31 + 128 * q^34 + 152 * q^37 + 60 * q^40 - 48 * q^43 + 24 * q^46 - 92 * q^49 - 328 * q^52 + 40 * q^55 + 328 * q^58 - 168 * q^61 - 44 * q^64 + 208 * q^67 - 120 * q^70 + 216 * q^73 - 160 * q^76 - 56 * q^79 - 152 * q^82 + 160 * q^85 + 384 * q^88 + 104 * q^91 - 256 * q^94 - 232 * q^97

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - \nu ) / 3$$ (v^3 - v) / 3 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 7\nu ) / 3$$ (-v^3 + 7*v) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$( \beta_{3} + 7\beta_1 ) / 2$$ (b3 + 7*b1) / 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 1.58114 + 0.707107i −1.58114 − 0.707107i −1.58114 + 0.707107i 1.58114 − 0.707107i
3.65028i 0 −9.32456 2.23607i 0 7.16228 19.4361i 0 −8.16228
26.2 0.821854i 0 3.32456 2.23607i 0 0.837722 6.01972i 0 −1.83772
26.3 0.821854i 0 3.32456 2.23607i 0 0.837722 6.01972i 0 −1.83772
26.4 3.65028i 0 −9.32456 2.23607i 0 7.16228 19.4361i 0 −8.16228
 $$n$$: 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

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 14T^{2} + 9$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$(T^{2} - 8 T + 6)^{2}$$
$11$ $$T^{4} + 236T^{2} + 6084$$
$13$ $$(T^{2} + 12 T - 214)^{2}$$
$17$ $$T^{4} + 704 T^{2} + 82944$$
$19$ $$(T^{2} - 40)^{2}$$
$23$ $$T^{4} + 1656 T^{2} + 219024$$
$29$ $$T^{4} + 2024T^{2} + 144$$
$31$ $$(T^{2} + 28 T - 1764)^{2}$$
$37$ $$(T^{2} - 76 T + 634)^{2}$$
$41$ $$T^{4} + 644 T^{2} + 101124$$
$43$ $$(T^{2} + 24 T - 856)^{2}$$
$47$ $$T^{4} + 2096 T^{2} + 788544$$
$53$ $$T^{4} + 2576 T^{2} + 419904$$
$59$ $$T^{4} + 5996 T^{2} + \cdots + 3392964$$
$61$ $$(T^{2} + 84 T + 1724)^{2}$$
$67$ $$(T^{2} - 104 T + 2064)^{2}$$
$71$ $$T^{4} + 19424 T^{2} + \cdots + 93083904$$
$73$ $$(T^{2} - 108 T + 2876)^{2}$$
$79$ $$(T^{2} + 28 T - 6564)^{2}$$
$83$ $$T^{4} + 7056 T^{2} + \cdots + 4981824$$
$89$ $$T^{4} + 14436 T^{2} + \cdots + 33385284$$
$97$ $$(T^{2} + 116 T + 124)^{2}$$