# Properties

 Label 45.5.g.d Level $45$ Weight $5$ Character orbit 45.g Analytic conductor $4.652$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,5,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, 5, names="a")

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

chi := DirichletCharacter("45.28");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$5$$ 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: $$4.65164833877$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 25$$ x^4 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - 11 \beta_{2} q^{4} + ( - 11 \beta_{3} + 2 \beta_1) q^{5} + ( - 5 \beta_{2} - 5) q^{7} - 27 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 - 11*b2 * q^4 + (-11*b3 + 2*b1) * q^5 + (-5*b2 - 5) * q^7 - 27*b3 * q^8 $$q + \beta_1 q^{2} - 11 \beta_{2} q^{4} + ( - 11 \beta_{3} + 2 \beta_1) q^{5} + ( - 5 \beta_{2} - 5) q^{7} - 27 \beta_{3} q^{8} + (10 \beta_{2} + 55) q^{10} + ( - 55 \beta_{3} + 55 \beta_1) q^{11} + (110 \beta_{2} - 110) q^{13} + ( - 5 \beta_{3} - 5 \beta_1) q^{14} - 41 q^{16} - 2 \beta_1 q^{17} - 198 \beta_{2} q^{19} + ( - 22 \beta_{3} - 121 \beta_1) q^{20} + (275 \beta_{2} + 275) q^{22} + 338 \beta_{3} q^{23} + ( - 585 \beta_{2} + 220) q^{25} + (110 \beta_{3} - 110 \beta_1) q^{26} + (55 \beta_{2} - 55) q^{28} + (315 \beta_{3} + 315 \beta_1) q^{29} - 1192 q^{31} - 473 \beta_1 q^{32} - 10 \beta_{2} q^{34} + (45 \beta_{3} - 65 \beta_1) q^{35} + (1810 \beta_{2} + 1810) q^{37} - 198 \beta_{3} q^{38} + ( - 1485 \beta_{2} + 270) q^{40} + ( - 490 \beta_{3} + 490 \beta_1) q^{41} + (1310 \beta_{2} - 1310) q^{43} + ( - 605 \beta_{3} - 605 \beta_1) q^{44} - 1690 q^{46} + 1414 \beta_1 q^{47} - 2351 \beta_{2} q^{49} + ( - 585 \beta_{3} + 220 \beta_1) q^{50} + (1210 \beta_{2} + 1210) q^{52} - 1024 \beta_{3} q^{53} + ( - 2475 \beta_{2} + 3575) q^{55} + (135 \beta_{3} - 135 \beta_1) q^{56} + (1575 \beta_{2} - 1575) q^{58} + ( - 315 \beta_{3} - 315 \beta_1) q^{59} - 1378 q^{61} - 1192 \beta_1 q^{62} - 1709 \beta_{2} q^{64} + (1430 \beta_{3} + 990 \beta_1) q^{65} + (1870 \beta_{2} + 1870) q^{67} + 22 \beta_{3} q^{68} + ( - 325 \beta_{2} - 225) q^{70} + (2180 \beta_{3} - 2180 \beta_1) q^{71} + (3905 \beta_{2} - 3905) q^{73} + (1810 \beta_{3} + 1810 \beta_1) q^{74} - 2178 q^{76} - 550 \beta_1 q^{77} - 612 \beta_{2} q^{79} + (451 \beta_{3} - 82 \beta_1) q^{80} + (2450 \beta_{2} + 2450) q^{82} + 3014 \beta_{3} q^{83} + ( - 20 \beta_{2} - 110) q^{85} + (1310 \beta_{3} - 1310 \beta_1) q^{86} + ( - 7425 \beta_{2} + 7425) q^{88} + ( - 3600 \beta_{3} - 3600 \beta_1) q^{89} + 1100 q^{91} + 3718 \beta_1 q^{92} + 7070 \beta_{2} q^{94} + ( - 396 \beta_{3} - 2178 \beta_1) q^{95} + ( - 7205 \beta_{2} - 7205) q^{97} - 2351 \beta_{3} q^{98}+O(q^{100})$$ q + b1 * q^2 - 11*b2 * q^4 + (-11*b3 + 2*b1) * q^5 + (-5*b2 - 5) * q^7 - 27*b3 * q^8 + (10*b2 + 55) * q^10 + (-55*b3 + 55*b1) * q^11 + (110*b2 - 110) * q^13 + (-5*b3 - 5*b1) * q^14 - 41 * q^16 - 2*b1 * q^17 - 198*b2 * q^19 + (-22*b3 - 121*b1) * q^20 + (275*b2 + 275) * q^22 + 338*b3 * q^23 + (-585*b2 + 220) * q^25 + (110*b3 - 110*b1) * q^26 + (55*b2 - 55) * q^28 + (315*b3 + 315*b1) * q^29 - 1192 * q^31 - 473*b1 * q^32 - 10*b2 * q^34 + (45*b3 - 65*b1) * q^35 + (1810*b2 + 1810) * q^37 - 198*b3 * q^38 + (-1485*b2 + 270) * q^40 + (-490*b3 + 490*b1) * q^41 + (1310*b2 - 1310) * q^43 + (-605*b3 - 605*b1) * q^44 - 1690 * q^46 + 1414*b1 * q^47 - 2351*b2 * q^49 + (-585*b3 + 220*b1) * q^50 + (1210*b2 + 1210) * q^52 - 1024*b3 * q^53 + (-2475*b2 + 3575) * q^55 + (135*b3 - 135*b1) * q^56 + (1575*b2 - 1575) * q^58 + (-315*b3 - 315*b1) * q^59 - 1378 * q^61 - 1192*b1 * q^62 - 1709*b2 * q^64 + (1430*b3 + 990*b1) * q^65 + (1870*b2 + 1870) * q^67 + 22*b3 * q^68 + (-325*b2 - 225) * q^70 + (2180*b3 - 2180*b1) * q^71 + (3905*b2 - 3905) * q^73 + (1810*b3 + 1810*b1) * q^74 - 2178 * q^76 - 550*b1 * q^77 - 612*b2 * q^79 + (451*b3 - 82*b1) * q^80 + (2450*b2 + 2450) * q^82 + 3014*b3 * q^83 + (-20*b2 - 110) * q^85 + (1310*b3 - 1310*b1) * q^86 + (-7425*b2 + 7425) * q^88 + (-3600*b3 - 3600*b1) * q^89 + 1100 * q^91 + 3718*b1 * q^92 + 7070*b2 * q^94 + (-396*b3 - 2178*b1) * q^95 + (-7205*b2 - 7205) * q^97 - 2351*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 20 q^{7}+O(q^{10})$$ 4 * q - 20 * q^7 $$4 q - 20 q^{7} + 220 q^{10} - 440 q^{13} - 164 q^{16} + 1100 q^{22} + 880 q^{25} - 220 q^{28} - 4768 q^{31} + 7240 q^{37} + 1080 q^{40} - 5240 q^{43} - 6760 q^{46} + 4840 q^{52} + 14300 q^{55} - 6300 q^{58} - 5512 q^{61} + 7480 q^{67} - 900 q^{70} - 15620 q^{73} - 8712 q^{76} + 9800 q^{82} - 440 q^{85} + 29700 q^{88} + 4400 q^{91} - 28820 q^{97}+O(q^{100})$$ 4 * q - 20 * q^7 + 220 * q^10 - 440 * q^13 - 164 * q^16 + 1100 * q^22 + 880 * q^25 - 220 * q^28 - 4768 * q^31 + 7240 * q^37 + 1080 * q^40 - 5240 * q^43 - 6760 * q^46 + 4840 * q^52 + 14300 * q^55 - 6300 * q^58 - 5512 * q^61 + 7480 * q^67 - 900 * q^70 - 15620 * q^73 - 8712 * q^76 + 9800 * q^82 - 440 * q^85 + 29700 * q^88 + 4400 * q^91 - 28820 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*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
 −1.58114 + 1.58114i 1.58114 − 1.58114i −1.58114 − 1.58114i 1.58114 + 1.58114i
−1.58114 + 1.58114i 0 11.0000i −20.5548 14.2302i 0 −5.00000 + 5.00000i −42.6907 42.6907i 0 55.0000 10.0000i
28.2 1.58114 1.58114i 0 11.0000i 20.5548 + 14.2302i 0 −5.00000 + 5.00000i 42.6907 + 42.6907i 0 55.0000 10.0000i
37.1 −1.58114 1.58114i 0 11.0000i −20.5548 + 14.2302i 0 −5.00000 5.00000i −42.6907 + 42.6907i 0 55.0000 + 10.0000i
37.2 1.58114 + 1.58114i 0 11.0000i 20.5548 14.2302i 0 −5.00000 5.00000i 42.6907 42.6907i 0 55.0000 + 10.0000i
 $$n$$: 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.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.5.g.d 4
3.b odd 2 1 inner 45.5.g.d 4
5.b even 2 1 225.5.g.k 4
5.c odd 4 1 inner 45.5.g.d 4
5.c odd 4 1 225.5.g.k 4
15.d odd 2 1 225.5.g.k 4
15.e even 4 1 inner 45.5.g.d 4
15.e even 4 1 225.5.g.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.5.g.d 4 1.a even 1 1 trivial
45.5.g.d 4 3.b odd 2 1 inner
45.5.g.d 4 5.c odd 4 1 inner
45.5.g.d 4 15.e even 4 1 inner
225.5.g.k 4 5.b even 2 1
225.5.g.k 4 5.c odd 4 1
225.5.g.k 4 15.d odd 2 1
225.5.g.k 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 25$$ acting on $$S_{5}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 25$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 440 T^{2} + 390625$$
$7$ $$(T^{2} + 10 T + 50)^{2}$$
$11$ $$(T^{2} - 30250)^{2}$$
$13$ $$(T^{2} + 220 T + 24200)^{2}$$
$17$ $$T^{4} + 400$$
$19$ $$(T^{2} + 39204)^{2}$$
$23$ $$T^{4} + 326292288400$$
$29$ $$(T^{2} + 992250)^{2}$$
$31$ $$(T + 1192)^{4}$$
$37$ $$(T^{2} - 3620 T + 6552200)^{2}$$
$41$ $$(T^{2} - 2401000)^{2}$$
$43$ $$(T^{2} + 2620 T + 3432200)^{2}$$
$47$ $$T^{4} + 99939609120400$$
$53$ $$T^{4} + 27487790694400$$
$59$ $$(T^{2} + 992250)^{2}$$
$61$ $$(T + 1378)^{4}$$
$67$ $$(T^{2} - 3740 T + 6993800)^{2}$$
$71$ $$(T^{2} - 47524000)^{2}$$
$73$ $$(T^{2} + 7810 T + 30498050)^{2}$$
$79$ $$(T^{2} + 374544)^{2}$$
$83$ $$T^{4} + 20\!\cdots\!00$$
$89$ $$(T^{2} + 129600000)^{2}$$
$97$ $$(T^{2} + 14410 T + 103824050)^{2}$$