# Properties

 Label 45.5.c.a Level $45$ Weight $5$ Character orbit 45.c Analytic conductor $4.652$ 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] = [45,5,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, 5, names="a")

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

chi := DirichletCharacter("45.26");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$5$$ 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: $$4.65164833877$$ 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: $$2\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{2} + ( - 2 \beta_{3} - 7) q^{4} + 5 \beta_{2} q^{5} + ( - 7 \beta_{3} - 12) q^{7} + ( - 27 \beta_{2} - \beta_1) q^{8}+O(q^{10})$$ q + (b2 + b1) * q^2 + (-2*b3 - 7) * q^4 + 5*b2 * q^5 + (-7*b3 - 12) * q^7 + (-27*b2 - b1) * q^8 $$q + (\beta_{2} + \beta_1) q^{2} + ( - 2 \beta_{3} - 7) q^{4} + 5 \beta_{2} q^{5} + ( - 7 \beta_{3} - 12) q^{7} + ( - 27 \beta_{2} - \beta_1) q^{8} + ( - 5 \beta_{3} - 25) q^{10} + ( - 26 \beta_{2} + 11 \beta_1) q^{11} + (5 \beta_{3} + 202) q^{13} + ( - 138 \beta_{2} - 47 \beta_1) q^{14} + ( - 4 \beta_{3} + 41) q^{16} + (112 \beta_{2} + 44 \beta_1) q^{17} + (2 \beta_{3} - 440) q^{19} + ( - 35 \beta_{2} - 50 \beta_1) q^{20} + (15 \beta_{3} - 68) q^{22} + (294 \beta_{2} - 2 \beta_1) q^{23} - 125 q^{25} + (292 \beta_{2} + 227 \beta_1) q^{26} + (73 \beta_{3} + 1344) q^{28} + (350 \beta_{2} - 72 \beta_1) q^{29} + (114 \beta_{3} - 414) q^{31} + ( - 463 \beta_{2} + 5 \beta_1) q^{32} + ( - 156 \beta_{3} - 1352) q^{34} + ( - 60 \beta_{2} - 175 \beta_1) q^{35} + (137 \beta_{3} + 86) q^{37} + ( - 404 \beta_{2} - 430 \beta_1) q^{38} + (5 \beta_{3} + 675) q^{40} + ( - 616 \beta_{2} + 267 \beta_1) q^{41} + ( - 90 \beta_{3} + 1444) q^{43} + ( - 214 \beta_{2} + 183 \beta_1) q^{44} + ( - 292 \beta_{3} - 1434) q^{46} + (124 \beta_{2} + 422 \beta_1) q^{47} + (168 \beta_{3} + 2153) q^{49} + ( - 125 \beta_{2} - 125 \beta_1) q^{50} + ( - 439 \beta_{3} - 2314) q^{52} + (568 \beta_{2} - 962 \beta_1) q^{53} + ( - 55 \beta_{3} + 650) q^{55} + (450 \beta_{2} + 957 \beta_1) q^{56} + ( - 278 \beta_{3} - 454) q^{58} + ( - 1346 \beta_{2} + 331 \beta_1) q^{59} + (18 \beta_{3} - 5042) q^{61} + (1638 \beta_{2} + 156 \beta_1) q^{62} + (394 \beta_{3} + 2881) q^{64} + (1010 \beta_{2} + 125 \beta_1) q^{65} + (176 \beta_{3} - 396) q^{67} + ( - 2368 \beta_{2} - 1428 \beta_1) q^{68} + (235 \beta_{3} + 3450) q^{70} + ( - 388 \beta_{2} - 892 \beta_1) q^{71} + (394 \beta_{3} + 22) q^{73} + (2552 \beta_{2} + 771 \beta_1) q^{74} + (866 \beta_{3} + 2720) q^{76} + ( - 1074 \beta_{2} + 778 \beta_1) q^{77} + (134 \beta_{3} + 4434) q^{79} + (205 \beta_{2} - 100 \beta_1) q^{80} + (349 \beta_{3} - 1726) q^{82} + (744 \beta_{2} - 282 \beta_1) q^{83} + ( - 220 \beta_{3} - 2800) q^{85} + ( - 176 \beta_{2} + 994 \beta_1) q^{86} + (271 \beta_{3} - 3312) q^{88} + (3876 \beta_{2} + 291 \beta_1) q^{89} + ( - 1474 \beta_{3} - 5574) q^{91} + ( - 1986 \beta_{2} - 2926 \beta_1) q^{92} + ( - 546 \beta_{3} - 8216) q^{94} + ( - 2200 \beta_{2} + 50 \beta_1) q^{95} + (274 \beta_{3} + 6854) q^{97} + (5177 \beta_{2} + 2993 \beta_1) q^{98}+O(q^{100})$$ q + (b2 + b1) * q^2 + (-2*b3 - 7) * q^4 + 5*b2 * q^5 + (-7*b3 - 12) * q^7 + (-27*b2 - b1) * q^8 + (-5*b3 - 25) * q^10 + (-26*b2 + 11*b1) * q^11 + (5*b3 + 202) * q^13 + (-138*b2 - 47*b1) * q^14 + (-4*b3 + 41) * q^16 + (112*b2 + 44*b1) * q^17 + (2*b3 - 440) * q^19 + (-35*b2 - 50*b1) * q^20 + (15*b3 - 68) * q^22 + (294*b2 - 2*b1) * q^23 - 125 * q^25 + (292*b2 + 227*b1) * q^26 + (73*b3 + 1344) * q^28 + (350*b2 - 72*b1) * q^29 + (114*b3 - 414) * q^31 + (-463*b2 + 5*b1) * q^32 + (-156*b3 - 1352) * q^34 + (-60*b2 - 175*b1) * q^35 + (137*b3 + 86) * q^37 + (-404*b2 - 430*b1) * q^38 + (5*b3 + 675) * q^40 + (-616*b2 + 267*b1) * q^41 + (-90*b3 + 1444) * q^43 + (-214*b2 + 183*b1) * q^44 + (-292*b3 - 1434) * q^46 + (124*b2 + 422*b1) * q^47 + (168*b3 + 2153) * q^49 + (-125*b2 - 125*b1) * q^50 + (-439*b3 - 2314) * q^52 + (568*b2 - 962*b1) * q^53 + (-55*b3 + 650) * q^55 + (450*b2 + 957*b1) * q^56 + (-278*b3 - 454) * q^58 + (-1346*b2 + 331*b1) * q^59 + (18*b3 - 5042) * q^61 + (1638*b2 + 156*b1) * q^62 + (394*b3 + 2881) * q^64 + (1010*b2 + 125*b1) * q^65 + (176*b3 - 396) * q^67 + (-2368*b2 - 1428*b1) * q^68 + (235*b3 + 3450) * q^70 + (-388*b2 - 892*b1) * q^71 + (394*b3 + 22) * q^73 + (2552*b2 + 771*b1) * q^74 + (866*b3 + 2720) * q^76 + (-1074*b2 + 778*b1) * q^77 + (134*b3 + 4434) * q^79 + (205*b2 - 100*b1) * q^80 + (349*b3 - 1726) * q^82 + (744*b2 - 282*b1) * q^83 + (-220*b3 - 2800) * q^85 + (-176*b2 + 994*b1) * q^86 + (271*b3 - 3312) * q^88 + (3876*b2 + 291*b1) * q^89 + (-1474*b3 - 5574) * q^91 + (-1986*b2 - 2926*b1) * q^92 + (-546*b3 - 8216) * q^94 + (-2200*b2 + 50*b1) * q^95 + (274*b3 + 6854) * q^97 + (5177*b2 + 2993*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 28 q^{4} - 48 q^{7}+O(q^{10})$$ 4 * q - 28 * q^4 - 48 * q^7 $$4 q - 28 q^{4} - 48 q^{7} - 100 q^{10} + 808 q^{13} + 164 q^{16} - 1760 q^{19} - 272 q^{22} - 500 q^{25} + 5376 q^{28} - 1656 q^{31} - 5408 q^{34} + 344 q^{37} + 2700 q^{40} + 5776 q^{43} - 5736 q^{46} + 8612 q^{49} - 9256 q^{52} + 2600 q^{55} - 1816 q^{58} - 20168 q^{61} + 11524 q^{64} - 1584 q^{67} + 13800 q^{70} + 88 q^{73} + 10880 q^{76} + 17736 q^{79} - 6904 q^{82} - 11200 q^{85} - 13248 q^{88} - 22296 q^{91} - 32864 q^{94} + 27416 q^{97}+O(q^{100})$$ 4 * q - 28 * q^4 - 48 * q^7 - 100 * q^10 + 808 * q^13 + 164 * q^16 - 1760 * q^19 - 272 * q^22 - 500 * q^25 + 5376 * q^28 - 1656 * q^31 - 5408 * q^34 + 344 * q^37 + 2700 * q^40 + 5776 * q^43 - 5736 * q^46 + 8612 * q^49 - 9256 * q^52 + 2600 * q^55 - 1816 * q^58 - 20168 * q^61 + 11524 * q^64 - 1584 * q^67 + 13800 * q^70 + 88 * q^73 + 10880 * q^76 + 17736 * q^79 - 6904 * q^82 - 11200 * q^85 - 13248 * q^88 - 22296 * q^91 - 32864 * q^94 + 27416 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu$$ v^3 - v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 7\nu$$ -v^3 + 7*v
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 6$$ (b3 + b1) / 6 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$( \beta_{3} + 7\beta_1 ) / 6$$ (b3 + 7*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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 1.58114 − 0.707107i −1.58114 − 0.707107i −1.58114 + 0.707107i 1.58114 + 0.707107i
6.47871i 0 −25.9737 11.1803i 0 −78.4078 64.6165i 0 −72.4342
26.2 2.00657i 0 11.9737 11.1803i 0 54.4078 56.1312i 0 22.4342
26.3 2.00657i 0 11.9737 11.1803i 0 54.4078 56.1312i 0 22.4342
26.4 6.47871i 0 −25.9737 11.1803i 0 −78.4078 64.6165i 0 −72.4342
 $$n$$: 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 45.5.c.a 4
3.b odd 2 1 inner 45.5.c.a 4
4.b odd 2 1 720.5.l.c 4
5.b even 2 1 225.5.c.b 4
5.c odd 4 2 225.5.d.b 8
12.b even 2 1 720.5.l.c 4
15.d odd 2 1 225.5.c.b 4
15.e even 4 2 225.5.d.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.5.c.a 4 1.a even 1 1 trivial
45.5.c.a 4 3.b odd 2 1 inner
225.5.c.b 4 5.b even 2 1
225.5.c.b 4 15.d odd 2 1
225.5.d.b 8 5.c odd 4 2
225.5.d.b 8 15.e even 4 2
720.5.l.c 4 4.b odd 2 1
720.5.l.c 4 12.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 46T^{2} + 169$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 125)^{2}$$
$7$ $$(T^{2} + 24 T - 4266)^{2}$$
$11$ $$T^{4} + 11116 T^{2} + \cdots + 1444804$$
$13$ $$(T^{2} - 404 T + 38554)^{2}$$
$17$ $$T^{4} + 195136 T^{2} + \cdots + 776848384$$
$19$ $$(T^{2} + 880 T + 193240)^{2}$$
$23$ $$T^{4} + 864504 T^{2} + \cdots + 186717323664$$
$29$ $$T^{4} + 1411624 T^{2} + \cdots + 269556179344$$
$31$ $$(T^{2} + 828 T - 998244)^{2}$$
$37$ $$(T^{2} - 172 T - 1681814)^{2}$$
$41$ $$T^{4} + 6360964 T^{2} + \cdots + 377091790084$$
$43$ $$(T^{2} - 2888 T + 1356136)^{2}$$
$47$ $$T^{4} + 6564784 T^{2} + \cdots + 9788338191424$$
$53$ $$T^{4} + \cdots + 226348173496384$$
$59$ $$T^{4} + 22061356 T^{2} + \cdots + 50218227136324$$
$61$ $$(T^{2} + 10084 T + 25392604)^{2}$$
$67$ $$(T^{2} + 792 T - 2631024)^{2}$$
$71$ $$T^{4} + \cdots + 184124057069824$$
$73$ $$(T^{2} - 44 T - 13970756)^{2}$$
$79$ $$(T^{2} - 8868 T + 18044316)^{2}$$
$83$ $$T^{4} + 8398224 T^{2} + \cdots + 1785558717504$$
$89$ $$T^{4} + 153282276 T^{2} + \cdots + 54\!\cdots\!84$$
$97$ $$(T^{2} - 13708 T + 40220476)^{2}$$