# Properties

 Label 450.4.c.e Level $450$ Weight $4$ Character orbit 450.c Analytic conductor $26.551$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(199,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, 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("450.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 450.c (of order $$2$$, degree $$1$$, not 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: $$26.5508595026$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 6) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - 4 q^{4} + 8 \beta q^{7} + 4 \beta q^{8} +O(q^{10})$$ q - b * q^2 - 4 * q^4 + 8*b * q^7 + 4*b * q^8 $$q - \beta q^{2} - 4 q^{4} + 8 \beta q^{7} + 4 \beta q^{8} - 12 q^{11} + 19 \beta q^{13} + 32 q^{14} + 16 q^{16} - 63 \beta q^{17} - 20 q^{19} + 12 \beta q^{22} - 84 \beta q^{23} + 76 q^{26} - 32 \beta q^{28} + 30 q^{29} - 88 q^{31} - 16 \beta q^{32} - 252 q^{34} - 127 \beta q^{37} + 20 \beta q^{38} - 42 q^{41} - 26 \beta q^{43} + 48 q^{44} - 336 q^{46} - 48 \beta q^{47} + 87 q^{49} - 76 \beta q^{52} - 99 \beta q^{53} - 128 q^{56} - 30 \beta q^{58} - 660 q^{59} - 538 q^{61} + 88 \beta q^{62} - 64 q^{64} - 442 \beta q^{67} + 252 \beta q^{68} - 792 q^{71} + 109 \beta q^{73} - 508 q^{74} + 80 q^{76} - 96 \beta q^{77} + 520 q^{79} + 42 \beta q^{82} + 246 \beta q^{83} - 104 q^{86} - 48 \beta q^{88} + 810 q^{89} - 608 q^{91} + 336 \beta q^{92} - 192 q^{94} - 577 \beta q^{97} - 87 \beta q^{98} +O(q^{100})$$ q - b * q^2 - 4 * q^4 + 8*b * q^7 + 4*b * q^8 - 12 * q^11 + 19*b * q^13 + 32 * q^14 + 16 * q^16 - 63*b * q^17 - 20 * q^19 + 12*b * q^22 - 84*b * q^23 + 76 * q^26 - 32*b * q^28 + 30 * q^29 - 88 * q^31 - 16*b * q^32 - 252 * q^34 - 127*b * q^37 + 20*b * q^38 - 42 * q^41 - 26*b * q^43 + 48 * q^44 - 336 * q^46 - 48*b * q^47 + 87 * q^49 - 76*b * q^52 - 99*b * q^53 - 128 * q^56 - 30*b * q^58 - 660 * q^59 - 538 * q^61 + 88*b * q^62 - 64 * q^64 - 442*b * q^67 + 252*b * q^68 - 792 * q^71 + 109*b * q^73 - 508 * q^74 + 80 * q^76 - 96*b * q^77 + 520 * q^79 + 42*b * q^82 + 246*b * q^83 - 104 * q^86 - 48*b * q^88 + 810 * q^89 - 608 * q^91 + 336*b * q^92 - 192 * q^94 - 577*b * q^97 - 87*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} - 24 q^{11} + 64 q^{14} + 32 q^{16} - 40 q^{19} + 152 q^{26} + 60 q^{29} - 176 q^{31} - 504 q^{34} - 84 q^{41} + 96 q^{44} - 672 q^{46} + 174 q^{49} - 256 q^{56} - 1320 q^{59} - 1076 q^{61} - 128 q^{64} - 1584 q^{71} - 1016 q^{74} + 160 q^{76} + 1040 q^{79} - 208 q^{86} + 1620 q^{89} - 1216 q^{91} - 384 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 - 24 * q^11 + 64 * q^14 + 32 * q^16 - 40 * q^19 + 152 * q^26 + 60 * q^29 - 176 * q^31 - 504 * q^34 - 84 * q^41 + 96 * q^44 - 672 * q^46 + 174 * q^49 - 256 * q^56 - 1320 * q^59 - 1076 * q^61 - 128 * q^64 - 1584 * q^71 - 1016 * q^74 + 160 * q^76 + 1040 * q^79 - 208 * q^86 + 1620 * q^89 - 1216 * q^91 - 384 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 1.00000i − 1.00000i
2.00000i 0 −4.00000 0 0 16.0000i 8.00000i 0 0
199.2 2.00000i 0 −4.00000 0 0 16.0000i 8.00000i 0 0
 $$n$$: 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 450.4.c.e 2
3.b odd 2 1 150.4.c.d 2
5.b even 2 1 inner 450.4.c.e 2
5.c odd 4 1 18.4.a.a 1
5.c odd 4 1 450.4.a.h 1
12.b even 2 1 1200.4.f.j 2
15.d odd 2 1 150.4.c.d 2
15.e even 4 1 6.4.a.a 1
15.e even 4 1 150.4.a.i 1
20.e even 4 1 144.4.a.c 1
35.f even 4 1 882.4.a.n 1
35.k even 12 2 882.4.g.f 2
35.l odd 12 2 882.4.g.i 2
40.i odd 4 1 576.4.a.q 1
40.k even 4 1 576.4.a.r 1
45.k odd 12 2 162.4.c.c 2
45.l even 12 2 162.4.c.f 2
55.e even 4 1 2178.4.a.e 1
60.h even 2 1 1200.4.f.j 2
60.l odd 4 1 48.4.a.c 1
60.l odd 4 1 1200.4.a.b 1
105.k odd 4 1 294.4.a.e 1
105.w odd 12 2 294.4.e.g 2
105.x even 12 2 294.4.e.h 2
120.q odd 4 1 192.4.a.c 1
120.w even 4 1 192.4.a.i 1
165.l odd 4 1 726.4.a.f 1
195.j odd 4 1 1014.4.b.d 2
195.s even 4 1 1014.4.a.g 1
195.u odd 4 1 1014.4.b.d 2
240.z odd 4 1 768.4.d.c 2
240.bb even 4 1 768.4.d.n 2
240.bd odd 4 1 768.4.d.c 2
240.bf even 4 1 768.4.d.n 2
255.o even 4 1 1734.4.a.d 1
285.j odd 4 1 2166.4.a.i 1
420.w even 4 1 2352.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 15.e even 4 1
18.4.a.a 1 5.c odd 4 1
48.4.a.c 1 60.l odd 4 1
144.4.a.c 1 20.e even 4 1
150.4.a.i 1 15.e even 4 1
150.4.c.d 2 3.b odd 2 1
150.4.c.d 2 15.d odd 2 1
162.4.c.c 2 45.k odd 12 2
162.4.c.f 2 45.l even 12 2
192.4.a.c 1 120.q odd 4 1
192.4.a.i 1 120.w even 4 1
294.4.a.e 1 105.k odd 4 1
294.4.e.g 2 105.w odd 12 2
294.4.e.h 2 105.x even 12 2
450.4.a.h 1 5.c odd 4 1
450.4.c.e 2 1.a even 1 1 trivial
450.4.c.e 2 5.b even 2 1 inner
576.4.a.q 1 40.i odd 4 1
576.4.a.r 1 40.k even 4 1
726.4.a.f 1 165.l odd 4 1
768.4.d.c 2 240.z odd 4 1
768.4.d.c 2 240.bd odd 4 1
768.4.d.n 2 240.bb even 4 1
768.4.d.n 2 240.bf even 4 1
882.4.a.n 1 35.f even 4 1
882.4.g.f 2 35.k even 12 2
882.4.g.i 2 35.l odd 12 2
1014.4.a.g 1 195.s even 4 1
1014.4.b.d 2 195.j odd 4 1
1014.4.b.d 2 195.u odd 4 1
1200.4.a.b 1 60.l odd 4 1
1200.4.f.j 2 12.b even 2 1
1200.4.f.j 2 60.h even 2 1
1734.4.a.d 1 255.o even 4 1
2166.4.a.i 1 285.j odd 4 1
2178.4.a.e 1 55.e even 4 1
2352.4.a.e 1 420.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{2} + 256$$ T7^2 + 256 $$T_{11} + 12$$ T11 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 256$$
$11$ $$(T + 12)^{2}$$
$13$ $$T^{2} + 1444$$
$17$ $$T^{2} + 15876$$
$19$ $$(T + 20)^{2}$$
$23$ $$T^{2} + 28224$$
$29$ $$(T - 30)^{2}$$
$31$ $$(T + 88)^{2}$$
$37$ $$T^{2} + 64516$$
$41$ $$(T + 42)^{2}$$
$43$ $$T^{2} + 2704$$
$47$ $$T^{2} + 9216$$
$53$ $$T^{2} + 39204$$
$59$ $$(T + 660)^{2}$$
$61$ $$(T + 538)^{2}$$
$67$ $$T^{2} + 781456$$
$71$ $$(T + 792)^{2}$$
$73$ $$T^{2} + 47524$$
$79$ $$(T - 520)^{2}$$
$83$ $$T^{2} + 242064$$
$89$ $$(T - 810)^{2}$$
$97$ $$T^{2} + 1331716$$