# Properties

 Label 450.4.c.d Level $450$ Weight $4$ Character orbit 450.c Analytic conductor $26.551$ Analytic rank $0$ Dimension $2$ 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 10) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 4 q^{4} + 2 \beta q^{7} - 4 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 4 * q^4 + 2*b * q^7 - 4*b * q^8 $$q + \beta q^{2} - 4 q^{4} + 2 \beta q^{7} - 4 \beta q^{8} - 12 q^{11} - 29 \beta q^{13} - 8 q^{14} + 16 q^{16} + 33 \beta q^{17} + 100 q^{19} - 12 \beta q^{22} - 66 \beta q^{23} + 116 q^{26} - 8 \beta q^{28} - 90 q^{29} + 152 q^{31} + 16 \beta q^{32} - 132 q^{34} + 17 \beta q^{37} + 100 \beta q^{38} + 438 q^{41} + 16 \beta q^{43} + 48 q^{44} + 264 q^{46} - 102 \beta q^{47} + 327 q^{49} + 116 \beta q^{52} - 111 \beta q^{53} + 32 q^{56} - 90 \beta q^{58} + 420 q^{59} + 902 q^{61} + 152 \beta q^{62} - 64 q^{64} + 512 \beta q^{67} - 132 \beta q^{68} - 432 q^{71} + 181 \beta q^{73} - 68 q^{74} - 400 q^{76} - 24 \beta q^{77} + 160 q^{79} + 438 \beta q^{82} - 36 \beta q^{83} - 64 q^{86} + 48 \beta q^{88} + 810 q^{89} + 232 q^{91} + 264 \beta q^{92} + 408 q^{94} - 553 \beta q^{97} + 327 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 4 * q^4 + 2*b * q^7 - 4*b * q^8 - 12 * q^11 - 29*b * q^13 - 8 * q^14 + 16 * q^16 + 33*b * q^17 + 100 * q^19 - 12*b * q^22 - 66*b * q^23 + 116 * q^26 - 8*b * q^28 - 90 * q^29 + 152 * q^31 + 16*b * q^32 - 132 * q^34 + 17*b * q^37 + 100*b * q^38 + 438 * q^41 + 16*b * q^43 + 48 * q^44 + 264 * q^46 - 102*b * q^47 + 327 * q^49 + 116*b * q^52 - 111*b * q^53 + 32 * q^56 - 90*b * q^58 + 420 * q^59 + 902 * q^61 + 152*b * q^62 - 64 * q^64 + 512*b * q^67 - 132*b * q^68 - 432 * q^71 + 181*b * q^73 - 68 * q^74 - 400 * q^76 - 24*b * q^77 + 160 * q^79 + 438*b * q^82 - 36*b * q^83 - 64 * q^86 + 48*b * q^88 + 810 * q^89 + 232 * q^91 + 264*b * q^92 + 408 * q^94 - 553*b * q^97 + 327*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} - 24 q^{11} - 16 q^{14} + 32 q^{16} + 200 q^{19} + 232 q^{26} - 180 q^{29} + 304 q^{31} - 264 q^{34} + 876 q^{41} + 96 q^{44} + 528 q^{46} + 654 q^{49} + 64 q^{56} + 840 q^{59} + 1804 q^{61} - 128 q^{64} - 864 q^{71} - 136 q^{74} - 800 q^{76} + 320 q^{79} - 128 q^{86} + 1620 q^{89} + 464 q^{91} + 816 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 - 24 * q^11 - 16 * q^14 + 32 * q^16 + 200 * q^19 + 232 * q^26 - 180 * q^29 + 304 * q^31 - 264 * q^34 + 876 * q^41 + 96 * q^44 + 528 * q^46 + 654 * q^49 + 64 * q^56 + 840 * q^59 + 1804 * q^61 - 128 * q^64 - 864 * q^71 - 136 * q^74 - 800 * q^76 + 320 * q^79 - 128 * q^86 + 1620 * q^89 + 464 * q^91 + 816 * 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 4.00000i 8.00000i 0 0
199.2 2.00000i 0 −4.00000 0 0 4.00000i 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.d 2
3.b odd 2 1 50.4.b.a 2
5.b even 2 1 inner 450.4.c.d 2
5.c odd 4 1 90.4.a.a 1
5.c odd 4 1 450.4.a.q 1
12.b even 2 1 400.4.c.c 2
15.d odd 2 1 50.4.b.a 2
15.e even 4 1 10.4.a.a 1
15.e even 4 1 50.4.a.c 1
20.e even 4 1 720.4.a.j 1
45.k odd 12 2 810.4.e.w 2
45.l even 12 2 810.4.e.c 2
60.h even 2 1 400.4.c.c 2
60.l odd 4 1 80.4.a.f 1
60.l odd 4 1 400.4.a.b 1
105.k odd 4 1 490.4.a.o 1
105.k odd 4 1 2450.4.a.b 1
105.w odd 12 2 490.4.e.a 2
105.x even 12 2 490.4.e.i 2
120.q odd 4 1 320.4.a.b 1
120.q odd 4 1 1600.4.a.bx 1
120.w even 4 1 320.4.a.m 1
120.w even 4 1 1600.4.a.d 1
165.l odd 4 1 1210.4.a.b 1
195.s even 4 1 1690.4.a.a 1
240.z odd 4 1 1280.4.d.g 2
240.bb even 4 1 1280.4.d.j 2
240.bd odd 4 1 1280.4.d.g 2
240.bf even 4 1 1280.4.d.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 15.e even 4 1
50.4.a.c 1 15.e even 4 1
50.4.b.a 2 3.b odd 2 1
50.4.b.a 2 15.d odd 2 1
80.4.a.f 1 60.l odd 4 1
90.4.a.a 1 5.c odd 4 1
320.4.a.b 1 120.q odd 4 1
320.4.a.m 1 120.w even 4 1
400.4.a.b 1 60.l odd 4 1
400.4.c.c 2 12.b even 2 1
400.4.c.c 2 60.h even 2 1
450.4.a.q 1 5.c odd 4 1
450.4.c.d 2 1.a even 1 1 trivial
450.4.c.d 2 5.b even 2 1 inner
490.4.a.o 1 105.k odd 4 1
490.4.e.a 2 105.w odd 12 2
490.4.e.i 2 105.x even 12 2
720.4.a.j 1 20.e even 4 1
810.4.e.c 2 45.l even 12 2
810.4.e.w 2 45.k odd 12 2
1210.4.a.b 1 165.l odd 4 1
1280.4.d.g 2 240.z odd 4 1
1280.4.d.g 2 240.bd odd 4 1
1280.4.d.j 2 240.bb even 4 1
1280.4.d.j 2 240.bf even 4 1
1600.4.a.d 1 120.w even 4 1
1600.4.a.bx 1 120.q odd 4 1
1690.4.a.a 1 195.s even 4 1
2450.4.a.b 1 105.k odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} + 12$$ T11 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 12)^{2}$$
$13$ $$T^{2} + 3364$$
$17$ $$T^{2} + 4356$$
$19$ $$(T - 100)^{2}$$
$23$ $$T^{2} + 17424$$
$29$ $$(T + 90)^{2}$$
$31$ $$(T - 152)^{2}$$
$37$ $$T^{2} + 1156$$
$41$ $$(T - 438)^{2}$$
$43$ $$T^{2} + 1024$$
$47$ $$T^{2} + 41616$$
$53$ $$T^{2} + 49284$$
$59$ $$(T - 420)^{2}$$
$61$ $$(T - 902)^{2}$$
$67$ $$T^{2} + 1048576$$
$71$ $$(T + 432)^{2}$$
$73$ $$T^{2} + 131044$$
$79$ $$(T - 160)^{2}$$
$83$ $$T^{2} + 5184$$
$89$ $$(T - 810)^{2}$$
$97$ $$T^{2} + 1223236$$