# Properties

 Label 450.4.a.k Level $450$ Weight $4$ Character orbit 450.a Self dual yes Analytic conductor $26.551$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(1,450)]

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, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 450.a (trivial)

## 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: yes Analytic conductor: $$26.5508595026$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} - 26 q^{7} + 8 q^{8}+O(q^{10})$$ q + 2 * q^2 + 4 * q^4 - 26 * q^7 + 8 * q^8 $$q + 2 q^{2} + 4 q^{4} - 26 q^{7} + 8 q^{8} + 28 q^{11} - 12 q^{13} - 52 q^{14} + 16 q^{16} - 64 q^{17} - 60 q^{19} + 56 q^{22} - 58 q^{23} - 24 q^{26} - 104 q^{28} - 90 q^{29} - 128 q^{31} + 32 q^{32} - 128 q^{34} - 236 q^{37} - 120 q^{38} - 242 q^{41} - 362 q^{43} + 112 q^{44} - 116 q^{46} + 226 q^{47} + 333 q^{49} - 48 q^{52} - 108 q^{53} - 208 q^{56} - 180 q^{58} + 20 q^{59} + 542 q^{61} - 256 q^{62} + 64 q^{64} + 434 q^{67} - 256 q^{68} + 1128 q^{71} - 632 q^{73} - 472 q^{74} - 240 q^{76} - 728 q^{77} - 720 q^{79} - 484 q^{82} - 478 q^{83} - 724 q^{86} + 224 q^{88} + 490 q^{89} + 312 q^{91} - 232 q^{92} + 452 q^{94} - 1456 q^{97} + 666 q^{98}+O(q^{100})$$ q + 2 * q^2 + 4 * q^4 - 26 * q^7 + 8 * q^8 + 28 * q^11 - 12 * q^13 - 52 * q^14 + 16 * q^16 - 64 * q^17 - 60 * q^19 + 56 * q^22 - 58 * q^23 - 24 * q^26 - 104 * q^28 - 90 * q^29 - 128 * q^31 + 32 * q^32 - 128 * q^34 - 236 * q^37 - 120 * q^38 - 242 * q^41 - 362 * q^43 + 112 * q^44 - 116 * q^46 + 226 * q^47 + 333 * q^49 - 48 * q^52 - 108 * q^53 - 208 * q^56 - 180 * q^58 + 20 * q^59 + 542 * q^61 - 256 * q^62 + 64 * q^64 + 434 * q^67 - 256 * q^68 + 1128 * q^71 - 632 * q^73 - 472 * q^74 - 240 * q^76 - 728 * q^77 - 720 * q^79 - 484 * q^82 - 478 * q^83 - 724 * q^86 + 224 * q^88 + 490 * q^89 + 312 * q^91 - 232 * q^92 + 452 * q^94 - 1456 * q^97 + 666 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
2.00000 0 4.00000 0 0 −26.0000 8.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.4.a.k 1
3.b odd 2 1 50.4.a.b 1
5.b even 2 1 450.4.a.j 1
5.c odd 4 2 90.4.c.b 2
12.b even 2 1 400.4.a.n 1
15.d odd 2 1 50.4.a.d 1
15.e even 4 2 10.4.b.a 2
20.e even 4 2 720.4.f.f 2
21.c even 2 1 2450.4.a.o 1
24.f even 2 1 1600.4.a.t 1
24.h odd 2 1 1600.4.a.bh 1
60.h even 2 1 400.4.a.h 1
60.l odd 4 2 80.4.c.a 2
105.g even 2 1 2450.4.a.bb 1
105.k odd 4 2 490.4.c.b 2
120.i odd 2 1 1600.4.a.u 1
120.m even 2 1 1600.4.a.bg 1
120.q odd 4 2 320.4.c.c 2
120.w even 4 2 320.4.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 15.e even 4 2
50.4.a.b 1 3.b odd 2 1
50.4.a.d 1 15.d odd 2 1
80.4.c.a 2 60.l odd 4 2
90.4.c.b 2 5.c odd 4 2
320.4.c.c 2 120.q odd 4 2
320.4.c.d 2 120.w even 4 2
400.4.a.h 1 60.h even 2 1
400.4.a.n 1 12.b even 2 1
450.4.a.j 1 5.b even 2 1
450.4.a.k 1 1.a even 1 1 trivial
490.4.c.b 2 105.k odd 4 2
720.4.f.f 2 20.e even 4 2
1600.4.a.t 1 24.f even 2 1
1600.4.a.u 1 120.i odd 2 1
1600.4.a.bg 1 120.m even 2 1
1600.4.a.bh 1 24.h odd 2 1
2450.4.a.o 1 21.c even 2 1
2450.4.a.bb 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(450))$$:

 $$T_{7} + 26$$ T7 + 26 $$T_{11} - 28$$ T11 - 28 $$T_{17} + 64$$ T17 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 26$$
$11$ $$T - 28$$
$13$ $$T + 12$$
$17$ $$T + 64$$
$19$ $$T + 60$$
$23$ $$T + 58$$
$29$ $$T + 90$$
$31$ $$T + 128$$
$37$ $$T + 236$$
$41$ $$T + 242$$
$43$ $$T + 362$$
$47$ $$T - 226$$
$53$ $$T + 108$$
$59$ $$T - 20$$
$61$ $$T - 542$$
$67$ $$T - 434$$
$71$ $$T - 1128$$
$73$ $$T + 632$$
$79$ $$T + 720$$
$83$ $$T + 478$$
$89$ $$T - 490$$
$97$ $$T + 1456$$