# Properties

 Label 450.4.a.h 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 6) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} + 16 q^{7} - 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + 4 * q^4 + 16 * q^7 - 8 * q^8 $$q - 2 q^{2} + 4 q^{4} + 16 q^{7} - 8 q^{8} - 12 q^{11} - 38 q^{13} - 32 q^{14} + 16 q^{16} - 126 q^{17} + 20 q^{19} + 24 q^{22} + 168 q^{23} + 76 q^{26} + 64 q^{28} - 30 q^{29} - 88 q^{31} - 32 q^{32} + 252 q^{34} - 254 q^{37} - 40 q^{38} - 42 q^{41} + 52 q^{43} - 48 q^{44} - 336 q^{46} - 96 q^{47} - 87 q^{49} - 152 q^{52} + 198 q^{53} - 128 q^{56} + 60 q^{58} + 660 q^{59} - 538 q^{61} + 176 q^{62} + 64 q^{64} - 884 q^{67} - 504 q^{68} - 792 q^{71} - 218 q^{73} + 508 q^{74} + 80 q^{76} - 192 q^{77} - 520 q^{79} + 84 q^{82} - 492 q^{83} - 104 q^{86} + 96 q^{88} - 810 q^{89} - 608 q^{91} + 672 q^{92} + 192 q^{94} - 1154 q^{97} + 174 q^{98}+O(q^{100})$$ q - 2 * q^2 + 4 * q^4 + 16 * q^7 - 8 * q^8 - 12 * q^11 - 38 * q^13 - 32 * q^14 + 16 * q^16 - 126 * q^17 + 20 * q^19 + 24 * q^22 + 168 * q^23 + 76 * q^26 + 64 * q^28 - 30 * q^29 - 88 * q^31 - 32 * q^32 + 252 * q^34 - 254 * q^37 - 40 * q^38 - 42 * q^41 + 52 * q^43 - 48 * q^44 - 336 * q^46 - 96 * q^47 - 87 * q^49 - 152 * q^52 + 198 * q^53 - 128 * q^56 + 60 * q^58 + 660 * q^59 - 538 * q^61 + 176 * q^62 + 64 * q^64 - 884 * q^67 - 504 * q^68 - 792 * q^71 - 218 * q^73 + 508 * q^74 + 80 * q^76 - 192 * q^77 - 520 * q^79 + 84 * q^82 - 492 * q^83 - 104 * q^86 + 96 * q^88 - 810 * q^89 - 608 * q^91 + 672 * q^92 + 192 * q^94 - 1154 * q^97 + 174 * 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 16.0000 −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.h 1
3.b odd 2 1 150.4.a.i 1
5.b even 2 1 18.4.a.a 1
5.c odd 4 2 450.4.c.e 2
12.b even 2 1 1200.4.a.b 1
15.d odd 2 1 6.4.a.a 1
15.e even 4 2 150.4.c.d 2
20.d odd 2 1 144.4.a.c 1
35.c odd 2 1 882.4.a.n 1
35.i odd 6 2 882.4.g.f 2
35.j even 6 2 882.4.g.i 2
40.e odd 2 1 576.4.a.r 1
40.f even 2 1 576.4.a.q 1
45.h odd 6 2 162.4.c.f 2
45.j even 6 2 162.4.c.c 2
55.d odd 2 1 2178.4.a.e 1
60.h even 2 1 48.4.a.c 1
60.l odd 4 2 1200.4.f.j 2
105.g even 2 1 294.4.a.e 1
105.o odd 6 2 294.4.e.h 2
105.p even 6 2 294.4.e.g 2
120.i odd 2 1 192.4.a.i 1
120.m even 2 1 192.4.a.c 1
165.d even 2 1 726.4.a.f 1
195.e odd 2 1 1014.4.a.g 1
195.n even 4 2 1014.4.b.d 2
240.t even 4 2 768.4.d.c 2
240.bm odd 4 2 768.4.d.n 2
255.h odd 2 1 1734.4.a.d 1
285.b even 2 1 2166.4.a.i 1
420.o odd 2 1 2352.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 15.d odd 2 1
18.4.a.a 1 5.b even 2 1
48.4.a.c 1 60.h even 2 1
144.4.a.c 1 20.d odd 2 1
150.4.a.i 1 3.b odd 2 1
150.4.c.d 2 15.e even 4 2
162.4.c.c 2 45.j even 6 2
162.4.c.f 2 45.h odd 6 2
192.4.a.c 1 120.m even 2 1
192.4.a.i 1 120.i odd 2 1
294.4.a.e 1 105.g even 2 1
294.4.e.g 2 105.p even 6 2
294.4.e.h 2 105.o odd 6 2
450.4.a.h 1 1.a even 1 1 trivial
450.4.c.e 2 5.c odd 4 2
576.4.a.q 1 40.f even 2 1
576.4.a.r 1 40.e odd 2 1
726.4.a.f 1 165.d even 2 1
768.4.d.c 2 240.t even 4 2
768.4.d.n 2 240.bm odd 4 2
882.4.a.n 1 35.c odd 2 1
882.4.g.f 2 35.i odd 6 2
882.4.g.i 2 35.j even 6 2
1014.4.a.g 1 195.e odd 2 1
1014.4.b.d 2 195.n even 4 2
1200.4.a.b 1 12.b even 2 1
1200.4.f.j 2 60.l odd 4 2
1734.4.a.d 1 255.h odd 2 1
2166.4.a.i 1 285.b even 2 1
2178.4.a.e 1 55.d odd 2 1
2352.4.a.e 1 420.o odd 2 1

## Hecke kernels

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

 $$T_{7} - 16$$ T7 - 16 $$T_{11} + 12$$ T11 + 12 $$T_{17} + 126$$ T17 + 126

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 16$$
$11$ $$T + 12$$
$13$ $$T + 38$$
$17$ $$T + 126$$
$19$ $$T - 20$$
$23$ $$T - 168$$
$29$ $$T + 30$$
$31$ $$T + 88$$
$37$ $$T + 254$$
$41$ $$T + 42$$
$43$ $$T - 52$$
$47$ $$T + 96$$
$53$ $$T - 198$$
$59$ $$T - 660$$
$61$ $$T + 538$$
$67$ $$T + 884$$
$71$ $$T + 792$$
$73$ $$T + 218$$
$79$ $$T + 520$$
$83$ $$T + 492$$
$89$ $$T + 810$$
$97$ $$T + 1154$$