# Properties

 Label 225.4.a.h Level $225$ Weight $4$ Character orbit 225.a Self dual yes Analytic conductor $13.275$ Analytic rank $0$ 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] = [225,4,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, 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("225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.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: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{2} + 17 q^{4} + 30 q^{7} + 45 q^{8}+O(q^{10})$$ q + 5 * q^2 + 17 * q^4 + 30 * q^7 + 45 * q^8 $$q + 5 q^{2} + 17 q^{4} + 30 q^{7} + 45 q^{8} - 50 q^{11} + 20 q^{13} + 150 q^{14} + 89 q^{16} - 10 q^{17} - 44 q^{19} - 250 q^{22} + 120 q^{23} + 100 q^{26} + 510 q^{28} + 50 q^{29} + 108 q^{31} + 85 q^{32} - 50 q^{34} + 40 q^{37} - 220 q^{38} - 400 q^{41} - 280 q^{43} - 850 q^{44} + 600 q^{46} - 280 q^{47} + 557 q^{49} + 340 q^{52} - 610 q^{53} + 1350 q^{56} + 250 q^{58} - 50 q^{59} - 518 q^{61} + 540 q^{62} - 287 q^{64} + 180 q^{67} - 170 q^{68} - 700 q^{71} + 410 q^{73} + 200 q^{74} - 748 q^{76} - 1500 q^{77} - 516 q^{79} - 2000 q^{82} + 660 q^{83} - 1400 q^{86} - 2250 q^{88} + 1500 q^{89} + 600 q^{91} + 2040 q^{92} - 1400 q^{94} + 1630 q^{97} + 2785 q^{98}+O(q^{100})$$ q + 5 * q^2 + 17 * q^4 + 30 * q^7 + 45 * q^8 - 50 * q^11 + 20 * q^13 + 150 * q^14 + 89 * q^16 - 10 * q^17 - 44 * q^19 - 250 * q^22 + 120 * q^23 + 100 * q^26 + 510 * q^28 + 50 * q^29 + 108 * q^31 + 85 * q^32 - 50 * q^34 + 40 * q^37 - 220 * q^38 - 400 * q^41 - 280 * q^43 - 850 * q^44 + 600 * q^46 - 280 * q^47 + 557 * q^49 + 340 * q^52 - 610 * q^53 + 1350 * q^56 + 250 * q^58 - 50 * q^59 - 518 * q^61 + 540 * q^62 - 287 * q^64 + 180 * q^67 - 170 * q^68 - 700 * q^71 + 410 * q^73 + 200 * q^74 - 748 * q^76 - 1500 * q^77 - 516 * q^79 - 2000 * q^82 + 660 * q^83 - 1400 * q^86 - 2250 * q^88 + 1500 * q^89 + 600 * q^91 + 2040 * q^92 - 1400 * q^94 + 1630 * q^97 + 2785 * 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
5.00000 0 17.0000 0 0 30.0000 45.0000 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
$$3$$ $$1$$
$$5$$ $$1$$

## 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 225.4.a.h 1
3.b odd 2 1 225.4.a.a 1
5.b even 2 1 45.4.a.a 1
5.c odd 4 2 225.4.b.a 2
15.d odd 2 1 45.4.a.e yes 1
15.e even 4 2 225.4.b.b 2
20.d odd 2 1 720.4.a.bc 1
35.c odd 2 1 2205.4.a.a 1
45.h odd 6 2 405.4.e.b 2
45.j even 6 2 405.4.e.n 2
60.h even 2 1 720.4.a.o 1
105.g even 2 1 2205.4.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 5.b even 2 1
45.4.a.e yes 1 15.d odd 2 1
225.4.a.a 1 3.b odd 2 1
225.4.a.h 1 1.a even 1 1 trivial
225.4.b.a 2 5.c odd 4 2
225.4.b.b 2 15.e even 4 2
405.4.e.b 2 45.h odd 6 2
405.4.e.n 2 45.j even 6 2
720.4.a.o 1 60.h even 2 1
720.4.a.bc 1 20.d odd 2 1
2205.4.a.a 1 35.c odd 2 1
2205.4.a.t 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2} - 5$$ T2 - 5 $$T_{7} - 30$$ T7 - 30

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 5$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 30$$
$11$ $$T + 50$$
$13$ $$T - 20$$
$17$ $$T + 10$$
$19$ $$T + 44$$
$23$ $$T - 120$$
$29$ $$T - 50$$
$31$ $$T - 108$$
$37$ $$T - 40$$
$41$ $$T + 400$$
$43$ $$T + 280$$
$47$ $$T + 280$$
$53$ $$T + 610$$
$59$ $$T + 50$$
$61$ $$T + 518$$
$67$ $$T - 180$$
$71$ $$T + 700$$
$73$ $$T - 410$$
$79$ $$T + 516$$
$83$ $$T - 660$$
$89$ $$T - 1500$$
$97$ $$T - 1630$$