# Properties

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 15.d odd 2 1
45.4.a.e yes 1 5.b even 2 1
225.4.a.a 1 1.a even 1 1 trivial
225.4.a.h 1 3.b odd 2 1
225.4.b.a 2 15.e even 4 2
225.4.b.b 2 5.c odd 4 2
405.4.e.b 2 45.j even 6 2
405.4.e.n 2 45.h odd 6 2
720.4.a.o 1 20.d odd 2 1
720.4.a.bc 1 60.h even 2 1
2205.4.a.a 1 105.g even 2 1
2205.4.a.t 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(225))$$:

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

## Hecke characteristic polynomials

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