# Properties

 Label 225.4.a.f Level $225$ Weight $4$ Character orbit 225.a Self dual yes Analytic conductor $13.275$ 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] = [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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 7 q^{4} + 24 q^{7} - 15 q^{8}+O(q^{10})$$ q + q^2 - 7 * q^4 + 24 * q^7 - 15 * q^8 $$q + q^{2} - 7 q^{4} + 24 q^{7} - 15 q^{8} - 52 q^{11} - 22 q^{13} + 24 q^{14} + 41 q^{16} - 14 q^{17} - 20 q^{19} - 52 q^{22} - 168 q^{23} - 22 q^{26} - 168 q^{28} - 230 q^{29} - 288 q^{31} + 161 q^{32} - 14 q^{34} + 34 q^{37} - 20 q^{38} - 122 q^{41} + 188 q^{43} + 364 q^{44} - 168 q^{46} + 256 q^{47} + 233 q^{49} + 154 q^{52} - 338 q^{53} - 360 q^{56} - 230 q^{58} - 100 q^{59} + 742 q^{61} - 288 q^{62} - 167 q^{64} + 84 q^{67} + 98 q^{68} + 328 q^{71} + 38 q^{73} + 34 q^{74} + 140 q^{76} - 1248 q^{77} - 240 q^{79} - 122 q^{82} + 1212 q^{83} + 188 q^{86} + 780 q^{88} - 330 q^{89} - 528 q^{91} + 1176 q^{92} + 256 q^{94} - 866 q^{97} + 233 q^{98}+O(q^{100})$$ q + q^2 - 7 * q^4 + 24 * q^7 - 15 * q^8 - 52 * q^11 - 22 * q^13 + 24 * q^14 + 41 * q^16 - 14 * q^17 - 20 * q^19 - 52 * q^22 - 168 * q^23 - 22 * q^26 - 168 * q^28 - 230 * q^29 - 288 * q^31 + 161 * q^32 - 14 * q^34 + 34 * q^37 - 20 * q^38 - 122 * q^41 + 188 * q^43 + 364 * q^44 - 168 * q^46 + 256 * q^47 + 233 * q^49 + 154 * q^52 - 338 * q^53 - 360 * q^56 - 230 * q^58 - 100 * q^59 + 742 * q^61 - 288 * q^62 - 167 * q^64 + 84 * q^67 + 98 * q^68 + 328 * q^71 + 38 * q^73 + 34 * q^74 + 140 * q^76 - 1248 * q^77 - 240 * q^79 - 122 * q^82 + 1212 * q^83 + 188 * q^86 + 780 * q^88 - 330 * q^89 - 528 * q^91 + 1176 * q^92 + 256 * q^94 - 866 * q^97 + 233 * 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
1.00000 0 −7.00000 0 0 24.0000 −15.0000 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.f 1
3.b odd 2 1 75.4.a.b 1
5.b even 2 1 45.4.a.c 1
5.c odd 4 2 225.4.b.e 2
12.b even 2 1 1200.4.a.t 1
15.d odd 2 1 15.4.a.a 1
15.e even 4 2 75.4.b.b 2
20.d odd 2 1 720.4.a.n 1
35.c odd 2 1 2205.4.a.l 1
45.h odd 6 2 405.4.e.g 2
45.j even 6 2 405.4.e.i 2
60.h even 2 1 240.4.a.e 1
60.l odd 4 2 1200.4.f.b 2
105.g even 2 1 735.4.a.e 1
120.i odd 2 1 960.4.a.b 1
120.m even 2 1 960.4.a.ba 1
165.d even 2 1 1815.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 15.d odd 2 1
45.4.a.c 1 5.b even 2 1
75.4.a.b 1 3.b odd 2 1
75.4.b.b 2 15.e even 4 2
225.4.a.f 1 1.a even 1 1 trivial
225.4.b.e 2 5.c odd 4 2
240.4.a.e 1 60.h even 2 1
405.4.e.g 2 45.h odd 6 2
405.4.e.i 2 45.j even 6 2
720.4.a.n 1 20.d odd 2 1
735.4.a.e 1 105.g even 2 1
960.4.a.b 1 120.i odd 2 1
960.4.a.ba 1 120.m even 2 1
1200.4.a.t 1 12.b even 2 1
1200.4.f.b 2 60.l odd 4 2
1815.4.a.e 1 165.d even 2 1
2205.4.a.l 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{7} - 24$$ T7 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 24$$
$11$ $$T + 52$$
$13$ $$T + 22$$
$17$ $$T + 14$$
$19$ $$T + 20$$
$23$ $$T + 168$$
$29$ $$T + 230$$
$31$ $$T + 288$$
$37$ $$T - 34$$
$41$ $$T + 122$$
$43$ $$T - 188$$
$47$ $$T - 256$$
$53$ $$T + 338$$
$59$ $$T + 100$$
$61$ $$T - 742$$
$67$ $$T - 84$$
$71$ $$T - 328$$
$73$ $$T - 38$$
$79$ $$T + 240$$
$83$ $$T - 1212$$
$89$ $$T + 330$$
$97$ $$T + 866$$