# Properties

 Label 225.4.a.b Level $225$ Weight $4$ Character orbit 225.a Self dual yes Analytic conductor $13.275$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

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 5) 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 8 q^{4} - 6 q^{7}+O(q^{10})$$ q - 4 * q^2 + 8 * q^4 - 6 * q^7 $$q - 4 q^{2} + 8 q^{4} - 6 q^{7} - 32 q^{11} + 38 q^{13} + 24 q^{14} - 64 q^{16} + 26 q^{17} + 100 q^{19} + 128 q^{22} - 78 q^{23} - 152 q^{26} - 48 q^{28} + 50 q^{29} - 108 q^{31} + 256 q^{32} - 104 q^{34} - 266 q^{37} - 400 q^{38} - 22 q^{41} - 442 q^{43} - 256 q^{44} + 312 q^{46} - 514 q^{47} - 307 q^{49} + 304 q^{52} + 2 q^{53} - 200 q^{58} - 500 q^{59} - 518 q^{61} + 432 q^{62} - 512 q^{64} - 126 q^{67} + 208 q^{68} - 412 q^{71} + 878 q^{73} + 1064 q^{74} + 800 q^{76} + 192 q^{77} + 600 q^{79} + 88 q^{82} + 282 q^{83} + 1768 q^{86} + 150 q^{89} - 228 q^{91} - 624 q^{92} + 2056 q^{94} - 386 q^{97} + 1228 q^{98}+O(q^{100})$$ q - 4 * q^2 + 8 * q^4 - 6 * q^7 - 32 * q^11 + 38 * q^13 + 24 * q^14 - 64 * q^16 + 26 * q^17 + 100 * q^19 + 128 * q^22 - 78 * q^23 - 152 * q^26 - 48 * q^28 + 50 * q^29 - 108 * q^31 + 256 * q^32 - 104 * q^34 - 266 * q^37 - 400 * q^38 - 22 * q^41 - 442 * q^43 - 256 * q^44 + 312 * q^46 - 514 * q^47 - 307 * q^49 + 304 * q^52 + 2 * q^53 - 200 * q^58 - 500 * q^59 - 518 * q^61 + 432 * q^62 - 512 * q^64 - 126 * q^67 + 208 * q^68 - 412 * q^71 + 878 * q^73 + 1064 * q^74 + 800 * q^76 + 192 * q^77 + 600 * q^79 + 88 * q^82 + 282 * q^83 + 1768 * q^86 + 150 * q^89 - 228 * q^91 - 624 * q^92 + 2056 * q^94 - 386 * q^97 + 1228 * 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
−4.00000 0 8.00000 0 0 −6.00000 0 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.b 1
3.b odd 2 1 25.4.a.c 1
5.b even 2 1 45.4.a.d 1
5.c odd 4 2 225.4.b.c 2
12.b even 2 1 400.4.a.m 1
15.d odd 2 1 5.4.a.a 1
15.e even 4 2 25.4.b.a 2
20.d odd 2 1 720.4.a.u 1
21.c even 2 1 1225.4.a.k 1
24.f even 2 1 1600.4.a.s 1
24.h odd 2 1 1600.4.a.bi 1
35.c odd 2 1 2205.4.a.q 1
45.h odd 6 2 405.4.e.l 2
45.j even 6 2 405.4.e.c 2
60.h even 2 1 80.4.a.d 1
60.l odd 4 2 400.4.c.k 2
105.g even 2 1 245.4.a.a 1
105.o odd 6 2 245.4.e.f 2
105.p even 6 2 245.4.e.g 2
120.i odd 2 1 320.4.a.g 1
120.m even 2 1 320.4.a.h 1
165.d even 2 1 605.4.a.d 1
195.e odd 2 1 845.4.a.b 1
240.t even 4 2 1280.4.d.l 2
240.bm odd 4 2 1280.4.d.e 2
255.h odd 2 1 1445.4.a.a 1
285.b even 2 1 1805.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 15.d odd 2 1
25.4.a.c 1 3.b odd 2 1
25.4.b.a 2 15.e even 4 2
45.4.a.d 1 5.b even 2 1
80.4.a.d 1 60.h even 2 1
225.4.a.b 1 1.a even 1 1 trivial
225.4.b.c 2 5.c odd 4 2
245.4.a.a 1 105.g even 2 1
245.4.e.f 2 105.o odd 6 2
245.4.e.g 2 105.p even 6 2
320.4.a.g 1 120.i odd 2 1
320.4.a.h 1 120.m even 2 1
400.4.a.m 1 12.b even 2 1
400.4.c.k 2 60.l odd 4 2
405.4.e.c 2 45.j even 6 2
405.4.e.l 2 45.h odd 6 2
605.4.a.d 1 165.d even 2 1
720.4.a.u 1 20.d odd 2 1
845.4.a.b 1 195.e odd 2 1
1225.4.a.k 1 21.c even 2 1
1280.4.d.e 2 240.bm odd 4 2
1280.4.d.l 2 240.t even 4 2
1445.4.a.a 1 255.h odd 2 1
1600.4.a.s 1 24.f even 2 1
1600.4.a.bi 1 24.h odd 2 1
1805.4.a.h 1 285.b even 2 1
2205.4.a.q 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2} + 4$$ T2 + 4 $$T_{7} + 6$$ T7 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 6$$
$11$ $$T + 32$$
$13$ $$T - 38$$
$17$ $$T - 26$$
$19$ $$T - 100$$
$23$ $$T + 78$$
$29$ $$T - 50$$
$31$ $$T + 108$$
$37$ $$T + 266$$
$41$ $$T + 22$$
$43$ $$T + 442$$
$47$ $$T + 514$$
$53$ $$T - 2$$
$59$ $$T + 500$$
$61$ $$T + 518$$
$67$ $$T + 126$$
$71$ $$T + 412$$
$73$ $$T - 878$$
$79$ $$T - 600$$
$83$ $$T - 282$$
$89$ $$T - 150$$
$97$ $$T + 386$$
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