# Properties

 Label 1225.4.a.k Level $1225$ Weight $4$ Character orbit 1225.a Self dual yes Analytic conductor $72.277$ 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] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1225.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: $$72.2773397570$$ 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 2 q^{3} + 8 q^{4} + 8 q^{6} - 23 q^{9}+O(q^{10})$$ q + 4 * q^2 + 2 * q^3 + 8 * q^4 + 8 * q^6 - 23 * q^9 $$q + 4 q^{2} + 2 q^{3} + 8 q^{4} + 8 q^{6} - 23 q^{9} + 32 q^{11} + 16 q^{12} - 38 q^{13} - 64 q^{16} + 26 q^{17} - 92 q^{18} - 100 q^{19} + 128 q^{22} + 78 q^{23} - 152 q^{26} - 100 q^{27} - 50 q^{29} + 108 q^{31} - 256 q^{32} + 64 q^{33} + 104 q^{34} - 184 q^{36} - 266 q^{37} - 400 q^{38} - 76 q^{39} - 22 q^{41} - 442 q^{43} + 256 q^{44} + 312 q^{46} - 514 q^{47} - 128 q^{48} + 52 q^{51} - 304 q^{52} - 2 q^{53} - 400 q^{54} - 200 q^{57} - 200 q^{58} - 500 q^{59} + 518 q^{61} + 432 q^{62} - 512 q^{64} + 256 q^{66} - 126 q^{67} + 208 q^{68} + 156 q^{69} + 412 q^{71} - 878 q^{73} - 1064 q^{74} - 800 q^{76} - 304 q^{78} + 600 q^{79} + 421 q^{81} - 88 q^{82} + 282 q^{83} - 1768 q^{86} - 100 q^{87} + 150 q^{89} + 624 q^{92} + 216 q^{93} - 2056 q^{94} - 512 q^{96} + 386 q^{97} - 736 q^{99}+O(q^{100})$$ q + 4 * q^2 + 2 * q^3 + 8 * q^4 + 8 * q^6 - 23 * q^9 + 32 * q^11 + 16 * q^12 - 38 * q^13 - 64 * q^16 + 26 * q^17 - 92 * q^18 - 100 * q^19 + 128 * q^22 + 78 * q^23 - 152 * q^26 - 100 * q^27 - 50 * q^29 + 108 * q^31 - 256 * q^32 + 64 * q^33 + 104 * q^34 - 184 * q^36 - 266 * q^37 - 400 * q^38 - 76 * q^39 - 22 * q^41 - 442 * q^43 + 256 * q^44 + 312 * q^46 - 514 * q^47 - 128 * q^48 + 52 * q^51 - 304 * q^52 - 2 * q^53 - 400 * q^54 - 200 * q^57 - 200 * q^58 - 500 * q^59 + 518 * q^61 + 432 * q^62 - 512 * q^64 + 256 * q^66 - 126 * q^67 + 208 * q^68 + 156 * q^69 + 412 * q^71 - 878 * q^73 - 1064 * q^74 - 800 * q^76 - 304 * q^78 + 600 * q^79 + 421 * q^81 - 88 * q^82 + 282 * q^83 - 1768 * q^86 - 100 * q^87 + 150 * q^89 + 624 * q^92 + 216 * q^93 - 2056 * q^94 - 512 * q^96 + 386 * q^97 - 736 * q^99

## 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 2.00000 8.00000 0 8.00000 0 0 −23.0000 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
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 1225.4.a.k 1
5.b even 2 1 245.4.a.a 1
7.b odd 2 1 25.4.a.c 1
15.d odd 2 1 2205.4.a.q 1
21.c even 2 1 225.4.a.b 1
28.d even 2 1 400.4.a.m 1
35.c odd 2 1 5.4.a.a 1
35.f even 4 2 25.4.b.a 2
35.i odd 6 2 245.4.e.f 2
35.j even 6 2 245.4.e.g 2
56.e even 2 1 1600.4.a.s 1
56.h odd 2 1 1600.4.a.bi 1
105.g even 2 1 45.4.a.d 1
105.k odd 4 2 225.4.b.c 2
140.c even 2 1 80.4.a.d 1
140.j odd 4 2 400.4.c.k 2
280.c odd 2 1 320.4.a.g 1
280.n even 2 1 320.4.a.h 1
315.z even 6 2 405.4.e.c 2
315.bg odd 6 2 405.4.e.l 2
385.h even 2 1 605.4.a.d 1
420.o odd 2 1 720.4.a.u 1
455.h odd 2 1 845.4.a.b 1
560.be even 4 2 1280.4.d.l 2
560.bf odd 4 2 1280.4.d.e 2
595.b odd 2 1 1445.4.a.a 1
665.g 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 35.c odd 2 1
25.4.a.c 1 7.b odd 2 1
25.4.b.a 2 35.f even 4 2
45.4.a.d 1 105.g even 2 1
80.4.a.d 1 140.c even 2 1
225.4.a.b 1 21.c even 2 1
225.4.b.c 2 105.k odd 4 2
245.4.a.a 1 5.b even 2 1
245.4.e.f 2 35.i odd 6 2
245.4.e.g 2 35.j even 6 2
320.4.a.g 1 280.c odd 2 1
320.4.a.h 1 280.n even 2 1
400.4.a.m 1 28.d even 2 1
400.4.c.k 2 140.j odd 4 2
405.4.e.c 2 315.z even 6 2
405.4.e.l 2 315.bg odd 6 2
605.4.a.d 1 385.h even 2 1
720.4.a.u 1 420.o odd 2 1
845.4.a.b 1 455.h odd 2 1
1225.4.a.k 1 1.a even 1 1 trivial
1280.4.d.e 2 560.bf odd 4 2
1280.4.d.l 2 560.be even 4 2
1445.4.a.a 1 595.b odd 2 1
1600.4.a.s 1 56.e even 2 1
1600.4.a.bi 1 56.h odd 2 1
1805.4.a.h 1 665.g even 2 1
2205.4.a.q 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2} - 4$$ T2 - 4 $$T_{3} - 2$$ T3 - 2 $$T_{19} + 100$$ T19 + 100

## Hecke characteristic polynomials

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