# Properties

 Label 1225.4.a.a 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 35) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{2} - 2 q^{3} + q^{4} + 6 q^{6} + 21 q^{8} - 23 q^{9}+O(q^{10})$$ q - 3 * q^2 - 2 * q^3 + q^4 + 6 * q^6 + 21 * q^8 - 23 * q^9 $$q - 3 q^{2} - 2 q^{3} + q^{4} + 6 q^{6} + 21 q^{8} - 23 q^{9} - 45 q^{11} - 2 q^{12} + 59 q^{13} - 71 q^{16} - 54 q^{17} + 69 q^{18} + 121 q^{19} + 135 q^{22} - 69 q^{23} - 42 q^{24} - 177 q^{26} + 100 q^{27} - 162 q^{29} + 88 q^{31} + 45 q^{32} + 90 q^{33} + 162 q^{34} - 23 q^{36} + 259 q^{37} - 363 q^{38} - 118 q^{39} - 195 q^{41} + 286 q^{43} - 45 q^{44} + 207 q^{46} + 45 q^{47} + 142 q^{48} + 108 q^{51} + 59 q^{52} - 597 q^{53} - 300 q^{54} - 242 q^{57} + 486 q^{58} + 360 q^{59} - 392 q^{61} - 264 q^{62} + 433 q^{64} - 270 q^{66} + 280 q^{67} - 54 q^{68} + 138 q^{69} + 48 q^{71} - 483 q^{72} + 668 q^{73} - 777 q^{74} + 121 q^{76} + 354 q^{78} + 782 q^{79} + 421 q^{81} + 585 q^{82} + 768 q^{83} - 858 q^{86} + 324 q^{87} - 945 q^{88} + 1194 q^{89} - 69 q^{92} - 176 q^{93} - 135 q^{94} - 90 q^{96} + 902 q^{97} + 1035 q^{99}+O(q^{100})$$ q - 3 * q^2 - 2 * q^3 + q^4 + 6 * q^6 + 21 * q^8 - 23 * q^9 - 45 * q^11 - 2 * q^12 + 59 * q^13 - 71 * q^16 - 54 * q^17 + 69 * q^18 + 121 * q^19 + 135 * q^22 - 69 * q^23 - 42 * q^24 - 177 * q^26 + 100 * q^27 - 162 * q^29 + 88 * q^31 + 45 * q^32 + 90 * q^33 + 162 * q^34 - 23 * q^36 + 259 * q^37 - 363 * q^38 - 118 * q^39 - 195 * q^41 + 286 * q^43 - 45 * q^44 + 207 * q^46 + 45 * q^47 + 142 * q^48 + 108 * q^51 + 59 * q^52 - 597 * q^53 - 300 * q^54 - 242 * q^57 + 486 * q^58 + 360 * q^59 - 392 * q^61 - 264 * q^62 + 433 * q^64 - 270 * q^66 + 280 * q^67 - 54 * q^68 + 138 * q^69 + 48 * q^71 - 483 * q^72 + 668 * q^73 - 777 * q^74 + 121 * q^76 + 354 * q^78 + 782 * q^79 + 421 * q^81 + 585 * q^82 + 768 * q^83 - 858 * q^86 + 324 * q^87 - 945 * q^88 + 1194 * q^89 - 69 * q^92 - 176 * q^93 - 135 * q^94 - 90 * q^96 + 902 * q^97 + 1035 * 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
−3.00000 −2.00000 1.00000 0 6.00000 0 21.0000 −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.a 1
5.b even 2 1 245.4.a.f 1
7.b odd 2 1 1225.4.a.b 1
7.d odd 6 2 175.4.e.b 2
15.d odd 2 1 2205.4.a.g 1
35.c odd 2 1 245.4.a.e 1
35.i odd 6 2 35.4.e.a 2
35.j even 6 2 245.4.e.a 2
35.k even 12 4 175.4.k.b 4
105.g even 2 1 2205.4.a.e 1
105.p even 6 2 315.4.j.b 2
140.s even 6 2 560.4.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a 2 35.i odd 6 2
175.4.e.b 2 7.d odd 6 2
175.4.k.b 4 35.k even 12 4
245.4.a.e 1 35.c odd 2 1
245.4.a.f 1 5.b even 2 1
245.4.e.a 2 35.j even 6 2
315.4.j.b 2 105.p even 6 2
560.4.q.b 2 140.s even 6 2
1225.4.a.a 1 1.a even 1 1 trivial
1225.4.a.b 1 7.b odd 2 1
2205.4.a.e 1 105.g even 2 1
2205.4.a.g 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2} + 3$$ T2 + 3 $$T_{3} + 2$$ T3 + 2 $$T_{19} - 121$$ T19 - 121

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T + 2$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 45$$
$13$ $$T - 59$$
$17$ $$T + 54$$
$19$ $$T - 121$$
$23$ $$T + 69$$
$29$ $$T + 162$$
$31$ $$T - 88$$
$37$ $$T - 259$$
$41$ $$T + 195$$
$43$ $$T - 286$$
$47$ $$T - 45$$
$53$ $$T + 597$$
$59$ $$T - 360$$
$61$ $$T + 392$$
$67$ $$T - 280$$
$71$ $$T - 48$$
$73$ $$T - 668$$
$79$ $$T - 782$$
$83$ $$T - 768$$
$89$ $$T - 1194$$
$97$ $$T - 902$$