# Properties

 Label 1225.4.a.i 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 25) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 7 q^{3} - 7 q^{4} - 7 q^{6} - 15 q^{8} + 22 q^{9}+O(q^{10})$$ q + q^2 - 7 * q^3 - 7 * q^4 - 7 * q^6 - 15 * q^8 + 22 * q^9 $$q + q^{2} - 7 q^{3} - 7 q^{4} - 7 q^{6} - 15 q^{8} + 22 q^{9} - 43 q^{11} + 49 q^{12} + 28 q^{13} + 41 q^{16} - 91 q^{17} + 22 q^{18} + 35 q^{19} - 43 q^{22} + 162 q^{23} + 105 q^{24} + 28 q^{26} + 35 q^{27} + 160 q^{29} - 42 q^{31} + 161 q^{32} + 301 q^{33} - 91 q^{34} - 154 q^{36} - 314 q^{37} + 35 q^{38} - 196 q^{39} + 203 q^{41} + 92 q^{43} + 301 q^{44} + 162 q^{46} - 196 q^{47} - 287 q^{48} + 637 q^{51} - 196 q^{52} + 82 q^{53} + 35 q^{54} - 245 q^{57} + 160 q^{58} + 280 q^{59} + 518 q^{61} - 42 q^{62} - 167 q^{64} + 301 q^{66} + 141 q^{67} + 637 q^{68} - 1134 q^{69} + 412 q^{71} - 330 q^{72} + 763 q^{73} - 314 q^{74} - 245 q^{76} - 196 q^{78} + 510 q^{79} - 839 q^{81} + 203 q^{82} - 777 q^{83} + 92 q^{86} - 1120 q^{87} + 645 q^{88} + 945 q^{89} - 1134 q^{92} + 294 q^{93} - 196 q^{94} - 1127 q^{96} - 1246 q^{97} - 946 q^{99}+O(q^{100})$$ q + q^2 - 7 * q^3 - 7 * q^4 - 7 * q^6 - 15 * q^8 + 22 * q^9 - 43 * q^11 + 49 * q^12 + 28 * q^13 + 41 * q^16 - 91 * q^17 + 22 * q^18 + 35 * q^19 - 43 * q^22 + 162 * q^23 + 105 * q^24 + 28 * q^26 + 35 * q^27 + 160 * q^29 - 42 * q^31 + 161 * q^32 + 301 * q^33 - 91 * q^34 - 154 * q^36 - 314 * q^37 + 35 * q^38 - 196 * q^39 + 203 * q^41 + 92 * q^43 + 301 * q^44 + 162 * q^46 - 196 * q^47 - 287 * q^48 + 637 * q^51 - 196 * q^52 + 82 * q^53 + 35 * q^54 - 245 * q^57 + 160 * q^58 + 280 * q^59 + 518 * q^61 - 42 * q^62 - 167 * q^64 + 301 * q^66 + 141 * q^67 + 637 * q^68 - 1134 * q^69 + 412 * q^71 - 330 * q^72 + 763 * q^73 - 314 * q^74 - 245 * q^76 - 196 * q^78 + 510 * q^79 - 839 * q^81 + 203 * q^82 - 777 * q^83 + 92 * q^86 - 1120 * q^87 + 645 * q^88 + 945 * q^89 - 1134 * q^92 + 294 * q^93 - 196 * q^94 - 1127 * q^96 - 1246 * q^97 - 946 * 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
1.00000 −7.00000 −7.00000 0 −7.00000 0 −15.0000 22.0000 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.i 1
5.b even 2 1 1225.4.a.h 1
7.b odd 2 1 25.4.a.b yes 1
21.c even 2 1 225.4.a.c 1
28.d even 2 1 400.4.a.c 1
35.c odd 2 1 25.4.a.a 1
35.f even 4 2 25.4.b.b 2
56.e even 2 1 1600.4.a.bs 1
56.h odd 2 1 1600.4.a.i 1
105.g even 2 1 225.4.a.e 1
105.k odd 4 2 225.4.b.f 2
140.c even 2 1 400.4.a.s 1
140.j odd 4 2 400.4.c.e 2
280.c odd 2 1 1600.4.a.bt 1
280.n even 2 1 1600.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 35.c odd 2 1
25.4.a.b yes 1 7.b odd 2 1
25.4.b.b 2 35.f even 4 2
225.4.a.c 1 21.c even 2 1
225.4.a.e 1 105.g even 2 1
225.4.b.f 2 105.k odd 4 2
400.4.a.c 1 28.d even 2 1
400.4.a.s 1 140.c even 2 1
400.4.c.e 2 140.j odd 4 2
1225.4.a.h 1 5.b even 2 1
1225.4.a.i 1 1.a even 1 1 trivial
1600.4.a.h 1 280.n even 2 1
1600.4.a.i 1 56.h odd 2 1
1600.4.a.bs 1 56.e even 2 1
1600.4.a.bt 1 280.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{3} + 7$$ T3 + 7 $$T_{19} - 35$$ T19 - 35

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 7$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 43$$
$13$ $$T - 28$$
$17$ $$T + 91$$
$19$ $$T - 35$$
$23$ $$T - 162$$
$29$ $$T - 160$$
$31$ $$T + 42$$
$37$ $$T + 314$$
$41$ $$T - 203$$
$43$ $$T - 92$$
$47$ $$T + 196$$
$53$ $$T - 82$$
$59$ $$T - 280$$
$61$ $$T - 518$$
$67$ $$T - 141$$
$71$ $$T - 412$$
$73$ $$T - 763$$
$79$ $$T - 510$$
$83$ $$T + 777$$
$89$ $$T - 945$$
$97$ $$T + 1246$$