# Properties

 Label 25.4.a.c Level $25$ Weight $4$ Character orbit 25.a Self dual yes Analytic conductor $1.475$ Analytic rank $0$ 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] = [25,4,Mod(1,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("25.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 25.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: $$1.47504775014$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} - 2 q^{3} + 8 q^{4} - 8 q^{6} - 6 q^{7} - 23 q^{9}+O(q^{10})$$ q + 4 * q^2 - 2 * q^3 + 8 * q^4 - 8 * q^6 - 6 * q^7 - 23 * q^9 $$q + 4 q^{2} - 2 q^{3} + 8 q^{4} - 8 q^{6} - 6 q^{7} - 23 q^{9} + 32 q^{11} - 16 q^{12} + 38 q^{13} - 24 q^{14} - 64 q^{16} - 26 q^{17} - 92 q^{18} + 100 q^{19} + 12 q^{21} + 128 q^{22} + 78 q^{23} + 152 q^{26} + 100 q^{27} - 48 q^{28} - 50 q^{29} - 108 q^{31} - 256 q^{32} - 64 q^{33} - 104 q^{34} - 184 q^{36} - 266 q^{37} + 400 q^{38} - 76 q^{39} + 22 q^{41} + 48 q^{42} - 442 q^{43} + 256 q^{44} + 312 q^{46} + 514 q^{47} + 128 q^{48} - 307 q^{49} + 52 q^{51} + 304 q^{52} - 2 q^{53} + 400 q^{54} - 200 q^{57} - 200 q^{58} + 500 q^{59} - 518 q^{61} - 432 q^{62} + 138 q^{63} - 512 q^{64} - 256 q^{66} - 126 q^{67} - 208 q^{68} - 156 q^{69} + 412 q^{71} + 878 q^{73} - 1064 q^{74} + 800 q^{76} - 192 q^{77} - 304 q^{78} + 600 q^{79} + 421 q^{81} + 88 q^{82} - 282 q^{83} + 96 q^{84} - 1768 q^{86} + 100 q^{87} - 150 q^{89} - 228 q^{91} + 624 q^{92} + 216 q^{93} + 2056 q^{94} + 512 q^{96} - 386 q^{97} - 1228 q^{98} - 736 q^{99}+O(q^{100})$$ q + 4 * q^2 - 2 * q^3 + 8 * q^4 - 8 * q^6 - 6 * q^7 - 23 * q^9 + 32 * q^11 - 16 * q^12 + 38 * q^13 - 24 * q^14 - 64 * q^16 - 26 * q^17 - 92 * q^18 + 100 * q^19 + 12 * q^21 + 128 * q^22 + 78 * q^23 + 152 * q^26 + 100 * q^27 - 48 * q^28 - 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 + 48 * q^42 - 442 * q^43 + 256 * q^44 + 312 * q^46 + 514 * q^47 + 128 * q^48 - 307 * q^49 + 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 + 138 * q^63 - 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 - 192 * q^77 - 304 * q^78 + 600 * q^79 + 421 * q^81 + 88 * q^82 - 282 * q^83 + 96 * q^84 - 1768 * q^86 + 100 * q^87 - 150 * q^89 - 228 * q^91 + 624 * q^92 + 216 * q^93 + 2056 * q^94 + 512 * q^96 - 386 * q^97 - 1228 * q^98 - 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 −6.00000 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
$$5$$ $$1$$

## 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 25.4.a.c 1
3.b odd 2 1 225.4.a.b 1
4.b odd 2 1 400.4.a.m 1
5.b even 2 1 5.4.a.a 1
5.c odd 4 2 25.4.b.a 2
7.b odd 2 1 1225.4.a.k 1
8.b even 2 1 1600.4.a.bi 1
8.d odd 2 1 1600.4.a.s 1
15.d odd 2 1 45.4.a.d 1
15.e even 4 2 225.4.b.c 2
20.d odd 2 1 80.4.a.d 1
20.e even 4 2 400.4.c.k 2
35.c odd 2 1 245.4.a.a 1
35.i odd 6 2 245.4.e.g 2
35.j even 6 2 245.4.e.f 2
40.e odd 2 1 320.4.a.h 1
40.f even 2 1 320.4.a.g 1
45.h odd 6 2 405.4.e.c 2
45.j even 6 2 405.4.e.l 2
55.d odd 2 1 605.4.a.d 1
60.h even 2 1 720.4.a.u 1
65.d even 2 1 845.4.a.b 1
80.k odd 4 2 1280.4.d.l 2
80.q even 4 2 1280.4.d.e 2
85.c even 2 1 1445.4.a.a 1
95.d odd 2 1 1805.4.a.h 1
105.g even 2 1 2205.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 5.b even 2 1
25.4.a.c 1 1.a even 1 1 trivial
25.4.b.a 2 5.c odd 4 2
45.4.a.d 1 15.d odd 2 1
80.4.a.d 1 20.d odd 2 1
225.4.a.b 1 3.b odd 2 1
225.4.b.c 2 15.e even 4 2
245.4.a.a 1 35.c odd 2 1
245.4.e.f 2 35.j even 6 2
245.4.e.g 2 35.i odd 6 2
320.4.a.g 1 40.f even 2 1
320.4.a.h 1 40.e odd 2 1
400.4.a.m 1 4.b odd 2 1
400.4.c.k 2 20.e even 4 2
405.4.e.c 2 45.h odd 6 2
405.4.e.l 2 45.j even 6 2
605.4.a.d 1 55.d odd 2 1
720.4.a.u 1 60.h even 2 1
845.4.a.b 1 65.d even 2 1
1225.4.a.k 1 7.b odd 2 1
1280.4.d.e 2 80.q even 4 2
1280.4.d.l 2 80.k odd 4 2
1445.4.a.a 1 85.c even 2 1
1600.4.a.s 1 8.d odd 2 1
1600.4.a.bi 1 8.b even 2 1
1805.4.a.h 1 95.d odd 2 1
2205.4.a.q 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 4$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(25))$$.

## Hecke characteristic polynomials

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