# Properties

 Label 25.4.b.a Level $25$ Weight $4$ Character orbit 25.b Analytic conductor $1.475$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,4,Mod(24,25)]

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([1]))

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

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

chi := DirichletCharacter("25.24");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$1.47504775014$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{2} + \beta q^{3} - 8 q^{4} - 8 q^{6} - 3 \beta q^{7} + 23 q^{9} +O(q^{10})$$ q + 2*b * q^2 + b * q^3 - 8 * q^4 - 8 * q^6 - 3*b * q^7 + 23 * q^9 $$q + 2 \beta q^{2} + \beta q^{3} - 8 q^{4} - 8 q^{6} - 3 \beta q^{7} + 23 q^{9} + 32 q^{11} - 8 \beta q^{12} - 19 \beta q^{13} + 24 q^{14} - 64 q^{16} - 13 \beta q^{17} + 46 \beta q^{18} - 100 q^{19} + 12 q^{21} + 64 \beta q^{22} - 39 \beta q^{23} + 152 q^{26} + 50 \beta q^{27} + 24 \beta q^{28} + 50 q^{29} - 108 q^{31} - 128 \beta q^{32} + 32 \beta q^{33} + 104 q^{34} - 184 q^{36} - 133 \beta q^{37} - 200 \beta q^{38} + 76 q^{39} + 22 q^{41} + 24 \beta q^{42} + 221 \beta q^{43} - 256 q^{44} + 312 q^{46} + 257 \beta q^{47} - 64 \beta q^{48} + 307 q^{49} + 52 q^{51} + 152 \beta q^{52} + \beta q^{53} - 400 q^{54} - 100 \beta q^{57} + 100 \beta q^{58} - 500 q^{59} - 518 q^{61} - 216 \beta q^{62} - 69 \beta q^{63} + 512 q^{64} - 256 q^{66} - 63 \beta q^{67} + 104 \beta q^{68} + 156 q^{69} + 412 q^{71} - 439 \beta q^{73} + 1064 q^{74} + 800 q^{76} - 96 \beta q^{77} + 152 \beta q^{78} - 600 q^{79} + 421 q^{81} + 44 \beta q^{82} + 141 \beta q^{83} - 96 q^{84} - 1768 q^{86} + 50 \beta q^{87} + 150 q^{89} - 228 q^{91} + 312 \beta q^{92} - 108 \beta q^{93} - 2056 q^{94} + 512 q^{96} - 193 \beta q^{97} + 614 \beta q^{98} + 736 q^{99} +O(q^{100})$$ q + 2*b * q^2 + b * q^3 - 8 * q^4 - 8 * q^6 - 3*b * q^7 + 23 * q^9 + 32 * q^11 - 8*b * q^12 - 19*b * q^13 + 24 * q^14 - 64 * q^16 - 13*b * q^17 + 46*b * q^18 - 100 * q^19 + 12 * q^21 + 64*b * q^22 - 39*b * q^23 + 152 * q^26 + 50*b * q^27 + 24*b * q^28 + 50 * q^29 - 108 * q^31 - 128*b * q^32 + 32*b * q^33 + 104 * q^34 - 184 * q^36 - 133*b * q^37 - 200*b * q^38 + 76 * q^39 + 22 * q^41 + 24*b * q^42 + 221*b * q^43 - 256 * q^44 + 312 * q^46 + 257*b * q^47 - 64*b * q^48 + 307 * q^49 + 52 * q^51 + 152*b * q^52 + b * q^53 - 400 * q^54 - 100*b * q^57 + 100*b * q^58 - 500 * q^59 - 518 * q^61 - 216*b * q^62 - 69*b * q^63 + 512 * q^64 - 256 * q^66 - 63*b * q^67 + 104*b * q^68 + 156 * q^69 + 412 * q^71 - 439*b * q^73 + 1064 * q^74 + 800 * q^76 - 96*b * q^77 + 152*b * q^78 - 600 * q^79 + 421 * q^81 + 44*b * q^82 + 141*b * q^83 - 96 * q^84 - 1768 * q^86 + 50*b * q^87 + 150 * q^89 - 228 * q^91 + 312*b * q^92 - 108*b * q^93 - 2056 * q^94 + 512 * q^96 - 193*b * q^97 + 614*b * q^98 + 736 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{4} - 16 q^{6} + 46 q^{9}+O(q^{10})$$ 2 * q - 16 * q^4 - 16 * q^6 + 46 * q^9 $$2 q - 16 q^{4} - 16 q^{6} + 46 q^{9} + 64 q^{11} + 48 q^{14} - 128 q^{16} - 200 q^{19} + 24 q^{21} + 304 q^{26} + 100 q^{29} - 216 q^{31} + 208 q^{34} - 368 q^{36} + 152 q^{39} + 44 q^{41} - 512 q^{44} + 624 q^{46} + 614 q^{49} + 104 q^{51} - 800 q^{54} - 1000 q^{59} - 1036 q^{61} + 1024 q^{64} - 512 q^{66} + 312 q^{69} + 824 q^{71} + 2128 q^{74} + 1600 q^{76} - 1200 q^{79} + 842 q^{81} - 192 q^{84} - 3536 q^{86} + 300 q^{89} - 456 q^{91} - 4112 q^{94} + 1024 q^{96} + 1472 q^{99}+O(q^{100})$$ 2 * q - 16 * q^4 - 16 * q^6 + 46 * q^9 + 64 * q^11 + 48 * q^14 - 128 * q^16 - 200 * q^19 + 24 * q^21 + 304 * q^26 + 100 * q^29 - 216 * q^31 + 208 * q^34 - 368 * q^36 + 152 * q^39 + 44 * q^41 - 512 * q^44 + 624 * q^46 + 614 * q^49 + 104 * q^51 - 800 * q^54 - 1000 * q^59 - 1036 * q^61 + 1024 * q^64 - 512 * q^66 + 312 * q^69 + 824 * q^71 + 2128 * q^74 + 1600 * q^76 - 1200 * q^79 + 842 * q^81 - 192 * q^84 - 3536 * q^86 + 300 * q^89 - 456 * q^91 - 4112 * q^94 + 1024 * q^96 + 1472 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 − 1.00000i 1.00000i
4.00000i 2.00000i −8.00000 0 −8.00000 6.00000i 0 23.0000 0
24.2 4.00000i 2.00000i −8.00000 0 −8.00000 6.00000i 0 23.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.4.b.a 2
3.b odd 2 1 225.4.b.c 2
4.b odd 2 1 400.4.c.k 2
5.b even 2 1 inner 25.4.b.a 2
5.c odd 4 1 5.4.a.a 1
5.c odd 4 1 25.4.a.c 1
15.d odd 2 1 225.4.b.c 2
15.e even 4 1 45.4.a.d 1
15.e even 4 1 225.4.a.b 1
20.d odd 2 1 400.4.c.k 2
20.e even 4 1 80.4.a.d 1
20.e even 4 1 400.4.a.m 1
35.f even 4 1 245.4.a.a 1
35.f even 4 1 1225.4.a.k 1
35.k even 12 2 245.4.e.g 2
35.l odd 12 2 245.4.e.f 2
40.i odd 4 1 320.4.a.g 1
40.i odd 4 1 1600.4.a.bi 1
40.k even 4 1 320.4.a.h 1
40.k even 4 1 1600.4.a.s 1
45.k odd 12 2 405.4.e.l 2
45.l even 12 2 405.4.e.c 2
55.e even 4 1 605.4.a.d 1
60.l odd 4 1 720.4.a.u 1
65.h odd 4 1 845.4.a.b 1
80.i odd 4 1 1280.4.d.e 2
80.j even 4 1 1280.4.d.l 2
80.s even 4 1 1280.4.d.l 2
80.t odd 4 1 1280.4.d.e 2
85.g odd 4 1 1445.4.a.a 1
95.g even 4 1 1805.4.a.h 1
105.k odd 4 1 2205.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 5.c odd 4 1
25.4.a.c 1 5.c odd 4 1
25.4.b.a 2 1.a even 1 1 trivial
25.4.b.a 2 5.b even 2 1 inner
45.4.a.d 1 15.e even 4 1
80.4.a.d 1 20.e even 4 1
225.4.a.b 1 15.e even 4 1
225.4.b.c 2 3.b odd 2 1
225.4.b.c 2 15.d odd 2 1
245.4.a.a 1 35.f even 4 1
245.4.e.f 2 35.l odd 12 2
245.4.e.g 2 35.k even 12 2
320.4.a.g 1 40.i odd 4 1
320.4.a.h 1 40.k even 4 1
400.4.a.m 1 20.e even 4 1
400.4.c.k 2 4.b odd 2 1
400.4.c.k 2 20.d odd 2 1
405.4.e.c 2 45.l even 12 2
405.4.e.l 2 45.k odd 12 2
605.4.a.d 1 55.e even 4 1
720.4.a.u 1 60.l odd 4 1
845.4.a.b 1 65.h odd 4 1
1225.4.a.k 1 35.f even 4 1
1280.4.d.e 2 80.i odd 4 1
1280.4.d.e 2 80.t odd 4 1
1280.4.d.l 2 80.j even 4 1
1280.4.d.l 2 80.s even 4 1
1445.4.a.a 1 85.g odd 4 1
1600.4.a.s 1 40.k even 4 1
1600.4.a.bi 1 40.i odd 4 1
1805.4.a.h 1 95.g even 4 1
2205.4.a.q 1 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 36$$
$11$ $$(T - 32)^{2}$$
$13$ $$T^{2} + 1444$$
$17$ $$T^{2} + 676$$
$19$ $$(T + 100)^{2}$$
$23$ $$T^{2} + 6084$$
$29$ $$(T - 50)^{2}$$
$31$ $$(T + 108)^{2}$$
$37$ $$T^{2} + 70756$$
$41$ $$(T - 22)^{2}$$
$43$ $$T^{2} + 195364$$
$47$ $$T^{2} + 264196$$
$53$ $$T^{2} + 4$$
$59$ $$(T + 500)^{2}$$
$61$ $$(T + 518)^{2}$$
$67$ $$T^{2} + 15876$$
$71$ $$(T - 412)^{2}$$
$73$ $$T^{2} + 770884$$
$79$ $$(T + 600)^{2}$$
$83$ $$T^{2} + 79524$$
$89$ $$(T - 150)^{2}$$
$97$ $$T^{2} + 148996$$