# Properties

 Label 725.2.d.a Level $725$ Weight $2$ Character orbit 725.d Analytic conductor $5.789$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(724,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("725.724");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.d (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: $$5.78915414654$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 29) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} - \beta_{3} q^{3} + 3 q^{4} + 5 q^{6} - 2 \beta_1 q^{7} - \beta_{3} q^{8} + 2 q^{9}+O(q^{10})$$ q - b3 * q^2 - b3 * q^3 + 3 * q^4 + 5 * q^6 - 2*b1 * q^7 - b3 * q^8 + 2 * q^9 $$q - \beta_{3} q^{2} - \beta_{3} q^{3} + 3 q^{4} + 5 q^{6} - 2 \beta_1 q^{7} - \beta_{3} q^{8} + 2 q^{9} - \beta_{2} q^{11} - 3 \beta_{3} q^{12} - \beta_1 q^{13} + 2 \beta_{2} q^{14} - q^{16} + 2 \beta_{3} q^{17} - 2 \beta_{3} q^{18} + 2 \beta_{2} q^{21} + 5 \beta_1 q^{22} + 6 \beta_1 q^{23} + 5 q^{24} + \beta_{2} q^{26} + \beta_{3} q^{27} - 6 \beta_1 q^{28} + (2 \beta_{2} + 3) q^{29} - 3 \beta_{2} q^{31} + 3 \beta_{3} q^{32} + 5 \beta_1 q^{33} - 10 q^{34} + 6 q^{36} + \beta_{2} q^{39} + 2 \beta_{2} q^{41} - 10 \beta_1 q^{42} - 3 \beta_{3} q^{43} - 3 \beta_{2} q^{44} - 6 \beta_{2} q^{46} - \beta_{3} q^{47} + \beta_{3} q^{48} + 3 q^{49} - 10 q^{51} - 3 \beta_1 q^{52} - 9 \beta_1 q^{53} - 5 q^{54} + 2 \beta_{2} q^{56} + ( - 3 \beta_{3} - 10 \beta_1) q^{58} - 6 q^{59} - 6 \beta_{2} q^{61} + 15 \beta_1 q^{62} - 4 \beta_1 q^{63} - 13 q^{64} - 5 \beta_{2} q^{66} - 8 \beta_1 q^{67} + 6 \beta_{3} q^{68} - 6 \beta_{2} q^{69} - 2 \beta_{3} q^{72} - 2 \beta_{3} q^{77} - 5 \beta_1 q^{78} - 3 \beta_{2} q^{79} - 11 q^{81} - 10 \beta_1 q^{82} - 6 \beta_1 q^{83} + 6 \beta_{2} q^{84} + 15 q^{86} + ( - 3 \beta_{3} - 10 \beta_1) q^{87} + 5 \beta_1 q^{88} - 2 \beta_{2} q^{89} - 2 q^{91} + 18 \beta_1 q^{92} + 15 \beta_1 q^{93} + 5 q^{94} - 15 q^{96} - 6 \beta_{3} q^{97} - 3 \beta_{3} q^{98} - 2 \beta_{2} q^{99}+O(q^{100})$$ q - b3 * q^2 - b3 * q^3 + 3 * q^4 + 5 * q^6 - 2*b1 * q^7 - b3 * q^8 + 2 * q^9 - b2 * q^11 - 3*b3 * q^12 - b1 * q^13 + 2*b2 * q^14 - q^16 + 2*b3 * q^17 - 2*b3 * q^18 + 2*b2 * q^21 + 5*b1 * q^22 + 6*b1 * q^23 + 5 * q^24 + b2 * q^26 + b3 * q^27 - 6*b1 * q^28 + (2*b2 + 3) * q^29 - 3*b2 * q^31 + 3*b3 * q^32 + 5*b1 * q^33 - 10 * q^34 + 6 * q^36 + b2 * q^39 + 2*b2 * q^41 - 10*b1 * q^42 - 3*b3 * q^43 - 3*b2 * q^44 - 6*b2 * q^46 - b3 * q^47 + b3 * q^48 + 3 * q^49 - 10 * q^51 - 3*b1 * q^52 - 9*b1 * q^53 - 5 * q^54 + 2*b2 * q^56 + (-3*b3 - 10*b1) * q^58 - 6 * q^59 - 6*b2 * q^61 + 15*b1 * q^62 - 4*b1 * q^63 - 13 * q^64 - 5*b2 * q^66 - 8*b1 * q^67 + 6*b3 * q^68 - 6*b2 * q^69 - 2*b3 * q^72 - 2*b3 * q^77 - 5*b1 * q^78 - 3*b2 * q^79 - 11 * q^81 - 10*b1 * q^82 - 6*b1 * q^83 + 6*b2 * q^84 + 15 * q^86 + (-3*b3 - 10*b1) * q^87 + 5*b1 * q^88 - 2*b2 * q^89 - 2 * q^91 + 18*b1 * q^92 + 15*b1 * q^93 + 5 * q^94 - 15 * q^96 - 6*b3 * q^97 - 3*b3 * q^98 - 2*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{4} + 20 q^{6} + 8 q^{9}+O(q^{10})$$ 4 * q + 12 * q^4 + 20 * q^6 + 8 * q^9 $$4 q + 12 q^{4} + 20 q^{6} + 8 q^{9} - 4 q^{16} + 20 q^{24} + 12 q^{29} - 40 q^{34} + 24 q^{36} + 12 q^{49} - 40 q^{51} - 20 q^{54} - 24 q^{59} - 52 q^{64} - 44 q^{81} + 60 q^{86} - 8 q^{91} + 20 q^{94} - 60 q^{96}+O(q^{100})$$ 4 * q + 12 * q^4 + 20 * q^6 + 8 * q^9 - 4 * q^16 + 20 * q^24 + 12 * q^29 - 40 * q^34 + 24 * q^36 + 12 * q^49 - 40 * q^51 - 20 * q^54 - 24 * q^59 - 52 * q^64 - 44 * q^81 + 60 * q^86 - 8 * q^91 + 20 * q^94 - 60 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2\beta_1$$ -b2 + 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$$.

 $$n$$ $$176$$ $$552$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
724.1
 0.618034i − 0.618034i − 1.61803i 1.61803i
−2.23607 −2.23607 3.00000 0 5.00000 2.00000i −2.23607 2.00000 0
724.2 −2.23607 −2.23607 3.00000 0 5.00000 2.00000i −2.23607 2.00000 0
724.3 2.23607 2.23607 3.00000 0 5.00000 2.00000i 2.23607 2.00000 0
724.4 2.23607 2.23607 3.00000 0 5.00000 2.00000i 2.23607 2.00000 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
29.b even 2 1 inner
145.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.d.a 4
5.b even 2 1 inner 725.2.d.a 4
5.c odd 4 1 29.2.b.a 2
5.c odd 4 1 725.2.c.c 2
15.e even 4 1 261.2.c.a 2
20.e even 4 1 464.2.e.a 2
29.b even 2 1 inner 725.2.d.a 4
35.f even 4 1 1421.2.b.b 2
40.i odd 4 1 1856.2.e.g 2
40.k even 4 1 1856.2.e.f 2
60.l odd 4 1 4176.2.o.k 2
145.d even 2 1 inner 725.2.d.a 4
145.e even 4 1 841.2.a.b 2
145.h odd 4 1 29.2.b.a 2
145.h odd 4 1 725.2.c.c 2
145.j even 4 1 841.2.a.b 2
145.o even 28 6 841.2.d.h 12
145.p odd 28 6 841.2.e.g 12
145.q odd 28 6 841.2.e.g 12
145.t even 28 6 841.2.d.h 12
435.i odd 4 1 7569.2.a.i 2
435.p even 4 1 261.2.c.a 2
435.t odd 4 1 7569.2.a.i 2
580.o even 4 1 464.2.e.a 2
1015.l even 4 1 1421.2.b.b 2
1160.bb even 4 1 1856.2.e.f 2
1160.be odd 4 1 1856.2.e.g 2
1740.v odd 4 1 4176.2.o.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.b.a 2 5.c odd 4 1
29.2.b.a 2 145.h odd 4 1
261.2.c.a 2 15.e even 4 1
261.2.c.a 2 435.p even 4 1
464.2.e.a 2 20.e even 4 1
464.2.e.a 2 580.o even 4 1
725.2.c.c 2 5.c odd 4 1
725.2.c.c 2 145.h odd 4 1
725.2.d.a 4 1.a even 1 1 trivial
725.2.d.a 4 5.b even 2 1 inner
725.2.d.a 4 29.b even 2 1 inner
725.2.d.a 4 145.d even 2 1 inner
841.2.a.b 2 145.e even 4 1
841.2.a.b 2 145.j even 4 1
841.2.d.h 12 145.o even 28 6
841.2.d.h 12 145.t even 28 6
841.2.e.g 12 145.p odd 28 6
841.2.e.g 12 145.q odd 28 6
1421.2.b.b 2 35.f even 4 1
1421.2.b.b 2 1015.l even 4 1
1856.2.e.f 2 40.k even 4 1
1856.2.e.f 2 1160.bb even 4 1
1856.2.e.g 2 40.i odd 4 1
1856.2.e.g 2 1160.be odd 4 1
4176.2.o.k 2 60.l odd 4 1
4176.2.o.k 2 1740.v odd 4 1
7569.2.a.i 2 435.i odd 4 1
7569.2.a.i 2 435.t odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 5$$ acting on $$S_{2}^{\mathrm{new}}(725, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 5)^{2}$$
$3$ $$(T^{2} - 5)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} + 5)^{2}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$(T^{2} - 20)^{2}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 36)^{2}$$
$29$ $$(T^{2} - 6 T + 29)^{2}$$
$31$ $$(T^{2} + 45)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 20)^{2}$$
$43$ $$(T^{2} - 45)^{2}$$
$47$ $$(T^{2} - 5)^{2}$$
$53$ $$(T^{2} + 81)^{2}$$
$59$ $$(T + 6)^{4}$$
$61$ $$(T^{2} + 180)^{2}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + 45)^{2}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T^{2} + 20)^{2}$$
$97$ $$(T^{2} - 180)^{2}$$