# Properties

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

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(157,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(4))

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("725.157");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.e (of order $$4$$, degree $$2$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ 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: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - 3 q^{4} + (\beta_{3} - \beta_{2}) q^{7} - \beta_{2} q^{8} - 3 q^{9}+O(q^{10})$$ q + b2 * q^2 - 3 * q^4 + (b3 - b2) * q^7 - b2 * q^8 - 3 * q^9 $$q + \beta_{2} q^{2} - 3 q^{4} + (\beta_{3} - \beta_{2}) q^{7} - \beta_{2} q^{8} - 3 q^{9} + ( - 4 \beta_1 + 4) q^{11} + (2 \beta_{3} - 2 \beta_{2}) q^{13} + (5 \beta_1 + 5) q^{14} - q^{16} - 2 \beta_{2} q^{17} - 3 \beta_{2} q^{18} + ( - 2 \beta_1 - 2) q^{19} + (4 \beta_{3} + 4 \beta_{2}) q^{22} + ( - \beta_{3} - \beta_{2}) q^{23} + (10 \beta_1 + 10) q^{26} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{28} + (5 \beta_1 + 2) q^{29} + (4 \beta_1 - 4) q^{31} - 3 \beta_{2} q^{32} + 10 q^{34} + 9 q^{36} - 2 \beta_{3} q^{37} + (2 \beta_{3} - 2 \beta_{2}) q^{38} + (\beta_1 + 1) q^{41} + 2 \beta_{3} q^{43} + (12 \beta_1 - 12) q^{44} + ( - 5 \beta_1 + 5) q^{46} + 2 \beta_{3} q^{47} - 3 \beta_1 q^{49} + ( - 6 \beta_{3} + 6 \beta_{2}) q^{52} + (4 \beta_{3} + 4 \beta_{2}) q^{53} + ( - 5 \beta_1 - 5) q^{56} + ( - 5 \beta_{3} + 2 \beta_{2}) q^{58} + (9 \beta_1 - 9) q^{61} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{62} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{63} + 13 q^{64} + ( - \beta_{3} - \beta_{2}) q^{67} + 6 \beta_{2} q^{68} - 12 \beta_1 q^{71} + 3 \beta_{2} q^{72} + 2 \beta_{2} q^{73} - 10 \beta_1 q^{74} + (6 \beta_1 + 6) q^{76} - 8 \beta_{2} q^{77} + ( - 2 \beta_1 - 2) q^{79} + 9 q^{81} + ( - \beta_{3} + \beta_{2}) q^{82} + (3 \beta_{3} + 3 \beta_{2}) q^{83} + 10 \beta_1 q^{86} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{88} + ( - 7 \beta_1 - 7) q^{89} - 20 \beta_1 q^{91} + (3 \beta_{3} + 3 \beta_{2}) q^{92} + 10 \beta_1 q^{94} + 2 \beta_{3} q^{97} + 3 \beta_{3} q^{98} + (12 \beta_1 - 12) q^{99}+O(q^{100})$$ q + b2 * q^2 - 3 * q^4 + (b3 - b2) * q^7 - b2 * q^8 - 3 * q^9 + (-4*b1 + 4) * q^11 + (2*b3 - 2*b2) * q^13 + (5*b1 + 5) * q^14 - q^16 - 2*b2 * q^17 - 3*b2 * q^18 + (-2*b1 - 2) * q^19 + (4*b3 + 4*b2) * q^22 + (-b3 - b2) * q^23 + (10*b1 + 10) * q^26 + (-3*b3 + 3*b2) * q^28 + (5*b1 + 2) * q^29 + (4*b1 - 4) * q^31 - 3*b2 * q^32 + 10 * q^34 + 9 * q^36 - 2*b3 * q^37 + (2*b3 - 2*b2) * q^38 + (b1 + 1) * q^41 + 2*b3 * q^43 + (12*b1 - 12) * q^44 + (-5*b1 + 5) * q^46 + 2*b3 * q^47 - 3*b1 * q^49 + (-6*b3 + 6*b2) * q^52 + (4*b3 + 4*b2) * q^53 + (-5*b1 - 5) * q^56 + (-5*b3 + 2*b2) * q^58 + (9*b1 - 9) * q^61 + (-4*b3 - 4*b2) * q^62 + (-3*b3 + 3*b2) * q^63 + 13 * q^64 + (-b3 - b2) * q^67 + 6*b2 * q^68 - 12*b1 * q^71 + 3*b2 * q^72 + 2*b2 * q^73 - 10*b1 * q^74 + (6*b1 + 6) * q^76 - 8*b2 * q^77 + (-2*b1 - 2) * q^79 + 9 * q^81 + (-b3 + b2) * q^82 + (3*b3 + 3*b2) * q^83 + 10*b1 * q^86 + (-4*b3 - 4*b2) * q^88 + (-7*b1 - 7) * q^89 - 20*b1 * q^91 + (3*b3 + 3*b2) * q^92 + 10*b1 * q^94 + 2*b3 * q^97 + 3*b3 * q^98 + (12*b1 - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{4} - 12 q^{9}+O(q^{10})$$ 4 * q - 12 * q^4 - 12 * q^9 $$4 q - 12 q^{4} - 12 q^{9} + 16 q^{11} + 20 q^{14} - 4 q^{16} - 8 q^{19} + 40 q^{26} + 8 q^{29} - 16 q^{31} + 40 q^{34} + 36 q^{36} + 4 q^{41} - 48 q^{44} + 20 q^{46} - 20 q^{56} - 36 q^{61} + 52 q^{64} + 24 q^{76} - 8 q^{79} + 36 q^{81} - 28 q^{89} - 48 q^{99}+O(q^{100})$$ 4 * q - 12 * q^4 - 12 * q^9 + 16 * q^11 + 20 * q^14 - 4 * q^16 - 8 * q^19 + 40 * q^26 + 8 * q^29 - 16 * q^31 + 40 * q^34 + 36 * q^36 + 4 * q^41 - 48 * q^44 + 20 * q^46 - 20 * q^56 - 36 * q^61 + 52 * q^64 + 24 * q^76 - 8 * q^79 + 36 * q^81 - 28 * q^89 - 48 * q^99

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 $$\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$$.

 $$n$$ $$176$$ $$552$$ $$\chi(n)$$ $$-\beta_{1}$$ $$-\beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1
 − 0.618034i 1.61803i − 1.61803i 0.618034i
2.23607i 0 −3.00000 0 0 2.23607 + 2.23607i 2.23607i −3.00000 0
157.2 2.23607i 0 −3.00000 0 0 −2.23607 2.23607i 2.23607i −3.00000 0
568.1 2.23607i 0 −3.00000 0 0 −2.23607 + 2.23607i 2.23607i −3.00000 0
568.2 2.23607i 0 −3.00000 0 0 2.23607 2.23607i 2.23607i −3.00000 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
145.e even 4 1 inner
145.j even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.e.a 4
5.b even 2 1 inner 725.2.e.a 4
5.c odd 4 2 725.2.j.a yes 4
29.c odd 4 1 725.2.j.a yes 4
145.e even 4 1 inner 725.2.e.a 4
145.f odd 4 1 725.2.j.a yes 4
145.j even 4 1 inner 725.2.e.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.e.a 4 1.a even 1 1 trivial
725.2.e.a 4 5.b even 2 1 inner
725.2.e.a 4 145.e even 4 1 inner
725.2.e.a 4 145.j even 4 1 inner
725.2.j.a yes 4 5.c odd 4 2
725.2.j.a yes 4 29.c odd 4 1
725.2.j.a yes 4 145.f 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

$p$ $F_p(T)$
$2$ $$(T^{2} + 5)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 100$$
$11$ $$(T^{2} - 8 T + 32)^{2}$$
$13$ $$T^{4} + 1600$$
$17$ $$(T^{2} + 20)^{2}$$
$19$ $$(T^{2} + 4 T + 8)^{2}$$
$23$ $$T^{4} + 100$$
$29$ $$(T^{2} - 4 T + 29)^{2}$$
$31$ $$(T^{2} + 8 T + 32)^{2}$$
$37$ $$(T^{2} - 20)^{2}$$
$41$ $$(T^{2} - 2 T + 2)^{2}$$
$43$ $$(T^{2} - 20)^{2}$$
$47$ $$(T^{2} - 20)^{2}$$
$53$ $$T^{4} + 25600$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 18 T + 162)^{2}$$
$67$ $$T^{4} + 100$$
$71$ $$(T^{2} + 144)^{2}$$
$73$ $$(T^{2} + 20)^{2}$$
$79$ $$(T^{2} + 4 T + 8)^{2}$$
$83$ $$T^{4} + 8100$$
$89$ $$(T^{2} + 14 T + 98)^{2}$$
$97$ $$(T^{2} - 20)^{2}$$
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