# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(146,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([6, 0]))

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

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

chi := DirichletCharacter("725.146");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.k (of order $$5$$, degree $$4$$, 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: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + \zeta_{10} - 1) q^{3} + 2 \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{5}+ \cdots + (3 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{9}+O(q^{10})$$ q + (-z^3 + z - 1) * q^3 + 2*z^3 * q^4 + (-2*z^3 + z^2 - 2*z) * q^5 - 3 * q^7 + (3*z^2 - 2*z + 3) * q^9 $$q + ( - \zeta_{10}^{3} + \zeta_{10} - 1) q^{3} + 2 \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{5}+ \cdots + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 5) q^{99}+O(q^{100})$$ q + (-z^3 + z - 1) * q^3 + 2*z^3 * q^4 + (-2*z^3 + z^2 - 2*z) * q^5 - 3 * q^7 + (3*z^2 - 2*z + 3) * q^9 + (-2*z^3 + z^2 - z + 2) * q^11 + (-2*z^2 + 4*z - 2) * q^12 + z * q^13 + (3*z^3 - 3*z^2 + 1) * q^15 - 4*z * q^16 + (2*z^3 + 2*z) * q^17 + (7*z^3 - 3*z^2 + 7*z) * q^19 + (-4*z^3 + 4*z^2 + 2) * q^20 + (3*z^3 - 3*z + 3) * q^21 + (-4*z^3 - 3*z^2 + 3*z + 4) * q^23 + (5*z^3 - 5*z^2 + 5*z - 5) * q^25 + (-z^3 - z^2 + z + 1) * q^27 - 6*z^3 * q^28 - z^3 * q^29 + (5*z^3 + z^2 + 5*z) * q^31 - z^2 * q^33 + (6*z^3 - 3*z^2 + 6*z) * q^35 + (2*z^3 + 4*z^2 - 4*z - 2) * q^36 + (8*z^2 - 2*z + 8) * q^37 + (-z^3 + 2*z^2 - 2*z + 1) * q^39 + (-z^2 + z - 1) * q^41 + (6*z^3 - 6*z^2 - 3) * q^43 + (2*z^3 + 2*z^2 + 2*z) * q^44 + (-7*z^3 + z - 1) * q^45 + (-3*z^3 - 3*z + 3) * q^47 + (4*z^3 - 8*z^2 + 8*z - 4) * q^48 + 2 * q^49 + (-2*z^3 + 2*z^2) * q^51 + (2*z^3 - 2*z^2 + 2*z - 2) * q^52 + (6*z^3 + 6*z - 6) * q^53 + (-3*z^2 - z - 3) * q^55 + (-10*z^3 + 10*z^2 - 3) * q^57 + (-2*z^2 + 11*z - 2) * q^59 + (2*z^3 - 6*z + 6) * q^60 + (-3*z^3 - 8*z^2 + 8*z + 3) * q^61 + (-9*z^2 + 6*z - 9) * q^63 + (-8*z^3 + 8*z^2 - 8*z + 8) * q^64 + (-z^3 - 2*z + 2) * q^65 + (-2*z^3 - 9*z^2 - 2*z) * q^67 + (4*z^3 - 4*z^2 - 4) * q^68 + (-10*z^3 + 13*z^2 - 10*z) * q^69 + (-2*z^3 - 7*z + 7) * q^71 + (6*z^3 - 3*z^2 + 3*z - 6) * q^73 + (-5*z^3 + 10*z^2 - 5*z) * q^75 + (14*z^3 - 14*z^2 - 8) * q^76 + (6*z^3 - 3*z^2 + 3*z - 6) * q^77 + (9*z^3 + 8*z - 8) * q^79 + (4*z^3 + 8*z - 8) * q^80 + (6*z^3 - 2*z^2 + 6*z) * q^81 - 9*z^2 * q^83 + (6*z^2 - 12*z + 6) * q^84 + (-6*z^3 + 4*z^2 - 4*z + 6) * q^85 + (z^2 - 2*z + 1) * q^87 + (2*z^3 + 2*z^2 - 2*z - 2) * q^89 - 3*z * q^91 + (14*z^3 - 6*z^2 + 14*z) * q^92 + (-4*z^3 + 4*z^2 + 1) * q^93 + (-18*z^3 + 17*z^2 - 17*z + 18) * q^95 - 3*z^3 * q^97 + (-4*z^3 + 4*z^2 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 2 q^{4} - 5 q^{5} - 12 q^{7} + 7 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 + 2 * q^4 - 5 * q^5 - 12 * q^7 + 7 * q^9 $$4 q - 4 q^{3} + 2 q^{4} - 5 q^{5} - 12 q^{7} + 7 q^{9} + 4 q^{11} - 2 q^{12} + q^{13} + 10 q^{15} - 4 q^{16} + 4 q^{17} + 17 q^{19} + 12 q^{21} + 18 q^{23} - 5 q^{25} + 5 q^{27} - 6 q^{28} - q^{29} + 9 q^{31} + q^{33} + 15 q^{35} - 14 q^{36} + 22 q^{37} - q^{39} - 2 q^{41} + 2 q^{44} - 10 q^{45} + 6 q^{47} + 4 q^{48} + 8 q^{49} - 4 q^{51} - 2 q^{52} - 12 q^{53} - 10 q^{55} - 32 q^{57} + 5 q^{59} + 20 q^{60} + 25 q^{61} - 21 q^{63} + 8 q^{64} + 5 q^{65} + 5 q^{67} - 8 q^{68} - 33 q^{69} + 19 q^{71} - 12 q^{73} - 20 q^{75} - 4 q^{76} - 12 q^{77} - 15 q^{79} - 20 q^{80} + 14 q^{81} + 9 q^{83} + 6 q^{84} + 10 q^{85} + q^{87} - 10 q^{89} - 3 q^{91} + 34 q^{92} - 4 q^{93} + 20 q^{95} - 3 q^{97} + 12 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 + 2 * q^4 - 5 * q^5 - 12 * q^7 + 7 * q^9 + 4 * q^11 - 2 * q^12 + q^13 + 10 * q^15 - 4 * q^16 + 4 * q^17 + 17 * q^19 + 12 * q^21 + 18 * q^23 - 5 * q^25 + 5 * q^27 - 6 * q^28 - q^29 + 9 * q^31 + q^33 + 15 * q^35 - 14 * q^36 + 22 * q^37 - q^39 - 2 * q^41 + 2 * q^44 - 10 * q^45 + 6 * q^47 + 4 * q^48 + 8 * q^49 - 4 * q^51 - 2 * q^52 - 12 * q^53 - 10 * q^55 - 32 * q^57 + 5 * q^59 + 20 * q^60 + 25 * q^61 - 21 * q^63 + 8 * q^64 + 5 * q^65 + 5 * q^67 - 8 * q^68 - 33 * q^69 + 19 * q^71 - 12 * q^73 - 20 * q^75 - 4 * q^76 - 12 * q^77 - 15 * q^79 - 20 * q^80 + 14 * q^81 + 9 * q^83 + 6 * q^84 + 10 * q^85 + q^87 - 10 * q^89 - 3 * q^91 + 34 * q^92 - 4 * q^93 + 20 * q^95 - 3 * q^97 + 12 * q^99

## Character values

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

 $$n$$ $$176$$ $$552$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
146.1
 −0.309017 − 0.951057i 0.809017 − 0.587785i 0.809017 + 0.587785i −0.309017 + 0.951057i
0 −2.11803 1.53884i 1.61803 + 1.17557i −1.80902 + 1.31433i 0 −3.00000 0 1.19098 + 3.66547i 0
291.1 0 0.118034 + 0.363271i −0.618034 1.90211i −0.690983 + 2.12663i 0 −3.00000 0 2.30902 1.67760i 0
436.1 0 0.118034 0.363271i −0.618034 + 1.90211i −0.690983 2.12663i 0 −3.00000 0 2.30902 + 1.67760i 0
581.1 0 −2.11803 + 1.53884i 1.61803 1.17557i −1.80902 1.31433i 0 −3.00000 0 1.19098 3.66547i 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
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.k.a 4
25.d even 5 1 inner 725.2.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.k.a 4 1.a even 1 1 trivial
725.2.k.a 4 25.d even 5 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 4 T^{3} + \cdots + 1$$
$5$ $$T^{4} + 5 T^{3} + \cdots + 25$$
$7$ $$(T + 3)^{4}$$
$11$ $$T^{4} - 4 T^{3} + \cdots + 1$$
$13$ $$T^{4} - T^{3} + T^{2} + \cdots + 1$$
$17$ $$T^{4} - 4 T^{3} + \cdots + 16$$
$19$ $$T^{4} - 17 T^{3} + \cdots + 3721$$
$23$ $$T^{4} - 18 T^{3} + \cdots + 3721$$
$29$ $$T^{4} + T^{3} + T^{2} + \cdots + 1$$
$31$ $$T^{4} - 9 T^{3} + \cdots + 361$$
$37$ $$T^{4} - 22 T^{3} + \cdots + 5776$$
$41$ $$T^{4} + 2 T^{3} + \cdots + 1$$
$43$ $$(T^{2} - 45)^{2}$$
$47$ $$T^{4} - 6 T^{3} + \cdots + 81$$
$53$ $$T^{4} + 12 T^{3} + \cdots + 1296$$
$59$ $$T^{4} - 5 T^{3} + \cdots + 9025$$
$61$ $$T^{4} - 25 T^{3} + \cdots + 21025$$
$67$ $$T^{4} - 5 T^{3} + \cdots + 9025$$
$71$ $$T^{4} - 19 T^{3} + \cdots + 3481$$
$73$ $$T^{4} + 12 T^{3} + \cdots + 81$$
$79$ $$T^{4} + 15 T^{3} + \cdots + 3025$$
$83$ $$T^{4} - 9 T^{3} + \cdots + 6561$$
$89$ $$T^{4} + 10 T^{3} + \cdots + 400$$
$97$ $$T^{4} + 3 T^{3} + \cdots + 81$$