# Properties

 Label 725.2.a.g Level $725$ Weight $2$ Character orbit 725.a Self dual yes 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(1,725)]

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([0, 0]))

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

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

chi := DirichletCharacter("725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.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: $$5.78915414654$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 7x^{2} + 4$$ x^4 - 7*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) 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)

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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_1 q^{3} + q^{4} + ( - \beta_{3} - 2) q^{6} + ( - 2 \beta_{2} + 2 \beta_1) q^{7} + \beta_{2} q^{8} + (\beta_{3} + 1) q^{9}+O(q^{10})$$ q - b2 * q^2 + b1 * q^3 + q^4 + (-b3 - 2) * q^6 + (-2*b2 + 2*b1) * q^7 + b2 * q^8 + (b3 + 1) * q^9 $$q - \beta_{2} q^{2} + \beta_1 q^{3} + q^{4} + ( - \beta_{3} - 2) q^{6} + ( - 2 \beta_{2} + 2 \beta_1) q^{7} + \beta_{2} q^{8} + (\beta_{3} + 1) q^{9} + (\beta_{3} + 4) q^{11} + \beta_1 q^{12} + ( - 2 \beta_{2} + \beta_1) q^{13} + ( - 2 \beta_{3} + 2) q^{14} - 5 q^{16} - 2 \beta_1 q^{17} + (\beta_{2} - 3 \beta_1) q^{18} + 4 q^{19} + 4 q^{21} + ( - 2 \beta_{2} - 3 \beta_1) q^{22} - 2 \beta_{2} q^{23} + (\beta_{3} + 2) q^{24} + ( - \beta_{3} + 4) q^{26} + (2 \beta_{2} - \beta_1) q^{27} + ( - 2 \beta_{2} + 2 \beta_1) q^{28} - q^{29} + \beta_{3} q^{31} + 3 \beta_{2} q^{32} + (2 \beta_{2} + 5 \beta_1) q^{33} + (2 \beta_{3} + 4) q^{34} + (\beta_{3} + 1) q^{36} + 4 \beta_1 q^{37} - 4 \beta_{2} q^{38} - \beta_{3} q^{39} + (2 \beta_{3} + 2) q^{41} - 4 \beta_{2} q^{42} + (4 \beta_{2} - 5 \beta_1) q^{43} + (\beta_{3} + 4) q^{44} + 6 q^{46} + (4 \beta_{2} - 5 \beta_1) q^{47} - 5 \beta_1 q^{48} + ( - 4 \beta_{3} + 5) q^{49} + ( - 2 \beta_{3} - 8) q^{51} + ( - 2 \beta_{2} + \beta_1) q^{52} + ( - 2 \beta_{2} + \beta_1) q^{53} + (\beta_{3} - 4) q^{54} + (2 \beta_{3} - 2) q^{56} + 4 \beta_1 q^{57} + \beta_{2} q^{58} + (2 \beta_{3} - 4) q^{59} + 6 q^{61} + (2 \beta_{2} - 3 \beta_1) q^{62} + (6 \beta_{2} - 2 \beta_1) q^{63} + q^{64} + ( - 5 \beta_{3} - 16) q^{66} + (2 \beta_{2} + 2 \beta_1) q^{67} - 2 \beta_1 q^{68} + ( - 2 \beta_{3} - 4) q^{69} - 2 \beta_{3} q^{71} + ( - \beta_{2} + 3 \beta_1) q^{72} + 4 \beta_{2} q^{73} + ( - 4 \beta_{3} - 8) q^{74} + 4 q^{76} + 4 \beta_1 q^{77} + ( - 2 \beta_{2} + 3 \beta_1) q^{78} + ( - \beta_{3} + 8) q^{79} + ( - 2 \beta_{3} - 3) q^{81} + (2 \beta_{2} - 6 \beta_1) q^{82} + (2 \beta_{2} - 8 \beta_1) q^{83} + 4 q^{84} + (5 \beta_{3} - 2) q^{86} - \beta_1 q^{87} + (2 \beta_{2} + 3 \beta_1) q^{88} + (2 \beta_{3} + 6) q^{89} + ( - 4 \beta_{3} + 8) q^{91} - 2 \beta_{2} q^{92} + (2 \beta_{2} + \beta_1) q^{93} + (5 \beta_{3} - 2) q^{94} + (3 \beta_{3} + 6) q^{96} + 4 \beta_{2} q^{97} + ( - 13 \beta_{2} + 12 \beta_1) q^{98} + (4 \beta_{3} + 12) q^{99}+O(q^{100})$$ q - b2 * q^2 + b1 * q^3 + q^4 + (-b3 - 2) * q^6 + (-2*b2 + 2*b1) * q^7 + b2 * q^8 + (b3 + 1) * q^9 + (b3 + 4) * q^11 + b1 * q^12 + (-2*b2 + b1) * q^13 + (-2*b3 + 2) * q^14 - 5 * q^16 - 2*b1 * q^17 + (b2 - 3*b1) * q^18 + 4 * q^19 + 4 * q^21 + (-2*b2 - 3*b1) * q^22 - 2*b2 * q^23 + (b3 + 2) * q^24 + (-b3 + 4) * q^26 + (2*b2 - b1) * q^27 + (-2*b2 + 2*b1) * q^28 - q^29 + b3 * q^31 + 3*b2 * q^32 + (2*b2 + 5*b1) * q^33 + (2*b3 + 4) * q^34 + (b3 + 1) * q^36 + 4*b1 * q^37 - 4*b2 * q^38 - b3 * q^39 + (2*b3 + 2) * q^41 - 4*b2 * q^42 + (4*b2 - 5*b1) * q^43 + (b3 + 4) * q^44 + 6 * q^46 + (4*b2 - 5*b1) * q^47 - 5*b1 * q^48 + (-4*b3 + 5) * q^49 + (-2*b3 - 8) * q^51 + (-2*b2 + b1) * q^52 + (-2*b2 + b1) * q^53 + (b3 - 4) * q^54 + (2*b3 - 2) * q^56 + 4*b1 * q^57 + b2 * q^58 + (2*b3 - 4) * q^59 + 6 * q^61 + (2*b2 - 3*b1) * q^62 + (6*b2 - 2*b1) * q^63 + q^64 + (-5*b3 - 16) * q^66 + (2*b2 + 2*b1) * q^67 - 2*b1 * q^68 + (-2*b3 - 4) * q^69 - 2*b3 * q^71 + (-b2 + 3*b1) * q^72 + 4*b2 * q^73 + (-4*b3 - 8) * q^74 + 4 * q^76 + 4*b1 * q^77 + (-2*b2 + 3*b1) * q^78 + (-b3 + 8) * q^79 + (-2*b3 - 3) * q^81 + (2*b2 - 6*b1) * q^82 + (2*b2 - 8*b1) * q^83 + 4 * q^84 + (5*b3 - 2) * q^86 - b1 * q^87 + (2*b2 + 3*b1) * q^88 + (2*b3 + 6) * q^89 + (-4*b3 + 8) * q^91 - 2*b2 * q^92 + (2*b2 + b1) * q^93 + (5*b3 - 2) * q^94 + (3*b3 + 6) * q^96 + 4*b2 * q^97 + (-13*b2 + 12*b1) * q^98 + (4*b3 + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} - 6 q^{6} + 2 q^{9}+O(q^{10})$$ 4 * q + 4 * q^4 - 6 * q^6 + 2 * q^9 $$4 q + 4 q^{4} - 6 q^{6} + 2 q^{9} + 14 q^{11} + 12 q^{14} - 20 q^{16} + 16 q^{19} + 16 q^{21} + 6 q^{24} + 18 q^{26} - 4 q^{29} - 2 q^{31} + 12 q^{34} + 2 q^{36} + 2 q^{39} + 4 q^{41} + 14 q^{44} + 24 q^{46} + 28 q^{49} - 28 q^{51} - 18 q^{54} - 12 q^{56} - 20 q^{59} + 24 q^{61} + 4 q^{64} - 54 q^{66} - 12 q^{69} + 4 q^{71} - 24 q^{74} + 16 q^{76} + 34 q^{79} - 8 q^{81} + 16 q^{84} - 18 q^{86} + 20 q^{89} + 40 q^{91} - 18 q^{94} + 18 q^{96} + 40 q^{99}+O(q^{100})$$ 4 * q + 4 * q^4 - 6 * q^6 + 2 * q^9 + 14 * q^11 + 12 * q^14 - 20 * q^16 + 16 * q^19 + 16 * q^21 + 6 * q^24 + 18 * q^26 - 4 * q^29 - 2 * q^31 + 12 * q^34 + 2 * q^36 + 2 * q^39 + 4 * q^41 + 14 * q^44 + 24 * q^46 + 28 * q^49 - 28 * q^51 - 18 * q^54 - 12 * q^56 - 20 * q^59 + 24 * q^61 + 4 * q^64 - 54 * q^66 - 12 * q^69 + 4 * q^71 - 24 * q^74 + 16 * q^76 + 34 * q^79 - 8 * q^81 + 16 * q^84 - 18 * q^86 + 20 * q^89 + 40 * q^91 - 18 * q^94 + 18 * q^96 + 40 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 5\nu ) / 2$$ (v^3 - 5*v) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4$$ b3 + 4 $$\nu^{3}$$ $$=$$ $$2\beta_{2} + 5\beta_1$$ 2*b2 + 5*b1

## 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
 −0.792287 2.52434 −2.52434 0.792287
−1.73205 −0.792287 1.00000 0 1.37228 −5.04868 1.73205 −2.37228 0
1.2 −1.73205 2.52434 1.00000 0 −4.37228 1.58457 1.73205 3.37228 0
1.3 1.73205 −2.52434 1.00000 0 −4.37228 −1.58457 −1.73205 3.37228 0
1.4 1.73205 0.792287 1.00000 0 1.37228 5.04868 −1.73205 −2.37228 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
$$5$$ $$-1$$
$$29$$ $$1$$

## 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 725.2.a.g 4
3.b odd 2 1 6525.2.a.bk 4
5.b even 2 1 inner 725.2.a.g 4
5.c odd 4 2 145.2.b.a 4
15.d odd 2 1 6525.2.a.bk 4
15.e even 4 2 1305.2.c.e 4
20.e even 4 2 2320.2.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.b.a 4 5.c odd 4 2
725.2.a.g 4 1.a even 1 1 trivial
725.2.a.g 4 5.b even 2 1 inner
1305.2.c.e 4 15.e even 4 2
2320.2.d.c 4 20.e even 4 2
6525.2.a.bk 4 3.b odd 2 1
6525.2.a.bk 4 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(725))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{3}^{4} - 7T_{3}^{2} + 4$$ T3^4 - 7*T3^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 3)^{2}$$
$3$ $$T^{4} - 7T^{2} + 4$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 28T^{2} + 64$$
$11$ $$(T^{2} - 7 T + 4)^{2}$$
$13$ $$T^{4} - 19T^{2} + 16$$
$17$ $$T^{4} - 28T^{2} + 64$$
$19$ $$(T - 4)^{4}$$
$23$ $$(T^{2} - 12)^{2}$$
$29$ $$(T + 1)^{4}$$
$31$ $$(T^{2} + T - 8)^{2}$$
$37$ $$T^{4} - 112T^{2} + 1024$$
$41$ $$(T^{2} - 2 T - 32)^{2}$$
$43$ $$T^{4} - 151T^{2} + 3844$$
$47$ $$T^{4} - 151T^{2} + 3844$$
$53$ $$T^{4} - 19T^{2} + 16$$
$59$ $$(T^{2} + 10 T - 8)^{2}$$
$61$ $$(T - 6)^{4}$$
$67$ $$T^{4} - 76T^{2} + 256$$
$71$ $$(T^{2} - 2 T - 32)^{2}$$
$73$ $$(T^{2} - 48)^{2}$$
$79$ $$(T^{2} - 17 T + 64)^{2}$$
$83$ $$T^{4} - 376 T^{2} + 26896$$
$89$ $$(T^{2} - 10 T - 8)^{2}$$
$97$ $$(T^{2} - 48)^{2}$$