# Properties

 Label 725.2.a.f Level $725$ Weight $2$ Character orbit 725.a Self dual yes Analytic conductor $5.789$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Learn more

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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 1$$ x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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.

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

Basis of coefficient ring in terms of $$\nu = \zeta_{24} + \zeta_{24}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu$$ v^3 - 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta_1$$ b3 + 4*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
 −1.93185 −0.517638 0.517638 1.93185
−1.93185 1.41421 1.73205 0 −2.73205 2.44949 0.517638 −1.00000 0
1.2 −0.517638 −1.41421 −1.73205 0 0.732051 2.44949 1.93185 −1.00000 0
1.3 0.517638 1.41421 −1.73205 0 0.732051 −2.44949 −1.93185 −1.00000 0
1.4 1.93185 −1.41421 1.73205 0 −2.73205 −2.44949 −0.517638 −1.00000 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.f 4
3.b odd 2 1 6525.2.a.bj 4
5.b even 2 1 inner 725.2.a.f 4
5.c odd 4 2 145.2.b.b 4
15.d odd 2 1 6525.2.a.bj 4
15.e even 4 2 1305.2.c.f 4
20.e even 4 2 2320.2.d.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.b.b 4 5.c odd 4 2
725.2.a.f 4 1.a even 1 1 trivial
725.2.a.f 4 5.b even 2 1 inner
1305.2.c.f 4 15.e even 4 2
2320.2.d.f 4 20.e even 4 2
6525.2.a.bj 4 3.b odd 2 1
6525.2.a.bj 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(725))$$:

 $$T_{2}^{4} - 4T_{2}^{2} + 1$$ T2^4 - 4*T2^2 + 1 $$T_{3}^{2} - 2$$ T3^2 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4T^{2} + 1$$
$3$ $$(T^{2} - 2)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 6)^{2}$$
$11$ $$(T^{2} + 6 T + 6)^{2}$$
$13$ $$T^{4} - 48T^{2} + 144$$
$17$ $$(T^{2} - 2)^{2}$$
$19$ $$(T^{2} + 10 T + 22)^{2}$$
$23$ $$T^{4} - 52T^{2} + 484$$
$29$ $$(T - 1)^{4}$$
$31$ $$(T^{2} + 14 T + 46)^{2}$$
$37$ $$T^{4} - 84T^{2} + 36$$
$41$ $$(T^{2} - 48)^{2}$$
$43$ $$T^{4} - 84T^{2} + 36$$
$47$ $$(T^{2} - 2)^{2}$$
$53$ $$T^{4} - 112T^{2} + 2704$$
$59$ $$(T^{2} - 108)^{2}$$
$61$ $$(T^{2} + 8 T - 32)^{2}$$
$67$ $$(T^{2} - 18)^{2}$$
$71$ $$(T^{2} - 12)^{2}$$
$73$ $$(T^{2} - 54)^{2}$$
$79$ $$(T^{2} - 2 T - 26)^{2}$$
$83$ $$T^{4} - 124T^{2} + 2116$$
$89$ $$(T^{2} - 108)^{2}$$
$97$ $$T^{4} - 156T^{2} + 4356$$
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