# Properties

 Label 725.2.a.b Level $725$ Weight $2$ Character orbit 725.a Self dual yes Analytic conductor $5.789$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 29) 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−0.414214 0.414214 −1.82843 0 −0.171573 −2.82843 1.58579 −2.82843 0
1.2 2.41421 −2.41421 3.82843 0 −5.82843 2.82843 4.41421 2.82843 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.a.b 2
3.b odd 2 1 6525.2.a.o 2
5.b even 2 1 29.2.a.a 2
5.c odd 4 2 725.2.b.b 4
15.d odd 2 1 261.2.a.d 2
20.d odd 2 1 464.2.a.h 2
35.c odd 2 1 1421.2.a.j 2
40.e odd 2 1 1856.2.a.w 2
40.f even 2 1 1856.2.a.r 2
55.d odd 2 1 3509.2.a.j 2
60.h even 2 1 4176.2.a.bq 2
65.d even 2 1 4901.2.a.g 2
85.c even 2 1 8381.2.a.e 2
145.d even 2 1 841.2.a.d 2
145.f odd 4 2 841.2.b.a 4
145.l even 14 6 841.2.d.f 12
145.n even 14 6 841.2.d.j 12
145.s odd 28 12 841.2.e.k 24
435.b odd 2 1 7569.2.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 5.b even 2 1
261.2.a.d 2 15.d odd 2 1
464.2.a.h 2 20.d odd 2 1
725.2.a.b 2 1.a even 1 1 trivial
725.2.b.b 4 5.c odd 4 2
841.2.a.d 2 145.d even 2 1
841.2.b.a 4 145.f odd 4 2
841.2.d.f 12 145.l even 14 6
841.2.d.j 12 145.n even 14 6
841.2.e.k 24 145.s odd 28 12
1421.2.a.j 2 35.c odd 2 1
1856.2.a.r 2 40.f even 2 1
1856.2.a.w 2 40.e odd 2 1
3509.2.a.j 2 55.d odd 2 1
4176.2.a.bq 2 60.h even 2 1
4901.2.a.g 2 65.d even 2 1
6525.2.a.o 2 3.b odd 2 1
7569.2.a.c 2 435.b odd 2 1
8381.2.a.e 2 85.c even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2} + 2T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 8$$
$11$ $$T^{2} - 2T - 1$$
$13$ $$T^{2} - 2T - 7$$
$17$ $$T^{2} - 4T - 4$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2} - 4T - 28$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2} - 6T - 41$$
$37$ $$(T - 4)^{2}$$
$41$ $$T^{2} - 8T - 56$$
$43$ $$T^{2} + 10T + 23$$
$47$ $$T^{2} + 2T - 17$$
$53$ $$T^{2} + 2T - 71$$
$59$ $$T^{2} - 4T - 28$$
$61$ $$T^{2} + 4T - 4$$
$67$ $$T^{2} - 32$$
$71$ $$T^{2} + 12T + 28$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} + 2T - 1$$
$83$ $$T^{2} + 4T - 28$$
$89$ $$T^{2} + 8T - 56$$
$97$ $$T^{2} - 8T - 56$$