# Properties

 Label 1425.2.a.k Level $1425$ Weight $2$ Character orbit 1425.a Self dual yes Analytic conductor $11.379$ Analytic rank $1$ 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] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.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: $$11.3786822880$$ Analytic rank: $$1$$ 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 285) 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.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} - q^{3} + ( - 2 \beta + 1) q^{4} + ( - \beta + 1) q^{6} - \beta q^{7} + (\beta - 3) q^{8} + q^{9} +O(q^{10})$$ q + (b - 1) * q^2 - q^3 + (-2*b + 1) * q^4 + (-b + 1) * q^6 - b * q^7 + (b - 3) * q^8 + q^9 $$q + (\beta - 1) q^{2} - q^{3} + ( - 2 \beta + 1) q^{4} + ( - \beta + 1) q^{6} - \beta q^{7} + (\beta - 3) q^{8} + q^{9} + (3 \beta + 2) q^{11} + (2 \beta - 1) q^{12} + ( - \beta + 2) q^{13} + (\beta - 2) q^{14} + 3 q^{16} + ( - 2 \beta - 4) q^{17} + (\beta - 1) q^{18} - q^{19} + \beta q^{21} + ( - \beta + 4) q^{22} + (4 \beta - 2) q^{23} + ( - \beta + 3) q^{24} + (3 \beta - 4) q^{26} - q^{27} + ( - \beta + 4) q^{28} - \beta q^{29} + ( - 2 \beta - 6) q^{31} + (\beta + 3) q^{32} + ( - 3 \beta - 2) q^{33} - 2 \beta q^{34} + ( - 2 \beta + 1) q^{36} + ( - \beta + 2) q^{37} + ( - \beta + 1) q^{38} + (\beta - 2) q^{39} + (3 \beta + 4) q^{41} + ( - \beta + 2) q^{42} + (3 \beta - 8) q^{43} + ( - \beta - 10) q^{44} + ( - 6 \beta + 10) q^{46} + ( - 4 \beta + 2) q^{47} - 3 q^{48} - 5 q^{49} + (2 \beta + 4) q^{51} + ( - 5 \beta + 6) q^{52} - 8 q^{53} + ( - \beta + 1) q^{54} + (3 \beta - 2) q^{56} + q^{57} + (\beta - 2) q^{58} + ( - 6 \beta + 4) q^{59} + ( - 8 \beta - 4) q^{61} + ( - 4 \beta + 2) q^{62} - \beta q^{63} + (2 \beta - 7) q^{64} + (\beta - 4) q^{66} + ( - 4 \beta + 4) q^{67} + (6 \beta + 4) q^{68} + ( - 4 \beta + 2) q^{69} + (2 \beta - 8) q^{71} + (\beta - 3) q^{72} + ( - 4 \beta + 2) q^{73} + (3 \beta - 4) q^{74} + (2 \beta - 1) q^{76} + ( - 2 \beta - 6) q^{77} + ( - 3 \beta + 4) q^{78} + q^{81} + (\beta + 2) q^{82} + (2 \beta - 10) q^{83} + (\beta - 4) q^{84} + ( - 11 \beta + 14) q^{86} + \beta q^{87} - 7 \beta q^{88} + ( - 7 \beta - 4) q^{89} + ( - 2 \beta + 2) q^{91} + (8 \beta - 18) q^{92} + (2 \beta + 6) q^{93} + (6 \beta - 10) q^{94} + ( - \beta - 3) q^{96} + (3 \beta + 14) q^{97} + ( - 5 \beta + 5) q^{98} + (3 \beta + 2) q^{99} +O(q^{100})$$ q + (b - 1) * q^2 - q^3 + (-2*b + 1) * q^4 + (-b + 1) * q^6 - b * q^7 + (b - 3) * q^8 + q^9 + (3*b + 2) * q^11 + (2*b - 1) * q^12 + (-b + 2) * q^13 + (b - 2) * q^14 + 3 * q^16 + (-2*b - 4) * q^17 + (b - 1) * q^18 - q^19 + b * q^21 + (-b + 4) * q^22 + (4*b - 2) * q^23 + (-b + 3) * q^24 + (3*b - 4) * q^26 - q^27 + (-b + 4) * q^28 - b * q^29 + (-2*b - 6) * q^31 + (b + 3) * q^32 + (-3*b - 2) * q^33 - 2*b * q^34 + (-2*b + 1) * q^36 + (-b + 2) * q^37 + (-b + 1) * q^38 + (b - 2) * q^39 + (3*b + 4) * q^41 + (-b + 2) * q^42 + (3*b - 8) * q^43 + (-b - 10) * q^44 + (-6*b + 10) * q^46 + (-4*b + 2) * q^47 - 3 * q^48 - 5 * q^49 + (2*b + 4) * q^51 + (-5*b + 6) * q^52 - 8 * q^53 + (-b + 1) * q^54 + (3*b - 2) * q^56 + q^57 + (b - 2) * q^58 + (-6*b + 4) * q^59 + (-8*b - 4) * q^61 + (-4*b + 2) * q^62 - b * q^63 + (2*b - 7) * q^64 + (b - 4) * q^66 + (-4*b + 4) * q^67 + (6*b + 4) * q^68 + (-4*b + 2) * q^69 + (2*b - 8) * q^71 + (b - 3) * q^72 + (-4*b + 2) * q^73 + (3*b - 4) * q^74 + (2*b - 1) * q^76 + (-2*b - 6) * q^77 + (-3*b + 4) * q^78 + q^81 + (b + 2) * q^82 + (2*b - 10) * q^83 + (b - 4) * q^84 + (-11*b + 14) * q^86 + b * q^87 - 7*b * q^88 + (-7*b - 4) * q^89 + (-2*b + 2) * q^91 + (8*b - 18) * q^92 + (2*b + 6) * q^93 + (6*b - 10) * q^94 + (-b - 3) * q^96 + (3*b + 14) * q^97 + (-5*b + 5) * q^98 + (3*b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 4 q^{11} - 2 q^{12} + 4 q^{13} - 4 q^{14} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{19} + 8 q^{22} - 4 q^{23} + 6 q^{24} - 8 q^{26} - 2 q^{27} + 8 q^{28} - 12 q^{31} + 6 q^{32} - 4 q^{33} + 2 q^{36} + 4 q^{37} + 2 q^{38} - 4 q^{39} + 8 q^{41} + 4 q^{42} - 16 q^{43} - 20 q^{44} + 20 q^{46} + 4 q^{47} - 6 q^{48} - 10 q^{49} + 8 q^{51} + 12 q^{52} - 16 q^{53} + 2 q^{54} - 4 q^{56} + 2 q^{57} - 4 q^{58} + 8 q^{59} - 8 q^{61} + 4 q^{62} - 14 q^{64} - 8 q^{66} + 8 q^{67} + 8 q^{68} + 4 q^{69} - 16 q^{71} - 6 q^{72} + 4 q^{73} - 8 q^{74} - 2 q^{76} - 12 q^{77} + 8 q^{78} + 2 q^{81} + 4 q^{82} - 20 q^{83} - 8 q^{84} + 28 q^{86} - 8 q^{89} + 4 q^{91} - 36 q^{92} + 12 q^{93} - 20 q^{94} - 6 q^{96} + 28 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9 + 4 * q^11 - 2 * q^12 + 4 * q^13 - 4 * q^14 + 6 * q^16 - 8 * q^17 - 2 * q^18 - 2 * q^19 + 8 * q^22 - 4 * q^23 + 6 * q^24 - 8 * q^26 - 2 * q^27 + 8 * q^28 - 12 * q^31 + 6 * q^32 - 4 * q^33 + 2 * q^36 + 4 * q^37 + 2 * q^38 - 4 * q^39 + 8 * q^41 + 4 * q^42 - 16 * q^43 - 20 * q^44 + 20 * q^46 + 4 * q^47 - 6 * q^48 - 10 * q^49 + 8 * q^51 + 12 * q^52 - 16 * q^53 + 2 * q^54 - 4 * q^56 + 2 * q^57 - 4 * q^58 + 8 * q^59 - 8 * q^61 + 4 * q^62 - 14 * q^64 - 8 * q^66 + 8 * q^67 + 8 * q^68 + 4 * q^69 - 16 * q^71 - 6 * q^72 + 4 * q^73 - 8 * q^74 - 2 * q^76 - 12 * q^77 + 8 * q^78 + 2 * q^81 + 4 * q^82 - 20 * q^83 - 8 * q^84 + 28 * q^86 - 8 * q^89 + 4 * q^91 - 36 * q^92 + 12 * q^93 - 20 * q^94 - 6 * q^96 + 28 * q^97 + 10 * q^98 + 4 * 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
−2.41421 −1.00000 3.82843 0 2.41421 1.41421 −4.41421 1.00000 0
1.2 0.414214 −1.00000 −1.82843 0 −0.414214 −1.41421 −1.58579 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
$$3$$ $$1$$
$$5$$ $$1$$
$$19$$ $$1$$

## 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 1425.2.a.k 2
3.b odd 2 1 4275.2.a.y 2
5.b even 2 1 285.2.a.g 2
5.c odd 4 2 1425.2.c.l 4
15.d odd 2 1 855.2.a.d 2
20.d odd 2 1 4560.2.a.bf 2
95.d odd 2 1 5415.2.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.g 2 5.b even 2 1
855.2.a.d 2 15.d odd 2 1
1425.2.a.k 2 1.a even 1 1 trivial
1425.2.c.l 4 5.c odd 4 2
4275.2.a.y 2 3.b odd 2 1
4560.2.a.bf 2 20.d odd 2 1
5415.2.a.n 2 95.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1425))$$:

 $$T_{2}^{2} + 2T_{2} - 1$$ T2^2 + 2*T2 - 1 $$T_{7}^{2} - 2$$ T7^2 - 2 $$T_{11}^{2} - 4T_{11} - 14$$ T11^2 - 4*T11 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 2$$
$11$ $$T^{2} - 4T - 14$$
$13$ $$T^{2} - 4T + 2$$
$17$ $$T^{2} + 8T + 8$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 4T - 28$$
$29$ $$T^{2} - 2$$
$31$ $$T^{2} + 12T + 28$$
$37$ $$T^{2} - 4T + 2$$
$41$ $$T^{2} - 8T - 2$$
$43$ $$T^{2} + 16T + 46$$
$47$ $$T^{2} - 4T - 28$$
$53$ $$(T + 8)^{2}$$
$59$ $$T^{2} - 8T - 56$$
$61$ $$T^{2} + 8T - 112$$
$67$ $$T^{2} - 8T - 16$$
$71$ $$T^{2} + 16T + 56$$
$73$ $$T^{2} - 4T - 28$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 20T + 92$$
$89$ $$T^{2} + 8T - 82$$
$97$ $$T^{2} - 28T + 178$$