# Properties

 Label 1425.2.a.l 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.

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

## 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 −3.41421 −4.41421 1.00000 0
1.2 0.414214 1.00000 −1.82843 0 0.414214 −0.585786 −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.l 2
3.b odd 2 1 4275.2.a.x 2
5.b even 2 1 285.2.a.f 2
5.c odd 4 2 1425.2.c.j 4
15.d odd 2 1 855.2.a.e 2
20.d odd 2 1 4560.2.a.bj 2
95.d odd 2 1 5415.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.f 2 5.b even 2 1
855.2.a.e 2 15.d odd 2 1
1425.2.a.l 2 1.a even 1 1 trivial
1425.2.c.j 4 5.c odd 4 2
4275.2.a.x 2 3.b odd 2 1
4560.2.a.bj 2 20.d odd 2 1
5415.2.a.p 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 $$T_{7}^{2} + 4T_{7} + 2$$ T7^2 + 4*T7 + 2 $$T_{11}^{2} - 2$$ T11^2 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4T + 2$$
$11$ $$T^{2} - 2$$
$13$ $$T^{2} + 8T + 14$$
$17$ $$T^{2} - 8T + 8$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 4T - 28$$
$29$ $$T^{2} + 4T - 46$$
$31$ $$T^{2} + 4T - 68$$
$37$ $$T^{2} + 8T - 34$$
$41$ $$T^{2} + 12T + 34$$
$43$ $$T^{2} + 4T + 2$$
$47$ $$T^{2} + 12T + 4$$
$53$ $$(T + 4)^{2}$$
$59$ $$T^{2} - 72$$
$61$ $$T^{2} - 32$$
$67$ $$(T + 12)^{2}$$
$71$ $$T^{2} - 8T - 56$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} - 128$$
$83$ $$T^{2} - 4T - 68$$
$89$ $$T^{2} + 4T - 158$$
$97$ $$T^{2} - 18$$