# Properties

 Label 1425.2.c.k Level $1425$ Weight $2$ Character orbit 1425.c Analytic conductor $11.379$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(799,1425)]

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, 1, 0]))

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

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

chi := DirichletCharacter("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$11.3786822880$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 285) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$$.

 $$n$$ $$476$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
1.73205i 1.00000i −1.00000 0 −1.73205 2.73205i 1.73205i −1.00000 0
799.2 1.73205i 1.00000i −1.00000 0 1.73205 0.732051i 1.73205i −1.00000 0
799.3 1.73205i 1.00000i −1.00000 0 1.73205 0.732051i 1.73205i −1.00000 0
799.4 1.73205i 1.00000i −1.00000 0 −1.73205 2.73205i 1.73205i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 1425.2.c.k 4
5.b even 2 1 inner 1425.2.c.k 4
5.c odd 4 1 285.2.a.e 2
5.c odd 4 1 1425.2.a.o 2
15.e even 4 1 855.2.a.f 2
15.e even 4 1 4275.2.a.t 2
20.e even 4 1 4560.2.a.bh 2
95.g even 4 1 5415.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.e 2 5.c odd 4 1
855.2.a.f 2 15.e even 4 1
1425.2.a.o 2 5.c odd 4 1
1425.2.c.k 4 1.a even 1 1 trivial
1425.2.c.k 4 5.b even 2 1 inner
4275.2.a.t 2 15.e even 4 1
4560.2.a.bh 2 20.e even 4 1
5415.2.a.r 2 95.g even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1425, [\chi])$$:

 $$T_{2}^{2} + 3$$ T2^2 + 3 $$T_{7}^{4} + 8T_{7}^{2} + 4$$ T7^4 + 8*T7^2 + 4 $$T_{11}^{2} - 6T_{11} + 6$$ T11^2 - 6*T11 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 3)^{2}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 8T^{2} + 4$$
$11$ $$(T^{2} - 6 T + 6)^{2}$$
$13$ $$T^{4} + 8T^{2} + 4$$
$17$ $$T^{4}$$
$19$ $$(T + 1)^{4}$$
$23$ $$(T^{2} + 12)^{2}$$
$29$ $$(T^{2} + 6 T - 18)^{2}$$
$31$ $$(T^{2} - 4 T - 44)^{2}$$
$37$ $$T^{4} + 56T^{2} + 676$$
$41$ $$(T^{2} - 6 T + 6)^{2}$$
$43$ $$T^{4} + 56T^{2} + 676$$
$47$ $$(T^{2} + 12)^{2}$$
$53$ $$T^{4} + 96T^{2} + 576$$
$59$ $$(T^{2} + 12 T + 24)^{2}$$
$61$ $$(T^{2} + 20 T + 88)^{2}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} - 12 T - 72)^{2}$$
$73$ $$T^{4} + 296T^{2} + 2704$$
$79$ $$(T^{2} - 8 T - 32)^{2}$$
$83$ $$T^{4} + 168T^{2} + 144$$
$89$ $$(T^{2} + 18 T + 78)^{2}$$
$97$ $$T^{4} + 56T^{2} + 676$$