# Properties

 Label 1425.2.c.j 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: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 1.00000i −3.82843 0 −2.41421 3.41421i 4.41421i −1.00000 0
799.2 0.414214i 1.00000i 1.82843 0 0.414214 0.585786i 1.58579i −1.00000 0
799.3 0.414214i 1.00000i 1.82843 0 0.414214 0.585786i 1.58579i −1.00000 0
799.4 2.41421i 1.00000i −3.82843 0 −2.41421 3.41421i 4.41421i −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.j 4
5.b even 2 1 inner 1425.2.c.j 4
5.c odd 4 1 285.2.a.f 2
5.c odd 4 1 1425.2.a.l 2
15.e even 4 1 855.2.a.e 2
15.e even 4 1 4275.2.a.x 2
20.e even 4 1 4560.2.a.bj 2
95.g even 4 1 5415.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.f 2 5.c odd 4 1
855.2.a.e 2 15.e even 4 1
1425.2.a.l 2 5.c odd 4 1
1425.2.c.j 4 1.a even 1 1 trivial
1425.2.c.j 4 5.b even 2 1 inner
4275.2.a.x 2 15.e even 4 1
4560.2.a.bj 2 20.e even 4 1
5415.2.a.p 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 $$S_{2}^{\mathrm{new}}(1425, [\chi])$$:

 $$T_{2}^{4} + 6T_{2}^{2} + 1$$ T2^4 + 6*T2^2 + 1 $$T_{7}^{4} + 12T_{7}^{2} + 4$$ T7^4 + 12*T7^2 + 4 $$T_{11}^{2} - 2$$ T11^2 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 12T^{2} + 4$$
$11$ $$(T^{2} - 2)^{2}$$
$13$ $$T^{4} + 36T^{2} + 196$$
$17$ $$T^{4} + 48T^{2} + 64$$
$19$ $$(T + 1)^{4}$$
$23$ $$T^{4} + 72T^{2} + 784$$
$29$ $$(T^{2} - 4 T - 46)^{2}$$
$31$ $$(T^{2} + 4 T - 68)^{2}$$
$37$ $$T^{4} + 132T^{2} + 1156$$
$41$ $$(T^{2} + 12 T + 34)^{2}$$
$43$ $$T^{4} + 12T^{2} + 4$$
$47$ $$T^{4} + 136T^{2} + 16$$
$53$ $$(T^{2} + 16)^{2}$$
$59$ $$(T^{2} - 72)^{2}$$
$61$ $$(T^{2} - 32)^{2}$$
$67$ $$(T^{2} + 144)^{2}$$
$71$ $$(T^{2} - 8 T - 56)^{2}$$
$73$ $$(T^{2} + 4)^{2}$$
$79$ $$(T^{2} - 128)^{2}$$
$83$ $$T^{4} + 152T^{2} + 4624$$
$89$ $$(T^{2} - 4 T - 158)^{2}$$
$97$ $$(T^{2} + 18)^{2}$$