# Properties

 Label 2475.2.c.j Level $2475$ Weight $2$ Character orbit 2475.c Analytic conductor $19.763$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} + \beta_{2} q^{7} + (4 \beta_{2} - \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 - 3) * q^4 + b2 * q^7 + (4*b2 - b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} + \beta_{2} q^{7} + (4 \beta_{2} - \beta_1) q^{8} + q^{11} + (3 \beta_{2} + \beta_1) q^{13} + ( - \beta_{3} + 1) q^{14} + ( - 3 \beta_{3} + 3) q^{16} + ( - 4 \beta_{2} - \beta_1) q^{17} - 3 q^{19} + \beta_1 q^{22} + (4 \beta_{2} - \beta_1) q^{23} + ( - 2 \beta_{3} - 2) q^{26} + ( - 2 \beta_{2} + \beta_1) q^{28} + ( - 2 \beta_{3} + 2) q^{29} - 3 \beta_{3} q^{31} + ( - 4 \beta_{2} + \beta_1) q^{32} + (3 \beta_{3} + 1) q^{34} + (2 \beta_{2} + 5 \beta_1) q^{37} - 3 \beta_1 q^{38} + ( - 3 \beta_{3} + 3) q^{41} + (3 \beta_{2} + 3 \beta_1) q^{43} + (\beta_{3} - 3) q^{44} + ( - 5 \beta_{3} + 9) q^{46} + ( - 8 \beta_{2} - \beta_1) q^{47} + 6 q^{49} - 2 \beta_{2} q^{52} + (4 \beta_{2} + 2 \beta_1) q^{53} + (\beta_{3} - 5) q^{56} + ( - 8 \beta_{2} + 2 \beta_1) q^{58} + ( - \beta_{3} + 9) q^{59} + (\beta_{3} - 6) q^{61} - 12 \beta_{2} q^{62} + ( - \beta_{3} - 3) q^{64} + (\beta_{2} + 3 \beta_1) q^{67} + (4 \beta_{2} - \beta_1) q^{68} + (5 \beta_{3} - 1) q^{71} + ( - 6 \beta_{2} + 4 \beta_1) q^{73} + (3 \beta_{3} - 23) q^{74} + ( - 3 \beta_{3} + 9) q^{76} + \beta_{2} q^{77} + ( - 3 \beta_{3} + 11) q^{79} + ( - 12 \beta_{2} + 3 \beta_1) q^{82} - 12 q^{86} + (4 \beta_{2} - \beta_1) q^{88} + (2 \beta_{3} - 14) q^{89} + ( - \beta_{3} - 2) q^{91} + ( - 12 \beta_{2} + 7 \beta_1) q^{92} + (7 \beta_{3} - 3) q^{94} + ( - 3 \beta_{2} - 2 \beta_1) q^{97} + 6 \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 - 3) * q^4 + b2 * q^7 + (4*b2 - b1) * q^8 + q^11 + (3*b2 + b1) * q^13 + (-b3 + 1) * q^14 + (-3*b3 + 3) * q^16 + (-4*b2 - b1) * q^17 - 3 * q^19 + b1 * q^22 + (4*b2 - b1) * q^23 + (-2*b3 - 2) * q^26 + (-2*b2 + b1) * q^28 + (-2*b3 + 2) * q^29 - 3*b3 * q^31 + (-4*b2 + b1) * q^32 + (3*b3 + 1) * q^34 + (2*b2 + 5*b1) * q^37 - 3*b1 * q^38 + (-3*b3 + 3) * q^41 + (3*b2 + 3*b1) * q^43 + (b3 - 3) * q^44 + (-5*b3 + 9) * q^46 + (-8*b2 - b1) * q^47 + 6 * q^49 - 2*b2 * q^52 + (4*b2 + 2*b1) * q^53 + (b3 - 5) * q^56 + (-8*b2 + 2*b1) * q^58 + (-b3 + 9) * q^59 + (b3 - 6) * q^61 - 12*b2 * q^62 + (-b3 - 3) * q^64 + (b2 + 3*b1) * q^67 + (4*b2 - b1) * q^68 + (5*b3 - 1) * q^71 + (-6*b2 + 4*b1) * q^73 + (3*b3 - 23) * q^74 + (-3*b3 + 9) * q^76 + b2 * q^77 + (-3*b3 + 11) * q^79 + (-12*b2 + 3*b1) * q^82 - 12 * q^86 + (4*b2 - b1) * q^88 + (2*b3 - 14) * q^89 + (-b3 - 2) * q^91 + (-12*b2 + 7*b1) * q^92 + (7*b3 - 3) * q^94 + (-3*b2 - 2*b1) * q^97 + 6*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{4}+O(q^{10})$$ 4 * q - 10 * q^4 $$4 q - 10 q^{4} + 4 q^{11} + 2 q^{14} + 6 q^{16} - 12 q^{19} - 12 q^{26} + 4 q^{29} - 6 q^{31} + 10 q^{34} + 6 q^{41} - 10 q^{44} + 26 q^{46} + 24 q^{49} - 18 q^{56} + 34 q^{59} - 22 q^{61} - 14 q^{64} + 6 q^{71} - 86 q^{74} + 30 q^{76} + 38 q^{79} - 48 q^{86} - 52 q^{89} - 10 q^{91} + 2 q^{94}+O(q^{100})$$ 4 * q - 10 * q^4 + 4 * q^11 + 2 * q^14 + 6 * q^16 - 12 * q^19 - 12 * q^26 + 4 * q^29 - 6 * q^31 + 10 * q^34 + 6 * q^41 - 10 * q^44 + 26 * q^46 + 24 * q^49 - 18 * q^56 + 34 * q^59 - 22 * q^61 - 14 * q^64 + 6 * q^71 - 86 * q^74 + 30 * q^76 + 38 * q^79 - 48 * q^86 - 52 * q^89 - 10 * q^91 + 2 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 4$$ (v^3 + 5*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$4\beta_{2} - 5\beta_1$$ 4*b2 - 5*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 2.56155i − 1.56155i 1.56155i 2.56155i
2.56155i 0 −4.56155 0 0 1.00000i 6.56155i 0 0
199.2 1.56155i 0 −0.438447 0 0 1.00000i 2.43845i 0 0
199.3 1.56155i 0 −0.438447 0 0 1.00000i 2.43845i 0 0
199.4 2.56155i 0 −4.56155 0 0 1.00000i 6.56155i 0 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 2475.2.c.j 4
3.b odd 2 1 2475.2.c.i 4
5.b even 2 1 inner 2475.2.c.j 4
5.c odd 4 1 2475.2.a.q yes 2
5.c odd 4 1 2475.2.a.u yes 2
15.d odd 2 1 2475.2.c.i 4
15.e even 4 1 2475.2.a.p 2
15.e even 4 1 2475.2.a.v yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.2.a.p 2 15.e even 4 1
2475.2.a.q yes 2 5.c odd 4 1
2475.2.a.u yes 2 5.c odd 4 1
2475.2.a.v yes 2 15.e even 4 1
2475.2.c.i 4 3.b odd 2 1
2475.2.c.i 4 15.d odd 2 1
2475.2.c.j 4 1.a even 1 1 trivial
2475.2.c.j 4 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2475, [\chi])$$:

 $$T_{2}^{4} + 9T_{2}^{2} + 16$$ T2^4 + 9*T2^2 + 16 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{29}^{2} - 2T_{29} - 16$$ T29^2 - 2*T29 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 9T^{2} + 16$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T - 1)^{4}$$
$13$ $$T^{4} + 21T^{2} + 4$$
$17$ $$T^{4} + 33T^{2} + 64$$
$19$ $$(T + 3)^{4}$$
$23$ $$T^{4} + 49T^{2} + 256$$
$29$ $$(T^{2} - 2 T - 16)^{2}$$
$31$ $$(T^{2} + 3 T - 36)^{2}$$
$37$ $$T^{4} + 213 T^{2} + 11236$$
$41$ $$(T^{2} - 3 T - 36)^{2}$$
$43$ $$T^{4} + 81T^{2} + 1296$$
$47$ $$T^{4} + 121T^{2} + 2704$$
$53$ $$T^{4} + 52T^{2} + 64$$
$59$ $$(T^{2} - 17 T + 68)^{2}$$
$61$ $$(T^{2} + 11 T + 26)^{2}$$
$67$ $$T^{4} + 77T^{2} + 1444$$
$71$ $$(T^{2} - 3 T - 104)^{2}$$
$73$ $$T^{4} + 264T^{2} + 16$$
$79$ $$(T^{2} - 19 T + 52)^{2}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} + 26 T + 152)^{2}$$
$97$ $$T^{4} + 42T^{2} + 169$$