# Properties

 Label 2475.2.c.p 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: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (2 \beta_{3} + 3 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 1) * q^4 + (2*b3 + 3*b1) * q^7 + (b3 + 2*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (2 \beta_{3} + 3 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} - q^{11} + ( - 3 \beta_{3} + 2 \beta_1) q^{13} + (\beta_{2} - 3) q^{14} + 3 \beta_{2} q^{16} + ( - \beta_{3} - \beta_1) q^{17} + ( - 6 \beta_{2} - 3) q^{19} - \beta_1 q^{22} + (4 \beta_{3} + 5 \beta_1) q^{23} + (5 \beta_{2} - 2) q^{26} + (5 \beta_{3} + 2 \beta_1) q^{28} + ( - \beta_{2} - 3) q^{29} - 3 q^{31} + (5 \beta_{3} + \beta_1) q^{32} + q^{34} + (7 \beta_{3} - 2 \beta_1) q^{37} + ( - 6 \beta_{3} + 3 \beta_1) q^{38} + 3 q^{41} + 6 \beta_{3} q^{43} + ( - \beta_{2} - 1) q^{44} + (\beta_{2} - 5) q^{46} + (\beta_{3} + 8 \beta_1) q^{47} + ( - 3 \beta_{2} - 6) q^{49} + ( - \beta_{3} - 3 \beta_1) q^{52} + ( - 2 \beta_{3} - 7 \beta_1) q^{53} + ( - \beta_{2} - 8) q^{56} + ( - \beta_{3} - 2 \beta_1) q^{58} + (4 \beta_{2} + 7) q^{59} + ( - 5 \beta_{2} - 8) q^{61} - 3 \beta_1 q^{62} + (2 \beta_{2} - 1) q^{64} + 8 \beta_{3} q^{67} + ( - 2 \beta_{3} - \beta_1) q^{68} + (10 \beta_{2} + 8) q^{71} + ( - 12 \beta_{3} - \beta_1) q^{73} + ( - 9 \beta_{2} + 2) q^{74} + ( - 3 \beta_{2} - 9) q^{76} + ( - 2 \beta_{3} - 3 \beta_1) q^{77} + (3 \beta_{2} - 1) q^{79} + 3 \beta_1 q^{82} + ( - 15 \beta_{3} - 3 \beta_1) q^{83} - 6 \beta_{2} q^{86} + ( - \beta_{3} - 2 \beta_1) q^{88} + ( - 5 \beta_{2} - 15) q^{89} + 11 \beta_{2} q^{91} + (9 \beta_{3} + 4 \beta_1) q^{92} + (7 \beta_{2} - 8) q^{94} - \beta_1 q^{97} + ( - 3 \beta_{3} - 3 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 1) * q^4 + (2*b3 + 3*b1) * q^7 + (b3 + 2*b1) * q^8 - q^11 + (-3*b3 + 2*b1) * q^13 + (b2 - 3) * q^14 + 3*b2 * q^16 + (-b3 - b1) * q^17 + (-6*b2 - 3) * q^19 - b1 * q^22 + (4*b3 + 5*b1) * q^23 + (5*b2 - 2) * q^26 + (5*b3 + 2*b1) * q^28 + (-b2 - 3) * q^29 - 3 * q^31 + (5*b3 + b1) * q^32 + q^34 + (7*b3 - 2*b1) * q^37 + (-6*b3 + 3*b1) * q^38 + 3 * q^41 + 6*b3 * q^43 + (-b2 - 1) * q^44 + (b2 - 5) * q^46 + (b3 + 8*b1) * q^47 + (-3*b2 - 6) * q^49 + (-b3 - 3*b1) * q^52 + (-2*b3 - 7*b1) * q^53 + (-b2 - 8) * q^56 + (-b3 - 2*b1) * q^58 + (4*b2 + 7) * q^59 + (-5*b2 - 8) * q^61 - 3*b1 * q^62 + (2*b2 - 1) * q^64 + 8*b3 * q^67 + (-2*b3 - b1) * q^68 + (10*b2 + 8) * q^71 + (-12*b3 - b1) * q^73 + (-9*b2 + 2) * q^74 + (-3*b2 - 9) * q^76 + (-2*b3 - 3*b1) * q^77 + (3*b2 - 1) * q^79 + 3*b1 * q^82 + (-15*b3 - 3*b1) * q^83 - 6*b2 * q^86 + (-b3 - 2*b1) * q^88 + (-5*b2 - 15) * q^89 + 11*b2 * q^91 + (9*b3 + 4*b1) * q^92 + (7*b2 - 8) * q^94 - b1 * q^97 + (-3*b3 - 3*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4}+O(q^{10})$$ 4 * q + 2 * q^4 $$4 q + 2 q^{4} - 4 q^{11} - 14 q^{14} - 6 q^{16} - 18 q^{26} - 10 q^{29} - 12 q^{31} + 4 q^{34} + 12 q^{41} - 2 q^{44} - 22 q^{46} - 18 q^{49} - 30 q^{56} + 20 q^{59} - 22 q^{61} - 8 q^{64} + 12 q^{71} + 26 q^{74} - 30 q^{76} - 10 q^{79} + 12 q^{86} - 50 q^{89} - 22 q^{91} - 46 q^{94}+O(q^{100})$$ 4 * q + 2 * q^4 - 4 * q^11 - 14 * q^14 - 6 * q^16 - 18 * q^26 - 10 * q^29 - 12 * q^31 + 4 * q^34 + 12 * q^41 - 2 * q^44 - 22 * q^46 - 18 * q^49 - 30 * q^56 + 20 * q^59 - 22 * q^61 - 8 * q^64 + 12 * q^71 + 26 * q^74 - 30 * q^76 - 10 * q^79 + 12 * q^86 - 50 * q^89 - 22 * q^91 - 46 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2\beta_1$$ b3 - 2*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
 − 1.61803i − 0.618034i 0.618034i 1.61803i
1.61803i 0 −0.618034 0 0 2.85410i 2.23607i 0 0
199.2 0.618034i 0 1.61803 0 0 3.85410i 2.23607i 0 0
199.3 0.618034i 0 1.61803 0 0 3.85410i 2.23607i 0 0
199.4 1.61803i 0 −0.618034 0 0 2.85410i 2.23607i 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.p 4
3.b odd 2 1 275.2.b.e 4
5.b even 2 1 inner 2475.2.c.p 4
5.c odd 4 1 2475.2.a.n 2
5.c odd 4 1 2475.2.a.s 2
12.b even 2 1 4400.2.b.x 4
15.d odd 2 1 275.2.b.e 4
15.e even 4 1 275.2.a.d 2
15.e even 4 1 275.2.a.g yes 2
60.h even 2 1 4400.2.b.x 4
60.l odd 4 1 4400.2.a.bg 2
60.l odd 4 1 4400.2.a.bv 2
165.l odd 4 1 3025.2.a.i 2
165.l odd 4 1 3025.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 15.e even 4 1
275.2.a.g yes 2 15.e even 4 1
275.2.b.e 4 3.b odd 2 1
275.2.b.e 4 15.d odd 2 1
2475.2.a.n 2 5.c odd 4 1
2475.2.a.s 2 5.c odd 4 1
2475.2.c.p 4 1.a even 1 1 trivial
2475.2.c.p 4 5.b even 2 1 inner
3025.2.a.i 2 165.l odd 4 1
3025.2.a.m 2 165.l odd 4 1
4400.2.a.bg 2 60.l odd 4 1
4400.2.a.bv 2 60.l odd 4 1
4400.2.b.x 4 12.b even 2 1
4400.2.b.x 4 60.h even 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 3T_{2}^{2} + 1$$ T2^4 + 3*T2^2 + 1 $$T_{7}^{4} + 23T_{7}^{2} + 121$$ T7^4 + 23*T7^2 + 121 $$T_{29}^{2} + 5T_{29} + 5$$ T29^2 + 5*T29 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 23T^{2} + 121$$
$11$ $$(T + 1)^{4}$$
$13$ $$T^{4} + 42T^{2} + 121$$
$17$ $$T^{4} + 3T^{2} + 1$$
$19$ $$(T^{2} - 45)^{2}$$
$23$ $$T^{4} + 67T^{2} + 841$$
$29$ $$(T^{2} + 5 T + 5)^{2}$$
$31$ $$(T + 3)^{4}$$
$37$ $$T^{4} + 138T^{2} + 3481$$
$41$ $$(T - 3)^{4}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$T^{4} + 178T^{2} + 5041$$
$53$ $$T^{4} + 127T^{2} + 3481$$
$59$ $$(T^{2} - 10 T + 5)^{2}$$
$61$ $$(T^{2} + 11 T - 1)^{2}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} - 6 T - 116)^{2}$$
$73$ $$T^{4} + 267 T^{2} + 17161$$
$79$ $$(T^{2} + 5 T - 5)^{2}$$
$83$ $$T^{4} + 387 T^{2} + 29241$$
$89$ $$(T^{2} + 25 T + 125)^{2}$$
$97$ $$T^{4} + 3T^{2} + 1$$