# Properties

 Label 2475.2.a.bb Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1 - 2) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + b1 + 1) * q^4 + (b2 - b1) * q^7 + (-b2 - 2*b1 - 2) * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1 - 2) q^{8} - q^{11} + (\beta_{2} + \beta_1) q^{13} + (\beta_{2} - \beta_1 + 4) q^{14} + (4 \beta_1 + 3) q^{16} + (\beta_{2} + 3 \beta_1 - 2) q^{17} + (2 \beta_{2} + 2) q^{19} + \beta_1 q^{22} + (2 \beta_{2} - 2 \beta_1) q^{23} + ( - \beta_{2} - 3 \beta_1 - 2) q^{26} + ( - \beta_{2} - 3 \beta_1 + 4) q^{28} + ( - 2 \beta_1 + 4) q^{29} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{31} + ( - 2 \beta_{2} - 3 \beta_1 - 8) q^{32} + ( - 3 \beta_{2} - 3 \beta_1 - 8) q^{34} + 2 q^{37} + ( - 6 \beta_1 + 2) q^{38} + (2 \beta_1 + 4) q^{41} + ( - 3 \beta_{2} - \beta_1) q^{43} + ( - \beta_{2} - \beta_1 - 1) q^{44} + (2 \beta_{2} - 2 \beta_1 + 8) q^{46} + (2 \beta_{2} + 2 \beta_1 - 4) q^{47} + ( - 4 \beta_1 + 5) q^{49} + (\beta_{2} + 5 \beta_1 + 8) q^{52} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{53} + (\beta_{2} + 3 \beta_1) q^{56} + (2 \beta_{2} - 2 \beta_1 + 6) q^{58} + (2 \beta_{2} - 2 \beta_1 - 4) q^{59} + (2 \beta_{2} - 2 \beta_1 - 2) q^{61} + (2 \beta_{2} + 2 \beta_1 + 4) q^{62} + (3 \beta_{2} + 7 \beta_1 + 1) q^{64} + (2 \beta_{2} + 2 \beta_1) q^{67} + (\beta_{2} + 11 \beta_1 + 10) q^{68} + (2 \beta_{2} + 2 \beta_1 - 4) q^{71} + ( - \beta_{2} + 3 \beta_1 + 4) q^{73} - 2 \beta_1 q^{74} + (2 \beta_{2} + 4 \beta_1 + 14) q^{76} + ( - \beta_{2} + \beta_1) q^{77} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{79} + ( - 2 \beta_{2} - 6 \beta_1 - 6) q^{82} + ( - 3 \beta_{2} - 3 \beta_1 + 2) q^{83} + (\beta_{2} + 7 \beta_1) q^{86} + (\beta_{2} + 2 \beta_1 + 2) q^{88} + (4 \beta_1 + 2) q^{89} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{91} + ( - 2 \beta_{2} - 6 \beta_1 + 8) q^{92} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{94} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{97} + (4 \beta_{2} - \beta_1 + 12) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b2 + b1 + 1) * q^4 + (b2 - b1) * q^7 + (-b2 - 2*b1 - 2) * q^8 - q^11 + (b2 + b1) * q^13 + (b2 - b1 + 4) * q^14 + (4*b1 + 3) * q^16 + (b2 + 3*b1 - 2) * q^17 + (2*b2 + 2) * q^19 + b1 * q^22 + (2*b2 - 2*b1) * q^23 + (-b2 - 3*b1 - 2) * q^26 + (-b2 - 3*b1 + 4) * q^28 + (-2*b1 + 4) * q^29 + (-2*b2 - 2*b1 + 4) * q^31 + (-2*b2 - 3*b1 - 8) * q^32 + (-3*b2 - 3*b1 - 8) * q^34 + 2 * q^37 + (-6*b1 + 2) * q^38 + (2*b1 + 4) * q^41 + (-3*b2 - b1) * q^43 + (-b2 - b1 - 1) * q^44 + (2*b2 - 2*b1 + 8) * q^46 + (2*b2 + 2*b1 - 4) * q^47 + (-4*b1 + 5) * q^49 + (b2 + 5*b1 + 8) * q^52 + (-2*b2 + 2*b1 - 2) * q^53 + (b2 + 3*b1) * q^56 + (2*b2 - 2*b1 + 6) * q^58 + (2*b2 - 2*b1 - 4) * q^59 + (2*b2 - 2*b1 - 2) * q^61 + (2*b2 + 2*b1 + 4) * q^62 + (3*b2 + 7*b1 + 1) * q^64 + (2*b2 + 2*b1) * q^67 + (b2 + 11*b1 + 10) * q^68 + (2*b2 + 2*b1 - 4) * q^71 + (-b2 + 3*b1 + 4) * q^73 - 2*b1 * q^74 + (2*b2 + 4*b1 + 14) * q^76 + (-b2 + b1) * q^77 + (-2*b2 - 4*b1 + 6) * q^79 + (-2*b2 - 6*b1 - 6) * q^82 + (-3*b2 - 3*b1 + 2) * q^83 + (b2 + 7*b1) * q^86 + (b2 + 2*b1 + 2) * q^88 + (4*b1 + 2) * q^89 + (-2*b2 - 2*b1 + 4) * q^91 + (-2*b2 - 6*b1 + 8) * q^92 + (-2*b2 - 2*b1 - 4) * q^94 + (-2*b2 - 2*b1 - 6) * q^97 + (4*b2 - b1 + 12) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 5 q^{4} - 9 q^{8}+O(q^{10})$$ 3 * q - q^2 + 5 * q^4 - 9 * q^8 $$3 q - q^{2} + 5 q^{4} - 9 q^{8} - 3 q^{11} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} + 8 q^{19} + q^{22} - 10 q^{26} + 8 q^{28} + 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34} + 6 q^{37} + 14 q^{41} - 4 q^{43} - 5 q^{44} + 24 q^{46} - 8 q^{47} + 11 q^{49} + 30 q^{52} - 6 q^{53} + 4 q^{56} + 18 q^{58} - 12 q^{59} - 6 q^{61} + 16 q^{62} + 13 q^{64} + 4 q^{67} + 42 q^{68} - 8 q^{71} + 14 q^{73} - 2 q^{74} + 48 q^{76} + 12 q^{79} - 26 q^{82} + 8 q^{86} + 9 q^{88} + 10 q^{89} + 8 q^{91} + 16 q^{92} - 16 q^{94} - 22 q^{97} + 39 q^{98}+O(q^{100})$$ 3 * q - q^2 + 5 * q^4 - 9 * q^8 - 3 * q^11 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 + 8 * q^19 + q^22 - 10 * q^26 + 8 * q^28 + 10 * q^29 + 8 * q^31 - 29 * q^32 - 30 * q^34 + 6 * q^37 + 14 * q^41 - 4 * q^43 - 5 * q^44 + 24 * q^46 - 8 * q^47 + 11 * q^49 + 30 * q^52 - 6 * q^53 + 4 * q^56 + 18 * q^58 - 12 * q^59 - 6 * q^61 + 16 * q^62 + 13 * q^64 + 4 * q^67 + 42 * q^68 - 8 * q^71 + 14 * q^73 - 2 * q^74 + 48 * q^76 + 12 * q^79 - 26 * q^82 + 8 * q^86 + 9 * q^88 + 10 * q^89 + 8 * q^91 + 16 * q^92 - 16 * q^94 - 22 * q^97 + 39 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 2$$ -v^2 + 2*v + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 2$$ b1 + 2

## 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
 2.17009 −1.48119 0.311108
−2.70928 0 5.34017 0 0 −1.07838 −9.04945 0 0
1.2 −0.193937 0 −1.96239 0 0 −3.35026 0.768452 0 0
1.3 1.90321 0 1.62222 0 0 4.42864 −0.719004 0 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$

## 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 2475.2.a.bb 3
3.b odd 2 1 825.2.a.k 3
5.b even 2 1 495.2.a.e 3
5.c odd 4 2 2475.2.c.r 6
15.d odd 2 1 165.2.a.c 3
15.e even 4 2 825.2.c.g 6
20.d odd 2 1 7920.2.a.cj 3
33.d even 2 1 9075.2.a.cf 3
55.d odd 2 1 5445.2.a.z 3
60.h even 2 1 2640.2.a.be 3
105.g even 2 1 8085.2.a.bk 3
165.d even 2 1 1815.2.a.m 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 15.d odd 2 1
495.2.a.e 3 5.b even 2 1
825.2.a.k 3 3.b odd 2 1
825.2.c.g 6 15.e even 4 2
1815.2.a.m 3 165.d even 2 1
2475.2.a.bb 3 1.a even 1 1 trivial
2475.2.c.r 6 5.c odd 4 2
2640.2.a.be 3 60.h even 2 1
5445.2.a.z 3 55.d odd 2 1
7920.2.a.cj 3 20.d odd 2 1
8085.2.a.bk 3 105.g even 2 1
9075.2.a.cf 3 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2}^{3} + T_{2}^{2} - 5T_{2} - 1$$ T2^3 + T2^2 - 5*T2 - 1 $$T_{7}^{3} - 16T_{7} - 16$$ T7^3 - 16*T7 - 16 $$T_{29}^{3} - 10T_{29}^{2} + 12T_{29} + 40$$ T29^3 - 10*T29^2 + 12*T29 + 40

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 5T - 1$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 16T - 16$$
$11$ $$(T + 1)^{3}$$
$13$ $$T^{3} - 2 T^{2} - 12 T + 8$$
$17$ $$T^{3} + 2 T^{2} - 52 T - 184$$
$19$ $$T^{3} - 8 T^{2} - 16 T + 160$$
$23$ $$T^{3} - 64T - 128$$
$29$ $$T^{3} - 10 T^{2} + 12 T + 40$$
$31$ $$T^{3} - 8 T^{2} - 32 T + 128$$
$37$ $$(T - 2)^{3}$$
$41$ $$T^{3} - 14 T^{2} + 44 T - 8$$
$43$ $$T^{3} + 4 T^{2} - 80 T - 400$$
$47$ $$T^{3} + 8 T^{2} - 32 T - 128$$
$53$ $$T^{3} + 6 T^{2} - 52 T + 8$$
$59$ $$T^{3} + 12 T^{2} - 16 T - 320$$
$61$ $$T^{3} + 6 T^{2} - 52 T - 248$$
$67$ $$T^{3} - 4 T^{2} - 48 T + 64$$
$71$ $$T^{3} + 8 T^{2} - 32 T - 128$$
$73$ $$T^{3} - 14 T^{2} + 4 T + 344$$
$79$ $$T^{3} - 12 T^{2} - 64 T + 800$$
$83$ $$T^{3} - 120T + 16$$
$89$ $$T^{3} - 10 T^{2} - 52 T + 200$$
$97$ $$T^{3} + 22 T^{2} + 108 T + 8$$