# Properties

 Label 2475.2.a.bi Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ 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] = [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: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 7x^{2} + 4$$ x^4 - 7*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + 2 \beta_{2} q^{7} + (2 \beta_{2} + \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 + 2) * q^4 + 2*b2 * q^7 + (2*b2 + b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + 2 \beta_{2} q^{7} + (2 \beta_{2} + \beta_1) q^{8} + q^{11} + (2 \beta_{3} + 4) q^{14} + (\beta_{3} + 4) q^{16} + (2 \beta_{2} - 2 \beta_1) q^{17} + 4 q^{19} + \beta_1 q^{22} + (\beta_{2} - \beta_1) q^{23} + 6 \beta_1 q^{28} + ( - 2 \beta_{3} - 4) q^{29} + (\beta_{3} + 1) q^{31} + ( - 2 \beta_{2} + 3 \beta_1) q^{32} - 4 q^{34} + ( - 5 \beta_{2} + 3 \beta_1) q^{37} + 4 \beta_1 q^{38} + ( - 2 \beta_{3} - 4) q^{41} + 2 \beta_{2} q^{43} + (\beta_{3} + 2) q^{44} - 2 q^{46} + ( - 2 \beta_{2} + 4 \beta_1) q^{47} + 5 q^{49} - 4 \beta_1 q^{53} + (2 \beta_{3} + 16) q^{56} + ( - 4 \beta_{2} - 6 \beta_1) q^{58} + (\beta_{3} + 5) q^{59} + ( - 2 \beta_{3} + 4) q^{61} + (2 \beta_{2} + 2 \beta_1) q^{62} - \beta_{3} q^{64} + (\beta_{2} + 3 \beta_1) q^{67} - 4 \beta_{2} q^{68} + (3 \beta_{3} + 3) q^{71} - 4 \beta_{2} q^{73} + ( - 2 \beta_{3} + 2) q^{74} + (4 \beta_{3} + 8) q^{76} + 2 \beta_{2} q^{77} + ( - 2 \beta_{3} + 6) q^{79} + ( - 4 \beta_{2} - 6 \beta_1) q^{82} + (2 \beta_{2} - 4 \beta_1) q^{83} + (2 \beta_{3} + 4) q^{86} + (2 \beta_{2} + \beta_1) q^{88} + ( - \beta_{3} + 1) q^{89} - 2 \beta_{2} q^{92} + (2 \beta_{3} + 12) q^{94} + ( - \beta_{2} + 3 \beta_1) q^{97} + 5 \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 + 2) * q^4 + 2*b2 * q^7 + (2*b2 + b1) * q^8 + q^11 + (2*b3 + 4) * q^14 + (b3 + 4) * q^16 + (2*b2 - 2*b1) * q^17 + 4 * q^19 + b1 * q^22 + (b2 - b1) * q^23 + 6*b1 * q^28 + (-2*b3 - 4) * q^29 + (b3 + 1) * q^31 + (-2*b2 + 3*b1) * q^32 - 4 * q^34 + (-5*b2 + 3*b1) * q^37 + 4*b1 * q^38 + (-2*b3 - 4) * q^41 + 2*b2 * q^43 + (b3 + 2) * q^44 - 2 * q^46 + (-2*b2 + 4*b1) * q^47 + 5 * q^49 - 4*b1 * q^53 + (2*b3 + 16) * q^56 + (-4*b2 - 6*b1) * q^58 + (b3 + 5) * q^59 + (-2*b3 + 4) * q^61 + (2*b2 + 2*b1) * q^62 - b3 * q^64 + (b2 + 3*b1) * q^67 - 4*b2 * q^68 + (3*b3 + 3) * q^71 - 4*b2 * q^73 + (-2*b3 + 2) * q^74 + (4*b3 + 8) * q^76 + 2*b2 * q^77 + (-2*b3 + 6) * q^79 + (-4*b2 - 6*b1) * q^82 + (2*b2 - 4*b1) * q^83 + (2*b3 + 4) * q^86 + (2*b2 + b1) * q^88 + (-b3 + 1) * q^89 - 2*b2 * q^92 + (2*b3 + 12) * q^94 + (-b2 + 3*b1) * q^97 + 5*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{4}+O(q^{10})$$ 4 * q + 6 * q^4 $$4 q + 6 q^{4} + 4 q^{11} + 12 q^{14} + 14 q^{16} + 16 q^{19} - 12 q^{29} + 2 q^{31} - 16 q^{34} - 12 q^{41} + 6 q^{44} - 8 q^{46} + 20 q^{49} + 60 q^{56} + 18 q^{59} + 20 q^{61} + 2 q^{64} + 6 q^{71} + 12 q^{74} + 24 q^{76} + 28 q^{79} + 12 q^{86} + 6 q^{89} + 44 q^{94}+O(q^{100})$$ 4 * q + 6 * q^4 + 4 * q^11 + 12 * q^14 + 14 * q^16 + 16 * q^19 - 12 * q^29 + 2 * q^31 - 16 * q^34 - 12 * q^41 + 6 * q^44 - 8 * q^46 + 20 * q^49 + 60 * q^56 + 18 * q^59 + 20 * q^61 + 2 * q^64 + 6 * q^71 + 12 * q^74 + 24 * q^76 + 28 * q^79 + 12 * q^86 + 6 * q^89 + 44 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7x^{2} + 4$$ :

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

## 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
 −2.52434 −0.792287 0.792287 2.52434
−2.52434 0 4.37228 0 0 −3.46410 −5.98844 0 0
1.2 −0.792287 0 −1.37228 0 0 3.46410 2.67181 0 0
1.3 0.792287 0 −1.37228 0 0 −3.46410 −2.67181 0 0
1.4 2.52434 0 4.37228 0 0 3.46410 5.98844 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

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.a.bi 4
3.b odd 2 1 275.2.a.h 4
5.b even 2 1 inner 2475.2.a.bi 4
5.c odd 4 2 495.2.c.a 4
12.b even 2 1 4400.2.a.cc 4
15.d odd 2 1 275.2.a.h 4
15.e even 4 2 55.2.b.a 4
33.d even 2 1 3025.2.a.ba 4
60.h even 2 1 4400.2.a.cc 4
60.l odd 4 2 880.2.b.h 4
165.d even 2 1 3025.2.a.ba 4
165.l odd 4 2 605.2.b.c 4
165.u odd 20 8 605.2.j.j 16
165.v even 20 8 605.2.j.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.b.a 4 15.e even 4 2
275.2.a.h 4 3.b odd 2 1
275.2.a.h 4 15.d odd 2 1
495.2.c.a 4 5.c odd 4 2
605.2.b.c 4 165.l odd 4 2
605.2.j.i 16 165.v even 20 8
605.2.j.j 16 165.u odd 20 8
880.2.b.h 4 60.l odd 4 2
2475.2.a.bi 4 1.a even 1 1 trivial
2475.2.a.bi 4 5.b even 2 1 inner
3025.2.a.ba 4 33.d even 2 1
3025.2.a.ba 4 165.d even 2 1
4400.2.a.cc 4 12.b even 2 1
4400.2.a.cc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2}^{4} - 7T_{2}^{2} + 4$$ T2^4 - 7*T2^2 + 4 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{29}^{2} + 6T_{29} - 24$$ T29^2 + 6*T29 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 7T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 12)^{2}$$
$11$ $$(T - 1)^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} - 28T^{2} + 64$$
$19$ $$(T - 4)^{4}$$
$23$ $$T^{4} - 7T^{2} + 4$$
$29$ $$(T^{2} + 6 T - 24)^{2}$$
$31$ $$(T^{2} - T - 8)^{2}$$
$37$ $$T^{4} - 123T^{2} + 144$$
$41$ $$(T^{2} + 6 T - 24)^{2}$$
$43$ $$(T^{2} - 12)^{2}$$
$47$ $$(T^{2} - 44)^{2}$$
$53$ $$T^{4} - 112T^{2} + 1024$$
$59$ $$(T^{2} - 9 T + 12)^{2}$$
$61$ $$(T^{2} - 10 T - 8)^{2}$$
$67$ $$T^{4} - 87T^{2} + 36$$
$71$ $$(T^{2} - 3 T - 72)^{2}$$
$73$ $$(T^{2} - 48)^{2}$$
$79$ $$(T^{2} - 14 T + 16)^{2}$$
$83$ $$(T^{2} - 44)^{2}$$
$89$ $$(T^{2} - 3 T - 6)^{2}$$
$97$ $$T^{4} - 51T^{2} + 576$$