# Properties

 Label 2475.2.a.o Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta + 1) q^{4} + (\beta + 2) q^{7} - 3 q^{8} +O(q^{10})$$ q - b * q^2 + (b + 1) * q^4 + (b + 2) * q^7 - 3 * q^8 $$q - \beta q^{2} + (\beta + 1) q^{4} + (\beta + 2) q^{7} - 3 q^{8} + q^{11} + 5 q^{13} + ( - 3 \beta - 3) q^{14} + (\beta - 2) q^{16} + (3 \beta - 3) q^{17} - q^{19} - \beta q^{22} + ( - \beta + 6) q^{23} - 5 \beta q^{26} + (4 \beta + 5) q^{28} + (3 \beta + 3) q^{29} + ( - 4 \beta + 5) q^{31} + (\beta + 3) q^{32} - 9 q^{34} + (2 \beta + 5) q^{37} + \beta q^{38} + ( - 2 \beta + 3) q^{41} + ( - 4 \beta + 2) q^{43} + (\beta + 1) q^{44} + ( - 5 \beta + 3) q^{46} + 3 q^{47} + 5 \beta q^{49} + (5 \beta + 5) q^{52} - \beta q^{53} + ( - 3 \beta - 6) q^{56} + ( - 6 \beta - 9) q^{58} + ( - 4 \beta + 9) q^{59} + (3 \beta - 4) q^{61} + ( - \beta + 12) q^{62} + ( - 6 \beta + 1) q^{64} - 4 q^{67} + (3 \beta + 6) q^{68} - 2 \beta q^{71} + (3 \beta - 4) q^{73} + ( - 7 \beta - 6) q^{74} + ( - \beta - 1) q^{76} + (\beta + 2) q^{77} + (3 \beta - 7) q^{79} + ( - \beta + 6) q^{82} + ( - 5 \beta - 3) q^{83} + (2 \beta + 12) q^{86} - 3 q^{88} + ( - \beta - 3) q^{89} + (5 \beta + 10) q^{91} + (4 \beta + 3) q^{92} - 3 \beta q^{94} + ( - \beta + 14) q^{97} + ( - 5 \beta - 15) q^{98} +O(q^{100})$$ q - b * q^2 + (b + 1) * q^4 + (b + 2) * q^7 - 3 * q^8 + q^11 + 5 * q^13 + (-3*b - 3) * q^14 + (b - 2) * q^16 + (3*b - 3) * q^17 - q^19 - b * q^22 + (-b + 6) * q^23 - 5*b * q^26 + (4*b + 5) * q^28 + (3*b + 3) * q^29 + (-4*b + 5) * q^31 + (b + 3) * q^32 - 9 * q^34 + (2*b + 5) * q^37 + b * q^38 + (-2*b + 3) * q^41 + (-4*b + 2) * q^43 + (b + 1) * q^44 + (-5*b + 3) * q^46 + 3 * q^47 + 5*b * q^49 + (5*b + 5) * q^52 - b * q^53 + (-3*b - 6) * q^56 + (-6*b - 9) * q^58 + (-4*b + 9) * q^59 + (3*b - 4) * q^61 + (-b + 12) * q^62 + (-6*b + 1) * q^64 - 4 * q^67 + (3*b + 6) * q^68 - 2*b * q^71 + (3*b - 4) * q^73 + (-7*b - 6) * q^74 + (-b - 1) * q^76 + (b + 2) * q^77 + (3*b - 7) * q^79 + (-b + 6) * q^82 + (-5*b - 3) * q^83 + (2*b + 12) * q^86 - 3 * q^88 + (-b - 3) * q^89 + (5*b + 10) * q^91 + (4*b + 3) * q^92 - 3*b * q^94 + (-b + 14) * q^97 + (-5*b - 15) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - q^2 + 3 * q^4 + 5 * q^7 - 6 * q^8 $$2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8} + 2 q^{11} + 10 q^{13} - 9 q^{14} - 3 q^{16} - 3 q^{17} - 2 q^{19} - q^{22} + 11 q^{23} - 5 q^{26} + 14 q^{28} + 9 q^{29} + 6 q^{31} + 7 q^{32} - 18 q^{34} + 12 q^{37} + q^{38} + 4 q^{41} + 3 q^{44} + q^{46} + 6 q^{47} + 5 q^{49} + 15 q^{52} - q^{53} - 15 q^{56} - 24 q^{58} + 14 q^{59} - 5 q^{61} + 23 q^{62} - 4 q^{64} - 8 q^{67} + 15 q^{68} - 2 q^{71} - 5 q^{73} - 19 q^{74} - 3 q^{76} + 5 q^{77} - 11 q^{79} + 11 q^{82} - 11 q^{83} + 26 q^{86} - 6 q^{88} - 7 q^{89} + 25 q^{91} + 10 q^{92} - 3 q^{94} + 27 q^{97} - 35 q^{98}+O(q^{100})$$ 2 * q - q^2 + 3 * q^4 + 5 * q^7 - 6 * q^8 + 2 * q^11 + 10 * q^13 - 9 * q^14 - 3 * q^16 - 3 * q^17 - 2 * q^19 - q^22 + 11 * q^23 - 5 * q^26 + 14 * q^28 + 9 * q^29 + 6 * q^31 + 7 * q^32 - 18 * q^34 + 12 * q^37 + q^38 + 4 * q^41 + 3 * q^44 + q^46 + 6 * q^47 + 5 * q^49 + 15 * q^52 - q^53 - 15 * q^56 - 24 * q^58 + 14 * q^59 - 5 * q^61 + 23 * q^62 - 4 * q^64 - 8 * q^67 + 15 * q^68 - 2 * q^71 - 5 * q^73 - 19 * q^74 - 3 * q^76 + 5 * q^77 - 11 * q^79 + 11 * q^82 - 11 * q^83 + 26 * q^86 - 6 * q^88 - 7 * q^89 + 25 * q^91 + 10 * q^92 - 3 * q^94 + 27 * q^97 - 35 * q^98

## 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
 2.30278 −1.30278
−2.30278 0 3.30278 0 0 4.30278 −3.00000 0 0
1.2 1.30278 0 −0.302776 0 0 0.697224 −3.00000 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.o 2
3.b odd 2 1 275.2.a.f yes 2
5.b even 2 1 2475.2.a.t 2
5.c odd 4 2 2475.2.c.k 4
12.b even 2 1 4400.2.a.bh 2
15.d odd 2 1 275.2.a.e 2
15.e even 4 2 275.2.b.c 4
33.d even 2 1 3025.2.a.h 2
60.h even 2 1 4400.2.a.bs 2
60.l odd 4 2 4400.2.b.y 4
165.d even 2 1 3025.2.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 15.d odd 2 1
275.2.a.f yes 2 3.b odd 2 1
275.2.b.c 4 15.e even 4 2
2475.2.a.o 2 1.a even 1 1 trivial
2475.2.a.t 2 5.b even 2 1
2475.2.c.k 4 5.c odd 4 2
3025.2.a.h 2 33.d even 2 1
3025.2.a.n 2 165.d even 2 1
4400.2.a.bh 2 12.b even 2 1
4400.2.a.bs 2 60.h even 2 1
4400.2.b.y 4 60.l odd 4 2

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} - 3$$ T2^2 + T2 - 3 $$T_{7}^{2} - 5T_{7} + 3$$ T7^2 - 5*T7 + 3 $$T_{29}^{2} - 9T_{29} - 9$$ T29^2 - 9*T29 - 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 3$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 5T + 3$$
$11$ $$(T - 1)^{2}$$
$13$ $$(T - 5)^{2}$$
$17$ $$T^{2} + 3T - 27$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 11T + 27$$
$29$ $$T^{2} - 9T - 9$$
$31$ $$T^{2} - 6T - 43$$
$37$ $$T^{2} - 12T + 23$$
$41$ $$T^{2} - 4T - 9$$
$43$ $$T^{2} - 52$$
$47$ $$(T - 3)^{2}$$
$53$ $$T^{2} + T - 3$$
$59$ $$T^{2} - 14T - 3$$
$61$ $$T^{2} + 5T - 23$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} + 2T - 12$$
$73$ $$T^{2} + 5T - 23$$
$79$ $$T^{2} + 11T + 1$$
$83$ $$T^{2} + 11T - 51$$
$89$ $$T^{2} + 7T + 9$$
$97$ $$T^{2} - 27T + 179$$