# Properties

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

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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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} + 2 q^{7} + (\beta + 3) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 + 2 * q^7 + (b + 3) * q^8 $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} + 2 q^{7} + (\beta + 3) q^{8} - q^{11} + ( - 2 \beta + 4) q^{13} + (2 \beta + 2) q^{14} + 3 q^{16} + (2 \beta + 4) q^{17} + ( - \beta - 1) q^{22} - 2 \beta q^{23} + 2 \beta q^{26} + (4 \beta + 2) q^{28} + (4 \beta - 2) q^{29} + (\beta - 3) q^{32} + (6 \beta + 8) q^{34} + (4 \beta + 2) q^{37} - 6 q^{41} + 6 q^{43} + ( - 2 \beta - 1) q^{44} + ( - 2 \beta - 4) q^{46} + 2 \beta q^{47} - 3 q^{49} + (6 \beta - 4) q^{52} + (4 \beta + 6) q^{53} + (2 \beta + 6) q^{56} + (2 \beta + 6) q^{58} + ( - 4 \beta + 4) q^{59} + ( - 8 \beta + 2) q^{61} + ( - 2 \beta - 7) q^{64} + ( - 6 \beta - 4) q^{67} + (10 \beta + 12) q^{68} - 8 \beta q^{71} + ( - 2 \beta + 4) q^{73} + (6 \beta + 10) q^{74} - 2 q^{77} + 4 q^{79} + ( - 6 \beta - 6) q^{82} - 6 q^{83} + (6 \beta + 6) q^{86} + ( - \beta - 3) q^{88} + (8 \beta + 2) q^{89} + ( - 4 \beta + 8) q^{91} + ( - 2 \beta - 8) q^{92} + (2 \beta + 4) q^{94} + ( - 4 \beta + 2) q^{97} + ( - 3 \beta - 3) q^{98} +O(q^{100})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 + 2 * q^7 + (b + 3) * q^8 - q^11 + (-2*b + 4) * q^13 + (2*b + 2) * q^14 + 3 * q^16 + (2*b + 4) * q^17 + (-b - 1) * q^22 - 2*b * q^23 + 2*b * q^26 + (4*b + 2) * q^28 + (4*b - 2) * q^29 + (b - 3) * q^32 + (6*b + 8) * q^34 + (4*b + 2) * q^37 - 6 * q^41 + 6 * q^43 + (-2*b - 1) * q^44 + (-2*b - 4) * q^46 + 2*b * q^47 - 3 * q^49 + (6*b - 4) * q^52 + (4*b + 6) * q^53 + (2*b + 6) * q^56 + (2*b + 6) * q^58 + (-4*b + 4) * q^59 + (-8*b + 2) * q^61 + (-2*b - 7) * q^64 + (-6*b - 4) * q^67 + (10*b + 12) * q^68 - 8*b * q^71 + (-2*b + 4) * q^73 + (6*b + 10) * q^74 - 2 * q^77 + 4 * q^79 + (-6*b - 6) * q^82 - 6 * q^83 + (6*b + 6) * q^86 + (-b - 3) * q^88 + (8*b + 2) * q^89 + (-4*b + 8) * q^91 + (-2*b - 8) * q^92 + (2*b + 4) * q^94 + (-4*b + 2) * q^97 + (-3*b - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 6 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8} - 2 q^{11} + 8 q^{13} + 4 q^{14} + 6 q^{16} + 8 q^{17} - 2 q^{22} + 4 q^{28} - 4 q^{29} - 6 q^{32} + 16 q^{34} + 4 q^{37} - 12 q^{41} + 12 q^{43} - 2 q^{44} - 8 q^{46} - 6 q^{49} - 8 q^{52} + 12 q^{53} + 12 q^{56} + 12 q^{58} + 8 q^{59} + 4 q^{61} - 14 q^{64} - 8 q^{67} + 24 q^{68} + 8 q^{73} + 20 q^{74} - 4 q^{77} + 8 q^{79} - 12 q^{82} - 12 q^{83} + 12 q^{86} - 6 q^{88} + 4 q^{89} + 16 q^{91} - 16 q^{92} + 8 q^{94} + 4 q^{97} - 6 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 6 * q^8 - 2 * q^11 + 8 * q^13 + 4 * q^14 + 6 * q^16 + 8 * q^17 - 2 * q^22 + 4 * q^28 - 4 * q^29 - 6 * q^32 + 16 * q^34 + 4 * q^37 - 12 * q^41 + 12 * q^43 - 2 * q^44 - 8 * q^46 - 6 * q^49 - 8 * q^52 + 12 * q^53 + 12 * q^56 + 12 * q^58 + 8 * q^59 + 4 * q^61 - 14 * q^64 - 8 * q^67 + 24 * q^68 + 8 * q^73 + 20 * q^74 - 4 * q^77 + 8 * q^79 - 12 * q^82 - 12 * q^83 + 12 * q^86 - 6 * q^88 + 4 * q^89 + 16 * q^91 - 16 * q^92 + 8 * q^94 + 4 * q^97 - 6 * 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
 −1.41421 1.41421
−0.414214 0 −1.82843 0 0 2.00000 1.58579 0 0
1.2 2.41421 0 3.82843 0 0 2.00000 4.41421 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.x 2
3.b odd 2 1 275.2.a.c 2
5.b even 2 1 495.2.a.b 2
5.c odd 4 2 2475.2.c.l 4
12.b even 2 1 4400.2.a.bn 2
15.d odd 2 1 55.2.a.b 2
15.e even 4 2 275.2.b.d 4
20.d odd 2 1 7920.2.a.ch 2
33.d even 2 1 3025.2.a.o 2
55.d odd 2 1 5445.2.a.y 2
60.h even 2 1 880.2.a.m 2
60.l odd 4 2 4400.2.b.q 4
105.g even 2 1 2695.2.a.f 2
120.i odd 2 1 3520.2.a.bn 2
120.m even 2 1 3520.2.a.bo 2
165.d even 2 1 605.2.a.d 2
165.o odd 10 4 605.2.g.f 8
165.r even 10 4 605.2.g.l 8
195.e odd 2 1 9295.2.a.g 2
660.g odd 2 1 9680.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 15.d odd 2 1
275.2.a.c 2 3.b odd 2 1
275.2.b.d 4 15.e even 4 2
495.2.a.b 2 5.b even 2 1
605.2.a.d 2 165.d even 2 1
605.2.g.f 8 165.o odd 10 4
605.2.g.l 8 165.r even 10 4
880.2.a.m 2 60.h even 2 1
2475.2.a.x 2 1.a even 1 1 trivial
2475.2.c.l 4 5.c odd 4 2
2695.2.a.f 2 105.g even 2 1
3025.2.a.o 2 33.d even 2 1
3520.2.a.bn 2 120.i odd 2 1
3520.2.a.bo 2 120.m even 2 1
4400.2.a.bn 2 12.b even 2 1
4400.2.b.q 4 60.l odd 4 2
5445.2.a.y 2 55.d odd 2 1
7920.2.a.ch 2 20.d odd 2 1
9295.2.a.g 2 195.e odd 2 1
9680.2.a.bn 2 660.g odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 2T_{2} - 1$$ T2^2 - 2*T2 - 1 $$T_{7} - 2$$ T7 - 2 $$T_{29}^{2} + 4T_{29} - 28$$ T29^2 + 4*T29 - 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 8T + 8$$
$17$ $$T^{2} - 8T + 8$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 8$$
$29$ $$T^{2} + 4T - 28$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 4T - 28$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T - 6)^{2}$$
$47$ $$T^{2} - 8$$
$53$ $$T^{2} - 12T + 4$$
$59$ $$T^{2} - 8T - 16$$
$61$ $$T^{2} - 4T - 124$$
$67$ $$T^{2} + 8T - 56$$
$71$ $$T^{2} - 128$$
$73$ $$T^{2} - 8T + 8$$
$79$ $$(T - 4)^{2}$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 4T - 124$$
$97$ $$T^{2} - 4T - 28$$
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