# Properties

 Label 1575.2.a.l Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ 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] = [1575,2,Mod(1,1575)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.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: $$12.5764383184$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 525) 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + 3 \beta q^{4} - q^{7} + ( - 4 \beta - 1) q^{8}+O(q^{10})$$ q + (-b - 1) * q^2 + 3*b * q^4 - q^7 + (-4*b - 1) * q^8 $$q + ( - \beta - 1) q^{2} + 3 \beta q^{4} - q^{7} + ( - 4 \beta - 1) q^{8} + (4 \beta - 1) q^{11} + ( - 2 \beta + 4) q^{13} + (\beta + 1) q^{14} + (3 \beta + 5) q^{16} + (6 \beta - 2) q^{17} - 2 \beta q^{19} + ( - 7 \beta - 3) q^{22} - 5 q^{23} - 2 q^{26} - 3 \beta q^{28} + ( - 6 \beta + 5) q^{29} + (4 \beta - 2) q^{31} + ( - 3 \beta - 6) q^{32} + ( - 10 \beta - 4) q^{34} + (4 \beta - 1) q^{37} + (4 \beta + 2) q^{38} + 8 q^{41} + (2 \beta + 5) q^{43} + (9 \beta + 12) q^{44} + (5 \beta + 5) q^{46} + ( - 2 \beta - 4) q^{47} + q^{49} + (6 \beta - 6) q^{52} + ( - 4 \beta + 6) q^{53} + (4 \beta + 1) q^{56} + (7 \beta + 1) q^{58} + ( - 2 \beta + 4) q^{59} + ( - 12 \beta + 4) q^{61} + ( - 6 \beta - 2) q^{62} + (6 \beta - 1) q^{64} + ( - 6 \beta + 7) q^{67} + (12 \beta + 18) q^{68} + ( - 4 \beta + 7) q^{71} + (2 \beta - 2) q^{73} + ( - 7 \beta - 3) q^{74} + ( - 6 \beta - 6) q^{76} + ( - 4 \beta + 1) q^{77} + ( - 2 \beta + 5) q^{79} + ( - 8 \beta - 8) q^{82} + ( - 4 \beta - 6) q^{83} + ( - 9 \beta - 7) q^{86} + ( - 16 \beta - 15) q^{88} + (6 \beta - 4) q^{89} + (2 \beta - 4) q^{91} - 15 \beta q^{92} + (8 \beta + 6) q^{94} + (4 \beta + 6) q^{97} + ( - \beta - 1) q^{98} +O(q^{100})$$ q + (-b - 1) * q^2 + 3*b * q^4 - q^7 + (-4*b - 1) * q^8 + (4*b - 1) * q^11 + (-2*b + 4) * q^13 + (b + 1) * q^14 + (3*b + 5) * q^16 + (6*b - 2) * q^17 - 2*b * q^19 + (-7*b - 3) * q^22 - 5 * q^23 - 2 * q^26 - 3*b * q^28 + (-6*b + 5) * q^29 + (4*b - 2) * q^31 + (-3*b - 6) * q^32 + (-10*b - 4) * q^34 + (4*b - 1) * q^37 + (4*b + 2) * q^38 + 8 * q^41 + (2*b + 5) * q^43 + (9*b + 12) * q^44 + (5*b + 5) * q^46 + (-2*b - 4) * q^47 + q^49 + (6*b - 6) * q^52 + (-4*b + 6) * q^53 + (4*b + 1) * q^56 + (7*b + 1) * q^58 + (-2*b + 4) * q^59 + (-12*b + 4) * q^61 + (-6*b - 2) * q^62 + (6*b - 1) * q^64 + (-6*b + 7) * q^67 + (12*b + 18) * q^68 + (-4*b + 7) * q^71 + (2*b - 2) * q^73 + (-7*b - 3) * q^74 + (-6*b - 6) * q^76 + (-4*b + 1) * q^77 + (-2*b + 5) * q^79 + (-8*b - 8) * q^82 + (-4*b - 6) * q^83 + (-9*b - 7) * q^86 + (-16*b - 15) * q^88 + (6*b - 4) * q^89 + (2*b - 4) * q^91 - 15*b * q^92 + (8*b + 6) * q^94 + (4*b + 6) * q^97 + (-b - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - 3 * q^2 + 3 * q^4 - 2 * q^7 - 6 * q^8 $$2 q - 3 q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8} + 2 q^{11} + 6 q^{13} + 3 q^{14} + 13 q^{16} + 2 q^{17} - 2 q^{19} - 13 q^{22} - 10 q^{23} - 4 q^{26} - 3 q^{28} + 4 q^{29} - 15 q^{32} - 18 q^{34} + 2 q^{37} + 8 q^{38} + 16 q^{41} + 12 q^{43} + 33 q^{44} + 15 q^{46} - 10 q^{47} + 2 q^{49} - 6 q^{52} + 8 q^{53} + 6 q^{56} + 9 q^{58} + 6 q^{59} - 4 q^{61} - 10 q^{62} + 4 q^{64} + 8 q^{67} + 48 q^{68} + 10 q^{71} - 2 q^{73} - 13 q^{74} - 18 q^{76} - 2 q^{77} + 8 q^{79} - 24 q^{82} - 16 q^{83} - 23 q^{86} - 46 q^{88} - 2 q^{89} - 6 q^{91} - 15 q^{92} + 20 q^{94} + 16 q^{97} - 3 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 + 3 * q^4 - 2 * q^7 - 6 * q^8 + 2 * q^11 + 6 * q^13 + 3 * q^14 + 13 * q^16 + 2 * q^17 - 2 * q^19 - 13 * q^22 - 10 * q^23 - 4 * q^26 - 3 * q^28 + 4 * q^29 - 15 * q^32 - 18 * q^34 + 2 * q^37 + 8 * q^38 + 16 * q^41 + 12 * q^43 + 33 * q^44 + 15 * q^46 - 10 * q^47 + 2 * q^49 - 6 * q^52 + 8 * q^53 + 6 * q^56 + 9 * q^58 + 6 * q^59 - 4 * q^61 - 10 * q^62 + 4 * q^64 + 8 * q^67 + 48 * q^68 + 10 * q^71 - 2 * q^73 - 13 * q^74 - 18 * q^76 - 2 * q^77 + 8 * q^79 - 24 * q^82 - 16 * q^83 - 23 * q^86 - 46 * q^88 - 2 * q^89 - 6 * q^91 - 15 * q^92 + 20 * q^94 + 16 * q^97 - 3 * 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
 1.61803 −0.618034
−2.61803 0 4.85410 0 0 −1.00000 −7.47214 0 0
1.2 −0.381966 0 −1.85410 0 0 −1.00000 1.47214 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
$$3$$ $$-1$$
$$5$$ $$+1$$
$$7$$ $$+1$$

## 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 1575.2.a.l 2
3.b odd 2 1 525.2.a.i yes 2
5.b even 2 1 1575.2.a.v 2
5.c odd 4 2 1575.2.d.f 4
12.b even 2 1 8400.2.a.cy 2
15.d odd 2 1 525.2.a.e 2
15.e even 4 2 525.2.d.e 4
21.c even 2 1 3675.2.a.bh 2
60.h even 2 1 8400.2.a.da 2
105.g even 2 1 3675.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.a.e 2 15.d odd 2 1
525.2.a.i yes 2 3.b odd 2 1
525.2.d.e 4 15.e even 4 2
1575.2.a.l 2 1.a even 1 1 trivial
1575.2.a.v 2 5.b even 2 1
1575.2.d.f 4 5.c odd 4 2
3675.2.a.r 2 105.g even 2 1
3675.2.a.bh 2 21.c even 2 1
8400.2.a.cy 2 12.b even 2 1
8400.2.a.da 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{2} + 3T_{2} + 1$$ T2^2 + 3*T2 + 1 $$T_{11}^{2} - 2T_{11} - 19$$ T11^2 - 2*T11 - 19 $$T_{13}^{2} - 6T_{13} + 4$$ T13^2 - 6*T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 2T - 19$$
$13$ $$T^{2} - 6T + 4$$
$17$ $$T^{2} - 2T - 44$$
$19$ $$T^{2} + 2T - 4$$
$23$ $$(T + 5)^{2}$$
$29$ $$T^{2} - 4T - 41$$
$31$ $$T^{2} - 20$$
$37$ $$T^{2} - 2T - 19$$
$41$ $$(T - 8)^{2}$$
$43$ $$T^{2} - 12T + 31$$
$47$ $$T^{2} + 10T + 20$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} - 6T + 4$$
$61$ $$T^{2} + 4T - 176$$
$67$ $$T^{2} - 8T - 29$$
$71$ $$T^{2} - 10T + 5$$
$73$ $$T^{2} + 2T - 4$$
$79$ $$T^{2} - 8T + 11$$
$83$ $$T^{2} + 16T + 44$$
$89$ $$T^{2} + 2T - 44$$
$97$ $$T^{2} - 16T + 44$$