# Properties

 Label 1575.2.a.n Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ Analytic rank $1$ 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: $$1$$ 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 175) 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta - 1) q^{4} + q^{7} + (2 \beta - 1) q^{8} +O(q^{10})$$ q - b * q^2 + (b - 1) * q^4 + q^7 + (2*b - 1) * q^8 $$q - \beta q^{2} + (\beta - 1) q^{4} + q^{7} + (2 \beta - 1) q^{8} + ( - 2 \beta - 1) q^{11} - 2 \beta q^{13} - \beta q^{14} - 3 \beta q^{16} + 4 \beta q^{17} + (4 \beta - 2) q^{19} + (3 \beta + 2) q^{22} + (2 \beta - 5) q^{23} + (2 \beta + 2) q^{26} + (\beta - 1) q^{28} - 5 q^{29} - 6 \beta q^{31} + ( - \beta + 5) q^{32} + ( - 4 \beta - 4) q^{34} + 3 q^{37} + ( - 2 \beta - 4) q^{38} + ( - 2 \beta - 6) q^{41} + (2 \beta + 3) q^{43} + ( - \beta - 1) q^{44} + (3 \beta - 2) q^{46} + 2 q^{47} + q^{49} - 2 q^{52} + (4 \beta - 6) q^{53} + (2 \beta - 1) q^{56} + 5 \beta q^{58} + (6 \beta - 8) q^{59} + (6 \beta - 6) q^{61} + (6 \beta + 6) q^{62} + (2 \beta + 1) q^{64} + (2 \beta - 3) q^{67} + 4 q^{68} + (6 \beta - 5) q^{71} + ( - 2 \beta - 10) q^{73} - 3 \beta q^{74} + ( - 2 \beta + 6) q^{76} + ( - 2 \beta - 1) q^{77} + (10 \beta - 5) q^{79} + (8 \beta + 2) q^{82} + ( - 6 \beta + 4) q^{83} + ( - 5 \beta - 2) q^{86} + ( - 4 \beta - 3) q^{88} + (2 \beta - 16) q^{89} - 2 \beta q^{91} + ( - 5 \beta + 7) q^{92} - 2 \beta q^{94} + ( - 2 \beta + 4) q^{97} - \beta q^{98} +O(q^{100})$$ q - b * q^2 + (b - 1) * q^4 + q^7 + (2*b - 1) * q^8 + (-2*b - 1) * q^11 - 2*b * q^13 - b * q^14 - 3*b * q^16 + 4*b * q^17 + (4*b - 2) * q^19 + (3*b + 2) * q^22 + (2*b - 5) * q^23 + (2*b + 2) * q^26 + (b - 1) * q^28 - 5 * q^29 - 6*b * q^31 + (-b + 5) * q^32 + (-4*b - 4) * q^34 + 3 * q^37 + (-2*b - 4) * q^38 + (-2*b - 6) * q^41 + (2*b + 3) * q^43 + (-b - 1) * q^44 + (3*b - 2) * q^46 + 2 * q^47 + q^49 - 2 * q^52 + (4*b - 6) * q^53 + (2*b - 1) * q^56 + 5*b * q^58 + (6*b - 8) * q^59 + (6*b - 6) * q^61 + (6*b + 6) * q^62 + (2*b + 1) * q^64 + (2*b - 3) * q^67 + 4 * q^68 + (6*b - 5) * q^71 + (-2*b - 10) * q^73 - 3*b * q^74 + (-2*b + 6) * q^76 + (-2*b - 1) * q^77 + (10*b - 5) * q^79 + (8*b + 2) * q^82 + (-6*b + 4) * q^83 + (-5*b - 2) * q^86 + (-4*b - 3) * q^88 + (2*b - 16) * q^89 - 2*b * q^91 + (-5*b + 7) * q^92 - 2*b * q^94 + (-2*b + 4) * q^97 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 2 q^{7}+O(q^{10})$$ 2 * q - q^2 - q^4 + 2 * q^7 $$2 q - q^{2} - q^{4} + 2 q^{7} - 4 q^{11} - 2 q^{13} - q^{14} - 3 q^{16} + 4 q^{17} + 7 q^{22} - 8 q^{23} + 6 q^{26} - q^{28} - 10 q^{29} - 6 q^{31} + 9 q^{32} - 12 q^{34} + 6 q^{37} - 10 q^{38} - 14 q^{41} + 8 q^{43} - 3 q^{44} - q^{46} + 4 q^{47} + 2 q^{49} - 4 q^{52} - 8 q^{53} + 5 q^{58} - 10 q^{59} - 6 q^{61} + 18 q^{62} + 4 q^{64} - 4 q^{67} + 8 q^{68} - 4 q^{71} - 22 q^{73} - 3 q^{74} + 10 q^{76} - 4 q^{77} + 12 q^{82} + 2 q^{83} - 9 q^{86} - 10 q^{88} - 30 q^{89} - 2 q^{91} + 9 q^{92} - 2 q^{94} + 6 q^{97} - q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 + 2 * q^7 - 4 * q^11 - 2 * q^13 - q^14 - 3 * q^16 + 4 * q^17 + 7 * q^22 - 8 * q^23 + 6 * q^26 - q^28 - 10 * q^29 - 6 * q^31 + 9 * q^32 - 12 * q^34 + 6 * q^37 - 10 * q^38 - 14 * q^41 + 8 * q^43 - 3 * q^44 - q^46 + 4 * q^47 + 2 * q^49 - 4 * q^52 - 8 * q^53 + 5 * q^58 - 10 * q^59 - 6 * q^61 + 18 * q^62 + 4 * q^64 - 4 * q^67 + 8 * q^68 - 4 * q^71 - 22 * q^73 - 3 * q^74 + 10 * q^76 - 4 * q^77 + 12 * q^82 + 2 * q^83 - 9 * q^86 - 10 * q^88 - 30 * q^89 - 2 * q^91 + 9 * q^92 - 2 * q^94 + 6 * q^97 - 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
−1.61803 0 0.618034 0 0 1.00000 2.23607 0 0
1.2 0.618034 0 −1.61803 0 0 1.00000 −2.23607 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.n 2
3.b odd 2 1 175.2.a.e yes 2
5.b even 2 1 1575.2.a.s 2
5.c odd 4 2 1575.2.d.k 4
12.b even 2 1 2800.2.a.bp 2
15.d odd 2 1 175.2.a.d 2
15.e even 4 2 175.2.b.c 4
21.c even 2 1 1225.2.a.u 2
60.h even 2 1 2800.2.a.bh 2
60.l odd 4 2 2800.2.g.s 4
105.g even 2 1 1225.2.a.n 2
105.k odd 4 2 1225.2.b.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.a.d 2 15.d odd 2 1
175.2.a.e yes 2 3.b odd 2 1
175.2.b.c 4 15.e even 4 2
1225.2.a.n 2 105.g even 2 1
1225.2.a.u 2 21.c even 2 1
1225.2.b.k 4 105.k odd 4 2
1575.2.a.n 2 1.a even 1 1 trivial
1575.2.a.s 2 5.b even 2 1
1575.2.d.k 4 5.c odd 4 2
2800.2.a.bh 2 60.h even 2 1
2800.2.a.bp 2 12.b even 2 1
2800.2.g.s 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1575))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} + 4T - 1$$
$13$ $$T^{2} + 2T - 4$$
$17$ $$T^{2} - 4T - 16$$
$19$ $$T^{2} - 20$$
$23$ $$T^{2} + 8T + 11$$
$29$ $$(T + 5)^{2}$$
$31$ $$T^{2} + 6T - 36$$
$37$ $$(T - 3)^{2}$$
$41$ $$T^{2} + 14T + 44$$
$43$ $$T^{2} - 8T + 11$$
$47$ $$(T - 2)^{2}$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} + 10T - 20$$
$61$ $$T^{2} + 6T - 36$$
$67$ $$T^{2} + 4T - 1$$
$71$ $$T^{2} + 4T - 41$$
$73$ $$T^{2} + 22T + 116$$
$79$ $$T^{2} - 125$$
$83$ $$T^{2} - 2T - 44$$
$89$ $$T^{2} + 30T + 220$$
$97$ $$T^{2} - 6T + 4$$