# Properties

 Label 1575.2.a.r 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: $$2$$ Twist minimal: no (minimal twist has level 105) 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + 3 q^{4} - q^{7} - \beta q^{8} +O(q^{10})$$ q - b * q^2 + 3 * q^4 - q^7 - b * q^8 $$q - \beta q^{2} + 3 q^{4} - q^{7} - \beta q^{8} + ( - 2 \beta - 2) q^{11} - 2 \beta q^{13} + \beta q^{14} - q^{16} - 2 q^{17} + ( - 2 \beta + 2) q^{19} + (2 \beta + 10) q^{22} + 4 q^{23} + 10 q^{26} - 3 q^{28} + 2 q^{29} + ( - 2 \beta + 6) q^{31} + 3 \beta q^{32} + 2 \beta q^{34} + (4 \beta - 2) q^{37} + ( - 2 \beta + 10) q^{38} + 2 q^{41} - 4 \beta q^{43} + ( - 6 \beta - 6) q^{44} - 4 \beta q^{46} + (4 \beta + 4) q^{47} + q^{49} - 6 \beta q^{52} + (2 \beta - 8) q^{53} + \beta q^{56} - 2 \beta q^{58} + 4 \beta q^{59} - 2 q^{61} + ( - 6 \beta + 10) q^{62} - 13 q^{64} + 4 q^{67} - 6 q^{68} + (2 \beta - 10) q^{71} + (2 \beta + 8) q^{73} + (2 \beta - 20) q^{74} + ( - 6 \beta + 6) q^{76} + (2 \beta + 2) q^{77} + (4 \beta + 4) q^{79} - 2 \beta q^{82} + ( - 4 \beta - 8) q^{83} + 20 q^{86} + (2 \beta + 10) q^{88} + 2 q^{89} + 2 \beta q^{91} + 12 q^{92} + ( - 4 \beta - 20) q^{94} + ( - 2 \beta - 4) q^{97} - \beta q^{98} +O(q^{100})$$ q - b * q^2 + 3 * q^4 - q^7 - b * q^8 + (-2*b - 2) * q^11 - 2*b * q^13 + b * q^14 - q^16 - 2 * q^17 + (-2*b + 2) * q^19 + (2*b + 10) * q^22 + 4 * q^23 + 10 * q^26 - 3 * q^28 + 2 * q^29 + (-2*b + 6) * q^31 + 3*b * q^32 + 2*b * q^34 + (4*b - 2) * q^37 + (-2*b + 10) * q^38 + 2 * q^41 - 4*b * q^43 + (-6*b - 6) * q^44 - 4*b * q^46 + (4*b + 4) * q^47 + q^49 - 6*b * q^52 + (2*b - 8) * q^53 + b * q^56 - 2*b * q^58 + 4*b * q^59 - 2 * q^61 + (-6*b + 10) * q^62 - 13 * q^64 + 4 * q^67 - 6 * q^68 + (2*b - 10) * q^71 + (2*b + 8) * q^73 + (2*b - 20) * q^74 + (-6*b + 6) * q^76 + (2*b + 2) * q^77 + (4*b + 4) * q^79 - 2*b * q^82 + (-4*b - 8) * q^83 + 20 * q^86 + (2*b + 10) * q^88 + 2 * q^89 + 2*b * q^91 + 12 * q^92 + (-4*b - 20) * q^94 + (-2*b - 4) * q^97 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{4} - 2 q^{7}+O(q^{10})$$ 2 * q + 6 * q^4 - 2 * q^7 $$2 q + 6 q^{4} - 2 q^{7} - 4 q^{11} - 2 q^{16} - 4 q^{17} + 4 q^{19} + 20 q^{22} + 8 q^{23} + 20 q^{26} - 6 q^{28} + 4 q^{29} + 12 q^{31} - 4 q^{37} + 20 q^{38} + 4 q^{41} - 12 q^{44} + 8 q^{47} + 2 q^{49} - 16 q^{53} - 4 q^{61} + 20 q^{62} - 26 q^{64} + 8 q^{67} - 12 q^{68} - 20 q^{71} + 16 q^{73} - 40 q^{74} + 12 q^{76} + 4 q^{77} + 8 q^{79} - 16 q^{83} + 40 q^{86} + 20 q^{88} + 4 q^{89} + 24 q^{92} - 40 q^{94} - 8 q^{97}+O(q^{100})$$ 2 * q + 6 * q^4 - 2 * q^7 - 4 * q^11 - 2 * q^16 - 4 * q^17 + 4 * q^19 + 20 * q^22 + 8 * q^23 + 20 * q^26 - 6 * q^28 + 4 * q^29 + 12 * q^31 - 4 * q^37 + 20 * q^38 + 4 * q^41 - 12 * q^44 + 8 * q^47 + 2 * q^49 - 16 * q^53 - 4 * q^61 + 20 * q^62 - 26 * q^64 + 8 * q^67 - 12 * q^68 - 20 * q^71 + 16 * q^73 - 40 * q^74 + 12 * q^76 + 4 * q^77 + 8 * q^79 - 16 * q^83 + 40 * q^86 + 20 * q^88 + 4 * q^89 + 24 * q^92 - 40 * q^94 - 8 * q^97

## 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
−2.23607 0 3.00000 0 0 −1.00000 −2.23607 0 0
1.2 2.23607 0 3.00000 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.r 2
3.b odd 2 1 525.2.a.g 2
5.b even 2 1 315.2.a.d 2
5.c odd 4 2 1575.2.d.d 4
12.b even 2 1 8400.2.a.cx 2
15.d odd 2 1 105.2.a.b 2
15.e even 4 2 525.2.d.c 4
20.d odd 2 1 5040.2.a.bw 2
21.c even 2 1 3675.2.a.y 2
35.c odd 2 1 2205.2.a.w 2
60.h even 2 1 1680.2.a.v 2
105.g even 2 1 735.2.a.k 2
105.o odd 6 2 735.2.i.k 4
105.p even 6 2 735.2.i.i 4
120.i odd 2 1 6720.2.a.cx 2
120.m even 2 1 6720.2.a.cs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.b 2 15.d odd 2 1
315.2.a.d 2 5.b even 2 1
525.2.a.g 2 3.b odd 2 1
525.2.d.c 4 15.e even 4 2
735.2.a.k 2 105.g even 2 1
735.2.i.i 4 105.p even 6 2
735.2.i.k 4 105.o odd 6 2
1575.2.a.r 2 1.a even 1 1 trivial
1575.2.d.d 4 5.c odd 4 2
1680.2.a.v 2 60.h even 2 1
2205.2.a.w 2 35.c odd 2 1
3675.2.a.y 2 21.c even 2 1
5040.2.a.bw 2 20.d odd 2 1
6720.2.a.cs 2 120.m even 2 1
6720.2.a.cx 2 120.i odd 2 1
8400.2.a.cx 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{2} - 5$$ T2^2 - 5 $$T_{11}^{2} + 4T_{11} - 16$$ T11^2 + 4*T11 - 16 $$T_{13}^{2} - 20$$ T13^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 4T - 16$$
$13$ $$T^{2} - 20$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} - 4T - 16$$
$23$ $$(T - 4)^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 12T + 16$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} - 80$$
$47$ $$T^{2} - 8T - 64$$
$53$ $$T^{2} + 16T + 44$$
$59$ $$T^{2} - 80$$
$61$ $$(T + 2)^{2}$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} + 20T + 80$$
$73$ $$T^{2} - 16T + 44$$
$79$ $$T^{2} - 8T - 64$$
$83$ $$T^{2} + 16T - 16$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} + 8T - 4$$