# Properties

 Label 1575.2.a.o 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: $$\Q(\sqrt{13})$$ 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 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta + 1) q^{4} + q^{7} - 3 q^{8} +O(q^{10})$$ q - b * q^2 + (b + 1) * q^4 + q^7 - 3 * q^8 $$q - \beta q^{2} + (\beta + 1) q^{4} + q^{7} - 3 q^{8} + 3 q^{11} + ( - 2 \beta + 2) q^{13} - \beta q^{14} + (\beta - 2) q^{16} - 2 \beta q^{17} + (2 \beta + 2) q^{19} - 3 \beta q^{22} + (4 \beta - 3) q^{23} + 6 q^{26} + (\beta + 1) q^{28} + (2 \beta + 3) q^{29} + ( - 4 \beta + 2) q^{31} + (\beta + 3) q^{32} + (2 \beta + 6) q^{34} + ( - 4 \beta + 5) q^{37} + ( - 4 \beta - 6) q^{38} + (2 \beta + 5) q^{43} + (3 \beta + 3) q^{44} + ( - \beta - 12) q^{46} + ( - 2 \beta - 6) q^{47} + q^{49} + ( - 2 \beta - 4) q^{52} + ( - 4 \beta + 6) q^{53} - 3 q^{56} + ( - 5 \beta - 6) q^{58} + (2 \beta + 6) q^{59} + ( - 4 \beta + 8) q^{61} + (2 \beta + 12) q^{62} + ( - 6 \beta + 1) q^{64} + (2 \beta + 11) q^{67} + ( - 4 \beta - 6) q^{68} + 3 q^{71} + (2 \beta - 4) q^{73} + ( - \beta + 12) q^{74} + (6 \beta + 8) q^{76} + 3 q^{77} + ( - 6 \beta - 1) q^{79} + (4 \beta - 6) q^{83} + ( - 7 \beta - 6) q^{86} - 9 q^{88} + ( - 6 \beta + 6) q^{89} + ( - 2 \beta + 2) q^{91} + (5 \beta + 9) q^{92} + (8 \beta + 6) q^{94} + (4 \beta - 10) q^{97} - \beta q^{98} +O(q^{100})$$ q - b * q^2 + (b + 1) * q^4 + q^7 - 3 * q^8 + 3 * q^11 + (-2*b + 2) * q^13 - b * q^14 + (b - 2) * q^16 - 2*b * q^17 + (2*b + 2) * q^19 - 3*b * q^22 + (4*b - 3) * q^23 + 6 * q^26 + (b + 1) * q^28 + (2*b + 3) * q^29 + (-4*b + 2) * q^31 + (b + 3) * q^32 + (2*b + 6) * q^34 + (-4*b + 5) * q^37 + (-4*b - 6) * q^38 + (2*b + 5) * q^43 + (3*b + 3) * q^44 + (-b - 12) * q^46 + (-2*b - 6) * q^47 + q^49 + (-2*b - 4) * q^52 + (-4*b + 6) * q^53 - 3 * q^56 + (-5*b - 6) * q^58 + (2*b + 6) * q^59 + (-4*b + 8) * q^61 + (2*b + 12) * q^62 + (-6*b + 1) * q^64 + (2*b + 11) * q^67 + (-4*b - 6) * q^68 + 3 * q^71 + (2*b - 4) * q^73 + (-b + 12) * q^74 + (6*b + 8) * q^76 + 3 * q^77 + (-6*b - 1) * q^79 + (4*b - 6) * q^83 + (-7*b - 6) * q^86 - 9 * q^88 + (-6*b + 6) * q^89 + (-2*b + 2) * q^91 + (5*b + 9) * q^92 + (8*b + 6) * q^94 + (4*b - 10) * q^97 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 3 q^{4} + 2 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - q^2 + 3 * q^4 + 2 * q^7 - 6 * q^8 $$2 q - q^{2} + 3 q^{4} + 2 q^{7} - 6 q^{8} + 6 q^{11} + 2 q^{13} - q^{14} - 3 q^{16} - 2 q^{17} + 6 q^{19} - 3 q^{22} - 2 q^{23} + 12 q^{26} + 3 q^{28} + 8 q^{29} + 7 q^{32} + 14 q^{34} + 6 q^{37} - 16 q^{38} + 12 q^{43} + 9 q^{44} - 25 q^{46} - 14 q^{47} + 2 q^{49} - 10 q^{52} + 8 q^{53} - 6 q^{56} - 17 q^{58} + 14 q^{59} + 12 q^{61} + 26 q^{62} - 4 q^{64} + 24 q^{67} - 16 q^{68} + 6 q^{71} - 6 q^{73} + 23 q^{74} + 22 q^{76} + 6 q^{77} - 8 q^{79} - 8 q^{83} - 19 q^{86} - 18 q^{88} + 6 q^{89} + 2 q^{91} + 23 q^{92} + 20 q^{94} - 16 q^{97} - q^{98}+O(q^{100})$$ 2 * q - q^2 + 3 * q^4 + 2 * q^7 - 6 * q^8 + 6 * q^11 + 2 * q^13 - q^14 - 3 * q^16 - 2 * q^17 + 6 * q^19 - 3 * q^22 - 2 * q^23 + 12 * q^26 + 3 * q^28 + 8 * q^29 + 7 * q^32 + 14 * q^34 + 6 * q^37 - 16 * q^38 + 12 * q^43 + 9 * q^44 - 25 * q^46 - 14 * q^47 + 2 * q^49 - 10 * q^52 + 8 * q^53 - 6 * q^56 - 17 * q^58 + 14 * q^59 + 12 * q^61 + 26 * q^62 - 4 * q^64 + 24 * q^67 - 16 * q^68 + 6 * q^71 - 6 * q^73 + 23 * q^74 + 22 * q^76 + 6 * q^77 - 8 * q^79 - 8 * q^83 - 19 * q^86 - 18 * q^88 + 6 * q^89 + 2 * q^91 + 23 * q^92 + 20 * q^94 - 16 * 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
 2.30278 −1.30278
−2.30278 0 3.30278 0 0 1.00000 −3.00000 0 0
1.2 1.30278 0 −0.302776 0 0 1.00000 −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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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.o 2
3.b odd 2 1 525.2.a.h yes 2
5.b even 2 1 1575.2.a.t 2
5.c odd 4 2 1575.2.d.g 4
12.b even 2 1 8400.2.a.cw 2
15.d odd 2 1 525.2.a.f 2
15.e even 4 2 525.2.d.d 4
21.c even 2 1 3675.2.a.bb 2
60.h even 2 1 8400.2.a.df 2
105.g even 2 1 3675.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.a.f 2 15.d odd 2 1
525.2.a.h yes 2 3.b odd 2 1
525.2.d.d 4 15.e even 4 2
1575.2.a.o 2 1.a even 1 1 trivial
1575.2.a.t 2 5.b even 2 1
1575.2.d.g 4 5.c odd 4 2
3675.2.a.w 2 105.g even 2 1
3675.2.a.bb 2 21.c even 2 1
8400.2.a.cw 2 12.b even 2 1
8400.2.a.df 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{2} + T_{2} - 3$$ T2^2 + T2 - 3 $$T_{11} - 3$$ T11 - 3 $$T_{13}^{2} - 2T_{13} - 12$$ T13^2 - 2*T13 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 3$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} - 2T - 12$$
$17$ $$T^{2} + 2T - 12$$
$19$ $$T^{2} - 6T - 4$$
$23$ $$T^{2} + 2T - 51$$
$29$ $$T^{2} - 8T + 3$$
$31$ $$T^{2} - 52$$
$37$ $$T^{2} - 6T - 43$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 12T + 23$$
$47$ $$T^{2} + 14T + 36$$
$53$ $$T^{2} - 8T - 36$$
$59$ $$T^{2} - 14T + 36$$
$61$ $$T^{2} - 12T - 16$$
$67$ $$T^{2} - 24T + 131$$
$71$ $$(T - 3)^{2}$$
$73$ $$T^{2} + 6T - 4$$
$79$ $$T^{2} + 8T - 101$$
$83$ $$T^{2} + 8T - 36$$
$89$ $$T^{2} - 6T - 108$$
$97$ $$T^{2} + 16T + 12$$