Properties

 Label 1575.2.a.m 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: $$\Q(\sqrt{2})$$ 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 315) 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.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} + q^{7} + (\beta - 3) q^{8}+O(q^{10})$$ q + (b - 1) * q^2 + (-2*b + 1) * q^4 + q^7 + (b - 3) * q^8 $$q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} + q^{7} + (\beta - 3) q^{8} + (2 \beta + 2) q^{11} + ( - 2 \beta + 2) q^{13} + (\beta - 1) q^{14} + 3 q^{16} + ( - 4 \beta - 2) q^{17} - 2 \beta q^{19} + 2 q^{22} + (4 \beta - 2) q^{23} + (4 \beta - 6) q^{26} + ( - 2 \beta + 1) q^{28} + 8 q^{29} + 6 \beta q^{31} + (\beta + 3) q^{32} + (2 \beta - 6) q^{34} + 6 q^{37} + (2 \beta - 4) q^{38} + ( - 4 \beta - 2) q^{41} + ( - 4 \beta + 4) q^{43} + ( - 2 \beta - 6) q^{44} + ( - 6 \beta + 10) q^{46} + 4 q^{47} + q^{49} + ( - 6 \beta + 10) q^{52} + (2 \beta - 8) q^{53} + (\beta - 3) q^{56} + (8 \beta - 8) q^{58} - 4 q^{59} + 6 q^{61} + ( - 6 \beta + 12) q^{62} + (2 \beta - 7) q^{64} + (8 \beta + 4) q^{67} + 14 q^{68} + (6 \beta + 2) q^{71} + (10 \beta - 2) q^{73} + (6 \beta - 6) q^{74} + ( - 2 \beta + 8) q^{76} + (2 \beta + 2) q^{77} + 4 \beta q^{79} + (2 \beta - 6) q^{82} + 8 q^{83} + (8 \beta - 12) q^{86} + ( - 4 \beta - 2) q^{88} + ( - 8 \beta + 6) q^{89} + ( - 2 \beta + 2) q^{91} + (8 \beta - 18) q^{92} + (4 \beta - 4) q^{94} + ( - 2 \beta + 6) q^{97} + (\beta - 1) q^{98} +O(q^{100})$$ q + (b - 1) * q^2 + (-2*b + 1) * q^4 + q^7 + (b - 3) * q^8 + (2*b + 2) * q^11 + (-2*b + 2) * q^13 + (b - 1) * q^14 + 3 * q^16 + (-4*b - 2) * q^17 - 2*b * q^19 + 2 * q^22 + (4*b - 2) * q^23 + (4*b - 6) * q^26 + (-2*b + 1) * q^28 + 8 * q^29 + 6*b * q^31 + (b + 3) * q^32 + (2*b - 6) * q^34 + 6 * q^37 + (2*b - 4) * q^38 + (-4*b - 2) * q^41 + (-4*b + 4) * q^43 + (-2*b - 6) * q^44 + (-6*b + 10) * q^46 + 4 * q^47 + q^49 + (-6*b + 10) * q^52 + (2*b - 8) * q^53 + (b - 3) * q^56 + (8*b - 8) * q^58 - 4 * q^59 + 6 * q^61 + (-6*b + 12) * q^62 + (2*b - 7) * q^64 + (8*b + 4) * q^67 + 14 * q^68 + (6*b + 2) * q^71 + (10*b - 2) * q^73 + (6*b - 6) * q^74 + (-2*b + 8) * q^76 + (2*b + 2) * q^77 + 4*b * q^79 + (2*b - 6) * q^82 + 8 * q^83 + (8*b - 12) * q^86 + (-4*b - 2) * q^88 + (-8*b + 6) * q^89 + (-2*b + 2) * q^91 + (8*b - 18) * q^92 + (4*b - 4) * q^94 + (-2*b + 6) * q^97 + (b - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 6 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 6 q^{8} + 4 q^{11} + 4 q^{13} - 2 q^{14} + 6 q^{16} - 4 q^{17} + 4 q^{22} - 4 q^{23} - 12 q^{26} + 2 q^{28} + 16 q^{29} + 6 q^{32} - 12 q^{34} + 12 q^{37} - 8 q^{38} - 4 q^{41} + 8 q^{43} - 12 q^{44} + 20 q^{46} + 8 q^{47} + 2 q^{49} + 20 q^{52} - 16 q^{53} - 6 q^{56} - 16 q^{58} - 8 q^{59} + 12 q^{61} + 24 q^{62} - 14 q^{64} + 8 q^{67} + 28 q^{68} + 4 q^{71} - 4 q^{73} - 12 q^{74} + 16 q^{76} + 4 q^{77} - 12 q^{82} + 16 q^{83} - 24 q^{86} - 4 q^{88} + 12 q^{89} + 4 q^{91} - 36 q^{92} - 8 q^{94} + 12 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 6 * q^8 + 4 * q^11 + 4 * q^13 - 2 * q^14 + 6 * q^16 - 4 * q^17 + 4 * q^22 - 4 * q^23 - 12 * q^26 + 2 * q^28 + 16 * q^29 + 6 * q^32 - 12 * q^34 + 12 * q^37 - 8 * q^38 - 4 * q^41 + 8 * q^43 - 12 * q^44 + 20 * q^46 + 8 * q^47 + 2 * q^49 + 20 * q^52 - 16 * q^53 - 6 * q^56 - 16 * q^58 - 8 * q^59 + 12 * q^61 + 24 * q^62 - 14 * q^64 + 8 * q^67 + 28 * q^68 + 4 * q^71 - 4 * q^73 - 12 * q^74 + 16 * q^76 + 4 * q^77 - 12 * q^82 + 16 * q^83 - 24 * q^86 - 4 * q^88 + 12 * q^89 + 4 * q^91 - 36 * q^92 - 8 * q^94 + 12 * q^97 - 2 * 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
−2.41421 0 3.82843 0 0 1.00000 −4.41421 0 0
1.2 0.414214 0 −1.82843 0 0 1.00000 −1.58579 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.m 2
3.b odd 2 1 1575.2.a.u 2
5.b even 2 1 315.2.a.f yes 2
5.c odd 4 2 1575.2.d.j 4
15.d odd 2 1 315.2.a.c 2
15.e even 4 2 1575.2.d.h 4
20.d odd 2 1 5040.2.a.bx 2
35.c odd 2 1 2205.2.a.y 2
60.h even 2 1 5040.2.a.bu 2
105.g even 2 1 2205.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.a.c 2 15.d odd 2 1
315.2.a.f yes 2 5.b even 2 1
1575.2.a.m 2 1.a even 1 1 trivial
1575.2.a.u 2 3.b odd 2 1
1575.2.d.h 4 15.e even 4 2
1575.2.d.j 4 5.c odd 4 2
2205.2.a.p 2 105.g even 2 1
2205.2.a.y 2 35.c odd 2 1
5040.2.a.bu 2 60.h even 2 1
5040.2.a.bx 2 20.d odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 4T - 4$$
$13$ $$T^{2} - 4T - 4$$
$17$ $$T^{2} + 4T - 28$$
$19$ $$T^{2} - 8$$
$23$ $$T^{2} + 4T - 28$$
$29$ $$(T - 8)^{2}$$
$31$ $$T^{2} - 72$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} + 4T - 28$$
$43$ $$T^{2} - 8T - 16$$
$47$ $$(T - 4)^{2}$$
$53$ $$T^{2} + 16T + 56$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T - 6)^{2}$$
$67$ $$T^{2} - 8T - 112$$
$71$ $$T^{2} - 4T - 68$$
$73$ $$T^{2} + 4T - 196$$
$79$ $$T^{2} - 32$$
$83$ $$(T - 8)^{2}$$
$89$ $$T^{2} - 12T - 92$$
$97$ $$T^{2} - 12T + 28$$