# Properties

 Label 675.2.a.m Level $675$ Weight $2$ Character orbit 675.a Self dual yes Analytic conductor $5.390$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(1,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 675.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: $$5.38990213644$$ 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 135) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 3 q^{7} - 2 \beta q^{8} +O(q^{10})$$ q + b * q^2 + 3 * q^7 - 2*b * q^8 $$q + \beta q^{2} + 3 q^{7} - 2 \beta q^{8} + 3 \beta q^{11} + 3 q^{13} + 3 \beta q^{14} - 4 q^{16} - 2 \beta q^{17} + q^{19} + 6 q^{22} + 5 \beta q^{23} + 3 \beta q^{26} - 3 \beta q^{29} + 2 q^{31} - 4 q^{34} + 9 q^{37} + \beta q^{38} - 3 \beta q^{41} + 6 q^{43} + 10 q^{46} - 2 \beta q^{47} + 2 q^{49} - 7 \beta q^{53} - 6 \beta q^{56} - 6 q^{58} - 6 \beta q^{59} - 13 q^{61} + 2 \beta q^{62} + 8 q^{64} - 3 q^{67} - 9 \beta q^{71} + 9 q^{73} + 9 \beta q^{74} + 9 \beta q^{77} - 5 q^{79} - 6 q^{82} - \beta q^{83} + 6 \beta q^{86} - 12 q^{88} + 9 q^{91} - 4 q^{94} + 3 q^{97} + 2 \beta q^{98} +O(q^{100})$$ q + b * q^2 + 3 * q^7 - 2*b * q^8 + 3*b * q^11 + 3 * q^13 + 3*b * q^14 - 4 * q^16 - 2*b * q^17 + q^19 + 6 * q^22 + 5*b * q^23 + 3*b * q^26 - 3*b * q^29 + 2 * q^31 - 4 * q^34 + 9 * q^37 + b * q^38 - 3*b * q^41 + 6 * q^43 + 10 * q^46 - 2*b * q^47 + 2 * q^49 - 7*b * q^53 - 6*b * q^56 - 6 * q^58 - 6*b * q^59 - 13 * q^61 + 2*b * q^62 + 8 * q^64 - 3 * q^67 - 9*b * q^71 + 9 * q^73 + 9*b * q^74 + 9*b * q^77 - 5 * q^79 - 6 * q^82 - b * q^83 + 6*b * q^86 - 12 * q^88 + 9 * q^91 - 4 * q^94 + 3 * q^97 + 2*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{7}+O(q^{10})$$ 2 * q + 6 * q^7 $$2 q + 6 q^{7} + 6 q^{13} - 8 q^{16} + 2 q^{19} + 12 q^{22} + 4 q^{31} - 8 q^{34} + 18 q^{37} + 12 q^{43} + 20 q^{46} + 4 q^{49} - 12 q^{58} - 26 q^{61} + 16 q^{64} - 6 q^{67} + 18 q^{73} - 10 q^{79} - 12 q^{82} - 24 q^{88} + 18 q^{91} - 8 q^{94} + 6 q^{97}+O(q^{100})$$ 2 * q + 6 * q^7 + 6 * q^13 - 8 * q^16 + 2 * q^19 + 12 * q^22 + 4 * q^31 - 8 * q^34 + 18 * q^37 + 12 * q^43 + 20 * q^46 + 4 * q^49 - 12 * q^58 - 26 * q^61 + 16 * q^64 - 6 * q^67 + 18 * q^73 - 10 * q^79 - 12 * q^82 - 24 * q^88 + 18 * q^91 - 8 * q^94 + 6 * 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
 −1.41421 1.41421
−1.41421 0 0 0 0 3.00000 2.82843 0 0
1.2 1.41421 0 0 0 0 3.00000 −2.82843 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
$$3$$ $$+1$$
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.a.m 2
3.b odd 2 1 inner 675.2.a.m 2
5.b even 2 1 675.2.a.l 2
5.c odd 4 2 135.2.b.b 4
15.d odd 2 1 675.2.a.l 2
15.e even 4 2 135.2.b.b 4
20.e even 4 2 2160.2.f.k 4
45.k odd 12 4 405.2.j.f 8
45.l even 12 4 405.2.j.f 8
60.l odd 4 2 2160.2.f.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.b.b 4 5.c odd 4 2
135.2.b.b 4 15.e even 4 2
405.2.j.f 8 45.k odd 12 4
405.2.j.f 8 45.l even 12 4
675.2.a.l 2 5.b even 2 1
675.2.a.l 2 15.d odd 2 1
675.2.a.m 2 1.a even 1 1 trivial
675.2.a.m 2 3.b odd 2 1 inner
2160.2.f.k 4 20.e even 4 2
2160.2.f.k 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(675))$$:

 $$T_{2}^{2} - 2$$ T2^2 - 2 $$T_{7} - 3$$ T7 - 3 $$T_{11}^{2} - 18$$ T11^2 - 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 3)^{2}$$
$11$ $$T^{2} - 18$$
$13$ $$(T - 3)^{2}$$
$17$ $$T^{2} - 8$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 50$$
$29$ $$T^{2} - 18$$
$31$ $$(T - 2)^{2}$$
$37$ $$(T - 9)^{2}$$
$41$ $$T^{2} - 18$$
$43$ $$(T - 6)^{2}$$
$47$ $$T^{2} - 8$$
$53$ $$T^{2} - 98$$
$59$ $$T^{2} - 72$$
$61$ $$(T + 13)^{2}$$
$67$ $$(T + 3)^{2}$$
$71$ $$T^{2} - 162$$
$73$ $$(T - 9)^{2}$$
$79$ $$(T + 5)^{2}$$
$83$ $$T^{2} - 2$$
$89$ $$T^{2}$$
$97$ $$(T - 3)^{2}$$