# Properties

 Label 3675.2.a.a Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, 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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3675 = 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3675.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: $$29.3450227428$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{6} + q^{9}+O(q^{10})$$ q - 2 * q^2 - q^3 + 2 * q^4 + 2 * q^6 + q^9 $$q - 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{6} + q^{9} - 2 q^{11} - 2 q^{12} - q^{13} - 4 q^{16} - 2 q^{18} + q^{19} + 4 q^{22} + 2 q^{26} - q^{27} + 4 q^{29} + 9 q^{31} + 8 q^{32} + 2 q^{33} + 2 q^{36} - 3 q^{37} - 2 q^{38} + q^{39} - 10 q^{41} - 5 q^{43} - 4 q^{44} + 6 q^{47} + 4 q^{48} - 2 q^{52} - 12 q^{53} + 2 q^{54} - q^{57} - 8 q^{58} - 12 q^{59} + 10 q^{61} - 18 q^{62} - 8 q^{64} - 4 q^{66} + 5 q^{67} - 6 q^{71} + 3 q^{73} + 6 q^{74} + 2 q^{76} - 2 q^{78} - q^{79} + q^{81} + 20 q^{82} - 6 q^{83} + 10 q^{86} - 4 q^{87} + 16 q^{89} - 9 q^{93} - 12 q^{94} - 8 q^{96} + 6 q^{97} - 2 q^{99}+O(q^{100})$$ q - 2 * q^2 - q^3 + 2 * q^4 + 2 * q^6 + q^9 - 2 * q^11 - 2 * q^12 - q^13 - 4 * q^16 - 2 * q^18 + q^19 + 4 * q^22 + 2 * q^26 - q^27 + 4 * q^29 + 9 * q^31 + 8 * q^32 + 2 * q^33 + 2 * q^36 - 3 * q^37 - 2 * q^38 + q^39 - 10 * q^41 - 5 * q^43 - 4 * q^44 + 6 * q^47 + 4 * q^48 - 2 * q^52 - 12 * q^53 + 2 * q^54 - q^57 - 8 * q^58 - 12 * q^59 + 10 * q^61 - 18 * q^62 - 8 * q^64 - 4 * q^66 + 5 * q^67 - 6 * q^71 + 3 * q^73 + 6 * q^74 + 2 * q^76 - 2 * q^78 - q^79 + q^81 + 20 * q^82 - 6 * q^83 + 10 * q^86 - 4 * q^87 + 16 * q^89 - 9 * q^93 - 12 * q^94 - 8 * q^96 + 6 * q^97 - 2 * q^99

## 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
 0
−2.00000 −1.00000 2.00000 0 2.00000 0 0 1.00000 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 3675.2.a.a 1
5.b even 2 1 147.2.a.c 1
7.b odd 2 1 3675.2.a.c 1
7.c even 3 2 525.2.i.e 2
15.d odd 2 1 441.2.a.b 1
20.d odd 2 1 2352.2.a.d 1
35.c odd 2 1 147.2.a.b 1
35.i odd 6 2 147.2.e.a 2
35.j even 6 2 21.2.e.a 2
35.l odd 12 4 525.2.r.e 4
40.e odd 2 1 9408.2.a.cv 1
40.f even 2 1 9408.2.a.bg 1
60.h even 2 1 7056.2.a.bp 1
105.g even 2 1 441.2.a.a 1
105.o odd 6 2 63.2.e.b 2
105.p even 6 2 441.2.e.e 2
140.c even 2 1 2352.2.a.w 1
140.p odd 6 2 336.2.q.f 2
140.s even 6 2 2352.2.q.c 2
280.c odd 2 1 9408.2.a.bz 1
280.n even 2 1 9408.2.a.k 1
280.bf even 6 2 1344.2.q.m 2
280.bi odd 6 2 1344.2.q.c 2
315.r even 6 2 567.2.h.f 2
315.v odd 6 2 567.2.g.f 2
315.bo even 6 2 567.2.g.a 2
315.br odd 6 2 567.2.h.a 2
420.o odd 2 1 7056.2.a.m 1
420.ba even 6 2 1008.2.s.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 35.j even 6 2
63.2.e.b 2 105.o odd 6 2
147.2.a.b 1 35.c odd 2 1
147.2.a.c 1 5.b even 2 1
147.2.e.a 2 35.i odd 6 2
336.2.q.f 2 140.p odd 6 2
441.2.a.a 1 105.g even 2 1
441.2.a.b 1 15.d odd 2 1
441.2.e.e 2 105.p even 6 2
525.2.i.e 2 7.c even 3 2
525.2.r.e 4 35.l odd 12 4
567.2.g.a 2 315.bo even 6 2
567.2.g.f 2 315.v odd 6 2
567.2.h.a 2 315.br odd 6 2
567.2.h.f 2 315.r even 6 2
1008.2.s.d 2 420.ba even 6 2
1344.2.q.c 2 280.bi odd 6 2
1344.2.q.m 2 280.bf even 6 2
2352.2.a.d 1 20.d odd 2 1
2352.2.a.w 1 140.c even 2 1
2352.2.q.c 2 140.s even 6 2
3675.2.a.a 1 1.a even 1 1 trivial
3675.2.a.c 1 7.b odd 2 1
7056.2.a.m 1 420.o odd 2 1
7056.2.a.bp 1 60.h even 2 1
9408.2.a.k 1 280.n even 2 1
9408.2.a.bg 1 40.f even 2 1
9408.2.a.bz 1 280.c odd 2 1
9408.2.a.cv 1 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3675))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{11} + 2$$ T11 + 2 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 2$$
$13$ $$T + 1$$
$17$ $$T$$
$19$ $$T - 1$$
$23$ $$T$$
$29$ $$T - 4$$
$31$ $$T - 9$$
$37$ $$T + 3$$
$41$ $$T + 10$$
$43$ $$T + 5$$
$47$ $$T - 6$$
$53$ $$T + 12$$
$59$ $$T + 12$$
$61$ $$T - 10$$
$67$ $$T - 5$$
$71$ $$T + 6$$
$73$ $$T - 3$$
$79$ $$T + 1$$
$83$ $$T + 6$$
$89$ $$T - 16$$
$97$ $$T - 6$$