# Properties

 Label 3549.2.a.c Level $3549$ Weight $2$ Character orbit 3549.a Self dual yes Analytic conductor $28.339$ 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] = [3549,2,Mod(1,3549)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3549, 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("3549.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3549 = 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3549.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: $$28.3389076774$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 + 2 * q^5 + q^6 + q^7 - 3 * q^8 + q^9 $$q + q^{2} + q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} - 3 q^{8} + q^{9} + 2 q^{10} - 4 q^{11} - q^{12} + q^{14} + 2 q^{15} - q^{16} - 6 q^{17} + q^{18} - 4 q^{19} - 2 q^{20} + q^{21} - 4 q^{22} - 3 q^{24} - q^{25} + q^{27} - q^{28} - 2 q^{29} + 2 q^{30} + 5 q^{32} - 4 q^{33} - 6 q^{34} + 2 q^{35} - q^{36} - 6 q^{37} - 4 q^{38} - 6 q^{40} - 2 q^{41} + q^{42} - 4 q^{43} + 4 q^{44} + 2 q^{45} - q^{48} + q^{49} - q^{50} - 6 q^{51} + 6 q^{53} + q^{54} - 8 q^{55} - 3 q^{56} - 4 q^{57} - 2 q^{58} - 12 q^{59} - 2 q^{60} - 2 q^{61} + q^{63} + 7 q^{64} - 4 q^{66} - 4 q^{67} + 6 q^{68} + 2 q^{70} - 3 q^{72} + 6 q^{73} - 6 q^{74} - q^{75} + 4 q^{76} - 4 q^{77} - 16 q^{79} - 2 q^{80} + q^{81} - 2 q^{82} + 12 q^{83} - q^{84} - 12 q^{85} - 4 q^{86} - 2 q^{87} + 12 q^{88} + 14 q^{89} + 2 q^{90} - 8 q^{95} + 5 q^{96} - 18 q^{97} + q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 + q^3 - q^4 + 2 * q^5 + q^6 + q^7 - 3 * q^8 + q^9 + 2 * q^10 - 4 * q^11 - q^12 + q^14 + 2 * q^15 - q^16 - 6 * q^17 + q^18 - 4 * q^19 - 2 * q^20 + q^21 - 4 * q^22 - 3 * q^24 - q^25 + q^27 - q^28 - 2 * q^29 + 2 * q^30 + 5 * q^32 - 4 * q^33 - 6 * q^34 + 2 * q^35 - q^36 - 6 * q^37 - 4 * q^38 - 6 * q^40 - 2 * q^41 + q^42 - 4 * q^43 + 4 * q^44 + 2 * q^45 - q^48 + q^49 - q^50 - 6 * q^51 + 6 * q^53 + q^54 - 8 * q^55 - 3 * q^56 - 4 * q^57 - 2 * q^58 - 12 * q^59 - 2 * q^60 - 2 * q^61 + q^63 + 7 * q^64 - 4 * q^66 - 4 * q^67 + 6 * q^68 + 2 * q^70 - 3 * q^72 + 6 * q^73 - 6 * q^74 - q^75 + 4 * q^76 - 4 * q^77 - 16 * q^79 - 2 * q^80 + q^81 - 2 * q^82 + 12 * q^83 - q^84 - 12 * q^85 - 4 * q^86 - 2 * q^87 + 12 * q^88 + 14 * q^89 + 2 * q^90 - 8 * q^95 + 5 * q^96 - 18 * q^97 + q^98 - 4 * 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
1.00000 1.00000 −1.00000 2.00000 1.00000 1.00000 −3.00000 1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$1$$

## 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 3549.2.a.c 1
13.b even 2 1 21.2.a.a 1
39.d odd 2 1 63.2.a.a 1
52.b odd 2 1 336.2.a.a 1
65.d even 2 1 525.2.a.d 1
65.h odd 4 2 525.2.d.a 2
91.b odd 2 1 147.2.a.a 1
91.r even 6 2 147.2.e.b 2
91.s odd 6 2 147.2.e.c 2
104.e even 2 1 1344.2.a.g 1
104.h odd 2 1 1344.2.a.s 1
117.n odd 6 2 567.2.f.b 2
117.t even 6 2 567.2.f.g 2
143.d odd 2 1 2541.2.a.j 1
156.h even 2 1 1008.2.a.l 1
195.e odd 2 1 1575.2.a.c 1
195.s even 4 2 1575.2.d.a 2
208.o odd 4 2 5376.2.c.l 2
208.p even 4 2 5376.2.c.r 2
221.b even 2 1 6069.2.a.b 1
247.d odd 2 1 7581.2.a.d 1
260.g odd 2 1 8400.2.a.bn 1
273.g even 2 1 441.2.a.f 1
273.w odd 6 2 441.2.e.a 2
273.ba even 6 2 441.2.e.b 2
312.b odd 2 1 4032.2.a.h 1
312.h even 2 1 4032.2.a.k 1
364.h even 2 1 2352.2.a.v 1
364.x even 6 2 2352.2.q.e 2
364.bl odd 6 2 2352.2.q.x 2
429.e even 2 1 7623.2.a.g 1
455.h odd 2 1 3675.2.a.n 1
728.b even 2 1 9408.2.a.m 1
728.l odd 2 1 9408.2.a.bv 1
1092.d odd 2 1 7056.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 13.b even 2 1
63.2.a.a 1 39.d odd 2 1
147.2.a.a 1 91.b odd 2 1
147.2.e.b 2 91.r even 6 2
147.2.e.c 2 91.s odd 6 2
336.2.a.a 1 52.b odd 2 1
441.2.a.f 1 273.g even 2 1
441.2.e.a 2 273.w odd 6 2
441.2.e.b 2 273.ba even 6 2
525.2.a.d 1 65.d even 2 1
525.2.d.a 2 65.h odd 4 2
567.2.f.b 2 117.n odd 6 2
567.2.f.g 2 117.t even 6 2
1008.2.a.l 1 156.h even 2 1
1344.2.a.g 1 104.e even 2 1
1344.2.a.s 1 104.h odd 2 1
1575.2.a.c 1 195.e odd 2 1
1575.2.d.a 2 195.s even 4 2
2352.2.a.v 1 364.h even 2 1
2352.2.q.e 2 364.x even 6 2
2352.2.q.x 2 364.bl odd 6 2
2541.2.a.j 1 143.d odd 2 1
3549.2.a.c 1 1.a even 1 1 trivial
3675.2.a.n 1 455.h odd 2 1
4032.2.a.h 1 312.b odd 2 1
4032.2.a.k 1 312.h even 2 1
5376.2.c.l 2 208.o odd 4 2
5376.2.c.r 2 208.p even 4 2
6069.2.a.b 1 221.b even 2 1
7056.2.a.p 1 1092.d odd 2 1
7581.2.a.d 1 247.d odd 2 1
7623.2.a.g 1 429.e even 2 1
8400.2.a.bn 1 260.g odd 2 1
9408.2.a.m 1 728.b even 2 1
9408.2.a.bv 1 728.l odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3549))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{5} - 2$$ T5 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T - 2$$
$7$ $$T - 1$$
$11$ $$T + 4$$
$13$ $$T$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T + 6$$
$41$ $$T + 2$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T + 2$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T - 6$$
$79$ $$T + 16$$
$83$ $$T - 12$$
$89$ $$T - 14$$
$97$ $$T + 18$$