# Properties

 Label 3549.2.a.q Level $3549$ Weight $2$ Character orbit 3549.a Self dual yes Analytic conductor $28.339$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + \beta_1) q^{4} + (\beta_{2} - \beta_1 + 1) q^{5} + \beta_1 q^{6} - q^{7} + (\beta_{2} + 1) q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 + q^3 + (b2 + b1) * q^4 + (b2 - b1 + 1) * q^5 + b1 * q^6 - q^7 + (b2 + 1) * q^8 + q^9 $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + \beta_1) q^{4} + (\beta_{2} - \beta_1 + 1) q^{5} + \beta_1 q^{6} - q^{7} + (\beta_{2} + 1) q^{8} + q^{9} + ( - \beta_{2} + \beta_1 - 3) q^{10} + ( - \beta_{2} - \beta_1 - 3) q^{11} + (\beta_{2} + \beta_1) q^{12} - \beta_1 q^{14} + (\beta_{2} - \beta_1 + 1) q^{15} + ( - 2 \beta_{2} - 1) q^{16} - 2 \beta_{2} q^{17} + \beta_1 q^{18} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{19} + ( - \beta_{2} - \beta_1 + 1) q^{20} - q^{21} + ( - \beta_{2} - 5 \beta_1 - 1) q^{22} + ( - 2 \beta_{2} - 2 \beta_1) q^{23} + (\beta_{2} + 1) q^{24} + (2 \beta_{2} - 4 \beta_1 + 3) q^{25} + q^{27} + ( - \beta_{2} - \beta_1) q^{28} + (4 \beta_1 - 2) q^{29} + ( - \beta_{2} + \beta_1 - 3) q^{30} + ( - 4 \beta_{2} - 4) q^{31} + ( - 2 \beta_{2} - 3 \beta_1) q^{32} + ( - \beta_{2} - \beta_1 - 3) q^{33} + ( - 2 \beta_1 + 2) q^{34} + ( - \beta_{2} + \beta_1 - 1) q^{35} + (\beta_{2} + \beta_1) q^{36} + (2 \beta_{2} + 2 \beta_1 - 2) q^{37} + ( - 2 \beta_{2} - 6 \beta_1 - 2) q^{38} + (\beta_{2} - 3 \beta_1 + 5) q^{40} + (3 \beta_{2} + \beta_1 - 5) q^{41} - \beta_1 q^{42} + (4 \beta_1 + 4) q^{43} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{44} + (\beta_{2} - \beta_1 + 1) q^{45} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{46} + (\beta_{2} + 3 \beta_1 - 1) q^{47} + ( - 2 \beta_{2} - 1) q^{48} + q^{49} + ( - 4 \beta_{2} + \beta_1 - 10) q^{50} - 2 \beta_{2} q^{51} - 6 q^{53} + \beta_1 q^{54} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{55} + ( - \beta_{2} - 1) q^{56} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{57} + (4 \beta_{2} + 2 \beta_1 + 8) q^{58} + ( - \beta_{2} + \beta_1 - 3) q^{59} + ( - \beta_{2} - \beta_1 + 1) q^{60} + (8 \beta_{2} + 2 \beta_1 + 4) q^{61} + ( - 8 \beta_1 + 4) q^{62} - q^{63} + (\beta_{2} - 5 \beta_1 - 2) q^{64} + ( - \beta_{2} - 5 \beta_1 - 1) q^{66} + (2 \beta_{2} + 2 \beta_1 - 6) q^{67} + (2 \beta_{2} - 4) q^{68} + ( - 2 \beta_{2} - 2 \beta_1) q^{69} + (\beta_{2} - \beta_1 + 3) q^{70} + ( - 5 \beta_{2} + 3 \beta_1 + 1) q^{71} + (\beta_{2} + 1) q^{72} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{73} + (2 \beta_{2} + 2 \beta_1 + 2) q^{74} + (2 \beta_{2} - 4 \beta_1 + 3) q^{75} + ( - 2 \beta_{2} - 6 \beta_1 - 6) q^{76} + (\beta_{2} + \beta_1 + 3) q^{77} + (2 \beta_1 - 6) q^{79} + ( - \beta_{2} + 5 \beta_1 - 9) q^{80} + q^{81} + (\beta_{2} - \beta_1 - 1) q^{82} + (5 \beta_{2} + 7 \beta_1 - 5) q^{83} + ( - \beta_{2} - \beta_1) q^{84} + (4 \beta_1 - 8) q^{85} + (4 \beta_{2} + 8 \beta_1 + 8) q^{86} + (4 \beta_1 - 2) q^{87} + ( - 3 \beta_{2} - \beta_1 - 5) q^{88} + (5 \beta_{2} + 3 \beta_1 - 3) q^{89} + ( - \beta_{2} + \beta_1 - 3) q^{90} + ( - 4 \beta_1 - 6) q^{92} + ( - 4 \beta_{2} - 4) q^{93} + (3 \beta_{2} + 3 \beta_1 + 5) q^{94} + (4 \beta_1 - 4) q^{95} + ( - 2 \beta_{2} - 3 \beta_1) q^{96} + 2 \beta_{2} q^{97} + \beta_1 q^{98} + ( - \beta_{2} - \beta_1 - 3) q^{99}+O(q^{100})$$ q + b1 * q^2 + q^3 + (b2 + b1) * q^4 + (b2 - b1 + 1) * q^5 + b1 * q^6 - q^7 + (b2 + 1) * q^8 + q^9 + (-b2 + b1 - 3) * q^10 + (-b2 - b1 - 3) * q^11 + (b2 + b1) * q^12 - b1 * q^14 + (b2 - b1 + 1) * q^15 + (-2*b2 - 1) * q^16 - 2*b2 * q^17 + b1 * q^18 + (-2*b2 - 2*b1 - 2) * q^19 + (-b2 - b1 + 1) * q^20 - q^21 + (-b2 - 5*b1 - 1) * q^22 + (-2*b2 - 2*b1) * q^23 + (b2 + 1) * q^24 + (2*b2 - 4*b1 + 3) * q^25 + q^27 + (-b2 - b1) * q^28 + (4*b1 - 2) * q^29 + (-b2 + b1 - 3) * q^30 + (-4*b2 - 4) * q^31 + (-2*b2 - 3*b1) * q^32 + (-b2 - b1 - 3) * q^33 + (-2*b1 + 2) * q^34 + (-b2 + b1 - 1) * q^35 + (b2 + b1) * q^36 + (2*b2 + 2*b1 - 2) * q^37 + (-2*b2 - 6*b1 - 2) * q^38 + (b2 - 3*b1 + 5) * q^40 + (3*b2 + b1 - 5) * q^41 - b1 * q^42 + (4*b1 + 4) * q^43 + (-3*b2 - 5*b1 - 3) * q^44 + (b2 - b1 + 1) * q^45 + (-2*b2 - 4*b1 - 2) * q^46 + (b2 + 3*b1 - 1) * q^47 + (-2*b2 - 1) * q^48 + q^49 + (-4*b2 + b1 - 10) * q^50 - 2*b2 * q^51 - 6 * q^53 + b1 * q^54 + (-2*b2 + 4*b1 - 4) * q^55 + (-b2 - 1) * q^56 + (-2*b2 - 2*b1 - 2) * q^57 + (4*b2 + 2*b1 + 8) * q^58 + (-b2 + b1 - 3) * q^59 + (-b2 - b1 + 1) * q^60 + (8*b2 + 2*b1 + 4) * q^61 + (-8*b1 + 4) * q^62 - q^63 + (b2 - 5*b1 - 2) * q^64 + (-b2 - 5*b1 - 1) * q^66 + (2*b2 + 2*b1 - 6) * q^67 + (2*b2 - 4) * q^68 + (-2*b2 - 2*b1) * q^69 + (b2 - b1 + 3) * q^70 + (-5*b2 + 3*b1 + 1) * q^71 + (b2 + 1) * q^72 + (-4*b2 + 2*b1 - 6) * q^73 + (2*b2 + 2*b1 + 2) * q^74 + (2*b2 - 4*b1 + 3) * q^75 + (-2*b2 - 6*b1 - 6) * q^76 + (b2 + b1 + 3) * q^77 + (2*b1 - 6) * q^79 + (-b2 + 5*b1 - 9) * q^80 + q^81 + (b2 - b1 - 1) * q^82 + (5*b2 + 7*b1 - 5) * q^83 + (-b2 - b1) * q^84 + (4*b1 - 8) * q^85 + (4*b2 + 8*b1 + 8) * q^86 + (4*b1 - 2) * q^87 + (-3*b2 - b1 - 5) * q^88 + (5*b2 + 3*b1 - 3) * q^89 + (-b2 + b1 - 3) * q^90 + (-4*b1 - 6) * q^92 + (-4*b2 - 4) * q^93 + (3*b2 + 3*b1 + 5) * q^94 + (4*b1 - 4) * q^95 + (-2*b2 - 3*b1) * q^96 + 2*b2 * q^97 + b1 * q^98 + (-b2 - b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 3 q^{3} + q^{4} + 2 q^{5} + q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + q^2 + 3 * q^3 + q^4 + 2 * q^5 + q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9 $$3 q + q^{2} + 3 q^{3} + q^{4} + 2 q^{5} + q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 8 q^{10} - 10 q^{11} + q^{12} - q^{14} + 2 q^{15} - 3 q^{16} + q^{18} - 8 q^{19} + 2 q^{20} - 3 q^{21} - 8 q^{22} - 2 q^{23} + 3 q^{24} + 5 q^{25} + 3 q^{27} - q^{28} - 2 q^{29} - 8 q^{30} - 12 q^{31} - 3 q^{32} - 10 q^{33} + 4 q^{34} - 2 q^{35} + q^{36} - 4 q^{37} - 12 q^{38} + 12 q^{40} - 14 q^{41} - q^{42} + 16 q^{43} - 14 q^{44} + 2 q^{45} - 10 q^{46} - 3 q^{48} + 3 q^{49} - 29 q^{50} - 18 q^{53} + q^{54} - 8 q^{55} - 3 q^{56} - 8 q^{57} + 26 q^{58} - 8 q^{59} + 2 q^{60} + 14 q^{61} + 4 q^{62} - 3 q^{63} - 11 q^{64} - 8 q^{66} - 16 q^{67} - 12 q^{68} - 2 q^{69} + 8 q^{70} + 6 q^{71} + 3 q^{72} - 16 q^{73} + 8 q^{74} + 5 q^{75} - 24 q^{76} + 10 q^{77} - 16 q^{79} - 22 q^{80} + 3 q^{81} - 4 q^{82} - 8 q^{83} - q^{84} - 20 q^{85} + 32 q^{86} - 2 q^{87} - 16 q^{88} - 6 q^{89} - 8 q^{90} - 22 q^{92} - 12 q^{93} + 18 q^{94} - 8 q^{95} - 3 q^{96} + q^{98} - 10 q^{99}+O(q^{100})$$ 3 * q + q^2 + 3 * q^3 + q^4 + 2 * q^5 + q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9 - 8 * q^10 - 10 * q^11 + q^12 - q^14 + 2 * q^15 - 3 * q^16 + q^18 - 8 * q^19 + 2 * q^20 - 3 * q^21 - 8 * q^22 - 2 * q^23 + 3 * q^24 + 5 * q^25 + 3 * q^27 - q^28 - 2 * q^29 - 8 * q^30 - 12 * q^31 - 3 * q^32 - 10 * q^33 + 4 * q^34 - 2 * q^35 + q^36 - 4 * q^37 - 12 * q^38 + 12 * q^40 - 14 * q^41 - q^42 + 16 * q^43 - 14 * q^44 + 2 * q^45 - 10 * q^46 - 3 * q^48 + 3 * q^49 - 29 * q^50 - 18 * q^53 + q^54 - 8 * q^55 - 3 * q^56 - 8 * q^57 + 26 * q^58 - 8 * q^59 + 2 * q^60 + 14 * q^61 + 4 * q^62 - 3 * q^63 - 11 * q^64 - 8 * q^66 - 16 * q^67 - 12 * q^68 - 2 * q^69 + 8 * q^70 + 6 * q^71 + 3 * q^72 - 16 * q^73 + 8 * q^74 + 5 * q^75 - 24 * q^76 + 10 * q^77 - 16 * q^79 - 22 * q^80 + 3 * q^81 - 4 * q^82 - 8 * q^83 - q^84 - 20 * q^85 + 32 * q^86 - 2 * q^87 - 16 * q^88 - 6 * q^89 - 8 * q^90 - 22 * q^92 - 12 * q^93 + 18 * q^94 - 8 * q^95 - 3 * q^96 + q^98 - 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## 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
 −1.48119 0.311108 2.17009
−1.48119 1.00000 0.193937 4.15633 −1.48119 −1.00000 2.67513 1.00000 −6.15633
1.2 0.311108 1.00000 −1.90321 −1.52543 0.311108 −1.00000 −1.21432 1.00000 −0.474572
1.3 2.17009 1.00000 2.70928 −0.630898 2.17009 −1.00000 1.53919 1.00000 −1.36910
 $$n$$: 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.q 3
13.b even 2 1 3549.2.a.k 3
13.d odd 4 2 273.2.c.b 6
39.f even 4 2 819.2.c.c 6
52.f even 4 2 4368.2.h.o 6
91.i even 4 2 1911.2.c.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.b 6 13.d odd 4 2
819.2.c.c 6 39.f even 4 2
1911.2.c.h 6 91.i even 4 2
3549.2.a.k 3 13.b even 2 1
3549.2.a.q 3 1.a even 1 1 trivial
4368.2.h.o 6 52.f even 4 2

## Hecke kernels

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

 $$T_{2}^{3} - T_{2}^{2} - 3T_{2} + 1$$ T2^3 - T2^2 - 3*T2 + 1 $$T_{5}^{3} - 2T_{5}^{2} - 8T_{5} - 4$$ T5^3 - 2*T5^2 - 8*T5 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 3T + 1$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3} - 2 T^{2} - 8 T - 4$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} + 10 T^{2} + 28 T + 20$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 16T - 16$$
$19$ $$T^{3} + 8T^{2} - 32$$
$23$ $$T^{3} + 2 T^{2} - 20 T - 8$$
$29$ $$T^{3} + 2 T^{2} - 52 T - 40$$
$31$ $$T^{3} + 12 T^{2} - 16 T - 320$$
$37$ $$T^{3} + 4 T^{2} - 16 T - 32$$
$41$ $$T^{3} + 14 T^{2} + 32 T + 20$$
$43$ $$T^{3} - 16 T^{2} + 32 T + 128$$
$47$ $$T^{3} - 28T - 52$$
$53$ $$(T + 6)^{3}$$
$59$ $$T^{3} + 8 T^{2} + 12 T + 4$$
$61$ $$T^{3} - 14 T^{2} - 172 T + 2392$$
$67$ $$T^{3} + 16 T^{2} + 64 T + 32$$
$71$ $$T^{3} - 6 T^{2} - 148 T + 740$$
$73$ $$T^{3} + 16 T^{2} - 8 T - 208$$
$79$ $$T^{3} + 16 T^{2} + 72 T + 80$$
$83$ $$T^{3} + 8 T^{2} - 172 T - 1252$$
$89$ $$T^{3} + 6 T^{2} - 88 T + 76$$
$97$ $$T^{3} - 16T + 16$$