# Properties

 Label 3549.2.a.h Level $3549$ Weight $2$ Character orbit 3549.a Self dual yes Analytic conductor $28.339$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

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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: 3.3.169.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x - 1$$ x^3 - x^2 - 4*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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{2} + 2 \beta_1 - 1) q^{5} + ( - \beta_1 + 1) q^{6} - q^{7} + ( - 2 \beta_{2} + \beta_1 - 2) q^{8} + q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 - q^3 + (b2 - b1 + 2) * q^4 + (-b2 + 2*b1 - 1) * q^5 + (-b1 + 1) * q^6 - q^7 + (-2*b2 + b1 - 2) * q^8 + q^9 $$q + (\beta_1 - 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{2} + 2 \beta_1 - 1) q^{5} + ( - \beta_1 + 1) q^{6} - q^{7} + ( - 2 \beta_{2} + \beta_1 - 2) q^{8} + q^{9} + (3 \beta_{2} - 2 \beta_1 + 6) q^{10} + ( - \beta_{2} - 3) q^{11} + ( - \beta_{2} + \beta_1 - 2) q^{12} + ( - \beta_1 + 1) q^{14} + (\beta_{2} - 2 \beta_1 + 1) q^{15} + (\beta_{2} - 2 \beta_1 - 1) q^{16} + (\beta_1 + 1) q^{17} + (\beta_1 - 1) q^{18} + ( - 2 \beta_{2} - 3) q^{19} + ( - 3 \beta_{2} + 5 \beta_1 - 7) q^{20} + q^{21} + (\beta_{2} - 4 \beta_1 + 2) q^{22} + (\beta_{2} + \beta_1 + 3) q^{23} + (2 \beta_{2} - \beta_1 + 2) q^{24} + (4 \beta_{2} - 3 \beta_1 + 6) q^{25} - q^{27} + ( - \beta_{2} + \beta_1 - 2) q^{28} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{29} + ( - 3 \beta_{2} + 2 \beta_1 - 6) q^{30} + (3 \beta_{2} - 4 \beta_1) q^{31} + (\beta_{2} - 2 \beta_1) q^{32} + (\beta_{2} + 3) q^{33} + (\beta_{2} + \beta_1 + 2) q^{34} + (\beta_{2} - 2 \beta_1 + 1) q^{35} + (\beta_{2} - \beta_1 + 2) q^{36} + (\beta_{2} - 2 \beta_1 + 1) q^{37} + (2 \beta_{2} - 5 \beta_1 + 1) q^{38} + (2 \beta_{2} - 6 \beta_1 + 7) q^{40} + ( - 4 \beta_1 + 2) q^{41} + (\beta_1 - 1) q^{42} + ( - \beta_{2} + \beta_1 - 7) q^{43} + ( - 3 \beta_{2} + 3 \beta_1 - 7) q^{44} + ( - \beta_{2} + 2 \beta_1 - 1) q^{45} + (4 \beta_1 + 1) q^{46} + ( - 5 \beta_{2} + 2 \beta_1 - 8) q^{47} + ( - \beta_{2} + 2 \beta_1 + 1) q^{48} + q^{49} + ( - 7 \beta_{2} + 10 \beta_1 - 11) q^{50} + ( - \beta_1 - 1) q^{51} + ( - 2 \beta_1 + 5) q^{53} + ( - \beta_1 + 1) q^{54} + (2 \beta_{2} - 7 \beta_1 + 3) q^{55} + (2 \beta_{2} - \beta_1 + 2) q^{56} + (2 \beta_{2} + 3) q^{57} + (5 \beta_{2} - 6 \beta_1 + 11) q^{58} + (\beta_{2} + \beta_1 - 1) q^{59} + (3 \beta_{2} - 5 \beta_1 + 7) q^{60} + ( - 2 \beta_1 + 5) q^{61} + ( - 7 \beta_{2} + 3 \beta_1 - 9) q^{62} - q^{63} + ( - 5 \beta_{2} + 5 \beta_1 - 3) q^{64} + ( - \beta_{2} + 4 \beta_1 - 2) q^{66} + ( - 2 \beta_{2} - 3 \beta_1 + 2) q^{67} + \beta_1 q^{68} + ( - \beta_{2} - \beta_1 - 3) q^{69} + ( - 3 \beta_{2} + 2 \beta_1 - 6) q^{70} + (5 \beta_{2} + \beta_1 + 4) q^{71} + ( - 2 \beta_{2} + \beta_1 - 2) q^{72} + (5 \beta_{2} - 3 \beta_1 + 2) q^{73} + ( - 3 \beta_{2} + 2 \beta_1 - 6) q^{74} + ( - 4 \beta_{2} + 3 \beta_1 - 6) q^{75} + ( - 3 \beta_{2} + 3 \beta_1 - 8) q^{76} + (\beta_{2} + 3) q^{77} + ( - 7 \beta_{2} - 2 \beta_1 - 2) q^{79} + ( - 2 \beta_{2} - \beta_1 - 9) q^{80} + q^{81} + ( - 4 \beta_{2} + 2 \beta_1 - 14) q^{82} + (7 \beta_{2} - 6 \beta_1 + 5) q^{83} + (\beta_{2} - \beta_1 + 2) q^{84} + (\beta_{2} + 2 \beta_1 + 4) q^{85} + (2 \beta_{2} - 8 \beta_1 + 9) q^{86} + (2 \beta_{2} - 3 \beta_1 + 4) q^{87} + (4 \beta_{2} - 2 \beta_1 + 9) q^{88} + (4 \beta_{2} - 6 \beta_1 - 3) q^{89} + (3 \beta_{2} - 2 \beta_1 + 6) q^{90} + (2 \beta_{2} - \beta_1 + 5) q^{92} + ( - 3 \beta_{2} + 4 \beta_1) q^{93} + (7 \beta_{2} - 13 \beta_1 + 9) q^{94} + (\beta_{2} - 8 \beta_1 + 3) q^{95} + ( - \beta_{2} + 2 \beta_1) q^{96} + ( - \beta_{2} + 2 \beta_1 + 8) q^{97} + (\beta_1 - 1) q^{98} + ( - \beta_{2} - 3) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 - q^3 + (b2 - b1 + 2) * q^4 + (-b2 + 2*b1 - 1) * q^5 + (-b1 + 1) * q^6 - q^7 + (-2*b2 + b1 - 2) * q^8 + q^9 + (3*b2 - 2*b1 + 6) * q^10 + (-b2 - 3) * q^11 + (-b2 + b1 - 2) * q^12 + (-b1 + 1) * q^14 + (b2 - 2*b1 + 1) * q^15 + (b2 - 2*b1 - 1) * q^16 + (b1 + 1) * q^17 + (b1 - 1) * q^18 + (-2*b2 - 3) * q^19 + (-3*b2 + 5*b1 - 7) * q^20 + q^21 + (b2 - 4*b1 + 2) * q^22 + (b2 + b1 + 3) * q^23 + (2*b2 - b1 + 2) * q^24 + (4*b2 - 3*b1 + 6) * q^25 - q^27 + (-b2 + b1 - 2) * q^28 + (-2*b2 + 3*b1 - 4) * q^29 + (-3*b2 + 2*b1 - 6) * q^30 + (3*b2 - 4*b1) * q^31 + (b2 - 2*b1) * q^32 + (b2 + 3) * q^33 + (b2 + b1 + 2) * q^34 + (b2 - 2*b1 + 1) * q^35 + (b2 - b1 + 2) * q^36 + (b2 - 2*b1 + 1) * q^37 + (2*b2 - 5*b1 + 1) * q^38 + (2*b2 - 6*b1 + 7) * q^40 + (-4*b1 + 2) * q^41 + (b1 - 1) * q^42 + (-b2 + b1 - 7) * q^43 + (-3*b2 + 3*b1 - 7) * q^44 + (-b2 + 2*b1 - 1) * q^45 + (4*b1 + 1) * q^46 + (-5*b2 + 2*b1 - 8) * q^47 + (-b2 + 2*b1 + 1) * q^48 + q^49 + (-7*b2 + 10*b1 - 11) * q^50 + (-b1 - 1) * q^51 + (-2*b1 + 5) * q^53 + (-b1 + 1) * q^54 + (2*b2 - 7*b1 + 3) * q^55 + (2*b2 - b1 + 2) * q^56 + (2*b2 + 3) * q^57 + (5*b2 - 6*b1 + 11) * q^58 + (b2 + b1 - 1) * q^59 + (3*b2 - 5*b1 + 7) * q^60 + (-2*b1 + 5) * q^61 + (-7*b2 + 3*b1 - 9) * q^62 - q^63 + (-5*b2 + 5*b1 - 3) * q^64 + (-b2 + 4*b1 - 2) * q^66 + (-2*b2 - 3*b1 + 2) * q^67 + b1 * q^68 + (-b2 - b1 - 3) * q^69 + (-3*b2 + 2*b1 - 6) * q^70 + (5*b2 + b1 + 4) * q^71 + (-2*b2 + b1 - 2) * q^72 + (5*b2 - 3*b1 + 2) * q^73 + (-3*b2 + 2*b1 - 6) * q^74 + (-4*b2 + 3*b1 - 6) * q^75 + (-3*b2 + 3*b1 - 8) * q^76 + (b2 + 3) * q^77 + (-7*b2 - 2*b1 - 2) * q^79 + (-2*b2 - b1 - 9) * q^80 + q^81 + (-4*b2 + 2*b1 - 14) * q^82 + (7*b2 - 6*b1 + 5) * q^83 + (b2 - b1 + 2) * q^84 + (b2 + 2*b1 + 4) * q^85 + (2*b2 - 8*b1 + 9) * q^86 + (2*b2 - 3*b1 + 4) * q^87 + (4*b2 - 2*b1 + 9) * q^88 + (4*b2 - 6*b1 - 3) * q^89 + (3*b2 - 2*b1 + 6) * q^90 + (2*b2 - b1 + 5) * q^92 + (-3*b2 + 4*b1) * q^93 + (7*b2 - 13*b1 + 9) * q^94 + (b2 - 8*b1 + 3) * q^95 + (-b2 + 2*b1) * q^96 + (-b2 + 2*b1 + 8) * q^97 + (b1 - 1) * q^98 + (-b2 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 2 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 - 3 * q^3 + 4 * q^4 + 2 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9 $$3 q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 2 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9} + 13 q^{10} - 8 q^{11} - 4 q^{12} + 2 q^{14} - 6 q^{16} + 4 q^{17} - 2 q^{18} - 7 q^{19} - 13 q^{20} + 3 q^{21} + q^{22} + 9 q^{23} + 3 q^{24} + 11 q^{25} - 3 q^{27} - 4 q^{28} - 7 q^{29} - 13 q^{30} - 7 q^{31} - 3 q^{32} + 8 q^{33} + 6 q^{34} + 4 q^{36} - 4 q^{38} + 13 q^{40} + 2 q^{41} - 2 q^{42} - 19 q^{43} - 15 q^{44} + 7 q^{46} - 17 q^{47} + 6 q^{48} + 3 q^{49} - 16 q^{50} - 4 q^{51} + 13 q^{53} + 2 q^{54} + 3 q^{56} + 7 q^{57} + 22 q^{58} - 3 q^{59} + 13 q^{60} + 13 q^{61} - 17 q^{62} - 3 q^{63} + q^{64} - q^{66} + 5 q^{67} + q^{68} - 9 q^{69} - 13 q^{70} + 8 q^{71} - 3 q^{72} - 2 q^{73} - 13 q^{74} - 11 q^{75} - 18 q^{76} + 8 q^{77} - q^{79} - 26 q^{80} + 3 q^{81} - 36 q^{82} + 2 q^{83} + 4 q^{84} + 13 q^{85} + 17 q^{86} + 7 q^{87} + 21 q^{88} - 19 q^{89} + 13 q^{90} + 12 q^{92} + 7 q^{93} + 7 q^{94} + 3 q^{96} + 27 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 - 3 * q^3 + 4 * q^4 + 2 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9 + 13 * q^10 - 8 * q^11 - 4 * q^12 + 2 * q^14 - 6 * q^16 + 4 * q^17 - 2 * q^18 - 7 * q^19 - 13 * q^20 + 3 * q^21 + q^22 + 9 * q^23 + 3 * q^24 + 11 * q^25 - 3 * q^27 - 4 * q^28 - 7 * q^29 - 13 * q^30 - 7 * q^31 - 3 * q^32 + 8 * q^33 + 6 * q^34 + 4 * q^36 - 4 * q^38 + 13 * q^40 + 2 * q^41 - 2 * q^42 - 19 * q^43 - 15 * q^44 + 7 * q^46 - 17 * q^47 + 6 * q^48 + 3 * q^49 - 16 * q^50 - 4 * q^51 + 13 * q^53 + 2 * q^54 + 3 * q^56 + 7 * q^57 + 22 * q^58 - 3 * q^59 + 13 * q^60 + 13 * q^61 - 17 * q^62 - 3 * q^63 + q^64 - q^66 + 5 * q^67 + q^68 - 9 * q^69 - 13 * q^70 + 8 * q^71 - 3 * q^72 - 2 * q^73 - 13 * q^74 - 11 * q^75 - 18 * q^76 + 8 * q^77 - q^79 - 26 * q^80 + 3 * q^81 - 36 * q^82 + 2 * q^83 + 4 * q^84 + 13 * q^85 + 17 * q^86 + 7 * q^87 + 21 * q^88 - 19 * q^89 + 13 * q^90 + 12 * q^92 + 7 * q^93 + 7 * q^94 + 3 * q^96 + 27 * q^97 - 2 * q^98 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3

## 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
 −1.37720 −0.273891 2.65109
−2.37720 −1.00000 3.65109 −4.02830 2.37720 −1.00000 −3.92498 1.00000 9.57608
1.2 −1.27389 −1.00000 −0.377203 1.10331 1.27389 −1.00000 3.02830 1.00000 −1.40550
1.3 1.65109 −1.00000 0.726109 2.92498 −1.65109 −1.00000 −2.10331 1.00000 4.82942
 $$n$$: 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.h 3
13.b even 2 1 3549.2.a.s 3
13.c even 3 2 273.2.k.d 6
39.i odd 6 2 819.2.o.d 6

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

## Hecke kernels

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

 $$T_{2}^{3} + 2T_{2}^{2} - 3T_{2} - 5$$ T2^3 + 2*T2^2 - 3*T2 - 5 $$T_{5}^{3} - 13T_{5} + 13$$ T5^3 - 13*T5 + 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} - 3 T - 5$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3} - 13T + 13$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} + 8 T^{2} + 17 T + 5$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 4T^{2} + T + 1$$
$19$ $$T^{3} + 7T^{2} - T - 47$$
$23$ $$T^{3} - 9 T^{2} + 14 T - 1$$
$29$ $$T^{3} + 7 T^{2} - 14 T + 5$$
$31$ $$T^{3} + 7 T^{2} - 40 T - 281$$
$37$ $$T^{3} - 13T - 13$$
$41$ $$T^{3} - 2 T^{2} - 68 T + 200$$
$43$ $$T^{3} + 19 T^{2} + 116 T + 229$$
$47$ $$T^{3} + 17 T^{2} + 14 T - 547$$
$53$ $$T^{3} - 13 T^{2} + 39 T + 13$$
$59$ $$T^{3} + 3 T^{2} - 10 T - 25$$
$61$ $$T^{3} - 13 T^{2} + 39 T + 13$$
$67$ $$T^{3} - 5 T^{2} - 74 T + 395$$
$71$ $$T^{3} - 8 T^{2} - 113 T + 515$$
$73$ $$T^{3} + 2 T^{2} - 81 T + 73$$
$79$ $$T^{3} + T^{2} - 290 T - 337$$
$83$ $$T^{3} - 2 T^{2} - 185 T - 229$$
$89$ $$T^{3} + 19 T^{2} - T - 1019$$
$97$ $$T^{3} - 27 T^{2} + 230 T - 599$$
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