# Properties

 Label 73.2.a.c Level $73$ Weight $2$ Character orbit 73.a Self dual yes Analytic conductor $0.583$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [73,2,Mod(1,73)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(73, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("73.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$73$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 73.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: $$0.582907934755$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( - \beta + 1) q^{3} + (\beta + 1) q^{4} - \beta q^{5} - 3 q^{6} - q^{7} + 3 q^{8} + ( - \beta + 1) q^{9} +O(q^{10})$$ q + b * q^2 + (-b + 1) * q^3 + (b + 1) * q^4 - b * q^5 - 3 * q^6 - q^7 + 3 * q^8 + (-b + 1) * q^9 $$q + \beta q^{2} + ( - \beta + 1) q^{3} + (\beta + 1) q^{4} - \beta q^{5} - 3 q^{6} - q^{7} + 3 q^{8} + ( - \beta + 1) q^{9} + ( - \beta - 3) q^{10} + (\beta + 3) q^{11} + ( - \beta - 2) q^{12} + (\beta - 1) q^{13} - \beta q^{14} + 3 q^{15} + (\beta - 2) q^{16} + (2 \beta - 3) q^{17} - 3 q^{18} - 7 q^{19} + ( - 2 \beta - 3) q^{20} + (\beta - 1) q^{21} + (4 \beta + 3) q^{22} + (\beta + 6) q^{23} + ( - 3 \beta + 3) q^{24} + (\beta - 2) q^{25} + 3 q^{26} + (2 \beta + 1) q^{27} + ( - \beta - 1) q^{28} + ( - 4 \beta + 3) q^{29} + 3 \beta q^{30} + (2 \beta + 2) q^{31} + ( - \beta - 3) q^{32} - 3 \beta q^{33} + ( - \beta + 6) q^{34} + \beta q^{35} + ( - \beta - 2) q^{36} + ( - 2 \beta + 5) q^{37} - 7 \beta q^{38} + (\beta - 4) q^{39} - 3 \beta q^{40} - 6 q^{41} + 3 q^{42} + ( - 4 \beta + 5) q^{43} + (5 \beta + 6) q^{44} + 3 q^{45} + (7 \beta + 3) q^{46} + 9 q^{47} + (2 \beta - 5) q^{48} - 6 q^{49} + ( - \beta + 3) q^{50} + (3 \beta - 9) q^{51} + (\beta + 2) q^{52} + (4 \beta - 3) q^{53} + (3 \beta + 6) q^{54} + ( - 4 \beta - 3) q^{55} - 3 q^{56} + (7 \beta - 7) q^{57} + ( - \beta - 12) q^{58} + (3 \beta + 3) q^{60} + ( - \beta - 4) q^{61} + (4 \beta + 6) q^{62} + (\beta - 1) q^{63} + ( - 6 \beta + 1) q^{64} - 3 q^{65} + ( - 3 \beta - 9) q^{66} + ( - 6 \beta + 5) q^{67} + (\beta + 3) q^{68} + ( - 6 \beta + 3) q^{69} + (\beta + 3) q^{70} + ( - 3 \beta + 3) q^{71} + ( - 3 \beta + 3) q^{72} + q^{73} + (3 \beta - 6) q^{74} + (2 \beta - 5) q^{75} + ( - 7 \beta - 7) q^{76} + ( - \beta - 3) q^{77} + ( - 3 \beta + 3) q^{78} + (3 \beta - 1) q^{79} + (\beta - 3) q^{80} + (2 \beta - 8) q^{81} - 6 \beta q^{82} + ( - 5 \beta + 6) q^{83} + (\beta + 2) q^{84} + (\beta - 6) q^{85} + (\beta - 12) q^{86} + ( - 3 \beta + 15) q^{87} + (3 \beta + 9) q^{88} + (6 \beta + 3) q^{89} + 3 \beta q^{90} + ( - \beta + 1) q^{91} + (8 \beta + 9) q^{92} + ( - 2 \beta - 4) q^{93} + 9 \beta q^{94} + 7 \beta q^{95} + 3 \beta q^{96} + ( - 3 \beta - 1) q^{97} - 6 \beta q^{98} - 3 \beta q^{99} +O(q^{100})$$ q + b * q^2 + (-b + 1) * q^3 + (b + 1) * q^4 - b * q^5 - 3 * q^6 - q^7 + 3 * q^8 + (-b + 1) * q^9 + (-b - 3) * q^10 + (b + 3) * q^11 + (-b - 2) * q^12 + (b - 1) * q^13 - b * q^14 + 3 * q^15 + (b - 2) * q^16 + (2*b - 3) * q^17 - 3 * q^18 - 7 * q^19 + (-2*b - 3) * q^20 + (b - 1) * q^21 + (4*b + 3) * q^22 + (b + 6) * q^23 + (-3*b + 3) * q^24 + (b - 2) * q^25 + 3 * q^26 + (2*b + 1) * q^27 + (-b - 1) * q^28 + (-4*b + 3) * q^29 + 3*b * q^30 + (2*b + 2) * q^31 + (-b - 3) * q^32 - 3*b * q^33 + (-b + 6) * q^34 + b * q^35 + (-b - 2) * q^36 + (-2*b + 5) * q^37 - 7*b * q^38 + (b - 4) * q^39 - 3*b * q^40 - 6 * q^41 + 3 * q^42 + (-4*b + 5) * q^43 + (5*b + 6) * q^44 + 3 * q^45 + (7*b + 3) * q^46 + 9 * q^47 + (2*b - 5) * q^48 - 6 * q^49 + (-b + 3) * q^50 + (3*b - 9) * q^51 + (b + 2) * q^52 + (4*b - 3) * q^53 + (3*b + 6) * q^54 + (-4*b - 3) * q^55 - 3 * q^56 + (7*b - 7) * q^57 + (-b - 12) * q^58 + (3*b + 3) * q^60 + (-b - 4) * q^61 + (4*b + 6) * q^62 + (b - 1) * q^63 + (-6*b + 1) * q^64 - 3 * q^65 + (-3*b - 9) * q^66 + (-6*b + 5) * q^67 + (b + 3) * q^68 + (-6*b + 3) * q^69 + (b + 3) * q^70 + (-3*b + 3) * q^71 + (-3*b + 3) * q^72 + q^73 + (3*b - 6) * q^74 + (2*b - 5) * q^75 + (-7*b - 7) * q^76 + (-b - 3) * q^77 + (-3*b + 3) * q^78 + (3*b - 1) * q^79 + (b - 3) * q^80 + (2*b - 8) * q^81 - 6*b * q^82 + (-5*b + 6) * q^83 + (b + 2) * q^84 + (b - 6) * q^85 + (b - 12) * q^86 + (-3*b + 15) * q^87 + (3*b + 9) * q^88 + (6*b + 3) * q^89 + 3*b * q^90 + (-b + 1) * q^91 + (8*b + 9) * q^92 + (-2*b - 4) * q^93 + 9*b * q^94 + 7*b * q^95 + 3*b * q^96 + (-3*b - 1) * q^97 - 6*b * q^98 - 3*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{3} + 3 q^{4} - q^{5} - 6 q^{6} - 2 q^{7} + 6 q^{8} + q^{9}+O(q^{10})$$ 2 * q + q^2 + q^3 + 3 * q^4 - q^5 - 6 * q^6 - 2 * q^7 + 6 * q^8 + q^9 $$2 q + q^{2} + q^{3} + 3 q^{4} - q^{5} - 6 q^{6} - 2 q^{7} + 6 q^{8} + q^{9} - 7 q^{10} + 7 q^{11} - 5 q^{12} - q^{13} - q^{14} + 6 q^{15} - 3 q^{16} - 4 q^{17} - 6 q^{18} - 14 q^{19} - 8 q^{20} - q^{21} + 10 q^{22} + 13 q^{23} + 3 q^{24} - 3 q^{25} + 6 q^{26} + 4 q^{27} - 3 q^{28} + 2 q^{29} + 3 q^{30} + 6 q^{31} - 7 q^{32} - 3 q^{33} + 11 q^{34} + q^{35} - 5 q^{36} + 8 q^{37} - 7 q^{38} - 7 q^{39} - 3 q^{40} - 12 q^{41} + 6 q^{42} + 6 q^{43} + 17 q^{44} + 6 q^{45} + 13 q^{46} + 18 q^{47} - 8 q^{48} - 12 q^{49} + 5 q^{50} - 15 q^{51} + 5 q^{52} - 2 q^{53} + 15 q^{54} - 10 q^{55} - 6 q^{56} - 7 q^{57} - 25 q^{58} + 9 q^{60} - 9 q^{61} + 16 q^{62} - q^{63} - 4 q^{64} - 6 q^{65} - 21 q^{66} + 4 q^{67} + 7 q^{68} + 7 q^{70} + 3 q^{71} + 3 q^{72} + 2 q^{73} - 9 q^{74} - 8 q^{75} - 21 q^{76} - 7 q^{77} + 3 q^{78} + q^{79} - 5 q^{80} - 14 q^{81} - 6 q^{82} + 7 q^{83} + 5 q^{84} - 11 q^{85} - 23 q^{86} + 27 q^{87} + 21 q^{88} + 12 q^{89} + 3 q^{90} + q^{91} + 26 q^{92} - 10 q^{93} + 9 q^{94} + 7 q^{95} + 3 q^{96} - 5 q^{97} - 6 q^{98} - 3 q^{99}+O(q^{100})$$ 2 * q + q^2 + q^3 + 3 * q^4 - q^5 - 6 * q^6 - 2 * q^7 + 6 * q^8 + q^9 - 7 * q^10 + 7 * q^11 - 5 * q^12 - q^13 - q^14 + 6 * q^15 - 3 * q^16 - 4 * q^17 - 6 * q^18 - 14 * q^19 - 8 * q^20 - q^21 + 10 * q^22 + 13 * q^23 + 3 * q^24 - 3 * q^25 + 6 * q^26 + 4 * q^27 - 3 * q^28 + 2 * q^29 + 3 * q^30 + 6 * q^31 - 7 * q^32 - 3 * q^33 + 11 * q^34 + q^35 - 5 * q^36 + 8 * q^37 - 7 * q^38 - 7 * q^39 - 3 * q^40 - 12 * q^41 + 6 * q^42 + 6 * q^43 + 17 * q^44 + 6 * q^45 + 13 * q^46 + 18 * q^47 - 8 * q^48 - 12 * q^49 + 5 * q^50 - 15 * q^51 + 5 * q^52 - 2 * q^53 + 15 * q^54 - 10 * q^55 - 6 * q^56 - 7 * q^57 - 25 * q^58 + 9 * q^60 - 9 * q^61 + 16 * q^62 - q^63 - 4 * q^64 - 6 * q^65 - 21 * q^66 + 4 * q^67 + 7 * q^68 + 7 * q^70 + 3 * q^71 + 3 * q^72 + 2 * q^73 - 9 * q^74 - 8 * q^75 - 21 * q^76 - 7 * q^77 + 3 * q^78 + q^79 - 5 * q^80 - 14 * q^81 - 6 * q^82 + 7 * q^83 + 5 * q^84 - 11 * q^85 - 23 * q^86 + 27 * q^87 + 21 * q^88 + 12 * q^89 + 3 * q^90 + q^91 + 26 * q^92 - 10 * q^93 + 9 * q^94 + 7 * q^95 + 3 * q^96 - 5 * q^97 - 6 * q^98 - 3 * 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
 −1.30278 2.30278
−1.30278 2.30278 −0.302776 1.30278 −3.00000 −1.00000 3.00000 2.30278 −1.69722
1.2 2.30278 −1.30278 3.30278 −2.30278 −3.00000 −1.00000 3.00000 −1.30278 −5.30278
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$73$$ $$-1$$

## 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 73.2.a.c 2
3.b odd 2 1 657.2.a.e 2
4.b odd 2 1 1168.2.a.c 2
5.b even 2 1 1825.2.a.b 2
7.b odd 2 1 3577.2.a.c 2
8.b even 2 1 4672.2.a.f 2
8.d odd 2 1 4672.2.a.i 2
11.b odd 2 1 8833.2.a.d 2
73.b even 2 1 5329.2.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
73.2.a.c 2 1.a even 1 1 trivial
657.2.a.e 2 3.b odd 2 1
1168.2.a.c 2 4.b odd 2 1
1825.2.a.b 2 5.b even 2 1
3577.2.a.c 2 7.b odd 2 1
4672.2.a.f 2 8.b even 2 1
4672.2.a.i 2 8.d odd 2 1
5329.2.a.c 2 73.b even 2 1
8833.2.a.d 2 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(73))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 3$$
$3$ $$T^{2} - T - 3$$
$5$ $$T^{2} + T - 3$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 7T + 9$$
$13$ $$T^{2} + T - 3$$
$17$ $$T^{2} + 4T - 9$$
$19$ $$(T + 7)^{2}$$
$23$ $$T^{2} - 13T + 39$$
$29$ $$T^{2} - 2T - 51$$
$31$ $$T^{2} - 6T - 4$$
$37$ $$T^{2} - 8T + 3$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} - 6T - 43$$
$47$ $$(T - 9)^{2}$$
$53$ $$T^{2} + 2T - 51$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 9T + 17$$
$67$ $$T^{2} - 4T - 113$$
$71$ $$T^{2} - 3T - 27$$
$73$ $$(T - 1)^{2}$$
$79$ $$T^{2} - T - 29$$
$83$ $$T^{2} - 7T - 69$$
$89$ $$T^{2} - 12T - 81$$
$97$ $$T^{2} + 5T - 23$$