# Properties

 Label 74.2.a.a Level $74$ Weight $2$ Character orbit 74.a Self dual yes Analytic conductor $0.591$ 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] = [74,2,Mod(1,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, 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("74.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 74.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: $$0.590892974957$$ 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + (\beta + 1) q^{3} + q^{4} - \beta q^{5} + ( - \beta - 1) q^{6} + ( - 2 \beta + 2) q^{7} - q^{8} + (3 \beta + 1) q^{9} +O(q^{10})$$ q - q^2 + (b + 1) * q^3 + q^4 - b * q^5 + (-b - 1) * q^6 + (-2*b + 2) * q^7 - q^8 + (3*b + 1) * q^9 $$q - q^{2} + (\beta + 1) q^{3} + q^{4} - \beta q^{5} + ( - \beta - 1) q^{6} + ( - 2 \beta + 2) q^{7} - q^{8} + (3 \beta + 1) q^{9} + \beta q^{10} - \beta q^{11} + (\beta + 1) q^{12} + (\beta - 1) q^{13} + (2 \beta - 2) q^{14} + ( - 2 \beta - 3) q^{15} + q^{16} - 6 q^{17} + ( - 3 \beta - 1) q^{18} + 2 q^{19} - \beta q^{20} + ( - 2 \beta - 4) q^{21} + \beta q^{22} + (3 \beta - 3) q^{23} + ( - \beta - 1) q^{24} + (\beta - 2) q^{25} + ( - \beta + 1) q^{26} + (4 \beta + 7) q^{27} + ( - 2 \beta + 2) q^{28} + ( - 3 \beta + 3) q^{29} + (2 \beta + 3) q^{30} + ( - \beta + 2) q^{31} - q^{32} + ( - 2 \beta - 3) q^{33} + 6 q^{34} + 6 q^{35} + (3 \beta + 1) q^{36} + q^{37} - 2 q^{38} + (\beta + 2) q^{39} + \beta q^{40} + (3 \beta + 3) q^{41} + (2 \beta + 4) q^{42} + (2 \beta - 4) q^{43} - \beta q^{44} + ( - 4 \beta - 9) q^{45} + ( - 3 \beta + 3) q^{46} + 2 \beta q^{47} + (\beta + 1) q^{48} + ( - 4 \beta + 9) q^{49} + ( - \beta + 2) q^{50} + ( - 6 \beta - 6) q^{51} + (\beta - 1) q^{52} - 6 q^{53} + ( - 4 \beta - 7) q^{54} + (\beta + 3) q^{55} + (2 \beta - 2) q^{56} + (2 \beta + 2) q^{57} + (3 \beta - 3) q^{58} + (2 \beta + 6) q^{59} + ( - 2 \beta - 3) q^{60} + (5 \beta - 4) q^{61} + (\beta - 2) q^{62} + ( - 2 \beta - 16) q^{63} + q^{64} - 3 q^{65} + (2 \beta + 3) q^{66} + ( - 5 \beta + 8) q^{67} - 6 q^{68} + (3 \beta + 6) q^{69} - 6 q^{70} + 6 q^{71} + ( - 3 \beta - 1) q^{72} + ( - \beta - 10) q^{73} - q^{74} + q^{75} + 2 q^{76} + 6 q^{77} + ( - \beta - 2) q^{78} + (7 \beta - 7) q^{79} - \beta q^{80} + (6 \beta + 16) q^{81} + ( - 3 \beta - 3) q^{82} + ( - 4 \beta + 12) q^{83} + ( - 2 \beta - 4) q^{84} + 6 \beta q^{85} + ( - 2 \beta + 4) q^{86} + ( - 3 \beta - 6) q^{87} + \beta q^{88} - 4 \beta q^{89} + (4 \beta + 9) q^{90} + (2 \beta - 8) q^{91} + (3 \beta - 3) q^{92} - q^{93} - 2 \beta q^{94} - 2 \beta q^{95} + ( - \beta - 1) q^{96} + ( - 8 \beta + 2) q^{97} + (4 \beta - 9) q^{98} + ( - 4 \beta - 9) q^{99} +O(q^{100})$$ q - q^2 + (b + 1) * q^3 + q^4 - b * q^5 + (-b - 1) * q^6 + (-2*b + 2) * q^7 - q^8 + (3*b + 1) * q^9 + b * q^10 - b * q^11 + (b + 1) * q^12 + (b - 1) * q^13 + (2*b - 2) * q^14 + (-2*b - 3) * q^15 + q^16 - 6 * q^17 + (-3*b - 1) * q^18 + 2 * q^19 - b * q^20 + (-2*b - 4) * q^21 + b * q^22 + (3*b - 3) * q^23 + (-b - 1) * q^24 + (b - 2) * q^25 + (-b + 1) * q^26 + (4*b + 7) * q^27 + (-2*b + 2) * q^28 + (-3*b + 3) * q^29 + (2*b + 3) * q^30 + (-b + 2) * q^31 - q^32 + (-2*b - 3) * q^33 + 6 * q^34 + 6 * q^35 + (3*b + 1) * q^36 + q^37 - 2 * q^38 + (b + 2) * q^39 + b * q^40 + (3*b + 3) * q^41 + (2*b + 4) * q^42 + (2*b - 4) * q^43 - b * q^44 + (-4*b - 9) * q^45 + (-3*b + 3) * q^46 + 2*b * q^47 + (b + 1) * q^48 + (-4*b + 9) * q^49 + (-b + 2) * q^50 + (-6*b - 6) * q^51 + (b - 1) * q^52 - 6 * q^53 + (-4*b - 7) * q^54 + (b + 3) * q^55 + (2*b - 2) * q^56 + (2*b + 2) * q^57 + (3*b - 3) * q^58 + (2*b + 6) * q^59 + (-2*b - 3) * q^60 + (5*b - 4) * q^61 + (b - 2) * q^62 + (-2*b - 16) * q^63 + q^64 - 3 * q^65 + (2*b + 3) * q^66 + (-5*b + 8) * q^67 - 6 * q^68 + (3*b + 6) * q^69 - 6 * q^70 + 6 * q^71 + (-3*b - 1) * q^72 + (-b - 10) * q^73 - q^74 + q^75 + 2 * q^76 + 6 * q^77 + (-b - 2) * q^78 + (7*b - 7) * q^79 - b * q^80 + (6*b + 16) * q^81 + (-3*b - 3) * q^82 + (-4*b + 12) * q^83 + (-2*b - 4) * q^84 + 6*b * q^85 + (-2*b + 4) * q^86 + (-3*b - 6) * q^87 + b * q^88 - 4*b * q^89 + (4*b + 9) * q^90 + (2*b - 8) * q^91 + (3*b - 3) * q^92 - q^93 - 2*b * q^94 - 2*b * q^95 + (-b - 1) * q^96 + (-8*b + 2) * q^97 + (4*b - 9) * q^98 + (-4*b - 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{5} - 3 q^{6} + 2 q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - q^5 - 3 * q^6 + 2 * q^7 - 2 * q^8 + 5 * q^9 $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{5} - 3 q^{6} + 2 q^{7} - 2 q^{8} + 5 q^{9} + q^{10} - q^{11} + 3 q^{12} - q^{13} - 2 q^{14} - 8 q^{15} + 2 q^{16} - 12 q^{17} - 5 q^{18} + 4 q^{19} - q^{20} - 10 q^{21} + q^{22} - 3 q^{23} - 3 q^{24} - 3 q^{25} + q^{26} + 18 q^{27} + 2 q^{28} + 3 q^{29} + 8 q^{30} + 3 q^{31} - 2 q^{32} - 8 q^{33} + 12 q^{34} + 12 q^{35} + 5 q^{36} + 2 q^{37} - 4 q^{38} + 5 q^{39} + q^{40} + 9 q^{41} + 10 q^{42} - 6 q^{43} - q^{44} - 22 q^{45} + 3 q^{46} + 2 q^{47} + 3 q^{48} + 14 q^{49} + 3 q^{50} - 18 q^{51} - q^{52} - 12 q^{53} - 18 q^{54} + 7 q^{55} - 2 q^{56} + 6 q^{57} - 3 q^{58} + 14 q^{59} - 8 q^{60} - 3 q^{61} - 3 q^{62} - 34 q^{63} + 2 q^{64} - 6 q^{65} + 8 q^{66} + 11 q^{67} - 12 q^{68} + 15 q^{69} - 12 q^{70} + 12 q^{71} - 5 q^{72} - 21 q^{73} - 2 q^{74} + 2 q^{75} + 4 q^{76} + 12 q^{77} - 5 q^{78} - 7 q^{79} - q^{80} + 38 q^{81} - 9 q^{82} + 20 q^{83} - 10 q^{84} + 6 q^{85} + 6 q^{86} - 15 q^{87} + q^{88} - 4 q^{89} + 22 q^{90} - 14 q^{91} - 3 q^{92} - 2 q^{93} - 2 q^{94} - 2 q^{95} - 3 q^{96} - 4 q^{97} - 14 q^{98} - 22 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - q^5 - 3 * q^6 + 2 * q^7 - 2 * q^8 + 5 * q^9 + q^10 - q^11 + 3 * q^12 - q^13 - 2 * q^14 - 8 * q^15 + 2 * q^16 - 12 * q^17 - 5 * q^18 + 4 * q^19 - q^20 - 10 * q^21 + q^22 - 3 * q^23 - 3 * q^24 - 3 * q^25 + q^26 + 18 * q^27 + 2 * q^28 + 3 * q^29 + 8 * q^30 + 3 * q^31 - 2 * q^32 - 8 * q^33 + 12 * q^34 + 12 * q^35 + 5 * q^36 + 2 * q^37 - 4 * q^38 + 5 * q^39 + q^40 + 9 * q^41 + 10 * q^42 - 6 * q^43 - q^44 - 22 * q^45 + 3 * q^46 + 2 * q^47 + 3 * q^48 + 14 * q^49 + 3 * q^50 - 18 * q^51 - q^52 - 12 * q^53 - 18 * q^54 + 7 * q^55 - 2 * q^56 + 6 * q^57 - 3 * q^58 + 14 * q^59 - 8 * q^60 - 3 * q^61 - 3 * q^62 - 34 * q^63 + 2 * q^64 - 6 * q^65 + 8 * q^66 + 11 * q^67 - 12 * q^68 + 15 * q^69 - 12 * q^70 + 12 * q^71 - 5 * q^72 - 21 * q^73 - 2 * q^74 + 2 * q^75 + 4 * q^76 + 12 * q^77 - 5 * q^78 - 7 * q^79 - q^80 + 38 * q^81 - 9 * q^82 + 20 * q^83 - 10 * q^84 + 6 * q^85 + 6 * q^86 - 15 * q^87 + q^88 - 4 * q^89 + 22 * q^90 - 14 * q^91 - 3 * q^92 - 2 * q^93 - 2 * q^94 - 2 * q^95 - 3 * q^96 - 4 * q^97 - 14 * q^98 - 22 * 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
−1.00000 −0.302776 1.00000 1.30278 0.302776 4.60555 −1.00000 −2.90833 −1.30278
1.2 −1.00000 3.30278 1.00000 −2.30278 −3.30278 −2.60555 −1.00000 7.90833 2.30278
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$37$$ $$-1$$

## 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 74.2.a.a 2
3.b odd 2 1 666.2.a.j 2
4.b odd 2 1 592.2.a.f 2
5.b even 2 1 1850.2.a.u 2
5.c odd 4 2 1850.2.b.i 4
7.b odd 2 1 3626.2.a.a 2
8.b even 2 1 2368.2.a.s 2
8.d odd 2 1 2368.2.a.ba 2
11.b odd 2 1 8954.2.a.p 2
12.b even 2 1 5328.2.a.bf 2
37.b even 2 1 2738.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 1.a even 1 1 trivial
592.2.a.f 2 4.b odd 2 1
666.2.a.j 2 3.b odd 2 1
1850.2.a.u 2 5.b even 2 1
1850.2.b.i 4 5.c odd 4 2
2368.2.a.s 2 8.b even 2 1
2368.2.a.ba 2 8.d odd 2 1
2738.2.a.l 2 37.b even 2 1
3626.2.a.a 2 7.b odd 2 1
5328.2.a.bf 2 12.b even 2 1
8954.2.a.p 2 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 3T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(74))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - 3T - 1$$
$5$ $$T^{2} + T - 3$$
$7$ $$T^{2} - 2T - 12$$
$11$ $$T^{2} + T - 3$$
$13$ $$T^{2} + T - 3$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 3T - 27$$
$29$ $$T^{2} - 3T - 27$$
$31$ $$T^{2} - 3T - 1$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} - 9T - 9$$
$43$ $$T^{2} + 6T - 4$$
$47$ $$T^{2} - 2T - 12$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 14T + 36$$
$61$ $$T^{2} + 3T - 79$$
$67$ $$T^{2} - 11T - 51$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 21T + 107$$
$79$ $$T^{2} + 7T - 147$$
$83$ $$T^{2} - 20T + 48$$
$89$ $$T^{2} + 4T - 48$$
$97$ $$T^{2} + 4T - 204$$