# Properties

 Label 74.2.c.b Level $74$ Weight $2$ Character orbit 74.c Analytic conductor $0.591$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,2,Mod(47,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("74.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 74.c (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$0.590892974957$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + 2 \zeta_{6} q^{3} - \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + 2 q^{6} + 4 \zeta_{6} q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} +O(q^{10})$$ q + (-z + 1) * q^2 + 2*z * q^3 - z * q^4 - 3*z * q^5 + 2 * q^6 + 4*z * q^7 - q^8 + (z - 1) * q^9 $$q + ( - \zeta_{6} + 1) q^{2} + 2 \zeta_{6} q^{3} - \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + 2 q^{6} + 4 \zeta_{6} q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} - 3 q^{10} - 6 q^{11} + ( - 2 \zeta_{6} + 2) q^{12} - 2 \zeta_{6} q^{13} + 4 q^{14} + ( - 6 \zeta_{6} + 6) q^{15} + (\zeta_{6} - 1) q^{16} + (3 \zeta_{6} - 3) q^{17} + \zeta_{6} q^{18} - 2 \zeta_{6} q^{19} + (3 \zeta_{6} - 3) q^{20} + (8 \zeta_{6} - 8) q^{21} + (6 \zeta_{6} - 6) q^{22} + 6 q^{23} - 2 \zeta_{6} q^{24} + (4 \zeta_{6} - 4) q^{25} - 2 q^{26} + 4 q^{27} + ( - 4 \zeta_{6} + 4) q^{28} + 3 q^{29} - 6 \zeta_{6} q^{30} + 2 q^{31} + \zeta_{6} q^{32} - 12 \zeta_{6} q^{33} + 3 \zeta_{6} q^{34} + ( - 12 \zeta_{6} + 12) q^{35} + q^{36} + (3 \zeta_{6} + 4) q^{37} - 2 q^{38} + ( - 4 \zeta_{6} + 4) q^{39} + 3 \zeta_{6} q^{40} - 3 \zeta_{6} q^{41} + 8 \zeta_{6} q^{42} - 4 q^{43} + 6 \zeta_{6} q^{44} + 3 q^{45} + ( - 6 \zeta_{6} + 6) q^{46} - 6 q^{47} - 2 q^{48} + (9 \zeta_{6} - 9) q^{49} + 4 \zeta_{6} q^{50} - 6 q^{51} + (2 \zeta_{6} - 2) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} + ( - 4 \zeta_{6} + 4) q^{54} + 18 \zeta_{6} q^{55} - 4 \zeta_{6} q^{56} + ( - 4 \zeta_{6} + 4) q^{57} + ( - 3 \zeta_{6} + 3) q^{58} - 6 q^{60} + \zeta_{6} q^{61} + ( - 2 \zeta_{6} + 2) q^{62} - 4 q^{63} + q^{64} + (6 \zeta_{6} - 6) q^{65} - 12 q^{66} - 2 \zeta_{6} q^{67} + 3 q^{68} + 12 \zeta_{6} q^{69} - 12 \zeta_{6} q^{70} + 12 \zeta_{6} q^{71} + ( - \zeta_{6} + 1) q^{72} - 10 q^{73} + ( - 4 \zeta_{6} + 7) q^{74} - 8 q^{75} + (2 \zeta_{6} - 2) q^{76} - 24 \zeta_{6} q^{77} - 4 \zeta_{6} q^{78} - 14 \zeta_{6} q^{79} + 3 q^{80} + 11 \zeta_{6} q^{81} - 3 q^{82} + (6 \zeta_{6} - 6) q^{83} + 8 q^{84} + 9 q^{85} + (4 \zeta_{6} - 4) q^{86} + 6 \zeta_{6} q^{87} + 6 q^{88} + (3 \zeta_{6} - 3) q^{89} + ( - 3 \zeta_{6} + 3) q^{90} + ( - 8 \zeta_{6} + 8) q^{91} - 6 \zeta_{6} q^{92} + 4 \zeta_{6} q^{93} + (6 \zeta_{6} - 6) q^{94} + (6 \zeta_{6} - 6) q^{95} + (2 \zeta_{6} - 2) q^{96} - 13 q^{97} + 9 \zeta_{6} q^{98} + ( - 6 \zeta_{6} + 6) q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 + 2*z * q^3 - z * q^4 - 3*z * q^5 + 2 * q^6 + 4*z * q^7 - q^8 + (z - 1) * q^9 - 3 * q^10 - 6 * q^11 + (-2*z + 2) * q^12 - 2*z * q^13 + 4 * q^14 + (-6*z + 6) * q^15 + (z - 1) * q^16 + (3*z - 3) * q^17 + z * q^18 - 2*z * q^19 + (3*z - 3) * q^20 + (8*z - 8) * q^21 + (6*z - 6) * q^22 + 6 * q^23 - 2*z * q^24 + (4*z - 4) * q^25 - 2 * q^26 + 4 * q^27 + (-4*z + 4) * q^28 + 3 * q^29 - 6*z * q^30 + 2 * q^31 + z * q^32 - 12*z * q^33 + 3*z * q^34 + (-12*z + 12) * q^35 + q^36 + (3*z + 4) * q^37 - 2 * q^38 + (-4*z + 4) * q^39 + 3*z * q^40 - 3*z * q^41 + 8*z * q^42 - 4 * q^43 + 6*z * q^44 + 3 * q^45 + (-6*z + 6) * q^46 - 6 * q^47 - 2 * q^48 + (9*z - 9) * q^49 + 4*z * q^50 - 6 * q^51 + (2*z - 2) * q^52 + (-6*z + 6) * q^53 + (-4*z + 4) * q^54 + 18*z * q^55 - 4*z * q^56 + (-4*z + 4) * q^57 + (-3*z + 3) * q^58 - 6 * q^60 + z * q^61 + (-2*z + 2) * q^62 - 4 * q^63 + q^64 + (6*z - 6) * q^65 - 12 * q^66 - 2*z * q^67 + 3 * q^68 + 12*z * q^69 - 12*z * q^70 + 12*z * q^71 + (-z + 1) * q^72 - 10 * q^73 + (-4*z + 7) * q^74 - 8 * q^75 + (2*z - 2) * q^76 - 24*z * q^77 - 4*z * q^78 - 14*z * q^79 + 3 * q^80 + 11*z * q^81 - 3 * q^82 + (6*z - 6) * q^83 + 8 * q^84 + 9 * q^85 + (4*z - 4) * q^86 + 6*z * q^87 + 6 * q^88 + (3*z - 3) * q^89 + (-3*z + 3) * q^90 + (-8*z + 8) * q^91 - 6*z * q^92 + 4*z * q^93 + (6*z - 6) * q^94 + (6*z - 6) * q^95 + (2*z - 2) * q^96 - 13 * q^97 + 9*z * q^98 + (-6*z + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 2 q^{3} - q^{4} - 3 q^{5} + 4 q^{6} + 4 q^{7} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 + 2 * q^3 - q^4 - 3 * q^5 + 4 * q^6 + 4 * q^7 - 2 * q^8 - q^9 $$2 q + q^{2} + 2 q^{3} - q^{4} - 3 q^{5} + 4 q^{6} + 4 q^{7} - 2 q^{8} - q^{9} - 6 q^{10} - 12 q^{11} + 2 q^{12} - 2 q^{13} + 8 q^{14} + 6 q^{15} - q^{16} - 3 q^{17} + q^{18} - 2 q^{19} - 3 q^{20} - 8 q^{21} - 6 q^{22} + 12 q^{23} - 2 q^{24} - 4 q^{25} - 4 q^{26} + 8 q^{27} + 4 q^{28} + 6 q^{29} - 6 q^{30} + 4 q^{31} + q^{32} - 12 q^{33} + 3 q^{34} + 12 q^{35} + 2 q^{36} + 11 q^{37} - 4 q^{38} + 4 q^{39} + 3 q^{40} - 3 q^{41} + 8 q^{42} - 8 q^{43} + 6 q^{44} + 6 q^{45} + 6 q^{46} - 12 q^{47} - 4 q^{48} - 9 q^{49} + 4 q^{50} - 12 q^{51} - 2 q^{52} + 6 q^{53} + 4 q^{54} + 18 q^{55} - 4 q^{56} + 4 q^{57} + 3 q^{58} - 12 q^{60} + q^{61} + 2 q^{62} - 8 q^{63} + 2 q^{64} - 6 q^{65} - 24 q^{66} - 2 q^{67} + 6 q^{68} + 12 q^{69} - 12 q^{70} + 12 q^{71} + q^{72} - 20 q^{73} + 10 q^{74} - 16 q^{75} - 2 q^{76} - 24 q^{77} - 4 q^{78} - 14 q^{79} + 6 q^{80} + 11 q^{81} - 6 q^{82} - 6 q^{83} + 16 q^{84} + 18 q^{85} - 4 q^{86} + 6 q^{87} + 12 q^{88} - 3 q^{89} + 3 q^{90} + 8 q^{91} - 6 q^{92} + 4 q^{93} - 6 q^{94} - 6 q^{95} - 2 q^{96} - 26 q^{97} + 9 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q + q^2 + 2 * q^3 - q^4 - 3 * q^5 + 4 * q^6 + 4 * q^7 - 2 * q^8 - q^9 - 6 * q^10 - 12 * q^11 + 2 * q^12 - 2 * q^13 + 8 * q^14 + 6 * q^15 - q^16 - 3 * q^17 + q^18 - 2 * q^19 - 3 * q^20 - 8 * q^21 - 6 * q^22 + 12 * q^23 - 2 * q^24 - 4 * q^25 - 4 * q^26 + 8 * q^27 + 4 * q^28 + 6 * q^29 - 6 * q^30 + 4 * q^31 + q^32 - 12 * q^33 + 3 * q^34 + 12 * q^35 + 2 * q^36 + 11 * q^37 - 4 * q^38 + 4 * q^39 + 3 * q^40 - 3 * q^41 + 8 * q^42 - 8 * q^43 + 6 * q^44 + 6 * q^45 + 6 * q^46 - 12 * q^47 - 4 * q^48 - 9 * q^49 + 4 * q^50 - 12 * q^51 - 2 * q^52 + 6 * q^53 + 4 * q^54 + 18 * q^55 - 4 * q^56 + 4 * q^57 + 3 * q^58 - 12 * q^60 + q^61 + 2 * q^62 - 8 * q^63 + 2 * q^64 - 6 * q^65 - 24 * q^66 - 2 * q^67 + 6 * q^68 + 12 * q^69 - 12 * q^70 + 12 * q^71 + q^72 - 20 * q^73 + 10 * q^74 - 16 * q^75 - 2 * q^76 - 24 * q^77 - 4 * q^78 - 14 * q^79 + 6 * q^80 + 11 * q^81 - 6 * q^82 - 6 * q^83 + 16 * q^84 + 18 * q^85 - 4 * q^86 + 6 * q^87 + 12 * q^88 - 3 * q^89 + 3 * q^90 + 8 * q^91 - 6 * q^92 + 4 * q^93 - 6 * q^94 - 6 * q^95 - 2 * q^96 - 26 * q^97 + 9 * q^98 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$$.

 $$n$$ $$39$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 1.00000 + 1.73205i −0.500000 0.866025i −1.50000 2.59808i 2.00000 2.00000 + 3.46410i −1.00000 −0.500000 + 0.866025i −3.00000
63.1 0.500000 + 0.866025i 1.00000 1.73205i −0.500000 + 0.866025i −1.50000 + 2.59808i 2.00000 2.00000 3.46410i −1.00000 −0.500000 0.866025i −3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.c.b 2
3.b odd 2 1 666.2.f.d 2
4.b odd 2 1 592.2.i.a 2
37.c even 3 1 inner 74.2.c.b 2
37.c even 3 1 2738.2.a.a 1
37.e even 6 1 2738.2.a.c 1
111.i odd 6 1 666.2.f.d 2
148.i odd 6 1 592.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.c.b 2 1.a even 1 1 trivial
74.2.c.b 2 37.c even 3 1 inner
592.2.i.a 2 4.b odd 2 1
592.2.i.a 2 148.i odd 6 1
666.2.f.d 2 3.b odd 2 1
666.2.f.d 2 111.i odd 6 1
2738.2.a.a 1 37.c even 3 1
2738.2.a.c 1 37.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$(T + 6)^{2}$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$(T - 6)^{2}$$
$29$ $$(T - 3)^{2}$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} - 11T + 37$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$(T + 4)^{2}$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2}$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$T^{2} - 12T + 144$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$T^{2} + 6T + 36$$
$89$ $$T^{2} + 3T + 9$$
$97$ $$(T + 13)^{2}$$