# Properties

 Label 74.2.e.a Level $74$ Weight $2$ Character orbit 74.e Analytic conductor $0.591$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,2,Mod(11,74)]

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([5]))

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

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

chi := DirichletCharacter("74.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 74.e (of order $$6$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{2} - 2) q^{5} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{6} - 4 \zeta_{12}^{2} q^{7} + \zeta_{12}^{3} q^{8} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 1) q^{9} +O(q^{10})$$ q + z * q^2 + (z^3 - z^2 + z) * q^3 + z^2 * q^4 + (z^2 - 2) * q^5 + (-z^3 + 2*z^2 - 1) * q^6 - 4*z^2 * q^7 + z^3 * q^8 + (-4*z^3 + z^2 + 2*z - 1) * q^9 $$q + \zeta_{12} q^{2} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{2} - 2) q^{5} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{6} - 4 \zeta_{12}^{2} q^{7} + \zeta_{12}^{3} q^{8} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 1) q^{9} + \cdots + ( - 10 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \cdots + 3) q^{99} +O(q^{100})$$ q + z * q^2 + (z^3 - z^2 + z) * q^3 + z^2 * q^4 + (z^2 - 2) * q^5 + (-z^3 + 2*z^2 - 1) * q^6 - 4*z^2 * q^7 + z^3 * q^8 + (-4*z^3 + z^2 + 2*z - 1) * q^9 + (z^3 - 2*z) * q^10 + (-z^3 + 2*z + 3) * q^11 + (2*z^3 - z^2 - z + 1) * q^12 + (6*z^3 - 6*z) * q^13 - 4*z^3 * q^14 + (z^2 - 3*z + 1) * q^15 + (z^2 - 1) * q^16 + (-z^2 - 1) * q^17 + (z^3 - 2*z^2 - z + 4) * q^18 + (3*z^3 - z^2 - 3*z + 2) * q^19 + (-z^2 - 1) * q^20 + (-8*z^3 + 4*z^2 + 4*z - 4) * q^21 + (z^2 + 3*z + 1) * q^22 + (-3*z^3 + 2*z^2 - 1) * q^23 + (-z^3 + z^2 + z - 2) * q^24 + (2*z^2 - 2) * q^25 - 6 * q^26 + 4 * q^27 + (-4*z^2 + 4) * q^28 + (6*z^3 - 2*z^2 + 1) * q^29 + (z^3 - 3*z^2 + z) * q^30 + (3*z^3 - 2*z^2 + 1) * q^31 + (z^3 - z) * q^32 + (2*z^3 + 2*z) * q^33 + (-z^3 - z) * q^34 + (4*z^2 + 4) * q^35 + (-2*z^3 + 4*z - 1) * q^36 + (7*z^2 - 4) * q^37 + (-z^3 + 2*z - 3) * q^38 + (-6*z^2 + 6*z - 6) * q^39 + (-z^3 - z) * q^40 + (-2*z^3 + 3*z^2 - 2*z) * q^41 + (4*z^3 - 4*z^2 - 4*z + 8) * q^42 + (-6*z^3 - 4*z^2 + 2) * q^43 + (z^3 + 3*z^2 + z) * q^44 + (6*z^3 - 2*z^2 + 1) * q^45 + (2*z^3 - 3*z^2 - z + 3) * q^46 + (-5*z^3 + 10*z + 3) * q^47 + (z^3 - 2*z + 1) * q^48 + (9*z^2 - 9) * q^49 + (2*z^3 - 2*z) * q^50 + (-3*z^3 + 2*z^2 - 1) * q^51 - 6*z * q^52 + (4*z^3 - 6*z^2 - 2*z + 6) * q^53 + 4*z * q^54 + (3*z^3 + 3*z^2 - 3*z - 6) * q^55 + (-4*z^3 + 4*z) * q^56 + (-4*z^2 + 6*z - 4) * q^57 + (-2*z^3 + 6*z^2 + z - 6) * q^58 + (2*z^2 - 6*z + 2) * q^59 + (-3*z^3 + 2*z^2 - 1) * q^60 + (-6*z^3 - 5*z^2 + 6*z + 10) * q^61 + (-2*z^3 + 3*z^2 + z - 3) * q^62 + (8*z^3 - 16*z + 4) * q^63 - q^64 + (-12*z^3 + 6*z) * q^65 + (4*z^2 - 2) * q^66 + (-3*z^3 - z^2 - 3*z) * q^67 + (-2*z^2 + 1) * q^68 + (6*z^3 - 4*z^2 - 6*z + 8) * q^69 + (4*z^3 + 4*z) * q^70 + (2*z^3 - 6*z^2 + 2*z) * q^71 + (2*z^2 - z + 2) * q^72 + (6*z^3 - 12*z - 2) * q^73 + (7*z^3 - 4*z) * q^74 + (2*z^3 - 4*z + 2) * q^75 + (z^2 - 3*z + 1) * q^76 + (-4*z^3 - 12*z^2 - 4*z) * q^77 + (-6*z^3 + 6*z^2 - 6*z) * q^78 + (3*z^3 + 3*z^2 - 3*z - 6) * q^79 + (-2*z^2 + 1) * q^80 + (-2*z^3 - z^2 - 2*z) * q^81 + (3*z^3 - 4*z^2 + 2) * q^82 + (10*z^3 + 3*z^2 - 5*z - 3) * q^83 + (-4*z^3 + 8*z - 4) * q^84 + 3 * q^85 + (-4*z^3 - 6*z^2 + 2*z + 6) * q^86 + (-9*z^3 + 7*z^2 + 9*z - 14) * q^87 + (3*z^3 + 2*z^2 - 1) * q^88 + (3*z^2 + 3) * q^89 + (-2*z^3 + 6*z^2 + z - 6) * q^90 + 24*z * q^91 + (-3*z^3 + z^2 + 3*z - 2) * q^92 + (-6*z^3 + 4*z^2 + 6*z - 8) * q^93 + (5*z^2 + 3*z + 5) * q^94 + (-6*z^3 + 3*z^2 + 3*z - 3) * q^95 + (-z^2 + z - 1) * q^96 + (-6*z^2 + 3) * q^97 + (9*z^3 - 9*z) * q^98 + (-10*z^3 - 3*z^2 + 5*z + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 2 q^{4} - 6 q^{5} - 8 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 2 * q^4 - 6 * q^5 - 8 * q^7 - 2 * q^9 $$4 q - 2 q^{3} + 2 q^{4} - 6 q^{5} - 8 q^{7} - 2 q^{9} + 12 q^{11} + 2 q^{12} + 6 q^{15} - 2 q^{16} - 6 q^{17} + 12 q^{18} + 6 q^{19} - 6 q^{20} - 8 q^{21} + 6 q^{22} - 6 q^{24} - 4 q^{25} - 24 q^{26} + 16 q^{27} + 8 q^{28} - 6 q^{30} + 24 q^{35} - 4 q^{36} - 2 q^{37} - 12 q^{38} - 36 q^{39} + 6 q^{41} + 24 q^{42} + 6 q^{44} + 6 q^{46} + 12 q^{47} + 4 q^{48} - 18 q^{49} + 12 q^{53} - 18 q^{55} - 24 q^{57} - 12 q^{58} + 12 q^{59} + 30 q^{61} - 6 q^{62} + 16 q^{63} - 4 q^{64} - 2 q^{67} + 24 q^{69} - 12 q^{71} + 12 q^{72} - 8 q^{73} + 8 q^{75} + 6 q^{76} - 24 q^{77} + 12 q^{78} - 18 q^{79} - 2 q^{81} - 6 q^{83} - 16 q^{84} + 12 q^{85} + 12 q^{86} - 42 q^{87} + 18 q^{89} - 12 q^{90} - 6 q^{92} - 24 q^{93} + 30 q^{94} - 6 q^{95} - 6 q^{96} + 6 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 + 2 * q^4 - 6 * q^5 - 8 * q^7 - 2 * q^9 + 12 * q^11 + 2 * q^12 + 6 * q^15 - 2 * q^16 - 6 * q^17 + 12 * q^18 + 6 * q^19 - 6 * q^20 - 8 * q^21 + 6 * q^22 - 6 * q^24 - 4 * q^25 - 24 * q^26 + 16 * q^27 + 8 * q^28 - 6 * q^30 + 24 * q^35 - 4 * q^36 - 2 * q^37 - 12 * q^38 - 36 * q^39 + 6 * q^41 + 24 * q^42 + 6 * q^44 + 6 * q^46 + 12 * q^47 + 4 * q^48 - 18 * q^49 + 12 * q^53 - 18 * q^55 - 24 * q^57 - 12 * q^58 + 12 * q^59 + 30 * q^61 - 6 * q^62 + 16 * q^63 - 4 * q^64 - 2 * q^67 + 24 * q^69 - 12 * q^71 + 12 * q^72 - 8 * q^73 + 8 * q^75 + 6 * q^76 - 24 * q^77 + 12 * q^78 - 18 * q^79 - 2 * q^81 - 6 * q^83 - 16 * q^84 + 12 * q^85 + 12 * q^86 - 42 * q^87 + 18 * q^89 - 12 * q^90 - 6 * q^92 - 24 * q^93 + 30 * q^94 - 6 * q^95 - 6 * q^96 + 6 * q^99

## Character values

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

 $$n$$ $$39$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i −1.36603 + 2.36603i 0.500000 0.866025i −1.50000 0.866025i 2.73205i −2.00000 + 3.46410i 1.00000i −2.23205 3.86603i 1.73205
11.2 0.866025 0.500000i 0.366025 0.633975i 0.500000 0.866025i −1.50000 0.866025i 0.732051i −2.00000 + 3.46410i 1.00000i 1.23205 + 2.13397i −1.73205
27.1 −0.866025 0.500000i −1.36603 2.36603i 0.500000 + 0.866025i −1.50000 + 0.866025i 2.73205i −2.00000 3.46410i 1.00000i −2.23205 + 3.86603i 1.73205
27.2 0.866025 + 0.500000i 0.366025 + 0.633975i 0.500000 + 0.866025i −1.50000 + 0.866025i 0.732051i −2.00000 3.46410i 1.00000i 1.23205 2.13397i −1.73205
 $$n$$: 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.e.a 4
3.b odd 2 1 666.2.s.e 4
4.b odd 2 1 592.2.w.d 4
37.e even 6 1 inner 74.2.e.a 4
37.g odd 12 1 2738.2.a.f 2
37.g odd 12 1 2738.2.a.j 2
111.h odd 6 1 666.2.s.e 4
148.j odd 6 1 592.2.w.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.e.a 4 1.a even 1 1 trivial
74.2.e.a 4 37.e even 6 1 inner
592.2.w.d 4 4.b odd 2 1
592.2.w.d 4 148.j odd 6 1
666.2.s.e 4 3.b odd 2 1
666.2.s.e 4 111.h odd 6 1
2738.2.a.f 2 37.g odd 12 1
2738.2.a.j 2 37.g odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 3T_{5} + 3$$ acting on $$S_{2}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} + 2 T^{3} + \cdots + 4$$
$5$ $$(T^{2} + 3 T + 3)^{2}$$
$7$ $$(T^{2} + 4 T + 16)^{2}$$
$11$ $$(T^{2} - 6 T + 6)^{2}$$
$13$ $$T^{4} - 36T^{2} + 1296$$
$17$ $$(T^{2} + 3 T + 3)^{2}$$
$19$ $$T^{4} - 6 T^{3} + \cdots + 36$$
$23$ $$T^{4} + 24T^{2} + 36$$
$29$ $$T^{4} + 78T^{2} + 1089$$
$31$ $$T^{4} + 24T^{2} + 36$$
$37$ $$(T^{2} + T + 37)^{2}$$
$41$ $$T^{4} - 6 T^{3} + \cdots + 9$$
$43$ $$T^{4} + 96T^{2} + 576$$
$47$ $$(T^{2} - 6 T - 66)^{2}$$
$53$ $$T^{4} - 12 T^{3} + \cdots + 576$$
$59$ $$T^{4} - 12 T^{3} + \cdots + 576$$
$61$ $$T^{4} - 30 T^{3} + \cdots + 1521$$
$67$ $$T^{4} + 2 T^{3} + \cdots + 676$$
$71$ $$T^{4} + 12 T^{3} + \cdots + 576$$
$73$ $$(T^{2} + 4 T - 104)^{2}$$
$79$ $$T^{4} + 18 T^{3} + \cdots + 324$$
$83$ $$T^{4} + 6 T^{3} + \cdots + 4356$$
$89$ $$(T^{2} - 9 T + 27)^{2}$$
$97$ $$(T^{2} + 27)^{2}$$