# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.590892974957$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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

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.

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

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 0.500000 + 0.866025i −2.00000 0 −1.00000 −0.500000 + 0.866025i 1.00000
63.1 0.500000 + 0.866025i −1.00000 + 1.73205i −0.500000 + 0.866025i 0.500000 0.866025i −2.00000 0 −1.00000 −0.500000 0.866025i 1.00000
 $$n$$: 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.a 2
3.b odd 2 1 666.2.f.b 2
4.b odd 2 1 592.2.i.d 2
37.c even 3 1 inner 74.2.c.a 2
37.c even 3 1 2738.2.a.b 1
37.e even 6 1 2738.2.a.d 1
111.i odd 6 1 666.2.f.b 2
148.i odd 6 1 592.2.i.d 2

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

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} - 6T + 36$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$(T + 6)^{2}$$
$29$ $$(T + 9)^{2}$$
$31$ $$(T - 10)^{2}$$
$37$ $$T^{2} + T + 37$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$(T + 8)^{2}$$
$47$ $$(T - 2)^{2}$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} + 8T + 64$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} - 6T + 36$$
$71$ $$T^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} + 6T + 36$$
$83$ $$T^{2} + 2T + 4$$
$89$ $$T^{2} - 13T + 169$$
$97$ $$(T - 3)^{2}$$