# 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: $$x^{2} - x + 1$$ 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.

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

## 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 $$T_{3}^{2} + 2 T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

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