# 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

## 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.

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

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

## Hecke characteristic polynomials

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