# Properties

 Label 76.2.e.a Level $76$ Weight $2$ Character orbit 76.e Analytic conductor $0.607$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 76.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.606863055362$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} -4 q^{11} + \zeta_{6} q^{13} -\zeta_{6} q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} + ( -5 + 2 \zeta_{6} ) q^{19} -5 \zeta_{6} q^{23} + 4 \zeta_{6} q^{25} + 5 q^{27} -7 \zeta_{6} q^{29} + 4 q^{31} + ( -4 + 4 \zeta_{6} ) q^{33} + 10 q^{37} + q^{39} + ( 5 - 5 \zeta_{6} ) q^{41} + ( 5 - 5 \zeta_{6} ) q^{43} + 2 q^{45} + 7 \zeta_{6} q^{47} -7 q^{49} + 3 \zeta_{6} q^{51} -11 \zeta_{6} q^{53} + ( -4 + 4 \zeta_{6} ) q^{55} + ( -3 + 5 \zeta_{6} ) q^{57} + ( -3 + 3 \zeta_{6} ) q^{59} -11 \zeta_{6} q^{61} + q^{65} + 3 \zeta_{6} q^{67} -5 q^{69} + ( -11 + 11 \zeta_{6} ) q^{71} + ( -15 + 15 \zeta_{6} ) q^{73} + 4 q^{75} + ( 13 - 13 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 3 \zeta_{6} q^{85} -7 q^{87} -3 \zeta_{6} q^{89} + ( 4 - 4 \zeta_{6} ) q^{93} + ( -3 + 5 \zeta_{6} ) q^{95} + ( 5 - 5 \zeta_{6} ) q^{97} -8 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + q^{5} + 2q^{9} + O(q^{10})$$ $$2q + q^{3} + q^{5} + 2q^{9} - 8q^{11} + q^{13} - q^{15} - 3q^{17} - 8q^{19} - 5q^{23} + 4q^{25} + 10q^{27} - 7q^{29} + 8q^{31} - 4q^{33} + 20q^{37} + 2q^{39} + 5q^{41} + 5q^{43} + 4q^{45} + 7q^{47} - 14q^{49} + 3q^{51} - 11q^{53} - 4q^{55} - q^{57} - 3q^{59} - 11q^{61} + 2q^{65} + 3q^{67} - 10q^{69} - 11q^{71} - 15q^{73} + 8q^{75} + 13q^{79} - q^{81} + 3q^{85} - 14q^{87} - 3q^{89} + 4q^{93} - q^{95} + 5q^{97} - 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
45.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0 1.00000 1.73205i 0
49.1 0 0.500000 0.866025i 0 0.500000 0.866025i 0 0 0 1.00000 + 1.73205i 0
 $$n$$: 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.2.e.a 2
3.b odd 2 1 684.2.k.b 2
4.b odd 2 1 304.2.i.a 2
5.b even 2 1 1900.2.i.a 2
5.c odd 4 2 1900.2.s.a 4
8.b even 2 1 1216.2.i.c 2
8.d odd 2 1 1216.2.i.g 2
12.b even 2 1 2736.2.s.g 2
19.b odd 2 1 1444.2.e.b 2
19.c even 3 1 inner 76.2.e.a 2
19.c even 3 1 1444.2.a.b 1
19.d odd 6 1 1444.2.a.c 1
19.d odd 6 1 1444.2.e.b 2
57.h odd 6 1 684.2.k.b 2
76.f even 6 1 5776.2.a.f 1
76.g odd 6 1 304.2.i.a 2
76.g odd 6 1 5776.2.a.k 1
95.i even 6 1 1900.2.i.a 2
95.m odd 12 2 1900.2.s.a 4
152.k odd 6 1 1216.2.i.g 2
152.p even 6 1 1216.2.i.c 2
228.m even 6 1 2736.2.s.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.e.a 2 1.a even 1 1 trivial
76.2.e.a 2 19.c even 3 1 inner
304.2.i.a 2 4.b odd 2 1
304.2.i.a 2 76.g odd 6 1
684.2.k.b 2 3.b odd 2 1
684.2.k.b 2 57.h odd 6 1
1216.2.i.c 2 8.b even 2 1
1216.2.i.c 2 152.p even 6 1
1216.2.i.g 2 8.d odd 2 1
1216.2.i.g 2 152.k odd 6 1
1444.2.a.b 1 19.c even 3 1
1444.2.a.c 1 19.d odd 6 1
1444.2.e.b 2 19.b odd 2 1
1444.2.e.b 2 19.d odd 6 1
1900.2.i.a 2 5.b even 2 1
1900.2.i.a 2 95.i even 6 1
1900.2.s.a 4 5.c odd 4 2
1900.2.s.a 4 95.m odd 12 2
2736.2.s.g 2 12.b even 2 1
2736.2.s.g 2 228.m even 6 1
5776.2.a.f 1 76.f even 6 1
5776.2.a.k 1 76.g odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$1 - T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$19 + 8 T + T^{2}$$
$23$ $$25 + 5 T + T^{2}$$
$29$ $$49 + 7 T + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$25 - 5 T + T^{2}$$
$43$ $$25 - 5 T + T^{2}$$
$47$ $$49 - 7 T + T^{2}$$
$53$ $$121 + 11 T + T^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$121 + 11 T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$121 + 11 T + T^{2}$$
$73$ $$225 + 15 T + T^{2}$$
$79$ $$169 - 13 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$9 + 3 T + T^{2}$$
$97$ $$25 - 5 T + T^{2}$$