# Properties

 Label 76.3.c.b Level $76$ Weight $3$ Character orbit 76.c Self dual yes Analytic conductor $2.071$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 76.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.07085000914$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 5 - \beta ) q^{5} + ( 1 + 3 \beta ) q^{7} + 9 q^{9} +O(q^{10})$$ $$q + ( 5 - \beta ) q^{5} + ( 1 + 3 \beta ) q^{7} + 9 q^{9} + ( 1 - 5 \beta ) q^{11} + ( -11 + 7 \beta ) q^{17} -19 q^{19} -30 q^{23} + ( 14 - 9 \beta ) q^{25} + ( -37 + 11 \beta ) q^{35} + ( 41 + 3 \beta ) q^{43} + ( 45 - 9 \beta ) q^{45} + ( -31 - 13 \beta ) q^{47} + ( 78 + 15 \beta ) q^{49} + ( 75 - 21 \beta ) q^{55} + ( -59 + 15 \beta ) q^{61} + ( 9 + 27 \beta ) q^{63} + ( 29 - 33 \beta ) q^{73} + ( -209 - 17 \beta ) q^{77} + 81 q^{81} + 90 q^{83} + ( -153 + 39 \beta ) q^{85} + ( -95 + 19 \beta ) q^{95} + ( 9 - 45 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 9q^{5} + 5q^{7} + 18q^{9} + O(q^{10})$$ $$2q + 9q^{5} + 5q^{7} + 18q^{9} - 3q^{11} - 15q^{17} - 38q^{19} - 60q^{23} + 19q^{25} - 63q^{35} + 85q^{43} + 81q^{45} - 75q^{47} + 171q^{49} + 129q^{55} - 103q^{61} + 45q^{63} + 25q^{73} - 435q^{77} + 162q^{81} + 180q^{83} - 267q^{85} - 171q^{95} - 27q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 4.27492 −3.27492
0 0 0 0.725083 0 13.8248 0 9.00000 0
37.2 0 0 0 8.27492 0 −8.82475 0 9.00000 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.b odd 2 1 CM by $$\Q(\sqrt{-19})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.c.b 2
3.b odd 2 1 684.3.h.a 2
4.b odd 2 1 304.3.e.e 2
5.b even 2 1 1900.3.e.a 2
5.c odd 4 2 1900.3.g.a 4
8.b even 2 1 1216.3.e.f 2
8.d odd 2 1 1216.3.e.e 2
12.b even 2 1 2736.3.o.c 2
19.b odd 2 1 CM 76.3.c.b 2
57.d even 2 1 684.3.h.a 2
76.d even 2 1 304.3.e.e 2
95.d odd 2 1 1900.3.e.a 2
95.g even 4 2 1900.3.g.a 4
152.b even 2 1 1216.3.e.e 2
152.g odd 2 1 1216.3.e.f 2
228.b odd 2 1 2736.3.o.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.b 2 1.a even 1 1 trivial
76.3.c.b 2 19.b odd 2 1 CM
304.3.e.e 2 4.b odd 2 1
304.3.e.e 2 76.d even 2 1
684.3.h.a 2 3.b odd 2 1
684.3.h.a 2 57.d even 2 1
1216.3.e.e 2 8.d odd 2 1
1216.3.e.e 2 152.b even 2 1
1216.3.e.f 2 8.b even 2 1
1216.3.e.f 2 152.g odd 2 1
1900.3.e.a 2 5.b even 2 1
1900.3.e.a 2 95.d odd 2 1
1900.3.g.a 4 5.c odd 4 2
1900.3.g.a 4 95.g even 4 2
2736.3.o.c 2 12.b even 2 1
2736.3.o.c 2 228.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{3}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$6 - 9 T + T^{2}$$
$7$ $$-122 - 5 T + T^{2}$$
$11$ $$-354 + 3 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-642 + 15 T + T^{2}$$
$19$ $$( 19 + T )^{2}$$
$23$ $$( 30 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$1678 - 85 T + T^{2}$$
$47$ $$-1002 + 75 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$-554 + 103 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$-15362 - 25 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( -90 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$