# Properties

 Label 76.7.c.a Level $76$ Weight $7$ Character orbit 76.c Self dual yes Analytic conductor $17.484$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 76.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.4841103551$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{57}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 27 + 7 \beta ) q^{5} + ( -305 + 9 \beta ) q^{7} + 729 q^{9} +O(q^{10})$$ $$q + ( 27 + 7 \beta ) q^{5} + ( -305 + 9 \beta ) q^{7} + 729 q^{9} + ( 531 - 70 \beta ) q^{11} + ( 4815 + 56 \beta ) q^{17} -6859 q^{19} + 20610 q^{23} + ( 29792 + 378 \beta ) q^{25} + ( 49221 - 1892 \beta ) q^{35} + ( 71315 - 2016 \beta ) q^{43} + ( 19683 + 5103 \beta ) q^{45} + ( 37575 + 5551 \beta ) q^{47} + ( 49248 - 5490 \beta ) q^{49} + ( -432543 + 1827 \beta ) q^{55} + ( 28531 - 12915 \beta ) q^{61} + ( -222345 + 6561 \beta ) q^{63} + ( -192025 - 19404 \beta ) q^{73} + ( -736515 + 26129 \beta ) q^{77} + 531441 q^{81} -1131030 q^{83} + ( 487509 + 35217 \beta ) q^{85} + ( -185193 - 48013 \beta ) q^{95} + ( 387099 - 51030 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 54q^{5} - 610q^{7} + 1458q^{9} + O(q^{10})$$ $$2q + 54q^{5} - 610q^{7} + 1458q^{9} + 1062q^{11} + 9630q^{17} - 13718q^{19} + 41220q^{23} + 59584q^{25} + 98442q^{35} + 142630q^{43} + 39366q^{45} + 75150q^{47} + 98496q^{49} - 865086q^{55} + 57062q^{61} - 444690q^{63} - 384050q^{73} - 1473030q^{77} + 1062882q^{81} - 2262060q^{83} + 975018q^{85} - 370386q^{95} + 774198q^{99} + O(q^{100})$$

## 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
 −3.27492 4.27492
0 0 0 −184.395 0 −576.794 0 729.000 0
37.2 0 0 0 238.395 0 −33.2060 0 729.000 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 76.7.c.a 2
3.b odd 2 1 684.7.h.a 2
4.b odd 2 1 304.7.e.b 2
19.b odd 2 1 CM 76.7.c.a 2
57.d even 2 1 684.7.h.a 2
76.d even 2 1 304.7.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.c.a 2 1.a even 1 1 trivial
76.7.c.a 2 19.b odd 2 1 CM
304.7.e.b 2 4.b odd 2 1
304.7.e.b 2 76.d even 2 1
684.7.h.a 2 3.b odd 2 1
684.7.h.a 2 57.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{7}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-43959 - 54 T + T^{2}$$
$7$ $$19153 + 610 T + T^{2}$$
$11$ $$-4186839 - 1062 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$20324193 - 9630 T + T^{2}$$
$19$ $$( 6859 + T )^{2}$$
$23$ $$( -20610 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$1379227753 - 142630 T + T^{2}$$
$47$ $$-26690123487 - 75150 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$-151305051239 - 57062 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$-306508276367 + 384050 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( 1131030 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$