# Properties

 Label 77.2.b.a Level 77 Weight 2 Character orbit 77.b Analytic conductor 0.615 Analytic rank 0 Dimension 2 CM discriminant -7 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$77 = 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 77.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.614848095564$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} -5 q^{4} -\beta q^{7} + 3 \beta q^{8} + 3 q^{9} +O(q^{10})$$ $$q -\beta q^{2} -5 q^{4} -\beta q^{7} + 3 \beta q^{8} + 3 q^{9} + ( 2 + \beta ) q^{11} -7 q^{14} + 11 q^{16} -3 \beta q^{18} + ( 7 - 2 \beta ) q^{22} -8 q^{23} + 5 q^{25} + 5 \beta q^{28} + 4 \beta q^{29} -5 \beta q^{32} -15 q^{36} -6 q^{37} -2 \beta q^{43} + ( -10 - 5 \beta ) q^{44} + 8 \beta q^{46} -7 q^{49} -5 \beta q^{50} + 10 q^{53} + 21 q^{56} + 28 q^{58} -3 \beta q^{63} -13 q^{64} -4 q^{67} -16 q^{71} + 9 \beta q^{72} + 6 \beta q^{74} + ( 7 - 2 \beta ) q^{77} -6 \beta q^{79} + 9 q^{81} -14 q^{86} + ( -21 + 6 \beta ) q^{88} + 40 q^{92} + 7 \beta q^{98} + ( 6 + 3 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 10q^{4} + 6q^{9} + O(q^{10})$$ $$2q - 10q^{4} + 6q^{9} + 4q^{11} - 14q^{14} + 22q^{16} + 14q^{22} - 16q^{23} + 10q^{25} - 30q^{36} - 12q^{37} - 20q^{44} - 14q^{49} + 20q^{53} + 42q^{56} + 56q^{58} - 26q^{64} - 8q^{67} - 32q^{71} + 14q^{77} + 18q^{81} - 28q^{86} - 42q^{88} + 80q^{92} + 12q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/77\mathbb{Z}\right)^\times$$.

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 0.5 + 1.32288i 0.5 − 1.32288i
2.64575i 0 −5.00000 0 0 2.64575i 7.93725i 3.00000 0
76.2 2.64575i 0 −5.00000 0 0 2.64575i 7.93725i 3.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
11.b odd 2 1 inner
77.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.b.a 2
3.b odd 2 1 693.2.c.a 2
4.b odd 2 1 1232.2.e.a 2
7.b odd 2 1 CM 77.2.b.a 2
7.c even 3 2 539.2.i.a 4
7.d odd 6 2 539.2.i.a 4
11.b odd 2 1 inner 77.2.b.a 2
11.c even 5 4 847.2.l.b 8
11.d odd 10 4 847.2.l.b 8
21.c even 2 1 693.2.c.a 2
28.d even 2 1 1232.2.e.a 2
33.d even 2 1 693.2.c.a 2
44.c even 2 1 1232.2.e.a 2
77.b even 2 1 inner 77.2.b.a 2
77.h odd 6 2 539.2.i.a 4
77.i even 6 2 539.2.i.a 4
77.j odd 10 4 847.2.l.b 8
77.l even 10 4 847.2.l.b 8
231.h odd 2 1 693.2.c.a 2
308.g odd 2 1 1232.2.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.b.a 2 1.a even 1 1 trivial
77.2.b.a 2 7.b odd 2 1 CM
77.2.b.a 2 11.b odd 2 1 inner
77.2.b.a 2 77.b even 2 1 inner
539.2.i.a 4 7.c even 3 2
539.2.i.a 4 7.d odd 6 2
539.2.i.a 4 77.h odd 6 2
539.2.i.a 4 77.i even 6 2
693.2.c.a 2 3.b odd 2 1
693.2.c.a 2 21.c even 2 1
693.2.c.a 2 33.d even 2 1
693.2.c.a 2 231.h odd 2 1
847.2.l.b 8 11.c even 5 4
847.2.l.b 8 11.d odd 10 4
847.2.l.b 8 77.j odd 10 4
847.2.l.b 8 77.l even 10 4
1232.2.e.a 2 4.b odd 2 1
1232.2.e.a 2 28.d even 2 1
1232.2.e.a 2 44.c even 2 1
1232.2.e.a 2 308.g odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 7$$ acting on $$S_{2}^{\mathrm{new}}(77, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + 2 T^{2} )( 1 + T + 2 T^{2} )$$
$3$ $$( 1 - 3 T^{2} )^{2}$$
$5$ $$( 1 - 5 T^{2} )^{2}$$
$7$ $$1 + 7 T^{2}$$
$11$ $$1 - 4 T + 11 T^{2}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$( 1 + 8 T + 23 T^{2} )^{2}$$
$29$ $$( 1 - 2 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} )$$
$31$ $$( 1 - 31 T^{2} )^{2}$$
$37$ $$( 1 + 6 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} )$$
$47$ $$( 1 - 47 T^{2} )^{2}$$
$53$ $$( 1 - 10 T + 53 T^{2} )^{2}$$
$59$ $$( 1 - 59 T^{2} )^{2}$$
$61$ $$( 1 + 61 T^{2} )^{2}$$
$67$ $$( 1 + 4 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 16 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 73 T^{2} )^{2}$$
$79$ $$( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} )$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 - 89 T^{2} )^{2}$$
$97$ $$( 1 - 97 T^{2} )^{2}$$