# Properties

 Label 77.2.b.b Level 77 Weight 2 Character orbit 77.b Analytic conductor 0.615 Analytic rank 0 Dimension 4 CM no 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: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + \beta_{3} q^{3} -\beta_{3} q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} -2 \beta_{2} q^{8} -2 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + \beta_{3} q^{3} -\beta_{3} q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} -2 \beta_{2} q^{8} -2 q^{9} + ( -2 \beta_{1} + \beta_{2} ) q^{10} + ( -3 + \beta_{2} ) q^{11} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{13} + ( 3 - \beta_{3} ) q^{14} + 5 q^{15} -4 q^{16} + 2 \beta_{2} q^{18} + ( 2 \beta_{1} - \beta_{2} ) q^{19} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{21} + ( 2 + 3 \beta_{2} ) q^{22} -3 q^{23} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{24} + 4 \beta_{3} q^{26} + \beta_{3} q^{27} -\beta_{2} q^{29} -5 \beta_{2} q^{30} -3 \beta_{3} q^{31} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{33} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{35} - q^{37} -2 \beta_{3} q^{38} -10 \beta_{2} q^{39} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{40} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{41} + ( 5 + 3 \beta_{3} ) q^{42} + 3 \beta_{2} q^{43} + 2 \beta_{3} q^{45} + 3 \beta_{2} q^{46} + 2 \beta_{3} q^{47} -4 \beta_{3} q^{48} + ( -2 + 3 \beta_{3} ) q^{49} + ( 2 \beta_{1} - \beta_{2} ) q^{54} + ( 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{55} + ( 6 - 2 \beta_{3} ) q^{56} + 5 \beta_{2} q^{57} -2 q^{58} -\beta_{3} q^{59} + ( 2 \beta_{1} - \beta_{2} ) q^{61} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{62} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{63} -8 q^{64} + 10 \beta_{2} q^{65} + ( -6 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{66} + 11 q^{67} -3 \beta_{3} q^{69} + ( -5 - 3 \beta_{3} ) q^{70} + 9 q^{71} + 4 \beta_{2} q^{72} + ( 2 \beta_{1} - \beta_{2} ) q^{73} + \beta_{2} q^{74} + ( -3 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{77} -20 q^{78} -6 \beta_{2} q^{79} + 4 \beta_{3} q^{80} -11 q^{81} -6 \beta_{3} q^{82} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{83} + 6 q^{86} + ( 2 \beta_{1} - \beta_{2} ) q^{87} + ( 4 + 6 \beta_{2} ) q^{88} -\beta_{3} q^{89} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{90} + ( -10 - 6 \beta_{3} ) q^{91} + 15 q^{93} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{94} -5 \beta_{2} q^{95} + 3 \beta_{3} q^{97} + ( 6 \beta_{1} - \beta_{2} ) q^{98} + ( 6 - 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{9} + O(q^{10})$$ $$4q - 8q^{9} - 12q^{11} + 12q^{14} + 20q^{15} - 16q^{16} + 8q^{22} - 12q^{23} - 4q^{37} + 20q^{42} - 8q^{49} + 24q^{56} - 8q^{58} - 32q^{64} + 44q^{67} - 20q^{70} + 36q^{71} - 12q^{77} - 80q^{78} - 44q^{81} + 24q^{86} + 16q^{88} - 40q^{91} + 60q^{93} + 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} + \beta_{1}$$

## 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
 −1.58114 + 0.707107i 1.58114 + 0.707107i 1.58114 − 0.707107i −1.58114 − 0.707107i
1.41421i 2.23607i 0 2.23607i −3.16228 −1.58114 + 2.12132i 2.82843i −2.00000 3.16228
76.2 1.41421i 2.23607i 0 2.23607i 3.16228 1.58114 + 2.12132i 2.82843i −2.00000 −3.16228
76.3 1.41421i 2.23607i 0 2.23607i 3.16228 1.58114 2.12132i 2.82843i −2.00000 −3.16228
76.4 1.41421i 2.23607i 0 2.23607i −3.16228 −1.58114 2.12132i 2.82843i −2.00000 3.16228
 $$n$$: 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 inner
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.b 4
3.b odd 2 1 693.2.c.b 4
4.b odd 2 1 1232.2.e.c 4
7.b odd 2 1 inner 77.2.b.b 4
7.c even 3 2 539.2.i.b 8
7.d odd 6 2 539.2.i.b 8
11.b odd 2 1 inner 77.2.b.b 4
11.c even 5 4 847.2.l.g 16
11.d odd 10 4 847.2.l.g 16
21.c even 2 1 693.2.c.b 4
28.d even 2 1 1232.2.e.c 4
33.d even 2 1 693.2.c.b 4
44.c even 2 1 1232.2.e.c 4
77.b even 2 1 inner 77.2.b.b 4
77.h odd 6 2 539.2.i.b 8
77.i even 6 2 539.2.i.b 8
77.j odd 10 4 847.2.l.g 16
77.l even 10 4 847.2.l.g 16
231.h odd 2 1 693.2.c.b 4
308.g odd 2 1 1232.2.e.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.b.b 4 1.a even 1 1 trivial
77.2.b.b 4 7.b odd 2 1 inner
77.2.b.b 4 11.b odd 2 1 inner
77.2.b.b 4 77.b even 2 1 inner
539.2.i.b 8 7.c even 3 2
539.2.i.b 8 7.d odd 6 2
539.2.i.b 8 77.h odd 6 2
539.2.i.b 8 77.i even 6 2
693.2.c.b 4 3.b odd 2 1
693.2.c.b 4 21.c even 2 1
693.2.c.b 4 33.d even 2 1
693.2.c.b 4 231.h odd 2 1
847.2.l.g 16 11.c even 5 4
847.2.l.g 16 11.d odd 10 4
847.2.l.g 16 77.j odd 10 4
847.2.l.g 16 77.l even 10 4
1232.2.e.c 4 4.b odd 2 1
1232.2.e.c 4 28.d even 2 1
1232.2.e.c 4 44.c even 2 1
1232.2.e.c 4 308.g odd 2 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} + 4 T^{4} )^{2}$$
$3$ $$( 1 - T^{2} + 9 T^{4} )^{2}$$
$5$ $$( 1 - 5 T^{2} + 25 T^{4} )^{2}$$
$7$ $$1 + 4 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 6 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{4}$$
$19$ $$( 1 + 28 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 3 T + 23 T^{2} )^{4}$$
$29$ $$( 1 - 56 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 17 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + T + 37 T^{2} )^{4}$$
$41$ $$( 1 - 8 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 68 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 74 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 53 T^{2} )^{4}$$
$59$ $$( 1 - 113 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 112 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 11 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 9 T + 71 T^{2} )^{4}$$
$73$ $$( 1 + 136 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 86 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 76 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 173 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 149 T^{2} + 9409 T^{4} )^{2}$$