# Properties

 Label 77.2.a.a Level $77$ Weight $2$ Character orbit 77.a Self dual yes Analytic conductor $0.615$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$77 = 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 77.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.614848095564$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 2 q^{4} - q^{5} - q^{7} + 6 q^{9}+O(q^{10})$$ q - 3 * q^3 - 2 * q^4 - q^5 - q^7 + 6 * q^9 $$q - 3 q^{3} - 2 q^{4} - q^{5} - q^{7} + 6 q^{9} - q^{11} + 6 q^{12} - 4 q^{13} + 3 q^{15} + 4 q^{16} + 2 q^{17} - 6 q^{19} + 2 q^{20} + 3 q^{21} - 5 q^{23} - 4 q^{25} - 9 q^{27} + 2 q^{28} + 10 q^{29} + q^{31} + 3 q^{33} + q^{35} - 12 q^{36} - 5 q^{37} + 12 q^{39} - 2 q^{41} - 8 q^{43} + 2 q^{44} - 6 q^{45} + 8 q^{47} - 12 q^{48} + q^{49} - 6 q^{51} + 8 q^{52} - 6 q^{53} + q^{55} + 18 q^{57} + 3 q^{59} - 6 q^{60} - 2 q^{61} - 6 q^{63} - 8 q^{64} + 4 q^{65} - 3 q^{67} - 4 q^{68} + 15 q^{69} + q^{71} + 10 q^{73} + 12 q^{75} + 12 q^{76} + q^{77} + 6 q^{79} - 4 q^{80} + 9 q^{81} + 12 q^{83} - 6 q^{84} - 2 q^{85} - 30 q^{87} - 15 q^{89} + 4 q^{91} + 10 q^{92} - 3 q^{93} + 6 q^{95} - 5 q^{97} - 6 q^{99}+O(q^{100})$$ q - 3 * q^3 - 2 * q^4 - q^5 - q^7 + 6 * q^9 - q^11 + 6 * q^12 - 4 * q^13 + 3 * q^15 + 4 * q^16 + 2 * q^17 - 6 * q^19 + 2 * q^20 + 3 * q^21 - 5 * q^23 - 4 * q^25 - 9 * q^27 + 2 * q^28 + 10 * q^29 + q^31 + 3 * q^33 + q^35 - 12 * q^36 - 5 * q^37 + 12 * q^39 - 2 * q^41 - 8 * q^43 + 2 * q^44 - 6 * q^45 + 8 * q^47 - 12 * q^48 + q^49 - 6 * q^51 + 8 * q^52 - 6 * q^53 + q^55 + 18 * q^57 + 3 * q^59 - 6 * q^60 - 2 * q^61 - 6 * q^63 - 8 * q^64 + 4 * q^65 - 3 * q^67 - 4 * q^68 + 15 * q^69 + q^71 + 10 * q^73 + 12 * q^75 + 12 * q^76 + q^77 + 6 * q^79 - 4 * q^80 + 9 * q^81 + 12 * q^83 - 6 * q^84 - 2 * q^85 - 30 * q^87 - 15 * q^89 + 4 * q^91 + 10 * q^92 - 3 * q^93 + 6 * q^95 - 5 * q^97 - 6 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −3.00000 −2.00000 −1.00000 0 −1.00000 0 6.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.a.a 1
3.b odd 2 1 693.2.a.c 1
4.b odd 2 1 1232.2.a.l 1
5.b even 2 1 1925.2.a.h 1
5.c odd 4 2 1925.2.b.e 2
7.b odd 2 1 539.2.a.c 1
7.c even 3 2 539.2.e.f 2
7.d odd 6 2 539.2.e.c 2
8.b even 2 1 4928.2.a.bj 1
8.d odd 2 1 4928.2.a.a 1
11.b odd 2 1 847.2.a.b 1
11.c even 5 4 847.2.f.i 4
11.d odd 10 4 847.2.f.h 4
21.c even 2 1 4851.2.a.j 1
28.d even 2 1 8624.2.a.a 1
33.d even 2 1 7623.2.a.j 1
77.b even 2 1 5929.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a 1 1.a even 1 1 trivial
539.2.a.c 1 7.b odd 2 1
539.2.e.c 2 7.d odd 6 2
539.2.e.f 2 7.c even 3 2
693.2.a.c 1 3.b odd 2 1
847.2.a.b 1 11.b odd 2 1
847.2.f.h 4 11.d odd 10 4
847.2.f.i 4 11.c even 5 4
1232.2.a.l 1 4.b odd 2 1
1925.2.a.h 1 5.b even 2 1
1925.2.b.e 2 5.c odd 4 2
4851.2.a.j 1 21.c even 2 1
4928.2.a.a 1 8.d odd 2 1
4928.2.a.bj 1 8.b even 2 1
5929.2.a.f 1 77.b even 2 1
7623.2.a.j 1 33.d even 2 1
8624.2.a.a 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(77))$$:

 $$T_{2}$$ T2 $$T_{3} + 3$$ T3 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 1$$
$7$ $$T + 1$$
$11$ $$T + 1$$
$13$ $$T + 4$$
$17$ $$T - 2$$
$19$ $$T + 6$$
$23$ $$T + 5$$
$29$ $$T - 10$$
$31$ $$T - 1$$
$37$ $$T + 5$$
$41$ $$T + 2$$
$43$ $$T + 8$$
$47$ $$T - 8$$
$53$ $$T + 6$$
$59$ $$T - 3$$
$61$ $$T + 2$$
$67$ $$T + 3$$
$71$ $$T - 1$$
$73$ $$T - 10$$
$79$ $$T - 6$$
$83$ $$T - 12$$
$89$ $$T + 15$$
$97$ $$T + 5$$