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

## 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

$p$ $F_p(T)$
$2$ $$T^{2} + 7$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 7$$
$11$ $$T^{2} - 4T + 11$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$(T + 8)^{2}$$
$29$ $$T^{2} + 112$$
$31$ $$T^{2}$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 28$$
$47$ $$T^{2}$$
$53$ $$(T - 10)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$(T + 4)^{2}$$
$71$ $$(T + 16)^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2} + 252$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$