# 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: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4x^{2} + 9$$ x^4 - 4*x^2 + 9 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_{3} q^{3} - \beta_{3} q^{5} + ( - \beta_{2} + 2 \beta_1) q^{6} + (\beta_{2} + \beta_1) q^{7} - 2 \beta_{2} q^{8} - 2 q^{9}+O(q^{10})$$ q - b2 * q^2 + b3 * q^3 - b3 * q^5 + (-b2 + 2*b1) * q^6 + (b2 + b1) * q^7 - 2*b2 * q^8 - 2 * q^9 $$q - \beta_{2} q^{2} + \beta_{3} q^{3} - \beta_{3} q^{5} + ( - \beta_{2} + 2 \beta_1) q^{6} + (\beta_{2} + \beta_1) q^{7} - 2 \beta_{2} q^{8} - 2 q^{9} + (\beta_{2} - 2 \beta_1) q^{10} + (\beta_{2} - 3) q^{11} + (2 \beta_{2} - 4 \beta_1) q^{13} + ( - \beta_{3} + 3) q^{14} + 5 q^{15} - 4 q^{16} + 2 \beta_{2} q^{18} + ( - \beta_{2} + 2 \beta_1) q^{19} + (4 \beta_{2} - 3 \beta_1) q^{21} + (3 \beta_{2} + 2) q^{22} - 3 q^{23} + ( - 2 \beta_{2} + 4 \beta_1) q^{24} + 4 \beta_{3} q^{26} + \beta_{3} q^{27} - \beta_{2} q^{29} - 5 \beta_{2} q^{30} - 3 \beta_{3} q^{31} + ( - 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{33} + ( - 4 \beta_{2} + 3 \beta_1) q^{35} - q^{37} - 2 \beta_{3} q^{38} - 10 \beta_{2} q^{39} + (2 \beta_{2} - 4 \beta_1) q^{40} + ( - 3 \beta_{2} + 6 \beta_1) q^{41} + (3 \beta_{3} + 5) q^{42} + 3 \beta_{2} q^{43} + 2 \beta_{3} q^{45} + 3 \beta_{2} q^{46} + 2 \beta_{3} q^{47} - 4 \beta_{3} q^{48} + (3 \beta_{3} - 2) q^{49} + ( - \beta_{2} + 2 \beta_1) q^{54} + (3 \beta_{3} - \beta_{2} + 2 \beta_1) q^{55} + ( - 2 \beta_{3} + 6) q^{56} + 5 \beta_{2} q^{57} - 2 q^{58} - \beta_{3} q^{59} + ( - \beta_{2} + 2 \beta_1) q^{61} + (3 \beta_{2} - 6 \beta_1) q^{62} + ( - 2 \beta_{2} - 2 \beta_1) q^{63} - 8 q^{64} + 10 \beta_{2} q^{65} + (2 \beta_{3} + 3 \beta_{2} - 6 \beta_1) q^{66} + 11 q^{67} - 3 \beta_{3} q^{69} + ( - 3 \beta_{3} - 5) q^{70} + 9 q^{71} + 4 \beta_{2} q^{72} + ( - \beta_{2} + 2 \beta_1) q^{73} + \beta_{2} q^{74} + (\beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{77} - 20 q^{78} - 6 \beta_{2} q^{79} + 4 \beta_{3} q^{80} - 11 q^{81} - 6 \beta_{3} q^{82} + (3 \beta_{2} - 6 \beta_1) q^{83} + 6 q^{86} + ( - \beta_{2} + 2 \beta_1) q^{87} + (6 \beta_{2} + 4) q^{88} - \beta_{3} q^{89} + ( - 2 \beta_{2} + 4 \beta_1) q^{90} + ( - 6 \beta_{3} - 10) q^{91} + 15 q^{93} + ( - 2 \beta_{2} + 4 \beta_1) q^{94} - 5 \beta_{2} q^{95} + 3 \beta_{3} q^{97} + ( - \beta_{2} + 6 \beta_1) q^{98} + ( - 2 \beta_{2} + 6) q^{99}+O(q^{100})$$ q - b2 * q^2 + b3 * q^3 - b3 * q^5 + (-b2 + 2*b1) * q^6 + (b2 + b1) * q^7 - 2*b2 * q^8 - 2 * q^9 + (b2 - 2*b1) * q^10 + (b2 - 3) * q^11 + (2*b2 - 4*b1) * q^13 + (-b3 + 3) * q^14 + 5 * q^15 - 4 * q^16 + 2*b2 * q^18 + (-b2 + 2*b1) * q^19 + (4*b2 - 3*b1) * q^21 + (3*b2 + 2) * q^22 - 3 * q^23 + (-2*b2 + 4*b1) * q^24 + 4*b3 * q^26 + b3 * q^27 - b2 * q^29 - 5*b2 * q^30 - 3*b3 * q^31 + (-3*b3 + b2 - 2*b1) * q^33 + (-4*b2 + 3*b1) * q^35 - q^37 - 2*b3 * q^38 - 10*b2 * q^39 + (2*b2 - 4*b1) * q^40 + (-3*b2 + 6*b1) * q^41 + (3*b3 + 5) * q^42 + 3*b2 * q^43 + 2*b3 * q^45 + 3*b2 * q^46 + 2*b3 * q^47 - 4*b3 * q^48 + (3*b3 - 2) * q^49 + (-b2 + 2*b1) * q^54 + (3*b3 - b2 + 2*b1) * q^55 + (-2*b3 + 6) * q^56 + 5*b2 * q^57 - 2 * q^58 - b3 * q^59 + (-b2 + 2*b1) * q^61 + (3*b2 - 6*b1) * q^62 + (-2*b2 - 2*b1) * q^63 - 8 * q^64 + 10*b2 * q^65 + (2*b3 + 3*b2 - 6*b1) * q^66 + 11 * q^67 - 3*b3 * q^69 + (-3*b3 - 5) * q^70 + 9 * q^71 + 4*b2 * q^72 + (-b2 + 2*b1) * q^73 + b2 * q^74 + (b3 - 3*b2 - 3*b1 - 3) * q^77 - 20 * q^78 - 6*b2 * q^79 + 4*b3 * q^80 - 11 * q^81 - 6*b3 * q^82 + (3*b2 - 6*b1) * q^83 + 6 * q^86 + (-b2 + 2*b1) * q^87 + (6*b2 + 4) * q^88 - b3 * q^89 + (-2*b2 + 4*b1) * q^90 + (-6*b3 - 10) * q^91 + 15 * q^93 + (-2*b2 + 4*b1) * q^94 - 5*b2 * q^95 + 3*b3 * q^97 + (-b2 + 6*b1) * q^98 + (-2*b2 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{9}+O(q^{10})$$ 4 * q - 8 * q^9 $$4 q - 8 q^{9} - 12 q^{11} + 12 q^{14} + 20 q^{15} - 16 q^{16} + 8 q^{22} - 12 q^{23} - 4 q^{37} + 20 q^{42} - 8 q^{49} + 24 q^{56} - 8 q^{58} - 32 q^{64} + 44 q^{67} - 20 q^{70} + 36 q^{71} - 12 q^{77} - 80 q^{78} - 44 q^{81} + 24 q^{86} + 16 q^{88} - 40 q^{91} + 60 q^{93} + 24 q^{99}+O(q^{100})$$ 4 * q - 8 * q^9 - 12 * q^11 + 12 * q^14 + 20 * q^15 - 16 * q^16 + 8 * q^22 - 12 * q^23 - 4 * q^37 + 20 * q^42 - 8 * q^49 + 24 * q^56 - 8 * q^58 - 32 * q^64 + 44 * q^67 - 20 * q^70 + 36 * q^71 - 12 * q^77 - 80 * q^78 - 44 * q^81 + 24 * q^86 + 16 * q^88 - 40 * q^91 + 60 * q^93 + 24 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - \nu ) / 3$$ (v^3 - v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2$$ b3 + 2 $$\nu^{3}$$ $$=$$ $$3\beta_{2} + \beta_1$$ 3*b2 + b1

## 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

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{2}$$
$3$ $$(T^{2} + 5)^{2}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$T^{4} + 4T^{2} + 49$$
$11$ $$(T^{2} + 6 T + 11)^{2}$$
$13$ $$(T^{2} - 40)^{2}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} - 10)^{2}$$
$23$ $$(T + 3)^{4}$$
$29$ $$(T^{2} + 2)^{2}$$
$31$ $$(T^{2} + 45)^{2}$$
$37$ $$(T + 1)^{4}$$
$41$ $$(T^{2} - 90)^{2}$$
$43$ $$(T^{2} + 18)^{2}$$
$47$ $$(T^{2} + 20)^{2}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} + 5)^{2}$$
$61$ $$(T^{2} - 10)^{2}$$
$67$ $$(T - 11)^{4}$$
$71$ $$(T - 9)^{4}$$
$73$ $$(T^{2} - 10)^{2}$$
$79$ $$(T^{2} + 72)^{2}$$
$83$ $$(T^{2} - 90)^{2}$$
$89$ $$(T^{2} + 5)^{2}$$
$97$ $$(T^{2} + 45)^{2}$$