# Properties

 Label 77.2.l.a Level $77$ Weight $2$ Character orbit 77.l Analytic conductor $0.615$ Analytic rank $0$ Dimension $8$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$77 = 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 77.l (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.614848095564$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.37515625.1 Defining polynomial: $$x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} + 6 x^{3} - 4 x^{2} - 8 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{2} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{4} + ( -2 \beta_{4} - \beta_{5} ) q^{7} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{8} + ( -3 - 3 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{2} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{4} + ( -2 \beta_{4} - \beta_{5} ) q^{7} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{8} + ( -3 - 3 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} ) q^{9} + ( -2 \beta_{2} - 3 \beta_{7} ) q^{11} + ( -\beta_{1} + 4 \beta_{3} + 4 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{14} + ( -2 + 5 \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{16} + ( -3 + 3 \beta_{2} + 3 \beta_{6} ) q^{18} + ( 3 - 3 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 7 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{22} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{23} + 5 \beta_{3} q^{25} + ( 2 + 3 \beta_{1} - \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{28} + ( -1 + 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{29} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} - 2 \beta_{5} - 6 \beta_{6} + 8 \beta_{7} ) q^{32} + ( -3 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} + 3 \beta_{7} ) q^{36} + ( -1 + 4 \beta_{2} - 4 \beta_{6} + 5 \beta_{7} ) q^{37} + ( 5 + 2 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 5 \beta_{5} - 10 \beta_{7} ) q^{43} + ( 4 \beta_{1} + 2 \beta_{3} - \beta_{4} - 5 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{44} + ( 7 - 6 \beta_{1} + 3 \beta_{3} + 5 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} - 3 \beta_{7} ) q^{46} + 7 \beta_{7} q^{49} + ( -5 + 5 \beta_{1} - 5 \beta_{3} - 5 \beta_{4} - 10 \beta_{5} + 5 \beta_{7} ) q^{50} + ( 3 + 4 \beta_{4} + 7 \beta_{5} + 4 \beta_{6} ) q^{53} + ( 1 - 7 \beta_{1} + 3 \beta_{2} + 7 \beta_{3} + 4 \beta_{4} + 9 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{56} + ( -\beta_{1} - 3 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{58} + ( 6 - 6 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} ) q^{63} + ( -1 + \beta_{1} + 2 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - 10 \beta_{7} ) q^{64} + ( -1 + 6 \beta_{1} - 6 \beta_{2} - \beta_{3} - \beta_{5} ) q^{67} + ( -11 + 2 \beta_{1} + 2 \beta_{2} - 9 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{71} + ( -3 - 3 \beta_{1} + 3 \beta_{2} + 12 \beta_{3} + 3 \beta_{4} + 9 \beta_{5} - 3 \beta_{7} ) q^{72} + ( -10 + 10 \beta_{1} + 9 \beta_{2} - 5 \beta_{3} - 9 \beta_{4} - 18 \beta_{5} + 10 \beta_{6} + \beta_{7} ) q^{74} + ( -9 + 4 \beta_{1} - 9 \beta_{3} - 9 \beta_{5} + 9 \beta_{7} ) q^{77} + ( -7 - 6 \beta_{4} + \beta_{5} + 6 \beta_{6} ) q^{79} + 9 \beta_{5} q^{81} + ( 8 - 5 \beta_{1} - 10 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - 12 \beta_{6} - \beta_{7} ) q^{86} + ( 3 - 10 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} - 9 \beta_{6} + 3 \beta_{7} ) q^{88} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 11 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{92} + ( -7 + 7 \beta_{1} + 7 \beta_{2} - 7 \beta_{5} + 7 \beta_{6} + 7 \beta_{7} ) q^{98} + ( 9 - 6 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 10q^{4} - 10q^{8} - 6q^{9} + O(q^{10})$$ $$8q + 10q^{4} - 10q^{8} - 6q^{9} - 4q^{11} - 21q^{14} + 8q^{16} - 15q^{18} - 14q^{22} + 16q^{23} - 10q^{25} + 35q^{28} + 30q^{36} - 18q^{37} + 25q^{44} + 15q^{46} + 14q^{49} + 30q^{53} - 42q^{56} + 19q^{58} - 34q^{64} + 8q^{67} - 48q^{71} - 75q^{72} - 14q^{77} - 40q^{79} - 18q^{81} + 23q^{86} - 8q^{88} + 25q^{92} + 48q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} + 6 x^{3} - 4 x^{2} - 8 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - \nu^{5} + 3 \nu^{4} - \nu^{3} + 6 \nu^{2} - 4 \nu - 8$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} + 3 \nu^{5} - \nu^{4} + 3 \nu^{3} - 4 \nu^{2} - 8 \nu + 16$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 3 \nu^{2}$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 7 \nu^{2}$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$-\nu^{5} - 5$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{7} + 3 \nu^{6} + 3 \nu^{5} - \nu^{4} + 3 \nu^{3} - 18 \nu^{2} + 12 \nu + 24$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 3 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-\beta_{6} - 5$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 5 \beta_{1} - 2$$ $$\nu^{7}$$ $$=$$ $$3 \beta_{5} - 7 \beta_{4}$$

## Character values

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

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$-1$$ $$-\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 1.10362 − 0.884319i −1.41264 − 0.0667372i 1.10362 + 0.884319i −1.41264 + 0.0667372i −0.373058 − 1.36412i 1.18208 + 0.776336i −0.373058 + 1.36412i 1.18208 − 0.776336i
−2.59471 + 0.843073i 0 4.40373 3.19950i 0 0 1.55513 + 2.14046i −5.52176 + 7.60006i 0.927051 + 2.85317i 0
6.2 1.47668 0.479802i 0 0.332338 0.241457i 0 0 −1.55513 2.14046i −1.45037 + 1.99627i 0.927051 + 2.85317i 0
13.1 −2.59471 0.843073i 0 4.40373 + 3.19950i 0 0 1.55513 2.14046i −5.52176 7.60006i 0.927051 2.85317i 0
13.2 1.47668 + 0.479802i 0 0.332338 + 0.241457i 0 0 −1.55513 + 2.14046i −1.45037 1.99627i 0.927051 2.85317i 0
41.1 0.0784543 + 0.107983i 0 0.612529 1.88517i 0 0 2.51626 + 0.817582i 0.505505 0.164249i −2.42705 + 1.76336i 0
41.2 1.03958 + 1.43086i 0 −0.348597 + 1.07287i 0 0 −2.51626 0.817582i 1.46663 0.476537i −2.42705 + 1.76336i 0
62.1 0.0784543 0.107983i 0 0.612529 + 1.88517i 0 0 2.51626 0.817582i 0.505505 + 0.164249i −2.42705 1.76336i 0
62.2 1.03958 1.43086i 0 −0.348597 1.07287i 0 0 −2.51626 + 0.817582i 1.46663 + 0.476537i −2.42705 1.76336i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 62.2 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.d odd 10 1 inner
77.l even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.l.a 8
3.b odd 2 1 693.2.bu.a 8
7.b odd 2 1 CM 77.2.l.a 8
7.c even 3 2 539.2.s.a 16
7.d odd 6 2 539.2.s.a 16
11.b odd 2 1 847.2.l.c 8
11.c even 5 1 847.2.b.b 8
11.c even 5 1 847.2.l.a 8
11.c even 5 1 847.2.l.c 8
11.c even 5 1 847.2.l.d 8
11.d odd 10 1 inner 77.2.l.a 8
11.d odd 10 1 847.2.b.b 8
11.d odd 10 1 847.2.l.a 8
11.d odd 10 1 847.2.l.d 8
21.c even 2 1 693.2.bu.a 8
33.f even 10 1 693.2.bu.a 8
77.b even 2 1 847.2.l.c 8
77.j odd 10 1 847.2.b.b 8
77.j odd 10 1 847.2.l.a 8
77.j odd 10 1 847.2.l.c 8
77.j odd 10 1 847.2.l.d 8
77.l even 10 1 inner 77.2.l.a 8
77.l even 10 1 847.2.b.b 8
77.l even 10 1 847.2.l.a 8
77.l even 10 1 847.2.l.d 8
77.n even 30 2 539.2.s.a 16
77.o odd 30 2 539.2.s.a 16
231.r odd 10 1 693.2.bu.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.l.a 8 1.a even 1 1 trivial
77.2.l.a 8 7.b odd 2 1 CM
77.2.l.a 8 11.d odd 10 1 inner
77.2.l.a 8 77.l even 10 1 inner
539.2.s.a 16 7.c even 3 2
539.2.s.a 16 7.d odd 6 2
539.2.s.a 16 77.n even 30 2
539.2.s.a 16 77.o odd 30 2
693.2.bu.a 8 3.b odd 2 1
693.2.bu.a 8 21.c even 2 1
693.2.bu.a 8 33.f even 10 1
693.2.bu.a 8 231.r odd 10 1
847.2.b.b 8 11.c even 5 1
847.2.b.b 8 11.d odd 10 1
847.2.b.b 8 77.j odd 10 1
847.2.b.b 8 77.l even 10 1
847.2.l.a 8 11.c even 5 1
847.2.l.a 8 11.d odd 10 1
847.2.l.a 8 77.j odd 10 1
847.2.l.a 8 77.l even 10 1
847.2.l.c 8 11.b odd 2 1
847.2.l.c 8 11.c even 5 1
847.2.l.c 8 77.b even 2 1
847.2.l.c 8 77.j odd 10 1
847.2.l.d 8 11.c even 5 1
847.2.l.d 8 11.d odd 10 1
847.2.l.d 8 77.j odd 10 1
847.2.l.d 8 77.l even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 7 T_{2}^{6} + 10 T_{2}^{5} + 19 T_{2}^{4} - 70 T_{2}^{3} + 67 T_{2}^{2} - 10 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(77, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 10 T + 67 T^{2} - 70 T^{3} + 19 T^{4} + 10 T^{5} - 7 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$2401 - 343 T^{2} + 49 T^{4} - 7 T^{6} + T^{8}$$
$11$ $$14641 + 5324 T + 605 T^{2} - 264 T^{3} - 151 T^{4} - 24 T^{5} + 5 T^{6} + 4 T^{7} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$( -619 + 408 T - 51 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$29$ $$259081 - 147610 T + 81757 T^{2} + 32480 T^{3} + 4714 T^{4} - 290 T^{5} - 112 T^{6} + T^{8}$$
$31$ $$T^{8}$$
$37$ $$2193361 - 2328132 T + 905868 T^{2} + 91026 T^{3} + 20350 T^{4} + 2436 T^{5} + 293 T^{6} + 18 T^{7} + T^{8}$$
$41$ $$T^{8}$$
$43$ $$16072081 + 2478498 T^{2} + 53459 T^{4} + 402 T^{6} + T^{8}$$
$47$ $$T^{8}$$
$53$ $$6027025 - 5646500 T + 2231500 T^{2} - 256350 T^{3} + 54590 T^{4} - 6300 T^{5} + 565 T^{6} - 30 T^{7} + T^{8}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$( 17341 + 1276 T - 319 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$71$ $$19321 - 128992 T + 2437888 T^{2} + 804336 T^{3} + 141290 T^{4} + 15456 T^{5} + 1123 T^{6} + 48 T^{7} + T^{8}$$
$73$ $$T^{8}$$
$79$ $$23338561 + 18164560 T + 4496132 T^{2} + 324040 T^{3} + 37254 T^{4} + 7680 T^{5} + 783 T^{6} + 40 T^{7} + T^{8}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$