# Properties

 Label 77.1.j.a Level 77 Weight 1 Character orbit 77.j Analytic conductor 0.038 Analytic rank 0 Dimension 4 Projective image $$D_{5}$$ CM discriminant -7 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$77 = 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 77.j (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.0384280059727$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{5}$$ Projective field Galois closure of 5.1.717409.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{2} + ( -\zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{4} -\zeta_{10} q^{7} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{8} -\zeta_{10}^{3} q^{9} +O(q^{10})$$ $$q + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{2} + ( -\zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{4} -\zeta_{10} q^{7} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{8} -\zeta_{10}^{3} q^{9} + \zeta_{10}^{2} q^{11} + ( 1 - \zeta_{10}^{3} ) q^{14} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{16} + ( 1 + \zeta_{10}^{2} ) q^{18} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{22} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{23} + \zeta_{10}^{4} q^{25} + ( 1 + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{28} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{29} + ( 2 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{32} + ( -\zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{36} + ( 1 + \zeta_{10}^{2} ) q^{37} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{43} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{44} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{46} + \zeta_{10}^{2} q^{49} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{50} + ( 1 - \zeta_{10} ) q^{53} + ( -\zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{56} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{58} + \zeta_{10}^{4} q^{63} + ( 1 - \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} + 2 \zeta_{10}^{4} ) q^{64} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{67} + ( 1 + \zeta_{10}^{4} ) q^{71} + ( 1 - \zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{72} + ( -\zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{4} ) q^{74} -\zeta_{10}^{3} q^{77} + ( 1 - \zeta_{10} ) q^{79} -\zeta_{10} q^{81} + ( 1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{86} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{88} + ( 2 + 2 \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{92} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{98} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 3q^{4} - q^{7} + q^{8} - q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 3q^{4} - q^{7} + q^{8} - q^{9} - q^{11} + 3q^{14} + 3q^{18} - 2q^{22} - 2q^{23} - q^{25} + 2q^{28} - 2q^{29} + 4q^{32} - 3q^{36} + 3q^{37} - 2q^{43} + 2q^{44} + q^{46} - q^{49} - 2q^{50} + 3q^{53} - 4q^{56} + q^{58} - q^{63} - 2q^{64} - 2q^{67} + 3q^{71} + q^{72} - 4q^{74} - q^{77} + 3q^{79} - q^{81} + q^{86} + q^{88} + 4q^{92} - 2q^{98} + 4q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
20.1
 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i
−0.500000 + 1.53884i 0 −1.30902 0.951057i 0 0 −0.809017 0.587785i 0.809017 0.587785i 0.309017 0.951057i 0
27.1 −0.500000 1.53884i 0 −1.30902 + 0.951057i 0 0 −0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i 0
48.1 −0.500000 0.363271i 0 −0.190983 0.587785i 0 0 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i 0
69.1 −0.500000 + 0.363271i 0 −0.190983 + 0.587785i 0 0 0.309017 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i 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.c even 5 1 inner
77.j odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.1.j.a 4
3.b odd 2 1 693.1.br.a 4
4.b odd 2 1 1232.1.cd.a 4
5.b even 2 1 1925.1.bn.a 4
5.c odd 4 2 1925.1.cb.a 8
7.b odd 2 1 CM 77.1.j.a 4
7.c even 3 2 539.1.u.a 8
7.d odd 6 2 539.1.u.a 8
11.b odd 2 1 847.1.j.b 4
11.c even 5 1 inner 77.1.j.a 4
11.c even 5 1 847.1.d.a 2
11.c even 5 2 847.1.j.c 4
11.d odd 10 1 847.1.d.b 2
11.d odd 10 2 847.1.j.a 4
11.d odd 10 1 847.1.j.b 4
21.c even 2 1 693.1.br.a 4
28.d even 2 1 1232.1.cd.a 4
33.h odd 10 1 693.1.br.a 4
35.c odd 2 1 1925.1.bn.a 4
35.f even 4 2 1925.1.cb.a 8
44.h odd 10 1 1232.1.cd.a 4
55.j even 10 1 1925.1.bn.a 4
55.k odd 20 2 1925.1.cb.a 8
77.b even 2 1 847.1.j.b 4
77.j odd 10 1 inner 77.1.j.a 4
77.j odd 10 1 847.1.d.a 2
77.j odd 10 2 847.1.j.c 4
77.l even 10 1 847.1.d.b 2
77.l even 10 2 847.1.j.a 4
77.l even 10 1 847.1.j.b 4
77.m even 15 2 539.1.u.a 8
77.p odd 30 2 539.1.u.a 8
231.u even 10 1 693.1.br.a 4
308.t even 10 1 1232.1.cd.a 4
385.y odd 10 1 1925.1.bn.a 4
385.bk even 20 2 1925.1.cb.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.1.j.a 4 1.a even 1 1 trivial
77.1.j.a 4 7.b odd 2 1 CM
77.1.j.a 4 11.c even 5 1 inner
77.1.j.a 4 77.j odd 10 1 inner
539.1.u.a 8 7.c even 3 2
539.1.u.a 8 7.d odd 6 2
539.1.u.a 8 77.m even 15 2
539.1.u.a 8 77.p odd 30 2
693.1.br.a 4 3.b odd 2 1
693.1.br.a 4 21.c even 2 1
693.1.br.a 4 33.h odd 10 1
693.1.br.a 4 231.u even 10 1
847.1.d.a 2 11.c even 5 1
847.1.d.a 2 77.j odd 10 1
847.1.d.b 2 11.d odd 10 1
847.1.d.b 2 77.l even 10 1
847.1.j.a 4 11.d odd 10 2
847.1.j.a 4 77.l even 10 2
847.1.j.b 4 11.b odd 2 1
847.1.j.b 4 11.d odd 10 1
847.1.j.b 4 77.b even 2 1
847.1.j.b 4 77.l even 10 1
847.1.j.c 4 11.c even 5 2
847.1.j.c 4 77.j odd 10 2
1232.1.cd.a 4 4.b odd 2 1
1232.1.cd.a 4 28.d even 2 1
1232.1.cd.a 4 44.h odd 10 1
1232.1.cd.a 4 308.t even 10 1
1925.1.bn.a 4 5.b even 2 1
1925.1.bn.a 4 35.c odd 2 1
1925.1.bn.a 4 55.j even 10 1
1925.1.bn.a 4 385.y odd 10 1
1925.1.cb.a 8 5.c odd 4 2
1925.1.cb.a 8 35.f even 4 2
1925.1.cb.a 8 55.k odd 20 2
1925.1.cb.a 8 385.bk even 20 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(77, [\chi])$$.

## Hecke characteristic polynomials

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