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 disc. -7
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.0384280059727\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.717409.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

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:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.b Odd 1 CM by \(\Q(\sqrt{-7}) \) yes
11.c Even 1 yes
77.j Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(77, [\chi])\).