Properties

Label 116.1.j.a
Level 116
Weight 1
Character orbit 116.j
Analytic conductor 0.058
Analytic rank 0
Dimension 6
Projective image \(D_{7}\)
CM disc. -4
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 116 = 2^{2} \cdot 29 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 116.j (of order \(14\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0578915414654\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.38068692544.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{14} q^{2} \) \( + \zeta_{14}^{2} q^{4} \) \( + ( \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{5} \) \( -\zeta_{14}^{3} q^{8} \) \( -\zeta_{14}^{3} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{14} q^{2} \) \( + \zeta_{14}^{2} q^{4} \) \( + ( \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{5} \) \( -\zeta_{14}^{3} q^{8} \) \( -\zeta_{14}^{3} q^{9} \) \( + ( -\zeta_{14}^{5} + \zeta_{14}^{6} ) q^{10} \) \( + ( \zeta_{14}^{2} + \zeta_{14}^{6} ) q^{13} \) \( + \zeta_{14}^{4} q^{16} \) \( + ( -\zeta_{14} + \zeta_{14}^{6} ) q^{17} \) \( + \zeta_{14}^{4} q^{18} \) \( + ( 1 + \zeta_{14}^{6} ) q^{20} \) \( + ( -\zeta_{14} + \zeta_{14}^{2} - \zeta_{14}^{3} ) q^{25} \) \( + ( 1 - \zeta_{14}^{3} ) q^{26} \) \( + \zeta_{14}^{4} q^{29} \) \( -\zeta_{14}^{5} q^{32} \) \( + ( 1 + \zeta_{14}^{2} ) q^{34} \) \( -\zeta_{14}^{5} q^{36} \) \( + ( -\zeta_{14} - \zeta_{14}^{5} ) q^{37} \) \( + ( 1 - \zeta_{14} ) q^{40} \) \( + ( \zeta_{14}^{2} - \zeta_{14}^{5} ) q^{41} \) \( + ( 1 - \zeta_{14} ) q^{45} \) \( -\zeta_{14}^{3} q^{49} \) \( + ( \zeta_{14}^{2} - \zeta_{14}^{3} + \zeta_{14}^{4} ) q^{50} \) \( + ( -\zeta_{14} + \zeta_{14}^{4} ) q^{52} \) \( + ( 1 + \zeta_{14}^{2} ) q^{53} \) \( -\zeta_{14}^{5} q^{58} \) \( + ( \zeta_{14}^{4} + \zeta_{14}^{6} ) q^{61} \) \( + \zeta_{14}^{6} q^{64} \) \( + ( 1 - \zeta_{14}^{3} + \zeta_{14}^{4} + \zeta_{14}^{6} ) q^{65} \) \( + ( -\zeta_{14} - \zeta_{14}^{3} ) q^{68} \) \( + \zeta_{14}^{6} q^{72} \) \( + ( 1 - \zeta_{14}^{5} ) q^{73} \) \( + ( \zeta_{14}^{2} + \zeta_{14}^{6} ) q^{74} \) \( + ( -\zeta_{14} + \zeta_{14}^{2} ) q^{80} \) \( + \zeta_{14}^{6} q^{81} \) \( + ( -\zeta_{14}^{3} + \zeta_{14}^{6} ) q^{82} \) \( + ( -\zeta_{14}^{3} + \zeta_{14}^{4} - \zeta_{14}^{5} + \zeta_{14}^{6} ) q^{85} \) \( + ( \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{89} \) \( + ( -\zeta_{14} + \zeta_{14}^{2} ) q^{90} \) \( + ( 1 + \zeta_{14}^{4} ) q^{97} \) \( + \zeta_{14}^{4} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut q^{8} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut q^{8} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut -\mathstrut 2q^{10} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut q^{18} \) \(\mathstrut +\mathstrut 5q^{20} \) \(\mathstrut -\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 5q^{26} \) \(\mathstrut -\mathstrut q^{29} \) \(\mathstrut -\mathstrut q^{32} \) \(\mathstrut +\mathstrut 5q^{34} \) \(\mathstrut -\mathstrut q^{36} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 5q^{40} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 5q^{45} \) \(\mathstrut -\mathstrut q^{49} \) \(\mathstrut -\mathstrut 3q^{50} \) \(\mathstrut -\mathstrut 2q^{52} \) \(\mathstrut +\mathstrut 5q^{53} \) \(\mathstrut -\mathstrut q^{58} \) \(\mathstrut -\mathstrut 2q^{61} \) \(\mathstrut -\mathstrut q^{64} \) \(\mathstrut +\mathstrut 3q^{65} \) \(\mathstrut -\mathstrut 2q^{68} \) \(\mathstrut -\mathstrut q^{72} \) \(\mathstrut +\mathstrut 5q^{73} \) \(\mathstrut -\mathstrut 2q^{74} \) \(\mathstrut -\mathstrut 2q^{80} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut -\mathstrut 2q^{82} \) \(\mathstrut -\mathstrut 4q^{85} \) \(\mathstrut -\mathstrut 2q^{89} \) \(\mathstrut -\mathstrut 2q^{90} \) \(\mathstrut +\mathstrut 5q^{97} \) \(\mathstrut -\mathstrut q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\) \(59\) \(89\)
\(\chi(n)\) \(-1\) \(\zeta_{14}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.222521 + 0.974928i
−0.623490 + 0.781831i
0.222521 0.974928i
0.900969 + 0.433884i
0.900969 0.433884i
−0.623490 0.781831i
−0.222521 0.974928i 0 −0.900969 + 0.433884i −0.277479 1.21572i 0 0 0.623490 + 0.781831i 0.623490 + 0.781831i −1.12349 + 0.541044i
23.1 0.623490 0.781831i 0 −0.222521 0.974928i −1.12349 + 1.40881i 0 0 −0.900969 0.433884i −0.900969 0.433884i 0.400969 + 1.75676i
83.1 −0.222521 + 0.974928i 0 −0.900969 0.433884i −0.277479 + 1.21572i 0 0 0.623490 0.781831i 0.623490 0.781831i −1.12349 0.541044i
103.1 −0.900969 0.433884i 0 0.623490 + 0.781831i 0.400969 + 0.193096i 0 0 −0.222521 0.974928i −0.222521 0.974928i −0.277479 0.347948i
107.1 −0.900969 + 0.433884i 0 0.623490 0.781831i 0.400969 0.193096i 0 0 −0.222521 + 0.974928i −0.222521 + 0.974928i −0.277479 + 0.347948i
111.1 0.623490 + 0.781831i 0 −0.222521 + 0.974928i −1.12349 1.40881i 0 0 −0.900969 + 0.433884i −0.900969 + 0.433884i 0.400969 1.75676i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.1
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
29.d Even 1 yes
116.j Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(116, [\chi])\).