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 discriminant -4
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.0578915414654\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Defining polynomial: \(x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.38068692544.1

$q$-expansion

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
29.d even 7 1 inner
116.j odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 116.1.j.a 6
3.b odd 2 1 1044.1.bb.a 6
4.b odd 2 1 CM 116.1.j.a 6
5.b even 2 1 2900.1.bj.a 6
5.c odd 4 2 2900.1.bd.a 12
8.b even 2 1 1856.1.bh.a 6
8.d odd 2 1 1856.1.bh.a 6
12.b even 2 1 1044.1.bb.a 6
20.d odd 2 1 2900.1.bj.a 6
20.e even 4 2 2900.1.bd.a 12
29.b even 2 1 3364.1.j.d 6
29.c odd 4 2 3364.1.h.e 12
29.d even 7 1 inner 116.1.j.a 6
29.d even 7 1 3364.1.b.c 3
29.d even 7 2 3364.1.j.a 6
29.d even 7 2 3364.1.j.b 6
29.e even 14 1 3364.1.b.b 3
29.e even 14 2 3364.1.j.c 6
29.e even 14 1 3364.1.j.d 6
29.e even 14 2 3364.1.j.e 6
29.f odd 28 2 3364.1.d.a 6
29.f odd 28 4 3364.1.h.c 12
29.f odd 28 4 3364.1.h.d 12
29.f odd 28 2 3364.1.h.e 12
87.j odd 14 1 1044.1.bb.a 6
116.d odd 2 1 3364.1.j.d 6
116.e even 4 2 3364.1.h.e 12
116.h odd 14 1 3364.1.b.b 3
116.h odd 14 2 3364.1.j.c 6
116.h odd 14 1 3364.1.j.d 6
116.h odd 14 2 3364.1.j.e 6
116.j odd 14 1 inner 116.1.j.a 6
116.j odd 14 1 3364.1.b.c 3
116.j odd 14 2 3364.1.j.a 6
116.j odd 14 2 3364.1.j.b 6
116.l even 28 2 3364.1.d.a 6
116.l even 28 4 3364.1.h.c 12
116.l even 28 4 3364.1.h.d 12
116.l even 28 2 3364.1.h.e 12
145.n even 14 1 2900.1.bj.a 6
145.p odd 28 2 2900.1.bd.a 12
232.p odd 14 1 1856.1.bh.a 6
232.s even 14 1 1856.1.bh.a 6
348.s even 14 1 1044.1.bb.a 6
580.v odd 14 1 2900.1.bj.a 6
580.bi even 28 2 2900.1.bd.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.1.j.a 6 1.a even 1 1 trivial
116.1.j.a 6 4.b odd 2 1 CM
116.1.j.a 6 29.d even 7 1 inner
116.1.j.a 6 116.j odd 14 1 inner
1044.1.bb.a 6 3.b odd 2 1
1044.1.bb.a 6 12.b even 2 1
1044.1.bb.a 6 87.j odd 14 1
1044.1.bb.a 6 348.s even 14 1
1856.1.bh.a 6 8.b even 2 1
1856.1.bh.a 6 8.d odd 2 1
1856.1.bh.a 6 232.p odd 14 1
1856.1.bh.a 6 232.s even 14 1
2900.1.bd.a 12 5.c odd 4 2
2900.1.bd.a 12 20.e even 4 2
2900.1.bd.a 12 145.p odd 28 2
2900.1.bd.a 12 580.bi even 28 2
2900.1.bj.a 6 5.b even 2 1
2900.1.bj.a 6 20.d odd 2 1
2900.1.bj.a 6 145.n even 14 1
2900.1.bj.a 6 580.v odd 14 1
3364.1.b.b 3 29.e even 14 1
3364.1.b.b 3 116.h odd 14 1
3364.1.b.c 3 29.d even 7 1
3364.1.b.c 3 116.j odd 14 1
3364.1.d.a 6 29.f odd 28 2
3364.1.d.a 6 116.l even 28 2
3364.1.h.c 12 29.f odd 28 4
3364.1.h.c 12 116.l even 28 4
3364.1.h.d 12 29.f odd 28 4
3364.1.h.d 12 116.l even 28 4
3364.1.h.e 12 29.c odd 4 2
3364.1.h.e 12 29.f odd 28 2
3364.1.h.e 12 116.e even 4 2
3364.1.h.e 12 116.l even 28 2
3364.1.j.a 6 29.d even 7 2
3364.1.j.a 6 116.j odd 14 2
3364.1.j.b 6 29.d even 7 2
3364.1.j.b 6 116.j odd 14 2
3364.1.j.c 6 29.e even 14 2
3364.1.j.c 6 116.h odd 14 2
3364.1.j.d 6 29.b even 2 1
3364.1.j.d 6 29.e even 14 1
3364.1.j.d 6 116.d odd 2 1
3364.1.j.d 6 116.h odd 14 1
3364.1.j.e 6 29.e even 14 2
3364.1.j.e 6 116.h odd 14 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(116, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$7$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$11$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$13$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$17$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$19$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$23$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$29$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$31$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$37$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$41$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$43$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$47$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$53$ \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$59$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$61$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$67$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$71$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$73$ \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$79$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$83$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$89$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$97$ \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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