# 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 $4$

# Related objects

## 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$1 - 3 T + 2 T^{2} + T^{3} + 4 T^{4} + 2 T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$1 - 3 T + 2 T^{2} + T^{3} + 4 T^{4} + 2 T^{5} + T^{6}$$
$17$ $$( -1 - 2 T + T^{2} + T^{3} )^{2}$$
$19$ $$T^{6}$$
$23$ $$T^{6}$$
$29$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$31$ $$T^{6}$$
$37$ $$1 - 3 T + 2 T^{2} + T^{3} + 4 T^{4} + 2 T^{5} + T^{6}$$
$41$ $$( -1 - 2 T + T^{2} + T^{3} )^{2}$$
$43$ $$T^{6}$$
$47$ $$T^{6}$$
$53$ $$1 - 3 T + 9 T^{2} - 13 T^{3} + 11 T^{4} - 5 T^{5} + T^{6}$$
$59$ $$T^{6}$$
$61$ $$1 - 3 T + 2 T^{2} + T^{3} + 4 T^{4} + 2 T^{5} + T^{6}$$
$67$ $$T^{6}$$
$71$ $$T^{6}$$
$73$ $$1 - 3 T + 9 T^{2} - 13 T^{3} + 11 T^{4} - 5 T^{5} + T^{6}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$1 - 3 T + 2 T^{2} + T^{3} + 4 T^{4} + 2 T^{5} + T^{6}$$
$97$ $$1 - 3 T + 9 T^{2} - 13 T^{3} + 11 T^{4} - 5 T^{5} + T^{6}$$