LMFDB - Newform orbit 114.2.i.c

Newspace parameters

Level:	$ N $	$=$	$ 114 = 2 \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	114.i (of order $9$, degree $6$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.910294583043$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/114\mathbb{Z}\right)^\times$.

$n$	$77$	$97$
$\chi(n)$	$1$	$\zeta_{18}^{2}$

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} $

25.1

−0.766044	+	0.642788i
0.939693	−	0.342020i
−0.173648	+	0.984808i
0.939693	+	0.342020i
−0.766044	−	0.642788i
−0.173648	−	0.984808i

0.766044

−

0.642788i

0.939693

0.342020i

0.173648

−

0.984808i

−0.613341

−

3.47843i

0.939693

−

0.342020i

−1.85844

3.21891i

−0.500000

−

0.866025i

0.766044

0.642788i

−2.70574

−

2.27038i

43.1

−0.939693

0.342020i

−0.173648

0.984808i

0.766044

−

0.642788i

−0.0923963

−

0.0775297i

−0.173648

−

0.984808i

2.14543

3.71599i

−0.500000

0.866025i

−0.939693

−

0.342020i

0.113341

0.0412527i

55.1

0.173648

−

0.984808i

−0.766044

−

0.642788i

−0.939693

−

0.342020i

2.20574

−

0.802823i

−0.766044

0.642788i

−1.78699

−

3.09516i

−0.500000

0.866025i

0.173648

0.984808i

−0.407604

−

2.31164i

61.1

−0.939693

−

0.342020i

−0.173648

−

0.984808i

0.766044

0.642788i

−0.0923963

0.0775297i

−0.173648

0.984808i

2.14543

−

3.71599i

−0.500000

−

0.866025i

−0.939693

0.342020i

0.113341

−

0.0412527i

73.1

0.766044

0.642788i

0.939693

−

0.342020i

0.173648

0.984808i

−0.613341

3.47843i

0.939693

0.342020i

−1.85844

−

3.21891i

−0.500000

0.866025i

0.766044

−

0.642788i

−2.70574

2.27038i

85.1

0.173648

0.984808i

−0.766044

0.642788i

−0.939693

0.342020i

2.20574

0.802823i

−0.766044

−

0.642788i

−1.78699

3.09516i

−0.500000

−

0.866025i

0.173648

−

0.984808i

−0.407604

2.31164i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	114.2.i.c	✓	6
3.b	odd	2	1		342.2.u.b		6
4.b	odd	2	1		912.2.bo.d		6
19.e	even	9	1	inner	114.2.i.c	✓	6
19.e	even	9	1		2166.2.a.r		3
19.f	odd	18	1		2166.2.a.p		3
57.j	even	18	1		6498.2.a.bu		3
57.l	odd	18	1		342.2.u.b		6
57.l	odd	18	1		6498.2.a.bp		3
76.l	odd	18	1		912.2.bo.d		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
114.2.i.c	✓	6	1.a	even	1	1	trivial
114.2.i.c	✓	6	19.e	even	9	1	inner
342.2.u.b		6	3.b	odd	2	1
342.2.u.b		6	57.l	odd	18	1
912.2.bo.d		6	4.b	odd	2	1
912.2.bo.d		6	76.l	odd	18	1
2166.2.a.p		3	19.f	odd	18	1
2166.2.a.r		3	19.e	even	9	1
6498.2.a.bp		3	57.l	odd	18	1
6498.2.a.bu		3	57.j	even	18	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} - 3T_{5}^{5} + 12T_{5}^{4} - 46T_{5}^{3} + 60T_{5}^{2} + 12T_{5} + 1 $ T5^6 - 3*T5^5 + 12*T5^4 - 46*T5^3 + 60*T5^2 + 12*T5 + 1 acting on $S_{2}^{\mathrm{new}}(114, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + T^{3} + 1 $ T^6 + T^3 + 1
$3$	$ T^{6} - T^{3} + 1 $ T^6 - T^3 + 1
$5$	$ T^{6} - 3 T^{5} + 12 T^{4} - 46 T^{3} + \cdots + 1 $ T^6 - 3T^5 + 12T^4 - 46T^3 + 60T^2 + 12*T + 1
$7$	$ T^{6} + 3 T^{5} + 27 T^{4} + \cdots + 3249 $ T^6 + 3T^5 + 27T^4 + 60T^3 + 495T^2 + 1026*T + 3249
$11$	$ T^{6} + 21 T^{4} + 74 T^{3} + \cdots + 1369 $ T^6 + 21T^4 + 74T^3 + 441T^2 + 777T + 1369
$13$	$ T^{6} - 9 T^{5} + 36 T^{4} - 28 T^{3} + \cdots + 1 $ T^6 - 9T^5 + 36T^4 - 28T^3 + 27T^2 - 9*T + 1
$17$	$ T^{6} + 3 T^{5} + 18 T^{4} + 24 T^{3} + \cdots + 9 $ T^6 + 3T^5 + 18T^4 + 24T^3 + 9T^2 - 27*T + 9
$19$	$ T^{6} + 107T^{3} + 6859 $ T^6 + 107*T^3 + 6859
$23$	$ T^{6} - 27 T^{5} + 378 T^{4} + \cdots + 157609 $ T^6 - 27T^5 + 378T^4 - 3556T^3 + 22626T^2 - 85752*T + 157609
$29$	$ T^{6} + 3 T^{5} + 12 T^{4} + 46 T^{3} + \cdots + 1 $ T^6 + 3T^5 + 12T^4 + 46T^3 + 60T^2 - 12*T + 1
$31$	$ T^{6} + 15 T^{5} + 153 T^{4} + \cdots + 11881 $ T^6 + 15T^5 + 153T^4 + 862T^3 + 3549T^2 + 7848*T + 11881
$37$	$ (T^{3} + 3 T^{2} - 45 T + 17)^{2} $ (T^3 + 3T^2 - 45T + 17)^2
$41$	$ T^{6} + 15 T^{5} + 87 T^{4} + \cdots + 2809 $ T^6 + 15T^5 + 87T^4 + 316T^3 + 1308T^2 + 2703*T + 2809
$43$	$ T^{6} - 3 T^{5} - 3 T^{4} + \cdots + 63001 $ T^6 - 3T^5 - 3T^4 + 280T^3 + 2406T^2 + 17319*T + 63001
$47$	$ T^{6} - 15 T^{5} + 51 T^{4} + \cdots + 5041 $ T^6 - 15T^5 + 51T^4 + 80T^3 + 570T^2 + 1491*T + 5041
$53$	$ T^{6} - 6 T^{5} - 30 T^{4} + \cdots + 395641 $ T^6 - 6T^5 - 30T^4 + 1205T^3 + 6186T^2 + 18870*T + 395641
$59$	$ T^{6} + 27 T^{5} + 198 T^{4} + \cdots + 81 $ T^6 + 27T^5 + 198T^4 - 72T^3 + 1701T^2 - 567*T + 81
$61$	$ T^{6} + 15 T^{5} + 84 T^{4} + \cdots + 289 $ T^6 + 15T^5 + 84T^4 + 215T^3 + 375T^2 + 408*T + 289
$67$	$ T^{6} + 3 T^{5} + 72 T^{4} + \cdots + 7017201 $ T^6 + 3T^5 + 72T^4 + 834T^3 + 10890T^2 + 619866*T + 7017201
$71$	$ T^{6} - 3 T^{5} - 87 T^{4} + \cdots + 26569 $ T^6 - 3T^5 - 87T^4 + 1304T^3 + 14946T^2 + 9291*T + 26569
$73$	$ T^{6} - 12 T^{5} + 54 T^{4} + \cdots + 3249 $ T^6 - 12T^5 + 54T^4 + 300T^3 + 522T^2 + 1539*T + 3249
$79$	$ T^{6} - 27 T^{5} + 351 T^{4} + \cdots + 32761 $ T^6 - 27T^5 + 351T^4 - 2152T^3 + 9918T^2 - 27693*T + 32761
$83$	$ T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 2601 $ T^6 - 3T^5 + 45T^4 + 210T^3 + 1143T^2 + 1836*T + 2601
$89$	$ T^{6} - 42 T^{5} + 756 T^{4} + \cdots + 239121 $ T^6 - 42T^5 + 756T^4 - 7104T^3 + 35406T^2 - 92421*T + 239121
$97$	$ T^{6} - 18 T^{5} + 171 T^{4} + \cdots + 218089 $ T^6 - 18T^5 + 171T^4 + 451T^3 + 19332T^2 + 121887*T + 218089
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	114.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{18} q^{2} - \zeta_{18}^{4} q^{3} + \zeta_{18}^{2} q^{4} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{5} + \zeta_{18}^{5} q^{6} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{7} - \zeta_{18}^{3} q^{8} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{9} +O(q^{10}) \) q - z * q^2 - z^4 * q^3 + z^2 * q^4 + (-z^5 + 2z^4 - z^3 + z^2 - 2z + 1) * q^5 + z^5 * q^6 + (-3z^5 - 3z^4 + z^3 + z^2 + 2z - 1) q^7 - z^3 * q^8 + (z^5 - z^2) * q^9	\( q - \zeta_{18} q^{2} - \zeta_{18}^{4} q^{3} + \zeta_{18}^{2} q^{4} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{5} + \zeta_{18}^{5} q^{6} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{7} - \zeta_{18}^{3} q^{8} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{9} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} - \zeta_{18} - 1) q^{10} + (2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{2} + 2 \zeta_{18}) q^{11} + ( - \zeta_{18}^{3} + 1) q^{12} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18} + 1) q^{13} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18} - 3) q^{14} + ( - \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18}) q^{15} + \zeta_{18}^{4} q^{16} + (\zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18}) q^{17} + q^{18} + (2 \zeta_{18}^{4} + 3 \zeta_{18}) q^{19} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18} - 2) q^{20} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18} - 2) q^{21} + (3 \zeta_{18}^{5} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{22} + (6 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 3 \zeta_{18} + 6) q^{23} + (\zeta_{18}^{4} - \zeta_{18}) q^{24} + ( - \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + \zeta_{18} - 4) q^{25} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18} - 2) q^{26} + \zeta_{18}^{3} q^{27} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{28} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2}) q^{29} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2}) q^{30} + (5 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} - 5) q^{31} - \zeta_{18}^{5} q^{32} + (\zeta_{18}^{5} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 1) q^{33} + (\zeta_{18}^{4} - 2 \zeta_{18}^{2} + 1) q^{34} + (4 \zeta_{18}^{5} - 7 \zeta_{18}^{4} + 4 \zeta_{18}^{3}) q^{35} - \zeta_{18} q^{36} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{2} - 4 \zeta_{18} - 1) q^{37} + ( - 2 \zeta_{18}^{5} - 3 \zeta_{18}^{2}) q^{38} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 2) q^{39} + ( - \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{40} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3) q^{41} + (\zeta_{18}^{4} + 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{42} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{43} + ( - \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18} + 3) q^{44} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18} + 2) q^{45} + ( - 6 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 6 \zeta_{18}) q^{46} + ( - 3 \zeta_{18}^{5} - \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4 \zeta_{18} + 4) q^{47} + ( - \zeta_{18}^{5} + \zeta_{18}^{2}) q^{48} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 8 \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18}) q^{49} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + 4 \zeta_{18} - 1) q^{50} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}) q^{51} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{52} + (2 \zeta_{18}^{3} + 9 \zeta_{18}^{2} + 2 \zeta_{18}) q^{53} - \zeta_{18}^{4} q^{54} + (7 \zeta_{18}^{5} - 7 \zeta_{18}^{3} + \zeta_{18}^{2} - 4 \zeta_{18} + 8) q^{55} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{2} - 3 \zeta_{18} + 1) q^{56} + ( - 5 \zeta_{18}^{5} + 2 \zeta_{18}^{2}) q^{57} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2) q^{58} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 6 \zeta_{18} - 4) q^{59} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3}) q^{60} + ( - 3 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} - 3) q^{61} + ( - 5 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 5 \zeta_{18}) q^{62} + ( - \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18} - 2) q^{63} + (\zeta_{18}^{3} - 1) q^{64} + ( - 5 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 5 \zeta_{18}) q^{65} + ( - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18} + 1) q^{66} + (4 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 11 \zeta_{18}^{3} - 4 \zeta_{18}^{2} - 6 \zeta_{18} - 6) q^{67} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{3} - \zeta_{18}) q^{68} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 3 \zeta_{18} + 2) q^{69} + (7 \zeta_{18}^{5} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 4) q^{70} + ( - 3 \zeta_{18}^{5} - 7 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + 7 \zeta_{18} + 3) q^{71} + \zeta_{18}^{2} q^{72} + (\zeta_{18}^{5} + 5 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3) q^{73} + (4 \zeta_{18}^{2} + \zeta_{18} + 4) q^{74} + ( - 4 \zeta_{18}^{5} + \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 1) q^{75} + (5 \zeta_{18}^{3} - 2) q^{76} + ( - 6 \zeta_{18}^{5} + 10 \zeta_{18}^{4} - 4 \zeta_{18}^{2} - 4 \zeta_{18} - 7) q^{77} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{78} + (4 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 5) q^{79} + (\zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18}) q^{80} + ( - \zeta_{18}^{4} + \zeta_{18}) q^{81} + ( - 3 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 3 \zeta_{18} - 2) q^{82} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{3} - 4 \zeta_{18}^{2} + \zeta_{18} + 1) q^{83} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18}) q^{84} + (5 \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{85} + (3 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 3 \zeta_{18} - 3) q^{86} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18}) q^{87} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 3 \zeta_{18}) q^{88} + ( - \zeta_{18}^{5} - 7 \zeta_{18} + 7) q^{89} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{90} + ( - 8 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + 2 \zeta_{18} - 8) q^{91} + ( - 3 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 6) q^{92} + (\zeta_{18}^{5} - \zeta_{18}^{3} + 5 \zeta_{18} + 1) q^{93} + (\zeta_{18}^{5} + 3 \zeta_{18}^{4} - 4 \zeta_{18}^{2} - 4 \zeta_{18} - 3) q^{94} + (6 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 10 \zeta_{18}^{2} + 5 \zeta_{18} + 3) q^{95} - q^{96} + (8 \zeta_{18}^{5} - 8 \zeta_{18}^{3} - \zeta_{18}^{2} - 4 \zeta_{18} + 7) q^{97} + (\zeta_{18}^{5} + 8 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{98} + ( - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 3 \zeta_{18} - 2) q^{99} +O(q^{100}) \) q - z * q^2 - z^4 * q^3 + z^2 * q^4 + (-z^5 + 2z^4 - z^3 + z^2 - 2z + 1) * q^5 + z^5 * q^6 + (-3z^5 - 3z^4 + z^3 + z^2 + 2z - 1) q^7 - z^3 * q^8 + (z^5 - z^2) * q^9 + (-2z^5 + z^4 + 2z^2 - z - 1) * q^10 + (2z^5 - 3z^4 - 3z^2 + 2z) * q^11 + (-z^3 + 1) * q^12 + (-2z^5 + z^4 + z^3 - 2z + 1) * q^13 + (3z^5 - z^4 + 2z^3 - 2z^2 + z - 3) q^14 + (-z^3 + 2z^2 - z) q^15 + z^4 * q^16 + (z^5 - z^3 - z^2 + 2z) q^17 + q^18 + (2z^4 + 3z) * q^19 + (-z^5 + z^2 + z - 2) * q^20 + (z^5 - z^3 - 3z^2 + z - 2) q^21 + (3z^5 + z^3 - 2z^2 + 2) * q^22 + (6z^4 - 3z^3 + 2z^2 - 3z + 6) * q^23 + (z^4 - z) * q^24 + (-z^5 + 3z^4 + 3z^3 + z - 4) * q^25 + (-z^5 - z^4 + 2z^3 + 2z^2 - z - 2) * q^26 + z^3 * q^27 + (z^5 - 2z^4 - z^3 - z^2 + 3z + 3) * q^28 + (2z^5 + z^4 - z^3 - 2z^2) * q^29 + (z^4 - 2z^3 + z^2) q^30 + (5z^3 + z^2 - z - 5) q^31 - z^5 * q^32 + (z^5 + 3z^3 - 3z^2 - 1) * q^33 + (z^4 - 2z^2 + 1) q^34 + (4z^5 - 7z^4 + 4z^3) q^35 - z * q^36 + (4z^5 - 4z^2 - 4z - 1) q^37 + (-2z^5 - 3z^2) * q^38 + (z^5 - 2z^4 + z^2 + z - 2) q^39 + (-z^2 + 2z - 1) q^40 + (-2z^5 + 3z^4 + z^3 + 3z^2 - 3) q^41 + (z^4 + 2z^3 - z^2 + 2z + 1) * q^42 + (-3z^5 - 3z^4 - 5z^3 + 5z^2 + 3z + 3) q^43 + (-z^4 - z^3 - 2z + 3) q^44 + (z^5 + z^4 - 2z^3 - z + 2) q^45 + (-6z^5 + 3z^4 - 2z^3 + 3z^2 - 6z) q^46 + (-3z^5 - z^4 - 3z^3 + 3z^2 + 4z + 4) * q^47 + (-z^5 + z^2) * q^48 + (-z^5 - z^4 - 8z^3 - z^2 - z) q^49 + (-3z^5 - 3z^4 + z^3 - z^2 + 4z - 1) q^50 + (-2z^5 + z^4 + z^3 - z) q^51 + (z^5 - 2z^4 - z^3 + z^2 + 2z - 1) * q^52 + (2z^3 + 9z^2 + 2z) q^53 - z^4 * q^54 + (7z^5 - 7z^3 + z^2 - 4z + 8) q^55 + (2z^5 + z^4 - 3z^2 - 3z + 1) q^56 + (-5z^5 + 2z^2) * q^57 + (-z^5 + z^4 + 2) * q^58 + (z^5 - z^3 - 5z^2 - 6z - 4) * q^59 + (-z^5 + 2z^4 - z^3) q^60 + (-3z^4 + z^3 + z^2 + z - 3) q^61 + (-5z^4 - z^3 + z^2 + 5z) * q^62 + (-z^5 + 3z^4 + 3z^3 - z - 2) * q^63 + (z^3 - 1) * q^64 + (-5z^5 + 2z^4 + 4z^3 + 2z^2 - 5z) q^65 + (-3z^4 + 2z^3 + z + 1) * q^66 + (4z^5 - 5z^4 + 11z^3 - 4z^2 - 6z - 6) q^67 + (-z^5 + 2z^3 - z) q^68 + (-3z^5 - 3z^4 - 2z^3 + 6z^2 - 3z + 2) q^69 + (7z^5 - 4z^4 - 4z^3 + 4) q^70 + (-3z^5 - 7z^4 - 5z^3 + 5z^2 + 7z + 3) q^71 + z^2 * q^72 + (z^5 + 5z^4 - 2z^3 - 3z^2 + 3) q^73 + (4z^2 + z + 4) q^74 + (-4z^5 + z^4 + 3z^2 + 3z - 1) q^75 + (5z^3 - 2) q^76 + (-6z^5 + 10z^4 - 4z^2 - 4z - 7) * q^77 + (2z^5 - 2z^3 - z^2 + 2z + 1) q^78 + (4z^5 + 5z^4 - z^3 - 5z^2 + 5) q^79 + (z^3 - 2z^2 + z) q^80 + (-z^4 + z) * q^81 + (-3z^5 - z^4 - z^3 + 3z - 2) * q^82 + (3z^5 + 3z^4 - z^3 - 4z^2 + z + 1) q^83 + (-z^5 - 2z^4 + z^3 - 2z^2 - z) * q^84 + (5z^5 - z^4 - 2z^3 - 5z^2 + 3z + 3) * q^85 + (3z^5 + 5z^4 - 2z^3 - 3z^2 - 3z - 3) q^86 + (-z^5 + z^4 + 2z^3 + z^2 - z) q^87 + (z^5 + z^4 + 2z^2 - 3z) * q^88 + (-z^5 - 7z + 7) q^89 + (-z^5 + 2z^4 - z^3 + z^2 - 2z + 1) * q^90 + (-8z^4 + 2z^3 + 5z^2 + 2z - 8) * q^91 + (-3z^5 + 2z^4 + 3z^3 + 6z^2 - 6) * q^92 + (z^5 - z^3 + 5z + 1) q^93 + (z^5 + 3z^4 - 4z^2 - 4z - 3) q^94 + (6z^5 - 3z^4 + 2z^3 - 10z^2 + 5z + 3) q^95 - q^96 + (8z^5 - 8z^3 - z^2 - 4z + 7) q^97 + (z^5 + 8z^4 + 2z^3 + z^2 - 1) * q^98 + (-2z^4 + 3z^3 + 3z - 2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 6 * q + 3 * q^5 - 3 * q^7 - 3 * q^8	\( 6 q + 3 q^{5} - 3 q^{7} - 3 q^{8} - 6 q^{10} + 3 q^{12} + 9 q^{13} - 12 q^{14} - 3 q^{15} - 3 q^{17} + 6 q^{18} - 12 q^{20} - 15 q^{21} + 15 q^{22} + 27 q^{23} - 15 q^{25} - 6 q^{26} + 3 q^{27} + 15 q^{28} - 3 q^{29} - 6 q^{30} - 15 q^{31} + 3 q^{33} + 6 q^{34} + 12 q^{35} - 6 q^{37} - 12 q^{39} - 6 q^{40} - 15 q^{41} + 12 q^{42} + 3 q^{43} + 15 q^{44} + 6 q^{45} - 6 q^{46} + 15 q^{47} - 24 q^{49} - 3 q^{50} + 3 q^{51} - 9 q^{52} + 6 q^{53} + 27 q^{55} + 6 q^{56} + 12 q^{58} - 27 q^{59} - 3 q^{60} - 15 q^{61} - 3 q^{62} - 3 q^{63} - 3 q^{64} + 12 q^{65} + 12 q^{66} - 3 q^{67} + 6 q^{68} + 6 q^{69} + 12 q^{70} + 3 q^{71} + 12 q^{73} + 24 q^{74} - 6 q^{75} + 3 q^{76} - 42 q^{77} + 27 q^{79} + 3 q^{80} - 15 q^{82} + 3 q^{83} + 3 q^{84} + 12 q^{85} - 24 q^{86} + 6 q^{87} + 42 q^{89} + 3 q^{90} - 42 q^{91} - 27 q^{92} + 3 q^{93} - 18 q^{94} + 24 q^{95} - 6 q^{96} + 18 q^{97} - 3 q^{99}+O(q^{100}) \) 6 * q + 3 * q^5 - 3 * q^7 - 3 * q^8 - 6 * q^10 + 3 * q^12 + 9 * q^13 - 12 * q^14 - 3 * q^15 - 3 * q^17 + 6 * q^18 - 12 * q^20 - 15 * q^21 + 15 * q^22 + 27 * q^23 - 15 * q^25 - 6 * q^26 + 3 * q^27 + 15 * q^28 - 3 * q^29 - 6 * q^30 - 15 * q^31 + 3 * q^33 + 6 * q^34 + 12 * q^35 - 6 * q^37 - 12 * q^39 - 6 * q^40 - 15 * q^41 + 12 * q^42 + 3 * q^43 + 15 * q^44 + 6 * q^45 - 6 * q^46 + 15 * q^47 - 24 * q^49 - 3 * q^50 + 3 * q^51 - 9 * q^52 + 6 * q^53 + 27 * q^55 + 6 * q^56 + 12 * q^58 - 27 * q^59 - 3 * q^60 - 15 * q^61 - 3 * q^62 - 3 * q^63 - 3 * q^64 + 12 * q^65 + 12 * q^66 - 3 * q^67 + 6 * q^68 + 6 * q^69 + 12 * q^70 + 3 * q^71 + 12 * q^73 + 24 * q^74 - 6 * q^75 + 3 * q^76 - 42 * q^77 + 27 * q^79 + 3 * q^80 - 15 * q^82 + 3 * q^83 + 3 * q^84 + 12 * q^85 - 24 * q^86 + 6 * q^87 + 42 * q^89 + 3 * q^90 - 42 * q^91 - 27 * q^92 + 3 * q^93 - 18 * q^94 + 24 * q^95 - 6 * q^96 + 18 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	114.2.i.c

Level	$114$
Weight	$2$
Character orbit	114.i
Analytic conductor	$0.910$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.910294583043\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{18})\)
Defining polynomial:	\( x^{6} - x^{3} + 1 \) x^6 - x^3 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 85.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} + T^{3} + 1 \) T^6 + T^3 + 1
$3$	\( T^{6} - T^{3} + 1 \) T^6 - T^3 + 1
$5$	\( T^{6} - 3 T^{5} + 12 T^{4} - 46 T^{3} + \cdots + 1 \) T^6 - 3T^5 + 12T^4 - 46T^3 + 60T^2 + 12*T + 1
$7$	\( T^{6} + 3 T^{5} + 27 T^{4} + \cdots + 3249 \) T^6 + 3T^5 + 27T^4 + 60T^3 + 495T^2 + 1026*T + 3249
$11$	\( T^{6} + 21 T^{4} + 74 T^{3} + \cdots + 1369 \) T^6 + 21T^4 + 74T^3 + 441T^2 + 777T + 1369
$13$	\( T^{6} - 9 T^{5} + 36 T^{4} - 28 T^{3} + \cdots + 1 \) T^6 - 9T^5 + 36T^4 - 28T^3 + 27T^2 - 9*T + 1
$17$	\( T^{6} + 3 T^{5} + 18 T^{4} + 24 T^{3} + \cdots + 9 \) T^6 + 3T^5 + 18T^4 + 24T^3 + 9T^2 - 27*T + 9
$19$	\( T^{6} + 107T^{3} + 6859 \) T^6 + 107*T^3 + 6859
$23$	\( T^{6} - 27 T^{5} + 378 T^{4} + \cdots + 157609 \) T^6 - 27T^5 + 378T^4 - 3556T^3 + 22626T^2 - 85752*T + 157609
$29$	\( T^{6} + 3 T^{5} + 12 T^{4} + 46 T^{3} + \cdots + 1 \) T^6 + 3T^5 + 12T^4 + 46T^3 + 60T^2 - 12*T + 1
$31$	\( T^{6} + 15 T^{5} + 153 T^{4} + \cdots + 11881 \) T^6 + 15T^5 + 153T^4 + 862T^3 + 3549T^2 + 7848*T + 11881
$37$	\( (T^{3} + 3 T^{2} - 45 T + 17)^{2} \) (T^3 + 3T^2 - 45T + 17)^2
$41$	\( T^{6} + 15 T^{5} + 87 T^{4} + \cdots + 2809 \) T^6 + 15T^5 + 87T^4 + 316T^3 + 1308T^2 + 2703*T + 2809
$43$	\( T^{6} - 3 T^{5} - 3 T^{4} + \cdots + 63001 \) T^6 - 3T^5 - 3T^4 + 280T^3 + 2406T^2 + 17319*T + 63001
$47$	\( T^{6} - 15 T^{5} + 51 T^{4} + \cdots + 5041 \) T^6 - 15T^5 + 51T^4 + 80T^3 + 570T^2 + 1491*T + 5041
$53$	\( T^{6} - 6 T^{5} - 30 T^{4} + \cdots + 395641 \) T^6 - 6T^5 - 30T^4 + 1205T^3 + 6186T^2 + 18870*T + 395641
$59$	\( T^{6} + 27 T^{5} + 198 T^{4} + \cdots + 81 \) T^6 + 27T^5 + 198T^4 - 72T^3 + 1701T^2 - 567*T + 81
$61$	\( T^{6} + 15 T^{5} + 84 T^{4} + \cdots + 289 \) T^6 + 15T^5 + 84T^4 + 215T^3 + 375T^2 + 408*T + 289
$67$	\( T^{6} + 3 T^{5} + 72 T^{4} + \cdots + 7017201 \) T^6 + 3T^5 + 72T^4 + 834T^3 + 10890T^2 + 619866*T + 7017201
$71$	\( T^{6} - 3 T^{5} - 87 T^{4} + \cdots + 26569 \) T^6 - 3T^5 - 87T^4 + 1304T^3 + 14946T^2 + 9291*T + 26569
$73$	\( T^{6} - 12 T^{5} + 54 T^{4} + \cdots + 3249 \) T^6 - 12T^5 + 54T^4 + 300T^3 + 522T^2 + 1539*T + 3249
$79$	\( T^{6} - 27 T^{5} + 351 T^{4} + \cdots + 32761 \) T^6 - 27T^5 + 351T^4 - 2152T^3 + 9918T^2 - 27693*T + 32761
$83$	\( T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 2601 \) T^6 - 3T^5 + 45T^4 + 210T^3 + 1143T^2 + 1836*T + 2601
$89$	\( T^{6} - 42 T^{5} + 756 T^{4} + \cdots + 239121 \) T^6 - 42T^5 + 756T^4 - 7104T^3 + 35406T^2 - 92421*T + 239121
$97$	\( T^{6} - 18 T^{5} + 171 T^{4} + \cdots + 218089 \) T^6 - 18T^5 + 171T^4 + 451T^3 + 19332T^2 + 121887*T + 218089
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\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(\zeta_{18}^{2}\)