LMFDB - Newform orbit 114.2.i.b

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

−0.0812519

−

0.460802i

−0.939693

0.342020i

2.20574

−

3.82045i

−0.500000

−

0.866025i

0.766044

0.642788i

−0.358441

−

0.300767i

43.1

−0.939693

0.342020i

0.173648

−

0.984808i

0.766044

−

0.642788i

−2.97178

−

2.49362i

0.173648

0.984808i

−0.613341

−

1.06234i

−0.500000

0.866025i

−0.939693

−

0.342020i

3.64543

1.32683i

55.1

0.173648

−

0.984808i

0.766044

0.642788i

−0.939693

−

0.342020i

1.55303

−

0.565258i

0.766044

−

0.642788i

−0.0923963

−

0.160035i

−0.500000

0.866025i

0.173648

0.984808i

−0.286989

−

1.62760i

61.1

−0.939693

−

0.342020i

0.173648

0.984808i

0.766044

0.642788i

−2.97178

2.49362i

0.173648

−

0.984808i

−0.613341

1.06234i

−0.500000

−

0.866025i

−0.939693

0.342020i

3.64543

−

1.32683i

73.1

0.766044

0.642788i

−0.939693

0.342020i

0.173648

0.984808i

−0.0812519

0.460802i

−0.939693

−

0.342020i

2.20574

3.82045i

−0.500000

0.866025i

0.766044

−

0.642788i

−0.358441

0.300767i

85.1

0.173648

0.984808i

0.766044

−

0.642788i

−0.939693

0.342020i

1.55303

0.565258i

0.766044

0.642788i

−0.0923963

0.160035i

−0.500000

−

0.866025i

0.173648

−

0.984808i

−0.286989

1.62760i

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.b	✓	6
3.b	odd	2	1		342.2.u.d		6
4.b	odd	2	1		912.2.bo.c		6
19.e	even	9	1	inner	114.2.i.b	✓	6
19.e	even	9	1		2166.2.a.t		3
19.f	odd	18	1		2166.2.a.n		3
57.j	even	18	1		6498.2.a.bt		3
57.l	odd	18	1		342.2.u.d		6
57.l	odd	18	1		6498.2.a.bo		3
76.l	odd	18	1		912.2.bo.c		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
114.2.i.b	✓	6	1.a	even	1	1	trivial
114.2.i.b	✓	6	19.e	even	9	1	inner
342.2.u.d		6	3.b	odd	2	1
342.2.u.d		6	57.l	odd	18	1
912.2.bo.c		6	4.b	odd	2	1
912.2.bo.c		6	76.l	odd	18	1
2166.2.a.n		3	19.f	odd	18	1
2166.2.a.t		3	19.e	even	9	1
6498.2.a.bo		3	57.l	odd	18	1
6498.2.a.bt		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} - 30T_{5}^{3} + 36T_{5}^{2} + 9 $ T5^6 + 3*T5^5 - 30*T5^3 + 36*T5^2 + 9 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} - 30 T^{3} + 36 T^{2} + \cdots + 9 $ T^6 + 3T^5 - 30T^3 + 36*T^2 + 9
$7$	$ T^{6} - 3 T^{5} + 15 T^{4} + 20 T^{3} + \cdots + 1 $ T^6 - 3T^5 + 15T^4 + 20T^3 + 33T^2 + 6*T + 1
$11$	$ T^{6} - 12 T^{5} + 99 T^{4} + \cdots + 2601 $ T^6 - 12T^5 + 99T^4 - 438T^3 + 1413T^2 - 2295*T + 2601
$13$	$ T^{6} + 21 T^{5} + 186 T^{4} + \cdots + 361 $ T^6 + 21T^5 + 186T^4 + 800T^3 + 1407T^2 - 57*T + 361
$17$	$ T^{6} - 3 T^{5} - 18 T^{4} + \cdots + 2601 $ T^6 - 3T^5 - 18T^4 - 24T^3 + 495T^2 + 459*T + 2601
$19$	$ T^{6} - 6 T^{5} - 12 T^{4} + \cdots + 6859 $ T^6 - 6T^5 - 12T^4 + 169T^3 - 228T^2 - 2166*T + 6859
$23$	$ T^{6} + 15 T^{5} + 108 T^{4} + 246 T^{3} + \cdots + 9 $ T^6 + 15T^5 + 108T^4 + 246T^3 + 846T^2 + 162*T + 9
$29$	$ T^{6} - 15 T^{5} + 72 T^{4} + \cdots + 2601 $ T^6 - 15T^5 + 72T^4 - 138T^3 + 576T^2 + 1836*T + 2601
$31$	$ T^{6} - 3 T^{5} + 15 T^{4} - 16 T^{3} + \cdots + 289 $ T^6 - 3T^5 + 15T^4 - 16T^3 + 87T^2 - 102*T + 289
$37$	$ (T^{3} - 3 T^{2} - 9 T + 19)^{2} $ (T^3 - 3T^2 - 9T + 19)^2
$41$	$ T^{6} + 9 T^{5} + 27 T^{4} - 216 T^{3} + \cdots + 729 $ T^6 + 9T^5 + 27T^4 - 216T^3 + 486T^2 - 729*T + 729
$43$	$ T^{6} + 9 T^{5} + 45 T^{4} + 80 T^{3} + \cdots + 1 $ T^6 + 9T^5 + 45T^4 + 80T^3 + 576T^2 + 45*T + 1
$47$	$ T^{6} + 21 T^{5} + 261 T^{4} + \cdots + 25281 $ T^6 + 21T^5 + 261T^4 + 1968T^3 + 10836T^2 + 27189*T + 25281
$53$	$ T^{6} - 30 T^{5} + 378 T^{4} + \cdots + 47961 $ T^6 - 30T^5 + 378T^4 - 2373T^3 + 7218T^2 - 11826*T + 47961
$59$	$ T^{6} - 27 T^{5} + 360 T^{4} + \cdots + 23409 $ T^6 - 27T^5 + 360T^4 - 3312T^3 + 19197T^2 - 31671*T + 23409
$61$	$ T^{6} + 9 T^{5} + 72 T^{4} + 323 T^{3} + \cdots + 1 $ T^6 + 9T^5 + 72T^4 + 323T^3 + 549T^2 - 36*T + 1
$67$	$ T^{6} + 15 T^{5} + 156 T^{4} + \cdots + 7921 $ T^6 + 15T^5 + 156T^4 + 1070T^3 + 4200T^2 + 8544*T + 7921
$71$	$ T^{6} - 9 T^{5} + 81 T^{4} + \cdots + 6561 $ T^6 - 9T^5 + 81T^4 - 648T^3 + 2916T^2 - 6561*T + 6561
$73$	$ T^{6} - 12 T^{5} + 246 T^{4} + \cdots + 546121 $ T^6 - 12T^5 + 246T^4 - 2404T^3 + 32586T^2 + 281559*T + 546121
$79$	$ T^{6} - 15 T^{5} + 105 T^{4} + \cdots + 5329 $ T^6 - 15T^5 + 105T^4 + 152T^3 + 3180T^2 + 7665*T + 5329
$83$	$ T^{6} + 3 T^{5} + 279 T^{4} + \cdots + 3583449 $ T^6 + 3T^5 + 279T^4 + 2976T^3 + 78579T^2 + 511110*T + 3583449
$89$	$ T^{6} + 48 T^{5} + 1044 T^{4} + \cdots + 3583449 $ T^6 + 48T^5 + 1044T^4 + 12174T^3 + 95616T^2 + 698517*T + 3583449
$97$	$ T^{6} - 18 T^{5} + 99 T^{4} + \cdots + 1630729 $ T^6 - 18T^5 + 99T^4 - 613T^3 + 3636T^2 + 34479*T + 1630729
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Overview	Random
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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)

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

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
$5$	\( T^{6} + 3 T^{5} - 30 T^{3} + 36 T^{2} + \cdots + 9 \) T^6 + 3T^5 - 30T^3 + 36*T^2 + 9
$7$	\( T^{6} - 3 T^{5} + 15 T^{4} + 20 T^{3} + \cdots + 1 \) T^6 - 3T^5 + 15T^4 + 20T^3 + 33T^2 + 6*T + 1
$11$	\( T^{6} - 12 T^{5} + 99 T^{4} + \cdots + 2601 \) T^6 - 12T^5 + 99T^4 - 438T^3 + 1413T^2 - 2295*T + 2601
$13$	\( T^{6} + 21 T^{5} + 186 T^{4} + \cdots + 361 \) T^6 + 21T^5 + 186T^4 + 800T^3 + 1407T^2 - 57*T + 361
$17$	\( T^{6} - 3 T^{5} - 18 T^{4} + \cdots + 2601 \) T^6 - 3T^5 - 18T^4 - 24T^3 + 495T^2 + 459*T + 2601
$19$	\( T^{6} - 6 T^{5} - 12 T^{4} + \cdots + 6859 \) T^6 - 6T^5 - 12T^4 + 169T^3 - 228T^2 - 2166*T + 6859
$23$	\( T^{6} + 15 T^{5} + 108 T^{4} + 246 T^{3} + \cdots + 9 \) T^6 + 15T^5 + 108T^4 + 246T^3 + 846T^2 + 162*T + 9
$29$	\( T^{6} - 15 T^{5} + 72 T^{4} + \cdots + 2601 \) T^6 - 15T^5 + 72T^4 - 138T^3 + 576T^2 + 1836*T + 2601
$31$	\( T^{6} - 3 T^{5} + 15 T^{4} - 16 T^{3} + \cdots + 289 \) T^6 - 3T^5 + 15T^4 - 16T^3 + 87T^2 - 102*T + 289
$37$	\( (T^{3} - 3 T^{2} - 9 T + 19)^{2} \) (T^3 - 3T^2 - 9T + 19)^2
$41$	\( T^{6} + 9 T^{5} + 27 T^{4} - 216 T^{3} + \cdots + 729 \) T^6 + 9T^5 + 27T^4 - 216T^3 + 486T^2 - 729*T + 729
$43$	\( T^{6} + 9 T^{5} + 45 T^{4} + 80 T^{3} + \cdots + 1 \) T^6 + 9T^5 + 45T^4 + 80T^3 + 576T^2 + 45*T + 1
$47$	\( T^{6} + 21 T^{5} + 261 T^{4} + \cdots + 25281 \) T^6 + 21T^5 + 261T^4 + 1968T^3 + 10836T^2 + 27189*T + 25281
$53$	\( T^{6} - 30 T^{5} + 378 T^{4} + \cdots + 47961 \) T^6 - 30T^5 + 378T^4 - 2373T^3 + 7218T^2 - 11826*T + 47961
$59$	\( T^{6} - 27 T^{5} + 360 T^{4} + \cdots + 23409 \) T^6 - 27T^5 + 360T^4 - 3312T^3 + 19197T^2 - 31671*T + 23409
$61$	\( T^{6} + 9 T^{5} + 72 T^{4} + 323 T^{3} + \cdots + 1 \) T^6 + 9T^5 + 72T^4 + 323T^3 + 549T^2 - 36*T + 1
$67$	\( T^{6} + 15 T^{5} + 156 T^{4} + \cdots + 7921 \) T^6 + 15T^5 + 156T^4 + 1070T^3 + 4200T^2 + 8544*T + 7921
$71$	\( T^{6} - 9 T^{5} + 81 T^{4} + \cdots + 6561 \) T^6 - 9T^5 + 81T^4 - 648T^3 + 2916T^2 - 6561*T + 6561
$73$	\( T^{6} - 12 T^{5} + 246 T^{4} + \cdots + 546121 \) T^6 - 12T^5 + 246T^4 - 2404T^3 + 32586T^2 + 281559*T + 546121
$79$	\( T^{6} - 15 T^{5} + 105 T^{4} + \cdots + 5329 \) T^6 - 15T^5 + 105T^4 + 152T^3 + 3180T^2 + 7665*T + 5329
$83$	\( T^{6} + 3 T^{5} + 279 T^{4} + \cdots + 3583449 \) T^6 + 3T^5 + 279T^4 + 2976T^3 + 78579T^2 + 511110*T + 3583449
$89$	\( T^{6} + 48 T^{5} + 1044 T^{4} + \cdots + 3583449 \) T^6 + 48T^5 + 1044T^4 + 12174T^3 + 95616T^2 + 698517*T + 3583449
$97$	\( T^{6} - 18 T^{5} + 99 T^{4} + \cdots + 1630729 \) T^6 - 18T^5 + 99T^4 - 613T^3 + 3636T^2 + 34479*T + 1630729
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\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(\zeta_{18}^{2}\)