LMFDB - Newform orbit 114.2.i.a

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

−0.386659

−

2.19285i

0.939693

−

0.342020i

1.32635

−

2.29731i

0.500000

0.866025i

0.766044

0.642788i

1.70574

1.43128i

43.1

0.939693

−

0.342020i

0.173648

−

0.984808i

0.766044

−

0.642788i

−0.907604

−

0.761570i

−0.173648

−

0.984808i

0.733956

1.27125i

0.500000

−

0.866025i

−0.939693

−

0.342020i

−1.11334

−

0.405223i

55.1

−0.173648

0.984808i

0.766044

0.642788i

−0.939693

−

0.342020i

−3.20574

1.16679i

−0.766044

0.642788i

2.43969

4.22567i

0.500000

−

0.866025i

0.173648

0.984808i

−0.592396

−

3.35965i

61.1

0.939693

0.342020i

0.173648

0.984808i

0.766044

0.642788i

−0.907604

0.761570i

−0.173648

0.984808i

0.733956

−

1.27125i

0.500000

0.866025i

−0.939693

0.342020i

−1.11334

0.405223i

73.1

−0.766044

−

0.642788i

−0.939693

0.342020i

0.173648

0.984808i

−0.386659

2.19285i

0.939693

0.342020i

1.32635

2.29731i

0.500000

−

0.866025i

0.766044

−

0.642788i

1.70574

−

1.43128i

85.1

−0.173648

−

0.984808i

0.766044

−

0.642788i

−0.939693

0.342020i

−3.20574

−

1.16679i

−0.766044

−

0.642788i

2.43969

−

4.22567i

0.500000

0.866025i

0.173648

−

0.984808i

−0.592396

3.35965i

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.a	✓	6
3.b	odd	2	1		342.2.u.e		6
4.b	odd	2	1		912.2.bo.a		6
19.e	even	9	1	inner	114.2.i.a	✓	6
19.e	even	9	1		2166.2.a.q		3
19.f	odd	18	1		2166.2.a.s		3
57.j	even	18	1		6498.2.a.bm		3
57.l	odd	18	1		342.2.u.e		6
57.l	odd	18	1		6498.2.a.br		3
76.l	odd	18	1		912.2.bo.a		6

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

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} + 9T_{5}^{5} + 36T_{5}^{4} + 90T_{5}^{3} + 162T_{5}^{2} + 162T_{5} + 81 $ T5^6 + 9*T5^5 + 36*T5^4 + 90*T5^3 + 162*T5^2 + 162*T5 + 81 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} + 9 T^{5} + 36 T^{4} + 90 T^{3} + \cdots + 81 $ T^6 + 9T^5 + 36T^4 + 90T^3 + 162T^2 + 162*T + 81
$7$	$ T^{6} - 9 T^{5} + 57 T^{4} - 178 T^{3} + \cdots + 361 $ T^6 - 9T^5 + 57T^4 - 178T^3 + 405T^2 - 456*T + 361
$11$	$ T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81 $ T^6 + 9T^4 + 18T^3 + 81T^2 + 81T + 81
$13$	$ T^{6} - 15 T^{5} + 96 T^{4} + \cdots + 1369 $ T^6 - 15T^5 + 96T^4 - 352T^3 + 885T^2 - 1443*T + 1369
$17$	$ T^{6} + 9 T^{5} + 36 T^{4} + 72 T^{3} + \cdots + 81 $ T^6 + 9T^5 + 36T^4 + 72T^3 + 81T^2 + 81*T + 81
$19$	$ T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 6859 $ T^6 + 12T^5 + 78T^4 + 385T^3 + 1482T^2 + 4332*T + 6859
$23$	$ T^{6} - 9 T^{5} + 36 T^{4} - 90 T^{3} + \cdots + 81 $ T^6 - 9T^5 + 36T^4 - 90T^3 + 162T^2 - 162*T + 81
$29$	$ T^{6} + 9 T^{5} - 18 T^{4} + \cdots + 29241 $ T^6 + 9T^5 - 18T^4 - 72T^3 + 2106T^2 - 6156*T + 29241
$31$	$ T^{6} + 9 T^{5} + 147 T^{4} + \cdots + 292681 $ T^6 + 9T^5 + 147T^4 + 488T^3 + 9225T^2 + 35706*T + 292681
$37$	$ (T^{3} - 9 T^{2} + 15 T + 1)^{2} $ (T^3 - 9T^2 + 15T + 1)^2
$41$	$ T^{6} - 27 T^{5} + 405 T^{4} + \cdots + 210681 $ T^6 - 27T^5 + 405T^4 - 4104T^3 + 27702T^2 - 111537*T + 210681
$43$	$ T^{6} + 21 T^{5} + 231 T^{4} + \cdots + 32041 $ T^6 + 21T^5 + 231T^4 + 1592T^3 + 7662T^2 + 23091*T + 32041
$47$	$ T^{6} - 27 T^{5} + 333 T^{4} + \cdots + 23409 $ T^6 - 27T^5 + 333T^4 - 2016T^3 + 6804T^2 - 15147*T + 23409
$53$	$ T^{6} + 90 T^{4} + 315 T^{3} + \cdots + 81 $ T^6 + 90T^4 + 315T^3 + 486T^2 + 324T + 81
$59$	$ T^{6} - 9 T^{5} + 162 T^{4} + \cdots + 6561 $ T^6 - 9T^5 + 162T^4 - 648T^3 + 729T^2 + 6561*T + 6561
$61$	$ T^{6} + 3 T^{5} - 12 T^{4} + \cdots + 94249 $ T^6 + 3T^5 - 12T^4 + 161T^3 + 1821T^2 + 12894*T + 94249
$67$	$ T^{6} + 3 T^{5} - 66 T^{4} + \cdots + 2809 $ T^6 + 3T^5 - 66T^4 + 224T^3 + 5502T^2 - 1272*T + 2809
$71$	$ T^{6} - 9 T^{5} - 27 T^{4} + \cdots + 263169 $ T^6 - 9T^5 - 27T^4 + 1728T^3 + 9720T^2 + 13851*T + 263169
$73$	$ T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 54289 $ T^6 + 12T^5 + 78T^4 - 244T^3 + 5754T^2 - 32853*T + 54289
$79$	$ T^{6} + 21 T^{5} + 231 T^{4} + \cdots + 2809 $ T^6 + 21T^5 + 231T^4 + 3356T^3 + 30144T^2 - 14151*T + 2809
$83$	$ T^{6} + 9 T^{5} + 225 T^{4} + \cdots + 408321 $ T^6 + 9T^5 + 225T^4 - 18T^3 + 26487T^2 + 92016*T + 408321
$89$	$ T^{6} + 90 T^{4} + 900 T^{3} + \cdots + 431649 $ T^6 + 90T^4 + 900T^3 - 4374T^2 - 5913T + 431649
$97$	$ T^{6} + 54 T^{5} + 1323 T^{4} + \cdots + 2679769 $ T^6 + 54T^5 + 1323T^4 + 17675T^3 + 141156T^2 + 751383*T + 2679769
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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)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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.a

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
$5$	\( T^{6} + 9 T^{5} + 36 T^{4} + 90 T^{3} + \cdots + 81 \) T^6 + 9T^5 + 36T^4 + 90T^3 + 162T^2 + 162*T + 81
$7$	\( T^{6} - 9 T^{5} + 57 T^{4} - 178 T^{3} + \cdots + 361 \) T^6 - 9T^5 + 57T^4 - 178T^3 + 405T^2 - 456*T + 361
$11$	\( T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81 \) T^6 + 9T^4 + 18T^3 + 81T^2 + 81T + 81
$13$	\( T^{6} - 15 T^{5} + 96 T^{4} + \cdots + 1369 \) T^6 - 15T^5 + 96T^4 - 352T^3 + 885T^2 - 1443*T + 1369
$17$	\( T^{6} + 9 T^{5} + 36 T^{4} + 72 T^{3} + \cdots + 81 \) T^6 + 9T^5 + 36T^4 + 72T^3 + 81T^2 + 81*T + 81
$19$	\( T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 6859 \) T^6 + 12T^5 + 78T^4 + 385T^3 + 1482T^2 + 4332*T + 6859
$23$	\( T^{6} - 9 T^{5} + 36 T^{4} - 90 T^{3} + \cdots + 81 \) T^6 - 9T^5 + 36T^4 - 90T^3 + 162T^2 - 162*T + 81
$29$	\( T^{6} + 9 T^{5} - 18 T^{4} + \cdots + 29241 \) T^6 + 9T^5 - 18T^4 - 72T^3 + 2106T^2 - 6156*T + 29241
$31$	\( T^{6} + 9 T^{5} + 147 T^{4} + \cdots + 292681 \) T^6 + 9T^5 + 147T^4 + 488T^3 + 9225T^2 + 35706*T + 292681
$37$	\( (T^{3} - 9 T^{2} + 15 T + 1)^{2} \) (T^3 - 9T^2 + 15T + 1)^2
$41$	\( T^{6} - 27 T^{5} + 405 T^{4} + \cdots + 210681 \) T^6 - 27T^5 + 405T^4 - 4104T^3 + 27702T^2 - 111537*T + 210681
$43$	\( T^{6} + 21 T^{5} + 231 T^{4} + \cdots + 32041 \) T^6 + 21T^5 + 231T^4 + 1592T^3 + 7662T^2 + 23091*T + 32041
$47$	\( T^{6} - 27 T^{5} + 333 T^{4} + \cdots + 23409 \) T^6 - 27T^5 + 333T^4 - 2016T^3 + 6804T^2 - 15147*T + 23409
$53$	\( T^{6} + 90 T^{4} + 315 T^{3} + \cdots + 81 \) T^6 + 90T^4 + 315T^3 + 486T^2 + 324T + 81
$59$	\( T^{6} - 9 T^{5} + 162 T^{4} + \cdots + 6561 \) T^6 - 9T^5 + 162T^4 - 648T^3 + 729T^2 + 6561*T + 6561
$61$	\( T^{6} + 3 T^{5} - 12 T^{4} + \cdots + 94249 \) T^6 + 3T^5 - 12T^4 + 161T^3 + 1821T^2 + 12894*T + 94249
$67$	\( T^{6} + 3 T^{5} - 66 T^{4} + \cdots + 2809 \) T^6 + 3T^5 - 66T^4 + 224T^3 + 5502T^2 - 1272*T + 2809
$71$	\( T^{6} - 9 T^{5} - 27 T^{4} + \cdots + 263169 \) T^6 - 9T^5 - 27T^4 + 1728T^3 + 9720T^2 + 13851*T + 263169
$73$	\( T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 54289 \) T^6 + 12T^5 + 78T^4 - 244T^3 + 5754T^2 - 32853*T + 54289
$79$	\( T^{6} + 21 T^{5} + 231 T^{4} + \cdots + 2809 \) T^6 + 21T^5 + 231T^4 + 3356T^3 + 30144T^2 - 14151*T + 2809
$83$	\( T^{6} + 9 T^{5} + 225 T^{4} + \cdots + 408321 \) T^6 + 9T^5 + 225T^4 - 18T^3 + 26487T^2 + 92016*T + 408321
$89$	\( T^{6} + 90 T^{4} + 900 T^{3} + \cdots + 431649 \) T^6 + 90T^4 + 900T^3 - 4374T^2 - 5913T + 431649
$97$	\( T^{6} + 54 T^{5} + 1323 T^{4} + \cdots + 2679769 \) T^6 + 54T^5 + 1323T^4 + 17675T^3 + 141156T^2 + 751383*T + 2679769
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\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(\zeta_{18}^{2}\)