LMFDB - Newform orbit 378.2.u.a

Newspace parameters

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.u (of order $9$, degree $6$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.01834519640$
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 $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$\zeta_{18}^{2}$	$1$

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} $

43.1

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

0.766044

0.642788i

1.11334

−

1.32683i

0.173648

0.984808i

2.37939

0.866025i

1.70574

−

0.300767i

0.173648

−

0.984808i

−0.500000

0.866025i

−0.520945

−

2.95442i

1.26604

2.19285i

85.1

−0.939693

−

0.342020i

0.592396

−

1.62760i

0.766044

0.642788i

0.152704

−

0.866025i

−1.11334

1.32683i

0.766044

−

0.642788i

−0.500000

−

0.866025i

−2.29813

−

1.92836i

−0.439693

0.761570i

169.1

−0.939693

0.342020i

0.592396

1.62760i

0.766044

−

0.642788i

0.152704

0.866025i

−1.11334

−

1.32683i

0.766044

0.642788i

−0.500000

0.866025i

−2.29813

1.92836i

−0.439693

−

0.761570i

211.1

0.766044

−

0.642788i

1.11334

1.32683i

0.173648

−

0.984808i

2.37939

−

0.866025i

1.70574

0.300767i

0.173648

0.984808i

−0.500000

−

0.866025i

−0.520945

2.95442i

1.26604

−

2.19285i

295.1

0.173648

−

0.984808i

−1.70574

−

0.300767i

−0.939693

−

0.342020i

−1.03209

0.866025i

−0.592396

1.62760i

−0.939693

0.342020i

−0.500000

0.866025i

2.81908

1.02606i

0.673648

1.16679i

337.1

0.173648

0.984808i

−1.70574

0.300767i

−0.939693

0.342020i

−1.03209

−

0.866025i

−0.592396

−

1.62760i

−0.939693

−

0.342020i

−0.500000

−

0.866025i

2.81908

−

1.02606i

0.673648

−

1.16679i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	378.2.u.a	✓	6
27.e	even	9	1	inner	378.2.u.a	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
378.2.u.a	✓	6	1.a	even	1	1	trivial
378.2.u.a	✓	6	27.e	even	9	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} - 3T_{5}^{5} + 3T_{5}^{3} + 9T_{5}^{2} + 9 $ T5^6 - 3*T5^5 + 3*T5^3 + 9*T5^2 + 9 acting on $S_{2}^{\mathrm{new}}(378, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + T^{3} + 1 $ T^6 + T^3 + 1
$3$	$ T^{6} + 9T^{3} + 27 $ T^6 + 9*T^3 + 27
$5$	$ T^{6} - 3 T^{5} + 3 T^{3} + 9 T^{2} + \cdots + 9 $ T^6 - 3T^5 + 3T^3 + 9*T^2 + 9
$7$	$ T^{6} + T^{3} + 1 $ T^6 + T^3 + 1
$11$	$ T^{6} - 6 T^{5} + 18 T^{4} - 3 T^{3} + \cdots + 9 $ T^6 - 6T^5 + 18T^4 - 3T^3 + 90T^2 + 54*T + 9
$13$	$ T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 $ T^6 + 3T^5 + 6T^4 + 8T^3 + 12T^2 + 6*T + 1
$17$	$ T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81 $ T^6 + 9T^4 + 18T^3 + 81T^2 + 81T + 81
$19$	$ T^{6} - 3 T^{5} + 33 T^{4} + \cdots + 2809 $ T^6 - 3T^5 + 33T^4 - 34T^3 + 735T^2 - 1272*T + 2809
$23$	$ T^{6} - 6 T^{5} - 36 T^{4} + \cdots + 3249 $ T^6 - 6T^5 - 36T^4 + 51T^3 + 2088T^2 + 1026*T + 3249
$29$	$ T^{6} + 3 T^{5} + 108 T^{4} + \cdots + 3249 $ T^6 + 3T^5 + 108T^4 - 84T^3 - 369T^2 + 1539*T + 3249
$31$	$ T^{6} + 3 T^{5} + 24 T^{4} + 17 T^{3} + \cdots + 361 $ T^6 + 3T^5 + 24T^4 + 17T^3 - 51T^2 - 228*T + 361
$37$	$ T^{6} - 12 T^{5} + 132 T^{4} + \cdots + 64 $ T^6 - 12T^5 + 132T^4 - 160T^3 + 240T^2 + 96*T + 64
$41$	$ T^{6} + 12 T^{5} + 144 T^{4} + \cdots + 36864 $ T^6 + 12T^5 + 144T^4 + 1536T^3 + 9216T^2 + 27648*T + 36864
$43$	$ T^{6} + 6 T^{5} - 6 T^{4} - 361 T^{3} + \cdots + 5329 $ T^6 + 6T^5 - 6T^4 - 361T^3 + 1122T^2 - 438*T + 5329
$47$	$ T^{6} - 24 T^{5} + 306 T^{4} + \cdots + 47961 $ T^6 - 24T^5 + 306T^4 - 2703T^3 + 15102T^2 - 39420*T + 47961
$53$	$ (T^{3} - 117 T + 153)^{2} $ (T^3 - 117*T + 153)^2
$59$	$ T^{6} + 24 T^{5} + 252 T^{4} + \cdots + 2601 $ T^6 + 24T^5 + 252T^4 + 1218T^3 + 2304T^2 + 459*T + 2601
$61$	$ T^{6} + 3 T^{5} + 114 T^{4} + \cdots + 16129 $ T^6 + 3T^5 + 114T^4 - 172T^3 - 1194T^2 - 1524*T + 16129
$67$	$ T^{6} - 3 T^{5} - 96 T^{4} + \cdots + 94249 $ T^6 - 3T^5 - 96T^4 - 514T^3 + 11742T^2 + 9210*T + 94249
$71$	$ T^{6} + 12 T^{5} + 189 T^{4} + \cdots + 106929 $ T^6 + 12T^5 + 189T^4 + 114T^3 + 5949T^2 + 14715*T + 106929
$73$	$ T^{6} - 3 T^{5} + 123 T^{4} + \cdots + 1369 $ T^6 - 3T^5 + 123T^4 + 416T^3 + 12885T^2 + 4218*T + 1369
$79$	$ T^{6} - 33 T^{5} + 492 T^{4} + \cdots + 73441 $ T^6 - 33T^5 + 492T^4 - 3691T^3 + 14925T^2 - 37398*T + 73441
$83$	$ T^{6} + 144 T^{4} - 720 T^{3} + \cdots + 5184 $ T^6 + 144T^4 - 720T^3 + 2592*T + 5184
$89$	$ T^{6} - 21 T^{5} + 405 T^{4} + \cdots + 12321 $ T^6 - 21T^5 + 405T^4 - 978T^3 + 3627T^2 + 3996*T + 12321
$97$	$ T^{6} + 15 T^{5} + 120 T^{4} + \cdots + 130321 $ T^6 + 15T^5 + 120T^4 - 190T^3 + 2166T^2 - 41154*T + 130321
show more
show less

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.u (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{18} q^{2} + ( - 2 \zeta_{18}^{4} + \zeta_{18}) q^{3} + \zeta_{18}^{2} q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 1) q^{5} + (2 \zeta_{18}^{5} - \zeta_{18}^{2}) q^{6} + ( - \zeta_{18}^{4} + \zeta_{18}) q^{7} - \zeta_{18}^{3} q^{8} - 3 \zeta_{18}^{2} q^{9} +O(q^{10}) \) q - z * q^2 + (-2z^4 + z) q^3 + z^2 * q^4 + (z^5 - z^4 - z^3 + 1) * q^5 + (2z^5 - z^2) q^6 + (-z^4 + z) * q^7 - z^3 * q^8 - 3z^2 q^9	\( q - \zeta_{18} q^{2} + ( - 2 \zeta_{18}^{4} + \zeta_{18}) q^{3} + \zeta_{18}^{2} q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 1) q^{5} + (2 \zeta_{18}^{5} - \zeta_{18}^{2}) q^{6} + ( - \zeta_{18}^{4} + \zeta_{18}) q^{7} - \zeta_{18}^{3} q^{8} - 3 \zeta_{18}^{2} q^{9} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18} + 1) q^{10} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{11} + ( - \zeta_{18}^{3} + 2) q^{12} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{13} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{14} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18} + 1) q^{15} + \zeta_{18}^{4} q^{16} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}) q^{17} + 3 \zeta_{18}^{3} q^{18} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3 \zeta_{18}) q^{19} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{20} + ( - \zeta_{18}^{5} - \zeta_{18}^{2}) q^{21} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}) q^{22} + (2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 2 \zeta_{18} + 2) q^{23} + (\zeta_{18}^{4} - 2 \zeta_{18}) q^{24} + (\zeta_{18}^{5} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{25} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{26} + (3 \zeta_{18}^{3} - 6) q^{27} + q^{28} + (5 \zeta_{18}^{5} - 5 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{29} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{30} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18} - 1) q^{31} - \zeta_{18}^{5} q^{32} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18} - 4) q^{33} + (2 \zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} - 2) q^{34} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2}) q^{35} - 3 \zeta_{18}^{4} q^{36} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 4 \zeta_{18} + 4) q^{37} + ( - 3 \zeta_{18}^{5} - \zeta_{18}^{4} + 3 \zeta_{18}^{2} - 3) q^{38} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{39} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} - \zeta_{18} - 1) q^{40} + (4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{2}) q^{41} + (2 \zeta_{18}^{3} - 1) q^{42} + ( - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2) q^{43} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 1) q^{44} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3 \zeta_{18} - 3) q^{45} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{46} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + \zeta_{18} + 2) q^{47} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{2}) q^{48} - \zeta_{18}^{5} q^{49} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{50} + ( - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2}) q^{51} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} - 1) q^{52} + ( - 7 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 5 \zeta_{18}^{2} + 5 \zeta_{18}) q^{53} + ( - 3 \zeta_{18}^{4} + 6 \zeta_{18}) q^{54} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 3) q^{55} - \zeta_{18} q^{56} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 2 \zeta_{18} + 3) q^{57} + (5 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 5) q^{58} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 5 \zeta_{18} - 3) q^{59} + (\zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{60} + ( - \zeta_{18}^{5} + 6 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 6 \zeta_{18} + 1) q^{61} + (\zeta_{18}^{5} - \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{62} - 3 q^{63} + (\zeta_{18}^{3} - 1) q^{64} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 2) q^{65} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 4 \zeta_{18} - 1) q^{66} + ( - 8 \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 8 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{67} + ( - \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18} + 2) q^{68} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18} - 10) q^{69} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3}) q^{70} + ( - 5 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} + 6 \zeta_{18} - 4) q^{71} + 3 \zeta_{18}^{5} q^{72} + (5 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 5 \zeta_{18}) q^{73} + ( - 2 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 4 \zeta_{18} + 2) q^{74} + ( - 4 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 3) q^{75} + (\zeta_{18}^{5} + 3 \zeta_{18} - 3) q^{76} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{2} - 2) q^{77} + ( - \zeta_{18}^{5} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 1) q^{78} + (\zeta_{18}^{5} - \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 3 \zeta_{18} + 6) q^{79} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{80} + 9 \zeta_{18}^{4} q^{81} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4) q^{82} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 6 \zeta_{18} + 2) q^{83} + ( - 2 \zeta_{18}^{4} + \zeta_{18}) q^{84} - 3 \zeta_{18}^{2} q^{85} + (5 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}) q^{86} + ( - 4 \zeta_{18}^{5} + \zeta_{18}^{4} + 8 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 8 \zeta_{18} - 1) q^{87} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} - 2) q^{88} + (4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 7 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4 \zeta_{18}) q^{89} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{90} + (\zeta_{18}^{2} - \zeta_{18}) q^{91} + ( - 2 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 2) q^{92} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} - 6) q^{93} + (\zeta_{18}^{5} - 4 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18} - 2) q^{94} + ( - 2 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{95} + ( - \zeta_{18}^{3} - 1) q^{96} + ( - 2 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3) q^{97} + (\zeta_{18}^{3} - 1) q^{98} + (6 \zeta_{18}^{5} + 6 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 3) q^{99} +O(q^{100}) \) q - z * q^2 + (-2z^4 + z) q^3 + z^2 * q^4 + (z^5 - z^4 - z^3 + 1) * q^5 + (2z^5 - z^2) q^6 + (-z^4 + z) * q^7 - z^3 * q^8 - 3z^2 q^9 + (z^5 + z^4 - z^3 - z + 1) * q^10 + (z^4 - 2z^3 - 2z^2 + 2) * q^11 + (-z^3 + 2) * q^12 + (z^4 - z^3) * q^13 + (z^5 - z^2) * q^14 + (z^5 - z^4 + z^3 - 2z^2 - z + 1) q^15 + z^4 * q^16 + (-2z^5 - 2z^4 + z^2 + z) * q^17 + 3z^3 q^18 + (-3z^5 + 3z^4 + z^3 + 3z^2 - 3z) * q^19 + (-z^5 + z^4 - z^3 + z^2 - z + 1) * q^20 + (-z^5 - z^2) * q^21 + (-z^5 + 2z^4 + 2z^3 - 2z) q^22 + (2z^4 - 2z^3 - 5z^2 - 2z + 2) * q^23 + (z^4 - 2z) q^24 + (z^5 - z^3 + z^2 + 2z + 2) q^25 + (-z^5 + z^4) * q^26 + (3z^3 - 6) q^27 + q^28 + (5z^5 - 5z^3 - 3z^2 + 2z + 2) * q^29 + (z^5 - z^4 + z^3 + z^2 - z + 1) * q^30 + (-z^4 + z^3 - 3z^2 + z - 1) q^31 - z^5 * q^32 + (-z^5 - 2z^4 + 2z^3 + 2z^2 - 2z - 4) * q^33 + (2z^5 + z^3 - z^2 - 2) q^34 + (-z^4 + z^3 - z^2) * q^35 - 3z^4 q^36 + (2z^5 + 2z^4 - 4z^3 + 2z^2 - 4z + 4) q^37 + (-3z^5 - z^4 + 3z^2 - 3) * q^38 + (-z^5 + z^4 + 2z^2 - 2z) * q^39 + (-z^5 + z^4 + z^2 - z - 1) * q^40 + (4z^5 + 4z^4 - 4z^3 - 4z^2) * q^41 + (2z^3 - 1) q^42 + (-5z^4 + 2z^3 + 2z^2 - 2) q^43 + (-2z^5 - 2z^4 + z^3 + 2z^2 - 1) q^44 + (3z^5 - 3z^4 + 3z^3 - 3z^2 + 3z - 3) q^45 + (-2z^5 + 2z^4 + 5z^3 + 2z^2 - 2z) q^46 + (-2z^5 - z^4 + 4z^3 - 4z^2 + z + 2) q^47 + (-z^5 + 2z^2) q^48 - z^5 * q^49 + (z^4 - 2z^3 - 2z^2 - 2z + 1) q^50 + (-3z^3 - 3z^2) * q^51 + (-z^5 + z^3 - 1) * q^52 + (-7z^5 + 2z^4 + 5z^2 + 5z) * q^53 + (-3z^4 + 6z) * q^54 + (z^5 - 2z^4 + z^2 + z - 3) q^55 - z * q^56 + (3z^5 - z^4 - 6z^3 + 3z^2 + 2z + 3) * q^57 + (5z^4 - 2z^3 - 2z^2 - 2z + 5) * q^58 + (z^5 - 2z^4 - 2z^3 + 5z - 3) q^59 + (z^5 - z^4 - 2z^3 + z^2 - z + 1) q^60 + (-z^5 + 6z^4 - 3z^3 + 3z^2 - 6z + 1) * q^61 + (z^5 - z^4 + 3z^3 - z^2 + z) q^62 - 3 * q^63 + (z^3 - 1) * q^64 + (-2z^5 + z^4 + 2z^2 - 2) * q^65 + (2z^5 - 2z^4 - z^3 + 2z^2 + 4z - 1) * q^66 + (-8z^5 + z^4 - 3z^3 + 8z^2 + 2z + 2) * q^67 + (-z^4 - z^3 + 2z + 2) q^68 + (2z^5 - 2z^4 + 5z^3 + 2z^2 - 2z - 10) q^69 + (z^5 - z^4 + z^3) * q^70 + (-5z^5 - 5z^4 + 4z^3 - z^2 + 6z - 4) * q^71 + 3z^5 q^72 + (5z^5 + 2z^4 + z^3 + 2z^2 + 5z) * q^73 + (-2z^5 + 4z^4 - 4z^3 + 4z^2 - 4z + 2) q^74 + (-4z^5 - 3z^4 + 2z^2 + 3) q^75 + (z^5 + 3z - 3) q^76 + (-2z^4 + z^2 - 2) q^77 + (-z^5 - z^3 + 2z^2 - 1) q^78 + (z^5 - z^3 + 5z^2 - 3z + 6) * q^79 + (-z^5 + z^2 + z - 1) * q^80 + 9z^4 q^81 + (-4z^5 + 4z^4 + 4) * q^82 + (4z^5 - 4z^3 - 2z^2 + 6z + 2) * q^83 + (-2z^4 + z) q^84 - 3z^2 q^85 + (5z^5 - 2z^4 - 2z^3 + 2z) * q^86 + (-4z^5 + z^4 + 8z^3 + 2z^2 - 8z - 1) * q^87 + (2z^5 - z^4 + z - 2) q^88 + (4z^5 + 3z^4 + 7z^3 + 3z^2 + 4z) q^89 + (3z^5 - 3z^4 - 3z^2 + 3z + 3) * q^90 + (z^2 - z) * q^91 + (-2z^5 - 5z^4 + 2z^2 - 2) q^92 + (-z^5 + z^4 + 3z^3 - z^2 + z - 6) q^93 + (z^5 - 4z^4 + 6z^3 - z^2 - 2z - 2) q^94 + (-2z^5 + 5z^4 - 6z^3 + 2z^2 + z + 1) * q^95 + (-z^3 - 1) * q^96 + (-2z^5 - 8z^4 + z^3 + 3z^2 - 3) q^97 + (z^3 - 1) * q^98 + (6z^5 + 6z^4 - 3z^3 - 6z^2 + 3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} - 3 q^{8}+O(q^{10}) \) 6 * q + 3 * q^5 - 3 * q^8	\( 6 q + 3 q^{5} - 3 q^{8} + 3 q^{10} + 6 q^{11} + 9 q^{12} - 3 q^{13} + 9 q^{15} + 9 q^{18} + 3 q^{19} + 3 q^{20} + 6 q^{22} + 6 q^{23} + 9 q^{25} - 27 q^{27} + 6 q^{28} - 3 q^{29} + 9 q^{30} - 3 q^{31} - 18 q^{33} - 9 q^{34} + 3 q^{35} + 12 q^{37} - 18 q^{38} - 6 q^{40} - 12 q^{41} - 6 q^{43} - 3 q^{44} - 9 q^{45} + 15 q^{46} + 24 q^{47} - 9 q^{51} - 3 q^{52} - 18 q^{55} + 24 q^{58} - 24 q^{59} - 3 q^{61} + 9 q^{62} - 18 q^{63} - 3 q^{64} - 12 q^{65} - 9 q^{66} + 3 q^{67} + 9 q^{68} - 45 q^{69} + 3 q^{70} - 12 q^{71} + 3 q^{73} + 18 q^{75} - 18 q^{76} - 12 q^{77} - 9 q^{78} + 33 q^{79} - 6 q^{80} + 24 q^{82} - 6 q^{86} + 18 q^{87} - 12 q^{88} + 21 q^{89} + 18 q^{90} - 12 q^{92} - 27 q^{93} + 6 q^{94} - 12 q^{95} - 9 q^{96} - 15 q^{97} - 3 q^{98} + 9 q^{99}+O(q^{100}) \) 6 * q + 3 * q^5 - 3 * q^8 + 3 * q^10 + 6 * q^11 + 9 * q^12 - 3 * q^13 + 9 * q^15 + 9 * q^18 + 3 * q^19 + 3 * q^20 + 6 * q^22 + 6 * q^23 + 9 * q^25 - 27 * q^27 + 6 * q^28 - 3 * q^29 + 9 * q^30 - 3 * q^31 - 18 * q^33 - 9 * q^34 + 3 * q^35 + 12 * q^37 - 18 * q^38 - 6 * q^40 - 12 * q^41 - 6 * q^43 - 3 * q^44 - 9 * q^45 + 15 * q^46 + 24 * q^47 - 9 * q^51 - 3 * q^52 - 18 * q^55 + 24 * q^58 - 24 * q^59 - 3 * q^61 + 9 * q^62 - 18 * q^63 - 3 * q^64 - 12 * q^65 - 9 * q^66 + 3 * q^67 + 9 * q^68 - 45 * q^69 + 3 * q^70 - 12 * q^71 + 3 * q^73 + 18 * q^75 - 18 * q^76 - 12 * q^77 - 9 * q^78 + 33 * q^79 - 6 * q^80 + 24 * q^82 - 6 * q^86 + 18 * q^87 - 12 * q^88 + 21 * q^89 + 18 * q^90 - 12 * q^92 - 27 * q^93 + 6 * q^94 - 12 * q^95 - 9 * q^96 - 15 * q^97 - 3 * q^98 + 9 * 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	378.2.u.a

Level	$378$
Weight	$2$
Character orbit	378.u
Analytic conductor	$3.018$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(3.01834519640\)
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 337.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} + T^{3} + 1 \) T^6 + T^3 + 1
$3$	\( T^{6} + 9T^{3} + 27 \) T^6 + 9*T^3 + 27
$5$	\( T^{6} - 3 T^{5} + 3 T^{3} + 9 T^{2} + \cdots + 9 \) T^6 - 3T^5 + 3T^3 + 9*T^2 + 9
$7$	\( T^{6} + T^{3} + 1 \) T^6 + T^3 + 1
$11$	\( T^{6} - 6 T^{5} + 18 T^{4} - 3 T^{3} + \cdots + 9 \) T^6 - 6T^5 + 18T^4 - 3T^3 + 90T^2 + 54*T + 9
$13$	\( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) T^6 + 3T^5 + 6T^4 + 8T^3 + 12T^2 + 6*T + 1
$17$	\( T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81 \) T^6 + 9T^4 + 18T^3 + 81T^2 + 81T + 81
$19$	\( T^{6} - 3 T^{5} + 33 T^{4} + \cdots + 2809 \) T^6 - 3T^5 + 33T^4 - 34T^3 + 735T^2 - 1272*T + 2809
$23$	\( T^{6} - 6 T^{5} - 36 T^{4} + \cdots + 3249 \) T^6 - 6T^5 - 36T^4 + 51T^3 + 2088T^2 + 1026*T + 3249
$29$	\( T^{6} + 3 T^{5} + 108 T^{4} + \cdots + 3249 \) T^6 + 3T^5 + 108T^4 - 84T^3 - 369T^2 + 1539*T + 3249
$31$	\( T^{6} + 3 T^{5} + 24 T^{4} + 17 T^{3} + \cdots + 361 \) T^6 + 3T^5 + 24T^4 + 17T^3 - 51T^2 - 228*T + 361
$37$	\( T^{6} - 12 T^{5} + 132 T^{4} + \cdots + 64 \) T^6 - 12T^5 + 132T^4 - 160T^3 + 240T^2 + 96*T + 64
$41$	\( T^{6} + 12 T^{5} + 144 T^{4} + \cdots + 36864 \) T^6 + 12T^5 + 144T^4 + 1536T^3 + 9216T^2 + 27648*T + 36864
$43$	\( T^{6} + 6 T^{5} - 6 T^{4} - 361 T^{3} + \cdots + 5329 \) T^6 + 6T^5 - 6T^4 - 361T^3 + 1122T^2 - 438*T + 5329
$47$	\( T^{6} - 24 T^{5} + 306 T^{4} + \cdots + 47961 \) T^6 - 24T^5 + 306T^4 - 2703T^3 + 15102T^2 - 39420*T + 47961
$53$	\( (T^{3} - 117 T + 153)^{2} \) (T^3 - 117*T + 153)^2
$59$	\( T^{6} + 24 T^{5} + 252 T^{4} + \cdots + 2601 \) T^6 + 24T^5 + 252T^4 + 1218T^3 + 2304T^2 + 459*T + 2601
$61$	\( T^{6} + 3 T^{5} + 114 T^{4} + \cdots + 16129 \) T^6 + 3T^5 + 114T^4 - 172T^3 - 1194T^2 - 1524*T + 16129
$67$	\( T^{6} - 3 T^{5} - 96 T^{4} + \cdots + 94249 \) T^6 - 3T^5 - 96T^4 - 514T^3 + 11742T^2 + 9210*T + 94249
$71$	\( T^{6} + 12 T^{5} + 189 T^{4} + \cdots + 106929 \) T^6 + 12T^5 + 189T^4 + 114T^3 + 5949T^2 + 14715*T + 106929
$73$	\( T^{6} - 3 T^{5} + 123 T^{4} + \cdots + 1369 \) T^6 - 3T^5 + 123T^4 + 416T^3 + 12885T^2 + 4218*T + 1369
$79$	\( T^{6} - 33 T^{5} + 492 T^{4} + \cdots + 73441 \) T^6 - 33T^5 + 492T^4 - 3691T^3 + 14925T^2 - 37398*T + 73441
$83$	\( T^{6} + 144 T^{4} - 720 T^{3} + \cdots + 5184 \) T^6 + 144T^4 - 720T^3 + 2592*T + 5184
$89$	\( T^{6} - 21 T^{5} + 405 T^{4} + \cdots + 12321 \) T^6 - 21T^5 + 405T^4 - 978T^3 + 3627T^2 + 3996*T + 12321
$97$	\( T^{6} + 15 T^{5} + 120 T^{4} + \cdots + 130321 \) T^6 + 15T^5 + 120T^4 - 190T^3 + 2166T^2 - 41154*T + 130321
show more
show less

\(n\)	\(29\)	\(325\)
\(\chi(n)\)	\(\zeta_{18}^{2}\)	\(1\)