LMFDB - Newform orbit 57.2.i.a

Newspace parameters

Level:	$ N $	$=$	$ 57 = 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	57.i (of order $9$, degree $6$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.455147291521$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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 $\left(\mathbb{Z}/57\mathbb{Z}\right)^\times$.

$n$	$20$	$40$
$\chi(n)$	$1$	$\zeta_{18}^{2}$

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} $

4.1

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

1.43969

0.524005i

−0.173648

−

0.984808i

0.266044

0.223238i

−1.93969

1.62760i

0.266044

−

1.50881i

−0.266044

0.460802i

−1.26604

−

2.19285i

−0.939693

0.342020i

−3.64543

1.32683i

16.1

−0.266044

−

0.223238i

0.939693

−

0.342020i

−0.326352

−

1.85083i

−0.233956

1.32683i

−0.326352

−

0.118782i

0.326352

0.565258i

−0.673648

1.16679i

0.766044

−

0.642788i

0.358441

−

0.300767i

25.1

−0.266044

0.223238i

0.939693

0.342020i

−0.326352

1.85083i

−0.233956

−

1.32683i

−0.326352

0.118782i

0.326352

−

0.565258i

−0.673648

−

1.16679i

0.766044

0.642788i

0.358441

0.300767i

28.1

0.326352

1.85083i

−0.766044

0.642788i

−1.43969

0.524005i

−0.826352

−

0.300767i

−1.43969

−

1.20805i

1.43969

−

2.49362i

0.439693

0.761570i

0.173648

−

0.984808i

0.286989

−

1.62760i

43.1

1.43969

−

0.524005i

−0.173648

0.984808i

0.266044

−

0.223238i

−1.93969

−

1.62760i

0.266044

1.50881i

−0.266044

−

0.460802i

−1.26604

2.19285i

−0.939693

−

0.342020i

−3.64543

−

1.32683i

55.1

0.326352

−

1.85083i

−0.766044

−

0.642788i

−1.43969

−

0.524005i

−0.826352

0.300767i

−1.43969

1.20805i

1.43969

2.49362i

0.439693

−

0.761570i

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	57.2.i.a	✓	6
3.b	odd	2	1		171.2.u.a		6
4.b	odd	2	1		912.2.bo.b		6
19.e	even	9	1	inner	57.2.i.a	✓	6
19.e	even	9	1		1083.2.a.m		3
19.f	odd	18	1		1083.2.a.n		3
57.j	even	18	1		3249.2.a.x		3
57.l	odd	18	1		171.2.u.a		6
57.l	odd	18	1		3249.2.a.w		3
76.l	odd	18	1		912.2.bo.b		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
57.2.i.a	✓	6	1.a	even	1	1	trivial
57.2.i.a	✓	6	19.e	even	9	1	inner
171.2.u.a		6	3.b	odd	2	1
171.2.u.a		6	57.l	odd	18	1
912.2.bo.b		6	4.b	odd	2	1
912.2.bo.b		6	76.l	odd	18	1
1083.2.a.m		3	19.e	even	9	1
1083.2.a.n		3	19.f	odd	18	1
3249.2.a.w		3	57.l	odd	18	1
3249.2.a.x		3	57.j	even	18	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} - 3T_{2}^{5} + 6T_{2}^{4} - 8T_{2}^{3} + 3T_{2}^{2} + 3T_{2} + 1 $ T2^6 - 3*T2^5 + 6*T2^4 - 8*T2^3 + 3*T2^2 + 3*T2 + 1 acting on $S_{2}^{\mathrm{new}}(57, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 $ T^6 - 3T^5 + 6T^4 - 8T^3 + 3T^2 + 3*T + 1
$3$	$ T^{6} - T^{3} + 1 $ T^6 - T^3 + 1
$5$	$ T^{6} + 6 T^{5} + 18 T^{4} + 30 T^{3} + \cdots + 9 $ T^6 + 6T^5 + 18T^4 + 30T^3 + 36T^2 + 27*T + 9
$7$	$ T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1 $ T^6 - 3T^5 + 9T^4 - 2T^3 + 3T^2 + 1
$11$	$ T^{6} - 3 T^{5} + 27 T^{4} + \cdots + 1369 $ T^6 - 3T^5 + 27T^4 - 20T^3 + 435T^2 - 666*T + 1369
$13$	$ T^{6} + 6 T^{5} + 30 T^{4} + 125 T^{3} + \cdots + 361 $ T^6 + 6T^5 + 30T^4 + 125T^3 + 366T^2 + 570*T + 361
$17$	$ T^{6} + 12 T^{5} + 114 T^{4} + \cdots + 2809 $ T^6 + 12T^5 + 114T^4 + 656T^3 + 2262T^2 + 3975*T + 2809
$19$	$ T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 6859 $ T^6 - 18T^5 + 162T^4 - 883T^3 + 3078T^2 - 6498*T + 6859
$23$	$ T^{6} + 9 T^{5} + 18 T^{4} + \cdots + 5329 $ T^6 + 9T^5 + 18T^4 + 8T^3 + 225T^2 - 657*T + 5329
$29$	$ T^{6} - 12 T^{5} + 60 T^{4} + \cdots + 2809 $ T^6 - 12T^5 + 60T^4 + 235T^3 + 1236T^2 + 3180*T + 2809
$31$	$ T^{6} + 24 T^{5} + 387 T^{4} + \cdots + 239121 $ T^6 + 24T^5 + 387T^4 + 3558T^3 + 23985T^2 + 92421*T + 239121
$37$	$ (T^{3} - 6 T^{2} - 9 T + 51)^{2} $ (T^3 - 6T^2 - 9T + 51)^2
$41$	$ T^{6} - 6 T^{5} + 81 T^{4} + \cdots + 2601 $ T^6 - 6T^5 + 81T^4 - 597T^3 + 2412T^2 - 4131*T + 2601
$43$	$ T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 361 $ T^6 - 18T^5 + 144T^4 - 1423T^3 + 11124T^2 + 3078*T + 361
$47$	$ T^{6} + 33 T^{5} + 477 T^{4} + \cdots + 45369 $ T^6 + 33T^5 + 477T^4 + 3216T^3 + 9108T^2 + 9585*T + 45369
$53$	$ T^{6} + 24 T^{5} + 375 T^{4} + \cdots + 83521 $ T^6 + 24T^5 + 375T^4 + 3655T^3 + 24276T^2 + 73695*T + 83521
$59$	$ T^{6} + 54 T^{4} - 224 T^{3} + \cdots + 7921 $ T^6 + 54T^4 - 224T^3 - 450T^2 + 801T + 7921
$61$	$ T^{6} + 21 T^{5} + 204 T^{4} + \cdots + 1369 $ T^6 + 21T^5 + 204T^4 + 323T^3 + 3801T^2 + 4218*T + 1369
$67$	$ T^{6} - 30 T^{5} + 372 T^{4} + \cdots + 87616 $ T^6 - 30T^5 + 372T^4 - 2368T^3 + 8736T^2 - 24864*T + 87616
$71$	$ T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 23409 $ T^6 - 18T^5 + 144T^4 - 819T^3 + 4212T^2 - 13770*T + 23409
$73$	$ T^{6} + 27 T^{5} + 324 T^{4} + \cdots + 11881 $ T^6 + 27T^5 + 324T^4 + 1918T^3 + 5544T^2 + 7848*T + 11881
$79$	$ T^{6} - 12 T^{5} + 168 T^{4} + \cdots + 2809 $ T^6 - 12T^5 + 168T^4 - 1061T^3 + 3144T^2 - 4452*T + 2809
$83$	$ T^{6} - 15 T^{5} + 189 T^{4} + \cdots + 2601 $ T^6 - 15T^5 + 189T^4 - 642T^3 + 2061T^2 + 1836*T + 2601
$89$	$ T^{6} - 45 T^{5} + 900 T^{4} + \cdots + 1265625 $ T^6 - 45T^5 + 900T^4 - 9000T^3 + 50625T^2 - 253125*T + 1265625
$97$	$ T^{6} - 21 T^{5} + 210 T^{4} + \cdots + 72361 $ T^6 - 21T^5 + 210T^4 - 1396T^3 + 7674T^2 - 32280*T + 72361
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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:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	57.i (of order \(9\), degree \(6\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

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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	57.2.i.a

Level	$57$
Weight	$2$
Character orbit	57.i
Analytic conductor	$0.455$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.455147291521\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{18})\)
Defining polynomial:	\( x^{6} - x^{3} + 1 \) x^6 - x^3 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
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 55.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) T^6 - 3T^5 + 6T^4 - 8T^3 + 3T^2 + 3*T + 1
$3$	\( T^{6} - T^{3} + 1 \) T^6 - T^3 + 1
$5$	\( T^{6} + 6 T^{5} + 18 T^{4} + 30 T^{3} + \cdots + 9 \) T^6 + 6T^5 + 18T^4 + 30T^3 + 36T^2 + 27*T + 9
$7$	\( T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1 \) T^6 - 3T^5 + 9T^4 - 2T^3 + 3T^2 + 1
$11$	\( T^{6} - 3 T^{5} + 27 T^{4} + \cdots + 1369 \) T^6 - 3T^5 + 27T^4 - 20T^3 + 435T^2 - 666*T + 1369
$13$	\( T^{6} + 6 T^{5} + 30 T^{4} + 125 T^{3} + \cdots + 361 \) T^6 + 6T^5 + 30T^4 + 125T^3 + 366T^2 + 570*T + 361
$17$	\( T^{6} + 12 T^{5} + 114 T^{4} + \cdots + 2809 \) T^6 + 12T^5 + 114T^4 + 656T^3 + 2262T^2 + 3975*T + 2809
$19$	\( T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 6859 \) T^6 - 18T^5 + 162T^4 - 883T^3 + 3078T^2 - 6498*T + 6859
$23$	\( T^{6} + 9 T^{5} + 18 T^{4} + \cdots + 5329 \) T^6 + 9T^5 + 18T^4 + 8T^3 + 225T^2 - 657*T + 5329
$29$	\( T^{6} - 12 T^{5} + 60 T^{4} + \cdots + 2809 \) T^6 - 12T^5 + 60T^4 + 235T^3 + 1236T^2 + 3180*T + 2809
$31$	\( T^{6} + 24 T^{5} + 387 T^{4} + \cdots + 239121 \) T^6 + 24T^5 + 387T^4 + 3558T^3 + 23985T^2 + 92421*T + 239121
$37$	\( (T^{3} - 6 T^{2} - 9 T + 51)^{2} \) (T^3 - 6T^2 - 9T + 51)^2
$41$	\( T^{6} - 6 T^{5} + 81 T^{4} + \cdots + 2601 \) T^6 - 6T^5 + 81T^4 - 597T^3 + 2412T^2 - 4131*T + 2601
$43$	\( T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 361 \) T^6 - 18T^5 + 144T^4 - 1423T^3 + 11124T^2 + 3078*T + 361
$47$	\( T^{6} + 33 T^{5} + 477 T^{4} + \cdots + 45369 \) T^6 + 33T^5 + 477T^4 + 3216T^3 + 9108T^2 + 9585*T + 45369
$53$	\( T^{6} + 24 T^{5} + 375 T^{4} + \cdots + 83521 \) T^6 + 24T^5 + 375T^4 + 3655T^3 + 24276T^2 + 73695*T + 83521
$59$	\( T^{6} + 54 T^{4} - 224 T^{3} + \cdots + 7921 \) T^6 + 54T^4 - 224T^3 - 450T^2 + 801T + 7921
$61$	\( T^{6} + 21 T^{5} + 204 T^{4} + \cdots + 1369 \) T^6 + 21T^5 + 204T^4 + 323T^3 + 3801T^2 + 4218*T + 1369
$67$	\( T^{6} - 30 T^{5} + 372 T^{4} + \cdots + 87616 \) T^6 - 30T^5 + 372T^4 - 2368T^3 + 8736T^2 - 24864*T + 87616
$71$	\( T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 23409 \) T^6 - 18T^5 + 144T^4 - 819T^3 + 4212T^2 - 13770*T + 23409
$73$	\( T^{6} + 27 T^{5} + 324 T^{4} + \cdots + 11881 \) T^6 + 27T^5 + 324T^4 + 1918T^3 + 5544T^2 + 7848*T + 11881
$79$	\( T^{6} - 12 T^{5} + 168 T^{4} + \cdots + 2809 \) T^6 - 12T^5 + 168T^4 - 1061T^3 + 3144T^2 - 4452*T + 2809
$83$	\( T^{6} - 15 T^{5} + 189 T^{4} + \cdots + 2601 \) T^6 - 15T^5 + 189T^4 - 642T^3 + 2061T^2 + 1836*T + 2601
$89$	\( T^{6} - 45 T^{5} + 900 T^{4} + \cdots + 1265625 \) T^6 - 45T^5 + 900T^4 - 9000T^3 + 50625T^2 - 253125*T + 1265625
$97$	\( T^{6} - 21 T^{5} + 210 T^{4} + \cdots + 72361 \) T^6 - 21T^5 + 210T^4 - 1396T^3 + 7674T^2 - 32280*T + 72361
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\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(\zeta_{18}^{2}\)