LMFDB - Newform orbit 3660.1.fu.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3660,1,Mod(59,3660)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3660, base_ring=CyclotomicField(60))

chi = DirichletCharacter(H, H._module([30, 30, 30, 31]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3660.59");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3660 = 2^{2} \cdot 3 \cdot 5 \cdot 61 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3660.fu (of order $60$, degree $16$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.82657794624$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{60})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 $ x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{60}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{60} + \cdots)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + \zeta_{60}^{13} q^{2} - \zeta_{60}^{6} q^{3} + \zeta_{60}^{26} q^{4} + \zeta_{60}^{7} q^{5} - \zeta_{60}^{19} q^{6} - \zeta_{60}^{9} q^{8} + \zeta_{60}^{12} q^{9} +O(q^{10}) $ q + z^13 * q^2 - z^6 * q^3 + z^26 * q^4 + z^7 * q^5 - z^19 * q^6 - z^9 * q^8 + z^12 * q^9	\( q + \zeta_{60}^{13} q^{2} - \zeta_{60}^{6} q^{3} + \zeta_{60}^{26} q^{4} + \zeta_{60}^{7} q^{5} - \zeta_{60}^{19} q^{6} - \zeta_{60}^{9} q^{8} + \zeta_{60}^{12} q^{9} + \zeta_{60}^{20} q^{10} + \zeta_{60}^{2} q^{12} - \zeta_{60}^{13} q^{15} - \zeta_{60}^{22} q^{16} + ( - \zeta_{60}^{27} - \zeta_{60}^{22}) q^{17} + \zeta_{60}^{25} q^{18} + (\zeta_{60}^{27} + \zeta_{60}^{25}) q^{19} - \zeta_{60}^{3} q^{20} + ( - \zeta_{60}^{28} + \zeta_{60}^{11}) q^{23} + \zeta_{60}^{15} q^{24} + \zeta_{60}^{14} q^{25} - \zeta_{60}^{18} q^{27} - \zeta_{60}^{26} q^{30} + (\zeta_{60}^{7} - \zeta_{60}^{6}) q^{31} + \zeta_{60}^{5} q^{32} + (\zeta_{60}^{10} + \zeta_{60}^{5}) q^{34} - \zeta_{60}^{8} q^{36} + ( - \zeta_{60}^{10} - \zeta_{60}^{8}) q^{38} - \zeta_{60}^{16} q^{40} + \zeta_{60}^{19} q^{45} + (\zeta_{60}^{24} + \zeta_{60}^{11}) q^{46} + \zeta_{60}^{5} q^{47} + \zeta_{60}^{28} q^{48} - \zeta_{60}^{23} q^{49} + \zeta_{60}^{27} q^{50} + (\zeta_{60}^{28} - \zeta_{60}^{3}) q^{51} + (\zeta_{60}^{16} - \zeta_{60}^{5}) q^{53} + \zeta_{60} q^{54} + (\zeta_{60}^{3} + \zeta_{60}) q^{57} + \zeta_{60}^{9} q^{60} + \zeta_{60}^{17} q^{61} + (\zeta_{60}^{20} - \zeta_{60}^{19}) q^{62} + \zeta_{60}^{18} q^{64} + (\zeta_{60}^{23} + \zeta_{60}^{18}) q^{68} + ( - \zeta_{60}^{17} - \zeta_{60}^{4}) q^{69} - \zeta_{60}^{21} q^{72} - \zeta_{60}^{20} q^{75} + ( - \zeta_{60}^{23} - \zeta_{60}^{21}) q^{76} + (\zeta_{60}^{9} - \zeta_{60}^{2}) q^{79} - \zeta_{60}^{29} q^{80} + \zeta_{60}^{24} q^{81} + ( - \zeta_{60}^{24} + \zeta_{60}^{8}) q^{83} + ( - \zeta_{60}^{29} + \zeta_{60}^{4}) q^{85} - \zeta_{60}^{2} q^{90} + (\zeta_{60}^{24} - \zeta_{60}^{7}) q^{92} + ( - \zeta_{60}^{13} + \zeta_{60}^{12}) q^{93} + \zeta_{60}^{18} q^{94} + ( - \zeta_{60}^{4} - \zeta_{60}^{2}) q^{95} - \zeta_{60}^{11} q^{96} + \zeta_{60}^{6} q^{98} +O(q^{100}) \) q + z^13 * q^2 - z^6 * q^3 + z^26 * q^4 + z^7 * q^5 - z^19 * q^6 - z^9 * q^8 + z^12 * q^9 + z^20 * q^10 + z^2 * q^12 - z^13 * q^15 - z^22 * q^16 + (-z^27 - z^22) * q^17 + z^25 * q^18 + (z^27 + z^25) * q^19 - z^3 * q^20 + (-z^28 + z^11) * q^23 + z^15 * q^24 + z^14 * q^25 - z^18 * q^27 - z^26 * q^30 + (z^7 - z^6) * q^31 + z^5 * q^32 + (z^10 + z^5) * q^34 - z^8 * q^36 + (-z^10 - z^8) * q^38 - z^16 * q^40 + z^19 * q^45 + (z^24 + z^11) * q^46 + z^5 * q^47 + z^28 * q^48 - z^23 * q^49 + z^27 * q^50 + (z^28 - z^3) * q^51 + (z^16 - z^5) * q^53 + z * q^54 + (z^3 + z) * q^57 + z^9 * q^60 + z^17 * q^61 + (z^20 - z^19) * q^62 + z^18 * q^64 + (z^23 + z^18) * q^68 + (-z^17 - z^4) * q^69 - z^21 * q^72 - z^20 * q^75 + (-z^23 - z^21) * q^76 + (z^9 - z^2) * q^79 - z^29 * q^80 + z^24 * q^81 + (-z^24 + z^8) * q^83 + (-z^29 + z^4) * q^85 - z^2 * q^90 + (z^24 - z^7) * q^92 + (-z^13 + z^12) * q^93 + z^18 * q^94 + (-z^4 - z^2) * q^95 - z^11 * q^96 + z^6 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{3} - 2 q^{4} - 4 q^{9}+O(q^{10}) $ 16 * q - 4 * q^3 - 2 * q^4 - 4 * q^9	$ 16 q - 4 q^{3} - 2 q^{4} - 4 q^{9} - 8 q^{10} - 2 q^{12} + 2 q^{16} + 2 q^{17} - 2 q^{23} - 2 q^{25} - 4 q^{27} + 2 q^{30} - 4 q^{31} + 8 q^{34} - 2 q^{36} - 10 q^{38} - 2 q^{40} - 4 q^{46} + 2 q^{48} + 2 q^{51} + 2 q^{53} - 8 q^{62} + 4 q^{64} + 4 q^{68} - 2 q^{69} + 8 q^{75} + 2 q^{79} - 4 q^{81} + 6 q^{83} + 2 q^{85} + 2 q^{90} - 4 q^{92} - 4 q^{93} + 4 q^{94} + 4 q^{98}+O(q^{100}) $ 16 * q - 4 * q^3 - 2 * q^4 - 4 * q^9 - 8 * q^10 - 2 * q^12 + 2 * q^16 + 2 * q^17 - 2 * q^23 - 2 * q^25 - 4 * q^27 + 2 * q^30 - 4 * q^31 + 8 * q^34 - 2 * q^36 - 10 * q^38 - 2 * q^40 - 4 * q^46 + 2 * q^48 + 2 * q^51 + 2 * q^53 - 8 * q^62 + 4 * q^64 + 4 * q^68 - 2 * q^69 + 8 * q^75 + 2 * q^79 - 4 * q^81 + 6 * q^83 + 2 * q^85 + 2 * q^90 - 4 * q^92 - 4 * q^93 + 4 * q^94 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3660\mathbb{Z}\right)^\times$.

$n$	$1831$	$2197$	$2441$	$3601$
$\chi(n)$	$-1$	$-1$	$-1$	$-\zeta_{60}^{17}$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

59.1

0.743145	−	0.669131i
0.406737	+	0.913545i
−0.207912	+	0.978148i
−0.406737	+	0.913545i
−0.994522	−	0.104528i
−0.994522	+	0.104528i
0.207912	+	0.978148i
−0.743145	−	0.669131i
0.743145	+	0.669131i
−0.207912	−	0.978148i
0.994522	−	0.104528i
0.994522	+	0.104528i
0.406737	−	0.913545i
0.207912	−	0.978148i
−0.406737	−	0.913545i
−0.743145	+	0.669131i

−0.994522

0.104528i

0.309017

−

0.951057i

0.978148

−

0.207912i

0.406737

0.913545i

−0.207912

0.978148i

−0.951057

0.309017i

−0.809017

−

0.587785i

−0.500000

−

0.866025i

299.1

−0.743145

0.669131i

−0.809017

−

0.587785i

0.104528

−

0.994522i

−0.207912

0.978148i

0.994522

−

0.104528i

0.587785

0.809017i

0.309017

0.951057i

−0.500000

−

0.866025i

359.1

−0.406737

−

0.913545i

0.309017

0.951057i

−0.669131

0.743145i

0.994522

−

0.104528i

0.743145

−

0.669131i

0.951057

0.309017i

−0.809017

0.587785i

−0.500000

−

0.866025i

539.1

0.743145

0.669131i

−0.809017

0.587785i

0.104528

0.994522i

0.207912

0.978148i

−0.994522

−

0.104528i

−0.587785

0.809017i

0.309017

−

0.951057i

−0.500000

0.866025i

959.1

−0.207912

−

0.978148i

−0.809017

−

0.587785i

−0.913545

0.406737i

−0.743145

−

0.669131i

−0.406737

0.913545i

0.587785

0.809017i

0.309017

0.951057i

−0.500000

0.866025i

1019.1

−0.207912

0.978148i

−0.809017

0.587785i

−0.913545

−

0.406737i

−0.743145

0.669131i

−0.406737

−

0.913545i

0.587785

−

0.809017i

0.309017

−

0.951057i

−0.500000

−

0.866025i

1499.1

0.406737

−

0.913545i

0.309017

−

0.951057i

−0.669131

−

0.743145i

−0.994522

−

0.104528i

−0.743145

−

0.669131i

−0.951057

0.309017i

−0.809017

−

0.587785i

−0.500000

0.866025i

1739.1

0.994522

0.104528i

0.309017

0.951057i

0.978148

0.207912i

−0.406737

0.913545i

0.207912

0.978148i

0.951057

0.309017i

−0.809017

0.587785i

−0.500000

0.866025i

1799.1

−0.994522

−

0.104528i

0.309017

0.951057i

0.978148

0.207912i

0.406737

−

0.913545i

−0.207912

−

0.978148i

−0.951057

−

0.309017i

−0.809017

0.587785i

−0.500000

0.866025i

2039.1

−0.406737

0.913545i

0.309017

−

0.951057i

−0.669131

−

0.743145i

0.994522

0.104528i

0.743145

0.669131i

0.951057

−

0.309017i

−0.809017

−

0.587785i

−0.500000

0.866025i

2519.1

0.207912

−

0.978148i

−0.809017

0.587785i

−0.913545

−

0.406737i

0.743145

−

0.669131i

0.406737

0.913545i

−0.587785

0.809017i

0.309017

−

0.951057i

−0.500000

−

0.866025i

2579.1

0.207912

0.978148i

−0.809017

−

0.587785i

−0.913545

0.406737i

0.743145

0.669131i

0.406737

−

0.913545i

−0.587785

−

0.809017i

0.309017

0.951057i

−0.500000

0.866025i

2999.1

−0.743145

−

0.669131i

−0.809017

0.587785i

0.104528

0.994522i

−0.207912

−

0.978148i

0.994522

0.104528i

0.587785

−

0.809017i

0.309017

−

0.951057i

−0.500000

0.866025i

3179.1

0.406737

0.913545i

0.309017

0.951057i

−0.669131

0.743145i

−0.994522

0.104528i

−0.743145

0.669131i

−0.951057

−

0.309017i

−0.809017

0.587785i

−0.500000

−

0.866025i

3239.1

0.743145

−

0.669131i

−0.809017

−

0.587785i

0.104528

−

0.994522i

0.207912

−

0.978148i

−0.994522

0.104528i

−0.587785

−

0.809017i

0.309017

0.951057i

−0.500000

−

0.866025i

3479.1

0.994522

−

0.104528i

0.309017

−

0.951057i

0.978148

−

0.207912i

−0.406737

−

0.913545i

0.207912

−

0.978148i

0.951057

−

0.309017i

−0.809017

−

0.587785i

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
244.w	even	60	1	inner
3660.fu	odd	60	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3660.1.fu.b	yes	16
3.b	odd	2	1		3660.1.fu.c	yes	16
4.b	odd	2	1		3660.1.fu.a	✓	16
5.b	even	2	1		3660.1.fu.c	yes	16
12.b	even	2	1		3660.1.fu.d	yes	16
15.d	odd	2	1	CM	3660.1.fu.b	yes	16
20.d	odd	2	1		3660.1.fu.d	yes	16
60.h	even	2	1		3660.1.fu.a	✓	16
61.l	odd	60	1		3660.1.fu.a	✓	16
183.x	even	60	1		3660.1.fu.d	yes	16
244.w	even	60	1	inner	3660.1.fu.b	yes	16
305.bj	odd	60	1		3660.1.fu.d	yes	16
732.bv	odd	60	1		3660.1.fu.c	yes	16
915.cw	even	60	1		3660.1.fu.a	✓	16
1220.db	even	60	1		3660.1.fu.c	yes	16
3660.fu	odd	60	1	inner	3660.1.fu.b	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3660.1.fu.a	✓	16	4.b	odd	2	1
3660.1.fu.a	✓	16	60.h	even	2	1
3660.1.fu.a	✓	16	61.l	odd	60	1
3660.1.fu.a	✓	16	915.cw	even	60	1
3660.1.fu.b	yes	16	1.a	even	1	1	trivial
3660.1.fu.b	yes	16	15.d	odd	2	1	CM
3660.1.fu.b	yes	16	244.w	even	60	1	inner
3660.1.fu.b	yes	16	3660.fu	odd	60	1	inner
3660.1.fu.c	yes	16	3.b	odd	2	1
3660.1.fu.c	yes	16	5.b	even	2	1
3660.1.fu.c	yes	16	732.bv	odd	60	1
3660.1.fu.c	yes	16	1220.db	even	60	1
3660.1.fu.d	yes	16	12.b	even	2	1
3660.1.fu.d	yes	16	20.d	odd	2	1
3660.1.fu.d	yes	16	183.x	even	60	1
3660.1.fu.d	yes	16	305.bj	odd	60	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3660, [\chi])$:

$ T_{7} $

$ T_{17}^{16} - 2 T_{17}^{15} - T_{17}^{14} + 8 T_{17}^{13} - 4 T_{17}^{12} + 26 T_{17}^{11} - 63 T_{17}^{10} + \cdots + 1 $

$ T_{23}^{16} + 2 T_{23}^{15} + 2 T_{23}^{14} + 8 T_{23}^{13} + 7 T_{23}^{12} - 10 T_{23}^{11} + 23 T_{23}^{10} + \cdots + 1 $

$ T_{29} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + T^{14} + \cdots + 1 $ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$3$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} $ (T^4 + T^3 + T^2 + T + 1)^4
$5$	$ T^{16} + T^{14} + \cdots + 1 $ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ T^{16} $ T^16
$17$	$ T^{16} - 2 T^{15} + \cdots + 1 $ T^16 - 2T^15 - T^14 + 8T^13 - 4T^12 + 26T^11 - 63T^10 + 4T^9 + 207T^8 - 208T^7 + 153T^6 - 98T^5 + 56T^4 - 26T^3 + 11T^2 - 4T + 1
$19$	$ T^{16} - 8 T^{14} + \cdots + 1 $ T^16 - 8T^14 + 30T^12 - 38T^10 - 21T^8 + 58T^6 + 35T^4 + 3*T^2 + 1
$23$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 2T^14 + 8T^13 + 7T^12 - 10T^11 + 23T^10 + 44T^9 + 106T^7 + 103T^6 + 50T^5 + 207T^4 - 38T^3 + 77T^2 + 18T + 1
$29$	$ T^{16} $ T^16
$31$	$ T^{16} + 4 T^{15} + \cdots + 1 $ T^16 + 4T^15 + 11T^14 + 16T^13 + 16T^12 - 2T^11 + 8T^10 + 58T^9 + 177T^8 + 206T^7 + 52T^6 - 176T^5 - 204T^4 - 28T^3 + 89T^2 + 12*T + 1
$37$	$ T^{16} $ T^16
$41$	$ T^{16} $ T^16
$43$	$ T^{16} $ T^16
$47$	$ (T^{4} - T^{2} + 1)^{4} $ (T^4 - T^2 + 1)^4
$53$	$ T^{16} - 2 T^{15} + \cdots + 1 $ T^16 - 2T^15 - 3T^14 + 12T^13 - 8T^12 - 10T^11 + 33T^10 - 24T^9 + 15T^8 - 36T^7 - 87T^6 + 20T^5 + 142T^4 + 48T^3 + 87T^2 - 8*T + 1
$59$	$ T^{16} $ T^16
$61$	$ T^{16} + T^{14} + \cdots + 1 $ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$67$	$ T^{16} $ T^16
$71$	$ T^{16} $ T^16
$73$	$ T^{16} $ T^16
$79$	$ T^{16} - 2 T^{15} + \cdots + 1 $ T^16 - 2T^15 - T^14 - 2T^13 + 16T^12 - 14T^11 + 32T^10 - 56T^9 + 42T^8 - 148T^7 + 283T^6 - 188T^5 + 36T^4 - 76T^3 + 86T^2 + 6T + 1
$83$	$ (T^{8} - 3 T^{7} + 6 T^{6} + \cdots + 1)^{2} $ (T^8 - 3T^7 + 6T^6 - 14T^5 + 19T^4 - 12T^3 + 4T^2 - T + 1)^2
$89$	$ T^{16} $ T^16
$97$	$ T^{16} $ T^16
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3660 = 2^{2} \cdot 3 \cdot 5 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3660.fu (of order \(60\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{60}^{13} q^{2} - \zeta_{60}^{6} q^{3} + \zeta_{60}^{26} q^{4} + \zeta_{60}^{7} q^{5} - \zeta_{60}^{19} q^{6} - \zeta_{60}^{9} q^{8} + \zeta_{60}^{12} q^{9} +O(q^{10}) \) q + z^13 * q^2 - z^6 * q^3 + z^26 * q^4 + z^7 * q^5 - z^19 * q^6 - z^9 * q^8 + z^12 * q^9	\( q + \zeta_{60}^{13} q^{2} - \zeta_{60}^{6} q^{3} + \zeta_{60}^{26} q^{4} + \zeta_{60}^{7} q^{5} - \zeta_{60}^{19} q^{6} - \zeta_{60}^{9} q^{8} + \zeta_{60}^{12} q^{9} + \zeta_{60}^{20} q^{10} + \zeta_{60}^{2} q^{12} - \zeta_{60}^{13} q^{15} - \zeta_{60}^{22} q^{16} + ( - \zeta_{60}^{27} - \zeta_{60}^{22}) q^{17} + \zeta_{60}^{25} q^{18} + (\zeta_{60}^{27} + \zeta_{60}^{25}) q^{19} - \zeta_{60}^{3} q^{20} + ( - \zeta_{60}^{28} + \zeta_{60}^{11}) q^{23} + \zeta_{60}^{15} q^{24} + \zeta_{60}^{14} q^{25} - \zeta_{60}^{18} q^{27} - \zeta_{60}^{26} q^{30} + (\zeta_{60}^{7} - \zeta_{60}^{6}) q^{31} + \zeta_{60}^{5} q^{32} + (\zeta_{60}^{10} + \zeta_{60}^{5}) q^{34} - \zeta_{60}^{8} q^{36} + ( - \zeta_{60}^{10} - \zeta_{60}^{8}) q^{38} - \zeta_{60}^{16} q^{40} + \zeta_{60}^{19} q^{45} + (\zeta_{60}^{24} + \zeta_{60}^{11}) q^{46} + \zeta_{60}^{5} q^{47} + \zeta_{60}^{28} q^{48} - \zeta_{60}^{23} q^{49} + \zeta_{60}^{27} q^{50} + (\zeta_{60}^{28} - \zeta_{60}^{3}) q^{51} + (\zeta_{60}^{16} - \zeta_{60}^{5}) q^{53} + \zeta_{60} q^{54} + (\zeta_{60}^{3} + \zeta_{60}) q^{57} + \zeta_{60}^{9} q^{60} + \zeta_{60}^{17} q^{61} + (\zeta_{60}^{20} - \zeta_{60}^{19}) q^{62} + \zeta_{60}^{18} q^{64} + (\zeta_{60}^{23} + \zeta_{60}^{18}) q^{68} + ( - \zeta_{60}^{17} - \zeta_{60}^{4}) q^{69} - \zeta_{60}^{21} q^{72} - \zeta_{60}^{20} q^{75} + ( - \zeta_{60}^{23} - \zeta_{60}^{21}) q^{76} + (\zeta_{60}^{9} - \zeta_{60}^{2}) q^{79} - \zeta_{60}^{29} q^{80} + \zeta_{60}^{24} q^{81} + ( - \zeta_{60}^{24} + \zeta_{60}^{8}) q^{83} + ( - \zeta_{60}^{29} + \zeta_{60}^{4}) q^{85} - \zeta_{60}^{2} q^{90} + (\zeta_{60}^{24} - \zeta_{60}^{7}) q^{92} + ( - \zeta_{60}^{13} + \zeta_{60}^{12}) q^{93} + \zeta_{60}^{18} q^{94} + ( - \zeta_{60}^{4} - \zeta_{60}^{2}) q^{95} - \zeta_{60}^{11} q^{96} + \zeta_{60}^{6} q^{98} +O(q^{100}) \) q + z^13 * q^2 - z^6 * q^3 + z^26 * q^4 + z^7 * q^5 - z^19 * q^6 - z^9 * q^8 + z^12 * q^9 + z^20 * q^10 + z^2 * q^12 - z^13 * q^15 - z^22 * q^16 + (-z^27 - z^22) * q^17 + z^25 * q^18 + (z^27 + z^25) * q^19 - z^3 * q^20 + (-z^28 + z^11) * q^23 + z^15 * q^24 + z^14 * q^25 - z^18 * q^27 - z^26 * q^30 + (z^7 - z^6) * q^31 + z^5 * q^32 + (z^10 + z^5) * q^34 - z^8 * q^36 + (-z^10 - z^8) * q^38 - z^16 * q^40 + z^19 * q^45 + (z^24 + z^11) * q^46 + z^5 * q^47 + z^28 * q^48 - z^23 * q^49 + z^27 * q^50 + (z^28 - z^3) * q^51 + (z^16 - z^5) * q^53 + z * q^54 + (z^3 + z) * q^57 + z^9 * q^60 + z^17 * q^61 + (z^20 - z^19) * q^62 + z^18 * q^64 + (z^23 + z^18) * q^68 + (-z^17 - z^4) * q^69 - z^21 * q^72 - z^20 * q^75 + (-z^23 - z^21) * q^76 + (z^9 - z^2) * q^79 - z^29 * q^80 + z^24 * q^81 + (-z^24 + z^8) * q^83 + (-z^29 + z^4) * q^85 - z^2 * q^90 + (z^24 - z^7) * q^92 + (-z^13 + z^12) * q^93 + z^18 * q^94 + (-z^4 - z^2) * q^95 - z^11 * q^96 + z^6 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 2 q^{4} - 4 q^{9}+O(q^{10}) \) 16 * q - 4 * q^3 - 2 * q^4 - 4 * q^9	\( 16 q - 4 q^{3} - 2 q^{4} - 4 q^{9} - 8 q^{10} - 2 q^{12} + 2 q^{16} + 2 q^{17} - 2 q^{23} - 2 q^{25} - 4 q^{27} + 2 q^{30} - 4 q^{31} + 8 q^{34} - 2 q^{36} - 10 q^{38} - 2 q^{40} - 4 q^{46} + 2 q^{48} + 2 q^{51} + 2 q^{53} - 8 q^{62} + 4 q^{64} + 4 q^{68} - 2 q^{69} + 8 q^{75} + 2 q^{79} - 4 q^{81} + 6 q^{83} + 2 q^{85} + 2 q^{90} - 4 q^{92} - 4 q^{93} + 4 q^{94} + 4 q^{98}+O(q^{100}) \) 16 * q - 4 * q^3 - 2 * q^4 - 4 * q^9 - 8 * q^10 - 2 * q^12 + 2 * q^16 + 2 * q^17 - 2 * q^23 - 2 * q^25 - 4 * q^27 + 2 * q^30 - 4 * q^31 + 8 * q^34 - 2 * q^36 - 10 * q^38 - 2 * q^40 - 4 * q^46 + 2 * q^48 + 2 * q^51 + 2 * q^53 - 8 * q^62 + 4 * q^64 + 4 * q^68 - 2 * q^69 + 8 * q^75 + 2 * q^79 - 4 * q^81 + 6 * q^83 + 2 * q^85 + 2 * q^90 - 4 * q^92 - 4 * q^93 + 4 * q^94 + 4 * q^98

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

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	3660.1.fu.b

Level	$3660$
Weight	$1$
Character orbit	3660.fu
Analytic conductor	$1.827$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{60}$
CM discriminant	-15
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.82657794624\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\Q(\zeta_{60})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \) x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{60}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{60} + \cdots)\)

\(n\)	\(1831\)	\(2197\)	\(2441\)	\(3601\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(-\zeta_{60}^{17}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 59.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + T^{14} + \cdots + 1 \) T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$3$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) (T^4 + T^3 + T^2 + T + 1)^4
$5$	\( T^{16} + T^{14} + \cdots + 1 \) T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( T^{16} \) T^16
$17$	\( T^{16} - 2 T^{15} + \cdots + 1 \) T^16 - 2T^15 - T^14 + 8T^13 - 4T^12 + 26T^11 - 63T^10 + 4T^9 + 207T^8 - 208T^7 + 153T^6 - 98T^5 + 56T^4 - 26T^3 + 11T^2 - 4T + 1
$19$	\( T^{16} - 8 T^{14} + \cdots + 1 \) T^16 - 8T^14 + 30T^12 - 38T^10 - 21T^8 + 58T^6 + 35T^4 + 3*T^2 + 1
$23$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 2T^14 + 8T^13 + 7T^12 - 10T^11 + 23T^10 + 44T^9 + 106T^7 + 103T^6 + 50T^5 + 207T^4 - 38T^3 + 77T^2 + 18T + 1
$29$	\( T^{16} \) T^16
$31$	\( T^{16} + 4 T^{15} + \cdots + 1 \) T^16 + 4T^15 + 11T^14 + 16T^13 + 16T^12 - 2T^11 + 8T^10 + 58T^9 + 177T^8 + 206T^7 + 52T^6 - 176T^5 - 204T^4 - 28T^3 + 89T^2 + 12*T + 1
$37$	\( T^{16} \) T^16
$41$	\( T^{16} \) T^16
$43$	\( T^{16} \) T^16
$47$	\( (T^{4} - T^{2} + 1)^{4} \) (T^4 - T^2 + 1)^4
$53$	\( T^{16} - 2 T^{15} + \cdots + 1 \) T^16 - 2T^15 - 3T^14 + 12T^13 - 8T^12 - 10T^11 + 33T^10 - 24T^9 + 15T^8 - 36T^7 - 87T^6 + 20T^5 + 142T^4 + 48T^3 + 87T^2 - 8*T + 1
$59$	\( T^{16} \) T^16
$61$	\( T^{16} + T^{14} + \cdots + 1 \) T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$67$	\( T^{16} \) T^16
$71$	\( T^{16} \) T^16
$73$	\( T^{16} \) T^16
$79$	\( T^{16} - 2 T^{15} + \cdots + 1 \) T^16 - 2T^15 - T^14 - 2T^13 + 16T^12 - 14T^11 + 32T^10 - 56T^9 + 42T^8 - 148T^7 + 283T^6 - 188T^5 + 36T^4 - 76T^3 + 86T^2 + 6T + 1
$83$	\( (T^{8} - 3 T^{7} + 6 T^{6} + \cdots + 1)^{2} \) (T^8 - 3T^7 + 6T^6 - 14T^5 + 19T^4 - 12T^3 + 4T^2 - T + 1)^2
$89$	\( T^{16} \) T^16
$97$	\( T^{16} \) T^16
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