LMFDB - Newform orbit 3660.1.fu.e

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:	$32$
Relative dimension:	$2$ over $\Q(\zeta_{60})$
Coefficient field:	$\Q(\zeta_{120})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 $ x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 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_{120}^{29} q^{2} + \zeta_{120}^{43} q^{3} + \zeta_{120}^{58} q^{4} - \zeta_{120}^{8} q^{5} + \zeta_{120}^{12} q^{6} + ( - \zeta_{120}^{59} + \zeta_{120}^{23}) q^{7} + \zeta_{120}^{27} q^{8} - \zeta_{120}^{26} q^{9} +O(q^{10}) $ q - z^29 * q^2 + z^43 * q^3 + z^58 * q^4 - z^8 * q^5 + z^12 * q^6 + (-z^59 + z^23) * q^7 + z^27 * q^8 - z^26 * q^9	\( q - \zeta_{120}^{29} q^{2} + \zeta_{120}^{43} q^{3} + \zeta_{120}^{58} q^{4} - \zeta_{120}^{8} q^{5} + \zeta_{120}^{12} q^{6} + ( - \zeta_{120}^{59} + \zeta_{120}^{23}) q^{7} + \zeta_{120}^{27} q^{8} - \zeta_{120}^{26} q^{9} + \zeta_{120}^{37} q^{10} - \zeta_{120}^{41} q^{12} + ( - \zeta_{120}^{52} - \zeta_{120}^{28}) q^{14} - \zeta_{120}^{51} q^{15} - \zeta_{120}^{56} q^{16} + \zeta_{120}^{55} q^{18} + \zeta_{120}^{6} q^{20} + (\zeta_{120}^{42} - \zeta_{120}^{6}) q^{21} + ( - \zeta_{120}^{47} - \zeta_{120}^{19}) q^{23} - \zeta_{120}^{10} q^{24} + \zeta_{120}^{16} q^{25} + \zeta_{120}^{9} q^{27} + (\zeta_{120}^{57} - \zeta_{120}^{21}) q^{28} + (\zeta_{120}^{38} - \zeta_{120}^{12}) q^{29} - \zeta_{120}^{20} q^{30} - \zeta_{120}^{25} q^{32} + ( - \zeta_{120}^{31} - \zeta_{120}^{7}) q^{35} + \zeta_{120}^{24} q^{36} - \zeta_{120}^{35} q^{40} + (\zeta_{120}^{40} + \zeta_{120}^{32}) q^{41} + (\zeta_{120}^{35} + \zeta_{120}^{11}) q^{42} + (\zeta_{120}^{49} - \zeta_{120}^{45}) q^{43} + \zeta_{120}^{34} q^{45} + (\zeta_{120}^{48} - \zeta_{120}^{16}) q^{46} + (\zeta_{120}^{19} + \zeta_{120}) q^{47} + \zeta_{120}^{39} q^{48} + ( - \zeta_{120}^{58} + \cdots + \zeta_{120}^{22}) q^{49} + \cdots + ( - \zeta_{120}^{51} + \cdots + \zeta_{120}^{15}) q^{98} +O(q^{100}) \) q - z^29 * q^2 + z^43 * q^3 + z^58 * q^4 - z^8 * q^5 + z^12 * q^6 + (-z^59 + z^23) * q^7 + z^27 * q^8 - z^26 * q^9 + z^37 * q^10 - z^41 * q^12 + (-z^52 - z^28) * q^14 - z^51 * q^15 - z^56 * q^16 + z^55 * q^18 + z^6 * q^20 + (z^42 - z^6) * q^21 + (-z^47 - z^19) * q^23 - z^10 * q^24 + z^16 * q^25 + z^9 * q^27 + (z^57 - z^21) * q^28 + (z^38 - z^12) * q^29 - z^20 * q^30 - z^25 * q^32 + (-z^31 - z^7) * q^35 + z^24 * q^36 - z^35 * q^40 + (z^40 + z^32) * q^41 + (z^35 + z^11) * q^42 + (z^49 - z^45) * q^43 + z^34 * q^45 + (z^48 - z^16) * q^46 + (z^19 + z) * q^47 + z^39 * q^48 + (-z^58 + z^46 + z^22) * q^49 - z^45 * q^50 - z^38 * q^54 + (z^50 + z^26) * q^56 + (z^41 + z^7) * q^58 + z^49 * q^60 - z^54 * q^61 + (-z^49 - z^25) * q^63 + z^54 * q^64 - z^53 * q^67 + (z^30 + z^2) * q^69 + (z^36 - 1) * q^70 - z^53 * q^72 + z^59 * q^75 - z^4 * q^80 + z^52 * q^81 + (z^9 + z) * q^82 + (-z^53 + z^35) * q^83 + (-z^40 + z^4) * q^84 + (z^18 - z^14) * q^86 + (-z^55 - z^21) * q^87 + (-z^44 - z^34) * q^89 + z^3 * q^90 + (z^45 + z^17) * q^92 + (-z^48 - z^30) * q^94 + z^8 * q^96 + (-z^51 - z^27 + z^15) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 4 q^{5} + 8 q^{6}+O(q^{10}) $ 32 * q - 4 * q^5 + 8 * q^6	$ 32 q - 4 q^{5} + 8 q^{6} + 8 q^{14} - 4 q^{16} + 4 q^{25} - 8 q^{29} - 16 q^{30} - 8 q^{36} - 12 q^{41} - 12 q^{46} - 24 q^{70} + 4 q^{80} - 4 q^{81} + 12 q^{84} + 4 q^{89} + 8 q^{94} + 4 q^{96}+O(q^{100}) $ 32 * q - 4 * q^5 + 8 * q^6 + 8 * q^14 - 4 * q^16 + 4 * q^25 - 8 * q^29 - 16 * q^30 - 8 * q^36 - 12 * q^41 - 12 * q^46 - 24 * q^70 + 4 * q^80 - 4 * q^81 + 12 * q^84 + 4 * q^89 + 8 * q^94 + 4 * q^96

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1831$	$2197$	$2441$	$3601$
$\chi(n)$	$-1$	$-1$	$-1$	$-\zeta_{120}^{58}$

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

0.0523360	+	0.998630i
−0.0523360	−	0.998630i
−0.358368	−	0.933580i
0.358368	+	0.933580i
−0.544639	+	0.838671i
0.544639	−	0.838671i
−0.933580	−	0.358368i
0.933580	+	0.358368i
0.629320	−	0.777146i
−0.629320	+	0.777146i
0.629320	+	0.777146i
−0.629320	−	0.777146i
−0.838671	+	0.544639i
0.838671	−	0.544639i
0.998630	+	0.0523360i
−0.998630	−	0.0523360i
0.0523360	−	0.998630i
−0.0523360	+	0.998630i
−0.544639	−	0.838671i
0.544639	+	0.838671i

−0.998630

−

0.0523360i

−0.777146

0.629320i

0.994522

0.104528i

−0.913545

0.406737i

0.809017

−

0.587785i

−0.881244

−

1.35700i

−0.987688

−

0.156434i

0.207912

−

0.978148i

0.933580

−

0.358368i

59.2

0.998630

0.0523360i

0.777146

−

0.629320i

0.994522

0.104528i

−0.913545

0.406737i

0.809017

−

0.587785i

0.881244

1.35700i

0.987688

0.156434i

0.207912

−

0.978148i

−0.933580

0.358368i

299.1

−0.933580

−

0.358368i

−0.0523360

−

0.998630i

0.743145

0.669131i

0.978148

0.207912i

−0.309017

0.951057i

0.480303

0.388941i

−0.453990

−

0.891007i

−0.994522

0.104528i

−0.838671

−

0.544639i

299.2

0.933580

0.358368i

0.0523360

0.998630i

0.743145

0.669131i

0.978148

0.207912i

−0.309017

0.951057i

−0.480303

−

0.388941i

0.453990

0.891007i

−0.994522

0.104528i

0.838671

0.544639i

359.1

−0.838671

0.544639i

−0.358368

−

0.933580i

0.406737

−

0.913545i

0.104528

0.994522i

0.809017

0.587785i

0.0846814

−

1.61582i

0.156434

0.987688i

−0.743145

0.669131i

−0.629320

−

0.777146i

359.2

0.838671

−

0.544639i

0.358368

0.933580i

0.406737

−

0.913545i

0.104528

0.994522i

0.809017

0.587785i

−0.0846814

1.61582i

−0.156434

−

0.987688i

−0.743145

0.669131i

0.629320

0.777146i

539.1

−0.358368

−

0.933580i

0.998630

0.0523360i

−0.743145

0.669131i

0.978148

−

0.207912i

−0.309017

−

0.951057i

−0.388941

−

0.480303i

0.891007

0.453990i

0.994522

0.104528i

−0.544639

−

0.838671i

539.2

0.358368

0.933580i

−0.998630

−

0.0523360i

−0.743145

0.669131i

0.978148

−

0.207912i

−0.309017

−

0.951057i

0.388941

0.480303i

−0.891007

−

0.453990i

0.994522

0.104528i

0.544639

0.838671i

959.1

−0.777146

0.629320i

0.838671

−

0.544639i

0.207912

−

0.978148i

−0.669131

0.743145i

−0.309017

0.951057i

0.576984

−

0.221484i

0.453990

0.891007i

0.406737

−

0.913545i

0.0523360

−

0.998630i

959.2

0.777146

−

0.629320i

−0.838671

0.544639i

0.207912

−

0.978148i

−0.669131

0.743145i

−0.309017

0.951057i

−0.576984

0.221484i

−0.453990

−

0.891007i

0.406737

−

0.913545i

−0.0523360

0.998630i

1019.1

−0.777146

−

0.629320i

0.838671

0.544639i

0.207912

0.978148i

−0.669131

−

0.743145i

−0.309017

−

0.951057i

0.576984

0.221484i

0.453990

−

0.891007i

0.406737

0.913545i

0.0523360

0.998630i

1019.2

0.777146

0.629320i

−0.838671

−

0.544639i

0.207912

0.978148i

−0.669131

−

0.743145i

−0.309017

−

0.951057i

−0.576984

−

0.221484i

−0.453990

0.891007i

0.406737

0.913545i

−0.0523360

−

0.998630i

1499.1

−0.544639

0.838671i

−0.933580

−

0.358368i

−0.406737

−

0.913545i

0.104528

−

0.994522i

0.809017

−

0.587785i

−1.61582

0.0846814i

0.987688

0.156434i

0.743145

0.669131i

0.777146

0.629320i

1499.2

0.544639

−

0.838671i

0.933580

0.358368i

−0.406737

−

0.913545i

0.104528

−

0.994522i

0.809017

−

0.587785i

1.61582

−

0.0846814i

−0.987688

−

0.156434i

0.743145

0.669131i

−0.777146

−

0.629320i

1739.1

−0.0523360

−

0.998630i

−0.629320

0.777146i

−0.994522

0.104528i

−0.913545

−

0.406737i

0.809017

0.587785i

1.35700

0.881244i

0.156434

0.987688i

−0.207912

−

0.978148i

−0.358368

0.933580i

1739.2

0.0523360

0.998630i

0.629320

−

0.777146i

−0.994522

0.104528i

−0.913545

−

0.406737i

0.809017

0.587785i

−1.35700

−

0.881244i

−0.156434

−

0.987688i

−0.207912

−

0.978148i

0.358368

−

0.933580i

1799.1

−0.998630

0.0523360i

−0.777146

−

0.629320i

0.994522

−

0.104528i

−0.913545

−

0.406737i

0.809017

0.587785i

−0.881244

1.35700i

−0.987688

0.156434i

0.207912

0.978148i

0.933580

0.358368i

1799.2

0.998630

−

0.0523360i

0.777146

0.629320i

0.994522

−

0.104528i

−0.913545

−

0.406737i

0.809017

0.587785i

0.881244

−

1.35700i

0.987688

−

0.156434i

0.207912

0.978148i

−0.933580

−

0.358368i

2039.1

−0.838671

−

0.544639i

−0.358368

0.933580i

0.406737

0.913545i

0.104528

−

0.994522i

0.809017

−

0.587785i

0.0846814

1.61582i

0.156434

−

0.987688i

−0.743145

−

0.669131i

−0.629320

0.777146i

2039.2

0.838671

0.544639i

0.358368

−

0.933580i

0.406737

0.913545i

0.104528

−

0.994522i

0.809017

−

0.587785i

−0.0846814

−

1.61582i

−0.156434

0.987688i

−0.743145

−

0.669131i

0.629320

−

0.777146i

See all 32 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5}) $
4.b	odd	2	1	inner
5.b	even	2	1	inner
183.x	even	60	1	inner
732.bv	odd	60	1	inner
915.cw	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.e	✓	32
3.b	odd	2	1		3660.1.fu.f	yes	32
4.b	odd	2	1	inner	3660.1.fu.e	✓	32
5.b	even	2	1	inner	3660.1.fu.e	✓	32
12.b	even	2	1		3660.1.fu.f	yes	32
15.d	odd	2	1		3660.1.fu.f	yes	32
20.d	odd	2	1	CM	3660.1.fu.e	✓	32
60.h	even	2	1		3660.1.fu.f	yes	32
61.l	odd	60	1		3660.1.fu.f	yes	32
183.x	even	60	1	inner	3660.1.fu.e	✓	32
244.w	even	60	1		3660.1.fu.f	yes	32
305.bj	odd	60	1		3660.1.fu.f	yes	32
732.bv	odd	60	1	inner	3660.1.fu.e	✓	32
915.cw	even	60	1	inner	3660.1.fu.e	✓	32
1220.db	even	60	1		3660.1.fu.f	yes	32
3660.fu	odd	60	1	inner	3660.1.fu.e	✓	32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3660.1.fu.e	✓	32	1.a	even	1	1	trivial
3660.1.fu.e	✓	32	4.b	odd	2	1	inner
3660.1.fu.e	✓	32	5.b	even	2	1	inner
3660.1.fu.e	✓	32	20.d	odd	2	1	CM
3660.1.fu.e	✓	32	183.x	even	60	1	inner
3660.1.fu.e	✓	32	732.bv	odd	60	1	inner
3660.1.fu.e	✓	32	915.cw	even	60	1	inner
3660.1.fu.e	✓	32	3660.fu	odd	60	1	inner
3660.1.fu.f	yes	32	3.b	odd	2	1
3660.1.fu.f	yes	32	12.b	even	2	1
3660.1.fu.f	yes	32	15.d	odd	2	1
3660.1.fu.f	yes	32	60.h	even	2	1
3660.1.fu.f	yes	32	61.l	odd	60	1
3660.1.fu.f	yes	32	244.w	even	60	1
3660.1.fu.f	yes	32	305.bj	odd	60	1
3660.1.fu.f	yes	32	1220.db	even	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}^{32} - 4T_{7}^{28} - 30T_{7}^{24} - 206T_{7}^{20} + 2159T_{7}^{16} + 514T_{7}^{12} + 75T_{7}^{8} + 11T_{7}^{4} + 1 $

$ T_{17} $

$ T_{23}^{32} + 2T_{23}^{28} + 193T_{23}^{24} - 1661T_{23}^{20} + 5490T_{23}^{16} + 989T_{23}^{12} + 1063T_{23}^{8} - 53T_{23}^{4} + 1 $

$ T_{29}^{16} + 4 T_{29}^{15} + 11 T_{29}^{14} + 26 T_{29}^{13} + 41 T_{29}^{12} + 48 T_{29}^{11} + \cdots + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{32} + T^{28} + \cdots + 1 $ T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$3$	$ T^{32} + T^{28} + \cdots + 1 $ T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$5$	$ (T^{8} + T^{7} - T^{5} + \cdots + 1)^{4} $ (T^8 + T^7 - T^5 - T^4 - T^3 + T + 1)^4
$7$	$ T^{32} - 4 T^{28} + \cdots + 1 $ T^32 - 4T^28 - 30T^24 - 206T^20 + 2159T^16 + 514T^12 + 75T^8 + 11*T^4 + 1
$11$	$ T^{32} $ T^32
$13$	$ T^{32} $ T^32
$17$	$ T^{32} $ T^32
$19$	$ T^{32} $ T^32
$23$	$ T^{32} + 2 T^{28} + \cdots + 1 $ T^32 + 2T^28 + 193T^24 - 1661T^20 + 5490T^16 + 989T^12 + 1063T^8 - 53*T^4 + 1
$29$	$ (T^{16} + 4 T^{15} + \cdots + 1)^{2} $ (T^16 + 4T^15 + 11T^14 + 26T^13 + 41T^12 + 48T^11 + 28T^10 - 52T^9 - 93T^8 - 84T^7 + 2T^6 + 144T^5 + 76T^4 + 22T^3 + 84T^2 - 18*T + 1)^2
$31$	$ T^{32} $ T^32
$37$	$ T^{32} $ T^32
$41$	$ (T^{8} + 3 T^{7} + 8 T^{6} + \cdots + 1)^{4} $ (T^8 + 3T^7 + 8T^6 + 11T^5 + 15T^4 + 11T^3 + 18T^2 - 7*T + 1)^4
$43$	$ T^{32} + 11 T^{28} + \cdots + 1 $ T^32 + 11T^28 + 195T^24 + 274T^20 + 17519T^16 - 147206T^12 + 273570T^8 + 956*T^4 + 1
$47$	$ (T^{16} - 8 T^{14} + \cdots + 1)^{2} $ (T^16 - 8T^14 + 45T^12 - 128T^10 + 264T^8 - 212T^6 + 125T^4 - 12*T^2 + 1)^2
$53$	$ T^{32} $ T^32
$59$	$ T^{32} $ T^32
$61$	$ (T^{8} - T^{6} + T^{4} + \cdots + 1)^{4} $ (T^8 - T^6 + T^4 - T^2 + 1)^4
$67$	$ T^{32} + T^{28} + \cdots + 1 $ T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$71$	$ T^{32} $ T^32
$73$	$ T^{32} $ T^32
$79$	$ T^{32} $ T^32
$83$	$ T^{32} + 4 T^{30} + \cdots + 1 $ T^32 + 4T^30 - T^28 - 76T^26 - 229T^24 - 388T^22 + 546T^20 + 9172T^18 + 34887T^16 + 39128T^14 + 31146T^12 + 22888T^10 + 9146T^8 - 824T^6 - 76T^4 - 4T^2 + 1
$89$	$ (T^{16} - 2 T^{15} + \cdots + 1)^{2} $ (T^16 - 2T^15 + 2T^14 + 2T^13 - 13T^12 - 10T^11 + 48T^10 - 104T^9 + 105T^8 + 104T^7 + 48T^6 + 10T^5 - 13T^4 - 2T^3 + 2T^2 + 2*T + 1)^2
$97$	$ T^{32} $ T^32
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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:	\( N \)	\(=\)	\( 3660 = 2^{2} \cdot 3 \cdot 5 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3660.fu (of order \(60\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{120}^{29} q^{2} + \zeta_{120}^{43} q^{3} + \zeta_{120}^{58} q^{4} - \zeta_{120}^{8} q^{5} + \zeta_{120}^{12} q^{6} + ( - \zeta_{120}^{59} + \zeta_{120}^{23}) q^{7} + \zeta_{120}^{27} q^{8} - \zeta_{120}^{26} q^{9} +O(q^{10}) \) q - z^29 * q^2 + z^43 * q^3 + z^58 * q^4 - z^8 * q^5 + z^12 * q^6 + (-z^59 + z^23) * q^7 + z^27 * q^8 - z^26 * q^9	\( q - \zeta_{120}^{29} q^{2} + \zeta_{120}^{43} q^{3} + \zeta_{120}^{58} q^{4} - \zeta_{120}^{8} q^{5} + \zeta_{120}^{12} q^{6} + ( - \zeta_{120}^{59} + \zeta_{120}^{23}) q^{7} + \zeta_{120}^{27} q^{8} - \zeta_{120}^{26} q^{9} + \zeta_{120}^{37} q^{10} - \zeta_{120}^{41} q^{12} + ( - \zeta_{120}^{52} - \zeta_{120}^{28}) q^{14} - \zeta_{120}^{51} q^{15} - \zeta_{120}^{56} q^{16} + \zeta_{120}^{55} q^{18} + \zeta_{120}^{6} q^{20} + (\zeta_{120}^{42} - \zeta_{120}^{6}) q^{21} + ( - \zeta_{120}^{47} - \zeta_{120}^{19}) q^{23} - \zeta_{120}^{10} q^{24} + \zeta_{120}^{16} q^{25} + \zeta_{120}^{9} q^{27} + (\zeta_{120}^{57} - \zeta_{120}^{21}) q^{28} + (\zeta_{120}^{38} - \zeta_{120}^{12}) q^{29} - \zeta_{120}^{20} q^{30} - \zeta_{120}^{25} q^{32} + ( - \zeta_{120}^{31} - \zeta_{120}^{7}) q^{35} + \zeta_{120}^{24} q^{36} - \zeta_{120}^{35} q^{40} + (\zeta_{120}^{40} + \zeta_{120}^{32}) q^{41} + (\zeta_{120}^{35} + \zeta_{120}^{11}) q^{42} + (\zeta_{120}^{49} - \zeta_{120}^{45}) q^{43} + \zeta_{120}^{34} q^{45} + (\zeta_{120}^{48} - \zeta_{120}^{16}) q^{46} + (\zeta_{120}^{19} + \zeta_{120}) q^{47} + \zeta_{120}^{39} q^{48} + ( - \zeta_{120}^{58} + \cdots + \zeta_{120}^{22}) q^{49} + \cdots + ( - \zeta_{120}^{51} + \cdots + \zeta_{120}^{15}) q^{98} +O(q^{100}) \) q - z^29 * q^2 + z^43 * q^3 + z^58 * q^4 - z^8 * q^5 + z^12 * q^6 + (-z^59 + z^23) * q^7 + z^27 * q^8 - z^26 * q^9 + z^37 * q^10 - z^41 * q^12 + (-z^52 - z^28) * q^14 - z^51 * q^15 - z^56 * q^16 + z^55 * q^18 + z^6 * q^20 + (z^42 - z^6) * q^21 + (-z^47 - z^19) * q^23 - z^10 * q^24 + z^16 * q^25 + z^9 * q^27 + (z^57 - z^21) * q^28 + (z^38 - z^12) * q^29 - z^20 * q^30 - z^25 * q^32 + (-z^31 - z^7) * q^35 + z^24 * q^36 - z^35 * q^40 + (z^40 + z^32) * q^41 + (z^35 + z^11) * q^42 + (z^49 - z^45) * q^43 + z^34 * q^45 + (z^48 - z^16) * q^46 + (z^19 + z) * q^47 + z^39 * q^48 + (-z^58 + z^46 + z^22) * q^49 - z^45 * q^50 - z^38 * q^54 + (z^50 + z^26) * q^56 + (z^41 + z^7) * q^58 + z^49 * q^60 - z^54 * q^61 + (-z^49 - z^25) * q^63 + z^54 * q^64 - z^53 * q^67 + (z^30 + z^2) * q^69 + (z^36 - 1) * q^70 - z^53 * q^72 + z^59 * q^75 - z^4 * q^80 + z^52 * q^81 + (z^9 + z) * q^82 + (-z^53 + z^35) * q^83 + (-z^40 + z^4) * q^84 + (z^18 - z^14) * q^86 + (-z^55 - z^21) * q^87 + (-z^44 - z^34) * q^89 + z^3 * q^90 + (z^45 + z^17) * q^92 + (-z^48 - z^30) * q^94 + z^8 * q^96 + (-z^51 - z^27 + z^15) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 4 q^{5} + 8 q^{6}+O(q^{10}) \) 32 * q - 4 * q^5 + 8 * q^6	\( 32 q - 4 q^{5} + 8 q^{6} + 8 q^{14} - 4 q^{16} + 4 q^{25} - 8 q^{29} - 16 q^{30} - 8 q^{36} - 12 q^{41} - 12 q^{46} - 24 q^{70} + 4 q^{80} - 4 q^{81} + 12 q^{84} + 4 q^{89} + 8 q^{94} + 4 q^{96}+O(q^{100}) \) 32 * q - 4 * q^5 + 8 * q^6 + 8 * q^14 - 4 * q^16 + 4 * q^25 - 8 * q^29 - 16 * q^30 - 8 * q^36 - 12 * q^41 - 12 * q^46 - 24 * q^70 + 4 * q^80 - 4 * q^81 + 12 * q^84 + 4 * q^89 + 8 * q^94 + 4 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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.e

Level	$3660$
Weight	$1$
Character orbit	3660.fu
Analytic conductor	$1.827$
Analytic rank	$0$
Dimension	$32$
Projective image	$D_{60}$
CM discriminant	-20
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.82657794624\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(2\) over \(\Q(\zeta_{60})\)
Coefficient field:	\(\Q(\zeta_{120})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 \) x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{60}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{60} - \cdots)\)

\(n\)	\(1831\)	\(2197\)	\(2441\)	\(3601\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(-\zeta_{120}^{58}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 59.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{32} + T^{28} + \cdots + 1 \) T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$3$	\( T^{32} + T^{28} + \cdots + 1 \) T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$5$	\( (T^{8} + T^{7} - T^{5} + \cdots + 1)^{4} \) (T^8 + T^7 - T^5 - T^4 - T^3 + T + 1)^4
$7$	\( T^{32} - 4 T^{28} + \cdots + 1 \) T^32 - 4T^28 - 30T^24 - 206T^20 + 2159T^16 + 514T^12 + 75T^8 + 11*T^4 + 1
$11$	\( T^{32} \) T^32
$13$	\( T^{32} \) T^32
$17$	\( T^{32} \) T^32
$19$	\( T^{32} \) T^32
$23$	\( T^{32} + 2 T^{28} + \cdots + 1 \) T^32 + 2T^28 + 193T^24 - 1661T^20 + 5490T^16 + 989T^12 + 1063T^8 - 53*T^4 + 1
$29$	\( (T^{16} + 4 T^{15} + \cdots + 1)^{2} \) (T^16 + 4T^15 + 11T^14 + 26T^13 + 41T^12 + 48T^11 + 28T^10 - 52T^9 - 93T^8 - 84T^7 + 2T^6 + 144T^5 + 76T^4 + 22T^3 + 84T^2 - 18*T + 1)^2
$31$	\( T^{32} \) T^32
$37$	\( T^{32} \) T^32
$41$	\( (T^{8} + 3 T^{7} + 8 T^{6} + \cdots + 1)^{4} \) (T^8 + 3T^7 + 8T^6 + 11T^5 + 15T^4 + 11T^3 + 18T^2 - 7*T + 1)^4
$43$	\( T^{32} + 11 T^{28} + \cdots + 1 \) T^32 + 11T^28 + 195T^24 + 274T^20 + 17519T^16 - 147206T^12 + 273570T^8 + 956*T^4 + 1
$47$	\( (T^{16} - 8 T^{14} + \cdots + 1)^{2} \) (T^16 - 8T^14 + 45T^12 - 128T^10 + 264T^8 - 212T^6 + 125T^4 - 12*T^2 + 1)^2
$53$	\( T^{32} \) T^32
$59$	\( T^{32} \) T^32
$61$	\( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{4} \) (T^8 - T^6 + T^4 - T^2 + 1)^4
$67$	\( T^{32} + T^{28} + \cdots + 1 \) T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$71$	\( T^{32} \) T^32
$73$	\( T^{32} \) T^32
$79$	\( T^{32} \) T^32
$83$	\( T^{32} + 4 T^{30} + \cdots + 1 \) T^32 + 4T^30 - T^28 - 76T^26 - 229T^24 - 388T^22 + 546T^20 + 9172T^18 + 34887T^16 + 39128T^14 + 31146T^12 + 22888T^10 + 9146T^8 - 824T^6 - 76T^4 - 4T^2 + 1
$89$	\( (T^{16} - 2 T^{15} + \cdots + 1)^{2} \) (T^16 - 2T^15 + 2T^14 + 2T^13 - 13T^12 - 10T^11 + 48T^10 - 104T^9 + 105T^8 + 104T^7 + 48T^6 + 10T^5 - 13T^4 - 2T^3 + 2T^2 + 2*T + 1)^2
$97$	\( T^{32} \) T^32
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