LMFDB - Newform orbit 186.2.m.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [186,2,Mod(7,186)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(186, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([0, 28])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("186.7"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 186 = 2 \cdot 3 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	186.m (of order $15$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,2,1,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$1.48521747760$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{15})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{15}$. We also show the integral $q$-expansion of the trace form.

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/186\mathbb{Z}\right)^\times$.

$n$	$125$	$127$
$\chi(n)$	$1$	$-1 + \zeta_{15} - \zeta_{15}^{3} + \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{7}$

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} $

7.1

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

−0.309017

−

0.951057i

−0.978148

0.207912i

−0.809017

0.587785i

−1.01807

1.76336i

0.500000

0.866025i

2.01807

0.898504i

0.809017

0.587785i

0.913545

−

0.406737i

1.99165

0.423339i

19.1

0.809017

0.587785i

0.913545

0.406737i

0.309017

0.951057i

−1.64728

2.85317i

0.500000

0.866025i

2.64728

−

2.94010i

−0.309017

0.951057i

0.669131

0.743145i

−3.00973

1.34002i

49.1

0.809017

−

0.587785i

0.913545

−

0.406737i

0.309017

−

0.951057i

−1.64728

−

2.85317i

0.500000

−

0.866025i

2.64728

2.94010i

−0.309017

−

0.951057i

0.669131

−

0.743145i

−3.00973

−

1.34002i

103.1

0.809017

0.587785i

−0.104528

−

0.994522i

0.309017

0.951057i

1.64728

2.85317i

0.500000

−

0.866025i

−0.647278

−

0.137583i

−0.309017

0.951057i

−0.978148

0.207912i

−0.344375

3.27651i

121.1

0.809017

−

0.587785i

−0.104528

0.994522i

0.309017

−

0.951057i

1.64728

−

2.85317i

0.500000

0.866025i

−0.647278

0.137583i

−0.309017

−

0.951057i

−0.978148

−

0.207912i

−0.344375

−

3.27651i

133.1

−0.309017

0.951057i

−0.978148

−

0.207912i

−0.809017

−

0.587785i

−1.01807

−

1.76336i

0.500000

−

0.866025i

2.01807

−

0.898504i

0.809017

−

0.587785i

0.913545

0.406737i

1.99165

−

0.423339i

169.1

−0.309017

0.951057i

0.669131

−

0.743145i

−0.809017

−

0.587785i

1.01807

−

1.76336i

0.500000

0.866025i

−0.0180739

0.171962i

0.809017

−

0.587785i

−0.104528

−

0.994522i

1.36245

1.51315i

175.1

−0.309017

−

0.951057i

0.669131

0.743145i

−0.809017

0.587785i

1.01807

1.76336i

0.500000

−

0.866025i

−0.0180739

−

0.171962i

0.809017

0.587785i

−0.104528

0.994522i

1.36245

−

1.51315i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.g	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	186.2.m.d	✓	8
3.b	odd	2	1		558.2.ba.b		8
31.g	even	15	1	inner	186.2.m.d	✓	8
31.g	even	15	1		5766.2.a.ba		4
31.h	odd	30	1		5766.2.a.x		4
93.o	odd	30	1		558.2.ba.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
186.2.m.d	✓	8	1.a	even	1	1	trivial
186.2.m.d	✓	8	31.g	even	15	1	inner
558.2.ba.b		8	3.b	odd	2	1
558.2.ba.b		8	93.o	odd	30	1
5766.2.a.x		4	31.h	odd	30	1
5766.2.a.ba		4	31.g	even	15	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} + 15T_{5}^{6} + 180T_{5}^{4} + 675T_{5}^{2} + 2025 $ T5^8 + 15*T5^6 + 180*T5^4 + 675*T5^2 + 2025 acting on $S_{2}^{\mathrm{new}}(186, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 + T^2 - T + 1)^2
$3$	$ T^{8} - T^{7} + T^{5} + \cdots + 1 $ T^8 - T^7 + T^5 - T^4 + T^3 - T + 1
$5$	$ T^{8} + 15 T^{6} + \cdots + 2025 $ T^8 + 15T^6 + 180T^4 + 675*T^2 + 2025
$7$	$ T^{8} - 8 T^{7} + \cdots + 1 $ T^8 - 8T^7 + 30T^6 - 38T^5 - 21T^4 + 58T^3 + 35T^2 + 3*T + 1
$11$	$ T^{8} - 6 T^{7} + \cdots + 77841 $ T^8 - 6T^7 + 126T^5 - 531T^4 + 1026T^3 + 4995T^2 - 27621T + 77841
$13$	$ T^{8} + 4 T^{7} + \cdots + 256 $ T^8 + 4T^7 + 16T^5 + 144T^4 + 256T^3 + 320T^2 + 384T + 256
$17$	$ T^{8} - 24 T^{7} + \cdots + 20736 $ T^8 - 24T^7 + 240T^6 - 1296T^5 + 4464T^4 - 12096T^3 + 25920T^2 - 31104*T + 20736
$19$	$ T^{8} + 4 T^{7} + \cdots + 256 $ T^8 + 4T^7 + 16T^5 + 144T^4 + 256T^3 + 320T^2 + 384T + 256
$23$	$ (T^{4} + 6 T^{3} + \cdots + 1296)^{2} $ (T^4 + 6T^3 + 36T^2 + 216*T + 1296)^2
$29$	$ T^{8} + 15 T^{7} + \cdots + 2025 $ T^8 + 15T^7 + 90T^6 + 45T^5 + 765T^4 + 675T^3 + 1350T^2 + 2025*T + 2025
$31$	$ T^{8} + 29 T^{7} + \cdots + 923521 $ T^8 + 29T^7 + 420T^6 + 3931T^5 + 25859T^4 + 121861T^3 + 403620T^2 + 863939*T + 923521
$37$	$ T^{8} - 8 T^{7} + \cdots + 256 $ T^8 - 8T^7 + 80T^6 + 112T^5 + 304T^4 + 128T^3 + 320T^2 + 128*T + 256
$41$	$ T^{8} + 48 T^{7} + \cdots + 17438976 $ T^8 + 48T^7 + 1080T^6 + 14688T^5 + 134064T^4 + 862272T^3 + 4034880T^2 + 12327552*T + 17438976
$43$	$ T^{8} + 24 T^{7} + \cdots + 215296 $ T^8 + 24T^7 + 200T^6 + 536T^5 + 384T^4 + 12416T^3 + 65280T^2 + 63104*T + 215296
$47$	$ T^{8} - 30 T^{7} + \cdots + 518400 $ T^8 - 30T^7 + 480T^6 - 4680T^5 + 33840T^4 - 151200T^3 + 518400T^2 - 777600*T + 518400
$53$	$ T^{8} - 33 T^{7} + \cdots + 12257001 $ T^8 - 33T^7 + 360T^6 - 1188T^5 + 4599T^4 - 123552T^3 + 689040T^2 - 1039797*T + 12257001
$59$	$ T^{8} + 3 T^{7} + \cdots + 641601 $ T^8 + 3T^7 - 45T^6 - 342T^5 + 1359T^4 + 19602T^3 + 64530T^2 + 201852*T + 641601
$61$	$ (T^{4} + 14 T^{3} + \cdots - 1424)^{2} $ (T^4 + 14T^3 - 4T^2 - 536*T - 1424)^2
$67$	$ T^{8} - 2 T^{7} + \cdots + 952576 $ T^8 - 2T^7 + 80T^6 - 152T^5 + 5104T^4 - 7648T^3 + 97280T^2 + 148352*T + 952576
$71$	$ T^{8} - 30 T^{7} + \cdots + 12960000 $ T^8 - 30T^7 + 360T^6 - 28800T^4 + 108000T^3 + 864000*T^2 + 12960000
$73$	$ T^{8} - 26 T^{7} + \cdots + 32761 $ T^8 - 26T^7 + 240T^6 - 1754T^5 + 9129T^4 + 52156T^3 + 96005T^2 + 83079*T + 32761
$79$	$ T^{8} + 110 T^{6} + \cdots + 126000625 $ T^8 + 110T^6 + 2450T^5 - 42675T^4 + 50000T^3 + 4432125T^2 - 40129375T + 126000625
$83$	$ T^{8} - 39 T^{7} + \cdots + 2653641 $ T^8 - 39T^7 + 660T^6 - 6471T^5 + 41589T^4 - 188811T^3 + 657720T^2 - 1744659*T + 2653641
$89$	$ T^{8} - 48 T^{7} + \cdots + 20736 $ T^8 - 48T^7 + 1308T^6 - 22536T^5 + 252000T^4 - 1649376T^3 + 5168448T^2 + 196992*T + 20736
$97$	$ T^{8} - 18 T^{7} + \cdots + 60855601 $ T^8 - 18T^7 + 443T^6 - 2476T^5 + 13725T^4 - 22876T^3 + 1934043T^2 + 15461582*T + 60855601
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 186 = 2 \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	186.m (of order \(15\), degree \(8\), minimal)

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

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	186.2.m.d

Level	$186$
Weight	$2$
Character orbit	186.m
Analytic conductor	$1.485$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(1.48521747760\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{15})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

\(n\)	\(125\)	\(127\)
\(\chi(n)\)	\(1\)	\(-1 + \zeta_{15} - \zeta_{15}^{3} + \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 - T^3 + T^2 - T + 1)^2
$3$	\( T^{8} - T^{7} + T^{5} + \cdots + 1 \) T^8 - T^7 + T^5 - T^4 + T^3 - T + 1
$5$	\( T^{8} + 15 T^{6} + \cdots + 2025 \) T^8 + 15T^6 + 180T^4 + 675*T^2 + 2025
$7$	\( T^{8} - 8 T^{7} + \cdots + 1 \) T^8 - 8T^7 + 30T^6 - 38T^5 - 21T^4 + 58T^3 + 35T^2 + 3*T + 1
$11$	\( T^{8} - 6 T^{7} + \cdots + 77841 \) T^8 - 6T^7 + 126T^5 - 531T^4 + 1026T^3 + 4995T^2 - 27621T + 77841
$13$	\( T^{8} + 4 T^{7} + \cdots + 256 \) T^8 + 4T^7 + 16T^5 + 144T^4 + 256T^3 + 320T^2 + 384T + 256
$17$	\( T^{8} - 24 T^{7} + \cdots + 20736 \) T^8 - 24T^7 + 240T^6 - 1296T^5 + 4464T^4 - 12096T^3 + 25920T^2 - 31104*T + 20736
$19$	\( T^{8} + 4 T^{7} + \cdots + 256 \) T^8 + 4T^7 + 16T^5 + 144T^4 + 256T^3 + 320T^2 + 384T + 256
$23$	\( (T^{4} + 6 T^{3} + \cdots + 1296)^{2} \) (T^4 + 6T^3 + 36T^2 + 216*T + 1296)^2
$29$	\( T^{8} + 15 T^{7} + \cdots + 2025 \) T^8 + 15T^7 + 90T^6 + 45T^5 + 765T^4 + 675T^3 + 1350T^2 + 2025*T + 2025
$31$	\( T^{8} + 29 T^{7} + \cdots + 923521 \) T^8 + 29T^7 + 420T^6 + 3931T^5 + 25859T^4 + 121861T^3 + 403620T^2 + 863939*T + 923521
$37$	\( T^{8} - 8 T^{7} + \cdots + 256 \) T^8 - 8T^7 + 80T^6 + 112T^5 + 304T^4 + 128T^3 + 320T^2 + 128*T + 256
$41$	\( T^{8} + 48 T^{7} + \cdots + 17438976 \) T^8 + 48T^7 + 1080T^6 + 14688T^5 + 134064T^4 + 862272T^3 + 4034880T^2 + 12327552*T + 17438976
$43$	\( T^{8} + 24 T^{7} + \cdots + 215296 \) T^8 + 24T^7 + 200T^6 + 536T^5 + 384T^4 + 12416T^3 + 65280T^2 + 63104*T + 215296
$47$	\( T^{8} - 30 T^{7} + \cdots + 518400 \) T^8 - 30T^7 + 480T^6 - 4680T^5 + 33840T^4 - 151200T^3 + 518400T^2 - 777600*T + 518400
$53$	\( T^{8} - 33 T^{7} + \cdots + 12257001 \) T^8 - 33T^7 + 360T^6 - 1188T^5 + 4599T^4 - 123552T^3 + 689040T^2 - 1039797*T + 12257001
$59$	\( T^{8} + 3 T^{7} + \cdots + 641601 \) T^8 + 3T^7 - 45T^6 - 342T^5 + 1359T^4 + 19602T^3 + 64530T^2 + 201852*T + 641601
$61$	\( (T^{4} + 14 T^{3} + \cdots - 1424)^{2} \) (T^4 + 14T^3 - 4T^2 - 536*T - 1424)^2
$67$	\( T^{8} - 2 T^{7} + \cdots + 952576 \) T^8 - 2T^7 + 80T^6 - 152T^5 + 5104T^4 - 7648T^3 + 97280T^2 + 148352*T + 952576
$71$	\( T^{8} - 30 T^{7} + \cdots + 12960000 \) T^8 - 30T^7 + 360T^6 - 28800T^4 + 108000T^3 + 864000*T^2 + 12960000
$73$	\( T^{8} - 26 T^{7} + \cdots + 32761 \) T^8 - 26T^7 + 240T^6 - 1754T^5 + 9129T^4 + 52156T^3 + 96005T^2 + 83079*T + 32761
$79$	\( T^{8} + 110 T^{6} + \cdots + 126000625 \) T^8 + 110T^6 + 2450T^5 - 42675T^4 + 50000T^3 + 4432125T^2 - 40129375T + 126000625
$83$	\( T^{8} - 39 T^{7} + \cdots + 2653641 \) T^8 - 39T^7 + 660T^6 - 6471T^5 + 41589T^4 - 188811T^3 + 657720T^2 - 1744659*T + 2653641
$89$	\( T^{8} - 48 T^{7} + \cdots + 20736 \) T^8 - 48T^7 + 1308T^6 - 22536T^5 + 252000T^4 - 1649376T^3 + 5168448T^2 + 196992*T + 20736
$97$	\( T^{8} - 18 T^{7} + \cdots + 60855601 \) T^8 - 18T^7 + 443T^6 - 2476T^5 + 13725T^4 - 22876T^3 + 1934043T^2 + 15461582*T + 60855601
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