LMFDB - Newform orbit 186.2.m.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [186,2,Mod(7,186)]

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

lf = mfeigenbasis(mf)

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")

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

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

Self dual:	no
Analytic conductor:	$1.48521747760$
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:	$\mathrm{SU}(2)[C_{15}]$

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

−1.28716

2.22943i

−0.500000

−

0.866025i

−4.52780

−

2.01591i

−0.809017

−

0.587785i

0.913545

−

0.406737i

−2.51807

−

0.535233i

19.1

−0.809017

−

0.587785i

0.913545

0.406737i

0.309017

0.951057i

1.72256

−

2.98357i

−0.500000

−

0.866025i

−0.784829

0.871641i

0.309017

−

0.951057i

0.669131

0.743145i

−3.14728

1.40126i

49.1

−0.809017

0.587785i

0.913545

−

0.406737i

0.309017

−

0.951057i

1.72256

2.98357i

−0.500000

0.866025i

−0.784829

−

0.871641i

0.309017

0.951057i

0.669131

−

0.743145i

−3.14728

−

1.40126i

103.1

−0.809017

−

0.587785i

−0.104528

−

0.994522i

0.309017

0.951057i

0.704489

1.22021i

−0.500000

0.866025i

3.13893

0.667200i

0.309017

−

0.951057i

−0.978148

0.207912i

0.147278

−

1.40126i

121.1

−0.809017

0.587785i

−0.104528

0.994522i

0.309017

−

0.951057i

0.704489

−

1.22021i

−0.500000

−

0.866025i

3.13893

−

0.667200i

0.309017

0.951057i

−0.978148

−

0.207912i

0.147278

1.40126i

133.1

0.309017

−

0.951057i

−0.978148

−

0.207912i

−0.809017

−

0.587785i

−1.28716

−

2.22943i

−0.500000

0.866025i

−4.52780

2.01591i

−0.809017

0.587785i

0.913545

0.406737i

−2.51807

0.535233i

169.1

0.309017

−

0.951057i

0.669131

−

0.743145i

−0.809017

−

0.587785i

0.360114

−

0.623735i

−0.500000

−

0.866025i

0.173699

−

1.65264i

−0.809017

0.587785i

−0.104528

−

0.994522i

−0.481926

−

0.535233i

175.1

0.309017

0.951057i

0.669131

0.743145i

−0.809017

0.587785i

0.360114

0.623735i

−0.500000

0.866025i

0.173699

1.65264i

−0.809017

−

0.587785i

−0.104528

0.994522i

−0.481926

0.535233i

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.a	✓	8
3.b	odd	2	1		558.2.ba.e		8
31.g	even	15	1	inner	186.2.m.a	✓	8
31.g	even	15	1		5766.2.a.bg		4
31.h	odd	30	1		5766.2.a.bc		4
93.o	odd	30	1		558.2.ba.e		8

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

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} - 3T_{5}^{7} + 15T_{5}^{6} - 18T_{5}^{5} + 99T_{5}^{4} - 162T_{5}^{3} + 270T_{5}^{2} - 162T_{5} + 81 $ T5^8 - 3*T5^7 + 15*T5^6 - 18*T5^5 + 99*T5^4 - 162*T5^3 + 270*T5^2 - 162*T5 + 81 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} - 3 T^{7} + \cdots + 81 $ T^8 - 3T^7 + 15T^6 - 18T^5 + 99T^4 - 162T^3 + 270T^2 - 162*T + 81
$7$	$ T^{8} + 4 T^{7} + \cdots + 961 $ T^8 + 4T^7 - 15T^6 - 74T^5 + 114T^4 + 16T^3 + 590T^2 + 744*T + 961
$11$	$ T^{8} - 3 T^{7} + \cdots + 81 $ T^8 - 3T^7 + 15T^6 - 18T^5 + 9T^4 + 108T^3 - 162T + 81
$13$	$ T^{8} - 11 T^{7} + \cdots + 841 $ T^8 - 11T^7 + 60T^6 - 224T^5 + 639T^4 - 1394T^3 + 2150T^2 - 2001*T + 841
$17$	$ T^{8} - 180 T^{5} + \cdots + 2025 $ T^8 - 180T^5 + 630T^4 - 2700T^3 + 8775T^2 + 8100*T + 2025
$19$	$ T^{8} - 2 T^{7} + \cdots + 73441 $ T^8 - 2T^7 - 272T^5 + 2169T^4 - 7958T^3 + 28895T^2 - 73983T + 73441
$23$	$ T^{8} + 6 T^{7} + \cdots + 6561 $ T^8 + 6T^7 + 27T^6 - 27T^5 - 243T^3 + 16767T^2 - 6561T + 6561
$29$	$ T^{8} - 3 T^{7} + \cdots + 81 $ T^8 - 3T^7 + 48T^6 + 99T^5 - 45T^4 - 621T^3 + 1458T^2 - 243*T + 81
$31$	$ T^{8} + 8 T^{7} + \cdots + 923521 $ T^8 + 8T^7 + 33T^6 + 16T^5 - 895T^4 + 496T^3 + 31713T^2 + 238328*T + 923521
$37$	$ T^{8} - 8 T^{7} + \cdots + 22801 $ T^8 - 8T^7 + 110T^6 + 142T^5 + 2869T^4 - 2782T^3 + 19715T^2 + 17063*T + 22801
$41$	$ T^{8} + 3 T^{7} + \cdots + 301401 $ T^8 + 3T^7 - 75T^6 + 18T^5 + 4329T^4 - 30078T^3 + 93150T^2 - 187758*T + 301401
$43$	$ T^{8} + 15 T^{7} + \cdots + 819025 $ T^8 + 15T^7 + 170T^6 + 1355T^5 + 4905T^4 + 3875T^3 + 86250T^2 + 538475*T + 819025
$47$	$ T^{8} + 12 T^{7} + \cdots + 20736 $ T^8 + 12T^7 + 108T^6 + 504T^5 + 1440T^4 + 864T^3 + 1728T^2 - 10368*T + 20736
$53$	$ T^{8} + 36 T^{7} + \cdots + 1846881 $ T^8 + 36T^7 + 600T^6 + 5904T^5 + 37674T^4 + 176904T^3 + 676755T^2 + 1663416*T + 1846881
$59$	$ T^{8} - 21 T^{7} + \cdots + 116834481 $ T^8 - 21T^7 + 210T^6 - 2574T^5 + 35784T^4 - 423819T^3 + 4900500T^2 - 37550466*T + 116834481
$61$	$ (T^{4} + 35 T^{3} + \cdots + 1945)^{2} $ (T^4 + 35T^3 + 395T^2 + 1555*T + 1945)^2
$67$	$ T^{8} - 17 T^{7} + \cdots + 23030401 $ T^8 - 17T^7 + 305T^6 - 2372T^5 + 27529T^4 - 184318T^3 + 1670900T^2 - 6344278*T + 23030401
$71$	$ T^{8} + 12 T^{7} + \cdots + 57289761 $ T^8 + 12T^7 - 30T^6 - 1458T^5 - 16191T^4 - 128412T^3 + 1703025T^2 + 13828563*T + 57289761
$73$	$ T^{8} + T^{7} + \cdots + 961 $ T^8 + T^7 - 15T^6 + 289T^5 + 2334T^4 + 1939T^3 + 4595T^2 + 3906T + 961
$79$	$ T^{8} + 33 T^{7} + \cdots + 3481 $ T^8 + 33T^7 + 425T^6 + 2498T^5 + 8259T^4 + 12872T^3 + 9390T^2 + 5192*T + 3481
$83$	$ T^{8} + 54 T^{7} + \cdots + 14220441 $ T^8 + 54T^7 + 1260T^6 + 16506T^5 + 136359T^4 + 768906T^3 + 3080295T^2 + 8518689*T + 14220441
$89$	$ T^{8} + 12 T^{7} + \cdots + 81 $ T^8 + 12T^7 + 153T^6 + 684T^5 + 1440T^4 + 864T^3 + 378T^2 + 162*T + 81
$97$	$ T^{8} + 9 T^{7} + \cdots + 1100401 $ T^8 + 9T^7 + 182T^6 - 43T^5 - 1875T^4 + 9143T^3 + 192462T^2 + 366101*T + 1100401
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

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

Introduction

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Newspace parameters

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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.a

Level	$186$
Weight	$2$
Character orbit	186.m
Analytic conductor	$1.485$
Analytic rank	$0$
Dimension	$8$
CM	no
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
$5$	\( T^{8} - 3 T^{7} + \cdots + 81 \) T^8 - 3T^7 + 15T^6 - 18T^5 + 99T^4 - 162T^3 + 270T^2 - 162*T + 81
$7$	\( T^{8} + 4 T^{7} + \cdots + 961 \) T^8 + 4T^7 - 15T^6 - 74T^5 + 114T^4 + 16T^3 + 590T^2 + 744*T + 961
$11$	\( T^{8} - 3 T^{7} + \cdots + 81 \) T^8 - 3T^7 + 15T^6 - 18T^5 + 9T^4 + 108T^3 - 162T + 81
$13$	\( T^{8} - 11 T^{7} + \cdots + 841 \) T^8 - 11T^7 + 60T^6 - 224T^5 + 639T^4 - 1394T^3 + 2150T^2 - 2001*T + 841
$17$	\( T^{8} - 180 T^{5} + \cdots + 2025 \) T^8 - 180T^5 + 630T^4 - 2700T^3 + 8775T^2 + 8100*T + 2025
$19$	\( T^{8} - 2 T^{7} + \cdots + 73441 \) T^8 - 2T^7 - 272T^5 + 2169T^4 - 7958T^3 + 28895T^2 - 73983T + 73441
$23$	\( T^{8} + 6 T^{7} + \cdots + 6561 \) T^8 + 6T^7 + 27T^6 - 27T^5 - 243T^3 + 16767T^2 - 6561T + 6561
$29$	\( T^{8} - 3 T^{7} + \cdots + 81 \) T^8 - 3T^7 + 48T^6 + 99T^5 - 45T^4 - 621T^3 + 1458T^2 - 243*T + 81
$31$	\( T^{8} + 8 T^{7} + \cdots + 923521 \) T^8 + 8T^7 + 33T^6 + 16T^5 - 895T^4 + 496T^3 + 31713T^2 + 238328*T + 923521
$37$	\( T^{8} - 8 T^{7} + \cdots + 22801 \) T^8 - 8T^7 + 110T^6 + 142T^5 + 2869T^4 - 2782T^3 + 19715T^2 + 17063*T + 22801
$41$	\( T^{8} + 3 T^{7} + \cdots + 301401 \) T^8 + 3T^7 - 75T^6 + 18T^5 + 4329T^4 - 30078T^3 + 93150T^2 - 187758*T + 301401
$43$	\( T^{8} + 15 T^{7} + \cdots + 819025 \) T^8 + 15T^7 + 170T^6 + 1355T^5 + 4905T^4 + 3875T^3 + 86250T^2 + 538475*T + 819025
$47$	\( T^{8} + 12 T^{7} + \cdots + 20736 \) T^8 + 12T^7 + 108T^6 + 504T^5 + 1440T^4 + 864T^3 + 1728T^2 - 10368*T + 20736
$53$	\( T^{8} + 36 T^{7} + \cdots + 1846881 \) T^8 + 36T^7 + 600T^6 + 5904T^5 + 37674T^4 + 176904T^3 + 676755T^2 + 1663416*T + 1846881
$59$	\( T^{8} - 21 T^{7} + \cdots + 116834481 \) T^8 - 21T^7 + 210T^6 - 2574T^5 + 35784T^4 - 423819T^3 + 4900500T^2 - 37550466*T + 116834481
$61$	\( (T^{4} + 35 T^{3} + \cdots + 1945)^{2} \) (T^4 + 35T^3 + 395T^2 + 1555*T + 1945)^2
$67$	\( T^{8} - 17 T^{7} + \cdots + 23030401 \) T^8 - 17T^7 + 305T^6 - 2372T^5 + 27529T^4 - 184318T^3 + 1670900T^2 - 6344278*T + 23030401
$71$	\( T^{8} + 12 T^{7} + \cdots + 57289761 \) T^8 + 12T^7 - 30T^6 - 1458T^5 - 16191T^4 - 128412T^3 + 1703025T^2 + 13828563*T + 57289761
$73$	\( T^{8} + T^{7} + \cdots + 961 \) T^8 + T^7 - 15T^6 + 289T^5 + 2334T^4 + 1939T^3 + 4595T^2 + 3906T + 961
$79$	\( T^{8} + 33 T^{7} + \cdots + 3481 \) T^8 + 33T^7 + 425T^6 + 2498T^5 + 8259T^4 + 12872T^3 + 9390T^2 + 5192*T + 3481
$83$	\( T^{8} + 54 T^{7} + \cdots + 14220441 \) T^8 + 54T^7 + 1260T^6 + 16506T^5 + 136359T^4 + 768906T^3 + 3080295T^2 + 8518689*T + 14220441
$89$	\( T^{8} + 12 T^{7} + \cdots + 81 \) T^8 + 12T^7 + 153T^6 + 684T^5 + 1440T^4 + 864T^3 + 378T^2 + 162*T + 81
$97$	\( T^{8} + 9 T^{7} + \cdots + 1100401 \) T^8 + 9T^7 + 182T^6 - 43T^5 - 1875T^4 + 9143T^3 + 192462T^2 + 366101*T + 1100401
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