LMFDB - Newform orbit 3627.1.jd.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3627,1,Mod(127,3627)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3627.127"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3627, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([0, 25, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3627 = 3^{2} \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3627.jd (of order $30$, degree $8$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.81010880082$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{30}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{30} + \cdots)$

$q$-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_{30}^{13} q^{4} + ( - \zeta_{30}^{14} - \zeta_{30}^{7}) q^{7} + \zeta_{30}^{4} q^{13} - \zeta_{30}^{11} q^{16} + (\zeta_{30}^{11} - \zeta_{30}^{7}) q^{19} - \zeta_{30}^{5} q^{25} + ( - \zeta_{30}^{12} - \zeta_{30}^{5}) q^{28} + \cdots + ( - \zeta_{30}^{10} - \zeta_{30}) q^{97} +O(q^{100}) $ q - z^13 * q^4 + (-z^14 - z^7) * q^7 + z^4 * q^13 - z^11 * q^16 + (z^11 - z^7) * q^19 - z^5 * q^25 + (-z^12 - z^5) * q^28 + z * q^31 + (z^11 + z^9) * q^37 + (-z^10 + z^8) * q^43 + (z^14 - z^13 - z^6) * q^49 + z^2 * q^52 + (-z^3 - z^2) * q^61 - z^9 * q^64 + (z^8 + z^7) * q^67 + (-z^7 + z^4) * q^73 + (z^9 - z^5) * q^76 + (z^4 - 1) * q^79 + (-z^11 + z^3) * q^91 + (-z^10 - z) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{4} + q^{13} + q^{16} - 4 q^{25} - 2 q^{28} - q^{31} + q^{37} + 5 q^{43} + 4 q^{49} + q^{52} - 3 q^{61} - 2 q^{64} + 2 q^{73} - 2 q^{76} - 7 q^{79} + 3 q^{91} + 5 q^{97}+O(q^{100}) $ 8 * q + q^4 + q^13 + q^16 - 4 * q^25 - 2 * q^28 - q^31 + q^37 + 5 * q^43 + 4 * q^49 + q^52 - 3 * q^61 - 2 * q^64 + 2 * q^73 - 2 * q^76 - 7 * q^79 + 3 * q^91 + 5 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3627\mathbb{Z}\right)^\times$.

$n$	$1522$	$1613$	$3070$
$\chi(n)$	$\zeta_{30}^{7}$	$1$	$-\zeta_{30}^{10}$

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} $

127.1

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

0.669131

0.743145i

−1.89169

−

0.614648i

199.1

−0.978148

0.207912i

0.773659

0.251377i

1531.1

−0.978148

−

0.207912i

0.773659

−

0.251377i

1603.1

−0.104528

0.994522i

0.244415

−

0.336408i

2305.1

0.913545

0.406737i

0.873619

1.20243i

2656.1

0.669131

−

0.743145i

−1.89169

0.614648i

3403.1

−0.104528

−

0.994522i

0.244415

0.336408i

3520.1

0.913545

−

0.406737i

0.873619

−

1.20243i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
403.br	odd	30	1	inner
1209.ds	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3627.1.jd.a	yes	8
3.b	odd	2	1	CM	3627.1.jd.a	yes	8
13.e	even	6	1		3627.1.it.a	✓	8
31.h	odd	30	1		3627.1.it.a	✓	8
39.h	odd	6	1		3627.1.it.a	✓	8
93.p	even	30	1		3627.1.it.a	✓	8
403.br	odd	30	1	inner	3627.1.jd.a	yes	8
1209.ds	even	30	1	inner	3627.1.jd.a	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3627.1.it.a	✓	8	13.e	even	6	1
3627.1.it.a	✓	8	31.h	odd	30	1
3627.1.it.a	✓	8	39.h	odd	6	1
3627.1.it.a	✓	8	93.p	even	30	1
3627.1.jd.a	yes	8	1.a	even	1	1	trivial
3627.1.jd.a	yes	8	3.b	odd	2	1	CM
3627.1.jd.a	yes	8	403.br	odd	30	1	inner
3627.1.jd.a	yes	8	1209.ds	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3627, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ T^{8} $ T^8
$7$	$ T^{8} - 3 T^{6} + \cdots + 1 $ T^8 - 3T^6 + 5T^5 + 4T^4 - 15T^3 + 13T^2 - 5T + 1
$11$	$ T^{8} $ T^8
$13$	$ T^{8} - T^{7} + T^{5} + \cdots + 1 $ T^8 - T^7 + T^5 - T^4 + T^3 - T + 1
$17$	$ T^{8} $ T^8
$19$	$ T^{8} - 3 T^{6} + \cdots + 1 $ T^8 - 3T^6 - 5T^5 + 4T^4 + 15T^3 + 13T^2 + 5T + 1
$23$	$ T^{8} $ T^8
$29$	$ T^{8} $ T^8
$31$	$ T^{8} + T^{7} - T^{5} + \cdots + 1 $ T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$37$	$ T^{8} - T^{7} + 5 T^{6} + \cdots + 1 $ T^8 - T^7 + 5T^6 - 4T^5 + 19T^4 - 14T^3 + 20T^2 + 4T + 1
$41$	$ T^{8} $ T^8
$43$	$ T^{8} - 5 T^{7} + \cdots + 1 $ T^8 - 5T^7 + 12T^6 - 15T^5 + 9T^4 - 5T^3 + 8T^2 - 5*T + 1
$47$	$ T^{8} $ T^8
$53$	$ T^{8} $ T^8
$59$	$ T^{8} $ T^8
$61$	$ T^{8} + 3 T^{7} + \cdots + 1 $ T^8 + 3T^7 + T^6 - 6T^5 - T^4 + 12T^3 + 14T^2 + 6*T + 1
$67$	$ T^{8} + 7 T^{6} + \cdots + 1 $ T^8 + 7T^6 + 14T^4 + 8*T^2 + 1
$71$	$ T^{8} $ T^8
$73$	$ T^{8} - 2 T^{7} + \cdots + 1 $ T^8 - 2T^7 - 2T^5 + 9T^4 - 8T^3 + 5T^2 - 3T + 1
$79$	$ T^{8} + 7 T^{7} + \cdots + 1 $ T^8 + 7T^7 + 21T^6 + 36T^5 + 39T^4 + 28T^3 + 14T^2 + 4*T + 1
$83$	$ T^{8} $ T^8
$89$	$ T^{8} $ T^8
$97$	$ T^{8} - 5 T^{7} + \cdots + 25 $ T^8 - 5T^7 + 15T^6 - 30T^5 + 45T^4 - 50T^3 + 50T^2 - 50*T + 25
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Level:	\( N \)	\(=\)	\( 3627 = 3^{2} \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3627.jd (of order \(30\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{30}^{13} q^{4} + ( - \zeta_{30}^{14} - \zeta_{30}^{7}) q^{7} + \zeta_{30}^{4} q^{13} - \zeta_{30}^{11} q^{16} + (\zeta_{30}^{11} - \zeta_{30}^{7}) q^{19} - \zeta_{30}^{5} q^{25} + ( - \zeta_{30}^{12} - \zeta_{30}^{5}) q^{28} + \cdots + ( - \zeta_{30}^{10} - \zeta_{30}) q^{97} +O(q^{100}) \) q - z^13 * q^4 + (-z^14 - z^7) * q^7 + z^4 * q^13 - z^11 * q^16 + (z^11 - z^7) * q^19 - z^5 * q^25 + (-z^12 - z^5) * q^28 + z * q^31 + (z^11 + z^9) * q^37 + (-z^10 + z^8) * q^43 + (z^14 - z^13 - z^6) * q^49 + z^2 * q^52 + (-z^3 - z^2) * q^61 - z^9 * q^64 + (z^8 + z^7) * q^67 + (-z^7 + z^4) * q^73 + (z^9 - z^5) * q^76 + (z^4 - 1) * q^79 + (-z^11 + z^3) * q^91 + (-z^10 - z) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{4} + q^{13} + q^{16} - 4 q^{25} - 2 q^{28} - q^{31} + q^{37} + 5 q^{43} + 4 q^{49} + q^{52} - 3 q^{61} - 2 q^{64} + 2 q^{73} - 2 q^{76} - 7 q^{79} + 3 q^{91} + 5 q^{97}+O(q^{100}) \) 8 * q + q^4 + q^13 + q^16 - 4 * q^25 - 2 * q^28 - q^31 + q^37 + 5 * q^43 + 4 * q^49 + q^52 - 3 * q^61 - 2 * q^64 + 2 * q^73 - 2 * q^76 - 7 * q^79 + 3 * q^91 + 5 * q^97

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