LMFDB - Newform orbit 147.3.n.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [147,3,Mod(2,147)] 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("147.2"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 26])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	147.n (of order $42$, degree $12$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$4.00545988610$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{42}]$

$q$-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 $\zeta_{21}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (3 \zeta_{21}^{9} + 3 \zeta_{21}^{2}) q^{3} + (4 \zeta_{21}^{11} - 4 \zeta_{21}^{10} + \cdots - 4) q^{4} + (3 \zeta_{21}^{10} - 5 \zeta_{21}^{3}) q^{7} + 9 \zeta_{21}^{11} q^{9} - 12 \zeta_{21}^{8} q^{12} + \cdots + (55 \zeta_{21}^{11} + \cdots + 57 \zeta_{21}^{3}) q^{97} +O(q^{100}) $ q + (3z^9 + 3z^2) * q^3 + (4z^11 - 4z^10 + 4z^8 - 4z^7 - 4z^6 + 4z^5 - 4z^3 + 4z^2 - 4) * q^4 + (3z^10 - 5z^3) * q^7 + 9z^11 q^9 - 12z^8 q^12 + (15z^11 - 15z^9 - 15z^6 + 7z^5 + 15z^4 - 15z^3 + 7z - 15) q^13 + 16z^5 q^16 + (-5z^11 + 5z^10 - 21z^4 - 16z^3) * q^19 + (-15z^11 + 15z^9 - 15z^8 + 15z^6 - 24z^5 - 15z^4 + 15z^3 - 15z + 15) * q^21 + 25z^4 q^25 - 27z^6 q^27 + (20z^9 + 32z^2) * q^28 + (11z^11 - 11z^10 + 46z^8 - 11z^7 + 24z^6 + 11z^5 - 11z^3 + 11z^2 + 11z - 11) q^31 + 36z^3 q^36 + (-7z^9 + 33z^7 - 40z^2 + 40) q^37 + (-24z^10 - 45z^7 + 21z^3 - 21) q^39 + (-48z^9 + 48z^8 - 35z^2 + 13z) * q^43 - 48 * q^48 + (-39z^11 + 39z^10 - 39z^8 + 39z^7 + 55z^6 - 39z^5 + 39z^3 - 39z^2 + 39) * q^49 + (-28z^11 + 32z^7 + 32z^4 - 28) q^52 + (-111z^11 + 63z^10 + 48z^9 - 111z^8 + 63z^7 + 63z^6 - 126z^5 - 48z^4 + 111z^3 - 63z^2 - 48z + 111) q^57 + (-56z^9 - 65z^7 + 9z^2 - 9) q^61 + (45z^7 + 72) q^63 + (-64z^11 - 64z^4) * q^64 + (-32z^11 + 77z^9 - 32z^8 + 32z^6 + 45z^5 - 32z^4 + 32z^3 - 32z^2 - 32z + 32) q^67 + (80z^11 - 80z^10 - 80z^9 + 80z^8 - 80z^6 + 17z^5 + 80z^4 - 143z^3 + 80z - 80) q^73 + (75z^11 - 75z^10 + 75z^8 - 75z^7 + 75z^5 - 75z^3 + 75z^2 - 75) q^75 + (84z^10 + 64z^9 + 64z^3 + 84z^2) * q^76 + (40z^11 - 40z^10 - 40z^7 - 91z^6 + 40z^5 - 40z^3 + 40z^2 - 91z - 40) * q^79 + 81z q^81 + (96z^11 + 36z^4) * q^84 + (96z^11 + 19z^8 + 85z^4 + 99z) * q^91 + (33z^10 - 33z^8 - 72z^3 - 105z) * q^93 + (55z^11 + 112z^10 - 57z^4 + 57z^3) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 4 q^{4} + 13 q^{7} + 9 q^{9} - 12 q^{12} - 46 q^{13} + 16 q^{16} + 11 q^{19} + 6 q^{21} + 25 q^{25} + 54 q^{27} - 8 q^{28} - 13 q^{31} - 72 q^{36} + 256 q^{37} - 48 q^{39} + 122 q^{43}+ \cdots - 4 q^{97}+O(q^{100}) $ 12 * q - 3 * q^3 + 4 * q^4 + 13 * q^7 + 9 * q^9 - 12 * q^12 - 46 * q^13 + 16 * q^16 + 11 * q^19 + 6 * q^21 + 25 * q^25 + 54 * q^27 - 8 * q^28 - 13 * q^31 - 72 * q^36 + 256 * q^37 - 48 * q^39 + 122 * q^43 - 576 * q^48 - 71 * q^49 - 524 * q^52 + 66 * q^57 + 403 * q^61 + 594 * q^63 - 128 * q^64 - 13 * q^67 - 97 * q^73 - 75 * q^75 - 88 * q^76 + 11 * q^79 + 81 * q^81 + 132 * q^84 + 299 * q^91 + 39 * q^93 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$.

$n$	$50$	$52$
$\chi(n)$	$-1$	$\zeta_{21}^{11}$

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} $

2.1

0.0747301	+	0.997204i
0.826239	−	0.563320i
0.365341	−	0.930874i
0.365341	+	0.930874i
−0.733052	+	0.680173i
−0.988831	+	0.149042i
0.955573	−	0.294755i
0.0747301	−	0.997204i
−0.988831	−	0.149042i
0.955573	+	0.294755i
0.826239	+	0.563320i
−0.733052	−	0.680173i

−1.09602

2.79262i

3.30496

2.25328i

−1.08655

6.91516i

−6.59747

−

6.12155i

11.1

2.96649

−

0.447127i

0.298920

−

3.98882i

3.97932

5.75891i

8.60016

−

2.65280i

23.1

−2.86672

0.884266i

−3.95532

−

0.596169i

6.98356

−

0.479459i

7.43615

−

5.06988i

32.1

−2.86672

−

0.884266i

−3.95532

0.596169i

6.98356

0.479459i

7.43615

5.06988i

44.1

−2.47872

−

1.68996i

3.82229

−

1.17902i

−2.02143

−

6.70178i

3.28807

8.37786i

53.1

2.19916

2.04052i

1.46136

3.72349i

4.72903

−

5.16103i

0.672571

8.97483i

65.1

−0.224190

−

2.99161i

−2.93221

2.72069i

−6.08394

3.46203i

−8.89948

1.34138i

74.1

−1.09602

−

2.79262i

3.30496

−

2.25328i

−1.08655

−

6.91516i

−6.59747

6.12155i

86.1

2.19916

−

2.04052i

1.46136

−

3.72349i

4.72903

5.16103i

0.672571

−

8.97483i

95.1

−0.224190

2.99161i

−2.93221

−

2.72069i

−6.08394

−

3.46203i

−8.89948

−

1.34138i

107.1

2.96649

0.447127i

0.298920

3.98882i

3.97932

−

5.75891i

8.60016

2.65280i

137.1

−2.47872

1.68996i

3.82229

1.17902i

−2.02143

6.70178i

3.28807

−

8.37786i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
49.g	even	21	1	inner
147.n	odd	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	147.3.n.a	✓	12
3.b	odd	2	1	CM	147.3.n.a	✓	12
49.g	even	21	1	inner	147.3.n.a	✓	12
147.n	odd	42	1	inner	147.3.n.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
147.3.n.a	✓	12	1.a	even	1	1	trivial
147.3.n.a	✓	12	3.b	odd	2	1	CM
147.3.n.a	✓	12	49.g	even	21	1	inner
147.3.n.a	✓	12	147.n	odd	42	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} $ T2 acting on $S_{3}^{\mathrm{new}}(147, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} + 3 T^{11} + \cdots + 531441 $ T^12 + 3T^11 - 27T^9 - 81T^8 + 729T^6 - 6561T^4 - 19683T^3 + 177147*T + 531441
$5$	$ T^{12} $ T^12
$7$	$ T^{12} + \cdots + 13841287201 $ T^12 - 13T^11 + 120T^10 - 923T^9 + 6119T^8 - 34320T^7 + 146329T^6 - 1681680T^5 + 14691719T^4 - 108590027T^3 + 691776120T^2 - 3672178237*T + 13841287201
$11$	$ T^{12} $ T^12
$13$	$ T^{12} + \cdots + 27888020989921 $ T^12 + 46T^11 + 1587T^10 + 21459T^9 + 147591T^8 - 5161223T^7 - 40792199T^6 + 1096767719T^5 + 43143535896T^4 + 485084005298T^3 + 2941168785148T^2 + 11192748279903*T + 27888020989921
$17$	$ T^{12} $ T^12
$19$	$ T^{12} + \cdots + 20\!\cdots\!21 $ T^12 - 11T^11 + 2527T^10 - 26466T^9 + 4546594T^8 - 46810148T^7 + 3916627758T^6 - 39000497459T^5 + 2451677280358T^4 - 18252636078116T^3 + 504970539200456T^2 + 2425205540627672*T + 20673734844082321
$23$	$ T^{12} $ T^12
$29$	$ T^{12} $ T^12
$31$	$ T^{12} + \cdots + 17\!\cdots\!61 $ T^12 + 13T^11 + 6727T^10 + 85254T^9 + 32294674T^8 + 405422836T^7 + 74363006166T^6 + 952994736421T^5 + 125196566035246T^4 + 1097912686718380T^3 + 73438217391500864T^2 - 647697159544162624*T + 17764354592401178161
$37$	$ T^{12} + \cdots + 11\!\cdots\!21 $ T^12 - 256T^11 + 31955T^10 - 2639272T^9 + 164723870T^8 - 7874422864T^7 + 283442850951T^6 - 8672242147954T^5 + 245527677861709T^4 - 4468206406831586T^3 + 22587223339621080T^2 + 248137258153876698*T + 1118207881398273121
$41$	$ T^{12} $ T^12
$43$	$ T^{12} + \cdots + 37\!\cdots\!01 $ T^12 - 122T^11 + 11163T^10 - 907924T^9 + 68931516T^8 - 2111603694T^7 + 89613715471T^6 - 2323693807710T^5 + 522759822875478T^4 + 13872678036120166T^3 + 558678745169866516T^2 + 7695425479477489769*T + 37984199118663798001
$47$	$ T^{12} $ T^12
$53$	$ T^{12} $ T^12
$59$	$ T^{12} $ T^12
$61$	$ T^{12} + \cdots + 70\!\cdots\!01 $ T^12 - 403T^11 + 96782T^10 - 15934099T^9 + 1973763935T^8 - 186034429168T^7 + 13166347006905T^6 - 628894339810408T^5 + 13713739743656503T^4 + 178986715957476505T^3 - 5839327008305498232T^2 - 293975866224654800181*T + 7052821630172203158001
$67$	$ T^{12} + \cdots + 34\!\cdots\!81 $ T^12 + 13T^11 + 31423T^10 + 406302T^9 + 705260674T^8 + 9099723724T^7 + 7597611456702T^6 + 104854604616853T^5 + 59771062124662678T^4 + 519749856476411548T^3 + 175293543481197979544T^2 - 2110444718829717754744*T + 343892836133901210728881
$71$	$ T^{12} $ T^12
$73$	$ T^{12} + \cdots + 44\!\cdots\!41 $ T^12 + 97T^11 - 912673T^9 - 453762954T^8 - 17139199077T^7 + 4667439774460T^6 + 593439281702400T^5 + 53164213990048629T^4 + 559904138585009107T^3 - 18585436796321081519T^2 - 854933305253274133069T + 44948810717821566541441
$79$	$ T^{12} + \cdots + 26\!\cdots\!21 $ T^12 - 11T^11 + 43687T^10 - 479226T^9 + 1363238194T^8 - 14937633788T^7 + 20418708413958T^6 - 240797293478819T^5 + 223193101566442558T^4 - 1639667314514320916T^3 + 918921031267549006256T^2 + 9719969124366044872592*T + 2677374935400633592335121
$83$	$ T^{12} $ T^12
$89$	$ T^{12} $ T^12
$97$	$ (T^{6} + \cdots - 5825847448511)^{2} $ (T^6 + 2T^5 - 65859T^4 - 131718T^3 + 1239146498T^2 + 2478292996*T - 5825847448511)^2
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Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	147.n (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\( q + (3 \zeta_{21}^{9} + 3 \zeta_{21}^{2}) q^{3} + (4 \zeta_{21}^{11} - 4 \zeta_{21}^{10} + \cdots - 4) q^{4} + (3 \zeta_{21}^{10} - 5 \zeta_{21}^{3}) q^{7} + 9 \zeta_{21}^{11} q^{9} - 12 \zeta_{21}^{8} q^{12} + \cdots + (55 \zeta_{21}^{11} + \cdots + 57 \zeta_{21}^{3}) q^{97} +O(q^{100}) \) q + (3z^9 + 3z^2) * q^3 + (4z^11 - 4z^10 + 4z^8 - 4z^7 - 4z^6 + 4z^5 - 4z^3 + 4z^2 - 4) * q^4 + (3z^10 - 5z^3) * q^7 + 9z^11 q^9 - 12z^8 q^12 + (15z^11 - 15z^9 - 15z^6 + 7z^5 + 15z^4 - 15z^3 + 7z - 15) q^13 + 16z^5 q^16 + (-5z^11 + 5z^10 - 21z^4 - 16z^3) * q^19 + (-15z^11 + 15z^9 - 15z^8 + 15z^6 - 24z^5 - 15z^4 + 15z^3 - 15z + 15) * q^21 + 25z^4 q^25 - 27z^6 q^27 + (20z^9 + 32z^2) * q^28 + (11z^11 - 11z^10 + 46z^8 - 11z^7 + 24z^6 + 11z^5 - 11z^3 + 11z^2 + 11z - 11) q^31 + 36z^3 q^36 + (-7z^9 + 33z^7 - 40z^2 + 40) q^37 + (-24z^10 - 45z^7 + 21z^3 - 21) q^39 + (-48z^9 + 48z^8 - 35z^2 + 13z) * q^43 - 48 * q^48 + (-39z^11 + 39z^10 - 39z^8 + 39z^7 + 55z^6 - 39z^5 + 39z^3 - 39z^2 + 39) * q^49 + (-28z^11 + 32z^7 + 32z^4 - 28) q^52 + (-111z^11 + 63z^10 + 48z^9 - 111z^8 + 63z^7 + 63z^6 - 126z^5 - 48z^4 + 111z^3 - 63z^2 - 48z + 111) q^57 + (-56z^9 - 65z^7 + 9z^2 - 9) q^61 + (45z^7 + 72) q^63 + (-64z^11 - 64z^4) * q^64 + (-32z^11 + 77z^9 - 32z^8 + 32z^6 + 45z^5 - 32z^4 + 32z^3 - 32z^2 - 32z + 32) q^67 + (80z^11 - 80z^10 - 80z^9 + 80z^8 - 80z^6 + 17z^5 + 80z^4 - 143z^3 + 80z - 80) q^73 + (75z^11 - 75z^10 + 75z^8 - 75z^7 + 75z^5 - 75z^3 + 75z^2 - 75) q^75 + (84z^10 + 64z^9 + 64z^3 + 84z^2) * q^76 + (40z^11 - 40z^10 - 40z^7 - 91z^6 + 40z^5 - 40z^3 + 40z^2 - 91z - 40) * q^79 + 81z q^81 + (96z^11 + 36z^4) * q^84 + (96z^11 + 19z^8 + 85z^4 + 99z) * q^91 + (33z^10 - 33z^8 - 72z^3 - 105z) * q^93 + (55z^11 + 112z^10 - 57z^4 + 57z^3) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 4 q^{4} + 13 q^{7} + 9 q^{9} - 12 q^{12} - 46 q^{13} + 16 q^{16} + 11 q^{19} + 6 q^{21} + 25 q^{25} + 54 q^{27} - 8 q^{28} - 13 q^{31} - 72 q^{36} + 256 q^{37} - 48 q^{39} + 122 q^{43}+ \cdots - 4 q^{97}+O(q^{100}) \) 12 * q - 3 * q^3 + 4 * q^4 + 13 * q^7 + 9 * q^9 - 12 * q^12 - 46 * q^13 + 16 * q^16 + 11 * q^19 + 6 * q^21 + 25 * q^25 + 54 * q^27 - 8 * q^28 - 13 * q^31 - 72 * q^36 + 256 * q^37 - 48 * q^39 + 122 * q^43 - 576 * q^48 - 71 * q^49 - 524 * q^52 + 66 * q^57 + 403 * q^61 + 594 * q^63 - 128 * q^64 - 13 * q^67 - 97 * q^73 - 75 * q^75 - 88 * q^76 + 11 * q^79 + 81 * q^81 + 132 * q^84 + 299 * q^91 + 39 * q^93 - 4 * q^97

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