LMFDB - Newform orbit 1008.3.cd.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(415,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 2])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.415"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.cd (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-1,0,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.1364138928.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 35x^{4} + 364x^{2} + 972 $ x^6 + 35x^4 + 364x^2 + 972
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 3) q^{7} + (\beta_{5} + \beta_{3} - \beta_{2}) q^{11} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \cdots - 5) q^{13} + (\beta_{5} + 3 \beta_{4} + \cdots - 2 \beta_1) q^{17}+ \cdots + ( - 2 \beta_{5} + \beta_{4} + \cdots - 44) q^{97}+O(q^{100}) $ q - b4 * q^5 + (-b5 + b4 - b3 + b2 + b1 - 3) * q^7 + (b5 + b3 - b2) * q^11 + (2b5 - b4 - b3 - b2 - b1 - 5) q^13 + (b5 + 3b4 - 2b3 + b2 - 2b1) q^17 + (-2b5 - b4 + 2b3 - 3b1 - 5) q^19 + (5b5 - 2b4 + 4b3 - 5b1) * q^23 + (-b5 + 4b4 - 5b3 - b2 - 3b1) q^25 + (2b5 - b4 + 7b3 - b2 - b1 - 6) * q^29 + (-3b5 - b4 - 4b3 + 3b2 + 5b1 - 10) * q^31 + (b5 + 6b4 + b3 + 2b2 - 9b1 + 18) q^35 + (3b5 - 3b4 - 6b2 - 4b1 + 7) * q^37 + (8b5 - 4b4 + 2b3 - 4b2 - 4b1 + 30) q^41 + (8b4 - 3b3 + 2b2 - 1) q^43 + (3b5 - 2b4 + 4b3 + 27b1 + 30) * q^47 + (-2b5 - 5b4 + 5b3 + b2 - 28b1 + 38) * q^49 + (b4 - b3 + 30b1) q^53 + (-7b4 + 3b3 - b2 + 37b1 - 18) q^55 + (2b5 - b4 + b3 - 2b2 + 6b1 - 12) q^59 + (4b5 + 8b4 - 8b2 + 18b1 - 14) * q^61 + (5b5 + 6b4 - 10b2 - 61b1 + 66) * q^65 + (-b4 - b3 - 39b1 + 78) q^67 + (3b4 + 2b3 + 7b2 - 19b1 + 6) * q^71 + (3b5 + 3b3 + 3b2 + 11b1) * q^73 + (3b5 - 3b4 - 4b3 - 6b2 + 33b1 - 48) q^77 + (9b5 - 10b4 + 20b3 - 6b1 + 3) * q^79 + (-12b4 + 7b3 + 2b2 - 2b1) * q^83 + (12b5 - 6b4 + 16b3 - 6b2 - 6b1 + 96) q^85 + (12b5 - 2b4 - 24b2 - 72b1 + 84) * q^89 + (6b5 + b4 + 20b3 - 2b2 + 51b1 - 39) * q^91 + (-11b5 + 7b4 - 4b3 + 11b2 + 54b1 - 108) q^95 + (-2b5 + b4 - 7b3 + b2 + b1 - 44) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{5} - 11 q^{7} - 3 q^{11} - 44 q^{13} - 8 q^{17} - 30 q^{19} - 24 q^{23} - 14 q^{25} - 34 q^{29} - 39 q^{31} + 90 q^{35} + 6 q^{37} + 136 q^{41} + 258 q^{47} + 157 q^{49} + 89 q^{53} - 63 q^{59}+ \cdots - 266 q^{97}+O(q^{100}) $ 6 * q - q^5 - 11 * q^7 - 3 * q^11 - 44 * q^13 - 8 * q^17 - 30 * q^19 - 24 * q^23 - 14 * q^25 - 34 * q^29 - 39 * q^31 + 90 * q^35 + 6 * q^37 + 136 * q^41 + 258 * q^47 + 157 * q^49 + 89 * q^53 - 63 * q^59 - 50 * q^61 + 184 * q^65 + 348 * q^67 + 36 * q^73 - 221 * q^77 + 3 * q^79 + 536 * q^85 + 202 * q^89 - 62 * q^91 - 432 * q^95 - 266 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 35x^{4} + 364x^{2} + 972 $ x^6 + 35*x^4 + 364*x^2 + 972 :

$\beta_{1}$	$=$	$ ( \nu^{5} + 17\nu^{3} + 22\nu + 36 ) / 72 $ (v^5 + 17v^3 + 22v + 36) / 72
$\beta_{2}$	$=$	$ ( -5\nu^{5} - 121\nu^{3} - 36\nu^{2} - 542\nu - 396 ) / 72 $ (-5v^5 - 121v^3 - 36v^2 - 542v - 396) / 72
$\beta_{3}$	$=$	$ \nu^{2} + 12 $ v^2 + 12
$\beta_{4}$	$=$	$ ( \nu^{5} + 23\nu^{3} + 6\nu^{2} + 118\nu + 72 ) / 12 $ (v^5 + 23v^3 + 6v^2 + 118*v + 72) / 12
$\beta_{5}$	$=$	$ ( \nu^{5} + 36\nu^{4} + 17\nu^{3} + 684\nu^{2} + 94\nu + 1980 ) / 72 $ (v^5 + 36v^4 + 17v^3 + 684v^2 + 94v + 1980) / 72

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{4} + \beta_{2} - \beta_1 ) / 2 $ (b4 + b2 - b1) / 2
$\nu^{2}$	$=$	$ \beta_{3} - 12 $ b3 - 12
$\nu^{3}$	$=$	$ -6\beta_{4} - \beta_{3} - 8\beta_{2} - 4\beta _1 + 6 $ -6b4 - b3 - 8b2 - 4*b1 + 6
$\nu^{4}$	$=$	$ 2\beta_{5} - \beta_{4} - 19\beta_{3} - \beta_{2} - \beta _1 + 174 $ 2b5 - b4 - 19b3 - b2 - b1 + 174
$\nu^{5}$	$=$	$ 91\beta_{4} + 17\beta_{3} + 125\beta_{2} + 151\beta _1 - 138 $ 91b4 + 17b3 + 125b2 + 151b1 - 138

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$-1 + \beta_{1}$	$1$	$1$

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} $

415.1

	−	2.02297i
	−	3.78298i
		4.07390i
		2.02297i
		3.78298i
	−	4.07390i

−3.95380

6.84819i

−6.90373

−

1.15694i

415.2

1.15548

−

2.00135i

6.36328

−

2.91696i

415.3

2.29832

−

3.98081i

−4.95955

4.93992i

991.1

−3.95380

−

6.84819i

−6.90373

1.15694i

991.2

1.15548

2.00135i

6.36328

2.91696i

991.3

2.29832

3.98081i

−4.95955

−

4.93992i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.g	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1008.3.cd.h		6
3.b	odd	2	1		336.3.be.d	✓	6
4.b	odd	2	1		1008.3.cd.i		6
7.c	even	3	1		1008.3.cd.i		6
12.b	even	2	1		336.3.be.f	yes	6
21.g	even	6	1		2352.3.m.o		6
21.h	odd	6	1		336.3.be.f	yes	6
21.h	odd	6	1		2352.3.m.n		6
28.g	odd	6	1	inner	1008.3.cd.h		6
84.j	odd	6	1		2352.3.m.o		6
84.n	even	6	1		336.3.be.d	✓	6
84.n	even	6	1		2352.3.m.n		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.3.be.d	✓	6	3.b	odd	2	1
336.3.be.d	✓	6	84.n	even	6	1
336.3.be.f	yes	6	12.b	even	2	1
336.3.be.f	yes	6	21.h	odd	6	1
1008.3.cd.h		6	1.a	even	1	1	trivial
1008.3.cd.h		6	28.g	odd	6	1	inner
1008.3.cd.i		6	4.b	odd	2	1
1008.3.cd.i		6	7.c	even	3	1
2352.3.m.n		6	21.h	odd	6	1
2352.3.m.n		6	84.n	even	6	1
2352.3.m.o		6	21.g	even	6	1
2352.3.m.o		6	84.j	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(1008, [\chi])$:

$ T_{5}^{6} + T_{5}^{5} + 45T_{5}^{4} - 212T_{5}^{3} + 1852T_{5}^{2} - 3696T_{5} + 7056 $

T5^6 + T5^5 + 45*T5^4 - 212*T5^3 + 1852*T5^2 - 3696*T5 + 7056

$ T_{11}^{6} + 3T_{11}^{5} - 109T_{11}^{4} - 336T_{11}^{3} + 12580T_{11}^{2} - 4032T_{11} + 432 $

T11^6 + 3*T11^5 - 109*T11^4 - 336*T11^3 + 12580*T11^2 - 4032*T11 + 432

$ T_{13}^{3} + 22T_{13}^{2} - 175T_{13} - 4312 $

T13^3 + 22*T13^2 - 175*T13 - 4312

$ T_{19}^{6} + 30T_{19}^{5} - 181T_{19}^{4} - 14430T_{19}^{3} + 249841T_{19}^{2} - 888888T_{19} + 1138368 $

T19^6 + 30*T19^5 - 181*T19^4 - 14430*T19^3 + 249841*T19^2 - 888888*T19 + 1138368

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + T^{5} + \cdots + 7056 $ T^6 + T^5 + 45T^4 - 212T^3 + 1852T^2 - 3696T + 7056
$7$	$ T^{6} + 11 T^{5} + \cdots + 117649 $ T^6 + 11T^5 - 18T^4 - 665T^3 - 882T^2 + 26411*T + 117649
$11$	$ T^{6} + 3 T^{5} + \cdots + 432 $ T^6 + 3T^5 - 109T^4 - 336T^3 + 12580T^2 - 4032*T + 432
$13$	$ (T^{3} + 22 T^{2} + \cdots - 4312)^{2} $ (T^3 + 22T^2 - 175T - 4312)^2
$17$	$ T^{6} + 8 T^{5} + \cdots + 20358144 $ T^6 + 8T^5 + 600T^4 + 4736T^3 + 323392T^2 + 2418432*T + 20358144
$19$	$ T^{6} + 30 T^{5} + \cdots + 1138368 $ T^6 + 30T^5 - 181T^4 - 14430T^3 + 249841T^2 - 888888*T + 1138368
$23$	$ T^{6} + 24 T^{5} + \cdots + 66382848 $ T^6 + 24T^5 - 1096T^4 - 30912T^3 + 1771840T^2 - 18176256*T + 66382848
$29$	$ (T^{3} + 17 T^{2} + \cdots - 35952)^{2} $ (T^3 + 17T^2 - 1664T - 35952)^2
$31$	$ T^{6} + 39 T^{5} + \cdots + 3756483 $ T^6 + 39T^5 - 1008T^4 - 59085T^3 + 2251584T^2 + 5085855*T + 3756483
$37$	$ T^{6} - 6 T^{5} + \cdots + 183115024 $ T^6 - 6T^5 + 1467T^4 - 18478T^3 + 2128953T^2 - 19364292*T + 183115024
$41$	$ (T^{3} - 68 T^{2} + \cdots + 37152)^{2} $ (T^3 - 68T^2 - 1196T + 37152)^2
$43$	$ T^{6} + 4010 T^{4} + \cdots + 7660812 $ T^6 + 4010T^4 + 3962713T^2 + 7660812
$47$	$ T^{6} + \cdots + 7468832448 $ T^6 - 258T^5 + 28968T^4 - 1749240T^3 + 58841568T^2 - 1014884640*T + 7468832448
$53$	$ T^{6} - 89 T^{5} + \cdots + 609892416 $ T^6 - 89T^5 + 5325T^4 - 181652T^3 + 4541272T^2 - 64110816*T + 609892416
$59$	$ T^{6} + 63 T^{5} + \cdots + 84672 $ T^6 + 63T^5 + 1535T^4 + 13356T^3 + 55528T^2 + 106848*T + 84672
$61$	$ T^{6} + \cdots + 73796982336 $ T^6 + 50T^5 + 10648T^4 + 135912T^3 + 79972704T^2 + 2213453088*T + 73796982336
$67$	$ T^{6} + \cdots + 84665952108 $ T^6 - 348T^5 + 53691T^4 - 4636404T^3 + 235964241T^2 - 6714552186*T + 84665952108
$71$	$ T^{6} + 6164 T^{4} + \cdots + 615874752 $ T^6 + 6164T^4 + 5578864T^2 + 615874752
$73$	$ T^{6} - 36 T^{5} + \cdots + 688642564 $ T^6 - 36T^5 + 2307T^4 - 16088T^3 + 1966833T^2 - 26530662*T + 688642564
$79$	$ T^{6} - 3 T^{5} + \cdots + 140726403 $ T^6 - 3T^5 - 10884T^4 + 32661T^3 + 118547316T^2 + 223695189*T + 140726403
$83$	$ T^{6} + \cdots + 22314082608 $ T^6 + 11123T^4 + 29227912T^2 + 22314082608
$89$	$ T^{6} + \cdots + 10788229287936 $ T^6 - 202T^5 + 54804T^4 - 3741088T^3 + 859477888T^2 - 45983616000*T + 10788229287936
$97$	$ (T^{3} + 133 T^{2} + \cdots + 35252)^{2} $ (T^3 + 133T^2 + 4136T + 35252)^2
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Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cd (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 3) q^{7} + (\beta_{5} + \beta_{3} - \beta_{2}) q^{11} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \cdots - 5) q^{13} + (\beta_{5} + 3 \beta_{4} + \cdots - 2 \beta_1) q^{17}+ \cdots + ( - 2 \beta_{5} + \beta_{4} + \cdots - 44) q^{97}+O(q^{100}) \) q - b4 * q^5 + (-b5 + b4 - b3 + b2 + b1 - 3) * q^7 + (b5 + b3 - b2) * q^11 + (2b5 - b4 - b3 - b2 - b1 - 5) q^13 + (b5 + 3b4 - 2b3 + b2 - 2b1) q^17 + (-2b5 - b4 + 2b3 - 3b1 - 5) q^19 + (5b5 - 2b4 + 4b3 - 5b1) * q^23 + (-b5 + 4b4 - 5b3 - b2 - 3b1) q^25 + (2b5 - b4 + 7b3 - b2 - b1 - 6) * q^29 + (-3b5 - b4 - 4b3 + 3b2 + 5b1 - 10) * q^31 + (b5 + 6b4 + b3 + 2b2 - 9b1 + 18) q^35 + (3b5 - 3b4 - 6b2 - 4b1 + 7) * q^37 + (8b5 - 4b4 + 2b3 - 4b2 - 4b1 + 30) q^41 + (8b4 - 3b3 + 2b2 - 1) q^43 + (3b5 - 2b4 + 4b3 + 27b1 + 30) * q^47 + (-2b5 - 5b4 + 5b3 + b2 - 28b1 + 38) * q^49 + (b4 - b3 + 30b1) q^53 + (-7b4 + 3b3 - b2 + 37b1 - 18) q^55 + (2b5 - b4 + b3 - 2b2 + 6b1 - 12) q^59 + (4b5 + 8b4 - 8b2 + 18b1 - 14) * q^61 + (5b5 + 6b4 - 10b2 - 61b1 + 66) * q^65 + (-b4 - b3 - 39b1 + 78) q^67 + (3b4 + 2b3 + 7b2 - 19b1 + 6) * q^71 + (3b5 + 3b3 + 3b2 + 11b1) * q^73 + (3b5 - 3b4 - 4b3 - 6b2 + 33b1 - 48) q^77 + (9b5 - 10b4 + 20b3 - 6b1 + 3) * q^79 + (-12b4 + 7b3 + 2b2 - 2b1) * q^83 + (12b5 - 6b4 + 16b3 - 6b2 - 6b1 + 96) q^85 + (12b5 - 2b4 - 24b2 - 72b1 + 84) * q^89 + (6b5 + b4 + 20b3 - 2b2 + 51b1 - 39) * q^91 + (-11b5 + 7b4 - 4b3 + 11b2 + 54b1 - 108) q^95 + (-2b5 + b4 - 7b3 + b2 + b1 - 44) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{5} - 11 q^{7} - 3 q^{11} - 44 q^{13} - 8 q^{17} - 30 q^{19} - 24 q^{23} - 14 q^{25} - 34 q^{29} - 39 q^{31} + 90 q^{35} + 6 q^{37} + 136 q^{41} + 258 q^{47} + 157 q^{49} + 89 q^{53} - 63 q^{59}+ \cdots - 266 q^{97}+O(q^{100}) \) 6 * q - q^5 - 11 * q^7 - 3 * q^11 - 44 * q^13 - 8 * q^17 - 30 * q^19 - 24 * q^23 - 14 * q^25 - 34 * q^29 - 39 * q^31 + 90 * q^35 + 6 * q^37 + 136 * q^41 + 258 * q^47 + 157 * q^49 + 89 * q^53 - 63 * q^59 - 50 * q^61 + 184 * q^65 + 348 * q^67 + 36 * q^73 - 221 * q^77 + 3 * q^79 + 536 * q^85 + 202 * q^89 - 62 * q^91 - 432 * q^95 - 266 * q^97

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