LMFDB - Newform orbit 1008.3.cd.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(415,1008)]

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

lf = mfeigenbasis(mf)

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")

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

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:	6.0.259470000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 14x^{4} - x^{3} + 176x^{2} - 91x + 49 $ x^6 - x^5 + 14x^4 - x^3 + 176x^2 - 91*x + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 112)
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_{3} q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 4) q^{7}+O(q^{10}) $ q - b3 * q^5 + (-b5 + b4 + b3 + b2 - b1 - 4) * q^7	$ q - \beta_{3} q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 4) q^{7}+ \cdots + (14 \beta_{5} + 14 \beta_{4} + \cdots + 21) q^{97}+O(q^{100}) $ q - b3 * q^5 + (-b5 + b4 + b3 + b2 - b1 - 4) * q^7 + (-b4 + b3 - 3b2 + b1 - 3) q^11 + (-2b5 - 2b4 + b1 + 5) * q^13 + (4b5 - 2b4 - b3 + b1 + 4) * q^17 + (b5 - b3 + 7b2 + 2b1 - 13) * q^19 + (7b5 + 6b2 - 5) * q^23 + (8b5 - 4b4 - 2b3 + 14b2 + 2b1 - 6) q^25 + (2b5 + 2b4 + b1 + 17) * q^29 + (3b4 + b3 + 7b2 + b1 + 7) * q^31 + (-3b5 - 4b4 + 3b3 + 3b2 - 3b1 + 30) q^35 + (-2b5 + 4b4 - 6b3 + b2 - 2) q^37 + (2b5 + 2b4 + 5b1 - 19) q^41 + (-4b5 + 4b4 - 8b3 + 16b2 + 4b1 - 12) q^43 + (-3b5 - 3b3 - 21b2 + 6b1 + 39) * q^47 + (-2b5 - 2b4 - 2b3 - 38b2 + 5b1) q^49 + (4b5 - 2b4 - 8b3 - 33b2 + 8b1 + 37) q^53 + (-13b5 + 13b4 + 8b3 - 84b2 - 4b1 + 29) q^55 + (13b4 + 4b3 + 4b1) q^59 + (-6b5 + 12b4 + 2b3 - 35b2 - 6) * q^61 + (-6b5 + 12b4 - 11b3 - 57b2 - 6) * q^65 + (-9b4 + 2b3 + 38b2 + 2b1 + 38) * q^67 + (-16b5 + 16b4 - 4b3 - 60b2 + 2b1 + 14) q^71 + (8b5 - 4b4 + 6b3 - 35b2 - 6b1 + 43) q^73 + (22b5 - 2b4 - 9b3 + 24b2 + 8b1 - 29) q^77 + (-3b5 + 4b3 - 22b2 - 8b1 + 41) * q^79 + (4b5 - 4b4 + 4b3 + 84b2 - 2b1 - 38) q^83 + (2b5 + 2b4 + 8b1 - 19) q^85 + (12b5 - 24b4 - 8b3 - 21b2 + 12) * q^89 + (24b5 - 8b4 + 6b3 + 10b2 - 11b1 - 27) q^91 + (11b4 + 10b3 - 36b2 + 10b1 - 36) * q^95 + (14b5 + 14b4 + b1 + 21) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{5} - 14 q^{7}+O(q^{10}) $ 6 * q + q^5 - 14 * q^7	$ 6 q + q^{5} - 14 q^{7} - 33 q^{11} + 28 q^{13} + 5 q^{17} - 63 q^{19} - 33 q^{23} - 32 q^{25} + 100 q^{29} + 69 q^{31} + 189 q^{35} + 15 q^{37} - 124 q^{41} + 171 q^{47} - 122 q^{49} + 97 q^{53} + 27 q^{59} - 89 q^{61} - 142 q^{65} + 309 q^{67} + 123 q^{73} - 181 q^{77} + 201 q^{79} - 130 q^{85} - 91 q^{89} - 212 q^{91} - 321 q^{95} + 124 q^{97}+O(q^{100}) $ 6 * q + q^5 - 14 * q^7 - 33 * q^11 + 28 * q^13 + 5 * q^17 - 63 * q^19 - 33 * q^23 - 32 * q^25 + 100 * q^29 + 69 * q^31 + 189 * q^35 + 15 * q^37 - 124 * q^41 + 171 * q^47 - 122 * q^49 + 97 * q^53 + 27 * q^59 - 89 * q^61 - 142 * q^65 + 309 * q^67 + 123 * q^73 - 181 * q^77 + 201 * q^79 - 130 * q^85 - 91 * q^89 - 212 * q^91 - 321 * q^95 + 124 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} + 14x^{4} - x^{3} + 176x^{2} - 91x + 49 $ x^6 - x^5 + 14*x^4 - x^3 + 176*x^2 - 91*x + 49 :

$\beta_{1}$	$=$	$ ( -2\nu^{5} + 28\nu^{4} - 392\nu^{3} + 352\nu^{2} - 182\nu + 175 ) / 2373 $ (-2v^5 + 28v^4 - 392v^3 + 352v^2 - 182*v + 175) / 2373
$\beta_{2}$	$=$	$ ( 26\nu^{5} - 25\nu^{4} + 350\nu^{3} + 170\nu^{2} + 4400\nu + 98 ) / 2373 $ (26v^5 - 25v^4 + 350v^3 + 170v^2 + 4400*v + 98) / 2373
$\beta_{3}$	$=$	$ ( -26\nu^{5} + 25\nu^{4} - 350\nu^{3} - 170\nu^{2} + 346\nu - 98 ) / 2373 $ (-26v^5 + 25v^4 - 350v^3 - 170v^2 + 346*v - 98) / 2373
$\beta_{4}$	$=$	$ ( 78\nu^{5} + 38\nu^{4} + 1050\nu^{3} + 1301\nu^{2} + 13878\nu + 8204 ) / 2373 $ (78v^5 + 38v^4 + 1050v^3 + 1301v^2 + 13878*v + 8204) / 2373
$\beta_{5}$	$=$	$ ( -91\nu^{5} + 144\nu^{4} - 1225\nu^{3} + 987\nu^{2} - 15061\nu + 13104 ) / 2373 $ (-91v^5 + 144v^4 - 1225v^3 + 987v^2 - 15061*v + 13104) / 2373

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

$\nu$	$=$	$ ( \beta_{3} + \beta_{2} ) / 2 $ (b3 + b2) / 2
$\nu^{2}$	$=$	$ 2\beta_{5} - \beta_{4} + 10\beta_{2} - 8 $ 2b5 - b4 + 10b2 - 8
$\nu^{3}$	$=$	$ ( 2\beta_{5} + 2\beta_{4} - 13\beta _1 - 17 ) / 2 $ (2b5 + 2b4 - 13*b1 - 17) / 2
$\nu^{4}$	$=$	$ -14\beta_{5} + 28\beta_{4} - 3\beta_{3} - 136\beta_{2} - 14 $ -14b5 + 28b4 - 3b3 - 136b2 - 14
$\nu^{5}$	$=$	$ ( -80\beta_{5} + 40\beta_{4} - 175\beta_{3} - 379\beta_{2} + 175\beta _1 + 299 ) / 2 $ (-80b5 + 40b4 - 175b3 - 379b2 + 175*b1 + 299) / 2

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

Character values

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

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$-\beta_{2}$	$1$	$1$

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

1.94170	−	3.36313i
0.264167	−	0.457551i
−1.70587	+	2.95466i
1.94170	+	3.36313i
0.264167	+	0.457551i
−1.70587	−	2.95466i

−3.38341

5.86024i

−5.88341

−

3.79282i

415.2

−0.0283339

0.0490758i

−2.52833

−

6.52744i

415.3

3.91174

−

6.77534i

1.41174

6.85616i

991.1

−3.38341

−

5.86024i

−5.88341

3.79282i

991.2

−0.0283339

−

0.0490758i

−2.52833

6.52744i

991.3

3.91174

6.77534i

1.41174

−

6.85616i

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.j		6
3.b	odd	2	1		112.3.r.b	✓	6
4.b	odd	2	1		1008.3.cd.k		6
7.c	even	3	1		1008.3.cd.k		6
12.b	even	2	1		112.3.r.c	yes	6
21.c	even	2	1		784.3.r.q		6
21.g	even	6	1		784.3.d.k		6
21.g	even	6	1		784.3.r.p		6
21.h	odd	6	1		112.3.r.c	yes	6
21.h	odd	6	1		784.3.d.l		6
24.f	even	2	1		448.3.r.d		6
24.h	odd	2	1		448.3.r.e		6
28.g	odd	6	1	inner	1008.3.cd.j		6
84.h	odd	2	1		784.3.r.p		6
84.j	odd	6	1		784.3.d.k		6
84.j	odd	6	1		784.3.r.q		6
84.n	even	6	1		112.3.r.b	✓	6
84.n	even	6	1		784.3.d.l		6
168.s	odd	6	1		448.3.r.d		6
168.v	even	6	1		448.3.r.e		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.3.r.b	✓	6	3.b	odd	2	1
112.3.r.b	✓	6	84.n	even	6	1
112.3.r.c	yes	6	12.b	even	2	1
112.3.r.c	yes	6	21.h	odd	6	1
448.3.r.d		6	24.f	even	2	1
448.3.r.d		6	168.s	odd	6	1
448.3.r.e		6	24.h	odd	2	1
448.3.r.e		6	168.v	even	6	1
784.3.d.k		6	21.g	even	6	1
784.3.d.k		6	84.j	odd	6	1
784.3.d.l		6	21.h	odd	6	1
784.3.d.l		6	84.n	even	6	1
784.3.r.p		6	21.g	even	6	1
784.3.r.p		6	84.h	odd	2	1
784.3.r.q		6	21.c	even	2	1
784.3.r.q		6	84.j	odd	6	1
1008.3.cd.j		6	1.a	even	1	1	trivial
1008.3.cd.j		6	28.g	odd	6	1	inner
1008.3.cd.k		6	4.b	odd	2	1
1008.3.cd.k		6	7.c	even	3	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} + 54T_{5}^{4} + 59T_{5}^{3} + 2806T_{5}^{2} + 159T_{5} + 9 $

$ T_{11}^{6} + 33T_{11}^{5} + 284T_{11}^{4} - 2607T_{11}^{3} - 28112T_{11}^{2} + 246717T_{11} + 3251043 $

$ T_{13}^{3} - 14T_{13}^{2} - 268T_{13} + 728 $

$ T_{19}^{6} + 63T_{19}^{5} + 1604T_{19}^{4} + 17703T_{19}^{3} + 99688T_{19}^{2} + 277347T_{19} + 324723 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - T^{5} + 54 T^{4} + \cdots + 9 $ T^6 - T^5 + 54T^4 + 59T^3 + 2806T^2 + 159T + 9
$7$	$ T^{6} + 14 T^{5} + \cdots + 117649 $ T^6 + 14T^5 + 159T^4 + 1204T^3 + 7791T^2 + 33614*T + 117649
$11$	$ T^{6} + 33 T^{5} + \cdots + 3251043 $ T^6 + 33T^5 + 284T^4 - 2607T^3 - 28112T^2 + 246717*T + 3251043
$13$	$ (T^{3} - 14 T^{2} + \cdots + 728)^{2} $ (T^3 - 14T^2 - 268T + 728)^2
$17$	$ T^{6} - 5 T^{5} + \cdots + 540225 $ T^6 - 5T^5 + 270T^4 + 2695T^3 + 56350T^2 + 180075*T + 540225
$19$	$ T^{6} + 63 T^{5} + \cdots + 324723 $ T^6 + 63T^5 + 1604T^4 + 17703T^3 + 99688T^2 + 277347*T + 324723
$23$	$ T^{6} + 33 T^{5} + \cdots + 231317883 $ T^6 + 33T^5 - 496T^4 - 28347T^3 + 448108T^2 + 22628637*T + 231317883
$29$	$ (T^{3} - 50 T^{2} + \cdots - 1560)^{2} $ (T^3 - 50T^2 + 580T - 1560)^2
$31$	$ T^{6} - 69 T^{5} + \cdots + 3479787 $ T^6 - 69T^5 + 1836T^4 - 17181T^3 - 12312T^2 + 804519*T + 3479787
$37$	$ T^{6} + \cdots + 1995855625 $ T^6 - 15T^5 + 2550T^4 - 54475T^3 + 6075750T^2 - 103869375*T + 1995855625
$41$	$ (T^{3} + 62 T^{2} + \cdots - 216)^{2} $ (T^3 + 62T^2 - 92T - 216)^2
$43$	$ T^{6} + \cdots + 10673651712 $ T^6 + 7424T^4 + 16187392T^2 + 10673651712
$47$	$ T^{6} - 171 T^{5} + \cdots + 5250987 $ T^6 - 171T^5 + 11196T^4 - 247779T^3 + 1873368T^2 + 5751081*T + 5250987
$53$	$ T^{6} + \cdots + 17908060041 $ T^6 - 97T^5 + 9606T^4 - 248533T^3 + 13019446T^2 - 26362737*T + 17908060041
$59$	$ T^{6} + \cdots + 1821240963 $ T^6 - 27T^5 - 4576T^4 + 130113T^3 + 22557508T^2 - 356206023*T + 1821240963
$61$	$ T^{6} + \cdots + 5424764409 $ T^6 + 89T^5 + 7414T^4 + 192429T^3 + 6812166T^2 - 37342071*T + 5424764409
$67$	$ T^{6} + \cdots + 7174510227 $ T^6 - 309T^5 + 39816T^4 - 2468601T^3 + 78935148T^2 - 1172058201*T + 7174510227
$71$	$ T^{6} + \cdots + 33942454272 $ T^6 + 17984T^4 + 47429632T^2 + 33942454272
$73$	$ T^{6} + \cdots + 20655150961 $ T^6 - 123T^5 + 13446T^4 - 494447T^3 + 20509926T^2 + 241879077*T + 20655150961
$79$	$ T^{6} - 201 T^{5} + \cdots + 2187 $ T^6 - 201T^5 + 15456T^4 - 399789T^3 + 3961548T^2 - 161109*T + 2187
$83$	$ T^{6} + \cdots + 72559411200 $ T^6 + 16640T^4 + 64384000T^2 + 72559411200
$89$	$ T^{6} + \cdots + 14363303409 $ T^6 + 91T^5 + 15654T^4 - 431249T^3 + 65267206T^2 + 883631931*T + 14363303409
$97$	$ (T^{3} - 62 T^{2} + \cdots - 195304)^{2} $ (T^3 - 62T^2 - 10252T - 195304)^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)

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 4) q^{7}+O(q^{10}) \) q - b3 * q^5 + (-b5 + b4 + b3 + b2 - b1 - 4) * q^7	\( q - \beta_{3} q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 4) q^{7}+ \cdots + (14 \beta_{5} + 14 \beta_{4} + \cdots + 21) q^{97}+O(q^{100}) \) q - b3 * q^5 + (-b5 + b4 + b3 + b2 - b1 - 4) * q^7 + (-b4 + b3 - 3b2 + b1 - 3) q^11 + (-2b5 - 2b4 + b1 + 5) * q^13 + (4b5 - 2b4 - b3 + b1 + 4) * q^17 + (b5 - b3 + 7b2 + 2b1 - 13) * q^19 + (7b5 + 6b2 - 5) * q^23 + (8b5 - 4b4 - 2b3 + 14b2 + 2b1 - 6) q^25 + (2b5 + 2b4 + b1 + 17) * q^29 + (3b4 + b3 + 7b2 + b1 + 7) * q^31 + (-3b5 - 4b4 + 3b3 + 3b2 - 3b1 + 30) q^35 + (-2b5 + 4b4 - 6b3 + b2 - 2) q^37 + (2b5 + 2b4 + 5b1 - 19) q^41 + (-4b5 + 4b4 - 8b3 + 16b2 + 4b1 - 12) q^43 + (-3b5 - 3b3 - 21b2 + 6b1 + 39) * q^47 + (-2b5 - 2b4 - 2b3 - 38b2 + 5b1) q^49 + (4b5 - 2b4 - 8b3 - 33b2 + 8b1 + 37) q^53 + (-13b5 + 13b4 + 8b3 - 84b2 - 4b1 + 29) q^55 + (13b4 + 4b3 + 4b1) q^59 + (-6b5 + 12b4 + 2b3 - 35b2 - 6) * q^61 + (-6b5 + 12b4 - 11b3 - 57b2 - 6) * q^65 + (-9b4 + 2b3 + 38b2 + 2b1 + 38) * q^67 + (-16b5 + 16b4 - 4b3 - 60b2 + 2b1 + 14) q^71 + (8b5 - 4b4 + 6b3 - 35b2 - 6b1 + 43) q^73 + (22b5 - 2b4 - 9b3 + 24b2 + 8b1 - 29) q^77 + (-3b5 + 4b3 - 22b2 - 8b1 + 41) * q^79 + (4b5 - 4b4 + 4b3 + 84b2 - 2b1 - 38) q^83 + (2b5 + 2b4 + 8b1 - 19) q^85 + (12b5 - 24b4 - 8b3 - 21b2 + 12) * q^89 + (24b5 - 8b4 + 6b3 + 10b2 - 11b1 - 27) q^91 + (11b4 + 10b3 - 36b2 + 10b1 - 36) * q^95 + (14b5 + 14b4 + b1 + 21) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{5} - 14 q^{7}+O(q^{10}) \) 6 * q + q^5 - 14 * q^7	\( 6 q + q^{5} - 14 q^{7} - 33 q^{11} + 28 q^{13} + 5 q^{17} - 63 q^{19} - 33 q^{23} - 32 q^{25} + 100 q^{29} + 69 q^{31} + 189 q^{35} + 15 q^{37} - 124 q^{41} + 171 q^{47} - 122 q^{49} + 97 q^{53} + 27 q^{59} - 89 q^{61} - 142 q^{65} + 309 q^{67} + 123 q^{73} - 181 q^{77} + 201 q^{79} - 130 q^{85} - 91 q^{89} - 212 q^{91} - 321 q^{95} + 124 q^{97}+O(q^{100}) \) 6 * q + q^5 - 14 * q^7 - 33 * q^11 + 28 * q^13 + 5 * q^17 - 63 * q^19 - 33 * q^23 - 32 * q^25 + 100 * q^29 + 69 * q^31 + 189 * q^35 + 15 * q^37 - 124 * q^41 + 171 * q^47 - 122 * q^49 + 97 * q^53 + 27 * q^59 - 89 * q^61 - 142 * q^65 + 309 * q^67 + 123 * q^73 - 181 * q^77 + 201 * q^79 - 130 * q^85 - 91 * q^89 - 212 * q^91 - 321 * q^95 + 124 * q^97

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