LMFDB - Newform orbit 432.4.i.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,4,Mod(145,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("432.145");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	432.i (of order $3$, degree $2$, not 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:	$25.4888251225$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{3} + 7 \beta_1) q^{5} + (3 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 2) q^{7} + (8 \beta_{3} - 8 \beta_{2} + 29 \beta_1 - 37) q^{11} + ( - 15 \beta_{3} + 13 \beta_1) q^{13} + (9 \beta_{2} - 45) q^{17}+ \cdots + ( - 102 \beta_{3} + 102 \beta_{2} + \cdots - 317) q^{97}+O(q^{100}) $ q + (b3 + 7b1) q^5 + (3b3 - 3b2 - 5b1 + 2) q^7 + (8b3 - 8b2 + 29b1 - 37) q^11 + (-15b3 + 13b1) * q^13 + (9b2 - 45) q^17 + (-27b2 + 25) q^19 + (-19b3 - 7b1) * q^23 + (15b3 - 15b2 - 68b1 + 53) q^25 + (b3 - b2 + 25b1 - 26) q^29 + (-3b3 + 23b1) * q^31 + (-19b2 - 8) q^35 + (-54b2 - 52) q^37 + (-98b3 + 115b1) * q^41 + (-6b3 + 6b2 - 41b1 + 47) q^43 + (-91b3 + 91b2 + 245b1 - 154) q^47 + (21b3 + 246b1) * q^49 + (-162b2 - 108) q^53 + (-93b2 - 360) q^55 + (-136b3 - 331b1) * q^59 + (105b3 - 105b2 + 167b1 - 272) q^61 + (-107b3 + 107b2 - 29b1 + 136) q^65 + (-66b3 + 527b1) * q^67 + (-144b2 + 612) q^71 + (-243b2 - 349) q^73 + (-71b3 - 47b1) * q^77 + (309b3 - 309b2 + 247b1 - 556) q^79 + (107b3 - 107b2 + 353b1 - 460) q^83 + (18b3 - 306b1) * q^85 + (72b2 + 234) q^89 + (-69b2 + 356) q^91 + (-164b3 + 148b1) * q^95 + (-102b3 + 102b2 + 419b1 - 317) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 15 q^{5} + 7 q^{7} - 66 q^{11} + 11 q^{13} - 198 q^{17} + 154 q^{19} - 33 q^{23} + 121 q^{25} - 51 q^{29} + 43 q^{31} + 6 q^{35} - 100 q^{37} + 132 q^{41} + 88 q^{43} - 399 q^{47} + 513 q^{49} - 108 q^{53}+ \cdots - 736 q^{97}+O(q^{100}) $ 4 * q + 15 * q^5 + 7 * q^7 - 66 * q^11 + 11 * q^13 - 198 * q^17 + 154 * q^19 - 33 * q^23 + 121 * q^25 - 51 * q^29 + 43 * q^31 + 6 * q^35 - 100 * q^37 + 132 * q^41 + 88 * q^43 - 399 * q^47 + 513 * q^49 - 108 * q^53 - 1254 * q^55 - 798 * q^59 - 439 * q^61 + 165 * q^65 + 988 * q^67 + 2736 * q^71 - 910 * q^73 - 165 * q^77 - 803 * q^79 - 813 * q^83 - 594 * q^85 + 792 * q^89 + 1562 * q^91 + 132 * q^95 - 736 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2*x^2 - 3*x + 9 :

$\beta_{1}$	$=$	$ ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 $ (v^3 + 2v^2 - 2v - 3) / 6
$\beta_{2}$	$=$	$ ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 $ (-v^3 + v^2 + 5*v) / 3
$\beta_{3}$	$=$	$ ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 $ (2v^3 + v^2 + 2v - 9) / 3

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

$\nu$	$=$	$ ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 $ (b3 + b2 - 2*b1 + 2) / 3
$\nu^{2}$	$=$	$ ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 $ (-b3 + 2b2 + 8b1 + 1) / 3
$\nu^{3}$	$=$	$ ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 $ (4b3 - 2b2 - 2*b1 + 11) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$-\beta_{1}$

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} $

145.1

−1.18614	+	1.26217i
1.68614	−	0.396143i
−1.18614	−	1.26217i
1.68614	+	0.396143i

2.31386

−

4.00772i

6.05842

10.4935i

145.2

5.18614

−

8.98266i

−2.55842

−

4.43132i

289.1

2.31386

4.00772i

6.05842

−

10.4935i

289.2

5.18614

8.98266i

−2.55842

4.43132i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	432.4.i.c		4
3.b	odd	2	1		144.4.i.c		4
4.b	odd	2	1		27.4.c.a		4
9.c	even	3	1	inner	432.4.i.c		4
9.c	even	3	1		1296.4.a.i		2
9.d	odd	6	1		144.4.i.c		4
9.d	odd	6	1		1296.4.a.u		2
12.b	even	2	1		9.4.c.a	✓	4
36.f	odd	6	1		27.4.c.a		4
36.f	odd	6	1		81.4.a.a		2
36.h	even	6	1		9.4.c.a	✓	4
36.h	even	6	1		81.4.a.d		2
60.h	even	2	1		225.4.e.b		4
60.l	odd	4	2		225.4.k.b		8
180.n	even	6	1		225.4.e.b		4
180.n	even	6	1		2025.4.a.g		2
180.p	odd	6	1		2025.4.a.n		2
180.v	odd	12	2		225.4.k.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.4.c.a	✓	4	12.b	even	2	1
9.4.c.a	✓	4	36.h	even	6	1
27.4.c.a		4	4.b	odd	2	1
27.4.c.a		4	36.f	odd	6	1
81.4.a.a		2	36.f	odd	6	1
81.4.a.d		2	36.h	even	6	1
144.4.i.c		4	3.b	odd	2	1
144.4.i.c		4	9.d	odd	6	1
225.4.e.b		4	60.h	even	2	1
225.4.e.b		4	180.n	even	6	1
225.4.k.b		8	60.l	odd	4	2
225.4.k.b		8	180.v	odd	12	2
432.4.i.c		4	1.a	even	1	1	trivial
432.4.i.c		4	9.c	even	3	1	inner
1296.4.a.i		2	9.c	even	3	1
1296.4.a.u		2	9.d	odd	6	1
2025.4.a.g		2	180.n	even	6	1
2025.4.a.n		2	180.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} - 15T_{5}^{3} + 177T_{5}^{2} - 720T_{5} + 2304 $ T5^4 - 15*T5^3 + 177*T5^2 - 720*T5 + 2304 acting on $S_{4}^{\mathrm{new}}(432, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} - 15 T^{3} + \cdots + 2304 $ T^4 - 15T^3 + 177T^2 - 720*T + 2304
$7$	$ T^{4} - 7 T^{3} + \cdots + 3844 $ T^4 - 7T^3 + 111T^2 + 434*T + 3844
$11$	$ T^{4} + 66 T^{3} + \cdots + 314721 $ T^4 + 66T^3 + 3795T^2 + 37026*T + 314721
$13$	$ T^{4} - 11 T^{3} + \cdots + 3334276 $ T^4 - 11T^3 + 1947T^2 + 20086*T + 3334276
$17$	$ (T^{2} + 99 T + 1782)^{2} $ (T^2 + 99*T + 1782)^2
$19$	$ (T^{2} - 77 T - 4532)^{2} $ (T^2 - 77*T - 4532)^2
$23$	$ T^{4} + 33 T^{3} + \cdots + 7322436 $ T^4 + 33T^3 + 3795T^2 - 89298*T + 7322436
$29$	$ T^{4} + 51 T^{3} + \cdots + 412164 $ T^4 + 51T^3 + 1959T^2 + 32742*T + 412164
$31$	$ T^{4} - 43 T^{3} + \cdots + 150544 $ T^4 - 43T^3 + 1461T^2 - 16684*T + 150544
$37$	$ (T^{2} + 50 T - 23432)^{2} $ (T^2 + 50*T - 23432)^2
$41$	$ T^{4} + \cdots + 5606565129 $ T^4 - 132T^3 + 92301T^2 + 9883764*T + 5606565129
$43$	$ T^{4} - 88 T^{3} + \cdots + 2686321 $ T^4 - 88T^3 + 6105T^2 - 144232*T + 2686321
$47$	$ T^{4} + 399 T^{3} + \cdots + 813276324 $ T^4 + 399T^3 + 187719T^2 - 11378682*T + 813276324
$53$	$ (T^{2} + 54 T - 215784)^{2} $ (T^2 + 54*T - 215784)^2
$59$	$ T^{4} + 798 T^{3} + \cdots + 43678881 $ T^4 + 798T^3 + 630195T^2 + 5273982*T + 43678881
$61$	$ T^{4} + \cdots + 1829786176 $ T^4 + 439T^3 + 235497T^2 - 18778664*T + 1829786176
$67$	$ T^{4} + \cdots + 43305193801 $ T^4 - 988T^3 + 768045T^2 - 205601812*T + 43305193801
$71$	$ (T^{2} - 1368 T + 296784)^{2} $ (T^2 - 1368*T + 296784)^2
$73$	$ (T^{2} + 455 T - 435398)^{2} $ (T^2 + 455*T - 435398)^2
$79$	$ T^{4} + \cdots + 392522298256 $ T^4 + 803T^3 + 1271325T^2 - 503092348*T + 392522298256
$83$	$ T^{4} + \cdots + 5010940944 $ T^4 + 813T^3 + 590181T^2 + 57550644*T + 5010940944
$89$	$ (T^{2} - 396 T - 3564)^{2} $ (T^2 - 396*T - 3564)^2
$97$	$ T^{4} + \cdots + 2459267281 $ T^4 + 736T^3 + 492105T^2 + 36498976*T + 2459267281
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Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	432.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{3} + 7 \beta_1) q^{5} + (3 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 2) q^{7} + (8 \beta_{3} - 8 \beta_{2} + 29 \beta_1 - 37) q^{11} + ( - 15 \beta_{3} + 13 \beta_1) q^{13} + (9 \beta_{2} - 45) q^{17}+ \cdots + ( - 102 \beta_{3} + 102 \beta_{2} + \cdots - 317) q^{97}+O(q^{100}) \) q + (b3 + 7b1) q^5 + (3b3 - 3b2 - 5b1 + 2) q^7 + (8b3 - 8b2 + 29b1 - 37) q^11 + (-15b3 + 13b1) * q^13 + (9b2 - 45) q^17 + (-27b2 + 25) q^19 + (-19b3 - 7b1) * q^23 + (15b3 - 15b2 - 68b1 + 53) q^25 + (b3 - b2 + 25b1 - 26) q^29 + (-3b3 + 23b1) * q^31 + (-19b2 - 8) q^35 + (-54b2 - 52) q^37 + (-98b3 + 115b1) * q^41 + (-6b3 + 6b2 - 41b1 + 47) q^43 + (-91b3 + 91b2 + 245b1 - 154) q^47 + (21b3 + 246b1) * q^49 + (-162b2 - 108) q^53 + (-93b2 - 360) q^55 + (-136b3 - 331b1) * q^59 + (105b3 - 105b2 + 167b1 - 272) q^61 + (-107b3 + 107b2 - 29b1 + 136) q^65 + (-66b3 + 527b1) * q^67 + (-144b2 + 612) q^71 + (-243b2 - 349) q^73 + (-71b3 - 47b1) * q^77 + (309b3 - 309b2 + 247b1 - 556) q^79 + (107b3 - 107b2 + 353b1 - 460) q^83 + (18b3 - 306b1) * q^85 + (72b2 + 234) q^89 + (-69b2 + 356) q^91 + (-164b3 + 148b1) * q^95 + (-102b3 + 102b2 + 419b1 - 317) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 15 q^{5} + 7 q^{7} - 66 q^{11} + 11 q^{13} - 198 q^{17} + 154 q^{19} - 33 q^{23} + 121 q^{25} - 51 q^{29} + 43 q^{31} + 6 q^{35} - 100 q^{37} + 132 q^{41} + 88 q^{43} - 399 q^{47} + 513 q^{49} - 108 q^{53}+ \cdots - 736 q^{97}+O(q^{100}) \) 4 * q + 15 * q^5 + 7 * q^7 - 66 * q^11 + 11 * q^13 - 198 * q^17 + 154 * q^19 - 33 * q^23 + 121 * q^25 - 51 * q^29 + 43 * q^31 + 6 * q^35 - 100 * q^37 + 132 * q^41 + 88 * q^43 - 399 * q^47 + 513 * q^49 - 108 * q^53 - 1254 * q^55 - 798 * q^59 - 439 * q^61 + 165 * q^65 + 988 * q^67 + 2736 * q^71 - 910 * q^73 - 165 * q^77 - 803 * q^79 - 813 * q^83 - 594 * q^85 + 792 * q^89 + 1562 * q^91 + 132 * q^95 - 736 * q^97

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