LMFDB - Newform orbit 448.4.i.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(65,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, 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("448.65");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.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:	$26.4328556826$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
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 primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (5 \zeta_{6} - 5) q^{3} - 9 \zeta_{6} q^{5} + (14 \zeta_{6} - 21) q^{7} + 2 \zeta_{6} q^{9} + (57 \zeta_{6} - 57) q^{11} + 70 q^{13} + 45 q^{15} + (51 \zeta_{6} - 51) q^{17} + 5 \zeta_{6} q^{19} + ( - 105 \zeta_{6} + 35) q^{21} + \cdots - 114 q^{99} +O(q^{100}) $ q + (5z - 5) q^3 - 9z q^5 + (14z - 21) q^7 + 2z q^9 + (57z - 57) q^11 + 70 * q^13 + 45 * q^15 + (51z - 51) q^17 + 5z q^19 + (-105z + 35) q^21 - 69z q^23 + (-44z + 44) q^25 - 145 * q^27 - 114 * q^29 + (23z - 23) q^31 - 285z q^33 + (63z + 126) q^35 - 253z q^37 + (350z - 350) q^39 - 42 * q^41 + 124 * q^43 + (-18z + 18) q^45 - 201z q^47 + (-392z + 245) q^49 - 255z q^51 + (393z - 393) q^53 + 513 * q^55 - 25 * q^57 + (-219z + 219) q^59 - 709z q^61 + (-14z - 28) q^63 - 630z q^65 + (-419z + 419) q^67 + 345 * q^69 - 96 * q^71 + (-313z + 313) q^73 + 220z q^75 + (-1197z + 399) q^77 - 461z q^79 + (-671z + 671) q^81 + 588 * q^83 + 459 * q^85 + (-570z + 570) q^87 + 1017z q^89 + (980z - 1470) q^91 - 115z q^93 + (-45z + 45) q^95 - 1834 * q^97 - 114 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{3} - 9 q^{5} - 28 q^{7} + 2 q^{9} - 57 q^{11} + 140 q^{13} + 90 q^{15} - 51 q^{17} + 5 q^{19} - 35 q^{21} - 69 q^{23} + 44 q^{25} - 290 q^{27} - 228 q^{29} - 23 q^{31} - 285 q^{33} + 315 q^{35}+ \cdots - 228 q^{99}+O(q^{100}) $ 2 * q - 5 * q^3 - 9 * q^5 - 28 * q^7 + 2 * q^9 - 57 * q^11 + 140 * q^13 + 90 * q^15 - 51 * q^17 + 5 * q^19 - 35 * q^21 - 69 * q^23 + 44 * q^25 - 290 * q^27 - 228 * q^29 - 23 * q^31 - 285 * q^33 + 315 * q^35 - 253 * q^37 - 350 * q^39 - 84 * q^41 + 248 * q^43 + 18 * q^45 - 201 * q^47 + 98 * q^49 - 255 * q^51 - 393 * q^53 + 1026 * q^55 - 50 * q^57 + 219 * q^59 - 709 * q^61 - 70 * q^63 - 630 * q^65 + 419 * q^67 + 690 * q^69 - 192 * q^71 + 313 * q^73 + 220 * q^75 - 399 * q^77 - 461 * q^79 + 671 * q^81 + 1176 * q^83 + 918 * q^85 + 570 * q^87 + 1017 * q^89 - 1960 * q^91 - 115 * q^93 + 45 * q^95 - 3668 * q^97 - 228 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-\zeta_{6}$	$1$

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} $

65.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−2.50000

−

4.33013i

−4.50000

7.79423i

−14.0000

−

12.1244i

1.00000

−

1.73205i

193.1

−2.50000

4.33013i

−4.50000

−

7.79423i

−14.0000

12.1244i

1.00000

1.73205i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	448.4.i.b		2
4.b	odd	2	1		448.4.i.e		2
7.c	even	3	1	inner	448.4.i.b		2
8.b	even	2	1		14.4.c.a	✓	2
8.d	odd	2	1		112.4.i.a		2
24.h	odd	2	1		126.4.g.d		2
28.g	odd	6	1		448.4.i.e		2
40.f	even	2	1		350.4.e.e		2
40.i	odd	4	2		350.4.j.b		4
56.h	odd	2	1		98.4.c.a		2
56.j	odd	6	1		98.4.a.f		1
56.j	odd	6	1		98.4.c.a		2
56.k	odd	6	1		112.4.i.a		2
56.k	odd	6	1		784.4.a.p		1
56.m	even	6	1		784.4.a.c		1
56.p	even	6	1		14.4.c.a	✓	2
56.p	even	6	1		98.4.a.d		1
168.i	even	2	1		882.4.g.u		2
168.s	odd	6	1		126.4.g.d		2
168.s	odd	6	1		882.4.a.f		1
168.ba	even	6	1		882.4.a.c		1
168.ba	even	6	1		882.4.g.u		2
280.bf	even	6	1		350.4.e.e		2
280.bf	even	6	1		2450.4.a.q		1
280.bk	odd	6	1		2450.4.a.d		1
280.bt	odd	12	2		350.4.j.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.4.c.a	✓	2	8.b	even	2	1
14.4.c.a	✓	2	56.p	even	6	1
98.4.a.d		1	56.p	even	6	1
98.4.a.f		1	56.j	odd	6	1
98.4.c.a		2	56.h	odd	2	1
98.4.c.a		2	56.j	odd	6	1
112.4.i.a		2	8.d	odd	2	1
112.4.i.a		2	56.k	odd	6	1
126.4.g.d		2	24.h	odd	2	1
126.4.g.d		2	168.s	odd	6	1
350.4.e.e		2	40.f	even	2	1
350.4.e.e		2	280.bf	even	6	1
350.4.j.b		4	40.i	odd	4	2
350.4.j.b		4	280.bt	odd	12	2
448.4.i.b		2	1.a	even	1	1	trivial
448.4.i.b		2	7.c	even	3	1	inner
448.4.i.e		2	4.b	odd	2	1
448.4.i.e		2	28.g	odd	6	1
784.4.a.c		1	56.m	even	6	1
784.4.a.p		1	56.k	odd	6	1
882.4.a.c		1	168.ba	even	6	1
882.4.a.f		1	168.s	odd	6	1
882.4.g.u		2	168.i	even	2	1
882.4.g.u		2	168.ba	even	6	1
2450.4.a.d		1	280.bk	odd	6	1
2450.4.a.q		1	280.bf	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(448, [\chi])$:

$ T_{3}^{2} + 5T_{3} + 25 $

$ T_{11}^{2} + 57T_{11} + 3249 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 5T + 25 $ T^2 + 5*T + 25
$5$	$ T^{2} + 9T + 81 $ T^2 + 9*T + 81
$7$	$ T^{2} + 28T + 343 $ T^2 + 28*T + 343
$11$	$ T^{2} + 57T + 3249 $ T^2 + 57*T + 3249
$13$	$ (T - 70)^{2} $ (T - 70)^2
$17$	$ T^{2} + 51T + 2601 $ T^2 + 51*T + 2601
$19$	$ T^{2} - 5T + 25 $ T^2 - 5*T + 25
$23$	$ T^{2} + 69T + 4761 $ T^2 + 69*T + 4761
$29$	$ (T + 114)^{2} $ (T + 114)^2
$31$	$ T^{2} + 23T + 529 $ T^2 + 23*T + 529
$37$	$ T^{2} + 253T + 64009 $ T^2 + 253*T + 64009
$41$	$ (T + 42)^{2} $ (T + 42)^2
$43$	$ (T - 124)^{2} $ (T - 124)^2
$47$	$ T^{2} + 201T + 40401 $ T^2 + 201*T + 40401
$53$	$ T^{2} + 393T + 154449 $ T^2 + 393*T + 154449
$59$	$ T^{2} - 219T + 47961 $ T^2 - 219*T + 47961
$61$	$ T^{2} + 709T + 502681 $ T^2 + 709*T + 502681
$67$	$ T^{2} - 419T + 175561 $ T^2 - 419*T + 175561
$71$	$ (T + 96)^{2} $ (T + 96)^2
$73$	$ T^{2} - 313T + 97969 $ T^2 - 313*T + 97969
$79$	$ T^{2} + 461T + 212521 $ T^2 + 461*T + 212521
$83$	$ (T - 588)^{2} $ (T - 588)^2
$89$	$ T^{2} - 1017 T + 1034289 $ T^2 - 1017*T + 1034289
$97$	$ (T + 1834)^{2} $ (T + 1834)^2
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Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (5 \zeta_{6} - 5) q^{3} - 9 \zeta_{6} q^{5} + (14 \zeta_{6} - 21) q^{7} + 2 \zeta_{6} q^{9} + (57 \zeta_{6} - 57) q^{11} + 70 q^{13} + 45 q^{15} + (51 \zeta_{6} - 51) q^{17} + 5 \zeta_{6} q^{19} + ( - 105 \zeta_{6} + 35) q^{21} + \cdots - 114 q^{99} +O(q^{100}) \) q + (5z - 5) q^3 - 9z q^5 + (14z - 21) q^7 + 2z q^9 + (57z - 57) q^11 + 70 * q^13 + 45 * q^15 + (51z - 51) q^17 + 5z q^19 + (-105z + 35) q^21 - 69z q^23 + (-44z + 44) q^25 - 145 * q^27 - 114 * q^29 + (23z - 23) q^31 - 285z q^33 + (63z + 126) q^35 - 253z q^37 + (350z - 350) q^39 - 42 * q^41 + 124 * q^43 + (-18z + 18) q^45 - 201z q^47 + (-392z + 245) q^49 - 255z q^51 + (393z - 393) q^53 + 513 * q^55 - 25 * q^57 + (-219z + 219) q^59 - 709z q^61 + (-14z - 28) q^63 - 630z q^65 + (-419z + 419) q^67 + 345 * q^69 - 96 * q^71 + (-313z + 313) q^73 + 220z q^75 + (-1197z + 399) q^77 - 461z q^79 + (-671z + 671) q^81 + 588 * q^83 + 459 * q^85 + (-570z + 570) q^87 + 1017z q^89 + (980z - 1470) q^91 - 115z q^93 + (-45z + 45) q^95 - 1834 * q^97 - 114 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{3} - 9 q^{5} - 28 q^{7} + 2 q^{9} - 57 q^{11} + 140 q^{13} + 90 q^{15} - 51 q^{17} + 5 q^{19} - 35 q^{21} - 69 q^{23} + 44 q^{25} - 290 q^{27} - 228 q^{29} - 23 q^{31} - 285 q^{33} + 315 q^{35}+ \cdots - 228 q^{99}+O(q^{100}) \) 2 * q - 5 * q^3 - 9 * q^5 - 28 * q^7 + 2 * q^9 - 57 * q^11 + 140 * q^13 + 90 * q^15 - 51 * q^17 + 5 * q^19 - 35 * q^21 - 69 * q^23 + 44 * q^25 - 290 * q^27 - 228 * q^29 - 23 * q^31 - 285 * q^33 + 315 * q^35 - 253 * q^37 - 350 * q^39 - 84 * q^41 + 248 * q^43 + 18 * q^45 - 201 * q^47 + 98 * q^49 - 255 * q^51 - 393 * q^53 + 1026 * q^55 - 50 * q^57 + 219 * q^59 - 709 * q^61 - 70 * q^63 - 630 * q^65 + 419 * q^67 + 690 * q^69 - 192 * q^71 + 313 * q^73 + 220 * q^75 - 399 * q^77 - 461 * q^79 + 671 * q^81 + 1176 * q^83 + 918 * q^85 + 570 * q^87 + 1017 * q^89 - 1960 * q^91 - 115 * q^93 + 45 * q^95 - 3668 * q^97 - 228 * q^99

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