LMFDB - Newform orbit 1368.3.o.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,3,Mod(721,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1368.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1368.o (of order $2$, degree $1$, 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:	$37.2753001645$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 34x^{6} + 345x^{4} + 1064x^{2} + 256 $ x^8 + 34x^6 + 345x^4 + 1064*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{4} + 2) q^{5} + (\beta_{3} - 1) q^{7}+O(q^{10}) $ q + (b4 + 2) * q^5 + (b3 - 1) * q^7	$ q + (\beta_{4} + 2) q^{5} + (\beta_{3} - 1) q^{7} + ( - 2 \beta_{6} + \beta_{4} + 4) q^{11} + (\beta_{7} - 2 \beta_1) q^{13} + ( - \beta_{6} + 2 \beta_{4} + 3) q^{17} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 2) q^{19}+ \cdots + ( - 4 \beta_{7} + 20 \beta_{5} - 26 \beta_1) q^{97}+O(q^{100}) $ q + (b4 + 2) * q^5 + (b3 - 1) * q^7 + (-2b6 + b4 + 4) q^11 + (b7 - 2b1) q^13 + (-b6 + 2b4 + 3) q^17 + (-b7 - b6 - b5 - b4 - b2 + 2) * q^19 + (-2b6 - b4 - b3 - 1) q^23 + (-4b6 + b4 + 2b3 + 5) * q^25 + (-b7 + 4b5 - b2 + 6b1) * q^29 + (-2b7 + b2 - 4b1) * q^31 + (-2b6 + b4 + 4b3 + 6) * q^35 + (6b5 + 3b2 + 2b1) q^37 + (2b7 + 4b5 - b2 + 8b1) q^41 + (5b4 - 4b3 + 10) * q^43 + (8b6 - 3b4 - 6b3 - 4) q^47 + (b6 + 6b4 - 8b3 + 6) * q^49 + (-b7 + 4b5 + b2 - 8b1) * q^53 + (-8b6 + 11b4 + 2b3 + 26) q^55 + (-20b5 - 3b2 - 13b1) q^59 + (-6b6 - 5b4 - 6b3 - 18) q^61 + (2b7 - 4b5 + 4b2 - 2b1) * q^65 + (-2b7 - 16b5 + 9b1) q^67 + (-2b7 + 16b5 + 16b1) q^71 + (13b6 + 8b4 - 6b3 - 21) q^73 + (-10b6 + 5b4 + 6b3 - 8) q^77 + (-16b5 + b2 + 14b1) * q^79 + (8b6 + 14b4 - 10b3 + 12) q^83 + (-10b6 + 5b4 + 4b3 + 54) q^85 + (4b7 + 5b2 + 12b1) q^89 + (2b7 - 18b5 + b2 - 19b1) q^91 + (-4b7 + 2b6 - 18b5 + 7b4 - 2b3 - 2b2 + 16b1 - 26) q^95 + (-4b7 + 20b5 - 26b1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{5} - 6 q^{7}+O(q^{10}) $ 8 * q + 14 * q^5 - 6 * q^7	$ 8 q + 14 q^{5} - 6 q^{7} + 26 q^{11} + 18 q^{17} + 16 q^{19} - 12 q^{23} + 34 q^{25} + 50 q^{35} + 62 q^{43} - 22 q^{47} + 22 q^{49} + 174 q^{55} - 158 q^{61} - 170 q^{73} - 82 q^{77} + 64 q^{83} + 410 q^{85} - 222 q^{95}+O(q^{100}) $ 8 * q + 14 * q^5 - 6 * q^7 + 26 * q^11 + 18 * q^17 + 16 * q^19 - 12 * q^23 + 34 * q^25 + 50 * q^35 + 62 * q^43 - 22 * q^47 + 22 * q^49 + 174 * q^55 - 158 * q^61 - 170 * q^73 - 82 * q^77 + 64 * q^83 + 410 * q^85 - 222 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 34x^{6} + 345x^{4} + 1064x^{2} + 256 $ x^8 + 34*x^6 + 345*x^4 + 1064*x^2 + 256 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -\nu^{7} - 30\nu^{5} - 205\nu^{3} + 16\nu ) / 40 $ (-v^7 - 30v^5 - 205v^3 + 16*v) / 40
$\beta_{3}$	$=$	$ ( -\nu^{6} - 20\nu^{4} - 35\nu^{2} + 216 ) / 40 $ (-v^6 - 20v^4 - 35v^2 + 216) / 40
$\beta_{4}$	$=$	$ ( \nu^{6} + 30\nu^{4} + 205\nu^{2} + 64 ) / 40 $ (v^6 + 30v^4 + 205v^2 + 64) / 40
$\beta_{5}$	$=$	$ ( \nu^{7} + 30\nu^{5} + 245\nu^{3} + 504\nu ) / 80 $ (v^7 + 30v^5 + 245v^3 + 504*v) / 80
$\beta_{6}$	$=$	$ ( \nu^{6} + 30\nu^{4} + 245\nu^{2} + 424 ) / 40 $ (v^6 + 30v^4 + 245v^2 + 424) / 40
$\beta_{7}$	$=$	$ ( -\nu^{7} - 40\nu^{5} - 495\nu^{3} - 1784\nu ) / 40 $ (-v^7 - 40v^5 - 495v^3 - 1784*v) / 40

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{6} - \beta_{4} - 9 $ b6 - b4 - 9
$\nu^{3}$	$=$	$ 2\beta_{5} + \beta_{2} - 13\beta_1 $ 2b5 + b2 - 13b1
$\nu^{4}$	$=$	$ -17\beta_{6} + 21\beta_{4} + 4\beta_{3} + 125 $ -17b6 + 21b4 + 4*b3 + 125
$\nu^{5}$	$=$	$ -4\beta_{7} - 58\beta_{5} - 25\beta_{2} + 197\beta_1 $ -4b7 - 58b5 - 25b2 + 197b1
$\nu^{6}$	$=$	$ 305\beta_{6} - 385\beta_{4} - 120\beta_{3} - 1969 $ 305b6 - 385b4 - 120*b3 - 1969
$\nu^{7}$	$=$	$ 120\beta_{7} + 1330\beta_{5} + 505\beta_{2} - 3229\beta_1 $ 120b7 + 1330b5 + 505b2 - 3229b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$1$	$-1$	$1$

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} $

721.1

		2.27869i
	−	2.27869i
		0.512197i
	−	0.512197i
	−	3.20945i
		3.20945i
		4.27138i
	−	4.27138i

−6.29008

−1.03740

721.2

−6.29008

−1.03740

721.3

2.30665

4.59559

721.4

2.30665

4.59559

721.5

3.06310

−12.3151

721.6

3.06310

−12.3151

721.7

7.92033

5.75693

721.8

7.92033

5.75693

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1368.3.o.b		8
3.b	odd	2	1		152.3.e.b	✓	8
4.b	odd	2	1		2736.3.o.p		8
12.b	even	2	1		304.3.e.g		8
19.b	odd	2	1	inner	1368.3.o.b		8
24.f	even	2	1		1216.3.e.n		8
24.h	odd	2	1		1216.3.e.m		8
57.d	even	2	1		152.3.e.b	✓	8
76.d	even	2	1		2736.3.o.p		8
228.b	odd	2	1		304.3.e.g		8
456.l	odd	2	1		1216.3.e.n		8
456.p	even	2	1		1216.3.e.m		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
152.3.e.b	✓	8	3.b	odd	2	1
152.3.e.b	✓	8	57.d	even	2	1
304.3.e.g		8	12.b	even	2	1
304.3.e.g		8	228.b	odd	2	1
1216.3.e.m		8	24.h	odd	2	1
1216.3.e.m		8	456.p	even	2	1
1216.3.e.n		8	24.f	even	2	1
1216.3.e.n		8	456.l	odd	2	1
1368.3.o.b		8	1.a	even	1	1	trivial
1368.3.o.b		8	19.b	odd	2	1	inner
2736.3.o.p		8	4.b	odd	2	1
2736.3.o.p		8	76.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} - 7T_{5}^{3} - 34T_{5}^{2} + 256T_{5} - 352 $ T5^4 - 7*T5^3 - 34*T5^2 + 256*T5 - 352 acting on $S_{3}^{\mathrm{new}}(1368, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ (T^{4} - 7 T^{3} + \cdots - 352)^{2} $ (T^4 - 7T^3 - 34T^2 + 256*T - 352)^2
$7$	$ (T^{4} + 3 T^{3} + \cdots + 338)^{2} $ (T^4 + 3T^3 - 99T^2 + 221*T + 338)^2
$11$	$ (T^{4} - 13 T^{3} + \cdots - 5912)^{2} $ (T^4 - 13T^3 - 174T^2 + 2260*T - 5912)^2
$13$	$ T^{8} + 778 T^{6} + \cdots + 246866944 $ T^8 + 778T^6 + 161977T^4 + 11290176*T^2 + 246866944
$17$	$ (T^{4} - 9 T^{3} + \cdots + 4814)^{2} $ (T^4 - 9T^3 - 195T^2 + 253*T + 4814)^2
$19$	$ T^{8} + \cdots + 16983563041 $ T^8 - 16T^7 + 1024T^6 - 16784T^5 + 479902T^4 - 6059024T^3 + 133448704T^2 - 752734096*T + 16983563041
$23$	$ (T^{4} + 6 T^{3} + \cdots + 27836)^{2} $ (T^4 + 6T^3 - 499T^2 + 504*T + 27836)^2
$29$	$ T^{8} + \cdots + 36014930176 $ T^8 + 2498T^6 + 1927129T^4 + 474986408*T^2 + 36014930176
$31$	$ T^{8} + \cdots + 2723080830976 $ T^8 + 5232T^6 + 10137808T^4 + 8628732032*T^2 + 2723080830976
$37$	$ T^{8} + \cdots + 5146582257664 $ T^8 + 8272T^6 + 22018512T^4 + 20060072576*T^2 + 5146582257664
$41$	$ T^{8} + \cdots + 5382400000000 $ T^8 + 7216T^6 + 17109456T^4 + 16332880000*T^2 + 5382400000000
$43$	$ (T^{4} - 31 T^{3} + \cdots + 258656)^{2} $ (T^4 - 31T^3 - 1982T^2 + 7264*T + 258656)^2
$47$	$ (T^{4} + 11 T^{3} + \cdots + 2613512)^{2} $ (T^4 + 11T^3 - 6302T^2 + 84772*T + 2613512)^2
$53$	$ T^{8} + \cdots + 13467138304 $ T^8 + 5410T^6 + 6302521T^4 + 2071871016*T^2 + 13467138304
$59$	$ T^{8} + \cdots + 247119406081024 $ T^8 + 20362T^6 + 126378201T^4 + 303682748480*T^2 + 247119406081024
$61$	$ (T^{4} + 79 T^{3} + \cdots + 17033600)^{2} $ (T^4 + 79T^3 - 7306T^2 - 298720*T + 17033600)^2
$67$	$ T^{8} + \cdots + 273018790144 $ T^8 + 15538T^6 + 72017257T^4 + 85706332456*T^2 + 273018790144
$71$	$ T^{8} + \cdots + 1596725395456 $ T^8 + 14696T^6 + 53362320T^4 + 25682937856*T^2 + 1596725395456
$73$	$ (T^{4} + 85 T^{3} + \cdots + 28567486)^{2} $ (T^4 + 85T^3 - 12111T^2 - 288325*T + 28567486)^2
$79$	$ T^{8} + \cdots + 787748552704 $ T^8 + 19312T^6 + 103889616T^4 + 110072696960*T^2 + 787748552704
$83$	$ (T^{4} - 32 T^{3} + \cdots - 454016)^{2} $ (T^4 - 32T^3 - 19300T^2 + 644096*T - 454016)^2
$89$	$ T^{8} + \cdots + 682745962430464 $ T^8 + 29768T^6 + 291139072T^4 + 992323117056*T^2 + 682745962430464
$97$	$ T^{8} + \cdots + 19\!\cdots\!96 $ T^8 + 58184T^6 + 1099281040T^4 + 7988311560704*T^2 + 19669519455748096
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1368.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{4} + 2) q^{5} + (\beta_{3} - 1) q^{7}+O(q^{10}) \) q + (b4 + 2) * q^5 + (b3 - 1) * q^7	\( q + (\beta_{4} + 2) q^{5} + (\beta_{3} - 1) q^{7} + ( - 2 \beta_{6} + \beta_{4} + 4) q^{11} + (\beta_{7} - 2 \beta_1) q^{13} + ( - \beta_{6} + 2 \beta_{4} + 3) q^{17} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 2) q^{19}+ \cdots + ( - 4 \beta_{7} + 20 \beta_{5} - 26 \beta_1) q^{97}+O(q^{100}) \) q + (b4 + 2) * q^5 + (b3 - 1) * q^7 + (-2b6 + b4 + 4) q^11 + (b7 - 2b1) q^13 + (-b6 + 2b4 + 3) q^17 + (-b7 - b6 - b5 - b4 - b2 + 2) * q^19 + (-2b6 - b4 - b3 - 1) q^23 + (-4b6 + b4 + 2b3 + 5) * q^25 + (-b7 + 4b5 - b2 + 6b1) * q^29 + (-2b7 + b2 - 4b1) * q^31 + (-2b6 + b4 + 4b3 + 6) * q^35 + (6b5 + 3b2 + 2b1) q^37 + (2b7 + 4b5 - b2 + 8b1) q^41 + (5b4 - 4b3 + 10) * q^43 + (8b6 - 3b4 - 6b3 - 4) q^47 + (b6 + 6b4 - 8b3 + 6) * q^49 + (-b7 + 4b5 + b2 - 8b1) * q^53 + (-8b6 + 11b4 + 2b3 + 26) q^55 + (-20b5 - 3b2 - 13b1) q^59 + (-6b6 - 5b4 - 6b3 - 18) q^61 + (2b7 - 4b5 + 4b2 - 2b1) * q^65 + (-2b7 - 16b5 + 9b1) q^67 + (-2b7 + 16b5 + 16b1) q^71 + (13b6 + 8b4 - 6b3 - 21) q^73 + (-10b6 + 5b4 + 6b3 - 8) q^77 + (-16b5 + b2 + 14b1) * q^79 + (8b6 + 14b4 - 10b3 + 12) q^83 + (-10b6 + 5b4 + 4b3 + 54) q^85 + (4b7 + 5b2 + 12b1) q^89 + (2b7 - 18b5 + b2 - 19b1) q^91 + (-4b7 + 2b6 - 18b5 + 7b4 - 2b3 - 2b2 + 16b1 - 26) q^95 + (-4b7 + 20b5 - 26b1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{5} - 6 q^{7}+O(q^{10}) \) 8 * q + 14 * q^5 - 6 * q^7	\( 8 q + 14 q^{5} - 6 q^{7} + 26 q^{11} + 18 q^{17} + 16 q^{19} - 12 q^{23} + 34 q^{25} + 50 q^{35} + 62 q^{43} - 22 q^{47} + 22 q^{49} + 174 q^{55} - 158 q^{61} - 170 q^{73} - 82 q^{77} + 64 q^{83} + 410 q^{85} - 222 q^{95}+O(q^{100}) \) 8 * q + 14 * q^5 - 6 * q^7 + 26 * q^11 + 18 * q^17 + 16 * q^19 - 12 * q^23 + 34 * q^25 + 50 * q^35 + 62 * q^43 - 22 * q^47 + 22 * q^49 + 174 * q^55 - 158 * q^61 - 170 * q^73 - 82 * q^77 + 64 * q^83 + 410 * q^85 - 222 * q^95

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