LMFDB - Newform orbit 396.2.j.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [396,2,Mod(37,396)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(396, base_ring=CyclotomicField(10))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("396.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 396 = 2^{2} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	396.j (of order $5$, degree $4$, 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:	$3.16207592004$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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 $\zeta_{10}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{10}^{2} - 1) q^{5} + (\zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{7}+O(q^{10}) $ q + (-z^2 - 1) * q^5 + (z^3 - 2z + 2) q^7	$ q + ( - \zeta_{10}^{2} - 1) q^{5} + (\zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{7} + ( - 2 \zeta_{10}^{3} + \cdots + \zeta_{10}) q^{11}+ \cdots + (7 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + \cdots - 7) q^{97}+O(q^{100}) $ q + (-z^2 - 1) * q^5 + (z^3 - 2z + 2) q^7 + (-2z^3 - 2z^2 + z) * q^11 + (-3z^3 + z^2 - z + 3) q^13 + (3z^2 - 5z + 3) * q^17 + (-3z^3 - 2z^2 - 3z) q^19 + 5 * q^23 + (z^3 - 4z^2 + z) q^25 + 2z^3 q^29 + (4z^3 - 5z^2 + 5z - 4) q^31 + (z^3 - 2z^2 + 2z - 1) * q^35 + (-z^3 - 2z + 2) q^37 + (6z^3 - z^2 + 6z) * q^41 + (-4z^3 + 4z^2 - 5) * q^43 + (-9z^3 + 5z^2 - 9z) q^47 + (8z^2 - 6z + 8) * q^49 + (-6z^3 + 5z^2 - 5z + 6) q^53 + (3z^3 + z - 4) q^55 + (5z^3 + 11z - 11) * q^59 + (3z^2 + 6z + 3) * q^61 + (3z^3 - 3z^2 - 5) * q^65 + (-z^3 + z^2 - 5) * q^67 + (-9z^2 - 9) q^71 + (12z^3 + 6z - 6) * q^73 + (5z^3 - 11z^2 + 9z - 3) q^77 + (-5z^3 - z^2 + z + 5) q^79 + (2z^2 - 5z + 2) * q^83 + (2z^3 - 3z^2 + 2z) q^85 + (-10z^3 + 10z^2 - 1) * q^89 - z^2 * q^91 + (8z^3 + 5z - 5) * q^95 + (7z^3 + 6z^2 - 6z - 7) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{5} + 7 q^{7}+O(q^{10}) $ 4 * q - 3 * q^5 + 7 * q^7	$ 4 q - 3 q^{5} + 7 q^{7} + q^{11} + 7 q^{13} + 4 q^{17} - 4 q^{19} + 20 q^{23} + 6 q^{25} + 2 q^{29} - 2 q^{31} + q^{35} + 5 q^{37} + 13 q^{41} - 28 q^{43} - 23 q^{47} + 18 q^{49} + 8 q^{53} - 12 q^{55} - 28 q^{59} + 15 q^{61} - 14 q^{65} - 22 q^{67} - 27 q^{71} - 6 q^{73} + 13 q^{77} + 17 q^{79} + q^{83} + 7 q^{85} - 24 q^{89} + q^{91} - 7 q^{95} - 33 q^{97}+O(q^{100}) $ 4 * q - 3 * q^5 + 7 * q^7 + q^11 + 7 * q^13 + 4 * q^17 - 4 * q^19 + 20 * q^23 + 6 * q^25 + 2 * q^29 - 2 * q^31 + q^35 + 5 * q^37 + 13 * q^41 - 28 * q^43 - 23 * q^47 + 18 * q^49 + 8 * q^53 - 12 * q^55 - 28 * q^59 + 15 * q^61 - 14 * q^65 - 22 * q^67 - 27 * q^71 - 6 * q^73 + 13 * q^77 + 17 * q^79 + q^83 + 7 * q^85 - 24 * q^89 + q^91 - 7 * q^95 - 33 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/396\mathbb{Z}\right)^\times$.

$n$	$145$	$199$	$353$
$\chi(n)$	$-\zeta_{10}^{3}$	$1$	$1$

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} $

37.1

0.809017	−	0.587785i
−0.309017	+	0.951057i
0.809017	+	0.587785i
−0.309017	−	0.951057i

−1.30902

0.951057i

0.0729490

0.224514i

181.1

−0.190983

0.587785i

3.42705

−

2.48990i

289.1

−1.30902

−

0.951057i

0.0729490

−

0.224514i

361.1

−0.190983

−

0.587785i

3.42705

2.48990i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	396.2.j.b		4
3.b	odd	2	1		132.2.i.a	✓	4
11.c	even	5	1	inner	396.2.j.b		4
11.c	even	5	1		4356.2.a.r		2
11.d	odd	10	1		4356.2.a.w		2
12.b	even	2	1		528.2.y.i		4
33.d	even	2	1		1452.2.i.g		4
33.f	even	10	1		1452.2.a.m		2
33.f	even	10	2		1452.2.i.f		4
33.f	even	10	1		1452.2.i.g		4
33.h	odd	10	1		132.2.i.a	✓	4
33.h	odd	10	1		1452.2.a.l		2
33.h	odd	10	2		1452.2.i.c		4
132.n	odd	10	1		5808.2.a.bn		2
132.o	even	10	1		528.2.y.i		4
132.o	even	10	1		5808.2.a.bq		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
132.2.i.a	✓	4	3.b	odd	2	1
132.2.i.a	✓	4	33.h	odd	10	1
396.2.j.b		4	1.a	even	1	1	trivial
396.2.j.b		4	11.c	even	5	1	inner
528.2.y.i		4	12.b	even	2	1
528.2.y.i		4	132.o	even	10	1
1452.2.a.l		2	33.h	odd	10	1
1452.2.a.m		2	33.f	even	10	1
1452.2.i.c		4	33.h	odd	10	2
1452.2.i.f		4	33.f	even	10	2
1452.2.i.g		4	33.d	even	2	1
1452.2.i.g		4	33.f	even	10	1
4356.2.a.r		2	11.c	even	5	1
4356.2.a.w		2	11.d	odd	10	1
5808.2.a.bn		2	132.n	odd	10	1
5808.2.a.bq		2	132.o	even	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(396, [\chi])$:

$ T_{5}^{4} + 3T_{5}^{3} + 4T_{5}^{2} + 2T_{5} + 1 $

$ T_{7}^{4} - 7T_{7}^{3} + 19T_{7}^{2} - 3T_{7} + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} + 3 T^{3} + \cdots + 1 $ T^4 + 3T^3 + 4T^2 + 2*T + 1
$7$	$ T^{4} - 7 T^{3} + \cdots + 1 $ T^4 - 7T^3 + 19T^2 - 3*T + 1
$11$	$ T^{4} - T^{3} + \cdots + 121 $ T^4 - T^3 + 21T^2 - 11T + 121
$13$	$ T^{4} - 7 T^{3} + \cdots + 1 $ T^4 - 7T^3 + 19T^2 - 3*T + 1
$17$	$ T^{4} - 4 T^{3} + \cdots + 1 $ T^4 - 4T^3 + 46T^2 + 11*T + 1
$19$	$ T^{4} + 4 T^{3} + \cdots + 1 $ T^4 + 4T^3 + 46T^2 - 11*T + 1
$23$	$ (T - 5)^{4} $ (T - 5)^4
$29$	$ T^{4} - 2 T^{3} + \cdots + 16 $ T^4 - 2T^3 + 4T^2 - 8*T + 16
$31$	$ T^{4} + 2 T^{3} + \cdots + 361 $ T^4 + 2T^3 + 24T^2 + 133*T + 361
$37$	$ T^{4} - 5 T^{3} + \cdots + 25 $ T^4 - 5T^3 + 15T^2 - 25*T + 25
$41$	$ T^{4} - 13 T^{3} + \cdots + 1681 $ T^4 - 13T^3 + 139T^2 - 697*T + 1681
$43$	$ (T^{2} + 14 T + 29)^{2} $ (T^2 + 14*T + 29)^2
$47$	$ T^{4} + 23 T^{3} + \cdots + 10201 $ T^4 + 23T^3 + 304T^2 + 2222*T + 10201
$53$	$ T^{4} - 8 T^{3} + \cdots + 841 $ T^4 - 8T^3 + 34T^2 - 87*T + 841
$59$	$ T^{4} + 28 T^{3} + \cdots + 22801 $ T^4 + 28T^3 + 454T^2 + 4077*T + 22801
$61$	$ T^{4} - 15 T^{3} + \cdots + 2025 $ T^4 - 15T^3 + 90T^2 + 2025
$67$	$ (T^{2} + 11 T + 29)^{2} $ (T^2 + 11*T + 29)^2
$71$	$ T^{4} + 27 T^{3} + \cdots + 6561 $ T^4 + 27T^3 + 324T^2 + 1458*T + 6561
$73$	$ T^{4} + 6 T^{3} + \cdots + 1296 $ T^4 + 6T^3 + 216T^2 - 864*T + 1296
$79$	$ T^{4} - 17 T^{3} + \cdots + 1681 $ T^4 - 17T^3 + 139T^2 - 533*T + 1681
$83$	$ T^{4} - T^{3} + \cdots + 121 $ T^4 - T^3 + 31T^2 + 99T + 121
$89$	$ (T^{2} + 12 T - 89)^{2} $ (T^2 + 12*T - 89)^2
$97$	$ T^{4} + 33 T^{3} + \cdots + 44521 $ T^4 + 33T^3 + 634T^2 + 6752*T + 44521
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Level:	\( N \)	\(=\)	\( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	396.j (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{10}^{2} - 1) q^{5} + (\zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{7}+O(q^{10}) \) q + (-z^2 - 1) * q^5 + (z^3 - 2z + 2) q^7	\( q + ( - \zeta_{10}^{2} - 1) q^{5} + (\zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{7} + ( - 2 \zeta_{10}^{3} + \cdots + \zeta_{10}) q^{11}+ \cdots + (7 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + \cdots - 7) q^{97}+O(q^{100}) \) q + (-z^2 - 1) * q^5 + (z^3 - 2z + 2) q^7 + (-2z^3 - 2z^2 + z) * q^11 + (-3z^3 + z^2 - z + 3) q^13 + (3z^2 - 5z + 3) * q^17 + (-3z^3 - 2z^2 - 3z) q^19 + 5 * q^23 + (z^3 - 4z^2 + z) q^25 + 2z^3 q^29 + (4z^3 - 5z^2 + 5z - 4) q^31 + (z^3 - 2z^2 + 2z - 1) * q^35 + (-z^3 - 2z + 2) q^37 + (6z^3 - z^2 + 6z) * q^41 + (-4z^3 + 4z^2 - 5) * q^43 + (-9z^3 + 5z^2 - 9z) q^47 + (8z^2 - 6z + 8) * q^49 + (-6z^3 + 5z^2 - 5z + 6) q^53 + (3z^3 + z - 4) q^55 + (5z^3 + 11z - 11) * q^59 + (3z^2 + 6z + 3) * q^61 + (3z^3 - 3z^2 - 5) * q^65 + (-z^3 + z^2 - 5) * q^67 + (-9z^2 - 9) q^71 + (12z^3 + 6z - 6) * q^73 + (5z^3 - 11z^2 + 9z - 3) q^77 + (-5z^3 - z^2 + z + 5) q^79 + (2z^2 - 5z + 2) * q^83 + (2z^3 - 3z^2 + 2z) q^85 + (-10z^3 + 10z^2 - 1) * q^89 - z^2 * q^91 + (8z^3 + 5z - 5) * q^95 + (7z^3 + 6z^2 - 6z - 7) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{5} + 7 q^{7}+O(q^{10}) \) 4 * q - 3 * q^5 + 7 * q^7	\( 4 q - 3 q^{5} + 7 q^{7} + q^{11} + 7 q^{13} + 4 q^{17} - 4 q^{19} + 20 q^{23} + 6 q^{25} + 2 q^{29} - 2 q^{31} + q^{35} + 5 q^{37} + 13 q^{41} - 28 q^{43} - 23 q^{47} + 18 q^{49} + 8 q^{53} - 12 q^{55} - 28 q^{59} + 15 q^{61} - 14 q^{65} - 22 q^{67} - 27 q^{71} - 6 q^{73} + 13 q^{77} + 17 q^{79} + q^{83} + 7 q^{85} - 24 q^{89} + q^{91} - 7 q^{95} - 33 q^{97}+O(q^{100}) \) 4 * q - 3 * q^5 + 7 * q^7 + q^11 + 7 * q^13 + 4 * q^17 - 4 * q^19 + 20 * q^23 + 6 * q^25 + 2 * q^29 - 2 * q^31 + q^35 + 5 * q^37 + 13 * q^41 - 28 * q^43 - 23 * q^47 + 18 * q^49 + 8 * q^53 - 12 * q^55 - 28 * q^59 + 15 * q^61 - 14 * q^65 - 22 * q^67 - 27 * q^71 - 6 * q^73 + 13 * q^77 + 17 * q^79 + q^83 + 7 * q^85 - 24 * q^89 + q^91 - 7 * q^95 - 33 * q^97

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