LMFDB - Newform orbit 403.2.c.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [403,2,Mod(311,403)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("403.311"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(403, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 403 = 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	403.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.21797120146$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 2 q^{3} + 2 q^{4} + 4 i q^{5} - 2 i q^{7} + q^{9} + i q^{11} + 4 q^{12} + ( - 3 i - 2) q^{13} + 8 i q^{15} + 4 q^{16} - 2 q^{17} - 2 i q^{19} + 8 i q^{20} - 4 i q^{21} - 4 q^{23} - 11 q^{25} - 4 q^{27} + \cdots + i q^{99} +O(q^{100}) $ q + 2 * q^3 + 2 * q^4 + 4i q^5 - 2i q^7 + q^9 + i * q^11 + 4 * q^12 + (-3i - 2) q^13 + 8i q^15 + 4 * q^16 - 2 * q^17 - 2i q^19 + 8i q^20 - 4i q^21 - 4 * q^23 - 11 * q^25 - 4 * q^27 - 4i q^28 + 8 * q^29 - i * q^31 + 2i q^33 + 8 * q^35 + 2 * q^36 - 3i q^37 + (-6i - 4) q^39 - 4i q^41 + 4 * q^43 + 2i q^44 + 4i q^45 - 2i q^47 + 8 * q^48 + 3 * q^49 - 4 * q^51 + (-6i - 4) q^52 - 4 * q^53 - 4 * q^55 - 4i q^57 + 6i q^59 + 16i q^60 + 10 * q^61 - 2i q^63 + 8 * q^64 + (-8i + 12) q^65 + 16i q^67 - 4 * q^68 - 8 * q^69 - 6i q^71 - 7i q^73 - 22 * q^75 - 4i q^76 + 2 * q^77 - 16 * q^79 + 16i q^80 - 11 * q^81 + 12i q^83 - 8i q^84 - 8i q^85 + 16 * q^87 - 15i q^89 + (4i - 6) q^91 - 8 * q^92 - 2i q^93 + 8 * q^95 + 2i q^97 + i * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 4 q^{4} + 2 q^{9} + 8 q^{12} - 4 q^{13} + 8 q^{16} - 4 q^{17} - 8 q^{23} - 22 q^{25} - 8 q^{27} + 16 q^{29} + 16 q^{35} + 4 q^{36} - 8 q^{39} + 8 q^{43} + 16 q^{48} + 6 q^{49} - 8 q^{51}+ \cdots + 16 q^{95}+O(q^{100}) $ 2 * q + 4 * q^3 + 4 * q^4 + 2 * q^9 + 8 * q^12 - 4 * q^13 + 8 * q^16 - 4 * q^17 - 8 * q^23 - 22 * q^25 - 8 * q^27 + 16 * q^29 + 16 * q^35 + 4 * q^36 - 8 * q^39 + 8 * q^43 + 16 * q^48 + 6 * q^49 - 8 * q^51 - 8 * q^52 - 8 * q^53 - 8 * q^55 + 20 * q^61 + 16 * q^64 + 24 * q^65 - 8 * q^68 - 16 * q^69 - 44 * q^75 + 4 * q^77 - 32 * q^79 - 22 * q^81 + 32 * q^87 - 12 * q^91 - 16 * q^92 + 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/403\mathbb{Z}\right)^\times$.

$n$	$249$	$313$
$\chi(n)$	$-1$	$1$

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} $

311.1

	−	1.00000i
		1.00000i

2.00000

−

4.00000i

2.00000i

1.00000

311.2

2.00000

4.00000i

−

2.00000i

1.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	403.2.c.a	✓	2
13.b	even	2	1	inner	403.2.c.a	✓	2
13.d	odd	4	1		5239.2.a.b		1
13.d	odd	4	1		5239.2.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
403.2.c.a	✓	2	1.a	even	1	1	trivial
403.2.c.a	✓	2	13.b	even	2	1	inner
5239.2.a.b		1	13.d	odd	4	1
5239.2.a.c		1	13.d	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} $ T2 acting on $S_{2}^{\mathrm{new}}(403, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ (T - 2)^{2} $ (T - 2)^2
$5$	$ T^{2} + 16 $ T^2 + 16
$7$	$ T^{2} + 4 $ T^2 + 4
$11$	$ T^{2} + 1 $ T^2 + 1
$13$	$ T^{2} + 4T + 13 $ T^2 + 4*T + 13
$17$	$ (T + 2)^{2} $ (T + 2)^2
$19$	$ T^{2} + 4 $ T^2 + 4
$23$	$ (T + 4)^{2} $ (T + 4)^2
$29$	$ (T - 8)^{2} $ (T - 8)^2
$31$	$ T^{2} + 1 $ T^2 + 1
$37$	$ T^{2} + 9 $ T^2 + 9
$41$	$ T^{2} + 16 $ T^2 + 16
$43$	$ (T - 4)^{2} $ (T - 4)^2
$47$	$ T^{2} + 4 $ T^2 + 4
$53$	$ (T + 4)^{2} $ (T + 4)^2
$59$	$ T^{2} + 36 $ T^2 + 36
$61$	$ (T - 10)^{2} $ (T - 10)^2
$67$	$ T^{2} + 256 $ T^2 + 256
$71$	$ T^{2} + 36 $ T^2 + 36
$73$	$ T^{2} + 49 $ T^2 + 49
$79$	$ (T + 16)^{2} $ (T + 16)^2
$83$	$ T^{2} + 144 $ T^2 + 144
$89$	$ T^{2} + 225 $ T^2 + 225
$97$	$ T^{2} + 4 $ T^2 + 4
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Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + 2 q^{3} + 2 q^{4} + 4 i q^{5} - 2 i q^{7} + q^{9} + i q^{11} + 4 q^{12} + ( - 3 i - 2) q^{13} + 8 i q^{15} + 4 q^{16} - 2 q^{17} - 2 i q^{19} + 8 i q^{20} - 4 i q^{21} - 4 q^{23} - 11 q^{25} - 4 q^{27} + \cdots + i q^{99} +O(q^{100}) \) q + 2 * q^3 + 2 * q^4 + 4i q^5 - 2i q^7 + q^9 + i * q^11 + 4 * q^12 + (-3i - 2) q^13 + 8i q^15 + 4 * q^16 - 2 * q^17 - 2i q^19 + 8i q^20 - 4i q^21 - 4 * q^23 - 11 * q^25 - 4 * q^27 - 4i q^28 + 8 * q^29 - i * q^31 + 2i q^33 + 8 * q^35 + 2 * q^36 - 3i q^37 + (-6i - 4) q^39 - 4i q^41 + 4 * q^43 + 2i q^44 + 4i q^45 - 2i q^47 + 8 * q^48 + 3 * q^49 - 4 * q^51 + (-6i - 4) q^52 - 4 * q^53 - 4 * q^55 - 4i q^57 + 6i q^59 + 16i q^60 + 10 * q^61 - 2i q^63 + 8 * q^64 + (-8i + 12) q^65 + 16i q^67 - 4 * q^68 - 8 * q^69 - 6i q^71 - 7i q^73 - 22 * q^75 - 4i q^76 + 2 * q^77 - 16 * q^79 + 16i q^80 - 11 * q^81 + 12i q^83 - 8i q^84 - 8i q^85 + 16 * q^87 - 15i q^89 + (4i - 6) q^91 - 8 * q^92 - 2i q^93 + 8 * q^95 + 2i q^97 + i * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 4 q^{4} + 2 q^{9} + 8 q^{12} - 4 q^{13} + 8 q^{16} - 4 q^{17} - 8 q^{23} - 22 q^{25} - 8 q^{27} + 16 q^{29} + 16 q^{35} + 4 q^{36} - 8 q^{39} + 8 q^{43} + 16 q^{48} + 6 q^{49} - 8 q^{51}+ \cdots + 16 q^{95}+O(q^{100}) \) 2 * q + 4 * q^3 + 4 * q^4 + 2 * q^9 + 8 * q^12 - 4 * q^13 + 8 * q^16 - 4 * q^17 - 8 * q^23 - 22 * q^25 - 8 * q^27 + 16 * q^29 + 16 * q^35 + 4 * q^36 - 8 * q^39 + 8 * q^43 + 16 * q^48 + 6 * q^49 - 8 * q^51 - 8 * q^52 - 8 * q^53 - 8 * q^55 + 20 * q^61 + 16 * q^64 + 24 * q^65 - 8 * q^68 - 16 * q^69 - 44 * q^75 + 4 * q^77 - 32 * q^79 - 22 * q^81 + 32 * q^87 - 12 * q^91 - 16 * q^92 + 16 * q^95

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