LMFDB - Newform orbit 315.2.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,2,Mod(1,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$2.51528766367$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 $\beta = \sqrt{2}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} - q^{5} - q^{7} + (\beta - 3) q^{8} + ( - \beta + 1) q^{10} + ( - 2 \beta - 2) q^{11} + (2 \beta - 2) q^{13} + ( - \beta + 1) q^{14} + 3 q^{16} + ( - 4 \beta - 2) q^{17}+ \cdots + (\beta - 1) q^{98}+O(q^{100}) $ q + (b - 1) * q^2 + (-2b + 1) q^4 - q^5 - q^7 + (b - 3) * q^8 + (-b + 1) * q^10 + (-2b - 2) q^11 + (2b - 2) q^13 + (-b + 1) * q^14 + 3 * q^16 + (-4b - 2) q^17 - 2b q^19 + (2b - 1) q^20 - 2 * q^22 + (4b - 2) q^23 + q^25 + (-4b + 6) q^26 + (2b - 1) q^28 - 8 * q^29 + 6b q^31 + (b + 3) * q^32 + (2b - 6) q^34 + q^35 - 6 * q^37 + (2b - 4) q^38 + (-b + 3) * q^40 + (4b + 2) q^41 + (4b - 4) q^43 + (2b + 6) q^44 + (-6b + 10) q^46 + 4 * q^47 + q^49 + (b - 1) * q^50 + (6b - 10) q^52 + (2b - 8) q^53 + (2b + 2) q^55 + (-b + 3) * q^56 + (-8b + 8) q^58 + 4 * q^59 + 6 * q^61 + (-6b + 12) q^62 + (2b - 7) q^64 + (-2b + 2) q^65 + (-8b - 4) q^67 + 14 * q^68 + (b - 1) * q^70 + (-6b - 2) q^71 + (-10b + 2) q^73 + (-6b + 6) q^74 + (-2b + 8) q^76 + (2b + 2) q^77 + 4b q^79 - 3 * q^80 + (-2b + 6) q^82 + 8 * q^83 + (4b + 2) q^85 + (-8b + 12) q^86 + (4b + 2) q^88 + (8b - 6) q^89 + (-2b + 2) q^91 + (8b - 18) q^92 + (4b - 4) q^94 + 2b q^95 + (2b - 6) q^97 + (b - 1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8} + 2 q^{10} - 4 q^{11} - 4 q^{13} + 2 q^{14} + 6 q^{16} - 4 q^{17} - 2 q^{20} - 4 q^{22} - 4 q^{23} + 2 q^{25} + 12 q^{26} - 2 q^{28} - 16 q^{29} + 6 q^{32}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8 + 2 * q^10 - 4 * q^11 - 4 * q^13 + 2 * q^14 + 6 * q^16 - 4 * q^17 - 2 * q^20 - 4 * q^22 - 4 * q^23 + 2 * q^25 + 12 * q^26 - 2 * q^28 - 16 * q^29 + 6 * q^32 - 12 * q^34 + 2 * q^35 - 12 * q^37 - 8 * q^38 + 6 * q^40 + 4 * q^41 - 8 * q^43 + 12 * q^44 + 20 * q^46 + 8 * q^47 + 2 * q^49 - 2 * q^50 - 20 * q^52 - 16 * q^53 + 4 * q^55 + 6 * q^56 + 16 * q^58 + 8 * q^59 + 12 * q^61 + 24 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^67 + 28 * q^68 - 2 * q^70 - 4 * q^71 + 4 * q^73 + 12 * q^74 + 16 * q^76 + 4 * q^77 - 6 * q^80 + 12 * q^82 + 16 * q^83 + 4 * q^85 + 24 * q^86 + 4 * q^88 - 12 * q^89 + 4 * q^91 - 36 * q^92 - 8 * q^94 - 12 * q^97 - 2 * q^98

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} $

1.1

−1.41421
1.41421

−2.41421

3.82843

−1.00000

−4.41421

2.41421

1.2

0.414214

−1.82843

−1.00000

−1.58579

−0.414214

Atkin-Lehner signs

$ p $	Sign
$3$	$ +1 $
$5$	$ +1 $
$7$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.2.a.c	✓	2
3.b	odd	2	1		315.2.a.f	yes	2
4.b	odd	2	1		5040.2.a.bu		2
5.b	even	2	1		1575.2.a.u		2
5.c	odd	4	2		1575.2.d.h		4
7.b	odd	2	1		2205.2.a.p		2
12.b	even	2	1		5040.2.a.bx		2
15.d	odd	2	1		1575.2.a.m		2
15.e	even	4	2		1575.2.d.j		4
21.c	even	2	1		2205.2.a.y		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.2.a.c	✓	2	1.a	even	1	1	trivial
315.2.a.f	yes	2	3.b	odd	2	1
1575.2.a.m		2	15.d	odd	2	1
1575.2.a.u		2	5.b	even	2	1
1575.2.d.h		4	5.c	odd	4	2
1575.2.d.j		4	15.e	even	4	2
2205.2.a.p		2	7.b	odd	2	1
2205.2.a.y		2	21.c	even	2	1
5040.2.a.bu		2	4.b	odd	2	1
5040.2.a.bx		2	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} + 2T_{2} - 1 $ T2^2 + 2*T2 - 1 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(315))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 2T - 1 $ T^2 + 2*T - 1
$3$	$ T^{2} $ T^2
$5$	$ (T + 1)^{2} $ (T + 1)^2
$7$	$ (T + 1)^{2} $ (T + 1)^2
$11$	$ T^{2} + 4T - 4 $ T^2 + 4*T - 4
$13$	$ T^{2} + 4T - 4 $ T^2 + 4*T - 4
$17$	$ T^{2} + 4T - 28 $ T^2 + 4*T - 28
$19$	$ T^{2} - 8 $ T^2 - 8
$23$	$ T^{2} + 4T - 28 $ T^2 + 4*T - 28
$29$	$ (T + 8)^{2} $ (T + 8)^2
$31$	$ T^{2} - 72 $ T^2 - 72
$37$	$ (T + 6)^{2} $ (T + 6)^2
$41$	$ T^{2} - 4T - 28 $ T^2 - 4*T - 28
$43$	$ T^{2} + 8T - 16 $ T^2 + 8*T - 16
$47$	$ (T - 4)^{2} $ (T - 4)^2
$53$	$ T^{2} + 16T + 56 $ T^2 + 16*T + 56
$59$	$ (T - 4)^{2} $ (T - 4)^2
$61$	$ (T - 6)^{2} $ (T - 6)^2
$67$	$ T^{2} + 8T - 112 $ T^2 + 8*T - 112
$71$	$ T^{2} + 4T - 68 $ T^2 + 4*T - 68
$73$	$ T^{2} - 4T - 196 $ T^2 - 4*T - 196
$79$	$ T^{2} - 32 $ T^2 - 32
$83$	$ (T - 8)^{2} $ (T - 8)^2
$89$	$ T^{2} + 12T - 92 $ T^2 + 12*T - 92
$97$	$ T^{2} + 12T + 28 $ T^2 + 12*T + 28
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Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\( q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} - q^{5} - q^{7} + (\beta - 3) q^{8} + ( - \beta + 1) q^{10} + ( - 2 \beta - 2) q^{11} + (2 \beta - 2) q^{13} + ( - \beta + 1) q^{14} + 3 q^{16} + ( - 4 \beta - 2) q^{17}+ \cdots + (\beta - 1) q^{98}+O(q^{100}) \) q + (b - 1) * q^2 + (-2b + 1) q^4 - q^5 - q^7 + (b - 3) * q^8 + (-b + 1) * q^10 + (-2b - 2) q^11 + (2b - 2) q^13 + (-b + 1) * q^14 + 3 * q^16 + (-4b - 2) q^17 - 2b q^19 + (2b - 1) q^20 - 2 * q^22 + (4b - 2) q^23 + q^25 + (-4b + 6) q^26 + (2b - 1) q^28 - 8 * q^29 + 6b q^31 + (b + 3) * q^32 + (2b - 6) q^34 + q^35 - 6 * q^37 + (2b - 4) q^38 + (-b + 3) * q^40 + (4b + 2) q^41 + (4b - 4) q^43 + (2b + 6) q^44 + (-6b + 10) q^46 + 4 * q^47 + q^49 + (b - 1) * q^50 + (6b - 10) q^52 + (2b - 8) q^53 + (2b + 2) q^55 + (-b + 3) * q^56 + (-8b + 8) q^58 + 4 * q^59 + 6 * q^61 + (-6b + 12) q^62 + (2b - 7) q^64 + (-2b + 2) q^65 + (-8b - 4) q^67 + 14 * q^68 + (b - 1) * q^70 + (-6b - 2) q^71 + (-10b + 2) q^73 + (-6b + 6) q^74 + (-2b + 8) q^76 + (2b + 2) q^77 + 4b q^79 - 3 * q^80 + (-2b + 6) q^82 + 8 * q^83 + (4b + 2) q^85 + (-8b + 12) q^86 + (4b + 2) q^88 + (8b - 6) q^89 + (-2b + 2) q^91 + (8b - 18) q^92 + (4b - 4) q^94 + 2b q^95 + (2b - 6) q^97 + (b - 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8} + 2 q^{10} - 4 q^{11} - 4 q^{13} + 2 q^{14} + 6 q^{16} - 4 q^{17} - 2 q^{20} - 4 q^{22} - 4 q^{23} + 2 q^{25} + 12 q^{26} - 2 q^{28} - 16 q^{29} + 6 q^{32}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8 + 2 * q^10 - 4 * q^11 - 4 * q^13 + 2 * q^14 + 6 * q^16 - 4 * q^17 - 2 * q^20 - 4 * q^22 - 4 * q^23 + 2 * q^25 + 12 * q^26 - 2 * q^28 - 16 * q^29 + 6 * q^32 - 12 * q^34 + 2 * q^35 - 12 * q^37 - 8 * q^38 + 6 * q^40 + 4 * q^41 - 8 * q^43 + 12 * q^44 + 20 * q^46 + 8 * q^47 + 2 * q^49 - 2 * q^50 - 20 * q^52 - 16 * q^53 + 4 * q^55 + 6 * q^56 + 16 * q^58 + 8 * q^59 + 12 * q^61 + 24 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^67 + 28 * q^68 - 2 * q^70 - 4 * q^71 + 4 * q^73 + 12 * q^74 + 16 * q^76 + 4 * q^77 - 6 * q^80 + 12 * q^82 + 16 * q^83 + 4 * q^85 + 24 * q^86 + 4 * q^88 - 12 * q^89 + 4 * q^91 - 36 * q^92 - 8 * q^94 - 12 * q^97 - 2 * q^98

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