LMFDB - Newform orbit 1215.2.a.n

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1215,2,Mod(1,1215)] 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("1215.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1215, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1215 = 3^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1215.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,1,0,7,2,0,-3,12,0,1,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.70182384559$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
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 $\beta = \frac{1}{2}(1 + \sqrt{21})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta q^{2} + (\beta + 3) q^{4} + q^{5} + ( - \beta - 1) q^{7} + (2 \beta + 5) q^{8} + \beta q^{10} + ( - \beta + 4) q^{11} + (2 \beta - 3) q^{13} + ( - 2 \beta - 5) q^{14} + (5 \beta + 4) q^{16}+ \cdots + (2 \beta + 15) q^{98}+O(q^{100}) $ q + b * q^2 + (b + 3) * q^4 + q^5 + (-b - 1) * q^7 + (2b + 5) q^8 + b * q^10 + (-b + 4) * q^11 + (2b - 3) q^13 + (-2b - 5) q^14 + (5b + 4) q^16 + (-b + 1) * q^17 + (-b + 6) * q^19 + (b + 3) * q^20 + (3b - 5) q^22 + (-2b + 1) q^23 + q^25 + (-b + 10) * q^26 + (-5b - 8) q^28 - 5 * q^29 + (5b + 15) q^32 - 5 * q^34 + (-b - 1) * q^35 + (-b - 5) * q^37 + (5b - 5) q^38 + (2b + 5) q^40 + (3b + 1) q^41 - 2 * q^43 + 7 * q^44 + (-b - 10) * q^46 + (-b + 1) * q^47 + (3b - 1) q^49 + b * q^50 + (5b + 1) q^52 + (b - 3) * q^53 + (-b + 4) * q^55 + (-9b - 15) q^56 - 5b q^58 + (-2b + 6) q^59 + (-6b + 4) q^61 + (10b + 17) q^64 + (2b - 3) q^65 + (-b + 2) * q^67 + (-3b - 2) q^68 + (-2b - 5) q^70 + (-4b - 3) q^71 + (-2b - 4) q^73 + (-6b - 5) q^74 + (2b + 13) q^76 + (-2b + 1) q^77 + (-b - 7) * q^79 + (5b + 4) q^80 + (4b + 15) q^82 + (3b - 6) q^83 + (-b + 1) * q^85 - 2b q^86 + (b + 10) * q^88 + 9 * q^89 + (-b - 7) * q^91 + (-7b - 7) q^92 - 5 * q^94 + (-b + 6) * q^95 + (b - 3) * q^97 + (2b + 15) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 7 q^{4} + 2 q^{5} - 3 q^{7} + 12 q^{8} + q^{10} + 7 q^{11} - 4 q^{13} - 12 q^{14} + 13 q^{16} + q^{17} + 11 q^{19} + 7 q^{20} - 7 q^{22} + 2 q^{25} + 19 q^{26} - 21 q^{28} - 10 q^{29} + 35 q^{32}+ \cdots + 32 q^{98}+O(q^{100}) $ 2 * q + q^2 + 7 * q^4 + 2 * q^5 - 3 * q^7 + 12 * q^8 + q^10 + 7 * q^11 - 4 * q^13 - 12 * q^14 + 13 * q^16 + q^17 + 11 * q^19 + 7 * q^20 - 7 * q^22 + 2 * q^25 + 19 * q^26 - 21 * q^28 - 10 * q^29 + 35 * q^32 - 10 * q^34 - 3 * q^35 - 11 * q^37 - 5 * q^38 + 12 * q^40 + 5 * q^41 - 4 * q^43 + 14 * q^44 - 21 * q^46 + q^47 + q^49 + q^50 + 7 * q^52 - 5 * q^53 + 7 * q^55 - 39 * q^56 - 5 * q^58 + 10 * q^59 + 2 * q^61 + 44 * q^64 - 4 * q^65 + 3 * q^67 - 7 * q^68 - 12 * q^70 - 10 * q^71 - 10 * q^73 - 16 * q^74 + 28 * q^76 - 15 * q^79 + 13 * q^80 + 34 * q^82 - 9 * q^83 + q^85 - 2 * q^86 + 21 * q^88 + 18 * q^89 - 15 * q^91 - 21 * q^92 - 10 * q^94 + 11 * q^95 - 5 * q^97 + 32 * 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

−1.79129
2.79129

−1.79129

1.20871

1.00000

0.791288

1.41742

−1.79129

1.2

2.79129

5.79129

1.00000

−3.79129

10.5826

2.79129

Atkin-Lehner signs

$ p $	Sign
$3$	$ +1 $
$5$	$ -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	1215.2.a.n	yes	2
3.b	odd	2	1		1215.2.a.l	✓	2
5.b	even	2	1		6075.2.a.bl		2
9.c	even	3	2		1215.2.e.k		4
9.d	odd	6	2		1215.2.e.m		4
15.d	odd	2	1		6075.2.a.bp		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1215.2.a.l	✓	2	3.b	odd	2	1
1215.2.a.n	yes	2	1.a	even	1	1	trivial
1215.2.e.k		4	9.c	even	3	2
1215.2.e.m		4	9.d	odd	6	2
6075.2.a.bl		2	5.b	even	2	1
6075.2.a.bp		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1215))$:

$ T_{2}^{2} - T_{2} - 5 $

T2^2 - T2 - 5

$ T_{7}^{2} + 3T_{7} - 3 $

T7^2 + 3*T7 - 3

$ T_{11}^{2} - 7T_{11} + 7 $

T11^2 - 7*T11 + 7

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - T - 5 $ T^2 - T - 5
$3$	$ T^{2} $ T^2
$5$	$ (T - 1)^{2} $ (T - 1)^2
$7$	$ T^{2} + 3T - 3 $ T^2 + 3*T - 3
$11$	$ T^{2} - 7T + 7 $ T^2 - 7*T + 7
$13$	$ T^{2} + 4T - 17 $ T^2 + 4*T - 17
$17$	$ T^{2} - T - 5 $ T^2 - T - 5
$19$	$ T^{2} - 11T + 25 $ T^2 - 11*T + 25
$23$	$ T^{2} - 21 $ T^2 - 21
$29$	$ (T + 5)^{2} $ (T + 5)^2
$31$	$ T^{2} $ T^2
$37$	$ T^{2} + 11T + 25 $ T^2 + 11*T + 25
$41$	$ T^{2} - 5T - 41 $ T^2 - 5*T - 41
$43$	$ (T + 2)^{2} $ (T + 2)^2
$47$	$ T^{2} - T - 5 $ T^2 - T - 5
$53$	$ T^{2} + 5T + 1 $ T^2 + 5*T + 1
$59$	$ T^{2} - 10T + 4 $ T^2 - 10*T + 4
$61$	$ T^{2} - 2T - 188 $ T^2 - 2*T - 188
$67$	$ T^{2} - 3T - 3 $ T^2 - 3*T - 3
$71$	$ T^{2} + 10T - 59 $ T^2 + 10*T - 59
$73$	$ T^{2} + 10T + 4 $ T^2 + 10*T + 4
$79$	$ T^{2} + 15T + 51 $ T^2 + 15*T + 51
$83$	$ T^{2} + 9T - 27 $ T^2 + 9*T - 27
$89$	$ (T - 9)^{2} $ (T - 9)^2
$97$	$ T^{2} + 5T + 1 $ T^2 + 5*T + 1
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Level:	\( N \)	\(=\)	\( 1215 = 3^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1215.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta q^{2} + (\beta + 3) q^{4} + q^{5} + ( - \beta - 1) q^{7} + (2 \beta + 5) q^{8} + \beta q^{10} + ( - \beta + 4) q^{11} + (2 \beta - 3) q^{13} + ( - 2 \beta - 5) q^{14} + (5 \beta + 4) q^{16}+ \cdots + (2 \beta + 15) q^{98}+O(q^{100}) \) q + b * q^2 + (b + 3) * q^4 + q^5 + (-b - 1) * q^7 + (2b + 5) q^8 + b * q^10 + (-b + 4) * q^11 + (2b - 3) q^13 + (-2b - 5) q^14 + (5b + 4) q^16 + (-b + 1) * q^17 + (-b + 6) * q^19 + (b + 3) * q^20 + (3b - 5) q^22 + (-2b + 1) q^23 + q^25 + (-b + 10) * q^26 + (-5b - 8) q^28 - 5 * q^29 + (5b + 15) q^32 - 5 * q^34 + (-b - 1) * q^35 + (-b - 5) * q^37 + (5b - 5) q^38 + (2b + 5) q^40 + (3b + 1) q^41 - 2 * q^43 + 7 * q^44 + (-b - 10) * q^46 + (-b + 1) * q^47 + (3b - 1) q^49 + b * q^50 + (5b + 1) q^52 + (b - 3) * q^53 + (-b + 4) * q^55 + (-9b - 15) q^56 - 5b q^58 + (-2b + 6) q^59 + (-6b + 4) q^61 + (10b + 17) q^64 + (2b - 3) q^65 + (-b + 2) * q^67 + (-3b - 2) q^68 + (-2b - 5) q^70 + (-4b - 3) q^71 + (-2b - 4) q^73 + (-6b - 5) q^74 + (2b + 13) q^76 + (-2b + 1) q^77 + (-b - 7) * q^79 + (5b + 4) q^80 + (4b + 15) q^82 + (3b - 6) q^83 + (-b + 1) * q^85 - 2b q^86 + (b + 10) * q^88 + 9 * q^89 + (-b - 7) * q^91 + (-7b - 7) q^92 - 5 * q^94 + (-b + 6) * q^95 + (b - 3) * q^97 + (2b + 15) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 7 q^{4} + 2 q^{5} - 3 q^{7} + 12 q^{8} + q^{10} + 7 q^{11} - 4 q^{13} - 12 q^{14} + 13 q^{16} + q^{17} + 11 q^{19} + 7 q^{20} - 7 q^{22} + 2 q^{25} + 19 q^{26} - 21 q^{28} - 10 q^{29} + 35 q^{32}+ \cdots + 32 q^{98}+O(q^{100}) \) 2 * q + q^2 + 7 * q^4 + 2 * q^5 - 3 * q^7 + 12 * q^8 + q^10 + 7 * q^11 - 4 * q^13 - 12 * q^14 + 13 * q^16 + q^17 + 11 * q^19 + 7 * q^20 - 7 * q^22 + 2 * q^25 + 19 * q^26 - 21 * q^28 - 10 * q^29 + 35 * q^32 - 10 * q^34 - 3 * q^35 - 11 * q^37 - 5 * q^38 + 12 * q^40 + 5 * q^41 - 4 * q^43 + 14 * q^44 - 21 * q^46 + q^47 + q^49 + q^50 + 7 * q^52 - 5 * q^53 + 7 * q^55 - 39 * q^56 - 5 * q^58 + 10 * q^59 + 2 * q^61 + 44 * q^64 - 4 * q^65 + 3 * q^67 - 7 * q^68 - 12 * q^70 - 10 * q^71 - 10 * q^73 - 16 * q^74 + 28 * q^76 - 15 * q^79 + 13 * q^80 + 34 * q^82 - 9 * q^83 + q^85 - 2 * q^86 + 21 * q^88 + 18 * q^89 - 15 * q^91 - 21 * q^92 - 10 * q^94 + 11 * q^95 - 5 * q^97 + 32 * q^98

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