LMFDB - Newform orbit 2299.2.a.g

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2299,2,Mod(1,2299)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2299, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("2299.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2299 = 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2299.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,6,-1,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.3576074247$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
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 $\beta = \frac{1}{2}(1 + \sqrt{5})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - 2 \beta + 1) q^{2} + 2 \beta q^{3} + 3 q^{4} - \beta q^{5} + ( - 2 \beta - 4) q^{6} + (\beta - 2) q^{7} + ( - 2 \beta + 1) q^{8} + (4 \beta + 1) q^{9} + (\beta + 2) q^{10} + 6 \beta q^{12} + ( - 4 \beta + 4) q^{13} + \cdots + (7 \beta + 4) q^{98} +O(q^{100}) $ q + (-2b + 1) q^2 + 2b q^3 + 3 * q^4 - b * q^5 + (-2b - 4) q^6 + (b - 2) * q^7 + (-2b + 1) q^8 + (4b + 1) q^9 + (b + 2) * q^10 + 6b q^12 + (-4b + 4) q^13 + (3b - 4) q^14 + (-2b - 2) q^15 - q^16 + (-5b + 5) q^17 + (-6b - 7) q^18 + q^19 - 3b q^20 + (-2b + 2) q^21 + (-b + 1) * q^23 + (-2b - 4) q^24 + (b - 4) * q^25 + (-4b + 12) q^26 + (4b + 8) q^27 + (3b - 6) q^28 + (4b + 2) q^29 + (6b + 2) q^30 + 8 * q^31 + (6b - 3) q^32 + (-5b + 15) q^34 + (b - 1) * q^35 + (12b + 3) q^36 - 2 * q^37 + (-2b + 1) q^38 - 8 * q^39 + (b + 2) * q^40 + (6b + 2) q^41 + (-2b + 6) q^42 + (-5b + 5) q^43 + (-5b - 4) q^45 + (-b + 3) * q^46 + (b + 7) * q^47 - 2b q^48 + (-3b - 2) q^49 + (7b - 6) q^50 - 10 * q^51 + (-12b + 12) q^52 + (-4b + 8) q^53 - 20b q^54 + (3b - 4) q^56 + 2b q^57 + (-8b - 6) q^58 + (4b + 6) q^59 + (-6b - 6) q^60 + (b + 7) * q^61 + (-16b + 8) q^62 + (-3b + 2) q^63 - 13 * q^64 + 4 * q^65 + (-4b - 2) q^67 + (-15b + 15) q^68 - 2 * q^69 + (b - 3) * q^70 + (6b + 2) q^71 + (-6b - 7) q^72 + (-4b + 10) q^73 + (4b - 2) q^74 + (-6b + 2) q^75 + 3 * q^76 + (16b - 8) q^78 + (8b - 10) q^79 + b * q^80 + (12b + 5) q^81 + (-10b - 10) q^82 + (7b - 6) q^83 + (-6b + 6) q^84 + 5 * q^85 + (-5b + 15) q^86 + (12b + 8) q^87 - 2b q^89 + (13b + 6) q^90 + (8b - 12) q^91 + (-3b + 3) q^92 + 16b q^93 + (-15b + 5) q^94 - b * q^95 + (6b + 12) q^96 + (-10b + 4) q^97 + (7b + 4) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{4} - q^{5} - 10 q^{6} - 3 q^{7} + 6 q^{9} + 5 q^{10} + 6 q^{12} + 4 q^{13} - 5 q^{14} - 6 q^{15} - 2 q^{16} + 5 q^{17} - 20 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + q^{23} - 10 q^{24}+ \cdots + 15 q^{98}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^4 - q^5 - 10 * q^6 - 3 * q^7 + 6 * q^9 + 5 * q^10 + 6 * q^12 + 4 * q^13 - 5 * q^14 - 6 * q^15 - 2 * q^16 + 5 * q^17 - 20 * q^18 + 2 * q^19 - 3 * q^20 + 2 * q^21 + q^23 - 10 * q^24 - 7 * q^25 + 20 * q^26 + 20 * q^27 - 9 * q^28 + 8 * q^29 + 10 * q^30 + 16 * q^31 + 25 * q^34 - q^35 + 18 * q^36 - 4 * q^37 - 16 * q^39 + 5 * q^40 + 10 * q^41 + 10 * q^42 + 5 * q^43 - 13 * q^45 + 5 * q^46 + 15 * q^47 - 2 * q^48 - 7 * q^49 - 5 * q^50 - 20 * q^51 + 12 * q^52 + 12 * q^53 - 20 * q^54 - 5 * q^56 + 2 * q^57 - 20 * q^58 + 16 * q^59 - 18 * q^60 + 15 * q^61 + q^63 - 26 * q^64 + 8 * q^65 - 8 * q^67 + 15 * q^68 - 4 * q^69 - 5 * q^70 + 10 * q^71 - 20 * q^72 + 16 * q^73 - 2 * q^75 + 6 * q^76 - 12 * q^79 + q^80 + 22 * q^81 - 30 * q^82 - 5 * q^83 + 6 * q^84 + 10 * q^85 + 25 * q^86 + 28 * q^87 - 2 * q^89 + 25 * q^90 - 16 * q^91 + 3 * q^92 + 16 * q^93 - 5 * q^94 - q^95 + 30 * q^96 - 2 * q^97 + 15 * 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

1.61803
−0.618034

−2.23607

3.23607

3.00000

−1.61803

−7.23607

−0.381966

−2.23607

7.47214

3.61803

1.2

2.23607

−1.23607

3.00000

0.618034

−2.76393

−2.61803

2.23607

−1.47214

1.38197

Atkin-Lehner signs

$ p $	Sign
$11$	$ +1 $
$19$	$ -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	2299.2.a.g		2
11.b	odd	2	1		2299.2.a.h		2
11.c	even	5	2		209.2.f.a	✓	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
209.2.f.a	✓	4	11.c	even	5	2
2299.2.a.g		2	1.a	even	1	1	trivial
2299.2.a.h		2	11.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2299))$:

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

T2^2 - 5

$ T_{3}^{2} - 2T_{3} - 4 $

T3^2 - 2*T3 - 4

$ T_{7}^{2} + 3T_{7} + 1 $

T7^2 + 3*T7 + 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - 5 $ T^2 - 5
$3$	$ T^{2} - 2T - 4 $ T^2 - 2*T - 4
$5$	$ T^{2} + T - 1 $ T^2 + T - 1
$7$	$ T^{2} + 3T + 1 $ T^2 + 3*T + 1
$11$	$ T^{2} $ T^2
$13$	$ T^{2} - 4T - 16 $ T^2 - 4*T - 16
$17$	$ T^{2} - 5T - 25 $ T^2 - 5*T - 25
$19$	$ (T - 1)^{2} $ (T - 1)^2
$23$	$ T^{2} - T - 1 $ T^2 - T - 1
$29$	$ T^{2} - 8T - 4 $ T^2 - 8*T - 4
$31$	$ (T - 8)^{2} $ (T - 8)^2
$37$	$ (T + 2)^{2} $ (T + 2)^2
$41$	$ T^{2} - 10T - 20 $ T^2 - 10*T - 20
$43$	$ T^{2} - 5T - 25 $ T^2 - 5*T - 25
$47$	$ T^{2} - 15T + 55 $ T^2 - 15*T + 55
$53$	$ T^{2} - 12T + 16 $ T^2 - 12*T + 16
$59$	$ T^{2} - 16T + 44 $ T^2 - 16*T + 44
$61$	$ T^{2} - 15T + 55 $ T^2 - 15*T + 55
$67$	$ T^{2} + 8T - 4 $ T^2 + 8*T - 4
$71$	$ T^{2} - 10T - 20 $ T^2 - 10*T - 20
$73$	$ T^{2} - 16T + 44 $ T^2 - 16*T + 44
$79$	$ T^{2} + 12T - 44 $ T^2 + 12*T - 44
$83$	$ T^{2} + 5T - 55 $ T^2 + 5*T - 55
$89$	$ T^{2} + 2T - 4 $ T^2 + 2*T - 4
$97$	$ T^{2} + 2T - 124 $ T^2 + 2*T - 124
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Level:	\( N \)	\(=\)	\( 2299 = 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2299.a (trivial)

\(f(q)\)	\(=\)	\( q + ( - 2 \beta + 1) q^{2} + 2 \beta q^{3} + 3 q^{4} - \beta q^{5} + ( - 2 \beta - 4) q^{6} + (\beta - 2) q^{7} + ( - 2 \beta + 1) q^{8} + (4 \beta + 1) q^{9} + (\beta + 2) q^{10} + 6 \beta q^{12} + ( - 4 \beta + 4) q^{13} + \cdots + (7 \beta + 4) q^{98} +O(q^{100}) \) q + (-2b + 1) q^2 + 2b q^3 + 3 * q^4 - b * q^5 + (-2b - 4) q^6 + (b - 2) * q^7 + (-2b + 1) q^8 + (4b + 1) q^9 + (b + 2) * q^10 + 6b q^12 + (-4b + 4) q^13 + (3b - 4) q^14 + (-2b - 2) q^15 - q^16 + (-5b + 5) q^17 + (-6b - 7) q^18 + q^19 - 3b q^20 + (-2b + 2) q^21 + (-b + 1) * q^23 + (-2b - 4) q^24 + (b - 4) * q^25 + (-4b + 12) q^26 + (4b + 8) q^27 + (3b - 6) q^28 + (4b + 2) q^29 + (6b + 2) q^30 + 8 * q^31 + (6b - 3) q^32 + (-5b + 15) q^34 + (b - 1) * q^35 + (12b + 3) q^36 - 2 * q^37 + (-2b + 1) q^38 - 8 * q^39 + (b + 2) * q^40 + (6b + 2) q^41 + (-2b + 6) q^42 + (-5b + 5) q^43 + (-5b - 4) q^45 + (-b + 3) * q^46 + (b + 7) * q^47 - 2b q^48 + (-3b - 2) q^49 + (7b - 6) q^50 - 10 * q^51 + (-12b + 12) q^52 + (-4b + 8) q^53 - 20b q^54 + (3b - 4) q^56 + 2b q^57 + (-8b - 6) q^58 + (4b + 6) q^59 + (-6b - 6) q^60 + (b + 7) * q^61 + (-16b + 8) q^62 + (-3b + 2) q^63 - 13 * q^64 + 4 * q^65 + (-4b - 2) q^67 + (-15b + 15) q^68 - 2 * q^69 + (b - 3) * q^70 + (6b + 2) q^71 + (-6b - 7) q^72 + (-4b + 10) q^73 + (4b - 2) q^74 + (-6b + 2) q^75 + 3 * q^76 + (16b - 8) q^78 + (8b - 10) q^79 + b * q^80 + (12b + 5) q^81 + (-10b - 10) q^82 + (7b - 6) q^83 + (-6b + 6) q^84 + 5 * q^85 + (-5b + 15) q^86 + (12b + 8) q^87 - 2b q^89 + (13b + 6) q^90 + (8b - 12) q^91 + (-3b + 3) q^92 + 16b q^93 + (-15b + 5) q^94 - b * q^95 + (6b + 12) q^96 + (-10b + 4) q^97 + (7b + 4) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{4} - q^{5} - 10 q^{6} - 3 q^{7} + 6 q^{9} + 5 q^{10} + 6 q^{12} + 4 q^{13} - 5 q^{14} - 6 q^{15} - 2 q^{16} + 5 q^{17} - 20 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + q^{23} - 10 q^{24}+ \cdots + 15 q^{98}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^4 - q^5 - 10 * q^6 - 3 * q^7 + 6 * q^9 + 5 * q^10 + 6 * q^12 + 4 * q^13 - 5 * q^14 - 6 * q^15 - 2 * q^16 + 5 * q^17 - 20 * q^18 + 2 * q^19 - 3 * q^20 + 2 * q^21 + q^23 - 10 * q^24 - 7 * q^25 + 20 * q^26 + 20 * q^27 - 9 * q^28 + 8 * q^29 + 10 * q^30 + 16 * q^31 + 25 * q^34 - q^35 + 18 * q^36 - 4 * q^37 - 16 * q^39 + 5 * q^40 + 10 * q^41 + 10 * q^42 + 5 * q^43 - 13 * q^45 + 5 * q^46 + 15 * q^47 - 2 * q^48 - 7 * q^49 - 5 * q^50 - 20 * q^51 + 12 * q^52 + 12 * q^53 - 20 * q^54 - 5 * q^56 + 2 * q^57 - 20 * q^58 + 16 * q^59 - 18 * q^60 + 15 * q^61 + q^63 - 26 * q^64 + 8 * q^65 - 8 * q^67 + 15 * q^68 - 4 * q^69 - 5 * q^70 + 10 * q^71 - 20 * q^72 + 16 * q^73 - 2 * q^75 + 6 * q^76 - 12 * q^79 + q^80 + 22 * q^81 - 30 * q^82 - 5 * q^83 + 6 * q^84 + 10 * q^85 + 25 * q^86 + 28 * q^87 - 2 * q^89 + 25 * q^90 - 16 * q^91 + 3 * q^92 + 16 * q^93 - 5 * q^94 - q^95 + 30 * q^96 - 2 * q^97 + 15 * q^98

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