LMFDB - Newform orbit 297.2.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [297,2,Mod(1,297)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(297, 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("297.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	297.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:	$2.37155694003$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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 $\beta = \sqrt{3}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta + 1) q^{2} + (2 \beta + 2) q^{4} + ( - \beta + 1) q^{5} + ( - \beta - 2) q^{7} + (2 \beta + 6) q^{8} - 2 q^{10} + q^{11} + (\beta - 2) q^{13} + ( - 3 \beta - 5) q^{14} + (4 \beta + 8) q^{16}+ \cdots + (4 \beta + 12) q^{98}+O(q^{100}) $ q + (b + 1) * q^2 + (2b + 2) q^4 + (-b + 1) * q^5 + (-b - 2) * q^7 + (2b + 6) q^8 - 2 * q^10 + q^11 + (b - 2) * q^13 + (-3b - 5) q^14 + (4b + 8) q^16 + (-3b + 1) q^17 - 3b q^19 - 4 * q^20 + (b + 1) * q^22 + 8 * q^23 + (-2b - 1) q^25 + (-b + 1) * q^26 + (-6b - 10) q^28 + (b - 3) * q^29 + (-2b + 4) q^31 + (8b + 8) q^32 + (-2b - 8) q^34 + (b + 1) * q^35 + (-2b - 3) q^37 + (-3b - 9) q^38 - 4b q^40 + (2b - 2) q^41 + 2b q^43 + (2b + 2) q^44 + (8b + 8) q^46 + (3b + 5) q^47 + 4b q^49 + (-3b - 7) q^50 + (-2b + 2) q^52 + (b + 3) * q^53 + (-b + 1) * q^55 + (-10b - 18) q^56 - 2b q^58 + (3b + 5) q^59 + (b - 6) * q^61 + (2b - 2) q^62 + (8b + 16) q^64 + (3b - 5) q^65 + (4b - 1) q^67 + (-4b - 16) q^68 + (2b + 4) q^70 - 8b q^71 + (3b - 2) q^73 + (-5b - 9) q^74 + (-6b - 18) q^76 + (-b - 2) * q^77 + (-b - 4) * q^79 + (-4b - 4) q^80 + 4 * q^82 + (-4b + 10) q^85 + (2b + 6) q^86 + (2b + 6) q^88 + (5b - 9) q^89 + q^91 + (16b + 16) q^92 + (8b + 14) q^94 + (-3b + 9) q^95 + (-6b + 5) q^97 + (4b + 12) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{7} + 12 q^{8} - 4 q^{10} + 2 q^{11} - 4 q^{13} - 10 q^{14} + 16 q^{16} + 2 q^{17} - 8 q^{20} + 2 q^{22} + 16 q^{23} - 2 q^{25} + 2 q^{26} - 20 q^{28} - 6 q^{29}+ \cdots + 24 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^7 + 12 * q^8 - 4 * q^10 + 2 * q^11 - 4 * q^13 - 10 * q^14 + 16 * q^16 + 2 * q^17 - 8 * q^20 + 2 * q^22 + 16 * q^23 - 2 * q^25 + 2 * q^26 - 20 * q^28 - 6 * q^29 + 8 * q^31 + 16 * q^32 - 16 * q^34 + 2 * q^35 - 6 * q^37 - 18 * q^38 - 4 * q^41 + 4 * q^44 + 16 * q^46 + 10 * q^47 - 14 * q^50 + 4 * q^52 + 6 * q^53 + 2 * q^55 - 36 * q^56 + 10 * q^59 - 12 * q^61 - 4 * q^62 + 32 * q^64 - 10 * q^65 - 2 * q^67 - 32 * q^68 + 8 * q^70 - 4 * q^73 - 18 * q^74 - 36 * q^76 - 4 * q^77 - 8 * q^79 - 8 * q^80 + 8 * q^82 + 20 * q^85 + 12 * q^86 + 12 * q^88 - 18 * q^89 + 2 * q^91 + 32 * q^92 + 28 * q^94 + 18 * q^95 + 10 * q^97 + 24 * 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

−1.73205
1.73205

−0.732051

−1.46410

2.73205

−0.267949

2.53590

−2.00000

1.2

2.73205

5.46410

−0.732051

−3.73205

9.46410

−2.00000

Atkin-Lehner signs

$ p $	Sign
$3$	$ +1 $
$11$	$ -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	297.2.a.f	yes	2
3.b	odd	2	1		297.2.a.e	✓	2
4.b	odd	2	1		4752.2.a.bf		2
5.b	even	2	1		7425.2.a.z		2
9.c	even	3	2		891.2.e.m		4
9.d	odd	6	2		891.2.e.p		4
11.b	odd	2	1		3267.2.a.l		2
12.b	even	2	1		4752.2.a.w		2
15.d	odd	2	1		7425.2.a.bl		2
33.d	even	2	1		3267.2.a.q		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
297.2.a.e	✓	2	3.b	odd	2	1
297.2.a.f	yes	2	1.a	even	1	1	trivial
891.2.e.m		4	9.c	even	3	2
891.2.e.p		4	9.d	odd	6	2
3267.2.a.l		2	11.b	odd	2	1
3267.2.a.q		2	33.d	even	2	1
4752.2.a.w		2	12.b	even	2	1
4752.2.a.bf		2	4.b	odd	2	1
7425.2.a.z		2	5.b	even	2	1
7425.2.a.bl		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} - 2T_{2} - 2 $ T2^2 - 2*T2 - 2 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(297))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - 2T - 2 $ T^2 - 2*T - 2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} - 2T - 2 $ T^2 - 2*T - 2
$7$	$ T^{2} + 4T + 1 $ T^2 + 4*T + 1
$11$	$ (T - 1)^{2} $ (T - 1)^2
$13$	$ T^{2} + 4T + 1 $ T^2 + 4*T + 1
$17$	$ T^{2} - 2T - 26 $ T^2 - 2*T - 26
$19$	$ T^{2} - 27 $ T^2 - 27
$23$	$ (T - 8)^{2} $ (T - 8)^2
$29$	$ T^{2} + 6T + 6 $ T^2 + 6*T + 6
$31$	$ T^{2} - 8T + 4 $ T^2 - 8*T + 4
$37$	$ T^{2} + 6T - 3 $ T^2 + 6*T - 3
$41$	$ T^{2} + 4T - 8 $ T^2 + 4*T - 8
$43$	$ T^{2} - 12 $ T^2 - 12
$47$	$ T^{2} - 10T - 2 $ T^2 - 10*T - 2
$53$	$ T^{2} - 6T + 6 $ T^2 - 6*T + 6
$59$	$ T^{2} - 10T - 2 $ T^2 - 10*T - 2
$61$	$ T^{2} + 12T + 33 $ T^2 + 12*T + 33
$67$	$ T^{2} + 2T - 47 $ T^2 + 2*T - 47
$71$	$ T^{2} - 192 $ T^2 - 192
$73$	$ T^{2} + 4T - 23 $ T^2 + 4*T - 23
$79$	$ T^{2} + 8T + 13 $ T^2 + 8*T + 13
$83$	$ T^{2} $ T^2
$89$	$ T^{2} + 18T + 6 $ T^2 + 18*T + 6
$97$	$ T^{2} - 10T - 83 $ T^2 - 10*T - 83
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Level:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	297.a (trivial)

\(f(q)\)	\(=\)	\( q + (\beta + 1) q^{2} + (2 \beta + 2) q^{4} + ( - \beta + 1) q^{5} + ( - \beta - 2) q^{7} + (2 \beta + 6) q^{8} - 2 q^{10} + q^{11} + (\beta - 2) q^{13} + ( - 3 \beta - 5) q^{14} + (4 \beta + 8) q^{16}+ \cdots + (4 \beta + 12) q^{98}+O(q^{100}) \) q + (b + 1) * q^2 + (2b + 2) q^4 + (-b + 1) * q^5 + (-b - 2) * q^7 + (2b + 6) q^8 - 2 * q^10 + q^11 + (b - 2) * q^13 + (-3b - 5) q^14 + (4b + 8) q^16 + (-3b + 1) q^17 - 3b q^19 - 4 * q^20 + (b + 1) * q^22 + 8 * q^23 + (-2b - 1) q^25 + (-b + 1) * q^26 + (-6b - 10) q^28 + (b - 3) * q^29 + (-2b + 4) q^31 + (8b + 8) q^32 + (-2b - 8) q^34 + (b + 1) * q^35 + (-2b - 3) q^37 + (-3b - 9) q^38 - 4b q^40 + (2b - 2) q^41 + 2b q^43 + (2b + 2) q^44 + (8b + 8) q^46 + (3b + 5) q^47 + 4b q^49 + (-3b - 7) q^50 + (-2b + 2) q^52 + (b + 3) * q^53 + (-b + 1) * q^55 + (-10b - 18) q^56 - 2b q^58 + (3b + 5) q^59 + (b - 6) * q^61 + (2b - 2) q^62 + (8b + 16) q^64 + (3b - 5) q^65 + (4b - 1) q^67 + (-4b - 16) q^68 + (2b + 4) q^70 - 8b q^71 + (3b - 2) q^73 + (-5b - 9) q^74 + (-6b - 18) q^76 + (-b - 2) * q^77 + (-b - 4) * q^79 + (-4b - 4) q^80 + 4 * q^82 + (-4b + 10) q^85 + (2b + 6) q^86 + (2b + 6) q^88 + (5b - 9) q^89 + q^91 + (16b + 16) q^92 + (8b + 14) q^94 + (-3b + 9) q^95 + (-6b + 5) q^97 + (4b + 12) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{7} + 12 q^{8} - 4 q^{10} + 2 q^{11} - 4 q^{13} - 10 q^{14} + 16 q^{16} + 2 q^{17} - 8 q^{20} + 2 q^{22} + 16 q^{23} - 2 q^{25} + 2 q^{26} - 20 q^{28} - 6 q^{29}+ \cdots + 24 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^7 + 12 * q^8 - 4 * q^10 + 2 * q^11 - 4 * q^13 - 10 * q^14 + 16 * q^16 + 2 * q^17 - 8 * q^20 + 2 * q^22 + 16 * q^23 - 2 * q^25 + 2 * q^26 - 20 * q^28 - 6 * q^29 + 8 * q^31 + 16 * q^32 - 16 * q^34 + 2 * q^35 - 6 * q^37 - 18 * q^38 - 4 * q^41 + 4 * q^44 + 16 * q^46 + 10 * q^47 - 14 * q^50 + 4 * q^52 + 6 * q^53 + 2 * q^55 - 36 * q^56 + 10 * q^59 - 12 * q^61 - 4 * q^62 + 32 * q^64 - 10 * q^65 - 2 * q^67 - 32 * q^68 + 8 * q^70 - 4 * q^73 - 18 * q^74 - 36 * q^76 - 4 * q^77 - 8 * q^79 - 8 * q^80 + 8 * q^82 + 20 * q^85 + 12 * q^86 + 12 * q^88 - 18 * q^89 + 2 * q^91 + 32 * q^92 + 28 * q^94 + 18 * q^95 + 10 * q^97 + 24 * q^98

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