LMFDB - Newform orbit 175.2.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,2,Mod(1,175)] 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("175.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	175.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$1.39738203537$
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:	$\Z[a_1, a_2]$
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 $\beta = \frac{1}{2}(1 + \sqrt{5})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta q^{2} + (2 \beta - 2) q^{3} + (\beta - 1) q^{4} + 2 q^{6} + q^{7} + ( - 2 \beta + 1) q^{8} + ( - 4 \beta + 5) q^{9} + (2 \beta + 1) q^{11} + ( - 2 \beta + 4) q^{12} - 2 \beta q^{13} + \beta q^{14} + \cdots + ( - 2 \beta - 3) q^{99} +O(q^{100}) $ q + b * q^2 + (2b - 2) q^3 + (b - 1) * q^4 + 2 * q^6 + q^7 + (-2b + 1) q^8 + (-4b + 5) q^9 + (2b + 1) q^11 + (-2b + 4) q^12 - 2b q^13 + b * q^14 - 3b q^16 - 4b q^17 + (b - 4) * q^18 + (4b - 2) q^19 + (2b - 2) q^21 + (3b + 2) q^22 + (-2b + 5) q^23 + (2b - 6) q^24 + (-2b - 2) q^26 + (4b - 12) q^27 + (b - 1) * q^28 + 5 * q^29 - 6b q^31 + (b - 5) * q^32 + (2b + 2) q^33 + (-4b - 4) q^34 + (5b - 9) q^36 + 3 * q^37 + (2b + 4) q^38 - 4 * q^39 + (2b + 6) q^41 + 2 * q^42 + (2b + 3) q^43 + (b + 1) * q^44 + (3b - 2) q^46 - 2 * q^47 - 6 * q^48 + q^49 - 8 * q^51 - 2 * q^52 + (-4b + 6) q^53 + (-8b + 4) q^54 + (-2b + 1) q^56 + (-4b + 12) q^57 + 5b q^58 + (-6b + 8) q^59 + (6b - 6) q^61 + (-6b - 6) q^62 + (-4b + 5) q^63 + (2b + 1) q^64 + (4b + 2) q^66 + (2b - 3) q^67 - 4 * q^68 + (10b - 14) q^69 + (-6b + 5) q^71 + (-6b + 13) q^72 + (-2b - 10) q^73 + 3b q^74 + (-2b + 6) q^76 + (2b + 1) q^77 - 4b q^78 + (10b - 5) q^79 + (-12b + 17) q^81 + (8b + 2) q^82 + (6b - 4) q^83 + (-2b + 4) q^84 + (5b + 2) q^86 + (10b - 10) q^87 + (-4b - 3) q^88 + (-2b + 16) q^89 - 2b q^91 + (5b - 7) q^92 - 12 * q^93 - 2b q^94 + (-10b + 12) q^96 + (-2b + 4) q^97 + b * q^98 + (-2b - 3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 2 q^{3} - q^{4} + 4 q^{6} + 2 q^{7} + 6 q^{9} + 4 q^{11} + 6 q^{12} - 2 q^{13} + q^{14} - 3 q^{16} - 4 q^{17} - 7 q^{18} - 2 q^{21} + 7 q^{22} + 8 q^{23} - 10 q^{24} - 6 q^{26} - 20 q^{27}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 - q^4 + 4 * q^6 + 2 * q^7 + 6 * q^9 + 4 * q^11 + 6 * q^12 - 2 * q^13 + q^14 - 3 * q^16 - 4 * q^17 - 7 * q^18 - 2 * q^21 + 7 * q^22 + 8 * q^23 - 10 * q^24 - 6 * q^26 - 20 * q^27 - q^28 + 10 * q^29 - 6 * q^31 - 9 * q^32 + 6 * q^33 - 12 * q^34 - 13 * q^36 + 6 * q^37 + 10 * q^38 - 8 * q^39 + 14 * q^41 + 4 * q^42 + 8 * q^43 + 3 * q^44 - q^46 - 4 * q^47 - 12 * q^48 + 2 * q^49 - 16 * q^51 - 4 * q^52 + 8 * q^53 + 20 * q^57 + 5 * q^58 + 10 * q^59 - 6 * q^61 - 18 * q^62 + 6 * q^63 + 4 * q^64 + 8 * q^66 - 4 * q^67 - 8 * q^68 - 18 * q^69 + 4 * q^71 + 20 * q^72 - 22 * q^73 + 3 * q^74 + 10 * q^76 + 4 * q^77 - 4 * q^78 + 22 * q^81 + 12 * q^82 - 2 * q^83 + 6 * q^84 + 9 * q^86 - 10 * q^87 - 10 * q^88 + 30 * q^89 - 2 * q^91 - 9 * q^92 - 24 * q^93 - 2 * q^94 + 14 * q^96 + 6 * q^97 + q^98 - 8 * q^99

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

−0.618034
1.61803

−0.618034

−3.23607

−1.61803

2.00000

1.00000

2.23607

7.47214

1.2

1.61803

1.23607

0.618034

2.00000

1.00000

−2.23607

−1.47214

Atkin-Lehner signs

$ p $	Sign
$5$	$ +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	175.2.a.e	yes	2
3.b	odd	2	1		1575.2.a.n		2
4.b	odd	2	1		2800.2.a.bp		2
5.b	even	2	1		175.2.a.d	✓	2
5.c	odd	4	2		175.2.b.c		4
7.b	odd	2	1		1225.2.a.u		2
15.d	odd	2	1		1575.2.a.s		2
15.e	even	4	2		1575.2.d.k		4
20.d	odd	2	1		2800.2.a.bh		2
20.e	even	4	2		2800.2.g.s		4
35.c	odd	2	1		1225.2.a.n		2
35.f	even	4	2		1225.2.b.k		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
175.2.a.d	✓	2	5.b	even	2	1
175.2.a.e	yes	2	1.a	even	1	1	trivial
175.2.b.c		4	5.c	odd	4	2
1225.2.a.n		2	35.c	odd	2	1
1225.2.a.u		2	7.b	odd	2	1
1225.2.b.k		4	35.f	even	4	2
1575.2.a.n		2	3.b	odd	2	1
1575.2.a.s		2	15.d	odd	2	1
1575.2.d.k		4	15.e	even	4	2
2800.2.a.bh		2	20.d	odd	2	1
2800.2.a.bp		2	4.b	odd	2	1
2800.2.g.s		4	20.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} - T_{2} - 1 $ T2^2 - T2 - 1 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(175))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - T - 1 $ T^2 - T - 1
$3$	$ T^{2} + 2T - 4 $ T^2 + 2*T - 4
$5$	$ T^{2} $ T^2
$7$	$ (T - 1)^{2} $ (T - 1)^2
$11$	$ T^{2} - 4T - 1 $ T^2 - 4*T - 1
$13$	$ T^{2} + 2T - 4 $ T^2 + 2*T - 4
$17$	$ T^{2} + 4T - 16 $ T^2 + 4*T - 16
$19$	$ T^{2} - 20 $ T^2 - 20
$23$	$ T^{2} - 8T + 11 $ T^2 - 8*T + 11
$29$	$ (T - 5)^{2} $ (T - 5)^2
$31$	$ T^{2} + 6T - 36 $ T^2 + 6*T - 36
$37$	$ (T - 3)^{2} $ (T - 3)^2
$41$	$ T^{2} - 14T + 44 $ T^2 - 14*T + 44
$43$	$ T^{2} - 8T + 11 $ T^2 - 8*T + 11
$47$	$ (T + 2)^{2} $ (T + 2)^2
$53$	$ T^{2} - 8T - 4 $ T^2 - 8*T - 4
$59$	$ T^{2} - 10T - 20 $ T^2 - 10*T - 20
$61$	$ T^{2} + 6T - 36 $ T^2 + 6*T - 36
$67$	$ T^{2} + 4T - 1 $ T^2 + 4*T - 1
$71$	$ T^{2} - 4T - 41 $ T^2 - 4*T - 41
$73$	$ T^{2} + 22T + 116 $ T^2 + 22*T + 116
$79$	$ T^{2} - 125 $ T^2 - 125
$83$	$ T^{2} + 2T - 44 $ T^2 + 2*T - 44
$89$	$ T^{2} - 30T + 220 $ T^2 - 30*T + 220
$97$	$ T^{2} - 6T + 4 $ T^2 - 6*T + 4
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta q^{2} + (2 \beta - 2) q^{3} + (\beta - 1) q^{4} + 2 q^{6} + q^{7} + ( - 2 \beta + 1) q^{8} + ( - 4 \beta + 5) q^{9} + (2 \beta + 1) q^{11} + ( - 2 \beta + 4) q^{12} - 2 \beta q^{13} + \beta q^{14} + \cdots + ( - 2 \beta - 3) q^{99} +O(q^{100}) \) q + b * q^2 + (2b - 2) q^3 + (b - 1) * q^4 + 2 * q^6 + q^7 + (-2b + 1) q^8 + (-4b + 5) q^9 + (2b + 1) q^11 + (-2b + 4) q^12 - 2b q^13 + b * q^14 - 3b q^16 - 4b q^17 + (b - 4) * q^18 + (4b - 2) q^19 + (2b - 2) q^21 + (3b + 2) q^22 + (-2b + 5) q^23 + (2b - 6) q^24 + (-2b - 2) q^26 + (4b - 12) q^27 + (b - 1) * q^28 + 5 * q^29 - 6b q^31 + (b - 5) * q^32 + (2b + 2) q^33 + (-4b - 4) q^34 + (5b - 9) q^36 + 3 * q^37 + (2b + 4) q^38 - 4 * q^39 + (2b + 6) q^41 + 2 * q^42 + (2b + 3) q^43 + (b + 1) * q^44 + (3b - 2) q^46 - 2 * q^47 - 6 * q^48 + q^49 - 8 * q^51 - 2 * q^52 + (-4b + 6) q^53 + (-8b + 4) q^54 + (-2b + 1) q^56 + (-4b + 12) q^57 + 5b q^58 + (-6b + 8) q^59 + (6b - 6) q^61 + (-6b - 6) q^62 + (-4b + 5) q^63 + (2b + 1) q^64 + (4b + 2) q^66 + (2b - 3) q^67 - 4 * q^68 + (10b - 14) q^69 + (-6b + 5) q^71 + (-6b + 13) q^72 + (-2b - 10) q^73 + 3b q^74 + (-2b + 6) q^76 + (2b + 1) q^77 - 4b q^78 + (10b - 5) q^79 + (-12b + 17) q^81 + (8b + 2) q^82 + (6b - 4) q^83 + (-2b + 4) q^84 + (5b + 2) q^86 + (10b - 10) q^87 + (-4b - 3) q^88 + (-2b + 16) q^89 - 2b q^91 + (5b - 7) q^92 - 12 * q^93 - 2b q^94 + (-10b + 12) q^96 + (-2b + 4) q^97 + b * q^98 + (-2b - 3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} - q^{4} + 4 q^{6} + 2 q^{7} + 6 q^{9} + 4 q^{11} + 6 q^{12} - 2 q^{13} + q^{14} - 3 q^{16} - 4 q^{17} - 7 q^{18} - 2 q^{21} + 7 q^{22} + 8 q^{23} - 10 q^{24} - 6 q^{26} - 20 q^{27}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 - q^4 + 4 * q^6 + 2 * q^7 + 6 * q^9 + 4 * q^11 + 6 * q^12 - 2 * q^13 + q^14 - 3 * q^16 - 4 * q^17 - 7 * q^18 - 2 * q^21 + 7 * q^22 + 8 * q^23 - 10 * q^24 - 6 * q^26 - 20 * q^27 - q^28 + 10 * q^29 - 6 * q^31 - 9 * q^32 + 6 * q^33 - 12 * q^34 - 13 * q^36 + 6 * q^37 + 10 * q^38 - 8 * q^39 + 14 * q^41 + 4 * q^42 + 8 * q^43 + 3 * q^44 - q^46 - 4 * q^47 - 12 * q^48 + 2 * q^49 - 16 * q^51 - 4 * q^52 + 8 * q^53 + 20 * q^57 + 5 * q^58 + 10 * q^59 - 6 * q^61 - 18 * q^62 + 6 * q^63 + 4 * q^64 + 8 * q^66 - 4 * q^67 - 8 * q^68 - 18 * q^69 + 4 * q^71 + 20 * q^72 - 22 * q^73 + 3 * q^74 + 10 * q^76 + 4 * q^77 - 4 * q^78 + 22 * q^81 + 12 * q^82 - 2 * q^83 + 6 * q^84 + 9 * q^86 - 10 * q^87 - 10 * q^88 + 30 * q^89 - 2 * q^91 - 9 * q^92 - 24 * q^93 - 2 * q^94 + 14 * q^96 + 6 * q^97 + q^98 - 8 * q^99

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