LMFDB - Newform orbit 275.2.h.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,2,Mod(26,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.26"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	275.h (of order $5$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,2,5] 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:	no
Analytic conductor:	$2.19588605559$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{5})$
Coefficient field:	8.0.13140625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 $ x^8 - 3x^7 + 5x^6 - 3x^5 + 4x^4 + 3x^3 + 5x^2 + 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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 a basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{6} q^{2} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots + 1) q^{3} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{4} + (\beta_{6} + \beta_{3} - \beta_{2} - 1) q^{6} + ( - \beta_{7} + 2 \beta_{6} + \cdots - 2 \beta_1) q^{7}+ \cdots + ( - 4 \beta_{7} - \beta_{6} - 5 \beta_{5} + \cdots - 6) q^{99}+O(q^{100}) $ q + b6 * q^2 + (b6 + b5 + b4 - b3 - b1 + 1) * q^3 + (-b7 + b6 - b2 - b1) * q^4 + (b6 + b3 - b2 - 1) * q^6 + (-b7 + 2b6 - b4 + b3 - b2 - 2b1) * q^7 + (-b6 - b5 - b4 + 2b1 - 1) q^8 + (-2b7 + 2b6 - b5 - b4 + b2 - 2) * q^9 + (2b7 + b5 + b4 - 2b3 + 2b2 - b1 + 1) q^11 + (-b7 - 2b5 + b3 - 2) q^12 + (2b7 - 3b6 + 2b4 + 2) q^13 + (-2b4 + b3 - 2b1 - 2) * q^14 + (-b7 + b6 - b5 - b4 - b2 + b1) * q^16 + (-2b7 + b6 - 2b4 - b3 - b2 + 1) * q^17 + (2b7 - b4 + b3) q^18 + (-b6 - b5 + 5b3 + 3b1) * q^19 + (-b7 - b5 + b3 - b2 - b1 - 2) * q^21 + (-b7 + b6 + b5 + 3b3 + b2 - 1) q^22 + (b2 + b1 - 2) * q^23 + (2b7 - b6 + b5 - 2b4 - b2 + 2) * q^24 + (-3b7 - b6 + b2 + b1) q^26 + (-3b7 - b6 - b5 - 3b4 + 4b3 + b2 + b1 - 4) q^27 + (-4b7 - b6 - b5 - 4b4 + 2b3 + b2 + b1 - 2) q^28 + (-3b7 + b6 + 2b4 - 2b3 - 2b2 - b1) * q^29 - 5b6 q^31 + (4b7 + 2b5 - 4b3 - 2b2 - 2b1 + 1) q^32 + (-3b7 + b6 + 2b4 - b3 + 3b2 - 2) q^33 + (b7 - 2b5 - b3 + 2b2 + 2b1) q^34 + (-b6 - b5 - 4b4 + 2b3 + 3b1 - 4) q^36 + (4b7 - 3b6 + 7b4 - 7b3 + 3b2 + 3b1) * q^37 + (2b7 + 6b6 + 4b5 + 2b4 - 3b3 - 6b2 - 4b1 + 3) q^38 + (2b7 - b6 + 2b5 + 2b4 - 5b3 + b2 - 2b1 + 5) q^39 + (3b6 + 3b5 + b4 + b3 - 4b1 + 1) q^41 + (-b7 - 4b6 - b5 - 2b4 + b2 - 1) * q^42 + (b5 - 2b2 - 2b1 + 6) * q^43 + (3b7 + b6 - b5 + 5b4 - b3 + b2 + b1) * q^44 + (b7 + b5 + b4 - b2 + 1) * q^46 + (-3b6 - 3b5 + 3b3 + 2b1) * q^47 + (b7 + b6 - b4 + b3 - 2b2 - b1) q^48 + (b6 + 4b5 - b3 - b2 - 4b1 + 1) * q^49 + (-2b7 + 2b6 + b4 - b3 + b2 - 2b1) q^51 + (b6 + b5 + 5b4 - 2b1 + 5) * q^52 + (-b7 - b6 - 5b5 + 5b4 + 5b2 - 1) q^53 + (4b5 + 1) q^54 + (4b7 - 2b5 - 4b3 + 3b2 + 3b1 + 3) q^56 + (8b7 + b6 + 7b5 - b4 - 7b2 + 8) q^57 + (-4b6 - 4b5 - b4 - b3 + 4b1 - 1) q^58 + (7b7 - 4b6 + 5b4 - 5b3 + b2 + 4b1) q^59 + (3b7 - 2b6 - 4b5 + 3b4 - 2b3 + 2b2 + 4b1 + 2) q^61 + (-5b7 - 5b6 + 5b2 + 5b1) * q^62 + (2b6 + 2b5 - 6b4 + 3b3 - b1 - 6) * q^63 + (-4b7 - 3b6 - 2b4 - 4) q^64 + (b7 + b6 + 2b5 + 3b3 - 2b2 + 3b1) * q^66 + (6b7 - b5 - 6b3 + 4b2 + 4b1 + 5) * q^67 + (-4b7 + 3b6 + b5 - 4b4 - b2 - 4) q^68 + (-b6 - b5 - 2b4 + 3b3 + 3b1 - 2) q^69 + (-b7 + 3b6 + 3b5 - b4 + 3b3 - 3b2 - 3b1 - 3) q^71 + (6b7 - b6 + b5 + 6b4 - 9b3 + b2 - b1 + 9) q^72 + (b7 + 2b6 - 3b2 - 2b1) q^73 + (-4b6 - 4b5 + 3b4 + 6b1 + 3) * q^74 + (2b7 + b5 - 2b3 - 7b2 - 7b1 + 6) * q^76 + (4b7 + 2b6 + 4b5 + 6b4 + b3 - b2 - 3b1 - 2) q^77 + (-3b7 - 2b5 + 3b3 + 4b2 + 4b1 - 2) q^78 + (6b7 - 4b6 - 3b5 - 3b4 + 3b2 + 6) q^79 + (4b7 - 2b6 + b4 - b3 + 6b2 + 2b1) * q^81 + (-b7 + 4b6 + 4b5 - b4 + 4b3 - 4b2 - 4b1 - 4) q^82 + (7b7 + b6 + 4b5 + 7b4 - b3 - b2 - 4b1 + 1) * q^83 + (-2b7 - 3b6 - 3b4 + 3b3 + 3b2 + 3b1) * q^84 + (-2b7 + 3b6 - b5 - b4 + b2 - 2) * q^86 + (-4b7 - 8b5 + 4b3 + 3b2 + 3b1 - 7) q^87 + (2b7 - b6 - 3b5 - 2b4 + 2b3 - b2 + b1 + 5) * q^88 + (6b5 - 6b2 - 6b1 + 1) q^89 + (-2b6 - 2b5 + 8b4 - b3 + 2b1 + 8) * q^91 + (4b7 - b6 + b4 - b3 + b1) q^92 + (-5b6 - 5b3 + 5b2 + 5) q^93 + (-b7 - b6 - b4 - 2b3 + b2 + 2) q^94 + (b6 + b5 + 3b4 - 9b3 - 7b1 + 3) q^96 + (-8b7 + 2b5 - 13b4 - 2b2 - 8) * q^97 + (-3b7 + 2b5 + 3b3 - 4) q^98 + (-4b7 - b6 - 5b5 + b4 + 11b3 + 3b2 + b1 - 6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 5 q^{3} - 2 q^{4} - 7 q^{6} + q^{7} - 4 q^{8} - 5 q^{9} + 3 q^{11} - 16 q^{12} + 2 q^{13} - 16 q^{14} + 4 q^{16} + 13 q^{17} + 15 q^{19} - 20 q^{21} + 7 q^{22} - 10 q^{23} + 13 q^{24} + 10 q^{26}+ \cdots - 20 q^{99}+O(q^{100}) $ 8 * q + 2 * q^2 + 5 * q^3 - 2 * q^4 - 7 * q^6 + q^7 - 4 * q^8 - 5 * q^9 + 3 * q^11 - 16 * q^12 + 2 * q^13 - 16 * q^14 + 4 * q^16 + 13 * q^17 + 15 * q^19 - 20 * q^21 + 7 * q^22 - 10 * q^23 + 13 * q^24 + 10 * q^26 - 10 * q^27 + 6 * q^28 - 9 * q^29 - 10 * q^31 - 16 * q^32 - 5 * q^33 + 4 * q^34 - 15 * q^36 - 24 * q^37 + 21 * q^39 + 8 * q^41 - 9 * q^42 + 38 * q^43 - 12 * q^44 + 3 * q^46 - 5 * q^48 + q^49 + q^51 + 28 * q^52 - 13 * q^53 + 16 * q^54 + 22 * q^56 + 45 * q^57 - 12 * q^58 - 27 * q^59 + 6 * q^61 + 30 * q^62 - 25 * q^63 - 26 * q^64 + 13 * q^66 + 38 * q^67 - 11 * q^68 - q^69 - 20 * q^71 + 30 * q^72 - 13 * q^73 + 20 * q^74 - 34 * q^77 + 16 * q^78 + 37 * q^79 + 8 * q^81 - 28 * q^82 - 27 * q^83 + 28 * q^84 - 3 * q^86 - 38 * q^87 + 36 * q^88 - 16 * q^89 + 44 * q^91 - 11 * q^92 + 35 * q^93 + 17 * q^94 - 17 * q^96 - 24 * q^97 - 16 * q^98 - 20 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 $ x^8 - 3*x^7 + 5*x^6 - 3*x^5 + 4*x^4 + 3*x^3 + 5*x^2 + 3*x + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 $ (-v^7 + 2v^6 - 3v^5 - 4v^3 - 7v^2 - 12*v - 7) / 8
$\beta_{3}$	$=$	$ ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 $ (v^7 - 7v^5 + 20v^4 - 16v^3 + 19v^2 + 6*v + 9) / 8
$\beta_{4}$	$=$	$ ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 $ (-v^7 + 4v^6 - 9v^5 + 12v^4 - 16v^3 + 13v^2 - 10v - 1) / 8
$\beta_{5}$	$=$	$ ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 $ (-3v^7 + 10v^6 - 17v^5 + 8v^4 - 4v^3 - 13v^2 - 8*v - 5) / 8
$\beta_{6}$	$=$	$ ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 $ (3v^7 - 12v^6 + 23v^5 - 20v^4 + 16v^3 + v^2 + 6v - 1) / 8
$\beta_{7}$	$=$	$ ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 $ (-5v^7 + 18v^6 - 35v^5 + 32v^4 - 28v^3 - 11v^2 - 12*v - 7) / 8

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} $ b6 + b5 + b4 - b2
$\nu^{3}$	$=$	$ \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 $ b7 + 3b6 + 2b5 + b4 - 3b2 - 2b1
$\nu^{4}$	$=$	$ 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 $ 3b7 + 4b6 + b4 - b3 - 5b2 - 4b1
$\nu^{5}$	$=$	$ 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 $ 4b7 - 6b5 - 4b3 - 6b2 - 6*b1 - 1
$\nu^{6}$	$=$	$ -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 $ -16b6 - 16b5 - 6b4 - 6b3 - 7*b1 - 6
$\nu^{7}$	$=$	$ -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 $ -16b7 - 51b6 - 29b5 - 23b4 + 29*b2 - 16

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$\beta_{7}$	$1$

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} $

26.1

−0.227943	+	0.701538i
0.418926	−	1.28932i
−0.386111	−	0.280526i
1.69513	+	1.23158i
−0.227943	−	0.701538i
0.418926	+	1.28932i
−0.386111	+	0.280526i
1.69513	−	1.23158i

−0.596764

−

0.433574i

0.868820

−

2.67395i

−0.449894

−

1.38463i

−1.67784

1.21902i

−0.318714

−

0.980901i

−0.787747

2.42443i

−3.96813

−

2.88301i

26.2

1.09676

0.796845i

−0.177837

0.547326i

−0.0501062

−

0.154211i

−0.631180

0.458579i

1.12773

3.47080i

0.905781

−

2.78771i

2.15911

1.56869i

126.1

−0.147481

−

0.453901i

0.261370

0.189896i

1.43376

−

1.04169i

0.0476470

−

0.146642i

2.17239

−

1.57833i

−1.45650

−

1.05821i

−0.894797

−

2.75390i

126.2

0.647481

1.99274i

1.54765

1.12443i

−1.93376

1.40496i

−1.23863

3.81211i

−2.48141

1.80285i

−0.661536

−

0.480634i

0.203814

0.627276i

201.1