LMFDB - Newform orbit 975.2.n.o

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [975,2,Mod(749,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("975.749"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$975 = 3 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	975.n (of order $4$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [8,6,2,0,0,-4,14,-12,4,0,-4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(12)] == traces)

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

Self dual:	no
Analytic conductor:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.619810816.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$ x^8 - 2x^5 + 14x^4 - 8x^3 + 2x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -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.

$f(q)$	$=$	$q + (\beta_{7} - \beta_{4} - \beta_{3} + 1) q^{2} + (\beta_{5} - \beta_{2} + \beta_1) q^{3} + (\beta_{7} + \beta_{6} + \cdots - 2 \beta_{3}) q^{4} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{2} + \cdots - 1) q^{6}+ \cdots + ( - 2 \beta_{7} + \beta_{6} + 3 \beta_{4} + \cdots + 3) q^{99}+O(q^{100})$ q + (b7 - b4 - b3 + 1) * q^2 + (b5 - b2 + b1) * q^3 + (b7 + b6 + b5 - b4 - 2b3) q^4 + (-2b7 - b6 - 2b2 + b1 - 1) * q^6 + (b7 + b6 + b4 - 2b3 + b2 + 2) q^7 + (2b6 + 2b5 - 2b3 - 2b2 + b1 - 2) * q^8 + (b7 - b5 + b1 + 1) * q^9 + (2b7 - b4) q^11 + (-3b7 - 3b6 - 2b5 + 3b3 - b2 - 2b1) q^12 + (b6 - b5 - b4 + 2) * q^13 + (4b7 + 3b6 + 4b5 - b4 - 4b3 + 3b1) q^14 + (-4b7 + 4b5 + b4 - 4b2 + 3b1 - 3) * q^16 + (-b7 - b5 - 3b3 - b1) q^17 + (b7 + b6 - b4 - 2b3 + b2 + 2b1 + 4) * q^18 + (-b7 + 2b4) q^19 + (-2b7 - 2b6 + 2b5 + 2b4 + b3 - b2 + b1 - 2) * q^21 + (2b6 + b4 - 3b3 + b1) * q^22 + (-b7 - b6 - b5 + 5b3 - b1) q^23 + (-2b7 - 3b6 - 5b5 + 2b4 + 4b3 + 2b2 - 4b1 + 5) q^24 + (2b7 + b6 + 2b5 - 3b4 - 3b3 + 4) * q^26 + (-b7 + 2b6 + 2b5 + 2b4 - 2b3 - b2 + 4b1 - 2) q^27 + (5b6 + 8b5 - 5b3 - 5b2 + 5b1 - 5) q^28 + (-3b7 - 2b6 - 3b5 + b4 - b3 - 2b1) * q^29 + (-2b7 - 2b6 - b4 + 2b3 - 2b2 - 2) * q^31 + (-7b7 - 4b6 - b4 + 4b3 - 4b2 - 4) * q^32 + (-b7 - 2b5 + b4 - b3 - b2 - 2) q^33 + (-b6 + 3b5 - 2b3 + b2 - 2b1 - 2) q^34 + (4b7 + 3b6 + 2b5 - 3b4 - 6b3 + b1 + 4) q^36 + (-b7 + 2b4) q^37 + (-b6 + b4 + 3b3 + b1) q^38 + (-3b7 + b6 + 3b5 + b4 + 2b3 - 3b2 + 2b1 + 1) q^39 + (-b6 - 3b5 - 4b3 + b2 - b1 - 4) * q^41 + (-4b7 - 5b6 - 5b5 + 2b4 + 7b3 - b2 - b1 - 4) q^42 + (3b7 - 3b5 - b4 + b2 - 2b1 + 1) q^43 + (2b6 + 3b5 - 2b3 - 2b2 + 2b1 - 2) q^44 + (-2b6 - 7b5 + 6b3 + 2b2 - 3b1 + 6) q^46 + (b5 - 3b3 + 4b1 - 3) * q^47 + (5b7 - b6 - 4b5 - 3b4 + 3b3 + 4b2 - b1 + 9) q^48 + (3b7 + 5b6 + 3b5 + b4 - 5b3 + 4b1) q^49 + (-4b7 - 3b6 + b5 - b1 + 2) * q^51 + (2b7 + 3b6 + 5b5 - 3b4 - 8b3 - b2 + 2b1 - 3) * q^52 + (b7 - b5 + 5b4 + 5b2 - 6b1 + 4) q^53 + (-4b7 + 6b5 + 3b3 - 5b2 + 7b1 - 4) q^54 + (-7b7 + 7b5 + 3b4 - 12b2 + 4b1 - 13) q^56 + (2b7 + b5 + b4 - b3 + 2b2 + 1) * q^57 + (-5b6 - 3b5 + 3b3 + 5b2 - 6b1 + 3) q^58 + (-5b7 - 4b6 - 3b4 + 3b3 - 4b2 - 3) q^59 + (5b4 - 5b1 - 1) * q^61 + (-6b7 - 6b6 - 6b5 + b4 + 5b3 - 5b1) q^62 + (7b7 - 2b5 - 3b4 - 3b3 + 4b2 + 1) q^63 + (-4b7 - 7b6 - 4b5 - 2b4 + 8b3 - 6b1) * q^64 + (-4b7 - 2b6 + b5 + 3b4 + 3b3 + b2 + 1) * q^66 + (-3b6 - 8b5 + 2b3 + 3b2 - 9b1 + 2) q^67 + (-2b7 + 2b5 + 3b4 - b2 - b1 - 6) q^68 + (5b7 + 5b6 + b5 - b4 - 2b3 + 1) q^69 + (-b6 + b5 + b3 + b2 + 3b1 + 1) q^71 + (4b7 + 5b6 + 10b5 - 9b3 - 3b2 + 5b1 - 1) * q^72 + (5b7 + 5b6 - 2b4 + 5b2) * q^73 + (-b6 + b4 + 3b3 + b1) q^74 + (-b6 - 3b5 + 4b3 + b2 - b1 + 4) * q^76 + (5b7 + 5b5 - 3b4 - b3 + 2b1) * q^77 + (-5b7 - 5b6 - 2b4 + 5b3 - 7b2 + b1 - 1) q^78 + (-4b7 + 4b5 + b4 - 3b2 + 3b1 + 5) * q^79 + (2b4 + 4b3 + 4b2 + 2b1 + 1) * q^81 + (-2b7 + 2b5 + 4b4 + 5b2 - 2b1 - 3) q^82 + (3b7 + 2b6 + b4 + 4b3 + 2b2 - 4) * q^83 + (-2b7 - 8b6 - 13b5 + 2b4 + 13b3 + 5b2 - 8b1 + 10) q^84 + (3b7 + 4b6 + 2b4 - 5b3 + 4b2 + 5) q^86 + (-b7 + 4b5 - b4 - 5b3 + b2 + b1 + 3) * q^87 + (-6b7 + 6b5 + 4b4 - 3b2 + 2b1 - 2) q^88 + (6b7 + 2b6 - 4b4 - 5b3 + 2b2 + 5) q^89 + (6b7 + 6b6 - 2b5 - b4 - 4b3 + b2 - 2b1 + 3) q^91 + (8b7 - 8b5 - 5b4 + 9b2 - 3b1 + 13) q^92 + (3b7 + 2b6 - 2b5 - 3b4 - 3b3 + b2 + 4) q^93 + (-3b7 + 3b5 - b4 - b2 + 4b1 - 4) q^94 + (7b7 + 4b6 - b5 - 8b4 - 4b3 + 3b2 + 11) q^96 + (-2b6 - 6b5 + 2b3 + 2b2 - 8b1 + 2) q^97 + (8b6 + 15b5 - 7b3 - 8b2 + 13b1 - 7) q^98 + (-2b7 + b6 + 3b4 - 3b3 + 3b2 + 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 6 q^{2} + 2 q^{3} - 4 q^{6} + 14 q^{7} - 12 q^{8} + 4 q^{9} - 4 q^{11} + 2 q^{12} + 14 q^{13} - 8 q^{16} + 30 q^{18} + 2 q^{19} - 8 q^{21} + 34 q^{24} + 32 q^{26} - 10 q^{27} - 24 q^{28} - 12 q^{31}+ \cdots + 28 q^{99}+O(q^{100})$ 8 * q + 6 * q^2 + 2 * q^3 - 4 * q^6 + 14 * q^7 - 12 * q^8 + 4 * q^9 - 4 * q^11 + 2 * q^12 + 14 * q^13 - 8 * q^16 + 30 * q^18 + 2 * q^19 - 8 * q^21 + 34 * q^24 + 32 * q^26 - 10 * q^27 - 24 * q^28 - 12 * q^31 - 18 * q^32 - 18 * q^33 - 10 * q^34 + 28 * q^36 + 2 * q^37 + 20 * q^39 - 38 * q^41 - 34 * q^42 - 4 * q^43 - 10 * q^44 + 34 * q^46 - 22 * q^47 + 54 * q^48 + 26 * q^51 - 18 * q^52 + 28 * q^53 - 12 * q^54 - 76 * q^56 + 6 * q^57 + 18 * q^58 - 14 * q^59 - 8 * q^61 - 10 * q^63 + 18 * q^66 - 40 * q^68 + 10 * q^71 + 4 * q^72 - 10 * q^73 + 26 * q^76 + 2 * q^78 + 56 * q^79 + 8 * q^81 - 16 * q^82 - 38 * q^83 + 58 * q^84 + 34 * q^86 + 34 * q^87 + 8 * q^88 + 28 * q^89 + 8 * q^91 + 72 * q^92 + 22 * q^93 - 20 * q^94 + 72 * q^96 + 4 * q^97 - 26 * q^98 + 28 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$ x^8 - 2*x^5 + 14*x^4 - 8*x^3 + 2*x^2 + 2*x + 1 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 16\nu^{7} + 4\nu^{6} + \nu^{5} + 48\nu^{4} + 236\nu^{3} - 69\nu^{2} - 65\nu + 574 ) / 319$ (16v^7 + 4v^6 + v^5 + 48v^4 + 236v^3 - 69v^2 - 65v + 574) / 319
$\beta_{3}$	$=$	$( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319$ (-63v^7 + 64v^6 + 16v^5 + 130v^4 - 1009v^3 + 1448v^2 - 402*v - 67) / 319
$\beta_{4}$	$=$	$( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 59\nu + 63 ) / 319$ (64v^7 + 16v^6 + 4v^5 - 127v^4 + 944v^3 - 276v^2 + 59*v + 63) / 319
$\beta_{5}$	$=$	$( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1582\nu + 268 ) / 319$ (-67v^7 + 63v^6 - 64v^5 + 118v^4 - 1068v^3 + 1545v^2 - 1582*v + 268) / 319
$\beta_{6}$	$=$	$( 126\nu^{7} - 128\nu^{6} - 32\nu^{5} - 260\nu^{4} + 2018\nu^{3} - 2577\nu^{2} + 804\nu + 134 ) / 319$ (126v^7 - 128v^6 - 32v^5 - 260v^4 + 2018v^3 - 2577v^2 + 804*v + 134) / 319
$\beta_{7}$	$=$	$( -255\nu^{7} + 16\nu^{6} + 4\nu^{5} + 511\nu^{4} - 3522\nu^{3} + 2276\nu^{2} - 579\nu - 575 ) / 319$ (-255v^7 + 16v^6 + 4v^5 + 511v^4 - 3522v^3 + 2276v^2 - 579*v - 575) / 319

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{6} + 2\beta_{3}$ b6 + 2*b3
$\nu^{3}$	$=$	$\beta_{7} + 3\beta_{4} - \beta_{3} + 1$ b7 + 3*b4 - b3 + 1
$\nu^{4}$	$=$	$-\beta_{4} + 4\beta_{2} + \beta _1 - 7$ -b4 + 4*b2 + b1 - 7
$\nu^{5}$	$=$	$\beta_{6} - 4\beta_{5} + 6\beta_{3} - \beta_{2} - 15\beta _1 + 6$ b6 - 4b5 + 6b3 - b2 - 15*b1 + 6
$\nu^{6}$	$=$	$\beta_{7} - 15\beta_{6} + \beta_{5} + 7\beta_{4} - 28\beta_{3} + 8\beta_1$ b7 - 15b6 + b5 + 7b4 - 28b3 + 8b1
$\nu^{7}$	$=$	$-15\beta_{7} + 8\beta_{6} - 43\beta_{4} + 30\beta_{3} + 8\beta_{2} - 30$ -15b7 + 8b6 - 43b4 + 30b3 + 8*b2 - 30

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$ .

$n$	$301$	$326$	$352$
$\chi(n)$	$-\beta_{3}$	$-1$	$-1$

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}

749.1

−0.252709	+	0.252709i
−1.49094	+	1.49094i
1.18254	−	1.18254i
0.561103	−	0.561103i
−0.252709	−	0.252709i
−1.49094	−	1.49094i
1.18254	+	1.18254i
0.561103	+	0.561103i

−0.978559

−

0.978559i

−0.146426

−

1.72585i

−

0.0848427i

−1.54556

1.83213i

2.39913

2.39913i

−2.04014

2.04014i

−2.95712

0.505418i

749.2

0.664640

0.664640i

1.29021

1.15558i

−

1.11651i

0.0894818

1.62557i

2.20073

2.20073i

2.07136

−

2.07136i

0.329281

2.98187i

749.3

1.42282

1.42282i

1.55654

−

0.759725i

2.04882i

3.29562

1.13372i

−0.739083

−

0.739083i

−0.0694623

0.0694623i

1.84564

−

2.36509i

749.4

1.89110

1.89110i

−1.70032

0.329998i

5.15253i

−3.83954

−

2.59143i

3.13922

3.13922i

−5.96175

5.96175i

2.78220

−

1.12221i

824.1