LMFDB - Newform orbit 961.2.d.q

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [961,2,Mod(374,961)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(961, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([8]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("961.374");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$961 = 31^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	961.d (of order $5$ , degree $4$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$7.67362363425$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{5})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1$ x^16 + 19x^14 + 140x^12 + 511x^10 + 979x^8 + 956x^6 + 410x^4 + 44*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$5^{2}$
Twist minimal:	no (minimal twist has level 31)
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_{14} + \beta_{12} + \beta_{11} + 1) q^{2} + (\beta_{13} - \beta_{11} - \beta_{8} + \cdots + 1) q^{3} + (\beta_{15} + 2 \beta_{11} + \cdots - \beta_{2}) q^{4} + (\beta_{15} - \beta_{14} + \cdots - 2 \beta_{3}) q^{5}+ \cdots + (\beta_{15} - \beta_{14} + 2 \beta_{13} + \cdots + 3) q^{99}+O(q^{100})$ q + (b14 + b12 + b11 + 1) * q^2 + (b13 - b11 - b8 + b6 + b5 + b4 + 1) * q^3 + (b15 + 2b11 - b3 - b2) q^4 + (b15 - b14 + 2b11 + b9 - b7 - b5 - b4 - 2b3) * q^5 + (-b15 + b14 + b11 - b9 - 2b7 - 2b5 - b4 + b3 + 2) * q^6 + (b15 + b14 + b13 + 2b12 + 2b11 + b10 + 2b9 + b6 - b3 + b2 - b1 + 2) q^7 + (-2b13 - b12 - b9 - b6 + b5 - b4 - b2 + b1 - 2) q^8 + (b15 + b11 + b10 + b9 + b6 + b4 - b3) * q^9 + (-3b12 - 2b11 + b10 + b9 + b7 - b6 + b5 - b3 + b2 - 1) * q^10 + (-b15 - b12 + b11 - b9 + b3) * q^11 + (2b14 + b12 + b11 - b10 - b9 - b8 + b7 - b5 + b4 + b3 - 2b2) * q^12 + (b13 - b12 - b11 - 2b9 - b7 + b4 + 2b2 - b1) * q^13 + (-b14 + 2b13 - 3b12 - 2b9 + b6 + b4 - b3 + 2b2 - 2b1 - 1) q^14 + (-3b12 - 2b11 - 2b10 - 2b9 + 2b7 - b5 - 2b2 - b1 - 4) * q^15 + (b15 - 2b13 + b12 + b5 - 2b4 - 2b2 + 3b1 - 2) * q^16 + (-b15 - 2b14 - b13 - 3b12 - b11 - b10 + b9 - b6 - b5 - b4 - b3 - b2 + b1 - 4) * q^17 + (b13 - 2b12 - 3b11 + b10 - 3b9 - b8 - b7 + b6 + b5 + b4 + b2 - 1) q^18 + (b14 + b11 + b10 + b9 + b8 - 2b7 - b4 - b3 + b2 + 1) q^19 + (b13 - 2b12 - 2b11 - 4b9 - b8 - 2b7 + b6 + b1 + 1) * q^20 + (b14 + b12 + 3b11 + 2b8 - 2b7 + b6 - 3b5 - 2b4 - b2) q^21 + (2b15 + b14 + 2b12 + 2b11 + 2b10 + 2b9 - b5 - b3 + 2) q^22 + (-b14 - b13 - 2b12 + 3b9 - b6 - b4 - b3 - 2) * q^23 + (b15 - b14 - b13 - b12 + 4b11 - b10 - 2b8 + b7 - b6 - b3 - 3b2 - 2) q^24 + (b15 - b14 - b13 - 3b11 - 3b9 + b8 - b3 - 1) * q^25 + (2b13 - b9 - 2b8 - b7 - b5 + 3b4 + 2b3 - 2b1 + 3) q^26 + (b15 - b13 + 2b11 - b9 - b7 - b6 - b3 + b2 + 2b1 - 1) * q^27 + (b13 - b12 + b11 + b10 + b9 - b8 - 2b7 + b6 - 2b5 + b4 + 2b2 - 3b1 + 4) * q^28 + (b14 + b12 - b11 + 2b8 - b7 - 2b4 + 2b2 - 1) q^29 + (2b15 - 2b14 - b13 + 5b11 + 6b9 + b8 + b7 + b5 - 2b3 - 3b1 + 3) * q^30 + (-b13 - 3b11 + b9 + b8 + 4b7 + 4b5 - 2b4 + b1 - 4) * q^32 + (-b14 - b12 - 2b11 + b10 + b9 - b8 + 3b7 - 2b6 + 2b5 + b4 - b3 + b2) * q^33 + (-2b15 - b13 - 3b12 - 3b11 + b10 + 4b9 + b8 - b6 - b4 + b1 + 3) * q^34 + (-b15 + b14 - b13 - 4b12 - 4b11 + b10 - b9 + 3b7 - b6 + b3 + b2 - 2b1) * q^35 + (-b15 + b14 - b13 - 3b11 - 5b9 + b8 - 2b7 - 2b5 + 3b3 + 3b1 - 2) * q^36 + (3b15 - 3b14 - b13 - b11 + b8 + b7 + b5 + b4 - 3b3 + b1 - 4) q^37 + (-2b15 + b14 - b13 - 4b12 - 3b11 + b10 - 4b9 - b8 - b6 + 2b3 + b1) q^38 + (-b14 - 2b12 + b9 - b6 - b4 - b3 + 3b2 - 3b1 + 1) q^39 + (2b14 + b13 - 2b6 - b5 - 2b4 + 2b3 - b2 + b1 - 1) * q^40 + (-2b12 + b11 + b10 + b9 - b7 + 2b6 - b3 + 1) * q^41 + (b15 - b14 - 2b13 - 3b12 - 2b11 - b10 + 2b9 + 5b7 - 2b6 - b3 + 2b2 - 3b1 - 3) * q^42 + (-b14 - 2b12 + b11 + 2b10 + 2b9 + 2b8 + 2b7 - 2b6 - b5 - 2b4 - 2b3 + b2) * q^43 + (b13 + 2b12 + b10 - b9 + 2b7 + 2b5 + b4 - b2 + b1 - 2) q^44 + (b14 - b13 - b12 - 3b9 + b6 - 4b5 + b4 + b3 - 2b2 + 2b1 - 3) * q^45 + (-3b15 - 4b12 - 6b11 + 2b10 + 2b5 + 2b1) * q^46 + (-b15 + b12 + b11 - b10 + 2b9 + 2b8 + 2b7 - 2b6) * q^47 + (b15 + b14 - b13 + b11 + b10 - b9 + b5 - 5b2 + 5b1 - 5) * q^48 + (-b15 + b13 + b12 + 2b11 - 2b9 - b5 + b4 + b2 - 2b1 - 1) q^49 + (-b14 + 3b12 - 4b11 - 3b10 - 3b9 - b8 + b6 + 3b5 + b4 + 3b3 + 2b2 - 1) q^50 + (-2b14 - 3b13 - 3b12 - 2b11 - 2b10 + b9 + 2b8 + 4b7 - 3b6 - b1 - 5) * q^51 + (b14 + b12 + b11 - b10 - b9 - 2b8 - 3b6 + 2b4 + b3 - 2b2 + 1) * q^52 + (-2b15 - b14 - b13 - 2b11 - 2b10 - b9 - b6 - b5 - b4 + b3 - 5b2 + 5b1 - 5) q^53 + (2b15 + b14 + b13 - b12 + 2b11 + 2b10 - b9 + b5 - b3 - 2b2 + 2b1 - 3) q^54 + (-2b15 + 2b14 + b13 + 3b12 + 5b11 + 2b10 + 3b9 + 2b8 + b6 + 2b3 - b2 - b1 + 3) * q^55 + (3b13 + b11 + 2b9 - 3b8 + b7 + b5 + 3b4 + 6) * q^56 + (-b13 + 4b11 + 3b9 + b8 - b7 - b5 - 2b4 - 3b3 - 3b1 - 1) q^57 + (-2b15 - 2b14 + b13 - 3b12 - 4b11 - 2b10 - b9 + b8 + 2b7 + b6 + 2b3 + 4b2 - 3b1 - 1) q^58 + (-2b15 - 3b13 - b12 + b10 - b9 + 2b8 - 4b7 - 2b6 - b5 - 3b4 + 3b2 + 2b1) * q^59 + (-7b12 - 6b11 + 2b10 + 2b9 - 3b8 + b7 + 2b6 + 3b5 + 3b4 - 2b3 - 3) q^60 + (-b15 + b14 + 2b13 - 2b11 - b9 - 2b8 + b7 + b5 + b3 - 2b1 + 5) * q^61 + (-b15 + b14 + b13 - 3b11 - 4b9 - b8 - b7 - b5 + 2b4 + 2b3 - 3) * q^63 + (-b14 + 2b12 + b10 + b9 + 2b8 - 4b7 + 5b6 + b5 - 2b4 - b3 + 3b2 + 1) * q^64 + (2b15 + 2b13 + b12 - b11 - b10 - 2b9 + b8 - b6 + 2b5 + 2b4 + 3b2) * q^65 + (b14 + 2b13 + 2b12 + 3b11 + b10 - 2b9 + 2b8 - 4b7 + 2b6 - b2 + 2b1 + 3) * q^66 + (-3b15 + 3b14 - 2b11 - 2b9 + b4 + b3 + b1 - 1) * q^67 + (-3b15 + 3b14 + b13 - b11 - 2b9 - b8 - b7 - b5 + b4 + 2b3 + 2b1 + 5) q^68 + (b15 - 2b14 - b13 + 4b11 - 2b10 + 3b9 + 2b8 + 3b7 - b6 - b3 - 2b2 - 2b1 - 3) * q^69 + (2b14 + 5b13 + 5b12 - 5b9 + 4b6 + 3b5 + 4b4 + 2b3 + b2 - b1 + 6) * q^70 + (-2b15 - 2b14 - 2b13 - b12 - 2b11 - 2b10 - b9 - b6 - 2b5 - b4 - 2b2 + 2b1 - 3) * q^71 + (b14 - 2b11 - 3b10 - 3b9 + 2b7 + b6 + 3b3 - 2) q^72 + (5b13 + 2b12 + 3b11 - 2b8 - 2b7 + 5b6 - 3b1 + 5) q^73 + (-3b14 - 2b12 - 8b11 - 2b7 + 2b6 + 4b5 + 4b2 - 4) q^74 + (b15 - b11 - 2b10 - 5b9 - 3b8 - 3b7 + 3b6 + b5 + b1 - 2) q^75 + (-b15 + 3b14 - b13 - b11 - b10 - 7b9 - 2b6 - b5 - 2b4 + 4b3 - 2b2 + 2b1 - 2) q^76 + (4b15 + 2b13 + 3b12 + 3b11 + 2b10 + 5b9 + 2b4 + 2b2 - 2b1 + 7) q^77 + (-b15 + 5b13 - 3b11 + 2b10 + b9 - 2b8 + 2b6 + 3b5 + 5b4 + 3b2 - 2b1 + 6) q^78 + (4b14 + 6b13 + 3b12 - b9 + 4b6 + 3b5 + 4b4 + 4b3 + 4b2 - 4b1 + 7) q^79 + (-4b13 - 6b12 - 3b11 - b10 - b9 + 2b8 + b7 - 2b6 - 3b5 - 4b4 - 3b2 + b1 - 6) * q^80 + (-4b14 - 5b12 - b11 - b8 - 2b6 + b4 - b2 - 1) q^81 + (-b15 + 3b14 + b13 - 2b12 - 4b11 + 3b10 - 3b9 - b8 - b7 + b6 + b3 + 3b2 + 4) * q^82 + (-2b14 + 2b12 - 4b11 + b10 + b9 - b8 + b7 + 3b6 - 2b5 + b4 - b3 - 3b2 - 6) * q^83 + (-b15 - 3b14 + 3b13 - 4b12 - b11 - b10 + 3b6 + 2b5 + 3b4 - 2b3 + 2b2 - 2b1 - 2) q^84 + (-2b15 + 2b14 - b13 + 8b12 - 2b11 - 2b10 - 3b9 + 3b6 + 3b5 + 3b4 + 4b3 - 2b2 + 2b1 + 6) * q^85 + (b15 + 2b14 + b13 - b12 - b11 + 2b10 + 4b8 + b7 + b6 - b3 + 4b2 - 2b1 + 3) q^86 + (-b15 + b14 - 2b13 - 2b9 + 2b8 - 2b7 - 2b5 - 5b4 - b1 - 3) * q^87 + (b15 - b14 - 3b13 + b11 - 2b9 + 3b8 - 3b7 - 3b5 - 4b4 - 3b3 + 3b1 - 6) * q^88 + (2b15 + 2b14 + b13 + 5b12 + 8b11 + 2b10 + 3b9 - 3b8 - 2b7 + b6 - 2b3 - 2b2 + b1 + 3) * q^89 + (3b15 + 2b13 + b12 + b11 - 3b10 - 2b9 + 4b7 + 2b4 - 2b2 - 2b1 - 4) * q^90 + (-5b14 - b12 - 5b11 - 3b10 - 3b9 - 2b8 + b7 - 2b6 + 2b4 + 3b3 - 2b2 - 5) q^91 + (-4b15 + 4b14 - 2b11 - 5b9 - 3b7 - 3b5 - 2b4 + 6b3 + 2b1 + 5) q^92 + (b15 - b14 - 2b13 + 10b11 + 9b9 + 2b8 - b7 - b5 - 4b3 - 3b1 + 6) * q^94 + (b14 - b11 - 4b10 - 4b9 + b7 - 2b6 + 2b5 + 4b3 + 2b2 + 1) * q^95 + (4b15 - 4b13 + 4b12 + 6b11 + 3b9 + 2b7 - 2b5 - 4b4 - 6b2 + 2b1 - 3) * q^96 + (b15 - 5b14 - b13 + 3b11 - 5b10 - b9 - 2b8 - b7 - b6 - b3 - b2 + 2b1 - 6) q^97 + (4b15 - 4b14 + 2b13 + 5b11 + 4b9 - 2b8 - b7 - b5 + 4b4 - 2b3 - 3b1 + 3) q^98 + (b15 - b14 + 2b13 + b11 + 2b9 - 2b8 + b7 + b5 + b3 - 4b1 + 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 4 q^{2} + 6 q^{3} + 6 q^{4} + 6 q^{5} + 22 q^{6} - 9 q^{7} - 8 q^{8} - 10 q^{9} - 6 q^{10} - 4 q^{11} + 5 q^{12} - 9 q^{13} - 18 q^{14} - 4 q^{15} - 2 q^{16} - 17 q^{17} - 14 q^{18} - 7 q^{19} + 36 q^{20}+ \cdots + 12 q^{99}+O(q^{100})$ 16 * q + 4 * q^2 + 6 * q^3 + 6 * q^4 + 6 * q^5 + 22 * q^6 - 9 * q^7 - 8 * q^8 - 10 * q^9 - 6 * q^10 - 4 * q^11 + 5 * q^12 - 9 * q^13 - 18 * q^14 - 4 * q^15 - 2 * q^16 - 17 * q^17 - 14 * q^18 - 7 * q^19 + 36 * q^20 - 2 * q^21 + 8 * q^22 - 21 * q^23 - 5 * q^24 + 26 * q^25 + 18 * q^26 - 9 * q^27 + 20 * q^28 - 26 * q^29 + 22 * q^30 - 42 * q^32 + 7 * q^33 + 56 * q^34 + 21 * q^35 - 2 * q^36 - 16 * q^37 + 24 * q^38 + 2 * q^39 - 13 * q^40 + 6 * q^41 - 12 * q^42 + 16 * q^43 - 37 * q^44 - 5 * q^45 + 16 * q^46 + 4 * q^47 - 37 * q^48 - 39 * q^49 - 21 * q^50 - 11 * q^51 + 18 * q^52 - 3 * q^53 - 39 * q^54 - 29 * q^55 + 60 * q^56 - 34 * q^57 - 10 * q^58 - 3 * q^59 - 35 * q^60 + 60 * q^61 - 46 * q^63 - 32 * q^64 - 9 * q^65 + 20 * q^66 - 26 * q^67 + 60 * q^68 - 21 * q^69 + 27 * q^70 + 18 * q^71 - 4 * q^72 + 9 * q^73 - 64 * q^74 - 19 * q^75 + 9 * q^76 + 42 * q^77 + 15 * q^78 - 14 * q^79 + 18 * q^80 + 29 * q^81 + 32 * q^82 - 67 * q^83 - 39 * q^84 + 63 * q^85 + 23 * q^86 - 30 * q^87 - 34 * q^88 - 26 * q^89 - 24 * q^90 - 8 * q^91 + 64 * q^92 + 44 * q^94 + 28 * q^95 - 4 * q^96 - 37 * q^97 + 20 * q^98 + 12 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1$ x^16 + 19*x^14 + 140*x^12 + 511*x^10 + 979*x^8 + 956*x^6 + 410*x^4 + 44*x^2 + 1 :

$\beta_{1}$	$=$	$( -4\nu^{14} - 65\nu^{12} - 358\nu^{10} - 641\nu^{8} + 691\nu^{6} + 3382\nu^{4} + 2839\nu^{2} + 255 ) / 93$ (-4v^14 - 65v^12 - 358v^10 - 641v^8 + 691v^6 + 3382v^4 + 2839*v^2 + 255) / 93
$\beta_{2}$	$=$	$( - 15 \nu^{15} - 4 \nu^{14} - 267 \nu^{13} - 65 \nu^{12} - 1792 \nu^{11} - 358 \nu^{10} - 5744 \nu^{9} + \cdots + 162 ) / 186$ (-15v^15 - 4v^14 - 267v^13 - 65v^12 - 1792v^11 - 358v^10 - 5744v^9 - 641v^8 - 9398v^7 + 691v^6 - 8413v^5 + 3382v^4 - 4846v^3 + 2839v^2 - 1485*v + 162) / 186
$\beta_{3}$	$=$	$( 13\nu^{14} + 219\nu^{12} + 1334\nu^{10} + 3517\nu^{8} + 3497\nu^{6} - 33\nu^{4} - 1035\nu^{2} + 140 ) / 93$ (13v^14 + 219v^12 + 1334v^10 + 3517v^8 + 3497v^6 - 33v^4 - 1035*v^2 + 140) / 93
$\beta_{4}$	$=$	$( -13\nu^{14} - 219\nu^{12} - 1334\nu^{10} - 3517\nu^{8} - 3497\nu^{6} + 33\nu^{4} + 1128\nu^{2} + 46 ) / 93$ (-13v^14 - 219v^12 - 1334v^10 - 3517v^8 - 3497v^6 + 33v^4 + 1128*v^2 + 46) / 93
$\beta_{5}$	$=$	$( - 8 \nu^{15} + 32 \nu^{14} - 192 \nu^{13} + 582 \nu^{12} - 1832 \nu^{11} + 4011 \nu^{10} - 8815 \nu^{9} + \cdots + 99 ) / 186$ (-8v^15 + 32v^14 - 192v^13 + 582v^12 - 1832v^11 + 4011v^10 - 8815v^9 + 13157v^8 - 22302v^7 + 21039v^6 - 27739v^5 + 14763v^4 - 13108v^3 + 3235v^2 - 172*v + 99) / 186
$\beta_{6}$	$=$	$( - 21 \nu^{15} + 43 \nu^{14} - 380 \nu^{13} + 753 \nu^{12} - 2608 \nu^{11} + 4887 \nu^{10} - 8581 \nu^{9} + \cdots - 52 ) / 186$ (-21v^15 + 43v^14 - 380v^13 + 753v^12 - 2608v^11 + 4887v^10 - 8581v^9 + 14478v^8 - 14174v^7 + 19069v^6 - 11369v^5 + 8206v^4 - 3765v^3 - 860v^2 - 33*v - 52) / 186
$\beta_{7}$	$=$	$( - 42 \nu^{15} + 49 \nu^{14} - 791 \nu^{13} + 897 \nu^{12} - 5743 \nu^{11} + 6261 \nu^{10} + \cdots + 573 ) / 186$ (-42v^15 + 49v^14 - 791v^13 + 897v^12 - 5743v^11 + 6261v^10 - 20448v^9 + 21097v^8 - 37617v^7 + 35904v^6 - 34611v^5 + 29514v^4 - 14009v^3 + 9839v^2 - 1616*v + 573) / 186
$\beta_{8}$	$=$	$( - 63 \nu^{15} + 34 \nu^{14} - 1140 \nu^{13} + 599 \nu^{12} - 7793 \nu^{11} + 3911 \nu^{10} + \cdots - 168 ) / 186$ (-63v^15 + 34v^14 - 1140v^13 + 599v^12 - 7793v^11 + 3911v^10 - 25216v^9 + 11602v^8 - 39298v^7 + 14881v^6 - 25706v^5 + 4857v^4 - 3080v^3 - 2571v^2 + 1389*v - 168) / 186
$\beta_{9}$	$=$	$( 50 \nu^{15} - 25 \nu^{14} + 983 \nu^{13} - 445 \nu^{12} + 7575 \nu^{11} - 2966 \nu^{10} + 29263 \nu^{9} + \cdots + 36 ) / 186$ (50v^15 - 25v^14 + 983v^13 - 445v^12 + 7575v^11 - 2966v^10 + 29263v^9 - 9222v^8 + 59919v^7 - 13483v^6 + 62350v^5 - 7987v^4 + 27117v^3 - 926v^2 + 1788*v + 36) / 186
$\beta_{10}$	$=$	$( 28 \nu^{15} + 38 \nu^{14} + 517 \nu^{13} + 664 \nu^{12} + 3653 \nu^{11} + 4300 \nu^{10} + 12516 \nu^{9} + \cdots + 11 ) / 186$ (28v^15 + 38v^14 + 517v^13 + 664v^12 + 3653v^11 + 4300v^10 + 12516v^9 + 12739v^8 + 21730v^7 + 16980v^6 + 18145v^5 + 7954v^4 + 6074v^3 - 109v^2 + 261*v + 11) / 186
$\beta_{11}$	$=$	$( - 50 \nu^{15} - 25 \nu^{14} - 983 \nu^{13} - 445 \nu^{12} - 7575 \nu^{11} - 2966 \nu^{10} - 29263 \nu^{9} + \cdots + 36 ) / 186$ (-50v^15 - 25v^14 - 983v^13 - 445v^12 - 7575v^11 - 2966v^10 - 29263v^9 - 9222v^8 - 59919v^7 - 13483v^6 - 62350v^5 - 7987v^4 - 27117v^3 - 926v^2 - 1788*v + 36) / 186
$\beta_{12}$	$=$	$( 59 \nu^{15} + 25 \nu^{14} + 1106 \nu^{13} + 445 \nu^{12} + 7993 \nu^{11} + 2966 \nu^{10} + 28357 \nu^{9} + \cdots - 129 ) / 186$ (59v^15 + 25v^14 + 1106v^13 + 445v^12 + 7993v^11 + 2966v^10 + 28357v^9 + 9222v^8 + 52048v^7 + 13483v^6 + 47440v^5 + 7987v^4 + 17792v^3 + 926v^2 + 1129*v - 129) / 186
$\beta_{13}$	$=$	$( - 63 \nu^{15} - 34 \nu^{14} - 1140 \nu^{13} - 599 \nu^{12} - 7793 \nu^{11} - 3911 \nu^{10} + \cdots + 168 ) / 186$ (-63v^15 - 34v^14 - 1140v^13 - 599v^12 - 7793v^11 - 3911v^10 - 25216v^9 - 11602v^8 - 39298v^7 - 14881v^6 - 25706v^5 - 4857v^4 - 3080v^3 + 2571v^2 + 1389*v + 168) / 186
$\beta_{14}$	$=$	$( 124 \nu^{15} - 23 \nu^{14} + 2325 \nu^{13} - 397 \nu^{12} + 16802 \nu^{11} - 2508 \nu^{10} + \cdots - 231 ) / 186$ (124v^15 - 23v^14 + 2325v^13 - 397v^12 + 16802v^11 - 2508v^10 + 59582v^9 - 7026v^8 + 109306v^7 - 8016v^6 + 99851v^5 - 1556v^4 + 37882v^3 + 1483v^2 + 2232*v - 231) / 186
$\beta_{15}$	$=$	$( 124 \nu^{15} + 23 \nu^{14} + 2325 \nu^{13} + 397 \nu^{12} + 16802 \nu^{11} + 2508 \nu^{10} + \cdots + 231 ) / 186$ (124v^15 + 23v^14 + 2325v^13 + 397v^12 + 16802v^11 + 2508v^10 + 59582v^9 + 7026v^8 + 109306v^7 + 8016v^6 + 99851v^5 + 1556v^4 + 37882v^3 - 1483v^2 + 2232*v + 231) / 186

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

$\nu$	$=$	$( 2 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} + \beta_{11} - 2 \beta_{10} - 3 \beta_{9} - \beta_{8} + \cdots + 2 ) / 5$ (2b15 + 2b14 + 3b13 + b11 - 2b10 - 3b9 - b8 + 4b6 + 2b4 + b3 + 2b2 - 3*b1 + 2) / 5
$\nu^{2}$	$=$	$\beta_{4} + \beta_{3} - 2$ b4 + b3 - 2
$\nu^{3}$	$=$	$( - 9 \beta_{15} - 9 \beta_{14} - 11 \beta_{13} + 12 \beta_{12} + 4 \beta_{10} + 13 \beta_{9} + 7 \beta_{8} + \cdots - 2 ) / 5$ (-9b15 - 9b14 - 11b13 + 12b12 + 4b10 + 13b9 + 7b8 - 3b7 - 18b6 + 3b5 - 9b4 - 2b3 - 2b2 + 13b1 - 2) / 5
$\nu^{4}$	$=$	$\beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + 2 \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + \cdots + 9$ b15 - b14 + b13 + b11 + 2b9 - b8 + b7 + b5 - 6b4 - 6b3 - 2b1 + 9
$\nu^{5}$	$=$	$( 42 \beta_{15} + 42 \beta_{14} + 53 \beta_{13} - 82 \beta_{12} - 11 \beta_{11} - 12 \beta_{10} - 60 \beta_{9} + \cdots - 5 ) / 5$ (42b15 + 42b14 + 53b13 - 82b12 - 11b11 - 12b10 - 60b9 - 51b8 + 23b7 + 104b6 - 23b5 + 52b4 + 6b3 - 20b2 - 65*b1 - 5) / 5
$\nu^{6}$	$=$	$- 8 \beta_{15} + 8 \beta_{14} - 11 \beta_{13} - 6 \beta_{11} - 15 \beta_{9} + 11 \beta_{8} - 9 \beta_{7} + \cdots - 50$ -8b15 + 8b14 - 11b13 - 6b11 - 15b9 + 11b8 - 9b7 - 9b5 + 33b4 + 33b3 + 22*b1 - 50
$\nu^{7}$	$=$	$( - 203 \beta_{15} - 203 \beta_{14} - 272 \beta_{13} + 500 \beta_{12} + 111 \beta_{11} + 68 \beta_{10} + \cdots + 97 ) / 5$ (-203b15 - 203b14 - 272b13 + 500b12 + 111b11 + 68b10 + 297b9 + 354b8 - 160b7 - 626b6 + 160b5 - 313b4 - 34b3 + 252b2 + 347*b1 + 97) / 5
$\nu^{8}$	$=$	$56 \beta_{15} - 56 \beta_{14} + 92 \beta_{13} + 25 \beta_{11} + 89 \beta_{9} - 92 \beta_{8} + 64 \beta_{7} + \cdots + 296$ 56b15 - 56b14 + 92b13 + 25b11 + 89b9 - 92b8 + 64b7 + 64b5 - 181b4 - 184b3 - 178*b1 + 296
$\nu^{9}$	$=$	$( 1016 \beta_{15} + 1016 \beta_{14} + 1439 \beta_{13} - 3038 \beta_{12} - 915 \beta_{11} - 516 \beta_{10} + \cdots - 937 ) / 5$ (1016b15 + 1016b14 + 1439b13 - 3038b12 - 915b11 - 516b10 - 1567b9 - 2373b8 + 1072b7 + 3812b6 - 1072b5 + 1906b4 + 258b3 - 2132b2 - 1912*b1 - 937) / 5
$\nu^{10}$	$=$	$- 385 \beta_{15} + 385 \beta_{14} - 688 \beta_{13} - 72 \beta_{11} - 495 \beta_{9} + 688 \beta_{8} + \cdots - 1792$ -385b15 + 385b14 - 688b13 - 72b11 - 495b9 + 688b8 - 423b7 - 423b5 + 1009b4 + 1060b3 + 1289*b1 - 1792
$\nu^{11}$	$=$	$( - 5253 \beta_{15} - 5253 \beta_{14} - 7797 \beta_{13} + 18688 \beta_{12} + 6944 \beta_{11} + \cdots + 7585 ) / 5$ (-5253b15 - 5253b14 - 7797b13 + 18688b12 + 6944b11 + 3988b10 + 8710b9 + 15599b8 - 7022b7 - 23396b6 + 7022b5 - 11698b4 - 1994b3 + 15890b2 + 10775*b1 + 7585) / 5
$\nu^{12}$	$=$	$2624 \beta_{15} - 2624 \beta_{14} + 4853 \beta_{13} - 9 \beta_{11} + 2712 \beta_{9} - 4853 \beta_{8} + \cdots + 10977$ 2624b15 - 2624b14 + 4853b13 - 9b11 + 2712b9 - 4853b8 + 2721b7 + 2721b5 - 5739b4 - 6294b3 - 8863*b1 + 10977
$\nu^{13}$	$=$	$( 27997 \beta_{15} + 27997 \beta_{14} + 43223 \beta_{13} - 116270 \beta_{12} - 50099 \beta_{11} + \cdots - 56363 ) / 5$ (27997b15 + 27997b14 + 43223b13 - 116270b12 - 50099b11 - 29572b10 - 50358b9 - 101401b8 + 45385b7 + 144624b6 - 45385b5 + 72312b4 + 14786b3 - 111508b2 - 61943*b1 - 56363) / 5
$\nu^{14}$	$=$	$- 17693 \beta_{15} + 17693 \beta_{14} - 33083 \beta_{13} + 2393 \beta_{11} - 14930 \beta_{9} + \cdots - 67810$ -17693b15 + 17693b14 - 33083b13 + 2393b11 - 14930b9 + 33083b8 - 17323b7 - 17323b5 + 33296b4 + 38231b3 + 59269*b1 - 67810
$\nu^{15}$	$=$	$( - 153509 \beta_{15} - 153509 \beta_{14} - 244981 \beta_{13} + 729212 \beta_{12} + 349095 \beta_{11} + \cdots + 398363 ) / 5$ (-153509b15 - 153509b14 - 244981b13 + 729212b12 + 349095b11 + 210574b10 + 299543b9 + 654727b8 - 291148b7 - 899708b6 + 291148b5 - 449854b4 - 105287b3 + 756648b2 + 362678*b1 + 398363) / 5

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/961\mathbb{Z}\right)^\times$ .

$n$	$3$
$\chi(n)$	$\beta_{9}$

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}

374.1

		0.333129i
	−	0.176392i
		2.16544i
	−	1.14660i
	−	0.333129i
		0.176392i
	−	2.16544i
		1.14660i
	−	1.83925i
	−	2.52368i
		1.03739i
		1.42343i
		1.83925i
		2.52368i
	−	1.03739i
	−	1.42343i

−1.67638

−

1.21796i

−1.73080

1.25750i

0.708788

2.18143i

2.34791

4.43308

−1.13550

−

3.49470i

0.188054

−

0.578772i

0.487319

−

1.49981i

−3.93600

−

2.85967i

374.2

−0.996848

−

0.724253i

1.67869

−

1.21964i

−0.148869

−

0.458173i

−1.54562

−2.55673

−1.17543

−

3.61759i

−0.944957

2.90828i

0.403428

−

1.24162i

1.54075

1.11942i

374.3

1.49685

1.08752i

0.403986

−

0.293513i

0.439812

1.35360i

1.20736

0.923909

1.15357

3.55033i

0.329747

−

1.01486i

−0.849996

2.61602i

1.80724

1.31303i

374.4

2.17638

1.58123i

1.14813

−

0.834164i

1.61830

4.98062i

0.608384

3.81777

−0.533635

−

1.64236i

−2.69088

8.28167i

−0.304682

0.937716i

1.32408

0.961997i

388.1

−1.67638

1.21796i

−1.73080

−

1.25750i

0.708788

−

2.18143i

2.34791

4.43308

−1.13550

3.49470i

0.188054

0.578772i

0.487319

1.49981i

−3.93600

2.85967i

388.2

−0.996848

0.724253i

1.67869

1.21964i

−0.148869

0.458173i

−1.54562

−2.55673

−1.17543

3.61759i

−0.944957

−

2.90828i

0.403428

1.24162i

1.54075

−

1.11942i

388.3

1.49685

−

1.08752i

0.403986

0.293513i

0.439812

−

1.35360i

1.20736

0.923909

1.15357

−

3.55033i

0.329747

1.01486i

−0.849996

−

2.61602i

1.80724

−

1.31303i

388.4

2.17638

−

1.58123i

1.14813

0.834164i

1.61830

−

4.98062i

0.608384

3.81777

−0.533635

1.64236i

−2.69088

−

8.28167i

−0.304682

−

0.937716i

1.32408

−

0.961997i

531.1

−0.213065

−

0.655747i

0.278857

−

0.858234i

1.23343

−

0.896137i

3.70752

−0.622199

−0.617599

0.448712i

−1.96606

−

1.42843i

1.76825

1.28471i

−0.789942

−

2.43119i

531.2

0.108599

0.334232i

0.894076

−

2.75168i

1.51812

−

1.10298i

2.97323

1.01680

0.875458

−

0.636058i

1.10215

0.800755i

−4.34534

−

3.15708i

0.322889

0.993749i

531.3

0.391401

1.20461i

−0.458679

1.41167i

0.320145

−

0.232599i

−3.80032

−1.88004

−1.77093

1.28666i

2.45490

1.78359i

0.644632

0.468353i

−1.48745

−

4.57790i

531.4

0.713065

2.19459i

0.785745

−

2.41827i

−2.68972

1.95420i

−2.49846

5.86740

−1.29595

0.941560i

−2.47295

−

1.79670i

−2.80360

−

2.03694i

−1.78156

−

5.48309i

628.1