LMFDB - Newform orbit 275.2.h.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,2,Mod(26,275)]

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

lf = mfeigenbasis(mf)

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")

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

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

Self dual:	no
Analytic conductor:	$2.19588605559$
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:	$x^{16} - 4 x^{15} + 10 x^{14} - 13 x^{13} + 53 x^{12} - 12 x^{11} + 136 x^{10} + 8 x^{9} + 300 x^{8} + \cdots + 16$ x^16 - 4x^15 + 10x^14 - 13x^13 + 53x^12 - 12x^11 + 136x^10 + 8x^9 + 300x^8 - 256x^7 + 472x^6 - 48x^5 + 432x^4 + 88x^3 + 128x^2 + 64*x + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$5$
Twist minimal:	yes
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.

$f(q)$	$=$	$q + ( - \beta_{9} + \beta_{5} + \beta_{2}) q^{2} - \beta_{14} q^{3} + (\beta_{11} - \beta_{10} - \beta_{9} + \cdots - 1) q^{4} + ( - \beta_{15} - \beta_{12} - \beta_{10}) q^{6} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots + 1) q^{7}+ \cdots + (\beta_{15} + \beta_{14} + \cdots - 3 \beta_1) q^{99}+O(q^{100})$ q + (-b9 + b5 + b2) * q^2 - b14 * q^3 + (b11 - b10 - b9 + b8 + b7 - b6 - b2 - b1 - 1) * q^4 + (-b15 - b12 - b10) * q^6 + (b9 - b8 - b7 + b6 + 1) * q^7 + (-b15 - b11 - b10 + b9 + b7 - b4 + b3 + 1) * q^8 + (b11 + b10 + b9 + b8 + b7 - b6 - b5 - b3 - b2) * q^9 + (-b13 + b12 - b11 - b8 + b7 - b6 + b5 - b4 + b3 - b1 - 1) * q^11 + (-2b11 + b9 + b6 - b5 + b1) q^12 + (-b11 - b9 - b8 + b4 + b1) * q^13 + (b14 - b11 + b9 - b8 - 2b7 - b5 + b4 + b3 + 1) q^14 + (b15 - 2b14 + 2b13 + b12 + b10 + b8 - b7 + 3b6 + b4 - b3 + b2 + 2b1 + 1) * q^16 + (-b15 - b12 + b10 - b8 - 2b6 - b4 + b3 - b2) q^17 + (-b15 - b13 - b12 + 2b9 - b8 - 2b7 + 3b6 + b5 - b4 + b2 + 3) q^18 + (b15 - 2b9 + 2b7 - 2b4 - 2) q^19 + (-b14 - b13 - b12 - b5 + b3 + b2 + b1 + 1) * q^21 + (-b13 + b12 + b11 - b10 + b9 + 2b8 + b7 + b6 - 2b5 + 2b4 + b2 + 2b1 + 2) * q^22 + (-b14 + b13 - b12 - 2b9 - 2b6 + b5 - b3 + 2b2 - b1) q^23 + (b10 + 3b9 - 5b7 + 5b6 + 3b4 - b3 + 3b1) q^24 + (b15 + b13 - b11 + b10 + b9 - 2b8 - b7 - 2b6 + b5 - b4 - b1 - 2) * q^26 + (b15 - 2b14 + 2b13 + b12 + b7 - b6 - 2b4 - 2b2 - 3b1 - 1) q^27 + (2b15 + 2b12 + b10 - b8 - 4b7 + b6 + b4 + b3 + b2 + b1 + 4) q^28 + (b15 + b13 + b12 - b11 + b10 - 2b9 - 2b8 + 2b7 - 2b6 + 2b5 - 2b4 + b2 - b1 - 2) * q^29 + (-2b15 + 2b14 - 2b13 - b11 + b9 - b8 - 2b7 + 2b6 - b5 - b2) q^31 + (-b14 - b13 - b12 + b11 - b9 - b6 + 3b5 - 2b3 - b2 - 3b1 - 4) q^32 + (b14 + b13 + b10 - b9 + b8 + 4b7 - b6 + b5 + b1 - 4) q^33 + (b14 - b13 + b12 - b11 + 2b5 + 5b2 - 2b1 + 3) q^34 + (-b15 + 2b14 + b11 + b8 + b7 - b5 + 2b4 - b3) * q^36 + (b15 + b13 + 2b11 - 2b10 - b9 + 2b8 + b7 - b6 - b5 + b4 + b1 - 1) q^37 + (b14 - b13 + b8 + 2b7 + 4b6 - b3 - 2) * q^38 + (-2b15 - 2b12 - 2b8 - b7 + b4 + 2b3 + b2 + b1 + 1) * q^39 + (2b15 + b14 + b11 + b9 + b8 + b7 + 2b5 - b4 - b3 + 1) * q^41 + (-2b15 + 2b14 - 2b13 + b11 - b10 - 5b9 + b8 - b7 + b6 + 2b5 + b4 + b3 + 2b2 + b1) * q^42 + (2b14 - b13 + 2b12 + b11 - b9 - b6 - b5 + b1 - 3) * q^43 + (-b14 - b13 + b11 - 5b9 + 2b8 + 3b5 - b4 - b3 + b1 - 2) q^44 + (2b13 - 2b12 - b10 - 4b9 + 3b7 - 3b6 - 2b5 + b3 - 2b2) q^46 + (-b15 + 2b14 - b11 + b10 - b9 - 2b8 + 3b7 + b5 - 3b4 + b3 - 1) * q^47 + (-b12 + 2b9 - 2b7 - 8b6 - 4b2 - 4b1 - 8) q^48 + (-b14 + b13 - 2b10 - b7 - b6 + b1 + 1) q^49 + (b15 + b13 + 2b12 + 2b11 - 2b10 + 2b9 + 2b8 - 2b7 + 4b6 + 4) q^51 + (2b15 + b14 - b11 - b10 + 5b9 - 4b7 - 3b5 + 3b4 + b3 + 5) q^52 + (b15 - b14 + 2b13 - b12 + b11 - b10 + b9 + b8 - 3b7 + 3b6 - 4b5 - b4 + b3 - 4b2 - b1) q^53 + (-2b14 + b13 - 2b12 - b11 + 5b9 + 5b6 - 2b5 - b3 + b2 + 2b1 + 7) * q^54 + (b14 - 2b13 + b12 + 2b11 - 6b9 - 6b6 + 3b5 + b3 + b2 - 3b1 - 4) * q^56 + (2b13 - 2b12 + 3b10 - 3b9 + 4b7 - 4b6 + b5 - b4 - 3b3 + b2 - b1) q^57 + (2b15 + b11 + b8 + 3b7 - 3b5 + 2b4 - b3) * q^58 + (b12 - b11 + b10 - 2b9 + 2b7 - b6 - 4b2 - 4b1 - 1) * q^59 + (-2b15 + b14 - b13 - 2b12 + 2b10 + 2b8 - 3b7 + 2b4 - 2b3 + 2b2 - b1 + 3) * q^61 + (b15 + b13 + 2b12 - 2b11 + 2b10 + 5b9 - 5b7 + 2b6 - b5 + b4 - b2 + 2) * q^62 + (3b11 + 2b10 + 5b9 + b8 + 3b7 - b4 - 3b3 + 5) q^63 + (-b15 + b14 - 2b13 + b12 - 2b11 - 4b10 + 4b9 - 2b8 + 4b7 - 4b6 - 5b5 - b4 + 4b3 - 5b2 - b1) * q^64 + (2b15 - 2b14 + b12 + 2b11 + b10 + b9 + b8 + 4b7 - b4 - 3b3 - 2b2 - 5) * q^66 + (b14 - 2b13 + b12 + 3b11 + 3b9 + 3b6 - 3b5 - 2b3 - 2b2 + 3b1 - 1) * q^67 + (-b15 + b14 - 2b13 + b12 + 4b11 + b10 - 8b9 + 4b8 + 3b7 - 3b6 - 2b5 + 4b4 - b3 - 2b2 + 4b1) * q^68 + (-2b15 - b14 + b11 + 5b9 + b8 - 4b7 - 2b5 + b4 - b3 + 5) * q^69 + (b15 + b12 - b10 - 2b8 + 4b7 - b4 + 2b3 - b2 - 2b1 - 4) * q^71 + (-b15 + b14 - b13 - b12 - b8 - 3b7 + 5b6 - b4 + b3 - b2 + b1 + 3) * q^72 + (2b15 + 2b13 + 4b9 + b8 - 4b7 + 3b6 + b5 - b4 - 3b2 - 4b1 + 3) q^73 + (-b15 - 2b9 + 3b7 + b5 - 4b4 - 2) q^74 + (-2b14 + 2b13 - 2b12 - b9 - b6 - 2b5 - 3b2 + 2b1 - 5) * q^76 + (2b15 - 2b14 + b13 - b11 + 2b10 - b8 - b7 - 2b6 - b5 + b4 - 3) * q^77 + (2b14 + 2b12 - b11 - 5b9 - 5b6 + b5 + 2b3 + 2b2 - b1) * q^78 + (b15 - b14 + 2b13 - b12 + b11 - b10 + 6b9 + b8 - 4b7 + 4b6 - 3b5 + 4b4 + b3 - 3b2 + 4b1) * q^79 + (b15 + b13 + b12 + 5b9 + 3b8 - 5b7 - b6 - 4b5 + 4b4 - 2b2 + 2b1 - 1) q^81 + (2b14 - 2b13 - 2b10 - b8 + 4b7 + b4 + b3 + b2 - b1 - 4) * q^82 + (-b15 - b12 - 2b10 + b8 - b7 - 5b6 - b4 - b3 - b2 - 2b1 + 1) q^83 + (-3b15 - 3b13 - b12 + b11 - b10 - 10b9 + 3b8 + 10b7 - 11b6 + 3b5 - 3b4 - 3b1 - 11) q^84 + (-2b13 + 2b12 + 2b11 + b10 + b9 + 2b8 - b5 - 2b4 - b3 - b2 - 2b1) * q^86 + (-b14 + 3b13 - b12 - 2b11 - 6b9 - 6b6 + b5 + 2b3 + b2 - b1 - 5) q^87 + (-b15 - b13 - 2b12 + b11 - 4b10 - 2b9 + 11b7 - 5b6 + 3b5 - 3b4 - b2 - 4b1 - 9) * q^88 + (-3b14 + b13 - 3b12 + b11 - b9 - b6 + 2b5 - b3 + 2b2 - 2b1) q^89 + (-2b15 - b11 - b10 - 4b9 + 3b7 + b5 - 2b4 + b3 - 4) * q^91 + (b12 - 2b11 + 2b10 + 3b9 - b8 - 3b7 + 3b6 + 2b5 - 2b4 + 4b2 + 2b1 + 3) q^92 + (-2b15 + 2b14 - 2b13 - 2b12 - 3b8 - 3b7 + 10b6 + b4 + 3b3 + b2 + 3b1 + 3) q^93 + (3b15 + 3b12 + 2b10 + 3b8 + b7 + 5b6 + 3b4 - 3b3 + 3b2 + b1 - 1) * q^94 + (b14 - 3b11 + 8b9 - 3b8 + 4b7 - 4b5 - 2b4 + 3b3 + 8) q^96 + (2b15 - 2b14 + 2b12 + b10 + 2b9 - 4b7 + 4b6 + b5 + b4 - b3 + b2 + b1) * q^97 + (-b14 + 3b13 - b12 - 2b11 + b9 + b6 - 2b3 + 3b2 - 1) * q^98 + (b15 + b14 + 2b13 + 2b12 - 2b11 - b10 - 3b9 - 2b8 - 4b7 + 3b6 + 3b5 - 3b4 + 2b3 - b2 - 3b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 2 q^{3} - 2 q^{4} - 3 q^{6} + 4 q^{7} + 16 q^{8} + 8 q^{9} - 5 q^{11} - 6 q^{12} + 7 q^{13} + 3 q^{14} - 4 q^{16} + 12 q^{17} + 16 q^{18} - 13 q^{19} + 10 q^{21} + 28 q^{22} - 4 q^{23} - 43 q^{24}+ \cdots - 10 q^{99}+O(q^{100})$ 16 * q - 2 * q^3 - 2 * q^4 - 3 * q^6 + 4 * q^7 + 16 * q^8 + 8 * q^9 - 5 * q^11 - 6 * q^12 + 7 * q^13 + 3 * q^14 - 4 * q^16 + 12 * q^17 + 16 * q^18 - 13 * q^19 + 10 * q^21 + 28 * q^22 - 4 * q^23 - 43 * q^24 - 34 * q^26 - 11 * q^27 + 47 * q^28 - 11 * q^29 - 10 * q^31 - 58 * q^32 - 34 * q^33 + 20 * q^34 + 3 * q^36 - 4 * q^37 - 36 * q^38 + 3 * q^39 + 25 * q^41 + 13 * q^42 - 28 * q^43 - q^44 + 40 * q^46 + 8 * q^47 - 106 * q^48 + 16 * q^49 + 35 * q^51 + 39 * q^52 - 22 * q^53 + 60 * q^54 - 20 * q^56 + 29 * q^57 + 6 * q^58 + 14 * q^59 + 16 * q^61 - 10 * q^62 + 73 * q^63 + 40 * q^64 - 55 * q^66 - 14 * q^67 + 83 * q^68 + 35 * q^69 - 46 * q^71 + 28 * q^72 + 7 * q^73 - 7 * q^74 - 62 * q^76 - 51 * q^77 + 34 * q^78 - 39 * q^79 - 43 * q^81 - 51 * q^82 + 28 * q^83 - 54 * q^84 + 2 * q^86 - 50 * q^87 - 76 * q^88 - 22 * q^89 - 34 * q^91 + 4 * q^92 + 3 * q^93 - 40 * q^94 + 108 * q^96 - 39 * q^97 - 52 * q^98 - 10 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} - 4 x^{15} + 10 x^{14} - 13 x^{13} + 53 x^{12} - 12 x^{11} + 136 x^{10} + 8 x^{9} + 300 x^{8} + \cdots + 16$ x^16 - 4*x^15 + 10*x^14 - 13*x^13 + 53*x^12 - 12*x^11 + 136*x^10 + 8*x^9 + 300*x^8 - 256*x^7 + 472*x^6 - 48*x^5 + 432*x^4 + 88*x^3 + 128*x^2 + 64*x + 16 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 26449018002 \nu^{15} - 236888516211 \nu^{14} + 788099178184 \nu^{13} - 1526842653922 \nu^{12} + \cdots + 6985303901728 ) / 25527087093000$ (26449018002v^15 - 236888516211v^14 + 788099178184v^13 - 1526842653922v^12 + 2602718491889v^11 - 6054632584655v^10 + 3736928471102v^9 - 11607551777402v^8 + 6418403294948v^7 - 31698255486024v^6 + 51848203982080v^5 - 31663186749736v^4 - 11824148418152v^3 - 11014842153296v^2 - 4678805932160*v + 6985303901728) / 25527087093000
$\beta_{3}$	$=$	$( 71383474437 \nu^{15} - 289509004196 \nu^{14} + 724686737314 \nu^{13} + \cdots - 26659476408112 ) / 25527087093000$ (71383474437v^15 - 289509004196v^14 + 724686737314v^13 - 880677312057v^12 + 3525945524039v^11 - 363156186260v^10 + 8887770758132v^9 + 2620938590578v^8 + 22607070780658v^7 - 13448782299344v^6 + 45541863076240v^5 + 9122657485544v^4 + 26729723248488v^3 + 16332507782304v^2 + 4415319120360*v - 26659476408112) / 25527087093000
$\beta_{4}$	$=$	$( - 104750151572 \nu^{15} + 673159972921 \nu^{14} - 2076349151074 \nu^{13} + \cdots + 2568824593792 ) / 25527087093000$ (-104750151572v^15 + 673159972921v^14 - 2076349151074v^13 + 3882890610742v^12 - 8663636557279v^11 + 14046913582755v^10 - 16746047375422v^9 + 29867025397622v^8 - 28381829490778v^7 + 93228468272964v^6 - 117638176881180v^5 + 106020546909796v^4 - 46024126428728v^3 + 58756715833256v^2 + 112672695560*v + 2568824593792) / 25527087093000
$\beta_{5}$	$=$	$( 64382245392 \nu^{15} - 136843233291 \nu^{14} + 153040707434 \nu^{13} + 333330036813 \nu^{12} + \cdots + 3970336916928 ) / 12763543546500$ (64382245392v^15 - 136843233291v^14 + 153040707434v^13 + 333330036813v^12 + 2058624775804v^11 + 4994405745285v^10 + 7807405615657v^9 + 13525078631538v^8 + 20679882229273v^7 + 11776548752296v^6 - 2491029501600v^5 + 38692712106364v^4 + 41906388720708v^3 + 29054194141544v^2 + 22146165625480*v + 3970336916928) / 12763543546500
$\beta_{6}$	$=$	$( - 160551537112 \nu^{15} + 537455996876 \nu^{14} - 932355398199 \nu^{13} + \cdots - 10162625679608 ) / 25527087093000$ (-160551537112v^15 + 537455996876v^14 - 932355398199v^13 + 10820831382v^12 - 4626340856194v^11 - 6737018111935v^10 - 7788095464477v^9 - 18030459672318v^8 - 18298435735978v^7 + 12719364009894v^6 + 17448142756100v^5 - 109931703099804v^4 + 36662282877412v^3 - 60152661694584v^2 + 38206119082920*v - 10162625679608) / 25527087093000
$\beta_{7}$	$=$	$( 335740577504 \nu^{15} - 1576476952372 \nu^{14} + 4304252214543 \nu^{13} + \cdots + 2834825497856 ) / 25527087093000$ (335740577504v^15 - 1576476952372v^14 + 4304252214543v^13 - 6747058073494v^12 + 21010481144828v^11 - 16809773038935v^10 + 49718820632729v^9 - 29674933986704v^8 + 103539041385746v^7 - 155691181790348v^6 + 228144923350260v^5 - 128771665598172v^4 + 173675052178796v^3 - 100291733049792v^2 + 43623121470680*v + 2834825497856) / 25527087093000
$\beta_{8}$	$=$	$( 169847619014 \nu^{15} - 638541728247 \nu^{14} + 1347321043353 \nu^{13} + \cdots - 8155768916224 ) / 12763543546500$ (169847619014v^15 - 638541728247v^14 + 1347321043353v^13 - 1028723196929v^12 + 6551150466593v^11 + 2629230531170v^10 + 12826968334994v^9 + 9453041104671v^8 + 28120030785466v^7 - 30526727810468v^6 + 23020958288750v^5 + 65689289665038v^4 - 1044461698464v^3 + 40415838627548v^2 - 18305565983140*v - 8155768916224) / 12763543546500
$\beta_{9}$	$=$	$( - 436581493858 \nu^{15} + 1772774993434 \nu^{14} - 4602703454791 \nu^{13} + \cdots - 32620021539072 ) / 25527087093000$ (-436581493858v^15 + 1772774993434v^14 - 4602703454791v^13 + 6463658598338v^12 - 24665661828396v^11 + 7841696418185v^10 - 65429715749343v^9 + 244276520238v^8 - 142581999934802v^7 + 118183265722596v^6 - 237764720587000v^5 + 72804115687264v^4 - 220266392096392v^3 - 50243319877656v^2 - 66897273367120*v - 32620021539072) / 25527087093000
$\beta_{10}$	$=$	$( 336455799554 \nu^{15} - 1405015474407 \nu^{14} + 3422183399388 \nu^{13} + \cdots - 6722455741504 ) / 12763543546500$ (336455799554v^15 - 1405015474407v^14 + 3422183399388v^13 - 4226220696494v^12 + 16839370278338v^11 - 5128034670295v^10 + 37992335976644v^9 - 3879946115949v^8 + 82597360451161v^7 - 104765990080148v^6 + 136018734259430v^5 + 4245458870298v^4 + 84700075330896v^3 - 17409829184032v^2 + 3284062557620*v - 6722455741504) / 12763543546500
$\beta_{11}$	$=$	$( - 784893186521 \nu^{15} + 3081039362473 \nu^{14} - 7454798472622 \nu^{13} + \cdots - 17909140090624 ) / 25527087093000$ (-784893186521v^15 + 3081039362473v^14 - 7454798472622v^13 + 8976897755181v^12 - 39074125561792v^11 + 3643508756785v^10 - 96378856115026v^9 - 17086544837664v^8 - 207570274117534v^7 + 185463996144352v^6 - 292774471048580v^5 - 29304998311712v^4 - 227244860243104v^3 - 134650637824912v^2 - 34023036048200*v - 17909140090624) / 25527087093000
$\beta_{12}$	$=$	$( - 825910307819 \nu^{15} + 3905169355947 \nu^{14} - 10779625022808 \nu^{13} + \cdots - 50705453705536 ) / 25527087093000$ (-825910307819v^15 + 3905169355947v^14 - 10779625022808v^13 + 16856514063109v^12 - 51345010395338v^11 + 40063718726465v^10 - 122216715017714v^9 + 60489618732604v^8 - 258153923168176v^7 + 354020688923328v^6 - 585912863151520v^5 + 266139300217932v^4 - 357424992722756v^3 + 69990320325232v^2 - 156868685551200*v - 50705453705536) / 25527087093000
$\beta_{13}$	$=$	$( 420153483844 \nu^{15} - 1344158135822 \nu^{14} + 2796519364033 \nu^{13} + \cdots + 30173885523636 ) / 12763543546500$ (420153483844v^15 - 1344158135822v^14 + 2796519364033v^13 - 2039811890584v^12 + 18041913947238v^11 + 11797528472210v^10 + 52012839132489v^9 + 41353563847396v^8 + 122166099037251v^7 - 24961931412903v^6 + 93546454294220v^5 + 115851367034918v^4 + 185751763890906v^3 + 121673581909168v^2 + 36369816748000*v + 30173885523636) / 12763543546500
$\beta_{14}$	$=$	$( 509735557264 \nu^{15} - 1869094610042 \nu^{14} + 4458813844393 \nu^{13} + \cdots + 14317509681756 ) / 12763543546500$ (509735557264v^15 - 1869094610042v^14 + 4458813844393v^13 - 5279241201079v^12 + 25987261338063v^11 + 434323779425v^10 + 71953266319074v^9 + 16904852793106v^8 + 162373708283871v^7 - 102372271874118v^6 + 210068455218140v^5 - 1446348459922v^4 + 285895050403086v^3 + 43812267340768v^2 + 105661989957340*v + 14317509681756) / 12763543546500
$\beta_{15}$	$=$	$( 1298485402504 \nu^{15} - 6312051157617 \nu^{14} + 17598840879833 \nu^{13} + \cdots + 5900772630536 ) / 25527087093000$ (1298485402504v^15 - 6312051157617v^14 + 17598840879833v^13 - 28484833417244v^12 + 84191619087023v^11 - 75232390015330v^10 + 195752053900259v^9 - 136321708178294v^8 + 399125050354776v^7 - 648937196489948v^6 + 946827630665600v^5 - 581097322310132v^4 + 654528357159496v^3 - 284467218881472v^2 + 147206829318760*v + 5900772630536) / 25527087093000

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{10} - \beta_{8} - 2\beta_{7} + \beta_{5} - 2\beta_{4}$ b10 - b8 - 2b7 + b5 - 2b4
$\nu^{3}$	$=$	$- \beta_{15} - \beta_{13} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} - 2 \beta_{7} + \cdots + 1$ -b15 - b13 - 2b11 + 2b10 + 2b9 - 3b8 - 2b7 + b6 + 8b5 - 8b4 + 3b2 - 5*b1 + 1
$\nu^{4}$	$=$	$- \beta_{15} + \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} + 3 \beta_{10} + 11 \beta_{9} + \cdots - 13 \beta_1$ -b15 + b14 - 3b13 + 2b12 - 9b11 + 3b10 + 11b9 - 9b8 - 3b7 + 3b6 + 23b5 - 13b4 - 3b3 + 23b2 - 13*b1
$\nu^{5}$	$=$	$9 \beta_{14} - 12 \beta_{13} + 9 \beta_{12} - 14 \beta_{11} + 14 \beta_{9} + 14 \beta_{6} + 42 \beta_{5} + \cdots - 11$ 9b14 - 12b13 + 9b12 - 14b11 + 14b9 + 14b6 + 42b5 - 25b3 + 79b2 - 42b1 - 11
$\nu^{6}$	$=$	$14 \beta_{15} + 25 \beta_{14} - 25 \beta_{13} + 14 \beta_{12} - 45 \beta_{10} + 88 \beta_{8} + 45 \beta_{7} + \cdots - 45$ 14b15 + 25b14 - 25b13 + 14b12 - 45b10 + 88b8 + 45b7 + 49b6 + 149b4 - 88b3 + 149b2 - 101b1 - 45
$\nu^{7}$	$=$	$88 \beta_{15} + 45 \beta_{14} + 163 \beta_{11} - 275 \beta_{10} - 165 \beta_{9} + 438 \beta_{8} + \cdots - 165$ 88b15 + 45b14 + 163b11 - 275b10 - 165b9 + 438b8 + 279b7 - 489b5 + 831b4 - 163b3 - 165
$\nu^{8}$	$=$	$275 \beta_{15} + 275 \beta_{13} - 163 \beta_{12} + 919 \beta_{11} - 919 \beta_{10} - 949 \beta_{9} + \cdots - 409$ 275b15 + 275b13 - 163b12 + 919b11 - 919b10 - 949b9 + 1453b8 + 949b7 - 409b6 - 2705b5 + 2705b4 - 1648b2 + 1057*b1 - 409
$\nu^{9}$	$=$	$534 \beta_{15} - 534 \beta_{14} + 1453 \beta_{13} - 919 \beta_{12} + 2980 \beta_{11} - 1811 \beta_{10} + \cdots + 5427 \beta_1$ 534b15 - 534b14 + 1453b13 - 919b12 + 2980b11 - 1811b10 - 3038b9 + 2980b8 + 1843b7 - 1843b6 - 8926b5 + 5427b4 + 1811b3 - 8926b2 + 5427*b1
$\nu^{10}$	$=$	$- 2980 \beta_{14} + 4791 \beta_{13} - 2980 \beta_{12} + 5961 \beta_{11} - 6063 \beta_{9} - 6063 \beta_{6} + \cdots + 4018$ -2980b14 + 4791b13 - 2980b12 + 5961b11 - 6063b9 - 6063b6 - 18007b5 + 9845b3 - 29297b2 + 18007b1 + 4018
$\nu^{11}$	$=$	$- 5961 \beta_{15} - 9845 \beta_{14} + 9845 \beta_{13} - 5961 \beta_{12} + 19818 \beta_{10} + \cdots + 20208$ -5961b15 - 9845b14 + 9845b13 - 5961b12 + 19818b10 - 32277b8 - 20208b7 - 12735b6 - 59328b4 + 32277b3 - 59328b2 + 37199b1 + 20208
$\nu^{12}$	$=$	$- 32277 \beta_{15} - 19818 \beta_{14} - 65289 \beta_{11} + 106372 \beta_{10} + 66561 \beta_{9} + \cdots + 66561$ -32277b15 - 19818b14 - 65289b11 + 106372b10 + 66561b9 - 171661b8 - 108682b7 + 195881b5 - 317602b4 + 65289b3 + 66561
$\nu^{13}$	$=$	$- 106372 \beta_{15} - 106372 \beta_{13} + 65289 \beta_{12} - 349879 \beta_{11} + 349879 \beta_{10} + \cdots + 137070$ -106372b15 - 106372b13 + 65289b12 - 349879b11 + 349879b10 + 357171b9 - 565578b8 - 357171b7 + 137070b6 + 1046139b5 - 1046139b4 + 645333b2 - 400806*b1 + 137070
$\nu^{14}$	$=$	$- 215699 \beta_{15} + 215699 \beta_{14} - 565578 \beta_{13} + 349879 \beta_{12} - 1152511 \beta_{11} + \cdots - 2127272 \beta_1$ -215699b15 + 215699b14 - 565578b13 + 349879b12 - 1152511b11 + 710622b10 + 1176821b9 - 1152511b8 - 725088b7 + 725088b6 + 3444661b5 - 2127272b4 - 710622b3 + 3444661b2 - 2127272*b1
$\nu^{15}$	$=$	$1152511 \beta_{14} - 1863133 \beta_{13} + 1152511 \beta_{12} - 2342971 \beta_{11} + 2391411 \beta_{9} + \cdots - 1482267$ 1152511b14 - 1863133b13 + 1152511b12 - 2342971b11 + 2391411b9 + 2391411b6 + 7007643b5 - 3794540b3 + 11345926b2 - 7007643b1 - 1482267

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

Character values

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

$n$	$101$	$177$
$\chi(n)$	$-1 - \beta_{6} + \beta_{7} - \beta_{9}$	$1$

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

2.66477	−	1.93607i
0.977523	−	0.710212i
−0.260198	+	0.189045i
−1.26407	+	0.918397i
0.545543	−	1.67901i
0.207669	−	0.639141i
−0.265939	+	0.818476i
−0.605307	+	1.86294i
2.66477	+	1.93607i
0.977523	+	0.710212i
−0.260198	−	0.189045i
−1.26407	−	0.918397i
0.545543	+	1.67901i
0.207669	+	0.639141i
−0.265939	−	0.818476i
−0.605307	−	1.86294i

−1.85576

−

1.34829i

0.0131041

−

0.0403304i

1.00792

3.10207i

−0.0786950

0.0571753i

−0.352180

−

1.08390i

0.894346

−

2.75251i

2.42560

1.76230i

26.2

−0.168506

−

0.122427i

−0.585361

1.80155i

−0.604628

−

1.86085i

0.319195

−

0.231909i

0.398265

1.22573i

−0.254662

0.783769i

−0.475901

−

0.345762i

26.3

1.06921

0.776830i

0.626199

−

1.92724i

−0.0782786

−

0.240917i

2.16668

−

1.57419i

−0.107618

−

0.331213i

0.920262

−

2.83228i

−0.895086

−

0.650318i

26.4

2.07308

1.50618i

−0.553942

1.70486i

1.41105

4.34277i

−3.71620

2.69998i

−1.17454

−

3.61485i

−2.03208

6.25410i

−0.172643

−

0.125433i

126.1

−0.854560

−

2.63006i

−1.32800

−

0.964848i

−4.56893

3.31953i

−1.40276

4.31724i

−1.53545

1.11557i

8.16046

5.92892i

−0.0944009

−

0.290536i

126.2

−0.516686

−

1.59020i

2.51599

1.82798i

−0.643729

0.467696i

1.60686

−

4.94542i

2.49985

−

1.81624i

−1.62907

−

1.18359i

2.06168

6.34519i

126.3

−0.0430779

−

0.132580i

−2.41558

−

1.75502i

1.60231

−

1.16415i

−0.128623

0.395861i

2.88786

−

2.09815i

−0.448926

−

0.326164i

1.82787

5.62561i

126.4

0.296290

0.911888i

0.727583

0.528620i

0.874283

−

0.635204i

−0.266466

0.820099i

−0.616184

0.447684i

2.38967

1.73620i

−0.677113

−

2.08394i

201.1