LMFDB - Newform orbit 927.2.f.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [927,2,Mod(46,927)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(927, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("927.46");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 927 = 3^{2} \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	927.f (of order $3$, degree $2$, 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:	$7.40213226737$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
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} - x^{15} + 10 x^{14} - 5 x^{13} + 64 x^{12} - 26 x^{11} + 216 x^{10} - 9 x^{9} + 477 x^{8} + \cdots + 1 $ x^16 - x^15 + 10x^14 - 5x^13 + 64x^12 - 26x^11 + 216x^10 - 9x^9 + 477x^8 - 17x^7 + 545x^6 + 120x^5 + 359x^4 - 16x^3 + 23x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 309)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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_1 q^{2} + (\beta_{13} - \beta_{3}) q^{4} + (\beta_{15} + \beta_{14} + \cdots + \beta_{2}) q^{5}+ \cdots + \beta_{12} q^{8}+O(q^{10}) $ q - b1 * q^2 + (b13 - b3) * q^4 + (b15 + b14 - b12 + b2) * q^5 + (b11 + b5) * q^7 + b12 * q^8	$ q - \beta_1 q^{2} + (\beta_{13} - \beta_{3}) q^{4} + (\beta_{15} + \beta_{14} + \cdots + \beta_{2}) q^{5}+ \cdots + (2 \beta_{14} + 2 \beta_{12} + \cdots - 1) q^{98}+O(q^{100}) $ q - b1 * q^2 + (b13 - b3) * q^4 + (b15 + b14 - b12 + b2) * q^5 + (b11 + b5) * q^7 + b12 * q^8 + (b7 + b3 + 1) * q^10 + (b15 - b10 - b7 - b6 - b1) * q^11 + (-b14 - b12 - b7 + b6 - b4) * q^13 + (b14 + b12 + b4) * q^14 + (b13 - b11) * q^16 + (b15 + b11 + b10 - b9 + b7 - b6 + b5 - b1 - 1) * q^17 + (b11 - b8 - b4) * q^19 + (b10 + b8 - 2b1) q^20 + (-b15 - 2b14 + b12 + b8 + b7 + b6 - 2b4 - b3 - 1) * q^22 + (-b14 + b12 + b6 + b5 - b4 - 1) * q^23 + (b10 + 2b4 - 2b2) * q^25 + (2b13 + 2b11 - 2b9 + b8 - 2b4 + b2 - b1) * q^26 + (-b13 - b10) * q^28 + (b10 + b2) * q^29 + (b15 + b12 - b8 - b6 + 2b3) q^31 + (-b14 + 2b12 - 2b6 - 2b2 - 2b1) * q^32 + (b15 + 3b14 - b8 - b7 - 2b6 + 3b4 - b3 - 1) q^34 + (b11 - b9 - 2b4 + b2) q^35 + (-2b15 - b14 + b12 + 2b8 + b6 - b4) * q^37 + (b14 + b12 + b11 + b10 - b9 + b7 + b5 - b2 - 1) * q^38 + (b15 + 2b14 - b12 + b10 - b9 + b7 - b6 + b2 - b1 - 1) q^40 + (-2b11 + b10 + 2b9 - b8 + 2b4 - 2b2 + b1) * q^41 + (b11 + b10 + 2b8 + 2b2) * q^43 + (b11 + b10 - b9 + b8 + 2b4 + 2b1) * q^44 + (b10 - 2b9 + b4 + b2 + b1) q^46 + (-b15 + b14 - 2b13 - b12 + b10 + b7 + 2b3 + b2) * q^47 + (2b11 - 2b9 + b8 - 2b4 + b2 - b1) q^49 + (b15 + 2b14 + 2b13 - b12 - 3b10 - 3b7 - b6 - 2b3 + b2 - b1) q^50 + (2b12 + b11 - b9 - 4b6 + b5 - 2b2 - 4b1 - 1) * q^52 + (b15 + 2b13 - 2b12 - 3b6 - 2b3 + 2b2 - 3b1) * q^53 + (-b11 + 2b10 + 2b9 + b8 + b2 - 3b1) q^55 + (-b15 + 2b12 + b10 + b7 + 3b6 - 2b2 + 3b1) * q^56 + (b15 + 2b14 - b13 - b12 - b11 - b10 - b7 - b6 - b5 + b3 + b2 - b1) q^58 + (-2b13 + b11 - 2b10 - b9 - b8 + 4b4 + 3b1) * q^59 + (b15 + b14 - b8 - b7 + 2b6 + b5 + b4 - 1) q^61 + (-2b13 - b11 + b9 - 2b2 - 4b1) q^62 + (-b7 + 3b5 - 2b3 - 4) * q^64 + (-b15 - 2b14 - 2b13 - b12 - b11 + b10 + b9 + b7 - b6 - b5 + 2b3 + b2 - b1 + 1) q^65 + (b15 - 3b14 - 2b13 + 2b12 - b11 - b5 + 2b3 - 2b2) q^67 + (-2b13 - b11 - 2b10 + b9 + 3b8 - 2b4 + 2b2) q^68 + (b14 - b13 + b12 + b11 + 2b10 + 2b7 - b6 + b5 + b3 - b2 - b1) * q^70 + (-4b15 - 3b14 + 2b13 + 2b12 - b11 - b10 + b9 - b7 + 3b6 - b5 - 2b3 - 2b2 + 3b1 + 1) * q^71 + (-b15 + 2b14 - b12 + b8 + 2b7 + 3b6 - 2b5 + 2b4 + 2b3 + 2) * q^73 + b10 * q^74 + (3b15 + b14 - 3b8 - 2b6 + b4 - b3) q^76 + (2b13 - b10 - 2b8 + 2b4 + b2 + 2b1) * q^77 + (b15 + 3b12 - b8 + b7 + b6 - 2b3 + 2) * q^79 + (-b15 + 2b14 - b12 + b8 - b7 + 2b6 + b5 + 2b4 - 1) q^80 + (b15 + b13 - 3b12 - 3b10 + b9 - 3b7 + b6 - b3 + 3b2 + b1 + 1) * q^82 + (-2b11 + 2b10 + 2b9 + b8 - 2b4 + 2b2 + 3b1) * q^83 + (4b13 - 2b10 + b9 + 2b8 - 3b4 + b2 + b1) * q^85 + (b15 + 3b14 - 2b13 - 2b11 - b10 + 2b9 - b7 - b6 - 2b5 + 2b3 - b1 + 2) * q^86 + (-b15 - b14 + 2b12 - 2b11 - b10 + 3b9 - b7 - 2b5 - 2b2 + 3) q^88 + (2b15 + b14 - b12 - 2b8 + b7 - b6 + b5 + b4 + 4b3 - 1) q^89 + (-2b14 + 2b13 - 2b12 - b11 - 3b10 + b9 - 3b7 - 4b6 - b5 - 2b3 + 2b2 - 4b1 + 1) q^91 + (b15 - 2b13 + b12 - 2b10 - 2b7 - b6 + 2b3 - b2 - b1) * q^92 + (b15 + 2b14 - 3b12 - b8 + 3b6 + 2b4 + b3 - 1) * q^94 + (3b15 - 2b12 - 3b8 - b6 - 3b5 - 2b3 + 3) q^95 + (-4b13 + b11 - 2b9 + 2b8 + 2b2) * q^97 + (2b14 + 2b12 + b11 + 2b10 - b9 + 2b7 - 2b6 + b5 - 2b2 - 2b1 - 1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - q^{2} - 3 q^{4} - q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) $ 16 * q - q^2 - 3 * q^4 - q^5 + 4 * q^7 + 6 * q^8	$ 16 q - q^{2} - 3 q^{4} - q^{5} + 4 q^{7} + 6 q^{8} + 24 q^{10} + 2 q^{11} - 10 q^{13} + 6 q^{14} + 7 q^{16} - 2 q^{19} - 5 q^{20} - 20 q^{22} - 4 q^{23} - 7 q^{25} + 14 q^{26} - 2 q^{28} + 2 q^{29} + 24 q^{31} + 8 q^{32} - 16 q^{34} + 7 q^{35} - 4 q^{37} - 7 q^{40} - 12 q^{41} - 3 q^{43} + 3 q^{44} + 19 q^{46} + 2 q^{47} + 8 q^{49} - 9 q^{50} + 6 q^{52} - 7 q^{53} - 16 q^{55} + 2 q^{56} - 2 q^{58} + 5 q^{59} - 10 q^{61} - 20 q^{62} - 54 q^{64} + 7 q^{65} + 10 q^{67} - 8 q^{68} + 13 q^{70} - 8 q^{71} + 16 q^{73} - q^{74} + 10 q^{76} + 16 q^{77} + 42 q^{79} - 24 q^{80} - 6 q^{82} - 3 q^{83} + 6 q^{85} + 16 q^{86} + 19 q^{88} + 22 q^{89} - 7 q^{91} + 10 q^{92} - 30 q^{94} + 14 q^{95} + 2 q^{97} + 6 q^{98}+O(q^{100}) $ 16 * q - q^2 - 3 * q^4 - q^5 + 4 * q^7 + 6 * q^8 + 24 * q^10 + 2 * q^11 - 10 * q^13 + 6 * q^14 + 7 * q^16 - 2 * q^19 - 5 * q^20 - 20 * q^22 - 4 * q^23 - 7 * q^25 + 14 * q^26 - 2 * q^28 + 2 * q^29 + 24 * q^31 + 8 * q^32 - 16 * q^34 + 7 * q^35 - 4 * q^37 - 7 * q^40 - 12 * q^41 - 3 * q^43 + 3 * q^44 + 19 * q^46 + 2 * q^47 + 8 * q^49 - 9 * q^50 + 6 * q^52 - 7 * q^53 - 16 * q^55 + 2 * q^56 - 2 * q^58 + 5 * q^59 - 10 * q^61 - 20 * q^62 - 54 * q^64 + 7 * q^65 + 10 * q^67 - 8 * q^68 + 13 * q^70 - 8 * q^71 + 16 * q^73 - q^74 + 10 * q^76 + 16 * q^77 + 42 * q^79 - 24 * q^80 - 6 * q^82 - 3 * q^83 + 6 * q^85 + 16 * q^86 + 19 * q^88 + 22 * q^89 - 7 * q^91 + 10 * q^92 - 30 * q^94 + 14 * q^95 + 2 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - x^{15} + 10 x^{14} - 5 x^{13} + 64 x^{12} - 26 x^{11} + 216 x^{10} - 9 x^{9} + 477 x^{8} + \cdots + 1 $ x^16 - x^15 + 10*x^14 - 5*x^13 + 64*x^12 - 26*x^11 + 216*x^10 - 9*x^9 + 477*x^8 - 17*x^7 + 545*x^6 + 120*x^5 + 359*x^4 - 16*x^3 + 23*x^2 + 2*x + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 141303024491 \nu^{15} + 5150947484 \nu^{14} + 1139284233334 \nu^{13} + 736364216107 \nu^{12} + \cdots + 410088315885 ) / 58572456733912 $ (141303024491v^15 + 5150947484v^14 + 1139284233334v^13 + 736364216107v^12 + 7289029468243v^11 + 5008228343361v^10 + 19737717148489v^9 + 25907782812018v^8 + 45955039631125v^7 + 46872994708778v^6 + 11115846738189v^5 + 61872126250865v^4 - 10447180118202v^3 + 8710167258542v^2 - 230804678436261*v + 410088315885) / 58572456733912
$\beta_{3}$	$=$	$ ( - 410088315885 \nu^{15} + 551391340376 \nu^{14} - 4095732211366 \nu^{13} + \cdots - 114479941600207 ) / 58572456733912 $ (-410088315885v^15 + 551391340376v^14 - 4095732211366v^13 + 3189725812759v^12 - 25509288000533v^11 + 17951325681253v^10 - 83570847887799v^9 + 23428511991454v^8 - 169704343865127v^7 + 52926541001170v^6 - 176625137448547v^5 - 38094751168011v^4 - 85349579151850v^3 - 3885767064042v^2 - 721864006813*v - 114479941600207) / 58572456733912
$\beta_{4}$	$=$	$ ( - 65502373395 \nu^{15} - 840482156 \nu^{14} - 528510105518 \nu^{13} - 338839824579 \nu^{12} + \cdots - 210370538485 ) / 5324768793992 $ (-65502373395v^15 - 840482156v^14 - 528510105518v^13 - 338839824579v^12 - 3341658539931v^11 - 2326802013273v^10 - 8936171417057v^9 - 12260299851378v^8 - 19302332154629v^7 - 22394210621658v^6 - 4228917692325v^5 - 29966072474233v^4 + 5464058020138v^3 - 4456367678430v^2 + 30087096397781*v - 210370538485) / 5324768793992
$\beta_{5}$	$=$	$ ( - 951993803725 \nu^{15} + 1241605345152 \nu^{14} - 9517890859134 \nu^{13} + \cdots - 51980678577115 ) / 29286228366956 $ (-951993803725v^15 + 1241605345152v^14 - 9517890859134v^13 + 7097132482307v^12 - 59432166511269v^11 + 39486446132349v^10 - 195337364703279v^9 + 47848028453094v^8 - 399623286600255v^7 + 99369201698222v^6 - 419104961965395v^5 - 96538132508307v^4 - 207888440054426v^3 - 9596828192058v^2 - 1740918883581*v - 51980678577115) / 29286228366956
$\beta_{6}$	$=$	$ ( 2664971867617 \nu^{15} - 2254883551732 \nu^{14} + 26098327335794 \nu^{13} + \cdots + 6051807742047 ) / 58572456733912 $ (2664971867617v^15 - 2254883551732v^14 + 26098327335794v^13 - 9229127126719v^12 + 167368473714729v^11 - 43779980557509v^10 + 557682597724019v^9 + 59586101079246v^8 + 1247763068861855v^7 + 124399822115638v^6 + 1399483126850095v^5 + 496421761562587v^4 + 994819651642514v^3 + 42710029269978v^2 + 6607663285321*v + 6051807742047) / 58572456733912
$\beta_{7}$	$=$	$ ( 1742896835455 \nu^{15} - 2074541331552 \nu^{14} + 17476449477546 \nu^{13} + \cdots + 44409672047737 ) / 29286228366956 $ (1742896835455v^15 - 2074541331552v^14 + 17476449477546v^13 - 11403217623041v^12 + 109913709572487v^11 - 60989214751407v^10 + 364511378269365v^9 - 53794055274210v^8 + 760910879820261v^7 - 89752148240470v^6 + 814233084375201v^5 + 218629913542017v^4 + 431024130558782v^3 + 20548592575902v^2 + 3524064103311*v + 44409672047737) / 29286228366956
$\beta_{8}$	$=$	$ ( 5495952198773 \nu^{15} - 11547759940820 \nu^{14} + 63676301597394 \nu^{13} + \cdots - 85489644399557 ) / 58572456733912 $ (5495952198773v^15 - 11547759940820v^14 + 63676301597394v^13 - 90252721966067v^12 + 408098306767501v^11 - 539439579785825v^10 + 1511841149942919v^9 - 1400434022628618v^8 + 3233718066217163v^7 - 2920557379256314v^6 + 4345937748807939v^5 - 2514321133447217v^4 + 2646313037163962v^3 - 1764112453012654v^2 + 1218055476087045*v - 85489644399557) / 58572456733912
$\beta_{9}$	$=$	$ ( 6051807742047 \nu^{15} - 8716779609664 \nu^{14} + 62772960972202 \nu^{13} + \cdots - 53076504535139 ) / 58572456733912 $ (6051807742047v^15 - 8716779609664v^14 + 62772960972202v^13 - 56357366046029v^12 + 396544822617727v^11 - 324715475007951v^10 + 1350970452839661v^9 - 612148867402442v^8 + 2827126191877173v^7 - 1350643800476654v^6 + 3173835397299977v^5 - 673266197804455v^4 + 1676177217832286v^3 - 1091648575515266v^2 + 96481548797103*v - 53076504535139) / 58572456733912
$\beta_{10}$	$=$	$ ( 1935399107537 \nu^{15} - 2904770157974 \nu^{14} + 20455068760914 \nu^{13} + \cdots - 19241402641523 ) / 14643114183478 $ (1935399107537v^15 - 2904770157974v^14 + 20455068760914v^13 - 19699626453989v^12 + 130062904543345v^11 - 114833257693157v^10 + 451097651444223v^9 - 240974455425684v^8 + 956142554331399v^7 - 527430092243878v^6 + 1107781299550611v^5 - 350960936506025v^4 + 604696674179090v^3 - 396059267651086v^2 + 46228615853009*v - 19241402641523) / 14643114183478
$\beta_{11}$	$=$	$ ( - 10053173904669 \nu^{15} + 14676602517448 \nu^{14} - 105067260887574 \nu^{13} + \cdots + 92828149732193 ) / 58572456733912 $ (-10053173904669v^15 + 14676602517448v^14 - 105067260887574v^13 + 96766103028263v^12 - 665543205232789v^11 + 559674321609637v^10 - 2284086666240327v^9 + 1107155174847614v^8 - 4805730664428711v^7 + 2436654895947458v^6 - 5464545107357219v^5 + 1478433694715053v^4 - 2925606539905322v^3 + 1909863702681182v^2 - 189353777560141*v + 92828149732193) / 58572456733912
$\beta_{12}$	$=$	$ ( 2664971867617 \nu^{15} - 2254883551732 \nu^{14} + 26098327335794 \nu^{13} + \cdots + 6051807742047 ) / 14643114183478 $ (2664971867617v^15 - 2254883551732v^14 + 26098327335794v^13 - 9229127126719v^12 + 167368473714729v^11 - 43779980557509v^10 + 557682597724019v^9 + 59586101079246v^8 + 1247763068861855v^7 + 124399822115638v^6 + 1399483126850095v^5 + 496421761562587v^4 + 980176537459036v^3 + 42710029269978v^2 + 6607663285321*v + 6051807742047) / 14643114183478
$\beta_{13}$	$=$	$ ( 11693527168209 \nu^{15} - 16882167878952 \nu^{14} + 121450189733038 \nu^{13} + \cdots - 103488037202661 ) / 58572456733912 $ (11693527168209v^15 - 16882167878952v^14 + 121450189733038v^13 - 109525006279299v^12 + 767580357234921v^11 - 631479624334649v^10 + 2618370057791523v^9 - 1200869222813430v^8 + 5484548039889219v^7 - 2648361059952138v^6 + 6171045657151407v^5 - 1384627146776921v^4 + 3267004856512722v^3 - 2128610461360662v^2 + 192241233587393*v - 103488037202661) / 58572456733912
$\beta_{14}$	$=$	$ ( - 3785503264689 \nu^{15} + 3390051748824 \nu^{14} - 37438564653690 \nu^{13} + \cdots - 8462579139243 ) / 14643114183478 $ (-3785503264689v^15 + 3390051748824v^14 - 37438564653690v^13 + 14948237014239v^12 - 240119166369729v^11 + 73182313351305v^10 - 806917320863295v^9 - 50517477400842v^8 - 1802712043846095v^7 - 116290241110290v^6 - 2067907805984381v^5 - 651824452967583v^4 - 1420041562540206v^3 - 50999793017154v^2 - 92542457279769*v - 8462579139243) / 14643114183478
$\beta_{15}$	$=$	$ \nu^{15} - \nu^{14} + 10 \nu^{13} - 5 \nu^{12} + 64 \nu^{11} - 26 \nu^{10} + 216 \nu^{9} - 9 \nu^{8} + \cdots + 2 $ v^15 - v^14 + 10v^13 - 5v^12 + 64v^11 - 26v^10 + 216v^9 - 9v^8 + 477v^7 - 17v^6 + 545v^5 + 120v^4 + 359v^3 - 16v^2 + 23*v + 2

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{13} - 2\beta_{9} - \beta_{3} - 2 $ b13 - 2*b9 - b3 - 2
$\nu^{3}$	$=$	$ -\beta_{12} + 4\beta_{6} $ -b12 + 4*b6
$\nu^{4}$	$=$	$ -5\beta_{13} - \beta_{11} + 8\beta_{9} $ -5b13 - b11 + 8b9
$\nu^{5}$	$=$	$ \beta_{14} + 6\beta_{12} - 18\beta_{6} - 6\beta_{2} - 18\beta_1 $ b14 + 6b12 - 18b6 - 6b2 - 18b1
$\nu^{6}$	$=$	$ -\beta_{7} - 7\beta_{5} + 24\beta_{3} + 36 $ -b7 - 7b5 + 24b3 + 36
$\nu^{7}$	$=$	$ \beta_{10} - \beta_{8} + 9\beta_{4} + 30\beta_{2} + 85\beta_1 $ b10 - b8 + 9b4 + 30b2 + 85*b1
$\nu^{8}$	$=$	$ - \beta_{15} - 2 \beta_{14} + 115 \beta_{13} + \beta_{12} + 39 \beta_{11} + 10 \beta_{10} - 169 \beta_{9} + \cdots - 169 $ -b15 - 2b14 + 115b13 + b12 + 39b11 + 10b10 - 169b9 + 10b7 + b6 + 39b5 - 115b3 - b2 + b1 - 169
$\nu^{9}$	$=$	$ -10\beta_{15} - 59\beta_{14} - 144\beta_{12} + 10\beta_{8} + 12\beta_{7} + 409\beta_{6} + \beta_{5} - 59\beta_{4} - 1 $ -10b15 - 59b14 - 144b12 + 10b8 + 12b7 + 409b6 + b5 - 59*b4 - 1
$\nu^{10}$	$=$	$ -553\beta_{13} - 203\beta_{11} - 71\beta_{10} + 808\beta_{9} + 12\beta_{8} - 25\beta_{4} + 11\beta_{2} - 13\beta_1 $ -553b13 - 203b11 - 71b10 + 808b9 + 12b8 - 25b4 + 11b2 - 13b1
$\nu^{11}$	$=$	$ 71 \beta_{15} + 345 \beta_{14} - 2 \beta_{13} + 685 \beta_{12} - 14 \beta_{11} - 96 \beta_{10} + \cdots + 14 $ 71b15 + 345b14 - 2b13 + 685b12 - 14b11 - 96b10 + 14b9 - 96b7 - 1985b6 - 14b5 + 2b3 - 685b2 - 1985*b1 + 14
$\nu^{12}$	$=$	$ 96 \beta_{15} + 206 \beta_{14} - 80 \beta_{12} - 96 \beta_{8} - 441 \beta_{7} - 114 \beta_{6} + \cdots + 3899 $ 96b15 + 206b14 - 80b12 - 96b8 - 441b7 - 114b6 - 1030b5 + 206b4 + 2670*b3 + 3899
$\nu^{13}$	$=$	$ 34 \beta_{13} + 126 \beta_{11} + 647 \beta_{10} - 132 \beta_{9} - 441 \beta_{8} + 1912 \beta_{4} + \cdots + 9680 \beta_1 $ 34b13 + 126b11 + 647b10 - 132b9 - 441b8 + 1912b4 + 3259b2 + 9680b1
$\nu^{14}$	$=$	$ - 647 \beta_{15} - 1420 \beta_{14} + 12939 \beta_{13} + 487 \beta_{12} + 5171 \beta_{11} + 2559 \beta_{10} + \cdots - 18919 $ -647b15 - 1420b14 + 12939b13 + 487b12 + 5171b11 + 2559b10 - 18919b9 + 2559b7 + 847b6 + 5171b5 - 12939b3 - 487b2 + 847*b1 - 18919
$\nu^{15}$	$=$	$ - 2559 \beta_{15} - 10289 \beta_{14} - 15551 \beta_{12} + 2559 \beta_{8} + 3979 \beta_{7} + 47356 \beta_{6} + \cdots - 1047 $ -2559b15 - 10289b14 - 15551b12 + 2559b8 + 3979b7 + 47356b6 + 933b5 - 10289b4 - 360*b3 - 1047

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/927\mathbb{Z}\right)^\times$.

$n$	$722$	$829$
$\chi(n)$	$1$	$\beta_{9}$

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} $

46.1

1.12538	+	1.94922i
1.01706	+	1.76159i
0.643600	+	1.11475i
0.139111	+	0.240948i
−0.0975889	−	0.169029i
−0.465833	−	0.806846i
−0.763528	−	1.32247i
−1.09820	−	1.90214i
1.12538	−	1.94922i
1.01706	−	1.76159i
0.643600	−	1.11475i
0.139111	−	0.240948i
−0.0975889	+	0.169029i
−0.465833	+	0.806846i
−0.763528	+	1.32247i
−1.09820	+	1.90214i

−1.12538

−

1.94922i

−1.53297

2.65517i

0.00533640

−

0.00924292i

−1.16700

2.02130i

2.39916

−0.0240219

46.2

−1.01706

−

1.76159i

−1.06881

1.85123i

−2.12070

3.67315i

0.784107

−

1.35811i

0.279926

8.62747

46.3

−0.643600

−

1.11475i

0.171558

−

0.297148i

0.349820

−

0.605907i

1.76958

−

3.06500i

−3.01606

−0.900577

46.4

−0.139111

−

0.240948i

0.961296

−

1.66501i

−0.480550

0.832336i

−0.809476

1.40205i

−1.09136

0.267400

46.5

0.0975889

0.169029i

0.980953

−

1.69906i

1.61176

−

2.79165i

−0.905490

1.56835i

0.773276

0.629160

46.6

0.465833

0.806846i

0.566000

−

0.980340i

−1.64222

2.84441i

0.793289

−

1.37402i

2.91798

−3.06000

46.7

0.763528

1.32247i

−0.165950

0.287433i

1.00356

−

1.73822i

2.11087

−

3.65614i

2.54728

3.06499

46.8

1.09820

1.90214i

−1.41208

2.44580i

0.772988

−

1.33885i

−0.575879

0.997451i

−1.81020

3.39558

262.1