LMFDB - Newform orbit 1008.2.df.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,2,Mod(689,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5, 1]))

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

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

chi := DirichletCharacter("1008.689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1008 = 2^{4} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1008.df (of order $6$ , degree $2$ , 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:	$8.04892052375$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561$ x^16 - 8x^15 + 23x^14 - 8x^13 - 131x^12 + 380x^11 - 289x^10 - 880x^9 + 2785x^8 - 2640x^7 - 2601x^6 + 10260x^5 - 10611x^4 - 1944x^3 + 16767x^2 - 17496*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$3^{4}$
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$ -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.

$f(q)$	$=$	$q - \beta_{3} q^{3} - \beta_{11} q^{5} + ( - \beta_{10} - \beta_{8} + \beta_{4} + \cdots + 1) q^{7} + ( - \beta_{14} + \beta_{13} - \beta_{11} + \cdots - 1) q^{9} + (\beta_{15} - \beta_{10} - \beta_{9} + \cdots + 1) q^{11}+ \cdots + (\beta_{15} - 3 \beta_{14} + 2 \beta_{13} + \cdots - 2) q^{99}+O(q^{100})$ q - b3 * q^3 - b11 * q^5 + (-b10 - b8 + b4 + b2 - b1 + 1) * q^7 + (-b14 + b13 - b11 + b10 + b9 + b8 + b6 - b4 - b2 - 1) * q^9 + (b15 - b10 - b9 + b7 + b4 + b3 - b2 - b1 + 1) * q^11 + (b15 + b14 - 2b9 - b6 + b5 + b3 + b2) q^13 + (b15 + b10 + b8 + b6 - b4 - 2b2 + b1) q^15 + (b15 + b14 - b12 - b10 - b9 - b8 - 2b7 + b3 - b2) q^17 + (-b15 - b11 - 2b10 - b7 + b3 + b1 - 2) q^19 + (b12 - b10 + b6 + b5 + b1 - 2) * q^21 + (-b15 + b8 + b4 + b3 + b2) * q^23 + (-2b15 - b13 + b12 + b11 - 3b10 + b9 + b8 - 2b7 + b4 + b1 - 2) q^25 + (2b14 - b12 - b11 + 2b10 + b7 - 2b4 + b3 + b1 + 2) q^27 + (b15 - b12 + b10 + b9 + b8 - b4 - b3 - b2) * q^29 + (b10 + b9 + 2b8 + 2b6 + b5 - 2b4 - 2b3 + b1 - 2) * q^31 + (2b14 - b13 - b12 + b11 - b10 - 2b9 - b8 + 2b7 + 2b6 + b5 - 2b4 + b2 - b1 + 2) q^33 + (b14 - b10 - b9 - 3b7 - b4 + 3b3 + b1 - 1) * q^35 + (-2b15 - 2b13 + b12 + b11 - 3b10 - b9 + b8 - 2b7 + 3b4 - 2b3 + b2 - b1 - 1) * q^37 + (2b15 - b14 + b12 + b11 - b9 - b8 - b7 - b6 - b4 - b3 - 2b1 + 3) * q^39 + (-b14 - b13 + b8 - b7 + b3 - b2 - b1) * q^41 + (-b15 - b14 - b12 - b11 - b9 + b7 - 2b4 - b2 - b1) q^43 + (b14 - 3b12 + b10 - 2b7 - 2b4 - b2 - 1) q^45 + (-2b15 - b14 - b13 + b12 + b10 + b9 + 2b8 - 3b7 - 2b4 - b3 + b2 + b1 - 2) * q^47 + (-b15 + b12 + 2b10 + b8 + b7 + b6 - b5 - 2b4 + 2b2 + 1) q^49 + (-b12 - 3b10 + 2b9 - b7 + 2b6 + 2b5 + b4 - b3 - b2 - b1 - 1) * q^51 + (b15 - b14 + 2b13 + b12 - b11 + 4b10 + b9 - b8 + 3b7 - 2b4 + 2b3 - 2b2 + b1 + 4) * q^53 + (-b15 + 4b14 - 2b13 + 2b11 - 2b9 - b8 + 2b7 + 2b6 + 4b5 + b4 - b3 + 3b2 + 2) * q^55 + (b15 - b13 + 2b6 + 3b5 + 2b4 + b3 - 2b2 + 1) * q^57 + (-b14 - b13 + 2b12 - b10 + b8 - 5b7 + b4 - b3 - 2b2 - 5) q^59 + (b15 + b13 - b12 - 2b11 + 4b10 - b8 - b7 - 2b4 + 2b3 - 2b2 + b1 - 4) q^61 + (2b15 + b14 - 2b12 + 2b10 - b9 - b8 - 2b7 + b6 + 3b5 - b4 + 3b3 - 2b2 - 3) q^63 + (2b12 + 3b11 - 3b10 + b7 - b6 + b5 + 2b4 - b3 + 3b2 - b1 + 4) q^65 + (-b14 - b11 - b9 + 3b8 - 4b7 + b4 + 3b3 - 2b2 + 2b1 - 5) q^67 + (-b13 + b12 - b9 - b8 + 2b7 - b6 - 2b5 + b4 - b3 + 2b2 + 5) q^69 + (b15 - b13 + 2b12 + 2b11 - b10 - b8 - b6 - 2b5 + 3b4 - 4b3 - b1 + 2) q^71 + (-b15 - b13 + b12 - b10 + b9 + 2b6 - 2b5 - 2b3 - b1) q^73 + (b14 - b13 - b12 + b11 + b10 + b9 - b8 + 4b7 + 2b6 + 4b5 + 3b4 - b3 + 2b2 - b1 + 3) q^75 + (-3b15 + 2b14 - 3b13 + b12 + 2b11 - b10 + 2b9 + 2b8 - 5b7 + b6 + 2b5 + 3b4 - b3 + 3b2 + b1 - 4) * q^77 + (-2b15 - 2b13 + b12 + b11 - b9 + b8 - 3b7 - 3b4 - 2b3 + b2 + 2b1 - 3) * q^79 + (2b15 - b12 - b10 - b9 + 2b8 - b7 - 2b6 - b5 + 2b4 - b3 - 2b2 + b1 + 2) q^81 + (b15 + 2b14 - b13 + b12 + b11 + b10 + b9 - b8 + b4 - b3 + b2 + b1 + 1) q^83 + (-2b14 + 4b13 - 3b11 + 3b9 - b8 + 2b7 + 3b6 + 3b5 + b4 + b2 + 1) q^85 + (-b15 - b14 + 2b13 + b10 + 2b9 + b8 - b4 - b3 + b2 + b1 + 2) * q^87 + (-2b15 - 3b14 - b13 + 2b12 - 4b10 - b9 + 3b8 - 3b5 + 3b4 - b3 - 2b2 - b1) * q^89 + (-2b15 - 3b13 + b12 + 2b11 - 5b10 - 4b9 + 2b8 - 2b7 - 2b6 - 4b5 + 4b4 + 3b3 + b1 - 2) q^91 + (b15 - 2b14 + 3b13 - 2b12 - 3b11 + 5b10 + 2b9 + b8 - b7 - b6 + b5 - 4b4 + 2b3 - 2b2 + 2b1) * q^93 + (-2b15 - b14 - 3b11 - 4b10 + b9 - b8 - 4b7 - 2b6 - b5 + b4 + 3b3 + 1) * q^95 + (b15 + b14 - 2b13 - 2b12 + 2b11 - 2b10 - b9 + b8 - b7 + 4b6 + 2b5 - b4 - 2b2 + b1 - 3) q^97 + (b15 - 3b14 + 2b13 - 2b12 - 4b11 - 2b10 - b9 + 2b7 - 2b6 + 2b5 - 2b4 + 3b3 - 3b2 - 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 2 q^{7} - 6 q^{9} - 6 q^{13} + 18 q^{15} + 18 q^{17} - 18 q^{21} + 16 q^{25} + 36 q^{27} + 6 q^{29} - 6 q^{31} + 18 q^{33} + 30 q^{35} - 2 q^{37} + 30 q^{39} + 6 q^{41} + 2 q^{43} + 12 q^{45} + 18 q^{47}+ \cdots - 18 q^{99}+O(q^{100})$ 16 * q - 2 * q^7 - 6 * q^9 - 6 * q^13 + 18 * q^15 + 18 * q^17 - 18 * q^21 + 16 * q^25 + 36 * q^27 + 6 * q^29 - 6 * q^31 + 18 * q^33 + 30 * q^35 - 2 * q^37 + 30 * q^39 + 6 * q^41 + 2 * q^43 + 12 * q^45 + 18 * q^47 + 10 * q^49 + 36 * q^53 + 6 * q^57 - 30 * q^59 - 60 * q^61 - 42 * q^63 + 42 * q^65 - 14 * q^67 + 42 * q^69 - 30 * q^75 - 18 * q^77 + 16 * q^79 + 54 * q^81 - 12 * q^85 + 48 * q^87 + 24 * q^89 + 12 * q^91 + 30 * q^93 + 66 * q^95 - 6 * q^97 - 18 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561$ x^16 - 8*x^15 + 23*x^14 - 8*x^13 - 131*x^12 + 380*x^11 - 289*x^10 - 880*x^9 + 2785*x^8 - 2640*x^7 - 2601*x^6 + 10260*x^5 - 10611*x^4 - 1944*x^3 + 16767*x^2 - 17496*x + 6561 :

$\beta_{1}$	$=$	$( - 154 \nu^{15} + 1325 \nu^{14} - 3608 \nu^{13} + 224 \nu^{12} + 22478 \nu^{11} - 55022 \nu^{10} + \cdots + 1285227 ) / 47385$ (-154v^15 + 1325v^14 - 3608v^13 + 224v^12 + 22478v^11 - 55022v^10 + 23518v^9 + 159688v^8 - 382978v^7 + 226785v^6 + 566733v^5 - 1355805v^4 + 871830v^3 + 949725v^2 - 2135727*v + 1285227) / 47385
$\beta_{2}$	$=$	$( 1342 \nu^{15} - 9134 \nu^{14} + 18833 \nu^{13} + 17821 \nu^{12} - 164858 \nu^{11} + 301448 \nu^{10} + \cdots - 6445089 ) / 142155$ (1342v^15 - 9134v^14 + 18833v^13 + 17821v^12 - 164858v^11 + 301448v^10 + 55817v^9 - 1253167v^8 + 2222275v^7 - 414219v^6 - 4778010v^5 + 8256384v^4 - 3144339v^3 - 8309871v^2 + 13120542*v - 6445089) / 142155
$\beta_{3}$	$=$	$( 2846 \nu^{15} - 22369 \nu^{14} + 55246 \nu^{13} + 17972 \nu^{12} - 402586 \nu^{11} + \cdots - 20783061 ) / 142155$ (2846v^15 - 22369v^14 + 55246v^13 + 17972v^12 - 402586v^11 + 878875v^10 - 166676v^9 - 3023495v^8 + 6386423v^7 - 2756469v^6 - 11333493v^5 + 23486949v^4 - 12667509v^3 - 19623951v^2 + 37934244*v - 20783061) / 142155
$\beta_{4}$	$=$	$( 2782 \nu^{15} - 15918 \nu^{14} + 26947 \nu^{13} + 42629 \nu^{12} - 270897 \nu^{11} + 425335 \nu^{10} + \cdots - 7405182 ) / 47385$ (2782v^15 - 15918v^14 + 26947v^13 + 42629v^12 - 270897v^11 + 425335v^10 + 220488v^9 - 2001225v^8 + 3082271v^7 + 70562v^6 - 7458951v^5 + 11186748v^4 - 2636253v^3 - 12673422v^2 + 17180343*v - 7405182) / 47385
$\beta_{5}$	$=$	$( - 5666 \nu^{15} + 35256 \nu^{14} - 67793 \nu^{13} - 74626 \nu^{12} + 609708 \nu^{11} + \cdots + 21264930 ) / 47385$ (-5666v^15 + 35256v^14 - 67793v^13 - 74626v^12 + 609708v^11 - 1074116v^10 - 260022v^9 + 4521984v^8 - 7792711v^7 + 1217186v^6 + 16859166v^5 - 28410246v^4 + 10140201v^3 + 28759374v^2 - 44393427*v + 21264930) / 47385
$\beta_{6}$	$=$	$( - 432 \nu^{15} + 2816 \nu^{14} - 5740 \nu^{13} - 5150 \nu^{12} + 49010 \nu^{11} - 90874 \nu^{10} + \cdots + 1853118 ) / 3645$ (-432v^15 + 2816v^14 - 5740v^13 - 5150v^12 + 49010v^11 - 90874v^10 - 11735v^9 + 363706v^8 - 657698v^7 + 149366v^6 + 1355028v^5 - 2395476v^4 + 962496v^3 + 2313684v^2 - 3756780*v + 1853118) / 3645
$\beta_{7}$	$=$	$( 4120 \nu^{15} - 25571 \nu^{14} + 48788 \nu^{13} + 55006 \nu^{12} - 441224 \nu^{11} + 771188 \nu^{10} + \cdots - 14963454 ) / 28431$ (4120v^15 - 25571v^14 + 48788v^13 + 55006v^12 - 441224v^11 + 771188v^10 + 200900v^9 - 3269857v^8 + 5585140v^7 - 796053v^6 - 12187116v^5 + 20322468v^4 - 7028856v^3 - 20790351v^2 + 31653180*v - 14963454) / 28431
$\beta_{8}$	$=$	$( 20555 \nu^{15} - 129232 \nu^{14} + 250807 \nu^{13} + 267434 \nu^{12} - 2233957 \nu^{11} + \cdots - 77795964 ) / 142155$ (20555v^15 - 129232v^14 + 250807v^13 + 267434v^12 - 2233957v^11 + 3963841v^10 + 897553v^9 - 16558319v^8 + 28688468v^7 - 4701822v^6 - 61695738v^5 + 104382702v^4 - 37593477v^3 - 105226533v^2 + 162783513*v - 77795964) / 142155
$\beta_{9}$	$=$	$( - 21860 \nu^{15} + 133486 \nu^{14} - 249901 \nu^{13} - 298607 \nu^{12} + 2298061 \nu^{11} + \cdots + 75541167 ) / 142155$ (-21860v^15 + 133486v^14 - 249901v^13 - 298607v^12 + 2298061v^11 - 3952333v^10 - 1174129v^9 + 17020292v^8 - 28631549v^7 + 3425376v^6 + 63403209v^5 - 104137596v^4 + 34463151v^3 + 108020304v^2 - 161740314*v + 75541167) / 142155
$\beta_{10}$	$=$	$( 2015 \nu^{15} - 12538 \nu^{14} + 24088 \nu^{13} + 26576 \nu^{12} - 216643 \nu^{11} + 381184 \nu^{10} + \cdots - 7453296 ) / 10935$ (2015v^15 - 12538v^14 + 24088v^13 + 26576v^12 - 216643v^11 + 381184v^10 + 93322v^9 - 1605386v^8 + 2760692v^7 - 423483v^6 - 5980032v^5 + 10047753v^4 - 3553308v^3 - 10193607v^2 + 15657462*v - 7453296) / 10935
$\beta_{11}$	$=$	$( 29357 \nu^{15} - 190006 \nu^{14} + 382390 \nu^{13} + 360905 \nu^{12} - 3304405 \nu^{11} + \cdots - 122535423 ) / 142155$ (29357v^15 - 190006v^14 + 382390v^13 + 360905v^12 - 3304405v^11 + 6053149v^10 + 947140v^9 - 24541961v^8 + 43845563v^7 - 9169941v^6 - 91502118v^5 + 159812406v^4 - 62353071v^3 - 156416184v^2 + 250528140*v - 122535423) / 142155
$\beta_{12}$	$=$	$( - 11745 \nu^{15} + 73063 \nu^{14} - 139508 \nu^{13} - 157621 \nu^{12} + 1262908 \nu^{11} + \cdots + 42448941 ) / 47385$ (-11745v^15 + 73063v^14 - 139508v^13 - 157621v^12 + 1262908v^11 - 2204834v^10 - 583027v^9 + 9367136v^8 - 15968257v^7 + 2214238v^6 + 34940067v^5 - 58105683v^4 + 19891683v^3 + 59661117v^2 - 90406692*v + 42448941) / 47385
$\beta_{13}$	$=$	$( - 45758 \nu^{15} + 290392 \nu^{14} - 571483 \nu^{13} - 583571 \nu^{12} + 5039353 \nu^{11} + \cdots + 181431333 ) / 142155$ (-45758v^15 + 290392v^14 - 571483v^13 - 583571v^12 + 5039353v^11 - 9054625v^10 - 1803427v^9 + 37411250v^8 - 65645234v^7 + 12014682v^6 + 139514814v^5 - 239326137v^4 + 89465067v^3 + 238163328v^2 - 374591547*v + 181431333) / 142155
$\beta_{14}$	$=$	$( - 17267 \nu^{15} + 108242 \nu^{14} - 209241 \nu^{13} - 227007 \nu^{12} + 1874096 \nu^{11} + \cdots + 65023155 ) / 47385$ (-17267v^15 + 108242v^14 - 209241v^13 - 227007v^12 + 1874096v^11 - 3311892v^10 - 782399v^9 + 13905028v^8 - 24003582v^7 + 3802087v^6 + 51849342v^5 - 87425352v^4 + 31201362v^3 + 88481403v^2 - 136401489*v + 65023155) / 47385
$\beta_{15}$	$=$	$( - 56402 \nu^{15} + 348301 \nu^{14} - 661000 \nu^{13} - 757115 \nu^{12} + 6004885 \nu^{11} + \cdots + 200862828 ) / 142155$ (-56402v^15 + 348301v^14 - 661000v^13 - 757115v^12 + 6004885v^11 - 10449439v^10 - 2822275v^9 + 44476256v^8 - 75645428v^7 + 10299831v^6 + 165686283v^5 - 275059071v^4 + 93864501v^3 + 282553839v^2 - 427711590*v + 200862828) / 142155

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

$\nu$	$=$	$( -\beta_{14} - \beta_{10} + \beta_{9} - \beta_{7} + 2\beta_{4} + \beta_{3} - 2\beta_{2} + 1 ) / 3$ (-b14 - b10 + b9 - b7 + 2b4 + b3 - 2b2 + 1) / 3
$\nu^{2}$	$=$	$( 2 \beta_{15} - \beta_{12} - \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + \cdots + 2 ) / 3$ (2b15 - b12 - b10 - b9 + 2b8 - b7 - 2b6 - b5 + 2b4 - b3 - 2*b2 + b1 + 2) / 3
$\nu^{3}$	$=$	$( 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots - 3 ) / 3$ (2b15 + 2b14 - 2b13 - 4b12 + b11 - 2b10 - b9 - 3b8 - b6 - 2b5 + b4 - 2b3 - 5*b2 - 3) / 3
$\nu^{4}$	$=$	$( 3 \beta_{15} + 10 \beta_{14} - 4 \beta_{13} - 6 \beta_{12} + 5 \beta_{11} + 4 \beta_{10} - 3 \beta_{9} + \cdots + 11 ) / 3$ (3b15 + 10b14 - 4b13 - 6b12 + 5b11 + 4b10 - 3b9 - b8 - b7 + 2b6 - 5b5 - 2b4 - 8b3 + 3b2 - 2*b1 + 11) / 3
$\nu^{5}$	$=$	$( - 3 \beta_{15} - 2 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} + 3 \beta_{11} - 8 \beta_{10} - 4 \beta_{9} + \cdots - 7 ) / 3$ (-3b15 - 2b14 - 6b13 - 6b12 + 3b11 - 8b10 - 4b9 - 3b8 - 20b7 + 15b6 - 3b5 - 5b4 + 2b3 - 13b2 - 9*b1 - 7) / 3
$\nu^{6}$	$=$	$( - 7 \beta_{15} - 3 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} - 2 \beta_{11} - 10 \beta_{10} + 7 \beta_{9} + \cdots - 31 ) / 3$ (-7b15 - 3b14 + 4b13 - 4b12 - 2b11 - 10b10 + 7b9 + 33b8 - 44b7 + 8b6 - 20b5 - b4 + 6b3 - 12b2 + 9b1 - 31) / 3
$\nu^{7}$	$=$	$( - 6 \beta_{15} - 35 \beta_{14} + \beta_{13} - 18 \beta_{12} - 38 \beta_{11} - 59 \beta_{10} + \beta_{9} + \cdots - 85 ) / 3$ (-6b15 - 35b14 + b13 - 18b12 - 38b11 - 59b10 + b9 + 22b8 - 36b7 + b6 - 34b5 - 35b4 + 21b3 - 68b2 + 11b1 - 85) / 3
$\nu^{8}$	$=$	$( - 10 \beta_{15} + 35 \beta_{14} - 18 \beta_{13} - 40 \beta_{12} - 48 \beta_{11} + 7 \beta_{10} + \cdots - 36 ) / 3$ (-10b15 + 35b14 - 18b13 - 40b12 - 48b11 + 7b10 + 84b9 + 38b8 + 13b7 - 17b6 - 70b5 - 41b4 - 21b3 - 22b2 + 34*b1 - 36) / 3
$\nu^{9}$	$=$	$( 32 \beta_{14} - 126 \beta_{13} - 72 \beta_{12} - 90 \beta_{11} - 46 \beta_{10} - 20 \beta_{9} + \cdots + 64 ) / 3$ (32b14 - 126b13 - 72b12 - 90b11 - 46b10 - 20b9 - 12b8 + 20b7 + 87b6 + 48b5 - 124b4 + 10b3 - 98b2 - 42b1 + 64) / 3
$\nu^{10}$	$=$	$( - 90 \beta_{15} + 171 \beta_{14} - 167 \beta_{13} - 36 \beta_{12} - 5 \beta_{11} + 153 \beta_{10} + \cdots - 225 ) / 3$ (-90b15 + 171b14 - 167b13 - 36b12 - 5b11 + 153b10 + 143b9 + 229b8 - 220b7 + 199b6 + 146b5 - 15b4 + 40b3 - 85b2 + 170*b1 - 225) / 3
$\nu^{11}$	$=$	$( 100 \beta_{15} + 83 \beta_{14} - 141 \beta_{13} - 113 \beta_{12} - 90 \beta_{11} - 27 \beta_{10} + \cdots - 325 ) / 3$ (100b15 + 83b14 - 141b13 - 113b12 - 90b11 - 27b10 - 412b9 + 403b8 - 42b7 + 335b6 + 286b5 - 360b4 + 272b3 - 489b2 + 383*b1 - 325) / 3
$\nu^{12}$	$=$	$( 251 \beta_{15} + 97 \beta_{14} + 298 \beta_{13} - 319 \beta_{12} - 128 \beta_{11} + 867 \beta_{10} + \cdots - 845 ) / 3$ (251b15 + 97b14 + 298b13 - 319b12 - 128b11 + 867b10 + 501b9 + 471b8 + 395b7 + 236b6 - 284b5 - 510b4 + 263b3 - 847b2 + 972*b1 - 845) / 3
$\nu^{13}$	$=$	$( 1440 \beta_{15} - 170 \beta_{14} + 597 \beta_{13} - 1254 \beta_{12} - 999 \beta_{11} + 1288 \beta_{10} + \cdots + 1901 ) / 3$ (1440b15 - 170b14 + 597b13 - 1254b12 - 999b11 + 1288b10 - 721b9 - 87b8 + 2173b7 + 198b6 - 549b5 - 1706b4 + 926b3 - 1453b2 + 99*b1 + 1901) / 3
$\nu^{14}$	$=$	$( 835 \beta_{15} - 918 \beta_{14} + 882 \beta_{13} - 1877 \beta_{12} - 1842 \beta_{11} + 2872 \beta_{10} + \cdots - 1013 ) / 3$ (835b15 - 918b14 + 882b13 - 1877b12 - 1842b11 + 2872b10 + 1198b9 - 389b8 - 1067b7 + 101b6 - 881b5 - 872b4 + 2104b3 - 2326b2 + 194*b1 - 1013) / 3
$\nu^{15}$	$=$	$( 1012 \beta_{15} + 838 \beta_{14} + 431 \beta_{13} - 2534 \beta_{12} - 3259 \beta_{11} + 1145 \beta_{10} + \cdots + 561 ) / 3$ (1012b15 + 838b14 + 431b13 - 2534b12 - 3259b11 + 1145b10 - 5435b9 + 951b8 - 2874b7 - 1079b6 - 214b5 - 1879b4 + 4997b3 - 1708b2 + 399*b1 + 561) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$ .

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$1 + \beta_{7}$	$1$	$-\beta_{7}$

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}

689.1

0.765614	+	1.55365i
0.320287	−	1.70218i
1.27866	+	1.16834i
1.58110	−	0.707199i
1.71298	−	0.256290i
−1.68301	−	0.409224i
−1.70672	+	0.295146i
1.73109	+	0.0577511i
0.765614	−	1.55365i
0.320287	+	1.70218i
1.27866	−	1.16834i
1.58110	+	0.707199i
1.71298	+	0.256290i
−1.68301	+	0.409224i
−1.70672	−	0.295146i
1.73109	−	0.0577511i

−1.47073

−

0.914855i

−3.64414

−2.62749

0.310282i

1.32608

2.69100i

689.2

−1.33318

1.10572i

−0.0676069

2.64192

0.142361i

0.554753

−

2.94826i

689.3

−0.626615

1.61473i

3.55225

1.49384

−

2.18368i

−2.21471

−

2.02363i

689.4

−0.361565

−

1.69389i

−0.900258

−1.05755

−

2.42520i

−2.73854

1.22490i

689.5

0.128499

1.72728i

−3.61932

−0.266972

2.63225i

−2.96698

0.443907i

689.6

0.206076

−

1.71975i

1.42985

2.43739

1.02913i

−2.91507

−

0.708796i

689.7

1.72571

−

0.148116i

0.967324

−2.40137

1.11060i

2.95612

−

0.511208i

689.8

1.73181

0.0288796i

2.28190

−1.21977

−

2.34780i

2.99833

0.100028i

929.1