LMFDB - Newform orbit 952.2.q.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [952,2,Mod(137,952)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("952.137"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(952, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 952 = 2^{3} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	952.q (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$7.60175827243$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 11 x^{12} - 6 x^{11} + 94 x^{10} - 39 x^{9} + 282 x^{8} - 47 x^{7} + 615 x^{6} - 134 x^{5} + \cdots + 16 $ x^14 + 11x^12 - 6x^11 + 94x^10 - 39x^9 + 282x^8 - 47x^7 + 615x^6 - 134x^5 + 348x^4 - 144x^3 + 168x^2 - 48x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{13}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{3} + ( - \beta_{10} + \beta_{3}) q^{5} + ( - \beta_{12} + \beta_{5} - \beta_{4}) q^{7} + (\beta_{10} + \beta_{7} + \beta_{5} + \cdots - \beta_1) q^{9} + ( - \beta_{13} - \beta_{6}) q^{11}+ \cdots + (\beta_{12} + \beta_{11} + 2 \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) $ q + b1 * q^3 + (-b10 + b3) * q^5 + (-b12 + b5 - b4) * q^7 + (b10 + b7 + b5 - b3 + b2 - b1) * q^9 + (-b13 - b6) * q^11 + (-b8 - b6 - b3 + 2) * q^13 + (b12 + b11 - b7 - b5 + b3 + 1) * q^15 + b4 * q^17 + (b13 - b12 + b11 - b10 + b9 - b7 - b4 + b3 + 1) * q^19 + (b11 + b8 + b6 + b2 - 2b1 - 1) q^21 + (-b12 + b11 + b10 + b8 + b5 + b4 - b3 + 2b2 - 2b1 - 1) * q^23 + (-b13 + b10 - b1) * q^25 + (b12 + b11 + b8 - b7 + b6 - b5 + 1) * q^27 + (b12 + b11 - b9 - b8 - b7 - b6 - b5 + b3 - b2 + 1) * q^29 + (b13 + b11 + b10 - b7 + b6 + b5 - b4) * q^31 + (b13 - b12 + b11 + b10 + b9 - b8 + b5 - b4 - b3 + b2 - b1 + 1) * q^33 + (b12 + b11 - b9 - b6 - b4 - b3 - b2 + b1 + 1) * q^35 + (-b12 + b11 + b10 - 2b8 - b7 - b3 + b2 - b1) q^37 + (b13 + b11 - b7 + b5 - b4) * q^39 + (-b12 - b11 + b7 + b5 - b3 - 2b2 + 1) q^41 + (b12 + 2b11 - b7 - 2b5 - b3 + 2) * q^43 + (-2b10 - b6 + 2b4 + 3b1) q^45 + (-b13 - 3b10 - b9 + b8 - b7 - b5 + b4 + 3b3 - b2 + b1 - 1) * q^47 + (2b12 - b10 - b9 + b6 - b5 - b4 + 2b3 - 3b2 + b1 - 2) q^49 + (-b2 + b1) * q^51 + (2b13 + b12 + b11 - b6 - b4) q^53 + (b12 + 2b11 - 2b8 - b7 - 2b6 - 2b5 - 3b3 + b2 + 3) q^55 + (-b12 + 2b8 + b7 + 2b6 + b3 + 3b2 - 1) q^57 + (-b13 - b12 - b11 + b10 + 3b6 + 2b1) * q^59 + (-2b13 + b12 - b11 - 2b9 + b8 - b5 - 3b4 + 3) q^61 + (-b13 - b11 - 2b10 + b8 - b7 + b6 - b5 - 3b4 + 2b3 - 3b2 + 3b1 + 2) q^63 + (b13 + b12 - b11 - 3b10 + b9 - b8 - b5 + 3b4 + 3b3 - 2b2 + 2b1 - 3) q^65 + (2b6 - 5b4 - b1) * q^67 + (-b12 + b11 + b9 + 2b8 + b7 + 2b6 - b5 - 2b2 + 3) q^69 + (-b12 - b11 + b9 + b7 + b5 - b3 + b2 + 1) * q^71 + (-2b13 - b11 - b10 + b7 - 2b6 - b5 + 3b4 + 3b1) * q^73 + (-b8 - b7 - b5 - 3b4 - 2b2 + 2b1 + 3) q^75 + (b12 + b11 - 2b10 + 2b9 - 2b8 + b7 - 2b6 - b5 + 2b4 + b3 - 2b2 + b1 - 1) * q^77 + (-b12 + b11 + 3b10 - b8 - b7 - 3b3 + 5b2 - 5b1) * q^79 + (-b13 + b12 + 2b10 + b7 - b6 - b5 + b4 + 2b1) * q^81 + (b9 - b3 + 4) * q^83 + b3 * q^85 + (b13 + b12 + 2b11 + b10 - b7 - 2b6 + b5 + b1) * q^87 + (b13 + 3b10 + b9 + b8 + 3b7 + 3b5 - 3b3 + 3b2 - 3b1) * q^89 + (-b12 + b11 + 3b10 + b9 - b8 + b6 + b5 - 3b4 - b3 - b2 + b1 + 1) * q^91 + (-b13 - b9 + 3b8 + 4b4 - b2 + b1 - 4) * q^93 + (b13 - b12 + b11 + 3b10 - 2b7 + 2b5 - 5b4 - 4b1) q^95 + (-b12 - 3b11 + b8 + b7 + b6 + 3b5 - 5b2 + 1) q^97 + (b12 + b11 + 2b9 - b7 - b5 + b3 - 2b2 - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + q^{5} - 7 q^{7} - q^{9} - 5 q^{11} + 22 q^{13} + 16 q^{15} + 7 q^{17} + 2 q^{19} - 13 q^{21} - 9 q^{23} - 2 q^{25} + 18 q^{27} + 18 q^{29} + 2 q^{31} - 2 q^{33} + 9 q^{35} - 8 q^{37} - q^{39}+ \cdots - 24 q^{99}+O(q^{100}) $ 14 * q + q^5 - 7 * q^7 - q^9 - 5 * q^11 + 22 * q^13 + 16 * q^15 + 7 * q^17 + 2 * q^19 - 13 * q^21 - 9 * q^23 - 2 * q^25 + 18 * q^27 + 18 * q^29 + 2 * q^31 - 2 * q^33 + 9 * q^35 - 8 * q^37 - q^39 + 12 * q^41 + 20 * q^43 + 10 * q^45 + q^47 - 21 * q^49 - 3 * q^53 + 22 * q^55 - 10 * q^57 + 4 * q^59 + 32 * q^61 + 13 * q^63 - 20 * q^65 - 31 * q^67 + 32 * q^69 + 6 * q^71 + 7 * q^73 + 19 * q^75 - 26 * q^77 - 8 * q^79 + q^81 + 48 * q^83 + 2 * q^85 + 3 * q^87 - 4 * q^89 - 15 * q^91 - 19 * q^93 - 23 * q^95 + 30 * q^97 - 24 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{14} + 11 x^{12} - 6 x^{11} + 94 x^{10} - 39 x^{9} + 282 x^{8} - 47 x^{7} + 615 x^{6} - 134 x^{5} + \cdots + 16 $ x^14 + 11*x^12 - 6*x^11 + 94*x^10 - 39*x^9 + 282*x^8 - 47*x^7 + 615*x^6 - 134*x^5 + 348*x^4 - 144*x^3 + 168*x^2 - 48*x + 16 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 1461811399 \nu^{13} + 2173843868 \nu^{12} + 11091037161 \nu^{11} + 13977387054 \nu^{10} + \cdots + 452274495472 ) / 1333059532968 $ (1461811399v^13 + 2173843868v^12 + 11091037161v^11 + 13977387054v^10 + 72641657926v^9 + 165127223747v^8 - 107427708470v^7 + 616163065155v^6 - 337170743703v^5 + 1014753138694v^4 - 2598797413564v^3 + 578350726588v^2 - 172238425216*v + 452274495472) / 1333059532968
$\beta_{3}$	$=$	$ ( 3336853395 \nu^{13} - 9060078011 \nu^{12} + 36361176440 \nu^{11} - 119151368385 \nu^{10} + \cdots - 1241095958984 ) / 222176588828 $ (3336853395v^13 - 9060078011v^12 + 36361176440v^11 - 119151368385v^10 + 363100774521v^9 - 974159395635v^8 + 1246069817105v^7 - 2644757662396v^6 + 2253176919570v^5 - 5834744995420v^4 + 1793516403349v^3 - 3289770673480v^2 + 981511928880*v - 1241095958984) / 222176588828
$\beta_{4}$	$=$	$ ( 28267155967 \nu^{13} - 1461811399 \nu^{12} + 308764871769 \nu^{11} - 180693972963 \nu^{10} + \cdots + 148474471768 ) / 1333059532968 $ (28267155967v^13 - 1461811399v^12 + 308764871769v^11 - 180693972963v^10 + 2643135273844v^9 - 1175060740639v^8 + 7806210758947v^7 - 1221128621979v^6 + 16768137854550v^5 - 3450628155875v^4 + 8822217137822v^3 - 1471673045684v^2 + 4170531475868*v + 148474471768) / 1333059532968
$\beta_{5}$	$=$	$ ( - 15392926629 \nu^{13} - 47772433816 \nu^{12} - 186463654336 \nu^{11} - 435916296356 \nu^{10} + \cdots - 3129422588932 ) / 666529766484 $ (-15392926629v^13 - 47772433816v^12 - 186463654336v^11 - 435916296356v^10 - 1342508016973v^9 - 3805916762277v^8 - 3996554761664v^7 - 12258433289690v^6 - 11386551352099v^5 - 26643365465279v^4 - 7749145429973v^3 - 12046092499548v^2 + 2525092874720*v - 3129422588932) / 666529766484
$\beta_{6}$	$=$	$ ( - 60456620389 \nu^{13} - 54766830803 \nu^{12} - 670608926121 \nu^{11} - 224200797663 \nu^{10} + \cdots - 729958684960 ) / 1333059532968 $ (-60456620389v^13 - 54766830803v^12 - 670608926121v^11 - 224200797663v^10 - 5415339957814v^9 - 2588786303399v^8 - 15538723757797v^7 - 10953907282533v^6 - 36712728936198v^5 - 20971206523189v^4 - 17512545248804v^3 - 476335243648v^2 - 2026385534324*v - 729958684960) / 1333059532968
$\beta_{7}$	$=$	$ ( 84349671011 \nu^{13} + 101392113228 \nu^{12} + 970824679427 \nu^{11} + 594813834838 \nu^{10} + \cdots + 2998828224856 ) / 1333059532968 $ (84349671011v^13 + 101392113228v^12 + 970824679427v^11 + 594813834838v^10 + 7775924450944v^9 + 5813411263891v^8 + 23758615024662v^7 + 21569656311109v^6 + 58179502882913v^5 + 44760496376844v^4 + 38197511992766v^3 + 7316865121376v^2 + 6261938539416*v + 2998828224856) / 1333059532968
$\beta_{8}$	$=$	$ ( 85612953886 \nu^{13} + 16264470305 \nu^{12} + 921482915118 \nu^{11} - 356051863797 \nu^{10} + \cdots - 1765736543264 ) / 1333059532968 $ (85612953886v^13 + 16264470305v^12 + 921482915118v^11 - 356051863797v^10 + 7735015026262v^9 - 1910875428688v^8 + 21793135917169v^7 - 597638611980v^6 + 47335300868943v^5 - 5855989975445v^4 + 15823864566002v^3 - 14619652891076v^2 + 6531823538900*v - 1765736543264) / 1333059532968
$\beta_{9}$	$=$	$ ( 46709339744 \nu^{13} - 62951064503 \nu^{12} + 458453892831 \nu^{11} - 976727739771 \nu^{10} + \cdots - 7560604912816 ) / 666529766484 $ (46709339744v^13 - 62951064503v^12 + 458453892831v^11 - 976727739771v^10 + 4175375410631v^9 - 7454443513898v^8 + 10619169619475v^7 - 18301069771911v^6 + 17709197603697v^5 - 44379886812946v^4 - 4700179341287v^3 - 24957614550052v^2 + 7449451669744*v - 7560604912816) / 666529766484
$\beta_{10}$	$=$	$ ( - 39935325561 \nu^{13} - 19332041111 \nu^{12} - 439038654861 \nu^{11} + 30071693133 \nu^{10} + \cdots - 361692704256 ) / 444353177656 $ (-39935325561v^13 - 19332041111v^12 - 439038654861v^11 + 30071693133v^10 - 3638117083110v^9 - 228826371659v^8 - 10533430389025v^7 - 3291370968441v^6 - 23951527160414v^5 - 5715034696625v^4 - 11935325570876v^3 + 735474593424v^2 - 5476449728332*v - 361692704256) / 444353177656
$\beta_{11}$	$=$	$ ( - 123613495075 \nu^{13} + 54480375356 \nu^{12} - 1304879782559 \nu^{11} + \cdots + 6893997976920 ) / 1333059532968 $ (-123613495075v^13 + 54480375356v^12 - 1304879782559v^11 + 1354391230958v^10 - 11342065992852v^9 + 9752975752393v^8 - 31911961042430v^7 + 20164783659695v^6 - 63673141851413v^5 + 50297053720696v^4 - 17572714992870v^3 + 34208750881508v^2 - 12349221034816*v + 6893997976920) / 1333059532968
$\beta_{12}$	$=$	$ ( 160791847725 \nu^{13} + 2912293946 \nu^{12} + 1718449387283 \nu^{11} - 967293005048 \nu^{10} + \cdots - 6277732261840 ) / 1333059532968 $ (160791847725v^13 + 2912293946v^12 + 1718449387283v^11 - 967293005048v^10 + 14549149288958v^9 - 6060972938487v^8 + 40778488133572v^7 - 7863469642523v^6 + 86433945635165v^5 - 25907572943612v^4 + 27700891860076v^3 - 32417978264976v^2 + 18122476767800*v - 6277732261840) / 1333059532968
$\beta_{13}$	$=$	$ ( 359879003875 \nu^{13} + 168905739005 \nu^{12} + 3981492790995 \nu^{11} + \cdots + 3234989159440 ) / 1333059532968 $ (359879003875v^13 + 168905739005v^12 + 3981492790995v^11 - 320037828447v^10 + 33028839156670v^9 + 1506419223905v^8 + 96766647886507v^7 + 28343111286855v^6 + 217175772946050v^5 + 50948881639255v^4 + 108419793802160v^3 - 7076071134920v^2 + 28171377563108*v + 3234989159440) / 1333059532968

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{10} + \beta_{7} + \beta_{5} + 3\beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 3 $ b10 + b7 + b5 + 3*b4 - b3 + b2 - b1 - 3
$\nu^{3}$	$=$	$ \beta_{12} + \beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - 6\beta_{2} + 1 $ b12 + b11 + b8 - b7 + b6 - b5 - 6*b2 + 1
$\nu^{4}$	$=$	$ -\beta_{13} - 8\beta_{12} - 9\beta_{11} - 7\beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - 17\beta_{4} + 11\beta_1 $ -b13 - 8b12 - 9b11 - 7b10 + b7 - b6 - b5 - 17b4 + 11*b1
$\nu^{5}$	$=$	$ \beta_{13} + 5 \beta_{10} + \beta_{9} - 11 \beta_{8} + 13 \beta_{7} + 13 \beta_{5} + 15 \beta_{4} + \cdots - 15 $ b13 + 5b10 + b9 - 11b8 + 13b7 + 13b5 + 15b4 - 5b3 + 46b2 - 46b1 - 15
$\nu^{6}$	$=$	$ 75 \beta_{12} + 64 \beta_{11} - 11 \beta_{9} + 14 \beta_{8} - 75 \beta_{7} + 14 \beta_{6} - 64 \beta_{5} + \cdots + 119 $ 75b12 + 64b11 - 11b9 + 14b8 - 75b7 + 14b6 - 64b5 + 52b3 - 108*b2 + 119
$\nu^{7}$	$=$	$ - 14 \beta_{13} - 134 \beta_{12} - 137 \beta_{11} - 70 \beta_{10} + 3 \beta_{7} - 97 \beta_{6} + \cdots + 383 \beta_1 $ -14b13 - 134b12 - 137b11 - 70b10 + 3b7 - 97b6 - 3b5 - 175b4 + 383*b1
$\nu^{8}$	$=$	$ 97 \beta_{13} - 94 \beta_{12} + 94 \beta_{11} + 410 \beta_{10} + 97 \beta_{9} - 154 \beta_{8} + \cdots - 925 $ 97b13 - 94b12 + 94b11 + 410b10 + 97b9 - 154b8 + 533b7 + 627b5 + 925b4 - 410b3 + 1009b2 - 1009b1 - 925
$\nu^{9}$	$=$	$ 1343 \beta_{12} + 1283 \beta_{11} - 154 \beta_{9} + 818 \beta_{8} - 1343 \beta_{7} + 818 \beta_{6} + \cdots + 1799 $ 1343b12 + 1283b11 - 154b9 + 818b8 - 1343b7 + 818b6 - 1283b5 + 753b3 - 3305*b2 + 1799
$\nu^{10}$	$=$	$ - 818 \beta_{13} - 4559 \beta_{12} - 5317 \beta_{11} - 3370 \beta_{10} + 758 \beta_{7} + \cdots + 9212 \beta_1 $ -818b13 - 4559b12 - 5317b11 - 3370b10 + 758b7 - 1557b6 - 758b5 - 7605b4 + 9212*b1
$\nu^{11}$	$=$	$ 1557 \beta_{13} - 799 \beta_{12} + 799 \beta_{11} + 7399 \beta_{10} + 1557 \beta_{9} - 6893 \beta_{8} + \cdots - 17373 $ 1557b13 - 799b12 + 799b11 + 7399b10 + 1557b9 - 6893b8 + 11898b7 + 12697b5 + 17373b4 - 7399b3 + 28989b2 - 28989b1 - 17373
$\nu^{12}$	$=$	$ 45717 \beta_{12} + 39623 \beta_{11} - 6893 \beta_{9} + 15053 \beta_{8} - 45717 \beta_{7} + 15053 \beta_{6} + \cdots + 64515 $ 45717b12 + 39623b11 - 6893b9 + 15053b8 - 45717b7 + 15053b6 - 39623b5 + 28483b3 - 83186*b2 + 64515
$\nu^{13}$	$=$	$ - 15053 \beta_{13} - 108580 \beta_{12} - 117539 \beta_{11} - 69756 \beta_{10} + 8959 \beta_{7} + \cdots + 256254 \beta_1 $ -15053b13 - 108580b12 - 117539b11 - 69756b10 + 8959b7 - 58704b6 - 8959b5 - 162371b4 + 256254*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/952\mathbb{Z}\right)^\times$.

$n$	$239$	$409$	$477$	$785$
$\chi(n)$	$1$	$-1 + \beta_{4}$	$1$	$1$

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} $

137.1

−1.49332	+	2.58650i
−0.759308	+	1.31516i
−0.412260	+	0.714055i
0.193047	−	0.334367i
0.297009	−	0.514435i
0.958831	−	1.66074i
1.21600	−	2.10617i
−1.49332	−	2.58650i
−0.759308	−	1.31516i
−0.412260	−	0.714055i
0.193047	+	0.334367i
0.297009	+	0.514435i
0.958831	+	1.66074i
1.21600	+	2.10617i

−1.49332

2.58650i

−0.658494

−

1.14054i

−2.62518

0.329326i

−2.96000

−

5.12687i

137.2

−0.759308

1.31516i

0.399186

0.691411i

1.00540

2.44728i

0.346902

0.600853i

137.3

−0.412260

0.714055i

1.30057

2.25265i

2.37291

1.17017i

1.16008

2.00932i

137.4

0.193047

−

0.334367i

−1.89700

−

3.28569i

−1.16458

2.37566i

1.42547

2.46898i

137.5

0.297009

−

0.514435i

−0.886441

−

1.53536i

−0.359879

−

2.62116i

1.32357

2.29249i

137.6

0.958831

−

1.66074i

0.937363

1.62356i

−0.860183

2.50202i

−0.338715

−

0.586672i

137.7

1.21600

−

2.10617i

1.30481

2.26001i

−1.86849

−

1.87316i

−1.45731

−

2.52413i

681.1