LMFDB - Newform orbit 364.2.bq.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [364,2,Mod(121,364)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(364, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

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

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

chi := DirichletCharacter("364.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 364 = 2^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	364.bq (of order $6$, 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:	$2.90655463357$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} + 17 x^{16} - 6 x^{15} + 188 x^{14} - 49 x^{13} + 1116 x^{12} - x^{11} + 4649 x^{10} + \cdots + 16 $ x^18 - x^17 + 17x^16 - 6x^15 + 188x^14 - 49x^13 + 1116x^12 - x^11 + 4649x^10 + 106x^9 + 9589x^8 + 3532x^7 + 12349x^6 + 753x^5 + 2622x^4 - 883x^3 + 541x^2 - 92*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{17}$ 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_{7} q^{7} + ( - \beta_{2} + 1) q^{9} + ( - \beta_{17} + \beta_{16} + \cdots - \beta_{13}) q^{11} + (\beta_{13} + \beta_{12} - \beta_{10} + \cdots - 1) q^{13}+ \cdots + (\beta_{17} + 3 \beta_{15} + \beta_{14} + \cdots + 2) q^{99}+O(q^{100}) $ q - b3 * q^3 + b11 * q^5 - b7 * q^7 + (-b2 + 1) * q^9 + (-b17 + b16 + b15 - b13) * q^11 + (b13 + b12 - b10 - b9 + 2b7 - b6 + b3 + b2 - 1) q^13 - b6 * q^15 + (-b17 - b16 - b15 - b14 - b12 - b11 + b9 + b4 - 1) * q^17 + (-b11 - b10 - b6 + b5 - 2b4 + 1) q^19 + (b16 + 2b15 + b14 + b12 + b11 - b10 - b9 + b5) q^21 + (-b17 + b16 + b14 - b11 - b10 + b7 - 2b6 + b5 - b4 + b3 - b1) q^23 + (-b14 - 2b13 - b12 + b10 + 2b9 - b7 + 2b6 - b5 + b4 - 2b3 + 2b1) q^25 + (-b17 - b16 - b11 - b10 - b9 + b7 - b6 - b5 + b3 + b2 - 2) * q^27 + (-b6 + 2b5 - b4 - b1 + 1) q^29 + (-b15 + b14 - b12 - b10 + b9 - 2b7 + 2b5 - b4 - b3 - b1 + 2) * q^31 + (b15 - b13 + b11 + b10 - b9 - b7 + 2b6 - 2b5) * q^33 + (-b17 + b16 + b15 - 2b13 - b12 - b10 + b9 - b8 - b7 + 2b6 - b5 + b4 - 2b3 + b2 + 2b1) * q^35 + (b17 + b16 - b15 + b12 + b11 + 2b10 + b9 + b8 - b7 + b5 - b4 - b3 - 2b2 - b1 + 2) * q^37 + (b17 - b15 - b14 + b10 - b8 - b5 + 2b4 + b3 + b1 - 1) q^39 + (-b16 + b13 + b12 - b10 - b9 + b8 + b7 - b6 - b4 + 2b3 + b2 - b1 - 1) q^41 + (-b15 - b14 - b12 + b10 + b9 + 2b6 - b5 - b4) q^43 + (b17 - b16 - b15 - b14 + 2b13 + b12 - b11 - b9 + 2b7 + b4 + 1) * q^45 + (b17 - b14 + b13 + b10 - b6 + b4 + 1) * q^47 + (2b17 - b15 + b13 + b12 + b9 - b3 + 2b1 - 1) * q^49 + (b17 - b15 + b14 + 3b13 + b11 + b10 + 2b9 + 2b8 + b7 - b6 + 2b5 - 2b4 - 2b2 - b1 + 2) * q^51 + (2b15 + b14 - b13 + b12 - b10 - b9 - b8 - b7 - b4 - b3 + b1) q^53 + (-b16 + b15 - 2b13 - b12 + b10 - 2b9 + b8 - b7 + b6 - 2b5 - b2 + b1) q^55 + (b17 - b16 - 2b15 + 2b13 + 2b11 + 2b10 + b6 - b5 + 2b4 + b3 - 2b1 - 1) * q^57 + (-2b17 + 2b15 + 2b14 + b13 - 2b10 - 3b9 + b7 - 2b6 + 2b3 - b1) q^59 + (-b16 - b14 + b13 + b12 - b11 - b9 + 2b7 + 2b3 + b2 - 3) * q^61 + (-2b17 + 2b15 + b14 - 3b13 - 2b12 - b11 + b10 - b9 - 2b8 - 2b7 + 2b6 - 3b5 + 3b4 - b3 + b2 + 2b1 - 3) * q^63 + (b17 + b16 - 2b15 + 2b14 - b11 - b10 + b9 - b7 - b6 + 2b5 - b4 - b3 - b2 - b1 + 3) q^65 + (b16 + 3b15 + b14 - 2b13 + b12 - b10 - b9 - 2b8 + b6 - b5 + 2b4 + b3 + b2 - 2b1 - 1) q^67 + (2b17 - 2b16 - 3b15 - 2b14 + 3b13 + 2b11 + 2b10 + b7 + 2b6 - b5) * q^69 + (b15 - b14 + b12 - 2b10 - b9 + 2b7 - b5 + 2b4 - 4) q^71 + (2b15 + b14 - b12 + 2b10 - 2b9 + b7 - b5 + b4 - 2) q^73 + (-b15 - b14 - b13 - b12 + b10 + 2b9 + 2b6 - b5 + 3b4 + b3 - b1) q^75 + (b16 - b14 - b13 + b11 + 2b10 + 4b9 - 2b8 - b7 + b6 + b5 + 2b4 - b3 + b1 - 1) * q^77 + (b17 - 2b15 + b14 + 2b13 + b11 + b10 + b9 + b8 - b7 + b4 - b2 - 2b1 - 1) q^79 + (2b17 + b16 + b15 - b14 + 2b13 + b12 + 2b11 + b10 + b7 + b6 + b5 + b3 + b2) q^81 + (-b16 - 2b15 - b14 + b13 - b12 + b10 + b9 + 4b8 - 4b4 - 2b2 + 2) * q^83 + (-2b17 - 2b16 - b13 - 2b12 - 2b11 + b10 - b9 - b5 + b4 + b3 + b1 - 2) * q^85 + (b17 - b16 - 3b15 + b14 + 2b13 - b12 + 2b11 + 4b10 + b9 + b8 - 2b7 - b4 - b2 - b1 + 1) q^87 + (-2b13 - 2b9 - b8 - b5 + b4 + b3 + 2b2 + b1 - 2) q^89 + (-b15 - 2b14 + b12 - 2b11 - b10 + b9 + b7 + b6 + b5 - 2b4 + b3 - b1 + 1) q^91 + (b17 + b16 - b15 + 2b14 + 2b13 - b12 + b11 + 2b10 + 3b9 + b8 - 3b7 + 3b5 - 2b4 - b3 - 2b2 - b1 + 4) * q^93 + (-b15 + b13 - b11 - 2b10 + b9 - 3b4 + 2b1 + 3) q^95 + (b15 + b14 - 2b13 - b12 + b10 - 3b9 - b8 - b5 + b4 + b3 + 2b2 + b1 - 2) q^97 + (b17 + 3b15 + b14 - 2b13 + b12 - 2b11 - 3b10 - 2b9 - b7 - 4b4 + 2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{3} + q^{7} + 12 q^{9} - 4 q^{13} + 6 q^{15} - 10 q^{17} - 7 q^{21} - 6 q^{23} + 5 q^{25} - 20 q^{27} + 2 q^{29} + 9 q^{31} + 3 q^{35} + 18 q^{37} + 11 q^{39} - 9 q^{41} - 14 q^{43} + 30 q^{45}+ \cdots - 18 q^{97}+O(q^{100}) $ 18 * q - 2 * q^3 + q^7 + 12 * q^9 - 4 * q^13 + 6 * q^15 - 10 * q^17 - 7 * q^21 - 6 * q^23 + 5 * q^25 - 20 * q^27 + 2 * q^29 + 9 * q^31 + 3 * q^35 + 18 * q^37 + 11 * q^39 - 9 * q^41 - 14 * q^43 + 30 * q^45 + 36 * q^47 - 11 * q^49 + 2 * q^51 - 13 * q^53 - q^55 - 42 * q^61 - 32 * q^63 + 30 * q^65 - q^69 - 45 * q^71 - 33 * q^73 + 25 * q^75 - 9 * q^77 - 14 * q^79 + 2 * q^81 - 30 * q^85 + 2 * q^87 - 9 * q^89 + 21 * q^93 + 30 * q^95 - 18 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{18} - x^{17} + 17 x^{16} - 6 x^{15} + 188 x^{14} - 49 x^{13} + 1116 x^{12} - x^{11} + 4649 x^{10} + \cdots + 16 $ x^18 - x^17 + 17*x^16 - 6*x^15 + 188*x^14 - 49*x^13 + 1116*x^12 - x^11 + 4649*x^10 + 106*x^9 + 9589*x^8 + 3532*x^7 + 12349*x^6 + 753*x^5 + 2622*x^4 - 883*x^3 + 541*x^2 - 92*x + 16 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 51\!\cdots\!75 \nu^{17} + \cdots + 19\!\cdots\!64 ) / 47\!\cdots\!71 $ (-51819620283698375v^17 + 2059632229889843775v^16 - 2780993119015092909v^15 + 33815319430162281148v^14 - 19716669488645619111v^13 + 371590790536576997759v^12 - 137629628199865653215v^11 + 2150703891466076233719v^10 - 142793393379230518920v^9 + 8892625685473055274899v^8 - 21345114540323625972v^7 + 17337355070915999628404v^6 + 6683477491031097942423v^5 + 23549398233566799817530v^4 + 64572050537397600002v^3 + 5252945082178456159547v^2 - 911227674933138092484*v + 19005549002757680306764) / 4746040363230798388371
$\beta_{3}$	$=$	$ ( 13\!\cdots\!80 \nu^{17} + \cdots + 78\!\cdots\!24 ) / 47\!\cdots\!71 $ (1336721864655422080v^17 - 1284902244371723705v^16 + 20664639469252331585v^15 - 5239338068917439571v^14 + 217488391125057069892v^13 - 45782701879470062809v^12 + 1120190810418874043521v^11 + 136292906335210231135v^10 + 4063716057316981016201v^9 + 284485911032705259400v^8 + 3925200274707787050221v^7 + 4742646740503274412532v^6 - 830176764286192362484v^5 - 5676925926945565116183v^4 - 20044513504440283123770v^3 - 1244897457028135296642v^2 + 216261809830925574104*v + 788249263384839261124) / 4746040363230798388371
$\beta_{4}$	$=$	$ ( - 19\!\cdots\!81 \nu^{17} + \cdots + 18\!\cdots\!68 ) / 18\!\cdots\!84 $ (-197062315846209815281v^17 + 202409203304831503601v^16 - 3355198978363053754597v^15 + 1265032452954268218026v^14 - 37068672731363115031112v^13 + 10526007040964509228337v^12 - 220104675291888034104832v^11 + 4677825557521705989365v^10 - 915597534743688590316829v^9 - 4633741250430316354982v^8 - 1888492603005175097691909v^7 - 680323298469981919371608v^6 - 2414551951422831911254941v^5 - 151708630889340760356529v^4 - 539405095856544396131514v^3 + 93827970874442134398043v^2 - 111590302700912051253589*v + 18994780297175005302268) / 18984161452923193553484
$\beta_{5}$	$=$	$ ( - 16\!\cdots\!85 \nu^{17} + \cdots - 11\!\cdots\!92 ) / 14\!\cdots\!13 $ (-167645750745686432085v^17 + 126666342420488663617v^16 - 2833459371212732594565v^15 + 334761672364525559662v^14 - 31674917294537295795470v^13 + 651486087349924425221v^12 - 189466876211595613927590v^11 - 44395243637877926218061v^10 - 804361744314134210613537v^9 - 209545286532662581555094v^8 - 1712202155899572144054949v^7 - 993278213963722985058574v^6 - 2396949861033495337178120v^5 - 751746795312133133946945v^4 - 665502461989616001943984v^3 + 7588972525525974803903v^2 - 40380374806760248966039*v - 11536916353080206823092) / 14238121089692395165113
$\beta_{6}$	$=$	$ ( 20\!\cdots\!81 \nu^{17} + \cdots - 22\!\cdots\!88 ) / 14\!\cdots\!13 $ (207404415944787318681v^17 - 234868214557738990054v^16 + 3517768669128498574245v^15 - 1663811519389338571993v^14 + 38543505109217442795050v^13 - 14922220460980756178282v^12 + 226118240624379976306575v^11 - 27038752045522039357660v^10 + 924605572772317541741193v^9 - 94063781986250040601822v^8 + 1824575835507140361929362v^7 + 510722944572844227164614v^6 + 2138892704206704932352632v^5 - 235697768450535871819314v^4 + 167485818470790813746344v^3 - 227242519887075112900253v^2 + 78492502987744308497599*v - 22350464033363397556888) / 14238121089692395165113
$\beta_{7}$	$=$	$ ( - 80\!\cdots\!19 \nu^{17} + \cdots - 28\!\cdots\!54 ) / 28\!\cdots\!26 $ (-808942959858078292819v^17 - 468499485767138198419v^16 - 12747722289292823438047v^15 - 16553083090036948191638v^14 - 149066557983517813281900v^13 - 198277348413312471728075v^12 - 891245654265081665482540v^11 - 1404652167088201682541169v^10 - 4060070724291727248515791v^9 - 5984753824924205440487382v^8 - 9125034558327265319273731v^7 - 14968939395178615091898450v^6 - 16952291419210193125878775v^5 - 17016483090752181402812867v^4 - 5977467262419235938645910v^3 - 2337374586491313000101977v^2 + 436814855150930976959247*v - 283351276857083453584054) / 28476242179384790330226
$\beta_{8}$	$=$	$ ( - 19\!\cdots\!81 \nu^{17} + \cdots + 18\!\cdots\!68 ) / 47\!\cdots\!71 $ (-197062315846209815281v^17 + 202409203304831503601v^16 - 3355198978363053754597v^15 + 1265032452954268218026v^14 - 37068672731363115031112v^13 + 10526007040964509228337v^12 - 220104675291888034104832v^11 + 4677825557521705989365v^10 - 915597534743688590316829v^9 - 4633741250430316354982v^8 - 1888492603005175097691909v^7 - 680323298469981919371608v^6 - 2414551951422831911254941v^5 - 151708630889340760356529v^4 - 539405095856544396131514v^3 + 98574011237672932786414v^2 - 111590302700912051253589*v + 18994780297175005302268) / 4746040363230798388371
$\beta_{9}$	$=$	$ ( - 12\!\cdots\!15 \nu^{17} + \cdots + 31\!\cdots\!14 ) / 28\!\cdots\!26 $ (-1286009689554228745915v^17 + 2187337398002621167723v^16 - 22366937726333556517201v^15 + 22666739988652310149548v^14 - 240484595621207534647360v^13 + 230555121789114091522433v^12 - 1405407367196293376959414v^11 + 993001916847706500109465v^10 - 5543110123351768017998287v^9 + 4086614719998853410850922v^8 - 10421756558670216255008217v^7 + 4279871026208408717668492v^6 - 9005030103225001206062495v^5 + 11860227993848615697911779v^4 + 2067529194670241601963288v^3 + 4097400567158179518029715v^2 - 679536308946915269251457*v + 313340186428784562644914) / 28476242179384790330226
$\beta_{10}$	$=$	$ ( 27\!\cdots\!49 \nu^{17} + \cdots + 22\!\cdots\!44 ) / 56\!\cdots\!52 $ (2703386920070797683449v^17 - 1203186253308902680385v^16 + 44511439413290650003949v^15 + 9184590640940138574546v^14 + 500242448206294507961792v^13 + 148407730531549867873079v^12 + 2954684976605687492315072v^11 + 1659637080423831563253211v^10 + 12637471757778561977796005v^9 + 7202428281873311563864514v^8 + 26384703194744529150405549v^7 + 23695474567316907441879112v^6 + 39400129710100549441804453v^5 + 20179020611774111733760105v^4 + 9152476611872020574513802v^3 + 944862471419112908275845v^2 + 523206354107897195929285*v + 226779914531504842977244) / 56952484358769580660452
$\beta_{11}$	$=$	$ ( 28\!\cdots\!73 \nu^{17} + \cdots - 42\!\cdots\!72 ) / 56\!\cdots\!52 $ (2884229088270051705773v^17 - 3554812091252797434113v^16 + 49538559870272833652609v^15 - 28639212014471240612898v^14 + 543574115284227822923972v^13 - 268026894503381716764757v^12 + 3221405606858777401343552v^11 - 760751733934652507416133v^10 + 13231200056815958675266433v^9 - 2911718693899911361642210v^8 + 26818691581290875480436921v^7 + 3338288516171534048570440v^6 + 31644232155191976574356481v^5 - 7415974940666632414265411v^4 + 4555461488195543036749026v^3 - 5208784132900919663973495v^2 + 1590723826856201872038805*v - 426870575347885752795272) / 56952484358769580660452
$\beta_{12}$	$=$	$ ( 13\!\cdots\!18 \nu^{17} + \cdots + 36\!\cdots\!22 ) / 14\!\cdots\!13 $ (1358094491405851926818v^17 + 436157428116211947400v^16 + 21654587565672789812921v^15 + 21981127199563578755400v^14 + 250684953563831376738515v^13 + 268417310386646587728563v^12 + 1495320359325242672119298v^11 + 1981042426106140922175604v^10 + 6711299028385306352145773v^9 + 8470824163802515465836143v^8 + 14866102571570441825755857v^7 + 21972230471739305888789455v^6 + 26453219609344194704978791v^5 + 24288860944251242995915795v^4 + 9080522625897302892157488v^3 + 3426104224069450498334361v^2 - 291381688126951878262040*v + 360180505865082752810722) / 14238121089692395165113
$\beta_{13}$	$=$	$ ( 34\!\cdots\!17 \nu^{17} + \cdots - 15\!\cdots\!64 ) / 18\!\cdots\!84 $ (3431383862944561351817v^17 - 2777833985174694416533v^16 + 57604887657766012459761v^15 - 9457233025556124592690v^14 + 639940758934935096717476v^13 - 45719088517997343026245v^12 + 3783311846923587741036516v^11 + 719714071240070287448643v^10 + 15867315425573571805675569v^9 + 3344593834195614932462362v^8 + 32603326857550793703388309v^7 + 18139791718913671579906020v^6 + 43872806553744749921623849v^5 + 9911342973559358663367153v^4 + 8215734451527360264527758v^3 - 2030758961494825810379327v^2 + 884663268253793241704077*v - 15220408695789011701464) / 18984161452923193553484
$\beta_{14}$	$=$	$ ( - 26\!\cdots\!98 \nu^{17} + \cdots - 43\!\cdots\!55 ) / 14\!\cdots\!13 $ (-2602243576899060058998v^17 + 1760750007886249784875v^16 - 43428749610267288120300v^15 + 1409257050200742852928v^14 - 484768821942839250320438v^13 - 29342107567108308631636v^12 - 2869002580493184109811583v^11 - 922040620720661263317113v^10 - 12132706780512477583788462v^9 - 4111769233091150975048420v^8 - 25170373326585922030224805v^7 - 16944421850425630841358421v^6 - 35320816149241084916590604v^5 - 11817098492620710219756201v^4 - 7424175486863380656161017v^3 + 917490219955818490583552v^2 - 502735460487028186019773*v - 43559419617840900025055) / 14238121089692395165113
$\beta_{15}$	$=$	$ ( - 10\!\cdots\!55 \nu^{17} + \cdots + 52\!\cdots\!40 ) / 56\!\cdots\!52 $ (-10471868114744633028255v^17 + 9715472553719092255727v^16 - 176796593202646450910619v^15 + 49628510027133858791018v^14 - 1956276799435271363824336v^13 + 370195097322441062581855v^12 - 11561393339788565345552460v^11 - 832882275558755578339849v^10 - 48158004274107294586556859v^9 - 4484073943834505948833786v^8 - 98272667706359871978775499v^7 - 43570508129486494656173468v^6 - 127307927172645588136187815v^5 - 14239848306447420812503287v^4 - 21463544758282210916150462v^3 + 9546199175135092701574633v^2 - 3253182358890573296225975*v + 528491096970737480427740) / 56952484358769580660452
$\beta_{16}$	$=$	$ ( 58\!\cdots\!98 \nu^{17} + \cdots - 28\!\cdots\!11 ) / 14\!\cdots\!13 $ (5810954692299075155698v^17 - 5544059391866606372370v^16 + 98210601229208855850871v^15 - 30112276516598151875918v^14 + 1085678234662629223596824v^13 - 234255084038458983934110v^12 + 6414090462415631814842629v^11 + 289894171147719252460185v^10 + 26670931920012713553320941v^9 + 1755171420388753392881558v^8 + 54292662893888463011035106v^7 + 22622613702004222169974003v^6 + 69646927996949875980641220v^5 + 5714546443269167622611705v^4 + 11147475212441957740571318v^3 - 5828731122441578582966467v^2 + 1900904345644812803945305*v - 283952065371760565790111) / 14238121089692395165113
$\beta_{17}$	$=$	$ ( - 14\!\cdots\!53 \nu^{17} + \cdots + 17\!\cdots\!40 ) / 28\!\cdots\!26 $ (-14058580563131991178753v^17 + 11078589677229294862521v^16 - 235832230401781912915231v^15 + 33750212393970719328506v^14 - 2622226840812873771081656v^13 + 131631774407272556440497v^12 - 15509798734539618505020586v^11 - 3275201671950396133869837v^10 - 65156416247258798139545467v^9 - 15075026865867583320031346v^8 - 134232932613595289390768249v^7 - 77084064100147533613557340v^6 - 182182177761002853720999153v^5 - 44400833511957766241772215v^4 - 35624636049712476362005082v^3 + 7862819610901581097699309v^2 - 3830602094380323610505095*v + 172466090394962102366640) / 28476242179384790330226

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{8} - 4\beta_{4} $ b8 - 4*b4
$\nu^{3}$	$=$	$ -\beta_{17} - \beta_{16} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - 5\beta_{3} + \beta_{2} - 2 $ -b17 - b16 - b11 - b10 - b9 + b7 - b6 - b5 - 5*b3 + b2 - 2
$\nu^{4}$	$=$	$ - \beta_{17} - 2 \beta_{16} - 2 \beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} + \cdots - 27 $ -b17 - 2b16 - 2b15 - b14 - b13 - 2b12 - b11 + b10 - 8b8 - 2b7 + b6 - 2b5 + 27b4 + 8b2 - b1 - 27
$\nu^{5}$	$=$	$ 2 \beta_{17} - 2 \beta_{16} - 2 \beta_{15} - 12 \beta_{14} - 6 \beta_{13} - 10 \beta_{12} + \cdots - 32 \beta_1 $ 2b17 - 2b16 - 2b15 - 12b14 - 6b13 - 10b12 + 2b11 + 12b10 + 8b9 - 12b8 - 10b7 + 22b6 - 11b5 + 26b4 + 32b3 - 32b1
$\nu^{6}$	$=$	$ 29 \beta_{17} + 15 \beta_{16} + 12 \beta_{15} - 14 \beta_{14} + 26 \beta_{13} + 14 \beta_{12} + \cdots + 207 $ 29b17 + 15b16 + 12b15 - 14b14 + 26b13 + 14b12 + 30b11 + 14b10 + 2b9 + 12b7 + 18b6 + 18b5 + 16b3 - 66b2 + 207
$\nu^{7}$	$=$	$ 89 \beta_{17} + 122 \beta_{16} + 63 \beta_{15} + 89 \beta_{14} + 89 \beta_{13} + 122 \beta_{12} + \cdots + 294 $ 89b17 + 122b16 + 63b15 + 89b14 + 89b13 + 122b12 + 85b11 - 41b10 + 126b8 + 63b7 - 108b6 + 216b5 - 294b4 - 126b2 + 229*b1 + 294
$\nu^{8}$	$=$	$ - 157 \beta_{17} + 157 \beta_{16} + 173 \beta_{15} + 327 \beta_{14} - 125 \beta_{13} + 170 \beta_{12} + \cdots + 209 \beta_1 $ -157b17 + 157b16 + 173b15 + 327b14 - 125b13 + 170b12 - 195b11 - 346b10 - 48b9 + 566b8 + 154b7 - 452b6 + 226b5 - 1703b4 - 209b3 + 209b1
$\nu^{9}$	$=$	$ - 1193 \beta_{17} - 789 \beta_{16} - 343 \beta_{15} + 404 \beta_{14} - 747 \beta_{13} - 404 \beta_{12} + \cdots - 3143 $ -1193b17 - 789b16 - 343b15 + 404b14 - 747b13 - 404b12 - 1265b11 - 717b10 - 441b9 + 37b7 - 1037b6 - 1037b5 - 1772b3 + 1272b2 - 3143
$\nu^{10}$	$=$	$ - 1771 \beta_{17} - 3399 \beta_{16} - 3008 \beta_{15} - 1771 \beta_{14} - 1589 \beta_{13} + \cdots - 14725 $ -1771b17 - 3399b16 - 3008b15 - 1771b14 - 1589b13 - 3399b12 - 1518b11 + 2134b10 + 182b9 - 5026b8 - 2826b7 + 2500b6 - 5000b5 + 14725b4 + 5026b2 - 2433b1 - 14725
$\nu^{11}$	$=$	$ 4427 \beta_{17} - 4427 \beta_{16} - 4544 \beta_{15} - 11550 \beta_{14} + 1002 \beta_{13} + \cdots - 14581 \beta_1 $ 4427b17 - 4427b16 - 4544b15 - 11550b14 + 1002b13 - 7123b12 + 6249b11 + 12461b10 + 3542b9 - 12629b8 - 7006b7 + 19830b6 - 9915b5 + 32454b4 + 14581b3 - 14581b1
$\nu^{12}$	$=$	$ 34133 \beta_{17} + 17845 \beta_{16} + 11667 \beta_{15} - 16288 \beta_{14} + 27955 \beta_{13} + \cdots + 132000 $ 34133b17 + 17845b16 + 11667b15 - 16288b14 + 27955b13 + 16288b12 + 37042b11 + 14936b10 + 4349b9 + 11939b7 + 26092b6 + 26092b5 + 26410b3 - 45883b2 + 132000
$\nu^{13}$	$=$	$ 65591 \beta_{17} + 111597 \beta_{16} + 83765 \beta_{15} + 65591 \beta_{14} + 63794 \beta_{13} + \cdots + 327660 $ 65591b17 + 111597b16 + 83765b15 + 65591b14 + 63794b13 + 111597b12 + 55515b11 - 66158b10 - 1797b9 + 124271b8 + 81968b7 - 94864b6 + 189728b5 - 327660b4 - 124271b2 + 125905b1 + 327660
$\nu^{14}$	$=$	$ - 160099 \beta_{17} + 160099 \beta_{16} + 177531 \beta_{15} + 337065 \beta_{14} - 113959 \beta_{13} + \cdots + 274743 \beta_1 $ -160099b17 + 160099b16 + 177531b15 + 337065b14 - 113959b13 + 176966b12 - 222239b11 - 368135b10 - 63572b9 + 427364b8 + 159534b7 - 528236b6 + 264118b5 - 1213942b4 - 274743b3 + 274743b1
$\nu^{15}$	$=$	$ - 1078369 \beta_{17} - 613705 \beta_{16} - 354497 \beta_{15} + 464664 \beta_{14} - 819161 \beta_{13} + \cdots - 3260085 $ -1078369b17 - 613705b16 - 354497b15 + 464664b14 - 819161b13 - 464664b12 - 1182953b11 - 509121b10 - 236193b9 - 228471b7 - 909458b6 - 909458b5 - 1127275b3 + 1216138b2 - 3260085
$\nu^{16}$	$=$	$ - 1739546 \beta_{17} - 3299254 \beta_{16} - 2823663 \beta_{15} - 1739546 \beta_{14} - 1576975 \beta_{13} + \cdots - 11361824 $ -1739546b17 - 3299254b16 - 2823663b15 - 1739546b14 - 1576975b13 - 3299254b12 - 1420778b11 + 2197244b10 + 162571b9 - 4035487b8 - 2661092b7 + 2627562b6 - 5255124b5 + 11361824b4 + 4035487b2 - 2784149b1 - 11361824
$\nu^{17}$	$=$	$ 4616962 \beta_{17} - 4616962 \beta_{16} - 4884083 \beta_{15} - 10425525 \beta_{14} + \cdots - 10358511 \beta_1 $ 4616962b17 - 4616962b16 - 4884083b15 - 10425525b14 + 2537345b13 - 5808563b12 + 6718136b11 + 11476112b10 + 2346738b9 - 11858436b8 - 5541442b7 + 17475252b6 - 8737626b5 + 32129056b4 + 10358511b3 - 10358511b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/364\mathbb{Z}\right)^\times$.

$n$	$157$	$183$	$197$
$\chi(n)$	$-1 + \beta_{4}$	$1$	$\beta_{4}$

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} $

121.1

1.55551	−	2.69422i
1.19055	−	2.06209i
0.893365	−	1.54735i
0.141496	−	0.245078i
0.118765	−	0.205706i
−0.302937	+	0.524702i
−0.591009	+	1.02366i
−1.23985	+	2.14749i
−1.26588	+	2.19257i
1.55551	+	2.69422i
1.19055	+	2.06209i
0.893365	+	1.54735i
0.141496	+	0.245078i
0.118765	+	0.205706i
−0.302937	−	0.524702i
−0.591009	−	1.02366i
−1.23985	−	2.14749i
−1.26588	−	2.19257i

−3.11102

1.03572

0.597971i

−1.67107

2.05123i

6.67841

121.2

−2.38110

−1.65948

−

0.958100i

2.32630

−

1.26029i

2.66964

121.3

−1.78673

1.22316

0.706193i

−0.507200

−

2.59668i

0.192402

121.4

−0.282992

−0.709183

−

0.409447i

−1.35281

2.27374i

−2.91992

121.5

−0.237529

−2.23233

−

1.28884i

2.37380

1.16836i

−2.94358

121.6

0.605874

3.48722

2.01335i

2.30111

−

1.30571i

−2.63292

121.7

1.18202

−3.36124

−

1.94061i

−1.68894

−

2.03654i

−1.60283

121.8

2.47971

0.484874

0.279942i

0.747784

2.53788i

3.14895

121.9

2.53177

1.73126

0.999545i

−2.02898

−

1.69801i

3.40984

361.1