Properties

 Label 3.19.b.b Level $3$ Weight $19$ Character orbit 3.b Analytic conductor $6.162$ Analytic rank $0$ Dimension $4$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3,19,Mod(2,3)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 19, names="a")

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

chi := DirichletCharacter("3.2");

S:= CuspForms(chi, 19);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$19$$ Character orbit: $$[\chi]$$ $$=$$ 3.b (of order $$2$$, degree $$1$$, 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: $$6.16158413129$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.601940665.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 123x^{2} - 1744x + 16016$$ x^4 - x^3 + 123*x^2 - 1744*x + 16016 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{11}\cdot 3^{11}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + 11 \beta_1 + 3969) q^{3} + (7 \beta_{3} + 7 \beta_{2} - 263384) q^{4} + ( - 9 \beta_{3} + 45 \beta_{2} + 706 \beta_1) q^{5} + ( - 297 \beta_{3} - 9 \beta_{2} + \cdots + 5674536) q^{6}+ \cdots + (567 \beta_{3} + 13041 \beta_{2} + \cdots - 221335335) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b2 + 11*b1 + 3969) * q^3 + (7*b3 + 7*b2 - 263384) * q^4 + (-9*b3 + 45*b2 + 706*b1) * q^5 + (-297*b3 - 9*b2 + 1467*b1 + 5674536) * q^6 + (-287*b3 - 287*b2 - 23936038) * q^7 + (-2016*b3 + 10080*b2 + 229552*b1) * q^8 + (567*b3 + 13041*b2 - 294354*b1 - 221335335) * q^9 $$q - \beta_1 q^{2} + (\beta_{2} + 11 \beta_1 + 3969) q^{3} + (7 \beta_{3} + 7 \beta_{2} - 263384) q^{4} + ( - 9 \beta_{3} + 45 \beta_{2} + 706 \beta_1) q^{5} + ( - 297 \beta_{3} - 9 \beta_{2} + \cdots + 5674536) q^{6}+ \cdots + (8675986498263 \beta_{3} + \cdots + 75\!\cdots\!40) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b2 + 11*b1 + 3969) * q^3 + (7*b3 + 7*b2 - 263384) * q^4 + (-9*b3 + 45*b2 + 706*b1) * q^5 + (-297*b3 - 9*b2 + 1467*b1 + 5674536) * q^6 + (-287*b3 - 287*b2 - 23936038) * q^7 + (-2016*b3 + 10080*b2 + 229552*b1) * q^8 + (567*b3 + 13041*b2 - 294354*b1 - 221335335) * q^9 + (-14230*b3 - 14230*b2 + 365284080) * q^10 + (14427*b3 - 72135*b2 + 3151562*b1) * q^11 + (11907*b3 - 156221*b2 - 10912336*b1 + 1780792776) * q^12 + (271492*b3 + 271492*b2 - 1356555382) * q^13 + (82656*b3 - 413280*b2 + 14575246*b1) * q^14 + (48573*b3 + 353115*b2 - 8111862*b1 - 17071955280) * q^15 + (-1852368*b3 - 1852368*b2 + 50306002048) * q^16 + (-1034892*b3 + 5174460*b2 - 45023208*b1) * q^17 + (-847098*b3 + 3725190*b2 + 307637271*b1 - 156016905840) * q^18 + (5826567*b3 + 5826567*b2 + 47854162370) * q^19 + (1738944*b3 - 8694720*b2 - 644336096*b1) * q^20 + (-488187*b3 - 28329721*b2 + 65323174*b1 - 210874853574) * q^21 + (-7172270*b3 - 7172270*b2 + 1665433191600) * q^22 + (15012522*b3 - 75062610*b2 + 253636396*b1) * q^23 + (32088528*b3 + 79740432*b2 - 1921812624*b1 - 4229325249408) * q^24 + (6957580*b3 + 6957580*b2 + 2851899863545) * q^25 + (-78189696*b3 + 390948480*b2 + 10211538454*b1) * q^26 + (-120138471*b3 - 103375116*b2 - 6667176663*b1 - 1158051922839) * q^27 + (-91961058*b3 - 91961058*b2 + 1437715245008) * q^28 + (117726021*b3 - 588630105*b2 - 7721532154*b1) * q^29 + (-24205230*b3 + 147437010*b2 + 20311702560*b1 - 4295374900560) * q^30 + (419221045*b3 + 419221045*b2 + 8932103771402) * q^31 + (4999680*b3 - 24998400*b2 - 50547157248*b1) * q^32 + (1194271641*b3 - 527493825*b2 + 6719743026*b1 + 3060717755760) * q^33 + (-752846088*b3 - 752846088*b2 - 24320836709568) * q^34 + (241316334*b3 - 1206581670*b2 + 1895023844*b1) * q^35 + (-2766764385*b3 + 356253471*b2 + 76079780064*b1 + 103164363505512) * q^36 + (147405468*b3 + 147405468*b2 - 118824958375558) * q^37 + (-1678051296*b3 + 8390256480*b2 + 142185146902*b1) * q^38 + (461807892*b3 + 2799715646*b2 - 325784793074*b1 + 104227386251274) * q^39 + (2574633760*b3 + 2574633760*b2 - 241750823650560) * q^40 + (1189354698*b3 - 5946773490*b2 + 420913326988*b1) * q^41 + (5808473118*b3 - 3053475810*b2 + 43605567558*b1 + 37287932487120) * q^42 + (-8960155673*b3 - 8960155673*b2 + 399901831143698) * q^43 + (5847565248*b3 - 29237826240*b2 - 1073200880992*b1) * q^44 + (-3148863741*b3 - 13668702615*b2 - 329533029486*b1 - 55310650325280) * q^45 + (13717467932*b3 + 13717467932*b2 + 142865492344992) * q^46 + (-8759224476*b3 + 43796122380*b2 + 564438621368*b1) * q^47 + (-3150877968*b3 + 21948100336*b2 + 2674357020416*b1 - 548206266335616) * q^48 + (13739285812*b3 + 13739285812*b2 - 855946861082061) * q^49 + (-2003783040*b3 + 10018915200*b2 - 2624971434265*b1) * q^50 + (-31897494468*b3 + 39468144996*b2 - 747624439944*b1 - 1246916675934144) * q^51 + (-81002536602*b3 - 81002536602*b2 + 4960980274354832) * q^52 + (-12279925917*b3 + 61399629585*b2 + 5144386753962*b1) * q^53 + (77582178189*b3 - 125189253459*b2 - 2669258849517*b1 - 3505569492615624) * q^54 + (49798073660*b3 + 49798073660*b2 - 21249758648160) * q^55 + (48152559168*b3 - 240762795840*b2 - 616303825312*b1) * q^56 + (9910990467*b3 + 137053076573*b2 - 6145116654002*b1 + 2542337906616162) * q^57 + (175543978750*b3 + 175543978750*b2 - 3982815471605040) * q^58 + (23776766127*b3 - 118883830635*b2 + 509087468002*b1) * q^59 + (-160238983968*b3 - 72798798240*b2 + 2312468383392*b1 + 6180817013902080) * q^60 + (-222731765300*b3 - 222731765300*b2 - 551817186470998) * q^61 + (-120735660960*b3 + 603678304800*b2 + 4741209832318*b1) * q^62 + (93742718391*b3 - 467582684373*b2 + 7105051038204*b1 + 3458295706395978) * q^63 + (-126597386496*b3 - 126597386496*b2 - 13373341897398272) * q^64 + (-12283923834*b3 + 61419619170*b2 - 18736071132844*b1) * q^65 + (-12200031270*b3 + 1555632910170*b2 + 26532129013920*b1 + 3714388572570480) * q^66 + (96122514523*b3 + 96122514523*b2 - 10332884019732478) * q^67 + (-54471055104*b3 + 272355275520*b2 - 12036555134592*b1) * q^68 + (344069165934*b3 - 576134732574*b2 + 11431360704348*b1 + 20355131369645856) * q^69 + (235773289780*b3 + 235773289780*b2 + 1149759107370720) * q^70 + (510738259374*b3 - 2553691296870*b2 - 18502672209756*b1) * q^71 + (-644855904000*b3 - 3327797753280*b2 - 95783160852432*b1 - 1248730460640000) * q^72 + (566482634352*b3 + 566482634352*b2 - 11347488690614062) * q^73 + (-42452774784*b3 + 212263873920*b2 + 123632735119846*b1) * q^74 + (11834843580*b3 + 2958413455765*b2 + 23404358077715*b1 + 14128227877345785) * q^75 + (-1199645386138*b3 - 1199645386138*b2 + 86196978217464464) * q^76 + (-740868875082*b3 + 3704344375410*b2 - 65467582908812*b1) * q^77 + (1633153172742*b3 + 3104474643270*b2 - 76456193495058*b1 - 171457484867426160) * q^78 + (563620174117*b3 + 563620174117*b2 - 44341346007473110) * q^79 + (-285640786944*b3 + 1428203934720*b2 + 156816236816896*b1) * q^80 + (-2229768066582*b3 - 2906838780930*b2 + 129206926869948*b1 - 18195631968179439) * q^81 + (-1718979240580*b3 - 1718979240580*b2 + 221960109520144800) * q^82 + (-10443077631*b3 + 52215388155*b2 - 110848762476802*b1) * q^83 + (-156425759658*b3 + 29883408086*b2 + 121111750482016*b1 - 31421853871205616) * q^84 + (-995855184240*b3 - 995855184240*b2 - 64609566788972160) * q^85 + (2580524833824*b3 - 12902624169120*b2 - 692146268574266*b1) * q^86 + (-185881547817*b3 - 4605359652495*b2 + 103888394420958*b1 + 214724639019436560) * q^87 + (11666925956000*b3 + 11666925956000*b2 - 123685199282960640) * q^88 + (-1262086677882*b3 + 6310433389410*b2 + 476882969842068*b1) * q^89 + (5527968516090*b3 - 2942981624070*b2 - 104578533570240*b1 - 172060869602900880) * q^90 + (-6109111434062*b3 - 6109111434062*b2 - 156280535783851388) * q^91 + (-15188197248*b3 + 75940986240*b2 + 371032701118144*b1) * q^92 + (713094997545*b3 + 15349958749307*b2 - 381761662576298*b1 + 204706858691706858) * q^93 + (-12990590008808*b3 - 12990590008808*b2 + 291043137989201472) * q^94 + (-956337029802*b3 + 4781685149010*b2 - 347761035769132*b1) * q^95 + (-14923006430976*b3 - 647557085952*b2 + 78083628086016*b1 + 294089939422046208) * q^96 + (-2928442015292*b3 - 2928442015292*b2 - 872891133628690078) * q^97 + (-3956914313856*b3 + 19784571569280*b2 + 1304067407126253*b1) * q^98 + (8675986498263*b3 + 5954898468645*b2 - 789455123862822*b1 + 756927064917779040) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 15876 q^{3} - 1053536 q^{4} + 22698144 q^{6} - 95744152 q^{7} - 885341340 q^{9}+O(q^{10})$$ 4 * q + 15876 * q^3 - 1053536 * q^4 + 22698144 * q^6 - 95744152 * q^7 - 885341340 * q^9 $$4 q + 15876 q^{3} - 1053536 q^{4} + 22698144 q^{6} - 95744152 q^{7} - 885341340 q^{9} + 1461136320 q^{10} + 7123171104 q^{12} - 5426221528 q^{13} - 68287821120 q^{15} + 201224008192 q^{16} - 624067623360 q^{18} + 191416649480 q^{19} - 843499414296 q^{21} + 6661732766400 q^{22} - 16917300997632 q^{24} + 11407599454180 q^{25} - 4632207691356 q^{27} + 5750860980032 q^{28} - 17181499602240 q^{30} + 35728415085608 q^{31} + 12242871023040 q^{33} - 97283346838272 q^{34} + 412657454022048 q^{36} - 475299833502232 q^{37} + 416909545005096 q^{39} - 967003294602240 q^{40} + 149151729948480 q^{42} + 15\!\cdots\!92 q^{43}+ \cdots + 30\!\cdots\!60 q^{99}+O(q^{100})$$ 4 * q + 15876 * q^3 - 1053536 * q^4 + 22698144 * q^6 - 95744152 * q^7 - 885341340 * q^9 + 1461136320 * q^10 + 7123171104 * q^12 - 5426221528 * q^13 - 68287821120 * q^15 + 201224008192 * q^16 - 624067623360 * q^18 + 191416649480 * q^19 - 843499414296 * q^21 + 6661732766400 * q^22 - 16917300997632 * q^24 + 11407599454180 * q^25 - 4632207691356 * q^27 + 5750860980032 * q^28 - 17181499602240 * q^30 + 35728415085608 * q^31 + 12242871023040 * q^33 - 97283346838272 * q^34 + 412657454022048 * q^36 - 475299833502232 * q^37 + 416909545005096 * q^39 - 967003294602240 * q^40 + 149151729948480 * q^42 + 1599607324574792 * q^43 - 221242601301120 * q^45 + 571461969379968 * q^46 - 2192825065342464 * q^48 - 3423787444328244 * q^49 - 4987666703736576 * q^51 + 19843921097419328 * q^52 - 14022277970462496 * q^54 - 84999034592640 * q^55 + 10169351626464648 * q^57 - 15931261886420160 * q^58 + 24723268055608320 * q^60 - 2207268745883992 * q^61 + 13833182825583912 * q^63 - 53493367589593088 * q^64 + 14857554290281920 * q^66 - 41331536078929912 * q^67 + 81420525478583424 * q^69 + 4599036429482880 * q^70 - 4994921842560000 * q^72 - 45389954762456248 * q^73 + 56512911509383140 * q^75 + 344787912869857856 * q^76 - 685829939469704640 * q^78 - 177365384029892440 * q^79 - 72782527872717756 * q^81 + 887840438080579200 * q^82 - 125687415484822464 * q^84 - 258438267155888640 * q^85 + 858898556077746240 * q^87 - 494740797131842560 * q^88 - 688243478411603520 * q^90 - 625122143135405552 * q^91 + 818827434766827432 * q^93 + 1164172551956805888 * q^94 + 1176359757688184832 * q^96 - 3491564534514760312 * q^97 + 3027708259671116160 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 123x^{2} - 1744x + 16016$$ :

 $$\beta_{1}$$ $$=$$ $$( -36\nu^{3} + 216\nu^{2} - 11592\nu + 59904 ) / 169$$ (-36*v^3 + 216*v^2 - 11592*v + 59904) / 169 $$\beta_{2}$$ $$=$$ $$( 882\nu^{3} + 22086\nu^{2} + 201870\nu + 229788 ) / 169$$ (882*v^3 + 22086*v^2 + 201870*v + 229788) / 169 $$\beta_{3}$$ $$=$$ $$( -4770\nu^{3} + 1242\nu^{2} - 139662\nu + 5911308 ) / 169$$ (-4770*v^3 + 1242*v^2 - 139662*v + 5911308) / 169
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 108\beta _1 + 1944 ) / 7776$$ (b3 + b2 - 108*b1 + 1944) / 7776 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 17\beta_{2} + 284\beta _1 - 158760 ) / 2592$$ (b3 + 17*b2 + 284*b1 - 158760) / 2592 $$\nu^{3}$$ $$=$$ $$( -38\beta_{3} - 2\beta_{2} + 423\beta _1 + 1181952 ) / 972$$ (-38*b3 - 2*b2 + 423*b1 + 1181952) / 972

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 −6.07949 − 12.9551i 6.57949 − 5.90892i 6.57949 + 5.90892i −6.07949 + 12.9551i
932.767i −4234.02 + 19222.2i −607911. 1.14247e6i 1.79299e7 + 3.94936e6i −9.81043e6 3.22520e8i −3.51567e8 1.62775e8i 1.06566e9
2.2 425.442i 12172.0 15468.1i 81143.0 787628.i −6.58078e6 5.17849e6i −3.80616e7 1.46049e8i −9.11041e7 3.76556e8i −3.35090e8
2.3 425.442i 12172.0 + 15468.1i 81143.0 787628.i −6.58078e6 + 5.17849e6i −3.80616e7 1.46049e8i −9.11041e7 + 3.76556e8i −3.35090e8
2.4 932.767i −4234.02 19222.2i −607911. 1.14247e6i 1.79299e7 3.94936e6i −9.81043e6 3.22520e8i −3.51567e8 + 1.62775e8i 1.06566e9
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.19.b.b 4
3.b odd 2 1 inner 3.19.b.b 4
4.b odd 2 1 48.19.e.b 4
12.b even 2 1 48.19.e.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.19.b.b 4 1.a even 1 1 trivial
3.19.b.b 4 3.b odd 2 1 inner
48.19.e.b 4 4.b odd 2 1
48.19.e.b 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 1051056T_{2}^{2} + 157480796160$$ acting on $$S_{19}^{\mathrm{new}}(3, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + \cdots + 157480796160$$
$3$ $$T^{4} + \cdots + 15\!\cdots\!21$$
$5$ $$T^{4} + \cdots + 80\!\cdots\!00$$
$7$ $$(T^{2} + \cdots + 373401093446500)^{2}$$
$11$ $$T^{4} + \cdots + 44\!\cdots\!00$$
$13$ $$(T^{2} + \cdots - 17\!\cdots\!40)^{2}$$
$17$ $$T^{4} + \cdots + 38\!\cdots\!40$$
$19$ $$(T^{2} + \cdots - 79\!\cdots\!64)^{2}$$
$23$ $$T^{4} + \cdots + 12\!\cdots\!60$$
$29$ $$T^{4} + \cdots + 21\!\cdots\!00$$
$31$ $$(T^{2} + \cdots - 34\!\cdots\!96)^{2}$$
$37$ $$(T^{2} + \cdots + 14\!\cdots\!40)^{2}$$
$41$ $$T^{4} + \cdots + 11\!\cdots\!00$$
$43$ $$(T^{2} + \cdots - 34\!\cdots\!00)^{2}$$
$47$ $$T^{4} + \cdots + 66\!\cdots\!40$$
$53$ $$T^{4} + \cdots + 14\!\cdots\!40$$
$59$ $$T^{4} + \cdots + 61\!\cdots\!00$$
$61$ $$(T^{2} + \cdots - 11\!\cdots\!96)^{2}$$
$67$ $$(T^{2} + \cdots + 84\!\cdots\!80)^{2}$$
$71$ $$T^{4} + \cdots + 58\!\cdots\!00$$
$73$ $$(T^{2} + \cdots - 64\!\cdots\!60)^{2}$$
$79$ $$(T^{2} + \cdots + 11\!\cdots\!36)^{2}$$
$83$ $$T^{4} + \cdots + 24\!\cdots\!40$$
$89$ $$T^{4} + \cdots + 66\!\cdots\!00$$
$97$ $$(T^{2} + \cdots + 74\!\cdots\!20)^{2}$$