# Properties

 Label 13.14.a.b Level $13$ Weight $14$ Character orbit 13.a Self dual yes Analytic conductor $13.940$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,14,Mod(1,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 14, names="a")

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

chi := DirichletCharacter("13.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 13.a (trivial)

## 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: yes Analytic conductor: $$13.9400207637$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 35596x^{5} + 594856x^{4} + 263632368x^{3} + 252644208x^{2} - 410033371968x - 5442114981888$$ x^7 - x^6 - 35596*x^5 + 594856*x^4 + 263632368*x^3 + 252644208*x^2 - 410033371968*x - 5442114981888 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}\cdot 3^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 18) q^{2} + (\beta_{2} + \beta_1 + 104) q^{3} + (\beta_{3} + 2 \beta_{2} + 12 \beta_1 + 2306) q^{4} + ( - \beta_{6} + \beta_{3} + \cdots + 4979) q^{5}+ \cdots + (\beta_{6} + 36 \beta_{5} + \cdots + 351176) q^{9}+O(q^{10})$$ q + (b1 + 18) * q^2 + (b2 + b1 + 104) * q^3 + (b3 + 2*b2 + 12*b1 + 2306) * q^4 + (-b6 + b3 + 6*b2 + 38*b1 + 4979) * q^5 + (2*b6 - b4 + b3 + 56*b2 + 419*b1 + 11812) * q^6 + (5*b6 + b5 + 6*b4 - 6*b3 + 82*b2 + 915*b1 + 18717) * q^7 + (-6*b6 - 10*b5 - 8*b4 - 23*b3 + 112*b2 + 4070*b1 + 17680) * q^8 + (b6 + 36*b5 - 8*b4 - 61*b3 - 134*b2 + 7774*b1 + 351176) * q^9 $$q + (\beta_1 + 18) q^{2} + (\beta_{2} + \beta_1 + 104) q^{3} + (\beta_{3} + 2 \beta_{2} + 12 \beta_1 + 2306) q^{4} + ( - \beta_{6} + \beta_{3} + \cdots + 4979) q^{5}+ \cdots + ( - 32222537 \beta_{6} + \cdots - 2486366775433) q^{99}+O(q^{100})$$ q + (b1 + 18) * q^2 + (b2 + b1 + 104) * q^3 + (b3 + 2*b2 + 12*b1 + 2306) * q^4 + (-b6 + b3 + 6*b2 + 38*b1 + 4979) * q^5 + (2*b6 - b4 + b3 + 56*b2 + 419*b1 + 11812) * q^6 + (5*b6 + b5 + 6*b4 - 6*b3 + 82*b2 + 915*b1 + 18717) * q^7 + (-6*b6 - 10*b5 - 8*b4 - 23*b3 + 112*b2 + 4070*b1 + 17680) * q^8 + (b6 + 36*b5 - 8*b4 - 61*b3 - 134*b2 + 7774*b1 + 351176) * q^9 + (-46*b6 - 44*b5 - 9*b4 + 47*b3 - 820*b2 + 16269*b1 + 474060) * q^10 + (-45*b6 - 33*b5 + 90*b4 + 150*b3 - 1989*b2 + 16954*b1 + 1270467) * q^11 + (230*b6 + 90*b5 - 46*b4 + 391*b3 - 744*b2 + 29544*b1 + 3618624) * q^12 + 4826809 * q^13 + (354*b6 + 108*b5 - 219*b4 + 123*b3 + 4668*b2 - 27139*b1 + 9621504) * q^14 + (-806*b6 - 198*b5 - 164*b4 - 2068*b3 + 8383*b2 - 158471*b1 + 11301758) * q^15 + (-398*b6 - 418*b5 + 480*b4 - 173*b3 + 5452*b2 - 267690*b1 + 22787548) * q^16 + (-1529*b6 - 704*b5 + 896*b4 - 3047*b3 + 10922*b2 - 475638*b1 + 31812375) * q^17 + (3598*b6 + 3420*b5 - 263*b4 + 2993*b3 + 3316*b2 - 314234*b1 + 85200554) * q^18 + (49*b6 + 209*b5 - 2010*b4 - 1490*b3 + 13705*b2 - 270630*b1 + 93279689) * q^19 + (554*b6 - 4826*b5 + 2018*b4 + 18639*b3 - 85324*b2 + 284156*b1 + 133588540) * q^20 + (3495*b6 + 1224*b5 - 4272*b4 - 2631*b3 + 19406*b2 + 663446*b1 + 173893141) * q^21 + (-13224*b6 - 8220*b5 + 102*b4 + 12130*b3 - 223396*b2 + 1827186*b1 + 196064116) * q^22 + (11668*b6 + 13616*b5 + 5248*b4 - 21516*b3 + 70324*b2 + 1217156*b1 + 256226132) * q^23 + (-1890*b6 + 11682*b5 + 4572*b4 - 17163*b3 - 158844*b2 + 3423822*b1 + 270253668) * q^24 + (-18875*b6 - 14524*b5 + 5400*b4 - 44105*b3 + 243178*b2 + 1188750*b1 + 35033878) * q^25 + (4826809*b1 + 86882562) * q^26 + (15338*b6 - 24642*b5 - 23020*b4 - 11552*b3 + 992813*b2 - 154549*b1 - 273525986) * q^27 + (2*b6 + 17182*b5 - 53610*b4 + 41*b3 + 161312*b2 + 5149800*b1 - 256372552) * q^28 + (70068*b6 - 2464*b5 + 69280*b4 + 194212*b3 + 554932*b2 - 14297420*b1 - 674487926) * q^29 + (-12230*b6 - 15336*b5 + 15199*b4 + 6929*b3 - 498464*b2 - 3974681*b1 - 1416391516) * q^30 + (-60351*b6 + 92853*b5 - 5730*b4 + 55586*b3 - 669713*b2 - 28766844*b1 - 1344009803) * q^31 + (-6954*b6 + 25498*b5 + 62624*b4 - 12691*b3 - 2013364*b2 - 9280638*b1 - 2459651044) * q^32 + (-28378*b6 - 113940*b5 - 29272*b4 + 25942*b3 + 140876*b2 - 33421480*b1 - 3657820646) * q^33 + (-107478*b6 - 99468*b5 + 10539*b4 - 237589*b3 - 2882228*b2 + 8846205*b1 - 4277447884) * q^34 + (-150800*b6 + 17316*b5 - 253032*b4 + 81788*b3 - 1248969*b2 - 22148837*b1 - 2979254588) * q^35 + (430070*b6 + 45306*b5 - 49666*b4 - 777644*b3 + 3602162*b2 + 50792360*b1 - 4537183382) * q^36 + (7055*b6 + 133012*b5 + 314136*b4 + 42629*b3 - 1793182*b2 - 31636362*b1 - 2114872741) * q^37 + (126120*b6 + 95516*b5 + 33874*b4 - 73994*b3 + 2612404*b2 + 88211894*b1 - 1079945540) * q^38 + (4826809*b2 + 4826809*b1 + 501988136) * q^39 + (-413922*b6 - 209502*b5 + 126636*b4 + 53793*b3 + 2824092*b2 + 145045662*b1 + 1487200860) * q^40 + (416578*b6 - 149920*b5 - 270944*b4 + 2287062*b3 + 5062912*b2 + 37071680*b1 + 8553501776) * q^41 + (469954*b6 + 367524*b5 + 48055*b4 + 748103*b3 + 11193292*b2 + 156577255*b1 + 9878053376) * q^42 + (-1073474*b6 - 139090*b5 + 52980*b4 - 468788*b3 - 11666159*b2 - 37602081*b1 + 10359387422) * q^43 + (-1511508*b6 - 879244*b5 - 335516*b4 + 2488642*b3 - 4453256*b2 + 94331344*b1 + 11780753320) * q^44 + (-5260*b6 + 130284*b5 + 277160*b4 - 4897880*b3 - 3226708*b2 - 202533640*b1 + 8615455042) * q^45 + (1864448*b6 + 1397056*b5 - 445704*b4 - 1928328*b3 + 7880656*b2 + 53698368*b1 + 16915868704) * q^46 + (1796245*b6 + 520825*b5 + 274934*b4 + 2894266*b3 - 35579914*b2 - 305953265*b1 + 27986857125) * q^47 + (-1309318*b6 + 41526*b5 + 240008*b4 - 2006309*b3 - 8610924*b2 - 215644698*b1 + 10007304804) * q^48 + (2631533*b6 - 1224728*b5 + 65040*b4 - 6531405*b3 - 15035858*b2 - 210288810*b1 + 7943503312) * q^49 + (-1342362*b6 - 1390180*b5 + 336901*b4 + 6129013*b3 - 5886812*b2 - 306142028*b1 + 12576130606) * q^50 + (-3799010*b6 - 333702*b5 + 378460*b4 - 6912280*b3 - 9334421*b2 - 606149147*b1 + 20407052234) * q^51 + (4826809*b3 + 9653618*b2 + 57921708*b1 + 11130621554) * q^52 + (1302870*b6 + 3172776*b5 - 1180272*b4 + 1844298*b3 + 24051768*b2 - 120466176*b1 + 58490341848) * q^53 + (1553366*b6 - 351288*b5 + 177617*b4 + 6985903*b3 + 111381872*b2 - 21036871*b1 - 6605853716) * q^54 + (-2474230*b6 - 1565666*b5 - 2512236*b4 + 13206704*b3 - 28270804*b2 + 1008860394*b1 + 59634781674) * q^55 + (546786*b6 + 2033182*b5 + 2365172*b4 + 5591027*b3 + 37281164*b2 + 59875234*b1 - 31106916820) * q^56 + (117566*b6 + 1401516*b5 + 970184*b4 + 343558*b3 + 107340040*b2 + 520893364*b1 + 34697233442) * q^57 + (112000*b6 - 1949248*b5 - 3390504*b4 - 30237064*b3 - 814896*b2 + 1295801046*b1 - 157181336852) * q^58 + (-1083453*b6 - 4388089*b5 + 2634634*b4 - 1450802*b3 + 49206271*b2 + 526081766*b1 - 22977857885) * q^59 + (3205574*b6 - 422982*b5 + 1901426*b4 + 15069307*b3 - 129415264*b2 - 203067976*b1 - 158374844024) * q^60 + (2054606*b6 - 103400*b5 + 5490096*b4 - 4924030*b3 - 195994192*b2 + 2306318952*b1 - 21385551064) * q^61 + (3005172*b6 + 4075276*b5 - 1770868*b4 - 47197396*b3 - 204333052*b2 - 882139012*b1 - 317096705476) * q^62 + (-1279957*b6 + 2003607*b5 - 8922838*b4 + 19824742*b3 + 135169331*b2 + 778328810*b1 + 36487400839) * q^63 + (-886326*b6 + 3095622*b5 - 3862464*b4 - 18468117*b3 - 267647292*b2 - 1078766466*b1 - 325074344012) * q^64 + (-4826809*b6 + 4826809*b3 + 28960854*b2 + 183418742*b1 + 24032682011) * q^65 + (-9746188*b6 - 8263368*b5 + 3395030*b4 - 6144698*b3 - 4498504*b2 - 3072005194*b1 - 405501893120) * q^66 + (4793585*b6 + 4235809*b5 + 10044486*b4 + 59080622*b3 + 137700197*b2 + 1938403014*b1 + 218745835241) * q^67 + (-4515566*b6 - 2811202*b5 - 355526*b4 + 66937887*b3 - 317204948*b2 - 3541665972*b1 - 247348706668) * q^68 + (11155092*b6 - 4377672*b5 - 10779600*b4 + 3598284*b3 + 865261308*b2 + 1606644948*b1 + 191492379276) * q^69 + (-1037282*b6 + 3364064*b5 + 6327129*b4 + 1990831*b3 + 2611048*b2 - 2079210855*b1 - 278839375548) * q^70 + (-3342629*b6 + 1667607*b5 + 7404042*b4 - 83262802*b3 + 194868762*b2 - 861747087*b1 + 184279828435) * q^71 + (11453712*b6 - 857088*b5 + 1842684*b4 + 29681562*b3 + 838995828*b2 - 8443853292*b1 - 264213526380) * q^72 + (4602208*b6 + 3527588*b5 - 15085128*b4 + 48907860*b3 - 386854216*b2 + 5019073404*b1 + 619301208850) * q^73 + (-2819382*b6 - 927932*b5 - 8853061*b4 - 91480261*b3 - 594386836*b2 - 2624504863*b1 - 360082518148) * q^74 + (-32998678*b6 + 6399270*b5 + 2874596*b4 - 80946416*b3 - 446044386*b2 - 2577722940*b1 + 498721347990) * q^75 + (18365444*b6 + 8918044*b5 + 10690476*b4 + 73991062*b3 + 271919496*b2 + 635929200*b1 + 113106962584) * q^76 + (15990798*b6 - 18606720*b5 + 2181984*b4 - 96958998*b3 - 434830944*b2 - 8147587456*b1 + 466958433546) * q^77 + (9653618*b6 - 4826809*b4 + 4826809*b3 + 270301304*b2 + 2022432971*b1 + 57014267908) * q^78 + (-10054490*b6 - 15765946*b5 + 10959396*b4 - 29361036*b3 - 694414006*b2 + 11167839888*b1 + 996431248318) * q^79 + (-40527886*b6 + 6205822*b5 - 15775480*b4 + 41068791*b3 + 590365172*b2 - 315877042*b1 + 407082380740) * q^80 + (16158112*b6 + 12748176*b5 + 31667200*b4 + 93951608*b3 - 1387332464*b2 + 4886182144*b1 + 1342256889161) * q^81 + (3627716*b6 - 10531160*b5 - 10176330*b4 - 3782202*b3 + 1199513512*b2 + 31802046864*b1 + 536401144420) * q^82 + (52300105*b6 + 11099265*b5 - 19895802*b4 + 34569878*b3 - 477043581*b2 - 5108586812*b1 + 1355775826777) * q^83 + (37431514*b6 + 22659318*b5 + 7291618*b4 + 48788513*b3 + 1075440360*b2 + 13797910404*b1 + 345123229680) * q^84 + (-87702071*b6 + 15467756*b5 - 32641272*b4 - 44854421*b3 + 1538957422*b2 - 1607940294*b1 + 1320454580031) * q^85 + (-83271930*b6 - 43241896*b5 + 20150089*b4 + 50492335*b3 - 1981935008*b2 + 2640003365*b1 - 196818913628) * q^86 + (56187076*b6 + 19175256*b5 - 18033680*b4 + 132196988*b3 - 3013604702*b2 - 117756902*b1 + 765883964684) * q^87 + (-59375804*b6 - 59750020*b5 + 23462040*b4 + 192640014*b3 + 736058408*b2 + 19349981652*b1 - 429161080216) * q^88 + (42894236*b6 - 10699636*b5 + 17678344*b4 - 173240056*b3 + 2040105892*b2 - 37490019984*b1 - 185597443098) * q^89 + (44087072*b6 + 49050720*b5 + 23144864*b4 - 84826808*b3 - 883673488*b2 - 37641886162*b1 - 1915232439956) * q^90 + (24134045*b6 + 4826809*b5 + 28960854*b4 - 28960854*b3 + 395798338*b2 + 4416530235*b1 + 90343384053) * q^91 + (157774960*b6 + 83188688*b5 - 71413616*b4 - 38420784*b3 + 2108076592*b2 - 9246109408*b1 - 1253494264720) * q^92 + (-88237976*b6 - 106358436*b5 - 23131448*b4 - 108128092*b3 + 2335648356*b2 - 36626766816*b1 - 1784275334880) * q^93 + (20027034*b6 + 59789532*b5 - 6764415*b4 - 622820985*b3 - 1014054660*b2 + 44572255185*b1 - 2591923539936) * q^94 + (-61727630*b6 + 16976410*b5 + 65400476*b4 - 55651724*b3 + 1136918336*b2 - 3964782374*b1 - 16363421382) * q^95 + (-48798266*b6 - 124926102*b5 - 19236944*b4 + 21181133*b3 - 1435217636*b2 - 38446703870*b1 - 4233702864532) * q^96 + (-52945540*b6 - 22504220*b5 - 54354120*b4 + 321972728*b3 + 3140572208*b2 + 50493643956*b1 + 212160807518) * q^97 + (59046942*b6 + 66973932*b5 + 76741737*b4 + 155986449*b3 + 2558036580*b2 - 57056879502*b1 - 1994404870350) * q^98 + (-32222537*b6 + 135004959*b5 - 34082054*b4 - 275948134*b3 - 6771709379*b2 - 34570624676*b1 - 2486366775433) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + 127 q^{2} + 728 q^{3} + 16153 q^{4} + 34886 q^{5} + 83045 q^{6} + 131864 q^{7} + 127671 q^{8} + 2466055 q^{9}+O(q^{10})$$ 7 * q + 127 * q^2 + 728 * q^3 + 16153 * q^4 + 34886 * q^5 + 83045 * q^6 + 131864 * q^7 + 127671 * q^8 + 2466055 * q^9 $$7 q + 127 q^{2} + 728 q^{3} + 16153 q^{4} + 34886 q^{5} + 83045 q^{6} + 131864 q^{7} + 127671 q^{8} + 2466055 q^{9} + 3335529 q^{10} + 8912632 q^{11} + 25360909 q^{12} + 33787663 q^{13} + 67318187 q^{14} + 78942892 q^{15} + 159240961 q^{16} + 222199706 q^{17} + 596088532 q^{18} + 652665968 q^{19} + 935513953 q^{20} + 1217880580 q^{21} + 1374511830 q^{22} + 1794723984 q^{23} + 1895354895 q^{24} + 246154813 q^{25} + 613004743 q^{26} - 1915909876 q^{27} - 1789780165 q^{28} - 4735865782 q^{29} - 9918164303 q^{30} - 9436127356 q^{31} - 17224649401 q^{32} - 25638368752 q^{33} - 29930612727 q^{34} - 20876359292 q^{35} - 31714020118 q^{36} - 14832967330 q^{37} - 7473991662 q^{38} + 3513916952 q^{39} + 10553061291 q^{40} + 59907995430 q^{41} + 69292649863 q^{42} + 72489466184 q^{43} + 82565539934 q^{44} + 60104811962 q^{45} + 118453633200 q^{46} + 195641345592 q^{47} + 69842813569 q^{48} + 55402933947 q^{49} + 87739798742 q^{50} + 142246774012 q^{51} + 77967445777 q^{52} + 409286178474 q^{53} - 46365876001 q^{54} + 418486272912 q^{55} - 217713137127 q^{56} + 243297441528 q^{57} - 1099013150598 q^{58} - 160361676600 q^{59} - 1108676787295 q^{60} - 147184998046 q^{61} - 2220407254292 q^{62} + 256048021580 q^{63} - 2276361582767 q^{64} + 168388058774 q^{65} - 2841576718138 q^{66} + 1533110763584 q^{67} - 1734599536391 q^{68} + 1341159298056 q^{69} - 1953936478521 q^{70} + 1288841132520 q^{71} - 1858742324166 q^{72} + 4340518042046 q^{73} - 2522725784507 q^{74} + 3488845434748 q^{75} + 792218810282 q^{76} + 3260905865264 q^{77} + 400842353405 q^{78} + 6986884509272 q^{79} + 2848664165317 q^{80} + 9402260691943 q^{81} + 3785376233100 q^{82} + 9485774126680 q^{83} + 2428655741171 q^{84} + 9239892384264 q^{85} - 1372999514421 q^{86} + 5364161696536 q^{87} - 2985250612134 q^{88} - 1338832432586 q^{89} - 13443400684582 q^{90} + 636482341976 q^{91} - 8786066700672 q^{92} - 12529067281768 q^{93} - 18098521583937 q^{94} - 119505100680 q^{95} - 29672968067657 q^{96} + 1532637634742 q^{97} - 14020062796872 q^{98} - 17432744537624 q^{99}+O(q^{100})$$ 7 * q + 127 * q^2 + 728 * q^3 + 16153 * q^4 + 34886 * q^5 + 83045 * q^6 + 131864 * q^7 + 127671 * q^8 + 2466055 * q^9 + 3335529 * q^10 + 8912632 * q^11 + 25360909 * q^12 + 33787663 * q^13 + 67318187 * q^14 + 78942892 * q^15 + 159240961 * q^16 + 222199706 * q^17 + 596088532 * q^18 + 652665968 * q^19 + 935513953 * q^20 + 1217880580 * q^21 + 1374511830 * q^22 + 1794723984 * q^23 + 1895354895 * q^24 + 246154813 * q^25 + 613004743 * q^26 - 1915909876 * q^27 - 1789780165 * q^28 - 4735865782 * q^29 - 9918164303 * q^30 - 9436127356 * q^31 - 17224649401 * q^32 - 25638368752 * q^33 - 29930612727 * q^34 - 20876359292 * q^35 - 31714020118 * q^36 - 14832967330 * q^37 - 7473991662 * q^38 + 3513916952 * q^39 + 10553061291 * q^40 + 59907995430 * q^41 + 69292649863 * q^42 + 72489466184 * q^43 + 82565539934 * q^44 + 60104811962 * q^45 + 118453633200 * q^46 + 195641345592 * q^47 + 69842813569 * q^48 + 55402933947 * q^49 + 87739798742 * q^50 + 142246774012 * q^51 + 77967445777 * q^52 + 409286178474 * q^53 - 46365876001 * q^54 + 418486272912 * q^55 - 217713137127 * q^56 + 243297441528 * q^57 - 1099013150598 * q^58 - 160361676600 * q^59 - 1108676787295 * q^60 - 147184998046 * q^61 - 2220407254292 * q^62 + 256048021580 * q^63 - 2276361582767 * q^64 + 168388058774 * q^65 - 2841576718138 * q^66 + 1533110763584 * q^67 - 1734599536391 * q^68 + 1341159298056 * q^69 - 1953936478521 * q^70 + 1288841132520 * q^71 - 1858742324166 * q^72 + 4340518042046 * q^73 - 2522725784507 * q^74 + 3488845434748 * q^75 + 792218810282 * q^76 + 3260905865264 * q^77 + 400842353405 * q^78 + 6986884509272 * q^79 + 2848664165317 * q^80 + 9402260691943 * q^81 + 3785376233100 * q^82 + 9485774126680 * q^83 + 2428655741171 * q^84 + 9239892384264 * q^85 - 1372999514421 * q^86 + 5364161696536 * q^87 - 2985250612134 * q^88 - 1338832432586 * q^89 - 13443400684582 * q^90 + 636482341976 * q^91 - 8786066700672 * q^92 - 12529067281768 * q^93 - 18098521583937 * q^94 - 119505100680 * q^95 - 29672968067657 * q^96 + 1532637634742 * q^97 - 14020062796872 * q^98 - 17432744537624 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 35596x^{5} + 594856x^{4} + 263632368x^{3} + 252644208x^{2} - 410033371968x - 5442114981888$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 969 \nu^{6} - 153125 \nu^{5} - 37273116 \nu^{4} + 5141065076 \nu^{3} + 205885624704 \nu^{2} + \cdots - 372250932068352 ) / 215581424640$$ (969*v^6 - 153125*v^5 - 37273116*v^4 + 5141065076*v^3 + 205885624704*v^2 - 20705918799936*v - 372250932068352) / 215581424640 $$\beta_{3}$$ $$=$$ $$( - 969 \nu^{6} + 153125 \nu^{5} + 37273116 \nu^{4} - 5141065076 \nu^{3} + \cdots - 724411775075328 ) / 107790712320$$ (-969*v^6 + 153125*v^5 + 37273116*v^4 - 5141065076*v^3 - 98094912384*v^2 + 23292895895616*v - 724411775075328) / 107790712320 $$\beta_{4}$$ $$=$$ $$( - 2191 \nu^{6} + 13092 \nu^{5} + 104657371 \nu^{4} - 1116041580 \nu^{3} + \cdots + 22\!\cdots\!04 ) / 26947678080$$ (-2191*v^6 + 13092*v^5 + 104657371*v^4 - 1116041580*v^3 - 1258280874924*v^2 + 13229222202768*v + 2266533377647104) / 26947678080 $$\beta_{5}$$ $$=$$ $$( 90703 \nu^{6} - 3354157 \nu^{5} - 3281844634 \nu^{4} + 139887596428 \nu^{3} + \cdots - 19\!\cdots\!80 ) / 215581424640$$ (90703*v^6 - 3354157*v^5 - 3281844634*v^4 + 139887596428*v^3 + 21174671657976*v^2 - 245859889036896*v - 19786700922439680) / 215581424640 $$\beta_{6}$$ $$=$$ $$( - 51142 \nu^{6} + 709161 \nu^{5} + 1685935189 \nu^{4} - 60895203816 \nu^{3} + \cdots + 92\!\cdots\!84 ) / 107790712320$$ (-51142*v^6 + 709161*v^5 + 1685935189*v^4 - 60895203816*v^3 - 9607215131556*v^2 + 201779155619760*v + 9214289011613184) / 107790712320
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2\beta_{2} - 24\beta _1 + 10174$$ b3 + 2*b2 - 24*b1 + 10174 $$\nu^{3}$$ $$=$$ $$-6\beta_{6} - 10\beta_{5} - 8\beta_{4} - 77\beta_{3} + 4\beta_{2} + 20778\beta _1 - 242636$$ -6*b6 - 10*b5 - 8*b4 - 77*b3 + 4*b2 + 20778*b1 - 242636 $$\nu^{4}$$ $$=$$ $$34\beta_{6} + 302\beta_{5} + 1056\beta_{4} + 28003\beta_{3} + 50428\beta_{2} - 1445466\beta _1 + 211264092$$ 34*b6 + 302*b5 + 1056*b4 + 28003*b3 + 50428*b2 - 1445466*b1 + 211264092 $$\nu^{5}$$ $$=$$ $$- 187182 \beta_{6} - 296962 \beta_{5} - 268640 \beta_{4} - 3095465 \beta_{3} - 3011468 \beta_{2} + \cdots - 14663379932$$ -187182*b6 - 296962*b5 - 268640*b4 - 3095465*b3 - 3011468*b2 + 523275462*b1 - 14663379932 $$\nu^{6}$$ $$=$$ $$3561850 \beta_{6} + 17744918 \beta_{5} + 40612416 \beta_{4} + 784047859 \beta_{3} + 1240168804 \beta_{2} + \cdots + 5319008263956$$ 3561850*b6 + 17744918*b5 + 40612416*b4 + 784047859*b3 + 1240168804*b2 - 56681468658*b1 + 5319008263956

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −175.134 −70.2059 −37.9195 −15.2660 48.8519 117.726 132.948
−157.134 286.918 16499.3 17641.6 −45084.7 −100427. −1.30536e6 −1.51200e6 −2.77210e6
1.2 −52.2059 −945.382 −5466.54 −30724.8 49354.5 −524401. 713057. −700576. 1.60402e6
1.3 −19.9195 1765.94 −7795.21 53134.3 −35176.8 17946.1 318458. 1.52424e6 −1.05841e6
1.4 2.73404 −42.7679 −8184.53 −45417.0 −116.929 591080. −44774.1 −1.59249e6 −124172.
1.5 66.8519 −2327.76 −3722.83 7472.85 −155615. −60830.3 −796528. 3.82413e6 499574.
1.6 135.726 2027.33 10229.5 −15679.1 275162. 295835. 276541. 2.51576e6 −2.12805e6
1.7 150.948 −36.2889 14593.4 48458.2 −5477.75 −87338.4 966274. −1.59301e6 7.31468e6
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.14.a.b 7
3.b odd 2 1 117.14.a.d 7
13.b even 2 1 169.14.a.b 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.14.a.b 7 1.a even 1 1 trivial
117.14.a.d 7 3.b odd 2 1
169.14.a.b 7 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{7} - 127 T_{2}^{6} - 28684 T_{2}^{5} + 3589516 T_{2}^{4} + 109262496 T_{2}^{3} + \cdots + 611898900480$$ acting on $$S_{14}^{\mathrm{new}}(\Gamma_0(13))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} + \cdots + 611898900480$$
$3$ $$T^{7} + \cdots - 35\!\cdots\!72$$
$5$ $$T^{7} + \cdots + 74\!\cdots\!00$$
$7$ $$T^{7} + \cdots - 87\!\cdots\!32$$
$11$ $$T^{7} + \cdots + 25\!\cdots\!84$$
$13$ $$(T - 4826809)^{7}$$
$17$ $$T^{7} + \cdots + 10\!\cdots\!12$$
$19$ $$T^{7} + \cdots - 59\!\cdots\!40$$
$23$ $$T^{7} + \cdots - 57\!\cdots\!12$$
$29$ $$T^{7} + \cdots + 17\!\cdots\!40$$
$31$ $$T^{7} + \cdots - 20\!\cdots\!00$$
$37$ $$T^{7} + \cdots + 12\!\cdots\!56$$
$41$ $$T^{7} + \cdots + 14\!\cdots\!68$$
$43$ $$T^{7} + \cdots - 24\!\cdots\!00$$
$47$ $$T^{7} + \cdots - 69\!\cdots\!20$$
$53$ $$T^{7} + \cdots - 44\!\cdots\!28$$
$59$ $$T^{7} + \cdots - 26\!\cdots\!20$$
$61$ $$T^{7} + \cdots - 14\!\cdots\!84$$
$67$ $$T^{7} + \cdots - 21\!\cdots\!68$$
$71$ $$T^{7} + \cdots - 39\!\cdots\!28$$
$73$ $$T^{7} + \cdots - 10\!\cdots\!28$$
$79$ $$T^{7} + \cdots + 45\!\cdots\!20$$
$83$ $$T^{7} + \cdots - 22\!\cdots\!92$$
$89$ $$T^{7} + \cdots + 65\!\cdots\!00$$
$97$ $$T^{7} + \cdots - 13\!\cdots\!20$$