# Properties

 Label 13.14.b.a Level $13$ Weight $14$ Character orbit 13.b Analytic conductor $13.940$ Analytic rank $0$ Dimension $14$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,14,Mod(12,13)]

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([1]))

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

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

chi := DirichletCharacter("13.12");

S:= CuspForms(chi, 14);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 13.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: $$13.9400207637$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{14} + 81921 x^{12} + 2522899104 x^{10} + 37246030694192 x^{8} + \cdots + 56\!\cdots\!64$$ x^14 + 81921*x^12 + 2522899104*x^10 + 37246030694192*x^8 + 270611051280428544*x^6 + 843436691228915970048*x^4 + 602910050859212473368576*x^2 + 56765934420683632215588864 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{15}\cdot 3^{6}\cdot 13^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} + 104) q^{3} + ( - \beta_{3} + \beta_{2} - 3511) q^{4} + ( - \beta_{7} + 24 \beta_1) q^{5} + ( - \beta_{8} - \beta_{7} + 207 \beta_1) q^{6} + (\beta_{9} + \beta_{7} - 444 \beta_1) q^{7} + (\beta_{10} + \beta_{9} + \cdots - 3944 \beta_1) q^{8}+ \cdots + ( - \beta_{4} + 283 \beta_{3} + \cdots + 367804) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b3 + 104) * q^3 + (-b3 + b2 - 3511) * q^4 + (-b7 + 24*b1) * q^5 + (-b8 - b7 + 207*b1) * q^6 + (b9 + b7 - 444*b1) * q^7 + (b10 + b9 + b8 - 4*b7 - 3944*b1) * q^8 + (-b4 + 283*b3 + 40*b2 + 367804) * q^9 $$q + \beta_1 q^{2} + (\beta_{3} + 104) q^{3} + ( - \beta_{3} + \beta_{2} - 3511) q^{4} + ( - \beta_{7} + 24 \beta_1) q^{5} + ( - \beta_{8} - \beta_{7} + 207 \beta_1) q^{6} + (\beta_{9} + \beta_{7} - 444 \beta_1) q^{7} + (\beta_{10} + \beta_{9} + \cdots - 3944 \beta_1) q^{8}+ \cdots + ( - 199232 \beta_{13} + \cdots - 63373502405 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b3 + 104) * q^3 + (-b3 + b2 - 3511) * q^4 + (-b7 + 24*b1) * q^5 + (-b8 - b7 + 207*b1) * q^6 + (b9 + b7 - 444*b1) * q^7 + (b10 + b9 + b8 - 4*b7 - 3944*b1) * q^8 + (-b4 + 283*b3 + 40*b2 + 367804) * q^9 + (-b6 - 385*b3 - 28*b2 - 283454) * q^10 + (b11 - b10 + 4*b8 + 18*b7 - 8117*b1) * q^11 + (b5 - b4 - 1688*b3 + 307*b2 - 1568508) * q^12 + (b12 + 5*b10 - 2*b9 - 3*b8 + 35*b7 + b6 + b4 + 15*b3 + 399*b2 - 832*b1 - 1025891) * q^13 + (b13 - b12 + b6 - b5 - 3*b4 + 3382*b3 - 940*b2 + 5203137) * q^14 + (b13 + b12 - 5*b11 + 7*b10 - 7*b9 - 34*b8 + 3*b7 + 68557*b1) * q^15 + (4*b13 - 4*b12 - 10*b6 - 9*b5 + 13*b4 + 10954*b3 - 6503*b2 + 17389088) * q^16 + (3*b13 - 3*b12 + 10*b6 + 8*b5 + 15*b4 - 15729*b3 - 586*b2 - 12750885) * q^17 + (4*b13 + 4*b12 - 20*b11 + 94*b10 - 82*b9 - 15*b8 - 257*b7 + 54892*b1) * q^18 + (4*b13 + 4*b12 + 15*b11 - 95*b10 - 270*b9 - 276*b8 + 836*b7 + 441037*b1) * q^19 + (6*b13 + 6*b12 - 17*b10 + 127*b9 + 217*b8 + 1902*b7 + 110358*b1) * q^20 + (5*b13 + 5*b12 + 56*b11 + 2*b10 - 188*b9 + 906*b8 + 1259*b7 - 550192*b1) * q^21 + (13*b13 - 13*b12 + 10*b6 + 9*b5 - 209*b4 + 47635*b3 - 2062*b2 + 95020487) * q^22 + (4*b13 - 4*b12 - 184*b6 + 40*b5 + 12*b4 - 147410*b3 - 800*b2 - 117263796) * q^23 + (10*b13 + 10*b12 - 68*b11 + 1015*b10 + 983*b9 - 1025*b8 - 1888*b7 - 2662484*b1) * q^24 + (12*b13 - 12*b12 - 296*b6 - 72*b5 - 207*b4 + 61829*b3 - 18216*b2 + 114384694) * q^25 + (13*b13 - 25*b12 + 104*b11 + 746*b10 + 2650*b9 - 1641*b8 - 8675*b7 + 183*b6 - 39*b5 + 495*b4 + 5020*b3 - 49326*b2 - 4416256*b1 + 9858801) * q^26 + (-18*b13 + 18*b12 + 468*b6 + 24*b5 - 1172*b4 + 484679*b3 + 90404*b2 + 454311740) * q^27 + (-8*b13 - 8*b12 - 180*b11 - 2267*b10 + 1365*b9 - 4569*b8 - 11566*b7 + 9910690*b1) * q^28 + (-85*b13 + 85*b12 - 102*b6 + 120*b5 + 954*b4 - 458614*b3 + 54366*b2 - 739165710) * q^29 + (-54*b13 + 54*b12 + 387*b6 - 56*b5 + 1916*b4 - 369751*b3 + 24094*b2 - 802172608) * q^30 + (-83*b13 - 83*b12 + 93*b11 + 2869*b10 - 9832*b9 + 15222*b8 - 30338*b7 - 11052929*b1) * q^31 + (-78*b13 - 78*b12 + 20*b11 - 8085*b10 - 20629*b9 - 42617*b8 + 3180*b7 + 41566800*b1) * q^32 + (-132*b13 - 132*b12 + 120*b11 - 4344*b10 + 18348*b9 + 37088*b8 + 169976*b7 - 8623176*b1) * q^33 + (-96*b13 - 96*b12 - 588*b11 + 1254*b10 - 7626*b9 + 64233*b8 - 51801*b7 - 9361743*b1) * q^34 + (-302*b13 + 302*b12 + 92*b6 - 128*b5 - 1500*b4 + 559099*b3 - 74588*b2 + 1445914740) * q^35 + (-72*b13 + 72*b12 + 828*b6 - 753*b5 + 745*b4 + 2263115*b3 - 447706*b2 + 2370092477) * q^36 + (-220*b13 - 220*b12 + 780*b11 + 8276*b10 + 18456*b9 - 58224*b8 + 118465*b7 - 46296244*b1) * q^37 + (-267*b13 + 267*b12 + 2812*b6 + 1229*b5 - 525*b4 - 3842259*b3 + 1526558*b2 - 5159625597) * q^38 + (-299*b13 + 277*b12 - 1547*b11 + 7885*b10 + 35807*b9 - 93690*b8 - 364211*b7 - 1296*b6 + 728*b5 - 1712*b4 - 2203362*b3 - 120760*b2 + 2181283*b1 + 221612108) * q^39 + (148*b13 - 148*b12 - 4382*b6 - 333*b5 + 4585*b4 - 300246*b3 + 12825*b2 - 3609180916) * q^40 + (-114*b13 - 114*b12 + 3476*b11 - 10512*b10 + 32996*b9 - 21068*b8 - 574904*b7 - 3208756*b1) * q^41 + (774*b13 - 774*b12 + 1539*b6 - 486*b5 - 5418*b4 + 9004605*b3 - 796104*b2 + 6437388936) * q^42 + (340*b13 - 340*b12 - 4920*b6 - 648*b5 - 11696*b4 - 332815*b3 + 2083280*b2 - 3020589616) * q^43 + (726*b13 + 726*b12 + 1396*b11 - 3904*b10 - 81312*b9 + 129400*b8 + 33620*b7 + 50648564*b1) * q^44 + (597*b13 + 597*b12 - 2940*b11 + 29478*b10 - 29028*b9 - 360534*b8 + 37494*b7 + 28716780*b1) * q^45 + (1264*b13 + 1264*b12 - 2352*b11 + 23440*b10 + 42832*b9 + 333930*b8 + 2121354*b7 - 125918822*b1) * q^46 + (702*b13 + 702*b12 + 1000*b11 - 3572*b10 - 28735*b9 + 465396*b8 - 594527*b7 - 68806468*b1) * q^47 + (3060*b13 - 3060*b12 - 1926*b6 + 541*b5 + 24863*b4 - 14049246*b3 - 10392377*b2 + 18318947860) * q^48 + (471*b13 - 471*b12 + 10226*b6 + 5904*b5 - 6915*b4 + 9417165*b3 - 3580922*b2 + 14477228354) * q^49 + (2076*b13 + 2076*b12 - 2700*b11 - 60758*b10 - 109862*b9 - 394649*b8 + 2406737*b7 + 274981286*b1) * q^50 + (2538*b13 - 2538*b12 - 5364*b6 - 10128*b5 + 19524*b4 - 35919273*b3 + 6908964*b2 - 32446500876) * q^51 + (3016*b13 - 1752*b12 + 8580*b11 - 78271*b10 - 151599*b9 - 276753*b8 - 1684682*b7 - 5548*b6 - 5499*b5 - 15805*b4 - 3451039*b3 - 10540646*b2 + 424554494*b1 + 43280506775) * q^52 + (-900*b13 + 900*b12 + 2936*b6 + 3256*b5 + 37848*b4 + 21845844*b3 + 6419104*b2 - 38662085682) * q^53 + (2168*b13 + 2168*b12 - 21568*b11 + 179072*b10 - 23744*b9 - 21831*b8 - 4895239*b7 - 266517691*b1) * q^54 + (-3812*b13 + 3812*b12 + 344*b6 + 72*b5 - 25916*b4 + 91134688*b3 + 1433392*b2 + 21056528908) * q^55 + (-220*b13 + 220*b12 + 11430*b6 + 9945*b5 - 38277*b4 - 36302662*b3 + 24652623*b2 - 73375755960) * q^56 + (-3302*b13 - 3302*b12 - 13100*b11 - 107024*b10 + 454088*b9 - 190092*b8 - 356030*b7 + 588946060*b1) * q^57 + (-1804*b13 - 1804*b12 + 27600*b11 + 152492*b10 + 670604*b9 + 551706*b8 + 2190318*b7 - 1247649964*b1) * q^58 + (-6744*b13 - 6744*b12 + 20351*b11 + 12449*b10 - 459050*b9 - 212964*b8 + 3954860*b7 - 17447859*b1) * q^59 + (-1698*b13 - 1698*b12 + 9120*b11 - 18411*b10 + 342501*b9 - 813873*b8 - 3856122*b7 - 480526314*b1) * q^60 + (-17148*b13 + 17148*b12 + 4872*b6 + 648*b5 - 44814*b4 - 70940506*b3 - 25679216*b2 + 120948509776) * q^61 + (-733*b13 + 733*b12 - 88962*b6 - 23421*b5 + 117*b4 + 130065713*b3 - 37738602*b2 + 129175048365) * q^62 + (-10523*b13 - 10523*b12 - 8045*b11 - 218657*b10 - 114250*b9 + 965166*b8 + 4230712*b7 - 798437687*b1) * q^63 + (-12732*b13 + 12732*b12 - 22398*b6 + 47637*b5 - 64521*b4 - 439277702*b3 + 86153951*b2 - 343931996092) * q^64 + (-17069*b13 + 6647*b12 - 16900*b11 + 44558*b10 - 538208*b9 + 1503594*b8 - 1845886*b7 + 83204*b6 + 18928*b5 + 66135*b4 - 179838989*b3 - 19725644*b2 + 617450340*b1 + 38212673619) * q^65 + (5652*b13 - 5652*b12 + 108648*b6 - 21932*b5 - 216964*b4 + 441803324*b3 + 27487432*b2 + 101243559548) * q^66 + (-15106*b13 - 15106*b12 + 46545*b11 - 392413*b10 - 925864*b9 + 751128*b8 - 1095522*b7 - 75431365*b1) * q^67 + (10152*b13 - 10152*b12 - 67732*b6 - 1649*b5 + 265161*b4 + 445784598*b3 - 31854197*b2 + 4644177354) * q^68 + (-9000*b13 + 9000*b12 + 83520*b6 - 49800*b5 + 451182*b4 - 140447670*b3 + 44957544*b2 - 300522851322) * q^69 + (7096*b13 + 7096*b12 + 26880*b11 + 170536*b10 + 1005544*b9 - 1730355*b8 + 404613*b7 + 2140608625*b1) * q^70 + (-3102*b13 - 3102*b12 - 116656*b11 - 546028*b10 + 56749*b9 + 5298564*b8 + 5261053*b7 + 2324552420*b1) * q^71 + (20878*b13 + 20878*b12 - 100700*b11 - 160130*b10 - 1428898*b9 - 6590178*b8 - 14232236*b7 + 6870146692*b1) * q^72 + (-7698*b13 - 7698*b12 - 18120*b11 - 133572*b10 - 189660*b9 - 7778124*b8 - 350478*b7 - 1818279768*b1) * q^73 + (53124*b13 - 53124*b12 + 53489*b6 - 9524*b5 - 327372*b4 - 366154635*b3 - 124904612*b2 + 542507028138) * q^74 + (-7182*b13 + 7182*b12 + 196092*b6 + 38512*b5 + 214772*b4 + 411427096*b3 - 47347916*b2 + 118719461220) * q^75 + (27506*b13 + 27506*b12 + 98244*b11 + 1769510*b10 + 1015206*b9 + 7783818*b8 - 14778152*b7 - 14942569576*b1) * q^76 + (29899*b13 - 29899*b12 + 32026*b6 - 129064*b5 - 156288*b4 - 1022378096*b3 + 44493518*b2 - 99424370490) * q^77 + (55874*b13 - 13282*b12 - 20800*b11 + 699368*b10 + 1767784*b9 + 5359082*b8 + 21711778*b7 - 299763*b6 - 6396*b5 + 214296*b4 - 877156389*b3 - 93325326*b2 + 1023760790*b1 - 25842635556) * q^78 + (-32164*b13 + 32164*b12 - 489176*b6 + 96624*b5 - 77752*b4 + 851453886*b3 + 71222152*b2 - 361448379480) * q^79 + (53922*b13 + 53922*b12 + 82820*b11 - 665561*b10 + 1349991*b9 - 1170613*b8 + 53812948*b7 - 2849765544*b1) * q^80 + (-10080*b13 + 10080*b12 + 360144*b6 + 69960*b5 - 706480*b4 + 2100377104*b3 + 60156568*b2 + 474138109393) * q^81 + (55708*b13 - 55708*b12 - 571320*b6 + 84564*b5 - 1061060*b4 - 363848516*b3 + 41783592*b2 + 36272436252) * q^82 + (294*b13 + 294*b12 + 108895*b11 + 1681333*b10 - 3607522*b9 - 1055376*b8 - 9368160*b7 + 10366103917*b1) * q^83 + (44290*b13 + 44290*b12 + 243688*b11 - 1459697*b10 - 7159201*b9 - 42375*b8 - 15937370*b7 + 9629221006*b1) * q^84 + (-32487*b13 - 32487*b12 + 302940*b11 - 298626*b10 + 8577156*b9 + 1252026*b8 - 59247207*b7 + 6755269836*b1) * q^85 + (69696*b13 + 69696*b12 - 259936*b11 + 2065032*b10 - 651256*b9 + 301735*b8 + 33671391*b7 - 20835754729*b1) * q^86 + (-56016*b13 + 56016*b12 - 438768*b6 - 10056*b5 + 1290108*b4 - 2236075920*b3 + 43705896*b2 - 930928681956) * q^87 + (44520*b13 - 44520*b12 + 239936*b6 + 48276*b5 - 1068276*b4 + 1330558924*b3 + 89226960*b2 + 184783754404) * q^88 + (-13398*b13 - 13398*b12 - 334724*b11 - 1858808*b10 + 1677152*b9 - 12694572*b8 - 73770002*b7 - 19199411260*b1) * q^89 + (19530*b13 - 19530*b12 + 480078*b6 + 154830*b5 + 1248738*b4 - 3307583172*b3 - 179841228*b2 - 334297176978) * q^90 + (-83746*b13 - 826*b12 + 80223*b11 + 177701*b10 + 3682186*b9 - 9597216*b8 + 39474008*b7 + 276568*b6 - 191880*b5 - 1496736*b4 - 977362959*b3 + 153559616*b2 + 16635494525*b1 + 246480548936) * q^91 + (123456*b13 - 123456*b12 + 613824*b6 - 250026*b5 + 2175210*b4 + 3016508016*b3 - 324132126*b2 + 517236443352) * q^92 + (-76912*b13 - 76912*b12 - 598420*b11 + 1168892*b10 + 2926120*b9 + 2043864*b8 + 53948096*b7 - 20406374332*b1) * q^93 + (-15027*b13 + 15027*b12 - 812383*b6 - 418293*b5 + 885489*b4 + 4098641306*b3 - 127394092*b2 + 801414098893) * q^94 + (-141288*b13 + 141288*b12 + 1297584*b6 + 208368*b5 + 186336*b4 - 189771918*b3 - 120820608*b2 + 852888137904) * q^95 + (-27210*b13 - 27210*b12 - 599844*b11 - 5566539*b10 - 12554187*b9 + 10823865*b8 + 47405316*b7 + 83618245248*b1) * q^96 + (-145002*b13 - 145002*b12 - 205320*b11 + 5161308*b10 + 5124772*b9 - 18300780*b8 - 83858612*b7 - 8577077640*b1) * q^97 + (-2040*b13 - 2040*b12 - 500820*b11 + 153114*b10 - 3449718*b9 + 26357259*b8 - 55769619*b7 + 45977821456*b1) * q^98 + (-199232*b13 - 199232*b12 + 1490881*b11 - 8296361*b10 - 10085020*b9 + 48306396*b8 + 128620090*b7 - 63373502405*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q + 1456 q^{3} - 49154 q^{4} + 5149262 q^{9}+O(q^{10})$$ 14 * q + 1456 * q^3 - 49154 * q^4 + 5149262 * q^9 $$14 q + 1456 q^{3} - 49154 q^{4} + 5149262 q^{9} - 3968358 q^{10} - 21959110 q^{12} - 14362478 q^{13} + 72843942 q^{14} + 243447170 q^{16} - 178512492 q^{17} + 1330288056 q^{22} - 1641693744 q^{23} + 1601386654 q^{25} + 138020766 q^{26} + 6360372232 q^{27} - 10348326348 q^{29} - 11230427010 q^{30} + 20242816056 q^{35} + 33181294876 q^{36} - 72234754500 q^{38} + 3102574280 q^{39} - 50528567766 q^{40} + 90123482634 q^{42} - 42288191696 q^{43} + 256465114846 q^{48} + 202681235282 q^{49} - 454251099624 q^{51} + 605927200580 q^{52} - 541269433788 q^{53} + 294791560608 q^{55} - 1027260370698 q^{56} + 1693279412900 q^{61} + 1808450592168 q^{62} - 4815047793506 q^{64} + 534977124552 q^{65} + 1417411440480 q^{66} + 65016763122 q^{68} - 4207322259360 q^{69} + 7595100503238 q^{74} + 1662071406584 q^{75} - 1391939668824 q^{77} - 361798757502 q^{78} - 5060278211056 q^{79} + 6637938210830 q^{81} + 507818992992 q^{82} - 13033010125344 q^{87} + 2586979258080 q^{88} - 4680167629284 q^{90} + 3450737986176 q^{91} + 7241299383420 q^{92} + 11219792119974 q^{94} + 11940434574336 q^{95}+O(q^{100})$$ 14 * q + 1456 * q^3 - 49154 * q^4 + 5149262 * q^9 - 3968358 * q^10 - 21959110 * q^12 - 14362478 * q^13 + 72843942 * q^14 + 243447170 * q^16 - 178512492 * q^17 + 1330288056 * q^22 - 1641693744 * q^23 + 1601386654 * q^25 + 138020766 * q^26 + 6360372232 * q^27 - 10348326348 * q^29 - 11230427010 * q^30 + 20242816056 * q^35 + 33181294876 * q^36 - 72234754500 * q^38 + 3102574280 * q^39 - 50528567766 * q^40 + 90123482634 * q^42 - 42288191696 * q^43 + 256465114846 * q^48 + 202681235282 * q^49 - 454251099624 * q^51 + 605927200580 * q^52 - 541269433788 * q^53 + 294791560608 * q^55 - 1027260370698 * q^56 + 1693279412900 * q^61 + 1808450592168 * q^62 - 4815047793506 * q^64 + 534977124552 * q^65 + 1417411440480 * q^66 + 65016763122 * q^68 - 4207322259360 * q^69 + 7595100503238 * q^74 + 1662071406584 * q^75 - 1391939668824 * q^77 - 361798757502 * q^78 - 5060278211056 * q^79 + 6637938210830 * q^81 + 507818992992 * q^82 - 13033010125344 * q^87 + 2586979258080 * q^88 - 4680167629284 * q^90 + 3450737986176 * q^91 + 7241299383420 * q^92 + 11219792119974 * q^94 + 11940434574336 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 81921 x^{12} + 2522899104 x^{10} + 37246030694192 x^{8} + \cdots + 56\!\cdots\!64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 171838599480073 \nu^{12} + \cdots + 42\!\cdots\!68 ) / 35\!\cdots\!00$$ (171838599480073*v^12 + 11986416837440757321*v^10 + 287487764024453840817888*v^8 + 2887415862757272861063410864*v^6 + 10984555997936659051378491357696*v^4 + 7459352546771357295500599571312640*v^2 + 4238517097169373950395979963871264768) / 358032775336243355172169580544000 $$\beta_{3}$$ $$=$$ $$( 171838599480073 \nu^{12} + \cdots + 48\!\cdots\!68 ) / 35\!\cdots\!00$$ (171838599480073*v^12 + 11986416837440757321*v^10 + 287487764024453840817888*v^8 + 2887415862757272861063410864*v^6 + 10984555997936659051378491357696*v^4 + 7101319771435113940328429990768640*v^2 + 48459527409317964816079362764832768) / 358032775336243355172169580544000 $$\beta_{4}$$ $$=$$ $$( 11\!\cdots\!77 \nu^{12} + \cdots - 16\!\cdots\!68 ) / 17\!\cdots\!00$$ (11881802872215577*v^12 + 797329779759570264729*v^10 + 17852269226832893443732512*v^8 + 157075791822081714166787132336*v^6 + 414003861289302419683277064943104*v^4 - 406264011913067438435813100965744640*v^2 - 166781842312860254402234488565287354368) / 179016387668121677586084790272000 $$\beta_{5}$$ $$=$$ $$( - 42\!\cdots\!29 \nu^{12} + \cdots - 10\!\cdots\!44 ) / 35\!\cdots\!00$$ (-421998551999132929*v^12 - 29703056692097686373313*v^10 - 725064984464729144360318304*v^8 - 7561484420102860763246930580272*v^6 - 31850497708622335371572377352974848*v^4 - 36459033729968383075507057154885283840*v^2 - 10043977231765832265138368102681615007744) / 358032775336243355172169580544000 $$\beta_{6}$$ $$=$$ $$( - 12\!\cdots\!89 \nu^{12} + \cdots - 55\!\cdots\!24 ) / 35\!\cdots\!00$$ (-1232010246602925989*v^12 - 86004189287817491239653*v^10 - 2066110195819231273341401184*v^8 - 20836620906085326036140373267952*v^6 - 80458267029470859481022100032291328*v^4 - 59550883606441488534331367961389445120*v^2 - 5565625545036299403927906252659987841024) / 358032775336243355172169580544000 $$\beta_{7}$$ $$=$$ $$( 12\!\cdots\!71 \nu^{13} + \cdots - 31\!\cdots\!84 \nu ) / 47\!\cdots\!00$$ (1251741136254920671*v^13 + 87056322375655838369247*v^11 + 2076809344125545483293628256*v^9 + 20645581308487268438543024734928*v^7 + 76694671635168010686021115140076032*v^5 + 43016892705901883294313630038777548800*v^3 - 313744419762143082569604739771042627584*v) / 4775934447036388251020190632273510400 $$\beta_{8}$$ $$=$$ $$( - 15\!\cdots\!51 \nu^{13} + \cdots + 71\!\cdots\!84 \nu ) / 21\!\cdots\!00$$ (-159478250918573810051*v^13 - 11112644914748751005207427*v^11 - 266027275572136502683648537056*v^9 - 2662286053650960774590691879226768*v^7 - 10044981621575994682960019945570314752*v^5 - 6198483593604483463432940679163662766080*v^3 + 7166057614263984883033205624712943632384*v) / 214917050116637471295908578452307968000 $$\beta_{9}$$ $$=$$ $$( 15\!\cdots\!07 \nu^{13} + \cdots - 95\!\cdots\!28 \nu ) / 20\!\cdots\!00$$ (1523930707480171007*v^13 + 104590630415259106126599*v^11 + 2431514141174954466185054952*v^9 + 22818649468669806486922502314576*v^7 + 69892704843957040767894526329193344*v^5 - 42648657944204049512860854511997184000*v^3 - 95744351370941280621850145868569934102528*v) / 2014847344843476293399142922990387200 $$\beta_{10}$$ $$=$$ $$( 66\!\cdots\!53 \nu^{13} + \cdots + 43\!\cdots\!48 \nu ) / 64\!\cdots\!00$$ (666717139939723870253*v^13 + 46879347094217491774503981*v^11 + 1141474347368218639850287284768*v^9 + 11833504237561669204770108253877104*v^7 + 49184401997652456773605213587010131456*v^5 + 56116894534495675627231381437626887434240*v^3 + 43553773663499880129265049249927426183528448*v) / 644751150349912413887725735356923904000 $$\beta_{11}$$ $$=$$ $$( 22\!\cdots\!89 \nu^{13} + \cdots + 15\!\cdots\!96 \nu ) / 12\!\cdots\!00$$ (2214777259889402165689*v^13 + 157834208140049013656193465*v^11 + 3942173099364483712885030774176*v^9 + 43025723599365740106312292529377712*v^7 + 201787786342503282582053488405377744384*v^5 + 329238794489286047651101297096894528892928*v^3 + 152861975637799943608484667588238914377220096*v) / 128950230069982482777545147071384780800 $$\beta_{12}$$ $$=$$ $$( 15\!\cdots\!69 \nu^{13} + \cdots + 90\!\cdots\!28 ) / 26\!\cdots\!00$$ (1521902424382047834769*v^13 + 161246845900121806081248*v^12 + 108030958066513478754287313*v^11 + 11268714156994438014920569056*v^10 + 2675672810420757994162915210464*v^9 + 271345959328375651975254566350848*v^8 + 28641954737118704235292253181330992*v^7 + 2751453892763024651672375878761368064*v^6 + 127586215420313486097239067115003058688*v^5 + 10791159863622952030677735788788678066176*v^4 + 173765168835830913218628006332974841425920*v^3 + 8756073393180508503479455691887389535764480*v^2 + 51050486519562722553845803945420911729967104*v + 908682020059092184180600761488902642496176128) / 26864631264579683911988572306538496000 $$\beta_{13}$$ $$=$$ $$( 15\!\cdots\!69 \nu^{13} + \cdots - 90\!\cdots\!28 ) / 26\!\cdots\!00$$ (1521902424382047834769*v^13 - 161246845900121806081248*v^12 + 108030958066513478754287313*v^11 - 11268714156994438014920569056*v^10 + 2675672810420757994162915210464*v^9 - 271345959328375651975254566350848*v^8 + 28641954737118704235292253181330992*v^7 - 2751453892763024651672375878761368064*v^6 + 127586215420313486097239067115003058688*v^5 - 10791159863622952030677735788788678066176*v^4 + 173765168835830913218628006332974841425920*v^3 - 8756073393180508503479455691887389535764480*v^2 + 51050486519562722553845803945420911729967104*v - 908682020059092184180600761488902642496176128) / 26864631264579683911988572306538496000
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} - 11703$$ -b3 + b2 - 11703 $$\nu^{3}$$ $$=$$ $$\beta_{10} + \beta_{9} + \beta_{8} - 4\beta_{7} - 20328\beta_1$$ b10 + b9 + b8 - 4*b7 - 20328*b1 $$\nu^{4}$$ $$=$$ $$4\beta_{13} - 4\beta_{12} - 10\beta_{6} - 9\beta_{5} + 13\beta_{4} + 35530\beta_{3} - 31079\beta_{2} + 237893152$$ 4*b13 - 4*b12 - 10*b6 - 9*b5 + 13*b4 + 35530*b3 - 31079*b2 + 237893152 $$\nu^{5}$$ $$=$$ $$- 78 \beta_{13} - 78 \beta_{12} + 20 \beta_{11} - 40853 \beta_{10} - 53397 \beta_{9} + \cdots + 506348112 \beta_1$$ -78*b13 - 78*b12 + 20*b11 - 40853*b10 - 53397*b9 - 75385*b8 + 134252*b7 + 506348112*b1 $$\nu^{6}$$ $$=$$ $$- 176572 \beta_{13} + 176572 \beta_{12} + 387202 \beta_{6} + 416277 \beta_{5} - 597001 \beta_{4} + \cdots - 5925541103548$$ -176572*b13 + 176572*b12 + 387202*b6 + 416277*b5 - 597001*b4 - 1491933318*b3 + 956496607*b2 - 5925541103548 $$\nu^{7}$$ $$=$$ $$3975030 \beta_{13} + 3975030 \beta_{12} - 2257220 \beta_{11} + 1402215925 \beta_{10} + \cdots - 14227924191120 \beta_1$$ 3975030*b13 + 3975030*b12 - 2257220*b11 + 1402215925*b10 + 1962416117*b9 + 3347301225*b8 - 3288981484*b7 - 14227924191120*b1 $$\nu^{8}$$ $$=$$ $$6180648732 \beta_{13} - 6180648732 \beta_{12} - 12275537154 \beta_{6} - 15210322677 \beta_{5} + \cdots + 16\!\cdots\!28$$ 6180648732*b13 - 6180648732*b12 - 12275537154*b6 - 15210322677*b5 + 22255396329*b4 + 55943785453430*b3 - 29796941287919*b2 + 166501245853943628 $$\nu^{9}$$ $$=$$ $$- 156075488310 \beta_{13} - 156075488310 \beta_{12} + 136854428100 \beta_{11} + \cdots + 42\!\cdots\!04 \beta_1$$ -156075488310*b13 - 156075488310*b12 + 136854428100*b11 - 45904107366917*b10 - 65550115580933*b9 - 124368173451257*b8 + 76663793518892*b7 + 426106565970841104*b1 $$\nu^{10}$$ $$=$$ $$- 202945672691804 \beta_{13} + 202945672691804 \beta_{12} + 376255147926434 \beta_{6} + \cdots - 49\!\cdots\!88$$ -202945672691804*b13 + 202945672691804*b12 + 376255147926434*b6 + 514877525958789*b5 - 773543571226553*b4 - 1944031283830589846*b3 + 937016857162289887*b2 - 4986475497582949117388 $$\nu^{11}$$ $$=$$ $$55\!\cdots\!74 \beta_{13} + \cdots - 13\!\cdots\!24 \beta_1$$ 5549473934634774*b13 + 5549473934634774*b12 - 6090119658329860*b11 + 1478606539656427477*b10 + 2122253092115327445*b9 + 4278188420486810633*b8 - 1845528413035187692*b7 - 13169203869598324183824*b1 $$\nu^{12}$$ $$=$$ $$65\!\cdots\!04 \beta_{13} + \cdots + 15\!\cdots\!92$$ 6527224483766949404*b13 - 6527224483766949404*b12 - 11575117317103343810*b6 - 16887118538970535509*b5 + 25924632192821954249*b4 + 64850537435565203151606*b3 - 29636731229595312123983*b2 + 154111544669809014154014092 $$\nu^{13}$$ $$=$$ $$- 18\!\cdots\!22 \beta_{13} + \cdots + 41\!\cdots\!84 \beta_1$$ -187788331972626562422*b13 - 187788331972626562422*b12 + 232500620454177289540*b11 - 47331943576630257403301*b10 - 67972064953865927628709*b9 - 141820644154633579606809*b8 + 47319846832413966566060*b7 + 413268102655710308606010384*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$$.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
12.1
 − 178.493i − 131.418i − 110.094i − 109.869i − 87.5010i − 28.8440i − 10.5212i 10.5212i 28.8440i 87.5010i 109.869i 110.094i 131.418i 178.493i
178.493i 850.060 −23667.6 1220.92i 151729.i 257415.i 2.76227e6i −871721. −217925.
12.2 131.418i −1774.35 −9078.57 27762.3i 233181.i 2614.44i 116510.i 1.55400e6 3.64846e6
12.3 110.094i 787.740 −3928.66 28789.1i 86725.3i 370728.i 469368.i −973789. 3.16950e6
12.4 109.869i −652.778 −3879.10 61212.2i 71719.8i 994.788i 473853.i −1.16820e6 −6.72530e6
12.5 87.5010i 2407.03 535.569 31938.7i 210617.i 191671.i 763671.i 4.19945e6 −2.79467e6
12.6 28.8440i 701.398 7360.02 35879.6i 20231.1i 318380.i 448583.i −1.10236e6 1.03491e6
12.7 10.5212i −1591.09 8081.30 9424.40i 16740.2i 484840.i 171215.i 937259. −99156.0
12.8 10.5212i −1591.09 8081.30 9424.40i 16740.2i 484840.i 171215.i 937259. −99156.0
12.9 28.8440i 701.398 7360.02 35879.6i 20231.1i 318380.i 448583.i −1.10236e6 1.03491e6
12.10 87.5010i 2407.03 535.569 31938.7i 210617.i 191671.i 763671.i 4.19945e6 −2.79467e6
12.11 109.869i −652.778 −3879.10 61212.2i 71719.8i 994.788i 473853.i −1.16820e6 −6.72530e6
12.12 110.094i 787.740 −3928.66 28789.1i 86725.3i 370728.i 469368.i −973789. 3.16950e6
12.13 131.418i −1774.35 −9078.57 27762.3i 233181.i 2614.44i 116510.i 1.55400e6 3.64846e6
12.14 178.493i 850.060 −23667.6 1220.92i 151729.i 257415.i 2.76227e6i −871721. −217925.
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 12.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.14.b.a 14
3.b odd 2 1 117.14.b.c 14
13.b even 2 1 inner 13.14.b.a 14
13.d odd 4 2 169.14.a.d 14
39.d odd 2 1 117.14.b.c 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.14.b.a 14 1.a even 1 1 trivial
13.14.b.a 14 13.b even 2 1 inner
117.14.b.c 14 3.b odd 2 1
117.14.b.c 14 39.d odd 2 1
169.14.a.d 14 13.d odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{14}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14} + \cdots + 56\!\cdots\!64$$
$3$ $$(T^{7} + \cdots + 20\!\cdots\!92)^{2}$$
$5$ $$T^{14} + \cdots + 41\!\cdots\!00$$
$7$ $$T^{14} + \cdots + 53\!\cdots\!00$$
$11$ $$T^{14} + \cdots + 70\!\cdots\!00$$
$13$ $$T^{14} + \cdots + 23\!\cdots\!37$$
$17$ $$(T^{7} + \cdots - 99\!\cdots\!00)^{2}$$
$19$ $$T^{14} + \cdots + 70\!\cdots\!36$$
$23$ $$(T^{7} + \cdots - 74\!\cdots\!16)^{2}$$
$29$ $$(T^{7} + \cdots - 15\!\cdots\!00)^{2}$$
$31$ $$T^{14} + \cdots + 84\!\cdots\!00$$
$37$ $$T^{14} + \cdots + 20\!\cdots\!64$$
$41$ $$T^{14} + \cdots + 21\!\cdots\!00$$
$43$ $$(T^{7} + \cdots + 65\!\cdots\!08)^{2}$$
$47$ $$T^{14} + \cdots + 50\!\cdots\!04$$
$53$ $$(T^{7} + \cdots - 20\!\cdots\!28)^{2}$$
$59$ $$T^{14} + \cdots + 23\!\cdots\!04$$
$61$ $$(T^{7} + \cdots - 13\!\cdots\!44)^{2}$$
$67$ $$T^{14} + \cdots + 18\!\cdots\!04$$
$71$ $$T^{14} + \cdots + 96\!\cdots\!00$$
$73$ $$T^{14} + \cdots + 22\!\cdots\!36$$
$79$ $$(T^{7} + \cdots + 83\!\cdots\!00)^{2}$$
$83$ $$T^{14} + \cdots + 11\!\cdots\!76$$
$89$ $$T^{14} + \cdots + 31\!\cdots\!24$$
$97$ $$T^{14} + \cdots + 10\!\cdots\!56$$