# Properties

 Label 37.8.a.a Level $37$ Weight $8$ Character orbit 37.a Self dual yes Analytic conductor $11.558$ Analytic rank $1$ Dimension $10$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,8,Mod(1,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("37.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 37.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: $$11.5582459429$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 4 x^{9} - 905 x^{8} + 4018 x^{7} + 291290 x^{6} - 1367036 x^{5} - 39566544 x^{4} + 175732512 x^{3} + 2143908736 x^{2} - 5929608704 x - 45399525376$$ x^10 - 4*x^9 - 905*x^8 + 4018*x^7 + 291290*x^6 - 1367036*x^5 - 39566544*x^4 + 175732512*x^3 + 2143908736*x^2 - 5929608704*x - 45399525376 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}\cdot 3$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 2) q^{2} + ( - \beta_{7} + \beta_1 - 10) q^{3} + (\beta_{7} + \beta_{6} + 3 \beta_1 + 59) q^{4} + (\beta_{7} - 2 \beta_{6} - \beta_{4} - 5 \beta_1 - 60) q^{5} + (9 \beta_{7} - 2 \beta_{6} + 3 \beta_{4} - \beta_{3} + 13 \beta_1 - 82) q^{6} + (6 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 4 \beta_{3} - \beta_{2} + 32 \beta_1 - 62) q^{7} + ( - \beta_{9} - 4 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + \cdots - 380) q^{8}+ \cdots + (6 \beta_{9} - \beta_{8} + 9 \beta_{7} + 8 \beta_{6} - 4 \beta_{5} - 6 \beta_{4} + \cdots + 587) q^{9}+O(q^{10})$$ q + (-b1 - 2) * q^2 + (-b7 + b1 - 10) * q^3 + (b7 + b6 + 3*b1 + 59) * q^4 + (b7 - 2*b6 - b4 - 5*b1 - 60) * q^5 + (9*b7 - 2*b6 + 3*b4 - b3 + 13*b1 - 82) * q^6 + (6*b7 - b6 - b5 + b4 + 4*b3 - b2 + 32*b1 - 62) * q^7 + (-b9 - 4*b6 + 5*b5 - 2*b4 - b3 + 4*b2 - 8*b1 - 380) * q^8 + (6*b9 - b8 + 9*b7 + 8*b6 - 4*b5 - 6*b4 - 8*b3 + b2 + 79*b1 + 587) * q^9 $$q + ( - \beta_1 - 2) q^{2} + ( - \beta_{7} + \beta_1 - 10) q^{3} + (\beta_{7} + \beta_{6} + 3 \beta_1 + 59) q^{4} + (\beta_{7} - 2 \beta_{6} - \beta_{4} - 5 \beta_1 - 60) q^{5} + (9 \beta_{7} - 2 \beta_{6} + 3 \beta_{4} - \beta_{3} + 13 \beta_1 - 82) q^{6} + (6 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 4 \beta_{3} - \beta_{2} + 32 \beta_1 - 62) q^{7} + ( - \beta_{9} - 4 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + \cdots - 380) q^{8}+ \cdots + ( - 21932 \beta_{9} - 15469 \beta_{8} + 53964 \beta_{7} + \cdots - 1723977) q^{99}+O(q^{100})$$ q + (-b1 - 2) * q^2 + (-b7 + b1 - 10) * q^3 + (b7 + b6 + 3*b1 + 59) * q^4 + (b7 - 2*b6 - b4 - 5*b1 - 60) * q^5 + (9*b7 - 2*b6 + 3*b4 - b3 + 13*b1 - 82) * q^6 + (6*b7 - b6 - b5 + b4 + 4*b3 - b2 + 32*b1 - 62) * q^7 + (-b9 - 4*b6 + 5*b5 - 2*b4 - b3 + 4*b2 - 8*b1 - 380) * q^8 + (6*b9 - b8 + 9*b7 + 8*b6 - 4*b5 - 6*b4 - 8*b3 + b2 + 79*b1 + 587) * q^9 + (-8*b9 + 7*b8 + 7*b7 + 9*b6 - 11*b5 - 6*b4 + 7*b3 - 13*b2 + 166*b1 + 807) * q^10 + (-12*b9 - 13*b8 + 17*b7 + 14*b6 + 14*b5 + 23*b4 - 16*b3 - 4*b2 + 87*b1 - 875) * q^11 + (28*b9 - 11*b8 - 120*b7 - 5*b5 - 32*b4 + 13*b3 + b2 + 163*b1 - 2052) * q^12 + (-6*b9 + 56*b8 - 17*b7 + 17*b6 + 15*b5 - b4 + 28*b3 + 43*b2 + 65*b1 - 1732) * q^13 + (25*b9 - 42*b8 - 134*b7 - 28*b6 - 5*b5 - 2*b4 - 17*b3 + 38*b2 + 19*b1 - 6596) * q^14 + (-76*b9 + 130*b7 + 7*b6 + 21*b5 + 98*b4 + 10*b3 - 35*b2 - 18*b1 - 5542) * q^15 + (-3*b9 + b8 - 34*b7 - 41*b6 - 54*b5 + 13*b4 + 11*b3 - 109*b2 + 388*b1 - 5801) * q^16 + (148*b9 - 72*b8 - 80*b7 - 16*b6 - 32*b5 - 62*b4 - 92*b3 - 106*b2 - 148*b1 - 7260) * q^17 + (-147*b9 + 126*b8 + 15*b7 - 130*b6 + 75*b5 - 15*b4 + 70*b3 + 60*b2 - 1306*b1 - 15026) * q^18 + (-8*b9 - 53*b8 + 360*b7 + 107*b6 - 83*b5 + 45*b4 - 50*b3 + 203*b2 - 824*b1 - 4475) * q^19 + (-28*b9 + 67*b8 - 67*b7 - 167*b6 + 93*b5 - 56*b4 - 57*b3 + 215*b2 - 1976*b1 - 21869) * q^20 + (118*b9 - 28*b8 + 58*b7 + 105*b6 - 31*b5 - 236*b4 + 58*b3 + 66*b2 - 1790*b1 - 5859) * q^21 + (303*b9 - 71*b8 - 613*b7 + 21*b6 + 50*b5 - 178*b4 + 42*b3 - 443*b2 + 801*b1 - 14281) * q^22 + (-204*b9 + 44*b8 + 581*b7 + 47*b6 + 107*b5 + 149*b4 + 116*b3 - 295*b2 - 1527*b1 - 14042) * q^23 + (-441*b9 - 25*b8 + 841*b7 + 125*b6 + 26*b5 + 402*b4 + 22*b3 + 55*b2 - 6681) * q^24 + (222*b9 - 72*b8 + 427*b7 + 425*b6 - 233*b5 - 19*b4 + 44*b3 + 103*b2 + 985*b1 + 10437) * q^25 + (-11*b9 - 200*b8 + 249*b7 + 124*b6 - 151*b5 + 265*b4 - 388*b3 + 840*b2 + 832*b1 - 6620) * q^26 + (-48*b9 + 129*b8 - 16*b7 + 51*b6 + 75*b5 - 126*b4 + 432*b3 - 477*b2 - 338*b1 - 8443) * q^27 + (-119*b9 + 231*b8 + 281*b7 + 238*b6 - 84*b5 + 477*b4 - 147*b3 - 127*b2 + 6359*b1 + 19648) * q^28 + (-402*b9 + 979*b8 + 315*b7 - 690*b6 + 226*b5 - 250*b4 + 130*b3 - 326*b2 - 2751*b1 - 14585) * q^29 + (1587*b9 - 659*b8 - 3682*b7 - 25*b6 - 38*b5 - 1197*b4 - 233*b3 - 487*b2 + 6046*b1 + 4601) * q^30 + (-16*b9 - 825*b8 - 295*b7 + 59*b6 - 73*b5 + 490*b4 - 500*b3 + 939*b2 + 321*b1 - 19779) * q^31 + (253*b9 - 133*b8 - 1692*b7 - 833*b6 - 264*b5 - 81*b4 - 81*b3 - 179*b2 + 6642*b1 - 5721) * q^32 + (-772*b9 + 229*b8 + 450*b7 - 713*b6 + 313*b5 - 361*b4 - 68*b3 + 1608*b2 - 9910*b1 - 14112) * q^33 + (-1856*b9 + 330*b8 + 784*b7 + 250*b6 + 934*b5 + 1574*b4 + 428*b3 - 930*b2 + 9102*b1 + 50102) * q^34 + (264*b9 - 339*b8 - 2496*b7 - 1389*b6 + 393*b5 - 183*b4 + 18*b3 - 921*b2 - 3316*b1 - 69413) * q^35 + (293*b9 - 20*b8 + 827*b7 + 716*b6 - 1073*b5 - 607*b4 + 438*b3 - 738*b2 + 12700*b1 + 187870) * q^36 + 50653 * q^37 + (192*b9 + 1792*b8 - 3044*b7 + 76*b6 + 40*b5 - 2012*b4 + 1864*b3 + 2464*b2 - 122*b1 + 125476) * q^38 + (2036*b9 - 895*b8 + 1119*b7 + 677*b6 - 919*b5 - 616*b4 - 792*b3 - 1645*b2 + 2227*b1 + 131377) * q^39 + (584*b9 + 257*b8 + 2411*b7 + 2359*b6 - 525*b5 + 728*b4 - 59*b3 + 533*b2 + 18096*b1 + 286381) * q^40 + (1240*b9 - 1844*b8 + 1457*b7 - 186*b6 - 1372*b5 + 167*b4 - 1762*b3 - 2607*b2 - 1565*b1 + 82413) * q^41 + (-3037*b9 + 907*b8 + 10314*b7 + 2125*b6 + 328*b5 + 1565*b4 + 295*b3 + 945*b2 - 5809*b1 + 345489) * q^42 + (-1388*b9 + 2320*b8 - 1034*b7 + 348*b6 + 2656*b5 + 1200*b4 + 1384*b3 + 2570*b2 - 17942*b1 - 52534) * q^43 + (-1905*b9 - 1550*b8 + 6432*b7 - 1173*b6 + 511*b5 + 3935*b4 - 776*b3 - 2684*b2 - 2085*b1 + 62707) * q^44 + (1342*b9 - 621*b8 + 4052*b7 + 134*b6 + 1092*b5 - 161*b4 - 2338*b3 + 4958*b2 - 38400*b1 - 11693) * q^45 + (3417*b9 - 3122*b8 - 5965*b7 + 2010*b6 + 443*b5 - 3055*b4 - 2034*b3 - 2870*b2 + 6642*b1 + 285542) * q^46 + (-808*b9 - 1028*b8 - 3374*b7 - 1047*b6 - 2067*b5 + 857*b4 + 1776*b3 - 535*b2 - 12132*b1 - 148346) * q^47 + (3613*b9 + 1006*b8 + 177*b7 - 1478*b6 - 23*b5 - 4175*b4 - 912*b3 - 596*b2 - 21052*b1 + 216944) * q^48 + (-2994*b9 + 2984*b8 + 3148*b7 - 1029*b6 + 271*b5 + 1156*b4 + 42*b3 - 1552*b2 - 7836*b1 - 15254) * q^49 + (-2243*b9 + 870*b8 - 3067*b7 - 2886*b6 + 2899*b5 + 855*b4 + 844*b3 + 7382*b2 - 42667*b1 - 216740) * q^50 + (-5284*b9 + 7072*b8 + 7110*b7 + 984*b6 - 1112*b5 + 1214*b4 + 5636*b3 - 3198*b2 - 5498*b1 + 80866) * q^51 + (2117*b9 - 358*b8 - 9069*b7 - 4168*b6 - 3399*b5 - 3931*b4 + 3352*b3 - 568*b2 + 12474*b1 + 10210) * q^52 + (1918*b9 - 5126*b8 - 14104*b7 + 3*b6 - 4573*b5 - 2944*b4 + 686*b3 - 6600*b2 + 11180*b1 - 256807) * q^53 + (516*b9 - 5733*b8 + 7212*b7 + 967*b6 + 933*b5 + 2445*b4 - 5290*b3 - 861*b2 - 12203*b1 + 112253) * q^54 + (5788*b9 - 529*b8 - 21521*b7 + 2730*b6 - 268*b5 - 1920*b4 + 38*b3 + 3800*b2 + 40179*b1 - 369589) * q^55 + (716*b9 + 4469*b8 - 9438*b7 - 5743*b6 + 3235*b5 - 3211*b4 + 904*b3 + 483*b2 - 35326*b1 - 350011) * q^56 + (-1504*b9 - 2121*b8 + 758*b7 - 4417*b6 + 5145*b5 + 4097*b4 + 2262*b3 + 25*b2 - 31410*b1 - 912793) * q^57 + (-961*b9 - 476*b8 - 1999*b7 + 2552*b6 - 3561*b5 - 5327*b4 - 5110*b3 - 144*b2 + 40982*b1 + 548316) * q^58 + (4028*b9 - 145*b8 - 12458*b7 + 4161*b6 - 593*b5 + 1555*b4 - 5810*b3 + 9015*b2 + 24254*b1 - 604161) * q^59 + (-12357*b9 - 419*b8 + 41186*b7 - 1587*b6 + 2620*b5 + 19383*b4 - 2757*b3 - 3589*b2 - 298*b1 - 122723) * q^60 + (3156*b9 - 3128*b8 - 12069*b7 + 3418*b6 + 536*b5 - 6493*b4 - 2892*b3 - 950*b2 + 54909*b1 - 281962) * q^61 + (5282*b9 + 6079*b8 - 8029*b7 + 577*b6 - 2625*b5 - 2390*b4 + 9885*b3 - 3289*b2 + 56362*b1 - 107033) * q^62 + (1232*b9 - 3874*b8 + 3642*b7 + 8728*b6 - 616*b5 + 5366*b4 - 3584*b3 - 5536*b2 + 74526*b1 - 314554) * q^63 + (-1817*b9 + 1391*b8 + 1528*b7 - 4025*b6 + 4962*b5 + 3695*b4 - 23*b3 + 13593*b2 + 9862*b1 - 417953) * q^64 + (-11174*b9 + 6621*b8 + 17832*b7 + 4450*b6 + 9052*b5 + 5429*b4 - 506*b3 - 5164*b2 - 5980*b1 - 669631) * q^65 + (2174*b9 + 11454*b8 + 24070*b7 + 13270*b6 - 13248*b5 - 11908*b4 + 8272*b3 - 3248*b2 + 74949*b1 + 1707968) * q^66 + (-9464*b9 - 595*b8 - 3599*b7 - 285*b6 - 2761*b5 - 1008*b4 + 10556*b3 + 3023*b2 + 82737*b1 - 729965) * q^67 + (12128*b9 - 7072*b8 - 31894*b7 - 5414*b6 + 1744*b5 - 10392*b4 + 16*b3 - 5496*b2 - 47842*b1 - 800138) * q^68 + (8504*b9 - 3707*b8 + 26229*b7 - 12785*b6 - 1085*b5 - 4864*b4 - 5378*b3 + 11217*b2 - 53029*b1 - 1136955) * q^69 + (-414*b9 - 7704*b8 + 18652*b7 + 10048*b6 - 4206*b5 + 13128*b4 - 3942*b3 - 21840*b2 + 175428*b1 + 832552) * q^70 + (-4336*b9 - 5260*b8 + 10252*b7 + 16857*b6 + 3805*b5 + 2077*b4 - 7712*b3 + 7293*b2 - 11966*b1 - 501746) * q^71 + (9645*b9 - 11312*b8 - 3361*b7 - 11438*b6 - 473*b5 - 2013*b4 - 9052*b3 + 12254*b2 - 144510*b1 - 709032) * q^72 + (3326*b9 + 5169*b8 - 11227*b7 - 17510*b6 + 1306*b5 - 12076*b4 + 6140*b3 - 8593*b2 + 54323*b1 - 1184710) * q^73 + (-50653*b1 - 101306) * q^74 + (14424*b9 + 755*b8 - 35820*b7 + 4011*b6 - 1693*b5 - 17448*b4 + 12504*b3 - 6533*b2 + 6674*b1 - 666161) * q^75 + (-18356*b9 + 10572*b8 + 55926*b7 - 9902*b6 - 2344*b5 + 15820*b4 - 3644*b3 + 12124*b2 - 70046*b1 + 532702) * q^76 + (-8306*b9 + 18050*b8 + 53128*b7 + 6599*b6 - 5769*b5 - 832*b4 - 1302*b3 + 9910*b2 - 44460*b1 + 166813) * q^77 + (-23410*b9 + 1707*b8 - 16283*b7 - 6487*b6 + 18903*b5 + 15140*b4 + 4167*b3 + 3943*b2 - 194702*b1 - 661109) * q^78 + (4656*b9 + 474*b8 + 4929*b7 - 15339*b6 - 6267*b5 - 411*b4 - 11148*b3 - 23523*b2 + 142017*b1 - 430558) * q^79 + (2360*b9 - 9729*b8 - 45327*b7 - 1711*b6 + 2189*b5 + 7228*b4 + 6191*b3 + 4963*b2 - 191744*b1 - 1134509) * q^80 + (12318*b9 - 5731*b8 + 7842*b7 - 9448*b6 - 5338*b5 - 5253*b4 + 13606*b3 - 38*b2 - 48446*b1 - 722314) * q^81 + (-11084*b9 + 10931*b8 - 47061*b7 - 15151*b6 + 19181*b5 - 7338*b4 + 15239*b3 - 14743*b2 - 83649*b1 + 91429) * q^82 + (-18588*b9 + 194*b8 - 32000*b7 + 217*b6 + 1349*b5 + 5937*b4 - 12676*b3 + 18083*b2 - 118966*b1 - 1523042) * q^83 + (18795*b9 + 8729*b8 - 96835*b7 - 15380*b6 + 2600*b5 - 38205*b4 - 2581*b3 + 1747*b2 - 300113*b1 + 591314) * q^84 + (12464*b9 - 17212*b8 + 57978*b7 + 15088*b6 - 8884*b5 + 23970*b4 - 19360*b3 + 19420*b2 + 187046*b1 + 1105324) * q^85 + (25256*b9 - 27536*b8 + 19402*b7 + 52188*b6 - 19920*b5 + 13382*b4 - 24258*b3 + 164*b2 + 137920*b1 + 3356616) * q^86 + (14276*b9 - 6490*b8 + 1859*b7 - 26472*b6 - 2266*b5 + 24983*b4 - 24106*b3 - 37126*b2 + 41607*b1 - 1270552) * q^87 + (17642*b9 - 14602*b8 - 111029*b7 + 1844*b6 - 4840*b5 - 22893*b4 - 1543*b3 + 11188*b2 - 27250*b1 + 1432874) * q^88 + (-19068*b9 + 17627*b8 + 38196*b7 - 6471*b6 - 12889*b5 + 1795*b4 + 37466*b3 - 11599*b2 + 27488*b1 + 10131) * q^89 + (-16443*b9 + 26459*b8 + 1216*b7 + 71941*b6 - 14518*b5 + 2655*b4 + 32927*b3 - 4697*b2 + 188150*b1 + 6398647) * q^90 + (-7580*b9 + 10560*b8 - 6906*b7 + 30164*b6 + 34008*b5 + 4080*b4 + 1968*b3 + 23894*b2 + 14858*b1 + 1210102) * q^91 + (-28251*b9 + 124*b8 + 92319*b7 - 732*b6 + 14647*b5 + 40373*b4 - 11438*b3 - 24278*b2 - 238260*b1 + 869662) * q^92 + (-50482*b9 + 4717*b8 + 45424*b7 + 3308*b6 + 24286*b5 + 21665*b4 + 5334*b3 + 39222*b2 - 259660*b1 + 560661) * q^93 + (21327*b9 - 2448*b8 + 10494*b7 - 18638*b6 - 3041*b5 - 8068*b4 + 1301*b3 + 29632*b2 + 181531*b1 + 2540854) * q^94 + (26360*b9 - 3808*b8 - 75150*b7 + 9956*b6 - 9248*b5 - 59698*b4 - 13988*b3 - 36276*b2 - 29614*b1 + 1187840) * q^95 + (-15827*b9 + 17690*b8 - 15531*b7 + 23016*b6 + 1229*b5 - 9101*b4 - 386*b3 - 8688*b2 - 158422*b1 + 4306082) * q^96 + (28712*b9 - 4143*b8 + 38456*b7 - 49813*b6 - 21403*b5 - 14837*b4 + 33118*b3 - 20481*b2 + 213504*b1 + 390501) * q^97 + (25305*b9 + 805*b8 - 69328*b7 - 11641*b6 - 4676*b5 - 51807*b4 - 13853*b3 + 5299*b2 + 23316*b1 + 1507581) * q^98 + (-21932*b9 - 15469*b8 + 53964*b7 - 39934*b6 - 10*b5 + 57229*b4 + 6164*b3 + 1332*b2 + 87688*b1 - 1723977) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 24 q^{2} - 95 q^{3} + 602 q^{4} - 624 q^{5} - 777 q^{6} - 501 q^{7} - 3810 q^{8} + 6181 q^{9}+O(q^{10})$$ 10 * q - 24 * q^2 - 95 * q^3 + 602 * q^4 - 624 * q^5 - 777 * q^6 - 501 * q^7 - 3810 * q^8 + 6181 * q^9 $$10 q - 24 q^{2} - 95 q^{3} + 602 q^{4} - 624 q^{5} - 777 q^{6} - 501 q^{7} - 3810 q^{8} + 6181 q^{9} + 8595 q^{10} - 8325 q^{11} - 19645 q^{12} - 17108 q^{13} - 65418 q^{14} - 55756 q^{15} - 56998 q^{16} - 72924 q^{17} - 156165 q^{18} - 47786 q^{19} - 226209 q^{20} - 65313 q^{21} - 138973 q^{22} - 148086 q^{23} - 68031 q^{24} + 108736 q^{25} - 60237 q^{26} - 87329 q^{27} + 219974 q^{28} - 164154 q^{29} + 78864 q^{30} - 189560 q^{31} - 30114 q^{32} - 179737 q^{33} + 532624 q^{34} - 705156 q^{35} + 1923693 q^{36} + 506530 q^{37} + 1256412 q^{38} + 1322800 q^{39} + 2936777 q^{40} + 814263 q^{41} + 3415826 q^{42} - 590572 q^{43} + 610311 q^{44} - 250574 q^{45} + 2903897 q^{46} - 1534185 q^{47} + 2082419 q^{48} - 214337 q^{49} - 2313525 q^{50} + 722138 q^{51} + 149159 q^{52} - 2518209 q^{53} + 1095990 q^{54} - 3482468 q^{55} - 3645834 q^{56} - 9225638 q^{57} + 5626023 q^{58} - 5894748 q^{59} - 1289832 q^{60} - 2569480 q^{61} - 863697 q^{62} - 2836574 q^{63} - 4093742 q^{64} - 6774600 q^{65} + 17251556 q^{66} - 6983232 q^{67} - 8114412 q^{68} - 11557564 q^{69} + 8982748 q^{70} - 5013963 q^{71} - 7567137 q^{72} - 11678449 q^{73} - 1215672 q^{74} - 6586901 q^{75} + 4912252 q^{76} + 1333113 q^{77} - 7352119 q^{78} - 3853378 q^{79} - 11975661 q^{80} - 7381718 q^{81} + 564093 q^{82} - 15677895 q^{83} + 4781738 q^{84} + 11909320 q^{85} + 34274010 q^{86} - 12611710 q^{87} + 14448317 q^{88} - 25836 q^{89} + 64591590 q^{90} + 12335744 q^{91} + 7579845 q^{92} + 4592632 q^{93} + 26251718 q^{94} + 11723664 q^{95} + 42299113 q^{96} + 4648834 q^{97} + 15230184 q^{98} - 16904018 q^{99}+O(q^{100})$$ 10 * q - 24 * q^2 - 95 * q^3 + 602 * q^4 - 624 * q^5 - 777 * q^6 - 501 * q^7 - 3810 * q^8 + 6181 * q^9 + 8595 * q^10 - 8325 * q^11 - 19645 * q^12 - 17108 * q^13 - 65418 * q^14 - 55756 * q^15 - 56998 * q^16 - 72924 * q^17 - 156165 * q^18 - 47786 * q^19 - 226209 * q^20 - 65313 * q^21 - 138973 * q^22 - 148086 * q^23 - 68031 * q^24 + 108736 * q^25 - 60237 * q^26 - 87329 * q^27 + 219974 * q^28 - 164154 * q^29 + 78864 * q^30 - 189560 * q^31 - 30114 * q^32 - 179737 * q^33 + 532624 * q^34 - 705156 * q^35 + 1923693 * q^36 + 506530 * q^37 + 1256412 * q^38 + 1322800 * q^39 + 2936777 * q^40 + 814263 * q^41 + 3415826 * q^42 - 590572 * q^43 + 610311 * q^44 - 250574 * q^45 + 2903897 * q^46 - 1534185 * q^47 + 2082419 * q^48 - 214337 * q^49 - 2313525 * q^50 + 722138 * q^51 + 149159 * q^52 - 2518209 * q^53 + 1095990 * q^54 - 3482468 * q^55 - 3645834 * q^56 - 9225638 * q^57 + 5626023 * q^58 - 5894748 * q^59 - 1289832 * q^60 - 2569480 * q^61 - 863697 * q^62 - 2836574 * q^63 - 4093742 * q^64 - 6774600 * q^65 + 17251556 * q^66 - 6983232 * q^67 - 8114412 * q^68 - 11557564 * q^69 + 8982748 * q^70 - 5013963 * q^71 - 7567137 * q^72 - 11678449 * q^73 - 1215672 * q^74 - 6586901 * q^75 + 4912252 * q^76 + 1333113 * q^77 - 7352119 * q^78 - 3853378 * q^79 - 11975661 * q^80 - 7381718 * q^81 + 564093 * q^82 - 15677895 * q^83 + 4781738 * q^84 + 11909320 * q^85 + 34274010 * q^86 - 12611710 * q^87 + 14448317 * q^88 - 25836 * q^89 + 64591590 * q^90 + 12335744 * q^91 + 7579845 * q^92 + 4592632 * q^93 + 26251718 * q^94 + 11723664 * q^95 + 42299113 * q^96 + 4648834 * q^97 + 15230184 * q^98 - 16904018 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4 x^{9} - 905 x^{8} + 4018 x^{7} + 291290 x^{6} - 1367036 x^{5} - 39566544 x^{4} + 175732512 x^{3} + 2143908736 x^{2} - 5929608704 x - 45399525376$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 6903850151 \nu^{9} - 44030216064 \nu^{8} + 5848661370847 \nu^{7} + 31740379054046 \nu^{6} + \cdots - 29\!\cdots\!24 ) / 52\!\cdots\!28$$ (-6903850151*v^9 - 44030216064*v^8 + 5848661370847*v^7 + 31740379054046*v^6 - 1706384992740926*v^5 - 7644109923112852*v^4 + 195061817824269920*v^3 + 729280338610839968*v^2 - 6958766180422642176*v - 29559102760876404224) / 5215682315440128 $$\beta_{3}$$ $$=$$ $$( - 62575071425 \nu^{9} - 287016653328 \nu^{8} + 51295267891465 \nu^{7} + 193693281740162 \nu^{6} + \cdots - 13\!\cdots\!64 ) / 31\!\cdots\!68$$ (-62575071425*v^9 - 287016653328*v^8 + 51295267891465*v^7 + 193693281740162*v^6 - 14323839873282866*v^5 - 41338331907724012*v^4 + 1531481356758599264*v^3 + 3209830508823277664*v^2 - 48031112437380639744*v - 137367854421988657664) / 31294093892640768 $$\beta_{4}$$ $$=$$ $$( 824744993 \nu^{9} + 5598640392 \nu^{8} - 698144499529 \nu^{7} - 3961595322746 \nu^{6} + 204042096531554 \nu^{5} + \cdots + 34\!\cdots\!60 ) / 372548736817152$$ (824744993*v^9 + 5598640392*v^8 - 698144499529*v^7 - 3961595322746*v^6 + 204042096531554*v^5 + 925765097462044*v^4 - 23393970265197440*v^3 - 84589566234085088*v^2 + 827380897448530176*v + 3413771101247720960) / 372548736817152 $$\beta_{5}$$ $$=$$ $$( 23822773327 \nu^{9} + 160382537808 \nu^{8} - 20325864531719 \nu^{7} - 115632112359166 \nu^{6} + \cdots + 10\!\cdots\!64 ) / 10\!\cdots\!56$$ (23822773327*v^9 + 160382537808*v^8 - 20325864531719*v^7 - 115632112359166*v^6 + 5984951136766894*v^5 + 27761794475288468*v^4 - 694102244390316064*v^3 - 2634884715364197280*v^2 + 25455716993854003200*v + 107093511603909912064) / 10431364630880256 $$\beta_{6}$$ $$=$$ $$( - 169096231607 \nu^{9} - 1115891967696 \nu^{8} + 141291187862959 \nu^{7} + 819419713942478 \nu^{6} + \cdots - 74\!\cdots\!88 ) / 31\!\cdots\!68$$ (-169096231607*v^9 - 1115891967696*v^8 + 141291187862959*v^7 + 819419713942478*v^6 - 40689248576248574*v^5 - 200785049314364788*v^4 + 4608856907815114016*v^3 + 19222062577785114272*v^2 - 165220982424296355840*v - 746947801971675289088) / 31294093892640768 $$\beta_{7}$$ $$=$$ $$( 169096231607 \nu^{9} + 1115891967696 \nu^{8} - 141291187862959 \nu^{7} - 819419713942478 \nu^{6} + \cdots + 74\!\cdots\!44 ) / 31\!\cdots\!68$$ (169096231607*v^9 + 1115891967696*v^8 - 141291187862959*v^7 - 819419713942478*v^6 + 40689248576248574*v^5 + 200785049314364788*v^4 - 4608856907815114016*v^3 - 19190768483892473504*v^2 + 165252276518188996608*v + 741220982789322028544) / 31294093892640768 $$\beta_{8}$$ $$=$$ $$( 81089504555 \nu^{9} + 473877845808 \nu^{8} - 67919217844483 \nu^{7} - 344607611020502 \nu^{6} + \cdots + 32\!\cdots\!72 ) / 78\!\cdots\!92$$ (81089504555*v^9 + 473877845808*v^8 - 67919217844483*v^7 - 344607611020502*v^6 + 19594548905919398*v^5 + 83375635956990244*v^4 - 2217074201499315104*v^3 - 7911295867676008736*v^2 + 78376745805633512448*v + 321736813219016156672) / 7823523473160192 $$\beta_{9}$$ $$=$$ $$( 396026017655 \nu^{9} + 2579512909920 \nu^{8} - 331845919248943 \nu^{7} + \cdots + 17\!\cdots\!04 ) / 15\!\cdots\!84$$ (396026017655*v^9 + 2579512909920*v^8 - 331845919248943*v^7 - 1889268355689566*v^6 + 95811394593348638*v^5 + 460959135883089172*v^4 - 10867738963822753760*v^3 - 43859905866542197664*v^2 + 388427153139967077888*v + 1710482873047796616704) / 15647046946320384
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta _1 + 183$$ b7 + b6 - b1 + 183 $$\nu^{3}$$ $$=$$ $$\beta_{9} - 6\beta_{7} - 2\beta_{6} - 5\beta_{5} + 2\beta_{4} + \beta_{3} - 4\beta_{2} + 258\beta _1 - 214$$ b9 - 6*b7 - 2*b6 - 5*b5 + 2*b4 + b3 - 4*b2 + 258*b1 - 214 $$\nu^{4}$$ $$=$$ $$- 11 \beta_{9} + \beta_{8} + 374 \beta_{7} + 335 \beta_{6} - 14 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 77 \beta_{2} - 532 \beta _1 + 46927$$ -11*b9 + b8 + 374*b7 + 335*b6 - 14*b5 - 3*b4 + 3*b3 - 77*b2 - 532*b1 + 46927 $$\nu^{5}$$ $$=$$ $$329 \beta_{9} + 123 \beta_{8} - 1888 \beta_{7} - 469 \beta_{6} - 1956 \beta_{5} + 1055 \beta_{4} + 523 \beta_{3} - 939 \beta_{2} + 74374 \beta _1 - 111261$$ 329*b9 + 123*b8 - 1888*b7 - 469*b6 - 1956*b5 + 1055*b4 + 523*b3 - 939*b2 + 74374*b1 - 111261 $$\nu^{6}$$ $$=$$ $$- 7185 \beta_{9} + 495 \beta_{8} + 128160 \beta_{7} + 102799 \beta_{6} - 4486 \beta_{5} - 785 \beta_{4} + 401 \beta_{3} - 39639 \beta_{2} - 201250 \beta _1 + 13564839$$ -7185*b9 + 495*b8 + 128160*b7 + 102799*b6 - 4486*b5 - 785*b4 + 401*b3 - 39639*b2 - 201250*b1 + 13564839 $$\nu^{7}$$ $$=$$ $$83577 \beta_{9} + 72597 \beta_{8} - 520180 \beta_{7} - 131975 \beta_{6} - 654390 \beta_{5} + 425293 \beta_{4} + 201703 \beta_{3} - 140629 \beta_{2} + 22416470 \beta _1 - 41517167$$ 83577*b9 + 72597*b8 - 520180*b7 - 131975*b6 - 654390*b5 + 425293*b4 + 201703*b3 - 140629*b2 + 22416470*b1 - 41517167 $$\nu^{8}$$ $$=$$ $$- 3247261 \beta_{9} + 23355 \beta_{8} + 43190140 \beta_{7} + 31563511 \beta_{6} - 902246 \beta_{5} + 66803 \beta_{4} - 265667 \beta_{3} - 15489787 \beta_{2} + \cdots + 4109646639$$ -3247261*b9 + 23355*b8 + 43190140*b7 + 31563511*b6 - 902246*b5 + 66803*b4 - 265667*b3 - 15489787*b2 - 71192182*b1 + 4109646639 $$\nu^{9}$$ $$=$$ $$17596401 \beta_{9} + 32119609 \beta_{8} - 138258172 \beta_{7} - 46361323 \beta_{6} - 211558658 \beta_{5} + 155327385 \beta_{4} + 70077991 \beta_{3} + \cdots - 14473144979$$ 17596401*b9 + 32119609*b8 - 138258172*b7 - 46361323*b6 - 211558658*b5 + 155327385*b4 + 70077991*b3 + 984871*b2 + 6901524294*b1 - 14473144979

## 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
 17.8339 16.8983 10.5631 10.4614 9.55485 −5.27803 −5.89746 −14.2020 −17.7200 −18.2140
−19.8339 −69.2191 265.384 −437.776 1372.88 1289.75 −2724.87 2604.28 8682.81
1.2 −18.8983 75.2961 229.145 −439.542 −1422.97 380.249 −1911.48 3482.50 8306.59
1.3 −12.5631 −77.5936 29.8308 318.311 974.814 −75.8943 1233.31 3833.76 −3998.96
1.4 −12.4614 49.0521 27.2868 −16.1140 −611.258 −704.986 1255.03 219.106 200.803
1.5 −11.5549 −22.2601 5.51464 118.339 257.212 925.681 1415.30 −1691.49 −1367.39
1.6 3.27803 −7.54665 −117.254 243.887 −24.7382 767.318 −803.953 −2130.05 799.468
1.7 3.89746 51.1724 −112.810 −91.5007 199.442 −1347.68 −938.547 431.610 −356.620
1.8 12.2020 8.53392 20.8894 −415.456 104.131 362.670 −1306.97 −2114.17 −5069.41
1.9 15.7200 −69.5612 119.119 316.724 −1093.50 −1289.70 −139.607 2651.76 4978.91
1.10 16.2140 −32.8739 134.894 −220.871 −533.017 −808.413 111.778 −1106.31 −3581.21
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$37$$ $$-1$$

## 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 37.8.a.a 10
3.b odd 2 1 333.8.a.c 10
4.b odd 2 1 592.8.a.f 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.8.a.a 10 1.a even 1 1 trivial
333.8.a.c 10 3.b odd 2 1
592.8.a.f 10 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + 24 T_{2}^{9} - 653 T_{2}^{8} - 16962 T_{2}^{7} + 139726 T_{2}^{6} + 4135692 T_{2}^{5} - 10527856 T_{2}^{4} - 394812000 T_{2}^{3} + 314897184 T_{2}^{2} + 11292902208 T_{2} - 26941953024$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(37))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 24 T^{9} + \cdots - 26941953024$$
$3$ $$T^{10} + 95 T^{9} + \cdots + 33\!\cdots\!44$$
$5$ $$T^{10} + 624 T^{9} + \cdots + 75\!\cdots\!00$$
$7$ $$T^{10} + 501 T^{9} + \cdots - 94\!\cdots\!24$$
$11$ $$T^{10} + 8325 T^{9} + \cdots - 11\!\cdots\!20$$
$13$ $$T^{10} + 17108 T^{9} + \cdots - 11\!\cdots\!12$$
$17$ $$T^{10} + 72924 T^{9} + \cdots + 53\!\cdots\!60$$
$19$ $$T^{10} + 47786 T^{9} + \cdots + 50\!\cdots\!60$$
$23$ $$T^{10} + 148086 T^{9} + \cdots - 83\!\cdots\!12$$
$29$ $$T^{10} + 164154 T^{9} + \cdots - 13\!\cdots\!32$$
$31$ $$T^{10} + 189560 T^{9} + \cdots + 26\!\cdots\!76$$
$37$ $$(T - 50653)^{10}$$
$41$ $$T^{10} - 814263 T^{9} + \cdots - 28\!\cdots\!98$$
$43$ $$T^{10} + 590572 T^{9} + \cdots + 30\!\cdots\!68$$
$47$ $$T^{10} + 1534185 T^{9} + \cdots - 32\!\cdots\!64$$
$53$ $$T^{10} + 2518209 T^{9} + \cdots + 44\!\cdots\!96$$
$59$ $$T^{10} + 5894748 T^{9} + \cdots - 91\!\cdots\!52$$
$61$ $$T^{10} + 2569480 T^{9} + \cdots - 19\!\cdots\!16$$
$67$ $$T^{10} + 6983232 T^{9} + \cdots + 31\!\cdots\!12$$
$71$ $$T^{10} + 5013963 T^{9} + \cdots + 38\!\cdots\!72$$
$73$ $$T^{10} + 11678449 T^{9} + \cdots + 32\!\cdots\!06$$
$79$ $$T^{10} + 3853378 T^{9} + \cdots - 64\!\cdots\!08$$
$83$ $$T^{10} + 15677895 T^{9} + \cdots - 30\!\cdots\!60$$
$89$ $$T^{10} + 25836 T^{9} + \cdots + 74\!\cdots\!12$$
$97$ $$T^{10} - 4648834 T^{9} + \cdots + 48\!\cdots\!24$$