# Properties

 Label 37.8.a.b Level $37$ Weight $8$ Character orbit 37.a Self dual yes Analytic conductor $11.558$ Analytic rank $0$ Dimension $11$ 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: $$0$$ Dimension: $$11$$ Coefficient field: $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{11} - 5 x^{10} - 1078 x^{9} + 4966 x^{8} + 379692 x^{7} - 1385588 x^{6} - 48765978 x^{5} + 87529978 x^{4} + 2159400643 x^{3} - 1763707223 x^{2} + \cdots + 6680404080$$ x^11 - 5*x^10 - 1078*x^9 + 4966*x^8 + 379692*x^7 - 1385588*x^6 - 48765978*x^5 + 87529978*x^4 + 2159400643*x^3 - 1763707223*x^2 - 25456347552*x + 6680404080 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{5}$$ 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 $$1,\beta_1,\ldots,\beta_{10}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{2} + (\beta_{2} + 11) q^{3} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 71) q^{4} + ( - \beta_{5} + \beta_{2} - 6 \beta_1 + 37) q^{5} + (\beta_{5} + \beta_{4} + 5 \beta_{2} + 19 \beta_1 + 38) q^{6} + (\beta_{10} - \beta_{8} - \beta_{7} - \beta_{5} - 2 \beta_{4} + 3 \beta_{2} + 5 \beta_1 + 202) q^{7} + ( - 2 \beta_{10} + 3 \beta_{8} + 3 \beta_{7} + 4 \beta_{5} + 2 \beta_{3} + \cdots + 297) q^{8}+ \cdots + ( - 3 \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} + \cdots + 864) q^{9}+O(q^{10})$$ q + (b1 + 1) * q^2 + (b2 + 11) * q^3 + (b3 + b2 + 2*b1 + 71) * q^4 + (-b5 + b2 - 6*b1 + 37) * q^5 + (b5 + b4 + 5*b2 + 19*b1 + 38) * q^6 + (b10 - b8 - b7 - b5 - 2*b4 + 3*b2 + 5*b1 + 202) * q^7 + (-2*b10 + 3*b8 + 3*b7 + 4*b5 + 2*b3 + 3*b2 + 122*b1 + 297) * q^8 + (-3*b10 - b9 - b8 - 2*b7 + b6 + 2*b5 + b4 - 3*b3 + 16*b2 + 66*b1 + 864) * q^9 $$q + (\beta_1 + 1) q^{2} + (\beta_{2} + 11) q^{3} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 71) q^{4} + ( - \beta_{5} + \beta_{2} - 6 \beta_1 + 37) q^{5} + (\beta_{5} + \beta_{4} + 5 \beta_{2} + 19 \beta_1 + 38) q^{6} + (\beta_{10} - \beta_{8} - \beta_{7} - \beta_{5} - 2 \beta_{4} + 3 \beta_{2} + 5 \beta_1 + 202) q^{7} + ( - 2 \beta_{10} + 3 \beta_{8} + 3 \beta_{7} + 4 \beta_{5} + 2 \beta_{3} + \cdots + 297) q^{8}+ \cdots + (17686 \beta_{10} + 10991 \beta_{9} - 17537 \beta_{8} + \cdots - 1304541) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^2 + (b2 + 11) * q^3 + (b3 + b2 + 2*b1 + 71) * q^4 + (-b5 + b2 - 6*b1 + 37) * q^5 + (b5 + b4 + 5*b2 + 19*b1 + 38) * q^6 + (b10 - b8 - b7 - b5 - 2*b4 + 3*b2 + 5*b1 + 202) * q^7 + (-2*b10 + 3*b8 + 3*b7 + 4*b5 + 2*b3 + 3*b2 + 122*b1 + 297) * q^8 + (-3*b10 - b9 - b8 - 2*b7 + b6 + 2*b5 + b4 - 3*b3 + 16*b2 + 66*b1 + 864) * q^9 + (5*b10 + 4*b9 - 2*b8 + 3*b7 - 4*b6 - b5 - 4*b4 - 13*b3 - 33*b2 + 73*b1 - 1177) * q^10 + (6*b10 - b9 - b8 - 8*b7 - 8*b5 - 5*b3 - 11*b2 - 12*b1 + 860) * q^11 + (-7*b10 - 10*b9 + b7 + 16*b6 + 7*b5 + 20*b4 + 12*b3 + 16*b2 + 93*b1 + 2488) * q^12 + (b10 + 3*b8 + b7 - 8*b6 - 13*b5 - 2*b4 - 14*b3 - 52*b2 + 29*b1 + 1129) * q^13 + (6*b10 + 13*b9 - 3*b8 - b7 - 28*b6 - 2*b5 - 17*b4 - 6*b3 - 108*b2 + 223*b1 + 1565) * q^14 + (-11*b10 + 16*b9 - 9*b8 + 21*b7 + 18*b6 - 10*b5 - 16*b4 - 14*b3 - 47*b2 - 495*b1 + 2588) * q^15 + (-19*b10 - 3*b9 + 29*b8 - 12*b7 + 16*b6 + 22*b5 + 8*b4 + 111*b3 + 40*b2 + 395*b1 + 15058) * q^16 + (14*b10 - 20*b9 - 6*b8 + 14*b7 - 4*b6 - 14*b5 - 8*b4 + 4*b3 - 22*b2 - 470*b1 + 5142) * q^17 + (-40*b9 - 7*b8 - 35*b7 - 8*b6 + 39*b5 + 47*b4 + 60*b3 + 158*b2 + 214*b1 + 14742) * q^18 + (-9*b10 - 31*b9 + 30*b8 + 19*b7 - 20*b6 + 8*b5 + 30*b4 - 25*b3 + 133*b2 - 157*b1 + 8895) * q^19 + (51*b10 + 64*b9 - 66*b8 - 7*b7 + 68*b6 - 29*b5 - 92*b4 + 51*b3 - 99*b2 - 2279*b1 + 8855) * q^20 + (24*b10 + 92*b9 - b8 - 37*b7 + 11*b6 + 27*b5 - 39*b4 - 92*b3 + 348*b2 - 1859*b1 + 9799) * q^21 + (7*b10 + 29*b9 - 13*b8 + 18*b7 - 92*b6 - 23*b5 - 29*b4 - 167*b3 - 189*b2 - 46*b1 - 1050) * q^22 + (-b10 - 12*b9 - 3*b8 - 15*b7 - 60*b6 + 9*b5 + 30*b4 - 140*b3 + 332*b2 - 1073*b1 + 10253) * q^23 + (-113*b10 - 218*b9 + 27*b8 + 64*b7 - 20*b6 + 79*b5 + 236*b4 - 79*b3 + 370*b2 + 2321*b1 + 13650) * q^24 + (-61*b10 - 28*b9 + 63*b8 + 57*b7 + 138*b6 - 27*b5 + 80*b4 - 34*b3 + 384*b2 - 1831*b1 + 15866) * q^25 + (78*b10 + 77*b9 - 43*b8 - 29*b7 + 36*b6 - 141*b5 - 156*b4 + 70*b3 - 619*b2 - 1168*b1 + 6185) * q^26 + (-111*b10 + 35*b9 + 146*b8 - 101*b7 - 98*b6 + 161*b5 + 76*b4 + 51*b3 + 853*b2 + 2529*b1 + 39441) * q^27 + (-39*b10 + 167*b9 + 93*b8 + 52*b7 + 304*b6 - 208*b5 - 204*b4 + 380*b3 - 923*b2 - 727*b1 + 20077) * q^28 + (102*b10 - 169*b9 - 169*b8 + 16*b7 - 18*b6 - 27*b5 - 42*b4 + 139*b3 - 229*b2 - 788*b1 + 4006) * q^29 + (337*b10 + 203*b9 - 197*b8 + 48*b7 - 216*b6 - 264*b5 - 298*b4 - 577*b3 - 1972*b2 + 321*b1 - 98218) * q^30 + (-59*b10 - 253*b9 - 84*b8 + 125*b7 - 88*b6 - 29*b5 + 226*b4 - 165*b3 - 486*b2 + 2395*b1 - 25984) * q^31 + (-129*b10 - 93*b9 + 251*b8 - 228*b7 - 412*b6 + 226*b5 + 28*b4 + 727*b3 - 478*b2 + 15137*b1 + 46788) * q^32 + (322*b10 + 387*b9 - 226*b8 - 309*b7 + 21*b6 + 211*b5 - 325*b4 + 135*b3 + 480*b2 - 2781*b1 - 18370) * q^33 + (-56*b10 + 34*b9 - 50*b8 + 62*b7 - 8*b6 - 60*b5 + 58*b4 - 360*b3 - 1380*b2 + 7800*b1 - 87404) * q^34 + (-305*b10 + 109*b9 - 72*b8 + 125*b7 + 554*b6 + 296*b5 + 112*b4 + 983*b3 + 705*b2 - 3135*b1 + 41039) * q^35 + (-322*b10 - 663*b9 + 421*b8 + 225*b7 + 336*b6 + 563*b5 + 964*b4 + 318*b3 + 2119*b2 + 15870*b1 - 49221) * q^36 - 50653 * q^37 + (-272*b10 - 266*b9 + 126*b8 - 190*b7 + 500*b6 + 292*b5 + 598*b4 + 656*b3 + 1792*b2 + 6444*b1 - 19296) * q^38 + (-13*b10 + 407*b9 - 90*b8 + 435*b7 - 83*b5 - 434*b4 - 745*b3 + 106*b2 - 9879*b1 - 139228) * q^39 + (19*b10 + 468*b9 - 100*b8 + 299*b7 - 1068*b6 - 553*b5 - 716*b4 - 2231*b3 - 2711*b2 + 7309*b1 - 293893) * q^40 + (367*b10 - 488*b9 + 690*b8 - 436*b7 - 547*b6 + 10*b5 + 73*b4 + 1430*b3 + 1156*b2 + 3654*b1 + 45360) * q^41 + (523*b10 + 706*b9 - 225*b8 - 78*b7 - 456*b6 - 400*b5 - 595*b4 - 2519*b3 + 1061*b2 - 9288*b1 - 341612) * q^42 + (588*b10 - 514*b9 - 286*b8 - 640*b7 - 348*b6 - 1462*b5 + 132*b4 - 846*b3 - 1634*b2 - 21048*b1 - 115802) * q^43 + (-132*b10 + 475*b9 - 577*b8 + 425*b7 + 1396*b6 - 361*b5 - 880*b4 + 913*b3 + 1064*b2 - 26738*b1 - 108834) * q^44 + (118*b10 + 123*b9 - 61*b8 + 1012*b7 - 458*b6 - 470*b5 - 1074*b4 - 1301*b3 - 620*b2 - 19974*b1 - 200535) * q^45 + (26*b10 - 397*b9 - 417*b8 - 603*b7 + 1212*b6 - 99*b5 + 740*b4 - 1090*b3 + 3975*b2 - 11908*b1 - 179893) * q^46 + (-1195*b10 + 590*b9 + 653*b8 + 199*b7 - 24*b6 + 375*b5 + 54*b4 + 2078*b3 - 89*b2 - 12819*b1 + 212272) * q^47 + (-642*b10 - 1969*b9 - 91*b8 - 1103*b7 + 844*b6 + 1101*b5 + 2276*b4 + 2022*b3 + 12415*b2 - 2534*b1 + 160819) * q^48 + (260*b10 + 158*b9 - 283*b8 - 1589*b7 - 397*b6 + 291*b5 - 343*b4 + 1130*b3 + 146*b2 - 5651*b1 + 209588) * q^49 + (182*b10 - 217*b9 + 73*b8 + 343*b7 - 1516*b6 + 1049*b5 + 1432*b4 - 1722*b3 - 475*b2 + 12609*b1 - 359234) * q^50 + (-736*b10 + 308*b9 + 738*b8 + 1862*b7 + 1310*b6 + 490*b5 + 234*b4 + 628*b3 - 1622*b2 - 20142*b1 + 336) * q^51 + (1006*b10 + 2037*b9 - 555*b8 + 721*b7 - 1568*b6 + 203*b5 - 2984*b4 - 702*b3 - 8705*b2 + 11382*b1 - 385453) * q^52 + (1280*b10 - 172*b9 - 1525*b8 + 223*b7 - 297*b6 + 313*b5 + 189*b4 + 476*b3 - 8382*b2 - 25675*b1 - 20905) * q^53 + (-1143*b10 - 904*b9 + 1724*b8 - 1295*b7 + 1288*b6 + 2322*b5 + 851*b4 + 5775*b3 + 12524*b2 + 34204*b1 + 560001) * q^54 + (386*b10 - 583*b9 + 935*b8 - 1424*b7 + 1448*b6 - 1233*b5 + 2516*b4 + 2571*b3 - 7953*b2 - 1710*b1 + 378030) * q^55 + (1207*b10 + 2111*b9 + 1770*b8 + 2485*b7 - 4220*b6 + 68*b5 - 3196*b4 + 299*b3 - 16325*b2 + 43553*b1 - 409953) * q^56 + (-2249*b10 - 1277*b9 + 774*b8 + 1601*b7 + 878*b6 + 2008*b5 + 1068*b4 + 681*b3 + 5295*b2 + 25625*b1 + 549279) * q^57 + (-960*b10 - 1077*b9 - 615*b8 + 429*b7 + 480*b6 + 99*b5 + 1440*b4 - 2420*b3 - 4001*b2 + 39986*b1 - 139557) * q^58 + (-1019*b10 + 105*b9 - 530*b8 + 759*b7 - 290*b6 - 422*b5 + 176*b4 - 2213*b3 - 11667*b2 + 4547*b1 + 416899) * q^59 + (4819*b10 + 2257*b9 - 2637*b8 - 2292*b7 - 708*b6 - 7396*b5 - 4612*b4 - 2111*b3 - 7432*b2 - 116775*b1 - 349706) * q^60 + (-1634*b10 + 792*b9 - 518*b8 + 690*b7 + 3368*b6 + 1367*b5 - 116*b4 - 2928*b3 - 151*b2 + 34348*b1 + 550979) * q^61 + (-845*b10 - 3252*b9 - 1546*b8 - 675*b7 + 4664*b6 - 257*b5 + 4150*b4 + 1653*b3 + 10659*b2 - 35139*b1 + 443481) * q^62 + (2526*b10 + 792*b9 - 3042*b8 - 5606*b7 - 4880*b6 - 4004*b5 - 2500*b4 - 7628*b3 - 96*b2 - 24374*b1 + 637806) * q^63 + (-1871*b10 - 1375*b9 + 1743*b8 - 166*b7 + 1580*b6 + 1414*b5 - 2708*b4 + 9825*b3 + 7620*b2 + 91351*b1 + 1092526) * q^64 + (-1782*b10 - 1519*b9 + 2955*b8 + 814*b7 - 1052*b6 - 312*b5 + 3536*b4 - 4239*b3 + 2204*b2 + 58692*b1 + 821611) * q^65 + (440*b10 + 2989*b9 - 178*b8 + 1314*b7 - 2508*b6 - 1986*b5 - 4031*b4 - 9152*b3 + 4443*b2 - 11841*b1 - 509110) * q^66 + (3495*b10 - 1967*b9 + 50*b8 + 861*b7 - 950*b6 - 1757*b5 + 1740*b4 - 2919*b3 - 15054*b2 + 78751*b1 + 162208) * q^67 + (-194*b10 + 2298*b9 - 1202*b8 - 2448*b7 + 2128*b6 - 1336*b5 + 104*b4 + 5578*b3 + 7760*b2 - 84302*b1 + 773708) * q^68 + (-1349*b10 + 1213*b9 + 450*b8 - 1045*b7 - 1444*b6 + 1311*b5 - 2482*b4 - 3069*b3 + 21046*b2 + 54097*b1 + 1206664) * q^69 + (-776*b10 - 1978*b9 + 2148*b8 + 3224*b7 - 4324*b6 + 5452*b5 + 2806*b4 - 5528*b3 + 730*b2 + 193098*b1 - 676912) * q^70 + (4213*b10 + 3674*b9 + 1549*b8 + 1459*b7 + 428*b6 + 867*b5 - 3422*b4 - 2414*b3 - 15483*b2 + 20977*b1 + 415922) * q^71 + (-8546*b10 - 5291*b9 + 5067*b8 + 4279*b7 + 5248*b6 + 6703*b5 + 13112*b4 + 14852*b3 + 51583*b2 + 6362*b1 + 1157127) * q^72 + (2917*b10 + 5091*b9 - 1261*b8 + 3470*b7 + 1785*b6 + 7814*b5 - 1751*b4 - 3839*b3 + 3076*b2 + 117250*b1 - 149207) * q^73 + (-50653*b1 - 50653) * q^74 + (-2861*b10 - 4481*b9 - 828*b8 + 2613*b7 + 954*b6 - 3071*b5 + 4*b4 + 7243*b3 + 4231*b2 - 16333*b1 + 1477657) * q^75 + (-3582*b10 - 2722*b9 - 374*b8 - 840*b7 - 688*b6 + 9224*b5 + 7392*b4 + 6134*b3 + 19520*b2 + 103542*b1 + 68188) * q^76 + (1142*b10 + 1828*b9 - 5421*b8 - 4765*b7 + 1037*b6 + 5191*b5 - 3725*b4 + 6832*b3 + 30860*b2 - 76479*b1 + 2038427) * q^77 + (6407*b10 + 5746*b9 - 4372*b8 - 777*b7 + 288*b6 - 7627*b5 - 6798*b4 - 7463*b3 - 23755*b2 - 250585*b1 - 2060251) * q^78 + (4113*b10 + 454*b9 - 1785*b8 + 1349*b7 - 4738*b6 - 7805*b5 - 5064*b4 + 266*b3 - 43446*b2 + 44655*b1 - 141215) * q^79 + (4693*b10 + 230*b9 - 552*b8 - 7223*b7 + 2644*b6 - 15249*b5 - 5308*b4 + 8423*b3 - 18633*b2 - 365221*b1 + 157481) * q^80 + (3408*b10 - 3247*b9 + 185*b8 - 7448*b7 - 7328*b6 + 2270*b5 + 712*b4 - 507*b3 + 41344*b2 + 16470*b1 + 859122) * q^81 + (-10941*b10 + 93*b9 + 11968*b8 + 1163*b7 - 1292*b6 + 10713*b5 + 795*b4 + 18789*b3 + 1162*b2 + 270376*b1 + 757531) * q^82 + (-6005*b10 - 8340*b9 + 2699*b8 - 5145*b7 + 3986*b6 - 2381*b5 + 4624*b4 + 6540*b3 + 3747*b2 - 34907*b1 + 2607670) * q^83 + (7363*b10 - 2015*b9 - 10017*b8 + 584*b7 + 1148*b6 - 19664*b5 - 5232*b4 - 4710*b3 - 47807*b2 - 504349*b1 - 3188659) * q^84 + (-7438*b10 + 4694*b9 - 1080*b8 - 1818*b7 + 7236*b6 - 2850*b5 + 1832*b4 + 8762*b3 + 25728*b2 - 61894*b1 + 1235628) * q^85 + (2532*b10 + 1266*b9 - 5066*b8 + 1170*b7 - 3584*b6 - 4838*b5 - 1516*b4 - 39292*b3 - 53302*b2 - 237566*b1 - 4221524) * q^86 + (-4126*b10 + 4564*b9 + 10190*b8 + 10284*b7 + 9630*b6 + 9667*b5 + 7490*b4 + 13666*b3 + 4647*b2 - 12038*b1 - 524529) * q^87 + (8198*b10 + 4695*b9 - 1400*b8 + 5146*b7 - 12648*b6 + 2983*b5 - 4360*b4 - 16184*b3 - 72218*b2 + 54354*b1 - 5358680) * q^88 + (2587*b10 - 7553*b9 - 8*b8 + 2351*b7 - 528*b6 - 3308*b5 - 1878*b4 - 17123*b3 + 1297*b2 - 12073*b1 - 809967) * q^89 + (9115*b10 + 10795*b9 - 6465*b8 - 3866*b7 + 1636*b6 - 13474*b5 - 13398*b4 - 2623*b3 - 83206*b2 - 369579*b1 - 4054242) * q^90 + (-3880*b10 - 1436*b9 - 1562*b8 + 4038*b7 + 398*b6 + 9644*b5 - 1918*b4 + 7400*b3 - 8970*b2 + 281710*b1 - 192360) * q^91 + (94*b10 - 6865*b9 - 5297*b8 + 2719*b7 - 6304*b6 + 7013*b5 + 16456*b4 - 20066*b3 - 8051*b2 - 215834*b1 - 3717519) * q^92 + (-12702*b10 + 21*b9 + 15579*b8 + 21098*b7 + 8384*b6 + 14804*b5 + 12820*b4 + 21077*b3 + 12428*b2 + 257144*b1 - 1015193) * q^93 + (596*b10 + 575*b9 + 10323*b8 + 387*b7 + 1260*b6 + 9090*b5 - 9937*b4 + 5260*b3 - 37348*b2 + 453033*b1 - 2581539) * q^94 + (-9396*b10 + 4420*b9 + 5504*b8 + 1544*b7 - 2996*b6 - 13498*b5 + 1460*b4 - 22232*b3 - 31738*b2 - 87244*b1 + 63190) * q^95 + (-9972*b10 - 6355*b9 + 6779*b8 - 8739*b7 + 4800*b6 + 34605*b5 + 26680*b4 - 5806*b3 + 105029*b2 + 222436*b1 - 1883591) * q^96 + (10529*b10 - 5367*b9 - 6214*b8 - 14865*b7 - 7178*b6 - 17120*b5 + 11840*b4 + 8111*b3 + 73887*b2 - 24069*b1 - 365415) * q^97 + (-5847*b10 - 2108*b9 + 6845*b8 - 2338*b7 - 6416*b6 + 7594*b5 - 5509*b4 - 14741*b3 - 13389*b2 + 276085*b1 - 783215) * q^98 + (17686*b10 + 10991*b9 - 17537*b8 - 22436*b7 - 14948*b6 + 1482*b5 - 18180*b4 - 8745*b3 - 8178*b2 + 35700*b1 - 1304541) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$11 q + 16 q^{2} + 121 q^{3} + 794 q^{4} + 376 q^{5} + 519 q^{6} + 2243 q^{7} + 3870 q^{8} + 9826 q^{9}+O(q^{10})$$ 11 * q + 16 * q^2 + 121 * q^3 + 794 * q^4 + 376 * q^5 + 519 * q^6 + 2243 * q^7 + 3870 * q^8 + 9826 * q^9 $$11 q + 16 q^{2} + 121 q^{3} + 794 q^{4} + 376 q^{5} + 519 q^{6} + 2243 q^{7} + 3870 q^{8} + 9826 q^{9} - 12629 q^{10} + 9415 q^{11} + 27955 q^{12} + 12512 q^{13} + 18260 q^{14} + 25714 q^{15} + 167866 q^{16} + 54312 q^{17} + 163911 q^{18} + 97192 q^{19} + 85625 q^{20} + 97795 q^{21} - 12345 q^{22} + 107342 q^{23} + 163119 q^{24} + 165051 q^{25} + 61531 q^{26} + 446611 q^{27} + 215454 q^{28} + 41748 q^{29} - 1080964 q^{30} - 272248 q^{31} + 593306 q^{32} - 216525 q^{33} - 923600 q^{34} + 436814 q^{35} - 456119 q^{36} - 557183 q^{37} - 175872 q^{38} - 1587326 q^{39} - 3206863 q^{40} + 525465 q^{41} - 3814396 q^{42} - 1376086 q^{43} - 1337377 q^{44} - 2315492 q^{45} - 2037327 q^{46} + 2269179 q^{47} + 1779791 q^{48} + 2282536 q^{49} - 3881347 q^{50} - 103604 q^{51} - 4200495 q^{52} - 346415 q^{53} + 6349248 q^{54} + 4169374 q^{55} - 4307934 q^{56} + 6170792 q^{57} - 1334849 q^{58} + 4598828 q^{59} - 4448200 q^{60} + 6208418 q^{61} + 4732115 q^{62} + 6882994 q^{63} + 12483426 q^{64} + 9330160 q^{65} - 5715150 q^{66} + 2199016 q^{67} + 8095824 q^{68} + 13516268 q^{69} - 6471708 q^{70} + 4653285 q^{71} + 12839097 q^{72} - 1080699 q^{73} - 810448 q^{74} + 16194855 q^{75} + 1331888 q^{76} + 22058153 q^{77} - 23968103 q^{78} - 1336084 q^{79} - 89443 q^{80} + 9585355 q^{81} + 9689125 q^{82} + 28551309 q^{83} - 37602282 q^{84} + 13256012 q^{85} - 47733694 q^{86} - 5826578 q^{87} - 58704117 q^{88} - 8994788 q^{89} - 46526086 q^{90} - 696642 q^{91} - 41894465 q^{92} - 9859184 q^{93} - 26180048 q^{94} + 124152 q^{95} - 19485621 q^{96} - 3968264 q^{97} - 7312590 q^{98} - 14172918 q^{99}+O(q^{100})$$ 11 * q + 16 * q^2 + 121 * q^3 + 794 * q^4 + 376 * q^5 + 519 * q^6 + 2243 * q^7 + 3870 * q^8 + 9826 * q^9 - 12629 * q^10 + 9415 * q^11 + 27955 * q^12 + 12512 * q^13 + 18260 * q^14 + 25714 * q^15 + 167866 * q^16 + 54312 * q^17 + 163911 * q^18 + 97192 * q^19 + 85625 * q^20 + 97795 * q^21 - 12345 * q^22 + 107342 * q^23 + 163119 * q^24 + 165051 * q^25 + 61531 * q^26 + 446611 * q^27 + 215454 * q^28 + 41748 * q^29 - 1080964 * q^30 - 272248 * q^31 + 593306 * q^32 - 216525 * q^33 - 923600 * q^34 + 436814 * q^35 - 456119 * q^36 - 557183 * q^37 - 175872 * q^38 - 1587326 * q^39 - 3206863 * q^40 + 525465 * q^41 - 3814396 * q^42 - 1376086 * q^43 - 1337377 * q^44 - 2315492 * q^45 - 2037327 * q^46 + 2269179 * q^47 + 1779791 * q^48 + 2282536 * q^49 - 3881347 * q^50 - 103604 * q^51 - 4200495 * q^52 - 346415 * q^53 + 6349248 * q^54 + 4169374 * q^55 - 4307934 * q^56 + 6170792 * q^57 - 1334849 * q^58 + 4598828 * q^59 - 4448200 * q^60 + 6208418 * q^61 + 4732115 * q^62 + 6882994 * q^63 + 12483426 * q^64 + 9330160 * q^65 - 5715150 * q^66 + 2199016 * q^67 + 8095824 * q^68 + 13516268 * q^69 - 6471708 * q^70 + 4653285 * q^71 + 12839097 * q^72 - 1080699 * q^73 - 810448 * q^74 + 16194855 * q^75 + 1331888 * q^76 + 22058153 * q^77 - 23968103 * q^78 - 1336084 * q^79 - 89443 * q^80 + 9585355 * q^81 + 9689125 * q^82 + 28551309 * q^83 - 37602282 * q^84 + 13256012 * q^85 - 47733694 * q^86 - 5826578 * q^87 - 58704117 * q^88 - 8994788 * q^89 - 46526086 * q^90 - 696642 * q^91 - 41894465 * q^92 - 9859184 * q^93 - 26180048 * q^94 + 124152 * q^95 - 19485621 * q^96 - 3968264 * q^97 - 7312590 * q^98 - 14172918 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{11} - 5 x^{10} - 1078 x^{9} + 4966 x^{8} + 379692 x^{7} - 1385588 x^{6} - 48765978 x^{5} + 87529978 x^{4} + 2159400643 x^{3} - 1763707223 x^{2} + \cdots + 6680404080$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 11\!\cdots\!61 \nu^{10} + \cdots - 13\!\cdots\!52 ) / 28\!\cdots\!48$$ (1193251898889487561*v^10 - 979319830519246352*v^9 - 1269263026164685271926*v^8 - 140411201316301302296*v^7 + 431075878661092539092820*v^6 + 777170088223649767887760*v^5 - 49563157170535999509005274*v^4 - 241969667191097153732668184*v^3 + 1542983972404003809468236899*v^2 + 8989389884685331331977369264*v - 13028334874733677645611554352) / 284489944984201786255208448 $$\beta_{3}$$ $$=$$ $$( - 11\!\cdots\!61 \nu^{10} + \cdots - 43\!\cdots\!52 ) / 28\!\cdots\!48$$ (-1193251898889487561*v^10 + 979319830519246352*v^9 + 1269263026164685271926*v^8 + 140411201316301302296*v^7 - 431075878661092539092820*v^6 - 777170088223649767887760*v^5 + 49563157170535999509005274*v^4 + 241969667191097153732668184*v^3 - 1258494027419802023213028451*v^2 - 8989389884685331331977369264*v - 43300674232138276032919718352) / 284489944984201786255208448 $$\beta_{4}$$ $$=$$ $$( - 23\!\cdots\!05 \nu^{10} + \cdots + 15\!\cdots\!88 ) / 28\!\cdots\!48$$ (-2378757807331832705*v^10 - 19078536412060677392*v^9 + 2615408266552203959654*v^8 + 18288805163588943482488*v^7 - 1032989775614370265286100*v^6 - 6172324928961353947469168*v^5 + 176402591084099943986891370*v^4 + 836445568813373873068645432*v^3 - 11605288507855592953099611179*v^2 - 28301527106644045465849847600*v + 156418088972719561846062848688) / 284489944984201786255208448 $$\beta_{5}$$ $$=$$ $$( 12\!\cdots\!57 \nu^{10} + \cdots - 59\!\cdots\!28 ) / 14\!\cdots\!24$$ (1296344937851036957*v^10 + 20029168286159990816*v^9 - 1802228146547479701086*v^8 - 19859740845189255095848*v^7 + 869605930636068583736724*v^6 + 5845291627616865098956256*v^5 - 162282471025654085164571058*v^4 - 451145922683955936916818760*v^3 + 8263644747927381968654192975*v^2 + 3707774094255093197141974640*v - 59978691421083689832060454128) / 142244972492100893127604224 $$\beta_{6}$$ $$=$$ $$( - 20\!\cdots\!65 \nu^{10} + \cdots + 17\!\cdots\!64 ) / 15\!\cdots\!36$$ (-200087722960213465*v^10 + 2276662861643336464*v^9 + 194517380831377392438*v^8 - 2206814150099815105000*v^7 - 54832791418693058308340*v^6 + 579366943780792361509744*v^5 + 3777946097081273109318426*v^4 - 23291058635390335574305000*v^3 - 66551183742887543160620467*v^2 - 498562981498597724047287984*v + 1729818755561091581836879664) / 15804996943566765903067136 $$\beta_{7}$$ $$=$$ $$( - 34\!\cdots\!99 \nu^{10} + \cdots - 28\!\cdots\!28 ) / 14\!\cdots\!24$$ (-3489565979032471999*v^10 + 5182423702528411376*v^9 + 3788644140835508511514*v^8 - 3903890688428478021208*v^7 - 1344154667437655813744556*v^6 - 3198884304638391257968*v^5 + 168178182663678857386021398*v^4 + 337244840342555497806566312*v^3 - 5067415942153572100824640021*v^2 - 21821890627941663081204215248*v - 28785593705921806838677361328) / 142244972492100893127604224 $$\beta_{8}$$ $$=$$ $$( - 16\!\cdots\!45 \nu^{10} + \cdots + 30\!\cdots\!60 ) / 31\!\cdots\!72$$ (-1690209891568367245*v^10 + 18470357161079567504*v^9 + 1755840354787973761678*v^8 - 18335873089401309642952*v^7 - 573778410489529112867748*v^6 + 5354090926443030909216240*v^5 + 63491898411314447735607810*v^4 - 414061430888909127816653640*v^3 - 2676526418627700169297161807*v^2 + 6581509861258658343245051536*v + 30311394644985248418579717360) / 31609993887133531806134272 $$\beta_{9}$$ $$=$$ $$( 25\!\cdots\!53 \nu^{10} + \cdots + 10\!\cdots\!88 ) / 35\!\cdots\!56$$ (2535573218605812853*v^10 - 16487801609311173332*v^9 - 2591151987645811332538*v^8 + 16302770487182668410268*v^7 + 821240211215688871406928*v^6 - 4490830208787365506797644*v^5 - 80794448675416221941361270*v^4 + 261814732872937859286006100*v^3 + 1649111221343516422458216475*v^2 - 4119777491194624330307853008*v + 10135743121113074052742167888) / 35561243123025223281901056 $$\beta_{10}$$ $$=$$ $$( - 68\!\cdots\!49 \nu^{10} + \cdots + 24\!\cdots\!08 ) / 71\!\cdots\!12$$ (-6876131443105370549*v^10 + 86131026502884933880*v^9 + 6806558278217977469222*v^8 - 84688781938404571319504*v^7 - 2021127720511697465113740*v^6 + 24010095602161571845103560*v^5 + 171940928463974319043995690*v^4 - 1661477073199019987441584640*v^3 - 4412882118128809925440579583*v^2 + 24013091811766942941900929296*v + 24652082294150393577913191408) / 71122486246050446563802112
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 198$$ b3 + b2 + 198 $$\nu^{3}$$ $$=$$ $$-2\beta_{10} + 3\beta_{8} + 3\beta_{7} + 4\beta_{5} - \beta_{3} + 375\beta _1 - 42$$ -2*b10 + 3*b8 + 3*b7 + 4*b5 - b3 + 375*b1 - 42 $$\nu^{4}$$ $$=$$ $$- 11 \beta_{10} - 3 \beta_{9} + 17 \beta_{8} - 24 \beta_{7} + 16 \beta_{6} + 6 \beta_{5} + 8 \beta_{4} + 493 \beta_{3} + 418 \beta_{2} - 341 \beta _1 + 74069$$ -11*b10 - 3*b9 + 17*b8 - 24*b7 + 16*b6 + 6*b5 + 8*b4 + 493*b3 + 418*b2 - 341*b1 + 74069 $$\nu^{5}$$ $$=$$ $$- 1078 \beta_{10} - 78 \beta_{9} + 1672 \beta_{8} + 1398 \beta_{7} - 492 \beta_{6} + 2204 \beta_{5} - 12 \beta_{4} - 714 \beta_{3} - 1042 \beta_{2} + 157471 \beta _1 - 91134$$ -1078*b10 - 78*b9 + 1672*b8 + 1398*b7 - 492*b6 + 2204*b5 - 12*b4 - 714*b3 - 1042*b2 + 157471*b1 - 91134 $$\nu^{6}$$ $$=$$ $$- 7358 \beta_{10} - 2782 \beta_{9} + 9956 \beta_{8} - 15934 \beta_{7} + 14532 \beta_{6} + 2100 \beta_{5} + 2364 \beta_{4} + 225215 \beta_{3} + 180643 \beta_{2} - 308154 \beta _1 + 31118420$$ -7358*b10 - 2782*b9 + 9956*b8 - 15934*b7 + 14532*b6 + 2100*b5 + 2364*b4 + 225215*b3 + 180643*b2 - 308154*b1 + 31118420 $$\nu^{7}$$ $$=$$ $$- 492764 \beta_{10} - 33482 \beta_{9} + 798175 \beta_{8} + 625297 \beta_{7} - 428524 \beta_{6} + 1070384 \beta_{5} - 14636 \beta_{4} - 466071 \beta_{3} - 1070842 \beta_{2} + \cdots - 72197176$$ -492764*b10 - 33482*b9 + 798175*b8 + 625297*b7 - 428524*b6 + 1070384*b5 - 14636*b4 - 466071*b3 - 1070842*b2 + 68467737*b1 - 72197176 $$\nu^{8}$$ $$=$$ $$- 3700033 \beta_{10} - 1519241 \beta_{9} + 4550125 \beta_{8} - 8591574 \beta_{7} + 9873860 \beta_{6} - 412598 \beta_{5} + 78260 \beta_{4} + 101903819 \beta_{3} + \cdots + 13538125867$$ -3700033*b10 - 1519241*b9 + 4550125*b8 - 8591574*b7 + 9873860*b6 - 412598*b5 + 78260*b4 + 101903819*b3 + 81054220*b2 - 203362295*b1 + 13538125867 $$\nu^{9}$$ $$=$$ $$- 213476112 \beta_{10} - 5774168 \beta_{9} + 365393820 \beta_{8} + 286873436 \beta_{7} - 279777448 \beta_{6} + 502567848 \beta_{5} - 13105640 \beta_{4} + \cdots - 45428759148$$ -213476112*b10 - 5774168*b9 + 365393820*b8 + 286873436*b7 - 279777448*b6 + 502567848*b5 - 13105640*b4 - 292497528*b3 - 741516812*b2 + 30296878529*b1 - 45428759148 $$\nu^{10}$$ $$=$$ $$- 1671109388 \beta_{10} - 693474316 \beta_{9} + 1862539160 \beta_{8} - 4372551324 \beta_{7} + 5957960584 \beta_{6} - 1034242168 \beta_{5} + \cdots + 5995121970338$$ -1671109388*b10 - 693474316*b9 + 1862539160*b8 - 4372551324*b7 + 5957960584*b6 - 1034242168*b5 - 443149704*b4 + 46185292301*b3 + 37209629605*b2 - 120286476708*b1 + 5995121970338

## 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
 −22.1395 −19.7203 −9.08402 −7.85832 −4.12221 0.259260 4.43681 7.16645 14.8861 20.2283 20.9474
−21.1395 34.2904 318.879 463.709 −724.881 1277.57 −4035.09 −1011.17 −9802.57
1.2 −18.7203 −14.9523 222.450 −4.96717 279.912 −1164.11 −1768.14 −1963.43 92.9870
1.3 −8.08402 72.5716 −62.6486 414.761 −586.670 49.3599 1541.21 3079.64 −3352.94
1.4 −6.85832 15.3600 −80.9635 −335.413 −105.343 735.210 1433.14 −1951.07 2300.37
1.5 −3.12221 −59.7866 −118.252 −342.676 186.666 −975.157 768.850 1387.44 1069.91
1.6 1.25926 −26.2543 −126.414 409.245 −33.0610 −873.009 −320.374 −1497.71 515.347
1.7 5.43681 85.9489 −98.4411 −143.945 467.288 1592.07 −1231.12 5200.21 −782.599
1.8 8.16645 −75.6574 −61.3091 −56.3977 −617.852 1198.36 −1545.98 3537.05 −460.569
1.9 15.8861 41.5699 124.370 303.592 660.386 157.297 −57.6738 −458.942 4822.91
1.10 21.2283 81.9649 322.641 −351.409 1739.98 −895.269 4131.89 4531.25 −7459.83
1.11 21.9474 −34.0550 353.689 19.5004 −747.419 1140.68 4953.28 −1027.26 427.983
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.11 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.b 11
3.b odd 2 1 333.8.a.d 11
4.b odd 2 1 592.8.a.g 11

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.8.a.b 11 1.a even 1 1 trivial
333.8.a.d 11 3.b odd 2 1
592.8.a.g 11 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{11} - 16 T_{2}^{10} - 973 T_{2}^{9} + 14278 T_{2}^{8} + 302086 T_{2}^{7} - 3815344 T_{2}^{6} - 32891120 T_{2}^{5} + 297768896 T_{2}^{4} + 1362254048 T_{2}^{3} - 7257649280 T_{2}^{2} + \cdots + 28348180480$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(37))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{11} - 16 T^{10} + \cdots + 28348180480$$
$3$ $$T^{11} - 121 T^{10} + \cdots + 67\!\cdots\!84$$
$5$ $$T^{11} - 376 T^{10} + \cdots - 75\!\cdots\!00$$
$7$ $$T^{11} - 2243 T^{10} + \cdots - 14\!\cdots\!64$$
$11$ $$T^{11} - 9415 T^{10} + \cdots + 11\!\cdots\!52$$
$13$ $$T^{11} - 12512 T^{10} + \cdots + 16\!\cdots\!72$$
$17$ $$T^{11} - 54312 T^{10} + \cdots + 27\!\cdots\!72$$
$19$ $$T^{11} - 97192 T^{10} + \cdots - 66\!\cdots\!60$$
$23$ $$T^{11} - 107342 T^{10} + \cdots + 10\!\cdots\!48$$
$29$ $$T^{11} - 41748 T^{10} + \cdots + 51\!\cdots\!60$$
$31$ $$T^{11} + 272248 T^{10} + \cdots - 68\!\cdots\!92$$
$37$ $$(T + 50653)^{11}$$
$41$ $$T^{11} - 525465 T^{10} + \cdots - 36\!\cdots\!24$$
$43$ $$T^{11} + 1376086 T^{10} + \cdots + 32\!\cdots\!04$$
$47$ $$T^{11} - 2269179 T^{10} + \cdots - 12\!\cdots\!12$$
$53$ $$T^{11} + 346415 T^{10} + \cdots + 14\!\cdots\!64$$
$59$ $$T^{11} - 4598828 T^{10} + \cdots - 44\!\cdots\!40$$
$61$ $$T^{11} - 6208418 T^{10} + \cdots - 16\!\cdots\!72$$
$67$ $$T^{11} - 2199016 T^{10} + \cdots + 19\!\cdots\!84$$
$71$ $$T^{11} - 4653285 T^{10} + \cdots + 32\!\cdots\!80$$
$73$ $$T^{11} + 1080699 T^{10} + \cdots + 83\!\cdots\!68$$
$79$ $$T^{11} + 1336084 T^{10} + \cdots + 35\!\cdots\!00$$
$83$ $$T^{11} - 28551309 T^{10} + \cdots - 64\!\cdots\!04$$
$89$ $$T^{11} + 8994788 T^{10} + \cdots + 13\!\cdots\!80$$
$97$ $$T^{11} + 3968264 T^{10} + \cdots + 31\!\cdots\!28$$