# Properties

 Label 37.6.a.a Level $37$ Weight $6$ Character orbit 37.a Self dual yes Analytic conductor $5.934$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,6,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, 6, names="a")

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

chi := DirichletCharacter("37.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$5.93420133308$$ Analytic rank: $$1$$ 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: $$x^{7} - x^{6} - 160x^{5} + 156x^{4} + 6495x^{3} - 2943x^{2} - 64880x + 53844$$ x^7 - x^6 - 160*x^5 + 156*x^4 + 6495*x^3 - 2943*x^2 - 64880*x + 53844 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 - 7) q^{3} + ( - \beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 + 15) q^{4} + (\beta_{6} + 3 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 9) q^{5} + ( - 3 \beta_{6} + \beta_{5} + \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 17 \beta_1 - 21) q^{6} + ( - 4 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} + 3 \beta_{3} + 2 \beta_1 - 45) q^{7} + (11 \beta_{6} + 4 \beta_{4} + 13 \beta_{3} + 9 \beta_{2} - 14 \beta_1 - 68) q^{8} + (10 \beta_{6} - 13 \beta_{5} + 11 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 17 \beta_1 + 48) q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^2 + (b2 + b1 - 7) * q^3 + (-b5 + b3 - 2*b2 + b1 + 15) * q^4 + (b6 + 3*b5 - 2*b3 - 2*b2 + 6*b1 - 9) * q^5 + (-3*b6 + b5 + b4 - 5*b3 - 2*b2 + 17*b1 - 21) * q^6 + (-4*b6 - 3*b5 - 5*b4 + 3*b3 + 2*b1 - 45) * q^7 + (11*b6 + 4*b4 + 13*b3 + 9*b2 - 14*b1 - 68) * q^8 + (10*b6 - 13*b5 + 11*b4 + 5*b3 - 5*b2 - 17*b1 + 48) * q^9 $$q + ( - \beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 - 7) q^{3} + ( - \beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 + 15) q^{4} + (\beta_{6} + 3 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 9) q^{5} + ( - 3 \beta_{6} + \beta_{5} + \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 17 \beta_1 - 21) q^{6} + ( - 4 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} + 3 \beta_{3} + 2 \beta_1 - 45) q^{7} + (11 \beta_{6} + 4 \beta_{4} + 13 \beta_{3} + 9 \beta_{2} - 14 \beta_1 - 68) q^{8} + (10 \beta_{6} - 13 \beta_{5} + 11 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 17 \beta_1 + 48) q^{9} + ( - 5 \beta_{6} + 5 \beta_{5} - 15 \beta_{4} - 15 \beta_{3} + 20 \beta_{2} + \cdots - 294) q^{10}+ \cdots + (2435 \beta_{6} + 2666 \beta_{5} - 4247 \beta_{4} - 2337 \beta_{3} + \cdots - 5281) q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^2 + (b2 + b1 - 7) * q^3 + (-b5 + b3 - 2*b2 + b1 + 15) * q^4 + (b6 + 3*b5 - 2*b3 - 2*b2 + 6*b1 - 9) * q^5 + (-3*b6 + b5 + b4 - 5*b3 - 2*b2 + 17*b1 - 21) * q^6 + (-4*b6 - 3*b5 - 5*b4 + 3*b3 + 2*b1 - 45) * q^7 + (11*b6 + 4*b4 + 13*b3 + 9*b2 - 14*b1 - 68) * q^8 + (10*b6 - 13*b5 + 11*b4 + 5*b3 - 5*b2 - 17*b1 + 48) * q^9 + (-5*b6 + 5*b5 - 15*b4 - 15*b3 + 20*b2 + 16*b1 - 294) * q^10 + (-23*b6 - 4*b5 + 13*b4 + 15*b3 - 6*b2 - 8*b1 - 210) * q^11 + (-15*b6 + 32*b5 - 33*b4 - 14*b3 + 24*b2 + 3*b1 - 571) * q^12 + (8*b6 - 7*b5 - b4 - 47*b3 - 19*b2 - 11*b1 - 58) * q^13 + (19*b6 + 8*b5 + 44*b4 - 9*b3 - 5*b2 + 13*b1 - 53) * q^14 + (7*b6 - 2*b5 - 11*b4 + 71*b3 - 37*b2 - 171*b1 - 29) * q^15 + (b6 - 36*b5 + 48*b4 - 21*b3 - 29*b2 + 68*b1 + 402) * q^16 + (40*b6 + 50*b5 - 50*b4 - 74*b3 - 68*b1 - 416) * q^17 + (76*b6 - 27*b5 - 15*b4 + 162*b3 + b2 - 232*b1 + 580) * q^18 + (-108*b6 - 30*b5 - 34*b4 + 82*b3 - 4*b2 + 4*b1 - 320) * q^19 + (-77*b6 - 25*b5 + 35*b4 - 147*b3 + 16*b2 + 352*b1 + 196) * q^20 + (-7*b6 + 12*b5 - 17*b4 - 21*b3 - 55*b2 - 11*b1 + 356) * q^21 + (-106*b6 - 35*b5 - 39*b4 + 132*b3 + 31*b2 + 215*b1 + 539) * q^22 + (132*b6 + 27*b5 + 17*b4 - 107*b3 + 167*b2 - 7*b1 - 538) * q^23 + (-64*b6 - 3*b5 + 55*b4 - 278*b3 - 81*b2 + 544*b1 + 1666) * q^24 + (196*b6 - 63*b5 + 243*b4 + 193*b3 + 19*b2 - 113*b1 + 1803) * q^25 + (128*b6 + 137*b5 - 179*b4 + 70*b3 + 145*b2 - 136*b1 + 76) * q^26 + (-119*b6 + 185*b5 - 136*b4 - 298*b3 + 49*b2 + 253*b1 - 1542) * q^27 + (3*b6 + 57*b5 - 142*b4 + 82*b3 + 125*b2 + 85*b1 + 833) * q^28 + (-226*b6 + 107*b5 + 175*b4 + 217*b3 - 41*b2 + 51*b1 - 1108) * q^29 + (193*b6 - 376*b5 + 324*b4 + 345*b3 - 363*b2 - 562*b1 + 7336) * q^30 + (59*b6 - 187*b5 - 2*b4 - 64*b3 + 76*b2 - 4*b1 + 841) * q^31 + (-273*b6 + 154*b5 - 456*b4 + 31*b3 + 137*b2 - 386*b1 - 2052) * q^32 + (88*b6 + 138*b5 - 130*b4 - 182*b3 - 230*b2 - 350*b1 + 535) * q^33 + (110*b6 + 64*b5 - 56*b4 - 506*b3 - 178*b2 + 704*b1 + 3516) * q^34 + (-216*b6 - 340*b5 - 96*b4 - 68*b3 + 148*b2 - 424*b1 - 2530) * q^35 + (176*b6 - 309*b5 + 517*b4 + 290*b3 - 711*b2 - 942*b1 + 8596) * q^36 - 1369 * q^37 + (-134*b6 - 40*b5 + 480*b4 + 74*b3 - 70*b2 + 176*b1 + 260) * q^38 + (-85*b6 + 47*b5 - 64*b4 + 430*b3 + 270*b2 - 490*b1 - 5097) * q^39 + (-211*b6 + 725*b5 - 329*b4 + 217*b3 + 466*b2 - 66*b1 - 6922) * q^40 + (319*b6 - 25*b5 - 186*b4 - 64*b3 - 118*b2 - 634*b1 - 3466) * q^41 + (145*b6 + 52*b5 - 68*b4 + 65*b3 + 201*b2 - 779*b1 - 825) * q^42 + (672*b6 - 166*b5 + 226*b4 + 254*b3 + 566*b2 + 430*b1 - 3274) * q^43 + (550*b6 + 162*b5 + 341*b4 - 707*b3 + 297*b2 - 245*b1 - 2889) * q^44 + (-1327*b6 + 252*b5 - 561*b4 - 1241*b3 - 121*b2 + 2535*b1 - 10147) * q^45 + (-176*b6 + 111*b5 - 285*b4 - 818*b3 - 593*b2 + 2188*b1 + 3540) * q^46 + (-370*b6 - 523*b5 + 165*b4 + 545*b3 + 246*b2 + 196*b1 - 8919) * q^47 + (262*b6 + 369*b5 - 525*b4 + 240*b3 + 1377*b2 - 1622*b1 - 10264) * q^48 + (403*b6 + 540*b5 - 187*b4 + 421*b3 + 69*b2 + 1149*b1 - 4459) * q^49 + (70*b6 - 1005*b5 - 345*b4 + 1760*b3 - 335*b2 - 2621*b1 + 3659) * q^50 + (-422*b6 - 472*b5 + 574*b4 + 838*b3 - 1044*b2 + 532*b1 + 3010) * q^51 + (-148*b6 - 345*b5 + 1385*b4 - 450*b3 - 1007*b2 + 1770*b1 + 11036) * q^52 + (1315*b6 + 158*b5 + 1179*b4 + 619*b3 - 103*b2 - 235*b1 - 1844) * q^53 + (-1004*b6 + 1032*b5 - 816*b4 - 2352*b3 + 568*b2 + 4361*b1 - 8783) * q^54 + (790*b6 - 757*b5 + 263*b4 + 251*b3 + 679*b2 + 97*b1 + 424) * q^55 + (-688*b6 - 392*b5 - 214*b4 - 1210*b3 - 678*b2 - 88*b1 - 496) * q^56 + (186*b6 + 324*b5 - 358*b4 - 674*b3 - 808*b2 - 140*b1 + 4114) * q^57 + (-2016*b6 - 763*b5 - 663*b4 + 670*b3 + 301*b2 + 2656*b1 - 380) * q^58 + (544*b6 + 1322*b5 - 1130*b4 - 514*b3 - 846*b2 + 502*b1 - 7840) * q^59 + (2221*b6 - 1298*b5 + 370*b4 + 3587*b3 + 1653*b2 - 9368*b1 + 11658) * q^60 + (-1153*b6 + 303*b5 - 1122*b4 + 936*b3 - 1342*b2 - 362*b1 + 6447) * q^61 + (951*b6 + 505*b5 + 265*b4 + 561*b3 - 60*b2 - 1878*b1 - 472) * q^62 + (586*b6 + 1216*b5 + 722*b4 - 386*b3 + 92*b2 - 656*b1 - 2528) * q^63 + (-481*b6 + 1094*b5 + 880*b4 - 2509*b3 - 1247*b2 + 2294*b1 + 10332) * q^64 + (-1931*b6 - 1700*b5 - 485*b4 - 1357*b3 - 737*b2 + 511*b1 + 3217) * q^65 + (872*b6 - 38*b5 - 366*b4 - 252*b3 + 98*b2 - 1971*b1 + 11445) * q^66 + (183*b6 + 643*b5 + 732*b4 - 2642*b3 + 1118*b2 + 186*b1 - 4871) * q^67 + (-402*b6 + 440*b5 - 284*b4 + 1270*b3 + 2846*b2 - 2140*b1 - 26760) * q^68 + (789*b6 - 2359*b5 + 2398*b4 + 876*b3 + 250*b2 + 854*b1 + 35527) * q^69 + (776*b6 + 772*b5 + 916*b4 + 984*b3 - 940*b2 + 1662*b1 + 23198) * q^70 + (-714*b6 + 1245*b5 + 173*b4 + 257*b3 - 412*b2 + 2634*b1 - 13613) * q^71 + (-118*b6 - 1023*b5 - 1379*b4 + 3022*b3 + 1515*b2 - 10796*b1 + 2578) * q^72 + (-454*b6 + 479*b5 - 2825*b4 + 217*b3 + 271*b2 - 2445*b1 + 9841) * q^73 + (1369*b1 + 1369) * q^74 + (-809*b6 + 1737*b5 - 1122*b4 - 4592*b3 + 3735*b2 + 7023*b1 - 23920) * q^75 + (1410*b6 + 648*b5 - 1620*b4 + 154*b3 + 1746*b2 + 380*b1 + 1320) * q^76 + (2167*b6 + 1010*b5 + 1603*b4 - 2717*b3 + 189*b2 - 2527*b1 + 1220) * q^77 + (-1129*b6 - 1725*b5 + 2195*b4 - 715*b3 - 3010*b2 + 7508*b1 + 33698) * q^78 + (-1602*b6 - 3301*b5 + 175*b4 + 3963*b3 + 2225*b2 - 3093*b1 + 10042) * q^79 + (-2087*b6 - 827*b5 + 527*b4 - 2147*b3 - 4114*b2 + 6266*b1 + 16226) * q^80 + (-215*b6 - 286*b5 - 151*b4 + 5669*b3 - 2483*b2 - 7259*b1 + 33474) * q^81 + (2499*b6 - 525*b5 + 1111*b4 + 237*b3 - 1334*b2 + 355*b1 + 29429) * q^82 + (-2886*b6 - 461*b5 - 737*b4 + 2319*b3 - 2520*b2 + 3566*b1 - 13181) * q^83 + (213*b6 - 1539*b5 + 1454*b4 + 112*b3 - 1065*b2 + 2741*b1 + 28917) * q^84 + (-14*b6 - 882*b5 - 360*b4 + 1252*b3 + 2788*b2 - 6236*b1 + 12418) * q^85 + (916*b6 - 898*b5 + 1230*b4 - 480*b3 - 1966*b2 + 4922*b1 - 7758) * q^86 + (1553*b6 + 821*b5 - 932*b4 - 1046*b3 - 3416*b2 - 5436*b1 + 16877) * q^87 + (2527*b6 + 1781*b5 - 3103*b4 - 4979*b3 - 1286*b2 + 674*b1 + 894) * q^88 + (2934*b6 + 286*b5 - 1788*b4 + 1860*b3 - 1942*b2 - 66*b1 + 25618) * q^89 + (-3961*b6 + 7642*b5 - 3550*b4 - 7357*b3 + 7989*b2 + 17312*b1 - 107770) * q^90 + (2620*b6 + 2546*b5 + 4302*b4 + 2586*b3 + 1142*b2 - 722*b1 - 1310) * q^91 + (-2564*b6 + 4017*b5 - 3097*b4 + 810*b3 + 2535*b2 - 6762*b1 - 98660) * q^92 + (161*b6 - 364*b5 + 367*b4 - 793*b3 + 4203*b2 + 2971*b1 - 6003) * q^93 + (-263*b6 - 188*b5 + 2112*b4 + 2805*b3 - 459*b2 + 7557*b1 + 3711) * q^94 + (1750*b6 - 2934*b5 - 76*b4 - 416*b3 + 2580*b2 - 2052*b1 - 31486) * q^95 + (-780*b6 - 2721*b5 + 3251*b4 + 780*b3 - 8321*b2 + 6844*b1 + 57626) * q^96 + (-5714*b6 - 3056*b5 + 2946*b4 - 2014*b3 + 2238*b2 - 170*b1 + 5890) * q^97 + (-547*b6 - 1410*b5 + 2198*b4 - 4639*b3 - 137*b2 + 6994*b1 - 44604) * q^98 + (2435*b6 + 2666*b5 - 4247*b4 - 2337*b3 - 263*b2 + 6079*b1 - 5281) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 8 q^{2} - 47 q^{3} + 106 q^{4} - 64 q^{5} - 141 q^{6} - 293 q^{7} - 474 q^{8} + 292 q^{9}+O(q^{10})$$ 7 * q - 8 * q^2 - 47 * q^3 + 106 * q^4 - 64 * q^5 - 141 * q^6 - 293 * q^7 - 474 * q^8 + 292 * q^9 $$7 q - 8 q^{2} - 47 q^{3} + 106 q^{4} - 64 q^{5} - 141 q^{6} - 293 q^{7} - 474 q^{8} + 292 q^{9} - 2017 q^{10} - 1457 q^{11} - 3917 q^{12} - 536 q^{13} - 488 q^{14} - 254 q^{15} + 2714 q^{16} - 3068 q^{17} + 4107 q^{18} - 1900 q^{19} + 1453 q^{20} + 2425 q^{21} + 4467 q^{22} - 3986 q^{23} + 11523 q^{24} + 12231 q^{25} + 911 q^{26} - 10697 q^{27} + 6486 q^{28} - 7436 q^{29} + 50276 q^{30} + 5776 q^{31} - 13366 q^{32} + 2973 q^{33} + 24128 q^{34} - 17714 q^{35} + 57889 q^{36} - 9583 q^{37} + 1248 q^{38} - 34826 q^{39} - 46751 q^{40} - 25089 q^{41} - 6232 q^{42} - 22538 q^{43} - 22817 q^{44} - 68648 q^{45} + 25485 q^{46} - 60861 q^{47} - 70825 q^{48} - 29182 q^{49} + 26797 q^{50} + 21508 q^{51} + 74493 q^{52} - 15681 q^{53} - 58620 q^{54} + 2930 q^{55} - 5542 q^{56} + 27032 q^{57} + 4979 q^{58} - 54536 q^{59} + 78104 q^{60} + 48694 q^{61} - 5601 q^{62} - 21062 q^{63} + 67074 q^{64} + 22480 q^{65} + 77598 q^{66} - 39724 q^{67} - 183104 q^{68} + 245960 q^{69} + 162468 q^{70} - 92187 q^{71} + 17685 q^{72} + 73251 q^{73} + 10952 q^{74} - 162813 q^{75} + 13504 q^{76} - 4605 q^{77} + 235693 q^{78} + 78604 q^{79} + 112473 q^{80} + 236431 q^{81} + 200777 q^{82} - 82223 q^{83} + 201198 q^{84} + 86716 q^{85} - 55686 q^{86} + 107506 q^{87} - 633 q^{88} + 181680 q^{89} - 732742 q^{90} - 14802 q^{91} - 684469 q^{92} - 37328 q^{93} + 34724 q^{94} - 222304 q^{95} + 397743 q^{96} + 39092 q^{97} - 318498 q^{98} - 29766 q^{99}+O(q^{100})$$ 7 * q - 8 * q^2 - 47 * q^3 + 106 * q^4 - 64 * q^5 - 141 * q^6 - 293 * q^7 - 474 * q^8 + 292 * q^9 - 2017 * q^10 - 1457 * q^11 - 3917 * q^12 - 536 * q^13 - 488 * q^14 - 254 * q^15 + 2714 * q^16 - 3068 * q^17 + 4107 * q^18 - 1900 * q^19 + 1453 * q^20 + 2425 * q^21 + 4467 * q^22 - 3986 * q^23 + 11523 * q^24 + 12231 * q^25 + 911 * q^26 - 10697 * q^27 + 6486 * q^28 - 7436 * q^29 + 50276 * q^30 + 5776 * q^31 - 13366 * q^32 + 2973 * q^33 + 24128 * q^34 - 17714 * q^35 + 57889 * q^36 - 9583 * q^37 + 1248 * q^38 - 34826 * q^39 - 46751 * q^40 - 25089 * q^41 - 6232 * q^42 - 22538 * q^43 - 22817 * q^44 - 68648 * q^45 + 25485 * q^46 - 60861 * q^47 - 70825 * q^48 - 29182 * q^49 + 26797 * q^50 + 21508 * q^51 + 74493 * q^52 - 15681 * q^53 - 58620 * q^54 + 2930 * q^55 - 5542 * q^56 + 27032 * q^57 + 4979 * q^58 - 54536 * q^59 + 78104 * q^60 + 48694 * q^61 - 5601 * q^62 - 21062 * q^63 + 67074 * q^64 + 22480 * q^65 + 77598 * q^66 - 39724 * q^67 - 183104 * q^68 + 245960 * q^69 + 162468 * q^70 - 92187 * q^71 + 17685 * q^72 + 73251 * q^73 + 10952 * q^74 - 162813 * q^75 + 13504 * q^76 - 4605 * q^77 + 235693 * q^78 + 78604 * q^79 + 112473 * q^80 + 236431 * q^81 + 200777 * q^82 - 82223 * q^83 + 201198 * q^84 + 86716 * q^85 - 55686 * q^86 + 107506 * q^87 - 633 * q^88 + 181680 * q^89 - 732742 * q^90 - 14802 * q^91 - 684469 * q^92 - 37328 * q^93 + 34724 * q^94 - 222304 * q^95 + 397743 * q^96 + 39092 * q^97 - 318498 * q^98 - 29766 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 160x^{5} + 156x^{4} + 6495x^{3} - 2943x^{2} - 64880x + 53844$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -589\nu^{6} - 7152\nu^{5} + 107452\nu^{4} + 892264\nu^{3} - 7034699\nu^{2} - 21137864\nu + 118170684 ) / 3673920$$ (-589*v^6 - 7152*v^5 + 107452*v^4 + 892264*v^3 - 7034699*v^2 - 21137864*v + 118170684) / 3673920 $$\beta_{3}$$ $$=$$ $$( 201\nu^{6} - 964\nu^{5} - 28196\nu^{4} + 117064\nu^{3} + 739263\nu^{2} - 1398164\nu - 1062180 ) / 244928$$ (201*v^6 - 964*v^5 - 28196*v^4 + 117064*v^3 + 739263*v^2 - 1398164*v - 1062180) / 244928 $$\beta_{4}$$ $$=$$ $$( 1051\nu^{6} - 2832\nu^{5} - 133648\nu^{4} + 417524\nu^{3} + 2519081\nu^{2} - 7485964\nu + 10836084 ) / 918480$$ (1051*v^6 - 2832*v^5 - 133648*v^4 + 417524*v^3 + 2519081*v^2 - 7485964*v + 10836084) / 918480 $$\beta_{5}$$ $$=$$ $$( 4193\nu^{6} - 156\nu^{5} - 637844\nu^{4} - 28568\nu^{3} + 21484423\nu^{2} + 17629348\nu - 83273748 ) / 3673920$$ (4193*v^6 - 156*v^5 - 637844*v^4 - 28568*v^3 + 21484423*v^2 + 17629348*v - 83273748) / 3673920 $$\beta_{6}$$ $$=$$ $$( -461\nu^{6} + 2706\nu^{5} + 60632\nu^{4} - 374656\nu^{3} - 1201555\nu^{2} + 7801838\nu - 4986408 ) / 367392$$ (-461*v^6 + 2706*v^5 + 60632*v^4 - 374656*v^3 - 1201555*v^2 + 7801838*v - 4986408) / 367392
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{3} - 2\beta_{2} - \beta _1 + 46$$ -b5 + b3 - 2*b2 - b1 + 46 $$\nu^{3}$$ $$=$$ $$-11\beta_{6} + 3\beta_{5} - 4\beta_{4} - 16\beta_{3} - 3\beta_{2} + 78\beta _1 - 7$$ -11*b6 + 3*b5 - 4*b4 - 16*b3 - 3*b2 + 78*b1 - 7 $$\nu^{4}$$ $$=$$ $$45\beta_{6} - 138\beta_{5} + 64\beta_{4} + 133\beta_{3} - 197\beta_{2} - 146\beta _1 + 3641$$ 45*b6 - 138*b5 + 64*b4 + 133*b3 - 197*b2 - 146*b1 + 3641 $$\nu^{5}$$ $$=$$ $$-1250\beta_{6} + 516\beta_{5} - 336\beta_{4} - 2210\beta_{3} - 254\beta_{2} + 7253\beta _1 - 2720$$ -1250*b6 + 516*b5 - 336*b4 - 2210*b3 - 254*b2 + 7253*b1 - 2720 $$\nu^{6}$$ $$=$$ $$6724\beta_{6} - 14953\beta_{5} + 9696\beta_{4} + 14917\beta_{3} - 19750\beta_{2} - 20489\beta _1 + 337886$$ 6724*b6 - 14953*b5 + 9696*b4 + 14917*b3 - 19750*b2 - 20489*b1 + 337886

## 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
 9.76658 7.47374 3.30071 0.860211 −4.74171 −5.27747 −10.3821
−10.7666 −23.6672 83.9191 70.6580 254.814 −64.6233 −558.991 317.134 −760.745
1.2 −8.47374 1.99035 39.8042 −10.3767 −16.8657 175.264 −66.1308 −239.039 87.9294
1.3 −4.30071 −0.151013 −13.5039 95.6487 0.649463 −203.420 195.699 −242.977 −411.358
1.4 −1.86021 19.8285 −28.5396 −83.4407 −36.8852 −62.6698 112.616 150.169 155.217
1.5 3.74171 −3.60982 −17.9996 −45.0957 −13.5069 −32.6141 −187.084 −229.969 −168.735
1.6 4.27747 −11.5828 −13.7033 12.1163 −49.5451 −81.5934 −195.494 −108.839 51.8273
1.7 9.38206 −29.8081 56.0230 −103.510 −279.661 −23.3438 225.385 645.520 −971.136
 $$n$$: 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
$$37$$ $$1$$

## 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 37.6.a.a 7
3.b odd 2 1 333.6.a.c 7
4.b odd 2 1 592.6.a.g 7
5.b even 2 1 925.6.a.a 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.6.a.a 7 1.a even 1 1 trivial
333.6.a.c 7 3.b odd 2 1
592.6.a.g 7 4.b odd 2 1
925.6.a.a 7 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{7} + 8T_{2}^{6} - 133T_{2}^{5} - 906T_{2}^{4} + 4326T_{2}^{3} + 19928T_{2}^{2} - 40920T_{2} - 109600$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(37))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} + 8 T^{6} - 133 T^{5} + \cdots - 109600$$
$3$ $$T^{7} + 47 T^{6} + 108 T^{5} + \cdots + 175797$$
$5$ $$T^{7} + 64 T^{6} + \cdots - 330953386760$$
$7$ $$T^{7} + 293 T^{6} + \cdots - 8969445254224$$
$11$ $$T^{7} + 1457 T^{6} + \cdots - 74\!\cdots\!65$$
$13$ $$T^{7} + 536 T^{6} + \cdots - 60\!\cdots\!84$$
$17$ $$T^{7} + 3068 T^{6} + \cdots - 26\!\cdots\!68$$
$19$ $$T^{7} + 1900 T^{6} + \cdots + 15\!\cdots\!88$$
$23$ $$T^{7} + 3986 T^{6} + \cdots - 21\!\cdots\!12$$
$29$ $$T^{7} + 7436 T^{6} + \cdots + 54\!\cdots\!08$$
$31$ $$T^{7} - 5776 T^{6} + \cdots - 15\!\cdots\!52$$
$37$ $$(T + 1369)^{7}$$
$41$ $$T^{7} + 25089 T^{6} + \cdots + 31\!\cdots\!45$$
$43$ $$T^{7} + 22538 T^{6} + \cdots - 26\!\cdots\!96$$
$47$ $$T^{7} + 60861 T^{6} + \cdots - 26\!\cdots\!76$$
$53$ $$T^{7} + 15681 T^{6} + \cdots - 80\!\cdots\!12$$
$59$ $$T^{7} + 54536 T^{6} + \cdots - 27\!\cdots\!04$$
$61$ $$T^{7} - 48694 T^{6} + \cdots + 41\!\cdots\!40$$
$67$ $$T^{7} + 39724 T^{6} + \cdots + 34\!\cdots\!52$$
$71$ $$T^{7} + 92187 T^{6} + \cdots - 58\!\cdots\!00$$
$73$ $$T^{7} - 73251 T^{6} + \cdots + 25\!\cdots\!45$$
$79$ $$T^{7} - 78604 T^{6} + \cdots + 54\!\cdots\!80$$
$83$ $$T^{7} + 82223 T^{6} + \cdots - 19\!\cdots\!68$$
$89$ $$T^{7} - 181680 T^{6} + \cdots - 14\!\cdots\!48$$
$97$ $$T^{7} - 39092 T^{6} + \cdots - 49\!\cdots\!20$$
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