# Properties

 Label 115.6.a.c Level $115$ Weight $6$ Character orbit 115.a Self dual yes Analytic conductor $18.444$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,6,Mod(1,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("115.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 115.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: $$18.4441392785$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - 3x^{6} - 196x^{5} + 464x^{4} + 11003x^{3} - 21041x^{2} - 142416x + 243340$$ x^7 - 3*x^6 - 196*x^5 + 464*x^4 + 11003*x^3 - 21041*x^2 - 142416*x + 243340 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} - \beta_1 + 26) q^{4} - 25 q^{5} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \cdots - 54) q^{6}+ \cdots + ( - 4 \beta_{5} + 2 \beta_{4} + \cdots + 70) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (-b2 + b1 - 1) * q^3 + (b3 - b1 + 26) * q^4 - 25 * q^5 + (3*b5 - 3*b4 - 3*b3 - 6*b2 - 3*b1 - 54) * q^6 + (-b6 + b5 + b4 - b2 - 3*b1 + 6) * q^7 + (-2*b5 - 4*b4 + b3 + 2*b2 - 23*b1 + 88) * q^8 + (-4*b5 + 2*b4 + 7*b3 + 5*b2 - 14*b1 + 70) * q^9 $$q + ( - \beta_1 + 1) q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} - \beta_1 + 26) q^{4} - 25 q^{5} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \cdots - 54) q^{6}+ \cdots + ( - 519 \beta_{6} - 552 \beta_{5} + \cdots + 32707) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (-b2 + b1 - 1) * q^3 + (b3 - b1 + 26) * q^4 - 25 * q^5 + (3*b5 - 3*b4 - 3*b3 - 6*b2 - 3*b1 - 54) * q^6 + (-b6 + b5 + b4 - b2 - 3*b1 + 6) * q^7 + (-2*b5 - 4*b4 + b3 + 2*b2 - 23*b1 + 88) * q^8 + (-4*b5 + 2*b4 + 7*b3 + 5*b2 - 14*b1 + 70) * q^9 + (25*b1 - 25) * q^10 + (2*b6 - b5 + 9*b4 + 3*b3 - 6*b2 - b1 + 196) * q^11 + (-3*b6 + 12*b5 + 3*b4 - 6*b3 - 58*b2 + 58*b1 - 73) * q^12 + (9*b6 - 5*b5 + 6*b4 + 6*b3 + 9*b2 - 19*b1 + 96) * q^13 + (4*b6 + 7*b5 - 9*b4 - 9*b3 - 14*b2 - 26*b1 + 197) * q^14 + (25*b2 - 25*b1 + 25) * q^15 + (-10*b6 - 24*b5 + 8*b4 + 23*b3 + 10*b2 - 47*b1 + 560) * q^16 + (-6*b6 - 20*b5 + 21*b4 + 16*b3 + 7*b2 - 39*b1 + 375) * q^17 + (-21*b5 - 21*b4 + 30*b3 + 96*b2 - 213*b1 + 1245) * q^18 + (-8*b6 - 11*b5 + 3*b4 - 36*b3 - 16*b2 - 78*b1 + 8) * q^19 + (-25*b3 + 25*b1 - 650) * q^20 + (-9*b6 + 11*b5 - 4*b4 + 22*b3 - 6*b2 - 24*b1 + 101) * q^21 + (15*b6 + 48*b5 - 39*b4 - 50*b3 + 42*b2 - 308*b1 + 563) * q^22 + 529 * q^23 + (21*b6 + 102*b5 - 63*b4 - 144*b3 - 282*b2 - 36*b1 - 1809) * q^24 + 625 * q^25 + (-2*b6 - 15*b5 + 31*b4 + 36*b3 + 188*b2 - 164*b1 + 1440) * q^26 + (-3*b6 + 114*b5 - 9*b4 - 111*b3 - 79*b2 + 274*b1 - 1882) * q^27 + (17*b6 - 8*b5 + 7*b4 + 19*b3 - 178*b2 + 57*b1 + 777) * q^28 + (-40*b6 - 52*b5 + 19*b4 + 82*b2 + 282*b1 + 232) * q^29 + (-75*b5 + 75*b4 + 75*b3 + 150*b2 + 75*b1 + 1350) * q^30 + (12*b6 - 45*b5 + 96*b4 - 4*b3 - 2*b2 + 132*b1 + 1874) * q^31 + (2*b6 + 20*b5 - 24*b4 + 81*b3 + 290*b2 - 233*b1 + 2168) * q^32 + (-30*b6 - 29*b5 - 11*b4 + 80*b3 - 194*b2 + 584*b1 + 1232) * q^33 + (28*b6 + 31*b5 - 135*b4 + 22*b3 + 376*b2 - 651*b1 + 4197) * q^34 + (25*b6 - 25*b5 - 25*b4 + 25*b2 + 75*b1 - 150) * q^35 + (-63*b6 - 304*b5 + 125*b4 + 370*b3 + 560*b2 - 1010*b1 + 11935) * q^36 + (48*b6 - 67*b5 - 62*b4 + 37*b3 + 147*b2 - 204*b1 + 1449) * q^37 + (3*b6 + 132*b5 + 39*b4 + 67*b3 - 72*b2 + 1063*b1 + 3665) * q^38 + (-6*b6 + 21*b5 - 69*b4 - 156*b3 - 303*b2 + 861*b1 - 3537) * q^39 + (50*b5 + 100*b4 - 25*b3 - 50*b2 + 575*b1 - 2200) * q^40 + (141*b6 + 66*b5 + 74*b4 - 30*b3 + 12*b2 + 524*b1 + 1743) * q^41 + (12*b6 - 42*b5 - 132*b4 - 21*b3 - 162*b2 - 903*b1 + 2130) * q^42 + (-85*b6 - 79*b5 - 128*b4 + 81*b3 + 212*b2 + 93*b1 + 10947) * q^43 + (-109*b6 - 150*b5 + 239*b4 + 314*b3 - 310*b2 + 682*b1 + 7573) * q^44 + (100*b5 - 50*b4 - 175*b3 - 125*b2 + 350*b1 - 1750) * q^45 + (-529*b1 + 529) * q^46 + (-91*b6 + 108*b5 + 43*b4 - 96*b3 - 335*b2 + 941*b1 + 8078) * q^47 + (51*b6 + 498*b5 - 33*b4 - 408*b3 - 1438*b2 + 2068*b1 - 5791) * q^48 + (74*b6 - 132*b5 - 39*b4 - 106*b3 - 251*b2 + 289*b1 - 7852) * q^49 + (-625*b1 + 625) * q^50 + (9*b6 + 304*b5 - 71*b4 - 193*b3 - 864*b2 + 2061*b1 - 4523) * q^51 + (-239*b6 - 352*b5 + 141*b4 + 251*b3 + 1240*b2 - 1027*b1 + 9345) * q^52 + (81*b6 - 144*b5 + 86*b4 + 262*b3 + 291*b2 + 1141*b1 + 6646) * q^53 + (99*b6 + 423*b5 + 324*b4 - 846*b3 - 1902*b2 + 3540*b1 - 24159) * q^54 + (-50*b6 + 25*b5 - 225*b4 - 75*b3 + 150*b2 + 25*b1 - 4900) * q^55 + (-139*b6 + 300*b5 - 259*b4 - 111*b3 - 392*b2 - 1083*b1 - 7345) * q^56 + (165*b6 - 5*b5 - 470*b4 - 124*b3 + 536*b2 - 1628*b1 - 327) * q^57 + (26*b6 - 170*b5 - 44*b4 - 45*b3 + 966*b2 + 510*b1 - 13763) * q^58 + (-91*b6 - 220*b5 + 96*b4 + 106*b3 + 113*b2 - 261*b1 + 6474) * q^59 + (75*b6 - 300*b5 - 75*b4 + 150*b3 + 1450*b2 - 1450*b1 + 1825) * q^60 + (136*b6 - 157*b5 - 41*b4 + 45*b3 - 1420*b2 - 221*b1 - 7138) * q^61 + (135*b6 + 398*b5 - 167*b4 - 424*b3 + 1054*b2 - 1427*b1 - 2922) * q^62 + (123*b6 + 94*b5 - 116*b4 + 23*b3 - 567*b2 + 888*b1 - 956) * q^63 + (290*b6 - 360*b5 + 368*b4 + 119*b3 + 1246*b2 - 2019*b1 - 1656) * q^64 + (-225*b6 + 125*b5 - 150*b4 - 150*b3 - 225*b2 + 475*b1 - 2400) * q^65 + (-21*b6 + 378*b5 - 1059*b4 - 831*b3 - 900*b2 - 4077*b1 - 27285) * q^66 + (10*b6 - 363*b5 + 896*b4 + 419*b3 - 955*b2 + 628*b1 - 285) * q^67 + (-75*b6 - 1072*b5 + 809*b4 + 1470*b3 + 1068*b2 - 2070*b1 + 24503) * q^68 + (-529*b2 + 529*b1 - 529) * q^69 + (-100*b6 - 175*b5 + 225*b4 + 225*b3 + 350*b2 + 650*b1 - 4925) * q^70 + (-160*b6 + 761*b5 - 78*b4 - 401*b3 - 1038*b2 - 2745*b1 + 8154) * q^71 + (9*b6 - 1248*b5 + 3*b4 + 1698*b3 + 4566*b2 - 10704*b1 + 52233) * q^72 + (-480*b6 + 215*b5 + 525*b4 + 99*b3 + 1235*b2 + 3554*b1 - 1839) * q^73 + (-239*b6 - 763*b5 + 590*b4 + 1124*b3 + 1446*b2 - 929*b1 + 13252) * q^74 + (-625*b2 + 625*b1 - 625) * q^75 + (463*b6 + 590*b5 - 511*b4 - 775*b3 - 860*b2 - 4921*b1 - 56821) * q^76 + (94*b6 + 472*b5 + 350*b4 + 137*b3 - 88*b2 + 29*b1 + 11166) * q^77 + (-111*b6 + 945*b5 - 156*b4 - 1212*b3 - 2778*b2 + 6738*b1 - 59013) * q^78 + (-267*b6 + 1233*b5 - 358*b4 - 523*b3 - 810*b2 - 565*b1 + 359) * q^79 + (250*b6 + 600*b5 - 200*b4 - 575*b3 - 250*b2 + 1175*b1 - 14000) * q^80 + (-324*b6 - 1204*b5 - 46*b4 + 1063*b3 + 5426*b2 - 5723*b1 + 20527) * q^81 + (73*b6 + 320*b5 + 779*b4 - 993*b3 + 360*b2 - 1276*b1 - 31992) * q^82 + (335*b6 - 454*b5 + 530*b4 + 402*b3 + 1219*b2 + 2185*b1 + 13402) * q^83 + (-30*b6 - 352*b5 + 8*b4 + 715*b3 - 1146*b2 - 639*b1 + 48248) * q^84 + (150*b6 + 500*b5 - 525*b4 - 400*b3 - 175*b2 + 975*b1 - 9375) * q^85 + (-250*b6 - 1310*b5 + 64*b4 + 1202*b3 + 1116*b2 - 11578*b1 + 8618) * q^86 + (255*b6 - 284*b5 + 913*b4 + 284*b3 + 1209*b2 + 4053*b1 - 7088) * q^87 + (-43*b6 - 278*b5 - 2111*b4 - 406*b3 - 78*b2 - 7218*b1 - 25769) * q^88 + (413*b6 + 915*b5 - 2466*b4 - 463*b3 - 2176*b2 - 3063*b1 - 19173) * q^89 + (525*b5 + 525*b4 - 750*b3 - 2400*b2 + 5325*b1 - 31125) * q^90 + (252*b6 + 1081*b5 + 409*b4 + 401*b3 + 748*b2 - 1253*b1 - 31088) * q^91 + (529*b3 - 529*b1 + 13754) * q^92 + (-6*b6 - 803*b5 + 91*b4 + 776*b3 - 277*b2 + 6991*b1 + 4305) * q^93 + (285*b6 + 1369*b5 - 1054*b4 - 2349*b3 - 3388*b2 - 8181*b1 - 47731) * q^94 + (200*b6 + 275*b5 - 75*b4 + 900*b3 + 400*b2 + 1950*b1 - 200) * q^95 + (-291*b6 + 1734*b5 + 153*b4 - 2112*b3 - 5394*b2 + 7764*b1 - 88737) * q^96 + (-450*b6 - 1475*b5 + 1493*b4 + 1443*b3 + 1956*b2 + 3657*b1 + 2812) * q^97 + (-284*b6 + 809*b5 - 13*b4 + 6*b3 - 336*b2 + 11674*b1 - 26300) * q^98 + (-519*b6 - 552*b5 - 105*b4 + 1251*b3 + 526*b2 + 3203*b1 + 32707) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9}+O(q^{10})$$ 7 * q + 4 * q^2 - 3 * q^3 + 178 * q^4 - 175 * q^5 - 381 * q^6 + 33 * q^7 + 546 * q^8 + 440 * q^9 $$7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9} - 100 q^{10} + 1373 q^{11} - 285 q^{12} + 605 q^{13} + 1317 q^{14} + 75 q^{15} + 3770 q^{16} + 2505 q^{17} + 7971 q^{18} - 115 q^{19} - 4450 q^{20} + 608 q^{21} + 2977 q^{22} + 3703 q^{23} - 12447 q^{24} + 4375 q^{25} + 9379 q^{26} - 12276 q^{27} + 5777 q^{28} + 2440 q^{29} + 9525 q^{30} + 13565 q^{31} + 14086 q^{32} + 10519 q^{33} + 26997 q^{34} - 825 q^{35} + 79889 q^{36} + 9414 q^{37} + 28717 q^{38} - 21738 q^{39} - 13650 q^{40} + 13725 q^{41} + 12426 q^{42} + 76694 q^{43} + 55203 q^{44} - 11000 q^{45} + 2116 q^{46} + 59692 q^{47} - 32985 q^{48} - 53608 q^{49} + 2500 q^{50} - 24725 q^{51} + 61195 q^{52} + 49536 q^{53} - 156168 q^{54} - 34325 q^{55} - 54461 q^{56} - 7580 q^{57} - 95562 q^{58} + 44536 q^{59} + 7125 q^{60} - 49097 q^{61} - 25763 q^{62} - 3578 q^{63} - 18654 q^{64} - 15125 q^{65} - 201873 q^{66} + 788 q^{67} + 163845 q^{68} - 1587 q^{69} - 32925 q^{70} + 49521 q^{71} + 328503 q^{72} - 3760 q^{73} + 88170 q^{74} - 1875 q^{75} - 411465 q^{76} + 77728 q^{77} - 389832 q^{78} + 918 q^{79} - 94250 q^{80} + 121235 q^{81} - 227459 q^{82} + 99202 q^{83} + 336602 q^{84} - 62625 q^{85} + 24584 q^{86} - 38666 q^{87} - 201275 q^{88} - 141676 q^{89} - 199275 q^{90} - 223605 q^{91} + 94162 q^{92} + 51412 q^{93} - 354292 q^{94} + 2875 q^{95} - 592095 q^{96} + 28731 q^{97} - 149557 q^{98} + 237333 q^{99}+O(q^{100})$$ 7 * q + 4 * q^2 - 3 * q^3 + 178 * q^4 - 175 * q^5 - 381 * q^6 + 33 * q^7 + 546 * q^8 + 440 * q^9 - 100 * q^10 + 1373 * q^11 - 285 * q^12 + 605 * q^13 + 1317 * q^14 + 75 * q^15 + 3770 * q^16 + 2505 * q^17 + 7971 * q^18 - 115 * q^19 - 4450 * q^20 + 608 * q^21 + 2977 * q^22 + 3703 * q^23 - 12447 * q^24 + 4375 * q^25 + 9379 * q^26 - 12276 * q^27 + 5777 * q^28 + 2440 * q^29 + 9525 * q^30 + 13565 * q^31 + 14086 * q^32 + 10519 * q^33 + 26997 * q^34 - 825 * q^35 + 79889 * q^36 + 9414 * q^37 + 28717 * q^38 - 21738 * q^39 - 13650 * q^40 + 13725 * q^41 + 12426 * q^42 + 76694 * q^43 + 55203 * q^44 - 11000 * q^45 + 2116 * q^46 + 59692 * q^47 - 32985 * q^48 - 53608 * q^49 + 2500 * q^50 - 24725 * q^51 + 61195 * q^52 + 49536 * q^53 - 156168 * q^54 - 34325 * q^55 - 54461 * q^56 - 7580 * q^57 - 95562 * q^58 + 44536 * q^59 + 7125 * q^60 - 49097 * q^61 - 25763 * q^62 - 3578 * q^63 - 18654 * q^64 - 15125 * q^65 - 201873 * q^66 + 788 * q^67 + 163845 * q^68 - 1587 * q^69 - 32925 * q^70 + 49521 * q^71 + 328503 * q^72 - 3760 * q^73 + 88170 * q^74 - 1875 * q^75 - 411465 * q^76 + 77728 * q^77 - 389832 * q^78 + 918 * q^79 - 94250 * q^80 + 121235 * q^81 - 227459 * q^82 + 99202 * q^83 + 336602 * q^84 - 62625 * q^85 + 24584 * q^86 - 38666 * q^87 - 201275 * q^88 - 141676 * q^89 - 199275 * q^90 - 223605 * q^91 + 94162 * q^92 + 51412 * q^93 - 354292 * q^94 + 2875 * q^95 - 592095 * q^96 + 28731 * q^97 - 149557 * q^98 + 237333 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3x^{6} - 196x^{5} + 464x^{4} + 11003x^{3} - 21041x^{2} - 142416x + 243340$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 11\nu^{6} - 631\nu^{5} + 1662\nu^{4} + 73276\nu^{3} - 253039\nu^{2} - 1466045\nu + 3713902 ) / 178848$$ (11*v^6 - 631*v^5 + 1662*v^4 + 73276*v^3 - 253039*v^2 - 1466045*v + 3713902) / 178848 $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 57$$ v^2 - v - 57 $$\beta_{4}$$ $$=$$ $$( 16\nu^{6} - 71\nu^{5} - 1986\nu^{4} + 13772\nu^{3} + 23512\nu^{2} - 708589\nu + 1154738 ) / 44712$$ (16*v^6 - 71*v^5 - 1986*v^4 + 13772*v^3 + 23512*v^2 - 708589*v + 1154738) / 44712 $$\beta_{5}$$ $$=$$ $$( -13\nu^{6} - 7\nu^{5} + 1950\nu^{4} + 5836\nu^{3} - 68887\nu^{2} - 377597\nu + 320206 ) / 19872$$ (-13*v^6 - 7*v^5 + 1950*v^4 + 5836*v^3 - 68887*v^2 - 377597*v + 320206) / 19872 $$\beta_{6}$$ $$=$$ $$( 343\nu^{6} - 707\nu^{5} - 64698\nu^{4} + 62828\nu^{3} + 3331141\nu^{2} - 191713\nu - 29553850 ) / 178848$$ (343*v^6 - 707*v^5 - 64698*v^4 + 62828*v^3 + 3331141*v^2 - 191713*v - 29553850) / 178848
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 + 57$$ b3 + b1 + 57 $$\nu^{3}$$ $$=$$ $$2\beta_{5} + 4\beta_{4} + 2\beta_{3} - 2\beta_{2} + 87\beta _1 + 20$$ 2*b5 + 4*b4 + 2*b3 - 2*b2 + 87*b1 + 20 $$\nu^{4}$$ $$=$$ $$-10\beta_{6} - 16\beta_{5} + 24\beta_{4} + 121\beta_{3} + 2\beta_{2} + 203\beta _1 + 4841$$ -10*b6 - 16*b5 + 24*b4 + 121*b3 + 2*b2 + 203*b1 + 4841 $$\nu^{5}$$ $$=$$ $$-52\beta_{6} + 136\beta_{5} + 616\beta_{4} + 386\beta_{3} - 516\beta_{2} + 8447\beta _1 + 6024$$ -52*b6 + 136*b5 + 616*b4 + 386*b3 - 516*b2 + 8447*b1 + 6024 $$\nu^{6}$$ $$=$$ $$-1472\beta_{6} - 3104\beta_{5} + 5064\beta_{4} + 13541\beta_{3} - 320\beta_{2} + 30613\beta _1 + 454473$$ -1472*b6 - 3104*b5 + 5064*b4 + 13541*b3 - 320*b2 + 30613*b1 + 454473

## 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
 10.7957 8.76815 4.33855 1.66253 −4.26829 −8.71037 −9.58627
−9.79570 20.6902 63.9557 −25.0000 −202.675 61.8409 −313.028 185.085 244.892
1.2 −7.76815 −8.56566 28.3442 −25.0000 66.5393 −61.0525 28.3987 −169.630 194.204
1.3 −3.33855 13.0297 −20.8541 −25.0000 −43.5004 −186.649 176.456 −73.2260 83.4637
1.4 −0.662532 −4.47473 −31.5611 −25.0000 2.96465 91.5985 42.1112 −222.977 16.5633
1.5 5.26829 −11.8410 −4.24511 −25.0000 −62.3818 72.8708 −190.950 −102.791 −131.707
1.6 9.71037 18.9751 62.2913 −25.0000 184.255 84.0805 294.140 117.055 −242.759
1.7 10.5863 −30.8137 80.0691 −25.0000 −326.202 −29.6892 508.872 706.483 −264.657
 $$n$$: 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
$$5$$ $$1$$
$$23$$ $$-1$$

## 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 115.6.a.c 7
3.b odd 2 1 1035.6.a.b 7
5.b even 2 1 575.6.a.d 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.6.a.c 7 1.a even 1 1 trivial
575.6.a.d 7 5.b even 2 1
1035.6.a.b 7 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{7} - 4T_{2}^{6} - 193T_{2}^{5} + 526T_{2}^{4} + 10874T_{2}^{3} - 12768T_{2}^{2} - 150624T_{2} - 91152$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(115))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} - 4 T^{6} + \cdots - 91152$$
$3$ $$T^{7} + 3 T^{6} + \cdots - 71539200$$
$5$ $$(T + 25)^{7}$$
$7$ $$T^{7} + \cdots + 11741977345536$$
$11$ $$T^{7} + \cdots - 12\!\cdots\!96$$
$13$ $$T^{7} + \cdots - 25\!\cdots\!28$$
$17$ $$T^{7} + \cdots - 25\!\cdots\!20$$
$19$ $$T^{7} + \cdots + 53\!\cdots\!40$$
$23$ $$(T - 529)^{7}$$
$29$ $$T^{7} + \cdots + 11\!\cdots\!00$$
$31$ $$T^{7} + \cdots - 11\!\cdots\!20$$
$37$ $$T^{7} + \cdots - 23\!\cdots\!56$$
$41$ $$T^{7} + \cdots + 67\!\cdots\!90$$
$43$ $$T^{7} + \cdots + 30\!\cdots\!00$$
$47$ $$T^{7} + \cdots + 14\!\cdots\!40$$
$53$ $$T^{7} + \cdots + 14\!\cdots\!12$$
$59$ $$T^{7} + \cdots - 13\!\cdots\!88$$
$61$ $$T^{7} + \cdots - 17\!\cdots\!52$$
$67$ $$T^{7} + \cdots + 20\!\cdots\!80$$
$71$ $$T^{7} + \cdots - 27\!\cdots\!00$$
$73$ $$T^{7} + \cdots + 72\!\cdots\!16$$
$79$ $$T^{7} + \cdots + 14\!\cdots\!64$$
$83$ $$T^{7} + \cdots - 77\!\cdots\!88$$
$89$ $$T^{7} + \cdots - 16\!\cdots\!20$$
$97$ $$T^{7} + \cdots - 11\!\cdots\!16$$