# Properties

 Label 7.10.a.b Level $7$ Weight $10$ Character orbit 7.a Self dual yes Analytic conductor $3.605$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7,10,Mod(1,7)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("7.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 7.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: $$3.60525085315$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 426x + 2016$$ x^3 - x^2 - 426*x + 2016 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 7) q^{2} + ( - \beta_{2} - \beta_1 + 28) q^{3} + ( - 8 \beta_{2} + 7 \beta_1 + 519) q^{4} + (43 \beta_{2} - 13 \beta_1 + 518) q^{5} + (36 \beta_{2} - 6 \beta_1 + 1638) q^{6} + 2401 q^{7} + ( - 470 \beta_{2} + 147 \beta_1 + 4685) q^{8} + (90 \beta_{2} - 126 \beta_1 - 8667) q^{9}+O(q^{10})$$ q + (-b2 + 7) * q^2 + (-b2 - b1 + 28) * q^3 + (-8*b2 + 7*b1 + 519) * q^4 + (43*b2 - 13*b1 + 518) * q^5 + (36*b2 - 6*b1 + 1638) * q^6 + 2401 * q^7 + (-470*b2 + 147*b1 + 4685) * q^8 + (90*b2 - 126*b1 - 8667) * q^9 $$q + ( - \beta_{2} + 7) q^{2} + ( - \beta_{2} - \beta_1 + 28) q^{3} + ( - 8 \beta_{2} + 7 \beta_1 + 519) q^{4} + (43 \beta_{2} - 13 \beta_1 + 518) q^{5} + (36 \beta_{2} - 6 \beta_1 + 1638) q^{6} + 2401 q^{7} + ( - 470 \beta_{2} + 147 \beta_1 + 4685) q^{8} + (90 \beta_{2} - 126 \beta_1 - 8667) q^{9} + (370 \beta_{2} - 470 \beta_1 - 32620) q^{10} + (650 \beta_{2} + 658 \beta_1 - 1148) q^{11} + ( - 700 \beta_{2} + 182 \beta_1 - 35462) q^{12} + (3017 \beta_{2} - 175 \beta_1 - 6594) q^{13} + ( - 2401 \beta_{2} + 16807) q^{14} + ( - 1392 \beta_{2} - 1848 \beta_1 + 66768) q^{15} + ( - 10614 \beta_{2} + 1617 \beta_1 + 160987) q^{16} + (3030 \beta_{2} + 1574 \beta_1 + 338898) q^{17} + (16947 \beta_{2} - 2268 \beta_1 - 91089) q^{18} + ( - 15371 \beta_{2} + 2437 \beta_1 + 74284) q^{19} + (41524 \beta_{2} - 2044 \beta_1 - 640696) q^{20} + ( - 2401 \beta_{2} - 2401 \beta_1 + 67228) q^{21} + ( - 40972 \beta_{2} + 4004 \beta_1 - 949016) q^{22} + ( - 24200 \beta_{2} - 3808 \beta_1 + 628544) q^{23} + (4500 \beta_{2} + 10338 \beta_1 - 483210) q^{24} + ( - 15338 \beta_{2} - 4802 \beta_1 + 1024407) q^{25} + (20986 \beta_{2} - 23394 \beta_1 - 2928352) q^{26} + (33930 \beta_{2} + 15786 \beta_1 + 183960) q^{27} + ( - 19208 \beta_{2} + 16807 \beta_1 + 1246119) q^{28} + (54866 \beta_{2} + 18914 \beta_1 + 1360606) q^{29} + (51960 \beta_{2} - 14280 \beta_1 + 2684400) q^{30} + (55698 \beta_{2} - 70302 \beta_1 + 956480) q^{31} + ( - 36066 \beta_{2} + 20055 \beta_1 + 8407317) q^{32} + ( - 76152 \beta_{2} + 65640 \beta_1 - 6753264) q^{33} + ( - 438178 \beta_{2} - 748 \beta_1 - 1327214) q^{34} + (103243 \beta_{2} - 31213 \beta_1 + 1243718) q^{35} + (209376 \beta_{2} - 83601 \beta_1 - 11798793) q^{36} + (209418 \beta_{2} + 60522 \beta_1 + 465206) q^{37} + ( - 248060 \beta_{2} + 139278 \beta_1 + 14493290) q^{38} + ( - 110012 \beta_{2} - 13748 \beta_1 - 2996896) q^{39} + (625640 \beta_{2} - 76600 \beta_1 - 27619760) q^{40} + ( - 163478 \beta_{2} - 131894 \beta_1 - 4806886) q^{41} + (86436 \beta_{2} - 14406 \beta_1 + 3932838) q^{42} + (121982 \beta_{2} + 65366 \beta_1 - 20543724) q^{43} + (314984 \beta_{2} + 1960 \beta_1 + 32337328) q^{44} + ( - 714681 \beta_{2} + 7551 \beta_1 + 9924894) q^{45} + ( - 405224 \beta_{2} + 119896 \beta_1 + 29915888) q^{46} + ( - 534778 \beta_{2} + 83238 \beta_1 - 3456320) q^{47} + (174140 \beta_{2} + 9710 \beta_1 + 5599594) q^{48} + 5764801 q^{49} + ( - 727615 \beta_{2} + 44940 \beta_1 + 24441685) q^{50} + ( - 587238 \beta_{2} - 186102 \beta_1 - 8715576) q^{51} + (2925244 \beta_{2} - 361424 \beta_1 - 26969348) q^{52} + (1553376 \beta_{2} - 450352 \beta_1 + 22500870) q^{53} + ( - 1176120 \beta_{2} - 32292 \beta_1 - 39293100) q^{54} + (1659356 \beta_{2} + 988364 \beta_1 - 35274344) q^{55} + ( - 1128470 \beta_{2} + 352947 \beta_1 + 11248685) q^{56} + (403894 \beta_{2} + 182350 \beta_1 + 2823704) q^{57} + ( - 2535150 \beta_{2} - 138180 \beta_1 - 53054610) q^{58} + ( - 2231195 \beta_{2} - 49659 \beta_1 - 14196700) q^{59} + ( - 991536 \beta_{2} + 396816 \beta_1 - 59850336) q^{60} + (589107 \beta_{2} - 844773 \beta_1 + 63915614) q^{61} + (3668848 \beta_{2} - 1303812 \beta_1 - 15661156) q^{62} + (216090 \beta_{2} - 302526 \beta_1 - 20809467) q^{63} + ( - 4312590 \beta_{2} - 314727 \beta_1 + 2617387) q^{64} + ( - 958090 \beta_{2} + 1035790 \beta_1 + 121427740) q^{65} + (2410512 \beta_{2} + 1386384 \beta_1 - 2685984) q^{66} + (3939816 \beta_{2} + 47712 \beta_1 - 85058596) q^{67} + ( - 613704 \beta_{2} + 2251634 \beta_1 + 247828602) q^{68} + (697656 \beta_{2} - 981336 \beta_1 + 85967952) q^{69} + (888370 \beta_{2} - 1128470 \beta_1 - 78320620) q^{70} + (3499356 \beta_{2} - 526260 \beta_1 + 98838168) q^{71} + (8765370 \beta_{2} - 1391229 \beta_1 - 203104755) q^{72} + ( - 9544844 \beta_{2} + 118516 \beta_1 + 114737770) q^{73} + ( - 4189718 \beta_{2} - 679140 \beta_1 - 230232154) q^{74} + ( - 4723 \beta_{2} - 1484467 \beta_1 + 93010372) q^{75} + ( - 15924468 \beta_{2} + 2299290 \beta_1 + 242946662) q^{76} + (1560650 \beta_{2} + 1579858 \beta_1 - 2756348) q^{77} + (3780504 \beta_{2} + 591360 \beta_1 + 93377592) q^{78} + ( - 7475532 \beta_{2} + 1679412 \beta_1 - 320137552) q^{79} + (11964112 \beta_{2} - 4328752 \beta_1 - 444444448) q^{80} + ( - 4598370 \beta_{2} + 3824982 \beta_1 - 11942559) q^{81} + (13216518 \beta_{2} - 570276 \beta_1 + 187558434) q^{82} + ( - 4479559 \beta_{2} - 1977367 \beta_1 - 366839060) q^{83} + ( - 1680700 \beta_{2} + 436982 \beta_1 - 85144262) q^{84} + (18558254 \beta_{2} - 1518034 \beta_1 + 146059804) q^{85} + (16416916 \beta_{2} - 4116 \beta_1 - 293660752) q^{86} + ( - 5138394 \beta_{2} + 457014 \beta_1 - 207273864) q^{87} + ( - 11172080 \beta_{2} - 4229456 \beta_1 + 402041600) q^{88} + ( - 10612104 \beta_{2} + 1815976 \beta_1 + 168938826) q^{89} + ( - 11130390 \beta_{2} + 5100930 \beta_1 + 767817540) q^{90} + (7243817 \beta_{2} - 420175 \beta_1 - 15832194) q^{91} + ( - 25723952 \beta_{2} + 6344912 \beta_1 + 230374496) q^{92} + (1921156 \beta_{2} - 7972076 \beta_1 + 564419504) q^{93} + ( - 2488928 \beta_{2} + 4825540 \beta_1 + 462668276) q^{94} + (10749416 \beta_{2} - 4098416 \beta_1 - 734357024) q^{95} + ( - 8360604 \beta_{2} - 6385806 \beta_1 + 111128598) q^{96} + (33276782 \beta_{2} - 864850 \beta_1 - 215832750) q^{97} + ( - 5764801 \beta_{2} + 40353607) q^{98} + ( - 7689150 \beta_{2} + 376362 \beta_1 - 633659724) q^{99}+O(q^{100})$$ q + (-b2 + 7) * q^2 + (-b2 - b1 + 28) * q^3 + (-8*b2 + 7*b1 + 519) * q^4 + (43*b2 - 13*b1 + 518) * q^5 + (36*b2 - 6*b1 + 1638) * q^6 + 2401 * q^7 + (-470*b2 + 147*b1 + 4685) * q^8 + (90*b2 - 126*b1 - 8667) * q^9 + (370*b2 - 470*b1 - 32620) * q^10 + (650*b2 + 658*b1 - 1148) * q^11 + (-700*b2 + 182*b1 - 35462) * q^12 + (3017*b2 - 175*b1 - 6594) * q^13 + (-2401*b2 + 16807) * q^14 + (-1392*b2 - 1848*b1 + 66768) * q^15 + (-10614*b2 + 1617*b1 + 160987) * q^16 + (3030*b2 + 1574*b1 + 338898) * q^17 + (16947*b2 - 2268*b1 - 91089) * q^18 + (-15371*b2 + 2437*b1 + 74284) * q^19 + (41524*b2 - 2044*b1 - 640696) * q^20 + (-2401*b2 - 2401*b1 + 67228) * q^21 + (-40972*b2 + 4004*b1 - 949016) * q^22 + (-24200*b2 - 3808*b1 + 628544) * q^23 + (4500*b2 + 10338*b1 - 483210) * q^24 + (-15338*b2 - 4802*b1 + 1024407) * q^25 + (20986*b2 - 23394*b1 - 2928352) * q^26 + (33930*b2 + 15786*b1 + 183960) * q^27 + (-19208*b2 + 16807*b1 + 1246119) * q^28 + (54866*b2 + 18914*b1 + 1360606) * q^29 + (51960*b2 - 14280*b1 + 2684400) * q^30 + (55698*b2 - 70302*b1 + 956480) * q^31 + (-36066*b2 + 20055*b1 + 8407317) * q^32 + (-76152*b2 + 65640*b1 - 6753264) * q^33 + (-438178*b2 - 748*b1 - 1327214) * q^34 + (103243*b2 - 31213*b1 + 1243718) * q^35 + (209376*b2 - 83601*b1 - 11798793) * q^36 + (209418*b2 + 60522*b1 + 465206) * q^37 + (-248060*b2 + 139278*b1 + 14493290) * q^38 + (-110012*b2 - 13748*b1 - 2996896) * q^39 + (625640*b2 - 76600*b1 - 27619760) * q^40 + (-163478*b2 - 131894*b1 - 4806886) * q^41 + (86436*b2 - 14406*b1 + 3932838) * q^42 + (121982*b2 + 65366*b1 - 20543724) * q^43 + (314984*b2 + 1960*b1 + 32337328) * q^44 + (-714681*b2 + 7551*b1 + 9924894) * q^45 + (-405224*b2 + 119896*b1 + 29915888) * q^46 + (-534778*b2 + 83238*b1 - 3456320) * q^47 + (174140*b2 + 9710*b1 + 5599594) * q^48 + 5764801 * q^49 + (-727615*b2 + 44940*b1 + 24441685) * q^50 + (-587238*b2 - 186102*b1 - 8715576) * q^51 + (2925244*b2 - 361424*b1 - 26969348) * q^52 + (1553376*b2 - 450352*b1 + 22500870) * q^53 + (-1176120*b2 - 32292*b1 - 39293100) * q^54 + (1659356*b2 + 988364*b1 - 35274344) * q^55 + (-1128470*b2 + 352947*b1 + 11248685) * q^56 + (403894*b2 + 182350*b1 + 2823704) * q^57 + (-2535150*b2 - 138180*b1 - 53054610) * q^58 + (-2231195*b2 - 49659*b1 - 14196700) * q^59 + (-991536*b2 + 396816*b1 - 59850336) * q^60 + (589107*b2 - 844773*b1 + 63915614) * q^61 + (3668848*b2 - 1303812*b1 - 15661156) * q^62 + (216090*b2 - 302526*b1 - 20809467) * q^63 + (-4312590*b2 - 314727*b1 + 2617387) * q^64 + (-958090*b2 + 1035790*b1 + 121427740) * q^65 + (2410512*b2 + 1386384*b1 - 2685984) * q^66 + (3939816*b2 + 47712*b1 - 85058596) * q^67 + (-613704*b2 + 2251634*b1 + 247828602) * q^68 + (697656*b2 - 981336*b1 + 85967952) * q^69 + (888370*b2 - 1128470*b1 - 78320620) * q^70 + (3499356*b2 - 526260*b1 + 98838168) * q^71 + (8765370*b2 - 1391229*b1 - 203104755) * q^72 + (-9544844*b2 + 118516*b1 + 114737770) * q^73 + (-4189718*b2 - 679140*b1 - 230232154) * q^74 + (-4723*b2 - 1484467*b1 + 93010372) * q^75 + (-15924468*b2 + 2299290*b1 + 242946662) * q^76 + (1560650*b2 + 1579858*b1 - 2756348) * q^77 + (3780504*b2 + 591360*b1 + 93377592) * q^78 + (-7475532*b2 + 1679412*b1 - 320137552) * q^79 + (11964112*b2 - 4328752*b1 - 444444448) * q^80 + (-4598370*b2 + 3824982*b1 - 11942559) * q^81 + (13216518*b2 - 570276*b1 + 187558434) * q^82 + (-4479559*b2 - 1977367*b1 - 366839060) * q^83 + (-1680700*b2 + 436982*b1 - 85144262) * q^84 + (18558254*b2 - 1518034*b1 + 146059804) * q^85 + (16416916*b2 - 4116*b1 - 293660752) * q^86 + (-5138394*b2 + 457014*b1 - 207273864) * q^87 + (-11172080*b2 - 4229456*b1 + 402041600) * q^88 + (-10612104*b2 + 1815976*b1 + 168938826) * q^89 + (-11130390*b2 + 5100930*b1 + 767817540) * q^90 + (7243817*b2 - 420175*b1 - 15832194) * q^91 + (-25723952*b2 + 6344912*b1 + 230374496) * q^92 + (1921156*b2 - 7972076*b1 + 564419504) * q^93 + (-2488928*b2 + 4825540*b1 + 462668276) * q^94 + (10749416*b2 - 4098416*b1 - 734357024) * q^95 + (-8360604*b2 - 6385806*b1 + 111128598) * q^96 + (33276782*b2 - 864850*b1 - 215832750) * q^97 + (-5764801*b2 + 40353607) * q^98 + (-7689150*b2 + 376362*b1 - 633659724) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 21 q^{2} + 84 q^{3} + 1557 q^{4} + 1554 q^{5} + 4914 q^{6} + 7203 q^{7} + 14055 q^{8} - 26001 q^{9}+O(q^{10})$$ 3 * q + 21 * q^2 + 84 * q^3 + 1557 * q^4 + 1554 * q^5 + 4914 * q^6 + 7203 * q^7 + 14055 * q^8 - 26001 * q^9 $$3 q + 21 q^{2} + 84 q^{3} + 1557 q^{4} + 1554 q^{5} + 4914 q^{6} + 7203 q^{7} + 14055 q^{8} - 26001 q^{9} - 97860 q^{10} - 3444 q^{11} - 106386 q^{12} - 19782 q^{13} + 50421 q^{14} + 200304 q^{15} + 482961 q^{16} + 1016694 q^{17} - 273267 q^{18} + 222852 q^{19} - 1922088 q^{20} + 201684 q^{21} - 2847048 q^{22} + 1885632 q^{23} - 1449630 q^{24} + 3073221 q^{25} - 8785056 q^{26} + 551880 q^{27} + 3738357 q^{28} + 4081818 q^{29} + 8053200 q^{30} + 2869440 q^{31} + 25221951 q^{32} - 20259792 q^{33} - 3981642 q^{34} + 3731154 q^{35} - 35396379 q^{36} + 1395618 q^{37} + 43479870 q^{38} - 8990688 q^{39} - 82859280 q^{40} - 14420658 q^{41} + 11798514 q^{42} - 61631172 q^{43} + 97011984 q^{44} + 29774682 q^{45} + 89747664 q^{46} - 10368960 q^{47} + 16798782 q^{48} + 17294403 q^{49} + 73325055 q^{50} - 26146728 q^{51} - 80908044 q^{52} + 67502610 q^{53} - 117879300 q^{54} - 105823032 q^{55} + 33746055 q^{56} + 8471112 q^{57} - 159163830 q^{58} - 42590100 q^{59} - 179551008 q^{60} + 191746842 q^{61} - 46983468 q^{62} - 62428401 q^{63} + 7852161 q^{64} + 364283220 q^{65} - 8057952 q^{66} - 255175788 q^{67} + 743485806 q^{68} + 257903856 q^{69} - 234961860 q^{70} + 296514504 q^{71} - 609314265 q^{72} + 344213310 q^{73} - 690696462 q^{74} + 279031116 q^{75} + 728839986 q^{76} - 8269044 q^{77} + 280132776 q^{78} - 960412656 q^{79} - 1333333344 q^{80} - 35827677 q^{81} + 562675302 q^{82} - 1100517180 q^{83} - 255432786 q^{84} + 438179412 q^{85} - 880982256 q^{86} - 621821592 q^{87} + 1206124800 q^{88} + 506816478 q^{89} + 2303452620 q^{90} - 47496582 q^{91} + 691123488 q^{92} + 1693258512 q^{93} + 1388004828 q^{94} - 2203071072 q^{95} + 333385794 q^{96} - 647498250 q^{97} + 121060821 q^{98} - 1900979172 q^{99}+O(q^{100})$$ 3 * q + 21 * q^2 + 84 * q^3 + 1557 * q^4 + 1554 * q^5 + 4914 * q^6 + 7203 * q^7 + 14055 * q^8 - 26001 * q^9 - 97860 * q^10 - 3444 * q^11 - 106386 * q^12 - 19782 * q^13 + 50421 * q^14 + 200304 * q^15 + 482961 * q^16 + 1016694 * q^17 - 273267 * q^18 + 222852 * q^19 - 1922088 * q^20 + 201684 * q^21 - 2847048 * q^22 + 1885632 * q^23 - 1449630 * q^24 + 3073221 * q^25 - 8785056 * q^26 + 551880 * q^27 + 3738357 * q^28 + 4081818 * q^29 + 8053200 * q^30 + 2869440 * q^31 + 25221951 * q^32 - 20259792 * q^33 - 3981642 * q^34 + 3731154 * q^35 - 35396379 * q^36 + 1395618 * q^37 + 43479870 * q^38 - 8990688 * q^39 - 82859280 * q^40 - 14420658 * q^41 + 11798514 * q^42 - 61631172 * q^43 + 97011984 * q^44 + 29774682 * q^45 + 89747664 * q^46 - 10368960 * q^47 + 16798782 * q^48 + 17294403 * q^49 + 73325055 * q^50 - 26146728 * q^51 - 80908044 * q^52 + 67502610 * q^53 - 117879300 * q^54 - 105823032 * q^55 + 33746055 * q^56 + 8471112 * q^57 - 159163830 * q^58 - 42590100 * q^59 - 179551008 * q^60 + 191746842 * q^61 - 46983468 * q^62 - 62428401 * q^63 + 7852161 * q^64 + 364283220 * q^65 - 8057952 * q^66 - 255175788 * q^67 + 743485806 * q^68 + 257903856 * q^69 - 234961860 * q^70 + 296514504 * q^71 - 609314265 * q^72 + 344213310 * q^73 - 690696462 * q^74 + 279031116 * q^75 + 728839986 * q^76 - 8269044 * q^77 + 280132776 * q^78 - 960412656 * q^79 - 1333333344 * q^80 - 35827677 * q^81 + 562675302 * q^82 - 1100517180 * q^83 - 255432786 * q^84 + 438179412 * q^85 - 880982256 * q^86 - 621821592 * q^87 + 1206124800 * q^88 + 506816478 * q^89 + 2303452620 * q^90 - 47496582 * q^91 + 691123488 * q^92 + 1693258512 * q^93 + 1388004828 * q^94 - 2203071072 * q^95 + 333385794 * q^96 - 647498250 * q^97 + 121060821 * q^98 - 1900979172 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 426x + 2016$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{2} + 25\nu + 276 ) / 6$$ (-v^2 + 25*v + 276) / 6 $$\beta_{2}$$ $$=$$ $$( \nu^{2} + 11\nu - 288 ) / 6$$ (v^2 + 11*v - 288) / 6
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 2 ) / 6$$ (b2 + b1 + 2) / 6 $$\nu^{2}$$ $$=$$ $$( 25\beta_{2} - 11\beta _1 + 1706 ) / 6$$ (25*b2 - 11*b1 + 1706) / 6

## 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
 18.2745 −22.2358 4.96128
−34.1627 −79.6469 655.088 1423.70 2720.95 2401.00 −4888.28 −13339.4 −48637.4
1.2 13.3607 163.415 −333.491 1922.19 2183.34 2401.00 −11296.4 7021.32 25681.8
1.3 41.8019 0.232339 1235.40 −1791.89 9.71222 2401.00 30239.6 −19682.9 −74904.4
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$

## 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 7.10.a.b 3
3.b odd 2 1 63.10.a.e 3
4.b odd 2 1 112.10.a.h 3
5.b even 2 1 175.10.a.d 3
5.c odd 4 2 175.10.b.d 6
7.b odd 2 1 49.10.a.c 3
7.c even 3 2 49.10.c.d 6
7.d odd 6 2 49.10.c.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.b 3 1.a even 1 1 trivial
49.10.a.c 3 7.b odd 2 1
49.10.c.d 6 7.c even 3 2
49.10.c.e 6 7.d odd 6 2
63.10.a.e 3 3.b odd 2 1
112.10.a.h 3 4.b odd 2 1
175.10.a.d 3 5.b even 2 1
175.10.b.d 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 21T_{2}^{2} - 1326T_{2} + 19080$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(7))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 21 T^{2} - 1326 T + 19080$$
$3$ $$T^{3} - 84 T^{2} - 12996 T + 3024$$
$5$ $$T^{3} - 1554 T^{2} + \cdots + 4903718400$$
$7$ $$(T - 2401)^{3}$$
$11$ $$T^{3} + \cdots + 108859759460352$$
$13$ $$T^{3} + 19782 T^{2} + \cdots - 41548412541440$$
$17$ $$T^{3} - 1016694 T^{2} + \cdots - 21\!\cdots\!32$$
$19$ $$T^{3} - 222852 T^{2} + \cdots - 43\!\cdots\!60$$
$23$ $$T^{3} - 1885632 T^{2} + \cdots + 97\!\cdots\!36$$
$29$ $$T^{3} - 4081818 T^{2} + \cdots + 44\!\cdots\!00$$
$31$ $$T^{3} - 2869440 T^{2} + \cdots - 74\!\cdots\!84$$
$37$ $$T^{3} - 1395618 T^{2} + \cdots - 34\!\cdots\!28$$
$41$ $$T^{3} + 14420658 T^{2} + \cdots - 19\!\cdots\!12$$
$43$ $$T^{3} + 61631172 T^{2} + \cdots + 68\!\cdots\!80$$
$47$ $$T^{3} + 10368960 T^{2} + \cdots - 43\!\cdots\!16$$
$53$ $$T^{3} - 67502610 T^{2} + \cdots + 23\!\cdots\!28$$
$59$ $$T^{3} + 42590100 T^{2} + \cdots + 42\!\cdots\!00$$
$61$ $$T^{3} - 191746842 T^{2} + \cdots + 51\!\cdots\!08$$
$67$ $$T^{3} + 255175788 T^{2} + \cdots - 20\!\cdots\!64$$
$71$ $$T^{3} - 296514504 T^{2} + \cdots + 16\!\cdots\!80$$
$73$ $$T^{3} - 344213310 T^{2} + \cdots + 19\!\cdots\!48$$
$79$ $$T^{3} + 960412656 T^{2} + \cdots - 11\!\cdots\!00$$
$83$ $$T^{3} + 1100517180 T^{2} + \cdots + 18\!\cdots\!48$$
$89$ $$T^{3} - 506816478 T^{2} + \cdots + 19\!\cdots\!40$$
$97$ $$T^{3} + 647498250 T^{2} + \cdots - 49\!\cdots\!16$$