# Properties

 Label 138.6.a.g Level $138$ Weight $6$ Character orbit 138.a Self dual yes Analytic conductor $22.133$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 138.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.1329671342$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 1383x - 16813$$ x^3 - x^2 - 1383*x - 16813 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

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.

 $$f(q)$$ $$=$$ $$q - 4 q^{2} - 9 q^{3} + 16 q^{4} + ( - \beta_{2} - 6) q^{5} + 36 q^{6} + (2 \beta_{2} + \beta_1 - 17) q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10})$$ q - 4 * q^2 - 9 * q^3 + 16 * q^4 + (-b2 - 6) * q^5 + 36 * q^6 + (2*b2 + b1 - 17) * q^7 - 64 * q^8 + 81 * q^9 $$q - 4 q^{2} - 9 q^{3} + 16 q^{4} + ( - \beta_{2} - 6) q^{5} + 36 q^{6} + (2 \beta_{2} + \beta_1 - 17) q^{7} - 64 q^{8} + 81 q^{9} + (4 \beta_{2} + 24) q^{10} + (\beta_{2} - 15 \beta_1 + 7) q^{11} - 144 q^{12} + (3 \beta_{2} + 19 \beta_1 + 257) q^{13} + ( - 8 \beta_{2} - 4 \beta_1 + 68) q^{14} + (9 \beta_{2} + 54) q^{15} + 256 q^{16} + ( - 10 \beta_{2} + 29 \beta_1 + 373) q^{17} - 324 q^{18} + (7 \beta_{2} - 32 \beta_1 + 908) q^{19} + ( - 16 \beta_{2} - 96) q^{20} + ( - 18 \beta_{2} - 9 \beta_1 + 153) q^{21} + ( - 4 \beta_{2} + 60 \beta_1 - 28) q^{22} - 529 q^{23} + 576 q^{24} + ( - 13 \beta_{2} - 77 \beta_1 + 1010) q^{25} + ( - 12 \beta_{2} - 76 \beta_1 - 1028) q^{26} - 729 q^{27} + (32 \beta_{2} + 16 \beta_1 - 272) q^{28} + (2 \beta_{2} + 96 \beta_1 - 2002) q^{29} + ( - 36 \beta_{2} - 216) q^{30} + (52 \beta_{2} - 138 \beta_1 - 1362) q^{31} - 1024 q^{32} + ( - 9 \beta_{2} + 135 \beta_1 - 63) q^{33} + (40 \beta_{2} - 116 \beta_1 - 1492) q^{34} + (73 \beta_{2} + 123 \beta_1 - 8161) q^{35} + 1296 q^{36} + ( - 11 \beta_{2} + 231 \beta_1 - 3533) q^{37} + ( - 28 \beta_{2} + 128 \beta_1 - 3632) q^{38} + ( - 27 \beta_{2} - 171 \beta_1 - 2313) q^{39} + (64 \beta_{2} + 384) q^{40} + (50 \beta_{2} - 190 \beta_1 - 12532) q^{41} + (72 \beta_{2} + 36 \beta_1 - 612) q^{42} + ( - 61 \beta_{2} + 274 \beta_1 - 9630) q^{43} + (16 \beta_{2} - 240 \beta_1 + 112) q^{44} + ( - 81 \beta_{2} - 486) q^{45} + 2116 q^{46} + (125 \beta_{2} - 325 \beta_1 - 14341) q^{47} - 2304 q^{48} + ( - 235 \beta_{2} - 223 \beta_1 + 1054) q^{49} + (52 \beta_{2} + 308 \beta_1 - 4040) q^{50} + (90 \beta_{2} - 261 \beta_1 - 3357) q^{51} + (48 \beta_{2} + 304 \beta_1 + 4112) q^{52} + ( - 161 \beta_{2} - 460 \beta_1 - 13702) q^{53} + 2916 q^{54} + ( - 258 \beta_{2} + 542 \beta_1 - 3166) q^{55} + ( - 128 \beta_{2} - 64 \beta_1 + 1088) q^{56} + ( - 63 \beta_{2} + 288 \beta_1 - 8172) q^{57} + ( - 8 \beta_{2} - 384 \beta_1 + 8008) q^{58} + (49 \beta_{2} + 867 \beta_1 - 15809) q^{59} + (144 \beta_{2} + 864) q^{60} + ( - 459 \beta_{2} - 749 \beta_1 + 2891) q^{61} + ( - 208 \beta_{2} + 552 \beta_1 + 5448) q^{62} + (162 \beta_{2} + 81 \beta_1 - 1377) q^{63} + 4096 q^{64} + (142 \beta_{2} - 358 \beta_1 - 15074) q^{65} + (36 \beta_{2} - 540 \beta_1 + 252) q^{66} + (507 \beta_{2} - 366 \beta_1 - 5550) q^{67} + ( - 160 \beta_{2} + 464 \beta_1 + 5968) q^{68} + 4761 q^{69} + ( - 292 \beta_{2} - 492 \beta_1 + 32644) q^{70} + (356 \beta_{2} + 1044 \beta_1 - 13108) q^{71} - 5184 q^{72} + (372 \beta_{2} + 1112 \beta_1 - 4274) q^{73} + (44 \beta_{2} - 924 \beta_1 + 14132) q^{74} + (117 \beta_{2} + 693 \beta_1 - 9090) q^{75} + (112 \beta_{2} - 512 \beta_1 + 14528) q^{76} + (394 \beta_{2} - 902 \beta_1 - 7546) q^{77} + (108 \beta_{2} + 684 \beta_1 + 9252) q^{78} + (668 \beta_{2} - 381 \beta_1 + 24437) q^{79} + ( - 256 \beta_{2} - 1536) q^{80} + 6561 q^{81} + ( - 200 \beta_{2} + 760 \beta_1 + 50128) q^{82} + (609 \beta_{2} - 1439 \beta_1 - 16221) q^{83} + ( - 288 \beta_{2} - 144 \beta_1 + 2448) q^{84} + ( - 41 \beta_{2} - 1669 \beta_1 + 36867) q^{85} + (244 \beta_{2} - 1096 \beta_1 + 38520) q^{86} + ( - 18 \beta_{2} - 864 \beta_1 + 18018) q^{87} + ( - 64 \beta_{2} + 960 \beta_1 - 448) q^{88} + ( - 988 \beta_{2} + 1575 \beta_1 + 14879) q^{89} + (324 \beta_{2} + 1944) q^{90} + ( - 330 \beta_{2} + 858 \beta_1 + 40294) q^{91} - 8464 q^{92} + ( - 468 \beta_{2} + 1242 \beta_1 + 12258) q^{93} + ( - 500 \beta_{2} + 1300 \beta_1 + 57364) q^{94} + ( - 1351 \beta_{2} + 1531 \beta_1 - 32061) q^{95} + 9216 q^{96} + (138 \beta_{2} + 3592 \beta_1 - 35078) q^{97} + (940 \beta_{2} + 892 \beta_1 - 4216) q^{98} + (81 \beta_{2} - 1215 \beta_1 + 567) q^{99}+O(q^{100})$$ q - 4 * q^2 - 9 * q^3 + 16 * q^4 + (-b2 - 6) * q^5 + 36 * q^6 + (2*b2 + b1 - 17) * q^7 - 64 * q^8 + 81 * q^9 + (4*b2 + 24) * q^10 + (b2 - 15*b1 + 7) * q^11 - 144 * q^12 + (3*b2 + 19*b1 + 257) * q^13 + (-8*b2 - 4*b1 + 68) * q^14 + (9*b2 + 54) * q^15 + 256 * q^16 + (-10*b2 + 29*b1 + 373) * q^17 - 324 * q^18 + (7*b2 - 32*b1 + 908) * q^19 + (-16*b2 - 96) * q^20 + (-18*b2 - 9*b1 + 153) * q^21 + (-4*b2 + 60*b1 - 28) * q^22 - 529 * q^23 + 576 * q^24 + (-13*b2 - 77*b1 + 1010) * q^25 + (-12*b2 - 76*b1 - 1028) * q^26 - 729 * q^27 + (32*b2 + 16*b1 - 272) * q^28 + (2*b2 + 96*b1 - 2002) * q^29 + (-36*b2 - 216) * q^30 + (52*b2 - 138*b1 - 1362) * q^31 - 1024 * q^32 + (-9*b2 + 135*b1 - 63) * q^33 + (40*b2 - 116*b1 - 1492) * q^34 + (73*b2 + 123*b1 - 8161) * q^35 + 1296 * q^36 + (-11*b2 + 231*b1 - 3533) * q^37 + (-28*b2 + 128*b1 - 3632) * q^38 + (-27*b2 - 171*b1 - 2313) * q^39 + (64*b2 + 384) * q^40 + (50*b2 - 190*b1 - 12532) * q^41 + (72*b2 + 36*b1 - 612) * q^42 + (-61*b2 + 274*b1 - 9630) * q^43 + (16*b2 - 240*b1 + 112) * q^44 + (-81*b2 - 486) * q^45 + 2116 * q^46 + (125*b2 - 325*b1 - 14341) * q^47 - 2304 * q^48 + (-235*b2 - 223*b1 + 1054) * q^49 + (52*b2 + 308*b1 - 4040) * q^50 + (90*b2 - 261*b1 - 3357) * q^51 + (48*b2 + 304*b1 + 4112) * q^52 + (-161*b2 - 460*b1 - 13702) * q^53 + 2916 * q^54 + (-258*b2 + 542*b1 - 3166) * q^55 + (-128*b2 - 64*b1 + 1088) * q^56 + (-63*b2 + 288*b1 - 8172) * q^57 + (-8*b2 - 384*b1 + 8008) * q^58 + (49*b2 + 867*b1 - 15809) * q^59 + (144*b2 + 864) * q^60 + (-459*b2 - 749*b1 + 2891) * q^61 + (-208*b2 + 552*b1 + 5448) * q^62 + (162*b2 + 81*b1 - 1377) * q^63 + 4096 * q^64 + (142*b2 - 358*b1 - 15074) * q^65 + (36*b2 - 540*b1 + 252) * q^66 + (507*b2 - 366*b1 - 5550) * q^67 + (-160*b2 + 464*b1 + 5968) * q^68 + 4761 * q^69 + (-292*b2 - 492*b1 + 32644) * q^70 + (356*b2 + 1044*b1 - 13108) * q^71 - 5184 * q^72 + (372*b2 + 1112*b1 - 4274) * q^73 + (44*b2 - 924*b1 + 14132) * q^74 + (117*b2 + 693*b1 - 9090) * q^75 + (112*b2 - 512*b1 + 14528) * q^76 + (394*b2 - 902*b1 - 7546) * q^77 + (108*b2 + 684*b1 + 9252) * q^78 + (668*b2 - 381*b1 + 24437) * q^79 + (-256*b2 - 1536) * q^80 + 6561 * q^81 + (-200*b2 + 760*b1 + 50128) * q^82 + (609*b2 - 1439*b1 - 16221) * q^83 + (-288*b2 - 144*b1 + 2448) * q^84 + (-41*b2 - 1669*b1 + 36867) * q^85 + (244*b2 - 1096*b1 + 38520) * q^86 + (-18*b2 - 864*b1 + 18018) * q^87 + (-64*b2 + 960*b1 - 448) * q^88 + (-988*b2 + 1575*b1 + 14879) * q^89 + (324*b2 + 1944) * q^90 + (-330*b2 + 858*b1 + 40294) * q^91 - 8464 * q^92 + (-468*b2 + 1242*b1 + 12258) * q^93 + (-500*b2 + 1300*b1 + 57364) * q^94 + (-1351*b2 + 1531*b1 - 32061) * q^95 + 9216 * q^96 + (138*b2 + 3592*b1 - 35078) * q^97 + (940*b2 + 892*b1 - 4216) * q^98 + (81*b2 - 1215*b1 + 567) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 12 q^{2} - 27 q^{3} + 48 q^{4} - 18 q^{5} + 108 q^{6} - 50 q^{7} - 192 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q - 12 * q^2 - 27 * q^3 + 48 * q^4 - 18 * q^5 + 108 * q^6 - 50 * q^7 - 192 * q^8 + 243 * q^9 $$3 q - 12 q^{2} - 27 q^{3} + 48 q^{4} - 18 q^{5} + 108 q^{6} - 50 q^{7} - 192 q^{8} + 243 q^{9} + 72 q^{10} + 6 q^{11} - 432 q^{12} + 790 q^{13} + 200 q^{14} + 162 q^{15} + 768 q^{16} + 1148 q^{17} - 972 q^{18} + 2692 q^{19} - 288 q^{20} + 450 q^{21} - 24 q^{22} - 1587 q^{23} + 1728 q^{24} + 2953 q^{25} - 3160 q^{26} - 2187 q^{27} - 800 q^{28} - 5910 q^{29} - 648 q^{30} - 4224 q^{31} - 3072 q^{32} - 54 q^{33} - 4592 q^{34} - 24360 q^{35} + 3888 q^{36} - 10368 q^{37} - 10768 q^{38} - 7110 q^{39} + 1152 q^{40} - 37786 q^{41} - 1800 q^{42} - 28616 q^{43} + 96 q^{44} - 1458 q^{45} + 6348 q^{46} - 43348 q^{47} - 6912 q^{48} + 2939 q^{49} - 11812 q^{50} - 10332 q^{51} + 12640 q^{52} - 41566 q^{53} + 8748 q^{54} - 8956 q^{55} + 3200 q^{56} - 24228 q^{57} + 23640 q^{58} - 46560 q^{59} + 2592 q^{60} + 7924 q^{61} + 16896 q^{62} - 4050 q^{63} + 12288 q^{64} - 45580 q^{65} + 216 q^{66} - 17016 q^{67} + 18368 q^{68} + 14283 q^{69} + 97440 q^{70} - 38280 q^{71} - 15552 q^{72} - 11710 q^{73} + 41472 q^{74} - 26577 q^{75} + 43072 q^{76} - 23540 q^{77} + 28440 q^{78} + 72930 q^{79} - 4608 q^{80} + 19683 q^{81} + 151144 q^{82} - 50102 q^{83} + 7200 q^{84} + 108932 q^{85} + 114464 q^{86} + 53190 q^{87} - 384 q^{88} + 46212 q^{89} + 5832 q^{90} + 121740 q^{91} - 25392 q^{92} + 38016 q^{93} + 173392 q^{94} - 94652 q^{95} + 27648 q^{96} - 101642 q^{97} - 11756 q^{98} + 486 q^{99}+O(q^{100})$$ 3 * q - 12 * q^2 - 27 * q^3 + 48 * q^4 - 18 * q^5 + 108 * q^6 - 50 * q^7 - 192 * q^8 + 243 * q^9 + 72 * q^10 + 6 * q^11 - 432 * q^12 + 790 * q^13 + 200 * q^14 + 162 * q^15 + 768 * q^16 + 1148 * q^17 - 972 * q^18 + 2692 * q^19 - 288 * q^20 + 450 * q^21 - 24 * q^22 - 1587 * q^23 + 1728 * q^24 + 2953 * q^25 - 3160 * q^26 - 2187 * q^27 - 800 * q^28 - 5910 * q^29 - 648 * q^30 - 4224 * q^31 - 3072 * q^32 - 54 * q^33 - 4592 * q^34 - 24360 * q^35 + 3888 * q^36 - 10368 * q^37 - 10768 * q^38 - 7110 * q^39 + 1152 * q^40 - 37786 * q^41 - 1800 * q^42 - 28616 * q^43 + 96 * q^44 - 1458 * q^45 + 6348 * q^46 - 43348 * q^47 - 6912 * q^48 + 2939 * q^49 - 11812 * q^50 - 10332 * q^51 + 12640 * q^52 - 41566 * q^53 + 8748 * q^54 - 8956 * q^55 + 3200 * q^56 - 24228 * q^57 + 23640 * q^58 - 46560 * q^59 + 2592 * q^60 + 7924 * q^61 + 16896 * q^62 - 4050 * q^63 + 12288 * q^64 - 45580 * q^65 + 216 * q^66 - 17016 * q^67 + 18368 * q^68 + 14283 * q^69 + 97440 * q^70 - 38280 * q^71 - 15552 * q^72 - 11710 * q^73 + 41472 * q^74 - 26577 * q^75 + 43072 * q^76 - 23540 * q^77 + 28440 * q^78 + 72930 * q^79 - 4608 * q^80 + 19683 * q^81 + 151144 * q^82 - 50102 * q^83 + 7200 * q^84 + 108932 * q^85 + 114464 * q^86 + 53190 * q^87 - 384 * q^88 + 46212 * q^89 + 5832 * q^90 + 121740 * q^91 - 25392 * q^92 + 38016 * q^93 + 173392 * q^94 - 94652 * q^95 + 27648 * q^96 - 101642 * q^97 - 11756 * q^98 + 486 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 1383x - 16813$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 19\nu - 916 ) / 5$$ (v^2 - 19*v - 916) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2} + 19\beta _1 + 916$$ 5*b2 + 19*b1 + 916

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −27.1335 42.6589 −14.5254
−4.00000 −9.00000 16.0000 −73.1526 36.0000 90.1716 −64.0000 81.0000 292.610
1.2 −4.00000 −9.00000 16.0000 −24.6530 36.0000 62.9650 −64.0000 81.0000 98.6122
1.3 −4.00000 −9.00000 16.0000 79.8056 36.0000 −203.137 −64.0000 81.0000 −319.222
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$23$$ $$1$$

## 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 138.6.a.g 3
3.b odd 2 1 414.6.a.n 3
4.b odd 2 1 1104.6.a.k 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.6.a.g 3 1.a even 1 1 trivial
414.6.a.n 3 3.b odd 2 1
1104.6.a.k 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} + 18T_{5}^{2} - 6002T_{5} - 143924$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(138))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 4)^{3}$$
$3$ $$(T + 9)^{3}$$
$5$ $$T^{3} + 18 T^{2} - 6002 T - 143924$$
$7$ $$T^{3} + 50 T^{2} - 25430 T + 1153340$$
$11$ $$T^{3} - 6 T^{2} - 314048 T + 41102592$$
$13$ $$T^{3} - 790 T^{2} + \cdots - 17724640$$
$17$ $$T^{3} - 1148 T^{2} + \cdots + 1251288504$$
$19$ $$T^{3} - 2692 T^{2} + \cdots + 566361936$$
$23$ $$(T + 529)^{3}$$
$29$ $$T^{3} + 5910 T^{2} + \cdots - 33999878760$$
$31$ $$T^{3} + 4224 T^{2} + \cdots - 140877524608$$
$37$ $$T^{3} + 10368 T^{2} + \cdots - 383183552304$$
$41$ $$T^{3} + 37786 T^{2} + \cdots + 1113643632728$$
$43$ $$T^{3} + 28616 T^{2} + \cdots - 163187296392$$
$47$ $$T^{3} + 43348 T^{2} + \cdots - 1511593331104$$
$53$ $$T^{3} + 41566 T^{2} + \cdots - 2969882763452$$
$59$ $$T^{3} + 46560 T^{2} + \cdots - 25961931296880$$
$61$ $$T^{3} - 7924 T^{2} + \cdots - 15216738764992$$
$67$ $$T^{3} + 17016 T^{2} + \cdots - 19674353996712$$
$71$ $$T^{3} + 38280 T^{2} + \cdots - 39250356203520$$
$73$ $$T^{3} + 11710 T^{2} + \cdots - 24824926300760$$
$79$ $$T^{3} - 72930 T^{2} + \cdots + 44957071020660$$
$83$ $$T^{3} + \cdots - 200825772591136$$
$89$ $$T^{3} + \cdots + 460268929768536$$
$97$ $$T^{3} + 101642 T^{2} + \cdots - 14\!\cdots\!36$$