# Properties

 Label 138.6.a.h Level $138$ Weight $6$ Character orbit 138.a Self dual yes Analytic conductor $22.133$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 1025x - 1873$$ x^3 - x^2 - 1025*x - 1873 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 9 q^{3} + 16 q^{4} + (\beta_1 - 1) q^{5} - 36 q^{6} + ( - \beta_{2} + \beta_1 - 11) q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10})$$ q - 4 * q^2 + 9 * q^3 + 16 * q^4 + (b1 - 1) * q^5 - 36 * q^6 + (-b2 + b1 - 11) * q^7 - 64 * q^8 + 81 * q^9 $$q - 4 q^{2} + 9 q^{3} + 16 q^{4} + (\beta_1 - 1) q^{5} - 36 q^{6} + ( - \beta_{2} + \beta_1 - 11) q^{7} - 64 q^{8} + 81 q^{9} + ( - 4 \beta_1 + 4) q^{10} + (\beta_{2} + 6 \beta_1 + 102) q^{11} + 144 q^{12} + (4 \beta_1 + 334) q^{13} + (4 \beta_{2} - 4 \beta_1 + 44) q^{14} + (9 \beta_1 - 9) q^{15} + 256 q^{16} + (\beta_{2} - 17 \beta_1 + 125) q^{17} - 324 q^{18} + ( - 33 \beta_1 + 755) q^{19} + (16 \beta_1 - 16) q^{20} + ( - 9 \beta_{2} + 9 \beta_1 - 99) q^{21} + ( - 4 \beta_{2} - 24 \beta_1 - 408) q^{22} - 529 q^{23} - 576 q^{24} + (10 \beta_{2} - 389) q^{25} + ( - 16 \beta_1 - 1336) q^{26} + 729 q^{27} + ( - 16 \beta_{2} + 16 \beta_1 - 176) q^{28} + ( - 10 \beta_{2} + 90 \beta_1 - 1248) q^{29} + ( - 36 \beta_1 + 36) q^{30} + (10 \beta_{2} + 30 \beta_1 + 3830) q^{31} - 1024 q^{32} + (9 \beta_{2} + 54 \beta_1 + 918) q^{33} + ( - 4 \beta_{2} + 68 \beta_1 - 500) q^{34} + (14 \beta_{2} - 146 \beta_1 + 1658) q^{35} + 1296 q^{36} + ( - \beta_{2} - 32 \beta_1 + 9418) q^{37} + (132 \beta_1 - 3020) q^{38} + (36 \beta_1 + 3006) q^{39} + ( - 64 \beta_1 + 64) q^{40} + ( - 10 \beta_{2} - 196 \beta_1 + 3934) q^{41} + (36 \beta_{2} - 36 \beta_1 + 396) q^{42} + (70 \beta_{2} + 27 \beta_1 + 11231) q^{43} + (16 \beta_{2} + 96 \beta_1 + 1632) q^{44} + (81 \beta_1 - 81) q^{45} + 2116 q^{46} + ( - 70 \beta_{2} + 146 \beta_1 + 10510) q^{47} + 2304 q^{48} + ( - 98 \beta_{2} - 224 \beta_1 + 20525) q^{49} + ( - 40 \beta_{2} + 1556) q^{50} + (9 \beta_{2} - 153 \beta_1 + 1125) q^{51} + (64 \beta_1 + 5344) q^{52} + ( - 22 \beta_{2} + 75 \beta_1 + 8073) q^{53} - 2916 q^{54} + (56 \beta_{2} + 244 \beta_1 + 17396) q^{55} + (64 \beta_{2} - 64 \beta_1 + 704) q^{56} + ( - 297 \beta_1 + 6795) q^{57} + (40 \beta_{2} - 360 \beta_1 + 4992) q^{58} + ( - 32 \beta_{2} + 78 \beta_1 - 4818) q^{59} + (144 \beta_1 - 144) q^{60} + ( - 111 \beta_{2} - 8 \beta_1 + 13738) q^{61} + ( - 40 \beta_{2} - 120 \beta_1 - 15320) q^{62} + ( - 81 \beta_{2} + 81 \beta_1 - 891) q^{63} + 4096 q^{64} + (40 \beta_{2} + 338 \beta_1 + 10606) q^{65} + ( - 36 \beta_{2} - 216 \beta_1 - 3672) q^{66} + (130 \beta_{2} + 493 \beta_1 + 6841) q^{67} + (16 \beta_{2} - 272 \beta_1 + 2000) q^{68} - 4761 q^{69} + ( - 56 \beta_{2} + 584 \beta_1 - 6632) q^{70} + (182 \beta_{2} - 600 \beta_1 - 16176) q^{71} - 5184 q^{72} + (152 \beta_{2} - 152 \beta_1 + 23914) q^{73} + (4 \beta_{2} + 128 \beta_1 - 37672) q^{74} + (90 \beta_{2} - 3501) q^{75} + ( - 528 \beta_1 + 12080) q^{76} + (98 \beta_{2} - 700 \beta_1 - 26804) q^{77} + ( - 144 \beta_1 - 12024) q^{78} + ( - 231 \beta_{2} + 1303 \beta_1 + 4075) q^{79} + (256 \beta_1 - 256) q^{80} + 6561 q^{81} + (40 \beta_{2} + 784 \beta_1 - 15736) q^{82} + (303 \beta_{2} - 434 \beta_1 - 19834) q^{83} + ( - 144 \beta_{2} + 144 \beta_1 - 1584) q^{84} + ( - 174 \beta_{2} + 244 \beta_1 - 45532) q^{85} + ( - 280 \beta_{2} - 108 \beta_1 - 44924) q^{86} + ( - 90 \beta_{2} + 810 \beta_1 - 11232) q^{87} + ( - 64 \beta_{2} - 384 \beta_1 - 6528) q^{88} + (41 \beta_{2} + 333 \beta_1 - 43269) q^{89} + ( - 324 \beta_1 + 324) q^{90} + ( - 282 \beta_{2} - 246 \beta_1 + 2914) q^{91} - 8464 q^{92} + (90 \beta_{2} + 270 \beta_1 + 34470) q^{93} + (280 \beta_{2} - 584 \beta_1 - 42040) q^{94} + ( - 330 \beta_{2} + 722 \beta_1 - 91010) q^{95} - 9216 q^{96} + ( - 208 \beta_{2} - 1422 \beta_1 + 7412) q^{97} + (392 \beta_{2} + 896 \beta_1 - 82100) q^{98} + (81 \beta_{2} + 486 \beta_1 + 8262) q^{99}+O(q^{100})$$ q - 4 * q^2 + 9 * q^3 + 16 * q^4 + (b1 - 1) * q^5 - 36 * q^6 + (-b2 + b1 - 11) * q^7 - 64 * q^8 + 81 * q^9 + (-4*b1 + 4) * q^10 + (b2 + 6*b1 + 102) * q^11 + 144 * q^12 + (4*b1 + 334) * q^13 + (4*b2 - 4*b1 + 44) * q^14 + (9*b1 - 9) * q^15 + 256 * q^16 + (b2 - 17*b1 + 125) * q^17 - 324 * q^18 + (-33*b1 + 755) * q^19 + (16*b1 - 16) * q^20 + (-9*b2 + 9*b1 - 99) * q^21 + (-4*b2 - 24*b1 - 408) * q^22 - 529 * q^23 - 576 * q^24 + (10*b2 - 389) * q^25 + (-16*b1 - 1336) * q^26 + 729 * q^27 + (-16*b2 + 16*b1 - 176) * q^28 + (-10*b2 + 90*b1 - 1248) * q^29 + (-36*b1 + 36) * q^30 + (10*b2 + 30*b1 + 3830) * q^31 - 1024 * q^32 + (9*b2 + 54*b1 + 918) * q^33 + (-4*b2 + 68*b1 - 500) * q^34 + (14*b2 - 146*b1 + 1658) * q^35 + 1296 * q^36 + (-b2 - 32*b1 + 9418) * q^37 + (132*b1 - 3020) * q^38 + (36*b1 + 3006) * q^39 + (-64*b1 + 64) * q^40 + (-10*b2 - 196*b1 + 3934) * q^41 + (36*b2 - 36*b1 + 396) * q^42 + (70*b2 + 27*b1 + 11231) * q^43 + (16*b2 + 96*b1 + 1632) * q^44 + (81*b1 - 81) * q^45 + 2116 * q^46 + (-70*b2 + 146*b1 + 10510) * q^47 + 2304 * q^48 + (-98*b2 - 224*b1 + 20525) * q^49 + (-40*b2 + 1556) * q^50 + (9*b2 - 153*b1 + 1125) * q^51 + (64*b1 + 5344) * q^52 + (-22*b2 + 75*b1 + 8073) * q^53 - 2916 * q^54 + (56*b2 + 244*b1 + 17396) * q^55 + (64*b2 - 64*b1 + 704) * q^56 + (-297*b1 + 6795) * q^57 + (40*b2 - 360*b1 + 4992) * q^58 + (-32*b2 + 78*b1 - 4818) * q^59 + (144*b1 - 144) * q^60 + (-111*b2 - 8*b1 + 13738) * q^61 + (-40*b2 - 120*b1 - 15320) * q^62 + (-81*b2 + 81*b1 - 891) * q^63 + 4096 * q^64 + (40*b2 + 338*b1 + 10606) * q^65 + (-36*b2 - 216*b1 - 3672) * q^66 + (130*b2 + 493*b1 + 6841) * q^67 + (16*b2 - 272*b1 + 2000) * q^68 - 4761 * q^69 + (-56*b2 + 584*b1 - 6632) * q^70 + (182*b2 - 600*b1 - 16176) * q^71 - 5184 * q^72 + (152*b2 - 152*b1 + 23914) * q^73 + (4*b2 + 128*b1 - 37672) * q^74 + (90*b2 - 3501) * q^75 + (-528*b1 + 12080) * q^76 + (98*b2 - 700*b1 - 26804) * q^77 + (-144*b1 - 12024) * q^78 + (-231*b2 + 1303*b1 + 4075) * q^79 + (256*b1 - 256) * q^80 + 6561 * q^81 + (40*b2 + 784*b1 - 15736) * q^82 + (303*b2 - 434*b1 - 19834) * q^83 + (-144*b2 + 144*b1 - 1584) * q^84 + (-174*b2 + 244*b1 - 45532) * q^85 + (-280*b2 - 108*b1 - 44924) * q^86 + (-90*b2 + 810*b1 - 11232) * q^87 + (-64*b2 - 384*b1 - 6528) * q^88 + (41*b2 + 333*b1 - 43269) * q^89 + (-324*b1 + 324) * q^90 + (-282*b2 - 246*b1 + 2914) * q^91 - 8464 * q^92 + (90*b2 + 270*b1 + 34470) * q^93 + (280*b2 - 584*b1 - 42040) * q^94 + (-330*b2 + 722*b1 - 91010) * q^95 - 9216 * q^96 + (-208*b2 - 1422*b1 + 7412) * q^97 + (392*b2 + 896*b1 - 82100) * q^98 + (81*b2 + 486*b1 + 8262) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 12 q^{2} + 27 q^{3} + 48 q^{4} - 4 q^{5} - 108 q^{6} - 34 q^{7} - 192 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q - 12 * q^2 + 27 * q^3 + 48 * q^4 - 4 * q^5 - 108 * q^6 - 34 * q^7 - 192 * q^8 + 243 * q^9 $$3 q - 12 q^{2} + 27 q^{3} + 48 q^{4} - 4 q^{5} - 108 q^{6} - 34 q^{7} - 192 q^{8} + 243 q^{9} + 16 q^{10} + 300 q^{11} + 432 q^{12} + 998 q^{13} + 136 q^{14} - 36 q^{15} + 768 q^{16} + 392 q^{17} - 972 q^{18} + 2298 q^{19} - 64 q^{20} - 306 q^{21} - 1200 q^{22} - 1587 q^{23} - 1728 q^{24} - 1167 q^{25} - 3992 q^{26} + 2187 q^{27} - 544 q^{28} - 3834 q^{29} + 144 q^{30} + 11460 q^{31} - 3072 q^{32} + 2700 q^{33} - 1568 q^{34} + 5120 q^{35} + 3888 q^{36} + 28286 q^{37} - 9192 q^{38} + 8982 q^{39} + 256 q^{40} + 11998 q^{41} + 1224 q^{42} + 33666 q^{43} + 4800 q^{44} - 324 q^{45} + 6348 q^{46} + 31384 q^{47} + 6912 q^{48} + 61799 q^{49} + 4668 q^{50} + 3528 q^{51} + 15968 q^{52} + 24144 q^{53} - 8748 q^{54} + 51944 q^{55} + 2176 q^{56} + 20682 q^{57} + 15336 q^{58} - 14532 q^{59} - 576 q^{60} + 41222 q^{61} - 45840 q^{62} - 2754 q^{63} + 12288 q^{64} + 31480 q^{65} - 10800 q^{66} + 20030 q^{67} + 6272 q^{68} - 14283 q^{69} - 20480 q^{70} - 47928 q^{71} - 15552 q^{72} + 71894 q^{73} - 113144 q^{74} - 10503 q^{75} + 36768 q^{76} - 79712 q^{77} - 35928 q^{78} + 10922 q^{79} - 1024 q^{80} + 19683 q^{81} - 47992 q^{82} - 59068 q^{83} - 4896 q^{84} - 136840 q^{85} - 134664 q^{86} - 34506 q^{87} - 19200 q^{88} - 130140 q^{89} + 1296 q^{90} + 8988 q^{91} - 25392 q^{92} + 103140 q^{93} - 125536 q^{94} - 273752 q^{95} - 27648 q^{96} + 23658 q^{97} - 247196 q^{98} + 24300 q^{99}+O(q^{100})$$ 3 * q - 12 * q^2 + 27 * q^3 + 48 * q^4 - 4 * q^5 - 108 * q^6 - 34 * q^7 - 192 * q^8 + 243 * q^9 + 16 * q^10 + 300 * q^11 + 432 * q^12 + 998 * q^13 + 136 * q^14 - 36 * q^15 + 768 * q^16 + 392 * q^17 - 972 * q^18 + 2298 * q^19 - 64 * q^20 - 306 * q^21 - 1200 * q^22 - 1587 * q^23 - 1728 * q^24 - 1167 * q^25 - 3992 * q^26 + 2187 * q^27 - 544 * q^28 - 3834 * q^29 + 144 * q^30 + 11460 * q^31 - 3072 * q^32 + 2700 * q^33 - 1568 * q^34 + 5120 * q^35 + 3888 * q^36 + 28286 * q^37 - 9192 * q^38 + 8982 * q^39 + 256 * q^40 + 11998 * q^41 + 1224 * q^42 + 33666 * q^43 + 4800 * q^44 - 324 * q^45 + 6348 * q^46 + 31384 * q^47 + 6912 * q^48 + 61799 * q^49 + 4668 * q^50 + 3528 * q^51 + 15968 * q^52 + 24144 * q^53 - 8748 * q^54 + 51944 * q^55 + 2176 * q^56 + 20682 * q^57 + 15336 * q^58 - 14532 * q^59 - 576 * q^60 + 41222 * q^61 - 45840 * q^62 - 2754 * q^63 + 12288 * q^64 + 31480 * q^65 - 10800 * q^66 + 20030 * q^67 + 6272 * q^68 - 14283 * q^69 - 20480 * q^70 - 47928 * q^71 - 15552 * q^72 + 71894 * q^73 - 113144 * q^74 - 10503 * q^75 + 36768 * q^76 - 79712 * q^77 - 35928 * q^78 + 10922 * q^79 - 1024 * q^80 + 19683 * q^81 - 47992 * q^82 - 59068 * q^83 - 4896 * q^84 - 136840 * q^85 - 134664 * q^86 - 34506 * q^87 - 19200 * q^88 - 130140 * q^89 + 1296 * q^90 + 8988 * q^91 - 25392 * q^92 + 103140 * q^93 - 125536 * q^94 - 273752 * q^95 - 27648 * q^96 + 23658 * q^97 - 247196 * q^98 + 24300 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 1025x - 1873$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$( 2\nu^{2} - 4\nu - 1366 ) / 5$$ (2*v^2 - 4*v - 1366) / 5
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 5\beta_{2} + 2\beta _1 + 1368 ) / 2$$ (5*b2 + 2*b1 + 1368) / 2

## 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
 −30.5473 −1.83665 33.3840
−4.00000 9.00000 16.0000 −63.0946 −36.0000 −197.588 −64.0000 81.0000 252.378
1.2 −4.00000 9.00000 16.0000 −5.67331 −36.0000 254.708 −64.0000 81.0000 22.6932
1.3 −4.00000 9.00000 16.0000 64.7679 −36.0000 −91.1204 −64.0000 81.0000 −259.072
 $$n$$: 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.h 3
3.b odd 2 1 414.6.a.m 3
4.b odd 2 1 1104.6.a.j 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} + 4T_{5}^{2} - 4096T_{5} - 23184$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(138))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 4)^{3}$$
$3$ $$(T - 9)^{3}$$
$5$ $$T^{3} + 4 T^{2} - 4096 T - 23184$$
$7$ $$T^{3} + 34 T^{2} - 55532 T - 4585832$$
$11$ $$T^{3} - 300 T^{2} + \cdots - 18434592$$
$13$ $$T^{3} - 998 T^{2} + \cdots - 16119176$$
$17$ $$T^{3} - 392 T^{2} + \cdots - 72900432$$
$19$ $$T^{3} - 2298 T^{2} + \cdots + 3608499640$$
$23$ $$(T + 529)^{3}$$
$29$ $$T^{3} + 3834 T^{2} + \cdots + 26873648952$$
$31$ $$T^{3} - 11460 T^{2} + \cdots - 22999096000$$
$37$ $$T^{3} - 28286 T^{2} + \cdots - 795432448904$$
$41$ $$T^{3} - 11998 T^{2} + \cdots + 1169089349784$$
$43$ $$T^{3} - 33666 T^{2} + \cdots + 3317925025400$$
$47$ $$T^{3} - 31384 T^{2} + \cdots + 2069618590080$$
$53$ $$T^{3} - 24144 T^{2} + \cdots - 90672025008$$
$59$ $$T^{3} + 14532 T^{2} + \cdots - 206257275840$$
$61$ $$T^{3} - 41222 T^{2} + \cdots + 54318607384$$
$67$ $$T^{3} - 20030 T^{2} + \cdots - 13525331979384$$
$71$ $$T^{3} + 47928 T^{2} + \cdots - 79612179492096$$
$73$ $$T^{3} - 71894 T^{2} + \cdots + 31081392542392$$
$79$ $$T^{3} + \cdots + 357863106252424$$
$83$ $$T^{3} + 59068 T^{2} + \cdots - 18660485323872$$
$89$ $$T^{3} + 130140 T^{2} + \cdots + 50621579393712$$
$97$ $$T^{3} + \cdots + 571541083003976$$