# Properties

 Label 138.6.a.e Level $138$ Weight $6$ Character orbit 138.a Self dual yes Analytic conductor $22.133$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{154})$$ Defining polynomial: $$x^{2} - 154$$ x^2 - 154 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{154}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 9 q^{3} + 16 q^{4} + (3 \beta - 2) q^{5} - 36 q^{6} + ( - 5 \beta - 20) q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10})$$ q - 4 * q^2 + 9 * q^3 + 16 * q^4 + (3*b - 2) * q^5 - 36 * q^6 + (-5*b - 20) * q^7 - 64 * q^8 + 81 * q^9 $$q - 4 q^{2} + 9 q^{3} + 16 q^{4} + (3 \beta - 2) q^{5} - 36 q^{6} + ( - 5 \beta - 20) q^{7} - 64 q^{8} + 81 q^{9} + ( - 12 \beta + 8) q^{10} + ( - 20 \beta - 80) q^{11} + 144 q^{12} + (12 \beta - 728) q^{13} + (20 \beta + 80) q^{14} + (27 \beta - 18) q^{15} + 256 q^{16} + ( - 135 \beta - 218) q^{17} - 324 q^{18} + (177 \beta - 224) q^{19} + (48 \beta - 32) q^{20} + ( - 45 \beta - 180) q^{21} + (80 \beta + 320) q^{22} + 529 q^{23} - 576 q^{24} + ( - 12 \beta - 1735) q^{25} + ( - 48 \beta + 2912) q^{26} + 729 q^{27} + ( - 80 \beta - 320) q^{28} + (282 \beta + 3182) q^{29} + ( - 108 \beta + 72) q^{30} + ( - 566 \beta - 1180) q^{31} - 1024 q^{32} + ( - 180 \beta - 720) q^{33} + (540 \beta + 872) q^{34} + ( - 50 \beta - 2270) q^{35} + 1296 q^{36} + (402 \beta - 9694) q^{37} + ( - 708 \beta + 896) q^{38} + (108 \beta - 6552) q^{39} + ( - 192 \beta + 128) q^{40} + (896 \beta - 8738) q^{41} + (180 \beta + 720) q^{42} + ( - 509 \beta - 11880) q^{43} + ( - 320 \beta - 1280) q^{44} + (243 \beta - 162) q^{45} - 2116 q^{46} + ( - 950 \beta - 17842) q^{47} + 2304 q^{48} + (200 \beta - 12557) q^{49} + (48 \beta + 6940) q^{50} + ( - 1215 \beta - 1962) q^{51} + (192 \beta - 11648) q^{52} + (923 \beta + 738) q^{53} - 2916 q^{54} + ( - 200 \beta - 9080) q^{55} + (320 \beta + 1280) q^{56} + (1593 \beta - 2016) q^{57} + ( - 1128 \beta - 12728) q^{58} + (2830 \beta + 11546) q^{59} + (432 \beta - 288) q^{60} + (238 \beta - 35442) q^{61} + (2264 \beta + 4720) q^{62} + ( - 405 \beta - 1620) q^{63} + 4096 q^{64} + ( - 2208 \beta + 7000) q^{65} + (720 \beta + 2880) q^{66} + (495 \beta - 40912) q^{67} + ( - 2160 \beta - 3488) q^{68} + 4761 q^{69} + (200 \beta + 9080) q^{70} + ( - 4392 \beta + 7776) q^{71} - 5184 q^{72} + ( - 4116 \beta + 24822) q^{73} + ( - 1608 \beta + 38776) q^{74} + ( - 108 \beta - 15615) q^{75} + (2832 \beta - 3584) q^{76} + (800 \beta + 17000) q^{77} + ( - 432 \beta + 26208) q^{78} + (911 \beta + 51340) q^{79} + (768 \beta - 512) q^{80} + 6561 q^{81} + ( - 3584 \beta + 34952) q^{82} + (5492 \beta + 19548) q^{83} + ( - 720 \beta - 2880) q^{84} + ( - 384 \beta - 61934) q^{85} + (2036 \beta + 47520) q^{86} + (2538 \beta + 28638) q^{87} + (1280 \beta + 5120) q^{88} + ( - 4791 \beta - 10614) q^{89} + ( - 972 \beta + 648) q^{90} + (3400 \beta + 5320) q^{91} + 8464 q^{92} + ( - 5094 \beta - 10620) q^{93} + (3800 \beta + 71368) q^{94} + ( - 1026 \beta + 82222) q^{95} - 9216 q^{96} + (7042 \beta - 12886) q^{97} + ( - 800 \beta + 50228) q^{98} + ( - 1620 \beta - 6480) q^{99}+O(q^{100})$$ q - 4 * q^2 + 9 * q^3 + 16 * q^4 + (3*b - 2) * q^5 - 36 * q^6 + (-5*b - 20) * q^7 - 64 * q^8 + 81 * q^9 + (-12*b + 8) * q^10 + (-20*b - 80) * q^11 + 144 * q^12 + (12*b - 728) * q^13 + (20*b + 80) * q^14 + (27*b - 18) * q^15 + 256 * q^16 + (-135*b - 218) * q^17 - 324 * q^18 + (177*b - 224) * q^19 + (48*b - 32) * q^20 + (-45*b - 180) * q^21 + (80*b + 320) * q^22 + 529 * q^23 - 576 * q^24 + (-12*b - 1735) * q^25 + (-48*b + 2912) * q^26 + 729 * q^27 + (-80*b - 320) * q^28 + (282*b + 3182) * q^29 + (-108*b + 72) * q^30 + (-566*b - 1180) * q^31 - 1024 * q^32 + (-180*b - 720) * q^33 + (540*b + 872) * q^34 + (-50*b - 2270) * q^35 + 1296 * q^36 + (402*b - 9694) * q^37 + (-708*b + 896) * q^38 + (108*b - 6552) * q^39 + (-192*b + 128) * q^40 + (896*b - 8738) * q^41 + (180*b + 720) * q^42 + (-509*b - 11880) * q^43 + (-320*b - 1280) * q^44 + (243*b - 162) * q^45 - 2116 * q^46 + (-950*b - 17842) * q^47 + 2304 * q^48 + (200*b - 12557) * q^49 + (48*b + 6940) * q^50 + (-1215*b - 1962) * q^51 + (192*b - 11648) * q^52 + (923*b + 738) * q^53 - 2916 * q^54 + (-200*b - 9080) * q^55 + (320*b + 1280) * q^56 + (1593*b - 2016) * q^57 + (-1128*b - 12728) * q^58 + (2830*b + 11546) * q^59 + (432*b - 288) * q^60 + (238*b - 35442) * q^61 + (2264*b + 4720) * q^62 + (-405*b - 1620) * q^63 + 4096 * q^64 + (-2208*b + 7000) * q^65 + (720*b + 2880) * q^66 + (495*b - 40912) * q^67 + (-2160*b - 3488) * q^68 + 4761 * q^69 + (200*b + 9080) * q^70 + (-4392*b + 7776) * q^71 - 5184 * q^72 + (-4116*b + 24822) * q^73 + (-1608*b + 38776) * q^74 + (-108*b - 15615) * q^75 + (2832*b - 3584) * q^76 + (800*b + 17000) * q^77 + (-432*b + 26208) * q^78 + (911*b + 51340) * q^79 + (768*b - 512) * q^80 + 6561 * q^81 + (-3584*b + 34952) * q^82 + (5492*b + 19548) * q^83 + (-720*b - 2880) * q^84 + (-384*b - 61934) * q^85 + (2036*b + 47520) * q^86 + (2538*b + 28638) * q^87 + (1280*b + 5120) * q^88 + (-4791*b - 10614) * q^89 + (-972*b + 648) * q^90 + (3400*b + 5320) * q^91 + 8464 * q^92 + (-5094*b - 10620) * q^93 + (3800*b + 71368) * q^94 + (-1026*b + 82222) * q^95 - 9216 * q^96 + (7042*b - 12886) * q^97 + (-800*b + 50228) * q^98 + (-1620*b - 6480) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{2} + 18 q^{3} + 32 q^{4} - 4 q^{5} - 72 q^{6} - 40 q^{7} - 128 q^{8} + 162 q^{9}+O(q^{10})$$ 2 * q - 8 * q^2 + 18 * q^3 + 32 * q^4 - 4 * q^5 - 72 * q^6 - 40 * q^7 - 128 * q^8 + 162 * q^9 $$2 q - 8 q^{2} + 18 q^{3} + 32 q^{4} - 4 q^{5} - 72 q^{6} - 40 q^{7} - 128 q^{8} + 162 q^{9} + 16 q^{10} - 160 q^{11} + 288 q^{12} - 1456 q^{13} + 160 q^{14} - 36 q^{15} + 512 q^{16} - 436 q^{17} - 648 q^{18} - 448 q^{19} - 64 q^{20} - 360 q^{21} + 640 q^{22} + 1058 q^{23} - 1152 q^{24} - 3470 q^{25} + 5824 q^{26} + 1458 q^{27} - 640 q^{28} + 6364 q^{29} + 144 q^{30} - 2360 q^{31} - 2048 q^{32} - 1440 q^{33} + 1744 q^{34} - 4540 q^{35} + 2592 q^{36} - 19388 q^{37} + 1792 q^{38} - 13104 q^{39} + 256 q^{40} - 17476 q^{41} + 1440 q^{42} - 23760 q^{43} - 2560 q^{44} - 324 q^{45} - 4232 q^{46} - 35684 q^{47} + 4608 q^{48} - 25114 q^{49} + 13880 q^{50} - 3924 q^{51} - 23296 q^{52} + 1476 q^{53} - 5832 q^{54} - 18160 q^{55} + 2560 q^{56} - 4032 q^{57} - 25456 q^{58} + 23092 q^{59} - 576 q^{60} - 70884 q^{61} + 9440 q^{62} - 3240 q^{63} + 8192 q^{64} + 14000 q^{65} + 5760 q^{66} - 81824 q^{67} - 6976 q^{68} + 9522 q^{69} + 18160 q^{70} + 15552 q^{71} - 10368 q^{72} + 49644 q^{73} + 77552 q^{74} - 31230 q^{75} - 7168 q^{76} + 34000 q^{77} + 52416 q^{78} + 102680 q^{79} - 1024 q^{80} + 13122 q^{81} + 69904 q^{82} + 39096 q^{83} - 5760 q^{84} - 123868 q^{85} + 95040 q^{86} + 57276 q^{87} + 10240 q^{88} - 21228 q^{89} + 1296 q^{90} + 10640 q^{91} + 16928 q^{92} - 21240 q^{93} + 142736 q^{94} + 164444 q^{95} - 18432 q^{96} - 25772 q^{97} + 100456 q^{98} - 12960 q^{99}+O(q^{100})$$ 2 * q - 8 * q^2 + 18 * q^3 + 32 * q^4 - 4 * q^5 - 72 * q^6 - 40 * q^7 - 128 * q^8 + 162 * q^9 + 16 * q^10 - 160 * q^11 + 288 * q^12 - 1456 * q^13 + 160 * q^14 - 36 * q^15 + 512 * q^16 - 436 * q^17 - 648 * q^18 - 448 * q^19 - 64 * q^20 - 360 * q^21 + 640 * q^22 + 1058 * q^23 - 1152 * q^24 - 3470 * q^25 + 5824 * q^26 + 1458 * q^27 - 640 * q^28 + 6364 * q^29 + 144 * q^30 - 2360 * q^31 - 2048 * q^32 - 1440 * q^33 + 1744 * q^34 - 4540 * q^35 + 2592 * q^36 - 19388 * q^37 + 1792 * q^38 - 13104 * q^39 + 256 * q^40 - 17476 * q^41 + 1440 * q^42 - 23760 * q^43 - 2560 * q^44 - 324 * q^45 - 4232 * q^46 - 35684 * q^47 + 4608 * q^48 - 25114 * q^49 + 13880 * q^50 - 3924 * q^51 - 23296 * q^52 + 1476 * q^53 - 5832 * q^54 - 18160 * q^55 + 2560 * q^56 - 4032 * q^57 - 25456 * q^58 + 23092 * q^59 - 576 * q^60 - 70884 * q^61 + 9440 * q^62 - 3240 * q^63 + 8192 * q^64 + 14000 * q^65 + 5760 * q^66 - 81824 * q^67 - 6976 * q^68 + 9522 * q^69 + 18160 * q^70 + 15552 * q^71 - 10368 * q^72 + 49644 * q^73 + 77552 * q^74 - 31230 * q^75 - 7168 * q^76 + 34000 * q^77 + 52416 * q^78 + 102680 * q^79 - 1024 * q^80 + 13122 * q^81 + 69904 * q^82 + 39096 * q^83 - 5760 * q^84 - 123868 * q^85 + 95040 * q^86 + 57276 * q^87 + 10240 * q^88 - 21228 * q^89 + 1296 * q^90 + 10640 * q^91 + 16928 * q^92 - 21240 * q^93 + 142736 * q^94 + 164444 * q^95 - 18432 * q^96 - 25772 * q^97 + 100456 * q^98 - 12960 * q^99

## 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
 −12.4097 12.4097
−4.00000 9.00000 16.0000 −39.2290 −36.0000 42.0484 −64.0000 81.0000 156.916
1.2 −4.00000 9.00000 16.0000 35.2290 −36.0000 −82.0484 −64.0000 81.0000 −140.916
 $$n$$: 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.e 2
3.b odd 2 1 414.6.a.g 2
4.b odd 2 1 1104.6.a.f 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4T_{5} - 1382$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(138))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 4)^{2}$$
$3$ $$(T - 9)^{2}$$
$5$ $$T^{2} + 4T - 1382$$
$7$ $$T^{2} + 40T - 3450$$
$11$ $$T^{2} + 160T - 55200$$
$13$ $$T^{2} + 1456 T + 507808$$
$17$ $$T^{2} + 436 T - 2759126$$
$19$ $$T^{2} + 448 T - 4774490$$
$23$ $$(T - 529)^{2}$$
$29$ $$T^{2} - 6364 T - 2121572$$
$31$ $$T^{2} + 2360 T - 47942424$$
$37$ $$T^{2} + 19388 T + 69086620$$
$41$ $$T^{2} + 17476 T - 47281020$$
$43$ $$T^{2} + 23760 T + 101235926$$
$47$ $$T^{2} + 35684 T + 179351964$$
$53$ $$T^{2} - 1476 T - 130652422$$
$59$ $$T^{2} - 23092 T - 1100060484$$
$61$ $$T^{2} + 70884 T + 1247412188$$
$67$ $$T^{2} + 81824 T + 1636057894$$
$71$ $$T^{2} - 15552 T - 2910142080$$
$73$ $$T^{2} - 49644 T - 1992852540$$
$79$ $$T^{2} - 102680 T + 2507987766$$
$83$ $$T^{2} - 39096 T - 4262833552$$
$89$ $$T^{2} + 21228 T - 3422209878$$
$97$ $$T^{2} + 25772 T - 7470774660$$