# Properties

 Label 138.8.a.a Level $138$ Weight $8$ Character orbit 138.a Self dual yes Analytic conductor $43.109$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 138.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.1091335168$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 8 q^{2} + 27 q^{3} + 64 q^{4} - 230 q^{5} + 216 q^{6} + 106 q^{7} + 512 q^{8} + 729 q^{9}+O(q^{10})$$ q + 8 * q^2 + 27 * q^3 + 64 * q^4 - 230 * q^5 + 216 * q^6 + 106 * q^7 + 512 * q^8 + 729 * q^9 $$q + 8 q^{2} + 27 q^{3} + 64 q^{4} - 230 q^{5} + 216 q^{6} + 106 q^{7} + 512 q^{8} + 729 q^{9} - 1840 q^{10} - 4326 q^{11} + 1728 q^{12} - 5426 q^{13} + 848 q^{14} - 6210 q^{15} + 4096 q^{16} - 3300 q^{17} + 5832 q^{18} - 4140 q^{19} - 14720 q^{20} + 2862 q^{21} - 34608 q^{22} + 12167 q^{23} + 13824 q^{24} - 25225 q^{25} - 43408 q^{26} + 19683 q^{27} + 6784 q^{28} - 150186 q^{29} - 49680 q^{30} - 307192 q^{31} + 32768 q^{32} - 116802 q^{33} - 26400 q^{34} - 24380 q^{35} + 46656 q^{36} - 55200 q^{37} - 33120 q^{38} - 146502 q^{39} - 117760 q^{40} + 270130 q^{41} + 22896 q^{42} + 36264 q^{43} - 276864 q^{44} - 167670 q^{45} + 97336 q^{46} + 494224 q^{47} + 110592 q^{48} - 812307 q^{49} - 201800 q^{50} - 89100 q^{51} - 347264 q^{52} + 646646 q^{53} + 157464 q^{54} + 994980 q^{55} + 54272 q^{56} - 111780 q^{57} - 1201488 q^{58} - 387948 q^{59} - 397440 q^{60} - 2060876 q^{61} - 2457536 q^{62} + 77274 q^{63} + 262144 q^{64} + 1247980 q^{65} - 934416 q^{66} - 17664 q^{67} - 211200 q^{68} + 328509 q^{69} - 195040 q^{70} - 3580320 q^{71} + 373248 q^{72} + 484550 q^{73} - 441600 q^{74} - 681075 q^{75} - 264960 q^{76} - 458556 q^{77} - 1172016 q^{78} - 2167314 q^{79} - 942080 q^{80} + 531441 q^{81} + 2161040 q^{82} + 381182 q^{83} + 183168 q^{84} + 759000 q^{85} + 290112 q^{86} - 4055022 q^{87} - 2214912 q^{88} - 628620 q^{89} - 1341360 q^{90} - 575156 q^{91} + 778688 q^{92} - 8294184 q^{93} + 3953792 q^{94} + 952200 q^{95} + 884736 q^{96} + 13964418 q^{97} - 6498456 q^{98} - 3153654 q^{99}+O(q^{100})$$ q + 8 * q^2 + 27 * q^3 + 64 * q^4 - 230 * q^5 + 216 * q^6 + 106 * q^7 + 512 * q^8 + 729 * q^9 - 1840 * q^10 - 4326 * q^11 + 1728 * q^12 - 5426 * q^13 + 848 * q^14 - 6210 * q^15 + 4096 * q^16 - 3300 * q^17 + 5832 * q^18 - 4140 * q^19 - 14720 * q^20 + 2862 * q^21 - 34608 * q^22 + 12167 * q^23 + 13824 * q^24 - 25225 * q^25 - 43408 * q^26 + 19683 * q^27 + 6784 * q^28 - 150186 * q^29 - 49680 * q^30 - 307192 * q^31 + 32768 * q^32 - 116802 * q^33 - 26400 * q^34 - 24380 * q^35 + 46656 * q^36 - 55200 * q^37 - 33120 * q^38 - 146502 * q^39 - 117760 * q^40 + 270130 * q^41 + 22896 * q^42 + 36264 * q^43 - 276864 * q^44 - 167670 * q^45 + 97336 * q^46 + 494224 * q^47 + 110592 * q^48 - 812307 * q^49 - 201800 * q^50 - 89100 * q^51 - 347264 * q^52 + 646646 * q^53 + 157464 * q^54 + 994980 * q^55 + 54272 * q^56 - 111780 * q^57 - 1201488 * q^58 - 387948 * q^59 - 397440 * q^60 - 2060876 * q^61 - 2457536 * q^62 + 77274 * q^63 + 262144 * q^64 + 1247980 * q^65 - 934416 * q^66 - 17664 * q^67 - 211200 * q^68 + 328509 * q^69 - 195040 * q^70 - 3580320 * q^71 + 373248 * q^72 + 484550 * q^73 - 441600 * q^74 - 681075 * q^75 - 264960 * q^76 - 458556 * q^77 - 1172016 * q^78 - 2167314 * q^79 - 942080 * q^80 + 531441 * q^81 + 2161040 * q^82 + 381182 * q^83 + 183168 * q^84 + 759000 * q^85 + 290112 * q^86 - 4055022 * q^87 - 2214912 * q^88 - 628620 * q^89 - 1341360 * q^90 - 575156 * q^91 + 778688 * q^92 - 8294184 * q^93 + 3953792 * q^94 + 952200 * q^95 + 884736 * q^96 + 13964418 * q^97 - 6498456 * q^98 - 3153654 * 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
 0
8.00000 27.0000 64.0000 −230.000 216.000 106.000 512.000 729.000 −1840.00
 $$n$$: 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 138.8.a.a 1
3.b odd 2 1 414.8.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.a.a 1 1.a even 1 1 trivial
414.8.a.a 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 230$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(138))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 8$$
$3$ $$T - 27$$
$5$ $$T + 230$$
$7$ $$T - 106$$
$11$ $$T + 4326$$
$13$ $$T + 5426$$
$17$ $$T + 3300$$
$19$ $$T + 4140$$
$23$ $$T - 12167$$
$29$ $$T + 150186$$
$31$ $$T + 307192$$
$37$ $$T + 55200$$
$41$ $$T - 270130$$
$43$ $$T - 36264$$
$47$ $$T - 494224$$
$53$ $$T - 646646$$
$59$ $$T + 387948$$
$61$ $$T + 2060876$$
$67$ $$T + 17664$$
$71$ $$T + 3580320$$
$73$ $$T - 484550$$
$79$ $$T + 2167314$$
$83$ $$T - 381182$$
$89$ $$T + 628620$$
$97$ $$T - 13964418$$