# Properties

 Label 138.4.a.b Level $138$ Weight $4$ Character orbit 138.a Self dual yes Analytic conductor $8.142$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 138.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.14226358079$$ 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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 3 q^{3} + 4 q^{4} + 2 q^{5} - 6 q^{6} - 32 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10})$$ q - 2 * q^2 + 3 * q^3 + 4 * q^4 + 2 * q^5 - 6 * q^6 - 32 * q^7 - 8 * q^8 + 9 * q^9 $$q - 2 q^{2} + 3 q^{3} + 4 q^{4} + 2 q^{5} - 6 q^{6} - 32 q^{7} - 8 q^{8} + 9 q^{9} - 4 q^{10} - 48 q^{11} + 12 q^{12} + 22 q^{13} + 64 q^{14} + 6 q^{15} + 16 q^{16} + 42 q^{17} - 18 q^{18} - 144 q^{19} + 8 q^{20} - 96 q^{21} + 96 q^{22} - 23 q^{23} - 24 q^{24} - 121 q^{25} - 44 q^{26} + 27 q^{27} - 128 q^{28} + 174 q^{29} - 12 q^{30} - 304 q^{31} - 32 q^{32} - 144 q^{33} - 84 q^{34} - 64 q^{35} + 36 q^{36} - 318 q^{37} + 288 q^{38} + 66 q^{39} - 16 q^{40} + 74 q^{41} + 192 q^{42} + 192 q^{43} - 192 q^{44} + 18 q^{45} + 46 q^{46} + 392 q^{47} + 48 q^{48} + 681 q^{49} + 242 q^{50} + 126 q^{51} + 88 q^{52} - 734 q^{53} - 54 q^{54} - 96 q^{55} + 256 q^{56} - 432 q^{57} - 348 q^{58} + 156 q^{59} + 24 q^{60} + 706 q^{61} + 608 q^{62} - 288 q^{63} + 64 q^{64} + 44 q^{65} + 288 q^{66} + 192 q^{67} + 168 q^{68} - 69 q^{69} + 128 q^{70} + 624 q^{71} - 72 q^{72} - 406 q^{73} + 636 q^{74} - 363 q^{75} - 576 q^{76} + 1536 q^{77} - 132 q^{78} + 696 q^{79} + 32 q^{80} + 81 q^{81} - 148 q^{82} - 800 q^{83} - 384 q^{84} + 84 q^{85} - 384 q^{86} + 522 q^{87} + 384 q^{88} - 102 q^{89} - 36 q^{90} - 704 q^{91} - 92 q^{92} - 912 q^{93} - 784 q^{94} - 288 q^{95} - 96 q^{96} - 918 q^{97} - 1362 q^{98} - 432 q^{99}+O(q^{100})$$ q - 2 * q^2 + 3 * q^3 + 4 * q^4 + 2 * q^5 - 6 * q^6 - 32 * q^7 - 8 * q^8 + 9 * q^9 - 4 * q^10 - 48 * q^11 + 12 * q^12 + 22 * q^13 + 64 * q^14 + 6 * q^15 + 16 * q^16 + 42 * q^17 - 18 * q^18 - 144 * q^19 + 8 * q^20 - 96 * q^21 + 96 * q^22 - 23 * q^23 - 24 * q^24 - 121 * q^25 - 44 * q^26 + 27 * q^27 - 128 * q^28 + 174 * q^29 - 12 * q^30 - 304 * q^31 - 32 * q^32 - 144 * q^33 - 84 * q^34 - 64 * q^35 + 36 * q^36 - 318 * q^37 + 288 * q^38 + 66 * q^39 - 16 * q^40 + 74 * q^41 + 192 * q^42 + 192 * q^43 - 192 * q^44 + 18 * q^45 + 46 * q^46 + 392 * q^47 + 48 * q^48 + 681 * q^49 + 242 * q^50 + 126 * q^51 + 88 * q^52 - 734 * q^53 - 54 * q^54 - 96 * q^55 + 256 * q^56 - 432 * q^57 - 348 * q^58 + 156 * q^59 + 24 * q^60 + 706 * q^61 + 608 * q^62 - 288 * q^63 + 64 * q^64 + 44 * q^65 + 288 * q^66 + 192 * q^67 + 168 * q^68 - 69 * q^69 + 128 * q^70 + 624 * q^71 - 72 * q^72 - 406 * q^73 + 636 * q^74 - 363 * q^75 - 576 * q^76 + 1536 * q^77 - 132 * q^78 + 696 * q^79 + 32 * q^80 + 81 * q^81 - 148 * q^82 - 800 * q^83 - 384 * q^84 + 84 * q^85 - 384 * q^86 + 522 * q^87 + 384 * q^88 - 102 * q^89 - 36 * q^90 - 704 * q^91 - 92 * q^92 - 912 * q^93 - 784 * q^94 - 288 * q^95 - 96 * q^96 - 918 * q^97 - 1362 * q^98 - 432 * 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
−2.00000 3.00000 4.00000 2.00000 −6.00000 −32.0000 −8.00000 9.00000 −4.00000
 $$n$$: 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 138.4.a.b 1
3.b odd 2 1 414.4.a.c 1
4.b odd 2 1 1104.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.4.a.b 1 1.a even 1 1 trivial
414.4.a.c 1 3.b odd 2 1
1104.4.a.c 1 4.b odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 3$$
$5$ $$T - 2$$
$7$ $$T + 32$$
$11$ $$T + 48$$
$13$ $$T - 22$$
$17$ $$T - 42$$
$19$ $$T + 144$$
$23$ $$T + 23$$
$29$ $$T - 174$$
$31$ $$T + 304$$
$37$ $$T + 318$$
$41$ $$T - 74$$
$43$ $$T - 192$$
$47$ $$T - 392$$
$53$ $$T + 734$$
$59$ $$T - 156$$
$61$ $$T - 706$$
$67$ $$T - 192$$
$71$ $$T - 624$$
$73$ $$T + 406$$
$79$ $$T - 696$$
$83$ $$T + 800$$
$89$ $$T + 102$$
$97$ $$T + 918$$