# Properties

 Label 154.4.a.d Level $154$ Weight $4$ Character orbit 154.a Self dual yes Analytic conductor $9.086$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 154.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.08629414088$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 18 q^{5} - 4 q^{6} + 7 q^{7} + 8 q^{8} - 23 q^{9}+O(q^{10})$$ q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 18 * q^5 - 4 * q^6 + 7 * q^7 + 8 * q^8 - 23 * q^9 $$q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 18 q^{5} - 4 q^{6} + 7 q^{7} + 8 q^{8} - 23 q^{9} + 36 q^{10} - 11 q^{11} - 8 q^{12} + 56 q^{13} + 14 q^{14} - 36 q^{15} + 16 q^{16} + 36 q^{17} - 46 q^{18} - 28 q^{19} + 72 q^{20} - 14 q^{21} - 22 q^{22} + 180 q^{23} - 16 q^{24} + 199 q^{25} + 112 q^{26} + 100 q^{27} + 28 q^{28} - 54 q^{29} - 72 q^{30} - 334 q^{31} + 32 q^{32} + 22 q^{33} + 72 q^{34} + 126 q^{35} - 92 q^{36} + 386 q^{37} - 56 q^{38} - 112 q^{39} + 144 q^{40} - 444 q^{41} - 28 q^{42} - 316 q^{43} - 44 q^{44} - 414 q^{45} + 360 q^{46} - 402 q^{47} - 32 q^{48} + 49 q^{49} + 398 q^{50} - 72 q^{51} + 224 q^{52} - 486 q^{53} + 200 q^{54} - 198 q^{55} + 56 q^{56} + 56 q^{57} - 108 q^{58} - 282 q^{59} - 144 q^{60} + 380 q^{61} - 668 q^{62} - 161 q^{63} + 64 q^{64} + 1008 q^{65} + 44 q^{66} + 176 q^{67} + 144 q^{68} - 360 q^{69} + 252 q^{70} - 324 q^{71} - 184 q^{72} + 800 q^{73} + 772 q^{74} - 398 q^{75} - 112 q^{76} - 77 q^{77} - 224 q^{78} - 1144 q^{79} + 288 q^{80} + 421 q^{81} - 888 q^{82} + 468 q^{83} - 56 q^{84} + 648 q^{85} - 632 q^{86} + 108 q^{87} - 88 q^{88} - 870 q^{89} - 828 q^{90} + 392 q^{91} + 720 q^{92} + 668 q^{93} - 804 q^{94} - 504 q^{95} - 64 q^{96} - 1330 q^{97} + 98 q^{98} + 253 q^{99}+O(q^{100})$$ q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 18 * q^5 - 4 * q^6 + 7 * q^7 + 8 * q^8 - 23 * q^9 + 36 * q^10 - 11 * q^11 - 8 * q^12 + 56 * q^13 + 14 * q^14 - 36 * q^15 + 16 * q^16 + 36 * q^17 - 46 * q^18 - 28 * q^19 + 72 * q^20 - 14 * q^21 - 22 * q^22 + 180 * q^23 - 16 * q^24 + 199 * q^25 + 112 * q^26 + 100 * q^27 + 28 * q^28 - 54 * q^29 - 72 * q^30 - 334 * q^31 + 32 * q^32 + 22 * q^33 + 72 * q^34 + 126 * q^35 - 92 * q^36 + 386 * q^37 - 56 * q^38 - 112 * q^39 + 144 * q^40 - 444 * q^41 - 28 * q^42 - 316 * q^43 - 44 * q^44 - 414 * q^45 + 360 * q^46 - 402 * q^47 - 32 * q^48 + 49 * q^49 + 398 * q^50 - 72 * q^51 + 224 * q^52 - 486 * q^53 + 200 * q^54 - 198 * q^55 + 56 * q^56 + 56 * q^57 - 108 * q^58 - 282 * q^59 - 144 * q^60 + 380 * q^61 - 668 * q^62 - 161 * q^63 + 64 * q^64 + 1008 * q^65 + 44 * q^66 + 176 * q^67 + 144 * q^68 - 360 * q^69 + 252 * q^70 - 324 * q^71 - 184 * q^72 + 800 * q^73 + 772 * q^74 - 398 * q^75 - 112 * q^76 - 77 * q^77 - 224 * q^78 - 1144 * q^79 + 288 * q^80 + 421 * q^81 - 888 * q^82 + 468 * q^83 - 56 * q^84 + 648 * q^85 - 632 * q^86 + 108 * q^87 - 88 * q^88 - 870 * q^89 - 828 * q^90 + 392 * q^91 + 720 * q^92 + 668 * q^93 - 804 * q^94 - 504 * q^95 - 64 * q^96 - 1330 * q^97 + 98 * q^98 + 253 * 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 −2.00000 4.00000 18.0000 −4.00000 7.00000 8.00000 −23.0000 36.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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 154.4.a.d 1
3.b odd 2 1 1386.4.a.a 1
4.b odd 2 1 1232.4.a.f 1
7.b odd 2 1 1078.4.a.g 1
11.b odd 2 1 1694.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.d 1 1.a even 1 1 trivial
1078.4.a.g 1 7.b odd 2 1
1232.4.a.f 1 4.b odd 2 1
1386.4.a.a 1 3.b odd 2 1
1694.4.a.c 1 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(154))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 2$$
$5$ $$T - 18$$
$7$ $$T - 7$$
$11$ $$T + 11$$
$13$ $$T - 56$$
$17$ $$T - 36$$
$19$ $$T + 28$$
$23$ $$T - 180$$
$29$ $$T + 54$$
$31$ $$T + 334$$
$37$ $$T - 386$$
$41$ $$T + 444$$
$43$ $$T + 316$$
$47$ $$T + 402$$
$53$ $$T + 486$$
$59$ $$T + 282$$
$61$ $$T - 380$$
$67$ $$T - 176$$
$71$ $$T + 324$$
$73$ $$T - 800$$
$79$ $$T + 1144$$
$83$ $$T - 468$$
$89$ $$T + 870$$
$97$ $$T + 1330$$