# Properties

 Label 154.4.a.c 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

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 10 q^{3} + 4 q^{4} - 14 q^{5} - 20 q^{6} + 7 q^{7} + 8 q^{8} + 73 q^{9}+O(q^{10})$$ q + 2 * q^2 - 10 * q^3 + 4 * q^4 - 14 * q^5 - 20 * q^6 + 7 * q^7 + 8 * q^8 + 73 * q^9 $$q + 2 q^{2} - 10 q^{3} + 4 q^{4} - 14 q^{5} - 20 q^{6} + 7 q^{7} + 8 q^{8} + 73 q^{9} - 28 q^{10} - 11 q^{11} - 40 q^{12} - 16 q^{13} + 14 q^{14} + 140 q^{15} + 16 q^{16} + 108 q^{17} + 146 q^{18} + 116 q^{19} - 56 q^{20} - 70 q^{21} - 22 q^{22} + 68 q^{23} - 80 q^{24} + 71 q^{25} - 32 q^{26} - 460 q^{27} + 28 q^{28} + 122 q^{29} + 280 q^{30} - 262 q^{31} + 32 q^{32} + 110 q^{33} + 216 q^{34} - 98 q^{35} + 292 q^{36} + 130 q^{37} + 232 q^{38} + 160 q^{39} - 112 q^{40} + 204 q^{41} - 140 q^{42} - 396 q^{43} - 44 q^{44} - 1022 q^{45} + 136 q^{46} + 166 q^{47} - 160 q^{48} + 49 q^{49} + 142 q^{50} - 1080 q^{51} - 64 q^{52} + 442 q^{53} - 920 q^{54} + 154 q^{55} + 56 q^{56} - 1160 q^{57} + 244 q^{58} + 702 q^{59} + 560 q^{60} + 196 q^{61} - 524 q^{62} + 511 q^{63} + 64 q^{64} + 224 q^{65} + 220 q^{66} - 416 q^{67} + 432 q^{68} - 680 q^{69} - 196 q^{70} + 492 q^{71} + 584 q^{72} + 408 q^{73} + 260 q^{74} - 710 q^{75} + 464 q^{76} - 77 q^{77} + 320 q^{78} + 600 q^{79} - 224 q^{80} + 2629 q^{81} + 408 q^{82} - 1212 q^{83} - 280 q^{84} - 1512 q^{85} - 792 q^{86} - 1220 q^{87} - 88 q^{88} + 1146 q^{89} - 2044 q^{90} - 112 q^{91} + 272 q^{92} + 2620 q^{93} + 332 q^{94} - 1624 q^{95} - 320 q^{96} - 482 q^{97} + 98 q^{98} - 803 q^{99}+O(q^{100})$$ q + 2 * q^2 - 10 * q^3 + 4 * q^4 - 14 * q^5 - 20 * q^6 + 7 * q^7 + 8 * q^8 + 73 * q^9 - 28 * q^10 - 11 * q^11 - 40 * q^12 - 16 * q^13 + 14 * q^14 + 140 * q^15 + 16 * q^16 + 108 * q^17 + 146 * q^18 + 116 * q^19 - 56 * q^20 - 70 * q^21 - 22 * q^22 + 68 * q^23 - 80 * q^24 + 71 * q^25 - 32 * q^26 - 460 * q^27 + 28 * q^28 + 122 * q^29 + 280 * q^30 - 262 * q^31 + 32 * q^32 + 110 * q^33 + 216 * q^34 - 98 * q^35 + 292 * q^36 + 130 * q^37 + 232 * q^38 + 160 * q^39 - 112 * q^40 + 204 * q^41 - 140 * q^42 - 396 * q^43 - 44 * q^44 - 1022 * q^45 + 136 * q^46 + 166 * q^47 - 160 * q^48 + 49 * q^49 + 142 * q^50 - 1080 * q^51 - 64 * q^52 + 442 * q^53 - 920 * q^54 + 154 * q^55 + 56 * q^56 - 1160 * q^57 + 244 * q^58 + 702 * q^59 + 560 * q^60 + 196 * q^61 - 524 * q^62 + 511 * q^63 + 64 * q^64 + 224 * q^65 + 220 * q^66 - 416 * q^67 + 432 * q^68 - 680 * q^69 - 196 * q^70 + 492 * q^71 + 584 * q^72 + 408 * q^73 + 260 * q^74 - 710 * q^75 + 464 * q^76 - 77 * q^77 + 320 * q^78 + 600 * q^79 - 224 * q^80 + 2629 * q^81 + 408 * q^82 - 1212 * q^83 - 280 * q^84 - 1512 * q^85 - 792 * q^86 - 1220 * q^87 - 88 * q^88 + 1146 * q^89 - 2044 * q^90 - 112 * q^91 + 272 * q^92 + 2620 * q^93 + 332 * q^94 - 1624 * q^95 - 320 * q^96 - 482 * q^97 + 98 * q^98 - 803 * 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 −10.0000 4.00000 −14.0000 −20.0000 7.00000 8.00000 73.0000 −28.0000
 $$n$$: 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.c 1
3.b odd 2 1 1386.4.a.g 1
4.b odd 2 1 1232.4.a.i 1
7.b odd 2 1 1078.4.a.h 1
11.b odd 2 1 1694.4.a.a 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 10$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(154))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 10$$
$5$ $$T + 14$$
$7$ $$T - 7$$
$11$ $$T + 11$$
$13$ $$T + 16$$
$17$ $$T - 108$$
$19$ $$T - 116$$
$23$ $$T - 68$$
$29$ $$T - 122$$
$31$ $$T + 262$$
$37$ $$T - 130$$
$41$ $$T - 204$$
$43$ $$T + 396$$
$47$ $$T - 166$$
$53$ $$T - 442$$
$59$ $$T - 702$$
$61$ $$T - 196$$
$67$ $$T + 416$$
$71$ $$T - 492$$
$73$ $$T - 408$$
$79$ $$T - 600$$
$83$ $$T + 1212$$
$89$ $$T - 1146$$
$97$ $$T + 482$$