# Properties

 Label 154.2.a.b Level $154$ Weight $2$ Character orbit 154.a Self dual yes Analytic conductor $1.230$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$1.22969619113$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} + 2 q^{3} + q^{4} + 2 q^{5} - 2 q^{6} - q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + 2 * q^3 + q^4 + 2 * q^5 - 2 * q^6 - q^7 - q^8 + q^9 $$q - q^{2} + 2 q^{3} + q^{4} + 2 q^{5} - 2 q^{6} - q^{7} - q^{8} + q^{9} - 2 q^{10} + q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 4 q^{15} + q^{16} - q^{18} + 4 q^{19} + 2 q^{20} - 2 q^{21} - q^{22} + 4 q^{23} - 2 q^{24} - q^{25} + 4 q^{26} - 4 q^{27} - q^{28} + 2 q^{29} - 4 q^{30} - 10 q^{31} - q^{32} + 2 q^{33} - 2 q^{35} + q^{36} - 6 q^{37} - 4 q^{38} - 8 q^{39} - 2 q^{40} + 2 q^{42} - 4 q^{43} + q^{44} + 2 q^{45} - 4 q^{46} + 10 q^{47} + 2 q^{48} + q^{49} + q^{50} - 4 q^{52} - 14 q^{53} + 4 q^{54} + 2 q^{55} + q^{56} + 8 q^{57} - 2 q^{58} + 10 q^{59} + 4 q^{60} - 8 q^{61} + 10 q^{62} - q^{63} + q^{64} - 8 q^{65} - 2 q^{66} + 8 q^{67} + 8 q^{69} + 2 q^{70} - 4 q^{71} - q^{72} + 4 q^{73} + 6 q^{74} - 2 q^{75} + 4 q^{76} - q^{77} + 8 q^{78} + 16 q^{79} + 2 q^{80} - 11 q^{81} + 4 q^{83} - 2 q^{84} + 4 q^{86} + 4 q^{87} - q^{88} + 10 q^{89} - 2 q^{90} + 4 q^{91} + 4 q^{92} - 20 q^{93} - 10 q^{94} + 8 q^{95} - 2 q^{96} + 6 q^{97} - q^{98} + q^{99}+O(q^{100})$$ q - q^2 + 2 * q^3 + q^4 + 2 * q^5 - 2 * q^6 - q^7 - q^8 + q^9 - 2 * q^10 + q^11 + 2 * q^12 - 4 * q^13 + q^14 + 4 * q^15 + q^16 - q^18 + 4 * q^19 + 2 * q^20 - 2 * q^21 - q^22 + 4 * q^23 - 2 * q^24 - q^25 + 4 * q^26 - 4 * q^27 - q^28 + 2 * q^29 - 4 * q^30 - 10 * q^31 - q^32 + 2 * q^33 - 2 * q^35 + q^36 - 6 * q^37 - 4 * q^38 - 8 * q^39 - 2 * q^40 + 2 * q^42 - 4 * q^43 + q^44 + 2 * q^45 - 4 * q^46 + 10 * q^47 + 2 * q^48 + q^49 + q^50 - 4 * q^52 - 14 * q^53 + 4 * q^54 + 2 * q^55 + q^56 + 8 * q^57 - 2 * q^58 + 10 * q^59 + 4 * q^60 - 8 * q^61 + 10 * q^62 - q^63 + q^64 - 8 * q^65 - 2 * q^66 + 8 * q^67 + 8 * q^69 + 2 * q^70 - 4 * q^71 - q^72 + 4 * q^73 + 6 * q^74 - 2 * q^75 + 4 * q^76 - q^77 + 8 * q^78 + 16 * q^79 + 2 * q^80 - 11 * q^81 + 4 * q^83 - 2 * q^84 + 4 * q^86 + 4 * q^87 - q^88 + 10 * q^89 - 2 * q^90 + 4 * q^91 + 4 * q^92 - 20 * q^93 - 10 * q^94 + 8 * q^95 - 2 * q^96 + 6 * q^97 - q^98 + 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
−1.00000 2.00000 1.00000 2.00000 −2.00000 −1.00000 −1.00000 1.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 154.2.a.b 1
3.b odd 2 1 1386.2.a.f 1
4.b odd 2 1 1232.2.a.c 1
5.b even 2 1 3850.2.a.o 1
5.c odd 4 2 3850.2.c.d 2
7.b odd 2 1 1078.2.a.b 1
7.c even 3 2 1078.2.e.h 2
7.d odd 6 2 1078.2.e.l 2
8.b even 2 1 4928.2.a.d 1
8.d odd 2 1 4928.2.a.bf 1
11.b odd 2 1 1694.2.a.i 1
21.c even 2 1 9702.2.a.bz 1
28.d even 2 1 8624.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.b 1 1.a even 1 1 trivial
1078.2.a.b 1 7.b odd 2 1
1078.2.e.h 2 7.c even 3 2
1078.2.e.l 2 7.d odd 6 2
1232.2.a.c 1 4.b odd 2 1
1386.2.a.f 1 3.b odd 2 1
1694.2.a.i 1 11.b odd 2 1
3850.2.a.o 1 5.b even 2 1
3850.2.c.d 2 5.c odd 4 2
4928.2.a.d 1 8.b even 2 1
4928.2.a.bf 1 8.d odd 2 1
8624.2.a.z 1 28.d even 2 1
9702.2.a.bz 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(154))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{5} - 2$$ T5 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 2$$
$5$ $$T - 2$$
$7$ $$T + 1$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T - 4$$
$23$ $$T - 4$$
$29$ $$T - 2$$
$31$ $$T + 10$$
$37$ $$T + 6$$
$41$ $$T$$
$43$ $$T + 4$$
$47$ $$T - 10$$
$53$ $$T + 14$$
$59$ $$T - 10$$
$61$ $$T + 8$$
$67$ $$T - 8$$
$71$ $$T + 4$$
$73$ $$T - 4$$
$79$ $$T - 16$$
$83$ $$T - 4$$
$89$ $$T - 10$$
$97$ $$T - 6$$
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