# Properties

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

# Related objects

## 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 4 q^{5} - q^{7} - q^{8} - 3 q^{9} + O(q^{10})$$ $$q - q^{2} + q^{4} - 4 q^{5} - q^{7} - q^{8} - 3 q^{9} + 4 q^{10} - q^{11} + 2 q^{13} + q^{14} + q^{16} - 4 q^{17} + 3 q^{18} - 6 q^{19} - 4 q^{20} + q^{22} + 4 q^{23} + 11 q^{25} - 2 q^{26} - q^{28} - 2 q^{29} - 2 q^{31} - q^{32} + 4 q^{34} + 4 q^{35} - 3 q^{36} + 10 q^{37} + 6 q^{38} + 4 q^{40} + 4 q^{41} - 8 q^{43} - q^{44} + 12 q^{45} - 4 q^{46} + 2 q^{47} + q^{49} - 11 q^{50} + 2 q^{52} + 6 q^{53} + 4 q^{55} + q^{56} + 2 q^{58} - 12 q^{59} - 14 q^{61} + 2 q^{62} + 3 q^{63} + q^{64} - 8 q^{65} - 12 q^{67} - 4 q^{68} - 4 q^{70} - 8 q^{71} + 3 q^{72} + 4 q^{73} - 10 q^{74} - 6 q^{76} + q^{77} - 4 q^{80} + 9 q^{81} - 4 q^{82} - 6 q^{83} + 16 q^{85} + 8 q^{86} + q^{88} - 6 q^{89} - 12 q^{90} - 2 q^{91} + 4 q^{92} - 2 q^{94} + 24 q^{95} - 14 q^{97} - q^{98} + 3 q^{99} + O(q^{100})$$

## 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 0 1.00000 −4.00000 0 −1.00000 −1.00000 −3.00000 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
$$2$$ $$1$$
$$7$$ $$1$$
$$11$$ $$1$$

## 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.a 1
3.b odd 2 1 1386.2.a.l 1
4.b odd 2 1 1232.2.a.e 1
5.b even 2 1 3850.2.a.u 1
5.c odd 4 2 3850.2.c.j 2
7.b odd 2 1 1078.2.a.d 1
7.c even 3 2 1078.2.e.j 2
7.d odd 6 2 1078.2.e.i 2
8.b even 2 1 4928.2.a.v 1
8.d odd 2 1 4928.2.a.w 1
11.b odd 2 1 1694.2.a.g 1
21.c even 2 1 9702.2.a.ba 1
28.d even 2 1 8624.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.a 1 1.a even 1 1 trivial
1078.2.a.d 1 7.b odd 2 1
1078.2.e.i 2 7.d odd 6 2
1078.2.e.j 2 7.c even 3 2
1232.2.a.e 1 4.b odd 2 1
1386.2.a.l 1 3.b odd 2 1
1694.2.a.g 1 11.b odd 2 1
3850.2.a.u 1 5.b even 2 1
3850.2.c.j 2 5.c odd 4 2
4928.2.a.v 1 8.b even 2 1
4928.2.a.w 1 8.d odd 2 1
8624.2.a.r 1 28.d even 2 1
9702.2.a.ba 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(154))$$:

 $$T_{3}$$ $$T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$4 + T$$
$7$ $$1 + T$$
$11$ $$1 + T$$
$13$ $$-2 + T$$
$17$ $$4 + T$$
$19$ $$6 + T$$
$23$ $$-4 + T$$
$29$ $$2 + T$$
$31$ $$2 + T$$
$37$ $$-10 + T$$
$41$ $$-4 + T$$
$43$ $$8 + T$$
$47$ $$-2 + T$$
$53$ $$-6 + T$$
$59$ $$12 + T$$
$61$ $$14 + T$$
$67$ $$12 + T$$
$71$ $$8 + T$$
$73$ $$-4 + T$$
$79$ $$T$$
$83$ $$6 + T$$
$89$ $$6 + T$$
$97$ $$14 + T$$