# Properties

 Label 154.2.e.b Level $154$ Weight $2$ Character orbit 154.e Analytic conductor $1.230$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 154.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.22969619113$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + q^{6} + ( 2 + \zeta_{6} ) q^{7} + q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + q^{6} + ( 2 + \zeta_{6} ) q^{7} + q^{8} + 2 \zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{11} -\zeta_{6} q^{12} + 5 q^{13} + ( 1 - 3 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + ( 2 - 2 \zeta_{6} ) q^{18} -2 \zeta_{6} q^{19} + ( -3 + 2 \zeta_{6} ) q^{21} - q^{22} -6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{24} + ( 5 - 5 \zeta_{6} ) q^{25} -5 \zeta_{6} q^{26} -5 q^{27} + ( -3 + 2 \zeta_{6} ) q^{28} + 3 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + \zeta_{6} q^{33} + 6 q^{34} -2 q^{36} -2 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{38} + ( -5 + 5 \zeta_{6} ) q^{39} -6 q^{41} + ( 2 + \zeta_{6} ) q^{42} -4 q^{43} + \zeta_{6} q^{44} + ( -6 + 6 \zeta_{6} ) q^{46} -6 \zeta_{6} q^{47} + q^{48} + ( 3 + 5 \zeta_{6} ) q^{49} -5 q^{50} -6 \zeta_{6} q^{51} + ( -5 + 5 \zeta_{6} ) q^{52} + ( 12 - 12 \zeta_{6} ) q^{53} + 5 \zeta_{6} q^{54} + ( 2 + \zeta_{6} ) q^{56} + 2 q^{57} -3 \zeta_{6} q^{58} + ( 3 - 3 \zeta_{6} ) q^{59} + 7 \zeta_{6} q^{61} + 8 q^{62} + ( -2 + 6 \zeta_{6} ) q^{63} + q^{64} + ( 1 - \zeta_{6} ) q^{66} + ( 13 - 13 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} + 6 q^{69} -12 q^{71} + 2 \zeta_{6} q^{72} + ( 10 - 10 \zeta_{6} ) q^{73} + ( -2 + 2 \zeta_{6} ) q^{74} + 5 \zeta_{6} q^{75} + 2 q^{76} + ( 3 - 2 \zeta_{6} ) q^{77} + 5 q^{78} + \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 6 \zeta_{6} q^{82} + 6 q^{83} + ( 1 - 3 \zeta_{6} ) q^{84} + 4 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{87} + ( 1 - \zeta_{6} ) q^{88} -6 \zeta_{6} q^{89} + ( 10 + 5 \zeta_{6} ) q^{91} + 6 q^{92} -8 \zeta_{6} q^{93} + ( -6 + 6 \zeta_{6} ) q^{94} -\zeta_{6} q^{96} -13 q^{97} + ( 5 - 8 \zeta_{6} ) q^{98} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} - q^{4} + 2q^{6} + 5q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} - q^{4} + 2q^{6} + 5q^{7} + 2q^{8} + 2q^{9} + q^{11} - q^{12} + 10q^{13} - q^{14} - q^{16} - 6q^{17} + 2q^{18} - 2q^{19} - 4q^{21} - 2q^{22} - 6q^{23} - q^{24} + 5q^{25} - 5q^{26} - 10q^{27} - 4q^{28} + 6q^{29} - 8q^{31} - q^{32} + q^{33} + 12q^{34} - 4q^{36} - 2q^{37} - 2q^{38} - 5q^{39} - 12q^{41} + 5q^{42} - 8q^{43} + q^{44} - 6q^{46} - 6q^{47} + 2q^{48} + 11q^{49} - 10q^{50} - 6q^{51} - 5q^{52} + 12q^{53} + 5q^{54} + 5q^{56} + 4q^{57} - 3q^{58} + 3q^{59} + 7q^{61} + 16q^{62} + 2q^{63} + 2q^{64} + q^{66} + 13q^{67} - 6q^{68} + 12q^{69} - 24q^{71} + 2q^{72} + 10q^{73} - 2q^{74} + 5q^{75} + 4q^{76} + 4q^{77} + 10q^{78} + q^{79} - q^{81} + 6q^{82} + 12q^{83} - q^{84} + 4q^{86} - 3q^{87} + q^{88} - 6q^{89} + 25q^{91} + 12q^{92} - 8q^{93} - 6q^{94} - q^{96} - 26q^{97} + 2q^{98} + 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/154\mathbb{Z}\right)^\times$$.

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 1.00000 2.50000 0.866025i 1.00000 1.00000 1.73205i 0
67.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 2.50000 + 0.866025i 1.00000 1.00000 + 1.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.2.e.b 2
3.b odd 2 1 1386.2.k.n 2
4.b odd 2 1 1232.2.q.c 2
7.b odd 2 1 1078.2.e.d 2
7.c even 3 1 inner 154.2.e.b 2
7.c even 3 1 1078.2.a.k 1
7.d odd 6 1 1078.2.a.i 1
7.d odd 6 1 1078.2.e.d 2
21.g even 6 1 9702.2.a.l 1
21.h odd 6 1 1386.2.k.n 2
21.h odd 6 1 9702.2.a.o 1
28.f even 6 1 8624.2.a.t 1
28.g odd 6 1 1232.2.q.c 2
28.g odd 6 1 8624.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.b 2 1.a even 1 1 trivial
154.2.e.b 2 7.c even 3 1 inner
1078.2.a.i 1 7.d odd 6 1
1078.2.a.k 1 7.c even 3 1
1078.2.e.d 2 7.b odd 2 1
1078.2.e.d 2 7.d odd 6 1
1232.2.q.c 2 4.b odd 2 1
1232.2.q.c 2 28.g odd 6 1
1386.2.k.n 2 3.b odd 2 1
1386.2.k.n 2 21.h odd 6 1
8624.2.a.l 1 28.g odd 6 1
8624.2.a.t 1 28.f even 6 1
9702.2.a.l 1 21.g even 6 1
9702.2.a.o 1 21.h odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} + T_{3} + 1$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$144 - 12 T + T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$169 - 13 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$100 - 10 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( 13 + T )^{2}$$