# Properties

 Label 154.2.e.e Level $154$ Weight $2$ Character orbit 154.e Analytic conductor $1.230$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{5} + ( -1 + \beta_{3} ) q^{6} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + q^{8} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{5} + ( -1 + \beta_{3} ) q^{6} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + q^{8} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{10} + ( -1 - \beta_{2} ) q^{11} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{12} + ( -1 + 2 \beta_{3} ) q^{13} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{14} -\beta_{3} q^{15} + \beta_{2} q^{16} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{17} -2 \beta_{1} q^{18} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{19} + ( -2 - \beta_{3} ) q^{20} + ( 1 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{21} + q^{22} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{23} + ( 1 + \beta_{1} + \beta_{2} ) q^{24} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{25} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{26} + ( -1 - \beta_{3} ) q^{27} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{28} + ( -3 - 4 \beta_{3} ) q^{29} + ( \beta_{1} + \beta_{3} ) q^{30} + ( 4 + 4 \beta_{2} ) q^{31} + ( -1 - \beta_{2} ) q^{32} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{33} + ( -2 - 4 \beta_{3} ) q^{34} + ( -3 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{35} -2 \beta_{3} q^{36} + ( -\beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{37} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{38} + ( -5 - 3 \beta_{1} - 5 \beta_{2} ) q^{39} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{40} + ( -4 + \beta_{3} ) q^{41} + ( -4 - 2 \beta_{1} - 3 \beta_{2} ) q^{42} + 4 \beta_{3} q^{43} + \beta_{2} q^{44} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{45} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{46} + ( -6 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{47} + ( -1 + \beta_{3} ) q^{48} + ( 5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{49} + ( 1 + 4 \beta_{3} ) q^{50} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{51} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{52} + ( 2 - 7 \beta_{1} + 2 \beta_{2} ) q^{53} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{54} + ( -2 - \beta_{3} ) q^{55} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{56} -\beta_{3} q^{57} + ( 4 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{58} + ( 7 - \beta_{1} + 7 \beta_{2} ) q^{59} -\beta_{1} q^{60} + ( 2 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} ) q^{61} -4 q^{62} + ( -4 - 2 \beta_{1} + 4 \beta_{2} ) q^{63} + q^{64} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{65} + ( 1 + \beta_{1} + \beta_{2} ) q^{66} + ( -7 + 3 \beta_{1} - 7 \beta_{2} ) q^{67} + ( 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{68} + ( 8 - 5 \beta_{3} ) q^{69} + ( -4 + 4 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{70} + ( -4 + 5 \beta_{3} ) q^{71} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{72} + ( 8 + \beta_{1} + 8 \beta_{2} ) q^{73} + ( 8 + \beta_{1} + 8 \beta_{2} ) q^{74} + ( 3 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} ) q^{75} + ( -2 - \beta_{3} ) q^{76} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{77} + ( 5 - 3 \beta_{3} ) q^{78} + ( -3 \beta_{1} - 9 \beta_{2} - 3 \beta_{3} ) q^{79} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{80} + ( 1 + 6 \beta_{1} + \beta_{2} ) q^{81} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{82} + ( 2 + 10 \beta_{3} ) q^{83} + ( 3 - \beta_{2} - 2 \beta_{3} ) q^{84} + ( 12 + 10 \beta_{3} ) q^{85} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{86} + ( 5 + \beta_{1} + 5 \beta_{2} ) q^{87} + ( -1 - \beta_{2} ) q^{88} + ( 6 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{89} + ( 4 + 4 \beta_{3} ) q^{90} + ( -8 - 3 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{91} + ( -2 + 3 \beta_{3} ) q^{92} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{93} + ( 2 + 6 \beta_{1} + 2 \beta_{2} ) q^{94} + ( -6 + 4 \beta_{1} - 6 \beta_{2} ) q^{95} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{96} + ( -1 + 2 \beta_{3} ) q^{97} + ( -5 - 4 \beta_{1} - 2 \beta_{3} ) q^{98} -2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 2q^{3} - 2q^{4} + 4q^{5} - 4q^{6} - 2q^{7} + 4q^{8} + O(q^{10})$$ $$4q - 2q^{2} + 2q^{3} - 2q^{4} + 4q^{5} - 4q^{6} - 2q^{7} + 4q^{8} + 4q^{10} - 2q^{11} + 2q^{12} - 4q^{13} - 2q^{14} - 2q^{16} + 4q^{17} + 4q^{19} - 8q^{20} - 4q^{21} + 4q^{22} + 4q^{23} + 2q^{24} - 2q^{25} + 2q^{26} - 4q^{27} + 4q^{28} - 12q^{29} + 8q^{31} - 2q^{32} + 2q^{33} - 8q^{34} - 8q^{35} + 16q^{37} + 4q^{38} - 10q^{39} + 4q^{40} - 16q^{41} - 10q^{42} - 2q^{44} - 8q^{45} + 4q^{46} + 4q^{47} - 4q^{48} + 10q^{49} + 4q^{50} + 12q^{51} + 2q^{52} + 4q^{53} + 2q^{54} - 8q^{55} - 2q^{56} + 6q^{58} + 14q^{59} - 18q^{61} - 16q^{62} - 24q^{63} + 4q^{64} + 4q^{65} + 2q^{66} - 14q^{67} + 4q^{68} + 32q^{69} - 8q^{70} - 16q^{71} + 16q^{73} + 16q^{74} - 14q^{75} - 8q^{76} + 4q^{77} + 20q^{78} + 18q^{79} + 4q^{80} + 2q^{81} + 8q^{82} + 8q^{83} + 14q^{84} + 48q^{85} + 10q^{87} - 2q^{88} - 8q^{89} + 16q^{90} - 22q^{91} - 8q^{92} - 8q^{93} + 4q^{94} - 12q^{95} + 2q^{96} - 4q^{97} - 20q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.500000 + 0.866025i −0.207107 0.358719i −0.500000 0.866025i 1.70711 2.95680i 0.414214 −2.62132 0.358719i 1.00000 1.41421 2.44949i 1.70711 + 2.95680i
23.2 −0.500000 + 0.866025i 1.20711 + 2.09077i −0.500000 0.866025i 0.292893 0.507306i −2.41421 1.62132 + 2.09077i 1.00000 −1.41421 + 2.44949i 0.292893 + 0.507306i
67.1 −0.500000 0.866025i −0.207107 + 0.358719i −0.500000 + 0.866025i 1.70711 + 2.95680i 0.414214 −2.62132 + 0.358719i 1.00000 1.41421 + 2.44949i 1.70711 2.95680i
67.2 −0.500000 0.866025i 1.20711 2.09077i −0.500000 + 0.866025i 0.292893 + 0.507306i −2.41421 1.62132 2.09077i 1.00000 −1.41421 2.44949i 0.292893 0.507306i
 $$n$$: 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.e 4
3.b odd 2 1 1386.2.k.t 4
4.b odd 2 1 1232.2.q.f 4
7.b odd 2 1 1078.2.e.m 4
7.c even 3 1 inner 154.2.e.e 4
7.c even 3 1 1078.2.a.t 2
7.d odd 6 1 1078.2.a.x 2
7.d odd 6 1 1078.2.e.m 4
21.g even 6 1 9702.2.a.ch 2
21.h odd 6 1 1386.2.k.t 4
21.h odd 6 1 9702.2.a.cx 2
28.f even 6 1 8624.2.a.bh 2
28.g odd 6 1 1232.2.q.f 4
28.g odd 6 1 8624.2.a.cc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 1.a even 1 1 trivial
154.2.e.e 4 7.c even 3 1 inner
1078.2.a.t 2 7.c even 3 1
1078.2.a.x 2 7.d odd 6 1
1078.2.e.m 4 7.b odd 2 1
1078.2.e.m 4 7.d odd 6 1
1232.2.q.f 4 4.b odd 2 1
1232.2.q.f 4 28.g odd 6 1
1386.2.k.t 4 3.b odd 2 1
1386.2.k.t 4 21.h odd 6 1
8624.2.a.bh 2 28.f even 6 1
8624.2.a.cc 2 28.g odd 6 1
9702.2.a.ch 2 21.g even 6 1
9702.2.a.cx 2 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 $$S_{2}^{\mathrm{new}}(154, [\chi])$$:

 $$T_{3}^{4} - 2 T_{3}^{3} + 5 T_{3}^{2} + 2 T_{3} + 1$$ $$T_{13}^{2} + 2 T_{13} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$4 - 8 T + 14 T^{2} - 4 T^{3} + T^{4}$$
$7$ $$49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$( 1 + T + T^{2} )^{2}$$
$13$ $$( -7 + 2 T + T^{2} )^{2}$$
$17$ $$784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$4 - 8 T + 14 T^{2} - 4 T^{3} + T^{4}$$
$23$ $$196 + 56 T + 30 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$( -23 + 6 T + T^{2} )^{2}$$
$31$ $$( 16 - 4 T + T^{2} )^{2}$$
$37$ $$3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4}$$
$41$ $$( 14 + 8 T + T^{2} )^{2}$$
$43$ $$( -32 + T^{2} )^{2}$$
$47$ $$4624 + 272 T + 84 T^{2} - 4 T^{3} + T^{4}$$
$53$ $$8836 + 376 T + 110 T^{2} - 4 T^{3} + T^{4}$$
$59$ $$2209 - 658 T + 149 T^{2} - 14 T^{3} + T^{4}$$
$61$ $$5329 + 1314 T + 251 T^{2} + 18 T^{3} + T^{4}$$
$67$ $$961 + 434 T + 165 T^{2} + 14 T^{3} + T^{4}$$
$71$ $$( -34 + 8 T + T^{2} )^{2}$$
$73$ $$3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4}$$
$79$ $$3969 - 1134 T + 261 T^{2} - 18 T^{3} + T^{4}$$
$83$ $$( -196 - 4 T + T^{2} )^{2}$$
$89$ $$3136 - 448 T + 120 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$( -7 + 2 T + T^{2} )^{2}$$