Properties

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

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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: \(1\)
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.

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

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
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i −1.50000 2.59808i −0.500000 0.866025i −1.00000 + 1.73205i 3.00000 −2.50000 + 0.866025i 1.00000 −3.00000 + 5.19615i −1.00000 1.73205i
67.1 −0.500000 0.866025i −1.50000 + 2.59808i −0.500000 + 0.866025i −1.00000 1.73205i 3.00000 −2.50000 0.866025i 1.00000 −3.00000 5.19615i −1.00000 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

Hecke kernels

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

\( T_{3}^{2} + 3 T_{3} + 9 \)
\( T_{13} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 9 + 3 T + T^{2} \)
$5$ \( 4 + 2 T + T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( 7 + T )^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 64 - 8 T + T^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 16 + 4 T + T^{2} \)
$41$ \( ( -4 + T )^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( 4 + 2 T + T^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( 9 + 3 T + T^{2} \)
$61$ \( 1 + T + T^{2} \)
$67$ \( 81 + 9 T + T^{2} \)
$71$ \( ( 2 + T )^{2} \)
$73$ \( 16 + 4 T + T^{2} \)
$79$ \( 81 + 9 T + T^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( -7 + T )^{2} \)
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