Properties

Label 184.1.e.b
Level $184$
Weight $1$
Character orbit 184.e
Analytic conductor $0.092$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -23
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 184 = 2^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 184.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0918279623245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.270848.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{2} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{3} -\zeta_{6} q^{4} + ( 1 + \zeta_{6} ) q^{6} - q^{8} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{2} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{3} -\zeta_{6} q^{4} + ( 1 + \zeta_{6} ) q^{6} - q^{8} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{9} + ( 1 - \zeta_{6}^{2} ) q^{12} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{13} + \zeta_{6}^{2} q^{16} + ( -1 + \zeta_{6} + \zeta_{6}^{2} ) q^{18} - q^{23} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{24} - q^{25} + ( -1 - \zeta_{6} ) q^{26} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{27} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{29} + q^{31} + \zeta_{6} q^{32} + ( 1 + \zeta_{6} + \zeta_{6}^{2} ) q^{36} + ( 2 + \zeta_{6} - \zeta_{6}^{2} ) q^{39} + q^{41} + \zeta_{6}^{2} q^{46} - q^{47} + ( -1 - \zeta_{6} ) q^{48} + q^{49} + \zeta_{6}^{2} q^{50} + ( -1 + \zeta_{6}^{2} ) q^{52} + ( -1 - \zeta_{6} ) q^{54} + ( 1 + \zeta_{6} ) q^{58} -\zeta_{6}^{2} q^{62} + q^{64} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{69} - q^{71} + ( 1 + \zeta_{6} - \zeta_{6}^{2} ) q^{72} + q^{73} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{75} + ( 1 - \zeta_{6} - 2 \zeta_{6}^{2} ) q^{78} + q^{81} -\zeta_{6}^{2} q^{82} + ( -2 - \zeta_{6} + \zeta_{6}^{2} ) q^{87} + \zeta_{6} q^{92} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{93} + \zeta_{6}^{2} q^{94} + ( -1 + \zeta_{6}^{2} ) q^{96} -\zeta_{6}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + 3q^{6} - 2q^{8} - 4q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + 3q^{6} - 2q^{8} - 4q^{9} + 3q^{12} - q^{16} - 2q^{18} - 2q^{23} - 2q^{25} - 3q^{26} + 2q^{31} + q^{32} + 2q^{36} + 6q^{39} + 2q^{41} - q^{46} - 2q^{47} - 3q^{48} + 2q^{49} - q^{50} - 3q^{52} - 3q^{54} + 3q^{58} + q^{62} + 2q^{64} - 2q^{71} + 4q^{72} + 2q^{73} + 3q^{78} + 2q^{81} + q^{82} - 6q^{87} + q^{92} - q^{94} - 3q^{96} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/184\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(93\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 1.73205i −0.500000 0.866025i 0 1.50000 + 0.866025i 0 −1.00000 −2.00000 0
45.2 0.500000 + 0.866025i 1.73205i −0.500000 + 0.866025i 0 1.50000 0.866025i 0 −1.00000 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
8.b even 2 1 inner
184.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 184.1.e.b 2
3.b odd 2 1 1656.1.l.c 2
4.b odd 2 1 736.1.e.b 2
8.b even 2 1 inner 184.1.e.b 2
8.d odd 2 1 736.1.e.b 2
23.b odd 2 1 CM 184.1.e.b 2
24.h odd 2 1 1656.1.l.c 2
69.c even 2 1 1656.1.l.c 2
92.b even 2 1 736.1.e.b 2
184.e odd 2 1 inner 184.1.e.b 2
184.h even 2 1 736.1.e.b 2
552.b even 2 1 1656.1.l.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.1.e.b 2 1.a even 1 1 trivial
184.1.e.b 2 8.b even 2 1 inner
184.1.e.b 2 23.b odd 2 1 CM
184.1.e.b 2 184.e odd 2 1 inner
736.1.e.b 2 4.b odd 2 1
736.1.e.b 2 8.d odd 2 1
736.1.e.b 2 92.b even 2 1
736.1.e.b 2 184.h even 2 1
1656.1.l.c 2 3.b odd 2 1
1656.1.l.c 2 24.h odd 2 1
1656.1.l.c 2 69.c even 2 1
1656.1.l.c 2 552.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3 \) acting on \(S_{1}^{\mathrm{new}}(184, [\chi])\).

Hecke characteristic polynomials

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