Properties

Label 184.2.a.e
Level $184$
Weight $2$
Character orbit 184.a
Self dual yes
Analytic conductor $1.469$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 184 = 2^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 184.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.46924739719\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + 2 q^{5} + ( 1 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} + 2 q^{5} + ( 1 + \beta ) q^{9} + 2 \beta q^{11} + ( 2 + \beta ) q^{13} -2 \beta q^{15} + ( 2 - 2 \beta ) q^{17} + 2 \beta q^{19} - q^{23} - q^{25} + ( -4 + \beta ) q^{27} + ( 2 - \beta ) q^{29} + ( -4 - \beta ) q^{31} + ( -8 - 2 \beta ) q^{33} + ( 2 - 4 \beta ) q^{37} + ( -4 - 3 \beta ) q^{39} + ( -2 + 5 \beta ) q^{41} -8 q^{43} + ( 2 + 2 \beta ) q^{45} + ( 4 + 3 \beta ) q^{47} -7 q^{49} + 8 q^{51} + 2 q^{53} + 4 \beta q^{55} + ( -8 - 2 \beta ) q^{57} + ( 4 - 4 \beta ) q^{59} + ( 2 + 4 \beta ) q^{61} + ( 4 + 2 \beta ) q^{65} -2 \beta q^{67} + \beta q^{69} + ( 12 - \beta ) q^{71} + ( -10 + 3 \beta ) q^{73} + \beta q^{75} -2 \beta q^{79} -7 q^{81} + ( 8 - 4 \beta ) q^{83} + ( 4 - 4 \beta ) q^{85} + ( 4 - \beta ) q^{87} + ( 2 - 6 \beta ) q^{89} + ( 4 + 5 \beta ) q^{93} + 4 \beta q^{95} + ( 2 - 6 \beta ) q^{97} + ( 8 + 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 4q^{5} + 3q^{9} + O(q^{10}) \) \( 2q - q^{3} + 4q^{5} + 3q^{9} + 2q^{11} + 5q^{13} - 2q^{15} + 2q^{17} + 2q^{19} - 2q^{23} - 2q^{25} - 7q^{27} + 3q^{29} - 9q^{31} - 18q^{33} - 11q^{39} + q^{41} - 16q^{43} + 6q^{45} + 11q^{47} - 14q^{49} + 16q^{51} + 4q^{53} + 4q^{55} - 18q^{57} + 4q^{59} + 8q^{61} + 10q^{65} - 2q^{67} + q^{69} + 23q^{71} - 17q^{73} + q^{75} - 2q^{79} - 14q^{81} + 12q^{83} + 4q^{85} + 7q^{87} - 2q^{89} + 13q^{93} + 4q^{95} - 2q^{97} + 20q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
0 −2.56155 0 2.00000 0 0 0 3.56155 0
1.2 0 1.56155 0 2.00000 0 0 0 −0.561553 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(1\)

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 184.2.a.e 2
3.b odd 2 1 1656.2.a.j 2
4.b odd 2 1 368.2.a.i 2
5.b even 2 1 4600.2.a.s 2
5.c odd 4 2 4600.2.e.o 4
7.b odd 2 1 9016.2.a.w 2
8.b even 2 1 1472.2.a.u 2
8.d odd 2 1 1472.2.a.p 2
12.b even 2 1 3312.2.a.t 2
20.d odd 2 1 9200.2.a.br 2
23.b odd 2 1 4232.2.a.o 2
92.b even 2 1 8464.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.a.e 2 1.a even 1 1 trivial
368.2.a.i 2 4.b odd 2 1
1472.2.a.p 2 8.d odd 2 1
1472.2.a.u 2 8.b even 2 1
1656.2.a.j 2 3.b odd 2 1
3312.2.a.t 2 12.b even 2 1
4232.2.a.o 2 23.b odd 2 1
4600.2.a.s 2 5.b even 2 1
4600.2.e.o 4 5.c odd 4 2
8464.2.a.bd 2 92.b even 2 1
9016.2.a.w 2 7.b odd 2 1
9200.2.a.br 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(184))\):

\( T_{3}^{2} + T_{3} - 4 \)
\( T_{5} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 + T + T^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -16 - 2 T + T^{2} \)
$13$ \( 2 - 5 T + T^{2} \)
$17$ \( -16 - 2 T + T^{2} \)
$19$ \( -16 - 2 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -2 - 3 T + T^{2} \)
$31$ \( 16 + 9 T + T^{2} \)
$37$ \( -68 + T^{2} \)
$41$ \( -106 - T + T^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( -8 - 11 T + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( -64 - 4 T + T^{2} \)
$61$ \( -52 - 8 T + T^{2} \)
$67$ \( -16 + 2 T + T^{2} \)
$71$ \( 128 - 23 T + T^{2} \)
$73$ \( 34 + 17 T + T^{2} \)
$79$ \( -16 + 2 T + T^{2} \)
$83$ \( -32 - 12 T + T^{2} \)
$89$ \( -152 + 2 T + T^{2} \)
$97$ \( -152 + 2 T + T^{2} \)
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