Properties

Label 1840.2.a.p
Level $1840$
Weight $2$
Character orbit 1840.a
Self dual yes
Analytic conductor $14.692$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + ( 1 + \beta ) q^{7} -2 q^{9} +O(q^{10})\) \( q + q^{3} - q^{5} + ( 1 + \beta ) q^{7} -2 q^{9} + ( 1 - \beta ) q^{11} + ( -4 + \beta ) q^{13} - q^{15} + ( -2 - 2 \beta ) q^{17} + ( -1 - 3 \beta ) q^{19} + ( 1 + \beta ) q^{21} + q^{23} + q^{25} -5 q^{27} + ( -5 - 2 \beta ) q^{29} + ( -2 + \beta ) q^{31} + ( 1 - \beta ) q^{33} + ( -1 - \beta ) q^{35} + ( -3 + 3 \beta ) q^{37} + ( -4 + \beta ) q^{39} + ( -3 + 2 \beta ) q^{41} + ( 3 + 3 \beta ) q^{43} + 2 q^{45} + ( -5 + 2 \beta ) q^{47} + ( -1 + 2 \beta ) q^{49} + ( -2 - 2 \beta ) q^{51} -6 q^{53} + ( -1 + \beta ) q^{55} + ( -1 - 3 \beta ) q^{57} + 4 \beta q^{59} + ( 1 - 5 \beta ) q^{61} + ( -2 - 2 \beta ) q^{63} + ( 4 - \beta ) q^{65} + ( -3 + 3 \beta ) q^{67} + q^{69} + ( 4 - \beta ) q^{71} + 3 \beta q^{73} + q^{75} -4 q^{77} + ( -11 - \beta ) q^{79} + q^{81} + ( 2 - 2 \beta ) q^{83} + ( 2 + 2 \beta ) q^{85} + ( -5 - 2 \beta ) q^{87} + ( 5 + \beta ) q^{89} + ( 1 - 3 \beta ) q^{91} + ( -2 + \beta ) q^{93} + ( 1 + 3 \beta ) q^{95} + ( 5 + 5 \beta ) q^{97} + ( -2 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} + 2q^{7} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} + 2q^{7} - 4q^{9} + 2q^{11} - 8q^{13} - 2q^{15} - 4q^{17} - 2q^{19} + 2q^{21} + 2q^{23} + 2q^{25} - 10q^{27} - 10q^{29} - 4q^{31} + 2q^{33} - 2q^{35} - 6q^{37} - 8q^{39} - 6q^{41} + 6q^{43} + 4q^{45} - 10q^{47} - 2q^{49} - 4q^{51} - 12q^{53} - 2q^{55} - 2q^{57} + 2q^{61} - 4q^{63} + 8q^{65} - 6q^{67} + 2q^{69} + 8q^{71} + 2q^{75} - 8q^{77} - 22q^{79} + 2q^{81} + 4q^{83} + 4q^{85} - 10q^{87} + 10q^{89} + 2q^{91} - 4q^{93} + 2q^{95} + 10q^{97} - 4q^{99} + O(q^{100}) \)

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
−0.618034
1.61803
0 1.00000 0 −1.00000 0 −1.23607 0 −2.00000 0
1.2 0 1.00000 0 −1.00000 0 3.23607 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(-1\)

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 1840.2.a.p 2
4.b odd 2 1 115.2.a.b 2
5.b even 2 1 9200.2.a.bm 2
8.b even 2 1 7360.2.a.bf 2
8.d odd 2 1 7360.2.a.bt 2
12.b even 2 1 1035.2.a.m 2
20.d odd 2 1 575.2.a.g 2
20.e even 4 2 575.2.b.c 4
28.d even 2 1 5635.2.a.n 2
60.h even 2 1 5175.2.a.bb 2
92.b even 2 1 2645.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.a.b 2 4.b odd 2 1
575.2.a.g 2 20.d odd 2 1
575.2.b.c 4 20.e even 4 2
1035.2.a.m 2 12.b even 2 1
1840.2.a.p 2 1.a even 1 1 trivial
2645.2.a.d 2 92.b even 2 1
5175.2.a.bb 2 60.h even 2 1
5635.2.a.n 2 28.d even 2 1
7360.2.a.bf 2 8.b even 2 1
7360.2.a.bt 2 8.d odd 2 1
9200.2.a.bm 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3} - 1 \)
\( T_{7}^{2} - 2 T_{7} - 4 \)
\( T_{11}^{2} - 2 T_{11} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -4 - 2 T + T^{2} \)
$11$ \( -4 - 2 T + T^{2} \)
$13$ \( 11 + 8 T + T^{2} \)
$17$ \( -16 + 4 T + T^{2} \)
$19$ \( -44 + 2 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 5 + 10 T + T^{2} \)
$31$ \( -1 + 4 T + T^{2} \)
$37$ \( -36 + 6 T + T^{2} \)
$41$ \( -11 + 6 T + T^{2} \)
$43$ \( -36 - 6 T + T^{2} \)
$47$ \( 5 + 10 T + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -80 + T^{2} \)
$61$ \( -124 - 2 T + T^{2} \)
$67$ \( -36 + 6 T + T^{2} \)
$71$ \( 11 - 8 T + T^{2} \)
$73$ \( -45 + T^{2} \)
$79$ \( 116 + 22 T + T^{2} \)
$83$ \( -16 - 4 T + T^{2} \)
$89$ \( 20 - 10 T + T^{2} \)
$97$ \( -100 - 10 T + T^{2} \)
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