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: \( x^{2} - x - 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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