Properties

Label 1805.2.a.d
Level $1805$
Weight $2$
Character orbit 1805.a
Self dual yes
Analytic conductor $14.413$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - \beta q^{2} + 3 q^{4} - q^{5} - 2 q^{7} - \beta q^{8} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + 3 q^{4} - q^{5} - 2 q^{7} - \beta q^{8} - 3 q^{9} + \beta q^{10} + 2 \beta q^{13} + 2 \beta q^{14} - q^{16} - 2 q^{17} + 3 \beta q^{18} - 3 q^{20} + 6 q^{23} + q^{25} - 10 q^{26} - 6 q^{28} + 4 \beta q^{29} - 4 \beta q^{31} + 3 \beta q^{32} + 2 \beta q^{34} + 2 q^{35} - 9 q^{36} + 2 \beta q^{37} + \beta q^{40} + 4 \beta q^{41} - 6 q^{43} + 3 q^{45} - 6 \beta q^{46} - 2 q^{47} - 3 q^{49} - \beta q^{50} + 6 \beta q^{52} - 2 \beta q^{53} + 2 \beta q^{56} - 20 q^{58} - 4 \beta q^{59} - 10 q^{61} + 20 q^{62} + 6 q^{63} - 13 q^{64} - 2 \beta q^{65} - 6 q^{68} - 2 \beta q^{70} - 4 \beta q^{71} + 3 \beta q^{72} + 14 q^{73} - 10 q^{74} - 4 \beta q^{79} + q^{80} + 9 q^{81} - 20 q^{82} + 14 q^{83} + 2 q^{85} + 6 \beta q^{86} + 4 \beta q^{89} - 3 \beta q^{90} - 4 \beta q^{91} + 18 q^{92} + 2 \beta q^{94} - 6 \beta q^{97} + 3 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{9} - 2 q^{16} - 4 q^{17} - 6 q^{20} + 12 q^{23} + 2 q^{25} - 20 q^{26} - 12 q^{28} + 4 q^{35} - 18 q^{36} - 12 q^{43} + 6 q^{45} - 4 q^{47} - 6 q^{49} - 40 q^{58} - 20 q^{61} + 40 q^{62} + 12 q^{63} - 26 q^{64} - 12 q^{68} + 28 q^{73} - 20 q^{74} + 2 q^{80} + 18 q^{81} - 40 q^{82} + 28 q^{83} + 4 q^{85} + 36 q^{92}+O(q^{100}) \) 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
1.61803
−0.618034
−2.23607 0 3.00000 −1.00000 0 −2.00000 −2.23607 −3.00000 2.23607
1.2 2.23607 0 3.00000 −1.00000 0 −2.00000 2.23607 −3.00000 −2.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(19\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.a.d 2
5.b even 2 1 9025.2.a.q 2
19.b odd 2 1 inner 1805.2.a.d 2
95.d odd 2 1 9025.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.a.d 2 1.a even 1 1 trivial
1805.2.a.d 2 19.b odd 2 1 inner
9025.2.a.q 2 5.b even 2 1
9025.2.a.q 2 95.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1805))\):

\( T_{2}^{2} - 5 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 80 \) Copy content Toggle raw display
$31$ \( T^{2} - 80 \) Copy content Toggle raw display
$37$ \( T^{2} - 20 \) Copy content Toggle raw display
$41$ \( T^{2} - 80 \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( (T + 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 20 \) Copy content Toggle raw display
$59$ \( T^{2} - 80 \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 80 \) Copy content Toggle raw display
$73$ \( (T - 14)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 80 \) Copy content Toggle raw display
$83$ \( (T - 14)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 80 \) Copy content Toggle raw display
$97$ \( T^{2} - 180 \) Copy content Toggle raw display
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