Properties

Label 1805.2.a.k
Level $1805$
Weight $2$
Character orbit 1805.a
Self dual yes
Analytic conductor $14.413$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{3} - 2 q^{4} - q^{5} + (\beta_{2} + 1) q^{7} - 4 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{3} - 2 q^{4} - q^{5} + (\beta_{2} + 1) q^{7} - 4 \beta_{2} q^{9} + (3 \beta_{2} - 1) q^{11} + (2 \beta_{3} - 2 \beta_1) q^{12} + (\beta_{3} - \beta_1) q^{13} + (\beta_{3} - \beta_1) q^{15} + 4 q^{16} + (3 \beta_{2} + 2) q^{17} + 2 q^{20} + \beta_{3} q^{21} + ( - 4 \beta_{2} - 1) q^{23} + q^{25} + ( - 5 \beta_{3} + \beta_1) q^{27} + ( - 2 \beta_{2} - 2) q^{28} + 2 \beta_{3} q^{29} + ( - \beta_{3} - 3 \beta_1) q^{31} + (7 \beta_{3} - 4 \beta_1) q^{33} + ( - \beta_{2} - 1) q^{35} + 8 \beta_{2} q^{36} + (5 \beta_{3} + \beta_1) q^{37} + (4 \beta_{2} - 3) q^{39} + (3 \beta_{3} - 5 \beta_1) q^{41} + ( - \beta_{2} - 9) q^{43} + ( - 6 \beta_{2} + 2) q^{44} + 4 \beta_{2} q^{45} + ( - 6 \beta_{2} - 5) q^{47} + ( - 4 \beta_{3} + 4 \beta_1) q^{48} + (\beta_{2} - 5) q^{49} + (4 \beta_{3} - \beta_1) q^{51} + ( - 2 \beta_{3} + 2 \beta_1) q^{52} + ( - \beta_{3} - 6 \beta_1) q^{53} + ( - 3 \beta_{2} + 1) q^{55} + ( - \beta_{3} + 2 \beta_1) q^{59} + ( - 2 \beta_{3} + 2 \beta_1) q^{60} + (2 \beta_{2} - 4) q^{61} - 4 q^{63} - 8 q^{64} + ( - \beta_{3} + \beta_1) q^{65} + ( - 2 \beta_{3} + 3 \beta_1) q^{67} + ( - 6 \beta_{2} - 4) q^{68} + ( - 7 \beta_{3} + 3 \beta_1) q^{69} + (3 \beta_{3} + 5 \beta_1) q^{71} - 11 q^{73} + ( - \beta_{3} + \beta_1) q^{75} + ( - \beta_{2} + 2) q^{77} + ( - 9 \beta_{3} - 3 \beta_1) q^{79} - 4 q^{80} + ( - 4 \beta_{2} + 7) q^{81} + ( - 6 \beta_{2} + 1) q^{83} - 2 \beta_{3} q^{84} + ( - 3 \beta_{2} - 2) q^{85} + (6 \beta_{2} - 2) q^{87} + 7 \beta_1 q^{89} - \beta_{3} q^{91} + (8 \beta_{2} + 2) q^{92} - 5 q^{93} + (5 \beta_{3} + \beta_1) q^{97} + (16 \beta_{2} - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9} - 10 q^{11} + 16 q^{16} + 2 q^{17} + 8 q^{20} + 4 q^{23} + 4 q^{25} - 4 q^{28} - 2 q^{35} - 16 q^{36} - 20 q^{39} - 34 q^{43} + 20 q^{44} - 8 q^{45} - 8 q^{47} - 22 q^{49} + 10 q^{55} - 20 q^{61} - 16 q^{63} - 32 q^{64} - 4 q^{68} - 44 q^{73} + 10 q^{77} - 16 q^{80} + 36 q^{81} + 16 q^{83} - 2 q^{85} - 20 q^{87} - 8 q^{92} - 20 q^{93} - 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{20} + \zeta_{20}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
−1.17557
−1.90211
1.90211
1.17557
0 −3.07768 −2.00000 −1.00000 0 −0.618034 0 6.47214 0
1.2 0 −0.726543 −2.00000 −1.00000 0 1.61803 0 −2.47214 0
1.3 0 0.726543 −2.00000 −1.00000 0 1.61803 0 −2.47214 0
1.4 0 3.07768 −2.00000 −1.00000 0 −0.618034 0 6.47214 0
\(n\): 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.k 4
5.b even 2 1 9025.2.a.bi 4
19.b odd 2 1 inner 1805.2.a.k 4
95.d odd 2 1 9025.2.a.bi 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.a.k 4 1.a even 1 1 trivial
1805.2.a.k 4 19.b odd 2 1 inner
9025.2.a.bi 4 5.b even 2 1
9025.2.a.bi 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1805))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{4} - 10T_{3}^{2} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 10T^{2} + 5 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 5 T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 10T^{2} + 5 \) Copy content Toggle raw display
$17$ \( (T^{2} - T - 11)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T - 19)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 20T^{2} + 80 \) Copy content Toggle raw display
$31$ \( T^{4} - 50T^{2} + 125 \) Copy content Toggle raw display
$37$ \( T^{4} - 130T^{2} + 4205 \) Copy content Toggle raw display
$41$ \( T^{4} - 170T^{2} + 4805 \) Copy content Toggle raw display
$43$ \( (T^{2} + 17 T + 71)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4 T - 41)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 185T^{2} + 4205 \) Copy content Toggle raw display
$59$ \( T^{4} - 25T^{2} + 125 \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 65T^{2} + 605 \) Copy content Toggle raw display
$71$ \( T^{4} - 170T^{2} + 5 \) Copy content Toggle raw display
$73$ \( (T + 11)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - 450 T^{2} + 49005 \) Copy content Toggle raw display
$83$ \( (T^{2} - 8 T - 29)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 245 T^{2} + 12005 \) Copy content Toggle raw display
$97$ \( T^{4} - 130T^{2} + 4205 \) Copy content Toggle raw display
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