Properties

Label 1805.2.a.l
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_1 q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{2} + 3) q^{6} + ( - 2 \beta_{2} - 3) q^{7} + (\beta_{3} - \beta_1) q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_1 q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{2} + 3) q^{6} + ( - 2 \beta_{2} - 3) q^{7} + (\beta_{3} - \beta_1) q^{8} + \beta_{2} q^{9} - \beta_1 q^{10} + ( - 3 \beta_{2} - 4) q^{11} + (\beta_{3} + \beta_1) q^{12} + ( - \beta_{3} - \beta_1) q^{13} + ( - 2 \beta_{3} - 3 \beta_1) q^{14} - \beta_1 q^{15} + ( - \beta_{2} - 4) q^{16} + (2 \beta_{2} + 4) q^{17} + \beta_{3} q^{18} + ( - \beta_{2} - 1) q^{20} + ( - 2 \beta_{3} - 3 \beta_1) q^{21} + ( - 3 \beta_{3} - 4 \beta_1) q^{22} + (5 \beta_{2} + 1) q^{23} + (\beta_{2} - 2) q^{24} + q^{25} + ( - 3 \beta_{2} - 4) q^{26} + (\beta_{3} - 3 \beta_1) q^{27} + ( - 3 \beta_{2} - 5) q^{28} + (4 \beta_{3} - \beta_1) q^{29} + ( - \beta_{2} - 3) q^{30} - \beta_1 q^{31} + ( - 3 \beta_{3} - 2 \beta_1) q^{32} + ( - 3 \beta_{3} - 4 \beta_1) q^{33} + (2 \beta_{3} + 4 \beta_1) q^{34} + (2 \beta_{2} + 3) q^{35} + q^{36} + (4 \beta_{3} - \beta_1) q^{37} + ( - 3 \beta_{2} - 4) q^{39} + ( - \beta_{3} + \beta_1) q^{40} + (\beta_{3} + 3 \beta_1) q^{41} + ( - 7 \beta_{2} - 11) q^{42} + ( - \beta_{2} + 1) q^{43} + ( - 4 \beta_{2} - 7) q^{44} - \beta_{2} q^{45} + (5 \beta_{3} + \beta_1) q^{46} + ( - 4 \beta_{2} + 1) q^{47} + ( - \beta_{3} - 4 \beta_1) q^{48} + (8 \beta_{2} + 6) q^{49} + \beta_1 q^{50} + (2 \beta_{3} + 4 \beta_1) q^{51} + ( - \beta_{3} - 2 \beta_1) q^{52} + ( - 3 \beta_{3} - 4 \beta_1) q^{53} + ( - \beta_{2} - 8) q^{54} + (3 \beta_{2} + 4) q^{55} + (\beta_{3} + \beta_1) q^{56} + (7 \beta_{2} + 1) q^{58} + (\beta_{3} - 8 \beta_1) q^{59} + ( - \beta_{3} - \beta_1) q^{60} + (8 \beta_{2} - 1) q^{61} + ( - \beta_{2} - 3) q^{62} + ( - \beta_{2} - 2) q^{63} + ( - 6 \beta_{2} - 1) q^{64} + (\beta_{3} + \beta_1) q^{65} + ( - 10 \beta_{2} - 15) q^{66} + (3 \beta_{3} - 5 \beta_1) q^{67} + (4 \beta_{2} + 6) q^{68} + (5 \beta_{3} + \beta_1) q^{69} + (2 \beta_{3} + 3 \beta_1) q^{70} + ( - 5 \beta_{3} + 3 \beta_1) q^{71} + ( - 2 \beta_{3} + \beta_1) q^{72} - q^{73} + (7 \beta_{2} + 1) q^{74} + \beta_1 q^{75} + (11 \beta_{2} + 18) q^{77} + ( - 3 \beta_{3} - 4 \beta_1) q^{78} + ( - 2 \beta_{3} + 4 \beta_1) q^{79} + (\beta_{2} + 4) q^{80} + ( - 4 \beta_{2} - 8) q^{81} + (5 \beta_{2} + 10) q^{82} + (2 \beta_{2} - 10) q^{83} + ( - 3 \beta_{3} - 5 \beta_1) q^{84} + ( - 2 \beta_{2} - 4) q^{85} + ( - \beta_{3} + \beta_1) q^{86} + (7 \beta_{2} + 1) q^{87} + (2 \beta_{3} + \beta_1) q^{88} + ( - 5 \beta_{3} - \beta_1) q^{89} - \beta_{3} q^{90} + (3 \beta_{3} + 5 \beta_1) q^{91} + (\beta_{2} + 6) q^{92} + ( - \beta_{2} - 3) q^{93} + ( - 4 \beta_{3} + \beta_1) q^{94} + ( - 8 \beta_{2} - 9) q^{96} + ( - \beta_{3} + 2 \beta_1) q^{97} + (8 \beta_{3} + 6 \beta_1) q^{98} + ( - \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{5} + 10 q^{6} - 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 4 q^{5} + 10 q^{6} - 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} + 12 q^{17} - 2 q^{20} - 6 q^{23} - 10 q^{24} + 4 q^{25} - 10 q^{26} - 14 q^{28} - 10 q^{30} + 8 q^{35} + 4 q^{36} - 10 q^{39} - 30 q^{42} + 6 q^{43} - 20 q^{44} + 2 q^{45} + 12 q^{47} + 8 q^{49} - 30 q^{54} + 10 q^{55} - 10 q^{58} - 20 q^{61} - 10 q^{62} - 6 q^{63} + 8 q^{64} - 40 q^{66} + 16 q^{68} - 4 q^{73} - 10 q^{74} + 50 q^{77} + 14 q^{80} - 24 q^{81} + 30 q^{82} - 44 q^{83} - 12 q^{85} - 10 q^{87} + 22 q^{92} - 10 q^{93} - 20 q^{96} - 10 q^{99}+O(q^{100}) \) 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
−1.90211
−1.17557
1.17557
1.90211
−1.90211 −1.90211 1.61803 −1.00000 3.61803 −4.23607 0.726543 0.618034 1.90211
1.2 −1.17557 −1.17557 −0.618034 −1.00000 1.38197 0.236068 3.07768 −1.61803 1.17557
1.3 1.17557 1.17557 −0.618034 −1.00000 1.38197 0.236068 −3.07768 −1.61803 −1.17557
1.4 1.90211 1.90211 1.61803 −1.00000 3.61803 −4.23607 −0.726543 0.618034 −1.90211
\(n\): 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.l 4
5.b even 2 1 9025.2.a.bl 4
19.b odd 2 1 inner 1805.2.a.l 4
95.d odd 2 1 9025.2.a.bl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.a.l 4 1.a even 1 1 trivial
1805.2.a.l 4 19.b odd 2 1 inner
9025.2.a.bl 4 5.b even 2 1
9025.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1805))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5T^{2} + 5 \) Copy content Toggle raw display
$3$ \( T^{4} - 5T^{2} + 5 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T - 1)^{2} \) 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
$17$ \( (T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 3 T - 29)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 85T^{2} + 605 \) Copy content Toggle raw display
$31$ \( T^{4} - 5T^{2} + 5 \) Copy content Toggle raw display
$37$ \( T^{4} - 85T^{2} + 605 \) Copy content Toggle raw display
$41$ \( T^{4} - 50T^{2} + 125 \) Copy content Toggle raw display
$43$ \( (T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T - 11)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 125T^{2} + 125 \) Copy content Toggle raw display
$59$ \( T^{4} - 325 T^{2} + 25205 \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T - 55)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 170T^{2} + 4805 \) Copy content Toggle raw display
$71$ \( T^{4} - 170T^{2} + 5 \) Copy content Toggle raw display
$73$ \( (T + 1)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - 100T^{2} + 2000 \) Copy content Toggle raw display
$83$ \( (T^{2} + 22 T + 116)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 130T^{2} + 4205 \) Copy content Toggle raw display
$97$ \( T^{4} - 25T^{2} + 125 \) Copy content Toggle raw display
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