Properties

Label 1805.2.a.e
Level $1805$
Weight $2$
Character orbit 1805.a
Self dual yes
Analytic conductor $14.413$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
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: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (\beta + 1) q^{3} + 3 \beta q^{4} - q^{5} + (3 \beta + 2) q^{6} + (2 \beta - 3) q^{7} + (4 \beta + 1) q^{8} + (3 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (\beta + 1) q^{3} + 3 \beta q^{4} - q^{5} + (3 \beta + 2) q^{6} + (2 \beta - 3) q^{7} + (4 \beta + 1) q^{8} + (3 \beta - 1) q^{9} + ( - \beta - 1) q^{10} + ( - \beta - 3) q^{11} + (6 \beta + 3) q^{12} + 5 q^{13} + (\beta - 1) q^{14} + ( - \beta - 1) q^{15} + (3 \beta + 5) q^{16} - 6 q^{17} + (5 \beta + 2) q^{18} - 3 \beta q^{20} + (\beta - 1) q^{21} + ( - 5 \beta - 4) q^{22} - \beta q^{23} + (9 \beta + 5) q^{24} + q^{25} + (5 \beta + 5) q^{26} + (2 \beta - 1) q^{27} + ( - 3 \beta + 6) q^{28} + ( - 3 \beta + 3) q^{29} + ( - 3 \beta - 2) q^{30} + ( - 3 \beta + 9) q^{31} + (3 \beta + 6) q^{32} + ( - 5 \beta - 4) q^{33} + ( - 6 \beta - 6) q^{34} + ( - 2 \beta + 3) q^{35} + (6 \beta + 9) q^{36} + ( - 3 \beta + 3) q^{37} + (5 \beta + 5) q^{39} + ( - 4 \beta - 1) q^{40} + (10 \beta - 5) q^{41} + \beta q^{42} + ( - 3 \beta + 2) q^{43} + ( - 12 \beta - 3) q^{44} + ( - 3 \beta + 1) q^{45} + ( - 2 \beta - 1) q^{46} + ( - 2 \beta - 7) q^{47} + (11 \beta + 8) q^{48} + ( - 8 \beta + 6) q^{49} + (\beta + 1) q^{50} + ( - 6 \beta - 6) q^{51} + 15 \beta q^{52} + (\beta + 4) q^{53} + (3 \beta + 1) q^{54} + (\beta + 3) q^{55} + ( - 2 \beta + 5) q^{56} - 3 \beta q^{58} + (5 \beta - 8) q^{59} + ( - 6 \beta - 3) q^{60} - 3 q^{61} + (3 \beta + 6) q^{62} + ( - 5 \beta + 9) q^{63} + (6 \beta - 1) q^{64} - 5 q^{65} + ( - 14 \beta - 9) q^{66} + ( - 6 \beta + 5) q^{67} - 18 \beta q^{68} + ( - 2 \beta - 1) q^{69} + ( - \beta + 1) q^{70} + ( - 2 \beta + 7) q^{71} + (11 \beta + 11) q^{72} - q^{73} - 3 \beta q^{74} + (\beta + 1) q^{75} + ( - 5 \beta + 7) q^{77} + (15 \beta + 10) q^{78} + 8 q^{79} + ( - 3 \beta - 5) q^{80} + ( - 6 \beta + 4) q^{81} + (15 \beta + 5) q^{82} + (4 \beta - 2) q^{83} + 3 q^{84} + 6 q^{85} + ( - 4 \beta - 1) q^{86} - 3 \beta q^{87} + ( - 17 \beta - 7) q^{88} + ( - 6 \beta + 3) q^{89} + ( - 5 \beta - 2) q^{90} + (10 \beta - 15) q^{91} + ( - 3 \beta - 3) q^{92} + (3 \beta + 6) q^{93} + ( - 11 \beta - 9) q^{94} + (12 \beta + 9) q^{96} + (5 \beta - 2) q^{97} + ( - 10 \beta - 2) q^{98} - 11 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 2 q^{5} + 7 q^{6} - 4 q^{7} + 6 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 2 q^{5} + 7 q^{6} - 4 q^{7} + 6 q^{8} + q^{9} - 3 q^{10} - 7 q^{11} + 12 q^{12} + 10 q^{13} - q^{14} - 3 q^{15} + 13 q^{16} - 12 q^{17} + 9 q^{18} - 3 q^{20} - q^{21} - 13 q^{22} - q^{23} + 19 q^{24} + 2 q^{25} + 15 q^{26} + 9 q^{28} + 3 q^{29} - 7 q^{30} + 15 q^{31} + 15 q^{32} - 13 q^{33} - 18 q^{34} + 4 q^{35} + 24 q^{36} + 3 q^{37} + 15 q^{39} - 6 q^{40} + q^{42} + q^{43} - 18 q^{44} - q^{45} - 4 q^{46} - 16 q^{47} + 27 q^{48} + 4 q^{49} + 3 q^{50} - 18 q^{51} + 15 q^{52} + 9 q^{53} + 5 q^{54} + 7 q^{55} + 8 q^{56} - 3 q^{58} - 11 q^{59} - 12 q^{60} - 6 q^{61} + 15 q^{62} + 13 q^{63} + 4 q^{64} - 10 q^{65} - 32 q^{66} + 4 q^{67} - 18 q^{68} - 4 q^{69} + q^{70} + 12 q^{71} + 33 q^{72} - 2 q^{73} - 3 q^{74} + 3 q^{75} + 9 q^{77} + 35 q^{78} + 16 q^{79} - 13 q^{80} + 2 q^{81} + 25 q^{82} + 6 q^{84} + 12 q^{85} - 6 q^{86} - 3 q^{87} - 31 q^{88} - 9 q^{90} - 20 q^{91} - 9 q^{92} + 15 q^{93} - 29 q^{94} + 30 q^{96} + q^{97} - 14 q^{98} - 11 q^{99}+O(q^{100}) \) 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
0.381966 0.381966 −1.85410 −1.00000 0.145898 −4.23607 −1.47214 −2.85410 −0.381966
1.2 2.61803 2.61803 4.85410 −1.00000 6.85410 0.236068 7.47214 3.85410 −2.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 15T + 45 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$41$ \( T^{2} - 125 \) Copy content Toggle raw display
$43$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$47$ \( T^{2} + 16T + 59 \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 19 \) Copy content Toggle raw display
$59$ \( T^{2} + 11T - 1 \) Copy content Toggle raw display
$61$ \( (T + 3)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 31 \) Copy content Toggle raw display
$73$ \( (T + 1)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 20 \) Copy content Toggle raw display
$89$ \( T^{2} - 45 \) Copy content Toggle raw display
$97$ \( T^{2} - T - 31 \) Copy content Toggle raw display
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