Properties

Label 1805.2.a.f
Level $1805$
Weight $2$
Character orbit 1805.a
Self dual yes
Analytic conductor $14.413$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1) q^{4} + q^{5} + ( - \beta_{2} - \beta_1 - 1) q^{6} + 2 \beta_{2} q^{7} + ( - \beta_{2} - 1) q^{8} + ( - 2 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1) q^{4} + q^{5} + ( - \beta_{2} - \beta_1 - 1) q^{6} + 2 \beta_{2} q^{7} + ( - \beta_{2} - 1) q^{8} + ( - 2 \beta_{2} + 1) q^{9} - \beta_1 q^{10} + ( - 2 \beta_1 - 2) q^{11} + ( - \beta_{2} + \beta_1 + 3) q^{12} + ( - \beta_{2} + \beta_1 - 3) q^{13} + ( - 2 \beta_1 + 2) q^{14} + (\beta_{2} + \beta_1 - 1) q^{15} + ( - 2 \beta_{2} - 1) q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + (\beta_1 - 2) q^{18} + (\beta_{2} + \beta_1) q^{20} + ( - 4 \beta_{2} + 4) q^{21} + (2 \beta_{2} + 4 \beta_1 + 4) q^{22} + (2 \beta_1 - 2) q^{23} + (\beta_{2} - \beta_1 - 1) q^{24} + q^{25} + ( - \beta_{2} + 3 \beta_1 - 3) q^{26} + (2 \beta_{2} - 2 \beta_1 - 2) q^{27} + ( - 2 \beta_{2} + 4) q^{28} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{29} + ( - \beta_{2} - \beta_1 - 1) q^{30} - 4 \beta_1 q^{31} + (2 \beta_{2} + 3 \beta_1) q^{32} + ( - 4 \beta_{2} - 4 \beta_1) q^{33} + ( - 2 \beta_{2} - 6) q^{34} + 2 \beta_{2} q^{35} + (3 \beta_{2} + \beta_1 - 4) q^{36} + ( - \beta_{2} + \beta_1 - 7) q^{37} + ( - 2 \beta_1 + 2) q^{39} + ( - \beta_{2} - 1) q^{40} + ( - 2 \beta_{2} + 2 \beta_1) q^{41} - 4 q^{42} + ( - 6 \beta_{2} - 4 \beta_1) q^{43} + ( - 4 \beta_{2} - 6 \beta_1 - 2) q^{44} + ( - 2 \beta_{2} + 1) q^{45} + ( - 2 \beta_{2} - 4) q^{46} + 2 \beta_{2} q^{47} + (3 \beta_{2} - \beta_1 - 3) q^{48} + ( - 4 \beta_{2} - 4 \beta_1 + 5) q^{49} - \beta_1 q^{50} + (6 \beta_{2} + 2 \beta_1 - 2) q^{51} + ( - \beta_{2} - \beta_1 - 1) q^{52} + (\beta_{2} - \beta_1 - 5) q^{53} + (2 \beta_{2} + 2 \beta_1 + 6) q^{54} + ( - 2 \beta_1 - 2) q^{55} + (2 \beta_1 - 6) q^{56} + (2 \beta_{2} + 2) q^{58} + (2 \beta_{2} + 2 \beta_1 + 6) q^{59} + ( - \beta_{2} + \beta_1 + 3) q^{60} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{61} + (4 \beta_{2} + 4 \beta_1 + 8) q^{62} + (6 \beta_{2} + 4 \beta_1 - 12) q^{63} + (\beta_{2} - 5 \beta_1 - 2) q^{64} + ( - \beta_{2} + \beta_1 - 3) q^{65} + (4 \beta_{2} + 8 \beta_1 + 4) q^{66} + ( - \beta_{2} - 5 \beta_1 + 1) q^{67} + (4 \beta_{2} + 4 \beta_1 - 2) q^{68} + 4 q^{69} + ( - 2 \beta_1 + 2) q^{70} + (4 \beta_{2} + 4 \beta_1) q^{71} + ( - \beta_{2} - 2 \beta_1 + 5) q^{72} + (2 \beta_{2} + 2 \beta_1) q^{73} + ( - \beta_{2} + 7 \beta_1 - 3) q^{74} + (\beta_{2} + \beta_1 - 1) q^{75} + ( - 4 \beta_{2} - 4 \beta_1 + 4) q^{77} + (2 \beta_{2} + 4) q^{78} + ( - 6 \beta_{2} - 6 \beta_1 + 2) q^{79} + ( - 2 \beta_{2} - 1) q^{80} + ( - 2 \beta_{2} - 4 \beta_1 + 1) q^{81} + ( - 2 \beta_{2} - 6) q^{82} + ( - 2 \beta_1 - 10) q^{83} + (8 \beta_{2} + 4 \beta_1 - 8) q^{84} + ( - 2 \beta_{2} + 2 \beta_1) q^{85} + (4 \beta_{2} + 10 \beta_1 + 2) q^{86} + (6 \beta_{2} + 2 \beta_1 - 10) q^{87} + (2 \beta_{2} + 4 \beta_1) q^{88} + ( - 6 \beta_{2} - 2 \beta_1) q^{89} + (\beta_1 - 2) q^{90} + ( - 4 \beta_{2} + 4 \beta_1 - 8) q^{91} + (2 \beta_1 + 2) q^{92} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{93} + ( - 2 \beta_1 + 2) q^{94} + ( - \beta_{2} + 3 \beta_1 + 7) q^{96} + (5 \beta_{2} + 7 \beta_1 - 9) q^{97} + (4 \beta_{2} + 3 \beta_1 + 4) q^{98} + (4 \beta_{2} + 2 \beta_1 - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} - 3 q^{8} + 3 q^{9} - q^{10} - 8 q^{11} + 10 q^{12} - 8 q^{13} + 4 q^{14} - 2 q^{15} - 3 q^{16} + 2 q^{17} - 5 q^{18} + q^{20} + 12 q^{21} + 16 q^{22} - 4 q^{23} - 4 q^{24} + 3 q^{25} - 6 q^{26} - 8 q^{27} + 12 q^{28} + 10 q^{29} - 4 q^{30} - 4 q^{31} + 3 q^{32} - 4 q^{33} - 18 q^{34} - 11 q^{36} - 20 q^{37} + 4 q^{39} - 3 q^{40} + 2 q^{41} - 12 q^{42} - 4 q^{43} - 12 q^{44} + 3 q^{45} - 12 q^{46} - 10 q^{48} + 11 q^{49} - q^{50} - 4 q^{51} - 4 q^{52} - 16 q^{53} + 20 q^{54} - 8 q^{55} - 16 q^{56} + 6 q^{58} + 20 q^{59} + 10 q^{60} - 2 q^{61} + 28 q^{62} - 32 q^{63} - 11 q^{64} - 8 q^{65} + 20 q^{66} - 2 q^{67} - 2 q^{68} + 12 q^{69} + 4 q^{70} + 4 q^{71} + 13 q^{72} + 2 q^{73} - 2 q^{74} - 2 q^{75} + 8 q^{77} + 12 q^{78} - 3 q^{80} - q^{81} - 18 q^{82} - 32 q^{83} - 20 q^{84} + 2 q^{85} + 16 q^{86} - 28 q^{87} + 4 q^{88} - 2 q^{89} - 5 q^{90} - 20 q^{91} + 8 q^{92} - 16 q^{93} + 4 q^{94} + 24 q^{96} - 20 q^{97} + 15 q^{98} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
2.17009
0.311108
−1.48119
−2.17009 1.70928 2.70928 1.00000 −3.70928 1.07838 −1.53919 −0.0783777 −2.17009
1.2 −0.311108 −2.90321 −1.90321 1.00000 0.903212 −4.42864 1.21432 5.42864 −0.311108
1.3 1.48119 −0.806063 0.193937 1.00000 −1.19394 3.35026 −2.67513 −2.35026 1.48119
\(n\): 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.f 3
5.b even 2 1 9025.2.a.bb 3
19.b odd 2 1 95.2.a.a 3
57.d even 2 1 855.2.a.i 3
76.d even 2 1 1520.2.a.p 3
95.d odd 2 1 475.2.a.f 3
95.g even 4 2 475.2.b.d 6
133.c even 2 1 4655.2.a.u 3
152.b even 2 1 6080.2.a.by 3
152.g odd 2 1 6080.2.a.bo 3
285.b even 2 1 4275.2.a.bk 3
380.d even 2 1 7600.2.a.bx 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.a 3 19.b odd 2 1
475.2.a.f 3 95.d odd 2 1
475.2.b.d 6 95.g even 4 2
855.2.a.i 3 57.d even 2 1
1520.2.a.p 3 76.d even 2 1
1805.2.a.f 3 1.a even 1 1 trivial
4275.2.a.bk 3 285.b even 2 1
4655.2.a.u 3 133.c even 2 1
6080.2.a.bo 3 152.g odd 2 1
6080.2.a.by 3 152.b even 2 1
7600.2.a.bx 3 380.d even 2 1
9025.2.a.bb 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1805))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 3T - 1 \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 16T + 16 \) Copy content Toggle raw display
$11$ \( T^{3} + 8 T^{2} + 8 T - 16 \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 36 T + 104 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 4 T^{2} - 8 T - 16 \) Copy content Toggle raw display
$29$ \( T^{3} - 10 T^{2} + 12 T + 40 \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} - 48 T - 64 \) Copy content Toggle raw display
$37$ \( T^{3} + 20 T^{2} + 124 T + 244 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} - 36 T + 104 \) Copy content Toggle raw display
$43$ \( T^{3} + 4 T^{2} - 144 T - 592 \) Copy content Toggle raw display
$47$ \( T^{3} - 16T + 16 \) Copy content Toggle raw display
$53$ \( T^{3} + 16 T^{2} + 76 T + 92 \) Copy content Toggle raw display
$59$ \( T^{3} - 20 T^{2} + 112 T - 160 \) Copy content Toggle raw display
$61$ \( T^{3} + 2 T^{2} - 84 T + 232 \) Copy content Toggle raw display
$67$ \( T^{3} + 2 T^{2} - 76 T + 116 \) Copy content Toggle raw display
$71$ \( T^{3} - 4 T^{2} - 80 T + 64 \) Copy content Toggle raw display
$73$ \( T^{3} - 2 T^{2} - 20 T + 8 \) Copy content Toggle raw display
$79$ \( T^{3} - 192T + 160 \) Copy content Toggle raw display
$83$ \( T^{3} + 32 T^{2} + 328 T + 1072 \) Copy content Toggle raw display
$89$ \( T^{3} + 2 T^{2} - 132 T - 680 \) Copy content Toggle raw display
$97$ \( T^{3} + 20 T^{2} - 60 T - 1748 \) Copy content Toggle raw display
show more
show less