Properties

Label 1859.2.a.d
Level $1859$
Weight $2$
Character orbit 1859.a
Self dual yes
Analytic conductor $14.844$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8441897358\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -1 + \beta ) q^{3} + q^{4} -\beta q^{5} + ( 3 - \beta ) q^{6} + ( 3 - \beta ) q^{7} -\beta q^{8} + ( 1 - 2 \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{2} + ( -1 + \beta ) q^{3} + q^{4} -\beta q^{5} + ( 3 - \beta ) q^{6} + ( 3 - \beta ) q^{7} -\beta q^{8} + ( 1 - 2 \beta ) q^{9} -3 q^{10} + q^{11} + ( -1 + \beta ) q^{12} + ( -3 + 3 \beta ) q^{14} + ( -3 + \beta ) q^{15} -5 q^{16} + ( -6 - \beta ) q^{17} + ( -6 + \beta ) q^{18} + ( -3 - \beta ) q^{19} -\beta q^{20} + ( -6 + 4 \beta ) q^{21} + \beta q^{22} + 2 \beta q^{23} + ( -3 + \beta ) q^{24} -2 q^{25} -4 q^{27} + ( 3 - \beta ) q^{28} + ( 6 + \beta ) q^{29} + ( 3 - 3 \beta ) q^{30} -6 q^{31} -3 \beta q^{32} + ( -1 + \beta ) q^{33} + ( -3 - 6 \beta ) q^{34} + ( 3 - 3 \beta ) q^{35} + ( 1 - 2 \beta ) q^{36} + \beta q^{37} + ( -3 - 3 \beta ) q^{38} + 3 q^{40} -9 q^{41} + ( 12 - 6 \beta ) q^{42} + ( -2 + 6 \beta ) q^{43} + q^{44} + ( 6 - \beta ) q^{45} + 6 q^{46} + ( 3 - \beta ) q^{47} + ( 5 - 5 \beta ) q^{48} + ( 5 - 6 \beta ) q^{49} -2 \beta q^{50} + ( 3 - 5 \beta ) q^{51} + ( 3 + 2 \beta ) q^{53} -4 \beta q^{54} -\beta q^{55} + ( 3 - 3 \beta ) q^{56} -2 \beta q^{57} + ( 3 + 6 \beta ) q^{58} + ( -3 - \beta ) q^{59} + ( -3 + \beta ) q^{60} + ( -2 + 3 \beta ) q^{61} -6 \beta q^{62} + ( 9 - 7 \beta ) q^{63} + q^{64} + ( 3 - \beta ) q^{66} + ( -3 + \beta ) q^{67} + ( -6 - \beta ) q^{68} + ( 6 - 2 \beta ) q^{69} + ( -9 + 3 \beta ) q^{70} + 2 \beta q^{71} + ( 6 - \beta ) q^{72} + ( -9 + 2 \beta ) q^{73} + 3 q^{74} + ( 2 - 2 \beta ) q^{75} + ( -3 - \beta ) q^{76} + ( 3 - \beta ) q^{77} + ( 1 + 3 \beta ) q^{79} + 5 \beta q^{80} + ( 1 + 2 \beta ) q^{81} -9 \beta q^{82} + ( -3 - 7 \beta ) q^{83} + ( -6 + 4 \beta ) q^{84} + ( 3 + 6 \beta ) q^{85} + ( 18 - 2 \beta ) q^{86} + ( -3 + 5 \beta ) q^{87} -\beta q^{88} + 2 \beta q^{89} + ( -3 + 6 \beta ) q^{90} + 2 \beta q^{92} + ( 6 - 6 \beta ) q^{93} + ( -3 + 3 \beta ) q^{94} + ( 3 + 3 \beta ) q^{95} + ( -9 + 3 \beta ) q^{96} -2 \beta q^{97} + ( -18 + 5 \beta ) q^{98} + ( 1 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} + 6 q^{6} + 6 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 2 q^{4} + 6 q^{6} + 6 q^{7} + 2 q^{9} - 6 q^{10} + 2 q^{11} - 2 q^{12} - 6 q^{14} - 6 q^{15} - 10 q^{16} - 12 q^{17} - 12 q^{18} - 6 q^{19} - 12 q^{21} - 6 q^{24} - 4 q^{25} - 8 q^{27} + 6 q^{28} + 12 q^{29} + 6 q^{30} - 12 q^{31} - 2 q^{33} - 6 q^{34} + 6 q^{35} + 2 q^{36} - 6 q^{38} + 6 q^{40} - 18 q^{41} + 24 q^{42} - 4 q^{43} + 2 q^{44} + 12 q^{45} + 12 q^{46} + 6 q^{47} + 10 q^{48} + 10 q^{49} + 6 q^{51} + 6 q^{53} + 6 q^{56} + 6 q^{58} - 6 q^{59} - 6 q^{60} - 4 q^{61} + 18 q^{63} + 2 q^{64} + 6 q^{66} - 6 q^{67} - 12 q^{68} + 12 q^{69} - 18 q^{70} + 12 q^{72} - 18 q^{73} + 6 q^{74} + 4 q^{75} - 6 q^{76} + 6 q^{77} + 2 q^{79} + 2 q^{81} - 6 q^{83} - 12 q^{84} + 6 q^{85} + 36 q^{86} - 6 q^{87} - 6 q^{90} + 12 q^{93} - 6 q^{94} + 6 q^{95} - 18 q^{96} - 36 q^{98} + 2 q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
−1.73205 −2.73205 1.00000 1.73205 4.73205 4.73205 1.73205 4.46410 −3.00000
1.2 1.73205 0.732051 1.00000 −1.73205 1.26795 1.26795 −1.73205 −2.46410 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(13\) \(-1\)

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 1859.2.a.d 2
13.b even 2 1 1859.2.a.c 2
13.f odd 12 2 143.2.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.j.a 4 13.f odd 12 2
1859.2.a.c 2 13.b even 2 1
1859.2.a.d 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1859))\):

\( T_{2}^{2} - 3 \)
\( T_{7}^{2} - 6 T_{7} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( -2 + 2 T + T^{2} \)
$5$ \( -3 + T^{2} \)
$7$ \( 6 - 6 T + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( 33 + 12 T + T^{2} \)
$19$ \( 6 + 6 T + T^{2} \)
$23$ \( -12 + T^{2} \)
$29$ \( 33 - 12 T + T^{2} \)
$31$ \( ( 6 + T )^{2} \)
$37$ \( -3 + T^{2} \)
$41$ \( ( 9 + T )^{2} \)
$43$ \( -104 + 4 T + T^{2} \)
$47$ \( 6 - 6 T + T^{2} \)
$53$ \( -3 - 6 T + T^{2} \)
$59$ \( 6 + 6 T + T^{2} \)
$61$ \( -23 + 4 T + T^{2} \)
$67$ \( 6 + 6 T + T^{2} \)
$71$ \( -12 + T^{2} \)
$73$ \( 69 + 18 T + T^{2} \)
$79$ \( -26 - 2 T + T^{2} \)
$83$ \( -138 + 6 T + T^{2} \)
$89$ \( -12 + T^{2} \)
$97$ \( -12 + T^{2} \)
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