Properties

Label 1859.2.a.f
Level $1859$
Weight $2$
Character orbit 1859.a
Self dual yes
Analytic conductor $14.844$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.361.1
Defining polynomial: \( x^{3} - x^{2} - 6x + 7 \) Copy content Toggle raw display
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{2} - 2 \beta_1 + 3) q^{6} + (\beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} - 2 \beta_1 + 3) q^{8} + ( - \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{2} - 2 \beta_1 + 3) q^{6} + (\beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} - 2 \beta_1 + 3) q^{8} + ( - \beta_{2} - \beta_1 + 2) q^{9} + ( - 2 \beta_{2} - \beta_1 - 1) q^{10} + q^{11} + (\beta_{2} - \beta_1 + 5) q^{12} + ( - 2 \beta_{2} - 3 \beta_1 - 1) q^{14} + ( - \beta_{2} + \beta_1 + 2) q^{15} + (\beta_{2} - \beta_1 + 1) q^{16} + ( - \beta_{2} - 2) q^{17} + (2 \beta_{2} + 1) q^{18} + (\beta_{2} - 2 \beta_1 + 1) q^{19} + (\beta_{2} + 3 \beta_1) q^{20} + (\beta_{2} + \beta_1 + 2) q^{21} - \beta_1 q^{22} + (\beta_{2} + 2 \beta_1) q^{23} + (2 \beta_{2} - 3 \beta_1 + 1) q^{24} + (\beta_1 - 1) q^{25} + ( - \beta_{2} - \beta_1 - 2) q^{27} + (3 \beta_{2} + 3 \beta_1 + 4) q^{28} + (3 \beta_{2} - \beta_1 - 2) q^{29} - 7 q^{30} + (2 \beta_{2} + 2 \beta_1 - 1) q^{31} + (2 \beta_{2} + \beta_1 + 1) q^{32} + \beta_{2} q^{33} + (\beta_{2} + 4 \beta_1 - 3) q^{34} + (2 \beta_{2} + 3 \beta_1 + 2) q^{35} + ( - 3 \beta_1 + 2) q^{36} + ( - \beta_{2} + \beta_1 + 4) q^{37} + (\beta_{2} - 3 \beta_1 + 11) q^{38} - 7 q^{40} + ( - \beta_{2} - 4 \beta_1 + 2) q^{41} + ( - 2 \beta_{2} - 4 \beta_1 - 1) q^{42} + (3 \beta_{2} + 5 \beta_1 - 4) q^{43} + (\beta_{2} + 2) q^{44} + (\beta_{2} - 5) q^{45} + ( - 3 \beta_{2} - 2 \beta_1 - 5) q^{46} + (4 \beta_{2} + 3 \beta_1 - 5) q^{47} + ( - \beta_{2} - 3 \beta_1 + 8) q^{48} + (4 \beta_{2} + 5 \beta_1 - 3) q^{49} + ( - \beta_{2} + \beta_1 - 4) q^{50} + ( - \beta_{2} + \beta_1 - 5) q^{51} + (\beta_{2} - \beta_1 - 3) q^{53} + (2 \beta_{2} + 4 \beta_1 + 1) q^{54} + (\beta_{2} + \beta_1 - 1) q^{55} + ( - 2 \beta_{2} - 4 \beta_1 - 1) q^{56} + ( - 2 \beta_{2} - 5 \beta_1 + 11) q^{57} + ( - 2 \beta_{2} - 4 \beta_1 + 13) q^{58} + ( - \beta_{2} + \beta_1 + 10) q^{59} + (2 \beta_{2} + 5 \beta_1 - 4) q^{60} + (3 \beta_1 - 7) q^{61} + ( - 4 \beta_{2} - 3 \beta_1 - 2) q^{62} + ( - \beta_{2} - 2 \beta_1 - 1) q^{63} + ( - 5 \beta_{2} - 3 \beta_1) q^{64} + ( - \beta_{2} - 2 \beta_1 + 3) q^{66} + (2 \beta_{2} + \beta_1 + 4) q^{67} + ( - 3 \beta_{2} + \beta_1 - 9) q^{68} + (\beta_{2} + 3 \beta_1 - 1) q^{69} + ( - 5 \beta_{2} - 6 \beta_1 - 6) q^{70} + (\beta_1 - 4) q^{71} + ( - \beta_{2} - 2 \beta_1 + 10) q^{72} - 2 q^{73} + ( - 2 \beta_1 - 7) q^{74} + (2 \beta_1 - 3) q^{75} + ( - 9 \beta_1 + 13) q^{76} + (\beta_{2} + \beta_1 + 1) q^{77} + (2 \beta_{2} - 2 \beta_1 + 4) q^{79} + ( - 2 \beta_{2} + \beta_1) q^{80} + (\beta_{2} + 2 \beta_1 - 8) q^{81} + (5 \beta_{2} + 13) q^{82} + ( - 3 \beta_{2} + \beta_1 - 4) q^{83} + (4 \beta_{2} + 3 \beta_1 + 6) q^{84} + ( - \beta_{2} - 3 \beta_1) q^{85} + ( - 8 \beta_{2} - 2 \beta_1 - 11) q^{86} + ( - 6 \beta_{2} - 5 \beta_1 + 18) q^{87} + ( - \beta_{2} - 2 \beta_1 + 3) q^{88} + ( - \beta_{2} - \beta_1 - 5) q^{89} + ( - \beta_{2} + 3 \beta_1 + 3) q^{90} + (3 \beta_{2} + 7 \beta_1 - 1) q^{92} + ( - \beta_{2} + 2 \beta_1 + 4) q^{93} + ( - 7 \beta_{2} - 3 \beta_1) q^{94} + ( - 4 \beta_{2} - 1) q^{95} + 7 q^{96} + ( - 4 \beta_{2} + 5) q^{97} + ( - 9 \beta_{2} - 5 \beta_1 - 8) q^{98} + ( - \beta_{2} - \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + q^{3} + 7 q^{4} - q^{5} + 6 q^{6} + 5 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + q^{3} + 7 q^{4} - q^{5} + 6 q^{6} + 5 q^{7} + 6 q^{8} + 4 q^{9} - 6 q^{10} + 3 q^{11} + 15 q^{12} - 8 q^{14} + 6 q^{15} + 3 q^{16} - 7 q^{17} + 5 q^{18} + 2 q^{19} + 4 q^{20} + 8 q^{21} - q^{22} + 3 q^{23} + 2 q^{24} - 2 q^{25} - 8 q^{27} + 18 q^{28} - 4 q^{29} - 21 q^{30} + q^{31} + 6 q^{32} + q^{33} - 4 q^{34} + 11 q^{35} + 3 q^{36} + 12 q^{37} + 31 q^{38} - 21 q^{40} + q^{41} - 9 q^{42} - 4 q^{43} + 7 q^{44} - 14 q^{45} - 20 q^{46} - 8 q^{47} + 20 q^{48} - 12 q^{50} - 15 q^{51} - 9 q^{53} + 9 q^{54} - q^{55} - 9 q^{56} + 26 q^{57} + 33 q^{58} + 30 q^{59} - 5 q^{60} - 18 q^{61} - 13 q^{62} - 6 q^{63} - 8 q^{64} + 6 q^{66} + 15 q^{67} - 29 q^{68} + q^{69} - 29 q^{70} - 11 q^{71} + 27 q^{72} - 6 q^{73} - 23 q^{74} - 7 q^{75} + 30 q^{76} + 5 q^{77} + 12 q^{79} - q^{80} - 21 q^{81} + 44 q^{82} - 14 q^{83} + 25 q^{84} - 4 q^{85} - 43 q^{86} + 43 q^{87} + 6 q^{88} - 17 q^{89} + 11 q^{90} + 7 q^{92} + 13 q^{93} - 10 q^{94} - 7 q^{95} + 21 q^{96} + 11 q^{97} - 38 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
2.28514
1.22188
−2.50702
−2.28514 1.22188 3.22188 2.50702 −2.79216 4.50702 −2.79216 −1.50702 −5.72889
1.2 −1.22188 −2.50702 −0.507019 −2.28514 3.06327 −0.285142 3.06327 3.28514 2.79216
1.3 2.50702 2.28514 4.28514 −1.22188 5.72889 0.778124 5.72889 2.22188 −3.06327
\(n\): 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.f 3
13.b even 2 1 1859.2.a.g 3
13.c even 3 2 143.2.e.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.e.b 6 13.c even 3 2
1859.2.a.f 3 1.a even 1 1 trivial
1859.2.a.g 3 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1859))\):

\( T_{2}^{3} + T_{2}^{2} - 6T_{2} - 7 \) Copy content Toggle raw display
\( T_{7}^{3} - 5T_{7}^{2} + 2T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 6T - 7 \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 6T + 7 \) Copy content Toggle raw display
$5$ \( T^{3} + T^{2} - 6T - 7 \) Copy content Toggle raw display
$7$ \( T^{3} - 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 7 T^{2} + 10 T - 7 \) Copy content Toggle raw display
$19$ \( T^{3} - 2 T^{2} - 43 T - 77 \) Copy content Toggle raw display
$23$ \( T^{3} - 3 T^{2} - 16 T - 1 \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 77 T - 49 \) Copy content Toggle raw display
$31$ \( T^{3} - T^{2} - 25 T - 31 \) Copy content Toggle raw display
$37$ \( T^{3} - 12 T^{2} + 29 T + 31 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} - 82 T - 31 \) Copy content Toggle raw display
$43$ \( T^{3} + 4 T^{2} - 115 T - 581 \) Copy content Toggle raw display
$47$ \( T^{3} + 8 T^{2} - 61 T - 259 \) Copy content Toggle raw display
$53$ \( T^{3} + 9 T^{2} + 8 T - 49 \) Copy content Toggle raw display
$59$ \( T^{3} - 30 T^{2} + 281 T - 791 \) Copy content Toggle raw display
$61$ \( T^{3} + 18 T^{2} + 51 T + 7 \) Copy content Toggle raw display
$67$ \( T^{3} - 15 T^{2} + 56 T - 11 \) Copy content Toggle raw display
$71$ \( T^{3} + 11 T^{2} + 34 T + 31 \) Copy content Toggle raw display
$73$ \( (T + 2)^{3} \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} - 28 T + 88 \) Copy content Toggle raw display
$83$ \( T^{3} + 14 T^{2} - 17 T - 341 \) Copy content Toggle raw display
$89$ \( T^{3} + 17 T^{2} + 90 T + 151 \) Copy content Toggle raw display
$97$ \( T^{3} - 11 T^{2} - 61 T + 7 \) Copy content Toggle raw display
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