Properties

Label 2001.2.a.e
Level $2001$
Weight $2$
Character orbit 2001.a
Self dual yes
Analytic conductor $15.978$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.9780654445\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} -2 q^{4} -\beta q^{5} + ( -2 + \beta ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -2 q^{4} -\beta q^{5} + ( -2 + \beta ) q^{7} + q^{9} -2 q^{12} + q^{13} -\beta q^{15} + 4 q^{16} + ( 1 + \beta ) q^{17} - q^{19} + 2 \beta q^{20} + ( -2 + \beta ) q^{21} - q^{23} + q^{25} + q^{27} + ( 4 - 2 \beta ) q^{28} + q^{29} + ( -8 + \beta ) q^{31} + ( -6 + 2 \beta ) q^{35} -2 q^{36} + ( 5 - 2 \beta ) q^{37} + q^{39} + ( -2 + 3 \beta ) q^{41} + ( -5 - 2 \beta ) q^{43} -\beta q^{45} + ( -10 + \beta ) q^{47} + 4 q^{48} + ( 3 - 4 \beta ) q^{49} + ( 1 + \beta ) q^{51} -2 q^{52} -2 q^{53} - q^{57} + ( -3 - \beta ) q^{59} + 2 \beta q^{60} + ( -8 - 2 \beta ) q^{61} + ( -2 + \beta ) q^{63} -8 q^{64} -\beta q^{65} + ( -4 + 2 \beta ) q^{67} + ( -2 - 2 \beta ) q^{68} - q^{69} + ( -5 + 3 \beta ) q^{71} + ( -2 + 5 \beta ) q^{73} + q^{75} + 2 q^{76} + ( 1 - 2 \beta ) q^{79} -4 \beta q^{80} + q^{81} + ( -10 - \beta ) q^{83} + ( 4 - 2 \beta ) q^{84} + ( -6 - \beta ) q^{85} + q^{87} + ( 3 - \beta ) q^{89} + ( -2 + \beta ) q^{91} + 2 q^{92} + ( -8 + \beta ) q^{93} + \beta q^{95} + ( 12 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{4} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{4} - 4q^{7} + 2q^{9} - 4q^{12} + 2q^{13} + 8q^{16} + 2q^{17} - 2q^{19} - 4q^{21} - 2q^{23} + 2q^{25} + 2q^{27} + 8q^{28} + 2q^{29} - 16q^{31} - 12q^{35} - 4q^{36} + 10q^{37} + 2q^{39} - 4q^{41} - 10q^{43} - 20q^{47} + 8q^{48} + 6q^{49} + 2q^{51} - 4q^{52} - 4q^{53} - 2q^{57} - 6q^{59} - 16q^{61} - 4q^{63} - 16q^{64} - 8q^{67} - 4q^{68} - 2q^{69} - 10q^{71} - 4q^{73} + 2q^{75} + 4q^{76} + 2q^{79} + 2q^{81} - 20q^{83} + 8q^{84} - 12q^{85} + 2q^{87} + 6q^{89} - 4q^{91} + 4q^{92} - 16q^{93} + 24q^{97} + O(q^{100}) \)

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
2.44949
−2.44949
0 1.00000 −2.00000 −2.44949 0 0.449490 0 1.00000 0
1.2 0 1.00000 −2.00000 2.44949 0 −4.44949 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(-1\)

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 2001.2.a.e 2
3.b odd 2 1 6003.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.e 2 1.a even 1 1 trivial
6003.2.a.e 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):

\( T_{2} \)
\( T_{5}^{2} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{2} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 + 4 T^{2} + 25 T^{4} \)
$7$ \( 1 + 4 T + 12 T^{2} + 28 T^{3} + 49 T^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( ( 1 - T + 13 T^{2} )^{2} \)
$17$ \( 1 - 2 T + 29 T^{2} - 34 T^{3} + 289 T^{4} \)
$19$ \( ( 1 + T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( 1 - T )^{2} \)
$31$ \( 1 + 16 T + 120 T^{2} + 496 T^{3} + 961 T^{4} \)
$37$ \( 1 - 10 T + 75 T^{2} - 370 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 4 T + 32 T^{2} + 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 10 T + 87 T^{2} + 430 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 20 T + 188 T^{2} + 940 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 2 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 6 T + 121 T^{2} + 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 16 T + 162 T^{2} + 976 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 8 T + 126 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 10 T + 113 T^{2} + 710 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 4 T + 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 2 T + 135 T^{2} - 158 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 20 T + 260 T^{2} + 1660 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 6 T + 181 T^{2} - 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 24 T + 314 T^{2} - 2328 T^{3} + 9409 T^{4} \)
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