Properties

Label 1911.2.a.o
Level $1911$
Weight $2$
Character orbit 1911.a
Self dual yes
Analytic conductor $15.259$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.2594118263\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} - q^{3} + \beta_{2} q^{4} + ( 1 + \beta_{1} ) q^{5} -\beta_{1} q^{6} + ( 1 - \beta_{1} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} - q^{3} + \beta_{2} q^{4} + ( 1 + \beta_{1} ) q^{5} -\beta_{1} q^{6} + ( 1 - \beta_{1} ) q^{8} + q^{9} + ( 2 + \beta_{1} + \beta_{2} ) q^{10} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} -\beta_{2} q^{12} + q^{13} + ( -1 - \beta_{1} ) q^{15} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{16} + ( 3 + \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( 1 + 2 \beta_{1} ) q^{19} + ( 1 + \beta_{1} + \beta_{2} ) q^{20} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{22} + ( -4 \beta_{1} + 5 \beta_{2} ) q^{23} + ( -1 + \beta_{1} ) q^{24} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{25} + \beta_{1} q^{26} - q^{27} + ( 2 + 5 \beta_{1} - 4 \beta_{2} ) q^{29} + ( -2 - \beta_{1} - \beta_{2} ) q^{30} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{31} + ( -3 - 3 \beta_{1} + \beta_{2} ) q^{32} + ( 1 - \beta_{1} + \beta_{2} ) q^{33} + ( 1 + 4 \beta_{1} ) q^{34} + \beta_{2} q^{36} + ( -1 + \beta_{2} ) q^{37} + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{38} - q^{39} + ( -1 - \beta_{2} ) q^{40} + 3 \beta_{1} q^{41} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{43} - q^{44} + ( 1 + \beta_{1} ) q^{45} + ( -3 + 5 \beta_{1} - 4 \beta_{2} ) q^{46} + ( 2 - 4 \beta_{1} + 4 \beta_{2} ) q^{47} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{48} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{50} + ( -3 - \beta_{2} ) q^{51} + \beta_{2} q^{52} + ( -1 - \beta_{1} + \beta_{2} ) q^{53} -\beta_{1} q^{54} -\beta_{1} q^{55} + ( -1 - 2 \beta_{1} ) q^{57} + ( 6 - 2 \beta_{1} + 5 \beta_{2} ) q^{58} + ( -1 - \beta_{1} - 8 \beta_{2} ) q^{59} + ( -1 - \beta_{1} - \beta_{2} ) q^{60} + ( 5 + 8 \beta_{1} - 3 \beta_{2} ) q^{61} + ( 6 \beta_{1} - \beta_{2} ) q^{62} + ( -1 - 4 \beta_{1} + 3 \beta_{2} ) q^{64} + ( 1 + \beta_{1} ) q^{65} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{66} + ( -3 + 5 \beta_{1} - 5 \beta_{2} ) q^{67} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{68} + ( 4 \beta_{1} - 5 \beta_{2} ) q^{69} + ( 7 - \beta_{1} + 6 \beta_{2} ) q^{71} + ( 1 - \beta_{1} ) q^{72} + ( -\beta_{1} - 6 \beta_{2} ) q^{73} + q^{74} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{75} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{76} -\beta_{1} q^{78} + ( -8 + \beta_{1} - 6 \beta_{2} ) q^{79} + ( -3 - 4 \beta_{1} - 2 \beta_{2} ) q^{80} + q^{81} + ( 6 + 3 \beta_{2} ) q^{82} + ( -4 + \beta_{1} - \beta_{2} ) q^{83} + ( 4 + 4 \beta_{1} + \beta_{2} ) q^{85} + ( 1 - 5 \beta_{1} + 2 \beta_{2} ) q^{86} + ( -2 - 5 \beta_{1} + 4 \beta_{2} ) q^{87} + ( -2 + 3 \beta_{1} - 2 \beta_{2} ) q^{88} + ( 5 - \beta_{1} - 3 \beta_{2} ) q^{89} + ( 2 + \beta_{1} + \beta_{2} ) q^{90} + ( 6 + \beta_{1} - 5 \beta_{2} ) q^{92} + ( -4 + \beta_{1} - 2 \beta_{2} ) q^{93} + ( -4 + 6 \beta_{1} - 4 \beta_{2} ) q^{94} + ( 5 + 3 \beta_{1} + 2 \beta_{2} ) q^{95} + ( 3 + 3 \beta_{1} - \beta_{2} ) q^{96} + ( 7 - 4 \beta_{1} - \beta_{2} ) q^{97} + ( -1 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{8} + 3 q^{9} + 6 q^{10} - 3 q^{11} + 3 q^{13} - 3 q^{15} - 6 q^{16} + 9 q^{17} + 3 q^{19} + 3 q^{20} + 3 q^{22} - 3 q^{24} - 6 q^{25} - 3 q^{27} + 6 q^{29} - 6 q^{30} + 12 q^{31} - 9 q^{32} + 3 q^{33} + 3 q^{34} - 3 q^{37} + 12 q^{38} - 3 q^{39} - 3 q^{40} - 6 q^{43} - 3 q^{44} + 3 q^{45} - 9 q^{46} + 6 q^{47} + 6 q^{48} + 15 q^{50} - 9 q^{51} - 3 q^{53} - 3 q^{57} + 18 q^{58} - 3 q^{59} - 3 q^{60} + 15 q^{61} - 3 q^{64} + 3 q^{65} - 3 q^{66} - 9 q^{67} + 6 q^{68} + 21 q^{71} + 3 q^{72} + 3 q^{74} + 6 q^{75} + 6 q^{76} - 24 q^{79} - 9 q^{80} + 3 q^{81} + 18 q^{82} - 12 q^{83} + 12 q^{85} + 3 q^{86} - 6 q^{87} - 6 q^{88} + 15 q^{89} + 6 q^{90} + 18 q^{92} - 12 q^{93} - 12 q^{94} + 15 q^{95} + 9 q^{96} + 21 q^{97} - 3 q^{99} + O(q^{100}) \)

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
−1.53209
−0.347296
1.87939
−1.53209 −1.00000 0.347296 −0.532089 1.53209 0 2.53209 1.00000 0.815207
1.2 −0.347296 −1.00000 −1.87939 0.652704 0.347296 0 1.34730 1.00000 −0.226682
1.3 1.87939 −1.00000 1.53209 2.87939 −1.87939 0 −0.879385 1.00000 5.41147
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(13\) \(-1\)

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 1911.2.a.o 3
3.b odd 2 1 5733.2.a.y 3
7.b odd 2 1 1911.2.a.p 3
7.c even 3 2 273.2.i.b 6
21.c even 2 1 5733.2.a.z 3
21.h odd 6 2 819.2.j.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.i.b 6 7.c even 3 2
819.2.j.e 6 21.h odd 6 2
1911.2.a.o 3 1.a even 1 1 trivial
1911.2.a.p 3 7.b odd 2 1
5733.2.a.y 3 3.b odd 2 1
5733.2.a.z 3 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1911))\):

\( T_{2}^{3} - 3 T_{2} - 1 \)
\( T_{5}^{3} - 3 T_{5}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 3 T + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 1 - 3 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( -1 + 3 T^{2} + T^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( -17 + 24 T - 9 T^{2} + T^{3} \)
$19$ \( 3 - 9 T - 3 T^{2} + T^{3} \)
$23$ \( 9 - 63 T + T^{3} \)
$29$ \( 289 - 51 T - 6 T^{2} + T^{3} \)
$31$ \( -19 + 39 T - 12 T^{2} + T^{3} \)
$37$ \( -1 + 3 T^{2} + T^{3} \)
$41$ \( -27 - 27 T + T^{3} \)
$43$ \( -51 - 9 T + 6 T^{2} + T^{3} \)
$47$ \( 24 - 36 T - 6 T^{2} + T^{3} \)
$53$ \( -3 + 3 T^{2} + T^{3} \)
$59$ \( -489 - 216 T + 3 T^{2} + T^{3} \)
$61$ \( 1007 - 72 T - 15 T^{2} + T^{3} \)
$67$ \( -73 - 48 T + 9 T^{2} + T^{3} \)
$71$ \( 597 + 54 T - 21 T^{2} + T^{3} \)
$73$ \( -71 - 129 T + T^{3} \)
$79$ \( -521 + 99 T + 24 T^{2} + T^{3} \)
$83$ \( 53 + 45 T + 12 T^{2} + T^{3} \)
$89$ \( 89 + 36 T - 15 T^{2} + T^{3} \)
$97$ \( 269 + 84 T - 21 T^{2} + T^{3} \)
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