Properties

Label 1911.2.a.j
Level $1911$
Weight $2$
Character orbit 1911.a
Self dual yes
Analytic conductor $15.259$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} - q^{5} + \beta q^{6} -2 \beta q^{8} + q^{9} +O(q^{10})\) \( q + \beta q^{2} + q^{3} - q^{5} + \beta q^{6} -2 \beta q^{8} + q^{9} -\beta q^{10} -2 q^{11} + q^{13} - q^{15} -4 q^{16} -\beta q^{17} + \beta q^{18} + ( -5 - \beta ) q^{19} -2 \beta q^{22} + ( -1 - \beta ) q^{23} -2 \beta q^{24} -4 q^{25} + \beta q^{26} + q^{27} + ( -3 - 3 \beta ) q^{29} -\beta q^{30} + ( 1 + \beta ) q^{31} -2 q^{33} -2 q^{34} -4 q^{37} + ( -2 - 5 \beta ) q^{38} + q^{39} + 2 \beta q^{40} + ( 8 + 2 \beta ) q^{41} + ( 1 - 2 \beta ) q^{43} - q^{45} + ( -2 - \beta ) q^{46} + ( 1 + 4 \beta ) q^{47} -4 q^{48} -4 \beta q^{50} -\beta q^{51} + ( 1 - \beta ) q^{53} + \beta q^{54} + 2 q^{55} + ( -5 - \beta ) q^{57} + ( -6 - 3 \beta ) q^{58} + ( 2 - 4 \beta ) q^{59} + ( -6 + 4 \beta ) q^{61} + ( 2 + \beta ) q^{62} + 8 q^{64} - q^{65} -2 \beta q^{66} + ( -6 - \beta ) q^{67} + ( -1 - \beta ) q^{69} + ( -4 + 2 \beta ) q^{71} -2 \beta q^{72} + ( -7 - \beta ) q^{73} -4 \beta q^{74} -4 q^{75} + \beta q^{78} + ( -7 - 6 \beta ) q^{79} + 4 q^{80} + q^{81} + ( 4 + 8 \beta ) q^{82} + ( 3 + 6 \beta ) q^{83} + \beta q^{85} + ( -4 + \beta ) q^{86} + ( -3 - 3 \beta ) q^{87} + 4 \beta q^{88} + ( -3 - 4 \beta ) q^{89} -\beta q^{90} + ( 1 + \beta ) q^{93} + ( 8 + \beta ) q^{94} + ( 5 + \beta ) q^{95} + ( -7 + 7 \beta ) q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} + 2q^{9} - 4q^{11} + 2q^{13} - 2q^{15} - 8q^{16} - 10q^{19} - 2q^{23} - 8q^{25} + 2q^{27} - 6q^{29} + 2q^{31} - 4q^{33} - 4q^{34} - 8q^{37} - 4q^{38} + 2q^{39} + 16q^{41} + 2q^{43} - 2q^{45} - 4q^{46} + 2q^{47} - 8q^{48} + 2q^{53} + 4q^{55} - 10q^{57} - 12q^{58} + 4q^{59} - 12q^{61} + 4q^{62} + 16q^{64} - 2q^{65} - 12q^{67} - 2q^{69} - 8q^{71} - 14q^{73} - 8q^{75} - 14q^{79} + 8q^{80} + 2q^{81} + 8q^{82} + 6q^{83} - 8q^{86} - 6q^{87} - 6q^{89} + 2q^{93} + 16q^{94} + 10q^{95} - 14q^{97} - 4q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
−1.41421 1.00000 0 −1.00000 −1.41421 0 2.82843 1.00000 1.41421
1.2 1.41421 1.00000 0 −1.00000 1.41421 0 −2.82843 1.00000 −1.41421
\(n\): 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.j yes 2
3.b odd 2 1 5733.2.a.r 2
7.b odd 2 1 1911.2.a.i 2
21.c even 2 1 5733.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.2.a.i 2 7.b odd 2 1
1911.2.a.j yes 2 1.a even 1 1 trivial
5733.2.a.q 2 21.c even 2 1
5733.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1911))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -2 + T^{2} \)
$19$ \( 23 + 10 T + T^{2} \)
$23$ \( -1 + 2 T + T^{2} \)
$29$ \( -9 + 6 T + T^{2} \)
$31$ \( -1 - 2 T + T^{2} \)
$37$ \( ( 4 + T )^{2} \)
$41$ \( 56 - 16 T + T^{2} \)
$43$ \( -7 - 2 T + T^{2} \)
$47$ \( -31 - 2 T + T^{2} \)
$53$ \( -1 - 2 T + T^{2} \)
$59$ \( -28 - 4 T + T^{2} \)
$61$ \( 4 + 12 T + T^{2} \)
$67$ \( 34 + 12 T + T^{2} \)
$71$ \( 8 + 8 T + T^{2} \)
$73$ \( 47 + 14 T + T^{2} \)
$79$ \( -23 + 14 T + T^{2} \)
$83$ \( -63 - 6 T + T^{2} \)
$89$ \( -23 + 6 T + T^{2} \)
$97$ \( -49 + 14 T + T^{2} \)
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