Properties

Label 950.2.a.l
Level $950$
Weight $2$
Character orbit 950.a
Self dual yes
Analytic conductor $7.586$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.993.1
Defining polynomial: \(x^{3} - x^{2} - 6 x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( 1 - \beta_{1} ) q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{6} + ( 1 - \beta_{2} ) q^{7} + q^{8} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 - \beta_{1} ) q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{6} + ( 1 - \beta_{2} ) q^{7} + q^{8} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{9} + 2 \beta_{1} q^{11} + ( 1 - \beta_{1} ) q^{12} + ( -2 + \beta_{1} - \beta_{2} ) q^{13} + ( 1 - \beta_{2} ) q^{14} + q^{16} + ( 4 - \beta_{1} + \beta_{2} ) q^{17} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{18} + q^{19} + ( 2 + \beta_{1} ) q^{21} + 2 \beta_{1} q^{22} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{23} + ( 1 - \beta_{1} ) q^{24} + ( -2 + \beta_{1} - \beta_{2} ) q^{26} + ( 6 - 3 \beta_{1} + 2 \beta_{2} ) q^{27} + ( 1 - \beta_{2} ) q^{28} + ( 2 + \beta_{1} - \beta_{2} ) q^{29} + ( 2 + 2 \beta_{1} ) q^{31} + q^{32} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{33} + ( 4 - \beta_{1} + \beta_{2} ) q^{34} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{36} + ( -1 + 2 \beta_{1} ) q^{37} + q^{38} + ( -5 + 5 \beta_{1} - \beta_{2} ) q^{39} + ( 2 + \beta_{1} ) q^{42} + ( -4 - 2 \beta_{1} ) q^{43} + 2 \beta_{1} q^{44} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{46} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 1 - \beta_{1} ) q^{48} + ( 3 + 3 \beta_{1} - 3 \beta_{2} ) q^{49} + ( 7 - 7 \beta_{1} + \beta_{2} ) q^{51} + ( -2 + \beta_{1} - \beta_{2} ) q^{52} + ( -5 + 4 \beta_{1} + \beta_{2} ) q^{53} + ( 6 - 3 \beta_{1} + 2 \beta_{2} ) q^{54} + ( 1 - \beta_{2} ) q^{56} + ( 1 - \beta_{1} ) q^{57} + ( 2 + \beta_{1} - \beta_{2} ) q^{58} + ( 1 - \beta_{1} + 4 \beta_{2} ) q^{59} -2 \beta_{2} q^{61} + ( 2 + 2 \beta_{1} ) q^{62} + ( -5 - \beta_{1} + 2 \beta_{2} ) q^{63} + q^{64} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{66} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{67} + ( 4 - \beta_{1} + \beta_{2} ) q^{68} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{69} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{72} + ( -5 + \beta_{1} + 2 \beta_{2} ) q^{73} + ( -1 + 2 \beta_{1} ) q^{74} + q^{76} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{77} + ( -5 + 5 \beta_{1} - \beta_{2} ) q^{78} + ( 8 - 4 \beta_{1} ) q^{79} + ( 10 - 7 \beta_{1} ) q^{81} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{83} + ( 2 + \beta_{1} ) q^{84} + ( -4 - 2 \beta_{1} ) q^{86} + ( -1 + \beta_{1} - \beta_{2} ) q^{87} + 2 \beta_{1} q^{88} + ( 6 - 4 \beta_{1} ) q^{89} + ( 6 + 2 \beta_{1} - \beta_{2} ) q^{91} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{92} + ( -6 - 2 \beta_{2} ) q^{93} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 1 - \beta_{1} ) q^{96} + ( -8 - 6 \beta_{1} + 2 \beta_{2} ) q^{97} + ( 3 + 3 \beta_{1} - 3 \beta_{2} ) q^{98} + ( -14 + 8 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 2q^{7} + 3q^{8} + 5q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 2q^{7} + 3q^{8} + 5q^{9} + 2q^{11} + 2q^{12} - 6q^{13} + 2q^{14} + 3q^{16} + 12q^{17} + 5q^{18} + 3q^{19} + 7q^{21} + 2q^{22} - 2q^{23} + 2q^{24} - 6q^{26} + 17q^{27} + 2q^{28} + 6q^{29} + 8q^{31} + 3q^{32} - 24q^{33} + 12q^{34} + 5q^{36} - q^{37} + 3q^{38} - 11q^{39} + 7q^{42} - 14q^{43} + 2q^{44} - 2q^{46} + 3q^{47} + 2q^{48} + 9q^{49} + 15q^{51} - 6q^{52} - 10q^{53} + 17q^{54} + 2q^{56} + 2q^{57} + 6q^{58} + 6q^{59} - 2q^{61} + 8q^{62} - 14q^{63} + 3q^{64} - 24q^{66} + 4q^{67} + 12q^{68} - 6q^{71} + 5q^{72} - 12q^{73} - q^{74} + 3q^{76} - 10q^{77} - 11q^{78} + 20q^{79} + 23q^{81} - 4q^{83} + 7q^{84} - 14q^{86} - 3q^{87} + 2q^{88} + 14q^{89} + 19q^{91} - 2q^{92} - 20q^{93} + 3q^{94} + 2q^{96} - 28q^{97} + 9q^{98} - 36q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)

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
2.77339
0.480031
−2.25342
1.00000 −1.77339 1.00000 0 −1.77339 −2.69168 1.00000 0.144903 0
1.2 1.00000 0.519969 1.00000 0 0.519969 4.76957 1.00000 −2.72963 0
1.3 1.00000 3.25342 1.00000 0 3.25342 −0.0778929 1.00000 7.58473 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(19\) \(-1\)

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 950.2.a.l yes 3
3.b odd 2 1 8550.2.a.ci 3
4.b odd 2 1 7600.2.a.bk 3
5.b even 2 1 950.2.a.j 3
5.c odd 4 2 950.2.b.h 6
15.d odd 2 1 8550.2.a.cp 3
20.d odd 2 1 7600.2.a.bz 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.j 3 5.b even 2 1
950.2.a.l yes 3 1.a even 1 1 trivial
950.2.b.h 6 5.c odd 4 2
7600.2.a.bk 3 4.b odd 2 1
7600.2.a.bz 3 20.d odd 2 1
8550.2.a.ci 3 3.b odd 2 1
8550.2.a.cp 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(950))\):

\( T_{3}^{3} - 2 T_{3}^{2} - 5 T_{3} + 3 \)
\( T_{7}^{3} - 2 T_{7}^{2} - 13 T_{7} - 1 \)
\( T_{11}^{3} - 2 T_{11}^{2} - 24 T_{11} + 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( 3 - 5 T - 2 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( -1 - 13 T - 2 T^{2} + T^{3} \)
$11$ \( 24 - 24 T - 2 T^{2} + T^{3} \)
$13$ \( -35 - 3 T + 6 T^{2} + T^{3} \)
$17$ \( 9 + 33 T - 12 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( 111 - 51 T + 2 T^{2} + T^{3} \)
$29$ \( 9 - 3 T - 6 T^{2} + T^{3} \)
$31$ \( 56 - 4 T - 8 T^{2} + T^{3} \)
$37$ \( -1 - 25 T + T^{2} + T^{3} \)
$41$ \( T^{3} \)
$43$ \( -24 + 40 T + 14 T^{2} + T^{3} \)
$47$ \( -45 - 57 T - 3 T^{2} + T^{3} \)
$53$ \( -867 - 105 T + 10 T^{2} + T^{3} \)
$59$ \( 1431 - 201 T - 6 T^{2} + T^{3} \)
$61$ \( -120 - 56 T + 2 T^{2} + T^{3} \)
$67$ \( 75 - 23 T - 4 T^{2} + T^{3} \)
$71$ \( -1512 - 192 T + 6 T^{2} + T^{3} \)
$73$ \( -317 - 27 T + 12 T^{2} + T^{3} \)
$79$ \( 320 + 32 T - 20 T^{2} + T^{3} \)
$83$ \( 168 - 108 T + 4 T^{2} + T^{3} \)
$89$ \( 312 - 36 T - 14 T^{2} + T^{3} \)
$97$ \( -2440 + 44 T + 28 T^{2} + T^{3} \)
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