Properties

Label 950.2.a.i
Level $950$
Weight $2$
Character orbit 950.a
Self dual yes
Analytic conductor $7.586$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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.

\(f(q)\) \(=\) \( q - q^{2} + ( -1 + \beta_{1} ) q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{6} + ( -1 - \beta_{1} ) q^{7} - q^{8} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q - q^{2} + ( -1 + \beta_{1} ) q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{6} + ( -1 - \beta_{1} ) q^{7} - q^{8} + ( 2 + \beta_{2} ) q^{9} + ( -\beta_{1} - \beta_{2} ) q^{11} + ( -1 + \beta_{1} ) q^{12} + ( -3 - \beta_{2} ) q^{13} + ( 1 + \beta_{1} ) q^{14} + q^{16} + ( 1 + \beta_{2} ) q^{17} + ( -2 - \beta_{2} ) q^{18} + q^{19} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{21} + ( \beta_{1} + \beta_{2} ) q^{22} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{23} + ( 1 - \beta_{1} ) q^{24} + ( 3 + \beta_{2} ) q^{26} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{27} + ( -1 - \beta_{1} ) q^{28} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{29} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{31} - q^{32} + ( -2 - \beta_{1} + \beta_{2} ) q^{33} + ( -1 - \beta_{2} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( -4 - 2 \beta_{1} ) q^{37} - q^{38} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{39} + ( 3 \beta_{1} + \beta_{2} ) q^{41} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{42} + ( -6 - \beta_{1} - \beta_{2} ) q^{43} + ( -\beta_{1} - \beta_{2} ) q^{44} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{46} + ( 4 - 2 \beta_{2} ) q^{47} + ( -1 + \beta_{1} ) q^{48} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{49} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{51} + ( -3 - \beta_{2} ) q^{52} + ( -5 + \beta_{2} ) q^{53} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{54} + ( 1 + \beta_{1} ) q^{56} + ( -1 + \beta_{1} ) q^{57} + ( 3 + 2 \beta_{1} + 3 \beta_{2} ) q^{58} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{59} + ( -10 - \beta_{1} - \beta_{2} ) q^{61} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{62} -2 \beta_{1} q^{63} + q^{64} + ( 2 + \beta_{1} - \beta_{2} ) q^{66} + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 1 + \beta_{2} ) q^{68} + ( -9 - 5 \beta_{2} ) q^{69} + ( -4 + 5 \beta_{1} + \beta_{2} ) q^{71} + ( -2 - \beta_{2} ) q^{72} + ( -5 + 2 \beta_{1} - 3 \beta_{2} ) q^{73} + ( 4 + 2 \beta_{1} ) q^{74} + q^{76} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{77} + ( -5 + 3 \beta_{1} - 2 \beta_{2} ) q^{78} + ( -2 + 6 \beta_{1} ) q^{79} + ( -5 - 2 \beta_{1} ) q^{81} + ( -3 \beta_{1} - \beta_{2} ) q^{82} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{84} + ( 6 + \beta_{1} + \beta_{2} ) q^{86} + ( 1 - 5 \beta_{1} + 4 \beta_{2} ) q^{87} + ( \beta_{1} + \beta_{2} ) q^{88} + ( -4 - \beta_{1} + \beta_{2} ) q^{89} + ( 1 + 3 \beta_{1} ) q^{91} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{92} + ( -4 + 3 \beta_{1} - 5 \beta_{2} ) q^{93} + ( -4 + 2 \beta_{2} ) q^{94} + ( 1 - \beta_{1} ) q^{96} + ( -2 + 4 \beta_{2} ) q^{97} + ( 2 - 4 \beta_{1} - \beta_{2} ) q^{98} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 4q^{7} - 3q^{8} + 5q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 4q^{7} - 3q^{8} + 5q^{9} - 2q^{12} - 8q^{13} + 4q^{14} + 3q^{16} + 2q^{17} - 5q^{18} + 3q^{19} - 10q^{21} + 2q^{24} + 8q^{26} - 2q^{27} - 4q^{28} - 8q^{29} + 4q^{31} - 3q^{32} - 8q^{33} - 2q^{34} + 5q^{36} - 14q^{37} - 3q^{38} + 10q^{39} + 2q^{41} + 10q^{42} - 18q^{43} + 14q^{47} - 2q^{48} - 3q^{49} - 6q^{51} - 8q^{52} - 16q^{53} + 2q^{54} + 4q^{56} - 2q^{57} + 8q^{58} + 2q^{59} - 30q^{61} - 4q^{62} - 2q^{63} + 3q^{64} + 8q^{66} - 2q^{67} + 2q^{68} - 22q^{69} - 8q^{71} - 5q^{72} - 10q^{73} + 14q^{74} + 3q^{76} + 8q^{77} - 10q^{78} - 17q^{81} - 2q^{82} + 6q^{83} - 10q^{84} + 18q^{86} - 6q^{87} - 14q^{89} + 6q^{91} - 4q^{93} - 14q^{94} + 2q^{96} - 10q^{97} + 3q^{98} - 12q^{99} + O(q^{100}) \)

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

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

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
−1.76156
−0.363328
3.12489
−1.00000 −2.76156 1.00000 0 2.76156 0.761557 −1.00000 4.62620 0
1.2 −1.00000 −1.36333 1.00000 0 1.36333 −0.636672 −1.00000 −1.14134 0
1.3 −1.00000 2.12489 1.00000 0 −2.12489 −4.12489 −1.00000 1.51514 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.i 3
3.b odd 2 1 8550.2.a.cl 3
4.b odd 2 1 7600.2.a.cd 3
5.b even 2 1 950.2.a.n 3
5.c odd 4 2 190.2.b.b 6
15.d odd 2 1 8550.2.a.ck 3
15.e even 4 2 1710.2.d.d 6
20.d odd 2 1 7600.2.a.bi 3
20.e even 4 2 1520.2.d.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 5.c odd 4 2
950.2.a.i 3 1.a even 1 1 trivial
950.2.a.n 3 5.b even 2 1
1520.2.d.j 6 20.e even 4 2
1710.2.d.d 6 15.e even 4 2
7600.2.a.bi 3 20.d odd 2 1
7600.2.a.cd 3 4.b odd 2 1
8550.2.a.ck 3 15.d odd 2 1
8550.2.a.cl 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(950))\):

\( T_{3}^{3} + 2 T_{3}^{2} - 5 T_{3} - 8 \)
\( T_{7}^{3} + 4 T_{7}^{2} - T_{7} - 2 \)
\( T_{11}^{3} - 10 T_{11} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( -8 - 5 T + 2 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( -2 - T + 4 T^{2} + T^{3} \)
$11$ \( -8 - 10 T + T^{3} \)
$13$ \( -2 + 13 T + 8 T^{2} + T^{3} \)
$17$ \( 4 - 7 T - 2 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -122 - 49 T + T^{3} \)
$29$ \( -410 - 51 T + 8 T^{2} + T^{3} \)
$31$ \( 232 - 62 T - 4 T^{2} + T^{3} \)
$37$ \( 16 + 40 T + 14 T^{2} + T^{3} \)
$41$ \( -100 - 50 T - 2 T^{2} + T^{3} \)
$43$ \( 148 + 98 T + 18 T^{2} + T^{3} \)
$47$ \( 64 + 32 T - 14 T^{2} + T^{3} \)
$53$ \( 106 + 77 T + 16 T^{2} + T^{3} \)
$59$ \( 80 - 29 T - 2 T^{2} + T^{3} \)
$61$ \( 892 + 290 T + 30 T^{2} + T^{3} \)
$67$ \( -64 - 61 T + 2 T^{2} + T^{3} \)
$71$ \( -1016 - 122 T + 8 T^{2} + T^{3} \)
$73$ \( 164 - 95 T + 10 T^{2} + T^{3} \)
$79$ \( -880 - 228 T + T^{3} \)
$83$ \( 8 - 28 T - 6 T^{2} + T^{3} \)
$89$ \( -20 + 46 T + 14 T^{2} + T^{3} \)
$97$ \( -488 - 100 T + 10 T^{2} + T^{3} \)
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