Properties

Label 1950.2.a.bc
Level $1950$
Weight $2$
Character orbit 1950.a
Self dual yes
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} + ( -1 - \beta ) q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} - q^{3} + q^{4} + q^{6} + ( -1 - \beta ) q^{7} - q^{8} + q^{9} + ( 1 + \beta ) q^{11} - q^{12} + q^{13} + ( 1 + \beta ) q^{14} + q^{16} + ( -3 + \beta ) q^{17} - q^{18} + ( 2 - \beta ) q^{19} + ( 1 + \beta ) q^{21} + ( -1 - \beta ) q^{22} + q^{24} - q^{26} - q^{27} + ( -1 - \beta ) q^{28} + ( 1 - 2 \beta ) q^{29} + ( 1 - 3 \beta ) q^{31} - q^{32} + ( -1 - \beta ) q^{33} + ( 3 - \beta ) q^{34} + q^{36} + ( -2 + \beta ) q^{37} + ( -2 + \beta ) q^{38} - q^{39} -\beta q^{41} + ( -1 - \beta ) q^{42} + ( 4 + 2 \beta ) q^{43} + ( 1 + \beta ) q^{44} -7 q^{47} - q^{48} + ( 4 + 3 \beta ) q^{49} + ( 3 - \beta ) q^{51} + q^{52} + ( 5 - 2 \beta ) q^{53} + q^{54} + ( 1 + \beta ) q^{56} + ( -2 + \beta ) q^{57} + ( -1 + 2 \beta ) q^{58} + ( -1 + \beta ) q^{59} + ( 3 + 3 \beta ) q^{61} + ( -1 + 3 \beta ) q^{62} + ( -1 - \beta ) q^{63} + q^{64} + ( 1 + \beta ) q^{66} + ( -1 + 2 \beta ) q^{67} + ( -3 + \beta ) q^{68} + ( 2 - \beta ) q^{71} - q^{72} + 12 q^{73} + ( 2 - \beta ) q^{74} + ( 2 - \beta ) q^{76} + ( -11 - 3 \beta ) q^{77} + q^{78} + ( -2 - \beta ) q^{79} + q^{81} + \beta q^{82} + ( 7 + \beta ) q^{83} + ( 1 + \beta ) q^{84} + ( -4 - 2 \beta ) q^{86} + ( -1 + 2 \beta ) q^{87} + ( -1 - \beta ) q^{88} + ( 4 + 2 \beta ) q^{89} + ( -1 - \beta ) q^{91} + ( -1 + 3 \beta ) q^{93} + 7 q^{94} + q^{96} + ( 10 - 2 \beta ) q^{97} + ( -4 - 3 \beta ) q^{98} + ( 1 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 3 q^{7} - 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 3 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{11} - 2 q^{12} + 2 q^{13} + 3 q^{14} + 2 q^{16} - 5 q^{17} - 2 q^{18} + 3 q^{19} + 3 q^{21} - 3 q^{22} + 2 q^{24} - 2 q^{26} - 2 q^{27} - 3 q^{28} - q^{31} - 2 q^{32} - 3 q^{33} + 5 q^{34} + 2 q^{36} - 3 q^{37} - 3 q^{38} - 2 q^{39} - q^{41} - 3 q^{42} + 10 q^{43} + 3 q^{44} - 14 q^{47} - 2 q^{48} + 11 q^{49} + 5 q^{51} + 2 q^{52} + 8 q^{53} + 2 q^{54} + 3 q^{56} - 3 q^{57} - q^{59} + 9 q^{61} + q^{62} - 3 q^{63} + 2 q^{64} + 3 q^{66} - 5 q^{68} + 3 q^{71} - 2 q^{72} + 24 q^{73} + 3 q^{74} + 3 q^{76} - 25 q^{77} + 2 q^{78} - 5 q^{79} + 2 q^{81} + q^{82} + 15 q^{83} + 3 q^{84} - 10 q^{86} - 3 q^{88} + 10 q^{89} - 3 q^{91} + q^{93} + 14 q^{94} + 2 q^{96} + 18 q^{97} - 11 q^{98} + 3 q^{99} + O(q^{100}) \)

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
3.70156
−2.70156
−1.00000 −1.00000 1.00000 0 1.00000 −4.70156 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 1.70156 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(1\)
\(13\) \(-1\)

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 1950.2.a.bc 2
3.b odd 2 1 5850.2.a.cj 2
5.b even 2 1 1950.2.a.bg yes 2
5.c odd 4 2 1950.2.e.p 4
15.d odd 2 1 5850.2.a.cg 2
15.e even 4 2 5850.2.e.bi 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.a.bc 2 1.a even 1 1 trivial
1950.2.a.bg yes 2 5.b even 2 1
1950.2.e.p 4 5.c odd 4 2
5850.2.a.cg 2 15.d odd 2 1
5850.2.a.cj 2 3.b odd 2 1
5850.2.e.bi 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1950))\):

\( T_{7}^{2} + 3 T_{7} - 8 \)
\( T_{11}^{2} - 3 T_{11} - 8 \)
\( T_{17}^{2} + 5 T_{17} - 4 \)
\( T_{23} \)
\( T_{31}^{2} + T_{31} - 92 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -8 + 3 T + T^{2} \)
$11$ \( -8 - 3 T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -4 + 5 T + T^{2} \)
$19$ \( -8 - 3 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( -41 + T^{2} \)
$31$ \( -92 + T + T^{2} \)
$37$ \( -8 + 3 T + T^{2} \)
$41$ \( -10 + T + T^{2} \)
$43$ \( -16 - 10 T + T^{2} \)
$47$ \( ( 7 + T )^{2} \)
$53$ \( -25 - 8 T + T^{2} \)
$59$ \( -10 + T + T^{2} \)
$61$ \( -72 - 9 T + T^{2} \)
$67$ \( -41 + T^{2} \)
$71$ \( -8 - 3 T + T^{2} \)
$73$ \( ( -12 + T )^{2} \)
$79$ \( -4 + 5 T + T^{2} \)
$83$ \( 46 - 15 T + T^{2} \)
$89$ \( -16 - 10 T + T^{2} \)
$97$ \( 40 - 18 T + T^{2} \)
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