Properties

Label 1950.2.e.a
Level $1950$
Weight $2$
Character orbit 1950.e
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.e.a 2
3.b odd 2 1 5850.2.e.be 2
5.b even 2 1 inner 1950.2.e.a 2
5.c odd 4 1 1950.2.a.l 1
5.c odd 4 1 1950.2.a.p yes 1
15.d odd 2 1 5850.2.e.be 2
15.e even 4 1 5850.2.a.k 1
15.e even 4 1 5850.2.a.bu 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.a.l 1 5.c odd 4 1
1950.2.a.p yes 1 5.c odd 4 1
1950.2.e.a 2 1.a even 1 1 trivial
1950.2.e.a 2 5.b even 2 1 inner
5850.2.a.k 1 15.e even 4 1
5850.2.a.bu 1 15.e even 4 1
5850.2.e.be 2 3.b odd 2 1
5850.2.e.be 2 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 \(S_{2}^{\mathrm{new}}(1950, [\chi])\):

\( T_{7}^{2} + 1 \)
\( T_{11} + 5 \)
\( T_{17}^{2} + 25 \)
\( T_{31} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 5 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 25 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -7 + T )^{2} \)
$31$ \( ( 9 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 81 + T^{2} \)
$53$ \( 121 + T^{2} \)
$59$ \( ( 1 + T )^{2} \)
$61$ \( ( 7 + T )^{2} \)
$67$ \( 225 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 81 + T^{2} \)
$89$ \( ( 16 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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