Properties

Label 1950.2.e.b
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.

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

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.b 2
3.b odd 2 1 5850.2.e.x 2
5.b even 2 1 inner 1950.2.e.b 2
5.c odd 4 1 1950.2.a.m 1
5.c odd 4 1 1950.2.a.q yes 1
15.d odd 2 1 5850.2.e.x 2
15.e even 4 1 5850.2.a.j 1
15.e even 4 1 5850.2.a.bt 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.a.m 1 5.c odd 4 1
1950.2.a.q yes 1 5.c odd 4 1
1950.2.e.b 2 1.a even 1 1 trivial
1950.2.e.b 2 5.b even 2 1 inner
5850.2.a.j 1 15.e even 4 1
5850.2.a.bt 1 15.e even 4 1
5850.2.e.x 2 3.b odd 2 1
5850.2.e.x 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 \(S_{2}^{\mathrm{new}}(1950, [\chi])\):

\( T_{7}^{2} + 1 \)
\( T_{11} + 3 \)
\( T_{17}^{2} + 1 \)
\( T_{31} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 1 + T^{2} \)
$19$ \( ( -8 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( -7 + T )^{2} \)
$31$ \( ( -1 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 144 + T^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( 25 + T^{2} \)
$59$ \( ( -9 + T )^{2} \)
$61$ \( ( -5 + T )^{2} \)
$67$ \( 121 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 49 + T^{2} \)
$89$ \( ( -8 + T )^{2} \)
$97$ \( 36 + T^{2} \)
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