Properties

Label 1950.2.e.f
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: no (minimal twist has level 390)
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.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + 2 i q^{7} -i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + 2 i q^{7} -i q^{8} - q^{9} + 4 q^{11} -i q^{12} + i q^{13} -2 q^{14} + q^{16} + 8 i q^{17} -i q^{18} + 6 q^{19} -2 q^{21} + 4 i q^{22} -6 i q^{23} + q^{24} - q^{26} -i q^{27} -2 i q^{28} + 4 q^{29} + i q^{32} + 4 i q^{33} -8 q^{34} + q^{36} -2 i q^{37} + 6 i q^{38} - q^{39} -2 q^{41} -2 i q^{42} + 4 i q^{43} -4 q^{44} + 6 q^{46} + i q^{48} + 3 q^{49} -8 q^{51} -i q^{52} + 10 i q^{53} + q^{54} + 2 q^{56} + 6 i q^{57} + 4 i q^{58} -4 q^{59} -10 q^{61} -2 i q^{63} - q^{64} -4 q^{66} + 12 i q^{67} -8 i q^{68} + 6 q^{69} -8 q^{71} + i q^{72} + 8 i q^{73} + 2 q^{74} -6 q^{76} + 8 i q^{77} -i q^{78} -8 q^{79} + q^{81} -2 i q^{82} -12 i q^{83} + 2 q^{84} -4 q^{86} + 4 i q^{87} -4 i q^{88} + 14 q^{89} -2 q^{91} + 6 i q^{92} - q^{96} -16 i q^{97} + 3 i q^{98} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + 8q^{11} - 4q^{14} + 2q^{16} + 12q^{19} - 4q^{21} + 2q^{24} - 2q^{26} + 8q^{29} - 16q^{34} + 2q^{36} - 2q^{39} - 4q^{41} - 8q^{44} + 12q^{46} + 6q^{49} - 16q^{51} + 2q^{54} + 4q^{56} - 8q^{59} - 20q^{61} - 2q^{64} - 8q^{66} + 12q^{69} - 16q^{71} + 4q^{74} - 12q^{76} - 16q^{79} + 2q^{81} + 4q^{84} - 8q^{86} + 28q^{89} - 4q^{91} - 2q^{96} - 8q^{99} + O(q^{100}) \)

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 2.00000i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 −1.00000 2.00000i 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.f 2
3.b odd 2 1 5850.2.e.i 2
5.b even 2 1 inner 1950.2.e.f 2
5.c odd 4 1 390.2.a.e 1
5.c odd 4 1 1950.2.a.h 1
15.d odd 2 1 5850.2.e.i 2
15.e even 4 1 1170.2.a.e 1
15.e even 4 1 5850.2.a.bi 1
20.e even 4 1 3120.2.a.o 1
60.l odd 4 1 9360.2.a.bh 1
65.f even 4 1 5070.2.b.e 2
65.h odd 4 1 5070.2.a.e 1
65.k even 4 1 5070.2.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.e 1 5.c odd 4 1
1170.2.a.e 1 15.e even 4 1
1950.2.a.h 1 5.c odd 4 1
1950.2.e.f 2 1.a even 1 1 trivial
1950.2.e.f 2 5.b even 2 1 inner
3120.2.a.o 1 20.e even 4 1
5070.2.a.e 1 65.h odd 4 1
5070.2.b.e 2 65.f even 4 1
5070.2.b.e 2 65.k even 4 1
5850.2.a.bi 1 15.e even 4 1
5850.2.e.i 2 3.b odd 2 1
5850.2.e.i 2 15.d odd 2 1
9360.2.a.bh 1 60.l odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 4 \)
\( T_{11} - 4 \)
\( T_{17}^{2} + 64 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 64 + T^{2} \)
$19$ \( ( -6 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( -4 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 64 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -14 + T )^{2} \)
$97$ \( 256 + T^{2} \)
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