Properties

Label 1950.2.f.l
Level $1950$
Weight $2$
Character orbit 1950.f
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{1} q^{6} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{7} - q^{8} - q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{1} q^{6} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{7} - q^{8} - q^{9} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + \beta_{1} q^{12} + ( -1 + \beta_{1} - \beta_{2} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{14} + q^{16} -2 \beta_{1} q^{17} + q^{18} + 6 \beta_{1} q^{19} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{21} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{22} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{23} -\beta_{1} q^{24} + ( 1 - \beta_{1} + \beta_{2} ) q^{26} -\beta_{1} q^{27} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{28} -2 q^{29} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{31} - q^{32} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{33} + 2 \beta_{1} q^{34} - q^{36} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{37} -6 \beta_{1} q^{38} + ( -2 - \beta_{1} - \beta_{3} ) q^{39} + ( 5 \beta_{1} + \beta_{2} + \beta_{3} ) q^{41} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{42} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{44} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{46} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{47} + \beta_{1} q^{48} + ( 9 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{49} + 2 q^{51} + ( -1 + \beta_{1} - \beta_{2} ) q^{52} + ( -\beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{53} + \beta_{1} q^{54} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{56} -6 q^{57} + 2 q^{58} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} + 10 q^{61} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{62} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + q^{64} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{66} + ( 8 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{67} -2 \beta_{1} q^{68} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{69} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{71} + q^{72} + ( 10 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{73} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{74} + 6 \beta_{1} q^{76} + 16 \beta_{1} q^{77} + ( 2 + \beta_{1} + \beta_{3} ) q^{78} -8 q^{79} + q^{81} + ( -5 \beta_{1} - \beta_{2} - \beta_{3} ) q^{82} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{84} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{86} -2 \beta_{1} q^{87} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{88} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{89} + ( 8 + 9 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{91} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{92} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{94} -\beta_{1} q^{96} + ( -10 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{97} + ( -9 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{98} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{7} - 4 q^{8} - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{7} - 4 q^{8} - 4 q^{9} - 6 q^{13} + 4 q^{14} + 4 q^{16} + 4 q^{18} + 6 q^{26} - 4 q^{28} - 8 q^{29} - 4 q^{32} - 4 q^{33} - 4 q^{36} - 4 q^{37} - 6 q^{39} + 8 q^{47} + 44 q^{49} + 8 q^{51} - 6 q^{52} + 4 q^{56} - 24 q^{57} + 8 q^{58} + 40 q^{61} + 4 q^{63} + 4 q^{64} + 4 q^{66} + 36 q^{67} + 4 q^{69} + 4 q^{72} + 36 q^{73} + 4 q^{74} + 6 q^{78} - 32 q^{79} + 4 q^{81} - 8 q^{83} + 40 q^{91} + 4 q^{93} - 8 q^{94} - 36 q^{97} - 44 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/4\)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu + 5 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 4 \nu^{2} + 9 \nu - 20 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + \beta_{1} - 10\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{3} - 5 \beta_{2} + 13 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.56155i
2.56155i
1.56155i
2.56155i
−1.00000 1.00000i 1.00000 0 1.00000i −5.12311 −1.00000 −1.00000 0
649.2 −1.00000 1.00000i 1.00000 0 1.00000i 3.12311 −1.00000 −1.00000 0
649.3 −1.00000 1.00000i 1.00000 0 1.00000i −5.12311 −1.00000 −1.00000 0
649.4 −1.00000 1.00000i 1.00000 0 1.00000i 3.12311 −1.00000 −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
65.d 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.f.l 4
5.b even 2 1 1950.2.f.o 4
5.c odd 4 1 390.2.b.d 4
5.c odd 4 1 1950.2.b.h 4
13.b even 2 1 1950.2.f.o 4
15.e even 4 1 1170.2.b.f 4
20.e even 4 1 3120.2.g.o 4
65.d even 2 1 inner 1950.2.f.l 4
65.f even 4 1 5070.2.a.bh 2
65.h odd 4 1 390.2.b.d 4
65.h odd 4 1 1950.2.b.h 4
65.k even 4 1 5070.2.a.bd 2
195.s even 4 1 1170.2.b.f 4
260.p even 4 1 3120.2.g.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.d 4 5.c odd 4 1
390.2.b.d 4 65.h odd 4 1
1170.2.b.f 4 15.e even 4 1
1170.2.b.f 4 195.s even 4 1
1950.2.b.h 4 5.c odd 4 1
1950.2.b.h 4 65.h odd 4 1
1950.2.f.l 4 1.a even 1 1 trivial
1950.2.f.l 4 65.d even 2 1 inner
1950.2.f.o 4 5.b even 2 1
1950.2.f.o 4 13.b even 2 1
3120.2.g.o 4 20.e even 4 1
3120.2.g.o 4 260.p even 4 1
5070.2.a.bd 2 65.k even 4 1
5070.2.a.bh 2 65.f even 4 1

Hecke kernels

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

\( T_{7}^{2} + 2 T_{7} - 16 \)
\( T_{11}^{4} + 36 T_{11}^{2} + 256 \)
\( T_{19}^{2} + 36 \)
\( T_{37}^{2} + 2 T_{37} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( -16 + 2 T + T^{2} )^{2} \)
$11$ \( 256 + 36 T^{2} + T^{4} \)
$13$ \( 169 + 78 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$17$ \( ( 4 + T^{2} )^{2} \)
$19$ \( ( 36 + T^{2} )^{2} \)
$23$ \( 256 + 36 T^{2} + T^{4} \)
$29$ \( ( 2 + T )^{4} \)
$31$ \( 256 + 36 T^{2} + T^{4} \)
$37$ \( ( -16 + 2 T + T^{2} )^{2} \)
$41$ \( 64 + 84 T^{2} + T^{4} \)
$43$ \( 4096 + 144 T^{2} + T^{4} \)
$47$ \( ( -64 - 4 T + T^{2} )^{2} \)
$53$ \( 23104 + 308 T^{2} + T^{4} \)
$59$ \( 64 + 52 T^{2} + T^{4} \)
$61$ \( ( -10 + T )^{4} \)
$67$ \( ( 64 - 18 T + T^{2} )^{2} \)
$71$ \( 4096 + 144 T^{2} + T^{4} \)
$73$ \( ( 64 - 18 T + T^{2} )^{2} \)
$79$ \( ( 8 + T )^{4} \)
$83$ \( ( -64 + 4 T + T^{2} )^{2} \)
$89$ \( 256 + 36 T^{2} + T^{4} \)
$97$ \( ( 64 + 18 T + T^{2} )^{2} \)
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