Properties

Label 1950.2.f.p
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: \(\Z[a_1, \ldots, a_{7}]\)
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 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.

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

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

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

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 0.438447 1.00000 −1.00000 0
649.2 1.00000 1.00000i 1.00000 0 1.00000i 4.56155 1.00000 −1.00000 0
649.3 1.00000 1.00000i 1.00000 0 1.00000i 0.438447 1.00000 −1.00000 0
649.4 1.00000 1.00000i 1.00000 0 1.00000i 4.56155 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.p 4
5.b even 2 1 1950.2.f.k 4
5.c odd 4 1 1950.2.b.i 4
5.c odd 4 1 1950.2.b.j yes 4
13.b even 2 1 1950.2.f.k 4
65.d even 2 1 inner 1950.2.f.p 4
65.h odd 4 1 1950.2.b.i 4
65.h odd 4 1 1950.2.b.j yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.b.i 4 5.c odd 4 1
1950.2.b.i 4 65.h odd 4 1
1950.2.b.j yes 4 5.c odd 4 1
1950.2.b.j yes 4 65.h odd 4 1
1950.2.f.k 4 5.b even 2 1
1950.2.f.k 4 13.b even 2 1
1950.2.f.p 4 1.a even 1 1 trivial
1950.2.f.p 4 65.d even 2 1 inner

Hecke kernels

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

\( T_{7}^{2} - 5 T_{7} + 2 \)
\( T_{11}^{4} + 9 T_{11}^{2} + 16 \)
\( T_{19}^{4} + 81 T_{19}^{2} + 1296 \)
\( T_{37}^{2} - 5 T_{37} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 2 - 5 T + T^{2} )^{2} \)
$11$ \( 16 + 9 T^{2} + T^{4} \)
$13$ \( 169 + 78 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$17$ \( 1444 + 77 T^{2} + T^{4} \)
$19$ \( 1296 + 81 T^{2} + T^{4} \)
$23$ \( 256 + 36 T^{2} + T^{4} \)
$29$ \( ( -1 + T )^{4} \)
$31$ \( 64 + 33 T^{2} + T^{4} \)
$37$ \( ( 2 - 5 T + T^{2} )^{2} \)
$41$ \( 4 + 21 T^{2} + T^{4} \)
$43$ \( 1024 + 132 T^{2} + T^{4} \)
$47$ \( ( -1 - 8 T + T^{2} )^{2} \)
$53$ \( ( 25 + T^{2} )^{2} \)
$59$ \( 2704 + 121 T^{2} + T^{4} \)
$61$ \( ( -8 - 11 T + T^{2} )^{2} \)
$67$ \( ( -17 + T^{2} )^{2} \)
$71$ \( 43264 + 417 T^{2} + T^{4} \)
$73$ \( ( -8 - 6 T + T^{2} )^{2} \)
$79$ \( ( 34 - 17 T + T^{2} )^{2} \)
$83$ \( ( -208 - T + T^{2} )^{2} \)
$89$ \( 256 + 36 T^{2} + T^{4} \)
$97$ \( ( 64 + 18 T + T^{2} )^{2} \)
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