Properties

Label 1950.2.b.k
Level $1950$
Weight $2$
Character orbit 1950.b
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + q^{3} - q^{4} + \beta_{1} q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} -\beta_{1} q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + q^{3} - q^{4} + \beta_{1} q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} -\beta_{1} q^{8} + q^{9} - q^{12} + \beta_{3} q^{13} + ( 1 - \beta_{3} ) q^{14} + q^{16} + ( -1 + \beta_{3} ) q^{17} + \beta_{1} q^{18} + ( \beta_{1} - \beta_{2} ) q^{19} + ( -\beta_{1} + \beta_{2} ) q^{21} + ( 5 + \beta_{3} ) q^{23} -\beta_{1} q^{24} + \beta_{2} q^{26} + q^{27} + ( \beta_{1} - \beta_{2} ) q^{28} + ( 1 - \beta_{3} ) q^{29} -6 \beta_{1} q^{31} + \beta_{1} q^{32} + ( -\beta_{1} + \beta_{2} ) q^{34} - q^{36} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -1 + \beta_{3} ) q^{38} + \beta_{3} q^{39} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 1 - \beta_{3} ) q^{42} -8 q^{43} + ( 5 \beta_{1} + \beta_{2} ) q^{46} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{47} + q^{48} + ( -7 + 2 \beta_{3} ) q^{49} + ( -1 + \beta_{3} ) q^{51} -\beta_{3} q^{52} -6 q^{53} + \beta_{1} q^{54} + ( -1 + \beta_{3} ) q^{56} + ( \beta_{1} - \beta_{2} ) q^{57} + ( \beta_{1} - \beta_{2} ) q^{58} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -4 + 2 \beta_{3} ) q^{61} + 6 q^{62} + ( -\beta_{1} + \beta_{2} ) q^{63} - q^{64} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 1 - \beta_{3} ) q^{68} + ( 5 + \beta_{3} ) q^{69} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{71} -\beta_{1} q^{72} + ( -5 \beta_{1} - \beta_{2} ) q^{73} + ( 2 - 2 \beta_{3} ) q^{74} + ( -\beta_{1} + \beta_{2} ) q^{76} + \beta_{2} q^{78} + 4 \beta_{3} q^{79} + q^{81} + ( -4 - 2 \beta_{3} ) q^{82} + ( 10 \beta_{1} + 2 \beta_{2} ) q^{83} + ( \beta_{1} - \beta_{2} ) q^{84} -8 \beta_{1} q^{86} + ( 1 - \beta_{3} ) q^{87} + ( -8 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 13 \beta_{1} - \beta_{2} ) q^{91} + ( -5 - \beta_{3} ) q^{92} -6 \beta_{1} q^{93} + ( 2 - 2 \beta_{3} ) q^{94} + \beta_{1} q^{96} + ( 5 \beta_{1} + \beta_{2} ) q^{97} + ( -7 \beta_{1} + 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{4} + 4q^{9} - 4q^{12} + 4q^{14} + 4q^{16} - 4q^{17} + 20q^{23} + 4q^{27} + 4q^{29} - 4q^{36} - 4q^{38} + 4q^{42} - 32q^{43} + 4q^{48} - 28q^{49} - 4q^{51} - 24q^{53} - 4q^{56} - 16q^{61} + 24q^{62} - 4q^{64} + 4q^{68} + 20q^{69} + 8q^{74} + 4q^{81} - 16q^{82} + 4q^{87} - 20q^{92} + 8q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 10 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{2} + 5 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
1.30278i
2.30278i
2.30278i
1.30278i
1.00000i 1.00000 −1.00000 0 1.00000i 2.60555i 1.00000i 1.00000 0
1351.2 1.00000i 1.00000 −1.00000 0 1.00000i 4.60555i 1.00000i 1.00000 0
1351.3 1.00000i 1.00000 −1.00000 0 1.00000i 4.60555i 1.00000i 1.00000 0
1351.4 1.00000i 1.00000 −1.00000 0 1.00000i 2.60555i 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
13.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.b.k 4
5.b even 2 1 390.2.b.c 4
5.c odd 4 1 1950.2.f.m 4
5.c odd 4 1 1950.2.f.n 4
13.b even 2 1 inner 1950.2.b.k 4
15.d odd 2 1 1170.2.b.d 4
20.d odd 2 1 3120.2.g.q 4
65.d even 2 1 390.2.b.c 4
65.g odd 4 1 5070.2.a.z 2
65.g odd 4 1 5070.2.a.bf 2
65.h odd 4 1 1950.2.f.m 4
65.h odd 4 1 1950.2.f.n 4
195.e odd 2 1 1170.2.b.d 4
260.g odd 2 1 3120.2.g.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.c 4 5.b even 2 1
390.2.b.c 4 65.d even 2 1
1170.2.b.d 4 15.d odd 2 1
1170.2.b.d 4 195.e odd 2 1
1950.2.b.k 4 1.a even 1 1 trivial
1950.2.b.k 4 13.b even 2 1 inner
1950.2.f.m 4 5.c odd 4 1
1950.2.f.m 4 65.h odd 4 1
1950.2.f.n 4 5.c odd 4 1
1950.2.f.n 4 65.h odd 4 1
3120.2.g.q 4 20.d odd 2 1
3120.2.g.q 4 260.g odd 2 1
5070.2.a.z 2 65.g odd 4 1
5070.2.a.bf 2 65.g odd 4 1

Hecke kernels

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

\( T_{7}^{4} + 28 T_{7}^{2} + 144 \)
\( T_{11} \)
\( T_{17}^{2} + 2 T_{17} - 12 \)
\( T_{19}^{4} + 28 T_{19}^{2} + 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( 144 + 28 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( -13 + T^{2} )^{2} \)
$17$ \( ( -12 + 2 T + T^{2} )^{2} \)
$19$ \( 144 + 28 T^{2} + T^{4} \)
$23$ \( ( 12 - 10 T + T^{2} )^{2} \)
$29$ \( ( -12 - 2 T + T^{2} )^{2} \)
$31$ \( ( 36 + T^{2} )^{2} \)
$37$ \( 2304 + 112 T^{2} + T^{4} \)
$41$ \( 1296 + 136 T^{2} + T^{4} \)
$43$ \( ( 8 + T )^{4} \)
$47$ \( 2304 + 112 T^{2} + T^{4} \)
$53$ \( ( 6 + T )^{4} \)
$59$ \( 2304 + 112 T^{2} + T^{4} \)
$61$ \( ( -36 + 8 T + T^{2} )^{2} \)
$67$ \( 1296 + 136 T^{2} + T^{4} \)
$71$ \( 2304 + 112 T^{2} + T^{4} \)
$73$ \( 144 + 76 T^{2} + T^{4} \)
$79$ \( ( -208 + T^{2} )^{2} \)
$83$ \( 2304 + 304 T^{2} + T^{4} \)
$89$ \( 144 + 232 T^{2} + T^{4} \)
$97$ \( 144 + 76 T^{2} + T^{4} \)
show more
show less