Properties

Label 1900.3.g.a
Level $1900$
Weight $3$
Character orbit 1900.g
Analytic conductor $51.771$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1900.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.7712502285\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{57})\)
Defining polynomial: \(x^{4} + 29 x^{2} + 196\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 \beta_{1} + \beta_{2} ) q^{7} -9 q^{9} +O(q^{10})\) \( q + ( 2 \beta_{1} + \beta_{2} ) q^{7} -9 q^{9} + ( -2 + \beta_{3} ) q^{11} + ( 6 \beta_{1} + \beta_{2} ) q^{17} + 19 q^{19} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{23} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{43} + ( -6 \beta_{1} - 7 \beta_{2} ) q^{47} + ( -87 + 3 \beta_{3} ) q^{49} + ( -50 - 3 \beta_{3} ) q^{61} + ( -18 \beta_{1} - 9 \beta_{2} ) q^{63} + ( 26 \beta_{1} + 7 \beta_{2} ) q^{73} + ( 9 \beta_{1} - 26 \beta_{2} ) q^{77} + 81 q^{81} + ( 9 \beta_{1} - 9 \beta_{2} ) q^{83} + ( 18 - 9 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 36q^{9} + O(q^{10}) \) \( 4q - 36q^{9} - 6q^{11} + 76q^{19} - 342q^{49} - 206q^{61} + 324q^{81} + 54q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 29 x^{2} + 196\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3 \nu^{3} + 59 \nu \)\()/14\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 13 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( 5 \nu^{2} + 73 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{2} + 7 \beta_{1}\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 73\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(-59 \beta_{2} - 91 \beta_{1}\)\()/10\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
4.27492i
3.27492i
3.27492i
4.27492i
0 0 0 0 0 13.8248i 0 −9.00000 0
949.2 0 0 0 0 0 8.82475i 0 −9.00000 0
949.3 0 0 0 0 0 8.82475i 0 −9.00000 0
949.4 0 0 0 0 0 13.8248i 0 −9.00000 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
5.b even 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.3.g.a 4
5.b even 2 1 inner 1900.3.g.a 4
5.c odd 4 1 76.3.c.b 2
5.c odd 4 1 1900.3.e.a 2
15.e even 4 1 684.3.h.a 2
19.b odd 2 1 CM 1900.3.g.a 4
20.e even 4 1 304.3.e.e 2
40.i odd 4 1 1216.3.e.f 2
40.k even 4 1 1216.3.e.e 2
60.l odd 4 1 2736.3.o.c 2
95.d odd 2 1 inner 1900.3.g.a 4
95.g even 4 1 76.3.c.b 2
95.g even 4 1 1900.3.e.a 2
285.j odd 4 1 684.3.h.a 2
380.j odd 4 1 304.3.e.e 2
760.t even 4 1 1216.3.e.f 2
760.y odd 4 1 1216.3.e.e 2
1140.w even 4 1 2736.3.o.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.b 2 5.c odd 4 1
76.3.c.b 2 95.g even 4 1
304.3.e.e 2 20.e even 4 1
304.3.e.e 2 380.j odd 4 1
684.3.h.a 2 15.e even 4 1
684.3.h.a 2 285.j odd 4 1
1216.3.e.e 2 40.k even 4 1
1216.3.e.e 2 760.y odd 4 1
1216.3.e.f 2 40.i odd 4 1
1216.3.e.f 2 760.t even 4 1
1900.3.e.a 2 5.c odd 4 1
1900.3.e.a 2 95.g even 4 1
1900.3.g.a 4 1.a even 1 1 trivial
1900.3.g.a 4 5.b even 2 1 inner
1900.3.g.a 4 19.b odd 2 1 CM
1900.3.g.a 4 95.d odd 2 1 inner
2736.3.o.c 2 60.l odd 4 1
2736.3.o.c 2 1140.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{3}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 14884 + 269 T^{2} + T^{4} \)
$11$ \( ( -354 + 3 T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 412164 + 1509 T^{2} + T^{4} \)
$19$ \( ( -19 + T )^{4} \)
$23$ \( ( 900 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( 2815684 + 3869 T^{2} + T^{4} \)
$47$ \( 1004004 + 7629 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -554 + 103 T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( 235991044 + 31349 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 8100 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
show more
show less