Properties

Label 1900.2.a.j
Level $1900$
Weight $2$
Character orbit 1900.a
Self dual yes
Analytic conductor $15.172$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.1715763840\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \(x^{4} - 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} + ( -3 - \beta_{2} ) q^{11} + ( -2 \beta_{1} + \beta_{3} ) q^{13} + 2 \beta_{3} q^{17} - q^{19} + ( -2 - 2 \beta_{2} ) q^{21} + ( 3 \beta_{1} + \beta_{3} ) q^{23} + ( -\beta_{1} + \beta_{3} ) q^{27} + 2 \beta_{2} q^{29} -4 q^{31} + ( -5 \beta_{1} - \beta_{3} ) q^{33} -3 \beta_{1} q^{37} + ( -7 - \beta_{2} ) q^{39} -6 q^{41} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{43} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{47} + q^{49} + ( -2 + 2 \beta_{2} ) q^{51} + 3 \beta_{1} q^{53} -\beta_{1} q^{57} + ( -6 - 2 \beta_{2} ) q^{59} + ( -5 + 3 \beta_{2} ) q^{61} + ( -3 \beta_{1} + \beta_{3} ) q^{63} -3 \beta_{3} q^{67} + ( 8 + 4 \beta_{2} ) q^{69} + ( -6 + 2 \beta_{2} ) q^{71} + 6 \beta_{1} q^{73} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{77} + ( 2 - 6 \beta_{2} ) q^{79} + ( -4 - 3 \beta_{2} ) q^{81} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{83} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{87} + ( -6 - 4 \beta_{2} ) q^{89} + ( -2 + 6 \beta_{2} ) q^{91} -4 \beta_{1} q^{93} + ( -\beta_{1} - 4 \beta_{3} ) q^{97} + ( -5 - 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 12 q^{11} - 4 q^{19} - 8 q^{21} - 16 q^{31} - 28 q^{39} - 24 q^{41} + 4 q^{49} - 8 q^{51} - 24 q^{59} - 20 q^{61} + 32 q^{69} - 24 q^{71} + 8 q^{79} - 16 q^{81} - 24 q^{89} - 8 q^{91} - 20 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} + 4\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.28825
−0.874032
0.874032
2.28825
0 −2.28825 0 0 0 2.82843 0 2.23607 0
1.2 0 −0.874032 0 0 0 −2.82843 0 −2.23607 0
1.3 0 0.874032 0 0 0 2.82843 0 −2.23607 0
1.4 0 2.28825 0 0 0 −2.82843 0 2.23607 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(19\) \(1\)

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 1900.2.a.j 4
4.b odd 2 1 7600.2.a.ce 4
5.b even 2 1 inner 1900.2.a.j 4
5.c odd 4 2 380.2.c.a 4
15.e even 4 2 3420.2.f.a 4
20.d odd 2 1 7600.2.a.ce 4
20.e even 4 2 1520.2.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.c.a 4 5.c odd 4 2
1520.2.d.f 4 20.e even 4 2
1900.2.a.j 4 1.a even 1 1 trivial
1900.2.a.j 4 5.b even 2 1 inner
3420.2.f.a 4 15.e even 4 2
7600.2.a.ce 4 4.b odd 2 1
7600.2.a.ce 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1900))\):

\( T_{3}^{4} - 6 T_{3}^{2} + 4 \)
\( T_{7}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 - 6 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -8 + T^{2} )^{2} \)
$11$ \( ( 4 + 6 T + T^{2} )^{2} \)
$13$ \( 484 - 46 T^{2} + T^{4} \)
$17$ \( 64 - 56 T^{2} + T^{4} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( 64 - 56 T^{2} + T^{4} \)
$29$ \( ( -20 + T^{2} )^{2} \)
$31$ \( ( 4 + T )^{4} \)
$37$ \( 324 - 54 T^{2} + T^{4} \)
$41$ \( ( 6 + T )^{4} \)
$43$ \( ( -72 + T^{2} )^{2} \)
$47$ \( ( -72 + T^{2} )^{2} \)
$53$ \( 324 - 54 T^{2} + T^{4} \)
$59$ \( ( 16 + 12 T + T^{2} )^{2} \)
$61$ \( ( -20 + 10 T + T^{2} )^{2} \)
$67$ \( 324 - 126 T^{2} + T^{4} \)
$71$ \( ( 16 + 12 T + T^{2} )^{2} \)
$73$ \( 5184 - 216 T^{2} + T^{4} \)
$79$ \( ( -176 - 4 T + T^{2} )^{2} \)
$83$ \( 5184 - 216 T^{2} + T^{4} \)
$89$ \( ( -44 + 12 T + T^{2} )^{2} \)
$97$ \( 3844 - 214 T^{2} + T^{4} \)
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