Properties

Label 1900.2.c.g
Level $1900$
Weight $2$
Character orbit 1900.c
Analytic conductor $15.172$
Analytic rank $0$
Dimension $6$
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.4227136.2
Defining polynomial: \( x^{6} + 9x^{4} + 22x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_1 q^{7} + \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_1 q^{7} + \beta_{5} q^{9} - \beta_{2} q^{11} + ( - \beta_{3} + \beta_1) q^{13} + \beta_1 q^{17} + q^{19} + ( - \beta_{5} + 3) q^{21} + (\beta_{4} + \beta_1) q^{23} + ( - \beta_{4} - \beta_{3} + 2 \beta_1) q^{27} + (\beta_{2} + 2) q^{29} + ( - \beta_{5} + \beta_{2} + 3) q^{31} + \beta_{4} q^{33} + (\beta_{4} + 3 \beta_{3} - 2 \beta_1) q^{37} + ( - \beta_{5} + 2) q^{39} + ( - \beta_{5} + 4) q^{41} + (\beta_{4} - \beta_{3} + \beta_1) q^{43} + ( - 2 \beta_{4} - \beta_{3} + \beta_1) q^{47} + (\beta_{5} + 4) q^{49} + ( - \beta_{5} + 3) q^{51} + (\beta_{4} - \beta_{3} + \beta_1) q^{53} - \beta_1 q^{57} + ( - 2 \beta_{5} - 2 \beta_{2} + 2) q^{59} + (\beta_{5} - \beta_{2} + 3) q^{61} + (\beta_{4} + \beta_{3} - 5 \beta_1) q^{63} + (\beta_{4} + 3 \beta_{3} + 2 \beta_1) q^{67} + (\beta_{2} + 4) q^{69} + (2 \beta_{5} - 1) q^{71} - 3 \beta_{3} q^{73} - \beta_{4} q^{77} + ( - 2 \beta_{5} + \beta_{2} + 5) q^{79} + ( - \beta_{2} + 4) q^{81} + ( - 5 \beta_{3} + 6 \beta_1) q^{83} + ( - \beta_{4} - 2 \beta_1) q^{87} + (3 \beta_{5} + 1) q^{89} + (\beta_{5} - 2) q^{91} + (\beta_{3} - 8 \beta_1) q^{93} + ( - 3 \beta_{3} + 3 \beta_1) q^{97} + (\beta_{5} - 2 \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 14 q^{29} + 22 q^{31} + 14 q^{39} + 26 q^{41} + 22 q^{49} + 20 q^{51} + 12 q^{59} + 14 q^{61} + 26 q^{69} - 10 q^{71} + 36 q^{79} + 22 q^{81} - 14 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9x^{4} + 22x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 6\nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} + 3\nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 9\nu^{3} + 19\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} - 5\nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{4} + 11\nu^{2} + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} - 4\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{2} - 14 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{5} + 11\beta_{2} + 57 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -26\beta_{4} - 4\beta_{3} - 59\beta_1 ) / 5 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1749.1
1.91223i
2.19869i
0.713538i
0.713538i
2.19869i
1.91223i
0 2.91223i 0 0 0 2.91223i 0 −5.48108 0
1749.2 0 1.19869i 0 0 0 1.19869i 0 1.56314 0
1749.3 0 0.286462i 0 0 0 0.286462i 0 2.91794 0
1749.4 0 0.286462i 0 0 0 0.286462i 0 2.91794 0
1749.5 0 1.19869i 0 0 0 1.19869i 0 1.56314 0
1749.6 0 2.91223i 0 0 0 2.91223i 0 −5.48108 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1749.6
Significant digits:
Format:

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.c.g 6
5.b even 2 1 inner 1900.2.c.g 6
5.c odd 4 1 1900.2.a.f 3
5.c odd 4 1 1900.2.a.h yes 3
20.e even 4 1 7600.2.a.bj 3
20.e even 4 1 7600.2.a.by 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.a.f 3 5.c odd 4 1
1900.2.a.h yes 3 5.c odd 4 1
1900.2.c.g 6 1.a even 1 1 trivial
1900.2.c.g 6 5.b even 2 1 inner
7600.2.a.bj 3 20.e even 4 1
7600.2.a.by 3 20.e even 4 1

Hecke kernels

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

\( T_{3}^{6} + 10T_{3}^{4} + 13T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} + 10T_{7}^{4} + 13T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 10 T^{4} + 13 T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 10 T^{4} + 13 T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 26 T + 15)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 17 T^{4} + 70 T^{2} + 9 \) Copy content Toggle raw display
$17$ \( T^{6} + 10 T^{4} + 13 T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 70 T^{4} + 153 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( (T^{3} - 7 T^{2} - 10 T + 25)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 11 T^{2} - 6 T + 241)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 169 T^{4} + 9234 T^{2} + \cdots + 164025 \) Copy content Toggle raw display
$41$ \( (T^{3} - 13 T^{2} + 36 T - 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 75 T^{4} + 1123 T^{2} + \cdots + 729 \) Copy content Toggle raw display
$47$ \( T^{6} + 225 T^{4} + 7018 T^{2} + \cdots + 3969 \) Copy content Toggle raw display
$53$ \( T^{6} + 75 T^{4} + 1123 T^{2} + \cdots + 729 \) Copy content Toggle raw display
$59$ \( (T^{3} - 6 T^{2} - 176 T + 856)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 7 T^{2} - 30 T - 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 265 T^{4} + 19978 T^{2} + \cdots + 358801 \) Copy content Toggle raw display
$71$ \( (T^{3} + 5 T^{2} - 73 T + 123)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 117 T^{4} + 810 T^{2} + \cdots + 729 \) Copy content Toggle raw display
$79$ \( (T^{3} - 18 T^{2} + T + 607)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 505 T^{4} + 65998 T^{2} + \cdots + 561001 \) Copy content Toggle raw display
$89$ \( (T^{3} - 183 T + 857)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 153 T^{4} + 5670 T^{2} + \cdots + 6561 \) Copy content Toggle raw display
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