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: \(x^{6} + 9 x^{4} + 22 x^{2} + 9\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 6 \nu^{3} + 4 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\( \nu^{4} + 3 \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 9 \nu^{3} + 19 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\( -\nu^{5} - 5 \nu^{3} - 4 \nu \)
\(\beta_{5}\)\(=\)\( 2 \nu^{4} + 11 \nu^{2} + 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} + \beta_{3} - 4 \beta_{1}\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - 2 \beta_{2} - 14\)\()/5\)
\(\nu^{3}\)\(=\)\(\beta_{4} + 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{5} + 11 \beta_{2} + 57\)\()/5\)
\(\nu^{5}\)\(=\)\((\)\(-26 \beta_{4} - 4 \beta_{3} - 59 \beta_{1}\)\()/5\)

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 \(S_{2}^{\mathrm{new}}(1900, [\chi])\):

\( T_{3}^{6} + 10 T_{3}^{4} + 13 T_{3}^{2} + 1 \)
\( T_{7}^{6} + 10 T_{7}^{4} + 13 T_{7}^{2} + 1 \)

Hecke characteristic polynomials

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