Properties

Label 1900.2.c.f
Level $1900$
Weight $2$
Character orbit 1900.c
Analytic conductor $15.172$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: 6.0.6594624.1
Defining polynomial: \(x^{6} + 9 x^{4} + 18 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( \beta_{4} - \beta_{5} ) q^{3} + ( 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{7} + ( -2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( \beta_{4} - \beta_{5} ) q^{3} + ( 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{7} + ( -2 + \beta_{2} ) q^{9} -\beta_{3} q^{11} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{13} + ( -2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{17} - q^{19} + ( -1 - 3 \beta_{2} - 2 \beta_{3} ) q^{21} + ( \beta_{1} + 5 \beta_{4} ) q^{23} + ( 2 \beta_{1} - \beta_{5} ) q^{27} + ( -2 + 2 \beta_{2} + \beta_{3} ) q^{29} + ( -1 + 3 \beta_{2} - \beta_{3} ) q^{31} + ( -\beta_{1} + 4 \beta_{4} + \beta_{5} ) q^{33} + ( 4 \beta_{4} + \beta_{5} ) q^{37} + ( -4 - \beta_{2} ) q^{39} -3 \beta_{2} q^{41} + ( 5 \beta_{4} - 2 \beta_{5} ) q^{43} + ( -5 \beta_{1} - \beta_{4} - \beta_{5} ) q^{47} + ( -6 - 3 \beta_{2} ) q^{49} + ( -5 + 5 \beta_{2} - 2 \beta_{3} ) q^{51} + ( -4 \beta_{1} + \beta_{4} ) q^{53} + ( -\beta_{4} + \beta_{5} ) q^{57} + ( 2 - 2 \beta_{2} + 2 \beta_{3} ) q^{59} + ( -3 + 5 \beta_{2} - \beta_{3} ) q^{61} + ( -2 \beta_{1} + 7 \beta_{4} ) q^{63} + ( -4 \beta_{1} - 4 \beta_{4} - 3 \beta_{5} ) q^{67} + ( -4 - 2 \beta_{2} + 5 \beta_{3} ) q^{69} + ( -1 - 2 \beta_{2} + 2 \beta_{3} ) q^{71} + ( \beta_{1} + 4 \beta_{4} + 2 \beta_{5} ) q^{73} + ( \beta_{1} + 2 \beta_{4} + 3 \beta_{5} ) q^{77} + ( 1 - 5 \beta_{3} ) q^{79} + ( -8 - \beta_{3} ) q^{81} + ( -\beta_{1} + 10 \beta_{4} ) q^{83} + ( 5 \beta_{1} - 8 \beta_{4} + \beta_{5} ) q^{87} + ( -3 - 3 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -6 - 5 \beta_{2} ) q^{91} + ( 5 \beta_{1} + 2 \beta_{5} ) q^{93} + ( -\beta_{1} - \beta_{4} + 5 \beta_{5} ) q^{97} + ( -1 + \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 10q^{9} + O(q^{10}) \) \( 6q - 10q^{9} - 2q^{11} - 6q^{19} - 16q^{21} - 6q^{29} - 2q^{31} - 26q^{39} - 6q^{41} - 42q^{49} - 24q^{51} + 12q^{59} - 10q^{61} - 18q^{69} - 6q^{71} - 4q^{79} - 50q^{81} - 28q^{89} - 46q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 9 x^{4} + 18 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} + 5 \nu^{2} + 1 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + 8 \nu^{2} + 10 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 8 \nu^{3} + 13 \nu \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} + 19 \nu^{3} + 41 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - \beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} - 2 \beta_{4} - 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-5 \beta_{3} + 8 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-8 \beta_{5} + 19 \beta_{4} + 27 \beta_{1}\)

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
1.69963i
0.239123i
2.46050i
2.46050i
0.239123i
1.69963i
0 2.58836i 0 0 0 2.81089i 0 −3.69963 0
1749.2 0 2.18194i 0 0 0 3.70370i 0 −1.76088 0
1749.3 0 1.59358i 0 0 0 4.51459i 0 0.460505 0
1749.4 0 1.59358i 0 0 0 4.51459i 0 0.460505 0
1749.5 0 2.18194i 0 0 0 3.70370i 0 −1.76088 0
1749.6 0 2.58836i 0 0 0 2.81089i 0 −3.69963 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.f 6
5.b even 2 1 inner 1900.2.c.f 6
5.c odd 4 1 1900.2.a.g 3
5.c odd 4 1 1900.2.a.i yes 3
20.e even 4 1 7600.2.a.bl 3
20.e even 4 1 7600.2.a.ca 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.a.g 3 5.c odd 4 1
1900.2.a.i yes 3 5.c odd 4 1
1900.2.c.f 6 1.a even 1 1 trivial
1900.2.c.f 6 5.b even 2 1 inner
7600.2.a.bl 3 20.e even 4 1
7600.2.a.ca 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 \(S_{2}^{\mathrm{new}}(1900, [\chi])\):

\( T_{3}^{6} + 14 T_{3}^{4} + 61 T_{3}^{2} + 81 \)
\( T_{7}^{6} + 42 T_{7}^{4} + 549 T_{7}^{2} + 2209 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 81 + 61 T^{2} + 14 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 2209 + 549 T^{2} + 42 T^{4} + T^{6} \)
$11$ \( ( 3 - 6 T + T^{2} + T^{3} )^{2} \)
$13$ \( 49 + 78 T^{2} + 21 T^{4} + T^{6} \)
$17$ \( 9801 + 1413 T^{2} + 66 T^{4} + T^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( 16641 + 2433 T^{2} + 94 T^{4} + T^{6} \)
$29$ \( ( 27 - 24 T + 3 T^{2} + T^{3} )^{2} \)
$31$ \( ( -109 - 40 T + T^{2} + T^{3} )^{2} \)
$37$ \( 3481 + 966 T^{2} + 69 T^{4} + T^{6} \)
$41$ \( ( -27 - 36 T + 3 T^{2} + T^{3} )^{2} \)
$43$ \( 4761 + 2755 T^{2} + 107 T^{4} + T^{6} \)
$47$ \( 385641 + 20394 T^{2} + 273 T^{4} + T^{6} \)
$53$ \( 9 + 4755 T^{2} + 139 T^{4} + T^{6} \)
$59$ \( ( 72 - 24 T - 6 T^{2} + T^{3} )^{2} \)
$61$ \( ( -489 - 98 T + 5 T^{2} + T^{3} )^{2} \)
$67$ \( 1750329 + 50950 T^{2} + 413 T^{4} + T^{6} \)
$71$ \( ( -27 - 33 T + 3 T^{2} + T^{3} )^{2} \)
$73$ \( 2401 + 3234 T^{2} + 141 T^{4} + T^{6} \)
$79$ \( ( 529 - 157 T + 2 T^{2} + T^{3} )^{2} \)
$83$ \( 741321 + 26238 T^{2} + 289 T^{4} + T^{6} \)
$89$ \( ( -489 - 9 T + 14 T^{2} + T^{3} )^{2} \)
$97$ \( 466489 + 22350 T^{2} + 309 T^{4} + T^{6} \)
show more
show less