Properties

Label 900.3.g.d
Level $900$
Weight $3$
Character orbit 900.g
Analytic conductor $24.523$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - \beta_{2} ) q^{7} +O(q^{10})\) \( q + ( 4 - \beta_{2} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{11} + ( -2 - \beta_{2} ) q^{13} -4 \beta_{1} q^{17} + ( 8 + 2 \beta_{2} ) q^{19} + ( 2 \beta_{1} - \beta_{3} ) q^{23} + ( -8 \beta_{1} + \beta_{3} ) q^{29} + ( 2 + 2 \beta_{2} ) q^{31} + ( 34 + 3 \beta_{2} ) q^{37} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{41} + ( -20 + 2 \beta_{2} ) q^{43} + ( -14 \beta_{1} - 2 \beta_{3} ) q^{47} + ( 57 - 8 \beta_{2} ) q^{49} + ( 10 \beta_{1} + 4 \beta_{3} ) q^{53} + ( -17 \beta_{1} - 3 \beta_{3} ) q^{59} + ( -10 - 6 \beta_{2} ) q^{61} + 76 q^{67} + ( -12 \beta_{1} + 2 \beta_{3} ) q^{71} + ( -38 + 6 \beta_{2} ) q^{73} + ( -34 \beta_{1} + 7 \beta_{3} ) q^{77} + ( 50 + 6 \beta_{2} ) q^{79} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{83} + ( -21 \beta_{1} - 2 \beta_{3} ) q^{89} + ( 82 - 2 \beta_{2} ) q^{91} + ( 106 - 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 16q^{7} + O(q^{10}) \) \( 4q + 16q^{7} - 8q^{13} + 32q^{19} + 8q^{31} + 136q^{37} - 80q^{43} + 228q^{49} - 40q^{61} + 304q^{67} - 152q^{73} + 200q^{79} + 328q^{91} + 424q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{3} + 7 \nu \)
\(\beta_{3}\)\(=\)\( 6 \nu^{2} - 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 12\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 7 \beta_{1}\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
1.58114 0.707107i
1.58114 + 0.707107i
−1.58114 + 0.707107i
−1.58114 0.707107i
0 0 0 0 0 −5.48683 0 0 0
701.2 0 0 0 0 0 −5.48683 0 0 0
701.3 0 0 0 0 0 13.4868 0 0 0
701.4 0 0 0 0 0 13.4868 0 0 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.g.d 4
3.b odd 2 1 inner 900.3.g.d 4
4.b odd 2 1 3600.3.l.n 4
5.b even 2 1 180.3.g.a 4
5.c odd 4 2 900.3.b.b 8
12.b even 2 1 3600.3.l.n 4
15.d odd 2 1 180.3.g.a 4
15.e even 4 2 900.3.b.b 8
20.d odd 2 1 720.3.l.c 4
20.e even 4 2 3600.3.c.k 8
40.e odd 2 1 2880.3.l.f 4
40.f even 2 1 2880.3.l.b 4
45.h odd 6 2 1620.3.o.f 8
45.j even 6 2 1620.3.o.f 8
60.h even 2 1 720.3.l.c 4
60.l odd 4 2 3600.3.c.k 8
120.i odd 2 1 2880.3.l.b 4
120.m even 2 1 2880.3.l.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.g.a 4 5.b even 2 1
180.3.g.a 4 15.d odd 2 1
720.3.l.c 4 20.d odd 2 1
720.3.l.c 4 60.h even 2 1
900.3.b.b 8 5.c odd 4 2
900.3.b.b 8 15.e even 4 2
900.3.g.d 4 1.a even 1 1 trivial
900.3.g.d 4 3.b odd 2 1 inner
1620.3.o.f 8 45.h odd 6 2
1620.3.o.f 8 45.j even 6 2
2880.3.l.b 4 40.f even 2 1
2880.3.l.b 4 120.i odd 2 1
2880.3.l.f 4 40.e odd 2 1
2880.3.l.f 4 120.m even 2 1
3600.3.c.k 8 20.e even 4 2
3600.3.c.k 8 60.l odd 4 2
3600.3.l.n 4 4.b odd 2 1
3600.3.l.n 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 8 T_{7} - 74 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -74 - 8 T + T^{2} )^{2} \)
$11$ \( 26244 + 396 T^{2} + T^{4} \)
$13$ \( ( -86 + 4 T + T^{2} )^{2} \)
$17$ \( ( 288 + T^{2} )^{2} \)
$19$ \( ( -296 - 16 T + T^{2} )^{2} \)
$23$ \( 11664 + 504 T^{2} + T^{4} \)
$29$ \( 944784 + 2664 T^{2} + T^{4} \)
$31$ \( ( -356 - 4 T + T^{2} )^{2} \)
$37$ \( ( 346 - 68 T + T^{2} )^{2} \)
$41$ \( 7387524 + 6084 T^{2} + T^{4} \)
$43$ \( ( 40 + 40 T + T^{2} )^{2} \)
$47$ \( 7884864 + 8496 T^{2} + T^{4} \)
$53$ \( 1166400 + 9360 T^{2} + T^{4} \)
$59$ \( 12830724 + 13644 T^{2} + T^{4} \)
$61$ \( ( -3140 + 20 T + T^{2} )^{2} \)
$67$ \( ( -76 + T )^{4} \)
$71$ \( 3504384 + 6624 T^{2} + T^{4} \)
$73$ \( ( -1796 + 76 T + T^{2} )^{2} \)
$79$ \( ( -740 - 100 T + T^{2} )^{2} \)
$83$ \( 7884864 + 5904 T^{2} + T^{4} \)
$89$ \( 52099524 + 17316 T^{2} + T^{4} \)
$97$ \( ( 10876 - 212 T + T^{2} )^{2} \)
show more
show less