Properties

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

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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: \(\Q(\sqrt{-2}, \sqrt{-23})\)
Defining polynomial: \(x^{4} - 2 x^{3} + 17 x^{2} - 16 x + 18\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{7} -7 \beta_{1} q^{11} + 3 \beta_{2} q^{13} -2 \beta_{3} q^{17} -12 q^{19} + \beta_{3} q^{23} -6 \beta_{1} q^{29} -38 q^{31} -\beta_{2} q^{37} -49 \beta_{1} q^{41} -10 \beta_{2} q^{43} + 8 \beta_{3} q^{47} -3 q^{49} -59 \beta_{1} q^{59} -70 q^{61} + 16 \beta_{2} q^{67} -84 \beta_{1} q^{71} + 2 \beta_{2} q^{73} -7 \beta_{3} q^{77} -30 q^{79} -14 \beta_{3} q^{83} + 23 \beta_{1} q^{89} -138 q^{91} -14 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 48q^{19} - 152q^{31} - 12q^{49} - 280q^{61} - 120q^{79} - 552q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} + 17 x^{2} - 16 x + 18\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} + 3 \nu^{2} - 25 \nu + 12 \)\()/15\)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu + 8 \)
\(\beta_{3}\)\(=\)\((\)\( 8 \nu^{3} - 12 \nu^{2} + 160 \nu - 78 \)\()/15\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 4 \beta_{1} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 4 \beta_{2} + 4 \beta_{1} - 30\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-11 \beta_{3} + 6 \beta_{2} - 74 \beta_{1} - 46\)\()/4\)

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
0.500000 0.983702i
0.500000 + 0.983702i
0.500000 + 3.81213i
0.500000 3.81213i
0 0 0 0 0 −6.78233 0 0 0
701.2 0 0 0 0 0 −6.78233 0 0 0
701.3 0 0 0 0 0 6.78233 0 0 0
701.4 0 0 0 0 0 6.78233 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
5.b even 2 1 inner
15.d 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.c 4
3.b odd 2 1 inner 900.3.g.c 4
4.b odd 2 1 3600.3.l.r 4
5.b even 2 1 inner 900.3.g.c 4
5.c odd 4 2 180.3.b.a 4
12.b even 2 1 3600.3.l.r 4
15.d odd 2 1 inner 900.3.g.c 4
15.e even 4 2 180.3.b.a 4
20.d odd 2 1 3600.3.l.r 4
20.e even 4 2 720.3.c.b 4
40.i odd 4 2 2880.3.c.c 4
40.k even 4 2 2880.3.c.f 4
45.k odd 12 4 1620.3.t.c 8
45.l even 12 4 1620.3.t.c 8
60.h even 2 1 3600.3.l.r 4
60.l odd 4 2 720.3.c.b 4
120.q odd 4 2 2880.3.c.f 4
120.w even 4 2 2880.3.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.b.a 4 5.c odd 4 2
180.3.b.a 4 15.e even 4 2
720.3.c.b 4 20.e even 4 2
720.3.c.b 4 60.l odd 4 2
900.3.g.c 4 1.a even 1 1 trivial
900.3.g.c 4 3.b odd 2 1 inner
900.3.g.c 4 5.b even 2 1 inner
900.3.g.c 4 15.d odd 2 1 inner
1620.3.t.c 8 45.k odd 12 4
1620.3.t.c 8 45.l even 12 4
2880.3.c.c 4 40.i odd 4 2
2880.3.c.c 4 120.w even 4 2
2880.3.c.f 4 40.k even 4 2
2880.3.c.f 4 120.q odd 4 2
3600.3.l.r 4 4.b odd 2 1
3600.3.l.r 4 12.b even 2 1
3600.3.l.r 4 20.d odd 2 1
3600.3.l.r 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 46 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -46 + T^{2} )^{2} \)
$11$ \( ( 98 + T^{2} )^{2} \)
$13$ \( ( -414 + T^{2} )^{2} \)
$17$ \( ( 368 + T^{2} )^{2} \)
$19$ \( ( 12 + T )^{4} \)
$23$ \( ( 92 + T^{2} )^{2} \)
$29$ \( ( 72 + T^{2} )^{2} \)
$31$ \( ( 38 + T )^{4} \)
$37$ \( ( -46 + T^{2} )^{2} \)
$41$ \( ( 4802 + T^{2} )^{2} \)
$43$ \( ( -4600 + T^{2} )^{2} \)
$47$ \( ( 5888 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( ( 6962 + T^{2} )^{2} \)
$61$ \( ( 70 + T )^{4} \)
$67$ \( ( -11776 + T^{2} )^{2} \)
$71$ \( ( 14112 + T^{2} )^{2} \)
$73$ \( ( -184 + T^{2} )^{2} \)
$79$ \( ( 30 + T )^{4} \)
$83$ \( ( 18032 + T^{2} )^{2} \)
$89$ \( ( 1058 + T^{2} )^{2} \)
$97$ \( ( -9016 + T^{2} )^{2} \)
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