Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.8405.1
Defining polynomial: \(x^{4} - 2 x^{3} + 8 x^{2} - 7 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 100)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{4} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{7} + ( -5 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{4} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{7} + ( -5 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{11} + ( -6 \beta_{2} - 2 \beta_{3} ) q^{13} + ( 8 + 2 \beta_{2} + 4 \beta_{3} ) q^{14} + ( -9 - 3 \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{16} + ( -5 - 6 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -3 \beta_{1} - 7 \beta_{2} + 4 \beta_{3} ) q^{19} + ( -10 - 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{22} + ( 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{23} + ( -20 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{26} + ( 30 + 2 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{28} + ( -12 - 6 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{31} + ( -5 + 9 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{32} + ( -20 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{34} + ( -30 + 6 \beta_{2} + 2 \beta_{3} ) q^{37} + ( 30 + 14 \beta_{1} + 11 \beta_{2} + 8 \beta_{3} ) q^{38} -17 q^{41} + ( 12 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -25 + \beta_{1} - 11 \beta_{2} - 3 \beta_{3} ) q^{44} + ( 12 - 4 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} ) q^{46} + ( -8 \beta_{1} - 14 \beta_{2} + 6 \beta_{3} ) q^{47} + ( -27 - 24 \beta_{2} - 8 \beta_{3} ) q^{49} + ( 20 - 4 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} ) q^{52} + ( 30 - 24 \beta_{2} - 8 \beta_{3} ) q^{53} + ( 18 - 10 \beta_{1} + 22 \beta_{2} - 6 \beta_{3} ) q^{56} + ( -20 + 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{58} + ( 24 \beta_{1} + 6 \beta_{2} + 18 \beta_{3} ) q^{59} + ( -38 - 30 \beta_{2} - 10 \beta_{3} ) q^{61} + ( -20 - 4 \beta_{1} - 6 \beta_{2} - 8 \beta_{3} ) q^{62} + ( -9 - 3 \beta_{1} + \beta_{2} + 15 \beta_{3} ) q^{64} + ( 11 \beta_{1} + \beta_{2} + 10 \beta_{3} ) q^{67} + ( 15 + \beta_{1} - 15 \beta_{2} + 9 \beta_{3} ) q^{68} + ( -32 \beta_{1} - 18 \beta_{2} - 14 \beta_{3} ) q^{71} + ( 55 - 6 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 20 - 4 \beta_{1} - 34 \beta_{2} - 4 \beta_{3} ) q^{74} + ( 25 - 17 \beta_{1} + 27 \beta_{2} + 11 \beta_{3} ) q^{76} + ( 70 + 30 \beta_{2} + 10 \beta_{3} ) q^{77} + ( -2 \beta_{1} + 12 \beta_{2} - 14 \beta_{3} ) q^{79} -17 \beta_{2} q^{82} + ( 17 \beta_{1} + 11 \beta_{2} + 6 \beta_{3} ) q^{83} + ( -12 - 20 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{86} + ( -35 + 7 \beta_{1} - 17 \beta_{2} + 9 \beta_{3} ) q^{88} + ( 53 - 6 \beta_{2} - 2 \beta_{3} ) q^{89} + ( 20 \beta_{2} - 20 \beta_{3} ) q^{91} + ( 70 + 10 \beta_{1} + 18 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 52 + 28 \beta_{1} + 20 \beta_{2} + 12 \beta_{3} ) q^{94} + ( 90 + 36 \beta_{2} + 12 \beta_{3} ) q^{97} + ( -80 + 16 \beta_{1} - 11 \beta_{2} + 16 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} + 5q^{4} - 19q^{8} + O(q^{10}) \) \( 4q - q^{2} + 5q^{4} - 19q^{8} + 8q^{13} + 26q^{14} - 31q^{16} - 12q^{17} - 35q^{22} - 84q^{26} + 110q^{28} - 40q^{29} - 11q^{32} - 79q^{34} - 128q^{37} + 115q^{38} - 68q^{41} - 85q^{44} + 34q^{46} - 76q^{49} + 92q^{52} + 152q^{53} + 46q^{56} - 72q^{58} - 112q^{61} - 70q^{62} - 55q^{64} + 67q^{68} + 228q^{73} + 114q^{74} + 45q^{76} + 240q^{77} + 17q^{82} - 64q^{86} - 125q^{88} + 220q^{89} + 270q^{92} + 204q^{94} + 312q^{97} - 309q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} + 8 x^{2} - 7 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{3} + \nu^{2} - 5 \nu + 1 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} + 7 \nu - 2 \)
\(\beta_{3}\)\(=\)\( -3 \nu^{3} + 5 \nu^{2} - 23 \nu + 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} - 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} - \beta_{2} - 6 \beta_{1} - 8\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 2.53999i
0.500000 + 2.53999i
0.500000 0.220086i
0.500000 + 0.220086i
−1.85078 0.758030i 0 2.85078 + 2.80590i 0 0 4.09573i −3.14922 7.35408i 0 0
451.2 −1.85078 + 0.758030i 0 2.85078 2.80590i 0 0 4.09573i −3.14922 + 7.35408i 0 0
451.3 1.35078 1.47492i 0 −0.350781 3.98459i 0 0 10.9190i −6.35078 4.86493i 0 0
451.4 1.35078 + 1.47492i 0 −0.350781 + 3.98459i 0 0 10.9190i −6.35078 + 4.86493i 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
4.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.c.l 4
3.b odd 2 1 100.3.b.e yes 4
4.b odd 2 1 inner 900.3.c.l 4
5.b even 2 1 900.3.c.m 4
5.c odd 4 2 900.3.f.d 8
12.b even 2 1 100.3.b.e yes 4
15.d odd 2 1 100.3.b.d 4
15.e even 4 2 100.3.d.a 8
20.d odd 2 1 900.3.c.m 4
20.e even 4 2 900.3.f.d 8
24.f even 2 1 1600.3.b.o 4
24.h odd 2 1 1600.3.b.o 4
60.h even 2 1 100.3.b.d 4
60.l odd 4 2 100.3.d.a 8
120.i odd 2 1 1600.3.b.p 4
120.m even 2 1 1600.3.b.p 4
120.q odd 4 2 1600.3.h.o 8
120.w even 4 2 1600.3.h.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.3.b.d 4 15.d odd 2 1
100.3.b.d 4 60.h even 2 1
100.3.b.e yes 4 3.b odd 2 1
100.3.b.e yes 4 12.b even 2 1
100.3.d.a 8 15.e even 4 2
100.3.d.a 8 60.l odd 4 2
900.3.c.l 4 1.a even 1 1 trivial
900.3.c.l 4 4.b odd 2 1 inner
900.3.c.m 4 5.b even 2 1
900.3.c.m 4 20.d odd 2 1
900.3.f.d 8 5.c odd 4 2
900.3.f.d 8 20.e even 4 2
1600.3.b.o 4 24.f even 2 1
1600.3.b.o 4 24.h odd 2 1
1600.3.b.p 4 120.i odd 2 1
1600.3.b.p 4 120.m even 2 1
1600.3.h.o 8 120.q odd 4 2
1600.3.h.o 8 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{4} + 136 T_{7}^{2} + 2000 \)
\( T_{13}^{2} - 4 T_{13} - 160 \)
\( T_{17}^{2} + 6 T_{17} - 155 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 4 T - 2 T^{2} + T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 2000 + 136 T^{2} + T^{4} \)
$11$ \( 125 + 130 T^{2} + T^{4} \)
$13$ \( ( -160 - 4 T + T^{2} )^{2} \)
$17$ \( ( -155 + 6 T + T^{2} )^{2} \)
$19$ \( 465125 + 1370 T^{2} + T^{4} \)
$23$ \( 147920 + 776 T^{2} + T^{4} \)
$29$ \( ( -64 + 20 T + T^{2} )^{2} \)
$31$ \( 2000 + 520 T^{2} + T^{4} \)
$37$ \( ( 860 + 64 T + T^{2} )^{2} \)
$41$ \( ( 17 + T )^{4} \)
$43$ \( 128000 + 3056 T^{2} + T^{4} \)
$47$ \( 5242880 + 4976 T^{2} + T^{4} \)
$53$ \( ( -1180 - 76 T + T^{2} )^{2} \)
$59$ \( 41472000 + 13680 T^{2} + T^{4} \)
$61$ \( ( -3316 + 56 T + T^{2} )^{2} \)
$67$ \( 1915805 + 3586 T^{2} + T^{4} \)
$71$ \( 67712000 + 19120 T^{2} + T^{4} \)
$73$ \( ( 3085 - 114 T + T^{2} )^{2} \)
$79$ \( 6962000 + 7720 T^{2} + T^{4} \)
$83$ \( 3621005 + 5426 T^{2} + T^{4} \)
$89$ \( ( 2861 - 110 T + T^{2} )^{2} \)
$97$ \( ( 180 - 156 T + T^{2} )^{2} \)
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