Properties

Label 900.3.l.h
Level $900$
Weight $3$
Character orbit 900.l
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.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Defining polynomial: \(x^{4} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{7} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{7} -\beta_{3} q^{11} + ( 6 + 6 \beta_{1} ) q^{13} + ( -\beta_{2} + \beta_{3} ) q^{17} + 14 \beta_{1} q^{19} + ( \beta_{2} + \beta_{3} ) q^{23} + \beta_{2} q^{29} + 16 q^{31} + ( 30 - 30 \beta_{1} ) q^{37} + 2 \beta_{3} q^{41} + ( 54 + 54 \beta_{1} ) q^{43} + ( 3 \beta_{2} - 3 \beta_{3} ) q^{47} + 47 \beta_{1} q^{49} + ( -4 \beta_{2} - 4 \beta_{3} ) q^{53} -5 \beta_{2} q^{59} + 54 q^{61} + ( 34 - 34 \beta_{1} ) q^{67} + 4 \beta_{3} q^{71} + ( 65 + 65 \beta_{1} ) q^{73} + ( \beta_{2} - \beta_{3} ) q^{77} + 108 \beta_{1} q^{79} + ( 3 \beta_{2} + 3 \beta_{3} ) q^{83} + 8 \beta_{2} q^{89} + 12 q^{91} + ( 69 - 69 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} + 24q^{13} + 64q^{31} + 120q^{37} + 216q^{43} + 216q^{61} + 136q^{67} + 260q^{73} + 48q^{91} + 276q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{2}\)\(=\)\( \nu^{3} + 5 \nu \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/10\)
\(\nu^{2}\)\(=\)\(5 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} + \beta_{2}\)\()/2\)

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
1.58114 1.58114i
−1.58114 + 1.58114i
1.58114 + 1.58114i
−1.58114 1.58114i
0 0 0 0 0 1.00000 + 1.00000i 0 0 0
757.2 0 0 0 0 0 1.00000 + 1.00000i 0 0 0
793.1 0 0 0 0 0 1.00000 1.00000i 0 0 0
793.2 0 0 0 0 0 1.00000 1.00000i 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.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.l.h 4
3.b odd 2 1 inner 900.3.l.h 4
5.b even 2 1 180.3.l.c 4
5.c odd 4 1 180.3.l.c 4
5.c odd 4 1 inner 900.3.l.h 4
15.d odd 2 1 180.3.l.c 4
15.e even 4 1 180.3.l.c 4
15.e even 4 1 inner 900.3.l.h 4
20.d odd 2 1 720.3.bh.h 4
20.e even 4 1 720.3.bh.h 4
60.h even 2 1 720.3.bh.h 4
60.l odd 4 1 720.3.bh.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.l.c 4 5.b even 2 1
180.3.l.c 4 5.c odd 4 1
180.3.l.c 4 15.d odd 2 1
180.3.l.c 4 15.e even 4 1
720.3.bh.h 4 20.d odd 2 1
720.3.bh.h 4 20.e even 4 1
720.3.bh.h 4 60.h even 2 1
720.3.bh.h 4 60.l odd 4 1
900.3.l.h 4 1.a even 1 1 trivial
900.3.l.h 4 3.b odd 2 1 inner
900.3.l.h 4 5.c odd 4 1 inner
900.3.l.h 4 15.e even 4 1 inner

Hecke kernels

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

\( T_{7}^{2} - 2 T_{7} + 2 \)
\( T_{11}^{2} - 250 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 2 - 2 T + T^{2} )^{2} \)
$11$ \( ( -250 + T^{2} )^{2} \)
$13$ \( ( 72 - 12 T + T^{2} )^{2} \)
$17$ \( 250000 + T^{4} \)
$19$ \( ( 196 + T^{2} )^{2} \)
$23$ \( 250000 + T^{4} \)
$29$ \( ( 250 + T^{2} )^{2} \)
$31$ \( ( -16 + T )^{4} \)
$37$ \( ( 1800 - 60 T + T^{2} )^{2} \)
$41$ \( ( -1000 + T^{2} )^{2} \)
$43$ \( ( 5832 - 108 T + T^{2} )^{2} \)
$47$ \( 20250000 + T^{4} \)
$53$ \( 64000000 + T^{4} \)
$59$ \( ( 6250 + T^{2} )^{2} \)
$61$ \( ( -54 + T )^{4} \)
$67$ \( ( 2312 - 68 T + T^{2} )^{2} \)
$71$ \( ( -4000 + T^{2} )^{2} \)
$73$ \( ( 8450 - 130 T + T^{2} )^{2} \)
$79$ \( ( 11664 + T^{2} )^{2} \)
$83$ \( 20250000 + T^{4} \)
$89$ \( ( 16000 + T^{2} )^{2} \)
$97$ \( ( 9522 - 138 T + T^{2} )^{2} \)
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