Properties

Label 900.4.d.d
Level $900$
Weight $4$
Character orbit 900.d
Analytic conductor $53.102$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(53.1017190052\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{7} - 30 q^{11} + 2 \beta q^{13} - 45 \beta q^{17} + 28 q^{19} + 60 \beta q^{23} + 210 q^{29} - 4 q^{31} + 100 \beta q^{37} - 240 q^{41} + 68 \beta q^{43} + 60 \beta q^{47} + 339 q^{49} - 15 \beta q^{53} - 450 q^{59} - 166 q^{61} + 454 \beta q^{67} + 1020 q^{71} + 125 \beta q^{73} - 30 \beta q^{77} + 916 q^{79} - 570 \beta q^{83} - 420 q^{89} - 8 q^{91} + 769 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 60 q^{11} + 56 q^{19} + 420 q^{29} - 8 q^{31} - 480 q^{41} + 678 q^{49} - 900 q^{59} - 332 q^{61} + 2040 q^{71} + 1832 q^{79} - 840 q^{89} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
0 0 0 0 0 2.00000i 0 0 0
649.2 0 0 0 0 0 2.00000i 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
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 900.4.d.d 2
3.b odd 2 1 900.4.d.i 2
5.b even 2 1 inner 900.4.d.d 2
5.c odd 4 1 180.4.a.b 1
5.c odd 4 1 900.4.a.i 1
15.d odd 2 1 900.4.d.i 2
15.e even 4 1 180.4.a.e yes 1
15.e even 4 1 900.4.a.j 1
20.e even 4 1 720.4.a.h 1
45.k odd 12 2 1620.4.i.i 2
45.l even 12 2 1620.4.i.c 2
60.l odd 4 1 720.4.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.4.a.b 1 5.c odd 4 1
180.4.a.e yes 1 15.e even 4 1
720.4.a.h 1 20.e even 4 1
720.4.a.w 1 60.l odd 4 1
900.4.a.i 1 5.c odd 4 1
900.4.a.j 1 15.e even 4 1
900.4.d.d 2 1.a even 1 1 trivial
900.4.d.d 2 5.b even 2 1 inner
900.4.d.i 2 3.b odd 2 1
900.4.d.i 2 15.d odd 2 1
1620.4.i.c 2 45.l even 12 2
1620.4.i.i 2 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} + 30 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 30)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 8100 \) Copy content Toggle raw display
$19$ \( (T - 28)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 14400 \) Copy content Toggle raw display
$29$ \( (T - 210)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 40000 \) Copy content Toggle raw display
$41$ \( (T + 240)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 18496 \) Copy content Toggle raw display
$47$ \( T^{2} + 14400 \) Copy content Toggle raw display
$53$ \( T^{2} + 900 \) Copy content Toggle raw display
$59$ \( (T + 450)^{2} \) Copy content Toggle raw display
$61$ \( (T + 166)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 824464 \) Copy content Toggle raw display
$71$ \( (T - 1020)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 62500 \) Copy content Toggle raw display
$79$ \( (T - 916)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1299600 \) Copy content Toggle raw display
$89$ \( (T + 420)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2365444 \) Copy content Toggle raw display
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