Properties

Label 900.4.d.k
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q + 8 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta q^{7} + 60 q^{11} + 43 \beta q^{13} + 9 \beta q^{17} - 44 q^{19} - 24 \beta q^{23} - 186 q^{29} + 176 q^{31} - 127 \beta q^{37} - 186 q^{41} - 50 \beta q^{43} + 84 \beta q^{47} + 87 q^{49} + 249 \beta q^{53} - 252 q^{59} - 58 q^{61} + 518 \beta q^{67} - 168 q^{71} + 253 \beta q^{73} + 480 \beta q^{77} - 272 q^{79} - 474 \beta q^{83} - 1014 q^{89} - 1376 q^{91} + 383 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 120 q^{11} - 88 q^{19} - 372 q^{29} + 352 q^{31} - 372 q^{41} + 174 q^{49} - 504 q^{59} - 116 q^{61} - 336 q^{71} - 544 q^{79} - 2028 q^{89} - 2752 q^{91}+O(q^{100}) \) 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 16.0000i 0 0 0
649.2 0 0 0 0 0 16.0000i 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.k 2
3.b odd 2 1 100.4.c.a 2
5.b even 2 1 inner 900.4.d.k 2
5.c odd 4 1 180.4.a.a 1
5.c odd 4 1 900.4.a.m 1
12.b even 2 1 400.4.c.j 2
15.d odd 2 1 100.4.c.a 2
15.e even 4 1 20.4.a.a 1
15.e even 4 1 100.4.a.a 1
20.e even 4 1 720.4.a.k 1
45.k odd 12 2 1620.4.i.j 2
45.l even 12 2 1620.4.i.d 2
60.h even 2 1 400.4.c.j 2
60.l odd 4 1 80.4.a.c 1
60.l odd 4 1 400.4.a.o 1
105.k odd 4 1 980.4.a.c 1
105.w odd 12 2 980.4.i.n 2
105.x even 12 2 980.4.i.e 2
120.q odd 4 1 320.4.a.k 1
120.q odd 4 1 1600.4.a.p 1
120.w even 4 1 320.4.a.d 1
120.w even 4 1 1600.4.a.bl 1
165.l odd 4 1 2420.4.a.d 1
240.z odd 4 1 1280.4.d.c 2
240.bb even 4 1 1280.4.d.n 2
240.bd odd 4 1 1280.4.d.c 2
240.bf even 4 1 1280.4.d.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 15.e even 4 1
80.4.a.c 1 60.l odd 4 1
100.4.a.a 1 15.e even 4 1
100.4.c.a 2 3.b odd 2 1
100.4.c.a 2 15.d odd 2 1
180.4.a.a 1 5.c odd 4 1
320.4.a.d 1 120.w even 4 1
320.4.a.k 1 120.q odd 4 1
400.4.a.o 1 60.l odd 4 1
400.4.c.j 2 12.b even 2 1
400.4.c.j 2 60.h even 2 1
720.4.a.k 1 20.e even 4 1
900.4.a.m 1 5.c odd 4 1
900.4.d.k 2 1.a even 1 1 trivial
900.4.d.k 2 5.b even 2 1 inner
980.4.a.c 1 105.k odd 4 1
980.4.i.e 2 105.x even 12 2
980.4.i.n 2 105.w odd 12 2
1280.4.d.c 2 240.z odd 4 1
1280.4.d.c 2 240.bd odd 4 1
1280.4.d.n 2 240.bb even 4 1
1280.4.d.n 2 240.bf even 4 1
1600.4.a.p 1 120.q odd 4 1
1600.4.a.bl 1 120.w even 4 1
1620.4.i.d 2 45.l even 12 2
1620.4.i.j 2 45.k odd 12 2
2420.4.a.d 1 165.l odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11} - 60 \) 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
$7$ \( T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T - 60)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 7396 \) Copy content Toggle raw display
$17$ \( T^{2} + 324 \) Copy content Toggle raw display
$19$ \( (T + 44)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2304 \) Copy content Toggle raw display
$29$ \( (T + 186)^{2} \) Copy content Toggle raw display
$31$ \( (T - 176)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64516 \) Copy content Toggle raw display
$41$ \( (T + 186)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 10000 \) Copy content Toggle raw display
$47$ \( T^{2} + 28224 \) Copy content Toggle raw display
$53$ \( T^{2} + 248004 \) Copy content Toggle raw display
$59$ \( (T + 252)^{2} \) Copy content Toggle raw display
$61$ \( (T + 58)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1073296 \) Copy content Toggle raw display
$71$ \( (T + 168)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 256036 \) Copy content Toggle raw display
$79$ \( (T + 272)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 898704 \) Copy content Toggle raw display
$89$ \( (T + 1014)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 586756 \) Copy content Toggle raw display
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