Properties

Label 900.3.f.b
Level $900$
Weight $3$
Character orbit 900.f
Analytic conductor $24.523$
Analytic rank $0$
Dimension $2$
CM discriminant -4
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{2} -4 q^{4} -8 i q^{8} +O(q^{10})\) \( q + 2 i q^{2} -4 q^{4} -8 i q^{8} -10 i q^{13} + 16 q^{16} + 16 i q^{17} + 20 q^{26} + 40 q^{29} + 32 i q^{32} -32 q^{34} + 70 i q^{37} + 80 q^{41} -49 q^{49} + 40 i q^{52} + 56 i q^{53} + 80 i q^{58} -22 q^{61} -64 q^{64} -64 i q^{68} + 110 i q^{73} -140 q^{74} + 160 i q^{82} + 160 q^{89} + 130 i q^{97} -98 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} + O(q^{10}) \) \( 2q - 8q^{4} + 32q^{16} + 40q^{26} + 80q^{29} - 64q^{34} + 160q^{41} - 98q^{49} - 44q^{61} - 128q^{64} - 280q^{74} + 320q^{89} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
2.00000i 0 −4.00000 0 0 0 8.00000i 0 0
199.2 2.00000i 0 −4.00000 0 0 0 8.00000i 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 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.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.f.b 2
3.b odd 2 1 900.3.f.a 2
4.b odd 2 1 CM 900.3.f.b 2
5.b even 2 1 inner 900.3.f.b 2
5.c odd 4 1 36.3.d.a 1
5.c odd 4 1 900.3.c.c 1
12.b even 2 1 900.3.f.a 2
15.d odd 2 1 900.3.f.a 2
15.e even 4 1 36.3.d.b yes 1
15.e even 4 1 900.3.c.b 1
20.d odd 2 1 inner 900.3.f.b 2
20.e even 4 1 36.3.d.a 1
20.e even 4 1 900.3.c.c 1
40.i odd 4 1 576.3.g.a 1
40.k even 4 1 576.3.g.a 1
45.k odd 12 2 324.3.f.f 2
45.l even 12 2 324.3.f.e 2
60.h even 2 1 900.3.f.a 2
60.l odd 4 1 36.3.d.b yes 1
60.l odd 4 1 900.3.c.b 1
80.i odd 4 1 2304.3.b.d 2
80.j even 4 1 2304.3.b.d 2
80.s even 4 1 2304.3.b.d 2
80.t odd 4 1 2304.3.b.d 2
120.q odd 4 1 576.3.g.c 1
120.w even 4 1 576.3.g.c 1
180.v odd 12 2 324.3.f.e 2
180.x even 12 2 324.3.f.f 2
240.z odd 4 1 2304.3.b.e 2
240.bb even 4 1 2304.3.b.e 2
240.bd odd 4 1 2304.3.b.e 2
240.bf even 4 1 2304.3.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.d.a 1 5.c odd 4 1
36.3.d.a 1 20.e even 4 1
36.3.d.b yes 1 15.e even 4 1
36.3.d.b yes 1 60.l odd 4 1
324.3.f.e 2 45.l even 12 2
324.3.f.e 2 180.v odd 12 2
324.3.f.f 2 45.k odd 12 2
324.3.f.f 2 180.x even 12 2
576.3.g.a 1 40.i odd 4 1
576.3.g.a 1 40.k even 4 1
576.3.g.c 1 120.q odd 4 1
576.3.g.c 1 120.w even 4 1
900.3.c.b 1 15.e even 4 1
900.3.c.b 1 60.l odd 4 1
900.3.c.c 1 5.c odd 4 1
900.3.c.c 1 20.e even 4 1
900.3.f.a 2 3.b odd 2 1
900.3.f.a 2 12.b even 2 1
900.3.f.a 2 15.d odd 2 1
900.3.f.a 2 60.h even 2 1
900.3.f.b 2 1.a even 1 1 trivial
900.3.f.b 2 4.b odd 2 1 CM
900.3.f.b 2 5.b even 2 1 inner
900.3.f.b 2 20.d odd 2 1 inner
2304.3.b.d 2 80.i odd 4 1
2304.3.b.d 2 80.j even 4 1
2304.3.b.d 2 80.s even 4 1
2304.3.b.d 2 80.t odd 4 1
2304.3.b.e 2 240.z odd 4 1
2304.3.b.e 2 240.bb even 4 1
2304.3.b.e 2 240.bd odd 4 1
2304.3.b.e 2 240.bf even 4 1

Hecke kernels

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

\( T_{7} \)
\( T_{29} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 100 + T^{2} \)
$17$ \( 256 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -40 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 4900 + T^{2} \)
$41$ \( ( -80 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 3136 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 22 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 12100 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -160 + T )^{2} \)
$97$ \( 16900 + T^{2} \)
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