Properties

Label 900.3.c.h
Level $900$
Weight $3$
Character orbit 900.c
Analytic conductor $24.523$
Analytic rank $0$
Dimension $2$
CM discriminant -20
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.c (of order \(2\), degree \(1\), 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 20)
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.

\(f(q)\) \(=\) \( q -2 i q^{2} -4 q^{4} + 4 i q^{7} + 8 i q^{8} +O(q^{10})\) \( q -2 i q^{2} -4 q^{4} + 4 i q^{7} + 8 i q^{8} + 8 q^{14} + 16 q^{16} -44 i q^{23} -16 i q^{28} -22 q^{29} -32 i q^{32} -62 q^{41} -76 i q^{43} -88 q^{46} -4 i q^{47} + 33 q^{49} -32 q^{56} + 44 i q^{58} -58 q^{61} -64 q^{64} -116 i q^{67} + 124 i q^{82} + 76 i q^{83} -152 q^{86} -142 q^{89} + 176 i q^{92} -8 q^{94} -66 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} + O(q^{10}) \) \( 2q - 8q^{4} + 16q^{14} + 32q^{16} - 44q^{29} - 124q^{41} - 176q^{46} + 66q^{49} - 64q^{56} - 116q^{61} - 128q^{64} - 304q^{86} - 284q^{89} - 16q^{94} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
1.00000i
1.00000i
2.00000i 0 −4.00000 0 0 4.00000i 8.00000i 0 0
451.2 2.00000i 0 −4.00000 0 0 4.00000i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
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 900.3.c.h 2
3.b odd 2 1 100.3.b.c 2
4.b odd 2 1 inner 900.3.c.h 2
5.b even 2 1 inner 900.3.c.h 2
5.c odd 4 1 180.3.f.a 1
5.c odd 4 1 180.3.f.b 1
12.b even 2 1 100.3.b.c 2
15.d odd 2 1 100.3.b.c 2
15.e even 4 1 20.3.d.a 1
15.e even 4 1 20.3.d.b yes 1
20.d odd 2 1 CM 900.3.c.h 2
20.e even 4 1 180.3.f.a 1
20.e even 4 1 180.3.f.b 1
24.f even 2 1 1600.3.b.f 2
24.h odd 2 1 1600.3.b.f 2
60.h even 2 1 100.3.b.c 2
60.l odd 4 1 20.3.d.a 1
60.l odd 4 1 20.3.d.b yes 1
120.i odd 2 1 1600.3.b.f 2
120.m even 2 1 1600.3.b.f 2
120.q odd 4 1 320.3.h.a 1
120.q odd 4 1 320.3.h.b 1
120.w even 4 1 320.3.h.a 1
120.w even 4 1 320.3.h.b 1
240.z odd 4 1 1280.3.e.b 2
240.z odd 4 1 1280.3.e.c 2
240.bb even 4 1 1280.3.e.b 2
240.bb even 4 1 1280.3.e.c 2
240.bd odd 4 1 1280.3.e.b 2
240.bd odd 4 1 1280.3.e.c 2
240.bf even 4 1 1280.3.e.b 2
240.bf even 4 1 1280.3.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.a 1 15.e even 4 1
20.3.d.a 1 60.l odd 4 1
20.3.d.b yes 1 15.e even 4 1
20.3.d.b yes 1 60.l odd 4 1
100.3.b.c 2 3.b odd 2 1
100.3.b.c 2 12.b even 2 1
100.3.b.c 2 15.d odd 2 1
100.3.b.c 2 60.h even 2 1
180.3.f.a 1 5.c odd 4 1
180.3.f.a 1 20.e even 4 1
180.3.f.b 1 5.c odd 4 1
180.3.f.b 1 20.e even 4 1
320.3.h.a 1 120.q odd 4 1
320.3.h.a 1 120.w even 4 1
320.3.h.b 1 120.q odd 4 1
320.3.h.b 1 120.w even 4 1
900.3.c.h 2 1.a even 1 1 trivial
900.3.c.h 2 4.b odd 2 1 inner
900.3.c.h 2 5.b even 2 1 inner
900.3.c.h 2 20.d odd 2 1 CM
1280.3.e.b 2 240.z odd 4 1
1280.3.e.b 2 240.bb even 4 1
1280.3.e.b 2 240.bd odd 4 1
1280.3.e.b 2 240.bf even 4 1
1280.3.e.c 2 240.z odd 4 1
1280.3.e.c 2 240.bb even 4 1
1280.3.e.c 2 240.bd odd 4 1
1280.3.e.c 2 240.bf even 4 1
1600.3.b.f 2 24.f even 2 1
1600.3.b.f 2 24.h odd 2 1
1600.3.b.f 2 120.i odd 2 1
1600.3.b.f 2 120.m even 2 1

Hecke kernels

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

\( T_{7}^{2} + 16 \)
\( T_{13} \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 1936 + T^{2} \)
$29$ \( ( 22 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( 62 + T )^{2} \)
$43$ \( 5776 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 58 + T )^{2} \)
$67$ \( 13456 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( 5776 + T^{2} \)
$89$ \( ( 142 + T )^{2} \)
$97$ \( T^{2} \)
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