Properties

Label 900.3.c.g
Level $900$
Weight $3$
Character orbit 900.c
Analytic conductor $24.523$
Analytic rank $0$
Dimension $2$
CM discriminant -15
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: \(\Q(\sqrt{-15}) \)
Defining polynomial: \(x^{2} - x + 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-15})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( -4 + \beta ) q^{4} + ( 4 + 3 \beta ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( -4 + \beta ) q^{4} + ( 4 + 3 \beta ) q^{8} + ( 12 - 7 \beta ) q^{16} -14 q^{17} + ( 8 - 16 \beta ) q^{19} + ( -8 + 16 \beta ) q^{23} + ( -16 + 32 \beta ) q^{31} + ( -28 - 5 \beta ) q^{32} + 14 \beta q^{34} + ( -64 + 8 \beta ) q^{38} + ( 64 - 8 \beta ) q^{46} + ( -24 + 48 \beta ) q^{47} + 49 q^{49} -86 q^{53} + 118 q^{61} + ( 128 - 16 \beta ) q^{62} + ( -20 + 33 \beta ) q^{64} + ( 56 - 14 \beta ) q^{68} + ( 32 + 56 \beta ) q^{76} + ( -32 + 64 \beta ) q^{79} + ( -16 + 32 \beta ) q^{83} + ( -32 - 56 \beta ) q^{92} + ( 192 - 24 \beta ) q^{94} -49 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 7q^{4} + 11q^{8} + O(q^{10}) \) \( 2q - q^{2} - 7q^{4} + 11q^{8} + 17q^{16} - 28q^{17} - 61q^{32} + 14q^{34} - 120q^{38} + 120q^{46} + 98q^{49} - 172q^{53} + 236q^{61} + 240q^{62} - 7q^{64} + 98q^{68} + 120q^{76} - 120q^{92} + 360q^{94} - 49q^{98} + O(q^{100}) \)

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
0.500000 + 1.93649i
0.500000 1.93649i
−0.500000 1.93649i 0 −3.50000 + 1.93649i 0 0 0 5.50000 + 5.80948i 0 0
451.2 −0.500000 + 1.93649i 0 −3.50000 1.93649i 0 0 0 5.50000 5.80948i 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
4.b odd 2 1 inner
60.h 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.g 2
3.b odd 2 1 900.3.c.i 2
4.b odd 2 1 inner 900.3.c.g 2
5.b even 2 1 900.3.c.i 2
5.c odd 4 2 180.3.f.f 4
12.b even 2 1 900.3.c.i 2
15.d odd 2 1 CM 900.3.c.g 2
15.e even 4 2 180.3.f.f 4
20.d odd 2 1 900.3.c.i 2
20.e even 4 2 180.3.f.f 4
60.h even 2 1 inner 900.3.c.g 2
60.l odd 4 2 180.3.f.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.f.f 4 5.c odd 4 2
180.3.f.f 4 15.e even 4 2
180.3.f.f 4 20.e even 4 2
180.3.f.f 4 60.l odd 4 2
900.3.c.g 2 1.a even 1 1 trivial
900.3.c.g 2 4.b odd 2 1 inner
900.3.c.g 2 15.d odd 2 1 CM
900.3.c.g 2 60.h even 2 1 inner
900.3.c.i 2 3.b odd 2 1
900.3.c.i 2 5.b even 2 1
900.3.c.i 2 12.b even 2 1
900.3.c.i 2 20.d odd 2 1

Hecke kernels

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

\( T_{7} \)
\( T_{13} \)
\( T_{17} + 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 14 + T )^{2} \)
$19$ \( 960 + T^{2} \)
$23$ \( 960 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 3840 + T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( 8640 + T^{2} \)
$53$ \( ( 86 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -118 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( 15360 + T^{2} \)
$83$ \( 3840 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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