Properties

Label 3600.3.l.g
Level $3600$
Weight $3$
Character orbit 3600.l
Analytic conductor $98.093$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3600.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(98.0928951697\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 900)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{7} +O(q^{10})\) \( q + q^{7} -\beta q^{11} -7 q^{13} -\beta q^{17} + 7 q^{19} + 7 \beta q^{23} -7 \beta q^{29} -17 q^{31} -16 q^{37} + 12 \beta q^{41} + 55 q^{43} + 11 \beta q^{47} -48 q^{49} -20 \beta q^{53} + 13 \beta q^{59} + 65 q^{61} + 49 q^{67} -12 \beta q^{71} -88 q^{73} -\beta q^{77} + 40 q^{79} + 37 \beta q^{83} -24 \beta q^{89} -7 q^{91} + 41 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{7} - 14q^{13} + 14q^{19} - 34q^{31} - 32q^{37} + 110q^{43} - 96q^{49} + 130q^{61} + 98q^{67} - 176q^{73} + 80q^{79} - 14q^{91} + 82q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
1.41421i
1.41421i
0 0 0 0 0 1.00000 0 0 0
1601.2 0 0 0 0 0 1.00000 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.3.l.g 2
3.b odd 2 1 inner 3600.3.l.g 2
4.b odd 2 1 900.3.g.a 2
5.b even 2 1 3600.3.l.f 2
5.c odd 4 2 3600.3.c.d 4
12.b even 2 1 900.3.g.a 2
15.d odd 2 1 3600.3.l.f 2
15.e even 4 2 3600.3.c.d 4
20.d odd 2 1 900.3.g.b yes 2
20.e even 4 2 900.3.b.a 4
60.h even 2 1 900.3.g.b yes 2
60.l odd 4 2 900.3.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.b.a 4 20.e even 4 2
900.3.b.a 4 60.l odd 4 2
900.3.g.a 2 4.b odd 2 1
900.3.g.a 2 12.b even 2 1
900.3.g.b yes 2 20.d odd 2 1
900.3.g.b yes 2 60.h even 2 1
3600.3.c.d 4 5.c odd 4 2
3600.3.c.d 4 15.e even 4 2
3600.3.l.f 2 5.b even 2 1
3600.3.l.f 2 15.d odd 2 1
3600.3.l.g 2 1.a even 1 1 trivial
3600.3.l.g 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7} - 1 \)
\( T_{11}^{2} + 18 \)
\( T_{13} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( 18 + T^{2} \)
$13$ \( ( 7 + T )^{2} \)
$17$ \( 18 + T^{2} \)
$19$ \( ( -7 + T )^{2} \)
$23$ \( 882 + T^{2} \)
$29$ \( 882 + T^{2} \)
$31$ \( ( 17 + T )^{2} \)
$37$ \( ( 16 + T )^{2} \)
$41$ \( 2592 + T^{2} \)
$43$ \( ( -55 + T )^{2} \)
$47$ \( 2178 + T^{2} \)
$53$ \( 7200 + T^{2} \)
$59$ \( 3042 + T^{2} \)
$61$ \( ( -65 + T )^{2} \)
$67$ \( ( -49 + T )^{2} \)
$71$ \( 2592 + T^{2} \)
$73$ \( ( 88 + T )^{2} \)
$79$ \( ( -40 + T )^{2} \)
$83$ \( 24642 + T^{2} \)
$89$ \( 10368 + T^{2} \)
$97$ \( ( -41 + T )^{2} \)
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