Properties

Label 3600.3.l.r
Level $3600$
Weight $3$
Character orbit 3600.l
Analytic conductor $98.093$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-23})\)
Defining polynomial: \( x^{4} - 2x^{3} + 17x^{2} - 16x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{7} - 7 \beta_1 q^{11} - 3 \beta_{2} q^{13} - 2 \beta_{3} q^{17} + 12 q^{19} - \beta_{3} q^{23} + 6 \beta_1 q^{29} + 38 q^{31} + \beta_{2} q^{37} + 49 \beta_1 q^{41} - 10 \beta_{2} q^{43} - 8 \beta_{3} q^{47} - 3 q^{49} - 59 \beta_1 q^{59} - 70 q^{61} + 16 \beta_{2} q^{67} - 84 \beta_1 q^{71} - 2 \beta_{2} q^{73} - 7 \beta_{3} q^{77} + 30 q^{79} + 14 \beta_{3} q^{83} - 23 \beta_1 q^{89} + 138 q^{91} + 14 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 48 q^{19} + 152 q^{31} - 12 q^{49} - 280 q^{61} + 120 q^{79} + 552 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 17x^{2} - 16x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} - 25\nu + 12 ) / 15 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu + 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{3} - 12\nu^{2} + 160\nu - 78 ) / 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 4\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 4\beta_{2} + 4\beta _1 - 30 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{3} + 6\beta_{2} - 74\beta _1 - 46 ) / 4 \) Copy content Toggle raw display

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
0.500000 0.983702i
0.500000 + 0.983702i
0.500000 + 3.81213i
0.500000 3.81213i
0 0 0 0 0 −6.78233 0 0 0
1601.2 0 0 0 0 0 −6.78233 0 0 0
1601.3 0 0 0 0 0 6.78233 0 0 0
1601.4 0 0 0 0 0 6.78233 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
5.b even 2 1 inner
15.d 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.r 4
3.b odd 2 1 inner 3600.3.l.r 4
4.b odd 2 1 900.3.g.c 4
5.b even 2 1 inner 3600.3.l.r 4
5.c odd 4 2 720.3.c.b 4
12.b even 2 1 900.3.g.c 4
15.d odd 2 1 inner 3600.3.l.r 4
15.e even 4 2 720.3.c.b 4
20.d odd 2 1 900.3.g.c 4
20.e even 4 2 180.3.b.a 4
40.i odd 4 2 2880.3.c.f 4
40.k even 4 2 2880.3.c.c 4
60.h even 2 1 900.3.g.c 4
60.l odd 4 2 180.3.b.a 4
120.q odd 4 2 2880.3.c.c 4
120.w even 4 2 2880.3.c.f 4
180.v odd 12 4 1620.3.t.c 8
180.x even 12 4 1620.3.t.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.b.a 4 20.e even 4 2
180.3.b.a 4 60.l odd 4 2
720.3.c.b 4 5.c odd 4 2
720.3.c.b 4 15.e even 4 2
900.3.g.c 4 4.b odd 2 1
900.3.g.c 4 12.b even 2 1
900.3.g.c 4 20.d odd 2 1
900.3.g.c 4 60.h even 2 1
1620.3.t.c 8 180.v odd 12 4
1620.3.t.c 8 180.x even 12 4
2880.3.c.c 4 40.k even 4 2
2880.3.c.c 4 120.q odd 4 2
2880.3.c.f 4 40.i odd 4 2
2880.3.c.f 4 120.w even 4 2
3600.3.l.r 4 1.a even 1 1 trivial
3600.3.l.r 4 3.b odd 2 1 inner
3600.3.l.r 4 5.b even 2 1 inner
3600.3.l.r 4 15.d 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 \(S_{3}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7}^{2} - 46 \) Copy content Toggle raw display
\( T_{11}^{2} + 98 \) Copy content Toggle raw display
\( T_{13}^{2} - 414 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 46)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 414)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 368)^{2} \) Copy content Toggle raw display
$19$ \( (T - 12)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 92)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$31$ \( (T - 38)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 46)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4802)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 5888)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 6962)^{2} \) Copy content Toggle raw display
$61$ \( (T + 70)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 11776)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 14112)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 184)^{2} \) Copy content Toggle raw display
$79$ \( (T - 30)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 18032)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1058)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 9016)^{2} \) Copy content Toggle raw display
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