Properties

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

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(98.0928951697\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
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 a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{7} +O(q^{10})\) \( q + \zeta_{8}^{2} q^{7} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{11} + 7 \zeta_{8}^{2} q^{13} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} -7 q^{19} + ( -21 \zeta_{8} + 21 \zeta_{8}^{3} ) q^{23} + ( -21 \zeta_{8} - 21 \zeta_{8}^{3} ) q^{29} -17 q^{31} -16 \zeta_{8}^{2} q^{37} + ( -36 \zeta_{8} - 36 \zeta_{8}^{3} ) q^{41} -55 \zeta_{8}^{2} q^{43} + ( 33 \zeta_{8} - 33 \zeta_{8}^{3} ) q^{47} + 48 q^{49} + ( 60 \zeta_{8} - 60 \zeta_{8}^{3} ) q^{53} + ( 39 \zeta_{8} + 39 \zeta_{8}^{3} ) q^{59} + 65 q^{61} + 49 \zeta_{8}^{2} q^{67} + ( 36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{71} + 88 \zeta_{8}^{2} q^{73} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{77} -40 q^{79} + ( -111 \zeta_{8} + 111 \zeta_{8}^{3} ) q^{83} + ( -72 \zeta_{8} - 72 \zeta_{8}^{3} ) q^{89} -7 q^{91} + 41 \zeta_{8}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 28q^{19} - 68q^{31} + 192q^{49} + 260q^{61} - 160q^{79} - 28q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 0 0 1.00000i 0 0 0
449.2 0 0 0 0 0 1.00000i 0 0 0
449.3 0 0 0 0 0 1.00000i 0 0 0
449.4 0 0 0 0 0 1.00000i 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.c.d 4
3.b odd 2 1 inner 3600.3.c.d 4
4.b odd 2 1 900.3.b.a 4
5.b even 2 1 inner 3600.3.c.d 4
5.c odd 4 1 3600.3.l.f 2
5.c odd 4 1 3600.3.l.g 2
12.b even 2 1 900.3.b.a 4
15.d odd 2 1 inner 3600.3.c.d 4
15.e even 4 1 3600.3.l.f 2
15.e even 4 1 3600.3.l.g 2
20.d odd 2 1 900.3.b.a 4
20.e even 4 1 900.3.g.a 2
20.e even 4 1 900.3.g.b yes 2
60.h even 2 1 900.3.b.a 4
60.l odd 4 1 900.3.g.a 2
60.l odd 4 1 900.3.g.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.b.a 4 4.b odd 2 1
900.3.b.a 4 12.b even 2 1
900.3.b.a 4 20.d odd 2 1
900.3.b.a 4 60.h even 2 1
900.3.g.a 2 20.e even 4 1
900.3.g.a 2 60.l odd 4 1
900.3.g.b yes 2 20.e even 4 1
900.3.g.b yes 2 60.l odd 4 1
3600.3.c.d 4 1.a even 1 1 trivial
3600.3.c.d 4 3.b odd 2 1 inner
3600.3.c.d 4 5.b even 2 1 inner
3600.3.c.d 4 15.d odd 2 1 inner
3600.3.l.f 2 5.c odd 4 1
3600.3.l.f 2 15.e even 4 1
3600.3.l.g 2 5.c odd 4 1
3600.3.l.g 2 15.e even 4 1

Hecke kernels

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

\( T_{7}^{2} + 1 \)
\( T_{13}^{2} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 18 + T^{2} )^{2} \)
$13$ \( ( 49 + T^{2} )^{2} \)
$17$ \( ( -18 + T^{2} )^{2} \)
$19$ \( ( 7 + T )^{4} \)
$23$ \( ( -882 + T^{2} )^{2} \)
$29$ \( ( 882 + T^{2} )^{2} \)
$31$ \( ( 17 + T )^{4} \)
$37$ \( ( 256 + T^{2} )^{2} \)
$41$ \( ( 2592 + T^{2} )^{2} \)
$43$ \( ( 3025 + T^{2} )^{2} \)
$47$ \( ( -2178 + T^{2} )^{2} \)
$53$ \( ( -7200 + T^{2} )^{2} \)
$59$ \( ( 3042 + T^{2} )^{2} \)
$61$ \( ( -65 + T )^{4} \)
$67$ \( ( 2401 + T^{2} )^{2} \)
$71$ \( ( 2592 + T^{2} )^{2} \)
$73$ \( ( 7744 + T^{2} )^{2} \)
$79$ \( ( 40 + T )^{4} \)
$83$ \( ( -24642 + T^{2} )^{2} \)
$89$ \( ( 10368 + T^{2} )^{2} \)
$97$ \( ( 1681 + T^{2} )^{2} \)
show more
show less