Properties

Label 3600.3.c.b
Level $3600$
Weight $3$
Character orbit 3600.c
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.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: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 18)
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.

\(f(q)\) \(=\) \( q + 4 \zeta_{8}^{2} q^{7} +O(q^{10})\) \( q + 4 \zeta_{8}^{2} q^{7} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{11} -8 \zeta_{8}^{2} q^{13} + ( 9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{17} -16 q^{19} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{23} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{29} -44 q^{31} -34 \zeta_{8}^{2} q^{37} + ( 33 \zeta_{8} + 33 \zeta_{8}^{3} ) q^{41} -40 \zeta_{8}^{2} q^{43} + ( -60 \zeta_{8} + 60 \zeta_{8}^{3} ) q^{47} + 33 q^{49} + ( 27 \zeta_{8} - 27 \zeta_{8}^{3} ) q^{53} + ( 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{59} + 50 q^{61} -8 \zeta_{8}^{2} q^{67} + ( 36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{71} + 16 \zeta_{8}^{2} q^{73} + ( 48 \zeta_{8} - 48 \zeta_{8}^{3} ) q^{77} -76 q^{79} + ( -84 \zeta_{8} + 84 \zeta_{8}^{3} ) q^{83} + ( -9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{89} + 32 q^{91} + 176 \zeta_{8}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 64q^{19} - 176q^{31} + 132q^{49} + 200q^{61} - 304q^{79} + 128q^{91} + O(q^{100}) \)

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 4.00000i 0 0 0
449.2 0 0 0 0 0 4.00000i 0 0 0
449.3 0 0 0 0 0 4.00000i 0 0 0
449.4 0 0 0 0 0 4.00000i 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.b 4
3.b odd 2 1 inner 3600.3.c.b 4
4.b odd 2 1 450.3.b.b 4
5.b even 2 1 inner 3600.3.c.b 4
5.c odd 4 1 144.3.e.b 2
5.c odd 4 1 3600.3.l.d 2
12.b even 2 1 450.3.b.b 4
15.d odd 2 1 inner 3600.3.c.b 4
15.e even 4 1 144.3.e.b 2
15.e even 4 1 3600.3.l.d 2
20.d odd 2 1 450.3.b.b 4
20.e even 4 1 18.3.b.a 2
20.e even 4 1 450.3.d.f 2
40.i odd 4 1 576.3.e.f 2
40.k even 4 1 576.3.e.c 2
45.k odd 12 2 1296.3.q.f 4
45.l even 12 2 1296.3.q.f 4
60.h even 2 1 450.3.b.b 4
60.l odd 4 1 18.3.b.a 2
60.l odd 4 1 450.3.d.f 2
80.i odd 4 1 2304.3.h.c 4
80.j even 4 1 2304.3.h.f 4
80.s even 4 1 2304.3.h.f 4
80.t odd 4 1 2304.3.h.c 4
120.q odd 4 1 576.3.e.c 2
120.w even 4 1 576.3.e.f 2
140.j odd 4 1 882.3.b.a 2
140.w even 12 2 882.3.s.b 4
140.x odd 12 2 882.3.s.d 4
180.v odd 12 2 162.3.d.b 4
180.x even 12 2 162.3.d.b 4
220.i odd 4 1 2178.3.c.d 2
240.z odd 4 1 2304.3.h.f 4
240.bb even 4 1 2304.3.h.c 4
240.bd odd 4 1 2304.3.h.f 4
240.bf even 4 1 2304.3.h.c 4
260.l odd 4 1 3042.3.d.a 4
260.p even 4 1 3042.3.c.e 2
260.s odd 4 1 3042.3.d.a 4
420.w even 4 1 882.3.b.a 2
420.bp odd 12 2 882.3.s.b 4
420.br even 12 2 882.3.s.d 4
660.q even 4 1 2178.3.c.d 2
780.u even 4 1 3042.3.d.a 4
780.w odd 4 1 3042.3.c.e 2
780.bn even 4 1 3042.3.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 20.e even 4 1
18.3.b.a 2 60.l odd 4 1
144.3.e.b 2 5.c odd 4 1
144.3.e.b 2 15.e even 4 1
162.3.d.b 4 180.v odd 12 2
162.3.d.b 4 180.x even 12 2
450.3.b.b 4 4.b odd 2 1
450.3.b.b 4 12.b even 2 1
450.3.b.b 4 20.d odd 2 1
450.3.b.b 4 60.h even 2 1
450.3.d.f 2 20.e even 4 1
450.3.d.f 2 60.l odd 4 1
576.3.e.c 2 40.k even 4 1
576.3.e.c 2 120.q odd 4 1
576.3.e.f 2 40.i odd 4 1
576.3.e.f 2 120.w even 4 1
882.3.b.a 2 140.j odd 4 1
882.3.b.a 2 420.w even 4 1
882.3.s.b 4 140.w even 12 2
882.3.s.b 4 420.bp odd 12 2
882.3.s.d 4 140.x odd 12 2
882.3.s.d 4 420.br even 12 2
1296.3.q.f 4 45.k odd 12 2
1296.3.q.f 4 45.l even 12 2
2178.3.c.d 2 220.i odd 4 1
2178.3.c.d 2 660.q even 4 1
2304.3.h.c 4 80.i odd 4 1
2304.3.h.c 4 80.t odd 4 1
2304.3.h.c 4 240.bb even 4 1
2304.3.h.c 4 240.bf even 4 1
2304.3.h.f 4 80.j even 4 1
2304.3.h.f 4 80.s even 4 1
2304.3.h.f 4 240.z odd 4 1
2304.3.h.f 4 240.bd odd 4 1
3042.3.c.e 2 260.p even 4 1
3042.3.c.e 2 780.w odd 4 1
3042.3.d.a 4 260.l odd 4 1
3042.3.d.a 4 260.s odd 4 1
3042.3.d.a 4 780.u even 4 1
3042.3.d.a 4 780.bn even 4 1
3600.3.c.b 4 1.a even 1 1 trivial
3600.3.c.b 4 3.b odd 2 1 inner
3600.3.c.b 4 5.b even 2 1 inner
3600.3.c.b 4 15.d odd 2 1 inner
3600.3.l.d 2 5.c odd 4 1
3600.3.l.d 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 \(S_{3}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7}^{2} + 16 \)
\( T_{13}^{2} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 16 + T^{2} )^{2} \)
$11$ \( ( 288 + T^{2} )^{2} \)
$13$ \( ( 64 + T^{2} )^{2} \)
$17$ \( ( -162 + T^{2} )^{2} \)
$19$ \( ( 16 + T )^{4} \)
$23$ \( ( -288 + T^{2} )^{2} \)
$29$ \( ( 18 + T^{2} )^{2} \)
$31$ \( ( 44 + T )^{4} \)
$37$ \( ( 1156 + T^{2} )^{2} \)
$41$ \( ( 2178 + T^{2} )^{2} \)
$43$ \( ( 1600 + T^{2} )^{2} \)
$47$ \( ( -7200 + T^{2} )^{2} \)
$53$ \( ( -1458 + T^{2} )^{2} \)
$59$ \( ( 1152 + T^{2} )^{2} \)
$61$ \( ( -50 + T )^{4} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( 2592 + T^{2} )^{2} \)
$73$ \( ( 256 + T^{2} )^{2} \)
$79$ \( ( 76 + T )^{4} \)
$83$ \( ( -14112 + T^{2} )^{2} \)
$89$ \( ( 162 + T^{2} )^{2} \)
$97$ \( ( 30976 + T^{2} )^{2} \)
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