Properties

Label 3600.2.f.r
Level $3600$
Weight $2$
Character orbit 3600.f
Analytic conductor $28.746$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.7461447277\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{11} - \beta q^{13} + \beta q^{17} - 4 q^{19} - 4 \beta q^{23} + 6 q^{29} - 8 q^{31} - 3 \beta q^{37} + 6 q^{41} - 2 \beta q^{43} + 7 q^{49} + \beta q^{53} - 4 q^{59} - 2 q^{61} - 2 \beta q^{67} + 8 q^{71} + 5 \beta q^{73} - 8 q^{79} - 2 \beta q^{83} - 6 q^{89} - \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{11} - 8 q^{19} + 12 q^{29} - 16 q^{31} + 12 q^{41} + 14 q^{49} - 8 q^{59} - 4 q^{61} + 16 q^{71} - 16 q^{79} - 12 q^{89}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
1.00000i
1.00000i
0 0 0 0 0 0 0 0 0
2449.2 0 0 0 0 0 0 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.2.f.r 2
3.b odd 2 1 1200.2.f.b 2
4.b odd 2 1 1800.2.f.c 2
5.b even 2 1 inner 3600.2.f.r 2
5.c odd 4 1 144.2.a.b 1
5.c odd 4 1 3600.2.a.v 1
12.b even 2 1 600.2.f.e 2
15.d odd 2 1 1200.2.f.b 2
15.e even 4 1 48.2.a.a 1
15.e even 4 1 1200.2.a.d 1
20.d odd 2 1 1800.2.f.c 2
20.e even 4 1 72.2.a.a 1
20.e even 4 1 1800.2.a.m 1
24.f even 2 1 4800.2.f.d 2
24.h odd 2 1 4800.2.f.bg 2
35.f even 4 1 7056.2.a.q 1
40.i odd 4 1 576.2.a.b 1
40.k even 4 1 576.2.a.d 1
45.k odd 12 2 1296.2.i.e 2
45.l even 12 2 1296.2.i.m 2
60.h even 2 1 600.2.f.e 2
60.l odd 4 1 24.2.a.a 1
60.l odd 4 1 600.2.a.h 1
80.i odd 4 1 2304.2.d.k 2
80.j even 4 1 2304.2.d.i 2
80.s even 4 1 2304.2.d.i 2
80.t odd 4 1 2304.2.d.k 2
105.k odd 4 1 2352.2.a.i 1
105.w odd 12 2 2352.2.q.r 2
105.x even 12 2 2352.2.q.l 2
120.i odd 2 1 4800.2.f.bg 2
120.m even 2 1 4800.2.f.d 2
120.q odd 4 1 192.2.a.d 1
120.q odd 4 1 4800.2.a.q 1
120.w even 4 1 192.2.a.b 1
120.w even 4 1 4800.2.a.cc 1
140.j odd 4 1 3528.2.a.d 1
140.w even 12 2 3528.2.s.j 2
140.x odd 12 2 3528.2.s.y 2
165.l odd 4 1 5808.2.a.s 1
180.v odd 12 2 648.2.i.g 2
180.x even 12 2 648.2.i.b 2
195.s even 4 1 8112.2.a.be 1
220.i odd 4 1 8712.2.a.u 1
240.z odd 4 1 768.2.d.e 2
240.bb even 4 1 768.2.d.d 2
240.bd odd 4 1 768.2.d.e 2
240.bf even 4 1 768.2.d.d 2
420.w even 4 1 1176.2.a.i 1
420.bp odd 12 2 1176.2.q.i 2
420.br even 12 2 1176.2.q.a 2
660.q even 4 1 2904.2.a.c 1
780.u even 4 1 4056.2.c.e 2
780.w odd 4 1 4056.2.a.i 1
780.bn even 4 1 4056.2.c.e 2
840.bm even 4 1 9408.2.a.h 1
840.bp odd 4 1 9408.2.a.cc 1
1020.x odd 4 1 6936.2.a.p 1
1140.w even 4 1 8664.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 60.l odd 4 1
48.2.a.a 1 15.e even 4 1
72.2.a.a 1 20.e even 4 1
144.2.a.b 1 5.c odd 4 1
192.2.a.b 1 120.w even 4 1
192.2.a.d 1 120.q odd 4 1
576.2.a.b 1 40.i odd 4 1
576.2.a.d 1 40.k even 4 1
600.2.a.h 1 60.l odd 4 1
600.2.f.e 2 12.b even 2 1
600.2.f.e 2 60.h even 2 1
648.2.i.b 2 180.x even 12 2
648.2.i.g 2 180.v odd 12 2
768.2.d.d 2 240.bb even 4 1
768.2.d.d 2 240.bf even 4 1
768.2.d.e 2 240.z odd 4 1
768.2.d.e 2 240.bd odd 4 1
1176.2.a.i 1 420.w even 4 1
1176.2.q.a 2 420.br even 12 2
1176.2.q.i 2 420.bp odd 12 2
1200.2.a.d 1 15.e even 4 1
1200.2.f.b 2 3.b odd 2 1
1200.2.f.b 2 15.d odd 2 1
1296.2.i.e 2 45.k odd 12 2
1296.2.i.m 2 45.l even 12 2
1800.2.a.m 1 20.e even 4 1
1800.2.f.c 2 4.b odd 2 1
1800.2.f.c 2 20.d odd 2 1
2304.2.d.i 2 80.j even 4 1
2304.2.d.i 2 80.s even 4 1
2304.2.d.k 2 80.i odd 4 1
2304.2.d.k 2 80.t odd 4 1
2352.2.a.i 1 105.k odd 4 1
2352.2.q.l 2 105.x even 12 2
2352.2.q.r 2 105.w odd 12 2
2904.2.a.c 1 660.q even 4 1
3528.2.a.d 1 140.j odd 4 1
3528.2.s.j 2 140.w even 12 2
3528.2.s.y 2 140.x odd 12 2
3600.2.a.v 1 5.c odd 4 1
3600.2.f.r 2 1.a even 1 1 trivial
3600.2.f.r 2 5.b even 2 1 inner
4056.2.a.i 1 780.w odd 4 1
4056.2.c.e 2 780.u even 4 1
4056.2.c.e 2 780.bn even 4 1
4800.2.a.q 1 120.q odd 4 1
4800.2.a.cc 1 120.w even 4 1
4800.2.f.d 2 24.f even 2 1
4800.2.f.d 2 120.m even 2 1
4800.2.f.bg 2 24.h odd 2 1
4800.2.f.bg 2 120.i odd 2 1
5808.2.a.s 1 165.l odd 4 1
6936.2.a.p 1 1020.x odd 4 1
7056.2.a.q 1 35.f even 4 1
8112.2.a.be 1 195.s even 4 1
8664.2.a.j 1 1140.w even 4 1
8712.2.a.u 1 220.i odd 4 1
9408.2.a.h 1 840.bm even 4 1
9408.2.a.cc 1 840.bp odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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