Properties

Label 3600.2.f.c
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: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q -4 q^{11} -6 i q^{13} + 6 i q^{17} -4 q^{19} -2 q^{29} + 8 q^{31} -2 i q^{37} + 6 q^{41} + 12 i q^{43} + 8 i q^{47} + 7 q^{49} + 6 i q^{53} -12 q^{59} + 14 q^{61} -4 i q^{67} + 8 q^{71} + 6 i q^{73} -8 q^{79} + 12 i q^{83} + 10 q^{89} + 2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 8q^{11} - 8q^{19} - 4q^{29} + 16q^{31} + 12q^{41} + 14q^{49} - 24q^{59} + 28q^{61} + 16q^{71} - 16q^{79} + 20q^{89} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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.c 2
3.b odd 2 1 1200.2.f.g 2
4.b odd 2 1 1800.2.f.j 2
5.b even 2 1 inner 3600.2.f.c 2
5.c odd 4 1 720.2.a.d 1
5.c odd 4 1 3600.2.a.t 1
12.b even 2 1 600.2.f.b 2
15.d odd 2 1 1200.2.f.g 2
15.e even 4 1 240.2.a.c 1
15.e even 4 1 1200.2.a.o 1
20.d odd 2 1 1800.2.f.j 2
20.e even 4 1 360.2.a.b 1
20.e even 4 1 1800.2.a.n 1
24.f even 2 1 4800.2.f.bc 2
24.h odd 2 1 4800.2.f.i 2
40.i odd 4 1 2880.2.a.bb 1
40.k even 4 1 2880.2.a.x 1
60.h even 2 1 600.2.f.b 2
60.l odd 4 1 120.2.a.b 1
60.l odd 4 1 600.2.a.c 1
120.i odd 2 1 4800.2.f.i 2
120.m even 2 1 4800.2.f.bc 2
120.q odd 4 1 960.2.a.c 1
120.q odd 4 1 4800.2.a.cd 1
120.w even 4 1 960.2.a.j 1
120.w even 4 1 4800.2.a.r 1
180.v odd 12 2 3240.2.q.g 2
180.x even 12 2 3240.2.q.q 2
240.z odd 4 1 3840.2.k.o 2
240.bb even 4 1 3840.2.k.j 2
240.bd odd 4 1 3840.2.k.o 2
240.bf even 4 1 3840.2.k.j 2
420.w even 4 1 5880.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.b 1 60.l odd 4 1
240.2.a.c 1 15.e even 4 1
360.2.a.b 1 20.e even 4 1
600.2.a.c 1 60.l odd 4 1
600.2.f.b 2 12.b even 2 1
600.2.f.b 2 60.h even 2 1
720.2.a.d 1 5.c odd 4 1
960.2.a.c 1 120.q odd 4 1
960.2.a.j 1 120.w even 4 1
1200.2.a.o 1 15.e even 4 1
1200.2.f.g 2 3.b odd 2 1
1200.2.f.g 2 15.d odd 2 1
1800.2.a.n 1 20.e even 4 1
1800.2.f.j 2 4.b odd 2 1
1800.2.f.j 2 20.d odd 2 1
2880.2.a.x 1 40.k even 4 1
2880.2.a.bb 1 40.i odd 4 1
3240.2.q.g 2 180.v odd 12 2
3240.2.q.q 2 180.x even 12 2
3600.2.a.t 1 5.c odd 4 1
3600.2.f.c 2 1.a even 1 1 trivial
3600.2.f.c 2 5.b even 2 1 inner
3840.2.k.j 2 240.bb even 4 1
3840.2.k.j 2 240.bf even 4 1
3840.2.k.o 2 240.z odd 4 1
3840.2.k.o 2 240.bd odd 4 1
4800.2.a.r 1 120.w even 4 1
4800.2.a.cd 1 120.q odd 4 1
4800.2.f.i 2 24.h odd 2 1
4800.2.f.i 2 120.i odd 2 1
4800.2.f.bc 2 24.f even 2 1
4800.2.f.bc 2 120.m even 2 1
5880.2.a.a 1 420.w even 4 1

Hecke kernels

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

\( T_{7} \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 36 \)
\( T_{17}^{2} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 144 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( ( -14 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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