Properties

Label 3600.2.f.i
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q + 4 i q^{7} +O(q^{10})\) \( q + 4 i q^{7} -2 i q^{13} -6 i q^{17} -4 q^{19} -6 q^{29} -8 q^{31} + 2 i q^{37} + 6 q^{41} -4 i q^{43} -9 q^{49} -6 i q^{53} -10 q^{61} + 4 i q^{67} -2 i q^{73} + 8 q^{79} -12 i q^{83} + 18 q^{89} + 8 q^{91} + 2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 8q^{19} - 12q^{29} - 16q^{31} + 12q^{41} - 18q^{49} - 20q^{61} + 16q^{79} + 36q^{89} + 16q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
1.00000i
1.00000i
0 0 0 0 0 4.00000i 0 0 0
2449.2 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
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.i 2
3.b odd 2 1 1200.2.f.e 2
4.b odd 2 1 450.2.c.b 2
5.b even 2 1 inner 3600.2.f.i 2
5.c odd 4 1 720.2.a.j 1
5.c odd 4 1 3600.2.a.f 1
12.b even 2 1 150.2.c.a 2
15.d odd 2 1 1200.2.f.e 2
15.e even 4 1 240.2.a.b 1
15.e even 4 1 1200.2.a.k 1
20.d odd 2 1 450.2.c.b 2
20.e even 4 1 90.2.a.c 1
20.e even 4 1 450.2.a.d 1
24.f even 2 1 4800.2.f.p 2
24.h odd 2 1 4800.2.f.w 2
40.i odd 4 1 2880.2.a.q 1
40.k even 4 1 2880.2.a.a 1
60.h even 2 1 150.2.c.a 2
60.l odd 4 1 30.2.a.a 1
60.l odd 4 1 150.2.a.b 1
120.i odd 2 1 4800.2.f.w 2
120.m even 2 1 4800.2.f.p 2
120.q odd 4 1 960.2.a.e 1
120.q odd 4 1 4800.2.a.cq 1
120.w even 4 1 960.2.a.p 1
120.w even 4 1 4800.2.a.d 1
140.j odd 4 1 4410.2.a.z 1
180.v odd 12 2 810.2.e.l 2
180.x even 12 2 810.2.e.b 2
240.z odd 4 1 3840.2.k.y 2
240.bb even 4 1 3840.2.k.f 2
240.bd odd 4 1 3840.2.k.y 2
240.bf even 4 1 3840.2.k.f 2
420.w even 4 1 1470.2.a.d 1
420.w even 4 1 7350.2.a.ct 1
420.bp odd 12 2 1470.2.i.o 2
420.br even 12 2 1470.2.i.q 2
660.q even 4 1 3630.2.a.w 1
780.u even 4 1 5070.2.b.k 2
780.w odd 4 1 5070.2.a.w 1
780.bn even 4 1 5070.2.b.k 2
1020.x odd 4 1 8670.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 60.l odd 4 1
90.2.a.c 1 20.e even 4 1
150.2.a.b 1 60.l odd 4 1
150.2.c.a 2 12.b even 2 1
150.2.c.a 2 60.h even 2 1
240.2.a.b 1 15.e even 4 1
450.2.a.d 1 20.e even 4 1
450.2.c.b 2 4.b odd 2 1
450.2.c.b 2 20.d odd 2 1
720.2.a.j 1 5.c odd 4 1
810.2.e.b 2 180.x even 12 2
810.2.e.l 2 180.v odd 12 2
960.2.a.e 1 120.q odd 4 1
960.2.a.p 1 120.w even 4 1
1200.2.a.k 1 15.e even 4 1
1200.2.f.e 2 3.b odd 2 1
1200.2.f.e 2 15.d odd 2 1
1470.2.a.d 1 420.w even 4 1
1470.2.i.o 2 420.bp odd 12 2
1470.2.i.q 2 420.br even 12 2
2880.2.a.a 1 40.k even 4 1
2880.2.a.q 1 40.i odd 4 1
3600.2.a.f 1 5.c odd 4 1
3600.2.f.i 2 1.a even 1 1 trivial
3600.2.f.i 2 5.b even 2 1 inner
3630.2.a.w 1 660.q even 4 1
3840.2.k.f 2 240.bb even 4 1
3840.2.k.f 2 240.bf even 4 1
3840.2.k.y 2 240.z odd 4 1
3840.2.k.y 2 240.bd odd 4 1
4410.2.a.z 1 140.j odd 4 1
4800.2.a.d 1 120.w even 4 1
4800.2.a.cq 1 120.q odd 4 1
4800.2.f.p 2 24.f even 2 1
4800.2.f.p 2 120.m even 2 1
4800.2.f.w 2 24.h odd 2 1
4800.2.f.w 2 120.i odd 2 1
5070.2.a.w 1 780.w odd 4 1
5070.2.b.k 2 780.u even 4 1
5070.2.b.k 2 780.bn even 4 1
7350.2.a.ct 1 420.w even 4 1
8670.2.a.g 1 1020.x odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 16 \)
\( T_{11} \)
\( T_{13}^{2} + 4 \)
\( T_{17}^{2} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 47 T^{2} )^{2} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 10 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 18 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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