Properties

Label 3600.2.f.o
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: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
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.

\(f(q)\) \(=\) \( q + 3 i q^{7} +O(q^{10})\) \( q + 3 i q^{7} + 2 q^{11} -3 i q^{13} + 6 i q^{17} -7 q^{19} + 6 i q^{23} -2 q^{29} + 5 q^{31} -10 i q^{37} -12 q^{41} -3 i q^{43} + 10 i q^{47} -2 q^{49} + 6 q^{59} -13 q^{61} + 7 i q^{67} -4 q^{71} -6 i q^{73} + 6 i q^{77} -8 q^{79} -6 i q^{83} + 16 q^{89} + 9 q^{91} + 7 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 4q^{11} - 14q^{19} - 4q^{29} + 10q^{31} - 24q^{41} - 4q^{49} + 12q^{59} - 26q^{61} - 8q^{71} - 16q^{79} + 32q^{89} + 18q^{91} + O(q^{100}) \)

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 3.00000i 0 0 0
2449.2 0 0 0 0 0 3.00000i 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.o 2
3.b odd 2 1 1200.2.f.c 2
4.b odd 2 1 1800.2.f.e 2
5.b even 2 1 inner 3600.2.f.o 2
5.c odd 4 1 3600.2.a.i 1
5.c odd 4 1 3600.2.a.bl 1
12.b even 2 1 600.2.f.d 2
15.d odd 2 1 1200.2.f.c 2
15.e even 4 1 1200.2.a.b 1
15.e even 4 1 1200.2.a.q 1
20.d odd 2 1 1800.2.f.e 2
20.e even 4 1 1800.2.a.e 1
20.e even 4 1 1800.2.a.t 1
24.f even 2 1 4800.2.f.k 2
24.h odd 2 1 4800.2.f.z 2
60.h even 2 1 600.2.f.d 2
60.l odd 4 1 600.2.a.b 1
60.l odd 4 1 600.2.a.i yes 1
120.i odd 2 1 4800.2.f.z 2
120.m even 2 1 4800.2.f.k 2
120.q odd 4 1 4800.2.a.bc 1
120.q odd 4 1 4800.2.a.bp 1
120.w even 4 1 4800.2.a.bd 1
120.w even 4 1 4800.2.a.bs 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.a.b 1 60.l odd 4 1
600.2.a.i yes 1 60.l odd 4 1
600.2.f.d 2 12.b even 2 1
600.2.f.d 2 60.h even 2 1
1200.2.a.b 1 15.e even 4 1
1200.2.a.q 1 15.e even 4 1
1200.2.f.c 2 3.b odd 2 1
1200.2.f.c 2 15.d odd 2 1
1800.2.a.e 1 20.e even 4 1
1800.2.a.t 1 20.e even 4 1
1800.2.f.e 2 4.b odd 2 1
1800.2.f.e 2 20.d odd 2 1
3600.2.a.i 1 5.c odd 4 1
3600.2.a.bl 1 5.c odd 4 1
3600.2.f.o 2 1.a even 1 1 trivial
3600.2.f.o 2 5.b even 2 1 inner
4800.2.a.bc 1 120.q odd 4 1
4800.2.a.bd 1 120.w even 4 1
4800.2.a.bp 1 120.q odd 4 1
4800.2.a.bs 1 120.w even 4 1
4800.2.f.k 2 24.f even 2 1
4800.2.f.k 2 120.m even 2 1
4800.2.f.z 2 24.h odd 2 1
4800.2.f.z 2 120.i odd 2 1

Hecke kernels

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

\( T_{7}^{2} + 9 \)
\( T_{11} - 2 \)
\( T_{13}^{2} + 9 \)
\( T_{17}^{2} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ \( 1 - 5 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 2 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 17 T^{2} + 169 T^{4} \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 10 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 2 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 5 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 12 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 77 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 6 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 13 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 85 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 4 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 16 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 145 T^{2} + 9409 T^{4} \)
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