Properties

Label 3600.2.f.b
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: \( 1 \)
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.

\(f(q)\) \(=\) \( q + 5 i q^{7} +O(q^{10})\) \( q + 5 i q^{7} -6 q^{11} + 3 i q^{13} + 2 i q^{17} + q^{19} + 2 i q^{23} + 6 q^{29} -3 q^{31} -6 i q^{37} -4 q^{41} + 11 i q^{43} -10 i q^{47} -18 q^{49} -8 i q^{53} + 6 q^{59} + 3 q^{61} + i q^{67} -12 q^{71} -10 i q^{73} -30 i q^{77} -8 q^{79} + 6 i q^{83} -16 q^{89} -15 q^{91} -7 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 12q^{11} + 2q^{19} + 12q^{29} - 6q^{31} - 8q^{41} - 36q^{49} + 12q^{59} + 6q^{61} - 24q^{71} - 16q^{79} - 32q^{89} - 30q^{91} + O(q^{100}) \)

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 5.00000i 0 0 0
2449.2 0 0 0 0 0 5.00000i 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.b 2
3.b odd 2 1 1200.2.f.i 2
4.b odd 2 1 1800.2.f.k 2
5.b even 2 1 inner 3600.2.f.b 2
5.c odd 4 1 3600.2.a.a 1
5.c odd 4 1 3600.2.a.bq 1
12.b even 2 1 600.2.f.a 2
15.d odd 2 1 1200.2.f.i 2
15.e even 4 1 1200.2.a.i 1
15.e even 4 1 1200.2.a.j 1
20.d odd 2 1 1800.2.f.k 2
20.e even 4 1 1800.2.a.a 1
20.e even 4 1 1800.2.a.x 1
24.f even 2 1 4800.2.f.bj 2
24.h odd 2 1 4800.2.f.a 2
60.h even 2 1 600.2.f.a 2
60.l odd 4 1 600.2.a.e 1
60.l odd 4 1 600.2.a.f yes 1
120.i odd 2 1 4800.2.f.a 2
120.m even 2 1 4800.2.f.bj 2
120.q odd 4 1 4800.2.a.b 1
120.q odd 4 1 4800.2.a.ct 1
120.w even 4 1 4800.2.a.a 1
120.w even 4 1 4800.2.a.cs 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.a.e 1 60.l odd 4 1
600.2.a.f yes 1 60.l odd 4 1
600.2.f.a 2 12.b even 2 1
600.2.f.a 2 60.h even 2 1
1200.2.a.i 1 15.e even 4 1
1200.2.a.j 1 15.e even 4 1
1200.2.f.i 2 3.b odd 2 1
1200.2.f.i 2 15.d odd 2 1
1800.2.a.a 1 20.e even 4 1
1800.2.a.x 1 20.e even 4 1
1800.2.f.k 2 4.b odd 2 1
1800.2.f.k 2 20.d odd 2 1
3600.2.a.a 1 5.c odd 4 1
3600.2.a.bq 1 5.c odd 4 1
3600.2.f.b 2 1.a even 1 1 trivial
3600.2.f.b 2 5.b even 2 1 inner
4800.2.a.a 1 120.w even 4 1
4800.2.a.b 1 120.q odd 4 1
4800.2.a.cs 1 120.w even 4 1
4800.2.a.ct 1 120.q odd 4 1
4800.2.f.a 2 24.h odd 2 1
4800.2.f.a 2 120.i odd 2 1
4800.2.f.bj 2 24.f even 2 1
4800.2.f.bj 2 120.m even 2 1

Hecke kernels

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

\( T_{7}^{2} + 25 \)
\( T_{11} + 6 \)
\( T_{13}^{2} + 9 \)
\( T_{17}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 25 + T^{2} \)
$11$ \( ( 6 + T )^{2} \)
$13$ \( 9 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 4 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( 4 + T )^{2} \)
$43$ \( 121 + T^{2} \)
$47$ \( 100 + T^{2} \)
$53$ \( 64 + T^{2} \)
$59$ \( ( -6 + T )^{2} \)
$61$ \( ( -3 + T )^{2} \)
$67$ \( 1 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( ( 16 + T )^{2} \)
$97$ \( 49 + T^{2} \)
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