Properties

Label 3600.2.f.q
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 360)
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.

\(f(q)\) \(=\) \( q + 2 i q^{7} +O(q^{10})\) \( q + 2 i q^{7} + 2 q^{11} + 4 i q^{13} -2 i q^{17} + 4 q^{19} + 8 i q^{23} -10 q^{29} -4 q^{31} + 8 i q^{43} -8 i q^{47} + 3 q^{49} -6 i q^{53} + 14 q^{59} -14 q^{61} -4 i q^{67} + 12 q^{71} + 6 i q^{73} + 4 i q^{77} -12 q^{79} + 4 i q^{83} -12 q^{89} -8 q^{91} + 14 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 4q^{11} + 8q^{19} - 20q^{29} - 8q^{31} + 6q^{49} + 28q^{59} - 28q^{61} + 24q^{71} - 24q^{79} - 24q^{89} - 16q^{91} + O(q^{100}) \)

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 2.00000i 0 0 0
2449.2 0 0 0 0 0 2.00000i 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.q 2
3.b odd 2 1 3600.2.f.g 2
4.b odd 2 1 1800.2.f.d 2
5.b even 2 1 inner 3600.2.f.q 2
5.c odd 4 1 720.2.a.i 1
5.c odd 4 1 3600.2.a.bh 1
12.b even 2 1 1800.2.f.h 2
15.d odd 2 1 3600.2.f.g 2
15.e even 4 1 720.2.a.a 1
15.e even 4 1 3600.2.a.bd 1
20.d odd 2 1 1800.2.f.d 2
20.e even 4 1 360.2.a.d yes 1
20.e even 4 1 1800.2.a.f 1
40.i odd 4 1 2880.2.a.e 1
40.k even 4 1 2880.2.a.n 1
60.h even 2 1 1800.2.f.h 2
60.l odd 4 1 360.2.a.c 1
60.l odd 4 1 1800.2.a.i 1
120.q odd 4 1 2880.2.a.bd 1
120.w even 4 1 2880.2.a.w 1
180.v odd 12 2 3240.2.q.n 2
180.x even 12 2 3240.2.q.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.a.c 1 60.l odd 4 1
360.2.a.d yes 1 20.e even 4 1
720.2.a.a 1 15.e even 4 1
720.2.a.i 1 5.c odd 4 1
1800.2.a.f 1 20.e even 4 1
1800.2.a.i 1 60.l odd 4 1
1800.2.f.d 2 4.b odd 2 1
1800.2.f.d 2 20.d odd 2 1
1800.2.f.h 2 12.b even 2 1
1800.2.f.h 2 60.h even 2 1
2880.2.a.e 1 40.i odd 4 1
2880.2.a.n 1 40.k even 4 1
2880.2.a.w 1 120.w even 4 1
2880.2.a.bd 1 120.q odd 4 1
3240.2.q.d 2 180.x even 12 2
3240.2.q.n 2 180.v odd 12 2
3600.2.a.bd 1 15.e even 4 1
3600.2.a.bh 1 5.c odd 4 1
3600.2.f.g 2 3.b odd 2 1
3600.2.f.g 2 15.d odd 2 1
3600.2.f.q 2 1.a even 1 1 trivial
3600.2.f.q 2 5.b even 2 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( ( 10 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( -14 + T )^{2} \)
$61$ \( ( 14 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 12 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( 12 + T )^{2} \)
$97$ \( 196 + T^{2} \)
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