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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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