Properties

Label 3600.2.x.c
Level $3600$
Weight $2$
Character orbit 3600.x
Analytic conductor $28.746$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.x (of order \(4\), degree \(2\), 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: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 i + 5) q^{13} + ( - 5 i - 5) q^{17} + 4 i q^{29} + ( - 5 i - 5) q^{37} - 8 q^{41} + 7 i q^{49} + ( - 5 i + 5) q^{53} - 12 q^{61} + (5 i - 5) q^{73} - 16 i q^{89} + ( - 5 i - 5) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{13} - 10 q^{17} - 10 q^{37} - 16 q^{41} + 10 q^{53} - 24 q^{61} - 10 q^{73} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(i\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2143.1
1.00000i
1.00000i
0 0 0 0 0 0 0 0 0
3007.1 0 0 0 0 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.2.x.c 2
3.b odd 2 1 400.2.n.a 2
4.b odd 2 1 CM 3600.2.x.c 2
5.b even 2 1 720.2.x.a 2
5.c odd 4 1 720.2.x.a 2
5.c odd 4 1 inner 3600.2.x.c 2
12.b even 2 1 400.2.n.a 2
15.d odd 2 1 80.2.n.a 2
15.e even 4 1 80.2.n.a 2
15.e even 4 1 400.2.n.a 2
20.d odd 2 1 720.2.x.a 2
20.e even 4 1 720.2.x.a 2
20.e even 4 1 inner 3600.2.x.c 2
24.f even 2 1 1600.2.n.g 2
24.h odd 2 1 1600.2.n.g 2
60.h even 2 1 80.2.n.a 2
60.l odd 4 1 80.2.n.a 2
60.l odd 4 1 400.2.n.a 2
120.i odd 2 1 320.2.n.d 2
120.m even 2 1 320.2.n.d 2
120.q odd 4 1 320.2.n.d 2
120.q odd 4 1 1600.2.n.g 2
120.w even 4 1 320.2.n.d 2
120.w even 4 1 1600.2.n.g 2
240.t even 4 1 1280.2.o.h 2
240.t even 4 1 1280.2.o.i 2
240.z odd 4 1 1280.2.o.i 2
240.bb even 4 1 1280.2.o.i 2
240.bd odd 4 1 1280.2.o.h 2
240.bf even 4 1 1280.2.o.h 2
240.bm odd 4 1 1280.2.o.h 2
240.bm odd 4 1 1280.2.o.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.n.a 2 15.d odd 2 1
80.2.n.a 2 15.e even 4 1
80.2.n.a 2 60.h even 2 1
80.2.n.a 2 60.l odd 4 1
320.2.n.d 2 120.i odd 2 1
320.2.n.d 2 120.m even 2 1
320.2.n.d 2 120.q odd 4 1
320.2.n.d 2 120.w even 4 1
400.2.n.a 2 3.b odd 2 1
400.2.n.a 2 12.b even 2 1
400.2.n.a 2 15.e even 4 1
400.2.n.a 2 60.l odd 4 1
720.2.x.a 2 5.b even 2 1
720.2.x.a 2 5.c odd 4 1
720.2.x.a 2 20.d odd 2 1
720.2.x.a 2 20.e even 4 1
1280.2.o.h 2 240.t even 4 1
1280.2.o.h 2 240.bd odd 4 1
1280.2.o.h 2 240.bf even 4 1
1280.2.o.h 2 240.bm odd 4 1
1280.2.o.i 2 240.t even 4 1
1280.2.o.i 2 240.z odd 4 1
1280.2.o.i 2 240.bb even 4 1
1280.2.o.i 2 240.bm odd 4 1
1600.2.n.g 2 24.f even 2 1
1600.2.n.g 2 24.h odd 2 1
1600.2.n.g 2 120.q odd 4 1
1600.2.n.g 2 120.w even 4 1
3600.2.x.c 2 1.a even 1 1 trivial
3600.2.x.c 2 4.b odd 2 1 CM
3600.2.x.c 2 5.c odd 4 1 inner
3600.2.x.c 2 20.e even 4 1 inner

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 10T_{13} + 50 \) Copy content Toggle raw display
\( T_{17}^{2} + 10T_{17} + 50 \) 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
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$17$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$41$ \( (T + 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 12)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 256 \) Copy content Toggle raw display
$97$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
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