Properties

Label 1600.2.f.b
Level $1600$
Weight $2$
Character orbit 1600.f
Analytic conductor $12.776$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 64)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 q^{3} + q^{9} +O(q^{10})\) \( q + 2 q^{3} + q^{9} -6 i q^{11} -6 i q^{17} -2 i q^{19} -4 q^{27} -12 i q^{33} -6 q^{41} + 10 q^{43} + 7 q^{49} -12 i q^{51} -4 i q^{57} + 6 i q^{59} + 14 q^{67} -2 i q^{73} -11 q^{81} + 18 q^{83} -18 q^{89} + 10 i q^{97} -6 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + 2q^{9} + O(q^{10}) \) \( 2q + 4q^{3} + 2q^{9} - 8q^{27} - 12q^{41} + 20q^{43} + 14q^{49} + 28q^{67} - 22q^{81} + 36q^{83} - 36q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
1.00000i
1.00000i
0 2.00000 0 0 0 0 0 1.00000 0
1249.2 0 2.00000 0 0 0 0 0 1.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
20.d odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.f.b 2
4.b odd 2 1 1600.2.f.a 2
5.b even 2 1 1600.2.f.a 2
5.c odd 4 1 64.2.b.a 2
5.c odd 4 1 1600.2.d.a 2
8.b even 2 1 1600.2.f.a 2
8.d odd 2 1 CM 1600.2.f.b 2
15.e even 4 1 576.2.d.a 2
20.d odd 2 1 inner 1600.2.f.b 2
20.e even 4 1 64.2.b.a 2
20.e even 4 1 1600.2.d.a 2
35.f even 4 1 3136.2.b.b 2
40.e odd 2 1 1600.2.f.a 2
40.f even 2 1 inner 1600.2.f.b 2
40.i odd 4 1 64.2.b.a 2
40.i odd 4 1 1600.2.d.a 2
40.k even 4 1 64.2.b.a 2
40.k even 4 1 1600.2.d.a 2
60.l odd 4 1 576.2.d.a 2
80.i odd 4 1 256.2.a.a 1
80.i odd 4 1 6400.2.a.a 1
80.j even 4 1 256.2.a.a 1
80.j even 4 1 6400.2.a.a 1
80.s even 4 1 256.2.a.d 1
80.s even 4 1 6400.2.a.x 1
80.t odd 4 1 256.2.a.d 1
80.t odd 4 1 6400.2.a.x 1
120.q odd 4 1 576.2.d.a 2
120.w even 4 1 576.2.d.a 2
140.j odd 4 1 3136.2.b.b 2
160.u even 8 2 1024.2.e.l 4
160.v odd 8 2 1024.2.e.l 4
160.ba even 8 2 1024.2.e.l 4
160.bb odd 8 2 1024.2.e.l 4
240.z odd 4 1 2304.2.a.h 1
240.bb even 4 1 2304.2.a.i 1
240.bd odd 4 1 2304.2.a.i 1
240.bf even 4 1 2304.2.a.h 1
280.s even 4 1 3136.2.b.b 2
280.y odd 4 1 3136.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 5.c odd 4 1
64.2.b.a 2 20.e even 4 1
64.2.b.a 2 40.i odd 4 1
64.2.b.a 2 40.k even 4 1
256.2.a.a 1 80.i odd 4 1
256.2.a.a 1 80.j even 4 1
256.2.a.d 1 80.s even 4 1
256.2.a.d 1 80.t odd 4 1
576.2.d.a 2 15.e even 4 1
576.2.d.a 2 60.l odd 4 1
576.2.d.a 2 120.q odd 4 1
576.2.d.a 2 120.w even 4 1
1024.2.e.l 4 160.u even 8 2
1024.2.e.l 4 160.v odd 8 2
1024.2.e.l 4 160.ba even 8 2
1024.2.e.l 4 160.bb odd 8 2
1600.2.d.a 2 5.c odd 4 1
1600.2.d.a 2 20.e even 4 1
1600.2.d.a 2 40.i odd 4 1
1600.2.d.a 2 40.k even 4 1
1600.2.f.a 2 4.b odd 2 1
1600.2.f.a 2 5.b even 2 1
1600.2.f.a 2 8.b even 2 1
1600.2.f.a 2 40.e odd 2 1
1600.2.f.b 2 1.a even 1 1 trivial
1600.2.f.b 2 8.d odd 2 1 CM
1600.2.f.b 2 20.d odd 2 1 inner
1600.2.f.b 2 40.f even 2 1 inner
2304.2.a.h 1 240.z odd 4 1
2304.2.a.h 1 240.bf even 4 1
2304.2.a.i 1 240.bb even 4 1
2304.2.a.i 1 240.bd odd 4 1
3136.2.b.b 2 35.f even 4 1
3136.2.b.b 2 140.j odd 4 1
3136.2.b.b 2 280.s even 4 1
3136.2.b.b 2 280.y odd 4 1
6400.2.a.a 1 80.i odd 4 1
6400.2.a.a 1 80.j even 4 1
6400.2.a.x 1 80.s even 4 1
6400.2.a.x 1 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3} - 2 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -2 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 36 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( 4 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( -10 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( ( -14 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( -18 + T )^{2} \)
$89$ \( ( 18 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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