Properties

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

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.c (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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 32)
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.

\(f(q)\) \(=\) \( q + 3 q^{9} +O(q^{10})\) \( q + 3 q^{9} -6 i q^{13} -2 i q^{17} -10 q^{29} -2 i q^{37} + 10 q^{41} + 7 q^{49} -14 i q^{53} + 10 q^{61} -6 i q^{73} + 9 q^{81} -10 q^{89} -18 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{9} + O(q^{10}) \) \( 2q + 6q^{9} - 20q^{29} + 20q^{41} + 14q^{49} + 20q^{61} + 18q^{81} - 20q^{89} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 0 0 0 0 0 0 3.00000 0
449.2 0 0 0 0 0 0 0 3.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.c.l 2
4.b odd 2 1 CM 1600.2.c.l 2
5.b even 2 1 inner 1600.2.c.l 2
5.c odd 4 1 64.2.a.a 1
5.c odd 4 1 1600.2.a.n 1
8.b even 2 1 800.2.c.e 2
8.d odd 2 1 800.2.c.e 2
15.e even 4 1 576.2.a.c 1
20.d odd 2 1 inner 1600.2.c.l 2
20.e even 4 1 64.2.a.a 1
20.e even 4 1 1600.2.a.n 1
24.f even 2 1 7200.2.f.m 2
24.h odd 2 1 7200.2.f.m 2
35.f even 4 1 3136.2.a.m 1
40.e odd 2 1 800.2.c.e 2
40.f even 2 1 800.2.c.e 2
40.i odd 4 1 32.2.a.a 1
40.i odd 4 1 800.2.a.d 1
40.k even 4 1 32.2.a.a 1
40.k even 4 1 800.2.a.d 1
55.e even 4 1 7744.2.a.v 1
60.l odd 4 1 576.2.a.c 1
80.i odd 4 1 256.2.b.b 2
80.j even 4 1 256.2.b.b 2
80.s even 4 1 256.2.b.b 2
80.t odd 4 1 256.2.b.b 2
120.i odd 2 1 7200.2.f.m 2
120.m even 2 1 7200.2.f.m 2
120.q odd 4 1 288.2.a.d 1
120.q odd 4 1 7200.2.a.v 1
120.w even 4 1 288.2.a.d 1
120.w even 4 1 7200.2.a.v 1
140.j odd 4 1 3136.2.a.m 1
160.u even 8 2 1024.2.e.j 4
160.v odd 8 2 1024.2.e.j 4
160.ba even 8 2 1024.2.e.j 4
160.bb odd 8 2 1024.2.e.j 4
220.i odd 4 1 7744.2.a.v 1
240.z odd 4 1 2304.2.d.j 2
240.bb even 4 1 2304.2.d.j 2
240.bd odd 4 1 2304.2.d.j 2
240.bf even 4 1 2304.2.d.j 2
280.s even 4 1 1568.2.a.e 1
280.y odd 4 1 1568.2.a.e 1
280.bp odd 12 2 1568.2.i.f 2
280.br even 12 2 1568.2.i.g 2
280.bt odd 12 2 1568.2.i.g 2
280.bv even 12 2 1568.2.i.f 2
360.bo even 12 2 2592.2.i.t 2
360.br even 12 2 2592.2.i.e 2
360.bt odd 12 2 2592.2.i.e 2
360.bu odd 12 2 2592.2.i.t 2
440.t even 4 1 3872.2.a.f 1
440.w odd 4 1 3872.2.a.f 1
520.bc even 4 1 5408.2.a.g 1
520.bg odd 4 1 5408.2.a.g 1
680.u even 4 1 9248.2.a.f 1
680.bi odd 4 1 9248.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 40.i odd 4 1
32.2.a.a 1 40.k even 4 1
64.2.a.a 1 5.c odd 4 1
64.2.a.a 1 20.e even 4 1
256.2.b.b 2 80.i odd 4 1
256.2.b.b 2 80.j even 4 1
256.2.b.b 2 80.s even 4 1
256.2.b.b 2 80.t odd 4 1
288.2.a.d 1 120.q odd 4 1
288.2.a.d 1 120.w even 4 1
576.2.a.c 1 15.e even 4 1
576.2.a.c 1 60.l odd 4 1
800.2.a.d 1 40.i odd 4 1
800.2.a.d 1 40.k even 4 1
800.2.c.e 2 8.b even 2 1
800.2.c.e 2 8.d odd 2 1
800.2.c.e 2 40.e odd 2 1
800.2.c.e 2 40.f even 2 1
1024.2.e.j 4 160.u even 8 2
1024.2.e.j 4 160.v odd 8 2
1024.2.e.j 4 160.ba even 8 2
1024.2.e.j 4 160.bb odd 8 2
1568.2.a.e 1 280.s even 4 1
1568.2.a.e 1 280.y odd 4 1
1568.2.i.f 2 280.bp odd 12 2
1568.2.i.f 2 280.bv even 12 2
1568.2.i.g 2 280.br even 12 2
1568.2.i.g 2 280.bt odd 12 2
1600.2.a.n 1 5.c odd 4 1
1600.2.a.n 1 20.e even 4 1
1600.2.c.l 2 1.a even 1 1 trivial
1600.2.c.l 2 4.b odd 2 1 CM
1600.2.c.l 2 5.b even 2 1 inner
1600.2.c.l 2 20.d odd 2 1 inner
2304.2.d.j 2 240.z odd 4 1
2304.2.d.j 2 240.bb even 4 1
2304.2.d.j 2 240.bd odd 4 1
2304.2.d.j 2 240.bf even 4 1
2592.2.i.e 2 360.br even 12 2
2592.2.i.e 2 360.bt odd 12 2
2592.2.i.t 2 360.bo even 12 2
2592.2.i.t 2 360.bu odd 12 2
3136.2.a.m 1 35.f even 4 1
3136.2.a.m 1 140.j odd 4 1
3872.2.a.f 1 440.t even 4 1
3872.2.a.f 1 440.w odd 4 1
5408.2.a.g 1 520.bc even 4 1
5408.2.a.g 1 520.bg odd 4 1
7200.2.a.v 1 120.q odd 4 1
7200.2.a.v 1 120.w even 4 1
7200.2.f.m 2 24.f even 2 1
7200.2.f.m 2 24.h odd 2 1
7200.2.f.m 2 120.i odd 2 1
7200.2.f.m 2 120.m even 2 1
7744.2.a.v 1 55.e even 4 1
7744.2.a.v 1 220.i odd 4 1
9248.2.a.f 1 680.u even 4 1
9248.2.a.f 1 680.bi odd 4 1

Hecke kernels

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

\( T_{3} \)
\( T_{7} \)
\( T_{11} \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 10 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 196 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 324 + T^{2} \)
show more
show less