Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.n (of order \(4\), degree \(2\), 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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q -3 i q^{9} +O(q^{10})\) \( q -3 i q^{9} + ( -1 + i ) q^{13} + ( -3 - 3 i ) q^{17} -4 i q^{29} + ( -7 - 7 i ) q^{37} -8 q^{41} + 7 i q^{49} + ( 9 - 9 i ) q^{53} -12 q^{61} + ( 11 - 11 i ) q^{73} -9 q^{81} -16 i q^{89} + ( -13 - 13 i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 2q^{13} - 6q^{17} - 14q^{37} - 16q^{41} + 18q^{53} - 24q^{61} + 22q^{73} - 18q^{81} - 26q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1343.1
1.00000i
1.00000i
0 0 0 0 0 0 0 3.00000i 0
1407.1 0 0 0 0 0 0 0 3.00000i 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 1600.2.n.h 2
4.b odd 2 1 CM 1600.2.n.h 2
5.b even 2 1 320.2.n.e 2
5.c odd 4 1 320.2.n.e 2
5.c odd 4 1 inner 1600.2.n.h 2
8.b even 2 1 100.2.e.b 2
8.d odd 2 1 100.2.e.b 2
20.d odd 2 1 320.2.n.e 2
20.e even 4 1 320.2.n.e 2
20.e even 4 1 inner 1600.2.n.h 2
24.f even 2 1 900.2.k.c 2
24.h odd 2 1 900.2.k.c 2
40.e odd 2 1 20.2.e.a 2
40.f even 2 1 20.2.e.a 2
40.i odd 4 1 20.2.e.a 2
40.i odd 4 1 100.2.e.b 2
40.k even 4 1 20.2.e.a 2
40.k even 4 1 100.2.e.b 2
80.i odd 4 1 1280.2.o.g 2
80.j even 4 1 1280.2.o.j 2
80.k odd 4 1 1280.2.o.g 2
80.k odd 4 1 1280.2.o.j 2
80.q even 4 1 1280.2.o.g 2
80.q even 4 1 1280.2.o.j 2
80.s even 4 1 1280.2.o.g 2
80.t odd 4 1 1280.2.o.j 2
120.i odd 2 1 180.2.k.c 2
120.m even 2 1 180.2.k.c 2
120.q odd 4 1 180.2.k.c 2
120.q odd 4 1 900.2.k.c 2
120.w even 4 1 180.2.k.c 2
120.w even 4 1 900.2.k.c 2
280.c odd 2 1 980.2.k.a 2
280.n even 2 1 980.2.k.a 2
280.s even 4 1 980.2.k.a 2
280.y odd 4 1 980.2.k.a 2
280.ba even 6 2 980.2.x.c 4
280.bf even 6 2 980.2.x.d 4
280.bi odd 6 2 980.2.x.d 4
280.bk odd 6 2 980.2.x.c 4
280.bp odd 12 2 980.2.x.c 4
280.br even 12 2 980.2.x.d 4
280.bt odd 12 2 980.2.x.d 4
280.bv even 12 2 980.2.x.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.e.a 2 40.e odd 2 1
20.2.e.a 2 40.f even 2 1
20.2.e.a 2 40.i odd 4 1
20.2.e.a 2 40.k even 4 1
100.2.e.b 2 8.b even 2 1
100.2.e.b 2 8.d odd 2 1
100.2.e.b 2 40.i odd 4 1
100.2.e.b 2 40.k even 4 1
180.2.k.c 2 120.i odd 2 1
180.2.k.c 2 120.m even 2 1
180.2.k.c 2 120.q odd 4 1
180.2.k.c 2 120.w even 4 1
320.2.n.e 2 5.b even 2 1
320.2.n.e 2 5.c odd 4 1
320.2.n.e 2 20.d odd 2 1
320.2.n.e 2 20.e even 4 1
900.2.k.c 2 24.f even 2 1
900.2.k.c 2 24.h odd 2 1
900.2.k.c 2 120.q odd 4 1
900.2.k.c 2 120.w even 4 1
980.2.k.a 2 280.c odd 2 1
980.2.k.a 2 280.n even 2 1
980.2.k.a 2 280.s even 4 1
980.2.k.a 2 280.y odd 4 1
980.2.x.c 4 280.ba even 6 2
980.2.x.c 4 280.bk odd 6 2
980.2.x.c 4 280.bp odd 12 2
980.2.x.c 4 280.bv even 12 2
980.2.x.d 4 280.bf even 6 2
980.2.x.d 4 280.bi odd 6 2
980.2.x.d 4 280.br even 12 2
980.2.x.d 4 280.bt odd 12 2
1280.2.o.g 2 80.i odd 4 1
1280.2.o.g 2 80.k odd 4 1
1280.2.o.g 2 80.q even 4 1
1280.2.o.g 2 80.s even 4 1
1280.2.o.j 2 80.j even 4 1
1280.2.o.j 2 80.k odd 4 1
1280.2.o.j 2 80.q even 4 1
1280.2.o.j 2 80.t odd 4 1
1600.2.n.h 2 1.a even 1 1 trivial
1600.2.n.h 2 4.b odd 2 1 CM
1600.2.n.h 2 5.c odd 4 1 inner
1600.2.n.h 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 \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3} \)
\( T_{7} \)
\( T_{11} \)
\( T_{13}^{2} + 2 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 2 + 2 T + T^{2} \)
$17$ \( 18 + 6 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 16 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 98 + 14 T + T^{2} \)
$41$ \( ( 8 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 162 - 18 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 12 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 242 - 22 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 256 + T^{2} \)
$97$ \( 338 + 26 T + T^{2} \)
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