Properties

Label 1600.2.n.q
Level $1600$
Weight $2$
Character orbit 1600.n
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

Related objects

Downloads

Learn more

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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 400)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - \beta_{2} q^{7} - 7 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - \beta_{2} q^{7} - 7 \beta_1 q^{9} - 10 q^{21} - 3 \beta_{3} q^{23} - 4 \beta_{2} q^{27} + 6 \beta_1 q^{29} - 12 q^{41} + \beta_{3} q^{43} - 3 \beta_{2} q^{47} + 3 \beta_1 q^{49} - 8 q^{61} + 7 \beta_{3} q^{63} + 5 \beta_{2} q^{67} - 30 \beta_1 q^{69} - 19 q^{81} - 3 \beta_{3} q^{83} + 6 \beta_{2} q^{87} - 6 \beta_1 q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 40 q^{21} - 48 q^{41} - 32 q^{61} - 76 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 2\nu^{2} + 4\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} + 4\nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} - \beta_{2} + 4\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(-\beta_{1}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1343.1
1.61803i
0.618034i
1.61803i
0.618034i
0 −2.23607 + 2.23607i 0 0 0 2.23607 + 2.23607i 0 7.00000i 0
1343.2 0 2.23607 2.23607i 0 0 0 −2.23607 2.23607i 0 7.00000i 0
1407.1 0 −2.23607 2.23607i 0 0 0 2.23607 2.23607i 0 7.00000i 0
1407.2 0 2.23607 + 2.23607i 0 0 0 −2.23607 + 2.23607i 0 7.00000i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.n.q 4
4.b odd 2 1 inner 1600.2.n.q 4
5.b even 2 1 inner 1600.2.n.q 4
5.c odd 4 2 inner 1600.2.n.q 4
8.b even 2 1 400.2.n.c 4
8.d odd 2 1 400.2.n.c 4
20.d odd 2 1 CM 1600.2.n.q 4
20.e even 4 2 inner 1600.2.n.q 4
24.f even 2 1 3600.2.x.f 4
24.h odd 2 1 3600.2.x.f 4
40.e odd 2 1 400.2.n.c 4
40.f even 2 1 400.2.n.c 4
40.i odd 4 2 400.2.n.c 4
40.k even 4 2 400.2.n.c 4
120.i odd 2 1 3600.2.x.f 4
120.m even 2 1 3600.2.x.f 4
120.q odd 4 2 3600.2.x.f 4
120.w even 4 2 3600.2.x.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.n.c 4 8.b even 2 1
400.2.n.c 4 8.d odd 2 1
400.2.n.c 4 40.e odd 2 1
400.2.n.c 4 40.f even 2 1
400.2.n.c 4 40.i odd 4 2
400.2.n.c 4 40.k even 4 2
1600.2.n.q 4 1.a even 1 1 trivial
1600.2.n.q 4 4.b odd 2 1 inner
1600.2.n.q 4 5.b even 2 1 inner
1600.2.n.q 4 5.c odd 4 2 inner
1600.2.n.q 4 20.d odd 2 1 CM
1600.2.n.q 4 20.e even 4 2 inner
3600.2.x.f 4 24.f even 2 1
3600.2.x.f 4 24.h odd 2 1
3600.2.x.f 4 120.i odd 2 1
3600.2.x.f 4 120.m even 2 1
3600.2.x.f 4 120.q odd 4 2
3600.2.x.f 4 120.w even 4 2

Hecke kernels

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

\( T_{3}^{4} + 100 \) Copy content Toggle raw display
\( T_{7}^{4} + 100 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 100 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 100 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 8100 \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T + 12)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 100 \) Copy content Toggle raw display
$47$ \( T^{4} + 8100 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T + 8)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 62500 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 8100 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
show more
show less