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 $8$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} -\beta_{2} q^{7} -7 \beta_{1} q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} -\beta_{2} q^{7} -7 \beta_{1} q^{9} -10 q^{21} -3 \beta_{3} q^{23} -4 \beta_{2} q^{27} + 6 \beta_{1} q^{29} -12 q^{41} + \beta_{3} q^{43} -3 \beta_{2} q^{47} + 3 \beta_{1} q^{49} -8 q^{61} + 7 \beta_{3} q^{63} + 5 \beta_{2} q^{67} -30 \beta_{1} q^{69} -19 q^{81} -3 \beta_{3} q^{83} + 6 \beta_{2} q^{87} -6 \beta_{1} q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 40q^{21} - 48q^{41} - 32q^{61} - 76q^{81} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 2 \nu^{2} + 4 \nu + 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} + 4 \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - 6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} - \beta_{2} + 4 \beta_{1}\)\()/2\)

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 \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{4} + 100 \)
\( T_{7}^{4} + 100 \)
\( T_{11} \)
\( T_{13} \)

Hecke characteristic polynomials

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