Properties

Label 3200.2.d.o
Level $3200$
Weight $2$
Character orbit 3200.d
Analytic conductor $25.552$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -2 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -2 q^{9} + \beta_{1} q^{11} -\beta_{2} q^{13} -3 q^{17} -\beta_{1} q^{19} + \beta_{3} q^{23} -\beta_{1} q^{27} + \beta_{2} q^{29} -\beta_{3} q^{31} + 5 q^{33} + 2 \beta_{2} q^{37} + \beta_{3} q^{39} -5 q^{41} -4 \beta_{1} q^{43} + \beta_{3} q^{47} -7 q^{49} + 3 \beta_{1} q^{51} + \beta_{2} q^{53} -5 q^{57} -4 \beta_{1} q^{59} + 2 \beta_{2} q^{61} -3 \beta_{1} q^{67} + 5 \beta_{2} q^{69} + \beta_{3} q^{71} + 9 q^{73} -11 q^{81} -3 \beta_{1} q^{83} -\beta_{3} q^{87} -15 q^{89} -5 \beta_{2} q^{93} -2 q^{97} -2 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{9} + O(q^{10}) \) \( 4q - 8q^{9} - 12q^{17} + 20q^{33} - 20q^{41} - 28q^{49} - 20q^{57} + 36q^{73} - 44q^{81} - 60q^{89} - 8q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{2}\)\(=\)\( -4 \nu^{3} - 8 \nu \)
\(\beta_{3}\)\(=\)\( 8 \nu^{2} + 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 4 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 12\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} - 2 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
1.61803i
0.618034i
0.618034i
1.61803i
0 2.23607i 0 0 0 0 0 −2.00000 0
1601.2 0 2.23607i 0 0 0 0 0 −2.00000 0
1601.3 0 2.23607i 0 0 0 0 0 −2.00000 0
1601.4 0 2.23607i 0 0 0 0 0 −2.00000 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 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.d.o 4
4.b odd 2 1 inner 3200.2.d.o 4
5.b even 2 1 3200.2.d.p yes 4
5.c odd 4 1 3200.2.f.m 4
5.c odd 4 1 3200.2.f.n 4
8.b even 2 1 inner 3200.2.d.o 4
8.d odd 2 1 inner 3200.2.d.o 4
16.e even 4 1 6400.2.a.bq 2
16.e even 4 1 6400.2.a.bs 2
16.f odd 4 1 6400.2.a.bq 2
16.f odd 4 1 6400.2.a.bs 2
20.d odd 2 1 3200.2.d.p yes 4
20.e even 4 1 3200.2.f.m 4
20.e even 4 1 3200.2.f.n 4
40.e odd 2 1 3200.2.d.p yes 4
40.f even 2 1 3200.2.d.p yes 4
40.i odd 4 1 3200.2.f.m 4
40.i odd 4 1 3200.2.f.n 4
40.k even 4 1 3200.2.f.m 4
40.k even 4 1 3200.2.f.n 4
80.k odd 4 1 6400.2.a.br 2
80.k odd 4 1 6400.2.a.bt 2
80.q even 4 1 6400.2.a.br 2
80.q even 4 1 6400.2.a.bt 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.o 4 1.a even 1 1 trivial
3200.2.d.o 4 4.b odd 2 1 inner
3200.2.d.o 4 8.b even 2 1 inner
3200.2.d.o 4 8.d odd 2 1 inner
3200.2.d.p yes 4 5.b even 2 1
3200.2.d.p yes 4 20.d odd 2 1
3200.2.d.p yes 4 40.e odd 2 1
3200.2.d.p yes 4 40.f even 2 1
3200.2.f.m 4 5.c odd 4 1
3200.2.f.m 4 20.e even 4 1
3200.2.f.m 4 40.i odd 4 1
3200.2.f.m 4 40.k even 4 1
3200.2.f.n 4 5.c odd 4 1
3200.2.f.n 4 20.e even 4 1
3200.2.f.n 4 40.i odd 4 1
3200.2.f.n 4 40.k even 4 1
6400.2.a.bq 2 16.e even 4 1
6400.2.a.bq 2 16.f odd 4 1
6400.2.a.br 2 80.k odd 4 1
6400.2.a.br 2 80.q even 4 1
6400.2.a.bs 2 16.e even 4 1
6400.2.a.bs 2 16.f odd 4 1
6400.2.a.bt 2 80.k odd 4 1
6400.2.a.bt 2 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3}^{2} + 5 \)
\( T_{7} \)
\( T_{11}^{2} + 5 \)
\( T_{13}^{2} + 16 \)
\( T_{17} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 5 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 5 + T^{2} )^{2} \)
$13$ \( ( 16 + T^{2} )^{2} \)
$17$ \( ( 3 + T )^{4} \)
$19$ \( ( 5 + T^{2} )^{2} \)
$23$ \( ( -80 + T^{2} )^{2} \)
$29$ \( ( 16 + T^{2} )^{2} \)
$31$ \( ( -80 + T^{2} )^{2} \)
$37$ \( ( 64 + T^{2} )^{2} \)
$41$ \( ( 5 + T )^{4} \)
$43$ \( ( 80 + T^{2} )^{2} \)
$47$ \( ( -80 + T^{2} )^{2} \)
$53$ \( ( 16 + T^{2} )^{2} \)
$59$ \( ( 80 + T^{2} )^{2} \)
$61$ \( ( 64 + T^{2} )^{2} \)
$67$ \( ( 45 + T^{2} )^{2} \)
$71$ \( ( -80 + T^{2} )^{2} \)
$73$ \( ( -9 + T )^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 45 + T^{2} )^{2} \)
$89$ \( ( 15 + T )^{4} \)
$97$ \( ( 2 + T )^{4} \)
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