Properties

Label 1600.2.f.d
Level $1600$
Weight $2$
Character orbit 1600.f
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{7} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{7} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{9} + ( -2 + 4 \zeta_{12}^{2} ) q^{11} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + ( -2 + 4 \zeta_{12}^{2} ) q^{17} -2 \zeta_{12}^{3} q^{19} + ( -4 + 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{21} + ( 3 - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{23} -4 q^{27} + ( -6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{33} -6 q^{37} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{39} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{41} + ( -5 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{43} + ( 3 - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{47} + ( -5 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{49} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{51} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{53} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{57} + 6 \zeta_{12}^{3} q^{59} + ( -4 + 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{61} + ( 7 - 14 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{63} + ( 5 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{67} -6 \zeta_{12}^{3} q^{69} + ( 6 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{71} + ( -6 + 12 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{73} + ( 6 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{77} + 12 q^{79} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{81} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{83} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( 6 - 12 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{91} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{93} + ( 6 - 12 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{97} + ( -2 + 4 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{9} - 16q^{27} - 24q^{31} - 24q^{37} - 24q^{39} + 24q^{41} - 20q^{43} - 20q^{49} + 20q^{67} + 24q^{71} + 24q^{77} + 48q^{79} + 4q^{81} - 12q^{83} + 24q^{89} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0 −2.73205 0 0 0 4.73205i 0 4.46410 0
1249.2 0 −2.73205 0 0 0 4.73205i 0 4.46410 0
1249.3 0 0.732051 0 0 0 1.26795i 0 −2.46410 0
1249.4 0 0.732051 0 0 0 1.26795i 0 −2.46410 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
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.f.d 4
4.b odd 2 1 1600.2.f.i 4
5.b even 2 1 1600.2.f.h 4
5.c odd 4 1 320.2.d.b yes 4
5.c odd 4 1 1600.2.d.b 4
8.b even 2 1 1600.2.f.h 4
8.d odd 2 1 1600.2.f.e 4
15.e even 4 1 2880.2.k.l 4
20.d odd 2 1 1600.2.f.e 4
20.e even 4 1 320.2.d.a 4
20.e even 4 1 1600.2.d.h 4
40.e odd 2 1 1600.2.f.i 4
40.f even 2 1 inner 1600.2.f.d 4
40.i odd 4 1 320.2.d.b yes 4
40.i odd 4 1 1600.2.d.b 4
40.k even 4 1 320.2.d.a 4
40.k even 4 1 1600.2.d.h 4
60.l odd 4 1 2880.2.k.e 4
80.i odd 4 1 1280.2.a.m 2
80.i odd 4 1 6400.2.a.ck 2
80.j even 4 1 1280.2.a.p 2
80.j even 4 1 6400.2.a.cd 2
80.s even 4 1 1280.2.a.b 2
80.s even 4 1 6400.2.a.y 2
80.t odd 4 1 1280.2.a.c 2
80.t odd 4 1 6400.2.a.bf 2
120.q odd 4 1 2880.2.k.e 4
120.w even 4 1 2880.2.k.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.d.a 4 20.e even 4 1
320.2.d.a 4 40.k even 4 1
320.2.d.b yes 4 5.c odd 4 1
320.2.d.b yes 4 40.i odd 4 1
1280.2.a.b 2 80.s even 4 1
1280.2.a.c 2 80.t odd 4 1
1280.2.a.m 2 80.i odd 4 1
1280.2.a.p 2 80.j even 4 1
1600.2.d.b 4 5.c odd 4 1
1600.2.d.b 4 40.i odd 4 1
1600.2.d.h 4 20.e even 4 1
1600.2.d.h 4 40.k even 4 1
1600.2.f.d 4 1.a even 1 1 trivial
1600.2.f.d 4 40.f even 2 1 inner
1600.2.f.e 4 8.d odd 2 1
1600.2.f.e 4 20.d odd 2 1
1600.2.f.h 4 5.b even 2 1
1600.2.f.h 4 8.b even 2 1
1600.2.f.i 4 4.b odd 2 1
1600.2.f.i 4 40.e odd 2 1
2880.2.k.e 4 60.l odd 4 1
2880.2.k.e 4 120.q odd 4 1
2880.2.k.l 4 15.e even 4 1
2880.2.k.l 4 120.w even 4 1
6400.2.a.y 2 80.s even 4 1
6400.2.a.bf 2 80.t odd 4 1
6400.2.a.cd 2 80.j even 4 1
6400.2.a.ck 2 80.i odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 2 T_{3} - 2 \)
\( T_{31}^{2} + 12 T_{31} + 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -2 + 2 T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 36 + 24 T^{2} + T^{4} \)
$11$ \( ( 12 + T^{2} )^{2} \)
$13$ \( ( -12 + T^{2} )^{2} \)
$17$ \( ( 12 + T^{2} )^{2} \)
$19$ \( ( 4 + T^{2} )^{2} \)
$23$ \( 324 + 72 T^{2} + T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( 24 + 12 T + T^{2} )^{2} \)
$37$ \( ( 6 + T )^{4} \)
$41$ \( ( 24 - 12 T + T^{2} )^{2} \)
$43$ \( ( -2 + 10 T + T^{2} )^{2} \)
$47$ \( 324 + 72 T^{2} + T^{4} \)
$53$ \( ( -108 + T^{2} )^{2} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( 144 + 168 T^{2} + T^{4} \)
$67$ \( ( -2 - 10 T + T^{2} )^{2} \)
$71$ \( ( -72 - 12 T + T^{2} )^{2} \)
$73$ \( 8464 + 248 T^{2} + T^{4} \)
$79$ \( ( -12 + T )^{4} \)
$83$ \( ( 6 + 6 T + T^{2} )^{2} \)
$89$ \( ( -12 - 12 T + T^{2} )^{2} \)
$97$ \( 8464 + 248 T^{2} + T^{4} \)
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