Properties

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

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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: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{3} + ( -2 + 4 \zeta_{12}^{2} ) q^{7} -2 q^{9} +O(q^{10})\) \( q + q^{3} + ( -2 + 4 \zeta_{12}^{2} ) q^{7} -2 q^{9} + 3 \zeta_{12}^{3} q^{11} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{13} -3 \zeta_{12}^{3} q^{17} -\zeta_{12}^{3} q^{19} + ( -2 + 4 \zeta_{12}^{2} ) q^{21} -5 q^{27} + ( -6 + 12 \zeta_{12}^{2} ) q^{29} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{31} + 3 \zeta_{12}^{3} q^{33} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{37} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{39} -9 q^{41} -4 q^{43} + ( -6 + 12 \zeta_{12}^{2} ) q^{47} -5 q^{49} -3 \zeta_{12}^{3} q^{51} -\zeta_{12}^{3} q^{57} + 12 \zeta_{12}^{3} q^{59} + ( -2 + 4 \zeta_{12}^{2} ) q^{61} + ( 4 - 8 \zeta_{12}^{2} ) q^{63} -11 q^{67} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{71} + 7 \zeta_{12}^{3} q^{73} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{77} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{79} + q^{81} + 15 q^{83} + ( -6 + 12 \zeta_{12}^{2} ) q^{87} + 3 q^{89} + 12 \zeta_{12}^{3} q^{91} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{93} -14 \zeta_{12}^{3} q^{97} -6 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 8q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 8q^{9} - 20q^{27} - 36q^{41} - 16q^{43} - 20q^{49} - 44q^{67} + 4q^{81} + 60q^{83} + 12q^{89} + O(q^{100}) \)

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 1.00000 0 0 0 3.46410i 0 −2.00000 0
1249.2 0 1.00000 0 0 0 3.46410i 0 −2.00000 0
1249.3 0 1.00000 0 0 0 3.46410i 0 −2.00000 0
1249.4 0 1.00000 0 0 0 3.46410i 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
8.d odd 2 1 inner
20.d odd 2 1 inner
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.g 4
4.b odd 2 1 1600.2.f.c 4
5.b even 2 1 1600.2.f.c 4
5.c odd 4 1 1600.2.d.e 4
5.c odd 4 1 1600.2.d.f yes 4
8.b even 2 1 1600.2.f.c 4
8.d odd 2 1 inner 1600.2.f.g 4
20.d odd 2 1 inner 1600.2.f.g 4
20.e even 4 1 1600.2.d.e 4
20.e even 4 1 1600.2.d.f yes 4
40.e odd 2 1 1600.2.f.c 4
40.f even 2 1 inner 1600.2.f.g 4
40.i odd 4 1 1600.2.d.e 4
40.i odd 4 1 1600.2.d.f yes 4
40.k even 4 1 1600.2.d.e 4
40.k even 4 1 1600.2.d.f yes 4
80.i odd 4 1 6400.2.a.ba 2
80.i odd 4 1 6400.2.a.bb 2
80.j even 4 1 6400.2.a.ba 2
80.j even 4 1 6400.2.a.bb 2
80.s even 4 1 6400.2.a.cf 2
80.s even 4 1 6400.2.a.cg 2
80.t odd 4 1 6400.2.a.cf 2
80.t odd 4 1 6400.2.a.cg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.d.e 4 5.c odd 4 1
1600.2.d.e 4 20.e even 4 1
1600.2.d.e 4 40.i odd 4 1
1600.2.d.e 4 40.k even 4 1
1600.2.d.f yes 4 5.c odd 4 1
1600.2.d.f yes 4 20.e even 4 1
1600.2.d.f yes 4 40.i odd 4 1
1600.2.d.f yes 4 40.k even 4 1
1600.2.f.c 4 4.b odd 2 1
1600.2.f.c 4 5.b even 2 1
1600.2.f.c 4 8.b even 2 1
1600.2.f.c 4 40.e odd 2 1
1600.2.f.g 4 1.a even 1 1 trivial
1600.2.f.g 4 8.d odd 2 1 inner
1600.2.f.g 4 20.d odd 2 1 inner
1600.2.f.g 4 40.f even 2 1 inner
6400.2.a.ba 2 80.i odd 4 1
6400.2.a.ba 2 80.j even 4 1
6400.2.a.bb 2 80.i odd 4 1
6400.2.a.bb 2 80.j even 4 1
6400.2.a.cf 2 80.s even 4 1
6400.2.a.cf 2 80.t odd 4 1
6400.2.a.cg 2 80.s even 4 1
6400.2.a.cg 2 80.t odd 4 1

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{31}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 12 + T^{2} )^{2} \)
$11$ \( ( 9 + T^{2} )^{2} \)
$13$ \( ( -12 + T^{2} )^{2} \)
$17$ \( ( 9 + T^{2} )^{2} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( ( 108 + T^{2} )^{2} \)
$31$ \( ( -48 + T^{2} )^{2} \)
$37$ \( ( -108 + T^{2} )^{2} \)
$41$ \( ( 9 + T )^{4} \)
$43$ \( ( 4 + T )^{4} \)
$47$ \( ( 108 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( ( 144 + T^{2} )^{2} \)
$61$ \( ( 12 + T^{2} )^{2} \)
$67$ \( ( 11 + T )^{4} \)
$71$ \( ( -108 + T^{2} )^{2} \)
$73$ \( ( 49 + T^{2} )^{2} \)
$79$ \( ( -108 + T^{2} )^{2} \)
$83$ \( ( -15 + T )^{4} \)
$89$ \( ( -3 + T )^{4} \)
$97$ \( ( 196 + T^{2} )^{2} \)
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