Properties

Label 1600.2.n.r
Level $1600$
Weight $2$
Character orbit 1600.n
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.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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + 3 \zeta_{12}^{3} q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + 3 \zeta_{12}^{3} q^{9} + ( 2 - 4 \zeta_{12}^{2} ) q^{11} + ( 1 - \zeta_{12}^{3} ) q^{13} + ( -1 - \zeta_{12}^{3} ) q^{17} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{19} + 6 q^{21} + ( -1 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{23} + 4 \zeta_{12}^{3} q^{29} + ( 2 - 4 \zeta_{12}^{2} ) q^{31} + ( -6 + 6 \zeta_{12}^{3} ) q^{33} + ( 5 + 5 \zeta_{12}^{3} ) q^{37} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{39} + 2 q^{41} + ( 1 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{43} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{47} + \zeta_{12}^{3} q^{49} + ( -2 + 4 \zeta_{12}^{2} ) q^{51} + ( -7 + 7 \zeta_{12}^{3} ) q^{53} + ( 12 + 12 \zeta_{12}^{3} ) q^{57} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{59} -6 q^{61} + ( 3 - 6 \zeta_{12} - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{63} + ( -3 - 6 \zeta_{12} + 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{67} -6 \zeta_{12}^{3} q^{69} + ( -6 + 12 \zeta_{12}^{2} ) q^{71} + ( 7 - 7 \zeta_{12}^{3} ) q^{73} + ( 6 + 6 \zeta_{12}^{3} ) q^{77} + 9 q^{81} + ( -7 + 14 \zeta_{12} + 14 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{83} + ( 4 + 8 \zeta_{12} - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{87} -8 \zeta_{12}^{3} q^{89} + ( -2 + 4 \zeta_{12}^{2} ) q^{91} + ( -6 + 6 \zeta_{12}^{3} ) q^{93} + ( 7 + 7 \zeta_{12}^{3} ) q^{97} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{13} - 4q^{17} + 24q^{21} - 24q^{33} + 20q^{37} + 8q^{41} - 28q^{53} + 48q^{57} - 24q^{61} + 28q^{73} + 24q^{77} + 36q^{81} - 24q^{93} + 28q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(\zeta_{12}\) \(1\) \(-1\)

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
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 −1.73205 + 1.73205i 0 0 0 −1.73205 1.73205i 0 3.00000i 0
1343.2 0 1.73205 1.73205i 0 0 0 1.73205 + 1.73205i 0 3.00000i 0
1407.1 0 −1.73205 1.73205i 0 0 0 −1.73205 + 1.73205i 0 3.00000i 0
1407.2 0 1.73205 + 1.73205i 0 0 0 1.73205 1.73205i 0 3.00000i 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
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.n.r 4
4.b odd 2 1 inner 1600.2.n.r 4
5.b even 2 1 320.2.n.i 4
5.c odd 4 1 320.2.n.i 4
5.c odd 4 1 inner 1600.2.n.r 4
8.b even 2 1 400.2.n.b 4
8.d odd 2 1 400.2.n.b 4
20.d odd 2 1 320.2.n.i 4
20.e even 4 1 320.2.n.i 4
20.e even 4 1 inner 1600.2.n.r 4
24.f even 2 1 3600.2.x.e 4
24.h odd 2 1 3600.2.x.e 4
40.e odd 2 1 80.2.n.b 4
40.f even 2 1 80.2.n.b 4
40.i odd 4 1 80.2.n.b 4
40.i odd 4 1 400.2.n.b 4
40.k even 4 1 80.2.n.b 4
40.k even 4 1 400.2.n.b 4
80.i odd 4 1 1280.2.o.r 4
80.j even 4 1 1280.2.o.q 4
80.k odd 4 1 1280.2.o.q 4
80.k odd 4 1 1280.2.o.r 4
80.q even 4 1 1280.2.o.q 4
80.q even 4 1 1280.2.o.r 4
80.s even 4 1 1280.2.o.r 4
80.t odd 4 1 1280.2.o.q 4
120.i odd 2 1 720.2.x.d 4
120.m even 2 1 720.2.x.d 4
120.q odd 4 1 720.2.x.d 4
120.q odd 4 1 3600.2.x.e 4
120.w even 4 1 720.2.x.d 4
120.w even 4 1 3600.2.x.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.n.b 4 40.e odd 2 1
80.2.n.b 4 40.f even 2 1
80.2.n.b 4 40.i odd 4 1
80.2.n.b 4 40.k even 4 1
320.2.n.i 4 5.b even 2 1
320.2.n.i 4 5.c odd 4 1
320.2.n.i 4 20.d odd 2 1
320.2.n.i 4 20.e even 4 1
400.2.n.b 4 8.b even 2 1
400.2.n.b 4 8.d odd 2 1
400.2.n.b 4 40.i odd 4 1
400.2.n.b 4 40.k even 4 1
720.2.x.d 4 120.i odd 2 1
720.2.x.d 4 120.m even 2 1
720.2.x.d 4 120.q odd 4 1
720.2.x.d 4 120.w even 4 1
1280.2.o.q 4 80.j even 4 1
1280.2.o.q 4 80.k odd 4 1
1280.2.o.q 4 80.q even 4 1
1280.2.o.q 4 80.t odd 4 1
1280.2.o.r 4 80.i odd 4 1
1280.2.o.r 4 80.k odd 4 1
1280.2.o.r 4 80.q even 4 1
1280.2.o.r 4 80.s even 4 1
1600.2.n.r 4 1.a even 1 1 trivial
1600.2.n.r 4 4.b odd 2 1 inner
1600.2.n.r 4 5.c odd 4 1 inner
1600.2.n.r 4 20.e even 4 1 inner
3600.2.x.e 4 24.f even 2 1
3600.2.x.e 4 24.h odd 2 1
3600.2.x.e 4 120.q odd 4 1
3600.2.x.e 4 120.w even 4 1

Hecke kernels

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

\( T_{3}^{4} + 36 \)
\( T_{7}^{4} + 36 \)
\( T_{11}^{2} + 12 \)
\( T_{13}^{2} - 2 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 36 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 36 + T^{4} \)
$11$ \( ( 12 + T^{2} )^{2} \)
$13$ \( ( 2 - 2 T + T^{2} )^{2} \)
$17$ \( ( 2 + 2 T + T^{2} )^{2} \)
$19$ \( ( -48 + T^{2} )^{2} \)
$23$ \( 36 + T^{4} \)
$29$ \( ( 16 + T^{2} )^{2} \)
$31$ \( ( 12 + T^{2} )^{2} \)
$37$ \( ( 50 - 10 T + T^{2} )^{2} \)
$41$ \( ( -2 + T )^{4} \)
$43$ \( 36 + T^{4} \)
$47$ \( 36 + T^{4} \)
$53$ \( ( 98 + 14 T + T^{2} )^{2} \)
$59$ \( ( -48 + T^{2} )^{2} \)
$61$ \( ( 6 + T )^{4} \)
$67$ \( 2916 + T^{4} \)
$71$ \( ( 108 + T^{2} )^{2} \)
$73$ \( ( 98 - 14 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( 86436 + T^{4} \)
$89$ \( ( 64 + T^{2} )^{2} \)
$97$ \( ( 98 - 14 T + T^{2} )^{2} \)
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