Properties

Label 1600.2.l.a
Level $1600$
Weight $2$
Character orbit 1600.l
Analytic conductor $12.776$
Analytic rank $0$
Dimension $2$
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.l (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + i ) q^{3} -2 i q^{7} + i q^{9} +O(q^{10})\) \( q + ( -1 + i ) q^{3} -2 i q^{7} + i q^{9} + ( -1 - i ) q^{11} + ( 1 - i ) q^{13} + 2 q^{17} + ( -3 + 3 i ) q^{19} + ( 2 + 2 i ) q^{21} + 6 i q^{23} + ( -4 - 4 i ) q^{27} + ( 3 - 3 i ) q^{29} + 8 q^{31} + 2 q^{33} + ( -3 - 3 i ) q^{37} + 2 i q^{39} + ( 5 + 5 i ) q^{43} + 8 q^{47} + 3 q^{49} + ( -2 + 2 i ) q^{51} + ( 5 + 5 i ) q^{53} -6 i q^{57} + ( 3 + 3 i ) q^{59} + ( -9 + 9 i ) q^{61} + 2 q^{63} + ( -5 + 5 i ) q^{67} + ( -6 - 6 i ) q^{69} + 10 i q^{71} + 4 i q^{73} + ( -2 + 2 i ) q^{77} + 5 q^{81} + ( -1 + i ) q^{83} + 6 i q^{87} + 4 i q^{89} + ( -2 - 2 i ) q^{91} + ( -8 + 8 i ) q^{93} + 2 q^{97} + ( 1 - i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{11} + 2q^{13} + 4q^{17} - 6q^{19} + 4q^{21} - 8q^{27} + 6q^{29} + 16q^{31} + 4q^{33} - 6q^{37} + 10q^{43} + 16q^{47} + 6q^{49} - 4q^{51} + 10q^{53} + 6q^{59} - 18q^{61} + 4q^{63} - 10q^{67} - 12q^{69} - 4q^{77} + 10q^{81} - 2q^{83} - 4q^{91} - 16q^{93} + 4q^{97} + 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(i\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.00000i
1.00000i
0 −1.00000 1.00000i 0 0 0 2.00000i 0 1.00000i 0
1201.1 0 −1.00000 + 1.00000i 0 0 0 2.00000i 0 1.00000i 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
16.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.l.a 2
4.b odd 2 1 400.2.l.c 2
5.b even 2 1 64.2.e.a 2
5.c odd 4 1 1600.2.q.a 2
5.c odd 4 1 1600.2.q.b 2
15.d odd 2 1 576.2.k.a 2
16.e even 4 1 inner 1600.2.l.a 2
16.f odd 4 1 400.2.l.c 2
20.d odd 2 1 16.2.e.a 2
20.e even 4 1 400.2.q.a 2
20.e even 4 1 400.2.q.b 2
40.e odd 2 1 128.2.e.b 2
40.f even 2 1 128.2.e.a 2
60.h even 2 1 144.2.k.a 2
80.i odd 4 1 1600.2.q.b 2
80.j even 4 1 400.2.q.b 2
80.k odd 4 1 16.2.e.a 2
80.k odd 4 1 128.2.e.b 2
80.q even 4 1 64.2.e.a 2
80.q even 4 1 128.2.e.a 2
80.s even 4 1 400.2.q.a 2
80.t odd 4 1 1600.2.q.a 2
120.i odd 2 1 1152.2.k.a 2
120.m even 2 1 1152.2.k.b 2
140.c even 2 1 784.2.m.b 2
140.p odd 6 2 784.2.x.f 4
140.s even 6 2 784.2.x.c 4
160.y odd 8 2 1024.2.a.b 2
160.y odd 8 2 1024.2.b.e 2
160.z even 8 2 1024.2.a.e 2
160.z even 8 2 1024.2.b.b 2
240.t even 4 1 144.2.k.a 2
240.t even 4 1 1152.2.k.b 2
240.bm odd 4 1 576.2.k.a 2
240.bm odd 4 1 1152.2.k.a 2
480.bs even 8 2 9216.2.a.d 2
480.bu odd 8 2 9216.2.a.s 2
560.be even 4 1 784.2.m.b 2
560.co even 12 2 784.2.x.c 4
560.cs odd 12 2 784.2.x.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 20.d odd 2 1
16.2.e.a 2 80.k odd 4 1
64.2.e.a 2 5.b even 2 1
64.2.e.a 2 80.q even 4 1
128.2.e.a 2 40.f even 2 1
128.2.e.a 2 80.q even 4 1
128.2.e.b 2 40.e odd 2 1
128.2.e.b 2 80.k odd 4 1
144.2.k.a 2 60.h even 2 1
144.2.k.a 2 240.t even 4 1
400.2.l.c 2 4.b odd 2 1
400.2.l.c 2 16.f odd 4 1
400.2.q.a 2 20.e even 4 1
400.2.q.a 2 80.s even 4 1
400.2.q.b 2 20.e even 4 1
400.2.q.b 2 80.j even 4 1
576.2.k.a 2 15.d odd 2 1
576.2.k.a 2 240.bm odd 4 1
784.2.m.b 2 140.c even 2 1
784.2.m.b 2 560.be even 4 1
784.2.x.c 4 140.s even 6 2
784.2.x.c 4 560.co even 12 2
784.2.x.f 4 140.p odd 6 2
784.2.x.f 4 560.cs odd 12 2
1024.2.a.b 2 160.y odd 8 2
1024.2.a.e 2 160.z even 8 2
1024.2.b.b 2 160.z even 8 2
1024.2.b.e 2 160.y odd 8 2
1152.2.k.a 2 120.i odd 2 1
1152.2.k.a 2 240.bm odd 4 1
1152.2.k.b 2 120.m even 2 1
1152.2.k.b 2 240.t even 4 1
1600.2.l.a 2 1.a even 1 1 trivial
1600.2.l.a 2 16.e even 4 1 inner
1600.2.q.a 2 5.c odd 4 1
1600.2.q.a 2 80.t odd 4 1
1600.2.q.b 2 5.c odd 4 1
1600.2.q.b 2 80.i odd 4 1
9216.2.a.d 2 480.bs even 8 2
9216.2.a.s 2 480.bu odd 8 2

Hecke kernels

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

\( T_{3}^{2} + 2 T_{3} + 2 \)
\( T_{7}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2 + 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( 2 + 2 T + T^{2} \)
$13$ \( 2 - 2 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 18 + 6 T + T^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( 18 - 6 T + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 18 + 6 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 50 - 10 T + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( 50 - 10 T + T^{2} \)
$59$ \( 18 - 6 T + T^{2} \)
$61$ \( 162 + 18 T + T^{2} \)
$67$ \( 50 + 10 T + T^{2} \)
$71$ \( 100 + T^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 2 + 2 T + T^{2} \)
$89$ \( 16 + T^{2} \)
$97$ \( ( -2 + T )^{2} \)
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