Properties

Label 1600.2.l.c
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 80)
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.

\(f(q)\) \(=\) \( q + ( 1 - i ) q^{3} + i q^{9} +O(q^{10})\) \( q + ( 1 - i ) q^{3} + i q^{9} + ( 3 + 3 i ) q^{11} + ( -3 + 3 i ) q^{13} -4 q^{17} + ( -1 + i ) q^{19} + 8 i q^{23} + ( 4 + 4 i ) q^{27} + ( -3 + 3 i ) q^{29} + 6 q^{33} + ( -3 - 3 i ) q^{37} + 6 i q^{39} + ( -3 - 3 i ) q^{43} + 2 q^{47} + 7 q^{49} + ( -4 + 4 i ) q^{51} + ( -9 - 9 i ) q^{53} + 2 i q^{57} + ( 9 + 9 i ) q^{59} + ( -5 + 5 i ) q^{61} + ( 3 - 3 i ) q^{67} + ( 8 + 8 i ) q^{69} -6 i q^{71} + 6 i q^{73} + 8 q^{79} + 5 q^{81} + ( 9 - 9 i ) q^{83} + 6 i q^{87} + 12 i q^{89} + 12 q^{97} + ( -3 + 3 i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + O(q^{10}) \) \( 2q + 2q^{3} + 6q^{11} - 6q^{13} - 8q^{17} - 2q^{19} + 8q^{27} - 6q^{29} + 12q^{33} - 6q^{37} - 6q^{43} + 4q^{47} + 14q^{49} - 8q^{51} - 18q^{53} + 18q^{59} - 10q^{61} + 6q^{67} + 16q^{69} + 16q^{79} + 10q^{81} + 18q^{83} + 24q^{97} - 6q^{99} + O(q^{100}) \)

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 0 0 1.00000i 0
1201.1 0 1.00000 1.00000i 0 0 0 0 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.c 2
4.b odd 2 1 400.2.l.a 2
5.b even 2 1 1600.2.l.b 2
5.c odd 4 1 320.2.q.a 2
5.c odd 4 1 320.2.q.b 2
16.e even 4 1 inner 1600.2.l.c 2
16.f odd 4 1 400.2.l.a 2
20.d odd 2 1 400.2.l.b 2
20.e even 4 1 80.2.q.a 2
20.e even 4 1 80.2.q.b yes 2
40.i odd 4 1 640.2.q.a 2
40.i odd 4 1 640.2.q.d 2
40.k even 4 1 640.2.q.b 2
40.k even 4 1 640.2.q.c 2
60.l odd 4 1 720.2.bm.a 2
60.l odd 4 1 720.2.bm.b 2
80.i odd 4 1 320.2.q.a 2
80.i odd 4 1 640.2.q.a 2
80.j even 4 1 80.2.q.b yes 2
80.j even 4 1 640.2.q.b 2
80.k odd 4 1 400.2.l.b 2
80.q even 4 1 1600.2.l.b 2
80.s even 4 1 80.2.q.a 2
80.s even 4 1 640.2.q.c 2
80.t odd 4 1 320.2.q.b 2
80.t odd 4 1 640.2.q.d 2
240.z odd 4 1 720.2.bm.b 2
240.bd odd 4 1 720.2.bm.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.q.a 2 20.e even 4 1
80.2.q.a 2 80.s even 4 1
80.2.q.b yes 2 20.e even 4 1
80.2.q.b yes 2 80.j even 4 1
320.2.q.a 2 5.c odd 4 1
320.2.q.a 2 80.i odd 4 1
320.2.q.b 2 5.c odd 4 1
320.2.q.b 2 80.t odd 4 1
400.2.l.a 2 4.b odd 2 1
400.2.l.a 2 16.f odd 4 1
400.2.l.b 2 20.d odd 2 1
400.2.l.b 2 80.k odd 4 1
640.2.q.a 2 40.i odd 4 1
640.2.q.a 2 80.i odd 4 1
640.2.q.b 2 40.k even 4 1
640.2.q.b 2 80.j even 4 1
640.2.q.c 2 40.k even 4 1
640.2.q.c 2 80.s even 4 1
640.2.q.d 2 40.i odd 4 1
640.2.q.d 2 80.t odd 4 1
720.2.bm.a 2 60.l odd 4 1
720.2.bm.a 2 240.bd odd 4 1
720.2.bm.b 2 60.l odd 4 1
720.2.bm.b 2 240.z odd 4 1
1600.2.l.b 2 5.b even 2 1
1600.2.l.b 2 80.q even 4 1
1600.2.l.c 2 1.a even 1 1 trivial
1600.2.l.c 2 16.e even 4 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2 - 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 18 - 6 T + T^{2} \)
$13$ \( 18 + 6 T + T^{2} \)
$17$ \( ( 4 + T )^{2} \)
$19$ \( 2 + 2 T + T^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( 18 + 6 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 18 + 6 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 18 + 6 T + T^{2} \)
$47$ \( ( -2 + T )^{2} \)
$53$ \( 162 + 18 T + T^{2} \)
$59$ \( 162 - 18 T + T^{2} \)
$61$ \( 50 + 10 T + T^{2} \)
$67$ \( 18 - 6 T + T^{2} \)
$71$ \( 36 + T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 162 - 18 T + T^{2} \)
$89$ \( 144 + T^{2} \)
$97$ \( ( -12 + T )^{2} \)
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