Properties

Label 1600.2.s.a
Level $1600$
Weight $2$
Character orbit 1600.s
Analytic conductor $12.776$
Analytic rank $1$
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.s (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q -2 q^{3} + ( -3 - 3 i ) q^{7} + q^{9} +O(q^{10})\) \( q -2 q^{3} + ( -3 - 3 i ) q^{7} + q^{9} + ( 1 - i ) q^{11} -2 i q^{13} + ( -1 - i ) q^{17} + ( 3 - 3 i ) q^{19} + ( 6 + 6 i ) q^{21} + ( -1 + i ) q^{23} + 4 q^{27} + ( -7 - 7 i ) q^{29} + 2 i q^{31} + ( -2 + 2 i ) q^{33} + 6 i q^{37} + 4 i q^{39} + 4 i q^{41} -4 i q^{43} + ( -7 + 7 i ) q^{47} + 11 i q^{49} + ( 2 + 2 i ) q^{51} + 8 q^{53} + ( -6 + 6 i ) q^{57} + ( -3 - 3 i ) q^{59} + ( -1 + i ) q^{61} + ( -3 - 3 i ) q^{63} + 4 i q^{67} + ( 2 - 2 i ) q^{69} + ( -3 - 3 i ) q^{73} -6 q^{77} -8 q^{79} -11 q^{81} -2 q^{83} + ( 14 + 14 i ) q^{87} -6 q^{89} + ( -6 + 6 i ) q^{91} -4 i q^{93} + ( 11 + 11 i ) q^{97} + ( 1 - i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} - 6q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} - 6q^{7} + 2q^{9} + 2q^{11} - 2q^{17} + 6q^{19} + 12q^{21} - 2q^{23} + 8q^{27} - 14q^{29} - 4q^{33} - 14q^{47} + 4q^{51} + 16q^{53} - 12q^{57} - 6q^{59} - 2q^{61} - 6q^{63} + 4q^{69} - 6q^{73} - 12q^{77} - 16q^{79} - 22q^{81} - 4q^{83} + 28q^{87} - 12q^{89} - 12q^{91} + 22q^{97} + 2q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
207.1
1.00000i
1.00000i
0 −2.00000 0 0 0 −3.00000 3.00000i 0 1.00000 0
943.1 0 −2.00000 0 0 0 −3.00000 + 3.00000i 0 1.00000 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
80.s 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.s.a 2
4.b odd 2 1 400.2.s.a 2
5.b even 2 1 320.2.s.a 2
5.c odd 4 1 320.2.j.a 2
5.c odd 4 1 1600.2.j.a 2
16.e even 4 1 400.2.j.a 2
16.f odd 4 1 1600.2.j.a 2
20.d odd 2 1 80.2.s.a yes 2
20.e even 4 1 80.2.j.a 2
20.e even 4 1 400.2.j.a 2
40.e odd 2 1 640.2.s.b 2
40.f even 2 1 640.2.s.a 2
40.i odd 4 1 640.2.j.b 2
40.k even 4 1 640.2.j.a 2
60.h even 2 1 720.2.z.d 2
60.l odd 4 1 720.2.bd.a 2
80.i odd 4 1 400.2.s.a 2
80.i odd 4 1 640.2.s.b 2
80.j even 4 1 320.2.s.a 2
80.k odd 4 1 320.2.j.a 2
80.k odd 4 1 640.2.j.b 2
80.q even 4 1 80.2.j.a 2
80.q even 4 1 640.2.j.a 2
80.s even 4 1 640.2.s.a 2
80.s even 4 1 inner 1600.2.s.a 2
80.t odd 4 1 80.2.s.a yes 2
240.bf even 4 1 720.2.z.d 2
240.bm odd 4 1 720.2.bd.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.a 2 20.e even 4 1
80.2.j.a 2 80.q even 4 1
80.2.s.a yes 2 20.d odd 2 1
80.2.s.a yes 2 80.t odd 4 1
320.2.j.a 2 5.c odd 4 1
320.2.j.a 2 80.k odd 4 1
320.2.s.a 2 5.b even 2 1
320.2.s.a 2 80.j even 4 1
400.2.j.a 2 16.e even 4 1
400.2.j.a 2 20.e even 4 1
400.2.s.a 2 4.b odd 2 1
400.2.s.a 2 80.i odd 4 1
640.2.j.a 2 40.k even 4 1
640.2.j.a 2 80.q even 4 1
640.2.j.b 2 40.i odd 4 1
640.2.j.b 2 80.k odd 4 1
640.2.s.a 2 40.f even 2 1
640.2.s.a 2 80.s even 4 1
640.2.s.b 2 40.e odd 2 1
640.2.s.b 2 80.i odd 4 1
720.2.z.d 2 60.h even 2 1
720.2.z.d 2 240.bf even 4 1
720.2.bd.a 2 60.l odd 4 1
720.2.bd.a 2 240.bm odd 4 1
1600.2.j.a 2 5.c odd 4 1
1600.2.j.a 2 16.f odd 4 1
1600.2.s.a 2 1.a even 1 1 trivial
1600.2.s.a 2 80.s even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 2 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( 18 + 6 T + T^{2} \)
$11$ \( 2 - 2 T + T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 2 + 2 T + T^{2} \)
$19$ \( 18 - 6 T + T^{2} \)
$23$ \( 2 + 2 T + T^{2} \)
$29$ \( 98 + 14 T + T^{2} \)
$31$ \( 4 + T^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( 16 + T^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 98 + 14 T + T^{2} \)
$53$ \( ( -8 + T )^{2} \)
$59$ \( 18 + 6 T + T^{2} \)
$61$ \( 2 + 2 T + T^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 18 + 6 T + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( ( 2 + T )^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 242 - 22 T + T^{2} \)
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