Properties

Label 1600.2.c.b
Level $1600$
Weight $2$
Character orbit 1600.c
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.c (of order \(2\), degree \(1\), 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 i q^{3} -2 i q^{7} -6 q^{9} +O(q^{10})\) \( q + 3 i q^{3} -2 i q^{7} -6 q^{9} + q^{11} + 4 i q^{13} + 5 i q^{17} - q^{19} + 6 q^{21} -2 i q^{23} -9 i q^{27} -8 q^{29} -10 q^{31} + 3 i q^{33} + 6 i q^{37} -12 q^{39} -3 q^{41} -4 i q^{43} -4 i q^{47} + 3 q^{49} -15 q^{51} + 6 i q^{53} -3 i q^{57} -8 q^{59} -10 q^{61} + 12 i q^{63} -i q^{67} + 6 q^{69} + 12 q^{71} -3 i q^{73} -2 i q^{77} + 6 q^{79} + 9 q^{81} + 13 i q^{83} -24 i q^{87} + 9 q^{89} + 8 q^{91} -30 i q^{93} -14 i q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 12q^{9} + O(q^{10}) \) \( 2q - 12q^{9} + 2q^{11} - 2q^{19} + 12q^{21} - 16q^{29} - 20q^{31} - 24q^{39} - 6q^{41} + 6q^{49} - 30q^{51} - 16q^{59} - 20q^{61} + 12q^{69} + 24q^{71} + 12q^{79} + 18q^{81} + 18q^{89} + 16q^{91} - 12q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 3.00000i 0 0 0 2.00000i 0 −6.00000 0
449.2 0 3.00000i 0 0 0 2.00000i 0 −6.00000 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
5.b 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.c.b 2
4.b odd 2 1 1600.2.c.a 2
5.b even 2 1 inner 1600.2.c.b 2
5.c odd 4 1 1600.2.a.b 1
5.c odd 4 1 1600.2.a.y 1
8.b even 2 1 400.2.c.a 2
8.d odd 2 1 200.2.c.a 2
20.d odd 2 1 1600.2.c.a 2
20.e even 4 1 1600.2.a.a 1
20.e even 4 1 1600.2.a.x 1
24.f even 2 1 1800.2.f.f 2
24.h odd 2 1 3600.2.f.n 2
40.e odd 2 1 200.2.c.a 2
40.f even 2 1 400.2.c.a 2
40.i odd 4 1 400.2.a.a 1
40.i odd 4 1 400.2.a.h 1
40.k even 4 1 200.2.a.a 1
40.k even 4 1 200.2.a.e yes 1
120.i odd 2 1 3600.2.f.n 2
120.m even 2 1 1800.2.f.f 2
120.q odd 4 1 1800.2.a.h 1
120.q odd 4 1 1800.2.a.r 1
120.w even 4 1 3600.2.a.m 1
120.w even 4 1 3600.2.a.bf 1
280.y odd 4 1 9800.2.a.c 1
280.y odd 4 1 9800.2.a.bq 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.a.a 1 40.k even 4 1
200.2.a.e yes 1 40.k even 4 1
200.2.c.a 2 8.d odd 2 1
200.2.c.a 2 40.e odd 2 1
400.2.a.a 1 40.i odd 4 1
400.2.a.h 1 40.i odd 4 1
400.2.c.a 2 8.b even 2 1
400.2.c.a 2 40.f even 2 1
1600.2.a.a 1 20.e even 4 1
1600.2.a.b 1 5.c odd 4 1
1600.2.a.x 1 20.e even 4 1
1600.2.a.y 1 5.c odd 4 1
1600.2.c.a 2 4.b odd 2 1
1600.2.c.a 2 20.d odd 2 1
1600.2.c.b 2 1.a even 1 1 trivial
1600.2.c.b 2 5.b even 2 1 inner
1800.2.a.h 1 120.q odd 4 1
1800.2.a.r 1 120.q odd 4 1
1800.2.f.f 2 24.f even 2 1
1800.2.f.f 2 120.m even 2 1
3600.2.a.m 1 120.w even 4 1
3600.2.a.bf 1 120.w even 4 1
3600.2.f.n 2 24.h odd 2 1
3600.2.f.n 2 120.i odd 2 1
9800.2.a.c 1 280.y odd 4 1
9800.2.a.bq 1 280.y odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 9 \)
\( T_{7}^{2} + 4 \)
\( T_{11} - 1 \)
\( T_{19} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( 25 + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 4 + T^{2} \)
$29$ \( ( 8 + T )^{2} \)
$31$ \( ( 10 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( 3 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( 1 + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 9 + T^{2} \)
$79$ \( ( -6 + T )^{2} \)
$83$ \( 169 + T^{2} \)
$89$ \( ( -9 + T )^{2} \)
$97$ \( 196 + T^{2} \)
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