Properties

Label 1600.2.c.i
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
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.

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

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 1.00000i 0 0 0 2.00000i 0 2.00000 0
449.2 0 1.00000i 0 0 0 2.00000i 0 2.00000 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.i 2
4.b odd 2 1 1600.2.c.h 2
5.b even 2 1 inner 1600.2.c.i 2
5.c odd 4 1 1600.2.a.j 1
5.c odd 4 1 1600.2.a.q 1
8.b even 2 1 50.2.b.a 2
8.d odd 2 1 400.2.c.c 2
20.d odd 2 1 1600.2.c.h 2
20.e even 4 1 1600.2.a.i 1
20.e even 4 1 1600.2.a.p 1
24.f even 2 1 3600.2.f.f 2
24.h odd 2 1 450.2.c.c 2
40.e odd 2 1 400.2.c.c 2
40.f even 2 1 50.2.b.a 2
40.i odd 4 1 50.2.a.a 1
40.i odd 4 1 50.2.a.b yes 1
40.k even 4 1 400.2.a.d 1
40.k even 4 1 400.2.a.f 1
56.h odd 2 1 2450.2.c.m 2
120.i odd 2 1 450.2.c.c 2
120.m even 2 1 3600.2.f.f 2
120.q odd 4 1 3600.2.a.l 1
120.q odd 4 1 3600.2.a.bc 1
120.w even 4 1 450.2.a.c 1
120.w even 4 1 450.2.a.g 1
280.c odd 2 1 2450.2.c.m 2
280.s even 4 1 2450.2.a.g 1
280.s even 4 1 2450.2.a.bd 1
440.t even 4 1 6050.2.a.h 1
440.t even 4 1 6050.2.a.bi 1
520.bg odd 4 1 8450.2.a.d 1
520.bg odd 4 1 8450.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.a.a 1 40.i odd 4 1
50.2.a.b yes 1 40.i odd 4 1
50.2.b.a 2 8.b even 2 1
50.2.b.a 2 40.f even 2 1
400.2.a.d 1 40.k even 4 1
400.2.a.f 1 40.k even 4 1
400.2.c.c 2 8.d odd 2 1
400.2.c.c 2 40.e odd 2 1
450.2.a.c 1 120.w even 4 1
450.2.a.g 1 120.w even 4 1
450.2.c.c 2 24.h odd 2 1
450.2.c.c 2 120.i odd 2 1
1600.2.a.i 1 20.e even 4 1
1600.2.a.j 1 5.c odd 4 1
1600.2.a.p 1 20.e even 4 1
1600.2.a.q 1 5.c odd 4 1
1600.2.c.h 2 4.b odd 2 1
1600.2.c.h 2 20.d odd 2 1
1600.2.c.i 2 1.a even 1 1 trivial
1600.2.c.i 2 5.b even 2 1 inner
2450.2.a.g 1 280.s even 4 1
2450.2.a.bd 1 280.s even 4 1
2450.2.c.m 2 56.h odd 2 1
2450.2.c.m 2 280.c odd 2 1
3600.2.a.l 1 120.q odd 4 1
3600.2.a.bc 1 120.q odd 4 1
3600.2.f.f 2 24.f even 2 1
3600.2.f.f 2 120.m even 2 1
6050.2.a.h 1 440.t even 4 1
6050.2.a.bi 1 440.t even 4 1
8450.2.a.d 1 520.bg odd 4 1
8450.2.a.v 1 520.bg odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \)
\( T_{7}^{2} + 4 \)
\( T_{11} - 3 \)
\( T_{19} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( -5 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 3 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 144 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 169 + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 121 + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 81 + T^{2} \)
$89$ \( ( 15 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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