Properties

Label 3200.2.c.e
Level $3200$
Weight $2$
Character orbit 3200.c
Analytic conductor $25.552$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.5521286468\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 128)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} + 4 i q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} + 4 i q^{7} - q^{9} -2 q^{11} + 2 i q^{13} + 2 i q^{17} -2 q^{19} -8 q^{21} + 4 i q^{23} + 4 i q^{27} + 6 q^{29} -4 i q^{33} -10 i q^{37} -4 q^{39} -6 q^{41} + 6 i q^{43} + 8 i q^{47} -9 q^{49} -4 q^{51} -6 i q^{53} -4 i q^{57} -14 q^{59} + 2 q^{61} -4 i q^{63} -10 i q^{67} -8 q^{69} + 12 q^{71} + 14 i q^{73} -8 i q^{77} + 8 q^{79} -11 q^{81} -6 i q^{83} + 12 i q^{87} + 2 q^{89} -8 q^{91} + 2 i q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 4q^{11} - 4q^{19} - 16q^{21} + 12q^{29} - 8q^{39} - 12q^{41} - 18q^{49} - 8q^{51} - 28q^{59} + 4q^{61} - 16q^{69} + 24q^{71} + 16q^{79} - 22q^{81} + 4q^{89} - 16q^{91} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2049.1
1.00000i
1.00000i
0 2.00000i 0 0 0 4.00000i 0 −1.00000 0
2049.2 0 2.00000i 0 0 0 4.00000i 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
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 3200.2.c.e 2
4.b odd 2 1 3200.2.c.k 2
5.b even 2 1 inner 3200.2.c.e 2
5.c odd 4 1 128.2.a.d yes 1
5.c odd 4 1 3200.2.a.h 1
8.b even 2 1 3200.2.c.l 2
8.d odd 2 1 3200.2.c.f 2
15.e even 4 1 1152.2.a.c 1
20.d odd 2 1 3200.2.c.k 2
20.e even 4 1 128.2.a.b yes 1
20.e even 4 1 3200.2.a.u 1
35.f even 4 1 6272.2.a.a 1
40.e odd 2 1 3200.2.c.f 2
40.f even 2 1 3200.2.c.l 2
40.i odd 4 1 128.2.a.a 1
40.i odd 4 1 3200.2.a.x 1
40.k even 4 1 128.2.a.c yes 1
40.k even 4 1 3200.2.a.e 1
60.l odd 4 1 1152.2.a.h 1
80.i odd 4 1 256.2.b.c 2
80.j even 4 1 256.2.b.a 2
80.s even 4 1 256.2.b.a 2
80.t odd 4 1 256.2.b.c 2
120.q odd 4 1 1152.2.a.r 1
120.w even 4 1 1152.2.a.m 1
140.j odd 4 1 6272.2.a.g 1
160.u even 8 2 1024.2.e.m 4
160.v odd 8 2 1024.2.e.i 4
160.ba even 8 2 1024.2.e.m 4
160.bb odd 8 2 1024.2.e.i 4
240.z odd 4 1 2304.2.d.b 2
240.bb even 4 1 2304.2.d.r 2
240.bd odd 4 1 2304.2.d.b 2
240.bf even 4 1 2304.2.d.r 2
280.s even 4 1 6272.2.a.h 1
280.y odd 4 1 6272.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 40.i odd 4 1
128.2.a.b yes 1 20.e even 4 1
128.2.a.c yes 1 40.k even 4 1
128.2.a.d yes 1 5.c odd 4 1
256.2.b.a 2 80.j even 4 1
256.2.b.a 2 80.s even 4 1
256.2.b.c 2 80.i odd 4 1
256.2.b.c 2 80.t odd 4 1
1024.2.e.i 4 160.v odd 8 2
1024.2.e.i 4 160.bb odd 8 2
1024.2.e.m 4 160.u even 8 2
1024.2.e.m 4 160.ba even 8 2
1152.2.a.c 1 15.e even 4 1
1152.2.a.h 1 60.l odd 4 1
1152.2.a.m 1 120.w even 4 1
1152.2.a.r 1 120.q odd 4 1
2304.2.d.b 2 240.z odd 4 1
2304.2.d.b 2 240.bd odd 4 1
2304.2.d.r 2 240.bb even 4 1
2304.2.d.r 2 240.bf even 4 1
3200.2.a.e 1 40.k even 4 1
3200.2.a.h 1 5.c odd 4 1
3200.2.a.u 1 20.e even 4 1
3200.2.a.x 1 40.i odd 4 1
3200.2.c.e 2 1.a even 1 1 trivial
3200.2.c.e 2 5.b even 2 1 inner
3200.2.c.f 2 8.d odd 2 1
3200.2.c.f 2 40.e odd 2 1
3200.2.c.k 2 4.b odd 2 1
3200.2.c.k 2 20.d odd 2 1
3200.2.c.l 2 8.b even 2 1
3200.2.c.l 2 40.f even 2 1
6272.2.a.a 1 35.f even 4 1
6272.2.a.b 1 280.y odd 4 1
6272.2.a.g 1 140.j odd 4 1
6272.2.a.h 1 280.s even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{7}^{2} + 16 \)
\( T_{11} + 2 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T^{2} + 9 T^{4} \)
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 2 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 30 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 50 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 14 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 34 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 50 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 2 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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