Properties

Label 3200.2.c.k
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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 \beta q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 2 \beta q^{7} - q^{9} + 2 q^{11} - \beta q^{13} - \beta q^{17} + 2 q^{19} - 8 q^{21} + 2 \beta q^{23} + 2 \beta q^{27} + 6 q^{29} + 2 \beta q^{33} + 5 \beta q^{37} + 4 q^{39} - 6 q^{41} + 3 \beta q^{43} + 4 \beta q^{47} - 9 q^{49} + 4 q^{51} + 3 \beta q^{53} + 2 \beta q^{57} + 14 q^{59} + 2 q^{61} - 2 \beta q^{63} - 5 \beta q^{67} - 8 q^{69} - 12 q^{71} - 7 \beta q^{73} + 4 \beta q^{77} - 8 q^{79} - 11 q^{81} - 3 \beta q^{83} + 6 \beta q^{87} + 2 q^{89} + 8 q^{91} - \beta q^{97} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 4 q^{11} + 4 q^{19} - 16 q^{21} + 12 q^{29} + 8 q^{39} - 12 q^{41} - 18 q^{49} + 8 q^{51} + 28 q^{59} + 4 q^{61} - 16 q^{69} - 24 q^{71} - 16 q^{79} - 22 q^{81} + 4 q^{89} + 16 q^{91} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.k 2
4.b odd 2 1 3200.2.c.e 2
5.b even 2 1 inner 3200.2.c.k 2
5.c odd 4 1 128.2.a.b yes 1
5.c odd 4 1 3200.2.a.u 1
8.b even 2 1 3200.2.c.f 2
8.d odd 2 1 3200.2.c.l 2
15.e even 4 1 1152.2.a.h 1
20.d odd 2 1 3200.2.c.e 2
20.e even 4 1 128.2.a.d yes 1
20.e even 4 1 3200.2.a.h 1
35.f even 4 1 6272.2.a.g 1
40.e odd 2 1 3200.2.c.l 2
40.f even 2 1 3200.2.c.f 2
40.i odd 4 1 128.2.a.c yes 1
40.i odd 4 1 3200.2.a.e 1
40.k even 4 1 128.2.a.a 1
40.k even 4 1 3200.2.a.x 1
60.l odd 4 1 1152.2.a.c 1
80.i odd 4 1 256.2.b.a 2
80.j even 4 1 256.2.b.c 2
80.s even 4 1 256.2.b.c 2
80.t odd 4 1 256.2.b.a 2
120.q odd 4 1 1152.2.a.m 1
120.w even 4 1 1152.2.a.r 1
140.j odd 4 1 6272.2.a.a 1
160.u even 8 2 1024.2.e.i 4
160.v odd 8 2 1024.2.e.m 4
160.ba even 8 2 1024.2.e.i 4
160.bb odd 8 2 1024.2.e.m 4
240.z odd 4 1 2304.2.d.r 2
240.bb even 4 1 2304.2.d.b 2
240.bd odd 4 1 2304.2.d.r 2
240.bf even 4 1 2304.2.d.b 2
280.s even 4 1 6272.2.a.b 1
280.y odd 4 1 6272.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 40.k even 4 1
128.2.a.b yes 1 5.c odd 4 1
128.2.a.c yes 1 40.i odd 4 1
128.2.a.d yes 1 20.e even 4 1
256.2.b.a 2 80.i odd 4 1
256.2.b.a 2 80.t odd 4 1
256.2.b.c 2 80.j even 4 1
256.2.b.c 2 80.s even 4 1
1024.2.e.i 4 160.u even 8 2
1024.2.e.i 4 160.ba even 8 2
1024.2.e.m 4 160.v odd 8 2
1024.2.e.m 4 160.bb odd 8 2
1152.2.a.c 1 60.l odd 4 1
1152.2.a.h 1 15.e even 4 1
1152.2.a.m 1 120.q odd 4 1
1152.2.a.r 1 120.w even 4 1
2304.2.d.b 2 240.bb even 4 1
2304.2.d.b 2 240.bf even 4 1
2304.2.d.r 2 240.z odd 4 1
2304.2.d.r 2 240.bd odd 4 1
3200.2.a.e 1 40.i odd 4 1
3200.2.a.h 1 20.e even 4 1
3200.2.a.u 1 5.c odd 4 1
3200.2.a.x 1 40.k even 4 1
3200.2.c.e 2 4.b odd 2 1
3200.2.c.e 2 20.d odd 2 1
3200.2.c.f 2 8.b even 2 1
3200.2.c.f 2 40.f even 2 1
3200.2.c.k 2 1.a even 1 1 trivial
3200.2.c.k 2 5.b even 2 1 inner
3200.2.c.l 2 8.d odd 2 1
3200.2.c.l 2 40.e odd 2 1
6272.2.a.a 1 140.j odd 4 1
6272.2.a.b 1 280.s even 4 1
6272.2.a.g 1 35.f even 4 1
6272.2.a.h 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 \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 14)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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