Properties

Label 3200.2.a.bc
Level $3200$
Weight $2$
Character orbit 3200.a
Self dual yes
Analytic conductor $25.552$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{3} + ( -2 - 2 \beta ) q^{7} -2 \beta q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{3} + ( -2 - 2 \beta ) q^{7} -2 \beta q^{9} + ( -1 + 3 \beta ) q^{11} + 4 \beta q^{13} + ( -3 - 2 \beta ) q^{17} + ( -3 - \beta ) q^{19} -2 q^{21} + ( -2 + 2 \beta ) q^{23} + ( -1 - \beta ) q^{27} + 8 q^{29} + ( 2 + 2 \beta ) q^{31} + ( 7 - 4 \beta ) q^{33} + ( -2 + 4 \beta ) q^{37} + ( 8 - 4 \beta ) q^{39} + ( 5 - 4 \beta ) q^{41} -10 q^{43} + ( -4 + 4 \beta ) q^{47} + ( 5 + 8 \beta ) q^{49} + ( -1 - \beta ) q^{51} + ( 2 - 4 \beta ) q^{53} + ( 1 - 2 \beta ) q^{57} + ( -2 - 4 \beta ) q^{59} + 6 q^{61} + ( 8 + 4 \beta ) q^{63} + ( 7 + 3 \beta ) q^{67} + ( 6 - 4 \beta ) q^{69} + ( -4 + 4 \beta ) q^{71} + ( 3 - 2 \beta ) q^{73} + ( -10 - 4 \beta ) q^{77} + ( 10 - 2 \beta ) q^{79} + ( -1 + 6 \beta ) q^{81} + ( 3 + 3 \beta ) q^{83} + ( -8 + 8 \beta ) q^{87} + ( 9 + 2 \beta ) q^{89} + ( -16 - 8 \beta ) q^{91} + 2 q^{93} + ( -6 + 8 \beta ) q^{97} + ( -12 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{3} - 4 q^{7} - 2 q^{11} - 6 q^{17} - 6 q^{19} - 4 q^{21} - 4 q^{23} - 2 q^{27} + 16 q^{29} + 4 q^{31} + 14 q^{33} - 4 q^{37} + 16 q^{39} + 10 q^{41} - 20 q^{43} - 8 q^{47} + 10 q^{49} - 2 q^{51} + 4 q^{53} + 2 q^{57} - 4 q^{59} + 12 q^{61} + 16 q^{63} + 14 q^{67} + 12 q^{69} - 8 q^{71} + 6 q^{73} - 20 q^{77} + 20 q^{79} - 2 q^{81} + 6 q^{83} - 16 q^{87} + 18 q^{89} - 32 q^{91} + 4 q^{93} - 12 q^{97} - 24 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −2.41421 0 0 0 0.828427 0 2.82843 0
1.2 0 0.414214 0 0 0 −4.82843 0 −2.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.a.bc 2
4.b odd 2 1 3200.2.a.bn yes 2
5.b even 2 1 3200.2.a.bm yes 2
5.c odd 4 2 3200.2.c.y 4
8.b even 2 1 3200.2.a.bj yes 2
8.d odd 2 1 3200.2.a.bg yes 2
20.d odd 2 1 3200.2.a.bd yes 2
20.e even 4 2 3200.2.c.ba 4
40.e odd 2 1 3200.2.a.bi yes 2
40.f even 2 1 3200.2.a.bh yes 2
40.i odd 4 2 3200.2.c.bb 4
40.k even 4 2 3200.2.c.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.a.bc 2 1.a even 1 1 trivial
3200.2.a.bd yes 2 20.d odd 2 1
3200.2.a.bg yes 2 8.d odd 2 1
3200.2.a.bh yes 2 40.f even 2 1
3200.2.a.bi yes 2 40.e odd 2 1
3200.2.a.bj yes 2 8.b even 2 1
3200.2.a.bm yes 2 5.b even 2 1
3200.2.a.bn yes 2 4.b odd 2 1
3200.2.c.y 4 5.c odd 4 2
3200.2.c.z 4 40.k even 4 2
3200.2.c.ba 4 20.e even 4 2
3200.2.c.bb 4 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3200))\):

\( T_{3}^{2} + 2 T_{3} - 1 \)
\( T_{7}^{2} + 4 T_{7} - 4 \)
\( T_{11}^{2} + 2 T_{11} - 17 \)
\( T_{13}^{2} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -1 + 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -4 + 4 T + T^{2} \)
$11$ \( -17 + 2 T + T^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( 1 + 6 T + T^{2} \)
$19$ \( 7 + 6 T + T^{2} \)
$23$ \( -4 + 4 T + T^{2} \)
$29$ \( ( -8 + T )^{2} \)
$31$ \( -4 - 4 T + T^{2} \)
$37$ \( -28 + 4 T + T^{2} \)
$41$ \( -7 - 10 T + T^{2} \)
$43$ \( ( 10 + T )^{2} \)
$47$ \( -16 + 8 T + T^{2} \)
$53$ \( -28 - 4 T + T^{2} \)
$59$ \( -28 + 4 T + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( 31 - 14 T + T^{2} \)
$71$ \( -16 + 8 T + T^{2} \)
$73$ \( 1 - 6 T + T^{2} \)
$79$ \( 92 - 20 T + T^{2} \)
$83$ \( -9 - 6 T + T^{2} \)
$89$ \( 73 - 18 T + T^{2} \)
$97$ \( -92 + 12 T + T^{2} \)
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