Properties

Label 3200.2.a.bh
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: \( x^{2} - 2 \) Copy content Toggle raw display
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.

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

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

Hecke kernels

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

\( T_{3}^{2} + 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 17 \) Copy content Toggle raw display
\( T_{13}^{2} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

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