Properties

Label 3528.2.a.bk
Level $3528$
Weight $2$
Character orbit 3528.a
Self dual yes
Analytic conductor $28.171$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1712218331\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} + \beta q^{11} + ( - \beta + 3) q^{13} + 4 q^{17} + (\beta - 3) q^{19} - 4 q^{23} + (\beta + 9) q^{25} + (\beta - 2) q^{29} - q^{31} + (\beta + 1) q^{37} + ( - 2 \beta - 2) q^{41} + ( - \beta - 3) q^{43} + 6 q^{47} + (\beta - 6) q^{53} + (\beta + 14) q^{55} + ( - \beta - 2) q^{59} + 10 q^{61} + (2 \beta - 14) q^{65} + (\beta + 3) q^{67} - 2 q^{71} + (\beta - 1) q^{73} + ( - 2 \beta + 5) q^{79} + (\beta - 4) q^{83} + 4 \beta q^{85} + ( - 2 \beta + 4) q^{89} + ( - 2 \beta + 14) q^{95} + (\beta + 12) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} + q^{11} + 5 q^{13} + 8 q^{17} - 5 q^{19} - 8 q^{23} + 19 q^{25} - 3 q^{29} - 2 q^{31} + 3 q^{37} - 6 q^{41} - 7 q^{43} + 12 q^{47} - 11 q^{53} + 29 q^{55} - 5 q^{59} + 20 q^{61} - 26 q^{65} + 7 q^{67} - 4 q^{71} - q^{73} + 8 q^{79} - 7 q^{83} + 4 q^{85} + 6 q^{89} + 26 q^{95} + 25 q^{97}+O(q^{100}) \) 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
−3.27492
4.27492
0 0 0 −3.27492 0 0 0 0 0
1.2 0 0 0 4.27492 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)

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 3528.2.a.bk 2
3.b odd 2 1 1176.2.a.k 2
4.b odd 2 1 7056.2.a.cu 2
7.b odd 2 1 3528.2.a.bd 2
7.c even 3 2 504.2.s.i 4
7.d odd 6 2 3528.2.s.bk 4
12.b even 2 1 2352.2.a.bf 2
21.c even 2 1 1176.2.a.n 2
21.g even 6 2 1176.2.q.l 4
21.h odd 6 2 168.2.q.c 4
24.f even 2 1 9408.2.a.dp 2
24.h odd 2 1 9408.2.a.ec 2
28.d even 2 1 7056.2.a.ch 2
28.g odd 6 2 1008.2.s.r 4
84.h odd 2 1 2352.2.a.ba 2
84.j odd 6 2 2352.2.q.bf 4
84.n even 6 2 336.2.q.g 4
168.e odd 2 1 9408.2.a.dw 2
168.i even 2 1 9408.2.a.dj 2
168.s odd 6 2 1344.2.q.w 4
168.v even 6 2 1344.2.q.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 21.h odd 6 2
336.2.q.g 4 84.n even 6 2
504.2.s.i 4 7.c even 3 2
1008.2.s.r 4 28.g odd 6 2
1176.2.a.k 2 3.b odd 2 1
1176.2.a.n 2 21.c even 2 1
1176.2.q.l 4 21.g even 6 2
1344.2.q.w 4 168.s odd 6 2
1344.2.q.x 4 168.v even 6 2
2352.2.a.ba 2 84.h odd 2 1
2352.2.a.bf 2 12.b even 2 1
2352.2.q.bf 4 84.j odd 6 2
3528.2.a.bd 2 7.b odd 2 1
3528.2.a.bk 2 1.a even 1 1 trivial
3528.2.s.bk 4 7.d odd 6 2
7056.2.a.ch 2 28.d even 2 1
7056.2.a.cu 2 4.b odd 2 1
9408.2.a.dj 2 168.i even 2 1
9408.2.a.dp 2 24.f even 2 1
9408.2.a.dw 2 168.e odd 2 1
9408.2.a.ec 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3528))\):

\( T_{5}^{2} - T_{5} - 14 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 14 \) Copy content Toggle raw display
\( T_{13}^{2} - 5T_{13} - 8 \) Copy content Toggle raw display
\( T_{23} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 14 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T - 14 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T - 8 \) Copy content Toggle raw display
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 5T - 8 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 12 \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 3T - 12 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 48 \) Copy content Toggle raw display
$43$ \( T^{2} + 7T - 2 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 11T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 5T - 8 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 7T - 2 \) Copy content Toggle raw display
$71$ \( (T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + T - 14 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 41 \) Copy content Toggle raw display
$83$ \( T^{2} + 7T - 2 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 25T + 142 \) Copy content Toggle raw display
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