Properties

Label 648.2.f.a
Level 648
Weight 2
Character orbit 648.f
Analytic conductor 5.174
Analytic rank 0
Dimension 4
CM discriminant -8
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 648.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -2 q^{4} -2 \beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} -2 q^{4} -2 \beta_{1} q^{8} + ( -\beta_{1} - \beta_{2} ) q^{11} + 4 q^{16} + ( 2 \beta_{1} - \beta_{2} ) q^{17} + ( -1 + \beta_{3} ) q^{19} + ( 2 + \beta_{3} ) q^{22} -5 q^{25} + 4 \beta_{1} q^{32} + ( -4 + \beta_{3} ) q^{34} + ( -\beta_{1} + 2 \beta_{2} ) q^{38} + ( -4 \beta_{1} - \beta_{2} ) q^{41} + ( 5 + \beta_{3} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{44} + 7 q^{49} -5 \beta_{1} q^{50} + ( 5 \beta_{1} - \beta_{2} ) q^{59} -8 q^{64} + ( -7 + \beta_{3} ) q^{67} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{68} + ( -1 - 2 \beta_{3} ) q^{73} + ( 2 - 2 \beta_{3} ) q^{76} + ( 8 + \beta_{3} ) q^{82} + 2 \beta_{1} q^{83} + ( 5 \beta_{1} + 2 \beta_{2} ) q^{86} + ( -4 - 2 \beta_{3} ) q^{88} -4 \beta_{1} q^{89} + ( 5 - 2 \beta_{3} ) q^{97} + 7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 16q^{16} - 4q^{19} + 8q^{22} - 20q^{25} - 16q^{34} + 20q^{43} + 28q^{49} - 32q^{64} - 28q^{67} - 4q^{73} + 8q^{76} + 32q^{82} - 16q^{88} + 20q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\( 3 \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{3} + 12 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 3\)\()/3\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1
−1.22474 0.707107i
1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.41421i 0 −2.00000 0 0 0 2.82843i 0 0
323.2 1.41421i 0 −2.00000 0 0 0 2.82843i 0 0
323.3 1.41421i 0 −2.00000 0 0 0 2.82843i 0 0
323.4 1.41421i 0 −2.00000 0 0 0 2.82843i 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
3.b odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.f.a 4
3.b odd 2 1 inner 648.2.f.a 4
4.b odd 2 1 2592.2.f.a 4
8.b even 2 1 2592.2.f.a 4
8.d odd 2 1 CM 648.2.f.a 4
9.c even 3 1 72.2.l.a 4
9.c even 3 1 216.2.l.a 4
9.d odd 6 1 72.2.l.a 4
9.d odd 6 1 216.2.l.a 4
12.b even 2 1 2592.2.f.a 4
24.f even 2 1 inner 648.2.f.a 4
24.h odd 2 1 2592.2.f.a 4
36.f odd 6 1 288.2.p.a 4
36.f odd 6 1 864.2.p.a 4
36.h even 6 1 288.2.p.a 4
36.h even 6 1 864.2.p.a 4
72.j odd 6 1 288.2.p.a 4
72.j odd 6 1 864.2.p.a 4
72.l even 6 1 72.2.l.a 4
72.l even 6 1 216.2.l.a 4
72.n even 6 1 288.2.p.a 4
72.n even 6 1 864.2.p.a 4
72.p odd 6 1 72.2.l.a 4
72.p odd 6 1 216.2.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.a 4 9.c even 3 1
72.2.l.a 4 9.d odd 6 1
72.2.l.a 4 72.l even 6 1
72.2.l.a 4 72.p odd 6 1
216.2.l.a 4 9.c even 3 1
216.2.l.a 4 9.d odd 6 1
216.2.l.a 4 72.l even 6 1
216.2.l.a 4 72.p odd 6 1
288.2.p.a 4 36.f odd 6 1
288.2.p.a 4 36.h even 6 1
288.2.p.a 4 72.j odd 6 1
288.2.p.a 4 72.n even 6 1
648.2.f.a 4 1.a even 1 1 trivial
648.2.f.a 4 3.b odd 2 1 inner
648.2.f.a 4 8.d odd 2 1 CM
648.2.f.a 4 24.f even 2 1 inner
864.2.p.a 4 36.f odd 6 1
864.2.p.a 4 36.h even 6 1
864.2.p.a 4 72.j odd 6 1
864.2.p.a 4 72.n even 6 1
2592.2.f.a 4 4.b odd 2 1
2592.2.f.a 4 8.b even 2 1
2592.2.f.a 4 12.b even 2 1
2592.2.f.a 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(648, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{2} \)
$3$ 1
$5$ \( ( 1 + 5 T^{2} )^{4} \)
$7$ \( ( 1 - 7 T^{2} )^{4} \)
$11$ \( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} ) \)
$13$ \( ( 1 - 13 T^{2} )^{4} \)
$17$ \( ( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} )( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} ) \)
$19$ \( ( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 - 37 T^{2} )^{4} \)
$41$ \( ( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} )( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} ) \)
$43$ \( ( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} )( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} ) \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( ( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( ( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2} \)
$89$ \( ( 1 - 18 T + 89 T^{2} )^{2}( 1 + 18 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} )^{2} \)
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