Properties

Label 648.4.a.c
Level $648$
Weight $4$
Character orbit 648.a
Self dual yes
Analytic conductor $38.233$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 648.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.2332376837\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{201}) \)
Defining polynomial: \(x^{2} - x - 50\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{201}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -4 - \beta ) q^{5} + ( -15 + \beta ) q^{7} +O(q^{10})\) \( q + ( -4 - \beta ) q^{5} + ( -15 + \beta ) q^{7} + ( 23 + 3 \beta ) q^{11} + ( 46 - \beta ) q^{13} + ( -11 - 8 \beta ) q^{17} + ( -43 + 5 \beta ) q^{19} + ( 29 + \beta ) q^{23} + ( 92 + 8 \beta ) q^{25} + ( -54 + \beta ) q^{29} + ( 68 + 16 \beta ) q^{31} + ( -141 + 11 \beta ) q^{35} + ( -90 - \beta ) q^{37} + ( -336 + 2 \beta ) q^{41} + ( -35 - 23 \beta ) q^{43} + ( -426 + 2 \beta ) q^{47} + ( 83 - 30 \beta ) q^{49} + ( -334 + 16 \beta ) q^{53} + ( -695 - 35 \beta ) q^{55} + ( -274 + 6 \beta ) q^{59} + ( 142 - 17 \beta ) q^{61} + ( 17 - 42 \beta ) q^{65} + ( 581 - 11 \beta ) q^{67} + ( 403 + 3 \beta ) q^{71} + ( -755 + 18 \beta ) q^{73} + ( 258 - 22 \beta ) q^{77} + ( -11 + 41 \beta ) q^{79} + ( -770 - 38 \beta ) q^{83} + ( 1652 + 43 \beta ) q^{85} -1323 q^{89} + ( -891 + 61 \beta ) q^{91} + ( -833 + 23 \beta ) q^{95} + ( 236 + 50 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{5} - 30q^{7} + O(q^{10}) \) \( 2q - 8q^{5} - 30q^{7} + 46q^{11} + 92q^{13} - 22q^{17} - 86q^{19} + 58q^{23} + 184q^{25} - 108q^{29} + 136q^{31} - 282q^{35} - 180q^{37} - 672q^{41} - 70q^{43} - 852q^{47} + 166q^{49} - 668q^{53} - 1390q^{55} - 548q^{59} + 284q^{61} + 34q^{65} + 1162q^{67} + 806q^{71} - 1510q^{73} + 516q^{77} - 22q^{79} - 1540q^{83} + 3304q^{85} - 2646q^{89} - 1782q^{91} - 1666q^{95} + 472q^{97} + O(q^{100}) \)

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
7.58872
−6.58872
0 0 0 −18.1774 0 −0.822553 0 0 0
1.2 0 0 0 10.1774 0 −29.1774 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 648.4.a.c 2
3.b odd 2 1 648.4.a.f yes 2
4.b odd 2 1 1296.4.a.m 2
9.c even 3 2 648.4.i.t 4
9.d odd 6 2 648.4.i.n 4
12.b even 2 1 1296.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.c 2 1.a even 1 1 trivial
648.4.a.f yes 2 3.b odd 2 1
648.4.i.n 4 9.d odd 6 2
648.4.i.t 4 9.c even 3 2
1296.4.a.m 2 4.b odd 2 1
1296.4.a.q 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 8 T_{5} - 185 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(648))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -185 + 8 T + T^{2} \)
$7$ \( 24 + 30 T + T^{2} \)
$11$ \( -1280 - 46 T + T^{2} \)
$13$ \( 1915 - 92 T + T^{2} \)
$17$ \( -12743 + 22 T + T^{2} \)
$19$ \( -3176 + 86 T + T^{2} \)
$23$ \( 640 - 58 T + T^{2} \)
$29$ \( 2715 + 108 T + T^{2} \)
$31$ \( -46832 - 136 T + T^{2} \)
$37$ \( 7899 + 180 T + T^{2} \)
$41$ \( 112092 + 672 T + T^{2} \)
$43$ \( -105104 + 70 T + T^{2} \)
$47$ \( 180672 + 852 T + T^{2} \)
$53$ \( 60100 + 668 T + T^{2} \)
$59$ \( 67840 + 548 T + T^{2} \)
$61$ \( -37925 - 284 T + T^{2} \)
$67$ \( 313240 - 1162 T + T^{2} \)
$71$ \( 160600 - 806 T + T^{2} \)
$73$ \( 504901 + 1510 T + T^{2} \)
$79$ \( -337760 + 22 T + T^{2} \)
$83$ \( 302656 + 1540 T + T^{2} \)
$89$ \( ( 1323 + T )^{2} \)
$97$ \( -446804 - 472 T + T^{2} \)
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