Properties

Label 63.4.a.e
Level $63$
Weight $4$
Character orbit 63.a
Self dual yes
Analytic conductor $3.717$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 7 + 3 \beta ) q^{4} + ( -2 - 2 \beta ) q^{5} + 7 q^{7} + ( 41 + 5 \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 7 + 3 \beta ) q^{4} + ( -2 - 2 \beta ) q^{5} + 7 q^{7} + ( 41 + 5 \beta ) q^{8} + ( -30 - 6 \beta ) q^{10} + ( 8 - 10 \beta ) q^{11} + ( 14 - 12 \beta ) q^{13} + ( 7 + 7 \beta ) q^{14} + ( 55 + 27 \beta ) q^{16} + ( 2 + 2 \beta ) q^{17} + ( 44 - 24 \beta ) q^{19} + ( -98 - 26 \beta ) q^{20} + ( -132 - 12 \beta ) q^{22} + ( -20 + 34 \beta ) q^{23} + ( -65 + 12 \beta ) q^{25} + ( -154 - 10 \beta ) q^{26} + ( 49 + 21 \beta ) q^{28} + ( 138 - 24 \beta ) q^{29} + ( -16 + 72 \beta ) q^{31} + ( 105 + 69 \beta ) q^{32} + ( 30 + 6 \beta ) q^{34} + ( -14 - 14 \beta ) q^{35} + ( -106 - 36 \beta ) q^{37} + ( -292 - 4 \beta ) q^{38} + ( -222 - 102 \beta ) q^{40} + ( 210 + 30 \beta ) q^{41} + ( 212 - 48 \beta ) q^{43} + ( -364 - 76 \beta ) q^{44} + ( 456 + 48 \beta ) q^{46} + ( 40 - 68 \beta ) q^{47} + 49 q^{49} + ( 103 - 41 \beta ) q^{50} + ( -406 - 78 \beta ) q^{52} + ( 554 - 4 \beta ) q^{53} + ( 264 + 24 \beta ) q^{55} + ( 287 + 35 \beta ) q^{56} + ( -198 + 90 \beta ) q^{58} + ( -460 + 116 \beta ) q^{59} + ( -250 + 72 \beta ) q^{61} + ( 992 + 128 \beta ) q^{62} + ( 631 + 27 \beta ) q^{64} + ( 308 + 20 \beta ) q^{65} + ( 20 + 108 \beta ) q^{67} + ( 98 + 26 \beta ) q^{68} + ( -210 - 42 \beta ) q^{70} + ( -492 + 30 \beta ) q^{71} + ( 530 + 12 \beta ) q^{73} + ( -610 - 178 \beta ) q^{74} + ( -700 - 108 \beta ) q^{76} + ( 56 - 70 \beta ) q^{77} + ( -232 - 108 \beta ) q^{79} + ( -866 - 218 \beta ) q^{80} + ( 630 + 270 \beta ) q^{82} + ( -924 - 96 \beta ) q^{83} + ( -60 - 12 \beta ) q^{85} + ( -460 + 116 \beta ) q^{86} + ( -372 - 420 \beta ) q^{88} + ( -254 + 142 \beta ) q^{89} + ( 98 - 84 \beta ) q^{91} + ( 1288 + 280 \beta ) q^{92} + ( -912 - 96 \beta ) q^{94} + ( 584 + 8 \beta ) q^{95} + ( 266 + 276 \beta ) q^{97} + ( 49 + 49 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} + 17q^{4} - 6q^{5} + 14q^{7} + 87q^{8} + O(q^{10}) \) \( 2q + 3q^{2} + 17q^{4} - 6q^{5} + 14q^{7} + 87q^{8} - 66q^{10} + 6q^{11} + 16q^{13} + 21q^{14} + 137q^{16} + 6q^{17} + 64q^{19} - 222q^{20} - 276q^{22} - 6q^{23} - 118q^{25} - 318q^{26} + 119q^{28} + 252q^{29} + 40q^{31} + 279q^{32} + 66q^{34} - 42q^{35} - 248q^{37} - 588q^{38} - 546q^{40} + 450q^{41} + 376q^{43} - 804q^{44} + 960q^{46} + 12q^{47} + 98q^{49} + 165q^{50} - 890q^{52} + 1104q^{53} + 552q^{55} + 609q^{56} - 306q^{58} - 804q^{59} - 428q^{61} + 2112q^{62} + 1289q^{64} + 636q^{65} + 148q^{67} + 222q^{68} - 462q^{70} - 954q^{71} + 1072q^{73} - 1398q^{74} - 1508q^{76} + 42q^{77} - 572q^{79} - 1950q^{80} + 1530q^{82} - 1944q^{83} - 132q^{85} - 804q^{86} - 1164q^{88} - 366q^{89} + 112q^{91} + 2856q^{92} - 1920q^{94} + 1176q^{95} + 808q^{97} + 147q^{98} + O(q^{100}) \)

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
−3.27492
4.27492
−2.27492 0 −2.82475 4.54983 0 7.00000 24.6254 0 −10.3505
1.2 5.27492 0 19.8248 −10.5498 0 7.00000 62.3746 0 −55.6495
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 63.4.a.e 2
3.b odd 2 1 21.4.a.c 2
4.b odd 2 1 1008.4.a.ba 2
5.b even 2 1 1575.4.a.p 2
7.b odd 2 1 441.4.a.r 2
7.c even 3 2 441.4.e.q 4
7.d odd 6 2 441.4.e.p 4
12.b even 2 1 336.4.a.m 2
15.d odd 2 1 525.4.a.n 2
15.e even 4 2 525.4.d.g 4
21.c even 2 1 147.4.a.i 2
21.g even 6 2 147.4.e.m 4
21.h odd 6 2 147.4.e.l 4
24.f even 2 1 1344.4.a.bo 2
24.h odd 2 1 1344.4.a.bg 2
84.h odd 2 1 2352.4.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 3.b odd 2 1
63.4.a.e 2 1.a even 1 1 trivial
147.4.a.i 2 21.c even 2 1
147.4.e.l 4 21.h odd 6 2
147.4.e.m 4 21.g even 6 2
336.4.a.m 2 12.b even 2 1
441.4.a.r 2 7.b odd 2 1
441.4.e.p 4 7.d odd 6 2
441.4.e.q 4 7.c even 3 2
525.4.a.n 2 15.d odd 2 1
525.4.d.g 4 15.e even 4 2
1008.4.a.ba 2 4.b odd 2 1
1344.4.a.bg 2 24.h odd 2 1
1344.4.a.bo 2 24.f even 2 1
1575.4.a.p 2 5.b even 2 1
2352.4.a.bz 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 T_{2} - 12 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(63))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -12 - 3 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -48 + 6 T + T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -1416 - 6 T + T^{2} \)
$13$ \( -1988 - 16 T + T^{2} \)
$17$ \( -48 - 6 T + T^{2} \)
$19$ \( -7184 - 64 T + T^{2} \)
$23$ \( -16464 + 6 T + T^{2} \)
$29$ \( 7668 - 252 T + T^{2} \)
$31$ \( -73472 - 40 T + T^{2} \)
$37$ \( -3092 + 248 T + T^{2} \)
$41$ \( 37800 - 450 T + T^{2} \)
$43$ \( 2512 - 376 T + T^{2} \)
$47$ \( -65856 - 12 T + T^{2} \)
$53$ \( 304476 - 1104 T + T^{2} \)
$59$ \( -30144 + 804 T + T^{2} \)
$61$ \( -28076 + 428 T + T^{2} \)
$67$ \( -160736 - 148 T + T^{2} \)
$71$ \( 214704 + 954 T + T^{2} \)
$73$ \( 285244 - 1072 T + T^{2} \)
$79$ \( -84416 + 572 T + T^{2} \)
$83$ \( 813456 + 1944 T + T^{2} \)
$89$ \( -253848 + 366 T + T^{2} \)
$97$ \( -922292 - 808 T + T^{2} \)
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