Properties

Label 147.4.a.i
Level $147$
Weight $4$
Character orbit 147.a
Self dual yes
Analytic conductor $8.673$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.67328077084\)
Analytic rank: \(1\)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{2} -3 q^{3} + ( 7 + 3 \beta ) q^{4} + ( -2 - 2 \beta ) q^{5} + ( 3 + 3 \beta ) q^{6} + ( -41 - 5 \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( -1 - \beta ) q^{2} -3 q^{3} + ( 7 + 3 \beta ) q^{4} + ( -2 - 2 \beta ) q^{5} + ( 3 + 3 \beta ) q^{6} + ( -41 - 5 \beta ) q^{8} + 9 q^{9} + ( 30 + 6 \beta ) q^{10} + ( -8 + 10 \beta ) q^{11} + ( -21 - 9 \beta ) q^{12} + ( -14 + 12 \beta ) q^{13} + ( 6 + 6 \beta ) q^{15} + ( 55 + 27 \beta ) q^{16} + ( 2 + 2 \beta ) q^{17} + ( -9 - 9 \beta ) q^{18} + ( -44 + 24 \beta ) q^{19} + ( -98 - 26 \beta ) q^{20} + ( -132 - 12 \beta ) q^{22} + ( 20 - 34 \beta ) q^{23} + ( 123 + 15 \beta ) q^{24} + ( -65 + 12 \beta ) q^{25} + ( -154 - 10 \beta ) q^{26} -27 q^{27} + ( -138 + 24 \beta ) q^{29} + ( -90 - 18 \beta ) q^{30} + ( 16 - 72 \beta ) q^{31} + ( -105 - 69 \beta ) q^{32} + ( 24 - 30 \beta ) q^{33} + ( -30 - 6 \beta ) q^{34} + ( 63 + 27 \beta ) q^{36} + ( -106 - 36 \beta ) q^{37} + ( -292 - 4 \beta ) q^{38} + ( 42 - 36 \beta ) q^{39} + ( 222 + 102 \beta ) q^{40} + ( 210 + 30 \beta ) q^{41} + ( 212 - 48 \beta ) q^{43} + ( 364 + 76 \beta ) q^{44} + ( -18 - 18 \beta ) q^{45} + ( 456 + 48 \beta ) q^{46} + ( 40 - 68 \beta ) q^{47} + ( -165 - 81 \beta ) q^{48} + ( -103 + 41 \beta ) q^{50} + ( -6 - 6 \beta ) q^{51} + ( 406 + 78 \beta ) q^{52} + ( -554 + 4 \beta ) q^{53} + ( 27 + 27 \beta ) q^{54} + ( -264 - 24 \beta ) q^{55} + ( 132 - 72 \beta ) q^{57} + ( -198 + 90 \beta ) q^{58} + ( -460 + 116 \beta ) q^{59} + ( 294 + 78 \beta ) q^{60} + ( 250 - 72 \beta ) q^{61} + ( 992 + 128 \beta ) q^{62} + ( 631 + 27 \beta ) q^{64} + ( -308 - 20 \beta ) q^{65} + ( 396 + 36 \beta ) q^{66} + ( 20 + 108 \beta ) q^{67} + ( 98 + 26 \beta ) q^{68} + ( -60 + 102 \beta ) q^{69} + ( 492 - 30 \beta ) q^{71} + ( -369 - 45 \beta ) q^{72} + ( -530 - 12 \beta ) q^{73} + ( 610 + 178 \beta ) q^{74} + ( 195 - 36 \beta ) q^{75} + ( 700 + 108 \beta ) q^{76} + ( 462 + 30 \beta ) q^{78} + ( -232 - 108 \beta ) q^{79} + ( -866 - 218 \beta ) q^{80} + 81 q^{81} + ( -630 - 270 \beta ) q^{82} + ( -924 - 96 \beta ) q^{83} + ( -60 - 12 \beta ) q^{85} + ( 460 - 116 \beta ) q^{86} + ( 414 - 72 \beta ) q^{87} + ( -372 - 420 \beta ) q^{88} + ( -254 + 142 \beta ) q^{89} + ( 270 + 54 \beta ) q^{90} + ( -1288 - 280 \beta ) q^{92} + ( -48 + 216 \beta ) q^{93} + ( 912 + 96 \beta ) q^{94} + ( -584 - 8 \beta ) q^{95} + ( 315 + 207 \beta ) q^{96} + ( -266 - 276 \beta ) q^{97} + ( -72 + 90 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} - 6q^{3} + 17q^{4} - 6q^{5} + 9q^{6} - 87q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 3q^{2} - 6q^{3} + 17q^{4} - 6q^{5} + 9q^{6} - 87q^{8} + 18q^{9} + 66q^{10} - 6q^{11} - 51q^{12} - 16q^{13} + 18q^{15} + 137q^{16} + 6q^{17} - 27q^{18} - 64q^{19} - 222q^{20} - 276q^{22} + 6q^{23} + 261q^{24} - 118q^{25} - 318q^{26} - 54q^{27} - 252q^{29} - 198q^{30} - 40q^{31} - 279q^{32} + 18q^{33} - 66q^{34} + 153q^{36} - 248q^{37} - 588q^{38} + 48q^{39} + 546q^{40} + 450q^{41} + 376q^{43} + 804q^{44} - 54q^{45} + 960q^{46} + 12q^{47} - 411q^{48} - 165q^{50} - 18q^{51} + 890q^{52} - 1104q^{53} + 81q^{54} - 552q^{55} + 192q^{57} - 306q^{58} - 804q^{59} + 666q^{60} + 428q^{61} + 2112q^{62} + 1289q^{64} - 636q^{65} + 828q^{66} + 148q^{67} + 222q^{68} - 18q^{69} + 954q^{71} - 783q^{72} - 1072q^{73} + 1398q^{74} + 354q^{75} + 1508q^{76} + 954q^{78} - 572q^{79} - 1950q^{80} + 162q^{81} - 1530q^{82} - 1944q^{83} - 132q^{85} + 804q^{86} + 756q^{87} - 1164q^{88} - 366q^{89} + 594q^{90} - 2856q^{92} + 120q^{93} + 1920q^{94} - 1176q^{95} + 837q^{96} - 808q^{97} - 54q^{99} + O(q^{100}) \)

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
4.27492
−3.27492
−5.27492 −3.00000 19.8248 −10.5498 15.8248 0 −62.3746 9.00000 55.6495
1.2 2.27492 −3.00000 −2.82475 4.54983 −6.82475 0 −24.6254 9.00000 10.3505
\(n\): 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 147.4.a.i 2
3.b odd 2 1 441.4.a.r 2
4.b odd 2 1 2352.4.a.bz 2
7.b odd 2 1 21.4.a.c 2
7.c even 3 2 147.4.e.m 4
7.d odd 6 2 147.4.e.l 4
21.c even 2 1 63.4.a.e 2
21.g even 6 2 441.4.e.q 4
21.h odd 6 2 441.4.e.p 4
28.d even 2 1 336.4.a.m 2
35.c odd 2 1 525.4.a.n 2
35.f even 4 2 525.4.d.g 4
56.e even 2 1 1344.4.a.bo 2
56.h odd 2 1 1344.4.a.bg 2
84.h odd 2 1 1008.4.a.ba 2
105.g even 2 1 1575.4.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 7.b odd 2 1
63.4.a.e 2 21.c even 2 1
147.4.a.i 2 1.a even 1 1 trivial
147.4.e.l 4 7.d odd 6 2
147.4.e.m 4 7.c even 3 2
336.4.a.m 2 28.d even 2 1
441.4.a.r 2 3.b odd 2 1
441.4.e.p 4 21.h odd 6 2
441.4.e.q 4 21.g even 6 2
525.4.a.n 2 35.c odd 2 1
525.4.d.g 4 35.f even 4 2
1008.4.a.ba 2 84.h odd 2 1
1344.4.a.bg 2 56.h odd 2 1
1344.4.a.bo 2 56.e even 2 1
1575.4.a.p 2 105.g even 2 1
2352.4.a.bz 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2}^{2} + 3 T_{2} - 12 \)
\( T_{5}^{2} + 6 T_{5} - 48 \)

Hecke characteristic polynomials

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