Properties

Label 147.4.e.g
Level 147
Weight 4
Character orbit 147.e
Analytic conductor 8.673
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -18 \zeta_{6} q^{5} -9 q^{6} + 21 q^{8} -9 \zeta_{6} q^{9} +O(q^{10})\) \( q + 3 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -18 \zeta_{6} q^{5} -9 q^{6} + 21 q^{8} -9 \zeta_{6} q^{9} + ( 54 - 54 \zeta_{6} ) q^{10} + ( 36 - 36 \zeta_{6} ) q^{11} -3 \zeta_{6} q^{12} + 34 q^{13} + 54 q^{15} + 71 \zeta_{6} q^{16} + ( 42 - 42 \zeta_{6} ) q^{17} + ( 27 - 27 \zeta_{6} ) q^{18} -124 \zeta_{6} q^{19} + 18 q^{20} + 108 q^{22} + ( -63 + 63 \zeta_{6} ) q^{24} + ( -199 + 199 \zeta_{6} ) q^{25} + 102 \zeta_{6} q^{26} + 27 q^{27} + 102 q^{29} + 162 \zeta_{6} q^{30} + ( -160 + 160 \zeta_{6} ) q^{31} + ( -45 + 45 \zeta_{6} ) q^{32} + 108 \zeta_{6} q^{33} + 126 q^{34} + 9 q^{36} -398 \zeta_{6} q^{37} + ( 372 - 372 \zeta_{6} ) q^{38} + ( -102 + 102 \zeta_{6} ) q^{39} -378 \zeta_{6} q^{40} + 318 q^{41} -268 q^{43} + 36 \zeta_{6} q^{44} + ( -162 + 162 \zeta_{6} ) q^{45} + 240 \zeta_{6} q^{47} -213 q^{48} -597 q^{50} + 126 \zeta_{6} q^{51} + ( -34 + 34 \zeta_{6} ) q^{52} + ( 498 - 498 \zeta_{6} ) q^{53} + 81 \zeta_{6} q^{54} -648 q^{55} + 372 q^{57} + 306 \zeta_{6} q^{58} + ( -132 + 132 \zeta_{6} ) q^{59} + ( -54 + 54 \zeta_{6} ) q^{60} + 398 \zeta_{6} q^{61} -480 q^{62} + 433 q^{64} -612 \zeta_{6} q^{65} + ( -324 + 324 \zeta_{6} ) q^{66} + ( -92 + 92 \zeta_{6} ) q^{67} + 42 \zeta_{6} q^{68} -720 q^{71} -189 \zeta_{6} q^{72} + ( -502 + 502 \zeta_{6} ) q^{73} + ( 1194 - 1194 \zeta_{6} ) q^{74} -597 \zeta_{6} q^{75} + 124 q^{76} -306 q^{78} + 1024 \zeta_{6} q^{79} + ( 1278 - 1278 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} + 954 \zeta_{6} q^{82} + 204 q^{83} -756 q^{85} -804 \zeta_{6} q^{86} + ( -306 + 306 \zeta_{6} ) q^{87} + ( 756 - 756 \zeta_{6} ) q^{88} + 354 \zeta_{6} q^{89} -486 q^{90} -480 \zeta_{6} q^{93} + ( -720 + 720 \zeta_{6} ) q^{94} + ( -2232 + 2232 \zeta_{6} ) q^{95} -135 \zeta_{6} q^{96} + 286 q^{97} -324 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} - 3q^{3} - q^{4} - 18q^{5} - 18q^{6} + 42q^{8} - 9q^{9} + O(q^{10}) \) \( 2q + 3q^{2} - 3q^{3} - q^{4} - 18q^{5} - 18q^{6} + 42q^{8} - 9q^{9} + 54q^{10} + 36q^{11} - 3q^{12} + 68q^{13} + 108q^{15} + 71q^{16} + 42q^{17} + 27q^{18} - 124q^{19} + 36q^{20} + 216q^{22} - 63q^{24} - 199q^{25} + 102q^{26} + 54q^{27} + 204q^{29} + 162q^{30} - 160q^{31} - 45q^{32} + 108q^{33} + 252q^{34} + 18q^{36} - 398q^{37} + 372q^{38} - 102q^{39} - 378q^{40} + 636q^{41} - 536q^{43} + 36q^{44} - 162q^{45} + 240q^{47} - 426q^{48} - 1194q^{50} + 126q^{51} - 34q^{52} + 498q^{53} + 81q^{54} - 1296q^{55} + 744q^{57} + 306q^{58} - 132q^{59} - 54q^{60} + 398q^{61} - 960q^{62} + 866q^{64} - 612q^{65} - 324q^{66} - 92q^{67} + 42q^{68} - 1440q^{71} - 189q^{72} - 502q^{73} + 1194q^{74} - 597q^{75} + 248q^{76} - 612q^{78} + 1024q^{79} + 1278q^{80} - 81q^{81} + 954q^{82} + 408q^{83} - 1512q^{85} - 804q^{86} - 306q^{87} + 756q^{88} + 354q^{89} - 972q^{90} - 480q^{93} - 720q^{94} - 2232q^{95} - 135q^{96} + 572q^{97} - 648q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 0.866025i
1.50000 + 2.59808i −1.50000 + 2.59808i −0.500000 + 0.866025i −9.00000 15.5885i −9.00000 0 21.0000 −4.50000 7.79423i 27.0000 46.7654i
79.1 1.50000 2.59808i −1.50000 2.59808i −0.500000 0.866025i −9.00000 + 15.5885i −9.00000 0 21.0000 −4.50000 + 7.79423i 27.0000 + 46.7654i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.e.g 2
3.b odd 2 1 441.4.e.d 2
7.b odd 2 1 147.4.e.i 2
7.c even 3 1 147.4.a.c 1
7.c even 3 1 inner 147.4.e.g 2
7.d odd 6 1 21.4.a.a 1
7.d odd 6 1 147.4.e.i 2
21.c even 2 1 441.4.e.b 2
21.g even 6 1 63.4.a.c 1
21.g even 6 1 441.4.e.b 2
21.h odd 6 1 441.4.a.j 1
21.h odd 6 1 441.4.e.d 2
28.f even 6 1 336.4.a.f 1
28.g odd 6 1 2352.4.a.r 1
35.i odd 6 1 525.4.a.g 1
35.k even 12 2 525.4.d.c 2
56.j odd 6 1 1344.4.a.ba 1
56.m even 6 1 1344.4.a.n 1
84.j odd 6 1 1008.4.a.v 1
105.p even 6 1 1575.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 7.d odd 6 1
63.4.a.c 1 21.g even 6 1
147.4.a.c 1 7.c even 3 1
147.4.e.g 2 1.a even 1 1 trivial
147.4.e.g 2 7.c even 3 1 inner
147.4.e.i 2 7.b odd 2 1
147.4.e.i 2 7.d odd 6 1
336.4.a.f 1 28.f even 6 1
441.4.a.j 1 21.h odd 6 1
441.4.e.b 2 21.c even 2 1
441.4.e.b 2 21.g even 6 1
441.4.e.d 2 3.b odd 2 1
441.4.e.d 2 21.h odd 6 1
525.4.a.g 1 35.i odd 6 1
525.4.d.c 2 35.k even 12 2
1008.4.a.v 1 84.j odd 6 1
1344.4.a.n 1 56.m even 6 1
1344.4.a.ba 1 56.j odd 6 1
1575.4.a.b 1 105.p even 6 1
2352.4.a.r 1 28.g odd 6 1

Hecke kernels

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

\( T_{2}^{2} - 3 T_{2} + 9 \)
\( T_{5}^{2} + 18 T_{5} + 324 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + T^{2} - 24 T^{3} + 64 T^{4} \)
$3$ \( 1 + 3 T + 9 T^{2} \)
$5$ \( 1 + 18 T + 199 T^{2} + 2250 T^{3} + 15625 T^{4} \)
$7$ 1
$11$ \( 1 - 36 T - 35 T^{2} - 47916 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 - 34 T + 2197 T^{2} )^{2} \)
$17$ \( 1 - 42 T - 3149 T^{2} - 206346 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 124 T + 8517 T^{2} + 850516 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 12167 T^{2} + 148035889 T^{4} \)
$29$ \( ( 1 - 102 T + 24389 T^{2} )^{2} \)
$31$ \( 1 + 160 T - 4191 T^{2} + 4766560 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 398 T + 107751 T^{2} + 20159894 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 - 318 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 + 268 T + 79507 T^{2} )^{2} \)
$47$ \( 1 - 240 T - 46223 T^{2} - 24917520 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 498 T + 99127 T^{2} - 74140746 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 132 T - 187955 T^{2} + 27110028 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 398 T - 68577 T^{2} - 90338438 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 92 T - 292299 T^{2} + 27670196 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 + 720 T + 357911 T^{2} )^{2} \)
$73$ \( 1 + 502 T - 137013 T^{2} + 195286534 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 1024 T + 555537 T^{2} - 504871936 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 - 204 T + 571787 T^{2} )^{2} \)
$89$ \( 1 - 354 T - 579653 T^{2} - 249559026 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 - 286 T + 912673 T^{2} )^{2} \)
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