# Properties

 Label 147.6.e.h Level $147$ Weight $6$ Character orbit 147.e Analytic conductor $23.576$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 147.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.5764215125$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) 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.

 $$f(q)$$ $$=$$ $$q + 6 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (4 \zeta_{6} - 4) q^{4} - 6 \zeta_{6} q^{5} - 54 q^{6} + 168 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q + 6*z * q^2 + (9*z - 9) * q^3 + (4*z - 4) * q^4 - 6*z * q^5 - 54 * q^6 + 168 * q^8 - 81*z * q^9 $$q + 6 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (4 \zeta_{6} - 4) q^{4} - 6 \zeta_{6} q^{5} - 54 q^{6} + 168 q^{8} - 81 \zeta_{6} q^{9} + ( - 36 \zeta_{6} + 36) q^{10} + ( - 564 \zeta_{6} + 564) q^{11} - 36 \zeta_{6} q^{12} + 638 q^{13} + 54 q^{15} + 1136 \zeta_{6} q^{16} + (882 \zeta_{6} - 882) q^{17} + ( - 486 \zeta_{6} + 486) q^{18} + 556 \zeta_{6} q^{19} + 24 q^{20} + 3384 q^{22} + 840 \zeta_{6} q^{23} + (1512 \zeta_{6} - 1512) q^{24} + ( - 3089 \zeta_{6} + 3089) q^{25} + 3828 \zeta_{6} q^{26} + 729 q^{27} + 4638 q^{29} + 324 \zeta_{6} q^{30} + (4400 \zeta_{6} - 4400) q^{31} + (1440 \zeta_{6} - 1440) q^{32} + 5076 \zeta_{6} q^{33} - 5292 q^{34} + 324 q^{36} + 2410 \zeta_{6} q^{37} + (3336 \zeta_{6} - 3336) q^{38} + (5742 \zeta_{6} - 5742) q^{39} - 1008 \zeta_{6} q^{40} - 6870 q^{41} + 9644 q^{43} + 2256 \zeta_{6} q^{44} + (486 \zeta_{6} - 486) q^{45} + (5040 \zeta_{6} - 5040) q^{46} + 18672 \zeta_{6} q^{47} - 10224 q^{48} + 18534 q^{50} - 7938 \zeta_{6} q^{51} + (2552 \zeta_{6} - 2552) q^{52} + (33750 \zeta_{6} - 33750) q^{53} + 4374 \zeta_{6} q^{54} - 3384 q^{55} - 5004 q^{57} + 27828 \zeta_{6} q^{58} + ( - 18084 \zeta_{6} + 18084) q^{59} + (216 \zeta_{6} - 216) q^{60} - 39758 \zeta_{6} q^{61} - 26400 q^{62} + 27712 q^{64} - 3828 \zeta_{6} q^{65} + (30456 \zeta_{6} - 30456) q^{66} + ( - 23068 \zeta_{6} + 23068) q^{67} - 3528 \zeta_{6} q^{68} - 7560 q^{69} - 4248 q^{71} - 13608 \zeta_{6} q^{72} + ( - 41110 \zeta_{6} + 41110) q^{73} + (14460 \zeta_{6} - 14460) q^{74} + 27801 \zeta_{6} q^{75} - 2224 q^{76} - 34452 q^{78} - 21920 \zeta_{6} q^{79} + ( - 6816 \zeta_{6} + 6816) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 41220 \zeta_{6} q^{82} + 82452 q^{83} + 5292 q^{85} + 57864 \zeta_{6} q^{86} + (41742 \zeta_{6} - 41742) q^{87} + ( - 94752 \zeta_{6} + 94752) q^{88} + 94086 \zeta_{6} q^{89} - 2916 q^{90} - 3360 q^{92} - 39600 \zeta_{6} q^{93} + (112032 \zeta_{6} - 112032) q^{94} + ( - 3336 \zeta_{6} + 3336) q^{95} - 12960 \zeta_{6} q^{96} + 49442 q^{97} - 45684 q^{99} +O(q^{100})$$ q + 6*z * q^2 + (9*z - 9) * q^3 + (4*z - 4) * q^4 - 6*z * q^5 - 54 * q^6 + 168 * q^8 - 81*z * q^9 + (-36*z + 36) * q^10 + (-564*z + 564) * q^11 - 36*z * q^12 + 638 * q^13 + 54 * q^15 + 1136*z * q^16 + (882*z - 882) * q^17 + (-486*z + 486) * q^18 + 556*z * q^19 + 24 * q^20 + 3384 * q^22 + 840*z * q^23 + (1512*z - 1512) * q^24 + (-3089*z + 3089) * q^25 + 3828*z * q^26 + 729 * q^27 + 4638 * q^29 + 324*z * q^30 + (4400*z - 4400) * q^31 + (1440*z - 1440) * q^32 + 5076*z * q^33 - 5292 * q^34 + 324 * q^36 + 2410*z * q^37 + (3336*z - 3336) * q^38 + (5742*z - 5742) * q^39 - 1008*z * q^40 - 6870 * q^41 + 9644 * q^43 + 2256*z * q^44 + (486*z - 486) * q^45 + (5040*z - 5040) * q^46 + 18672*z * q^47 - 10224 * q^48 + 18534 * q^50 - 7938*z * q^51 + (2552*z - 2552) * q^52 + (33750*z - 33750) * q^53 + 4374*z * q^54 - 3384 * q^55 - 5004 * q^57 + 27828*z * q^58 + (-18084*z + 18084) * q^59 + (216*z - 216) * q^60 - 39758*z * q^61 - 26400 * q^62 + 27712 * q^64 - 3828*z * q^65 + (30456*z - 30456) * q^66 + (-23068*z + 23068) * q^67 - 3528*z * q^68 - 7560 * q^69 - 4248 * q^71 - 13608*z * q^72 + (-41110*z + 41110) * q^73 + (14460*z - 14460) * q^74 + 27801*z * q^75 - 2224 * q^76 - 34452 * q^78 - 21920*z * q^79 + (-6816*z + 6816) * q^80 + (6561*z - 6561) * q^81 - 41220*z * q^82 + 82452 * q^83 + 5292 * q^85 + 57864*z * q^86 + (41742*z - 41742) * q^87 + (-94752*z + 94752) * q^88 + 94086*z * q^89 - 2916 * q^90 - 3360 * q^92 - 39600*z * q^93 + (112032*z - 112032) * q^94 + (-3336*z + 3336) * q^95 - 12960*z * q^96 + 49442 * q^97 - 45684 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{2} - 9 q^{3} - 4 q^{4} - 6 q^{5} - 108 q^{6} + 336 q^{8} - 81 q^{9}+O(q^{10})$$ 2 * q + 6 * q^2 - 9 * q^3 - 4 * q^4 - 6 * q^5 - 108 * q^6 + 336 * q^8 - 81 * q^9 $$2 q + 6 q^{2} - 9 q^{3} - 4 q^{4} - 6 q^{5} - 108 q^{6} + 336 q^{8} - 81 q^{9} + 36 q^{10} + 564 q^{11} - 36 q^{12} + 1276 q^{13} + 108 q^{15} + 1136 q^{16} - 882 q^{17} + 486 q^{18} + 556 q^{19} + 48 q^{20} + 6768 q^{22} + 840 q^{23} - 1512 q^{24} + 3089 q^{25} + 3828 q^{26} + 1458 q^{27} + 9276 q^{29} + 324 q^{30} - 4400 q^{31} - 1440 q^{32} + 5076 q^{33} - 10584 q^{34} + 648 q^{36} + 2410 q^{37} - 3336 q^{38} - 5742 q^{39} - 1008 q^{40} - 13740 q^{41} + 19288 q^{43} + 2256 q^{44} - 486 q^{45} - 5040 q^{46} + 18672 q^{47} - 20448 q^{48} + 37068 q^{50} - 7938 q^{51} - 2552 q^{52} - 33750 q^{53} + 4374 q^{54} - 6768 q^{55} - 10008 q^{57} + 27828 q^{58} + 18084 q^{59} - 216 q^{60} - 39758 q^{61} - 52800 q^{62} + 55424 q^{64} - 3828 q^{65} - 30456 q^{66} + 23068 q^{67} - 3528 q^{68} - 15120 q^{69} - 8496 q^{71} - 13608 q^{72} + 41110 q^{73} - 14460 q^{74} + 27801 q^{75} - 4448 q^{76} - 68904 q^{78} - 21920 q^{79} + 6816 q^{80} - 6561 q^{81} - 41220 q^{82} + 164904 q^{83} + 10584 q^{85} + 57864 q^{86} - 41742 q^{87} + 94752 q^{88} + 94086 q^{89} - 5832 q^{90} - 6720 q^{92} - 39600 q^{93} - 112032 q^{94} + 3336 q^{95} - 12960 q^{96} + 98884 q^{97} - 91368 q^{99}+O(q^{100})$$ 2 * q + 6 * q^2 - 9 * q^3 - 4 * q^4 - 6 * q^5 - 108 * q^6 + 336 * q^8 - 81 * q^9 + 36 * q^10 + 564 * q^11 - 36 * q^12 + 1276 * q^13 + 108 * q^15 + 1136 * q^16 - 882 * q^17 + 486 * q^18 + 556 * q^19 + 48 * q^20 + 6768 * q^22 + 840 * q^23 - 1512 * q^24 + 3089 * q^25 + 3828 * q^26 + 1458 * q^27 + 9276 * q^29 + 324 * q^30 - 4400 * q^31 - 1440 * q^32 + 5076 * q^33 - 10584 * q^34 + 648 * q^36 + 2410 * q^37 - 3336 * q^38 - 5742 * q^39 - 1008 * q^40 - 13740 * q^41 + 19288 * q^43 + 2256 * q^44 - 486 * q^45 - 5040 * q^46 + 18672 * q^47 - 20448 * q^48 + 37068 * q^50 - 7938 * q^51 - 2552 * q^52 - 33750 * q^53 + 4374 * q^54 - 6768 * q^55 - 10008 * q^57 + 27828 * q^58 + 18084 * q^59 - 216 * q^60 - 39758 * q^61 - 52800 * q^62 + 55424 * q^64 - 3828 * q^65 - 30456 * q^66 + 23068 * q^67 - 3528 * q^68 - 15120 * q^69 - 8496 * q^71 - 13608 * q^72 + 41110 * q^73 - 14460 * q^74 + 27801 * q^75 - 4448 * q^76 - 68904 * q^78 - 21920 * q^79 + 6816 * q^80 - 6561 * q^81 - 41220 * q^82 + 164904 * q^83 + 10584 * q^85 + 57864 * q^86 - 41742 * q^87 + 94752 * q^88 + 94086 * q^89 - 5832 * q^90 - 6720 * q^92 - 39600 * q^93 - 112032 * q^94 + 3336 * q^95 - 12960 * q^96 + 98884 * q^97 - 91368 * q^99

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
3.00000 + 5.19615i −4.50000 + 7.79423i −2.00000 + 3.46410i −3.00000 5.19615i −54.0000 0 168.000 −40.5000 70.1481i 18.0000 31.1769i
79.1 3.00000 5.19615i −4.50000 7.79423i −2.00000 3.46410i −3.00000 + 5.19615i −54.0000 0 168.000 −40.5000 + 70.1481i 18.0000 + 31.1769i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 147.6.e.h 2
7.b odd 2 1 147.6.e.k 2
7.c even 3 1 3.6.a.a 1
7.c even 3 1 inner 147.6.e.h 2
7.d odd 6 1 147.6.a.a 1
7.d odd 6 1 147.6.e.k 2
21.g even 6 1 441.6.a.i 1
21.h odd 6 1 9.6.a.a 1
28.g odd 6 1 48.6.a.a 1
35.j even 6 1 75.6.a.e 1
35.l odd 12 2 75.6.b.b 2
56.k odd 6 1 192.6.a.l 1
56.p even 6 1 192.6.a.d 1
63.g even 3 1 81.6.c.c 2
63.h even 3 1 81.6.c.c 2
63.j odd 6 1 81.6.c.a 2
63.n odd 6 1 81.6.c.a 2
77.h odd 6 1 363.6.a.d 1
84.n even 6 1 144.6.a.f 1
91.r even 6 1 507.6.a.b 1
105.o odd 6 1 225.6.a.a 1
105.x even 12 2 225.6.b.b 2
112.u odd 12 2 768.6.d.h 2
112.w even 12 2 768.6.d.k 2
119.j even 6 1 867.6.a.a 1
133.r odd 6 1 1083.6.a.c 1
168.s odd 6 1 576.6.a.s 1
168.v even 6 1 576.6.a.t 1
231.l even 6 1 1089.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.6.a.a 1 7.c even 3 1
9.6.a.a 1 21.h odd 6 1
48.6.a.a 1 28.g odd 6 1
75.6.a.e 1 35.j even 6 1
75.6.b.b 2 35.l odd 12 2
81.6.c.a 2 63.j odd 6 1
81.6.c.a 2 63.n odd 6 1
81.6.c.c 2 63.g even 3 1
81.6.c.c 2 63.h even 3 1
144.6.a.f 1 84.n even 6 1
147.6.a.a 1 7.d odd 6 1
147.6.e.h 2 1.a even 1 1 trivial
147.6.e.h 2 7.c even 3 1 inner
147.6.e.k 2 7.b odd 2 1
147.6.e.k 2 7.d odd 6 1
192.6.a.d 1 56.p even 6 1
192.6.a.l 1 56.k odd 6 1
225.6.a.a 1 105.o odd 6 1
225.6.b.b 2 105.x even 12 2
363.6.a.d 1 77.h odd 6 1
441.6.a.i 1 21.g even 6 1
507.6.a.b 1 91.r even 6 1
576.6.a.s 1 168.s odd 6 1
576.6.a.t 1 168.v even 6 1
768.6.d.h 2 112.u odd 12 2
768.6.d.k 2 112.w even 12 2
867.6.a.a 1 119.j even 6 1
1083.6.a.c 1 133.r odd 6 1
1089.6.a.b 1 231.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{2} - 6T_{2} + 36$$ T2^2 - 6*T2 + 36 $$T_{5}^{2} + 6T_{5} + 36$$ T5^2 + 6*T5 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 6T + 36$$
$3$ $$T^{2} + 9T + 81$$
$5$ $$T^{2} + 6T + 36$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 564T + 318096$$
$13$ $$(T - 638)^{2}$$
$17$ $$T^{2} + 882T + 777924$$
$19$ $$T^{2} - 556T + 309136$$
$23$ $$T^{2} - 840T + 705600$$
$29$ $$(T - 4638)^{2}$$
$31$ $$T^{2} + 4400 T + 19360000$$
$37$ $$T^{2} - 2410 T + 5808100$$
$41$ $$(T + 6870)^{2}$$
$43$ $$(T - 9644)^{2}$$
$47$ $$T^{2} - 18672 T + 348643584$$
$53$ $$T^{2} + 33750 T + 1139062500$$
$59$ $$T^{2} - 18084 T + 327031056$$
$61$ $$T^{2} + 39758 T + 1580698564$$
$67$ $$T^{2} - 23068 T + 532132624$$
$71$ $$(T + 4248)^{2}$$
$73$ $$T^{2} - 41110 T + 1690032100$$
$79$ $$T^{2} + 21920 T + 480486400$$
$83$ $$(T - 82452)^{2}$$
$89$ $$T^{2} - 94086 T + 8852175396$$
$97$ $$(T - 49442)^{2}$$