# Properties

 Label 147.6.a.h Level $147$ Weight $6$ Character orbit 147.a Self dual yes Analytic conductor $23.576$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(1,147)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$23.5764215125$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 48$$ x^2 - x - 48 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{193})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} - 9 q^{3} + (3 \beta + 17) q^{4} + 36 q^{5} + (9 \beta + 9) q^{6} + (9 \beta - 129) q^{8} + 81 q^{9}+O(q^{10})$$ q + (-b - 1) * q^2 - 9 * q^3 + (3*b + 17) * q^4 + 36 * q^5 + (9*b + 9) * q^6 + (9*b - 129) * q^8 + 81 * q^9 $$q + ( - \beta - 1) q^{2} - 9 q^{3} + (3 \beta + 17) q^{4} + 36 q^{5} + (9 \beta + 9) q^{6} + (9 \beta - 129) q^{8} + 81 q^{9} + ( - 36 \beta - 36) q^{10} + (8 \beta + 236) q^{11} + ( - 27 \beta - 153) q^{12} + (72 \beta + 612) q^{13} - 324 q^{15} + (15 \beta - 847) q^{16} + ( - 216 \beta + 576) q^{17} + ( - 81 \beta - 81) q^{18} + ( - 288 \beta + 36) q^{19} + (108 \beta + 612) q^{20} + ( - 252 \beta - 620) q^{22} + ( - 56 \beta - 224) q^{23} + ( - 81 \beta + 1161) q^{24} - 1829 q^{25} + ( - 756 \beta - 4068) q^{26} - 729 q^{27} + (640 \beta + 2866) q^{29} + (324 \beta + 324) q^{30} + ( - 576 \beta + 5256) q^{31} + (529 \beta + 4255) q^{32} + ( - 72 \beta - 2124) q^{33} + ( - 144 \beta + 9792) q^{34} + (243 \beta + 1377) q^{36} + ( - 1056 \beta + 6090) q^{37} + (540 \beta + 13788) q^{38} + ( - 648 \beta - 5508) q^{39} + (324 \beta - 4644) q^{40} + (216 \beta + 10368) q^{41} + ( - 2400 \beta - 1932) q^{43} + (868 \beta + 5164) q^{44} + 2916 q^{45} + (336 \beta + 2912) q^{46} + (3456 \beta + 2232) q^{47} + ( - 135 \beta + 7623) q^{48} + (1829 \beta + 1829) q^{50} + (1944 \beta - 5184) q^{51} + (3276 \beta + 20772) q^{52} + ( - 1184 \beta + 1702) q^{53} + (729 \beta + 729) q^{54} + (288 \beta + 8496) q^{55} + (2592 \beta - 324) q^{57} + ( - 4146 \beta - 33586) q^{58} + (864 \beta + 14436) q^{59} + ( - 972 \beta - 5508) q^{60} + (4680 \beta - 10980) q^{61} + ( - 4104 \beta + 22392) q^{62} + ( - 5793 \beta - 2543) q^{64} + (2592 \beta + 22032) q^{65} + (2268 \beta + 5580) q^{66} + (6480 \beta - 13580) q^{67} + ( - 2592 \beta - 21312) q^{68} + (504 \beta + 2016) q^{69} + (2552 \beta - 47416) q^{71} + (729 \beta - 10449) q^{72} + ( - 1872 \beta + 29232) q^{73} + ( - 3978 \beta + 44598) q^{74} + 16461 q^{75} + ( - 5652 \beta - 40860) q^{76} + (6804 \beta + 36612) q^{78} + ( - 6480 \beta - 24808) q^{79} + (540 \beta - 30492) q^{80} + 6561 q^{81} + ( - 10800 \beta - 20736) q^{82} + (9504 \beta - 40428) q^{83} + ( - 7776 \beta + 20736) q^{85} + (6732 \beta + 117132) q^{86} + ( - 5760 \beta - 25794) q^{87} + (1164 \beta - 26988) q^{88} + ( - 3672 \beta + 63432) q^{89} + ( - 2916 \beta - 2916) q^{90} + ( - 1792 \beta - 11872) q^{92} + (5184 \beta - 47304) q^{93} + ( - 9144 \beta - 168120) q^{94} + ( - 10368 \beta + 1296) q^{95} + ( - 4761 \beta - 38295) q^{96} + (19584 \beta + 8136) q^{97} + (648 \beta + 19116) q^{99}+O(q^{100})$$ q + (-b - 1) * q^2 - 9 * q^3 + (3*b + 17) * q^4 + 36 * q^5 + (9*b + 9) * q^6 + (9*b - 129) * q^8 + 81 * q^9 + (-36*b - 36) * q^10 + (8*b + 236) * q^11 + (-27*b - 153) * q^12 + (72*b + 612) * q^13 - 324 * q^15 + (15*b - 847) * q^16 + (-216*b + 576) * q^17 + (-81*b - 81) * q^18 + (-288*b + 36) * q^19 + (108*b + 612) * q^20 + (-252*b - 620) * q^22 + (-56*b - 224) * q^23 + (-81*b + 1161) * q^24 - 1829 * q^25 + (-756*b - 4068) * q^26 - 729 * q^27 + (640*b + 2866) * q^29 + (324*b + 324) * q^30 + (-576*b + 5256) * q^31 + (529*b + 4255) * q^32 + (-72*b - 2124) * q^33 + (-144*b + 9792) * q^34 + (243*b + 1377) * q^36 + (-1056*b + 6090) * q^37 + (540*b + 13788) * q^38 + (-648*b - 5508) * q^39 + (324*b - 4644) * q^40 + (216*b + 10368) * q^41 + (-2400*b - 1932) * q^43 + (868*b + 5164) * q^44 + 2916 * q^45 + (336*b + 2912) * q^46 + (3456*b + 2232) * q^47 + (-135*b + 7623) * q^48 + (1829*b + 1829) * q^50 + (1944*b - 5184) * q^51 + (3276*b + 20772) * q^52 + (-1184*b + 1702) * q^53 + (729*b + 729) * q^54 + (288*b + 8496) * q^55 + (2592*b - 324) * q^57 + (-4146*b - 33586) * q^58 + (864*b + 14436) * q^59 + (-972*b - 5508) * q^60 + (4680*b - 10980) * q^61 + (-4104*b + 22392) * q^62 + (-5793*b - 2543) * q^64 + (2592*b + 22032) * q^65 + (2268*b + 5580) * q^66 + (6480*b - 13580) * q^67 + (-2592*b - 21312) * q^68 + (504*b + 2016) * q^69 + (2552*b - 47416) * q^71 + (729*b - 10449) * q^72 + (-1872*b + 29232) * q^73 + (-3978*b + 44598) * q^74 + 16461 * q^75 + (-5652*b - 40860) * q^76 + (6804*b + 36612) * q^78 + (-6480*b - 24808) * q^79 + (540*b - 30492) * q^80 + 6561 * q^81 + (-10800*b - 20736) * q^82 + (9504*b - 40428) * q^83 + (-7776*b + 20736) * q^85 + (6732*b + 117132) * q^86 + (-5760*b - 25794) * q^87 + (1164*b - 26988) * q^88 + (-3672*b + 63432) * q^89 + (-2916*b - 2916) * q^90 + (-1792*b - 11872) * q^92 + (5184*b - 47304) * q^93 + (-9144*b - 168120) * q^94 + (-10368*b + 1296) * q^95 + (-4761*b - 38295) * q^96 + (19584*b + 8136) * q^97 + (648*b + 19116) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 18 q^{3} + 37 q^{4} + 72 q^{5} + 27 q^{6} - 249 q^{8} + 162 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 18 * q^3 + 37 * q^4 + 72 * q^5 + 27 * q^6 - 249 * q^8 + 162 * q^9 $$2 q - 3 q^{2} - 18 q^{3} + 37 q^{4} + 72 q^{5} + 27 q^{6} - 249 q^{8} + 162 q^{9} - 108 q^{10} + 480 q^{11} - 333 q^{12} + 1296 q^{13} - 648 q^{15} - 1679 q^{16} + 936 q^{17} - 243 q^{18} - 216 q^{19} + 1332 q^{20} - 1492 q^{22} - 504 q^{23} + 2241 q^{24} - 3658 q^{25} - 8892 q^{26} - 1458 q^{27} + 6372 q^{29} + 972 q^{30} + 9936 q^{31} + 9039 q^{32} - 4320 q^{33} + 19440 q^{34} + 2997 q^{36} + 11124 q^{37} + 28116 q^{38} - 11664 q^{39} - 8964 q^{40} + 20952 q^{41} - 6264 q^{43} + 11196 q^{44} + 5832 q^{45} + 6160 q^{46} + 7920 q^{47} + 15111 q^{48} + 5487 q^{50} - 8424 q^{51} + 44820 q^{52} + 2220 q^{53} + 2187 q^{54} + 17280 q^{55} + 1944 q^{57} - 71318 q^{58} + 29736 q^{59} - 11988 q^{60} - 17280 q^{61} + 40680 q^{62} - 10879 q^{64} + 46656 q^{65} + 13428 q^{66} - 20680 q^{67} - 45216 q^{68} + 4536 q^{69} - 92280 q^{71} - 20169 q^{72} + 56592 q^{73} + 85218 q^{74} + 32922 q^{75} - 87372 q^{76} + 80028 q^{78} - 56096 q^{79} - 60444 q^{80} + 13122 q^{81} - 52272 q^{82} - 71352 q^{83} + 33696 q^{85} + 240996 q^{86} - 57348 q^{87} - 52812 q^{88} + 123192 q^{89} - 8748 q^{90} - 25536 q^{92} - 89424 q^{93} - 345384 q^{94} - 7776 q^{95} - 81351 q^{96} + 35856 q^{97} + 38880 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 - 18 * q^3 + 37 * q^4 + 72 * q^5 + 27 * q^6 - 249 * q^8 + 162 * q^9 - 108 * q^10 + 480 * q^11 - 333 * q^12 + 1296 * q^13 - 648 * q^15 - 1679 * q^16 + 936 * q^17 - 243 * q^18 - 216 * q^19 + 1332 * q^20 - 1492 * q^22 - 504 * q^23 + 2241 * q^24 - 3658 * q^25 - 8892 * q^26 - 1458 * q^27 + 6372 * q^29 + 972 * q^30 + 9936 * q^31 + 9039 * q^32 - 4320 * q^33 + 19440 * q^34 + 2997 * q^36 + 11124 * q^37 + 28116 * q^38 - 11664 * q^39 - 8964 * q^40 + 20952 * q^41 - 6264 * q^43 + 11196 * q^44 + 5832 * q^45 + 6160 * q^46 + 7920 * q^47 + 15111 * q^48 + 5487 * q^50 - 8424 * q^51 + 44820 * q^52 + 2220 * q^53 + 2187 * q^54 + 17280 * q^55 + 1944 * q^57 - 71318 * q^58 + 29736 * q^59 - 11988 * q^60 - 17280 * q^61 + 40680 * q^62 - 10879 * q^64 + 46656 * q^65 + 13428 * q^66 - 20680 * q^67 - 45216 * q^68 + 4536 * q^69 - 92280 * q^71 - 20169 * q^72 + 56592 * q^73 + 85218 * q^74 + 32922 * q^75 - 87372 * q^76 + 80028 * q^78 - 56096 * q^79 - 60444 * q^80 + 13122 * q^81 - 52272 * q^82 - 71352 * q^83 + 33696 * q^85 + 240996 * q^86 - 57348 * q^87 - 52812 * q^88 + 123192 * q^89 - 8748 * q^90 - 25536 * q^92 - 89424 * q^93 - 345384 * q^94 - 7776 * q^95 - 81351 * q^96 + 35856 * q^97 + 38880 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 7.44622 −6.44622
−8.44622 −9.00000 39.3387 36.0000 76.0160 0 −61.9840 81.0000 −304.064
1.2 5.44622 −9.00000 −2.33867 36.0000 −49.0160 0 −187.016 81.0000 196.064
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 147.6.a.h 2
3.b odd 2 1 441.6.a.q 2
7.b odd 2 1 147.6.a.j yes 2
7.c even 3 2 147.6.e.n 4
7.d odd 6 2 147.6.e.m 4
21.c even 2 1 441.6.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.h 2 1.a even 1 1 trivial
147.6.a.j yes 2 7.b odd 2 1
147.6.e.m 4 7.d odd 6 2
147.6.e.n 4 7.c even 3 2
441.6.a.q 2 3.b odd 2 1
441.6.a.r 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2}^{2} + 3T_{2} - 46$$ T2^2 + 3*T2 - 46 $$T_{5} - 36$$ T5 - 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T - 46$$
$3$ $$(T + 9)^{2}$$
$5$ $$(T - 36)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 480T + 54512$$
$13$ $$T^{2} - 1296 T + 169776$$
$17$ $$T^{2} - 936 T - 2032128$$
$19$ $$T^{2} + 216 T - 3990384$$
$23$ $$T^{2} + 504T - 87808$$
$29$ $$T^{2} - 6372 T - 9612604$$
$31$ $$T^{2} - 9936 T + 8672832$$
$37$ $$T^{2} - 11124 T - 22869468$$
$41$ $$T^{2} - 20952 T + 107495424$$
$43$ $$T^{2} + 6264 T - 268110576$$
$47$ $$T^{2} - 7920 T - 560613312$$
$53$ $$T^{2} - 2220 T - 66407452$$
$59$ $$T^{2} - 29736 T + 185038992$$
$61$ $$T^{2} + 17280 T - 982141200$$
$67$ $$T^{2} + 20680 T - 1919121200$$
$71$ $$T^{2} + 92280 T + 1814661632$$
$73$ $$T^{2} - 56592 T + 631577088$$
$79$ $$T^{2} + 56096 T - 1239346496$$
$83$ $$T^{2} + 71352 T - 3085453296$$
$89$ $$T^{2} - 123192 T + 3143484288$$
$97$ $$T^{2} - 35856 T - 18184056768$$