# Properties

 Label 147.6.a.j Level $147$ Weight $6$ Character orbit 147.a Self dual yes Analytic conductor $23.576$ Analytic rank $1$ 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: $$1$$ 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.j yes 2
3.b odd 2 1 441.6.a.r 2
7.b odd 2 1 147.6.a.h 2
7.c even 3 2 147.6.e.m 4
7.d odd 6 2 147.6.e.n 4
21.c even 2 1 441.6.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.h 2 7.b odd 2 1
147.6.a.j yes 2 1.a even 1 1 trivial
147.6.e.m 4 7.c even 3 2
147.6.e.n 4 7.d odd 6 2
441.6.a.q 2 21.c even 2 1
441.6.a.r 2 3.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 $$S_{6}^{\mathrm{new}}(\Gamma_0(147))$$:

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

## Hecke characteristic polynomials

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