# 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 $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,4,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, 4, names="a")

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

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$8.67328077084$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ 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

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 $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} - 3 q^{3} + (3 \beta + 7) q^{4} + ( - 2 \beta - 2) q^{5} + (3 \beta + 3) q^{6} + ( - 5 \beta - 41) q^{8} + 9 q^{9}+O(q^{10})$$ q + (-b - 1) * q^2 - 3 * q^3 + (3*b + 7) * q^4 + (-2*b - 2) * q^5 + (3*b + 3) * q^6 + (-5*b - 41) * q^8 + 9 * q^9 $$q + ( - \beta - 1) q^{2} - 3 q^{3} + (3 \beta + 7) q^{4} + ( - 2 \beta - 2) q^{5} + (3 \beta + 3) q^{6} + ( - 5 \beta - 41) q^{8} + 9 q^{9} + (6 \beta + 30) q^{10} + (10 \beta - 8) q^{11} + ( - 9 \beta - 21) q^{12} + (12 \beta - 14) q^{13} + (6 \beta + 6) q^{15} + (27 \beta + 55) q^{16} + (2 \beta + 2) q^{17} + ( - 9 \beta - 9) q^{18} + (24 \beta - 44) q^{19} + ( - 26 \beta - 98) q^{20} + ( - 12 \beta - 132) q^{22} + ( - 34 \beta + 20) q^{23} + (15 \beta + 123) q^{24} + (12 \beta - 65) q^{25} + ( - 10 \beta - 154) q^{26} - 27 q^{27} + (24 \beta - 138) q^{29} + ( - 18 \beta - 90) q^{30} + ( - 72 \beta + 16) q^{31} + ( - 69 \beta - 105) q^{32} + ( - 30 \beta + 24) q^{33} + ( - 6 \beta - 30) q^{34} + (27 \beta + 63) q^{36} + ( - 36 \beta - 106) q^{37} + ( - 4 \beta - 292) q^{38} + ( - 36 \beta + 42) q^{39} + (102 \beta + 222) q^{40} + (30 \beta + 210) q^{41} + ( - 48 \beta + 212) q^{43} + (76 \beta + 364) q^{44} + ( - 18 \beta - 18) q^{45} + (48 \beta + 456) q^{46} + ( - 68 \beta + 40) q^{47} + ( - 81 \beta - 165) q^{48} + (41 \beta - 103) q^{50} + ( - 6 \beta - 6) q^{51} + (78 \beta + 406) q^{52} + (4 \beta - 554) q^{53} + (27 \beta + 27) q^{54} + ( - 24 \beta - 264) q^{55} + ( - 72 \beta + 132) q^{57} + (90 \beta - 198) q^{58} + (116 \beta - 460) q^{59} + (78 \beta + 294) q^{60} + ( - 72 \beta + 250) q^{61} + (128 \beta + 992) q^{62} + (27 \beta + 631) q^{64} + ( - 20 \beta - 308) q^{65} + (36 \beta + 396) q^{66} + (108 \beta + 20) q^{67} + (26 \beta + 98) q^{68} + (102 \beta - 60) q^{69} + ( - 30 \beta + 492) q^{71} + ( - 45 \beta - 369) q^{72} + ( - 12 \beta - 530) q^{73} + (178 \beta + 610) q^{74} + ( - 36 \beta + 195) q^{75} + (108 \beta + 700) q^{76} + (30 \beta + 462) q^{78} + ( - 108 \beta - 232) q^{79} + ( - 218 \beta - 866) q^{80} + 81 q^{81} + ( - 270 \beta - 630) q^{82} + ( - 96 \beta - 924) q^{83} + ( - 12 \beta - 60) q^{85} + ( - 116 \beta + 460) q^{86} + ( - 72 \beta + 414) q^{87} + ( - 420 \beta - 372) q^{88} + (142 \beta - 254) q^{89} + (54 \beta + 270) q^{90} + ( - 280 \beta - 1288) q^{92} + (216 \beta - 48) q^{93} + (96 \beta + 912) q^{94} + ( - 8 \beta - 584) q^{95} + (207 \beta + 315) q^{96} + ( - 276 \beta - 266) q^{97} + (90 \beta - 72) q^{99}+O(q^{100})$$ q + (-b - 1) * q^2 - 3 * q^3 + (3*b + 7) * q^4 + (-2*b - 2) * q^5 + (3*b + 3) * q^6 + (-5*b - 41) * q^8 + 9 * q^9 + (6*b + 30) * q^10 + (10*b - 8) * q^11 + (-9*b - 21) * q^12 + (12*b - 14) * q^13 + (6*b + 6) * q^15 + (27*b + 55) * q^16 + (2*b + 2) * q^17 + (-9*b - 9) * q^18 + (24*b - 44) * q^19 + (-26*b - 98) * q^20 + (-12*b - 132) * q^22 + (-34*b + 20) * q^23 + (15*b + 123) * q^24 + (12*b - 65) * q^25 + (-10*b - 154) * q^26 - 27 * q^27 + (24*b - 138) * q^29 + (-18*b - 90) * q^30 + (-72*b + 16) * q^31 + (-69*b - 105) * q^32 + (-30*b + 24) * q^33 + (-6*b - 30) * q^34 + (27*b + 63) * q^36 + (-36*b - 106) * q^37 + (-4*b - 292) * q^38 + (-36*b + 42) * q^39 + (102*b + 222) * q^40 + (30*b + 210) * q^41 + (-48*b + 212) * q^43 + (76*b + 364) * q^44 + (-18*b - 18) * q^45 + (48*b + 456) * q^46 + (-68*b + 40) * q^47 + (-81*b - 165) * q^48 + (41*b - 103) * q^50 + (-6*b - 6) * q^51 + (78*b + 406) * q^52 + (4*b - 554) * q^53 + (27*b + 27) * q^54 + (-24*b - 264) * q^55 + (-72*b + 132) * q^57 + (90*b - 198) * q^58 + (116*b - 460) * q^59 + (78*b + 294) * q^60 + (-72*b + 250) * q^61 + (128*b + 992) * q^62 + (27*b + 631) * q^64 + (-20*b - 308) * q^65 + (36*b + 396) * q^66 + (108*b + 20) * q^67 + (26*b + 98) * q^68 + (102*b - 60) * q^69 + (-30*b + 492) * q^71 + (-45*b - 369) * q^72 + (-12*b - 530) * q^73 + (178*b + 610) * q^74 + (-36*b + 195) * q^75 + (108*b + 700) * q^76 + (30*b + 462) * q^78 + (-108*b - 232) * q^79 + (-218*b - 866) * q^80 + 81 * q^81 + (-270*b - 630) * q^82 + (-96*b - 924) * q^83 + (-12*b - 60) * q^85 + (-116*b + 460) * q^86 + (-72*b + 414) * q^87 + (-420*b - 372) * q^88 + (142*b - 254) * q^89 + (54*b + 270) * q^90 + (-280*b - 1288) * q^92 + (216*b - 48) * q^93 + (96*b + 912) * q^94 + (-8*b - 584) * q^95 + (207*b + 315) * q^96 + (-276*b - 266) * q^97 + (90*b - 72) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 6 q^{3} + 17 q^{4} - 6 q^{5} + 9 q^{6} - 87 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 6 * q^3 + 17 * q^4 - 6 * q^5 + 9 * q^6 - 87 * q^8 + 18 * q^9 $$2 q - 3 q^{2} - 6 q^{3} + 17 q^{4} - 6 q^{5} + 9 q^{6} - 87 q^{8} + 18 q^{9} + 66 q^{10} - 6 q^{11} - 51 q^{12} - 16 q^{13} + 18 q^{15} + 137 q^{16} + 6 q^{17} - 27 q^{18} - 64 q^{19} - 222 q^{20} - 276 q^{22} + 6 q^{23} + 261 q^{24} - 118 q^{25} - 318 q^{26} - 54 q^{27} - 252 q^{29} - 198 q^{30} - 40 q^{31} - 279 q^{32} + 18 q^{33} - 66 q^{34} + 153 q^{36} - 248 q^{37} - 588 q^{38} + 48 q^{39} + 546 q^{40} + 450 q^{41} + 376 q^{43} + 804 q^{44} - 54 q^{45} + 960 q^{46} + 12 q^{47} - 411 q^{48} - 165 q^{50} - 18 q^{51} + 890 q^{52} - 1104 q^{53} + 81 q^{54} - 552 q^{55} + 192 q^{57} - 306 q^{58} - 804 q^{59} + 666 q^{60} + 428 q^{61} + 2112 q^{62} + 1289 q^{64} - 636 q^{65} + 828 q^{66} + 148 q^{67} + 222 q^{68} - 18 q^{69} + 954 q^{71} - 783 q^{72} - 1072 q^{73} + 1398 q^{74} + 354 q^{75} + 1508 q^{76} + 954 q^{78} - 572 q^{79} - 1950 q^{80} + 162 q^{81} - 1530 q^{82} - 1944 q^{83} - 132 q^{85} + 804 q^{86} + 756 q^{87} - 1164 q^{88} - 366 q^{89} + 594 q^{90} - 2856 q^{92} + 120 q^{93} + 1920 q^{94} - 1176 q^{95} + 837 q^{96} - 808 q^{97} - 54 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 - 6 * q^3 + 17 * q^4 - 6 * q^5 + 9 * q^6 - 87 * q^8 + 18 * q^9 + 66 * q^10 - 6 * q^11 - 51 * q^12 - 16 * q^13 + 18 * q^15 + 137 * q^16 + 6 * q^17 - 27 * q^18 - 64 * q^19 - 222 * q^20 - 276 * q^22 + 6 * q^23 + 261 * q^24 - 118 * q^25 - 318 * q^26 - 54 * q^27 - 252 * q^29 - 198 * q^30 - 40 * q^31 - 279 * q^32 + 18 * q^33 - 66 * q^34 + 153 * q^36 - 248 * q^37 - 588 * q^38 + 48 * q^39 + 546 * q^40 + 450 * q^41 + 376 * q^43 + 804 * q^44 - 54 * q^45 + 960 * q^46 + 12 * q^47 - 411 * q^48 - 165 * q^50 - 18 * q^51 + 890 * q^52 - 1104 * q^53 + 81 * q^54 - 552 * q^55 + 192 * q^57 - 306 * q^58 - 804 * q^59 + 666 * q^60 + 428 * q^61 + 2112 * q^62 + 1289 * q^64 - 636 * q^65 + 828 * q^66 + 148 * q^67 + 222 * q^68 - 18 * q^69 + 954 * q^71 - 783 * q^72 - 1072 * q^73 + 1398 * q^74 + 354 * q^75 + 1508 * q^76 + 954 * q^78 - 572 * q^79 - 1950 * q^80 + 162 * q^81 - 1530 * q^82 - 1944 * q^83 - 132 * q^85 + 804 * q^86 + 756 * q^87 - 1164 * q^88 - 366 * q^89 + 594 * q^90 - 2856 * q^92 + 120 * q^93 + 1920 * q^94 - 1176 * q^95 + 837 * q^96 - 808 * q^97 - 54 * 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
 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: 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 147.4.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2}^{2} + 3T_{2} - 12$$ T2^2 + 3*T2 - 12 $$T_{5}^{2} + 6T_{5} - 48$$ T5^2 + 6*T5 - 48

## Hecke characteristic polynomials

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