# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(67,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("147.67");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

## 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: no Analytic conductor: $$23.5764215125$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + ( - 28 \zeta_{6} + 28) q^{4} + 11 \zeta_{6} q^{5} - 18 q^{6} + 120 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q + 2*z * q^2 + (9*z - 9) * q^3 + (-28*z + 28) * q^4 + 11*z * q^5 - 18 * q^6 + 120 * q^8 - 81*z * q^9 $$q + 2 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + ( - 28 \zeta_{6} + 28) q^{4} + 11 \zeta_{6} q^{5} - 18 q^{6} + 120 q^{8} - 81 \zeta_{6} q^{9} + (22 \zeta_{6} - 22) q^{10} + (269 \zeta_{6} - 269) q^{11} + 252 \zeta_{6} q^{12} + 308 q^{13} - 99 q^{15} - 656 \zeta_{6} q^{16} + ( - 1896 \zeta_{6} + 1896) q^{17} + ( - 162 \zeta_{6} + 162) q^{18} - 164 \zeta_{6} q^{19} + 308 q^{20} - 538 q^{22} + 3264 \zeta_{6} q^{23} + (1080 \zeta_{6} - 1080) q^{24} + ( - 3004 \zeta_{6} + 3004) q^{25} + 616 \zeta_{6} q^{26} + 729 q^{27} + 2417 q^{29} - 198 \zeta_{6} q^{30} + ( - 2841 \zeta_{6} + 2841) q^{31} + ( - 5152 \zeta_{6} + 5152) q^{32} - 2421 \zeta_{6} q^{33} + 3792 q^{34} - 2268 q^{36} + 11328 \zeta_{6} q^{37} + ( - 328 \zeta_{6} + 328) q^{38} + (2772 \zeta_{6} - 2772) q^{39} + 1320 \zeta_{6} q^{40} + 16856 q^{41} - 7894 q^{43} + 7532 \zeta_{6} q^{44} + ( - 891 \zeta_{6} + 891) q^{45} + (6528 \zeta_{6} - 6528) q^{46} + 21102 \zeta_{6} q^{47} + 5904 q^{48} + 6008 q^{50} + 17064 \zeta_{6} q^{51} + ( - 8624 \zeta_{6} + 8624) q^{52} + ( - 29691 \zeta_{6} + 29691) q^{53} + 1458 \zeta_{6} q^{54} - 2959 q^{55} + 1476 q^{57} + 4834 \zeta_{6} q^{58} + (8163 \zeta_{6} - 8163) q^{59} + (2772 \zeta_{6} - 2772) q^{60} + 15166 \zeta_{6} q^{61} + 5682 q^{62} - 10688 q^{64} + 3388 \zeta_{6} q^{65} + ( - 4842 \zeta_{6} + 4842) q^{66} + ( - 32078 \zeta_{6} + 32078) q^{67} - 53088 \zeta_{6} q^{68} - 29376 q^{69} - 38274 q^{71} - 9720 \zeta_{6} q^{72} + ( - 34866 \zeta_{6} + 34866) q^{73} + (22656 \zeta_{6} - 22656) q^{74} + 27036 \zeta_{6} q^{75} - 4592 q^{76} - 5544 q^{78} - 13529 \zeta_{6} q^{79} + ( - 7216 \zeta_{6} + 7216) q^{80} + (6561 \zeta_{6} - 6561) q^{81} + 33712 \zeta_{6} q^{82} + 68103 q^{83} + 20856 q^{85} - 15788 \zeta_{6} q^{86} + (21753 \zeta_{6} - 21753) q^{87} + (32280 \zeta_{6} - 32280) q^{88} - 114922 \zeta_{6} q^{89} + 1782 q^{90} + 91392 q^{92} + 25569 \zeta_{6} q^{93} + (42204 \zeta_{6} - 42204) q^{94} + ( - 1804 \zeta_{6} + 1804) q^{95} + 46368 \zeta_{6} q^{96} - 154959 q^{97} + 21789 q^{99} +O(q^{100})$$ q + 2*z * q^2 + (9*z - 9) * q^3 + (-28*z + 28) * q^4 + 11*z * q^5 - 18 * q^6 + 120 * q^8 - 81*z * q^9 + (22*z - 22) * q^10 + (269*z - 269) * q^11 + 252*z * q^12 + 308 * q^13 - 99 * q^15 - 656*z * q^16 + (-1896*z + 1896) * q^17 + (-162*z + 162) * q^18 - 164*z * q^19 + 308 * q^20 - 538 * q^22 + 3264*z * q^23 + (1080*z - 1080) * q^24 + (-3004*z + 3004) * q^25 + 616*z * q^26 + 729 * q^27 + 2417 * q^29 - 198*z * q^30 + (-2841*z + 2841) * q^31 + (-5152*z + 5152) * q^32 - 2421*z * q^33 + 3792 * q^34 - 2268 * q^36 + 11328*z * q^37 + (-328*z + 328) * q^38 + (2772*z - 2772) * q^39 + 1320*z * q^40 + 16856 * q^41 - 7894 * q^43 + 7532*z * q^44 + (-891*z + 891) * q^45 + (6528*z - 6528) * q^46 + 21102*z * q^47 + 5904 * q^48 + 6008 * q^50 + 17064*z * q^51 + (-8624*z + 8624) * q^52 + (-29691*z + 29691) * q^53 + 1458*z * q^54 - 2959 * q^55 + 1476 * q^57 + 4834*z * q^58 + (8163*z - 8163) * q^59 + (2772*z - 2772) * q^60 + 15166*z * q^61 + 5682 * q^62 - 10688 * q^64 + 3388*z * q^65 + (-4842*z + 4842) * q^66 + (-32078*z + 32078) * q^67 - 53088*z * q^68 - 29376 * q^69 - 38274 * q^71 - 9720*z * q^72 + (-34866*z + 34866) * q^73 + (22656*z - 22656) * q^74 + 27036*z * q^75 - 4592 * q^76 - 5544 * q^78 - 13529*z * q^79 + (-7216*z + 7216) * q^80 + (6561*z - 6561) * q^81 + 33712*z * q^82 + 68103 * q^83 + 20856 * q^85 - 15788*z * q^86 + (21753*z - 21753) * q^87 + (32280*z - 32280) * q^88 - 114922*z * q^89 + 1782 * q^90 + 91392 * q^92 + 25569*z * q^93 + (42204*z - 42204) * q^94 + (-1804*z + 1804) * q^95 + 46368*z * q^96 - 154959 * q^97 + 21789 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 9 q^{3} + 28 q^{4} + 11 q^{5} - 36 q^{6} + 240 q^{8} - 81 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 9 * q^3 + 28 * q^4 + 11 * q^5 - 36 * q^6 + 240 * q^8 - 81 * q^9 $$2 q + 2 q^{2} - 9 q^{3} + 28 q^{4} + 11 q^{5} - 36 q^{6} + 240 q^{8} - 81 q^{9} - 22 q^{10} - 269 q^{11} + 252 q^{12} + 616 q^{13} - 198 q^{15} - 656 q^{16} + 1896 q^{17} + 162 q^{18} - 164 q^{19} + 616 q^{20} - 1076 q^{22} + 3264 q^{23} - 1080 q^{24} + 3004 q^{25} + 616 q^{26} + 1458 q^{27} + 4834 q^{29} - 198 q^{30} + 2841 q^{31} + 5152 q^{32} - 2421 q^{33} + 7584 q^{34} - 4536 q^{36} + 11328 q^{37} + 328 q^{38} - 2772 q^{39} + 1320 q^{40} + 33712 q^{41} - 15788 q^{43} + 7532 q^{44} + 891 q^{45} - 6528 q^{46} + 21102 q^{47} + 11808 q^{48} + 12016 q^{50} + 17064 q^{51} + 8624 q^{52} + 29691 q^{53} + 1458 q^{54} - 5918 q^{55} + 2952 q^{57} + 4834 q^{58} - 8163 q^{59} - 2772 q^{60} + 15166 q^{61} + 11364 q^{62} - 21376 q^{64} + 3388 q^{65} + 4842 q^{66} + 32078 q^{67} - 53088 q^{68} - 58752 q^{69} - 76548 q^{71} - 9720 q^{72} + 34866 q^{73} - 22656 q^{74} + 27036 q^{75} - 9184 q^{76} - 11088 q^{78} - 13529 q^{79} + 7216 q^{80} - 6561 q^{81} + 33712 q^{82} + 136206 q^{83} + 41712 q^{85} - 15788 q^{86} - 21753 q^{87} - 32280 q^{88} - 114922 q^{89} + 3564 q^{90} + 182784 q^{92} + 25569 q^{93} - 42204 q^{94} + 1804 q^{95} + 46368 q^{96} - 309918 q^{97} + 43578 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 9 * q^3 + 28 * q^4 + 11 * q^5 - 36 * q^6 + 240 * q^8 - 81 * q^9 - 22 * q^10 - 269 * q^11 + 252 * q^12 + 616 * q^13 - 198 * q^15 - 656 * q^16 + 1896 * q^17 + 162 * q^18 - 164 * q^19 + 616 * q^20 - 1076 * q^22 + 3264 * q^23 - 1080 * q^24 + 3004 * q^25 + 616 * q^26 + 1458 * q^27 + 4834 * q^29 - 198 * q^30 + 2841 * q^31 + 5152 * q^32 - 2421 * q^33 + 7584 * q^34 - 4536 * q^36 + 11328 * q^37 + 328 * q^38 - 2772 * q^39 + 1320 * q^40 + 33712 * q^41 - 15788 * q^43 + 7532 * q^44 + 891 * q^45 - 6528 * q^46 + 21102 * q^47 + 11808 * q^48 + 12016 * q^50 + 17064 * q^51 + 8624 * q^52 + 29691 * q^53 + 1458 * q^54 - 5918 * q^55 + 2952 * q^57 + 4834 * q^58 - 8163 * q^59 - 2772 * q^60 + 15166 * q^61 + 11364 * q^62 - 21376 * q^64 + 3388 * q^65 + 4842 * q^66 + 32078 * q^67 - 53088 * q^68 - 58752 * q^69 - 76548 * q^71 - 9720 * q^72 + 34866 * q^73 - 22656 * q^74 + 27036 * q^75 - 9184 * q^76 - 11088 * q^78 - 13529 * q^79 + 7216 * q^80 - 6561 * q^81 + 33712 * q^82 + 136206 * q^83 + 41712 * q^85 - 15788 * q^86 - 21753 * q^87 - 32280 * q^88 - 114922 * q^89 + 3564 * q^90 + 182784 * q^92 + 25569 * q^93 - 42204 * q^94 + 1804 * q^95 + 46368 * q^96 - 309918 * q^97 + 43578 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 + 1.73205i −4.50000 + 7.79423i 14.0000 24.2487i 5.50000 + 9.52628i −18.0000 0 120.000 −40.5000 70.1481i −11.0000 + 19.0526i
79.1 1.00000 1.73205i −4.50000 7.79423i 14.0000 + 24.2487i 5.50000 9.52628i −18.0000 0 120.000 −40.5000 + 70.1481i −11.0000 19.0526i
 $$n$$: 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.g 2
7.b odd 2 1 21.6.e.a 2
7.c even 3 1 147.6.a.d 1
7.c even 3 1 inner 147.6.e.g 2
7.d odd 6 1 21.6.e.a 2
7.d odd 6 1 147.6.a.c 1
21.c even 2 1 63.6.e.a 2
21.g even 6 1 63.6.e.a 2
21.g even 6 1 441.6.a.g 1
21.h odd 6 1 441.6.a.h 1
28.d even 2 1 336.6.q.b 2
28.f even 6 1 336.6.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.a 2 7.b odd 2 1
21.6.e.a 2 7.d odd 6 1
63.6.e.a 2 21.c even 2 1
63.6.e.a 2 21.g even 6 1
147.6.a.c 1 7.d odd 6 1
147.6.a.d 1 7.c even 3 1
147.6.e.g 2 1.a even 1 1 trivial
147.6.e.g 2 7.c even 3 1 inner
336.6.q.b 2 28.d even 2 1
336.6.q.b 2 28.f even 6 1
441.6.a.g 1 21.g even 6 1
441.6.a.h 1 21.h odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 2T_{2} + 4$$ T2^2 - 2*T2 + 4 $$T_{5}^{2} - 11T_{5} + 121$$ T5^2 - 11*T5 + 121

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} + 9T + 81$$
$5$ $$T^{2} - 11T + 121$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 269T + 72361$$
$13$ $$(T - 308)^{2}$$
$17$ $$T^{2} - 1896 T + 3594816$$
$19$ $$T^{2} + 164T + 26896$$
$23$ $$T^{2} - 3264 T + 10653696$$
$29$ $$(T - 2417)^{2}$$
$31$ $$T^{2} - 2841 T + 8071281$$
$37$ $$T^{2} - 11328 T + 128323584$$
$41$ $$(T - 16856)^{2}$$
$43$ $$(T + 7894)^{2}$$
$47$ $$T^{2} - 21102 T + 445294404$$
$53$ $$T^{2} - 29691 T + 881555481$$
$59$ $$T^{2} + 8163 T + 66634569$$
$61$ $$T^{2} - 15166 T + 230007556$$
$67$ $$T^{2} - 32078 T + 1028998084$$
$71$ $$(T + 38274)^{2}$$
$73$ $$T^{2} - 34866 T + 1215637956$$
$79$ $$T^{2} + 13529 T + 183033841$$
$83$ $$(T - 68103)^{2}$$
$89$ $$T^{2} + 114922 T + 13207066084$$
$97$ $$(T + 154959)^{2}$$