# Properties

 Label 147.6.e.d Level $147$ Weight $6$ Character orbit 147.e Analytic conductor $23.576$ Analytic rank $0$ Dimension $2$ 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.

 $$f(q)$$ $$=$$ $$q - 5 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} + 94 \zeta_{6} q^{5} - 45 q^{6} - 195 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q - 5*z * q^2 + (-9*z + 9) * q^3 + (-7*z + 7) * q^4 + 94*z * q^5 - 45 * q^6 - 195 * q^8 - 81*z * q^9 $$q - 5 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} + 94 \zeta_{6} q^{5} - 45 q^{6} - 195 q^{8} - 81 \zeta_{6} q^{9} + ( - 470 \zeta_{6} + 470) q^{10} + (52 \zeta_{6} - 52) q^{11} - 63 \zeta_{6} q^{12} + 770 q^{13} + 846 q^{15} + 751 \zeta_{6} q^{16} + (2022 \zeta_{6} - 2022) q^{17} + (405 \zeta_{6} - 405) q^{18} + 1732 \zeta_{6} q^{19} + 658 q^{20} + 260 q^{22} + 576 \zeta_{6} q^{23} + (1755 \zeta_{6} - 1755) q^{24} + (5711 \zeta_{6} - 5711) q^{25} - 3850 \zeta_{6} q^{26} - 729 q^{27} + 5518 q^{29} - 4230 \zeta_{6} q^{30} + ( - 6336 \zeta_{6} + 6336) q^{31} + (2485 \zeta_{6} - 2485) q^{32} + 468 \zeta_{6} q^{33} + 10110 q^{34} - 567 q^{36} + 7338 \zeta_{6} q^{37} + ( - 8660 \zeta_{6} + 8660) q^{38} + ( - 6930 \zeta_{6} + 6930) q^{39} - 18330 \zeta_{6} q^{40} + 3262 q^{41} + 5420 q^{43} + 364 \zeta_{6} q^{44} + ( - 7614 \zeta_{6} + 7614) q^{45} + ( - 2880 \zeta_{6} + 2880) q^{46} + 864 \zeta_{6} q^{47} + 6759 q^{48} + 28555 q^{50} + 18198 \zeta_{6} q^{51} + ( - 5390 \zeta_{6} + 5390) q^{52} + (4182 \zeta_{6} - 4182) q^{53} + 3645 \zeta_{6} q^{54} - 4888 q^{55} + 15588 q^{57} - 27590 \zeta_{6} q^{58} + (11220 \zeta_{6} - 11220) q^{59} + ( - 5922 \zeta_{6} + 5922) q^{60} - 45602 \zeta_{6} q^{61} - 31680 q^{62} + 36457 q^{64} + 72380 \zeta_{6} q^{65} + ( - 2340 \zeta_{6} + 2340) q^{66} + (1396 \zeta_{6} - 1396) q^{67} + 14154 \zeta_{6} q^{68} + 5184 q^{69} + 18720 q^{71} + 15795 \zeta_{6} q^{72} + ( - 46362 \zeta_{6} + 46362) q^{73} + ( - 36690 \zeta_{6} + 36690) q^{74} + 51399 \zeta_{6} q^{75} + 12124 q^{76} - 34650 q^{78} - 97424 \zeta_{6} q^{79} + (70594 \zeta_{6} - 70594) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 16310 \zeta_{6} q^{82} + 81228 q^{83} - 190068 q^{85} - 27100 \zeta_{6} q^{86} + ( - 49662 \zeta_{6} + 49662) q^{87} + ( - 10140 \zeta_{6} + 10140) q^{88} - 3182 \zeta_{6} q^{89} - 38070 q^{90} + 4032 q^{92} - 57024 \zeta_{6} q^{93} + ( - 4320 \zeta_{6} + 4320) q^{94} + (162808 \zeta_{6} - 162808) q^{95} + 22365 \zeta_{6} q^{96} - 4914 q^{97} + 4212 q^{99} +O(q^{100})$$ q - 5*z * q^2 + (-9*z + 9) * q^3 + (-7*z + 7) * q^4 + 94*z * q^5 - 45 * q^6 - 195 * q^8 - 81*z * q^9 + (-470*z + 470) * q^10 + (52*z - 52) * q^11 - 63*z * q^12 + 770 * q^13 + 846 * q^15 + 751*z * q^16 + (2022*z - 2022) * q^17 + (405*z - 405) * q^18 + 1732*z * q^19 + 658 * q^20 + 260 * q^22 + 576*z * q^23 + (1755*z - 1755) * q^24 + (5711*z - 5711) * q^25 - 3850*z * q^26 - 729 * q^27 + 5518 * q^29 - 4230*z * q^30 + (-6336*z + 6336) * q^31 + (2485*z - 2485) * q^32 + 468*z * q^33 + 10110 * q^34 - 567 * q^36 + 7338*z * q^37 + (-8660*z + 8660) * q^38 + (-6930*z + 6930) * q^39 - 18330*z * q^40 + 3262 * q^41 + 5420 * q^43 + 364*z * q^44 + (-7614*z + 7614) * q^45 + (-2880*z + 2880) * q^46 + 864*z * q^47 + 6759 * q^48 + 28555 * q^50 + 18198*z * q^51 + (-5390*z + 5390) * q^52 + (4182*z - 4182) * q^53 + 3645*z * q^54 - 4888 * q^55 + 15588 * q^57 - 27590*z * q^58 + (11220*z - 11220) * q^59 + (-5922*z + 5922) * q^60 - 45602*z * q^61 - 31680 * q^62 + 36457 * q^64 + 72380*z * q^65 + (-2340*z + 2340) * q^66 + (1396*z - 1396) * q^67 + 14154*z * q^68 + 5184 * q^69 + 18720 * q^71 + 15795*z * q^72 + (-46362*z + 46362) * q^73 + (-36690*z + 36690) * q^74 + 51399*z * q^75 + 12124 * q^76 - 34650 * q^78 - 97424*z * q^79 + (70594*z - 70594) * q^80 + (6561*z - 6561) * q^81 - 16310*z * q^82 + 81228 * q^83 - 190068 * q^85 - 27100*z * q^86 + (-49662*z + 49662) * q^87 + (-10140*z + 10140) * q^88 - 3182*z * q^89 - 38070 * q^90 + 4032 * q^92 - 57024*z * q^93 + (-4320*z + 4320) * q^94 + (162808*z - 162808) * q^95 + 22365*z * q^96 - 4914 * q^97 + 4212 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{2} + 9 q^{3} + 7 q^{4} + 94 q^{5} - 90 q^{6} - 390 q^{8} - 81 q^{9}+O(q^{10})$$ 2 * q - 5 * q^2 + 9 * q^3 + 7 * q^4 + 94 * q^5 - 90 * q^6 - 390 * q^8 - 81 * q^9 $$2 q - 5 q^{2} + 9 q^{3} + 7 q^{4} + 94 q^{5} - 90 q^{6} - 390 q^{8} - 81 q^{9} + 470 q^{10} - 52 q^{11} - 63 q^{12} + 1540 q^{13} + 1692 q^{15} + 751 q^{16} - 2022 q^{17} - 405 q^{18} + 1732 q^{19} + 1316 q^{20} + 520 q^{22} + 576 q^{23} - 1755 q^{24} - 5711 q^{25} - 3850 q^{26} - 1458 q^{27} + 11036 q^{29} - 4230 q^{30} + 6336 q^{31} - 2485 q^{32} + 468 q^{33} + 20220 q^{34} - 1134 q^{36} + 7338 q^{37} + 8660 q^{38} + 6930 q^{39} - 18330 q^{40} + 6524 q^{41} + 10840 q^{43} + 364 q^{44} + 7614 q^{45} + 2880 q^{46} + 864 q^{47} + 13518 q^{48} + 57110 q^{50} + 18198 q^{51} + 5390 q^{52} - 4182 q^{53} + 3645 q^{54} - 9776 q^{55} + 31176 q^{57} - 27590 q^{58} - 11220 q^{59} + 5922 q^{60} - 45602 q^{61} - 63360 q^{62} + 72914 q^{64} + 72380 q^{65} + 2340 q^{66} - 1396 q^{67} + 14154 q^{68} + 10368 q^{69} + 37440 q^{71} + 15795 q^{72} + 46362 q^{73} + 36690 q^{74} + 51399 q^{75} + 24248 q^{76} - 69300 q^{78} - 97424 q^{79} - 70594 q^{80} - 6561 q^{81} - 16310 q^{82} + 162456 q^{83} - 380136 q^{85} - 27100 q^{86} + 49662 q^{87} + 10140 q^{88} - 3182 q^{89} - 76140 q^{90} + 8064 q^{92} - 57024 q^{93} + 4320 q^{94} - 162808 q^{95} + 22365 q^{96} - 9828 q^{97} + 8424 q^{99}+O(q^{100})$$ 2 * q - 5 * q^2 + 9 * q^3 + 7 * q^4 + 94 * q^5 - 90 * q^6 - 390 * q^8 - 81 * q^9 + 470 * q^10 - 52 * q^11 - 63 * q^12 + 1540 * q^13 + 1692 * q^15 + 751 * q^16 - 2022 * q^17 - 405 * q^18 + 1732 * q^19 + 1316 * q^20 + 520 * q^22 + 576 * q^23 - 1755 * q^24 - 5711 * q^25 - 3850 * q^26 - 1458 * q^27 + 11036 * q^29 - 4230 * q^30 + 6336 * q^31 - 2485 * q^32 + 468 * q^33 + 20220 * q^34 - 1134 * q^36 + 7338 * q^37 + 8660 * q^38 + 6930 * q^39 - 18330 * q^40 + 6524 * q^41 + 10840 * q^43 + 364 * q^44 + 7614 * q^45 + 2880 * q^46 + 864 * q^47 + 13518 * q^48 + 57110 * q^50 + 18198 * q^51 + 5390 * q^52 - 4182 * q^53 + 3645 * q^54 - 9776 * q^55 + 31176 * q^57 - 27590 * q^58 - 11220 * q^59 + 5922 * q^60 - 45602 * q^61 - 63360 * q^62 + 72914 * q^64 + 72380 * q^65 + 2340 * q^66 - 1396 * q^67 + 14154 * q^68 + 10368 * q^69 + 37440 * q^71 + 15795 * q^72 + 46362 * q^73 + 36690 * q^74 + 51399 * q^75 + 24248 * q^76 - 69300 * q^78 - 97424 * q^79 - 70594 * q^80 - 6561 * q^81 - 16310 * q^82 + 162456 * q^83 - 380136 * q^85 - 27100 * q^86 + 49662 * q^87 + 10140 * q^88 - 3182 * q^89 - 76140 * q^90 + 8064 * q^92 - 57024 * q^93 + 4320 * q^94 - 162808 * q^95 + 22365 * q^96 - 9828 * q^97 + 8424 * 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
−2.50000 4.33013i 4.50000 7.79423i 3.50000 6.06218i 47.0000 + 81.4064i −45.0000 0 −195.000 −40.5000 70.1481i 235.000 407.032i
79.1 −2.50000 + 4.33013i 4.50000 + 7.79423i 3.50000 + 6.06218i 47.0000 81.4064i −45.0000 0 −195.000 −40.5000 + 70.1481i 235.000 + 407.032i
 $$n$$: 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.d 2
7.b odd 2 1 147.6.e.c 2
7.c even 3 1 147.6.a.f 1
7.c even 3 1 inner 147.6.e.d 2
7.d odd 6 1 21.6.a.c 1
7.d odd 6 1 147.6.e.c 2
21.g even 6 1 63.6.a.b 1
21.h odd 6 1 441.6.a.c 1
28.f even 6 1 336.6.a.i 1
35.i odd 6 1 525.6.a.b 1
35.k even 12 2 525.6.d.c 2
84.j odd 6 1 1008.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 7.d odd 6 1
63.6.a.b 1 21.g even 6 1
147.6.a.f 1 7.c even 3 1
147.6.e.c 2 7.b odd 2 1
147.6.e.c 2 7.d odd 6 1
147.6.e.d 2 1.a even 1 1 trivial
147.6.e.d 2 7.c even 3 1 inner
336.6.a.i 1 28.f even 6 1
441.6.a.c 1 21.h odd 6 1
525.6.a.b 1 35.i odd 6 1
525.6.d.c 2 35.k even 12 2
1008.6.a.a 1 84.j odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} + 5T_{2} + 25$$ T2^2 + 5*T2 + 25 $$T_{5}^{2} - 94T_{5} + 8836$$ T5^2 - 94*T5 + 8836

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 5T + 25$$
$3$ $$T^{2} - 9T + 81$$
$5$ $$T^{2} - 94T + 8836$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 52T + 2704$$
$13$ $$(T - 770)^{2}$$
$17$ $$T^{2} + 2022 T + 4088484$$
$19$ $$T^{2} - 1732 T + 2999824$$
$23$ $$T^{2} - 576T + 331776$$
$29$ $$(T - 5518)^{2}$$
$31$ $$T^{2} - 6336 T + 40144896$$
$37$ $$T^{2} - 7338 T + 53846244$$
$41$ $$(T - 3262)^{2}$$
$43$ $$(T - 5420)^{2}$$
$47$ $$T^{2} - 864T + 746496$$
$53$ $$T^{2} + 4182 T + 17489124$$
$59$ $$T^{2} + 11220 T + 125888400$$
$61$ $$T^{2} + \cdots + 2079542404$$
$67$ $$T^{2} + 1396 T + 1948816$$
$71$ $$(T - 18720)^{2}$$
$73$ $$T^{2} + \cdots + 2149435044$$
$79$ $$T^{2} + \cdots + 9491435776$$
$83$ $$(T - 81228)^{2}$$
$89$ $$T^{2} + 3182 T + 10125124$$
$97$ $$(T + 4914)^{2}$$