# Properties

 Label 147.6.e.f 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + ( - 31 \zeta_{6} + 31) q^{4} + 34 \zeta_{6} q^{5} - 9 q^{6} - 63 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^2 + (-9*z + 9) * q^3 + (-31*z + 31) * q^4 + 34*z * q^5 - 9 * q^6 - 63 * q^8 - 81*z * q^9 $$q - \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + ( - 31 \zeta_{6} + 31) q^{4} + 34 \zeta_{6} q^{5} - 9 q^{6} - 63 q^{8} - 81 \zeta_{6} q^{9} + ( - 34 \zeta_{6} + 34) q^{10} + ( - 340 \zeta_{6} + 340) q^{11} - 279 \zeta_{6} q^{12} + 454 q^{13} + 306 q^{15} - 929 \zeta_{6} q^{16} + ( - 798 \zeta_{6} + 798) q^{17} + (81 \zeta_{6} - 81) q^{18} - 892 \zeta_{6} q^{19} + 1054 q^{20} - 340 q^{22} + 3192 \zeta_{6} q^{23} + (567 \zeta_{6} - 567) q^{24} + ( - 1969 \zeta_{6} + 1969) q^{25} - 454 \zeta_{6} q^{26} - 729 q^{27} - 8242 q^{29} - 306 \zeta_{6} q^{30} + ( - 2496 \zeta_{6} + 2496) q^{31} + (2945 \zeta_{6} - 2945) q^{32} - 3060 \zeta_{6} q^{33} - 798 q^{34} - 2511 q^{36} - 9798 \zeta_{6} q^{37} + (892 \zeta_{6} - 892) q^{38} + ( - 4086 \zeta_{6} + 4086) q^{39} - 2142 \zeta_{6} q^{40} + 19834 q^{41} - 17236 q^{43} - 10540 \zeta_{6} q^{44} + ( - 2754 \zeta_{6} + 2754) q^{45} + ( - 3192 \zeta_{6} + 3192) q^{46} - 8928 \zeta_{6} q^{47} - 8361 q^{48} - 1969 q^{50} - 7182 \zeta_{6} q^{51} + ( - 14074 \zeta_{6} + 14074) q^{52} + (150 \zeta_{6} - 150) q^{53} + 729 \zeta_{6} q^{54} + 11560 q^{55} - 8028 q^{57} + 8242 \zeta_{6} q^{58} + ( - 42396 \zeta_{6} + 42396) q^{59} + ( - 9486 \zeta_{6} + 9486) q^{60} - 14758 \zeta_{6} q^{61} - 2496 q^{62} - 26783 q^{64} + 15436 \zeta_{6} q^{65} + (3060 \zeta_{6} - 3060) q^{66} + ( - 1676 \zeta_{6} + 1676) q^{67} - 24738 \zeta_{6} q^{68} + 28728 q^{69} + 14568 q^{71} + 5103 \zeta_{6} q^{72} + (78378 \zeta_{6} - 78378) q^{73} + (9798 \zeta_{6} - 9798) q^{74} - 17721 \zeta_{6} q^{75} - 27652 q^{76} - 4086 q^{78} + 2272 \zeta_{6} q^{79} + ( - 31586 \zeta_{6} + 31586) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 19834 \zeta_{6} q^{82} - 37764 q^{83} + 27132 q^{85} + 17236 \zeta_{6} q^{86} + (74178 \zeta_{6} - 74178) q^{87} + (21420 \zeta_{6} - 21420) q^{88} + 117286 \zeta_{6} q^{89} - 2754 q^{90} + 98952 q^{92} - 22464 \zeta_{6} q^{93} + (8928 \zeta_{6} - 8928) q^{94} + ( - 30328 \zeta_{6} + 30328) q^{95} + 26505 \zeta_{6} q^{96} + 10002 q^{97} - 27540 q^{99} +O(q^{100})$$ q - z * q^2 + (-9*z + 9) * q^3 + (-31*z + 31) * q^4 + 34*z * q^5 - 9 * q^6 - 63 * q^8 - 81*z * q^9 + (-34*z + 34) * q^10 + (-340*z + 340) * q^11 - 279*z * q^12 + 454 * q^13 + 306 * q^15 - 929*z * q^16 + (-798*z + 798) * q^17 + (81*z - 81) * q^18 - 892*z * q^19 + 1054 * q^20 - 340 * q^22 + 3192*z * q^23 + (567*z - 567) * q^24 + (-1969*z + 1969) * q^25 - 454*z * q^26 - 729 * q^27 - 8242 * q^29 - 306*z * q^30 + (-2496*z + 2496) * q^31 + (2945*z - 2945) * q^32 - 3060*z * q^33 - 798 * q^34 - 2511 * q^36 - 9798*z * q^37 + (892*z - 892) * q^38 + (-4086*z + 4086) * q^39 - 2142*z * q^40 + 19834 * q^41 - 17236 * q^43 - 10540*z * q^44 + (-2754*z + 2754) * q^45 + (-3192*z + 3192) * q^46 - 8928*z * q^47 - 8361 * q^48 - 1969 * q^50 - 7182*z * q^51 + (-14074*z + 14074) * q^52 + (150*z - 150) * q^53 + 729*z * q^54 + 11560 * q^55 - 8028 * q^57 + 8242*z * q^58 + (-42396*z + 42396) * q^59 + (-9486*z + 9486) * q^60 - 14758*z * q^61 - 2496 * q^62 - 26783 * q^64 + 15436*z * q^65 + (3060*z - 3060) * q^66 + (-1676*z + 1676) * q^67 - 24738*z * q^68 + 28728 * q^69 + 14568 * q^71 + 5103*z * q^72 + (78378*z - 78378) * q^73 + (9798*z - 9798) * q^74 - 17721*z * q^75 - 27652 * q^76 - 4086 * q^78 + 2272*z * q^79 + (-31586*z + 31586) * q^80 + (6561*z - 6561) * q^81 - 19834*z * q^82 - 37764 * q^83 + 27132 * q^85 + 17236*z * q^86 + (74178*z - 74178) * q^87 + (21420*z - 21420) * q^88 + 117286*z * q^89 - 2754 * q^90 + 98952 * q^92 - 22464*z * q^93 + (8928*z - 8928) * q^94 + (-30328*z + 30328) * q^95 + 26505*z * q^96 + 10002 * q^97 - 27540 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 9 q^{3} + 31 q^{4} + 34 q^{5} - 18 q^{6} - 126 q^{8} - 81 q^{9}+O(q^{10})$$ 2 * q - q^2 + 9 * q^3 + 31 * q^4 + 34 * q^5 - 18 * q^6 - 126 * q^8 - 81 * q^9 $$2 q - q^{2} + 9 q^{3} + 31 q^{4} + 34 q^{5} - 18 q^{6} - 126 q^{8} - 81 q^{9} + 34 q^{10} + 340 q^{11} - 279 q^{12} + 908 q^{13} + 612 q^{15} - 929 q^{16} + 798 q^{17} - 81 q^{18} - 892 q^{19} + 2108 q^{20} - 680 q^{22} + 3192 q^{23} - 567 q^{24} + 1969 q^{25} - 454 q^{26} - 1458 q^{27} - 16484 q^{29} - 306 q^{30} + 2496 q^{31} - 2945 q^{32} - 3060 q^{33} - 1596 q^{34} - 5022 q^{36} - 9798 q^{37} - 892 q^{38} + 4086 q^{39} - 2142 q^{40} + 39668 q^{41} - 34472 q^{43} - 10540 q^{44} + 2754 q^{45} + 3192 q^{46} - 8928 q^{47} - 16722 q^{48} - 3938 q^{50} - 7182 q^{51} + 14074 q^{52} - 150 q^{53} + 729 q^{54} + 23120 q^{55} - 16056 q^{57} + 8242 q^{58} + 42396 q^{59} + 9486 q^{60} - 14758 q^{61} - 4992 q^{62} - 53566 q^{64} + 15436 q^{65} - 3060 q^{66} + 1676 q^{67} - 24738 q^{68} + 57456 q^{69} + 29136 q^{71} + 5103 q^{72} - 78378 q^{73} - 9798 q^{74} - 17721 q^{75} - 55304 q^{76} - 8172 q^{78} + 2272 q^{79} + 31586 q^{80} - 6561 q^{81} - 19834 q^{82} - 75528 q^{83} + 54264 q^{85} + 17236 q^{86} - 74178 q^{87} - 21420 q^{88} + 117286 q^{89} - 5508 q^{90} + 197904 q^{92} - 22464 q^{93} - 8928 q^{94} + 30328 q^{95} + 26505 q^{96} + 20004 q^{97} - 55080 q^{99}+O(q^{100})$$ 2 * q - q^2 + 9 * q^3 + 31 * q^4 + 34 * q^5 - 18 * q^6 - 126 * q^8 - 81 * q^9 + 34 * q^10 + 340 * q^11 - 279 * q^12 + 908 * q^13 + 612 * q^15 - 929 * q^16 + 798 * q^17 - 81 * q^18 - 892 * q^19 + 2108 * q^20 - 680 * q^22 + 3192 * q^23 - 567 * q^24 + 1969 * q^25 - 454 * q^26 - 1458 * q^27 - 16484 * q^29 - 306 * q^30 + 2496 * q^31 - 2945 * q^32 - 3060 * q^33 - 1596 * q^34 - 5022 * q^36 - 9798 * q^37 - 892 * q^38 + 4086 * q^39 - 2142 * q^40 + 39668 * q^41 - 34472 * q^43 - 10540 * q^44 + 2754 * q^45 + 3192 * q^46 - 8928 * q^47 - 16722 * q^48 - 3938 * q^50 - 7182 * q^51 + 14074 * q^52 - 150 * q^53 + 729 * q^54 + 23120 * q^55 - 16056 * q^57 + 8242 * q^58 + 42396 * q^59 + 9486 * q^60 - 14758 * q^61 - 4992 * q^62 - 53566 * q^64 + 15436 * q^65 - 3060 * q^66 + 1676 * q^67 - 24738 * q^68 + 57456 * q^69 + 29136 * q^71 + 5103 * q^72 - 78378 * q^73 - 9798 * q^74 - 17721 * q^75 - 55304 * q^76 - 8172 * q^78 + 2272 * q^79 + 31586 * q^80 - 6561 * q^81 - 19834 * q^82 - 75528 * q^83 + 54264 * q^85 + 17236 * q^86 - 74178 * q^87 - 21420 * q^88 + 117286 * q^89 - 5508 * q^90 + 197904 * q^92 - 22464 * q^93 - 8928 * q^94 + 30328 * q^95 + 26505 * q^96 + 20004 * q^97 - 55080 * 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
−0.500000 0.866025i 4.50000 7.79423i 15.5000 26.8468i 17.0000 + 29.4449i −9.00000 0 −63.0000 −40.5000 70.1481i 17.0000 29.4449i
79.1 −0.500000 + 0.866025i 4.50000 + 7.79423i 15.5000 + 26.8468i 17.0000 29.4449i −9.00000 0 −63.0000 −40.5000 + 70.1481i 17.0000 + 29.4449i
 $$n$$: 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.f 2
7.b odd 2 1 147.6.e.e 2
7.c even 3 1 21.6.a.b 1
7.c even 3 1 inner 147.6.e.f 2
7.d odd 6 1 147.6.a.e 1
7.d odd 6 1 147.6.e.e 2
21.g even 6 1 441.6.a.d 1
21.h odd 6 1 63.6.a.c 1
28.g odd 6 1 336.6.a.l 1
35.j even 6 1 525.6.a.c 1
35.l odd 12 2 525.6.d.d 2
84.n even 6 1 1008.6.a.t 1

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

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{5}^{2} - 34T_{5} + 1156$$ T5^2 - 34*T5 + 1156

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - 9T + 81$$
$5$ $$T^{2} - 34T + 1156$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 340T + 115600$$
$13$ $$(T - 454)^{2}$$
$17$ $$T^{2} - 798T + 636804$$
$19$ $$T^{2} + 892T + 795664$$
$23$ $$T^{2} - 3192 T + 10188864$$
$29$ $$(T + 8242)^{2}$$
$31$ $$T^{2} - 2496 T + 6230016$$
$37$ $$T^{2} + 9798 T + 96000804$$
$41$ $$(T - 19834)^{2}$$
$43$ $$(T + 17236)^{2}$$
$47$ $$T^{2} + 8928 T + 79709184$$
$53$ $$T^{2} + 150T + 22500$$
$59$ $$T^{2} + \cdots + 1797420816$$
$61$ $$T^{2} + 14758 T + 217798564$$
$67$ $$T^{2} - 1676 T + 2808976$$
$71$ $$(T - 14568)^{2}$$
$73$ $$T^{2} + \cdots + 6143110884$$
$79$ $$T^{2} - 2272 T + 5161984$$
$83$ $$(T + 37764)^{2}$$
$89$ $$T^{2} + \cdots + 13756005796$$
$97$ $$(T - 10002)^{2}$$