# Properties

 Label 294.6.e.h Level $294$ Weight $6$ Character orbit 294.e Analytic conductor $47.153$ 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] = [294,6,Mod(67,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, 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("294.67");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 294.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: $$47.1528430250$$ 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 42) 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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 54 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q + 4*z * q^2 + (9*z - 9) * q^3 + (16*z - 16) * q^4 - 54*z * q^5 - 36 * q^6 - 64 * q^8 - 81*z * q^9 $$q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 54 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + ( - 216 \zeta_{6} + 216) q^{10} + (216 \zeta_{6} - 216) q^{11} - 144 \zeta_{6} q^{12} - 998 q^{13} + 486 q^{15} - 256 \zeta_{6} q^{16} + ( - 1302 \zeta_{6} + 1302) q^{17} + ( - 324 \zeta_{6} + 324) q^{18} + 884 \zeta_{6} q^{19} + 864 q^{20} - 864 q^{22} + 2268 \zeta_{6} q^{23} + ( - 576 \zeta_{6} + 576) q^{24} + ( - 209 \zeta_{6} + 209) q^{25} - 3992 \zeta_{6} q^{26} + 729 q^{27} - 1482 q^{29} + 1944 \zeta_{6} q^{30} + ( - 8360 \zeta_{6} + 8360) q^{31} + ( - 1024 \zeta_{6} + 1024) q^{32} - 1944 \zeta_{6} q^{33} + 5208 q^{34} + 1296 q^{36} + 4714 \zeta_{6} q^{37} + (3536 \zeta_{6} - 3536) q^{38} + ( - 8982 \zeta_{6} + 8982) q^{39} + 3456 \zeta_{6} q^{40} + 9786 q^{41} + 19436 q^{43} - 3456 \zeta_{6} q^{44} + (4374 \zeta_{6} - 4374) q^{45} + (9072 \zeta_{6} - 9072) q^{46} + 22200 \zeta_{6} q^{47} + 2304 q^{48} + 836 q^{50} + 11718 \zeta_{6} q^{51} + ( - 15968 \zeta_{6} + 15968) q^{52} + (26790 \zeta_{6} - 26790) q^{53} + 2916 \zeta_{6} q^{54} + 11664 q^{55} - 7956 q^{57} - 5928 \zeta_{6} q^{58} + ( - 28092 \zeta_{6} + 28092) q^{59} + (7776 \zeta_{6} - 7776) q^{60} - 38866 \zeta_{6} q^{61} + 33440 q^{62} + 4096 q^{64} + 53892 \zeta_{6} q^{65} + ( - 7776 \zeta_{6} + 7776) q^{66} + (23948 \zeta_{6} - 23948) q^{67} + 20832 \zeta_{6} q^{68} - 20412 q^{69} - 20628 q^{71} + 5184 \zeta_{6} q^{72} + ( - 290 \zeta_{6} + 290) q^{73} + (18856 \zeta_{6} - 18856) q^{74} + 1881 \zeta_{6} q^{75} - 14144 q^{76} + 35928 q^{78} + 99544 \zeta_{6} q^{79} + (13824 \zeta_{6} - 13824) q^{80} + (6561 \zeta_{6} - 6561) q^{81} + 39144 \zeta_{6} q^{82} - 19308 q^{83} - 70308 q^{85} + 77744 \zeta_{6} q^{86} + ( - 13338 \zeta_{6} + 13338) q^{87} + ( - 13824 \zeta_{6} + 13824) q^{88} + 36390 \zeta_{6} q^{89} - 17496 q^{90} - 36288 q^{92} + 75240 \zeta_{6} q^{93} + (88800 \zeta_{6} - 88800) q^{94} + ( - 47736 \zeta_{6} + 47736) q^{95} + 9216 \zeta_{6} q^{96} + 79078 q^{97} + 17496 q^{99} +O(q^{100})$$ q + 4*z * q^2 + (9*z - 9) * q^3 + (16*z - 16) * q^4 - 54*z * q^5 - 36 * q^6 - 64 * q^8 - 81*z * q^9 + (-216*z + 216) * q^10 + (216*z - 216) * q^11 - 144*z * q^12 - 998 * q^13 + 486 * q^15 - 256*z * q^16 + (-1302*z + 1302) * q^17 + (-324*z + 324) * q^18 + 884*z * q^19 + 864 * q^20 - 864 * q^22 + 2268*z * q^23 + (-576*z + 576) * q^24 + (-209*z + 209) * q^25 - 3992*z * q^26 + 729 * q^27 - 1482 * q^29 + 1944*z * q^30 + (-8360*z + 8360) * q^31 + (-1024*z + 1024) * q^32 - 1944*z * q^33 + 5208 * q^34 + 1296 * q^36 + 4714*z * q^37 + (3536*z - 3536) * q^38 + (-8982*z + 8982) * q^39 + 3456*z * q^40 + 9786 * q^41 + 19436 * q^43 - 3456*z * q^44 + (4374*z - 4374) * q^45 + (9072*z - 9072) * q^46 + 22200*z * q^47 + 2304 * q^48 + 836 * q^50 + 11718*z * q^51 + (-15968*z + 15968) * q^52 + (26790*z - 26790) * q^53 + 2916*z * q^54 + 11664 * q^55 - 7956 * q^57 - 5928*z * q^58 + (-28092*z + 28092) * q^59 + (7776*z - 7776) * q^60 - 38866*z * q^61 + 33440 * q^62 + 4096 * q^64 + 53892*z * q^65 + (-7776*z + 7776) * q^66 + (23948*z - 23948) * q^67 + 20832*z * q^68 - 20412 * q^69 - 20628 * q^71 + 5184*z * q^72 + (-290*z + 290) * q^73 + (18856*z - 18856) * q^74 + 1881*z * q^75 - 14144 * q^76 + 35928 * q^78 + 99544*z * q^79 + (13824*z - 13824) * q^80 + (6561*z - 6561) * q^81 + 39144*z * q^82 - 19308 * q^83 - 70308 * q^85 + 77744*z * q^86 + (-13338*z + 13338) * q^87 + (-13824*z + 13824) * q^88 + 36390*z * q^89 - 17496 * q^90 - 36288 * q^92 + 75240*z * q^93 + (88800*z - 88800) * q^94 + (-47736*z + 47736) * q^95 + 9216*z * q^96 + 79078 * q^97 + 17496 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} - 54 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 - 54 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 $$2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} - 54 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} + 216 q^{10} - 216 q^{11} - 144 q^{12} - 1996 q^{13} + 972 q^{15} - 256 q^{16} + 1302 q^{17} + 324 q^{18} + 884 q^{19} + 1728 q^{20} - 1728 q^{22} + 2268 q^{23} + 576 q^{24} + 209 q^{25} - 3992 q^{26} + 1458 q^{27} - 2964 q^{29} + 1944 q^{30} + 8360 q^{31} + 1024 q^{32} - 1944 q^{33} + 10416 q^{34} + 2592 q^{36} + 4714 q^{37} - 3536 q^{38} + 8982 q^{39} + 3456 q^{40} + 19572 q^{41} + 38872 q^{43} - 3456 q^{44} - 4374 q^{45} - 9072 q^{46} + 22200 q^{47} + 4608 q^{48} + 1672 q^{50} + 11718 q^{51} + 15968 q^{52} - 26790 q^{53} + 2916 q^{54} + 23328 q^{55} - 15912 q^{57} - 5928 q^{58} + 28092 q^{59} - 7776 q^{60} - 38866 q^{61} + 66880 q^{62} + 8192 q^{64} + 53892 q^{65} + 7776 q^{66} - 23948 q^{67} + 20832 q^{68} - 40824 q^{69} - 41256 q^{71} + 5184 q^{72} + 290 q^{73} - 18856 q^{74} + 1881 q^{75} - 28288 q^{76} + 71856 q^{78} + 99544 q^{79} - 13824 q^{80} - 6561 q^{81} + 39144 q^{82} - 38616 q^{83} - 140616 q^{85} + 77744 q^{86} + 13338 q^{87} + 13824 q^{88} + 36390 q^{89} - 34992 q^{90} - 72576 q^{92} + 75240 q^{93} - 88800 q^{94} + 47736 q^{95} + 9216 q^{96} + 158156 q^{97} + 34992 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 - 54 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 + 216 * q^10 - 216 * q^11 - 144 * q^12 - 1996 * q^13 + 972 * q^15 - 256 * q^16 + 1302 * q^17 + 324 * q^18 + 884 * q^19 + 1728 * q^20 - 1728 * q^22 + 2268 * q^23 + 576 * q^24 + 209 * q^25 - 3992 * q^26 + 1458 * q^27 - 2964 * q^29 + 1944 * q^30 + 8360 * q^31 + 1024 * q^32 - 1944 * q^33 + 10416 * q^34 + 2592 * q^36 + 4714 * q^37 - 3536 * q^38 + 8982 * q^39 + 3456 * q^40 + 19572 * q^41 + 38872 * q^43 - 3456 * q^44 - 4374 * q^45 - 9072 * q^46 + 22200 * q^47 + 4608 * q^48 + 1672 * q^50 + 11718 * q^51 + 15968 * q^52 - 26790 * q^53 + 2916 * q^54 + 23328 * q^55 - 15912 * q^57 - 5928 * q^58 + 28092 * q^59 - 7776 * q^60 - 38866 * q^61 + 66880 * q^62 + 8192 * q^64 + 53892 * q^65 + 7776 * q^66 - 23948 * q^67 + 20832 * q^68 - 40824 * q^69 - 41256 * q^71 + 5184 * q^72 + 290 * q^73 - 18856 * q^74 + 1881 * q^75 - 28288 * q^76 + 71856 * q^78 + 99544 * q^79 - 13824 * q^80 - 6561 * q^81 + 39144 * q^82 - 38616 * q^83 - 140616 * q^85 + 77744 * q^86 + 13338 * q^87 + 13824 * q^88 + 36390 * q^89 - 34992 * q^90 - 72576 * q^92 + 75240 * q^93 - 88800 * q^94 + 47736 * q^95 + 9216 * q^96 + 158156 * q^97 + 34992 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/294\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$199$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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
2.00000 + 3.46410i −4.50000 + 7.79423i −8.00000 + 13.8564i −27.0000 46.7654i −36.0000 0 −64.0000 −40.5000 70.1481i 108.000 187.061i
79.1 2.00000 3.46410i −4.50000 7.79423i −8.00000 13.8564i −27.0000 + 46.7654i −36.0000 0 −64.0000 −40.5000 + 70.1481i 108.000 + 187.061i
 $$n$$: 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 294.6.e.h 2
7.b odd 2 1 294.6.e.r 2
7.c even 3 1 294.6.a.h 1
7.c even 3 1 inner 294.6.e.h 2
7.d odd 6 1 42.6.a.a 1
7.d odd 6 1 294.6.e.r 2
21.g even 6 1 126.6.a.k 1
21.h odd 6 1 882.6.a.o 1
28.f even 6 1 336.6.a.j 1
35.i odd 6 1 1050.6.a.n 1
35.k even 12 2 1050.6.g.o 2
84.j odd 6 1 1008.6.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.a 1 7.d odd 6 1
126.6.a.k 1 21.g even 6 1
294.6.a.h 1 7.c even 3 1
294.6.e.h 2 1.a even 1 1 trivial
294.6.e.h 2 7.c even 3 1 inner
294.6.e.r 2 7.b odd 2 1
294.6.e.r 2 7.d odd 6 1
336.6.a.j 1 28.f even 6 1
882.6.a.o 1 21.h odd 6 1
1008.6.a.x 1 84.j odd 6 1
1050.6.a.n 1 35.i odd 6 1
1050.6.g.o 2 35.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{2} + 54T_{5} + 2916$$ T5^2 + 54*T5 + 2916 $$T_{11}^{2} + 216T_{11} + 46656$$ T11^2 + 216*T11 + 46656

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 4T + 16$$
$3$ $$T^{2} + 9T + 81$$
$5$ $$T^{2} + 54T + 2916$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 216T + 46656$$
$13$ $$(T + 998)^{2}$$
$17$ $$T^{2} - 1302 T + 1695204$$
$19$ $$T^{2} - 884T + 781456$$
$23$ $$T^{2} - 2268 T + 5143824$$
$29$ $$(T + 1482)^{2}$$
$31$ $$T^{2} - 8360 T + 69889600$$
$37$ $$T^{2} - 4714 T + 22221796$$
$41$ $$(T - 9786)^{2}$$
$43$ $$(T - 19436)^{2}$$
$47$ $$T^{2} - 22200 T + 492840000$$
$53$ $$T^{2} + 26790 T + 717704100$$
$59$ $$T^{2} - 28092 T + 789160464$$
$61$ $$T^{2} + \cdots + 1510565956$$
$67$ $$T^{2} + 23948 T + 573506704$$
$71$ $$(T + 20628)^{2}$$
$73$ $$T^{2} - 290T + 84100$$
$79$ $$T^{2} + \cdots + 9909007936$$
$83$ $$(T + 19308)^{2}$$
$89$ $$T^{2} + \cdots + 1324232100$$
$97$ $$(T - 79078)^{2}$$