# Properties

 Label 294.6.e.r 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.r 2
7.b odd 2 1 294.6.e.h 2
7.c even 3 1 42.6.a.a 1
7.c even 3 1 inner 294.6.e.r 2
7.d odd 6 1 294.6.a.h 1
7.d odd 6 1 294.6.e.h 2
21.g even 6 1 882.6.a.o 1
21.h odd 6 1 126.6.a.k 1
28.g odd 6 1 336.6.a.j 1
35.j even 6 1 1050.6.a.n 1
35.l odd 12 2 1050.6.g.o 2
84.n even 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.c even 3 1
126.6.a.k 1 21.h odd 6 1
294.6.a.h 1 7.d odd 6 1
294.6.e.h 2 7.b odd 2 1
294.6.e.h 2 7.d odd 6 1
294.6.e.r 2 1.a even 1 1 trivial
294.6.e.r 2 7.c even 3 1 inner
336.6.a.j 1 28.g odd 6 1
882.6.a.o 1 21.g even 6 1
1008.6.a.x 1 84.n even 6 1
1050.6.a.n 1 35.j even 6 1
1050.6.g.o 2 35.l odd 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

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