# Properties

 Label 294.6.e.i 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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 26 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q + 4*z * q^2 + (9*z - 9) * q^3 + (16*z - 16) * q^4 - 26*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} - 26 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + ( - 104 \zeta_{6} + 104) q^{10} + (664 \zeta_{6} - 664) q^{11} - 144 \zeta_{6} q^{12} + 318 q^{13} + 234 q^{15} - 256 \zeta_{6} q^{16} + (1582 \zeta_{6} - 1582) q^{17} + ( - 324 \zeta_{6} + 324) q^{18} - 236 \zeta_{6} q^{19} + 416 q^{20} - 2656 q^{22} - 2212 \zeta_{6} q^{23} + ( - 576 \zeta_{6} + 576) q^{24} + ( - 2449 \zeta_{6} + 2449) q^{25} + 1272 \zeta_{6} q^{26} + 729 q^{27} - 4954 q^{29} + 936 \zeta_{6} q^{30} + ( - 7128 \zeta_{6} + 7128) q^{31} + ( - 1024 \zeta_{6} + 1024) q^{32} - 5976 \zeta_{6} q^{33} - 6328 q^{34} + 1296 q^{36} - 4358 \zeta_{6} q^{37} + ( - 944 \zeta_{6} + 944) q^{38} + (2862 \zeta_{6} - 2862) q^{39} + 1664 \zeta_{6} q^{40} + 10542 q^{41} - 8452 q^{43} - 10624 \zeta_{6} q^{44} + (2106 \zeta_{6} - 2106) q^{45} + ( - 8848 \zeta_{6} + 8848) q^{46} - 5352 \zeta_{6} q^{47} + 2304 q^{48} + 9796 q^{50} - 14238 \zeta_{6} q^{51} + (5088 \zeta_{6} - 5088) q^{52} + ( - 33354 \zeta_{6} + 33354) q^{53} + 2916 \zeta_{6} q^{54} + 17264 q^{55} + 2124 q^{57} - 19816 \zeta_{6} q^{58} + ( - 15436 \zeta_{6} + 15436) q^{59} + (3744 \zeta_{6} - 3744) q^{60} + 36762 \zeta_{6} q^{61} + 28512 q^{62} + 4096 q^{64} - 8268 \zeta_{6} q^{65} + ( - 23904 \zeta_{6} + 23904) q^{66} + (40972 \zeta_{6} - 40972) q^{67} - 25312 \zeta_{6} q^{68} + 19908 q^{69} - 9092 q^{71} + 5184 \zeta_{6} q^{72} + ( - 73454 \zeta_{6} + 73454) q^{73} + ( - 17432 \zeta_{6} + 17432) q^{74} + 22041 \zeta_{6} q^{75} + 3776 q^{76} - 11448 q^{78} - 89400 \zeta_{6} q^{79} + (6656 \zeta_{6} - 6656) q^{80} + (6561 \zeta_{6} - 6561) q^{81} + 42168 \zeta_{6} q^{82} - 6428 q^{83} + 41132 q^{85} - 33808 \zeta_{6} q^{86} + ( - 44586 \zeta_{6} + 44586) q^{87} + ( - 42496 \zeta_{6} + 42496) q^{88} + 122658 \zeta_{6} q^{89} - 8424 q^{90} + 35392 q^{92} + 64152 \zeta_{6} q^{93} + ( - 21408 \zeta_{6} + 21408) q^{94} + (6136 \zeta_{6} - 6136) q^{95} + 9216 \zeta_{6} q^{96} + 21370 q^{97} + 53784 q^{99} +O(q^{100})$$ q + 4*z * q^2 + (9*z - 9) * q^3 + (16*z - 16) * q^4 - 26*z * q^5 - 36 * q^6 - 64 * q^8 - 81*z * q^9 + (-104*z + 104) * q^10 + (664*z - 664) * q^11 - 144*z * q^12 + 318 * q^13 + 234 * q^15 - 256*z * q^16 + (1582*z - 1582) * q^17 + (-324*z + 324) * q^18 - 236*z * q^19 + 416 * q^20 - 2656 * q^22 - 2212*z * q^23 + (-576*z + 576) * q^24 + (-2449*z + 2449) * q^25 + 1272*z * q^26 + 729 * q^27 - 4954 * q^29 + 936*z * q^30 + (-7128*z + 7128) * q^31 + (-1024*z + 1024) * q^32 - 5976*z * q^33 - 6328 * q^34 + 1296 * q^36 - 4358*z * q^37 + (-944*z + 944) * q^38 + (2862*z - 2862) * q^39 + 1664*z * q^40 + 10542 * q^41 - 8452 * q^43 - 10624*z * q^44 + (2106*z - 2106) * q^45 + (-8848*z + 8848) * q^46 - 5352*z * q^47 + 2304 * q^48 + 9796 * q^50 - 14238*z * q^51 + (5088*z - 5088) * q^52 + (-33354*z + 33354) * q^53 + 2916*z * q^54 + 17264 * q^55 + 2124 * q^57 - 19816*z * q^58 + (-15436*z + 15436) * q^59 + (3744*z - 3744) * q^60 + 36762*z * q^61 + 28512 * q^62 + 4096 * q^64 - 8268*z * q^65 + (-23904*z + 23904) * q^66 + (40972*z - 40972) * q^67 - 25312*z * q^68 + 19908 * q^69 - 9092 * q^71 + 5184*z * q^72 + (-73454*z + 73454) * q^73 + (-17432*z + 17432) * q^74 + 22041*z * q^75 + 3776 * q^76 - 11448 * q^78 - 89400*z * q^79 + (6656*z - 6656) * q^80 + (6561*z - 6561) * q^81 + 42168*z * q^82 - 6428 * q^83 + 41132 * q^85 - 33808*z * q^86 + (-44586*z + 44586) * q^87 + (-42496*z + 42496) * q^88 + 122658*z * q^89 - 8424 * q^90 + 35392 * q^92 + 64152*z * q^93 + (-21408*z + 21408) * q^94 + (6136*z - 6136) * q^95 + 9216*z * q^96 + 21370 * q^97 + 53784 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} - 26 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 - 26 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 $$2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} - 26 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} + 104 q^{10} - 664 q^{11} - 144 q^{12} + 636 q^{13} + 468 q^{15} - 256 q^{16} - 1582 q^{17} + 324 q^{18} - 236 q^{19} + 832 q^{20} - 5312 q^{22} - 2212 q^{23} + 576 q^{24} + 2449 q^{25} + 1272 q^{26} + 1458 q^{27} - 9908 q^{29} + 936 q^{30} + 7128 q^{31} + 1024 q^{32} - 5976 q^{33} - 12656 q^{34} + 2592 q^{36} - 4358 q^{37} + 944 q^{38} - 2862 q^{39} + 1664 q^{40} + 21084 q^{41} - 16904 q^{43} - 10624 q^{44} - 2106 q^{45} + 8848 q^{46} - 5352 q^{47} + 4608 q^{48} + 19592 q^{50} - 14238 q^{51} - 5088 q^{52} + 33354 q^{53} + 2916 q^{54} + 34528 q^{55} + 4248 q^{57} - 19816 q^{58} + 15436 q^{59} - 3744 q^{60} + 36762 q^{61} + 57024 q^{62} + 8192 q^{64} - 8268 q^{65} + 23904 q^{66} - 40972 q^{67} - 25312 q^{68} + 39816 q^{69} - 18184 q^{71} + 5184 q^{72} + 73454 q^{73} + 17432 q^{74} + 22041 q^{75} + 7552 q^{76} - 22896 q^{78} - 89400 q^{79} - 6656 q^{80} - 6561 q^{81} + 42168 q^{82} - 12856 q^{83} + 82264 q^{85} - 33808 q^{86} + 44586 q^{87} + 42496 q^{88} + 122658 q^{89} - 16848 q^{90} + 70784 q^{92} + 64152 q^{93} + 21408 q^{94} - 6136 q^{95} + 9216 q^{96} + 42740 q^{97} + 107568 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 - 26 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 + 104 * q^10 - 664 * q^11 - 144 * q^12 + 636 * q^13 + 468 * q^15 - 256 * q^16 - 1582 * q^17 + 324 * q^18 - 236 * q^19 + 832 * q^20 - 5312 * q^22 - 2212 * q^23 + 576 * q^24 + 2449 * q^25 + 1272 * q^26 + 1458 * q^27 - 9908 * q^29 + 936 * q^30 + 7128 * q^31 + 1024 * q^32 - 5976 * q^33 - 12656 * q^34 + 2592 * q^36 - 4358 * q^37 + 944 * q^38 - 2862 * q^39 + 1664 * q^40 + 21084 * q^41 - 16904 * q^43 - 10624 * q^44 - 2106 * q^45 + 8848 * q^46 - 5352 * q^47 + 4608 * q^48 + 19592 * q^50 - 14238 * q^51 - 5088 * q^52 + 33354 * q^53 + 2916 * q^54 + 34528 * q^55 + 4248 * q^57 - 19816 * q^58 + 15436 * q^59 - 3744 * q^60 + 36762 * q^61 + 57024 * q^62 + 8192 * q^64 - 8268 * q^65 + 23904 * q^66 - 40972 * q^67 - 25312 * q^68 + 39816 * q^69 - 18184 * q^71 + 5184 * q^72 + 73454 * q^73 + 17432 * q^74 + 22041 * q^75 + 7552 * q^76 - 22896 * q^78 - 89400 * q^79 - 6656 * q^80 - 6561 * q^81 + 42168 * q^82 - 12856 * q^83 + 82264 * q^85 - 33808 * q^86 + 44586 * q^87 + 42496 * q^88 + 122658 * q^89 - 16848 * q^90 + 70784 * q^92 + 64152 * q^93 + 21408 * q^94 - 6136 * q^95 + 9216 * q^96 + 42740 * q^97 + 107568 * 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 −13.0000 22.5167i −36.0000 0 −64.0000 −40.5000 70.1481i 52.0000 90.0666i
79.1 2.00000 3.46410i −4.50000 7.79423i −8.00000 13.8564i −13.0000 + 22.5167i −36.0000 0 −64.0000 −40.5000 + 70.1481i 52.0000 + 90.0666i
 $$n$$: 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.i 2
7.b odd 2 1 294.6.e.p 2
7.c even 3 1 42.6.a.d 1
7.c even 3 1 inner 294.6.e.i 2
7.d odd 6 1 294.6.a.b 1
7.d odd 6 1 294.6.e.p 2
21.g even 6 1 882.6.a.s 1
21.h odd 6 1 126.6.a.i 1
28.g odd 6 1 336.6.a.h 1
35.j even 6 1 1050.6.a.k 1
35.l odd 12 2 1050.6.g.i 2
84.n even 6 1 1008.6.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.d 1 7.c even 3 1
126.6.a.i 1 21.h odd 6 1
294.6.a.b 1 7.d odd 6 1
294.6.e.i 2 1.a even 1 1 trivial
294.6.e.i 2 7.c even 3 1 inner
294.6.e.p 2 7.b odd 2 1
294.6.e.p 2 7.d odd 6 1
336.6.a.h 1 28.g odd 6 1
882.6.a.s 1 21.g even 6 1
1008.6.a.j 1 84.n even 6 1
1050.6.a.k 1 35.j even 6 1
1050.6.g.i 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 $$S_{6}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{2} + 26T_{5} + 676$$ T5^2 + 26*T5 + 676 $$T_{11}^{2} + 664T_{11} + 440896$$ T11^2 + 664*T11 + 440896

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 4T + 16$$
$3$ $$T^{2} + 9T + 81$$
$5$ $$T^{2} + 26T + 676$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 664T + 440896$$
$13$ $$(T - 318)^{2}$$
$17$ $$T^{2} + 1582 T + 2502724$$
$19$ $$T^{2} + 236T + 55696$$
$23$ $$T^{2} + 2212 T + 4892944$$
$29$ $$(T + 4954)^{2}$$
$31$ $$T^{2} - 7128 T + 50808384$$
$37$ $$T^{2} + 4358 T + 18992164$$
$41$ $$(T - 10542)^{2}$$
$43$ $$(T + 8452)^{2}$$
$47$ $$T^{2} + 5352 T + 28643904$$
$53$ $$T^{2} + \cdots + 1112489316$$
$59$ $$T^{2} - 15436 T + 238270096$$
$61$ $$T^{2} + \cdots + 1351444644$$
$67$ $$T^{2} + \cdots + 1678704784$$
$71$ $$(T + 9092)^{2}$$
$73$ $$T^{2} + \cdots + 5395490116$$
$79$ $$T^{2} + \cdots + 7992360000$$
$83$ $$(T + 6428)^{2}$$
$89$ $$T^{2} + \cdots + 15044984964$$
$97$ $$(T - 21370)^{2}$$