# Properties

 Label 294.6.e.f 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.

 $$f(q)$$ $$=$$ $$q - 4 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + (16 \zeta_{6} - 16) q^{4} + 24 \zeta_{6} q^{5} - 36 q^{6} + 64 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q - 4*z * q^2 + (-9*z + 9) * q^3 + (16*z - 16) * q^4 + 24*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} + 24 \zeta_{6} q^{5} - 36 q^{6} + 64 q^{8} - 81 \zeta_{6} q^{9} + ( - 96 \zeta_{6} + 96) q^{10} + (66 \zeta_{6} - 66) q^{11} + 144 \zeta_{6} q^{12} - 98 q^{13} + 216 q^{15} - 256 \zeta_{6} q^{16} + (216 \zeta_{6} - 216) q^{17} + (324 \zeta_{6} - 324) q^{18} - 340 \zeta_{6} q^{19} - 384 q^{20} + 264 q^{22} + 1038 \zeta_{6} q^{23} + ( - 576 \zeta_{6} + 576) q^{24} + ( - 2549 \zeta_{6} + 2549) q^{25} + 392 \zeta_{6} q^{26} - 729 q^{27} - 2490 q^{29} - 864 \zeta_{6} q^{30} + (7048 \zeta_{6} - 7048) q^{31} + (1024 \zeta_{6} - 1024) q^{32} + 594 \zeta_{6} q^{33} + 864 q^{34} + 1296 q^{36} + 12238 \zeta_{6} q^{37} + (1360 \zeta_{6} - 1360) q^{38} + (882 \zeta_{6} - 882) q^{39} + 1536 \zeta_{6} q^{40} - 6468 q^{41} - 15412 q^{43} - 1056 \zeta_{6} q^{44} + ( - 1944 \zeta_{6} + 1944) q^{45} + ( - 4152 \zeta_{6} + 4152) q^{46} + 20604 \zeta_{6} q^{47} - 2304 q^{48} - 10196 q^{50} + 1944 \zeta_{6} q^{51} + ( - 1568 \zeta_{6} + 1568) q^{52} + (32490 \zeta_{6} - 32490) q^{53} + 2916 \zeta_{6} q^{54} - 1584 q^{55} - 3060 q^{57} + 9960 \zeta_{6} q^{58} + ( - 34224 \zeta_{6} + 34224) q^{59} + (3456 \zeta_{6} - 3456) q^{60} + 35654 \zeta_{6} q^{61} + 28192 q^{62} + 4096 q^{64} - 2352 \zeta_{6} q^{65} + ( - 2376 \zeta_{6} + 2376) q^{66} + (12680 \zeta_{6} - 12680) q^{67} - 3456 \zeta_{6} q^{68} + 9342 q^{69} - 42642 q^{71} - 5184 \zeta_{6} q^{72} + ( - 33734 \zeta_{6} + 33734) q^{73} + ( - 48952 \zeta_{6} + 48952) q^{74} - 22941 \zeta_{6} q^{75} + 5440 q^{76} + 3528 q^{78} + 85108 \zeta_{6} q^{79} + ( - 6144 \zeta_{6} + 6144) q^{80} + (6561 \zeta_{6} - 6561) q^{81} + 25872 \zeta_{6} q^{82} + 106764 q^{83} - 5184 q^{85} + 61648 \zeta_{6} q^{86} + (22410 \zeta_{6} - 22410) q^{87} + (4224 \zeta_{6} - 4224) q^{88} + 34884 \zeta_{6} q^{89} - 7776 q^{90} - 16608 q^{92} + 63432 \zeta_{6} q^{93} + ( - 82416 \zeta_{6} + 82416) q^{94} + ( - 8160 \zeta_{6} + 8160) q^{95} + 9216 \zeta_{6} q^{96} - 18662 q^{97} + 5346 q^{99} +O(q^{100})$$ q - 4*z * q^2 + (-9*z + 9) * q^3 + (16*z - 16) * q^4 + 24*z * q^5 - 36 * q^6 + 64 * q^8 - 81*z * q^9 + (-96*z + 96) * q^10 + (66*z - 66) * q^11 + 144*z * q^12 - 98 * q^13 + 216 * q^15 - 256*z * q^16 + (216*z - 216) * q^17 + (324*z - 324) * q^18 - 340*z * q^19 - 384 * q^20 + 264 * q^22 + 1038*z * q^23 + (-576*z + 576) * q^24 + (-2549*z + 2549) * q^25 + 392*z * q^26 - 729 * q^27 - 2490 * q^29 - 864*z * q^30 + (7048*z - 7048) * q^31 + (1024*z - 1024) * q^32 + 594*z * q^33 + 864 * q^34 + 1296 * q^36 + 12238*z * q^37 + (1360*z - 1360) * q^38 + (882*z - 882) * q^39 + 1536*z * q^40 - 6468 * q^41 - 15412 * q^43 - 1056*z * q^44 + (-1944*z + 1944) * q^45 + (-4152*z + 4152) * q^46 + 20604*z * q^47 - 2304 * q^48 - 10196 * q^50 + 1944*z * q^51 + (-1568*z + 1568) * q^52 + (32490*z - 32490) * q^53 + 2916*z * q^54 - 1584 * q^55 - 3060 * q^57 + 9960*z * q^58 + (-34224*z + 34224) * q^59 + (3456*z - 3456) * q^60 + 35654*z * q^61 + 28192 * q^62 + 4096 * q^64 - 2352*z * q^65 + (-2376*z + 2376) * q^66 + (12680*z - 12680) * q^67 - 3456*z * q^68 + 9342 * q^69 - 42642 * q^71 - 5184*z * q^72 + (-33734*z + 33734) * q^73 + (-48952*z + 48952) * q^74 - 22941*z * q^75 + 5440 * q^76 + 3528 * q^78 + 85108*z * q^79 + (-6144*z + 6144) * q^80 + (6561*z - 6561) * q^81 + 25872*z * q^82 + 106764 * q^83 - 5184 * q^85 + 61648*z * q^86 + (22410*z - 22410) * q^87 + (4224*z - 4224) * q^88 + 34884*z * q^89 - 7776 * q^90 - 16608 * q^92 + 63432*z * q^93 + (-82416*z + 82416) * q^94 + (-8160*z + 8160) * q^95 + 9216*z * q^96 - 18662 * q^97 + 5346 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 9 q^{3} - 16 q^{4} + 24 q^{5} - 72 q^{6} + 128 q^{8} - 81 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 + 9 * q^3 - 16 * q^4 + 24 * q^5 - 72 * q^6 + 128 * q^8 - 81 * q^9 $$2 q - 4 q^{2} + 9 q^{3} - 16 q^{4} + 24 q^{5} - 72 q^{6} + 128 q^{8} - 81 q^{9} + 96 q^{10} - 66 q^{11} + 144 q^{12} - 196 q^{13} + 432 q^{15} - 256 q^{16} - 216 q^{17} - 324 q^{18} - 340 q^{19} - 768 q^{20} + 528 q^{22} + 1038 q^{23} + 576 q^{24} + 2549 q^{25} + 392 q^{26} - 1458 q^{27} - 4980 q^{29} - 864 q^{30} - 7048 q^{31} - 1024 q^{32} + 594 q^{33} + 1728 q^{34} + 2592 q^{36} + 12238 q^{37} - 1360 q^{38} - 882 q^{39} + 1536 q^{40} - 12936 q^{41} - 30824 q^{43} - 1056 q^{44} + 1944 q^{45} + 4152 q^{46} + 20604 q^{47} - 4608 q^{48} - 20392 q^{50} + 1944 q^{51} + 1568 q^{52} - 32490 q^{53} + 2916 q^{54} - 3168 q^{55} - 6120 q^{57} + 9960 q^{58} + 34224 q^{59} - 3456 q^{60} + 35654 q^{61} + 56384 q^{62} + 8192 q^{64} - 2352 q^{65} + 2376 q^{66} - 12680 q^{67} - 3456 q^{68} + 18684 q^{69} - 85284 q^{71} - 5184 q^{72} + 33734 q^{73} + 48952 q^{74} - 22941 q^{75} + 10880 q^{76} + 7056 q^{78} + 85108 q^{79} + 6144 q^{80} - 6561 q^{81} + 25872 q^{82} + 213528 q^{83} - 10368 q^{85} + 61648 q^{86} - 22410 q^{87} - 4224 q^{88} + 34884 q^{89} - 15552 q^{90} - 33216 q^{92} + 63432 q^{93} + 82416 q^{94} + 8160 q^{95} + 9216 q^{96} - 37324 q^{97} + 10692 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 + 9 * q^3 - 16 * q^4 + 24 * q^5 - 72 * q^6 + 128 * q^8 - 81 * q^9 + 96 * q^10 - 66 * q^11 + 144 * q^12 - 196 * q^13 + 432 * q^15 - 256 * q^16 - 216 * q^17 - 324 * q^18 - 340 * q^19 - 768 * q^20 + 528 * q^22 + 1038 * q^23 + 576 * q^24 + 2549 * q^25 + 392 * q^26 - 1458 * q^27 - 4980 * q^29 - 864 * q^30 - 7048 * q^31 - 1024 * q^32 + 594 * q^33 + 1728 * q^34 + 2592 * q^36 + 12238 * q^37 - 1360 * q^38 - 882 * q^39 + 1536 * q^40 - 12936 * q^41 - 30824 * q^43 - 1056 * q^44 + 1944 * q^45 + 4152 * q^46 + 20604 * q^47 - 4608 * q^48 - 20392 * q^50 + 1944 * q^51 + 1568 * q^52 - 32490 * q^53 + 2916 * q^54 - 3168 * q^55 - 6120 * q^57 + 9960 * q^58 + 34224 * q^59 - 3456 * q^60 + 35654 * q^61 + 56384 * q^62 + 8192 * q^64 - 2352 * q^65 + 2376 * q^66 - 12680 * q^67 - 3456 * q^68 + 18684 * q^69 - 85284 * q^71 - 5184 * q^72 + 33734 * q^73 + 48952 * q^74 - 22941 * q^75 + 10880 * q^76 + 7056 * q^78 + 85108 * q^79 + 6144 * q^80 - 6561 * q^81 + 25872 * q^82 + 213528 * q^83 - 10368 * q^85 + 61648 * q^86 - 22410 * q^87 - 4224 * q^88 + 34884 * q^89 - 15552 * q^90 - 33216 * q^92 + 63432 * q^93 + 82416 * q^94 + 8160 * q^95 + 9216 * q^96 - 37324 * q^97 + 10692 * 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 12.0000 + 20.7846i −36.0000 0 64.0000 −40.5000 70.1481i 48.0000 83.1384i
79.1 −2.00000 + 3.46410i 4.50000 + 7.79423i −8.00000 13.8564i 12.0000 20.7846i −36.0000 0 64.0000 −40.5000 + 70.1481i 48.0000 + 83.1384i
 $$n$$: 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.f 2
7.b odd 2 1 294.6.e.b 2
7.c even 3 1 294.6.a.i 1
7.c even 3 1 inner 294.6.e.f 2
7.d odd 6 1 42.6.a.f 1
7.d odd 6 1 294.6.e.b 2
21.g even 6 1 126.6.a.b 1
21.h odd 6 1 882.6.a.i 1
28.f even 6 1 336.6.a.g 1
35.i odd 6 1 1050.6.a.a 1
35.k even 12 2 1050.6.g.m 2
84.j odd 6 1 1008.6.a.k 1

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

 $$T_{5}^{2} - 24T_{5} + 576$$ T5^2 - 24*T5 + 576 $$T_{11}^{2} + 66T_{11} + 4356$$ T11^2 + 66*T11 + 4356

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4T + 16$$
$3$ $$T^{2} - 9T + 81$$
$5$ $$T^{2} - 24T + 576$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 66T + 4356$$
$13$ $$(T + 98)^{2}$$
$17$ $$T^{2} + 216T + 46656$$
$19$ $$T^{2} + 340T + 115600$$
$23$ $$T^{2} - 1038 T + 1077444$$
$29$ $$(T + 2490)^{2}$$
$31$ $$T^{2} + 7048 T + 49674304$$
$37$ $$T^{2} - 12238 T + 149768644$$
$41$ $$(T + 6468)^{2}$$
$43$ $$(T + 15412)^{2}$$
$47$ $$T^{2} - 20604 T + 424524816$$
$53$ $$T^{2} + \cdots + 1055600100$$
$59$ $$T^{2} + \cdots + 1171282176$$
$61$ $$T^{2} + \cdots + 1271207716$$
$67$ $$T^{2} + 12680 T + 160782400$$
$71$ $$(T + 42642)^{2}$$
$73$ $$T^{2} + \cdots + 1137982756$$
$79$ $$T^{2} + \cdots + 7243371664$$
$83$ $$(T - 106764)^{2}$$
$89$ $$T^{2} + \cdots + 1216893456$$
$97$ $$(T + 18662)^{2}$$