# Properties

 Label 294.6.e.k 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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} + 44 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q + 4*z * q^2 + (9*z - 9) * q^3 + (16*z - 16) * q^4 + 44*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} + 44 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + (176 \zeta_{6} - 176) q^{10} + ( - 470 \zeta_{6} + 470) q^{11} - 144 \zeta_{6} q^{12} + 1158 q^{13} - 396 q^{15} - 256 \zeta_{6} q^{16} + ( - 1204 \zeta_{6} + 1204) q^{17} + ( - 324 \zeta_{6} + 324) q^{18} - 2644 \zeta_{6} q^{19} - 704 q^{20} + 1880 q^{22} + 1190 \zeta_{6} q^{23} + ( - 576 \zeta_{6} + 576) q^{24} + ( - 1189 \zeta_{6} + 1189) q^{25} + 4632 \zeta_{6} q^{26} + 729 q^{27} + 3614 q^{29} - 1584 \zeta_{6} q^{30} + ( - 5616 \zeta_{6} + 5616) q^{31} + ( - 1024 \zeta_{6} + 1024) q^{32} + 4230 \zeta_{6} q^{33} + 4816 q^{34} + 1296 q^{36} + 6478 \zeta_{6} q^{37} + ( - 10576 \zeta_{6} + 10576) q^{38} + (10422 \zeta_{6} - 10422) q^{39} - 2816 \zeta_{6} q^{40} - 2856 q^{41} - 13492 q^{43} + 7520 \zeta_{6} q^{44} + ( - 3564 \zeta_{6} + 3564) q^{45} + (4760 \zeta_{6} - 4760) q^{46} - 18372 \zeta_{6} q^{47} + 2304 q^{48} + 4756 q^{50} + 10836 \zeta_{6} q^{51} + (18528 \zeta_{6} - 18528) q^{52} + ( - 4374 \zeta_{6} + 4374) q^{53} + 2916 \zeta_{6} q^{54} + 20680 q^{55} + 23796 q^{57} + 14456 \zeta_{6} q^{58} + ( - 30248 \zeta_{6} + 30248) q^{59} + ( - 6336 \zeta_{6} + 6336) q^{60} + 19542 \zeta_{6} q^{61} + 22464 q^{62} + 4096 q^{64} + 50952 \zeta_{6} q^{65} + (16920 \zeta_{6} - 16920) q^{66} + (54328 \zeta_{6} - 54328) q^{67} + 19264 \zeta_{6} q^{68} - 10710 q^{69} - 10730 q^{71} + 5184 \zeta_{6} q^{72} + ( - 35374 \zeta_{6} + 35374) q^{73} + (25912 \zeta_{6} - 25912) q^{74} + 10701 \zeta_{6} q^{75} + 42304 q^{76} - 41688 q^{78} + 49956 \zeta_{6} q^{79} + ( - 11264 \zeta_{6} + 11264) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 11424 \zeta_{6} q^{82} + 26948 q^{83} + 52976 q^{85} - 53968 \zeta_{6} q^{86} + (32526 \zeta_{6} - 32526) q^{87} + (30080 \zeta_{6} - 30080) q^{88} + 100776 \zeta_{6} q^{89} + 14256 q^{90} - 19040 q^{92} + 50544 \zeta_{6} q^{93} + ( - 73488 \zeta_{6} + 73488) q^{94} + ( - 116336 \zeta_{6} + 116336) q^{95} + 9216 \zeta_{6} q^{96} - 77134 q^{97} - 38070 q^{99} +O(q^{100})$$ q + 4*z * q^2 + (9*z - 9) * q^3 + (16*z - 16) * q^4 + 44*z * q^5 - 36 * q^6 - 64 * q^8 - 81*z * q^9 + (176*z - 176) * q^10 + (-470*z + 470) * q^11 - 144*z * q^12 + 1158 * q^13 - 396 * q^15 - 256*z * q^16 + (-1204*z + 1204) * q^17 + (-324*z + 324) * q^18 - 2644*z * q^19 - 704 * q^20 + 1880 * q^22 + 1190*z * q^23 + (-576*z + 576) * q^24 + (-1189*z + 1189) * q^25 + 4632*z * q^26 + 729 * q^27 + 3614 * q^29 - 1584*z * q^30 + (-5616*z + 5616) * q^31 + (-1024*z + 1024) * q^32 + 4230*z * q^33 + 4816 * q^34 + 1296 * q^36 + 6478*z * q^37 + (-10576*z + 10576) * q^38 + (10422*z - 10422) * q^39 - 2816*z * q^40 - 2856 * q^41 - 13492 * q^43 + 7520*z * q^44 + (-3564*z + 3564) * q^45 + (4760*z - 4760) * q^46 - 18372*z * q^47 + 2304 * q^48 + 4756 * q^50 + 10836*z * q^51 + (18528*z - 18528) * q^52 + (-4374*z + 4374) * q^53 + 2916*z * q^54 + 20680 * q^55 + 23796 * q^57 + 14456*z * q^58 + (-30248*z + 30248) * q^59 + (-6336*z + 6336) * q^60 + 19542*z * q^61 + 22464 * q^62 + 4096 * q^64 + 50952*z * q^65 + (16920*z - 16920) * q^66 + (54328*z - 54328) * q^67 + 19264*z * q^68 - 10710 * q^69 - 10730 * q^71 + 5184*z * q^72 + (-35374*z + 35374) * q^73 + (25912*z - 25912) * q^74 + 10701*z * q^75 + 42304 * q^76 - 41688 * q^78 + 49956*z * q^79 + (-11264*z + 11264) * q^80 + (6561*z - 6561) * q^81 - 11424*z * q^82 + 26948 * q^83 + 52976 * q^85 - 53968*z * q^86 + (32526*z - 32526) * q^87 + (30080*z - 30080) * q^88 + 100776*z * q^89 + 14256 * q^90 - 19040 * q^92 + 50544*z * q^93 + (-73488*z + 73488) * q^94 + (-116336*z + 116336) * q^95 + 9216*z * q^96 - 77134 * q^97 - 38070 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} + 44 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 + 44 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 $$2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} + 44 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} - 176 q^{10} + 470 q^{11} - 144 q^{12} + 2316 q^{13} - 792 q^{15} - 256 q^{16} + 1204 q^{17} + 324 q^{18} - 2644 q^{19} - 1408 q^{20} + 3760 q^{22} + 1190 q^{23} + 576 q^{24} + 1189 q^{25} + 4632 q^{26} + 1458 q^{27} + 7228 q^{29} - 1584 q^{30} + 5616 q^{31} + 1024 q^{32} + 4230 q^{33} + 9632 q^{34} + 2592 q^{36} + 6478 q^{37} + 10576 q^{38} - 10422 q^{39} - 2816 q^{40} - 5712 q^{41} - 26984 q^{43} + 7520 q^{44} + 3564 q^{45} - 4760 q^{46} - 18372 q^{47} + 4608 q^{48} + 9512 q^{50} + 10836 q^{51} - 18528 q^{52} + 4374 q^{53} + 2916 q^{54} + 41360 q^{55} + 47592 q^{57} + 14456 q^{58} + 30248 q^{59} + 6336 q^{60} + 19542 q^{61} + 44928 q^{62} + 8192 q^{64} + 50952 q^{65} - 16920 q^{66} - 54328 q^{67} + 19264 q^{68} - 21420 q^{69} - 21460 q^{71} + 5184 q^{72} + 35374 q^{73} - 25912 q^{74} + 10701 q^{75} + 84608 q^{76} - 83376 q^{78} + 49956 q^{79} + 11264 q^{80} - 6561 q^{81} - 11424 q^{82} + 53896 q^{83} + 105952 q^{85} - 53968 q^{86} - 32526 q^{87} - 30080 q^{88} + 100776 q^{89} + 28512 q^{90} - 38080 q^{92} + 50544 q^{93} + 73488 q^{94} + 116336 q^{95} + 9216 q^{96} - 154268 q^{97} - 76140 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 + 44 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 - 176 * q^10 + 470 * q^11 - 144 * q^12 + 2316 * q^13 - 792 * q^15 - 256 * q^16 + 1204 * q^17 + 324 * q^18 - 2644 * q^19 - 1408 * q^20 + 3760 * q^22 + 1190 * q^23 + 576 * q^24 + 1189 * q^25 + 4632 * q^26 + 1458 * q^27 + 7228 * q^29 - 1584 * q^30 + 5616 * q^31 + 1024 * q^32 + 4230 * q^33 + 9632 * q^34 + 2592 * q^36 + 6478 * q^37 + 10576 * q^38 - 10422 * q^39 - 2816 * q^40 - 5712 * q^41 - 26984 * q^43 + 7520 * q^44 + 3564 * q^45 - 4760 * q^46 - 18372 * q^47 + 4608 * q^48 + 9512 * q^50 + 10836 * q^51 - 18528 * q^52 + 4374 * q^53 + 2916 * q^54 + 41360 * q^55 + 47592 * q^57 + 14456 * q^58 + 30248 * q^59 + 6336 * q^60 + 19542 * q^61 + 44928 * q^62 + 8192 * q^64 + 50952 * q^65 - 16920 * q^66 - 54328 * q^67 + 19264 * q^68 - 21420 * q^69 - 21460 * q^71 + 5184 * q^72 + 35374 * q^73 - 25912 * q^74 + 10701 * q^75 + 84608 * q^76 - 83376 * q^78 + 49956 * q^79 + 11264 * q^80 - 6561 * q^81 - 11424 * q^82 + 53896 * q^83 + 105952 * q^85 - 53968 * q^86 - 32526 * q^87 - 30080 * q^88 + 100776 * q^89 + 28512 * q^90 - 38080 * q^92 + 50544 * q^93 + 73488 * q^94 + 116336 * q^95 + 9216 * q^96 - 154268 * q^97 - 76140 * 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 22.0000 + 38.1051i −36.0000 0 −64.0000 −40.5000 70.1481i −88.0000 + 152.420i
79.1 2.00000 3.46410i −4.50000 7.79423i −8.00000 13.8564i 22.0000 38.1051i −36.0000 0 −64.0000 −40.5000 + 70.1481i −88.0000 152.420i
 $$n$$: 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.k 2
7.b odd 2 1 294.6.e.o 2
7.c even 3 1 294.6.a.f 1
7.c even 3 1 inner 294.6.e.k 2
7.d odd 6 1 42.6.a.b 1
7.d odd 6 1 294.6.e.o 2
21.g even 6 1 126.6.a.h 1
21.h odd 6 1 882.6.a.v 1
28.f even 6 1 336.6.a.o 1
35.i odd 6 1 1050.6.a.o 1
35.k even 12 2 1050.6.g.l 2
84.j odd 6 1 1008.6.a.g 1

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

 $$T_{5}^{2} - 44T_{5} + 1936$$ T5^2 - 44*T5 + 1936 $$T_{11}^{2} - 470T_{11} + 220900$$ T11^2 - 470*T11 + 220900

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 4T + 16$$
$3$ $$T^{2} + 9T + 81$$
$5$ $$T^{2} - 44T + 1936$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 470T + 220900$$
$13$ $$(T - 1158)^{2}$$
$17$ $$T^{2} - 1204 T + 1449616$$
$19$ $$T^{2} + 2644 T + 6990736$$
$23$ $$T^{2} - 1190 T + 1416100$$
$29$ $$(T - 3614)^{2}$$
$31$ $$T^{2} - 5616 T + 31539456$$
$37$ $$T^{2} - 6478 T + 41964484$$
$41$ $$(T + 2856)^{2}$$
$43$ $$(T + 13492)^{2}$$
$47$ $$T^{2} + 18372 T + 337530384$$
$53$ $$T^{2} - 4374 T + 19131876$$
$59$ $$T^{2} - 30248 T + 914941504$$
$61$ $$T^{2} - 19542 T + 381889764$$
$67$ $$T^{2} + \cdots + 2951531584$$
$71$ $$(T + 10730)^{2}$$
$73$ $$T^{2} + \cdots + 1251319876$$
$79$ $$T^{2} + \cdots + 2495601936$$
$83$ $$(T - 26948)^{2}$$
$89$ $$T^{2} + \cdots + 10155802176$$
$97$ $$(T + 77134)^{2}$$