# Properties

 Label 294.4.e.g Level $294$ Weight $4$ Character orbit 294.e Analytic conductor $17.347$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("294.67");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$17.3465615417$$ 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 6) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (4 \zeta_{6} - 4) q^{4} + 6 \zeta_{6} q^{5} - 6 q^{6} - 8 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q + 2*z * q^2 + (3*z - 3) * q^3 + (4*z - 4) * q^4 + 6*z * q^5 - 6 * q^6 - 8 * q^8 - 9*z * q^9 $$q + 2 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (4 \zeta_{6} - 4) q^{4} + 6 \zeta_{6} q^{5} - 6 q^{6} - 8 q^{8} - 9 \zeta_{6} q^{9} + (12 \zeta_{6} - 12) q^{10} + (12 \zeta_{6} - 12) q^{11} - 12 \zeta_{6} q^{12} - 38 q^{13} - 18 q^{15} - 16 \zeta_{6} q^{16} + (126 \zeta_{6} - 126) q^{17} + ( - 18 \zeta_{6} + 18) q^{18} + 20 \zeta_{6} q^{19} - 24 q^{20} - 24 q^{22} - 168 \zeta_{6} q^{23} + ( - 24 \zeta_{6} + 24) q^{24} + ( - 89 \zeta_{6} + 89) q^{25} - 76 \zeta_{6} q^{26} + 27 q^{27} + 30 q^{29} - 36 \zeta_{6} q^{30} + (88 \zeta_{6} - 88) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 36 \zeta_{6} q^{33} - 252 q^{34} + 36 q^{36} - 254 \zeta_{6} q^{37} + (40 \zeta_{6} - 40) q^{38} + ( - 114 \zeta_{6} + 114) q^{39} - 48 \zeta_{6} q^{40} - 42 q^{41} - 52 q^{43} - 48 \zeta_{6} q^{44} + ( - 54 \zeta_{6} + 54) q^{45} + ( - 336 \zeta_{6} + 336) q^{46} - 96 \zeta_{6} q^{47} + 48 q^{48} + 178 q^{50} - 378 \zeta_{6} q^{51} + ( - 152 \zeta_{6} + 152) q^{52} + (198 \zeta_{6} - 198) q^{53} + 54 \zeta_{6} q^{54} - 72 q^{55} - 60 q^{57} + 60 \zeta_{6} q^{58} + (660 \zeta_{6} - 660) q^{59} + ( - 72 \zeta_{6} + 72) q^{60} - 538 \zeta_{6} q^{61} - 176 q^{62} + 64 q^{64} - 228 \zeta_{6} q^{65} + ( - 72 \zeta_{6} + 72) q^{66} + (884 \zeta_{6} - 884) q^{67} - 504 \zeta_{6} q^{68} + 504 q^{69} + 792 q^{71} + 72 \zeta_{6} q^{72} + ( - 218 \zeta_{6} + 218) q^{73} + ( - 508 \zeta_{6} + 508) q^{74} + 267 \zeta_{6} q^{75} - 80 q^{76} + 228 q^{78} + 520 \zeta_{6} q^{79} + ( - 96 \zeta_{6} + 96) q^{80} + (81 \zeta_{6} - 81) q^{81} - 84 \zeta_{6} q^{82} + 492 q^{83} - 756 q^{85} - 104 \zeta_{6} q^{86} + (90 \zeta_{6} - 90) q^{87} + ( - 96 \zeta_{6} + 96) q^{88} + 810 \zeta_{6} q^{89} + 108 q^{90} + 672 q^{92} - 264 \zeta_{6} q^{93} + ( - 192 \zeta_{6} + 192) q^{94} + (120 \zeta_{6} - 120) q^{95} + 96 \zeta_{6} q^{96} - 1154 q^{97} + 108 q^{99} +O(q^{100})$$ q + 2*z * q^2 + (3*z - 3) * q^3 + (4*z - 4) * q^4 + 6*z * q^5 - 6 * q^6 - 8 * q^8 - 9*z * q^9 + (12*z - 12) * q^10 + (12*z - 12) * q^11 - 12*z * q^12 - 38 * q^13 - 18 * q^15 - 16*z * q^16 + (126*z - 126) * q^17 + (-18*z + 18) * q^18 + 20*z * q^19 - 24 * q^20 - 24 * q^22 - 168*z * q^23 + (-24*z + 24) * q^24 + (-89*z + 89) * q^25 - 76*z * q^26 + 27 * q^27 + 30 * q^29 - 36*z * q^30 + (88*z - 88) * q^31 + (-32*z + 32) * q^32 - 36*z * q^33 - 252 * q^34 + 36 * q^36 - 254*z * q^37 + (40*z - 40) * q^38 + (-114*z + 114) * q^39 - 48*z * q^40 - 42 * q^41 - 52 * q^43 - 48*z * q^44 + (-54*z + 54) * q^45 + (-336*z + 336) * q^46 - 96*z * q^47 + 48 * q^48 + 178 * q^50 - 378*z * q^51 + (-152*z + 152) * q^52 + (198*z - 198) * q^53 + 54*z * q^54 - 72 * q^55 - 60 * q^57 + 60*z * q^58 + (660*z - 660) * q^59 + (-72*z + 72) * q^60 - 538*z * q^61 - 176 * q^62 + 64 * q^64 - 228*z * q^65 + (-72*z + 72) * q^66 + (884*z - 884) * q^67 - 504*z * q^68 + 504 * q^69 + 792 * q^71 + 72*z * q^72 + (-218*z + 218) * q^73 + (-508*z + 508) * q^74 + 267*z * q^75 - 80 * q^76 + 228 * q^78 + 520*z * q^79 + (-96*z + 96) * q^80 + (81*z - 81) * q^81 - 84*z * q^82 + 492 * q^83 - 756 * q^85 - 104*z * q^86 + (90*z - 90) * q^87 + (-96*z + 96) * q^88 + 810*z * q^89 + 108 * q^90 + 672 * q^92 - 264*z * q^93 + (-192*z + 192) * q^94 + (120*z - 120) * q^95 + 96*z * q^96 - 1154 * q^97 + 108 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 6 q^{5} - 12 q^{6} - 16 q^{8} - 9 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 3 * q^3 - 4 * q^4 + 6 * q^5 - 12 * q^6 - 16 * q^8 - 9 * q^9 $$2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 6 q^{5} - 12 q^{6} - 16 q^{8} - 9 q^{9} - 12 q^{10} - 12 q^{11} - 12 q^{12} - 76 q^{13} - 36 q^{15} - 16 q^{16} - 126 q^{17} + 18 q^{18} + 20 q^{19} - 48 q^{20} - 48 q^{22} - 168 q^{23} + 24 q^{24} + 89 q^{25} - 76 q^{26} + 54 q^{27} + 60 q^{29} - 36 q^{30} - 88 q^{31} + 32 q^{32} - 36 q^{33} - 504 q^{34} + 72 q^{36} - 254 q^{37} - 40 q^{38} + 114 q^{39} - 48 q^{40} - 84 q^{41} - 104 q^{43} - 48 q^{44} + 54 q^{45} + 336 q^{46} - 96 q^{47} + 96 q^{48} + 356 q^{50} - 378 q^{51} + 152 q^{52} - 198 q^{53} + 54 q^{54} - 144 q^{55} - 120 q^{57} + 60 q^{58} - 660 q^{59} + 72 q^{60} - 538 q^{61} - 352 q^{62} + 128 q^{64} - 228 q^{65} + 72 q^{66} - 884 q^{67} - 504 q^{68} + 1008 q^{69} + 1584 q^{71} + 72 q^{72} + 218 q^{73} + 508 q^{74} + 267 q^{75} - 160 q^{76} + 456 q^{78} + 520 q^{79} + 96 q^{80} - 81 q^{81} - 84 q^{82} + 984 q^{83} - 1512 q^{85} - 104 q^{86} - 90 q^{87} + 96 q^{88} + 810 q^{89} + 216 q^{90} + 1344 q^{92} - 264 q^{93} + 192 q^{94} - 120 q^{95} + 96 q^{96} - 2308 q^{97} + 216 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 3 * q^3 - 4 * q^4 + 6 * q^5 - 12 * q^6 - 16 * q^8 - 9 * q^9 - 12 * q^10 - 12 * q^11 - 12 * q^12 - 76 * q^13 - 36 * q^15 - 16 * q^16 - 126 * q^17 + 18 * q^18 + 20 * q^19 - 48 * q^20 - 48 * q^22 - 168 * q^23 + 24 * q^24 + 89 * q^25 - 76 * q^26 + 54 * q^27 + 60 * q^29 - 36 * q^30 - 88 * q^31 + 32 * q^32 - 36 * q^33 - 504 * q^34 + 72 * q^36 - 254 * q^37 - 40 * q^38 + 114 * q^39 - 48 * q^40 - 84 * q^41 - 104 * q^43 - 48 * q^44 + 54 * q^45 + 336 * q^46 - 96 * q^47 + 96 * q^48 + 356 * q^50 - 378 * q^51 + 152 * q^52 - 198 * q^53 + 54 * q^54 - 144 * q^55 - 120 * q^57 + 60 * q^58 - 660 * q^59 + 72 * q^60 - 538 * q^61 - 352 * q^62 + 128 * q^64 - 228 * q^65 + 72 * q^66 - 884 * q^67 - 504 * q^68 + 1008 * q^69 + 1584 * q^71 + 72 * q^72 + 218 * q^73 + 508 * q^74 + 267 * q^75 - 160 * q^76 + 456 * q^78 + 520 * q^79 + 96 * q^80 - 81 * q^81 - 84 * q^82 + 984 * q^83 - 1512 * q^85 - 104 * q^86 - 90 * q^87 + 96 * q^88 + 810 * q^89 + 216 * q^90 + 1344 * q^92 - 264 * q^93 + 192 * q^94 - 120 * q^95 + 96 * q^96 - 2308 * q^97 + 216 * 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
1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 3.00000 + 5.19615i −6.00000 0 −8.00000 −4.50000 7.79423i −6.00000 + 10.3923i
79.1 1.00000 1.73205i −1.50000 2.59808i −2.00000 3.46410i 3.00000 5.19615i −6.00000 0 −8.00000 −4.50000 + 7.79423i −6.00000 10.3923i
 $$n$$: 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 294.4.e.g 2
3.b odd 2 1 882.4.g.f 2
7.b odd 2 1 294.4.e.h 2
7.c even 3 1 294.4.a.e 1
7.c even 3 1 inner 294.4.e.g 2
7.d odd 6 1 6.4.a.a 1
7.d odd 6 1 294.4.e.h 2
21.c even 2 1 882.4.g.i 2
21.g even 6 1 18.4.a.a 1
21.g even 6 1 882.4.g.i 2
21.h odd 6 1 882.4.a.n 1
21.h odd 6 1 882.4.g.f 2
28.f even 6 1 48.4.a.c 1
28.g odd 6 1 2352.4.a.e 1
35.i odd 6 1 150.4.a.i 1
35.k even 12 2 150.4.c.d 2
56.j odd 6 1 192.4.a.i 1
56.m even 6 1 192.4.a.c 1
63.i even 6 1 162.4.c.c 2
63.k odd 6 1 162.4.c.f 2
63.s even 6 1 162.4.c.c 2
63.t odd 6 1 162.4.c.f 2
77.i even 6 1 726.4.a.f 1
84.j odd 6 1 144.4.a.c 1
91.s odd 6 1 1014.4.a.g 1
91.bb even 12 2 1014.4.b.d 2
105.p even 6 1 450.4.a.h 1
105.w odd 12 2 450.4.c.e 2
112.v even 12 2 768.4.d.c 2
112.x odd 12 2 768.4.d.n 2
119.h odd 6 1 1734.4.a.d 1
133.o even 6 1 2166.4.a.i 1
140.s even 6 1 1200.4.a.b 1
140.x odd 12 2 1200.4.f.j 2
168.ba even 6 1 576.4.a.q 1
168.be odd 6 1 576.4.a.r 1
231.k odd 6 1 2178.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 7.d odd 6 1
18.4.a.a 1 21.g even 6 1
48.4.a.c 1 28.f even 6 1
144.4.a.c 1 84.j odd 6 1
150.4.a.i 1 35.i odd 6 1
150.4.c.d 2 35.k even 12 2
162.4.c.c 2 63.i even 6 1
162.4.c.c 2 63.s even 6 1
162.4.c.f 2 63.k odd 6 1
162.4.c.f 2 63.t odd 6 1
192.4.a.c 1 56.m even 6 1
192.4.a.i 1 56.j odd 6 1
294.4.a.e 1 7.c even 3 1
294.4.e.g 2 1.a even 1 1 trivial
294.4.e.g 2 7.c even 3 1 inner
294.4.e.h 2 7.b odd 2 1
294.4.e.h 2 7.d odd 6 1
450.4.a.h 1 105.p even 6 1
450.4.c.e 2 105.w odd 12 2
576.4.a.q 1 168.ba even 6 1
576.4.a.r 1 168.be odd 6 1
726.4.a.f 1 77.i even 6 1
768.4.d.c 2 112.v even 12 2
768.4.d.n 2 112.x odd 12 2
882.4.a.n 1 21.h odd 6 1
882.4.g.f 2 3.b odd 2 1
882.4.g.f 2 21.h odd 6 1
882.4.g.i 2 21.c even 2 1
882.4.g.i 2 21.g even 6 1
1014.4.a.g 1 91.s odd 6 1
1014.4.b.d 2 91.bb even 12 2
1200.4.a.b 1 140.s even 6 1
1200.4.f.j 2 140.x odd 12 2
1734.4.a.d 1 119.h odd 6 1
2166.4.a.i 1 133.o even 6 1
2178.4.a.e 1 231.k odd 6 1
2352.4.a.e 1 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{2} - 6T_{5} + 36$$ T5^2 - 6*T5 + 36 $$T_{11}^{2} + 12T_{11} + 144$$ T11^2 + 12*T11 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} - 6T + 36$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 12T + 144$$
$13$ $$(T + 38)^{2}$$
$17$ $$T^{2} + 126T + 15876$$
$19$ $$T^{2} - 20T + 400$$
$23$ $$T^{2} + 168T + 28224$$
$29$ $$(T - 30)^{2}$$
$31$ $$T^{2} + 88T + 7744$$
$37$ $$T^{2} + 254T + 64516$$
$41$ $$(T + 42)^{2}$$
$43$ $$(T + 52)^{2}$$
$47$ $$T^{2} + 96T + 9216$$
$53$ $$T^{2} + 198T + 39204$$
$59$ $$T^{2} + 660T + 435600$$
$61$ $$T^{2} + 538T + 289444$$
$67$ $$T^{2} + 884T + 781456$$
$71$ $$(T - 792)^{2}$$
$73$ $$T^{2} - 218T + 47524$$
$79$ $$T^{2} - 520T + 270400$$
$83$ $$(T - 492)^{2}$$
$89$ $$T^{2} - 810T + 656100$$
$97$ $$(T + 1154)^{2}$$