# Properties

 Label 294.4.e.e 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 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (4 \zeta_{6} - 4) q^{4} - 15 \zeta_{6} q^{5} - 6 q^{6} - 8 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q + 2*z * q^2 + (3*z - 3) * q^3 + (4*z - 4) * q^4 - 15*z * q^5 - 6 * q^6 - 8 * q^8 - 9*z * q^9 $$q + 2 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (4 \zeta_{6} - 4) q^{4} - 15 \zeta_{6} q^{5} - 6 q^{6} - 8 q^{8} - 9 \zeta_{6} q^{9} + ( - 30 \zeta_{6} + 30) q^{10} + ( - 9 \zeta_{6} + 9) q^{11} - 12 \zeta_{6} q^{12} + 88 q^{13} + 45 q^{15} - 16 \zeta_{6} q^{16} + (84 \zeta_{6} - 84) q^{17} + ( - 18 \zeta_{6} + 18) q^{18} + 104 \zeta_{6} q^{19} + 60 q^{20} + 18 q^{22} + 84 \zeta_{6} q^{23} + ( - 24 \zeta_{6} + 24) q^{24} + (100 \zeta_{6} - 100) q^{25} + 176 \zeta_{6} q^{26} + 27 q^{27} + 51 q^{29} + 90 \zeta_{6} q^{30} + ( - 185 \zeta_{6} + 185) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} + 27 \zeta_{6} q^{33} - 168 q^{34} + 36 q^{36} - 44 \zeta_{6} q^{37} + (208 \zeta_{6} - 208) q^{38} + (264 \zeta_{6} - 264) q^{39} + 120 \zeta_{6} q^{40} + 168 q^{41} + 326 q^{43} + 36 \zeta_{6} q^{44} + (135 \zeta_{6} - 135) q^{45} + (168 \zeta_{6} - 168) q^{46} - 138 \zeta_{6} q^{47} + 48 q^{48} - 200 q^{50} - 252 \zeta_{6} q^{51} + (352 \zeta_{6} - 352) q^{52} + (639 \zeta_{6} - 639) q^{53} + 54 \zeta_{6} q^{54} - 135 q^{55} - 312 q^{57} + 102 \zeta_{6} q^{58} + ( - 159 \zeta_{6} + 159) q^{59} + (180 \zeta_{6} - 180) q^{60} + 722 \zeta_{6} q^{61} + 370 q^{62} + 64 q^{64} - 1320 \zeta_{6} q^{65} + (54 \zeta_{6} - 54) q^{66} + ( - 166 \zeta_{6} + 166) q^{67} - 336 \zeta_{6} q^{68} - 252 q^{69} + 1086 q^{71} + 72 \zeta_{6} q^{72} + ( - 218 \zeta_{6} + 218) q^{73} + ( - 88 \zeta_{6} + 88) q^{74} - 300 \zeta_{6} q^{75} - 416 q^{76} - 528 q^{78} + 583 \zeta_{6} q^{79} + (240 \zeta_{6} - 240) q^{80} + (81 \zeta_{6} - 81) q^{81} + 336 \zeta_{6} q^{82} + 597 q^{83} + 1260 q^{85} + 652 \zeta_{6} q^{86} + (153 \zeta_{6} - 153) q^{87} + (72 \zeta_{6} - 72) q^{88} - 1038 \zeta_{6} q^{89} - 270 q^{90} - 336 q^{92} + 555 \zeta_{6} q^{93} + ( - 276 \zeta_{6} + 276) q^{94} + ( - 1560 \zeta_{6} + 1560) q^{95} + 96 \zeta_{6} q^{96} + 169 q^{97} - 81 q^{99} +O(q^{100})$$ q + 2*z * q^2 + (3*z - 3) * q^3 + (4*z - 4) * q^4 - 15*z * q^5 - 6 * q^6 - 8 * q^8 - 9*z * q^9 + (-30*z + 30) * q^10 + (-9*z + 9) * q^11 - 12*z * q^12 + 88 * q^13 + 45 * q^15 - 16*z * q^16 + (84*z - 84) * q^17 + (-18*z + 18) * q^18 + 104*z * q^19 + 60 * q^20 + 18 * q^22 + 84*z * q^23 + (-24*z + 24) * q^24 + (100*z - 100) * q^25 + 176*z * q^26 + 27 * q^27 + 51 * q^29 + 90*z * q^30 + (-185*z + 185) * q^31 + (-32*z + 32) * q^32 + 27*z * q^33 - 168 * q^34 + 36 * q^36 - 44*z * q^37 + (208*z - 208) * q^38 + (264*z - 264) * q^39 + 120*z * q^40 + 168 * q^41 + 326 * q^43 + 36*z * q^44 + (135*z - 135) * q^45 + (168*z - 168) * q^46 - 138*z * q^47 + 48 * q^48 - 200 * q^50 - 252*z * q^51 + (352*z - 352) * q^52 + (639*z - 639) * q^53 + 54*z * q^54 - 135 * q^55 - 312 * q^57 + 102*z * q^58 + (-159*z + 159) * q^59 + (180*z - 180) * q^60 + 722*z * q^61 + 370 * q^62 + 64 * q^64 - 1320*z * q^65 + (54*z - 54) * q^66 + (-166*z + 166) * q^67 - 336*z * q^68 - 252 * q^69 + 1086 * q^71 + 72*z * q^72 + (-218*z + 218) * q^73 + (-88*z + 88) * q^74 - 300*z * q^75 - 416 * q^76 - 528 * q^78 + 583*z * q^79 + (240*z - 240) * q^80 + (81*z - 81) * q^81 + 336*z * q^82 + 597 * q^83 + 1260 * q^85 + 652*z * q^86 + (153*z - 153) * q^87 + (72*z - 72) * q^88 - 1038*z * q^89 - 270 * q^90 - 336 * q^92 + 555*z * q^93 + (-276*z + 276) * q^94 + (-1560*z + 1560) * q^95 + 96*z * q^96 + 169 * q^97 - 81 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} - 15 q^{5} - 12 q^{6} - 16 q^{8} - 9 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 3 * q^3 - 4 * q^4 - 15 * q^5 - 12 * q^6 - 16 * q^8 - 9 * q^9 $$2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} - 15 q^{5} - 12 q^{6} - 16 q^{8} - 9 q^{9} + 30 q^{10} + 9 q^{11} - 12 q^{12} + 176 q^{13} + 90 q^{15} - 16 q^{16} - 84 q^{17} + 18 q^{18} + 104 q^{19} + 120 q^{20} + 36 q^{22} + 84 q^{23} + 24 q^{24} - 100 q^{25} + 176 q^{26} + 54 q^{27} + 102 q^{29} + 90 q^{30} + 185 q^{31} + 32 q^{32} + 27 q^{33} - 336 q^{34} + 72 q^{36} - 44 q^{37} - 208 q^{38} - 264 q^{39} + 120 q^{40} + 336 q^{41} + 652 q^{43} + 36 q^{44} - 135 q^{45} - 168 q^{46} - 138 q^{47} + 96 q^{48} - 400 q^{50} - 252 q^{51} - 352 q^{52} - 639 q^{53} + 54 q^{54} - 270 q^{55} - 624 q^{57} + 102 q^{58} + 159 q^{59} - 180 q^{60} + 722 q^{61} + 740 q^{62} + 128 q^{64} - 1320 q^{65} - 54 q^{66} + 166 q^{67} - 336 q^{68} - 504 q^{69} + 2172 q^{71} + 72 q^{72} + 218 q^{73} + 88 q^{74} - 300 q^{75} - 832 q^{76} - 1056 q^{78} + 583 q^{79} - 240 q^{80} - 81 q^{81} + 336 q^{82} + 1194 q^{83} + 2520 q^{85} + 652 q^{86} - 153 q^{87} - 72 q^{88} - 1038 q^{89} - 540 q^{90} - 672 q^{92} + 555 q^{93} + 276 q^{94} + 1560 q^{95} + 96 q^{96} + 338 q^{97} - 162 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 3 * q^3 - 4 * q^4 - 15 * q^5 - 12 * q^6 - 16 * q^8 - 9 * q^9 + 30 * q^10 + 9 * q^11 - 12 * q^12 + 176 * q^13 + 90 * q^15 - 16 * q^16 - 84 * q^17 + 18 * q^18 + 104 * q^19 + 120 * q^20 + 36 * q^22 + 84 * q^23 + 24 * q^24 - 100 * q^25 + 176 * q^26 + 54 * q^27 + 102 * q^29 + 90 * q^30 + 185 * q^31 + 32 * q^32 + 27 * q^33 - 336 * q^34 + 72 * q^36 - 44 * q^37 - 208 * q^38 - 264 * q^39 + 120 * q^40 + 336 * q^41 + 652 * q^43 + 36 * q^44 - 135 * q^45 - 168 * q^46 - 138 * q^47 + 96 * q^48 - 400 * q^50 - 252 * q^51 - 352 * q^52 - 639 * q^53 + 54 * q^54 - 270 * q^55 - 624 * q^57 + 102 * q^58 + 159 * q^59 - 180 * q^60 + 722 * q^61 + 740 * q^62 + 128 * q^64 - 1320 * q^65 - 54 * q^66 + 166 * q^67 - 336 * q^68 - 504 * q^69 + 2172 * q^71 + 72 * q^72 + 218 * q^73 + 88 * q^74 - 300 * q^75 - 832 * q^76 - 1056 * q^78 + 583 * q^79 - 240 * q^80 - 81 * q^81 + 336 * q^82 + 1194 * q^83 + 2520 * q^85 + 652 * q^86 - 153 * q^87 - 72 * q^88 - 1038 * q^89 - 540 * q^90 - 672 * q^92 + 555 * q^93 + 276 * q^94 + 1560 * q^95 + 96 * q^96 + 338 * q^97 - 162 * 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 −7.50000 12.9904i −6.00000 0 −8.00000 −4.50000 7.79423i 15.0000 25.9808i
79.1 1.00000 1.73205i −1.50000 2.59808i −2.00000 3.46410i −7.50000 + 12.9904i −6.00000 0 −8.00000 −4.50000 + 7.79423i 15.0000 + 25.9808i
 $$n$$: 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.e 2
3.b odd 2 1 882.4.g.l 2
7.b odd 2 1 42.4.e.b 2
7.c even 3 1 294.4.a.g 1
7.c even 3 1 inner 294.4.e.e 2
7.d odd 6 1 42.4.e.b 2
7.d odd 6 1 294.4.a.a 1
21.c even 2 1 126.4.g.a 2
21.g even 6 1 126.4.g.a 2
21.g even 6 1 882.4.a.r 1
21.h odd 6 1 882.4.a.h 1
21.h odd 6 1 882.4.g.l 2
28.d even 2 1 336.4.q.d 2
28.f even 6 1 336.4.q.d 2
28.f even 6 1 2352.4.a.u 1
28.g odd 6 1 2352.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 7.b odd 2 1
42.4.e.b 2 7.d odd 6 1
126.4.g.a 2 21.c even 2 1
126.4.g.a 2 21.g even 6 1
294.4.a.a 1 7.d odd 6 1
294.4.a.g 1 7.c even 3 1
294.4.e.e 2 1.a even 1 1 trivial
294.4.e.e 2 7.c even 3 1 inner
336.4.q.d 2 28.d even 2 1
336.4.q.d 2 28.f even 6 1
882.4.a.h 1 21.h odd 6 1
882.4.a.r 1 21.g even 6 1
882.4.g.l 2 3.b odd 2 1
882.4.g.l 2 21.h odd 6 1
2352.4.a.q 1 28.g odd 6 1
2352.4.a.u 1 28.f even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 15T_{5} + 225$$ T5^2 + 15*T5 + 225 $$T_{11}^{2} - 9T_{11} + 81$$ T11^2 - 9*T11 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} + 15T + 225$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 9T + 81$$
$13$ $$(T - 88)^{2}$$
$17$ $$T^{2} + 84T + 7056$$
$19$ $$T^{2} - 104T + 10816$$
$23$ $$T^{2} - 84T + 7056$$
$29$ $$(T - 51)^{2}$$
$31$ $$T^{2} - 185T + 34225$$
$37$ $$T^{2} + 44T + 1936$$
$41$ $$(T - 168)^{2}$$
$43$ $$(T - 326)^{2}$$
$47$ $$T^{2} + 138T + 19044$$
$53$ $$T^{2} + 639T + 408321$$
$59$ $$T^{2} - 159T + 25281$$
$61$ $$T^{2} - 722T + 521284$$
$67$ $$T^{2} - 166T + 27556$$
$71$ $$(T - 1086)^{2}$$
$73$ $$T^{2} - 218T + 47524$$
$79$ $$T^{2} - 583T + 339889$$
$83$ $$(T - 597)^{2}$$
$89$ $$T^{2} + 1038 T + 1077444$$
$97$ $$(T - 169)^{2}$$