# Properties

 Label 294.3.b.e Level $294$ Weight $3$ Character orbit 294.b Analytic conductor $8.011$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 294.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.01091977219$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - 2x^{3} + 11x^{2} - 10x + 3$$ x^4 - 2*x^3 + 11*x^2 - 10*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{3} - 2 q^{4} + (2 \beta_{3} - 2 \beta_{2} - 1) q^{5} + (2 \beta_{3} - \beta_{2} - \beta_1) q^{6} - 2 \beta_{2} q^{8} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 3) q^{9}+O(q^{10})$$ q + b2 * q^2 + (-b3 - b2 - b1) * q^3 - 2 * q^4 + (2*b3 - 2*b2 - 1) * q^5 + (2*b3 - b2 - b1) * q^6 - 2*b2 * q^8 + (-b3 - 4*b2 + 2*b1 + 3) * q^9 $$q + \beta_{2} q^{2} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{3} - 2 q^{4} + (2 \beta_{3} - 2 \beta_{2} - 1) q^{5} + (2 \beta_{3} - \beta_{2} - \beta_1) q^{6} - 2 \beta_{2} q^{8} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 3) q^{9} + (\beta_{2} + 2 \beta_1 + 4) q^{10} + ( - 10 \beta_{3} + \beta_{2} + 5) q^{11} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{12} + ( - 2 \beta_{2} - 4 \beta_1 + 8) q^{13} + ( - 5 \beta_{3} + 7 \beta_{2} + \beta_1 + 6) q^{15} + 4 q^{16} + ( - 2 \beta_{3} + 2 \beta_{2} + 1) q^{17} + ( - 4 \beta_{3} + 2 \beta_{2} - \beta_1 + 12) q^{18} + (3 \beta_{2} + 6 \beta_1 + 1) q^{19} + ( - 4 \beta_{3} + 4 \beta_{2} + 2) q^{20} + ( - 5 \beta_{2} - 10 \beta_1 - 2) q^{22} + ( - 10 \beta_{3} + 7 \beta_{2} + 5) q^{23} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{24} + ( - 4 \beta_{2} - 8 \beta_1 + 6) q^{25} + (8 \beta_{3} + 8 \beta_{2} - 4) q^{26} + ( - 10 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 15) q^{27} + ( - 20 \beta_{3} - 10 \beta_{2} + 10) q^{29} + ( - 2 \beta_{3} + \beta_{2} - 5 \beta_1 - 12) q^{30} + ( - 5 \beta_{2} - 10 \beta_1 + 5) q^{31} + 4 \beta_{2} q^{32} + (7 \beta_{3} - 26 \beta_{2} + 4 \beta_1 - 30) q^{33} + ( - \beta_{2} - 2 \beta_1 - 4) q^{34} + (2 \beta_{3} + 8 \beta_{2} - 4 \beta_1 - 6) q^{36} - q^{37} + ( - 12 \beta_{3} + \beta_{2} + 6) q^{38} + ( - 12 \beta_{3} - 18 \beta_{2} - 6 \beta_1 + 24) q^{39} + ( - 2 \beta_{2} - 4 \beta_1 - 8) q^{40} + ( - 12 \beta_{3} - 6 \beta_{2} + 6) q^{41} + (6 \beta_{2} + 12 \beta_1 + 26) q^{43} + (20 \beta_{3} - 2 \beta_{2} - 10) q^{44} + (13 \beta_{3} - 20 \beta_{2} - 8 \beta_1 - 21) q^{45} + ( - 5 \beta_{2} - 10 \beta_1 - 14) q^{46} + (6 \beta_{3} + 45 \beta_{2} - 3) q^{47} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{48} + (16 \beta_{3} + 6 \beta_{2} - 8) q^{50} + (5 \beta_{3} - 7 \beta_{2} - \beta_1 - 6) q^{51} + (4 \beta_{2} + 8 \beta_1 - 16) q^{52} + (6 \beta_{3} + 36 \beta_{2} - 3) q^{53} + ( - 4 \beta_{3} - 25 \beta_{2} - 10 \beta_1 - 6) q^{54} + (11 \beta_{2} + 22 \beta_1 + 59) q^{55} + (5 \beta_{3} + 14 \beta_{2} - 4 \beta_1 - 36) q^{57} + ( - 10 \beta_{2} - 20 \beta_1 + 20) q^{58} + (14 \beta_{3} + 55 \beta_{2} - 7) q^{59} + (10 \beta_{3} - 14 \beta_{2} - 2 \beta_1 - 12) q^{60} + (4 \beta_{2} + 8 \beta_1 + 53) q^{61} + (20 \beta_{3} + 5 \beta_{2} - 10) q^{62} - 8 q^{64} + 6 \beta_{2} q^{65} + ( - 8 \beta_{3} - 23 \beta_{2} + 7 \beta_1 + 60) q^{66} + (13 \beta_{2} + 26 \beta_1 - 39) q^{67} + (4 \beta_{3} - 4 \beta_{2} - 2) q^{68} + (19 \beta_{3} - 32 \beta_{2} - 2 \beta_1 - 30) q^{69} + (8 \beta_{3} - 68 \beta_{2} - 4) q^{71} + (8 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 24) q^{72} + ( - 4 \beta_{2} - 8 \beta_1 + 33) q^{73} - \beta_{2} q^{74} + ( - 14 \beta_{3} - 26 \beta_{2} - 2 \beta_1 + 48) q^{75} + ( - 6 \beta_{2} - 12 \beta_1 - 2) q^{76} + (12 \beta_{3} + 12 \beta_{2} - 12 \beta_1 + 24) q^{78} + (17 \beta_{2} + 34 \beta_1 + 13) q^{79} + (8 \beta_{3} - 8 \beta_{2} - 4) q^{80} + (35 \beta_{3} - 4 \beta_{2} + 20 \beta_1 - 42) q^{81} + ( - 6 \beta_{2} - 12 \beta_1 + 12) q^{82} + (16 \beta_{3} + 44 \beta_{2} - 8) q^{83} + (4 \beta_{2} + 8 \beta_1 + 19) q^{85} + ( - 24 \beta_{3} + 26 \beta_{2} + 12) q^{86} + ( - 10 \beta_{3} - 40 \beta_{2} + 20 \beta_1 - 60) q^{87} + (10 \beta_{2} + 20 \beta_1 + 4) q^{88} + ( - 66 \beta_{3} + 36 \beta_{2} + 33) q^{89} + (16 \beta_{3} - 8 \beta_{2} + 13 \beta_1 + 24) q^{90} + (20 \beta_{3} - 14 \beta_{2} - 10) q^{92} + ( - 15 \beta_{3} - 30 \beta_{2} + 60) q^{93} + (3 \beta_{2} + 6 \beta_1 - 90) q^{94} + (26 \beta_{3} - 35 \beta_{2} - 13) q^{95} + (8 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{96} + ( - 10 \beta_{2} - 20 \beta_1 - 72) q^{97} + ( - 29 \beta_{3} + 82 \beta_{2} + 49 \beta_1 - 3) q^{99}+O(q^{100})$$ q + b2 * q^2 + (-b3 - b2 - b1) * q^3 - 2 * q^4 + (2*b3 - 2*b2 - 1) * q^5 + (2*b3 - b2 - b1) * q^6 - 2*b2 * q^8 + (-b3 - 4*b2 + 2*b1 + 3) * q^9 + (b2 + 2*b1 + 4) * q^10 + (-10*b3 + b2 + 5) * q^11 + (2*b3 + 2*b2 + 2*b1) * q^12 + (-2*b2 - 4*b1 + 8) * q^13 + (-5*b3 + 7*b2 + b1 + 6) * q^15 + 4 * q^16 + (-2*b3 + 2*b2 + 1) * q^17 + (-4*b3 + 2*b2 - b1 + 12) * q^18 + (3*b2 + 6*b1 + 1) * q^19 + (-4*b3 + 4*b2 + 2) * q^20 + (-5*b2 - 10*b1 - 2) * q^22 + (-10*b3 + 7*b2 + 5) * q^23 + (-4*b3 + 2*b2 + 2*b1) * q^24 + (-4*b2 - 8*b1 + 6) * q^25 + (8*b3 + 8*b2 - 4) * q^26 + (-10*b3 + 5*b2 + 2*b1 - 15) * q^27 + (-20*b3 - 10*b2 + 10) * q^29 + (-2*b3 + b2 - 5*b1 - 12) * q^30 + (-5*b2 - 10*b1 + 5) * q^31 + 4*b2 * q^32 + (7*b3 - 26*b2 + 4*b1 - 30) * q^33 + (-b2 - 2*b1 - 4) * q^34 + (2*b3 + 8*b2 - 4*b1 - 6) * q^36 - q^37 + (-12*b3 + b2 + 6) * q^38 + (-12*b3 - 18*b2 - 6*b1 + 24) * q^39 + (-2*b2 - 4*b1 - 8) * q^40 + (-12*b3 - 6*b2 + 6) * q^41 + (6*b2 + 12*b1 + 26) * q^43 + (20*b3 - 2*b2 - 10) * q^44 + (13*b3 - 20*b2 - 8*b1 - 21) * q^45 + (-5*b2 - 10*b1 - 14) * q^46 + (6*b3 + 45*b2 - 3) * q^47 + (-4*b3 - 4*b2 - 4*b1) * q^48 + (16*b3 + 6*b2 - 8) * q^50 + (5*b3 - 7*b2 - b1 - 6) * q^51 + (4*b2 + 8*b1 - 16) * q^52 + (6*b3 + 36*b2 - 3) * q^53 + (-4*b3 - 25*b2 - 10*b1 - 6) * q^54 + (11*b2 + 22*b1 + 59) * q^55 + (5*b3 + 14*b2 - 4*b1 - 36) * q^57 + (-10*b2 - 20*b1 + 20) * q^58 + (14*b3 + 55*b2 - 7) * q^59 + (10*b3 - 14*b2 - 2*b1 - 12) * q^60 + (4*b2 + 8*b1 + 53) * q^61 + (20*b3 + 5*b2 - 10) * q^62 - 8 * q^64 + 6*b2 * q^65 + (-8*b3 - 23*b2 + 7*b1 + 60) * q^66 + (13*b2 + 26*b1 - 39) * q^67 + (4*b3 - 4*b2 - 2) * q^68 + (19*b3 - 32*b2 - 2*b1 - 30) * q^69 + (8*b3 - 68*b2 - 4) * q^71 + (8*b3 - 4*b2 + 2*b1 - 24) * q^72 + (-4*b2 - 8*b1 + 33) * q^73 - b2 * q^74 + (-14*b3 - 26*b2 - 2*b1 + 48) * q^75 + (-6*b2 - 12*b1 - 2) * q^76 + (12*b3 + 12*b2 - 12*b1 + 24) * q^78 + (17*b2 + 34*b1 + 13) * q^79 + (8*b3 - 8*b2 - 4) * q^80 + (35*b3 - 4*b2 + 20*b1 - 42) * q^81 + (-6*b2 - 12*b1 + 12) * q^82 + (16*b3 + 44*b2 - 8) * q^83 + (4*b2 + 8*b1 + 19) * q^85 + (-24*b3 + 26*b2 + 12) * q^86 + (-10*b3 - 40*b2 + 20*b1 - 60) * q^87 + (10*b2 + 20*b1 + 4) * q^88 + (-66*b3 + 36*b2 + 33) * q^89 + (16*b3 - 8*b2 + 13*b1 + 24) * q^90 + (20*b3 - 14*b2 - 10) * q^92 + (-15*b3 - 30*b2 + 60) * q^93 + (3*b2 + 6*b1 - 90) * q^94 + (26*b3 - 35*b2 - 13) * q^95 + (8*b3 - 4*b2 - 4*b1) * q^96 + (-10*b2 - 20*b1 - 72) * q^97 + (-29*b3 + 82*b2 + 49*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 8 q^{4} + 4 q^{6} + 10 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 8 * q^4 + 4 * q^6 + 10 * q^9 $$4 q - 2 q^{3} - 8 q^{4} + 4 q^{6} + 10 q^{9} + 16 q^{10} + 4 q^{12} + 32 q^{13} + 14 q^{15} + 16 q^{16} + 40 q^{18} + 4 q^{19} - 8 q^{22} - 8 q^{24} + 24 q^{25} - 80 q^{27} - 52 q^{30} + 20 q^{31} - 106 q^{33} - 16 q^{34} - 20 q^{36} - 4 q^{37} + 72 q^{39} - 32 q^{40} + 104 q^{43} - 58 q^{45} - 56 q^{46} - 8 q^{48} - 14 q^{51} - 64 q^{52} - 32 q^{54} + 236 q^{55} - 134 q^{57} + 80 q^{58} - 28 q^{60} + 212 q^{61} - 32 q^{64} + 224 q^{66} - 156 q^{67} - 82 q^{69} - 80 q^{72} + 132 q^{73} + 164 q^{75} - 8 q^{76} + 120 q^{78} + 52 q^{79} - 98 q^{81} + 48 q^{82} + 76 q^{85} - 260 q^{87} + 16 q^{88} + 128 q^{90} + 210 q^{93} - 360 q^{94} + 16 q^{96} - 288 q^{97} - 70 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 8 * q^4 + 4 * q^6 + 10 * q^9 + 16 * q^10 + 4 * q^12 + 32 * q^13 + 14 * q^15 + 16 * q^16 + 40 * q^18 + 4 * q^19 - 8 * q^22 - 8 * q^24 + 24 * q^25 - 80 * q^27 - 52 * q^30 + 20 * q^31 - 106 * q^33 - 16 * q^34 - 20 * q^36 - 4 * q^37 + 72 * q^39 - 32 * q^40 + 104 * q^43 - 58 * q^45 - 56 * q^46 - 8 * q^48 - 14 * q^51 - 64 * q^52 - 32 * q^54 + 236 * q^55 - 134 * q^57 + 80 * q^58 - 28 * q^60 + 212 * q^61 - 32 * q^64 + 224 * q^66 - 156 * q^67 - 82 * q^69 - 80 * q^72 + 132 * q^73 + 164 * q^75 - 8 * q^76 + 120 * q^78 + 52 * q^79 - 98 * q^81 + 48 * q^82 + 76 * q^85 - 260 * q^87 + 16 * q^88 + 128 * q^90 + 210 * q^93 - 360 * q^94 + 16 * q^96 - 288 * q^97 - 70 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} + 11x^{2} - 10x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 8\nu + 3 ) / 3$$ (v^3 + 8*v + 3) / 3 $$\beta_{2}$$ $$=$$ $$( -2\nu^{3} + 3\nu^{2} - 19\nu + 9 ) / 3$$ (-2*v^3 + 3*v^2 - 19*v + 9) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} - 3\nu^{2} + 22\nu - 9 ) / 3$$ (2*v^3 - 3*v^2 + 22*v - 9) / 3
 $$\nu$$ $$=$$ $$\beta_{3} + \beta_{2}$$ b3 + b2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 2\beta _1 - 5$$ b3 + 2*b2 + 2*b1 - 5 $$\nu^{3}$$ $$=$$ $$-8\beta_{3} - 8\beta_{2} + 3\beta _1 - 3$$ -8*b3 - 8*b2 + 3*b1 - 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/294\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$199$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 0.5 + 0.244099i 0.5 − 3.07253i 0.5 − 0.244099i 0.5 + 3.07253i
1.41421i −2.84521 0.951206i −2.00000 6.14505i −1.34521 + 4.02373i 0 2.82843i 7.19042 + 5.41276i 8.69042
197.2 1.41421i 1.84521 + 2.36542i −2.00000 0.488198i 3.34521 2.60952i 0 2.82843i −2.19042 + 8.72938i −0.690416
197.3 1.41421i −2.84521 + 0.951206i −2.00000 6.14505i −1.34521 4.02373i 0 2.82843i 7.19042 5.41276i 8.69042
197.4 1.41421i 1.84521 2.36542i −2.00000 0.488198i 3.34521 + 2.60952i 0 2.82843i −2.19042 8.72938i −0.690416
 $$n$$: 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.3.b.e 4
3.b odd 2 1 inner 294.3.b.e 4
7.b odd 2 1 294.3.b.i 4
7.c even 3 2 294.3.h.h 8
7.d odd 6 2 42.3.h.b 8
21.c even 2 1 294.3.b.i 4
21.g even 6 2 42.3.h.b 8
21.h odd 6 2 294.3.h.h 8
28.f even 6 2 336.3.bn.g 8
84.j odd 6 2 336.3.bn.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.h.b 8 7.d odd 6 2
42.3.h.b 8 21.g even 6 2
294.3.b.e 4 1.a even 1 1 trivial
294.3.b.e 4 3.b odd 2 1 inner
294.3.b.i 4 7.b odd 2 1
294.3.b.i 4 21.c even 2 1
294.3.h.h 8 7.c even 3 2
294.3.h.h 8 21.h odd 6 2
336.3.bn.g 8 28.f even 6 2
336.3.bn.g 8 84.j odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{4} + 38T_{5}^{2} + 9$$ T5^4 + 38*T5^2 + 9 $$T_{13}^{2} - 16T_{13} - 24$$ T13^2 - 16*T13 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{2}$$
$3$ $$T^{4} + 2 T^{3} - 3 T^{2} + 18 T + 81$$
$5$ $$T^{4} + 38T^{2} + 9$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 554 T^{2} + 74529$$
$13$ $$(T^{2} - 16 T - 24)^{2}$$
$17$ $$T^{4} + 38T^{2} + 9$$
$19$ $$(T^{2} - 2 T - 197)^{2}$$
$23$ $$T^{4} + 746 T^{2} + 31329$$
$29$ $$T^{4} + 2600 T^{2} + 810000$$
$31$ $$(T^{2} - 10 T - 525)^{2}$$
$37$ $$(T + 1)^{4}$$
$41$ $$T^{4} + 936 T^{2} + 104976$$
$43$ $$(T^{2} - 52 T - 116)^{2}$$
$47$ $$T^{4} + 8298 T^{2} + \cdots + 15610401$$
$53$ $$T^{4} + 5382 T^{2} + \cdots + 6215049$$
$59$ $$T^{4} + 13178 T^{2} + \cdots + 30371121$$
$61$ $$(T^{2} - 106 T + 2457)^{2}$$
$67$ $$(T^{2} + 78 T - 2197)^{2}$$
$71$ $$T^{4} + 18848 T^{2} + \cdots + 82301184$$
$73$ $$(T^{2} - 66 T + 737)^{2}$$
$79$ $$(T^{2} - 26 T - 6189)^{2}$$
$83$ $$T^{4} + 9152 T^{2} + \cdots + 10036224$$
$89$ $$T^{4} + 29142 T^{2} + \cdots + 88115769$$
$97$ $$(T^{2} + 144 T + 2984)^{2}$$