Properties

 Label 693.4.a.l Level $693$ Weight $4$ Character orbit 693.a Self dual yes Analytic conductor $40.888$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 693.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$40.8883236340$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.522072.1 Defining polynomial: $$x^{4} - x^{3} - 12x^{2} + 5x + 1$$ x^4 - x^3 - 12*x^2 + 5*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - 2 \beta_{3} - \beta_{2} + 6) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{5} - 7 q^{7} + ( - 5 \beta_{2} + 4 \beta_1 + 4) q^{8}+O(q^{10})$$ q - b2 * q^2 + (-2*b3 - b2 + 6) * q^4 + (b3 + b2 - 2*b1 - 3) * q^5 - 7 * q^7 + (-5*b2 + 4*b1 + 4) * q^8 $$q - \beta_{2} q^{2} + ( - 2 \beta_{3} - \beta_{2} + 6) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{5} - 7 q^{7} + ( - 5 \beta_{2} + 4 \beta_1 + 4) q^{8} + (7 \beta_{3} + 9 \beta_{2} - 6 \beta_1 + 1) q^{10} + 11 q^{11} + ( - 10 \beta_{3} - 10 \beta_{2} - 3 \beta_1 + 8) q^{13} + 7 \beta_{2} q^{14} + ( - 6 \beta_{3} - 5 \beta_{2} + 8 \beta_1 + 2) q^{16} + ( - 9 \beta_{3} - 13 \beta_{2} + 5 \beta_1 - 5) q^{17} + ( - 7 \beta_{3} - 15 \beta_{2} - 4 \beta_1 + 55) q^{19} + (21 \beta_{3} + 27 \beta_{2} - 10 \beta_1 - 37) q^{20} - 11 \beta_{2} q^{22} + (6 \beta_{3} + 2 \beta_{2} - 18 \beta_1 - 10) q^{23} + (2 \beta_{3} - 26 \beta_{2} - 4 \beta_1 + 5) q^{25} + ( - \beta_{3} - 45 \beta_{2} + 14 \beta_1 + 105) q^{26} + (14 \beta_{3} + 7 \beta_{2} - 42) q^{28} + (8 \beta_{3} - 4 \beta_{2} - 14 \beta_1 + 90) q^{29} + ( - 25 \beta_{3} + 7 \beta_{2} - 9 \beta_1 - 15) q^{31} + ( - 28 \beta_{3} + 7 \beta_{2} - 4 \beta_1 - 32) q^{32} + ( - 32 \beta_{3} - 40 \beta_{2} + 28 \beta_1 + 112) q^{34} + ( - 7 \beta_{3} - 7 \beta_{2} + 14 \beta_1 + 21) q^{35} + ( - 32 \beta_{3} + 14 \beta_{2} - 16 \beta_1 + 20) q^{37} + ( - 11 \beta_{3} - 87 \beta_{2} + 6 \beta_1 + 195) q^{38} + (7 \beta_{3} + 65 \beta_{2} - 14 \beta_1 - 231) q^{40} + ( - 23 \beta_{3} - 51 \beta_{2} + 23 \beta_1 - 27) q^{41} + ( - 74 \beta_{2} + 2 \beta_1 + 66) q^{43} + ( - 22 \beta_{3} - 11 \beta_{2} + 66) q^{44} + (52 \beta_{3} + 48 \beta_{2} - 48 \beta_1 + 92) q^{46} + ( - \beta_{3} + 15 \beta_{2} + 39 \beta_1 + 25) q^{47} + 49 q^{49} + ( - 42 \beta_{3} - 21 \beta_{2} - 12 \beta_1 + 394) q^{50} + ( - 51 \beta_{3} - 87 \beta_{2} + 54 \beta_1 + 491) q^{52} + ( - 10 \beta_{3} + 4 \beta_{2} + 8 \beta_1 - 150) q^{53} + (11 \beta_{3} + 11 \beta_{2} - 22 \beta_1 - 33) q^{55} + (35 \beta_{2} - 28 \beta_1 - 28) q^{56} + (26 \beta_{3} - 56 \beta_{2} - 44 \beta_1 + 166) q^{58} + (24 \beta_{3} - 124 \beta_{2} - 3 \beta_1 - 4) q^{59} + (120 \beta_{3} + 24 \beta_{2} + 29 \beta_1 - 14) q^{61} + (66 \beta_{3} - 44 \beta_{2} + 32 \beta_1 - 178) q^{62} + (102 \beta_{3} - \beta_{2} - 16 \beta_1 - 234) q^{64} + (142 \beta_{3} + 146 \beta_{2} - 54 \beta_1 + 54) q^{65} + (124 \beta_{3} + 120 \beta_{2} - 62 \beta_1 + 364) q^{67} + ( - 60 \beta_{3} - 172 \beta_{2} + 80 \beta_1 + 300) q^{68} + ( - 49 \beta_{3} - 63 \beta_{2} + 42 \beta_1 - 7) q^{70} + ( - 96 \beta_{3} - 92 \beta_{2} + 26 \beta_1 - 480) q^{71} + (61 \beta_{3} + 217 \beta_{2} - 47 \beta_1 + 449) q^{73} + (108 \beta_{3} - 86 \beta_{2} + 32 \beta_1 - 276) q^{74} + ( - 125 \beta_{3} - 201 \beta_{2} + 66 \beta_1 + 693) q^{76} - 77 q^{77} + ( - 114 \beta_{3} + 28 \beta_{2} - 6 \beta_1 - 320) q^{79} + ( - 3 \beta_{3} + 115 \beta_{2} + 38 \beta_1 - 509) q^{80} + ( - 148 \beta_{3} - 116 \beta_{2} + 92 \beta_1 + 484) q^{82} + (43 \beta_{3} - 25 \beta_{2} + 76 \beta_1 - 87) q^{83} + (82 \beta_{3} + 186 \beta_{2} - 2 \beta_1 - 342) q^{85} + ( - 154 \beta_{3} - 142 \beta_{2} + 4 \beta_1 + 1026) q^{86} + ( - 55 \beta_{2} + 44 \beta_1 + 44) q^{88} + (104 \beta_{3} - 44 \beta_{2} + 48 \beta_1 + 770) q^{89} + (70 \beta_{3} + 70 \beta_{2} + 21 \beta_1 - 56) q^{91} + (140 \beta_{3} + 144 \beta_{2} - 56 \beta_1 - 92) q^{92} + ( - 86 \beta_{3} - 52 \beta_{2} + 80 \beta_1 - 410) q^{94} + (220 \beta_{3} + 216 \beta_{2} - 196 \beta_1 + 4) q^{95} + (32 \beta_{3} - 176 \beta_{2} + 154 \beta_1 - 174) q^{97} - 49 \beta_{2} q^{98}+O(q^{100})$$ q - b2 * q^2 + (-2*b3 - b2 + 6) * q^4 + (b3 + b2 - 2*b1 - 3) * q^5 - 7 * q^7 + (-5*b2 + 4*b1 + 4) * q^8 + (7*b3 + 9*b2 - 6*b1 + 1) * q^10 + 11 * q^11 + (-10*b3 - 10*b2 - 3*b1 + 8) * q^13 + 7*b2 * q^14 + (-6*b3 - 5*b2 + 8*b1 + 2) * q^16 + (-9*b3 - 13*b2 + 5*b1 - 5) * q^17 + (-7*b3 - 15*b2 - 4*b1 + 55) * q^19 + (21*b3 + 27*b2 - 10*b1 - 37) * q^20 - 11*b2 * q^22 + (6*b3 + 2*b2 - 18*b1 - 10) * q^23 + (2*b3 - 26*b2 - 4*b1 + 5) * q^25 + (-b3 - 45*b2 + 14*b1 + 105) * q^26 + (14*b3 + 7*b2 - 42) * q^28 + (8*b3 - 4*b2 - 14*b1 + 90) * q^29 + (-25*b3 + 7*b2 - 9*b1 - 15) * q^31 + (-28*b3 + 7*b2 - 4*b1 - 32) * q^32 + (-32*b3 - 40*b2 + 28*b1 + 112) * q^34 + (-7*b3 - 7*b2 + 14*b1 + 21) * q^35 + (-32*b3 + 14*b2 - 16*b1 + 20) * q^37 + (-11*b3 - 87*b2 + 6*b1 + 195) * q^38 + (7*b3 + 65*b2 - 14*b1 - 231) * q^40 + (-23*b3 - 51*b2 + 23*b1 - 27) * q^41 + (-74*b2 + 2*b1 + 66) * q^43 + (-22*b3 - 11*b2 + 66) * q^44 + (52*b3 + 48*b2 - 48*b1 + 92) * q^46 + (-b3 + 15*b2 + 39*b1 + 25) * q^47 + 49 * q^49 + (-42*b3 - 21*b2 - 12*b1 + 394) * q^50 + (-51*b3 - 87*b2 + 54*b1 + 491) * q^52 + (-10*b3 + 4*b2 + 8*b1 - 150) * q^53 + (11*b3 + 11*b2 - 22*b1 - 33) * q^55 + (35*b2 - 28*b1 - 28) * q^56 + (26*b3 - 56*b2 - 44*b1 + 166) * q^58 + (24*b3 - 124*b2 - 3*b1 - 4) * q^59 + (120*b3 + 24*b2 + 29*b1 - 14) * q^61 + (66*b3 - 44*b2 + 32*b1 - 178) * q^62 + (102*b3 - b2 - 16*b1 - 234) * q^64 + (142*b3 + 146*b2 - 54*b1 + 54) * q^65 + (124*b3 + 120*b2 - 62*b1 + 364) * q^67 + (-60*b3 - 172*b2 + 80*b1 + 300) * q^68 + (-49*b3 - 63*b2 + 42*b1 - 7) * q^70 + (-96*b3 - 92*b2 + 26*b1 - 480) * q^71 + (61*b3 + 217*b2 - 47*b1 + 449) * q^73 + (108*b3 - 86*b2 + 32*b1 - 276) * q^74 + (-125*b3 - 201*b2 + 66*b1 + 693) * q^76 - 77 * q^77 + (-114*b3 + 28*b2 - 6*b1 - 320) * q^79 + (-3*b3 + 115*b2 + 38*b1 - 509) * q^80 + (-148*b3 - 116*b2 + 92*b1 + 484) * q^82 + (43*b3 - 25*b2 + 76*b1 - 87) * q^83 + (82*b3 + 186*b2 - 2*b1 - 342) * q^85 + (-154*b3 - 142*b2 + 4*b1 + 1026) * q^86 + (-55*b2 + 44*b1 + 44) * q^88 + (104*b3 - 44*b2 + 48*b1 + 770) * q^89 + (70*b3 + 70*b2 + 21*b1 - 56) * q^91 + (140*b3 + 144*b2 - 56*b1 - 92) * q^92 + (-86*b3 - 52*b2 + 80*b1 - 410) * q^94 + (220*b3 + 216*b2 - 196*b1 + 4) * q^95 + (32*b3 - 176*b2 + 154*b1 - 174) * q^97 - 49*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 26 q^{4} - 10 q^{5} - 28 q^{7} + 18 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 + 26 * q^4 - 10 * q^5 - 28 * q^7 + 18 * q^8 $$4 q + 2 q^{2} + 26 q^{4} - 10 q^{5} - 28 q^{7} + 18 q^{8} - 2 q^{10} + 44 q^{11} + 58 q^{13} - 14 q^{14} + 2 q^{16} - 4 q^{17} + 258 q^{19} - 182 q^{20} + 22 q^{22} - 8 q^{23} + 80 q^{25} + 482 q^{26} - 182 q^{28} + 396 q^{29} - 56 q^{31} - 134 q^{32} + 472 q^{34} + 70 q^{35} + 84 q^{37} + 942 q^{38} - 1026 q^{40} - 52 q^{41} + 408 q^{43} + 286 q^{44} + 368 q^{46} - 8 q^{47} + 196 q^{49} + 1642 q^{50} + 2030 q^{52} - 624 q^{53} - 110 q^{55} - 126 q^{56} + 864 q^{58} + 238 q^{59} - 162 q^{61} - 688 q^{62} - 902 q^{64} + 32 q^{65} + 1340 q^{67} + 1384 q^{68} + 14 q^{70} - 1788 q^{71} + 1456 q^{73} - 996 q^{74} + 3042 q^{76} - 308 q^{77} - 1324 q^{79} - 2342 q^{80} + 1984 q^{82} - 450 q^{83} - 1736 q^{85} + 4380 q^{86} + 198 q^{88} + 3072 q^{89} - 406 q^{91} - 544 q^{92} - 1696 q^{94} - 24 q^{95} - 652 q^{97} + 98 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 + 26 * q^4 - 10 * q^5 - 28 * q^7 + 18 * q^8 - 2 * q^10 + 44 * q^11 + 58 * q^13 - 14 * q^14 + 2 * q^16 - 4 * q^17 + 258 * q^19 - 182 * q^20 + 22 * q^22 - 8 * q^23 + 80 * q^25 + 482 * q^26 - 182 * q^28 + 396 * q^29 - 56 * q^31 - 134 * q^32 + 472 * q^34 + 70 * q^35 + 84 * q^37 + 942 * q^38 - 1026 * q^40 - 52 * q^41 + 408 * q^43 + 286 * q^44 + 368 * q^46 - 8 * q^47 + 196 * q^49 + 1642 * q^50 + 2030 * q^52 - 624 * q^53 - 110 * q^55 - 126 * q^56 + 864 * q^58 + 238 * q^59 - 162 * q^61 - 688 * q^62 - 902 * q^64 + 32 * q^65 + 1340 * q^67 + 1384 * q^68 + 14 * q^70 - 1788 * q^71 + 1456 * q^73 - 996 * q^74 + 3042 * q^76 - 308 * q^77 - 1324 * q^79 - 2342 * q^80 + 1984 * q^82 - 450 * q^83 - 1736 * q^85 + 4380 * q^86 + 198 * q^88 + 3072 * q^89 - 406 * q^91 - 544 * q^92 - 1696 * q^94 - 24 * q^95 - 652 * q^97 + 98 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 12x^{2} + 5x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - 12\nu - 3$$ v^3 - 12*v - 3 $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 11\nu + 3$$ v^3 - v^2 - 11*v + 3 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 13\nu - 4$$ -v^3 + v^2 + 13*v - 4
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + 1 ) / 2$$ (b3 + b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - \beta_{2} + 2\beta _1 + 13 ) / 2$$ (b3 - b2 + 2*b1 + 13) / 2 $$\nu^{3}$$ $$=$$ $$6\beta_{3} + 6\beta_{2} + \beta _1 + 9$$ 6*b3 + 6*b2 + b1 + 9

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.148103 3.79597 0.555307 −3.20317
−4.60395 0 13.1964 −1.84418 0 −7.00000 −23.9238 0 8.49053
1.2 −1.53253 0 −5.65135 −8.69995 0 −7.00000 20.9211 0 13.3330
1.3 3.24550 0 2.53327 16.0955 0 −7.00000 −17.7423 0 52.2379
1.4 4.89098 0 15.9217 −15.5514 0 −7.00000 38.7449 0 −76.0614
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.4.a.l 4
3.b odd 2 1 77.4.a.d 4
12.b even 2 1 1232.4.a.s 4
15.d odd 2 1 1925.4.a.p 4
21.c even 2 1 539.4.a.g 4
33.d even 2 1 847.4.a.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.d 4 3.b odd 2 1
539.4.a.g 4 21.c even 2 1
693.4.a.l 4 1.a even 1 1 trivial
847.4.a.d 4 33.d even 2 1
1232.4.a.s 4 12.b even 2 1
1925.4.a.p 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(693))$$:

 $$T_{2}^{4} - 2T_{2}^{3} - 27T_{2}^{2} + 40T_{2} + 112$$ T2^4 - 2*T2^3 - 27*T2^2 + 40*T2 + 112 $$T_{5}^{4} + 10T_{5}^{3} - 240T_{5}^{2} - 2648T_{5} - 4016$$ T5^4 + 10*T5^3 - 240*T5^2 - 2648*T5 - 4016

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} - 27 T^{2} + 40 T + 112$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 10 T^{3} - 240 T^{2} + \cdots - 4016$$
$7$ $$(T + 7)^{4}$$
$11$ $$(T - 11)^{4}$$
$13$ $$T^{4} - 58 T^{3} - 4926 T^{2} + \cdots - 4947656$$
$17$ $$T^{4} + 4 T^{3} - 6186 T^{2} + \cdots - 2705024$$
$19$ $$T^{4} - 258 T^{3} + \cdots - 14423904$$
$23$ $$T^{4} + 8 T^{3} - 22392 T^{2} + \cdots - 17449856$$
$29$ $$T^{4} - 396 T^{3} + \cdots + 22336464$$
$31$ $$T^{4} + 56 T^{3} - 31890 T^{2} + \cdots - 11250248$$
$37$ $$T^{4} - 84 T^{3} - 63516 T^{2} + \cdots + 11157312$$
$41$ $$T^{4} + 52 T^{3} + \cdots - 659233664$$
$43$ $$T^{4} - 408 T^{3} + \cdots + 1210397376$$
$47$ $$T^{4} + 8 T^{3} - 121650 T^{2} + \cdots - 318931592$$
$53$ $$T^{4} + 624 T^{3} + \cdots + 403923072$$
$59$ $$T^{4} - 238 T^{3} + \cdots + 17599820728$$
$61$ $$T^{4} + 162 T^{3} + \cdots + 6668930664$$
$67$ $$T^{4} - 1340 T^{3} + \cdots - 140865466496$$
$71$ $$T^{4} + 1788 T^{3} + \cdots - 72982082688$$
$73$ $$T^{4} - 1456 T^{3} + \cdots - 322052228384$$
$79$ $$T^{4} + 1324 T^{3} + \cdots - 59537293568$$
$83$ $$T^{4} + 450 T^{3} + \cdots + 21951092064$$
$89$ $$T^{4} - 3072 T^{3} + \cdots - 109303561968$$
$97$ $$T^{4} + 652 T^{3} + \cdots + 868634650768$$