# Properties

 Label 207.6.a.g Level $207$ Weight $6$ Character orbit 207.a Self dual yes Analytic conductor $33.199$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$207 = 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 207.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.1994507013$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2x^{5} - 149x^{4} + 215x^{3} + 6182x^{2} - 4625x - 79150$$ x^6 - 2*x^5 - 149*x^4 + 215*x^3 + 6182*x^2 - 4625*x - 79150 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2\cdot 3^{2}\cdot 5$$ Twist minimal: no (minimal twist has level 23) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 20) q^{4} + (\beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 - 6) q^{5} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{2} + 11 \beta_1 + 48) q^{7} + ( - 3 \beta_{5} - 8 \beta_{2} + 7 \beta_1 - 69) q^{8}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b3 + b2 - 2*b1 + 20) * q^4 + (b4 + 2*b3 + b2 - b1 - 6) * q^5 + (-2*b5 + b4 + 3*b2 + 11*b1 + 48) * q^7 + (-3*b5 - 8*b2 + 7*b1 - 69) * q^8 $$q + (\beta_1 - 1) q^{2} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 20) q^{4} + (\beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 - 6) q^{5} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{2} + 11 \beta_1 + 48) q^{7} + ( - 3 \beta_{5} - 8 \beta_{2} + 7 \beta_1 - 69) q^{8} + ( - 10 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 22 \beta_1 - 4) q^{10} + ( - \beta_{4} + 8 \beta_{3} + 11 \beta_{2} - 11 \beta_1 + 20) q^{11} + (14 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} + 8 \beta_{2} - 22 \beta_1 + 138) q^{13} + (12 \beta_{4} + 14 \beta_{3} - 6 \beta_{2} + 66 \beta_1 + 474) q^{14} + (6 \beta_{5} + \beta_{4} + 20 \beta_{3} + 45 \beta_{2} - 84 \beta_1 - 271) q^{16} + (18 \beta_{5} + 13 \beta_{4} - 26 \beta_{3} + 23 \beta_{2} - 53 \beta_1 + 70) q^{17} + (30 \beta_{5} + 2 \beta_{4} + 24 \beta_{3} + 40 \beta_{2} + 6 \beta_1 + 460) q^{19} + ( - 2 \beta_{4} + 50 \beta_{3} + 40 \beta_{2} + 24 \beta_1 + 1210) q^{20} + ( - 20 \beta_{5} - 2 \beta_{3} - 86 \beta_{2} + 192 \beta_1 - 412) q^{22} - 529 q^{23} + ( - 90 \beta_{4} - 50 \beta_{2} + 250 \beta_1 + 1591) q^{25} + ( - 27 \beta_{5} - 51 \beta_{4} - 111 \beta_{3} - 116 \beta_{2} + \cdots - 885) q^{26}+ \cdots + ( - 1764 \beta_{5} + 852 \beta_{4} + 2200 \beta_{3} + \cdots + 55455) q^{98}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b3 + b2 - 2*b1 + 20) * q^4 + (b4 + 2*b3 + b2 - b1 - 6) * q^5 + (-2*b5 + b4 + 3*b2 + 11*b1 + 48) * q^7 + (-3*b5 - 8*b2 + 7*b1 - 69) * q^8 + (-10*b5 + 2*b4 + 4*b3 - 2*b2 + 22*b1 - 4) * q^10 + (-b4 + 8*b3 + 11*b2 - 11*b1 + 20) * q^11 + (14*b5 - 4*b4 + 5*b3 + 8*b2 - 22*b1 + 138) * q^13 + (12*b4 + 14*b3 - 6*b2 + 66*b1 + 474) * q^14 + (6*b5 + b4 + 20*b3 + 45*b2 - 84*b1 - 271) * q^16 + (18*b5 + 13*b4 - 26*b3 + 23*b2 - 53*b1 + 70) * q^17 + (30*b5 + 2*b4 + 24*b3 + 40*b2 + 6*b1 + 460) * q^19 + (-2*b4 + 50*b3 + 40*b2 + 24*b1 + 1210) * q^20 + (-20*b5 - 2*b3 - 86*b2 + 192*b1 - 412) * q^22 - 529 * q^23 + (-90*b4 - 50*b2 + 250*b1 + 1591) * q^25 + (-27*b5 - 51*b4 - 111*b3 - 116*b2 + 264*b1 - 885) * q^26 + (-26*b5 - 16*b4 + 130*b3 + 82*b2 + 138*b1 + 1638) * q^28 + (2*b5 + 68*b4 - 43*b3 + 20*b2 + 458*b1 - 2058) * q^29 + (-84*b5 + 87*b4 - 31*b3 + 69*b2 - 93*b1 + 2348) * q^31 + (20*b5 + 10*b4 - 163*b3 - 157*b2 + 156*b1 - 1272) * q^32 + (-10*b5 + 34*b4 - 404*b3 - 286*b2 + 148*b1 - 2990) * q^34 + (-8*b5 - 36*b4 + 220*b3 + 204*b2 + 724*b1 + 632) * q^35 + (28*b5 - 91*b4 + 50*b3 + 193*b2 + 219*b1 + 4262) * q^37 + (-140*b5 - 68*b4 - 208*b3 - 320*b2 + 1010*b1 + 918) * q^38 + (178*b5 - 80*b4 + 30*b3 - 138*b2 + 1238*b1 + 1194) * q^40 + (98*b5 - 50*b4 - 181*b3 - 342*b2 + 604*b1 + 1970) * q^41 + (-34*b5 + 150*b4 - 322*b3 - 256*b2 + 470*b1 - 3396) * q^43 + (46*b5 + 8*b4 + 324*b3 + 564*b2 - 1042*b1 + 9142) * q^44 + (-529*b1 + 529) * q^46 + (-220*b5 + 245*b4 + 91*b3 + 15*b2 + 153*b1 + 3860) * q^47 + (-52*b5 + 350*b4 + 156*b3 - 198*b2 + 1162*b1 + 5745) * q^49 + (360*b5 - 320*b4 + 580*b3 + 380*b2 + 711*b1 + 11249) * q^50 + (143*b5 + 51*b4 + 188*b3 + 615*b2 - 2479*b1 + 7140) * q^52 + (278*b5 + 144*b4 - 382*b3 + 386*b2 - 1140*b1 - 2726) * q^53 + (-100*b5 - 102*b4 + 176*b3 + 318*b2 + 294*b1 + 8080) * q^55 + (-274*b5 - 402*b4 + 308*b3 - 62*b2 + 1134*b1 - 7516) * q^56 + (-147*b5 + 261*b4 + 33*b3 + 412*b2 - 2544*b1 + 24483) * q^58 + (116*b5 - 114*b4 + 596*b3 + 210*b2 - 2238*b1 + 7060) * q^59 + (-310*b5 - 450*b4 + 278*b3 - 572*b2 - 1730*b1 + 5294) * q^61 + (-87*b5 + 613*b4 - 41*b3 - 278*b2 + 2988*b1 - 9425) * q^62 + (217*b5 - 56*b4 - 988*b3 - 364*b2 - 1437*b1 + 14847) * q^64 + (-774*b5 + 529*b4 - 230*b3 - 325*b2 - 4353*b1 + 292) * q^65 + (376*b5 - 49*b4 + 616*b3 - 509*b2 + 869*b1 + 5764) * q^67 + (520*b5 - 166*b4 - 180*b3 + 1014*b2 - 7180*b1 - 410) * q^68 + (-500*b5 - 100*b4 + 1340*b3 - 560*b2 + 3432*b1 + 40796) * q^70 + (-264*b5 - 17*b4 - 437*b3 - 667*b2 - 721*b1 + 26436) * q^71 + (586*b5 + 614*b4 - 145*b3 + 498*b2 - 268*b1 + 18414) * q^73 + (158*b5 - 214*b4 - 124*b3 - 1554*b2 + 5998*b1 + 8580) * q^74 + (216*b5 + 40*b4 + 1278*b3 + 1950*b2 - 4964*b1 + 28912) * q^76 + (184*b5 - 112*b4 + 1180*b3 - 96*b2 - 1168*b1 + 15656) * q^77 + (-456*b5 - 784*b4 - 1074*b3 - 1300*b2 + 1556*b1 + 5320) * q^79 + (-126*b5 - 878*b4 - 884*b3 + 526*b2 - 1974*b1 + 27316) * q^80 + (547*b5 - 605*b4 + 139*b3 + 2632*b2 - 3260*b1 + 26945) * q^82 + (-252*b5 - 223*b4 - 1290*b3 - 227*b2 - 3985*b1 + 13748) * q^83 + (-188*b5 - 188*b4 - 1844*b3 - 592*b2 - 8692*b1 - 14780) * q^85 + (434*b5 + 618*b4 - 434*b3 + 2392*b2 - 8446*b1 + 19808) * q^86 + (-456*b5 + 126*b4 - 1388*b3 - 2222*b2 + 11724*b1 - 41430) * q^88 + (1352*b5 - 246*b4 - 1866*b3 - 2674*b2 - 542*b1 - 4610) * q^89 + (310*b5 - 33*b4 + 60*b3 + 1529*b2 - 1475*b1 - 40888) * q^91 + (-529*b3 - 529*b2 + 1058*b1 - 10580) * q^92 + (-813*b5 + 1319*b4 + 1613*b3 + 1390*b2 + 5060*b1 + 1429) * q^94 + (-2340*b5 + 626*b4 + 172*b3 - 578*b2 - 6754*b1 + 34760) * q^95 + (-154*b5 - 969*b4 + 198*b3 - 1011*b2 - 6679*b1 + 45594) * q^97 + (-1764*b5 + 852*b4 + 2200*b3 + 4212*b2 + 4749*b1 + 55455) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8}+O(q^{10})$$ 6 * q - 4 * q^2 + 112 * q^4 - 42 * q^5 + 300 * q^7 - 393 * q^8 $$6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8} - 10 q^{10} + 58 q^{11} + 792 q^{13} + 2984 q^{14} - 1904 q^{16} + 400 q^{17} + 2738 q^{19} + 7124 q^{20} - 1972 q^{22} - 3174 q^{23} + 9966 q^{25} - 4511 q^{26} + 9570 q^{28} - 11244 q^{29} + 13748 q^{31} - 6600 q^{32} - 16226 q^{34} + 4296 q^{35} + 25426 q^{37} + 8028 q^{38} + 10230 q^{40} + 14268 q^{41} - 18082 q^{43} + 51146 q^{44} + 2116 q^{46} + 23084 q^{47} + 37422 q^{49} + 67436 q^{50} + 36807 q^{52} - 17522 q^{53} + 47576 q^{55} - 44946 q^{56} + 141001 q^{58} + 36392 q^{59} + 27062 q^{61} - 48971 q^{62} + 89451 q^{64} - 7108 q^{65} + 37138 q^{67} - 17260 q^{68} + 248380 q^{70} + 158556 q^{71} + 112228 q^{73} + 66878 q^{74} + 157816 q^{76} + 89760 q^{77} + 36844 q^{79} + 158530 q^{80} + 150039 q^{82} + 76350 q^{83} - 102132 q^{85} + 100578 q^{86} - 219028 q^{88} - 16100 q^{89} - 250592 q^{91} - 59248 q^{92} + 12887 q^{94} + 190096 q^{95} + 259432 q^{97} + 325816 q^{98}+O(q^{100})$$ 6 * q - 4 * q^2 + 112 * q^4 - 42 * q^5 + 300 * q^7 - 393 * q^8 - 10 * q^10 + 58 * q^11 + 792 * q^13 + 2984 * q^14 - 1904 * q^16 + 400 * q^17 + 2738 * q^19 + 7124 * q^20 - 1972 * q^22 - 3174 * q^23 + 9966 * q^25 - 4511 * q^26 + 9570 * q^28 - 11244 * q^29 + 13748 * q^31 - 6600 * q^32 - 16226 * q^34 + 4296 * q^35 + 25426 * q^37 + 8028 * q^38 + 10230 * q^40 + 14268 * q^41 - 18082 * q^43 + 51146 * q^44 + 2116 * q^46 + 23084 * q^47 + 37422 * q^49 + 67436 * q^50 + 36807 * q^52 - 17522 * q^53 + 47576 * q^55 - 44946 * q^56 + 141001 * q^58 + 36392 * q^59 + 27062 * q^61 - 48971 * q^62 + 89451 * q^64 - 7108 * q^65 + 37138 * q^67 - 17260 * q^68 + 248380 * q^70 + 158556 * q^71 + 112228 * q^73 + 66878 * q^74 + 157816 * q^76 + 89760 * q^77 + 36844 * q^79 + 158530 * q^80 + 150039 * q^82 + 76350 * q^83 - 102132 * q^85 + 100578 * q^86 - 219028 * q^88 - 16100 * q^89 - 250592 * q^91 - 59248 * q^92 + 12887 * q^94 + 190096 * q^95 + 259432 * q^97 + 325816 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 149x^{4} + 215x^{3} + 6182x^{2} - 4625x - 79150$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -3\nu^{5} + 45\nu^{4} + 274\nu^{3} - 5031\nu^{2} - 2171\nu + 101838 ) / 412$$ (-3*v^5 + 45*v^4 + 274*v^3 - 5031*v^2 - 2171*v + 101838) / 412 $$\beta_{3}$$ $$=$$ $$( 3\nu^{5} - 45\nu^{4} - 274\nu^{3} + 5443\nu^{2} + 2171\nu - 122850 ) / 412$$ (3*v^5 - 45*v^4 - 274*v^3 + 5443*v^2 + 2171*v - 122850) / 412 $$\beta_{4}$$ $$=$$ $$( 27\nu^{5} + 7\nu^{4} - 3290\nu^{3} - 2513\nu^{2} + 75571\nu + 106866 ) / 412$$ (27*v^5 + 7*v^4 - 3290*v^3 - 2513*v^2 + 75571*v + 106866) / 412 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 30\nu^{4} - 217\nu^{3} + 3457\nu^{2} + 3782\nu - 72424 ) / 103$$ (2*v^5 - 30*v^4 - 217*v^3 + 3457*v^2 + 3782*v - 72424) / 103
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 51$$ b3 + b2 + 51 $$\nu^{3}$$ $$=$$ $$-3\beta_{5} + 3\beta_{3} - 5\beta_{2} + 68\beta _1 + 21$$ -3*b5 + 3*b3 - 5*b2 + 68*b1 + 21 $$\nu^{4}$$ $$=$$ $$-6\beta_{5} + \beta_{4} + 122\beta_{3} + 115\beta_{2} + 3474$$ -6*b5 + b4 + 122*b3 + 115*b2 + 3474 $$\nu^{5}$$ $$=$$ $$-364\beta_{5} + 15\beta_{4} + 427\beta_{3} - 546\beta_{2} + 5487\beta _1 + 2447$$ -364*b5 + 15*b4 + 427*b3 - 546*b2 + 5487*b1 + 2447

## 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
 −9.33646 −5.29078 −4.81709 5.64307 6.03655 9.76471
−10.3365 0 74.8423 32.5264 0 43.6495 −442.838 0 −336.208
1.2 −6.29078 0 7.57397 45.2266 0 −206.967 153.659 0 −284.511
1.3 −5.81709 0 1.83858 −83.3054 0 84.5782 175.452 0 484.596
1.4 4.64307 0 −10.4419 −9.65155 0 199.091 −197.061 0 −44.8128
1.5 5.03655 0 −6.63314 −108.846 0 −33.8609 −194.578 0 −548.208
1.6 8.76471 0 44.8202 82.0499 0 213.509 112.365 0 719.144
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$23$$ $$1$$

## 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 207.6.a.g 6
3.b odd 2 1 23.6.a.b 6
12.b even 2 1 368.6.a.h 6
15.d odd 2 1 575.6.a.c 6
69.c even 2 1 529.6.a.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.6.a.b 6 3.b odd 2 1
207.6.a.g 6 1.a even 1 1 trivial
368.6.a.h 6 12.b even 2 1
529.6.a.c 6 69.c even 2 1
575.6.a.c 6 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 4T_{2}^{5} - 144T_{2}^{4} - 381T_{2}^{3} + 5928T_{2}^{2} + 7784T_{2} - 77528$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(207))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 4 T^{5} - 144 T^{4} + \cdots - 77528$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 42 T^{5} + \cdots - 10563094144$$
$7$ $$T^{6} - 300 T^{5} + \cdots + 1099779388928$$
$11$ $$T^{6} - 58 T^{5} + \cdots + 97362234604672$$
$13$ $$T^{6} - 792 T^{5} + \cdots - 83\!\cdots\!88$$
$17$ $$T^{6} - 400 T^{5} + \cdots - 22\!\cdots\!32$$
$19$ $$T^{6} - 2738 T^{5} + \cdots - 18\!\cdots\!48$$
$23$ $$(T + 529)^{6}$$
$29$ $$T^{6} + 11244 T^{5} + \cdots + 15\!\cdots\!00$$
$31$ $$T^{6} - 13748 T^{5} + \cdots + 76\!\cdots\!36$$
$37$ $$T^{6} - 25426 T^{5} + \cdots - 24\!\cdots\!28$$
$41$ $$T^{6} - 14268 T^{5} + \cdots - 63\!\cdots\!16$$
$43$ $$T^{6} + 18082 T^{5} + \cdots - 11\!\cdots\!28$$
$47$ $$T^{6} - 23084 T^{5} + \cdots - 16\!\cdots\!00$$
$53$ $$T^{6} + 17522 T^{5} + \cdots - 60\!\cdots\!96$$
$59$ $$T^{6} - 36392 T^{5} + \cdots - 35\!\cdots\!00$$
$61$ $$T^{6} - 27062 T^{5} + \cdots - 14\!\cdots\!32$$
$67$ $$T^{6} - 37138 T^{5} + \cdots - 47\!\cdots\!72$$
$71$ $$T^{6} - 158556 T^{5} + \cdots + 63\!\cdots\!32$$
$73$ $$T^{6} - 112228 T^{5} + \cdots + 30\!\cdots\!00$$
$79$ $$T^{6} - 36844 T^{5} + \cdots + 19\!\cdots\!92$$
$83$ $$T^{6} - 76350 T^{5} + \cdots + 75\!\cdots\!36$$
$89$ $$T^{6} + 16100 T^{5} + \cdots - 71\!\cdots\!00$$
$97$ $$T^{6} - 259432 T^{5} + \cdots + 29\!\cdots\!00$$