Properties

 Label 207.8.a.a Level $207$ Weight $8$ Character orbit 207.a Self dual yes Analytic conductor $64.664$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$207 = 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 207.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$64.6637002752$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - 455x^{3} - 474x^{2} + 42284x + 127016$$ x^5 - 455*x^3 - 474*x^2 + 42284*x + 127016 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 69) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{4} + \beta_{3} + 2 \beta_1 + 54) q^{4} + (\beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 54) q^{5} + ( - 4 \beta_{4} - 12 \beta_{3} - \beta_{2} + 3 \beta_1 - 96) q^{7} + (\beta_{4} - 11 \beta_{2} - 9 \beta_1 - 284) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b4 + b3 + 2*b1 + 54) * q^4 + (b4 - b3 + b2 - 2*b1 + 54) * q^5 + (-4*b4 - 12*b3 - b2 + 3*b1 - 96) * q^7 + (b4 - 11*b2 - 9*b1 - 284) * q^8 $$q - \beta_1 q^{2} + (\beta_{4} + \beta_{3} + 2 \beta_1 + 54) q^{4} + (\beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 54) q^{5} + ( - 4 \beta_{4} - 12 \beta_{3} - \beta_{2} + 3 \beta_1 - 96) q^{7} + (\beta_{4} - 11 \beta_{2} - 9 \beta_1 - 284) q^{8} + (4 \beta_{4} - 10 \beta_{3} + 8 \beta_{2} - 72 \beta_1 + 296) q^{10} + (5 \beta_{4} + 29 \beta_{3} - 4 \beta_{2} + 117 \beta_1 + 220) q^{11} + ( - 27 \beta_{4} + 82 \beta_{3} - 13 \beta_{2} - 372 \beta_1 - 172) q^{13} + (6 \beta_{4} + 23 \beta_{3} + 125 \beta_{2} + 589 \beta_1 - 1158) q^{14} + ( - 59 \beta_{4} + 83 \beta_{3} + 10 \beta_{2} - 108 \beta_1 - 4578) q^{16} + ( - 160 \beta_{4} - 76 \beta_{3} - 65 \beta_{2} + 1103 \beta_1 + 1126) q^{17} + (125 \beta_{4} + 127 \beta_{3} + 121 \beta_{2} - 598 \beta_1 - 1208) q^{19} + ( - 56 \beta_{4} + 92 \beta_{3} - 40 \beta_{2} + 196 \beta_1 + 5416) q^{20} + ( - 130 \beta_{4} - 83 \beta_{3} - 291 \beta_{2} - 1461 \beta_1 - 19598) q^{22} - 12167 q^{23} + ( - 19 \beta_{4} - 280 \beta_{3} + 257 \beta_{2} - 1198 \beta_1 - 52391) q^{25} + (138 \beta_{4} + 334 \beta_{3} - 780 \beta_{2} + 316 \beta_1 + 71120) q^{26} + ( - 718 \beta_{4} - 1325 \beta_{3} - 233 \beta_{2} + \cdots - 101182) q^{28}+ \cdots + (8554 \beta_{4} + 17220 \beta_{3} - 34650 \beta_{2} + \cdots - 2349564) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b4 + b3 + 2*b1 + 54) * q^4 + (b4 - b3 + b2 - 2*b1 + 54) * q^5 + (-4*b4 - 12*b3 - b2 + 3*b1 - 96) * q^7 + (b4 - 11*b2 - 9*b1 - 284) * q^8 + (4*b4 - 10*b3 + 8*b2 - 72*b1 + 296) * q^10 + (5*b4 + 29*b3 - 4*b2 + 117*b1 + 220) * q^11 + (-27*b4 + 82*b3 - 13*b2 - 372*b1 - 172) * q^13 + (6*b4 + 23*b3 + 125*b2 + 589*b1 - 1158) * q^14 + (-59*b4 + 83*b3 + 10*b2 - 108*b1 - 4578) * q^16 + (-160*b4 - 76*b3 - 65*b2 + 1103*b1 + 1126) * q^17 + (125*b4 + 127*b3 + 121*b2 - 598*b1 - 1208) * q^19 + (-56*b4 + 92*b3 - 40*b2 + 196*b1 + 5416) * q^20 + (-130*b4 - 83*b3 - 291*b2 - 1461*b1 - 19598) * q^22 - 12167 * q^23 + (-19*b4 - 280*b3 + 257*b2 - 1198*b1 - 52391) * q^25 + (138*b4 + 334*b3 - 780*b2 + 316*b1 + 71120) * q^26 + (-718*b4 - 1325*b3 - 233*b2 - 161*b1 - 101182) * q^28 + (-496*b4 - 540*b3 - 286*b2 + 3746*b1 + 33850) * q^29 + (-430*b4 - 1400*b3 - 744*b2 + 4906*b1 - 39540) * q^31 + (-531*b4 - 474*b3 + 627*b2 + 7289*b1 + 56936) * q^32 + (-1426*b4 - 421*b3 + 985*b2 + 6707*b1 - 205950) * q^34 + (109*b4 - 1166*b3 + 517*b2 - 2746*b1 + 28046) * q^35 + (177*b4 + 1431*b3 - 660*b2 + 13985*b1 - 40902) * q^37 + (364*b4 - 1334*b3 - 1516*b2 - 6430*b1 + 111664) * q^38 + (-972*b4 + 1396*b3 - 1848*b2 + 3920*b1 - 69128) * q^40 + (434*b4 - 422*b3 + 2772*b2 - 8482*b1 - 107914) * q^41 + (2975*b4 - 1083*b3 + 5109*b2 + 25824*b1 - 180296) * q^43 + (1792*b4 + 2633*b3 + 1763*b2 + 14067*b1 + 246870) * q^44 + 12167*b1 * q^46 + (-3311*b4 + 7350*b3 - 99*b2 + 9026*b1 - 20306) * q^47 + (4893*b4 + 3080*b3 - 1043*b2 + 14966*b1 + 170843) * q^49 + (378*b4 - 2944*b3 + 2562*b2 + 64865*b1 + 189612) * q^50 + (7062*b4 + 3112*b3 - 1034*b2 - 46762*b1 + 30968) * q^52 + (5451*b4 - 8805*b3 - 2343*b2 + 10232*b1 + 61330) * q^53 + (-570*b4 + 4700*b3 - 3110*b2 + 19000*b1 - 188820) * q^55 + (-382*b4 + 1189*b3 - 1799*b2 + 95305*b1 + 107358) * q^56 + (-3716*b4 + 498*b3 + 6182*b2 - 3676*b1 - 706228) * q^58 + (5535*b4 + 6800*b3 - 12851*b2 + 39800*b1 + 634970) * q^59 + (4461*b4 + 20597*b3 + 7862*b2 + 50535*b1 + 23102) * q^61 + (-536*b4 + 9566*b3 + 15174*b2 + 80282*b1 - 925332) * q^62 + (-4579*b4 - 30375*b3 + 3364*b2 - 11466*b1 - 818206) * q^64 + (5208*b4 + 6662*b3 - 3432*b2 + 13664*b1 - 776048) * q^65 + (-6131*b4 + 1711*b3 - 32053*b2 - 38948*b1 - 96120) * q^67 + (2560*b4 - 19571*b3 + 12971*b2 + 148751*b1 - 1493762) * q^68 + (3038*b4 - 3792*b3 + 11034*b2 + 4088*b1 + 421372) * q^70 + (-2758*b4 + 14726*b3 - 7316*b2 + 18550*b1 - 322352) * q^71 + (2604*b4 + 11284*b3 - 14012*b2 - 15000*b1 - 11110) * q^73 + (-12662*b4 - 4259*b3 - 13827*b2 - 37087*b1 - 2436334) * q^74 + (2498*b4 + 21586*b3 - 996*b2 - 23908*b1 + 1370252) * q^76 + (-12636*b4 - 8018*b3 - 15122*b2 - 69014*b1 - 2068324) * q^77 + (-8456*b4 - 7904*b3 + 21401*b2 + 122297*b1 - 782888) * q^79 + (4836*b4 + 10888*b3 - 6020*b2 + 39524*b1 - 1269376) * q^80 + (-2364*b4 - 38834*b3 + 1014*b2 + 136084*b1 + 1374460) * q^82 + (-27507*b4 + 25355*b3 + 178*b2 - 72749*b1 - 3176088) * q^83 + (10887*b4 + 6172*b3 - 9729*b2 + 106802*b1 - 1702262) * q^85 + (-34328*b4 - 103720*b3 + 2746*b2 + 36996*b1 - 4947060) * q^86 + (-2548*b4 - 33275*b3 + 7363*b2 - 233881*b1 + 22934) * q^88 + (37888*b4 + 33662*b3 + 12543*b2 - 375585*b1 - 221606) * q^89 + (-22260*b4 + 30604*b3 + 43414*b2 + 131090*b1 - 5495056) * q^91 + (-12167*b4 - 12167*b3 - 24334*b1 - 657018) * q^92 + (-39786*b4 - 35188*b3 - 70090*b2 + 7068*b1 - 1432588) * q^94 + (-4877*b4 - 1866*b3 + 11495*b2 - 66138*b1 + 1212362) * q^95 + (-16134*b4 - 782*b3 + 4426*b2 + 490412*b1 - 3153854) * q^97 + (8554*b4 + 17220*b3 - 34650*b2 - 553197*b1 - 2349564) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8}+O(q^{10})$$ 5 * q + 270 * q^4 + 266 * q^5 - 496 * q^7 - 1422 * q^8 $$5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8} + 1452 q^{10} + 1148 q^{11} - 642 q^{13} - 5756 q^{14} - 22606 q^{16} + 5798 q^{17} - 6036 q^{19} + 27376 q^{20} - 97896 q^{22} - 60835 q^{23} - 262477 q^{25} + 355992 q^{26} - 507124 q^{28} + 169162 q^{29} - 199640 q^{31} + 284794 q^{32} - 1027740 q^{34} + 137680 q^{35} - 202002 q^{37} + 554924 q^{38} - 340904 q^{40} - 541282 q^{41} - 909596 q^{43} + 1236032 q^{44} - 80208 q^{47} + 850589 q^{49} + 941416 q^{50} + 146940 q^{52} + 278138 q^{53} - 933560 q^{55} + 539932 q^{56} - 3522712 q^{58} + 3177380 q^{59} + 147782 q^{61} - 4606456 q^{62} - 4142622 q^{64} - 3877332 q^{65} - 464916 q^{67} - 7513072 q^{68} + 2093200 q^{70} - 1576792 q^{71} - 38190 q^{73} - 12164864 q^{74} + 6889436 q^{76} - 10332384 q^{77} - 3913336 q^{79} - 6334776 q^{80} + 6799360 q^{82} - 15774716 q^{83} - 8520740 q^{85} - 24874084 q^{86} + 53216 q^{88} - 1116482 q^{89} - 27369552 q^{91} - 3285090 q^{92} - 7153744 q^{94} + 6067832 q^{95} - 15738566 q^{97} - 11730488 q^{98}+O(q^{100})$$ 5 * q + 270 * q^4 + 266 * q^5 - 496 * q^7 - 1422 * q^8 + 1452 * q^10 + 1148 * q^11 - 642 * q^13 - 5756 * q^14 - 22606 * q^16 + 5798 * q^17 - 6036 * q^19 + 27376 * q^20 - 97896 * q^22 - 60835 * q^23 - 262477 * q^25 + 355992 * q^26 - 507124 * q^28 + 169162 * q^29 - 199640 * q^31 + 284794 * q^32 - 1027740 * q^34 + 137680 * q^35 - 202002 * q^37 + 554924 * q^38 - 340904 * q^40 - 541282 * q^41 - 909596 * q^43 + 1236032 * q^44 - 80208 * q^47 + 850589 * q^49 + 941416 * q^50 + 146940 * q^52 + 278138 * q^53 - 933560 * q^55 + 539932 * q^56 - 3522712 * q^58 + 3177380 * q^59 + 147782 * q^61 - 4606456 * q^62 - 4142622 * q^64 - 3877332 * q^65 - 464916 * q^67 - 7513072 * q^68 + 2093200 * q^70 - 1576792 * q^71 - 38190 * q^73 - 12164864 * q^74 + 6889436 * q^76 - 10332384 * q^77 - 3913336 * q^79 - 6334776 * q^80 + 6799360 * q^82 - 15774716 * q^83 - 8520740 * q^85 - 24874084 * q^86 + 53216 * q^88 - 1116482 * q^89 - 27369552 * q^91 - 3285090 * q^92 - 7153744 * q^94 + 6067832 * q^95 - 15738566 * q^97 - 11730488 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 455x^{3} - 474x^{2} + 42284x + 127016$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{4} + 142\nu^{3} + 467\nu^{2} - 37904\nu - 76396 ) / 1552$$ (-v^4 + 142*v^3 + 467*v^2 - 37904*v - 76396) / 1552 $$\beta_{3}$$ $$=$$ $$( 11\nu^{4} - 10\nu^{3} - 3585\nu^{2} + 2560\nu + 117124 ) / 1552$$ (11*v^4 - 10*v^3 - 3585*v^2 + 2560*v + 117124) / 1552 $$\beta_{4}$$ $$=$$ $$( -11\nu^{4} + 10\nu^{3} + 5137\nu^{2} - 5664\nu - 399588 ) / 1552$$ (-11*v^4 + 10*v^3 + 5137*v^2 - 5664*v - 399588) / 1552
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + 2\beta _1 + 182$$ b4 + b3 + 2*b1 + 182 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + 11\beta_{2} + 265\beta _1 + 284$$ -b4 + 11*b2 + 265*b1 + 284 $$\nu^{4}$$ $$=$$ $$325\beta_{4} + 467\beta_{3} + 10\beta_{2} + 660\beta _1 + 48926$$ 325*b4 + 467*b3 + 10*b2 + 660*b1 + 48926

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
 18.2528 12.4672 −3.24502 −9.64907 −17.8260
−18.2528 0 205.166 55.2224 0 −1258.24 −1408.51 0 −1007.97
1.2 −12.4672 0 27.4323 40.8788 0 1126.70 1253.80 0 −509.647
1.3 3.24502 0 −117.470 −168.083 0 149.817 −796.555 0 −545.434
1.4 9.64907 0 −34.8955 306.939 0 733.069 −1571.79 0 2961.68
1.5 17.8260 0 189.767 31.0428 0 −1247.35 1101.05 0 553.369
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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 207.8.a.a 5
3.b odd 2 1 69.8.a.a 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.8.a.a 5 3.b odd 2 1
207.8.a.a 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} - 455T_{2}^{3} + 474T_{2}^{2} + 42284T_{2} - 127016$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(207))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 455 T^{3} + 474 T^{2} + \cdots - 127016$$
$3$ $$T^{5}$$
$5$ $$T^{5} - 266 T^{4} + \cdots + 3615360000$$
$7$ $$T^{5} + \cdots - 194204974150720$$
$11$ $$T^{5} + \cdots - 154691619430400$$
$13$ $$T^{5} + 642 T^{4} + \cdots + 16\!\cdots\!84$$
$17$ $$T^{5} - 5798 T^{4} + \cdots - 11\!\cdots\!80$$
$19$ $$T^{5} + 6036 T^{4} + \cdots - 14\!\cdots\!36$$
$23$ $$(T + 12167)^{5}$$
$29$ $$T^{5} - 169162 T^{4} + \cdots - 24\!\cdots\!88$$
$31$ $$T^{5} + 199640 T^{4} + \cdots - 23\!\cdots\!28$$
$37$ $$T^{5} + 202002 T^{4} + \cdots + 82\!\cdots\!20$$
$41$ $$T^{5} + 541282 T^{4} + \cdots - 16\!\cdots\!52$$
$43$ $$T^{5} + 909596 T^{4} + \cdots + 66\!\cdots\!00$$
$47$ $$T^{5} + 80208 T^{4} + \cdots + 24\!\cdots\!00$$
$53$ $$T^{5} - 278138 T^{4} + \cdots - 10\!\cdots\!60$$
$59$ $$T^{5} - 3177380 T^{4} + \cdots - 75\!\cdots\!64$$
$61$ $$T^{5} - 147782 T^{4} + \cdots - 14\!\cdots\!84$$
$67$ $$T^{5} + 464916 T^{4} + \cdots - 79\!\cdots\!40$$
$71$ $$T^{5} + 1576792 T^{4} + \cdots - 49\!\cdots\!52$$
$73$ $$T^{5} + 38190 T^{4} + \cdots + 11\!\cdots\!64$$
$79$ $$T^{5} + 3913336 T^{4} + \cdots - 36\!\cdots\!20$$
$83$ $$T^{5} + 15774716 T^{4} + \cdots - 18\!\cdots\!20$$
$89$ $$T^{5} + 1116482 T^{4} + \cdots + 25\!\cdots\!96$$
$97$ $$T^{5} + 15738566 T^{4} + \cdots + 11\!\cdots\!20$$