# Properties

 Label 201.4.a.b Level 201 Weight 4 Character orbit 201.a Self dual Yes Analytic conductor 11.859 Analytic rank 1 Dimension 6 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 201.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$11.8593839112$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( -3 + 3 \beta_{1} ) q^{6} + ( -10 - \beta_{1} - \beta_{2} - 3 \beta_{5} ) q^{7} + ( -13 - \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( -3 + 3 \beta_{1} ) q^{6} + ( -10 - \beta_{1} - \beta_{2} - 3 \beta_{5} ) q^{7} + ( -13 - \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{8} + 9 q^{9} + ( -17 + 4 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{10} + ( -12 + \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{11} + ( 6 - 6 \beta_{1} + 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} ) q^{12} + ( -34 + 2 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{13} + ( 4 - 16 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - \beta_{5} ) q^{14} + ( -3 - 6 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{15} + ( 2 - 11 \beta_{1} + 2 \beta_{2} - 11 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{16} + ( -24 - 3 \beta_{1} + 3 \beta_{2} + 7 \beta_{3} + 15 \beta_{4} + 7 \beta_{5} ) q^{17} + ( -9 + 9 \beta_{1} ) q^{18} + ( -44 + 15 \beta_{1} - 13 \beta_{2} + 5 \beta_{3} - 7 \beta_{4} + 11 \beta_{5} ) q^{19} + ( 50 - 30 \beta_{1} + 11 \beta_{2} - 13 \beta_{3} + 6 \beta_{5} ) q^{20} + ( -30 - 3 \beta_{1} - 3 \beta_{2} - 9 \beta_{5} ) q^{21} + ( 14 - 29 \beta_{1} + 17 \beta_{2} - 11 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{22} + ( 7 - 17 \beta_{1} + 11 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 11 \beta_{5} ) q^{23} + ( -39 - 3 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} - 9 \beta_{5} ) q^{24} + ( -11 - 6 \beta_{1} + 24 \beta_{2} + 21 \beta_{4} + 15 \beta_{5} ) q^{25} + ( 78 - 34 \beta_{1} - 4 \beta_{2} + \beta_{3} + 20 \beta_{4} + 13 \beta_{5} ) q^{26} + 27 q^{27} + ( -57 + 37 \beta_{1} - 22 \beta_{2} + 10 \beta_{3} - 13 \beta_{4} - 4 \beta_{5} ) q^{28} + ( -42 + 7 \beta_{1} + 25 \beta_{2} + 4 \beta_{3} - 15 \beta_{4} - 6 \beta_{5} ) q^{29} + ( -51 + 12 \beta_{1} - 27 \beta_{2} + 6 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} ) q^{30} + ( -68 + 38 \beta_{1} - 48 \beta_{2} - 11 \beta_{3} + \beta_{4} - 14 \beta_{5} ) q^{31} + ( 13 + 37 \beta_{1} - 52 \beta_{2} - 2 \beta_{3} - 12 \beta_{4} - 16 \beta_{5} ) q^{32} + ( -36 + 3 \beta_{1} - 15 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 15 \beta_{5} ) q^{33} + ( -38 - 3 \beta_{1} + 53 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{34} + ( -65 + 31 \beta_{1} + 17 \beta_{2} + 22 \beta_{3} + 20 \beta_{4} - 8 \beta_{5} ) q^{35} + ( 18 - 18 \beta_{1} + 9 \beta_{2} + 9 \beta_{4} + 9 \beta_{5} ) q^{36} + ( -100 + 52 \beta_{1} - 30 \beta_{2} - 10 \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{37} + ( 132 - 67 \beta_{1} + 11 \beta_{2} - 19 \beta_{3} - 17 \beta_{4} + 37 \beta_{5} ) q^{38} + ( -102 + 6 \beta_{1} + 24 \beta_{2} + 9 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} ) q^{39} + ( -163 + 97 \beta_{1} - 14 \beta_{2} + 27 \beta_{3} + 5 \beta_{4} - 37 \beta_{5} ) q^{40} + ( -11 - \beta_{1} + 81 \beta_{2} + 20 \beta_{4} + 8 \beta_{5} ) q^{41} + ( 12 - 48 \beta_{1} + 9 \beta_{2} - 12 \beta_{3} - 3 \beta_{5} ) q^{42} + ( -54 - 37 \beta_{1} + 43 \beta_{2} - 37 \beta_{3} + 16 \beta_{4} - 20 \beta_{5} ) q^{43} + ( -128 + 77 \beta_{1} - 51 \beta_{2} + 31 \beta_{3} - 17 \beta_{4} - 19 \beta_{5} ) q^{44} + ( -9 - 18 \beta_{1} - 9 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{45} + ( -157 + 55 \beta_{1} - 18 \beta_{2} + 20 \beta_{3} - 3 \beta_{4} - 12 \beta_{5} ) q^{46} + ( -78 + 32 \beta_{1} - 58 \beta_{2} - 53 \beta_{3} - 60 \beta_{4} - 49 \beta_{5} ) q^{47} + ( 6 - 33 \beta_{1} + 6 \beta_{2} - 33 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} ) q^{48} + ( -45 + 55 \beta_{1} - 13 \beta_{2} - 17 \beta_{3} - 20 \beta_{4} + 58 \beta_{5} ) q^{49} + ( -43 + 70 \beta_{1} + 18 \beta_{2} + 60 \beta_{3} + 27 \beta_{4} - 27 \beta_{5} ) q^{50} + ( -72 - 9 \beta_{1} + 9 \beta_{2} + 21 \beta_{3} + 45 \beta_{4} + 21 \beta_{5} ) q^{51} + ( -190 + 136 \beta_{1} - 44 \beta_{2} + 3 \beta_{3} - 38 \beta_{4} - 59 \beta_{5} ) q^{52} + ( 135 - 25 \beta_{1} - 19 \beta_{2} - 24 \beta_{3} - 12 \beta_{4} + 64 \beta_{5} ) q^{53} + ( -27 + 27 \beta_{1} ) q^{54} + ( -196 + 73 \beta_{1} + 25 \beta_{2} + 33 \beta_{3} + 11 \beta_{4} - 19 \beta_{5} ) q^{55} + ( 326 - 29 \beta_{1} + 30 \beta_{2} - 27 \beta_{3} + 7 \beta_{4} + 88 \beta_{5} ) q^{56} + ( -132 + 45 \beta_{1} - 39 \beta_{2} + 15 \beta_{3} - 21 \beta_{4} + 33 \beta_{5} ) q^{57} + ( 222 - 59 \beta_{1} - 45 \beta_{2} - 4 \beta_{3} + 67 \beta_{4} + 34 \beta_{5} ) q^{58} + ( -63 + 35 \beta_{1} - 103 \beta_{2} + 12 \beta_{3} + 22 \beta_{4} - 58 \beta_{5} ) q^{59} + ( 150 - 90 \beta_{1} + 33 \beta_{2} - 39 \beta_{3} + 18 \beta_{5} ) q^{60} + ( -240 - 44 \beta_{1} + 58 \beta_{2} - 3 \beta_{3} + 18 \beta_{4} + 39 \beta_{5} ) q^{61} + ( 292 - 159 \beta_{1} + 70 \beta_{2} - 39 \beta_{3} - 55 \beta_{4} + 4 \beta_{5} ) q^{62} + ( -90 - 9 \beta_{1} - 9 \beta_{2} - 27 \beta_{5} ) q^{63} + ( 204 - 28 \beta_{1} + 47 \beta_{2} + 12 \beta_{3} - 21 \beta_{4} + 27 \beta_{5} ) q^{64} + ( -64 + 85 \beta_{1} - 105 \beta_{2} + 29 \beta_{3} - \beta_{4} - 77 \beta_{5} ) q^{65} + ( 42 - 87 \beta_{1} + 51 \beta_{2} - 33 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{66} -67 q^{67} + ( 350 + 29 \beta_{1} - 41 \beta_{2} - 19 \beta_{3} - 9 \beta_{4} - 29 \beta_{5} ) q^{68} + ( 21 - 51 \beta_{1} + 33 \beta_{2} + 9 \beta_{3} + 12 \beta_{4} + 33 \beta_{5} ) q^{69} + ( 371 - 119 \beta_{1} + 148 \beta_{2} - 15 \beta_{3} + 95 \beta_{4} + 77 \beta_{5} ) q^{70} + ( -110 - 74 \beta_{1} - 16 \beta_{2} - 9 \beta_{3} - 98 \beta_{4} - 61 \beta_{5} ) q^{71} + ( -117 - 9 \beta_{2} + 27 \beta_{3} - 9 \beta_{4} - 27 \beta_{5} ) q^{72} + ( 12 - 109 \beta_{1} + 11 \beta_{2} + 15 \beta_{3} - 98 \beta_{4} + 58 \beta_{5} ) q^{73} + ( 486 - 171 \beta_{1} + 60 \beta_{2} - 14 \beta_{3} - 13 \beta_{4} + 21 \beta_{5} ) q^{74} + ( -33 - 18 \beta_{1} + 72 \beta_{2} + 63 \beta_{4} + 45 \beta_{5} ) q^{75} + ( -390 + 185 \beta_{1} - 119 \beta_{2} + 29 \beta_{3} - 45 \beta_{4} - 195 \beta_{5} ) q^{76} + ( 468 + 29 \beta_{1} - \beta_{2} - 35 \beta_{3} - 21 \beta_{4} + 83 \beta_{5} ) q^{77} + ( 234 - 102 \beta_{1} - 12 \beta_{2} + 3 \beta_{3} + 60 \beta_{4} + 39 \beta_{5} ) q^{78} + ( -194 - 87 \beta_{1} + 75 \beta_{2} + 102 \beta_{3} + 85 \beta_{4} + 16 \beta_{5} ) q^{79} + ( 658 - 157 \beta_{1} + 156 \beta_{2} + 4 \beta_{3} + 133 \beta_{4} + 125 \beta_{5} ) q^{80} + 81 q^{81} + ( 189 + 107 \beta_{1} - 30 \beta_{2} + 109 \beta_{3} + 153 \beta_{4} - 21 \beta_{5} ) q^{82} + ( 293 + 101 \beta_{1} - 41 \beta_{2} + 12 \beta_{3} + 142 \beta_{4} - 24 \beta_{5} ) q^{83} + ( -171 + 111 \beta_{1} - 66 \beta_{2} + 30 \beta_{3} - 39 \beta_{4} - 12 \beta_{5} ) q^{84} + ( -126 - 83 \beta_{1} - 31 \beta_{2} - 77 \beta_{3} - 91 \beta_{4} - 85 \beta_{5} ) q^{85} + ( -142 + 76 \beta_{1} - 123 \beta_{2} + 113 \beta_{3} + 32 \beta_{4} - 164 \beta_{5} ) q^{86} + ( -126 + 21 \beta_{1} + 75 \beta_{2} + 12 \beta_{3} - 45 \beta_{4} - 18 \beta_{5} ) q^{87} + ( 628 - 141 \beta_{1} + 53 \beta_{2} - 61 \beta_{3} + 49 \beta_{4} + 163 \beta_{5} ) q^{88} + ( 82 + 19 \beta_{1} + 89 \beta_{2} - 15 \beta_{3} - 69 \beta_{4} - 191 \beta_{5} ) q^{89} + ( -153 + 36 \beta_{1} - 81 \beta_{2} + 18 \beta_{3} - 36 \beta_{4} - 36 \beta_{5} ) q^{90} + ( 228 - 77 \beta_{1} + 31 \beta_{2} - 27 \beta_{3} - 5 \beta_{4} + 109 \beta_{5} ) q^{91} + ( 572 - 161 \beta_{1} + 48 \beta_{2} - 97 \beta_{3} + 19 \beta_{4} + 30 \beta_{5} ) q^{92} + ( -204 + 114 \beta_{1} - 144 \beta_{2} - 33 \beta_{3} + 3 \beta_{4} - 42 \beta_{5} ) q^{93} + ( 410 - 220 \beta_{1} - 200 \beta_{2} - 61 \beta_{3} - 88 \beta_{4} - 67 \beta_{5} ) q^{94} + ( 96 + 353 \beta_{1} - 73 \beta_{2} + 51 \beta_{3} - 11 \beta_{4} - 85 \beta_{5} ) q^{95} + ( 39 + 111 \beta_{1} - 156 \beta_{2} - 6 \beta_{3} - 36 \beta_{4} - 48 \beta_{5} ) q^{96} + ( 284 - 51 \beta_{1} - 47 \beta_{2} - 71 \beta_{3} - 87 \beta_{4} + 117 \beta_{5} ) q^{97} + ( 425 + 17 \beta_{1} - 101 \beta_{2} + 59 \beta_{3} - 46 \beta_{4} + 24 \beta_{5} ) q^{98} + ( -108 + 9 \beta_{1} - 45 \beta_{2} + 9 \beta_{3} + 9 \beta_{4} - 45 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 5q^{2} + 18q^{3} + 13q^{4} - 12q^{5} - 15q^{6} - 62q^{7} - 75q^{8} + 54q^{9} + O(q^{10})$$ $$6q - 5q^{2} + 18q^{3} + 13q^{4} - 12q^{5} - 15q^{6} - 62q^{7} - 75q^{8} + 54q^{9} - 111q^{10} - 72q^{11} + 39q^{12} - 192q^{13} + 3q^{14} - 36q^{15} - 27q^{16} - 100q^{17} - 45q^{18} - 266q^{19} + 255q^{20} - 186q^{21} + 44q^{22} + 50q^{23} - 225q^{24} - 6q^{25} + 472q^{26} + 162q^{27} - 333q^{28} - 242q^{29} - 333q^{30} - 438q^{31} + 35q^{32} - 216q^{33} - 150q^{34} - 258q^{35} + 117q^{36} - 596q^{37} + 664q^{38} - 576q^{39} - 831q^{40} + 54q^{41} + 9q^{42} - 360q^{43} - 714q^{44} - 108q^{45} - 871q^{46} - 720q^{47} - 81q^{48} - 302q^{49} + 4q^{50} - 300q^{51} - 1118q^{52} + 694q^{53} - 135q^{54} - 990q^{55} + 1917q^{56} - 798q^{57} + 1354q^{58} - 378q^{59} + 765q^{60} - 1396q^{61} + 1475q^{62} - 558q^{63} + 1225q^{64} - 348q^{65} + 132q^{66} - 402q^{67} + 2032q^{68} + 150q^{69} + 2415q^{70} - 964q^{71} - 675q^{72} - 192q^{73} + 2751q^{74} - 18q^{75} - 2306q^{76} + 2724q^{77} + 1416q^{78} - 802q^{79} + 4221q^{80} + 486q^{81} + 1735q^{82} + 2126q^{83} - 999q^{84} - 1206q^{85} - 609q^{86} - 726q^{87} + 3656q^{88} + 432q^{89} - 999q^{90} + 1258q^{91} + 3163q^{92} - 1314q^{93} + 1742q^{94} + 936q^{95} + 105q^{96} + 1290q^{97} + 2492q^{98} - 648q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 28 x^{4} + 22 x^{3} + 202 x^{2} - 96 x - 384$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 9 \nu^{4} + 20 \nu^{3} - 182 \nu^{2} - 26 \nu + 560$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} - 5 \nu^{4} + 92 \nu^{3} + 94 \nu^{2} - 590 \nu - 240$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$11 \nu^{5} - 3 \nu^{4} - 252 \nu^{3} + 82 \nu^{2} + 990 \nu - 336$$$$)/128$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{5} - 15 \nu^{4} + 212 \nu^{3} + 410 \nu^{2} - 938 \nu - 1936$$$$)/128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{2} + 9$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 13 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$20 \beta_{5} + 22 \beta_{4} + \beta_{3} + 28 \beta_{2} - 3 \beta_{1} + 119$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{5} + 56 \beta_{4} + 69 \beta_{3} + 46 \beta_{2} + 207 \beta_{1} - 27$$

## 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
 −4.17870 −2.57710 −1.43201 2.04345 2.81400 4.33036
−5.17870 3.00000 18.8190 17.1025 −15.5361 −19.8930 −56.0282 9.00000 −88.5686
1.2 −3.57710 3.00000 4.79568 −7.96859 −10.7313 0.508092 11.4622 9.00000 28.5045
1.3 −2.43201 3.00000 −2.08534 −2.10555 −7.29602 −2.84912 24.5276 9.00000 5.12071
1.4 1.04345 3.00000 −6.91121 −6.71512 3.13035 8.77104 −15.5591 9.00000 −7.00690
1.5 1.81400 3.00000 −4.70941 5.30362 5.44200 −31.2342 −23.0548 9.00000 9.62075
1.6 3.33036 3.00000 3.09132 −17.6168 9.99109 −17.3029 −16.3477 9.00000 −58.6704
 $$n$$: 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

## Inner twists

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

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$67$$ $$1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{6} + 5 T_{2}^{5} - 18 T_{2}^{4} - 80 T_{2}^{3} + 105 T_{2}^{2} + 263 T_{2} - 284$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(201))$$.