Properties

Label 201.4.a.c
Level 201
Weight 4
Character orbit 201.a
Self dual Yes
Analytic conductor 11.859
Analytic rank 1
Dimension 7
CM No
Inner twists 1

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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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -3 q^{3} + ( 3 + \beta_{1} + \beta_{2} ) q^{4} + ( 1 - \beta_{2} + \beta_{5} ) q^{5} + 3 \beta_{1} q^{6} + ( -4 + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{7} + ( -6 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -3 q^{3} + ( 3 + \beta_{1} + \beta_{2} ) q^{4} + ( 1 - \beta_{2} + \beta_{5} ) q^{5} + 3 \beta_{1} q^{6} + ( -4 + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{7} + ( -6 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{8} + 9 q^{9} + ( -7 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{10} + ( -19 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{11} + ( -9 - 3 \beta_{1} - 3 \beta_{2} ) q^{12} + ( 1 + 8 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{13} + ( -3 + 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{14} + ( -3 + 3 \beta_{2} - 3 \beta_{5} ) q^{15} + ( 10 + 20 \beta_{1} + 5 \beta_{2} + 7 \beta_{4} - 5 \beta_{5} ) q^{16} + ( 12 + 13 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{17} -9 \beta_{1} q^{18} + ( -20 + 3 \beta_{1} + 5 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} ) q^{19} + ( -52 - 3 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} ) q^{20} + ( 12 - 3 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} ) q^{21} + ( -33 + 35 \beta_{1} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{22} + ( -60 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 8 \beta_{4} + 6 \beta_{5} - 5 \beta_{6} ) q^{23} + ( 18 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} ) q^{24} + ( -21 - \beta_{1} + 3 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + 8 \beta_{6} ) q^{25} + ( -87 - 20 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{26} -27 q^{27} + ( -37 + 7 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{28} + ( -33 - 15 \beta_{1} + 10 \beta_{2} - 13 \beta_{3} - 10 \beta_{4} + \beta_{5} + \beta_{6} ) q^{29} + ( 21 - 6 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} ) q^{30} + ( -71 + 7 \beta_{1} + 6 \beta_{2} - 10 \beta_{3} - 8 \beta_{4} - 7 \beta_{5} - 5 \beta_{6} ) q^{31} + ( -151 - 43 \beta_{1} - 21 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} ) q^{32} + ( 57 - 9 \beta_{1} + 6 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} ) q^{33} + ( -110 - 53 \beta_{1} - 11 \beta_{2} - 3 \beta_{3} - 14 \beta_{4} + \beta_{5} + 12 \beta_{6} ) q^{34} + ( -102 - 11 \beta_{1} + 9 \beta_{2} + 13 \beta_{4} - 18 \beta_{5} - 4 \beta_{6} ) q^{35} + ( 27 + 9 \beta_{1} + 9 \beta_{2} ) q^{36} + ( 51 + 11 \beta_{1} + 16 \beta_{2} - \beta_{3} + 11 \beta_{4} - 21 \beta_{5} + 12 \beta_{6} ) q^{37} + ( 22 + 17 \beta_{1} + 25 \beta_{2} + 19 \beta_{3} + 12 \beta_{4} - 39 \beta_{5} + 12 \beta_{6} ) q^{38} + ( -3 - 24 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 12 \beta_{4} - 9 \beta_{5} + 3 \beta_{6} ) q^{39} + ( 28 + 57 \beta_{1} - \beta_{2} + 5 \beta_{3} + 6 \beta_{4} + 10 \beta_{5} - 2 \beta_{6} ) q^{40} + ( -110 - 33 \beta_{1} - 15 \beta_{2} + 14 \beta_{3} + 5 \beta_{4} + 8 \beta_{5} - 14 \beta_{6} ) q^{41} + ( 9 - 15 \beta_{1} + 6 \beta_{2} - 15 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{42} + ( -35 - 34 \beta_{1} - 7 \beta_{2} + 9 \beta_{3} - 9 \beta_{4} + 3 \beta_{5} - 16 \beta_{6} ) q^{43} + ( -253 - 45 \beta_{1} - 32 \beta_{2} - 6 \beta_{3} - 15 \beta_{4} + 17 \beta_{5} - 18 \beta_{6} ) q^{44} + ( 9 - 9 \beta_{2} + 9 \beta_{5} ) q^{45} + ( -82 + 25 \beta_{1} - 39 \beta_{2} - 13 \beta_{3} - 17 \beta_{4} + 23 \beta_{5} - 11 \beta_{6} ) q^{46} + ( -138 - 60 \beta_{1} + 8 \beta_{2} + \beta_{3} + 7 \beta_{4} - 16 \beta_{5} + 29 \beta_{6} ) q^{47} + ( -30 - 60 \beta_{1} - 15 \beta_{2} - 21 \beta_{4} + 15 \beta_{5} ) q^{48} + ( 32 - 26 \beta_{1} - 25 \beta_{2} - 5 \beta_{3} - 29 \beta_{4} + 45 \beta_{5} - 12 \beta_{6} ) q^{49} + ( 77 + 39 \beta_{1} + 35 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 33 \beta_{5} + 14 \beta_{6} ) q^{50} + ( -36 - 39 \beta_{1} - 15 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} ) q^{51} + ( 175 + 82 \beta_{1} + \beta_{2} - 3 \beta_{3} + 32 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} ) q^{52} + ( 30 - 31 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 27 \beta_{4} + 16 \beta_{5} - 24 \beta_{6} ) q^{53} + 27 \beta_{1} q^{54} + ( 165 + \beta_{1} + 24 \beta_{2} + 20 \beta_{3} + 27 \beta_{4} - 41 \beta_{5} + 28 \beta_{6} ) q^{55} + ( -42 - 4 \beta_{1} + 12 \beta_{2} - 39 \beta_{3} - 14 \beta_{4} - 12 \beta_{5} + 11 \beta_{6} ) q^{56} + ( 60 - 9 \beta_{1} - 15 \beta_{2} - 24 \beta_{3} + 6 \beta_{4} + 12 \beta_{5} - 6 \beta_{6} ) q^{57} + ( 225 - 5 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} - 28 \beta_{4} + 32 \beta_{5} + 21 \beta_{6} ) q^{58} + ( -35 - 71 \beta_{1} + 12 \beta_{2} + 14 \beta_{3} - 2 \beta_{4} + 11 \beta_{5} + 20 \beta_{6} ) q^{59} + ( 156 + 9 \beta_{1} + 21 \beta_{2} + 12 \beta_{3} + 15 \beta_{4} - 18 \beta_{5} + 12 \beta_{6} ) q^{60} + ( 61 - 48 \beta_{1} - 35 \beta_{2} - 29 \beta_{3} + 46 \beta_{4} + 15 \beta_{5} - 29 \beta_{6} ) q^{61} + ( -52 + 40 \beta_{1} - 14 \beta_{2} + 15 \beta_{3} - 4 \beta_{4} + 26 \beta_{5} + 7 \beta_{6} ) q^{62} + ( -36 + 9 \beta_{4} - 18 \beta_{5} - 9 \beta_{6} ) q^{63} + ( 251 + 125 \beta_{1} + \beta_{2} - 28 \beta_{3} + 2 \beta_{4} + 30 \beta_{5} - 48 \beta_{6} ) q^{64} + ( 27 - 13 \beta_{1} + 12 \beta_{2} + 18 \beta_{3} - 55 \beta_{4} + 33 \beta_{5} - 2 \beta_{6} ) q^{65} + ( 99 - 105 \beta_{1} + 6 \beta_{3} - 3 \beta_{4} - 9 \beta_{5} + 6 \beta_{6} ) q^{66} + 67 q^{67} + ( 515 + 183 \beta_{1} + 58 \beta_{2} - 12 \beta_{3} + 47 \beta_{4} - 7 \beta_{5} - 26 \beta_{6} ) q^{68} + ( 180 - 3 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} - 24 \beta_{4} - 18 \beta_{5} + 15 \beta_{6} ) q^{69} + ( 156 + 75 \beta_{1} - 5 \beta_{2} + 49 \beta_{3} - 22 \beta_{4} + 4 \beta_{5} + 14 \beta_{6} ) q^{70} + ( -161 + 32 \beta_{1} + 17 \beta_{2} - 45 \beta_{3} - 40 \beta_{4} + 31 \beta_{5} - 23 \beta_{6} ) q^{71} + ( -54 - 18 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} - 18 \beta_{4} - 9 \beta_{5} + 18 \beta_{6} ) q^{72} + ( 136 - 16 \beta_{1} + 55 \beta_{3} + 44 \beta_{4} - 50 \beta_{5} + 16 \beta_{6} ) q^{73} + ( 38 - 82 \beta_{1} + 10 \beta_{2} + 45 \beta_{3} - 47 \beta_{4} - \beta_{5} + 44 \beta_{6} ) q^{74} + ( 63 + 3 \beta_{1} - 9 \beta_{2} - 21 \beta_{3} + 6 \beta_{4} - 24 \beta_{6} ) q^{75} + ( 231 - 65 \beta_{1} + 20 \beta_{2} + 46 \beta_{3} + 7 \beta_{4} - 57 \beta_{5} + 46 \beta_{6} ) q^{76} + ( 35 + 17 \beta_{1} + 22 \beta_{2} - 24 \beta_{3} - 61 \beta_{4} + 41 \beta_{5} + 28 \beta_{6} ) q^{77} + ( 261 + 60 \beta_{1} + 21 \beta_{2} + 15 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{78} + ( -326 - 67 \beta_{1} - 21 \beta_{2} - 71 \beta_{3} - 51 \beta_{4} - 16 \beta_{5} - 15 \beta_{6} ) q^{79} + ( -253 - 98 \beta_{1} - 17 \beta_{2} + 18 \beta_{3} + 32 \beta_{4} - 61 \beta_{5} + 28 \beta_{6} ) q^{80} + 81 q^{81} + ( 206 + 171 \beta_{1} + 15 \beta_{2} - 3 \beta_{3} + 54 \beta_{4} - 36 \beta_{5} - 44 \beta_{6} ) q^{82} + ( -396 + 101 \beta_{1} - 13 \beta_{2} + 4 \beta_{3} - 29 \beta_{4} + 32 \beta_{5} - 22 \beta_{6} ) q^{83} + ( 111 - 21 \beta_{1} - 6 \beta_{3} - 12 \beta_{4} + 3 \beta_{5} - 9 \beta_{6} ) q^{84} + ( -83 - 45 \beta_{1} - 22 \beta_{2} - 4 \beta_{3} - 39 \beta_{4} + 31 \beta_{5} + 4 \beta_{6} ) q^{85} + ( 280 + 81 \beta_{1} + 35 \beta_{2} + 12 \beta_{3} + 63 \beta_{4} - 38 \beta_{5} - 30 \beta_{6} ) q^{86} + ( 99 + 45 \beta_{1} - 30 \beta_{2} + 39 \beta_{3} + 30 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{87} + ( 499 + 129 \beta_{1} + 10 \beta_{2} - 38 \beta_{3} + 75 \beta_{4} + 17 \beta_{5} - 66 \beta_{6} ) q^{88} + ( -214 + 7 \beta_{1} + 45 \beta_{2} + 16 \beta_{3} + 52 \beta_{4} - 56 \beta_{5} + 68 \beta_{6} ) q^{89} + ( -63 + 18 \beta_{1} - 9 \beta_{2} - 27 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} - 18 \beta_{6} ) q^{90} + ( -249 - 15 \beta_{1} + 30 \beta_{2} - 36 \beta_{3} + 53 \beta_{4} - 91 \beta_{5} + 26 \beta_{6} ) q^{91} + ( -70 + 202 \beta_{1} - 38 \beta_{2} - 63 \beta_{3} + 10 \beta_{4} + 26 \beta_{5} - 49 \beta_{6} ) q^{92} + ( 213 - 21 \beta_{1} - 18 \beta_{2} + 30 \beta_{3} + 24 \beta_{4} + 21 \beta_{5} + 15 \beta_{6} ) q^{93} + ( 864 + 272 \beta_{1} + 122 \beta_{2} + 12 \beta_{3} - 41 \beta_{4} - 5 \beta_{5} + 45 \beta_{6} ) q^{94} + ( -481 + 187 \beta_{1} - 44 \beta_{2} + 6 \beta_{3} - 33 \beta_{4} + 41 \beta_{5} - 48 \beta_{6} ) q^{95} + ( 453 + 129 \beta_{1} + 63 \beta_{2} - 21 \beta_{3} + 9 \beta_{4} - 30 \beta_{5} + 18 \beta_{6} ) q^{96} + ( -113 + 59 \beta_{1} - 46 \beta_{2} - 6 \beta_{3} - 53 \beta_{4} - 17 \beta_{5} + 82 \beta_{6} ) q^{97} + ( 64 + 26 \beta_{1} + 5 \beta_{2} - 108 \beta_{3} + 65 \beta_{4} + 16 \beta_{5} - 62 \beta_{6} ) q^{98} + ( -171 + 27 \beta_{1} - 18 \beta_{2} + 9 \beta_{4} - 9 \beta_{5} + 18 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - q^{2} - 21q^{3} + 21q^{4} + 11q^{5} + 3q^{6} - 33q^{7} - 45q^{8} + 63q^{9} + O(q^{10}) \) \( 7q - q^{2} - 21q^{3} + 21q^{4} + 11q^{5} + 3q^{6} - 33q^{7} - 45q^{8} + 63q^{9} - 51q^{10} - 130q^{11} - 63q^{12} + 16q^{13} + 5q^{14} - 33q^{15} + 77q^{16} + 90q^{17} - 9q^{18} - 132q^{19} - 359q^{20} + 99q^{21} - 192q^{22} - 399q^{23} + 135q^{24} - 132q^{25} - 638q^{26} - 189q^{27} - 245q^{28} - 302q^{29} + 153q^{30} - 555q^{31} - 1031q^{32} + 390q^{33} - 832q^{34} - 775q^{35} + 189q^{36} + 297q^{37} + 98q^{38} - 48q^{39} + 305q^{40} - 717q^{41} - 15q^{42} - 245q^{43} - 1766q^{44} + 99q^{45} - 497q^{46} - 1072q^{47} - 231q^{48} + 314q^{49} + 454q^{50} - 270q^{51} + 1344q^{52} + 265q^{53} + 27q^{54} + 1096q^{55} - 477q^{56} + 396q^{57} + 1610q^{58} - 255q^{59} + 1077q^{60} + 418q^{61} - 191q^{62} - 297q^{63} + 1889q^{64} + 262q^{65} + 576q^{66} + 469q^{67} + 3720q^{68} + 1197q^{69} + 1309q^{70} - 1194q^{71} - 405q^{72} + 995q^{73} + 259q^{74} + 396q^{75} + 1506q^{76} + 230q^{77} + 1914q^{78} - 2640q^{79} - 1949q^{80} + 567q^{81} + 1535q^{82} - 2579q^{83} + 735q^{84} - 562q^{85} + 1991q^{86} + 906q^{87} + 3624q^{88} - 1604q^{89} - 459q^{90} - 2116q^{91} - 351q^{92} + 1665q^{93} + 6178q^{94} - 3028q^{95} + 3093q^{96} - 808q^{97} + 258q^{98} - 1170q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 38 x^{5} + 18 x^{4} + 373 x^{3} - 151 x^{2} - 956 x + 498\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 11 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 47 \nu^{5} + 27 \nu^{4} + 1339 \nu^{3} - 408 \nu^{2} - 6314 \nu + 1966 \)\()/466\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{6} + 10 \nu^{5} + 956 \nu^{4} + 310 \nu^{3} - 9139 \nu^{2} - 5250 \nu + 17954 \)\()/932\)
\(\beta_{5}\)\(=\)\((\)\( -35 \nu^{6} + 14 \nu^{5} + 1152 \nu^{4} + 434 \nu^{3} - 7389 \nu^{2} - 4554 \nu + 4818 \)\()/932\)
\(\beta_{6}\)\(=\)\((\)\( -87 \nu^{6} + 128 \nu^{5} + 3010 \nu^{4} - 2556 \nu^{3} - 23919 \nu^{2} + 13418 \nu + 32134 \)\()/1864\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 11\)
\(\nu^{3}\)\(=\)\(-2 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 18 \beta_{1} + 6\)
\(\nu^{4}\)\(=\)\(-5 \beta_{5} + 7 \beta_{4} + 29 \beta_{2} + 44 \beta_{1} + 210\)
\(\nu^{5}\)\(=\)\(-58 \beta_{6} + 22 \beta_{5} + 67 \beta_{4} - 39 \beta_{3} + 53 \beta_{2} + 427 \beta_{1} + 343\)
\(\nu^{6}\)\(=\)\(-48 \beta_{6} - 170 \beta_{5} + 282 \beta_{4} - 28 \beta_{3} + 777 \beta_{2} + 1501 \beta_{1} + 4939\)

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
5.41567
3.01477
1.85609
0.535294
−2.11972
−3.26131
−4.44080
−5.41567 −3.00000 21.3295 −5.98460 16.2470 0.240383 −72.1883 9.00000 32.4106
1.2 −3.01477 −3.00000 1.08886 14.7190 9.04432 −32.2394 20.8355 9.00000 −44.3744
1.3 −1.85609 −3.00000 −4.55494 −4.35854 5.56826 17.6123 23.3031 9.00000 8.08983
1.4 −0.535294 −3.00000 −7.71346 12.7037 1.60588 −8.67573 8.41132 9.00000 −6.80020
1.5 2.11972 −3.00000 −3.50678 1.76139 −6.35917 22.2326 −24.3912 9.00000 3.73365
1.6 3.26131 −3.00000 2.63612 7.83356 −9.78392 −27.8825 −17.4933 9.00000 25.5476
1.7 4.44080 −3.00000 11.7207 −15.6745 −13.3224 −4.28763 16.5228 9.00000 −69.6071
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

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 \( T_{2}^{7} + T_{2}^{6} - 38 T_{2}^{5} - 18 T_{2}^{4} + 373 T_{2}^{3} + 151 T_{2}^{2} - 956 T_{2} - 498 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(201))\).