Properties

Label 11.6.a.b
Level 11
Weight 6
Character orbit 11.a
Self dual Yes
Analytic conductor 1.764
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 11 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 11.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.76422201794\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{2} q^{2} \) \( + ( 11 + \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 30 - 6 \beta_{1} - 4 \beta_{2} ) q^{4} \) \( + ( 5 + 9 \beta_{1} - \beta_{2} ) q^{5} \) \( + ( -72 + 10 \beta_{1} + 13 \beta_{2} ) q^{6} \) \( + ( 38 - 30 \beta_{1} - 10 \beta_{2} ) q^{7} \) \( + ( -188 + 26 \beta_{2} ) q^{8} \) \( + ( -6 + 11 \beta_{1} - 19 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{2} \) \( + ( 11 + \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 30 - 6 \beta_{1} - 4 \beta_{2} ) q^{4} \) \( + ( 5 + 9 \beta_{1} - \beta_{2} ) q^{5} \) \( + ( -72 + 10 \beta_{1} + 13 \beta_{2} ) q^{6} \) \( + ( 38 - 30 \beta_{1} - 10 \beta_{2} ) q^{7} \) \( + ( -188 + 26 \beta_{2} ) q^{8} \) \( + ( -6 + 11 \beta_{1} - 19 \beta_{2} ) q^{9} \) \( + ( -152 + 42 \beta_{1} - 9 \beta_{2} ) q^{10} \) \( + 121 q^{11} \) \( + ( 354 - 70 \beta_{1} - 112 \beta_{2} ) q^{12} \) \( + ( 156 + 18 \beta_{1} + 70 \beta_{2} ) q^{13} \) \( + ( -320 - 60 \beta_{1} + 138 \beta_{2} ) q^{14} \) \( + ( 523 + 85 \beta_{1} + 27 \beta_{2} ) q^{15} \) \( + ( 652 + 36 \beta_{1} - 164 \beta_{2} ) q^{16} \) \( + ( 382 - 60 \beta_{1} + 132 \beta_{2} ) q^{17} \) \( + ( -1288 + 158 \beta_{1} + 48 \beta_{2} ) q^{18} \) \( + ( 480 - 60 \beta_{1} - 60 \beta_{2} ) q^{19} \) \( + ( -1138 - 66 \beta_{1} - 168 \beta_{2} ) q^{20} \) \( + ( -182 - 362 \beta_{1} - 318 \beta_{2} ) q^{21} \) \( + 121 \beta_{2} q^{22} \) \( + ( -1189 + 501 \beta_{1} - 21 \beta_{2} ) q^{23} \) \( + ( -3940 + 72 \beta_{1} + 526 \beta_{2} ) q^{24} \) \( + ( -104 + 255 \beta_{1} + 265 \beta_{2} ) q^{25} \) \( + ( 4160 - 348 \beta_{1} - 160 \beta_{2} ) q^{26} \) \( + ( -887 - 329 \beta_{1} + 57 \beta_{2} ) q^{27} \) \( + ( 7940 - 108 \beta_{1} - 432 \beta_{2} ) q^{28} \) \( + ( -1096 - 138 \beta_{1} - 182 \beta_{2} ) q^{29} \) \( + ( 824 + 178 \beta_{1} + 245 \beta_{2} ) q^{30} \) \( + ( -1389 + 69 \beta_{1} - 101 \beta_{2} ) q^{31} \) \( + ( -4512 + 1128 \beta_{1} + 404 \beta_{2} ) q^{32} \) \( + ( 1331 + 121 \beta_{1} - 121 \beta_{2} ) q^{33} \) \( + ( 8784 - 1032 \beta_{1} - 26 \beta_{2} ) q^{34} \) \( + ( -7770 - 918 \beta_{1} - 818 \beta_{2} ) q^{35} \) \( + ( 1588 - 8 \beta_{1} - 1188 \beta_{2} ) q^{36} \) \( + ( 5711 + 591 \beta_{1} + 937 \beta_{2} ) q^{37} \) \( + ( -3120 + 120 \beta_{1} + 840 \beta_{2} ) q^{38} \) \( + ( -2532 + 1036 \beta_{1} + 844 \beta_{2} ) q^{39} \) \( + ( -4892 - 600 \beta_{1} - 46 \beta_{2} ) q^{40} \) \( + ( 1904 + 282 \beta_{1} - 1378 \beta_{2} ) q^{41} \) \( + ( -16096 + 460 \beta_{1} + 1814 \beta_{2} ) q^{42} \) \( + ( -8366 - 1110 \beta_{1} + 1190 \beta_{2} ) q^{43} \) \( + ( 3630 - 726 \beta_{1} - 484 \beta_{2} ) q^{44} \) \( + ( 6334 - 544 \beta_{1} + 496 \beta_{2} ) q^{45} \) \( + ( -6312 + 2130 \beta_{1} - 2107 \beta_{2} ) q^{46} \) \( + ( -5320 - 1272 \beta_{1} - 600 \beta_{2} ) q^{47} \) \( + ( 20564 - 628 \beta_{1} - 2604 \beta_{2} ) q^{48} \) \( + ( 15437 + 2220 \beta_{1} + 340 \beta_{2} ) q^{49} \) \( + ( 13880 - 570 \beta_{1} - 1674 \beta_{2} ) q^{50} \) \( + ( -7942 + 1102 \beta_{1} + 1034 \beta_{2} ) q^{51} \) \( + ( -11432 - 1008 \beta_{1} + 3256 \beta_{2} ) q^{52} \) \( + ( 17402 - 1620 \beta_{1} - 476 \beta_{2} ) q^{53} \) \( + ( 6824 - 1658 \beta_{1} - 457 \beta_{2} ) q^{54} \) \( + ( 605 + 1089 \beta_{1} - 121 \beta_{2} ) q^{55} \) \( + ( -15464 + 4080 \beta_{1} + 5468 \beta_{2} ) q^{56} \) \( + ( 6960 - 720 \beta_{1} - 1560 \beta_{2} ) q^{57} \) \( + ( -9904 + 540 \beta_{1} - 92 \beta_{2} ) q^{58} \) \( + ( -1495 + 747 \beta_{1} + 3141 \beta_{2} ) q^{59} \) \( + ( -3326 - 3478 \beta_{1} - 1376 \beta_{2} ) q^{60} \) \( + ( 7508 - 4038 \beta_{1} - 1466 \beta_{2} ) q^{61} \) \( + ( -6952 + 882 \beta_{1} - 1123 \beta_{2} ) q^{62} \) \( + ( -4268 + 308 \beta_{1} - 3332 \beta_{2} ) q^{63} \) \( + ( -7096 + 936 \beta_{1} - 3136 \beta_{2} ) q^{64} \) \( + ( -4172 + 4848 \beta_{1} - 264 \beta_{2} ) q^{65} \) \( + ( -8712 + 1210 \beta_{1} + 1573 \beta_{2} ) q^{66} \) \( + ( -15011 - 2721 \beta_{1} - 6575 \beta_{2} ) q^{67} \) \( + ( -3516 - 2052 \beta_{1} + 6728 \beta_{2} ) q^{68} \) \( + ( 10477 + 3611 \beta_{1} + 3421 \beta_{2} ) q^{69} \) \( + ( -41536 + 1236 \beta_{1} - 2662 \beta_{2} ) q^{70} \) \( + ( 13985 - 2673 \beta_{1} - 3935 \beta_{2} ) q^{71} \) \( + ( -32360 + 2040 \beta_{1} + 4820 \beta_{2} ) q^{72} \) \( + ( 6316 - 3522 \beta_{1} + 5370 \beta_{2} ) q^{73} \) \( + ( 52184 - 3258 \beta_{1} + 781 \beta_{2} ) q^{74} \) \( + ( -9004 + 5096 \beta_{1} + 4824 \beta_{2} ) q^{75} \) \( + ( 35520 - 2640 \beta_{1} - 4800 \beta_{2} ) q^{76} \) \( + ( 4598 - 3630 \beta_{1} - 1210 \beta_{2} ) q^{77} \) \( + ( 41968 - 920 \beta_{1} - 7980 \beta_{2} ) q^{78} \) \( + ( 41262 + 1362 \beta_{1} + 3902 \beta_{2} ) q^{79} \) \( + ( 39564 - 12 \beta_{1} + 1868 \beta_{2} ) q^{80} \) \( + ( -26879 - 6280 \beta_{1} + 4600 \beta_{2} ) q^{81} \) \( + ( -88256 + 9396 \beta_{1} + 6852 \beta_{2} ) q^{82} \) \( + ( -51726 + 11250 \beta_{1} + 3150 \beta_{2} ) q^{83} \) \( + ( 113692 + 2540 \beta_{1} - 14096 \beta_{2} ) q^{84} \) \( + ( -37114 + 7302 \beta_{1} - 3310 \beta_{2} ) q^{85} \) \( + ( 84880 - 11580 \beta_{1} - 10906 \beta_{2} ) q^{86} \) \( + ( -5024 - 4296 \beta_{1} - 1960 \beta_{2} ) q^{87} \) \( + ( -22748 + 3146 \beta_{2} ) q^{88} \) \( + ( -34085 - 4569 \beta_{1} - 5167 \beta_{2} ) q^{89} \) \( + ( 36192 - 5152 \beta_{1} + 5438 \beta_{2} ) q^{90} \) \( + ( -33032 - 10536 \beta_{1} + 6840 \beta_{2} ) q^{91} \) \( + ( -113886 + 5130 \beta_{1} - 1472 \beta_{2} ) q^{92} \) \( + ( -4971 - 1709 \beta_{1} + 421 \beta_{2} ) q^{93} \) \( + ( -24480 - 1488 \beta_{1} - 376 \beta_{2} ) q^{94} \) \( + ( -7440 + 120 \beta_{1} - 1680 \beta_{2} ) q^{95} \) \( + ( -29088 + 10808 \beta_{1} + 15404 \beta_{2} ) q^{96} \) \( + ( 1085 + 6429 \beta_{1} + 123 \beta_{2} ) q^{97} \) \( + ( -1120 + 6840 \beta_{1} + 9637 \beta_{2} ) q^{98} \) \( + ( -726 + 1331 \beta_{1} - 2299 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 34q^{3} \) \(\mathstrut +\mathstrut 84q^{4} \) \(\mathstrut +\mathstrut 24q^{5} \) \(\mathstrut -\mathstrut 206q^{6} \) \(\mathstrut +\mathstrut 84q^{7} \) \(\mathstrut -\mathstrut 564q^{8} \) \(\mathstrut -\mathstrut 7q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 34q^{3} \) \(\mathstrut +\mathstrut 84q^{4} \) \(\mathstrut +\mathstrut 24q^{5} \) \(\mathstrut -\mathstrut 206q^{6} \) \(\mathstrut +\mathstrut 84q^{7} \) \(\mathstrut -\mathstrut 564q^{8} \) \(\mathstrut -\mathstrut 7q^{9} \) \(\mathstrut -\mathstrut 414q^{10} \) \(\mathstrut +\mathstrut 363q^{11} \) \(\mathstrut +\mathstrut 992q^{12} \) \(\mathstrut +\mathstrut 486q^{13} \) \(\mathstrut -\mathstrut 1020q^{14} \) \(\mathstrut +\mathstrut 1654q^{15} \) \(\mathstrut +\mathstrut 1992q^{16} \) \(\mathstrut +\mathstrut 1086q^{17} \) \(\mathstrut -\mathstrut 3706q^{18} \) \(\mathstrut +\mathstrut 1380q^{19} \) \(\mathstrut -\mathstrut 3480q^{20} \) \(\mathstrut -\mathstrut 908q^{21} \) \(\mathstrut -\mathstrut 3066q^{23} \) \(\mathstrut -\mathstrut 11748q^{24} \) \(\mathstrut -\mathstrut 57q^{25} \) \(\mathstrut +\mathstrut 12132q^{26} \) \(\mathstrut -\mathstrut 2990q^{27} \) \(\mathstrut +\mathstrut 23712q^{28} \) \(\mathstrut -\mathstrut 3426q^{29} \) \(\mathstrut +\mathstrut 2650q^{30} \) \(\mathstrut -\mathstrut 4098q^{31} \) \(\mathstrut -\mathstrut 12408q^{32} \) \(\mathstrut +\mathstrut 4114q^{33} \) \(\mathstrut +\mathstrut 25320q^{34} \) \(\mathstrut -\mathstrut 24228q^{35} \) \(\mathstrut +\mathstrut 4756q^{36} \) \(\mathstrut +\mathstrut 17724q^{37} \) \(\mathstrut -\mathstrut 9240q^{38} \) \(\mathstrut -\mathstrut 6560q^{39} \) \(\mathstrut -\mathstrut 15276q^{40} \) \(\mathstrut +\mathstrut 5994q^{41} \) \(\mathstrut -\mathstrut 47828q^{42} \) \(\mathstrut -\mathstrut 26208q^{43} \) \(\mathstrut +\mathstrut 10164q^{44} \) \(\mathstrut +\mathstrut 18458q^{45} \) \(\mathstrut -\mathstrut 16806q^{46} \) \(\mathstrut -\mathstrut 17232q^{47} \) \(\mathstrut +\mathstrut 61064q^{48} \) \(\mathstrut +\mathstrut 48531q^{49} \) \(\mathstrut +\mathstrut 41070q^{50} \) \(\mathstrut -\mathstrut 22724q^{51} \) \(\mathstrut -\mathstrut 35304q^{52} \) \(\mathstrut +\mathstrut 50586q^{53} \) \(\mathstrut +\mathstrut 18814q^{54} \) \(\mathstrut +\mathstrut 2904q^{55} \) \(\mathstrut -\mathstrut 42312q^{56} \) \(\mathstrut +\mathstrut 20160q^{57} \) \(\mathstrut -\mathstrut 29172q^{58} \) \(\mathstrut -\mathstrut 3738q^{59} \) \(\mathstrut -\mathstrut 13456q^{60} \) \(\mathstrut +\mathstrut 18486q^{61} \) \(\mathstrut -\mathstrut 19974q^{62} \) \(\mathstrut -\mathstrut 12496q^{63} \) \(\mathstrut -\mathstrut 20352q^{64} \) \(\mathstrut -\mathstrut 7668q^{65} \) \(\mathstrut -\mathstrut 24926q^{66} \) \(\mathstrut -\mathstrut 47754q^{67} \) \(\mathstrut -\mathstrut 12600q^{68} \) \(\mathstrut +\mathstrut 35042q^{69} \) \(\mathstrut -\mathstrut 123372q^{70} \) \(\mathstrut +\mathstrut 39282q^{71} \) \(\mathstrut -\mathstrut 95040q^{72} \) \(\mathstrut +\mathstrut 15426q^{73} \) \(\mathstrut +\mathstrut 153294q^{74} \) \(\mathstrut -\mathstrut 21916q^{75} \) \(\mathstrut +\mathstrut 103920q^{76} \) \(\mathstrut +\mathstrut 10164q^{77} \) \(\mathstrut +\mathstrut 124984q^{78} \) \(\mathstrut +\mathstrut 125148q^{79} \) \(\mathstrut +\mathstrut 118680q^{80} \) \(\mathstrut -\mathstrut 86917q^{81} \) \(\mathstrut -\mathstrut 255372q^{82} \) \(\mathstrut -\mathstrut 143928q^{83} \) \(\mathstrut +\mathstrut 343616q^{84} \) \(\mathstrut -\mathstrut 104040q^{85} \) \(\mathstrut +\mathstrut 243060q^{86} \) \(\mathstrut -\mathstrut 19368q^{87} \) \(\mathstrut -\mathstrut 68244q^{88} \) \(\mathstrut -\mathstrut 106824q^{89} \) \(\mathstrut +\mathstrut 103424q^{90} \) \(\mathstrut -\mathstrut 109632q^{91} \) \(\mathstrut -\mathstrut 336528q^{92} \) \(\mathstrut -\mathstrut 16622q^{93} \) \(\mathstrut -\mathstrut 74928q^{94} \) \(\mathstrut -\mathstrut 22200q^{95} \) \(\mathstrut -\mathstrut 76456q^{96} \) \(\mathstrut +\mathstrut 9684q^{97} \) \(\mathstrut +\mathstrut 3480q^{98} \) \(\mathstrut -\mathstrut 847q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(52\) \(x\mathstrut -\mathstrut \) \(38\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 3 \nu - 34 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(34\)

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
−0.749680
8.04796
−6.29828
−10.3963 20.6466 76.0833 8.64919 −214.649 164.454 −458.304 183.283 −89.9197
1.2 2.20859 16.8394 −27.1221 75.2230 37.1913 −225.525 −130.577 40.5643 166.137
1.3 8.18772 −3.48600 35.0388 −59.8722 −28.5424 145.071 24.8808 −230.848 −490.217
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{3} \) \(\mathstrut -\mathstrut 90 T_{2} \) \(\mathstrut +\mathstrut 188 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(11))\).