Properties

Label 11.4.a.a
Level 11
Weight 4
Character orbit 11.a
Self dual Yes
Analytic conductor 0.649
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 11 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 11.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.649021010063\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 + \beta ) q^{2} \) \( + ( -1 - 4 \beta ) q^{3} \) \( + ( -4 + 2 \beta ) q^{4} \) \( + ( 1 + 8 \beta ) q^{5} \) \( + ( -13 - 5 \beta ) q^{6} \) \( + ( 10 - 4 \beta ) q^{7} \) \( + ( -6 - 10 \beta ) q^{8} \) \( + ( 22 + 8 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 + \beta ) q^{2} \) \( + ( -1 - 4 \beta ) q^{3} \) \( + ( -4 + 2 \beta ) q^{4} \) \( + ( 1 + 8 \beta ) q^{5} \) \( + ( -13 - 5 \beta ) q^{6} \) \( + ( 10 - 4 \beta ) q^{7} \) \( + ( -6 - 10 \beta ) q^{8} \) \( + ( 22 + 8 \beta ) q^{9} \) \( + ( 25 + 9 \beta ) q^{10} \) \( -11 q^{11} \) \( + ( -20 + 14 \beta ) q^{12} \) \( + ( 40 - 20 \beta ) q^{13} \) \( + ( -2 + 6 \beta ) q^{14} \) \( + ( -97 - 12 \beta ) q^{15} \) \( + ( -4 - 32 \beta ) q^{16} \) \( + ( -62 + 12 \beta ) q^{17} \) \( + ( 46 + 30 \beta ) q^{18} \) \( + ( 36 + 60 \beta ) q^{19} \) \( + ( 44 - 30 \beta ) q^{20} \) \( + ( 38 - 36 \beta ) q^{21} \) \( + ( -11 - 11 \beta ) q^{22} \) \( + ( -49 - 36 \beta ) q^{23} \) \( + ( 126 + 34 \beta ) q^{24} \) \( + ( 68 + 16 \beta ) q^{25} \) \( + ( -20 + 20 \beta ) q^{26} \) \( + ( -91 + 12 \beta ) q^{27} \) \( + ( -64 + 36 \beta ) q^{28} \) \( + ( 72 - 56 \beta ) q^{29} \) \( + ( -133 - 109 \beta ) q^{30} \) \( + ( -17 + 28 \beta ) q^{31} \) \( + ( -52 + 44 \beta ) q^{32} \) \( + ( 11 + 44 \beta ) q^{33} \) \( + ( -26 - 50 \beta ) q^{34} \) \( + ( -86 + 76 \beta ) q^{35} \) \( + ( -40 + 12 \beta ) q^{36} \) \( + ( 27 - 8 \beta ) q^{37} \) \( + ( 216 + 96 \beta ) q^{38} \) \( + ( 200 - 140 \beta ) q^{39} \) \( + ( -246 - 58 \beta ) q^{40} \) \( + ( 268 - 4 \beta ) q^{41} \) \( + ( -70 + 2 \beta ) q^{42} \) \( + ( -30 - 16 \beta ) q^{43} \) \( + ( 44 - 22 \beta ) q^{44} \) \( + ( 214 + 184 \beta ) q^{45} \) \( + ( -157 - 85 \beta ) q^{46} \) \( + ( -136 - 120 \beta ) q^{47} \) \( + ( 388 + 48 \beta ) q^{48} \) \( + ( -195 - 80 \beta ) q^{49} \) \( + ( 116 + 84 \beta ) q^{50} \) \( + ( -82 + 236 \beta ) q^{51} \) \( + ( -280 + 160 \beta ) q^{52} \) \( + ( -246 - 56 \beta ) q^{53} \) \( + ( -55 - 79 \beta ) q^{54} \) \( + ( -11 - 88 \beta ) q^{55} \) \( + ( 60 - 76 \beta ) q^{56} \) \( + ( -756 - 204 \beta ) q^{57} \) \( + ( -96 + 16 \beta ) q^{58} \) \( + ( 317 - 132 \beta ) q^{59} \) \( + ( 316 - 146 \beta ) q^{60} \) \( + ( 420 + 184 \beta ) q^{61} \) \( + ( 67 + 11 \beta ) q^{62} \) \( + ( 124 - 8 \beta ) q^{63} \) \( + ( 112 + 248 \beta ) q^{64} \) \( + ( -440 + 300 \beta ) q^{65} \) \( + ( 143 + 55 \beta ) q^{66} \) \( + ( 377 - 20 \beta ) q^{67} \) \( + ( 320 - 172 \beta ) q^{68} \) \( + ( 481 + 232 \beta ) q^{69} \) \( + ( 142 - 10 \beta ) q^{70} \) \( + ( -339 + 76 \beta ) q^{71} \) \( + ( -372 - 268 \beta ) q^{72} \) \( + ( -200 - 468 \beta ) q^{73} \) \( + ( 3 + 19 \beta ) q^{74} \) \( + ( -260 - 288 \beta ) q^{75} \) \( + ( 216 - 168 \beta ) q^{76} \) \( + ( -110 + 44 \beta ) q^{77} \) \( + ( -220 + 60 \beta ) q^{78} \) \( + ( 158 + 656 \beta ) q^{79} \) \( + ( -772 - 64 \beta ) q^{80} \) \( + ( -647 + 136 \beta ) q^{81} \) \( + ( 256 + 264 \beta ) q^{82} \) \( + ( 234 + 120 \beta ) q^{83} \) \( + ( -368 + 220 \beta ) q^{84} \) \( + ( 226 - 484 \beta ) q^{85} \) \( + ( -78 - 46 \beta ) q^{86} \) \( + ( 600 - 232 \beta ) q^{87} \) \( + ( 66 + 110 \beta ) q^{88} \) \( + ( -921 - 328 \beta ) q^{89} \) \( + ( 766 + 398 \beta ) q^{90} \) \( + ( 640 - 360 \beta ) q^{91} \) \( + ( -20 + 46 \beta ) q^{92} \) \( + ( -319 + 40 \beta ) q^{93} \) \( + ( -496 - 256 \beta ) q^{94} \) \( + ( 1476 + 348 \beta ) q^{95} \) \( + ( -476 + 164 \beta ) q^{96} \) \( + ( 1097 + 144 \beta ) q^{97} \) \( + ( -435 - 275 \beta ) q^{98} \) \( + ( -242 - 88 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 26q^{6} \) \(\mathstrut +\mathstrut 20q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 44q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 26q^{6} \) \(\mathstrut +\mathstrut 20q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 44q^{9} \) \(\mathstrut +\mathstrut 50q^{10} \) \(\mathstrut -\mathstrut 22q^{11} \) \(\mathstrut -\mathstrut 40q^{12} \) \(\mathstrut +\mathstrut 80q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut -\mathstrut 194q^{15} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 124q^{17} \) \(\mathstrut +\mathstrut 92q^{18} \) \(\mathstrut +\mathstrut 72q^{19} \) \(\mathstrut +\mathstrut 88q^{20} \) \(\mathstrut +\mathstrut 76q^{21} \) \(\mathstrut -\mathstrut 22q^{22} \) \(\mathstrut -\mathstrut 98q^{23} \) \(\mathstrut +\mathstrut 252q^{24} \) \(\mathstrut +\mathstrut 136q^{25} \) \(\mathstrut -\mathstrut 40q^{26} \) \(\mathstrut -\mathstrut 182q^{27} \) \(\mathstrut -\mathstrut 128q^{28} \) \(\mathstrut +\mathstrut 144q^{29} \) \(\mathstrut -\mathstrut 266q^{30} \) \(\mathstrut -\mathstrut 34q^{31} \) \(\mathstrut -\mathstrut 104q^{32} \) \(\mathstrut +\mathstrut 22q^{33} \) \(\mathstrut -\mathstrut 52q^{34} \) \(\mathstrut -\mathstrut 172q^{35} \) \(\mathstrut -\mathstrut 80q^{36} \) \(\mathstrut +\mathstrut 54q^{37} \) \(\mathstrut +\mathstrut 432q^{38} \) \(\mathstrut +\mathstrut 400q^{39} \) \(\mathstrut -\mathstrut 492q^{40} \) \(\mathstrut +\mathstrut 536q^{41} \) \(\mathstrut -\mathstrut 140q^{42} \) \(\mathstrut -\mathstrut 60q^{43} \) \(\mathstrut +\mathstrut 88q^{44} \) \(\mathstrut +\mathstrut 428q^{45} \) \(\mathstrut -\mathstrut 314q^{46} \) \(\mathstrut -\mathstrut 272q^{47} \) \(\mathstrut +\mathstrut 776q^{48} \) \(\mathstrut -\mathstrut 390q^{49} \) \(\mathstrut +\mathstrut 232q^{50} \) \(\mathstrut -\mathstrut 164q^{51} \) \(\mathstrut -\mathstrut 560q^{52} \) \(\mathstrut -\mathstrut 492q^{53} \) \(\mathstrut -\mathstrut 110q^{54} \) \(\mathstrut -\mathstrut 22q^{55} \) \(\mathstrut +\mathstrut 120q^{56} \) \(\mathstrut -\mathstrut 1512q^{57} \) \(\mathstrut -\mathstrut 192q^{58} \) \(\mathstrut +\mathstrut 634q^{59} \) \(\mathstrut +\mathstrut 632q^{60} \) \(\mathstrut +\mathstrut 840q^{61} \) \(\mathstrut +\mathstrut 134q^{62} \) \(\mathstrut +\mathstrut 248q^{63} \) \(\mathstrut +\mathstrut 224q^{64} \) \(\mathstrut -\mathstrut 880q^{65} \) \(\mathstrut +\mathstrut 286q^{66} \) \(\mathstrut +\mathstrut 754q^{67} \) \(\mathstrut +\mathstrut 640q^{68} \) \(\mathstrut +\mathstrut 962q^{69} \) \(\mathstrut +\mathstrut 284q^{70} \) \(\mathstrut -\mathstrut 678q^{71} \) \(\mathstrut -\mathstrut 744q^{72} \) \(\mathstrut -\mathstrut 400q^{73} \) \(\mathstrut +\mathstrut 6q^{74} \) \(\mathstrut -\mathstrut 520q^{75} \) \(\mathstrut +\mathstrut 432q^{76} \) \(\mathstrut -\mathstrut 220q^{77} \) \(\mathstrut -\mathstrut 440q^{78} \) \(\mathstrut +\mathstrut 316q^{79} \) \(\mathstrut -\mathstrut 1544q^{80} \) \(\mathstrut -\mathstrut 1294q^{81} \) \(\mathstrut +\mathstrut 512q^{82} \) \(\mathstrut +\mathstrut 468q^{83} \) \(\mathstrut -\mathstrut 736q^{84} \) \(\mathstrut +\mathstrut 452q^{85} \) \(\mathstrut -\mathstrut 156q^{86} \) \(\mathstrut +\mathstrut 1200q^{87} \) \(\mathstrut +\mathstrut 132q^{88} \) \(\mathstrut -\mathstrut 1842q^{89} \) \(\mathstrut +\mathstrut 1532q^{90} \) \(\mathstrut +\mathstrut 1280q^{91} \) \(\mathstrut -\mathstrut 40q^{92} \) \(\mathstrut -\mathstrut 638q^{93} \) \(\mathstrut -\mathstrut 992q^{94} \) \(\mathstrut +\mathstrut 2952q^{95} \) \(\mathstrut -\mathstrut 952q^{96} \) \(\mathstrut +\mathstrut 2194q^{97} \) \(\mathstrut -\mathstrut 870q^{98} \) \(\mathstrut -\mathstrut 484q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−1.73205
1.73205
−0.732051 5.92820 −7.46410 −12.8564 −4.33975 16.9282 11.3205 8.14359 9.41154
1.2 2.73205 −7.92820 −0.535898 14.8564 −21.6603 3.07180 −23.3205 35.8564 40.5885
\(n\): 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

There are no other newforms in \(S_{4}^{\mathrm{new}}(\Gamma_0(11))\).