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: \( 1 \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 34q^{3} + 84q^{4} + 24q^{5} - 206q^{6} + 84q^{7} - 564q^{8} - 7q^{9} + O(q^{10}) \) \( 3q + 34q^{3} + 84q^{4} + 24q^{5} - 206q^{6} + 84q^{7} - 564q^{8} - 7q^{9} - 414q^{10} + 363q^{11} + 992q^{12} + 486q^{13} - 1020q^{14} + 1654q^{15} + 1992q^{16} + 1086q^{17} - 3706q^{18} + 1380q^{19} - 3480q^{20} - 908q^{21} - 3066q^{23} - 11748q^{24} - 57q^{25} + 12132q^{26} - 2990q^{27} + 23712q^{28} - 3426q^{29} + 2650q^{30} - 4098q^{31} - 12408q^{32} + 4114q^{33} + 25320q^{34} - 24228q^{35} + 4756q^{36} + 17724q^{37} - 9240q^{38} - 6560q^{39} - 15276q^{40} + 5994q^{41} - 47828q^{42} - 26208q^{43} + 10164q^{44} + 18458q^{45} - 16806q^{46} - 17232q^{47} + 61064q^{48} + 48531q^{49} + 41070q^{50} - 22724q^{51} - 35304q^{52} + 50586q^{53} + 18814q^{54} + 2904q^{55} - 42312q^{56} + 20160q^{57} - 29172q^{58} - 3738q^{59} - 13456q^{60} + 18486q^{61} - 19974q^{62} - 12496q^{63} - 20352q^{64} - 7668q^{65} - 24926q^{66} - 47754q^{67} - 12600q^{68} + 35042q^{69} - 123372q^{70} + 39282q^{71} - 95040q^{72} + 15426q^{73} + 153294q^{74} - 21916q^{75} + 103920q^{76} + 10164q^{77} + 124984q^{78} + 125148q^{79} + 118680q^{80} - 86917q^{81} - 255372q^{82} - 143928q^{83} + 343616q^{84} - 104040q^{85} + 243060q^{86} - 19368q^{87} - 68244q^{88} - 106824q^{89} + 103424q^{90} - 109632q^{91} - 336528q^{92} - 16622q^{93} - 74928q^{94} - 22200q^{95} - 76456q^{96} + 9684q^{97} + 3480q^{98} - 847q^{99} + O(q^{100}) \)

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.6.a.b 3
3.b odd 2 1 99.6.a.g 3
4.b odd 2 1 176.6.a.i 3
5.b even 2 1 275.6.a.b 3
5.c odd 4 2 275.6.b.b 6
7.b odd 2 1 539.6.a.e 3
8.b even 2 1 704.6.a.q 3
8.d odd 2 1 704.6.a.t 3
11.b odd 2 1 121.6.a.d 3
33.d even 2 1 1089.6.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.b 3 1.a even 1 1 trivial
99.6.a.g 3 3.b odd 2 1
121.6.a.d 3 11.b odd 2 1
176.6.a.i 3 4.b odd 2 1
275.6.a.b 3 5.b even 2 1
275.6.b.b 6 5.c odd 4 2
539.6.a.e 3 7.b odd 2 1
704.6.a.q 3 8.b even 2 1
704.6.a.t 3 8.d odd 2 1
1089.6.a.r 3 33.d even 2 1

Atkin-Lehner signs

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T^{2} + 188 T^{3} + 192 T^{4} + 32768 T^{6} \)
$3$ \( 1 - 34 T + 946 T^{2} - 15312 T^{3} + 229878 T^{4} - 2007666 T^{5} + 14348907 T^{6} \)
$5$ \( 1 - 24 T + 5004 T^{2} - 111046 T^{3} + 15637500 T^{4} - 234375000 T^{5} + 30517578125 T^{6} \)
$7$ \( 1 - 84 T + 4473 T^{2} + 2556872 T^{3} + 75177711 T^{4} - 23727920916 T^{5} + 4747561509943 T^{6} \)
$11$ \( ( 1 - 121 T )^{3} \)
$13$ \( 1 - 486 T + 767415 T^{2} - 196760188 T^{3} + 284935817595 T^{4} - 66999227038614 T^{5} + 51185893014090757 T^{6} \)
$17$ \( 1 - 1086 T + 2690223 T^{2} - 2752177348 T^{3} + 3819731958111 T^{4} - 2189369375887614 T^{5} + 2862423051509815793 T^{6} \)
$19$ \( 1 - 1380 T + 7644297 T^{2} - 6777009240 T^{3} + 18928036157403 T^{4} - 8460871435765380 T^{5} + 15181127029874798299 T^{6} \)
$23$ \( 1 + 3066 T + 8993526 T^{2} + 22463329348 T^{3} + 57885418115418 T^{4} + 127013683381047834 T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \)
$29$ \( 1 + 3426 T + 62121159 T^{2} + 136513203828 T^{3} + 1274176348301691 T^{4} + 1441342981286488626 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} \)
$31$ \( 1 + 4098 T + 90136878 T^{2} + 235738865996 T^{3} + 2580542290930578 T^{4} + 3358836720047322498 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} \)
$37$ \( 1 - 17724 T + 229846956 T^{2} - 1916316420702 T^{3} + 15938497433444892 T^{4} - 85227349416733955676 T^{5} + \)\(33\!\cdots\!93\)\( T^{6} \)
$41$ \( 1 - 5994 T + 174379803 T^{2} - 1186954316020 T^{3} + 20202981506708403 T^{4} - 80455419905053491594 T^{5} + \)\(15\!\cdots\!01\)\( T^{6} \)
$43$ \( 1 + 26208 T + 443706117 T^{2} + 5261719449744 T^{3} + 65228545409745831 T^{4} + \)\(56\!\cdots\!92\)\( T^{5} + \)\(31\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 + 17232 T + 689784333 T^{2} + 7833971382112 T^{3} + 158198592680375331 T^{4} + \)\(90\!\cdots\!68\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \)
$53$ \( 1 - 50586 T + 1969881291 T^{2} - 44160585727452 T^{3} + 823795477641221463 T^{4} - \)\(88\!\cdots\!14\)\( T^{5} + \)\(73\!\cdots\!57\)\( T^{6} \)
$59$ \( 1 + 3738 T + 1293303186 T^{2} + 13104411496384 T^{3} + 924613873645516614 T^{4} + \)\(19\!\cdots\!38\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} \)
$61$ \( 1 - 18486 T + 1754869767 T^{2} - 15992539689564 T^{3} + 1482156513944931867 T^{4} - \)\(13\!\cdots\!86\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} \)
$67$ \( 1 + 47754 T + 997454514 T^{2} - 18340812610856 T^{3} + 1346688382441882998 T^{4} + \)\(87\!\cdots\!46\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \)
$71$ \( 1 - 39282 T + 4433022990 T^{2} - 143037873283668 T^{3} + 7998190192215779490 T^{4} - \)\(12\!\cdots\!82\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} \)
$73$ \( 1 - 15426 T + 2562304635 T^{2} - 98498106053188 T^{3} + 5311840951430733555 T^{4} - \)\(66\!\cdots\!74\)\( T^{5} + \)\(89\!\cdots\!57\)\( T^{6} \)
$79$ \( 1 - 125148 T + 13122635793 T^{2} - 768895025227784 T^{3} + 40379090438597089407 T^{4} - \)\(11\!\cdots\!48\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} \)
$83$ \( 1 + 143928 T + 12127124157 T^{2} + 722278658611584 T^{3} + 47769234937130112951 T^{4} + \)\(22\!\cdots\!72\)\( T^{5} + \)\(61\!\cdots\!07\)\( T^{6} \)
$89$ \( 1 + 106824 T + 17674467768 T^{2} + 1102702152985302 T^{3} + 98695278745946339832 T^{4} + \)\(33\!\cdots\!24\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} \)
$97$ \( 1 - 9684 T + 23649435576 T^{2} - 176541508624682 T^{3} + \)\(20\!\cdots\!32\)\( T^{4} - \)\(71\!\cdots\!16\)\( T^{5} + \)\(63\!\cdots\!93\)\( T^{6} \)
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