Properties

Label 6.11.b.a
Level 6
Weight 11
Character orbit 6.b
Analytic conductor 3.812
Analytic rank 0
Dimension 4
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6 = 2 \cdot 3 \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 6.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(3.81214351604\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{85})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 21 - 3 \beta_{1} + \beta_{2} ) q^{3} \) \( -512 q^{4} \) \( + ( 24 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{5} \) \( + ( 1344 + 21 \beta_{1} - 4 \beta_{3} ) q^{6} \) \( + ( -11278 - 18 \beta_{1} + 48 \beta_{2} + 3 \beta_{3} ) q^{7} \) \( -512 \beta_{1} q^{8} \) \( + ( 39753 + 1404 \beta_{1} + 42 \beta_{2} + 21 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 21 - 3 \beta_{1} + \beta_{2} ) q^{3} \) \( -512 q^{4} \) \( + ( 24 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{5} \) \( + ( 1344 + 21 \beta_{1} - 4 \beta_{3} ) q^{6} \) \( + ( -11278 - 18 \beta_{1} + 48 \beta_{2} + 3 \beta_{3} ) q^{7} \) \( -512 \beta_{1} q^{8} \) \( + ( 39753 + 1404 \beta_{1} + 42 \beta_{2} + 21 \beta_{3} ) q^{9} \) \( + ( -13440 + 144 \beta_{1} - 384 \beta_{2} - 24 \beta_{3} ) q^{10} \) \( + ( -3696 \beta_{1} - 210 \beta_{2} + 105 \beta_{3} ) q^{11} \) \( + ( -10752 + 1536 \beta_{1} - 512 \beta_{2} ) q^{12} \) \( + ( 68810 - 360 \beta_{1} + 960 \beta_{2} + 60 \beta_{3} ) q^{13} \) \( + ( -11422 \beta_{1} + 384 \beta_{2} - 192 \beta_{3} ) q^{14} \) \( + ( -295200 + 41436 \beta_{1} + 1134 \beta_{2} - 105 \beta_{3} ) q^{15} \) \( + 262144 q^{16} \) \( + ( -74304 \beta_{1} + 1416 \beta_{2} - 708 \beta_{3} ) q^{17} \) \( + ( -726912 + 38745 \beta_{1} + 2688 \beta_{2} - 168 \beta_{3} ) q^{18} \) \( + ( -392182 + 1746 \beta_{1} - 4656 \beta_{2} - 291 \beta_{3} ) q^{19} \) \( + ( -12288 \beta_{1} - 3072 \beta_{2} + 1536 \beta_{3} ) q^{20} \) \( + ( 2407002 + 75144 \beta_{1} - 11278 \beta_{2} + 567 \beta_{3} ) q^{21} \) \( + ( 1932672 - 5040 \beta_{1} + 13440 \beta_{2} + 840 \beta_{3} ) q^{22} \) \( + ( -90336 \beta_{1} - 4956 \beta_{2} + 2478 \beta_{3} ) q^{23} \) \( + ( -688128 - 10752 \beta_{1} + 2048 \beta_{3} ) q^{24} \) \( + ( -8433095 + 7560 \beta_{1} - 20160 \beta_{2} - 1260 \beta_{3} ) q^{25} \) \( + ( 65930 \beta_{1} + 7680 \beta_{2} - 3840 \beta_{3} ) q^{26} \) \( + ( 8654877 - 247779 \beta_{1} + 33579 \beta_{2} - 4797 \beta_{3} ) q^{27} \) \( + ( 5774336 + 9216 \beta_{1} - 24576 \beta_{2} - 1536 \beta_{3} ) q^{28} \) \( + ( 757128 \beta_{1} + 17682 \beta_{2} - 8841 \beta_{3} ) q^{29} \) \( + ( -21432960 - 290160 \beta_{1} - 13440 \beta_{2} - 4536 \beta_{3} ) q^{30} \) \( + ( -5446462 - 54810 \beta_{1} + 146160 \beta_{2} + 9135 \beta_{3} ) q^{31} \) \( + 262144 \beta_{1} q^{32} \) \( + ( 6493536 - 1510236 \beta_{1} - 39690 \beta_{2} + 15099 \beta_{3} ) q^{33} \) \( + ( 37771776 + 33984 \beta_{1} - 90624 \beta_{2} - 5664 \beta_{3} ) q^{34} \) \( + ( 1956288 \beta_{1} - 57588 \beta_{2} + 28794 \beta_{3} ) q^{35} \) \( + ( -20353536 - 718848 \beta_{1} - 21504 \beta_{2} - 10752 \beta_{3} ) q^{36} \) \( + ( -17753542 + 102312 \beta_{1} - 272832 \beta_{2} - 17052 \beta_{3} ) q^{37} \) \( + ( -378214 \beta_{1} - 37248 \beta_{2} + 18624 \beta_{3} ) q^{38} \) \( + ( 54321810 + 619770 \beta_{1} + 68810 \beta_{2} + 11340 \beta_{3} ) q^{39} \) \( + ( 6881280 - 73728 \beta_{1} + 196608 \beta_{2} + 12288 \beta_{3} ) q^{40} \) \( + ( -4121328 \beta_{1} + 80484 \beta_{2} - 40242 \beta_{3} ) q^{41} \) \( + ( -36308352 + 2379786 \beta_{1} + 72576 \beta_{2} + 45112 \beta_{3} ) q^{42} \) \( + ( -117672166 - 122094 \beta_{1} + 325584 \beta_{2} + 20349 \beta_{3} ) q^{43} \) \( + ( 1892352 \beta_{1} + 107520 \beta_{2} - 53760 \beta_{3} ) q^{44} \) \( + ( 78079680 + 4602096 \beta_{1} - 236106 \beta_{2} - 157743 \beta_{3} ) q^{45} \) \( + ( 47203584 - 118944 \beta_{1} + 317184 \beta_{2} + 19824 \beta_{3} ) q^{46} \) \( + ( -4185600 \beta_{1} + 86760 \beta_{2} - 43380 \beta_{3} ) q^{47} \) \( + ( 5505024 - 786432 \beta_{1} + 262144 \beta_{2} ) q^{48} \) \( + ( -12514605 + 406008 \beta_{1} - 1082688 \beta_{2} - 67668 \beta_{3} ) q^{49} \) \( + ( -8372615 \beta_{1} - 161280 \beta_{2} + 80640 \beta_{3} ) q^{50} \) \( + ( -177144192 + 8099568 \beta_{1} + 267624 \beta_{2} + 295092 \beta_{3} ) q^{51} \) \( + ( -35230720 + 184320 \beta_{1} - 491520 \beta_{2} - 30720 \beta_{3} ) q^{52} \) \( + ( -9081864 \beta_{1} - 356034 \beta_{2} + 178017 \beta_{3} ) q^{53} \) \( + ( 120415680 + 8885133 \beta_{1} - 614016 \beta_{2} - 134316 \beta_{3} ) q^{54} \) \( + ( 675339840 - 675864 \beta_{1} + 1802304 \beta_{2} + 112644 \beta_{3} ) q^{55} \) \( + ( 5848064 \beta_{1} - 196608 \beta_{2} + 98304 \beta_{3} ) q^{56} \) \( + ( -264688302 - 2830524 \beta_{1} - 392182 \beta_{2} - 54999 \beta_{3} ) q^{57} \) \( + ( -391044480 + 424368 \beta_{1} - 1131648 \beta_{2} - 70728 \beta_{3} ) q^{58} \) \( + ( 17198448 \beta_{1} + 139638 \beta_{2} - 69819 \beta_{3} ) q^{59} \) \( + ( 151142400 - 21215232 \beta_{1} - 580608 \beta_{2} + 53760 \beta_{3} ) q^{60} \) \( + ( -296009686 - 353592 \beta_{1} + 942912 \beta_{2} + 58932 \beta_{3} ) q^{61} \) \( + ( -5884942 \beta_{1} + 1169280 \beta_{2} - 584640 \beta_{3} ) q^{62} \) \( + ( -226251774 - 28899450 \beta_{1} + 1979652 \beta_{2} - 390171 \beta_{3} ) q^{63} \) \( -134217728 q^{64} \) \( + ( 46190640 \beta_{1} + 614460 \beta_{2} - 307230 \beta_{3} ) q^{65} \) \( + ( 780861312 + 5768784 \beta_{1} + 1932672 \beta_{2} + 158760 \beta_{3} ) q^{66} \) \( + ( -74341462 + 898506 \beta_{1} - 2396016 \beta_{2} - 149751 \beta_{3} ) q^{67} \) \( + ( 38043648 \beta_{1} - 724992 \beta_{2} + 362496 \beta_{3} ) q^{68} \) \( + ( 149067072 - 35706888 \beta_{1} - 936684 \beta_{2} + 368778 \beta_{3} ) q^{69} \) \( + ( -990562560 - 1382112 \beta_{1} + 3685632 \beta_{2} + 230352 \beta_{3} ) q^{70} \) \( + ( -14146272 \beta_{1} - 1281588 \beta_{2} + 640794 \beta_{3} ) q^{71} \) \( + ( 372178944 - 19837440 \beta_{1} - 1376256 \beta_{2} + 86016 \beta_{3} ) q^{72} \) \( + ( 1633567250 + 832032 \beta_{1} - 2218752 \beta_{2} - 138672 \beta_{3} ) q^{73} \) \( + ( -16935046 \beta_{1} - 2182656 \beta_{2} + 1091328 \beta_{3} ) q^{74} \) \( + ( -1287507795 + 7949085 \beta_{1} - 8433095 \beta_{2} - 238140 \beta_{3} ) q^{75} \) \( + ( 200797184 - 893952 \beta_{1} + 2383872 \beta_{2} + 148992 \beta_{3} ) q^{76} \) \( + ( -35848848 \beta_{1} + 918876 \beta_{2} - 459438 \beta_{3} ) q^{77} \) \( + ( -330533760 + 53777490 \beta_{1} + 1451520 \beta_{2} - 275240 \beta_{3} ) q^{78} \) \( + ( 49820642 + 2886534 \beta_{1} - 7697424 \beta_{2} - 481089 \beta_{3} ) q^{79} \) \( + ( 6291456 \beta_{1} + 1572864 \beta_{2} - 786432 \beta_{3} ) q^{80} \) \( + ( 364741137 + 70979328 \beta_{1} + 10971828 \beta_{2} + 1192590 \beta_{3} ) q^{81} \) \( + ( 2094667008 + 1931616 \beta_{1} - 5150976 \beta_{2} - 321936 \beta_{3} ) q^{82} \) \( + ( -24958608 \beta_{1} + 2330154 \beta_{2} - 1165077 \beta_{3} ) q^{83} \) \( + ( -1232385024 - 38473728 \beta_{1} + 5774336 \beta_{2} - 290304 \beta_{3} ) q^{84} \) \( + ( -3220128000 - 9731232 \beta_{1} + 25949952 \beta_{2} + 1621872 \beta_{3} ) q^{85} \) \( + ( -118648918 \beta_{1} + 2604672 \beta_{2} - 1302336 \beta_{3} ) q^{86} \) \( + ( 52567200 + 136526292 \beta_{1} + 3341898 \beta_{2} - 3055035 \beta_{3} ) q^{87} \) \( + ( -989528064 + 2580480 \beta_{1} - 6881280 \beta_{2} - 430080 \beta_{3} ) q^{88} \) \( + ( -118832112 \beta_{1} - 3522372 \beta_{2} + 1761186 \beta_{3} ) q^{89} \) \( + ( -2310940800 + 85651344 \beta_{1} - 20191104 \beta_{2} + 944424 \beta_{3} ) q^{90} \) \( + ( 2079308020 + 2821500 \beta_{1} - 7524000 \beta_{2} - 470250 \beta_{3} ) q^{91} \) \( + ( 46252032 \beta_{1} + 2537472 \beta_{2} - 1268736 \beta_{3} ) q^{92} \) \( + ( 7936117098 + 142128336 \beta_{1} - 5446462 \beta_{2} + 1726515 \beta_{3} ) q^{93} \) \( + ( 2126369280 + 2082240 \beta_{1} - 5552640 \beta_{2} - 347040 \beta_{3} ) q^{94} \) \( + ( -225427488 \beta_{1} - 3330852 \beta_{2} + 1665426 \beta_{3} ) q^{95} \) \( + ( 352321536 + 5505024 \beta_{1} - 1048576 \beta_{3} ) q^{96} \) \( + ( -9794088766 + 1435608 \beta_{1} - 3828288 \beta_{2} - 239268 \beta_{3} ) q^{97} \) \( + ( -9266541 \beta_{1} - 8661504 \beta_{2} + 4330752 \beta_{3} ) q^{98} \) \( + ( -656728128 - 271729080 \beta_{1} + 586782 \beta_{2} + 6000813 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 84q^{3} \) \(\mathstrut -\mathstrut 2048q^{4} \) \(\mathstrut +\mathstrut 5376q^{6} \) \(\mathstrut -\mathstrut 45112q^{7} \) \(\mathstrut +\mathstrut 159012q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 84q^{3} \) \(\mathstrut -\mathstrut 2048q^{4} \) \(\mathstrut +\mathstrut 5376q^{6} \) \(\mathstrut -\mathstrut 45112q^{7} \) \(\mathstrut +\mathstrut 159012q^{9} \) \(\mathstrut -\mathstrut 53760q^{10} \) \(\mathstrut -\mathstrut 43008q^{12} \) \(\mathstrut +\mathstrut 275240q^{13} \) \(\mathstrut -\mathstrut 1180800q^{15} \) \(\mathstrut +\mathstrut 1048576q^{16} \) \(\mathstrut -\mathstrut 2907648q^{18} \) \(\mathstrut -\mathstrut 1568728q^{19} \) \(\mathstrut +\mathstrut 9628008q^{21} \) \(\mathstrut +\mathstrut 7730688q^{22} \) \(\mathstrut -\mathstrut 2752512q^{24} \) \(\mathstrut -\mathstrut 33732380q^{25} \) \(\mathstrut +\mathstrut 34619508q^{27} \) \(\mathstrut +\mathstrut 23097344q^{28} \) \(\mathstrut -\mathstrut 85731840q^{30} \) \(\mathstrut -\mathstrut 21785848q^{31} \) \(\mathstrut +\mathstrut 25974144q^{33} \) \(\mathstrut +\mathstrut 151087104q^{34} \) \(\mathstrut -\mathstrut 81414144q^{36} \) \(\mathstrut -\mathstrut 71014168q^{37} \) \(\mathstrut +\mathstrut 217287240q^{39} \) \(\mathstrut +\mathstrut 27525120q^{40} \) \(\mathstrut -\mathstrut 145233408q^{42} \) \(\mathstrut -\mathstrut 470688664q^{43} \) \(\mathstrut +\mathstrut 312318720q^{45} \) \(\mathstrut +\mathstrut 188814336q^{46} \) \(\mathstrut +\mathstrut 22020096q^{48} \) \(\mathstrut -\mathstrut 50058420q^{49} \) \(\mathstrut -\mathstrut 708576768q^{51} \) \(\mathstrut -\mathstrut 140922880q^{52} \) \(\mathstrut +\mathstrut 481662720q^{54} \) \(\mathstrut +\mathstrut 2701359360q^{55} \) \(\mathstrut -\mathstrut 1058753208q^{57} \) \(\mathstrut -\mathstrut 1564177920q^{58} \) \(\mathstrut +\mathstrut 604569600q^{60} \) \(\mathstrut -\mathstrut 1184038744q^{61} \) \(\mathstrut -\mathstrut 905007096q^{63} \) \(\mathstrut -\mathstrut 536870912q^{64} \) \(\mathstrut +\mathstrut 3123445248q^{66} \) \(\mathstrut -\mathstrut 297365848q^{67} \) \(\mathstrut +\mathstrut 596268288q^{69} \) \(\mathstrut -\mathstrut 3962250240q^{70} \) \(\mathstrut +\mathstrut 1488715776q^{72} \) \(\mathstrut +\mathstrut 6534269000q^{73} \) \(\mathstrut -\mathstrut 5150031180q^{75} \) \(\mathstrut +\mathstrut 803188736q^{76} \) \(\mathstrut -\mathstrut 1322135040q^{78} \) \(\mathstrut +\mathstrut 199282568q^{79} \) \(\mathstrut +\mathstrut 1458964548q^{81} \) \(\mathstrut +\mathstrut 8378668032q^{82} \) \(\mathstrut -\mathstrut 4929540096q^{84} \) \(\mathstrut -\mathstrut 12880512000q^{85} \) \(\mathstrut +\mathstrut 210268800q^{87} \) \(\mathstrut -\mathstrut 3958112256q^{88} \) \(\mathstrut -\mathstrut 9243763200q^{90} \) \(\mathstrut +\mathstrut 8317232080q^{91} \) \(\mathstrut +\mathstrut 31744468392q^{93} \) \(\mathstrut +\mathstrut 8505477120q^{94} \) \(\mathstrut +\mathstrut 1409286144q^{96} \) \(\mathstrut -\mathstrut 39176355064q^{97} \) \(\mathstrut -\mathstrut 2626912512q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(2\) \(x^{3}\mathstrut -\mathstrut \) \(37\) \(x^{2}\mathstrut +\mathstrut \) \(38\) \(x\mathstrut +\mathstrut \) \(531\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -32 \nu^{3} + 48 \nu^{2} + 464 \nu - 240 \)\()/93\)
\(\beta_{2}\)\(=\)\((\)\( -36 \nu^{3} - 132 \nu^{2} + 2196 \nu + 2520 \)\()/31\)
\(\beta_{3}\)\(=\)\((\)\( -64 \nu^{3} + 3072 \nu^{2} + 928 \nu - 58512 \)\()/31\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(16\) \(\beta_{2}\mathstrut -\mathstrut \) \(60\) \(\beta_{1}\mathstrut +\mathstrut \) \(432\)\()/864\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(1872\)\()/96\)
\(\nu^{3}\)\(=\)\((\)\(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(116\) \(\beta_{2}\mathstrut -\mathstrut \) \(1731\) \(\beta_{1}\mathstrut +\mathstrut \) \(12528\)\()/432\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−4.10977 + 1.41421i
5.10977 + 1.41421i
−4.10977 1.41421i
5.10977 1.41421i
22.6274i −200.269 + 137.627i −512.000 3630.47i 3114.15 + 4531.57i −23226.5 11585.2i 21166.4 55125.0i 82148.2
5.2 22.6274i 242.269 18.8335i −512.000 4818.41i −426.153 5481.92i 670.530 11585.2i 58339.6 9125.53i −109028.
5.3 22.6274i −200.269 137.627i −512.000 3630.47i 3114.15 4531.57i −23226.5 11585.2i 21166.4 + 55125.0i 82148.2
5.4 22.6274i 242.269 + 18.8335i −512.000 4818.41i −426.153 + 5481.92i 670.530 11585.2i 58339.6 + 9125.53i −109028.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{11}^{\mathrm{new}}(6, [\chi])\).