# 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

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.81214351604$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{85})$$ Defining polynomial: $$x^{4} - 2 x^{3} - 37 x^{2} + 38 x + 531$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{11}\cdot 3^{4}$$ Twist minimal: yes 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 84q^{3} - 2048q^{4} + 5376q^{6} - 45112q^{7} + 159012q^{9} + O(q^{10})$$ $$4q + 84q^{3} - 2048q^{4} + 5376q^{6} - 45112q^{7} + 159012q^{9} - 53760q^{10} - 43008q^{12} + 275240q^{13} - 1180800q^{15} + 1048576q^{16} - 2907648q^{18} - 1568728q^{19} + 9628008q^{21} + 7730688q^{22} - 2752512q^{24} - 33732380q^{25} + 34619508q^{27} + 23097344q^{28} - 85731840q^{30} - 21785848q^{31} + 25974144q^{33} + 151087104q^{34} - 81414144q^{36} - 71014168q^{37} + 217287240q^{39} + 27525120q^{40} - 145233408q^{42} - 470688664q^{43} + 312318720q^{45} + 188814336q^{46} + 22020096q^{48} - 50058420q^{49} - 708576768q^{51} - 140922880q^{52} + 481662720q^{54} + 2701359360q^{55} - 1058753208q^{57} - 1564177920q^{58} + 604569600q^{60} - 1184038744q^{61} - 905007096q^{63} - 536870912q^{64} + 3123445248q^{66} - 297365848q^{67} + 596268288q^{69} - 3962250240q^{70} + 1488715776q^{72} + 6534269000q^{73} - 5150031180q^{75} + 803188736q^{76} - 1322135040q^{78} + 199282568q^{79} + 1458964548q^{81} + 8378668032q^{82} - 4929540096q^{84} - 12880512000q^{85} + 210268800q^{87} - 3958112256q^{88} - 9243763200q^{90} + 8317232080q^{91} + 31744468392q^{93} + 8505477120q^{94} + 1409286144q^{96} - 39176355064q^{97} - 2626912512q^{99} + O(q^{100})$$

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

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.11.b.a 4
3.b odd 2 1 inner 6.11.b.a 4
4.b odd 2 1 48.11.e.d 4
5.b even 2 1 150.11.d.a 4
5.c odd 4 2 150.11.b.a 8
8.b even 2 1 192.11.e.g 4
8.d odd 2 1 192.11.e.h 4
9.c even 3 2 162.11.d.d 8
9.d odd 6 2 162.11.d.d 8
12.b even 2 1 48.11.e.d 4
15.d odd 2 1 150.11.d.a 4
15.e even 4 2 150.11.b.a 8
24.f even 2 1 192.11.e.h 4
24.h odd 2 1 192.11.e.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.11.b.a 4 1.a even 1 1 trivial
6.11.b.a 4 3.b odd 2 1 inner
48.11.e.d 4 4.b odd 2 1
48.11.e.d 4 12.b even 2 1
150.11.b.a 8 5.c odd 4 2
150.11.b.a 8 15.e even 4 2
150.11.d.a 4 5.b even 2 1
150.11.d.a 4 15.d odd 2 1
162.11.d.d 8 9.c even 3 2
162.11.d.d 8 9.d odd 6 2
192.11.e.g 4 8.b even 2 1
192.11.e.g 4 24.h odd 2 1
192.11.e.h 4 8.d odd 2 1
192.11.e.h 4 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{11}^{\mathrm{new}}(6, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 512 + T^{2} )^{2}$$
$3$ $$3486784401 - 4960116 T - 75978 T^{2} - 84 T^{3} + T^{4}$$
$5$ $$306009247334400 + 36397440 T^{2} + T^{4}$$
$7$ $$( -15574076 + 22556 T + T^{2} )^{2}$$
$11$ $$21\!\cdots\!24$$$$+ 58313211264 T^{2} + T^{4}$$
$13$ $$( -52372127900 - 137620 T + T^{2} )^{2}$$
$17$ $$32\!\cdots\!84$$$$+ 7560967182336 T^{2} + T^{4}$$
$19$ $$( -1189491369116 + 784364 T + T^{2} )^{2}$$
$23$ $$61\!\cdots\!24$$$$+ 33055507478016 T^{2} + T^{4}$$
$29$ $$20\!\cdots\!00$$$$+ 907304099736960 T^{2} + T^{4}$$
$31$ $$( -1294078582786556 + 10892924 T + T^{2} )^{2}$$
$37$ $$( -4297319054834396 + 35507084 T + T^{2} )^{2}$$
$41$ $$28\!\cdots\!04$$$$+ 23561404561257984 T^{2} + T^{4}$$
$43$ $$( 7278142478596516 + 235344332 T + T^{2} )^{2}$$
$47$ $$26\!\cdots\!00$$$$+ 25124763600230400 T^{2} + T^{4}$$
$53$ $$37\!\cdots\!04$$$$+ 212636466457531776 T^{2} + T^{4}$$
$59$ $$20\!\cdots\!04$$$$+ 324064557407447424 T^{2} + T^{4}$$
$61$ $$( 32529703648081636 + 592019372 T + T^{2} )^{2}$$
$67$ $$( -350207761464045596 + 148682924 T + T^{2} )^{2}$$
$71$ $$49\!\cdots\!00$$$$+ 1847488216292328960 T^{2} + T^{4}$$
$73$ $$( 2363496913262627140 - 3267134500 T + T^{2} )^{2}$$
$79$ $$( -3668964988480567676 - 99641284 T + T^{2} )^{2}$$
$83$ $$57\!\cdots\!44$$$$+ 5977139602070968704 T^{2} + T^{4}$$
$89$ $$15\!\cdots\!84$$$$+ 27084125311735371264 T^{2} + T^{4}$$
$97$ $$( 95016028790224257796 + 19588177532 T + T^{2} )^{2}$$